Datasets:

haritzpuerto

/

the_pile_arxiv_1k_sample

like 0

--- abstract: 'A geometric interpretation of curvature and torsion of linear transports along paths is presented. A number of (Bianchi type) identities satisfied by these quantities are derived. The obtained results contain as special cases the corresponding classical ones concerning curvature and torsion of linear connections.' author: - 'Bozhidar Z. Iliev [^1] [^2] [^3]' bibliography: - 'bozhopub.bib' - 'bozhoref.bib' date: | Ended: November 28, 1995\ Revised: March 25, November 6, 1996\ Updated: July 9, 1998\ Produced:\ Submitted to JINR communication: January 13, 1997\ Published: Communication JINR, E5-97-1, Dubna, 1997\ LANL xxx archive E-print No. dg-ga/9709017\ title: | **Linear transports along paths\ in vector bundles\ V. Properties of curvature and torsion** --- [l-tran-5.bbl]{} [1]{} J. A. Schouten. . Springer Verlag, Berlin-G[ö]{}ttingen-Heidelberg, second edition, 1954. S. Helgason. . Academic Press, New York-San Francisco-London, 1978. Bozhidar Z. Iliev. Linear transports along paths in vector bundles. [III]{}. [Curvature]{} and torsion. JINR Communication E5-93-261, Dubna, 1993. Bozhidar Z. Iliev. Linear transports along paths in vector bundles. [I]{}. [General]{} theory. JINR Communication E5-93-239, Dubna, 1993. Bozhidar Z. Iliev. Some generalizations of the [Jacobi]{} identity with applications to the curvature- and torsion-depending hamiltonians of particle systems. In J. [Ł]{}awrynowicz, editor, [*Hurwitz-type structures and applications to surface physics*]{}, number [II]{} in Deformations of Mathematical Structures, pages 161–188. Kluwer Academic Publishers Group, Dordrecht-Boston-London, 1993. (Papers from the Seminars on Deformations 1988–1992). Bozhidar Z. Iliev. Deviation equations in spaces with a transport along paths. JINR Communication E5-94-40, Dubna, 1994. J. A. Schouten. . Clarendon Press, Oxford, 1951. [l-5pic\_1.pic]{} 0.65mm (124.00,143.01) (45.34,62.01)[(0,1)[0.2]{}]{} (65.34,42.01)[(1,0)[0.2]{}]{} (48.67,58.34)[(0,0)\[lt\][s]{}]{} (61.01,46.01)[(0,0)\[lt\][t]{}]{} (86.00,134.00)[(0,0)\[cc\][$\eta(\cdot,t+\varepsilon)$]{}]{} (80.00,112.34)[(0,0)\[lc\][$\eta(s+\delta,t+\varepsilon)$]{}]{} (92.99,43.34)[(0,0)\[lc\][$\eta(s,t+\varepsilon)$]{}]{} (11.66,105.00)[(0,0)\[lc\][$\eta(s+\delta,t)$]{}]{} (19.00,37.34)[(0,0)\[lc\][$\eta(s,t)$]{}]{} (92.67,84.67)[(0,0)\[cc\][$\eta(s+\delta,\cdot)$]{}]{} (124.00,39.00)[(0,0)\[cc\][$\eta(s,\cdot)$]{}]{} (23.67,133.00)[(0,0)\[cc\][$\eta(\cdot,t)$]{}]{} (64.67,32.00)[(3,-1)[0.2]{}]{} (46.66,27.66)[(0,0)\[lc\][$B=\varepsilon\eta''(s,t)$]{}]{} (39.33,72.33)[(1,4)[0.2]{}]{} (41.67,70.67)[(0,0)\[lc\][$A=\delta\eta'(s,t)$]{}]{} (59.33,111.00)[(4,1)[0.2]{}]{} (37.67,102.34)[(0,0)\[lc\][$B_1 = L_{s\to s+\delta}^{\eta(\cdot,t)}B$]{}]{} (82.33,63.33)[(-1,3)[0.2]{}]{} (86.00,60.00)[(0,0)\[lc\][$A_1 = L_{t\to t+\varepsilon}^{\eta(s,\cdot)}A$]{}]{} (110.33,89.33)[(3,-2)[0.2]{}]{} (98.00,100.67)[(0,0)\[lc\] [$B_2=L_{t\to t+\varepsilon}^{\eta(s+\delta,\cdot)}B_1$]{}]{} (54.33,140.33)[(-2,3)[0.2]{}]{} (55.66,143.00)[(0,0)\[lc\] [$A_2=L_{s\to s+\delta}^{\eta(\cdot,t+\varepsilon)}A_1$]{}]{} [l-5pic\_2.pic]{} 0.45mm (134.33,143.01) (41.67,71.34)[(0,1)[0.2]{}]{} (61.67,51.34)[(1,0)[0.2]{}]{} (45.00,67.67)[(0,0)\[lt\][s]{}]{} (56.34,58.01)[(0,0)\[lt\][t]{}]{} (86.33,134.67)[(0,0)\[cc\][$\eta(\cdot,t+\varepsilon)$]{}]{} (81.00,110.67)[(0,0)\[lc\][$\eta(s+\delta,t+\varepsilon)$]{}]{} (90.66,45.01)[(0,0)\[lc\][$\eta(s,t+\varepsilon)$]{}]{} (-0.01,106.34)[(0,0)\[lc\][$\eta(s+\delta,t)$]{}]{} (13.33,37.34)[(0,0)\[lc\][$\eta(s,t)$]{}]{} (93.67,85.34)[(0,0)\[cc\][$\eta(s+\delta,\cdot)$]{}]{} (134.33,40.01)[(0,0)\[cc\][$\eta(s,\cdot)$]{}]{} (21.00,133.33)[(0,0)\[cc\][$\eta(\cdot,t)$]{}]{} **Introduction** {#I} ================ The properties of curvature and torsion tensors of a linear connection and the satisfied by them Bianchi identities are well-known [@Schouten/Ricci; @Helgason]. Looking over the given in [@f-LTP-Cur+Tor] definitions and properties of the curvature and torsion of linear transports along paths, one can expect to find out similar results in this more general case too. To their derivation is devoted this paper. Sect. \[II\] reviews some definitions and results from [@f-LTP-general; @f-LTP-Cur+Tor] and also contains new ones needed for our investigation. Sect. \[III\] proposes a geometrical interpretation of the torsion of a linear transport along paths on the basis of the question of the existence of an ‘infinitesimal’ parallelogram. Sect. \[IV\] deals with the geometrical meaning of the curvature of a linear transport along paths. It is shown that the curvature governs the main change of a vector after a suitable transportation along a ‘small’ (infinitesimal) close path. Sect. \[V\] gives the derivation of the generalizations of the Bianchi identities in the case of linear transports along paths. This is done by using the developed in [@f-Jacobi] method for obtaining many-point generalizations of the Jacobi identity. Sect. \[VI\] closes the paper with some concluding remarks, including a criterion for flatness of a linear transport along paths. **Some preliminary definitions and results** {#II} ============================================ Below are summarized some needed for this investigation definitions and results for linear transports along paths in vector bundles and their curvature and torsion. Let $(E,\pi,B)$ be a real[^4] vector bundle with base $B$, total space $E$ and projection $\pi:E\to B$. The fibres $\pi^{-1}(x)$, $x\in B$ are supposed to be isomorphic real vector spaces. Let $\gamma:J\to B$, with $J$ being a real interval, be an arbitrary path in $B$. According to [@f-LTP-general definition 2.5] a linear transport (L-transport) in $(E,\pi,B)$ is a map $L:\gamma\mapsto L^\gamma$, where the L-transport along $\gamma$ is $L^\gamma:(s,t)\mapsto L^\gamma_{s\to t}$, $s,t\in J$. Here $L^\gamma_{s\to t}: \pi^{-1}(\gamma(s))\to \pi^{-1}(\gamma(t))$ is the L-transport along $\gamma$ from $s$ to $t$. It satisfies the equations: $$\begin{aligned} & & \!\!\! \!\!\! \!\!\! \!\!\! \! L^\gamma_{s\to t}(\lambda u + \mu v) = \lambda L^\gamma_{s\to t}u + \mu L^\gamma_{s\to t}v, \ \lambda,\mu \in {\mathbb{R}}, \ u,v \in \pi^{-1}(\gamma(s)), \label{2.1} \\ & & L^\gamma_{t\to r} \circ L^\gamma_{s\to t} = L^\gamma_{s\to r}, \quad r,s,t \in J, \label{2.2} \\ & & L^\gamma_{s\to s} = id_{\pi^{-1}(\gamma(s))} \label{2.3}\end{aligned}$$ with $id_U$ being the identity map of the set $U$. Propositions 2.1 and 2.3 of [@f-LTP-general] state that the general structure of $L^\gamma_{s\to t}$ is $$L^\gamma_{s\to t}= \left( F^\gamma_t\right) ^{-1} \circ F^\gamma_s, \ s,t\in J \label{2.4}$$ where the map $F^\gamma_s:\pi^{-1}(\gamma(s)) \to V$ is a linear isomorphism on a vector space $V$. The map $F^\gamma_s$ is defined up to a left composition with a linear isomorphism $D^\gamma:V\to \underline{V}$, with $\underline{V}$ being a vector space, i.e. up to the change $F^\gamma_s \to D^\gamma \circ F^\gamma_s $. Let $\{e_i(s)\}$ be a basis in $\pi^{-1}(\gamma(s))$. Here and below the indices $i,j,k,...$ run from 1 to $\dim(\pi^{-1}(x))=\mathrm{const}=:n$. The matrix of the L-transport $L$ (see [@f-LTP-general p. 5]), $H(t,s;\gamma)= \left[ H^i_j(t,s;\gamma) \right] = H^{-1}(s,t;\gamma)$, is defined by $L^\gamma_{s\to t}e_j(s)=H^i_j(t,s;\gamma)e_i(t) $, where hereafter summation over repeated indices is assumed. The matrix of the coefficients of $L$ [@f-LTP-general p. 13] is $\Gamma_\gamma(s) = \left[ \Gamma^i_j(s;\gamma) \right] = {\partial H(s,t;\gamma)/\partial t }\left.\right|_{t=s}$. Therefore for a $C^2$ L-transport, we have $$\begin{aligned} & & H^{\pm 1}(s+\varepsilon,s;\gamma) = H^{\mp 1}(s,s+\varepsilon;\gamma)= \nonumber \\ & & = \openone \mp \varepsilon \Gamma_\gamma(s) + {\varepsilon^2 \over 2} \left(\Gamma_\gamma(s) \Gamma_\gamma(s) \mp {\partial \Gamma_\gamma(s) \over \partial s} \right) + O(\varepsilon ^2), \label{2.5}\end{aligned}$$ with $\openone$ being the unit matrix. Here we have used $$\label{2.6} \begin{array}{c} \rule[-1.123456789em]{0em}{0em} \left. {\partial^2 H(s,t;\gamma) \over \partial t^2} \right|_{t=s} = \Gamma_\gamma(s) \Gamma_\gamma(s) + {\partial \Gamma_\gamma(s) \over \partial s}, \\ \left. {\partial^2 H(t,s;\gamma) \over \partial t^2}\right|_{t=s} = \Gamma_\gamma(s) \Gamma_\gamma(s) - {\partial \Gamma_\gamma(s) \over \partial s}. \end{array}$$ These equations follow from the fact that the general form of the matrix $H$ is $H(t,s;\gamma)=F^{-1}(t;\gamma) F(s;\gamma)$ for some nondegenerate matrix function $F$ [@f-LTP-general]. Let $\eta:J\times J^\prime\to M$, $J$, with $J^\prime$ being ${\mathbb R} $-intervals, be a $C^2$ map on the real differentiable manifold $M$ with a tangent bundle $(T(M),\pi,M)$. Let $\eta(\cdot,t):s\mapsto \eta(s,t)$ and $\eta(s,\cdot):t\mapsto \eta(s,t)$, $(s,t)\in J\times J^\prime$. Here by $\eta^\prime(\cdot,t)$ and $\eta^{\prime\prime}(s,\cdot)$ we denote the tangent to $\eta(\cdot,t)$ and $\eta(s,\cdot)$, respectively, vector fields. By [@f-LTP-Cur+Tor definition 2.1] the torsion (operator) of a $C^1$ L-transport $L$ in $(T(M),\pi,M)$ is a map $${\mathcal T}:\eta\mapsto{\mathcal T}^{\,\eta}:J\times J^\prime\to T(M)$$ such that $${\mathcal T}^{\,\eta} (s,t):= {\mathcal D}^{\eta(\cdot,t)}_s \eta^{\prime\prime}(\cdot,t) - {\mathcal D}^{\eta(s,\cdot)}_t \eta^{\prime}(s,\cdot) \in T_{\eta(s,t)}(M), \label{2.7}$$ where ${\mathcal D}^\gamma_s$ is the associated with $L$ differentiation along paths [@f-LTP-general], defined by $${\mathcal D}^\gamma_s \sigma := \left( {\mathcal D}^\gamma \sigma \right) (\gamma(s)) := \left. \left[ {\partial\over \partial \varepsilon}\left( L^\gamma_{s+\varepsilon \to s}\sigma(s+\varepsilon) \right) \right] \right|_{\varepsilon=0}$$ for a $C^1$ section $\sigma$. Analogously [@f-LTP-Cur+Tor], for $\eta:J\times J^\prime \to B$ the curvature (operator) of an sport $L$ in the vector bundle $(E,\pi,B)$ is a map $${\mathcal R}: \eta\mapsto {\mathcal R}^\eta:(s,t) \mapsto {\mathcal R}^\eta(s,t):\mathrm{Sec}^2(E,\pi,B)\to\mathrm{Sec}(E,\pi,B)$$ such that $$\label{2.8} {\mathcal R}^\eta(s,t):= {\mathcal D}^{\eta(\cdot,t)}\circ {\mathcal D}^{\eta(s,\cdot)} - {\mathcal D}^{\eta(s,\cdot)}\circ {\mathcal D}^{\eta(\cdot,t)}.$$ In terms of the coefficient matrix $\Gamma$ the components of torsion and curvature are respectively [@f-LTP-Cur+Tor] $$\begin{aligned} & & \left({\mathcal T}^{\,\eta}(s,t) \right)^i = \Gamma^i_j(s;\eta(\cdot,t)) \left( \eta^{\prime\prime}(s,t)\right)^j - \Gamma^i_j(t;\eta(s,\cdot)) \left( \eta^{\prime}(s,t) \right)^j , \label{2.9} \\ & & \left[ \left( {\mathcal R}^\eta(s,t)\right) ^i_j \right] = {\partial\over\partial s}\Gamma_{\eta(s,\cdot)}(t) - {\partial\over{\partial t}} \Gamma_{\eta(\cdot,t)}(s) \> + \nonumber\\ & & \hspace{24.3mm} + \> \Gamma_{\eta(\cdot,t)}(s)\Gamma_{\eta(s,\cdot)}(t) - \Gamma_{\eta(s,\cdot)}(t)\Gamma_{\eta(\cdot,t)}(s). \label{2.10}\end{aligned}$$ Below we shall need the following definitions: \[d2.1\] The torsion vector field (operator) of an L-transport in the tangent to a manifold bundle is a section $T^{\,\eta}\in$$ \mathrm{Sec} \left(\left.\left(T(M),\pi,M\right)\right|_{\eta(J,J')}\right) $ defined by $$T^{\,\eta}(\eta(s,t)):={\mathcal T}^{\,\eta}(s,t). \label{2.11}$$ Defining $ ({\mathcal D}^\gamma\sigma)(\gamma(s)) := {\mathcal D}_{s}^{\gamma}\sigma, $ from (\[2.7\]) we get $$\label{2.12} T^{\,\eta}(\eta(s,t)) = \left( {\mathcal D}^{\eta(\cdot,t)}\eta ''(\cdot,t) - {\mathcal D}^{\eta(s,\cdot)}\eta '(s,\cdot) \right)(\eta(s,t)).$$ \[d2.2\] The curvature vector field (operator) of an L-transport is a $C^2$ section $ R^\eta\in \mathrm{Sec}^2\left( \left. (E,\pi,B) \right|_{\eta(J,J')}\right) $ defined by $$R^\eta(\eta(s,t)):={\mathcal R}^\eta(s,t). \label{2.13}$$ \[d2.3\] An L-transport along paths is called flat ($\equiv$curvature free) on a set $U\subseteq B$ if its curvature operator vanishes on $U$. It is called flat if it is flat on $B$, i.e. in the case $U=B$. **Geometrical interpretation of the torsion** {#III} ============================================= Let $\eta:J\times J'\to M$ be a $C^1$ map into the manifold $M$, $(s,t)\in J\times J'$, and $\delta,\varepsilon\in{\mathbb{R}}$ be such that $(s+\delta,t+\varepsilon)\in J\times J'$. Below we consider $\delta$ and $\varepsilon$ as ‘small’ (infinitesimal) parameters with respect to which expansions like (\[2.5\]) will be used. Consider the following two paths from $\eta(s,t)$ to $\eta(s+\delta,t+\varepsilon)$ (see figure \[Fig1\]): the first, through $\eta(s+\delta,t)$, being a product of $\eta(\cdot,t):[s,s+\delta]\to M$ and $\eta(s+\delta,\cdot):[t,t+\varepsilon]\to M$, and the second one, through $\eta(s,t+\varepsilon)$, being a product of $\eta(s,\cdot):[t,t+\varepsilon]\to M$ and $\eta(\cdot,t+\varepsilon):[s,s+\delta]\to M$. (Here $\delta$ and $\varepsilon$ are considered as positive, but this is inessential.) Up to $O(\delta^2)$ and $O(\varepsilon^2)$ the vectors $A:=\delta\eta'(s,t)$ and $B:=\varepsilon\eta''(s,t)$ are the displacement vectors [@f-DE-TP] (linear elements [@Schouten/Ricci]), respectively, of $\eta(s+\delta,t)$ and $ \eta(s,t+\varepsilon)$ with respect to $\eta(s,t)$. Using (\[2.5\]) and keeping only the first order in $\varepsilon$ and $\delta$ terms in it, we get the following *component* relation: $$\left( L_{s\to s+\delta}^{\eta(\cdot,t)} B \right)^i - \left( L_{t\to t+\varepsilon}^{\eta(s,\cdot)} A \right)^i = (B-A)^i - \delta\varepsilon\left( {\mathcal T}^{\,\eta}(s,t) \right)^i + O(\delta\varepsilon^2) + O(\delta^2\varepsilon). \label{3.1}$$ According to [@Schouten/physics ch. V, sect. 1] this result has the following interpretation. *After the ‘L-transportation’ of two linear elements $A$ and $B$ along each other we get, up to second order terms, a pentagon with a closure vector $-\delta\varepsilon{\mathcal T}^{\,\eta}(s,t)$.* This implies the existence of an infinitesimal parallelogram only in the torsion free case. Using again (\[2.5\]) and keeping only first order terms, after some algebra, we find $$\begin{aligned} & & \nonumber \left( L_{t\to t+\varepsilon}^{\eta(s+\delta,\cdot)} \circ L_{s\to s+\delta}^{\eta(\cdot,t)} B - L_{s\to s+\delta}^{\eta(\cdot,t+\varepsilon)} \circ L_{t\to t+\varepsilon}^{\eta(s,\cdot)} A \right)^i \> = \\ & & \nonumber = \> \left[ \left( L_{t\to t+\varepsilon}^{\eta(s,\cdot)} B \right)^i - \left( L_{s\to s+\delta}^{\eta(\cdot,t)} A \right)^i \right] - \delta\varepsilon\left({\mathcal T}^{\,\eta}(s,t)\right)^i \> + \\ & & \label{3.2} \quad\> + \> O(\delta^3) + O(\delta^2\varepsilon) + O(\delta\varepsilon^2) + O(\varepsilon^3).\end{aligned}$$ Note that if $\eta$ is a family of L-paths, i.e. $ L_{s_1\to s_2}^{\eta(\cdot,t)}\eta'(s_1,t)=\eta^\prime(s_2,t) $ and $ L_{t_1\to t_2}^{\eta(s,\cdot)}\eta''(s,t_1) = \eta{^\prime}{^\prime}(s,t_2), $ for all $s,s_1,s_2\in J$ and $t,t_1,t_2\in J^\prime$, the expression in the square brackets in (\[3.2\]) is simply $(B-A)^i$. So, the torsion describes the first order correction to the difference of two (infinitesimal) displacement vectors when they are (L-)transported in the above-described way. **Geometrical interpretation of curvature** {#IV} =========================================== Let $(E,\pi,B)$ be a vector bundle, $\eta:J\times J^\prime\to B$ be a $C^1$ map, and $L$ be a $C^2$ L-transport along paths in $(E,\pi,B)$. Let $(s,t)\in J\times J^\prime$ and $\delta, \varepsilon \in {\mathbb{R}}$, $\delta,\varepsilon >0$ (this condition is insignificant for the final result) be such that $(s+\delta,t+\varepsilon)\in J\times J^\prime$. Consider the paths on figure \[Fig2\]. The result of an L-transport of a vector from $\eta(s,t)$ to $\eta(s+\delta,t)$ along $\left.\eta(\cdot,t)\right|_{[s,s+\delta]}$, then from $\eta(s+\delta,t)$ to $\eta(s+\delta,t+\varepsilon)$ along $\left.\eta(s+\delta,\cdot)\right|_{[t,t+\varepsilon]}$, then from $\eta(s+\delta,t+\varepsilon)$ to $\eta(s,t+\varepsilon)$ along $\left.\eta(\cdot,t+\varepsilon)\right|_{[s,s+\delta]}$, and, at last, from $\eta(s,t+\varepsilon)$ to $\eta(s,t)$ along $\left.\eta(s,\cdot)\right|_{[t,t+\varepsilon]}$ is expressed by \[p4.1\] For any $C^2$ L-transport, we have $$\begin{aligned} & & \!\!\!\! \!\!\!\! \!\!\!\!\! L_{t+\varepsilon\to t}^{\eta(s,\cdot)} \circ L_{s+\delta\to s}^{\eta(\cdot,t+\varepsilon)} \circ L_{t\to t+\varepsilon}^{\eta(s+\delta,\cdot)} \circ L_{s\to s+\delta}^{\eta(\cdot,t)} = \nonumber \\ & & \!\!\!\!\!\!\! \!\!\!\! =id_{\pi^{-1}(\eta(s,t))} - \delta\varepsilon{\mathcal R}^{\eta}(s,t) + O(\delta^3) + O(\delta^2\varepsilon) +O(\delta\varepsilon^2) + O(\varepsilon^3). \label{4.1} \end{aligned}$$ *Proof.* In a field $ \{e_i(s,t),\ (s,t)\in J\times J^\prime\}$ of bases in $\pi^{-1}(\eta(J,J^\prime))$ the matrix of the linear map standing in the l.h.s. of (\[4.1\]) is $$H(t,t+\varepsilon;\eta(s,\cdot)) H(s,s+\delta;\eta(\cdot,t+\varepsilon)) H(t+\varepsilon,t;\eta(s+\delta,\cdot)) H(s+\delta,s;\eta(\cdot,t)).$$ Substituting here (\[2.5\]) and using the expressions $$\Gamma_{\eta(s+\delta,\cdot)}(t)=\Gamma_{\eta(s,\cdot)}(t) + \delta{\partial\over\partial s}\Gamma_{\eta(s,\cdot)}(t) +O(\delta^2),$$ $$\Gamma_{\eta(\cdot,t+\varepsilon)}(s)=\Gamma_{\eta(\cdot,t)}(s) + \varepsilon{\partial\over\partial t}\Gamma_{\eta(\cdot,t)}(s) + O(\varepsilon^2),$$ we find this matrix to be $$\begin{aligned} &\!\!\!& \!\!\!\!\!\!\!\!\!\!\!\openone \! + \! \delta\varepsilon\left( {\partial\over{\partial t}}\Gamma_{\eta(\cdot,t)}(s) - {\partial\over{\partial s}}\Gamma_{\eta(s,\cdot)}(t) - \Gamma_{\eta(\cdot,t)}(s) \Gamma_{\eta(s,\cdot)}(t) + \Gamma_{\eta(s,\cdot)}(t) \Gamma_{\eta(\cdot,t)}(s) \right) \!\!+ \\ & & + \> O(\delta^3) +O(\delta^2\varepsilon) +O(\delta\varepsilon^2) +O(\varepsilon^3). \end{aligned}$$ Taking into account (\[2.10\]), we get this expression as $$\label{4.1a} \openone - \delta\varepsilon \left[ \left({\mathcal R}^\eta(s,t)\right)_{j}^{i} \right] +O(\delta^3)+O(\delta^2\varepsilon) +O(\delta\varepsilon^2) + O(\varepsilon^3)$$ which is simply the matrix form of (\[4.1\]).  Proposition \[p4.1\] shows that *up to third order terms the result of the above-described transportation of a vector $A\in\pi^{-1}(\eta(s,t))$ is* $$\label{4.2} A -\delta\varepsilon\left( {\mathcal R}^\eta(s,t)\right) (A).$$ Another corollary of (\[4.1\]) is the (equivalent to the definition of the curvature) equality $$\label{4.3} {\mathcal R}^\eta(s,t) = - \lim_{ \stackrel{\delta\to 0}{\varepsilon\to 0} } \left[{1\over \delta\varepsilon} \left( L_{t+\varepsilon\to t}^{\eta(s,\cdot)}\!\circ\! L_{s+\delta\to s}^{\eta(\cdot,t+\varepsilon)}\!\circ\! L_{t\to t+\varepsilon}^{\eta(s+\delta,\cdot)}\!\circ\! L_{s\to s+\delta}^{\eta(\cdot,t)} - id_{\pi^{-1}(\eta(s,t))} \right) \right].$$ **Bianchi-type identities** {#V} =========================== The curvature operator (\[2.8\]) is simply a commutator of two derivations along paths. As we shall see below, the torsion (\[2.7\]) is also a skewsymmetric expression. All this allows one to apply the developed in [@f-Jacobi] method for obtaining Jacobi-type identities. This can be done as follows. Let us take an arbitrary map $\tau^k:J^k\to B$, with $J^k:=J\times\cdots\times J$, where $J$ appears $k$ times, $k\in \mathbb{N}$, and $B$ being the base of the vector bundle $(E,\pi,B)$. Let $s:=(s^1,...,s^k)\in J^k$. We define the $C^1$ paths $\tau_a:J\to B$ by $\tau_a(\sigma):=\left.\tau^k(s)\right|_{s^a=\sigma}$, $\sigma\in J$ and the maps (families of paths) $\tau_{ab}:J\times J\to B$ by $\tau_{ab}(\sigma_1,\sigma_2):=\left. \tau^k(s)\right|_{s^a=\sigma_1, s^b=\sigma_2}$, $ \sigma_1,\sigma_2\in J$ which depend implicitly on s. Hereafter $a,b,c,d=1,\ldots,k$. We write $\dot\tau_a$ for the tangent to $\tau_a$ vector field in the case when $ (E,\pi,B)=(T(M),\pi,M) $ for some manifold $M$. \[p5.1\] The following properties of antisymmetry are valid: $$\begin{aligned} & & {\mathcal R}^{\tau_{ab}}(s_a,s_b) + {\mathcal R}^{\tau_{ba}}(s_b,s_a)=0, \label{5.1}\\ & & T^{\tau_{ab}} + T^{\tau_{ba}} =0 \ {\mathrm{or}}\ {\mathcal T}^{\,\tau_{ab}}(s_a,s_b) + {\mathcal T}^{\,\tau_{ba}}(s_b,s_a) =0. \label{5.2} \end{aligned}$$ [**Remark**]{} These equalities are analogues of the usual skewsymmetry of curvature and torsion tensors in the tensor analysis [@Schouten/Ricci]. *Proof.* The two-point Jacobi-type identity is $ \left( (A_{ab})_{[a,b]} \right)_{<a,b>} \equiv 0 $ (see [@f-Jacobi eq. (5.1)]), where $ A_{ab} $ are elements of an Abelian group, $ (A_{ab})_{[a,b]}:=A_{ab} - A_{ba} $ and $ \left(A_{ab}\right)_{<a,b>}:=A_{ab} + A_{ba} $. Substituting here $ A_{ab}={\mathcal D}^{\tau_a} \circ {\mathcal D}^{\tau_b} $ in the case of a vector bundle $ (E,\pi,B) $ and $ A_{ab}={\mathcal D}^{\tau_a}({\dot\tau_b})$ in the case of the tangent to a manifold $M$ bundle $(T(M),\pi,M)$ and using (\[2.8\]) and (\[2.12\]) (or (\[2.7\])), one gets respectively (\[5.1\]) and (\[5.2\]).  \[p5.2\] The following identities are valid: $$\begin{aligned} & & \begin{array}{l} \left\{ {\mathcal D}^{\tau_a} \circ {\mathcal R}^{\tau_{bc}}(s_b,s_c) - {\mathcal R}^{\tau_{bc}}(s_b,s_c) \circ {\mathcal D}^{\tau_a} \right\} _{<a,b,c>} \equiv 0 \ \mathrm{or}\ \\ \left\{ {\mathcal D}^{\tau_a} \left( {\mathcal R}^{\tau_{bc}}(s_b,s_c) \right) \right\}_{<a,b,c>} \equiv 0, \end{array} \label{5.3} \\ & & \ \left\{ \left( {\mathcal R}^{\tau_{ab}}(s_a,s_b) \right)(\dot\tau_c) \right\} _{<a,b,c>} \equiv \left\{ {\mathcal D}^{\tau_a} \left( {\mathcal T}^{\,\tau_{bc}} \right) \right\}_{<a,b,c>}, \label{5.4}\end{aligned}$$ where $<\ldots>$ means summation over the cyclic permutations of the corresponding indices. [**Remark.**]{} These identities are analogous, respectively, of the second and first Bianchi identities in tensor analysis [@Schouten/physics; @Helgason]. This is clear from the fact that due to the antisymmetries (\[5.1\]) and (\[5.2\]) the cyclization over the indices $a,\ b\ \mathrm{and}\ c$, i.e. the operation $<\ldots>$, in (\[5.3\]) and (\[5.4\]) may be replaced with antisymmetrization over the indices $a,\ b \ \mathrm{and}\ c$. (E.g. if $A_{abc}=-A_{acb}$ and $ \left(A_{abc}\right)_{[a,b,c]} := \left(A_{abc}+A_{bca}+A_{cab}\right)_{[b,c]}, $ then $2\left(A_{abc}\right)_{<abc>} = \left(A_{abc}\right)_{[abc]}$.) *Proof.* The (3-point) generalized Jacobi identity (see [@f-Jacobi eq. (5.2)]) is $\left(\left(A_{abc}\right)_{[a,[b,c]]}\right)_{<a,b,c>} \equiv 0,$ with $A_{abc}$ being elements of an Abelian group, $ \left(A_{abc}\right)_{[a,[b,c]]} := \left(A_{abc} - A_{bca}\right)_{[b,c]} $ and $\left(A_{abc}\right)_{<a,b,c>}:=A_{abc} + A_{bca} + A_{cab}$. We put $A_{abc}=\mathcal{D}^{\tau_a}\circ\mathcal{D}^{\tau_b}\circ \mathcal{D}^{\tau_c} $ in the vector bundle case and $A_{abc}=\left(\mathcal{D}^{\tau_a} \circ \mathcal{D}^{\tau_b}\right) (\dot\tau_c)$ in the tangent bundle case. In this way, after some simple algebra (see (\[2.8\]), (\[2.7\]) and (\[2.1\])-(\[2.3\])), we get respectively (\[5.3\]) and (\[5.4\]).  The 4-point generalized Jacobi-type identity $$\left\{ \left( A_{abcd} \right) _ {[a,[b,[c,d]]]} + \left( A_{adcb} \right) _ {[a,[d,[c,b]]]} \right\}_{<a,b,c,d>}\equiv 0$$ with $ \left( A_{abcd} \right) _ {[a,[b,[c,d]]]} := \left( A_{abcd} - A_{bcda} \right) _ {[b,[c,d]]} $ and $ \left( A_{abcd} \right) _ {<a,b,c,d>} := A_{abcd} + A_{bcda} + A_{cdab} + A_{dabc}$ also produces an interesting identity in our case. In fact, putting $ A_{abcd} = \mathcal{D}^{\tau_a}\circ\mathcal{D}^{\tau_b}\circ \mathcal{D}^{\tau_c}\circ\mathcal{D}^{\tau_d}$ in the vector bundle case, one can easily prove after some simple calculations \[p5.3\] The identity $$\left\{ \mathcal{R}^{\tau_{ab}}(s_a,s_b) \left(R^{\tau_{cd}} \right) \right\}_{<a,b,c,d>} \equiv 0, \label{5.5}$$ where $R^{\tau_{cd}}$ is the curvature vector field on $\tau^k(J,\ldots,J)$ is valid. [**Remark.**]{} This result generalizes eq. (6.5) of [@f-Jacobi] in the classical tensor case. The last result also follows from the evident chain identity $$\begin{aligned} 0 &\equiv& \left\{ \mathcal{R}^{\tau_{ab}}(s_a,s_b) \circ \mathcal{R}^{\tau_{cd}}(s_c,s_d) - \mathcal{R}^{\tau_{ab}}(s_a,s_b) \circ \mathcal{R}^{\tau_{cd}}(s_c,s_d) \right\}_{<a,b,c,d>} \equiv \\ &\equiv& \left\{ \mathcal{R}^{\tau_{ab}}(s_a,s_b) \circ \mathcal{R}^{\tau_{cd}}(s_c,s_d) - \mathcal{R}^{\tau_{cd}}(s_c,s_d) \circ \mathcal{R}^{\tau_{ab}}(s_a,s_b) \right\}_{<a,b,c,d>} \equiv \\ &\equiv& \left\{ \left( \mathcal{R}^{\tau_{ab}}(s_a,s_b) \left( R^{\tau_{cd}} \right) \right) \left( \tau_{cd}(s_c,s_d) \right) \right\}_{<a,b,c,d>} \equiv \\ &\equiv& \left( \left\{ \mathcal{R}^{\tau_{ab}}(s_a,s_b) \left( R^{\tau_{cd}} \right) \right\}_{<a,b,c,d>} \right) \left(\tau^k(s)\right). \end{aligned}$$ Note that in the tangent bundle case the substitution $$A_{abcd} = \left( \mathcal{D}^{\tau_a}\circ\mathcal{D}^{\tau_b}\circ \mathcal{D}^{\tau_c} \right) \bigl( \dot\tau_d \bigr)$$ leads to the trivial identity $0\equiv 0$. **Conclusion** {#VI} ============== In this paper we have examined some natural properties of the curvature (resp. the torsion) of linear transports along paths in vector bundles (resp. in the tangent bundle to a manifold). These properties are similar to the ones in the theory of linear connections. The cause for this similarity is that in the case of the parallel transport assigned to a linear connection our results reproduce the corresponding ones in the classical tensor analysis. The reduction to the known classical results can easily be proved by applying the used in [@f-LTP-Cur+Tor] method for introduction of curvature and torsion of a linear connection by means of its parallel transport. In connection with this below is presented the generalization of the theorem that a linear connection is flat iff the assigned to it parallel transport is independent of the path (curve) along which it acts and depends only on the initial and final points of the transportation. \[th6.1\] An L-transport in $ (E,\pi,B) $ is flat on $ U\subseteq B $ if and only if in $U$ it is independent of the path (lying in $U$) along which it acts and depends only on its initial and final points, i.e. the set $\{ L_{s\to t}^{\gamma}\}$ forms a flat L-transport in $U\subseteq B$ iff $L_{s\to t}^{\gamma}$ for $\gamma:J\to U$ depends only on the points $\gamma(s)$ and $\gamma(t)$, but not on the path $\gamma$ itself. [**Remark.**]{} In this theorem we implicitly suppose $U$ to be linearly connected, i.e. its every two points can be connected by a path lying entirely in $U$. Otherwise the theorem may not be true. *Proof.* Let the L-transport $L$ be flat, i.e. $ \mathcal{R}^\eta(s,t)\equiv 0 \ \mathrm{for}\ \eta:J\times J^\prime\to U\subseteq B. $ By [@f-LTP-Cur+Tor theorem 3.1] there is a field of bases $\{e_i\}$ on $U$ in which the matrix of $L$ is unit, i.e. $H(t,s;\gamma)=\openone$, $\gamma:J\to U$. In these bases for $u\in\pi^{-1}(\gamma(s))$, we have $L_{s\to t}^{\gamma}u= H_j^i(t,s;\gamma)u^j\left(e_i\left.\right|_{\gamma(t)}\right)= u^i\left(e_i\left.\right|_{\gamma(t)}\right)$, which evidently depends on the points $\gamma(s)$ and $\gamma(t)$ but not on the path $\gamma$ itself. Conversely, let for $\gamma:J\to U$ the transport $L_{s\to t}^{\gamma}$ depends only on the points $\gamma(s)$ and $\gamma(t)$ and not on the path $\gamma$ connecting them. For fixed $x_0\in U$ and basis $\{ e_{i}^{0} \}$ in $\pi^{-1}(x)$ we define on $U$ the field of bases $\{e_i\}$ by $e_i\left.\right|_x := L_{a\to b}^{\beta} e_{i}^{0}$, where $\beta$ is any path in $U$ joining $x_0$ and $x\in U$, and such that $\beta(a):=x_0\ \mathrm{and}\ \beta(b):=x$. By assumption $\{ e_i\left.\right|_x \}$ depends only on $x$ but not on $\beta$. Using that $L_{s\to t}^{\gamma}$ depends only on $\gamma(s)$ and $\gamma(t)$, we have $$\begin{aligned} L_{s\to t}^{\gamma} \left( { e_i\left.\right|_{\gamma(s)} }\right) &=& L_{a\to b}^{\alpha} \left( { e_i\left.\right|_{\alpha(a)} }\right)\>=\\ &=&\> L_{a\to b}^{\alpha} \left( L_{c\to a}^{\alpha} { e_i^0} \right) = L_{c\to b}^{\alpha} { e_i^0} = { e_i\left.\right|_{\alpha(b)} } = { e_i\left.\right|_{\gamma(t)} }, \end{aligned}$$ where $\alpha$ is any path in $U$ such that $\alpha(a)=\gamma(s),\ \alpha(b)=\gamma(t)$, and $\alpha(c)=x_0$. As $L_{s\to t}^{\gamma} \left( { e_i\left.\right|_{\gamma(s)} } \right)= H_{i}^{j}(t,s;\gamma) { e_j\left.\right|_{\gamma(t)} }$, we see that in $\{e_i\}$ the matrix of $L$ is $H(t,s;\gamma)=\openone$, which, again by [@f-LTP-Cur+Tor theorem 3.1], implies the flatness of $L$ in $U$.  In conclusion we have to note that all of the results of the present paper remain true in the complex case. For this purpose one has simply to replace in it the word ‘real’ with ‘complex’ and the symbols $\mathbb{R}$ and $\dim$ with $\mathbb{C}$ and $\dim_\mathbb{C}$ respectively. **Acknowledgement** {#VII .unnumbered} =================== This work was partially supported by the National Science Foundation of Bulgaria under Grant No. F642. [^1]: Permanent address: Department Mathematical Modeling, Institute for Nuclear Research and Energy, Bulgarian Academy of Sciences, Boul. Tzarigradsko chaussée 72, 1784 Sofia, Bulgaria. [^2]: E-mail address: bozho@inrne.bas.bg [^3]: URL: http://www.inrne.bas.bg/mathmod/bozhome/ [^4]: All of the results of this work are valid mutatis mutandis in the complex case too.

--- abstract: 'Splines are a popular and attractive way of smoothing noisy data. Computing splines involves minimizing a functional which is a linear combination of a fitting term and a regularization term. The former is classically computed using a (weighted) L2 norm while the latter ensures smoothness. Thus, when dealing with grid data, the optimization can be solved very efficiently using the DCT. In this work we propose to replace the L2 norm in the fitting term with an L1 norm, leading to automatic robustness to outliers. To solve the resulting minimization problem we propose an extremely simple and efficient numerical scheme based on split-Bregman iteration combined with DCT. Experimental validation shows the high-quality results obtained in short processing times.' address: ' Department of Electrical and Computer Engineering, Duke University.[^1]\' author: - - bibliography: - 'mtepper.bib' title: 'L1 Splines for Robust, Simple, and Fast Smoothing of Grid Data' --- Introduction ============ Smoothing a dataset consists in finding an approximating function that captures important patterns in the data, while disregarding noise or other fine-scale structures. Let $y \in {{\mathbb R}}^{n_1 \times \dots \times n_m} \rightarrow {{\mathbb C}}$ be an $m$-dimensional discrete signal, where $n_j$ ($1 \leq j \leq d$) is the domain of $y$ along the $j$-th dimension. We can model $y$ by $$y = \hat{y} + r , \label{eq:model}$$ where $r$ represents some noise and $\hat{y}$ is a smooth function. A very common regularization choice is to enforce $C^2$ continuity, in which case $\hat{y}$ is called a (cubic) spline. Smoothing $y$ relies upon finding the best estimate of $\hat{y}$ under the proper smoothness and noise assumptions. We can approximate $\hat{y}$ by minimizing an objective functional $${\mathrm{F}}(z) = {\mathrm{R}}_y (z) + s {\mathrm{P}}(z) ,$$ where ${\mathrm{R}}_y (z)$ is a data fitting term, defined by the distribution of $r$, ${\mathrm{P}}(z)$ is a regularization term, and $s$ is a scalar that determines the balance between both terms. Such scalar parameter can be automatically derived using Bayesian or MDL techniques, as will be later shown. For clarity, we will describe in depth the case $m=1$, where we have $n=n_1$ samples. We extend the results later to the general $m$-dimensional case. Let us begin by explaining the smoothing term. The $C^2$ continuity requirement leads to define ${\mathrm{P}}(z) = {\left\| Dz \right\|_{2}}^ 2$, $D$ being a discrete *second-order* differential operator, defined $\forall i, 2 \leq i \leq n-1$, by $$\begin{aligned} D_{i, i-1} &= \tfrac{2}{h_{i-1} (h_{i-1} + h_{i})} ,\\ D_{i, i} &= \tfrac{-2}{h_{i-1} \ h_{i}} ,\\ D_{i-1, i} &= \tfrac{2}{h_{i} (h_{i-1} + h_{i})} ,\end{aligned}$$ where $h_i$ represents the step, or sampling rate, between $y_i$ and $y_{i+1}$. Assuming repeating border elements, that is, $y_0 = y_1$ and $y_{n+1} =y_n$, gives $D_{1, 1} = -D_{1, 2} = -1/h_1^2$, and $D_{n, n-1} = -D_{n, n} = -1/h_{n-1}^2$. Regarding the fitting term, the classical assumption is that the noise $r$ in Equation () has Gaussian distribution with zero mean and unknown variance, which leads to setting ${\mathrm{R}}_y (z) = {\left\| z - y \right\|_{2}}^2$. Smoothing then can be formulated as the least-squares regression $$\hat{y} = {\underset{z}{\operatorname{argmin}}\ } {\left\| z - y \right\|_{2}}^2 + s {\left\| Dz \right\|_{2}}^ 2 .$$ For clarity, we call the estimate $\hat{y}$ obtained with this method an L2 spline. It is a well known fact that least squares estimates for regression models are highly non-robust to outliers. Although there is no agreement on a universal and formal definition of an outlier, it is usually regarded as an observation that does not follow the patterns in the data. Notice that smoothing should produce an estimate $\hat{y}$ taking into account only important patterns, that is, the inliers, in the data. In this sense, the L2 formulation cannot correctly handle outliers by itself. In order to solve this problem, we propose to take a different assumption on the distribution of the noise $r$ in Equation (). By choosing a distribution with fatter tails than the Gaussian distribution, the derived estimator will correctly handle outliers. We thus assume that $r$ follows a Laplace distribution with zero mean and unknown scale parameter, a common practice in other problems as we will further discuss below. This leads to the fitting term ${\mathrm{R}}_ y(z) = {\left\| z - y \right\|_{1}}$ and the regression then becomes $$\hat{y} = {\underset{z}{\operatorname{argmin}}\ } \ {\left\| z - y \right\|_{1}} + s {\left\| Dz \right\|_{2}}^2 .$$ We call the estimate $\hat{y}$ obtained with this formulation an L1 spline. Let us point out that the use of L1 fitting terms for solving inverse problems is not new. For example, in 2001, Nikolova proved the theoretic pertinence of using L1 fitting terms for image denoising [@nikolova02]. Other interesting works have addressed this approach for total variation image denoising [@alliney97; @chan2004; @aujol06; @nikolova12] or total variation optical flow [@zach07; @wedel09; @raket11] (this robustness-to-outliers type of ideas was previously introduced in the context of optical flow by Black and Anandan [@black91]). Recall that the total-variation regularization term involves first-order derivatives, while the proposed L1 splines, on the other hand, involve second-order derivatives. Regularly sampled signals are extremely common in practice, and their analysis becomes easier and faster. In particular, we follow the most common choice when dealing with discrete $m$-dimensional data, which is assuming a “rectangular” Cartesian sampling pattern. When the sampling is isotropic, i.e., “square,” we refer to this type of data as grid data. We developed an iterative algorithm for computing L1 splines, based on split-Bregman iteration [@goldstein09], that is specially suited for the case of grid data. This algorithm is extremely fast, both in running time and in the number of iterations until convergence. It is also outstandingly simple, making the implementation completely straightforward. The remainder of the paper is structured as follows. In Section \[sec:L2splines\], we overview a fast algorithm to compute L2 splines and robust L2 splines, a modification of the least squares regression that allows to handle outliers. In Section \[sec:L1splines\], we present the proposed algorithm for computing L1 splines. Then, in Section \[sec:results\], we show results obtained with L1 splines, systematically outperforming its L2 and robust L2 counterparts in the presence of outliers. We also show that the proposed computational algorithm is very efficient. Finally, in Section \[sec:conclusions\] we provide some concluding remarks. Smoothing splines {#sec:L2splines} ================= As aforementioned, the classical assumption is that the noise $r$ in Equation () follows a Gaussian distribution with zero mean and unknown variance. This leads to solve the least-squares regression $$\hat{y} = {\underset{z}{\operatorname{argmin}}\ } {\left\| z - y \right\|_{2}}^2 + s {\left\| Dz \right\|_{2}}^ 2 . \label{eq:LSformulation}$$ Since both terms are differentiable, we obtain $$\hat{y} = (I + s {D^\mathrm{T}} D)^{-1} y . \label{eq:LSsolution}$$ Garcia proposed a very efficient method for dealing with regularly sampled data [@garcia10]. Assuming that the data are equally spaced, that is, without loss of generality $\forall i, h_i = 1$, we obtain $$D = \begin{pmatrix} -1 & 1 \\ 1 & -2 & 1 \\ & \ddots & \ddots & \ddots \\ && 1 & -2 & 1 \\ &&& 1 & -1 \end{pmatrix} .$$ An eigendecomposition of $D$ yields $D = U \varLambda {U^\mathrm{T}}$, where $\varLambda$ is a diagonal matrix containing the eigenvalues of $D$, given by [@yueh05] $$\varLambda_{i,j} = \begin{cases} -2 + 2 \cos((i-1) \pi / n) , & \text{if $i=j$;} \\ 0 , & \text{otherwise.} \end{cases}$$ Since $U$ is a unitary matrix, we can write Equation () as $$\hat{y} = U (I + s \varLambda^2)^{-1} {U^\mathrm{T}} y . \label{eq:LSsolutionDecomposed}$$ Let us define the matrix $\varGamma = (I + s \varLambda^2)^{-1}$. Trivially, $$\varGamma_{i, j} = \begin{cases} \left[ 1 + s (-2 + 2 \cos((i-1) \pi / n))^2 \right]^{-1} , & \text{if $i=j$;} \\ 0 , & \text{otherwise.} \end{cases} \label{eq:varGamma}$$ Following Strang [@strang99] and Garcia [@garcia10], let us observe that ${U^\mathrm{T}}$ is a DCT-II matrix and $U$ is an inverse DCT-II matrix. Then, Equation () can be expressed as $$\hat{y} = {\mathrm{DCT^{-1}}} (\varGamma \ {\mathrm{DCT}} (y)) , \label{eq:LSsolutionDCT}$$ where ${\mathrm{DCT}}(\cdot)$ and ${\mathrm{DCT^{-1}}}(\cdot)$ stand for the DCT-II and inverse DCT-II functions. Equation () provides a fast and simple algorithm for computing L2 splines. Robust estimation. {#sec:robustL2} ------------------ Often in practice there are in $y$ some values $y_i$ that could not be observed (or recorded) for some reason. We would like to be able to handle such cases in such a way that the missing values are inferred from the ones that can be observed. Let $W$ be an $n \times n$ diagonal matrix such that $W_{i, i}$ represents a weight assigned to observation $i$. $W$ is defined by $$W_{i, i} = \begin{cases} 0 &\text{if datapoint $i$ is missing;} \\ \rho &\text{otherwise.} \end{cases}$$ where $\rho$ is some arbitrary constant in $(0, 1]$; in practice, and without loss of generality, we set $\rho=1$. We can then solve $$\hat{y} = {\underset{z}{\operatorname{argmin}}\ } {\left\| W^{1/2} (z - y) \right\|_{2}}^2 + s {\left\| Dz \right\|_{2}}^ 2 , \label{eq:WLSformulation}$$ which will simply omit the missing points from the computation of the residual while the regularizer will still have a smoothing effect over both present and missing points. Equation () acts as an impainting algorithm, filling the missing values in such a way that continuity between filled values and smoothed ones is preserved. The minimization of Equation () gives $$(I + s {D^\mathrm{T}}D) \hat{y} = W (y - \hat{y}) + \hat{y} .$$ This leads to the iterative procedure $$\hat{y}^{k+1} = (I + s {D^\mathrm{T}}D)^{-1} \left( W \left( y - \hat{y}^k \right) + \hat{y}^k \right) ,$$ which, similarly to Equation (), becomes $$\hat{y}^{k+1} = {\mathrm{DCT}}^{-1} \left( \varGamma \ {\mathrm{DCT}} \left( W \left(y - \hat{y}^k \right) + \hat{y}^k \right) \right) . \label{eq:WLSsolutionDCT}$$ On a different note, real data often present observations that lie abnormally far from their “true” value, i.e., that do not appear to follow the pattern of the other data points. The main drawback of the penalized least squares formulation Equation () is its sensitivity to these outliers. To address this issue, weights can be assigned to every point, as in Equation (), such that outliers exert less influence during the estimation process. In this case, the weights are iteratively refined during the estimation process using robust estimators for the mean and variance of the data. Defining these estimators is a complex problem by itself. For details about how $W$ can be set and updated for added robustness to outliers, refer to Garcia’s work [@garcia10]. L1 splines {#sec:L1splines} ========== In this section we introduce a different splines formulation in order to handle outliers in the data. We assume that the noise $r$ in Equation () follows a Laplace distribution with zero mean and unknown scale parameter, which leads to ${\mathrm{R}}_ y(z) = {\left\| z - y \right\|_{1}}$. The regression then becomes $$\min_{z} \ {\left\| z - y \right\|_{1}} + s {\left\| Dz \right\|_{2}}^2 . \label{eq:L1formulation}$$ Goldstein and Osher [@goldstein09] proposed a very elegant and efficient algorithm for solving the L1 constrained problem (related to a number of very efficient optimization algorithms, e.g., see [@combettes11]) $$\min_u {\left\| \Phi(u) \right\|_{1}} + {\mathrm{H}}(u) .$$ For this, they consider the equivalent problem $$\min_u {\left\| d \right\|_{1}} + {\mathrm{H}}(u) \quad \text{s.t.} \quad d = \Phi(u) ,$$ which they first convert it into the unconstrained problem $$\min_u {\left\| d \right\|_{1}} + {\mathrm{H}}(u) + \tfrac{\lambda}{2} {\left\| d - \Phi(u) \right\|_{2}}^2 .$$ In this form, the penalty function does not accurately enforce the constraint for small $\lambda$. The constraint is enforced by letting $\lambda \rightarrow \infty$. However, another solution for this new formulation is found by using the following two-phase algorithm $$\begin{aligned} \left( u^{k+1}, d^{k+1} \right) &= {\underset{u}{\operatorname{argmin}}\ } {\left\| d \right\|_{1}} + {\mathrm{H}}(u) + \tfrac{\lambda}{2} {\left\| d - \Phi(u) - b^k \right\|_{2}}^2 ,\\ b^{k+1} &= b^k + \left( \Phi(u^{k+1}) - d^k \right) .\end{aligned}$$ This algorithm is often denoted in the literature as split-Bregman iteration. This class of algorithms has several nice theoretical properties and has successfully been applied to several problems in practice such as image restoration [@osher05], image denoising [@xu07], compressed sensing [@yin08], and image segmentation [@goldstein10]; see also [@combettes11] and references therein. We use this technique for solving Equation (). We begin by setting $\Phi(z) = z-y$ and $H(z) = s {\left\| Dz \right\|_{2}}^2$, which leads to the problem $$\min_{z, d} \ {\left\| d \right\|_{1}} + s {\left\| Dz \right\|_{2}}^ 2 \quad \text{s.t.} \quad d = z - y .$$ We then transform it into the unconstrained form $$\min_{z, d} \ {\left\| d \right\|_{1}} + s {\left\| Dz \right\|_{2}}^ 2 + \tfrac{\lambda}{2} {\left\| d - z + y \right\|_{2}}^2 , \label{eq:L1formulationExtended}$$ and the Bregman iteration simply takes the form $$\begin{aligned} \left( z^{k+1}, d^{k+1} \right) & = {\underset{z, d}{\operatorname{argmin}}\ } \ {\left\| d \right\|_{1}} + s {\left\| Dz \right\|_{2}}^ 2 + \tfrac{\lambda}{2} {\left\| d - z + y - b^k \right\|_{2}}^2 , \label{eq:splitBregman1}\\ b^{k+1} & = b^k + (z^{k+1} - y - d^{k+1}) . \label{eq:splitBregman2}\end{aligned}$$ Because of the splitting of the L1 and L2 components in the functional (), we can perform this minimization efficiently by iteratively minimizing with respect to $z$ and $d$ separately, $$\begin{aligned} z^{k+1} &= {\underset{z}{\operatorname{argmin}}\ } \ s {\left\| Dz \right\|_{2}}^ 2 + \tfrac{\lambda}{2} {\left\| d^k - z + y - b^k \right\|_{2}}^2 \label{eq:splitBregman1_1} , \\ d^{k+1} &= {\underset{d}{\operatorname{argmin}}\ } \ {\left\| d \right\|_{1}} + \tfrac{\lambda}{2} {\left\| d - z^{k+1} + y - b^k \right\|_{2}}^2 \label{eq:splitBregman1_2} .\end{aligned}$$ For minimizing Equation () we set $\tilde{y} = d^k + y - b^k$ and $\tilde{s} = 2s / \lambda$. We obtain $$\min_{z} \ \tilde{s} {\left\| Dz \right\|_{2}}^ 2 + {\left\| \tilde{y} - z \right\|_{2}}^2 .$$ This is a classical L2 spline and can be minimized using Equation (), as already explained. The optimal value of $d$ in Equation () can be explicitly computed using shrinkage operators, $$\begin{aligned} d^{k+1} &= {\underset{d}{\operatorname{argmin}}\ } \ {\left\| d \right\|_{1}} + \tfrac{\lambda}{2} {\left\| d - z^{k+1} + y - b^k \right\|_{2}}^2 \nonumber \\ &= {\mathrm{Shrink}} (z^{k+1} - y + b^k , 1/\lambda) ,\end{aligned}$$ where $$\begin{aligned} {\mathrm{Shrink}}(v, \gamma) = \begin{pmatrix} {\mathrm{shrink}}(v_1, \gamma) \\ \vdots \\ {\mathrm{shrink}}(v_j, \gamma) \\ \vdots \\ {\mathrm{shrink}}(v_m, \gamma) \end{pmatrix} \intertext{and} {\mathrm{shrink}}(x, \gamma) = \frac{x}{|x|} \max(|x|-\gamma, 0) . \label{eq:shrinkage}\end{aligned}$$ We thus obtain a very efficient algorithm for computing L1 splines, combining DCT and shrinkage operators. On a different note let us mention that Equation () can also be interpreted [@mateos12] as a relaxation of $$\min_{z, d} {\left\| d \right\|_{0}} + s {\left\| Dz \right\|_{2}}^2 + \tfrac{\lambda}{2} {\left\| d-z+y \right\|_{2}}^2 , \label{eq:L0formulation}$$ where the L0 norm is replaced by its (convex) L1 counterpart. In this case the underlying model for $y$ is $y = \hat{y} + r + d$, where $r$ is zero-mean Gaussian noise and $d$ represents the “oulier” noise. Under this assumptions, $d$ practically becomes an “indicator function” of the presence of (sparse) outliers (see also [@black91]). Besides the different angle in the derivation of the model, our approach differs from [@mateos12] in two very important points. First, we use split-Bregman iteration by introducing the variable $b^k$ in the optimization procedure, see equations () and (). In [@mateos12] Equation () is first solved using direct alternate minimization over $z$ and $d$, and then Equation () is solved via non-convex minimization using the previous solution as a starting point. Second, considering the grid structure, we use the DCT approach to solve Equation (), instead of the classical Cholesky decomposition. The combination between split-Bregman and DCT results in a sound and fast algorithm for computing L1 splines on grid data. Handling missing data --------------------- In the classical L2 formulation, a diagonal binary weighting matrix $W$ is used to cope with missing values (see Section \[sec:robustL2\] for details). Let us denote by $w$ the diagonal of $W$. Let us first define $$\begin{aligned} {\left\| z \right\|_{1, w}} &= \sum_{\substack{i=1 \\ w_i = 1}}^{m} |z_i| & \text{and} && {\left\| z \right\|_{2, w}} &= \left( \sum_{\substack{i=1 \\ w_i = 1}}^{m} z_i^2 \right)^{1/2} .\end{aligned}$$ We then pose Equation () as $$\begin{gathered} \hat{y} = {\underset{z}{\operatorname{argmin}}\ } {\left\| (z - y) \right\|_{2, w}}^2 + s {\left\| Dz \right\|_{2}}^ 2 ,\end{gathered}$$ and equivalently extend Equation () as $$\begin{gathered} \hat{y} = {\underset{z}{\operatorname{argmin}}\ } {\left\| (z - y) \right\|_{1, w}} + s {\left\| Dz \right\|_{2}}^ 2 .\end{gathered}$$ This leads to $$\min_{z, d} \ {\left\| d \right\|_{1, w}} + s {\left\| Dz \right\|_{2}}^ 2 + \tfrac{\lambda}{2} {\left\| d - z + y \right\|_{2, w}}^2 .$$ Then the split-Bregman iteration can be written as $$\begin{aligned} z^{k+1} &= {\underset{z}{\operatorname{argmin}}\ } \, s {\left\| Dz \right\|_{2}}^ 2 + \tfrac{\lambda}{2} {\left\| d^k - z + y - b^k \right\|_{2,w}}^2 , \label{eq:WsplitBregman1_1} \\ d^{k+1} &= {\underset{d}{\operatorname{argmin}}\ } \, {\left\| d \right\|_{1,w}} + \tfrac{\lambda}{2} {\left\| d - z^{k+1} + y - b^k \right\|_{2,w}}^2 , \label{eq:WsplitBregman1_2} \\ b^{k+1} &= b^k + (z^{k+1} - y - d^{k+1}) . \label{eq:WsplitBregman2}\end{aligned}$$ Equation () can be solved using Equation (). Solving Equation () amounts to performing a shrinkage operation on the dimensions where $w$ equals 1. In Equation (), it suffices to update the dimensions of $b^k$ where $w$ equals 1. Handling multidimensional data ------------------------------ Let us now return to the general case of $m$-dimensional data. Following Garcia [@garcia10], we extend Equation () as $$\hat{y} = {\mathrm{DCT}}^{-1}_m \left( \varGamma^m \circ {\mathrm{DCT}}_m (y) \right) , \label{eq:LSsolutionDCTd}$$ where ${\mathrm{DCT}}_m(\cdot)$ and ${\mathrm{DCT}}^{-1}_m(\cdot)$ stand for the $m$-dimensional DCT-II and inverse DCT-II functions, and $\circ$ denotes the Schur (element-wise) product. Notice that the multidimensional DCT is simply a composition of one-dimensional DCTs along each dimension. Extending Equation (), $\varGamma^m$ is an $m$-th order tensor defined by $$\varGamma^m = 1^m \div \left( 1^m + s \varLambda^m \circ \varLambda^m \right) , \label{eq:varGammad}$$ where $1^m$ is an $m$-th order tensor of ones, and $\div$ denotes the element-wise division. Finally, $\varLambda^m$ is an $m$-th order tensor, defined by $$\varLambda^m_{i_1, \dots, i_m} = \sum_{j=1}^{d} \left( -2 + 2 \cos \frac{(i_j - 1) \pi}{n_j} \right) . \label{eq:varLambdad}$$ where $n_j$ denotes the size of $\varLambda^m$ along the $j$-th dimension. **The algorithm and its complexity.** The pseudocode for the general $m$-dimensional case is presented in Algorithm \[algo:l1spline\]. Let us analyze its complexity. The DCT and inverse DCT require $O(n \log n)$ operations, where $n=\prod_{1 \leq j \leq m} \, n_j$. The remaining operations are linear in $m$. The overall complexity of the algorithm is then $O \left( N_o N_i (m + n \log n) \right)$, where $N_o$ and $N_i$ are, respectively, the number of outer-loop and inner-loop iterations in Algorithm \[algo:l1spline\]. Notice that Goldstein and Osher [@goldstein09] recommend to perform only one inner-loop iteration for achieving optimal efficiency. Thus, we set $N_i = 1$ for all experiments. We will later see that in many cases the algorithm converges quickly ($N_o$ can be very small then). The algorithm’s complexity is thus dominated by the computation of the DCT and inverse DCT. Of course, these standard operations can be easily computed using GPU, speeding-up the execution by several orders of magnitude. compute $\varGamma^m$ according to equations () and (). $d^1 \gets 0$, $b^1 \gets 0$, $k \gets 1$ $z^{k+1} \gets {\mathrm{DCT}}^{-1}_m (\varGamma^m \circ {\mathrm{DCT}}_m (d^k + y - b^k))$ $\displaystyle d^{k+1} \gets {\mathrm{Shrink}} (z^{k+1} - y + b^k , 1/ \lambda)$ $b^{k+1} \gets b^k + (z^{k+1} - y - d^{k+1})$ $k \gets k+1$ $z^k$ Experimental results {#sec:results} ==================== For all experiments we adhere to the following setup: 1. using generalized cross validation, we find the best estimate $\hat{s}$ for $s$ for the robust L2 formulation (problem ()); 2. we then find L2 splines (problem ()), robust L2 splines (problem ()),[^2] and/or L1 splines (problem ()), setting $s = \hat{s}$. This protocol allows us to show that, even when $s$ is chosen to fit optimally the robust L2 formulation, the proposed method provides better estimates. For the L1 formulation, in Equation (), we simply set $\lambda=\min(s,1)$ for all examples. We recall that $N_i$ is set to 1. We also set $\varepsilon = 10^{-3}$ (see Algorithm \[algo:l1spline\]) and additionally limit the maximum number of outer iterations to a hundred. The algorithm stops when any of the two conditions is met. Fig. \[fig:spline\_1D\_16\] presents two one-dimensional examples. We depict the original signal $\hat{y} \in [1, \dots, n] \rightarrow {{\mathbb R}}$, where $n=2^{16}$. We observe the signal $y = \hat{y} + r_1$, where $r_1$ is Gaussian noise. Some points $y_j$ ($1 \leq j \leq n$) are further contaminated with uniform noise $r_2$, where $r_2 \in [ a, \dots, b ]$, such that $y_j = \min (\max (\hat{y}_j + r_1 + r_2, a), b)$. The points affected by $r_2$ are depicted in red and the remaining ones in green. In the top row, $a = -5, b = 5$; and in the bottom row, $a = 0, b = 5$. In both examples, only the L1 spline is correct. The classical and robust L2 splines are both unable to correctly recover the original data in the corrupted part. Fig. \[fig:errorPlot\] shows the evolution of the relative error as the number of iterations increases for the example in Fig. \[fig:spline\_1D\_16\] (top row). As we can observe, the proposed algorithm is able to converge quickly, reaching a precision of $10^{-3}$ in less than 20 iterations. ![The relative error (in logarithmic scale) as the iterations progress when computing the first example in Figure \[fig:spline\_1D\_16\]. The algorithm is able to quickly decrease the error during the first 20 iterations.[]{data-label="fig:errorPlot"}](errorPlot.pdf){width=".5\columnwidth"} Fig. \[fig:timeDistribution\] depicts the relative time-cost of each operation during the execution of the proposed method (Fig. \[fig:spline\_1D\_16\], top row). Computing the DCT and the inverse DCT covers more than 84% of the total running time. Implementing these standard operations in GPU would boost the performance of the algorithm by orders of magnitude. ![Percentage of the execution time spent in each operation when computing the first example in Figure \[fig:spline\_1D\_16\]. Clearly, the vast majority of time is spent in DCT or inverse DCT operations.[]{data-label="fig:timeDistribution"}](timeChart.pdf){width=".5\columnwidth"} In Fig. \[fig:spline\_2D\_n30\] we present a two-dimensional example. We depict the original signal $\hat{y} \in [1, \dots, 256]^2\rightarrow [-6.5497, 8.1054]$ in Fig. \[fig:spline\_2D\_n30\_original\], and we add two types of noise: first, Gaussian noise $r_1$ with zero mean and variance $\sigma^2=2$ (Fig. \[fig:spline\_2D\_n30\_noisy\]), and then uniform noise $r_2$ in the interval $[ -5 \cdot \max (\hat{y} + r_1), \dots, 5 \cdot \max (\hat{y} + r_1) ]$ (Fig. \[fig:spline\_2D\_n30\_corrupted\]). Again, only the L1 spline correctly recovers the original signal. -- -- -- -- -- -- We next test the proposed algorithm with a climate time-series provided by the Met Office Hadley Centre [@brohan06].[^3] The dataset contains the evolution of global average land temperature anomaly (in ) with respect to the 1961-1990 average temperature. The results, which confirm an upward trend in the second half of the 20th century, are shown in Fig. \[fig:temp\]. We also test on this dataset the effect of varying the parameters $s$ and $\lambda$, see Fig. \[fig:params\]. As in the classical L2 formulation, $s$ has a direct impact on the obtained result, controlling the degree of smoothness of the solution, see Fig. \[fig:params\_s\]. On the contrary, Fig. \[fig:params\_lambda\] shows that the newly introduced parameter $\lambda$ is very stable and provides very similar results in a wide range $(\lambda \in [0.1, 100])$. This stability allows us to fix its value to $\lambda=1$ for all the experiments in this work. Another interesting example is presented in [@mateos12]. The dataset consists of power consumption measurements (in kW) for a government building, collected every fifteen minutes from July 2005 to October 2010. As in [@mateos12], we downsample the data by a factor of four, yielding one measurement per hour, and use only a subset of the whole data. The results are displayed in Fig. \[fig:load\]. ![Power consumption measurements (in kW) for a government building [@mateos12] with zoom-in details on the bottom.[]{data-label="fig:load"}](spline_1D_load-a.pdf "fig:"){width=".2\columnwidth"} ![Power consumption measurements (in kW) for a government building [@mateos12] with zoom-in details on the bottom.[]{data-label="fig:load"}](spline_1D_load-b.pdf "fig:"){width=".2\columnwidth"} ![Power consumption measurements (in kW) for a government building [@mateos12] with zoom-in details on the bottom.[]{data-label="fig:load"}](spline_1D_load-c.pdf "fig:"){width=".2\columnwidth"} We also test in a synthetic example the ability to recover signals with sharp transitions, see Fig. \[fig:square\]. In this case we use a simple piece-wise constant function. We can observe clear overshoot (plus ringing) effects on the L2 and robust L2 splines. The robust L2 spline also results in transitions with less vertical slopes, creating a bluring effect. With the L1 spline we obtain a much better reconstruction, with almost non-existent overshooting. This very same effect can be observed in real examples, see Fig. \[fig:119082\]. When approximating images with splines, some structure is lost by blur and some structure is artificially created by overshooting and ringing. This can be observed in Figs. \[fig:119082\_original\] and \[fig:119082\_noise\], were the difference between the original image and the image estimated by robust L2 splines exhibits structure. Observe, however, that almost no structure in the difference is visible when the reconstruction is performed using L1 splines. [@c@c@]{} Application to range data ------------------------- In this section we perform smoothing of depth data obtained with a Kinect camera. This kind of data is particularly challenging because: - it presents relatively smooth areas separated by sharp transitions, - edges are highly noisy, that is, edge pixels oscillate over time between foreground and background, and - it contains missing data, which appear for two different reasons: (1) the disparity between the IR projector and the IR camera produces “shadows,” and (2) the depth cannot be recovered in areas where the IR pattern is not clearly observable (e.g., because they receive direct sunlight or interference from another Kinect). We use splines to interpolate and denoise these data, showing the advantage of L1 splines over its robust L2 counterpart. The displayed images are part of the LIRIS human activities dataset [@harl2012]. In the first example, shown in Fig. \[fig:kinect2D\], we use a single depth frame (with standard Kinect resolution of $640 \times 480$). The missing data are represented in black, while depth data goes from red to yellow as depth increases. Both, the L1 spline and the robust L2 spline are able to interpolate the missing data with reasonable values. Notice, however, that the latter exhibits, as aforementioned, overshooting and ringing (clearly perceived in the 1D profile). These effects are much milder in the L1 reconstruction. [ccc]{} In Fig. \[fig:kinect3D\_original\] we can clearly observe that the position of the missing data is not consistent across frames. We can integrate data from several frames to achieve more accurate interpolations, by performing 3D reconstructions. Thus, in this example, we treat depth data as a 3D signal (2D + time), by considering three consecutive frames. The data dimensionality is then $640 \times 480 \times 3$. A full depth video can be smoothed by using 3D splines as a sliding-window type of filter. The robust L2 spline again presents a noisier behavior and with significative overshooting. On the other hand, the L1 spline is much smoother in smooth areas while correctly preserving abrupt transitions. **Running times.** We present in Table \[tab:runningTime\] the running-time and number of iterations until convergence for every example in this work. The time of the robust L2 and the L1 splines is comparable. All code is written in pure Matlab, with no C++ or mex optimizations. All experiments were run on a MacBook Pro with a 2.7GHz Intel Core i7 processor. Finally, note that in most cases the algorithm converges in less than twenty outer-iterations (recall that $\varepsilon=10^{-3}$). In the example in Fig. \[fig:spline\_2D\_n30\], the maximum number of iterations (100) is reached with a final error of $10^{-2.8}$. -- ------------------------------- --------------------------- ------------------ ------- -------- Robust L2 spline Time Time Iters. Fig. \[fig:spline\_1D\_16\] $2^{20}$ 4.931 3.590 7 Fig. \[fig:spline\_2D\_n30\] $256 \times 256$ 0.541 0.982 100 Fig. \[fig:temp\] 163 0.007 0.045 72 Fig. \[fig:load\] 501 0.010 0.010 11 Fig. \[fig:square\] $2^{10}$ 0.035 0.007 6 Fig. \[fig:119082\_original\] $321 \times 481$ 1.167 0.758 17 Fig. \[fig:119082\_noise\] $321 \times 481$ 1.143 0.873 20 Fig. \[fig:kinect2D\] $480 \times 640$ 0.911 0.266 2 Fig. \[fig:kinect3D\] $480 \times 640 \times 3$ 7.511 2.202 3 -- ------------------------------- --------------------------- ------------------ ------- -------- : Execution times (in seconds) and number of iterations until convergence of the proposed algorithm for the different experiments performed in this work.[]{data-label="tab:runningTime"} Conclusions {#sec:conclusions} =========== We have presented a new method for robustly smoothing regularly sampled data. We do this with modified splines, where we replace the classical L2-norm in the fitting term by an L1-norm. This automatically handles outliers, thus obtaining a robust approximation. We also presented a new technique, using split-Bregman iteration, for solving the resulting optimization problem. The algorithm is extremely simple and easy to code. The method converges very quickly and has a small memory footprint. It also makes extensive use of the DCT, thus being straightforward to implement in GPU. These characteristics make this method very suitable for large-scale problems. Acknowledgment {#acknowledgment .unnumbered} ============== Work partially supported by NSF, ONR, NGA, ARO, DARPA, and NSSEFF. We thank Dr. Gonzalo Mateos for kindly providing the power consumption dataset. [^1]: This work was partially done while the authors were with the Department of Electrical and Computer Engineering, University of Minnesota. [^2]: Code available at <http://www.mathworks.com/matlabcentral/fileexchange/25634-robust-spline-smoothing-for-1-d-to-n-d-data>. [^3]: Data are available in <http://hadobs.metoffice.com/crutem3/diagnostics/global/nh+sh/annual>.

--- abstract: 'In this paper, we propose a new inexact version of the projected subgradient method to solve nondifferentiable constrained convex optimization problems. The method combine $\epsilon$-subgradient method with a procedure to obtain a feasible inexact projection onto the constraint set. Asymptotic convergence results and iteration-complexity bounds for the sequence generated by the method employing the well known exogenous stepsizes, Polyak’s stepsizes, and dynamic stepsizes are stablished.' author: - 'A. A. Aguiar [^1]' - 'O. P. Ferreira' - 'L. F. Prudente' bibliography: - 'SubgradInexP.bib' title: Subgradient method with feasible inexact projections for constrained convex optimization problems --- [**Keywords:**]{} Subgradient method, feasible inexact projection, constrained convex optimization. [**AMS subject classification:**]{} 49J52, 49M15, 65H10, 90C30. Introduction ============ The Subgradient method is one of the most interesting iterative method for solving nondifferentiable convex optimization problems, which has its origin and development in the 60’s, see [@Ermolev1966; @Shor1985]. Since then, the subgradient method has attracted the attention of the scientific community working on optimization. One of the factors that explains this interest is its simplicity and ease of implementation. In particular, allowing a low cost of storage and ready exploitation of separability and sparsity. For these reasons, several variants of this method have emerged and properties of it have been discovered throughout the years, resulting in a wide literature on the subject; see, for exemple [@AlberIusemSolodov1998; @Yunier2013; @Bertsekas1999; @GoffinKiwiel1999; @KiwielBook1985; @NedicBertsekas2010] and the references therein. The aim of this paper is to present an inexact version of the projected subgradient method, which consists in using an inexact projection instead of the exact one, for minimizing a convex function $f: \mathbb{R}^n \to \mathbb{R}$ onto a closed and convex subset $C$ of $\mathbb{R}^n$. The proposed method, that we call [*Subgradient-InexP method*]{}, generates a sequence $\{x_k\}$ where each iteration consists of two stages. The first stage performs a step from the current iterate $x_k$ in the opposite direction of a $\epsilon$-subgradient of $f$ at $ x_k $ and the second inexactly projects the resulting vector onto the feasible set $C$. From the theoretical point of view, considering methods that use inexact projections are particularly interesting for the following reasons. Even when the projection onto a convex set is an easy problem, iterative methods provide only approximated solutions with small errors, due to round-off errors in floating-point arithmetics. Therefore, the study of inexact methods gives theoretical support for real computational implementations of exact schemes. On the other hand, in general, one drawback of methods that use exact projections is having to solve a quadratic problem at each stage, which may substantially increasing the cost per iteration if the number of unknowns is large. In fact, it may not be justified to compute exact projection when the current iterate $x_k$ is far from the solution of the problem in consideration. Moreover, a procedure for computing a feasible inexact projection may present a low computation cost per iteration in comparison with one that computes the exact projection. Thus, it seems reasonable to consider versions of projected subgradient method that compute the projection only approximately. In order to present formally and analyze the Subgradient-InexP method, we use the concept of feasible inexact projection with relative error, which was appeared in [@VillaSalzBaldassarre2013] (see also[@OrizonFabianaGilson2018]). It is worth noting that the concept of feasible inexact projection also accepts an exact projection when it is easy to obtain. For instance, the exact projections onto a box or a second order cone is very easy to obtain; see, respectively, [@NocedalWright2006 p. 520] and [@FukushimaTseng2002 Proposition 3.3]. A feasible inexact projection onto a polyhedral closed convex set can be obtained using quadratic programming methods that generate feasible iterates, such as feasible active set methods and interior point methods; see, for example, [@NicholasPhilippe2002; @NocedalWright2006; @Robert1996]. It is worth mentioning that, if the exact projection is used, then Subgradient-InexP method becomes the projected subgradient method considered in [@AlberIusemSolodov1998]. Several methods similar to the projected subgradient method have been studied in different papers, see [@GoffinKiwiel1999; @Mainge2008]. However, as far as we know, none of them use the concept of feasible inexact projection. The main tool used in our analysis of Subgradient-InexP method is a version of the inequality obtained in [@CorreaLemarecha1993 Lemma 1.1]; see also a variant of it in [@nedic_bertsekas2001 Lemma 2.1]. By using this inequality, we establish asymptotic convergence results and iteration-complexity bounds for the sequence generated by our method employing the well known exogenous stepsizes, Polyak’s stepsizes, and dynamic stepsizes. We point out that these stepsizes have been discussed extensively in the related literature, including [@AlberIusemSolodov1998; @GoffinKiwiel1999; @nedic_bertsekas2001rate; @nedic_bertsekas2001; @NedicBertsekas2010; @xmwang2018], where many of our results were inspired. Let us describe the results in the present and their relationship with the literature on the subject. With respect to the exogenous stepsize we establish convergence results without any compactness assumption, existence of a solution, and the iteration-complexity bound, which are similar to the well known bound presented in [@AlberIusemSolodov1998; @nedic_bertsekas2001]. In particular, for $C = \mathbb{R}^n$, the convergence results merge into the ones presented in [@CorreaLemarecha1993] and the iteration-complexity bound into [@Nesterov2004 Theorem 3.2.2]. The asymptotic convergence result and the iteration-complexity bound obtained using Polyak’s stepsizes are similar to the corespondent ones in [@nedic_bertsekas2001; @Nesterov2014; @Polyak1969] and [@Nesterov2014], respectively. Regarding to the dynamic stepsize, we establish global convergence in objective values as address, for example, in [@GoffinKiwiel1999; @nedic_bertsekas2001]. In [@nedic_bertsekas2001rate Proposition 2.15], the authors presented the rate of convergence for another variant of subgradient method, known as incremental subgradient algorithms. This study allowed us to estimate an iteraction-complexity bound for the dynamic stepsize. The organization of the paper is as follows. In Section \[sec:int.1\], we present some notation and basic results used in our presentation. In Section \[Sec:SubInexProj\] we describe the Subgradient-InexP method with different choices for the stepsize. The main results of the present paper, including the converge theorems and iteration-complexity, are presented in Section \[Sec:aca\]. Some numerical experiments are provided in Section \[Sec:NumExp\]. We conclude the paper with some remarks in Section \[Sec:Conclusions\]. Notation and definitions {#sec:int.1} ======================== In this section, we present some notations, definitions, and results used throughout the paper. We are interested in $$\label{eq:OptP} \min \{ f(x) :~ x\in C\},$$ where $C$ is a closed and convex subset of $\mathbb{R}^n$, $f:\mathbb{R}^n \to \mathbb{R}$ is a convex function. We denote by $$\label{eq:ValueOpt} f^*:= \inf_{x\in C} f(x),$$ its infimal value (possibly $-\infty$) and by $\Omega^*$ its solution set (possibly $\Omega^*= \varnothing$). The next concept will be useful in the analysis of the sequence generated by the subgradient method to solve . \[def:QuasiFejer\] A sequence $\{y_k\}\subset \mathbb{R}^n$ is said to be quasi-Fejér convergent to a nonempty set $W\subset \mathbb{R}^n$ if, for every $w\in W$, there exists a sequence $\{\delta_k\}\subset\mathbb{R}$ such that $\delta_k\geq 0$, $\sum_{k=1}^{\infty}\delta_k<+\infty$, and $$\|y_{k+1}-w\|^2\leq \|y_k-w\|^2+\delta_k, \qquad \forall~k=0, 1, \ldots.$$ When, $\delta_k= 0$, for all $k=0, 1, \ldots.$, $\{y_k\}$ is called Fejér convergent to a set $W$. The main property of the quasi-Fejér convergent sequence is stated in the next result, and its proof can be found in [@burachik1995full]. \[teo.qf\] Let $\{y_k\}$ be a sequence in $\mathbb{R}^n$. If $\{y_k\}$ is quasi-Fejér convergent to a nomempty set $W\subset \mathbb{R}^n$, then $\{y_k\}$ is bounded. If furthermore, a cluster point $y$ of $\{y_k\}$ belongs to $W$, then $\lim_{k\rightarrow\infty}y_k=y$. To describe the method for solving the problem  we need to define, for each $\epsilon \geq 0$, the $\epsilon$-subdifferential $\partial_{\epsilon} f(x)$ of a convex function $f$ at $x\in {\mathbb R}^n$, $$\label{eq:e-subdif} \partial_{\epsilon} f(x):=\{ s\in {\mathbb R}^n:~f(y)\geq f(x)+\langle s, y-x\rangle -\epsilon, ~\forall y\in {\mathbb R}^n\}.$$ We end this section by presenting important properties of the set $\epsilon$-subdifferential of a convex function, which proofs follow by combining [@Bertsekas2003 Proposition 4.3.1(a)] and [@UrrutyLemarechal1993_II Proposition 4.1.1, Proposition 4.1.2]. \[pr:CompE-subdif\] Let $f:\mathbb{R}^n \to \mathbb{R}$ be a convex function and $\epsilon \geq 0$. The set $\partial_{\epsilon} f(x)$ is nonempty, convex, and compact. Moreover, if $B\subset \mathbb{R}^n$ is a bounded set, then there exists a real number $L>0$ such that $\|s\|<L$, for all $s\in \cup_{x\in B} \partial_{\epsilon} f(x)$. In addition, if $\{\epsilon_k\}$ is a bounded sequence of nonnegative real numbers, the sequence $\{x_k\}$ converges to $x \in \mathbb{R}^n$, and $s_k \in \partial_{\epsilon_k} f(x_k)$ for all $k$, then the sequence $\{s_k\}$ is bounded. Subgradient-InexP method {#Sec:SubInexProj} ======================== Next, we present the subgradient method with a feasible inexact projections, which will be called [*Subgradient-InexP method*]{}. We begin by presenting the concept of relative feasible inexact projection, which is a variation of those presented in [@OrizonFabianaGilson2018; @VillaSalzBaldassarre2013]. \[def:InexactProj\] Let $C\subset {\mathbb R}^n$ be a closed convex set and $\varphi_{\gamma, \theta, \lambda}: {\mathbb R}^n\times {\mathbb R}^n\times {\mathbb R}^n \to {\mathbb R}_{+}$ be a relative error tolerance function such that $$\label{eq:fphi} \varphi_{\gamma, \theta, \lambda}(u, v, w)\leq \gamma \|v-u\|^2 + \theta \|w-v\|^2 + \lambda \|w-u\|^2, \qquad \forall~ u, v, w \in \mathbb{R}^n,$$ where $ \gamma, \theta, \lambda \geq 0$ are given forcing parameters. The [*feasible inexact projection mapping*]{} relative to $u \in C$ with relative error tolerance function $\varphi_{\gamma, \theta, \lambda}$, denoted by ${\cal P}_C(\varphi_{\gamma, \theta, \lambda},u, \cdot): {\mathbb R}^n \rightrightarrows C$ is the set-valued mapping defined as follows $$\label{eq:ProjI} {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v) := \left\{w\in C:~\left\langle v-w, z-w \right\rangle \leq \varphi_{\gamma, \theta, \lambda}(u, v, w), \quad \forall~ z \in C \right\}.$$ Each point $w\in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$ is called a [*feasible inexact projection of $v$ onto $C$ relative to $u$ and with relative error tolerance function $\varphi_{\gamma, \theta, \lambda}$*]{}. In the following, we present some remarks about the definition of the feasible inexact projection mapping onto the convex set $C$. \[rem: welldef\] Let $C\subset {\mathbb R}^n$, $u\in C$ and $\varphi_{\gamma, \theta, \lambda}$ be as in Definition \[def:InexactProj\]. Therefore, for all $v\in {\mathbb R}^n$, it follows from that ${\cal P}_C(0, u, v)$ is the exact projection of $u$ onto $C$; see [@Bertsekas1999 Proposition 2.1.3, p. 201]. Moreover, ${\cal P}_C(0, u, v) \in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$ concluding that ${\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)\neq \varnothing$, for all $u\in C$ and $v\in {\mathbb R}^n$. Consequently, the set-valued mapping ${\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, \cdot) $ is well-defined. Next lemma is a variation of [@Reiner_Orizon_Leandro2019 Lemma 6]. It will play an important role in the remainder of this paper. \[pr:cond\] Let $v \in {\mathbb R}^n$, $u \in C$, $\gamma, \theta, \lambda \geq 0$ and $w\in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$. Then, there holds $$\|w-x\|^2 \leq \|v-x\|^2 + \frac{2\gamma+2\lambda}{1-2\lambda}\|v-u\|^2, \qquad \forall ~x \in C,$$ for all $ \lambda, \theta \in [0, 1/2)$. Let $x \in C$. First note that $\|w-x\|^2 = \|v-x\|^2 - \|w-v\|^2 + 2 \langle v-w, x-w \rangle$. Since $w \in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$ and $0\leq \theta < 1/2$, combining the last equality with and we obtain $$\begin{aligned} \|w-x\|^2 &\leq \|v-x\|^2 - (1-2\theta)\|v-w\|^2 + 2\gamma \|v-u\|^2 + 2\lambda \|w-u\|^2\notag\\ &\leq \|v-x\|^2 + 2\gamma \|v-u\|^2 + 2\lambda \|w-u\|^2 \label{eq:fg}. \end{aligned}$$ On the other hand, we also have $$\begin{aligned} \|w-u\|^2 &= \|v-u\|^2 + \|w-v\|^2 + 2 \langle v-w,u-v \rangle \\ &= \|v-u\|^2 + \|w-v\|^2 + 2 \langle v-w,u-w \rangle - 2 \|w-v\|^2 \\ &= \|v-u\|^2 - \|w-v\|^2 + 2 \langle v-w,u-w \rangle. \end{aligned}$$ Thus, due to $w\in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$ and $u \in C$, using , , $0\leq \theta < 1/2$ and $0 \leq \lambda <1/2$, we have $$\|w-u\|^2 \leq \frac{1+2\gamma}{1-2\lambda}\|v-u\|^2 - \frac{1-2\theta}{1-2\lambda} \|w-v\|^2\leq \frac{1+2\gamma}{1-2\lambda}\|v-u\|^2.$$ Therefore, combining the last inequality with , we obtain the desired inequality. The conceptual subgradient method with feasible inexact projections for solving the Problem  is formally defined as follows:\ \[H\] Step 0. : Let $\{\epsilon_k\}$, $\{\theta_k\}$, and $\{\lambda_k\}$ be sequences of nonnegative real numbers. Let $x_0\in C$ and set $k=0$. Step 1. : If $0\in \partial f(x_k)$, then [**stop**]{}. Otherwise, choose a non-null element $s_k \in \partial_{\epsilon_k} f(x_k)$, compute a stepsize $t_k>0$, (to be specified later), and take the next iterate as any point such that $$x_{k+1} \in {\cal P}_C\left(\varphi_{ \gamma_k, \theta_k, \lambda_k}, x_k, x_k-t_ks_k\right).$$ Step 2. : Set $k\gets k+1$, and go to **Step 1**. Let us describe the main features of the subgradient-InexP method. Firstly, we check if the current iterate $x_k$ is a solution of Problem . If $x_k$ is not a a solution, then we choose a non-null element $s_k \in \partial_{\epsilon_k} f(x_k)$, compute a stepsize $t_k>0$, and take the next iterate $x_{k+1}\in C$ as any feasible inexact projection of $x_k-t_ks_k$ onto $C$ relative to $x_{k}$ with error tolerance given by $\varphi_{ \gamma_k, \theta_k, \lambda_k}(x_k, x_k-t_ks_k,x_{k+1}) $, i.e., $x_{k+1} \in {\cal P}_C\left(\varphi_{ \gamma_k, \theta_k, \lambda_k}, x_k, x_k-t_ks_k\right)$. We remark that if $\varphi_{ \gamma_k, \theta_k, \lambda_k}\equiv~0$, then ${\cal P}_C\left(0, x_k, x_k-t_ks_k\right)$ is the exact projection of $x_k-t_ks_k$ onto $C$, and Algorithm \[Alg:INP\] amounts to the projected subgradient method studied in [@AlberIusemSolodov1998]. Among the several possible choices that appeared in literature on subject, see for example [@Bertsekas1999; @nedic_bertsekas2001; @Shor1985], we studied three well known strategies, beginning with exogenous stepsize. \[Exogenous.Step\] Let $ \mu\geq 0$. Take exogenous sequences $\{\alpha_k\}$ and $\{\epsilon_k\}$ of nonnegative real numbers satisfying the following conditions: the sequence $\{\epsilon_k\}$ is nonincreasing and $$\label{eq:ExogSeq} \sum_{k=0}^{\infty}\alpha_k=+\infty, \qquad \qquad \sum_{k=0}^{\infty}\alpha_k^2<+\infty, \qquad \qquad \epsilon_k\leq \mu \alpha_ k, \qquad \qquad ~k=0, 1, \ldots.$$ Given $s_k \in \partial_{\epsilon_k} f(x_k)$, define the stepsize $t_k $ as the following nonnegative real number $$\label{eq:StepSize1} t_k:=\frac{\alpha_k}{\eta_k}, \qquad \qquad \eta_k:= \max\left\{1, \| s_k\|\right\}, \qquad \qquad ~k=0, 1, \ldots.$$ The stepsize in Rule \[Exogenous.Step\] is one the most popular. It have been used in several paper for analyzing subgradient method; see for example, [@AlberIusemSolodov1998; @CorreaLemarecha1993; @nedic_bertsekas2001rate; @nedic_bertsekas2001; @xmwang2018]. [*From now on we assume that there exist $0\leq \bar{\theta} < 1/2$ and $0\leq \bar{\lambda} < 1/2$, such that $\{\theta_k\}\subset [0, {\bar \theta})$, $\{\gamma_k\}\subset [0, {\bar \gamma})$ and $\{\lambda_k\}\subset [0, {\bar \lambda})$. For future references define*]{} $$\label{eq:nu} \nu := \frac{1+2{\bar \gamma}}{1-2{\bar \lambda}}> 0.$$ To define the next stepsize, we need to known the optimum value $f^*$ given in . In [@PolyakBook p.142], is present some examples of problems for which the optimum value are known. The statement of the Polyak’s stepsize is as follows. \[Poliak.Step\] Assume that $\Omega^*\neq\varnothing$ and the optimal value $f^*>-\infty$ is known. Let $\mu\geq 0$, $ \underline{\beta} >0$, ${\bar \beta} >0$ and take exogenous sequences $\{\beta_k\}$ and $\{\epsilon_k\}$ of real numbers satisfying the following conditions: the sequence $\{\epsilon_k\}$ is nonincreasing and $$\label{beta} 0< \underline{\beta} \leq \beta_k \leq \bar{\beta} < \frac{1}{2\mu +\nu}, \qquad\qquad 0<\epsilon_k\leq \mu \beta_ k[f(x_k)-f^*], \quad \qquad ~k=0, 1, \ldots.$$ Given $s_k \in \partial_{\epsilon_k} f(x_k)$, $s_k\neq 0$, define the stepsize $t_k $ as the following nonnegative real number $$\label{StepsizePolyak} t_k=\beta_k\frac{f(x_k)-f^*}{\left\|s_k\right\|^2}, \qquad \qquad ~k=0, 1, \ldots.$$ The stepsize in Rule \[Poliak.Step\] was introduced in [@Polyak1969] and has been used in several papers, including the ones [@nedic_bertsekas2001rate; @nedic_bertsekas2001; @xmwang2018]. In general, in practical problems the optimum value is not known. In this case, we may modify the stepsize by replacing the optimum value with a suitable estimate in each iteration. This leads to the dynamic stepsize rule as follows. \[Dynamic.Step\] Let $\mu\geq 0$, $ \underline{\beta} >0$, ${\bar\beta}>0$, and take exogenous sequences $\{\beta_k\}$ and $\{\epsilon_k\}$ of real numbers satisfying the following conditions: the sequence $\{\epsilon_k\}$ is nonincreasing and $$\label{betaDinamic} 0< \underline{\beta} \leq \beta_k \leq \bar{\beta} < \frac{2}{2\mu+\nu}, \qquad \qquad 0<\epsilon_k\leq \mu \beta_ k[f(x_k)-f_{k}^{lev}], \quad \qquad ~k=0, 1, \ldots,$$ where $f_{k}^{lev}$ will be specified later (see Section \[Sec:Analyisdynamic\]). Given $s_k \in \partial_{\epsilon_k} f(x_k)$ such that $s_k\neq 0$, define the stepsize $t_k $ as the following nonnegative real number $$\label{eq.dynamicstep} t_k= \frac{\tilde{t}_k}{\left\|s_k\right\|}, \qquad\qquad \tilde{t}_k = \beta_k\frac{f(x_k)-f_{k}^{lev}}{\left\|s_k\right\|}, \qquad\qquad ~k=0, 1, \ldots.$$ The dynamic stepsize in Rule \[Dynamic.Step\] is based on the ideas of [@brannlund1993]; see also [@GoffinKiwiel1999]. This rule has been used in several papers, see for example [@nedic_bertsekas2001; @NedicBertsekas2010; @xmwang2018]. [*From now on we assume that the sequence $\{x_k\}$ is generated by Algorithm \[Alg:INP\], with one of the three above strategies for choosing the stepsize, is infinite.*]{} Analysis of the subgradient-InexP method {#Sec:aca} ======================================== In the following, we state and prove our first result to analyze the sequence $\{x_k\}$ generated by Algorithm \[Alg:INP\]. The obtained inequality in next lemma is its counterpart for unconstrained optimization provided in [@CorreaLemarecha1993 Lemma 1.1)]. As we shall see, this inequality will be the main tool in our asymptotic convergence analysis, as well as in the iteration-complexity analysis. \[Le:FejerConv\] Let $\nu>0$ be as defined . For all $x \in C$, the following inequality holds $$\label{eq:MainIneq} \|x_{k+1}-x\|^2 \leq \|x_k-x\|^2 + \nu t_k^2\|s_k\|^2 - 2 t_k \left[f(x_k) - f(x) -\epsilon_k\right], \qquad k=0, 1, \ldots.$$ Let $x \in C$. To simply the notations we set $z_k:= x_k-t_ks_k$. Due to $x_{k+1} \in {\cal P}_C\left(\varphi_{\gamma_k, \theta_k, \lambda_k}, x_k, z_k\right)$ and $x_k \in C$, we apply Lemma \[pr:cond\] with $w=x_{k+1}$, $v=z_k$, $u=x_k$, and $\varphi_{ \gamma, \theta, \lambda}=\varphi_{ \gamma_k, \theta_k, \lambda_k}$ to conclude $$\label{eq:mip} \|x_{k+1}-x\|^2 \leq \|z_k-x\|^2+ \frac{2\gamma_k+2\lambda_k}{1-2\lambda_k}t_k ^2\|s_k\| ^2.$$ On the other hand, due to $z_k= x_k-t_ks_k$, after some algebraic manipulations, we obtain $$\|z_k-x\|^2 = \|x_k-x\|^2 + t_k ^2\|s_k\| ^2 + 2t_k \langle s_k, x - x_k \rangle.$$ Since $s_k \in \partial_{\epsilon_k} f(x_k)$, the definition implies that $\langle s_{k}, z-x_k\rangle\leq f(z)- f(x_k)+\epsilon_k$. Thus, $$\|z_k-x\|^2 \leq \|x_k-x\|^2 + t_k ^2\|s_k\| ^2 + 2t_k\left[f(x) - f(x_k) + \epsilon_k\right].$$ Therefore, combining last inequality with we conclude that $$\begin{aligned} \|x_{k+1}-x\|^2 &\leq \|x_k-x\|^2 + t_k ^2\|s_k\|^2 + 2t_k\left[f(x) - f(x_k) + \epsilon_k\right] + \frac{2\gamma_k+2\lambda_k}{1-2\lambda_k}t_k^2 \|s_k\|^2\\ &= \|x_k-x\|^2 + \frac{1+2\gamma_k}{1-2\lambda_k}t_k^2 \|s_k\|^2- 2 t_k \left[f(x_k) - f(x) -\epsilon_k\right]. \end{aligned}$$ Considering that $0 \leq \lambda_k < \bar{\lambda}<1/2$ and $0 \leq \gamma_k < \bar{\gamma}$, and using , we obtain . Analysis of the subgradient-InexP method with exogenous stepsize {#Sec:AnalyisExog} ---------------------------------------------------------------- In this section we will analyze the subgradient-InexP method with stepsizes satisfying Rule \[Exogenous.Step\]. For that, [*throughout this section we assume also that $\{x_k\}$ is a sequence generated by Algorithm \[Alg:INP\] with the stepsize given by Rule \[Exogenous.Step\] and, define $$\label{eq:rho} \rho:= \nu + 2\mu > 0.$$* ]{} First of all, note that under the above assumptions, Lemma \[Le:FejerConv\] becomes as follows. \[Le:FejerConvExog\] Let $\rho>0$ be as in . For all $x\in C$, the following inequality holds $$\|x_{k+1}-x\|^2\leq \|x_k-x\|^2 + \rho \alpha_k^2 - 2 \frac{\alpha_k}{\eta_k} \left[f(x_k)-f(x)\right], \qquad k=0, 1, \ldots.$$ The definition of $t_k$ in implies $t_k\leq \alpha_k$, which combined with the last inequality in yields $2t_k\epsilon_k\leq 2\mu \alpha_k^2$. Moreover, also implies that $t_k^2\|s_k\|^2\leq \alpha_k^2$. Therefore, using , the desired inequality follows directly from . To proceed with the analysis of Algorithm \[Alg:INP\], we also need the following auxiliary set $$\label{eq:DefOmega} \Omega:=\left\{x\in C:~f(x)\leq \inf_kf(x_k)\right\}.$$ It is worth mentioning that, in principle, set $\Omega$ can be empty and, in such case, $f^*=-\infty$. In the next lemma we analyze the behavior of the sequence $\{x_k\}$ under the hypothesis that $\Omega \neq \varnothing$. \[Le:BoundExog\] If $\Omega \neq \varnothing$, then $\{x_k\}$ is quasi-Féjer convergent to $\Omega$. Consequently, $\{x_k\}$ is bounded. Since $\Omega \neq \varnothing$, take $x\in \Omega$. Thus, by using the definition of $\Omega$ in and Lemma \[Le:FejerConvExog\], we conclude that $\|x_{k+1}-x\|^2\leq \|x_k-x\|^2 + \rho \alpha_k^2$, for all $k=0, 1, \ldots$. Hence, using the first inequality in , the first statement of the lemma follows from Definition \[def:QuasiFejer\]. The second statement of the lemma follows from the first part of Theorem \[teo.qf\]. Now, we are ready to prove the main result of this section, which refers to the asymptotic convergence of $\{x_k\}$. We remark that in the first part of the next theorem we do not assume neither $\Omega^* \neq \varnothing$ nor that $f^*$ is finite. \[teo.Main\] The following equality holds $$\label{eq:linfs} \liminf_k f(x_k)=f^*.$$ In addition, if $\Omega^* \neq \varnothing$ then the sequence $\{x_k\}$ converges to a point $x_*\in\Omega^*$. Assume by contradiction that $\liminf_k f(x_k) > f^*$. In this case, we have $\Omega \neq \varnothing$. Consequently by Lemma \[Le:BoundExog\], we conclude that $\{x_k\}$ is bounded. Letting $x\in \Omega$, there exist $\tau>0$ and $k_0\in\mathbb{N}$ such that $f(x)<f(x_k)-\tau,$ for all $k\geq k_0$. Hence, using Lemma \[Le:FejerConvExog\], we have $$\label{eq:bXk} \|x_{k+1}-x\|^2 \leq \|x_{k}-x\|^2 + \rho \alpha_k^2- 2 \frac{\alpha_ k}{\eta_k} \tau, \qquad k=k_0, k_0+1, \dots.$$ On the other hand, it follows from that the sequence $\left\lbrace \epsilon_k \right\rbrace$ is bounded. Thus, considering that $\{x_k\}$ is bounded, Proposition \[pr:CompE-subdif\] implies that $\{s_k\}$ is also bounded. Let $c > 0$ be such that $\|s_k\| \leq c$, for all $k \geq 0$. Hence, using second equality in , we have $\eta_k = \max\left\{1, \| s_k\|\right\} \leq \max\left\{1, c \right\} =: \varGamma$. Thus, letting $\ell \in \mathbb{N}$ and using , we conclude that $$\frac{2\tau}{\varGamma}\sum_{j=k_0}^{\ell+k_0}\alpha_j \leq \|x_{k_0}-x\|^2 - \|x_{k_0+ \ell+1}-x\| + \rho \sum_{j=k_0}^{\ell+k_0}\alpha^2_j \leq \|x_{k_0}-x\|^2 + \rho \sum_{j=k_0}^{\ell+k_0}\alpha^2_j.$$ Since the last inequality holds for all $\ell \in \mathbb{N}$ then, by using the first two conditions on $\{\alpha_k\}$ in , we have a contraction. Therefore, holds. For proving the last statement, let us assume that $\Omega^*\neq\varnothing$. In this case, we also have $\Omega\neq\varnothing$ and, from Lemma \[Le:BoundExog\], the sequence $\{x_k\}$ is bounded and quasi-Féjer convergent to $\Omega$. The equality implies that $\{f(x_k)\}$ has a decreasing monotonous subsequence $\{f(x_{k_j})\}$ such that $\lim_{j\rightarrow \infty}f(x_{k_j})= f^*.$ Without lose of generality, we can assume that $\{f(x_k)\}$ is decreasing, is monotonous, and converges to $f^*$. Being bounded, the sequence $\{x_k\}$ has a convergent subsequence $\{x_{k_\ell}\}$. Let us say that $\lim_{\ell\rightarrow\infty}x_{k_\ell}=x_*,$ which by the continuity of $f$ implies $f(x_*)=\lim_{\ell\rightarrow\infty}f(x_{k_\ell})=f^*,$ and then $x_*\in\Omega^*$. Hence, $\{x_k\}$ has an cluster point $x_*\in\Omega$, and due to $\{x_k\}$ be quasi-Féjer convergent to $\Omega$, Theorem \[teo.qf\] implies that $\{x_k\}$ converges to $x_*$. Next theorem presents an iteration-complexity bound; similar bound can be found in [@Nesterov2004 Theorem 3.2.2]. \[teo:complrule1\] Assume that the sequence $\{x_k\}$ converges to a point $x_*\in\Omega^*$. Then, for every $N \in \mathbb{N}$, the following inequality holds $$\min \{f(x_k) - f^{*}:~ \, k = 0, 1,\ldots, N \} \leq \varGamma \frac{\|x_0 - x_{*}\|^2 + \rho\sum_{k=0}^{N}\alpha_k^{2}}{2\sum_{k=0}^{N}\alpha_k}.$$ Since $\left\lbrace \epsilon_k \right\rbrace $ and $\left\lbrace x_k\right\rbrace $ are bounded sequences, then using Proposition \[pr:CompE-subdif\], it follows that $\left\lbrace s_k\right\rbrace$ is also bounded, i.e. there exists $c > 0$ such that $\|s_k\| \leq c$, for all $k \geq 0$. Therefore, using the definition of $\eta_k$ in , we have $\eta_k = \max\left\{1, \| s_k\|\right\} \leq \max\left\{1, c \right\} =: \varGamma$. Now, applying Lemma \[Le:FejerConvExog\] with $x = x_*$ and due to $f^* = f(x_*)$, we obtain $$\frac{2 \alpha_k}{\varGamma} [f(x_k)-f^*] \leq \|x_k-x_*\|^2 -\|x_{k+1}-x_*\|^2+ \rho\alpha_k^2 , \qquad k = 0,1, \ldots.$$ Hence, performing the sum of the above inequality for $k = 0, 1, \ldots, N,$ we have $$\frac{2}{\varGamma} \sum_{k=0}^{N} \alpha_k [f(x_k)-f^*] \leq \|x_0-x_*\|^2-\|x_{N+1}-x_*\|^2+\rho\sum_{k=0}^{N}\alpha_k^2.$$ Therefore, $$\frac{2}{\varGamma} \, \min \left\{f(x_k) - f^{*}: ~ \, k = 0, 1,\ldots, N \right\} \sum_{k=0}^{N}\alpha_k \leq \|x_0 - x_{*}\|^2 +\rho\sum_{k=0}^{N}\alpha_k^{2},$$ which is equivalent to the desired inequality. Analysis of the subgradient-InexP method with Polyak’s stepsize rule {#Sec:AnalysisPolyak} -------------------------------------------------------------------- In this section we will analyze the subgradient-InexP method with Polyak’s step sizes. [*Throughout this section, we assume also that $\Omega^* \neq \varnothing$ and $\{x_k\}$ is a sequence generated by Algorithm \[Alg:INP\] with the stepsize given by Rule \[Poliak.Step\]*]{}. \[Le:FejerConvPolyak\] Let $x\in \Omega^*$. Then, the following inequality holds $$\label{eq:MainIneqPolyak} \|x_{k+1}-x\|^2\leq \|x_k-x\|^2 - \underline{\beta} \frac{\left[f(x_k)-f^*\right]^2}{\|s_k\|^2}, \qquad\qquad k=0, 1, \ldots.$$ Considering that $x\in \Omega^*$ we have $f^*= f(x)$. The combination of with implies $2t_k\epsilon_k\leq 2\mu \beta_k^2 [f(x_k)-f^*]^2/\|s_k\|^2$. Moreover, also implies that $t_k^2\|s_k\|^2 = \beta_k^2[f(x_k)-f^*]^2/\|s_k\|^2$. Thus, we conclude form that $$\label{eq:mdpss} \|x_{k+1}-x\|^2\leq \|x_k-x\|^2 -\left(2-\nu\beta_k-2\mu \beta_k\right)\beta_k\frac{\left[f(x_k)-f^*\right]^2}{\|s_k\|^2}, \qquad k=0, 1, \ldots.$$ On the other hand, gives us $\beta_k<1/(2\mu+\nu)$, which is equivalent to $2-\nu\beta_k-2\mu \beta_k>1$. Therefore, since also gives $\underline{\beta}\leq \beta_k$, we conclude that implies . In the following theorem we present our main result about the asymptotic convergence of $\{x_k\}$. It has as correspondent result in [@Polyak1969 Theorem 1]; see also [@nedic_bertsekas2001]. \[teo.MainConvPolyak\] The sequence $\{x_k\}$ converges to a point $x_*\in\Omega^*$. Let $x\in \Omega^*$. Then, Lemma \[Le:FejerConvPolyak\] implies $\|x_{k+1}-x\|^2\leq \|x_k-x\|^2 $, for all $k=0, 1, \ldots$. Thus, $\{x_k\}$ is Fejér convergent to $\Omega^*$. Since $\Omega^* \neq \varnothing$, Theorem \[teo.qf\] implies that $\{x_k\}$ is bounded. By using Proposition \[pr:CompE-subdif\], we conclude that there exists $c>0$ such that $\left\|s_k\right\| \leq c$, for $k=0,1,\ldots$. Then, from , after some algebra, we have $$\left[f(x_k)-f^*\right]^2 \leq \frac{c^2}{\underline{\beta}}\left(\|x_k-x\|^2 - \|x_{k+1}-x\|^2\right), \qquad \qquad k=0, 1, \ldots.$$ Thus, performing the sum of the this inequality for $j=0, 1, \ldots, \ell$, we obtain $$\sum_{j=0}^{\ell}\left[f(x_j)-f^*\right]^2 \leq \frac{c^2}{\underline{\beta}}\left(\|x_{0}-x\|^2 - \|x_{ \ell+1}-x\|^2\right)\leq \frac{c^2}{\underline{\beta}} \|x_{0}-x\|^2.$$ Considering that this inequality holds for all $\ell\in \mathbb{N}$, we conclude that $\lim_{k\to +\infty}f(x_k)=f^*$. Let $x_*$ be a cluster point of $\{x_k\}$ and $\{x_{k_j}\}$ a subsequence of $\{x_k\}$ such that $\lim_{j\to +\infty}x_{k_j}=x_*$. Since $f$ is continuous, we have $f(x_*)= \lim_{j\to +\infty}f(x_{k_j})=f^*$. Therefore, $x_*\in\Omega^*$. Since $\{x_k\}$ is quasi-Fejér convergent to a set $\Omega^*$, it follows from Theorem \[teo.qf\] that $\{x_k\}$ converges $x_*$. The next result presents an iteration-complexity bound, which is a version of [@Nesterov2014 Theorem 1]. Assume that the sequence $\{x_k\}$ converges to a point $x_*\in\Omega^*$. Then, for every $N \in \mathbb{N}$, the following inequality holds $$\min \{f(x_k) - f^{*}: ~ \, k = 0, 1,\ldots, N \} \leq \frac{c}{\sqrt{\underline{\beta} (N+1)}} \|x_0 - x_{*}\|,$$ where $c\geq \max\{\|s_k\|:~ k=0, 1,\ldots \}$. Applying Lemma \[Le:FejerConvPolyak\] with $x = x_*$ , where $f^* = f(x_*)$, we obtain $$\underline{\beta} \frac{\left[f(x_k)-f^*\right]^2}{\|s_k\|^2} \leq \|x_k-x_*\|^2 - \|x_{k+1}-x_*\|^2, \qquad\qquad k=0, 1, \ldots.$$ Performing the sum of the above inequality for $k = 0, 1, \ldots, N,$ we conclude that $$\sum_{k=0}^{N} \frac{\left[f(x_k)-f^*\right]^2}{\|s_k\|^2} \leq \frac{1}{\underline{\beta}} \|x_0-x_*\|^2.$$ Since $\{x_k\}$ is bounded, by using Proposition \[pr:CompE-subdif\], we conclude that there exists $c>0$ such that $\left\|s_k\right\| \leq c$, for $k=0,1,\ldots$. Thus, we have $$\sum_{k=0}^{N} \left[f(x_k)-f^*\right]^2 \leq \frac{c^2}{\underline{\beta}} \|x_0-x_*\|^2.$$ Therefore, $$(N+1) \min \{\left[f(x_k)-f^*\right]^2:~ \, k = 0, 1,\ldots, N \} \leq \frac{c^2}{\underline{\beta}} \|x_0-x_*\|^2,$$ which is equivalent to the desired inequality. Analysis of the subgradient-InexP method with dynamic stepsize {#Sec:Analyisdynamic} -------------------------------------------------------------- Next we consider the Subgradient-InexP method employing the dynamic stepsize Rule \[Dynamic.Step\], which guarantees that $\{f_{k}^{lev}\} $ converges to the optimum value $f^*$. In the following we present formally the algorithm which compute $f_{k}^{lev}$. This scheme was introduced in [@brannlund1993]; see also [@GoffinKiwiel1999]. Step 0. : Select $x_0\in C, \delta_0 > 0$, and $R > 0$. Set $k=0, \sigma_0 = 0, f_{-1}^{rec} = \infty, \ell=0, k(\ell) = 0$. Step 1. : If $f(x_k) < f_{k-1}^{rec},$ set $f_{k}^{rec} = f(x_k)$ and $x_{k}^{rec}= x_k,$ else set $f_{k}^{rec} = f_{k-1}^{rec}$ and $x_{k}^{rec}=x_{k-1}^{rec}$ Step 2. : If $0\in \partial f(x_k)$, then [**stop**]{}. Step 3. : If $f(x_k) \leq f_{k(\ell)}^{rec} - \frac{1}{2} \delta_\ell $, set $k(\ell+1) = k, \sigma_k = 0, \delta_{\ell+1} = \delta_\ell$, replace $\ell$ by $\ell+1,$ and go to Step 5. Step 4. : If $\sigma_k > R$, set $k(\ell+1) = k, \sigma_k = 0, \delta_{\ell+1} = \frac{1}{2} \delta_\ell$, $x_k=x_k^{rec}$, and $\ell\leftarrow\ell+1$. Step 5. : Set $f_k^{lev} := f_{k(\ell)}^{rec} - \delta_\ell$. Select $\beta_k \in [\underline{\beta}, \bar{\beta}]$ and calculate $x_{k+1}$ via Algorithm \[Alg:INP\] with the stepsize given by Rule \[Dynamic.Step\]. Step 6. : Set $\sigma_{k+1}:= \sigma_k + \tilde{t}_k$, $k\leftarrow k+1$, and go to Step 1. Following [@GoffinKiwiel1999; @nedic_bertsekas2001rate; @nedic_bertsekas2001; @xmwang2018], we describe the main features of the Subgradient-InexP method. \[eq:bfkrec\] Note that in Step 1, $f_{k}^{rec}$ keeps the record of the smallest functional value attained by the iterates generated so far, i.e., $f_{k}^{rec}:=\min\{f(x_j):~j=0, \ldots,k\}$. Splitting the iterations into groups $$K_\ell := \{k(\ell), k(\ell) + 1, \ldots, k(\ell+1)-1\}, \quad \ell = 0, 1, \dots,$$ Algorithm \[Alg:INPDyn\] uses the same target level $f_k^{lev} = f_{k(\ell)}^{rec} - \delta_\ell$, for $k \in K_\ell$. Also, note that the target level is update only if sufficient descent or oscillation is detected (Step 3 or Step 4, respectively). Whenever $\sigma_k$ exceeds the upper bound $R$, the parameter $\delta_\ell$ is decreased, which increases the target level $f_k^{lev}$. From now on, we assume that Algorithm \[Alg:INPDyn\] generates an infinite sequence. In the next theorem we present the result about the asymptotic convergence of the sequence $\{x_k\}$. It is the versions of [@GoffinKiwiel1999 Theorem 1] and [@nedic_bertsekas2001 Proposition 2.7] by using inexact projections. \[teo.MainConvDynamic\] There holds $\inf_{k \geq 0} f(x_k)= f^*$. Since $x_k \in C$ and $x_{k+1} \in {\cal P}_C\left(\varphi_{\gamma_k, \theta_k, \lambda_k}, x_k, x_k - t_ks_k\right)$, by the first equality in and applying Lemma \[pr:cond\] with $w=x_{k+1}$, $v=x_k - t_ks_k$, $x=x_k$ , $u=x_k$, and $\varphi_{ \gamma, \theta, \lambda}=\varphi_{ \gamma_k, \theta_k, \lambda_k}$, we conclude that $$\|x_{k+1}-x_{k}\| \leq \sqrt{\frac{1+2 \bar{\gamma}}{1-2 \bar{\lambda}}} \,\,t_k\|s_k\| \leq \sqrt{\frac{1+2 \bar{\gamma}}{1-2 \bar{\lambda}}} \,\, \tilde{t}_k, \qquad k=0, 1, \ldots.$$ We claim that the index $\ell$ goes to $+\infty$ and either $\inf_{k \geq 0} f(x_k) =-\infty$ or $\lim_{l\to\infty} \delta_\ell = 0$. Indeed, assume that $\ell$ takes only a finite number of values, i.e., $\ell < \infty$. Since $\sigma_k + \tilde{t}_k = \sigma_{k+1} \leq R$, for all $k \geq k(\ell)$, then we conclude that $$\label{eq.boundxk} \|x_k - x_{k(\ell)}\| \leq \sum_{j=k(\ell)}^k \|x_{j+1} - x_j\| \leq \sqrt{\frac{1+2 \bar{\gamma}}{1-2 \bar{\lambda}}} \sum_{j=k(\ell)}^k \tilde{t}_j= \sqrt{\frac{1+2 \bar{\gamma}}{1-2 \bar{\lambda}}} \,\, \sigma_{k+1} \leq \sqrt{\frac{1+2 \bar{\gamma}}{1-2 \bar{\lambda}}} \,\, R.$$ Hence, $\{x_k\}$ is bounded. Besides, from the last condition in , the sequence $\{\epsilon_k\}$ is bounded and, by using Proposition \[pr:CompE-subdif\], $\{s_k\}$ is also bounded. Moreover, by we also conclude $\sum_{j=k(\ell)}^{+\infty} \tilde{t}_j<+\infty$, which implies $\lim_{k\to \infty} \tilde{t}_k = 0$. Thus, due to $\beta_k \in [\underline{\beta}, \bar{\beta}]$, it follows from second equality in that $$\label{eq.boundfk} \lim_{k \to \infty} [f(x_k) - f_k^{lev}] = 0.$$ On the other hand, Steps 3 and 5 of Algorithm \[Alg:INPDyn\] yield $$f(x_k) > f_{k(\ell)}^{rec} - \frac{1}{2} \delta_\ell = f_k^{lev} + \delta_\ell - \frac{1}{2} \delta_\ell = f_k^{lev} + \frac{1}{2} \delta_\ell \qquad \quad k= k(\ell), k(\ell)+1, \ldots,$$ contradicting . Therefore, $\ell$ goes to $+ \infty$. Now, suppose that $ \lim_{\ell\to\infty} \delta_\ell=\delta > 0$. Then, from Steps 3 and 4 of Algorithm \[Alg:INPDyn\], it follows that for all $\ell$ large enough, we have $\delta_\ell = \delta$ and $ f_{k(\ell+1)}^{rec} \leq f_{k(\ell)}^{rec} -\frac{1}{2} \delta, $ implying that $\displaystyle\inf_{k \geq 0} f(x_k) = -\infty$, which concludes the claim. If $\lim_{\ell\to\infty} \delta_\ell > 0$ then, according to above claim, we have $\inf_{k \geq 0} f(x_k) = -\infty$, obtain the desired result. Now, we assume by contradiction that $\lim_{\ell\to\infty} \delta_\ell = 0$ and $\inf_{k \geq 0} f(x_k) > f^*$. Thus, it follows from Remark \[eq:bfkrec\] that $\inf_{k \geq 0} f_{k}^{rec} =\inf_{k \geq 0} f(x_k) $. Hence, we conclude that $\inf_{k \geq 0} f_{k}^{rec}> f^*$. In this case, by using the definition of $\{f_k^{lev}\}$ in Step 5 and taking into account that $\lim_{\ell\to\infty} \delta_\ell = 0$, we conclude that $$\displaystyle\inf_{k \geq 0} f_k^{lev} =\displaystyle\inf_{\ell \geq 0} (f_{k(\ell)}^{rec} - \delta_\ell) = \displaystyle\inf_{\ell \geq 0} f_{k(\ell)}^{rec} > f^*.$$ Therefore, there exist $\bar{\delta}>0$, ${\bar x}\in C$ and $\bar{k} \in \mathbb{N}$ such that $$\label{eq:ahc} f_k^{lev} - f({\bar x}) \geq \bar{\delta}, \qquad k =\bar{k}, \bar{k}+1, \ldots.$$ Hence, by using the definition of $\tilde{t}_k$ in , it follows from that $$\label{eq:limtildtk} \tilde{t}_k = \beta_k\frac{f(x_k)-f_{k}^{lev}}{\left\|s_k\right\|} < \bar{\beta} \frac{f(x_k)-f({\bar x})-\bar{\delta}}{\left\|s_k\right\|}, \qquad k =\bar{k}, \bar{k}+1, \ldots.$$ Now, applying Lemma \[Le:FejerConv\] with $x={\bar x}$ and then using and , we obtain $$\|x_{k+1} - {\bar x}\|^2 \leq \|x_k-{\bar x}\|^2+ \tilde{t}_k \left(\nu \tilde{t}_k - \frac{2}{\|s_k\|} \left[f(x_k)-f({\bar x})-\epsilon_k\right] \right), \qquad k =\bar{k}, \bar{k}+1, \ldots.$$ Thus, the combination of the last inequality with , , and the last inequality in yields $$\|x_{k+1} - {\bar x}\|^2 \leq \|x_k-{\bar x}\|^2+ \frac{\tilde{t}_k}{\|s_k\|} \bigg(\left[(2\mu+\nu)\bar{\beta}-2\right]\left[f(x_k)-f({\bar x})\right] - \left( 2\mu+\nu\right)\bar{\beta}\bar{\delta}\bigg), \qquad k =\bar{k}, \bar{k}+1, \ldots.$$ It follows from that $\bar{\beta} < 2/(2\mu+\nu)$, which implies that $(2\mu+\nu)\bar{\beta}-2 < 0$. Thus, by using that $f(x_k)\geq f_k^{lev}$ for all $k=0, 1, \ldots$, the last inequality implies $$\label{eq:xk_fejerdynamic} \|x_{k+1} - {\bar x}\|^2 \leq \|x_k-{\bar x}\|^2 - \frac{\tilde{t}_k}{\|s_k\|} \left( 2\mu+\nu\right)\bar{\beta}\bar{\delta}, \qquad k =\bar{k}, \bar{k}+1, \ldots.$$ Hence, $\|x_{k+1}-{\bar x}\| \leq \|x_{\bar k}-{\bar x}\|$, for all $k \geq \bar{k}$, which implies that $\{x_k\}$ is bounded. Besides, by using , it follows from the last condition in that the sequence $\left\lbrace \epsilon_k \right\rbrace$ is also bounded. Thus, using Proposition \[pr:CompE-subdif\], we conclude that there exists $c>0$ such that $\left\|s_k\right\| \leq c$, for all $k \geq 0$, which together , yield $$\frac{{\bar \beta}\bar{\delta}}{c} \left(2\mu+\nu\right) \displaystyle\sum_{k = \bar{k}}^\infty \tilde{t}_k \leq \|x_{\bar{k}} - \bar{x}\|^2<+\infty.$$ Since $\sigma_{k} = \sum_{j= {k(\ell)}}^{ {k(\ell+1)}-1} \tilde{t}_j $, the last inequality implies that there exists $\ell_0 \in \mathbb{N}$ such that $$\sigma_{{k(\ell+1)}} \leq \displaystyle\sum_{k = {k(\ell)}}^\infty \tilde{t}_k < R, \qquad \quad ~\ell = \ell_0, \ell_0+1\ldots .$$ Hence, Step 4 in Algorithm \[Alg:INPDyn\] cannot occur infinitely to decrease $\delta_\ell$, contradicting the fact that $\displaystyle\lim_{\ell\to\infty} \delta_\ell = 0$. Therefore, the result follows and the proof is concluded. The next result presents an iteration-complexity bound for the subgradient-InexP method with the stepsize given by Rule \[Dynamic.Step\], which is a version of [@nedic_bertsekas2001rate Proposition 2.15] for our algorithm. Assume that the sequence $\{x_k\}$ converges to a point $x_*\in\Omega^*$. Let $ \delta_0>0$ be given in Algorithm \[Alg:INPDyn\] and $c\geq \max\{\|s_k\|:~k=0, 1,\ldots \}$. Then, $$\label{eq:minfxkdynamic} \min \{f(x_k) - f^{*}:~ \, k = 0, 1,\ldots, N \} \leq \delta_0,$$ where $N$ is the largest positive integer such that $$\label{eq:defNdynamic} \sum_{k=0}^{N-1}\left( \beta_k \left[2 - (2 \mu+\nu) \beta_k\right]\delta_k^2\right) \leq \left(c\|x_0 - x_*\|\right)^2.$$ Assume by contradiction that does not holds. Thus, for all $k$ with $0 \leq k \leq N$ we have $ f(x_k) > f^* + \delta_0. $ Hence, considering that $\delta_\ell \leq \delta_0$ for all $\ell$, we have $$\label{eq:fklevfstar} f_k^{lev} = f_{k(\ell)}^{rec}-\delta_\ell > f^* + \delta_0 - \delta_\ell \geq f^*, \quad \qquad k=0, \ldots, N.$$ The combination of the last inequality in with gives $2t_k\epsilon_k \leq 2\mu \beta_k^2 \left[f(x_k)-f_k^{lev}\right]^2 / \|s_k\|^2$. Moreover, implies that $t_k^2\|s_k\|^2 = \beta_k^2 \left[f(x_k)-f_k^{lev}\right]^2/\|s_k\|^2$. Now, using , Lemma \[Le:FejerConv\] with $x = x_* \in \Omega^*$, and since $\beta_k \in [\underline{\beta}, \bar{\beta}]$, we obtain $$\label{eq:cdinq} \|x_{k+1}-x_*\|^2 \leq \|x_k-x_*\|^2 - \beta_k \left[2 -(2 \mu+\nu) \beta_k\right]\frac{\left[f(x_k)-f_k^{lev}\right]^2}{\|s_k\|^2}.$$ Since $\{x_k\}$ converges to $x_*\in\Omega^*$, Proposition \[pr:CompE-subdif\] implies that there exists $c>0$ such that $\left\|s_k\right\| \leq c$, for $k=0,1,\ldots$. Furthermore, using the fact $f(x_k)-f_k^{lev} \geq \delta_k$, $0 \leq k \leq N$, yields $$\ \|x_{k+1}-x_*\|^2 \leq \|x_k-x_*\|^2 - \beta_k \left[2 -(2 \mu+\nu) \beta_k\right] \frac{\delta_k^2}{c^2}.$$ Performing the sum of the above inequality for $k = 0, 1, \ldots, N,$ we conclude that $$\sum_{k=0}^{N} \left(\beta_k \left[2 -(2 \mu+\nu) \beta_k\right]\frac{\delta_k^2}{c^2}\right) \leq \|x_0 - x_*\|^2,$$ which contradicts . Numerical results {#Sec:NumExp} ================= Our intention in this section is to report some numerical results in order to illustrate the practical behavior of SInexPD Algorithm when $C$ is a compact convex set. We implemented SInexPD Algorithm in Fortran 90 considering set $C$ in the general form $C= \left\{x\in\mathbb{R}^n:~ h(x)=0, g(x)\leq 0 \right\}$, where $h: \mathbb{R}^n \to \mathbb{R}^m $ and $g: \mathbb{R}^n \to \mathbb{R}^p$ are smooth functions. At each iteration $k$, the Frank-Wolfe algorithm is used to compute a feasible inexact projection as explained below. The algorithm codes are freely available at <https://orizon.ime.ufg.br/>. Frank-Wolfe algorithm to find an approximated projection {#Sec:CondG} -------------------------------------------------------- In this section we use the [*Frank-Wolfe algorithm*]{} also known [*conditional gradient method*]{} to find an inexact projection onto a compact convex set $C\subset \mathbb{R}^n$; papers dealing with this method include [@BeckTeboulle2004; @FrankWolfe1956; @JAGGI2013; @Konnov2018; @LanZhou2016; @Ravi2017]. The exact projection of $v\in \mathbb{R}^n$ onto $C$ is the solution of the following convex quadratic optimization problem $$\label{eq:ProbCond} {\min}_{w \in C} \psi(w) := \frac{1}{2}\|w-v\|^2.$$ Assume that $v\notin C$. Let us describe the subroutine, which we nominate [*FW-Procedure*]{}, for finding an approximated solution of relative to a point $u \in C$, i.e., a point belonging to the set ${\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$, where the error tolerance mapping $\varphi_{\gamma, \theta, \lambda}$ and the set valued mapping ${\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, .)$ are given in Definition \[def:InexactProj\]. Step 0. : Set $w_1 = u$ and $k=1$. Step 1. : Call the linear optimization oracle (or simply LO oracle) to compute $$\label{eq:condG} z_k := \arg\min_{z \in C} \langle w_k-v, z-w_k \rangle, \qquad g_k^*:= \langle w_k - v, z_k-w_k \rangle.$$ Step 2. : If $g^*_k \geq - \varphi_{\gamma, \theta, \lambda}(u, v, w_k)$, set $w_{+}:=w_k$ and [**stop**]{}; otherwise, compute $$\label{eq:stepsize} \tau_k: = \min\left\{1, \frac{-g^*_k}{\|z_k-w_k\|^2} \right\}, \qquad w_{k+1}:=w_k + \tau_k(z_k-w_k).$$ Step 3. : Set $k \gets k+1$, and go to [**Step 1**]{}. Since $\psi$ is strictly convex, we conclude from that $\psi(z) > \psi(w_k) + g_k^*$, for all $z \in C$ such that $z\neq w_k$. Setting $\psi^*:=\min_{w \in C} \psi(w)$ we have $\psi(w_k) \geq \psi^* \geq \psi(w_k) + g_k^*$, which implies $g_k^* <0.$ Thus, the stepsize $\tau_k$ given by is computed using exact minimization, i.e., $0<\tau_k := \arg\min_{\tau \in [0,1]} \psi(w_k + \tau(z_k - w_k))$. Since $C$ is convex and $z_k$, $w_k \in C$, we have from that $w_{k+1} \in C$, which implies that all points generated by [*FW-Procedure*]{} are in $C$. Moreover, implies that $g_k^* = \langle w_k - v, z_k - w_k \rangle \leq \langle w_k - v, z - w_k \rangle$, for all $z \in C$. Hence, if the stopping criteria $g_k^* = \langle w_k - v, z_k - w_k \rangle \geq - \varphi_{\gamma, \theta, \lambda}(u, v, w_k)$ in Step 2 of [*FW-Procedure*]{} is satisfied, then $ \langle v-w_k , z - w_k \rangle \leq \varphi_{\gamma, \theta, \lambda}(u, v, w_k)$, for all $z \in C$. Therefore, from Definition \[def:InexactProj\], we conclude that $w_{+}=w_k\in {\cal P}_C(\varphi_{\gamma, \theta, \lambda}, u, v)$, i.e., the output of [*FW-Procedure*]{}, is a feasible inexact projection of $v \in \mathbb{R}^n$ relative to $u \in C$. Finally, [@BeckTeboulle2004 Proposition A.2] implies that $\lim_{k\to +\infty} g_k^* =0$. Thus, the stopping criteria $g_k^* \geq - \varphi_{\gamma, \theta, \lambda}(u, v, w_k)$ in Step 2 of [*FW-Procedure*]{} is satisfied in a finite number of iterations if and only if $\varphi_{\gamma, \theta, \lambda}(u, v, w_k)\neq 0$, for all $k=0, 1, \ldots$. The following theorem is an import result about the convergence rate of the conditional gradient method applied to problem , which its proof can be found in [@GarberHazan2015]. For stating the theorem, we first note that $$\label{eq:PropPsi} \psi(w) - \psi(w^*) \leq \frac{1}{2} \|w-w^*\|, \qquad \forall z \in C;$$ see also [@Nesterov2004 Theorem 2.1.8]. \[th:fcr\] Let $d_C := \max_{z,w \in C} \|z-w\|$ be the diameter of C. For $k \geq 1$, the iterate $w_k$ of *FW-Procedure* satisfies $\psi(w_k) -\psi(w_*) \leq 8d_{C}^2/k$. Consequently, using , we have $\|w_k - w_*\| \leq 4d_{C}/\sqrt{k}$, for all $k \geq 1$. Examples -------- Consider the problem $$\label{numprob} \min_{x \in C} \, f(x) :=\|x\|_1,$$ where $C:=\left\{x\in\mathbb{R}^n \colon x\geq0 \mbox{ and } (x-\bar{x})^TQ(x-\bar{x})\leq 1 \right\}$ for a given vector $\bar{x}\in{\mathbb{R}}^n$ and a symmetric positive definite matrix $Q\in\mathbb{R}^{n\times n}$. Since the $\ell_1$ norm tends to promote sparse solutions, we formulated instances of Problem where there are vectors in $C$ with only one non-null component. Thus we can verify the ability of SInexPD Algorithm to recover sparsity. Let us describe the main characteristics of the considered instances. Consider the spectral decomposition of $Q$ given by $$Q=\sum_{i=1}^n\lambda_iv^i(v^i)^T,$$ where $\lambda_1 \geq \ldots \geq \lambda_{n-1} > \lambda_n>0$ are the eigenvalues of $Q$ and $\{v_1, v_2,\dots,v_n\}$ is an orthonormal system of corresponding eigenvectors. We assume that there exists $u\in \mathbb{R}^n_{++}$ such that $$\label{eq:id2} v_n=u/\|u\|, \quad \lambda_ n<1/\|u\|^2, \qquad \mbox{and} \qquad \bar{x}=u+ \xi e_n,$$ where $\xi\geq 1/\sqrt{\lambda_n}$ and $e_n\in{\mathbb{R}}^n$ is such that $e_n=[0,\ldots,0,1]^T$. We claim that $\tilde{x}:= \xi e_n\in C$ and $0\notin C$. Indeed, using we have $\tilde{x}-\bar{x}=-\|u\|v_n$ , which implies $$(\tilde{x}-\bar{x})^TQ (\tilde{x}-\bar{x})= \|u\|^2 v_n^TQv_n= \|u\|^2\lambda_ n<1,$$ concluding that $\tilde{x}\in C$. Now note that $0\in C$ if and only if $\bar{x}^TQ\bar{x} \leq 1$. Since $$\xi\geq \frac{1}{\sqrt{\lambda_n}} > -\langle u, e_n\rangle + \frac{1}{\sqrt{\lambda_n}} >\left(-\langle u, e_n\rangle+\sqrt{\langle u, e_n\rangle^2-(\|u\|^2-1/\lambda_n)} \right)> 0$$ and $\|u+ \xi e_n\|^2=\xi^2+2\langle u, e_n\rangle \xi +\|u\|^2$, we have $$\bar{x}^TQ\bar{x}\geq \lambda_n\|\bar{x}\|^2=\lambda_n\|u+ \xi e_n\|^2>1,$$ implying that $0\notin C$. For Problem , given $x\in{\mathbb{R}}^n$ we can get $s\in\partial f(x)$ by taking $$[s]_i := \left\{ \begin{array}{rl} -1, & \mbox{ if } [x]_i < 0 \\ 0, & \mbox{ if } [x]_i = 0 \\ 1, & \mbox{ if } [x]_i > 0, \\ \end{array} \right.$$ where $[\cdot]_i$ stands for the $i$-th component of the corresponding vector. For computing the optimal solution $z_k$ at Step 1 of the FW-Procedure, we use the software Algencan [@algencan], an augmented Lagrangian code for general nonlinear optimization programming. We set $R=\|x_1-x_0\|$ and $\delta_0=\|s_0\|/2$ as suggested in [@nedic_bertsekas2001] and [@GoffinKiwiel1999], respectively. Our implementation uses the stopping criterion $$\delta_{\ell}\leq 10^{-3}(1+|f_k^{rec}|),$$ also suggested in [@GoffinKiwiel1999]. Thus, in Algorithm \[Alg:INP\], we have $\epsilon_k=0$ for all $k$. In our tests, we set $x_0=\bar{x}$ and, for all $k$, $\theta_k=0.25$, $\lambda_k=0.025$, $\gamma_k=0.025$, and defined $\beta_k := 2 (1-2\lambda_k)/(1+2\gamma_k)-10^{-6}$ satisfying . Figure \[fig:Behavior\] shows the behavior of SInexPD Algorithm on a two-dimensional instance of Problem . The hatched region represents set $C$ and only the iterates for which the target level was updated are plotted. As can be seen, the algorithm successfully found the [*solution*]{} for $\ell = 6$ iterations. We point out that the algorithm performed a total of $\ell = 14$ ($k=189$) iterations until it met the stopping criterion. The highlight of the figure is that, before finding the solution, the iterates belong to the interior of set $C$. This is mostly due to the fact that SInexPD Algorithm performs inexact projections. ![Behavior of SInexPD Algorithm on a two-dimensional instance of Problem .[]{data-label="fig:Behavior"}](example.eps "fig:")\ Finally, we considered six instances of Problem varying the dimension $n$. Without attempting to go into details, we mention that the problems were randomly generated such that $\lambda_n\in(10^{-2},10^{-6})$, $\lambda_i\in(10,10^{3})$ for $i = 1,\ldots,n-1$, vector $u\in{\mathbb{R}}^n_{++}$ in is such that $\|u\| \in(0.8/\sqrt{\lambda_n},1/\sqrt{\lambda_n})$, and $\xi = 1\sqrt{\lambda_n}$. These imply that, with respect to the ellipsoid that makes up set $C$, the axis corresponding to the eigenvector $v_n$ is [*much larger*]{} than the others ones. Moreover, the vectors of $C$ that have only one non-null component are [*far*]{} from the center $\bar{x}$. These characteristics make problems more challenging for the algorithm. Table \[tab:Performance\] shows the performance of SInexPD Algorithm. In the table, column “$n$" informs the considered dimension, “$k$" and “$\ell$" are the number of iterations according to SInexPD Algorithm, “$\|x_k^{rec}\|_0$" is the number of non-null elements at the final iterate, and “$f_k^{rec}$" and “$\delta_{\ell}$" are their corresponding values at the final iterate. $n$ $k$ $\ell$ $\|x_k^{rec}\|_0$ $f_k^{rec}$ $\delta_{\ell}$ ------ ----- -------- ------------------- ------------- ----------------- 10 91 19 1 1.12D+01 6.18D-03 100 85 21 1 1.07D+01 9.77D-03 200 63 36 1 2.11D+01 1.38D-02 500 58 22 1 1.01D+01 1.09D-02 800 575 26 1 1.20D+01 6.91D-03 1000 669 24 1 1.15D+01 7.72D-03 : Performance of SInexPD Algorithm on six instances of Problem varying the dimension.[]{data-label="tab:Performance"} As showed in Table \[tab:Performance\], the algorithm found vectors with only one non-null component in all instances, showing its ability to recover sparsity in this class of problems. Remembering that the table data corresponds to the values when the stop criterion was met, we reported that the [*final*]{} iterates were found with $\ell = 10, 16, 27, 12, 15$ and $12$ iterations, respectively. We point out that, due to the inexact projections and mimicking the behavior of SInexPD Algorithm in the two-dimensional case, in each instance the iterates remained in the interior of $C$ before the corresponding solution was found. Conclusions {#Sec:Conclusions} =========== It is well known that the application of the subgradient method is only suitable for certain specific classes of non-differentiable convex optimization problems. However, this method is basic in the sense that it is the first step towards designing more efficient methods for solving that problems. Indeed, it is intrinsically related to cutting-plane and bundle methods; see [@UrrutyLemarechal1993_II]. These considerations lead us to conclude that the knowledge of new properties of the subgradient method has great theoretical value. In particular, our inexact version of the projected subgradient method will be useful in this theoretical context. Finally, one issue we believe deserves attention is the construction of inexact projected versions of cutting-plane and bundle methods. [^1]: Instituto de Matemática e Estatística, Universidade Federal de Goiás, CEP 74001-970 - Goiânia, GO, Brazil, E-mails: [ademiraguia@gmail.com]{}, [orizon@ufg.br]{}, [lfprudente@ufg.br]{}. The authors was supported in part by CNPq grants 305158/2014-7 and 302473/2017-3, FAPEG/PRONEM- 201710267000532 and CAPES.

--- abstract: 'A large-$N_{\rm c}$ expansion is combined with the Kubo formalism to study the shear viscosity $\eta$ of strongly interacting matter in the two-flavor NJL model. We discuss analytical and numerical approaches to $\eta$ and investigate systematically its strong dependence on the spectral width and the momentum-space cutoff. Thermal effects on the constituent quark mass from spontaneous chiral symmetry breaking are included. The ratio $\eta/s$ and its thermal dependence are derived for different parameterizations of the spectral width and for an explicit one-loop calculation including mesonic modes within the NJL model.' author: - Robert Lang - Wolfram Weise date: 'March 26, 2014' title: 'Shear viscosity from Kubo formalism: NJL-model study' --- Introduction ============ Heavy-ion collisions at RHIC [@BRAHMSatRHIC05; @PHENIXatRHIC05; @STARatRHIC05; @PHOBOSatRHIC05] and at the LHC [@AamodtALICE10; @AamodtALICE11; @CaffarriALICE12] explore strongly interacting matter under extreme conditions. The quark-gluon matter created in such collisions at temperatures exceeding $T_{\rm c}\approx 0.2$ GeV leaves its indirect signatures in the produced particles at lower temperatures long after thermalization. Transport properties such as the shear viscosity $\eta$ of the highly excited matter are of prime interest in this context. Inertial anisotropies in the collision plane translate into non-trivial particle flow patterns [@DerendarzATLAS2013; @RoyMohantyChaudhuri2013; @DuslingTeaney2008; @LuzumGombeaudOllitrault2010]. In particular the elliptic-flow parameter, $v_2$, features a strong dependence on the ratio of shear viscosity to entropy density, $\eta/s$, of the dissipative quark-gluon matter formed in the collision. It is known that the temperature dependence of $\eta/s$ is crucial in order to describe the elliptic flow of hadrons in ultrarelativistic heavy-ion collisions at RHIC and LHC [@Niemi2011]. At the very high LHC energies, the temperature dependence of $\eta/s$ for $T>T_{\rm c}$ becomes dominant for the elliptic flow. Two basic approaches are commonly used to deal with non-equilibrium systems and transport properties: the Boltzmann equation [@HuiDefuLic2006; @SasakiRedlich2010; @KhvorostukhinToneevVoskresensky2011] (most frequently applied in relaxation-time approximation) and the Kubo formalism using retarded correlators of the energy-momentum tensor [@Alberico2008; @HidakaKunihiro2011; @HidakaKunihiro2011NJL; @NamKao2013]. When examined in comparison, the kinetic approach seems generally to underestimate the shear viscosity [@PlumariEtAl2012; @PlumariEtAl2013]. The calculation in relaxation-time approximation makes use of the thermal cross sections of the colliding particles, whereas the Kubo formalism asks for the spectral functions of the basic degrees of freedom. These two approaches are connected via the optical theorem. In the present paper we choose the Kubo formalism, realizing at the same time that, in this approach, a perturbative treatment of transport coefficients is insufficient [@JeonSkeleton1995; @JeonYaffeSkeleton1996] and requires resummation techniques even in a weak-coupling situation. We use the Nambu–Jona-Lasinio (NJL) model [@Nambu1961tp; @Nambu1961; @Klevansky1992; @KLVW1990; @KLVW1990:2; @VoglWeise1991; @BuballaHabil2005] as a schematic, non-perturbative approach to the thermodynamics of quark matter. Gluonic degrees of freedom are integrated out and hidden in point vertices of the effective interaction between quarks, while all relevant chiral and flavor symmetries and symmetry-breaking patterns of QCD are taken properly into account. The applicability of such a model is supposed to cover a temperature range $T_{\rm c} \lesssim T < \Lambda$, where $T_{\rm c}\approx 0.2$ GeV is the transition temperature from the hadronic to the quark phase and $\Lambda\approx 0.6\;{\rm GeV}$ is the characteristic NJL cutoff scale. We combine the NJL model with a large-${N_{\rm c}}$ expansion [@tHooft1974; @QuackKlevansky94; @BuballaMuellerWambach2010] to study the shear viscosity of quark matter. In Section \[Sec:EtaLO\] the shear viscosity $\eta$ in leading order is deduced. We follow mainly the developments in [@Fukutome2006; @Fukutome2008Nucl; @Fukutome2008Prog] but do not restrict ourselves to the chiral limit and arrive at results under more general assumptions. Taking only the dominant scalar and pseudoscalar channels into account, an infinite number of ring diagrams reduces to just one single generic diagram. Corrections to correlation functions by ladder diagrams are suppressed in a large-${N_{\rm c}}$ expansion. However, for the shear viscosity itself a resummation of these subleading diagrams is potentially important, depending on the ${N_{\rm c}}$ scaling of the spectral width [@HidakaKunihiro2011; @HidakaKunihiro2011NJL]. This effect has been studied first in [@JeonSkeleton1995] for a bosonic field theory. In the present work resummations in the Kubo sector will not be included. A discussion concerning the conditions under which such resummations are necessary will however be given. The general derivation is followed by a detailed parameter study in Section \[Sec:ParStud\]: for $\eta[\Gamma]$ as a functional of the quasiparticle spectral width $\Gamma(p)$ of the quarks. First an analytical result is derived assuming a constant spectral width to start with. Furthermore, the implementation of different parameterizations for $\Gamma(p)$ teaches us about the general dependence of the shear viscosity on the spectral width for a variety of examples. A strong dependence on the pertinent momentum-space cutoff, $\Lambda$, is found, reflecting the sensitivity to physical scales: the characteristic cutoff fixed by the NJL gap equation excludes up to $90\%$ of the mathematically accessible high-momentum contributions to $\eta$, a feature that actually turns out to be a prerequisite for achieving physically meaningful results within this framework. We also investigate to what extent the functional $\eta[\Gamma(p)]$ can be treated perturbatively by expanding in a Laurent series for a “small” spectral width and comparing with the full result. The impact of thermal constituent quark masses on the shear viscosity is investigated in Section \[Sec:ThermoMass\]. The thermal quark masses are generated dynamically by the spontaneous chiral symmetry breaking mechanism through the NJL gap equation. Not surprisingly, we find that thermal effects are crucial to obtain physically relevant results for the shear viscosity. In Section \[Sec:GammaNJL\] an explicit calculation of the spectral width is performed within the NJL model, using the one-loop mesonic contributions to the quark self-energy at next-to-leading order in the large-${N_{\rm c}}$ expansion. Two different physical effects contribute to this width: Landau damping and mesonic recombination. For temperatures well above the critical/crossover temperature the resulting spectral width decreases, implying an increasing shear viscosity $\eta(T)$ in this temperature range. Shear viscosity at leading order {#Sec:EtaLO} ================================ In this work we model quark matter starting from the two-flavor NJL Lagrangian $$\label{NJL2} \mathcal{L}={\bar{\psi}}\left({{\rm i}}{\slashed{\partial}}-\hat{m}\right)\psi+\frac G2\left[({\bar{\psi}}\psi)^2+({\bar{\psi}}{{\rm i}}\gamma_5{\mbox{\boldmath $\tau$}}\psi)^2\right],$$ where $\psi = (u,d)^{\rm T}$ is the isospin doublet quark field, $\hat{m} = {\rm diag}(m_{\rm u},m_{\rm d})$ is the current quark mass matrix (we work in the isospin limit, $m_{\rm u}= m_{\rm d} \equiv m$), $G$ denotes the scalar/pseudoscalar coupling, and ${\mbox{\boldmath $\tau$}}$ collects the three Pauli isospin matrices. Vector or axialvector terms are not considered in this work. The large masses of the corresponding quark-antiquark modes make their contributions to the relevant correlation functions far less important than those of pseudoscalar and scalar modes. In the Kubo formalism [@Kubo57] transport coefficients are related to retarded correlators of energy-momentum tensors, i.e. to four-point functions in Matsubara space. The energy momentum tensor of the NJL model is simply $$\label{Tmunu} T_{\mu\nu}={{\rm i}}{\bar{\psi}}\gamma_\mu\partial_\nu\psi-g_{\mu\nu}\mathcal{L}$$ in terms of the quark fields $\psi$. The Kubo formula for the shear viscosity reads $$\begin{aligned} \label{KuboEta} \eta(\omega) = \frac{\beta}{15}\int_0^\infty{\text{d}}t\;{{\rm e}}^{{{\rm i}}\omega t}\int{\text{d}}^3r\; (T_{\mu\nu}({\mbox{\boldmath $r$}},t),T^{\mu\nu}(0))\;, \end{aligned}$$ with $\beta = 1/T$ the inverse temperature. The correlator $(X,Y)$ is defined by[^1] $$(X,Y)=T\int_0^\beta{\text{d}}\xi\;\langle{{\rm e}}^{\xi H}X{{\rm e}}^{-\xi H}Y\rangle\;,$$ where $H$ is the NJL Hamiltonian and $\langle\mathcal{\cdot}\rangle={{\rm Tr}}\left(\cdot\,{{\rm e}}^{-\beta H}\right)$ denotes the thermal expectation value. An equivalent reduced expression for the shear viscosity is also frequently used in the literature [@Alberico2008]: $$\label{KuboEtaAlternative} \eta(\omega)=\beta\int_0^\infty{\text{d}}t\;{{\rm e}}^{{{\rm i}}\omega t}\int{\text{d}}^3r\; (T_{21}({\mbox{\boldmath $r$}},t),T_{21}(0))\;,$$ written in terms of only one component of the energy-momentum tensor. The relative factor $15$ in comparison with Eq. results from the following identity ($i,j\in\{1,2,3\}$ with $i\neq j$): $$\int{\text{d}}^3x\; x_i^2\,x_j^2\,f(x^2)=\frac{1}{15}\int{\text{d}}^3x\;x^4f(x^2)\;.$$ The (classical) components of the energy-momentum tensor are real quantities, $T_{\mu\nu}\in\mathds{R}$. It follows that the *static* shear viscosity $\eta(\omega = 0)$ is also real: $$\eta(\omega)^*=\eta(-\omega)\;\;\;\Rightarrow\;\;\;\eta:=\eta(0)\in\mathds{R}\;.$$ Neglecting surface terms at infinite time, one derives $$\label{DefRetGreenFctFromDifference} \eta(\omega)=\frac{{{\rm i}}}{\omega}\left[\Pi^{\rm R}(\omega)-\Pi^{\rm R}(0)\right],$$ with the retarded correlation function $$\label{DefPiROmega} \Pi^{\rm R}(\omega)=-{{\rm i}}\int_0^\infty{\text{d}}t\;{{\rm e}}^{{{\rm i}}\omega t}\int{\text{d}}^3r\,\langle[T_{21}({\mbox{\boldmath $r$}},t),T_{21}(0)]\rangle\;.$$ The static shear viscosity ($\omega\to 0$) follows as $$\eta=-\left.\frac{{\text{d}}}{{\text{d}}\omega}\,{{\rm Im}\,}\Pi^{\rm R}(\omega)\right|_{\omega=0}\;.$$ The calculation of the retarded correlator can be performed switching to the Matsubara formalism and calculating $$\Pi(\omega_n)=\int_0^\beta{\text{d}}\tau\;{{\rm e}}^{{{\rm i}}\omega_n\tau}\int{\text{d}}^3 r\; \langle \mathcal{T}_\tau \big(T_{21}({\mbox{\boldmath $r$}},\tau)T_{21}(0)\big)\rangle\;,$$ where have applied a Wick rotation $\tau={{\rm i}}t$ and introduced the time-ordering symbol in imaginary time, $\mathcal{T}_\tau$. Note that whereas the underlying (quark) degrees of freedom are fermionic, the Matsubara frequencies relevant for the correlator $\Pi$ are bosonic, $\omega_n=2\pi n T$, since the fermion fields under the integral group together to form quantities of bosonic character: ${\bar{\psi}}(\cdot)\psi$. The global sign of $\Pi(\omega_n)$ is fixed by the sign convention for analytical continuations: $$\left.\Pi(\omega_n)\right|_{i\omega_n=\omega\pm{{\rm i}}{\varepsilon}}= -\,\Pi^{\rm R/A}(\omega)\;,$$ where the upper and lower sign in $\pm{{\rm i}}{\varepsilon}$ corresponds to the retarded and advanced correlation function, respectively. The correlator $\Pi(\omega_n)$ is governed by non-perturbative physics resulting from the underlying interactions of the NJL model. We now apply a large-${N_{\rm c}}$ expansion and organize this correlator in ring diagrams, ladder diagrams and higher-order terms: $$\label{KuboNcExpansionFourPointFunction} \begin{minipage}{0.08\textwidth} \mbox{$\Pi(\omega_n)=$} \end{minipage} \begin{minipage}{0.15\textwidth} \includegraphics[width=\textwidth]{NJLsk4pointfctFull.pdf} \end{minipage} \hspace{-0.2cm} \begin{minipage}{0.2\textwidth} \mbox{$= \mathcal{O}(N_{\rm c}^1)+\mathcal{O}(N_{\rm c}^0)+ \ldots $} \end{minipage}$$ The four-point coupling of the NJL Lagrangian effectively incorporates gluonic degrees of freedom resulting in the scaling $G\sim 1/{N_{\rm c}}$. At leading order $\mathcal{O}(N_{\rm c}^1)$ there is just a one-loop diagram contributing to the four-point correlator, given that the NJL Lagrangian in its simplest form takes into account only scalar and pseudoscalar interactions: $\Gamma\in\{{{\mathds 1}},{{\rm i}}\gamma_5\}$. Iterating these interaction kernels in ring diagrams at leading order in $1/{N_{\rm c}}$ to $\Pi(\omega_n)$ does not affect the correlator: $$\label{Chain} \hspace{0.5cm} \begin{minipage}{0.04\textwidth} \end{minipage} \begin{minipage}{0.3\textwidth} \includegraphics[width=\textwidth]{NJLsk4pointfctABsnd.pdf} \end{minipage} \begin{minipage}{0.05\textwidth} \!=\;0\;, \end{minipage}$$ because the trace (in momentum and Dirac space) in the first ring vanishes due to the orthogonal operator structure involving the combination of $\gamma_2$ and $\Gamma$: $$\begin{aligned} \label{KuboRingAZero1} &{T\sum_{n\in\mathds{Z}}\int\frac{{\text{d}}^3 p}{(2\pi)^3}}\; {{\rm Tr}}\left[\gamma_2\, G_\beta({\mbox{\boldmath $p$}},\nu_n)\, \Gamma\, G_\beta({\mbox{\boldmath $p$}},\nu_n)\right] =\\ & ={T\sum_{n\in\mathds{Z}}\int\frac{{\text{d}}^3 p}{(2\pi)^3}}\; \frac{1}{(\nu_n^2+{\mbox{\boldmath $p$}}^2+M^2)^2}\; \\ &\hspace{0.5cm} \times\, {{\rm Tr}}\left[\gamma_2\Gamma M^2 +\gamma_2\slashed{p}\Gamma\slashed{p}+\gamma_2\slashed{p}\Gamma M+\gamma_2\Gamma\slashed{p} M\right]=0\;, \end{aligned}$$ where we have used the notation $\slashed{p}=\nu_n\gamma_4-{\mbox{\boldmath $p$}}\cdot{\mbox{\boldmath $\gamma$}}$ and the full Matsubara propagator $$G_\beta({\mbox{\boldmath $p$}},\nu_n)=\frac{\slashed{p}+M}{\nu_n^2+{\mbox{\boldmath $p$}}^2+M^2}\;,$$ with frequencies $\nu_n=(2n+1)\pi T-{{\rm i}}\mu$. Exchange (ladder diagram) corrections to the chain in Eq.  are non-vanishing but of subleading order in $1/{N_{\rm c}}$, because each rank in the ladder gives rise to a suppression factor $G^2{N_{\rm c}}\sim 1/{N_{\rm c}}$. Note that adding one rank introduces two additional momentum integrations but only one additional color trace. The shear viscosity in the NJL model has been deduced previously in Refs. [@Fukutome2006; @Fukutome2008Nucl; @Fukutome2008Prog] using the Kubo formula, but assuming the quarks to be in the chiral limit, $m=0$. We point out that this result can in fact be derived without assuming to work in the chiral limit. Setting the current quark masses to zero is *not necessary* to ensure the absence of iterated ring-diagram contributions when taking only scalar and pseudoscalar interactions of the NJL model into account. Iterated ring diagrams involving these interactions vanish naturally. (Note that even in the chiral limit and the Nambu-Goldstone phase, the second term of the trace in Eq. , ${{\rm Tr}}\left[\gamma_2\slashed{p}\Gamma\slashed{p}\right]$ would survive in the presence of vector interactions, but their contribution to the correlators would be small as mentioned before). Collecting all arguments, we can summarize in general: for purely fermionic theories $\mathcal{L}=\mathcal{L}_{\rm kin}+\mathcal{L}_{\rm int}$ with momentum-independent pseudoscalar/scalar interactions and $2n$-vertices that scale as $G_{2n}\sim 1/{N_{\rm c}}^{n-1}$, the dominant contribution to the correlation function $\Pi^{\rm R}(\omega)$ in Matsubara space is: $$\label{PiRingLO} \begin{minipage}{0.08\textwidth} \mbox{$\Pi(\omega_n)=$} \end{minipage} \begin{minipage}{0.125\textwidth} \includegraphics[width=\textwidth]{NJLsk4pointfctA0snd.pdf} \end{minipage} \begin{minipage}{0.1\textwidth} \mbox{$\;+\;\mathcal{O}({N_{\rm c}}^0)\;.$} \end{minipage}$$ With the definition of the spectral function, $$\rho(\omega,{\mbox{\boldmath $p$}})=-\frac{1}{\pi}{{\rm Im}\,}G^{\rm R}(\omega,{\mbox{\boldmath $p$}})=\frac{1}{2\pi{{\rm i}}}\left(G^{\rm A}(\omega,{\mbox{\boldmath $p$}})-G^{\rm R}(\omega,{\mbox{\boldmath $p$}})\right),$$ and using residue calculus one derives: $$\begin{aligned} \label{EtaLeadingNcAsTwoSpectralDensityPap} \eta &={\pi\over T} \int_{-\infty}^\infty {\text{d}}{\varepsilon}\int\frac{{\text{d}}^3 p}{(2\pi)^3}\;p_x^2\, n_{\rm F}^+({\varepsilon})\big(1-n_{\rm F}^+({\varepsilon})\big) \\ &\hspace{2.5cm} \times\,{{\rm Tr}}\left[\gamma_2\,\rho({\varepsilon},{\mbox{\boldmath $p$}})\,\gamma_2\,\rho({\varepsilon},{\mbox{\boldmath $p$}})\right], \end{aligned}$$ with the Fermi-Dirac distribution $$n_{\rm F}^+(E)=\frac{1}{1+{{\rm e}}^{\beta (E-\mu)}}\;.$$ As in [@Fukutome2006] the dressed quark propagator is written as $$\label{KuboQuasiPartApprQuarkProp} G^{\rm R/A}(p_0,{\mbox{\boldmath $p$}})=\frac{1}{\slashed{p}-M\pm{{\rm i}}\,{\rm sgn}(p_0)\Gamma(p)}\;,$$ with the quasiparticle mass $M$ and width $\Gamma(p)$. The next step is to relate this spectral width to the shear viscosity $\eta$. Even in the chiral limit the dynamical NJL mechanism of spontaneous chiral symmetry breaking generates a large constituent quark mass in the vacuum: $M\approx 0.3\;{\rm GeV}$, see the brief discussion in Section \[Sec:ThermoMass\]. Apart from this mechanism, the thermal environment at temperature $T$ and baryo-chemical potential $\mu$ of the quarks affects parameterically both the dynamical quark mass $M(T,\mu)$ and the spectral width $\Gamma(p;T,\mu)$. The spectral function $\rho$ is represented in the standard form of a generalized Breit-Wigner shape, to be inserted in Eq.: $$\begin{aligned} \label{EtaLeadingNcAsSpectralWidth} \eta[\Gamma(p)] &=\frac{16{N_{\rm c}}{N_{\rm f}}}{15\pi^3 T}\int_{-\infty}^\infty{\text{d}}{\varepsilon}\int_0^\infty{\text{d}}p\,p^6 \\ &\times\,\frac{M^2\,\Gamma^2(p)\,n_{\rm F}^+({\varepsilon})(1-n_{\rm F}^+({\varepsilon}))}{\left[({\varepsilon}^2-p^2-M^2+\Gamma^2(p))^2+4M^2\Gamma^2(p)\right]^2}\;. \end{aligned}$$ Even though we started from the NJL model in our derivation, the expression is generic for a system of strongly interacting Fermions, with real and imaginary parts of their self-energies encoded in $M$ and $\Gamma$, respectively. In general, the non-perturbative origin of $\Gamma(p)$ does not permit expanding the functional $\eta[\Gamma]$ in a Laurent series. In perturbative approaches (e.g. in chiral perturbation theory at low temperatures) such a treatment is possible: $\eta[\Gamma]\sim 1/\Gamma$ [@LangKaiserWeise2012]. In this context we comment briefly on the issue of ladder resummation and its correction to Eq. . If one assumes the spectral width $\Gamma\sim 1/{N_{\rm c}}$ to be suppressed for large ${N_{\rm c}}$ as suggested by hot-QCD calculations, then the superficial ${N_{\rm c}}$ counting of Eqs.  and , $\eta\sim{N_{\rm c}}$, is spoiled: in this case, the integrand becomes highly singular in the large-${N_{\rm c}}$ limit (“pinch poles” as described in [@JeonSkeleton1995; @HidakaKunihiro2011]), resulting in an additional factor ${N_{\rm c}}$, therefore $\eta\sim{N_{\rm c}}^2$. We find in the limit ${N_{\rm c}}\to\infty$, i.e. $\Gamma\to 0$: $$\label{EtaSmallGamma} \eta[\Gamma]\to\frac{2{N_{\rm c}}{N_{\rm f}}}{15\pi^2 T}\int_{|{\varepsilon}|>M}{\text{d}}{\varepsilon}\,\frac{({\varepsilon}^2-M^2)^{5/2}\,n_{\rm F}^+({\varepsilon})(1-n_{\rm F}^+({\varepsilon}))}{M\,\Gamma(\sqrt{{\varepsilon}^2-M^2})}\;.$$ In contrast to the non-perturbative result in Eq.  the ${\varepsilon}$-integration excludes the region $|{\varepsilon}|<M$. This is due to the delta functions appearing in the limit of small $\Gamma$. The momentum integration of the integrand involving $\delta({\varepsilon}^2-p^2-M^2)$ is readily carried out. Exploratory studies of the shear viscosity {#Sec:ParStud} ========================================== Analytical results ------------------ Consider now first the case of a constant spectral width, $\Gamma={\rm const}$. This rough schematic approximation allows for an analytical treatment of the momentum integral in Eq.  and one is left with the numerical ${\varepsilon}$-integration only. For the momentum integral we use the following identity: $$\begin{aligned} \label{KuboAnalyitcalPIntegral} \int_0^\infty &{\text{d}}p\;\frac{p^6}{[(A-p^2)^2+B^2]^2}= \\ &\hspace{-0.5cm}=\frac{\pi}{8\sqrt{2}}\frac{\sqrt{\sqrt{A^2+B^2}-A}}{B^4} \left[(2A^2+3B^2)\sqrt{A^2+B^2} \right.\\ &\hspace{3.7cm} \left. +2A(A^2+2B^2)\right], \end{aligned}$$ where we have introduced $A={\varepsilon}^2-M^2+\Gamma^2$ and $B=2M\Gamma$. This result is found by extending the integration region to negative $p$ (the integrand is an even function of $p$) and using residue calculus. Having performed the $p$-integration analytically reduces the computation time by roughly one order of magnitude. Furthermore, it helps finding an appropriate approximation scheme for the whole $({\varepsilon},p)$-integration when the spectral width is momentum dependent. Fig. \[Fig1\] shows the results for $\eta$ assuming $\Gamma={\rm const.}$ For $\Gamma\to 0$ the shear viscosity diverges, as it follows from Eq. . This limit describes a system of free quarks for which the mean free path is infinite. With increasing temperature and chemical potential, the shear viscosity increases, but the dependence on temperature is more pronounced. Compare these figures to those in Ref. [@Fukutome2006], where $\eta(\Gamma)$ has been evaluated numerically without a momentum-space cutoff, equivalent to our analytical approach based on Eq. . Inspecting the detailed behavior of the integrand in Eq. , a convergence criterion for the shear viscosity in the absence of a momentum-space cutoff can be derived: > In order for the shear viscosity $\eta[\Gamma]$ as functional of $\Gamma(p)$ to be convergent, the asymptotic $\Gamma(p)$ should not converge too rapidly to zero: $$\label{CriterionEtaGamma} > \eta[\Gamma(p)]<\infty \;\;\; \Leftrightarrow \;\;\; p^3{{\rm e}}^{-\beta p/2}\in {\rm o}(\Gamma(p))\;,$$ where ${\rm o}(\cdot)$ denotes the little Landau symbol[^2]. Possible parameterizations of $\Gamma(p)$ satisfying this constraint are: $$\begin{aligned} \label{ModelsMomentumDepGamma} {\rm constant:}~~~~~\Gamma_{\rm const} &= 100\;{\rm MeV}\;,\\ {\rm exponential:}~~~\Gamma_{\rm exp}(p) &= \Gamma_{\rm const}\,{{\rm e}}^{-\beta p/8}\;,\\ {\rm Lorentzian:}~~~\Gamma_{\rm Lor}(p) &= \Gamma_{\rm const}\,\frac{\beta p}{1+(\beta p)^2}\;,\\ {\rm divergent:}~~~\Gamma_{\rm div}(p) &= \Gamma_{\rm const}\,\sqrt{\beta p}\;. \end{aligned}$$ Note that all these parameterizations lead to a finite shear viscosity and no mathematical regularization must be applied, compare the cutoff discussion in Section \[SectionCutOff\]. The particular shapes of these prototype widths have been chosen because of their different behavior at small and large momenta: vanishing or non-vanishing $\Gamma(p=0)$, convergent or divergent $\Gamma(p)$ for $p\to\infty$. These prototypes represent physical spectral widths in several theories [@LangKaiserWeise2012]: $\Gamma(p)$ in $\phi^4$ theory, for instance, is a monotonous function and converges to zero for large momenta. This can be described by the Lorentz parameterization for large momenta: $\lim_{p\to\infty}\Gamma_{\rm Lor}(p)\sim T/p$. In contrast, the spectral width of an interacting pion gas diverges for $p\to\infty$. Numerical approximation scheme ------------------------------ Our numerical approximation of ![Accuracy of the numerical approximation scheme for the ${\varepsilon}$-integral, Eq. : to reach an accuracy of $10^{-4}$ it is sufficient to restrict its range to $|{\varepsilon}(p)|<1.3\,{\varepsilon}^*(p)$, see Eq. .[]{data-label="Fig2"}](Fig2.pdf){width="45.00000%"} $\eta[\Gamma(p)]$ is based on the observation that its integrand typically ranges over many orders of magnitude. For every momentum $p$ there is a maximum of the integrand in Eq. , located at the denominator’s minimizer $$\label{EpsStarMaximizer} {\varepsilon}^*(p)=\sqrt{p^2+M^2(T,\mu)-\Gamma^2(p;T,\mu)}\;.$$ Adaptive methods do not work when facing a sharp peak structure: either the step size becomes too small for fast convergence (or convergence at all), or the most important contribution in the vicinity of the peak is not sampled by a step size that is too coarse. We overcome this numerical issue by cutting the ${\varepsilon}$-integration and allowing only $|{\varepsilon}(p)|<x{\varepsilon}^*(p)$ for some $x\gtrsim 1$. In comparison with the analytical result for $\Gamma(p)={\rm const.}$ we find that $x=1.3$ is sufficient to produce accurate results within a relative error of $10^{-4}$, see Fig. \[Fig2\]. For the momentum-dependent parameterizations of $\Gamma(p)$ the integrands for $\eta$ in Eq.  look qualitatively the same as for a constant spectral width. We therefore expect the described numerical scheme to work well also in these and more physical cases, where full momentum dependence and effects from the thermal environment are (parameterically) taken into account. Cutoff dependence {#SectionCutOff} ----------------- Generally, the shear viscosity increases when the spectral width decreases, compare Eq. . This behavior is also visible in Fig. \[Fig3\](a) when comparing our different parameterizations of $\Gamma(p)$: the “more divergent” the spectral width as $p\rightarrow \infty$, the smaller the corresponding shear viscosity: $$\eta_{\rm Lor}>\eta_{\rm exp}>\eta_{\rm const}>\eta_{\rm div}\,,$$ using notations as in Eq. . This sequence is implied by the corresponding (inverse) order for the spectral widths. These arguments hold also for non-vanishing chemical potentials. Assuming the spectral width to be independent of the chemical potential as in our parameterizations of $\Gamma(p)$ in Eq. , the shear viscosity increases for increasing $\mu$, but the qualitative shape of $\eta(T)$ does not change. We note that the results in Fig. \[Fig3\] have been derived using a constant constituent quark mass $M=325\;{\rm MeV}$, see the brief discussion in Section \[Sec:ThermoMass\]. The integrand of $\eta[\Gamma(p)]$, Eq. , is sizable for unphysically large momenta, so we expect a strong cutoff dependence. In the NJL model the quasiparticle interactions are restricted to quark momenta $p\le\Lambda=650\;{\rm MeV}$. Quarks with momenta $p>\Lambda$ do not interact and have infinite mean free paths. Retricting the momentum integration to the interval $p \le \Lambda$, we find a shear viscosity as shown in Fig. \[Fig3\](b). Excluding $p>\Lambda$ reduces the shear viscosity by one order of magnitude at low temperatures and even by two orders of magnitude at high $T$. As expected, this expresses a very strong cutoff dependence. In addition to these quantitative differences, the qualitative behavior of the shear viscosity also changes strongly and flattens for high temperatures. This strong cutoff dependence is investigated in more detail in Fig. \[Fig4\]: the contributions taken into account (compared to the analytical result for $\eta$) depend strongly on temperature and just weakly on the chemical potential. At $T=200\,{\rm MeV}$ the momentum cutoff excludes about $90\%$ of the full integral extended to infinity, see Fig. \[Fig4\](a). As shown in Fig. \[Fig4\](b), varying the cutoff by up to $\pm 20\%$ implies for $\eta$ a change of up to $100\%$. To assess the order of magnitude of the NJL shear viscosity, a comparison with $\eta(T)$ for other systems is instructive. For example, an interacting pion gas treated within the framework of chiral perturbation theory [@LangKaiserWeise2012] has a typical shear viscosity of order $\eta(T)\approx 40\,{\rm MeV}/{\rm fm}^2 \approx 1.6\cdot 10^{-3}\,{\rm GeV}^3$ at $T\approx 100\;{\rm MeV}$. This is a similar order of magnitude as the results shown in Fig. \[Fig3\](b) when applying the NJL cutoff $\Lambda=650\,{\rm MeV}$. We recall that this cutoff is fixed by reproducing physical observables such as the pion decay constant in vacuum. A physically meaningful order of magnitude for $\eta$ then follows naturally. Perturbative aspects of [$\eta[\Gamma]$]{} {#Sec:PertEta} ------------------------------------------ ![Scaling of $\eta\cdot\Gamma$ for different $T$ and $\mu$ as function of the inverse width expressed in units of the pion mass $m_\pi$. Solid horizontal lines correspond to the residues $A_{-1}$ of $\eta[\Gamma]$ in Eq. . A constituent quark mass $M=100\;{\rm MeV}$ has been used for convenience.[]{data-label="Fig5"}](Fig5.pdf){width="46.00000%"} We have already mentioned that the shear viscosity $\eta$ diverges for non-interacting systems, i.e. for a vanishing spectral width, corresponding to infinite mean free path. Close to this limit $\eta$ can be expanded in a Laurent series (as realized for example analytically in ChPT and $\lambda\phi^4$ theory [@LangKaiserWeise2012]): $$\label{LaurentExpEtaGamma} \eta[\Gamma]=\frac{A_{-1}}{\Gamma}+A_0+A_1\Gamma+A_2\Gamma^2+\ldots$$ For small $\Gamma$, the combination $\eta\cdot\Gamma$ is just the residue $A_{-1}$. What does “small” mean in this context? In contrast to $\lambda\phi^4$ theory where $\Gamma\sim\lambda^2$, the NJL model is generically non-perturbative in its coupling, even though the scaling $G\sim 1/{N_{\rm c}}$ applies. The spectral width is therefore not expected to be sufficiently small in order to permit an expansion as in Eq. . Fig. \[Fig5\] shows results of the fully non-perturbative calculation of $\eta\cdot\Gamma$ as a function of the inverse width, conveniently written as $x=m_\pi/\Gamma$, at different $T$ and $\mu$ in comparison with the residue $A_{-1}$. As it can be seen from the figure, corrections to the leading term of the Laurent series are small for $x>1.5$ (demanding $10\%$ accuracy or better). From these considerations we conclude that a perturbative approach is justified only for spectral widths $\Gamma\ll\,m_\pi=140\;{\rm MeV}$. The discussion of a perturbative treatment of $\eta[\Gamma(p)]$ is closely related to the resummation of ladder diagrams: if in the large-${N_{\rm c}}$ limit the spectral width decreases, i.e. $\Gamma\sim 1/{N_{\rm c}}$ as suggested by hot-QCD calcuations where the coupling $\alpha_{\rm s}\sim 1/{N_{\rm c}}$ becomes small, then the perturbative regime is reached in this limit and the Laurent series expansion in can be restricted to its leading-order term. As seen from Eq. , for a constant but small spectral width $\Gamma$ the residue $A_{-1}$ can be identified with the remaining ${\varepsilon}$-integral. While only this residue term of $\eta[\Gamma(p)]$ is relevant in this case, ladder diagrams now become sizable corrections and contribute also at leading order. Furthermore, the shear viscosity now scales as $\eta\sim{N_{\rm c}}^2$ and no longer linearly with ${N_{\rm c}}$ as Eqs.  and do for ${N_{\rm c}}$-independent spectral function $\ rho$ and width $\Gamma$, respectively. We conclude that ladder diagram resummations are necessary in the perturbative regime of $\eta[\Gamma(p)]$ in Eq. , i.e. when the spectral width is small, $\Gamma\ll m_\pi$. In the NJL model with its genuine non-perturbative structure, the physical spectral width is large and outside the perturbative regime. Ladder diagram resummations are subleading corrections, while the shear viscosity functional is valid also for large spectral width when including all orders of the Laurent series expansion . Effects of thermal quark masses and the ratio [$\eta/\lowercase{s}$]{} {#Sec:ThermoMass} ====================================================================== The constituent quark mass has so far been treated as a constant. We now proceed to incorporate its explicit $T$ and $\mu$ dependence. One of the key features of the NJL model is the spontaneous breaking of chiral symmetry: ${\rm SU}(2)_{\rm L}\times{\rm SU}(2)_{\rm R}\to{\rm SU}(2)_{\rm V}$. In addition, chiral symmetry is explicitly broken by the non-zero current quark mass $m$. Solving the NJL gap equation results in a dynamically generated constituent quark mass [@Nambu1961tp; @Nambu1961; @VoglWeise1991]: $$\begin{aligned} \label{ThermalGapEquation} &M(T,\mu)=m-G\langle\bar{\psi}\psi\rangle =\,\\ &\!\!=m+\frac{2G{N_{\rm c}}M(T,\mu)} {\pi^2}\int_0^\Lambda{\text{d}}p\,\frac{p^2}{E_p}\,\big[1-n_{\rm F}^+(E_p)-n_{\rm F}^-(E_p)\big], \end{aligned}$$ with $E_p(T,\mu)=\sqrt{p^2 + M^2(T,\mu)}$ and the Fermi-Dirac distribution for quarks and antiquarks $$\label{DefFermiDistrMuT} n_{\rm F}^\pm(E)=n_{\rm F}(E\mp\mu)=\frac{1}{{{\rm e}}^{\beta (E\mp\mu)}+1}\;.$$ In the vacuum, $(T,\mu)=(0,0)$, the mass is determined to be about one third of the nucleon mass, $M=325\;{\rm MeV}$, where the input parameters of the NJL Lagrangian are chosen as $G=10.08\;{\rm GeV^{-2}}$, $m=5.5\;{\rm MeV}$ and $\Lambda=650\;{\rm MeV}$. This parameter set produces physical (vacuum) values of the pion mass, $m_\pi=140\;{\rm MeV}$, the pion decay constant, $f_\pi=94\;{\rm MeV}$, and the chiral condensate $\langle{\bar{\psi}}\psi\rangle=-(316.4\;{\rm MeV})^3$. Fig. \[Fig6\](a) shows the shear viscosity $\eta$ for varying constituent quark mass $M$ treated as a parameter, assuming a constant spectral width $\Gamma_{\rm const}=100\;{\rm MeV}$. For $M\to 0$ the shear viscosity becomes divergent, again due to “pinch poles” appearing in Eq.  in this limit. In fact, the origin of this divergence is the same as for $\Gamma\to 0$, since $M$ and $\Gamma$ are formally (almost) interchangeable in the integrand of Eq. . For large constituent quark masses, two effects occur: first, the maximizer ${\varepsilon}^*(p)\sim M$ in Eq.  moves to larger values, and second, the integrand scales as $M^{-6}$. Both features result in a decreasing function $\eta(M)$. Taking the full thermal dependence of the constituent quark mass into account has an essential influence on the shear viscosity, see Fig. \[Fig6\](b): for small $T$, a constant mass $M=325\;{\rm MeV}$ approximates the thermal constituent quark mass. In contrast, at large $T$, with a melting chiral condensate, the dropping dynamical quark mass implies a strongly increasing shear viscosity, qualitatively different from the case with constant quark mass. (We have chosen to compare thermal and non-thermal results for constant and exponential parameterizations of the spectral width. For $\Gamma_{\rm Lor}$ and $\Gamma_{\rm div}$ the results are qualitatively similar and therefore not shown.) The shear viscosity itself is a dimensionful quantity. One usually compares $\eta$ to the entropy density $s$, in terms of the dimensionless ratio $\eta/s$. The corresponding ratio extracted from heavy-ion collisions is the smallest value of $\eta/s$ found so far in nature [@HeinzShenSong2012]. The entropy density of quark matter is given by [@Kapusta]: $$\begin{aligned} s(T,\mu)=\frac{{N_{\rm c}}{N_{\rm f}}}{\pi^2}\int_0^\infty{\text{d}}p\,p^2\, &\left[\ln\left(1+{{\rm e}}^{-\beta E_p^+}\right) \right.\\ & \left. \hspace{-4cm}+\ln\left(1+{{\rm e}}^{-\beta E_p^-}\right)+\frac{\beta E_p^+}{1+{{\rm e}}^{\beta E_p^+}}+\frac{\beta E_p^-}{1+{{\rm e}}^{\beta E_p^-}}\right], \end{aligned}$$ with $E_p^\pm=E_p\mp\mu$. The momentum integration ranges up to infinity. It can be performed without regularization. In the NJL model the thermal constituent quark mass, for momenta above the cutoff scale $\Lambda=650\;{\rm MeV}$, reduces to the bare current quark mass, $M\rightarrow m=5.5\;{\rm MeV}$. At low temperatures, $T\lesssim 150$ MeV, confinement implies that quarks are not the relevant physical degrees of freedom any more and the NJL model cannot be expected to give a realistic description of transport properties. The evaluation of the ratio $\eta/s$ is shown in Fig. \[Fig7\] for ${N_{\rm f}}=2$ and ${N_{\rm c}}=3$, for the cases of a constant and an exponentially damped spectral width. The comparison between the panels (a) and (b) clearly demonstrates that implementing thermal masses is crucial in order to avoid an unphysical, continuously decreasing ratio $\eta/s(T)$. From experimental data [@Lacey07] and lattice calculations [@Nakamura05; @Meyer07] it is in fact suggested that this ratio increases for $T> 200\;{\rm MeV}$. We also compare to the benchmark $\eta/s=1/4\pi$ from AdS/CFT correspondence [@Maldacena99; @Kovtun05]. This value is known not to be a universal bound; it can be undershot in some field theories [@Cohen07; @Rebhan12; @Mamo12]. This is also found in Fig. \[Fig7\](a), additionally to the unphysical evolution of $\eta/s$ with increasing temperature. A constant quark mass $M=100\;{\rm MeV}$ has been chosen there for convenience. Its vacuum value, $M=325\;{\rm MeV}$, would reduce the scale of the $\eta/s$ ratio even more, compare Fig. \[Fig6\](a). However, taking the thermal constituent quark masses into account leads to an increasing function $\eta/s(T)$, see Fig. \[Fig7\](b). Despite the fact that $\eta$ itself rapidly increases at high $T$, the ratio $\eta/s$ flattens in that region. This flattening is expected to be shifted to higher temperatures for more realistic forms of the width $\Gamma(p)$ such the Lorentzian. In the considered temperature range $\eta/s$ stays above the AdS/CFT benchmark for all parameterizations of the width $\Gamma$. Spectral width at one-loop level {#Sec:GammaNJL} ================================ So far we have discussed the impact of the shape of the spectral width on the shear viscosity of quark matter, its strong sensitivity to the NJL cutoff and to the thermal constituent quark masses. In this section an explicit calculation is performed including one-loop mesonic contributions at next-to-leading order in the large-${N_{\rm c}}$ expansion [@BuballaMuellerWambach2010]. The leading-order gap equation is modified by the mesonic insertions $$\label{NJLMesonInsertion} \Sigma^{\rm S/P}_\beta({\mbox{\boldmath $p$}},\nu_n) = \includegraphics[width=0.15\textwidth]{NJLMesonInsertion.pdf}\;,$$ where $\nu_n=(2n+1)\pi T-{{\rm i}}\mu$ are the Matsubara frequencies for quarks. The corresponding expression for antiquarks is easily obtained by inserting $\bar{\nu}_n=\nu_n^*$. The spectral width is extracted from the imaginary part of this self-energy[^3]: $$\Gamma^{\rm S/P}_{\rm q}(p_0,{\mbox{\boldmath $p$}})=-\lim_{{\varepsilon}\to 0}\,{\rm Im}\,\Sigma^{\rm S/P}_\beta({\mbox{\boldmath $p$}},-{{\rm i}}p_0+{\varepsilon})\;.$$ It is of next-to-leading order using the mesonic modes generated by the Bethe-Salpeter equation (BSE) in the pertinent quark-antiquark channels (e.g. [@HatsudaKunihiro87; @KLVW1990; @BuballaMuellerWambach2010]). Whereas the gap equation is of leading order, $\mathcal{O}(1)$ in $N_c$, the BSE is of order $\mathcal{O}(1/{N_{\rm c}})$. In the two-flavor case the mesonic loop involves the three pions and the sigma mode which contribute with positive and negative signs, respectively, to the spectral width $\Gamma(p)$. This “antiscreening” by the sigma mode [@QuackKlevansky94] cancels roughly one of the three pionic contributions to the spectral width, as seen in Fig. \[Fig9\]a. The effective spectral width (for $N_f = 2$) reads: $$\label{QuarkSpectralWidthOneLoopAllMesons} \Gamma(p)=3\Gamma_{\rm q}^{\rm P}(p)+\Gamma_{\rm q}^{\rm S}(p)\;,$$ with the scalar and pseudoscalar contributions, $\widetilde{N}^{\rm S}=-2$ and $\widetilde{N}^{\rm P}=2$, respectively: $$\label{QuarkSpectralWidthOneLoop} \Gamma^{\rm S/P}_{\rm q}(p)=\frac{M{g_{\rm Mqq}}^2 \widetilde{N}^{\rm S/P}}{4\pi |{\mbox{\boldmath $p$}}|}\int_{E_{\rm min}}^{E_{\rm max}} {\text{d}}E_f \left[n_{\rm B}(E_b)+n_{\rm F}^-(E_f)\right],$$ involving the Fermi distribution for antiquarks, $n_{\rm F}^{-}$ in Eq. , and the Bose distribution for mesons at zero (quark-)chemical potential: $$\label{BoseDistribution} n_{\rm B}(E)=\frac{1}{{{\rm e}}^{\beta E}-1}\;.$$ Energy-momentum conservation implies $E_b=E_f+p_0$ in Eq.  and leads to the restricted range of integration: $$\begin{aligned} \label{QuarkSpectralWidthEminEmax} E_{{\rm min},{\rm max}}(p)&=\frac{1}{2M^2}\left[\left(m_{\rm M}^2-2M^2\right)\sqrt{M^2+p^2}\right. \\ &\hspace{2cm}\left.\mp\, p\,m_{\rm M}\sqrt{m_{\rm M}^2-4M^2}\right]. \end{aligned}$$ Here $m_{\rm M}$ denotes the $T$- and $\mu$-dependent (pseuodoscalar or scalar) meson mass and $M(T,\mu)$ is the dynamical quark mass in the thermal medium, c.f. Eq. . The quark-meson vertex is of Yukawa type with a (thermal) coupling constant $g_{\rm Mqq}(T,\mu)$ arising from the renormalized meson propagator $D_{\rm M}(p_0,{\mbox{\boldmath $p$}})$ at $(p_0,{\mbox{\boldmath $p$}})=(m_{\rm M},{\mbox{\boldmath $0$}})$: $$g_{\rm Mqq}=\left({\rm Res}\, D_{\rm M}\right)^{-1/2}\;.$$ The details of the derivation of Eqs.  and are presented in the Appendix \[Sec:App\]. The spectral width is a function of the energy $p_0$ and the momentum $p=|{\mbox{\boldmath $p$}}|$ of the quark propagating in the (isotropic) thermal medium. In the on-shell case, $p_0=\sqrt{p^2+M^2}$, the width $\Gamma$ becomes a function of $p$ only. Taking the imaginary part of the mesonic insertion means cutting the loop diagram and forcing it on-shell. Two different dissipative mechanisms contribute to the width, as sketched in Fig. \[Fig8\]. The term involving the Bose distribution describes Landau damping, a process also known from high-$T$ QCD calculations [@Boyanovsky1998], and recently applied to the calculation of the shear viscosity from kinetic theory [@Ghosh2013]. In the NJL model Landau damping arises from the scattering of quarks on the mesonic collective modes in the thermal medium. This scattering process dissipates energy from the quark sector and contributes to the shear viscosity. The second mechanism is a recombination process: a collective mesonic mode is created by quark-antiquark rescattering. This is described by the term involving the Fermi distribution of thermal antiquarks in Eq. . Results for the calculated on-shell spectral width including all mesons, $\Gamma(p)$ in Eq. , are shown in Fig. \[Fig9\]b. Due to the “antiscreening” caused by the sigma mode, the total $\Gamma(p)$ is rougly twice the single pion contribution, c.f. Fig. \[Fig9\]a. For small quark momenta the spectral width can become quite large. At $p\approx 200\;{\rm MeV}$ it is of the order of $100\;{\rm MeV}$, roughly as large as the dynamical (constituent) quark mass. We recall from Fig. \[Fig5\] that the shear viscosity $\eta[\Gamma]$ can be treated perturbatively only if $\Gamma\ll m_\pi$. Hence for the one-loop spectral width calculated in the NJL model, the perturbative regime is not reached, given the large values of $\Gamma(p)$. With rising temperature the spectral width decreases leading to an increasing shear viscosity $\eta(T)$, as explored parametrically in Fig. \[Fig1\]. One important reason for this behavior is the range of integration, $E_f(p)\in[E_{\rm min}(p),E_{\rm max}(p)]$, which has its support for $m_\pi>2M$, the kinematic threshold condition for a pion to decay on-shell into two constituent quarks. As the temperature increases this range of integration is shifted to higher values of $E_f$ and $p$, but these are exponentially suppressed by the Bose and Fermi distributions in Eq. . For more details, see also Fig. \[Fig11\] in the Appendix. ![Shear viscosity per entropy density at zero chemical potential from the two-flavor NJL model at one-loop level in a large-${N_{\rm c}}$ expansion (solid line). Shown for comparison are results using the kinetic approach (open circles and triangles), those of a related calculation using the Kubo formalism (solid triangles) and pure-gauge lattice data (solid squares). The dashed horizontal line is the AdS/CFT benchmark $1/4\pi$.[]{data-label="Fig10"}](Fig10.pdf){width="49.00000%"} Finally the resulting shear viscosity $\eta(T,\mu = 0)$ (its ratio $\eta/s$) is shown in Fig. \[Fig10\]. Only temperatures large enough to satisfy $m_\pi>2M$ give rise to a finite shear viscosity. Therefore the one-loop results are restricted to $T\gtrsim 210\;{\rm MeV}$. We compare our results to those from a kinetic approach using the two-favor NJL model [@SasakiRedlich2010] (open circles) and [@Ghosh2013] (open triangles). Our values based on the Kubo formalism are larger but feature the same order of magnitude and the same qualitative behavior. These findings are consistent with the general observation that the kinetic approach seems to underestimate the shear viscosity as pointed out in Refs. [@PlumariEtAl2012; @PlumariEtAl2013]. Their calculations have also been performed using the Kubo formalism (c.f. Fig. 8 in [@PlumariEtAl2012]) and lie slightly above our result for $\eta/s$ as shown in Fig. \[Fig10\] (filled triangles). A flattening of $\eta/s$ is observed at higher temperatures around $T\approx 300\;{\rm MeV}$, indicating that the shear viscosity scales roughly as $T^3$ in this range. The same behavior has been found in our preceding parameter study, c.f. Fig. \[Fig6\]. All results for $\eta/s$, both from kinetic and Kubo approaches, exceed those from pure-gauge lattice QCD [@Nakamura05; @Meyer07] that are close to the AdS/CFT bound as shown by the solid squares in Fig. \[Fig10\]. Summary and Conclusions ======================= In the present work we have used the Kubo formalism to derive a general functional $\eta[\Gamma(p)]$ for a class of fermionic theories, in particular the two-flavor NJL model with scalar and pseudoscalar interactions. At leading order in $1/{N_{\rm c}}$ the retarded correlation function $\Pi^{\rm R}(\omega)$ reduces to a single generic diagram at $\mathcal{O}({N_{\rm c}}^1)$. We have found that it is not necessary to work in the chiral limit to obtain this result derived previously in [@Fukutome2006; @Fukutome2008Nucl; @Fukutome2008Prog] using stronger assumptions. The detailed study of the functional $\eta[\Gamma(p)]$ leads to a convergence criterion, Eq. , to be fulfilled by the spectral width $\Gamma(p)$ in order for the shear viscosity $\eta$ to be finite without regularization. Four different schematic parameterizations have been chosen for $\Gamma(p)$ and a numerical approximation scheme suitable for arbitrary, momentum-dependent spectral widths has been introduced. The results for $\eta$ show a strong cutoff dependence: restricting the momentum integration to the typical NJL cutoff, $p<\Lambda=650\;{\rm MeV}$, changes the shear viscosity drastically as shown in Fig. \[Fig3\]. Such a cutoff places $\eta$ at physically meaningful values by excluding up to $90\%$ of the numerically accessible high-momentum region, as demonstrated in Fig. \[Fig4\]. An exploration has been performed determining the range of $\Gamma(p)$ for which a perturbative treatment of the (generally non-perturbative) shear viscosity is adequate. The functional $\eta[\Gamma(p)]$ as given in Eq.  is valid in particular for a large spectral width as it is expected in the NJL model. For a small spectral width, $\Gamma\ll m_\pi$, a Laurent series expansion of $\eta$ with restriction to the leading term $\eta \sim 1/\Gamma$ is justified. In this limit, subleading effects from resummations of ladder diagrams are expected to become non-negligible as known from studies in bosonic field theories [@JeonSkeleton1995; @HidakaKunihiro2011]. On the other hand, for the NJL model with its comparatively large spectral width, it is consistent to omit ladder diagram resummations while taking all orders of the Laurent expansion in the width $\Gamma(p)$ into account. Including thermal quasiparticle masses (the temperature and density dependence of the dynamically generated constituent quark masses in NJL-like models) is of crucial importance. As experimental data indicate, the ratio $\eta/s$ is expected to increase for high temperatures $T > T_{\rm c}$ where $T_{\rm c}\approx 180-200\;{\rm MeV}$ is the typical temperature interval of the chiral crossover. A constant quark mass would instead lead to a continuously decreasing ratio $\eta/s$ with increasing temperature, as seen in Fig. \[Fig7\]. In comparison with other approaches to the shear viscosity using NJL-type models, such as the Boltzmann equation in relaxation time approximation, one can find different results: an increasing ratio $\eta/s(T)$ for two flavors and restricting to scalar and pseudoscalar interactions [@SasakiRedlich2010], but also a decreasing behavior [@MartyFrankfurt2013; @HidakaKunihiro2011NJL]. From AdS/CFT correspondence, the scaling of the ratio $\eta/s$ is expected to be of $\mathcal{O}({N_{\rm c}}^0)$ in the large-${N_{\rm c}}$ limit. With the entropy density of quarks and the functional $\eta[\Gamma(p)]\sim{N_{\rm c}}$ in Eq. , we find indeed $\eta/s\sim \mathcal{O}(1)$ in the NJL model. Its applicability is, however, limited to the temperature range $T_{\rm c} \lesssim T < \Lambda$. At much higher temperatures gluonic degrees of freedom are dominant and contribute to the entropy density as $s_{\rm gluon}\sim{N_{\rm c}}^2$ due to their adjoint representation instead of the fundamental representation for fermions. At the same time the spectral width is expected to became small ($\Gamma\to 0$) and scales as $1/N_c$, leading to “pinch poles” and a scaling of the shear viscosity as $\eta \sim N_c^2$, as seen from Eq. . Consequently the ratio still scales as $\eta/s\sim\mathcal{O}(1)$ at large $T$ in the limit ${N_{\rm c}}\to\infty$. From the systematic parameter study the conclusion can be drawn that the momentum dependence of the spectral width $\Gamma(p)$ determines primarily the overall scale of the ratio $\eta/s$. Its actual behavior as a function of temperature is largely governed by the thermal properties of the quark propagator. A consistent one-loop NJL model calculation of the quark self-energy in the thermal medium has been performed at next-to-leading order in the large-${N_{\rm c}}$ expansion. At this order the gap equation for the thermal constituent quark mass is corrected by the mesonic insertions . Two generic dissipative contributions to the shear viscosity emerge: Landau damping and a quark-antiquark recombination process, resulting in the spectral width $\Gamma(p)$ given in Eq.. The restricted range of the energy integration leads to a decreasing function $\Gamma(T)$ as the temperature increases, which implies an increasing shear viscosity for high temperatures as seen in Fig. \[Fig10\]. Additional contributions to $\Gamma$ from quark-quark scattering via exchange of mesonic quark-antiquark modes still have to be included. Such processes appear at order ${N_{\rm c}}^{-1}$ in the large-${N_{\rm c}}$ expansion and are calculated in ongoing work. Results will be reported elsewhere. Acknowledgments {#acknowledgments .unnumbered} =============== This work is partially supported by BMBF and by the DFG Cluster of Excellence “Origin and Structure of the Universe”. R. L. thanks the ECT\* for kind hospitality. Valuable discussions with Y. Hidaka and T. Hatsuda are acknowledged. Thanks go to N. Kaiser for useful advices and reading of the manuscript. R.L. has been supported by the TUM Graduate School (TUM-GS), by the RIKEN IPA and the RIKEN iTHES projects. Derivation of the NJL spectral width {#Sec:App} ==================================== In this appendix we focus on the analytical treatment of the mesonic insertion $\Sigma_\beta^{\rm S/P}$ in Eq.  and the extraction of the spectral width $\Gamma(p)$ as given in Eqs.  and . The Matsubara frequencies for a thermal quark are $\nu_n=(2n+1)\pi T-{{\rm i}}\mu$, whereas they read for an antiquark: $\bar{\nu}_n=(2n+1)\pi T+{{\rm i}}\mu=\nu_n^*$. The diagram for the mesonic insertion represents both pions (pseudoscalar channel) and the sigma meson (scalar channel) contributing the quark spectral width: $$\begin{aligned} \Sigma^{\rm S/P}_\beta({\mbox{\boldmath $p$}},\nu_n)\; &=-4M g_{\rm Mqq}^2 \widetilde{N}^{\rm S/P} \\ & \hspace{-1.5cm}\times{T\sum_{m\in\mathds{Z}}\int\frac{{\text{d}}^3 q}{(2\pi)^3}}\,\frac{1}{\nu_m^2+E_f^2}\,\,\frac{1}{(\nu_m-\nu_n)^2+E_b^2}\;, \end{aligned}$$ with $E_f^2={\mbox{\boldmath $q$}}^2+M^2$ and $E_b^2=({\mbox{\boldmath $q$}}-{\mbox{\boldmath $p$}})^2+m_{\rm M}^2$. We have introduced $\widetilde{N}^{\rm P}=2$ and $\widetilde{N}^{\rm S}=-{N_{\rm f}}=-2$ reflecting the opposite parities of pion and sigma boson, leading to screening and antiscreening of the quark mass, respectively. Carrying out the Matsubara sum introduces combinations of Fermi and Bose distribution functions, Eq.  and , respectively: $$\begin{aligned} Z_1 &=1+n_{\rm B}(E_b)-\frac 12\left(n_{\rm F}^+(E_f)+n_{\rm F}^-(E_f)\right), \\ Z_2 &= n_{\rm B}(E_b)+\frac 12\left(n_{\rm F}^+(E_f)+n_{\rm F}^-(E_f)\right), \\ Z_3 &=n_{\rm F}^+(E_f)-n_{\rm F}^-(E_f)\;. \end{aligned}$$ The thermal self-energy of the quark is given by: $$\label{WidthFirstItThermalSelfEnergy} \Sigma^{\rm S/P}_\beta\!=\!-4Mg_{\rm Mqq}^2\widetilde{N}^{\rm S/P}\!\!\int\frac{{\text{d}}^3q}{(2\pi)^3}\left[\frac{1}{2E_bE_f}\left[\frac{(E_f+E_b)Z_1}{(E_f+E_b)^2+\nu_n^2}+\frac{(E_f-E_b)Z_2}{(E_f-E_b)^2+\nu_n^2}\right]+\frac{{{\rm i}}\nu_n Z_3}{[(E_f+E_b)^2+\nu_n^2][(E_f-E_b)^2+\nu_n^2]} \right].$$ The analytical continuation of the thermal self-energy to the retarded self-energy in Minkowskian spacetime, $$\Sigma^{\rm S/P}_{\rm R}(p_0,{\mbox{\boldmath $p$}}):=\Sigma^{\rm S/P}_\beta({\mbox{\boldmath $p$}},-{{\rm i}}p_0+{\varepsilon})\;,$$ leads to the spectral width $$\label{IterationOneSpectralWidthPart1} \Gamma^{\rm S/P}_{\rm q}(p_0,{\mbox{\boldmath $p$}}) :=-\lim_{{\varepsilon}\to 0}{{\rm Im}\,}\Sigma^{\rm S/P}_{\rm R}(p_0,{\mbox{\boldmath $p$}})\;.$$ The non-vanishing spectral width is induced by the pole structure of the propagators: $$\begin{aligned} \label{DeltaFctArrImPart} \lim_{{\varepsilon}\to 0}{{\rm Im}\,}\left.\frac{Z}{x^2+\nu_n^2}\right|_{\nu_n\mapsto -{{\rm i}}p_0+{\varepsilon}} &= Z\pi\delta(x^2-(p_0)^2) \\ & \hspace{-1.5cm}=\frac{\pi Z}{2p_0}\left[\delta(x-p_0)+\delta(x+p_0)\right]. \end{aligned}$$ This means for the $Z_1$ term: $E_f+E_b\pm p_0=0$, where only the minus sign can be realized. For the $Z_2$ term, $E_f-E_b\pm p_0=0$, both signs can be realized for the time being. We will see that only the plus sign contributes to the (on-shell) spectral width, so there is just one contribution from $Z_2$. The $Z_3$ term is considered later. From the identity in Eq.  the following structure is found for the $Z_1$ and $Z_2$ terms: $$\label{CalcZ1Z2PhaseSpace} \begin{aligned} &\int\frac{{\text{d}}^3q}{(2\pi)^3}\frac{\pi Z}{2p_0}\frac{1}{2E_bE_f}\,\delta(E_b-(*))= \\ &=\int\frac{{\text{d}}^3q}{(2\pi)^3}\frac{\pi Z}{2p_0E_f}\,\delta(E_b^2-(*)^2)= \\ \end{aligned}$$ $$\begin{aligned} &=2\pi\int_{-1}^1{\text{d}}\xi\int_0^\infty \frac{{\text{d}}q\, q^2}{(2\pi)^3}\frac{\pi Z}{2p_0E_f }\delta(E_b^2(\xi)-(*)^2)=\\ &=2\pi\int_{M}^\infty\frac{{\text{d}}E_f}{(2\pi)^3}\frac{\pi Z}{4p_0p}\,\Theta(1-\xi^2)\;, \end{aligned}$$ with $\xi =\cos\theta$. In order to carry out this integral over the delta function we have used $$E_b^2 =m_{\rm M}^2+({\mbox{\boldmath $p$}}-{\mbox{\boldmath $q$}})^2=m_{\rm M}^2+p^2+q^2-2pq\xi\;,$$ from which follows $$\left|\frac{\partial E_b^2}{\partial\xi}\right|=2pq\;.$$ In addition the integral over momentum has been translated into an energy integral using $q\,{\text{d}}q=E_f\,{\text{d}}E_f$. The ill-conditioned $\Theta$ term can be removed by the following consideration: The condition $|\xi|\leq 1$ is equivalent to $$\begin{aligned} \label{DefCapitalFfunctionPhaseSpace} -1\leq & \frac{E_b^2-m_{\rm M}^2-p^2-q^2}{2pq}\leq 1 \\ & \Leftrightarrow F(E_f,p)\geq 0\;, \end{aligned}$$ where we have defined $$F(E_f,p):=4p^2(E_f^2-M^2)-\left[E_b^2-m_{\rm M}^2-p^2+M^2-E_f^2\right]^2.$$ ![Range of integration for $E_f\in[E_{\rm min},E_{\rm max}]$ as function of absolute momentum $p=|{\mbox{\boldmath $p$}}|$ for different temperatures. The solid line displays the orbit of minimal values.[]{data-label="Fig11"}](Fig11.pdf){width="49.00000%"} At given momentum $p$ the roots of $F(\,\cdot\, ,p)$ read, for the plus-sign case of both the $Z_1$ and $Z_2$ terms, where $E_b^2=(E_f+p_0)^2$: $$\begin{aligned} E_{{\rm max},{\rm min}} &=\frac{1}{2M^2}\left[\left(m_{\rm M}^2-2M^2\right)\sqrt{M^2+p^2}\right. \\ & \hspace{1.5cm} \left.\pm pm_{\rm M}\sqrt{m_{\rm M}^2-4M^2}\right]. \end{aligned}$$ From this we find $$\label{EmaxEminDifferenceZeroValue} \begin{aligned} & E_{{\rm max}}-E_{{\rm min}}=\frac{p\,m_{\rm M}}{M^2}\sqrt{m_{\rm M}^2-4M^2}\sim p\;, \\ & E_{{\rm max},{\rm min}}(p=0)=\frac{m_{\rm M}^2}{2M}-M>M\;. \end{aligned}$$ The momentum-dependent phase space for $E_f$ is shown in Fig. \[Fig11\] for different temperatures. Due to the larger meson mass at large $T$, the curves $E_{\rm min,max}$ are shifted to higher energies and momenta when increasing the temperature. There is an exact linear dependence of the integration range, $E_{\rm max}-E_{\rm min}$, on the incoming momentum $p$. At zero momentum the phase space collapses to one single point. Under the condition $m_{\rm M}>2M$ the phase space is always non-empty and compact with $M$ as minimal value of $E_{\rm min}$: $\emptyset\neq [E_{\rm min},E_{\rm max}]\subseteq [M,\infty)$. We also have derived the following substitution rule $$\int_{M}^\infty\frac{{\text{d}}E_f}{(2\pi)^3} \big(\cdot\big) \Theta(1-\xi^2) = \int_{E_{\rm min}}^{E_{\rm max}} \frac{{\text{d}}E_f}{(2\pi)^3} \big(\cdot\big)\;,$$ which leads finally to a well-conditioned one-dimensional numerical integral. Inserting the minus-sign case into Eq., $E_b^2=(E_f-p_0)^2$, the phase space vanishes for any incoming momentum, since both the minimal and maximal energy of the loop fermion need to be negative: $$E_{\rm min}'=-E_{\rm max}\, ,\;\;\; E_{\rm max}'=-E_{\rm min}\;.$$ We conclude that only the case $E_b=E_f+p_0$ leads to an on-shell contribution to the mesonic insertion. With this in mind the third term, involving $Z_3$, in Eq. becomes: $$\begin{aligned} \lim_{{\varepsilon}\to 0} &\;{{\rm Im}\,}\left.\frac{{{\rm i}}\nu_n Z_3}{[(E_f+E_b)^2+\nu_n^2][(E_f-E_b)^2+\nu_n^2]}\right|_{{{\rm i}}\nu_n\mapsto p_0+{{\rm i}}{\varepsilon}}= \\ &=\frac{p_0\pi Z_3}{2p_0}\,\delta\big(\underbrace{[(E_f+E_b)^2-p_0^2]}_{=4E_fE_b}[E_b-E_f-p_0]\big)=\\ &=\frac{\pi Z_3}{4E_f}\,\delta\left(E_b^2-(E_f+p_0)^2\right). \end{aligned}$$ Note that due to the ${{\rm i}}\nu_n$ factor in the first line, the $p_0$ terms cancel in the final result. As done in the calculation the momentum integral can be performed: $$\begin{aligned} \int\frac{{\text{d}}^3q}{(2\pi)^3} \frac{\pi Z_3}{4E_f}\, &\delta(E_b^2-(E_f+p_0)^2)= \\ &=2\pi\int_{M}^\infty\frac{{\text{d}}E_f}{(2\pi)^3}\frac{\pi Z_3}{8p}\,\Theta(1-\xi^2)\;. \end{aligned}$$ Combining all contributions, we find for the spectral width of an incoming quark: $$\begin{aligned} &\Gamma^{\rm S/P}_{\rm q}(p) = \\ &=-8\pi Mg_{\rm Mqq}^2\widetilde{N}^{\rm S/P}\!\int_{E_{\rm min}}^{E_{\rm max}}\frac{{\text{d}}E_f}{(2\pi)^3}\frac{\pi}{4|{\mbox{\boldmath $p$}}|}\left(\frac{E_f-E_b}{p_0}Z_2+\frac{Z_3}{2}\right)=\\ &=\frac{Mg_{\rm Mqq}^2\widetilde{N}^{\rm S/P}}{4\pi|{\mbox{\boldmath $p$}}|}\int_{E_{\rm min}}^{E_{\rm max}}{\text{d}}E_f\left[n_{\rm B}(E_b)+n_{\rm F}^-(E_f)\right], \end{aligned}$$ using $E_b=E_f+p_0$. The effective spectral width in Eq.  is determined by summing the pseudoscalar (pion) and scalar (sigma boson) channels weighted by their respective multiplicities, $3=N_{\rm f}^2-1$ and $1$. [9]{} I. Arsene et al. (BRAHMS Collaboration), Nucl. Phys. A [**757**]{}, 1 (2005). K. Adcox et al. (PHENIX Collaboration), Nucl. Phys. A [**757**]{}, 184 (2005). B. B. Back et al. (PHOBOS Collaboration), Nucl. Phys. A [**757**]{}, 28 (2005). J. Adams et al. (STAR Collaboration), Nucl. Phys. A [**757**]{}, 102 (2005). K. Aamodt et al. (ALICE Collaboration), Phys. Rev. Lett. [**108**]{}, 252302 (2010). K. Aamodt et al. (ALICE Collaboration), Phys. Lett. B [**696**]{}, 30 (2011). D. Caffarri et al. (ALICE Collaboration), Nucl. Phys. A [**904–905**]{}, 643c (2013). D. Derendarz et al. (ATLAS Collaboration), Nucl. Phys. A [**904–905**]{}, 373c (2013). V. Roy, B. Mohanty, and A.K. Chaudhuri, J. Phys. G [**40**]{}, 065103 (2013). K. Dusling, and D. Teaney, Phys. Rev. C [**77**]{}, 034905 (2008). M. Luzum, C. Gombeaud, and J.Y.  Ollitrault, Phys. Rev. C [**81**]{}, 054910 (2010). H. Niemi, G.S. Denicol, P. Huovinen, E. Molnár, and D.H. Rischke, Phys. Rev. Lett. [**106**]{}, 212302 (2011). H. Liu, D. Hou, and J. Li, Eur. Phys. J. C [**45**]{}, 459 (2006). Ch. Sasaki and K. Redlich, Nucl. Phys. A [**832**]{}, 62 (2010). A.S. Khvorostukhin, V.D. Toneev, and D.N. Voskresensky, Phys. Rev. C [**84**]{}, 9 (2011).  W.M. Alberico, S. Chiacchiera, H. Hansen, A. Molinari, and M. Nardi, Eur. Phys. J. A [**38**]{}, 97 (2008). Y. Hidaka and T. Kunihiro, Phys. Rev. D [**83**]{}, 076004 (2011). Y. Hidaka and T. Kunihiro, J. Phys.: Conf. Ser. [**270**]{}, 012050 (2011). S.-I. Nam and C.-W. Kao, Phys. Rev. D [**87**]{}, 114003 (2013). S. Plumari et al., Phys. Rev. C [**86**]{} 054902, (2012).  S. Plumari et al., J. Phys.: Conf. Ser. [**420**]{}, 012029 (2013). S. Jeon, Phys. Rev. D [**52**]{}, 3591 (1995). S. Jeon and L.G. Yaffe, Phys. Rev. D [**53**]{}, 5799 (1996). Y. Nambu, G. Jona-Lasinio, Phys. Rev. [**122**]{}, 345 (1961). Y. Nambu, G. Jona-Lasinio, Phys. Rev. [**124**]{}, 246 (1961). S.P. Klevansky, Rev. Mod. Phys. [**64**]{}, 649 (1992). S. Klimt, M. Lutz, U. Vogl, and W. Weise, Nucl. Phys. A [**516**]{}, 429 (1990). S. Klimt, M. Lutz, U. Vogl, and W. Weise, Nucl. Phys. A [**516**]{}, 469 (1990). U. Vogl and W. Weise, Prog. Part. Nucl. Phys. [**27**]{}, 195 (1991). M. Buballa, Phys. Rept. [**407**]{}, 205 (2005).  G. ’t Hooft, Nucl. Phys. B [**72**]{}, 461 (1974). E. Quack and S.P. Klevansky, Phys. Rev. C [**49**]{}(6) 3283 (1994). D. Müller, M. Buballa, and J. Wambach, Phys. Rev. D [**81**]{}, 1 (2010). M. Iwasaki, and H. Ohnishi, and T. Fukutome, arXiv:hep-ph/0606192 (2006). T. Fukutome and M. Iwasaki, Prog. Theor. Phys. [**119**]{}, 991 (2008). M. Iwasaki, H. Ohnishi, and T. Fukutome, arXiv:hep-ph/0703271 (2007). R. Kubo, J. Phys. Soc. Jpn. [**12**]{}, 570 (1957). R. Lang, N. Kaiser, and W. Weise, Eur. Phys. J. A [**48**]{}, 109 (2012). U. Heinz, Ch. Shen, and H. Song, AIP. Conf. Proc. [**1441**]{}, 766 (2012). J.I. Kapusta and Ch. Gale, *Finite-Temperature Field Theory* (Cambridge, 2006). R.A. Lacey et al., Phys. Rev. Lett. [**98**]{}, 092301 (2007). A. Nakamura and S. Sakai, Phys. Rev. Lett. [**94**]{} 072305 (2005).  H.B. Meyer., Phys. Rev. D [**76**]{}, 101701(R) (2007). J. Maldacena, Int. J. Theor. Phys. [**38**]{}, 1113 (1999). P.K. Kovtun, D.T. Son, and A.O. Starinets, Phys. Rev. Lett. [**94**]{}, 111601 (2005). T.D. Cohen, Phys. Rev. Lett. [**99**]{}, 021602 (2007). A. Rebhan and D. Steineder, Phys. Rev. Lett. [**108**]{}, 021601 (2012). K.A. Mamo, JHEP [**1210**]{}, 070 (2012).  T. Hatsuda and T.Kunihiro, Prog. Theor. Phys. [**91**]{}, 284 (1987). D. Boyanovsky et al., Phys. Rev. D [**58**]{} 125009 (1998). S. Ghosh et al., Phys. Rev. C [**88**]{}(6) 068201 (2013). R. Marty, E. Bratkovskaya, W. Cassing, J Aichelin, and H. Berrehrah, Phys. Rev. C [**88**]{}(4) 045204 (2013). [^1]: Due to ${{\rm e}}^{-\beta H}X(t){{\rm e}}^{\beta H}=X(t+{{\rm i}}\beta)$, which implies $\langle X(t)Y(t'+{{\rm i}}\beta)\rangle=\langle Y(t')X(t)\rangle$, it follows that this correlator is symmetric in its arguments: $(X,Y)=(Y,X)$. [^2]: The notation $f\in{\rm o}(g)$ is used to express accurately that “$f$ is growing less fast than $g$”, meaning that $f(x)/g(x)\to 0$ for $x\to\infty$. More intuitively, this also means that “$g$ grows much faster than $f$”. [^3]: Note that in this definition as well as in previous expressions the spectral width $\Gamma$ corresponds to half the FWHM (full width at half maximum) of the resulting quark spectral function, $\pi\rho(p)=-{{\rm Im}\,}\,G^{\rm R}(p)$, where the retarded quark propagator has been defined in Eq. .

--- author: - | Remco F. J. van der Burg[^1], Sean McGee, Hervé Aussel, Håkon Dahle,\ Monique Arnaud, Gabriel W. Pratt, Adam Muzzin bibliography: - 'MasterRefs.bib' date: 'Submitted 5 June 2018; accepted 29 June 2018' title: 'The stellar mass function of galaxies in *Planck*-selected clusters at $0.5 < z < 0.7$: new constraints on the timescale and location of satellite quenching' --- Introduction ============ Over the past decades we have obtained an increasingly clear picture of the formation and evolution of galaxies in the Universe. The galaxy population can be broadly divided in two distinct types. Star-forming galaxies have blue colours, typically disk-like morphologies and a relatively high star formation rate, whereas quiescent galaxies have redder colours, more spheroidal morphologies, and a (near) absence of star formation. Generally speaking, star-forming galaxies are found to dominate in abundance at relatively early times and at low stellar masses [@kauffmann04; @baldry06; @peng10; @muzzin13b]. A central question is how galaxies transform from actively star-forming systems to passive quiescent galaxies. An important factor in this quest is to understand how the environment of a galaxy affects this transformation process, since at fixed stellar mass, galaxies in groups and clusters are found to have a higher probability of being quiescent than galaxies in the field [@dressler80; @blanton05; @woo13]. In recent years it has been shown that the effect of the environment is largely separable from the quenching processes that act internally, at least in the local universe [@baldry06; @peng10; @kovac2014]. Specifically, @peng10 introduced terms of “mass quenching”, which would be operating independently of the environment and most effectively quench massive galaxies, and “environmental quenching”, which would be operating independently of stellar mass and quench galaxies preferentially in overdense regions [also see e.g. @davies16; @kawinwanichakij17]. The total quenching effect would simply be the product of these two contributions. What the physics are behind the two separate quenching routes is a matter of debate. “Mass quenching” is often associated with feedback, either from supernovae and galactic winds [@oppenheimer10], or from active galactic nuclei [@bower06; @croton06; @tremonti07]. Many mechanisms have been proposed to be responsible for “environmental quenching”. In the overdensities of groups and clusters stripping of cold gas [through ram-pressure: @Gunn1972], or hot gas [“strangulation”/“starvation”: @Larson1980], as well as galaxy harassment or (dry) mergers [@Moore1996], could be responsible for the observed trends. To make progress in understanding these environmental processes, it is essential to understand in which environments and on which timescales they operate [@balogh04]. Most studies that aim to understand the process of environmental quenching are focussed on large cosmological volumes in for instance the COSMOS field [@peng10; @darvish16], VIPERS [@davidzon16], 3D-HST [@fossati17], the Subaru Hyper Suprime-Cam survey [@jian18], and ZFOURGE [@kawinwanichakij17; @papovich18]. Most of these studies separate the galaxy population in four density quartiles, so that the environmental effects can be studied between the different quartiles. At least at high redshift ($z\gtrsim 1$) it is found that the environmental quenching process is not working completely independently of stellar mass [@papovich18]. A starvation/strangulation scenario in which the supply of hot gas is cut off from a galaxy would be highly effective at these redshifts, where star formation rates and outflows deplete the cold gas supply [the “overconsumption model”, @mcgee14]. This would likely introduce a mass-dependent effect, since the gas depletion time of higher-mass galaxies ought to be shorter, and this could explain the measured trends. Also for lower mass galaxies ($M_{\star} \lesssim 10^{9.5}\,\mathrm{M_{\odot}}$), measured quenching time scales are comparable to the gas depletion timescale [@davies16]. Whereas such a division in density quartiles allows for a study of environmental quenching in reasonably overdense regions, the most extreme environments are either not probed, or are washed out by more moderate overdensities. And yet it is in these regions where the physics of the quenching may be notably different [@balogh16; @kawinwanichakij17]. Quenching processes may be more violent, leading to very high quenched fractions of cluster galaxies compared to the field at the same redshift [@delucia04; @vdB13; @annunziatella14; @balogh16; @nantais16; @nantais17; @wagner17]. Dynamical processes in clusters may become more important at later time to quench star formation, and these may act in a mass-independent fashion. A notable example of this is stripping by ram pressure, which would directly remove the cold gas supply from a galaxy and quench its star formation on a very short timescale [@zinger18; @fossati16; @bellhouse17; @jaffe18]. To understand where and when the most extreme environmental quenching is taking place in clusters, several studies have focussed on a clustercentric-distance dependent study, some even focussed on projected phase space of different galaxy populations [such as those of galaxies in the “transition” phase; @Oman2013; @muzzin14; @poggianti16; @jaffe18]. To study the relative excess of already-quenched galaxies in clusters compared to the field, single values of the environmental quenching efficiency are typically reported for satellite galaxies [cf. Fig. 7 in @nantais16 and references therein], even though there is likely to be a substantial trend with radius. Such a study as a function of radius can probe the “backsplash” of already-quenched ejected cluster satellites [@wetzel14], and pre-processing/quenching of future cluster satellites in the surrounding large-scale overdensity [@fujita04]. While new surveys push the frontiers of these studies to higher redshifts [e.g. @balogh17], their samples are thus-far limited in size, and clusters are of moderate mass and over-density. This renders a radial-dependent study of environmental quenching difficult. In this paper we focus instead on highly massive, and thus over-dense, clusters at intermediate redshifts. This allows us to specifically study the environmental quenching excess as a function of radial distance from the cluster centres. The sample we study is composed of 21 massive clusters detected with the *Planck* SZ survey at redshifts $0.5<z<0.7$. Since *Planck* is an all-sky survey (even though we only consider the northernmost 2/3 here), we probe the highest-density environments at these intermediate redshifts, where environmental effects are expected to be substantial. We concentrate on a photometric data set spanning $u$- to the $\mathrm{K_s}$-band for each cluster, using which we estimate stellar masses for individual galaxies, and separate them by type based on their best-fit SED. Our starting point is a measurement of the galaxy Stellar Mass Function (SMF), which describes the number density of galaxies as a function of their stellar mass, and which is a key observable to study the formation and evolution of galaxies. By comparing the SMF to the underlying halo mass function, the efficiency with which galaxies form can be measured, and this is an essential test and diagnostic tool for large hydrodynamical simulations such as Illustris [@genel14] and EAGLE [@schaye15]. Measuring the SMF in these massive clusters provides further constraints for the next generation of hydrodynamical simulations, in which large overdensities can be specifically focussed on to study the influence of such environments on the evolution of galaxies [e.g. @bahe17]. These SMFs are the main ingredients to estimate environmental quenching locations and timescales, which we describe and discuss in the remainder of this work. The structure of this paper is as follows. Section \[sec:sample\] describes the cluster sample and photometric data set we utilise. Section \[sec:analysis\] lays out the main analysis, ranging from photometric redshift measurements to a statistical accounting of fore- and background galaxies in our cluster galaxy sample. The main results, measurements of the SMF and environmental quenching efficiency, are presented in Sect. \[sec:SMF\] & \[sec:EQE\], respectively. We discuss our findings in Sect. \[sec:discussion\], and conclude in Sect. \[sec:summary\]. All magnitudes we quote are in the AB magnitudes system, and we adopt $\Lambda$CDM cosmology with $\Omega_{\mathrm{m}}=0.3$, $\Omega_{\Lambda}=0.7$ and $\mathrm{H_0=70\, km\, s^{-1}\, Mpc^{-1}}$. Uncertainties are given at the 1-$\sigma$ level, unless explicitly stated otherwise. Cluster Sample & Data {#sec:sample} ===================== The clusters we study are drawn from a sample of 33 clusters that were detected with *Planck*, and confirmed by autumn 2011 to be at $z>0.5$. This sample was the target of an XMM-Newton Large Programme ‘Unveiling the most massive galaxy clusters at $z > 0.5$ with Planck and XMM-Newton’ (PI M. Arnaud), in AO-11. The properties of the sample, in particular regarding their morphological properties, Intra Cluster Medium and the cluster scaling relations, are outlined in Arnaud et al., in prep. In this paper, we study the galaxy content in a sub-sample of 21 clusters that make up the northernmost ($\mathrm{Dec} > -25^{\circ}$) part of this parent sample. Several of these clusters were already priorly studied, particularly as part of X-ray selected samples of clusters in this redshift range [@bohringer00; @ebeling07; @piffaretti11]. Several are in optical catalogues constructed using SDSS data [@wen12; @rykoff14]. Table \[tab:dataoverview\] presents the main characteristics of the sample. It makes a comparison of the mass estimated from the *Planck* SZ signal, and mass based on the deep X-ray maps (M-$Y_X$ relation). Even though the SZ mass proxy is blindly extracted [i.e. without prior knowledge on the location of the cluster, cf. @psz2], both proxies are consistent at the massive end (within $\sim$10% in mass). They slightly diverge at the low-mass end due to Eddington bias in the SZ mass proxy [cf. e.g. @vdB16]. The differences between $Y_X$ and $Y_{SZ}$ will be discussed in more detail in Arnaud et al., in prep. The current paper refers to cluster masses as $M_{500}$[^2], based on the $Y_X$ scaling relation. To support our analysis, we combine different sources of spectroscopic information in the 21 fields we study. Fourteen of the clusters are covered in DR14 of SDSS [@sdssDR14]. For nine clusters we have obtained redshifts with the Nordic Optical Telescope [@psz1 Dahle et al., in prep.] or Gemini [@planckxmmvalidation13]. @ebeling14 publish a catalogue with hundreds of spectroscopic redshifts in two of the fields we study (`PSZ2 G180.25+21.03` and `PSZ2 G228.16+75.20`). For `PSZ2 G046.13+30.72` and `PSZ2 G155.27-68.42` we have obtained spectroscopic redshifts from a program undertaken with the Canary Islands observatories [@planckcanary16]. We searched the NED database[^3] for any spec-$z$s we may have missed in the literature. The only two clusters that remain without a previously-measured spectroscopic redshift (`PSZ2 G193.31-46.13` and `PSZ2 G219.89-34.39`) have been observed using VLT/FORS2 multi-object spectroscopy (PID=090.A-0925, PI=Bohringer). Using our own custom pipeline we reduced these spectra and measured the redshifts of several member galaxies. The number of (unique) spectroscopic targets and cluster members for all clusters are listed in Table \[tab:dataoverview\]. [-0.30cm]{} ----------------------------------- ------------------------- ------------------------------------ ------------------------------------- ------------------------------------ ------------------------------------ --------------------------- ---------------------- $\mathrm{Y_{X}}-M_{500}$ $\mathrm{Y_{SZ}}-M_{500}$ $R_{500,\mathrm{Y_{X}}}$ Name Redshift$^{\mathrm{a}}$ RA$_\mathrm{J2000}^{\mathrm{BCG}}$ Dec$_\mathrm{J2000}^{\mathrm{BCG}}$ \[$10^{14}\, \mathrm{M_{\odot}}$\] \[$10^{14}\, \mathrm{M_{\odot}}$\] \[kpc\] Alternative Name `PSZ2 G044.77-51.30` 0.503(3/1) 22:14:57.27 $-$14:00:12.7 $7.95^{+0.44}_{-0.43}$ $8.36^{+0.61}_{-0.62}$ ${1175}^{+21}_{-22}$ MACSJ2214.9-1359 `PSZ2 G045.32-38.46` 0.589(10/1) 21:29:26.13 $-$07:41:27.6 $7.36^{+0.66}_{-0.64}$ $7.63^{+0.64}_{-0.68}$ ${1107}^{+32}_{-33}$ MACSJ2129.4-0741 `PSZ2 G045.87+57.70` 0.609(87/22) 15:18:20.56 +29:27:40.2 $5.82^{+0.22}_{-0.22}$ $7.03^{+0.66}_{-0.71}$ ${1016}^{+13}_{-13}$ `PSZ2 G046.13+30.72` 0.569(67/17) 17:17:05.55 +24:04:23.7 $3.17^{+0.22}_{-0.22}$ $6.39^{+0.80}_{-0.84}$ ${\,\,\,843}^{+19}_{-20}$ `PSZ2 G070.89+49.26` 0.602(94/34) 15:56:25.24 +44:40:42.6 $5.02^{+0.20}_{-0.21}$ $6.46^{+0.65}_{-0.72}$ ${\,\,\,970}^{+13}_{-14}$ `PSZ2 G073.31+67.52` 0.609(110/35) 14:20:40.11 +39:55:06.9 $6.15^{+0.26}_{-0.25}$ $6.74^{+0.55}_{-0.63}$ ${1035}^{+14}_{-14}$ WHL J215.168+39.91 `PSZ2 G094.56+51.03` 0.539(47/20) 15:08:21.98 +57:55:14.9 $6.15^{+0.25}_{-0.24}$ $5.87^{+0.44}_{-0.43}$ ${1064}^{+14}_{-14}$ WHL J227.050+57.90 `PSZ2 G099.86+58.45` 0.615(104/16) 14:14:47.20 +54:47:03.5 $7.09^{+0.42}_{-0.42}$ $6.85^{+0.48}_{-0.49}$ ${1082}^{+21}_{-22}$ WHL J213.697+54.78 `PSZ2 G111.61-45.71` 0.546(388/187) 00:18:33.58 +16:26:15.9 $9.21^{+0.24}_{-0.24}$ $9.79^{+0.53}_{-0.53}$ ${1214}^{+11}_{-11}$ RXC J0018.5+1626 `PSZ2 G144.83+25.11` 0.591(4/1) 06:47:50.65 +70:14:54.0 $7.78^{+0.21}_{-0.20}$ $8.25^{+0.71}_{-0.73}$ ${1127}^{+10}_{-10}$ MACSJ0647.7+7015 `PLCK G147.30-16.60^{\mathrm{b}}` 0.645(8/8) 02:56:23.45 +40:17:28.9 $6.51^{+0.29}_{-0.28}$ $7.41^{+0.80}_{-0.86}$ ${1040}^{+15}_{-15}$ RXC J0254.4+4134 `PSZ2 G155.27-68.42` 0.567(68/24) 01:37:24.98 $-$08:27:22.9 $8.01^{+0.46}_{-0.38}$ $8.93^{+0.65}_{-0.70}$ ${1149}^{+22}_{-18}$ WHL J24.3324-8.477 `PSZ2 G180.25+21.03` 0.546(1151/529) 07:17:35.63 +37:45:17.4 $12.83^{+0.17}_{-0.17}$ $11.49^{+0.53}_{-0.55}$ ${1356}^{+6}_{-6}$ MACSJ0717.5+3745 `PSZ2 G183.90+42.99` 0.559(94/25) 09:10:51.05 +38:50:22.3 $8.44^{+0.60}_{-0.53}$ $6.95^{+0.73}_{-0.75}$ ${1173}^{+27}_{-25}$ WHL J137.713+38.83 `PSZ2 G193.31-46.13` 0.634(45/2) 03:35:52.00 $-$06:59:23.4 $5.49^{+0.30}_{-0.32}$ $6.07^{+0.75}_{-0.83}$ ${\,\,\,986}^{+18}_{-19}$ `PSZ2 G201.50-27.31` 0.538(1181/278) 04:54:10.83 $-$03:00:51.5 $7.90^{+0.30}_{-0.29}$ $8.30^{+0.70}_{-0.73}$ ${1157}^{+14}_{-14}$ MACSJ0454.1-0300 `PSZ2 G208.61-74.39` 0.718(10/5) 02:00:16.38 $-$24:54:51.5 $5.23^{+0.23}_{-0.23}$ $6.25^{+0.72}_{-0.79}$ ${\,\,\,939}^{+14}_{-14}$ `PSZ2 G211.21+38.66` 0.505(46/18) 09:11:11.52 +17:46:29.1 $5.48^{+0.22}_{-0.22}$ $6.99^{+0.73}_{-0.79}$ ${1038}^{+14}_{-14}$ RXC J0911.1+1746 `PSZ2 G212.44+63.19` 0.529(56/14) 10:52:48.75 +24:16:11.3 $4.15^{+0.23}_{-0.23}$ $5.62^{+0.80}_{-0.90}$ ${\,\,\,937}^{+17}_{-18}$ RMJ105252.4+241530.0 `PSZ2 G219.89-34.39` 0.734(15/5) 04:54:45.32 $-$20:16:58.8 $6.77^{+0.33}_{-0.29}$ $7.97^{+0.61}_{-0.67}$ ${1016}^{+16}_{-15}$ `PSZ2 G228.16+75.20` 0.544(585/285) 11:49:35.68 +22:23:54.7 $9.36^{+0.64}_{-0.62}$ $10.42^{+0.52}_{-0.55}$ ${1221}^{+27}_{-27}$ RXC J1149.5+2224 ----------------------------------- ------------------------- ------------------------------------ ------------------------------------- ------------------------------------ ------------------------------------ --------------------------- ---------------------- In parentheses the number of spectroscopic redshifts overlapping with the region for which we have photometry, and the number of spectroscopic cluster members (within 3000 km/s from the cluster mean redshift), respectively. Cluster detected at a significance slightly below the cut-off value used for the PSZ2 catalogue. The $\mathrm{Y_{SZ}}$ is measured on the final Planck maps. Cluster Photometry ------------------ [-0.90cm]{} ---------------------- ---------------------- ---------- ------------------------------------ ------------------- ------------------- -------------------- ------------------- -------------------- ------------------- -------------------- ------------------- -------------------- ------------------- ------------------- Name $\mathrm{K_{s,det}}$ PSF $\mathrm{M_{\star,det}/M_{\odot}}$ $u$ $B$ $g$ $V$ $r$ $R_c$ $i$ $I_c$ $z$ J $\mathrm{K_s}$ FWHM dex `PSZ2 G044.77-51.30` 23.14 0.55$''$ 9.25 25.4$^\mathrm{a}$ 26.0$^\mathrm{e}$ $-$ 25.9$^\mathrm{e}$ 24.8$^\mathrm{a}$ 26.0$^\mathrm{e}$ $-$ 25.4$^\mathrm{e}$ 24.9$^\mathrm{f}$ 23.4$^\mathrm{g}$ 23.3$^\mathrm{g}$ `PSZ2 G045.32-38.46` 23.19 0.57$''$ 9.36 25.5$^\mathrm{a}$ 25.6$^\mathrm{e}$ $-$ 25.3$^\mathrm{e}$ $-$ 25.2$^\mathrm{e}$ $-$ 24.9$^\mathrm{e}$ 24.9$^\mathrm{d}$ 23.8$^\mathrm{g}$ 23.3$^\mathrm{g}$ `PSZ2 G045.87+57.70` 22.66 0.54$''$ 9.61 25.5$^\mathrm{b}$ $-$ 26.1$^\mathrm{d}$ $-$ 25.9$^\mathrm{d}$ $-$ 25.4$^\mathrm{d}$ $-$ 23.8$^\mathrm{b}$ 23.0$^\mathrm{g}$ 22.5$^\mathrm{g}$ `PSZ2 G046.13+30.72` 22.38 0.66$''$ 9.68 25.2$^\mathrm{b}$ $-$ 25.7$^\mathrm{d}$ $-$ 25.8$^\mathrm{d}$ $-$ 24.7$^\mathrm{d}$ $-$ 24.1$^\mathrm{d}$ 22.8$^\mathrm{g}$ 22.3$^\mathrm{g}$ `PSZ2 G070.89+49.26` 22.27 0.64$''$ 9.75 25.5$^\mathrm{b}$ $-$ 26.2$^\mathrm{d}$ $-$ 25.8$^\mathrm{d}$ $-$ 25.1$^\mathrm{d}$ $-$ 24.0$^\mathrm{b}$ 23.1$^\mathrm{g}$ 22.4$^\mathrm{g}$ `PSZ2 G073.31+67.52` 22.44 0.55$''$ 9.69 25.7$^\mathrm{b}$ $-$ 26.3$^\mathrm{a}$ $-$ 25.4$^\mathrm{a}$ $-$ 25.0$^\mathrm{b}$ $-$ 23.9$^\mathrm{b}$ 23.2$^\mathrm{g}$ 22.6$^\mathrm{g}$ `PSZ2 G094.56+51.03` 22.38 0.66$''$ 9.62 25.3$^\mathrm{b}$ $-$ 26.1$^\mathrm{ad}$ $-$ 25.8$^\mathrm{d}$ $-$ 25.0$^\mathrm{ad}$ $-$ 23.5$^\mathrm{b}$ 23.0$^\mathrm{g}$ 22.5$^\mathrm{g}$ `PSZ2 G099.86+58.45` 22.31 0.57$''$ 9.75 25.6$^\mathrm{a}$ $-$ 26.0$^\mathrm{ad}$ $-$ 25.4$^\mathrm{ad}$ $-$ 24.7$^\mathrm{cd}$ $-$ 23.6$^\mathrm{a}$ 23.1$^\mathrm{g}$ 22.4$^\mathrm{g}$ `PSZ2 G111.61-45.71` 22.25 0.72$''$ 9.68 25.5$^\mathrm{a}$ 26.5$^\mathrm{e}$ 25.5$^\mathrm{a}$ 26.3$^\mathrm{e}$ 25.5$^\mathrm{a}$ 26.2$^\mathrm{e}$ 26.0$^\mathrm{cd}$ 25.7$^\mathrm{e}$ 25.0$^\mathrm{af}$ 23.5$^\mathrm{g}$ 22.6$^\mathrm{g}$ `PSZ2 G144.83+25.11` 22.95 0.78$''$ 9.46 25.0$^\mathrm{a}$ 25.7$^\mathrm{e}$ $-$ 25.4$^\mathrm{e}$ $-$ 25.5$^\mathrm{e}$ $-$ 25.2$^\mathrm{e}$ 25.0$^\mathrm{d}$ 23.3$^\mathrm{g}$ 23.2$^\mathrm{g}$ `PLCK G147.30-16.60` 22.57 0.52$''$ 9.68 24.5$^\mathrm{b}$ $-$ 25.8$^\mathrm{d}$ $-$ 25.4$^\mathrm{d}$ $-$ 25.1$^\mathrm{d}$ $-$ 23.6$^\mathrm{b}$ 23.2$^\mathrm{g}$ 22.7$^\mathrm{g}$ `PSZ2 G155.27-68.42` 22.36 0.59$''$ 9.69 24.6$^\mathrm{b}$ $-$ 25.7$^\mathrm{b}$ $-$ 25.3$^\mathrm{b}$ $-$ 24.3$^\mathrm{b}$ $-$ 23.8$^\mathrm{b}$ 23.3$^\mathrm{g}$ 22.6$^\mathrm{g}$ `PSZ2 G180.25+21.03` 23.13 0.65$''$ 9.33 25.6$^\mathrm{a}$ 25.9$^\mathrm{e}$ 25.6$^\mathrm{a}$ 25.7$^\mathrm{e}$ 25.3$^\mathrm{a}$ 25.5$^\mathrm{e}$ 24.8$^\mathrm{d}$ $-$ 25.0$^\mathrm{d}$ 23.1$^\mathrm{g}$ 23.3$^\mathrm{g}$ `PSZ2 G183.90+42.99` 22.51 0.64$''$ 9.58 25.4$^\mathrm{b}$ $-$ 25.9$^\mathrm{d}$ $-$ 25.9$^\mathrm{d}$ $-$ 25.5$^\mathrm{d}$ $-$ 23.9$^\mathrm{b}$ 23.5$^\mathrm{g}$ 22.9$^\mathrm{g}$ `PSZ2 G193.31-46.13` 22.45 0.59$''$ 9.73 24.8$^\mathrm{b}$ $-$ 25.8$^\mathrm{b}$ $-$ 25.1$^\mathrm{b}$ $-$ 24.8$^\mathrm{b}$ $-$ 23.6$^\mathrm{b}$ 23.1$^\mathrm{g}$ 22.7$^\mathrm{g}$ `PSZ2 G201.50-27.31` 23.08 0.56$''$ 9.34 25.7$^\mathrm{a}$ 26.2$^\mathrm{e}$ 25.6$^\mathrm{a}$ 26.0$^\mathrm{e}$ 25.6$^\mathrm{a}$ 26.0$^\mathrm{e}$ 24.7$^\mathrm{c}$ 25.5$^\mathrm{e}$ 25.1$^\mathrm{ad}$ 23.5$^\mathrm{g}$ 23.4$^\mathrm{g}$ `PSZ2 G208.61-74.39` 21.98 0.69$''$ 10.00 25.0$^\mathrm{b}$ $-$ 25.8$^\mathrm{b}$ $-$ 25.2$^\mathrm{b}$ $-$ 24.5$^\mathrm{b}$ $-$ 23.8$^\mathrm{b}$ 23.3$^\mathrm{g}$ 22.4$^\mathrm{g}$ `PSZ2 G211.21+38.66` 22.94 0.78$''$ 9.33 25.9$^\mathrm{a}$ 26.1$^\mathrm{e}$ 25.2$^\mathrm{a}$ 26.1$^\mathrm{e}$ 24.9$^\mathrm{a}$ 26.1$^\mathrm{e}$ 24.0$^\mathrm{c}$ 25.4$^\mathrm{e}$ 25.1$^\mathrm{ad}$ 23.0$^\mathrm{g}$ 23.3$^\mathrm{g}$ `PSZ2 G212.44+63.19` 22.54 0.67$''$ 9.52 25.4$^\mathrm{b}$ $-$ 26.0$^\mathrm{d}$ $-$ 26.0$^\mathrm{d}$ $-$ 25.5$^\mathrm{d}$ $-$ 23.7$^\mathrm{b}$ 23.7$^\mathrm{g}$ 22.7$^\mathrm{g}$ `PSZ2 G219.89-34.39` 22.08 0.74$''$ 9.96 24.9$^\mathrm{b}$ $-$ 25.7$^\mathrm{b}$ $-$ 25.4$^\mathrm{b}$ $-$ 24.4$^\mathrm{b}$ $-$ 23.7$^\mathrm{b}$ 22.9$^\mathrm{g}$ 22.4$^\mathrm{g}$ `PSZ2 G228.16+75.20` 22.82 0.73$''$ 9.45 26.0$^\mathrm{a}$ 26.6$^\mathrm{e}$ $-$ 26.2$^\mathrm{e}$ $-$ 26.3$^\mathrm{e}$ 25.3$^\mathrm{d}$ $-$ 25.4$^\mathrm{d}$ 23.1$^\mathrm{g}$ 23.2$^\mathrm{g}$ ---------------------- ---------------------- ---------- ------------------------------------ ------------------- ------------------- -------------------- ------------------- -------------------- ------------------- -------------------- ------------------- -------------------- ------------------- ------------------- CFHT/MegaCam filters used until Jan 2015, $i$-band after Oct 2007 CFHT/MegaCam filters used after Feb 2015 CFHT/MegaCam $i$-band filter used until June 2007 Subaru/Suprime-Cam fully-depleted back-illuminated CCDs (installed Jan 2009) with SDSS-like filters Subaru/Suprime-Cam MIT/LL CCDs (used until Dec 2008) with Johnson/Bessell-like filters Subaru/Suprime-Cam MIT/LL CCDs (used until Dec 2008) with standard $z$-band filter CFHT/WIRCam Deep archival follow-up imaging data are available for many of these clusters [in particular data retrieved from the SMOKA science archive; @SMOKA], but these tend to be drawn from X-ray selected samples. Our analysis benefits from a homogeneous wavelength coverage of the full sample of 21 clusters. Studying the full sample not only statistically enhances the results presented in this study, but also ensures that we sample the full range of cluster dynamical properties, approximating a mass-selected sample. Since the spectroscopic data is of varying quality and completeness, we base our analysis almost entirely on the photometry. To be able to measure accurate and precise photometric redshifts of galaxies in the cluster fields, we require photometric coverage with at least 7 filters per cluster, ranging from the $u$-band ($\sim3000\AA$) to the $\mathrm{K_s}$-band ($\sim22,000\AA$). Accounting for the deep archival data for a sub-set of the sample, we obtained the remaining photometry through different time allocations on the wide-field imagers at CFHT (PI vdBurg, PIDs: 15AF006, 15BF005, 16BF013) and Subaru (PI Dahle, PIDs: S12B-164S, S13A-120, S15A-118). An overview of all imaging data is given in Table \[tab:photometry\]. The photometric data we use have been taken over more than 10 years, and some of the instruments and filter sets have been upgraded over this time span. These differences are indicated in the footnotes of Table \[tab:photometry\], and the different wavelength-responses of each filter and instrument are taken into account in our analysis. We perform basic steps to reduce the optical imaging data, such as bias, flat-field correction and cosmic-ray removal. As an additional step, we remove background patterns, particularly fringe residuals, by using the dithered pattern of observations to differentiate signals that are fixed in position on the ccd array from sky-bound signals. This is explained and illustrated in Fig. 1 of @vdB16. For the near infrared data from our own WIRCam program (PI vdBurg) we have followed an extended dither pattern, where the cluster centre is dithered from chip to chip. This ensures a very clean background subtraction, even on scales of the intra-cluster light (ICL). Astrometric registering has been performed with `SCAMP` [@scamp] using the USNO-B1 catalogue as reference, or with external catalogues from Pan-STARRS [@chambers16] for clusters that have been observed after the public release of their data. The astrometric precision between filters is better than 0.10$''$, ensuring that colour measurements can be done accurately. We automatically place masks on bright stars based on their locations in the guide star catalogue 2.3 [@lasker08]. After this the images are inspected manually, and masks are placed on additional diffraction spikes, reflective haloes, and image artefacts. During our analysis we take account of the reduced effective areas after masking. ### Object detection and colours {#sec:objectdetection} We perform object detection in the original $\mathrm{K_s}$-band. Since the range in M/L between the different galaxy types is smallest in this band, this ensures a catalogue that is close to being stellar-mass selected. We use `SExtractor` to detect objects, following the criterion that at least 5 adjacent pixels have a flux density that is $>$ 1.5$\sigma$ relative to the local background RMS. To be able to measure colours of the same intrinsic part of the galaxies we study, differences in the PSF between clusters and filters need to be accounted for. We use `PSFEx` [@psfex] to determine convolution kernels that homogenise the PSF for each cluster between different filters. We then measure colours for the $\mathrm{K_s}$-band-selected sources by performing photometry in circular apertures with a 2$''$ diameter on these PSF-matched images. A benefit of these wide-field images is that they contain a large population of Galactic stars that can be used to calibrate the flux scale [e.g. @SLR; @gilbank11; @vdB13; @kelly14]. We use the effective wavelength-response curves of each detector, filter[^4] and atmosphere model in combination with the stellar spectral library of @pickles98 to create a reference stellar locus. The stellar spectral library we use[^5] is updated with additional information in the near-infrared from @ivanov04, in addition to the original library from @pickles98. After applying magnitude offsets to our instrumental magnitudes to match this reference locus, we reach a photometric calibration with an uncertainty that depends on the filter, but is generally $\lesssim0.05$ magnitude. While this calibrates the colours of sources in our catalogues, we perform absolute flux calibration in the $\mathrm{K_s}$-band with respect to the 2MASS all-sky reference catalogue [@2MASS]. Uncertainties on aperture flux measurements of faint galaxies are dominated by fluctuations in the sky background. We estimate this noise component by randomly placing apertures on sky positions that do not overlap with sources detected in the $\mathrm{K_s}$-band. This procedure takes into account the correlated noise properties between adjacent pixels that originated from the convolution and the re-binning of the data to a common grid and PSF. UltraVISTA Reference Field -------------------------- There are two reasons to study a reference field, i.e. a field without a massive cluster, in our work. First, to study the impact on the evolution of galaxies by their massive host haloes, properties of galaxies at the same redshift between cluster and field are compared. Second, to study the properties of galaxies that are part of the massive clusters in our sample, we have to consider projection effects; i.e. fore- and background galaxies that enter our sample of cluster galaxies. By performing the same selection criteria on galaxies in a parallel field, we can take these projection effects into account statistically. We refer to this as “background subtraction”. The more information (photometric redshift, distance from cluster centre) one can use, the more cleanly the background can be accounted for. We make use of the COSMOS/UltraVISTA field, which has been extensively studied and for which a multi-band photometric catalogue is publicly available [@muzzin13a]. How we use this catalogue for the background correction, taking cosmic variance uncertainties due to this single field into account, is described in the Sect. \[sec:bgcorrection\]. Analysis {#sec:analysis} ======== Photometric Redshifts --------------------- We use the template-fitting code `EAZY` [@brammer08] to estimate photometric redshifts (photo-$z$s) for each object. The photo-$z$s correspond to the peak ($z_\mathrm{peak}$) of the posterior probability distribution P($z$) given by `EAZY`. Figure \[fig:speczphotz\]a shows a comparison between the spectroscopic redshift and photometric redshift. We define a relative scatter $\Delta z = \frac{z_{\mathrm{phot}}-z_{\mathrm{spec}}}{1+z_{\mathrm{spec}}}$ for each object with a reliable spectroscopic redshift $z_{\mathrm{spec}}$. There are $\sim3.5 \%$ outliers, defined as objects for which $|\Delta z |> 0.15$. For the remaining galaxies we measure the mean of $\Delta z$ and the scatter around this mean, $\sigma_z$, finding a bias of $|\Delta z |= 0.008$ and scatter of $\sigma_z = 0.029$. Since the galaxies for which spectroscopic redshifts have been measured are generally bright, and preferentially have emission lines, it is not immediately clear if the reported performance is representative for the entire galaxy sample down to the detection limit. In Fig. \[fig:speczphotz\]b we separate the galaxies by class (star-forming versus quiescent, cf. Sect. \[sec:rfcolours\]), and plot the differences as a function of $\mathrm{K_s}$-band magnitude, restricted to the redshift range where our clusters are located, $0.5<z_\mathrm{spec}<0.7$. Within this spectroscopic redshift range, the photo-$z$ scatter is $\sigma_z = 0.028$ for star-forming galaxies, and $\sigma_z = 0.023$ for quiescent galaxies. This Figure shows that the success rate of measuring spectroscopic redshifts of faint galaxies is higher for star-forming than for quiescent galaxies, since the former have typically strong emission lines. The scatter, shown in the solid curve, based on a running bin width of 1 magnitude, does not significantly depend on magnitude. This suggest that the flux measurements that define the SED are precise, and the photo-$z$ determination is limited by the representativity of the templates, the accuracy of the filter curves, and the accuracy of the flux calibration. Since photo-$z$s for quiescent galaxies are slightly more precise than for star-forming galaxies (due to a stronger spectral feature around the characteristic 4000$\AA$ break), we expect the photo-$z$ performance of quiescent galaxies to be at least similar to that of star-forming galaxies at faint magnitudes. We use the broad-band colours to identify and flag stars in our catalogue [e.g. @whitaker11; @vdB13], without having to make a selection based on size or morphology. That is because galaxies have very different Spectral Energy Distributions (SEDs) from stars, particularly towards near-IR wavelengths. We use the following colour criterion, which is similar to the ones used in aforementioned studies, to select the sample of galaxies: $$\mathrm{J-K_s} >0.18\cdot(u- \mathrm{J} )-0.60 \cup \mathrm{J-K_s} >0.08\cdot(u- \mathrm{J} )-0.30$$ Identifying BCGs ---------------- We select the brightest cluster galaxy (BCG) in the $\mathrm{K_s}$-band that is located within 1$'$ from the X-ray peak, and that has a photometric redshift within 0.10 from the cluster redshift. In all but one case this automatic identification corresponds to what we would have selected by hand based on the colour images in Appendix \[sec:colourimages\]. For `PSZ2 G219.89-34.39` there is a mis-identification since the apparent BCG has a blue core, and is likely contaminated by blue light from a nearby source (cf. \[fig:gallery4\], as is also apparent from the VLT/FORS2 spectrum of this galaxy). The photo-$z$ is 0.56, which is 0.17 lower than the cluster redshift. We select only this BCG by hand, and all the others following the criteria above. The BCGs are marked in the Figures of Appendix \[sec:colourimages\], and their positions are reported in Table \[tab:dataoverview\]. Stellar masses and background correction {#sec:bgcorrection} ---------------------------------------- We measure stellar masses for each galaxy using the SED-fitting code FAST [@kriek09]. We use stellar population synthesis models from @bc03, and assume a @chabrier03 IMF, solar metallicity, and the @calzetti00 dust law. The star formation history is parametrised as $SFR \propto e^{-t/\tau}$, where the time-scale $\tau$ is allowed to range between 10 Myr and 10 Gyr. These settings are identical to those used to measure stellar masses of the UltraVISTA sample, which we use to provide a field comparison. For an appropriate analysis that relies on a statistical subtraction of galaxies in the clusters’ fore- and background, as described next, we fix the redshift of each individual galaxy to the clusters’ mean redshift (cf. Table \[tab:dataoverview\]). We also construct tailored catalogues from the UltraVISTA main catalogue for each cluster, where we select only 8 filters: $uBVriz\mathrm{JK_s}$, and add artificial noise to the aperture flux measurements to match the depth of the data in the cluster fields. We verify that the performance of EAZY (scatter in photo-$z$s versus spec-$z$s) is then similar to that of the cluster fields for similarly bright galaxies. To perform the statistical field subtraction of fore- and background galaxies, we run FAST on the tailored catalogues to estimate stellar masses, while also fixing the redshift of each galaxy to the cluster’s mean redshift. Identical settings are used when running FAST on the cluster fields, as on the reference field. Cluster galaxies are initially selected to have a photometric redshift within 0.07 from the cluster mean spectroscopic redshift, which is several times larger than the photo-$z$ scatter. In Appendix \[sec:appphotozsel\] we test the effect of this choice, and show that the reported results are robust. The uncertainty due to cosmic variance in the reference field is taken into account following the prescription of @moster11, based on the volume subtended by the UltraVISTA area in the redshift range $0.5<z<0.7$. We find that this has no significant effect on the cluster SMF, except in the outskirts, where the overdensity of the cluster field with respect to the background decreases. The estimated uncertainties are shown in the main figures presented in this work. Completeness correction {#sec:completeness} ----------------------- We identify and correct for two observational effects that affect galaxies around the detection limit in the $\mathrm{K_s}$-band stack. First, the detection rate of objects drops towards fainter magnitudes due to noise fluctuations. Second, the objects that *are* detected may have a flux measurement that is biased compared to their intrinsic brightness. To measure the influence of these effects, we study the recovered fraction and fluxes of simulated sources in the detection band. We inject sources with an exponential (i.e. Sérsic-$n$=1) profile and half-light radii between 1-3 kpc (uniform distribution), ellipticities uniformly drawn between 0.0 and 0.2, and a uniform magnitude distribution. These values are appropriate for sources around our detection limit. Since the depth of the detection image is not uniform, we consider the region within 6$'$ radial distance from the cluster centres. This corresponds to 2.2 Mpc (2.6 Mpc) at $z=0.50$ ($z=0.70$) and covers the parts of the clusters that are relevant for this study. In one occasion we will probe the properties of galaxies up to larger cluster-centric distances; in Appendix \[sec:appraddependence\] we study the effect of the reduced depth in the cluster outskirts on this result. We run exactly the same source detection algorithm as for the main analysis (cf. Sect. \[sec:objectdetection\]) on the $\mathrm{K_s}$-band stacks that include the simulated sources. The $\mathrm{K_s}$-band magnitude limits at which 80% of the simulated sources are recovered, are listed in Table \[tab:photometry\] and also indicated in the right hand panel of Fig. \[fig:speczphotz\]. We find that around this limit, sources are measured to be 0.12 magnitudes fainter than they are intrinsically, while for the brightest sources this difference is negligible ($\sim 0.01$). We correct the measured fluxes by these magnitude-dependent corrections. Stellar mass limits that correspond to the 80% completeness limit in the $\mathrm{K_s}$-band are also listed in Table \[tab:photometry\]. We base these on a @bc03 template with a formation redshift of $z$=3.0. Since such an old stellar population is relatively faint for their stellar mass, this corresponds to a conservative limit when galaxies with more recent star formation are also considered. Star-forming versus Quiescent galaxies {#sec:rfcolours} -------------------------------------- We measure rest-frame U-V and V-J colours of the best-fit SEDs from `EAZY`, while fixing the redshift to the cluster mean redshift. These colours have been shown to be effective to separate star-forming from quiescent galaxies, even in the presence of dust reddening [e.g. @wuyts07; @williams09; @patel12]. There are small offsets in the UVJ colour distribution between the different clusters, and with respect to the UltraVISTA field sample. Such differences are not uncommon, and several studies have applied corrections to the selection criteria to rectify this [@whitaker11; @muzzin13b; @skelton14 also see the discussion in Appendix A of @leebrown17]. Rather than adapting the selection criteria of quiescent galaxies for each cluster, we applied offsets to the rest-frame colours to shift them back to the UltraVISTA reference in redshift range $0.5<z<0.7$. The mean absolute shift applied is 0.06 magnitude to the U-V colour, and 0.04 to the V-J colour. This brings the colour distribution of cluster galaxies in good agreement with those of field galaxies in UltraVISTA, see Fig. \[fig:UVJ\]. In Appendix \[sec:appuvjdiv\] we test the effect of a possible residual shift in colour between the cluster fields and the reference field to the main results presented in this paper. Inspecting the bimodal galaxy distribution by-eye, we select a sample of quiescent galaxies following the following criteria: $$\mathrm{U-V} > 1.3 \,\,\,\cap\,\,\, \mathrm{V-J} < 1.6 \,\,\,\cap\,\,\, \mathrm{U-V} > 0.55+(\mathrm{V-J})$$ Note that Fig. \[fig:UVJ\] shows the UVJ colour distribution of the UltraVISTA galaxies based on all the photometric information. For the purpose of a statistical background subtraction we run a similar analysis on a subset of eight UltraVISTA filters, with noise added to resemble the photometric quality of the cluster fields (cf. Sect. \[sec:bgcorrection\]). The Stellar Mass Function {#sec:SMF} ========================= -- ------------------------------ ---------------------------------------- ---------------------------------------- ---------------------------------------- ---------------------------------------- ---------------------------------------- ------------------------------ log\[M$_\star$/$M_{\odot}$\] R$<$2R$_{500}$ R$<$0.5R$_{500}$ 0.5R$_{500}<$R$<$R$_{500}$ R$_{500}<$R$<$1.5R$_{500}$ 1.5R$_{500}<$R$<$2R$_{500}$ 9.55 $270.4^{+23.3+10.2}_{-23.2-10.2}$ $67.73^{+9.76+0.63}_{-7.06-0.63}$ $86.75^{+15.26+1.90}_{-8.97-1.90}$ $45.34^{+9.68+3.17}_{-8.13-3.17}$ $64.16^{+12.98+4.44}_{-8.35-4.44}$ $577.8^{+16.9}_{-14.2}$ 9.65 $272.0^{+17.4+10.3}_{-18.6-10.3}$ $65.95^{+8.54+0.64}_{-7.99-0.65}$ $98.68^{+11.22+1.94}_{-10.19-1.94}$ $55.37^{+10.82+3.23}_{-8.44-3.23}$ $51.56^{+15.92+4.53}_{-9.64-4.53}$ $639.3^{+18.9}_{-20.3}$ 9.75 $258.8^{+15.9+8.7}_{-14.3-8.7}$ $68.02^{+6.58+0.54}_{-5.32-0.54}$ $84.51^{+8.48+1.63}_{-8.44-1.63}$ $63.29^{+6.33+2.72}_{-9.62-2.72}$ $43.49^{+8.61+3.81}_{-7.83-3.81}$ $506.2^{+21.6}_{-18.2}$ 9.85 $238.5^{+14.8+9.5}_{-14.5-9.5}$ $68.28^{+8.60+0.60}_{-6.34-0.60}$ $70.43^{+5.75+1.79}_{-9.00-1.79}$ $56.29^{+5.65+2.98}_{-6.85-2.98}$ $41.40^{+8.14+4.17}_{-5.79-4.17}$ $522.4^{+25.0}_{-20.3}$ 9.95 $248.8^{+13.7+8.1}_{-15.2-8.1}$ $69.72^{+8.28+0.51}_{-5.98-0.51}$ $79.01^{+7.48+1.52}_{-8.22-1.52}$ $50.37^{+6.21+2.53}_{-6.99-2.53}$ $42.79^{+6.85+3.55}_{-5.77-3.55}$ $458.9^{+14.2}_{-16.9}$ 10.05 $249.0^{+11.1+7.8}_{-13.3-7.8}$ $67.56^{+4.22+0.49}_{-6.63-0.49}$ $79.23^{+8.03+1.47}_{-6.96-1.47}$ $52.95^{+6.29+2.45}_{-6.02-2.45}$ $45.16^{+6.99+3.43}_{-5.49-3.43}$ $417.6^{+10.1}_{-14.9}$ 10.15 $255.4^{+14.1+8.6}_{-14.5-8.6}$ $77.62^{+6.52+0.54}_{-6.26-0.54}$ $74.86^{+6.95+1.61}_{-5.67-1.61}$ $56.22^{+5.47+2.68}_{-6.60-2.68}$ $41.55^{+7.43+3.75}_{-5.25-3.75}$ $435.2^{+14.9}_{-16.9}$ 10.25 $240.0^{+10.7+7.6}_{-11.5-7.6}$ $82.06^{+4.90+0.47}_{-7.44-0.47}$ $76.57^{+8.50+1.42}_{-6.20-1.42}$ $40.23^{+6.51+2.37}_{-5.71-2.37}$ $37.70^{+6.63+3.02}_{-5.40-2.10}$ $374.4^{+16.9}_{-16.2}$ 10.35 $252.9^{+17.0+7.7}_{-15.2-7.7}$ $85.57^{+6.23+0.48}_{-5.77-0.48}$ $67.31^{+6.20+1.44}_{-6.65-1.44}$ $52.52^{+5.59+2.39}_{-6.36-2.22}$ $44.74^{+6.50+3.35}_{-5.47-3.35}$ $364.3^{+18.9}_{-16.2}$ 10.45 $236.7^{+12.6+6.4}_{-13.7-6.4}$ $73.37^{+7.55+0.40}_{-6.77-0.40}$ $62.11^{+6.20+1.19}_{-4.32-1.19}$ $61.17^{+5.71+1.99}_{-5.99-1.99}$ $34.88^{+5.68+2.78}_{-5.48-2.78}$ $281.8^{+14.9}_{-11.5}$ 10.55 $197.1^{+11.0+6.2}_{-12.6-6.2}$ $61.12^{+5.39+0.17}_{-6.06-0.20}$ $63.43^{+4.28+1.15}_{-5.78-1.15}$ $39.87^{+7.27+1.92}_{-5.73-1.92}$ $30.35^{+4.67+2.69}_{-4.67-2.69}$ $287.9^{+11.5}_{-14.2}$ 10.65 $191.1^{+6.9+6.1}_{-10.3-6.1}$ $55.54^{+7.38+0.38}_{-6.72-0.34}$ $60.70^{+5.20+1.15}_{-4.99-1.15}$ $47.44^{+6.30+1.48}_{-4.68-1.28}$ $25.73^{+5.49+2.68}_{-4.37-2.68}$ $270.3^{+16.2}_{-14.2}$ 10.75 $135.3^{+8.4+4.5}_{-8.1-4.5}$ $44.03^{+5.78+0.28}_{-4.35-0.28}$ $35.68^{+3.86+0.85}_{-4.40-0.85}$ $31.59^{+4.16+1.42}_{-3.41-1.42}$ $21.96^{+3.73+1.93}_{-2.82-1.76}$ $206.8^{+8.8}_{-8.8}$ 10.85 $116.8^{+7.4+4.3}_{-9.6-4.3}$ $41.62^{+5.69+0.27}_{-2.60-0.27}$ $30.98^{+4.58+0.66}_{-3.42-0.46}$ $24.82^{+3.62+1.35}_{-4.57-1.35}$ $15.00^{+2.80+1.81}_{-3.46-1.74}$ $166.9^{+12.8}_{-9.5}$ 10.95 $92.26^{+8.30+2.93}_{-7.31-2.93}$ $27.66^{+3.55+0.18}_{-3.93-0.18}$ $32.27^{+4.50+0.55}_{-3.46-0.55}$ $14.46^{+3.64+0.91}_{-3.01-0.90}$ $16.20^{+4.26+1.03}_{-4.04-0.87}$ $138.5^{+10.8}_{-7.4}$ 11.05 $54.26^{+6.71+2.02}_{-5.49-1.73}$ $18.72^{+3.42+0.13}_{-3.11-0.12}$ $11.42^{+2.56+0.38}_{-2.95-0.38}$ $12.21^{+2.50+0.44}_{-2.25-0.44}$ $13.09^{+3.34+0.85}_{-2.91-0.62}$ $99.34^{+6.76}_{-8.11}$ 11.15 $36.58^{+5.66+1.20}_{-6.54-1.27}$ $11.57^{+3.04+0.09}_{-2.78-0.09}$ $10.10^{+2.81+0.21}_{-2.18-0.21}$ $\,\,\,7.01^{+2.47+0.35}_{-2.15-0.35}$ $\,\,\,6.80^{+2.92+0.57}_{-2.63-0.48}$ $65.55^{+6.76}_{-5.41}$ 11.25 $20.02^{+2.51+0.64}_{-4.76-0.64}$ $\,\,\,7.63^{+1.93+0.04}_{-2.04-0.04}$ $\,\,\,6.98^{+2.16+0.14}_{-1.83-0.14}$ $\,\,\,3.18^{+1.19+0.20}_{-1.44-0.20}$ $\,\,\,1.29^{+1.47+0.28}_{-1.07-0.28}$ $33.79^{+6.08}_{-4.05}$ 11.35 $16.69^{+3.46+0.54}_{-3.62-0.54}$ $\,\,\,5.84^{+1.71+0.03}_{-2.09-0.03}$ $\,\,\,5.25^{+1.81+0.10}_{-1.45-0.09}$ $\,\,\,2.53^{+1.48+0.14}_{-1.48-0.14}$ $\,\,\,2.76^{+1.07+0.20}_{-1.48-0.20}$ $24.33^{+4.73}_{-4.05}$ 11.45 $\,\,\,2.40^{+1.55+0.18}_{-1.43-0.18}$ $\,\,\,0.44^{+0.51+0.01}_{-0.44-0.01}$ $\,\,\,1.77^{+1.01+0.03}_{-1.03-0.03}$ $\,\,\,0.74^{+0.55+0.06}_{-0.60-0.06}$ - $14.87^{+3.38}_{-3.38}$ 11.55 $\,\,\,2.96^{+1.65+0.10}_{-1.44-0.10}$ $\,\,\,3.41^{+1.33+0.01}_{-1.15-0.01}$ - - - $\,\,\,6.76^{+2.03}_{-1.35}$ 11.65 $\,\,\,2.25^{+1.09+0.05}_{-0.99-0.05}$ $\,\,\,0.47^{+0.49+0.00}_{-0.47-0.00}$ $\,\,\,0.93^{+0.98+0.01}_{-0.50-0.01}$ $\,\,\,0.91^{+0.99+0.01}_{-0.91-0.02}$ - $\,\,\,4.73^{+2.03}_{-2.03}$ 11.75 $\,\,\,0.55^{+0.55+0.00}_{-0.55-0.00}$ $\,\,\,0.67^{+0.67+0.00}_{-0.67-0.00}$ - - - - 9.55 $208.9^{+15.5+1.0}_{-14.3-1.0}$ $61.36^{+7.11+0.06}_{-6.14-0.06}$ $76.56^{+10.23+0.18}_{-8.45-0.18}$ $31.98^{+8.31+0.30}_{-6.93-0.30}$ $36.66^{+7.53+0.42}_{-5.74-0.42}$ $79.74^{+7.43}_{-5.41}$ 9.65 $196.0^{+15.3+1.2}_{-13.2-1.2}$ $59.39^{+6.04+0.07}_{-8.69-0.07}$ $68.76^{+7.72+0.22}_{-8.66-0.22}$ $38.39^{+8.51+0.36}_{-6.05-0.36}$ $28.49^{+9.56+0.51}_{-5.77-0.51}$ $93.94^{+6.76}_{-8.11}$ 9.75 $189.4^{+13.2+1.1}_{-12.5-1.1}$ $56.59^{+4.75+0.07}_{-5.68-0.07}$ $66.27^{+7.49+0.22}_{-7.63-0.22}$ $36.91^{+4.40+0.36}_{-4.17-0.36}$ $27.87^{+4.69+0.50}_{-4.82-0.50}$ $83.80^{+7.43}_{-7.43}$ 9.85 $185.4^{+13.0+1.5}_{-11.6-1.5}$ $58.36^{+6.96+0.09}_{-6.11-0.09}$ $54.21^{+6.41+0.27}_{-6.09-0.27}$ $45.99^{+6.66+0.46}_{-4.08-0.46}$ $24.60^{+5.13+0.64}_{-4.35-0.64}$ $108.1^{+8.1}_{-8.8}$ 9.95 $184.5^{+11.4+1.5}_{-10.1-1.5}$ $61.24^{+6.61+0.09}_{-5.01-0.09}$ $64.33^{+4.62+0.28}_{-7.37-0.28}$ $33.14^{+5.62+0.47}_{-4.14-0.47}$ $22.77^{+5.86+0.65}_{-4.73-0.65}$ $109.5^{+8.8}_{-10.1}$ 10.05 $195.5^{+13.9+1.6}_{-11.4-1.7}$ $54.46^{+4.36+0.10}_{-5.43-0.10}$ $64.31^{+5.31+0.31}_{-6.57-0.31}$ $40.65^{+6.06+0.52}_{-5.30-0.52}$ $33.50^{+5.39+0.72}_{-4.21-0.72}$ $110.8^{+8.1}_{-7.4}$ 10.15 $197.6^{+10.3+2.3}_{-11.7-2.3}$ $67.70^{+6.61+0.14}_{-5.56-0.14}$ $57.65^{+7.12+0.43}_{-5.08-0.43}$ $39.00^{+5.20+0.72}_{-5.50-0.72}$ $27.86^{+4.97+1.00}_{-4.82-1.00}$ $150.0^{+8.1}_{-8.8}$ 10.25 $203.3^{+12.6+2.4}_{-9.0-2.4}$ $72.08^{+4.81+0.15}_{-5.76-0.15}$ $59.82^{+6.77+0.45}_{-6.26-0.45}$ $31.61^{+4.76+0.76}_{-3.54-0.76}$ $35.54^{+4.95+1.06}_{-4.57-1.06}$ $140.6^{+9.5}_{-10.8}$ 10.35 $209.9^{+11.1+2.8}_{-12.4-2.8}$ $76.81^{+5.61+0.17}_{-5.41-0.17}$ $55.44^{+5.46+0.52}_{-5.53-0.52}$ $48.01^{+5.22+0.86}_{-6.06-0.86}$ $27.57^{+3.98+1.20}_{-3.51-1.20}$ $148.7^{+10.1}_{-9.5}$ 10.45 $209.2^{+11.1+2.5}_{-11.9-2.5}$ $71.39^{+6.97+0.16}_{-7.24-0.16}$ $55.60^{+5.05+0.47}_{-5.27-0.47}$ $50.44^{+6.26+0.78}_{-3.96-0.78}$ $26.87^{+3.99+1.09}_{-3.85-1.09}$ $138.5^{+8.8}_{-9.5}$ 10.55 $171.3^{+12.1+2.7}_{-9.8-2.7}$ $60.52^{+5.64+0.17}_{-6.16-0.17}$ $56.03^{+4.08+0.50}_{-5.23-0.50}$ $33.58^{+4.78+0.84}_{-4.82-0.84}$ $19.98^{+3.95+1.17}_{-3.95-1.17}$ $150.0^{+8.8}_{-9.5}$ 10.65 $171.8^{+9.3+2.8}_{-7.6-2.8}$ $52.72^{+7.25+0.18}_{-5.05-0.18}$ $50.89^{+5.19+0.53}_{-3.07-0.53}$ $46.88^{+5.27+0.88}_{-5.24-0.88}$ $19.83^{+4.11+1.23}_{-4.21-1.23}$ $152.1^{+12.2}_{-12.2}$ 10.75 $126.0^{+6.0+2.4}_{-6.6-2.4}$ $42.84^{+4.95+0.15}_{-5.22-0.15}$ $34.77^{+3.35+0.45}_{-4.92-0.45}$ $27.09^{+2.92+0.74}_{-3.12-0.74}$ $19.31^{+3.92+1.04}_{-3.34-1.04}$ $134.5^{+7.4}_{-6.8}$ 10.85 $107.3^{+7.0+2.4}_{-8.9-2.4}$ $38.84^{+4.71+0.15}_{-3.35-0.15}$ $30.28^{+4.23+0.46}_{-3.29-0.46}$ $21.01^{+3.54+0.76}_{-3.28-0.76}$ $13.05^{+3.05+1.07}_{-2.13-1.07}$ $123.0^{+10.1}_{-10.1}$ 10.95 $86.82^{+7.28+1.98}_{-8.12-1.98}$ $25.63^{+3.61+0.12}_{-2.85-0.12}$ $29.82^{+4.57+0.37}_{-3.20-0.37}$ $12.61^{+3.73+0.62}_{-3.40-0.62}$ $15.77^{+4.59+0.87}_{-3.62-0.87}$ $103.4^{+9.5}_{-8.1}$ 11.05 $52.98^{+6.69+1.42}_{-5.66-1.42}$ $18.30^{+3.22+0.09}_{-3.08-0.09}$ $\,\,\,9.85^{+2.78+0.27}_{-2.26-0.27}$ $11.96^{+2.52+0.44}_{-2.11-0.44}$ $12.32^{+3.11+0.62}_{-2.20-0.62}$ $77.04^{+6.76}_{-6.76}$ 11.15 $35.66^{+5.50+1.11}_{-5.70-1.11}$ $10.77^{+2.95+0.07}_{-2.43-0.07}$ $\,\,\,9.97^{+2.64+0.21}_{-2.27-0.21}$ $\,\,\,7.01^{+2.47+0.35}_{-2.15-0.35}$ $\,\,\,6.67^{+2.83+0.48}_{-2.52-0.48}$ $59.47^{+6.08}_{-7.43}$ 11.25 $19.90^{+2.50+0.64}_{-4.78-0.64}$ $\,\,\,7.63^{+1.93+0.04}_{-2.04-0.04}$ $\,\,\,6.19^{+2.41+0.12}_{-1.42-0.12}$ $\,\,\,3.18^{+1.19+0.20}_{-1.44-0.20}$ $\,\,\,1.29^{+1.47+0.28}_{-1.07-0.28}$ $31.76^{+6.08}_{-3.38}$ 11.35 $15.64^{+3.93+0.46}_{-3.26-0.46}$ $\,\,\,5.19^{+1.46+0.03}_{-1.92-0.03}$ $\,\,\,5.01^{+1.41+0.09}_{-1.61-0.09}$ $\,\,\,2.05^{+1.46+0.14}_{-1.46-0.14}$ $\,\,\,2.76^{+1.07+0.20}_{-1.48-0.20}$ $22.30^{+4.73}_{-4.05}$ 11.45 $\,\,\,2.40^{+1.55+0.18}_{-1.43-0.18}$ $\,\,\,0.44^{+0.51+0.01}_{-0.44-0.01}$ $\,\,\,1.77^{+1.01+0.03}_{-1.03-0.03}$ $\,\,\,0.74^{+0.55+0.06}_{-0.60-0.06}$ - $14.87^{+2.70}_{-3.38}$ 11.55 $\,\,\,2.96^{+1.65+0.10}_{-1.44-0.10}$ $\,\,\,3.41^{+1.33+0.01}_{-1.15-0.01}$ - - - $\,\,\,6.76^{+2.03}_{-1.35}$ 11.65 $\,\,\,2.25^{+1.09+0.05}_{-0.99-0.05}$ $\,\,\,0.47^{+0.49+0.00}_{-0.47-0.00}$ $\,\,\,0.93^{+0.98+0.01}_{-0.50-0.01}$ $\,\,\,0.91^{+0.99+0.01}_{-0.91-0.02}$ - $\,\,\,4.73^{+2.03}_{-2.03}$ 11.75 $\,\,\,0.55^{+0.55+0.00}_{-0.55-0.00}$ $\,\,\,0.67^{+0.67+0.00}_{-0.67-0.00}$ - - - - 9.55 $59.59^{+14.93+9.19}_{-14.11-9.19}$ $\,\,\,7.59^{+2.97+0.57}_{-4.42-0.57}$ $10.71^{+5.75+1.72}_{-4.77-1.72}$ $13.26^{+6.21+2.87}_{-5.77-2.87}$ $28.35^{+7.58+4.02}_{-9.32-4.02}$ $497.4^{+17.6}_{-13.5}$ 9.65 $75.72^{+13.39+9.18}_{-12.83-9.18}$ $\,\,\,7.92^{+4.18+0.57}_{-3.73-0.57}$ $30.11^{+8.43+1.72}_{-6.42-1.72}$ $17.61^{+5.90+2.87}_{-6.77-2.87}$ $23.35^{+7.92+4.02}_{-6.99-4.02}$ $545.4^{+16.9}_{-18.9}$ 9.75 $70.13^{+8.25+7.56}_{-11.11-7.56}$ $12.19^{+3.50+0.47}_{-4.03-0.47}$ $19.02^{+4.43+1.42}_{-4.95-1.42}$ $25.68^{+5.21+2.36}_{-6.45-2.36}$ $15.47^{+5.73+3.31}_{-4.47-3.31}$ $423.0^{+19.6}_{-17.6}$ 9.85 $53.06^{+9.29+8.08}_{-8.34-8.09}$ $10.30^{+2.51+0.51}_{-2.59-0.51}$ $15.02^{+4.76+1.52}_{-3.84-1.52}$ $\,\,\,8.79^{+4.00+2.53}_{-4.40-2.53}$ $17.56^{+4.30+3.54}_{-5.73-3.54}$ $417.6^{+18.2}_{-19.6}$ 9.95 $63.89^{+8.45+6.61}_{-8.22-6.61}$ $\,\,\,8.60^{+2.65+0.41}_{-2.08-0.41}$ $15.27^{+3.66+1.24}_{-3.76-1.24}$ $16.63^{+4.13+2.07}_{-3.72-2.07}$ $19.82^{+5.03+2.89}_{-5.55-2.89}$ $348.0^{+14.2}_{-14.9}$ 10.05 $51.46^{+6.05+6.19}_{-7.61-6.19}$ $11.91^{+3.03+0.39}_{-2.34-0.39}$ $15.32^{+3.13+1.16}_{-2.72-1.16}$ $11.96^{+2.92+1.93}_{-3.55-1.94}$ $11.57^{+4.26+2.71}_{-3.29-2.71}$ $304.1^{+14.9}_{-10.8}$ 10.15 $58.46^{+5.74+6.29}_{-7.40-6.29}$ $\,\,\,9.26^{+2.45+0.39}_{-2.44-0.39}$ $16.94^{+3.67+1.18}_{-2.85-1.18}$ $17.33^{+3.88+1.97}_{-4.12-1.97}$ $14.74^{+3.58+2.75}_{-3.98-2.75}$ $283.8^{+12.8}_{-14.2}$ 10.25 $35.01^{+6.63+5.15}_{-5.68-5.15}$ $\,\,\,9.59^{+1.91+0.32}_{-2.21-0.32}$ $17.04^{+3.50+0.97}_{-3.89-0.97}$ $\,\,\,8.64^{+2.71+1.61}_{-3.62-1.61}$ $\,\,\,1.42^{+2.98+2.25}_{-1.42-1.42}$ $233.8^{+12.2}_{-12.8}$ 10.35 $43.59^{+6.49+4.91}_{-6.04-4.91}$ $\,\,\,9.09^{+2.42+0.31}_{-2.11-0.31}$ $12.25^{+3.50+0.92}_{-3.60-0.92}$ $\,\,\,5.10^{+2.60+1.53}_{-3.22-1.54}$ $16.68^{+4.51+2.15}_{-3.92-2.15}$ $213.6^{+14.2}_{-10.1}$ 10.45 $27.28^{+4.98+3.86}_{-5.58-3.86}$ $\,\,\,1.80^{+2.28+0.24}_{-1.16-0.24}$ $\,\,\,7.08^{+2.58+0.72}_{-1.92-0.72}$ $\,\,\,9.79^{+2.78+1.21}_{-3.01-1.21}$ $\,\,\,7.68^{+3.65+1.69}_{-2.69-1.69}$ $143.9^{+10.1}_{-7.4}$ 10.55 $24.65^{+4.92+3.47}_{-4.61-3.47}$ $\,\,\,0.00^{+0.86+0.10}_{-0.00-0.00}$ $\,\,\,6.95^{+3.83+0.65}_{-1.99-0.65}$ $\,\,\,6.67^{+2.68+1.09}_{-2.98-1.09}$ $10.12^{+3.30+1.52}_{-3.25-1.52}$ $138.5^{+8.8}_{-10.1}$ 10.65 $18.31^{+4.38+3.33}_{-5.26-3.33}$ $\,\,\,2.18^{+1.38+0.21}_{-1.70-0.21}$ $\,\,\,9.67^{+2.52+0.62}_{-2.30-0.62}$ $\,\,\,0.00^{+2.76+0.80}_{-0.00-0.00}$ $\,\,\,6.41^{+2.41+1.46}_{-2.92-1.46}$ $119.6^{+8.1}_{-10.8}$ 10.75 $\,\,\,9.30^{+4.17+2.17}_{-3.86-2.17}$ $\,\,\,1.52^{+1.01+0.14}_{-0.96-0.14}$ $\,\,\,1.08^{+1.03+0.41}_{-1.08-0.41}$ $\,\,\,4.49^{+2.37+0.68}_{-2.20-0.68}$ $\,\,\,2.48^{+2.67+0.95}_{-2.01-0.95}$ $71.63^{+6.76}_{-6.76}$ 10.85 $\,\,\,8.52^{+3.52+1.88}_{-2.70-1.88}$ $\,\,\,3.10^{+1.48+0.12}_{-1.47-0.12}$ $\,\,\,1.04^{+1.03+0.35}_{-1.04-0.35}$ $\,\,\,3.30^{+2.51+0.59}_{-2.07-0.59}$ $\,\,\,1.05^{+1.55+0.82}_{-1.05-0.82}$ $43.93^{+5.41}_{-4.05}$ 10.95 $\,\,\,5.43^{+3.48+0.95}_{-2.14-0.95}$ $\,\,\,1.44^{+1.17+0.06}_{-1.17-0.06}$ $\,\,\,2.24^{+1.46+0.18}_{-1.45-0.18}$ $\,\,\,2.13^{+1.47+0.30}_{-1.58-0.30}$ $\,\,\,0.00^{+0.82+0.21}_{-0.00-0.00}$ $35.14^{+6.08}_{-4.73}$ 11.05 $\,\,\,1.54^{+1.47+0.60}_{-1.51-0.60}$ $\,\,\,0.26^{+0.50+0.04}_{-0.26-0.04}$ $\,\,\,1.21^{+1.00+0.11}_{-0.97-0.11}$ $\,\,\,0.00^{+0.70+0.00}_{-0.00-0.00}$ $\,\,\,0.41^{+1.08+0.26}_{-0.41-0.26}$ $21.63^{+3.38}_{-2.70}$ 11.15 $\,\,\,0.12^{+1.14+0.37}_{-0.12-0.12}$ $\,\,\,0.81^{+0.49+0.02}_{-0.49-0.02}$ $\,\,\,0.04^{+0.50+0.07}_{-0.04-0.04}$ - $\,\,\,0.00^{+1.00+0.14}_{-0.00-0.00}$ $\,\,\,7.43^{+1.35}_{-2.03}$ 11.25 $\,\,\,0.00^{+0.32+0.00}_{-0.00-0.00}$ - $\,\,\,0.36^{+0.48+0.02}_{-0.36-0.02}$ - - $\,\,\,2.03^{+0.68}_{-1.35}$ 11.35 $\,\,\,1.04^{+1.10+0.08}_{-1.00-0.08}$ $\,\,\,0.64^{+0.67+0.00}_{-0.64-0.00}$ $\,\,\,0.41^{+0.50+0.01}_{-0.41-0.01}$ $\,\,\,0.33^{+0.48+0.02}_{-0.33-0.02}$ - $\,\,\,2.03^{+0.68}_{-0.68}$ 11.45 - - - - - $\,\,\,0.68^{+0.68}_{-0.68}$ -- ------------------------------ ---------------------------------------- ---------------------------------------- ---------------------------------------- ---------------------------------------- ---------------------------------------- ------------------------------ The galaxy stellar mass function (SMF) is a fundamental measure of any population of galaxies, and a critical measurement against which galaxy-formation models are tested. The SMF is measured for the cluster galaxies up to a cluster-centric radius of 2$\times R_{500}$ and shown in the left panel of Fig. \[fig:SMF\_clustervsfield\]. For this measurement all 21 clusters are stacked, excluding the BCGs. The background is subtracted and an incompleteness correction is applied. For each bin we only use clusters down to their 80% stellar mass completeness limit (cf. Table \[tab:photometry\]). To compensate for clusters missing in the lowest-mass bins, we increase the weight of galaxies in the remaining clusters. To weigh each cluster properly, we base these on the richnesses measured for each cluster. Richnesses are measured following the definition given in @rykoff14, and we discuss this mass proxy in van der Burg et al., in prep.. We measure the “average” cluster all the way down to the stellar mass limit of $10^{9.5}\,\mathrm{M_{\odot}}$. Error bars denote uncertainties estimated from 100 bootstrap re-samplings of all galaxies in which we draw galaxies with replacement. The number of galaxies we draw in each realisation follow a Poisson distribution with mean equal to the number of galaxies in the stack. To make sure that the uncertainties are not dominated by the (perhaps too low) number of clusters in our sample, we perform 25 additional re-samplings of the clusters themselves. We find that both bootstrap procedures result in comparable uncertainties; the sample of clusters is thus large enough that we would have obtained the same results as presented here, if we would have observed 21 different clusters taken from a similar parent sample. From the separation between quiescent and star-forming galaxies in the left panel of Fig. \[fig:SMF\_clustervsfield\], we find that the galaxy population in these massive clusters is completely dominated by quiescent galaxies, all the way down to the stellar mass limit ($10^{9.5}\,\mathrm{M_{\odot}}$). To interpret our findings of the cluster SMF further, we make a comparison with the SMF of field galaxies at the same redshift as the clusters, in the right hand panel of Fig. \[fig:SMF\_clustervsfield\]. From a comparison of cluster and field, it is clear that the fraction of quenched galaxies in the clusters is substantially higher than in the field, at each stellar mass. This is quantified further in the lower panels, where the relative fractions of star-forming and quiescent galaxies are presented as a function of stellar mass. We model the SMF by fitting a Schechter [@schechter76] function to the data. This function is parameterized as $$\Phi(M)= \ln (10) \Phi^{*} \left[ 10^{(M-M^{*})(1+\alpha)}\right] \exp \left[ -10^{(M-M^{*})}\right],$$ with $M^{*}$ being the characteristic mass, $\alpha$ the low-mass slope, and $\Phi^{*}$ the overall normalisation. We follow a maximum likelihood approach to estimate the parameters that define the shape of the Schechter functions, $M^{*}$ and $\alpha$, along with their uncertainties. For this we use the un-binned data points, and include the completeness correction to the individual galaxies, by setting their weights in the likelihood maximisation. For low stellar masses, these weights also compensate for clusters that are not complete down to these limits. For this purpose, each cluster is scaled by its richness. The background galaxies, from the reference field, are included in the same likelihood and have a negative weight. The normalisation of the Schechter function, $\Phi^{*}$, is evaluated by requiring that the integral over the considered stellar mass range (i.e. stellar masses larger than $10^{9.5}\,\mathrm{M_\odot}$) equals the number of all cluster galaxies (or more specifically, the sum of all weights). The best-fitting Schechter parameters are listed in Table \[tab:Schechter\], and the corresponding functions are shown in the top panels of Fig. \[fig:SMF\_clustervsfield\]. The reported Goodness-of-Fit values indicate that the Schechter functions provide reasonable fits to the data points. However, there seem to be some systematic residuals, especially towards the low-mass end of the SMF. Indeed, some literature studies fit double Schechter functions, but since we primarily work with the data points from now on, this paper does not discuss whether a fit can be improved with more degrees of freedom. The data points themselves are listed in Table \[tab:SMFpoints\]. Environment $\mathrm{log_{10}[}M^{*}/\mathrm{M_{\odot}}]$ $\alpha$ $\Phi^{*a}$ GoF$^b$ -- ------------------------- ----------------------------------------------- ------------------------------------ ----------------------------------- --------- $R<2R_{500}$ $10.81^{+0.02+0.00}_{-0.02-0.00}$ $-0.91^{+0.02+0.00}_{-0.02-0.00}$ $355.97\pm 4.22^{+9.26}_{-8.28}$ 1.19 $R<0.5R_{500}$ $10.81^{+0.02+0.02}_{-0.02--0.02}$ $-0.81^{+0.03+-0.02}_{-0.02-0.02}$ $119.89\pm 2.63^{+-4.29}_{-5.75}$ 1.12 $0.5R_{500}<R<R_{500}$ $10.85^{+0.03+0.00}_{-0.03-0.01}$ $-1.00^{+0.04+0.00}_{-0.03-0.00}$ $93.39\pm 2.00^{+2.36}_{-1.85}$ 1.01 $R_{500}<R<1.5R_{500}$ $10.76^{+0.03+-0.00}_{-0.04-0.01}$ $-0.85^{+0.05+0.02}_{-0.03--0.01}$ $86.44\pm 2.22^{+6.20}_{-2.33}$ 0.88 $1.5R_{500}<R<2R_{500}$ $10.79^{+0.06+0.01}_{-0.03-0.00}$ $-1.00^{+0.04+0.00}_{-0.06-0.00}$ $55.48\pm 1.58^{+3.98}_{-4.41}$ 0.73 Average field $10.98^{+0.02}_{-0.02}$ $-1.20^{+0.02}_{-0.02}$ $320.59\pm 3.49$ 0.61 $R<2R_{500}$ $10.81^{+0.01+0.00}_{-0.02-0.00}$ $-0.83^{+0.03+0.00}_{-0.02-0.00}$ $325.41\pm 4.27^{+3.99}_{-4.87}$ 1.22 $R<0.5R_{500}$ $10.82^{+0.03+-0.01}_{-0.03-0.01}$ $-0.78^{+0.04+0.02}_{-0.03--0.02}$ $111.98\pm 2.58^{+3.29}_{--2.61}$ 1.22 $0.5R_{500}<R<R_{500}$ $10.85^{+0.04+0.00}_{-0.03-0.00}$ $-0.95^{+0.04+0.00}_{-0.04--0.00}$ $82.55\pm 1.95^{+1.18}_{-0.26}$ 0.88 $R_{500}<R<1.5R_{500}$ $10.75^{+0.03+0.00}_{-0.03--0.00}$ $-0.71^{+0.05+0.00}_{-0.05-0.01}$ $80.44\pm 2.33^{+1.32}_{-2.82}$ 1.05 $1.5R_{500}<R<2R_{500}$ $10.80^{+0.05+0.01}_{-0.05-0.01}$ $-0.85^{+0.06+0.01}_{-0.06-0.01}$ $47.16\pm 1.62^{+1.79}_{-2.30}$ 0.86 Average field $10.86^{+0.02}_{-0.02}$ $-0.55^{+0.03}_{-0.03}$ $313.85\pm 5.83$ 0.72 $R<2R_{500}$ $10.50^{+0.04+0.02}_{-0.04-0.05}$ $-1.02^{+0.06+0.04}_{-0.06-0.02}$ $76.58\pm 2.12^{+5.14}_{-2.79}$ 0.80 $R<0.5R_{500}$ $10.69^{+0.13+0.00}_{-0.11-0.01}$ $-1.11^{+0.15+0.00}_{-0.14-0.01}$ $8.65\pm 0.61^{+0.35}_{-0.31}$ 1.32 $0.5R_{500}<R<R_{500}$ $10.54^{+0.09+0.01}_{-0.07-0.01}$ $-1.00^{+0.12+0.02}_{-0.11-0.00}$ $22.24\pm 1.13^{+1.46}_{-1.01}$ 1.03 $R_{500}<R<1.5R_{500}$ $10.42^{+0.09+0.06}_{-0.07-0.08}$ $-0.92^{+0.14+0.11}_{-0.14-0.07}$ $23.10\pm 1.29^{+0.31}_{--0.19}$ 1.09 $1.5R_{500}<R<2R_{500}$ $10.35^{+0.09+0.09}_{-0.07-0.17}$ $-0.95^{+0.14+0.23}_{-0.15-0.10}$ $28.89\pm 1.48^{+4.80}_{-1.30}$ 0.91 Average field $10.69^{+0.03}_{-0.03}$ $-1.33^{+0.03}_{-0.03}$ $238.72\pm 3.20$ 0.75 Normalisation is reported in the same units as the data points were presented in Table \[tab:SMFpoints\], i.e. \[cluster$^{-1}$\] for the cluster data, and \[$10^{-5}$ Mpc$^{-3}$\] for the average field. Even though we perform a maximum likelihood fit to the unbinned data, we report Goodness of Fits as $\chi^2/\mathrm{d.o.f.}$, where the best-fit models are compared to the binned data. Radial-dependence of SMF ------------------------ A study of the SMF of galaxies at different radial distances from the cluster centres would allow a more detailed understanding of what happens to the galaxies as they are accreted by the clusters. With the current sample we have the statistics to do this, and Fig. \[fig:SMF\_clusterradbins\] shows the former cluster SMF, split in four radial bins. The best-fitting Schechter functions are overplotted. Qualitatively, a strong trend is immediately visible; the quenched fraction of galaxies drops with radial distance from the cluster centre, at each stellar mass. Realising that the uncertainties on the best-fitting Schechter parameter are not independent of each other, we plot the 2-dimensional 68% and 95% confidence regions on $\alpha$ and $M^*$ in Fig. \[fig:ellipses\_master\]. We note that for star-forming galaxies the over-density compared to the reference field is low, especially in the outskirts. This results in large uncertainties on their SMF, and on the fitted Schechter parameters. The cosmic variance uncertainty is visualised by the lines that are superimposed on the ellipses; these connect the $\pm 1\sigma$ systematic uncertainties due to cosmic variance with the nominal measurement. These systematics are so large that the shape of the SMF of star-forming galaxies is consistent with being independent of environment. The quiescent galaxies have a much higher over-density compared to the reference field, and their SMF can thus be measured more accurately since cosmic variance plays a negligible role. The shape of the SMF of quiescent galaxies does not vary significantly with the radial distance from the cluster centre. Also the shape of the total SMF, which is always completely dominated by quiescent galaxies, does not vary significantly between radial bins. There is a significant difference, however, between the SMF of quiescent galaxies in the clusters and in the average field; there are relatively more low-mass quiescent galaxies in the clusters. Some quenching models [e.g. @peng10] interpret this as indicative of another quenching mechanism of galaxies in the field as in clusters. If galaxies are “environmentally” quenched, one would expect a steeper SMF of quenched galaxies at low masses. That is because the SMF of star-forming galaxies is steep, and because quenching due to environment is supposed to be largely mass-independent. Our findings are broadly consistent with that picture. Normalisation of the SMF ------------------------ The cluster SMFs presented in Figs. \[fig:SMF\_clustervsfield\] & \[fig:SMF\_clusterradbins\] are normalised in number of galaxies per cluster. To understand and compare the efficiency of galaxy formation in clusters to the field, we make a more direct comparison in the normalisations between the different environments. Since clusters have, by definition, a very high volume density of galaxies compared to the field, normalising the SMF over the total volume may not be insightful. Instead, following @vdB13, we normalise the SMF of field and clusters to the total amount of matter associated with each galaxy population in Fig. \[fig:SMF\_massnorm\]. For the field sample we take the total comoving volume in the redshift range $0.5<z<0.7$ covered by the UltraVISTA unmasked survey area of 1.62 square degrees [@muzzin13a], amounting to $1.5\cdot 10^6\,\mathrm{Mpc^{3}}$. Multiplying this with the average matter density of the Universe in our cosmology, we find a total amount of matter (i.e. dark matter + baryonic) of $6.1\cdot 10^{16}\,\mathrm{M_{\odot}}$. For the clusters we take the total mass within a projected radius of $R_{500}$, but integrated along the line-of-sight. For this we assume that the galaxies follow the total mass, approximated by an NFW profile with concentration $c_{500}\approx 2-3$ [e.g. @dutton14; @klypin16; @vdB18b]. To go from the mass within a sphere of radius $R_{500}$ (which equals $M_{500}$, by definition) to the mass within this cylinder, one multiplies $M_{500}$ with a constant factor of $\sim 1.43$. The total mass associated with the 21 clusters, within a projected radius of $R_{500}$ is then $2.0\cdot 10^{16}\,\mathrm{M_{\odot}}$. Fig. \[fig:SMF\_massnorm\] shows the resulting SMF, normalised by the total mass. Per unit total mass there is a clear overdensity of galaxies in the clusters compared to the field [qualitatively similar to what was found by @vdB13]. This shows that it is impossible to create clusters from simply accreting an average field population of galaxies, since the latter includes low-density regions such as voids, where the star formation efficiency is very low. Since groups are typically found in the vicinity of clusters, it is likely that the accretion of these systems caused a galaxy excess in clusters compared to the field [cf. discussions in @hennig17; @chiu18]. Literature comparison --------------------- The general trends we observe are in line with previous measurements. @vulcani13 study the SMF of cluster galaxies in the same redshift range as we do. They do, for each galaxy type, find no significant difference in the shape of their SMF between cluster and field. Contrary to this work, however, we do find a significant difference between the SMF of quiescent cluster and field galaxies. It is possible that this is owing to our data being substantially deeper ($\sim$ 1 dex), allowing us to probe the low-mass regions where the differences are most pronounced. @papovich18 also study the SMF over a redshift range that overlaps with ours, in different environments probed with ZFOURGE and in the NMBS. They have data with similar depth, or even slighly deeper than ours, and base their density estimates on a nearest-neighbour approach. Similar to this work, they find a steepening in the low-mass slope of the SMF of quiescent galaxies in overdensities compared to the field. However, @papovich18 also find an increase in $M^*$ towards higher densities [also see @davidzon16 who base their study on the VIPERS data set], which we do not find in the cluster environments. A possible explanation for the apparent discrepancy with these studies is that they probe more moderate overdensities, and include galaxies that are central to their own haloes, while we have deliberately not taken the BCGs into account. Central galaxies are expected to grow from the merging of in-falling satellites, and this may explain the increase in $M^*$. Also in the more moderate environments probed by ZFOURGE and VIPERS, mergers between satellites may be more frequent than in the cluster environment, where relative velocities are expected to be too high for mergers to occur. @annunziatella14 study the SMF of a massive CLASH cluster at $z=0.44$. They find that the galaxy population is dominated by quiescent galaxies, but only for stellar masses $M_{\star} \gtrsim 10^{10}\,\mathrm{M_\odot}$. This apparent lack of cluster quiescent galaxies may be the result of a different way of separating star-forming from quiescent galaxies, compared to our method. Also cluster-to-cluster variations may play a role here, as suggested by a similar study of the more local cluster Abell 209 [@annunziatella16]. At higher redshift, $z\sim 1$, @vdB13 measure the SMF of 10 clusters, and find that these systems are already dominated by quiescent galaxies down to stellar masses of $\sim10^{10}\,\mathrm{M_\odot}$. Contrarily to more local studies, they find that the shape of the SMF of quiescent is similar between clusters and the field. A likely explanation, apart from a possible evolution with redshift, is that a stellar mass limit of $10^{10}\,\mathrm{M_\odot}$ may not be low enough to probe any differences in the SMF were they are expected [cf. @nantais16 for a study at even higher redshift]. Environmental Quenching Efficiency {#sec:EQE} ================================== A significant result of this work is that the fraction of quiescent galaxies is much higher in the cluster environment than in the field, at the same redshift, and at a given stellar mass. Here we quantify this using the environmental quenching efficiency ($f_{\mathrm{EQ}}$), which can be thought of as the fraction of galaxies that would normally be star-forming in the field, but are quenched by their environment. Specifically, $$\label{eq:EQE} f_{\mathrm{EQ}}=\frac{f_\mathrm{q,cluster}-f_\mathrm{q,field}}{1-f_\mathrm{q,field}},$$ where $f_\mathrm{q,cluster}$ and $f_\mathrm{q,field}$ are the quiescent fraction of galaxies in the cluster and field environment, respectively. The quenched fractions are a function of both stellar mass and environment, and therefore also $f_{\mathrm{EQ}}$ may depend on stellar mass and environment. Such a term was already introduced by @vandenbosch2008, and sometimes the term “conversion fraction” is used for a similarly defined quantity [@balogh16; @fossati17]. A compilation of $f_{\mathrm{EQ}}$ in groups and clusters is shown in Figure 7 of @nantais16. The general trend is that $f_{\mathrm{EQ}}$ increases with increasing halo mass (broadly speaking, from group to cluster environments). There is also a hint that $f_{\mathrm{EQ}}$ increases towards the local universe, at a given halo mass. This is in line with results from e.g. @darvish16, who find no evidence for environmental quenching in more moderate overdensities in the COSMOS field at redshift $z\gtrsim 1$. Environmental quenching efficiency versus stellar mass {#sec:eqemass} ------------------------------------------------------ Figure \[fig:eqe\_vsstelmass\] shows the $f_{\mathrm{EQ}}$ as a function of stellar mass, in four radial bins from the cluster centres. Error bars are the 68% confidence regions from 100 bootstrap resamplings, where galaxies within the clusters are drawn with replacement. In addition, we perform 25 bootstrap resamplings in which the clusters themselves are drawn with replacement. The two bootstrap runs lead to similar uncertainties. We also show a cosmic variance uncertainty (indicated by the shaded regions) increases towards larger clustercentric radii, where the cluster over-density is lower. In none of the radial bins does the $f_{\mathrm{EQ}}$ show a systematic trend with stellar mass (either increasing or decreasing). However, there are significant wiggles around the mean values (indicated by the dotted line); in Sect. \[sec:discussion\] we quantify these as signatures of merging of cluster galaxies compared to the field, which likely happened in their pre-processing environment. Since there is no clear stellar-mass dependence of the $f_{\mathrm{EQ}}$ in any of the radial bins, it suggests that the main quenching process happening in clusters is mass-independent [also see e.g. @kawinwanichakij17]. A candidate would be a stripping process of the star-forming gas that is so rapid that it is essentially mass-independent, namely ram pressure stripping. Environmental quenching efficiency versus radius ------------------------------------------------ The $f_{\mathrm{EQ}}$ being independent of stellar mass, we study its dependence on clustercentric radius. We combine the $f_{\mathrm{EQ}}$ of all galaxies with stellar masses in the range $10^{9.5} \leq M_{\star}/\mathrm{M_{\odot}}\leq 10^{11}$ in Figure \[fig:eqe\_vsradius\]. Using logarithmic bins, we study the $f_{\mathrm{EQ}}$ from $0.01\times R_{500}$ to $4\times R_{500}$. There is a clear signature of environmental quenching that depends on radial distance (which scales with local density). The $f_{\mathrm{EQ}}$ in the cluster centres (where $R/R_{500}<0.1$) are extremely high, $\sim90\%$, even in projection where galaxies on the cluster periphery are mixed along the line-of-sight. There is a steep drop in the $f_{\mathrm{EQ}}$ with radial distance, especially in the range $0.2<R/R_{500}<2.0$. Interestingly, the $f_{\mathrm{EQ}}$ does not drop all the way to zero towards the cluster periphery, but rather converges to a value of $\sim 0.35$ at the largest clustercentric distances we probe. In Appendix \[sec:appraddependence\] we demonstrate that this measurement is robust, even though we study galaxies detected in a slightly shallower region of the $\mathrm{K_s}$-band stack. Using a cosmological N-body simulation, @wetzel14 show that this excess quenching at large radii may be the result of ejected cluster satellites, which orbit even beyond the clusters’ virial radii. Another possible contributor to this observed excess quenching on larger scales comes from galaxies that have been pre-processed in the rich group environment that surrounds galaxy clusters [e.g. @haines15; @bianconi18]. If we re-define the environmental quenching efficiency in Eq. \[eq:EQE\] with respect to the quenched fraction in the cluster periphery, i.e. substitute $f_\mathrm{q,field}$ with $f_\mathrm{q,periphery}$, we obtain the dashed curve shown in Fig. \[fig:eqe\_vsradius\]. This curve is based on a pre-processed value of 0.35, and illustrates the effect of the main quenching mechanism in the cluster. At first glance, the measured strong dependence of $f_{\mathrm{EQ}}$ on radius suggest that, whatever physical process is responsible, quenching must happen on a reasonably rapid timescale, at least when galaxies approach the cluster centres. If quenching were a slow process, freshly accreted star-forming galaxies would have time to migrate to the cluster centres while still forming stars, and this would lower the observed $f_{\mathrm{EQ}}$ in the cluster centres. We quantify these statements in the following subsection, in which we employ a simple quenching model to put the observations into context. A simple quenching model ------------------------ We consider a model to identify the approximate timescale over which a galaxy is environmentally quenched in the cluster, and the location where this quenching process is triggered. Our basis is a set of N-body simulations of four galaxy clusters, introduced in @taranu14. The four most massive clusters were identified from a large cosmological N-body simulation with 256$^3$ particles in a cosmological box of side length 512$h^{-1}$ Mpc. Particles in the re-simulation have masses of $6.16\times 10^8\, \mathrm{M_{\odot}}$, meaning that subhaloes down to relatively low halo mass can be resolved and traced in time from $z=3$ to $z=0$. Using this simulation, we investigate at which distances from the cluster centres a quenching transformation process is likely to start, and how long it would take for a galaxy to show a signature of quenched star formation. Following a similar approach as in @muzzin14, in which phase-space distribution of specific subhaloes in these simulations were tracked, we now only consider the projected clustercentric distances of a population of subhaloes in the simulation. Subhaloes are marked that have passed, for the first time, a clustercentric distance $r_{\mathrm{3D,quench}}/R_{500}$ at least a time of $T_{\mathrm{quench}}$ Gyr ago. Projecting each cluster in three directions (x,y,z), we mark the fraction of subhaloes that satisfy these criteria, as a function of projected clustercentric radius. The results are in Figs. \[fig:eqe\_sim\_varT\] & \[fig:eqe\_sim\_varR\], where one parameter in the model is kept constant, while the other is varied. We note that $T_{\mathrm{quench}}$ has to be interpreted as a delay time + quenching time, and that the quenching time itself is supposed to be a rapid process due to the absence of a significant fraction of green valley (transition) galaxies [@peng10; @wetzel13]. The similarity between Figs. \[fig:eqe\_sim\_varT\] & \[fig:eqe\_sim\_varR\] indicate that there is a degeneracy between the quenching radius and time scale. While any ejected satellites that have been quenched by the same (cluster-specific) mechanism would show up in the projected distributions, we note that any possible pre-processing of galaxies in the large-scale overdensity surrounding the clusters is by definition not shown in this simplistic model. We perform a maximum likelihood comparison between the results from the simulation and the actual data, as a function of quenching timescale and -radius, while we include the pre-processed fraction of satellites separately as an extra free parameter. For each value of $r_{\mathrm{3D,quench}}/R_{500}$ and $T_{\mathrm{quench}}$ in the model, we thus marginalise over this pre-processed fraction of galaxies. As an example, the data points for a pre-processed fraction of 0.35 are shown in Fig. \[fig:eqe\_vsradius\]. The resulting confidence regions on the parameters that represent the best model, are shown in Fig. \[fig:maxlikewithmodel\] and these confirm the strong degeneracy between $r_{\mathrm{3D,quench}}/R_{500}$ and $T_{\mathrm{quench}}$. The contours shown in Fig. \[fig:maxlikewithmodel\] follow an intriguing degeneracy; quenching appears to happen either on a short timescale at small clustercentric radii, or over a longer time scale at larger clustercentric radii. Both scenarios reproduce the data similarly well. We note that the bimodality in the 1-$\sigma$ contour is likely due to the limited number of four clusters that we used in the simulation (albeit as studied from 3 orthogonal sight lines). The pre-processed fractions of galaxies are not shown in Fig. \[fig:maxlikewithmodel\], and this fraction gradually decreases when going from small radii (and corresponding short times), to larger radii (and longer times). For example, the pre-processing level at $r_{\mathrm{3D,quench}}=1\times R_{500}$ and $T_{\mathrm{quench}}$=1 Gyr is $31.7^{+2.1}_{-2.4}$, while it decreases to $19.8^{+3.2}_{-2.9}$ at $r_{\mathrm{3D,quench}}=7\times R_{500}$ and $T_{\mathrm{quench}}$=3 Gyr. Even though there is a strong degeneracy between the parameters, we can put a firm lower limit to the radial distance from the cluster centre where quenching is triggered; $r_{\mathrm{quench}}> 0.67R_{500}$ (95%CL). We can link this to an ICM density where quenching occurs, using our deep XMM-Newton observations. Leaving a more detailed study of the radial gas density distribution of this cluster sample to @vdB18b, the gas density at $R=0.7R_{500}$ is around $\rho_{\mathrm{ICM}}\approx 2\cdot10^4\,\mathrm{M_{\odot}\,kpc^{-3}}$. Following their Equation 62, @Gunn1972 estimate at which ICM density ram pressure stripping is expected to become effective in the stripping of the interstellar material of a typical infalling galaxy. They find this to happen at an ICM density of $\sim 5\times 10^{-4}\,\mathrm{atoms\,cm^{-3}}$. The density at $R=0.7R_{500}$ is already about twice this value, making ram pressure stripping a likely contributor given our constraints on location and time scale of the main stripping process. Similarly, based on a cosmological simulation @zinger18 find that a significant removal of star-forming gas happens at $r \lesssim 0.5R_{\mathrm{vir}}$, which is a similar fraction of $R_{500}$. Another likely contributor to the quenching process is strangulation/starvation, which is a cut-off from the cosmological accretion of hot gas after a galaxy is accreted by the cluster main halo. Star formation is then expected to quench after the reservoir of molecular gas is depleted. This time scale is on the order of 1 Gyr for local low-mass galaxies [@tacconi18], and likely shorter at higher redshift. This is a slower process than ram pressure stripping, but its time scale is also consistent with our quenching constraints in Fig. \[fig:maxlikewithmodel\]. Since this process will be triggered at larger distance (or equivalently, at earlier times) than ram pressure stripping, both processes may be contributing to the observed elevated fraction of quenched galaxies. Caveats {#sec:caveats} ------- Our modelling has shown that the radial distribution of $f_{\mathrm{EQ}}$ combined with a realistic simulation of the orbits of cluster galaxies can provide meaningful constraints on the $r_{\mathrm{3D,quench}}/R_{500}$ and $T_{\mathrm{quench}}$ of cluster galaxies. There are several caveats in our analysis that could play a role in the interpretation, and these are highlighted below. In this paper we measure the $f_{\mathrm{EQ}}$ as the excess quenching with respect to the field, i.e. a representative section of the universe at the same redshift. The field sample therefore already includes regions of higher densities, such as groups. Several studies have quantified the $f_{\mathrm{EQ}}$ (or conversion fraction) as the excess quenching with respect to the lowest mass density found in the field [typically the lowest-density quartile, cf.  @papovich18]. The quenched fractions we report for the field include some “pre-quenching” in moderate overdensities, and the reported $f_{\mathrm{EQ}}$ values would have been even higher if defined with respect to the lowest-density regions. However, since our modelling marginalises over this pre-processing component, this assumption does not affect our estimated parameters that describe the main quenching process in the clusters. We have defined $f_{\mathrm{EQ}}$ with respect to the field *at the same epoch as the clusters are observed at*. An alternative approach is to consider the field quenched fraction at the time of galaxy accretion [see the discussion in @balogh16], as e.g. @foltz18 have done in their modelling of the quenching time scales. Both approaches have their uses; we have chosen the former so that we can isolate what happens in clusters separately, and in addition to, what would have happened to the star-forming properties of the galaxies if they had remained centrals in their own haloes. Furthermore, our definition of $f_{\mathrm{EQ}}$ is interpreted in the absence of mergers between galaxies. We know mergers are happening, especially for galaxies in group scales, and mergers may even contribute to the quenching of satellites [e.g. @peng10; @darvish16]. Mergers may also have affected the observed $f_{\mathrm{EQ}}$ dependence of clustercentric radius. Moreover, mergers lead eventually to growth of the central BCGs. In the next section we discuss these limitations and present a “transformation function” that describes how additional processes, such as mergers, are affecting the galaxy population to lead to the SMF we observe for cluster galaxies. Discussion {#sec:discussion} ========== The previous Section explored the stellar-mass dependence, and the radial dependence of $f_{\mathrm{EQ}}$ to infer a simple quenching scenario for which we have constrained the approximate location and time scale. In this Section we study the impact of a basic environmental-quenching scenario on the SMF, which we measured and studied in Sect. \[sec:SMF\]. The simplest environmental quenching model that works in a mass-independent fashion (as suggested by Fig. \[fig:eqe\_vsstelmass\]) leaves the shape of the SMF of star-forming galaxies independent of environment [@peng10]. This is consistent with our findings, considering the measurement uncertainties associated with measuring the SMF of star-forming galaxies in the cluster periphery (Fig. \[fig:ellipses\_master\]). The population of quenched cluster galaxies is then a combination of galaxies that would also have quenched outside of the cluster (mass-quenched galaxies), and the environmentally-quenched galaxies, which in principle follow a similar mass distribution as the star-forming galaxies. We start by employing this quenching model in its basic form, following @vdB13. Figure \[fig:simplemodel\] shows the SMF of cluster galaxies within $R \leq R_{500}$, i.e. a combination of the first two panels of Fig. \[fig:SMF\_clusterradbins\], or a differently-normalised version of Fig. \[fig:SMF\_massnorm\]. The plotted curves are not fits to the plotted data, but rather adapted versions (only in normalisation) of the best-fitting Schechter functions to the field data from the UltraVISTA survey. The total normalisation of the (red+blue) curve is set by the total stellar mass in cluster galaxies (all data points, including the BCGs). Since we assume that the environmental quenching process affects the star-forming population in a mass-independent manner, we set this by parameter $f_{\mathrm{EQ}}$, so that the relative normalisation of the star-forming SMF is (1-$f_{\mathrm{EQ}}$) compared to the one in the field. Again requiring that the normalisation (total stellar mass) of the blue Schechter function is the same as that of the blue cluster galaxies (data points), we find that $f_{\mathrm{EQ}}=0.80$. The quenched part of the star-forming population gives the “environmentally quenched galaxies”, and when these are combined with the “mass quenched galaxies”, we arrive at the total population of quenched cluster members, as shown by the solid red curve in Fig. \[fig:simplemodel\]. In its basic form, this quenching model over-predicts the abundance of quiescent low-mass galaxies with $M_{\star} \lesssim 10^{10.2}\,\mathrm{M_{\odot}}$. Interestingly, a similar trend is revealed in Fig. 10 of @vdB13, in which clusters at slightly higher redshift were studied. However, contrary to this earlier work, we now have the statistics to explore this regime in more detail. That low-mass galaxies show a deficit in clusters compared to this simple quenching model may have to do with their destruction, potentially leading to a build-up of the intra-cluster light (ICL). Given the negative colour gradients that are observed in the ICL of massive clusters, dwarf galaxies are likely contributors to the ICL at large clustercentric distances [@demaio15]. Furthermore, milder interactions and mergers between galaxies [likely also in their pre-processing environment, cf. @tomczak17] may also affect the SMF of galaxies, and make it diverge from the field. For instance, @rudnick12 invoke a model that includes mergers between galaxies to reproduce the luminosity function of clusters in the local universe (starting from a distant cluster at $z=1.62$). We attempt to encompass all these processes in a “transformation function”, shown in the lower panel of Fig. \[fig:simplemodel\]. Plotted is the difference between the data points for the quiescent galaxies, and the simple quenching model (solid red line). The plot is normalised in stellar mass per bin and per cluster, and highlights again the relative destruction of low-mass galaxies, in favour of the growth of more massive galaxies such as the central BCGs. Since this plot sums up to 0, by construction, it does not include the build-up of ICL. How to build a massive cluster of galaxies? ------------------------------------------- Given our study of 21 massive clusters at $0.5<z<0.7$, we summarise some of the steps required to assemble the galaxy population observed within these systems, as opposed to the general field: - As shown in Fig. \[fig:SMF\_massnorm\], where we have normalised the SMFs of clusters and field with respect to the total amount of mass associated with the respective galaxy populations, galaxies form relatively efficiently in (future) cluster environments compared to the average Universe. The galaxy abundance in the clusters we study, per unit total mass, is about twice average. - The quenched fraction is much higher in these clusters than in the field, at the same redshift, for each stellar mass we probe. It is also elevated compared to the pre-processing we find happening in the cluster surroundings. This quenching process happens in a largely mass-independent fashion (cf. Fig. \[fig:eqe\_vsstelmass\]). - There is a significant and strong radial trend in the quenched fraction of cluster galaxies, which we describe as the environmental quenching efficiency $f_{\mathrm{EQ}}$ (cf. Fig. \[fig:eqe\_vsradius\]). A comparison with a model that is based on orbits taken from an N-body simulation suggests that the quenching process likely involves strangulation/starvation after cut-off from cosmological accretion, and ram pressure stripping at smaller clustercentric distances to “finish the job”. Each of these processes ought to happen on time scales that are roughly consistent with what we find in the model. - Additional transformations are required to reproduce the observed cluster SMF. These are likely largely caused by merger events, the possible destruction of low-mass galaxies in the clusters, and an effective build-up of the BCGs. Mergers are happening likely *before* galaxies are being accreted into the clusters, since relative velocities in the cluster environments are too high for galaxies to merge there. We have quantified the combination of these effects in the lower panel of Fig. \[fig:simplemodel\]. Summary and conclusions {#sec:summary} ======================= We have studied the galaxy population in a sample of 21 high-mass clusters at $0.5<z<0.7$, found in the *Planck* SZ survey. Using multi-band photometry spanning $u$- to the $\mathrm{K_s}$-band for each cluster, we have defined a sample of cluster galaxies, which are highly overdense compared to the back- and foreground. The data allow for a precise measurement of the galaxy SMF in clusters of intermediate redshift. We have identified differences in the SMF between the cluster population, and galaxies in the field at the same redshift. Normalising the SMF to the total amount of matter associated with each galaxy population, we find that clusters have a higher galaxy content, per unit total mass, than the average field. The most significant differences between the galaxy population in cluster and field arise when we separate the galaxy population between star-forming and quiescent galaxies by means of their rest-frame U-V and V-J colour distributions. The shape of the SMF of star-forming galaxies does not depend on environment. On the contrary, the SMF of quiescent galaxies is significantly different between the cluster and field; there is a relatively higher fraction of low-mass quiescent galaxies in the clusters. Moreover, the fraction of passive galaxies is much higher in the cluster than in the field, and we quantify how this fraction rises steeply with decreasing cluster-centric radius. We measured the environmental quenching efficiency ($f_{\mathrm{EQ}}$), which describes the fraction of galaxies that would be forming stars in the field, but are quenched solely due to their environment. At fixed radial distance from the cluster centre, the $f_{\mathrm{EQ}}$ does not depend on stellar mass. Contrarily, the $f_{\mathrm{EQ}}$ shows a strong radial dependence within the cluster environment. We interpret the observed radial-dependence of the $f_{\mathrm{EQ}}$ with a simple quenching model based on an N-body simulation using which we constrain the characteristic location and time scale of the main environmental quenching process. We find a strong degeneracy between those two parameters. According to the model, quenching may already be triggered at $r_{\mathrm{3D,quench}}\approx 7\times R_{500}$, and would then happen on a long time scale $T_{\mathrm{quench}}\approx$3 Gyr. If quenching is triggered at shorter radial distances $r_{\mathrm{3D,quench}}\approx 1\times R_{500}$, it happens on roughly the molecular gas depletion time scale, $T_{\mathrm{quench}}\approx$ 1 Gyr. Part of the observed quenching excess is thus likely due to “starvation”/“strangulation” of galaxies using up their cold gas supply. Interestingly, the model rules out a quenching location $r_{\mathrm{quench}}< 0.67R_{500}$ at 95% confidence. Our *XMM-Newton* data show that the gas density at this clustercentric distance is so large that ram pressure stripping should be effective, and is likely responsible for the satellite quenching there. This process may thus “finish the job” whenever the starvation mechanism does not operate rapidly enough. We thank Andrea Biviano and Gabriella De Lucia for insightful discussions, and Michael Balogh for constructive feedback on the manuscript. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n$^{\circ}$ 340519. HD acknowledges financial support from the Research Council of Norway. Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institute National des Sciences de l’Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. This work is also based on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan. Based in part on data collected at Subaru Telescope and obtained from the SMOKA, which is operated by the Astronomy Data Center, National Astronomical Observatory of Japan. Based on observations obtained with *XMM-Newton*, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. The development of Planck has been supported by: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN and JA (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). This research made use of the following databases: the NED and IRSA databases, operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the NASA; SIMBAD, operated at CDS, Strasbourg, France; SZ-cluster database operated by IDOC at IAS under contract with CNES and CNRS. Robustness tests ================ In this Section we study the effect of several assumptions we had to make in our analysis on the final results. Photo-$z$ selection {#sec:appphotozsel} ------------------- One of the choices in our analysis is the initial photo-$z$ selection window of cluster members, before the background subtraction is performed. If such a window is chosen too small, uncertainties in photometric redshifts of cluster members may cause them to scatter out of the selection window. On the other hand, if the photo-$z$ selection window is chosen larger than necessary, we introduce additional noise while performing the statistical background subtraction. Since we use a single reference field to do the statistical background subtraction, especially the contribution from cosmic variance in that field would increase with a larger photo-$z$ cut (since the overdensity will go down). From all 1527 spectroscopically-confirmed cluster members in our sample, we find that 89.6% satisfy the photo-$z$ cut of 0.07 (i.e. only 10.4% have scattered out). So to first order, the normalisation of the cluster SMF is higher by 1/0.896 compared to our measurement. The scattering may also bias the shape of the SMF, since the photo-$z$ scatter depends (in principle, though only slightly in practise) on the galaxy stellar mass (cf. Fig. \[fig:speczphotz\]). To test the effect of this, we have performed the analysis with photo-$z$ selections of $|\Delta z|<$0.10 and 0.13. This leads to percentages of galaxies that scatter out, of 5.7% and 4.5% respectively, approaching the percentage of catastrophic outliers. We checked that the main results in this paper, i.e. the SMF and the behaviour of the $f_{\mathrm{EQ}}$ do not change significantly (i.e. by more than the reported uncertainties) when a broader photo-$z$ selection is chosen. The only exception is the SMF of star-forming galaxies in the outskirts ($R>R_{500}$) of the clusters, where the overdensity is very low. We note, however, that since the overdensity of cluster galaxies with respect to the background drops for a broader photo-$z$ selection, uncertainties of all measurements grow substantially. We have therefore chosen a cut of 0.07 in $|\Delta z|$. UVJ division {#sec:appuvjdiv} ------------ The division between star-forming and quiescent galaxies is a critical part of the analysis. Our analysis makes use of the U-V and V-J rest-frame colours, which ensures that we separate the effect of dust reddening from the reddening due to lack of star formation. In Sect. \[sec:rfcolours\] we described the small corrections (on average 0.04-0.06) we made to the rest-frame U-V and V-J colours of the cluster galaxies, to match them to the colour distribution of the reference field. Here we perform a test to check how a residual systematic colour offset between field and cluster data would impact our results. We increase all U-V rest-frame colours of cluster galaxies, and reference background galaxies, by $\pm$0.05, and re-measure the quenched fractions of galaxies in the cluster. Even such a large offset, compared to the residuals we expect, changes that quenched fractions of cluster galaxies by at most 10%. When we measure $f_{\mathrm{EQ}}$ this has a larger effect, since we have not changed the rest-frame colours of the field galaxies. The result on the radial dependence of $f_{\mathrm{EQ}}$ is shown in Fig. \[fig:eqe\_vsradius\_testUVJdiff\]. Even though a significant change is notable, we note that this is based on a rather extreme systematic between rest-frame colours of cluster and field galaxies. Radial dependence of $f_{\mathrm{EQ}}$ {#sec:appraddependence} -------------------------------------- The main analysis of this paper studies the properties of galaxies within $2\times R_{500}$ from the cluster centres. These regions fall on the deep part of the $\mathrm{K_s}$-band stacks, and we have characterised the completeness in Sect. \[sec:completeness\]. Figure \[fig:eqe\_vsradius\], however, shows the environmental quenching efficiency up to $4\times R_{500}$, which covers part of the shallower regions. Due to the dither strategy we chose for the $\mathrm{K_s}$-band imaging, the depth in the $\mathrm{K_s}$-band drops by a maximum amount of 0.7 magnitudes towards a distance of $4\times R_{500}$ from the cluster centres, corresponding to 0.3`dex` in stellar mass. We note that the optical data, which are essential for precise estimates of photometric redshifts, extend to a larger region around the clusters at uniform depth. We studied the impact of a slight decrease in depth of the detection band on the results plotted in Fig. \[fig:eqe\_vsradius\]. The measurements move within the plotted uncertainties, when galaxies in the range $10^{9.8} \leq M_{\star}/\mathrm{M_{\odot}}\leq 10^{11}$ are considered (instead of $10^{9.5} \leq M_{\star}/\mathrm{M_{\odot}}\leq 10^{11}$). In particular, we verified that the “plateau” in $f_{\mathrm{EQ}}$ at radii $R \gtrsim 2\times R_{500}$ is robust, and not an effect of this decrease in depth. Cluster gallery {#sec:colourimages} =============== Colour-composite images of all 21 clusters are shown in Figs. \[fig:gallery1\]-\[fig:gallery4\]. They are composed of $g$- or $B$-, $i$- or $I_\mathrm{c}$-, and $\mathrm{K_s}$-band imaging. Regions around bright stars, and their diffraction spikes, are clearly visible here, but these are all masked and not considered in our analysis. Overplotted are X-ray surface brightness contours from the deep *XMM-Newton* observations, which we have available for all clusters. They are based on (adaptively-) smoothed surface brightness maps, which are background subtracted, exposure corrected, and from which point sources are excised. Contours are logarithmically spaced with 0.2dex increments. These data form the basis of the X-ray morphological analysis, which is presented in Arnaud et al., in prep. {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} {width=".90\linewidth"} [^1]: [^2]: All quoted masses in this paper are defined with respect to the critical density at the cluster redshift. $R_{500}$ is thus defined as the radius at which the mean interior density is 500 times the critical density, and $M_{500}$ is the mass contained within this radius. We will occasionally use an overdensity of 200 in an analogous fashion. [^3]: https://ned.ipac.caltech.edu/ [^4]: Most of these are compiled at http://svo2.cab.inta-csic.es/theory/fps/index.php [^5]: see http://www.eso.org/sci/facilities/paranal/decommissioned/isaac/ tools/lib.html

--- abstract: | The rapid growth of research in explainable artificial intelligence (XAI) follows on two substantial developments. First, the enormous application success of modern machine learning methods, especially deep and reinforcement learning, which have created high expectations for industrial, commercial and social value. Second, the emergence of concern for creating trusted AI systems, including the creation of regulatory principles to ensure transparency and trust of AI systems. These two threads have created a kind of “perfect storm” of research activity, all eager to create and deliver [*any*]{} set of tools and techniques to address the XAI demand. As some surveys of current XAI suggest, there is yet to appear a principled framework that respects the literature of explainability in the history of science, and which provides a basis for the development of a framework for transparent XAI. Here we intend to provide a strategic inventory of XAI requirements, demonstrate their connection to a history of XAI ideas, and synthesize those ideas into a simple framework to calibrate five successive levels of XAI. address: - | XAI Lab,\ Alberta Machine Intelligence Institute,\ Department of Computing Science,\ University of Alberta,\ Edmonton, Alberta, Canada T6G 2E8 - | Department of Science,\ Augustana Faculty, University of Alberta,\ Camrose, Alberta, Canada, T4V 2R3 author: - 'S. Atakishiyev, H. Babiker, N. Farruque, R. Goebel[^1], M-Y. Kim$^{a}$, M.H. Motallebi, J. Rabelo, T. Syed, O. R. Zaïane' bibliography: - 'aij.bib' title: 'A multi-component framework for the analysis and design of explainable artificial intelligence' --- interpretation explanation reproducibility causal explainee-specific measurable evaluable Introduction {#section:intro} ============ Fueled by a growing need for trust and ethical artificial intelligence (AI) by design, the wake of the last decade of machine learning is crowded with a broad spectrum of research on explainable AI (XAI). It is therefore not surprising that the rapid and eclectic flurry of activities in XAI have exposed confusion and controversy about foundational concepts like [*interpretability*]{}, [*explanation*]{}, and [*causality*]{}. Perhaps confusion and disagreement is not surprising, given some of the complexity of modern learning methods. For example, when some deep learning methods can build predictive models based on more than 50 million distinct parameters, it is not a surprise that humans will debate what has been captured (e.g., see [@Chollet2017][^2]). Note also the confusion regarding misconceptions on a specious trade off between predictive accuracy and explainability (cf. [@rudin2018stopexplaining]), which have precipitated scientific workshops to address such misconceptions[^3]. Another example of yet-to-be-resolved issues includes the strange anomaly where syntactic reduction of the parameter space of some deep learning created models actually results in improved predictive accuracy (e.g., [@molchanov2019pruning; @cohen2018overparamterization]). The reality is that the foundations for scientific understanding of general machine learning, and thus XAI, is not yet sufficiently developed. Even though the long history of science and more recent history of scientific explanation and causality have considerable contributions to make (e.g., [@woodward2003scientific; @pearl2018book]), it seems like the demand created by potential industrial value has induced brittleness in identifying and confirming a robust trajectory from the history of formal systems to modern applications of AI. However, we believe that one can re-establish some important scientific momentum by exploiting what Newell and Simon’s Turing Award paper ([@newellsimon1976]) identified as the physical symbol systems hypothesis: “A physical symbol system has the necessary and sufficient means for general intelligent action." (p. 116). The challenge is to clarify connections between the recent vocabulary of XAI and their historical roots, in order to distinguish between scientifically valuable history and potentially new extensions. In what follows, we hope to articulate and connect a broader historical fabric of concepts essential to XAI, including interpretation, explanation, causality, evaluation, system debugging, expressivity, semantics, inference, abstraction, and prediction. +\[line width=1pt,solid, black, fill=yellow, postaction=[ pattern=north east lines ]{} \] table\[y=Explicit-explanation-representation, meta=Explicit-explanation-representation, col sep=comma\] [temptable.txt]{}; +\[line width=1pt,solid, black, fill=lime, postaction=[ pattern=grid ]{} \] table\[y=Alternative-explanations, meta=Alternative-explanations, col sep=comma\] [temptable.txt]{}; +\[line width=1pt,solid, black, fill=cyan, postaction=[ pattern=vertical lines ]{} \] table\[y=Knowledge-of-the-explainee, meta=Knowledge-of-the-explainee, col sep=comma\] [temptable.txt]{}; +\[line width=1pt,solid, black, fill=gray, postaction=[ pattern=dots ]{} \] table\[y=Interactive, meta=Interactive, col sep=comma\] [temptable.txt]{}; To do so, we need to articulate a general, if incomplete XAI framework, which is immediately challenged to be comprehensive enough to acknowledge a broad spectrum of components, yet but still avoid becoming a naïve long and unstructured survey of overlapping – sometimes competing – concepts and methods. Our approach is to start with a simple structural idea first illustrated by Figure \[fig:explainability-diagram\]. One can conceptualize the content of Figure \[fig:explainability-diagram\] by comparing an accepted framework around the articulation levels of autonomous driving ([@SAE2016]). The simple distinction of degree of human control creates the basis for discussion about how one could achieve Level 0 (no automation) to Level 5 (full automation), then discuss the details of those levels and the components necessary to achieve them. In our case for explainability, we intend the x-axis of Figure \[fig:explainability-diagram\] to distinguish what we consider a kind of quality of explanatory system, with the intuition that explanations at a higher level can be confirmed as “better” by some evaluation measure, e.g., we expect a causal explanation to provide the basis for recreating a causal inference chain to a prediction. We would distinguish, for example “explanation by authority,” towards the left end of the scale, to be something like an explanation to the question “why can’t I go out tonight?” to be something like “Because I said so,” from a parent to a teenager, which we might just say is explanation by authority. Just these simple distinctions help frame and motivate a more detailed analysis of distinctions across our informal scale of levels of explanation, which will be further articulated in what follows. Note that an alternative to Figure \[fig:explainability-diagram\], might be a simple abstract comparison of levels of explanation as a check list of possible components, like in Table \[tab:tabular\_level\_explainability\]. Please note that both representations are intended only to begin to consider how such components may obtain in any particular XAI system. For example, our figure and table do not intend the reader to draw the inference that an XAI system is somehow better with a dialogue component than without, nor that any anticipated evaluation of performance is higher with an integration of more of the components. The idea is only to suggest that there will emerge a foundation of what XAI components are essential, orthogonal, and have distinct and value-contributing roles in future XAI systems. Similarly, our desire for some kind of sensible measures to clarify a level $n$ explanation from a level $n+1$ explanation creates our speculation on a kind of y-axis, which is not intended to imply the existence of any measure. Rather our y-axis is a kind of independent set of plausible orthogonal explanatory system attributes, which should be distinguished clearly enough to be able to use each attribute as a check list of attributes that any explanatory system may or may not have (cf. Table \[tab:tabular\_level\_explainability\]). For example, most surveys of XAI note the requirement for a system to produce alternative explanations; simply put, producing a single explanation may be completely insufficient for multiple explainees [@miller-explanation-2019]. Similarly, many have noted the value of interactive XAI systems and dialogue systems [@Bex2016combining], which provide a basis for an explainee to submit and receive responses to questions about a model prediction, and thus build deeper trust of the system. In what follows, we will provide increasing detail and precision about how we believe existing XAI concepts align with this simple framework, in order to consider how to articulate choices in the design of an XAI system. The rest of our paper is organized as follows. Section \[section:principlecomponents\] presents what we consider the principal components of XAI, including that explanations need to be explainee-specific, that there can always be multiple explanations, and that the evaluation of the quality of explanation has more to do with the explainee than the explanation system. This will help provide sufficient detail to articulate the relationship between current explanatory concepts and their relationship to historical roots, e.g., to consider the emerging demands on the properties of a formal definition of [*interpretability*]{} by assessing the classical formal systems view of interpretability. Section \[section:history\] considers the more general history of explanation, as an attempt to connect the formal philosophy and scientific explanation foundations. This concludes with the articulation of the explanatory role of recent theories of causal representation. Section \[section:trends\] summarizes important emerging trends and components in proposed XAI architectures, including those that apply to both machine-learned predictive models and general AI systems. Section \[section:components\] provides a synopsis of current XAI research threads, and how they might integrate into an emerging XAI architecture. The opinions include the description of important XAI ideas like pre-hoc versus post-hoc explanation creation, and the evaluation of explanations, in order to sketch an architecture of how necessary components for XAI are connected. Finally Section \[section:conclusions\] provides a brief summary, and what we believe the future architectures of XAI systems will require to ensure the trust of future AI systems. Principal Components at the foundations of XAI {#section:principlecomponents} ============================================== Explainability and Interpretability ----------------------------------- There is a confusion in the literature regarding the definitions of interpretability and explainability of models. Many recent papers use those terms interchangeably [@koh2017understanding], [@vaughan2018explainable]. Some papers do make a distinction between those terms, but we do not agree with those definitions as well. For example, Gilpin et al. [@gilpin2018explaining] define interpretability as a rudimentary form of explainability. Rudin [@rudin2019stop] finds that there is no single definition on interpretability. However, the author defines a spectrum which extends from fully interpretable models such as rule-based models (that provide explanations by definition) to deep models that cannot provide explanations out of the box. We note that there is no confusion about interpretation, explainability and semantics in the case of the history of mathematical logic (e.g., [@mendelson2015]). When the vocabulary of the representation (well-formed formulae) is precise, interpretability is obtained by ensuring that each component is assigned a fixed interpretation (e.g., constants to individuals in a world, variables range over constants, truth values to logical connectives, etc.). And the semantic interpretation of [*[any]{}*]{} expression is determined compositionally by interpretation of an expression’s components. But the manner in which representations emerge in the context of empirical developments in machine learning has not typically been guided by any adaption of extension of the systems of interpretability and semantics of logic. Our perspective is that the principles of mathematical logic can be easily adopted to a broad range of machine learned representations, in order to help humans understand learned representations. In this context, an interpretable model is one that a human user can read or inspect, and analyze in terms of composable parts. In this way, interpretability refers to a static property of the model, and can vary from fully interpretable (models such as a small decision tree), to deep neural network models in which interpretability is more complex and typically limited. For instance, consider what each layer learns in a convolutional neural network (CNN): early layers are responsible for extracting low-level features such as edges and simple shapes, while later layers usually extract high-level features whose semantics are understood with respect to an application domain. In fact, with this perspective, models such as deep neural networks could hardly be classified as interpretable. It is important to point out that interpretability applies to the interpretation a learned model before considering the inference the model can do. Note that we are against classifying models as interpretable or non-interpretable, but rather we believe there should be a spectrum allowing an interpretability score to be assigned to each model. On the other hand, explainability has to deal with what kind of output the system provides to the user, rather than how a human user directly interprets the meaning of each model component. In other words, explanation has to do with clarifying the reason or reasons a prediction was made or an action was taken. Thus, we define an explainable model as a system which is capable of providing explanations without doing any extra computation. [*Explainability*]{} is, thus a dynamic property of a model, in the sense that it requires runtime information to produce explanations. Explainability pertains to the mechanism of justification provided for an inference or prediction using a learned model, whether the used model is clearly interpretable or loosely interpretable. Figure \[fig:explain\] illustrates the distinction between interpretability, which concerns the rendition or comprehension of a predictive model learned from data during training, and explainability, which pertains to the elucidation and justification of a prediction or decision made in the presence of a new observation or case. Both may revert to and rely on the original training data for analogy or grounds for justification. ![Interpretability of a model vs. Explainability of a prediction.[]{data-label="fig:explain"}](Interpret-explain.jpg){width="95.00000%"} Based on the above definitions, models such as decision trees and rule-based systems that are considered transparent (i.e., they are generally considered toward the fully-interpretable end of the transparency spectrum) are also explainable, while deep models are not. For example, once we add an explanation module to the deep neural model (e.g., [@babiker2017]), they become explainable systems as well but, interpretation of their meaning can be either in terms of how models are learned or what predictions mean (i.e., their interpretation in their application domain). Notice that these distinctions between explainability and interpretability do not comply with a reasonable assumption that is true in a common sense usage of the terms: outside of the AI arena, it is reasonable to expect explainability requires interpretability; one can only explain something they fully understand and can interpret. That stems from our definition of explainability: in AI, an explanation carries a completely different meaning from the one of its usual usage. Even a saliency map which highlights areas of an image is considered to be (to some extent) an explanation of an image classification system. That “explanation” does not consider image semantics, it is just an objective identification of which pixels contribute more to the final activations in a neural network. Still, they can be especially useful as debugging tools for machine learning practitioners (see Subsection \[subsec:debug\_vs\_explanation\]). Alternative explanations: who are explanations for? {#subsec:explainee} --------------------------------------------------- According to [@Lent2012], a person’s background knowledge, often called prior knowledge, is a collection of “abstracted residue” that has been formed from all of life’s experiences, and is brought by everyone, at any age, to every subsequent life experience, as well as used to connect new information to old. As such, it becomes clear that, in the context of XAI, systems should be able to effectively take background knowledge into consideration in order to connect predictions and predictive models, and to shape explanations to the appropriate level of detail, i.e., adjusting explanations to conform to the knowledge of the corresponding [*explainee*]{}. However, the most common current approaches to explainability in AI systems attempt to provide information on a model’s inner functioning without regard for the consumer of that information (see Subsection \[subsec:debug\_vs\_explanation\]). To illustrate the importance of considering the explainee (and hence his/her background knowledge, expectations, etc.), consider an interview by Richard Feynman with the British Broadcasting Corporation (BBC) in 1983, in which he was asked why magnets with the same poles repel each other when placed close enough [@Feynman_why]. Feynman argues that, to properly explain that behaviour, he would need to consider the reporter’s background on that matter, and any answer provided could unfold a new round of questions and explanations; and this process could continue indefinitely as new details are provided to explain the previous answer. The point is, of course, that an explanation’s satisfaction with this iterative dialogue is the foundation of how XAI systems should be evaluated (cf. Subsections \[subsec:explainee\], and \[subsec:measuring\]). Debugging versus explanation {#subsec:debug_vs_explanation} ---------------------------- As mentioned above, some approaches to explainability provide information related to how the model works internally. However, not all information provided by those approaches can really be considered domain explanatory information. Of course, the information provided by, e.g., rule-based systems can be understood as a detailed explanation on how the system operates and could be applicable in scenarios where an end user needs to understand how the system generated a prediction. However, other approaches (especially those applicable to the so called opaque or black-box systems) are way less informative, and can be considered superficial “hints” rather than actual explanations, e.g., saliency maps on convolutional neural networks. This is really an observation about understanding internal components so as to debug those mechanisms, not as explanation. Although explanation-wise constrained, these approaches are still useful on helping to understand how a model behaves, especially if the consumer of the information has the necessary background. Hence, they may be considered debugging techniques for opaque AI systems rather than production of explanations based on a user’s understanding of the semantics of an application domain. One example of a debugging tool to augment a model is the work of [@lecue_kgxai], in which the authors used a ResNet [@rcnn] to recognize objects in a scene. Applying a saliency map to figure out what area in the image was contributing more to the final activations is not really helpful for a (lay) human consuming the model output to understand misclassifications, but it may help a researcher at design time to figure out alternatives to overcome the model limitations. In this case, the authors augmented the model by post-processing the final results using an external knowledge graph to add semantic context and modify the confidence score of the recognized objects. An alternative, perhaps more foundational model, is presented by Evans and Greffenstette in [@evans2018], who articulate an inductive logic programming (ILP) framework in which explanatory models are identified in the space of all possible inductive logic programs. This framework requires the development of a measure space for all such ILP instances, in which a gradient can be determined. But the positive consequences of that technical maneuver is that an instance of an inductive logic program can be interpreted at the level of the semantics of an application domain, all the way down to instructions for a Turing machine. This framework does not resolve the challenge of what an appropriate level of explanation should be for a particular explainee; but it does provide a rich and mathematically elegant space in which to identify everything from descriptions of computation to arrive at a predictive model all the way to rule-based specifications at the level of an application domain. Is there a trade off between explanatory models and classification accuracy? ---------------------------------------------------------------------------- Deep learning-based systems became prevalent in AI especially after its successful applications in image classification problems. Deep learning-based systems achieve impressive accuracy rates on standard datasets (e.g., ImageNet [@imagenet_cvpr09]) without requiring much effort on designing and implementing handcrafted rules or feature extractors. In fact, by leveraging transfer learning techniques and well known architectures based on convolutional neural networks, a deep learning practitioner can quickly build an image classifier outperforming image classification methods which were state of the art before the “deep learning revolution.” Nevertheless, despite their excellent overall accuracy, deep learning systems are considered black-boxes unable to provide explanations as to why they make a given prediction. In some applications, that limitation does not translate into a serious practical problem: a mobile phone picture classification application which misclassifies two animals will not bring consequences to a user other than a few giggles and a funny discussion topic at friends gatherings. If those errors are seldom, nobody would really care or lose confidence on the application. Errors may come up in random images and could be induced. Figure \[panda-gibbon\] shows an example of a panda picture being classified as a gibbon after some adversarial noise is added [@goodfellow2014explaining]. ![A panda image mistakenly classified as a gibbon after noise is added [@goodfellow2014explaining].[]{data-label="panda-gibbon"}](panda-gibbon.png){width="70.00000%"} The above example illustrates it is possible to intentionally fool a classifier through addition of appropriate noise. Depending on the image classification application, that kind of error may produce more serious consequences than the hypothetical phone application mentioned above. For example, recently hackers were able to fool Tesla’s autopilot by tampering speed limit signs with adhesive tape (see Figure \[fig:teslaspeedlimit\]), making the car to accelerate to 85 mph. ![A modified speed limit sign reads as 85 mph on the Tesla’s heads-up display [@teslaspeedlimit].[]{data-label="fig:teslaspeedlimit"}](tesla-speed-sign.png){width="70.00000%"} This is a simple example which illustrates predictions from AI models cannot be blindly accepted in many practical applications. Moreover, techniques unable to explain how they arrive at a prediction make them even more sensitive to random errors or deliberate attacks. That observation raises an important question around a potential trade off between model accuracy and explanatory capabilities: it is true that a deep learning-based model can achieve accuracy in many practical applications. That allows practitioners to quickly build accurate models with not so much effort. However, some preconditions do exist, the main one being the availability of potentially large labelled datasets (a problem potentially alleviated by transfer learning, but still common in machine learning in general and in deep learning techniques in particular). In some cases, training large state of the art deep learning networks requires thousands of even millions of dollars (the estimated cost of training just one of models developed in [@adiwardana2020humanlike] was estimated in US\$1.4 million [@meenacost]). All considered, it is not appropriate to claim there is necessarily a trade off between accuracy and explainability (or more generally, model performance). In some cases, deep learning methods will not be able to provide state of the art results (e.g., when there is not enough labelled data, when the model is so large it will be impractical to deploy on the target platforms, or even train due to prohibitive costs, etc.) so more explanation capable techniques might even provide better results. But as previously noted, there is no reason in principle that induced models like decision trees should in principle be less accurate than deep learned models. Assessing the quality of explanations {#subsection:assessing quality} ------------------------------------- Whereas a factually wrong explanation is obviously inappropriate, determining if an explanation is good transcends its correctness. The quality of an explanation is a little like beauty; it is in the eye of the beholder. It is very clear (and quite common) that two factually correct, but different explanations could be considered good or bad depending on to whom they were provided. This means that, to assess quality of explanations, one (again) needs to consider the explainee, the person who receives the explanation (see Subsection \[subsec:explainee\]). The explainee’s background, expectations, goals, context, etc., will play a determinant role in the evaluation process. From the above paragraph, it is clear that assessing the quality of explanations is subjective, and a quite complicated task, even if done manually. Thus, coming up with an effective technique to evaluate explanation capabilities is beyond the reach of currently available methods. In fact, automatic evaluation of any generative model is a difficult task. Metrics commonly used for translation systems such as BLEU [@Papineni2002bleu] or for automatic summarization such as ROUGE [@lin2004rouge] are not appropriate for more sophisticated tasks such as explainability or even dialogue systems, since they assume that valid responses have significant word overlap with the ground truth responses [@liu2016_diagsys]. For that reason, most evaluation methods for explainability systems require human intervention. For example, the organizers of a fake news detection competition[^4] which requires an explanation of why a given statement is considered fake news or not, split the competition in two phases and limited the explanations assessment to the second phase to which only 10 teams would be qualified, thus making it manually tractable. The history of evaluation in the field of data visualization is also relevant to the question of how to evaluate explanations. The initial focus on alternative visual renderings of data have, over a decade, transformed from whether a visualization was “interesting” to consideration for what human inferences are enabled by alternative visualization techniques (e.g., [@Spence2014]). The simplest conceptual alignment is that a visualization is a visual explanation. The compression of a large volume of data to a visualization picture is lossy and inductive, so the choice of how to create that lossy inductive picture or explanation is about what inferences to imply for the human visual system. The evaluation of alternative visualizations has evolved to a framework where evaluation is about what inferences are easily observed (e.g., [@Lam2012]). Furthermore, interactive visual explanation is easily considered as our suggestion of explanation being interactive and driven by the semantics of the application domain (e.g., [@Goebel2013]). Evaluation of what we can consider as a more general explanatory framework, which produces alternative explanations in terms of text, rules, pictures, and various media, can similarly be aligned with the evolution of how to evaluate visual explanations. But of course there is yet no clear explanation evaluation framework, but only a broad scope of important components (e.g., [@miller-explanation-2019]). Even specific instances of proposals for explanation evaluation beg the need for increased precision. For example, [@adiwardana2020humanlike] suggest explanation quality is dependent on two main factors: sensibleness and specificity. A measure which takes those factors into account (Sensibleness and Specificity Average - SSA). This suggestion arose from work on the topic of dialogue systems, and has been characterized in terms of a high correlation with another measure called “perplexity:” a measurement of how well a probability distribution or probability model predicts a sample. A low perplexity indicates the probability distribution is good at predicting the sample. In the context of conversational systems, perplexity measures the uncertainty of a language model, which is a probability distribution over entire sentences or texts. The lower the perplexity, the more confident the model is in generating the next token (character, subword, or word). Thus, perplexity can be understood as a representation of the number of choices the model is trying to choose from when producing the next token. This measure is commonly used to assess the quality of conversational agents and as a metric which must be optimized by machine learning based dialogue models. Thus, although not ideal and lacking specific experiments on the domain of explainability, perplexity could potentially be effectively used to evaluate text-based XAI systems as a reasonable approximation of human evaluation. While we have more to say about evaluation below, what is clear is that evaluation of explanatory systems is based on how the explainee confirms their own understanding of an explanation or the conclusion of an explanatory dialogue. A brief history of explanation {#section:history} ============================== Abduction --------- Explanations have always been an indispensable component of decision making, learning, understanding, and communication in the human-in-the-loop environments. After the emergence and rapid growth of artificial intelligence as a science in the 1950s, an interest in interpreting underlying decisions of intelligent systems also proliferated. Especially, C.S. Peirce’s hypothesis of abduction [@peirce1891architecture] stimulated the AI community’s attention to exploiting this conceptual framework for the design and development of complex expert systems in a variety of domains. Abduction or abductive reasoning is a form of reasoning that starts with a set of observations and then uses them to find the most likely explanations for the observations. A compressed historical journey of Peirce’s ideas can be traced in four projects, beginning with Pople [@Pople1973], Poole et al. [@poole1987], Muggleton [@muggleton1991], to Evans et al. [@evans2018]. In 1973, Pople provided a description of an algorithm to implement abductive and showed its application to medical diagnosis. Poole et al. extended abductive ideas to a full first order implementation and showed its application to guide the creation of explanatory hypothesis for any application domain. Muggleton produced a further refined system called inductive logic programming, in which creation of hypotheses are generally identified by inductive constraints in any general logic. Finally, the adoption of this thread of mechanisms based on abductive reasoning have been generalized to the full scope of explanation generation based on inductive logic programming by Evans et al. Every instance of these contributions relies on a logical architecture in which explanations arise as rational connections between hypotheses and observations (cf. scientific explanation). The most recent work by Evans et al. extends the framework in a manner that supports modern heuristics of inductive model construction – or learning of predictive models – by providing the definition of a gradient measure to guide search over alternative inductive logic programs. In fact, that thread of exploiting abduction in Artificial Intelligence is aligned with perspectives from other disciplines. For example, Eriksson and Lindstr[ö]{}m describe abductive reasoning as an initial step of inquiry to develop hypotheses where the corresponding outcomes are explained logically through deductive reasoning and experimentally through inductive reasoning [@eriksson1997abduction]. Their application to “care science” is just another example that confirms the generality of abductive reasoning. The block diagram of Figure \[fig: abduction components\], partially inspired by a figure in [@abductionfigure], is intended only to confirm the connection between abductive, deductive, and inductive reasoning. We see that abductive reasoning entails justification of ideas that support the articulation of new knowledge by integrating deductive and inductive reasoning. In Artificial Intelligence studies, the process involving these reasoning steps are as follows: 1) identify observations that require explanation as they cannot be confirmed with already accepted hypotheses; 2) identify a new covering hypothesis using abductive reasoning; 3) empirical consequences of the hypothesis, including consistency with already known knowledge, is established through deduction; 4) after an accepted level of verification, the hypothesis is accepted as the most [*scientifically plausible*]{}. ![The process steps of the reasoning methods[]{data-label="fig: abduction components"}](abduction_deduction_induction.png){width="70.00000%"} Scientific explanation ---------------------- The connection between Artificial Intelligence frameworks for abductive explanation have suggested a direct connection between “scientific explanation,” and is the subject of many debatable issues in the community of science and philosophy [@woodward2003scientific]. Some of the discussions imply that there is only one form of explanation that may be considered scientific. There are also some proponents of the idea that a theory of explanation should include both scientific and other simpler forms of explanation. Consequently, it has been a common goal to formulate principles that can confirm an explanation a scientific explanation. As far back in history as Aristotle, generally is considered to be the first philosopher to articulate an opinion that knowledge becomes scientific when it tries to explain the causes of “why”. His view urges that science should not only keep facts, but also describe them in an appropriate explanatory framework [@falcon2006aristotle]. In addition to this theoretical view, empiricists also maintain a belief that the components of ideas should be acquired from perceptions with which humans become familiar through sensory experience. The development of the principles of scientific explanation from this perspective prospered with the so-called Deductive-Nomological (DN) model that was described by Hempel in [@hempel1942function], [@Hempel1958-HEMTTD], [@hempel1965aspects], and by Hempel and Oppenheim in [@hempel1948studies]. The DN model is based on the idea that two main elements form a scientific explanation: An *explanandum*, a sentence that outlines a phenomenon to be explained, and an *explanan*, a sentence that is specified as explanations of that phenomenon. For instance, one might constitute an explanandum by asking “Why did the dish washer stop working?” and another person may provide an explanan by answering “Because the electricity went off.” We may infer that the explanan is rationally or even causally connected to the explanandum, or at least that the explanandum is the reasonable consequence of the explanans, otherwise speaking [@mcgrew2009philosophy]. In this way, the explanation delivered as an explanans becomes a form of deductive argument and constitutes the “deductive” part of the model. Note that a series of statements comprises an explanan should comply with “laws of nature.” This is a vital property, because derivation of the explanandum from the explanan loses its validity if this property is violated [@woodward2003scientific]. This is the nomological component of the model, where the term “nomological” means “lawful.” Hempel and Oppenheim’s DN model formulation states that a scientific explanation is an answer to a so-called “why” question, and there may be multiple such answers. There may also be several types of questions (e.g., “How does an airplane fly?”) that cannot be converted into why-questions. Consequently, answers to such questions are not considered to be scientific explanations. This does not mean that such answers are not part of a scientific discipline; these answers just become *descriptive* rather than being *explanatory* [@bunzl1993context], and is related to our next section on causality. Another aspect of the DN model is that the elements of an explanation are statements or sentences describing phenomenon, not the phenomenon itself. Finally, the sentences in the explanans must be accurate and verified, urging the arguments of the scientific explanation to be *valid* and *sound*. Thus, the DN model can be summarized as a model of a scientific explanation outlining a *conception* of explanation and a *connection* in the flow of an explanation. Causality --------- As hinted in the summary of scientific explanation, perhaps the most strict form of explanation is causal explanation. Informally, a casual explanation is one that arises from the construction of causal models, which require that explanations for arising predictions are in face “recipes” for reconstruction that prediction. Causal models typically facilitate the creation of explanation for a phenomena or an answer to a query by constructing a formal expression. That formal expression is derived from some causal representation, which typically captures directed causal relationships in a graphical model of cause and effect (or causality). This representation encodes an incomplete set of assumptions built upon prior knowledge. The causal explanation expression is, as with abduction and scientific explanation, revised continuously until a suitable explanation is obtained, and can answer the particular query. The most relevant and recent framework of causal representation and reasoning is given by the culmination of Pearl’s research in [@pearl2018book]. In that work, an abductive explanation is called an “estimand.” This idea of a formal expression that best explains a particular phenomena (or a query) has its root in formal philosophy, as noted about, and especially in abductive reasoning. As noted above, abductive reasoning is given a set of incomplete observations (or assumptions as described above) and seeks to construct an explanation which best describes it (or the estimand). An important point here is that the overall information architectures of abduction, scientific reasoning, and causal reasoning are similar, but their mechanism and the evaluation of an explanation are successfully refined. Explaining mechanism/syntax versus semantics --------------------------------------------- A lingering unaddressed distinction is about the content or meaning of an explanation, especially in the context of what counts as an explanation to a user. Again a principled distinction exists in the realm of mathematical logic (cf. [@mendelson2015]; any logic textbook will suffice). In the context of predictions from domain models (whether learned or fabricated by hand), a prediction has at least two kinds of explanation. For example, consider the simple familiar syllogism - All men are mortal. - Socrates is a man. - Socrates is mortal. Consider “Socrates is mortal” as a prediction of the very simple model. From the perspective of formal logic, there are (at least) two explanations. One is the explanation of the deductive mechanism that produced “Socrates is moral” from the first two expressions. This a so-called proof-theoretic explanation as it amounts to a description of how two premises are combined by deductive inference to derive the prediction. In an analogy with programming language debuggers, this kind of explanation is about the mechanism that produced the prediction, and is akin to how current work in explaining image classification (e.g., [@babiker2017]). This kind of explanation is appropriate when the explainee has interest in understanding and debugging the mechanism. But note an alternative explanation is not about mechanism but about the meaning of the expressions. Logically, the proof theory or deductive chain explanation is about mechanism. But the semantic explanation is about what it means to be mortal and what it means to be a man. That kind of explanation is semantic, and is intended to be appropriate for an explainee who is not interested in mechanism but in meaning. If a prediction was “Socrates is a duck” obtained from the same system, it can immediately be viewed with suspicion because of its meaning, not because of the mechanism that produced it from a presumably faulty model. So distinguishing syntax from semantics or meaning has more to do with the internal rules that a system has to follow to compute something. We all know that symbolic debuggers for programming languages create labels and traces which become the basis for producing mechanism explanations. The computation rules themselves might not be sufficient to provide a clear picture on why a system came to a conclusion (or an answer to a query). But interpretation of syntactic expressions is what creates asemantic interpretation. Returning to the idea of estimands, an estimand can be viewed as a well-constructed expression if it makes sense semantically. As with the simple syllogism above, the form of the explanation can be based on that of the causal (or any) model. In this era of deep learned models, we can consider these relationships between syntax and semantics as the internal representations of each layer and their composition at the final layer respectively. Interestingly, this notion of construction of semantics (whole) as a function of semantics of its parts and their careful combination that obeys a particular syntax is very familiar in the logics to interpret natural language, developed by a famous linguist-logician, Richard Montague [@stanford-encyclo-phil]. At the syntactic level we might infer the correlation among different variables (in the intermediate layers) of a deep learned system but in semantic level we know what combination of those variables (in the final layer) provide an interpretation for a particular query. Ontology driven explanation for a ResNet model [@lecue_kgxai] described in Section \[section:principlecomponents\] is one good example of the use of semantics to explain an opaque system. Classification of Current Research Trends {#section:trends} ========================================= In the last five years, there has been a surge in the papers attempting to introduce new explanation methods. This intensity of work in XAI is, in fact, a side effect of widespread use of AI in sensitive domains such as legal reasoning and the medical field. In this section, we review some of the various explanation approaches popular in the literature, and classify in our framework based on how the explanations are built, and compare that with the levels of explanation introduced in Section \[section:intro\]. Concurrently constructed explanations ------------------------------------- Some have focused on creating models that try to build explanations concurrently together with the main task (e.g., learning a classifier). As an example, consider the work of [@lei-etal-2016-rationalizing] who seek to identify segments of text that support an explanation of text review classification. Their approach proposes a neural architecture that is made up of a generator followed by an encoder component. The generator extracts portions of the input text as salient words, then forwards them to the encoder to predict the target class. The final output of the system comprises the class label, together with the extracted “justification” from the input text. Other similar work, applied beyond text to images and text, has relied on learning attention weights from the input. In the related work, some authors referred to this category as learning interpretable representations. In natural language processing (NLP) text classifications for instance, attention layers attempt to learn the weight of each latent representation produced by the recurrent layer. The attention weights are then used to explain the prediction made by the classifier [@yang2016hierarchical]. There is a debate in the literature on whether attention weights could be used as an explanation or not [@serrano2019attention; @jain2019attention]. An interesting connection is to our discussion above regarding the difference between debugging explanations and semantic explanations; much of this research is motivated to equate a mechanism behaviour to semantic interpretability. Post-hoc explanations --------------------- Another approach is to use a post-hoc technique. The basic idea is to approximate explanations from a trained model. As mentioned earlier, concurrently constructed explanations need to be computed within the model, which means they need to have access to the internals of the model, or what many refer to as model-dependent (this further creates confusion about whether a model is syntactic or semantic). However, some post-hoc approaches can create approximate explanations without having access to the internals of the model, thus could be classified either as model-dependent or model-independent[^5]. In the next subsection, we will briefly discuss the difference between model-dependent and model-independent. ### Model-dependent explanations To describe model-dependent explanations, consider the case of non-linear deep networks. One can use a back-propagation algorithm to learn feature importance (e.g., which pixels contributed most in classifying the image as a cat rather than a dog) then use that learned feature ranking as the basis for explaining predictions. The simplest general approach is to compute a gradient with respect to the predicted class and use the back-propagation to propagate the gradient to the input. Finally, one can combine the input with the gradient to capture the salient pixels which can be used to explain the predicted class (e.g., Grad-CAM [@selvaraju2017grad]). ### Model-independent explanations The goal of this group of methods is to focus more on explaining individual instances without the target model being exposed. In fact, the target model is now a black-box model. Ribeiro et al. introduced LIME [@ribeiro2016should] to approach the explanation problem using a perturbation method. They perturb the original data point to create a new dataset in the vicinity of that instance. The black-box model is queried to get the labels associated with the aforementioned points. This labelled dataset is then used to frame a near enough justification. While LIME is the most cited model-independent method, there are other approaches which can be classified as model-independent [@lundberg2017unified; @guidotti2018local]. Application-dependent vs. generic explanations {#subsection:application_dependent} ---------------------------------------------- Another way to classify explanation methods is to consider how an explanation mechanism is related to the application domain of the task. An application-dependent method implicitly assumes the explainee is knowledgeable about the application and thus it employs the domain’s vocabulary. In a medical application, for instance, a system can explain the prediction using medical terms. A generic explanation, on the other hand, can only provide explanations based on the mechanism of model building, combined with information available in the training set (e.g., correlation between features). Note that a model-dependent method is not necessarily taking into account the knowledge of the explainee (i.e., it will provide the same explanation irrespective of the customers’ knowledge), but it must take advantage of the application’s vocabulary (see Subsection \[subsec:debug\_vs\_explanation\]). It is also noteworthy that the system needs to go beyond correlative features — which is how most current machine learning methods work — to be capable of providing such application-dependent explanations. Many explainees (e.g., physicians, lawyers) would prefer having application-dependent explanations. This will not be achieved without moving the machine learning research on explanation toward scientific and casual explanation. Classification based on levels of explanation --------------------------------------------- As described briefly in Section 1, different levels of explanation could be introduced as shown in Figure \[fig:explainability-diagram\]. Here we want to further elaborate those abstract levels and classify the related work accordingly. Table \[tab: classification\_of\_research\_threads\] classifies some of the most prominent existing work based on the levels of explanation. Most of recent research have focused on Level 1, and only a few have worked on Level 2. To the best of our knowledge there is as yet no existing work on Levels 3 and 4. However, there are some conceptual approaches that aim to achieve such levels [@madumal2018towards]. In the subsections below, we provide details. ### Level 0 Models classified as Level 0, provide no explanation at all. They are, in essence, black-box models that cannot provide any explanatory information to a user. In other words, the explainee is expected to accept or reject a system’s decision without any further information. Most off-the-shelf methods for learning classifiers (e.g., deep learned models, support vector machines, or random forests) belong to this level. ### Level 1 The explainee is provided with a single type of explanation in models falling into this category. For example, a framework that provides heat-maps to explain image classification belongs to this level. Most of this approach focuses on providing a post-hoc explanation, which transitions a black-box model —that originally belonged to Level 0— to a Level 1 model. Recently, however, a few methods have been proposed to look at building concurrently constructed explanation algorithms [@lei-etal-2016-rationalizing; @bastings2019interpretable] to make models that by definition belong to Level 1. ### Level 2 Level 2 adds another type of explanation to enrich the knowledge communicated with the explainees. At this level, the system not only provides a heat-map to explain a classified animal image as a cat, but it also contains another type of explanation such as a textual explanation as an alternative description of the predicted classification. In this way, the alternative explanations allows the user to grasp more insights about the reasoning process employed by the system to make the prediction. If one explanation is not well understood by the explainee, then they have the opportunity to understand from an alternative explanation. Note that in the case of the abductive systems described above, there can be a large number of alternative explanations. ### Level 3 An explainee and their familiarity with the domain plays a vital role in this level. The explanatory system includes some model of the explainee’s domain model, and is capable of deciding the right type of explanation according to the knowledge of the explainee. For instance, a patient is diagnosed with some disease and an AI system is used to provide a potential treatment therapy. While the therapist requires a detailed medical explanation by the AI system, the patient would strongly prefer to have a lay person’s explanation for any alternative treatment recommendations. In the current context of the COVID-19 pandemic as an example, Hydroxychloroquine is alleged to be a potential cure and has attracted many ordinary people’s attention around the globe. People are interested to understand why this drug is a potential treatment. As a result, many medical researchers provide interviews to the media explaining how this drug works, typically with very shallow detail. As we can expect, however, the same experts would use a different level of granularity to explain the drug to other experts. Please note that, none of the existing explanation methods take into account the knowledge of the explainee. ### Level 4 While previous levels (e.g., Level 0, 1, and 2) do not include the capability of interaction between the explainee and the system except perhaps for at most one interaction (Level 3), methods classified as Level 4 can interact with the user. They are expected to support a conversation sort of capability which allows the explainee to refine their questions and concerns regarding the decision. In other words, each interaction in the conversation allows the explainee to get clarifications. Here, the system is capable of adapting its explanation to the vocabulary of the explainee. Take the Richard Feynman’s interview [@Feynman_why] with the BBC as an example. He could provide the reporter with what he thought the reporter would understand most. Once the reporter understood that explanation, if he had further questions, or wanted more in-depth explanation, the reporter could ask, and an appropriate explanation could be provided by Richard Feynman. To the best of our knowledge, existing systems lack this interaction capability. Priority Components for a synthesis of an XAI architecture {#section:components} ========================================================== XAI architecture ---------------- As noted, much of the work on the explainability has focused on deep supervised learning, which describe methods that answer the following two questions: (1) which input features are used to create an output prediction, and (2) which input features are semantically correlated with the outcome prediction. The answers for these two questions contribute to the trust in the system, but explanation additionally requires a social process of transferring knowledge to the explainee considering the background knowledge of the explainee. While the answers to questions (1) and (2) may acknowledge the importance of features that a model uses to arrive at a prediction, it may not necessarily align with a human explanation; prior knowledge, experience and other forms of approximate reasoning (e.g., metaphorical inference) may further shape an explanation, while the predictions of a machine learning model may be restricted to the dataset and the semantics around it. Generally, an explanation system (for example, a human) is not restricted to the knowledge on which they make predictions and explanations and can draw parallels with different events, semantics and knowledge. So merely responding to questions (1) and (2) do not satisfy the multiple purposes that XAI researchers aim to achieve: to increase societal acceptance of algorithmic decision outcomes, to generate human-level transparency about why a decision outcome is achieved, and to have a fruitful conversation among different stakeholders concerning the justification of using these algorithms for decision-making [@Kasirzadeh2019mathematical]. To incorporate an interactive “explainer” in XAI, an emerging XAI architecture needs to embed both an explainable model and an explanation interface. The explainable model includes all types of the pre-hoc, post-hoc and concurrent explanation models. As examples of the explainable model, there can be a causal model, an explainable deep adaptive program, an explainable reinforcement learning model, etc. An explanation interface can be also a variety of types, such as a visualization system, or a dialogue manager with a query manager and a natural language generator that corresponds to Level-3 and Level-4 of Figure \[fig:explainability-diagram\]. User-guided explanation ----------------------- As Miller [@miller-explanation-2019] notes, the process of explanation involves two processes: (a) a cognitive process, namely the process of determining an explanation for a given event, called, as with Hempel, the explanandum. This identifies causes for the event, and a subset of these causes are selected as the explanation (or explanans); and (b) a social process of transferring knowledge between explainer and explainee, generally an interaction between a group of people, in which the goal is that the explainee has enough information to understand the causes of the event. This is one kind of blueprint for the Level 4 interactive explanation process noted above. Miller provided an in-depth survey on explanation research in philosophy, psychology, and cognitive science. He noted that the latter could be a valuable resource for the progress of the field of XAI, and highlighted three major findings: (i) Explanations are contrastive: people do not ask why event $E$ happened, but rather why event $E$ happened instead of some other event $F$; (ii) Explanations are selective and focus on one or two possible causes and not all causes for the recommendation; and (iii) Explanations are social conversation and interaction for transfer of knowledge, implying that the explainer must be able to leverage the mental model of the explainee while engaging in the explanation process. He asserted that it is imperative to take into account these three points if the goal is to build a useful XAI. One should note that it is plausible, given the study of explanation based on cognitive norms, that an explanation may not be required to be factual, but rather only to be judged to be satisfactory to the explainee (cf. Subsections \[subsection:assessing quality\], and  \[subsection:application\_dependent\]). As we described in Figure \[fig:explainability-diagram\], a dialogue system that can process a question of “what if another condition” from an explainee and produce a new prediction output based on the new condition will achieve another higher level of explanation. The explanation that can deal with “What would the outcome be if the data looked like this instead?” or “How could I alter the data to get outcome X?” is called contrastive explanation. Contrastive explanation is a human-friendly explanation as it mimics human explanations that are contrastive, selective, and social. To accommodate the communication aspects of explanations, several dialogue models have been proposed. Bex and Walton [@Bex2016combining] introduce a dialogue system for argumentation and explanation that consists of a communication language that defines the speech acts and protocols that allow transitions in the dialogue. This allows the explainee to challenge and interrogate the given explanations to gain further understanding. Madumal et al. [@madumal2018towards] also proposed a grounded, data-driven approach for explanation interaction protocol between explainer and explainee. Measuring value of explanations {#subsec:measuring} ------------------------------- The production of explanations about decisions made by AI systems is not the end of the AI explainability debate. The practical value of these explanations, partly, depends on the audience who consumes them: an explanation must result in an appropriate level of understanding for the receivers of explanations. In other words, explanations are required to be interpreted and judged against different points, about whether they are good or bad, satisfactory or unsatisfactory, effective or ineffective, acceptable or unacceptable. Again the previously mentioned evolution of the evaluation of visualization systems is highly relevant, as that evolution ultimately requires the design of cognitive experiments to confirm the quality and value of alternative explanations, visual or not (see Subsection \[subsection:assessing quality\]). It is clearly the case that quality of a “visual explanation” is about how well it leads the reader to the intended inferences from the visualized data domain. Naturally, the background knowledge of a viewer is like the background knowledge of an explainee; their knowledge and experience determines what preferred inferences obtain. Looking forward to how to evaluate XAI systems, among those background assumptions that impact the judgements of explanations are what are returned to as cognitive “norms.” It has been empirically shown that norms influence causal judgements [@adam2017norm]. To put it simply, norms are informal rules that are held by people, and can have statistical or prescriptive content. The empirical and mathematical aspects for why a decision outcome is achieved are interpreted against some background assumptions held by the audiences of explanations. Some disagreements with an explanation for a decision outcome in a sensitive context due to the background assumptions of the audience of explanations reveal some moral or social mismatch about algorithmic decision-making between the receiver of an explanation and its producer. If one does not have an appropriate level of knowledge about the relevant precedent assumptions, one might not have the capacity to judge and interpret an explanation of a decision. In that case, iteratively refined question-answer dialogue (cf. Fenyman’s point made in Section \[subsec:explainee\]) may lead to an improved understanding by the explainee. In general the interpretability of explanations has significant practical value for revealing the explicit and the implicit reasons about why a decision-making procedure and process is chosen. A schema for the interpretability of explanations aims to capture various precedent assumptions that become relevant in context-dependent evaluation of each kind of AI explanation for why a decision outcome is achieved. Finally, in another elaboration of how to evaluate explanations [@DARPA], there are proposed five measures of explanation effectiveness: (1) User satisfaction, (2) Mental model, (3) Task performance, (4) Trust assessment, and (5) Correctability. User satisfaction is measured in terms of clarity of the explanation, and utility of the explanation. Task performance is to check if the explanation improved user’s decision and task performance. Trust assessment is to assess trust and measure if it can be appropriate for future use. Assessment of a mental model is related to strength/weakness assessment, and it also assesses the predictions of “what will it do” or what if questions, and “how do I intervene” to adjust or guide explanatory outputs. Finally, Correctability is to measure if interaction with the system helps to identify and correct errors. As far as we are aware, there is also no Level 4 system that has confirmed any experiments that demonstrate this kind of richly faceted evaluation. Finally, to measure contrastive explanation that is close to human explanation, we need additional evaluation metrics for contrast, selection, and social explanation. Contrast can be measured in terms of the clear justification of the output through the comparison. Contrastive explanation should be able to explain why the output has been produced between the probable output candidates. Selection can be measured in terms of the importance (salience) of the reasons (features) that were mentioned during the contrastive explanation. Lastly, social explanation can be measured in terms of the clarity, understandability and utility of the explanation to the explainee. The measure of the social explanation corresponds to the measure of user satisfaction in [@DARPA]. But as noted, we know of no existing explanation systems that have been so considered with this rich palette of evaluation parameters. Summary and Conclusions {#section:conclusions} ======================= In summary, our goal has been to articulate a set of required components of an XAI architecture, and describe a high level framework to understand their connections. In two alternative graphical depictions (Figure  \[fig:explainability-diagram\], Table \[tab:tabular\_level\_explainability\]), we distinguish what we believe are mostly orthogonal components of an explanation system, and suggest an information framework related to levels of autonomous driving, where a richer set of components provides a more sophisticated explanation system. That framework is descriptive and informal, but it allows us to factor some components (e.g., interpretability, explanation quality) into separate analyses, which we hope creates some line of sight to historical work on explanation. No where is this more important than the history of abductive reasoning and its connection to the history of scientific reasoning, culminating in the construction and use of causal models as a basis for causal explanations. We then try and consider more recent research in the context of these components and their relationship to the analysis of a deeper background literature, and provide some description of how those early ideas fit, and what they lack. This culminates with a considering of how to evaluate explanatory systems, and connects recent work that addresses the cognitive properties of explanations. Overall, we hope that our framework and analysis provides some connective tissue between historical threads of explanation mechanisms and modern reinterpretation of those mechanisms in the context of cognitive evaluation. We conclude that there is much still to do to inform a principled design of a high level explanation system, but that there are many components and integrating them with the appropriate knowledge representations within machine learning models, and respecting the cognitive aspects of evaluation, are a minimal requirement for progress. Acknowledgements {#acknowledgements .unnumbered} ================ We acknowledge support from the Alberta Machine Intelligence Institute (AMII), from the Computing Science Department of the University of Alberta, and the Natural Sciences and Engineering Research Council of Canada (NSERC). [^1]: Authors listed in alphabetical order; R. Goebel (rgoebel@ualberta.ca) is the corresponding author. [^2]: Especially see Section 2, Chapter 9, [*The limitations of deep learning.*]{} [^3]: Note the 32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Workshop on Critiquing and Correcting Trends in Machine Learning, https://ml-critique-correct.github.io [^4]: https://leadersprize.truenorthwaterloo.com/en/ [^5]: Sometimes, this is called model-agnostic post-hoc explanations.

--- abstract: 'We present updated predictions for the total cross section of top-quark pair production at Tevatron and LHC. For the LHC we also provide results at $\sqrt{s}$ = 10 TeV, in view of the anticipated run in 2008 and quote numbers for the production of new heavy-quark pairs with mass in the range 0.5 – 2 TeV. Our two-loop results incorporate all logarithmically enhanced terms near threshold including Coulomb corrections as well as the exact dependence on the renormalization and factorization scale through next-to-next-to-leading order in QCD.' address: - 'DESY, Platanenallee 6, D–15738, Zeuthen, Germany' - 'Institut für Theoretische Teilchenphysik, Universität Karlsruhe, D–76128 Karlsruhe, Germany' author: - 'S. Moch and P. Uwer' title: ' Heavy-quark pair production at two loops in QCD[[^1]]{}' --- Introduction {#sec:intro} ============ Research on top-quark physics at hadron colliders has received great interest in the past years in view of the steadily improving measurements at Tevatron and the upcoming LHC (see Ref. [@Bernreuther:2008ju] for a recent review). In this respect, the total cross section for top-quark pair production is a quantity of great importance for experimental analyses and even allows for measurements of the top-quark mass. Moreover, on the theory side, the total cross section has been subject to numerous studies the motivation being improved predictions beyond the long-known next-to-leading order (NLO) corrections in QCD [@Nason:1988xz; @Beenakker:1989bq; @Bernreuther:2004jv]. Recent work in this direction has aimed at completing the next-to-next-to-leading order (NNLO) QCD predictions [@Dittmaier:2007wz; @Czakon:2007ej; @Czakon:2007wk; @Korner:2008bn; @Czakon:2008zk], at resumming large Sudakov logarithms to next-to-next-to-leading logarithmic accuracy [@Moch:2008qy] and, at estimating bound state effects [@Hagiwara:2008df]. Also our knowledge on the parton distribution functions (PDFs) and the precision of the top-quark mass determination has continuously improved over the last years. In order to study the impact of the various improvements on Tevatron and LHC predictions we build on the recent results of Ref. [@Moch:2008qy]. These approximate NNLO results for the total cross section are based on the complete logarithmic dependence on the heavy quark velocity $\beta = \sqrt{1-4m^2/s}$ near threshold $s \simeq 4m^2$. Moreover, they include the complete two-loop Coulomb corrections as well as the exact dependence on the renormalization and factorization scale at NNLO [@Kidonakis:2001nj]. Recently, similar studies have appeared in Refs. [@Kidonakis:2008mu; @Cacciari:2008zb; @Nadolsky:2008zw]. While Ref. [@Kidonakis:2008mu] largely follows our approach [@Moch:2008qy] to describe the total top-quark pair cross section at NNLO, Ref. [@Cacciari:2008zb] has limited itself to updating older predictions based on threshold resummation to next-to-leading logarithmic accuracy only. Thus, Ref. [@Cacciari:2008zb] necessarily arrives at larger theoretical uncertainties. The interesting study of Ref. [@Nadolsky:2008zw] on the other hand applied consistently predictions to NLO accuracy in QCD. In doing so, it has investigated correlations of rates for top-quark pair production with many other cross sections at LHC to quantify a potential sensitivity to the gluon luminosity. Total cross section {#sec:total} =================== The total hadronic cross section for top-quark pair production depends on the hadronic center-of-mass energy squared $s$ and the top-quark mass $\mt$. It is given by $$\begin{aligned} \label{eq:totalcrs} \sigma(s, \mt^2) &=& \sum\limits_{i,j = q,{\bar{q}},g} f_{i/p}\left(\mufs\right) \otimes f_{j/p}\left(\mufs\right) \nonumber\\ & & \otimes \,\, \hat{\sigma}(\mt^2,\mufs,\murs)\, ,\end{aligned}$$ where $f_{i/p}$ are the PDFs of the proton. The partonic cross section is given by $\hat{\sigma}$ and $\otimes$ denotes the standard convolution (see e.g. Ref. [@Moch:2008qy]). The generally adopted procedure to estimate the theoretical uncertainty for $\sigma$ in Eq. (\[eq:totalcrs\]) exploits the residual dependence on the renormalization and factorization scale, $\mur$ and $\muf$, which are identified throughout this article (i.e. $\mur = \muf = \mu$). The NLO QCD corrections for the parton cross section $\hat{\sigma}$ and the PDFs $f_{i/p}$ provide the first instance where a meaningful error can be determined in this way. We define the range as $$\begin{aligned} \label{eq:range} {\lefteqn{ \sigma(\mu=2\mt)-\Delta\sigma_{PDF}(\mu=2\mt) \, \le \, \sigma(\mu) }} \nonumber \\ && \, \le \, \sigma(\mu=\mt/2)+\Delta\sigma_{PDF}(\mu=\mt/2) \, ,\end{aligned}$$ where $\Delta\sigma_{PDF}$ is computed from the variation of the cross section with respect to the parameters of the global fit (see e.g. Refs. [@Nadolsky:2008zw; @Martin:2007bv; @Tung:2006tb]). In this contribution we employ the approximate NNLO result [@Moch:2008qy] to predict cross sections (\[eq:totalcrs\]) and the associated uncertainty ranges (\[eq:range\]) at Tevatron and LHC. Let us therefore briefly comment on the anticipated accuracy. Our cross section  takes along all logarithmically enhanced terms $\ln^k\beta$, $k=1,\dots,4$ as well as the complete Coulomb corrections ($\sim 1/\beta, 1/\beta^2$) at two loops for the dominant parton channels $q{\bar q}$ and $gg$ and adds them on top of the exact NLO predictions. In this way, our predictions rely on exact expressions in the region of phase space $s \simeq 4m^2$, where perturbative corrections receive the largest weight from the convolution with the parton luminosities, cf. Eq. (\[eq:totalcrs\]). The effect of new parton channels opening at NNLO ($qq$ and ${\bar q}{\bar q}$) is expected to be small, cf. the $qg$ and ${\bar q}g$ channels at NLO. The region of large energies $s \gg 4m^2$ on the other hand is inaccessible within our approach [@Moch:2008qy]. However, it is expected to give only small contributions in a full NNLO calculation in line with the observed small corrections for top-quark pair production together with an additional jet at NLO [@Dittmaier:2007wz] which are part of the full NNLO correction for top-quark pair production. Moreover, we have also included the exact $\mur$ and $\muf$ scale dependence at NNLO [@Kidonakis:2001nj] which can be constructed using renormalization group methods. For the time being, we have chosen a common value $\mu$ for the scales, and we will address the independent variation of $\mur$ and $\muf$ in a future publication. However, based on preliminary studies we do not expect large modifications here. In summary, we have accounted for all numerically dominant contributions and are confident that this provides a very good approximation to the unknown full NNLO result as experience from other reactions, e.g. Higgs-production in gluon fusion [@Moch:2005ky] shows. Let us next present our results for Tevatron and LHC. In Figs. \[fig:tev\] and \[fig:lhc\] we plot the uncertainty range (\[eq:range\]) comparing NLO and NNLO accuracy. ![ \[fig:tev\] The  and  QCD prediction for the $t{\bar t}$ total cross section at Tevatron for $\sqrt{s}=1.96$ TeV. The bands denote the total uncertainty from PDF and scale variations for the MRST06nnlo set [@Martin:2007bv] according to Eq. (\[eq:range\]). ](./Pics/ll-mass-tev){width="7.0cm"} ![ \[fig:lhc\] Same as Fig. \[fig:tev\] for LHC with $\sqrt{s}=14$ TeV. ](./Pics/ll-mass-lhc){width="7.0cm"} At Tevatron (Fig. \[fig:tev\]) the central value at NNLO increases typically by 8% with respect to NLO. The residual scale dependence of  is 3%, which corresponds to a reduction by a factor of two compared to NLO. The overall uncertainty according to Eq. (\[eq:range\]) is at  about 8% for the CTEQ6.6 and 6% for the MRST06nnlo PDF set. At LHC (Fig. \[fig:lhc\]) our  leads only to a small shift of a few percent in the central value and the  band is about 6% for CTEQ6.6 and about 4% for MRST06nnlo, which exhibits again a drastic reduction of the scale uncertainty as compared to the prediction based on NLO QCD. For phenomenological applications, the results of Eqs. (\[eq:totalcrs\]), (\[eq:range\]) are best presented by means of simple formulae for the mass dependence of the total cross section. To that end we make the ansatz following Ref. [@Cacciari:2008zb] $$\begin{aligned} \label{eq:mass-dep} \sigma(\mt) = a + b x + c x^2 + d x^3 + e x^4 \, ,\end{aligned}$$ where $x=(\mt/\mbox{GeV}-171)$. The parameters $a,b,c,d,e$ are fitted to reproduce $\sigma$ in the mass range $150~\mbox{GeV} \le \mt \le 190~\mbox{GeV}$ with a typical accuracy of better than $0.1$ per mille. For Tevatron and LHC the respective results for various PDF sets are given in Tabs. \[tab:mass-fit-tev\]–\[tab:mass-fit-lhc\]. Note, that Eq.(\[eq:mass-dep\]) uses a polynomial of degree four and also determines the parameter $a$ for the central value $\sigma(\mt=171~\mbox{GeV})$ from the fit. [|l|l|l|l|l|l|l|]{} &a\[pb\]&b\[pb\]&c\[pb\] $\times 10^{2}$&d\[pb\] $\times 10^{5}$&e\[pb\] $\times 10^{7}$\ [|l|l|l|l|l|l|l|]{} &a\[pb\]&b\[pb\]&c\[pb\]&d\[pb\] $\times 10^{2}$&e\[pb\] $\times 10^{5}$\ [|l|l|l|l|l|l|l|]{} &a\[pb\]&b\[pb\]&c\[pb\]&d\[pb\] $\times 10^{2}$&e\[pb\] $\times 10^{5}$\ Finally, we briefly quote some  rates for the pair-production of new heavy quarks in the fundamental representation of the color $SU(3)$ gauge group at LHC with $\sqrt{s}=14$ TeV (see also Ref. [@Cacciari:2008zb]). Such particles with a mass $m_T$ appear in certain extensions of the Standard Model and we focus on a production model which is entirely dominated by QCD effects. Thus, our cross section $\sigmaNNLO$ provides a meaningful and accurate prediction because its numerical values arises largely from the threshold region where the logarithms $\ln^k \beta$ dominate. In Tabs. \[tab:new-hvq-mrst06\], \[tab:new-hvq-cteq65\] we quote the corresponding numbers in the mass range $0.5~\mbox{TeV} \le m_T \le 2~\mbox{TeV}$ (see Ref. [@Cacciari:2008zb] for results to NLO accuracy). We observe that the scale dependence at NNLO accuracy is rather small, showing the expected good stability of the perturbative prediction. The relative variation of $\sigma$ with respect to the PDFs, though, is dominating by far. Note there is the usual factor of two between the PDF uncertainty quoted by MRST06nnlo [@Martin:2007bv] and the CTEQ6.5 [@Tung:2006tb] PDF sets due to the definition of the tolerance criteria in the respective fits. The reason for the large observed PDF uncertainty is the gluon PDF being poorly constrained in the relevant region of large momentum fraction $x \simeq 0.1 \dots 0.3$. This is a fact well-known to influence many searches for high-mass particles in gluon fusion channels (see e.g. Ref. [@Nadolsky:2008zw] for the correlation of top-quark pair production rate with the high mass Higgs cross section). -------------- ------- ------- --------------- ------- ------- --------------- ------- ------- --------------- $m_T$\[TeV\] min max $\delta [\%]$ min max $\delta [\%]$ min max $\delta [\%]$ 0.5 4345. 4472. 1 4287. 4656. 4 4160. 4656. 6 0.6 1561. 1601. 1 1526. 1676. 5 1486. 1676. 6 0.7 634.1 649.2 1 616.0 682.5 5 600.8 682.5 6 0.8 282.3 288.5 1 272.6 304.4 6 266.4 304.4 7 0.9 134.5 137.2 1 129.3 145.1 6 126.6 145.1 7 1.0 67.64 68.94 1 64.81 73.08 6 63.50 73.08 7 1.1 35.45 36.17 1 33.93 38.41 6 33.22 38.41 7 1.2 19.23 19.65 1 18.38 20.91 6 17.97 20.91 8 1.3 10.74 10.99 1 10.26 11.72 7 10.01 11.72 8 1.4 6.147 6.301 1 5.862 6.741 7 5.708 6.741 8 1.5 3.589 3.687 1 3.417 3.957 7 3.319 3.957 9 1.6 2.130 2.192 1 2.021 2.363 8 1.959 2.363 9 1.7 1.282 1.322 2 1.212 1.432 8 1.172 1.432 10 1.8 0.781 0.806 2 0.735 0.878 9 0.710 0.878 11 1.9 0.480 0.497 2 0.450 0.544 9 0.433 0.544 11 2.0 0.298 0.309 2 0.277 0.340 10 0.266 0.340 12 -------------- ------- ------- --------------- ------- ------- --------------- ------- ------- --------------- -------------- ------- ------- --------------- ------- ------- --------------- ------- ------- --------------- $m_T$\[TeV\] min max $\delta [\%]$ min max $\delta [\%]$ min max $\delta [\%]$ 0.5 3921. 4037. 1 3639. 4436. 10 3522. 4436. 11 0.6 1402. 1440. 1 1275. 1604. 11 1238. 1604. 13 0.7 568.0 582.5 1 508.6 656.5 13 494.1 656.5 14 0.8 252.4 258.2 1 222.5 294.0 14 216.6 294.0 15 0.9 120.3 122.8 1 104.5 141.0 15 102.0 141.0 16 1.0 60.48 61.66 1 51.94 71.37 16 50.76 71.37 17 1.1 31.78 32.30 1 26.95 37.64 17 26.44 37.64 17 1.2 17.25 17.57 1 14.50 20.57 17 14.21 20.57 18 1.3 9.626 9.802 1 8.023 11.58 18 7.847 11.58 19 1.4 5.503 5.614 1 4.555 6.673 19 4.444 6.673 20 1.5 3.208 3.277 1 2.630 3.925 20 2.560 3.925 21 1.6 1.902 1.946 1 1.545 2.347 21 1.502 2.347 22 1.7 1.144 1.173 1 0.921 1.426 22 0.892 1.426 23 1.8 0.696 0.715 1 0.554 0.876 23 0.535 0.876 24 1.9 0.428 0.441 1 0.337 0.544 24 0.324 0.544 25 2.0 0.265 0.274 2 0.206 0.342 25 0.198 0.342 27 -------------- ------- ------- --------------- ------- ------- --------------- ------- ------- --------------- Conclusion {#sec:concl} ========== We have presented updated predictions for cross sections of top-quark pair production based on the (approximate) NNLO results of Ref. [@Moch:2008qy]. These represent the best present estimates for hadro-production of top-quark pairs, both at Tevatron and LHC. We have argued that the neglected contributions (i.e. power suppressed terms away from threshold and new parton channels) are numerically small. We have found good convergence properties of the higher order corrections and greatly improved stability of the total cross section with respect to scale variations by our  result. For applications, we have presented simple formulae (\[eq:mass-dep\]) with 0.1 per mille accuracy for the mass dependence of the total cross section in the range $150~\mbox{GeV} \le \mt \le 190~\mbox{GeV}$. Finally, we have applied our results to estimate the pair-production rates of new quarks heavier than the top-quark in the range up to $2~\mbox{TeV}$. The results of Tabs. \[tab:mass-fit-tev\]–\[tab:mass-fit-lhc\] for the fit of the mass dependence of $\sigma$ have also been coded in a C-program, which is available from the authors upon request. Acknowledgments {#acknowledgments .unnumbered} --------------- S.M. is supported by the Helmholtz Gemeinschaft under contract VH-NG-105 and and P.U. is a Heisenberg fellow of Deutsche Forschungsgemeinschaft (DFG). This work is also partly supported by DFG in SFB/TR 9. [10]{} W. Bernreuther, J. Phys. G35 (2008) 083001, arXiv:0805.1333 \[hep-ph\] P. Nason, S. Dawson and R.K. Ellis, Nucl. Phys. B303 (1988) 607 W. Beenakker et al., Phys. Rev. D40 (1989) 54 W. Bernreuther et al., Nucl. Phys. B690 (2004) 81, hep-ph/0403035 S. Dittmaier, P. Uwer and S. Weinzierl, Phys. Rev. Lett. 98 (2007) 262002, hep-ph/0703120 M. Czakon, A. Mitov and S. Moch, Phys. Lett. B651 (2007) 147, arXiv:0705.1975 \[hep-ph\] M. Czakon, A. Mitov and S. Moch, Nucl. Phys. B798 (2008) 210, arXiv:0707.4139 \[hep-ph\] J.G. Körner, Z. Merebashvili and M. Rogal, (2008), arXiv:0802.0106 \[hep-ph\] M. Czakon, Phys. Lett. B664 (2008) 307, arXiv:0803.1400 \[hep-ph\] S. Moch and P. Uwer, Phys. Rev. D (2008) in press, arXiv:0804.1476 \[hep-ph\] K. Hagiwara, Y. Sumino and H. Yokoya, (2008), arXiv:0804.1014 \[hep-ph\] N. Kidonakis et al., Phys. Rev. D64 (2001) 114001, hep-ph/0105041 N. Kidonakis and R. Vogt, (2008), arXiv:0805.3844 \[hep-ph\] M. Cacciari et al., (2008), arXiv:0804.2800 \[hep-ph\] P.M. Nadolsky et al., (2008), arXiv:0802.0007 \[hep-ph\] A.D. Martin et al., Phys. Lett. B652 (2007) 292, arXiv:0706.0459 \[hep-ph\] W.K. Tung et al., JHEP 02 (2007) 053, hep-ph/0611254 S. Moch and A. Vogt, Phys. Lett. B631 (2005) 48, hep-ph/0508265 [^1]: Presented by S.M. at [*[Loops and Legs in Quantum Field Theory]{}*]{}, 20–25 April 2008, Sondershausen (Germany).

Recent achievements of nanolithography in semiconductor technology allow for the fabrication of devices in which a definite number of electrons are confined within two-dimensional islands of size as small as tenths of nanometers \[Quantum Dots (QD)\][@dots]. In the last few years there has been growing interest in the study of these devices in view of improving our understanding of correlated electron systems. In fact, QD are unique with respect to other structures, e.g., macromolecules and clusters, because a dot can be connected to sources and/or a measuring apparatus via contacts. This possibility allows investigation of the system with probes changing the number of particles [@tunnel]. Indeed, quite recently Tarucha [*et al.*]{}[@tarucha] have measured the tunneling current in gated vertical quantum dots as a function of a magnetic field $B$ applied parallel to the current. In the Coulomb blockade regime and in presence of a very small voltage bias, the current shows a sequence of peaks that occur whenever the gate voltage $V_g$ is proportional (via a voltage-to-energy conversion coefficient) to the chemical potential $\mu_N = E(N)-E(N-1)$ for adding one more particle to the dot. Here $E(N)$ is the ground-state (GS) energy of the dot once $N$ electrons are localized in it. General features of the current peaks are qualitatively reproduced by assuming that electrons are confined by a parabolic potential of frequency $ \omega_0$ and by adding the charging energy $E_{\rm ch}=V_0 N(N-1)/2$ to account for the electron-electron repulsion \[Constant Interaction (CI) model\][@tarucha; @mceuen]. The GS energy is obtained, for any value of $B$, by filling the lowest one-particle free harmonic oscillator levels with electrons of both spin. In this model the observed increasing of the addition energy $\Delta_N = E(N+1) + E(N-1) - 2 E(N)$ for $N=N_p$, where $N_p=2,6,10,20,...$, is easily related to the shell structure of the two dimensional harmonic potential spectrum, which leads to a marked increase of $\mu_N$ whenever a new shell is opened. The CI model then reproduces the oscillations of the current peaks with the field $B$ as well as their shift in pairs. Because the CI GS wave function has minimum total spin, its agreement with the experimental pattern also confirms that in some QD, like the one in Ref.[@tarucha], the effective $g$-factor $g_*$ can be very small[@wagner]. However, there are some evident features of the experiments that cannot be reproduced by the simple independent electron CI Hamiltonian. Because the harmonic single-particle levels are equally spaced, according to the CI model (at $B=0$) $\Delta_N$ should be constant ($\Delta_N=V_0$) within a unfilled shell. For the same reason one should then find $\Delta_{N_p} = V_0 + \hbar\omega_0$ for any $N_p$. While the experimental pattern in Ref.[@tarucha] allows a clear identification of a shell structure for dots of less than $\approx 20$ electrons, it also clearly shows a smooth shrinkage of the spacing $\Delta_N$ within a unfilled shell with growing $N$, as it was previously noticed in single electron capacitance spectroscopy[@ashoori2]. Then, one observes that the expected peaks at $\Delta_{N_p}$ decrease in height each time a new shell opens. In general, as discussed by Schmidt [*et al*]{}.[@schmidt], fittings of capacitance data within the CI model indicate that the confinement strength of the potential appears to decrease rapidly with increasing energy. Question arises whether such an effect can be attributed to electron-electron correlation. In this letter we consider the model of harmonic interactions (HI) between $N$ electrons[@green] in a parabolic confining potential. Although there is tenuous justification for this model interaction, when the magnetic field is not too small and the electron number not too large we show that it embodies correlation effects which correctly reproduce interesting experimental features. The model is well known and its popularity relies on the fact that for fully spin polarized electrons the Laughlin trial wave function, which is successful in describing the Coulomb gas in a quantizing magnetic field, has a form similar to one of its eigenstates[@girvin]. Successively, the model has been discussed, again in the fully polarized spin sector, in connection with the QD problem in a large magnetic field orthogonal to the dot plane[@payne]. In view of the finding $g_*\approx 0$, results about the maximum spin sector are not relevant to the present discussion. We present some details of the solution because little information can be found in literature about the GS wave function in an arbitrary spin sector. We do not expect the HI model useful in discussing the region of small magnetic field. In fact, there are evidences[@tarucha] that in this case the spin sector of the QD GS obeys Hund’s rule, whereas the GS of the HI model at $B=0$ has necessarily lowest spin[@lieb]. Instead, for higher magnetic field our results closely reproduce features of the experimental data of Ref.[@tarucha] and are summarized in Fig. (1). The narrowing, compared with the CI model, of the distance between current peaks is transparent. The shift, at fixed $V_0$, of the orbital angular momentum transitions which take place with increasing $B$ to lower values of the magnetic field also clearly appears in our results. Then, close to the point where the oscillations in the peak location drop, for large enough $N$, the peaks making up a pair have an intriguing “out of phase” behaviour, similar to the experimental pattern. The HI Hamiltonian in a constant magnetic field $B$ is $H_B = H_0 + \hbar {\omega}_c(L_3 +g_*S_3)/2$, where $\hbar {L}_3=\sum_{i=1}^N (x_{1,i}p_{2,i}-x_{2,i}p_{1,i})$ and $S_3=\sum_{i=1}^N S_{3,i}$ are the third component of the total angular momentum and total spin operators, respectively, ${{\mbox{\boldmath${r}$}}}_i=(x_{1,i},x_{2,i})$ and ${{\mbox{\boldmath${p}$}}}_i=-i\hbar\partial/\partial {{\mbox{\boldmath${r}$}}}_i$ are the coordinates and momentum of the $i$-th electron, ${\omega}_c = e B/(m_*c)$ is the cyclotron frequency, and $$\begin{aligned} H_0 = & - & {\hbar^2\over{2m_*}} \sum_{i=1}^N {{\partial}\over{\partial {{\mbox{\boldmath${r}$}}}_i}} \cdot {{\partial}\over{\partial {{\mbox{\boldmath${r}$}}}_i}} + m_*{{\omega^2}\over2} \sum_{i=1}^N \vert {{\mbox{\boldmath${r}$}}}_i\vert ^2 \nonumber\\ & + & \sum_{1\le i<j\le N}(V_0- {U\over2} \vert {{\mbox{\boldmath${r}$}}}_i-{{\mbox{\boldmath${r}$}}}_j\vert^2). \eqnum{1} \label{H2} \end{aligned}$$ Here $m_*$, $g_*$ denote effective parameters and for the time being we set $m_*=e=\hbar=1$. The frequency $\omega^2=\omega_0^2 +{\omega}_c^2/4$ enters $H_0$ upon choosing the gauge ${{\mbox{\boldmath${A}$}}}=(B/2)(-x_{2},x_{1})$ for the vector potential and represents the frequency of the effective parabolic confining potential, whereas $U\ge0$ is the strength of the interaction. For $U=0$ then $H_B$ reduces to the CI model. Introducing $\Lambda_{ij} = \Omega^2 \delta_{ij}+U{\cal J}_{ij}$, where $\Omega^2=\omega^2-NU$ and ${\cal J}_{ij}$ denotes the matrix with all unit entries, (i.e., ${\cal J}_{ij}=1$ $\forall\,i,j$), the potential energy entering Eq. (\[H2\]) can be compactly written as $V = \sum_{i,j=1}^N\Lambda_{ij} {{\mbox{\boldmath${r}$}}}_i\cdot{{\mbox{\boldmath${r}$}}}_j/2$, so that $H_0$ is bound from below if $\Lambda_{ij}$ is positive definite. The symmetric matrix $\Lambda_{ij}$ is diagonalized by any unitary matrix ${\cal U}_{i \nu}$ satisfying to $\sum_{i=1}^N{\cal U}_{i \nu} = 0$, $\forall\nu\ne N$, and ${\cal U}_{i N} = N^{-{1\over2}}$, $\forall i$. Its eigenvalues $\lambda_1=\ldots=\lambda_{N-1}=\Omega^2$, $\lambda_N = \omega^2$ are readily evaluated, so that positivity is ensured whenever $\omega^2>NU$. Hence, if we limit the discussion to $N$ not too large, the unphysical feature of dealing with an interaction unbounded at large distances is compensated by the presence of the confining potential. We then introduce normal coordinates ${{\mbox{\boldmath${y}$}}}_{\nu} = \sum_{i=1}^N {\cal U}^{\dagger}_{\nu i} {{\mbox{\boldmath${r}$}}}_i$. The Laplacian $\Delta = \sum_{i=1}^N {{\partial}/{\partial {{\mbox{\boldmath${r}$}}}_i}}\cdot {{\partial}/{\partial {{\mbox{\boldmath${r}$}}}_i}}$, the angular momentum ${L}_3$, and the operator $R^2 = \sum_{i=1}^N \vert {{\mbox{\boldmath${r}$}}}_i\vert^2$ are invariant under unitary transformations and from the invariance of $R^2$ one gets the simple but key identity $\sum_{{\nu}=1}^{N-1} \vert {{\mbox{\boldmath${y}$}}}_{\nu}\vert^2 = \sum_{ i<j } \vert {{\mbox{\boldmath${r}$}}}_i-{{\mbox{\boldmath${r}$}}}_j\vert^2/N$. In the new basis the equation $H_B\Psi=E\Psi$ is immediately solved, because the key identity allows to write $H_0$ as a sum of separated harmonic oscillators of frequencies $\sqrt{\lambda_{\nu}}$. However, the straightforward normal-mode approach is of no much practical help, because the main problem one faces is to account for the identity of the particles. Eigenfunctions of $H_B$ with definite symmetry under particle permutations can be factorized in the form $\Psi=\Psi_{\rm cm}\Psi_r$, in the usual way[@firsov], where $\Psi_{\rm cm}$ is the completely symmetric harmonic wave function of the center of mass (c.m.) coordinate ${{\mbox{\boldmath${r}$}}} = N^{-{1\over 2}} {{\mbox{\boldmath${y}$}}}_N$, with energy $E_{\rm cm}$ and angular momentum $L_3^{\rm cm}$, while the relative motion wave function $\Psi_r$ must be solution of the equation $$\begin{aligned} - {1\over2} \Delta \Psi_r & + & {\Omega^2\over{2N}} \sum_{i<j} \vert {{\mbox{\boldmath${r}$}}}_i-{{\mbox{\boldmath${r}$}}}_j\vert^2 \Psi_r + {{\omega}_c\over2}(L_3 +g_*S_3) \Psi_r \nonumber\\ & = & (E-E_{\rm ch}-E_{\rm cm}-{{\omega}_c\over2}L_3^{\rm cm})\Psi_r, \eqnum{2} \label{calogero_mod}\end{aligned}$$ of same symmetry as $\Psi$ and of zero total linear momentum ${{\mbox{\boldmath${P}$}}} = \sum_{i=1}^N {{\mbox{\boldmath${p}$}}}_i$. -0.truecm =7.7cm [ FIG. 1 Chemical potential $\mu _{N}$ versus magnetic field $B$ for $N \le 22 $: $(a)$ HI model for $\hbar U /(m_*\omega_0) = 0.07$ meV; $(b)$ CI model. Parameter values are $\hbar \omega_0 = 3$ meV, $\hbar\omega_c/B=1.63$ meV/T, $V_0 =1.5$ meV, $g_* = -0.03 $. ]{} Henceforth we need only consider the lowest energy c.m. wave function and for the time being we thus set $\Psi_{\rm cm}=\exp\{-\omega\sum_{i,j=1}^N{{\mbox{\boldmath${r}$}}}_i\cdot{{\mbox{\boldmath${r}$}}}_j/(2N)\}$, for which one has $E_{\rm cm}=\omega$ and $L_3^{\rm cm}=0$. If one looks for eigenstates of Eq. (\[calogero\_mod\]) in the form $\Psi_r = \Phi\Psi_0$, where $\Psi_0=\exp\{-\Omega \sum_{i<j}\vert {{\mbox{\boldmath${r}$}}}_i-{{\mbox{\boldmath${r}$}}}_j\vert^2/(2N)\}$, and employes holomorphic coordinates $z_i=x_{1,i}+ix_{2,i}$, and ${\bar z}_i=x_{1,i}-ix_{2,i}$, one gets that the unknown function $\Phi$ must satisfy the couple of equations $$\begin{aligned} & - & \sum_{i=1}^N (2{\bar\partial}_i{\partial}_i - \Omega_+ z_i{\partial}_i - \Omega_- {\bar z}_i{\bar\partial}_i) \Phi + {{\omega}_c\over2}g_*S_3 \Phi \nonumber\\ & = & (E-{\cal E}_0-E_{\rm ch})\Phi, \qquad {{\mbox{\boldmath${P}$}}}\Phi=0, \eqnum{3$a$,$b$} \label{hermite-gen}\end{aligned}$$ where ${\cal E}_0=\Omega(N-1)+\omega$ is the zero-point energy, $\Omega_{\pm}=\Omega\pm\omega_c/2$, and we have used the shorthand notation $\partial_i={\partial/\partial z_i}$, and ${\bar\partial}_i={\partial/\partial {\bar z}_i}$. Because the function $\Psi_0$ is completely symmetric, $\Psi_r$ and $\Phi$ must have same symmetry. Eq. (3$a$) is a sum of $N$ separated Hamiltonians and its solutions are built up in terms of single-particle orbital wave functions $f_{n_1,n_2}$, with eigenvalues $\varepsilon_{n_1,n_2}$, $$\begin{aligned} f_{n_1,n_2} & = & e^{\Omega{\bar z} z} {\bar\partial}^{n_1}{\partial}^{n_2} e^{-\Omega{\bar z} z}, \nonumber\\ \varepsilon_{n_1,n_2} & = & \Omega (n_1 + n_2) + {{\omega_c}\over2} (n_1-n_2), \eqnum{4} \label{levels}\end{aligned}$$ where $n_1$ and $n_2$ are non negative integers. Apparently, the original interacting problem reduces to a free problem and the main effect of the interaction seems just related to a redefinition of the effective frequency $\omega\to \Omega$. However, Eq. (3$b$) is highly non trivial and already the two-particle problem shows that it can be satisfied in general only by taking linear combinations of degenerate $N$-particle solutions. Moreover, the frequency $\Omega=\sqrt{\omega^2-NU}$ depends on $N$, so that energy differences like $E(N)-E(N-1)$ cannot be analyzed in terms of one-particle levels. In these respects the system is correlated. Fortunately, as expected on general ground, one can easily check that the ordinary Slater determinant $Z$ obtained by filling the $N$ lowest states (\[levels\]) (so that $S_3=N/2$) is a solution of [*both*]{} Eqs. (\[hermite-gen\]). Fig. (2) shows some typical shape of the set of occupied levels. By increasing $B$ one observes a depletion of the levels in the $SE$ side and an extra filling in the $NW$ side, because the contribution $H_L=\omega_c L_z/2$ favours decrease of the orbital angular momentum. Noticing that the functions $f_{n_1,n_2}$ are polynomials and using standard properties of determinants under column addition and multiplication, it is easy to see that for any set of the form depicted in Fig. (2) the Slater determinant can be rearranged into the compact form $Z=\det[{z}_{i}^{n_1}{\bar z}_{i}^{n_2}]$. This expression is invariant under translation $z_i\to z_i+z_0$, $\forall i$, so that ${{\mbox{\boldmath${P}$}}}Z=0$ as requested. Introducing the spin index $\sigma=1,2$ for up and down electrons, respectively, it follows that the GS wave function $\Psi_{\rm GS}$ of $H_B$ in the spin sector $S_3=(M_1-M_2)/2$, $N=M_1+M_2$, is obtained, up to a normalization constant, by antisymmetrizing the product of the spin-up and spin-down wave functions. Denoting ${\cal A}$ the antisymmetrization, we formally have $$\begin{aligned} \Psi_{\rm GS} = \exp\{-{1\over2}&&\sum_{i,j=1}^N{\bar z}_{i}\Gamma_{ij}z_{j}\} \nonumber\\ \times && {\cal A}\left\{\prod_{\sigma=1}^2 \det[z^{n_1^{\sigma}}_{i_{\sigma}} {\bar z}^{n_2^{\sigma}}_{i_{\sigma}}]\chi_{\sigma}\right\}, \eqnum{5} \label{ground-state}\end{aligned}$$ where $z_{i_{\sigma}}$, with $i_{1}=1,\ldots,M_1$ and $i_{2}=M_1+1,\ldots,N$, are the coordinates of spin-up and spin-down electrons, respectively, $\chi_{1}$, $\chi_{2}$, are the totally symmetric spin functions of spin-up and spin-down electrons, and $(n_1^{\sigma},n_2^{\sigma})$ are the labels of the (lowest energy) levels occupied by spin-$\sigma$ electrons. The matrix $\Gamma_{ij} = \Omega \delta_{ij}+N^{-1}(\omega-\Omega){\cal J}_{ij}$ is the square root of $\Lambda_{ij}$ and enters Eq. (\[ground-state\]) upon combination of $\Psi_0$ with the c.m. GS wave function. For zero magnetic field the GS is obtained by filling consecutively the levels along successive diagonals $n_1+n_2 =k$, $ k=0,1,...$ with two electrons of opposite spin, so that $\Psi_{\rm GS}$ has $S_z=0$ or $S_z=\pm 1/2$. Completion of a diagonal corresponds to one more shell filled. When the magnetic field is turned on, for a wide range of $B$ the GS remains in the lowest spin sector due to the smallness of $g_*$, whereas as previously seen the shape of the Fermi sea modifies in order to lower the orbital angular momentum. -0.truecm =7.7cm [ FIG. 2 Occupied single-particle quantum numbers $(n_1,n_2)$ of $15$ spinless fermions for six different values of the magnetic field $B$ (T). Parameter values are the same as in Fig. (1a). ]{} The lowest attainable value of the orbital angular momentum $L^{\rm min}_3=-\sum_{\sigma}M_{\sigma}(M_{\sigma}-1)/2$ is obtained by filling the levels $(n_1^{\sigma},n_2^{\sigma })=(0,m_{\sigma })$, where $m_{\sigma}=0,1,\ldots,M_{\sigma}-1$. In this case the Vandermonde determinant $Z =\det[{\bar z}_i^m] = \prod_{i<j}({\bar z}_j-{\bar z}_i)$ allows a closed expression for the GS wave function $\Psi_{\rm GS}^L$. Denoting $\{{{\mbox{\boldmath${r}$}}}_i,\sigma_i\}$ orbital and spin coordinates of the $i$-th electron, Eq. (\[ground-state\]) gives the Laughlin-like[@johnson] state $$\begin{aligned} \Psi_{\rm GS}^L = \exp\{-{1\over2}&&\sum_{i,j=1}^N{\bar z}_i\Gamma_{ij}z_j\} \nonumber\\ \times && \prod_{i<j}({\bar z}_j-{\bar z}_i)^{\delta_{\sigma_i,\sigma_j}} e^{i{\pi\over2}{\rm sign}(\sigma_i-\sigma_j)}. \eqnum{6} \label{laughlin}\end{aligned}$$ Our results for the chemical potential vs. magnetic field are shown in Fig. (1). The curves are splitted by the charging energy $V_0$ but the $N$-dependence of $\Omega$ leads to a sizeable reduction of the energy scale of the HI model compared with the CI model. This behaviour is in agreement with the experimental pattern of Ref.[@tarucha] and with the discussion in Ref.[@schmidt]. Although one cannot overestimate the model nature of the harmonic interaction, our results indicate that the rapid decrease of the confinement strength with increasing electron number can be due to pure correlation effects. The curves (for $N\ge5$) perform oscillations that signal transitions to states of lower $L_3$, and drop at the field value where the minimum angular momentum state (\[laughlin\]) becomes energetically favourable. It is easy to see that this happens when $2\Omega/\omega_c\le M_1/(M_1-2)$ (with $M_1\ge M_2$). In the CI model the curves are perfectly paired because of the spin degeneracy of the one-particle levels. For example, the transition to the state of lowest $L_3$ takes place when $2\omega/\omega_c = M_1/(M_1-2)$. This relation gives the same value of the magnetic field at which the transition takes place both for $N=2M$ and for $N=2M-1$. Instead, in the HI model the $N$-dependence of $\Omega$ breaks this perfect pairing, in particular for large N where $\Omega$ becomes small. While usually the angular momentum of the GS is a decreasing function of the number of particles, at the field values where the paired curves come out of phase one instead finds $L_z(2N-1) \le L_z(2N)$. For instance, at $B=3.6$ T one has $L_z(21)= -89$, whereas $L_z(22)=-88$. Hence, the strong reduction of the escillator frequency at large $N$ allows for level crossing even within a single pair of particles which are added with opposite spin. This in turn leads to the oscillations out of phase of the peaks forming a pair, a feature which is absent in the CI model. We conclude that, in spite of the simplicity of the model for including electron-electron correlations, many features of the experiment of Ref.[@tarucha] are nicely reproduced. We would like to thank G. Faini, R. Fazio, A. Mastellone, and U. Merkt for useful discussions. A.A. also acknowledges the University of Udine for warm hospitality. This work is partly supported by Consiglio Nazionale delle Ricerche under Contract $\#$ 115.22594. U. Meirav, M.A. Kastner, and S.J. Wind, , 771 (1990); H. van Houten, C.W. Benakker, and A.A.M. Staring, in [*Single Charge tunneling*]{}, H. Grabert and M.H. Devoret editors, NATO ASI (Plenum Press, New York, 1992). J. Weis [*et al.*]{}, , 4019 (1993); R.C. Ashoori, Nature, 379 (1996). S. Tarucha [*et al.*]{}, , 3613 (1996). P.L. McEuen [*et al.*]{}, , 1926 (1991). M. Wagner, U. Merkt, and A.V. Chaplik, , 1951 (1992). R.C. Ashoori [*et al.*]{}, , 613 (1993). T. Schmidt [*et al.*]{}, , 5570 (1995). H.S. Green, Nuovo Cimento IX, 880 (1958); R.G. Storer, , 5, (1970). S.M. Girvin and T. Jach, , 4506 (1983). N.F. Johnson and M.C. Payne, , 1157 (1991). E.H. Lieb and D. Mattis, Phys. Rev. [**125**]{}, 164 (1962). The result applies to any separately symmetric potential, i.e., such that $V({{\mbox{\boldmath${r}$}}}_i)=W(x_{1,i}) + W(x_{2,i})$. See, e.g., Yu A. Firsov and V.L. Gurevitch, Zh. Eksp. Teor. Fiz.[**41**]{}, 512 (1961) \[Sov. Phys. JETP [**14**]{}, 367, (1962)\]. We denote as ’Laughlin-like ’ any state written as a Vandermonde determinant times a gaussian factor. For a discussion of the relationship between the present state in the fully polarized sector and the standard Laughlin wave-function see: N.F. Johnson, , 2636 (1992).

--- abstract: 'We propose an attention-injective deformable convolutional network called for crowd understanding that can address the accuracy degradation problem of highly congested noisy scenes. contains two concatenated networks. An attention-aware network called Attention Map Generator (AMG) first detects crowd regions in images and computes the congestion degree of these regions. Based on detected crowd regions and congestion priors, a multi-scale deformable network called Density Map Estimator (DME) then generates high-quality density maps. With the attention-aware training scheme and multi-scale deformable convolutional scheme, the proposed achieves the capability of being more effective to capture the crowd features and more resistant to various noises. We have evaluated our method on four popular crowd counting datasets (ShanghaiTech, UCF\_CC\_50, WorldEXPO’10, and UCSD) and an extra vehicle counting dataset TRANCOS, and our approach beats existing state-of-the-art approaches on all of these datasets.' author: - | Ning Liu$^{1, 2}$     Yongchao Long$^{1, 2}$     Changqing Zou$^3$     Qun Niu$^{1, 2}$     Li Pan$^4$     Hefeng Wu$^{1, 5, 6\,}$[^1]\ $^1$School of Data and Computer Science, Sun Yat-sen University\ $^2$Guangdong Key Laboratory of Information Security Technology $^3$University of Maryland\ $^4$Shanghai Jiao Tong University $^5$Guangdong University of Foreign Studies $^6$WINNER Technology\ [liuning2@mail.sysu.edu.cn, {longych3, niuqun}@mail2.sysu.edu.cn, ]{}\ [cqzou@umiacs.umd.edu, panli@sjtu.edu.cn, wuhefeng@gmail.com]{} bibliography: - 'mybib.bib' title: 'ADCrowdNet: An Attention-Injective Deformable Convolutional Network for Crowd Understanding' --- Introduction {#sec:intro} ============ Crowd understanding has attracted much attention recently because of its wide range of applications like public safety, congestion avoidance, and flow analysis. The current research trend for crowd understanding has developed from counting the number of people to displaying distribution of crowd through density map. Generally, generating accurate crowd density maps and performing precise crowd counting for highly congested noisy scenes is challenging due to the complexity of crowd scenes caused by various factors including background noises, occlusions, and diversified crowd distributions. {width="79.00000%"} Researchers recently have leveraged deep neural networks (DNN) for accurate crowd density map generation and precise crowd counting. Although these DNNs-based methods [@zhang2016single; @sam2017switching; @sindagi2017generating; @liu2018crowd; @CaoWZS18eccv; @li2018csrnet] have made significant success in solving the above issues, they still have the problem of accuracy degradation when applied in highly congested noisy scenes. As shown in Figure \[fig:sample\], the state-of-the-art approach [@li2018csrnet], which has achieved much lower Mean Absolute Error (MAE) than the previous state-of-the-art methods, is still severely affected by background noises, occlusions, and non-uniform crowd distributions. In this paper, we aim at an approach which is capable of dealing with highly congested noisy scenes for the crowd understanding problem. To achieve this, we designed an attention-injective deformable convolutional neural network called which is empowered by a visual attention mechanism and a multi-scale deformable convolution scheme. The visual attention mechanism is delicately designed for alleviating the effects from various noises in the input. The multi-scale deformable convolution scheme is specially introduced for the congested environments. The basic principle of visual attention mechanism is to use the pertinent information rather than all available information in the input image to compute the neural response. This principle of focusing on specific parts of the input has been successfully applied in various deep learning models for images classification [@hu2017squeeze], semantic segmentation [@ren2017end], image deblurring [@qian2018attentive], and visual pose estimation [@chu2017multi], which also suits our problem where the interest regions containing the crowd need to be recognized and highlighted out from noisy scenes. The multi-scale deformable convolution scheme takes as input the information of the dynamic sampling locations, other than evenly distributed locations, which has the capability of modeling complex geometric transformation and diverse crowd distribution. This scheme fits well the nature of the distortion caused by the perspective view of the camera and diverse crowd distributions in real world, therefore guaranteeing more accurate crowd density maps for the congested scenes. To incorporate the visual attention mechanism and deformable convolution scheme, we leverage an architecture consisting of two neural networks as shown in Figure \[fig:overviewArch\]. Our training contains two stages. The first stage generates an attention map for a target image via a network called Attention Map Generator (AMG). The second stage takes the output of AMG as input and generates the crowd density map via a network called Density Map Estimator (DME). The attention map generator AMG mainly provides two types of priors for the DME network: 1) candidate crowd regions and 2) the congestion degree of crowd regions. The former prior enables the multi-scale deformable convolution scheme empowered DME network to pay more attention to those regions having people crowds, and thus improving the capacity of being resistant to various noises. The latter prior indicates each crowd region with congestion degree (i.e., how crowded each crowd region is), which provides fine-grained congestion context prior for the subsequent DME network and boosts the performance of the DME network on the scenes containing diverse crowd distribution. The main contributions of this paper are summarized as follows. First, a novel attention-injective deformable convolutional network framework is proposed for crowd understanding. Second, our AMG model that attends the crowd regions in images, is innovatively formulated as a binary classification network by introducing third party negative data (i.e., background images with no crowds). Third, our DME model can estimate the crowds effectively by using the proposed structure of aggregating multi-scale deformable convolution representations. Furthermore, extensive experiments conducted on all popular datasets demonstrate the superior performance of our approach over existing leading ones. In particular, the proposed model outperforms the state-of-the-art crowd counting solution CSRNet [@li2018csrnet] with 3.0%, 18.8%, 3.0%, 13.9% and 5.1% lower Mean Absolute Error (MAE) on ShanghaiTech Part\_A, Part\_B, UCF\_CC\_50, WorldExpo’10, UCSD datasets, respectively. Apart from crowd counting, is also general for other counting tasks. We have evaluated on a popular vehicle counting dataset named TRANCOS [@guerrero2015extremely], and achieves 32.8% lower MAE than CSRNet. {width="99.00000%"} \[fig:overviewArch\] Related Work {#sec:related} ============ **Counting by detection:** Early approaches of crowd understanding mostly focus on the number of people in crowds [@DollarWSP12]. The major characteristics of these approaches are the sliding window based detection scheme and hand crafted features extracted from the whole human body or particular body parts with low-level descriptors like Haar wavelets [@viola2004robust] and HOG [@dalal2005histograms]. Generally, approaches in these groups deliver accurate counts when their underlying assumptions are met but are not applicable in more challenging congested scenes. **Counting by regression:** Counting by regression approaches differs depending on the target of regression: object count [@ChenLGX12; @ChenGXL13], or object density [@lempitsky2010learning]. This group of approaches avoid solving the hard detection problem. Instead, they deploy regression model to learn the mapping between image characteristics (mainly histograms of lower level or middle level features) and object count or density. These approaches that directly regress the total object count discard the information of the location of the objects and only use 1-dimensional object count for learning. As a result, a large number of training images with the supplied counts are needed in training. Lempitsky  [@lempitsky2010learning] propose a method to solve counting problem by modeling the crowd density at each pixel and cast the problem as that of estimating an image density whose integral over any image region gives the count of objects within that region. Since the ideal linear mapping is hard to obtain, Pham  [@pham2015count] use random forest regression to learn a non-linear mapping instead of the linear one. **Crowd understanding by CNN:** Inspired by the great success in visual classification and recognition, literature also focuses on the CNN-based approaches to predict crowd density map and count the number of crowds [@walach2016learning; @onoro2016towards; @LiHH2018arXiv; @KangC18bmvc; @qiu2019crowd]. Walach  [@walach2016learning] use CNN with a layered training structure. Shang [@shang2016end] adapt an end-to-end CNN which uses the entire images as input to learn the local and global count of the images and ultimately outputs the crowd count. A dual-column network combining shallow and deep layers is used in [@boominathan2016crowdnet] to generate density maps. In [@zhang2016single], a multi-column CNN is proposed to estimate density map by exacting features at different scales. Similar idea is used in [@onoro2016towards]. Marsden  [@marsden2016fully] try a single-column fully convolutional network to generate density map while Sindagi  [@sindagi2017cnn] present a CNN that uses high-level prior to boost accuracy. More recently, Sindagi  [@sindagi2017generating] propose a multi-column CNN called CP-CNN that uses context at various levels to improve generate high-quality density maps. Li  [@li2018csrnet] propose a model called CSRNet that uses dilated convolution to enlarge receptive fields and extract deeper features for boosting performance. These two approaches have achieved the state-of-the-art performances. Attention-Injective Deformable Convolutional Network {#sec:detail} ==================================================== The architecture of the proposed method is illustrated in Figure \[fig:overviewArch\]. It employs two concatenated networks: AMG and DME. AMG is a classification network based on fully convolutional architecture for attention map generation, while DME is a multi-scale network based on deformable convolutional layers for density map generation. Before training DME, we train the AMG module with crowd images (positive training examples) and background images (negative training examples). We then use the well-trained AMG to generate the attention map of the input image. Afterward, we train the DME module using the pixel-wise product of input images and the corresponding attention maps. In the following sections, we will detail the architectures of the AMG and DME netwroks. \[fig:ddlevel\] Attention Map Generator ----------------------- ### Attention map Attention map is an image-sized weight map where crowd regions have higher values. In our work, attention map is a feature map from a two-category classification network AMG which classifies an input image into crowd image or background image. The idea of using feature map to find the crowd regions in the input is motivated by an object localization work [@zhou2016learning] which points out that the feature maps of classification network contain the location information of target objects. The pipeline of the attention map generation is shown in Figure \[fig:AMG\]. $F_c$ and $F_b$ are the feature maps from the last convolution layer of AMG. $W_c$ and $W_b$ are the spatial average of the $F_c$ and $F_b$ after global average pooling (i.e., GAP in Figure \[fig:AMG\]). $P_c$ and $P_b$ are confidence scores of the predicted two class. They are generated by softmax from $W_c$ and $W_b$. The attention map is obtained by up-sampling the linear weighted fusion of the two feature maps $F_c$ and $F_b$ (i.e., $F_c \cdot P_c + F_b \cdot P_b$) to the same size as the input image. We also normalize the attention map such that all element values fall in the range $[0, 1]$. The attention map highlights the regions of crowds. In addition, it also indicates the degree of congestion in individual regions, i.e., higher congestion degree values indicate more congested crowds and lower values indicate less congested ones. Figure \[fig:ddlevel\] illustrates the effect of attention maps at different density levels. The pixel-wise product between the attention map and the input image produces the input data used by the DME network. ### Architecture of attention map generator The architecture of AMG is shown in Figure \[fig:AMG\], we use the first 10 layers of trained VGG-16 model [@simonyan2014very] as the front end to extract low-level features. We build the back end by adopting multiple dilated convolution layers of different dilation rates with an architecture similar to the inception module in [@szegedy2015going]. The multiple dilated convolution architecture is motivated from [@wei2018revisiting]. It has the capability of localizing people clusters with enlarged receptive fields. The inception module was originally proposed in [@szegedy2015going] to process and aggregate visual information of various scales. We use this module to deal with the diversified crowd distribution in congested scenes. {width="46.00000%"} \[fig:AMG\] Density Map Estimator --------------------- The DME network consists of two components: the front end and the back end. We remove the fully-connected layers of VGG-16 [@simonyan2014very] and leave 10 convolutional layers to as the front end of the DME. The back end is a multi-scale deformable convolution based CNN network [@dai2017deformable]. The architecture of DME is shown in Figure \[fig:DMEArchi\]. The front end uses the first 10 layers of trained VGG-16 model [@simonyan2014very] to extract low-level features. The back end uses multi-scale deformable convolutional layers with a structure similar to the inception module in [@szegedy2015going], which enables DME to cope with various occlusion, diversified crowd distribution, and the distortion caused by perspective view. ![Architecture of DME. The convolutional layers’ parameters are denoted as “Conv-(kernel size)-(number of filters)-(stride)”, max-pooling layers are conducted over a 2$\times$2 pixel window, with stride 2. The deformable convolutional layers’ parameters are denoted as [“Dconv-(kernel size)-(number of filters)-(stride)”]{}.[]{data-label="fig:DMEArchi"}](DME.pdf){width="46.00000%"} The deformable convolution scheme was originally proposed in [@dai2017deformable]. Beneficial from the adaptive (deformable) sampling location selection scheme, deformable convolution has shown its effectiveness on various tasks, such as object detection, in the wild environment. The deformable convolution treats the offsets of sampling locations as learning parameters. Rather than uniform sampling, the sampling locations in the deformable convolution can be adjusted and optimized via training (see Figure \[fig:dcov\_example\] for the deformed sampling points by the deformable convolution on an example form ShanghaiTech Part\_B dataset  [@zhang2016single]). Compared to the uniform sampling scheme, this kind of dynamic sampling scheme is more suitable for the crowd understanding problem of congested noisy scenes. We will show the comparative advantages of the deformable convolution in our experimental section. ![Illustration of the deformed sampling locations. Left: standard convolution; right: deformable convolution; top: activation units on the feature map; bottom: the sampling locations of the $3 \times 3$ filter.[]{data-label="fig:dcov_example"}](DCNN.pdf){width="40.00000%"} Experiments {#sec:exp} =========== Datasets and Settings --------------------- We evaluate on four challenging datasets for crowd counting: ShanghaiTech dataset [@zhang2016single], the UCF\_CC\_50 dataset [@idrees2013multi], the WorldExpo’10 dataset [@zhang2015cross], and the UCSD dataset [@chan2008privacy]. **ShanghaiTech dataset** [@zhang2016single]. The ShanghaiTech dataset contains 1,198 images with a total of 330,165 people. It is divided into two parts: Part\_A and Part\_B. Part\_A contains 482 pictures of congested scenes, in which 300 are used as training dataset and 182 are used as testing dataset; Part\_B contains 716 images of sparse scene, 400 of which are used as training dataset and 316 are used as testing dataset. **UCF\_CC\_50 dataset** [@idrees2013multi]. This dataset contains 50 images downloaded from the Internet. The number of persons per image ranges from 94 to 4543 with an average of 1280 individuals. It is a very challenging dataset with two problems: the limited number of the images and the large span in person count between images. We used 5-fold-cross-validation setting described in [@idrees2013multi]. **WorldExpo’10 dataset** [@zhang2015cross]. It contains 3980 from 5 different scenes. Among 3980 images, 3380 images are used as training dataset and the remaining 600 images are used as testing dataset. Region-of-Interest (ROI) regions are provided in this dataset. **UCSD dataset** [@chan2008privacy]. The UCSD dataset contains 2000 images in sparse scene. The dataset also provides ROI region information. We created the ground truth in the same way as we did for the WorldExpo’10 dataset. Since the size of each image is too small to support the generation of high-quality density maps, we therefore enlarge each image to 952$\times$632 size by bilinear interpolation. Among the 2000 images, 800 images were used as training dataset, and the rest were used as testing dataset. Region-of-Interest (ROI) regions are also provided in this dataset. We show a representative example for each crowd counting dataset in Figure \[fig:examples\]. These four crowd counting datasets have their own characteristics. In general, the scenes in ShanghaiTech Part\_A dataset are congested and noisy. Examples in ShanghaiTech Part\_B are noisy but not highly congested. The UCF\_CC\_50 dataset consists of extremely congested scenes which have hardly any background noises. Both WorldExpo’10 dataset and UCSD dataset provide example with sparse crowd scenes in the form of ROI regions. Scenes in the ROI regions of the WorldExpo’10 dataset are generally noisier than the only one scene in the UCSD dataset. Following [@sindagi2017generating; @li2018csrnet], we use the mean absolute error (MAE) and the mean square error (MSE) for quantitative evaluation of the estimated density maps. PSNR (Peak Signal-to-Noise Ratio) and SSIM [@wang2004image] are used to measure the quality of the generated density map. For fair comparison, we follow the measurement procedure in [@li2018csrnet] and resize the density map and ground truth to the size of the original input image by linear interpolation. {width="90.00000%"} Training -------- ### AMG Training Training data for the binary classification network AMG consists of two groups of samples: positive and negative samples. The positive samples are from the training sets of the four crowd counting datasets. The negative samples are 650 background images downloaded from the Internet. These negative samples are shared by the training of each individual dataset. These 650 negative samples contain various outdoor scenes where people appear, such as streets, squares, etc., ensuring that the biggest difference between positive sample and negative samples is whether the image contains people. Adam [@kingma2014adam] is selected as the optimization method with the learning rate at 1e-5 and Standard cross-entropy loss is used as the loss function. ### DME training We simply crop 9 patches from each image where each patch is 1/4 of the original image size. The first four patches contain four quarters of the image without overlapping. The other five patches are randomly cropped from the image. After that, we mirror the patches so that we double the training dataset. We generate the ground truth for DME training following the procedure in [@li2018csrnet]. We select Adam [@kingma2014adam] as the optimization method with the learning rate at 1e-5. As previous works [@zhang2016single; @sam2017switching; @li2018csrnet], we use the euclidean distance to measure the difference between the generating density map and ground truth and define the loss function as $$L(\Theta)=\frac{1}{2N}\sum_{i=1}^N ||F(X_i;\Theta)-F_i||_{2}^2\eqno(2),$$ where $N$ is the batch size, $F(X_i;\Theta)$ is the estimated density map generated by DME with the parameter$\Theta$, $X_i$ is the input image, and $F_i$ is the ground truth of $X_i$. Results and Analyses -------------------- In this section, we first study several alternative network design of . After that, we evaluate the overall performance of and compare it with previous state-of-the-art methods. ### Alternative study \[fig:DMEVSAMG\] **DME**. Our first study is to investigate the influence of the AMG network, we compared two network designs on all the four datasets. The first one named AMG-DME has the architecture shown in Figure \[fig:overviewArch\]. The other one named DME uses the only DME network. Our quantitative experimental results in Table \[tab:ablation\] show that AMG-DME is significantly superior than DME on the those datasets which are characteristic of noisy scenes: ShanghaiTech Part\_A, Part\_B and WorldExpo’10. In Figure \[fig:DMEVSAMG\], we illustrate two representative samples from the testing set of ShanghaiTech Part\_A. On the top example which contains a congested noisy scene, estimated people number of AMG-DME is 198 that is much closer to the groud truth 171 than that estimated by DME. From the density map in the 3rd column of Figure \[fig:DMEVSAMG\], we can see the trees in the distance have been recognized as people by the single DME model. However, AMG-DME does not suffer this problem due to the help from the AMG network. On the middle-row example containing a noisy and more congested scene, the performances of AMG-DME and DME agree with those on the top example. The comparison results indicate that AMG-DME is more effective than DME on those noisy examples. On the UCF\_CC\_50 dataset, AMG-DME has approximate performance (slightly higher AME but lower MSE) with DME. It may due to the fact most of examples in the UCF\_CC\_50 dataset have a large regions of congested crowds while rarely have background noises. On the UCSD dataset where scenes are neither congested nor noisy, both MSE and MAE of AMG-DME is slightly higher than DME. This might because the examples in the UCSD dataset have already provide the accurate information of ROI regions. The attention map generated by the AMG network may destroy the ROI regions, which degrades the performance of the DME network since some ROI regions may be erased from its input. **AMG-bAttn-DME**. Since the AMG network has shown its strength in coping with noise background of scenes, our second study is to explore if a hard binary attention mask is more effective than the soft attention employed by AMG-DME. We therefore set up an variant of AMG-DME called AMG-bAttn-DME in Table \[tab:ablation\]. AMG-bAttn-DME has the same architecture as AMG-DME while differing with AMG-DME on the attention map (i.e., the attention maps of AMG-bAttn-DME contain either 0 or 1, other than a floating point within $[0,1]$ in the attention maps of AMG-DME). We first conducted the experiments on the ShanghaiTech dataset to find out the optimal binarization threshold for AMG-bAttn-DME. We set three different threshold attention values,$\{0.2, 0.1, 0.0\}$, for the binarization of attention maps. The ROI regions are gradually enlarged with the decreasing the threshold values as shown in Figure \[fig:partA\_seg\]. The results shown in Table \[tab:sh\_thre\] indicates AMG-bAttn-DME with attention threshold of 0.1 achieved the best performance. We then evaluated AMG-bAttn-DME with this optimal attention threshold on the rest three datasets and reported the results in Table \[tab:ablation\]. It is observed that AMG-bAttn-DME is superior than AMG-DME only on ShanghaiTech Part\_A while AMG-DME outperforms AMG-bAttn-DME on all other datasets. It may be due to the AMG network can learn more accurate attention maps on ShanghaiTech Part\_A and the binarization process does not destroy too much information of the crowd regions. ----------- ------ ------- ----- ------ Threshold MAE MSE MAE MSE t = 0.2 68.0 104.1 9.2 17.8 t = 0.1 63.2 98.9 8.2 15.7 t = 0.0 63.2 100.6 8.6 15.0 ----------- ------ ------- ----- ------ : Performance of AMG-bAttn-DME under different binarization thresholds on the ShanghaiTech dataset.[]{data-label="tab:sh_thre"} {width="45.00000%"} \[fig:model\_AMG-attn-DME\] **AMG-attn-DME**. Complement to the above experiments, we stretched the design choice exploration to studying an alternative way of injecting the learned attention from the AMG network to the DME network. In our proposed architecture, the DME network directly takes the crowd images as input. An alternative architecture is to weigh intermediate the feature map of a certain layer of the DME network with the attention map from the AMG network. In our implementation, we inject the attention map into the output of the front end of the DME network as shown in Figure \[fig:model\_AMG-attn-DME\]. Following the same training procedures as those in Table \[tab:ablation\], this alternative architecture, named AMG-attn-DME, performs slightly worse than AMG-DME on the datasets with congested noisy scenes like ShanghaiTech Part\_A and ShanghaiTech Part\_B. This may be due to some non-crowd pixels in the attention map from the AMG network having an attention value of zero, which, during the injection, would make convolution features at those corresponding locations vanish, reducing the feature information learned by previous convolutional lays from the input. On the UCF\_CC\_50 dataset and UCSD dataset, AMG-attn-DME is worse than the the only DME network as AMG-bAttn-DME and AMG-DME. This is because the scenes of these two datasets have less noisy background, AMG-attn-DME may reduce the information of the ROI regions through the injected attention map. On the UCSD and WorldExpo’10 datasets, AMG-attn-DME achieved higher effectiveness. Maybe it is because the convolution feature vanishing problem has been alleviated by the black regions around the ROI regions in the input. ### Quantitative results In this section, we study the overall performance of and compare it with existing methods on each individual crowd counting dataset. **Comparison on MAE and MSE**. We first compare the variants of the proposed network with the state-of-the-art work CSRNet [@li2018csrnet] along with several previous methods including CP-CNN [@sindagi2017generating], MCNN [@zhang2016single], Cascaded-MTL [@sindagi2017cnn], Switching-CNN [@sam2017switching] on the ShanghaiTech dataset and the UCF\_CC\_50 dataset. These two datasets are characteristic of congested and/or noisy scenes. The comparison results were summarized in Table \[tab:SHTPA\]. On the ShanghaiTech dataset, two of our approach variants (AMG-DME) and (AMG-bAttn-DME) achieved better performances than existing approaches. The only DME network achieved the performance generally close to the state-of-the-art approach CSRNet [@li2018csrnet]. On the two relatively less challenging datasets WorldExpo’10 and UCSD, we compared with recent state-of-the-art recent approaches including Switching-CNN [@sam2017switching], MCNN [@zhang2016single], and CSRNet [@li2018csrnet]. The comparison results are shown in Table \[tab:UCSD\]. Our method achieved the best accuracy in scenes 1, 4, 5 as well as the best average accuracy on the WorldExpo’10 dataset . On the UCSD dataset, our DME model achieved the best accuracy on terms of both MAE and MSE. **Comparison on PSNR and SSIM.** To study the quality of the density maps generated by , another experiment was conducted on all the five datasets for both and the state-of-the-art method CSRNet [@li2018csrnet]. The comparison results are shown in Table \[tab:qulaityCSR\]. Our method outperforms CSRNet on all the five datasets. On UCF\_CC\_50 dataset, our method improves 7.03% on PSNR and 55.76% on SSIM. On USCD dataset, our method improves 31.81% on PSNR and 8.13% on SSIM. **[Evaluation on vehicle counting dataset.]{}** We conducted experiments on the TRANCOS [@guerrero2015extremely] dataset for vehicle counting to evaluate the generalization capability of the proposed approach. The positive samples for training are from the training set of TRANCOS [@guerrero2015extremely]. The negative samples use 250 background images downloaded from the Internet, including various road scenes without vehicle. As previous work CSRNet [@li2018csrnet], we use the Grid Average Mean Absolute Error (GAME) to measure the counting accuracy. The comparison results are shown in Table \[tab:TRANCOS\]. It clearly shows that the approach achieved the best performance at all levels of GAMEs. ### Qualitative results In this section, we further investigate the general performance of the proposed by qualitative results. We mainly compared with the state-of-the-art approach CSRNet [@li2018csrnet] which have demonstrated the best performance on the datasets including the ShanghaiTech, UCF\_CC\_50, the WorlExpo’10, and UCSD datasets. In general, CSRNet has a front-end and back-end architecture as the DME network of the proposed . It is empowered by a dilated convolution design in the back-end of its architecture. Apart from the additional AMG netwok, differs from CSRNet by two additional features in its DME network: 1) the multiple-scale convolution scheme different from the single scale scheme of CSRNet, and 2) the deformable sampling scheme different from the evenly fixed-offset sampling in the dilated convolution of CSRNet. Figure \[fig:Qualitative\] shows some qualitative comparisons between the proposed (the variant AMG-DME is used) and the state-of-the-art approach CSRNet [@li2018csrnet]. Through visualization, it is observed that CSRNet is much less effective on those examples with various noises than . We can see the evidence from the noise regions marked by red boxes of the 1st column where noises exist in the background regions, as well as the marked regions of the 3rd column where noises can be found in the crowd regions. This may be due to CSRNet directly takes the crowd image as input while the DME network of takes as the input the crowd information highlighted by its AMG network. On the example of the 2nd column where there is not much noise but a significantly non-uniform crowd distribution, also clearly outperforms CSRNet. This indicates that the multi-scale deformable convolution scheme in is more effective than the single-scale fixed-offset dilated convolution scheme in CSRNet. On the rightmost example of Figure \[fig:Qualitative\] which have highly occluded crowd regions (see the regions within the two green dotted bordered rectangle), only recognized part of the severely occluded crowd regions. It may because the AMG network of cannot highlight out the whole occluded crowd regions for the DME network. Nevertheless, still achieved better performance in terms of all the measurement parameters: estimated number, PSNR and SSIM. ![ From top to bottom: representative samples from the testing set of the ShanghaiTech dataset, ground truth density maps, estimated density maps generated by the state-of-the-art approach CSRNet [@li2018csrnet] and (AMG-DME) respectively. []{data-label="fig:Qualitative"}](quanlity_result.png){width="49.00000%"} Conclusion {#sec:conclusion} ========== We propose a convolutional neural network based architecture named for crowd understanding of congested noisy scenes. Benefiting from the multi-scale deformable convolutional layers and [[attention-aware]{}]{} training scheme, generally achieved more accurate crowd counting and density map estimation than existing methods by suppressing the problems caused by noises, occlusions, and diversified crowd distributions commonly presented in highly congested noisy environments. On four popular crowd counting datasets (ShanghaiTech, UCF\_CC\_50, WorldEXPO’10, UCSD) and an extra vehicle counting dataset TRANCOS, achieved significant improvements over recent state-of-the-art approaches. [^1]: The corresponding author is Hefeng Wu. This research is supported by the National Natural Science Foundation of China (Grant No. 91746204 and 61876045), and Opening Project of Guangdong Province Key Laboratory of Information Security Technology (Grant No. 2017B030314131).

--- abstract: | The Ring Learning-With-Errors (LWE) problem, whose security is based on hard ideal lattice problems, has proven to be a promising primitive with diverse applications in cryptography. There are however recent discoveries of faster algorithms for the principal ideal SVP problem, and attempts to generalize the attack to non-principal ideals. In this work, we study the LWE problem on group rings, and build cryptographic schemes based on this new primitive. One can regard the LWE on cyclotomic integers as a special case when the underlying group is cyclic, while our proposal utilizes non-commutative groups, which eliminates the weakness associated with the principal ideal lattices. In particular, we show how to build public key encryption schemes from dihedral group rings, which maintains the efficiency of the ring-LWE and improves its security. -LWE, Non-commutative group ring, Dihedral group ring author: - Qi Cheng - Jun Zhang - Jincheng Zhuang bibliography: - 'non-commutative.bib' title: 'LWE from Non-commutative Group Rings' --- Introduction ============ The LWE problem --------------- Regev [@Regev05] introduced the learning with errors (LWE) problem as a generalization of the classic learning parity with noise (LPN) problem. To be precise, let $q$ be a prime, $\mathbf{s} \in{{\mathbb F}}_q^n$ be a fixed private vector, $\mathbf{a_i}\in{{\mathbb F}}_q^n,1\leq i\leq m$ be randomly chosen, $e_i\in{{\mathbb F}}_q,1\leq i\leq m$ be chosen independently accordingly to an error distribution ${{\mathbb F}}_q\mapsto {{\mathbb R}}^+$, which is a discrete Gaussian distribution that centers around $ 0 $ with width $ q n^{-0.5 - \epsilon} $, and $b_i=\langle \mathbf{a_i},\mathbf{s} \rangle + e_i$. Given a list of pairs $({{\textbf{a}}}_i,b_i),1\leq i\leq m$, the LWE problem asks to solve for $\mathbf{s}$, and the LPN problem is the special case when $ q=2 $. Informally speaking, it is believed that LWE is hard in the sense that even though $ e_i $ tends to be small, when $ \mathbf{s} $ is hidden, $(\mathbf{a_i},b_i) $ can not be distinguished from a random vector in $ {{\mathbb F}}_q^{n+1} $. In fact, Regev [@Regev05] proved the hardness for certain parameters $q$ and error distributions by showing quantum reductions from approx-SVP and approx-SIVP problems for lattices. Later, Peikert [@Peikert09] showed a classical reduction from approx-SVP to the LWE problem under more restrictive constraints. Lyubashevsky, Peikert, and Regev [@LPR10] introduced an analogous version of standard LWE over rings, and coined it ring-LWE. Furthermore, they established the hardness of ring-LWE by showing the reduction from a certain ideal lattice problem to the ring-LWE problem. The cryptography systems based on ring-LWE are much more efficient in terms of key sizes and encryption and decryption complexity. However, the security of systems is based on conjecturally hard problems on ideal lattices rather than on general lattices. The LWE problem and ring-LWE problem have proven to be versatile primitives for cryptographic purposes. Besides many other schemes, these applications include public key encryption schemes proposed by Regev [@Regev05], Peikert and Waters [@PeikertW08], Peikert [@Peikert09], Lindner and Peikert [@LindnerP11], Stehl[é]{} and Steinfeld [@StehleS11], Micciancio and Peikert [@MicciancioP12]; identity-based encryption (IBE) schemes proposed by Gentry, Peikert, and Vaikuntanathan [@GentryPV08], Cash, Hofheinz, Kiltz, and Peikert [@CashHKP10], Agrawal, Boneh, and Boyen [@AgrawalBB10a; @AgrawalBB10b]; fully homomorphic encryption (FHE) schemes proposed by Brakerski and Vaikuntanathan [@BrakerskiV11a; @BrakerskiV11b], Brakerski, Gentry, and Vaikuntanathan [@BrakerskiGV12], Fan and Vercauteren [@FV12]. Our results ----------- The main contribution of the paper is to propose a general framework of generating LWE instances from group rings. In particular, we demonstrate our approach by generating LWE instances from dihedral group rings. Recall that given a finite group $G=\{g_1,\ldots,g_n \}$ and a commutative ring $R$, the elements in group ring $R[G]$ are formal sums $$\sum_{i=1}^n r_i g_i,r_i\in R.$$ If $ R ={{\mathbb Z}}$, and we provide a $ {{\mathbb Z}}$-module homomorphism from $ {{\mathbb Z}}[G] $ to $ {{\mathbb R}}^n $ (otherwise known as an embedding), then (one-side) ideals in group rings naturally correspond to integral lattices. We can generalize LWE to the group ring setting. In particular, let $ n $ be a power of two, $ D_{2n} $ be the dihedral group of order $ 2n $, and $ r\in D_{2n} $ be an element that generates the cyclic subgroup of order $ n $, then we should use the ring $${{\mathbb Z}}[D_{2n}]/( (r^{n/2}+1) {{\mathbb Z}}[D_{2n}]),$$ which is also a free $ {{\mathbb Z}}$-module of rank $ n $. Note that $ (r^{n/2}+1) {{\mathbb Z}}[D_{2n}] $ is a two-sided ideal, thus the quotient ring is well defined. In ring-LWE, there are two types of embeddings of rings of algebraic integers into Euclidean spaces: canonical embedding and coefficient embedding. If using canonical embedding, multiplication is component-wise. This is the main reason that the original ring-LWE paper preferred canonical embedding. Nevertheless, the whole ring is embedded as a lattice that is not self-dual, which complicates the implementation [@Peikert16]. Note that the canonical embedding of cyclotomic integers is basically the combined map: $${{\mathbb Z}}[x]/(x^n+1) \hookrightarrow {{\mathbb C}}[x]/(x^n +1) \rightarrow \bigoplus_{0\leq k\leq n, 2\nmid k}{{\mathbb C}}[x]/(x- e^{2\pi \sqrt{-1} k /(2n)}),$$ where the first map is an inclusion, and the second one is an isomorphism. A component of the canonical embedding of $ {{\mathbb Z}}[x]/(x^n+1) $ corresponds to a group representation of the cyclic group $ \langle x \rangle $ of order $ 2n $: $$\rho_k (x^j) = e^{2\pi \sqrt{-1} k j/(2n)}, 2\nmid k.$$ If a group is not commutative, we can use irreducible group representations to find a canonical embedding of the group ring. However, some irreducible representations will have dimensions larger than one, thus multiplication in the group ring is not component-wise under these representations. We should use coefficient embedding to make implementation simpler. There are recent discoveries of faster SVP algorithms for principal ideal lattices, and attempts to generalize the idea to non-principal ideal lattices. See [@CDPR16; @CDW16] and references therein. First observe that the ratio between two generators of a principal ideal is an integral unit. The main idea of the attacks comes from the Dirichlet unit theorem: the group of integral units in a number field is a direct product of a finite group with a free abelian group, whose generators are known as fundamental units. If taking logarithms of complex norms of their conjugates, the units are sent to the so-called log-unit lattice, whose SVP is not hard in many cases. Nevertheless, the ring-LWE cryptosystems are not under direct threat, since lattice problems in ideal lattices form lower bounds for their security, and the approximation factors in the attack are too large. The principal ideals from non-commutative integral group rings do not appear to suffer from the weakness, since multiplications of units may not commute [@Sehgal93]. A few remarks are in order: 1. The group ring LWE includes LWE on cyclotomic integers as a special case, thus has security no less than the ring-LWE. Indeed, the ring $R = {{\mathbb Z}}[x]/(x^n+1) $, used in many ring-LWE cryptosystems, is a direct summand of a group ring from $ C_{2n} $ ( the cyclic group of order $ 2n $ ): $${{\mathbb Z}}[C_{2n}] = {{\mathbb Z}}[x]/(x^{2n}-1) \equiv {{\mathbb Z}}[x]/(x^n+1) \oplus {{\mathbb Z}}[x]/(x^n-1)$$ One should avoid using the ring $ {{\mathbb Z}}[x]/(x^{2n}-1) $, as the map $${{\mathbb Z}}[x]/(x^{2n}-1) \rightarrow {{\mathbb Z}}[x]/(x-1)$$ may leak secret information. 2. We regard one-dimensional representations over finite fields as security risks that should be eliminated. Many attacks on the ring-LWE (implicitly) explores a one-dimensional representation that sends $ x $ to a small order element [@CLS15; @CLS16; @EHL14; @ELOS15], for example, $${{\mathbb F}}_q[x]/(f(x)) \rightarrow {{\mathbb F}}_q [x]/(x-1),$$ if $ (x-1) | f(x) $ over $ {{\mathbb F}}_q $. 3. Even though rings of algebraic integers in number fields may not be principal ideal domains (PID), their reductions modulo primes are always principal ideal rings. The group ring $ {{\mathbb F}}_p [G] $, however, is not necessarily a principal ideal rings if $ G $ is non-commutative. We believe that this property provides an extra protection against attacks. The proof of security is largely similar to the case of ring-LWE. There is, however, an important difference: unlike the ring of algebraic integers in a number field, group rings have ideals that are not invertible. The security of group-ring-LWE should be based on lattice problems of invertible ideals. We note that there have been attempts to use non-commutative algebraic structures, especially the group structures, in designing cryptographical systems [@MSU11]. The approaches that relate closely to ours include using group rings to replace $ ({{\mathbb Z}}/q{{\mathbb Z}})[x]/(x^n-1) $ in NTRU [@YDS15; @Coppersmith97; @Truman07] and using the learning problem of non-commutative groups. The former approach has no security proof from lattice problems. The latter approach is not based on lattice problems. Paper organization ------------------ The paper is organized as follows. In Section \[sec:math-prel\], we review the mathematical background. In Section \[sec:previous-works\] we briefly discuss previous works. In Section \[sec:pkc-from-dihedral\], we propose generating LWE instances from non-commutative group rings and establish public key cryptosystem from dihedral group rings. In Section \[sec:secur-analys-new\] we analyse the security of the new approach. Section \[sec:conclusion\] concludes the paper. We will not try to optimize the parameters in this paper, leaving it to future work. Mathematical preliminary {#sec:math-prel} ======================== In this section, we review the mathematical background on lattices and group rings. Efficiency of cryptographic schemes ----------------------------------- To use a cryptography algorithm, one should first establish a security level $ n $. It is expected that the cryptosystem cannot be broken in $ 2^n $ bit operations. In terms of efficiency, the most important parameters for an encryption algorithm are block size, public/secret key sizes, cipher-text expansion factor and time complexity per bit in encryption and decryption. Ideally these parameters should have sizes that grow slowly with the security level. Let us first calculate the parameters for the popular public key cryptosystem RSA, whose security is based on the integer factorization problem. To factor a number of $ l $ bits, the best algorithm – Number Field Sieve – takes heuristic time at most $ 2^{l^{1/3 + \epsilon}} $. Thus for security level $ n $, the RSA-OAEP system, a practical implementation of RSA, should have key size $ l = n^{3-\epsilon}$. To encrypt a block of $ O(l) $ bits, it adds some padding into the message and computes an exponentiation modulo a number of $ l $ bits. Thus it has cipher-text expansion $ O(1) $. The public exponent is small (e.g. $ e=65537 $), but the private exponent has $ l $ bits. Therefore, encryption takes time $ \tilde{O}(l) $ and decryption takes time $ \tilde{O}(l^2) $, assuming that we use the fast multiplication algorithm for each modular multiplication. This results in bit complexity $ n^{3-\epsilon} $ per ciphertext bit for decryption, and $ (\log n)^{O(1)} $ per message bit for encryption if using small encryption exponent. Asymptotically the key size for RSA is not so good. However, the $ \epsilon $ part has played an important role in its favor when $ n $ is small. To achieve a security level $ n= 80 $, one can use a public modulus of size $ 1000 $ bits rather than $ 80^3= 512000 $ bits, although a public modulus of $ 2000 $-bits is recommended now. Lattices and ring-LWE --------------------- Given a list of linearly independent column vectors ${{\mathbf B}}=({{\mathbf b}}_1,\ldots,{{\mathbf b}}_n)\in {{\mathbb R}}^{n\times n}$, the (full rank) lattice ${{\mathcal L}}({{\mathbf B}})$ is the set $${{\mathcal L}}({{\mathbf B}})=\left\{\sum_{i=1}^{n}x_i{{\mathbf b}}_i\,|\,x_i\in {{\mathbb Z}}\right\}.$$ The determinant of the lattice is $$\det({{\mathcal L}}):=|\det({{\mathbf B}})|.$$ The minimum distance of the lattice is $$\lambda_1({{\mathcal L}}):=\min_{0\neq v\in{{\mathcal L}}}||v||$$ where $||\cdot||$ is the Euclidean norm. The dual lattice is $${{\mathcal L}}^{*}:=\{u\in{{\mathbb R}}^n\,\vert\, \forall v\in{{\mathcal L}},\langle u,v \rangle\in {{\mathbb Z}}\}.$$ Let ${{\mathcal L}}\in{{\mathbb R}}^n$ be a full rank lattice. The Shortest Vector Problem (SVP) is to find a vector $v\in{{\mathcal L}}$ such that $$||v||=\lambda_1.$$ Given a target vector $t\in {{\mathbb R}}^n$, the Closest Vector Problem (CVP) is to find a vector $v\in{{\mathcal L}}$ such that $$||v-t||\leq ||v'-t||,\forall v'\in {{\mathcal L}}.$$ Let $ 0< \beta < 1/2 $ be a constant, and $ {{\mathcal L}}$ be a lattice. Let $ y = x +e $ where $ x\in {{\mathcal L}}$, and $ ||e|| < \beta \lambda_1 ({{\mathcal L}}) $. Given $ y $, the $\beta$-BDD problem is to find $ x $. Let $ 0< \beta < 1/2 $ be a constant, and $ {{\mathcal L}}$ be a lattice. Let $ y = x +e $ where $ x\in {{\mathcal L}}$, and $ ||e|| < \beta \lambda_1 ({{\mathcal L}}) $. Given $ y $, the $(q, \beta)$-BDD problem is to find any $ x' $ such that $ x \equiv x' \pmod{q {{\mathcal L}}} $. The $ \beta $-BDD problem can be reduced to $ (q, \beta) $-BDD problem. In fact, if $ x - x' \in q {{\mathcal L}}$, then $ (x-x')/q\in {{\mathcal L}}$. The distance between $ (y-x')/q $ and $ (x-x')/q $ is $ ||e/q|| $ . So we have a new BDD problem on the same lattice but with smaller error. Repeating the procedure will give us a BDD problem that can be solved by lattice reduction algorithms such as LLL. Dihedral groups and group rings ------------------------------- Let $ G =\{g_1,g_2,\ldots,g_n \}$ be a finite group of order $ n $. The elements in group ring $R[G]$ are formal sums $$\sum_{i=1}^nr_ig_i,r_i\in R.$$ Addition is defined by $$\sum_{i=1}^na_ig_i+\sum_{i=1}^nb_ig_i=\sum_{i=1}^n(a_i+b_i)g_i.$$ Multiplication is defined by $$\label{eq:mul} (\sum_{i=1}^na_ig_i)(\sum_{i=1}^nb_ig_i)=\sum_{l=1}^n(\sum_{g_ig_j=g_l}a_ib_j)g_l.$$ If $ R={{\mathbb Z}}$, a (one-side) ideal of $ {{\mathbb Z}}[G] $ is mapped to a lattice, under an embedding of $ {{\mathbb Z}}[G] $ to $ {{\mathbb R}}^n $. Here we use coefficient embedding, i.e. a group element is sent to a unit vector in $ {{\mathbb Z}}^n $. The whole group ring $ {{\mathbb Z}}[G] $ corresponds to $ {{\mathbb Z}}^n $. Denote the length of a group ring element $ X $ in the Euclidean norm under the embedding by $ ||X|| $. The following lemma shows that lengths of group ring elements behave nicely under multiplication. \[lem:smallmul\] Let $ X, Y \in {{\mathbb R}}[G] $ be two elements. Then $$||X Y|| \leq \sqrt{n} ||X||\cdot ||Y||$$ From Equation (\[eq:mul\]), the $ l_\infty $ norm of $ X Y $ is less than $ |X| |Y| $ by the Cauchy-Schwarz inequality. Next, we introduce a new norm of elements in the group ring $\mathbb{R}(G)$. For any element $\mathfrak{h}=\sum_{i=1}^{n}a_ig_i\in \mathbb{R}[G]$, by the multiplication law (1), it defines a linear transformation from $\mathbb{R}^n=\mathbb{R}[G]$ to itself, denoted by $A(\mathfrak{h})$. Indeed, it corresponds the regular representation of the finite group $G$. Then we define the matrix-norm $|\mathfrak{h}|_{\rm Mat}$ of $\mathfrak{h}$ to be the square root of the norm of the matrix $A(\mathfrak{h})A(\mathfrak{h})^T$, i.e., $$|\mathfrak{h}|_{\rm Mat}=\sqrt{{\rm Norm}(A(\mathfrak{h})A(\mathfrak{h})^T)}=\sqrt{{\rm Largest\,Eigenvalue\,of\,} A(\mathfrak{h})A(\mathfrak{h})^T}.$$ This definition should be the right definition for ring-LWE under any given embedding. In particular, if the transformation matrix $A$ is diagonal, then it reduces to the case $\ell_\infty$-norm used in [@LPR10] for caninocal embedding. Let $ I $ be a right ideal, the left dual of $ I $ is defined as $$I^{-1} = \{ x\in {{\mathbb Q}}[G] \mid \forall y\in I, x y \in {{\mathbb Z}}[G] \}$$ It can be verified that the left dual is a left $ {{\mathbb Z}}[G] $ module, and $$I \subseteq {{\mathbb Z}}[G] \subseteq I^{-1}.$$ We call an ideal invertible if $I^{-1} I = {{\mathbb Z}}[G] $. If $ I $ is invertible, then $ I^{-1} $ is a left fractional ideal, namely, there is an integer $ t $ such that $ t I^{-1} \subseteq {{\mathbb Z}}[G] $. A dihedral group of order $ 2n $, denoted by $ D_{2n} $, is the set $$\{ \mathfrak{r}^i \mathfrak{s}^j \mid 0\leq i\leq n-1, 0\leq j\leq 1 \}$$ satisfying the relations $$\mathfrak{r}^n = \mathfrak{s}^2 =1, \mathfrak{s}\mathfrak{r}\mathfrak{s}=\mathfrak{r}^{-1}.$$ In some sense, the dihedral group is the non-commutative group that is the closest to the commutative one, since the dimension of any irreducible representation is bounded by $ 2 $, while commutative groups only have one-dimensional irreducible representations. If $ n $ is odd, there are $ (n+1)/2 $ irreducible representations for $ D_{2n} $. Two of them are one-dimensional: $$\rho_0 (\mathfrak{r}^i) =1, \rho_0(\mathfrak{s} \mathfrak{r}^j) = 1$$ and $$\rho_1 (\mathfrak{r}^i) =1, \rho_1(\mathfrak{s} \mathfrak{r}^j) = -1.$$ The rest are two-dimensional: for $ 2 \leq k\leq (n+1)/2 $, $$\begin{aligned} \rho_k (\mathfrak{r}^i) &= \begin{pmatrix} e^{2\pi \sqrt{-1} i (k-1)/n} & 0 \\ 0 & e^{-2\pi \sqrt{-1} i (k-1)/n} \end{pmatrix},\\ \rho_k (\mathfrak{s} \mathfrak{r}^i) &= \begin{pmatrix} 0 & e^{2\pi \sqrt{-1} i (k-1) /n} \\ e^{-2\pi \sqrt{-1} i (k-1)/n} & 0 \end{pmatrix}. \end{aligned}$$ By the Wedderburn theorem, the group ring $ {{\mathbb C}}[D_{2n}] $ can be decomposed into $${{\mathbb C}}[D_{2n}] \equiv {{\mathbb C}}\oplus {{\mathbb C}}\oplus \bigoplus_{i=2}^{ (n+1)/2} {{\mathbb C}}^{2\times 2},$$ where the first two copies of $ {{\mathbb C}}$ correspond to $ \rho_0 $ and $ \rho_1 $, the last $ (n-1)/2 $ copies of $ 2\times 2 $ matrix algebras corresponds to the two-dimensional representations $ \rho_i $ ($2 \leq k\leq (n+1)/2 $ ). To guarantee the hardness results of ring-LWE based on the group ring of dihedral group, we need to study the matrix-norm of any element in $\mathbb{R}(D_{2n})$. \[eigenvalue\] For any element $\mathfrak{h}=f(\mathfrak{r})+\mathfrak{s}g(\mathfrak{r})\in \mathbb{R}[D_{2n}]$ where $$f(x)=\sum_{i=0}^{n-1}a_ix^i {\rm \ and \ } g(x)=\sum_{i=0}^{n-1}b_ix^i$$ are two polynomials over $\mathbb{R}$. Then the eigenvalues of the matrix $A(\mathfrak{h})\cdot A(\mathfrak{h})^T$ are $(|f(\xi^i)|\pm |g(\xi^i)|)^2$ for $i=0,1,\cdots,n-1,$ where $\xi=e^{2\pi \sqrt{-1}/n}$ is the $n$-th root of unity and $|*|$ is the complex norm. So the matrix-norm of $\mathfrak{h}$ is bounded from above by $\max\{|f(\xi^i)|+ |g(\xi^i)|\,|\,i=0,1,\cdots,n-1\}$. By representation theory, we have decomposition of the regular representation $\rho_{\rm reg}$: $$\begin{CD} \mathbb{C}[D_{2n}] @>\rho_{\rm reg}>> \mathbb{C}[D_{2n}]\\ @V\cong V\psi V @V\cong V\psi V\\ \oplus_i {\rm dim}(V_i)V_i @>\rho_{\rm bd}>> \oplus_i {\rm dim}(V_i)V_i, \end{CD}$$ where $V_i$ runs over all irreducible representations of $D_{2n}$ such that $\rho_{\rm bd}(g) \,(\forall g\in D_{2n})$ is block-diagonal. One can show the isomorphism $\psi$ is unitary, i.e., $\psi\cdot\bar{\psi}^T=I_{2n}$. Then $$\begin{aligned} A(\mathfrak{h})\cdot A(\mathfrak{h})^T=\rho_{\rm reg}(\mathfrak{h})\cdot \overline{\rho_{\rm reg}(\mathfrak{h})}^T =(\psi^{-1}\cdot\rho_{\rm bd}(\mathfrak{h})\cdot\psi)\cdot \overline{(\psi^{-1}\cdot\rho_{\rm bd}(\mathfrak{h})\cdot\psi)}^T\\ =\psi^{-1}\cdot\rho_{\rm bd}(\mathfrak{h})\cdot\psi\cdot\bar{\psi}^T\cdot\overline{\rho_{\rm bd}(\mathfrak{h})}^T\cdot\bar{\psi}^{-T}=\psi^{-1}\cdot\rho_{\rm bd}(\mathfrak{h})\cdot\overline{\rho_{\rm bd}(\mathfrak{h})}^T\cdot\bar{\psi}^{-T}. \end{aligned}$$ So $A(\mathfrak{h})\cdot A(\mathfrak{h})^T$ have the same eigenvalues as $\rho_{\rm bd}(\mathfrak{h})\cdot\overline{\rho_{\rm bd}(\mathfrak{h})}^T$. Moreover, $\rho_{\rm bd}(\mathfrak{h})\cdot\overline{\rho_{\rm bd}(\mathfrak{h})}^T$ is block-diagonal with blocks of size at most $2\times 2$. By direct computation of eigenvalues of each block, it is easy to obtain the eigenvalues of $\rho_{\rm bd}(\mathfrak{h})\cdot\overline{\rho_{\rm bd}(\mathfrak{h})}^T$ are $(|f(\xi^i)|\pm |g(\xi^i)|)^2$ for $i=0,1,\cdots,n-1$. And hence eigenvalues of the matrix $A(\mathfrak{h})\cdot A(\mathfrak{h})^T$ are $(|f(\xi^i)|\pm |g(\xi^i)|)^2$ for $i=0,1,\cdots,n-1$. For any invertible (right) ideal $I$ of $\mathbb{Z}[D_{2n}]$, let $I^{-1}$ be the left inverse of $I$. Let $\Lambda$ and $\Lambda^{-1}$ be the lattices defined by coefficients embedding of $I$ and $I^{-1}$ respectively. Then $\Lambda^*$ and $\Lambda^{-1}$ are the same under a permutation of coordinates. For any $(x_0,x_1,\cdots,x_{n-1})\in \mathbb{Q}^n$, let $$(z_0,z_1,\cdots,z_{n-1})=(x_0,x_{n-1},x_{n-2}\cdots,x_{1}).$$ We claim that $$(x_0,x_1,\cdots,x_{n-1},y_0,y_1,\cdots,y_{n-1})\in \Lambda^{-1}$$ if and only if $$(z_0,z_1,\cdots,z_{n-1},y_0,y_1,\cdots,y_{n-1})\in \Lambda^{*}.$$ And hence, we finish the proof. On one hand, if $\sum_{i=0}^{n-1}x_i \mathfrak{r}^i+\sum_{j=0}^{n-1}y_j\mathfrak{s}\mathfrak{r}^j\in I^{-1}$, then $$(\sum_{i=0}^{n-1}x_i \mathfrak{r}^i+\sum_{j=0}^{n-1}y_j\mathfrak{s}\mathfrak{r}^j)(\sum_{k=0}^{n-1}w_k \mathfrak{r}^k+\sum_{l=0}^{n-1}v_l\mathfrak{s}\mathfrak{r}^l)\in \mathbb{Z}[D_{2n}]$$ for any $\sum_{k=0}^{n-1}w_k \mathfrak{r}^k+\sum_{l=0}^{n-1}v_l\mathfrak{s}\mathfrak{r}^l\in I$. Expending the product, this is equivalent to that for any $a,b=0,1,\cdots,n-1$, $$\sum_{i=0}^{n-1}x_{i}w_{a-i\mod n}+\sum_{j=0}^{n-1}y_jv_{a+j\mod n}\in \mathbb{Z},$$ and $$\sum_{i=0}^{n-1}x_{i}v_{b+i\mod n}+\sum_{j=0}^{n-1}y_jw_{b-j\mod n}\in \mathbb{Z}.$$ So $\sum_{i=0}^{n-1}x_i \mathfrak{r}^i+\sum_{j=0}^{n-1}y_j\mathfrak{s}\mathfrak{r}^j\in I^{-1}$ if and only if for any $\sum_{k=0}^{n-1}w_k \mathfrak{r}^k+\sum_{l=0}^{n-1}v_l\mathfrak{s}\mathfrak{r}^l\in I$ and for any $a,b=0,1,\cdots,n-1$, $$\sum_{i=0}^{n-1}z_{i}w_{a+i\mod n}+\sum_{j=0}^{n-1}y_jv_{a+j\mod n}\in \mathbb{Z},$$ and $$\sum_{i=0}^{n-1}z_{i}v_{b-i\mod n}+\sum_{j=0}^{n-1}y_jw_{b-j\mod n}\in \mathbb{Z}.$$ On the other hand, we have $$(z_0,z_1,\cdots,z_{n-1},y_0,y_1,\cdots,y_{n-1})\in \Lambda^{*}$$ if and only if for any $\sum_{k=0}^{n-1}w_k \mathfrak{r}^k+\sum_{l=0}^{n-1}v_l\mathfrak{s}\mathfrak{r}^l\in I$, $$\sum_{i=0}^{n-1}z_iw_i+\sum_{j=0}^{n-1}y_jv_j\in \mathbb{Z}.$$ Note that $I$ is a right ideal of $\mathbb{Z}[D_{2n}]$, so for any $a,b=0,1,\cdots,n-1$, $$(\sum_{k=0}^{n-1}w_k \mathfrak{r}^k+\sum_{l=0}^{n-1}v_l\mathfrak{s}\mathfrak{r}^l)\mathfrak{r}^{-a}=\sum_{k=0}^{n-1}w_{k+a\mod n} \mathfrak{r}^k+\sum_{l=0}^{n-1}v_{l+a\mod n}\mathfrak{s}\mathfrak{r}^l\in I$$ and $$(\sum_{k=0}^{n-1}w_k \mathfrak{r}^k+\sum_{l=0}^{n-1}v_l\mathfrak{s}\mathfrak{r}^l)\mathfrak{s}\mathfrak{r}^{b}=\sum_{k=0}^{n-1}v_{b-k} \mathfrak{r}^k+\sum_{l=0}^{n-1}w_{b-k}\mathfrak{s}\mathfrak{r}^l\in I.$$ So we have $$(z_0,z_1,\cdots,z_{n-1},y_0,y_1,\cdots,y_{n-1})\in \Lambda^{*}$$ if and only if for any $\sum_{k=0}^{n-1}w_k \mathfrak{r}^k+\sum_{l=0}^{n-1}v_l\mathfrak{s}\mathfrak{r}^l\in I$ for any $a,b=0,1,\cdots,n-1$, $$\sum_{i=0}^{n-1}z_{i}w_{a+i\mod n}+\sum_{j=0}^{n-1}y_jv_{a+j\mod n}\in \mathbb{Z},$$ and $$\sum_{i=0}^{n-1}z_{i}v_{b-i\mod n}+\sum_{j=0}^{n-1}y_jw_{b-j\mod n}\in \mathbb{Z}.$$ So the claim is proved. To eliminate the influence of one-dimensional representations, one can let $ n $ be a prime, and use the direct summand of the ring $ {{\mathbb Z}}[D_{2n}] $: $${{\mathbb Z}}[D_{2n}]/ ((\mathfrak{r}^{n-1} + \mathfrak{r}^{n-2} + \cdots + 1) {{\mathbb Z}}[D_{2n}]).$$ Note that $ (\mathfrak{r}^{n-1} +\mathfrak{r}^{n-2} + \cdots + 1) {{\mathbb Z}}[D_{2n}] $ is a two-sided ideal, so the above ring is well defined, and it can be regarded as a projection of $ {{\mathbb Z}}[D_{2n}] $ to $ \bigoplus_{i=2}^{ (n+1)/2} {{\mathbb C}}^{2\times 2} $. In this paper we assume that $ n $ is a power of two, and let $${\mathbf R} = {{\mathbb Z}}[D_{2n}]/ ((\mathfrak{r}^{n/2}+1) {{\mathbb Z}}[D_{2n}]),$$ which is also without one-dimensional component. Denote $${\mathbf R}_\mathbb{R}={\mathbf R}\otimes_{\mathbb{Z}} \mathbb{R}$$ which is $\mathbb{R}^n$ under coefficients embedding, and let $ \mathbb{T}={\mathbf R}_\mathbb{R}/{\mathbf R}$. Let $ q $ be a prime such that $ \gcd(q,2n)=1 $. Define $${\mathbf R}_q = {{\mathbb F}}_q [D_{2n}]/ ((\mathfrak{r}^{n/2} + 1) {{\mathbb F}}_q [D_{2n}]).$$ Let $ \chi_{\alpha_1, \alpha_2, \cdots, \alpha_n} $ be a Gauss distribution in $ {{\mathbb R}}^n $ such that $$\chi_{\alpha_1, \alpha_2, \cdots, \alpha_n} (x_1, x_2, \cdots, x_n) = e^{-\pi ((x_1/\alpha_1)^2 + (x_2/\alpha_2)^2 + \cdots + (x_n/\alpha_n)^2)}$$ Let $\Psi_{\leq \alpha}$ be the set of all the Gaussian distributions $ \chi_{ \alpha_1, \alpha_2, \cdots, \alpha_n} $ such that $\alpha_i \leq \alpha$ for all $ 1\leq i \leq n $ . The $ {\mathbf R}_q $-LWE problem is to find the secret $ s \in {\mathbf R}_q$, given a sequence of $ (a_i, b_i)\in {\mathbf R}_q \times \mathbb{T}$, where $ a_i $ is selected uniformly and independently from $ {\mathbf R}_q $, $ b_i = (a_i s)/q + e_i \mod {\mathbf R}$, $ e_i $ is selected independently according to some fixed distribution $ \chi\in \Psi_{\leq \alpha} $. Not every ideal is invertible. For example, $ 1+\mathfrak{s} \in {\mathbf R} $ generates an ideal that is not invertible. It is very important to have an ideal that is invertible in order to have hard lattice problems. In the later proof, we need an onto $ {\mathbf R} $-module morphism $ I \rightarrow {\mathbf R}_q $, which requires $ I $ to be invertible. The element $ \sum_{0\leq i\leq (n/2)-1} a_i \mathfrak{r}^i + \sum_{0\leq i\leq (n/2)-1} b_i \mathfrak{s} \mathfrak{r}^i \in {\mathbf{R}}$ is invertible in $ {\mathbf R} \otimes {{\mathbb Q}}$ iff for all odd $1 \leq k\leq n/2 $, $$| \sum_{0\leq i\leq (n/2)-1} a_i e^{ 2\pi \sqrt{-1} k i/n} | - | \sum_{0\leq i\leq (n/2)-1} b_i e^{ 2\pi \sqrt{-1} k i/n} | \not= 0,$$ where $ |*| $ is the complex norm. It is easy from Lemma \[eigenvalue\]. Previous works {#sec:previous-works} ============== Lattice-based cryptography has attracted much attention recently. It has a few advantages over classical number theoretic cryptosystems such as RSA or Diffie-Hellman. First, it resists quantum attacks, in contrast to the traditional hard problems such as integer factorization, or discrete logarithms [@Shor94]. Second, it enjoys the worst case to the average case reduction, shown in the pioneering work of Ajtai [@Ajtai96]. Third, computation can be done on small numbers. No large number exponentiations are needed, which tend to slow down the other public key cryptosystems. It does have a major drawback in key sizes. The NTRU cryptosystem [@HoffsteinPS98] is the first successful cryptosystem based on lattices. Regev’s scheme -------------- Regev [@Regev05] introduced the Learning With Errors (LWE) problem as a generalization of the classic learning parity with noise (LPN) problem to higher moduli and proposed a public key encryption system based on the LWE problem. In the following description of Regev’s scheme, $ n $ is the security parameter, $ q \in [n^2, 2n^2] $ is a prime number and $ m = O( n\log q) , \alpha = o(\frac{1}{\sqrt{n}\log n})$. The distribution $\Psi_\alpha = \chi_\alpha \pmod{{{\mathbb Z}}}$ is defined to be a normal distribution on $ R/{{\mathbb Z}}$ with mean $0$ and standard deviation $\frac{\alpha}{\sqrt{2\pi}}$. And $\bar{\Psi}_\alpha$ is the discrete distribution of the random variable $\lfloor q\cdot {{\mathbf X}}\rceil \mod q$ over ${{\mathbb F}}_q$, where $a \mod b = a - \lfloor a/b \rfloor b $ and ${{\mathbf X}}$ is from the distribution $\Psi_{\alpha}$. - [**Private key:**]{} Choose a random $ \mathbf{s}\in ({{\mathbb Z}}/q{{\mathbb Z}})^n $ uniformly. - [**Public key:**]{} Choose a random matrix $ {{\mathbf A}}\in ({{\mathbb Z}}/q{{\mathbb Z}})^{n\times m } $ uniformly. Choose an error vector $\mathbf{x} $ from $ ({{\mathbb Z}}/q{{\mathbb Z}})^m $, where each component of $\mathbf{x}$ is chosen according to the distribution $ \bar{\Psi}_{\alpha} $. Announce the public key $({{\mathbf A}}, {{\mathbf P}}) $ where $ {{\mathbf P}}\in ({{\mathbb Z}}/q{{\mathbb Z}})^m $ should be calculated as $ \mathbf{s} {{\mathbf A}}+ \mathbf{x} $. - [**Encryption:**]{} First select a random vector $ \mathbf{e}^{T} \in \{0,1 \}^m $. For a message bit $ v \in \{0, 1\} $, the encryption is $ ({{\mathbf A}}\mathbf{e}, v \lfloor \frac{q}{2} \rfloor + {{\mathbf P}}\mathbf{e} ) $. - [**Decryption:**]{} For the cipher-text $ (\mathbf{a},b) $, output $ 0 $ if $ b - \langle \mathbf{a},\mathbf{s} \rangle $ is closer to $ 0 $ than to $ q/2 $; Otherwise de-crypt to $ 1 $. For security level $ n $, the private key has $ \tilde{O}(n) $ bits. The public key has $ \tilde{O}(n^2) $ bits, and can be reduced to $ \tilde{O}(n) $. The cipher-text expansion is $ \tilde{O}( n) $. The encoding and decoding complexity is $ \tilde{O}( n^2) $ per bit. Hence this system is not efficient, especially in terms of cipher-text expansion and encryption/decryption complexity. To find the private key from the public key, one can solve a CVP problem in the lattice $ {{\mathcal L}}= \{ v {{\mathbf A}}\mid v \in ({{\mathbb Z}}/q{{\mathbb Z}})^n \} $, which is a sub-lattice of $ q {{\mathbb Z}}^m $. Note that $ q^{m-n} \mid \det({{\mathcal L}}). $ The shortest vector of $ {{\mathcal L}}$ has length $ \tilde{O}(q\sqrt{m}). $ This means that the secret key is likely unique. PVW improvement --------------- Peikert, Vaikuntanathan, and Waters [@PVW08] proposed a more efficient system based on LWE. They made two important changes: first the secret and the error in the public key are matrices, and the message space consists of vectors; secondly the alphabet of the message is $ {{\mathbb Z}}/p{{\mathbb Z}}$ for some $ p $ that may be greater than $ 2 $. The latter idea has also been utilized by Kawachi, Tanaka, and Xagawa [@KTX07] to improve the efficiency of several single-bit cryptosystems based on lattice problems. Suppose that $ p = poly(n) $, $ l = poly(n) $, $m=O(n\log n)$, $\alpha = 1/(p\sqrt{m}\log n)$ and $ q>p $ is a prime. Let $ t $ be a function from $ {{\mathbb Z}}/p{{\mathbb Z}}$ to $ {{\mathbb Z}}/q{{\mathbb Z}}$ defined by $ t(x) = [ x \times \frac{q}{p} ] $ and extended to act component-wise on vector spaces over $ {{\mathbb Z}}/p{{\mathbb Z}}$. - [**Private key:**]{} Choose a random matrix $ {{\mathbf S}}\in ({{\mathbb Z}}/q{{\mathbb Z}})^{n\times l} $ uniformly. - [**Public key:**]{} Choose a random matrix $ {{\mathbf A}}\in ({{\mathbb Z}}/q{{\mathbb Z}})^{n\times m } $ uniformly. Find an error matrix $ {{\mathbf X}}\in ({{\mathbb Z}}/q{{\mathbb Z}})^{ l\times m } $ where each entry is chosen independently according to the error distribution $ \chi = \bar{\Psi}_{\alpha} $. The public key is $ ({{\mathbf A}}, {{\mathbf P}}) $ where $ {{\mathbf P}}={{\mathbf S}}^T {{\mathbf A}}+ {{\mathbf X}}\in ({{\mathbb Z}}/q{{\mathbb Z}})^{l \times m } $. - [**Encryption:**]{} The message is assumed to be a vector $ \mathbf{v} \in ({{\mathbb Z}}/p{{\mathbb Z}})^l $. First convert it to a vector $ t(\mathbf{v}) $ in $ ({{\mathbb Z}}/q{{\mathbb Z}})^l $. Then select $ \mathbf{e}^T \in \{0,1\}^m $ uniformly at random. The encryption is $ ({{\mathbf A}}\mathbf{e}, {{\mathbf P}}\mathbf{e} + t(\mathbf{v})) \in ({{\mathbb Z}}/q{{\mathbb Z}})^n \times ({{\mathbb Z}}/q{{\mathbb Z}})^l $. - [**Decryption:**]{} For the cipher-text $ (\mathbf{u},\mathbf{c}) $, compute $ \mathbf{d} = \mathbf{c} - {{\mathbf S}}^T \mathbf{u} $, and output $ \mathbf{v} \in ({{\mathbb Z}}/p{{\mathbb Z}})^l $, where $ v_i $ is the element in $ {{\mathbb Z}}/p{{\mathbb Z}}$ that makes $ d_i - t(v_i) $ closest to $ 0 \pmod{q} $. Note that one may set $ l=n $ in the cryptosystem. In this case, the public key size and secure key size are $ \tilde{O}(n^2) $. The algorithm has cipher-text expansion $ O(1) $. The encryption and decryption complexity is $ \tilde{O}(n) $ per bit. The security of the cryptosystem comes from the fact that if $ {{\mathbf S}}$ is hidden, the public key $ ({{\mathbf A}},{{\mathbf P}}) $ is computationally indistinguishable from uniform distribution over $({{\mathbb Z}}/q{{\mathbb Z}})^{n\times m } \times ({{\mathbb Z}}/q{{\mathbb Z}})^{l \times m } $, for suitable parameters, under the hypothesis that LWE is hard. PKC based on ideal lattices --------------------------- To improve the efficiency of the LWE-based system, Lyubashevsky, Peikert, and Regev [@LPR10] proposed the primitive of ring-LWE. Let $R = {{\mathbb Z}}[x]/(x^n+1) $, where $ n $ is a power of two. Let $ R_q = ({{\mathbb Z}}/q{{\mathbb Z}})[x]/(x^n+1) $. - [**Private key:**]{} The private key is $ s,e \in R_q $ from an error distribution. - [**Public key:**]{} Select a random $ a \in R_q $ uniformly. Output $ (a, b) \in R_q^2 $, where $ b = as + e $. - [**Encryption:**]{} To encrypt a bit string $ z $ of length $ n $, we view it as an element in $ R_q $ so that bits in $ z $ become coefficients of a polynomial. The cipher-text is $ (u,v) $ obtained by $$u = a r + e_1, v = b r + e_2 + \lfloor q/2 \rfloor z,$$ where $ r, e_1, e_2 $ are chosen from an error distribution. - [**Decryption:**]{} For cipher-text $ (u,v) $, computes $ v - us $, which equals $$(r e - s e_1 - e_2) + \lfloor q/2 \rfloor z.$$ One can read $ z $ from $ v-us $, since $ r, e, e_1 $ and $ e_2 $ have small coefficients. The algorithm is very efficient. Public and private key size is $ \tilde{O}(n) $. Cipher-text expansion is $ O(1) $, and encryption/decryption complexity per bit is $ (\log n)^{O(1)} $, assuming that we use the fast multiplication algorithm. The parameters are optimal asymptotically, however, the security is based on approx-SVP of ideal lattices, rather than general lattices. PKC from dihedral group rings {#sec:pkc-from-dihedral} ============================== In this section, we describe a cryptosystem based on the dihedral group ring. The protocol is identical to one based on the ideal lattice, except that since multiplication is not commutative, one needs to pay attention to the order of multiplication. The discretization $\bar{\chi}\,:\,{{\mathbb Z}}/q{{\mathbb Z}}\rightarrow \mathbb{R}$ of a Gaussian $\chi$ on $\mathbb{R}$. First, reduce $\chi$ by modulo ${{\mathbb Z}}$ to obtain a distribution $\chi\mod{{\mathbb Z}}$ on $[0,1)$. Then divide $[0,1)$ into $q$ parts $[1-1/2q,1)\cup [0,1/2q)$, $[1/2q,3/2q),\,\cdots,$ and $[1-3/2q,1-1/2q)$, and integrate the distribution ($\chi\mod{{\mathbb Z}}$) on each part to define $\bar{\chi}(0),\bar{\chi}(1),\cdots,\bar{\chi}(q-1)$. Let $ n $ be a power of two, let $ q $ be a prime such that $ \gcd(q,2n)=1 $, and $ q \in [n^2, 2n^2] $. Recall $${\mathbf R} = {{\mathbb Z}}[D_{2n}]/ ((\mathfrak{r}^{n/2}+1) {{\mathbb Z}}[D_{2n}]),$$ $${\mathbf R}_\mathbb{R} = \mathbb{R}[D_{2n}]/ ((\mathfrak{r}^{n/2}+1) \mathbb{R}[D_{2n}]),$$ $${\mathbf R}_q = {{\mathbb F}}_q [D_{2n}]/ ((\mathfrak{r}^{n/2}+1) {{\mathbb F}}_q [D_{2n}]),$$ and the error distribution $ \bar{\chi} $ on $ {\mathbf R}_q $ is to select coefficients independently according to the discretization of a Guassian of width $ \tilde{O}(1/\sqrt{n}) $. - [**Private key:**]{} The private key is $ s,e \in {\mathbf R}_q$ from the error distribution. - [**Public key:**]{} Select a random $ a \in {\mathbf R}_q $ uniformly. Output $ (a, b) \in {\mathbf R}_q^2 $, where $ b = s a + e $. - [**Encryption:**]{} To encrypt a bit string $ z $ of length $ n $, we view it as an element in $ {\mathbf R}_q $ so that bits in $ z $ become coefficients of a polynomial. The cipher-text is $ (u,v) $ obtained by $$u = a r + e_1, v = b r + e_2 + \lfloor q/2 \rfloor z,$$ where $ r, e_1, e_2 $ are chosen from an error distribution. - [**Decryption:**]{} For cipher-text $ (u,v) $, one computes $ v - s u $, which equals $$(r e - s e_1 - e_2) + \lfloor q/2 \rfloor z.$$ One can read $ z $ from $ v-us $, since $ r, e, e_1 $ and $ e_2 $ have small coefficients. One can verify that the public and private key sizes are linear in the security level, and the ciphertext expansion is almost a constant. The following theorem shows that the encryption/decryption complexity is logarithmic per bit. The multiplication in $ ({{\mathbb Z}}/q {{\mathbb Z}}) [D_{2n}] $ can be done in $ \tilde{O} (n \log q) $ time. In this theorem, we use the whole group ring for generality. One can check that it applies to $ \mathbf{R} $ as well. The main idea is to separate the terms in $ ({{\mathbb Z}}/q{{\mathbb Z}})[D_{2n}] $ into two parts. Let $ f_1 + \mathfrak{s} f_2 $ and $ f_3+\mathfrak{s} f_4 $ be two elements where $ f_1, f_2, f_3 $ and $ f_4 $ are polynomials in $ \mathfrak{r} $. We have $$\begin{aligned} & (f_1 + \mathfrak{s} f_2 ) (f_3 + \mathfrak{s} f_4)\\ =& f_1 f_3 + \mathfrak{s} f_2 f_3 + f_1 \mathfrak{s} f_4 + \mathfrak{s} f_2 \mathfrak{s} f_4\\ =& f_1 f_3 + \mathfrak{s} f_2 f_3 + \mathfrak{s} (\mathfrak{s} f_1 \mathfrak{s}) f_4 + (\mathfrak{s} f_2 \mathfrak{s}) f_4\\ =& (f_1 f_3 + (\mathfrak{s} f_2 \mathfrak{s}) f_4 ) + \mathfrak{s} ( f_2 f_3 + (\mathfrak{s} f_1 \mathfrak{s}) f_4 )\end{aligned}$$ where $ \mathfrak{s}f_1\mathfrak{s} $ and $ \mathfrak{s} f_2 \mathfrak{s} $ are polynomials in $ \mathfrak{r} $ that can be calculated in linear time. To find the product, we need to compute four polynomial multiplications in $ ({{\mathbb Z}}/q{{\mathbb Z}})[\mathfrak{r}] $, that can be done in time $ \tilde{O} (n \log q) $. In the normal version of group ring LWE, $ s $ and $ e $ are selected according to error distribution, while in the regular version, only $ e $ is selected according to error distribution. The following theorem shows that these two versions are equivalent. The regular version of dihedral GR-LWE can be reduced to the normal version of dihedral GR-LWE. (Sketch) Suppose that the input of the LWE problem is $ (a_1,b_1) $ and $ (a_2, b_2) $. With high probability, $ a_1 $ is invertible, we construct the input for normal version of LWE as $$(a_2 a_1^{-1}, a_2 a_1^{-1} b_1- b_2).$$ Note that $$a_2 a_1^{-1} b_1- b_2 = a_2 a_1^{-1} (a_1 s + e_1) - (a_2 s + e_2) = a_2 a_1^{-1} e_1 - e_2.$$ Security analysis of the group ring LWE {#sec:secur-analys-new} ======================================= In this section, we prove the main theorem \[MainTheorem\] Let $\alpha=\alpha(n)\in (0,1)$, and let $q=q(n)$ be a prime such that $\alpha q\geq \sqrt{n}\omega(\sqrt{\log n})$. Given an average case of search version of dihedral GR-LWE$_{q,\Psi_{\leq \alpha}}$ oracle with error distributions $\Psi_{\leq \alpha}$, there is a quantum polynomial time algorithm that solves the search version of the SVP problem for any invertible ideal $I$ of $ {\mathbf R} $ with approximate factor $ \tilde{O}({n}/\alpha) $. It is from Lemmas \[iter1\] and \[iter2\] that with dihedral GR-LWE$_{q,\Psi_{\leq \alpha}}$ oracle one can sample a discrete Gaussian on the ideal $I$ of width $\lambda_n\sqrt{n}\omega(\log n)/\alpha$, starting with a sufficiently large value of width $r\geq 2^{2n}\lambda_n(I)$ where any polynomial number of samples can be generated classically [@Regev05]. As a sample from the discrete Gaussian has Euclidean length at most $\sqrt{n}\cdot\lambda_n(I)\sqrt{n}\omega(\log n)/\alpha$ with overwhelming probability. So the sample solves the search version of the SVP problem for the ideal $I$ with approximate factor $ \tilde{O}({n}/\alpha) $. Let us first review the main ideas in Regev’s reduction from approx-SVP to LWE, which inspires our proof. The reduction can be divided into iterative steps. We will solve the [*Discrete Gaussian Sampling*]{} problem (DGS) for a lattice, that has a comparable hardness as approx-SVP. The DGS$_{{{\mathcal L}}, r}$ problem is to sample lattice points of a lattice $ {{\mathcal L}}$ according to a Gaussian centering at $ O $ with width $ r $. For precise definition, see [@Regev05]. The DGS will be reduced, by a quantum algorithm, to a $ \beta $-BDD problem on its dual lattice $ {{\mathcal L}}^* $, which will then be reduced to a $ (q,\beta) $-BDD problem. The $ (q,\beta) $-BDD will be reduced to a DGS problem of larger width. This step needs help from the search LWE oracle. After a few iterations, we arrive at DGS with a width that allows a polynomial time algorithm. The only step that needs an LWE oracle is the reduction from $ (q,\beta) $-BDD to DGS. Suppose we have a $ (q, \beta) $-BDD instance $ y ( = x +e) $, where $ x\in {{\mathcal L}}^* $ and $ ||e|| \leq \lambda_1 ({{\mathcal L}}^*) \beta $. We wish to find $ x \pmod{q {{\mathcal L}}^*} $. We are able to sample a random element $ z\in {{\mathcal L}}$ by the DGS algorithm, such that $ ||z|| \leq m/ \lambda_1({{\mathcal L}}^*) $, where $ m \geq q \sqrt{n} $. So we have $$m/\lambda_1({{\mathcal L}}^*) \geq q \sqrt{n } /\lambda_1({{\mathcal L}}^*) \approx q \eta( {{\mathcal L}}) = \eta(q {{\mathcal L}}),$$ where $ \eta ( * ) $ is the smoothing parameter of a lattice. Let $ a $ be $ z \mod q {{\mathcal L}}$. Then $ a $ is a random element in $ {{\mathbb F}}_q^n $ by the definition of a smoothing parameter. We compute $ a $ by writing down the coefficients of $ z $ in the base $ B $ and modulo them by $ q $. There is a map from $ {{\mathbb F}}_q^n $ to $ {{\mathcal L}}\pmod{q {{\mathcal L}}} $ given by the base matrix $ {{\mathbf B}}$, such that $ \psi {{\mathbf B}}=1 $, where $ \psi $ is a map in the $ {{\mathbb Z}}$-module exact sequence: $$0 \rightarrow q {{\mathcal L}}\rightarrow {{\mathcal L}}\stackrel{\psi}{\rightarrow} {{\mathbb F}}_q^n \rightarrow 0$$ Note that the map given by $ {{\mathbf B}}$ is not a $ {{\mathbb Z}}$-module homomorphism, since the exact sequence is not splitting. Let $ b= z(x+e)^T = z x^T + z e^T \pmod{ q {{\mathbb Z}}}$, and $ s = x {{\mathbf B}}^T $. Note that $ |z e^T|_\infty \leq m \beta $, and $ z x^T = a {{\mathbf B}}x^T = a {{\mathbf B}}(s ({{\mathbf B}}^{-1})^T)^T = a s $. Call the search LWE oracle, we will get $ s $, which gives us $ x \pmod{q {{\mathcal L}}^*} $, and completes the reduction. We can see that working with the dual lattice is very important. Here the transformation by $ {{\mathbf B}}$ is important. We can not just mod $ z $ by $ q {{\mathbb Z}}^n $, since it may be the case that $ {{\mathcal L}}\subseteq q {{\mathbb Z}}^n $, or $ {{\mathcal L}}$ is not even an integral lattice. For LWE on the ring $ R = {{\mathbb Z}}[x]/(x^n+1) $, the idea is similar. Any ideal in the number field $ {{\mathbb Q}}[x]/(x^n+1) $ is a $ {{\mathbb Z}}$-module thus corresponds to a lattice if we provide an embedding. There are two ways of embedding: canonical and coefficient. If we use canonical embedding, then the dual is $ I^{\vee} $ [@LPR10], instead of $ I^{-1} $. To keep the multiplicative structure of the ring, we need a $ {R} $-module isomorphism from $ I/(q I) $ to $ {R}/(q {R}) = {{\mathbb F}}_q[x]/(f(x)) $, and from $ I^\vee/(q I^\vee) $ to $ {R}^\vee/(q {R}^\vee) = {{\mathbb F}}_q[x]/(f(x)) $, so we can recover $ I^{\vee}/(q I^{\vee}) $ from a polynomial in $ {R}/(q {R}) $. As pointed out in [@LPR10], it is important to clear ideals while preserving the $ R $-module structure. Let $ R={{\mathbb Z}}$, $ q =5 $ and $ I = (3) $. Suppose that $ z = 24 \in I $, $ z \pmod{qI} $ should be $ 9 $ in the parallelepiped $ [0,15) $. Dividing by $ t=3 $, we send $ z $ to $ 3 $ in $ {{\mathbb Z}}/q{{\mathbb Z}}$. Hence multiplying by $ 3 $ is a ${{\mathbb Z}}$-module isomorphism from $ {{\mathbb Z}}/5{{\mathbb Z}}$ to $ I/ 5I $. On the other hand, ${{\mathbb Z}}$-module isomorphism is not unique. If we can just use the inclusion $ I \hookrightarrow R $, we have $ z = 4 \pmod{5} $. This is another $ {{\mathbb Z}}$-module isomorphism. If $ \psi: I \rightarrow R $ is a R-module isomorphism, so is $ t \psi $ for any $ t\in R $. To complete the reduction, one needs to send an element in $ {{\mathbb Z}}/5{{\mathbb Z}}$ back to $ I^{-1}/5 I^{-1} $. Here $ I^{-1} = (1/3){{\mathbb Z}}$. One can see that the inclusion $ {{\mathbb Z}}\subseteq I^{-1} $ induces an isomorphism $ {{\mathbb Z}}/5{{\mathbb Z}}\rightarrow I^{-1}/5I^{-1} $. Now we will extend the idea to non-commutative group ring LWE. We should use coefficient embedding to map ideals to lattices. In the following discussion, we will use the same symbol for an ideal and its corresponding lattice under coefficient embedding. The precise error distribution in the definition of ring-LWE to ensure the hardness result is one important issue. In [@LPR10], the authors generalized one dimensional Gaussian error distribution in plain-LWE [@Regev05] to $n$-dimensional (ellipitcal) Gaussian which is described by an $n\times n$-covariance matrix. However, in [@LPR10] they chose the canonical embedding which makes the Gaussian error distributions during the reduction always diagonal. In our case, the error distributions in the reduction do not appear as diagonal any more. Let $L$ be a lattice, let $u\in \mathbb{R}^n$ be a vector, let $r,s>0$ be two reals, let $A\in \mathbb{R}^{n\times n}$ be a non-singular matrix. Assume that smooth property $\sum_{y\in L^*\setminus \{0\}}\exp(-\pi y^T(\frac{1}{r^2}I_n+\frac{1}{s^2}A^T\cdot A)^{-1}y) \leq \epsilon$ holds for some $\epsilon$, where $I_n$ denotes the $n\times n$ identity matrix. The distribution of $Av+e$ where $v$ is distributed according to $DGS_{L+u,r}$ and $e$ is the $n$ dimensional Gaussian multivariable with mean vector $0$ and diagonal covariance matrix $\frac{s^2}{2\pi}I_{n}$ is within statistical distance $4\epsilon$ of a Gaussian multivariable with mean vector $0$ and covariance matrix $\frac{r^2}{2\pi}A\cdot A^T+\frac{s^2}{2\pi}I_n$. Note that non-singular linear transformation of Gausssian multivariable is still Gaussian, and $Av+e=A(v+A^{-1}e)$. Let $Y=v+A^{-1}e$. One can directly compute the distribution of $Y$ $$Y(x)=\frac{\exp(-\pi x^T\Sigma^{-1}x)}{\det(\Sigma)^{1/2}}\frac{\sum_{y\in L^*}e^{-2\pi \sqrt{-1}<c_0,y>}\exp(-\pi y^T(\frac{1}{r^2}I_n+\frac{1}{s^2}A^T\cdot A)^{-1}y)}{\sum_{y\in L^*}e^{2\pi \sqrt{-1}<u,y>}\exp(-\pi r^2 ||y||^2)}$$ where $\Sigma=r^2I_n+s^2A^{-1}\cdot A^{-T}$ and $c_0$ is a certain vector computed from $u$ and $x$. Since we have $$\begin{aligned} &|1-\sum_{y\in L^*}e^{-2\pi \sqrt{-1}<c_0,y>}\exp(-\pi y^T(\frac{1}{r^2}I_n+\frac{1}{s^2}A^T\cdot A)^{-1}y)|\\ \leq& \sum_{y\in L^*\setminus \{0\}}\exp(-\pi y^T(\frac{1}{r^2}I_n+\frac{1}{s^2}A^T\cdot A)^{-1}y)\\ \leq & \epsilon \end{aligned}$$ and $$\begin{aligned} & |1-\sum_{y\in L^*}e^{2\pi \sqrt{-1}<u,y>}\exp(-\pi r^2 ||y||^2)|\\ \leq & |\sum_{y\in L^*\setminus \{0\}}\exp(-\pi r^2 ||y||^2)|\\ \leq& \sum_{y\in L^*\setminus \{0\}}\exp(-\pi y^T(\frac{1}{r^2}I_n+\frac{1}{s^2}A^T\cdot A)^{-1}y)\\ \leq & \epsilon, \end{aligned}$$ we immediately have $$\begin{aligned} |Y(x)-\frac{1}{\det(\Sigma)^{1/2}}\exp(-\pi x^T\Sigma^{-1}x)|\leq 4\epsilon. \end{aligned}$$ So by integrating over $\mathbb{R}^n$, the statistical distance between $Y=v+A^{-1}e$ and the Gaussian distribution $\frac{1}{\det(\Sigma)^{n/2}}\exp(-\pi x^T\Sigma^{-1}x)$ is at most $4\epsilon$. Finally, since non-singular linear transformation of Gausssian multivariable is still Gaussian, $Av+e=AY$ has statistical distance at most $4\epsilon$ with the Gaussian distribution with mean vector $0$ and covariance matrix $$\frac{1}{2\pi}A\Sigma A^T=\frac{1}{2\pi}(r^2A\cdot A^T+s^2I_n).$$ - If the transformation matrix $A$ is diagonal, then it reduces to the case in [@LPR10]. - The proof relies on the invertibility of the matrix $A$. In the application to BDD problem, the errors in BDD are invertible with very high probability except a zero-measure set. Applying the above lemma to the group ring considered in this paper, together with Lemma \[eigenvalue\], the following corollay is immediate. \[ErrorInReducion\] Let $L$ be the ideal lattice obtained by coefficients embedding of $I\subset \mathbf{R}$ to $\mathbb{R}^{n}$. Let $\mathfrak{h}=f(\mathfrak{r})+\mathfrak{s}g(\mathfrak{r})\in \mathbf{R}_\mathbb{R}$ for some polynomials of degree at most $\frac{n}{2}-1$ over $\mathbb{R}$, and let $\lambda=|\mathfrak{h}|_{\rm Mat}$. Let $r,s>0$ be two reals, denote $t=1/\sqrt{\frac{1}{r^2}+\frac{\lambda^2}{s^2}}$. Assume that smooth property $\sum_{y\in L^*\setminus \{0\}}\exp(-\pi t^2||y||^2)\leq \epsilon$ holds for some $\epsilon$. The distribution of $\mathfrak{h}v+e$ where $v$ is distributed according to $DGS_{L,r}$ and $e$ is the $n$ dimensional Gaussian multivariable with mean vector $0$ and diagonal covariance matrix $\frac{s^2}{2\pi}I_{n}$ is within statistical distance $4\epsilon$ of a Gaussian multivariable that is equivalent to the diagonal Gaussian $$\prod_i \chi_{\sqrt{r^2(|f(\xi^i)|+ |g(\xi^i)|)^2+s^2}}\times \prod_i \chi_{\sqrt{r^2(|f(\xi^i)|- |g(\xi^i)|)^2+s^2}}$$ up to certain unitary base change. Now we can prove the first part of the iteration algorithm in our scenario. \[iter1\]\[First part of iteration\] Let $\alpha=\alpha(n)\in (0,1)$, prime $q=q(n)>2$, let $I$ be a right ideal of $\mathbf{R}$ and integer $r>0$ such that $$\sum_{y\in I^{-1}\setminus \{0\}}\exp(-\pi \frac{r^2}{2q^2}||y||^2)\leq \epsilon$$ for some negligible $\epsilon=\epsilon(n)$. There is a probabilistic polynomial time classical reduction from BDD$_{I^{-1},\alpha q/\sqrt{2}r}$ in the matrix norm to GR-LWE$_{q,\Psi_{\leq \alpha}}$. Suppose $ y=x+\mathfrak{h}\in \mathfrak{h} +I^{-1} $, where the error $\mathfrak{h}$ has matrix-norm $\leq q/\sqrt{2}r$. We want to recover $x$. We sample a $ v \in I $ according to the Gaussian distribution $DGS_{I,r}$, and let $ a = \phi_1 (v) \pmod{q {\mathbf R}} \in {\mathbf R}/(q {\mathbf R}) $, where $\phi_1 $ is the inclusion $ I \rightarrow {\mathbf R} $, which is also a left ${\mathbf R}$-module homomorphism. Note that $ q {\mathbf R} $ is a two-sided ideal, $ {\mathbf R} /q {\mathbf R}$ is a direct summand of the ring $ {{\mathbb F}}_q [D_{2n}] $. Since $ det(I)$ is not divisible by $q$, $\phi_1$ induces a natural left ${\mathbf R}$-module surjective homomorphism $ I \rightarrow {\mathbf R} / (q {\mathbf R})$. We then calculate $ b= y v + e$ (in $ {{\mathbb R}}_{\mathbf R} $), where $e$ is a Gaussian $\chi_{\alpha/\sqrt{2}}$ on ${{\mathbb R}}_{\mathbf R}$. We have $ b \equiv y v = x v+ \mathfrak{h} v+e \pmod{q {\mathbf R}}$, where $ x v \in {\mathbf R} $ and the distribution of $ \mathfrak{h} v+e $ has statistic distance within $4\epsilon$ to the Gaussian $\prod_i \chi_{\sqrt{(r/q)^2(|f(\xi^i)|+ |g(\xi^i)|)^2+(\alpha/\sqrt{2})^2}}\times \prod_i \chi_{\sqrt{(r/q)^2(|f(\xi^i)|- |g(\xi^i)|)^2+(\alpha/\sqrt{2})^2}}$ by Corollary \[ErrorInReducion\]. We generate several instances of $ (a,b) $, and send them to the GR-LWE$_{q,\Phi_{\leq \alpha}}$ oracle. Then the oracle answers $ s $ in $ {\mathbf R}/q {\mathbf R}$, as long as $$\sqrt{(r/q)^2|\mathfrak{h}|_{\rm Mat}^2+(\alpha/\sqrt{2})^2}\leq \alpha, \mbox{\,or\,}|\mathfrak{h}|_{\rm Mat}\leq \alpha q/\sqrt{2}r.$$ Let $\phi_2 $ be the inclusion $ {\mathbf R} \rightarrow I^{-1} $, which is also a right ${\mathbf R}$-module homomorphism. It induces a natural right module homomorphism $ I^{-1} \rightarrow {\mathbf R}/ (q {\mathbf R}) $, since $ q \nmid det(I) $. So pulling $s$ back along the homomorphism gives us the residue class of $x \pmod{q I^{-1}} $. \[GaussBall\] If $\mathfrak{h}=f(\mathfrak{r})+\mathfrak{s}g(\mathfrak{r})\in \mathbf{R}_\mathbb{R}$ is taken from the Gaussian distribution $\chi_\sigma$, then $\mathfrak{h}$ has matrix-norm at most $\sigma \sqrt{n} \omega(\sqrt{\log n})$ except with negligible probability. Let $\theta=2\pi/n$ and $\xi=e^{\theta\sqrt{-1}}$. By Lemma \[eigenvalue\], the eigenvaules of $A(\mathfrak{h})A(\mathfrak{h})^T$ is contained in $\{(|f(\xi^i)|\pm |g(\xi^i)|)^2\,|\,i\neq 0,n/2\}$ as $\xi^0=1,\xi^{n/2}=-1$ appear in the one dimensional irreducible representations. So $$|\mathfrak{h}|_{\rm Mat}\leq \max_{i=1}^{n/2-1}\{|f(\xi^i)|+ |g(\xi^i)|\}.$$ Next, we give an upper bound for $|f(\xi^i)|$ and $|g(\xi^i)|$ for any $i=1,2,\cdots,n/2-1$. We can rewrite $$|f(\xi^i)|=\sqrt{\left(\sum_{j=0}^{n/2-1}a_j\cos(j i \theta)\right)^2+\left(\sum_{j=0}^{n/2-1}a_j\sin(j i \theta)\right)^2}.$$ Since $a_0,a_1,\cdots,a_{n/2-1}$ are independently distributed from Gaussian $\chi_\sigma$, the sum $\sum_{j=0}^{n/2-1}\cos(j i \theta)a_j$ is Gaussian $\chi_{\sqrt{\sum_{j=0}^{n/2-1}\cos^2(j i \theta)}\cdot\sigma}$. Because $i=1,2,\cdots,n/2-1$, we have $$\sum_{j=0}^{n/2-1}\cos^2(j i \theta)=\frac{n}{2}+\frac{1}{2}\sum_{j=0}^{n/2-1}\cos(j 2 i \theta)=\frac{n}{2}+\frac{1}{2}{\rm Re}(\sum_{j=0}^{n/2-1}e^{j 2 i \theta\sqrt{-1}})=\frac{n}{2}.$$ So the sum $\sum_{j=0}^{n/2-1}\cos(j i \theta)a_j$ is one dimensional Gaussian $\chi_{\sqrt{n/2}\cdot\sigma}$. It is well-known that a sample from $\chi_{\sqrt{n/2}\cdot\sigma}$ has length at most $\omega(\sqrt{\log n})\sqrt{n}\cdot\sigma$ except with negligible probability. Similarly, the sum $\sum_{j=0}^{n/2-1}a_j\sin(j i \theta)$ is bounded by $\omega(\sqrt{\log n})\sqrt{n}\cdot\sigma$ except with negligible probability. And hence, $|f(\xi^i)|$ is bounded by $\omega(\sqrt{\log n})\sqrt{n}\cdot\sigma$ except with negligible probability. By the same reason, $|g(\xi^i)|$ is bounded by $\omega(\sqrt{\log n})\sqrt{n}\cdot\sigma$ except with negligible probability. Then the lemma is proved. The second (quantum) part of the iteration algorithm in [@Regev05] was improved by [@LPR10] using BDD for error distributed from a Gaussian. By the above lemma, samples from a Gaussian $\chi_{d/\sqrt{2n}}$ are distributed in the ball $B_{d\omega(\sqrt{\log n})}$ under the matrix norm except with a negligible probability. So it is enough to have a BDD oracle which can solve errors of matrix-norm $\leq d\omega(\sqrt{\log n})$. \[iter2\]\[Second part of iteration\] There is an efficient quantum algorithm that, given any $n$-dimensional lattice $\Lambda$, a number $d<\lambda_1(\Lambda^{*})/2$ (here, $\lambda_1$ is under Euclidean norm), and an oracle that solves BDD$_{\Lambda^*,d\omega(\sqrt{\log n})}$ in matrix-norm, outputs a sample from $DGS_{\Lambda, \sqrt{n}/d}$. Conclusion {#sec:conclusion} ========== We propose generating LWE instances from non-commutative group rings and illustrate the approach by presenting a public key scheme based on dihedral group rings. We believe that LWE on dihedral group rings achieves the right trade-off between security and efficiency. As with the original LWE and ring-LWE, we hope that the new approach is a versatile primitive, so we can build various cryptographic schemes based on this primitive besides public-key encryption. There is one open problem that we find very interesting: Can we generalize the approach to other non-commutative groups and keep the efficiency of ring-LWE?

--- abstract: 'The spin structure of the system of quasifree fermions having total angular momentum $J=1/2$ is studied in a consistently covariant approach. Within this model the relations between the spin functions are obtained. Their particular cases are the sum rules Wanzura - Wilczek, Efremov - Leader - Teryaev, Burkhardt - Cottingham and also the expression for the Wanzura - Wilczek twist 2 term $g_{2}^{WW}$. With the use of the proton valence quark distributions as an input, the corresponding spin functions are obtained. The resulting structure functions $g_{1}$ and $g_{2}$ are well compatible with the experimental data. Comparison with the basic formulas following from the standard quark-parton model reveals the importance of the quark intrinsic motion inside the target for the correct evaluation of the spin structure functions.' address: | Institute of Physics, Academy of Sciences of the Czech Republic,\ Na Slovance 2, CZ-182 21 Prague 8\ E-mail: zavada@fzu.cz\ author: - Petr Závada title: 'Proton spin structure and intrinsic motion of the constituents [^1]' --- Introduction ============ In this talk some results following from the covariant quark-parton model (QPM) will be shortly discussed, details of the model can be found in Refs. [@zav4], [@zav5]. In this version of QPM valence quarks are considered as quasifree fermions on mass shell. Momenta distributions describing the quark intrinsic motion have spherical symmetry corresponding to the constraint $J=1/2$, which represents the total angular momentum - generally consisting of spin and orbital parts. I shall mention the following items: 1\. What sum rules follow from this approach for the spin structure functions $g_{1}$ and $g_{2}$? 2\. How can these structure functions be obtained from the valence quark distributions $u_{V}$ and $d_{V}$ - if the $\ SU(6)$ symmetry is assumed? The results are compared with the existing experimental data. 3\. Why the first moment $\Gamma _{1}$ calculated in this approach can be substantially less, than the corresponding moment calculated within the standard, non covariant QPM, which is based on the infinite momentum frame? Recently, this model was generalized to include also the transversity distribution, for details I refer to [@tra]. Model ===== The model is based on the set of distribution functions $G_{k,\lambda }(% \frac{pP}{M})$, which measure probability to find a quark in the state:$$u\left( p,\lambda {\bf n}\right) =\frac{1}{\sqrt{N}}\left( \begin{array}{c} \phi _{\lambda {\bf n}} \\ \frac{{\bf p}{\bf \sigma }}{p_{0}+m}\phi _{\lambda {\bf n}}% \end{array}% \right) ;\qquad \frac{1}{2}{\bf n\sigma }\phi _{\lambda {\bf n}}=\lambda \phi _{\lambda {\bf n}},\qquad \lambda =\pm \frac{1}{2},$$where$\ {\bf n}$ coincides with the direction of the proton polarization $% {\bf J}$. Correspondingly, $m$ and $p$ are quark mass and momentum, similarly $M$ and $P$ for the proton.[** **]{} With the use of these distribution functions one can define the function $H$, which in the target rest frame reads:$$H(p_{0})=\sum_{k=1}^{3}e_{k}^{2}\Delta G_{k}(p_{0});\qquad \Delta G_{k}(p_{0})=G_{k,+1/2}(p_{0})-G_{k,-1/2}(p_{0}), \label{t4}$$where $e_{k}$ represent the charges of the proton valence quarks. In the paper [@zav4] I shown, how from the generic function $H$ the spin structure functions can be obtained. If one assume $Q^{2}\gg 4M^{2}x^{2},$ then:$$g_{1}(x)=\frac{1}{2}\int H(p_{0})\left( m+p_{1}+\frac{p_{1}^{2}}{p_{0}+m}% \right) \delta \left( \frac{p_{0}+p_{1}}{M}-x\right) \frac{d^{3}p}{p_{0}}% ;\quad x=\frac{Q^{2}}{2M\nu },$$$$g_{2}(x)=-\frac{1}{2}\int H(p_{0})\left( p_{1}+\frac{p_{1}^{2}-p_{T}^{2}/2}{% p_{0}+m}\right) \delta \left( \frac{p_{0}+p_{1}}{M}-x\right) \frac{d^{3}p}{% p_{0}},$$which implies$$g_{T}(x)\equiv g_{1}(x)+g_{2}(x)=\frac{1}{2}\int H(p_{0})\left( m+\frac{% p_{T}^{2}/2}{p_{0}+m}\right) \delta \left( \frac{p_{0}+p_{1}}{M}-x\right) \frac{d^{3}p}{p_{0}}.$$Let me remark, that procedure for obtaining the functions $g_{1},g_{2}$ from the distribution $H$ is rather complex, nevertheless the task is well-defined and unambiguous. Resulting structure functions are related to a naive QPM, in which the relativistic kinematics and spheric symmetry (which follows from the requirement $J=1/2$) are consistently applied. Both these requirements are very important. Sum rules ========= One can observe, that the functions above have the same general form$$\int H(p_{0})f(p_{0},p_{1},p_{T})\delta \left( \frac{p_{0}+p_{1}}{M}% -x\right) d^{3}p \label{s1}$$and differ only in kinematic term $\ f$. This integral, due to spheric symmetry and presence of the $\delta -$function term, can be expressed as a combination of the momenta: $$V_{n}(x)=\int H(p_{0})\left( \frac{p_{0}}{M}\right) ^{n}\delta \left( \frac{% p_{0}+p_{1}}{M}-x\right) d^{3}p. \label{s2}$$One can prove [@zav5], that these functions satisfy$$\frac{V_{j}^{\prime }(x)}{V_{k}^{\prime }(x)}=\left( \frac{x}{2}+\frac{% x_{0}^{2}}{2x}\right) ^{j-k};\qquad x_{0}=\frac{m}{M}.$$This relation then gives possibility to obtain integral relations between different functions having form (\[s2\]) or (\[s1\]), in particular for $% g_{1}(x)$ and $g_{2}(x)$ one gets: $$g_{2}(x)=-\frac{x-x_{0}}{x}g_{1}(x)+\frac{x\left( x+2x_{0}\right) }{\left( x+x_{0}\right) ^{2}}\int_{x}^{1}\frac{y^{2}-x_{0}^{2}}{y^{3}}g_{1}(y)dy,$$$$g_{1}(x)=-\frac{x}{x-x_{0}}g_{2}(x)-\frac{x+2x_{0}}{x^{2}-x_{0}^{2}}% \int_{x}^{1}g_{2}(y)dy$$and for limiting case $m\rightarrow 0$:$$g_{2}(x)=-g_{1}(x)+\int_{x}^{1}\frac{g_{1}(y)}{y}dy,$$$$g_{1}(x)=-g_{2}(x)-\frac{1}{x}\int_{x}^{1}g_{2}(y)dy.$$Obviously, the first relation is the known expression for Wanzura - Wilczek twist-2 term for $g_{2}$ approximation [@wawi]. Further, if one define $$\left\langle x^{\alpha }\right\rangle =\int_{0}^{1}x^{\alpha }V_{0}(x)dx,$$then one can prove that $$\int_{0}^{1}x^{\alpha }\left[ g_{1}(x)+g_{2}(x)\right] dx=\left\langle x^{\alpha }\right\rangle \frac{\alpha +1}{\left( \alpha +2\right) \left( \alpha +3\right) },$$$$\int_{0}^{1}x^{\alpha }g_{2}(x)dx=-\left\langle x^{\alpha }\right\rangle \frac{\alpha \left( \alpha +1\right) }{\left( \alpha +2\right) \left( \alpha +3\right) }$$for [*any*]{} $\alpha $, for which the integrals exist. Apparently these relations imply$$\int_{0}^{1}x^{\alpha }\left[ \frac{\alpha }{\alpha +1}g_{1}(x)+g_{2}(x)% \right] dx=0,$$which for $\alpha =2,4,6,...$ corresponds to the Wanzura - Wilczek sum rules [@wawi]. Other special cases correspond to the Burkhardt - Cottingham ($% \alpha =0$)  [@buco] and the Efremov - Leader - Teryaev (ELT, $\alpha =1 $) [@elt] sum rules. Let me point out, that all these rules here were obtained only on the basis of covariant kinematics and requirement of rotational symmetry. Valence quarks ============== Now I shall try to apply the suggested approach to the description of the real proton. For simplicity I assume: 1\) Spin contribution from the sea of quark-antiquark pairs and gluons can be neglected, so the proton spin is generated only by the valence quarks. 2\) In accordance with the non-relativistic [*SU(6)*]{} approach, the spin contribution of individual valence terms is given by fractions:$$s_{u}=4/3,\qquad s_{d}=-1/3. \label{t69}$$If the symbols $h_{u}$ and $h_{d}$ denote momenta distributions of the valence quarks in the proton rest frame, which are normalized as$$\frac{1}{2}\int h_{u}(p_{0})d^{3}p=\int h_{d}(p_{0})d^{3}p=1, \label{t70}$$then the generic distribution (\[t4\]) reads$$H(p_{0})=\sum e_{j}^{2}\Delta h_{j}(p_{0})=\left( \frac{2}{3}\right) ^{2}% \frac{2}{3}h_{u}(p_{0})-\left( \frac{1}{3}\right) ^{2}\frac{1}{3}% h_{d}(p_{0}). \label{t71}$$ In the paper [@zav1], using a similar approach, I studied also the unpolarized structure functions. Structure function $F_{2}$ can be expressed as$$F_{2}(x)=x^{2}\int G(p_{0})\frac{M}{p_{0}}\delta \left( \frac{p_{0}+p_{1}}{M}% -x\right) d^{3}p;\qquad G(p_{0})=\sum_{q}e_{q}^{2}h_{q}(p_{0}). \label{t72}$$On the other hand, for proton valence quarks one can write$$F_{2}(x)=\frac{4}{9}xu_{V}(x)+\frac{1}{9}xd_{V}(x), \label{t73}$$so combination of the last two relations gives:$$q_{V}(x)=x\int h_{q}(p_{0})\frac{M}{p_{0}}\delta \left( \frac{p_{0}+p_{1}}{M}% -x\right) d^{3}p;\qquad q=u,d. \label{t74}$$Since this is again the integral having the structure (\[s1\]), one can apply the technique of integral transforms and (instead of relation between $% g_{1}$ and $g_{2}$) obtain the relations between $g_{j}^{q}$ and $q_{V}$. For $m\rightarrow 0$ these relations read: $$g_{1}^{q}(x)=\frac{1}{2}\left[ \allowbreak q_{V}(x)-2x^{2}\int_{x}^{1}\frac{% q_{V}(y)}{y^{3}}dy\right] ,$$$$g_{2}^{q}(x)=\frac{1}{2}\left[ -\allowbreak \allowbreak q_{V}(x)+3x^{2}\int_{x}^{1}\frac{q_{V}(y)}{y^{3}}dy\right] .$$Now, taking quark charges and corresponding $SU(6)$ factors as in Eq. ([t71]{}), one can directly calculate $g_{1},g_{2}$ only using the input on the valence quark distribution $q_{V}=u_{V},d_{V}$. In Fig. \[gps1\] the results of $g_{1}$ and $g_{2}$ calculation are shown. Experimental data on $% g_{1}$ are represented by the new parameterization of the world data [e155g1]{} and the $g_{2}$ points are data of the E155 Collaboration [e155g2]{}. More detailed discussion of these figures is done in [@zav5], in this talk I want concentrate on the discussion and explanation, why intrinsic quark motion substantially reduces the first moment of the spin function $g_{1}$. In [@zav4] it is shown, that $$\Gamma _{1}\equiv \int g_{1}(x)dx=\frac{1}{2}\int H(p_{0})\left( \frac{1}{3}+% \frac{2m}{3p_{0}}\right) d^{3}p, \label{v1}$$ which, in the $SU(6)$ approach gives$$\frac{5}{18}\geq \Gamma _{1}\geq \frac{5}{54},$$where left limit is valid for the static ($p_{0}\rightarrow m$) and right one for massless quarks ($m\rightarrow 0$). In other words, it means: $$more\ intrinsic\ motion\Leftrightarrow less\ spin$$This is a mathematical result, but how to understand it from the point of view of physics? First, forget structure functions for a while and calculate completely another task. Let me remind general rules concerning angular momentum in quantum mechanics: 1\) Angular momentum consist of orbital and spin part: [**j=l+s**]{} 2\) In the relativistic case [**l**]{} and [**s**]{} are not conserved separately, only total angular momentum [**j**]{} is conserved. So, one can have pure states of $j(j^{2},j_{z})$ only, which are for fermions with $s=1/2 $ represented by the relativistic spheric waves, see e.g. [@lali]: $$\psi _{jlj_{z}}\left( {\bf p}\right) =\frac{1}{\sqrt{2p_{0}}}\left( \begin{array}{c} i^{-l}\sqrt{p_{0}+m}\Omega _{jlj_{z}}\left( \frac{{\bf p}}{p}\right) \\ i^{-l^{\prime }}\sqrt{p_{0}-m}\Omega _{jl^{\prime }j_{z}}\left( \frac{{\bf p}% }{p}\right)% \end{array}% \right) ;\qquad j=l\pm \frac{1}{2},\qquad l^{\prime }=2j-l,$$$$\begin{aligned} \Omega _{l+1/2,l,j_{z}}\left( \frac{{\bf p}}{p}\right) &=&\left( \begin{array}{c} \sqrt{\frac{j+j_{z}}{2j}}Y_{l,j_{z}-1/2} \\ \sqrt{\frac{j-j_{z}}{2j}}Y_{l,j_{z}+1/2}% \end{array}% \right) ,\qquad \\ \Omega _{l^{\prime }-1/2,l^{\prime },j_{z}}\left( \frac{{\bf p}}{p}\right) &=&\left( \begin{array}{c} -\sqrt{\frac{j-j_{z}+1}{2j+2}}Y_{l^{\prime },j_{z}-1/2} \\ \sqrt{\frac{j+j_{z}+1}{2j+2}}Y_{l^{\prime },j_{z}+1/2}% \end{array}% \right) .\end{aligned}$$This wavefunction is simplified for the state with total angular momentum (spin) equal 1/2: $$j=j_{z}=\frac{1}{2},\qquad l=0\qquad \Rightarrow \qquad l^{\prime }=1,$$$$Y_{00}=\frac{1}{\sqrt{4\pi }},\qquad Y_{10}=i\sqrt{\frac{3}{4\pi }}\cos \theta ,\qquad Y_{11}=-i\sqrt{\frac{3}{8\pi }}\sin \theta \exp \left( i\varphi \right) ,$$which gives$$\psi _{jlm}\left( {\bf p}\right) =\frac{1}{\sqrt{8\pi p_{0}}}\left( \begin{array}{c} \sqrt{p_{0}+m}\left( \begin{array}{c} 1 \\ 0% \end{array}% \right) \\ -\sqrt{p_{0}-m}\left( \begin{array}{c} \cos \theta \\ \sin \theta \exp \left( i\varphi \right)% \end{array}% \right)% \end{array}% \right) .$$Let me remark, that $j=1/2$ is minimum angular momentum for particle with $% s=1/2.$ Now, one can easily calculate the average contribution of the spin operator to the total angular momentum: $$\Sigma _{3}=\frac{1}{2}\left( \begin{array}{cc} \sigma _{3} & \cdot \\ \cdot & \sigma _{3}% \end{array}% \right) \Rightarrow$$$$\psi _{jlm}^{\dagger }\left( {\bf p}\right) \Sigma _{3}\psi _{jlm}\left( {\bf p}\right) =\frac{1}{16\pi p_{0}}\left[ \left( p_{0}+m\right) +\left( p_{0}-m\right) \left( \cos ^{2}\theta -\sin ^{2}\theta \right) \right]$$If $a_{p}$ is the probability amplitude of the state $\psi _{jlm}$, then $$\left\langle \Sigma _{3}\right\rangle =\int a_{p}^{\star }a_{p}\psi _{jlm}^{\dagger }\left( {\bf p}\right) \Sigma _{3}\psi _{jlm}\left( {\bf p}% \right) d^{3}p=\frac{1}{2}\int a_{p}^{\star }a_{p}\left( \frac{1}{3}+\frac{2m% }{3p_{0}}\right) p^{2}dp, \label{v2}$$which means, that: [*i)*]{} For the fermion at rest ($p_{0}=m$) we have $j=s=1/2,$ which is quite comprehensible, since without kinetic energy no orbital momentum can be generated. [*ii)*]{} For the state in which $p_{0}\geq m$, we have in general: $$\frac{1}{3}\leq \frac{\left\langle s\right\rangle }{j}\leq 1.$$where left limit is valid for the energetic fermion, $p_{0}\gg m$. In other words, in the states $\psi _{jlm}$ with $p_{0}>m$ part of the total angular momentum $j=1/2$ is [*necessarily* ]{}created by orbital momentum. This is a simple consequence of quantum mechanics. Now, one can compare integrals (\[v1\]) and (\[v2\]). Since both integrals involve the same kinematic term, the interpretation of dependence on ratio $m/p_{0}$ in (\[v2\]) is valid also for (\[v1\]). Otherwise, the comparison is a rigorous illustration of the statement, that $\Gamma _{1} $ measures contributions from quark spins (and not their total angular momenta). In which point the present approach differ from standard QPM? Standard approach is closely connected with the preferred reference frame - infinite momentum frame. The basic relations like $$g_{1}(x)=\frac{1}{2}\sum e_{j}^{2}\Delta q_{j}(x),\qquad F_{2}(x)=x\sum e_{i}^{2}q_{i}(x)$$are derived with the use of approximation$$p_{\alpha }=xP_{\alpha }.$$In the covariant formulation this relation is equivalent to the assumption, that the quarks are static. In the presented covariant approach quarks are not static, so this approximation cannot be used. As a result, different relations between the distribution and structure functions and also different behavior of $\Gamma _{1}$ are obtained. Summary ======= I have studied spin functions in system of quasifree fermions having fixed effective mass $x_{0}=m/M$ and total spin $J=1/2$ - representing a covariant version of naive QPM. The main results are: 1\) Spin functions $g_{1}$ and $g_{2}$ depend on intrinsic motion. In particular, the momenta $\Gamma _{1}$ corresponding to the static (massive) fermions and massless fermions, can differ significantly: $\Gamma _{1}(m\ll p_{0})/\Gamma _{1}(p_{0}\approx m)=1/3$. It is due to splitting of angular momentum into spin and orbital part, as soon as intrinsic motion is present. 2\) $g_{1}$ and $g_{2}$ are connected by a simple transformation, which is for $m\rightarrow 0$ identical to Wanzura - Wilczek relation for twist-2 term of the $g_{2}$ approximation. Relations for the $n-th$ momenta of the structure functions have been obtained, their particular cases are identical to known sum rules: Wanzura - Wilczek ($n=2,4,6...$), Efremov - Leader - Teryaev ($n=1$) and Burkhardt - Cottingham ($n=0$). 3\) Model has been applied to the proton spin structure, assuming proton spin is generated only by spins and orbital momenta of the valence quarks with [*SU(6)*]{} symmetry and for quark effective mass $m\rightarrow 0$. As an input I used  known parameterization of the valence terms, then without any other free parameter, the functions $g_{1},$ $g_{2}$ were obtained. Comparison with the proton data demonstrates a good agreement. 4\) Comparison with the corresponding relations for the structure functions following from the usual naive QPM was done. Both the approaches are equivalent for the static quarks. Differences for quarks with internal motion inside the proton are result of the conflict with the assumption $% p_{\alpha }=xP_{\alpha }$, which is crucial for derivation of the relations in the standard QPM. P. Zavada, Phys. Rev. D[** 65**]{}, 054040 (2002). P. Zavada, Phys. Rev. D[** 67**]{}, 014019 (2003). A.V. Efremov, O.V. Teryaev and P. Zavada, hep-ph/0405225, will be published in Phys. Rev. D. S. Wanzura and W. Wilczek, Phys. Lett. B[** 72**]{}, 195 (1977). H. Burkhardt, W.N. Cottingham, Ann. Phys. [**56**]{}, 453 (1970). A.V. Efremov, O.V. Teryaev, E. Leader, Phys. Rev. D[** 55**]{}, 4307 (1997). P. Zavada, Phys. Rev. D[** 55**]{}, 4290 (1997). E155 Collaboration, P. Anthony [*et al.*]{}, Phys. Lett. B [**493**]{}, 19 (2000). E155 Collaboration, P. Anthony [*et al.*]{}, Phys. Lett. B [**553**]{}, 80 (2003). L.D. Landau, E.M. Lishitz et al., Quantum Electrodynamics (Course of Theoretical Physics, vol. 4), Elsevier Science Ltd., 1982. [^1]: Talk prepared for the conference DIS2004

--- abstract: 'This paper investigates an intelligent reflecting surface (IRS)-aided multi-cell multiple-input single-output (MISO) system consisting of several multi-antenna base stations (BSs) each communicating with a single-antenna user, in which an IRS is dedicatedly deployed for assisting the wireless transmission and suppressing the inter-cell interference. Under this setup, we jointly optimize the coordinated transmit beamforming at the BSs and the reflective beamforming at the IRS, for the purpose of maximizing the minimum weighted received signal-to-interference-plus-noise ratio (SINR) at users, subject to the individual maximum transmit power constraints at the BSs and the reflection constraints at the IRS. To solve the difficult non-convex minimum SINR maximization problem, we propose efficient algorithms based on alternating optimization, in which the transmit and reflective beamforming vectors are optimized in an alternating manner. In particular, we use the second-order-cone programming (SOCP) for optimizing the coordinated transmit beamforming, and develop two efficient designs for updating the reflective beamforming based on the techniques of semi-definite relaxation (SDR) and successive convex approximation (SCA), respectively. Numerical results show that the use of IRS leads to significantly higher SINR values than benchmark schemes without IRS or without proper reflective beamforming optimization; while the developed SCA-based solution outperforms the SDR-based one with lower implementation complexity.' author: - | Hailiang Xie$^1$, Jie Xu$^1$, and Ya-Feng Liu$^2$\ $^1$School of Information Engineering, Guangdong University of Technology, Guangzhou, China\ $^2$LSEC, ICMSEC, AMSS, Chinese Academy of Sciences, Beijing, China\ E-mail: hailiang.gdut@gmail.com, jiexu@gdut.edu.cn, yafliu@lsec.cc.ac.cn title: 'Max-Min Fairness in IRS-Aided Multi-Cell MISO Systems via Joint Transmit and Reflective Beamforming [^1]' --- Intelligent reflecting surface (IRS), multi-cell systems, multiple-input single-output (MISO), coordinated transmit beamforming, reflective beamforming, optimization. \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] Introduction ============ To enable emerging Internet of things (IoT) and artificial intelligence (AI) applications, the fifth-generation (5G)-and-beyond cellular networks need to support massive wireless devices with diverse quality of service (QoS) requirements, such as significantly increased spectrum efficiency, ultra-low transmission latency, and extremely-high communication reliability [@5G; @6G]. Towards this end, small base stations (BSs) are densely deployed to shorten the distances with cellular subscribers [@small_cells], and device-to-device (D2D) communications are enabled underlying conventional cellular transmissions to create more spectrum reuse opportunities [@D2D]. However, the emergence of small BSs and D2D communications in 5G-and-beyond cellular networks also introduces severe co-channel interference among different cells and different D2D links, which needs to be carefully dealt with from technical perspectives. In the literature, various approaches have been proposed to mitigate or even utilize the co-channel interference, some examples including coordinated beamforming [@Co_beam_2010; @Co_beam_2011; @ICIC; @Net_MIMO3] and network multiple-input multiple-output (MIMO) [@Net_MIMO1; @Net_MIMO2; @Net_MIMO4]. Recently, intelligent reflecting surface (IRS) has emerged as a promising technology for beyond-5G cellular networks [@IRS_wu; @IRS_survey], which can also be used to tackle the critical co-channel interference issue in a cost-effective manner. IRS is a passive meta-material panel consisting of a large number of reflecting units, each of which can introduce an independent phase shift on radio-frequency (RF) signals to change the signal transmission environment. By jointly controlling these phase shifts, the IRS can form reflective signal beams, such that the reflected signals can be coherently combined with the directly transmitted signals at intended receivers for enhancing the desirable signal strength, or destructively combined at unintended receivers for suppressing the undesirable interference. As the IRS is a passive device with no dedicated power consumption, it is envisioned as a green and cost-effective solution to enhance the spectrum- and energy-efficiency of future cellular networks [@IRS_wu; @IRS_survey]. There have been some prior works [@IRS_single1; @IRS_single2; @IRS_multiuser; @IRS_NOMA1; @IRS_NOMA2; @IRS_OFDM; @IRS_OFDM_est; @IRS_OFDM_protocol] investigating the joint transmit and reflective beamforming design in IRS-aided wireless communication systems. The authors in [@IRS_single1; @IRS_single2] investigated the received signal-to-noise ratio (SNR) maximization problem in a point-to-point IRS-aided multiple-input single-output (MISO) communication system, which is solved by using the techniques of semi-definite relaxation (SDR) [@IRS_single1] and manifold optimization [@IRS_single2], respectively. Furthermore, [@IRS_multiuser] considered the signal-to-interference-plus-noise ratio (SINR)-constrained power minimization problem in IRS-aided multiuser MISO downlink communication systems, in which alternating optimization is employed to update the transmit and reflective beamforming vectors in an alternating manner, and SDR is employed to optimize the reflective beamforming. In addition, prior works also studied other communication setups aided by the IRS such as IRS-aided orthogonal frequency division multiplexing (OFDM) [@IRS_OFDM_protocol; @IRS_OFDM_est; @IRS_OFDM], non-orthogonal multiple access (NOMA) [@IRS_NOMA1; @IRS_NOMA2] and simultaneous wireless information and power transfer (SWIPT) systems [@SWIPT]. Nevertheless, all the above prior works [@IRS_single1; @IRS_single2; @IRS_multiuser; @IRS_NOMA1; @IRS_NOMA2; @IRS_OFDM; @IRS_OFDM_est; @IRS_OFDM_protocol; @SWIPT] focused on a single-cell setup. This thus motivates us to use IRSs to facilitate the multi-cell communications in this work. In this paper, we consider an IRS-aided multi-cell MISO system, where an IRS is dedicatedly deployed at the cell boundary to assist the wireless transmission from BSs to users and suppress their inter-cell interference. We assume that there is one multi-antenna BS serving one single-antenna user in each cell. Our objective is to jointly optimize the coordinated transmit beamforming at the multiple BSs and the reflective beamforming at the IRS, to maximize the minimum weighted received SINR at users, subject to the individual maximum transmit power constraints at BSs, and the reflection constraints at the IRS. However, the formulated minimum SINR maximization problem is highly non-convex due to the coupling between the transmit and reflective beamforming vectors. To solve this difficult problem, we propose efficient algorithms based on alternating optimization, in which the transmit and reflective beamforming vectors are optimized in an alternating manner. In particular, under any given reflective beamforming, we obtain the optimal coordinated transmit beamforming via second-order cone programming (SOCP); while under any given coordinated transmit beamforming, we develop two efficient designs to update the reflective beamforming by using the techniques of SDR and successive convex approximation (SCA), respectively. It is observed that the performance of the SDR-based solution generally depends on the randomization procedure, while the SCA-based solution can always converge towards a stationary point. Numerical results show that the use of IRS leads to significant performance gains over benchmark schemes without IRS or without proper reflective beamforming design at the IRS, and the developed SCA-based solution outperforms the SDR-based one with lower implementation complexity. It is worth noting that there is only one existing work [@IRS_multicell] that studied the weighted sum-rate maximization in IRS-aided multi-cell networks by applying the alternating-optimization-based approaches. Nevertheless, this paper is different from [@IRS_multicell] in the following two aspects. First, while [@IRS_multicell] focsed on the weighted sum-rate maximization, this paper considers a different objective of the min-weighted-SINR maximization. Second, while [@IRS_multicell] only optimized the reflection phases at the IRS by considering unit amplitudes, this paper further exploits the optimization of reflection amplitudes to enhance the communication performance. System Model and Problem Formulation ==================================== We consider an IRS-aided multi-cell MISO system, where an IRS is dedicatedly deployed to assist the multi-cell communication and suppress the inter-cell interference, especially for cell-edge users. Suppose that in each cell there is a BS with $M\ge1$ antennas communicating with a user with one single antenna. Let $\mathcal K \triangleq\{1,\ldots,K\}$ denote the set of BSs or users in the system, and $\mathcal N \triangleq\{1,\ldots,N\}$ denote the set of reflecting units at the IRS. The IRS can adaptively adjust the reflecting phases to form reflective signal beams, such that the reflected signal can be coherently combined with the directly transmitted signal at the intended user or destructively combined at the unintended users. We consider a quasi-static narrow-band channel model, where the wireless channels remain unchanged within each transmission block of our interest but may change over different blocks. Let ${\mbox{\boldmath{$ G $}}}_i\in {\mathbb C}^{N\times M}$ denote the channel matrix from BS $i$ to the IRS, ${{\mbox{\boldmath{$ f $}}}_{i}}\in {\mathbb C}^{N\times 1}$ denote the channel vector from the IRS to user $i$, and ${{\mbox{\boldmath{$ h $}}}_{i,k}}\in {\mathbb C}^{M\times 1}$ denote that from BS $k$ to user $i$, where $\mathbb{C}^{x\times y}$ denotes the space of $x\times y$ complex matrices. Let $s_i$ denote the transmitted signal by each BS $i$ and ${\mbox{\boldmath{$ w $}}}_i\in\mathbb C^{M\times1}$ the corresponding transmit beamforming vector. We assume that $s_i$’s are independent and identically distributed (i.i.d.) circularly symmetric complex Gaussian (CSCG) random variables with zero mean and unit variance, i.e., $s_i\!\sim\!\mathcal{CN}(0,1)$. The transmitted signal by each BS $i$ is thus given by ${\mbox{\boldmath{$ x $}}}_i = {\mbox{\boldmath{$ w $}}}_i s_i, \forall i\in\mathcal K.$ Suppose that each BS has a maximum power budget denoted by $P_i$. Then we have $\mathbb{E}(\|{\mbox{\boldmath{$ x $}}}_i\|^2)\!=\!\|{\mbox{\boldmath{$ w $}}}_i\|^2\!\le\!P_i, \forall i\!\in\!\mathcal K$, where $\mathbb{E}(\cdot)$ denotes the stochastic expectation. As for the reflection at the IRS, let $\theta_n\!\in\![0, 2\pi)$ and $\beta_n\in[0, 1]$ denote the phase shift and the reflection amplitude imposed by the $n$-th reflecting unit on the incident signal, respectively. Accordingly, let ${\mbox{\boldmath{$ \Theta $}}}=\mathrm{diag} (\beta_1 e^{j\theta_{1}},\ldots,\beta_N e^{j\theta_{N}} ) $ represent the reflection coefficient matrix at the IRS, where $j\triangleq \sqrt{-1}$, and $\mathrm{diag}(a_1,\ldots, a_N)$ denotes a diagonal matrix with its diagonal elements being $a_1,\ldots, a_N$. Furthermore, let ${\mbox{\boldmath{$ v $}}}=[\beta_1 e^{j\theta_{1}},\ldots,\beta_N e^{j\theta_{N}}]^H$ denote the reflective beamforming vector, where each element $n$, denoted by $v_n$, must satisfy $|v_n|\le 1, \forall n\in\mathcal N$. Here, the superscript $H$ denotes the conjugate transpose of a vector or matrix. As a consequence, we have the combined reflective channel from BS $k$ to user $i$ as ${\mbox{\boldmath{$ f $}}}^H_{i}{\mbox{\boldmath{$ \Theta $}}}{\mbox{\boldmath{$ G $}}}_k = {\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ \Phi $}}}_{i,k}$, where ${\mbox{\boldmath{$ \Phi $}}}_{i,k} = \mathrm{diag}({\mbox{\boldmath{$ f $}}}^H_{i}){\mbox{\boldmath{$ G $}}}_k$. Notice that this transformation separates the reflective beamforming vector ${\mbox{\boldmath{$ v $}}}$ from the reflective channels, which will significantly facilitate our derivation later. By combining the directly transmitted and reflected signals, the signal received at user $i$ is accordingly expressed as $$\begin{aligned} y_{i}\!=\!({\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ \Phi $}}}_{i,i}\!+\!{\mbox{\boldmath{$ h $}}}^H_{i,i}){\mbox{\boldmath{$ w $}}}_i s_i\!+\!\sum\limits_{k\ne i,k\in\mathcal K}( {\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ \Phi $}}}_{i,k}\!+\!{\mbox{\boldmath{$ h $}}}^H_{i,k}){\mbox{\boldmath{$ w $}}}_k s_k\!+\!n_{i}, \label{fm:1}\end{aligned}$$ where $n_{i}$ denotes the additive white Gaussian noise (AWGN) at the receiver of user $i$ with zero mean and variance $\sigma_i^2$, i.e., $n_{i}\sim\mathcal{CN}(0,\sigma_i^2), \forall i\in\mathcal K$. By treating the interference as noise, the received SINR at user $i$ is given by $$\begin{aligned} \mathrm{\gamma}_i({\mbox{\boldmath{$ v $}}}, \{{\mbox{\boldmath{$ w $}}}_i\})\!=\!\frac{|({\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ \Phi $}}}_{i,i}+ {\mbox{\boldmath{$ h $}}}^H_{i,i}){\mbox{\boldmath{$ w $}}}_i|^2}{\sum\limits_{k\ne i,k\in\mathcal K}|({\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ \Phi $}}}_{i,k}+ {\mbox{\boldmath{$ h $}}}^H_{i,k}){\mbox{\boldmath{$ w $}}}_k|^2+\sigma_i^2}.\label{fm:2}\end{aligned}$$ Our objective is to maximize the users’ communication performance in a fair manner. As a result, we consider the max-min fairness problem with the objective of maximizing the minimum weighted SINR of all users, by jointly optimizing the transmit beamforming $\{{\mbox{\boldmath{$ w $}}}_i\}$ at the BSs and the reflective beamforming ${\mbox{\boldmath{$ v $}}}$ at the IRS, subject to the individual transmit power constraints at BSs and the reflection constraints at the IRS. Let $\alpha_i>0$ denote a weight parameter for user $i\in\mathcal K$ characterizing the fairness among the $K$ users, where a larger value of $\alpha_i$ indicates that user $i$ has a higher priority in transmission. Therefore, the minimum SINR maximization problem is formulated as $$\begin{aligned} \mathtt{(P1)}:&\mathop\mathtt{max}_{{\mbox{\boldmath{$ v $}}},\{{\mbox{\boldmath{$ w $}}}_i\}}\mathop\mathtt{min}_{i\in\mathcal K}~\frac{\mathrm{\gamma}_i({\mbox{\boldmath{$ v $}}}, \{{\mbox{\boldmath{$ w $}}}_i\})}{\alpha_i} \label{Problem:ori:1} \\ &~~~{\mathtt{s.t.}}~~\|{\mbox{\boldmath{$ w $}}}_i\|^2 \le P_i, \forall i\in\mathcal K \label{Problem:ori:2} \\ &~~~~~~~~~~|v_n|\le 1, \forall n\in\mathcal N. \label{Problem:ori:3}\end{aligned}$$ To facilitate the derivation, we first introduce an auxiliary variable $t$ and reformulate problem (P1) as the following equivalent problem: $$\begin{aligned} &\mathtt{(P1.1)}:\mathop\mathtt{max}_{{\mbox{\boldmath{$ v $}}},\{{\mbox{\boldmath{$ w $}}}_i\},t} t \nonumber \\ &~~~~~~~~~~~{\mathtt{s.t.}} ~~~\mathrm{\gamma}_i({\mbox{\boldmath{$ v $}}}, \{{\mbox{\boldmath{$ w $}}}_i\})\ge\alpha_i t, \forall i\in\mathcal K \label{Problem:ori_epi:1} \\ &~~~~~~~~~~~~~~~~~~~(\ref{Problem:ori:2})~\text{and}~(\ref{Problem:ori:3}). \nonumber\end{aligned}$$ Notice that problem (P1.1) or (P1) is difficult to be optimally solved due to the coupling between the transmit beamforming $\{{\mbox{\boldmath{$ w $}}}_i\}$ and the reflective beamforming ${\mbox{\boldmath{$ v $}}}$ at the SINR terms. To tackle this difficulty, we propose alternating-optimization-based algorithms to solve problem (P1.1) or (P1), in which the transmit beamforming vector $\{{\mbox{\boldmath{$ w $}}}_i\}$ and the reflective beamforming vector ${\mbox{\boldmath{$ v $}}}$ are optimized in an alternating manner, with the other being fixed. In particular, the alternating-optimization-based algorithms are implemented in an iterative manner. For notational convenience, suppose that at each iteration $l \ge 0$, the obtained beamforming vectors are denoted by ${\mbox{\boldmath{$ v $}}}^{(l)}$ and $\{{\mbox{\boldmath{$ w $}}}^{(l)}_i\}$, where ${\mbox{\boldmath{$ v $}}}^{(0)}$ and $\{{\mbox{\boldmath{$ w $}}}^{(0)}_i\}$ denote the initial beamforming vectors. In Sections \[sec:III\] and \[sec:IV\], we present efficient approaches for updating $\{{\mbox{\boldmath{$ w $}}}_i\}$ and ${\mbox{\boldmath{$ v $}}}$, respectively. Coordinated Transmit Beamforming Optimization {#sec:III} ============================================= In this section, we optimize the coordinated transmit beamforming $\{{\mbox{\boldmath{$ w $}}}_i\}$ under any given reflective beamforming ${\mbox{\boldmath{$ v $}}}$. For notational convenience, we define ${\mbox{\boldmath{$ a $}}}_{i,k}={\mbox{\boldmath{$ \Phi $}}}_{i,k}^H{\mbox{\boldmath{$ v $}}} + {\mbox{\boldmath{$ h $}}}_{i,k}$ as the effective or combined channel from BS $k\in\mathcal K$ to user $i\in\mathcal K$. Accordingly, the coordinated transmit beamforming optimization problem becomes $$\begin{aligned} \mathrm{(P2):}&\mathop\mathtt{max}_{\{{\mbox{\boldmath{$ w $}}}_i\},t} ~ t \nonumber \\ &~~~{\mathtt{s.t.}} ~\frac{|{\mbox{\boldmath{$ a $}}}^H_{i,i}{\mbox{\boldmath{$ w $}}}_i|^2}{\sum\limits_{k\ne i,k\in\mathcal K}|{\mbox{\boldmath{$ a $}}}^H_{i,k}{\mbox{\boldmath{$ w $}}}_k|^2+\sigma_i^2}\ge\alpha_i t, \forall i\in\mathcal K, \label{Problem:given_phase:1} \\ &~~~~~~~~~(\ref{Problem:ori:2}). \nonumber\end{aligned}$$ It is observed that problem (P2) is still not a convex optimization problem. To tackle this issue, we introduce the following feasibility problem (P2.1), which is obtained based on problem (P2) by fixing $t$. $$\begin{aligned} \mathrm{(P2.1):}&\mathop\mathtt{find}~ \{{\mbox{\boldmath{$ w $}}}_i\} \nonumber \\ &~~~{\mathtt{s.t.}} ~(\ref{Problem:ori:2})~\mathrm{and}~(\ref{Problem:given_phase:1}). \nonumber\end{aligned}$$ In particular, suppose that the optimal solution of $t$ to problem (P2) is given by $t^\star$. It is thus clear that if problem (P2.1) is feasible under any given $t$, then we have $t \le t^\star$; while if (P2.1) is infeasible, then it follows that $t > t^\star$. Therefore, problem (P2) can be equivalently solved by checking the feasibility of problem (P2.1) under any given $t > 0$, together with a bisection search over $t > 0$. Therefore, to solve problem (P2), we only need to solve problem (P2.1) under any fixed $t > 0$, by using SOCP as follows [@SOCP]. Towards this end, we notice that the SINR constraints in (\[Problem:given\_phase:1\]) can be reformulated as $$\begin{aligned} \big(1+\frac{1}{\alpha_i t}\big)|{\mbox{\boldmath{$ a $}}}^H_{i,i}{\mbox{\boldmath{$ w $}}}_i|^2 \ge \sum\limits_{k\in\mathcal K}|{\mbox{\boldmath{$ a $}}}^H_{i,k}{\mbox{\boldmath{$ w $}}}_k|^2+\sigma_i^2, \forall i\in\mathcal K. \label{tf_socp:1}\end{aligned}$$ Based on (\[tf\_socp:1\]), it is evident that if $\{{\mbox{\boldmath{$ w $}}}_i\}$ is a feasible solution to problem (P2.1), then any phase rotation of $\{{\mbox{\boldmath{$ w $}}}_i\}$ will still be feasible. Without loss of optimality, we choose the solution of $\{{\mbox{\boldmath{$ w $}}}_i\}$ such that ${\mbox{\boldmath{$ a $}}}^H_{i,i}{\mbox{\boldmath{$ w $}}}_i$ becomes a non-negative value for any user $k\in \mathcal K$. As a result, we have the following constraints: $$\begin{aligned} {\mbox{\boldmath{$ a $}}}^H_{i,i}{\mbox{\boldmath{$ w $}}}_i \ge 0, \forall i\in\mathcal K,\label{tf_socp:2}\end{aligned}$$ where ${\mbox{\boldmath{$ a $}}}^H_{i,i}{\mbox{\boldmath{$ w $}}}_i$ has a non-negative real part and a zero imaginary part, i.e., $\mathrm{Re}({\mbox{\boldmath{$ a $}}}_{i,i}^H {\mbox{\boldmath{$ w $}}}_i) \ge 0$ and $\mathrm{Im}({\mbox{\boldmath{$ a $}}}^H_{i,i}{\mbox{\boldmath{$ w $}}}_i)=0$, with $\mathrm{Re}(x)$ and $\mathrm{Im}(x)$ denoting the real and imaginary parts of a complex number $x$. Accordingly, (\[tf\_socp:1\]) can be further re-expressed as $$\begin{aligned} \sqrt{1+\frac{1}{\alpha_i t}}{\mbox{\boldmath{$ a $}}}^H_{i,i}{\mbox{\boldmath{$ w $}}}_i \ge \begin{Vmatrix} {\mbox{\boldmath{$ A $}}}^H {\mbox{\boldmath{$ e $}}}_i \\ \sigma_i \end{Vmatrix}_2, \forall i\in\mathcal K, \label{tf_socp:3}\end{aligned}$$ where ${\mbox{\boldmath{$ A $}}}\in\mathbb C^{K\times K}$ denotes a matrix with the element in its $i$-th row and $j$-th column being ${\mbox{\boldmath{$ a $}}}^H_{i,j}{\mbox{\boldmath{$ w $}}}_j$, ${\mbox{\boldmath{$ e $}}}_i\in\mathbb C^{K\times 1}$ denotes a vector with the $i$-th element being one and others being zero, and $\|\cdot\|_2$ denotes the Euclidean norm of a vector. Therefore, problem (P2.1) is reformulated as the following equivalent form: $$\begin{aligned} \mathrm{(P2.2):}\mathop\mathtt{find}~& \{{\mbox{\boldmath{$ w $}}}_i\} \nonumber \\ {\mathtt{s.t.}} ~& (\ref{Problem:ori:2}),(\ref{tf_socp:2}),~\mathrm{and}~(\ref{tf_socp:3}). \nonumber\end{aligned}$$ Problem (P2.1) is an SOCP problem that can be optimally solved by standard convex optimization solvers such as CVX [@CVX]. Therefore, the optimal coordinated transmit beamforming solution to problem (P2) is finally obtained. Reflectve Beamforming Optimization {#sec:IV} ================================== In this section, we optimize the reflective beamforming vector ${\mbox{\boldmath{$ v $}}}$ under given transmit beamforming $\{{\mbox{\boldmath{$ w $}}}_i\}$. For notational convenience, we define ${\mbox{\boldmath{$ c $}}}_{i,k} = {\mbox{\boldmath{$ \Phi $}}}_{i,k}{\mbox{\boldmath{$ w $}}}_k$ and $d_{i,k} = {\mbox{\boldmath{$ h $}}}^H_{i,k}{\mbox{\boldmath{$ w $}}}_k, \forall i, k\in\mathcal K$. Then, we have $$\begin{aligned} |({\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ \Phi $}}}_{i,k}\!+\!{\mbox{\boldmath{$ h $}}}^H_{i,k}){\mbox{\boldmath{$ w $}}}_k|^2\!=\!{\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ C $}}}_{i,k}{\mbox{\boldmath{$ v $}}}\!+\!2\mathrm{Re}\{{\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ u $}}}_{i,k}\}\!+\!|d_{i,k}|^2, \label{fm:3}\end{aligned}$$ where ${\mbox{\boldmath{$ C $}}}_{i,k}={\mbox{\boldmath{$ c $}}}_{i,k}{\mbox{\boldmath{$ c $}}}^H_{i,k}$ and ${\mbox{\boldmath{$ u $}}}_{i,k}={\mbox{\boldmath{$ c $}}}_{i,k}d_{i,k}^H$. Accordingly, the reflective beamforming optimization problem is given by $$\begin{aligned} &\mathtt{(P3):}\mathop\mathtt{max}_{{\mbox{\boldmath{$ v $}}},t}~t \nonumber \\ &~~~~~~~~{\mathtt{s.t.}} ~\frac{{\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ C $}}}_{i,i}{\mbox{\boldmath{$ v $}}}\!+\!2\mathrm{Re}\{{\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ u $}}}_{i,i}\}\!+\!|d_{i,i}|^2}{\sum\limits_{k\ne i,k\in\mathcal K}{\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ C $}}}_{i,k}{\mbox{\boldmath{$ v $}}}\!+\!2\mathrm{Re}\{{\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ u $}}}_{i,k}\}\!+\!|d_{i,k}|^2\!+\!\sigma_i^2}\nonumber \\ &~~~~~~~~~~~~~~~~~~~~~~~~\ge\alpha_i t, \forall i\in\mathcal K, \label{Problem:given_beam:1} \\ &~~~~~~~~~~~~~~(\ref{Problem:ori:3}). \nonumber\end{aligned}$$ Notice that problem (P3) is also a non-convex optimization problem. In the following, we propose two solutions by leveraging the SDR and SCA techniques, respectively. SDR-based Solution to Problem (P3) ---------------------------------- In this subsection, we use the well-established SDR technique to solve problem (P3). This is motivated by the wide application of SDR in reflective beamforming optimization (see, e.g., [@IRS_multiuser]). Towards this end, we first define $|({\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ \Phi $}}}_{i,k} + {\mbox{\boldmath{$ h $}}}^H_{i,k}){\mbox{\boldmath{$ w $}}}_k|^2 = \bar{{\mbox{\boldmath{$ v $}}}}^{\rm{H}}{{\mbox{\boldmath{$ R $}}}}_{i,k}\bar{{\mbox{\boldmath{$ v $}}}} + |d_{i,k}|^2$, where $$\begin{aligned} {{\mbox{\boldmath{$ R $}}}}_{i,k} = \begin{bmatrix} {\mbox{\boldmath{$ C $}}}_{i,k} & {\mbox{\boldmath{$ u $}}}_{i,k} \\ {\mbox{\boldmath{$ u $}}}^H_{i,k} & 0 \end{bmatrix} ~\mathrm{and} ~\bar{{\mbox{\boldmath{$ v $}}}} = \begin{bmatrix} {\mbox{\boldmath{$ v $}}} \\ 1 \end{bmatrix}.\end{aligned}$$ Accordingly, problem (P3) is re-expressed as $$\begin{aligned} &\mathtt{(P3.1):}\mathop\mathtt{max}_{\bar{{\mbox{\boldmath{$ v $}}}},t}~t \nonumber \\ &~~~{\mathtt{s.t.}} ~\frac{\bar{{\mbox{\boldmath{$ v $}}}}^{H}{{\mbox{\boldmath{$ R $}}}}_{i,i}\bar{{\mbox{\boldmath{$ v $}}}} + |d_{i,i}|^2}{\sum\limits_{k\ne i,k\in\mathcal K}\bar{{\mbox{\boldmath{$ v $}}}}^{H}{{\mbox{\boldmath{$ R $}}}}_{i,k}\bar{{\mbox{\boldmath{$ v $}}}} + |d_{i,k}|^2 +\sigma_i^2}\ge\alpha_i t, \forall i\in\mathcal K \label{Problem:given_beam:2} \\ &~~~~~~~~~|\bar v_n|\le 1,|\bar v_{N+1}| = 1, \forall n\in\mathcal N \label{Problem:given_beam:3}\end{aligned}$$ Furthermore, we define ${\mbox{\boldmath{$ V $}}}= \bar{{\mbox{\boldmath{$ v $}}}}\bar{{\mbox{\boldmath{$ v $}}}}^H$ with ${\mbox{\boldmath{$ V $}}}$ being positive semi-definite (i.e., ${\mbox{\boldmath{$ V $}}} \succeq 0$) and $\mathrm{rank}({\mbox{\boldmath{$ V $}}})\le 1$. Then problem (P3.1) or (P3) is further reformulated as the following equivalent form: $$\begin{aligned} &\mathtt{(P3.2):}\mathop\mathtt{max}_{{\mbox{\boldmath{$ V $}}},t} ~ t \nonumber \\ &~~{\mathtt{s.t.}} ~\frac{\mathrm{Tr}({{\mbox{\boldmath{$ R $}}}_{i,i}}{\mbox{\boldmath{$ V $}}})\!+\!|d_{i,i}|^2}{\sum\limits_{k\ne i,k\in\mathcal K}\mathrm{Tr}({{\mbox{\boldmath{$ R $}}}_{i,k}}{\mbox{\boldmath{$ V $}}})\!+\!|d_{i,k}|^2\!+\!\sigma^2_i}\ge\alpha_i t, \forall i\in\mathcal K \label{Problem:given_beam_SDR:1}\\ &~~~~~~ V_{n,n}\le 1, V_{N+1,N+1}=1, \forall n\in\mathcal N \label{Problem:given_beam_SDR:2}\\ &~~~~~~{\mbox{\boldmath{$ V $}}}\succeq 0 \label{Problem:given_beam_SDR:3}\\ &~~~~~~\mathrm{rank}({\mbox{\boldmath{$ V $}}})\le 1, \label{Problem:given_beam_SDR:5}\end{aligned}$$ where $V_{m,n}$ denotes the element in the $m$-th row and $n$-th column of the matrix ${\mbox{\boldmath{$ V $}}}$, and $\mathrm{Tr}({\mbox{\boldmath{$ A $}}})$ denotes the trace of matrix ${\mbox{\boldmath{$ A $}}}$. However, problem (P3.2) is still challenging to be optimally solved due to the non-convex rank-one constraint in (\[Problem:given\_beam\_SDR:5\]). Motivated by the idea of SDR, we relax this constraint, and obtain a relaxed version of (P3.2) as $$\begin{aligned} &\mathtt{(P3.3):}\mathop\mathtt{max}_{{\mbox{\boldmath{$ V $}}},t} ~ t \nonumber \\ &~~~~~~~~~~{\mathtt{s.t.}}~(\ref{Problem:given_beam_SDR:1}),(\ref{Problem:given_beam_SDR:2}),~\mathrm{and}~(\ref{Problem:given_beam_SDR:3}).\nonumber\end{aligned}$$ Although problem (P3.3) is non-convex, it can be shown, similarly as for problem (P2.1), that (P3.3) can be solved equivalently by solving the following feasibility problem (P3.4) together with a bisection search over $t$. $$\begin{aligned} &\mathtt{(P3.4):}\mathop\mathtt{find}~ {\mbox{\boldmath{$ V $}}} \nonumber \\ &~~~~~~~~~~~{\mathtt{s.t.}} ~\mathrm{Tr}({{\mbox{\boldmath{$ R $}}}_{i,i}}{\mbox{\boldmath{$ V $}}})\!+\!|d_{i,i}|^2\ge \nonumber \\ &~~~~~~~~~~~\alpha_i t(\sum\limits_{k\ne i,k\in\mathcal K}\mathrm{Tr}({{\mbox{\boldmath{$ R $}}}_{i,k}}V)\!+\!|d_{i,k}|^2\!+\!\sigma^2_i), \forall i\in\mathcal K \label{Problem:given_beam_f:1}\\ &~~~~~~~~~~~(\ref{Problem:given_beam_SDR:2})~\mathrm{and}~(\ref{Problem:given_beam_SDR:3}).\nonumber\end{aligned}$$ Notice that problem (P3.4) is a convex semi-definite program (SDP) and thus can be solved optimally by using CVX [@CVX]. As a result, we have obtained the optimal solution to problem (P3.3), denoted by ${\mbox{\boldmath{$ V $}}}^\star$ and $t^\star$. Now, it remains to reconstruct the solution to problem (P3.2) or equivalently (P3.1)/(P3) based on ${\mbox{\boldmath{$ V $}}}^\star$ and $t^\star$. In particular, if rank$({\mbox{\boldmath{$ V $}}}^\star)\le 1$, then ${\mbox{\boldmath{$ V $}}}^\star$ and $t^\star$ are also the optimal solution to problem (P3.2). In this case, we have ${\mbox{\boldmath{$ V $}}}^\star=\bar{{\mbox{\boldmath{$ v $}}}}^\star \bar{{\mbox{\boldmath{$ v $}}}}^{\star H}$, where $\bar{{\mbox{\boldmath{$ v $}}}}^\star$ becomes the optimal solution to problem (P3.1). However, if rank$({\mbox{\boldmath{$ V $}}}^\star)>1$, then the following Gaussian randomization procedure [@Random] needs to be further adopted to produce a high-quality rank-one solution to problem (P3.2) and (P3.1). Specifically, suppose that the eigenvalue decomposition of ${\mbox{\boldmath{$ V $}}}^\star$ is ${\mbox{\boldmath{$ V $}}}^\star={\mbox{\boldmath{$ U $}}}{\mbox{\boldmath{$ \Sigma $}}}{\mbox{\boldmath{$ U $}}}^H$. Then, we set $\tilde{{\mbox{\boldmath{$ v $}}}} = {\mbox{\boldmath{$ U $}}}{\mbox{\boldmath{$ \Sigma $}}}^{\frac{1}{2}}{\mbox{\boldmath{$ r $}}}$, where ${\mbox{\boldmath{$ r $}}}$ corresponds to a CSCG random vector with zero mean and covariance matrix ${\mbox{\boldmath{$ I $}}}$, i.e., ${\mbox{\boldmath{$ r $}}}\sim\mathcal{CN}(0,{\mbox{\boldmath{$ I $}}})$. Accordingly, we construct a feasible solution $\bar{{\mbox{\boldmath{$ v $}}}}$ to problem (P3.1) as $\bar v_n= e^{j\mathrm{arg}(\tilde{v}_n/\tilde{v}_{N+1})}$, where $\bar v_n$ and $\tilde v_n$ denote the $n$-th element of vector $\bar{{\mbox{\boldmath{$ v $}}}}$ and $\tilde{{\mbox{\boldmath{$ v $}}}}$, respectively, and $\mathrm{arg}(x)$ denotes the phase of a complex number $x$. To guarantee the performance, the randomization process needs to be implemented multiple times and the best solution among them is selected as the obtained solution to problem (P3.1), denoted by ${\bar{{\mbox{\boldmath{$ v $}}}}}^\star$. In this case, the obtained solution to problem (P3.2) is ${\bar{{\mbox{\boldmath{$ v $}}}}}^\star {\bar{{\mbox{\boldmath{$ v $}}}}}^{\star H}$. Based on the solution of ${\bar{{\mbox{\boldmath{$ v $}}}}}^\star$ to problem (P3.1), we can accordingly obtain the solution of (P3) as ${\mbox{\boldmath{$ v $}}}^\star$. Therefore, the SDR-based algorithm for solving problem (P3) is complete. By alternately implementing the SDR-based solution to (P3) and the SOCP-based solution to (P2), we can obtain an efficient solution to the original problem (P1). We refer to this algorithm as alternating optimization with SDR. In summary, the algorithm of alternating optimization with SDR is presented as Algorithm 1. It is worth noticing that the performance of the SDR-based solution to problem (P3) critically depends on the performance of the Gaussian randomization when the rank of the obtained ${\mbox{\boldmath{$ V $}}}^\star$ is larger than one. This results in the following drawbacks for the algorithm of alternating optimization with SDR for solving (P1). On one hand, the SDR may introduce increased implementation complexity, which is due to the fact that solving the SDR is generally time-consuming (especially when the dimension of the matrix becomes large) and a large number of randomizations are generally needed in order to get a better solution. On the other hand, the alternating optimization with SDR may lead to compromised performance, as alternating optimization may terminate if the SDR leads to a highly suboptimal solution due to the uncertainty in randomizations. Therefore, this motivates us to further develop an alternative algorithm with performance guarantee. ------------------------------------------------------------------------ **Algorithm 1**: Alternating optimization with SDR ------------------------------------------------------------------------ - Initialize: $l=0$, ${\mbox{\boldmath{$ v $}}}^{(0)}$ and accuracy threshold $\epsilon > 0$. - [${\mbox{\boldmath{$ \mathrm{Repeat} $}}}$:]{} -    $l=l+1$; -    Under given ${\mbox{\boldmath{$ v $}}}^{(l-1)}$, solve problem (P2) to obtain $\{{\mbox{\boldmath{$ w $}}}_i^\star\}$, and set ${\mbox{\boldmath{$ w $}}}_i^{(l)}={\mbox{\boldmath{$ w $}}}_i^\star, \forall i\in \mathcal K$; -    Under given $\{{\mbox{\boldmath{$ w $}}}_i^{(l)}\}$, solve problem (P3) to obtain ${\mbox{\boldmath{$ v $}}}^\star$, and set ${\mbox{\boldmath{$ v $}}}^{(l)}={\mbox{\boldmath{$ v $}}}^\star$; - ${\mbox{\boldmath{$ \mathrm{Until} $}}}$ the increase of the objective function in (P1) is smaller than $\epsilon$. ------------------------------------------------------------------------ \[Table:1\] SCA-based Design for Updating Reflective Beamforming ---------------------------------------------------- To overcome the above drawbacks of the SDR-based solution, in this subsection, we propose an efficient design for updating the reflective beamforming vector ${\mbox{\boldmath{$ v $}}}$, by applying the SCA technique. Recall that the update of ${\mbox{\boldmath{$ v $}}}$ in problem (P3) is implemented iteratively in the alternating-optimization-based algorithm for solving the original problem (P1). Therefore, instead of directly solving (P3), in the SCA-based design we aim to find an updated ${\mbox{\boldmath{$ v $}}}$ to increase the users’ minimum SINR. Towards this end, we consider a particular iteration $l\ge 1$, the local point of ${\mbox{\boldmath{$ v $}}}$ as ${{\mbox{\boldmath{$ v $}}}}^{(l-1)}$, which corresponds to the obtained ${\mbox{\boldmath{$ v $}}}$ in the previous iteration. Under given $\{{\mbox{\boldmath{$ w $}}}_i^{(l)}\}$ together with ${{\mbox{\boldmath{$ v $}}}}^{(l-1)}$, we denote the achieved minimum SINR at users as $ t^{(l)} = \min_{i\in\mathcal K} \gamma_i({{\mbox{\boldmath{$ v $}}}}^{(l-1)}, \{{\mbox{\boldmath{$ w $}}}_i^{(l)}\})$. In the following, we explain how to update ${\mbox{\boldmath{$ v $}}}$ to increase the minimum SINR at users based on SCA. For notational convenience, we first define an auxiliary function for user $i\in \mathcal K$ as $$\begin{aligned} &\mathcal{F}_i({\mbox{\boldmath{$ v $}}}, \{{\mbox{\boldmath{$ w $}}}_i\}, t) \nonumber\\ &=\alpha_i t [\sum\limits_{k\ne i,k\in\mathcal K} ({\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ C $}}}_{i,k}{\mbox{\boldmath{$ v $}}}\!+\!2\mathrm{Re}\{{\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ u $}}}_{i,k}\}\!+\!|d_{i,k}|^2 )+\sigma^2_i ] \nonumber \\ &~~~~-\! ({\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ C $}}}_{i,i}{\mbox{\boldmath{$ v $}}}\!+\!2\mathrm{Re}\{{\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ u $}}}_{i,i}\}\!+\!|d_{i,i}|^2 ),\label{function}\end{aligned}$$ where ${\mbox{\boldmath{$ C $}}}_{i,k}$, ${\mbox{\boldmath{$ u $}}}_{i,k}$, and $d_{i,k}$, $i, k\in\mathcal K$ are defined at the beginning of Section IV. Note that after the update of $\{{\mbox{\boldmath{$ w $}}}_i\}$ at each iteration $l$, it must hold that $\min_{i\in\mathcal K} \mathcal F_i({\mbox{\boldmath{$ v $}}}^{(l-1)},\{{\mbox{\boldmath{$ w $}}}_i^{(l)}\}, t^{(l)})\!=\!0$. Accordingly, we update the reflective beamforming vector ${\mbox{\boldmath{$ v $}}}$ at the IRS by solving the following problem: $$\begin{aligned} \mathtt{(P4):}&\mathop\mathtt{min}_{{\mbox{\boldmath{$ v $}}}} ~\mathop\mathtt{max}_{i\in \mathcal K} ~\mathcal{F}_i ({\mbox{\boldmath{$ v $}}}, \{{\mbox{\boldmath{$ w $}}}_i^{(l)}\}, t^{(l)} ) \label{Problem:given_beam2:1} \\ &~{\mathtt{s.t.}} ~(\ref{Problem:ori:3}). \nonumber\end{aligned}$$ As ${\mbox{\boldmath{$ v $}}}^{(l-1)}$ is a feasible solution to problem (P4) leading to an objective value of zero, the optimal solution to problem (P4) should be non-positive. Suppose that the obtained solution to (P4) as ${\mbox{\boldmath{$ v $}}}^{(l)}$. If $\min_{i\in\mathcal K}\mathcal F({\mbox{\boldmath{$ v $}}}^{(l)}, \{{\mbox{\boldmath{$ w $}}}_i^{(l)}\}, t^{(l)}) < 0$, then it can be easily shown that $\min_{i\in\mathcal K} \gamma_i({\mbox{\boldmath{$ v $}}}^{(l)}, \{{\mbox{\boldmath{$ w $}}}_i^{(l)}\}) >\min_{i\in\mathcal K} \gamma_i({\mbox{\boldmath{$ v $}}}^{(l-1)}, \{{\mbox{\boldmath{$ w $}}}_i^{(l)}\})$, i.e., the minimum SINR is increased. Therefore, we focus on solving problem (P4) next. Problem (P4) is still non-convex as the objective function is non-convex with respect to ${\mbox{\boldmath{$ v $}}}$. To address this issue, we apply the SCA technique to approximate the second convex term in the right-hand-side of (\[function\]) by its first-order Taylor expansion. Note that a convex function is lower bounded by its first-order Taylor expansion at any given point. Thus, at the local point of ${\mbox{\boldmath{$ v $}}}^{(l-1)}$, we have $$\begin{aligned} &\mathcal{F}_i({\mbox{\boldmath{$ v $}}}, \{{\mbox{\boldmath{$ w $}}}_i^{(l)}\}, t^{(l)})\le \nonumber \\ &~~~\alpha_i t^{(l)}[\sum\limits_{k\ne i,k\in\mathcal K} ({\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ C $}}}_{i,k}{\mbox{\boldmath{$ v $}}}\!+\! 2\mathrm{Re}\{{\mbox{\boldmath{$ v $}}}^H{\mbox{\boldmath{$ u $}}}_{i,k}\}\!+\!|d_{i,k}|^2 )+\sigma^2_i] \nonumber \\ &~~~-\! ({{\mbox{\boldmath{$ v $}}}^{(l-1)}}^H{\mbox{\boldmath{$ C $}}}_{i,i}{{\mbox{\boldmath{$ v $}}}^{(l-1)}}\!+\!2\mathrm{Re}\{{{\mbox{\boldmath{$ v $}}}^{(l-1)}}^H{\mbox{\boldmath{$ u $}}}_{i,i}\}\!+\!|d_{i,i}|^2 ) \nonumber \\ &~~~-\!2 ({\mbox{\boldmath{$ C $}}}^H_{i,i}{{\mbox{\boldmath{$ v $}}}^{(l-1)}}\!+\!{\mbox{\boldmath{$ u $}}}_{i,i} )^H ({\mbox{\boldmath{$ v $}}}\!-\!{\mbox{\boldmath{$ v $}}}^{(l-1)} )\nonumber \\ &~~~\triangleq\mathcal{F}^{\mathtt{up}}_i ({\mbox{\boldmath{$ v $}}}, \{{\mbox{\boldmath{$ w $}}}_i^{(l)}\}, t^{(l)},{\mbox{\boldmath{$ v $}}}^{(l-1)} ).\label{upper_bound}\end{aligned}$$ By introducing an auxiliary variable $z$ and replacing $\mathcal{F}_i ({\mbox{\boldmath{$ v $}}}, \{{\mbox{\boldmath{$ w $}}}_i^{(l)}\}, t^{(l)} )$ by $\mathcal{F}^{\mathtt{up}}_i ({\mbox{\boldmath{$ v $}}}, \{{\mbox{\boldmath{$ w $}}}_i^{(l)}\}, t^{(l)},{\mbox{\boldmath{$ v $}}}^{(l-1)} )$, problem (P4) is approximated as the following problem: $$\begin{aligned} \mathtt{(P4.1):}\mathop\mathtt{min}_{{\mbox{\boldmath{$ v $}}}, z} ~& z \nonumber \\ {\mathtt{s.t.}} ~&\mathcal{F}^{\mathtt{up}}_i ({\mbox{\boldmath{$ v $}}}, \{{\mbox{\boldmath{$ w $}}}_i^{(l)}\}, t^{(l)},{\mbox{\boldmath{$ v $}}}^{(l-1)} )\le z, \forall i\in\mathcal K, \label{Problem:given_beam_SCA:1} \\ ~&(\ref{Problem:ori:3}). \nonumber\end{aligned}$$ Problem (P4.1) is a convex problem that can be solved optimally by CVX [@CVX]. Suppose that the optimal solution to problem (P4.1) is denoted as ${\mbox{\boldmath{$ v $}}}^{\star\star}$ and $t^{\star\star}$. By substituting ${\mbox{\boldmath{$ v $}}}^{\star\star}$ into $\mathcal{F}_i ({\mbox{\boldmath{$ v $}}}, \{{\mbox{\boldmath{$ w $}}}_i^{(l)}\}, t^{(l)} )$, it is evident that as any feasible solution problem (P4.1) is also feasible for problem (P4), the obtained value of (P4.1) by ${\mbox{\boldmath{$ v $}}}^{\star\star}$ is smaller than that of (P4). Therefore, ${\mbox{\boldmath{$ v $}}}^{\star\star}$ leads to an increased SINR value. Therefore, we can directly update ${\mbox{\boldmath{$ v $}}}$ as ${\mbox{\boldmath{$ v $}}}^{\star\star}$, i.e., ${\mbox{\boldmath{$ v $}}}^{(l)} = {\mbox{\boldmath{$ v $}}}^{\star\star}$. In summary, the alternating optimization with SCA is presented as Algorithm 2. ------------------------------------------------------------------------ **Algorithm 2**: Alternating optimization with SCA ------------------------------------------------------------------------ - Initialize: $l=0$, ${\mbox{\boldmath{$ v $}}}^{(0)}$ and accuracy threshold $\epsilon > 0$. - [${\mbox{\boldmath{$ \mathrm{Repeat} $}}}$:]{} -    $l=l+1$; -    Under given ${\mbox{\boldmath{$ v $}}}^{(l-1)}$, solve problem (P2) to obtain $\{{\mbox{\boldmath{$ w $}}}_i^\star\}$ and $t^\star$, and set ${\mbox{\boldmath{$ w $}}}_i^{(l)}={\mbox{\boldmath{$ w $}}}_i^\star, \forall i\in \mathcal K,$ and $t^{(l)}=t^\star$; -    Under given $\{{\mbox{\boldmath{$ w $}}}_i^{(l)}\}$, $t^{(l)}$, and ${\mbox{\boldmath{$ v $}}}^{(l-1)}$, solve problem (P4.1) to obtain ${\mbox{\boldmath{$ v $}}}^{\star\star}$, and set ${\mbox{\boldmath{$ v $}}}^{(l)}={\mbox{\boldmath{$ v $}}}^{\star\star}$; - ${\mbox{\boldmath{$ \mathrm{Until} $}}}$ the increase of the objective function in (P1) is smaller than $\epsilon$. ------------------------------------------------------------------------ \[Table:2\] It is worth noting that for the algorithm of alternating optimization with SCA, it is ensured that after each update of ${\mbox{\boldmath{$ v $}}}$ by SCA, the minimum SINR among all users is always non-decreasing. Therefore, the objective value of (P1) is ensured to be non-decreasing at each iteration. As a result, this algorithm will converge towards a stationary solution to problem (P1). Numerical Results ================= In this section, we provide numerical results to validate the performance of the proposed alternating-optimization-based algorithms in the IRS-aided multi-cell MISO system. In the simulation, there are $K=3$ BSs located at $(-100~\text{m},0)$, $(100~\text{m},0)$ and $(0,100~\text{m})$, respectively, each of which is equipped with $M=2$ antennas. We consider a scenario with symmetrically distributed users unless otherwise stated, where the three users are located at $(-d_\text{user},0)$, $(d_\text{user},0)$ and $(0,d_\text{user})$, with $d_\text{user}\!=\!5~\text{m}$. An IRS with $N\!=\!20$ reflecting units is deployed at $(0,-d_\text{IRS})$, with $d_\text{IRS} = 10~\text{m}$. Furthermore, we set the maximum transmit power at all BSs to be identical, i.e., $P_i=P_\text{max},\forall i\in\mathcal K$, and we are interested in the minimum SINR at users by setting $\alpha_i=1,\forall i\in\mathcal K$. In addition, we consider the distance-dependent path loss model as $$\begin{aligned} P_L=C_0 (\frac{d}{d_0} )^{-\alpha}, \label{fm:16}\end{aligned}$$ where $C_0=-30~$dB denotes the path loss at the reference distance of $d_0=1~$m, $\alpha$ denotes the path loss exponent, $d$ denotes the distance between the transmitter and receiver. For the BS-user, BS-IRS and IRS-user links, we set the path-loss exponents $\alpha$ to be $3.6$, $2$, and $2.5$, respectively. Furthermore, we consider line-of-sight (LOS) channels from BSs to the IRS, and Rayleigh fading for the BS-user and IRS-user links. The noise power at each user $i$ is set as $\sigma^2_i=-80~{\text{dBm}}, \forall i\in\mathcal K$. First, Fig. \[fig:Iteration\] shows the convergence behaviour of the two alternating-optimization-based algorithms, where $P_\text{max} = 35~$dBm. It is observed that the alternating optimization with SCA leads to monotonically increasing SINR values over iterations, thus converging towards a stationary solution; while the alternating optimization with SDR results in fluctuated SINR values due to the randomization process. Furthermore, it is also observed that running on a computer with E5-2667v4 CPU and 32G memory, the average run time of the alternating optimization with SDR is $895.1628$ seconds, while that of the alternating optimization with SCA is $432.7232$ seconds. This shows the advantage of SCA again in terms of the implementation complexity. Next, we evaluate the performance of the proposed two alternating-optimization-based algorithms, as compared with the following two benchmark schemes. - [**Benchmark scheme with random reflective phases**]{}: We set the phase shift $\theta_n$ of each unit $n$ at the IRS as a random value uniformly distributed in $[0, 2\pi)$, and set $\beta = 1$. Under such given reflective beamforming, we solve problem (P2) to obtain the corresponding coordinated transmit beamforming. - [**Benchmark scheme without IRS**]{}: Without IRS deployed, we only need to optimize the coordinated transmit beamforming by solving problem (P2), in which $\{{\mbox{\boldmath{$ a $}}}_{i,k}\}$ is replaced as $\{{\mbox{\boldmath{$ h $}}}_{i,k}\}$. Fig. \[fig:Power\] shows the minimum SINR at users versus the maximum transmit power $P_\text{max}$ at each BS, in which the results are averaged over $100$ random channel realizations. First, it is observed that the two proposed alternating-optimization-based algorithms considerably outperform the two benchmark schemes, and the performance gains become more significant when $P_\text{max}$ gets large. This shows the benefit of IRS in both signal enhancement and interference suppression, especially when the interference (or transmit power) becomes strong. Next, the alternating optimization with SCA is observed to lead to higher minimum SINR values than that with SDR. This is consistent with the observation in Fig. \[fig:Iteration\]. Furthermore, the benchmark scheme with random reflective phases is observed to have a similar performance as that without IRS. This shows that the benefit of IRS can only be gained via proper reflective beamforming optimization. To further reveal the practical performance, Fig. \[fig:Random\] shows the minimum SINR at users versus $P_\text{max}$, in another scenario with the three users uniformly distributed within a triangle area whose vertexes correspond to the three BSs. The results in Fig. \[fig:Random\] are obtained by averaging over $100$ random user realizations. Similar observations are made in Fig. \[fig:Random\] as in Fig. \[fig:Power\]. Nevertheless, the performance gain brought by IRS becomes less significant in this scenario with randomly distributed users, as the users are likely to be located at the cell center, such that the direct BS-user links become strong but the IRS-user links become weak. Conclusion ========== In this paper, we investigated the IRS-aided multi-cell MISO system, with the objective of maximizing the minimum weighted SINR at all users by jointly optimizing the coordinated transmit beamforming at BSs and reflective beamforming at the IRS, subject to individual power constraints at BSs. We proposed efficient alternating-optimization-based algorithms to update the transmit and reflective beamforming vectors in an alternating manner. In particular, we used the SOCP to optimize the transmit beamforming, and proposed two designs based on SDR and SCA for updating the reflective beamforming. Numerical results demonstrated that the dedicatedly deployed IRS considerably improves the performance of the multi-cell MISO system by not only enhancing the received signal strength but also suppressing the inter-cell interference, especially for cell-edge users. It was also shown that the SCA-based design is an efficient algorithm for optimizing the reflective beamforming at the IRS with guaranteed convergence, which outperforms the conventionally adopted SDR-based reflective beamforming optimization. J. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. Soong, and J. C. Zhang, “What will 5G be?” [*IEEE J. Sel. Areas Commun.*]{}, vol. 32, no. 6, pp. 1065–1082, Jun. 2014. K. B. Letaief, W. Chen, Y. Shi, J. Zhang, and Y. A. Zhang, “The roadmap to 6G: AI empowered wireless networks,” [*IEEE Commun. Magazine*]{}, vol. 57, no. 8, pp. 84–90, Aug. 2019. V. Chandrasekhar, J. Andrews, and A. Gatherer, “Femtocell networks: A survey,” [*IEEE Commun. Mag.*]{}, vol. 46, no. 9, pp. 59–67, Sep. 2008. D. Feng, L. Lu, Y. Yuan-Wu, G. Y. Li, S. Li, and G. Feng, “Device-to-device communications in cellular networks,” [*IEEE Commun. Mag.*]{}, vol. 52, no. 4, pp. 49-55, Apr. 2014. H. Dahrouj and W. Yu, “Coordinated beamforming for the multicell multi-antenna wireless system,” [*IEEE Trans. Wireless Commun.*]{}, vol. 9, no. 5, pp. 1748–1759, May 2010. Y.-F. Liu, Y.-H. Dai, and Z.-Q. Luo, “Coordinated beamforming for MISO interference channel: Complexity analysis and efficient algorithms,” [*IEEE Trans. Signal Process.*]{}, vol. 59, no. 3, pp. 1142–1157, Mar. 2011. R. Irmer, H. Droste, P. Marsch, M. Grieger, G. Fettweis, S. Brueck, H.P. Mayer, L. Thiele, and V. Jungnickel, “Coordinated multipoint: Concepts, performance, and field trial results,” [*IEEE Commun. Mag.*]{}, vol. 49, no. 2, pp. 102–111, Feb. 2011. Y.-F. Liu, Y.-H. Dai, and Z.-Q. Luo, “Max-min fairness linear transceiver design for a multi-user MIMO interference channel,” [*IEEE Trans. Signal Process.*]{}, vol. 61, no. 9, pp. 2413–2423, May 2013. J. Zhang, R. Chen, J. Andrews, A. Ghosh, and R. Heath, “Networked MIMO with clustered linear precoding,” [*IEEE Trans. Wireless Commun.*]{}, vol. 8, no. 4, pp. 1910–1921, Apr. 2009. R. Zhang, “Cooperative multi–cell block diagonalization with per-base-station power constraints,” [*IEEE J. Sel. Areas in Commun.*]{}, vol. 28, no. 9, pp. 1435–1445, Dec. 2010. M. Hong, Q. Li, and Y. Liu, “Decomposition by successive convex approximation: A unifying approach for linear transceiver design in heterogeneous networks," [*IEEE Trans. Wireless Commun.*]{}, vol. 15, no. 2, pp. 1377–1392, Feb. 2016. Q. Wu and R. Zhang, “Towards smart and reconfigurable environment: Intelligent reflecting surface aided wireless network," [*IEEE Commun. Mag.*]{}, vol. 58, no. 1, pp. 106–112, Jan. 2020. E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M. Alouini, and R. Zhang, “Wireless communications through reconfigurable intelligent surfaces," [*IEEE Access*]{}, vol. 7, pp. 116753–116773, Aug. 2019. Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network: Joint active and passive beamforming design,” in [*Proc. IEEE Globlecom*]{}, 2018, pp. 1–6, Dec. 2018. X. Yu, D. Xu, and R. Schober, “MISO wireless communication systems via intelligent reflecting surfaces : (Invited paper)," [*2019 IEEE/CIC ICCC*]{}, Changchun, China, pp. 735–740, Aug. 2019. Q. Wu and R. Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming," [*IEEE Trans. Wireless Commun.*]{}, vol. 18, no. 11, pp. 5394–5409, Nov. 2019. Y. Yang, B. Zheng, S. Zhang, and R. Zhang, “Intelligent reflecting surface meets OFDM: Protocol design and rate maximization.” \[Online\] Available: [<https://arxiv.org/abs/1906.09956>]{}. B. Zheng and R. Zhang, “Intelligent reflecting surface–enhanced OFDM: Channel estimation and reflection optimization,” [*IEEE Wireless Commun. Letters*]{}., pp. 1–1, Dec. 2019. Y. Yang, S. Zhang, and R. Zhang, “IRS–enhanced OFDM: Power allocation and passive array optimization,” in [*Proc. IEEE Globecom*]{}, Dec. 2019. \[Online\] Available: [<https://arxiv.org/abs/1905.00604>]{}. M. Fu, Y. Zhou, and Y. Shi, “Intelligent reflecting surface for downlink non-orthogonal multiple access networks,” in [*Proc. IEEE Globecom*]{}, Dec. 2019. \[Online\] Available: [<https://arxiv.org/abs/1906.09434>]{}. G. Yang, X. Xu, and Y. C. Liang, “Intelligent reflecting surface assisted non–orthogonal multiple access.” \[Online\] Available: [<https://arxiv.org/abs/1907.03133>]{}. Y. Tang, G. Ma, H. Xie, J. Xu, and X. Han, “Joint transmit and reflective beamforming design for IRS-assisted multiuser MISO SWIPT systems,” to appear in [*Proc. IEEE ICC 2020.*]{} \[Online\] Available: <https://arxiv.org/pdf/1910.07156>. C. Pan, H. Ren, K. Wang, W. Xu, M. Elkashlan, A. Nallanathan, and L. Hanzo, “Intelligent reflecting surface for multicell MIMO communications.” \[Online\] Available: [<https://arxiv.org/abs/1907.10864>]{}. A. Wiesel, Y. C. Eldar, and S. Shamai, “Linear precoding via conic optimization for fixed MIMO receivers,” [*IEEE Trans. Signal Process.*]{}, vol. 54, no. 1, pp. 161–176, Jan. 2006. Z. Luo, W. Ma, A. M. So, Y. Ye, and S. Zhang, “Semidefinite relaxation of quadratic optimization problems,” [*IEEE Signal Process. Mag.*]{}, vol. 27, no. 3, pp. 20–34, May 2010. M. Grant and S. Boyd, “CVX: MATLAB software for disciplined convex programming,” 2016. \[Online\] Available: [<http://cvxr.com/cvx>]{}. [^1]: This work was supported in part by the National Key R&D Program of China (No. 2018YFB1800800), the Natural Science Foundation of China (Nos. 61871137 and 11671419), the Guangdong Province Key Area R&D Program (No. 2018B030338001), and the Guangdong Province Basic Research Program (Natural Science) (No. 2018KZDXM028). J. Xu is the corresponding author.

--- abstract: 'Surface effects in strange-quark matter play an important role for certain observables which have been proposed in order to identify strange stars, and color superconductivity can strongly modify these effects. We study the surface of color superconducting strange-quark matter by solving the Hartree-Fock-Bogoliubov equations for finite systems (“strangelets”) within the MIT bag model, supplemented with a pairing interaction. Due to the bag-model boundary condition, the strange-quark density is suppressed at the surface. This leads to a positive surface charge, concentrated in a layer of $\sim$ 1 fm below the surface, even in the color-flavor locked (CFL) phase. However, since in the CFL phase all quarks are paired, this positive charge is compensated by a negative charge, which turns out to be situated in a layer of a few tens of fm below the surface, and the total charge of CFL strangelets is zero. We also study the surface and curvature contributions to the total energy. Due to the strong pairing, the energy as a function of the mass number is very well reproduced by a liquid-drop type formula with curvature term.' author: - Micaela Oertel - Michael Urban title: 'Surface effects in color superconducting strange-quark matter' --- Introduction ============ From rather general arguments it is expected that at low temperatures and high densities quark matter is in a color superconducting state [@colorsup]. More recently [@Alford97; @Rapp97] it has been suggested that the diquark pairing gaps for quark matter at densities of several times nuclear matter saturation density could be of the order of $\sim 100$ MeV. Since this could have important phenomenological consequences in particular for the interior of compact stars, this has triggered much work on color superconductivity in dense quark matter (for reviews, see, e.g., Ref. [@reviews]). These investigations of the QCD phase diagram have revealed a very rich phase structure with many different possible pairing patterns, depending on external conditions such as, for instance, electrical neutrality or quark masses. The largest diquark pairing gaps arise from scalar condensates, leading either to the two-flavor color superconducting (2SC) phase or to the color-flavor-locked (CFL) phase [@CFL; @weakCFL]. The latter pairing pattern involves strange ($s$) quarks, in addition to the two light quark flavors, up ($u$) and down ($d$). If color superconducting quark matter exists in nature, the most likely place to find it is the interior of compact stars because matter is compressed there to densities much higher than nuclear matter saturation density. However, it has been argued that strange-quark matter (SQM) might be absolutely stable [@Witten]. Under this hypothesis, even pure strange stars should exist [@strangestars], i.e., stars entirely composed of SQM. Also small lumps of SQM, called “strangelets,” might be stable. Because of their low charge to baryon number ratio $Z/A$, strangelets have been proposed to populate ultra-high energy cosmic rays [@cosmicrays]. In SQM without pairing, the density of strange quarks is supposed to be smaller than that of light quarks because of their higher mass. Consequently, SQM and strangelets are positively charged and the charge neutrality of strange stars has to be achieved via the presence of electrons. At the surface an atmosphere of electrons forms [@strangestars] which can potentially be detected [@Usov; @UsovPage] via the emission of electron-positron pairs from an extremely strong electric field at the surface. Recently another possible picture of the surface of a strange star has been proposed [@nuggets]: there could be a “crust” composed of strangelets immersed in an electron gas. Similar to an ordinary neutron star, there could be an interface between the crust and the interior in form of the famous “pasta phases.” Within this scenario the electric field at the surface would be strongly reduced. Obviously, surface effects for the strangelets play an important role for the description of this scenario. For instance, there is a critical surface tension deciding whether a homogeneous phase or the droplet phase is favored [@stablenuggets]. Another question for which surface effects should be considered is the formation of a strange star in a supernova explosion. Before the explosion the original star contains hadronic matter. During the formation of the star, nucleation of strangelets sets in, leading then to a conversion of the entire star to SQM. For the nucleation process the properties of small strangelets are important. Pairing tends to reduce the differences in density of different quark species. For bulk quark matter in the CFL state, requiring color neutrality, all quarks are paired. The densities are thus equal and CFL quark matter is electrically neutral on its own, i.e., without any electrons [@RW01]. This would suggest dramatic changes in the properties of strangelets and SQM inside compact stars. For instance, the electrosphere at the surface of a strange star could completely disappear. But, the presence of the surface can modify this picture since it can lead to a non-zero surface charge which remains even for large objects. For example, the boundary condition of the MIT bag model suppresses the density of the massive strange quarks at the surface, resulting in a positive surface charge [@Madsen01]. Within this scenario, the total charge of a strangelet, following roughly $Z \approx 0.3 A^{2/3}$, is drastically reduced with respect to “normal” strangelets. For strange stars, this requires the presence of electrons [@Usov04]. However, pairing has not been treated self-consistently in previous work (see, e.g., Ref. [@Madsen01]). In this paper we will therefore reinvestigate finite-size strangelets with pairing by considering quark matter in a color superconducting spherical bag, solving the Hartree-Fock-Bogoliubov (HFB) equations. We will show, in particular, that there exist CFL type solutions where all quarks are paired and the total charge of the strangelet strictly vanishes. The outline of the paper is as follows. In Section \[sec:model\] we will present our model for treating color superconducting quark matter in a finite volume. In Section \[sec:results\] we will show numerical results. In Section \[sec:solutions\] we discuss the possibility of qualitatively different configurations. In Section \[sec:chdens\] we concentrate on the charge-density distributions of the CFL like solutions. In Section \[sec:liquiddrop\] we discuss a liquid-drop like mass formua for the CFL-like solutions and calculate the surface tension. Finally, in Section \[sec:sum\] we will summarize our results. Model {#sec:model} ===== Lagrangian ---------- Since it is not possible to describe strangelets or SQM with a surface from first principles (QCD), we will use a quark model which allows to describe finite-size objects. For this purpose we will use here the MIT bag model [@MITbag]. The idea of this model is that confinement can be simulated by the existence of a “bag” which consists of a “hole” in the non-perturbative QCD vacuum. Inside this “bag”, the vacuum is supposed to be perturbative, i.e., inside the bag the interactions of the quarks can be treated perturbatively. To create this “hole” in the non-perturbative QCD vacuum, an energy per volume, $B$, is necessary. In the present work we will consider a static spherical bag with radius $R$. On the surface of the bag, the quark field $\psi$ has to satisfy an appropriate boundary condition. In the simplest version of the MIT bag model, the boundary condition reads $$-i {\bm{\mathrm{e}}}_r\cdot {\bm{\mathrm{\gamma}}} \psi = \psi|_{r = R}\,. \label{boundarycondition}$$ which ensures that there is no particle flux across the surface. By $r = |{\bm{\mathrm{r}}}|$ we denote the radial coordinate, measured from the center of the bag, and ${\bm{\mathrm{e}}}_r = {\bm{\mathrm{r}}}/r$ is the radial unit vector. The boundary condition (\[boundarycondition\]) leads to a suppression of the wave functions of massive particles at the surface. This means that the strange-quark density will a priori be suppressed at the surface with respect to the light quark densities. The MIT bag model can be expressed in terms of a Lagrangian density as follows [@Bhaduri]: $$\begin{gathered} \mathcal{L}_{\mathit{bag}} = [\bar{\psi}(i\gamma^\mu\partial_\mu-m)\psi-B]\theta(R-r)\\ -\frac{1}{2}\bar{\psi}\psi\delta(R-r)\,, \label{Lbag}\end{gathered}$$ where $m$ is the matrix of quark masses. Due to the second term, the boundary condition (\[boundarycondition\]) follows immediately from the Euler-Lagrange equation for the quark field [@Bhaduri]. In order to include pairing, we will supplement the bag model with a pairing interaction. In principle, perturbative one-gluon exchange generates an attractive paring interaction in certain channels, in particular in the scalar color antitriplet channel. For simplicity, we will use here a four-point pairing interaction acting only in this dominant channel. The corresponding Lagrangian reads (see any of the standard review articles on color superconductivity [@reviews]) $$\mathcal{L}_\mathit{pair} = H \sum_{A,A^\prime} (\bar{\psi} i \gamma_5 \tau_A \lambda_{A^\prime} C\bar{\psi}^T) (\psi^T C i \gamma_5 \tau_A \lambda_{A^\prime} \psi)\,,$$ where $H$ is a dimensionful coupling constant, $C$ denotes the charge conjugation matrix, and $\tau_A, \lambda_{A^\prime}$ represent $SU(3)$ matrices in flavor and color space, respectively. We follow the convention that capital letters $A, A^\prime$ indicate that we are restricting $\tau_A$ and $\lambda_{A^\prime}$ to be antisymmetric, i.e., in terms of the Gell-Mann matrices, $A,A^\prime \in \{2,5,7\}$. In addition to the strong interaction, the quarks will exhibit electromagnetic interactions which, due to their long range, become particularly important for large objects. The corresponding Lagrangian reads $$\mathcal{L}_{\mathit{e.m.}} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} -e\bar{\psi}Q A_\mu\gamma^\mu\psi,$$ where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ and $A_\mu$ denote respectively the electromagnetic field strength tensor and four-potential, and $Q$ is the matrix of quark charges in units of $e$, $Q_u = 2/3$, $Q_d = Q_s = -1/3$. It would be in the spirit of the bag model to include also the gluon exchange in a perturbative way, i.e., in the same way as the photon. However, this goes beyond the scope of the present paper and will be postponed to a future study. Solution in the framework of HFB theory {#sec:hfb} --------------------------------------- The model described by $\mathcal{L} = \mathcal{L}_{\mathit{bag}} + \mathcal{L}_{\mathit{pair}} + \mathcal{L}_{\mathit{e.m.}}$ will be treated in the framework of HFB theory. By minimizing the energy in mean field approximation (for more details see Appendix \[app:hfb\] and Ref. [@CH] where the “Dirac-Hartree-Bogoliubov” approximation was developped for finite nuclei), one obtains the following HFB equations: $$\begin{pmatrix} h & \Delta \\ \Delta & -h \end{pmatrix} \begin{pmatrix} U_\alpha({\bm{\mathrm{r}}}) \\ \gamma^0 V_\alpha({\bm{\mathrm{r}}}) \end{pmatrix} = \epsilon_\alpha \begin{pmatrix} U_\alpha({\bm{\mathrm{r}}}) \\ \gamma^0 V_\alpha({\bm{\mathrm{r}}}) \end{pmatrix}\,. \label{hfbequations}$$ The single-particle Hamiltonian $$h = -i {\bm{\mathrm{\alpha}}} \cdot{\bm{\mathrm{\nabla}}} + m \gamma^0 + \Sigma - \mu$$ includes besides the free Dirac Hamiltonian the quark self-energy $\Sigma$ (in our case due to Coulomb interaction) and the matrix of chemical potentials $\mu$ which depend on flavor $f \in \{u,d,s\}$ and color $c \in \{r,g,b\}$ (we will denote the three colors by red, green, and blue). $\Delta$ denotes the pairing field (gap). The spinors $U_\alpha$ and $V_\alpha$ describe the particle- and hole-like components of the quark fields, respectively \[see Eq. (\[greensfunctions\])\], where $\alpha$ is a multi-index containing all quantum numbers characterizing a single-particle state (see Appendix \[app:basis\]). In writing Eq. (\[hfbequations\]), we implicitly assumed that the pairing field $\Delta$ can be chosen real, which is the case for the pairing pattern we consider, and that the self-energy $\Sigma$ is local, which is the case since we neglect the exchange (Fock) term (see below). The pairing field $\Delta$ and the self-energy $\Sigma$ depend themselves on the wave functions $U$ and $V$, such that we have to solve a self-consistency problem. To be specific, the pairing field $\Delta$ depends on the diquark condensates $$s_{AA^\prime}(x) = - \langle \bar{\psi_T}(x) \tau_A \lambda_{A^\prime} \psi(x) \rangle\,, \label{saa}$$ where $\psi_T$ denotes the time-reversed conjugate of $\psi$, $$\psi_T = \gamma_5 C \bar{\psi}^T\,.$$ The diquark condensates can be expressed in terms of the $U$ and $V$ functions as $$s_{AA^\prime}(r) = -\sum_{\beta, \epsilon_\beta < 0} \bar{V}_\beta({\bm{\mathrm{r}}}) \tau_A\lambda_{A^\prime} U_\beta({\bm{\mathrm{r}}}) \label{saadivergent}$$ (since we are dealing with a static problem, the condensates do not depend on time, and due to spherical symmetry, they depend only on the radial coordinate $r$). We will limit our investigations here to diagonal condensates, i.e., only condensates with $A = A^\prime$ are non-zero[^1]. In uniform infinite matter and for an exact $SU(3)$ flavor symmetry, the CFL phase is characterized by nonzero values $s_{22} = s_{55} = s_{77}$, whereas the 2SC state has only $s_{22} \neq 0$. The relation between the condensates $s_{AA}$ and the pairing field $\Delta$ reads $$\begin{gathered} \Delta(r) = \sum_{A=2,5,7} \Delta_A(r) \tau_A \lambda_A\,,\\ \Delta_A(r) = 2\, H s_{AA}(r)\,. \label{gapequation}\end{gathered}$$ In practice, the expression (\[saadivergent\]) is divergent and it is necessary to introduce a cutoff in order to obtain a finite result. Since in a finite system the levels are discrete, a sharp cutoff would generate discontinuities as a function of the system’s size. We therefore introduce a smooth cutoff function $f(p/\Lambda)$ (see Appendix \[app:cutoff\] for details). Another practical problem arises from antiparticle contributions. However, since the chemical potentials $\mu_{fc}$ are large and positive and pairing involves mostly the states near the Fermi surface, we assume that the antiparticle contributions are not important and can be neglected. We checked this approximation (analogous to the “no-sea approximation” in nuclear physics [@CH]) in infinite matter and found that the effect of antiparticle states can be absorbed in a readjustment of the coupling constant by $\sim 20\,\%$. For the normal self-energy $\Sigma$ we employ the Hartree approximation, i.e., we neglect the Coulomb exchange (Fock) term as well as exchange contributions from the magnetic field. We also disregard the contribution of $\mathcal{L}_{\mathit{pair}}$ to the normal self-energy. Hence, the self-energy is simply proportional to the static Coulomb potential $$\Sigma(r) = e Q A_0(r)\gamma^0\,.$$ The Coulomb potential is related to the quark densities by $$A_0(r) = e \int d^3 r^\prime \frac{\rho_\mathit{ch}({\bm{\mathrm{r}}}^\prime)}{|{\bm{\mathrm{r}}}-{\bm{\mathrm{r}}}^\prime|}\,, \label{vcoulomb}$$ where $$\rho_\mathit{ch}(r) = \sum_f Q_f\rho_f(r)$$ is the charge density (divided by $e$), $\rho_f$ being the number density of quarks of flavor $f$. As it was the case for the diquark condensates, the quark number densities can be expressed in terms of the $U$ and $V$ functions. Denoting by $\tilde{\beta}$ all single-particle quantum numbers except flavor, we can write the number density of quarks of flavor $f$ as $$\rho_f(r) = \sum_{\tilde{\beta}, \epsilon_{f\tilde{\beta}} < 0} U^\dagger_{f\tilde{\beta}}({\bm{\mathrm{r}}}) U_{f\tilde{\beta}}({\bm{\mathrm{r}}})\,. \label{rhof}$$ Let us now summarize the procedure how the HFB equations are solved. We start with an initial guess for the pairing fields $\Delta_A(r)$ and for the Coulomb potential $A_0(r)$. Then we solve the eigenvalue problem (\[hfbequations\]) in order to find the $U$ and $V$ functions. From these functions the diquark condensates $s_{AA}(r)$ and the quark densities $\rho_f(r)$ are computed according to Eqs. (\[saadivergent\]) and (\[rhof\]), which are then used to update the pairing fields $\Delta_A$ and the Coulomb field $A_0$ according to Eqs. (\[gapequation\]) and (\[vcoulomb\]). These steps are iterated until convergence (i.e., self-consistency) is reached. The crucial difference to the BCS formalism in homogeneous infinite matter is that in our case the wave functions adapt themselves to the pairing field and to the Coulomb potential, whereas in the case of homogeneous infinite matter the wave functions always stay plane waves, and the $U$ and $V$ factors are just coefficients multiplying them. Determination of chemical potentials and bag radius --------------------------------------------------- In Section \[sec:hfb\] we described how the HFB equations are solved for given values of the chemical potentials $\mu_{fc}$ and of the bag radius $R$. However, in reality, only one quantity is given, namely the baryon number $A$. Even the fractions of different quark flavors cannot be fixed, unless one allows for $\beta$ unstable strangelets. Let us now describe how we determine the chemical potentials $\mu_{fc}$ and the bag radius $R$ for given baryon number $A$. The first step consists in fixing the quark numbers, $N_{fc}$, for each flavor $f$ and color $c$, and to adjust the chemical potentials $\mu_{fc}$ in order to obtain these quark numbers. Before we address the question how the nine quark numbers $N_{fc}$ are determined, let us discuss the issue of the bag radius $R$. Until now, the radius was imposed from outside, but in reality the system will choose its radius such that it minimizes its total energy for given quark numbers $N_{fc}$. Within the bag model, the total energy is given by $$E = E_q+BV\,, \label{Etotal}$$ where $V = 4\pi R^3/3$ is the volume of the bag. By $E_q$ we denote the energy of the quarks inside the bag, including the interaction energy, which in our case comes from pairing and Coulomb interactions. It can be obtained from the solution of the HFB equations as follows [@CH]: $$\begin{gathered} E_q = \int_{r<R} d^3 r \sum_{\beta,\epsilon_\beta < 0} \left(U^\dagger_\beta({\bm{\mathrm{r}}}) (\epsilon_\beta+\mu)U_\beta({\bm{\mathrm{r}}})\phantom{\frac{1}{2}}\right.\\ \left.+\frac{1}{2}\left[\bar{U}_\beta({\bm{\mathrm{r}}})\Delta(r)V_\beta({\bm{\mathrm{r}}}) -U^\dagger_\beta({\bm{\mathrm{r}}})e Q A_0(r)U_\beta({\bm{\mathrm{r}}})\right]\right)\,. \label{quarkenergy}\end{gathered}$$ Minimizing the total energy $E$ is of course completely equivalent to saying that the quark pressure in the bag is counterbalanced by the bag pressure $B$, i.e., $$\left.\frac{dE_q}{dV}\right|_N = -B\,. \label{dEdV}$$ This equation determines the radius of the strangelet for given bag pressure $B$, interaction strength $H$ and quark numbers $N_{fc}$. In practice, however, we find it more convenient to minimize $E$ rather than solve Eq. (\[dEdV\]). Let us now turn to the determination of the quark numbers. The nine quark numbers $N_{fc}$ cannot be chosen arbitrarily, but they have to fulfil certain requirements. Imposing the total baryon number $A$ and color neutrality, i.e., equal numbers of quarks for each color, we have to satisfy the constraint $$\sum_f N_{fc} = A \qquad \mbox{for all $c$}\,. \label{numberconstraint}$$ Of course, these three equations are not sufficient for determining all the nine quark numbers. In order to get unique values for the $N_{fc}$, it is necessary to impose $\beta$ stability, as we will describe now. In an infinite homogeneous system the condition for $\beta$ equilibrium gives just a relation between the chemical potentials[^2] $$\mu_{dc} = \mu_{sc} = \mu_{uc} + \mu_{e}\qquad\mbox{for all $c$}\,. \label{betainfinite}$$ In a small system this is slightly different. First, even if there are electrons (i.e., if the strangelet is charged), they are not localized inside the strangelet, but they form a large cloud like in ordinary atoms and hence their chemical potential $\mu_e$ is approximately equal to the electron mass and can be neglected. Without pairing, it has been estimated in Ref. [@FarhiJaffe] that this may be still the case for strangelets with charge $Z \lesssim 1000$, corresponding roughly to $A\lesssim 10^6$. The second difference to bulk matter comes from the fact that, due to the discrete levels, particle numbers are discontinuous functions of the chemical potentials. The term $\beta$ equilibrium should now be replaced by $\beta$ stability, which means that the system does not gain energy by performing a $\beta$ decay, inverse $\beta$ decay, or electron capture, i.e., transforming an up into a down or strange quark, or vice versa, accompanied by the corresponding leptons. To achieve $\beta$ stability, we therefore compare the energies of adjacent strangelets with the same total quark number per color, differing only in the number of up, down, and strange quarks, respectively, in order to find the configuration with the lowest total energy $E$. Of course, in the case of large particle numbers, the minimum-energy configuration fulfils approximately the condition (\[betainfinite\]). Choice of the model parameters ------------------------------ Besides the quark masses, which we take as $m_u = m_d = 0$ and $m_s = 120$ MeV, our model contains three parameters: the bag constant $B$, the coupling constant of the pairing interaction, $H$, and the cutoff $\Lambda$ which is necessary to avoid the divergence of the gap equation (\[saadivergent\]), see below Eq. (\[gapequation\]). In fact, a change of the cutoff in reasonable limits can to very good approximation be compensated by a change of the coupling constant. We therefore choose rather arbitrarily $\Lambda = 600$ MeV and give instead of the dimensionful coupling constant $H$ the dimensionless combination $H\Lambda^2$. So we are left with two parameters, $B$ and $H\Lambda^2$ We can get an idea of the value of the bag pressure by looking at the stability of bulk quark matter. Non-strange quark matter should be energetically less favored than normal hadronic matter, whereas SQM should be stable if for some baryon number $A > A_c$ strangelets become stable and consequently strange stars can exist. This means that we want the energy per baryon of SQM to be less than 931 MeV, the energy per baryon of the most stable nucleus, $^{57}$Fe. On the other hand, the energy per baryon of non-strange quark matter should be larger than the nucleon mass. Without interaction the window for the values of the bag constant is then $148$ MeV $< B^{1/4} < 157$ MeV. These values change as a function of the interaction strength $H$. To better compare the results, we will readjust for each coupling strength the bag constant in order to get $E/A = 900$ MeV. The corresponding values are listed in Table \[tab:bagvalue\], together with other properties of infinite matter. Non-strange quark matter is unstable with these parameter values. Note that for the weakest non-vanishing coupling constant given in Table \[tab:bagvalue\], SQM is in the 2SC phase and not in the CFL phase. $H \Lambda^2$ $\begin{matrix}B^{1/4}\\ \mbox{(MeV)}\end{matrix}$ $\begin{matrix}\rho_B\\ \mbox{(fm$^{-3}$)}\end{matrix}$ $\begin{matrix}\rho_e\\ \mbox{(fm$^{-3}$)}\end{matrix}$ $\begin{matrix}\Delta_2\\ \mbox{(MeV)}\end{matrix}$ $\begin{matrix}\Delta_5=\Delta_7\\ \mbox{(MeV)}\end{matrix}$ --------------- ---------------------------------------------------- --------------------------------------------------------- --------------------------------------------------------- ----------------------------------------------------- -------------------------------------------------------------- -- 0 152.03 0.329 7.3$\times 10^{-6}$ 0 0 1.5 152.44 0.339 9.7$\times 10^{-5}$ 27.7 0 1.75 153.97 0.367 0 35.1 34.5 2 156.26 0.395 0 50.6 49.7 2.25 159.46 0.427 0 67.2 66.0 2.5 163.46 0.463 0 84.6 83.1 : \[tab:bagvalue\] Values of the bag constants for different values of the coupling constant $H$, resulting in color and electrically neutral SQM with electrons in $\beta$ equilibrium with an energy per baryon of $E/A = 900$ MeV. The corresponding baryon densities $\rho_B$, electron densities $\rho_e$, and pairing gaps in infinite matter are also displayed. For the larger coupling constants, the CFL phase is preferred. Note that, due to the mass difference of light and strange quarks, the flavor $SU(3)$ symmetry is not exact and the gap $\Delta_2$ is different from $\Delta_5$ and $\Delta_7$. However, since the CFL phase is electrically neutral, and we have $m_u= m_d = 0$, the isospin $SU(2)$ symmetry in the up- and down-quark sector is exact and therefore $\Delta_5 = \Delta_7$. Results {#sec:results} ======= Different types of solutions {#sec:solutions} ---------------------------- We will first discuss the qualitatively different configurations we find. Let us start by discussing a small strangelet ($A = 108$, $Z = 24$) without any pairing interaction ($H\Lambda^2 = 0$). The mass number has been chosen such that the minimum-energy configuration is a closed-shell configuration. The quark numbers and other relevant information are listed in Table \[tab:strangelets\]. [cccccccccc]{} $\begin{matrix}B^{1/4}\\\mbox{(MeV)}\end{matrix}$ & $\begin{matrix}H\Lambda^2\\ \mbox{(MeV)}\end{matrix}$ & $A$ & $Z$ & ------------------------------------------------------------------------ $\left(\begin{smallmatrix} N_{ur}& N_{ug}& N_{ub}\\ N_{dr}& N_{dg}& N_{db}\\ N_{sr}& N_{sg}& N_{sb}\end{smallmatrix}\right)$ & $\begin{matrix} E/A\\ \mbox{(MeV)}\end{matrix}$ & $\begin{matrix} R\\ \mbox{(fm)}\end{matrix}$ & $\begin{matrix} \Delta_2(0)\\ \mbox{(MeV)}\end{matrix}$ & $\begin{matrix} \Delta_5(0)\\ \mbox{(MeV)}\end{matrix}$ & $\begin{matrix} \Delta_7(0)\\ \mbox{(MeV)}\end{matrix}$\ 152.03 & 0 & 108 & 24 & ------------------------------------------------------------------------ $\left(\begin{smallmatrix} 44&44&44\\44&44&44\\20&20&20\end{smallmatrix}\right)$ & 932.5 & 4.36 & 0 & 0 & 0\ 152.44 & 1.5 & 108 & 24 & ------------------------------------------------------------------------ $\left(\begin{smallmatrix} 44&44&44\\44&44&44\\20&20&20\end{smallmatrix}\right)$ & 930.7 & 4.31 & 32.9 & 0 & 0\ 153.97 & 1.75 & 108 & 24 & ------------------------------------------------------------------------ $\left(\begin{smallmatrix} 44&44&44\\44&44&44\\20&20&20\end{smallmatrix}\right)$ & 934.0 & 4.24 & 49.5 & 0 & 0\ 153.97 & 1.75 & 108 & 10 & ------------------------------------------------------------------------ $\left(\begin{smallmatrix} 38&39&41\\39&38&41\\31&31&26\end{smallmatrix}\right)$ & 934.8 & 4.21 & 41.6 & 24.9 & 24.9\ 153.97 & 1.75 & 108 & 0 & ------------------------------------------------------------------------ $\left(\begin{smallmatrix} 38&35&35\\35&38&35\\35&35&38\end{smallmatrix}\right)$ & 934.8 & 4.17 & 33.9 & 37.0 & 36.9 Due to the finite size of the bag, the energy per baryon ($E/A = 932.5$ MeV, including 1.0 MeV due to Coulomb) is much higher than that of color neutral infinite matter with $\mu_{uc} = \mu_{dc} = \mu_{sc}$[^3] ($E/A = 899.5$ MeV). This effect will be discussed in more detail in Section \[sec:liquiddrop\]. The density profiles of light and strange quarks are shown in Fig. \[fig:rho\_H0\_A108\_Z24\]. ![Quark number density profiles of the strangelet $A = 108$, $Z = 24$ in the case of vanishing pairing interaction (free quarks in a bag) and $B^{1/4} = 152.03$ MeV.[]{data-label="fig:rho_H0_A108_Z24"}](figure1.eps){width="7cm"} As expected, due to the boundary condition, the strange-quark density is strongly suppressed at the surface, contrary to the densities of the light quarks. For comparison we mention that for the same value of the bag constant, the densities in color neutral infinite matter with $\mu_{uc} = \mu_{dc} = \mu_{sc}$ are: $\rho_u = \rho_d = 0.355$ fm$^{-3}$, $\rho_s = 0.274$ fm$^{-3}$. We see that not only the strange-quark density, but also the densities of the light quarks are quite different from these values and depend strongly on $r$ because of the existence of discrete levels in the bag. Let us mention that, due to the Coulomb potential, the density profiles of up and down quarks are slightly different, but the difference is too small to be visible in Fig. \[fig:rho\_H0\_A108\_Z24\]. Now we switch on the pairing interaction. In the case of $H\Lambda^2 = 1.5$, SQM is in the 2SC phase, i.e., only up and down quarks of two colors (red and green in our notation) are paired. This is also true in a finite strangelet. Therefore it is clear that the strange-quark density profile remains the same as without pairing. The oscillations of the densities of the light quarks, however, are much weaker now than in the case without pairing, since pairing washes out the occupation numbers. This can be seen in Fig. \[fig:rho\_H1.5\_A108\_Z24\]. ![Quark number density profiles of the strangelet $A = 108$, $Z = 24$ in the case of $H\Lambda^2 = 1.5$ and $B^{1/4} = 152.44$ MeV.[]{data-label="fig:rho_H1.5_A108_Z24"}](figure2.eps){width="7cm"} In this 2SC-like solution, only one of the gaps, $\Delta_2$, is non-zero. Since $\Delta_2$ involves only the wave functions of up and down quarks, which are not suppressed at the surface, it extends up to the surface of the bag, as shown in Fig. \[fig:delta\_H1.5\_A108\_Z24\]. ![Gap $\Delta_2(r)$ of the strangelet $A = 108$, $Z = 24$ in the case of $H\Lambda^2 = 1.5$ and $B^{1/4} = 152.44$ MeV.[]{data-label="fig:delta_H1.5_A108_Z24"}](figure3.eps){width="7cm"} As a function of $r$, it is almost constant and quite close to the corresponding value in infinite matter with $\mu_{uc} = \mu_{dc} = \mu_{sc}$, which is $\Delta_2 = 29.2$ MeV. If we increase the coupling constant to $H\Lambda^2 = 1.75$, we obtain three qualitatively different solutions which have comparable energies. The most stable one is still of the 2SC type, although in infinite matter the CFL phase is preferred. In this case, the strangelet still has $Z = 24$ and the density profiles are almost identical to those shown in Fig. \[fig:rho\_H1.5\_A108\_Z24\]. The main difference is that now the value of the gap is larger. In the two other solutions, also strange quarks participate in pairing ($\Delta_5 \approx \Delta_7\neq 0$ – note that $\Delta_5$ and $\Delta_7$ are not exactly equal because the isospin symmetry is broken by the Coulomb interaction). These two solutions have charge $Z = 10$ and $Z = 0$, respectively. Let us first discuss the case $Z = 10$. In this case, there are a couple of up and down quarks which remain unpaired. The wave function of the unpaired level is mainly localized near the surface of the bag, as can be seen in Fig. \[fig:rho\_H1.75\_A108\_Z10\], where the density profiles are shown. ![Density profiles of the strangelet $A = 108$, $Z = 10$ in the case of $H\Lambda^2 = 1.75$ and $B^{1/4} = 153.97$ MeV.[]{data-label="fig:rho_H1.75_A108_Z10"}](figure4.eps){width="7cm"} In the inner part, the densities of up, down, and strange quarks are almost equal, while near the surface, where the strange-quark density is suppressed due to the boundary condition, there is an excess of up and down quarks. This excess is due to the unpaired quarks. The fact that one level of up and down quarks (in the present case the $1g_{9/2}$ level, i.e., the lowest level with $j = 9/2, \kappa = -5$ in the notation of Appendix \[app:basis\]) does not participate in pairing means that the occupation number of this level is equal to 1. At the same time, the corresponding level of the strange quarks has an occupation number equal to 0. In a certain sense this situation is analogous to the “breached pairing” phase of infinite matter [@LiuWilczek]. The charge $Z$ is equal to the degeneracy $2j+1$ of the unpaired level. The gaps $\Delta_A$ as functions of $r$ corresponding to this solution are displayed in Fig. \[fig:delta\_H1.75\_A108\_Z10\]. ![Gaps $\Delta_A$ as functions of $r$ for the strangelet $A = 108$, $Z = 10$ in the case of $H\Lambda^2 = 1.75$ and $B^{1/4} = 153.97$ MeV.[]{data-label="fig:delta_H1.75_A108_Z10"}](figure5.eps){width="7cm"} In the third solution, all quarks are paired. As a consequence, the numbers of up, down, and strange quarks are equal, and the total charge is $Z = 0$. This is analogous to the CFL phase in the infinite system. Since the strange-quark density is suppressed near the surface, but the number of strange quarks is equal to that of up and down quarks, it is clear that the strange-quark density must be larger than the up- and down-quark densities in some other part of the system. This is indeed the case, as can be seen in Fig. \[fig:rho\_H1.75\_A108\_Z0\]. ![Density profiles of the strangelet $A = 108$, $Z = 0$ in the case of $H\Lambda^2 = 1.75$ and $B^{1/4} = 153.97$ MeV.[]{data-label="fig:rho_H1.75_A108_Z0"}](figure6.eps){width="7cm"} We also see that the excess of the light-quark densities over the strange-quark density is reduced as compared with the case $Z=10$ discussed above (cf. Fig. \[fig:rho\_H1.75\_A108\_Z10\]). We will discuss the charge-density distribution in detail in Section \[sec:chdens\]. The gaps, shown in Fig. \[fig:delta\_H1.75\_A108\_Z0\], are much closer to the gaps in infinite matter (cf. Table \[tab:bagvalue\]) than in the case $Z=10$. ![Gaps $\Delta_A$ as functions of $r$ for the strangelet $A = 108$, $Z = 0$ in the case of $H\Lambda^2 = 1.75$ and $B^{1/4} = 153.97$ MeV.[]{data-label="fig:delta_H1.75_A108_Z0"}](figure7.eps){width="7cm"} For the larger values of the coupling constant we considered ($H\Lambda^2 = 2, 2.25, 2.5$), it is always the CFL-type solution ($Z = 0$) which has the lowest energy. We do not show any figures because in all these cases the results are analogous to those shown in Figs. \[fig:rho\_H1.75\_A108\_Z0\] and \[fig:delta\_H1.75\_A108\_Z0\] (just the values of the gaps change, they are close to those given in Table \[tab:bagvalue\] for infinite matter). It should be mentioned that the fully paired solutions with $Z = 0$ are very robust as soon as the coupling constant is sufficiently large, i.e., we find this type of solution for arbitrary numbers of quarks[^4]. This solution is in contrast to previous findings (see, e.g., Ref. [@Madsen01]), where it was supposed that the CFL matter should be neutral in the bulk with just a thin positively charged surface layer with an excess of up and down quarks because of the boundary condition. In fact, this idea corresponds roughly to our solution with unpaired up and down quarks near the surface. This solution is, however, very fragile and exists only for certain values of parameters and mass numbers, since it requires the existence of a suitable level of light and strange quarks near the respective Fermi surfaces which can serve as unpaired level. Charge density distribution {#sec:chdens} --------------------------- We have seen in Section \[sec:solutions\] that in all cases except the 2SC phase, pairing drastically reduces the total charge $Z$. Because of surface effects, the local charge density does, however, not vanish, even within the CFL-type solution which has $Z=0$. Due to the suppression of the strange-quark wave function at the surface, a positively charged surface layer remains with an extension of $\sim 1$ fm, as has already been pointed out in Ref. [@Madsen01]. Within the configuration with some unpaired light quarks at the surface, the total charge of the strangelet results from this positive surface charge, the interior of the strangelet has almost zero charge density. The total charge is here reduced compared with a strangelet without pairing, for example the $A = 108$ strangelet has $Z = 10$ within this paired configuration, whereas the corresponding unpaired strangelet has $Z = 24$. A systematic study of the total charge of strangelets in this configuration will not be discussed here since this configuration is rather fragile with respect to the details of the single-particle spectra and thus difficult to realize for many different particle numbers. Let us therefore concentrate on the CFL-type solution, which exists for arbitrary particle numbers. We consider different mass numbers $A$ from $A = 108$ to $A = 90000$, for one particular value of the coupling constant, $H\Lambda^2 = 2$. In order to reduce the considerable numerical effort, we use for the large strangelets (starting from $A = 15000$) the condition (\[betainfinite\]) with $\mu_e = 0$ (as a consequence, the quark numbers for each flavor and color are not integers) instead of looking for the true energy minimum with respect to $\beta$ decay. In addition, we do not minimize the energy with respect to the radius, but we simply estimate the volume of the bag by dividing the mass number $A$ by the baryon density $\rho_{B\,\mathit{bulk}}$ of infinite matter. These two approximations are very accurate for such large strangelets. Already in the case of $A = 3000$, the quark numbers and the radius are very well reproduced within these approximations: the full minimization results in quark numbers $N_{ur} = N_{dg} = N_{sb} = 1052$, $N_{ug} = N_{dr} = N_{ub} = N_{sr} = N_{db} = N_{sg} = 974$, and a radius $R = 12.23$ fm, while the approximations lead to $N_{ur} = 1051.8$, $N_{dg} = 1051.7$, $N_{sb} = 1051.1$, $N_{ug} = N_{dr} = 973.8$, $N_{ub} = N_{sr} = 974.4$, $N_{db} = N_{sg} = 974.5$, and $R = 12.19$ fm. Our results for the charge densities for $A = $108, 3000, 15000, 45000, and 90000 are shown in Fig. \[fig:coulomb1\]. ![Charge density profiles of the fully paired ($H\Lambda^2 = 2$) strangelets $A = $108, 3000, 15000, 45000, and 90000 (from left to right).[]{data-label="fig:coulomb1"}](figure8.eps){width="7cm"} Since all quarks are paired, we have equal numbers of up, down, and strange quarks such that the total charge of these strangelets is zero. The positive surface charge is mostly compensated by an excess of negative charge concentrated at around 1-3 fm below the surface. We stress that this concentration of negative charge in a thin layer is a consequence of pairing and the effect persists if Coulomb interaction is switched off. In fact, since also the strange quarks are paired, the “missing” strange-quark density at the surface must be compensated by an “overshooting” of the strange-quark density within a distance corresponding to the size of the Cooper pairs, i.e., the coherence length $\xi$. Due to the strong gap, the coherence length is very small: Using the estimate $\xi \sim 1/(\pi\Delta)$, one finds that it is of the same order as the Fermi wavelength and, strictly speaking, one might therefore question that mean-field results are quantitatively correct [@Abuki]. The smallness of $\xi$ explains why the compensation of the negative surface charge is mostly concentrated in such a thin layer at a small distance from the surface. Below this strongly negatively charged layer, the charge density stays negative but much smaller. Due to Coulomb interaction, which tries to push the charge towards the surface, this negative charge density decreases with increasing distance from the surface, especially for large strangelets. Actually, if Coulomb interaction is switched off, the remaining charge is distributed more or less homogeneously over the whole volume. The behaviour of the charge density far away from the surface in the presence of Coulomb interaction can easily be interpreted in terms of Debye screening (similar considerations can be found in Ref. [@Tatsumi] for the case of hadron-quark mixed phases): We know that in a uniform medium with Debye screening the Laplace equation for the Coulomb potential is replaced by $$\left({\bm{\mathrm{\nabla}}}^2-\frac{1}{\lambda^2}\right)A_0 = 0\,, \label{poissonscreened}$$ where $\lambda$ is the screening length, which can be obtained from the limit $\Pi^{00}(q^0 = 0,{\bm{\mathrm{q}}}\to 0)$, where $\Pi^{\mu\nu}(q)$ is the polarization tensor in the uniform system. This is equivalent to the expression [@Tatsumi] $$\frac{1}{\lambda^2} = 4\pi e^2 \sum_{fc} Q_f \frac{\partial\rho_{\mathit{ch}}}{\partial \mu_{fc}}\,.$$ Computing numerically this derivative within our model for the case of bulk CFL matter with $B = 156.26$ MeV and $H\Lambda^2 = 2$, we obtain $\lambda = 7.74$ fm. Taking the Laplacian of Eq. (\[poissonscreened\]), we see that the charge density obeys the analogous equation $$\left({\bm{\mathrm{\nabla}}}^2-\frac{1}{\lambda^2}\right)\rho_{\mathit{ch}} = 0\,.$$ In the case of half-infinite matter with a surface at $z = 0$, the solution of this equation shows that the charge density goes to zero as $\rho_{\mathit{ch}}\propto\exp(z/\lambda)$ if one goes away from the surface ($z\to -\infty$). In the case of a sphere, the corresponding solution reads $$\rho_{\mathit{ch}} \propto \frac{\sinh(r/\lambda)}{r/\lambda}\,. \label{chargecenter}$$ Far away from the surface, the charge densities which we obtain are very well described by Eq. (\[chargecenter\]). To show this, we display in Fig. \[fig:coulomb2\] the same charge densities as in Fig. \[fig:coulomb1\], but divided by their value at $r = 0$. Far away from the surface, all curves follow exactly Eq. (\[chargecenter\]) with the value $\lambda = 7.74$ fm calculated for bulk CFL matter. Near the surface, i.e., at distances which are of the order of a couple of Fermi wavelengths, there are strong deviations from this behavior due to Friedel-type oscillations [@GarciaMoliner]. This is because Eq. (\[poissonscreened\]) is not exact, but it is only valid in a uniform medium and in the long-wavelength limit. It is interesting to notice that the value of the Debye screeing length we obtain is in reasonable agreement with the photon Debye mass calculated from perturbative QCD, which reads, for the CFL phase, $m_{D,\gamma\gamma}^2 = 1/\lambda^2 = 4 \frac{21-8\ln 2}{54} e^2 N_f \mu^2/(6 \pi^2)$ [@SWR04] ($m_{D,\gamma\gamma}$ denotes the Debye mass without gluon-photon mixing, see below). For typical values of the chemical potential this gives $\lambda \sim 10$ fm. ![Zoom into the part of Fig. \[fig:coulomb1\] where the charge density behaves as given by Eq. (\[chargecenter\]). For a better visibility, the charge densities have been divided by their respective values at $r = 0$. The thin dotted curve corresponds to Eq. (\[chargecenter\]) with $\lambda = 7.74$ fm.[]{data-label="fig:coulomb2"}](figure9.eps){width="7cm"} In principle, in color superconducting phases, the photon can mix with one of the gluons. In the CFL phase, in bulk matter, one linear combination of photon and gluon stays massless. This means that at large distances $d\gg \xi$, the Debye screening for the “rotated” photon [@ABR00; @LitimManuel] does not work, since the Cooper pairs are neutral with respect to the rotated charge $\tilde{Q}$. Within the simple model we use for the moment, there are no gluons, such that the mixing cannot be studied. It could be taken into account, as mentioned at the end of Section \[sec:model\], by including the gluons in the same way as the photon, i.e., on the Hartree level. We expect that if we included the gluons in this way, we would find an even faster decrease of the charge if we go away from the surface, since in addition to the electromagnetic force we would have the color forces, which try to push the color charges to the surface, and in the CFL phase color neutrality goes hand in hand with electrical neutrality. Therefore, this is not in contradiction with the fact that the rotated photon is massless, but it is just a consequence of the fact that the combination of photon and gluon which is orthogonal to the rotated photon is massive (in fact, it is even heavier than the other gluons [@LitimManuel]). This means that in a large object, like a strange star, all the negative charge will be concentrated within a layer of a thickness of at most a few tens of fm below the surface. However, before drawing any firm conclusion, one should study this problem in more detail. This will be left for future work. Liquid-drop type expansion {#sec:liquiddrop} -------------------------- The advantage of the present approach is that finite size effects are correctly implemented. For large numbers of particles, this becomes, however, rather cumbersome and asymptotic expansions such as a liquid-drop type approach can be very useful. We will discuss here the determination of the parameters, such as the surface tension, of a liquid-drop type formula for the energy per baryon as a function of the baryon number $A$, including a surface and a curvature term, $$\frac{E}{A} = \left(\frac{E}{A}\right)_{\mathit{bulk}} + \frac{a_S}{A^{1/3}} + \frac{a_C}{A^{2/3}}\,, \label{liquiddrop}$$ from our results. As in Section \[sec:chdens\], we will restrict our discussion to the CFL-type solutions with $Z = 0$, such that we do not need to include a Coulomb term $\propto Z/A^{1/3}$. As explained after Eq. (\[betainfinite\]), $(E/A)_{\mathit{bulk}}$ should be the energy per baryon of infinite matter with $\mu_e = 0$ rather than that of $\beta$ stable infinite matter. However, since we consider only the CFL-type solution, this distinction is irrelevant. Hence, for our chosen parameter sets, we have $(E/A)_\mathit{bulk} = 900$ MeV. Since for the neutral strangelets the Coulomb interaction has only a negligible effect on the total energies (for example, in the case of the strangelets considered in Section \[sec:chdens\], the Coulomb interaction changes the total energy per baryon by less than 5 keV) it will be neglected here in order to reduce the numerical effort. The result of the fitted coefficients $a_S$ and $a_C$ for the different parameter sets are listed in Table \[tab:liquiddrop\]. $\begin{matrix}B^{1/4}\\\mbox{(MeV)}\end{matrix}$ $\begin{matrix}H\Lambda^2\\ \mbox{(MeV)}\end{matrix}$ $\begin{matrix}a_S\\ \mbox{(MeV)}\end{matrix}$ $\begin{matrix}a_C\\ \mbox{(MeV)}\end{matrix}$ $\begin{matrix}\sigma\\ \mbox{(MeV$/$fm$^2$)}\end{matrix}$ --------------------------------------------------- ------------------------------------------------------- ------------------------------------------------ ------------------------------------------------ ------------------------------------------------------------ 156.26 2 107 289 11.9 159.46 2.25 109 297 12.8 163.46 2.5 112 306 13.9 : \[tab:liquiddrop\] Fitted liquid-drop parameters for the CFL-type neutral strangelets ($Z = 0$). The surface tension $\sigma$ corresponding to the fitted value of $a_S$ is also given. As an example, in order to show the accuracy of the asymptotic expansion, we display in Fig. \[fig:ea\] some results for the energy per baryon together with the liquid-drop formula, Eq. (\[liquiddrop\]). ![Energy per baryon as a function of baryon number for $H\Lambda^2 = 2$ and $B^{1/4} = 156.26$ MeV. The exact results are indicated by the crosses, the fitted liquid-drop formula by the solid line. The dashed line corresponds to the liquid-drop formula without the curvature term.[]{data-label="fig:ea"}](figure10.eps){width="7cm"} The dashed line corresponds to the liquid-drop formula without the curvature term ($a_C = 0$). From this figure it becomes clear that the liquid-drop formula with curvature term works extremely well, much better than in the case without pairing [@GilsonJaffe]. The reason is that shell effects are completely washed out because, contrary to the situation in ordinary nuclei, the pairing gap is much larger than the spacing between neighboring shells. Another interesting observation is that the curvature term is very important, even for rather large mass numbers $A$. The coefficient $a_S$ is closely related to a very interesting quantity, namely the surface tension. As explained in Ref. [@Parija], the surface tension is obtained as $$\sigma = \frac{E_S}{4\pi R_0^2}\,, \label{defsigma}$$ where $$E_S = E - A\left(\frac{E}{A}\right)_\mathit{bulk}$$ is the energy excess due to the surface and $R_0$ is an effective radius defined by $$A = \rho_{B\,\mathit{bulk}}\frac{4\pi R_0^3}{3}\,,$$ which is actually very close to $R$ for not too small strangelets. On the one hand, using the liquid-drop formula (\[liquiddrop\]) for $E$ in Eq. (\[defsigma\]), one would obtain a surface tension which depends on $A$ because of the curvature term. Therefore it is clear that one has to use Eq. (\[defsigma\]) in the limit $A\to\infty$, where the curvature term vanishes, i.e., $$\sigma = \frac{a_S\rho_{B\,\mathit{bulk}}^{2/3}}{(36\pi)^{1/3}}\,.$$ The corresponding numbers are given in the last column of Table \[tab:liquiddrop\]. They are of the same order of magnitude as the estimate $\sigma \sim (70$ MeV$)^3 = 8.8$ MeV$/$fm$^2$ for SQM without color superconductivity [@FarhiJaffe]. On the other hand, the fact that the curvature term is very strong implies that the knowledge of the surface tension alone might not be sufficient in order to determine, e.g., the possibility of mixed phases, the size of droplets, etc. Before we conclude, let us comment on the physical meaning of the surface tension we obtain. In the MIT bag model, it is supposed that the energy needed to create a bag with volume $V$ is simply given by $B V$. In principle one could imagine that there is an explicit dependence of the bag energy on, e.g., the surface or the curvature of the bag boundary. In Ref. [@FarhiJaffe], this contribution to the surface tension was called “intrinsic surface tension”, $\sigma_I$, and it was argued that it should be small. What we calculate here is the “dynamical surface tension”, $\sigma_D$, which has its origin in the change of the level density of the quarks inside the bag as a function of the bag geometry. Summary {#sec:sum} ======= In this paper we have investigated finite lumps of color superconducting SQM. To that end we have treated the MIT bag model, supplemented with a pairing interaction, in the framework of HFB theory. This allows us to correctly include finite size effects for pairing, too. The calculation is numerically rather involved, since in addition to solving self-consistently the HFB equations, we have to determine the bag radius and the fractions of the different quark species by minimizing the total energy of the system. As expected from previous MIT bag-model studies, we find a suppression of the strange-quark densities at the surface, resulting in a positive surface charge. Our main result is that, in spite of this surface charge, the total charge of the CFL type solution is zero due to pairing, as in bulk matter. Most of the positive surface charge is compensated in a negatively charged layer situated at about 1-3 fm below the surface. The origin of this concentration of the negative charge is pairing: Since all quarks are paired, the positive surface charge must be compensated on a length scale corresponding to the coherence length. The remaining negative charge, which is necessary to compensate all of the positive surface charge, is situated below this layer. With increasing distance from the surface, the charge density decreases on a length scale of $\sim 8$ fm, corresponding to the Debye screening length. This number will probably be strongly decreased if the gluons are included in a perturbative way similar to the photon. In any case, in the biggest part of a large object, such as a strange star, one finds vanishing charge density if one goes more than a few tens of fm away from the surface. It remains to be investigated in which way our results change the traditional picture of the surface of a strange star and the detectability of smaller strangelets in current experiments such as AMS-02 or LSSS [@Madsen06]. We have also compared our results for the energy per baryon of finite strangelets with a liquid-drop like formula. We obtain a surface tension of the order of 12-14 MeV, in reasonable agreement with previous studies where color superconductivity was not considered, and a strong curvature term which is crucial to reproduce the correct energies up to baryon numbers of several thousands. An interesting result is that, in the presence of color superconductivity, the liquid-drop formula describes very accurately the total energies even for $A\lesssim 100$, at least for strangelets with even baryon number. The reason is that, since the gap $\Delta$ is much larger than the spacing between the energy levels, shell effects are strongly suppressed. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Michael Buballa for useful discussions and for the critical reading of the first version of this manuscript. Spinors in a spherical cavity {#app:basis} ============================= In this appendix we recall basic properties of free Dirac spinors in a spherical cavity (see, e.g., Ref. [@Bhaduri]). They can be written as $$\psi_{fj\kappa mn}({\bm{\mathrm{r}}}) = \begin{pmatrix} g_{fj\kappa n}(r) \mathcal{Y}^m_{jl}(\Omega)\\ i\,f_{fj\kappa n}(r) \mathcal{Y}^m_{jl'}(\Omega)\end{pmatrix}\,,$$ where $\mathcal{Y}$ are spinor spherical harmonics [@Varshalovich]. We have the following relations between the angular momentum quantum numbers $$\begin{aligned} {3} &\kappa = j+\frac{1}{2} &\quad\rightarrow\quad l = j+\frac{1}{2},\quad & l' = j - \frac{1}{2} \nonumber\\ &\kappa = -(j+\frac{1}{2}) &\quad\rightarrow\quad l = j-\frac{1}{2},\quad & l' = j + \frac{1}{2}\,.\end{aligned}$$ For the solutions of the free Dirac equation, the functions $f$ and $g$ are given as follows in terms of the spherical Bessel functions ($\xi_{fj\kappa n} = \sqrt{p_{fj\kappa n}^2 + m_f^2}$) $$\begin{aligned} g_{fj\kappa n}(r) =& C_{fj\kappa n} \, j_l(p_{fj\kappa n}r) \nonumber\\ f_{fj\kappa n} (r) =& C_{fj\kappa n} \mathrm{sgn}(\kappa n) \sqrt{\frac{\xi_{fj\kappa n} - m_f}{\xi_{fj\kappa n} + m_f}}\, j_{l'}(p_{fj\kappa n} r)\,,\end{aligned}$$ where the $C_{fj\kappa n}$ are normalisation coefficients which can be determined from the normalization $$\int_0^R dr r^2 \int d\Omega \psi^\dagger({\bm{\mathrm{r}}}) \psi({\bm{\mathrm{r}}}) = 1\,.$$ The momenta $p_{fj\kappa n}$ are obtained from the boundary condition. The boundary condition of the MIT bag model, Eq. (\[boundarycondition\]), translates into the following equation $$f_{fj\kappa n}(R) = -g_{fj\kappa n}(R)\,,$$ or, explicitly, $$j_l(p_{fj\kappa n} R) = \mathrm{sgn}(\kappa n) \sqrt{\frac{ \xi_{fj\kappa n}- m_f}{\xi_{fj\kappa n} + m_f} } \, j_{l'}(p_{fj\kappa n} R)\,,$$ where we number by $n > 0$ the positive-energy (particle) states and by $n < 0$ the negative-energy (antiparticle) states. In practice, we will keep only the states with positive eigenvalues and neglect the antiparticle contributions. The latter can approximately be absorbed into a redefintion of the coupling constant. HFB equations {#app:hfb} ============= In this appendix we will give some more details about the HFB equations. Their derivation is analogous to the derivation of the Dirac-Hartree-Bogoliubov equations in finite nuclei, which is given in Ref. [@CH]. The HFB equations are derived from the Lagrangian by minimizing the energy in the mean field approximation, i.e., linearizing the interaction under the assumption of nonzero expectation values for the condensates $s_{AA^\prime}(x)$, Eq. (\[saa\]). Due to the inhomogeneities of a finite system, the Green’s functions become nondiagonal in momentum. In the stationary case, it is convenient to work in ${\bm{\mathrm{r}}}$ space for the spatial coordinates but to perform the Fourier transformation for the time variable. Then the Green’s functions, $$S(x,y) = -i \langle T(\Psi(x)\bar{\Psi}(y))\rangle~,$$ with $$\Psi(x) = \begin{pmatrix} \psi(x)\\\psi_T(x)\end{pmatrix}$$ take the following general form in Nambu-Gorkov space: $$\begin{gathered} S({\bm{\mathrm{r}}},{\bm{\mathrm{r}}}^\prime;\omega) = \begin{pmatrix} G({\bm{\mathrm{r}}},{\bm{\mathrm{r}}}^\prime;\omega)& F({\bm{\mathrm{r}}},{\bm{\mathrm{r}}}^\prime;\omega)\\ \tilde{F}({\bm{\mathrm{r}}},{\bm{\mathrm{r}}}^\prime;\omega)& \tilde{G}({\bm{\mathrm{r}}},{\bm{\mathrm{r}}}^\prime;\omega)\end{pmatrix}\\ = \sum_{\alpha (\epsilon_\alpha >0)} \begin{pmatrix} U_\alpha({\bm{\mathrm{r}}})\\V_\alpha({\bm{\mathrm{r}}})\end{pmatrix} \frac{1}{\omega-\epsilon_\alpha + i\eta} (\bar{U}_\alpha({\bm{\mathrm{r}}}^\prime),\bar{V}_\alpha({\bm{\mathrm{r}}}^\prime))\\ + \sum_{\beta (\epsilon_\beta < 0)} \begin{pmatrix} U_\beta({\bm{\mathrm{r}}})\\V_\beta({\bm{\mathrm{r}}})\end{pmatrix} \frac{1}{\omega+\epsilon_\beta - i\eta} (\bar{U}_\beta({\bm{\mathrm{r}}}^\prime),\bar{V}_\beta({\bm{\mathrm{r}}}^\prime))\,, \label{greensfunctions}\end{gathered}$$ where $G,\tilde{G}$ and $F,\tilde{F}$ are normal and anomalous Green’s functions, respectively. The spinors $U_{\alpha,\beta}$ and $V_{\alpha,\beta}$ correspond to the particle- and hole-like components, respectively. The energy in mean-field approximation can now be written as [@CH] $$\begin{gathered} E_q = \int d^3 x \left(i\, \mathrm{Tr}[(i{\bm{\mathrm{\gamma}}}\cdot{\bm{\mathrm{\nabla}}}-m) G(x,x^+)]\phantom{\int}\right. \\ -\left.\frac{i}{2}\int d^4 y \,\mathrm{Tr}[\Sigma(x,y) G(y,x^+) - \Delta(x,y) \tilde{F}(y,x^+)]\right)~, \label{energygeneral}\end{gathered}$$ where the derivative in the first term acts only on $x$ and not on $x^+$, and $x^+$ means the four vector $(x^0+t,{\bm{\mathrm{x}}})$ in the limit $t \to 0^+$. In our case, the normal and anomalous self-energies $\Sigma$ and $\Delta$ are local and time-independent: $\Sigma(x,y) = eQA_0({\bm{\mathrm{x}}})\gamma^0\delta(x-y)$ and $\Delta(x,y) = \Delta({\bm{\mathrm{x}}})\delta(x-y)$, and Eq. (\[energygeneral\]) can be reduced to Eq. (\[quarkenergy\]). As mentioned in Section \[sec:hfb\], the expectation values (like condensates, densities, etc.) which are needed for calculating self-consistently the self-energy $\Sigma$ and the pairing field $\Delta$ can be expressed in terms of the $U$ and $V$ functions. To that end, it is sufficient to express them in terms of the Green’s functions, e.g. $$s_{AA^\prime} = - \langle\bar{\psi}_T(x) \tau_A \lambda_{A^\prime} \psi(x)\rangle = i\, \mathrm{Tr}\, F(x,x^+) \tau_A \lambda_A\,,$$ which leads to Eq. (\[saadivergent\]). By minimizing the total energy with respect to the $U$ and $V$ functions, one obtains the HFB equations, \[see Eq. (\[hfbequations\])\]: $$\mathcal{H} W_\alpha = \epsilon_\alpha W_\alpha\,,$$ with $W_\alpha = (U_\alpha,V_\alpha)^T$ and $\mathcal{H}$ being the matrix on the left-hand side of Eq. (\[hfbequations\]). For homogeneous infinite systems the matrix elements of $\mathcal{H}$ are diagonal in momentum space and solutions to the HFB equations are known for many cases. For finite systems, in general, these equations are solved numerically by diagonalizing the matrix $\mathcal{H}$ in some conveniently chosen basis. Here, we are working in the basis which diagonalizes the Dirac hamiltonian (i.e., $h_{fc}$ without the Coulomb potential), see Appendix \[app:basis\], and the eigenvectors $U_\alpha({\bm{\mathrm{r}}})$ and $V_\alpha({\bm{\mathrm{r}}})$ are developped within this basis. The matrix elements of the pairing fields $\Delta_A(r)$ and of the Coulomb field $A_0(r)$ are computed in the usual way. For illustration, we give here the explicit expression for the matrix elements of $\Delta_2(r)$, which connects up and down quarks, in the basis described in Appendix \[app:basis\]: $$\begin{gathered} (\Delta_2)_{j\kappa nn'} = \int_{r<R} d^3 r\, \psi^\dagger_{uj\kappa mn}({\bm{\mathrm{r}}})\Delta_2(r)\psi_{dj\kappa mn'}({\bm{\mathrm{r}}})\\ = \int_0^R dr\, r^2 \Delta_2(r) (g_{uj\kappa n}(r) g_{dj\kappa n'}(r)\\ +f_{uj\kappa n}(r) f_{dj\kappa n'}(r))\,.\end{gathered}$$ Note that, due to spherical symmetry, all matrices are diagonal in $j$ and $\kappa$ and proportional to the unit matrix with respect to $m$. In spite of the spherical symmetry, the matrix to be diagonalized is still huge, limiting the baryon number which can be calculated with reasonable computational effort. It is therefore important to reduce the size of the actual matrix to be diagonalized. By means of an orthogonal transformation $$\tilde{\mathcal{H}} = S \mathcal{H} S^T, \quad \tilde{W} = S W, \quad S S^T = 1$$ in color, flavor, and Nambu-Gorkov space, the matrix can actually be block-diagonalized (see, e.g., Ref. [@ABR99]) containing seven blocks. Six of them, $\tilde{\mathcal{H}}_{B,\dots G}$, are $2\times 2$ matrices in Nambu-Gorkov space, describing mutual pairing of two particles, such as, e.g., red down quarks ($dr$) with green up quarks ($ug$): $$\tilde{\mathcal{H}}_B = \begin{pmatrix} h_{ug} & \Delta_2 \\ \Delta_2 & -h_{dr} \end{pmatrix}\,,$$ where $h_{fc}$ is the single particle Hamiltonian for flavor $f$ and color $c$. The second and third $2\times 2$ blocks are $$\tilde{\mathcal{H}}_C = \begin{pmatrix} h_{ub} & \Delta_5 \\ \Delta_5 & -h_{sr} \end{pmatrix}\,,\quad \tilde{\mathcal{H}}_D = \begin{pmatrix} h_{db} & \Delta_7 \\ \Delta_7 & -h_{sg} \end{pmatrix}\,.$$ Since we have in addition the pairwise relations $\tilde{\mathcal{H}}_{E,F,G} = -\tilde{\mathcal{H}}_{B,C,D}$, only three of the six $2\times 2$ blocks have to be diagonalized in practice. The seventh block, $\tilde{\mathcal{H}}_A$, is $6\times 6$ in Nambu-Gorkov space and describes pairing between red up, green down and blue strange quarks $$\tilde{\mathcal{H}}_A = \begin{pmatrix} h_{ur} &0&0&0&\Delta_2&\Delta_5\\ 0&h_{dg}&0&\Delta_2&0&\Delta_7\\ 0&0&h_{sb}&\Delta_5&\Delta_7&0\\ 0&\Delta_2&\Delta_5&-h_{ur}&0&0\\ \Delta_2&0&\Delta_7&0&-h_{dg}&0\\ \Delta_5&\Delta_7&0&0&0&-h_{sb}\end{pmatrix}\,.$$ Cutoff for the gap equation {#app:cutoff} =========================== As mentioned in Section \[sec:hfb\], the divergent gap equation is regularized with the help of a smooth cutoff function $$f(p/\Lambda) = \frac{1}{1 + c_1 \exp(c_2 a (p/\Lambda - 1))}\,,$$ where $c_1 = \sqrt{2}-1$, $c_2 = 1/(4-2\sqrt{2})$, and $a = 22.58$ have been chosen such that $f^2(p/\Lambda)$ approximates the cutoff function $g(p/\Lambda)$ used in Ref. [@YasuiHosaka], but our function has the advantage to fall off more rapidly at very high momenta, which allows us to truncate the basis at a lower energy. This function is used as a form factor multiplying each of the four legs of the four-point vertex. In practice, this means that the form factor is used in two places: First, when calculating $s_{AA}(r)$, and second, when calculating the matrix elements of $\Delta_A(r)$ in the basis of the spinors defined in Appendix \[app:basis\]. It should be noted that the diagonalization of the HFB matrix does not directly provide us with the eigenfunctions $U_\alpha({\bm{\mathrm{r}}})$ and $V_\alpha({\bm{\mathrm{r}}})$, but with their respective expansion coefficients in the basis of the spinors defined in Appendix \[app:basis\]. When calculating $s_{AA}(r)$ according to Eq. (\[saadivergent\]), the coefficients have to be multiplied with the corresponding basis functions, and in this step the factor $f(p_{fj\kappa n}/\Lambda)$ is attached to each basis function. Second, when calculating the matrix elements of the gap $\Delta_A$, we again attach a factor $f(p_{fj\kappa n}/\Lambda)$ to each basis function. [99]{} J.C. Collins and M.J. Perry, Phys. Rev. Lett. [ **34**]{}, 1353 (1975); B. Barrois, Nucl. Phys. **B129**, 390 (1977); S.C. Frautschi, Asymptotic freedom and color superconductivity in dense quark matter, in: Proc. of the Workshop on Hadronic Matter at Extreme Energy Density, N. Cabibbo (ed.), Erice 1978; D. Bailin and A. Love, Phys. Rep. **107**, 325 (1984). M.G. Alford, K. Rajagopal, and F. Wilczek, Phys. Lett. B **422**, 247 (1998). R. Rapp, T. Schäfer, E.V. Shuryak, and M. Velkovsky, Phys. Rev. Lett. **81**, 53 (1998). K. Rajagopal and F. Wilczek, in: M. Shifman (Editor), *At the Frontier of Particle Physics, Handbook of QCD, Boris Ioffe Festschrift*, vol. 3, p. 2061 (World Scientific, Singapore 2001) \[hep-ph/0011333\]; M. Alford, Ann. Rev. Nucl. Part. Sci. **51**, 131 (2001); T. Schäfer, hep-ph/0304281; D.H. Rischke, Prog. Part. Nucl. Phys. **52**, 197 (2004); M. Buballa, Phys. Rep. **407**, 205 (2005); H.-C. Ren, hep-ph/0404074; M. Huang, Int. J. Mod. Phys. E **14**, 675 (2005); I. A. Shovkovy, Found. Phys. **35**, 1309 (2005). M.G. Alford, K. Rajagopal, and F. Wilczek, Nucl. Phys. **B 537**, 443 (1999). I.A. Shovkovy and L.C.R. Wijewardhana, Phys. Lett. B **470**, 189 (1999); T. Schäfer, Nucl. Phys. **B 575**, 269 (2000); N.J. Evans, J. Hormuzdiar, S.D.H. Hsu, and M. Schwetz, Nucl. Phys. **B 581**, 391 (2000). A.R. Bodmer, Phys. Rev. D **4**, 1601 (1971); E. Witten, Phys. Rev. D**30**, 272 (1984). C. Alcock, E. Farhi, and A. Olinto, Astrophys. J. **310**, 261 (1986); P. Haensel, J.L. Zdunik, and R. Schaeffer, Astron. Astrophys. **160**, 121 (1986). J. Madsen and J.M. Larsen, Phys. Rev. Lett. **90**, 121102 (2003); M. Rybczynski, Z. Wlodarczyk, and G. Wilk, Nucl. Phys. Proc. Suppl. **151**, 341 (2006). V.V. Usov, Phys. Rev. Lett. **80**, 230 (1998). D. Page and V.V. Usov, Phys. Rev. Lett.**89**, 131101 (2002). P. Jaikumar, S. Reddy and A.W. Steiner, Phys. Rev. Lett. **96**,041101 (2006). M.G. Alford, K. Rajagopal, S. Reddy, and A.W. Steiner, Phys. Rev. D **73**, 114016 (2006). K. Rajagopal and F. Wilczek, Phys. Rev. Lett. **86**, 3492 (2001). J. Madsen, Phys. Rev. Lett. **87**, 172003 (2001). V.V. Usov, Phys. Rev. D **70**, 067301 (2004). A. Chodos, R.L. Jaffe, K. Johnson, C.B. Thorn, and V.F. Weisskopf, Phys. Rev. D **9**, 3471 (1974). R.K. Bhaduri, *Models of the Nucleon: From Quarks to Soliton* (Addison-Wesley, Redwood City 1988) B.V. Carlson and D. Hirata, Phys. Rev. C **62**, 054310 (2000). E. Farhi and R.L. Jaffe, Phys. Rev. D **30**, 2379 (1984). W.V. Liu and F. Wilczek, Phys. Rev. Lett. **90**, 047002 (2003). H. Abuki, T. Hatsuda, K. Itakura, Phys. Rev. D **65**, 074014 (2002). T. Tatsumi, M. Yasuhira, and D.N. Voskresensky, Nucl. Phys. **A718**, 359c (2003). F. García-Moliner and F. Flores, *Introduction to the theory of solid surfaces* (Cambridge University Press, Cambridge, 1979). A. Schmitt, Q. Wang and D.H. Rischke, Phys. Rev. D **69**, 094017 (2004). M.G. Alford, J. Berges, and K. Rajagopal, Nucl. Phys. **B 571**, 269 (2000) D.F. Litim and C. Manuel, Phys. Rev. D **64**, 094013 (2001). E.P. Gilson and R.L. Jaffe, Phys. Rev. Lett. **71**, 332 (1993). B.C. Parija, Phys. Rev. C **48**, 2483 (1993). J. Madsen, arXiv:astro-ph/0612784 (2006). D.A. Varshalovich, A.N. Moskalev, and V.K. Khersonskii, *Quantum Theory of Angular Momentum* (World Scientific, Singapore 1988). M.G. Alford, J. Berges, and K. Rajagopal, Nucl. Phys. B **558**, 219 (1999). S. Yasui and A. Hosaka, Phys. Rev. D **74**, 054036 (2006). [^1]: In uniform infinite matter it can be shown [@CFL] that for the energetically favored solution the arbitrary orientation in color can be chosen in such a way that only the diagonal condensates with $A = A^\prime$ are non-zero. [^2]: Here we assume that neutrinos are not trapped, i.e., they can freely leave the system [^3]: As discussed below Eq. (\[betainfinite\]), it is more appropriate to compare a small strangelet with this kind of matter rather than electrically neutral matter with electrons in $\beta$ equilibrium. [^4]: If the number of quarks is odd, it it impossible to pair all quarks and one or several state(s) should be “blocked” by the unpaired quark(s). At present, we have not included this effect in our calculation, and we restrict ourselves to even quark numbers

--- abstract: 'We prove an analogue of Fujino and Mori’s “bounding the denominators” [@fm Theorem 3.1] in the log canonical bundle formula (see also [@shok Theorem 8.1]) for Kawamata log terminal pairs of relative dimension one. As an application we prove that for a klt pair $(X,\Delta)$ of Kodaira codimension one and dimension at most three such that the coefficients of $\Delta$ are in a DCC set $\mathcal{A}$, there is a natural number $N$ that depends only on $\mathcal{A}$ for which ${\lfloor N(K_X+\Delta) \rfloor}$ induces the Iitaka fibration. We also prove a birational boundedness result for klt surfaces of general type.' author: - 'Gueorgui Tomov Todorov[^1]' title: 'Effective log Iitaka fibrations for surfaces and threefolds.' --- Introduction. ============= Let us start by recalling Kodaira’s canonical bundle formula for a minimal elliptic surface $f\colon S\to C$ defined over the complex number field: $$K_S=f^*(K_C+B_C+M_C).$$ The [*moduli part*]{} $M_C$ is a ${\mathbb{Q}}$-divisor such that $12M_C$ is integral and ${{\mathcal O}}_C(12M_C)\simeq J^*{{\mathcal O}}_{{\mathbb P}^1}(1)$, where $J\colon C\to {\mathbb P}^1$ is the $J$-invariant function. The [*discriminant*]{} $B_C=\sum_P b_PP$, supported by the singular locus of $f$, is computed in terms of the local monodromies around the singular fibers $S_P$. Kawamata [@kaw97; @kaw98] proposed an equivalent definition, which does not require classification of the fibers: $1-b_P$ is the [*log canonical threshold*]{} of the log pair $(S,S_P)$ in a neighborhood of the fiber $S_P$. A higher dimensional analogue consists of a log klt pair $(X,\Delta)$ and surjective morphism such that the Kodaira dimension of $K_X+\Delta$ restricted to the general fibre is zero. For now let us assume that $K_X+\Delta=f^*D$ for some ${\mathbb{Q}}$-divisor $D$ on $Y$. Then we can define the *discriminant* or divisorial part on $Y$ for $K_X+\Delta$ to be the ${\mathbb{Q}}$-Weil divisor $B_Y:=\sum_P b_P P$, where $1-b_P$ is the maximal real number $t$ such that the log pair $(X,\Delta+tf^*(P))$ has log canonical singularities over the generic point of $P$. The sum runs over all codimension one points of $Y$, but it has finite support. The *moduli part* or *J-part* is the unique ${\mathbb{Q}}$-Weil divisor $M_Y$ on $Y$ satisfying $$K_X+\Delta= f^*(K_Y+B_Y+M_Y).$$ According to Kawamata [@kaw Theorem 2](see also Ambro [@ambroSBP Theorem 0.2 (ii)] and Fujino [@fujino]) we know that on some birational model $\mu:Y'{\longrightarrow}Y$ the moduli divisor $M_{Y'}$ is nef. Some of the main questions concerning the moduli part are the following. \[conj\] ([@shok Conjecture 7.12]) Let $(X,\Delta)$ and $f:X{\longrightarrow}Y$ be as above and let us write as before $$K_X+\Delta=f^*(K_Y+B_Y+M_Y).$$ Then we have the following (1) : (Log Canonical Adjunction) There exists a birational contraction $\mu:Y'{\longrightarrow}Y$ such that after base change the induced moduli divisor $M_{Y'}$ on $Y'$ is semiample. (2) : (Particular Case of Effective Log Abundance Conjecture). Let $X_\eta$ be the generic fibre of $f$. Then $I(K_{X_\eta}+\Delta_{X_\eta})\sim 0$, where $I$ depends only on $\dim X_\eta$ and the horizontal multiplicities of $\Delta$. (3) : (Effective Adjunction) There exist a positive integer depending only on the dimension of $X$ and the horizontal multiplicities of $\Delta$ such that $IM_{Y'}$ is base point free on some model $Y'/Y$. There is a proof of the above conjecture by Shokurov and Prokhorov in the case in which the relative dimension of $f$ is one (Theorem 8.1 of [@shok]). For results towards [**(1)**]{} see Ambro [@ambro3]. Here we prove that there exist a positive integer $I$ depending only on the dimension of $X$ and the horizontal multiplicities of $\Delta$ such that $IM$ is integral when the relative dimension is one using ideas of Mori and Fujino [@fm] (see also [@ko]). The main advantage of our proof is that the number $I$ that we produce is explicitly computable. Our main interest in Conjecture \[conj\] is because of its applications towards boundedness results for Iitaka fibrations. When $X$ is of general type the existence of a natural number $N$ such that $|NK_X|$ induces the Iitaka fibration is know by results of C. Hacon and J. M$^\textrm{c}$Kernan(cf. [@chris1]) and Takayama(cf. [@tak]) following ideas by Tsuji. Similar results in low dimension when $X$ is not of general type appear in the recent preprints [@vie; @ringler; @pacienza]. Here we address the boundedness of Iitaka fibrations in the log case. \[appl\] Let $(X,\Delta)$ be a klt log pair of Kodaira codimension one and dimension at most three. Then there is a natural number $ N$ depending only on the coefficients of $\Delta$ such that $|{\lfloor N(K_X+\Delta) \rfloor}|$ induces the Iitaka fibration. The proof of the above Theorem in dimension two relies on the existence of $I$ as in the Conjecture and follows the strategy in Section 6 of [@fm]. For the proof of the Theorem in dimension three we need to bound the smallest positive number $N$ such that $|N(K_X+\Delta)|$ induces a birational map for any log surface of general type with the coefficients of $\Delta$ in a DCC set $\mathcal{A}$ as a function of the DCC set only (i.e. $N=N(\mathcal{A})$). This is an interesting question in its own right (cf. [@vie]) and we address it in the last section. We can show that: Let $(X,\Delta)$ be a klt surface and assume that the coefficients of $\Delta$ are in a DCC set $\mathcal{A}$ . Then there is a number $N$ depending only on $\mathcal{A}$ such that ${\lceil m(K_X+\Delta) \rceil}$ (and ${\lfloor m(K_X+\Delta) \rfloor}$) defines a birational map for $m\ge N$. The above two Theorems complete the boundedness of Iitaka fibrations of klt pairs of dimension two ( for the case of Kodaira dimension zero see [@alex]). The proof is based on the fact that by a result of [@alex] (see also [@almo]) for these surfaces we have a lower bound of the volume which allows us to produce centres of log canonical singularities of a controlled multiple of $K_X+\Delta$. Using standard techniques we reduce to the case where the centres are isolated points. In order to achieve this, using ideas of M$^\textrm{c}$Kernan [@mac] and Tsuji, we produce a morphism to a curve and we use this morphism to produce the required sections (cf. [@tod]). Preliminaries. ============== Notations and Conventions. -------------------------- We will work over the field of complex numbers $\mathbb{C}$. A ${\mathbb{Q}}$-Cartier divisor $D$ is nef if $D\cdot C \ge0$ for any curve $C$ on $X$. We call two ${\mathbb{Q}}$-divisors $D_1, D_2$ ${\mathbb{Q}}$-linearly equivalent $D_1\sim_{\mathbb{Q}}D_2$ if there exists an integer $m>0$ such that $mD_i$ are integral and linearly equivalent. We call two ${\mathbb{Q}}$-Cartier divisors $D_1, D_2$ numerically equivalent $D_1\equiv D_2$ if $(D_1-D_2)\cdot C=0$ for any curve $C$ on $X$. A log pair $(X,\Delta)$ is a normal variety $X$ and an effective ${\mathbb{Q}}$-Weil divisor $\Delta$ such that ${K_X}+\Delta$ is ${\mathbb{Q}}$-Cartier. A projective morphism $\mu:Y {\longrightarrow}X$ is a log resolution of the pair $(X,\Delta)$ if $Y$ is smooth and $\mu^{-1}(\Delta)\cup\{\textrm{exceptional set of } \mu\}$ is a divisor with simple normal crossing support. For such $\mu$ we write $\mu^*({K_X}+\Delta) =K_Y+\Gamma$, and $\Gamma=\Sigma a_i\Gamma_i$ where $\Gamma_i$ are distinct integral divisors. A pair is called klt (resp. lc) if there is a log resolution $\mu:Y {\longrightarrow}X$ such that in the above notation we have $a_i <1$ (resp. $a_i\le 1$). The number $1-a_i$ is called log discrepancy of $\Gamma_i$ with respect to the pair $(X,\Delta)$. We say that a subvariety $V \subset X$ is a log canonical centre if it is the image of a divisor of log discrepancy at most zero. A log canonical place is a valuation corresponding to a divisor of log discrepancy at most zero. A log canonical centre is pure if ${K_X}+\Delta$ is log canonical at the generic point of $V$. If moreover there is a unique log canonical place lying over the generic point of V, then we say that $V$ is exceptional. LCS$(X,\Delta,x)$ is the union of all log canonical centres of $(X,\Delta)$ through the point $x$. We will denote by LLC$(X,\Delta,x)$ the set of all log canonical centres containing a point $x \in X$. Generalities on cyclic covers. ------------------------------ \[cyc\] Let $X$ be a smooth variety and $L$ a line bundle on $X$ and $D$ an integral divisor. Assume that $L^m\sim {\mathcal{O}_X}(D)$. Let $s$ be any rational section and $1_D$ the constant section of ${\mathcal{O}_X}(D)$. Then $1_D/s^m$ is a rational function which gives a well defined element of the quotient group $k(X)^*/(k(X)^*)^m$, thus a well defined degree $m$ field extension $k(X)(^m\sqrt{1_D/s^m})$. Let $\pi:X'{\longrightarrow}X$ denote the normalization of $X$ in the field $k(X)(^m\sqrt{1_D/s^m})$. Then (1) : $\pi_*{\mathcal{O}}_{X'}=\sum_{i=0}^{m-1}L^{-i}({\lfloor iD/m \rfloor})$, and (2) : $\pi_*\omega_{X'}=\sum_{i=0}^{m-1}\omega_X\otimes L^i(-{\lfloor iD/m \rfloor})$. In particular, if $E$ is any integral divisor then the normalized cyclic cover obtained from $L^m\sim {\mathcal{O}_X}(D)$ is the same as the normalized cyclic cover obtained from $(L(E))^m\sim {\mathcal{O}_X}(D+mE)$. If $D$ has simple normal crossing support then $X'$ has only rational singularities. DCC sets -------- A subset $\mathcal{A}$ of $\mathbb{R}$ is said to satisfy the descending chain condition if any strictly decreasing subsequence of elements of $\mathcal{A}$ is finite. In this case we also say that $\mathcal{A}$ is a DCC set. For the general properties of DCC sets we refer to Section 2 of [@almo]. A sum of $n$ sets $\mathcal{A}_1,\mathcal{A}_2,...,\mathcal{A}_n$ is defined as $$\sum_{i=1}^n\mathcal{A}_i=\{a_1+a_2+...+a_n|a_i\in\mathcal{A}_i\}.$$ Define also $$\mathcal{A}_\infty=\{0\}\cup\bigcup_{n=1}^{\infty}\sum_{i=1}^n\mathcal{A}.$$ If $\mathcal{A}$ is a DCC set and it contains only non-negative numbers then it is easy to see that $\mathcal{A}_\infty$ is also a DCC set. For $\mathcal{A}\subset [0,1]$ we define the derivative set $$\mathcal{A}'=\{\frac{n-1+a_\infty}{n}|n\in\mathbb{N},a_\infty\in\mathcal{A}_\infty\cap[0,1]\}\cup\{1\}.$$ It is easy to verify that if $\mathcal{A}$ is a DCC set then so is $\mathcal{A}'$. Bounding the moduli part. ========================= We start by describing the moduli part as it appears in [@ko]. Let $f:(X,R){\longrightarrow}Y$ be a proper morphism of normal varieties with generic fibre $F$ and $R$ a ${\mathbb{Q}}$-divisor such that $K_X+R$ is ${\mathbb{Q}}$-Cartier and assume that $(F,R_{|F})$ is lc and that $K_F+R_{|F}\sim_{{\mathbb{Q}}}0$. Let $Y^0\subset Y$ and $X^0=f^{-1}(Y^0)$ be open subsets such that $K_{X^0}+R^0\sim_{{\mathbb{Q}}} 0$ where $R^0:=R_{|X^0}$ (cf. [@ko Lemma 8.3.4]). Write $R^0=D^0+\Delta^0$ with $D$ integral and ${\lfloor \Delta \rfloor}=0$. Assume that $X^0, Y^0$ are smooth and $R^0$ is relative simple normal crossing over $Y^0$. Define $V={\mathcal{O}}_{X^0}(-K_{X^0}-D^0)$. Let $m$ be (the smallest) positive integer such that $m\Delta^0$ is an integral divisor. Then we have an isomorphism $$V^{\otimes m}\cong {\mathcal{O}}_{X^0}(m\Delta^0),$$ which defines a local system $\mathbb{V}$ on $X^0\setminus R^0$ (cf. [@ko Definition 8.4.6]). Assume also that $Y$ is smooth, $Y\setminus Y^0$ is a simple normal crossing divisor and that $R^{\dim F}f_{*}\mathbb{V}$ has only unipotent monodromies. Then the bottom piece of the Hodge filtration of $R^{\dim F}f_{*}\mathbb{V}$ has a natural extension to a line bundle $J$. Set $J(X/Y,R)$ to be the divisor class corresponding to $J$. If the smoothness, normal crossing, and unipotency assumptions above are not satisfied, take a generically finite morphism $\pi:Y'{\longrightarrow}Y$ and a resolution of the main component $f':X'{\longrightarrow}X\times_{Y} Y'{\longrightarrow}Y'$ for which the assumptions hold and $R'$ the corresponding divisor. Then define $$J(X/Y,R)=\frac{1}{\deg \pi}\pi_*J(X'/Y',R').$$ We need the following definition. ([@ko Definition 8.4.2]) Assume that $(X,R)$ is lc and $K_X+R\sim_{\mathbb{Q}}0$ and write $R=R_{\ge 0}-R_{\le 0}$ as the difference of its positive and negative parts. Define $$p_g^+(X,R):=h^0(X,{\mathcal{O}}_X(\lceil R_{\le 0} \rceil)).$$ \[kodform\]([@ko Theorem 8.5.1]) Let $X, Y$ be normal projective varieties and let $f:X{\longrightarrow}Y$ a dominant morphism with generic fibre $F$. Let $R$ be a ${\mathbb{Q}}$-divisor on $X$ such that $K_X+R$ is ${\mathbb{Q}}$-Cartier and $B$ a reduced divisor on $Y$. Assume that (1) : $K_X+R\sim f^*($some ${\mathbb{Q}}$-Cartier divisor on $Y),$ (2) : $p_g^+(F,R_{|F})=1$, and (3) : $f$ has slc fibres in codimension 1 over $Y\setminus B$ (cf. [@ko]). Then one can write $$K_X+R\sim_{\mathbb{Q}}f^*(K_Y+J(X/Y,R)+B_R),\textrm{ where}$$ (i) : $J(X/Y,R)$ is the moduli part defined above, (ii) : $B_R$ is the unique ${\mathbb{Q}}$-divisor supported on $B$ for which there is a codimension $\ge 2$ closed subset $Z \subset Y$ such that $(X\setminus f^{-1}(Z),R+f^*(B-B_R))$ is lc and every irreducible component of $B$ is dominated by a log canonical centre of $(X,R+f^*(B-B_R))$. Let $(X_1,R_1)$, $f_1:X_1{\longrightarrow}Y_1$ and $B_1$ be a pair satisfying the assumptions of Theorem \[kodform\] and $R_1$ effective on the general fibre. Assume furthermore that the relative dimension of $f_1$ is one and that $(X_1,R_1)$ is klt. Then the following holds. \[main\] There exist an integer $N$ depending only on the horizontal multiplicities of $R_1$ such that the divisor $NJ(X_1/Y_1,R_1)$ is integral. We are going to prove the theorem in the case when the restriction of $\Delta$ to the general fibre of $f$ is non-trivial. When the restriction is trivial the Theorem follows as in [@fm Theorem 3.1]. [**Step I.**]{} We start with some harmless reductions. Cutting by hyperplanes we can reduce to the case when $Y_1$ is a curve. From Step 1 of the proof of [@ko Theorem 8.5.1], we can reduce to the case with normal crossing assumptions, that is we can assume that $X_1, Y_1$ are smooth, $R_1+f_1^*B_1$ and $B_1$ are snc divisors, $f_1$ is smooth over $Y_1\setminus B_1$ and $R_1$ is relative snc divisor over $Y_1\setminus B_1$. By Step 2 of the same proof we can assume also that $B_1=B_{R_1}$. [**Step II - Galois cover of $Y_1$.**]{} By [@mo (4.6) and (4.7)] there is a finite Galois cover $\pi:Y{\longrightarrow}Y_1$ with Galois group $G$, such that for the induced morphism $f:X {\longrightarrow}Y$ every possible local system $R^{\dim F}f_{*}\mathbb{V}_j$ (the difference being given by a choice of isomorphism between two line bundles, compare [@ko Remark 8.4.7]) has unipotent monodromies around every irreducible component of $B$. Note that we can also arrange that $G$ acts on $X$. Thus by [@ko Theorem 8.5.1] we have that the moduli part $L:=J(X/Y,R)$ is integral. Here $\pi_{X}^*(K_{X_1}+R_1)=K_X+R$ and $R=(\pi_{X})_*R_1$. [**Step III - Constructing the right cyclic cover.**]{} There is a unique way to write $$R-f^*B=\Delta-G+E,$$ where $\Delta$ is effective, ${\lfloor \Delta \rfloor}=0$, the divisors $E$ and $G$ are integral and vertical. Let $M=f^*(K_Y+L+B)-(K_X+E-G+f^*B)$. Notice that $M$ is integral and that $\Delta\sim_{\mathbb{Q}}M$. Pick $m>0$ such that $m\Delta$ is integral on the general fibre $F$. Such $m$ depends only on the horizontal multiplicities of $\Delta$. Since $m\Delta_{|F}\sim m M_{|F}$ there is an integral divisor $D$ on $X$ such that $D\sim mM$ and $D_{|F}=m\Delta_{|F}$. Construct the cyclic covering $h:Z{\longrightarrow}X$ corresponding to $D \sim mM$ and let $X^0$ be as in Definition \[cyc\]. After possibly changing the birational model of $X$ we can assume that $D$ is simple normal crossing and $Z$ has rational singularities. We have the following diagram. $$\xymatrix{ & &Z\ar[dl]_h \ar@/^/[ddl]^{h'}\\ X_1\ar[d]^{f_1}&X\ar[l]_{\pi_X} \ar[d]^f&\\ Y_1 &\ar[l]_{\pi} Y& }$$ The restriction of $\pi_X$ to $X^0$ gives one of the cyclic covers used in the construction of the local systems $R^{\dim F}f_*\mathbb{V}_j$. We have that $$h_*\omega_{Z}=\sum_{i=0}^{m-1}{\mathcal{O}}_X(K_X+iM-{\lfloor iD/m \rfloor}).$$ [**Step IV - The G-action on $h'_*(\omega_{Z/Y})$.**]{} We now proceed as [@fm 3.8]. By the pull-back property [@ko Proposition 8.4.9 (3)] we have that $L=\pi^*J(X_1/Y_1,R_1)$. Let $P\in Y$ and localize everything in a neighborhood of $P_1=\pi(P)$ and $P$, and let $e$ be the ramification index at $P$. Let $z_1$ be a local coordinate for the germ $(Y_1,P_1)$ and $z=(z_1)^{1/e}$ for $(Y,P)$. Since the divisor $D$ is $\mu_e$-equivariant over an open set $Y_0\subset Y$ there is a group $G_0$ acting on $Z_{|Y_0}$ which fits in the sequence $0\rightarrow\mu_m\rightarrow G_0\rightarrow\mu_e\rightarrow 0$. In fact if locally $X$ is Spec$A$ then $Z$ is Spec$A[\phi^{1/m}]$ where $\phi$ is a local equation of $D$. Since locally $D$ is $\mu_e$-equivariant $\mu_e$ acts on $\phi$ by multiplication by $e$-th root of unity $\epsilon$ and $\mu_m$ acts on $\phi^{1/m}$ by a multiplication by an $m$-th root of unity $\varepsilon$ there is $\mu_m\rtimes\mu_e$ action on $Z$. Thus we can define a $\mu_{er}$-action on the local systems $R^1h'_*\mathbb{V}_j$ where $r=m/(m,e)$ and hence on the canonical extension $h'_*\omega_{Z/Y}\otimes\mathbb{C}(P)$. The action on the summand $ L\otimes\mathbb{C}(P)\subset h'_*\omega_{Z/Y}\otimes\mathbb{C}(P)$ is by a character $\chi_{P}$.By Lemma \[trivial\], $NJ(X_1/Y_1,R_1)$ is a divisor if and only if the character $\chi_P^N$ is trivial for every $P\in Y$. Let $E$ be the general fibre of $h'$. Then by \[cyc\] we have that $$h^0(E,\omega_E)=h^0(F,\sum_{i=0}^{m-1}\omega_F^{1-i}(-{\lfloor i\Delta_{|F} \rfloor}))\le (m-1)^2.$$ Reasoning as in [@fm 3.8] if $l$ is the order of $\chi_P$, then $\varphi(l)\le (m-1)^2$, where $\varphi(l)$ is the Euler function. Set $N(x)=\textrm{lcm}\{l|\varphi(l)\le x\}$. Then for $N_1=N((m-1)^2)$, the divisor $N_1J(X_1/Y_1,R_1)$ is integral. [*Remark.*]{} Note that the number above is easy to compute explicitly. This is then main advantage of our approach. Auxiliary Lemma. ---------------- Let $Y$ be a smooth curve and let $h:Y'{\longrightarrow}Y$ be a finite Galois cover with group $G$. Let $D$ be a ${\mathbb{Q}}$-divisor on $Y$ such that $D'=h^*D$ is Cartier. For $p'\in Y'$ let $G_{p'}$ be the stabilizer. We have that $G_{p'}$ acts on ${\mathcal{O}}_{Y'}(D')\otimes {\mathcal{O}}_{P'}$ via a character $\chi_{p'}:G_{p'}{\longrightarrow}\mathbb{C}$. In this setting we have the following lemma due to Fujino and Mori [@fm]. \[trivial\](cf. [@fm]) For an integer $N$ the divisor $ND$ is integral if and only if for each $p'\in Y'$ the character $\chi_{p'}^{N}$ is trivial. Iitaka fibrations for surfaces of log Kodaira dimension one. ============================================================ In this section we prove Theorem \[appl\] in dimension two. We start with the following lemma. \[simDCC\] Let $(X,\Delta)$ be a klt pair of dimension $n$ where the coefficients of $\Delta$ are in a DCC set $\mathcal{A}\subset [0,1]$. Let $f:X{\longrightarrow}Y$ be a surjective projective morphism such that for the general fibre $F\cong{\mathbb{P}^1}$ we have that $(K_X+\Delta)_{|F}\sim_{\mathbb{Q}}0$. Then the set $\mathcal{B}$ of coefficients of the horizontal components of $\Delta$ is finite. In particular there is an integer $m=m(\mathcal{B})$ that clears all the denominators of the horizontal components. We can describe $\mathcal{B}$ as the set $\{b\in\mathcal{A}|b+a=2,\textrm{for some }a\in\mathcal{A}_\infty\}$. $\mathcal{B}$ is a subset of a bounded DCC set, so it is itself a bounded DCC set. If $\mathcal{B}$ is infinite, then there is an increasing infinite sequence. But this would give a decreasing infinite sequence in $\mathcal{A}_\infty$, which is impossible since $\mathcal{A}_\infty$ is a DCC set. \[surf\] Let $(X,\Delta)$ be a klt pair of dimension two and assume that the coefficients of $\Delta$ are in a DCC set of rational numbers $\mathcal{A}\subset [0,1]$. Assume that $\kappa(K_X+\Delta)=1$. Then there is an explicitly computable constant $N$ depending only on the set $\mathcal{A}$ such that ${\lfloor N(K_X+\Delta) \rfloor}$ induces the Iitaka fibration. To prove the theorem we are free to change the birational model of $(X,\Delta)$ (without changing the coefficients of $\Delta$). So after running the Log Minimal Model Program we can assume that $K_X+\Delta$ is nef. Log abundance for surfaces implies that $K_X+\Delta$ is semiample. Therefore there exists a positive integer $k$ such that $|{\lfloor k(K_X+\Delta) \rfloor}| $ defines the Iitaka fibration $f:X{\longrightarrow}Y$. The morphism $f:X{\longrightarrow}Y$ for $K_X+\Delta$ satisfies the hypothesis of Theorem \[kodform\], and hence we can write $$K_X+\Delta\sim_{\mathbb{Q}}f^*(K_Y+B+J).$$ By replacing the morphism $f:X {\longrightarrow}Y$ by an appropriate model we can assume that we have an isomorphism $$H^0(X,{\lfloor n(K_X+\Delta) \rfloor})\cong H^0(Y,{\lfloor n(K_Y+B+J) \rfloor})$$ for every natural number $n$ divisible by $m$ as of Lemma \[simDCC\] and $\Delta$ is simple normal crossing over the generic point of $Y$ ( cf. [@fm Theorem 4.5]). Here $Y$ is a smooth curve. The coefficients of $B$ are in a DCC set depending only on $\mathcal{A}$ (cf. [@ambroAC Remark 3.1.4]). We follow the argument in Section 6 of [@fm] to compute an integer $N$ depending only on $\mathcal{A}$ for which ${\lfloor N(K_Y+B+J) \rfloor}$ is an ample divisor. By Theorem \[main\] there is an integer $m$, depending only on the DCC set $\mathcal{A}$ by Lemma \[simDCC\], for which $mJ$ is integral. Also note that ${\lfloor B \rfloor}\ge 0$. We treat three cases. Case 1 : ($g\ge2$). For $N=3m$ we obtain that deg${\lfloor N(K_Y+B+J) \rfloor}\ge 2g+1$ and so the divisor in question is ample. Case 2 : ($g=1$). We have that $\deg(J+B)>0$ and the coefficients of $m(J+B)$ are of the form integer plus an element in a fixed DCC set. Hence there is a positive constant $c=c(\mathcal{A})$ such that the multiplicity at of $m(J+B)$ at some point is greater than $c$. Then for $N>\frac{3}{c}$ we have that $\deg{\lfloor N(J+B) \rfloor}\ge3$. Case 3 : ($g=0$). In this case we have to find an integer $N$ such that deg${\lfloor N(J+B) \rfloor}-2N>0$. This follows immediately form Lemma \[DCCr\]. \[DCCr\] For any set of elements $a_i$ in a DCC set $\mathcal{A}\subset (0,1)$ such that $-2+\sum_{i=1}^na_i>0$ there is an integer $N=N(\mathcal{A})$ such that $-2N+\sum_{i=1}^n{\lfloor Na_i \rfloor}>0$. We proceed by induction on $n$. Let $c$ be any number $0<c<\min\mathcal{A}$ and $k$ such that $0<k<\min\{\mathcal{A}_\infty\cap(2,\infty)\}-2$. The base case is $n=3$ and then it is enough to take $N>\frac{4}{k}$. In fact $${\lfloor Na_1 \rfloor}+{\lfloor Na_2 \rfloor}+{\lfloor Na_3 \rfloor}\ge {\lfloor Na_1 \rfloor}+{\lfloor Na_2 \rfloor}+{\lfloor 2N \rfloor}-{\lfloor N(2-a_3) \rfloor}-1.$$ But $N(a_1+a_2+a_3-2)>4$ hence ${\lfloor Na_1 \rfloor}+{\lfloor Na_2 \rfloor}-{\lfloor N(2-a_3) \rfloor}>2$ and so the desired inequality follows. For the inductive step suppose that $\sum_{i=1}^na_i\ge 3$ and order the $a_i$ so that $a_i\le a_{i+1}$. Then $\sum_{i=1}^{n-1}a_i>2$ and the assertion follows by induction. If not we have that $\sum_{i=1}^na_i < 3$ and hence $n<\frac{3}{c}$. It suffices to take $N>\frac{3+c}{ck}>\frac{n+1}{k}$ since then $$\sum_{i=1}^n{\lfloor Na_i \rfloor}-2N\ge {\lfloor \sum_{i=1}^nNa_i-2N \rfloor}-n+1\ge 2.$$ Iitaka fibration for threefolds of log Kodaira dimension two. ============================================================= In this section we complete the proof of Theorem \[appl\] by proving it in dimension three. \[3folds\] Let $(X,\Delta)$ be a klt pair of dimension three and assume that the coefficients of $\Delta$ are in a DCC set of rational number $\mathcal{A}\subset [0,1]$. Assume that $\kappa (K_X+\Delta)=2$. Then there is a constant $N$ depending only on the set $\mathcal{A}$ such that ${\lfloor N(K_X+\Delta) \rfloor}$ induces the Iitaka fibration. Performing the same type of reductions in the proof of Theorem \[surf\] we assume that we are in the case when we have a morphism $f:X{\longrightarrow}Y$ where $Y$ is a surface, $\Delta_{|F}$ is non-trivial and we have an isomorphism $ H^0(X,{\lfloor n(K_X+\Delta) \rfloor})\cong H^0(Y,{\lfloor n(K_Y+B+M) \rfloor}) $ for every $n$ sufficiently divisible. Here the divisor $K_Y+B+M$ is big, the coefficients of $B$ are in a DCC set depending only on $\mathcal{A}$ (cf. [@ambroAC Remark 3.1.4]), $M$ is nef. Now take $n$ also divisible by by $l$ where $l$ is an integer such that $lM$ is integral and $|lM|$ is base point free. The integer $l$ depends only on the DCC set $\mathcal{A}$. Such $l$ exists by the case of Conjecture \[conj\] that is proven in [@shok Theorem 8.1]. Notice that $(Y,B)$ is klt by [@ambroSBP Theorem 3.1] and also by [@shok Corollary 7.17]. The divisor $lM$ is base point free so we can replace it with a linearly equivalent divisor in $M'$, such that the the pair $(Y,B+\frac{1}{l}M')$ is klt and $H^0(Y,{\lfloor n (K_Y+B+\frac{1}{l}M' \rfloor}) = H^0(Y,{\lfloor n(K_Y+B+M) \rfloor})$ for every natural number $n$ divisible by $l$. Now define the DCC set $\mathcal{B}=\mathcal{A'}\cup\{\frac{1}{l}\}$. Observe that $\mathcal{B}$ depends only on $\mathcal{A}$. Define $B_1=B+\frac{1}{l}M'=\sum_i b_iB_i$ where $B_i$ are distinct irreducible divisors. By [@almo Theorem 4.6] there is a computable constant $\beta$ that depends only on $\mathcal{B}$ such that $K_Y+(1-\beta)B_1$ is a big divisor. Let $b$ be the minimum of the set $\mathcal{B}$ and let $k={\lceil \frac{1}{b\beta} \rceil}$. Then define $B'=\sum_ib_i'B_i$ where $b_i'=\frac{{\lfloor kb_i \rfloor}}{k}$. We have that the divisor $K_Y+B'$ is big with coefficients in the DCC set $\mathcal{C}=\{\frac{i}{k}|i=1,\ldots, k-1\}$. Also we have the inclusion $H^0(Y,{\lfloor m(K_Y+B') \rfloor}) \subset H^0(Y, {\lfloor m(K_Y+B+\frac{1}{l}M') \rfloor})$ for every $m$. Now Theorem \[birsurf\] implies that there is a number $N'$ depending only on $\mathcal{A}$ such that ${\lceil m(K_Y+B') \rceil}$ defines a birational map for $m\ge N'$. Define $N=kN'$. Then we have that $H^0(Y,{\lceil N(K_Y+B') \rceil})=H^0(Y,{\lfloor N(K_Y+B') \rfloor})\subset H^0(Y,{\lfloor N(K_Y+B+\frac{1}{l}M') \rfloor})$ and hence the theorem follows. Birational boundedness for log surfaces of general type. ======================================================== In this section we prove that for a surface pair $(X,\Delta)$ of log general type with the coefficients of $\Delta$ in a DCC set $\mathcal{A}$ there is a number $N$ depending only on $\mathcal{A}$ such that the linear system $|{\lceil N(K_X+\Delta) \rceil}|$ gives a birational map. Again by [@alex] or [@almo Theorem 4.8] we have that vol$(K_X+\Delta)>\alpha^2$ for some $\alpha$ depending only on the DCC set $\mathcal{A}$. We are going to use this lower bound of the volume to create a log canonical centre. The good case is when the volume of the restriction of $K_X+\Delta$ to the log canonical centre is large. Then we can proceed by cutting down the log canonical centre to a point and we generate a section of an appropriate multiple of $K_X+\Delta$. If the volume of the restriction is smaller then we are going to proceed as in [@tod]. \[birsurf\] Let $(X,\Delta)$ be a klt surface of log general type and assume that the coefficients of $\Delta$ are in a DCC set $\mathcal{A}\subset{\mathbb{Q}}$. Then there is a number $N$ depending only on $\mathcal{A}$ such that ${\lceil m(K_X+\Delta) \rceil}$ defines a birational map for $m\ge N$. Consider a log resolution $f:X'{\longrightarrow}X$ of $(X,\Delta)$ and write $f^*(K_X+\Delta)=K_{X'}+(f^{-1})_*\Delta+\sum_ie_iE_i$ with $E_i$ exceptional. There is a natural number $n$ such that $e_i<1-\frac{1}{n}$ for every $i$. Define $\Delta'=(f^{-1})_*\Delta+\sum(1-\frac{1}{n})E_i$. Since we have the inclusion $H^0(X',{\lceil m(K_{X'}+\Delta') \rceil})\subset H^0(X,{\lceil m(K_X+\Delta) \rceil})$ by replacing the $\mathcal{A}$ with the DCC set $\mathcal{A}\cup\{1-\frac{1}{n}|n\in\mathbb{N}\}$ we can assume that $X$ is smooth. By [@alex] or [@almo Theorem 4.8] we have that vol$(K_X+\Delta)>\alpha^2$ for some $\alpha$ depending only on the DCC set $\mathcal{A}$. Take a Zariski decomposition $K_X+\Delta \sim_{\mathbb{Q}}A +E $ with $A$ nef and $E$ effective and $A$ orthogonal to each component of $E$. We have that vol$(K_X+D)=$vol$(A)>\alpha^2$ Choose two general points $x_1,x_2 \in X$. Arguing as in [@tak Lemma 5.4 and Lemma 5.5] we can produce a divisor $D_1 \sim a_1 A$, with $a_1 < \frac {\sqrt{2}}{\alpha}$ such that there is a non-empty subset $I_1$ of $\{1,2\}$ with the following property: (\*) : $(X,D_1)$ is lc but not klt at $x_i$ for $i \in I_1$ and not lc at $x_i$ for $i \notin I_1$. With this choice of $a_1$ we can furthermore assume that either codim Nklt$(X,D_1)=2$ at $x_i$ for $i\in I_1$ or Nklt$(X,D_1)=Z\cup Z_+$ such that $Z$ is irreducible curve and $x_i$ is in $Z$ but not in $Z_+$ for $i \in I_1$. Assuming that $Z\cdot A > c$ for some constant $c$ and still following [@tak Lemma 5.8] we can produce a divisor $D_2\sim a_2 A$ with $a_2 <c+\epsilon+a_1$ such that there is a subset $I_2$ of $\{1,2\}$ with the property that $(X,D_2)$ is lc but not klt at $x_i\in I_2$ and not lc at $x_i$ for $i \notin I_2$ and codim Nklt$(X,D_2)=2$ at $x_i$ for $i \in I_2$. Now if we set $G=D_2+(m-1-a_2-\epsilon)A+(m-1)E+F$ where $0<\epsilon\ll 1$ and $F={\lceil (m-1)K_X+m\Delta \rceil}-(m-1)K_X-(m-1)\Delta$ we observe that ${\lceil (m-1)K_X+m\Delta \rceil}-G\sim_{\mathbb{Q}}\epsilon A$. Since $A$ is an ample divisor Kawamata-Viehweg vanishing implies that $H^1(X,{\lceil m(K_X+\Delta) \rceil}\otimes{\mathcal{J}}(G))=0$ for $m>a_2+1$ and hence the linear system $|{\lceil m(K_X+\Delta) \rceil}|$ gives a birational map onto its image (cf. [@raz Chapter 9]). Thus we can now assume that for every general point $x\in X$ we have a pair $(D_x,V_x)$, such that $D_x \sim a_1 A$, $V_x$ is a pure log canonical centre of $D_x$, and dim $V_x = 1$. By [@mac Lemma 3.2] we have a diagram $$\xymatrix{ X'\ar[d]^f \ar[r]^\pi & X \\ B& }$$ where $\pi$ is dominant and generically finite morphism of normal projective varieties, and the image of the general fibre of $f$ is $V_x$ for some $x\in X$. Arguing as in section 3 of [@tod] we can assume that the map $\pi$ is birational. In fact if $\pi$ is not birational we have at least two centres of log canonical singularities through a general point. Replacing each such pair of centres with a *minimal* centre we may assume that the dimension of the centres is zero and this way ${\lceil m(K_X+\Delta) \rceil}$ gives a birational map onto its image for $m>3a_1+1$ (compare [@tod page 11]). Thus we consider the case when $\pi$ is birational. We replace $X$ with a model on which $K_X+\Delta$ is nef and big. To complete the proof we will show that the degree of the restriction of $K_X+\Delta$ to a log canonical centre through a general point on an appropriate model is bounded from below by a constant that depends only on the DCC set $\mathcal{A}$. This is enough since we can apply Kawamata-Viehweg vanishing as before before to produce sections with the desired properties and hence a birational map. If $X {\longrightarrow}B$ is not a morphism (over a general point $b \in B$ then there is a point $x\in X$ such that we have at least two pairs $(D_1,V_1)$ and $(D_2,V_2)$, such that $D_i \sim a_1 (K_X+\Delta)$, $x\in V_i$ a pure log canonical centre of $K_X+\Delta+D_i$ of dimension 1 and $V_1\neq V_2$ corresponding to two general fibres of $f$. If $x$ is a smooth point, we have that $$(K_X+\Delta)\cdot V_1=\frac{1}{a_1}D_2\cdot V_1 \ge \frac{1}{a_1} V_2\cdot V_1 \ge\frac{1}{a_1}$$ since $V_1^2\ge 0$. If $x$ is not smooth then $(X,\Delta)$ is not terminal at $x$ and so there is a projective birational morphism $\pi:X'{\longrightarrow}X$ extracting a divisor of discrepancy less than or equal to zero. Therefore $\pi^*(K_X+\Delta)=K_{X'}+\Delta'$ where $\Delta'\ge 0$ and $K_{X'}+\Delta'$ is still nef and big. Since there are only finitely many divisors of non-positive discrepancy after finitely many extractions as above we may assume that there is a morphism $f:X'{\longrightarrow}B$. Thus we may write $\pi^*(K_X+\Delta)=K_{X'}+\Delta'$. Here $\Delta'$ is effective and $K_{X'}+\Delta'$ is nef and big. Now let $\beta=\beta(\mathcal{A})$ be as defined in 3.5 of [@almo]. We can assume that every $\pi$-exceptional divisor that dominates $B$ appears with a coefficient grater than $1-\beta$ in $K_{X'}+\Delta'$. In fact suppose that this is not the case for an exceptional divisor $E$. Away from the intersection of $E$ with the other components of $\Delta'$ the divisor $E$ intersects two general fibres $F_1$ and $F_2$ corresponding to two log canonical centres as before. With this choice there are no other log canonical places of $K_X+\Delta+D_1+D_2$ lying over $\pi(E)$ connecting the intersection of $E$ with $F_1$ and $F_2$. Then by the Connectedness Principle $E$ is a log canonical place for $K_X+\Delta+D_1+D_2$. In particular mult$_E\pi^*D_i\ge \frac{\beta}{2}$ for at least one of the $D_i$, say $D_1$, and hence $(K_X+\Delta)\cdot V_1>\frac{\beta}{2a_1}$. Now take a log resolution $g:X''{\longrightarrow}X'$ of $(X',\Delta')$ and let $f'=f\circ g$ and write $K_{X''}+\Delta''+\sum e_iE_i+N_1=g^*(K_{X'}+\Delta')+N_2$ where $\Delta'''+\sum e_iE_i+N_1$ and $N_2$ are effective with no common components, $\Delta''$ is the strict transform of $\Delta$ and the $E_i$ are the strict transforms of the $\pi$-exceptional divisors that dominate $B$ with $g_*(\Delta''+\sum e_iE_i+N_1)=\Delta'$ ($N_1$ and $N_2$ do not intersect the general fibre of $f'$). The divisor $K_{X''}+\Delta''+\sum E_i+{\lceil N_1 \rceil}$ is big and $(X'', \Delta''+\sum E_i+{\lceil N_1 \rceil})$ is lc with the coefficients in a DCC set that depends only on $\mathcal{A}$ and so by [@almo Theorem 4.6] the divisor $K_{X''}+\Delta''+(1-\beta)\sum E_i+{\lceil N_1 \rceil}$ is still big. Hence for the general fibre $F'$ of $f'$ we have that deg$(K_{X''}+\Delta''+(1-\beta)\sum E_i+{\lceil N_1 \rceil})_{|F'}=\deg (K_{X''}+\Delta''+(1-\beta)\sum E_i)_{|F'}\ge c>0$ where $c$ depends only on $\mathcal{A}$. Now $g_*(\Delta''+(1-\beta)\sum E_i)\le \Delta'$ and so $(K_{X'}+\Delta')_{|F}\ge c$. Since $K_{X'}+\Delta'=\pi_*(K_X+\Delta)$ it follows that $(K_X+\Delta)\cdot V_1\ge c$. Let $(X,\Delta)$ be a surface klt pair of log general type and assume that the coefficients of $\Delta$ are in a DCC set $\mathcal{A}$. Then there is a number $N$ depending only on $\mathcal{A}$ such that ${\lfloor N(K_X+\Delta) \rfloor}$ defines a birational map. Change the coefficients as in the last part of the proof of Theorem \[3folds\] and reduce to the case in which all the denominators of the coefficients of $\Delta$ are the same. Then by taking an appropriate multiple proceed with integral divisors only. [\[G-K-MP98\]]{} V. Alexeev, *Boundedness and $K^2$ for singular surfaces.* arXiv:alg-geom/9402007. V. Alexeev, S. Mori, *Bounding singular surfaces of general type*, Algebra, arithmetic and geometry with applications (2000), 143-174. F. Ambro, *Shokurov’s boundary property*, J. Diff. Geom. [**67**]{}(2004), no. 2, 309-322. F. Ambro, *The adjunction conjecture and its applications.* arXiv:math/9903060. F. Ambro, *The moduli b-divisor of an lc-trivial fibration*, Compos. Math. [**141**]{} (2005), no. 2, 385-403. O. Fujino, *Application of Kawamata’s positivity theorem*, Poc. Japan Acad. Ser. A Math. [**75(6)**]{} (1999), 129-171. O. Fujino and S. Mori, *A canonical bundle formula*, J. Diff. Geom. [**50**]{} (2000), 167-188. C. Hacon and J. M$^\textrm{c}$Kernan, *Boundedness of pluricanonical maps of varieties of general type*, Invent. Math. [**166**]{} (2006), no. 1, 1-25. Y. Kawamata, *Kodaira dimension of certain algebraic fiber spaces*, J. Fac. Sci. Univ. Tokyo [**30**]{} (1983), 1-24. Y.Kawamata, *Subadjunction of log canonical divisors for a variety of codimension 2*, Contmp. Math. [**207**]{} (1997), 78-88. Y. Kawamata, *Subadjunction of log canonical divisors II*, Amer. J. Math, [**120**]{} (1998), 79-88. J. Kollár, *Kodaira’s canonical bundle formula and subadjanction*, Flips for 3-folds and 4-folds, A. Corti et. al., Oxfrod Univ. Press (2007), 134-162. J. Kollár et. al., *Flips and abundance for algebraic threefolds.*, Astérisque No. 211 (1992). R. Lazarsfeld, Positivity in algebraic geometry II, Springer-Verlag, Berlin (2004). J. M$^\textrm{c}$Kernan, *Boundedness of log terminal Fano pairs of bounded index*, arXiv:math.AG/0205214 v1. S. Mori, *Classification of higher dimensional varieties*, Proc. Symp. Pure Math. [**46**]{} (1987), 269-331. G. Pacienza, *On the uniformity of the Iitaka fibration*, arXiv:0709.0310. A. Ringler , *On a conjecture of Hacon and McKernan in dimension three*, arXiv:0708.3662. Y. G. Prokhorov and V. V. Shokurov, *Toward the second main theorem on the complements*, arXiv:math.AG/0606242. S. Takayama, *Pluricanonical systems on algebraic varieties of general type*, Invent. Math. [**165**]{} (2006), no. 3, 551-587. G. Todorov, *Pluricanonical maps on threefolds of general type* Ann. Inst Fourier, [**57**]{}(4) (2007) 1315-1330. E. Viehweg, De-Qi Zhang, *Effective Iitaka fibrations*, arXiv:0707.4287. [^1]: The author would like to thank Professor Christopher Hacon for suggesting the problem and many useful conversations and suggestions.

--- abstract: 'We introduce the [*diffusion $K$-means*]{} clustering method on Riemannian submanifolds, which maximizes the within-cluster connectedness based on the diffusion distance. The diffusion $K$-means constructs a random walk on the similarity graph with vertices as data points randomly sampled on the manifolds and edges as similarities given by a kernel that captures the local geometry of manifolds. Thus the diffusion $K$-means is a multi-scale clustering tool that is suitable for data with non-linear and non-Euclidean geometric features in mixed dimensions. Given the number of clusters, we propose a polynomial-time convex relaxation algorithm via the semidefinite programming (SDP) to solve the diffusion $K$-means. In addition, we also propose a nuclear norm (i.e., trace norm) regularized SDP that is adaptive to the number of clusters. In both cases, we show that exact recovery of the SDPs for diffusion $K$-means can be achieved under suitable between-cluster separability and within-cluster connectedness of the submanifolds, which together quantify the hardness of the manifold clustering problem. We further propose the [*localized diffusion $K$-means*]{} by using the local adaptive bandwidth estimated from the nearest neighbors. We show that exact recovery of the localized diffusion $K$-means is fully adaptive to the local probability density and geometric structures of the underlying submanifolds.' address: - 'Department of StatisticsUniversity of Illinois at Urbana-ChampaignS. Wright Street, Champaign, IL 61820: <xhchen@illinois.edu>: <http://publish.illinois.edu/xiaohuichen/> ' - 'Department of StatisticsUniversity of Illinois at Urbana-ChampaignS. Wright Street, Champaign, IL 61820: <yy84@illinois.edu>: <https://sites.google.com/site/yunyangstat/> ' author: - Xiaohui Chen - Yun Yang bibliography: - 'clustering\_sdp.bib' date: 'First arXiv version: March 11, 2019. This version: ' title: 'Diffusion $K$-means clustering on manifolds: provable exact recovery via semidefinite relaxations' --- [^1] Introduction {#sec:introduction} ============ This article studies the clustering problem of partitioning $n$ data points to $K$ disjoint (smooth) Riemannian submanifolds with $1 {\leqslant}K {\leqslant}n$. Problem formulation ------------------- Let ${\mathcal{D}}_{k},k=1,\dots,K$ be compact and connected Riemannian manifolds of dimension $q_{k}$. Suppose that ${\mathcal{D}}_{k}$ can be embedded as a [*submanifold*]{} of an ambient Euclidean space ${\mathbb{R}}^{p}$ equipped with the Euclidean metric $\|\cdot\|$ (i.e., there is an immersion $\varphi_{k}: {\mathcal{D}}_{k} \to {\mathbb{R}}^{p}$ such that the differential ${\mathrm{d}}\varphi_{x}$ is injective for all $x \in {\mathcal{D}}_{k}$ and $\varphi_{k}$ is a homeomorphism onto $\varphi_{k}({\mathcal{D}}_{k}) \subset {\mathbb{R}}^{p}$; cf. [@doCarmo1992_RG]). In our clustering setting, we work with [*disjoint*]{} submanifolds ${\mathcal{D}}_{1},\dots,{\mathcal{D}}_{K}$ in ${\mathbb{R}}^{p}$ and denote $S = \bigsqcup_{k=1}^{K} {\mathcal{D}}_{k}$ as their disjoint union. Each smooth submanifold ${\mathcal{D}}_{k}$ is endowed with the Riemannian metric $\rho_{k}$ induced from $\|\cdot\|$, and we denote ${\mathscr{D}}_{k}$ as the Borel $\sigma$-algebra on ${\mathcal{D}}_{k}$ (i.e., the $\sigma$-algebra generated by the open balls in ${\mathcal{D}}_{k}$ with respect to $\rho_{k}$). Let $X_{1}^{n} := \{X_{1},\dots,X_{n}\}$ be a sequence of independent random variables taking values in $S$. Suppose that there exists a clustering structure $G_{1}^{*},\dots,G_{K}^{*}$ (i.e., a partition on $[n] := \{1,\dots,n\}$ satisfying $\bigsqcup_{k=1}^{K} G_{k}^{*} = [n]$) such that each of the $n$ data points belongs to one of the $K$ clusters: if $i \in G_{k}^{*}$, then $X_{i} \sim \mu_{k}$ for some probability distribution $\mu_{k}$ supported on ${\mathcal{D}}_{k}$. Given the observations $X_{1}^{n}$, the task of this paper is to develop computationally tractable algorithms with strong theoretical guarantees for recovering the true clustering structure $G_{1}^{*},\dots,G_{K}^{*}$. ![Comparison of the $K$-means and the SDP relaxed diffusion $K$-means clustering methods on a synthetic data sampled from three clusters with one disk and two annuli.[]{data-label="fig:kmeans_demo"}](kmeans_demo.pdf "fig:")\ ![Comparison of the $K$-means and the SDP relaxed diffusion $K$-means clustering methods on a synthetic data sampled from three clusters with one disk and two annuli.[]{data-label="fig:kmeans_demo"}](diffusion_kmeans_demo.pdf "fig:") Classical clustering methods such as $K$-means [@MacQueen1967_kmeans] and mixture models [@FraleyRaftery2002_JASA] assume that data points from each cluster are sampled in the neighborhood (with the same dimension) of a [*centroid*]{}, where ${\mathcal{D}}_{k}$ contains only one point in ${\mathbb{R}}^{p}$. Such methods are effective for partitioning data with ellipsoidal contours, which implicitly implies that the similarity (or affinity) criteria of centroid-based clustering methods target on some notions of “compactness". In modern applications such as image processing and computer vision [@ShiMalik2000_IEEEPAMI; @SouvenirPless2005_ICCV; @ElhmifarVidal2011_NIPS], structured data with geometric features are commonly seen as clusters without necessarily being close together and having the same dimension. Figure \[fig:kmeans\_demo\] is an illustration for such observation on a synthetic data sampled from a noisy version of three clusters with one disk and two annuli. In this example, it is visually clear to distinguish the three clusters, however the $K$-means method fails to correctly cluster the data points. There are two main reasons for the failure of $K$-means. First, the north pole and south pole in the outer annulus have the largest Euclidean distance among all data points, even though they belong to the same cluster. Second, the annuli and the disk live in different dimensions. In particular, the annulus is a one-dimensional circle in ${\mathbb{R}}^{2}$ that is locally isometric to the real line and the disk has dimension two. Thus these geometric concerns motivate us to seek a more natural and flexible notion of closeness for clustering analysis. In this paper, we shall focus on the clustering criterion based on the [*connectedness*]{}, which is suitable for simultaneously addressing the two issues. First, connectedness is a graph property that does not rely on the physical distance: two vertices are connected if there is a path joining them. This extends the closeness from the local neighborhood to the global sense. Second, connectivity is a viable notion for clustering components of mixed dimensions, as long as all clusters live in the same ambient space where the graph connectivity weights can be computed. In the population version, a clustering component can be viewed as a smooth submanifold, embedded in ${\mathbb{R}}^{p}$. In Riemannian geometry, a Riemannian submanifold ${\mathcal{M}}$ in ${\mathbb{R}}^{p}$ is said to be [*connected*]{} if for any $x, y \in {\mathcal{M}}$, there is a parameterized regular curve joining $x$ and $y$. Thus an appealing notion of ${\mathcal{M}}$ for being a cluster is that ${\mathcal{M}}$ is a compact and connected component in ${\mathbb{R}}^{p}$. In our setting, a clustering model is the union of $K$ disjoint submanifolds ${\mathcal{D}}_{1},\dots,{\mathcal{D}}_{K}$, and each ${\mathcal{D}}_{k}$ is equipped with a probability distribution $\mu_{k}$. Thus for the underlying true clustering model, there are $K$ connected graphs that do not overlap. In the sample version, data points are randomly generated from $\{({\mathcal{D}}_{k}, \mu_{k})\}_{i=1}^{k}$ with a clustering structure $\{G_{k}^{*}\}_{k=1}^{K}$. Typically, weighted graphs computed from the observed data points are fully connected (e.g., based on the Gaussian kernel). Thus a fundamental challenge of clustering analysis is to recover the true clustering structure from a noisy and fully connected weighted graph on the data. Our contributions ----------------- In this paper, we propose a new clustering method, termed as the [*diffusion $K$-means*]{}, for manifold clustering. The diffusion $K$-means contains two key ingredients. First, it constructs a random walk (i.e., a Markov chain) on the weighted random graph with data points $X_{1}^{n}$ as vertices and edge weights computed from a kernel representing the similarity of data in a local neighborhood. By running the Markov chain forward in time, the local geometry (specified by the kernel bandwidth) will be integrated at multiple time scales to reveal global (topological) structures such as the connectedness properties of the graph. In the limiting case as the sample size tends to infinity and the bandwidth tends to zero, the random walk becomes a diffusion process over the the manifold. By looking at the spectral decomposition of this limiting diffusion process, the evaluations of the eigenfunctions at vertices $X_{1}^{n}$ can be viewed as a continuous embedding of the data, called [*diffusion map*]{}, into a higher-dimensional Euclidean space. Second, once the diffusion map is obtained, we can compute the diffusion distance [@coifman2006diffusion] and the $K$-means algorithm (with the Euclidean metric) can be naturally extended with the diffusion affinity as the similarity measure. Since the diffusion distance/affinity captures the connectedness among vertices on the weighted random graph, the diffusion $K$-means aims to maximize the within-cluster connectedness, which can be recast as an [*assignment*]{} problem via a 0-1 integer program. Because 0-1 integer programming problems with a non-linear objective function is generally $\mathsf{NP}$-hard, solving the diffusion $K$-means is computationally intractable, i.e., polynomial-time algorithms with exact solutions only exists in special cases. This motivates us to consider semidefinite programming (SDP) relaxations. We propose two versions of SDP relaxations of the diffusion $K$-means. The first one requires the knowledge of the number of clusters, and it can be viewed as an extension from Peng and Wei’s SDP relaxation [@PengWei2007_SIAMJOPTIM] for the $K$-means (as well as Chen and Yang’s SDP relaxation [@ChenYang2018] for the generalized $K$-means for non-Euclidean data in an inner product space) to the manifold clustering setting with diffusion distances. Figure \[fig:kmeans\_demo\] (on the right) shows that the SDP relaxed diffusion $K$-means can correctly identify the three clusters in the previous example. The second SDP relaxation does not require the number of clusters as an input. The idea is to drop the constraint on the trace of the clustering membership matrix (which involves number of clusters $K$), and to add a penalization term on the diffusion $K$-means objective function. Thus it can be seen as a nuclear norm [*regularized*]{} version of the SDP for diffusion $K$-means that is adaptive to the number of clusters. For both SDP relaxations of the diffusion $K$-means, we show that exact recovery can be achieved when the underlying submanifolds are well separated and subsamples within each submanifold are well connected. Since the diffusion $K$-means and its regularized version have only one (non-adaptive) bandwidth parameter to control the local geometry, they may fail for clustering problems with unbalanced sizes, mixed dimensions, and different densities. In such situations, a random walk on the vertices sampled from regions of low density mixes slower than that from regions of high density. This motivates us to consider a variant of diffusion $K$-means, termed as the [*localized diffusion $K$-means*]{}, by using data-dependent local bandwidth. We adopt the self-tuning procedure from [@zelnik2005self] where local adaptive bandwidth is estimated from the nearest neighbors and we show that the localized diffusion $K$-means is adaptive to the local geometry and the local sampling density for the purpose of exact recovery of the true clustering structure. To summarize, our contributions are listed as below. 1. We introduce the diffusion $K$-means clustering method for manifold clustering, which integrates the nonlinear embedding via the diffusion maps and the $K$-means clustering. 2. We propose two versions of the SDP relaxations of the diffusion $K$-means: one requires to know the number of clusters, and the other one does not require such knowledge as an input (and thus it is adaptive to the unknown number of clusters). 3. We derive the exact recovery property of the SDP relaxed diffusion $K$-means in terms of two hardness parameters of the clustering problem: one reflects the separation of the submanifolds, and the other one quantifies the degree of connectedness of the submanifolds. 4. We combine the local scaling procedures with the diffusion $K$-means and its regularized version, and derive their adaptivity when the clustering problems have unbalanced sizes, mixed dimensions, and different densities. Related work ------------ There is a large collection of clustering methods and algorithms in literature, which can be broadly classified into two categories: hierarchical clustering and partition-based clustering. Hierarchical clustering recursively divides data points into groups in either a top-down or bottom-up way. Such algorithms are greedy and they often get stuck into local optimal solutions. Partition-based clustering methods such as $K$-means clustering [@MacQueen1967_kmeans] and spectral clustering [@vanLuxburg2007_spectralclustering] directly assign each data point with a group membership. Perhaps one of the most widely used clustering methods is the $K$-means method, due to the existence of algorithms with linear sample complexity (such as Lloyd’s algorithm [@Lloyd1982_TIT]). However, the $K$-means clustering converges locally to a stationary point that depends on the initial partition. Recent theoretical studies in [@LuZhou2016] show that, given a proper initialization (such as spectral clustering), Lloyd’s algorithm for optimizing the $K$-means objective function can consistently recover the clustering structures. Exact and approximate recovery of various convex relaxations for the $K$-means and mixture models are studied in literature [@PengWei2007_SIAMJOPTIM; @LiLiLingStohmerWei2017; @FeiChen2018; @Royer2017_NIPS; @BuneaGiraudRoyerVerzelen2016]. To the best of our knowledge, existing theoretical guarantees developed for the convex relaxed $K$-means clustering assumes that the clusters are sampled in a neighborhood of a centroid. Thus results derived for $K$-means in literature cannot be directly compared with our results. On the other hand, spectral clustering methods [@ShiMalik2000_IEEEPAMI; @NgJordanWeiss2001_NIPS] take the similarity matrix as the input and solve the clustering problem by applying $K$-means to top eigenvectors of the graph Laplacian matrix or its normalized versions [@Chung1996_SpectralGraphTheory]. In essence, spectral clustering contains two steps: (i) the Laplacian eigenmaps embed data into feature spaces, and (ii) $K$-means on top eigenvectors serves as a rounding procedure to obtain the true clustering structure [@vanLuxburg2007_spectralclustering]. Conventional intuition for spectral clustering is that the embedding step (i) often “magnifies" the cluster structure from the dataset to the feature space such that it can be revealed by a relatively simple algorithm (such as $K$-means) in step (ii). However, theoretical guarantees (such as exact recovery) for the spectral clustering is rather vague in literature, partially due to its two-step nature which complicates its theoretical analysis. For instance, [@vonLuxburgBelkinBousquet2008_AoS] study the convergence of spectral properties of random graph Laplacian matrices constructed from sample points and they establish the consistency of the spectral clustering in terms of eigenvectors. However, they do not address the problem of the exact recovery property of the clustering structure. Similar results along this direction can be found in [@Rosasco2010_JMLR; @schiebinger2015; @TrillosHoffmannHosseini2019]. [@LingStrohmer2019] propose similar SDP relaxations for the spectral clustering as in the present paper with the diffusion distance metric replaced with the graph Laplacian. Specifically, it is shown in [@LingStrohmer2019] that those SDP relaxations can exactly recover the true clustering structure under a spectral proximity condition. Such condition is deterministic and difficult to check for general data generation models. (A particular checkable random model is the stochastic ball model [@LingStrohmer2019].) During the preparation of this work, we notice a recent work [@maggioni2018learning] which proposes a similar idea of applying the diffusion distance as the similarity metric for clustering based on fast search and find of density peaks clustering (FSFDPS) [@rodriguez2014clustering]. To prove exact recovery, [@maggioni2018learning] requires strong deterministic assumptions on the Markov transition matrix associated with the diffusion process that could be difficult to check under their stochastic clustering model. Literature on theoretical guarantees for manifold clustering is rather scarce, with a few exceptions [@Arias-Castro2011_IEEETIT; @LittleMaggioniMurphy2017]. Near-optimal recovery of some emblematic clustering methods based on pairwise distances of data is derived under a condition that the [*minimal*]{} signal separation strength over all pairs of submanifolds is larger than a threshold. Compared with our diffusion $K$-means with local scaling, results established in [@Arias-Castro2011_IEEETIT] are non-adaptive to the local density and (geometric) structures of the submanifolds (cf. Theorem \[thm:main\_adaptive\_h\] and \[thm:main\_adaptive\] ahead). [@LittleMaggioniMurphy2017] derive recovery guarantees for manifold clusters using a data-dependent metric called the longest-leg path distance (LLPD) that adapts to the geometry of data, where the data points are drawn from a mixture of uniform distributions on disjoint low-dimensional geometric objects. Notation -------- For a matrix $A\in{\mathbb{R}}^{n\times n}$ and index subsets $G,G'\subset [n]$, we use notation $A_{GG'}$ to denote the submatrix of $A$ with rows being selected by $G$ and columns by $G'$, and $\mbox{diag}(A)$ the $n$-dimensional vector composed of all diagonal entries of $A$. Let $\|A\|_{\infty} =\max_{1 {\leqslant}i,j {\leqslant}n} |A_{ij}|$ and $\|A\|_1=\sum_{i,j=1}^{n} |A_{ij}|$ denote the $\ell_\infty$ and the $\ell_1$ norm of the vectorization $\mbox{vec}(A)$ of matrix $A$. Let ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert A \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast}$ and ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert A \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mbox{\scriptsize op}}$ denote the nuclear norm and the operator norm of matrix $A$. We shall use $c, c', c_{1}, c_{2},\dots,C,C', C_{1},C_{2},\dots$ to denote positive and finite (non-random) constants whose values may depend on the submanifolds $\{{\mathcal{D}}_{k}\}_{k=1}^{K}$ and the probability distributions $\{\mu_{k}\}_{k=1}^{K}$ supported on $\{{\mathcal{D}}_{k}\}_{k=1}^{K}$ and whose values may vary from place to place. The rest of the paper is organized as follows. In Section \[sec:prelims\], we discuss some related background on diffusion distances and nonlinear embeddings in Euclidean spaces, as well as the Laplace-Beltrami operator for the heat diffusion process on Riemannian manifolds. In Section \[sec:diffusion\_Kmeans\], we introduce the diffusion $K$-means and its SDP relaxations. Regularized and localized diffusion $K$-means clustering methods are also proposed in this section. In Section \[sec:main\_results\], we present our main results on the exact recovery property of the SDP relaxed diffusion $K$-means. Simulation studies are presented in Section \[sec:simulations\]. Proofs are relegated to Section \[sec:proofs\]. Preliminaries {#sec:prelims} ============= Let $S \subset {\mathbb{R}}^{p}$ and $\mu$ be a probability distribution on $S$. Let $S_n=\{X_1,X_2,\ldots,X_n\}$ be $n$ i.i.d. random variables in $S$ sampled from $\mu$, and $\mu_n =n^{-1} \sum_{i=1}^n \delta_{X_i}$ the empirical distribution. In Section \[sec:prelims\], we discuss the Euclidean embedding via the diffusion distance in the general setting. Thus in this section, we do not assume $S$ to be a disjoint union of $K$ submanifolds in ${\mathbb{R}}^{p}$ and the sample $S_{n}$ does not necessarily have a clustering structure. Euclidean embedding and diffusion distances {#subsec:Euclidean_embedding_diffusion_dist} ------------------------------------------- Let $\kappa : S \times S \to {\mathbb{R}}$ be a positive semidefinite kernel that satisfies: symmetry: $\kappa(x, y) = \kappa(y, x)$, positivity preserving: $\kappa(x, y) {\geqslant}0$. A kernel is a similarity measure between points of $S$. A widely used example is the Gaussian kernel: $$\label{eqn:gaussian_kernel} \kappa(x, y) = \exp \left( -{ \|x-y\|^{2} \over 2 h^{2}} \right),$$ where $h > 0$ is the bandwidth parameter that captures the local similarity of points in $S$. Given a kernel $\kappa$ with property (i) and (ii), we can define a reversible Markov chain on $S$ via the normalized graph Laplacian constructed as follows. Specifically, for any $x \in S$, let $$d(x) = \int_{S} \kappa (x, y)\, {\mathrm{d}}\mu(y)$$ be the degree function of the graph on $S$. For simplicity, we assume $d(x)>0$ for all $x\in S$. Define $$\label{eqn:transition_kernel} p(x, y) = {\kappa(x, y) \over d(x)},$$ which satisfies the positivity preserving property (ii) and the conservation property $$\int_{S} p(x, y) \, {\mathrm{d}}\mu(y) = 1.$$ Thus $p(x, y)$ can be viewed as the one-step transition probability of a (stationary) Markov chain on $S$ from $x$ to $y$. We shall write this Markov chain (i.e., random walk) as ${\mathcal{W}}= (S, \mu, p)$, where $p(\cdot, \cdot)$ is called the [*transition kernel*]{} of ${\mathcal{W}}$. Equivalently, we can describe ${\mathcal{W}}$ by the bounded linear operator $P : L^{2}({\mathrm{d}}\mu) \to L^{2}({\mathrm{d}}\mu)$ defined as $$Pf(x) = \int_{S} p(x, y) f(y) \, {\mathrm{d}}\mu(y).$$ Here $L^{2}({\mathrm{d}}\mu) := L^{2}(S, {\mathrm{d}}\mu)$ is the class of squared integrable functions on $S$ with respect to $\mu$. In literature, $P$ is often called the [*diffusion operator*]{} for the following reason. If we denote $p_{t}(x, y)$ as the $t$-step transition probability of the Markov chain ${\mathcal{W}}$ from $x$ to $y$ in $S$, then $$P_{t}f(x) = \int_{S} p_{t}(x, y) f(y) \, {\mathrm{d}}\mu(y),\qquad t=1,2,\ldots,$$ form a semi-group of bounded linear operators on $L^{2}({\mathrm{d}}\mu)$. Let $\Pi$ be a stationary distribution of the Markov chain ${\mathcal{W}}$ over $S$. Then $\Pi$ is absolutely continuous with respect to $\mu$, and the probability density function $\pi$ of $\Pi$ with respect to $\mu$ is given by the Radon-Nikodym derivative $$\label{eqn:stationary_dist} \pi(x) = \frac{{\mathrm{d}}\Pi}{{\mathrm{d}}\mu} (x) = { d(x) \over \int_{S} d(y)\, {\mathrm{d}}\mu(y)}.$$ Since $\Pi$ is the stationary measure of the Markov chain ${\mathcal{W}}$ with transition $P$, we have $$\Pi(P_{t}f) = \Pi(f)$$ for all bounded measurable functions $f$, where $\Pi(f) := \int_{S} f(x) \, {\mathrm{d}}\Pi(x)$. Note that, since the kernel $\kappa$ is symmetric, ${\mathcal{W}}$ is reversible and satisfies the detailed balance condition: $$\pi(x)\, p(x, y) = \pi(y) \,p(y, x),\quad\forall x,y\in S.$$ \[lem:spectral\_decomposition\_Markov\_chain\] Let $$R(x,y) = {\kappa(x,y) \over \sqrt{\pi(x)} \, \sqrt{\pi(y)}},\quad\forall x,y\in S.$$ If $$\label{eqn:kernel_integrability_condition} \int_{S} \int_{S} R(x,y)^{2} \, {\mathrm{d}}\mu(x) \, {\mathrm{d}}\mu(y) < \infty,$$ then the following statements hold. 1. There exists a sequence of nonnegative eigenvalues $\lambda_{0} {\geqslant}\lambda_{1} {\geqslant}\cdots {\geqslant}0$ such that $$R(x,y) = \sum_{j=0}^{\infty} \lambda_{j} \phi_{j}(x) \phi_{j}(y),$$ where $\{\phi_{j}\}_{j=0}^{\infty}$ is the set of associated eigenfunctions to $\{\lambda_{j}\}_{j=0}^{\infty}$, and $\{\phi_{j}\}_{j=0}^{\infty}$ forms an orthonormal basis of $L^{2}({\mathrm{d}}\mu)$. 2. The transition probability $p(x,y)$ admits the following decomposition $$p(x,y) = \sum_{j=0}^{\infty} \lambda_{j} \psi_{j}(x) \varphi_{j}(y),$$ where $\psi_{j}(x) = \phi_{j}(x) / \sqrt{\pi(x)}$ and $\varphi_{j}(x) = \phi_{j}(x) \sqrt{\pi(x)}$. 3. The diffusion operator $P$ satisfies $$P\, \psi_{j} = \lambda_{j} \,\psi_{j}, \quad j = 0, 1, \dots.$$ In addition, $\lambda_0=1$ and $\psi_0\equiv 1$. The proof of Lemma \[lem:spectral\_decomposition\_Markov\_chain\] is given in Appendix \[app:A\], and our argument is similar to Lemma 12.2 in [@levin2017markov] in the finite-dimensional setting. If ${\mathcal{W}}$ is irreducible (i.e., the graph on $S$ is connected in that for all $x,y \in S$, there is some $t>0$ such that $p_{t}(x,y) > 0$), then the stationary distribution $\pi$ is unique. Thus if we run this Markov chain ${\mathcal{W}}$ forward in time, then the local geometry (captured by the kernel $\kappa$ which is parameterized by the bandwidth $h$) will be integrated to reveal global structures of $S$ at multiple (time) scales. In particular, we can define a class $\{D_{t}\}_{t \in {\mathbb{N}}_{+}}$ of [*diffusion distances*]{} [@coifman2006diffusion] on $S$ by $$D_{t}(x, y) := \|\, p_{t}(x, \cdot) - p_{t}(y, \cdot)\,\|_{L^{2}({\mathrm{d}}\mu/\pi)} = \left\{ \int_{S} [\, p_{t}(x, z) - p_{t}(y, z)]^{2}\, {{\mathrm{d}}\mu(z) \over \pi(z)} \right\}^{1/2}.$$ Roughly speaking, for each $t\in {\mathbb{N}}_{+}$ and $x,y\in S$, the diffusion distance $D_{t}(x,y)$ quantifies the the total number of paths with length $t$ connecting $x$ and $y$ (see Figure \[fig:diffussion\_dist\]), thereby reflecting the local connectivity at the time scale $t$. ![Illustration of the diffusion distance between two red dots as the total number of paths connecting them. The region on the left panel (the Cheeger dumbbell) is “less" connected than the region on the right as there are fewer paths in the former due to the narrow bottleneck in the middle. In particular, the second smallest eigenvalue associated with the Laplace-Beltrami operator (or the Cheeger isoperimetric constant) of the left region is smaller that of the right.[]{data-label="fig:diffussion_dist"}](examples.pdf "fig:") ![Illustration of the diffusion distance between two red dots as the total number of paths connecting them. The region on the left panel (the Cheeger dumbbell) is “less" connected than the region on the right as there are fewer paths in the former due to the narrow bottleneck in the middle. In particular, the second smallest eigenvalue associated with the Laplace-Beltrami operator (or the Cheeger isoperimetric constant) of the left region is smaller that of the right.[]{data-label="fig:diffussion_dist"}](examples_2.pdf "fig:") \[lem:spectral\_representation\_diffusion\_distances\] If the Markov chain ${\mathcal{W}}= (S, \mu, p)$ is irreducible, then we have $$D_{t}^2(x, y) = \sum_{j=0}^{\infty} \lambda_{j}^{2t} \, [\psi_{j}(x) - \psi_{j}(y)]^{2}$$ for all $t\in{\mathbb{N}}_{+}$ and $x,y\in S$. The proof of Lemma \[lem:spectral\_representation\_diffusion\_distances\] is given in Appendix \[app:A\]. For an irreducible Markov chain, the spectral gap is strictly positive (i.e., $|\lambda_{j}| < 1$ for all $j > 0$). Based on the spectral decomposition in Lemma \[lem:spectral\_representation\_diffusion\_distances\] and noting that $\psi_{0} \equiv 1$, we see that the diffusion distance can be written as $$\begin{aligned} \label{Eqn:Spectral_rep} D_{t}(x, y) = \left\{ \sum_{j=1}^{\infty} \lambda_{j}^{2t} \, [\psi_{j}(x) - \psi_{j}(y)]^{2} \right\}^{1/2}.\end{aligned}$$ In this case, the diffusion distance $D_{t}(x, y)$ decays to zero as $t$ increases, provided that $x$ and $y$ belong to a connected component of the graph on $S$. In particular, the decay rate of the spectrum quantifies the connectivity of points in the graph on $S$. Given a positive integer $q \in {\mathbb{N}}_{+}$, the diffusion maps $\{\Psi_{t}\}_{t \in {\mathbb{N}}}$ are defined as $$\Psi_{t}^{(q)}(x) = (\lambda_{1}^{t} \psi_{1}(x), \dots, \lambda_{q}^{t} \psi_{q}(x))^{T},$$ where the $\ell$-th component $\Psi_{t\ell}^{(q)}(x)$ is the $\ell$-th diffusion coordinate in ${\mathbb{R}}^{q}$. Thus we obtain an embedding of $(S, \mu)$ into the Euclidean space ${\mathbb{R}}^{q}$ in the limiting sense that $$D_{t}(x, y) = \lim_{q \to \infty} \|\Psi_{t}^{(q)}(x) - \Psi_{t}^{(q)}(y)\|_{2}.$$ Empirical diffusion embedding {#subsec:empirical_diffusion_embedding} ----------------------------- Recall that $S_n=\{X_1,X_2,\ldots,X_n\}$ are $n$ i.i.d. random variables in $S$ sampled from $\mu$, and $\mu_n =n^{-1} \sum_{i=1}^n \delta_{X_i}$ is the empirical distribution. Given $S_n$, we can consider finite sample approximations $\{D_{n,t}\}_{t\in{\mathbb{N}}_{+}}$ to the underlying population level quantities $\{D_{t}\}_{t\in{\mathbb{N}}_{+}}$. More precisely, consider a weighted graph with nodes corresponding to the elements in $S_n$, where the weight between a pair $(X_i,X_j)$ of nodes is $\kappa(X_i,X_j)$, for $i,j\in [n]$. Define the (rescaled) empirical degree function $d_n:\, S_n\to {\mathbb{R}}_{+}$ by $$\begin{aligned} d_n(x) = n\,\int_{S_n} \kappa(x, y)\, {\mathrm{d}}\mu_n (y) =\sum_{i=1}^n \kappa(x, X_i),\quad \forall x\in S_n,\end{aligned}$$ where we added an extra $n$-factor so that $d_n(X_i)$ is also the degree of node $X_i$ in the weighted graph. Let $D_n$ denote the $n$-by-$n$ diagonal matrix whose $i$-th diagonal entry is $d_n(X_i)$. Consider the (empirical) random walk ${\mathcal{W}}_n=(S_n, \mu_n, P_n)$ over $S_n$ with transition probability $$\begin{aligned} P_n (x, y) = \frac{\kappa(x,y)}{d_n(x)}\quad\forall x,y\in S_n.\end{aligned}$$ The (empirical) stationary distribution $\pi_n$ of the random walk ${\mathcal{W}}_n$ over $S_n$ becomes $$\begin{aligned} \pi_n(x) = \frac{d_n(x)}{\sum_{i=1}^{n} d_n(X_i)} \quad \forall x\in S_n.\end{aligned}$$ For any vector $v\in{\mathbb{R}}^n_{+}$, let $L^2(v)=\{u=(u_1,\ldots,u_n)\in{\mathbb{R}}^n:\, \|u\|_{L^2(v)} = \sum_{i=1}^n v_i\, u_i^2\}$ denote a weighted $L^2$ space over $S_n$. We define the [*empirical diffusion distances*]{} $\{D_{n,t}\}_{t\in {\mathbb{N}}_{+}}$ as $$\begin{aligned} D_{n,t}(x,y) =\|P^t_n(x,\cdot) - P^t_n(y,\cdot)\|_{L^2(\mbox{diag}(D_n^{-1}))} = \left\{ \sum_{i=1}^n [\, P_{n}^t(x, X_i) - P^t_n(y, X_i)]^{2}\, {1 \over d_n(X_i)} \right\}^{1/2},\end{aligned}$$ for all $x,y\in S_n$ and $t\in{\mathbb{N}}_{+}$. Roughly speaking, $\sqrt{n^{-1}\sum_{i=1}^{n} d_n(X_i)}\,D_{n,t}$ provides an empirical estimate to $D_t$. Similar to the spectral representation  for $D_t$, we also have the following spectral representation of $D_{n,t}$ (see Appendix \[app:B\]), $$\begin{aligned} D_{n,t}(x, y)& = \left\{ \sum_{j=0}^{n-1} \lambda_{n, j}^{2t} \, [\psi_{n,j}(x) - \psi_{n,j}(y)]^{2} \right\}^{1/2}, \quad\forall t\in{\mathbb{N}}_{+} \mbox{ and }x,y\in S_n,\end{aligned}$$ where $1=\lambda_{n,0}{\geqslant}\lambda_{n,1} {\geqslant}\cdots {\geqslant}\lambda_{n,n-1} {\geqslant}0$ are the nonnegative eigenvalues (due to the positive semidefiniteness of the kernel $\kappa$) of the transition probability operator $P_n$, which can be identified with a matrix in ${\mathbb{R}}^{n\times n}$ with $[P_{n}]_{ij} = P_n(X_i,X_j)$ as its $(i,j)$-th element, and $\psi_{n,0},\psi_{n,1},\ldots,\psi_{n,n-1}:\,S_n \to{\mathbb{R}}$ are the associated eigen-functions on $S_n$ with unit $L^2(\mbox{diag}(D_n))$ norm, which can be identified with vectors in ${\mathbb{R}}^n$ with $[\psi_{n,j}]_i = \psi_{n,j}(X_i)$ as the $i$-th element of $\psi_{n,j}$ for $i\in [n]$ and $j =0,1\ldots,n-1$. The empirical diffusion distance $D_{n,t}(X_i, X_j)$ between two nodes $X_i$ and $X_j$ is also the Euclidean distance between their embeddings $\Psi_{n,t}(X_i)$ and $\Psi_{n,t}(X_j)$ via the empirical diffusion map $$\begin{aligned} \Psi_{n,t}:\, S_n \to {\mathbb{R}}^{n},\quad x\mapsto \big(\lambda_{n,1}^t \psi_{n,1}(x),\ldots,\lambda_{n,n}^t\psi_{n,n}(x)\big)^T.\end{aligned}$$ The Laplace-Beltrami operator on Riemannian manifolds {#subsec:Laplace-Beltrami_operator} ----------------------------------------------------- The Laplace-Beltrami operator on Riemannian manifolds is a generalization of the Laplace operator on Euclidean spaces. Let $f : {\mathcal{M}}\to {\mathbb{R}}$ be an (infinitely) differentiable function with continuous derivatives on a $q$-dimensional compact and smooth Riemannian manifold and $\nabla_{{\mathcal{M}}} f$ be the gradient vector field on ${\mathcal{M}}$ (i.e., $\nabla_{{\mathcal{M}}} f(x)$ is the deepest direction of ascent for $f$ at the point $x \in {\mathcal{M}}$). The Laplace-Beltrami operator $\Delta_{{\mathcal{M}}}$ is defined as the divergence of the gradient vector $$\Delta_{{\mathcal{M}}} f = -\mbox{div}(\nabla_{{\mathcal{M}}} f),$$ where the $\mbox{div}$ operator is relative to the volume form $\mbox{Vol}_{{\mathcal{M}}}$ of ${\mathcal{M}}$. Here we adopt the convention with the minus sign of the divergence such that $\Delta_{{\mathcal{M}}}$ is a positive-definite operator. With integration-by-parts, we have for any two differentiable functions $f$ and $g$, $$\int_{{\mathcal{M}}} g(x) \Delta_{{\mathcal{M}}} f \,{\mathrm{d}}\mbox{Vol}_{{\mathcal{M}}}(x) = \int_{{\mathcal{M}}} \langle \nabla_{{\mathcal{M}}} g(x), \nabla_{{\mathcal{M}}} f(x) \rangle \,{\mathrm{d}}\mbox{Vol}_{{\mathcal{M}}}(x),$$ where the inner product is taken in the $q$-dimensional tangent space of ${\mathcal{M}}$ (at the point $x$). In a Euclidean space (i.e., ${\mathcal{M}}= {\mathbb{R}}^{q}$), the Laplace-Beltrami operator is the usual Laplace operator $$\Delta f = -\sum_{j=1}^{q}{\partial^{2}f \over \partial x_{j}^{2}}.$$ On a general $q$-dimensional Riemannian manifold ${\mathcal{M}}$, the Laplace-Beltrami operator in a local coordinate system $(e^{1},\dots,e^{q})$ with a metric tensor ${\mathbf{G}}= (g_{ij})_{i,j=1}^{q}$ is given by $$\Delta_{{\mathcal{M}}} f = -{1 \over \sqrt{\det({\mathbf{G}})}} \sum_{j=1}^{q} {\partial \over \partial e^{j}} \left( \sqrt{\det({\mathbf{G}})} \sum_{i=1}^{q} g^{ij} {\partial f \over \partial e^{i}} \right),$$ where $g^{ij}$ are the entries of ${\mathbf{G}}^{-1}$. In the special case ${\mathcal{M}}= {\mathbb{R}}^{q}$, ${\mathbf{G}}$ is the $q \times q$ identity matrix. Note that $\Delta_{{\mathcal{M}}}$ is a self-adjoint positive-definite compact operator, its spectrum contains a sequence of nonnegative eigenvalues $0 {\leqslant}\lambda_{0} {\leqslant}\lambda_{1} {\leqslant}\cdots$. If in addition ${\mathcal{M}}$ is connected, then the second smallest eigenvalue $\lambda_{1} > 0$. As we will show, $\lambda_{1}$ depends on the connectivity of the manifold (Figure \[fig:diffussion\_dist\]), thus characterizing the limiting mixing time of the empirical random walk ${\mathcal{W}}_n$ over the $S_n$ as $n\to \infty$ and $h\to 0^{+}$, when $S_n$ is sampled from the manifold ${\mathcal{M}}$. Diffusion $K$-means {#sec:diffusion_Kmeans} =================== Recall that in our clustering model, $S_n=\{X_{1},X_2,\ldots,X_n\}$ is a sample of independent random variables taking values in $S$, where $S$ is the union of $K$ disjoint Riemannian submanifolds ${\mathcal{D}}_{1},\dots,{\mathcal{D}}_{K}$ embedded in the ambient space ${\mathbb{R}}^p$. The clustering problem is to divide these $n$ data points into $K$ clusters, so that points in the same cluster belongs to the same connected component in $S$, based on certain similarity measures between the points. In particular, the (classical) $K$-means clustering method minimizes the total intra-cluster squared Euclidean distances in ${\mathbb{R}}^p$ $$\min_{G_{1},\dots,G_{K}} \sum_{k=1}^{K} {1 \over |G_{k}|} \sum_{i,j \in G_{k}} \|X_{i}-X_{j}\|^{2}$$ over all possible partitions on $[n]$, where $|G_{k}|$ is the cardinality of $G_{k}$. Dropping the sum of squared norms $\sum_{i=1}^{n} \|X_{i}\|^{2}$, we see that the $K$-means clustering is equivalent to the maximization of the total within-cluster covariances $$\max_{G_{1},\dots,G_{K}} \sum_{k=1}^{K} {1 \over |G_{k}|} \sum_{i,j \in G_{k}} a_{ij}, \quad\mbox{with }a_{ij} = X_{i}^{T} X_{j}.$$ Here, $a_{ij}=X_{i}^{T} X_{j}$ can be viewed as a similarity measure specified by the Euclidean space inner product $\langle X_{i}, X_{j}\rangle_{{\mathbb{R}}^p}$. In general, we can replace the Euclidean inner product with any other inner product over $S_n$ [@ChenYang2018]. For manifold clustering, we replace it with the inner product induced from the empirical diffusion distance, that is, $$\begin{aligned} \langle x,\, y\rangle_{D_{n,t}} = \langle \Psi_{n,t}(x), \,\Psi_{n,t}(y) \rangle_{{\mathbb{R}}^{n}} =\sum_{j=1}^{n} \lambda_{n,j}^{2t} \,\psi_{n,j}(x) \, \psi_{n,j}(y),\quad \forall x,y\in S_n.\end{aligned}$$ Henceforth, we will refer to $\langle \cdot,\, \cdot\rangle_{D_{n,t}}$ as the [*diffusion affinity*]{}. Interestingly, we can obtain this diffusion affinity value without explicitly conducting eigen-decomposition (spectral decomposition) to the transition probability matrix $P_n=D_n^{-1} K_n$ (or the symmetrized matrix $D_n^{-1/2} K_n D_n^{-1/2}$), where recall that $D_n=\mbox{diag}\big(d_n(X_1),\ldots,d_n(X_n)\big)\in{\mathbb{R}}^n$ is the degree diagonal matrix, and $K_n=\big[\kappa(X_i,X_j)\big]_{n\times n}\in{\mathbb{R}}^{n\times n}$ is the empirical kernel matrix. In fact, we may use the following relation that links the empirical diffusion affinity with entries in matrix $P_n$ raising to power $2t$ (see Appendix \[app:B\] for details), $$\begin{aligned} \langle x,\, y\rangle_{D_{n,t}} = \sum_{j=1}^{n} \lambda_{n,j}^{2t} \,\psi_{n,j}(x) \, \psi_{n,j}(y)=[P_n^{2t}D_n^{-1}](x,y).\end{aligned}$$ This motivates a $K$-means clustering method via diffusion distances, referred to as the *diffusion $K$-means* as $$\begin{aligned} \label{Eqn:Diffussion_K_Means} \max_{G_{1},\dots,G_{K}} \sum_{k=1}^{K} {1 \over |G_{k}|} \sum_{i,j \in G_{k}} [P_n^{2t}D_n^{-1}]_{ij},\end{aligned}$$ for the tuning parameter $t$ interpreted as the number of steps in the empirical random walk ${\mathcal{W}}_n$. Note that here the affinity matrix $P_n^{2t}D_n^{-1}= D_n^{-1/2} (D_n^{-1/2} K_n D_n^{-1/2})^{2t} D_n^{-1/2} \in{\mathbb{R}}^{n\times n}$ is symmetric. In light of the connections between the diffusion distance and the random walk ${\mathcal{W}}_n$ over $S_n$ in Section \[subsec:empirical\_diffusion\_embedding\], the diffusion $K$-means attempts to maximize the total within-cluster connectedness. \[rem:intution\_DKM\] In Section \[subsec:Euclidean\_embedding\_diffusion\_dist\], we see that, on a connected submanifold ${\mathcal{D}}_{k}$, the (population) diffusion process  converges to the stationary distribution : $$p_{t}(x,y) \to \pi(y) = {\int_{{\mathcal{D}}_{k}} \kappa(x,y) \, {\mathrm{d}}\mu(x) \over \iint_{{\mathcal{D}}_{k} \times {\mathcal{D}}_{k}} \kappa(x,z) \, {\mathrm{d}}\mu(x)\, {\mathrm{d}}\mu(z)} \quad \text{as } t \to \infty.$$ In fact, since the kernel $\kappa$ is positive semidefinite, this convergence holds at a geometric rate governed by the spectral gap of the Laplace-Beltrami operator on ${\mathcal{D}}_{k}$ (cf.  in the proof of Lemma \[Lem:T\_2\]). Thus the empirical version of the diffusion (i.e., the Markov chain on the random graph generated by $X_{1}^{n}$ and $\kappa$) obeys $${P^{t}_{n}(X_{i}, X_{j}) \over \sum_{\ell \in G_{k}^{*}} \kappa(X_{\ell}, X_{j})} \approx {1 \over \sum_{\ell,\ell' \in G_{k}^{*}} \kappa(X_{\ell}, X_{\ell'})}$$ for any two data points $X_{i}, X_{j} \in {\mathcal{D}}_{k}$. On the other hand, if the separation between the submanifolds is large enough and $t$ is not so large, then the probability to diffuse from one cluster to another one is small (cf. Lemma \[lem:between\_cluster\_random\_walk\]). Thus we expect that $${P^{t}_{n}(X_{i}, X_{j}) \over \sum_{\ell \in G_{k}^{*}} \kappa(X_{\ell}, X_{j})} \approx 0$$ for any two data points $X_{i} \in {\mathcal{D}}_{k}$ and $X_{j} \in {\mathcal{D}}_{m}$ such that $k \neq m$. This means that the within-cluster entries of the empirical diffusion affinity are larger than the between-cluster entries. In particular, for suitably large $t \in {\mathbb{N}}_{+}$, the empirical diffusion affinity matrix $A_{n} := A_{n,t} = P_{n}^{2t} K_{n}^{-1}$ tends to become close to a block-diagonal matrix $$\begin{aligned} \label{Eqn:approx_form_A_n} A_n \approx \begin{pmatrix} \displaystyle \frac{1}{N_1} \mathbf{1}_{G_1^\ast} \mathbf{1}^T_{G_1^\ast} & 0 & \cdots & 0\\ 0 & \displaystyle \frac{1}{N_2} \mathbf{1}_{G_2^\ast} \mathbf{1}^T_{G_2^\ast} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & \cdots & 0 & \displaystyle \frac{1}{N_K} \mathbf{1}_{G_K^\ast} \mathbf{1}^T_{G_K^\ast} \end{pmatrix},\end{aligned}$$ where $N_{k} = \sum_{\ell,\ell' \in G_{k}^{*}} \kappa(X_{\ell}, X_{\ell'})$. Since each diagonal block of $A_{n}$ tends to be a constant matrix, if we run the Markov chain for a suitably long time, then this block-diagonal structure in the limit precisely conveys the true clustering structure in that $i, j \in G_{k}^{*}$ if and only if $\lim_{t \to \infty} [A_{n,t}]_{i,j} = N_{k}^{-1} > 0$. The trade-off regime of $t$ (cf.  in Theorem \[thm:main\]) is determined by the non-asymptotic bounds on the convergence of the empirical diffusion maps to its population version (cf. Lemma \[lem:within\_cluster\_random\_walk\] and \[lem:between\_cluster\_random\_walk\]), as well as the submanifolds separation. Note that, for every partition $G_{1},\dots,G_{K}$, there is a one-to-one $n \times K$ [*assignment matrix*]{} $H = (h_{ik}) \in \{0,1\}^{n \times K}$ such that $h_{ij} = 1$ if $i \in G_{k}$ and $h_{ij} = 0$ if $i \notin G_{k}$. Thus the diffusion $K$-means clustering problem can be recast as a 0-1 integer program: $$\label{eqn:kernel_Kmeans_integer_program} \max \left\{ \langle P_n^{2t}D_n^{-1}, H B H^{T} \rangle : H \in \{0,1\}^{n \times K}, H {\mathbf{1}}_{K} = {\mathbf{1}}_{n} \right\},$$ where ${\mathbf{1}}_{n}$ denotes the $n \times 1$ vector of all ones and $B = {\text{diag}}(n_{1}^{-1},\dots,n_{K}^{-1})$, where $n_{k}=|G_{k}|$ for $k=1,\ldots,K$ is the size of the $k$-th cluster. The diffusion $K$-means clustering problem (\[eqn:kernel\_Kmeans\_integer\_program\]) is often computationally intractable, namely, polynomial-time algorithms with exact solutions only exist in special cases [@SongSmolaGrettonBorgwardt2007_ICML]. For instances, the (classical) $K$-means clustering is an $\mathsf{NP}$-hard integer programming problem with a non-linear objective function [@PengWei2007_SIAMJOPTIM]. Exact and approximate recovery of various SDP relaxations for the $K$-means [@PengWei2007_SIAMJOPTIM; @LiLiLingStohmerWei2017; @FeiChen2018; @Royer2017_NIPS; @GiraudVerzelen2018] are studied in literature. However, it remains a challenging task to provide statistical guarantees for clustering methods that can capture non-linear features of data taking values on manifolds. Semidefinite programming relaxations ------------------------------------ We consider the SDP relaxations for the diffusion $K$-means clustering. Note that every partition $G_{1},\dots,G_{K}$ of $[n]$ can be represented by a partition function $\sigma : [n] \to [K]$ via $G_{k}=\sigma^{-1}(k), k=1,\dots,n$. If we change the variable $Z = H B H^{T}$ in the 0-1 integer program formulation (\[eqn:kernel\_Kmeans\_integer\_program\]) of the diffusion $K$-means, then $Z$ satisfies the following properties: $$\label{eqn:constraints_clustering_generic_integer_program} Z^{T} = Z, \quad Z \succeq 0, \quad \operatorname{tr}(Z) = \sum_{k=1}^{K} n_{k} b_{kk}, \quad (Z {\mathbf{1}}_{n})_{i} = \sum_{k=1}^{K} n_{k} b_{\sigma(i)k}, \; i=1,\dots,n.$$ For the diffusion $K$-means $B = {\text{diag}}(n_{1}^{-1},\dots,n_{K}^{-1})$, the last constraint in (\[eqn:constraints\_clustering\_generic\_integer\_program\]) reduces to $Z {\mathbf{1}}_{n} = {\mathbf{1}}_{n}$, which does not depend on the partition function $\sigma$. Thus we can relax the diffusion $K$-means clustering to the SDP problem: $$\label{eqn:clustering_Kmeans_sdp} \begin{gathered} \hat{Z} = \operatorname{argmax}\left\{ \langle A, Z \rangle : Z \in {\mathscr{C}}_K \right\} \\ \qquad \mbox{with } {\mathscr{C}}_K = \{Z \in {\mathbb{R}}^{n \times n} : Z^{T} = Z, Z \succeq 0, \operatorname{tr}(Z) = K, Z {\mathbf{1}}_{n} = {\mathbf{1}}_{n}, Z {\geqslant}0 \}, \end{gathered}$$ where $Z \succeq 0$ means that $Z$ is positive semidefinite and $Z {\geqslant}0$ means that all entries of $Z$ are non-negative, and matrix $A=[a_{ij}]:=A_{n}=P_n^{2t}D_n^{-1}\in{\mathbb{R}}^{n\times n}$. We shall use $\hat{Z}$ to estimate the true “membership matrix" $Z^{*}$, where $$\label{eqn:Kmeans_true_membership_matrix} Z_{ij}^{*} = \left\{ \begin{array}{cc} 1/n_{k} & \text{if } i, j \in G_{k}^{*} \\ 0 & \text{otherwise} \\ \end{array} \right. .$$ Note that $Z^{*}$ is a block diagonal matrix (up to a permutation) of rank $K$. If $X_{1},\dots,X_{n} \in {\mathbb{H}}$ (i.e., ${\mathbb{S}}= {\mathbb{H}}$) for some Hilbert space ${\mathbb{H}}$ and $a_{ij} = \langle X_{i}, X_{j} \rangle_{{\mathbb{H}}}$ is the inner product between $X_i$ and $X_j$, then (\[eqn:clustering\_Kmeans\_sdp\]) is the SDP for kernel $K$-means proposed in [@ChenYang2018]. In particular, [@PengWei2007_SIAMJOPTIM] consider the special case for the (Euclidean) $K$-means, where ${\mathbb{H}}= {\mathbb{R}}^{p}$ and $a_{ij} = X_{i}^{T} X_{j}$. Observe that the SDP relaxation (\[eqn:clustering\_Kmeans\_sdp\]) does not require the knowledge of the cluster sizes other than the number of clusters $K$. Thus it can handle the general case for unequal cluster sizes. Regularized diffusion $K$-means {#subsec:regularized_diffusion_Kmeans} ------------------------------- In practice, the number $K$ of clusters is rarely known. Note that the SDP problem  depends on $K$ only through the constraint $\operatorname{tr}(Z) = K$. Therefore we propose a [*regularized diffusion $K$-means*]{} estimator by dropping the constraint on the trace and penalizing $\operatorname{tr}(Z)$ as follows, $$\label{eqn:clustering_Kmeans_sdp_unknown_K} \begin{gathered} \tilde{Z} := \tilde{Z}_{\lambda} = \operatorname{argmax}\left\{ \langle A, Z \rangle - n\,\lambda \operatorname{tr}(Z) : Z \in {\mathscr{C}}\right\} \\ \mbox{with } {\mathscr{C}}= \{Z \in {\mathbb{R}}^{n \times n} : Z^{T} = Z, Z \succeq 0,\, Z {\mathbf{1}}_{n} = {\mathbf{1}}_{n}, Z {\geqslant}0 \}, \end{gathered}$$ where $\lambda>0$ is the regularization parameter. Recall that ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert Z \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast}$ denotes the nuclear norm of a matrix $Z$ (i.e., ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert Z \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast}$ is the sum of the singular values of $Z$). Since ${{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert Z \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast} = \operatorname{tr}(Z)$ for $Z \in {\mathscr{C}}$, it is interesting to note that  is the same as the nuclear norm penalized diffusion $K$-means $$\label{eqn:clustering_Kmeans_sdp_unknown_K_nuclear-norm_form} \begin{gathered} \tilde{Z} = \operatorname{argmax}\left\{ \langle A, Z \rangle - n\,\lambda {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert Z \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast} : Z \in {\mathscr{C}}\right\}. \end{gathered}$$ Recall that the true membership matrix $Z^{*}$ has a block diagonal structure with rank $K$, the nuclear norm penalized diffusion $K$-means  can be thought as an $\ell^{1}$ norm convex relaxation of the $\mathsf{NP}$-hard rank minimization problem (i.e., minimizing the number of non-zero eigenvalues). Thus the parameter $K$ in the clustering problem plays a similar role as the sparsity (or low-rankness) parameter in the matrix completion context [@CandesRecht2009_FoCM]. Hence the SDP problem  can be viewed as a (further) convex relaxation of the infeasible SDP problem  when $K$ is unknown. Note that similar regularizations have been considered in [@BuneaGiraudRoyerVerzelen2016] for the $G$-latent clustering models and in [@YanSarkarCheng2018_AISTATS] for stochastic block models. It remains a question to choose the value of $\lambda$. Larger values of $\lambda$ will lead to solutions containing less number of clusters (with larger sizes). In particular, when matrix $A$ is positive-definite, the following Lemma \[lem:feasibility\_SDP\_lambda\_infinity\] shows that the solution reduces to a rank one matrix that assigns all points into a giant cluster when $\lambda$ is large enough, and becomes the identity matrix that assigns $n$ points into $n$ distinct clusters when $\lambda$ is small enough. In addition, the trace $\operatorname{tr}(\tilde Z_\lambda)$ of the solution is nonincreasing in $\lambda>0$. \[lem:feasibility\_SDP\_lambda\_infinity\] Suppose $A$ is a positive definite matrix, and let $\lambda_{\max}(A)$ and $\lambda_{\min}(A)$ denote its respective largest and smallest eigenvalues. (1) If $n \lambda > \lambda_{\max}(A)$, then $Z^{\diamond} = n^{-1} J_{n}$, where $J_{n}$ is the $n \times n$ matrix of all ones, is the unique solution of the SDP . (2) If $n \lambda < \lambda_{\min}(A)$, then $Z^{\dagger} = I_n$, the $n \times n$ identity matrix, is the unique solution of the SDP . (3) If $\tilde Z_{\lambda_1}$ and $\tilde Z_{\lambda_2}$ are two solutions of the SDP  with the regularization parameter taking values $\lambda_1$ and $\lambda_2$, respectively. If $\lambda_1 < \lambda_2$, then $\operatorname{tr}(\tilde Z_{\lambda_1}){\geqslant}\operatorname{tr}(\tilde Z_{\lambda_2})$. Furthermore, if at least one of the two SDPs has a unique solution, then $\operatorname{tr}(\tilde Z_{\lambda_1})> \operatorname{tr}(\tilde Z_{\lambda_2})$. According to the interpretation of the SDP , the trace $\operatorname{tr}(Z)$ of the solution can be viewed as the fitted number of clusters. Consequently, Lemma \[lem:feasibility\_SDP\_lambda\_infinity\] implies that smaller values of $\lambda$ will result in more clusters (with smaller sizes) in $\tilde{Z}_\lambda$. In practice, we need to properly select the tuning parameter $\lambda$. We propose the following decision rule for this purpose. For each $\lambda$, we run the SDP problem  and extract the value of $\operatorname{tr}(\tilde{Z}_\lambda)$. Then we plot the solution path $\operatorname{tr}(\tilde{Z}_\lambda)$ versus $\log(\lambda)$ and pick the values of $\lambda$ which spend the longest time (on the logarithmic scale) with a flat value of $\operatorname{tr}(\tilde{Z}_\lambda)$. Here we recommend using the logarithmic scale since values of $\lambda$ with non-trivial solutions $\operatorname{tr}(\tilde Z_\lambda)$ tends to be close to zero. Algorithm \[alg1\] below summarizes this decision rule. \[alg1\] Set an increasing sequence $\{\lambda_j\}_{j=1}^J$ of candidate values for $\lambda$, for example, a geometric sequence in the interval $[n^{-1}\lambda_{\min}(A),\,n^{-1}\lambda_{\max}(A)]$. Set an upper bound $K_{\max}$ of $K$ and a tolerance level $\varepsilon\in (0,1/2)$.\ Figure \[fig:diffusion\_kmeans\_lambda\_demo\] shows the empirical result on the three clusters (one disk and two annuli) example in Section \[sec:introduction\]. According to Lemma \[lem:feasibility\_SDP\_lambda\_infinity\], the estimated number of clusters, proxied by $\operatorname{tr}(\tilde{Z}_\lambda)$, is a non-increasing function of $\lambda$. In particular, the trace $\operatorname{tr}(\tilde Z_\lambda)$ in the solution path in the upper left panel of Figure \[fig:diffusion\_kmeans\_lambda\_demo\] stabilizes around $2$ and $3$, indicating that both $2$ and $3$ are candidate values for the number of clusters. In particular, the interval of $\lambda$ (on the logarithmic scale) corresponding to value $3$ is much larger than that to value $2$, indicating that $3$ is more likely to be the true number of clusters (cf. the (correct) case in Figure \[fig:diffusion\_kmeans\_lambda\_demo\]). In Section \[sec:main\_results\], we will use our theory to explain this stabilization phenomenon, which partially justifies our $\lambda$ selection rule. In addition, as we can see from the rest three panels in Figure \[fig:diffusion\_kmeans\_lambda\_demo\], by gradually increasing $\lambda$, the adaptive diffusion $K$-means method produces a hierarchical clustering structure. Unlike the top-down or bottom-up clustering procedures which are based on certain greedy rule and can incur inconsistency, the hierarchical clustering structure produced by our approach is consistent — it does not depend on the order of partitioning or merging due to the uniqueness of the global solution from the convex optimization via the SDP. Similar observations can be drawn on another example shown in Figure \[fig:diffusion\_kmeans\_lambda\_demo\_DGP2\] containing a uniform sample on three rectangles (see DGP 2 in our simulation studies Section \[sec:simulations\] for details). Further, it is interesting to observe in Figure \[fig:diffusion\_kmeans\_lambda\_demo\] that the regularized diffusion $K$-means tuned with two clusters yields a merge between the outer annulus and the disk, which gives the largest total diffusion affinity in the objective function  among the three possible combinations of the true clusters. Since diffusion affinity decays exponentially fast to zero in the squared Euclidean distance (for the Gaussian kernel), the diffusion affinity matrix $A = P_{n}^{2t} D_{n}^{-1}$ tends to have a block diagonal structure, as weights between points belonging to different clusters are exponentially small (cf.  and Lemma \[Lemma:total\_degree\]). Thus running SDP for examples with relatively well separated clusters, such as the one in Figure \[fig:diffusion\_kmeans\_lambda\_demo\], tends to merge two clusters with largest within-cluster diffusion affinities that is irrespective of the between-cluster Euclidean distances. This may lead to a visually less appealing merge as in the Euclidean distance case (cf. the (under) case in Figure \[fig:diffusion\_kmeans\_lambda\_demo\]). On the other hand, the regularized diffusion $K$-means is able to produce more reasonable partition in splitting the clusters (cf. the bottom-left panel in Figure \[fig:diffusion\_kmeans\_lambda\_demo\]). In particular, if the regularization parameter $\lambda$ is chosen such that the corresponding number $\hat{K}$ of clusters in the SDP solution is greater than $K$, then this will cause a split in one of the true clustering structures that minimizes the between-cluster diffusion affinities after the splitting. Moreover, in our simulation studies (setup DGP 3 in Section \[sec:simulations\]), we observe that the SDP relaxed regularized diffusion $K$-means performs much better in harder cases than the spectral clustering methods when the true clusters are not well separated. ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  on the synthetic data in Figure \[fig:kmeans\_demo\]. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo"}](diffusion_kmeans_lambda_demo.pdf "fig:") ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  on the synthetic data in Figure \[fig:kmeans\_demo\]. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo"}](diffusion_kmeans_adaptive_correct_demo "fig:")\ ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  on the synthetic data in Figure \[fig:kmeans\_demo\]. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo"}](diffusion_kmeans_adaptive_over_demo "fig:") ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  on the synthetic data in Figure \[fig:kmeans\_demo\]. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo"}](diffusion_kmeans_adaptive_under_demo "fig:") ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  for DGP=2. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo_DGP2"}](diffusion_kmeans_lambda_demo_DGP2.pdf "fig:") ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  for DGP=2. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo_DGP2"}](diffusion_kmeans_adaptive_correct_demo_DGP2 "fig:")\ ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  for DGP=2. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo_DGP2"}](diffusion_kmeans_adaptive_over_demo_DGP2 "fig:") ![Plot of the estimated trace norms along the path of tuning parameter $\lambda$ (on the log-scale) in the SDP relaxed regularized diffusion $K$-means clustering method  for DGP=2. The clustered data are shown for estimated number of clusters 2 (under), 3 (correct), 4 (over).[]{data-label="fig:diffusion_kmeans_lambda_demo_DGP2"}](diffusion_kmeans_adaptive_under_demo_DGP2 "fig:") ![Plots of the SDP diffusion $K$-means clustering method without (left) and with (right) local scaling for the data generation mechanism DGP=3 in the simulation studies Section \[sec:simulations\].[]{data-label="fig:diffusion_kmeans_local_scaling_demo"}](diffusion_kmeans_no_local_scaling_demo.pdf "fig:") ![Plots of the SDP diffusion $K$-means clustering method without (left) and with (right) local scaling for the data generation mechanism DGP=3 in the simulation studies Section \[sec:simulations\].[]{data-label="fig:diffusion_kmeans_local_scaling_demo"}](diffusion_kmeans_local_scaling_demo.pdf "fig:") Localized diffusion $K$-means {#subsec:localized_diffusion_Kmeans} ----------------------------- For clustering problems with different sizes, dimensions and densities, the diffusion $K$-means may have limitations since only one bandwidth parameter $h$ is used to control the local geometry on the domain. More precisely, according to our theory (for example, Theorem \[thm:main\] below), the optimal choice of the bandwidth parameter for a $q$-dimensional submanifold as one connected component in our clustering model is $h\asymp (\log \tilde{n}/\tilde{n})^{1/q}$, where $\tilde n$ corresponds to the sample size within this cluster and thus depends on the local cluster size or density level. Figure \[fig:diffusion\_kmeans\_local\_scaling\_demo\] demonstrates such an example for a mixture of three bivariate Gaussians, which consist of one larger Gaussian component with low density and two smaller Gaussian components with high density. Empirically, the diffusion $K$-means fails on this example (even after tuning) for the reason that the larger and smaller clusters have very different local densities. This motivates us to consider a variant of diffusion $K$-means, termed as *localized diffusion $K$-means*, by using local adaptive bandwidth $h_i=h(X_i)$ for each $X_i$, $i\in[n]$. In particular, we adopt the self-tuning procedure from [@zelnik2005self] by setting $h_i$ to be $\|X_i-X_i^{(k_0)}\|$, where $X_i^{(k_0)}$ denotes the $k_0$-th nearest neighbor to $X_i$, and replacing $K_n$ with $K_n^\dagger=[K^\dagger(X_i,X_j)]_{n\times n}$ (and accordingly replacing $D_n$ with $D_n^\dagger$ corresponding to the diagonal degree matrix associated with $K^\dagger_n$) given by $$\begin{aligned} K^\dagger_n (X_i,X_j) = \exp\Big(-\frac{\|X_i-X_j\|^2}{2h_ih_j}\Big),\end{aligned}$$ in the SDP . Note that $K^\dagger_n$ is generally no longer a positive semidefinite matrix. Intuitively for $i \in G_{k}^{*}$, the local scaling $h_{i}$ automatically adapts to the local density $p_{k}(X_{i})$ about $X_{i}$, the cluster size $n_k$ and the dimension $q_{k}$ for the $k$-th Riemannian submanifold. Specifically, for each cluster $k=1,\ldots,K$, the $n_k$-by-$n_k$ submatrix $[K_n^\dagger]_{G^\ast_kG^\ast_k}$ resembles the a Gaussian kernel matrix with a homogeneous bandwidth $h_k \asymp (\log n_k/n_k)^{1/q_k}$ that adapts to the local geometry in ${\mathcal{D}}_k$. For points $X_i$ and $X_j$ belonging to distinct clusters that are properly separated, $K^\dagger_n (X_i,X_j)$ tends to be close to zero and is less affected by the choice of $h_i$ and $h_j$. Heuristically, $h_{i}$ is larger for lower density regions where the degree function of $X_{i}$ is smaller so that the random walk can speed up mixing at such lower density regions. Overall, such a locally adaptive choice of bandwidth improves the mixing time of the random walk within each cluster, while leaves the between cluster jumping probabilities remaining small. As a consequence, the pairwise diffusion affinity matrix $A$ in our SDP formulation  tends to exhibit a clearer block form reflecting the clustering structure. To compute $h_{i}$, we only need to specify $k_{0}$ to replace the (non-adaptive) bandwidth parameter $h$ whose value depend on the unknown cluster sizes $n_{k}$, dimensions $q_{k}$ of submanifolds ${\mathcal{D}}_{k}$, and the underlying probability density functions $p_{k}$ on ${\mathcal{D}}_{k}$. In contrast, the simple choice $k_{0} = \lfloor C \log{n} \rfloor$ guarantees that the local scaling $h_{i}$ adapts to the local density (cf. Theorem \[thm:main\_adaptive\_h\] in Section \[sec:main\_results\]). Main results {#sec:main_results} ============ In this section, we assume each ${\mathcal{D}}_k$ is a compact connected $q_k$-dimensional Riemannian submanifold embedded in ${\mathbb{R}}^p$ with bounded diameter, absolute sectional curvature value, and injectivity radius. Throughout the rest of the paper, we assume that each $\mu_k$ has a Lipschitz density function $p_k$ with respect to the Riemannian volume measure on ${\mathcal{D}}_k$, such that $$\begin{aligned} \label{Eqn:density_condition} c{\leqslant}p_k(x) {\leqslant}\frac{1}{c}\quad \mbox{for all }x\in {\mathcal{D}}_k,\end{aligned}$$ for some constant $c>0$. Exact recovery of diffusion $K$-means ------------------------------------- Let $\delta = \min_{1{\leqslant}k\neq k'{\leqslant}K} \|{\mathcal{D}}_k-{\mathcal{D}}_{k'}\|$, where $\|A-B\|=\inf_{x\in A, y\in B} \|x-y\|$ denote the (Euclidean) distance between two disjoint sets $A$ and $B$. Recall that the size of the true cluster $n_{k} =|G_{k}^{*}|$ for $k=1,\ldots,K$ and let $\underline{n} = \min_{1 \le k \le K} n_{k}$ denote the minimal cluster size. \[thm:main\] Let $c_1,c_2, c_3,c_4,C_1,C_2$ be some positive constants only depending on $S=\bigsqcup_{k=1}^K {\mathcal{D}}_k$, and $c$ some constant that depends on $\max_{1 \le k \le K}q_k$. If $$\label{eqn:bandwidth_condition_nonadaptive} c_1 (\log n_k/ n_k)^{1/q_k}\,(\log n_k)^{1(q_k=2)/4}{\leqslant}h {\leqslant}c_2 \quad \mbox{for each } k\in[K]$$ for some sufficiently large constant $c_{1}$ and sufficiently small constant $c_{2}$, then we can achieve exact recovery, that is $\hat Z= Z^\ast$, with probability at least $1- c_3K\,\underline{n}^{-c_4}$ as long as $$\begin{aligned} \label{eqn:exact_recovery_condition_SDP_diffusion_Kmeans_LB_eigenval} C_1\,nt\, \exp\Big\{-\frac{\delta^2}{2h^2}\Big\} +C_1\,\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\}\, h^2 t\Big\} < \frac{C_2}{n\, \max_{1 {\leqslant}k {\leqslant}K} \{n_k h^{q_k}\}},\end{aligned}$$ where $\lambda_1({\mathcal{D}}_k)>0$ denotes the second smallest eigenvalue of the Laplace-Beltrami operator (with the minus sign) on ${\mathcal{D}}_k$. A proof of this theorem is provided in Section \[Sec:proof\_thm\_main\]. A crucial step in the proof is bounding from below the absolute spectral gap of the transition matrix associated with the restricted random walk $\mathcal W_n$ onto each submanifold $\mathcal D_k$ for $k=1,\ldots,K$. We do so by applying a comparison theorem of Markov chains to connect this spectral gap with the eigensystem of the Laplace-Beltrami operator over the submanifold ${\mathcal{D}}_k$ that captures the diffusion geometry. In particular, our proof borrows existing results [@burago2014graph; @trillos2018error] on error estimates of using the spectrum of a random geometric graph to approximate the eigensystem of the Laplace-Beltrami operator in the numerical analysis literature. \[rem:comments\_condition\_main\_thm\] We begin with comments on the second term on the left hand side of . We point out that this is essentially a mixing condition on random walks over the $K$ submanifolds. In particular, it is due to a relation between the mixing times of the heat diffusion process on each submanifold and its discretized random walk ${\mathcal{W}}_n$ over vertices sampled from the submanifold. For simplicity, we illustrate this relation for $K = 1$ and ${\mathcal{D}}_{1} = {\mathbb{S}}^{1}$, where ${\mathbb{S}}^{1}$ is the unit circle in ${\mathbb{R}}^{2}$. As a one-dimensional compact smooth manifold (without boundary), ${\mathbb{S}}^{1}$ can be parametrized by the angle $\theta \in [-\pi, \pi)$. Under this parametrization, the density function $u(\tau,\theta)$ of the heat diffusion process on ${\mathbb{S}}^{1}$, as a function of time $\tau$ and location $\theta$, is determined by the corresponding heat equation, $$\label{eqn:heat_eq} {\partial \over \partial \tau}u(\tau,\theta) + \Delta u(\tau,\theta) = 0, \quad (\tau, \theta) \in (0,\infty) \times {\mathbb{S}}^{1},$$ where $\Delta$ is the Laplace-Beltrami operator on ${\mathbb{S}}^{1}$. Under the same parametrization, the Laplace operator $\Delta = -{{\mathrm{d}}^{2} \over {\mathrm{d}}\theta^{2}}$ on ${\mathbb{S}}^{1}$ (with the minus sign) admits the following eigen-decomposition $$\Delta e^{\iota n \theta} = n^{2} e^{\iota n \theta}, \quad n=0,\pm1,\pm2,\dots,$$ where $\iota = \sqrt{-1}$. That is, $(\lambda_{n}({\mathbb{S}}^{1}), e_{n}) := (n^{2}, e^{\iota n \theta})$ is an eigen-pair of $\Delta$, which implies that $\Delta$ is a positive semidefinite and unbounded operator on $L^{2}({\mathbb{S}}^{1})$ functions. Now we can solve the heat equation  by expanding it with respect to this orthonormal basis. More precisely, for any $f \in L^{2}({\mathbb{S}}^{1})$, the Fourier transform of $f$ is given by $$f(\theta) = \sum_{n=-\infty}^{\infty} a_{n} e_{n} := \sum_{n=-\infty}^{\infty} \langle f, e_{n} \rangle e_{n},$$ where $\langle f, g \rangle = (2\pi)^{-1} \int_{{\mathbb{S}}^{1}} f(\theta) \overline{g(\theta)} \, {\mathrm{d}}\theta$ is the standard inner product on ${\mathbb{S}}^{1}$. Then the solution to heat equation with the initial distribution $u(0,\theta) = f(\theta)$ is given by $$u(\tau,\theta) = \sum_{n=-\infty}^{\infty} a_{n} e^{-n^{2}\tau} e_{n}.$$ So if ${\mathbb{S}}^{1}$ is insulated, then as $\tau \to \infty$ the heat flow has a constant equilibrium state with the value equal to the average of the initial heat distribution, namely $\lim_{\tau\to\infty} u(\tau,\theta) = (2\pi)^{-1} \int_{{\mathbb{S}}^{1}} f(\theta)\,{\mathrm{d}}\theta$. In particular, the second smallest eigenvalue $\lambda_{1}({\mathbb{S}}^{1})=1$ characterizes the mixing rate of the heat diffusion process. Now we consider the “inverse Fourier transform" by expressing the solution $u$ in terms of Green’s function, also called the heat kernel, on ${\mathbb{S}}^{1}$ as $$\label{eqn:heat_kernel_unit_circle} K_{{\mathbb{S}}^{1}}(\tau, \theta, \varphi) = \sum_{n=-\infty}^{\infty} e^{-n^{2}\tau} {\tilde{e}}^{\iota n (\theta-\varphi)},$$ where $ {\tilde{e}}^{\iota n \theta} = e_{n} / \sqrt{2\pi}$ is the rescaled orthonormal basis of $L^{2}({\mathbb{S}}^{1})$. Then we obtain that $$u(\tau,\theta) = {\mathcal{K}}_{{\mathbb{S}}^{1}}^{\tau} f (\theta), \mbox{ where } {\mathcal{K}}_{{\mathbb{S}}^{1}}^{\tau} f (\theta) := \int_{{\mathbb{S}}^{1}} K_{{\mathbb{S}}^{1}}(\tau, \theta, \varphi) f(\varphi) \,{\mathrm{d}}\varphi$$ defines an (integral) heat diffusion operator on ${\mathbb{S}}^{1}$. Then the Laplace operator on ${\mathbb{S}}^{1}$ can be seen as the generator of the heat diffusion process: $$\Delta f(\theta) = - \left.{\partial \over \partial \tau}u(\tau,\theta) \right|_{\tau=0} = - \left.{\partial \over \partial \tau}{\mathcal{K}}_{{\mathbb{S}}^{1}}^{\tau}f(\theta) \right|_{\tau=0} = \lim_{\tau \to 0^{+}} {f(x) - {\mathcal{K}}_{{\mathbb{S}}^{1}}^{\tau} f(x) \over \tau}.$$ Similarly, the normalized graph Laplacian $L_n = I_n - P_n = I_n- D_n^{-1} K_n$ corresponding to the random walk ${\mathcal{W}}_n$ over a random sample of $S_n$ on ${\mathbb{S}}^{1}$ can also be seen as a discrete generator of ${\mathcal{W}}_n$. Recall that in the heat diffusion process, the second smallest eigenvalue $\lambda_{1}({\mathbb{S}}^{1})$ of its generator, i.e., the Laplace-Beltrami operator, characterizes the mixing rate of the heat diffusion process. Similarly, in the random walk ${\mathcal{W}}_n$ (as a discretization of the heat process), the second smallest eigenvalue $\lambda_{j}(L_n)$ of its discrete generator, i.e., the normalized graph Laplacian operator $L_n$, characterizes its mixing rate. From Lemma \[lem:eigenval\_convergence\_normalized\_graph\_Laplacian\], the spectrum of these two operators are related in the sense that for each $j=1,2\dots,$ with probability at least $1-c_{1} n^{-c_{2}}$, $$\lambda_{j}({\mathbb{S}}^{1}) \asymp h^{2} \lambda_{j}(L_n),$$ where $\lambda_{j}({\mathbb{S}}^{1})$ is the $j$-th eigenvalue of $\Delta$ and $\lambda_{j}(L_n)$ is the $j$-th eigenvalue of the normalized graph Laplacian. This means that we must change the time clock unit of the random walk on the graph by multiplying a factor of $h^{2}$ to approximate its underlying heat diffusion process. Thus the term $h^{2}t$ in the second term of is the right time scale $\tau$ for running the heat diffusion process on the manifold, and we need $h^{2}t \to \infty$ for the heat diffusion process converges to an equilibrium distribution. On finite data, this means that the random walk converges to its stationary distribution over the points $S_n$ sampled from the submanifold. Using this correspondence, the second term on the left hand side of is a mixing condition on random walks over the $K$ submanifolds. The first term on the left hand side of can be seen as a separation requirement of the $K$ disjoint submanifolds ${\mathcal{D}}_{1},\dots,{\mathcal{D}}_{K}$. In particular, if $t = n^{\epsilon}$ for some $\epsilon > 0$ (i.e., we run the random walk in polynomial times/steps), then the minimal separation should obey $$\label{eqn:lower_bound_Delta} \delta \gtrsim h \sqrt{\log n}$$ in order to achieve the exact recovery for the manifold clustering problem. Combining the two terms of , we see that steps of the random walk must be properly balanced: we would like the random walk on the similarity graph to sufficiently mix within each cluster (second term of ), while it does not overly mix to merge the true clusters (first term of ). This reflects the [*multi-scale*]{} property of the diffusion $K$-means. \[rem:thresholding\] In view of the approximation property  of the empirical diffusion affinity matrix $A_{n} = P_{n}^{2t} K_{n}^{-1}$ to a block-diagonal matrix in Remark \[rem:intution\_DKM\], one can show that a simple thresholding of the matrix $A_{n}$ also yields the exact recovery for a properly chosen threshold. Indeed, by the triangle inequality, we have for any $i,j \in G_k^\ast$ in the same cluster, $$[A_n]_{ij} {\geqslant}N_{k}^{-1} -\max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty.$$ Choosing a threshold value $\gamma$ such that $$\label{eqn:thresholding_value} \max_{1 {\leqslant}k\neq m {\leqslant}K} \|[A_n]_{G_k^\ast G_m^\ast}\|_\infty < \gamma < \min_{1{\leqslant}k{\leqslant}K}\Big\{\frac{1}{N_k}\Big\} - \max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty,$$ one can completely separate the block-diagonal entries from the off-diagonal ones, thus achieving exact recovery. This argument leads to the following lemma. \[lem:thresholding\_master\_bound\] If $$\label{eqn:thresholding_exact_recover_master_condition} \max_{1 {\leqslant}k\neq m {\leqslant}K} \|[A_n]_{G_k^\ast G_m^\ast}\|_\infty + \max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty < \min_{1{\leqslant}k{\leqslant}K}\Big\{\frac{1}{N_k}\Big\},$$ then thresholded estimator on the empirical diffusion affinity matrix $A_{n}$ with the threshold value satisfying  yields exact recovery. Note that condition  in Lemma \[lem:thresholding\_master\_bound\] is slightly weaker than the master condition  of the SDP relaxed diffusion $K$-means in Lemma \[lem:DKM\_SDP\_master\_bound\] (up to a factor of $K^{-1}$ for balanced clusters, say). On the other hand, thresholding has a tuning parameter $\gamma$ and it is unclear how to develop a principled procedure to choose the threshold value satisfying . On the contrary, our SDP relaxed diffusion $K$-means only requires the knowledge of the number of cluster $K$ and it is tuning-free in that sense. It is worthy to note that the second smallest eigenvalue of the Laplace-Beltrami operator in condition of Theorem \[thm:main\] can be regarded as characterizations of the connectedness of the submanifolds, where the latter can be formally quantified by the Cheeger isoperimetric constant defined as follows. Let ${\mathcal{M}}$ be a $q$-dimensional compact Riemannian manifold. Let $\mbox{Vol}({\mathcal{A}})$ denote the volume of a $q$-dimensional submanifold ${\mathcal{A}}\subset {\mathcal{M}}$ and $\mbox{Area}({\mathcal{E}})$ denote the $(q-1)$-dimensional area of a submanifold ${\mathcal{E}}$. The [*Cheeger isoperimetric constant*]{} of ${\mathcal{M}}$ is defined to be $${\mathfrak{h}}({\mathcal{M}}) = \inf_{{\mathcal{E}}} \left\{ \frac{\mbox{Area}({\mathcal{E}})}{\min(\mbox{Vol}({\mathcal{M}}_{1}), \mbox{Vol}({\mathcal{M}}_{2}))} \right\},$$ where the infimum of the normalized manifold cut (in the curly brackets) is taken over all smooth $(q-1)$-dimensional submanifolds ${\mathcal{E}}$ of ${\mathcal{M}}$ that cut ${\mathcal{M}}$ into two disjoint submanifolds ${\mathcal{M}}_{1}$ and ${\mathcal{M}}_{2}$ such that ${\mathcal{M}}= {\mathcal{M}}_{1} \bigsqcup {\mathcal{M}}_{2}$. In words, ${\mathfrak{h}}({\mathcal{M}})$ quantifies the minimal area of a hypersurface that bisects ${\mathcal{M}}$ into two disjoint pieces (cf. Figure \[fig:diffussion\_dist\]). Smaller values of ${\mathfrak{h}}({\mathcal{M}})$ mean that ${\mathcal{M}}$ is less connected – in particular, ${\mathfrak{h}}({\mathcal{M}}) = 0$ implies that there are two disconnected components in ${\mathcal{M}}$. The Cheeger isoperimetric constant may also be analogously defined for a graph and its value (i.e., the conductance of the graph) is closely related to the normalized graph cut problem. Suppose we have an i.i.d. sample $X_{1},\dots,X_{n}$ drawn from the uniform distribution on ${\mathcal{M}}$ and ${\mathcal{G}}_{n}$ is the neighborhood random graph with an edge between $X_{i}$ and $X_{j}$ if $\|X_{i}-X_{j}\| \le h$. It is shown in [@Arias-CastroPelletierPudlo2012_AAP] that the normalized graph cut (after a suitable normalization) converges to the normalized manifold cut, yielding an asymptotic upper bound on the conductance of ${\mathcal{G}}_{n}$ based on ${\mathfrak{h}}({\mathcal{M}})$. See also [@Trillos:2016_JMLR] for improved results. \[cor:main\] Under the setting of Theorem \[thm:main\], if $$\begin{aligned} \label{eqn:exact_recovery_condition_SDP_diffusion_Kmeans_Cheeger} C_1\,nt\, \exp\Big\{-\frac{\delta^2}{2h^2}\Big\} +C_1\,\exp\Big\{-c\, \underline{{\mathfrak{h}}}^{2}\, h^2 t\Big\} < \frac{C_2}{n\,\max_{1 {\leqslant}k {\leqslant}K} \{n_kh^{q_k}\}},\end{aligned}$$ where $\underline{{\mathfrak{h}}} = \min_{1 {\leqslant}k {\leqslant}K} {\mathfrak{h}}({\mathcal{D}}_k)$, then $\hat Z= Z^\ast$ with probability at least $1- c_3K\,\underline{n}^{-c_4}$. Since $\delta$ reflects the separation of the submanifolds of $S = \bigsqcup_{k=1}^{K} {\mathcal{D}}_{k}$ and ${\mathfrak{h}}({\mathcal{D}}_k)$ reflects the degree of connectedness of the submanifold ${\mathcal{D}}_{k}$, the (overall) hardness of the manifold clustering problem is determined by $(\delta, \underline{{\mathfrak{h}}})$. In particular, if $t = n^{\epsilon}$ for some $\epsilon > 0$, then we require that $$\underline{{\mathfrak{h}}} \gtrsim {1 \over h} \sqrt{\log n \over n^{\epsilon}},$$ in addition to . Our results in the rest subsections can also be stated via this geometric quantity of the Cheeger isoperimetric constant. \[rem:kernel\_choice\] In Theorem \[thm:main\] and Corollary \[cor:main\], the kernel $k$ is assumed to be the Gaussian kernel in . However, these exact recovery results do not rely on the particular choice of the Gaussian kernel. Specifically, Theorem \[thm:main\] and Corollary \[cor:main\] still hold, as long as the kernel is isotropic and satisfies the exponential decay in the squared Euclidean distance. For the heat kernel on ${\mathbb{R}}$ (i.e., Green’s function associated with the heat equation on ${\mathbb{R}}$) $$\label{eqn:heat_kernel_real_line} H(\tau,x,y) = (4\pi \tau)^{-1/2} e^{-{(x-y)^{2} \over 4\tau}}, \quad x,y \in {\mathbb{R}},$$ it can be viewed as an approximation to the short time dynamics of the heat kernel  on ${\mathbb{S}}^{1}$ in Remark \[rem:comments\_condition\_main\_thm\] as $\tau \to 0^{+}$ (cf. Chapter 1 in [@Rosenberg1997]). Hence, we can approximate the short time behavior of the heat flow on the compact manifold ${\mathbb{S}}^{1}$ by that of the non-compact manifold ${\mathbb{R}}$, where the latter is governed by the Gaussian heat kernel on ${\mathbb{R}}$. Setting $h^{2} = 2\tau$ in and noticing that the normalization $(4\pi \tau)^{-1/2}$ does not affect the results in Theorem \[thm:main\] and Corollary \[cor:main\] since the SDP solution in is invariant under scaling. Thus the bandwidth parameter in the Gaussian kernel in has the time scale interpretation in terms of the heat flow dynamics, in addition to capturing the local neighborhood geometry of the submanifolds. Exact recovery of the regularized and localized diffusion $K$-means ------------------------------------------------------------------- In this subsection, we extend the exact recovery results to the two variants of the diffusion $K$-means. First, we consider the regularized diffusion $K$-means  that does not require knowledge of the true number of clusters $K$. \[thm:main\_adaptive\_lambda\] Suppose all conditions in Theorem \[thm:main\] are true. In addition, if the regularization parameter satisfies $$\begin{aligned} \label{Eqn:lambda_condition} C_1\,nt\, \exp\Big\{-\frac{\delta^2}{2h^2}\Big\} &+C_1\,\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\}\, h^2 t\Big\} < \lambda {\leqslant}\frac{C_2}{n\max_{1 {\leqslant}k {\leqslant}K}\{n_kh^{q_k}\}},\end{aligned}$$ then we can achieve exact recovery for $\tilde{Z}$ from the regularized diffusion $K$-means with probability at least $1- c_3K\,\underline{n}^{-c_4}$. Condition , as a sufficient condition for the exact recovery, provides some justification of our $\lambda$ selection Algorithm \[alg1\], in particular, the reason of why using the logarithmic scale. More precisely, we observe from Lemma \[lem:feasibility\_SDP\_lambda\_infinity\] that an upper bound for $\lambda$ to produce non-trivial clustering is $n^{-1}$ times the largest eigenvalue of the affinity matrix $A$ in the SDP, which is of order $n^{-1}\max_{k}\{n_kh^{q_k}\}^{-1} =\mathcal O((n\log n)^{-1})$ (from the approximating form  and Lemma \[Lemma:total\_degree\] in the proof of Theorem \[thm:main\]). As a consequence, in the original scale, the interval length of those $\lambda$ that underestimates $K$ is of order $(n\log n)^{-1}$, which is comparable to the range of $\lambda$ corresponding to exact recovery (correct $K$) implied by  as $(n\log n)^{-1}$. On the other hand, the range of $\lambda$ corresponding to exact recovery will dominate if we instead consider the logarithmic scale. Precisely, on the logarithmic scale, the interval length for underestimating $K$ is of order $\log n$, while the range of $\log \lambda$ implied by  becomes $\log n - C'\,\big(\log n - \min\big\{\delta^2/(2h^2), \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\}\, h^2 t\big\}\big)$, which is of order $\Omega(n^{\iota})$ for some constant $C'>0$ and $\iota>0$ as long as both $\delta^2/h^2$ and $h^2t$ are bounded below by $n^{\iota}$. (The latter requirement can be easily satisfied, for example, in our simulations the choice of $t=n^{1.2}$ is good enough to produce robust results.) In particular, this suggests that the interval length of $\log \lambda$ for exact recovery is proportional to $\delta^2/h^2$, which can be viewed as a signal-to-noise ratio characteristic. Now let us turn to the localized diffusion $K$-means that locally selects the node-wise bandwidth adapting to the local geometric structure. \[thm:main\_adaptive\_h\] Let $\delta_{kk'}=\|{\mathcal{D}}_k-{\mathcal{D}}_{k'}\|$. If the number of neighbor parameter $k_0$ satisfies $k_0=\lfloor C\log n\rfloor$ for some constant $C>0$, and $$\delta_{kk'} {\geqslant}C'\, \max\{(\log n/ n_k)^{1/q_k},(\log n/ n_{k'})^{1/q_{k'}}\}$$ for each distinct $k,k'\in[K]$, then with probability at least $1- c_3K\,\underline{n}^{-c_4}$, the followings are true. (1) For each $i\in G^\ast_k$, its local bandwidth parameter $h_i$ satisfies $$\begin{aligned} \label{Eqn:local_h_nounds} c_1 (\log n/ n_k)^{1/q_k}{\leqslant}h_i {\leqslant}c_2 (\log n/ n_k)^{1/q_k}.\end{aligned}$$ (2) We can achieve exact recovery for $\tilde{Z}$ from the localized diffusion $K$-means as long as $$\begin{aligned} &C_1\,nt\, \exp\Big\{-c\,\Big(\min_{k,k'\in[K]}\frac{\delta_{kk'}}{\max\{(\log n/ n_k)^{1/q_k},(\log n/ n_{k'})^{1/q_{k'}}\}}\Big)^2\Big\}\notag \\ &\qquad\qquad\qquad+C_1\,\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\, (\log n/ n_k)^{2/q_k}\}\, t\Big\} < \frac{C_2}{n\log n}.\label{eqn:local_condition}\end{aligned}$$ \[rem:localization\] The first result  Part (1) in Theorem \[thm:main\_adaptive\_h\] shows that our localized selection scheme via nearest neighbors truly leads to bandwidth adaptation to the unknown submanifold dimension $q_k$ and the unknown true cluster size $n_k$, by only sacrificing a $\log n$ term (from $\log n_k$ to $\log n$ as compared with the optimal bandwidth choice $(\log n_k/ n_k)^{1/q_k}$ from Theorem \[thm:main\]). The second result Part (2) in Theorem \[thm:main\_adaptive\_h\] indicates the advantages of using the localized node-wise bandwidth, by comparing the condition   with those in Theorem \[thm:main\]. In particular, in order for the lower bound condition on the global bandwidth $h$ in Theorem \[thm:main\] to hold, the smallest $h$ would be $\max_{k} h_k$, where $h_k=(\log n_k/n_k)^{q_k}$ denotes the optimal bandwidth in the $k$-th cluster ${\mathcal{D}}_k$. Note that this lower bound on $h$ is uniformly larger than the magnitudes of localized bandwidth provided in . As a consequence, this large $h$ would require the same separation condition as $\delta_{kk'} {\geqslant}\max_{k}h_k$ for each pair $({\mathcal{D}}_k,{\mathcal{D}}_{k'})$ of distinct clusters. In comparison, the new sufficient condition  for exact recovery only needs a cluster-dependent separation condition as $\delta_{kk'} {\geqslant}\max\{h_k,h_{k'}\}$, which can be substantially weaker than $\delta_{kk'} {\geqslant}\max_{k}h_k$ if clusters are highly unbalanced with unequal sizes and mixed dimensions. Finally, we can further combine the regularized diffusion $K$-means with local adaptive bandwidths into the *localized and regularized diffusion $K$-means*. The following result is an immediate consequence by combining the proofs of Theorem \[thm:main\_adaptive\_lambda\] and Theorem \[thm:main\_adaptive\_h\], and thus its proof is omitted. \[thm:main\_adaptive\] Suppose all conditions in Theorem \[thm:main\] and Theorem \[thm:main\_adaptive\_h\] are true. In addition, if the regularization parameter satisfies $$\begin{aligned} &C_1\,nt\, \exp\Big\{-c\,\Big(\min_{k,k'\in[K]}\frac{\delta_{kk'}}{\max\{(\log n/ n_k)^{1/q_k},(\log n/ n_{k'})^{1/q_{k'}}\}}\Big)^2\Big\} \notag\\ &\qquad\qquad\qquad+C_1\,\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\, (\log n/ n_k)^{2/q_k}\}\, t\Big\}< \lambda {\leqslant}\frac{C_2}{n\log n},\label{Eqn:local_lambda_condition}\end{aligned}$$ then we can achieve exact recovery for $\tilde{Z}$ from the localized and regularized diffusion $K$-means with probability at least $1- c_3K\,\underline{n}^{-c_4}$. \[rem:bandwidth\_adaptivity\] It is interesting to note the signal separation and random walk mixing of the localized diffusion $K$-means and its regularized version are adaptive to local probability density and local geometric structures of the Riemannian submanifolds. In [@Arias-Castro2011_IEEETIT], nearly-optimal exact recovery of a collection of clustering methods based on pairwise distances of data is derived under a condition that the [*minimal*]{} signal separation strength over all pairs of submanifolds is larger than a threshold (even for their local scaling version, cf. Proposition 3 therein). Thus results established in [@Arias-Castro2011_IEEETIT] are non-adaptive to the local density and (geometric) structures of the submanifolds. In addition, the $\max_k\{n_k h_k^{q_k}\}$ on the right hand side of now reduces to $\log(n)$ as in . This means that the localized diffusion $K$-means tends to increase the signal-to-noise ratio (cf. Remark \[rem:localization\]) as well as the upper bound on $\lambda$ for exact recovery, thereby widening the interval length of $\log(\lambda)$ corresponding to the true clustering structure and improves the performance of the $\lambda$ selection Algorithm \[alg1\]. Simulations {#sec:simulations} =========== In this section, we assess the empirical performance of the diffusion $K$-means on some simulation examples. We generate $n=768$ data points from the following three data generation mechanisms (DGPs). The clustering structure contains three disjoint submanifolds: - ${\mathcal{D}}_{1} = \mbox{unit disk}$ and $n/4$ data points are uniformly sampled on ${\mathcal{D}}_{1}$, - ${\mathcal{D}}_{2} = \mbox{annulus with radius } 2.5$ and $n/4$ data points are uniformly sampled on ${\mathcal{D}}_{2}$, - ${\mathcal{D}}_{3} = \mbox{annulus with radius } 4$ and $n/2$ data points are uniformly sampled on ${\mathcal{D}}_{3}$, where all ${\mathcal{D}}_{1},{\mathcal{D}}_{2},{\mathcal{D}}_{3}$ are centered at the origin $(0,0)$. The clustering structure contains three disjoint rectangles: - ${\mathcal{D}}_{1} = \{(-15,-8), (-15,8), (-8,8), (8,8)\}$, - ${\mathcal{D}}_{2} = \{(10,3), (10,8), (15,3), (15,8)\}$, - ${\mathcal{D}}_{3} = \{(10,-8), (10,-3), (15,-8), (15,-3)\}$, where data points are uniformly distributed on ${\mathcal{D}}_{1} \bigsqcup {\mathcal{D}}_{2} \bigsqcup {\mathcal{D}}_{3}$. The clustering structure is a mixture of three bivariate Gaussians: $$\alpha_{1} N(\mu_{1}, \sigma_{1}^{2} {\text{Id}}_{2}) + \alpha_{2} N(\mu_{2}, \sigma_{2}^{2} {\text{Id}}_{2}) + \alpha_{3} N(\mu_{3}, \sigma_{3}^{2} {\text{Id}}_{2}),$$ where $(\alpha_{1}, \alpha_{2}, \alpha_{3}) = (1/3, 1/3, 1/3)$, $\mu_{1} = (-6,0), \mu_{2} = (0,0), \mu_{3} = (2.5,0)$, $\sigma_{1} = 2$, and $\sigma_{2} = \sigma_{3} = 0.5$. Our simulation setups are similar to [@zelnik2005self; @NadlerGalun2006_NIPS]. Note that the sampling density in DGP 1 and 2 is uniform on the disjoint submanifolds, the hardness of the problems is mainly determined by the geometry, and we thus expect the diffusion $K$-means and its localized version can both succeed in these two cases. In addition, since DGP 1 contains two annuli that are less connected than the rectangles and ellipsoids, we expect that, for the localized diffusion $K$-means (with self-tuned bandwidths), more random walk steps are needed for DGP 1 to correctly identify the clusters than those for DGP 2 and DGP 3. In our simulation studies, we use $t = n^{2}$ for DPG 1 and $t = n^{1.2}$ for both DGP 2 and DGP 3 (all with local scaling). Further, DGP 3 has a mixture of Gaussian densities, the local scaling is expected to improve the performance of the diffusion $K$-means. In fact, we have observed in Figure \[fig:diffusion\_kmeans\_local\_scaling\_demo\] that the diffusion $K$-means without local scaling does not work for DGP 2. It is also known that spectral clustering methods fail on such setup [@NadlerGalun2006_NIPS]. Thus we do not report results on DGP 2 without local scaling for all competing methods since it does not provide meaningful comparisons with other setups. For the SDP relaxed diffusion $K$-means clustering methods, we report the $\ell^{1}$ estimation error for estimating the true clustering membership $Z^{*}$ and the (normalized) Hamming distance error for classifying the clustering labels. In each setup, our results are reported on 1,000 simulations. For brevity, DKM stands for the diffusion $K$-means, RDKM for the (nuclear norm) regularized diffusion $K$-means, LDKM for the localized diffusion $K$-means, and LRDKM for the localized and regularized diffusion $K$-means. In the cases of no local scaling, the steps of random walks is fixed as $t = n^{1.2}$ in all setups. In the cases of local scaling, the nearest neighborhood size is chosen as $\lfloor \log{n} \rfloor$ for DPG 1 and DGP 3, and as $\lfloor 0.5 \log{n} \rfloor$ for DGP 2. For the comparison purpose, we also include three spectral clustering methods: the unnormalized spectral clustering (SC-UN), the random walk normalized spectral clustering (SC-RWN) [@ShiMalik2000_IEEEPAMI], a symmetrically normalized spectral clustering (SC-NJW) proposed in [@NgJordanWeiss2001_NIPS]. For each spectral clustering method, we also consider their localized versions (LSC-UN, LSC-RWN, LSC-NJW) by replacing the kernel matrix $K_{n}$ with $K^{\dagger}_{n}$. We can draw several observations from the simulation studies. First, the estimation error agrees well with our exact recovery theory for SDP relaxed DKM and LDKM, given the number of clusters (cf. Table \[tab:estimation\_errors\]). Second, all methods works relatively better for DGP 1 and DGP 2 since the separation signal strength is stronger than DGP 3 (cf. Table \[tab:classification\_errors\]). Third, the RDKM and LRDKM perform well in selecting the true number of clusters (cf. Table \[tab:percentages\_correctly\_estimated\_noc\]). We also modify DGP 3 to make the problem harder. We consider the mixture of three Gaussian with parameters $(\alpha_{1}, \alpha_{2}, \alpha_{3}) = (1/4, 1/4, 1/2)$, $\mu_{1} = (-6,0), \mu_{2} = (0,0), \mu_{3} = (1.45,0)$, $\sigma_{1} = 2$, and $\sigma_{2} = \sigma_{3} = 0.5$. This setup is denoted as DGP 3’. For DGP 3’, LDKM has much smaller classification errors than all spectral methods with local scaling (i.e., LSC-UN, LSC-RWN, LSC-NJW); see last column of Table \[tab:classification\_errors\]. ------ ------------------------- ------------------------- ----------- DGP=1 DGP=2 DGP=3 DKM $4.7642 \times 10^{-6}$ $3.3258 \times 10^{-4}$ [**–**]{} LDKM $5.2835 \times 10^{-5}$ 0.0049 0.0451 ------ ------------------------- ------------------------- ----------- : $\ell^{1}$ estimation errors of the SDP solutions of various diffusion $K$-means clustering methods.[]{data-label="tab:estimation_errors"} --------- ------- ------------------------- ----------- ----------- DGP=1 DGP=2 DGP=3 DGP=3’ DKM 0 $1.3021 \times 10^{-4}$ [**–**]{} [**–**]{} SC-UN 0 0 [**–**]{} [**–**]{} SC-RWN 0 0 [**–**]{} [**–**]{} SC-NJW 0 0 [**–**]{} [**–**]{} LDKM 0 0.0018 0.0086 0.0594 LSC-UN 0 $4.7917 \times 10^{-4}$ 0.0098 0.0801 LSC-RWN 0 0.0016 0.0105 0.0802 LSC-NJW 0 0.0399 0.0084 0.0884 --------- ------- ------------------------- ----------- ----------- : Classification errors of various diffusion $K$-means and spectral clustering methods.[]{data-label="tab:classification_errors"} Method DGP=1 DGP=2 DGP=3 -------- -------- -------- ----------- RDKM 94.30% 95.10% [**–**]{} LRDKM 99.20% 83.10% 97.70% : Percentages of correctly estimated number of clusters by the regularized diffusion $K$-means and its local scaling version.[]{data-label="tab:percentages_correctly_estimated_noc"} Proofs {#sec:proofs} ====== Recall that $n_k=|G_k^\ast|$ is the size of $k$-th true cluster index set $G_k^\ast$, and let $N_k = \sum_{i,j\in G_k^\ast} \kappa(X_i,\,X_j)$ denote the total within-weight in $G_k^\ast$. For any subset $G\subset [n]$, we use $\mathbf{1}_{G}$ to denote the all-one vector whose size equal to the size of $G$. Proof of Theorem \[thm:main\] {#Sec:proof_thm_main} ----------------------------- For simplicity of notation, we use $A:=A_n$ to denote the empirical diffusion affinity matrix $P_n^{2t}D_n^{-1}$ in the proof, and recall $$\begin{gathered} \hat{Z} = \operatorname{argmax}\left\{ \langle A, Z \rangle : Z \in {\mathscr{C}}_K \right\} \\ \qquad \mbox{with } {\mathscr{C}}_K = \{Z \in {\mathbb{R}}^{n \times n} : Z^{T} = Z, Z \succeq 0, \operatorname{tr}(Z) = K, Z {\mathbf{1}}_{n} = {\mathbf{1}}_{n}, Z {\geqslant}0 \}, \end{gathered}$$ At a high level, our strategy is to show that for suitably large $t\in{\mathbb{N}}_{+}$, the matrix $A_n$ tends to become close to a block-diagonal matrix, where each diagonal block tends to be a constant matrix (cf. equation ). Based on this approximation, we expect the global optimum $\hat Z$ to share a similar block-diagonal structure, thereby recovers the true membership matrix $Z^\ast$ in  which takes the form of $$\begin{aligned} Z^\ast = \begin{pmatrix} \displaystyle \frac{1}{n_1} \mathbf{1}_{G_1^\ast} \mathbf{1}^T_{G_1^\ast} & 0 & \cdots & 0\\ 0 & \displaystyle \frac{1}{n_2} \mathbf{1}_{G_2^\ast} \mathbf{1}^T_{G_2^\ast} & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots \\ 0 & \cdots & 0 & \displaystyle \frac{1}{n_K} \mathbf{1}_{G_K^\ast} \mathbf{1}^T_{G_K^\ast} \end{pmatrix}.\end{aligned}$$ To put this intuition in a technical form, since $Z^\ast$ defined in  is also a feasible solution belonging to the convex set ${\mathscr{C}}_K$, we have by the optimality of $\hat Z$ that $$\begin{aligned} \label{Eqn:basic_ineq} 0{\leqslant}\langle A_n,\, \hat Z - Z^\ast\rangle =\sum_{1{\leqslant}k\neq m{\leqslant}K} \big\langle\, [A_n]_{G_k^\ast G_m^\ast},\, [\hat Z - Z^\ast]_{G_k^\ast G_m^\ast}\,\big\rangle + \sum_{k=1}^K \big\langle\, [A_n]_{G_k^\ast G_k^\ast},\, [\hat Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle.\end{aligned}$$ We analyze the two sums separately as follows. [**The first sum:**]{} By noticing that $Z^\ast_{G_k^\ast G_m^\ast}$ is a zero matrix for each pair $k\neq m \in [K]$, we have the following bound $$\begin{aligned} &\sum_{1{\leqslant}k\neq m{\leqslant}K} \big\langle\, [A_n]_{G_k^\ast G_m^\ast},\, [\hat Z - Z^\ast]_{G_k^\ast G_m^\ast}\,\big\rangle \notag \\ = &\,\sum_{1{\leqslant}k\neq m{\leqslant}K} \big\langle\, [A_n]_{G_k^\ast G_m^\ast},\, \hat Z_{G_k^\ast G_m^\ast}\,\big\rangle {\leqslant}\max_{1{\leqslant}k\neq m{\leqslant}K} \|[A_n]_{G_k^\ast G_m^\ast}\|_\infty \, \sum_{1{\leqslant}k\neq m{\leqslant}K} \|\hat Z_{G_k^\ast G_m^\ast}\|_1,\label{eqn:k_neq_m}\end{aligned}$$ where the leading factor $\max_{k\neq m} \|[A_n]_{G_k^\ast G_m^\ast}\|_\infty$ is expected to be small due to the approximating structure . [**The second sum:**]{} Since we expect the $k$th block $[A_n]_{G_k^\ast G_k^\ast}$ of $A_n$ in the diagonal to be close to $N_k^{-1} \mathbf{1}_{G_k^\ast} \mathbf{1}^T_{G_k^\ast}$, we can subtract and add the same term to decompose it into $$\begin{aligned} &\sum_{k=1}^K \big\langle\, [A_n]_{G_k^\ast G_k^\ast},\, [\hat Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle\\ = &\, \sum_{k=1}^K \big\langle\, [A_n]_{G_k^\ast G_k^\ast} - N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T,\, [\hat Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle + \sum_{k=1}^K N_k^{-1} \,\big\langle\, \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T,\, [\hat Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle.\end{aligned}$$ The first term on the right hand side can be bounded by applying Hölder’s inequality, $$\begin{aligned} \sum_{k=1}^K& \big\langle\, [A_n]_{G_k^\ast G_k^\ast} - N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T,\, [\hat Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle \\ &\,{\leqslant}\max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} 1_{G_k^\ast}1_{G_k^\ast}^T\,\big\|_\infty \, \sum_{k=1}^K \big\|\, [\hat Z -Z^\ast]_{G_k^\ast G_k^\ast}\big\|_1,\end{aligned}$$ where again the leading factor $\max_{k} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty $ is expected to be small due to the approximating structure . Now consider the second term. By the definition of ${\mathscr{C}}_K$, the sums of entries in each row of $\hat Z$ and $Z^\ast$ are equal (to one), we have for fixed $k\in[K]$ and each $i\in G^\ast_k$, $$\begin{aligned} \sum_{j\in G_k^\ast} [\hat Z - Z^\ast]_{ij} + \sum_{j\not\in G_k^\ast} [\hat Z - Z^\ast]_{ij}=0.\end{aligned}$$ Since $Z^\ast_{ij}=0$ for each pair $(i,j)$ with $i\in G_k^\ast$ and $j\notin G_k^\ast$, and $\hat Z$ has nonnegative entries, we can sum up the preceding display over all $i\in G_k^\ast$ to obtain $$\begin{aligned} \big\langle\, \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T,\, [\hat Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle = - \sum_{m:\,m\neq k} \|\hat Z_{G_k^\ast G_m^\ast}\|_1,\quad \forall k\in[K].\end{aligned}$$ Putting pieces together, we obtain $$\label{Eqn:k_eq_m} \begin{aligned} &\sum_{k=1}^K \big\langle\, [A_n]_{G_k^\ast G_k^\ast},\, [\hat Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle \\ {\leqslant}&\, \max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty \, \sum_{k=1}^K \big\|\, [\hat Z -Z^\ast]_{G_k^\ast G_k^\ast}\big\|_1- \sum_{1{\leqslant}k\neq m{\leqslant}K} \frac{1}{N_k}\|\hat Z_{G_k^\ast G_m^\ast}\|_1. \end{aligned}$$ Now by combining inequalities , and , we can reach the following inequality $$\label{Eqn:key_ineq} \begin{aligned} &\sum_{1{\leqslant}k\neq m{\leqslant}K} \frac{1}{N_k}\|\hat Z_{G_k^\ast G_m^\ast}\|_1 {\leqslant}\max_{1 {\leqslant}k\neq m {\leqslant}K} \|[A_n]_{G_k^\ast G_m^\ast}\|_\infty \, \sum_{1{\leqslant}k\neq m{\leqslant}K} \|\hat Z_{G_k^\ast G_m^\ast}\|_1\\ &\qquad \qquad\qquad +\max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty \, \sum_{k=1}^K \big\|\, [\hat Z -Z^\ast]_{G_k^\ast G_k^\ast}\big\|_1. \end{aligned}$$ Since $\hat{Z} \in {\mathscr{C}}_{K} \subset {\mathscr{C}}$, according to inequalities - in Lemma \[lem:some\_ineq\_feasible\_set\] in Appendix C, $$\begin{aligned} \label{eqn:DKM_SDP_another_core_inequality} \sum_{1{\leqslant}k\neq m{\leqslant}K} \frac{1}{n_k}\|\hat Z_{G_k^\ast G_m^\ast}\|_1 {\geqslant}\frac{1}{n}\,\big\|\hat Z - Z^\ast\big\|_1.\end{aligned}$$ The last two displays  and  imply the exact recovery $\hat Z=Z^\ast$ as long as $$\label{eqn:DKM_SDP_exact_recover_master_condition} \max_{1 {\leqslant}k\neq m {\leqslant}K} \|[A_n]_{G_k^\ast G_m^\ast}\|_\infty + \max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty < \frac{1}{n}\,\min_{1{\leqslant}k{\leqslant}K}\Big\{\frac{n_k}{N_k}\Big\}.$$ \[lem:DKM\_SDP\_master\_bound\] If  holds, then we can achieve exact recovery $\hat{Z} = Z^{*}$. To further proceed, we will make use of following two lemmas to provide high probability bounds for the empirical diffusion affinity entries deviating from their expectations. Proofs of Lemma \[lem:within\_cluster\_random\_walk\] and \[lem:between\_cluster\_random\_walk\] are deferred to the following subsections. \[lem:within\_cluster\_random\_walk\] Let $\kappa = \max_{1{\leqslant}k\neq k'{\leqslant}K}\sup_{x\in {\mathcal{D}}_k,\,x'\in {\mathcal{D}}_{k'}} \kappa(x,\,x')$ and $\tau = \inf_{x,y\in S:\, \|x-y\| {\leqslant}h} \kappa(x,y)$. If $c_1 (\log n_k/ n_k)^{1/q_k}\,(\log n_k)^{1(q_k=2)/4}{\leqslant}h$, then it holds with probability at least $1-c_2\,n_k^{-c_3}$ that $$\begin{aligned} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty {\leqslant}C_0\, t\,(n_k\, h^{q_k})^{-2} \,n\,\kappa+ C_0\, (n_k\,h^{q_k})^{-1}\, e^{-2t\,\gamma(P_{n,k})},\end{aligned}$$ where the spectral gap $\gamma(P_{n,k})$, defined as one minus the second largest eigenvalue of $P_{n,k}$, satisfies $$\begin{aligned} \label{Eqn:Spectral_gap_lower} \gamma(P_{n,k}) {\geqslant}\bigg(1- C\,\Big(c_1^{-1}+(\sqrt{\lambda_1({\mathcal{D}}_k)} + 1)\,h+ h^2\Big)\bigg)\,\tau \, C_k \, \lambda_1({\mathcal{D}}_k)\,h^2,\end{aligned}$$ for some constants $C_0,C>0$ that only depends on ${\mathcal{D}}_k$ and $p_k$ and $C_k$ only depends on $q_k$. Here $\lambda_1({\mathcal{D}}_k)>0$ is the second smallest eigenvalue of the Laplace-Beltrami operator on ${\mathcal{D}}_k$ (cf. Section \[subsec:Laplace-Beltrami\_operator\]). \[lem:between\_cluster\_random\_walk\] Let $\kappa = \max_{1{\leqslant}k\neq k'{\leqslant}K}\sup_{x\in {\mathcal{D}}_k,\,x'\in {\mathcal{D}}_{k'}} \kappa(x,\,x')$. Suppose conditions  and  in Theorem \[thm:main\] are satisfied. Then it holds with probability at least $1 - c_2 K\, \underline{n}^{-c_3}$ that $$\begin{aligned} \|[A_n]_{G_k^\ast G_m^\ast}\|_\infty {\leqslant}C\, t \, (n_k h^{q_k}\,n_m h^{q_m})^{-1}\,n\,\kappa, \quad\forall k\neq m \in[K],\end{aligned}$$ where recall $\underline n = \min_{k\in [K]} n_k$ and $C$ is a constant only depending on $\{{\mathcal{D}}_k, \mu_k\}_{k = 1}^{K}$. Recall that $\kappa(x,y) = \exp\{-\|x-y\|^2/(2h^2)\}$ is the Gaussian kernel. Consequently, we may choose $\kappa = \exp\{-\delta^2/(2h^2)\}$ and $\tau = e^{-1/2}$ in these two lemmas. By Weyl’s law for the growth of eigenvalues of the Laplace-Beltrami operator (cf. Remark 6 in [@trillos2018error]), we have $\lambda_{j}({\mathcal{D}}_{k}) \sim j^{2/q_{k}}$. Thus $\lambda_1({\mathcal{D}}_k)$ is bounded. Since the bandwidth parameter satisfies $c_1 (\log n_k/ n_k)^{1/q_k}\,(\log n_k)^{1(q_k=2)/4}{\leqslant}h {\leqslant}c_2$ for some sufficient large constant $c_1$ and sufficiently small constant $c_2$ for all $k \in[K]$, inequality  in Lemma \[lem:within\_cluster\_random\_walk\] becomes $$\begin{aligned} \gamma(P_{n,k}) {\geqslant}C_k \, \lambda_1({\mathcal{D}}_k)\,h^2,\quad\forall k\in[K],\end{aligned}$$ for some constant $C_k$ only depending on $q_k$. Therefore, the following two bounds hold with probability at least $1 - c_2 K\, \underline{n}^{-c_3}$, $$\begin{aligned} \max_{1 {\leqslant}k {\leqslant}K} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty &{\leqslant}C\, nt\,\exp\Big\{-\frac{\delta^2}{2h^2}\Big\}+ C\, \exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\}\, h^2 t\Big\},\\ \max_{1{\leqslant}k\neq m{\leqslant}K}\|[A_n]_{G_k^\ast G_m^\ast}\|_\infty & {\leqslant}C\,nt\,\exp\Big\{-\frac{\delta^2}{2h^2}\Big\},\end{aligned}$$ for some constant $C$ only depending on $\{{\mathcal{D}}_k, \mu_k\}_{k = 1}^{K}$ and $c$ only on $\{q_k\}_{k=1}^K$. The last two inequalities combining with Lemma \[lem:DKM\_SDP\_master\_bound\] imply the exact recovery $\hat Z=Z^\ast$ as long as $$\begin{aligned} C\,nt\, \exp\Big\{-\frac{\delta^2}{2h^2}\Big\} +C\,\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\} \, h^2 t\Big\} < \frac{1}{n}\,\min_{1{\leqslant}k{\leqslant}K}\Big\{\frac{n_k}{N_k}\Big\}. \end{aligned}$$ Finally, the claimed result follows from the preceding display and the following lemma of a high probability bound on $N_k$, whose proof is postponed to Section \[Sec:Proof\_total\_degree\]. \[Lemma:total\_degree\] Suppose the density $p_{k}$ satisfies  and the bandwidth $h {\leqslant}c$ for some constant $c>0$. Then it holds with probability at least $1-c_2\,n_k^{-c_3}$ that $$\begin{aligned} \bigg| \frac{N_k}{n_k} - (\sqrt{2\pi}\, h)^{q_k} \,n_k \beta_k\bigg| {\leqslant}C \left( n_{k} h^{q_{k}+1} + \sqrt{n_kh^{q_k}\log n_k} \right), \end{aligned}$$ where $\beta_k = \mathbb E_{X\sim p_k}[p_k(X)]$ and $C>0$ is a constant depends only on $p_{k}, {\mathcal{D}}_{k}$, and $c$. Proof of Corollary \[cor:main\] ------------------------------- Corollary \[cor:main\] follows from Theorem \[thm:main\] and the Cheeger inequality [@Cheeger1970]: $$\lambda_{1}({\mathcal{D}}_{k}) {\geqslant}\frac{{\mathfrak{h}}({\mathcal{D}}_{k})^{2}}{4}.$$ Proof of Theorem \[thm:main\_adaptive\_lambda\] {#Sec:proof_thm_main_adaptive} ----------------------------------------------- Similar to the proof of Theorem \[thm:main\] in Section \[Sec:proof\_thm\_main\], we can use the optimality of $\tilde Z$ and the feasibility of $Z^\ast$ for the SDP program  to obtain the following basic inequality, $$\begin{aligned} 0&{\leqslant}\langle A_n,\, \tilde Z - Z^\ast\rangle + n \lambda\big(\operatorname{tr}(Z^\ast) - \operatorname{tr}(\tilde Z)\big) \notag \\ &=\sum_{1{\leqslant}k\neq m{\leqslant}K} \big\langle\, [A_n]_{G_k^\ast G_m^\ast},\, [\tilde Z - Z^\ast]_{G_k^\ast G_m^\ast}\,\big\rangle + \sum_{k=1}^K \big\langle\, [A_n]_{G_k^\ast G_k^\ast},\, [\tilde Z - Z^\ast]_{G_k^\ast G_k^\ast}\,\big\rangle \notag\\ &\qquad\qquad\qquad+ n\,\lambda\,\big(\operatorname{tr}(Z^\ast) - \operatorname{tr}(\tilde Z)\big).\label{Eqn:basic_inequality_adaptive}\end{aligned}$$ Since the only place where we used the constraint $\operatorname{tr}(Z) =K$ in Section \[Sec:proof\_thm\_main\] is Lemma \[lem:some\_ineq\_feasible\_set\] in Appendix C, the analysis of the first two sums in  still apply, leading to $$\label{Eqn:final_bound_adaptive} \begin{aligned} \sum_{1{\leqslant}k\neq m{\leqslant}K} \frac{1}{N_K}\|\tilde Z_{G_k^\ast G_m^\ast}\|_1 {\leqslant}&\, C\,\bigg( nt\, \exp\Big\{-\frac{\delta^2}{2h^2}\Big\} +\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\}\, h^2 t\Big\}\bigg)\, \big\|\tilde Z - Z^\ast\big\|_1\\ &\,\qquad + n\,\lambda\,\big(\operatorname{tr}(Z^\ast) - \operatorname{tr}(\tilde Z)\big), \end{aligned}$$ which holds with probability at least $1-c_3K\,\underline{n}^{-c_3}$. Now we apply Lemma \[lem:some\_ineq\_feasible\_set\_adaptive\] in Appendix C to obtain $$\begin{aligned} n\, \lambda\, \big(\operatorname{tr}(Z^\ast) - \operatorname{tr}(\tilde Z)\big) {\leqslant}4n\,\lambda\sum_{1{\leqslant}k\neq m{\leqslant}K} \frac{1}{n_k}\|\tilde Z_{G_k^\ast G_m^\ast}\|_1-\lambda\,\big\|\tilde Z - Z^\ast\big\|_1.\end{aligned}$$ Combining this inequality with , we obtain $$\begin{aligned} \bigg(\lambda - C\,\Big( nt\, \exp\Big\{-\frac{\delta^2}{2h^2}\Big\} &+\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\}\, h^2 t\Big\}\Big)\bigg)\, \big\|\tilde Z - Z^\ast\big\|_1\\ &\qquad\qquad+\,\bigg(1-4n\lambda \max_{1{\leqslant}k{\leqslant}K}\Big\{\frac {N_k}{n_k}\Big\}\bigg)\,\sum_{1{\leqslant}k\neq m{\leqslant}K} \frac{1}{N_k}\|\tilde Z_{G_k^\ast G_m^\ast}\|_1 {\leqslant}0.\end{aligned}$$ This implies the exact recovering $\tilde Z=Z^\ast$ provided that $$\begin{aligned} C_1\, \Big( nt\, \exp\Big\{-\frac{\delta^2}{2h^2}\Big\} &+\exp\Big\{-c\, \min_{1 {\leqslant}k {\leqslant}K}\{\lambda_1({\mathcal{D}}_k)\}\, h^2 t\Big\}\Big) < \lambda {\leqslant}\frac{C_2}{n}\,\min_{1{\leqslant}k{\leqslant}K}\Big\{\frac{n_k}{N_k}\Big\}.\end{aligned}$$ Finally, the claimed result follows by combining the above with Lemma \[Lemma:total\_degree\]. Proof of Lemma \[lem:within\_cluster\_random\_walk\] ---------------------------------------------------- We consider a fixed $k\in[K]$ throughout this proof. Recall that $P_n=D_n^{-1} K_n$ defines the random walk ${\mathcal{W}}_n$ over $S_n=\{X_1,X_2,\ldots,X_n\}$, and $A_n = P_n^{2t} D_n^{-1}$. Now consider a new random walk ${\mathcal{W}}_{n,k}$ over the $k$th cluster $G_k^\ast$ defined in the following way. For simplicity of notation, we may rearrange the nodes order so that $G_k^\ast = \{1,2,\ldots,n_k\}$. Then the transition probability matrix $P_{n,k}\in{\mathbb{R}}^{n_k\times n_k}$ of ${\mathcal{W}}_{n,k}$ is defined as $$\begin{aligned} [P_{n,k}]_{ij} =\frac{\kappa(X_i,\,X_j)}{d_{n,k}(X_i)},\quad \forall i,j\in G_k^\ast,\quad \mbox{where } d_{n,k}(x) = \sum_{j\in G_k^\ast} \kappa(x,\,X_j)\end{aligned}$$ is the induced degree function within cluster $G_k^\ast$. Similar to the diagonal degree matrix $D_n$, we denote by $D_{n,k}\in{\mathbb{R}}^{n_k\times n_k}$ the diagonal matrix whose $i$th diagonal entry is $d_{n,k}(X_i)$ for $i\in[n_k]$. Note that $N_k=\sum_{i\in G_k^\ast} d_{n,k}(X_i)$ the total degrees within $G_k^\ast$ so that $N_k {\geqslant}n_k \min_{i\in G^\ast_k} d_{n,k}(X_i)$. It is easy to see that the limiting distribution of ${\mathcal{W}}_{n,k}$ is $\pi_{n,k} = N_k^{-1} \mbox{diag}(D_{n,k})\in{\mathbb{R}}^{n_k}$. Under the separation condition on $\delta$ in the lemma, the probability of moving out from $G_k^\ast$ is exponentially small, suggesting that we may approximate the sub-matrix $[P_n^{2t}]_{G_k^\ast G_k^\ast} $ of $P_n^{2t}$ with $P_{n,k}^{2t}$. We will formalize this statement in the rest of the proof. First, we apply the triangle inequality to obtain $$\begin{aligned} &\big\|\,[A_n]_{G_k^\ast G_k^\ast} - N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty \notag\\ & {\leqslant}\big\| \,[P_n^{2t}]_{G_k^\ast G_k^\ast} [D_{n}]_{G_k^\ast G_k^\ast}^{-1} - P_{n,k}^{2t} D_{n,k}^{-1}\big\|_\infty + \big\| \,P_{n,k}^{2t} D_{n,k}^{-1} - N_k^{-1} \mathbf{1}_{G_k^\ast} \mathbf{1}_{G_k^\ast}^T \big\|_\infty =: T_1 + T_2,\label{Eqn:within_random_walk}\end{aligned}$$ where $T_1$ captures the difference between $[P_{n}^{2t}]_{G_k^\ast G_k^\ast}$ and $P_{n,k}^{2t}$, and $T_2$ characterizes the convergence of the Markov chain ${\mathcal{W}}_{n,k}$ to its limiting distribution $\pi_{n,k}$ after $2t$ steps. Recall that $\kappa = \max_{1{\leqslant}k\neq k'{\leqslant}K}\sup_{x\in {\mathcal{D}}_k,\,x'\in {\mathcal{D}}_{k'}} \kappa(x,\,x')$ is the minimal between-cluster affinity. The first term $T_1$ and the second term $T_2$ can be bounded via two lemmas below. \[Lem:T\_1\] If $n \kappa {\leqslant}\min_{i \in G_{k}^{*}} d_{n,k}(X_{i})$, then $$\begin{aligned} T_1 {\leqslant}(2t+1)\,n\,\kappa\,\max_{i\in[n]}d_{n,k}^{-2}(X_i). \end{aligned}$$ \[Lem:T\_2\] Let $\tau = \inf_{x,y\in S:\, \|x-y\| {\leqslant}h} \kappa(x,y)$. For each $i\in G_k^\ast$, let $d_{k}^\dagger(X_{i})$ denote total number of points in $\{X_j\}_{j\in G_k^\ast}$ inside the $d$-ball centered at $X_i$ with radius $h$. If $c_1 (\log n_k/ n_k)^{1/q_k}\,(\log n_k)^{1(q_k=2)/4}{\leqslant}h$, then it holds with probability at least $1-c_2\,n_k^{-c_3}$ that $$\begin{aligned} T_2 {\leqslant}e^{-2t\,\gamma(P_{n,k})} \,\max_{i\in G_k^\ast} d_{n,k}^{-1}(X_i),\quad\forall t=1,2,\ldots,\end{aligned}$$ where $$\begin{aligned} \label{Eqn:spectral_gap_bound} \gamma(P_{n,k}) {\geqslant}\bigg(1- C\,\Big(c_1^{-1}+(\sqrt{\lambda_1({\mathcal{D}}_k)} + 1)\,h+ h^2\Big)\bigg) \,\bigg[\min_{i\in G_k^\ast} \frac{d_{k}^\dagger(X_{i})}{d_{n,k}(X_i)}\bigg]\,\tau\, C_k \, \lambda_1({\mathcal{D}}_k)\,h^2,\end{aligned}$$ where recall that $\lambda_1({\mathcal{D}}_k)>0$ denotes the second smallest eigenvalue of the Laplace-Beltrami operator on ${\mathcal{D}}_k$. Proofs of these two lemmas are provided in Sections \[Sec:Proof\_T\_1\] and \[Sec:Proof\_T\_2\]. Combining upper bounds for the two terms in inequality  together, we can reach $$\begin{aligned} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty {\leqslant}(2t+1)\,n\,\kappa\, \max_{i\in[n]}d_{n,k}^{-2}(X_i) + e^{-2t\,\gamma(P_{n,k})}\,\max_{i\in[n]}d_{n,k}^{-1}(X_i),\end{aligned}$$ where the spectral gap $\gamma(P_{n,k})$ satisfies  . It remains to prove some high probability bounds for $d_{n,k}(X_i)$ and $d_{k}^\dagger(X_i)$, which are summarized in the following lemma. \[lem:node\_degree\] Suppose the density $p_{k}$ satisfies  and the bandwidth $h {\leqslant}c$ for some constant $c>0$. Then it holds with probability at least $1-c_2\,n_k^{-c_3}$ that $$\begin{aligned} \max_{i\in G_k^\ast} \bigg|\,\frac{d_{n,k}(X_i)}{n_k\,(\sqrt{2\pi} \,h)^{q_k}} - p_k(X_i) \bigg| &{\leqslant}C\bigg( h + \sqrt{\frac{\log n_k}{n_k h^{q_{k}}}}\,\bigg), \quad \mbox{and} \\ \max_{i\in G_k^\ast} \bigg|\,\frac{d_{k}^\dagger(X_i)}{n_k\,\nu_{q_k}\,h^{q_k}} - p_k(X_i) \bigg| &{\leqslant}C\bigg( h^2 + \sqrt{\frac{\log n_k}{n_k h^{q_{k}}}}\,\bigg),\end{aligned}$$ where $\nu_{q_k}$ denotes the volume of an unit ball in ${\mathbb{R}}^{q_k}$, and $C>0$ is some constant depends only on $p_{k}, {\mathcal{D}}_{k}$, and $c$. A proof of this lemma is deferred to Section \[Sec:Proof\_lem:node\_degree\]. Finally, by combining this lemma with the last display, and applying the uniform boundedness condition  on $p_k$, we obtain $$\begin{aligned} \big\|\,[A_n]_{G_k^\ast G_k^\ast}- N_k^{-1} \mathbf{1}_{G_k^\ast}\mathbf{1}_{G_k^\ast}^T\,\big\|_\infty {\leqslant}C_0\,n\,t\,\kappa\, (n_k\,h^{q_k})^{-2}+ C_0\, (n_k\,h^{q_k})^{-1}\, e^{-2t\,\gamma(P_{n,k})}\end{aligned}$$ with probability at least $1-c_{2} \, n_{k}^{-c_{3}}$, where the spectral gap $\gamma(P_{n,k})$ satisfies $$\begin{aligned} \gamma(P_{n,k}) {\geqslant}\bigg(1- C\,\Big(c_1^{-1}+(\sqrt{\lambda_1(D_k)} + 1)\,h+ h^2\Big)\bigg)\,\tau \, C_1 \, \lambda_1(D_k)\,h^2,\end{aligned}$$ for some constants $C_1,C_2,C>0$ that only depends on ${\mathcal{D}}_k$ and $p_k$. Proof of Lemma \[lem:between\_cluster\_random\_walk\] ----------------------------------------------------- For each indices $i$ and $j$ that belong to two difference clusters $G_k^\ast$ and $G_m^\ast$ with $k \neq m$, we have $$\begin{aligned} [A_n]_{ij}= [P_{n}^{2t}]_{ij} \cdot d_n(X_j)^{-1}.\end{aligned}$$ Let ${\mathcal{W}}_n=\{Y_t:\,t{\geqslant}0\}$, with $Y_t$ denote the state of the Markov chain ${\mathcal{W}}_n$ at time $t$. Define $T_k(i)=\min\big\{t \in {\mathbb{N}}_{+}:\, Y_t \not\in G_k^\ast, \, Y_{0} = i\big\}$ denote the first exit time from $G_k^\ast$ of ${\mathcal{W}}_n$ starting from $Y_0=i$. Then it is easy to see that $$\begin{aligned} [P_{n}^{2t}]_{ij} = {\mathds{P}}(Y_0=i,\,Y_{2t} = j) {\leqslant}{\mathds{P}}(T_k(i) {\leqslant}2t) = 1 - {\mathds{P}}(T_k(i) > 2t).\end{aligned}$$ Since for each $i\in G_k^\ast$, the one-step transition probability of moving out from $G_k^\ast$ is bounded by $n\,\kappa\,\max_{i\in G_k^\ast} d_{n}^{-1}(X_i)$, we have $$\begin{aligned} {\mathds{P}}(T_k(i) > 2t) {\geqslant}(1-n\,\kappa\,\max_{i\in G_k^\ast} d_{n}^{-1}(X_i))^{2t} {\geqslant}1-2n\,t \,\kappa\,\max_{i\in G_k^\ast} d_{n,k}^{-1}(X_i), \end{aligned}$$ provided that $n\,\kappa {\leqslant}\min_{i\in G_k^\ast} d_{n}(X_i)$. Therefore, for each $i\in G_k^\ast$ and $j\in G_m^\ast$ with $k\neq m$, we have $$\begin{aligned} \big|[A_n]_{ij}\big| {\leqslant}2n\,t \,\kappa\,\max_{i'\in G_k^\ast} d_{n,k}^{-1}(X_{i'}) \,\max_{j'\in G_m^\ast} d_{n,m}^{-1}(X_{j'}).\end{aligned}$$ By Lemma \[lem:node\_degree\] and , we have that with probability at least $1-c_{2}n^{-c_{3}}$, $C_{1} n_{k} h^{q_{k}} {\leqslant}d_{n,k}(X_{i}) {\leqslant}C_{2} n_{k} h^{q_{k}}$ for some constants $C_1$ and $C_{2}$. Note that condition  in Theorem \[thm:main\] yields that $n \kappa = n e^{-\delta^{2} / (2h^{2})} {\leqslant}C n_{k} h^{q_{k}}$. Then the claimed bound is implied by the above combined with the union bound. Proof of Lemma \[Lem:T\_1\] {#Sec:Proof_T_1} --------------------------- By adding and subtracting the same term we obtain $$\begin{aligned} T_1 {\leqslant}&\, \big\| \big([P_n^{2t}]_{G_k^\ast G_k^\ast} - P_{n,k}^{2t} \big)\,D_{n,k}^{-1}\big\|_\infty +\big\| \,[P_n^{2t}]_{G_k^\ast G_k^\ast}\,\big( [D_{n}]_{G_k^\ast G_k^\ast}^{-1} - D_{n,k}^{-1}\big)\big\|_\infty \notag \\ {\leqslant}&\, \big\| \,[P_n^{2t}]_{G_k^\ast G_k^\ast} - P_{n,k}^{2t} \big\|_\infty \,\|D_{n,k}^{-1}\|_\infty+ \big\| [D_{n}]_{G_k^\ast G_k^\ast}^{-1} - D_{n,k}^{-1}\big\|_\infty \notag \\ = &\, \big\| \,[P_n^{2t}]_{G_k^\ast G_k^\ast} - P_{n,k}^{2t} \big\|_\infty \,\max_{i\in G_k^\ast} d_{n,k}^{-1}(X_i)+\max_{i\in G_k^\ast} |d_n^{-1}(X_i)-d_{n,k}^{-1}(X_i)| \notag\\ {\leqslant}&\, \big\| \,[P_n^{2t}]_{G_k^\ast G_k^\ast} - P_{n,k}^{2t} \big\|_\infty \,\max_{i\in G_k^\ast}d_{n,k}^{-1}(X_i) +\max_{i\in G_k^\ast} |d_n(X_i)-d_{n,k}(X_i)|\,\max_{i\in G_k^\ast}d_{n,k}^{-2}(X_i),\label{Eqn:T_1_bound}\end{aligned}$$ where the second inequality is due to the fact that each row sum of $[P_n^{2t}]_{G_k^\ast G_k^\ast}$ is at most one. Now we consider the first term in . Recall that ${\mathcal{W}}_n=\{Y_t:\,t{\geqslant}0\}$, where $Y_t$ is the state of the Markov chain ${\mathcal{W}}_n$ at time $t$. Note that for each $i,j\in G_k^\ast$, we have $$\begin{aligned} [P_{n,k}]_{ij} =\frac{\kappa(X_i, X_j)}{\sum_{j\in G_k^\ast} \kappa(X_i,X_j)}= \frac{{\mathds{P}}(Y_{2t}=j\,|\, Y_{2t-1}=i)}{{\mathds{P}}(Y_{2t}\in G_k^\ast\,|\, Y_{2t-1}=i)} = {\mathds{P}}(Y_{2t}=j\, |\,Y_{2t-1}=i,\,Y_{2t}\in G_k^\ast).\end{aligned}$$ As a consequence, we have by the law of total probability and the Markov property of ${\mathcal{W}}_n$ that for each $i,j\in G_k^\ast$ and $s\in{\mathbb{N}}_{+}$, $$\begin{aligned} [P_n^{s}]_{ij} &= {\mathds{P}}(Y_{s} = j \,|\, Y_0 =i) \\ &=\sum_{\ell\in G_k^\ast} {\mathds{P}}(Y_{s} = j \,|\, Y_{s-1} =\ell,\, Y_0=i, \,Y_{s}\in G_k^\ast) \\ & \qquad \ \ \ \ \cdot {\mathds{P}}(Y_{s}\in G_k^\ast \,|\, Y_{s-1} =\ell,\, Y_0=i)\cdot {\mathds{P}}(Y_{s-1} = \ell\,|\, Y_0=i) \\ & \ \ \ \ + \sum_{\ell\not\in G_k^\ast} {\mathds{P}}(Y_{s} = j\,|\, Y_{s-1} =\ell,\, Y_0=i) \cdot {\mathds{P}}(Y_{s-1} =\ell\,|\, Y_0=i)\\ &= \sum_{\ell\in G_k^\ast} [P_n^{s-1}]_{i\ell} \cdot [P_{n,k}]_{\ell j}\cdot {\mathds{P}}(Y_{s}\in G_k^\ast \,|\, Y_{s-1} =\ell)\\ & \ \ \ \ + \sum_{\ell\not\in G_k^\ast} {\mathds{P}}(Y_{s} = j\,|\, Y_{s-1} =\ell) \cdot {\mathds{P}}(Y_{s-1} =\ell\,|\, Y_0=i).\end{aligned}$$ For each pair $(j,\ell)$ belonging to different clusters, noting that $d_{n,k}(X_i) {\leqslant}d_n(X_i)$, we have $$\begin{aligned} {\mathds{P}}(Y_{s} = j\,|\, Y_{s-1} =\ell) = [P_n]_{\ell j} =\frac{\kappa(X_\ell, X_j)}{d_n(X_\ell)} {\leqslant}\kappa\, \max_{i\in[n]} d_n^{-1}(X_i) {\leqslant}\max_{i\in[n]} d_{n,k}^{-1}(X_i),\end{aligned}$$ which implies that for each $\ell \in G_k^\ast$, $$\begin{aligned} 0{\leqslant}1- {\mathds{P}}(Y_{s}\in G_k^\ast \,|\, Y_{s-1} =\ell) = {\mathds{P}}(Y_{s} \not\in G_{k}^{\ast} \,|\, Y_{s-1}=\ell) {\leqslant}n \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i) {\leqslant}1.\end{aligned}$$ Combining the last three displays, we obtain for each $i,j\in G_k^\ast$ and $s\in{\mathbb{N}}^{+}$, $$\begin{aligned} \big(1- n \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i)\big)\,\sum_{\ell\in G_k^\ast} [P_n^{s-1}]_{i\ell} & \cdot [P_{n,k}]_{\ell j} {\leqslant}[P_n^{s}]_{ij} \\ {\leqslant}& \sum_{\ell\in G_k^\ast} [P_n^{s-1}]_{i\ell} \cdot [P_{n,k}]_{\ell j} + n \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i),\end{aligned}$$ which can be further simplified into $$\begin{aligned} \big(1- n \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i)\big)\,\big[ [P_n^{s-1}]_{G_k^\ast G_k^\ast} P_{n,k}\big]_{ij} {\leqslant}[P_n^{s}]_{ij}{\leqslant}\big[ [P_n^{s-1}]_{G_k^\ast G_k^\ast} P_{n,k}\big]_{ij} + n \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i).\end{aligned}$$ Now we can recursively apply this two-sided inequality to get $$\begin{aligned} \big(1- n \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i)\big)^s\,\big[ P_{n,k}^s]_{ij} {\leqslant}[P_n^{s}]_{ij}{\leqslant}\big[ P_{n,k}^s ]_{ij} + n\, s \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i),\quad\forall i,j\in G_k^\ast.\end{aligned}$$ By taking $s=2t$ and applying the inequality $(1-x)^s{\geqslant}1-xs$ for $s\in{\mathbb{N}}_{+}$ and $x\in[0,1]$, the above can be further reduced into $$\begin{aligned} \label{Eqn:T_1_first_term} \big\| \,[P_n^{2t}]_{G_k^\ast G_k^\ast} - P_{n,k}^{2t} \big\|_\infty {\leqslant}2n\, t \,\kappa\, \max_{i\in[n]} d_{n,k}^{-1}(X_i).\end{aligned}$$ Then we get $$\big\| \,[P_n^{2t}]_{G_k^\ast G_k^\ast} - P_{n,k}^{2t} \big\|_\infty \,\max_{i\in G_k^\ast}d_{n,k}^{-1}(X_i) {\leqslant}2n\, t \,\kappa\, \max_{i\in[n]} d_{n,k}^{-2}(X_i).$$ which is an upper bound to the first term in inequality  for $T_1$. The second term in inequality  can be bounded as $$\begin{aligned} \label{Eqn:T_1_second_term} & \max_{i\in G_k^\ast} |d_n(X_i)-d_{n,k}(X_i)| =\max_{i\in G_k^\ast} \sum_{j\not\in G_k^\ast} \kappa(X_i,X_j) {\leqslant}n \kappa.\end{aligned}$$ Finally, by combining , and , we obtain $$\begin{aligned} T_1 {\leqslant}(2t+1) \, n\,\kappa\,\max_{i\in[n]}d_n^{-2}(X_i).\end{aligned}$$ Proof of Lemma \[Lem:T\_2\] {#Sec:Proof_T_2} --------------------------- Recall that $\gamma(P_{n,k}) = 1-\lambda_1(P_{n,k})$ denote spectral gap of the transition matrix $P_{n,k}$, where $\lambda_1(P_{n,k})$ denotes the second largest eigenvalue of $P_{n,k}\in {\mathbb{R}}^{n_k\times n_k}$ (due to similar arguments as in Appendix B, $P_{n,k}$ has $n_k$ real eigenvalues with $1$ as the largest one). In addition, since the kernel function $k$ is positive semidefinite, all eigenvalues of $P_{n,k}$ are nonnegative, meaning that $\gamma(P_{n,k})$ is equal to the absolute spectral gap $1-\max\{\lambda_1(P_{n,k}),\, \lambda_{n_k-1}(P_{n,k})\}$ where $\lambda_{n_k-1}(P_{n,k})$ is the $n_k$th (smallest) eigenvalue of $P_{n,k}$. Therefore, according to the relationship between the mixing time of a Markov chain and its absolute spectral gap (see, for example, equation (12.11) in [@levin2017markov]), we have for each $i,j\in G_k^\ast$, $$\label{eqn:mixing_T2} \bigg|\frac{[P_{n,k}^{2t}]_{ij}}{[\pi_{n,k}]_j} - 1\bigg| {\leqslant}\frac{e^{-2t\,\gamma(P_{n,k})}}{\min_{\ell \in G_k^\ast} [\pi_{n,k}]_\ell},\quad\forall t=1,2,\ldots,$$ where $\pi_{n,k} =\big(d_{n,k}(X_1)/ N_k,\ldots, d_{n,k}(X_{n_k})/ N_k\big)^T\in{\mathbb{R}}^{n_k}$ is the limiting distribution of induced Markov chain ${\mathcal{W}}_{n,k}$ over $G_k^\ast$ with transition probability matrix $P_{n,k}$. This leads to a bound on $T_2$ as $$\begin{aligned} T_2 & = \big\| \,P_{n,k}^{2t} D_{n,k}^{-1} - N_k^{-1} \mathbf{1}_{G_k^\ast} \mathbf{1}_{G_k^\ast}^T \big\|_\infty= \max_{i,j\in G_k^\ast}\frac{1}{N_k}\, \bigg|\frac{[P_{n,k}^{2t}]_{ij}}{[\pi_{n,k}]_j} - 1\bigg| \\ &{\leqslant}\frac{1}{N_k} \, \frac{e^{-2t\,\gamma(P_{n,k})}}{\min_{\ell \in G_k^\ast} [\pi_{n,k}]_\ell}=e^{-2t\,\gamma(P_{n,k})} \,\max_{i\in G_k^\ast} d_{n,k}^{-1}(X_i),\quad\forall t=1,2,\ldots.\end{aligned}$$ Therefore, it remains to provide a lower bound on the spectral gap $\gamma(P_{n,k}) = 1-\lambda_1(P_{n,k})$. We do so by applying a comparison theorem of Markov chains (Lemma 13.22 in [@levin2017markov]), where we compare the spectral gap of $P_{n,k}$ with that of a standard random walk on a random geometric graph over $\{X_i\}_{i\in G_k^\ast}$, where each pair of nodes are connected if and only if they are at most $h$ far away from each other. The spectrum of the normalized graph Laplacian of the latter is known to behave like the eigensystem of the Laplace-Beltrami operator over the submanifold corresponding to the $k$-th connected subset ${\mathcal{D}}_k$. In particular, we will use existing results [@burago2014graph; @trillos2018error] on error estimates by using the spectrum of a random geometric graph to approximate the eigensystem of the Laplace-Beltrami operator in the numerical analysis literature. Let us first formally define a random geometric graph over $\{X_i\}_{i\in G_k^\ast}$ as i.i.d. samples from the compact connected $q_k$-dimensional Riemannian submanifold ${\mathcal{D}}_k$ in ${\mathbb{R}}^{p}$ with bounded diameter, absolute sectional curvature value, and injective radius. Recall that $\mu_k$ is a probability measure on ${\mathcal{D}}_k$ that has a Lipschitz density $p_k$ with respect to the Riemannian volume measure on ${\mathcal{D}}_k$ satisfying . $\{X_i\}_{i\in G_k^\ast}$ can be viewed as a sequence of i.i.d. samples from $\mu_k$, and without loss of generality, we may assume $G_k^\ast=\{1,2,\ldots,n_{k}\}$. Consider the random geometric graph ${\mathcal{G}}_k^{\dagger}=(V_k, E_k)$, with $V_k=\{X_i\}_{i\in G_k^\ast}$ being its set of vertices and $E_k$ set of edges, constructed by putting an edge between $X_i$ and $X_j$ (write $i\sim j$ and call $X_i$ to be a neighbor of $X_j$) if and only if $\|X_i-X_j\| {\leqslant}h$. We define the natural random walk ${\mathcal{W}}_{k}^\dagger$ as a reversible Markov chain on $V_k$ that moves to neighbors of the current state with equal probabilities. In other words, the transition probability matrix ${\mathcal{P}}_k^{\dagger}\in {\mathbb{R}}^{n_k\times n_k}$ satisfies $$\begin{aligned} [{\mathcal{P}}_k^{\dagger}]_{ij} = \begin{cases} \displaystyle (d^{\dagger}_{k,i})^{-1}, & \quad\mbox{if } j\sim i\\ \displaystyle 0, &\quad\mbox{otherwise}, \end{cases}\end{aligned}$$ where $d^\dagger_{k,i} := d^\dagger_{k}(X_{i}) = \sum_{j=1}^{n_k} 1(j\sim i)$ denotes the degree of vertex $X_i$. It is easy to see that $\pi^\dagger_k = (d^\dagger_{k,1}/d^\dagger_k,d^\dagger_{k,2}/d^\dagger_k,\ldots,d^\dagger_{k,n_k}/d^\dagger_k)^T$, where $d^\dagger_k=\sum_{i=1}^{n_k}d^{\dagger}_{k,i}$ denotes the total degree, is the stationary distribution of this random walk. Let $1= \lambda_0( {\mathcal{P}}_k^{\dagger}){\geqslant}\lambda_1({\mathcal{P}}_k^{\dagger}){\geqslant}\ldots{\geqslant}\lambda_{n-1}({\mathcal{P}}_k^{\dagger}){\geqslant}-1$ denote the eigenvalues of matrix ${\mathcal{P}}_k^{\dagger}$, and $\gamma({\mathcal{P}}_k^{\dagger})=1-\lambda_1({\mathcal{P}}_k^{\dagger})$ denote its spectral gap. Let $L_{{\mathcal{G}}_k^\dagger}=D_k^\dagger - A_k^\dagger\in{\mathbb{R}}^{n_k\times n_k}$ denote the graph Laplacian matrix associated with graph ${\mathcal{G}}_k^\dagger=(V_k,E_k)$, where $D_k^\dagger\in{\mathbb{R}}^{n_k\times n_k}$ is a diagonal matrix with $[D^\dagger_k]_{ii} = d_{k,i}^\dagger$, and $A_k^\dagger\in {\mathbb{R}}^{n_k\times n_k}$ is the adjacency matrix with $[A_k^\dagger]_{ij}=1(i\sim j)$ for all distinct pair $(i,j)\in [n_k]^2$. Define the normalized Laplacian of ${\mathcal{G}}_k^\dagger$ as $L^N_{{\mathcal{G}}_k^\dagger}=(D_k^\dagger)^{-1/2}L_{{\mathcal{G}}_k^\dagger} (D_k^\dagger)^{-1/2}= I - (D_k^\dagger)^{-1/2} A_k^\dagger (D_k^\dagger)^{-1/2}$, and denote its ordered eigenvalues by $0 {\leqslant}\lambda_0(L^N_{{\mathcal{G}}_k^\dagger}) {\leqslant}\lambda_1(L^N_{{\mathcal{G}}_k^\dagger}){\leqslant}\cdots{\leqslant}\lambda_{n_k-1}(L^N_{{\mathcal{G}}_k^\dagger})$. Since $(D_k^\dagger)^{-1/2} A_k^\dagger (D_k^\dagger)^{-1/2}= (D_k^\dagger)^{1/2} {\mathcal{P}}_k^\dagger (D_k^\dagger)^{-1/2}$ is a similarity transformation of ${\mathcal{P}}_k^\dagger$, they share the same eigenvalues. Therefore, we have the relation $\lambda_j(L^N_{{\mathcal{G}}_k^\dagger}) = 1-\lambda_j({\mathcal{P}}_k^\dagger)$ for all $j=0,1,\ldots,n_k-1$. In particular, by taking $j=1$, we can relate the spectral gap of ${\mathcal{P}}_k^\dagger$ with the second smallest eigenvalue of the normalized Laplacian matrix $L^N_{{\mathcal{G}}_k^\dagger}$ as $\gamma({\mathcal{P}}_k^\dagger) = \lambda_1(L^N_{{\mathcal{G}}_k^\dagger})$. It is known that the eigenvalues of the normalized Laplacian $L^N_{{\mathcal{G}}_k^\dagger}$ of the geometric random graph ${\mathcal{G}}_k^{\dagger}$ approaches (up to a scaling factor) the eigenvalues of the Laplace-Beltrami operator on ${\mathcal{D}}_k$. More concretely, let $L^2({\mathcal{D}}_k, {\mathrm{d}}\mu_k)$ be the space of all square integrable functions on ${\mathcal{D}}_k$, and $\Delta_{{\mathcal{D}}_k}$ denote the Laplace-Beltrami operator on ${\mathcal{D}}_k$ (cf. Section \[subsec:Laplace-Beltrami\_operator\]). Let $0=\lambda_0({\mathcal{D}}_k){\leqslant}\lambda_1({\mathcal{D}}_k){\leqslant}\cdots$ denote the sequence of nonnegative eigenvalues of $\Delta_{{\mathcal{D}}_k}$. The connectedness of ${\mathcal{D}}_k$ implies that its second smallest eigenvalue $\lambda_1({\mathcal{D}}_k)$ is strictly positive. We will invoke Corollary 2 of [@trillos2018error], which generalizes Theorem 1 of [@burago2014graph] from the uniform density to any Lipschitz continuous density satisfying , for relating the spectrum of the Laplace-Beltrami operator $\Delta_{{\mathcal{D}}_k}$ on ${\mathcal{D}}_k$ with the spectrum of the discrete normalized graph Laplacian $L^N_{{\mathcal{G}}_k^\dagger}$, as summarized in the following. \[lem:eigenval\_convergence\_normalized\_graph\_Laplacian\] Let $\nu_{q_k}$ to denote the volume of an unit ball in ${\mathbb{R}}^{q_k}$. For each $j=0,1,\ldots$, suppose the radius $h$ and the $\varepsilon_{n,k}$ to be defined below satisfy $(\sqrt{\lambda_j({\mathcal{D}}_k)} + 1)\,h +\varepsilon_{n,k}/h {\leqslant}c_1$. Then it holds with probability at least $1-c_2\,n_k^{-c_3}$ $$\begin{aligned} \bigg| \frac{2(q_k+2)}{\nu_{q_k} h^2}\cdot\frac{\lambda_j(L^N_{{\mathcal{G}}_k^\dagger})}{\lambda_j({\mathcal{D}}_k)} - 1 \bigg| {\leqslant}C\, \bigg(\frac{\varepsilon_{n,k}}{h}+(\sqrt{\lambda_j({\mathcal{D}}_k)} + 1)\,h+ h^2\bigg),\end{aligned}$$ where constants $c_1,c_2, c_3,C>0$ depend only on the submanifold ${\mathcal{D}}_k$ and the density $p_k$, and $$\begin{aligned} \varepsilon_{n,k} = \begin{cases} \displaystyle\frac{(\log n_k)^{3/4}}{n_k^{1/2}} , & \quad\mbox{if } q_k=2\\[2ex] \displaystyle \Big(\frac{\log n_k}{n_k}\Big)^{1/q_k}, & \quad\mbox{otherwise} \end{cases}.\end{aligned}$$ In particular, this lemma (taking $j=1$) implies a lower bound on the spectral gap of ${\mathcal{P}}_k^\dagger$ as $$\begin{aligned} \label{Eqn:RGG_spectral_lower_bound} \gamma({\mathcal{P}}_k^\dagger) {\geqslant}\bigg(1- C\,\Big(c_1^{-1}+(\sqrt{\lambda_1({\mathcal{D}}_k)} + 1)\,h+ h^2\Big)\bigg) C_k \, \lambda_1({\mathcal{D}}_k)\,h^2\end{aligned}$$ as long as $h {\geqslant}c_1 \varepsilon_{n,k}$, where constant $C_k$ only depends on $q_k$. Next, we will apply the following comparison theorem to relate the spectral gaps of Markov chains ${\mathcal{W}}_{n,k}$ and ${\mathcal{W}}_{k}^\dagger$. \[lem:comparison\_thm\] Let $P$ and $P'$ be transition matrices of two reversible Markov chains on the same state space $\Omega$, whose stationary distributions are denoted by $\pi$ and $\pi'$, respectively. Let ${\mathcal{E}}(f)$ and ${\mathcal{E}}'(f)$ denote the Dirichlet forms associated to the pairs $(P,\pi)$ and $(P',\pi')$, where $$\begin{aligned} \label{Eqn:Dirichlet_form} {\mathcal{E}}(f) =\frac{1}{2} \sum_{x,y\in\Omega} [\,f(x) - f(y)]^2\,\pi(x)\, P(x,y),\quad \forall f\in L^2(\Omega),\end{aligned}$$ and ${\mathcal{E}}'(f)$ can be similarly defined. If there exists some constant $B>0$ such that ${\mathcal{E}}'(f) {\leqslant}B \,{\mathcal{E}}(f)$ for all $f$, then $$\begin{aligned} \gamma(P') {\leqslant}\bigg[\max_{x\in\Omega} \frac{\pi(x)}{\pi'(x)}\bigg] \, B\,\gamma(P),\end{aligned}$$ where $\gamma(P)$ and $\gamma(P')$ denote the spectral gaps associated with $P$ and $P'$, respectively. We will apply this comparison theorem with $P_{n,k}\to P, {\mathcal{P}}_{k}^\dagger \to P'$, and $\Omega = G_{k}^{*}$. Let us find the constant $B$ such that ${\mathcal{E}}'(f){\leqslant}B\, {\mathcal{E}}(f)$ for all $f$, where in our setting, $$\begin{aligned} {\mathcal{E}}(f) &=\frac{1}{2 N_k} \sum_{1{\leqslant}i,j{\leqslant}n_k} (\,f_i - f_j)^2\,\kappa(X_i,X_j),\quad\mbox{and}\\ {\mathcal{E}}'(f) &=\frac{1}{2 d_k^\dagger} \sum_{(i,j):\, \|X_i-X_j\| {\leqslant}h} (\,f_i - f_j)^2,\quad \forall f=(f_1,\ldots,f_{n_k})^T\in{\mathbb{R}}^{n_k}, \end{aligned}$$ According to the definition of $\tau$ as $\inf_{x,y\in S:\, \|x-y\| {\leqslant}h} \kappa(x,y)$, we can simply choose $B = (N_k/d_k^\dagger)\, \tau^{-1}$. In addition, we have the bound $$\begin{aligned} \max_{i\in G_k^\ast} \frac{[\pi_{n,k}]_i}{[\pi_k^\dagger]_i} = \frac{d_k^\dagger}{\tilde N_k}\, \max_{i\in G_k^\ast} \frac{d_{n,k}(X_i)}{d_{k,i}^\dagger}.\end{aligned}$$ Therefore, we can apply Lemma \[lem:comparison\_thm\] to get $$\begin{aligned} \gamma(P_{n,k}) {\geqslant}\tau \bigg[\min_{i\in G_k^\ast} \frac{d_{k,i}^\dagger}{d_{n,k}(X_i)}\bigg]\, \gamma({\mathcal{P}}_{k}^\dagger),\end{aligned}$$ which combined with inequality  leads to $$\begin{aligned} \gamma(P_{n,k}) {\geqslant}\bigg(1- C\,\Big(c_1^{-1}+(\sqrt{\lambda_1({\mathcal{D}}_k)} + 1)\,h+ h^2\Big)\bigg) \,\bigg[\min_{i\in G_k^\ast} \frac{d_{k,i}^\dagger}{d_{n,k}(X_i)}\bigg]\,\tau\, C_k \, \lambda_1({\mathcal{D}}_k)\,h^2.\end{aligned}$$ Proof of Lemma \[lem:node\_degree\] {#Sec:Proof_lem:node_degree} ----------------------------------- Recall that $\kappa(x,y)=\exp\{-\|x-y\|^2/(2h^2)\}$ is the Gaussian kernel with bandwidth parameter $h>0$. For $x \in S$, define $d_{n,k}(x) = \sum_{j \in G_{k}^{*}} \kappa(x, X_{j})$ as the induced degree function of $x$ within cluster $G_{k}^{*}$. Then for each $i\in G_k^\ast$ we have $d_{n,k}(X_{i}) = 1+\sum_{j\in G_k^\ast,\,j\neq i} \kappa(X_i,X_j) =: 1+\tilde{d}_{n,k}(X_{i})$. Denote $\alpha_{k}(x) = \operatorname{\mathds{E}}_{X \sim p_{k}} \kappa(x,\, X)$ and $v_{k}(x) = {\text{Var}}_{X \sim p_{k}}[\kappa(x,\, X)]$. Applying Lemma \[lem:expectation\_variance\_bound\] with ${\mathcal{M}}={\mathcal{D}}_k$ and $f(x)=p_k(x)$, we have $v_{k}(x) {\leqslant}\alpha_{k}(x) {\leqslant}C h^{q_{k}}$ for all $x \in {\mathcal{D}}_{k}$. Then for any fixed $x \in {\mathcal{D}}_{k}$, using the bound in  and the boundedness of $\kappa$, we may apply Bernstein inequality (cf. Lemma 2.2.9 in [@vandervaartwellner1996]) to obtain that for all $u>0$, $$\begin{aligned} {\mathds{P}}\left( \Big|d_{n,k}(x) - n_k \alpha_{k}(x) \Big| {\geqslant}u \right) {\leqslant}2 \exp\left( -{u^{2} \over 2 C n_{k} \alpha_{k}(x) + {2 \over 3} u} \right). \end{aligned}$$ Choosing $u = t n_{k} \alpha_{k}(x)$ for $t \in (0, C]$, we have $$\label{Eqn:degree_con} {\mathds{P}}\left( \Big|d_{n,k}(x) - n_k \alpha_{k}(x) \Big| {\geqslant}t n_{k} \alpha_{k}(x) \right) {\leqslant}2 \exp \left( -{t^{2} n_{k} \alpha_{k}(x) \over 2 C + {2 \over 3} t} \right) \le 2 \exp \left( -C n_{k} \alpha_{k}(x) t^{2} \right).$$ Now choosing $t = c_{1} \sqrt{\log(n_{k})/(\alpha_{k}(x) n_{k})}$ for some large enough constant $c_{1} > 0$, we get $${\mathds{P}}\left( \Big|d_{n,k}(x) - n_k \alpha_{k}(x) \Big| {\geqslant}c_{1} \sqrt{\alpha_{k}(x) n_{k} \log{n_{k}}} \right) {\leqslant}2 n_{k}^{-C c_{1}^{2}},$$ provided that $\log(n_{k}) {\leqslant}C \alpha_{k}(x) n_{k} {\leqslant}C n_{k} h^{q_{k}}$ in view of the uniform bounds  and . But this is ensured by our bandwidth assumption . Thus for any fixed $x \in {\mathcal{D}}_{k}$, we have with probability at least $1 - c_{2} n_{k}^{-c_{3}}$, $$\left| {d_{n,k}(x) \over n_{k}} - \alpha_{k}(x) \right| {\leqslant}c_{1} \sqrt{\alpha_{k}(x) \log{n_{k}} \over n_{k}}.$$ Then it follows that with probability at least $1 - c_{2} n_{k}^{-c_{3}}$, $$\left| {d_{n,k}(x) \over n_{k} (\sqrt{2\pi} h)^{q_{k}}} - p_{k}(x) \right| {\leqslant}{C \over (\sqrt{2\pi})^{q_{k}}} h + {c_{1} \over (\sqrt{2\pi} h)^{q_{k}}} \sqrt{\alpha_{k}(x) \log{n_{k}} \over n_{k}} {\leqslant}C \left( h + \sqrt{\log{n_{k}} \over n_{k} h^{q_{k}}} \right).$$ This implies that the rescaled degree function $n_k^{-1}\,\big(\sqrt{2\pi} h\big)^{-q_k}\,d_{n,k}(x)$ provides a good estimate of the density $p_k(x)$ at $x$. Since $X_{i} \in {\mathcal{D}}_{k}$ are i.i.d. for $i \in G_{k}^{*}$, we have with probability at least $1 - c_{2} n_{k}^{-c_{3}}$, $$\left| {\tilde{d}_{n,k}(X_{i}) \over (n_{k}-1) (\sqrt{2\pi} h)^{q_{k}}} - p_{k}(X_{i}) \right| {\leqslant}C \left( h + \sqrt{\log{n_{k}} \over n_{k} h^{q_{k}}} \right).$$ Then union bound implies that $$\max_{i \in G_{k}^{*}} \left| {d_{n,k}(X_{i}) \over n_{k} (\sqrt{2\pi} h)^{q_{k}}} - p_{k}(X_{i}) \right| {\leqslant}C \left( h + \sqrt{\log{n_{k}} \over n_{k} h^{q_{k}}} + {1 \over n_{k} h^{q_{k}}} \right) {\leqslant}C \left( h + \sqrt{\log{n_{k}} \over n_{k} h^{q_{k}}} \right)$$ with probability at least $1-c_{2}n_{k}^{-c_{3}}$ (by choosing the constant $c_{1}$ large enough). The second part about the concentration of $d_{k}^\dagger(X_{i})$ can be analogously proved by applying the Chernoff bound for sum of i.i.d. Bernoulli random variables. Let $$\begin{aligned} d_{k}^\dagger(x) = \sum_{j \in G_{k}^{*}} 1(\|x-X_j\|{\leqslant}h)\end{aligned}$$ so that $d_{k}^\dagger(X_{i}) = 1+\tilde{d}_{k}^\dagger(X_{i})$ where $\tilde{d}_{k}^\dagger(X_{i}) = \sum_{j \in G_{k}^{*}, j\neq i} 1(\|X_{i}-X_j\|{\leqslant}h)$. Note that Section 2.2 of [@burago2014graph] provides a uniform estimate of the expectation ${\mathbb{E}}_{X\sim p_k} [1(\|x-X\|{\leqslant}h)]$ in terms of the density $p(x)$ as $$\begin{aligned} \label{Eqn:Expected-degree} \sup_{x \in {\mathcal{D}}_{k}} \Big|{\mathbb{E}}_{X\sim p_k} [1(\|x-X\|{\leqslant}h)] - \nu_{q_k} \, h^{q_k}\,p_k(x) \Big| {\leqslant}C\, h^{q_k+2},\end{aligned}$$ where recall that $\nu_{q_k}$ denotes the volume of unit ball in ${\mathbb{R}}^{q_k}$. The rest of the proof follows a similar line as the proof of the first part, and we omit the details. \[lem:expectation\_variance\_bound\] Let ${\mathcal{M}}$ be a $q$-dimensional compact submanifold in ${\mathbb{R}}^p$ with bounded absolute sectional curvature and injective radius, and $\mbox{Vol}_{{\mathcal{M}}}$ denote its volume form. Let $f$ be a Lipschitz probability density function on ${\mathcal{M}}$ such that $c {\leqslant}f(x) {\leqslant}c^{-1}$ for some constant $c > 0$. Let $\alpha(x) = \operatorname{\mathds{E}}_{X \sim f} [\kappa(x,\, X)]$ and $v(x) = {\text{Var}}_{X \sim f}[\kappa(x,\, X)]$. Then we have $$\label{Eqn:expectation_bound} \sup_{x\in {\mathcal{M}}} \Big| \alpha(x) - \big(\sqrt{2\pi} h\big)^{q} \,f(x)\Big| {\leqslant}C \, h^{q+1}$$ and $v(x) {\leqslant}C \alpha(x)$ for all $x \in {\mathcal{M}}$, where the constant $C$ only depends on $f$, ${\mathcal{M}}$, and $c$. Consequently, we have $\sup_{x \in {\mathcal{M}}} \alpha(x) {\leqslant}C h^{q}$ and $\sup_{x \in {\mathcal{M}}} v(x) {\leqslant}C h^{q}$. Note that for each $x\in {\mathcal{M}}$ and any Lipschitz probability density function $f$ on ${\mathcal{M}}$, the expectation ${\mathbb{E}}[\kappa(x,\, X)^2]$ takes the form $$\begin{aligned} {\mathbb{E}}_{X \sim f} [\kappa(x,\, X)] = \int_{{\mathcal{M}}} \exp\{-\|x-y\|^2/(2h^2)\} \, f(y)\, {\mathrm{d}}\mbox{Vol}_{{\mathcal{M}}} (y).\end{aligned}$$ Then  follows from Lemma \[Lem:convolution\_bound\]. Similarly, we can bounded the variance of $\kappa(x,\, X)$ via $\mbox{Var}_{X \sim f} [\kappa(x,\, X)] {\leqslant}{\mathbb{E}}_{X \sim f} [\kappa(x,\, X)^2]$, where $$\begin{aligned} {\mathbb{E}}_{X \sim f} [\kappa(x,\, X)^2]= \int_{{\mathcal{M}}} \exp\{-\|x-y\|^2/(h^2)\} \, f(y)\, {\mathrm{d}}\mbox{Vol}_{{\mathcal{M}}} (y).\end{aligned}$$ By a second application of Lemma \[Lem:convolution\_bound\] with $h/\sqrt{2} \to h$ to obtain $$\begin{aligned} \Big| {\mathbb{E}}_{X \sim f} [\kappa(x,\, X)^2] - \big(\sqrt{\pi} h\big)^{q} \,f(x)\Big| {\leqslant}C' \, h^{q+1}.\end{aligned}$$ This together with the uniform boundedness condition on $f$ and inequality  imply an upper bound to the variance by the expectation, $$\begin{aligned} \label{Eqn:variance_bound} \mbox{Var}_{X \sim f} [\kappa(x,\, X)] {\leqslant}C \,h^{q}{\leqslant}C \,{\mathbb{E}}_{X \sim f} [\kappa(x,\, X)],\quad \mbox{for some $C>0$}.\end{aligned}$$ The bounds $\sup_{x \in {\mathcal{M}}} \alpha(x) {\leqslant}C h^{q}$ and $\sup_{x \in {\mathcal{M}}} v(x) {\leqslant}C h^{q}$ follow from the fact that $h {\leqslant}c$. \[Lem:convolution\_bound\] Let ${\mathcal{M}}$ be a $q$-dimensional compact submanifold in ${\mathbb{R}}^p$ with bounded absolute sectional curvature and injective radius, and $\mbox{Vol}_{{\mathcal{M}}}$ denote its volume form. Then there exists some constant $h_0>0$ only depending on ${\mathcal{M}}$, such that for all $h\in(0, h_0]$ and any Lipschitz function $f$ on ${\mathcal{M}}$, we have $$\begin{aligned} \bigg| \frac{1}{\big(\sqrt{2\pi} h\big)^q}\,\int_{{\mathcal{M}}} \exp\{-\|x-y\|^2/(2h^2)\} \, f(y)\, {\mathrm{d}}\mbox{Vol}_{{\mathcal{M}}} (y) - f(x)\bigg| {\leqslant}C_{f}\,h,\end{aligned}$$ where constant $C_f$ only depends on $f$ and ${\mathcal{M}}$. This lemma follows from Lemma 5.2 on pages 895–898 in [@yang2016bayesian]. Proof of Lemma \[Lemma:total\_degree\] {#Sec:Proof_total_degree} -------------------------------------- Recall that $N_k=\sum_{i,j\in G^\ast_k} \kappa(X_i,X_j)$ is the total within-weight in $G^\ast_k$. According to Lemma \[lem:node\_degree\] (note that $N_k=\sum_{i\in G_k^\ast} d_{n,k}(X_i)$ using the notation therein), it holds with probability at least $1-c_2 n_k^{-c_3}$ that $$\begin{aligned} \bigg| \frac{N_k}{n_k} - (\sqrt{2\pi}\, h)^{q_k}\, \sum_{i\in G_k^\ast} p_k(X_i)\bigg| {\leqslant}C\, n_{k} h^{q_k}\,\bigg(h + \sqrt{\frac{\log n_k}{n_kh^{q_{k}}}} \, \bigg).\end{aligned}$$ According to the sandwiched bound on the density function $p_k$, $\{p_k(X_i):\,i\in G_k^\ast\}$ are independent and bounded random variables. Therefore, we may apply Hoeffding’s inequality to obtain that $$\begin{aligned} \bigg|\sum_{i\in G_k^\ast} p_k(X_i) - n_k \,\beta_k\bigg| {\leqslant}C\,\sqrt{n_k\log n_k}\end{aligned}$$ holds with probability at least $1-c_2 n_k^{-c_3}$, where $\beta_k=\mathbb E[p_k(X_i)]$. Combining the two preceding inequalities, we obtain that $$\begin{aligned} \bigg| \frac{N_k}{n_k} - (\sqrt{2\pi}\, h)^{q_k} \,n_k \beta_k\bigg| {\leqslant}C \left( n_{k} h^{q_{k}+1} + \sqrt{n_kh^{q_k}\log n_k} \right),\end{aligned}$$ which completes the proof. Proof of Theorem \[thm:main\_adaptive\_h\] {#Sec:Proof_thm:main_adaptive_h} ------------------------------------------ #### *Proof of Part (1):* Consider $X_i$, where $i\in G_k^\ast$ for some $k\in[K]$. Fix $h_U=c_U(\log n/n_k)^{1/q_k}$ and $h_L=c_L(\log n/n_k)^{1/q_k}$ for two sufficiently large constant $c_U$ and $c_L$ with $c_U=2c_L$. We use notation $N(X_i,h)$ to denote the number of points from $\{X_i\}_{i=1}^n$ that is within $h$ distance from $X_i$. Recall that $d_k^\dagger(X_i) := d_{k,h}^\dagger(X_i)$ in Lemma \[lem:node\_degree\] is the number of points from $\{X_i\}_{i\in G_k^\ast}$ that is within $h$ distance to $X_i$ (we will choose $h=h_L$ and $h=h_U$ later). Here we put an subscript $h$ in $d_{k,h}^\dagger(X_i)$ to indicate the dependence of $d_k^\dagger$ on $h$. The condition on $\delta_{kk'}$ implies that any point outside the $k$th cluster $D_k$ has distance at least $C'(\log n/n_k)^{1/q_k} {\geqslant}\max\{h_L,h_U\}$. Therefore all points that are within $h_L(h_U)$ distance to $X_i$ must belong to $\mathcal D_k$, implying $N(X_i,h) = d_{k,h}^\dagger(X_i)$. Consequently, from the proof of Lemma \[lem:node\_degree\] (take $h=h_U$ and $h=h_L$ respectively), we have (a concentration inequality as  plus the expectation bound ), $$\begin{aligned} &\mathbb P \big( N(x,h_L) {\leqslant}c_1 (1-t)\, n_k h_L^{q_k} \big) {\leqslant}\exp\{- c_1' n_k h_L^{q_k} t^2\},\\ &\mathbb P \big( N(x,h_U) {\geqslant}c_2 (1+t)\, n_k h_U^{q_k} \big) {\leqslant}\exp\{- c_2' n_k h_U^{q_k} t^2\},\quad t>0, x \in {\mathcal{D}}_{k}, \end{aligned}$$ where constants $(c_1,c_1')$ and $(c_2,c_2')$ only depend on the constant $c$ in the two-sided bound on density $p_k(\cdot)$ on the $k$th region $\mathcal D_k$. Let $c_U$ be sufficiently large such that $c_2 c_U^{q_k} = C$, where $C$ is the constant appearing in the neighborhood parameter $k_0=\lfloor C\log n\rfloor$. By taking $t$ such that $(1-t) c_1 c_L^{q_k} = C/2$ in the first inequality of the preceding display, and $t=1$ in the second inequality, we obtain that $$\begin{aligned} &\mathbb P \big( N(X_{i},h_L) {\leqslant}k_0/2 \big) {\leqslant}\exp\{- c_1'' \,C \log n\},\\ &\mathbb P \big( N(X_{i},h_U) {\geqslant}2k_0 \big) {\leqslant}\exp\{- c_2''\, C\log n\},\end{aligned}$$ where $c_1''$ and $c_2''$ are two constants independent of $C$. For large enough $C$, we can make these two probabilities smaller than $1/n^3$. Since the event $\{N(X_i,h_L) {\leqslant}k_0/2\}\cap\{N(X_i,h_U) {\geqslant}2k_0\}$ implies $h_i \in[h_L,h_U]$ for each $i\in[n]$, a union bound argument over $i=1,2,\ldots,n$ leads to the claimed two sided bound for $h_i$ (with probability at least $1-n^{-1}$). #### *Proof of Part (2):* Using the two sided bound in the proof of Part (1), the proof follows same steps in the proof of Theorem \[thm:main\], with the only exception in proving a counterpart of Lemma \[Lem:T\_2\] for bounding the spectral gap $\gamma(P_{n,k})$ of chain $P_{n,k}$ on $\{X_i\}_{i\in G_k^\ast}$ in cluster $G_k^\ast$ for $k\in [K]$, since $P_{n,k}$ now has different bandwidth parameter $h_i$ at each observed point $X_i$ (that is, a bandwidth parameter inhomogeneous chain). More precisely, it remains to show that with high probability, it holds that $$\begin{aligned} \gamma(P_{n,k}){\geqslant}C' \lambda_1(\mathcal D_k) \,(\log n /n_k)^{2/q_k}, \quad \mbox{for some constant $C'>0$.}\end{aligned}$$ Here, we assume without loss of generality that the absolute spectral gap of $P_{n,k}$ is dominated by one minus its second largest eigenvalue. Otherwise, we can always consider the lazy random walk by replacing $P_n$ with $P_n/2 +I_n/2$ in the diffusion $K$-mean SDP, whose absolute spectral gap is $\gamma(P_{n,k})/2$. Our proof strategy is again based on the Markov chains comparison theorem (Lemma \[lem:comparison\_thm\]) by comparing this bandwidth parameter inhomogeneous chain with a bandwidth parameter homogeneous chain with $h_i \equiv h_L$, for each $i\in G_k^\ast$, where $h_L$ is the lower bound of $h_i$ in the proof of Part (1). In particular, a lower bound on the spectral gap of the latter is already derived in Lemma \[Lem:T\_2\] as of order $\lambda_1(\mathcal D_k) h_L^2$. Fix the cluster index $k\in [K]$, and without loss of generality assume $G_k^\ast=\{1,2,\ldots,n_k\}$. To avoid confusion of notation, we put an superscript “IH" indicate the bandwidth parameter inhomogeneous chain, and “H" to denote the bandwidth parameter homogeneous chain with $h_i \equiv h_L$. For example, $P_{n,k}^{IH}$ and $\mathcal E^{IH}$ denote the transition probability matrix and the Dirichlet form  (defined in Lemma \[lem:comparison\_thm\]), respectively, associated with the bandwidth parameter inhomogeneous chain. Due to the fact that $h_i {\geqslant}h_L$ for all $i\in G^\ast_k$, we immediately have $$\begin{aligned} 2 N_k^H \mathcal E^{H}(f) &= \sum_{1{\leqslant}i,j{\leqslant}n_k} (f_i-f_j)^2 \kappa^{H}(X_i,X_j) \\ &{\leqslant}\sum_{1{\leqslant}i,j{\leqslant}n_k} (f_i-f_j)^2 \kappa^{IH}(X_i,X_j) = 2 N_k^{IH}\mathcal E^{IH}(f), \quad\mbox{for all $f\in\mathbb R^{n_k}$,}\end{aligned}$$ where $k^{H}(X_i,X_j) = \exp\{-\|X_i-X_j\|^2/(2h_L^2)\}$ and $\kappa^{IH}(X_i,X_j) = \exp\{-\|X_i-X_j\|^2/(2h_ih_j)\}$, and $(N_k^H,N_k^{IH})$ are the respective total degrees within $G_k^\ast$. Moreover, recall that the stationary distributions of $P_{n,k}^{IH}$ and $P_{n,k}^{H}$ are $\pi_{n,k}^{IH} = d_{n,k}^{IH}(X_i) / N_k^{IH}$ and $\pi_{n,k}^{H} = d_{n,k}^{H}(X_i) / N_k^{H}$, for $i\in G_k^\ast$, respectively, where $d_{n,k}^{IH}(X_i) =\sum_{j\in G_k^\ast} \kappa^{IH}(X_i,X_j)$, $d_{n,k}^{H}(X_i) =\sum_{j\in G_k^\ast} \kappa^{H}(X_i,X_j)$ are the node degrees, and $N_k^{IH}= \sum_{i\in G_k^\ast} d_{n,k}^{IH}(X_i)$, $N_k^{H}= \sum_{i\in G_k^\ast} d_{n,k}^{H}(X_i)$ are the total degrees. Now we can apply Lemma \[lem:comparison\_thm\] with $B= N_k^{IH}/ N_k^{H}$ and Lemma \[Lem:T\_2\] (to the homogeneous chain $P_{n,k}^H$) to obtain $$\begin{aligned} \gamma(P_{n,k}^{IH}) {\geqslant}\min_{i\in G_k^\ast} \bigg[\frac{d_{n,k}^{H}(X_i)}{d_{n,k}^{IH}(X_i)} \bigg]\, \gamma(P_{n,k}^{H}) \quad\mbox{and}\quad \gamma(P_{n,k}^{H}){\geqslant}C \lambda_1(\mathcal D_k) \,h_L^2,\end{aligned}$$ for some constant $C>0$. Similar to the proof of Lemma \[lem:node\_degree\], we can apply the concentration inequality to the nodes degree with bandwidth $h=h_U$ and $h=h_L$ to obtain that with probability at least $1-c_2\, n^{-c_3}$, $$\begin{aligned} \max_{i\in G_k^\ast} \bigg|\,\frac{\sum_{j\in G_k^\ast}\exp\{-\|X_i-X_j\|^2/(2h_L^2)\} }{n_k(\sqrt{2\pi} h_L)^{q_k}} - p_k(X_i) \bigg| {\leqslant}C'\bigg( h_L + \sqrt{\frac{\log n_k}{n_k h_{L}^{q_{k}}}}\,\bigg),\\ \max_{i\in G_k^\ast} \bigg|\,\frac{\sum_{j\in G_k^\ast}\exp\{-\|X_i-X_j\|^2/(2h_U^2)\} }{n_k(\sqrt{2\pi} h_U)^{q_k}} - p_k(X_i) \bigg| {\leqslant}C'\bigg( h_U+ \sqrt{\frac{\log n_k}{n_k h_{U}^{q_{k}}}}\,\bigg), \end{aligned}$$ for all $i\in G_k^\ast$. Combining this with the sandwiched bound for $h_i$ in Part (1), we obtain $$\begin{aligned} c_1\, n_k h_L^{q_k} &{\leqslant}d_{n,k}^{H}(X_i)=\sum_{j\in G_k^\ast}\exp\{-\|X_i-X_j\|^2/(2h_L^2)\} \\ &{\leqslant}d_{n,k}^{IH}(X_i) = \sum_{j\in G_k^\ast} \exp\{-\|X_i-X_j\|^2/(2h_ih_j)\} \\ &{\leqslant}\sum_{j\in G_k^\ast} \exp\{-\|X_i-X_j\|^2/(2h_U^2)\} {\leqslant}c_1'\, n_k h_U^{q_k},\ \ \ \mbox{for all $i\in G_k^\ast$.}\end{aligned}$$ Putting all pieces together, we obtain that it holds with probability at least $1-c_2\, n^{-c_3}$ that $$\begin{aligned} \gamma(P_{n,k}^{IH}){\geqslant}C' \lambda_1(\mathcal D_k) \,(\log n /n_k)^{2/q_k}, \quad \mbox{for some constant $C'>0$.}\end{aligned}$$ Proof of Lemma \[lem:feasibility\_SDP\_lambda\_infinity\] --------------------------------------------------------- *Proof of Part (1):* Since $n\lambda >\lambda_{\max}(A)$, the matrix $n\lambda I_n - A$ is positive definite. For any $Z \in {\mathscr{C}}$, from the constraint $Z {\mathbf{1}}_{n} = {\mathbf{1}}_{n}$, we see that $(1, n^{-1/2} {\mathbf{1}}_{n})$ is an eigen-pair of $Z$. In addition, since $Z \succeq 0$, all eigenvalues $\lambda_{1},\dots,\lambda_{n}$ of $Z$ are non-negative. Let $U_1=n^{-1/2}{\mathbf{1}}_{n}, U_2,\ldots, U_n$ denote the corresponding eigenvectors of $Z$. Thus the objective function $$\begin{aligned} &\langle A ,Z\rangle - n\lambda \operatorname{tr}(Z) = - \langle n\lambda I_n - A, Z\rangle \\ &= -\frac{1}{n}\,{\mathbf{1}}_{n}^T(n\lambda I_n - A){\mathbf{1}}_{n} - \sum_{i=2}^n \lambda_i\, U_i^T(n\lambda I_n - A)U_i {\leqslant}-\frac{1}{n}{\mathbf{1}}_{n}^T(n\lambda I_n - A){\mathbf{1}}_{n},\end{aligned}$$ where the equality holds if and only if $\lambda_2=\cdots=\lambda_n=0$. Note that $Z^{\diamond} \in {\mathscr{C}}$ is a feasible solution for  and $Z^{\diamond}$ has a non-zero eigenvalue equal to $1$ and $(n-1)$ zero eigenvalues. Therefore, $Z^{\diamond}$ is the unique solution of the SDP . *Proof of Part (2):* For any $Z\in {\mathscr{C}}$, since $Z$ is a symmetric matrix satisfying $Z {\geqslant}0$ and $Z {\mathbf{1}}_{n} = {\mathbf{1}}_{n}$, $Z$ is a stochastic matrix and all its eigenvalues have absolute values less than or equal to one. Moreover, from the positive semi-definiteness of $Z$, all eigenvalues of $Z$ lie in the $[0,1]$ interval. Now since $n\lambda <\lambda_{\min}(A)$, the matrix $A- n\lambda I_n$ is positive definite. From matrix Hölder’s inequality, the objective function satisfies $$\begin{aligned} &\langle A ,Z\rangle - n\lambda \operatorname{tr}(Z) = \langle A- n\lambda I_n, Z\rangle \\ &{\leqslant}{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert A- n\lambda I_n \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast}\, {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert Z \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mbox{\scriptsize op}} = {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert A- n\lambda I_n \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast},\end{aligned}$$ where the equality holds if and only if all eigenvalues of $Z$ equal to one (since $A- n\lambda I_n$ is strictly positive definite). Note that $Z^{\dagger} = I_{n} \in {\mathscr{C}}$ is a feasible solution for . Therefore, $Z^{\dagger}$ is the unique solution of the SDP . *Proof of Part (3):* By the optimality and feasibility of the solutions $\tilde Z_{\lambda_1}$ and $\tilde Z_{\lambda_2}$, we have $$\label{Eqn:optimality} \begin{aligned} \langle A ,\tilde Z_{\lambda_2}\rangle - \lambda_1 \operatorname{tr}(\tilde Z_{\lambda_2}) &{\leqslant}\langle A ,\tilde Z_{\lambda_1}\rangle - \lambda_1 \operatorname{tr}(\tilde Z_{\lambda_1}),\ \ \mbox{and}\\ \langle A ,\tilde Z_{\lambda_1}\rangle - \lambda_2 \operatorname{tr}(\tilde Z_{\lambda_1}) &{\leqslant}\langle A ,\tilde Z_{\lambda_2}\rangle - \lambda_2 \operatorname{tr}(\tilde Z_{\lambda_2}). \end{aligned}$$ Adding these two inequalities together yields $$\begin{aligned} (\lambda_1-\lambda_2) \big( \operatorname{tr}(\tilde Z_{\lambda_1}) - \operatorname{tr}(\tilde Z_{\lambda_2})\big) {\leqslant}0,\end{aligned}$$ which implies $\operatorname{tr}(\tilde Z_{\lambda_1}) {\geqslant}\operatorname{tr}(\tilde Z_{\lambda_2})$ when $\lambda_1> \lambda_2$. Moreover, if at least one of the SDPs has a unique solution, then at least one of the two inequalities in  is strict, implying $$(\lambda_1-\lambda_2) \big( \operatorname{tr}(\tilde Z_{\lambda_1}) - \operatorname{tr}(\tilde Z_{\lambda_2})\big) < 0,$$ and $\operatorname{tr}(\tilde Z_{\lambda_1}) > \operatorname{tr}(\tilde Z_{\lambda_2})$. Proofs on spectral decompositions {#app:A} ================================= Since $k$ is symmetric and positive semidefinite, so is $R$. Thus the corresponding operator $R$ is self-adjoint in $L^2({\mathrm{d}}\mu)$ and is also compact if holds. Then $R$ has a discrete set of nonnegative eigenvalues $\lambda_0{\geqslant}\lambda_1{\geqslant}\cdots {\geqslant}0$, and has the following eigen-decomposition $$\begin{aligned} R(x,y) = \sum_{j=0}^\infty \lambda_j\, \phi_j(x)\,\phi_j(y),\quad\forall x,y\in S,\end{aligned}$$ where $\{\phi_j\}_{j=0}^\infty$ is an orthonormal basis of $L^2({\mathrm{d}}\mu)$. Note that $$\begin{aligned} R(x,y) = \sqrt{\frac{\pi(x)}{\pi(y)}}\,p(x,y), \quad\forall x,y\in S.\end{aligned}$$ This implies a decomposition of the transition probability $p(x,y)$ as $$\begin{aligned} p(x,y) = \sum_{j=0}^\infty \lambda_j\, \psi_j(x)\,\varphi_j(y),\quad\forall x,y\in S,\end{aligned}$$ where $\psi_j(x) = \phi_j(x)/\sqrt{\pi(x)}$ and $\varphi_j(x) = \phi_j(x)\sqrt{\pi(x)}$. In particular, for each $j=0,1,\ldots$, $$\begin{aligned} P\psi_j(x) &= \sum_{l=0}^\infty \lambda_l \psi_l(x)\, \int_S \varphi_l(y)\, \psi_j(y)\,{\mathrm{d}}\mu(y)=\sum_{l=0}^\infty \lambda_l \psi_l(x)\,\delta_{lj} = \lambda_j \psi_j(x),\quad\forall x\in S,\end{aligned}$$ implying that $\{\psi_j\}_{j=0}^\infty$ are the corresponding (right) eigenfunctions of $P$, with unit $L^2(\pi {\mathrm{d}}\mu)$ norm, associated with the same eigenvalues $\lambda_0{\geqslant}\lambda_1{\geqslant}\cdots {\geqslant}0$. Since $P$ is the transition operator of a Markov chain, $\lambda_0=1$ and $\psi_0\equiv 1$. For $t\in{\mathbb{N}}_{+}$ and $x,y\in S$, let $p_{t}(x,y)$ be the $t$-step transition probability from $x$ to $y$. By Lemma \[lem:spectral\_decomposition\_Markov\_chain\], we have $$\begin{aligned} p_t(x,y) = \sum_{j=0}^\infty \lambda_j^t\, \psi_j(x)\,\varphi_j(y) \end{aligned}$$ and $\{\varphi_j\}_{j=0}^\infty$ forms an orthonormal basis of $L^2({\mathrm{d}}\mu/\pi)$. Consequently, by viewing $\lambda_j^t\, \psi_j(x)$ as the coefficient associated with $\varphi_j$ in the orthogonal expansion of function $p_t(x,\cdot)$, we have $$\begin{aligned} D_{t}^2(x, y) =\|\, p_{t}(x, \cdot) - p_{t}(y, \cdot)\,\|^2_{L^{2}({\mathrm{d}}\mu/\pi)}= \sum_{j=0}^{\infty} \lambda_{j}^{2t} \, [\psi_{j}(x) - \psi_{j}(y)]^{2}. \end{aligned}$$ Empirical diffusion affinity {#app:B} ============================ Similar to the derivations in Appendix \[app:A\], if we define matrix $R_n\in{\mathbb{R}}^{n\times n}$ with $$\begin{aligned} [R_n]_{ij}=\frac{\kappa(X_i,X_j)}{\sqrt{d_n(X_i)}\sqrt{d_n(X_j)}},\end{aligned}$$ then $\{\lambda_{n,j}\}_{j=0}^{n-1}$ are also the eigenvalues of $R_n$. Let $\phi_{n,j}\in{\mathbb{R}}^n$ denote the unit Euclidean norm eigenvector associated with $\lambda_{n,j}$. Then the empirical probability transition matrix $P_n$ has the decomposition $$\begin{aligned} P_{n}^t = \sum_{j=0}^{n-1} \lambda_{n,j}^t\, \psi_{n,j}\,\varphi_{n,j}^T,\end{aligned}$$ where $\psi_{n,j} = D_n^{-1/2}\,\phi_{n,j}\in{\mathbb{R}}^n$ and $\varphi_{n,j} = D_n^{1/2}\phi_{n,j}\in{\mathbb{R}}^n$, so that $\psi_{n,j}=D_n^{-1} \varphi_{n,j}$ for each $j\in\{0,1,\dots,n-1\}$. In particular, $\psi_{n,j}$ has unit $L^2(\mbox{diag}(D_n))$ norm, and $\varphi_{n,j}$ has unit $L^2(\mbox{diag}(D_n^{-1}))$ norm, for each $j\in\{0,1,\dots,n-1\}$. In addition, we have the following relation between the diffusion affinity and $P_n^{2t}$, $$\begin{aligned} \langle X_i,\, X_j\rangle_{D_{n,t}} = \sum_{l=0}^{n-1} \lambda_{n,l}^{2t} \,[\psi_{n,l}]_i \, [\psi_{n,l}]_j =\sum_{l=0}^{n-1} \lambda_{n,l}^{2t} \, [\psi_{n,l}]_i \, [\varphi_{n,l}]_j d_n^{-1}(X_j) =[P_n^{2t}D_n^{-1}]_{ij}.\end{aligned}$$ Technical proofs ================ In this appendix, we collect some technical lemmas used in the proofs of our main results. \[lem:some\_ineq\_feasible\_set\] Let $Z^{*}$ be defined in (\[eqn:Kmeans\_true\_membership\_matrix\]). Then for any $Z \in {\mathscr{C}}$ defined in , we have $$\begin{aligned} \label{eqn:ineq_1_feasible_set} \|Z^{*} - Z^{*} Z Z^{*}\|_{1} = \|Z^{*} - Z^{*} Z\|_{1} =&\, 2\sum_{k=1}^K \sum_{m\neq k} \|Z_{G_{k}^{*} G_{m}^{*}}\|_{1}.\end{aligned}$$ In addition, if $Z$ also satisfies $\operatorname{tr}(Z)=\operatorname{tr}(Z^\ast)$, or $Z\in {\mathscr{C}}_K$, where ${\mathscr{C}}_K$ is defined in , then $$\begin{aligned} \label{eqn:ineq_2_feasible_set} {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (I-Z^{*}) Z (I-Z^{*}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast} = &\, \sum_{k=1}^K \sum_{m\neq k} \frac{1}{n_k}\, \|Z_{G_{k}^{*} G_{m}^{*}}\|_{1} {\leqslant}{\|Z^{*} - Z^{*} Z\|_{1} \over 2 \underline{n}}, \\ \label{eqn:ineq_3_feasible_set} \|Z^{*} - Z^{*} Z\|_{1} {\leqslant}\|Z^{*} - Z\|_{1} {\leqslant}& \, n\,{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (I-Z^{*}) Z (I-Z^{*}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast} {\leqslant}{2 n \over \underline{n}} \|Z^{*} - Z^{*} Z\|_{1}.\end{aligned}$$ Inequalities  and  follows from Lemma 1 in [@GiraudVerzelen2018]. Inequality  is due to inequality (57) in [@BuneaGiraudRoyerVerzelen2016]. \[lem:some\_ineq\_feasible\_set\_adaptive\] Let $Z^{*}$ be defined in (\[eqn:Kmeans\_true\_membership\_matrix\]). Then for any $Z \in {\mathscr{C}}$ defined in (\[eqn:clustering\_Kmeans\_sdp\_unknown\_K\]), we have $$\begin{aligned} {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert (I-Z^{*}) Z (I-Z^{*}) \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast}{\leqslant}&\, \sum_{k=1}^K \sum_{m\neq k} \frac{1}{n_k}\, \|Z_{G_{k}^{*} G_{m}^{*}}\|_{1} + \operatorname{tr}(Z) - \operatorname{tr}(Z^\ast),\\ \|Z^{*} - Z\|_{1} {\leqslant}& \,4n\,\sum_{k=1}^K \sum_{m\neq k} \frac{1}{n_k}\, \|Z_{G_{k}^{*} G_{m}^{*}}\|_{1}+ n \big(\operatorname{tr}(Z) - \operatorname{tr}(Z^\ast)\big),\end{aligned}$$ and  holds for $Z \in {\mathscr{C}}$. The first inequality follows from inequality (57) in [@BuneaGiraudRoyerVerzelen2016]. The second one follows from the first,  , and the following decomposition, $$\begin{aligned} Z - Z^\ast = (I-Z^{*}) Z (I-Z^{*}) + (Z^\ast Z - Z^\ast) + (Z Z^\ast - Z^\ast) + (Z^\ast - Z^\ast ZZ^\ast),\end{aligned}$$ with inequality $\|U\|_1 {\leqslant}n {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert U \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\mbox{\scriptsize op}} {\leqslant}n {{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert U \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}_{\ast}$ for any matrix $U\in {\mathbb{R}}^{n\times n}$. [^1]: X. Chen’s research is supported in part by NSF DMS-1404891, NSF CAREER Award DMS-1752614, and UIUC Research Board Awards (RB17092, RB18099). Y. Yang’s research is supported in part by NSF DMS-1810831. This work is completed in part with the high-performance computing resource provided by the Illinois Campus Cluster Program at UIUC. Authors are listed in alphabetical order

--- abstract: | In this work we present a theoretical study on the propagation of light in heterogeneous systems with fluctuating optical properties. To understand the consequences of the fluctuations we perform numerical calculations with uniform and non uniforms systems using Monte Carlo simulations. We consider two distributions to represent a non-uniform medium: delta function and an exponential negative distributions.The results show that even with finite moments distributions, may require a large number of interactions for a convergence towards Gaussian statistics. This can be important when estimating the optical properties of thin films. *\ OCIS codes:* 290.0290, 030.6600 author: - 'Emiliano Terán-Bobadilla' - Eugenio Rafael Méndez Méndez title: A study of the fluctuations of the optical properties of a turbid media through Monte Carlo method --- Introduction\[sec:intro\] ========================= Inhomogeneous medium is characterized by having variations in refractive index or have inclusions of particles with another index. Normally, it is assumed that the media has properties which do not vary with average position. However, many of these systems, for example, composed of a suspension of particles, tend to form lumps or regions where the particle density is higher (by sedimentation, for example). This breaks down with the assumption of uniformity of the system (see Figure 1) and involves certain difficulties to study. A major consequence is that, depending on the type of fluctuation, random process may not be stationary and hence not ergodic. This means that sample averages aren’t equivalent to ensemble averages (ensemble). This class of disordered systems are known as superdifusive media [@3; @6] and have raised recent interest in different areas of science, in particular in optics [@4]. ![Schematic view of a non homogeneous media with a uniform (a) and non-uniform particle distribution. \[fig:no\_uni\] ](no_uni2) Figure \[fig:fli\] illustrates the type of paths we can expect in a medium with particles of many sizes. We see that, in regions with small particles, the light follows a zigzag path with step short while to encounter a big particles, abruptaly, the paths are so long. This means that in the media there is a non-negligible probability that the light suddenly run, a much longer than the other. The paper is organized in the following way. In the first section we describe in broad terms the problem we will address. Section II discusses the stochastic properties of a medium where the statistics are Gaussian and non-Gaussian. Section III presents a study of the resultant of flight on uniform and disordered media. Finally, the section provides a summary and conclusions of this work. ![Illustration of possible optical paths in a non-homogeneous media with fractal characteristics due to the dispersity in the particle sizes [@4]. \[fig:fli\] ](FLIGHTS) Statistical fluctuations of the optical properties ================================================== In simplest theories, which are approximations to the radiative transport equation, inhomogeneous medium can be characterized by their interaction coefficients $\mu_a$ and $\mu_s$, and also by the anisotropy parameter $g$. These coefficients represent, respectively, the scattering probability per unit length ($\mu_s$), the probability of absorption per unit length ($\mu_a$) and average scattering angle after an interaction ($g = <\cos \theta_s>$). The total interaction coefficient $\mu_t = \mu_s + \mu_a$ and the mean free path is $l = 1/\mu_s$. In these simple theories it’s has, moreover, that $\mu_s$ and $g$ always appear as $\mu_s' = \mu_s (1 - g)$, which is known as the reduced scattering coefficient. Henceforth, we assume that the medium is not absorbing, so that we can $\mu_a = 0$ and $\mu_t = \mu_s$ and also it’s assume that $g = 0$ (isotropic scattering). Generally it’s assumed that the parameter $\mu_s$ is constant. However, this is not always true in experimental systems. Under the assumption of independent scattering of the particles, the interaction coefficient can be written as $$\label{eq:mut} \mu_t=\rho C_t,\mbox{ }[\rm{cm}^{-1}]$$ where $C_s$ denotes the scattering cross section of particles and $\rho$ indicates the bulk density, *i.e.*, the number of particles per unit volume. We can see that if there are changes in the density, or if the particles have different properties, $\mu_s$ could be a function of position. We can see from equation (\[eq:mut\]) that $\mu_t$ depends on the optical size of the particles $C_t$ and of the density $\rho$. We start the study considering fluctuations with respect of one the parameter these parameters. Then we will consider the effect of the fluctuations of both of them in the total interaction coefficient. Uniform systems --------------- We consider first the case of a non-homogeneous system on a microscopic level but that, beyond a certain level, not changed in their optical properties. We model the media using identical particles with a uniform particle density. The probability of interaction per unit length along a line is then constant. Let $\mu_s$ the probability of interaction per unit length of a photon in the media. This probability can be written as the sum of the probabilities of scattering and absorption: $\mu_t = \mu_s + \mu_a$. For simplicity, we consider that there is no absorption in the media so that $\mu_t = \mu_s$. We also define the mean free path between interactions as $l = (\mu_s)^{-1}$. Let $F(s)$ the probability that a photon, which begins at $s = 0$, does not scattered on the length $s$. The probability of scattering in a differential length $ds$ is $\mu_s ds$, so that the probability of no scattering at $ds$ is $(1 - \mu_sds)$. We then have $$F(s+ds)=F(s)(1 - \mu_sds),$$ so we can write the differential equation $$dF(s)=-F(s)\mu_sds.$$ The solution, gives the probability that a photon is not scattered in the length $s$, $$F(s)=\exp(-\mu_ss),$$ so that the probability of scattering at this length is given by $$P(s)=1-\exp(-\mu_ss),$$ The probability density function (PDF), $p_s (s) = dP (s) / ds$, which governs the interaction of photons with the media may then be written in the form $$\label{eq_6} p_s(s)=\mu_s\exp(-\mu_ss)=\dfrac{1}{l}\exp(-s/l),$$ The moments of the distribution are given by \[7\] $$\label{eq_7} \langle s^n\rangle=\int s^np(s)ds=n!l^n,$$ so that $$\label{eq_8} \langle s\rangle=l, \mbox{ \hspace{1cm} }<s^2>=2l^2,\mbox{ }\dots,$$ and standard deviation $$\label{eq_9} \sigma_s=\sqrt{\langle s^2\rangle-\langle s\rangle^2}=l.$$ Systems with two types of particles ----------------------------------- In a non-uniform system it’s present variations of the interaction parameter. We call this parameter fluctuating $\nu$, which its average is $\mu_s$. The probability density function of displacement in a given region will depend on the specific value of the random variable $\nu$ takes. From the equation (\[eq\_6\]), then the conditional PDF can be writen as $$\label{eq_11} p_s(s | \nu) =\nu \exp(-\nu s).$$ Denoting by $p_{\nu} (\nu)$ the PDF to scattering coefficients, we can write an expression for the new PDF to displacements, $$\label{eq_12} p_s(s) =\int p_s(s | \nu)p_{\nu}(\nu)d\nu.$$ In the case where the scattering coefficient, $\nu$, take just two random values, we can write the PDF as, $$\label{eq_13} p_{\nu} (\nu) = a \delta(\nu-\mu_1) + b \delta(\nu- \mu_2),$$ where $a$ is the probability of the coefficient $\mu_1$ and $b$ represents the probability of occurring $\mu_2$. It’s necessaryy that $a + b = 1$ and $\mu_s=a\mu_1 + b\mu_2$. At this point, it is necessary to mention that the modeled system is not a homogeneous mixing of two components (in which case, $\mu_s$ would be equal to the sum $\mu_1 + \mu_2$), but a system with regions with properties $\mu_1$ and other with properties $\mu_2$. It’s convenient to define also a parameter $\alpha$, by the relationship $$\label{eq_14} \mu_1 = \dfrac{\alpha}{a}\mu_s,$$ implies that, $$\label{eq_15} \mu_2 = \dfrac{1-\alpha}{1-a}\mu_s.$$ This means that the PDF (\[eq\_13\]) can be specified by the parameters $a$, $\mu_1$ and $\mu_2$ or, alternatively, by $\mu_s$, $a$ and $\alpha$. The PDF for the displacements, or flights, can be determined by equations (\[eq\_11\]), (\[eq\_12\]) and (\[eq\_13\]), and is given by $$\begin{aligned} \label{eq_16} p_s(s)&=&\int \nu e^{-\nu s}\left[a\delta (\nu-\mu_1)+b\delta (\nu-\mu_2)\right]d\nu,\nonumber\\ &=&a\mu_1 e^{-\mu_1 s}+b\mu_2 e^{-\mu_2 s}\end{aligned}$$ We can verify that this FDP is normalized properly and that the first moments are $$\label{eq_17} \langle s\rangle = \dfrac{a}{\mu_1}+\dfrac{b}{\mu_2},\mbox{\hspace{2cm}} \langle s^2\rangle = \dfrac{2a}{\mu^2_1}+\dfrac{2b}{\mu^2_2}$$ Systems with negative exponential fluctuations ---------------------------------------------- For a medium with a distribution of scattering coefficients negative exponential type, we have that the PDF for $\nu$ can be written as, $$\label{eq_18} p_{\nu}(\nu)=\beta\exp(-\beta\nu),$$ where $\beta = 1/\mu_s$ and $\mu_s$ is the average scattering coefficient of the system. As in the previous case, the equations (\[eq\_11\]), (\[eq\_12\]) and (\[eq\_18\]) we have that the PDF for flight is given by, $$\begin{aligned} \label{eq_19} p_s(s)&=&\int_0^{\infty}\left[ \nu \exp(-\nu s)\right]\left[ \beta \exp(-\beta\nu)\right]d\nu,\nonumber\\ &=&\beta\int_0^{\infty}\nu\exp\left[-(s+\beta)\nu\right]d\nu.\end{aligned}$$ Evaluating the integral, we have $$\label{eq_20} p_{s}(s)=\dfrac{\beta}{(s+\beta)^2}=\dfrac{\mu_s}{(1+\mu_ss)^2}.$$ ![Comparing the probability density function defined by equations (\[eq\_16\]) and (\[eq\_20\]). The solid blue curve represents the distribution for flights governed by a variation in $\nu$ double-delta type. The dotted red curve for the negative exponential distribution. \[fig\_4\] ](PP) We see that for large s arguments, the probability density behaves as a Lorentzian Lévy flight \[equation (\[eq\_10\])\]. It is worth mentioning that the PDF defined by equation (\[eq\_20\]) has no definite time, which is characteristic of Lévy flights. Figure \[fig\_4\] shows the behavior of the density (\[eq\_16\]) and (\[eq\_20\]). The curves were scaled independently to illustrate the differences. We can see that, although the curves appear similar, they have important differences. In particular, it should be noted that the decay of the curve corresponding to negative exponential fluctuations is very slow. To better understand the consequences of adopting these PDF, in the next section we present calculations of random walks using Monte Carlo simulations. Fluctuations in density and optical size ---------------------------------------- A realistic system must consider fluctuations in the optical size $C_x$ and density $\rho$. Random walks\[sec:medio\] ========================= ![Random walk trajectories. (a) Random walk in a uniform media. (b) Lévy random walk. \[fig\_5\] ](VV) In the context of this study, it is interesting to see the result of random walks with different probability density laws described in the previous section, focusing on situations in which the number of interactions is large. The types of situations that may occur in a case with uniform distribution and one in which they occur Lévy flight is illustrated in Figure 5. We consider the distance after N number of displacements. For simplicity, we illustrate the method by considering a two-dimensional space and we’ll write the total displacement in polar coordinates $(a, \theta)$ (see Figure \[fig\_5\]): Evaluating the integral, we have $$\label{eq_21} \mathbf a=ae^{i\theta}=\dfrac{1}{\sqrt{N}}\sum_{k=1}^Ns_ke^{\phi_k}.$$ We assume that: 1. Amplitudes $s_k / \sqrt{N}$ and phases $\phi_k$ are statistically independent. 2. The variables $s_k$ follow the distribution (\[eq\_6\]) with moments given by equation (\[eq\_7\]). 3. The phases $\phi_k$ are uniformly distributed in the interval $(-\pi, \pi)$. This means that the scattering of the particles is isotropic. We then have that the $x$ and $y$ components are given by: $$\begin{aligned} a_x=&a\cos{\theta}=\dfrac{1}{\sqrt{N}}\sum_{k=1}^Ns_k\cos({\phi_k}),\\ a_y=&a\sin{\theta}=\dfrac{1}{\sqrt{N}}\sum_{k=1}^Ns_k\sin({\phi_k}),\end{aligned}$$ and, with our assumptions, we find that $$\begin{aligned} \langle a_x\rangle=0,& \mbox{\hspace{2cm}}\langle a_x^2\rangle=\dfrac{l}{2},\\ \langle a_y\rangle=0&\hspace{2cm} \langle a_y^2\rangle=\dfrac{l}{2}.\end{aligned}$$ ![Random walk. \[fig\_5\] ](RR) When the number of steps, $N$, is very large, the displacement of the photon statistics are Gaussian. That is, both $a_x$ and $a_y$ follow as Gaussian distributions.In this case, the variables have zero mean, second moment $l / 2$ and they are not correlated. It is then circulated Gaussian aletorio process \[7\]. The joint probability density is then, Evaluating the integral, we have $$\label{eq_24} p_{x,y}(a_x,a_y)=\dfrac{1}{\pi l}\exp\left \{-\dfrac{a_x^2-a_y^2}{l}\right\}.$$ layer $n$ \[-\] $\mu_a$ \[cm$^{-1}$\] $\mu_s$ \[cm$^{-1}$\] $g$ \[-\] $d$ \[cm\] ---------------- ----------- ----------------------- ----------------------- ----------- ------------ sup. 1.4 optical system 1.4 0.003 1867 0.4 10 inf. 1.4 : The system parameters which were performed the Monte Carlo calculations. \[tab:aa\] \[default\] The length statistics are founded with a probability transformation to express (\[eq\_24\]) in terms of $(a, \theta)$, and integrating over phase \[7\], It is found that $$\label{eq_25 } p_{a}(a)=\dfrac{a}{2 l}\exp\left \{-\dfrac{a^2}{l}\right\},$$ for $a> 0$. Then the steps follow a a Rayleigh distribution. We note, however, that our assumptions are valid, moments of the distribution that governs the movement must be finite. In the next section, we will use Monte Carlo simulations to study three cases, corresponding to the displacement PDF given by equations (\[eq\_6\]), (\[eq\_16\]) and (\[eq\_20\]), starting with the case of the uniform system. Flights ---------------------- ------------------ ------------------------------------------------------------- $\rho$ \[cm$^{-3}$\] $C_t$ \[cm$^2$\] $p_S(s)$ cte. cte. $\langle\mu_t \rangle\exp\big[-\langle\mu_t \rangle s\big]$ cte. deltas $a\mu_1 e^{-\mu_1 s}+b\mu_2 e^{-\mu_2 s}$ cte. exp. neg. $\dfrac{\mu_s}{(1+\mu_ss)^2}$ delta cte. $\langle\mu_t \rangle\exp(-\langle\mu_t \rangle s)$ delta deltas $a\mu_1 e^{-\mu_1 s}+b\mu_2 e^{-\mu_2 s}$ delta exp. neg. $\dfrac{\mu_s}{(1+\mu_ss)^2}$ exp. neg. cte. $\langle\mu_t \rangle\exp(-\langle\mu_t \rangle s)$ exp. neg. deltas $a\mu_1 e^{-\mu_1 s}+b\mu_2 e^{-\mu_2 s}$ exp. neg. exp. neg. $\dfrac{\mu_s}{(1+\mu_ss)^2}$ : The system parameters which were performed the Monte Carlo calculations. \[tab:aa\] \[default\] MONTE CARLO SIMULATIONs\[sec:medio\] ==================================== To explore the convergence to Gaussian statistics for the three PDF considered, we present calculations based on the MCML (Monte Carlo Multi Layered) simulation \[9\] using the values for the average properties of the medium shown in Table I. Table defines the parameters of the hypothetical medium that we study. Assume that the medium is highly scatterer ($\mu_s$ = 1867 cm$^{-1}$) and with low absorption ($\mu_a = 0.003$ cm$^{-1}$) and optically thick $(\mu_sd \gg 1)$, so that a great number interactions occur before the photon is lost. To simplify the system, we assumed that the refractive index does not change, we can visualize as if we were immersed in the environment. To encourage the development of flight we move the point of initial interaction “photons” to the center of the sample and we count $N$ interactions from the origin to obtain the components $a_x$, $a_y$ and $a_z$ of the resultant $\mathbf a$ (see Fig. 6). As discussed earlier, for a large number of interactions, applying the central limit theorem, these components must follow Gaussian statistics. Uniform System -------------- We study first the uniform system. In this case, the PDF for movement is given by equation (6). Figure \[fig\_6\] shows histograms of the components resulting in, after 10 interactions. The vertical bars represent the histograms of displacement, and red curves, Gaussian functions that adjust data. We can see that although there are slight changes in the heights of the curves all have the same width $\omega_o = 22.5\ $ m. Clearly the components of the displacements resulting in good agreement with the expected Gaussian distribution, which is consistent with the central limit theorem. This, despite the fact that we considered only 10 interactions. Figure \[fig\_7\] shows the histogram of the magnitude, $ a$, of the resultant. As expected, the result fits very well to a Rayleigh PDF. System with two types of particles ---------------------------------- ![Histogram of the components $a_x$, $a_y$, and $a_z$ of the resultant of the random walks for a uniform system. \[fig\_6\] ](axayaz.png) ![ Histogram of the magnitude of the resultant $a$ of the random walk. \[fig\_7\] ](aa.png) Consider now the case of the medium with two types of particles. FDP for displacement is given by equation (16). Figure \[fig\_6\] shows histograms of the resultant components after 10 interactions. As in previous cases, vertical bars represent histograms and red curves Gaussian functions adjusted in height. This figure was generated by taking $a = 0.1$ and $\alpha = 0.01$, so that $\mu_1 = 0.1\mu_s$ (long steps with a low probability) and $\mu_2 = 1.1\mu_s$ (short step with high probability). As the difference between the values of the coefficients of scattering is great, it has the possibility of abrupt fluctuations. Unlike the previous case, Figure \[fig\_8\] shows that it does not have a good fit to Gaussian curves. This means that after 10 steps, the statistics do not converge to such statistics. Should be noted that, if we increase the number of interactions or flights eventually expected convergence is obtained based on the central limit theorem. On the other hand, keeping the number of flights 10, but considering larger values of $\alpha$ (which implies that the two deltas distribution of the scattering coefficients are closer) is also obtained convergence Gaussian statistics. ![ Histogram of the components $a_x$, $a_y$, and $a_z$ of the resultant of the random walk, considering the PDF given by equation (\[eq\_16\]) with $a = 0.1$ and $\alpha = 0.01$. \[fig\_8\] ](G_2p.png) System with a negative exponential distribution of particles ------------------------------------------------------------ ![ Histogram of the components $a_x$, $a_y$, and $a_z$ of the resultant of the random walk, considering the PDF given by equation (\[eq\_20\]). \[fig\_9\] ](Levy2.png) Now we assume that PDF governing displacement is given by equation (\[eq\_20\]). Figure 10 shows the histogram of the components of the resultant, $\mathbf a$, after 10 flights. As in previous figures, vertical bars represent histograms and red curves Gaussian functions adjusted in height .We see that the histograms of the components do not fit Gaussian curves.We can also see that the range of values ??taken by these components is much broader than in the previous cases, indicating that the fluctuations are much larger in flight, and can be up to an order of magnitude larger. This is not surprising, then flight statistics given by equation (\[eq\_20\]) represent a statistical approach to type Lévy, and Lévy type processes are caracterizazdos by violent fluctuations that make the resulting not converge to Gaussian statistics. We studied, however, the possibility of convergence after a very large number of interactions. Figure \[fig\_10\] shows the histogram of the resultant components after $1,000$ flights. We see that, after such a large number of flights, the statistics converge if Gaussian statistics appear, although we should mention that this does not necessarily mean that the central limit theorem is valid in this type of situation. The statistics themselves seem to converge to Gaussian statistics, although we should mention that this does not necessarily mean that the central limit theorem is valid in this type of situation. The above results show that for this medium, if they occur a sufficiently large number of interactions, it will appear to be Gaussian statistics. However, such large fluctuations have important implications finite system, as in films, in which the number of interactions is limited by the film thickness. ![Histogram of the components $a_x$, $a_y$, and $a_z$ the resultant of random walks for $1,000$ flights. \[fig\_10\] ](G_exn_1k.png) COMMENTS AND CONCLUSIONS ======================== We have seen that the FDP adopted for the flights can determine the convergence or lack of it to Gaussian statistics. The fact of having non-Gaussian statistics and the ability to take big steps involve major changes in the properties of a film of this type of media. In these superdifusivos media, for example, the opacity of a film can be lowered considerably. The results show that in uniform media, after 10 steps, has a good convergence to Gaussian statistics. In the other two types of media considered more interactions are required to have these statistics. In particular, for the medium with negative exponential fluctuations are required in the order of 1,000 interactions to approach these statistics. Acknowledgements ================ E. T. is grateful to the authorities of the UAS and CICESE for their support to perform this study. This work has been supported by PROMEP under grant 2012. [xx]{} Lévy, P. Théorie de lÕAddition des Variables Aléatoires (Gauthier-Villars, Paris, 1954). S. Chandrasekhar, “Stochastic Problems in Physics and Astronomy”, Rev. Mod. Phys. 15, 1Ð89 (1943) A. B. Davis, A. Marshak, K. P. Pfeilsticker “Anomalous Lévy Photon Diffusion Theory: Toward a New Parameterization of Shortwave Transport in Cloudy Columns”, Nineth ARM Science Team Meeting Proceedings, San Antonio, Texas, March 22-26, 1999. P. Barthelemy, J. Bertolotti and D. S. Wiersma, ÓA Levy flight for lightÓ, Nature 453, 495-498 (2008). N. Mercadier, W. Guerin, M. Chevrollier, and R. Kaiser, ÒLévy flights of photons in hot atomic vapours,Ó Nature Physics 5, 602 - 605 (2009). D. Wiersma, P. Barthelemy y J. Bertolotti, “Optical Lévy flights and super diffusion of light,” in Quantum Sensing and Nanophotonic Devices VI, edited by Manijeh Razeghi, Rengarajan Sudharsanan and Gail J. Brown. Proc. of SPIE 7222, 72221A (2009). J. W. Goodman, Statistical Optics, John Wiley & Sons (New York, 1985). E. Pereira, J. M. G. Martinho, and M. N. Berberan-Santos, ÓPhoton trajectories in incoher- ent atomic radiation trapping as Lévy flights,Ó Phys. Rev. Lett. 93, 120201 (2004). L-H. Wang, S.L. Jacques y L-Q Zheng ÒMCML - Monte Carlo modeling of photon transport in multi-layered tissues,Ó Computer Methods and Programs in Biomedicine 47, 131-146, (1995).

--- abstract: 'Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with perfect residue field. Let $(\pi_n)_{n\ge 0}$ be a system of $p$-power roots of a uniformizer $\pi=\pi_0$ of $K$ with $\pi^p_{n+1}=\pi_n$, and define $G_s$ (resp. $G_{\infty}$) the absolute Galois group of $K(\pi_s)$ (resp. $K_{\infty}:=\bigcup_{n\ge 0} K(\pi_n)$). In this paper, we study $G_s$-equivatiantness properties of $G_{\infty}$-equivariant homomorphisms between torsion (potentially) crystalline representations.' author: - 'Yoshiyasu Ozeki[^1]' title: On Galois equivariance of homomorphisms between torsion potentially crystalline representations --- Introduction ============ Let $p$ be a prime number and $r\ge 0 $ an integer. Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with perfect residue field and absolute ramification index $e$. Let $\pi=\pi_0$ be a uniformizer of $K$ and $\pi_n$ a $p^n$-th root of $\pi$ such that $\pi^p_{n+1}=\pi_n$ for all $n\ge 0$. For any integer $s\ge 0$, we put $K_{(s)}=K(\pi_s)$. We also put $K_{\infty}=\bigcup_{n\ge 0}K_{(n)}$. We denote by $G_K, G_s$ and $G_{\infty}$ absolute Galois groups of $K$, $K_{(s)}$ and $K_{\infty}$, respectively. By definition we have the following decreasing sequence of Galois groups: $$G_K=G_0\supset G_1\supset G_2\supset\cdots \supset G_{\infty}.$$ Since $K_{\infty}$ is a strict APF extension of $K$, the theory of fields of norm implies that $G_{\infty}$ is isomorphic to the absolute Galois group of some field of characteristic $p$. Therefore, representations of $G_{\infty}$ has easy interpretations via Fontaine’s étale ${\varphi}$-modules. Hence it seems natural to have the following question: \[que1\] Let $T$ be a representation of $G_K$. For a “small” integer $s\ge 0$, can we reconstruct various information of the $G_s$-action on $T$ from that of the $G_{\infty}$-action? Nowadays there is an interesting insight of Breuil; he showed that representations of $G_K$ arising from finite flat group schemes or $p$-divisible groups over the integer ring of $K$ is “determined” by its restriction to $G_{\infty}$. Moreover, for ${\mathbb}{Q}_p$-representations, Kisin proved the following theorem in [@Kis] (which was a conjecture of Breuil): the restriction functor from the category of crystalline ${\mathbb}{Q}_p$-representations of $G_K$ into the category of ${\mathbb}{Q}_p$-representations of $G_{\infty}$ is fully faithful. In this paper, we give some partial answers to Question \[que1\] for [*torsion crystalline representations*]{}, moreover, [*torsion potentially crystalline representations*]{}. Our first main result is as follows. Let ${\mathrm}{Rep}^{r,{\mathrm}{ht},{\mathrm}{pcris}(s)}_{{\mathrm}{tor}}(G_K)$ be the category of torsion ${\mathbb}{Z}_p$-representations $T$ of $G_K$ which satisfy the following: there exist free ${\mathbb}{Z}_p$-representations $L$ and $L'$ of $G_K$, of height $\le r$, such that - $L|_{G_s}$ is a subrepresentation of $L'|_{G_s}$. Furthermore, $L|_{G_s}$ and $L'|_{G_s}$ are lattices in some crystalline ${\mathbb}{Q}_p$-representation of $G_s$ with Hodge-Tate weights in $[0,r]$; - $T|_{G_s} \simeq (L'|_{G_s})/(L|_{G_s})$. \[Main1\] Suppose that $p$ is odd and $e(r-1)<p-1$. Let $T$ and $T'$ be objects of ${\mathrm}{Rep}^{r,{\mathrm}{ht},{\mathrm}{pcris}(s)}_{{\mathrm}{tor}}(G_K)$. Then any $G_{\infty}$-equivariant homomorphism $T\to T'$ is in fact $G_s$-equivariant. We should remark that the condition $e(r-1)<p-1$ in the above does not depend on $s$. We put ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)={\mathrm}{Rep}^{r, {\mathrm}{ht}, {\mathrm}{pcris}(0)}_{{\mathrm}{tor}}(G_K)$. By definition, a torsion ${\mathbb}{Z}_p$-representation $T$ of $G_K$ is contained in this category if and only if it can be written as the quotient of lattices in some crystalline ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$. We call the objects in ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ [*torsion crystalline representations with Hodge-Tate weights in $[0,r]$*]{}. In the case $r=1$, such representations are equivalent to finite flat representations. (Here, a torsion ${\mathbb}{Z}_p$-representation of $G_K$ is finite flat if it arises from the generic fiber of some $p$-power order finite flat commutative group scheme over the integer ring of $K$.) Combining Theorem \[Main1\] with results of [@Kim], [@La], [@Li3] (the case $p=2$) we obtain the following full faithfulness theorem for torsion crystalline representations. \[FFTHMtorcris\] Suppose $e(r-1)<p-1$. Then the restriction functor ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is fully faithful. Before this work, some results are already known. First, the full faithfulness theorem was proved by Breuil for $e=1$ and $r<p-1$ via the Fontaine-Laffaille theory ([@Br2], the proof of Théorèm 5.2). He also proved the theorem under the assumptions $p>2$ and $r\le 1$ as a consequence of a study of the category of finite flat group schemes ([@Br3 Theorem 3.4.3]). Later, his result was extended to the case $p=2$ in [@Kim], [@La], [@Li3] (proved independently). In particular, the full faithfulness theorem for $p=2$ is a consequence of their works. On the other hand, Abrashkin proved the full faithfulness in the case where $p>2, r<p$ and $K$ is a finite unramified extension of ${\mathbb}{Q}_p$ ([@Ab2 Section 8.3.3]). His proof is based on calculations of ramification bounds for torsion crystalline representations. In [@Oz2], a proof of Corollary \[FFTHMtorcris\] under the assumption $er<p-1$ is given via $({\varphi},\hat{G})$-modules (which was introduced by Tong Liu [@Li2] to classify lattices in semi-stable representations). Our proof of Theorem \[Main1\] is similar to the proof for the main result of [@Oz2], but we need more careful considerations for $({\varphi},\hat{G})$-modules. Indeed we need special base change arguments to study some potential crystalline representations. In fact, we prove a full faithfulness theorem for torsion representations arising from certain classes of $({\varphi},\hat{G})$-modules (cf. Theorem \[FFTHM\]), which immediately gives our main theorem. In addition, our study gives a result as below which is the second main result of this paper (here, we define ${\mathrm}{log}_p(x):=-\infty$ for any real number $x\le 0$). \[Main2\] Suppose that $p$ is odd and $s> n-1 + {\mathrm}{log}_p(r-(p-1)/e)$. Let $T$ and $T'$ be objects of ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ which are killed by $p^n$. Then any $G_{\infty}$-equivariant homomorphism $T\to T'$ is in fact $G_s$-equivariant. For torsion semi-stable representations, a similar result was shown in Theorem 3 of [@CL2], which was a consequence of a study of ramification bounds. The bound appearing in their theorem was $n-1 + {\mathrm}{log}_p(nr)$. By applying our arguments given in this paper, we can obtain a generalization of their result; our refined condition is $n-1 + {\mathrm}{log}_pr$ (see Theorem \[Main3\]). Some other consequences of our study are described in subsection \[consequences\]. Motivated by the full faithfulness theorem (= Corollary \[FFTHMtorcris\]) and Theorem \[Main2\], we raise the following question. Is any $G_{\infty}$-equivariant homomorphism in the category ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ in fact $G_s$-equivariant for $s>{\mathrm}{log}_p(r-(p-1)/e)$? On the other hand, there exist counter examples of the full faithfulness theorem when we ignore the condition $e(r-1)< p-1$. Let ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_1)$ be the category of torsion ${\mathbb}{Z}_p$-representations of $G_1$. \[nonfull\] Suppose that $K$ is a finite extension of ${\mathbb}{Q}_p$, and also suppose $e\mid (p-1)$ or $(p-1)\mid e$. If $e(r-1)\ge p-1$, the restriction functor ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_1)$ is not full $($in particular, the restriction functor ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is not full$)$. In particular, if $p=2$, then the full faithfulness never hold for any finite extension $K$ of ${\mathbb}{Q}_2$ and any $r\ge 2$. Theorem \[nonfull\] implies that the condition “$e(r-1)<p-1$” in Corollary \[FFTHMtorcris\] is the best possible for many finite extensions $K$ of ${\mathbb}{Q}_p$. Now we describe the organization of this paper. In Section 2, we setup notations and summarize facts we need later. In Section 3, we define variant notions of $({\varphi},\hat{G})$-modules and give some basic properties. They are needed to study certain classes of potentially crystalline representations and restrictions of semi-stable representations. In Section 4, we study technical torsion $({\varphi},\hat{G})$-modules which are related with torsion (potentially) crystalline representations. The key result in this section is the full faithfulness result Theorem \[FFTHM\] on them, which allows us to prove our main results immediately. Finally, in Section 5, we calculate the smallest integer $r$ for a given torsion representation $T$ such that $T$ admits a crystalline lift with Hodge-Tate weights in $[0,r]$. We mainly study the rank two case. We use our full faithfulness theorem to assure the non-existence of crystalline lifts with small Hodge-Tate weights. Theorem \[nonfull\] is a consequence of studies of this section. The author would like to thank Shin Hattori, Naoki Imai and Yuichiro Taguchi who gave him many valuable advice. This work was supported by JSPS KAKENHI Grant Number 25$\cdot$173. [**Notation and convention:**]{} Throughout this paper, we fix a prime number $p$. Except Section 5, we always assume that [*$p$ is odd*]{}. For any topological group $H$, we denote by ${\mathrm}{Rep}_{{\mathrm}{tor}}(H)$ (resp. ${\mathrm}{Rep}_{{\mathbb}{Z}_p}(H)$, resp. ${\mathrm}{Rep}_{{\mathbb}{Q}_p}(H)$) the category of torsion ${\mathbb}{Z}_p$-representations of $H$ (resp. the category of free ${\mathbb}{Z}_p$-representations of $H$, resp. the category of ${\mathbb}{Q}_p$-representations of $H$). All ${\mathbb}{Z}_p$-representations (resp. ${\mathbb}{Q}_p$-representations) in this paper are always assumed to be finitely generated over ${\mathbb}{Z}_p$ (resp. ${\mathbb}{Q}_p$). For any field $F$, we denote by $G_F$ the absolute Galois group of $F$ (for a fixed separable closure of $F$). Preliminaries ============= In this section, we recall definitions and basic properties for Kisin modules and $({\varphi},\hat{G})$-modules. Throughout Section 2, 3 and 4, we always assume that $p$ is an odd prime. Basic notations --------------- Let $k$ be a perfect field of characteristic $p$, $W(k)$ the ring of Witt vectors with coefficients in $k$, $K_0=W(k)[1/p]$, $K$ a finite totally ramified extension of $K_0$ of degree $e$, $\overline{K}$ a fixed algebraic closure of $K$. Throughout this paper, we fix a uniformizer $\pi$ of $K$. Let $E(u)$ be the minimal polynomial of $\pi$ over $K_0$. Then $E(u)$ is an Eisenstein polynomial. For any integer $n\ge 0$, we fix a system $(\pi_n)_{n\ge 0}$ of a $p^n$-th root of $\pi$ in $\overline{K}$ such that $\pi^p_{n+1}=\pi_n$. Let $R={\varprojlim}{\mathcal{O}}_{\overline{K}}/p$, where ${\mathcal{O}}_{\overline{K}}$ is the integer ring of $\overline{K}$ and the transition maps are given by the $p$-th power map. For any integer $s\ge 0$, we write $\underline{\pi_s}:=(\pi_{s+n})_{n\ge 0}\in R$ and $\underline{\pi}:=\underline{\pi_0}\in R$. Note that we have $\underline{\pi_s}^{p^s}=\underline{\pi}$. Let $L$ be the completion of an unramified algebraic extension of $K$ with residue field $k_L$. Then $\pi_s$ is a uniformizer of $L_{(s)}:=L(\pi_s)$ and $L_{(s)}$ is a totally ramified degree $ep^s$ extension of $L_0:=W(k_L)[1/p]$. We set $L_{\infty}:=\bigcup_{n\ge 0}L_{(n)}$. We put $G_{L,s}:=G_{L_{(s)}}={\mathrm}{Gal}(\overline{L}/L_{(s)})$ and $G_{L,\infty}:=G_{L_{\infty}}={\mathrm}{Gal}(\overline{L}/L_{\infty})$. By definitions, we have $L=L_{(0)}$ and $G_{L,0}=G_L$. Put ${\mathfrak{S}}_{L,s}=W(k_L)[\![u_s]\!]$ (resp. ${\mathfrak{S}}_L=W(k_L)[\![u]\!]$) with an indeterminate $u_s$ (resp. $u$). We equip a Frobenius endomorphism $\varphi$ of ${\mathfrak{S}}_{L,s}$ (resp. ${\mathfrak{S}}_L$) by $u_s\mapsto u_s^p$ (resp. $u\mapsto u^p$) and the Frobenius on $W(k_L)$. We embed the $W(k_L)$-algebra $W(k_L)[u_s]$ (resp. $W(k_L)[u]$) into $W(R)$ via the map $u_s\mapsto [\underline{\pi_s}]$ (resp. $u\mapsto [\underline{\pi}]$), where $[\ast]$ stands for the Teichmüller representative. This embedding extends to an embedding ${\mathfrak{S}}_{L,s}\hookrightarrow W(R)$ (resp. ${\mathfrak{S}}_L\hookrightarrow W(R)$). By identifying $u$ with $u_s^{p^s}$, we regard ${\mathfrak{S}}_L$ as a subalgebra of ${\mathfrak{S}}_{L,s}$. It is readily seen that the embedding ${\mathfrak{S}}_L\hookrightarrow {\mathfrak{S}}_{L,s} \hookrightarrow W(R)$ is compatible with the Frobenius endomorphisms. If we denote by $E_s(u_s)$ the minimal polynomial of $\pi_s$ over $K_0$, with indeterminate $u_s$, then we have $E_s(u_s)=E(u_s^{p^s})$. Therefore, we have $E_s(u_s)=E(u)$ in ${\mathfrak{S}}_{L,s}$. We note that the minimal polynomial of $\pi_s$ over $L_0$ is $E_s(u_s)$. Let $S^{{\mathrm}{int}}_{L_0,s}$ (resp. $S^{{\mathrm}{int}}_{L_0})$) be the $p$-adic completion of the divided power envelope of $W(k_L)[u_s]$ (resp. $W(k_L)[u]$) with respect to the ideal generated by $E_s(u_s)$ (resp. $E(u)$). There exists a unique Frobenius map ${\varphi}\colon S^{{\mathrm}{int}}_{L_0,s}\to S^{{\mathrm}{int}}_{L_0,s}$ (resp. ${\varphi}\colon S^{{\mathrm}{int}}_{L_0}\to S^{{\mathrm}{int}}_{L_0}$) and monodromy $N\colon S^{{\mathrm}{int}}_{L_0,s}\to S^{{\mathrm}{int}}_{L_0,s}$ defined by $\varphi(u_s)=u_s^p$ (resp. $\varphi(u)=u^p$) and $N(u_s)=-u_s$ (resp. $N(u)=-u$). Put $S_{L_0,s}=S^{{\mathrm}{int}}_{L_0,s}[1/p]=L_0\otimes_{W(k_L)} S^{{\mathrm}{int}}_{L_0,s}$ (resp. $S_{L_0}=S^{{\mathrm}{int}}_{L_0}[1/p]=L_0\otimes_{W(k_L)} S^{{\mathrm}{int}}_{L_0}$). We equip $S^{{\mathrm}{int}}_{L_0,s}$ and $S_{L_0,s}$ (resp. $S^{{\mathrm}{int}}_{L_0}$ and $S_{L_0}$) with decreasing filtrations ${\mathrm}{Fil}^iS^{{\mathrm}{int}}_{L_0,s}$ and ${\mathrm}{Fil}^iS_{L_0,s}$ (resp. ${\mathrm}{Fil}^iS^{{\mathrm}{int}}_{L_0,s}$ and ${\mathrm}{Fil}^iS_{L_0,s}$) by the $p$-adic completion of the ideal generated by $E^j_s(u_s)/j!$ (resp. $E^j(u)/j!$) for all $j\ge 0$. The inclusion $W(k_L)[u_s]\hookrightarrow W(R)$ (resp. $W(k_L)[u]\hookrightarrow W(R)$) via the map $u_s\mapsto [\underline{\pi_s}]$ (resp. $u\mapsto [\underline{\pi}]$) induces ${\varphi}$-compatible inclusions ${\mathfrak{S}}_{L,s}\hookrightarrow S^{{\mathrm}{int}}_{L_0,s}\hookrightarrow A_{{\mathrm}{cris}}$ and $S_{L_0,s}\hookrightarrow B^+_{{\mathrm}{cris}}$ (resp. ${\mathfrak{S}}_L\hookrightarrow S^{{\mathrm}{int}}_{L_0}\hookrightarrow A_{{\mathrm}{cris}}$ and $S_{L_0}\hookrightarrow B^+_{{\mathrm}{cris}}$). By these inclusions, we often regard these rings as subrings of $B^+_{{\mathrm}{cris}}$. By identifying $u$ with $u_s^{p^s}$ as before, we regard $S^{{\mathrm}{int}}_{L_0}$ (resp. $S_{L_0}$) as a ${\varphi}$-stable (but not $N$-stable) subalgebra of $S^{{\mathrm}{int}}_{L_0,s}$ (resp. $S_{L_0,s}$). By definitions, we have ${\mathfrak{S}}_L={\mathfrak{S}}_{L,0},\ S^{{\mathrm}{int}}_{L_0,0}=S^{{\mathrm}{int}}_{L_0}$ and $S_{L_0,0}= S_{L_0}$.\ [**Convention:**]{} For simplicity, if $L=K$, then we often omit the subscript “$L$” from various notations (e.g. $G_{K_s}=G_s$, $G_{K_{\infty}}=G_{\infty}$, ${\mathfrak{S}}_K={\mathfrak{S}}, {\mathfrak{S}}_{K,s}={\mathfrak{S}}_s$).\ $\displaystyle \xymatrix{ & W(R) \ar@{-}[rr] & & A_{{\mathrm}{cris}} \ar@{-}[rr] & & B^+_{{\mathrm}{cris}} \\ & {\mathfrak{S}}_{L,s} \ar@{-}[rr] \ar@{-}[u] & & S^{{\mathrm}{int}}_{L_0,s}\ar@{-}[rr] \ar@{-}[u] & & S_{L_0,s} \ar@{-}[u] \\ {\mathfrak{S}}_L \ar@{-}[rr] \ar@{-}[ru] & \ar@{-}[u] & S^{{\mathrm}{int}}_{L_0}\ar@{-}[rr] \ar@{-}[ru] & \ar@{-}[u] & S_{L_0} \ar@{-}[ru] & \\ & {\mathfrak{S}}_s \ar@{-}[r] \ar@{-}[u] & \ar@{-}[r] & S^{{\mathrm}{int}}_{K_0,s}\ar@{-}[r] \ar@{-}[u] & \ar@{-}[r] & S_{K_0,s} \ar@{-}[uu] \\ {\mathfrak{S}}\ar@{-}[rr] \ar@{-}[uu] \ar@{-}[ru] & & S^{{\mathrm}{int}}_{K_0}\ar@{-}[rr] \ar@{-}[uu] \ar@{-}[ru] & & S_{K_0} \ar@{-}[uu] \ar@{-}[ru] & }$ Kisin modules ------------- Let $r, s\ge 0$ be integers. A [*${\varphi}$-module*]{} [*over ${\mathfrak{S}}_{L,s}$*]{} is an ${\mathfrak{S}}_{L,s}$-module ${\mathfrak{M}}$ equipped with a ${\varphi}$-semilinear map ${\varphi}\colon {\mathfrak{M}}\to {\mathfrak{M}}$. A morphism between two ${\varphi}$-modules $({\mathfrak{M}}_1,{\varphi}_1)$ and $({\mathfrak{M}}_2,{\varphi}_2)$ over ${\mathfrak{S}}_{L,s}$ is an ${\mathfrak{S}}_{L,s}$-linear map ${\mathfrak{M}}_1\to {\mathfrak{M}}_2$ compatible with ${\varphi}_1$ and ${\varphi}_2$. Denote by $'{\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ the category of ${\varphi}$-modules $({\mathfrak{M}},{\varphi})$ over ${\mathfrak{S}}_{L,s}$ [*of height $\le r$*]{} in the sense that ${\mathfrak{M}}$ is of finite type over ${\mathfrak{S}}_{L,s}$ and the cokernel of $1\otimes {\varphi}\colon {\mathfrak{S}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}}{\mathfrak{M}}\to {\mathfrak{M}}$ is killed by $E_s(u_s)^r$. Let ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ be the full subcategory of $'{\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ consisting of finite free ${\mathfrak{S}}_{L,s}$-modules. We call an object of ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ a [*free Kisin module*]{} [*of height $\le r$ $($over ${\mathfrak{S}}_{L,s})$*]{}. Let ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s,\infty}}$ be the full subcategory of $'{\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ consisting of finite ${\mathfrak{S}}_{L,s}$-modules which are killed by some power of $p$ and have projective dimension $1$ in the sense that ${\mathfrak{M}}$ has a two term resolution by finite free ${\mathfrak{S}}_{L,s}$-modules. We call an object of ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s,\infty}}$ a [*torsion Kisin module of height $\le r$ $($over ${\mathfrak{S}}_{L,s})$*]{}. For any free or torsion Kisin module ${\mathfrak{M}}$ over ${\mathfrak{S}}_{L,s}$, we define a ${\mathbb}{Z}_p[G_{L,\infty}]$-module $T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}})$ by $$\begin{aligned} T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}}):= \left\{ \begin{array}{ll} {\mathrm}{Hom}_{{\mathfrak{S}}_{L,s},{\varphi}}({\mathfrak{M}},W(R))\hspace{21mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm free}, \cr {\mathrm}{Hom}_{{\mathfrak{S}}_{L,s},{\varphi}}({\mathfrak{M}},{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))\quad {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm torsion}. \end{array} \right.\end{aligned}$$ Here a $G_{L,\infty}$-action on $T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}})$ is given by $(\sigma.g)(x)=\sigma(g(x))$ for $\sigma\in G_{L,\infty}, g\in T_{{\mathfrak{S}}}({\mathfrak{M}}), x\in {\mathfrak{M}}$.\ [**Convention:**]{} For simplicity, if $L=K$, then we often omit the subscript “$L$” from various notations (e.g. ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{K,s,\infty}}={\mathrm}{Mod}^r_{/{\mathfrak{S}}_{s,\infty}}$, $T_{{\mathfrak{S}}_{K,s}}=T_{{\mathfrak{S}}_s}$ ). Also, if $s=0$, we often omit the subscript “$s$” from various notations (e.g. ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,0,\infty}}={\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,\infty}}$, $T_{{\mathfrak{S}}_{L,0}}=T_{{\mathfrak{S}}_L}$, ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{K,0,\infty}}={\mathrm}{Mod}^r_{/{\mathfrak{S}}_{\infty}}$, $T_{{\mathfrak{S}}_{K,0}}=T_{{\mathfrak{S}}}$ ).\ \[Kisinfunctor\] $(1)$ [([@Kis Corollary 2.1.4 and Proposition 2.1.12])]{}  The functor $T_{{\mathfrak{S}}_{L,s}}\colon {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}\to {\mathrm}{Rep}_{{\mathbb}{Z}_p}(G_{\infty})$ is exact and fully faithful. $(2)$ [([@CL1 Corollary 2.1.6, 3.3.10 and 3.3.15])]{}  The functor $T_{{\mathfrak{S}}_{L,s}}\colon {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s,\infty}}\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is exact and faithful. Furthermore, it is full if $er<p-1$. $({\varphi},\hat{G})$-modules {#Liumodule:section} ----------------------------- The notion of $({\varphi},\hat{G})$-modules are introduced by Tong Liu in [@Li2] to classify lattices in semi-stable representations. We recall definitions and properties of them. We continue to use same notations as above. Let $L_{p^{\infty}}$ be the field obtained by adjoining all $p$-power roots of unity to $L$. We denote by $\hat{L}$ the composite field of $L_{\infty}$ and $L_{p^{\infty}}$. We define $H_L:={\mathrm}{Gal}(\hat{L}/L_{\infty})$, $H_{L,\infty}:={\mathrm}{Gal}(\overline{K}/\hat{L})$ $G_{L,p^{\infty}}:={\mathrm}{Gal}(\hat{L}/L_{p^{\infty}})$ and $\hat{G}_L:={\mathrm}{Gal}(\hat{L}/L)$. Furthermore, putting $L_{(s),p^{\infty}}=L_{(s)}L_{p^{\infty}}$, we define $\hat{G}_{L,s}={\mathrm}{Gal}(\hat{L}/L_{(s)})$ and $G_{L,s,p^{\infty}}:={\mathrm}{Gal}(\hat{L}/L_{(s), p^{\infty}})$. $$\xymatrix{ & & & & & \bar{K} \\ & & & & & \hat{L} \ar@{-}[u] \ar@{-}[u] \ar@/_1pc/@{-}[u]_{H_{L,\infty}} \\ & & & L_{p^{\infty}} \ar@/^1pc/@{-}[rru] ^{G_{L,p^{\infty}}} \ar@{-}[rru] & & & & \\ & & & & & L_{\infty} \ar@/_1pc/@{-}[uu] _{H_L} \ar@{-}[uu] \ar@/_4pc/@{-}[uuu] _{G_{L,\infty}} & & & & \\ & & & L \ar@{-}[rru] \ar@/^1pc/@{-}[rruuu] ^{\hat{G}_L} \ar@{-}[rruuu] \ar@/^6pc/@{-}[rruuuu] ^{G_L} \ar@{-}[uu] & & & & \\ }$$ Since $p>2$, it is known that $L_{(s),p^{\infty}}\cap L_{\infty}=L_{(s)}$ and thus $\hat{G}_{L,s}=G_{L,s,p^{\infty}}\rtimes H_{L,s}$. Furthermore, $G_{L,s,p^{\infty}}$ is topologically isomorphic to ${\mathbb}{Z}_p$. \[easylemma\] A natural map $G_{L,s,p^{\infty}}\to G_{K,s,p^{\infty}}$ defined by $g\mapsto g|_{\hat{K}}$ is bijective. By replacing $L_s$ with $L$, we may assume $s=0$. It suffices to prove $\hat{K}\cap L_{p^{\infty}}=K_{p^{\infty}}$. Since $G_{K,p^{\infty}}$ is isomorphic to ${\mathbb}{Z}_p$, we know that any finite subextension of $\hat{K}/K_{p^{\infty}}$ is of the form $K_{(s),p^{\infty}}$ for some $s\ge 0$. Assume that we have $\hat{K}\cap L_{p^{\infty}}\not=K_{p^{\infty}}$. Then we have $K_{(1)}\subset \hat{K}\cap L_{p^{\infty}}\subset L_{p^{\infty}}$. Thus $\pi_1$ is contained in $L_{p^{\infty}}\cap L_{\infty}=L$. However, since $L$ is unramified over $K$, this contradicts the fact that $\pi$ is a uniformizer of $L$. We fix a topological generator $\tau$ of $G_{K,p^{\infty}}$. We also denote by $\tau$ the pre-image of $\tau\in G_{K,p^{\infty}}$ for the bijection $G_{L,p^{\infty}}\simeq G_{K,p^{\infty}}$ of the above lemma. Note that $\tau^{p^s}$ is a topological generator of $G_{L,s,p^{\infty}}$. For any $g\in G_K$, we put $\underline{{\varepsilon}}(g)=g(\underline{\pi})/\underline{\pi}\in R$, and define $\underline{{\varepsilon}}:=\underline{{\varepsilon}}(\tilde{\tau})$. Here, $\tilde{\tau}\in G_K$ is any lift of $\tau\in \hat{G}_K$ and then $\underline{{\varepsilon}}(\tilde{\tau})$ is independent of the choice of the lift of $\tau$. With these notation, we also note that we have $g(u)=[\underline{{\varepsilon}}(g)]u$ (recall that ${\mathfrak{S}}$ is embedded in $W(R)$). An easy computation shows that $\tau(\underline{\pi})/\underline{\pi}=\tau^{p^s}(\underline{\pi_s})/\underline{\pi_s}=\underline{{\varepsilon}}$. Therefore, we have $\tau(u)/u=\tau^{p^s}(u_s)/u_s=[\underline{{\varepsilon}}]$. We put $t=-{\mathrm}{log}([\underline{{\varepsilon}}])\in A_{{\mathrm}{cris}}$. Denote by $\nu\colon W(R)\to W(\overline{k})$ a unique lift of the projection $R\to \overline{k}$, which extends to a map $\nu \colon B^+_{{\mathrm}{cris}}\to W(\overline{k})[1/p]$. For any subring $A\subset B^+_{{\mathrm}{cris}}$, we put $I_+A={\mathrm}{Ker}(\nu\ {\mathrm}{on}\ B^+_{{\mathrm}{cris}})\cap A$. For any integer $n\ge 0$, let $t^{\{n\}}:=t^{r(n)}\gamma_{\tilde{q}(n)}(\frac{t^{p-1}}{p})$ where $n=(p-1)\tilde{q}(n)+r(n)$ with $\tilde{q}(n)\ge 0,\ 0\le r(n) <p-1$ and $\gamma_i(x)=\frac{x^i}{i!}$ is the standard divided power. We define a subring ${\mathcal}{R}_{L_0,s}$ (resp. ${\mathcal}{R}_{L_0}$) of $B^+_{{\mathrm}{cris}}$ as below: $${\mathcal}{R}_{L_0,s}:=\{\sum^{\infty}_{i=0} f_it^{\{i\}}\mid f_i\in S_{L_0,s}\ {\mathrm}{and}\ f_i\to 0\ {\mathrm}{as}\ i\to \infty\}$$ $$({\rm resp}.\quad {\mathcal}{R}_{L_0}:=\{\sum^{\infty}_{i=0} f_it^{\{i\}}\mid f_i\in S_{L_0}\ {\mathrm}{and}\ f_i\to 0\ {\mathrm}{as}\ i\to \infty\}).$$ Put ${\widehat}{{\mathcal}{R}}_{L,s}={\mathcal}{R}_{L_0,s}\cap W(R)$ (resp. ${\widehat}{{\mathcal}{R}}_{L}={\mathcal}{R}_{L_0}\cap W(R)$) and $I_{+,L,s}=I_+{\widehat}{{\mathcal}{R}}_{L,s}$ (resp. $I_{+,L}=I_+{\widehat}{{\mathcal}{R}}_L$). By definitions, we have ${\mathcal}{R}_{L_0,0}={\mathcal}{R}_{L_0}$, ${\widehat}{{\mathcal}{R}}_{L,0}={\widehat}{{\mathcal}{R}}_{L}$ and $I_{+,L,0}=I_{+,L}$. Lemma 2.2.1 in [@Li2] shows that ${\widehat}{{\mathcal}{R}}_{L,s}$ $($resp. ${\mathcal}{R}_{L_0,s})$ is a ${\varphi}$-stable ${\mathfrak{S}}_{L,s}$-subalgebra of $W(R)$ $($resp. $B^+_{{\mathrm}{cris}})$, and $\nu$ induces ${\mathcal}{R}_{L_0,s}/I_+{\mathcal}{R}_{L_0,s}\simeq L_0$ and ${\widehat}{{\mathcal}{R}}_{L,s}/I_{+,L,s}\simeq S^{{\mathrm}{int}}_{L_0,s}/I_+S^{{\mathrm}{int}}_{L_0,s} \simeq {\mathfrak{S}}_{L,s}/I_+{\mathfrak{S}}_{L,s}\simeq W(k_L)$. Furthermore, ${\widehat}{{\mathcal}{R}}_{L,s}, I_{+,L,s}, {\mathcal}{R}_{L_0,s}$ and $I_+{\mathcal}{R}_{L_0,s}$ are $G_{L,s}$-stable, and $G_{L,s}$-actions on them factors through $\hat{G}_{L,s}$. For any torsion Kisin module ${\mathfrak{M}}$ over ${\mathfrak{S}}_{L,s}$, we equip ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}$ with a Frobenius by ${\varphi}_{{\widehat}{{\mathcal}{R}}_{L,s}}\otimes {\varphi}_{{\mathfrak{M}}}$. It is known that the natural map $ {\mathfrak{M}}\rightarrow {\widehat}{{\mathcal}{R}}_{L,s}\otimes_{{\varphi}, {\mathfrak{S}}_{L,s}} {\mathfrak{M}}$ given by $x\mapsto 1\otimes x$ is an injection (cf. [@Oz1 Corollary 2.12]). By this injection, we regard ${\mathfrak{M}}$ as a ${\varphi}({\mathfrak{S}}_{L,s})$-stable submodule of ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}$. \[Liumod\] A [*free*]{} (resp. [*torsion*]{}) [*$({\varphi}, \hat{G}_{L,s})$-module of height*]{} $\le r$ over ${\mathfrak{S}}_{L,s}$ is a triple $\hat{{\mathfrak{M}}}=({\mathfrak{M}}, {\varphi}_{{\mathfrak{M}}}, \hat{G}_{L,s})$ where 1. $({\mathfrak{M}}, {\varphi}_{{\mathfrak{M}}})$ is a free (resp. torsion) Kisin module of height $\le r$ over ${\mathfrak{S}}_{L,s}$, 2. $\hat{G}_{L,s}$ is an ${\widehat}{{\mathcal}{R}}_{L,s}$-semilinear $\hat{G}_{L,s}$-action on ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi}, {\mathfrak{S}}_{L,s}} {\mathfrak{M}}$ which induces a continuous $G_{L,s}$-action on $W({\mathrm}{Fr}R)\otimes_{{\varphi}, {\mathfrak{S}}_{L,s}} {\mathfrak{M}}$ for the weak topology[^2], 3. the $\hat{G}_{L,s}$-action commutes with ${\varphi}_{{\widehat}{{\mathcal}{R}}_{L,s}}\otimes {\varphi}_{{\mathfrak{M}}}$, 4. ${\mathfrak{M}}\subset ({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}})^{H_L}$, 5. $\hat{G}_{L,s}$ acts on the $W(k_L)$-module $({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}})/ I_{+,L,s}({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}})$ trivially. A morphism between two $({\varphi}, \hat{G}_{L,s})$-modules $\hat{{\mathfrak{M}}}_1=({\mathfrak{M}}_1, {\varphi}_1, \hat{G})$ and $\hat{{\mathfrak{M}}}_2=({\mathfrak{M}}_2, {\varphi}_2, \hat{G})$ is a morphism $f\colon {\mathfrak{M}}_1\to {\mathfrak{M}}_2$ of ${\varphi}$-modules over ${\mathfrak{S}}_{L,s}$ such that ${\widehat}{{\mathcal}{R}}_{L,s}\otimes f\colon {\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}_1\to {\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}_2$ is $\hat{G}_{L,s}$-equivariant. We denote by ${\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s}}$ (resp. ${\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s,\infty}}$) the category of free (resp. torsion) $({\varphi}, \hat{G}_{L,s})$-modules of height $\le r$ over ${\mathfrak{S}}_{L,s}$. We often regard ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}$ as a $G_{L,s}$-module via the projection $G_{L,s}\twoheadrightarrow \hat{G}_{L,s}$. For any free or torsion $({\varphi}, \hat{G}_{L,s})$-module $\hat{{\mathfrak{M}}}$ over ${\mathfrak{S}}_{L,s}$, we define a ${\mathbb}{Z}_p[G_{L,s}]$-module $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ by $$\begin{aligned} \hat{T}_{L,s}(\hat{{\mathfrak{M}}})= \left\{ \begin{array}{ll} {\mathrm}{Hom}_{{\widehat}{{\mathcal}{R}}_{L,s},{\varphi}} ({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}, W(R)) \hspace{21mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm free}, \cr {\mathrm}{Hom}_{{\widehat}{{\mathcal}{R}}_{L,s},{\varphi}} ({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}, {\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p}W(R)) \hspace{3.5mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm torsion}. \end{array} \right.\end{aligned}$$ Here, $G_{L,s}$ acts on $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ by $(\sigma.f)(x)=\sigma(f(\sigma^{-1}(x)))$ for $\sigma\in G_{L,s},\ f\in \hat{T}_{L,s}(\hat{{\mathfrak{M}}}),\ x\in {\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}}{\mathfrak{M}}$.\ Then, there exists a natural $G_{L,\infty}$-equivariant map $$\theta_{L,s}\colon T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}})\to \hat{T}_{L,s}(\hat{{\mathfrak{M}}})$$ defined by $\theta(f)(a\otimes x)=a{\varphi}(f(x))$ for $f\in T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}}),\ a\in {\widehat}{{\mathcal}{R}}_{L,s}, x\in {\mathfrak{M}}$. We have The map $\theta_{L,s}$ is an isomorphism of ${\mathbb}{Z}_p[G_{L,\infty}]$-modules. [**Convention:**]{} For simplicity, if $L=K$, then we often omit the subscript “$L$” from various notations (e.g. “a $({\varphi}, \hat{G}_{K,s})$-module” = “a $({\varphi}, \hat{G}_s)$-module”, ${\mathrm}{Mod}^{r,\hat{G}_{K,s}}_{/{\mathfrak{S}}_{K,s}} ={\mathrm}{Mod}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$, ${\mathrm}{Mod}^{r,\hat{G}_{K,s}}_{/{\mathfrak{S}}_{K,s,\infty}} ={\mathrm}{Mod}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{s,\infty}}$, $\hat{T}_{K,s}=\hat{T}_s$, $\theta_{K,s}=\theta_s$). Furthermore, if $s=0$, we often omit the subscript “$s$” from various notations (e.g. ${\mathrm}{Mod}^{r,\hat{G}_{L,0}}_{/{\mathfrak{S}}_{L,0}} ={\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}$, ${\mathrm}{Mod}^{r,\hat{G}_{L,0}}_{/{\mathfrak{S}}_{L,0,\infty}} ={\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_{L,\infty}}$, $\hat{T}_{L,0}=\hat{T}_L$, ${\mathrm}{Mod}^{r,\hat{G}_{K,0}}_{/{\mathfrak{S}}_{K,0}} ={\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}}$, “a $({\varphi}, \hat{G}_{K,0})$-module” = “a $({\varphi}, \hat{G})$-module”, $\hat{T}_{K,0}=\hat{T}$, $\theta_{K,0}=\theta$).\ Let ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Q}_p}(G_{L,s})$ (resp. ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathbb}{Q}_p}(G_{L,s})$, resp. ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_{L,s})$, resp. ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathbb}{Z}_p}(G_{L,s})$) be the categories of semi-stable ${\mathbb}{Q}_p$-representations of $G_{L,s}$ with Hodge-Tate weights in $[0,r]$ (resp. the categories of crystalline ${\mathbb}{Q}_p$-representations of $G_{L,s}$ with Hodge-Tate weights in $[0,r]$, resp. the categories of lattices in semi-stable ${\mathbb}{Q}_p$-representations of $G_{L,s}$ with Hodge-Tate weights in $[0,r]$, resp. the categories of lattices in crystalline ${\mathbb}{Q}_p$-representations of $G_{L,s}$ with Hodge-Tate weights in $[0,r]$). There exists ${\mathfrak{t}}\in W(R)\smallsetminus pW(R)$ such that ${\varphi}({\mathfrak{t}})=pE(0)^{-1}E(u){\mathfrak{t}}$. Such ${\mathfrak{t}}$ is unique up to units of ${\mathbb}{Z}_p$ (cf. [@Li2 Example 2.3.5]). Now we define the full subcategory ${\mathrm}{Mod}^{r,\hat{G},{\mathrm}{cris}}_{/{\mathfrak{S}}}$ (resp. ${\mathrm}{Mod}^{r,\hat{G},{\mathrm}{cris}}_{/{\mathfrak{S}}_{\infty}}$) of ${\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}}$ (resp. ${\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}_{\infty}}$) consisting of objects $\hat{{\mathfrak{M}}}$ which satisfy the following condition; $ \tau(x)-x\in u^p{\varphi}({\mathfrak{t}})(W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}) $ for any $x\in {\mathfrak{M}}$. The following results are important properties for the functor $\hat{T}_{L,s}$. \[Thm1\] $(1)$ [([@Li2 Theorem 2.3.1 (2)])]{}The functor $\hat{T}$ induces an anti-equivalence of categories between ${\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}}$ and ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K)$. $(2)$ [([@GLS Proposition 5.9], [@Oz2 Theorem 19])]{}  The functor $\hat{T}$ induces an anti-equivalence of categories between ${\mathrm}{Mod}^{r,\hat{G},{\mathrm}{cris}}_{/{\mathfrak{S}}}$ and ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathbb}{Z}_p}(G_K)$. $(3)$ [([@Oz1 Corollary 2.8 and 5.34])]{}The functor $\hat{T}_{L,s}\colon {\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s,\infty}} \to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{L,s})$ is exact and faithful. Furthermore, it is full if $er<p-1$. $({\varphi}, \hat{G})$-modules, Breuil modules and filtered $({\varphi},N)$-modules {#relations} ----------------------------------------------------------------------------------- We recall some relations between Breuil modules and $({\varphi},\hat{G})$-modules. Here we give a rough sketch only. For more precise information, see [@Br1 Section 6], [@Li1 Section 5] and the proof of [@Li2 Theorem 2.3.1 (2)]. Let $\hat{{\mathfrak{M}}}$ be a free $({\varphi},\hat{G}_{L,s})$-module over ${\mathfrak{S}}_{L,s}$. If we put ${\mathcal}{D}:=S_{L_0,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}$, then ${\mathcal}{D}$ has a structure of a Breuil module over $S_{L_0,s}$ which corresponds to the semi-stable representation ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ of $G_{L,s}$ (for the definition and properties of Breuil modules, see [@Br1]). Thus ${\mathcal}{D}$ is equipped with a Frobenius ${\varphi}_{{\mathcal}{D}}(={\varphi}_{S_{L_0,s}}\otimes {\varphi}_{{\mathfrak{M}}})$, a decreasing filtration $({\mathrm}{Fil}^i{\mathcal}{D})_{i\ge 0}$ of $S_{L_0,s}$-submodules of ${\mathcal}{D}$ and a $L_0$-linear monodromy operator $N\colon {\mathcal}{D}\to {\mathcal}{D}$ which satisfy certain properties (for example, Griffiths transversality). Putting $D={\mathcal}{D}/I_+S_{L_0,s}{\mathcal}{D}$, we can associate a filtered $({\varphi},N)$-module over $L_{(s)}$ as following: ${\varphi}_{D}:={\varphi}_{{\mathcal}{D}}\ {\mathrm}{mod}\ I_+S_{L_0,s}{\mathcal}{D}$, $N_D:= N_{{\mathcal}{D}}\ {\mathrm}{mod}\ I_+S_{L_0,s}{\mathcal}{D}$ and ${\mathrm}{Fil}^iD_{L_{(s)}}:=f_{\pi_s}(Fil^i({\mathcal}{D}))$. Here, $f_{\pi_s}\colon {\mathcal}{D}\to D_{L_{(s)}}$ is the projection defined by ${\mathcal}{D}\twoheadrightarrow {\mathcal}{D}/{\mathrm}{Fil}^1S_{L_0,s}{\mathcal}{D}\simeq D_{L_{(s)}}$. Proposition 6.2.1.1 of [@Br1] implies that there exists a unique ${\varphi}$-compatible section $s\colon D\hookrightarrow {\mathcal}{D}$ of ${\mathcal}{D}\twoheadrightarrow D$. By this embedding, we regard $D$ as a submodule of ${\mathcal}{D}$. Then we have $N_{{\mathcal}{D}}|_{D}=N_D$ and $N_{{\mathcal}{D}}=N_{S_{L_0,s}}\otimes {\mathrm}{Id}_D + {\mathrm}{Id}_{S_{L_0,s}}\otimes N_D$ under the identification ${\mathcal}{D}=S_{L_0,s} \otimes_{L_{(s)}} D$. The $G_{L,s}$-action on ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}}{\mathfrak{M}}$ extends to $B^+_{{\mathrm}{cris}}\otimes_{{\widehat{\mathcal{R}}}_{L,s}} ({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}}{\mathfrak{M}}) \simeq B^+_{{\mathrm}{cris}}\otimes_{S_{L_0,s}}{\mathcal}{D}$. This action is in fact explicitly written as follows: $$\label{explicit} g(a\otimes x)=\sum^{\infty}_{i=0}g(a)\gamma_i (-{\mathrm}{log}(\frac{g[\underline{\pi_s}]}{[\underline{\pi_s}]}))\otimes N^i_{{\mathcal}{D}}(x)\quad {\rm for}\ g\in G_{L,s}, a\in B^+_{{\mathrm}{cris}}, x\in {\mathcal}{D}.$$ By this explicit formula, we can obtain an easy relation between $N_{{\mathcal}{D}}$ and $\tau^{p^s}$-action on $\hat{{\mathfrak{M}}}$ as follows: first we recall that $t=-{\mathrm}{log}(\tau([\underline{\pi}])/[\underline{\pi}]) =-{\mathrm}{log}(\tau^{p^s}([\underline{\pi_s}])/[\underline{\pi_s}])$. By the formula, for any $n\ge 0$ and $x\in {\mathcal}{D}$, an induction on $n$ shows that we have $$(\tau^{p^s}-1)^n(x)=\sum^{\infty}_{m=n}(\sum_{i_1+\cdots i_n=m, i_j\ge 0}\frac{m!}{i_1!\cdots i_n!}) \gamma_m(t)\otimes N^m_{{\mathcal}{D}}(x) \in B^{+}_{{\mathrm}{cris}}\otimes_{S_{L_0,s}} {\mathcal}{D}$$ and in particular we see $\frac{(\tau^{p^s}-1)^n}{n}(x)\to 0$ $p$-adically as $n\to \infty$. Hence we can define $${\mathrm}{log}(\tau^{p^s})(x):= \sum^{\infty}_{n=1}(-1)^{n-1}\frac{(\tau^{p^s}-1)^n}{n}(x) \in B^{+}_{{\mathrm}{cris}}\otimes_{S_{L_0,s}} {\mathcal}{D}.$$ It is not difficult to check the equation $$\label{eq1} {\mathrm}{log}(\tau^{p^s})(x)=t\otimes N_{{\mathcal}{D}}(x).$$ Base changes for Kisin modules ------------------------------ Let ${\mathfrak{M}}$ be a free or torsion Kisin module of height $\le r$ over ${\mathfrak{S}}_L$ (resp. over ${\mathfrak{S}}$). We put ${\mathfrak{M}}_{L,s}={\mathfrak{S}}_{L,s}\otimes_{{\mathfrak{S}}_L} {\mathfrak{M}}$ (resp. ${\mathfrak{S}}_L={\mathfrak{S}}_L\otimes_{{\mathfrak{S}}} {\mathfrak{M}}$) and equip ${\mathfrak{M}}_{L,s}$ (resp. ${\mathfrak{M}}_L$) with a Frobenius by ${\varphi}={\varphi}_{{\mathfrak{S}}_{L,s}}\otimes {\varphi}_{{\mathfrak{M}}}$ (resp. ${\varphi}={\varphi}_{{\mathfrak{S}}_L}\otimes {\varphi}_{{\mathfrak{M}}}$). Then it is not difficult to check that ${\mathfrak{M}}_{L,s}$ (resp. ${\mathfrak{M}}_L$) is a free or torsion Kisin module of height $\le r$ over ${\mathfrak{S}}_{L,s}$ (resp. over ${\mathfrak{S}}_L$) (here we recall that $E_s(u_s)=E(u^{p^s}_s)=E(u)$). Hence we obtained natural functors $${\mathrm}{Mod}^r_{/{\mathfrak{S}}_L}\to {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}} \quad {\rm and}\quad {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,\infty}}\to {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s,\infty}}$$ $${\rm (resp.}\quad {\mathrm}{Mod}^r_{/{\mathfrak{S}}}\to {\mathrm}{Mod}^r_{/{\mathfrak{S}}_L} \quad {\rm and}\quad {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,\infty}}).$$ By definition, we immediately see that we have $T_{{\mathfrak{S}}_L}({\mathfrak{M}})\simeq T_{{\mathfrak{S}}_{L,s}}({\mathfrak{M}}_{L,s})$ (resp. $T_{{\mathfrak{S}}}({\mathfrak{M}})|_{G_{L_{\infty}}}\simeq T_{{\mathfrak{S}}_L}({\mathfrak{M}}_L)$). In particular, it follows from Proposition \[Kisinfunctor\] (1) that the following holds: \[basechange1:Kisin\] The functor ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_L}\to {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{L,s}}$ is fully faithful. Base changes for $({\varphi},\hat{G})$-modules ---------------------------------------------- Let $\hat{{\mathfrak{M}}}$ be a free or torsion $({\varphi},\hat{G}_L)$-module (resp. $({\varphi},\hat{G})$-module) of height $\le r$ over ${\mathfrak{S}}_L$ (resp. over ${\mathfrak{S}}$). The $G_{L,s}$ action on ${\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}$ (resp. the $G_L$ action on ${\widehat{\mathcal{R}}}\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$) extends to ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\widehat{\mathcal{R}}}_L}({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}) \simeq {\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_{L,s}} {\mathfrak{M}}_{L,s}$ (resp.  ${\widehat{\mathcal{R}}}_L\otimes_{{\widehat{\mathcal{R}}}}({\widehat{\mathcal{R}}}\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}) \simeq {\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$), which factors through $\hat{G}_{L,s}$ (resp. $\hat{G}_L$). Then it is not difficult to check that ${\mathfrak{M}}_{L,s}$ (resp. ${\mathfrak{M}}_L$) has a structure of a $({\varphi},\hat{G}_{L,s})$-module (resp. $({\varphi},\hat{G}_L)$-module). Hence we obtained natural functors $${\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}\to {\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s}} \quad {\rm and}\quad {\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_{L,\infty}}\to {\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s,\infty}}$$ $${\rm (resp.}\quad {\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}}\to {\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L} \quad {\rm and}\quad {\mathrm}{Mod}^{r,\hat{G}}_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_{L,\infty}}).$$ By definition, we immediately see that we have $\hat{T}_L(\hat{{\mathfrak{M}}})|_{G_{L,s}}\simeq \hat{T}_{L,s}(\hat{{\mathfrak{M}}}_{L,s})$ (resp. $\hat{T}(\hat{{\mathfrak{M}}})|_{G_L}\simeq \hat{T}_L(\hat{{\mathfrak{M}}}_L)$). Similar to Proposition \[basechange1:Kisin\], we can prove the following. The functor ${\mathrm}{Mod}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}\to {\mathrm}{Mod}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s}} $ is fully faithful. The proposition immediately follows from the full faithfulness property of Theorem \[Thm1\] (1) and the lemma below. \[totst\] Let $K'$ is a finite totally ramified extension of $K$. Then the restriction functor from the category of semi-stable ${\mathbb}{Q}_p$-representations of $G_K$ into the category of semi-stable ${\mathbb}{Q}_p$-representations of $G_{K'}$ is fully faithful. Let $V$ and $V'$ be semi-stable ${\mathbb}{Q}_p$-representations of $G_K$ and let $f\colon V\to V'$ be a $G_{K'}$-equivariant homomorphism. Considering the morphism of filtered $({\varphi}, N)$-modules over $K'$ corresponding to $f$, we can check without difficulty that $f$ is in fact a morphism of filtered $({\varphi}, N)$-modules over $K$. This is because $K'$ is totally ramified over $K_0$ as same as $K$. This gives the desired result. Variants of free $({\varphi},\hat{G})$-modules ============================================== In this section, we define some variant notions of $({\varphi},\hat{G})$-modules. We continue to use same notation as in the previous section. In particular, $p$ is odd. Definitions {#vardef} ----------- We start with some definitions which are our main concern in this and the next section. \[varLiumod\] We define the category ${{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$ (resp. ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$) as follows. An object is a triple $\hat{{\mathfrak{M}}}=({\mathfrak{M}}, {\varphi}_{{\mathfrak{M}}}, \hat{G}_{L,s})$ where 1. $({\mathfrak{M}}, {\varphi}_{{\mathfrak{M}}})$ is a free Kisin module of height $\le r$ over ${\mathfrak{S}}_L$, 2. $\hat{G}_{L,s}$ is an ${\widehat}{{\mathcal}{R}}_L$-semilinear $\hat{G}_{L,s}$-action on ${\widehat{\mathcal{R}}}_L\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}$ (resp. an ${\widehat}{{\mathcal}{R}}_{L,s}$-semilinear $\hat{G}_{L,s}$-action on ${\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}$) which induces a continuous $G_{L,s}$-action on $W({\mathrm}{Fr}R)\otimes_{{\varphi}, {\mathfrak{S}}_L} {\mathfrak{M}}$ for the weak topology, 3. the $\hat{G}_{L,s}$-action commutes with ${\varphi}_{{\widehat}{{\mathcal}{R}}_L}\otimes {\varphi}_{{\mathfrak{M}}}$ (resp. ${\varphi}_{{\widehat}{{\mathcal}{R}}_{L,s}}\otimes {\varphi}_{{\mathfrak{M}}}$), 4. ${\mathfrak{M}}\subset ({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})^{H_L}$ (resp. ${\mathfrak{M}}\subset ({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})^{H_L}$), 5. $\hat{G}_{L,s}$ acts on the $W(k_L)$-module $({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})/ I_{+,L}({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})$ (resp. $({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})/ I_{+,L,s}({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}})$) trivially. Morphisms are defined by the obvious way. By replacing “free” of (1) with “torsion”[^3], we define the category ${{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$ (resp. ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$). For any object $\hat{{\mathfrak{M}}}$ of ${{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$ or ${{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$, we define a ${\mathbb}{Z}_p[G_{L,s}]$-module $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ by $$\begin{aligned} \hat{T}_{L,s}(\hat{{\mathfrak{M}}})= \left\{ \begin{array}{ll} {\mathrm}{Hom}_{{\widehat}{{\mathcal}{R}}_L,{\varphi}} ({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}, W(R)) \hspace{21mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm free}, \cr {\mathrm}{Hom}_{{\widehat{\mathcal{R}}}_L,{\varphi}} ({\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}, {\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p}W(R)) \hspace{3.5mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm torsion}. \end{array} \right.\end{aligned}$$ Here, $G_{L,s}$ acts on $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ by $(\sigma.f)(x)=\sigma(f(\sigma^{-1}(x)))$ for $\sigma\in G_{L,s},\ f\in \hat{T}_{L,s}(\hat{{\mathfrak{M}}}),\ x\in {\widehat{\mathcal{R}}}_L\otimes_{{\varphi},{\mathfrak{S}}_L}{\mathfrak{M}}$. Similar to the above, for any object $\hat{{\mathfrak{M}}}$ of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$ or ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$, we define a ${\mathbb}{Z}_p[G_{L,s}]$-module $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ by $$\begin{aligned} \hat{T}_{L,s}(\hat{{\mathfrak{M}}})= \left\{ \begin{array}{ll} {\mathrm}{Hom}_{{\widehat{\mathcal{R}}}_{L,s},{\varphi}} ({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}, W(R)) \hspace{21mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm free}, \cr {\mathrm}{Hom}_{{\widehat}{{\mathcal}{R}}_{L,s},{\varphi}} ({\widehat{\mathcal{R}}}_{L,s}\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}, {\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p}W(R)) \hspace{3.5mm} {\rm if}\ {\mathfrak{M}}\ {\rm is}\ {\rm torsion}. \end{array} \right.\end{aligned}$$ On the other hand, we obtain functors $ {{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}\to {{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}\to {{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s}} $ and $ {{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_{L,\infty}}\to {{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}\to {{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s,\infty}} $ by natural manners and it is readily seen that these functors are compatible with $\hat{T}_L$ and $\hat{T}_{L,s}$. In particular, the essential images of the functors $\hat{T}_{L,s}$ on ${{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$ and ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_L}$ has values in ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_{L,s})$ since we have an equivalence of categories $\hat{T}_{L,s}\colon {{\mathrm}{Mod}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,s}} \overset{\sim}{\rightarrow} {\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_{L,s})$ by Theorem \[Thm1\]. In the rest of this section, we study free cases. We leave studies for torsion cases to the next section.\ [**Convention:**]{} For simplicity, if $L=K$, then we often omit the subscript “$L$” from various notations (e.g. ${{\mathrm}{Mod}}^{r,\hat{G}_{K,s}}_{/{\mathfrak{S}}_K}={{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}, {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{K,s}}_{/{\mathfrak{S}}_K}={\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$). Furthermore, if $s=0$, we often omit the subscript “$s$” from various notations (e.g. ${{\mathrm}{Mod}}^{r,\hat{G}_{L,0}}_{/{\mathfrak{S}}_L}={{\mathrm}{Mod}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}, {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,0}}_{/{\mathfrak{S}}_{L,0}}={\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_L}_{/{\mathfrak{S}}_L}$).\ The functors $ {{\mathrm}{Mod}}^{r,\hat{G}}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s} $ ------------------------------------------------------------------------------------------------------------------ Now we consider the functors $ {{\mathrm}{Mod}}^{r,\hat{G}}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}. $ At first, by Proposition \[basechange1:Kisin\], we see that the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$ is fully faithful. It follows from this fact and Theorem \[Thm1\] (1) that the functor $\hat{T}_s\colon {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}} \to {\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_s)$ is fully faithful. It is clear that the functor ${{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$ is fully faithful and thus so is $\hat{T}_s\colon {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}} \to {\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_s)$. Combining this with Theorem \[Thm1\] (1) and Lemma \[totst\], we obtain that the functor ${{\mathrm}{Mod}}^{r,\hat{G}}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$ is also fully faithful. Furthermore, we prove the following. \[equal\] The functor ${{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$ is an equivalence of categories. Summary, we obtained the following commutative diagram. $\displaystyle \xymatrix{ {{\mathrm}{Mod}}^{r,\hat{G}}_{/{\mathfrak{S}}} \ar@{^{(}->}[r] \ar[d]_{\wr} \ar^{\hat{T}}[d] & {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}} \ar^{\sim}[r] \ar@{^{(}->}[rrd] \ar^{\hat{T}_s}[rrd] & {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}} \ar@{^{(}->}[r] \ar@{^{(}->}[rd] \ar^{\hat{T}_s}[rd] & {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s} \ar[d]_{\wr} \ar^{\hat{T}_s}[d]\\ {\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_K) \ar@{^{(}->}[rrr] \ar^{{\mathrm}{restriction}}[rrr] & & & {\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathbb}{Z}_p}(G_s). }$ The functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\hookrightarrow {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$ may not be possibly essentially surjective. In fact, under some conditions, there exists a representation of $G_K$ which is crystalline over $K_s$ but not of finite height. For more precise information, see [@Li2 Example 4.2.3]. Before a proof of Proposition \[equal\], we give an explicit formula such as (\[explicit\]) for an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$. The argument below follows the method of [@Li2]. Let $\hat{{\mathfrak{M}}}$ be an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$. Let $\hat{{\mathfrak{M}}}_s$ be the image of $\hat{{\mathfrak{M}}}$ for the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$. Put ${\mathcal}{D}=S_{K_0}\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$ and also put ${\mathcal}{D}_s=S_{K_0,s}\otimes_{{\varphi},{\mathfrak{S}}_s} {\mathfrak{M}}_s =S_{K_0,s}\otimes_{S_{K_0}} {\mathcal}{D}$. Then ${\mathcal}{D}_s$ has a structure of a Breuil module and also $D={\mathcal}{D}_s/I_+S_{K_0,s}{\mathcal}{D}_s$ has a structure of a filtered $({\varphi},N)$-module corresponding to ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}}_s)$ (see subsection \[relations\]), which is isomorphic to ${\mathcal}{D}/I_+S_{K_0}{\mathcal}{D}$ as a ${\varphi}$-module over $K_0$. By [@Li1 Lemma 7.3.1], we have a unique ${\varphi}$-compatible section $D\hookrightarrow {\mathcal}{D}$ and we regard $D$ as a submodule of ${\mathcal}{D}\subset {\mathcal}{D}_s$ by this section. Then we have ${\mathcal}{D}=S_{K_0} \otimes_{K_0} D$ and ${\mathcal}{D}_s=S_{K_0,s} \otimes_{K_0} D$. By the explicit formula (\[explicit\]) for $\hat{{\mathfrak{M}}}_s$, we know that $$\hat{G}_s(D)\subset (K_0[\![t]\!]\cap {\mathcal}{R}_{K_0,s})\otimes_{K_0} D.$$ Hence, taking any $K_0$-basis $e_1,\dots ,e_d$ of $D$, there exist $A_s(t)\in M_{d\times d}(K_0[\![t]\!])$ such that $\tau^{p^s}(e_1,\cdots ,e_d)=(e_1,\dots ,e_d)A_s(t)$. Since $A_s(0)={\mathrm}{I}_d$, we see that ${\mathrm}{log}(A_s(t))\in M_{d\times d}(K_0[\![t]\!])$ is well-defined. On the other hand, choose $g_0\in G_s$ such that $\chi_p(g_0)\not=1$, where $\chi_p$ is the $p$-adic cyclotomic character. Since $g_0\tau^{p^s}=(\tau^{p^s})^{\chi_p(g_0)}g_0$, we have $A_s(\chi_p(g_0)t)=A_s(t)^{\chi_p(g_0)}$ and thus we also have ${\mathrm}{log}(A_s(\chi_p(g_0)t))=\chi_p(g_0){\mathrm}{log}(A_s(t))$. Since ${\mathrm}{log}(A_s(0))={\mathrm}{log}(I_d)=0$, we can write ${\mathrm}{log}(A_s(t))$ as $tB(t)$ for some $B(t)\in M_{d\times d}(K_0[\![t]\!])$. Then we have $\chi_p(g_0)tB(\chi_p(g_0)t)=\chi_p(g_0)tB(t)$, that is, $B(\chi_p(g_0)t)=B(t)$. Hence the assumption $\chi_p(g_0)\not= 1$ implies that $B(t)$ is a constant. Putting $N_s=B(t)\in M_{d\times d}(K_0)$, we obtain $$\tau^{p^s}(e_1,\cdots ,e_d)=(e_1,\cdots ,e_d)(\sum_{i\ge 0}N^i_s\gamma_i(t)).$$ Now we define $N_D\colon D\to D$ by $N(e_1,\cdots ,e_d)=(e_1,\cdots ,e_d)p^{-s}N_s$ and also define $N_{{\mathcal}{D}}:=N_{S_{K_0}}\otimes {\mathrm}{Id}_D+{\mathrm}{Id}_{S_{K_0}}\otimes N_D$. (Note that we have $N_D{\varphi}_D=p{\varphi}_D N_D$ and thus $N_D$ is nilpotent.) It is a routine work to check the following: $$\label{explicit''} g(a\otimes x)=\sum^{\infty}_{i=0}g(a)\gamma_i (-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))\otimes N^i_D(x)\quad {\rm for}\ g\in G_s, a\in B^+_{{\mathrm}{cris}}, x\in D.$$ Since we have $$\label{easyeq} g(f)=\sum_{i\ge 0} \gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))N^i_{S_{K_0}}(f)$$ for any $g\in G_K$ and $f\in S_{K_0}$, we obtain the following explicit formula: $$\label{explicit'} g(a\otimes x)=\sum^{\infty}_{i=0}g(a)\gamma_i (-{\mathrm}{log}([\underline{{\varepsilon}}(g)]))\otimes N^i_{{\mathcal}{D}}(x)\quad {\rm for}\ g\in G_s, a\in B^+_{{\mathrm}{cris}}, x\in {\mathcal}{D}.$$ In particular, as in subsection \[relations\], we can show that $$\label{eq2} {\mathrm}{log}(\tau^{p^s})(x)=p^st\otimes N_{{\mathcal}{D}}(x)$$ for any $x\in {\mathcal}{D}$. We continue to use the above notation. It suffices to prove that the $G_s$-action on ${\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$ preserves ${\widehat{\mathcal{R}}}\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}$. Take any $g\in G_s$. We know that $g({\mathfrak{M}})\subset {\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}\subset W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$. Hence it is enough to prove that $g({\mathcal}{D})\in {\mathcal}{R}_{K_0}\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$. Let $s\in S^{{\mathrm}{int}}_{K_0}$ and $y\in D$ and put $x=s\otimes y\in S^{{\mathrm}{int}}_{K_0}\otimes_{W(k)} D={\mathcal}{D}$. By (\[explicit”\]) or (\[explicit’\]), we have $$g(x)=\sum_{i\ge 0}\sum_{0\le j\le i} \gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)])) \binom{i}{j}N^{i-j}_{S_{K_0}}(s)\otimes N^j_D(y)$$ On the other hand, we know that $N_D$ is nilpotent, that is, there exists $j_0>0$ such that $N^{j_0}_D=0$. Then we obtain $$g(x)=\sum_{0\le j\le j_0} \sum^{\infty}_{i=j} \gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)])) \binom{i}{j}N^{i-j}_{S_{K_0}}(s)\otimes N^j_D(y).$$ Therefore, it suffices to show that $\sum^{\infty}_{i=j} \gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)])) \binom{i}{j}N^{i-j}_{S_{K_0}}(s)$ is contained in ${\mathcal}{R}_{K_0}$ for each $0\le j\le j_0$. Taking $\alpha(g)\in {\mathbb}{Z}_p$ such that ${\mathrm}{log}([\underline{{\varepsilon}}(g)])=-\alpha(g)t$, we have $$\sum^{\infty}_{i=j} \gamma_i(-{\mathrm}{log}([\underline{{\varepsilon}}(g)])) \binom{i}{j}N^{i-j}_{S_{K_0}}(s) = \sum^{\infty}_{i=j} (\alpha(g))^i\frac{\tilde{q}(i)!p^{\tilde{q}(i)}}{i!} \binom{i}{j}N^{i-j}_{S_{K_0}}(s)t^{\{i\}}.$$ Since $\frac{\tilde{q}(i)p^{\tilde{q}(i)}}{i!}\to 0$ ($p$-adically) as $i\to \infty$, we finish a proof. Relations with crystalline representations ------------------------------------------ We know that ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}})$ is semi-stable over $K_s$ for any object $\hat{{\mathfrak{M}}}$ of ${{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$ or ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$. This subsection is devoted to prove a criterion, for $\hat{{\mathfrak{M}}}$, that describes when ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}})$ becomes crystalline. Following [@Fo2 Section 5] we set $I^{[m]}B^+_{{\mathrm}{cris}}:=\{x\in B^+_{{\mathrm}{cris}} \mid {\varphi}^n(x) \in {\mathrm}{Fil}^mB^+_{{\mathrm}{cris}}\ {\rm for\ all}\ n\ge0 \}$. For any subring $A\subset B^+_{{\mathrm}{cris}}$, we put $I^{[m]}A=A\cap I^{[m]}B^+_{{\mathrm}{cris}}$. Furthermore, we also put $I^{[m+]}A=I^{[m]}A.I_+A$ (here, $I_+A$ is defined in Subsection \[Liumodule:section\]). By [@Fo2 Proposition 5.1.3] and the proof of [@Li2 Lemma 3.2.2], we know that $I^{[m]}W(R)$ is a principal ideal which is generated by ${\varphi}({\mathfrak{t}})^m$. Now we recall Theorem \[Thm1\] (2): if ${\mathfrak{M}}$ is an object of ${{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$, then ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}})$ is crystalline if and only if $\tau^{p^s}(x)-x\in u_s^p(I^{[1]}W(R)\otimes_{{\varphi},{\mathfrak{S}}_s} {\mathfrak{M}})$ for any $x\in {\mathfrak{M}}$. However, if such ${\mathfrak{M}}$ descends to a Kisin module over ${\mathfrak{S}}$, then we can show the following. \[cris\] Let $\hat{{\mathfrak{M}}}$ be an object of ${{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$ or ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$. Then the following is equivalent: 1. ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}})$ is crystalline, 2. $\tau^{p^s}(x)-x\in u^p(I^{[1]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}})$ for any $x\in {\mathfrak{M}}$, 3. $\tau^{p^s}(x)-x\in I^{[1+]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$ for any $x\in {\mathfrak{M}}$. \(1) $\Rightarrow$ (2): The proof here mainly follows that of [@GLS Proposition 4.7]. We may suppose $\hat{{\mathfrak{M}}}$ is an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$. Put ${\mathcal}{D}=S_{K_0}\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$ and $D={\mathcal}{D}/I_+S_{K_0} {\mathcal}{D}$ as in the previous subsection. We fix a ${\varphi}({\mathfrak{S}})$-basis $(\hat{e}_1,\dots ,\hat{e}_d)$ of ${\mathfrak{M}}\subset {\mathcal}{D}$ and denote by $(e_1,\dots ,e_d)$ the image of $(\hat{e}_1,\dots ,\hat{e}_d)$ for the projection ${\mathcal}{D}\to D$. Then $(e_1,\dots ,e_d)$ is a $K_0$-basis of $D$. As described before the proof of Proposition \[equal\], we regard $D$ as a ${\varphi}$-stable submodule of ${\mathcal}{D}$, and we have $N_D\colon D\to D$ and $N_{{\mathcal}{D}}\colon D_{{\mathcal}{D}}\to D_{{\mathcal}{D}}$. Now we consider a matrix $X\in GL_{d\times d}(S_{K_0})$ such that $(\hat{e}_1,\dots ,\hat{e}_d)=(e_1,\dots ,e_d)X$. We define $\tilde{S}=W(k)[\![u^p, u^{ep}/p]\!]$ as in Section 4 of [@GLS], which is a sub $W(k)$-algebra of $S^{{\mathrm}{int}}_{K_0}$ with the property $N_{S_{K_0}}(\tilde{S})\subset u^p\tilde{S}$. By an easy computation we have $U=X^{-1}BX+X^{-1}N_{S_{K_0}}(X)$. Here, $B\in M_{d\times d}(K_0)$ and $U\in M_{d\times d}(S_{K_0})$ are defined by $N_D(e_1,\dots ,e_d)=(e_1,\dots ,e_d)B$ and $N_{\mathcal}{D}(\hat{e}_1,\dots ,\hat{e}_d)=(\hat{e}_1,\dots ,\hat{e}_d)U$. By the same proof as in the former half part of the proof of [@GLS Proposition 4.7], we obtain $X,X^{-1}\in M_{d\times d}(\tilde{S}[1/p])$. On the other hand, let $\hat{{\mathfrak{M}}}_s$ be the image of $\hat{{\mathfrak{M}}}$ for the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{{\mathfrak{S}}_s}$. Now we recall that ${\mathcal}{D}_s=S_{K_0,s}\otimes_{{\varphi},{\mathfrak{S}}_s} {\mathfrak{M}}_s$ has a structure of the Breuil module corresponding to ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}}_s)$ Denote by $N_{{\mathcal}{D}_s}$ its monodromy operator. By the formula (\[eq1\]) for $\hat{{\mathfrak{M}}}_s$ and the formula (\[eq2\]) for $\hat{{\mathfrak{M}}}$, we see that $p^sN_{{\mathcal}{D}}=N_{{\mathcal}{D}_s}$ on ${\mathcal}{D}$. Therefore, ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}})$ is crystalline if and only if $N_{{\mathcal}{D}_s}$ mod $I_+S_{K_0,s}{\mathcal}{D}_s$ is zero, which is equivalent to say that $N_D=(N_{{\mathcal}{D}}$ mod $I_+S_{K_0}{\mathcal}{D})$ is zero, that is, $B=0$. Therefore, the latter half part of the proof [@GLS Proposition] gives the assertion (2). \(2) $\Rightarrow$ (3): This is clear. \(3) $\Rightarrow$ (1): Suppose that (3) holds. We denote by $\hat{{\mathfrak{M}}}_s$ the image of $\hat{{\mathfrak{M}}}$ for the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{{\mathfrak{S}}_s}$ as above. We claim that, for any $x\in {\mathfrak{M}}_s$, we have $\tau^{p^s}(x)-x\in I^{[1+]}W(R)\otimes_{{\varphi}_s} {\mathfrak{M}}_s$. Let $x=a\otimes y\in {\mathfrak{M}}_s={\mathfrak{S}}_s\otimes_{{\mathfrak{S}}} {\mathfrak{M}}$ where $a\in {\mathfrak{S}}_s$ and $y\in {\mathfrak{M}}$. Then $$\tau^{p^s}(x)-x=\tau^{p^s}({\varphi}(a))(\tau^{p^s}(y)-y)+(\tau^{p^s}({\varphi}(a))-{\varphi}(a))y$$ and thus it suffices to show $\tau^{p^s}({\varphi}(a))-{\varphi}(a)\in I^{[1+]}W(R)$. This follows from the lemma below and thus we obtained the claim. By the claim and Theorem \[Thm1\] (2), we know that ${\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}}_s)\simeq {\mathbb}{Q}_p\otimes_{{\mathbb}{Z}_p} \hat{T}_s(\hat{{\mathfrak{M}}})$ is crystalline. \[cryslem\] $(1)$ We have $I^{[1]}W(R)\cap u^{\ell}B^+_{{\mathrm}{cris}} =u^{\ell}I^{[1]}W(R)$ for $\ell\ge 0$. $(2)$ We have $g(a)-a\in uI^{[1]}W(R)$ for $g\in G$ and $a\in {\mathfrak{S}}$. This is due to [@GLS the proof of Proposition 7] but we write a proof here. \(1) Take $x=u^{\ell}y\in I^{[1]}W(R)$ with $y\in B^+_{{\mathrm}{cris}}$. By Lemma 3.2.2 of [@Li3] we have $y\in W(R)$. Now we remark that $uz\in {\mathrm}{Fil}^nW(R)$ with $z\in W(R)$ implies $z\in {\mathrm}{Fil}^nW(R)$ since $u$ is a unit of $B^+_{{\mathrm}{dR}}$. Hence $u^{\ell}y\in I^{[1]}W(R)$ implies $y\in I^{[1]}W(R)$. \(2) By the relation (\[easyeq\]), we see that $g(a)-a\in I^{[1]}W(R)$. On the other hand, if $i>0$, we can write $N^i_{S_{K_0}}(a)=ub_i$ for some $b_i\in {\mathfrak{S}}$. Thus by the relation (\[easyeq\]) again we obtain $g(a)-a\in uB^+_{{\mathrm}{cris}}$. Then the result follows from (1). Variants of torsion $({\varphi},\hat{G})$-modules ================================================= In this section, we mainly study full subcategories of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ defined below and also study representations associated with them. As a consequence, we prove theorems in Introduction. We use same notation as in Section 2 and 3. In particular, $p$ is odd. In below, let $v_R$ be the valuation of $R$ normalized such that $v_R(\underline{\pi})=1/e$ and, for any real number $x\ge 0$, we denote by ${\mathfrak{m}}^{\ge x}_R$ the ideal of $R$ consisting of elements $a$ with $v_R(a)\ge x$. Let $J$ be an ideal of $W(R)$ which satisfies the following conditions: - $J\not\subset pW(R)$, - $J$ is a principal ideal, - $J$ is ${\varphi}$-stable and $G_s$-stable in $W(R)$. By the above first and second assumptions for $J$, the image of $J$ under the projection $W(R)\twoheadrightarrow R$ is of the form ${\mathfrak{m}}^{\ge c_J}_{R}$ for some real number $c_J\ge 0$. We denote by ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ the full subcategory of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ consisting of objects $\hat{{\mathfrak{M}}}$ which satisfy the following condition: $$\tau^{p^s}(x)-x\in JW(R)\otimes_{{\varphi}, {\mathfrak{S}}} {\mathfrak{M}}\quad {\rm for\ any}\ x\in {\mathfrak{M}}.$$ Also, we denote by ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s, J}_{{\mathrm}{tor}}(G_s)$ the essential image of the functor $\hat{T}_s\colon {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$ restricted to ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. By definition, we have ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}\subset {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J'}_{/{\mathfrak{S}}_{\infty}}$ and ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s) \subset {\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J'}_{{\mathrm}{tor}}(G_s)$ for $J\subset J'$. Full faithfullness for ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ ------------------------------------------------------------------------------------------------ For the beginning of a study of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$, we prove the following full faithfullness result. \[FFTHMMOD\] Let $r$ and $r'$ be non-negative integers with $c_J> pr/(p-1)$. Let $\hat{{\mathfrak{M}}}$ and $\hat{{\mathfrak{N}}}$ be objects of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ and ${\widetilde{{\mathrm}{Mod}}}^{r',\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$, respectively. Then we have ${\mathrm}{Hom}(\hat{{\mathfrak{M}}}, \hat{{\mathfrak{N}}})={\mathrm}{Hom}({\mathfrak{M}}, {\mathfrak{N}})$. $($Here, two “${\mathrm}{Hom}$”s are defined by obvious manners.$)$ In particular, if $c_J> pr/(p-1)$, then the forgetful functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}\to {{\mathrm}{Mod}}^r_{/{\mathfrak{S}}_{\infty}}$ is fully faithful. A very similar proof of [@Oz2 Lemma 7] proceeds, and hence we only give a sketch here. Let $\hat{{\mathfrak{M}}}$ and $\hat{{\mathfrak{N}}}$ be objects of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ and ${\widetilde{{\mathrm}{Mod}}}^{r',\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$, respectively. Let $f\colon {\mathfrak{M}}\to {\mathfrak{N}}$ be a morphism of Kisin modules over ${\mathfrak{S}}$. Put $\hat{f}=W(R)\otimes f\colon W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}\to W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}$. Choose any lift of $\tau\in \hat{G}$ to $G_K$; we denote it also by $\tau$. Since the $\hat{G}_s$-action for $\hat{{\mathfrak{M}}}$ is continuous, it suffices to prove that $\Delta(1\otimes x)=0$ for any $x\in {\mathfrak{M}}$ where $\Delta:=\tau^{p^s}\circ \hat{f}-\hat{f}\circ \tau^{p^s}$. We use induction on $n$ such that $p^n{\mathfrak{N}}=0$. Suppose $n=1$. Since $\Delta=(\tau^{p^s}-1)\circ \hat{f}-\hat{f}\circ (\tau^{p^s}-1)$, we obtain the following: $(0)$:For any $x\in {\mathfrak{M}}$, $\Delta(1\otimes x)\in {\mathfrak{m}}^{\ge c(0)}_R(R\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{N}})$ where $c(0)=c_J$. Since ${\mathfrak{M}}$ is of height $\le r$, we further obtain the following for any $i\ge 1$ inductively: $(i)$:For any $x\in {\mathfrak{M}}$, $\Delta(1\otimes x)\in {\mathfrak{m}}^{\ge c(i)}_R(R\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{N}})$ where $c(i)=pc(i-1)-pr =(c_J-pr/(p-1))p^i+pr/(p-1)$. The condition $c_J>pr/(p-1)$ implies that $c(i)\to \infty$ as $i\to \infty$ and thus $\Delta(1\otimes x)=0$. Suppose $n>1$. Consider the exact sequence $0\to {\mathrm}{Ker}(p)\to {\mathfrak{N}}\overset{p}{\to} p{\mathfrak{N}}\to 0$ of ${\varphi}$-modules over ${\mathfrak{S}}$. It is not difficult to check that ${\mathfrak{N}}':={\mathrm}{Ker}(p)$ and ${\mathfrak{N}}'':=p{\mathfrak{N}}$ are torsion Kisin modules of height $\le r'$ over ${\mathfrak{S}}$ (cf. [@Li1 Lemma 2.3.1]). Moreover, we can check that ${\mathfrak{N}}'$ and ${\mathfrak{N}}''$ have natural structures of objects of ${\widetilde{{\mathrm}{Mod}}}^{r',\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ (which are denoted by $\hat{{\mathfrak{N}}}'$ and $\hat{{\mathfrak{N}}}''$, respectively) such that the sequence $0\to {\mathfrak{N}}'\to {\mathfrak{N}}\overset{p}{\to} {\mathfrak{N}}''\to 0$ induces an exact sequence $0\to \hat{{\mathfrak{N}}}'\to \hat{{\mathfrak{N}}}\to \hat{{\mathfrak{N}}}''\to 0$. By the lemma below, we know that $\hat{{\mathfrak{N}}}'$ and $\hat{{\mathfrak{N}}}''$ are in fact contained in ${\widetilde{{\mathrm}{Mod}}}^{r',\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. By the induction hypothesis, we see that $\Delta(1\otimes x)$ has values in $(W(R)\otimes_{{\varphi}, {\mathfrak{S}}} {\mathfrak{N}}')\cap (JW(R)\otimes_{{\varphi}, {\mathfrak{S}}} {\mathfrak{N}})$. By Lemma 6 of [@Oz2] and the assumption that $J\not\subset pW(R)$ is principal, we obtain that $\Delta(1\otimes x)\in JW(R)\otimes_{{\varphi}, {\mathfrak{S}}} {\mathfrak{N}}'$. Since $p{\mathfrak{N}}'=0$, an analogous argument in the case $n=1$ proceeds and we have $\Delta(1\otimes x)=0$ as desired. \[speciallemma\] Let $0\to \hat{{\mathfrak{M}}}'\to \hat{{\mathfrak{M}}}\to \hat{{\mathfrak{M}}}''\to 0$ be an exact sequence in ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$. Suppose that $\hat{{\mathfrak{M}}}$ is an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. Then $\hat{{\mathfrak{M}}}'$ and $\hat{{\mathfrak{M}}}''$ are also objects of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. The fact $\hat{{\mathfrak{M}}}''\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ is clear. Take any $x\in {\mathfrak{M}}'$. Then we have $\tau^{p^s}(x)-x\in (JW(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}})\cap (W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}')$. Since $J$ is a principal ideal which is not contained in $pW(R)$, we obtain $\tau^{p^s}(x)-x\in JW(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}'$ by Lemma 6 of [@Oz2]. This implies $\hat{{\mathfrak{M}}}'\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. The category ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ --------------------------------------------------------------------------------- In this subsection, we study some categorical properties of ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$. Let $\hat{{\mathfrak{M}}}$ be an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$. Following Section 3.2 of [@Li2] (note that arguments in [@Li2] is the “free case”), we construct a map $\hat{\iota}_s$ which connects $\hat{{\mathfrak{M}}}$ and $\hat{T}_s(\hat{{\mathfrak{M}}})$ as follows. Observe that there exists a natural isomorphism of ${\mathbb}{Z}_p[G_s]$-modules $$\hat{T}_s(\hat{{\mathfrak{M}}}) \simeq {\mathrm}{Hom}_{W(R),{\varphi}}(W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}, {\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))$$ where $G_s$ acts on ${\mathrm}{Hom}_{W(R),{\varphi}}(W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}},{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))$ by $(\sigma.f)(x)=\sigma(f(\sigma^{-1}(x)))$ for $\sigma\in G_s, f\in {\mathrm}{Hom}_{W(R),{\varphi}}(W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}},{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R)), x\in W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}=W(R)\otimes_{{\widehat{\mathcal{R}}}_s} ({\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}})$. Thus we can define a morphism $\hat{\iota}'_s\colon W(R)\otimes_{{\varphi}, {\mathfrak{S}}}{\mathfrak{M}}\to {\mathrm}{Hom}_{{\mathbb}{Z}_p}(\hat{T}_s(\hat{{\mathfrak{M}}}),{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))$ by $$x\mapsto (f\mapsto f(x)),\quad x\in W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}, f\in \hat{T}_s(\hat{{\mathfrak{M}}}).$$ Since $\hat{T}_s(\hat{{\mathfrak{M}}})\simeq \oplus_{i\in I}{\mathbb}{Z}_p/p^{n_i}{\mathbb}{Z}_p$ as ${\mathbb}{Z}_p$-modules, we have a natural isomorphism ${\mathrm}{Hom}_{{\mathbb}{Z}_p}(\hat{T}_s(\hat{{\mathfrak{M}}}),{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))\simeq W(R)\otimes_{{\mathbb}{Z}_p}\hat{T}_s^{\vee}(\hat{{\mathfrak{M}}})$ where $\hat{T}_s^{\vee}(\hat{{\mathfrak{M}}})={\mathrm}{Hom}_{{\mathbb}{Z}_p}(\hat{T}_s(\hat{{\mathfrak{M}}}),{\mathbb}{Q}_p/{\mathbb}{Z}_p)$ is the dual representation of $\hat{T}_s(\hat{{\mathfrak{M}}})$. Composing this isomorphism with $\hat{\iota}'_s$, we obtain the desired map $$\hat{\iota}_s\colon W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}\to W(R)\otimes_{{\mathbb}{Z}_p}\hat{T}_s^{\vee}(\hat{{\mathfrak{M}}}).$$ It follows from a direct calculation that $\hat{\iota}_s$ is ${\varphi}$-equivariant and $G_s$-equivariant. If we denote by $\hat{{\mathfrak{M}}}_s$ the image of $\hat{{\mathfrak{M}}}$ for the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{s,\infty}}$ (cf. Section \[vardef\]), then the above $\hat{\iota}_s$ is isomorphic to “$\hat{\iota}$ for $\hat{{\mathfrak{M}}}_s$ in Section 4.1 of [@Oz1]”. Hence Lemma 4.2 (4) in [*loc*]{}. [*cit*]{}. implies that $$W({\mathrm}{Fr}\ R)\otimes \hat{\iota}_s\colon W({\mathrm}{Fr}\ R)\otimes_{W(R)}(W(R)\otimes_{{\varphi},{\mathfrak{S}}}{\mathfrak{M}}) \to W({\mathrm}{Fr}\ R)\otimes_{W(R)}(W(R)\otimes_{{\mathbb}{Z}_p}\hat{T}_s^{\vee}(\hat{{\mathfrak{M}}}))$$ is bijective. Let $(R) \colon 0\to T'\to T\to T''\to 0$ be an exact sequence in ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$. Assume that there exists $\hat{{\mathfrak{M}}}\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ such that $\hat{T}_s(\hat{{\mathfrak{M}}})\simeq T$. Then there exists an exact sequence $(M) \colon 0\to \hat{{\mathfrak{M}}}''\to \hat{{\mathfrak{M}}}\to \hat{{\mathfrak{M}}}'\to 0$ in ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ such that $\hat{T}_s((M))\simeq (R)$. The same proof as [@Oz1 Theorem 4.5], except using not $\hat{\iota}$ in the proof of [*loc.*]{} [*cit.*]{} but $\hat{\iota}_s$ as above, gives an exact sequence $(M) \colon 0\to \hat{{\mathfrak{M}}}''\to \hat{{\mathfrak{M}}}\to \hat{{\mathfrak{M}}}'\to 0$ in ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ such that $\hat{T}_s((M))\simeq (R)$. Therefore, Lemma \[speciallemma\], gives the desired result. \[stability\] The full subcategory ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ of ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$ is stable under subquotients. Let $L$ be as in Section 2, that is, the completion of an unramified algebraic extension of $K$ with residue field $k_L$. We prove the following base change lemma. \[bc\] Assume that $J\supset u^pI^{[1]}W(R)$ or $L$ is a finite unramified extension of $K$. If $T$ is an object of ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$, then $T|_{G_{L,s}}$ is an object of ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_{L,s},J}_{{\mathrm}{tor}}(G_{L,s})$. By an obvious way, we define a functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$. The underlying Kisin module of the image of $\hat{{\mathfrak{M}}}\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ for this functor is ${\mathfrak{M}}_L={\mathfrak{S}}_L\otimes_{{\mathfrak{S}}} {\mathfrak{M}}$. Lemma \[bc\] immediately follows from the lemma below. Assume that $J\supset u^pI^{[1]}W(R)$ or $L$ is a finite unramified extension of $K$. Then the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$ induces a functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s},J}_{/{\mathfrak{S}}_{L,\infty}}$. Let $\hat{{\mathfrak{M}}}$ be an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ and let $\hat{{\mathfrak{M}}}_L$ be the image of $\hat{{\mathfrak{M}}}$ for the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}\to {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_{L,s}}_{/{\mathfrak{S}}_{L,\infty}}$. In the rest of this proof, to avoid confusions, we denote the image of $x\in {\mathfrak{M}}_L$ in $W(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$ by $1\otimes x$. Recall that we abuse notations by writing $\tau$ for the pre-image of $\tau\in G_{K,p^{\infty}}$ via the bijection $G_{L,p^{\infty}}\simeq G_{K,p^{\infty}}$ of lemma \[easylemma\]. Then $\tau^{p^s}$ is a topological generator of $G_{L,s,p^{\infty}}$. It suffices to show the following: if $\hat{{\mathfrak{M}}}$ is an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$, then we have $\tau^{p^s}(1\otimes x)-(1\otimes x)\in JW(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$ for any $x\in {\mathfrak{M}}_L$. Now we suppose $\hat{{\mathfrak{M}}}\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. Take any $a\in {\mathfrak{S}}_L$ and $x\in {\mathfrak{M}}$. Note that we have $\tau^{p^s}(1\otimes ax)-(1\otimes ax) =\tau^{p^s}({\varphi}(a))(\tau^{p^s}(1\otimes x)-(1\otimes x)) +(\tau^{p^s}({\varphi}(a))-{\varphi}(a))(1\otimes x)$ in $W(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$. Since $\hat{{\mathfrak{M}}}$ is an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$, we have $\tau^{p^s}({\varphi}(a))(\tau^{p^s}(1\otimes x)-(1\otimes x))\in JW(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$. Therefore, it is enough to show $(\tau^{p^s}({\varphi}(a))-{\varphi}(a))(1\otimes x)\in JW(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$. This follows from Lemma \[cryslem\] immediately in the case where $J\supset u^pI^{[1]}W(R)$. Next we consider the case where $L$ is a finite unramified extension of $K$. Let $c_1,\dots ,c_{\ell}\in W(k_L)$ be generators of $W(k_L)$ as a $W(k)$-module. Then we have ${\mathfrak{S}}_L=\sum^{\ell}_{j=1} c_j {\mathfrak{S}}$ and thus we can write $a=\sum^{\ell}_{j=1}a_jc_j$ for some $a_j\in {\mathfrak{S}}$. Hence it suffices to show $(\tau^{p^s}({\varphi}(a_j))-{\varphi}(a_j))(1\otimes x)\in JW(R)\otimes_{{\varphi},{\mathfrak{S}}_L} {\mathfrak{M}}_L$ but this in fact immediately follows from the equation $(\tau^{p^s}({\varphi}(a_j))-{\varphi}(a_j))(1\otimes x) =(\tau^{p^s}(1\otimes a_jx)-(1\otimes a_jx))- (\tau^{p^s}({\varphi}(a_j))(\tau^{p^s}(1\otimes x)-(1\otimes x)))$. Full faithfulness theorem for ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ -------------------------------------------------------------------------------------------------- Our goal in this subsection is to prove the following full faithfulness theorem, which plays an important roll in our proofs of main theorems. \[FFTHM\] Assume that $J\supset u^pI^{[1]}W(R)$ or $k$ is algebraically closed. If $p^{s+2}/(p-1)\ge c_J>pr/(p-1)$, then the restriction functor ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is fully faithful. First we give a very rough sketch of the theory of maximal models for Kisin modules (cf. [@CL1]). For any ${\mathfrak{M}}\in {\mathrm}{Mod}^r_{/{\mathfrak{S}}_{\infty}}$, put ${\mathfrak{M}}[1/u]={\mathfrak{S}}[1/u]\otimes_{{\mathfrak{S}}} {\mathfrak{M}}$ and denote by $F^r_{{\mathfrak{S}}}({\mathfrak{M}}[1/u])$ the (partially) ordered set (by inclusion) of torsion Kisin modules ${\mathfrak{N}}$ of height $\le r$ which are contained in ${\mathfrak{M}}[1/u]$ and ${\mathfrak{N}}[1/u]={\mathfrak{M}}[1/u]$ as ${\varphi}$-modules. The set $F^r_{{\mathfrak{S}}}({\mathfrak{M}}[1/u])$ has a greatest element (cf. [*loc*]{}. [*cit*]{}., Corollary 3.2.6). We denote this element by ${\mathrm{Max}}^r({\mathfrak{M}})$. We say that ${\mathfrak{M}}$ is [*maximal of height $\le r$*]{} (or, [*maximal*]{} for simplicity) if it is the greatest element of $F^r_{{\mathfrak{S}}}({\mathfrak{M}}[1/u])$. The implication ${\mathfrak{M}}\mapsto {\mathrm{Max}}^r({\mathfrak{M}})$ defines a functor “${\mathrm{Max}}^r$” from the category ${\mathrm}{Mod}^r_{/{\mathfrak{S}}_{\infty}}$ of torsion Kisin modules of height $\le r$ into the category ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$ of maximal Kisin modules of height $\le r$. The category ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$ is abelian (cf. [*loc*]{}. [*cit*]{}., Theorem 3.3.8). Furthermore, the functor $T_{{\mathfrak{S}}}\colon {\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$, defined by $T_{{\mathfrak{S}}}({\mathfrak{M}})={\mathrm}{Hom}_{{\mathfrak{S}},{\varphi}}({\mathfrak{M}},{\mathbb}{Q}_p/{\mathbb}{Z}_p\otimes_{{\mathbb}{Z}_p} W(R))$, is exact and fully faithful (cf. [*loc*]{}. [*cit*]{}., Corollary 3.3.10). It is not difficult to check that $T_{{\mathfrak{S}}}({\mathrm{Max}}^r({\mathfrak{M}}))$ is canonically isomorphic to $T_{{\mathfrak{S}}}({\mathfrak{M}})$ as representations of $G_{\infty}$ for any torsion Kisin module ${\mathfrak{M}}$ of height $\le r$. \[Def1\] Let $d$ be a positive integer. Let ${\mathfrak{n}}=(n_i)_{i\in {\mathbb}{Z}/d{\mathbb}{Z}}$ be a sequence of non-negative integers of smallest period $d$. We define a torsion Kisin module ${\mathfrak{M}}({\mathfrak{n}})$ as below: - as a ${k[\![u]\!]}$-module, ${\mathfrak{M}}({\mathfrak{n}})=\bigoplus_{i\in {\mathbb}{Z}/d{\mathbb}{Z}} {k[\![u]\!]}e_i$; - for all $i\in {\mathbb}{Z}/d{\mathbb}{Z}$, ${\varphi}(e_i)=u^{n_i}e_{i+1}$. We denote by ${\mathcal}{S}^r_{{\mathrm}{max}}$ the set of sequences ${\mathfrak{n}}=(n_i)_{i\in {\mathbb}{Z}/d{\mathbb}{Z}}$ of integers $0\le n_i\le {\mathrm}{min}\{er, p-1\}$ with smallest period $d$ for some integer $d$ except the constant sequence with value $p-1$ (if necessary). By definition, we see that ${\mathfrak{M}}({\mathfrak{n}})$ is of height $\le r$ for any ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$. Putting $r_0={\mathrm}{max}\{r'\in {\mathbb}{Z}_{\ge 0};e(r'-1)<p-1 \}$, we also see that ${\mathfrak{M}}({\mathfrak{n}})$ is of height $\le r_0$ for any ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$. It is known that ${\mathfrak{M}}({\mathfrak{n}})$ is maximal for any ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$ ([@CL1 Proposition 3.6.7]). If $k$ is algebraically closed, then ${\mathfrak{M}}({\mathfrak{n}})$ is simple in ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$ for any ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$ (cf. [*loc. cit.*]{}, Propositions 3.6.7 and 3.6.12) and furthermore, the converse holds; any simple object in ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$ is of the form ${\mathfrak{M}}({\mathfrak{n}})$ for some ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$ (cf. [*loc. cit.*]{}, Propositions 3.6.8 and 3.6.12). \[Lem1\] Assume that $p^{s+2}/(p-1)\ge c_J$. Let $d$ be a positive integer. Let ${\mathfrak{n}}=(n_i)_{i\in {\mathbb}{Z}/d{\mathbb}{Z}}$ be a sequence of non-negative integers of smallest period $d$. If ${\mathfrak{M}}({\mathfrak{n}})$ is of height $\le r$, then ${\mathfrak{M}}({\mathfrak{n}})$ has a structure of an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. Choose any $(p^d-1)$-th root $\eta\in R$ of $\underline{{\varepsilon}}$. Since $[\eta]\cdot {\mathrm}{exp}(t/(p^d-1))$ is a $(p^d-1)$-th root of unity, it is of the form $[a]$ for some $a\in {\mathbb}{F}^{\times}_{p^d}$. Replacing $\eta a^{-1}$ with $\eta$, we obtain $[\eta]={\mathrm}{exp}(-t/(p^d-1))\in {\widehat{\mathcal{R}}}^{\times}$. Put $x_i=[\eta]^{m_i}\in {\widehat{\mathcal{R}}}^{\times}$ and $\bar{x}_i=\eta^{m_i}\in ({\widehat{\mathcal{R}}}/p{\widehat{\mathcal{R}}})^{\times}\subset R^{\times}$ for any $i\in {\mathbb}{Z}/d{\mathbb}{Z}$, where $m_i=\sum^{d-1}_{j=0}n_{i+j}p^{d-j}$. We see that $x_i-1$ is contained in $I_+{\widehat{\mathcal{R}}}$. In the rest of this proof, to avoid confusions, we denote the image of $x\in {\mathfrak{M}}({\mathfrak{n}})$ in ${\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}({\mathfrak{n}})\subset R\otimes_{{\varphi},k[\![u]\!]} {\mathfrak{M}}({\mathfrak{n}})$ by $1\otimes x$. Now we define a $\hat{G}_s$-action on ${\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}({\mathfrak{n}})$ by $\tau^{p^s}(1\otimes e_i):=x^{p^s}_i(1\otimes e_i)$ for the basis $\{e_i\}_{i\in {\mathbb}{Z}/d{\mathbb}{Z}}$ of ${\mathfrak{M}}({\mathfrak{n}})$ as in Definition \[Def1\]. It is not difficult to check that ${\mathfrak{M}}({\mathfrak{n}})$ has a structure of an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$ via this $\hat{G}_s$-action; we denote it by $\hat{{\mathfrak{M}}}({\mathfrak{n}})$. It suffices to prove that $\hat{{\mathfrak{M}}}({\mathfrak{n}})$ is in fact an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. Recall that $v_R$ is the valuation of $R$ normalized such that $v_R(\underline{\pi})=1/e$. Define $\tilde{{\mathfrak{t}}}={\mathfrak{t}}\ {\mathrm}{mod}\ pW(R)$ an element of $R$. We denote by $v_p$ the usual $p$-adic valuation normalized by $v_p(p)=1$. Note that we have $v_R(\underline{{\varepsilon}}-1)=p/(p-1)$ and $v_R(\tilde{{\mathfrak{t}}})=1/(p-1)$ (here, the latter equation follows from the relation ${\varphi}({\mathfrak{t}})=pE(0)^{-1}E(u){\mathfrak{t}}$). We see that $$v_R(\bar{x}^{p^s}_i-1)=p^{s+v_p(m_i)}\cdot p/(p-1)\ge p^{s+2}/(p-1).$$ Since $p^{s+2}/(p-1)\ge c_J$ and the image of $J$ in $R$ is ${\mathfrak{m}}_R^{\ge c_J}$, we obtain $$\tau^{p^s}(1\otimes e_i)-(1\otimes e_i)\in {\mathfrak{m}}_R^{\ge c_J}R\otimes_{{\varphi},k[\![u]\!]} {\mathfrak{M}}({\mathfrak{n}}) \simeq JW(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}({\mathfrak{n}}).$$ Finally we have to show that $\tau^{p^s}(1\otimes ae_i)-(1\otimes ae_i)\in {\mathfrak{m}}_R^{\ge c_J}R\otimes_{{\varphi},k[\![u]\!]} {\mathfrak{M}}({\mathfrak{n}})$ for any $a\in k[\![u]\!]$. Since $\tau^{p^s}(1\otimes ae_i)-(1\otimes ae_i)=\tau^{p^s}({\varphi}(a)) (\tau^{p^s}(1\otimes e_i)-(1\otimes e_i))+ (\tau^{p^s}({\varphi}(a))-{\varphi}(a))(1\otimes e_i)$, it suffices to show $\tau^{p^s}({\varphi}(a))-{\varphi}(a)\in {\mathfrak{m}}_R^{\ge c_J}$. Write ${\varphi}(a)=\sum_{i\ge 0}a_iu^{pi}$ for some $a_i\in k$. Then we have $\tau^{p^s}({\varphi}(a))-{\varphi}(a)= \sum_{i\ge 1}a_i(\underline{{\varepsilon}}^{p^{s+1}i}-1)u^{pi}$. Since we have $$v_R((\underline{{\varepsilon}}^{p^{s+1}i}-1)u^{pi}) =p^{s+1}v_R(\underline{{\varepsilon}}^i-1)+v_R(u^{pi}) > p^{s+2}/(p-1)\ge c_J$$ for any $i\ge 1$, we have done. Recall that $r_0={\mathrm}{max}\{r'\in {\mathbb}{Z}_{\ge 0}; e(r'-1)<p-1\}$. Put $r_1:={\mathrm}{min}\{r,r_0\}$. \[Cor1\] Assume that $p^{s+2}/(p-1)\ge c_J$. If ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$, then ${\mathfrak{M}}({\mathfrak{n}})$ has a structure of an object of ${\widetilde{{\mathrm}{Mod}}}^{r',\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ for any $r'\ge r_1$. Furthermore, if $c_J>pr_1/(p-1)$, it is uniquely determined. We denote this object by $\hat{{\mathfrak{M}}}({\mathfrak{n}})$. We should remark that ${\mathfrak{M}}({\mathfrak{n}})$ is of height $\le r_1$ for any ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$. The uniqueness assertion follows from Proposition \[FFTHMMOD\]. \[tameres\] The functor from tamely ramified ${\mathbb}{Z}_p$-representations of $G_K$ to ${\mathbb}{Z}_p$-representations of $G_{\infty}$, obtained by restricting the action of $G_K$ to $G_{\infty}$, is fully faithful. The result immediately follows from the fact that $G_K$ is topologically generated by $G_{\infty}$ and the wild inertia subgroup of $G_K$. We remark that any semi-simple ${\mathbb}{F}_p$-representation of $G_K$ is automatically tame. \[FFLEM\] Assume that $J\supset u^pI^{[1]}W(R)$ or $k$ is algebraically closed. Let $T\in {\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$ and $T'\in {\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$. Suppose that $T$ is tame, $pT=0$ and $T|_{G_{\infty}}\simeq T_{{\mathfrak{S}}}({\mathfrak{M}})$ for some ${\mathfrak{M}}\in {{\mathrm}{Mod}}^r_{/{\mathfrak{S}}_{\infty}}$. Furthermore, we suppose $p^{s+2}/(p-1)\ge c_J>pr/(p-1)$. Then all $G_{\infty}$-equivariant homomorphisms $T\to T'$ are in fact $G_s$-equivariant. Let $L$ be the completion of the maximal unramified extension $K^{{\mathrm}{ur}}$ of $K$. By identifying $G_L$ with $G_{K^{{\mathrm}{ur}}}$, we may regard $G_L$ as a subgroup of $G_K$. Note that $L_{(s)}=K_{(s)}L$ is the completion of the maximal unramified extension of $K_{(s)}$, and $G_s$ is topologically generated by $G_{L,s}$ and $G_{\infty}$. Consider the following commutative diagram: $\displaystyle \xymatrix{ {\mathrm}{Hom}_{G_{L,s}}(T,T')\ar@{^{(}->}[rr] & & {\mathrm}{Hom}_{G_{L,\infty}}(T,T') \\ {\mathrm}{Hom}_{G_s}(T,T') \ar@{^{(}->}[u] \ar@{^{(}->}[rr] & & {\mathrm}{Hom}_{G_{\infty}}(T,T'). \ar@{^{(}->}[u] }$ Since $T'|_{G_{L,s}}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_{L,s},J}_{{\mathrm}{tor}}(G_{L,s})$ if $J\supset u^pI^{[1]}W(R)$ (cf. Lemma \[bc\]), the above diagram allows us to reduce a proof to the case where $k$ is algebraically closed. In the rest of this proof, we assume that $k$ is algebraically closed. Under this assumption, an ${\mathbb}{F}_p$-representation of $G_s$ is tame if and only if it is semi-simple by Maschke’s theorem. Thus we may also assume that $T$ is irreducible (here, we remark that any subquotient of $T$ is tame and, also remark that the essential image of $T_{{\mathfrak{S}}}\colon {{\mathrm}{Mod}}^r_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is stable under subquotients in ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$). We claim that $T|_{G_{\infty}}$ is also irreducible. If not, there exists a non-zero irreducible ${\mathbb}{F}_p[G_{\infty}]$-submodule $W$ of $T|_{G_{\infty}}$. Let $K^{{\mathrm}{t}}_{(s)}$ be the maximal tamely ramified extension of $K_{(s)}$ and $I_{p,s}:={\mathrm}{Gal}(\overline{K}/K^{{\mathrm}{t}}_{(s)})$ the wild inertia subgroup of $G_s$. We see that $K^{{\mathrm}{t}}_{(s)}\cap K_{\infty}=K_{(s)}$. Since $G_{\infty}\cap I_{p,s}$ acts on $W$ trivially, the $G_{\infty}$-action on $W$ extends to $G_s$ via the composition map $G_s\twoheadrightarrow {\mathrm}{Gal}(K^{{\mathrm}{t}}_{(s)}/K_{(s)}) \simeq G_{\infty}/(G_{\infty}\cap I_{p,s})$. Thus we can regard $W$ as an irreducible ${\mathbb}{F}_p[G_s]$-module. By Lemma \[tameres\], we see that $W$ is a sub ${\mathbb}{F}_p[G_s]$-module of $T$. This contradicts the irreducibility of $T$ and the claim follows. By the assumption on $T$, we have $T|_{G_{\infty}}\simeq T_{{\mathfrak{S}}}({\mathfrak{M}})\simeq T_{{\mathfrak{S}}}({\mathrm{Max}}^r({\mathfrak{M}}))$ for some ${\mathfrak{M}}\in {{\mathrm}{Mod}}^r_{/{\mathfrak{S}}_{\infty}}$. Since $T|_{G_{\infty}}$ is irreducible and $T_{{\mathfrak{S}}}\colon {\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$ is exact and fully faithful, we know that ${\mathrm{Max}}^r({\mathfrak{M}})$ is a simple object in the abelian category ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$. Therefore, since $k$ is algebraically closed, we have ${\mathrm{Max}}^r({\mathfrak{M}})\simeq {\mathfrak{M}}({\mathfrak{n}})$ for some ${\mathfrak{n}}\in {\mathcal}{S}^r_{{\mathrm}{max}}$ (cf. [@CL1 Propositions 3.6.8 and 3.6.12]). Let $\hat{{\mathfrak{M}}}({\mathfrak{n}})$ be the object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G},J}_{/{\mathfrak{S}}_{\infty}}$ as in Corollary \[Cor1\]. We recall that $T_{{\mathfrak{S}}}({\mathfrak{M}}({\mathfrak{n}}))$ is isomorphic to $\hat{T}_s(\hat{{\mathfrak{M}}}({\mathfrak{n}}))|_{G_{\infty}}$ (see Theorem \[Thm1\] (1)), and hence we have an isomorphism $T|_{G_{\infty}}\simeq \hat{T}_s(\hat{{\mathfrak{M}}}({\mathfrak{n}}))|_{G_{\infty}}$. Here, we note that $T$ and $\hat{T}_s(\hat{{\mathfrak{M}}}({\mathfrak{n}}))$ are irreducible as representations of $G_s$ (cf. [@CL1 Theorem 3.6.11]). Applying Lemma \[tameres\] again, we obtain an isomorphism $T\simeq \hat{T}_s(\hat{{\mathfrak{M}}}({\mathfrak{n}}))$ as representations of $G_s$. On the other hand, we can take $\hat{{\mathfrak{M}}}'=({\mathfrak{M}}',{\varphi},\hat{G}_s)\in {\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ such that $T'\simeq \hat{T}_s(\hat{{\mathfrak{M}}}')$. We consider the following commutative diagram: $\displaystyle \xymatrix{ {\mathrm}{Hom}_{G_s}(T,T')\ar@{^{(}->}[rr] & & {\mathrm}{Hom}_{G_{\infty}}(T,T') \\ {\mathrm}{Hom}(\hat{{\mathfrak{M}}}',\hat{{\mathfrak{M}}}({\mathfrak{n}})) \ar^{\hat{T}_s}[u] \ar^{{\mathrm}{forgetful}\ }[r] & {\mathrm}{Hom}({\mathfrak{M}}',{\mathfrak{M}}({\mathfrak{n}})) \ar^{{\mathrm{Max}}^r\quad \ \ }[r] & {\mathrm}{Hom}({\mathrm{Max}}^r({\mathfrak{M}}'),{\mathfrak{M}}({\mathfrak{n}})). \ar^{T_{{\mathfrak{S}}}}[u]. }$ Here, ${\mathrm}{Hom}(\hat{{\mathfrak{M}}}',\hat{{\mathfrak{M}}}({\mathfrak{n}}))$ (resp. ${\mathrm}{Hom}({\mathfrak{M}}',{\mathfrak{M}}({\mathfrak{n}}))$, resp. ${\mathrm}{Hom}({\mathrm{Max}}^r({\mathfrak{M}}'),{\mathfrak{M}}({\mathfrak{n}}))$) is the set of morphisms $\hat{{\mathfrak{M}}}'\to \hat{{\mathfrak{M}}}({\mathfrak{n}})$ (resp. ${\mathfrak{M}}'\to {\mathfrak{M}}({\mathfrak{n}})$, resp. ${\mathrm{Max}}^r({\mathfrak{M}}')\to {\mathfrak{M}}({\mathfrak{n}})$) in ${\widetilde{{\mathrm}{Mod}}}^{r_1,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$ (resp. ${{\mathrm}{Mod}}^{r_1}_{/{\mathfrak{S}}_{\infty}}$, resp. ${\mathrm{Max}}^r_{/{\mathfrak{S}}_{\infty}}$). The first bottom horizontal arrow is bijective by Theorem \[Thm1\] (3) and so is the second (this follows from the fact that ${\mathfrak{M}}({\mathfrak{n}})$ is maximal by [@CL1 Proposition 3.6.7]). Since the right vertical arrow is bijective, the top horizontal arrow must be bijective. Now we are ready to prove Theorem \[FFTHM\]. Let $T$ and $T'$ be objects of ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$. Take any Jordan-Höllder sequence $0=T_0\subset T_1\subset \cdots \subset T_n=T$ of $T$ in ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$. By Corollary \[stability\], we know that $T_i$ and $T_i/T_{i-1}$ are contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ for any $i$. By Corollary \[stability\] again, the category ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ is an exact category in the sense of Quillen ([@Qu Section 2]). Hence short exact sequences in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ give rise to exact sequences of Hom’s and Ext’s in the usual way. (This property holds for any exact category.) On the other hand, by Lemma \[FFLEM\], if an exact sequence $0\to T'\to V\to T_i/T_{i-1}\to 0$ in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ splits as representation of $G_{\infty}$, then it splits as a sequence of representations of $G_s$. Therefore, comparing exact sequences of Hom’s and Ext’s arising from $0\to T_{i-1}\to T_i\to T_i/T_{i-1}\to 0$ in the category ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ with that in the category ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_{\infty})$, we obtain the following implication (here, use Lemma \[FFLEM\] again): if we have ${\mathrm}{Hom}_{G_s}(T_{i-1}, T')={\mathrm}{Hom}_{G_{\infty}}(T_{i-1}, T')$, then it gives the equality ${\mathrm}{Hom}_{G_s}(T_i, T')={\mathrm}{Hom}_{G_{\infty}}(T_i, T')$. Hence a dévissage argument works and the desired full faithfulness follows. Proof of Theorem \[Main1\] -------------------------- Now we are ready to prove our main theorems. First we prove Theorem \[Main1\]. Recall that ${\mathrm}{Rep}^{r,{\mathrm}{ht},{\mathrm}{pcris}(s)}_{{\mathrm}{tor}}(G_K)$ is the category of torsion ${\mathbb}{Z}_p$-representations $T$ of $G_K$ which satisfy the following: there exist free ${\mathbb}{Z}_p$-representations $L$ and $L'$ of $G_K$, of height $\le r$, such that - $L|_{G_s}$ is a subrepresentation of $L'|_{G_s}$. Furthermore, $L|_{G_s}$ and $L'|_{G_s}$ are lattices in some crystalline ${\mathbb}{Q}_p$-representation of $G_s$ with Hodge-Tate weights in $[0,r]$; - $T|_{G_s} \simeq (L'|_{G_s})/(L|_{G_s})$. We apply our arguments given in previous subsections with the following $J$: $$J=u^pI^{[1]}W(R)=u^p{\varphi}({\mathfrak{t}})W(R).$$ Then we have $c_J=p/e+p/(p-1)$ and thus the inequalities $p^{s+2}/(p-1)\ge c_J>pr/(p-1)$ are satisfied if $e(r-1)<p-1$. Therefore, Theorem \[Main1\] is an easy consequence of the following proposition and Theorem \[FFTHM\]. \[proofMain1\] If $T$ is an object of ${\mathrm}{Rep}^{r,{\mathrm}{ht},{\mathrm}{pcris}(s)}_{{\mathrm}{tor}}(G_K)$, then $T|_{G_s}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$. Take free ${\mathbb}{Z}_p$-representations $L$ and $L'$ of $G_K$, of height $\le r$, such that - $L|_{G_s}$ is a subrepresentation of $L'|_{G_s}$. Furthermore, $L|_{G_s}$ and $L'|_{G_s}$ are lattices in some crystalline ${\mathbb}{Q}_p$-representation of $G_s$ with Hodge-Tate weights in $[0,r]$; - $T|_{G_s} \simeq (L'|_{G_s})/(L|_{G_s})$. By Theorem \[Thm1\] (1), there exists an injection $\hat{{\mathfrak{L}}}'\hookrightarrow \hat{{\mathfrak{L}}}$ of $({\varphi},\hat{G}_s)$-modules over ${\mathfrak{S}}_s$ which corresponds to the injection $L|_{G_s}\hookrightarrow L'|_{G_s}$. On the other hand, there exist ${\mathfrak{N}}$ and ${\mathfrak{N}}'$ in ${{\mathrm}{Mod}}^r_{/{\mathfrak{S}}}$ such that $T_{{\mathfrak{S}}}({\mathfrak{N}})\simeq L|_{G_{\infty}}$ and $T_{{\mathfrak{S}}}({\mathfrak{N}}')\simeq L'|_{G_{\infty}}$. Then Proposition \[Kisinfunctor\] (1) implies that ${\mathfrak{S}}_s\otimes_{{\mathfrak{S}}} {\mathfrak{N}}\simeq {\mathfrak{L}}$ and ${\mathfrak{S}}_s\otimes_{{\mathfrak{S}}} {\mathfrak{N}}'\simeq {\mathfrak{L}}'$ as ${\varphi}$-modules over ${\mathfrak{S}}_s$. Therefore, we see that ${\mathfrak{N}}$ and ${\mathfrak{N}}'$ have structures of objects of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}$; denote them by $\hat{{\mathfrak{N}}}$ and $\hat{{\mathfrak{N}}}'$, respectively. Since the functor ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}}\to {{\mathrm}{Mod}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_s}$ is fully faithful (cf. Section \[vardef\]), the injection $\hat{{\mathfrak{L}}}'\hookrightarrow \hat{{\mathfrak{L}}}$ descends to an injection $\hat{{\mathfrak{N}}}'\hookrightarrow \hat{{\mathfrak{N}}}$. Now we put ${\mathfrak{M}}={\mathfrak{N}}/{\mathfrak{N}}'$. Since $(L'|_{G_s})/(L|_{G_s})$ is killed by a power of $p$, it is an object of ${{\mathrm}{Mod}}^r_{/{\mathfrak{S}}_{\infty}}$. We equip a $\hat{G}_s$-action with ${\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$ by a natural isomorphism ${\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}\simeq ({\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{N}})/({\widehat{\mathcal{R}}}_s\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{N}}')$. Then we see that ${\mathfrak{M}}$ has a structure of an object of ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s}_{/{\mathfrak{S}}_{\infty}}$; denote it by $\hat{{\mathfrak{M}}}$. Moreover, Theorem \[cris\] implies that $\hat{{\mathfrak{M}}}$ is in fact contained in ${\widetilde{{\mathrm}{Mod}}}^{r,\hat{G}_s,J}_{/{\mathfrak{S}}_{\infty}}$. By a similar argument to the proof of Lemma 3.1.4 of [@CL2], we have an exact sequence $0\to \hat{T}_s(\hat{{\mathfrak{N}}})\to \hat{T}_s(\hat{{\mathfrak{N}}}')\to \hat{T}_s(\hat{{\mathfrak{M}}})\to 0$ in ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_s)$, which is isomorphic to $0\to L|_{G_s}\to L'|_{G_s}\to T|_{G_s}\to 0$. This finishes a proof. Proof of Theorem \[Main2\] -------------------------- We give a proof of Theorem \[Main2\]. If $s\ge n-1$, then we put $$J=u^pI^{[p^{s-n+1}]}W(R)=u^p{\varphi}({\mathfrak{t}})^{p^{s-n+1}}W(R).$$ Note that we have $c_J=p/e+p^{s-n+2}/(p-1)$ and thus the inequalities $p^{s+2}/(p-1)\ge c_J>pr/(p-1)$ are satisfied if $s>n-1+{\mathrm}{log}_p(r-e/(p-1))$. \[Main2Lem\] Suppose $s\ge n-1$. If $T$ is an object of ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ which is killed by $p^n$, then $T|_{G_s}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$. Let $L$ be an object of ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathbb}{Z}_p}(G_K)$. Take a $({\varphi},\hat{G})$-module $\hat{{\mathfrak{L}}}$ over ${\mathfrak{S}}$ such that $L\simeq \hat{T}(\hat{{\mathfrak{L}}})$. It is known that $(\tau-1)^i(x)\in u^pI^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{L}}$ for any $i\ge 1$ and any $x\in {\mathfrak{L}}$ (cf. the latter half part of the proof of [@GLS Proposition 4.7]). Take any $x\in {\mathfrak{L}}$. Since $(\tau^{p^s}-1)(x)=\sum^{p^s}_{i=1}\binom{p^s}{i}(\tau-1)^i(x)$, we obtain that $$\label{relat} (\tau^{p^s}-1)(x)\in \sum^{p^s}_{i=1} p^{s-v_p(i)}u^pI^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{L}}.$$ Now let $T$ be an object of ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ which is killed by $p^n$. Take an exact sequence $(R)\colon 0\to L_1\to L_2\to T\to 0$ of ${\mathbb}{Z}_p$-representations of $G_K$ with $L_1,L_2\in {\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathbb}{Z}_p}(G_K)$. By Theorem 3.1.3 and Lemma 3.1.4 of [@CL2], there exists an exact sequence $(M)\colon 0\to \hat{{\mathfrak{L}}}_2\to \hat{{\mathfrak{L}}}_1\to \hat{{\mathfrak{M}}}\to 0$ of $({\varphi},\hat{G})$-modules over ${\mathfrak{S}}$ such that $\hat{T}((M))\simeq (R)$. By (\[relat\]), we see that $$(\tau^{p^s}-1)(x)\in \sum^{p^s}_{i=1} p^{s-v_p(i)}u^pI^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}$$ for any $x\in {\mathfrak{M}}$. Since ${\mathfrak{M}}$ is killed by $p^n$ and $s\ge n-1$, we have $$\begin{aligned} \sum^{p^s}_{i=1} p^{s-v_p(i)}u^pI^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}&= \sum_{i=1,\dots ,p^s, s-v_p(i)<n} p^{s-v_p(i)}u^pI^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}\\ &= \sum^{n-1}_{\ell=0} p^{\ell}u^pI^{[p^{s-\ell}]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}\\ & \subset u^pI^{[p^{s-n+1}]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{M}}.\end{aligned}$$ Therefore, we obtained the desired result. By Corollary \[FFTHMtorcris\], we may suppose ${\mathrm}{log}_p(r-(p-1)/e)\ge 0$, that is, $e(r-1)\ge p-1$ . Suppose $s> n-1+{\mathrm}{log}_p(r-(p-1)/e)$. Note that the condition $s\ge n-1$ is now satisfied. Let $T$ and $T'$ be as in the statement of Theorem \[Main2\]. Let $f\colon T\to T'$ be a $G_{\infty}$-equivariant homomorphism. Denote by $L$ the completion of $K^{{\mathrm}{ur}}$ and identify $G_L$ with the inertia subgroup of $G_K$. We note that $T|_{G_L}$ and $T'|_{G_L}$ are object of ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_L)$. By Proposition \[Main2Lem\], $T|_{G_{L,s}}$ and $T'|_{G_{L,s}}$ are objects of ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_{L,s},J}_{{\mathrm}{tor}}(G_{L,s})$. Hence we have that $f$ is $G_{L,s}$-equivariant by Theorem \[FFTHM\]. Since $G_s$ is topologically generated by $G_{L,s}$ and $G_{\infty}$, we see that $f$ is $G_s$-equivariant. Galois equivariance for torsion semi-stable representations {#torsemi} ----------------------------------------------------------- In this subsection, we prove a Galois equivariance theorem for torsion semi-stable representations. A torsion ${\mathbb}{Z}_p$-representation $T$ of $G_K$ is [*torsion semi-stable with Hodge-Tate weights in $[0,r]$*]{} if it can be written as the quotient of lattices in some semi-stable ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$. We denote by ${\mathrm}{Rep}^{r, {\mathrm}{st}}_{{\mathrm}{tor}}(G_K)$ the category of them. Note that ${\mathrm}{Rep}^{0, {\mathrm}{st}}_{{\mathrm}{tor}}(G_K)= {\mathrm}{Rep}^{0, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. Similar to Theorem \[Main2\], we show the following, which is the main result of this subsection. \[Main3\] Suppose that $s> n-1 + {\mathrm}{log}_pr$. Let $T$ and $T'$ be objects of ${\mathrm}{Rep}^{r, {\mathrm}{st}}_{{\mathrm}{tor}}(G_K)$ which are killed by $p^n$. Then any $G_{\infty}$-equivariant homomorphism $T\to T'$ is in fact $G_s$-equivariant. If $s\ge n-1$, then we put $$J=I^{[p^{s-n+1}]}W(R)={\varphi}({\mathfrak{t}})^{p^{s-n+1}}W(R).$$ Then we have $c_J=p^{s-n+2}/(p-1)$. To show Theorem \[Main3\], we use similar arguments to those in the proof of Theorem \[Main2\]. \[Main3Lem\] Suppose $s\ge n-1$. If $T$ is an object of ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathrm}{tor}}(G_K)$ which is killed by $p^n$, then $T|_{G_s}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$. Let $L$ be a lattice in a semi-stable ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$. Take a $({\varphi},\hat{G})$-module $\hat{{\mathfrak{L}}}$ over ${\mathfrak{S}}$ such that $L\simeq \hat{T}(\hat{{\mathfrak{L}}})$. It is known that $(\tau-1)^i(x)\in I^{[i]}W(R)\otimes_{{\varphi},{\mathfrak{S}}} {\mathfrak{L}}$ for any $i\ge 1$ and any $x\in {\mathfrak{L}}$ (cf. the proof of [@Li4 Proposition 2.4.1]). Thus the same proof proceeds as that of Proposition \[Main2Lem\]. We have the equality ${\mathrm}{Rep}^{0, {\mathrm}{st}}_{{\mathrm}{tor}}(G_K)={\mathrm}{Rep}^{0, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ and thus Theorem \[Main2\] for $r=0$ is an easy consequence of Corollary \[FFTHMtorcris\]. Hence we may assume $r\ge 1$. The rest of a proof is similar to the proof of Theorem \[Main2\]. Some consequences {#consequences} ----------------- In this subsection, we generalize some results proved in Section 3.4 of [@Br3]. First of all, we show the following elementary lemma, which should be well-known to experts, but we include a proof here for the sake of completeness. \[stability’\] The full subcategories ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ and ${\mathrm}{Rep}^{r,{\mathrm}{st}}_{{\mathrm}{tor}}(G_K)$ of ${\mathrm}{Rep}_{{\mathrm}{tor}}(G_K)$ are stable under formation of subquotients, direct sums and the association $T\mapsto T^{\vee}(r)$. Here $T^{\vee}={\mathrm}{Hom}_{{\mathbb}{Z}_p}(T,{\mathbb}{Q}_p/{\mathbb}{Z}_p)$ is the dual representation of $T$. We prove the statement only for ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. Let $T\in {\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ be killed by $p^n$ for some $n>0$. Assertions for quotients and direct sums are clear. We prove that $T^{\vee}(r)$ is contained in ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. There exist lattices $L_1\subset L_2$ in some crystalline ${\mathbb}{Q}_p$-representation of $G_K$ and an exact sequence $0\to L_1\to L_2\to T\to 0$ of ${\mathbb}{Z}_p[G_K]$-modules. This exact sequence induces an exact sequence $0\to T\to L_1/p^nL_1\to L_2/p^nL_2\to T\to 0$ of finite ${\mathbb}{Z}_p[G_K]$-modules. By duality, we obtain an exact sequence $0\to T^{\vee}\to (L_2/p^nL_2)^{\vee}\to (L_1/p^nL_1)^{\vee}\to T^{\vee}\to 0$ of finite ${\mathbb}{Z}_p[G_K]$-modules. Then we obtain a $G_K$-equivariant surjection $L_1^{\vee}\twoheadrightarrow T^{\vee}$ by the composite $L_1^{\vee}\twoheadrightarrow L_1^{\vee}/p^nL_1^{\vee}\overset{\sim}{\to} (L_1/p^nL_1)^{\vee}\twoheadrightarrow T^{\vee}$ of natural maps (here, for any free ${\mathbb}{Z}_p$-representation $L$ of $G_K$, $L^{\vee}:={\mathrm}{Hom}_{{\mathbb}{Z}_p}(L,{\mathbb}{Z}_p)$ stands for the dual of $L$). Therefore, we obtain $L_1^{\vee}(r)\twoheadrightarrow T^{\vee}(r)$ and thus $T^{\vee}(r)\in {\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. Finally, we prove the stability assertion for subobjects. Let $T'$ be a $G_K$-stable submodule of $T$. We have a $G_K$-equivariant surjection $f\colon L_1^{\vee}\twoheadrightarrow T^{\vee}\twoheadrightarrow (T')^{\vee}$. Let $L'_2$ be a free ${\mathbb}{Z}_p$-representation of $G_K$ such that its dual is the kernel of $f$. We have an exact sequence $0\to (L'_2)^{\vee}\to L^{\vee}_1\overset{f}{\to} (T')^{\vee}\to 0$ of ${\mathbb}{Z}_p[G_K]$-modules. Repeating the construction of the surjection $L_1^{\vee}\twoheadrightarrow T^{\vee}$, we obtain a $G_K$-equivariant surjection $L'_2=(L'_2)^{\vee \vee}\twoheadrightarrow (T')^{\vee \vee}=T'$ and thus we have $T'\in {\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. In the case where $r=1$, the assertion (1) of the following corollary was shown in Theorem 3.4.3 of [@Br3]. \[imagestable\] Let $T$ be an object of ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ which is killed by $p^n$ for some $n>0$. Let $T'$ be a $G_{\infty}$-stable subquotient of $T$. $(1)$ If $e(r-1)<p-1$, then $T'$ is $G_K$-stable $($with respect to $T)$. $(2)$ If $s>n-1+{\mathrm}{log}_p(r-(p-1)/e)$, then $T'$ is $G_s$-stable $($with respect to $T)$. By the duality assertion of Lemma \[stability’\], it is enough to show the case where $T'$ is a $G_{\infty}$-stable submodule of $T$. Take any sequence $T'=T_0\subset T_1\subset \cdots \subset T_m=T$ of torsion $G_{\infty}$-stable submodules of $T$ such that $T_i/T_{i-1}$ is irreducible for any $i$. As explained in the proof of Proposition \[FFLEM\], the $G_{\infty}$-action on $T_i/T_{i-1}$ can be (uniquely) extended to $G_K$. By Theorem \[tamelift\] given in the next section, we know that $T_i/T_{i-1}$ is an object of ${\mathrm}{Rep}^{r_0,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ where $r_0:={\mathrm}{max}\{ r'\in {\mathbb}{Z}_{\ge 0}; e(r'-1)<p-1 \}$. \(1) We may suppose $r=r_0$. The $G_{\infty}$-equivariant projection $T=T_m\twoheadrightarrow T_m/T_{m-1}$ is $G_K$-stable by the full faithfulness theorem (= Corollary \[FFTHMtorcris\]). Thus we know that $T_{m-1}$ is $G_K$-stable in $T$, and also know that $T_{m-1}$ is contained in ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ by Lemma \[stability’\]. By the same argument for the $G_{\infty}$-equivariant projection $T_{m-1}\twoheadrightarrow T_{m-1}/T_{m-2}$, we know that $T_{m-2}$ is $G_K$-stable in $T$, and also know that $T_{m-2}$ is contained in ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$. Repeating this argument, we have that $T'=T_0$ is $G_K$-stable in $T$. \(2) Put $J=u^pI^{[p^{s-n+1}]}W(R)$. By (1) we may assume $e(r-1)\ge p-1$. Under this assumption we have $r\ge r_0$ and $s>n-1+{\mathrm}{log}_p(r-(p-1)/e)\ge n-1$. In particular, $T|_{G_s}$ and $(T_i/T_{i-1})|_{G_s}$, for any $i$, are contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ by Proposition \[Main2Lem\]. First we consider the case where $k$ is algebraically closed. By Theorem \[FFTHM\], the $G_{\infty}$-equivariant projection $T=T_m\twoheadrightarrow T_m/T_{m-1}$ is $G_s$-stable. Thus we know that $T_{m-1}$ is $G_s$-stable in $T$, and also know that $T_{m-1}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$ by Corollary \[stability\]. By the same argument for the $G_{\infty}$-equivariant projection $T_{m-1}\twoheadrightarrow T_{m-1}/T_{m-2}$, we know that $T_{m-2}$ is $G_s$-stable in $T$, and also know that $T_{m-2}$ is contained in ${\widetilde}{{\mathrm}{Rep}}^{r,\hat{G}_s,J}_{{\mathrm}{tor}}(G_s)$. Repeating this argument, we have that $T'=T_0$ is $G_s$-stable in $T$. Next we consider the case where $k$ is not necessary algebraically closed. Let $L$ be the completion of the maximal unramified extension $K^{{\mathrm}{ur}}$ of $K$, and we identify $G_L$ with the inertia subgroup of $G_K$. Clearly $T|_{G_L}$ is contained in ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_L)$ and $T'$ is $G_{L_{\infty}}$-stable submodule of $T$. We have already shown that $T'$ is $G_{L,s}$-stable in $T$. Since $G_s$ is topologically generated by $G_{L,s}$ and $G_{\infty}$, we conclude that $T'$ is $G_s$-stable in $T$. Now let $V$ be a ${\mathbb}{Q}_p$-representation of $G_K$ and $T$ a ${\mathbb}{Z}_p$-lattice of $V$ which is stable under $G_{\infty}$. Then we know that $T$ is automatically $G_s$-stable for some $s\ge 0$. Indeed we can check this as follows. Take any $G_K$-stable ${\mathbb}{Z}_p$-lattice $T'$ of $V$ which contains $T$, and take an integer $n>0$ with the property that $p^nT'\subset T$. Furthermore, we take a finite extension $K'$ of $K$ such that $G_{K'}$ acts trivially on $T'/p^nT'$. Then $T/p^nT'$ is $G_{\infty}$-stable and also $G_{K'}$-stable in $T'/p^nT'$. If we take any integer $s\ge 0$ with the property $K'\cap K_{\infty}\subset K_{(s)}$, we know that $T/p^nT'$ is $G_s$-stable. This implies that $T$ is $G_s$-stable in $T'$. The following corollary, which was shown in Corollary 3.4.4 of [@Br3] in the case where $r=1$, is related with the above property. \[stablelattice\] Let $V$ be a crystalline ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$ and $T$ a finitely generated ${\mathbb}{Z}_p$-submodule of $V$ which is stable under $G_{\infty}$. If $e(r-1)<p-1$, then $T$ is stable under $G_K$. We completely follow the method of the proof of [@Br3 Corollary 3.4.4]. Take any $G_K$-stable ${\mathbb}{Z}_p$-lattice $T'$ of $V$ which contains $T$. Since $T'/p^nT'$ is contained in ${\mathrm}{Rep}^{r,{\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)$ for any $n>0$, Corollary \[imagestable\] (1) implies that any $G_{\infty}$-stable submodule of $T'/p^nT'$ is in fact $G_K$-stable. Thus $(T+p^nT')/p^nT'$ is $G_K$-stable in $T'/p^nT'$. Therefore, we obtain $g(T)\subset \bigcap_{n>0}\ (T+p^nT')=T$ for any $g\in G_K$. Crystalline lifts and c-weights =============================== We continue to use the same notation except for that we may allow $p=2$. We remark that a torsion ${\mathbb}{Z}_p$-representation of $G_K$ is torsion crystalline with Hodge-Tate weights in $[0,r]$ if there exist a lattice $L$ in some crystalline ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$ and a $G_K$-equivariant surjection $f\colon L\twoheadrightarrow T$. We call $f$ a [*crystalline lift*]{} ([*of $T$*]{}) [*of weight $\le r$*]{}. Our interest in this section is to determine the minimum integer $r$ (if it exists) such that $T$ admits crystalline lifts of weight $\le r$. We call this minimum integer the [*c-weight of $T$*]{} and denote it by $w_c(T)$. If $T$ does not have crystalline lifts of weight $\le r$ for any integer $r$, then we define the c-weight $w_c(T)$ of $T$ to be $\infty$. Motivated by [@CL2 Question 5.5], we raise the following question. For a torsion ${\mathbb}{Z}_p$-representation $T$ of $G_K$, is the c-weight $w_c(T)$ of $T$ finite? Furthermore, can we calculate $w_c(T)$? General properties of c-weights ------------------------------- We study general properties of c-weights. At first, by ramification estimates, it is known that c-weights may have infinitely large values ([@CL2 Theorem 5.4]); for any $c>0$, there exists a torsion ${\mathbb}{Z}_p$-extension $T$ of $G_K$ with $w_c(T)>c$. In this paper, we mainly consider representations with “small” c-weights. If c-weights are “small”, they are closely related with [*tame inertia weights*]{}. Now we recall the definition of tame inertia weights. Let $I_K$ be the inertia subgroup of $G_K$. Let $T$ be a $d$-dimensional irreducible ${\mathbb}{F}_p$-representation of $I_K$. Then $T$ is isomorphic to $${\mathbb}{F}_{p^d}(\theta^{n_1}_{d,1}\cdots \theta^{n_d}_{d,d})$$ for one sequence of integers between $0$ and $p-1$, periodic of period $d$. Here, $\theta_{d,1},\dots , \theta_{d,d}$ are the fundamental characters of level $d$. The integers $n_1/e,\dots ,n_d/e$ are called the tame inertia weights of $T$. For any ${\mathbb}{F}_p$-representation $T$ of $G_K$, the tame inertia weights of $T$ are the tame inertia weights of the Jordan-Hölder quotients of $T|_{I_K}$. Let $\chi_p\colon G_K\to {\mathbb}{Z}^{\times}_p$ be the $p$-adic cyclotomic character and $\bar{\chi}_p\colon G_K\to {\mathbb}{F}^{\times}_p$ the mod $p$ cyclotomic character. It is well-known that $\bar{\chi}_p|_{I_K}=\theta_1^e$ where $\theta_1\colon I_K\twoheadrightarrow {\mathbb}{F}^{\times}_p$ is the fundamental character of level $1$. In particular, denoting by $K^{{\mathrm}{ur}}$ the maximal unramified extension of $K$, we have $[K^{{\mathrm}{ur}}(\mu_p):K^{{\mathrm}{ur}}]=(p-1)/{\mathrm}{gcd}(e,p-1)$. \[poly\] $(1)$ Minimum c-weights are invariant under finite unramified extensions of the base field $K$. $(2)$ The c-weight of an unramified torsion ${\mathbb}{Z}_p$-representation of $G_K$ is $0$. $(3)$ Put $\nu=(p-1)/{\mathrm}{gcd}(e,p-1)$. If $\nu (s-1)<w_c(T)\le \nu s$, then we have $\nu (s-1)<w_c(T^{\vee})\le \nu s$. In particular, if $(p-1)\mid e$, then we have $w_c(T)=w_c(T^{\vee})$. $(4)$ Let $T$ be an ${\mathbb}{F}_p$-representation of $G_K$ and $i$ the largest tame inertia weight of $T$. Then we have $w_c(T)\geq i$. \(1) Let $T$ be a torsion ${\mathbb}{Z}_p$-representations of $G_K$. Let $K'$ be a finite unramified extension of $K$. It suffices to prove that $T$ has crystalline lifts of weight $\le r$ if and only if $T|_{G_{K'}}$ has crystalline lifts of weight $\le r$. The “only if” assertion is clear and thus it is enough to prove the “if” assertion. Let $f\colon L\twoheadrightarrow T|_{G_{K'}}$ be a crystalline lift of $T|_{G_{K'}}$ of weight $\le r$. Since $K'/K$ is unramified, ${\mathrm}{Ind}^{G_K}_{G_{K'}}L$ is a lattice in some crystalline ${\mathbb}{Q}_p$-representation of $G_K$ with Hodge-Tate weights in $[0,r]$. Furthermore, the map $${\mathrm}{Ind}^{G_K}_{G_{K'}}L={\mathbb}{Z}_p[G_K]\otimes_{{\mathbb}{Z}_p[G_{K'}]} L\to T,\quad \sigma\otimes x\mapsto \sigma(f(x))$$ is a $G_K$-equivariant surjection and hence we have done. \(2) The result follows from (1) immediately. \(3) Taking a finite unramified extension $K'$ of $K$ with the property $[K^{{\mathrm}{ur}}(\mu_p):K^{{\mathrm}{ur}}]=[K'(\mu_p):K']$, it follows from Lemma \[stability’\] that we have $\nu (s-1)<w_c(T|_{G_K'})\le \nu s$ if and only if we have $\nu (s-1)<w_c((T^{\vee})|_{G_K'})\le \nu s$. Thus the result follows from the assertion (1). \(4) If $ew_c(T)\ge p-1$, then there is nothing to prove, and thus we may suppose that $ew_c(T)< p-1$. Let $L\twoheadrightarrow T$ be a crystalline lift of $T$ of weight $\le w_c(T)$. Since the tame inertia polygon of $L$ lies on the Hodge polygon of $L$ ([@CS Théorème 1]), the largest slope of the former polygon is less than or equal to that of the latter polygon. This implies $w_c(T)\geq i$. \[tamelift\] Let $T$ be a tamely ramified ${\mathbb}{F}_p$-representation of $G_K$. Let $i$ be the largest tame inertia weight of $T$. Then we have $w_c(T)={\mathrm}{min}\{ h\in {\mathbb}{Z}_{\ge 0}; h\ge i \}$. The proof below is essentially due to Caruso and Liu [@CL2 Theorem 5.7], but we give a proof here for the sake of completeness. Put $i_0={\mathrm}{min}\{ h\in {\mathbb}{Z}_{\ge 0}; h\ge i \}$. By Proposition \[poly\] (4), we have $w_c(T)\ge i_0$. Thus it suffices to show $w_c(T)\le i_0$. We note that $T|_{I_K}$ is semi-simple. Any irreducible component $T_0$ of $T|_{I_K}$ is of the form $ {\mathbb}{F}_{p^d}(\theta^{n_1}_{d,1}\cdots \theta^{n_d}_{d,d}) $ for one sequence of integers between $0$ and $p-1$, periodic of period $d$. We decompose $n_j=em_j+n'_j$ by integers $0\le m_j\le i_0$ and $0\le n'_j<e$. Now we define an integer $k_{j, \ell}$ by $$\begin{aligned} k_{j, \ell}:= \left\{ \begin{array}{ll} e\quad \hspace{2.0mm} {\rm if}\ 1\le \ell\le m_j, \cr n'_j\quad {\rm if}\ \ell=m_j+1, \cr 0\quad \hspace{2.0mm} {\rm if}\ \ell>m_j+1. \end{array} \right.\end{aligned}$$ Note that we have $n_j=\sum^{i_0}_{\ell=1} k_{j, \ell}$, and also have an $I_K$-equivariant surjection $$T_0={\mathbb}{F}_{p^d}(\theta^{n_1}_{d,1}\dots \theta^{n_d}_{d,d}) =\bigotimes_{\ell=1,\dots i_0, {\mathbb}{F}_{p^d}} {\mathbb}{F}_{p^d}(\theta^{k_{1,\ell}}_{d,1}\dots \theta^{k_{d,\ell}}_{d,d}) \twoheadleftarrow \bigotimes_{\ell=1,\dots i_0, {\mathbb}{F}_p} {\mathbb}{F}_{p^d}(\theta^{k_{1,\ell}}_{d,1}\dots \theta^{k_{d,\ell}}_{d,d}).$$ By a classical result of Raynaud, each ${\mathbb}{F}_{p^d}(\theta^{k_{1,\ell}}_{d,1}\cdots \theta^{k_{d,\ell}}_{d,d})$ comes from a finite flat group scheme defined over $K^{{\mathrm}{ur}}$. We should remark that such a finite flat group scheme is in fact defined over a finite unramified extension of $K$. Since any finite flat group scheme can be embedded in a $p$-divisible group, the above observation implies the following: there exist a finite unramified extension $K'$ over $K$, a lattice $L$ in some crystalline ${\mathbb}{Q}_p$-representation of $G_{K'}$ with Hodge-Tate weights in $[0, i_0]$ and an $I_K$-equivariant surjection $f\colon L\twoheadrightarrow T$. The map $f$ induces an $I_K$-equivariant surjection $\tilde{f}\colon L/pL\twoheadrightarrow T$. Since $L/pL$ and $T$ is finite, we see that $\tilde{f}$ is in fact $G_{K''}$-equivariant for some finite unramified extension $K''$ over $K'$, and then so is $f$. Therefore, we obtain $w_c(T|_{G_{K''}})\le i_0$. By Proposition \[poly\] (1), we obtain $w_c(T)\le i_0$. Rank $2$ cases -------------- We give some computations of c-weights related with torsion representations of rank $2$. We prove the following lemma by an almost identical method with [@GLS Lemma 9.4]. \[2liftlem\] Let $K$ be a finite extension of ${\mathbb}{Q}_p$. Let $E$ be a finite extension of ${\mathbb}{Q}_p$ with residue field ${\mathbb}{F}$. Let $i$ and $\nu$ be integers such that $\nu$ is divisible by $[K(\mu_p):K]$. Suppose that $T$ is an ${\mathbb}{F}$-representation of $G_K$ which sits in an exact sequence $(\ast)\colon 0\to {\mathbb}{F}(i)\to T\to {\mathbb}{F}\to 0$ of ${\mathbb}{F}$-representations of $G_K$. Then there exist a ramified degree at most $2$ extension $E'$ over $E$, with integer ring ${\mathcal{O}}_{E'}$, and an unramified continuous character $\chi\colon G_K\to {\mathbb}{F}^{\times}$ with trivial reduction such that $(\ast)$ is the reduction of some exact sequence $0\to {\mathcal{O}}_{E'}(\chi \chi_p^{i+\nu})\to \Lambda\to {\mathcal{O}}_{E'}\to 0$ of free ${\mathcal{O}}_{E'}$-representations of $G_K$. Furthermore, we have the followings: $(1)$ If $i+\nu=1$ or $\bar{\chi}_p^{1-i}\not= 1$, then we can take $E'=E$ and $\chi=1$. $(2)$ If $i+\nu=0$ and $T$ is unramified, then we can take $E'=E$, $\chi=1$ and $\Lambda$ to be unramified. Suppose $i+\nu=1$ (resp. $\bar{\chi}_p^{1-i}\not= 1$). Then the map $H^1(K, {\mathcal{O}}_E(i+\nu))\to H^1(K,{\mathbb}{F}(i))$ arising from the exact sequence $0\to {\mathcal{O}}_E(i+\nu)\overset{\varpi}{\to} {\mathcal{O}}_E(i+\nu)\to {\mathbb}{F}(i)\to 0$ is surjective since $H^2(K, {\mathcal{O}}_E(1))\simeq {\mathcal{O}}_E$ (resp. $H^2(K, {\mathcal{O}}_E(i+\nu))=0$), where $\varpi$ is a uniformizer of $E$. Hence we obtained a proof of (1). The assertion (2) follows immediately from the fact that the natural map $H^1(G_K/I_K, {\mathcal{O}}_E)\to H^1(G_K/I_K, {\mathbb}{F})$ is surjective. In the rest of this proof, we always assume that $i+\nu\not=1$ and $\bar{\chi}_p^{1-i}= 1$. Let $L\in H^1(K,{\mathbb}{F}(i))$ be a $1$-cocycle corresponding to $(\ast)$. We may suppose $L \not=0$. For any unramified continuous character $\chi\colon G_K\to {\mathbb}{F}^{\times}$ with trivial reduction, we denote by $$\begin{aligned} &\delta^1_{\chi}\colon H^1(K,{\mathbb}{F}(i))\to H^2(K, {\mathcal{O}}_E(\chi\chi_p^{i+\nu}))\\ ({\mathrm}{resp.}\ &\delta^0_{\chi}\colon H^0(K, E/{\mathcal{O}}_E(\chi^{-1}\chi_p^{1-i-\nu}))\to H^1(K, {\mathbb}{F}))\end{aligned}$$ the connection map arising from the exact sequence $0\to {\mathcal{O}}_E(\chi\chi_p^{i+\nu})\overset{\varpi}{\to} {\mathcal{O}}_E(\chi\chi_p^{i+\nu}) \to {\mathbb}{F}(i)\to 0$ (resp.  $0\to {\mathbb}{F}\to E/{\mathcal{O}}_E(\chi^{-1}\chi_p^{1-i-\nu})\overset{\varpi}{\to} E/{\mathcal{O}}_E(\chi^{-1}\chi_p^{1-i-\nu})\to 0$) of ${\mathcal{O}}_E[G_K]$-modules. Consider the following commutative diagram: $\displaystyle \xymatrix{ H^1(K,{\mathbb}{F}(i)) \ar_{\delta^1_{\chi}}[d] & \times & H^1(K,{\mathbb}{F}) \ar[rrr] & & & E/{\mathcal{O}}_E \ar@{=}[d] \\ H^2(K, {\mathcal{O}}_E(\chi\chi_p^{i+\nu})) & \times & H^0(K, E/{\mathcal{O}}_E(\chi^{-1}\chi_p^{1-i-\nu})) \ar^{\delta^0_{\chi}}[u] \ar[rrr] & & & E/{\mathcal{O}}_E }$ Since the above two pairings are perfect, we see that $L$ lifts to $H^1(G_K, {\mathcal{O}}_E(\chi\chi_p^{i+\nu}))$ if and only if $H$ is contained in the image of $\delta^0_{\chi}$. Here, $H\subset H^1(K,{\mathbb}{F})$ is the annihilator of $L$ under the local Tate pairing $H^1(K,{\mathbb}{F}(i)) \times H^1(K,{\mathbb}{F}) \to E/{\mathcal{O}}_E$. Let $n\ge 1$ be the largest integer with the property that $\chi^{-1}\chi_p^{1-i-\nu}\equiv 1\ {\mathrm}{mod}\ \varpi^n$ (such $n$ exists since $\bar{\chi}_p^{1-i}=1$ and $1-i-\nu\not=0$). We define $\alpha_{\chi}\colon G_K\to {\mathcal{O}}_E$ by the relation $\chi^{-1}\chi_p^{1-i-\nu}=1+\varpi^n\alpha_{\chi}$, and denote $(\alpha_{\chi}\ {\mathrm}{mod}\ \varpi)\colon G_K\to {\mathbb}{F}$ by $\bar{\alpha}_{\chi}$. By definition, $\bar{\alpha}_{\chi}$ is a non-zero element of $H^1(K,{\mathbb}{F})$, and it is not difficult to check that the image of $\delta^0_{\chi}$ is generated by $\bar{\alpha}_{\chi}$. If $\bar{\alpha}_{\chi}$ is contained in $H$ for some $\chi$, we are done. Suppose this is not the case. Suppose that $H$ is not contained in the unramified line in $H^1(K,{\mathbb}{F})$. We claim that we can choose $\chi$ such that $\bar{\alpha}_{\chi}$ is ramified. Let $m$ be the largest integer with the property that $(\chi^{-1}\chi_p^{1-i-\nu})|_{I_K}\equiv 1\ {\mathrm}{mod}\ \varpi^n$. Clearly, we have $m\ge n$. If $m=n$, then we are done and thus we may assume $m>n$. Fix a lift $g\in G_K$ of the Frobenius of $K$. We see that $\bar{\alpha}_{\chi}(g)\not= 0$. Let $\chi'$ be the unramified character sending $g$ to $1+\varpi^n\alpha_{\chi}(g)$. Then $\chi'$ has trivial reduction. After replacing $\chi$ with $\chi\chi'$, we reduce the case where $m=n$ and thus the claim follows. Suppose $\bar{\alpha}_{\chi}$ is ramified. Then there exists a unique $\bar{x}\in {\mathbb}{F}^{\times}$ such that $\bar{\alpha}_{\chi}+u_{\bar x}\in H$ where $u_{\bar x}\colon G_K\to {\mathbb}{F}$ is the unramified character sending $g$ to $\bar x$. Denote by $\chi''$ the unramified character sending $g$ to $1+\varpi^n\alpha_{\chi}(g)$. Replacing $\chi$ with $\chi\chi''$, we have done. Suppose that $H$ is contained in the unramified line in $H^1(K,{\mathbb}{F})$ (thus $H$ and the unramified line coincide with each other). By replacing $E$ with $E(\sqrt{\varpi})$, we may assume that $n>1$. Let $\chi_0$ be a character defined by $\chi$ times the unramified character sending our fixed $g$ to $1+\varpi$. Since $n>1$, we see that $\chi_0^{-1}\chi_p^{1-i-\nu}\equiv 1\ {\mathrm}{mod}\ \varpi$ and $\chi_0^{-1}\chi_p^{1-i-\nu}\not\equiv 1\ {\mathrm}{mod}\ \varpi^2$. We define $\alpha_{\chi_0}\colon G_K\to {\mathcal{O}}_E$ by the relation $\chi_0^{-1}\chi_p^{1-i-\nu}=1+\varpi\alpha_{\chi_0}$, and denote $(\alpha_{\chi_0}\ {\mathrm}{mod}\ \varpi)\colon G_K\to {\mathbb}{F}$ by $\bar{\alpha}_{\chi_0}$. By definition and the assumption $n>1$, $\bar{\alpha}_{\chi_0}$ is a non-zero unramified element of $H^1(K,{\mathbb}{F})$, hence it is contained in $H$. Therefore, we have done. Let $K$ be a finite extension of ${\mathbb}{Q}_p$, $n\ge 2$ an integer and $\chi\colon G_K\to E^{\times}$ an unramified character. Since any $E$-representation of $G_K$ which is an extension of $E$ by $E(\chi\chi_p^n)$ is automatically crystalline, we obtain the following. \[rank2\] Suppose $p>2$. Let $K$ be a finite unramified extension of ${\mathbb}{Q}_p$. Let $T\in {\mathrm}{Rep}_{{\mathrm}{tor}}(G_K)$ be killed by $p$ and sit in an exact sequence $0\to {\mathbb}{F}_p(i)\to T\to {\mathbb}{F}_p\to 0$ of ${\mathbb}{F}_p$-representations of $G_K$. Then we have the followings: $(1)$ If $i=0$ and $T$ is unramified, then we have $w_c(T)=0$. $(2)$ If $i=0$ and $T$ is not unramified, then we have $w_c(T)=p-1$. $(3)$ If $i=2,\dots, p-2$, then we have $w_c(T)=i$. (1), (2) By Lemma \[2liftlem\] (2), it suffices to prove that $T$ is not torsion crystalline with Hodge-Tate weights in $[0,p-2]$ if $T$ is not unramified. Let $K_T$ be the definition field of the representation $T$ of $G_K$ and put $G={\mathrm}{Gal}(K_T/K)$. Let $G^j$ be the upper numbering $j$-th ramification subgroup of $G$ (in the sense of [@Se]). Since $T$ is not unramified and killed by $p$, we see that $K_T$ is a totally ramified degree $p$ extension over $K$. Thus $G^1$ is the wild inertia subgroup of $G$ and $G^1=G$, which does not act on $T$ trivial by the definition of $G$. Thus we obtain the desired result by ramification estimates of [@Fo1] (or [@Ab1]) for torsion crystalline representations with Hodge-Tate weights in $[0,p-2]$: if $T$ is torsion crystalline with Hodge-Tate weights in $[0,p-2]$, then $G^j$ acts on $T$ trivial for any $j>(p-2)/(p-1)$. \(3) The result follows immediately from Proposition \[poly\] (4) and Lemma \[2liftlem\]. \[2lift\] Let $K$ be a finite unramified extension of ${\mathbb}{Q}_p$. Then any $2$-dimensional ${\mathbb}{F}_p$-representation of $G_K$ is torsion crystalline with Hodge-Tate weights in $[0,2p-2]$. If $T$ is irreducible, the result follows from Theorem \[tamelift\]. Assume that $T$ is reducible. Since $K$ is unramified over ${\mathbb}{Q}_p$, any continuous character $G_K\to {\mathbb}{F}^{\times}_p$ is of the form $\chi \bar{\chi}^i_p$ for some unramified character $\chi$ and some integer $i$. Replacing $K$ with its finite unramified extension, we may assume that $T$ sits in an exact sequence $0\to {\mathbb}{F}_p(i)\to T\to {\mathbb}{F}_p(j)\to 0$ of ${\mathbb}{F}_p$-representations of $G_K$, where $i$ and $j$ are integers in the range $[0,p-2]$ (we remark that $w_c(T)$ is invariant under unramified extensions of $K$ by Proposition \[poly\] (1)). It follows from Lemma \[2liftlem\] that $w_c(T(-j))\le p$. Therefore, we obtain $w_c(T)=w_c(T(-j)\otimes_{{\mathbb}{F}_p} {\mathbb}{F}_p(j)) \le w_c(T(-j))+ w_c({\mathbb}{F}_p(j))\le p+(p-2) =2p-2$. Extensions of ${\mathbb}{F}_p$ by ${\mathbb}{F}_p(1)$ and non-fullness theorems ------------------------------------------------------------------------------- By Lemma \[2liftlem\], we know that the c-weight $w_c(T)$ of an ${\mathbb}{F}_p$-representation $T$ of $G_K$ which sits in an exact sequence $0\to {\mathbb}{F}_p(1)\to T\to {\mathbb}{F}_p\to 0$ of ${\mathbb}{F}_p$-representations of $G_K$, is less than or equal to $p$. Let us calculate $w_c(T)$ for such $T$ more precisely. We should remark that such $T$ is written as $p$-torsion points of a Tate curve. Hence we consider torsion representations coming from Tate curves. Let $v_K$ be the valuation of $K$ normalized such that $v_K(K^{\times})={\mathbb}{Z}$, and take any $q\in K^{\times}$ with $v_K(q)>0$. Let $E_q$ be the Tate curve over $K$ associated with $q$ and $E_q[p^n]$ the module of $p^n$-torsion points of $E_q$ for any integer $n>0$. It is well-known that there exists an exact sequence $$(\#)\quad 0\to \mu_{p^n}\to E_q[p^n]\to {\mathbb}{Z}/p^n{\mathbb}{Z}\to 0$$ of ${\mathbb}{Z}_p[G_K]$-modules. Here, $\mu_{p^n}$ is the group of $p^n$-th roots of unity in $\overline{K}$. Let $x_n\colon G_K\to \mu_{p^n}$ be the $1$-cocycle defined to be the image of $1$ for the connection map $H^0(K, {\mathbb}{Z}/p^n{\mathbb}{Z})\to H^1(K, \mu_{p^n})$ arising from the exact sequence $(\#)$. Then $x_n$ corresponds to $q$ mod $(K^{\times})^{p^n}$ via the isomorphism $K^{\times}/(K^{\times})^{p^n}\simeq H^1(K,\mu_{p^n})$ of Kummer theory. Thus the exact sequence $(\#)$ splits if and only if $q\in (K^{\times})^{p^n}$. First we consider the case $p\mid v_K(q)$ (i.e.  [*peu ramifié*]{} case). \[easycase\] Let $K$ be a finite extension of ${\mathbb}{Q}_p$. If $p\mid v_K(q)$, then $E_q[p]$ is the reduction modulo $p$ of a lattice in some $2$-dimensional crystalline ${\mathbb}{Q}_p$-representation with Hodge-Tate weights in $[0,1]$. Since $p\mid v_K(q)$, there exists $q'\in K^{\times}$ such that $v_K(q'-1)>0$ and $q\equiv q'$ mod $(K^{\times})^p$. Considering the exact sequence $0\to {\mathbb}{Z}_p(1)\to L\to {\mathbb}{Z}\to 0$ of ${\mathbb}{Z}_p$-representations of $G_K$ corresponding to $q'$ via the isomorphism $H^1(K,{\mathbb}{Z}_p(1))\simeq {\varprojlim}_{n} K^{\times}/(K^{\times})^{p^n}$ of Kummer theory, we obtain the desired result. \[minwt\] Suppose that $K$ is a finite extension of ${\mathbb}{Q}_p$, $(p-1)\nmid e$ and $p\mid v_K(q)$. Then we have $w_c(E_q[p])=1$. By the assumption $(p-1)\nmid e$, we know that the largest tame inertia weight of $E_q[p]$ is positive. Thus Proposition \[poly\] (4) shows $w_c(E_q[p])\ge 1$. The inequality $w_c(E_q[p])\le 1$ follows from Lemma \[easycase\]. Next we consider the case $p\nmid v_K(q)$ (i.e. [*très ramifié*]{} case). \[Tatecurve\] If $e(r-1)<p-1$ and $p\nmid v_K(q)$, then $E_q[p^n]$ is not torsion crystalline with Hodge-Tate weights in $[0,r]$ for any $n>0$. If $e=1$, the fact that $E_{\pi}[p^n]$ is not torsion crystalline with Hodge-Tate weights in $[0,p-1]$ immediately follows from the theory of ramification bound as below. We may suppose $n=1$. Suppose $E_{\pi}[p]$ is torsion crystalline with Hodge-Tate weights in $[0,p-1]$. Then the upper numbering $j$-th ramification subgroup $G^j_K$ of $G_K$ (in the sense of [@Se]) acts trivially on $E_{\pi}[p]$ for any $j>1$ ([@Ab1 Section 6, Theorem 3.1]). However, this contradicts the fact that the upper bound of the ramification of $E_{\pi}[p]$ is $1+1/(p-1)$. We may suppose $n=1$. We choose any uniformizer $\pi'$ of $K$. Putting $v_K(q)=m$, we can write $q=(\pi')^mx$ with some unit $x$ of the integer ring of $K$. Since $m$ is prime to $p$, we have a decomposition $x=\zeta_{\ell}y^m$ in $K^{\times}$ for some $\ell>0$ prime to $p$ and $y\in K$ with $v_K(y-1)>0$. Here $\zeta_{\ell}$ is a (not necessary primitive) $\ell$-th root of unity. Since $\ell$ is prime to $p$, we have $\zeta_{\ell}=\zeta^{ps}_{\ell}$ for some integer $s$. We put $\pi=\pi'y$. This is a uniformizer of $K$. Choose any $p$-th root $\pi_1$ of $\pi$ and put $q_1=\zeta^s_{\ell}\pi^m_1\in K(\pi_1)^{\times}$. Then we have $q=q^p_1\in (K(\pi_1)^{\times})^p$ and in particular, the exact sequence $(\#)$ (for $n=1$) splits as representations of ${\mathrm}{Gal}(\overline{K}/K(\pi_1))$. Now assume that $E_q[p]$ is torsion crystalline with Hodge-Tate weights in $[0,r]$. Then $(\#)$ (for $n=1$) splits as representations of $G_K$ by Corollary \[FFTHMtorcris\]. This contradicts the assumption $p\nmid v_K(q)$ (and hence $q\notin (K^{\times})^p$). Now we put $r'_0={\mathrm}{min}\{r\in {\mathbb}{Z}_{\ge 0} ; e(r-1)\ge p-1\}$. Recall that we have $[K^{{\mathrm}{ur}}(\mu_p):K^{{\mathrm}{ur}}]=(p-1)/{\mathrm}{gcd}(e,p-1)$. \[roughbound\] Let $K$ be a finite extension of ${\mathbb}{Q}_p$. Then $E_q[p]$ is torsion crystalline with Hodge-Tate weights in $[0,1+(p-1)/{\mathrm}{gcd}(e,p-1)]$. Taking a finite unramified extension $K'$ of $K$ such that $[K^{{\mathrm}{ur}}(\mu_p):K^{{\mathrm}{ur}}]=[K'(\mu_p):K']$, we obtain $w_c((E_q[p])|_{G_{K'}})\le 1+(p-1)/{\mathrm}{gcd}(e,p-1)$ by Lemma \[2liftlem\]. Thus we have $w_c(E_q[p])\le 1+(p-1)/{\mathrm}{gcd}(e,p-1)$ by Proposition \[poly\] (1). \[wctr1\] Suppose that $K$ is a finite extension of ${\mathbb}{Q}_p$, and also suppose $e\mid (p-1)$ or $(p-1)\mid e$. We further suppose that $p\nmid v_K(q)$. Then we have $w_c(E_q[p])=r'_0$. We have $w_c(E_q[p])\le r'_0$ by Lemma \[roughbound\]. In addition, we also have $w_c(E_q[p])\ge r'_0$ by Proposition \[Tatecurve\]. Lemma \[roughbound\] gives some non-fullness results on torsion crystalline representations. \[nonfullthm\] Suppose that $K$ is a finite extension of ${\mathbb}{Q}_p$. If $r\ge 1+(p-1)/{\mathrm}{gcd}(e,p-1)$, then the restriction functor ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_1)$ is not full. Two representations $E_{\pi}[p]$ and ${\mathbb}{F}_p(1)\oplus {\mathbb}{F}_p$ are objects of ${\mathrm}{Rep}^r_{{\mathrm}{tor}}(G_K)$ by Lemma \[roughbound\]. They are not isomorphic as representations of $G_K$ but isomorphic as representations of $G_1$. Thus the desired non-fullness follows. \[p2\] Suppose that any one of the following holds: - $p=2$ and $K$ is a finite extension of ${\mathbb}{Q}_2$ $($in this case $r'_0=2)$; - $K$ is a finite unramified extension of ${\mathbb}{Q}_p$ $($in this case $r'_0=p)$; - $K$ is a finite extension of ${\mathbb}{Q}_p(\mu_p)$ $($in this case $r'_0=2)$. Then the restriction functor ${\mathrm}{Rep}^{r, {\mathrm}{cris}}_{{\mathrm}{tor}}(G_K)\to {\mathrm}{Rep}_{{\mathrm}{tor}}(G_1)$ is not full. [1000]{} Victor Abrashkin, *Modular representations of the Galois group of a local field and a generalization of a conjecture of Shafarevich*, Izv. Akad. Nauk SSSR Ser. Mat. [**53**]{} (1989), no. 6, 1135–1182 (Russian), Math. USSR-Izv. [**35**]{}, no. 3, 469–518 (English). Victor Abrashkin, *Group schemes of period $p>2$*, Proc. Lond. Math. Soc. (3), [**101**]{} (2010), 207–259. Christophe Breuil, *Représentations [$p$]{}-adiques semi-stables et transversalité de Griffiths*, Math. Ann. [**307**]{} (1997), 191–224. Christophe Breuil, *Une application du corps des normes*, Compos. Math. [**117**]{} (1999) 189–203. Christophe Breuil, *Integral $p$-adic Hodge theory*, in Algebraic geometry 2000, Azumino (Hotaka), Adv. Stud. Pure Math., vol. 36, Math. Soc. Japan, 2002, 51–80. Xavier Caruso, [Représentations semi-stables de torsion dans le case $er < p-1$]{}, J. Reine Angew. Math. [**594**]{} (2006), 35–92. Xavier Caruso and Tong Liu, *Quasi-semi-stable representations*, Bull. Soc. Math. France, [**137**]{} (2009), no. 2, 185–223. Xavier Caruso and Tong Liu, *Some bounds for ramification of $p^n$-torsion semi-stable representations*, J. Algebra, [**325**]{} (2011), 70–96. Xavier Caruso and David Savitt, *Polygons de Hodge, de Newton et de línertie moderee des representations semi-stables*, Math. Ann. [**343**]{} (2009): 773–789. Jean-Marc Fontaine, *Scémas propres et lisses sur ${\mathbb}{Z}$*, Proceedings of the Indo-French Conference on Geometry Bombay, 1989, (Hindustan Book Agency 1993), 43–56. Jean-Marc Fontaine, *Le corps des périodes $p$-adiques*, Astérisque (1994), no. 223, 59–111, With an appendix by Pierre Colmez, Périodes $p$-adiques (Bures-sur-Yvette, 1988). Toby Gee, Tong Liu and David Savitt, *The Buzzard-Diamond-Jarvis conjecture for unitary groups*, J. Amer. Math. Soc. [**27**]{} (2014), 389-435. Mark Kisin, *Crystalline representations and [$F$]{}-crystals*, Algebraic geometry and number theory, Progr. Math. [**253**]{}, Birkhäuser Boston, Boston, MA (2006), 459–496. Wansu Kim, *The classification of $p$-divisible groups over 2-adic discrete valuation rings*, Math. Res. Lett. [**19**]{} (2012), no. 1, 121–141. Eike Lau, *A relation between Dieudonné displays and crystalline Dieudonné theory*, preprint, arXiv:1006.2720v3 Tong Liu, *Torsion $p$-adic Galois representations and a conjecture of Fontaine*, Ann. Sci. École Norm. Sup. (4) [**40**]{} (2007), no. 4, 633–674. Tong Liu, *A note on lattices in semi-stable representations*, Math. Ann. [**346**]{} (2010), 117–138. Tong Liu, *Lattices in filtered $({\varphi},N)$-modules*, J. Inst. Math. Jussieu 2, Volume 11, Issue 03 (2012), 659-693. Tong Liu, *The correspondence between Barsotti-Tate groups and Kisin modules when $p=2$*, appear at Journal de Théroie des Nombres de Bordeaux. Daniel Quillen, *Higher algebraic $K$-theory: I*, in Algebraic $K$-theory, I: Higher $K$-theories (Seattle, 1972), Lecture Notes in Math. [**341**]{}, Springer-Verlag, New York, 1973, 85–147. Jean-Pierre Serre, *Corps locaux*, Hermann, Paris, 1968. Yoshiyasu Ozeki, *Torsion representations arising from $({\varphi},\hat{G})$-modules*, J. Number Theory [**133**]{} (2013), 3810–3861. Yoshiyasu Ozeki, *Full faithfulness theorem for torsion crystalline representations*, preprint, arXiv:1206.4751v4 [^1]: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, JAPAN. e-mail: [yozeki@kurims.kyoto-u.ac.jp]{} Supported by the JSPS Fellowships for Young Scientists. [^2]: This continuity condition is needed to assure that the $G_{L,s}$-action on $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$, defined after Definition \[Liumod\], is continuous (here, note that $\hat{T}_{L,s}(\hat{{\mathfrak{M}}})$ is the dual of $(W({\mathrm}{Fr}R)\otimes_{{\varphi}, {\mathfrak{S}}_{L,s}} {\mathfrak{M}})^{{\varphi}=1}$; see Corollary 3.12 of [@Oz1]). [^3]: As explained in the footnote of Definition \[Liumod\], we may ignore the continuity condition of (2) in the case where ${\mathfrak{M}}$ is torsion.

--- abstract: 'In this note, we study the $n \times n$ random Euclidean matrix whose entry $(i,j)$ is equal to $f ( \| X_i - X_j \| )$ for some function $f$ and the $X_i$’s are i.i.d. isotropic vectors in ${\mathbb{R}}^p$. In the regime where $n$ and $p$ both grow to infinity and are proportional, we give some sufficient conditions for the empirical distribution of the eigenvalues to converge weakly. We illustrate our result on log-concave random vectors.' address: 'CNRS & Université de Toulouse, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France' author: - Charles Bordenave bibliography: - 'mat.bib' title: On Euclidean random matrices in high dimension --- Introduction ============ Let $Y$ be an *isotropic* random vector in ${\mathbb{R}}^p$, i.e. ${\mathbb{E}}Y = 0$, ${\mathbb{E}}[ Y Y^T ] = I / p $, where $I$ is the identity matrix. Let $(X_1, \cdots, X_n)$ be independent copies of $Y$. We define the $n \times n$ matrix $A$ by, for all $1 {\leqslant}i , j {\leqslant}n$, $$A_{i j } =f ( \| X_i - X_j \| ^2 ),$$ where $f : [ 0, \infty) \to {\mathbb{R}}$ is a measurable function and $\| \cdot \|$ denotes the Euclidean norm. The matrix $A$ is a random Euclidean matrix. It has already attracted some attention see e.g. Mézard, Parisi and Zhee [@MR1724455], Vershik [@MR2086637] or Bordenave [@MR2462254] and references therein. If $B$ is a symmetric matrix of size $n$, then its eigenvalues, say $\lambda_1(B), \cdots, \lambda_n(B)$ are real. The empirical spectral distribution (ESD) of $B$ is classically defined as $$\mu_B = \frac 1 n \sum_{i=1} ^ n \delta_{\lambda_i(B)},$$ where $\delta_x$ is the Dirac delta function at $x$. In this note, we are interested in the asymptotic convergence of $\mu_A$ as $p$ and $n$ converge to $+\infty$. This regime has notably been previously considered in El Karoui [@MR2589315] and Do and Vu [@DoVu]. More precisely, we fix a sequence $p(n)$ such that $$\begin{aligned} \label{eq:povern} \lim_{n \to \infty} \frac{ p(n) }{ n } = y \in (0, \infty).\end{aligned}$$ Throughout this note, we consider, on a common probability space, an array of random variables $(X_k(n))_{ 1 {\leqslant}k {\leqslant}n}$ such that $(X_1(n), \cdots, X_n(n))$ are independent copies of $Y(n)$, an isotropic vector in ${\mathbb{R}}^{p(n)}$. For each $n$, we define the Euclidean matrix $A(n)$ associated. For ease of notation, we will often remove the explicit dependence in $n$: we write $p$, $Y$, $X_k$ or $A$ in place of $p(n)$, $Y(n)$, $X_k (n)$ or $A(n)$. The Marcenko-Pastur probability distribution with parameter $1/y$ is given by $$\nu_{MP} (dx) = (1 - y )^+ \delta_0 (dx) + \frac{ y } { 2 \pi x } \sqrt { (y_+ - x) ( x - y_-) } {\mathbf{1}}_{ [ y_- , y_+] } (x) dx,$$ where $x^+ = ( x \vee 0)$, $y_ {\pm} = ( 1 \pm \frac 1 {\sqrt y }) ^2$ and $dx$ denotes the Lebesgue measure. Since the celebrated paper of Marcenko and Pastur [@MR0208649], this distribution is known to be closely related to empirical covariance matrices in high-dimension. We say that $Y$ has a *log-concave distribution*, if $Y$ has a density on ${\mathbb{R}}^p$ which is log-concave. Log-concave random vectors have an increasing importance in convex geometry, probability and statistics (see e.g. Barthe [@MR2648682]). We will prove the following result. \[th:main\] If $Y$ has a log-concave distribution and $f$ is three times differentiable at $2$, then, almost surely, as $n \to \infty$, $ \mu_{A}$ converges weakly to $\mu$, the law of $f(0 ) - f(2) +2 f'(2) - 2 f'(2) S$, where $S$ has distribution $\nu_{MP}$. With the weaker assumption that $f$ is differentiable at $2$, Theorem \[th:main\] is conjectured in Do and Vu [@DoVu]. Their conjecture has motivated this note. It would follow from the thin-shell hypothesis which asserts that there exists $c > 0$, such that for any isotropic log-concave vector $Y$ in ${\mathbb{R}}^p$, ${\mathbb{E}}(\| Y\| -1 )^2 {\leqslant}c/p$ (see Anttila, Ball and Perissinaki [@MR1997580] and Bobkov and Koldobsky [@MR2083387]). Klartag [@MR2520120] has proved the thin-shell hypothesis for isotropic unconditional log-concave vectors. The proof of Theorem \[th:main\] will rely on two recent results on log-concave vectors. Let $X = X(n) $ be the $n \times n$ matrix with columns given by $(X_1(n), \cdots, X_n (n))$. Pajor and Pastur have proved the following : \[th:PP\] If $Y$ has a log-concave distribution, then, in probability, as $n \to \infty$, $ \mu_{ X ^T X }$ converges weakly to $ \nu_{MP}$. We will also rely on a theorem due to Guédon and Millman. \[th:GM\] There exist positive constants $c_0, c_1$ such that if $Y$ is an isotropic log-concave vector in ${\mathbb{R}}^p$, for any $t {\geqslant}0$, $${\mathbb{P}}{{{\left( {{{\left| \| Y \| - 1 \right|}}} {\geqslant}t \right)}}} {\leqslant}c_1 \exp {{{\left( - c_0 \sqrt p {{{\left( t \wedge t^3 \right)}}} \right)}}}.$$ With Theorems \[th:PP\] and \[th:GM\] in hand, the heuristic behind Theorem \[th:main\] is simple. Theorem \[th:GM\] implies that $\|X_i \|^2 \simeq 1$ with high probability. Hence, since $\| X_i - X_j \|^2 = \|X_i \|^2 + \|X_j \|^2 - 2 X_i ^T X_j $, a Taylor expansion of $f$ around $2$ gives $$A_{i j } \simeq \left\{ \begin{array}{ll} f ( 2 ) - 2 f' ( 2) X_i ^T X_j & \hbox{ if } i \ne j \\ f(0) & \hbox{ if } i = j. \end{array} \right.$$ In other words, the matrix $A$ is close to the matrix $$\label{eq:defM} M = ( f(0) - f(2) + 2 f'(2) ) I + f ( 2 ) J - 2 f'(2) X^T X,$$ where $I$ is the identity matrix and $J$ is the matrix with all entries equal to $1$. From Theorem \[th:PP\], $\mu_{X^TX}$ converges weakly to $\nu_{MP}$. Moreover, since $J $ has rank one, it is negligible for the weak convergence of ESD. It follows that $\mu_{M}$ is close to $\mu$. The actual proof of Theorem \[th:main\] will be elementary and it will follow this heuristic. We shall use some standard perturbation inequalities for the eigenvalues. The idea to perform a Taylor expansion was already central in [@MR2589315; @DoVu]. Beyond Theorems \[th:PP\]-\[th:GM\], the proof of Theorem \[th:main\] is not related to log-concave vectors. In fact, it is nearly always possible to linearize $f$ as soon as the norms of the vectors concentrate around their mean. More precisely, let us say that two sequences of probability measures $(\mu_n)$, $(\nu_n)$, are asymptotically weakly equal, if for any bounded continuous function $f$, $\int f d \mu_n - \int f d \nu_n$ converges to $0$. \[th:main2\] Assume that there exists an integer $\ell {\geqslant}1$ such that ${\mathbb{E}}{{{\left| \| Y \| -1 \right|}}}^{2 \ell} = O ( p^{-1} )$, and that for any ${\varepsilon}> 0$, $$\label{eq:boundEE} \lim_{n \to \infty} {\mathbb{P}}{{{\left( \max_{1 {\leqslant}i , j {\leqslant}n } {{{\left\{ {{{\left| \| X_i - X_j \|^2 - 2 \right|}}} \vee {{{\left| \| X_i\| ^2 - 1 \right|}}} \right\}}}} {\leqslant}{\varepsilon}\right)}}} = 1.$$ Then, if $f$ is $\ell$ times differentiable at $2$, almost surely, $ \mu_{A}$ is asymptotically weakly equal to the law of $f(0 ) - f(2) +2 f'(2) - 2 f'(2) S$, where $S$ has distribution ${\mathbb{E}}\mu_{X^T X}$. The case $\ell =1$ of the above statement is contained in Do and Vu [@DoVu Theorem 5]. Besides Theorem \[th:PP\], some general conditions on the matrix $X$ guarantee the convergence of $\mu_{X^T X}$, see Yin and Krishnaiah [@MR816299], G[ö]{}tze and Tikhomirov [@MR2092202] or Adamczak [@MR2820070]. Proofs ====== Perturbation inequalities ------------------------- We first recall some basic perturbation inequalities of eigenvalues and introduce a good notion of distances for ESD. For $\mu$, $\nu$ two real probability measures, the *Kolmogorov-Smirnov distance* can be defined as $$\label{eq:KSdual} d_{KS} ( \mu , \nu) = \sup \left\{ \int f d \mu - \int f d \nu : \| f \|_{BV} {\leqslant}1 \right\},$$ where, for $f : {\mathbb{R}}\to {\mathbb{R}}$, the bounded variation norm is $ \| f \|_{BV} = \sup \sum_{ k \in {\mathbb{Z}}} | f ( x_{k+1}) - f (x_k) |, $ and the supremum is over all real increasing sequence $(x_k)_{ k \in {\mathbb{Z}}}$. The following inequality is a classical consequence of the interlacing of eigenvalues (see e.g. Bai and Silverstein [@MR2567175 Theorem A.43]). \[le:rank\] If $B$, $C$ are $n \times n$ Hermitian matrices, then, $$d_{KS} (\mu_B, \mu_C) {\leqslant}\frac{ {\mathrm{rank}}( B - C ) }n.$$ For $p {\geqslant}1$, let $\mu$, $\nu$ be two real probability measures such that $\int |x|^p d \mu$ and $\int |x|^p d \nu$ are finite. We define the *$L^p$-Wasserstein distance* as $$\label{eq:defWp} W_p ( \mu, \nu) = {{{\left( \inf_\pi \int_{{\mathbb{R}}\times {\mathbb{R}}} | x - y| ^p d \pi \right)}}} ^{\frac 1 p}$$ where the infimum is over all coupling $\pi$ of $\mu$ and $\nu$ (i.e. $\pi$ is probability measure on ${\mathbb{R}}\times {\mathbb{R}}$ whose first marginal is equal to $\mu$ and second marginal is equal to $\nu$). Hölder inequality implies that for $1 {\leqslant}p {\leqslant}q$, $W_p {\leqslant}W_{q}$. Moreover, the Kantorovich-Rubinstein duality gives a variational expression for $W_1$: $$\label{eq:KRdual} W_1 (\mu, \nu) = \sup \left\{ \int f d \mu - \int f d \nu : \| f \|_{L} {\leqslant}1 \right\},$$ where $\| f \|_L = \sup_{ x \ne y } |f(x) - f(y) | / |x- y| $ is the Lipschitz constant of $f$. The next classical inequality is particularly useful (see e.g. Anderson, Guionnet and Zeitouni [@AGZ Lemma 2.1.19]). \[le:HW\] If $B$, $C$ are $n \times n$ Hermitian matrices, then $$W_2 ( \mu_B, \mu_C) {\leqslant}\sqrt{ \frac 1 n {\mathrm{tr}}( B - C)^2}.$$ We finally introduce the distance $$d ( \mu, \nu) = \sup {{{\left\{ \int f d \mu - \int f d \nu : \| f \|_L {\leqslant}1 \hbox{ and } \| f \|_{BV} {\leqslant}1\right\}}}}.$$ By Lemmas \[le:rank\] and \[le:HW\], we obtain that for any $n \times n$ Hermitian matrices $B$, $C$, $$\label{eq:hehe} d ( \mu_B , \mu _C) {\leqslant}\sqrt{ \frac 1 n {\mathrm{tr}}( B - C)^2} \wedge \frac{ {\mathrm{rank}}( B- C ) }n.$$ Notice that $d( \mu_n , \mu) \to 0$ implies that $\mu_n$ converges weakly to $\mu$. Concentration inequality ------------------------ For $x = ( x_1, \cdots, x_n ) \in {\mathcal{M}}_{p , n} ({\mathbb{R}})$, define $a(x)$ as the Euclidean matrix obtained from the columns of $x$ : $a(x) _{ij} = f ( \| x_i - x_j \|^2)$. In particular, we have $A = a (X)$. Let $i \in \{1, \cdots, n\}$, $x' = (x'_1, \cdots, x'_n) \in {\mathcal{M}}_{p , n} ({\mathbb{R}})$ and assume that $x'_j = x_j$ for all $j \ne i$. Then $a(x)$ and $a(x')$ have all entries equal but the entries on the $i$-th row or column. We get $${\mathrm{rank}}( a(x) - a(x') ) {\leqslant}2.$$ It thus follows from Lemma \[le:rank\] that for any function $f$ with $\| f \|_{BV} < \infty$, $$\left| \int f d \mu_{a(x) } - \int f d \mu_{a(x') } \right| {\leqslant}\frac { 2 \| f \|_{BV} } { n}.$$ Using Azuma-Hoeffding’s inequality, it is then straightforward to check that for any $t {\geqslant}0$, $$\label{eq:concineq} {\mathbb{P}}{{{\left( \int f d \mu_{A } - {\mathbb{E}}\int f d \mu_{A } {\geqslant}t \right)}}} {\leqslant}\exp{{{\left( - \frac { n t ^2 }{ 8 \| f \|^2_{BV} } \right)}}}.$$ (For a proof, see [@MR2837123 proof of Lemma C.2] or Guntuboyina and Leeb [@MR2535081]). Using the Borel-Cantelli Lemma, this shows that for any such function $f$, a.s. $$\int f d \mu_{A } - \int f d {\mathbb{E}}\mu_{A } \to 0.$$ Now, recall that $M$ was defined by . Since the matrix $J$ has rank one, from Theorem \[th:PP\] and Lemma \[le:rank\], ${\mathbb{E}}\mu_M$ converges weakly to $\mu$. Hence our Theorem \[th:main\] is a corollary of the following proposition. \[prop:main\] Under the assumptions of Theorem \[th:main\], we have $$\lim_{n \to \infty} d {{{\left( {\mathbb{E}}\mu_{A} , {\mathbb{E}}\mu_{M} \right)}}} = 0.$$ Proof of Proposition \[prop:main\] ---------------------------------- The idea is to perform a multiple Taylor expansion which takes the best out of . ### Step 1 : concentration of norms {#step-1-concentration-of-norms .unnumbered} By assumption, there exists an open interval $K = (2 - \delta , 2 + \delta)$ such that $f$ is $C^1$ in $K$ and, for any $x \in K$, $$f( x ) = f(2) + f'(2) (x-2) + \frac{f'' (2)}{2} ( x - 2)^2 + \frac{f''' (2)}{6} ( x - 2)^3 ( 1 + o ( 1) ).$$ For any $i \ne j$, $(X_i- X_j)/\sqrt 2$ is an isotropic log-concave vector. Define the sequence ${\varepsilon}(n) = n ^ { - \kappa } \wedge ( \delta /2)$ with $0 < \kappa < 1/6$. It follows from Theorem \[th:GM\] and the union bound that the event $${\mathcal{E}}= {{{\left\{ \max_{i , j } {{{\left\{ {{{\left| \| X_i - X_j \|^2 - 2 \right|}}} \vee {{{\left| \| X_i\|^2 - 1 \right|}}} \right\}}}} {\leqslant}{\varepsilon}(n) \right\}}}}$$ has probability tending to $1$ as $n$ goes to infinity. ### Step 2 : Taylor expansion around $\| X_i \|^2 + \| X_j \|^2$ {#step-2-taylor-expansion-around-x_i-2-x_j-2 .unnumbered} We consider the matrix $$B_{i j } = \left\{ \begin{array}{ll} f ( \| X_i \|^2 + \|X_j\|^2 ) - 2 f' ( \| X_i \|^2 + \|X_j\|^2 ) X_i ^T X_j & \hbox{ if } i \ne j \\ f(0) & \hbox{ if } i = j. \end{array} \right.$$ On the event ${\mathcal{E}}$, $\| X_i \|^2 + \|X_j\|^2 \in K$. Since $f$ is $C^1$ in $K$, we may perform a Taylor expansion of $f ( \| X_i - X_j \|^2)$ around $\|X_i \|^2 + \| X_j \|^2$. It follows that for $i \ne j$, $${{{\left| A_{ij} - B_{ij} \right|}}} = o {{{\left( \| X_i - X_j \|^2 - \|X_i \|^2 - \|X_j \|^2 \right)}}} {\leqslant}\delta(n) {{{\left| X_i ^T X_j \right|}}},$$ where $\delta(n)$ is a sequence going to $0$. From and Jensen’s inequality, we get $$\begin{aligned} d ( {\mathbb{E}}\mu_A , {\mathbb{E}}\mu_B ) {\leqslant}{\mathbb{E}}d ( \mu_A , \mu_B ) & {\leqslant}& {\mathbb{P}}( {\mathcal{E}}^c ) +{{{\left( \frac 1 n \sum_{ i \ne j } {\mathbb{E}}|A_{ij} - B_{ij} |^2 {\mathbf{1}}_{\mathcal{E}}\right)}}}^{1/2} \\ & {\leqslant}& {\mathbb{P}}( {\mathcal{E}}^c ) + \delta(n) {{{\left( n {\mathbb{E}}{{{\left| X_1 ^T X_2 \right|}}}^2 \right)}}}^{1/2}.\end{aligned}$$ Now, from the assumption that $X_1$ and $X_2$ are independent and isotropic, we find $$\begin{aligned} {\mathbb{E}}{{{\left| X_1 ^T X_2 \right|}}}^2 & = & {\mathbb{E}}{{{\left( \sum_{k=1} ^ p X_{k 1} X_ {k 2} \right)}}} ^2 = \sum_{k=1} ^ p {{{\left( {\mathbb{E}}X_{k 1}^2 \right)}}} ^2 = \frac 1 p.\end{aligned}$$ By assumption , we deduce that $$\lim_{ n \to \infty} d ( {\mathbb{E}}\mu_A , {\mathbb{E}}\mu_B ) = 0.$$ It thus remains to compare ${\mathbb{E}}\mu_B$ and ${\mathbb{E}}\mu_{M}$. ### Step 3 : Taylor expansion around $2$ {#step-3-taylor-expansion-around-2 .unnumbered} We define the matrix $$C_{i j } = \left\{ \begin{array}{ll} f ( \| X_i \|^2 + \|X_j\|^2 ) - 2 f' ( 2 ) X_i ^T X_j & \hbox{ if } i \ne j \\ f(0) & \hbox{ if } i = j. \end{array} \right.$$ We now use the fact that $f'$ is locally Lipschitz at 2. It follows that if ${\mathcal{E}}$ holds, for $i \ne j$, $${{{\left| B_{ij} - C_{ij} \right|}}} = O {{{\left( X_i^ T X_j ( \|X_i \|^2 + \|X_j \|^2 -2 ) \right)}}} {\leqslant}c\, {\varepsilon}(n) {{{\left| X_i ^T X_j \right|}}}.$$ The argument of step 2 implies that $$\lim_{ n \to \infty} d ( {\mathbb{E}}\mu_B , {\mathbb{E}}\mu_C ) = 0.$$ It thus remains to compare ${\mathbb{E}}\mu_C$ and ${\mathbb{E}}\mu_{M}$. ### Step 4 : Taylor expansion around $2$ again {#step-4-taylor-expansion-around-2-again .unnumbered} We now consider the matrix $$D_{i j } = \left\{ \begin{array}{ll} f ( 2 ) + f'(2) ( \| X_i \| ^2 + \| X_j \| ^2 -2 ) + \frac{ f''(2) } {2} ( \| X_i \| ^2 + \| X_j \| ^2 -2 )^2 & \\ \quad \quad +\frac{ f'''(2) } {6} ( \| X_i \| ^2 + \| X_j \| ^2 -2 )^3 - 2 f' ( 2 ) X_i ^T X_j & \hbox{ if } i \ne j \\ f(0) & \hbox{ if } i = j. \end{array} \right.$$ We are going to prove that $$\label{eq:T3} \lim_{ n \to \infty} d ( {\mathbb{E}}\mu_C , {\mathbb{E}}\mu_D ) = 0.$$ We perform a Taylor expansion of order $3$ of $f( \| X_i \| ^2+ \| X_j \|^2)$ around $2$. It follows that if ${\mathcal{E}}$ holds, for $i \ne j$, $${{{\left| C_{ij} - D_{ij} \right|}}} = o {{{\left( \|X_i \|^2 + \|X_j \|^2 -2 \right)}}}^3 {\leqslant}\delta (n) {{{\left| \|X_i \|^2 + \|X_j \|^2 -2 \right|}}}^3,$$ where $\delta(n)$ is a sequence going to $0$. Using and arguing as in step 2, in order to prove , it thus suffices to show that $$\frac 1 n \sum_{ i \ne j } {\mathbb{E}}| \|X_i \|^2 + \|X_j \|^2 -2 |^6 {\mathbf{1}}_{\mathcal{E}}= O (1).$$ Since, for $\ell {\geqslant}1$, $| x + y | ^\ell {\leqslant}2^{\ell-1} ( |x|^\ell + |y|^\ell)$, it is sufficient to show that $$n {\mathbb{E}}{{{\left( \| X_1\|^2 - 1 \right)}}}^6 {\mathbf{1}}_{\mathcal{E}}= O (1).$$ To this end, for integer $\ell {\geqslant}1$, we write $$\begin{aligned} {\mathbb{E}}{{{\left| \| X_1\|^2 - 1 \right|}}}^{\ell} {\mathbf{1}}_{\mathcal{E}}& = & {\mathbb{E}}{{{\left| \| X_1\| - 1 \right|}}}^{\ell} {{{\left| \| X_1\| + 1 \right|}}}^{\ell} {\mathbf{1}}_{\mathcal{E}}{\leqslant}3^\ell {\mathbb{E}}{{{\left| \| X_1\| - 1 \right|}}}^{\ell}.\end{aligned}$$ Then, Theorem \[th:GM\] implies that there exists $c_\ell$ such that $${\mathbb{E}}{{{\left| \| X_1\| - 1 \right|}}}^{\ell} {\leqslant}c_\ell \, p^{ - \ell/ 6}.$$ It follows that $$\begin{aligned} \label{eq:var6} {\mathbb{E}}{{{\left| \| X_1\|^2 - 1 \right|}}}^{\ell} {\mathbf{1}}_{\mathcal{E}}= O {{{\left( p^{ - \ell / 6} \right)}}}. \end{aligned}$$ This proves . It finally remains to compare ${\mathbb{E}}\mu_D$ and ${\mathbb{E}}\mu_{M}$. ### Step 5 : End of proof {#step-5-end-of-proof .unnumbered} We set $$z_i = ( \| X_i \|^ 2 -1 ).$$ We note that for $i \ne j$, $$D_{ij} = M_{ij} + \sum_{1 {\leqslant}k + \ell {\leqslant}3} c_{k\ell} z_i ^{k} z_j ^{\ell},$$ for some coefficients $c_{k \ell}$ depending on $f'(2),f''(2),f'''(2)$. Note that $c_{10} = c_{01} = f'(2)$. Similarly, $$D_{ii} = M_{ii} + 2 f'(2) z_i = M_{ii} + c_{10} z_i + c_{01} z_i .$$ Define the matrix $E$, for all $ 1 {\leqslant}i , j {\leqslant}n$, $$E_{ij} = M_{ij} + \sum_{1 {\leqslant}k + \ell {\leqslant}3} c_{k\ell} z_i ^{k} z_j ^{\ell}.$$ If ${\mathcal{E}}$ holds, then $\max_i |z_i |{\leqslant}{\varepsilon}(n)$ and we find $${{{\left| E_{ij} - D_{ij} \right|}}} = {\mathbf{1}}( i = j ){{{\left| \sum_{2 {\leqslant}k + \ell {\leqslant}3} c_{k\ell} z_i ^{k} z_i ^{\ell} \right|}}} {\leqslant}c {\mathbf{1}}( i = j ) {\varepsilon}(n)^2.$$ It follows from that $$\begin{aligned} d ( {\mathbb{E}}\mu_D , {\mathbb{E}}\mu_E ) {\leqslant}{\mathbb{E}}d ( \mu_D , \mu_E ) & {\leqslant}& {\mathbb{P}}( {\mathcal{E}}^c) + {{{\left( \frac 1 n \sum_{i,j} {\mathbb{E}}{{{\left| E_{ij} - D_{ij} \right|}}}^2 {\mathbf{1}}_{\mathcal{E}}\right)}}}^{1/2} \\ & {\leqslant}& {\mathbb{P}}( {\mathcal{E}}^c) + c {\varepsilon}(n)^2. \end{aligned}$$ We deduce that $$\lim_{n \to \infty} d ( {\mathbb{E}}\mu_D , {\mathbb{E}}\mu_E ) = 0.$$ We notice finally that the matrix $E - M$ is equal to $$\sum_{1 {\leqslant}k + \ell {\leqslant}3} c_{k\ell} {Z_k} {Z_\ell}^T,$$ where $Z_{k}$ is the vector with coordinates $(z_i ^{k})_{1 {\leqslant}i {\leqslant}n}$. It implies in particular that ${\mathrm{rank}}( E-M ) {\leqslant}9$, indeed the ${\mathrm{rank}}$ is subadditive and ${\mathrm{rank}}( {Z_{k}} {Z_{\ell}}^T) {\leqslant}1$. In particular, it follows from that $$d ( {\mathbb{E}}\mu_E , {\mathbb{E}}\mu_M ) {\leqslant}{\mathbb{E}}d ( \mu_E , \mu_M ) {\leqslant}\frac 9 n .$$ This concludes the proof of Proposition \[prop:main\] and of Theorem \[th:main\]. Proof of Theorem \[th:main2\] ----------------------------- The concentration inequality holds. It is thus sufficient to prove the analog of Proposition \[prop:main\]. If $\ell {\geqslant}2$, the proof is essentially unchanged. In step $1$, the assumption implies the existence of a sequence ${\varepsilon}= {\varepsilon}(n)$ going to $0$ such that ${\mathbb{P}}( {\mathcal{E}}) \to 1$. Then, in step $4$, it suffices to extend the Taylor expansion up to $\ell$. For the case $\ell =1$ : in step $2$, we perform directly the Taylor expansion around $2$, for $i \ne j$ we write $f(\| X_i - X_j \|^2) = f (2) - 2 f'(2) X_i ^T X_j ( 1 + o(1))$. We then move directly to step $5$. (As already pointed, this case is treated in [@DoVu]).

--- abstract: 'In 1984, Johnson and Lindenstrauss proved that any finite set of data in a high-dimensional space can be projected to a lower-dimensional space while preserving the pairwise Euclidean distance between points up to a bounded relative error. If the desired dimension of the image is too small, however, Kane, Meka, and Nelson (2011) and Jayram and Woodruff (2013) independently proved that such a projection does not exist. In this paper, we provide a precise asymptotic threshold for the dimension of the image, above which, there exists a projection preserving the Euclidean distance, but, below which, there does not exist such a projection.' author: - 'Michael Burr[^1]' - 'Shuhong Gao[^2]' - 'Fiona Knoll[^3]' bibliography: - 'JLT\_PaperBibliography.bib' title: 'Optimal Bounds for Johnson-Lindenstrauss Transformations ' --- [**Keywords**]{} Johnson-Lindenstrauss transformation, Dimension reduction, Phase transition, Uniform measure of spheres, Asymptotic threshold Introduction ============ In 1984, Johnson and Lindenstrauss [@JohnsonLindenstrauss], in establishing a bound on the Lipschitz constant for the Lipschitz extension problem, proved that any finite set of data in a high-dimensional space can be projected into a lower-dimensional space while preserving the pairwise Euclidean distance within any desired relative error. In particular, for any finite set of vectors $x_1, \ldots, x_N \in {{\mathbb R}}^d$ and for any error factor $0 < \epsilon < \frac{1}{2}$, there exists an absolute constant $c$ such that for all $k \geq c \epsilon^{-2} \log N$, there exists a linear map $A: {{\mathbb R}}^d \rightarrow {{\mathbb R}}^k$ such that for all pairs $1\leq i,j \leq N$, $$(1-\epsilon) \|x_i-x_j\|_2 \leq \|Ax_i - Ax_j\|_2 \leq (1+\epsilon) \|x_i-x_j\|_2,$$ where $\|\cdot \|_2$ denotes the Euclidean norm. These inequalities are implied by the following theorem (by setting $\delta = \frac{1}{N^2}$ and taking the union bound): \[Thm: Main JL 2\] For any real numbers $0< \epsilon, \delta <\frac{1}{2}$, there exists an absolute constant $c>0$ such that for any integer $k \geq c \epsilon^{-2} \log \frac{1}{\delta}$, there exists a probability distribution ${\mathcal{D}}$ on $k \times d$ real matrices such that for any fixed $x \in {{\mathbb R}}^d$, $$\label{Ineq: JL} {\mbox{Prob}_{A \sim {\mathcal{D}}} \left[ (1-\epsilon)\|x\|_2^2 \leq \|Ax\|_2^2 \leq (1+\epsilon) \|x\|_2^2 \right] } > 1-\delta,$$ where $A\sim {\mathcal{D}}$ means that the matrix $A$ is a random matrix with distribution ${\mathcal{D}}$. Note that, in order to project a large number of vectors, $\delta$ must be sufficiently small. For instance, suppose we wish to project a set of $N=2^{20}$ vectors to a smaller dimensional space. To apply the union bound to Inequality (\[Ineq: JL\]), we use $\delta=2^{-40}$. In this case, Inequality (\[Ineq: JL\]) implies that the probability of preserving all pairwise distances between $N$ points (up to a relative error of $\epsilon$) is at least $1-\delta N^2/2 = 1/2$. Since the probability is nonzero, such a projection exists. A probability distribution ${\mathcal{D}}$ satisfying Inequality (\[Ineq: JL\]) is called an $(\epsilon,\delta)$-JL distribution, or simply a JL distribution. Since these transformations are linear, without loss of generality, we assume for the rest of the paper that $\|x\|_2=1$. When a JL-distribution is specified via an explicit construction, we may call a random projection $x\mapsto Ax$ generated in this way a JL transformation. Since the introduction of JL distributions, there has been considerable work on explicit constructions of JL distributions, see, e.g., [@JohnsonLindenstrauss; @FranklMaehara; @IndykMotwani; @Achlioptas; @AilonChazelle_ANN; @Matousek; @DasguptaKumarSarlos; @KaneNelson_Sparser] and the references therein. A simple and easily described JL distribution is that of Achlioptas [@Achlioptas]. In this construction, the entries of $A$ are distributed as follows: $$a_{ij} = \sqrt{\frac{3}{k}}\cdot \left\lbrace \begin{matrix} 1, & \text{ with probability } 1/6, \\ 0, & \text{ with probability } 1/3, \\ - 1, & \text{ with probability } 1/6. \end{matrix} \right.$$ The recent constructions in [@AilonChazelle_ANN; @Matousek; @DasguptaKumarSarlos; @KaneNelson_Sparser] have focused on the complexity of computing the projection for the purpose of applications. We note that the ability to project a vector to a smaller dimensional space, independent of the original dimension, while preserving the Euclidean norm up to a prescribed relative error, is highly desirable. In particular, dimension reduction has applications to many fields, including machine learning [@MachineLearning_Vempala; @MachineLearning_Weinberger], low rank approximation [@LowRank_ClarksonWoodruff; @LowRank_Nguyen; @LowRank_Ubaru], approximate nearest neighbors [@AilonChazelle_ANN; @IndykMotwani], data storage [@RIP_Candes; @Streaming], and document similarity [@DocSim_Bingham; @DocSim_Lin]. For both practical and theoretical purposes, it is important to know the smallest possible dimension $k$ of a potential image space for any given $\epsilon$ and $\delta$. Note that, for any $d_1 < d$, each $(\epsilon, \delta)$-JL distribution ${\mathcal{D}}$ on ${{\mathbb R}}^{k\times d}$ induces an $(\epsilon, \delta)$-JL distribution ${\mathcal{D}}_1$ on ${{\mathbb R}}^{k\times d_1}$ in a natural way: the matrices of ${\mathcal{D}}_1$ are obtained from ${\mathcal{D}}$ by deleting the last $d-d_1$ columns, together with the induced probability distribution. This construction is a JL distribution since ${{\mathbb R}}^{d_1}$ can be naturally embedded into ${{\mathbb R}}^d$ by extending a vector in ${{\mathbb R}}^{d_1}$ by $d-d_1$ zeros. Hence, if there exists an $(\epsilon, \delta)$-JL distribution on ${{\mathbb R}}^{k\times d}$, then there is an $(\epsilon, \delta)$-JL distribution on ${{\mathbb R}}^{k\times d_1}$ for all $1 \leq d_1 \leq d$. Similarly, if an $(\epsilon,\delta)$-JL distribution does not exist on $\mathbb{R}^{k\times d}$, then, for any $k_1<k$, then there cannot be an $(\epsilon,\delta)$-JL distribution on $\mathbb{R}^{k_1\times d}$. In particular, since $\mathbb{R}^{k_1}$ can be naturally embedded into $\mathbb{R}^k$ by extending a vector in $\mathbb{R}^{k_1}$ by $k-k_1$ zeros, if an $(\epsilon,\delta)$-JL distribution existed for $\mathbb{R}^{k_1\times d}$, it could be extended to an $(\epsilon,\delta)$-JL distribution existed for $\mathbb{R}^{k\times d}$. For any $\epsilon$ and $\delta$, we define $$k_0(\epsilon,\delta) = \min \{k: \mbox{there exists an $(\epsilon, \delta)$-JL distribution on ${{\mathbb R}}^{k\times d}$ for every $d\geq 1$} \}.$$ By our definition, $k_0= k_0(\epsilon,\delta)$ is independent of $d$, and, by Theorem \[Thm: Main JL 2\], we have $k_0 \leq c \epsilon^{-2} \log(1/\delta)$ for some absolute constant $c>0$. Frankl and Maehara [@FranklMaehara] show that $c \leq 9$. Achlioptas [@Achlioptas] further improves this bound by providing a JL distribution with $$k> 2 \log(2/\delta) \left(\frac{\epsilon^2}{2} - \frac{\epsilon^3}{3} \right)^{-1} ,$$ resulting in the following upper bound: $$k_0 \leq 2 \log(2/\delta) \left(\frac{\epsilon^2}{2} - \frac{\epsilon^3}{3} \right)^{-1} = 4\epsilon^{-2} \log(1/\delta) \left[ 1+ o(1) \right],$$ where $o(1)$ approaches zero as both $\epsilon$ and $\delta$ approach zero. A lower bound on $k_0$ was not given until 2003 when Alon [@Alon] proved that $$k_0 \geq c \epsilon^{-2} \log(1/\delta) \Big/ \log(1/\epsilon)$$ for some absolute constant $c>0$. Improving Alon’s work, Jayram and Woodruff [@WoodruffJayram] and Kane, Meka, and Nelson [@KaneNelsonMeka] showed, through different methods, that, for some absolute constant $c_1>0$, there is no $(\epsilon,\delta)$-JL distribution for $k \leq c_1\epsilon^{-2} \log \frac{1}{\delta}$. Hence, there is a lower bound of the form $k_0 \geq c_1 \epsilon^{-2} \log \frac{1}{\delta}$. This situation is summarized in Figure \[Figure:Line\]. (-5.5,0) – (5.5,0) ; (-5.5,0) – (5.5,0) ; in [-4,-2,0,2,4]{} (0pt,3pt) – (0pt,-3pt); (0pt,0pt) – (0pt,-3pt) node\[below\] [$c_1 \epsilon^{-2} \log \frac{1}{\delta}$]{}; (0pt,0pt) – (0pt,-3pt) node\[below\] [$4 \epsilon^{-2} \log \frac{1}{\delta}\left[ 1+ o(1) \right]$]{}; (2,.5)–(5.5,.5); (3.75, .5) node\[above\] [JLD Exists]{}; (-5.5,.5)–(-2,.5); (-3.75, .5) node\[above\] [No JLD Exists]{}; The goal of the current paper is to close the gap between the upper and lower bounds in the limit. In particular, we prove an optimal lower bound that asymptotically matches the known upper bound when $\epsilon$ and $\delta$ approach $0$, see Theorem \[Thm:IntroThm\]. This means that $4 \epsilon^{-2} \log(1/\delta)$ is an asymptotic threshold for $k_0$ where a phase change phenomenon occurs. \[Thm:IntroThm\] For $\epsilon$ and $\delta$ sufficiently small, $k_0 \approx 4\epsilon^{-2} \log(1/\delta)$. More precisely, $$\lim_{\epsilon, \delta \rightarrow 0} \ \frac{k_0(\epsilon, \delta)}{4\epsilon^{-2} \log(1/\delta)} = 1.$$ The rest of the paper is organized as follows: To prove Theorem \[Thm:IntroThm\], we follow the approach of Kane, Meka and Nelson [@KaneNelsonMeka]. To make their constant $c_1$ explicit, however, we must use a more careful argument. In Section \[Sec:Main:New\], we provide explicit conditions under which we prove the main result, Theorem \[Thm:IntroThm\]. We delay the proofs of the explicit conditions until Sections \[Sec:UniformMeasure\] and \[Concentration\] in order to make the main result more accessible since only the statements of these results (which are of independent interest) are needed, and not their more technical proofs. In Section \[Sec:UniformMeasure\], we study uniform distributions and surface areas (or hypervolumes) on high-dimensional spheres. More precisely, for any $d\geq 1$, let $S^{d-1}$ denote the unit sphere of dimension $d-1$, i.e., $S^0 = \{1, -1\}$ has two points, $S^1$ is the unit circle, $S^2$ is the unit sphere in ${{\mathbb R}}^3$, and, in general, $$S^{d-1} = \left\lbrace x \in {{\mathbb R}}^{d}: \sum_{i=1}^d x_i^2 = 1 \right\rbrace,$$ and $d\Omega_{d-1}$ be the surface area measure for $S^{d-1}$. We show that, for any $1\leq k \leq d$, $$d\Omega_{d-1} = \frac{1}{2} f(s)ds\, d\Omega_{k-1} d\Omega_{d-k-1},$$ where $s \in [0,1]$ and $f(s) = s^{\frac{k-2}{2}}(1-s)^{\frac{d-k-2}{2}}$. This is a more precise version of a result in [@KaneNelsonMeka], replacing an unspecified constant by $1/2$. This formula is of independent interest since it shows that the uniform distribution on $S^{d-1}$ is a product of uniform distributions on $S^{k-1}$ and $S^{d-k-1}$ with a distribution on $[0,1]$, see Theorem \[Thm\_ud\]. In Section \[Concentration\], we prove probabilistic bounds on $s =x_1^2 + \cdots + x_k^2$ where $x=(x_1, \cdots, x_k, \cdots, x_d)$ is a random variable uniformly distributed on $S^{d-1}$. These bounds can be viewed as explicit bounds for concentration theorems for laws of large numbers in probability theory. Asymptotic Threshold Bound {#Sec:Main:New} ========================== In this section, we prove the asymptotic threshold bound for JL transformations. In particular, we provide specific conditions that result in the asymptotic threshold bound of $4\epsilon^{-2}\log(1/\delta)$. In Sections \[Sec:UniformMeasure\] and \[Concentration\], we prove that these specific conditions hold, but the details of these proofs are more technical, and only the statements are needed for the asymptotic bound. The Uniform Distribution on $S^{d-1}$ ------------------------------------- There is a unique probability distribution, called the uniform distribution, on $S^{d-1}$ that is invariant under the orthonormal group. From a sampling point of view, a uniform random point on $S^{d-1}$ can be obtained as follows: Let $x_1, x_2, \ldots, x_d$ be independent random variables on ${{\mathbb R}}$ distributed according to the Gaussian distribution $N(0,1)$ (i.e., the standard normal distribution with mean $0$ and variance $1$), and let $X=(x_1,x_2, \ldots, x_d)^t$. Then, $x= \frac{X}{\|X\|_2}$ is a random point uniformly distributed on $S^{d-1}$. The uniform distribution may also be defined in terms of the surface area as follows: Let ${\text{Vol}_{d-1}\left( S^{d-1} \right)}$ denote the $(d-1)$-dimensional surface area (or hypervolume) of $S^{d-1}$, and, similarly, let ${\text{Vol}_{d-1}\left( V \right)}$ denote the surface area of $V$ for any (measurable) subset $V$ of $S^{d-1}$. For example, $${\text{Vol}_{0}\left( S^0 \right)} =2, \quad {\text{Vol}_{1}\left( S^1 \right)} = 2 \pi, \quad\text{and}\quad {\text{Vol}_{d-1}\left( S^{d-1} \right)} = \frac{2 \pi^{d/2}}{\Gamma(d/2)},$$ where $\Gamma(z)$ denotes the Gamma function $$\Gamma(z) = \int_0^\infty x^{z-1}e^{-x}dx.$$ The probability that a random point $x$ from $S^{d-1}$ drawn from the uniform distribution is in $V$ equals ${\text{Vol}_{d-1}\left( V \right)}/{\text{Vol}_{d-1}\left( S^{d-1} \right)}$, hence the probability is invariant under orthonormal transformations. We express the uniform distribution on $S^{d-1}$ in term of the surface area differential form[^4] $d \Omega_{d-1}$, which means that, for any measurable subset $V \subset S^{d-1}$, the $(d-1)$-dimensional surface area of $V$ is equal to the integral with respect to $d\Omega_{d-1}$, i.e., ${\text{Vol}_{d-1}\left( V \right)}=\int_V d \Omega_{d-1}$. For example, $d \Omega_{0} = \delta_{1}+\delta_{-1}$ consists of two point measures and $d \Omega_{1} = \frac{dx_1}{x_2} = - \frac{dx_2}{x_1}$ at any point $(x_1,x_2)^t \in S^1$. Thus, the uniform distribution on $S^{d-1}$ is defined in terms of $d \Omega_{d-1}/{\text{Vol}_{d-1}\left( S^{d-1} \right)}$, i.e., for any measurable subset $V \subset S^{d-1}$, $${\mbox{Prob}_{x \sim S^{d-1}} \left[ x \in V \right] } = \frac{1}{{\text{Vol}_{d-1}\left( S^{d-1} \right)}}\int_V d \Omega_{d-1}=\frac{{\text{Vol}_{d-1}\left( V \right)}}{{\text{Vol}_{d-1}\left( S^{d-1} \right)}}$$ where $x \sim S^{d-1}$ means that $x$ is a random variable uniformly distributed on $S^{d-1}$. As we are interested in reducing a $d$-dimensional vector to a $k$-dimensional vector for $1 \leq k < d$, we derive a relationship between the uniform distribution on $S^{d-1}$ and the uniform distributions on $S^{k-1}$ and $S^{d-k-1}$. Following the approach of Kane, Meka and Nelson [@KaneNelsonMeka], for $1\leq k<d$, we define an injective map $$\Psi: S^{d-1} \rightarrow [0,1] \times S^{k-1} \times S^{d-k-1}$$ as follows: For any $x= (x_1,x_2, \ldots, x_d)^t \in S^{d-1}$, we define $s$ in $\Psi(x) = (s, u , v)$ as $s=x_1^2+\dots+x_k^2$. In the case where $0<s<1$, we define $$u = (x_1, \ldots, x_k)^t/\sqrt{s} \quad\text{and}\quad v = (x_{k+1}, \ldots, x_d)^t/\sqrt{1-s}.$$ When $s=0$, i.e., $x_1=\dots=x_k=0$, we define $u=(1,0, \ldots, 0)^t$ (or any point in $S^{k-1}$) and $v = (x_{k+1}, \ldots, x_d)^t$. Similarly, for $s=1$, we define $u=(x_1, \ldots, x_k)^t$ and $v=(1,0, \ldots, 0)^t$ (or any point in $S^{d-k-1}$). It is straight-forward to check that $\Psi$ is injective. In addition, the complement of the image of $\Psi$ is a subset of $\{0,1\} \times S^{k-1} \times S^{d-k-1}$ which has $(d-1)$-dimensional surface area $0$. Therefore, when necessary, we assume that $s\in(0,1)$. For $s \in [0,1]$, we define $$f(s) = s^{\frac{k-2}{2}}(1-s)^{\frac{d-k-2}{2}}.$$ In Theorem \[Thm\_ud\], we prove that, via the map $\Psi$, $$d\Omega_{d-1} = \frac{1}{2} f(s)ds\, d\Omega_{k-1} d\Omega_{d-k-1}.$$ Equivalently, in term of probability distributions, $$\label{eq:equalitymeasures} \frac{d\Omega_{d-1}}{{\text{Vol}_{d-1}\left( S^{d-1} \right)}} = B f(s) ds \frac{d\Omega_{k-1}}{{\text{Vol}_{k-1}\left( S^{k-1} \right)}} \frac{d\Omega_{d-k-1}}{{\text{Vol}_{d-k-1}\left( S^{d-k-1} \right)}},$$ where $B$ is an appropriate scaling constant depending on $d$ and $k$, for more details, see Equation (\[Eqn:B\]). Moreover, in this situation, $Bf(s)$ is a probability distribution on $[0,1]$. This implies that the uniform distribution on $S^{d-1}$ is a direct product of the distributions on the factors. In other words, a uniformly distributed random variable $X_{d-1}$ on $S^{d-1}$ can be decomposed into three random variables $\Psi(X_{d-1})=(S,X_{k-1},X_{d-k-1})$ with the following properties: (i) $S$ is a random variable on $[0,1]$ with density function $Bf(s)$, (ii) $X_{k-1}$ and $X_{d-k-1}$ are uniformly distributed on $S^{k-1}$ and $S^{d-k-1}$, and (iii) The random variables $S$, $X_{k-1}$, and $X_{d-k-1}$ are [*independent*]{}. The independence of these three random variables is a key property in our proof as it allows us to study the three spaces independently. Upper Bound: Explicit JL Distribution {#Sec: OrthogProj} ------------------------------------- We recall that Achlioptas [@Achlioptas] proved that $$k_0(\epsilon, \delta) \leq 2 \log(2/\delta) \left(\frac{\epsilon^2}{2} - \frac{\epsilon^3}{3} \right)^{-1} = 4\epsilon^{-2} \log(1/\delta) \left[ 1+ o(1) \right].$$ In this section, we give an alternate proof of this result using the approach and bounds from this paper. We recall the following construction by Gupta and Dasgupta [@DasguptaGupta]: A distribution ${\mathcal{D}}$ on $k \times d$ matrices is formed by picking a $d\times d$ orthonormal matrix $V = (v_1, \ldots, v_d)^t$ uniformly at random with respect to the Haar measure on orthonormal matrices, and then letting $A = \frac{1}{\sqrt{s_0}}(v_1, \ldots, v_k)^t$ where $s_0= k/d$. From a sampling perspective, $A$ can be constructed by drawing $v_1$ from a uniform distribution on $S^{d-1}$, and then drawing each $v_i$ from a uniform distribution on the $(d-i)$-dimensional sphere perpendicular to $v_1$, $\dots$, $v_{i-1}$. The following theorem shows that $k_0(\epsilon, \delta) \leq 4\epsilon^{-2} \log(1/\delta) \left[ 1+ o(1)\right]$, which, in turn, implies that the limit appearing in Theorem \[Thm:IntroThm\] (if it exists) is at most $1$: \[Thm: exists\] Let $0<\epsilon,\delta<\frac{1}{2}$ and $s_0=k/d$. Suppose that there is some constant $C$ so that $$\max\{{\mbox{Prob}_{x\sim S^{d-1}} \left[ s < s_0(1-\epsilon) \right] },{\mbox{Prob}_{x\sim S^{d-1}} \left[ s> s_0(1+\epsilon) \right] } \}\leq Ce^{-\frac{k-2}{4}\epsilon^2\left(1-\frac{2}{3}\epsilon\right)},$$ where $s$ is defined as in $\Psi(x)=(s,u,v)$. Then, there exists an $o(1)$ function, which approaches zero as both $\epsilon$ and $\delta$ approach zero so that if $k> 4\epsilon^{-2} \log \left(\frac{1}{\delta} \right) \left[1+o(1)\right]$, then the distribution on $k\times d$ random matrices defined as above is an $(\epsilon, \delta)$-JL distribution, that is, for any $w \in S^{d-1}$, $${\mbox{Prob}_{A \sim {\mathcal{D}}} \left[ \left|\|Aw\|^2_2 -1\right| < \epsilon \right] } \geq 1- \delta.$$ Let $V$ be the random orthogonal matrix as defined above, and let $x =(x_1, \ldots, x_d)^t = Vw$. Then $Aw = \sqrt{s_0^{-1}} (x_1, \ldots, x_k)^t$, and $$||Aw||_2^2 = \frac{1}{s_0} (x_1^2 + \cdots+ x_k^2).$$ Since $V$ is orthonormal and $||w||_2 =1$, we have $||x||_2 =1$, hence $x \in S^{d-1}$. We observe that since $V$ is a random orthogonal matrix, for fixed $w\in S^{d-1}$, $x=Vw$ is a random variable, uniformly distributed on $S^{d-1}$. Hence, $${\mbox{Prob}_{A\sim{\mathcal{D}}} \left[ \left| \|Aw\|^2_2-1 \right|> \epsilon \right] } = {\mbox{Prob}_{x\sim S^{d-1}} \left[ \, \vline \frac{1}{s_0}\sum_{i=1}^k x_i^2 -1 \vline > \epsilon \right] },$$ where $x\sim S^{d-1}$ means that $x$ is a random variable uniformly distributed on $S^{d-1}$. Let $s= \sum_{i=1}^k x_i^2$. Then, $s \in [0,1]$ and the probability above becomes $$\label{eq:exteriorbounds} {\mbox{Prob}_{x\sim S^{d-1}} \left[ s < s_0(1-\epsilon) \right] } + {\mbox{Prob}_{x\sim S^{d-1}} \left[ s> s_0(1+\epsilon) \right] }\leq 2Ce^{-\frac{k-2}{4}\epsilon^2\left(1-\frac{2}{3}\epsilon\right)},$$ by assumption. We observe that when $$\label{eq:form:o1}k> 4\epsilon^{-2} \log \left(\frac{1}{\delta} \right)\left[1+ \frac{2\epsilon}{3-2\epsilon} + \frac{\log(2C)}{\log\left( \frac{1}{\delta}\right)} \cdot \frac{1} {1-2\epsilon/3 }+ \frac{2\epsilon^2}{4\log \left(\frac{1}{\delta}\right)}\right] = 4\epsilon^{-2} \log \left(\frac{1}{\delta} \right)\left[1+o(1)\right] ,$$ the right-hand-side of Inequality (\[eq:exteriorbounds\]) is less than $\delta$. In this case, the $o(1)$ term needed in the theorem statement appears in Inequality (\[eq:form:o1\]). Therefore, when $k> 4\epsilon^{-2} \log \left(\frac{1}{\delta} \right)\left[1+o(1)\right] $, the distribution ${\mathcal{D}}$ is an $(\epsilon, \delta)$-JL distribution. Lower Bound for Arbitrary Distributions --------------------------------------- In this section, we prove an optimal lower bound on the limit in Theorem \[Thm:IntroThm\] that matches the upper bound from the previous section. The proof of this lower bound is the main challenge in this paper. We begin with the following key lemma: \[lemma\_c\] Let $x =(x_1, \ldots, x_d)^t$ be a random variable, uniformly distributed on $S^{d-1}$, $\Psi(x)=(s,u,v)$, and $s_0=k/d$. Suppose that $$\min\{ {\mbox{Prob}\left[ s>s_0(1+\epsilon) \right] }, {\mbox{Prob}\left[ s<s_0(1-\epsilon) \right] }\} \geq L,$$ where $s$ is a random variable with probability distribution $Bf(s)$ on $[0,1]$. For any function $c(u,v)>0$ depending only on $u \in S^{k-1}$ and $v \in S^{d-k-1}$ (i.e., independent of $s$), we have $${\mbox{Prob}_{x\sim S^{d-1}} \left[ |s c -1| > \epsilon \right] } \geq L.$$ By the equality of differential forms in Equation (\[eq:equalitymeasures\]), $$\begin{aligned} {\mbox{Prob}\left[ |sc -1| > \epsilon \right] } & =& \int_{|s c -1| > \epsilon} Bf(s) ds \frac{d\Omega_{k-1}}{{\text{Vol}_{k-1}\left( S^{k-1} \right)}} \frac{d\Omega_{d-k-1}}{{\text{Vol}_{d-k-1}\left( S^{d-k-1} \right)}}\\ & = & \int_{S^{k-1}\times S^{d-k-1}} \left( \int_{|s c -1| > \epsilon} Bf(s) ds \right) \frac{d\Omega_{k-1}}{{\text{Vol}_{k-1}\left( S^{k-1} \right)}} \frac{d\Omega_{d-k-1}}{{\text{Vol}_{d-k-1}\left( S^{d-k-1} \right)}}.\end{aligned}$$ Our goal is to find a lower bound on the integral $\int_{|s c -1| > \epsilon} Bf(s) ds$. Due to the independence of $u$, $v$, and $s$, $c(u,v)$ is a fixed positive constant within this integral. We observe that $|sc-1|>\epsilon$ consists of two intervals, $s<(1-\epsilon)/c$ and $s>(1+\epsilon)/c$ and consider two cases depending on the value of $c$. We begin by recalling that $${\mbox{Prob}\left[ s>s_0 (1+\epsilon) \right] } = \int_{s>s_0 (1+\epsilon)} B f(s)ds\quad\text{and}\quad{\mbox{Prob}\left[ s<s_0 (1-\epsilon) \right] } = \int_{s<s_0(1-\epsilon)} Bf(s)ds.$$ If $c \geq s_0$, then $(1+\epsilon)/c \leq (1+\epsilon)/s_0$, and, hence $$\int_{|s c -1| > \epsilon} Bf(s) ds \geq \int_{s > (1+ \epsilon)/c} Bf(s) ds \geq \int_{s > (1+ \epsilon)/s_0} Bf(s) ds \geq L.$$ On the other hand, if $c < s_0$, then $(1-\epsilon)/s_0 < (1-\epsilon)/c$, then $$\int_{|s c -1| > \epsilon} Bf(s) ds \geq \int_{s < (1- \epsilon)/c} Bf(s) ds \geq \int_{s < (1- \epsilon)/s_0} Bf(s) ds \geq L.$$ Therefore, the integral $\int_{|s c -1| > \epsilon} Bf(s) ds$ is bounded from below by $L$, and $${\mbox{Prob}\left[ |sc -1| > \epsilon \right] } \geq \int_{S^{k-1}\times S^{d-k-1}} L \frac{d\Omega_{k-1}}{{\text{Vol}_{k-1}\left( S^{k-1} \right)}} \frac{d\Omega_{d-k-1}}{{\text{Vol}_{d-k-1}\left( S^{d-k-1} \right)}} = L.\vspace{-.37in}$$ We now show that when $k\leq\eta\epsilon^{-2}\log(1/\delta)$ with $\eta<4$, and $\epsilon$ and $\delta$ are sufficiently small, there does not exist an $(\epsilon, \delta)$-JL distribution on ${{\mathbb R}}^{k\times d}$. This fact, combined with the results in Section \[Sec: OrthogProj\], shows that the limit appearing in Theorem \[Thm:IntroThm\] exists and equals $1$. It is challenging to show this directly; instead, we consider the following related problem: By definition, for a probability distribution ${\mathcal{D}}$ on ${{\mathbb R}}^{k\times d}$ to be an $(\epsilon, \delta)$-JL distribution, the following inequality must hold for every $w \in S^{d-1}$: $${\mbox{Prob}_{A \sim {\mathcal{D}}} \left[ |\|Aw\|_2^2 -1| > \epsilon \right] } < \delta.$$ Hence, $$\label{eq4.2.1} {\mbox{Prob}_{A \sim {\mathcal{D}}, \ w \sim S^{d-1}} \left[ |\|Aw\|_2^2 -1| > \epsilon \right] } < \delta,$$ where $w \in S^{d-1}$ is a random variable distributed uniformly on $S^{d-1}$. Our approach is to prove that, for every $A \in {{\mathbb R}}^{k \times d}$, $$\label{eq:oppositedir} {\mbox{Prob}_{w \sim S^{d-1}} \left[ |\|Aw\|_2^2 -1| > \epsilon \right] } > \delta.$$ When Inequality (\[eq:oppositedir\]) holds for all $A$, then Inequality (\[eq4.2.1\]) can not hold for any distribution ${\mathcal{D}}$ on ${{\mathbb R}}^{k \times d}$. Therefore, an $(\epsilon, \delta)$-JL distribution does not exist. We make this precise in the following theorem: \[thm\_lowbound\] Suppose that $\eta<4$ and let $k(\epsilon,\delta)=\left\lfloor\eta\epsilon^{-2} \log \left( \frac{1}{\delta} \right)\right\rfloor$. Let $s_0=k/d$, and suppose that, for every $\epsilon$, $\delta$, and $s_0$ sufficiently small (to make $s_0$ sufficiently small, $d$ must be sufficiently large), $$\min\{{\mbox{Prob}\left[ s>s_0(1+\epsilon) \right] },{\mbox{Prob}\left[ s<s_0(1-\epsilon) \right] }\} \geq C\delta^{\frac{\eta}{4}\gamma},$$ where $C>0$ is an absolute constant, and $\gamma$ approaches $1$ as $\epsilon$, $\delta$, and $s_0$ approach $0$. Then, by decreasing $\epsilon$, $\delta$, and $s_0$ as needed, for every matrix $A \in {{\mathbb R}}^{k(\epsilon,\delta) \times d}$, $${\mbox{Prob}_{w \sim S^{d-1}} \left[ \left|\|Aw\|^2_2-1\right|> \epsilon \right] } > \delta.$$ We assume that $A$ has rank $k=k(\epsilon,\delta)$ since, if not, we may reduce $k$ (and decrease $\eta$ correspondingly) to the rank of $A$. Let $A=U\Sigma V^t$ be the singular value decomposition of $A$ where $U$ is a $k \times k$ orthonormal matrix, $V= (v_1, \ldots, v_d)$ is a $d \times d$ orthonormal matrix, and $\Sigma$ is a $k \times d$ diagonal matrix with $\lambda_i> 0$ its entry at $(i,i)$ for $1 \leq i \leq k$. Let $$x = (x_1, \ldots, x_d)^t = V^t w.$$ Since $V$ is orthonormal, we have $x \in S^{d-1}$. We observe that since $w$ is a uniformly distributed random variable on $S^{d-1}$, $V^tw$ is also a uniformly distributed random variable on $S^{d-1}$. Therefore, since $U$ is orthonormal, we have $$\|Aw\|^2_2 = \|U \Sigma x\|^2_2 = \|\Sigma x\|^2_2 = \sum_{i=1}^k \lambda_i^2x_i^2.$$ Let $\Psi(x)=(s,u,v)$ where $s = x_1^2 + \cdots + x_k^2$. We restrict our attention to the case where $s\in(0,1)$ since the complement has zero measure. Let $$c = \sum_{i=1}^k \lambda_i^2x_i^2/s = \|\Sigma u \|^2_2,$$ then $${\mbox{Prob}_{w\sim S^{d-1}} \left[ \left|\|Aw\|^2_2-1\right|> \epsilon \right] } = {\mbox{Prob}_{x\sim S^{d-1}} \left[ \left|s c -1\right|>\epsilon \right] }.$$ Due to the independence of $u$, $v$, and $s$, it follows that $c$ depends only on $u$. Therefore, by Lemma \[lemma\_c\], it follows that $${\mbox{Prob}_{w\sim S^{d-1}} \left[ \left| \|Aw\|_2^2 -1\right| > \epsilon \right] } \geq C\delta^{\frac{\eta}{4}\gamma}.$$ It follows that for $\epsilon$, $\delta$, and $s_0$ sufficiently small, $C\delta^{\frac{\eta}{4}\gamma}>\delta$. Since $d$ grows as $s_0$ approaches $0$, it follows from Theorem \[thm\_lowbound\], that for $d$ sufficiently large, there is no $(\epsilon, \delta)$-JL distribution when $k< \eta\epsilon^{-2} \log \left( \frac{1}{\delta} \right)$ for $\eta<4$. Therefore, $k_0(\epsilon,\delta)>\eta\epsilon^{-2}\log\left(\frac{1}{\delta}\right)$. We collect the results of Theorems \[Thm: exists\] and \[thm\_lowbound\] in the following corollary: Assume the hypotheses on ${\mbox{Prob}\left[ s>s_0(1+\epsilon) \right] }$ and ${\mbox{Prob}\left[ s<s_0(1-\epsilon) \right] }$ from Theorems \[Thm: exists\] and \[thm\_lowbound\] hold. (a) There exists an $o(1)$ function that approaches $0$ as $\epsilon$ and $\delta$ approach zero such that if $k>4\epsilon^{-2} \log \left( \frac{1}{\delta} \right)\left[1+o(1)\right]$, then there exists a JL distribution. (b) If $k(\epsilon,\delta)=\left\lfloor\eta\epsilon^{-2} \log \left( \frac{1}{\delta} \right)\right\rfloor$, then, by decreasing $\epsilon$ and $\delta$, and increasing $d$, there is no $(\epsilon, \delta)$-JL distribution for any $k'\leq k(\epsilon,\delta)$. This proves the main result in the paper. In the following sections, we provide the more technical results that verify the assumptions in Theorems \[Thm: exists\] and \[thm\_lowbound\]. Uniform Distributions on Unit Spheres in High Dimensions {#Sec:UniformMeasure} ======================================================== In this section, we prove the explicit relationship between the surface area differential forms $d\Omega_{d-1}$, $d\Omega_{k-1}$, and $d\Omega_{d-k-1}$. In particular, we prove that \[Thm\_ud\] Under the almost bijective map $\Psi: S^{d-1} \rightarrow [0,1] \times S^{k-1} \times S^{d-k-1}$, we have equality of the surface area differential forms on $S^{d-1}$, $S^{k-1}$, and $S^{d-k-1}$, i.e., $$d\Omega_{d-1} = \frac{1}{2} f(s)ds\, d\Omega_{k-1} d\Omega_{d-k-1},$$ where $f(s)=s^{(k-2)/2}(1-s)^{(d-k-2)/2}$. Equivalently, in terms of probability distribution measures, $$\frac{d\Omega_{d-1}}{{\text{Vol}_{d-1}\left( S^{d-1} \right)}} = B f(s) ds \frac{d\Omega_{k-1}}{{\text{Vol}_{k-1}\left( S^{k-1} \right)}} \frac{d\Omega_{d-k-1}}{{\text{Vol}_{d-k-1}\left( S^{d-k-1} \right)}}.$$ Hence, the uniform distribution on $S^{d-1}$ can be identified with the product distribution on $[0,1] \times S^{k-1} \times S^{d-k-1}$ where the distribution of $s$ on $ [0,1]$ has density function $Bf(s)$ and $$\label{Eqn:B} B = \frac{1}{2} \cdot \frac{{\text{Vol}_{k-1}\left( S^{k-1} \right)} \cdot {\text{Vol}_{d-k-1}\left( S^{d-k-1} \right)}}{{\text{Vol}_{k-1}\left( S^{k-1} \right)}} = \frac{\Gamma(\frac{d}{2})}{\Gamma(\frac{k}{2}) \Gamma(\frac{d-k}{2})}.$$ This theorem is based on the following lemma, which is well-known to experts, but is included here for completeness. \[Lemma:SAofSphere\] Let $x=(x_1, \ldots, x_d)$ with $x_d>0$ be a point on the upper hemisphere of $S^{d-1}$. Then the surface area measure of the unit sphere $S^{d-1}$ at $x$ is $$d\Omega_{d-1} = \frac{1}{x_d}dx_1 \dots dx_{d-1}.$$ Before we begin the proof, we recall the approach for $S^2$ in $3$-dimensional space. We consider the upper hemisphere of $S^2$ as the graph of a function over $D^2$, where $D^{d-1}$ denotes $(d-1)$-dimensional disk, namely $$D^{d-1}=\left\lbrace \hat{x} \in {{\mathbb R}}^{d-1}: \sum_{i=1}^{d-1} \hat{x}_i^2 \leq 1\right\rbrace.$$ We then integrate over the disk $D^2$ to calculate the surface area of $S^2$. In particular, the integrand is the limit of the ratios of the area of a square in $D^2$ to the area of the corresponding parallelogram above the square in the tangent space of $S^2$ as the square shrinks a point. In the case of the sphere, the parallelogram’s area is calculated using the cross product, but we must replace the use of the cross product in higher dimensions. In $d$-dimensional space, we consider the upper hemisphere of $S^{d-1}$ as the graph of a function over the $(d-1)$-dimensional disk $D^{d-1}$. We construct a pair of $(d-1)$-dimensional parallelepipeds as follows: $P^{d-1}$ is in the tangent space of $D^{d-1}$ and $Q^{d-1}$ is in the tangent space of $S^{d-1}$. Then, we take the limit of their $(d-1)$-dimensional volumes as $P^{d-1}$ approaches a point. Due to complications in taking the $(d-1)$-dimensional volume in $d$-dimensional space, we extend both $P^{d-1}$ and $Q^{d-1}$ to associated, full-dimensional parallelepipeds. Let $(\hat{x}_1,\dots,\hat{x}_{d-1})\in D^{d-1}$ and define $\phi: D^{d-1} \rightarrow {{\mathbb R}}_{\geq 0}$ as $$\phi(\hat{x}_1, \ldots, \hat{x}_{d-1}) = \sqrt{1-\sum_{i=1}^{d-1} \hat{x}_i^2}.$$ We observe that the graph of this function is the upper hemisphere of $S^{d-1}$. We now extend this map to $D^{d-1}\times\mathbb{R}$ as $\Phi_d: D^{d-1} \times {{\mathbb R}}\rightarrow {{\mathbb R}}^d$ defined by $$(\hat{x}_1, \ldots, \hat{x}_{d-1}, \hat{x}_d) \mapsto \left( (1+\hat{x}_d)\hat{x}_1, (1+\hat{x}_d)\hat{x}_2, \ldots, (1+\hat{x}_d)\hat{x}_{d-1}, (1+\hat{x}_d)\phi(\hat{x}_1, \ldots, \hat{x}_{d-1}) \right).$$ We observe that $\Phi_d|_{D^{d-1}\times\{0\}}$ maps the disk $D^{d-1}\times\{0\}$ surjectively onto the graph of $\phi$, i.e., the upper hemisphere of $S^{d-1}$, see Figure \[Figure: MapPhi\]. \[ hide axis, view=[0]{}[10]{}, z buffer=sort, height = 2.5in, width = 2in \] 3 \[ domain=0:360, y domain=0:90, surf, shader=flat, blue, opacity=0.4 \] ([sin(y)\*cos(x)]{},[sin(y)\*sin(x)]{},[cos(y)]{}); 3 \[ domain = 0:360, y domain = 0:1, surf, shader=flat, black , opacity =0.4 \] ([cos(x)\*y]{},[sin(x)\*y]{},[-2]{}); (-.4,0,-.5) –node\[left\][$\Phi_d$]{} (-.4,0,-1.5); (.4,0,-.5) –node\[right\][$\pi$]{} (.4,0,-1.5); We recall that, at $(\hat{x}_1,\dots,\hat{x}_{d-1})\in D^{d-1}$, the tangent space is $\mathbb{R}^{d-1}$ and we define the parallelpiped $P^{d-1}$ in the tangent space by the vectors $\Delta \hat{x}_i e_i$ of length $\Delta \hat{x}_i$ in the direction of the $i^{\text{th}}$ standard basis vector $e_i$ of ${{\mathbb R}}^{d-1}$. Similarly, the tangent space at $(\hat{x}_1,\dots,\hat{x}_{d-1},0)\in D^{d-1}\times{{\mathbb R}}$ is $\mathbb{R}^d$, and we define the parallelepiped $P^d$ in this tangent space by the vectors $\Delta \hat{x}_i e_i$ for $1\leq i\leq d-1$ and $he_d$ for the final direction. Then, as the vector $he_d$ is perpendicular to the tangent vectors of the disk $D^{d-1}$, the $d$-dimensional volume of $P^d$ can be computed in terms of the $(d-1)$-dimensional volume of $P^{d-1}$ and the height $h$, i.e., $$ {\text{Vol}_{d}\left( P^d \right)} = h {\text{Vol}_{d-1}\left( P^{d-1} \right)}.$$ Next, we let $Q^{d-1}$ and $Q^d$ be the images of $P^{d-1}$ and $P^d$ under the Jacobian of $\Phi_d$, i.e., $\operatorname{Jac}\Phi_d$, respectively. Note that the Jacobian of $\Phi_d$, when restricted to $D^{d-1} \times \{0\}$, is $$\begin{aligned} \left(\operatorname{Jac}\Phi_d \right)|_{D^{d-1} \times \{0\}} & = \left[ \begin{array}{ccc|c} &&&\hat{x}_1 \\[.5cm] &I&&\vdots\\[.5cm] &&& \hat{x}_{d-1} \\[.3cm]\hline -\frac{ \hat{x}_1}{\phi(\hat{x}_1, \ldots, \hat{x}_{d-1})}&\dots&-\frac{ \hat{x}_{d-1}}{\phi(\hat{x}_1, \ldots, \hat{x}_{d-1})}& \phi(\hat{x}_1, \ldots, \hat{x}_{d-1}) \end{array} \right]. \end{aligned}$$ Since the Jacobian acts on tangent vectors, $Q^{d-1}$ is defined by the vectors $$\Delta \hat{x}_i \left(f_i - \frac{\hat{x}_i}{\phi(\hat{x}_1, \ldots, \hat{x}_{d-1})}f_d\right),$$ for $1\leq i\leq d-1$, where the $f_i$ is the $i^{\text{th}}$ standard basis vector of $\mathbb{R}^d$. Moreover, $Q^d$ is defined by these vectors as well as the image of $he_d$, i.e., $$h\left( \hat{x}_1, \ldots,\hat{x}_{d-1}, \phi(\hat{x}_1, \ldots, \hat{x}_{d-1} )\right) .$$ We observe that the vectors $\Delta \hat{x}_i \left(f_i - \frac{\hat{x}_i}{\phi(\hat{x}_1, \ldots, \hat{x}_{d-1})}f_d\right)$ for $1\leq i \leq d-1$ are tangent vectors to the sphere $S^{d-1}$, and that $h\left( \hat{x}_1, \ldots,\hat{x}_{d-1}, \phi(\hat{x}_1, \ldots, \hat{x}_{d-1} )\right)$ is the outward pointing surface normal with length $h$. Since the tangent vectors are perpendicular to the outward point normal, the volumes of $Q^{d-1}$ and $Q^d$ have a similar relationship as the volumes of $P^{d-1}$ and $P^d$, i.e., $$ {\text{Vol}_{d}\left( Q^d \right)} = h {\text{Vol}_{d-1}\left( Q^{d-1} \right)}.$$ Therefore, the ratio between the $d$-dimensional volumes of $Q^d$ and $P^d$ is the same as the ratio of the $(d-1)$-dimensional volumes of $Q^{d-1}$ and $P^{d-1}$. Since $Q^d$ is the image of $P^d$ under the linear map $\operatorname{Jac}\, \Phi_d \vline_{D^{d-1} \times \{0\}}$, it follows that $$ {\text{Vol}_{d}\left( Q^d \right)} = \, \left\vert \det \left(\operatorname{Jac}\Phi_d \right)\vline_{\left(\hat{x}_1, \ldots, \hat{x}_{d-1}, 0 \right)} \, \right\vert {\text{Vol}_{d}\left( P^d \right)}.$$ Therefore, the ratio of the volumes of $Q^{d-1}$ and $P^{d-1}$ is $\left| \det \left(\operatorname{Jac}\Phi_d \right)\vline_{\left(\hat{x}_1, \ldots, \hat{x}_{d-1}, 0 \right)} \, \right|$. It is straight-forward to compute the determinant of $\operatorname{Jac}(\Phi_d)$ at the point $\left( \hat{x}_1, \ldots,\hat{x}_{d-1}, 0\right)$ via a few row reductions to eliminate the first $d-1$ entries in the last row and turn the matrix into an upper triangular matrix whose lower right corner is $\frac{1}{\phi(\hat{x}_1, \ldots, \hat{x}_{d-1})}$. Hence, the determinant of $\operatorname{Jac}(\Phi_d)$ is $\frac{1}{\phi(\hat{x}_1, \ldots, \hat{x}_{d-1})}$, which is the desired scaling factor. Therefore, $\frac{1}{\phi(\hat{x}_1, \ldots, \hat{x}_{d-1})}$ is the local factor in the stretching of the surface area in the map from $D^{d-1}$ to $S^{d-1}$. We recall that the coordinates $x_1,\dots,x_d$ are the coordinates on the upper hemisphere and $\hat{x}_1,\dots,\hat{x}_{d-1}$ are the coordinates on $D^{d-1}$. Since, under the map $\Phi_d|_{D^{d-1}\times\{0\}}$, $x_i=\hat{x}_i$ for $1\leq i\leq d-1$, it follows that $$d\hat{x}_1\dots d\hat{x}_{d-1}=dx_1\dots dx_{d-1}\qquad\text{and}\qquad \phi(\hat{x}_1,\dots,\hat{x}_{d-1})=x_d.$$ From here, the result follows directly. [*Proof of Theorem \[Thm\_ud\]*]{}. Let $(w_1, \ldots, w_d)$, $(x_1, \ldots,x_k)$, and $(y_1, \ldots, y_{d-k})$ be coordinates of points on the $(d-1)$-dimensional unit sphere, the $(k-1)$-dimensional unit sphere, and the $(d-k-1)$-dimensional unit sphere, respectively. Let $(\hat{w}_1, \ldots, \hat{w}_{d-1})$, $(\hat{x}_1, \ldots,\hat{x}_{k-1})$, and $(\hat{y}_1, \ldots, \hat{y}_{d-k-1})$ be the coordinates of points on the disks $D^{d-1}$, $D^{k-1}$ and $D^{d-k-1}$, respectively. Let $$\varphi: [0,1] \times D^{k-1} \times D^{d-k-1} \rightarrow D^{d-1}$$ be defined by $$\begin{aligned} s \times (\hat{x}_1, \ldots, \hat{x}_{k-1}) \times & (\hat{y}_1, \ldots, \hat{y}_{d-k-1}) \mapsto \\ & \left( \sqrt{s}\hat{x}_1, \ldots, \sqrt{s}\hat{x}_{k-1},\sqrt{s} \sqrt{1-\sum_{i=1}^{k-1} \hat{x}_i^2} , \sqrt{1-s} \hat{y}_1, \ldots, \sqrt{1-s} \hat{y}_{d-k-1} \right). \end{aligned}$$ We observe that $\varphi$ maps the disks $D^{k-1}$ and $D^{d-k-1}$ onto the half of the disk $D^{d-1}$ whose $k^{th}$ coordinate is nonnegative. As the measure of the image is the measure of the preimage scaled by the determinant of the Jacobian of $\varphi$, the surface area measure of the disk $D^{d-1}$ is $$\label{Eq: det_disks} d\hat{w}_1 \dots d\hat{w}_{d-1} = |\det \operatorname{Jac}( \varphi) | ds\,d\hat{x}_1 \dots d\hat{x}_{k-1} d\hat{y}_1 \dots d\hat{y}_{d-k-1}.$$ The Jacobian of $\varphi$ for $s\in(0,1)$ is $$\text{Jac }\varphi = \left[ \begin{array}{c|ccc|ccc} \frac{\hat{x}_1}{2\sqrt{s}} & \sqrt{s} & \dots & 0 & 0& \dots & 0 \\[.3cm] \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\[.4cm] \frac{\hat{x}_{k-1}}{2\sqrt{s}} & 0 & \dots & \sqrt{s} & 0 & \dots & 0\\ [.3cm] \hline \frac{\sqrt{1-\sum_{i=1}^{k-1}\hat{x}_i^2}}{2\sqrt{s}} & \frac{-\sqrt{s}\cdot \hat{x}_1}{\sqrt{1-\sum_{i=1}^{k-1}\hat{x}_i^2}} & \dots & \frac{-\sqrt{s}\cdot \hat{x}_{k-1}} {\sqrt{1-\sum_{i=1}^{k-1}\hat{x}_i^2}} & 0 & \dots & 0 \\[.35cm] \hline \frac{-\hat{y}_1}{2\sqrt{1-s}} & 0 & \dots & 0 & \sqrt{1-s} & \dots & 0 \\[.3cm] \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ [.4cm] \frac{-\hat{y}_{d-k-1}}{2\sqrt{1-s}} & 0 & \dots & 0 & 0 & \dots & \sqrt{1-s}\\ [.3cm] \end{array} \right].$$ Eliminating all but the $k^{th}$ entry of the first column by adding multiples of the other columns to the first column, we obtain $$\det \operatorname{Jac}( \varphi ) = \frac{1}{2\hat{x}_k} s^{\frac{k-2}{2}} (1-s)^{\frac{d-k-1}{2}}.$$ Substituting this value into Expression (\[Eq: det\_disks\]), we have the surface area measure of $D^{d-1}$ in terms of the disks $D^{k-1}$ and $D^{d-k-1}$. That is, $$\label{Eqn: SA of D} d\hat{w}_1 \dots d\hat{w}_{d-1} = \frac{1}{2\hat{x}_{k}} s^{(k-2)/2} (1-s)^{(d-k-1)/2} ds \, d\hat{x}_1 \dots d\hat{x}_{k-1} d\hat{y}_1 \dots d\hat{y}_{d-k-1}.$$ We observe that the coordinates of the disk $D^{t-1}$ correspond to the first $t-1$ entries of coordinates of the unit sphere $S^{t-1}$. Therefore, we may extend $\varphi$ to the map $\Psi$, as defined above, where $$\Psi^{-1}=\Phi_d\circ\varphi\circ (id_s\times(\Phi_k)^{-1}\times(\Phi_{d-k})^{-1}).$$ Employing the results of Lemma \[Lemma:SAofSphere\] in various dimensions, we rewrite the surface measure of a unit sphere in terms of the surface measure of the corresponding disks: $$\begin{aligned} d\hat{x}_1 \dots d\hat{x}_{k-1} &= dx_1 \dots dx_{k-1} = x_k d\Omega_{k-1} \\ d\hat{y}_1 \dots d\hat{y}_{d-k-1} &= dy_1 \dots dy_{d-k-1} = y_{d-k} d\Omega_{d-k-1} \\ d\hat{w}_1 \dots d\hat{w}_{d-1} &= dw_1 \dots dw_{d-1} = w_d d\Omega_{d-1}. \end{aligned}$$ By applying the $\Psi$, we can substitute these three equalities into Equation (\[Eqn: SA of D\]) to obtain $$\begin{aligned} d\Omega_{d-1} = \frac{1}{w_d} dw_1 \dots dw_{d-1} &= \frac{y_{d-k}}{2w_d} s^{(k-2)/2} (1-s)^{(d-k-1)/2} ds\, d\Omega_{k-1} d\Omega_{d-k-1}\\ &\hspace{-1in}= \frac{1}{2} s^{(k-2)/2} (1-s)^{(d-k-2)/2} ds \, d\Omega_{k-1} d\Omega_{d-k-1}=\frac{1}{2} f(s)ds \, d\Omega_{k-1} d\Omega_{d-k-1},\end{aligned}$$ where the third equality follows from the fact that $w_d = \sqrt{1-s}y_{d-k}$ by the map $\Psi$. Since the cases where $s=0$ or $s=1$ have measure $0$, the result follows. Explicit Concentration Bounds {#Concentration} ============================= Throughout this section, we assume that $s$ is a random variable with probability distribution $Bf(s)$. We define $s_0=\frac{k}{d}$, and further assume that $0\leq \epsilon,\delta\leq 1/2$, $k-4\geq\epsilon^{-2}$, and $s_0<0.4$. We derive lower and upper bounds for the following probabilities: $${\mbox{Prob}\left[ s>s_0(1+\epsilon) \right] }\quad\text{and}\quad {\mbox{Prob}\left[ s<s_0(1-\epsilon) \right] },$$ using the probability density $Bf(s)$ for $s$ and $f(s)=s^{(k-2)/2}(1-s)^{(d-k-2)/2}$. These bounds are instances of explicit concentration theorems (or explicit laws of large numbers) from probability theory. Our goal is to formulate these bounds as precisely as possible so that the lower and upper bounds are asymptotically the same when $\epsilon$ and $\delta$ approach $0$. Bounds for B ------------ We recall that $\Gamma(1/2) = \sqrt{\pi}$, $\Gamma(1) = 1$, and $\Gamma(1+z) = z\Gamma(z)$, hence $$\Gamma\left(\frac{d}{2}\right) = \left(\frac{d}{2} -1\right)! \mbox{ if $d$ is even}, \quad \quad \Gamma\left(\frac{d}{2}\right) = \left(\frac{d}{2} -1\right)\cdot \left(\frac{d}{2} -2\right) \cdots \frac{3}{2}\cdot \frac{1}{2} \cdot \sqrt{\pi} \mbox{ if $d$ is odd}.$$ In this section, we derive lower and upper bounds for $B$, see Equation (\[Eqn:B\]), by using the following form of Stirling’s approximation of $n!$ due to Robbins [@Robbins]: $$\sqrt{2\pi}n^{n+1/2}e^{-n}e^{\frac{1}{12n+1}} < \Gamma(n+1) = n! < \sqrt{2\pi} n^{n+1/2}e^{-n}e^{\frac{1}{12n}}.$$ Since we are interested in the asymptotic behavior, we focus on the case where $d$ is even. This choice does not affect the asymptotic results of our paper, but the calculations are more straight-forward in this case. We leave the details for the case where $d$ is odd to the interested reader. \[Lemma:B\_Bound\] Suppose $k$ and $d$ are both even. Then we have the following inequality[^5]: $$\frac{e^{-2}}{2\sqrt{\pi}} \cdot \frac{\left(d-2\right)^{(d-1)/2}}{ \left(k-2 \right)^{(k-1)/2}\left(d-k-2\right)^{(d-k-1)/2}} \leq B \leq \frac{e^{-1}}{2\sqrt{\pi}} \cdot \frac{\left(d-2\right)^{(d-1)/2}} {\left(k-2 \right)^{(k-1)/2}\left(d-k-2\right)^{(d-k-1)/2}}.$$ Using the bound on $n!$ from Robbins [@Robbins], we obtain $$C_0 \frac{\left(d-2\right)^{(d-1)/2}}{ \left(k-2 \right)^{(k-1)/2}\left(d-k-2\right)^{(d-k-1)/2}} \leq B \leq C_1 \frac{\left(d-2\right)^{(d-1)/2}} {\left(k-2 \right)^{(k-1)/2}\left(d-k-2\right)^{(d-k-1)/2}},$$ where $$C_0 = \frac{1}{2\sqrt{\pi}}e^{-1}e^{\frac{1}{6(d-2)+1}}e^{\frac{-1}{6(k-2)}}e^{\frac{-1}{6(d-k-2)}} \geq \frac{e^{-2}}{2\sqrt{\pi}}, \quad \text{and}$$ $$C_1 = \frac{1}{2\sqrt{\pi}}e^{-1}e^{\frac{1}{6(d-2)}}e^{\frac{-1}{6(k-2)+1}}e^{\frac{-1}{6(d-k-2)+1}} \leq \frac{e^{-1}}{2\sqrt{\pi}}.\vspace{-.25in}$$ \[Corollary:Bsf\_Bound\] With $s_0 = k/d$, we have $$\frac{e^{-2}}{2\sqrt{\pi}} \sqrt{k} \leq Bs_0f(s_0) \leq \frac{9e^{-1}}{\sqrt{2\pi}} \sqrt{k}.$$ By evaluating $f$ at $s_0$ and replacing $B$ by its lower bound found in Lemma \[Lemma:B\_Bound\], we obtain the lower bound $$\begin{aligned} Bs_0f(s_0) \geq \frac{e^{-2}}{2\sqrt{\pi}} \sqrt{k} \left(\frac{d-2}{d-k}\right)^{\frac{1}{2}} \left( \frac{k(d-2)}{d(k-2)} \right)^{\frac{k-1}{2}} \left( \frac{(d-k)(d-2)}{d(d- k-2)} \right)^{\frac{d-k-1}{2}}\geq \frac{e^{-2}}{2\sqrt{\pi}} \sqrt{k} . \end{aligned}$$ Similarly, by using the upper bound in Lemma \[Lemma:B\_Bound\], we obtain the upper bound $$Bs_0f(s_0) \leq \frac{e^{-1}}{2\sqrt{\pi}} \sqrt{k} \left(\frac{d-2}{d}\right)^{(d-1)/2} \left(\frac{k}{k-2}\right)^{(k-1)/2} \left( \frac{d- k}{d-k-2} \right)^{(d-k-1)/2} \frac{\sqrt{d}}{\sqrt{d-k}} \leq \frac{9e^{-1}}{\sqrt{2\pi}} \sqrt{k},$$ where the last inequality follows from $\frac{d}{d-k} \leq 2$ since $s_0<0.4$, and $\left(\frac{x}{x-2}\right)^{\frac{x-1}{2}} \leq 3$ for $x \geq 3$. Bounds on ${\mbox{Prob}\left[ s>s_0(1+\epsilon) \right] }$ ----------------------------------------------------------- We begin by mentioning the following inequalities which are used in our arguments below: $$\begin{aligned} & \log (1+x) \geq x-\frac{x^2}{2} \quad \text{for $0<x<1$}, \label{Bound:Ln_plus} \\ & \log (1-x) \geq -x -x^2 \quad \text{for $0 <x<0.68$,} \label{Bound:Ln_minus2}\\ & \log(1+x) \leq x \quad \text{for $x>-1$}, \quad \text{and} \label{Bound:Ln_plus_upper}\\ & \log(1+x) \leq x - \frac{x^2}{2} + \frac{x^3}{3} \quad \text{for $x>-1$}. \label{Bound:Ln_plus_upper2} \end{aligned}$$ These bounds can be verified by employing basic calculus techniques (e.g., derivatives and Taylor expansions) as well as sufficiently accurate approximations. Using these inequalities, we derive the following bounds: \[Lemma:lower+\] $${\mbox{Prob}\left[ s>s_0(1+\epsilon) \right] } \geq \frac{e^{-2}}{4 \sqrt{\pi}} e^{-\frac{1}{4}(\sqrt{k}\epsilon+1)^2 \frac{1+s_0}{1-s_0}}.$$ Moreover, when $k<\eta\epsilon^{-2}\log\frac{1}{\delta}$, $${\mbox{Prob}\left[ s>s_0(1+\epsilon) \right] }\geq \frac{e^{-2}}{4\pi}\delta^{\frac{\eta}{4}\gamma_1},$$ where $$\gamma_1=\left(1+(\eta\log(1/\delta))^{-1/2}\right)^2\left(\frac{1+s_0}{1-s_0}\right).$$ Additionally, $\gamma_1$ approaches $1$ as $\epsilon$, $\delta$, and $s_0$ approach $0$. Note that $$\label{Eqn: plus B int} {\mbox{Prob}\left[ s>s_0(1+\epsilon) \right] } = B \int_{s>s_0(1+\epsilon)} f(s)ds=Bs_0 \int_\epsilon^{\frac{1}{s_0}-1} f(s_0(1+x)) dx,$$ via the substitution $s=s_0(1+x)$. Let $g(s) = s^{k/2}(1-s)^{(d-k)/2}$, then $f(s_0(1+x))$ can be expressed in terms of $g(s)$, namely, $$\label{Eqn: f=g} f(s_0(1+x)) = \frac{g(s_0(1+x))}{s_0(1+x)\left(1-s_0(1+x)\right)}.$$ To find a lower bound on ${\mbox{Prob}\left[ s>s_0(1+\epsilon) \right] }$, we compute a bound on $g(s_0(1+x))$ from below. Taking the logarithm of $g(s_0(1+x))$, we find $$\label{Eqn: log g} \log \left( g(s_0(1+x)) \right) = \log g(s_0) + \frac{d}{2}\left( s_0 \log (1+x) + (1-s_0)\log\left(1-\frac{s_0}{1-s_0}x\right)\right).$$ Restricting $x$ to the interval $0\leq x<1$, it then follows that $0< \frac{s_0}{1-s_0}x<0.68$ from the assumption that $s_0<0.4$. We now bound the second term in Equation (\[Eqn: log g\]) using Inequalities (\[Bound:Ln\_plus\]) and (\[Bound:Ln\_minus2\]), as follows: $$\begin{aligned} s_0 \log (1+x) + & (1-s_0)\log \left(1-\frac{s_0}{1-s_0}x\right) \geq -\left(\frac{s_0(1+s_0)}{2(1-s_0)}\right)x^2. \label{Ineq: innerstatement} \end{aligned}$$ Hence, by substituting Inequality (\[Ineq: innerstatement\]) into Equation (\[Eqn: log g\]) and exponentiating, we obtain the following lower bound for $g(s_0(1+x))$: $$\label{Ineq: g term} g(s_0(1+x)) \geq g(s_0) e^{-\frac{k}{4}x^2\frac{1+s_0}{1-s_0} }.$$ We also observe that since $s_0<0.4$ and $0\leq x<1$, the denominator of Equation (\[Eqn: f=g\]) is bounded from below as follows: $$\label{Ineq: remaining term} \frac{1}{s_0(1+x)(1-s_0(1+x))} \geq \frac{1}{2s_0(1-s_0)}.$$ Therefore, by substituting Inequalities (\[Ineq: g term\]) and (\[Ineq: remaining term\]) into Equation (\[Eqn: f=g\]) when $0\leq x<1$, we have $$\label{ineq:f-lowerbound} f(s_0(1+x)) \geq \frac{g(s_0)}{2s_0(1-s_0)} e^{-\frac{k} {4}x^2\frac{1+s_0}{1-s_0}} = \frac{1}{2}f(s_0)e^{-\frac{k}{4}x^2\frac{1+s_0}{1-s_0}}.$$ Since $s_0<0.4$, we observe that $\frac{1}{s_0}-1=\frac{d-k}{k}>1.5$. Since we assumed that $\epsilon<\frac{1}{2}$ and $k\geq 4+\epsilon^{-2}>4$, it follows that $\epsilon+k^{-1/2}<1$ and so $\epsilon+k^{-1/2}<1<\frac{1}{s_0}-1$. Therefore, we further restrict $x$ to the interval $(\epsilon,\epsilon+k^{-1/2})$ and observe that Inequality (\[ineq:f-lowerbound\]) applies in this range. Therefore, $$\label{ineq:lower:restrictingrange} Bs_0 \int_\epsilon^{\frac{1}{s_0}-1} f(s_0(1+x)) dx \geq Bs_0 \int_\epsilon^{\epsilon + k^{-1/2}}f(s_0(1+x)) dx\geq \frac{1}{2} Bs_0 f(s_0) \int_\epsilon^{\epsilon + k^{-1/2}}e^{-\frac{k}{4}x^2\frac{1+s_0}{1-s_0}} dx.$$ Replacing $Bs_0f(s_0)$ with its lower bound given in Corollary \[Corollary:Bsf\_Bound\] and observing that the integrand is decreasing over an interval of width $k^{-1/2}$, Inequality (\[ineq:lower:restrictingrange\]) is bounded from below by $$\begin{aligned} \frac{1}{2} Bs_0 f(s_0) \int_\epsilon^{\epsilon + k^{-1/2}}e^{-\frac{k}{4}x^2\frac{1+s_0}{1-s_0}} dx & \geq \frac{e^{-2}}{4\sqrt{\pi}} e^{-\frac{1}{4}(\sqrt{k}\epsilon+1)^2\frac{1+s_0}{1-s_0}} , \end{aligned}$$ which completes the first inequality. The second inequality follows by replacing $k$ by the given upper bound and simplifying. \[Lemma:Upper+\] $${\mbox{Prob}\left[ s>s_0(1+\epsilon) \right] } \leq \frac{27e^{-1}}{\sqrt{2\pi}}e^{-\frac{k-2}{4}\epsilon^2(1-\frac{2}{3}\epsilon) } .$$ To derive an upper bound, we start with the expression for ${\mbox{Prob}\left[ s>(1+\epsilon)s_0 \right] }$ from Equation (\[Eqn: plus B int\]). We first find an upper bound on $f(s_0(1+x))$. We bound $f(s_0(1+x))$ using the inequality $1-x\leq e^{-x}$ for all $x$ as follows: $$\label{eqn:f:bound:first} f(s_0(1+x)) = f(s_0) (1+x)^{\frac{k-2}{2}} \left( 1- \frac{s_0}{1-s_0}x\right)^{\frac{d-k-2}{2}} \leq f(s_0) (1+x)^{\frac{k-2}{2}} \left( e^{\frac{-s_0}{1-s_0}x}\right)^{\frac{d-k-2}{2}},$$ Moreover, since $s_0<0.4$, $$\label{eqn:exponent:bound} \frac{s_0}{1-s_0}\frac{d-k-2}{2}=\frac{k}{d-k}\frac{d-k-2}{2}>\frac{k-2}{2}.$$ By applying Inequality (\[eqn:exponent:bound\]) to Inequality (\[eqn:f:bound:first\]), we derive the upper bound $$f(s_0) (1+x)^{\frac{k-2}{2}} e^{-\frac{k-2}{2}x}.$$ Therefore, by extending the region of integration in Inequality (\[Eqn: plus B int\]), we find the following upper bound on the probability: $$\label{eq:upperbound:firstpart} Bs_0\int_\epsilon^{\frac{1}{s_0}-1}f(s_0(1+x))dx\leq Bs_0f(s_0)\int_\epsilon^\infty(1+x)^{\frac{k-2}{2}} e^{-\frac{k-2}{2}x}dx.$$ By integrating by parts, we observe that for any $\ell$ and $m$ with $1\leq\ell\leq m$, $$\begin{aligned} \int_{\epsilon}^\infty (1+x)^{\ell} e^{-mx} dx \leq \frac{1}{m}(1+\epsilon)^\ell e^{-m\epsilon} + \int_\epsilon^\infty (1+x)^{\ell-1} e^{-mx}dx. \label{Equation:ineq} \end{aligned}$$ Applying Inequality (\[Equation:ineq\]) $\frac{k-2}{2}$ times to the integral in Inequality (\[eq:upperbound:firstpart\]) and bounding the resulting geometric series from above gives $$\begin{aligned} \int_\epsilon^\infty (1+x)^{\frac{k-2}{2}} e^{-\frac{k-2}{2}x} dx \leq \frac{2e^{-\frac{k-2}{2}\epsilon}}{k-2} \left( (1+ \epsilon)^{\frac{k-2}{2}} + \dots + (1+\epsilon)^0 \right) \leq \frac{2(1+\epsilon)}{\epsilon(k-2)} (1+\epsilon)^{\frac{k-2}{2}} e^{-\frac{k-2}{2}\epsilon}. \end{aligned}$$ By applying Inequality (\[Bound:Ln\_plus\_upper2\]) to $(1+\epsilon)^{\frac{k-2}{2}} = e^{\frac{k-2}{2}\log (1+\epsilon)}$, we obtain the upper bound $$\frac{2(1+\epsilon)}{\epsilon(k-2)} (1+\epsilon)^{\frac{k-2}{2}} e^{-\frac{k-2}{2}\epsilon} \leq \frac{2(1+\epsilon)}{\epsilon(k-2)} e^{-\frac{k-2} {4}\epsilon^2(1-\frac{2}{3}\epsilon)}.$$ Since $k-4\geq \epsilon^{-2}$, it follows that $\frac{(k-2)^2}{k}\geq k-4\geq\epsilon^{-2}$, and, hence, that $\epsilon(k-2)\geq \sqrt{k}$. Therefore, we can further simplify our bound to $$\frac{2(1+\epsilon)}{\epsilon(k-2)} e^{-\frac{k-2}{4}\epsilon^2(1-\frac{2}{3}\epsilon)} \leq \frac{2(1+\epsilon)}{\sqrt{k}} e^{-\frac{k-2}{4}\epsilon^2(1-\frac{2}{3}\epsilon)}. \label{Ineq: int_(1+x)}$$ By combining Inequalities (\[eq:upperbound:firstpart\]) and (\[Ineq: int\_(1+x)\]), we find an upper bound on the probability as follows: $$Bs_0\int_\epsilon^{\frac{1}{s_0}-1}f(s_0(1+x))dx\leq B s_0 f(s_0) \frac{2(1+\epsilon)}{\sqrt{k}} e^{-\frac{k-2}{4}\epsilon^2(1-\frac{2}{3}\epsilon)}.$$ By applying the upper bound on $Bs_0f(s_0)$ from Corollary \[Corollary:Bsf\_Bound\] and the assumption that $\epsilon\leq\frac{1}{2}$, which completes the inequality. Bounds on ${\mbox{Prob}\left[ s<s_0(1-\epsilon) \right] }$ ----------------------------------------------------------- We begin this section by including two additional inequalities on $\log(1-x)$. $$\begin{aligned} & & \log (1-x) \geq -x -\frac{x^2}{2}-x^3 \quad \text{for $0<x<0.815$}, \label{Bound:Ln_minus} \\ && \log(1-x)\leq -x-\frac{x^2}{2}\quad\text{ for }0\leq x<1. \label{Bound:Ln:minus:upper} \end{aligned}$$ These bounds can be justified using a similar approach as for Inequalities (\[Bound:Ln\_plus\]-\[Bound:Ln\_plus\_upper2\]). Using these inequalities, we derive the following bounds: $${\mbox{Prob}\left[ s<s_0(1-\epsilon) \right] } \geq \frac{e^{-2}}{2\sqrt{\pi}} e^{ - \frac{1}{4}\left(\frac{(\sqrt{k} \epsilon + 1)^2}{1-s_0} + 2(\sqrt[3]{k} \epsilon + k^{-1/6})^3\right)}.$$ Moreover, when $k<\eta\epsilon^{-2}\log\frac{1}{\delta}$, $${\mbox{Prob}\left[ s<s_0(1-\epsilon) \right] } \geq \frac{e^{-2}}{2\sqrt{\pi}} \delta^{\frac{\eta}{4}\gamma_2},$$ where $$\gamma_2=\frac{1}{1-s_0}\left(1+\frac{1}{\sqrt{\eta\log\frac{1}{\delta}}}\right)^2+2\left(\epsilon^{1/3}+\frac{1}{\sqrt[3]{\eta\log\frac{1}{\delta}}}\right)^3.$$ Additionally, $\gamma_2$ approaches $1$ as $\epsilon$, $\delta$, and $s_0$ approach $0$. The proof of this lemma is very similar to the proof of Lemma \[Lemma:lower+\], so we focus on the new details. The probability can be rewritten, using the substitution $s=s_0(1-x)$, as $$\label{Eqn:minus int} {\mbox{Prob}\left[ s<(1-\epsilon)s_0 \right] } = B\int_{s<s_0(1-\epsilon)} f(s)ds=Bs_0\int_{\epsilon}^1 f(s_0(1-x))dx.$$ Using $g(s)$ as in Lemma \[Lemma:lower+\], it follows that $$\label{Eq: f=g minus} f(s_0(1-x))=\frac{g(s_0(1-x))}{s_0(1-x)(1-s_0(1-x))}$$ and $$\label{Eqn: g lowerbound} \log g(s_0(1-x)) = \log g(s_0) + \frac{d}{2}\left[ s_0 \log (1-x) + (1-s_0) \log \left( 1+ \frac{s_0}{1-s_0}x \right) \right].$$ Since $0< x\leq 1$, it follows that $0<\frac{s_0}{1-s_0}x<0.68$ from the assumption that $s_0<0.4$. Therefore, we can bound the second term in Equation (\[Eqn: g lowerbound\]) using Inequalities (\[Bound:Ln\_plus\]) and (\[Bound:Ln\_minus\]), as follows: $$s_0 \log (1-x) + (1-s_0) \log \left( 1+ \frac{s_0}{1-s_0}x \right) \geq \frac{-s_0}{2(1-s_0)}x^2 -s_0x^3. \label{Ineq: taylor}$$ Substituting Inequality (\[Ineq: taylor\]) into Expression (\[Eqn: g lowerbound\]) and exponentiating, we get $$\label{Ineq: g_lower} g(s_0(1-x)) \geq g(s_0)e^{ - \frac{k}{4}\left[\frac{x^2}{1-s_0} + 2x^3 \right]}.$$ Since $s_0<0.4$, it follows that $\frac{s_0}{1-s_0}<1$ and, hence, that $(1-x)\left(1+\frac{s_0}{1-s_0}x\right)<1$. Therefore, the denominator in Equation (\[Eq: f=g minus\]) can be bounded from below by $$\label{Ineq: s0_2} \frac{1}{s_0(1-x)\left(1-s_0(1-x)\right)} = \frac{1}{s_0(1-s_0)} \cdot \frac{1}{(1-x)\left(1+\frac{s_0x}{1-s_0} \right)} \geq \frac{1}{s_0(1-s_0)}.$$ Therefore, by substituting Inequalities (\[Ineq: g\_lower\]) and (\[Ineq: s0\_2\]) into Expression (\[Eq: f=g minus\]), when $\epsilon<x<1$, we have $$\label{Ineq: fs_0_lower f_s0} f(s_0(1-x)) \geq f(s_0)e^{ - \frac{k}{4}\left(\frac{x^2}{1-s_0} + 2x^3 \right)}.$$ Since $\epsilon<\frac{1}{2}$ and $k\geq 4+\epsilon^{-2}>4$, it follows that $\epsilon+k^{-1/2}<1$. Therefore, we restrict $x$ to the interval $(\epsilon,\epsilon+k^{-1/2})$, and observe that Inequality (\[Ineq: fs\_0\_lower f\_s0\]) applies in this range. Therefore, $$\label{ineq:lower:restrict:second} Bs_0 \int_{\epsilon}^1 f(s_0(1-x))dx \geq Bs_0 f(s_0) \int_\epsilon^{\epsilon+k^{-1/2}} e^{ - \frac{k}{4}\left(\frac{x^2}{1-s_0} + 2x^3 \right)} dx.$$ Replacing $Bs_0f(s_0)$ with its lower bound given in Corollary \[Corollary:Bsf\_Bound\] and observing that the integrand is decreasing on an interval of width $k^{-1/2}$, Inequality (\[ineq:lower:restrict:second\]) is bounded from below by $$Bs_0 f(s_0)\int_\epsilon^{\epsilon+k^{-1/2}} e^{ - \frac{k}{4}\left(\frac{x^2}{1-s_0} + 2x^3 \right)} dx \geq \frac{e^{-2}}{2\sqrt{\pi}} e^{ - \frac{1}{4}\left(\frac{(\sqrt{k} \epsilon + 1)^2}{1-s_0} + 2(\sqrt[3]{k} \epsilon + k^{-1/6})^3\right)},$$ which completes the first inequality. The second inequality follows by replacing $k\geq 1$ by the given upper bound and simplifying. $${\mbox{Prob}\left[ s<s_0(1-\epsilon) \right] } \leq \frac{18\sqrt{2}e^{1/2}}{\sqrt{\pi}} e^{-\left(\frac{k}{4}\right)\epsilon^2}\leq \frac{18\sqrt{2}e^{1/2}}{\sqrt{\pi}} e^{-\left(\frac{k-2}{4}\right)\epsilon^2\left(1-\frac{2}{3}\epsilon\right)}.$$ The proof of this lemma is very similar to the proof of Lemma \[Lemma:Upper+\], so we focus on the new details. To prove an upper bound, we start with the bound on ${\mbox{Prob}\left[ s<s_0(1-\epsilon) \right] }$ from Equation (\[Eqn:minus int\]). We first observe that $$\label{eq:fs:minus} f(s_0(1-x)) = f(s_0) (1-x)^{\frac{k-2}{2}} \left( 1+ \frac{s_0}{1-s_0}x\right)^{\frac{d-k-2}{2}}.$$ We now bound the logarithm of the second and third factors in Equation (\[eq:fs:minus\]) using Inequalities (\[Bound:Ln\_plus\_upper\]) and (\[Bound:Ln:minus:upper\]) as follows: $$\begin{aligned} \log \left( (1-x)^{\frac{k-2}{2}} \left( 1+ \frac{s_0}{1-s_0}x\right)^{\frac{d-k-2}{2}} \right) & \leq \frac{k-2}{2} \left(-x-\frac{x^2}{2}\right) + \frac{d-k-2}{2} \left( \frac{s_0}{1-s_0}x \right)\label{Ineq:First:Half} \end{aligned}$$ Since $\frac{s_0}{1-s_0}=\frac{k}{d-k}$ and $\epsilon<x<1$, Inequality (\[Ineq:First:Half\]) further simplifies to $$\frac{k-2}{2} \left(-x-\frac{x^2}{2}\right) + \frac{d-k-2}{2} \left( \frac{k}{d-k}x \right) \leq x -\left(\frac{k-2}{4}\right)x^2 \leq \frac{3}{2} - \left(\frac{k}{4}\right)x^2.$$ Hence, for $\epsilon \leq x < 1$, we have $$\label{Ineq:fs0_lower} f(s_0(1-x)) \leq f(s_0) e^{3/2} e^{-\left(\frac{k}{4}\right)x^2} \leq f(s_0) e^{3/2} e^{-\left(\frac{k}{4}\right)\epsilon x }.$$ Substituting Inequality (\[Ineq:fs0\_lower\]) into the integral of Equation (\[Eqn:minus int\]), we find $$\begin{aligned} Bs_0\int_\epsilon^1f(s_0(1-x))dx &\leq Bs_0f(s_0)e^{3/2}\int_\epsilon^1 e^{-\left(\frac{k}{4}\right)\epsilon x}dx\notag\\ &\leq Bs_0f(s_0)e^{3/2}\int_\epsilon^\infty e^{-\left(\frac{k}{4}\right)\epsilon x}dx \leq Bs_0f(s_0) \frac{4e^{3/2}}{k\epsilon} e^{-\frac{k}{4}\epsilon^2}.\label{Ineq:Almost} \end{aligned}$$ Since $k\geq\epsilon^{-2}$, Inequality (\[Ineq:Almost\]) can be further simplified to $$Bs_0f(s_0) \frac{4e^{3/2}}{\sqrt{k}} e^{-\frac{k}{4}\epsilon^2}$$ Applying the upper bound on $Bs_0f(s_0)$ from Corollary \[Corollary:Bsf\_Bound\] completes the first inequality of the proof. The final inequality follows from the fact that $k\geq (k-2)\left(1-\frac{2}{3}\epsilon\right)$. This completes the proof all of the conditions in Section \[Sec:Main:New\], and, therefore, completes the proof of our main theorem, Theorem \[Thm:IntroThm\]. [^1]: Partially supported by a grant from the Simons Foundation (\#282399 to Michael Burr) and National Science Foundation Grant CCF-1527193. [^2]: Partially supported by the National Science Foundation under Grants CCF-1407623, DMS-1403062, and DMS-1547399. [^3]: Most of the work was done while the author was at Clemson University. [^4]: In this paper, we suppress the pullback maps on equalities for differential forms since there is a unique (almost) bijective map under consideration in each case. We leave the details to the interested reader. [^5]: It is possible, to derive tighter bounds on constants, but the ones appearing here are sufficient for our proofs. We leave the details of the tighter bounds to the interested reader

**** Conformal scattering on the Schwarzschild metric \ [Jean-Philippe NICOLAS]{}\ \ \ \ [**Abstract.**]{} We show that existing decay results for scalar fields on the Schwarzschild metric are sufficient to obtain a conformal scattering theory. Then we re-interpret this as an analytic scattering theory defined in terms of wave operators, with an explicit comparison dynamics associated with the principal null geodesic congruences. The case of the Kerr metric is also discussed. [**Keywords.**]{} Conformal scattering, black holes, wave equation, Schwarzschild metric, Goursat problem. [**Mathematics subject classification.**]{} 35L05, 35P25, 35Q75, 83C57. Introduction ============ Conformal time dependent scattering originates from the combination of the ideas of R. Penrose on spacetime conformal compactification [@Pe1963; @Pe1964; @Pe1965; @PeRi], the Lax-Phillips theory of scattering [@LaPhi] and F.G. Friedlander’s notion of radiation fields [@Fri1962; @Fri1964; @Fri1967]. The Lax-Phillips scattering theory for the wave equation is a construction on flat spacetime. It is based on a translation representer of the solution, which is re-interpreted as an asymptotic profile of the field along outgoing radial null geodesics, analogous to Friedlander’s radiation field[^1]. Observing this, Friedlander formulated the first version of conformal time-dependent scattering in 1980 [@Fri1980]. The framework was a static spacetime with a metric approaching the flat metric fast enough at infinity (like $1/r^2$) so as to ensure that the conformal spacetime has a regular null infinity (denoted ${{\mathscr I}}$). This allowed him to construct radiation fields as traces on ${{\mathscr I}}$ of conformally rescaled fields. The scattering theory as such was obtained by the resolution of a Goursat (characteristic Cauchy) problem on null infinity, whose data are the radiation fields. Then he went on to recover the analytically explicit aspects of the Lax-Phillips theory, in particular the translation representation of the propagator, a feature which is tied in with the staticity of the geometry[^2]. His ideas were taken up by J.C. Baez, I.E. Segal and Zhou Z.F. in 1989-1990 [@Ba1989a; @Ba1989b; @Ba1990; @BaSeZho1990; @BaZho1989] to develop conformal scattering theories on flat spacetime for non linear equations. Note that the resolution of the characteristic Cauchy problem was the object of a short paper by L. Hörmander in 1990 [@Ho1990], in which he described a method of resolution based entirely on energy estimates and weak compactness, for the wave equation on a general spatially compact spacetime. Friedlander himself came back to conformal scattering just before his death in a paper published posthumously in 2001 [@Fri2001]. It is on the whole quite surprising that his idea did not entail more active research in the domain. It is even more puzzling that the research it did entail remained strictly focused on static geometries. In fact, the observation that a complete scattering theory in the physical spacetime, amounts to the resolution of a Goursat problem on the compactified spacetime, is the door open to the development of scattering theories on generic non stationary geometries. Probably Friedlander’s wish to recover all the analytic richness of the Lax-Phillips theory prevented him from pushing his theory this far. However, the door being open, somebody had to go through it one day. This was done in 2004 by L.J. Mason and the author in [@MaNi2004], a paper in which a conformal scattering theory was developed for scalar waves[^3], Dirac and Maxwell fields, on generically non stationary asymptotically simple spacetimes. A conformal scattering theory for a non linear wave equation on non stationary backgrounds was then obtained by J. Joudioux in 2012 [@Jo2012]. The purpose of the present work is to show how existing decay results can be used to obtain conformal scattering constructions on black hole backgrounds. We treat the case of the wave equation on the Schwarzschild metric, for which the analytic scattering theory is already known (see J. Dimock and B.S. Kay in 1985-1987 [@Di1985; @DiKa1986; @DiKa1987]). The staticity of the exterior of the black hole gives a positive definite conserved quantity on spacelike slices, which can be extended to the conformally rescaled spacetime ; the known decay results (we use those of M. Dafermos and I. Rodnianski, see for example their lecture notes [@DaRoLN]) are then enough to obtain a complete scattering theory. It is in some sense unsatisfactory to use decay results, because they require a precise understanding of the trapping by the photon sphere, which is much more information than is needed for a scattering theory. However, such results should by nature be fairly robust under small perturbations. So the conformal scattering theories on stationary black hole backgrounds obtained using them can in principle be extended to non stationary perturbations. Not that this is at all trivial. This work is to be considered a first step in the developent of conformal scattering theories on black hole backgrounds, to be followed by extensions to other equations and to more general, non stationary situations. The paper is organized as follows. Section \[GeomFrame\] contains the description of the geometrical framework for the case of the wave equation on the Schwarzschild metric. We describe the conformal compactification of the geometry and the corresponding rescaling of the wave equation. In section \[EnIdent\], we derive the main energy estimates on the compactified spacetime. Section \[Scattering\] is devoted to the conformal scattering construction and to its re-interpretation in terms of wave operators associated to a comparison dynamics. This type of structure, contrary to the translation representation, would survive in a non stationary situation (see [@MaNi2004] for an analogous construction on non stationary asymptotically simple spacetimes). This re-interpretation concerns the most difficult aspects of analytic scattering theory : the existence of inverse wave operators and asymptotic completeness. For the existence of direct wave operators, which is the easy part, we keep the analytic approach using Cook’s method ; this is explained in appendix \[AppendixCook\]. The reason for this choice is the simplicity of the method and its easy entendibility to fairly general geometries, using a geometric transport equation as comparison dynamics, provided we have a precise knowledge of the asymptotic behaviour of the metric and good uniform energy estimates (which are in any case crucial for developing a conformal scattering theory). Some technical aspects of the resolution of the Goursat problem on the conformal boundary, which is at the core of the conformal scattering theory, are explained in appendix \[HormGP\]. Section \[Kerr\] is devoted to remarks concerning the extension of these results to the Kerr metric and some concluding comments. Since the first version of this work, this last section has been entirely re-written in order to take the new results by M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman [@DaRoShla] into account. [**Notations and conventions.**]{} Given a smooth manifold $M$ without boundary, we denote by ${\cal C}^\infty_0 (M)$ the space of smooth compactly supported scalar functions on $M$ and by ${\cal D}' (M)$ its topological dual, the space of distributions on $M$. Concerning differential forms and Hodge duality, following R. Penrose and W. Rindler [@PeRi], we adopt the following convention : on a spacetime $({\cal M},g)$ (i.e. a $4$-dimensional Lorentzian manifold that is oriented and time-oriented), the Hodge dual of a $1$-form $\alpha$ is given by $$(*\alpha)_{abc} = e_{abcd} \alpha^d\, ,$$ where $e_{abcd}$ is the volume form on $({\cal M} , g)$, which in this paper we simply denote ${\mathrm{dVol}}$. We shall use two important properties of the Hodge star : - given two $1$-forms $\alpha$ and $\beta$, we have $$\label{HStarP1} \alpha \wedge * \beta = -\frac{1}{4} \alpha_a \beta^a\, {\mathrm{dVol}}\, ;$$ - for a $1$-form $\alpha$ that is differentiable, $$\label{HStarP2} {\mathrm{d}}* \alpha = -\frac{1}{4} (\nabla_a \alpha^a ) {\mathrm{dVol}}\, .$$ Throughout this work, we shall talk about analytic and conformal scattering as two different approaches to scattering theory. In most cases, we mean that the former is based on spectral techniques and the latter relies on a conformal compactification. The truly significant difference however is that conformal scattering understands the scattering construction as the resolution of a Goursat problem on the conformal boundary, described as a finite hypersurface, whereas analytic scattering sees the scattering channels as asymptotic regions. Geometrical framework {#GeomFrame} ===================== The Schwarzschild metric is given on ${\mathbb{R}}_t \times ]0,+\infty [_r \times S^2_\omega$ by $$g = F {\mathrm{d}}t^2 - F^{-1} {\mathrm{d}}r^2 - r^2 {\mathrm{d}}\omega^2 \, ,~ F = F(r) = 1 -\frac{2M}{r} \, ,$$ where ${\mathrm{d}}\omega^2$ (also denoted $e_{S^2}$ below) is the euclidean metric on $S^2$ and $M>0$ is the mass of the black hole. We work on the exterior of the black hole $\{ r>2M \}$, which is the only region of spacetime perceived by static observers at infinity (think for instance of a distant telescope pointed at the black hole). Introducing the Regge-Wheeler coordinate $r_* = r + 2M \log (r-2M)$, such that ${\mathrm{d}}r = F {\mathrm{d}}r_*$, the metric $g$ takes the form $$g = F ({\mathrm{d}}t^2 - {\mathrm{d}}r_*^2 ) - r^2 {\mathrm{d}}\omega^2 \, .$$ The Schwarzschild metric has a four-dimensional space of global Killing vector fields, generated by $$\label{KVF} K:=\partial_t \, ,~ X:=\sin \varphi \, \partial_\theta + \cot \theta \cos \varphi \, \partial_\varphi \, ,~Y:=\cos \varphi \, \partial_\theta - \cot \theta \sin \varphi \, \partial_\varphi \, ,~Z:=\partial_\varphi \, ,$$ which are the timelike (outside the black hole) Killing vector field $\partial_t$ and the three generators of the rotation group. Some other essential vector fields are the principal null vector fields (the vectors we give here are “unnormalized”, they are not the first two vectors of a normalized Newman-Penrose tetrad) $$\label{PND} l = \partial_t + \partial_{r_*} \, ,~ n = \partial_t - \partial_{r_*} \, .$$ We perform a conformal compactification of the exterior region using the conformal factor $\Omega = 1/r$, i.e. we put $$\hat{g} = \Omega^2 g \, .$$ To express the rescaled Schwarzschild metric, we use coordinates $u = t-r_*$, $R=1/r$, $\omega$ : $$\label{ghatu} \hat{g} = R^2 (1-2MR) {\mathrm{d}}u^2 - 2 {\mathrm{d}}u {\mathrm{d}}R - {\mathrm{d}}\omega^2 \, .$$ The inverse metric is $$\label{InvRescSchwaMet} \hat{g}^{-1} = - \partial_u \otimes \partial_R - \partial_R \otimes \partial_u - R^2 (1-2MR) \partial_R\otimes \partial_R - e^{-1}_{S^2} \, .$$ The non-zero Christoffel symbols for $\hat{g}$ in the coordinates $u,R,\omega$ are : $$\begin{gathered} {\hat\Gamma}^0_{00} = R (1-3MR) \, ,~ {\hat\Gamma}^1_{00} = R^3 (1-2MR)(1-3MR) \, ,~ {\hat\Gamma}^1_{01} = -R(1-3MR) \, , \\ {\hat\Gamma}^2_{33} = -\sin \theta \cos \theta \, ,~ {\hat\Gamma}^3_{23} = \cot \theta \, .\end{gathered}$$ If we use the coordinates $(t,r,\theta , \varphi )$, we get instead (still for the metric $\hat{g}$) $$\begin{gathered} {\hat\Gamma}^0_{01} = \frac{3M-r}{r(r-2M)} \, ,~ {\hat\Gamma}^1_{00} = \frac{(r-2M)(3M-r)}{r^3} \, ,~ {\hat\Gamma}^1_{11} = \frac{M-r}{r(r-2M)} \, , \\ {\hat\Gamma}^2_{33} = -\sin \theta \cos \theta \, ,~ {\hat\Gamma}^3_{23} = \cot \theta \, ,\end{gathered}$$ the others being zero. Future null infinity ${{\mathscr I}}^+$ and the past horizon ${{\mathscr H}}^-$ are null hupersurfaces of the rescaled spacetime $${{\mathscr I}}^+ = {\mathbb{R}}_u \times \{ 0\}_R \times S^2_\omega \, ,~ {{\mathscr H}}^- = {\mathbb{R}}_u \times \{ 1/2M \}_R \times S^2_\omega \, .$$ If instead of $u,R,\omega$ we use the coordinates $v=t+r_*,R,\omega$, the metric $\hat{g}$ takes the form $$\label{ghatv} \hat{g} = R^2 (1-2MR) {\mathrm{d}}v^2 + 2 {\mathrm{d}}v {\mathrm{d}}R - {\mathrm{d}}\omega^2 \, .$$ In these coordinates we have access to past null infinity ${{\mathscr I}}^-$ and the future horizon ${{\mathscr H}}^+$ described as the null hypersurfaces $${{\mathscr I}}^- = {\mathbb{R}}_v \times \{ 0\}_R \times S^2_\omega \, ,~ {{\mathscr H}}^+ = {\mathbb{R}}_v \times \{ 1/2M \}_R \times S^2_\omega \, .$$ The compactification is not complete ; spacelike infinity $i^0$ and the timelike infinities $i^\pm$ remain at infinity for $\hat{g}$. The crossing sphere $S^2_\mathrm{c}$, which is the boundary of all level hypersurfaces of $t$ outside the black hole and the place where the future and past horizons meet, is not at infinity but it is not described by the coordinate systems $\{u,R,\omega \}$ and $\{v,R,\omega \}$ ; it is the only place in $\{ r\geq 2M \} \cup {{\mathscr I}}^\pm$ where $\partial_t$ vanishes. See Figure \[PenD\] for a Carter-Penrose diagram of the compactified exterior. ![Carter-Penrose diagram of the conformal compactification of the exterior of the black hole.[]{data-label="PenD"}](PenroseD.jpg){width="4in"} A crucial feature of the conformal compactification using the conformal factor $1/r$ is that it preserves the symmetries : the vector fields are still Killing for $\hat{g}$. In particular, the vector field $\partial_t$ becomes $\partial_u$ in the $(u,R,\omega )$ coordinate system, respectively $\partial_v$ in the $(v,R,\omega )$ coordinate system ; thus it extends as the future-oriented null generator of null infinities ${{\mathscr I}}^\pm$ and the future and past horizons ${{\mathscr H}}^\pm$. We shall denote by $\cal M$ the exterior of the black hole, ${\cal M}={\mathbb{R}}_t \times ]2M , +\infty [_r \times S^2$, and by $\bar{\cal M}$ its conformal compactification, i.e. $$\bar{\cal M} = {\cal M} \cup {{\mathscr I}}^+ \cup {{\mathscr H}}^+ \cup {{\mathscr I}}^- \cup {{\mathscr H}}^- \cup S^2_c \, .$$ The constructions of the horizons and of null infinities are of a very different nature. Understanding the horizons as smooth null hypersurfaces of the analytically extended Schwarzschild exterior only requires a change of coordinates, for instance the advanced and retarded Eddington-Finkelstein coordinates $(u,R,\omega)$ and $(v,R,\omega)$. For the construction of null infinities however, the conformal rescaling is necessary and ${{\mathscr I}}^\pm$ are boundaries of the exterior of the black hole endowed with the metric $\hat{g}$, not of the physical exterior $({\cal M},g)$. The main hypersurfaces that we shall use in this paper are the following : $$\begin{aligned} \Sigma_t &=& \{ t \} \times \Sigma \, ,~ \Sigma = ]2M , +\infty [_r \times S^2_\omega = {\mathbb{R}}_{ r_*} \times S^2_\omega \, , \label{Sigt} \\ S_T &=& \left\{ (t,r_* , \omega) \in {\mathbb{R}}\times {\mathbb{R}}\times S^2 \, ;~ t = T+ \sqrt{1+r_*^2} \right\} \label{ST} \, , \\ {{\mathscr I}}^+_T &=& {{\mathscr I}}^+ \cap \{ u \leq T\} = ]-\infty , T]_u \times \{ 0 \}_R \times S^2_\omega \, , \label{scriT} \\ {{\mathscr H}}^+_T &=& S^2_{\mathrm{c}} \cup ({{\mathscr H}}^+ \cap \{ v \leq T\} ) = S^2_{\mathrm{c}} \cup (]-\infty , T]_v \times \{ 1/2M \}_R \times S^2_\omega ) \, . \label{scrhT}\end{aligned}$$ For $T>0$, the hypersurfaces $\Sigma_0$, ${{\mathscr H}}^+_T$, $S_T$ and ${{\mathscr I}}^+_T$ form a closed — except for the part where ${{\mathscr I}}^+$ and $\Sigma_0$ touch $i^0$ — hypersurface on the compactified exterior (see Figure \[3surface\]). We make such an explicit choice for the hypersurface $S_T$ for the sake of clarity but it is not strictly necessary, all that is required of $S_T$ is that it is uniformly spacelike for the rescaled metric, or even achronal, and forms a closed hypersurface with $\Sigma_0$, ${{\mathscr H}}^+_T$, and ${{\mathscr I}}^+_T$. ![The main hypersurfaces represented on the compactified exterior.[]{data-label="3surface"}](3Surface.jpg){width="4in"} The scalar curvature of the rescaled metric $\hat{g}$ is $$\mathrm{Scal}_{\hat{g}} = 12MR \, .$$ So $\phi \in {\cal D}' ({\mathbb{R}}_t \times ]0,+\infty [_r \times S^2_\omega)$ satisfies $$\label{WEqPhys} \square_g \phi =0$$ if and only if $\hat{\phi} = \Omega^{-1} \phi$ satisfies $$\label{WEqResc} (\square_{\hat{g}} + 2MR ) \hat{\phi} =0 \, .$$ By the classic theory of hyperbolic partial differential equations (see Leray [@Le1953]), for smooth and compactly supported initial data $\hat{\phi}_0$ and $\hat{\phi}_1$ on $\Sigma_0$, we have the following properties : - there exists a unique $\hat{\phi} \in {\cal C}^\infty ({\cal M})$ solution of such that $$\hat{\phi} \vert_{\Sigma_0} = \hat{\phi}_0 \mbox{ and } \partial_t \hat{\phi} \vert_{\Sigma_0} = \hat{\phi}_1 \, ,$$ - $\hat{\phi}$ extends as a smooth function on $\bar{\cal M}$ and therefore has a smooth trace on ${{\mathscr H}}^\pm \cup {{\mathscr I}}^\pm$. The D’Alembertians for the metrics $g$ and $\hat{g}$ have the following expressions in variables $(t, r_*, \omega)$ : $$\begin{aligned} \square_g &=& \frac{1}{F} \left( \frac{\partial^2}{\partial t^2} - \frac{1}{r^2} \frac{\partial}{\partial r_*} r^2 \frac{\partial}{\partial r_*} \right) - \frac{1}{r^2} \Delta_{S^2} \, ,\\ \square_{\hat{g}} &=& \frac{r^2}{F} \left( \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial r_*^2} \right) - \Delta_{S^2} \, .\end{aligned}$$ The volume forms associated with $g$ and $\hat{g}$ are $$\begin{aligned} {\mathrm{dVol}}_g &=& r^2 \sin \theta {\mathrm{d}}t \wedge {\mathrm{d}}r \wedge {\mathrm{d}}\theta \wedge {\mathrm{d}}\varphi = r^2 {\mathrm{d}}t \wedge {\mathrm{d}}r \wedge {\mathrm{d}}^2 \omega = r^2 F {\mathrm{d}}t \wedge {\mathrm{d}}r_* \wedge {\mathrm{d}}^2 \omega \, ,\\ {\mathrm{dVol}}_{\hat{g}} &=& \Omega^4 {\mathrm{dVol}}_g = R^2{\mathrm{d}}t \wedge {\mathrm{d}}r \wedge {\mathrm{d}}^2 \omega = R^2 F {\mathrm{d}}t \wedge {\mathrm{d}}r_* \wedge {\mathrm{d}}^2 \omega \, ,\end{aligned}$$ ${\mathrm{d}}^2 \omega$ being the euclidean area element on $S^2$. Energy identities {#EnIdent} ================= The usual stress-energy tensor for the wave equation is not conformally invariant. We have therefore two possible approaches to establish energy identities or inequalities. 1. Work with the rescaled quantities $\hat{\phi}$ and $\hat{g}$. The main advantage is that for all $T>0$, the four hypersurfaces $\Sigma_0$, ${{\mathscr H}}^+_T$, $S_T$ and ${{\mathscr I}}^+_T$ are finite hypersurfaces in our rescaled spacetime (except for the part of $\Sigma_0$ and ${{\mathscr I}}^+$ near $i^0$, but we will work with solutions supported away from $i^0$ to establish our energy identities). However, we encounter a problem of a different kind : equation does not admit a conserved stress-energy tensor. Fortunately, it turns out that if we use the stress-energy tensor for the wave equation on the rescaled spacetime, and contract it with $\partial_t$, the error term is a divergence. Therefore, we recover an exact conservation law. 2. Work with the physical quantities $\phi$ and $g$. We have an immediate conserved stress energy tensor associated with the equation. The drawback here is that ${{\mathscr I}}$ is at infinity. So we must use our conservation law to get energy identities on finite closed hypersurfaces, then take the limit of these identities as some parts of the hypersurfaces approach ${{\mathscr I}}$. Both methods are in principle absolutely fine. We choose the first one since, thanks to the stationarity of Schwarzschild’s spacetime, it gives energy identities in a more direct manner[^4]. By the finite propagation speed, we know that for smooth compactly supported data on $\Sigma_0$, i.e. supported away from $i^0$, the associated solution of vanishes in a neighbourhood of $i^0$. For such solutions, the singularity of the conformal metric at $i^0$ can be ignored and we obtain energy identities for all $T>0$ between the hypersurfaces $\Sigma_0$, ${{\mathscr H}}^+_T$, $S_T$ and ${{\mathscr I}}^+_T$. Then we show, using known decay results, that the energy flux through $S_T$ tends to zero as $T\rightarrow +\infty$. This yields an energy identity between $\Sigma_0$, ${{\mathscr H}}^+$ and ${{\mathscr I}}^+$, which carries over by density to initial data in a Hilbert space on $\Sigma_0$ (see section \[EnEstTInfinite\] for details). Conserved energy current for the rescaled field ----------------------------------------------- The stress-energy tensor for the wave equation associated with $\hat{g}$ is given by $$\label{SET} \hat{T}_{ab}= \hat\nabla_a \hat\phi \hat\nabla_b \hat\phi - \frac12 \langle \hat\nabla \hat\phi \, ,~ \hat\nabla \hat\phi \rangle_{\hat{g}} \, \hat{g}_{ab}\, .$$ When $\hat\phi$ is a solution of , the divergence of $\hat{T}$ is $$\hat\nabla^a \hat{T}_{ab} = (\square_{\hat{g}} \hat\phi ) \hat\nabla_b \hat\phi = -2MR \hat\phi \hat\nabla_b \hat\phi \, .$$ The energy current $1$-form associated with static observers is obtained by contracting $\hat{T}$ with the timelike Killing vector $K=\partial_t$ : $$\hat{J}_a = K^b \hat{T}_{ab} \, .$$ This is not conserved since $$\label{DivCurrent} \hat\nabla^a \hat{J}_a = -2MR \hat\phi \partial_t \hat\phi \, .$$ Putting $$V = MR\hat\phi^2 \partial_t \, ,$$ it is easy to see that $$2MR \hat\phi \partial_t \hat\phi = \mathrm{div} V \, .$$ Indeed $$\mathrm{div} V = \hat\nabla_a V^a = \frac{\partial}{\partial t} \left( MR \hat\phi^2 \right) + {\hat\Gamma}^{\mathbf{a}}_{{\mathbf{a}}0} V^0$$ and in the coordinate system $(t,r,\theta,\varphi)$, all the Christoffel symbols ${\hat\Gamma}^{\mathbf{a}}_{{\mathbf{a}}0}$ are zero. So can be written as an exact conservation law $$\label{ConsLaw} \hat\nabla_a \left( \hat{J}^a + V^a \right) =0 \, ,~ \mbox{with } V = MR\hat\phi^2 \partial_t \, .$$ The vector $V$ is causal and future oriented on $\bar{\cal M}$, timelike on $\cal M$, and the stress-energy tensor $\hat{T}_{ab}$ satisfies the dominant energy condition. Therefore, the energy flux across achronal hypersurfaces will be non negative and that across spacelike hypersurfaces will be positive definite. We will observe these properties on the explicit expressions of the fluxes that we calculate in the next section. Energy identity up to $S_T$ {#EnIdST} --------------------------- The conservation law gives an exact energy identity between the hypersurfaces $\Sigma_0$, ${{\mathscr H}}^+_T$, $S_T$ and ${{\mathscr I}}^+_T$, for solutions of the rescaled equation associated with smooth and compactly supported initial data. We denote by $\hat{\cal E}_{\partial_t , S}$ the rescaled energy flux, associated with $\partial_t$, across an oriented hypersurface $S$, i.e.[^5] $$\label{RescEnS} \hat{\cal E}_{\partial_t, S} = -4 \int_{S} * (\hat{J}_a + V_a ){\mathrm{d}}x^a \, .$$ For any $T>0$, we have $$\label{EnIdentityT} \hat{\cal E}_{\partial_t, \Sigma_0} = \hat{\cal E}_{\partial_t, {{\mathscr I}}^+_T} + \hat{\cal E}_{\partial_t, {{\mathscr H}}^+_T} + \hat{\cal E}_{\partial_t, S_T} \, .$$ The property of the Hodge star gives us an easy way to express the energy flux across an oriented $3$-surface $S$ $$\hat{\cal E}_{\partial_t, S} = -4\int_{S} * (\hat{J}_a + V_a ){\mathrm{d}}x^a = \int_S (\hat{J}_a+V_a)\hat{N}^a \, \hat{L}\lrcorner {\mathrm{dVol}}_{\hat{g}} \, ,$$ where $\hat{L}$ is a vector field transverse to $S$ and compatible with the orientation of the hypersurface, and $\hat{N}$ is the normal vector field to $S$ such that $\hat{g} (\hat{L},\hat{N})=1$. On $\Sigma_0$, we take $$\hat{L}= \frac{r^2}{F} \partial_t \, ,~\hat{N} = \partial_t \, .$$ On ${{\mathscr I}}^+$, we take for $\hat{L}$ the future-oriented null vector $\hat{L}_{{{\mathscr I}}^+} =-\partial_R$ in coordinates $u,R,\omega$. The vector field $-\partial_R$ in the exterior of the black hole is equal to $r^2 F^{-1} l$, with $l$ being the first principal null vector field given in , and extends smoothly to ${{\mathscr I}}^+$ : $$\hat{L}_{{{\mathscr I}}^+} = \left. r^2 F^{-1} l \right\vert_{{{\mathscr I}}^+} \, .$$ On ${{\mathscr H}}^+$, we choose $\hat{L}_{{{\mathscr H}}^+} = \partial_R$ (in coordinates $v,R,\omega$), i.e. $$\hat{L}_{{{\mathscr H}}^+} = \left. r^2 F^{-1} n \right\vert_{{{\mathscr H}}^+}\, ,$$ where $n$ is the second principal null vector field in . On both ${{\mathscr I}}^+$ and ${{\mathscr H}}^+$, we therefore have $\hat{N} = \partial_t$ (i.e. $\partial_v$ on ${{\mathscr H}}^+$ and $\partial_u$ on ${{\mathscr I}}^+$). Since $V \propto \partial_t$ and on ${{\mathscr I}}$ and ${{\mathscr H}}$ the vector field $\partial_t$ is null, we have $\hat{g} (V,\hat{N})=0$. The energy identity reads $$\begin{gathered} \int_{S_T} ((\hat{J}_a+V_a)\hat{N}^a) \, \hat{L}\lrcorner {\mathrm{dVol}}_{\hat{g}} + \int_{{{\mathscr I}}^+_T} (\hat{J}_a K^a) \, \hat{L}_{{{\mathscr I}}^+} \lrcorner {\mathrm{dVol}}_{\hat{g}} + \int_{{{\mathscr H}}^+_T} (\hat{J}_a K^a) \, \hat{L}_{{{\mathscr H}}^+} \lrcorner {\mathrm{dVol}}_{\hat{g}} \nonumber \\ = \int_{\Sigma_0} ((\hat{J}_a+V_a)K^a) \, r^2 F^{-1} \partial_t \lrcorner {\mathrm{dVol}}_{\hat{g}} \, . \label{EIT}\end{gathered}$$ We calculate the explicit expressions of the energy fluxes through ${{\mathscr I}}^+_T$, ${{\mathscr H}}^+_T$ and $\Sigma_0$ : $$\begin{aligned} \hat{\cal E}_{\partial_t, \Sigma_0} &=& \int_{\Sigma_0} (\hat{J}_a+V_a)K^a \, r^2 F^{-1} \partial_t \lrcorner {\mathrm{dVol}}_{\hat{g}} \\ &=& \frac12 \int_{\Sigma_0} \left( (\partial_t \hat\phi )^2 + (\partial_{r_*} \hat\phi )^2 + R^2 F \vert \nabla_{S^2} \hat\phi \vert^2 + 2 MFR^3\hat\phi^2 \right) {\mathrm{d}}r_* {\mathrm{d}}^2 \omega \, ; \\ \hat{\cal E}_{\partial_t, {{\mathscr I}}^+_T} &=& \int_{{{\mathscr I}}^+_T} \hat{J}_a K^a \hat{L}_{{{\mathscr I}}^+} \lrcorner {\mathrm{dVol}}_{\hat{g}} = \int_{{{\mathscr I}}^+_T} (\hat\nabla_K \hat\phi )^2 \hat{L}_{{{\mathscr I}}^+} \lrcorner {\mathrm{dVol}}_{\hat{g}} \\ &=& \int_{{{\mathscr I}}^+_T} (\partial_u (\hat\phi \vert_{{{\mathscr I}}^+} ) )^2 {\mathrm{d}}u {\mathrm{d}}^2 \omega\, ; \\ \hat{\cal E}_{\partial_t, {{\mathscr H}}^+_T} &=& \int_{{{\mathscr H}}^+_T} \hat{J}_a K^a \hat{L}_{{{\mathscr H}}^+} \lrcorner {\mathrm{dVol}}_{\hat{g}} = \int_{{{\mathscr H}}^+_T} (\hat\nabla_K \hat\phi )^2 \hat{L}_{{{\mathscr H}}^+} \lrcorner {\mathrm{dVol}}_{\hat{g}} \\ &=& \int_{{{\mathscr H}}^+_T} (\partial_v (\hat\phi \vert_{{{\mathscr H}}^+}) )^2 {\mathrm{d}}v {\mathrm{d}}^2 \omega \, .\end{aligned}$$ We observe that the first flux defines a positive definite quadratic form and the two others non-negative quadratic forms. We now calculate the flux through $S_T$. To this purpose, we make explicit choices of vectors $\hat{L}$ and $\hat{N}$ on $S_T$. Let us denote $$\Psi (t,r_*,\omega ) = t - \sqrt{1+r_*^2} \, ,$$ so the hypersurface $S_T$ is $$S_T = \{ (t,r_*,\omega) \, ; ~\Psi (t,r_*,\omega ) = T \} \, .$$ A co-normal to $S_T$ is given by $$N_a {\mathrm{d}}x^a = {\mathrm{d}}\Psi = {\mathrm{d}}t - \frac{r_*}{\sqrt{1+r_*^2}} {\mathrm{d}}r_*$$ and the associated normal vector field is $$\hat{N}^a = \hat{g}^{ab} N_b \, ,~ \mbox{i.e. } \hat{N}^a \frac{\partial}{\partial x^a} = r^2 F^{-1} \left( \frac{\partial}{\partial t} + \frac{r_*}{\sqrt{1+r_*^2}} \frac{\partial}{\partial r_*} \right) \, .$$ For the transverse vector $\hat{L}$, we can take $$\hat{L}^a \frac{\partial}{\partial x^a} = \frac{1+r_*^2}{1+2r_*^2} \left( \frac{\partial}{\partial t} - \frac{r_*}{\sqrt{1+r_*^2}} \frac{\partial}{\partial r_*} \right) \, ,$$ which is future-oriented and satisfies $\hat{L}_a \hat{N}^a =1$. We can now calculate the energy flux through $S_T$. First we have $$(\hat{J}_a+V_a )\hat{N}^a = MR \hat\phi^2 + \frac{r^2}{2F} \left( (\partial_t \hat\phi )^2 + (\partial_{r_*} \hat\phi )^2 + \frac{2r_*}{\sqrt{1+r_*^2}} \partial_t \hat\phi \partial_{r_*} \hat\phi + R^2F \vert \nabla_{S^2} \hat\phi \vert^2 \right) \, .$$ The contraction of $\hat{L}$ into the volume form for $\hat{g}$ is as follows $$\hat{L}\lrcorner {\mathrm{dVol}}_{\hat{g}} = \frac{1+r_*^2}{1+2r_*^2} R^2 F \sin \theta \left( {\mathrm{d}}r_* \wedge {\mathrm{d}}\theta \wedge {\mathrm{d}}\varphi + \frac{r_*}{\sqrt{1+r_*^2}} {\mathrm{d}}t \wedge {\mathrm{d}}\theta \wedge {\mathrm{d}}\varphi \right) \, .$$ On $S_T$, we have $${\mathrm{d}}t = \frac{r_*}{\sqrt{1+r_*^2}} {\mathrm{d}}r_* \, ,$$ and therefore $$\begin{aligned} \hat{L}\lrcorner {\mathrm{dVol}}_{\hat{g}} \vert_{S_T} &=& \frac{1+r_*^2}{1+2r_*^2} R^2 F \sin \theta \left( 1+ \frac{r_*^{2}}{1+r_*^2} \right) {\mathrm{d}}r_* \wedge {\mathrm{d}}\theta \wedge {\mathrm{d}}\varphi \\ &=& R^2 F \sin \theta {\mathrm{d}}r_* \wedge {\mathrm{d}}\theta \wedge {\mathrm{d}}\varphi \, .\end{aligned}$$ So we obtain $$\begin{aligned} \hat{\cal E}_{\partial_t, S_T} &:=& \int_{S_T} ((\hat{J}_a+V_a)N^a) \, \hat{L}\lrcorner {\mathrm{dVol}}_{\hat{g}} \nonumber \\ &=& \int_{S_T} \bigg[ MR \hat\phi^2 + \frac{r^2}{2F} \bigg( (\partial_t \hat\phi )^2 + (\partial_{r_*} \hat\phi )^2 \nonumber \\ && \hspace{0.3in}+ \frac{2r_*}{\sqrt{1+r_*^2}} \partial_t \hat\phi \partial_{r_*} \hat\phi + R^2F \vert \nabla_{S^2} \hat\phi \vert^2 \bigg)\bigg] R^2 F {\mathrm{d}}r_* {\mathrm{d}}^2 \omega \, . \label{FluxST}\end{aligned}$$ This is positive definite since $\vert r_* \vert < \sqrt{1+r_*^2}$ (and degenerates asymptotically as $\vert r_* \vert \rightarrow +\infty$). The energy fluxes across ${{\mathscr I}}^+_T$ and ${{\mathscr H}}^+_T$ are increasing non negative functions of $T$ and their sum is bounded by ${\cal E}_{\Sigma_0}$, by and the positivity of ${\cal E}_{S_T}$. Therefore they admit limits as $T \rightarrow +\infty$ and these limits are $\hat{\cal E}_{\partial_t, {{\mathscr I}}^+}$ and $\hat{\cal E}_{\partial_t, {{\mathscr H}}^+}$. We have the following result : For smooth and compactly supported initial data on $\Sigma_0$, the energy fluxes of the rescaled solution across ${{\mathscr I}}^+$ and ${{\mathscr H}}^+$ are finite and satisfy $$\hat{\cal E}_{\partial_t, {{\mathscr I}}^+} + \hat{\cal E}_{\partial_t, {{\mathscr H}}^+} \leq \hat{\cal E}_{\partial_t, \Sigma_0} \, .$$ We have equality in the estimate above if any only if $$\label{LidVanishing} \lim_{T \rightarrow +\infty} \hat{\cal E}_{\partial_t, S_T} =0 \, .$$ In order to construct a conformal scattering theory, we merely need to prove for a dense class of data, say smooth and compactly supported. The final identity will extend to minimum regularity initial data by density. Moreover, we can allow any loss of derivatives in the proof of for smooth compactly supported data, since we do not need to prove that $\hat{\cal E}_{\partial_t, S_T}$ tends to zero uniformly in terms of the data. This is the object of subsection \[EnEstTInfinite\]. Function space of initial data ------------------------------ The scattering theory we are about to construct will be valid for a function space of initial data defined by the finiteness of the rescaled energy $\hat{\cal E}_{\partial_t, \Sigma_0}$. The analytic scattering theory constructed in [@Di1985] was valid for a function space of initial data defined by the finiteness of the energy of the physical field. It is interesting to notice that although the stress-energy tensor is not conformally invariant, the physical energy and the rescaled energy on $\Sigma_0$ are the same. Therefore, the function space of initial data for our conformal scattering theory is the same as in the analytic scattering theory of Dimock. Let us prove this. Consider the stress-energy tensor for the wave equation on the Schwarzschild metric $$\label{PhysicalSET} T_{ab} = \nabla_a \phi \nabla_b \phi - \frac12 \langle \nabla \phi \, ,~ \nabla \phi \rangle_g g_{ab} \, ,$$ which satisfies $$\nabla^a T_{ab} =0$$ for $\phi$ solution to the wave equation. The physical energy current $1$-form associated with static observers is $$J_a = K^b T_{ab}$$ where $K$ is the timelike Killing vector field $K=\partial_t$. This is conserved $$\nabla^a J_a = 0 \, .$$ The associated energy flux through an oriented hypersurface $S$ is given by[^6] $$\label{PhysEnS} {\cal E}_{\partial_t , S} = -4 \int_S * J_a {\mathrm{d}}x^a \, .$$ Similarly to what we saw for the rescaled energy fluxes, can be expressed more explicitely as $${\cal E}_{\partial_t, S} = \int_S J_a N^a \, L\lrcorner {\mathrm{dVol}}_{g} \, ,$$ where $L$ is a vector field transverse to $S$ and compatible with the orientation of the hypersurface, and $N$ is the normal vector field to $S$ such that $g (L,N)=1$. The energy fluxes $\hat{\cal E}_{\partial_t, \Sigma_0}$ and ${\cal E}_{\partial_t, \Sigma_0}$ are the same. [**Proof.**]{} A direct calculation shows that the physical energy flux across $\Sigma_0$ can be expressed in terms of $\hat\phi$ as follows $$\begin{aligned} {\cal E}_{\partial_t , \Sigma_0} &=& \frac12 \int_{\Sigma_0} \left( (\partial_t \hat\phi )^2 + (\partial_{r_*} \hat\phi )^2 + \frac{F}{r^2} \vert \nabla_{S^2} \hat\phi \vert^2 + \frac{FF'}{r} \hat\phi^2 \right) {\mathrm{d}}r_* \wedge {\mathrm{d}}^2 \omega \, , \\ &=& \frac12 \int_{\Sigma_0} \left( (\partial_t \hat\phi )^2 + (\partial_{r_*} \hat\phi )^2 + \frac{F}{r^2} \vert \nabla_{S^2} \hat\phi \vert^2 + F\frac{2M}{r^3} \hat\phi^2 \right) {\mathrm{d}}r_* \wedge {\mathrm{d}}^2 \omega \, ,\end{aligned}$$ which is exactly the expression of the rescaled energy flux $\hat{\cal E}_{\partial_t , \Sigma_0}$. We denote by $\cal H$ the completion of ${\cal C}^\infty_0 (\Sigma ) \times {\cal C}^\infty_0 (\Sigma )$ in the norm $$\Vert (\hat{\phi}_0 \, ,~ \hat{\phi}_1 ) \Vert_{\cal H} = \frac{1}{\sqrt{2}} \left( \int_{\Sigma} \left( (\hat\phi_1 )^2 + (\partial_{r_*} \hat\phi_0 )^2 + \frac{F}{r^2} \vert \nabla_{S^2} \hat\phi_0 \vert^2 + F\frac{2M}{r^3} \hat\phi_0^2 \right) {\mathrm{d}}r_* \wedge {\mathrm{d}}^2 \omega \right)^{1/2} \, .$$ The following result is classic. Its second part can be proved by Leray’s theorem combined with energy identities. Its first part may be established by either the same method or by a spectral approach (showing that the Hamiltonian for is self-adjoint on $\cal H$ as this was done in [@Di1985] and [@Ni1995]). \[CauchyPb\] The Cauchy problem for on $\cal M$ (and therefore also for ) is well-posed in $\cal H$, i.e. for any $(\hat{\phi}_0 \, ,~ \hat{\phi}_1 ) \in {\cal H}$, there exists a unique $\phi \in {\cal D}' ({\cal M})$ solution of such that : $$(r \phi \, ,~ r\partial_t \phi ) \in {\cal C} ({\mathbb{R}}_t \, ;~ {\cal H}) \, ; ~ r \phi \vert_{t=0} = \hat{\phi}_0 \, ;~ r \partial_t \phi \vert_{t=0} = \hat{\phi}_1 \, .$$ Moreover, $\hat{\phi}=r\phi$ belongs to $H^1_{\mathrm{loc}} (\bar{\cal M} )$ (see Remark \[Hsloc1\] and Definition \[Hsloc2\] below). \[Hsloc1\] The notation $H^s_\mathrm{loc} (\bar{\cal M})$ is a perhaps not ideal, Sobolev spaces being defined on open sets. What we mean by this notation is merely that the conformal boundary is seen as a finite boundary : only the neighbourhoods of $i^\pm$ and $i^0$ are considered as asymptotic regions in $\cal M$. With this in mind the definition of $H^s_\mathrm{loc} (\bar{\cal M})$, $s\in [0,+\infty [$ is unambiguous and goes as follows. \[Hsloc2\] Let $s \in [0,+\infty [$, a scalar function $u$ on ${\cal M}$ is said to belong to $H^s_{\mathrm{loc}} (\bar{\cal M})$ if for any local chart $(\Omega , \zeta )$, such that $\Omega \subset \cal M$ is an open set with smooth compact boundary in $\bar{\cal M}$ (note that this excludes neighbourhoods of either $i^\pm$ or $i^0$ but allows open sets whose boundary contains parts of the conformal boundary) and $\zeta$ is a smooth diffeomorphism from $\Omega$ onto a bounded open set $U \subset {\mathbb{R}}^4$ with smooth compact boundary, we have $u \circ \zeta^{-1} \in H^s ( U )$. Energy identity up to $i^+$ and trace operator {#EnEstTInfinite} ---------------------------------------------- Here, we prove for smooth and compactly supported data, using the estimates obtained in M. Dafermos and I. Rodnianski [@DaRoLN]. Theorem 4.1 in [@DaRoLN] contains sufficient information : an estimate giving decay of energy with a loss of 3 angular derivatives and one order of fall-off, as well as uniform decay estimates for more regular solutions with sufficiently fast fall-off at infinity. These are expressed in terms of quantities on the physical spacetime, i.e. unrescaled quantities. We need to make sure that they give the correct information for our energy on $S_T$, which is entirely expressed in terms of rescaled quantities ; this is not completely direct since the usual stress-energy tensor for the wave equation is not conformally invariant. We start by translating their estimates using the notations we have adopted here. Theorem 4.1 in [@DaRoLN] is expressed for a spacelike hypersurface for the metric $\hat{g}$ that crosses ${{\mathscr H}}^+$ and ${{\mathscr I}}^+$, i.e. an asymptotically hyperbolic hypersurface for $g$, defined by translation along $\partial_t$ of a reference asymptotically hyperbolic hypersurface. Our hypersurface $S_T$ fits in this framework. The content of the theorem is the following. (i) : Consider the stress-energy tensor for the wave equation on the Schwarzschild metric : $T_{ab}$ given by and let $\phi$ be a solution to the wave equation associated with smooth compactly supported data. Consider also a timelike vector field $\tau$ that is transverse to the horizon and equal to $\partial_t$ for $r$ large enough ; the vector $\tau^a$ is of the form $$\tau^a \partial_a = \alpha \partial_t + \beta \frac{1}{F} (\partial_t - \partial_{r_*} ) \, ,$$ where $\alpha \geq 1$, $\alpha =1$ for $r$ large enough and $\beta \geq 0$, $\beta =0$ for $r$ large enough. Denote by $j_a$ the unrescaled energy current $1$-form associated with $\tau$, $$j_a = \tau^b T_{ab} \, .$$ The physical energy flux, associated with $\tau$, of the solution $\phi$ across $S_T$ is given by $${\cal E}_{\tau ,S_T} = \int_{S_T} j_a N^a L \lrcorner {\mathrm{dVol}}_g \, ,$$ where $N^a$ is the normal vector field to $S_T$ associated via the metric $g$ to the co-normal ${\mathrm{d}}\Psi$, $$N_a {\mathrm{d}}x^a = {\mathrm{d}}\Psi \, ,~ N^a \frac{\partial}{\partial x^a} = g^{ab} N_b \frac{\partial}{\partial x^a} = F^{-1} \left( \frac{\partial}{\partial t} + \frac{r_*}{\sqrt{1+r_*^2}} \frac{\partial}{\partial r_*} \right) = \frac{1}{r^2} \hat{N}^a \frac{\partial}{\partial x^a} \, ,$$ and $$L^a \frac{\partial}{\partial x^a} = \hat{L}^a \frac{\partial}{\partial x^a} = \frac{1+r_*^2}{1+2r_*^2} \left( \frac{\partial}{\partial t} - \frac{r_*}{\sqrt{1+r_*^2}} \frac{\partial}{\partial r_*} \right) \, ,$$ so that $L_a N^a = g_{ab} L^a N^b = \hat{g}_{ab} \hat{L}^a \hat{N}^b = 1$. The energy flux ${\cal E}_{\tau ,S_T}$ decays as follows : $$\label{31} {\cal E}_{\tau ,S_T} \lesssim 1/T^2 \, .$$ (ii) : The solution also satisfies the following uniform decay estimates : $$\label{32} \sup_{S_T} \sqrt{r} \phi \lesssim 1/T\, ,~ \sup_{S_T} r \phi \lesssim 1/\sqrt{T} \, .$$ The constants in front of the powers of $1/T$ in the estimates of Theorem 4.1 in [@DaRoLN] involve some higher order weighted energy norms (third order for (i) and sixth order for (ii)) of the data, which are all finite in our case. The details of these norms are not important to us here. We merely need to establish that for any smooth and compactly supported data, the energy of the rescaled field on $S_T$ tends to zero as $T\rightarrow +\infty$. For smooth and compactly supported data $\phi$ and $\partial_t \phi$ at $t =0$ there exists $K>0$ such that for $T \geq 1$ large enough, $${\cal E}_{\partial_t ,S_T} \leq \frac{K}{T} \, .$$ [**Proof.**]{} First note that since $\alpha \geq 1$ and $\beta \geq 0$, thanks to the dominant energy condition, we have $$T_{ab} \tau^a N^b = \alpha T_{ab} (\partial_t)^a N^b + \beta T_{ab} (\partial_u)^a N^b \geq T_{ab} (\partial_t)^a N^b \, .$$ Hence the physical energy on $S_T$ associated with the vector field $\tau^a$ controls the physical energy on $S_T$ associated with the vector field $\partial_t$ : $$\label{EstEnDtTau} {\cal E}_{\tau ,S_T} \geq {\cal E}_{\partial_t ,S_T} \, .$$ Let us now compare the physical energy flux ${\cal E}_{\partial_t ,S_T}$ and the rescaled energy flux $\hat{\cal E}_{\partial_t ,S_T}$ using the relation $\hat\phi = r \phi$. First, we have $$\begin{aligned} T_{ab} (\partial_t)^a N^b &=& \frac{1}{2F} \left( (\partial_t \phi )^2 + (\partial_{r_*} \phi )^2 + 2 \frac{r_*}{\sqrt{1+r_*^2}} \partial_t \phi \partial_{r_*} \phi + \frac{F}{r^2} \vert \nabla_{S^2} \phi \vert^2 \right) \\ &=& \frac{1}{2r^2F} \left( (\partial_t \hat\phi )^2 + F^2(\partial_{r} \hat\phi - \frac{\hat\phi}{r})^2 + 2 \frac{Fr_*}{\sqrt{1+r_*^2}} \partial_t \hat\phi (\partial_{r} \hat\phi - \frac{\hat\phi}{r}) + \frac{F}{r^2} \vert \nabla_{S^2} \hat\phi \vert^2 \right) \end{aligned}$$ and since $L^a = \hat{L}^a$, $$L \lrcorner {\mathrm{dVol}}_g = r^4 \hat{L} \lrcorner {\mathrm{dVol}}_{\hat{g}} \, .$$ Therefore $${\cal E}_{\partial_t , S_T} = \frac12 \int_{S_T} \left( (\partial_t \hat\phi )^2 + F^2(\partial_{r} \hat\phi - \frac{\hat\phi}{r})^2 + 2 \frac{Fr_*}{\sqrt{1+r_*^2}} \partial_t \hat\phi (\partial_{r} \hat\phi - \frac{\hat\phi}{r}) + \frac{F}{r^2} \vert \nabla_{S^2} \hat\phi \vert^2 \right) \frac{r^4}{r^2 F} \hat{L} \lrcorner {\mathrm{dVol}}_{\hat{g}}$$ and comparing with , we obtain $$\begin{aligned} \hat{\cal E}_{\partial_t , S_T} &=& {\cal E}_{\partial_t , S_T} + \int_{S_T} \frac{M}{r} \hat\phi^2 \hat{L} \lrcorner {\mathrm{dVol}}_{\hat{g}} - \frac12 \int_{S_T} \left( F \frac{\hat\phi^2}{r^2} - 2 \frac{\hat\phi}{r} \partial_{r_*} \hat\phi - 2 \frac{r_*}{\sqrt{1+r_*^2}} \partial_t \hat\phi \frac{\hat\phi}{r} \right) r^2 \hat{L} \lrcorner {\mathrm{dVol}}_{\hat{g}} \\ &=& {\cal E}_{\partial_t , S_T} + \frac12 \int_{S_T} (\frac{2M}{r} -F) \hat\phi^2 \hat{L} \lrcorner {\mathrm{dVol}}_{\hat{g}} + \frac12 \int_{S_T} 2 \frac{\hat\phi}{r} \left( \partial_{r_*} \hat\phi + \frac{r_*}{\sqrt{1+r_*^2}} \partial_t \hat\phi \right) r^2 \hat{L} \lrcorner {\mathrm{dVol}}_{\hat{g}} \\ & \leq & {\cal E}_{\partial_t , S_T} + \frac12 \int_{S_T} (\frac{2M}{r} -F) \hat\phi^2 \hat{L} \lrcorner {\mathrm{dVol}}_{\hat{g}} \\ && + \frac12 \int_{S_T} \left( 2F \frac{\hat\phi^2}{r^2} + \frac{1}{2F} \left( \partial_{r_*} \hat\phi + \frac{r_*}{\sqrt{1+r_*^2}} \partial_t \hat\phi \right)^2 \right)r^2 \hat{L} \lrcorner {\mathrm{dVol}}_{\hat{g}} \\ &\leq & {\cal E}_{\partial_t , S_T} + \frac12 \int_{S_T} (\frac{2M}{r} +F) \hat\phi^2 \hat{L} \lrcorner {\mathrm{dVol}}_{\hat{g}} \\ && + \frac12 \int_{S_T} \frac{1}{2F} \left( ( \partial_{r_*} \hat\phi )^2+ \frac{2r_*}{\sqrt{1+r_*^2}} \partial_t \hat\phi \partial_{r_*} \hat\phi + ( \partial_{t} \hat\phi )^2 \right)r^2 \hat{L} \lrcorner {\mathrm{dVol}}_{\hat{g}} \\ &\leq & {\cal E}_{\partial_t , S_T} + \frac12 \int_{S_T} \hat\phi^2 \hat{L} \lrcorner {\mathrm{dVol}}_{\hat{g}} + \frac12 \hat{\cal E}_{\partial_t , S_T} \, .\end{aligned}$$ This gives us $$\hat{\cal E}_{\partial_t , S_T} \leq 2 {\cal E}_{\partial_t , S_T} + \int_{S_T} \hat\phi^2 \hat{L} \lrcorner {\mathrm{dVol}}_{\hat{g}} \, .$$ The second estimate in says exactly that $$\sup_{S_T} \hat\phi^2 \lesssim \frac{1}{T} \, .$$ Since moreover $$\int_{S_T} \hat{L} \lrcorner {\mathrm{dVol}}_{\hat{g}} = \int_{{\mathbb{R}}\times S^2} \frac{F}{r^2} \sin \theta {\mathrm{d}}r_* {\mathrm{d}}\theta {\mathrm{d}}\varphi = \int_{[2M , +\infty [ \times S^2} \frac{1}{r^2} \sin \theta {\mathrm{d}}r {\mathrm{d}}\theta {\mathrm{d}}\varphi =\frac{2\pi}{M} <\infty$$ and ${\cal E}_{\partial_t , S_T} \lesssim 1/T^2$ by and , this concludes the proof of the proposition. The finiteness of the last integral in the proof is strongly related to the finiteness of the volume of $S_T$ for the measure $\hat{\mu}_{S_T}$ induced by $\hat{g}$. As one can readily guess from the definitions of $S_T$ and $\hat{g}$, the volume of $S_T$ for the measure $\hat{\mu}_{S_T}$ is independent of $T$. Figure \[3surface\] may be a little misleading in giving the impression that $S_T$ shrinks to a point, we must not forget that due to the way $\hat{g}$ is rescaled, $i^+$ is still at infinity. The volume of $S_T$ for $\hat{\mu}_{S_T}$ is easy to calculate. First we restrict $\hat{g}$ to $S_T$ using the explicit dependence of $t$ on $r_*$ on $S_T$ : $$\hat{g}\vert_{S_T} = - \left[ \frac{R^2 F}{1+r_*^2} {\mathrm{d}}r_*^2 +{\mathrm{d}}\omega^2 \right] \, .$$ Then we calculate $\hat{\mu}_{S_T}$ : $${\mathrm{d}}\hat{\mu}_{S_T} = \frac{R\sqrt{F}}{\sqrt{1+r_*^2}} {\mathrm{d}}r_* {\mathrm{d}}^2 \omega = \frac{R}{\sqrt{F} \sqrt{1+r_*^2}} {\mathrm{d}}r {\mathrm{d}}^2 \omega\, .$$ So the volume of $S_T$ for $\hat{\mu}_{S_T}$ is $$\mathrm{Vol}_{\hat{g}} (S_T) = 4\pi \int_{2M}^{+\infty} \frac{1}{r\sqrt{F} \sqrt{1+r_*^2}} {\mathrm{d}}r = 4\pi \int_{2M}^{+\infty} \frac{{\mathrm{d}}r}{\sqrt{r^2-2Mr} \sqrt{1+r_*^2}} < +\infty \, .$$ Note that $${\mathrm{d}}\hat{\mu}_{S_T} = \sqrt{\hat{g}_{ab} \hat{N}^a \hat{N}^b} \hat{L} \lrcorner {\mathrm{dVol}}_{\hat{g}} \, .$$ The two measures $\hat{\mu}_{S_T}$ and $\hat{L} \lrcorner {\mathrm{dVol}}_{\hat{g}}$ on $S_T$ are not uniformly equivalent since $$\sqrt{\hat{g}_{ab} \hat{N}^a \hat{N}^b} = \frac{r}{\sqrt{F} \sqrt{1+r_*^2}} \left\{ \begin{array}{ccl} { \simeq 1} & {\mbox{as}} & { r\rightarrow +\infty \, ,} \\ {\rightarrow +\infty} & {\mbox{as}} & { R \rightarrow 2M \, ,} \end{array} \right.$$ but this is integrable in the neighbourhood of $2M$. If we had normalized $N$ to start with, $$\tilde{N}^a = \frac{1}{\sqrt{\hat{g}_{cd} \hat{N}^c \hat{N}^d}} \hat{N}^a \, ,$$ and put $$\tilde{L}^a = \sqrt{\hat{g}_{cd} \hat{N}^c \hat{N}^d} \hat{L}^a \, ,$$ so that $\hat{g}_{ab} \tilde{N}^a \tilde{L}^b =1$, then we would have $${\mathrm{d}}\hat{\mu}_{S_T} = \tilde{L} \lrcorner {\mathrm{dVol}}_{\hat{g}} \, .$$ So we have the following result : \[PropEnergyIdentityFuture\] For smooth and compactly supported initial data on $\Sigma_0$, we have $$\label{EnergyIdentityFuture} \hat{\cal E}_{\partial_t, {{\mathscr I}}^+} + \hat{\cal E}_{\partial_t, {{\mathscr H}}^+} = \hat{\cal E}_{\partial_t, \Sigma_0} \, ,$$ with $$\begin{aligned} \hat{\cal E}_{\partial_t, \Sigma_0} &=& \int_{\Sigma_0} (\hat{J}_a+V_a)K^a \, r^2 F^{-1} \partial_t \lrcorner {\mathrm{dVol}}_{\hat{g}} \\ &=& \frac12 \int_{\Sigma_0} \left( (\partial_t \hat\phi )^2 + (\partial_{r_*} \hat\phi )^2 + R^2 F \vert \nabla_{S^2} \hat\phi \vert^2 + 2 MFR^3\hat\phi^2 \right) {\mathrm{d}}r_* {\mathrm{d}}^2 \omega \, ; \\ \hat{\cal E}_{\partial_t, {{\mathscr I}}^+} &=& \int_{{{\mathscr I}}^+} (\hat\nabla_K \hat\phi )^2 \hat{L}_{{{\mathscr I}}^+} \lrcorner {\mathrm{dVol}}_{\hat{g}} = \int_{{{\mathscr I}}^+} (\partial_u (\hat\phi \vert_{{{\mathscr I}}^+} ) )^2 {\mathrm{d}}u {\mathrm{d}}^2 \omega\, ; \\ \hat{\cal E}_{\partial_t, {{\mathscr H}}^+} &=& \int_{{{\mathscr H}}^+} (\hat\nabla_K \hat\phi )^2 \hat{L}_{{{\mathscr H}}^+} \lrcorner {\mathrm{dVol}}_{\hat{g}} = \int_{{{\mathscr H}}^+} (\partial_v (\hat\phi \vert_{{{\mathscr H}}^+}) )^2 {\mathrm{d}}v {\mathrm{d}}^2 \omega \, .\end{aligned}$$ We can extend this result to minimum regularity initial data (i.e. data in $\cal H$) by standard density arguments, provided we give a meaning to the energy fluxes across ${{\mathscr I}}$ and the horizon. We define a trace operator that to smooth and compactly supported initial data associates the future scattering data : Let $(\hat\phi_0 , \hat\phi_1 ) \in {\cal C}^\infty_0 (\Sigma_0 ) \times {\cal C}^\infty_0 (\Sigma_0 )$. Consider the solution of $\hat{\phi} \in {\cal C}^\infty ( \bar{\cal M} )$ such that $$\hat{\phi} \vert_{\Sigma_0} = \hat{\phi}_0 \, ,~ \partial_t \hat{\phi} \vert_{\Sigma_0} = \hat{\phi}_1 \, .$$ We define the trace operator ${\cal T}^+$ from ${\cal C}^\infty_0 (\Sigma_0 ) \times {\cal C}^\infty_0 (\Sigma_0 )$ to ${\cal C}^\infty ({{\mathscr H}}^+ ) \times {\cal C}^\infty ({{\mathscr I}}^+ )$ as follows $${\cal T}^+ (\hat\phi_0 , \hat\phi_1 ) = (\hat\phi \vert_{{{\mathscr H}}^+} , \hat\phi \vert_{{{\mathscr I}}^+} ) \, .$$ Then we extend this trace operator by density to $\cal H$ with values in the natural function space on ${{\mathscr H}}^+ \cup {{\mathscr I}}^+$ inherited from . \[FuncSpaceScatt\] We define on ${{\mathscr H}}^+ \cup {{\mathscr I}}^+$ the function space ${\cal H}^+$, completion of ${\cal C}^\infty_0 ({{\mathscr H}}^+ ) \times {\cal C}^\infty_0 ({{\mathscr I}}^+ )$ in the norm $$\begin{aligned} \Vert (\xi , \zeta ) \Vert_{{\cal H}^+} &=& \sqrt{ \int_{{{\mathscr H}}^+} \left( \hat\nabla_K \xi \right)^2 \hat{L}_{{{\mathscr H}}^+} \lrcorner {\mathrm{dVol}}_{\hat{g}} + \int_{{{\mathscr I}}^+} \left( \hat\nabla_K \zeta \right)^2 \hat{L}_{{{\mathscr I}}^+} \lrcorner {\mathrm{dVol}}_{\hat{g}}} \\ &=& \sqrt{ \int_{{{\mathscr H}}^+} \left( \partial_v \xi \right)^2 {\mathrm{d}}v {\mathrm{d}}^2 \omega + \int_{{{\mathscr I}}^+} \left( \partial_u \zeta \right)^2 {\mathrm{d}}u {\mathrm{d}}^2 \omega} \, ,\end{aligned}$$ i.e. $${\cal H}^+\simeq \dot{H}^1 ({\mathbb{R}}_v \, ;~ L^2 (S^2_\omega)) \times \dot{H}^1 ({\mathbb{R}}_u \, ;~ L^2 (S^2_\omega)) \, .$$ The homogeneous Sobolev space (also referred to as the Beppo-Levi space) of order one on ${\mathbb{R}}$ is a delicate object. It is not a function space in the usual sense that its elements should belong to $L^1_\mathrm{loc}$, nor is it even a distribution space (see for example [@So1983] for a precise study of the one and two-dimensional cases). It is a space of classes of equivalence modulo constants. The reason is that constants have zero $\dot{H}^1$ norm and can in addition be approached in $\dot{H}^1$ norm by smooth and compactly supported functions on ${\mathbb{R}}$ (using a simple dilation of a given smooth compactly supported function whose value at the origin is the constant we wish to approach). The definition of $\dot{H}^1 ({\mathbb{R}})$ by completion of ${\cal C}^\infty_0 ({\mathbb{R}})$ in the $\dot{H}^1$ norm makes it the space of the limits of Cauchy sequences where indistiguishable limits are identified, i.e. a space of classes of equivalence modulo constants. If one is reluctant to using classes of equivalence as scattering data, a more comfortable solution is to consider that the scattering data are in fact the traces of $\partial_t \hat{\phi}$ on ${{\mathscr H}}^+$ and ${{\mathscr I}}^+$ and the function space in each case is then merely $L^2 ({\mathbb{R}}\times S^2 )$. This is what Friedlander did in is 1980 paper [@Fri1980]. It is however not clear to me that he did so for precisely this reason. He had, in my opinion, deeper motives for making this choice, guided as he was by the desire to recover the Lax-Phillips translation representer. Whether one chooses to consider the scattering data as the traces of $\hat{\phi}$ (in $\dot{H}^1 ({\mathbb{R}}\, ;~ L^2 (S^2 ))$), or as the traces of $\partial_t \hat{\phi}$ (in $L^2 ({\mathbb{R}}\times S^2 )$) is purely a matter of taste, the two choices are equivalent. We infer from Proposition \[PropEnergyIdentityFuture\] the following theorem : \[ThmPartialIsometry\] The trace operator ${\cal T}^+$ extends uniquely as a bounded linear map from $\cal H$ to ${\cal H}^+$. It is a partial isometry, i.e. for any $(\hat{\phi}_0 , \hat{\phi}_1 ) \in {\cal H}$, $$\Vert {\cal T}^+ (\hat{\phi}_0 , \hat{\phi}_1 ) \Vert_{{\cal H}^+} = \Vert (\hat{\phi}_0 , \hat{\phi}_1 ) \Vert_{{\cal H}} \, .$$ An interesting property of second order equations is that once extended to act on minimal regularity solutions, the operator ${\cal T}^+$ can still be understood as a trace operator acting on the solution. We have seen in Proposition \[CauchyPb\] that finite energy solutions of belong to $H^1_\mathrm{loc} (\bar{\cal M})$ (see Remark \[Hsloc1\] and Definition \[Hsloc2\] for the definition of this function space). Elements of $H^1_\mathrm{loc} (\bar{\cal M})$ admit a trace at the conformal boundary that is locally $H^{1/2}$ on ${{\mathscr H}}^+ \cup {{\mathscr I}}^+$. This is a consequence of a standard property of elements of $H^s (\Omega )$ for $\Omega$ a bounded open set of ${\mathbb{R}}^n$ with smooth boundary ; a function $f \in H^s (\Omega )$, $s > 1/2$, admits a trace on the boundary $\partial \Omega$ of $\Omega$ that is in $H^{s-1/2} (\partial \Omega )$. Hence the extended operator ${\cal T}^+$ is still a trace operator in a usual sense, i.e. $${\cal T}^+ (\hat{\phi}_0 , \hat{\phi}_1 ) = (\hat{\phi} \vert_{{{\mathscr H}}^+} , \hat{\phi} \vert_{ {{\mathscr I}}^+ } ) \, .$$ This is in sharp contrast with what happens when working with first order equations like Dirac or Maxwell. In this case, finite energy solutions are in $L^2_\mathrm{loc} (\bar{\cal M})$ but in general not in $H^s_\mathrm{loc} (\bar{\cal M})$ for $s>0$. The density argument used in Theorem \[ThmPartialIsometry\] would still give us an extension of the operator ${\cal T}^+$, whose range would be $L^2 ({{\mathscr H}}^+) \times L^2 ({{\mathscr I}}^+ )$. This extended operator could not however be understood as a trace operator in the usual sense mentionned above, the regularity of the solutions being too weak. Scattering theory {#Scattering} ================= The construction of a conformal scattering theory on the Schwarzschild spacetime consists in solving a Goursat problem for the rescaled field on ${{\mathscr H}}^- \cup {{\mathscr I}}^-$ and on ${{\mathscr H}}^+ \cup {{\mathscr I}}^+$. In this section, we first solve the Goursat problem on ${{\mathscr H}}^+ \cup {{\mathscr I}}^+$, the construction being similar in the past. Then we show that the conformal scattering theory entails a conventional analytic scattering theory defined in terms of wave operators. Since the exterior of a Schwarzschild black hole is static and the global timelike Killing vector $\partial_t$ extends as the null generator of ${{\mathscr I}}^\pm$ and ${{\mathscr H}}^\pm$, it is easy to show that the past (resp. future) scattering data, i.e. the trace of the rescaled field on ${{\mathscr H}}^- \cup {{\mathscr I}}^-$ (resp. on ${{\mathscr H}}^+ \cup {{\mathscr I}}^+$) is a translation representer of the scalar field. We have a natural link between the conformal scattering theory and the Lax-Phillips approach, analogous to the one Friedlander established in his class of spacetimes. The difference is that in our case, the scattering data consist of a pair of data : the trace of the rescaled field on null infinity (which is exactly the radiation field) and on the horizon. The Goursat problem and the scattering operator ----------------------------------------------- We solve the Goursat problem on ${{\mathscr H}}^+ \cup {{\mathscr I}}^+$ following Hormander [@Ho1990] : the principle is to show that the trace operator ${\cal T}^+$ is an isomorphism between $\cal H$ and ${\cal H}^+$. Theorem \[ThmPartialIsometry\] entails that ${\cal T}^+$ is one-to-one and that its range is a closed subspace of ${\cal H}^+$. Therefore, all we need to do is to show that its range is dense in ${\cal H}^+$. Let $(\xi , \zeta) \in {\cal C}^\infty_0 ({{\mathscr H}}^+) \times {\cal C}^\infty_0 ({{\mathscr I}}^+)$, i.e. the support of $\xi$ remains away from both the crossing sphere and $i^+$ and the support of $\zeta$ remains away from both $i^+$ and $i^0$ ; in other words, the values of $v$ remain bounded on the support of $\xi$ and the values of $u$ remain bounded on the support of $\zeta$. We wish to show the existence of $\hat\phi$ solution of such that $$( \hat{\phi } , \partial_t \hat{\phi} ) \in {\cal C} ({\mathbb{R}}_t \, ;~ {\cal H} ) \mbox{ and } {\cal T}^+ (\hat{\phi} \vert_{\Sigma_0} \, ,~ \partial_t \hat{\phi} \vert_{\Sigma_0} ) = (\xi , \zeta ) \, .$$ For such data the singularity at $i^+$ is not seen. We must however deal with the singularity at $i^0$. We proceed in two steps. First, we consider $\cal S$ a spacelike hypersurface for $\hat{g}$ on $\bar{\cal M}$ that crosses ${{\mathscr I}}^+$ in the past of the support of $\zeta$ and meets the horizon at the crossing sphere. The compact support of the data on ${{\mathscr I}}^+$ allows us to apply the results of Hörmander [@Ho1990] even though we are not working with a spatially compact spacetime with a product structure (see Appendix \[HormGP\] for details). We know from [@Ho1990] that there exists a unique solution $\hat{\Phi}$ of such that : - $\hat{\Phi} \in H^1 ({\cal I}^+ ({\cal S}))$, where ${\cal I}^+ ({\cal S})$ is the causal future of $\cal S$ in $\bar{\cal M}$ ; here we do not need to distinguish between $H^1 ({\cal I}^+ ({\cal S}))$ and $H^1_\mathrm{loc} ({\cal I}^+ ({\cal S}))$ because, due to the compact support of the Goursat data, the solution vanishes in a neighbourhood of $i^+$ ; - given any foliation of ${\cal I}^+ ({\cal S})$ by $\hat{g}$-spacelike hypersurfaces $\{ {\cal S}_{\tau} \}_{\tau \geq 0}$, such that ${\cal S}_0 = {\cal S}$ (see figure \[Foliation\]), $\hat{\Phi}$ is continuous in $\tau$ with values in $H^1$ of the slices and ${\cal C}^1$ in $\tau$ with values in $L^2$ of the slices ; in fact what we have a slightly stronger property, see Appendix \[HormGP\] for a precise statement ; - $\hat{\Phi} \vert_{{{{\mathscr I}}^+}} = \zeta$, $\hat{\Phi} \vert_{{{{\mathscr H}}^+}} = \xi$. ![A foliation $\{ {\cal S}_{\tau} \}_{\tau \geq 0}$ of $\overline{{\cal I}^+ ({\cal S})}$.[]{data-label="Foliation"}](Foliation.jpg){width="4in"} Second, we extend the solution down to $\Sigma_0$ in a manner that avoids the singularity at $i^0$. The crucial remark is that the restriction of $\hat{\Phi}$ to $\cal S$ is in $H^1 ({\cal S})$ and its trace on ${\cal S} \cap {{\mathscr I}}^+$ is also the trace of $\zeta$ on ${\cal S} \cap {{\mathscr I}}^+$, which is zero because of the way we have chosen $\cal S$. It follows that $\hat{\Phi} \vert_{\cal S}$ can be approached by a sequence $\{ \hat{\phi}^n_{0,{\cal S}} \}_{n\in {\mathbb{N}}}$ of smooth functions on $\cal S$ supported away from ${{\mathscr I}}^+$ that converge towards $\hat{\Phi} \vert_{\cal S}$ in $H^1 ({\cal S})$. And of course $\partial_t \hat{\Phi} \vert_{{\cal S}}$ can be approached by a sequence $\{ \hat{\phi}^n_{1,{\cal S}} \}_{n\in {\mathbb{N}}}$ of smooth functions on $\cal S$ supported away from ${{\mathscr I}}^+$ that converge towards $\partial_t \hat{\Phi} \vert_{{\cal S}}$ in $L^2 ({\cal S})$. Consider $\hat{\phi}^n$ the smooth solution of on $\overline{\cal M}$ with data $( \hat{\phi}^n_{0,{\cal S}}\, ,~ \hat{\phi}^n_{1,{\cal S}})$ on $\cal S$. This solution vanishes in the neighbourhood of $i^0$ and we can therefore perform energy estimates for $\hat{\phi}^n$ between $\cal S$ and $\Sigma_0$ : we have the energy identity $$\label{EnIdentSSigma0} {\cal E}_{\partial_t} ({\cal S} , \hat{\phi}^n ) = {\cal E}_{\partial_t} (\Sigma_0 , \hat{\phi}^n ) \, .$$ The $H^1 \times L^2$ norm on $S$ is that induced by the rescaled metric $\hat{g}$. This is not equivalent to the norm induced by the energy ${\cal E}_{\partial_t}$ on $\cal S$, but the $H^1 \times L^2$ norm controls the other, which degenerates near null infinity and the crossing sphere. Consequently, $( \hat{\phi}^n_{0,{\cal S}}\, ,~ \hat{\phi}^n_{1,{\cal S}})$ is a Cauchy sequence also in the energy norm on $\cal S$. Similar energy identities between $\cal S$ and the hypersurfaces $\Sigma_t$ entail that $(\hat{\phi}^n , \partial_t \hat{\phi}^n )$ converges in ${\cal C} ({\mathbb{R}}_t \, ;~{\cal H} ) $ towards $(\hat{\phi} \, ,~ \partial_t \hat{\phi} )$, where $\hat{\phi}$ is a solution of . By local uniqueness $\hat{\phi}$ coincides with $\hat{\Phi}$ in the future of $\cal S$. Hence if we denote $$\hat{\phi}_0 = \hat{\phi} \vert_{\Sigma_0} \, ,~ \hat{\phi}_1 = \partial_t \hat{\phi} \vert_{\Sigma_0} \, ,$$ we have $$( \hat{\phi}_0 \, ,~ \hat{\phi}_1 ) \in {\cal H}$$ and $$(\xi , \zeta ) = {\cal T}^+ ( \hat{\phi}_0 \, ,~ \hat{\phi}_1 ) \, .$$ This shows that the range of ${\cal T}^+$ contains ${\cal C}^\infty_0 ({{\mathscr H}}^+) \times {\cal C}^\infty_0 ({{\mathscr I}}^+)$ and is therefore dense in ${\cal H}^+$. We have proved the following theorem. \[ThmGoursatPb\] The trace operator ${\cal T}^+$ is an isometry from $\cal H$ onto ${\cal H}^+$. We introduce in a similar manner the past trace operator ${\cal T}^-$ and the space ${\cal H}^-$ of past scattering data[^7]. We define the scattering operator $S$ as the operator that to the past scattering data associates the future scattering data, i.e. $$S := {\cal T}^+ ({\cal T}^-)^{-1} \, .$$ The scattering operator is an isometry from ${\cal H}^-$ onto ${\cal H}^+$. Wave operators {#WaveOps} -------------- A conformal scattering construction such as the one we have just established can be re-interpreted as a scattering theory defined in terms of wave operators. This re-interpretation is more an a posteriori embellishment than a fundamental aspect of the theory, but it is interesting to realize that such fundamental objects of analytic scattering as wave operators, can be recovered from a purely geometrical construction which remains valid in time dependent geometries. To be completely precise, it is the inverse wave operators and the asymptotic completeness that we recover from the conformal scattering theory ; the direct wave operators are obtained in the classic analytic manner involving Cook’s method. This choice is guided by simplicity and the flexibility of the method. The proof of existence of direct wave operators using Cook’s method is the simplest part of analytic time-dependent scattering theory. Moreover, provided we have sufficiently explicit asymptotic information on our spacetime and good uniform energy estimates (without which we have in any case little hope of constructing a conformal scattering theory), it can be easily extended to fairly general non-stationary geometries, using a comparison dynamics that is defined geometrically, namely the flow of a family of null geodesics in the neighbourhood of the conformal boundary. The existence of inverse wave operators and asymptotic completeness, that we deduce from the conformal scattering construction in a direct manner, are the difficult aspects of analytic scattering. When constructing wave operators using a conformal scattering theory, there is, just as for analytic scattering, some freedom in the choice of comparison dynamics, as well as some complications inherent to the fact that the full and simplified dynamics often act on different function spaces, defined on different manifolds that may not have the same topology. The freedom of choice is two-fold. First we may choose different types of dynamics : for the wave equation, we may wish to compare near infinity with the wave equation on flat spacetime or with a geometrically defined transport equation. In analytic scattering, the choice of comparison dynamics essentially fixes the space of scattering data as the finite energy space for the simplified Hamiltonian. In contrast, in conformal scattering, the energy space of scattering data is imposed by the energy estimates ; that is to say, the choice of vector field that we contract the stress-energy tensor with in order to get an energy current, fixes the functional framework, for both the scattering data and the initial data in fact. The comparison dynamics is then an additional choice, not completely determined by the space of scattering data. For instance, with a rather strong control on scattering data that seems to indicate the full flat spacetime wave equation as a natural simplified dynamics, we may yet choose a transport equation. All we really need is that the function space and the dynamics are compatible : the comparison dynamics can usually be expressed as an evolution equation on the space of scattering data, whose coefficients are independent of the time parameter ; this compatibility then simply means that the Hamiltonian should be self-adjoint. Second, for a given type of dynamics, there may still be some freedom. Say, if we choose a transport equation along a family of curves whose end-points span the conformal boundary, two different families of curves with the same end-points would work just as well. In [@MaNi2004], a conformal scattering construction on asymptotically simple spacetimes was re-interpreted as an analytic scattering theory defined in terms of wave operators. The comparison dynamics was determined by a null geodesic congruence in the neighbourhood of ${{\mathscr I}}$, for which there are many choices. Also, some cut-off was required in a compact region in space, in order to avoid caustics. In the case we are considering here, the Schwarzschild geometry is sufficiently special that it singles out two congruences of null geodesics. Moreover, the topology of the spacetime (or equivalently the fact that the scattering data are specified on two disjoint null hypersurfaces instead of one in the asymptotically simple case) means that no cut-off is required. The Schwarzschild spacetime is algebraically special of Petrov type D ; the four roots of the Weyl tensor are grouped at each point as two double principal null directions : $\partial_t \pm \partial_{r_*}$. The two principal null congruences provide two preferred families of null curves along which to define a comparison dynamics. We now proceed to introduce the full and the comparison dynamics as well as the other ingredients of the wave operators. We denote by ${\cal U} (t)$ the propagator for the wave equation on the finite energy space $\cal H$, i.e. for data $(\hat{\phi}_0 \, ,~ \hat{\phi}_1 ) \in {\cal H}$ at $t=0$, given $(\hat{\phi} \, ,~ \partial_t \hat{\phi} ) \in {\cal C} ({\mathbb{R}}_t \, ;~ {\cal H} )$ the associated solution of , we have $${\cal U} (t) (\hat{\phi}_0 \, ,~ \hat{\phi}_1 ) = (\hat{\phi} (t) \, ,~ \partial_t \hat{\phi} (t) ) \, .$$ The propagator ${\cal U} (t) $ is a strongly continuous one-parameter group of isometries on $\cal H$. The comparison dynamics, denoted by ${\cal U}_0 (t)$, acts on pairs of functions on $\Sigma_0$ as the push-forward along the flow of the incoming principal null geodesics on the first function, and the push-forward along the flow of the outgoing principal null geodesics on the second function. Considered as an operator on pairs of functions on the generic slice $\Sigma$, it acts as a translation to the left on the first function and a translation to the right on the second : $${\cal U}_0 (t) (\xi , \zeta ) (r_*,\omega) = \left( \xi (r_*+t , \omega ) , \zeta (r_*-t , \omega ) \right) \, .$$ It is a strongly continuous one-parameter group of isometries on $$\label{EnSpaceFree} {\cal H}_0 = \dot{H}^1 ({\mathbb{R}}_{r_*} \, ;~ L^2 (S^2_\omega )) \times \dot{H}^1 ({\mathbb{R}}_{r_*} \, ;~ L^2 (S^2_\omega )) \, .$$ For our definition of direct and inverse wave operators, we need, in addition to the two dynamics ${\cal U} (t)$ and ${\cal U}_0 (t)$, an identifying operator, two cut-off functions and a pull-back operator between functions on the future conformal boundary and pairs of functions on $\Sigma_0$. 1. In order to obtain explicit formulae, we use on ${{\mathscr H}}^+$ the coordinates $(v,\omega)$, on ${{\mathscr I}}^+$ the coordinates $(-u,\omega)$ and on $\Sigma_0$ we use $(r_* , \omega )$. Both for functions on ${{\mathscr H}}^+$ and ${{\mathscr I}}^+$, we shall denote by $\partial_s$ the partial derivative with respect to their first variable, i.e. for $\xi$ a function on ${{\mathscr H}}^+$, $$\partial_s \xi = \partial_v \xi$$ and for a function $\zeta$ on ${{\mathscr I}}^+$, $$\partial_s \zeta = - \partial_u \zeta \, .$$ 2. We define the identifying operator $${\cal J} \, :~ {\cal C}^\infty_0 (\Sigma ) \times {\cal C}^\infty_0 (\Sigma ) \rightarrow {\cal C}^\infty_0 (\Sigma ) \times {\cal C}^\infty_0 (\Sigma )$$ by $${\cal J} (\xi, \zeta ) (r_* , \omega ) = \left( \xi (r_* , \omega ) + \zeta (r_* , \omega ) \, ,~ \partial_s \xi (r_* , \omega ) - \partial_s\zeta (r_* , \omega ) \right) \, .$$ It combines pairs of functions on $\Sigma$ into initial data for equation . 3. We also define two cut-off functions $\chi_\pm \in {\cal C}^\infty ({\mathbb{R}}) $, $\chi_+$ non decreasing on ${\mathbb{R}}$, $\chi_+ \equiv 0$ on $]-\infty , -1]$, $\chi_+ \equiv 1$ on $[1,+\infty [$, $\chi_- = 1-\chi_+$. They will be used with the variable $r*$ in order to isolate a part of the field living in a neighbourhood of either null infinity or the horizon. They can also be understood as functions on the exterior of the black hole ; we shall usually simply denote $\chi_\pm$ without referring explicitely to their argument. 4. We introduce the operator $$P^+ \, : ~ {\cal C}^\infty_0 ({{\mathscr H}}^+) \times {\cal C}^\infty_0 ({{\mathscr I}}^+) \longrightarrow {\cal C}^\infty_0 (\Sigma_0) \times {\cal C}^\infty_0 (\Sigma_0)$$ that pulls back the first function along the flow of incoming principal null geodesics and the second along the flow of outgoing principal null geodesics. By the definition of the variables $u=t-r_*$ and $v=t+r_*$, in terms of coordinates $(r_*,\omega)$ on $\Sigma_0$, $(v,\omega )$ on ${{\mathscr H}}^+$ and $(-u,\omega)$ on ${{\mathscr I}}^+$, the action of $P^+$ can be described very simply : take $(\xi (v,\omega) , \zeta (-u,\omega)) \in {\cal C}^\infty_0 ({{\mathscr H}}^+) \times {\cal C}^\infty_0 ({{\mathscr I}}^+)$, $$P^+ (\xi , \zeta ) (r_* , \omega ) = (\xi (r_*, \omega) , \zeta (r_*, \omega) ) \, .$$ The operator $P^+$ is an isometry from ${\cal H}^+$ onto ${\cal H}_0$ (see and Definition \[FuncSpaceScatt\]). The operator $P^+$ provides an identification between the conformal scattering data (that are functions defined on the conformal boundary) and initial data for the comparison dynamics (seen as acting between the slices $\Sigma_t$). \[WaveOps\] The direct future wave operator, defined for smooth compactly supported scattering data $$(\xi , \zeta) \in {\cal C}^\infty_0 ({{\mathscr H}}^+ ) \times {\cal C}^\infty_0 ({{\mathscr I}}^+ )$$ by $$W^+ (\xi , \zeta) := \lim_{t\rightarrow +\infty} {\cal U} (-t) {\cal J} \, {\cal U}_0 (t) P^+ (\xi , \zeta) \, ,$$ extends as an isometry from ${\cal H}^+$ onto $\cal H$. The inverse future wave operator, defined for smooth compactly supported initial data for $$(\hat{\phi}_0 , \hat{\phi}_1 ) \in {\cal C}^\infty_0 (\Sigma_0 ) \times {\cal C}^\infty_0 (\Sigma_0 )$$ by $$\tilde{W}^+ (\hat{\phi}_0 , \hat{\phi}_1 ) = \lim_{t\rightarrow +\infty} (P^+)^{-1} \, {\cal U}_0 (-t) \left( \begin{array}{cc} \chi_- & 0 \\ \chi_+ & 0 \end{array} \right) {\cal U} (t) (\hat{\phi}_0 , \hat{\phi}_1 ) \, ,$$ extends as an isometry from $\cal H$ onto ${\cal H}^+$. Moreover, we have $$\begin{gathered} \tilde{W}^+ = {\cal T}^+ = \operatorname*{\mathrm{s}\,--\lim ~}_{t\rightarrow +\infty} (P^+)^{-1} \, {\cal U}_0 (-t) \left( \begin{array}{cc} \chi_- & 0 \\ \chi_+ & 0 \end{array} \right) {\cal U} (t) \, , \label{InvWOpSLim} \\ \tilde{W}^+ = (W^+)^{-1} \, .\label{InvWOp}\end{gathered}$$ It is important to understand that as soon as we have proved that $\tilde{W}^+ = {\cal T}^+$, we have established the asymptotic completeness, since ${\cal T}^+$ is an isometry from $\cal H$ onto ${\cal H}^+$. The proof of only relies on the conformal scattering construction. Once is established, all that remains to prove is the existence of the direct wave operator, which we do using Cook’s method. The fact that $W^+$ is the inverse of $\tilde{W}^+$ is an immediate consequence of as we shall see. The expressions of the wave operators can be simplified a little if we consider ${{\mathscr I}}^+$ as the family of outgoing principal null geodesics and ${{\mathscr H}}^+$ as the family of incoming principal null geodesics. With this viewpoint, the comparison dynamics seen as acting on functions on ${{\mathscr H}}^+ \cup {{\mathscr I}}^+$ reduces to the identity. We introduce a family of projections ${\cal P}_t$ that to a pair of functions $(\xi , \zeta) \in {\cal C}^\infty_0 ({{\mathscr H}}^+ ) \times {\cal C}^\infty_0 ({{\mathscr I}}^+)$ associates its realization as a pair of functions on $\Sigma_t$, which as functions of $(r_*,\omega)$ have the following expression : $$( \xi (r_*+t,\omega ) , \zeta (r_*-t , \omega )) \, .$$ The direct and inverse wave operators acting on $(\xi , \zeta)$ then become : $$\begin{aligned} W^+ (\xi , \zeta) &=& \lim_{t\rightarrow +\infty} {\cal U} (-t) {\cal J}{\cal P}_t (\xi , \zeta) \, ;\\ \tilde{W}^+ (\hat{\phi}_0 , \hat{\phi}_1 ) &=& \lim_{t\rightarrow +\infty} {\cal P}_t^{-1} \left( \begin{array}{cc} \chi_- & 0 \\ \chi_+ & 0 \end{array} \right) {\cal U} (t) (\hat{\phi}_0 , \hat{\phi}_1 ) \, .\end{aligned}$$ We keep the version of the theorem however in order to get a closer similarity with the usual analytic expression of wave operators. [**Proof of Theorem \[WaveOps\].**]{} All we need to do is prove that on a dense subspace of $\cal H$, $\tilde{W}^+$ is well-defined and coincides with ${\cal T}^+$, and that similarly, on a dense subspace of ${\cal H}^+$, $W^+$ is well-defined and coincides with $({\cal T}^+)^{-1}$. Let us consider $(\hat{\phi}_0 , \hat{\phi}_1 ) \in {\cal C}^\infty_0 (\Sigma_0 ) \times {\cal C}^\infty_0 (\Sigma_0 ) \subset {\cal H}$. We denote by $\hat{\phi}$ the associated solution of such that $(\hat{\phi} , \partial_t \hat{\phi}) \in {\cal C} ({\mathbb{R}}_t ; {\cal H})$ and put $(\xi , \zeta ) = {\cal T}^+ (\hat{\phi}_0 , \hat{\phi}_1 )$. For $t>0$, the operator $$(P^+)^{-1} \, {\cal U}_0 (-t) \left( \begin{array}{cc} \chi_- & 0 \\ \chi_+ & 0 \end{array} \right) {\cal U} (t)$$ first propagates the solution $\hat{\phi}$ up to the slice $\Sigma_t$, then cuts-off using $\chi_-$ (resp. $\chi_+$) the part of $\hat{\phi} (t)$ near infinity (resp. near the horizon) and puts the result in the first (resp. second) slot. Finally, the combination $(P^+)^{-1} \, {\cal U}_0 (-t)$ is the push-forward of the function in the first slot onto ${{\mathscr H}}^+$ along the flow of incoming principal null geodesics, and the push-forward of the function in the second slot onto ${{\mathscr I}}^+$ along the flow of outgoing principal null geodesics. Since the support of the non constant part of the cut-off functions $\chi_\pm$ on $\Sigma_t$ remains away from both ${{\mathscr I}}^+$ and ${{\mathscr H}}^+$ and accumulates at $i^+$ as $t \rightarrow +\infty$ (see figure \[SuppDerivativeCutOff\]), we have the following pointwise limit ![The support of the derivatives of the cut-off functions $\chi_\pm$[]{data-label="SuppDerivativeCutOff"}](SupportCutOff.jpg){width="4in"} $$\lim_{t\rightarrow +\infty} (P^+ )^{-1} {\cal U}_0 (-t) \left( \begin{array}{cc} \chi_- & 0 \\ \chi_+ & 0 \end{array} \right) {\cal U} (t) (\hat{\phi}_0 , \hat{\phi}_1 ) = (\xi , \zeta ) \, .$$ This already proves that $\tilde{W}^+$ is well-defined on smooth compactly supported initial data and coincides with ${\cal T}^+$ on this dense subset of $\cal H$. Therefore $\tilde{W}^+$ extends as the isometry ${\cal T}^+$ from $\cal H$ onto ${\cal H}^+$. Let us now prove that the convergence above takes place in ${\cal H}^+$. This means that $$\begin{aligned} \lim_{t \rightarrow +\infty} \int_{{\mathbb{R}}\times S^2} \left( \frac{\partial}{\partial v} \left( \chi_- (-t+v)\hat{\phi} (t, -t+v, \omega) - \xi (v , \omega) \right) \right)^2 {\mathrm{d}}v {\mathrm{d}}\omega &=&0 \, , \label{CvHorCInfty} \\ \lim_{t \rightarrow +\infty} \int_{{\mathbb{R}}\times S^2} \left( \frac{\partial}{\partial u} \left( \chi_+ (t-u)\hat{\phi} (t, t-u, \omega) - \zeta (-u , \omega) \right) \right)^2 {\mathrm{d}}u {\mathrm{d}}\omega &=&0 \, . \label{CvInftyCInfty}\end{aligned}$$ We prove , the proof of is similar. Since $\hat{\phi} \in {\cal C}^\infty ( \bar{\cal M})$, we have $$\frac{\partial}{\partial v} \left( \chi_- (-t+v)\hat{\phi} (t, -t+v, \omega) - \xi (v , \omega) \right) \rightarrow 0 \mbox{ in } L^2_\mathrm{loc} ({\mathbb{R}}_{v} \, ;~ L^2 (S^2 )) \, .$$ In particular due to the compact support of the initial data, for any $v_0 \in {\mathbb{R}}$, $$\lim_{t \rightarrow +\infty} \int_{]-\infty , v_0 [ \times S^2} \left( \frac{\partial}{\partial v} \left( \chi_- (-t+v)\hat{\phi} (t, -t+v, \omega) - \xi (v , \omega) \right) \right)^2 {\mathrm{d}}v {\mathrm{d}}\omega =0 \, . \label{ConvCompact}$$ Let $\varepsilon >0$, consider $T>0$ large enough such that $\hat{\cal E}_{\partial_t , S_T} < \varepsilon$. As a consequence, we also have that the energy flux across the part of ${{\mathscr H}}^+$ in the future of $S_T$ is lower than $\varepsilon$ : $$\hat{\cal E}_{\partial_t , ({{\mathscr H}}^+ \setminus {{\mathscr H}}^+_T)} < \varepsilon \, .$$ We choose $t_0 >0$ large enough such that for all $t>t_0$, the intersection of $\Sigma_t$ with the support of $\chi_-'$ is entirely in the future of $S_T$ ; we also choose $v_0 >0$ such that the null hypersurface $\{ v=v_0 \}$ intersects all $\Sigma_t$, $t>t_0$, entirely in the future of $S_T$ (see figure \[StrongLimit\] for an illustration of both choices). Then we have $$\begin{aligned} \int_{] v_0 , +\infty [ \times S^2} \left( \frac{\partial \xi}{\partial v} (v , \omega) \right)^2 {\mathrm{d}}v {\mathrm{d}}\omega &<& \varepsilon \, , \label{Small1}\\ \int_{] v_0 , +\infty [ \times S^2} \left( \frac{\partial}{\partial v} \left( \chi_- (-t+v)\hat{\phi} (t, -t+v, \omega) \right) \right)^2 {\mathrm{d}}v {\mathrm{d}}\omega &<& \varepsilon \, ,\hspace{0.1in} \mbox{ for all } t > t_0 \, . \label{Small2}\end{aligned}$$ Now thanks to , we can choose $t_1 > t_0$ such that for all $t>t_1$ we have $$\label{Small3} \int_{]-\infty , v_0 [ \times S^2} \left( \frac{\partial}{\partial v} \left( \chi_- (-t+v)\hat{\phi} (t, -t+v, \omega) - \xi (v , \omega) \right) \right)^2 {\mathrm{d}}v {\mathrm{d}}\omega < \varepsilon \, .$$ Putting , and together, we obtain that for $t>t_1$ $$\int_{{\mathbb{R}}\times S^2} \left( \frac{\partial}{\partial v} \left( \chi_- (-t+v)\hat{\phi} (t, -t+v, \omega) - \xi (v , \omega) \right) \right)^2 {\mathrm{d}}v {\mathrm{d}}\omega < 5 \varepsilon \, .$$ This proves for data in ${\cal C}^\infty_0 (\Sigma_0 ) \times {\cal C}^\infty_0 (\Sigma_0 ) \subset {\cal H}$. ![The ingredients of the proof of .[]{data-label="StrongLimit"}](StrongLimit.jpg){width="4in"} Let us now consider initial data $(\hat{\phi}_0 , \hat{\phi}_1 ) \in {\cal H}$. Still denoting $\hat\phi$ the associated solution of and $(\xi , \zeta ) = {\cal T}^+ (\hat{\phi}_0 , \hat{\phi}_1 )$, we prove for such data. Let $\varepsilon >0$, consider $(\hat{\Phi}_0 , \hat{\Phi}_1 ) \in {\cal C}^\infty_0 (\Sigma_0 ) \times {\cal C}^\infty_0 (\Sigma_0 )$, $\hat{\Phi}$ the associated solution and $(\Xi , \mathrm{Z} ) = {\cal T}^+ (\hat{\Phi}_0 , \hat{\Phi}_1 )$, such that $$\Vert (\hat{\phi}_0 , \hat{\phi}_1 ) - (\hat{\Phi}_0 , \hat{\Phi}_1 ) \Vert_{{\cal H}}^2 < \varepsilon \, .$$ Then the energy fluxes, on ${{\mathscr H}}^+$ and $\Sigma_t$ for all $t$, of $\hat{\phi} - \hat{\Phi}$, are all lower than $\varepsilon$. Since is valid for $\hat\Phi$, we can find $t_0 >0$ such that for all $t >t_0$ we have $$\int_{{\mathbb{R}}\times S^2} \left( \frac{\partial}{\partial v} \left( \chi_- (-t+v)\hat{\Phi} (t, -t+v, \omega) - \Xi (v , \omega) \right) \right)^2 {\mathrm{d}}v {\mathrm{d}}\omega < \varepsilon \, .$$ It follows that for $t>t_0$, we have $$\int_{{\mathbb{R}}\times S^2} \left( \frac{\partial}{\partial v} \left( \chi_- (-t+v)\hat{\phi} (t, -t+v, \omega) - \xi (v , \omega) \right) \right)^2 {\mathrm{d}}v {\mathrm{d}}\omega < 9 \varepsilon \, .$$ This proves for finite energy data. We have therefore established . Let us now consider $(\xi , \zeta ) \in {\cal C}^\infty_0 ({{\mathscr H}}^+ ) \times {\cal C}^\infty_0 ({{\mathscr I}}^+ ) \subset {\cal H}^+$. For $t>0$, the operator $${\cal U} (-t) {\cal J} \, {\cal U}_0 (t) P^+$$ first (by the combination ${\cal U}_0 (t) P^+$) pulls back $\xi$ along the flow of incoming principal null geodesics and $\zeta$ along the flow of outgoing principal null geodesics, as a pair functions on $\Sigma_t$. Then ${\cal J}$ combines these two functions to obtain the initial data on $\Sigma_t$ for the wave equation : $$\hat{\phi} \vert_{\Sigma_t} (r_*,\omega)= \xi (t+r_* , \omega) + \zeta (r_*-t , \omega) \, ,~ \partial_t \hat{\phi} \vert_{\Sigma_t} (r_* , \omega)= \partial_s \xi (t+r_* , \omega) - \partial_s \zeta (r_*-t , \omega)\, .$$ After which ${\cal U} (-t)$ propagates the corresponding solution of down to $\Sigma_0$. In order to prove that ${\cal U} (-t) {\cal J} \, {\cal U}_0 (t) P^+ (\xi , \zeta )$ converges in $\cal H$ as $t \rightarrow +\infty$, we use Cook’s method ; the details of the proof can be found in Appendix \[AppendixCook\]. Then it is easy to conclude that $W^+$ is the inverse of $\tilde{W}^+$. Let us consider for $(\xi , \zeta ) \in {\cal C}^\infty_0 ({{\mathscr H}}^+ ) \times {\cal C}^\infty_0 ({{\mathscr I}}^+ )$ the quantity $$\label{CandidateInverse} (P^+)^{-1} \, {\cal U}_0 (-t) \left( \begin{array}{cc} \chi_- & 0 \\ \chi_+ & 0 \end{array} \right) {\cal U} (t) {\cal U} (-t) {\cal J} \, {\cal U}_0 (t) P^+ (\xi , \zeta ) \, .$$ By the strong convergence part of and the convergence in $\cal H$ of $${\cal U} (-t) {\cal J} \, {\cal U}_0 (t) P^+ (\xi , \zeta ) \, ,$$ converges in ${\cal H}^+$ towards $\tilde{W}^+ W^+ (\xi , \zeta )$. But simplifies as $$(P^+)^{-1} \, {\cal U}_0 (-t) \left( \begin{array}{cc} \chi_- & 0 \\ \chi_+ & 0 \end{array} \right) {\cal J} \, {\cal U}_0 (t) P^+ (\xi , \zeta ) = (P^+)^{-1} \, {\cal U}_0 (-t) \left( \begin{array}{cc} \chi_- & \chi_- \\ \chi_+ & \chi_+ \end{array} \right) \, {\cal U}_0 (t) P^+ (\xi , \zeta ) \, .$$ Thanks to the compact support of $\xi$ and $\zeta$, this is equal to $(\xi , \zeta)$ for $t$ large enough. This concludes the proof. Let us define similarly the past wave operators $W^-$ and $\tilde{W}^-$. We have $$\tilde{W}^- = (W^-)^{-1} ={\cal T}^- \, .$$ The scattering operator is related to the wave operators as follows $$S = \tilde{W}^+ W^- = (W^+)^{-1} W^- \, .$$ Translation representer, scattering data, radiation field --------------------------------------------------------- The conformal scattering theory we have constructed allows us, using the staticity of the exterior of a Schwarzschild black hole, to re-interpret immediately the scattering data as the crucial structure of the Lax-Phillips theory : the translation representer. This is expressed in the following theorem. The scattering data are a translation representer of the associated scalar field. More precisely, consider $(\hat\phi , \partial_t \hat\phi ) \in {\cal C} ({\mathbb{R}}_t \, ;~ {\cal H} ) $ a solution to , put $\hat\phi_0 := \hat\phi \vert_{\Sigma_0}$, $\hat\phi_1 := \partial_t \hat\phi \vert_{\Sigma_0}$ and $$(\xi , \zeta ) := {\cal T}^+ (\hat\phi_0 , \hat\phi_1 ) \, .$$ Then (expressing the functions using variables $(v,\omega)$ on ${{\mathscr H}}^+$ and $(-u,\omega )$ on ${{\mathscr I}}^+$), $${\cal T}^+ (\hat\phi \vert_{\Sigma_t} , \partial_t \hat\phi \vert_{\Sigma_t} ) = (\xi (v+t , \omega ) , \zeta (-u-t , \omega )) \, .$$ [**Proof.**]{} If instead of $(\hat\phi_0 , \hat\phi_1 ) $ we take $(\hat\phi \vert_{\Sigma_t} , \partial_t \hat\phi \vert_{\Sigma_t} )$ for initial data, since $\partial_t$ is Killing, this is equivalent to pulling back the whole solution $\hat\phi$ of a time interval $t$ along the flow of $\partial_t$. Moreover $\partial_t$ extends as $\partial_v$ on ${{\mathscr H}}^+$ and as $\partial_u$ on ${{\mathscr I}}^+$. This concludes the proof. Note that the part of the scattering data on ${{\mathscr I}}^+$ is the trace of $\hat\phi = r \phi$ on ${{\mathscr I}}^+$ and is therefore exactly the future radiation field. The essential difference from the theory of Lax-Phillips and the construction of Friedlander in 1980 [@Fri1980] is that we have a scattering theory with two scattering channels and therefore we need one extra scattering data. The important thing to understand here is that the translation representer is intimately related to the stationarity of the spacetime. If we give up stationarity, we also have to give up the translation representer but the conformal scattering construction would still be valid provided we have good estimates and a well-defined conformal boundary. Extension to the Kerr metric and concluding remarks {#Kerr} =================================================== At the time when this paper first appeared on the arXiv, the analysis in the Kerr framework was not as advanced as in the Schwarzschild setting. A variety of decay results were available for scalar waves and one for Maxwell fields, some of them establishing Price’s law (which is the decay generically expected both in timelike directions and up the generators of null infinity, see R. Price [@Pri1972a] for scalar fields and [@Pri1972b] for zero-rest-mass fields) : these results were due to L. Andersson and P. Blue [@ABlu], M. Dafermos and I. Rodnianski [@DaRoLN], F. Finster, N. Kamran, J. Smoller and S.T. Yau [@FiKaSmoYa1; @FiKaSmoYa2], F. Finster and J. Smoller [@FiSmo], J. Metcalfe, D. Tataru and M. Tohaneanu [@MeTaTo], D. Tataru and M. Tohaneanu [@TaTo] for the wave equation, and to L. Andersson and P. Blue [@ABlu2] for Maxwell fields. All these papers, except [@FiKaSmoYa1; @FiKaSmoYa2; @FiSmo], deal with slowly rotating Kerr black holes. Decay estimates are useful in our conformal scattering construction in order to prove that the energy on the far future hypersurface $S_T$ tends to zero as $T \rightarrow +\infty$ (see subsection \[EnEstTInfinite\]). This step however relies on the solidity of the foundations laid in subsection \[EnIdST\] : uniform energy estimates both ways, without loss, between a Cauchy hypersurface and the union of $S_T$ and the parts of ${{\mathscr H}}^+$ and ${{\mathscr I}}^+$ in the past of $S_T$. Among the works we have just cited, the only one providing, if not exactly this kind of estimate, at least a way of obtaining them using the symmetry of the Kerr metric and the decay estimates, is [@ABlu]. They are the only ones establishing uniform estimates, for a positive definite energy, on a foliation by Cauchy hypersurfaces of the Kerr exterior. Many of the other works use the redshift effect near the horizon, see M. Dafermos and I. Rodnianski [@DaRo2009]. This is perfectly fine for proving decay, but the estimates cannot be reversed because when we go backwards in time, it is a blueshift effect that we have to deal with. The works of F. Finster, N. Kamran, J. Smoller and S.T. Yau rely on a different technique, which is an integral representation of the propagator for the wave equation ; they do not however obtain the type of estimate we need. The main drawback of the energy used by L. Andersson and P. Blue is that it is of too high order to be convenient for scattering theory. In fact, this is rather an aesthetic consideration than any serious scientific objection and it would be interesting to try an develop a conformal scattering theory based on their energy. Since then, M. dafermos, I. Rodnianski and Y. Shlapentokh-Rothman [@DaRoShla] have obtained the missing uniform energy equivalence, without slow rotation assumption, and used it to construct a complete analytic scattering theory for the wave equation on the Kerr metric. They make the comment that it is crucial to chose an energy that does not see the redshift effect. Such an energy is based on a vector field that reduces at ${{\mathscr H}}^+$ to the null generator of the horizon, i.e. that is timelike outside the black hole but tangent to the horizon. This has interesting connections with our comments in section \[WaveOps\]. It appears that with the results of [@DaRoShla], our construction in the Schwarzschild case can now be extended to Kerr black holes essentially without change. It could be interesting to write this in detail provided we use only the relevent estimates and not the full scattering theory. Indeed, the re-interpretation of an analytic scattering theory as a conformal one is in many cases easy and purely a matter of understanding the scattering data as radiation fields (see A. Bachelot [@Ba1991] and D. Häfner and J.-P. Nicolas [@HaNi2004]). In the case of [@DaRoShla] the re-interpretation would be totally trivial since their scattering data are already described as radiation fields. The question of inferring an analytic scattering, defined in terms of wave operators, from a conformal scattering theory is more delicate however. It has been addressed in [@MaNi2004] and in the present work but still needs to be understood precisely in general. As far as the development of conformal scattering theory per se is concerned, it appears essential to find a way of replacing pointwise decay estimates, such as Price’s law, by integrated decay estimates requiring a less precise knowledge of the local geometry and that are closer in nature to the minimal velocity estimates one obtains in the spectral approach to scattering theory (involving Mourre estimates and commutator methods). Acknowledgments =============== I would like to thank Dean Baskin, Fang Wang and Jérémie Joudioux for interesting discussions that contributed to improve this paper. I am also indebted to the anonymous referee for his/her useful comments. This research was partly supported by the ANR funding ANR-12-BS01-012-01. Cook’s method for the direct wave operator {#AppendixCook} ========================================== In this proof we represent the free dynamics in a slightly different but equivalent manner. The space ${\mathbb H}_0 = \dot{H}^1 ({\mathbb{R}}_{r_*} \, ;~ L^2 (S^2 ) ) \times L^2 ({\mathbb{R}}_{r_*} \times S^2 )$ is the direct orthogonal sum of two supplementary subspaces : $${\mathbb H}_0^\pm = \{ (\psi_0 , \psi_1 ) \, ;~ \psi_1 = \mp \partial_{r_*} \psi_0 \} \, ;$$ via the operator $P^+$, ${\mathbb H}^+_0$ corresponds to $\dot{H}^1 ({\mathbb{R}}_{u} \, ;~ L^2 (S^2 ) )$ on ${{\mathscr I}}^+$ and ${\mathbb H}^-_0$ to $\dot{H}^1 ({\mathbb{R}}_{v} \, ;~ L^2 (S^2 ) )$ on ${{\mathscr H}}^+$. On ${\mathbb H}_0$, we consider the free Hamiltonian $$H_0 = -i \left( \begin{array}{cc} 0 & 1 \\ {\partial_{r_*}^2 } & 0 \end{array} \right) \, .$$ The equation $\partial_t U = iH_0 U$ is the Hamiltonian form of the free equation $$\partial_t^2 \psi - \partial_{r_*}^2 \psi =0 \, .$$ The operator $H_0$ is self-adjoint on ${\mathbb H}_0$ and the free propagator ${\cal U}_0 (t)$ is just the group $e^{itH_0}$ conjugated by the identifying operator : $${\cal J} {\cal U}_0 (t) = e^{itH_0} {\cal J} \, .$$ With this description of the comparison dynamics, we need neither $P^+$ nor the identifying operator in the expression of the limit defining the direct wave operator $W^+$. On $\cal H$ we consider the operator $$H = -i \left( \begin{array}{cc} 0 & 1 \\ {\partial_{r_*}^2 + \frac{F}{r^2} \Delta_{S^2} - \frac{FF'}{r}} & 0 \end{array} \right) \, ;$$ the equation $\partial_t U = iHU$ is the Hamiltonian form of . The operator $H$ is self-adjoint on $\cal H$ and the propagator ${\cal U} (t)$ is equal to $e^{itH}$. For all $(U^h , U^\infty ) \in {\mathbb H}_0^- \times {\mathbb H}_0^+$, smooth and compactly supported, the following limits exist in $\cal H$ : $$\begin{gathered} \lim_{t\rightarrow +\infty} e^{-itH} e^{itH_0} U^h \, , \label{LimHorizon} \\ \lim_{t\rightarrow +\infty} e^{-itH} e^{itH_0} U^\infty \, .\label{LimInfinity}\end{gathered}$$ [**Proof.**]{} Take $$U^h = \left( \begin{array}{c} \psi_0 \\ \psi_1 = \partial_{r_*} \psi_0 \end{array} \right) \, ,~ \psi_0 \in {\cal C}^\infty_0 ( {\mathbb{R}}\, ;~ {\cal C}^\infty (S^2) ) \, .$$ A sufficient condition for the limit to exist is that $$\frac{{\mathrm{d}}}{{\mathrm{d}}t} e^{-itH} e^{itH_0} U^h = e^{-itH} \left( -i H + i H_0 \right) e^{itH_0} U^h \in L^1 ({\mathbb{R}}_t^+ \, ;~ {\cal H} ) \, .$$ Since $e^{-itH}$ is a group of unitary operators on $\cal H$, the condition is equivalent to $$\left( -i H + i H_0 \right) e^{itH_0} U^h \in L^1 ({\mathbb{R}}_t^+ \, ;~ {\cal H} ) \, .$$ This is easy to check : $$\Vert \left( -i H + i H_0 \right) e^{itH_0} U^h \Vert^2_{{\cal H}} = \frac12 \int_{{\mathbb{R}}\times S^2} \left( -\frac{F}{r^2} \Delta_{S^2} \psi_0 (t+r_*) + \frac{FF'}{r} \psi_0 (t+r_*) \right)^2 {\mathrm{d}}r_* {\mathrm{d}}^2 \omega$$ and this falls off exponentially fast as $t \rightarrow +\infty$ thanks to the compact support and the smoothness of $\psi_0$ and to the fact that $$F(r) = 1 - \frac{2M}{r} = \frac{1}{r} e^{(r_*-r)/2M}$$ fall-off exponentially fast as $r_* \rightarrow -\infty$. The proof of the existence of the other limit is similar, but we do not get exponential decay in this case. Take $$U^\infty = \left( \begin{array}{c} \psi_0 \\ \psi_1 = -\partial_{r_*} \psi_0 \end{array} \right) \, ,~ \psi_0 \in {\cal C}^\infty_0 ( {\mathbb{R}}\, ;~ {\cal C}^\infty (S^2) ) \, .$$ This time we have $$\Vert \left( -i H + i H_0 \right) e^{itH_0} U^\infty \Vert^2_{{\cal H}} = \frac12 \int_{{\mathbb{R}}\times S^2} \left( -\frac{F}{r^2} \Delta_{S^2} \psi_0 (r_*-t) + \frac{FF'}{r} \psi_0 (r_*-t) \right)^2 {\mathrm{d}}r_* {\mathrm{d}}^2 \omega$$ and this falls-off like $1/t^4$ as $t\rightarrow +\infty$, thanks to the compact support and the smoothness of $\psi_0$ and to the fact that $$\frac{F}{r^2} \simeq \frac{1}{r^2} \mbox{ and } r_* \simeq r \mbox{ at infinity.}$$ The other term falls off faster since $$\frac{FF'}{r} \simeq \frac{2M}{r^3} \mbox{ at infinity.}$$ So we still obtain the integrability in time of $\Vert \left( -i H + i H_0 \right) e^{itH_0} U^\infty \Vert_{{\cal H}}$ and this concludes the proof. As a consequence, for all $U_0 \in \mathbb{H}_0$, smooth and compactly supported, the limit $$\lim_{t\rightarrow +\infty} e^{-itH} e^{itH_0} U_0$$ exists in $\cal H$. This is equivalent to the existence for smooth and compactly supported scattering data of the limit defining $W^+$. Applying L. Hörmander’s results in the Schwarzschild setting {#HormGP} ============================================================ ![A cut-extend construction for the solution of the Goursat problem from ${{\mathscr I}}^+$.[]{data-label="CutExtend"}](CutExtendBis.jpg){width="80.00000%"} The work of L. Hörmander [@Ho1990] is a proof of the well-posedness of a weakly spacelike Cauchy problem, for a wave equation $$\label{ModWEq} \partial_t^2 u - \Delta u + L_1 u = f \, ,$$ on a Lorentzian spacetime that is a product ${\mathbb{R}}_t \times X$, with metric ${\mathrm{d}}t^2 - g$, $X$ being a compact manifold without boundary of dimension $n$ and $g(t)$ a Riemannian metric on $X$ smoothly varying with $t$. In , $\Delta$ is a modified version of the Laplace-Beltrami operator in which the volume density associated with the metric is replaced by a given smooth density on $X$ ; the operator $L_1$ is a first order differential operator with smooth coefficients and $f$ is a source. Any non degenerate change in the metric or the volume density can be absorbed in the operator $L_1$ so the results of [@Ho1990] are valid for the wave equation on any spatially compact globally hyperbolic spacetime. The data for the Cauchy problem are set on a hypersurface $\Sigma$ that is assumed Lipschitz and achronal (i.e. weakly spacelike), meaning that the normal vector field (which in the case of a Lipschitz hypersurface is defined almost everywhere) is causal wherever it is defined. In the present work, we are not dealing with a spatially compact spacetime, but an easy construction allows us to understand the resolution of the Goursat problem for compactly supported data on the future conformal boundary as a Goursat problem on a cylindrical spacetime, for which Hörmander’s results are valid. The construction, described schematically in figure \[CutExtend\], goes as follows. The data on ${{\mathscr H}}^+ \cup {{\mathscr I}}^+$ are compactly supported, which guarantees that the past of their support remains away from $i^+$. We simply consider the future ${\cal I}^+ ({\cal S})$ of the hypersurface $\cal S$ in $\bar{\cal M}$ (recall that $\cal S$ is a spacelike hypersurface on ${\bar{\cal M}}$ whose intersection with the horizon is the crossing sphere and which crosses ${{\mathscr I}}^+$ strictly in the past of the support of the data) and we cut off the future $\cal V$ of a point in $\cal M$ lying in the future of the past of the support of the Goursat data (see figure \[CutExtend\]). We denote by $\mathfrak{M}$ the resulting spacetime. Then we extend $\mathfrak{M}$ as a cylindrical globally hyperbolic spacetime $({\mathbb{R}}_t \times S^3 \, ,~ \mathfrak{g})$. We also extend the part of ${{\mathscr I}}^+ \cup {{\mathscr H}}^+$ inside ${\cal I}^+ ({\cal S}) \setminus {\cal V}$ as a null hypersurface $\cal C$ (see figure \[CutExtend\]) that is the graph of a Lipschitz function over $S^3$ and the data by zero on the rest of the extended hypersurface. The Goursat problem for equation $$\square_\mathfrak{g} \psi + \frac16 \mathrm{Scal}_\mathfrak{g} \psi =0 \, ,$$ with the data we have just constructed has a unique solution (see [@Ho1990]) $$\psi \in {\cal C} ({\mathbb{R}}\, ;~ H^1 (S^3 )) \cap {\cal C}^1 ({\mathbb{R}}\, ;~ L^2 (S^3 )) \, .$$ Then by local uniqueness and causality, using in particular the fact that as a consequence of the finite propagation speed, the solution $\psi$ vanishes in ${\cal I}^+ ({\cal S}) \setminus \mathfrak{M}$, the Goursat problem that we are studying has a unique smooth solution in the future of $\cal S$, that is the restriction of $\psi$ to $\mathfrak{M}$. [100]{} L. Andersson, P. Blue, [*Hidden symmetries and decay for the wave equation on the Kerr spacetime*]{}, arXiv:0908.2265. L. Andersson, P. Blue, Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior, arxiv:1310.2664. A. Bachelot, [*Gravitational scattering of electromagnetic field by Schwarzschild black hole*]{}, Annales de l’I.H.P. A [**54**]{} (1991), 261–320. J.C. Baez, [*Scattering and the geometry of the solution manifold of $\square f+ \lambda f^3 =0$*]{}, J. Funct. Anal. [**83**]{} (1989), 317–332. J.C. Baez, [*Scattering for the Yang-Mills equations*]{}, Trans. Amer. Math. Soc. [**315**]{} (1989), 2, 823–832. J.C. Baez, [*Conserved quantities for the Yang-Mills equations*]{}, Adv. Math. [**82**]{} (1990), 1, 126–131. J.C. Baez, I.E. Segal, Zhou Z.F., [*The global Goursat problem and scattering for nonlinear wave equations*]{}, J. Funct. Anal. [**93**]{} (1990), 2, 239–269. J.C. Baez, Zhou Z.F., [*The global Goursat problem on ${\mathbb{R}}\times S^1$*]{}, J. Funct. Anal. [**83**]{} (1989), 364–382. M. Dafermos, I. Rodnianski, [*Lectures on black holes and linear waves*]{}, Evolution equations, 97–205, Clay Math. Proc., 17, Amer. Math. Soc., Providence, RI, 2013, arXiv:0811.0354, 2008. M. Dafermos, I. Rodnianski, [*The red-shift effect and radiation decay on black hole spacetimes*]{}, Comm. Pure Appl. Math. [**62**]{} (2009), 7, 859–919. M. Dafermos, I. Rodnianski, Y. Shlapentokh-Rothman, [*A scattering theory for the wave equation on Kerr black hole exteriors*]{}, arXiv:1412.8379. J. Dimock, [*Scattering for the wave equation on the Schwarzschild metric*]{}, Gen. Rel. Grav. [**17**]{} (1985), 4, 353–369. J. Dimock, B.S. Kay, [*Classical and quantum scattering theory for linear scalar fields on the Schwarzschild metric. II*]{}, J. Math. Phys. [**27**]{} (1986), 10, 2520–2525. J. Dimock, B.S. Kay, [*Classical and Quantum Scattering theory for linear scalar fields on the Schwarzschild metric I*]{}, Ann. Phys. [**175**]{} (1987), 366–426. F. Finster, N. Kamran, J. Smoller, Yau, S.-T., [*Decay of solutions of the wave equation in the Kerr geometry*]{}, Comm. Math. Phys. [**264**]{} (2006), 2, 465–503. F. Finster, N. Kamran, J. Smoller, Yau, S.-T., [*Erratum: "Decay of solutions of the wave equation in the Kerr geometry”*]{} \[Comm. Math. Phys. [**264**]{} (2006), no. 2, 465–503; MR2215614\]. Comm. Math. Phys. [**280**]{} (2008), no. 2, 563–573. F. Finster, J. Smoller, [*A time-independent energy estimate for outgoing scalar waves in the Kerr geometry*]{}, J. Hyperbolic Differ. Equ. [**5**]{} (2008), 1, 221–255. F.G. Friedlander, [*On the radiation field of pulse solutions of the wave equation*]{}, Proc. Roy. Soc. Ser. A [**269**]{} (1962), 53–65. F.G. Friedlander, [*On the radiation field of pulse solutions of the wave equation II*]{}, Proc. Roy. Soc. Ser. A [**279**]{} (1964), 386–394. F.G. Friedlander, [*On the radiation field of pulse solutions of the wave equation III*]{}, Proc. Roy. Soc. Ser. A [**299**]{} (1967), 264–278. F.G. Friedlander, [*Radiation fields and hyperbolic scattering theory*]{}, Math. Proc. Camb. Phil. Soc. [**88**]{} (1980), 483–515. F.G. Friedlander, [*Notes on the Wave Equation on Asymptotically Euclidean Manifolds*]{}, J. Funct. Anal. [**184**]{} (2001), 1–18. D. Häfner, J.-P. Nicolas, [*Scattering of massless Dirac fields by a Kerr black hole*]{}, Rev. Math. Phys. [**16**]{} (2004), 1, 29?123. L. Hörmander, [*A remark on the characteristic Cauchy problem*]{}, J. Funct. Ana. [**93**]{} (1990), 270–277. J. Joudioux, [*Conformal scattering for a nonlinear wave equation*]{}, J. Hyperbolic Differ. Equ. [*9*]{} (2012), 1, 1–65. P.D. Lax, R.S. Phillips, [*Scattering theory*]{}, Academic Press 1967. J. Leray, [*Hyperbolic differential equations*]{}, lecture notes, Princeton Institute for Advanced Studies, 1953. L.J. Mason, [*On Ward’s integral formula for the wave equation in plane-wave spacetimes*]{}, Twistor Newsletter [**28**]{} (1989), 17–19. L.J. Mason, J.-P. Nicolas, [*Conformal scattering and the Goursat problem*]{}, J. Hyperbolic Differ. Equ. [**1**]{} (2004), 2, 197–233. L.J. Mason, J.-P. Nicolas, [*Regularity an space-like and null infinity*]{}, J. Inst. Math. Jussieu [**8**]{} (2009), 1, 179–208. J. Metcalfe, D. Tataru, M. Tohaneanu, [*Price’s law on nonstationary space-times*]{}, Adv. Math. [**230**]{} (2012), 3, 995–1028. J.-P. Nicolas, [*Non linear Klein-Gordon equation on Schwarzschild-like metrics*]{}, J. Math. Pures Appl. 74 (1995), p. 35-58. J.-P. Nicolas, [*On Lars Hörmander’s remark on the characteristic Cauchy problem*]{}, Annales de l’Institut Fourier, [**56**]{} (2006), 3, 517–543. R. Penrose, [*Asymptotic properties of fields and spacetime*]{}, Phys. Rev. Lett. [**10**]{} (1963), 66–68. R. Penrose, [*Conformal approach to infinity*]{}, in Relativity, groups and topology, Les Houches 1963, ed. B.S. De Witt and C.M. De Witt, Gordon and Breach, New-York, 1964. R. Penrose, [*Zero rest-mass fields including gravitation : asymptotic behaviour*]{}, Proc. Roy. Soc. London [**A284**]{} (1965), 159–203. R. Penrose, W. Rindler, [*Spinors and space-time*]{}, Vol. I & II, Cambridge monographs on mathematical physics, Cambridge University Press 1984 & 1986. R. Price, [*Nonspherical Perturbations of Relativistic Gravitational Collapse. I. Scalar and Gravitational Perturbations*]{}, Phys. Rev. D [**5**]{} (1972), 10, p. 2419–2438. R. Price, [*Nonspherical Perturbations of Relativistic Gravitational Collapse. II. Integer-Spin, Zero-Rest-Mass Fields*]{}, Phys. Rev. D [**5**]{} (1972), 10, p. 2439–2454. H. Soga, [*Singularities of the scattering kernel for convex obstacles*]{}, J. Math. Kyoto Univ. [**22**]{}-4 (1983), 729–765. D. Tataru, M. Tohaneanu, [*A local energy estimate on Kerr black hole backgrounds*]{}, Int. Math. Res. Not. IMRN 2011, 2, 248–292. R. Ward, [*Progressing waves in flat spacetime and in plane-wave spacetimes*]{}, Class. Quantum Grav. [**4**]{} (1987), 775–778. E.T. Whittaker, [*On the partial differential equations of mathematical physics*]{}, Mathematische Annalen [**57**]{} (1903), p. 333-355. [^1]: It is interesting to note that the integral formula, obtained by Lax and Phillips, that recovers the field in terms of its scattering data, was in fact discovered by E.T. Whittaker in 1903 [@Whi]. This does not seem to have been known to them or to Friedlander. The Lax-Phillips theory gave Whittaker’s formula its rightful interpretation as a scattering representation of the solutions of the wave equation. There is an interesting extension of this formula to plane wave spacetimes due to R.S. Ward [@Wa], developed further by L.J. Mason [@Ma]. [^2]: More precisely, the existence of a translation representation of the propagator is tied in with the existence of a timelike Killing vector field that extends as the null generator of null infinity. [^3]: The treatment of the wave equation was not completed in this paper, the additional ingredients required can be found in another work by the same authors, dealing with the peeling of scalar fields, published in 2009 [@MaNi2009]. [^4]: We will still however work with both the rescaled and the physical field when comparing our energy norms with those used by other authors. Of course the indices of vectors and $1$-forms will have to be raised and lowered using the rescaled metric $\hat{g}$ when working with rescaled quantities and using the physical metric $g$ when working with unrescaled quantities. [^5]: The factor $-4$ comes form the identity applied to $\hat{J}_a + V_a$, i.e. $${\mathrm{d}}* ((\hat{J}_a + V_a ){\mathrm{d}}x^a) = -(1/4) \nabla_a (\hat{J}^a + V^a ) {\mathrm{dVol}}\, .$$ [^6]: Of course the Hodge star in equation is associated with the physical metric, whereas in the expression of the rescaled energy, it is associated with the rescaled metric. [^7]: Note that the spaces ${\cal H}^\pm$ are naturally identified via a time reflexion $t \mapsto -t$.

--- abstract: | The air entrainment due to the turbulence in a free surface boundary layer shear flow created by a horizontally moving vertical surface-piercing wall is studied through experiments and direct numerical simulations. In the experiments, a laboratory-scale device was built that utilizes a surface-piercing stainless steel belt that travels in a loop around two vertical rollers, with one length of the belt between the rollers acting as a horizontally-moving flat wall. The belt is accelerated suddenly from rest until reaching constant speed in order to create a temporally-evolving boundary layer analogous to the spatially-evolving boundary layer that would exist along a surface-piercing towed flat plate. To complement the experiments, Direct Numerical Simulations (DNS) of the two-phase boundary layer problem were carried out with the domain including a streamwise belt section simulated with periodic boundary conditions. Cinematic Laser-Induced Fluorescence (LIF) measurements of water surface profiles in two vertical planes oriented parallel to the belt surface (wall-parallel profiles) are presented and compared to previous measurements of profiles in a vertical plane oriented normal to the belt surface (wall-normal profiles). Additionally, photographic observations of air entrainment and measurements of air bubble size distributions and motions are reported herein. The bubble entrainment mechanisms are studied in detail through the results obtained by the DNS simulations. Free surface features resembling breaking waves and traveling parallel to the belt are observed in the wall-parallel LIF movies. These free surface features travel up to 3 times faster than the free surface features moving away from the belt in the wall-normal LIF movies. These breaking events are thought to be one of the mechanisms by which the air is entrained into the underlying flow. The bubble size distribution is found to have a characteristic break in slope, similar to the Hinze scale previously observed in breaking waves [@Deane2002]. The number of bubbles, their velocity, and size are reported versus depth from the experimental data. These results are qualitatively similar to results obtained by the simulations. Finally, several entrainment mechanisms are found in the simulations and their prevalence in the free surface boundary layer is assessed. author: - | Naeem Masnadi$^{1a}$, Martin A. Erinin$^1$, Nathan Washuta$^{1b}$, Farshad Nasiri$^2$,\ Elias Balaras$^2$ and James H. Duncan$^1$, bibliography: - 'Library.bib' title: Air Entrainment and Surface Fluctuations in a Turbulent Ship Hull Boundary Layer --- INTRODUCTION ============ Turbulent boundary layers near the free surface along ship hulls and surface-piercing flat plates have been explored by a number of authors, see for example [@Stern1989], [@Longo1998], [@Sreedhar1998], and [@Stern1993]. Air entrainment and bubble distributions in these free surface flows have been explored by [@CarricaEtAl1999], [@MoragaEtAl2008], [@PerretCarrica2015], [@CastroEtAl2016] and [@LiEtAl2016]. One obvious region of two-phase flow in the vicinity of a ship is the layer of white water next to the hull, see for example the photograph in Figure \[fig:ship\]. The mechanisms by which air enters this region of the flow is poorly understood. In particular, it is not known to what degree this white water is the result of active spray generation and air entrainment due to turbulence in the boundary layer along the ship hull and to what degree the white water is the result of spray and air bubbles that are generated upstream in the breaking bow wave and then swept downstream with the flow. In the free surface boundary layer, the air entrainment process is controlled by the ratios of the turbulent kinetic energy to the gravitational potential energy and the turbulent kinetic energy to the surface tension energy. The ratio of the turbulent kinetic energy to the gravitational potential energy is given by the square of the turbulent Froude number ($Fr^2 = q^2 / (g L)$) and the ratio of turbulent kinetic energy to surface tension energy is given by the Weber number ($We = \rho q^2 L/ \sigma$), where $g$ is the acceleration of gravity, $\rho$ is the density of water, $\sigma$ is the surface tension of water, $q$ is the characteristic magnitude of the turbulent velocity fluctuations and $L$ is the length scale of this turbulence. Several authors have applied theory and numerical methods to explore the interaction of turbulence and a free surface, see for example [@Shen2001], [@Guo2009], [@kim:2013] and [@broc:2001]. [@broc:2001] have used scaling arguments to predict the critical Froude and Weber numbers above which air entrainment and spray generation will occur due to strong free-surface turbulence. Figure \[fig:brocchiniperegrine\], which is from their paper, shows the boundaries of various types of surface undulations on a plot of $q$ versus $L$. The upper region of the plot is the region of air entrainment and droplet generation. We have used classical boundary layer correlations to make estimates of $q$ (taken as the root-mean-square vertical component of the turbulent velocity fluctuations) and $L$ (taken as the boundary layer thickness) at three streamwise positions in a ship boundary layer and plotted these points on the $q$-$L$ map in Figure \[fig:brocchiniperegrine\]. As can be seen from the figure, the points are clearly in the air entrainment region of the plot, especially the points near the bow. Thus, air entrainment due to strong turbulent fluid motions in the hull boundary layer at the free surface is a likely cause of the layer of white water. ![Regions of various types of surface motions for free surface turbulence with velocity fluctuation magnitude $q$ (vertical axis) and length scale $L$ (horizontal axis), from [@broc:2001]. Air entrainment and spray production occur in the upper region, above the uppermost curved line. The three data points are values obtained for the turbulent boundary layer on a flat plate with $q$ taken as the rms of the spanwise (which is vertical for the boundary layer along a ship hull) velocity fluctuation ($w'$) and L taken as the boundary layer thickness ($\delta$).[]{data-label="fig:brocchiniperegrine"}](figure2.pdf){width="3in"} The difficulty with laboratory experiments on bubble entrainment and spray stems from the fact that the experiments are performed in the same gravitational field as found in ship flows and that the only practical liquid available is water, as is also found in the ocean. Thus, with $g$, $\rho$ , and $\sigma$ the same in the field and in the laboratory, one must attempt to achieve full-scale flow speeds in order to obtain Froude, Weber, and Reynolds similarity with field conditions. Also, even if full scale-values of $q$ and $L$ were obtained by towing a surface piercing flat plate with the length of the ship at high speed in a ship model basin, the free surface flow would include a bow wave which would obfuscate the source of the bubbles and spray. Another problem is that in order to obtain realistic entrainment/spray conditions and bubble/droplet size distributions, these experiments should be performed in salt water which is not typically used in ship model basins (note that though the experiments presented in this paper were performed in fresh water, we hope to repeat the experiments in salt water in the furture). In view of the above difficulties in simulating air entrainment due to the turbulent boundary layer, we have built a novel device that produces an approximation of a full scale ship boundary layer in the laboratory. This device, called the Ship Boundary Layer (SBL) simulator generates a temporally evolving boundary layer on a vertical, surface-piercing flat wall. This vertical wall consists of a stainless steel belt loop that is 1.0 m wide and about 15 m long. The belt is mounted on two vertically oriented rollers as shown in Figure \[fig:TankSchem\]. The rollers are driven by hydraulic motors and the entire device is placed in a large open-surface water tank as shown in the figure. Before each experimental run, the belt and the water in the tank are stationary. The water level is set below the top edge of the belt and the flow outside the belt loop on one of the long lengths between the rollers is studied. The belt is accelerated from rest using a hydraulic control system, which is able to create a highly repeatable belt motion. In the experiments discussed herein, the belt is accelerated suddenly from rest until it reaches a pre-defined speed which is held steady for a short time. The flow on the surface of the belt in this case is a simulation of the flow seen by a stationary observer in the ocean as a ship, that makes no waves, passes by at constant speed. The temporally-evolving boundary layer created along the belt can be considered equivalent to the spatially-developing boundary layer along a flat ship hull, with the distance along the ship hull corresponding to the distance traveled by the belt at any time $t$. The primary objective of the experimental study is to gain insight into the different entrainment mechanisms via quantitative and qualitative observations of the water free surface as the belt is launched from rest and to quantify the statistics of the entrained bubbles including their diameters, positions and velocities. In addition to the experiments, a Direct Numerical Simulation (DNS) that reproduces the main features of the above-described experiments is performed. The computations consider a small streamwise section of the flow along the belt in the experiments, and apply periodic boundary conditions along the direction of motion of the belt. The Navier-Stokes equations for incompressible flow are solved in both the air and water portions of the flow, allowing us to examine the entire three-dimensional velocity field. The primary objective of the computational study is to identify and understand the behavior of turbulent structures immediately below the free surface and their impact on air entrainment. The remainder of the paper is divided into four sections. The experimental setup for the ship boundary layer, surface profile measurements, and bubble measurements, are described in the second section of the paper and the numerical setup for the DNS is reported in the third section. This is followed by the presentation and discussion of the results in the forth section of the paper. Finally, the conclusions of this study are presented in the fifth section. EXPERIMENTAL DETAILS ==================== The experiments were performed in the same tank and with similar measurement techniques as those described in [@WashutaThesis] and [@Washuta2016]. A brief overview of these facilities and techniques are given below; the interested reader is referred to the original references for further details. The experiments were performed in an open-surface water tank that is 13.34 m long, 2.37 m wide and 1.32 m deep, see Figure \[fig:TankSchem\]. The tank walls and bottom are made of clear plastic panels for optical access. The top of the tank is open, offering an unobstructed view of the water surface. The main functional component of the Ship Boundary Layer (SBL) simulator is a one-meter-wide 0.8-mm-thick stainless steel belt loop that is driven by two 0.46-meter-diameter, 1.1-meter-long rollers whose rotation axes are vertically oriented and separated by a horizontal distance of approximately 7.5 meters. The rollers are each driven by two bent-axis hydraulic motors via toothed-belt-and-pulley systems. Each roller along with the motors and drive systems form single drive units that are attached to a welded steel frame that maintains the separation between, and relative parallel orientation of the rollers. The assembled SBL device is placed in a stainless steel sheet metal box (called the dry box). The dry box keeps the assembly essentially dry, while one of the two straight sections of the belt exits the dry box through a set of seals near the roller on the left and travels through the water to the second set of seals near the opposite roller where the belt re-enters the dry box. The lone straight section exposed to water is approximately 6 meters long and pierces the free surface with approximately 0.33 meters of freeboard for the water level used in the experiments presented in this paper. At the location where the belt leaves the dry box and enters the water, a sheet metal fairing is installed to reduce the flow separation caused by the backwards-facing step associated with the shape of the dry box at this location. When performing experiments, the belt is launched from rest and accelerates until reaching constant speed. Throughout these transient experiments, the belt travel is analogous to the passage of a flat-sided ship that makes no bow waves; the length along the hull is equivalent to the total distance traveled by the belt. Belt speeds ranging from 3 to 5 m/s were used and measurements were continued until a specified belt length had passed by the measurement site. The time to accelerate varies depending on the final belt speeds and independent measurements of the belt travel show that during launch the belt travels 0.85, 1.45, and 2.29 m at belt speeds of 3, 4, and 5 m/s, respectively. [cc]{} A cinematic Laser Induced Fluorescence (LIF) technique, see Figure \[fig:LIFSchem\], was used to measure the temporally evolving water surface deformation pattern. In this technique, a continuous-wave Argon Ion laser beam is converted to a thin sheet using a system of spherical and cylindrical lenses. This sheet is projected vertically down onto the water surface in two orientations; one with the plane of the light sheet parallel to the plane of the belt and one with the light sheet perpendicular to the belt. The laser emits light primarily at wavelengths of 488 nm and 512 nm. The water in the tank is mixed with fluorescein dye at a concentration of about 5 ppm and dye within the light sheet fluoresces. High-speed cameras viewed the intersection of the light sheet and the water surface from the side with viewing angles of approximately 20 degrees from horizontal. The images seen by the cameras show a sharp line at the intersection of the light sheet with the free surface. Using image processing, instantaneous surface profiles are extracted from these images. The present research was focused on preliminary measurements of bubbles under the above-described experimental conditions. In these measurements, a single camera viewed the boundary layer region of the flow from underwater as shown in Figure \[fig:bubble\_setup\_planar\]. A sample image from this setup is shown in Figure \[fig:BubblePhotos\]. Analysis of the images allows for quantitative measurement of the bubble diameters, their two-dimensional positions, and trajectories for radii ranging down to 0.5 mm. In future experiments, more accurate measurements of bubbles are planned using cinematic stereo photography and cinematic inline holography. In the stereo measurements, the cameras and lights are mounted in underwater boxes close to the water free surface and the surface of the belt. Each box contains a camera, that is mounted on a Scheimpflug mount and oriented so that the camera looks down at a mirror that turns the camera’s line of sight to horizontal. The lines of sight of the two cameras are oriented to view the belt at $\pm45$ degrees from the normal to the belt surface. Both cameras are calibrated and focused to look at the same portion of the belt. The system is calibrated through images of a known 3D target and yields the 3D positions and equivalent diameters of the bubbles in any image pair. Illumination is provided by LED light sources that are placed in each underwater box. In future experiments we are also planning to use a digital inline holography system to measure the size, velocity, and position of bubbles down to a radius of 20 $\mu$m. The experimental setup consists of a camera fitted with a long distance microfocus lens, oriented vertically next to the belt, looking down into a dry box, as seen in Figure \[fig:bubble\_holography\]. The dry box is always partially submerged so as to negate the light distorting properties of the rough water free surface. A collimated laser beam from a pulsed ND:YLF laser is directed upwards from the bottom of the tank into the camera lens and sensor. When a bubble is in the path of the collimated laser beam a hologram is recorded by the camera sensor. This hologram can then be reconstructed digitally and the size and three-dimensional position of the bubble can be measured. Once the size and location of the bubbles is obtained, the bubbles are tracked in time and their velocities can be obtained from the resulting trajectories. This system has been successfully implemented in our laboratory to measure droplets generated by breaking waves [@Erinin2017]. An example of a hologram from these droplet measurements is shown in Figure \[fig:droplet\_hologram\_example\]. COMPUTATIONAL SETUP =================== The computational simulations were designed to mimic the conditions in the experiments. The main challenges for the computations are to properly resolve the boundary layer along the moving belt and, at the same time, capture the complex free-surface deformations and air-entrainment phenomena. We only simulate a small part of the moving belt and apply periodic boundary conditions along the direction of motion of the belt. A schematic of a typical computational box is shown in Figure \[fig:domain\], where a portion of the air above the free-surface is also considered. We define the Cartesian domain $(x,y,z)$ in such way that $x$ is the streamwise direction, making $y-z$ the cross-stream plane. The moving wall is located at $y=0$, the undisturbed free surface is at $z=0$ (parallel to $x-y$ plane) and gravity is imposed in the $-z$ direction. The Navier-Stokes equations for incompressible flow are solved in both the air and water portions of the domain and the interface is implicitly advected and tracked using a geometric reconstruction approach [@QIN2015219]. The governing equations are solved on a block-structured Cartesian grid with Adaptive Mesh Refinement (AMR) [@Vanella2010JCP; @Vanella2014]. AMR allows us to cluster grid points at the dynamically evolving interface, as well as the boundary layer in a cost-efficient manner. The equations are advanced in time using an exact projection method. All spatial derivatives are discretized using second-order, central finite-differences. The jump conditions at the interface are imposed in a sharp manner using a variant of the ghost-fluid method [@FEDKIW1999457]. Details on the overall formulation together with a detailed validation in a series of problems of increasing complexity can be found in @DelaneyPhD2014. In the experiments, the belt starts from rest and quickly reaches its terminal speed. The boundary layer, undergoes transition and gradually thickens as a function of time. The critical Froude number based on the local momentum thickness, $\theta$, and the belt velocity when air entrainment is initiated is $Fr \sim O(10)$. The corresponding Reynolds number at this time instant is $Re_\theta\sim O(10^4)$. Simulating this process starting from the belt at rest and arriving to post-entrainment Reynolds and Froude number has the advantage of well defined initial and boundary conditions but it is prohibitively expensive even on leadership parallel computing platforms. Due to this limitation, and in order to keep the computational cost at reasonable levels, we considered significantly lower Reynolds numbers, but kept the Froude number in the same regime as in the experiments, where high deformations of the free-surface are observed, leading to air-entrainment. This is a significant advantage of the simulations where we can independently change the surface tension and gravity to replicate the conditions in the air entrainment regime in the experiment at lower Reynolds numbers. In particular we consider the boundary layer to evolve from $Re_\theta=900$ to $1400$ and will discuss two Froude numbers: $Fr=4$ and $Fr=12$, for the same Reynolds number, $Re_\theta=1400$. As an additional cost reduction, we started from the fully turbulent regime, bypassing the transition phase. Despite these approximations, which only enable qualitative comparisons to the experimental results, the computations are well positioned to quantify the effects of the Froude number on the flow physics. [figure9.pdf]{} (80,6) (41,90) (8,25) In addition to these considerations the grid was dynamically refined to capture the dynamics of the triple contact point, while the resolution at any location in the domain containing the interface was kept at the highest refinement level. We use periodic boundary conditions in the streamwise direction. At the moving wall the impermeability condition is enforced and the velocity in the streamwise direction is set to the reference value. We impose a Sommerfeld radiation condition on the boundary opposite the moving wall in order to convect the surface waves out of the domain. The convective velocity is calculated using the average water phase wall-normal velocity at the boundary. Details can be found in @DelaneyPhD2014. Slip-wall conditions were used at the two remaining boundaries. The domain dimensions were driven by the maximum Reynolds number we wanted to achieve, and were selected based on prior computations of turbulent boundary layers in the literature. In all computations, we define the *midsection* as the depth range where the effects of the free-surface are not felt and the velocity statistics are identical to the ones in a zero pressure gradient boundary layer. Also, unless otherwise stated, the midsection quantities used to normalize the results are taken from the flow field at $Re_\theta=1400$. [figure10a.pdf]{} (20,70) (48,3) [figure10b.pdf]{} (80,70) (48,2) Quantitative comparisons at the midsection to reference data in the literature are shown in Figure \[fig:valid\]. The velocity statistics at $Re_\theta=1400$ are compared to the DNS by @Spalart1988. The agreement for both the mean velocity and the turbulent intensities is excellent. The numerical investigation is primarily deployed to study the mechanisms of air entrainment due to the turbulence field beneath the surface. In the remainder of this study, we report on the numerical findings where comparison to experiments is possible, giving us greater confidence in the turbulent entrainment analysis. RESULTS AND DISCUSSION ====================== Surface Profiles ---------------- Surface profile measurements were performed at belt speeds of $U=3$, $4$, and $5$ m/s using the cinematic Laser Induced Fluorescence (LIF) technique. Through initial trials, it was determined that a frame rate of 1000 fps was necessary to provide a sufficient temporal resolution so that surface features perpendicular to the belt could be identified and tracked smoothly in successive frames. LIF images of the water free surface with the light sheet oriented perpendicular to the belt in an experimental run with $U$ = 5 m/s are shown in Figure \[fig:overall\]. The five images in the figure are spaced out equally by distance of belt travel, with the first image (a) taken at 0.0 s, the time when the belt first starts to move. The instantaneous belt speed from the beginning of belt motion through the acceleration portion until the belt reaches constant speed has been measured separately and is used to correlate the time of each frame to the belt travel distance. Here and in the following, rather than refer to images and data by the time after the belt has started moving, we refer to them by the distance, $x$, from the leading edge of an equivalent flat plate, which is also the distance that the belt has traveled $$x = \int_0^t U(t^\prime )dt^\prime,$$ where $t=0$ is the instant that the belt starts to move. This integral is performed numerically with the measured function $U(t)$, which includes an initial phase of nearly constant acceleration, i.e., $dU/dt\approx$ a constant, followed by a longer period of constant speed, i.e., $U = $ a constant. Thus, the images in Figure \[fig:overall\] depict a portion of a run, with images (a), (b), (c), (d) and (e) captured at 0 s, 1.35 s, 2.35 s, 3.35 s, and 4.35 s, respectively, corresponding to $x=$ 0.0, 5.0, 10.0, 15.0 and 20.0 m. In this subsection, we will quantitatively examine the free surface profiles parallel and perpendicular to the belt surface for $U= 5$ m/s and report some observed entrainment events from the parallel free surface profiles. Then we will look at processed free surface profiles perpendicular and parallel to the belt at belt speed $3$ m/s and compare them to the computational results qualitatively. Finally, the free surface height from the experimental results is analyzed qualitatively for $3$, $4$, and $5$ m/s. [c]{}\ (a)\ \ (b)\ \ (c)\ \ (d)\ \ (e)\ [c]{}\ (a)\ \ (b)\ \ (c)\ \ (d)\ \ (e)\ As discussed in the experimental details section, in our previously reported measurements [@Washuta2016] the plane of the vertical light sheet was oriented normal to the belt surface and two cameras, from upstream and downstream, looked down at an angle of approximately 20 degrees at the intersection of the light sheet and the free surface. The images in Figure \[fig:overall\] are from the downstream camera in one of these wall-normal water surface profile measurements, and these images have been flipped horizontally for convenience in order to match the coordinate system of later plots, so that the belt is near the left side of each image and is moving out of the page. The position of the belt is marked on the left side of image (a) and the intensity pattern to the left of this location is a reflection of the light pattern on the right. This line of symmetry gives a good indication of the position of the belt in each image. The sharp boundary between the upper dark and lower bright region of each image is the intersection of the light sheet and the water surface. The upper regions of the later images contain light scattered from roughness features on the water surface behind the light sheet. These roughness features include bubbles that appear to be floating on the water surface and moving primarily in the direction of the belt motion. The bright area below the boundary is created by the glowing fluorescent in the underwater portion of the light sheet. The complex light intensity pattern here is created by a combination of the refraction of the laser light sheet as it passes down through the water surface and the refraction of the light from the glowing underwater dye as the light passes up through the water surface between the light sheet and the camera, on its way to the lens. It can be seen from these images that surface height fluctuations (ripples) are created close to the belt surface, at the left side of each image, and propagate away from the belt (to the right). As time passes, the surface height fluctuations grow dramatically and eventually surface breaking and air entrainment events begin to occur, resulting in bubble and droplet production. From these images, it is observed that the free surface remains nearly quiescent during a period of belt travel at the beginning of each run; during this time period, the LIF images appear similar to what is seen in Figure \[fig:overall\] (a). After a short time, the surface suddenly bursts with activity near the belt surface, creating free surface ripples. After this point, see Figure \[fig:overall\] (b), the free surface fluctuations are continually generated close to the belt and this generation region grows in time. As the belt travel length continues to increase, free surface ripples begin to appear to the right side of the image, away from the belt, see Figure \[fig:overall\] (c-d). Qualitatively, from looking at Figure \[fig:overall\] (c-d) it is evident that the free surface ripples are most intense closest to the belt and decay in intensity as they move away from the belt. In more recent experiments with the light sheet perpendicular to the belt, the two cameras were both placed downstream in a side-by-side configuration with an overlap in their fields of view. By combining the profiles from the two cameras, a higher resolution was achieved (approximately 15 pixels/mm) while viewing a similar horizontal distance away from the belt (approximately 30 cm). Results shown in Figures \[fig:surface\_profiles\] (a) and Figure \[fig:rms\_height\] are obtained using this new configuration. A corresponding set of images with the plane of the light sheet parallel to the belt are shown in Figure \[fig:LighSheetParallel\]. In these images the laser light sheet was located at a distance of approximately $y = 1.25$ cm away from the belt surface and the camera is looking toward the belt and down at the free surface at a small angle from horizontal. The belt is in the black background traveling from left to right in the series of images shown in the figure. As in the images of the surface profiles perpendicular to the belt, shown in Figure \[fig:overall\], the sharp boundary in the images in Figure \[fig:LighSheetParallel\] is the intersection of the light sheet with the water surface. The images in Figure \[fig:LighSheetParallel\] (a-e) correspond the the same lengths of belt travel as the images in Figure \[fig:overall\]. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13a.jpg "fig:"){height=".43in"} ![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13f.jpg "fig:"){height=".43in"} (a) (f) ![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13b.jpg "fig:"){height=".43in"} ![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13g.jpg "fig:"){height=".43in"} (b) (g) ![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13c.jpg "fig:"){height=".43in"} ![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13h.jpg "fig:"){height=".43in"} (c) (h) ![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13d.jpg "fig:"){height=".43in"} ![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13i.jpg "fig:"){height=".43in"} (d) (i) ![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13e.jpg "fig:"){height=".43in"} ![Two series of images capturing entrainment events parallel to the motion of the belt $y = 1.25$ cm away from the belt at a belt speed of $5$ m/s with the belt moving from the left to the right. Images (a) through (e) correspond to $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. In this series of images we see a jet of water, indicated by an arrow in image (b), overtake a surface depression, indicated by an arroaw in image (a), the jet eventually overtakes the air cavity below (c) and entrains a pocket of air in the process (d-e). In images (f-j), corresponding to belt travel of $15.71$ m, $15.73$ m, $15.75$ m, $15.78$ m, and $15.80$ m respectively, we see a similar event where a water jet overtakes a cavity in the free surface, presumably entraining air into the water below. The surface depression is indicated by an arrow in image (f) while the jet is indicated by the arrow in image (g). Both of these series of images evolve on the order of tens of milliseconds[]{data-label="fig:parallel_entrainment_events"}](figure13j.jpg "fig:"){height=".43in"} (e) (j) ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ From watching the movies of the free surface profiles parallel to the belt, it is clear that the free surface experiences a sudden burst of activity somewhere between images (a) and (b) in Figure \[fig:LighSheetParallel\]. This burst of activity on the free surface is associated with the growing size of the turbulent boundary layer in the water. Once the free surface become rough, wave-like breaking events can be observed moving from left to right (in the same direction as the motion of the belt) in the midst of other surface features moving parallel and perpendicular to the light sheet. These wave-like breaking events, presented in Figure \[fig:LighSheetParallel\] (b - d), are persistent and can be observed frequently once the free surface becomes rough. Sometimes, the above-mentioned breaking events appear to entrap pockets of air into the water below. Two of these breaking events are shown in the images in Figure \[fig:parallel\_entrainment\_events\], which were taken from a wall parallel LIF movie for a belt speed of 5 m/s and with the light sheet 1.25 cm from the belt surface. The non-uniformity of the light intensity at the free surface is partially due to the curvature of the free surface, which reflects laser light and focuses it underneath the surface, and partly because water has obstructed the laser light from reaching the areas below. In Figure \[fig:parallel\_entrainment\_events\] (a) to (e) we see a a sequence of images taken at $12.98$ m, $13.03$ m, $13.07$ m, $13.10$ m, and $13.13$ m of belt travel. To the left in Figure \[fig:parallel\_entrainment\_events\] (a) we see a jet forming with an air cavity directly below. The jet then proceeds to plunge forward and above the air cavity, from left to right in the direction of the belt motion, in images (b-c). The jet then splashes on the free surface and closes off the pocket of air in (d) . Finally, air is presumably entrained in the flow by image (e) and the jet is no longer present on the free surface. The time span between images (a) through (e) is 30 milliseconds. Images (f-j) in Figure \[fig:parallel\_entrainment\_events\] tell a similar story. A jet of water is moving from left to right over a cavity in (f). The jet proceeds to overtake the air cavity in (g-h) and the jet splashes on the free surface in (i-j), again, presumably entraining air. The sequence of images in (f-j) takes place over a time span of 18 milliseconds. There are many similar wave-like breaking events in the movies of the free surface profiles parallel to the belt surface. ----- ----- ----- (a) (b) (c) ----- ----- ----- In addition to qualitative observations of free surface motions, quantitative surface profiles can be extracted from each frame of the LIF movies through the use of gradient-based image processing techniques. Figure \[fig:surface\_profiles\] shows an example of the surface profiles extracted from LIF images using image processing in MATLAB for a belt speed of $3$ m/s. In these plots the horizontal axis is the horizontal distance in each set of movies. The surface profiles in Figure \[fig:surface\_profiles\] (a) come from movies of the laser light sheet perpendicular to the belt, similar to the images show in Figure \[fig:overall\], hence the horizontal axis is in the y direction with the belt located at $y = 0$ mm. The surface profiles in Figure \[fig:surface\_profiles\] (b) and (c) come from movies of the surface profile parallel to the belt at two different distances from the belt (b, $y = 1.25$ cm and c, $y = 2.5$ cm), similar to the surface profile images shown in Figure \[fig:LighSheetParallel\]; hence the horizontal axis is in the $x$ direction. The profiles are spaced in time by $4$ ms in (a) and by $1$ ms in (b) and (c), Each new surface profile is shifted 1 mm up from the previous profile to reduce overlap and show the propagation of surface features through time. The earliest profile in time is shown at the bottom. Using this plotting technique surface features like ripple crests can be tracked over a number of successive profiles and the slopes of imaginary lines connecting these features indicates their horizontal speed. Analyzing \[fig:surface\_profiles\] (a) we can estimate the speed of surface features moving away from the belt at a speed of $0.34$ m/s, which is much lower than the belt speed of $3$ m/s. It should be noted that there is a constant train of surface features propagating outwards in plot (a). Plot (b) shows the parallel surface profiles at $y = 1.25$ cm away from the belt. The location of the light sheet is shown as a blue dashed line in plot (a). Surface features propagating along the direction of the belt can be seen and their speed is estimated to be around $1$ m/s in the $x$ direction. Similarly, plot (c) shows parallel surface profiles at $y = 2.5$ cm away from the belt (its location shown in a red dashed line on plot (a)), with surface features speed estimated to be about $0.75$ m/s. Theses surface feature speed estimates from plots (b) and (c) are taken fairly close to the surface of the belt, yet their speeds are significantly less than the belt speed of 3 m/s. It’s interesting to note that surface features traveling parallel to the belt ($1$ m/s), measured at a distance of $y = 1.25$ cm away from the belt, travel about three times faster than features moving away from the belt ($0.34$ m/s). It should be kept in mind these velocity estimates are the $y$ or $x$ components of the phase speed. Comparison of computational results to experiments can be made by considering a succession of surface profiles at the mid-streamwise location of the numerical domain for $Fr=12$ (Figure \[fig:instant\_DNS\_profiles\]), analogous to the experimental data in figure \[fig:surface\_profiles\]. The profiles are plotted in the same manner in both figures. In Figure \[fig:instant\_DNS\_profiles\], the lowermost profile corresponds to $Re_\theta=900$ and the uppermost profile corresponds to $Re_\theta=1400$. The surface disturbances have greater amplitude closer to the moving wall as compared to the outer regions, in agreement with the experiments. In the immediate vicinity of the moving wall ($0<y/\delta<0.25$) the disturbances appear to be uncorrelated and persist for only a few profiles, suggesting that in this region waves are heavily influenced by the underlying turbulent boundary layer flow. In the regions away from the wall however, the waves persist for much longer periods and maintain their shape, similar to the results in the experiments as discussed in the previous paragraph. The straight black lines track the crests of a few of the outer region waves. This shows that the propagation speed of these waves is very nearly constant and that the waves exhibit the behavior of freely moving waves. On average the propagation speed normalized by wall speed is 0.08 which is on the same range as the experiments (the corresponding experimental value for a belt speed of 3 m/s is 0.11). Overall from a qualitative point of view, the computations are in agreement with the experimental results and capture the surface dynamics of the two-phase turbulent boundary layer. [figure15.png]{} (30,1) The profile data from the experiments discussed above was used to obtain distributions of the root-mean-square (RMS) water surface height fluctuation as a function of time and space dimensions. The RMS height at any $x$ or $y$ location is obtained as the square root of the average of the squares of the differences between the height and the average height, where the average is taken over the run time and over all experimental runs with the same belt speed. Figure \[fig:rms\_height\] (a) is a plot of the RMS free surface height versus $y$ at belt speeds $U = $5, 4, and 3 m/s averaged over 20 runs for each speed. The belt is located at $y = 0$ mm. The RMS height reaches a maximum near the belt region where the free surface fluctuations were visibly the most violent as seen in Figure \[fig:surface\_profiles\] (a). Further away from the belt the free surface RMS fluctuations decay for all three belt speeds. Figure \[fig:rms\_height\] (b) shows the RMS height, averaged over all $y$, versus time for all three belt speeds. The belt starts to move at $t = 0$ s, however the RMS height for all three speeds does not change until a little bit before $t = 1$ s. The RMS height then increases at similar constant rate for all three belt speeds until about $t = 1.5$ s when the three curves start to diverge. After approximately $t = 5$ s, all three height RMS curves reach a constant value. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ (a) ![RMS of surface height fluctuations for light sheet perpendicular to the belt. (a) Height RMS in time versus distance from the belt for speeds of $U$ = 3, 4, and 5 m/s, and (b) RMS surface height averaged over $0\leq y \leq 30$ cm versus time for the same three speeds. Each curve is an ensemble average over 20 identical runs for each belt velocity.[]{data-label="fig:rms_height"}](figure16a.pdf "fig:"){width="3in"} (b) ![RMS of surface height fluctuations for light sheet perpendicular to the belt. (a) Height RMS in time versus distance from the belt for speeds of $U$ = 3, 4, and 5 m/s, and (b) RMS surface height averaged over $0\leq y \leq 30$ cm versus time for the same three speeds. Each curve is an ensemble average over 20 identical runs for each belt velocity.[]{data-label="fig:rms_height"}](figure16b.pdf "fig:"){width="3in"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ A similar analysis was carried out on the surface profiles parallel to the belt. Figure \[fig:rms\_height\_parallel\] shows the RMS height versus $x$ for belt speeds of $U =$ 3, 4, and 5 m/s at two different distances from the belt for each speed. The RMS height varies with $x$ in a random manner with a relatively small amplitude; it is thought that this variation would decrease with increasing numbers of runs. In agreement with Figure \[fig:rms\_height\] (a), the data in Figure \[fig:rms\_height\_parallel\] indicates a strong increase in the RMS height with belt speed and little change (except for the $U=$ 3 m/s case) between the two measurement locations, $y = $ 12.5 and 25 mm. ![RMS of surface height fluctuations in time versus x for light sheet parallel to the belt at $y = 12.5$ and $25.0$ mm away from the belt surface.[]{data-label="fig:rms_height_parallel"}](figure17.pdf "fig:"){width="3in"}\ Bubble Statistics ----------------- In this section, preliminary estimates of bubble statistics are reported for a belt speed of 5 m/s. A single camera set up, as shown in Figure \[fig:bubble\_setup\_planar\], was used to record images of entrained bubbles as the belt starts from rest and travels a distance of $24.88$ m. The camera has a field of view of 47 by 47 mm at the plane of the belt surface and a resolution of 67 $\mu$m per pixel. The belt is illuminated with diffuse white light and the movies are recorded at 1,000 fps with a total of 5,000 images taken for each run. The resulting series of images were processed using a MATLAB code which identifies all the bubbles in each image. Bubbles are measured down to a diameter of  0.5 mm. With the lens f-number used in the measurements, bubbles are in focus over a horizontal distance of about 35 mm from the belt surface along the line of sight of the camera lens and this region contains all the bubbles present in the imaged region of the flow. Since most of the larger bubbles are not spherical, an equivalent radius is calculated based on the two-dimensional projected shape imaged by the camera. Each bubble is then subsequently tracked based on the series of frames in which it appears and its two-dimensional position and average equivalent radius are recorded. Figure \[fig:prob\_dist\_bubbles\_5mps\] is a log-log plot of the experimentally measured number of bubbles per radius bin width versus bubble radius for a belt speed of 5 m/s. The sample population is all of the bubbles that passed through the upstream side of the measurement region during the belt travel of 24.88 m. A uniform bin spacing of $dr = $ 0.118 mm is used and the centers of the bins range from $ r = $ 0.369 to 3.669 mm. Separate linear regions are observed for small-diameter and large-diameter bubbles. The two linear regions are fitted separately using linear regression to a function of the form $Ar^{\alpha}$ for the region of smaller bubbles and $Br^{\beta}$ for the region of larger bubbles. These functions plot as straight lines in Figure \[fig:prob\_dist\_bubbles\_5mps\] and the optimum position for the break in slope between the two regions was determined by an iterative bisection-like routine outlined as follows: First, an initial guess, $r_0$, is estimated as the radius where the break in slope is to occur, the data is split into two distinct sets and a power law, of the form described above, is fitted to each set. The intersection, $r_i$, of the two fitted lines is then found. If the difference between $r_0$ and $r_i$ does not fall within a specific tolerance, a new guess for $r_0$ between the previous values of $r_0$ and $r_i$ is assigned and the processes is repeated until the tolerance is reached. Using this method with an initial guess of $r_0 = $ 1.3 mm, the break in slope is estimated to be approximately $r_i = $ 1.265 mm. The break in slope in the bubble size distribution has long been observed and identified as the Hinze scale, see for example the work of [@Deane2002] on bubble size distributions in breaking waves. Generally speaking, the Hinze scale implies that different physical mechanisms influence the two different sides of the bubble size spectrum. Dean and Stokes suggest that bubbles that were larger than the Hinze scale were fragmented by turbulent flow with a -10/3 power-law scaling, while bubbles smaller than the Hinze scale are stabilized by surface tension and show a -3/2 power-law scaling with the radius [@Deane2002]. The Hinze scale is defined as $$r_H=2^{-8/5}\epsilon^{-2/5}(\sigma We_c/\rho)^{3/5}$$ where $\epsilon$ is the turbulent dissipation rate and $We_c$ is the critical Weber number and typically takes on a value of 4.7 (see for example [@Deane2002]). \ In our numerical simulations we found that $r_H /\delta \approx3.5\times10^{-3} $. The largest bubbles in our simulations have a radius of about $\delta/20 \sim 0.16$ with 25-30 computational points across the bubble whereas the smallest observed bubbles have a radius of about $\delta/160\sim 0.02$ with 3-4 points across. The latter is an order of magnitude greater than the Hinze scale. Further refinement in the numerical resolution to capture smaller bubbles is out of the scope of the present DNS, which focused on the primary entrainment events as a result of the turbulent boundary layer interacting with the free surface. Figure \[fig:NvsRad\] shows the number of observed bubbles against bubble radius from the DNS simulation. The radius of a bubble is calculated by considering the equivalent spherical bubble with the same volume. The vertical axis has been normalized with the total number of observations and the radius has been normalized by the largest radius in the data set. A line with a slope of $-10/3$ is plotted to the top right of the data for reference. The scaling agrees fairly well with the $-10/3$ law and shows further qualitative agreement to the experiments (Figure \[fig:prob\_dist\_bubbles\_5mps\]). ![Relative bubble population versus bubble radius from a DNS simulation of the problem. Horizontal axis has been normalized by the maximum bubble radius and the vertical axis by the total number of bubbles observed. The bubble radius is the radius of a spherical bubble with the same volume as the irregularly shaped bubble in the calculations.[]{data-label="fig:NvsRad"}](figure19.pdf){width="\linewidth"} Figure \[fig:num\_of\_bubbles\_vs\_z\_5mps\] shows the mean number of unique bubbles vs. depth measured in the experiments for a belt travel of $24.88$ m at a belt speed of 5 m/s. Before the launch of the belt, the calm free surface is positioned at $z = $ 0 mm. Once the belt is launched, the free surface fluctuates dramatically in the $z$ direction, making it difficult to measure bubbles close to $z = $ 0 mm. Generally, the free surface does not fluctuate more than 15 mm below it’s original depth, which is the reason why the measurements presented start at $z \approx 14$ mm. Bubbles are tracked in the series of images in which they appear, the average depth of the bubble is obtained by averaging the $z$ position over all the tracked particle trajectories. The $z$ direction is divided from $z = -14.2$ mm to $z = -45.3$ mm by increments of $dz = 1.072$ mm. Each bubble’s mean $z$ position is then placed into each appropriate bin. From Figure \[fig:num\_of\_bubbles\_vs\_z\_5mps\], it can be seen that the number of bubbles slowly decreases from a few hundred bubbles in the area around $z = 15$ mm to tens of bubbles near $z = 50$ mm indicating that the majority of these large bubbles in the boundary layer tend to stay near the surface. \ Figure \[fig:numBubbleVsDepth\] shows depth of observed bubbles in DNS against their relative population and is qualitatively analogous to figure experimental data in \[fig:num\_of\_bubbles\_vs\_z\_5mps\]. The depth of the bubbles has been normalized by the average bubble radius and has been broken up into twenty equally sized bins. The overall trend is similar to that of the experiment where the majority of the bubbles are found closer to the free surface. It must be noted that the two plots are not directly comparable given the limitations explained earlier ![Relative bubble population with respect to depth (DNS). Depth has been normalized by average bubble radius.[]{data-label="fig:numBubbleVsDepth"}](figure21.pdf){width="\linewidth"} Given that most of the large bubbles reside near the free surface, it may be of interest to look at the average equivalent bubble radius versus depth. Figure \[fig:mean\_bubble\_radius\_vs\_z\_5mps\] shows the experimentally measured average bubble radius versus depth from $z = -13.68$ mm to $z = -45.84$ mm in increments of $dz = 2.14$ mm. The average depth of each unique bubble is calculated, in a similar way as described in the previous paragraph, and the bubble is placed into the appropriate $z$ bin. Once all bubbles are assigned the the proper bin, the average radius of the bubbles in each bin is calculated. It should be noted that data points close to the free surface are averaged over significantly more bubbles than ones farther away, perhaps accounting for the noisier data at greater depths. The average bubble radius increases by about 0.4 mm (about 30%) from the deepest and shallowest measurement positions. \ Finally, from the tracked bubble trajectories, we can fit a second order polynomial to estimate the speed of the bubbles as they enter the camera’s field of view. Second order polynomials were fitted to the $x$ and $z$ positions versus time data for each unique bubble. Then the $u$ and $w$ components of velocity were computed for each bubble as it entered the upstream side of the camera’s field of view. The speed was calculated as $|\vec{u}| = \sqrt{u^2+w^2}$. The results of these calculations, including the bubble speeds and the values of the $u$ and $w$ velocity components, are shown in Figure \[fig:mean\_bubble\_speed\_vs\_z\_5mps\]. It is interesting to note that the mean bubble speed does not seem to change dramatically over the range of depths in which the measurements were taken. The $u$ component of velocity is 3 to 4 times larger then the $w$ component. \ Analysis of turbulent structures -------------------------------- In this section we will discuss the mechanisms of air entrainment in the context of the numerical simulations. In general, there are three different entrainment mechanism. Water droplets, for example, can break off ligaments and entrain air upon impact with the free-surface (see Figure \[fig:ent\_mech\]a). This type of air entrainment has been studied extensively primarily in simplified configurations [@Esmailizadeh1986; @Oguz1990; @Hasan1990; @TOMITA2007; @Ray2015; @Hendrix2016]. In such case, a droplet falling towards the free surface traps air between it and the water surface. A crater forms on the surface and upon impact of the droplet, the air inside the crater is entrapped. In the numerical study, about 12% of the air entrainment incidents are from surface impact. Alternatively, entrainment is also caused by turbulent motions underneath the surface. There are two types of vortices that result in entrainment and are distinguished by their orientation with respect to the surface. The first are the vortices that are mainly oriented parallel to the free surface (Figure \[fig:ent\_mech\]b). The second are those that are perpendicular to the free surface (Figure \[fig:ent\_mech\]c). In our numerical study, we found that the latter type of vortices are rare ($< 1\%$) and most of the turbulent entrainment comes from the former type ($\sim 88\%$). A small portion of the entraining vortices lie between the two where the orientation with respect to the free surface is not clear. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Proposed air entrainment mechanism schematics, a breaking wave (a), vortices oriented parallel to the free surface (b), and vortices oriented perpendicular to the free surface (c). (d-f) shows an example sequence of a single bubble being entrained from the DNS results.[]{data-label="fig:ent_mech"}](figure24a.png "fig:"){height=".65in"} ![Proposed air entrainment mechanism schematics, a breaking wave (a), vortices oriented parallel to the free surface (b), and vortices oriented perpendicular to the free surface (c). (d-f) shows an example sequence of a single bubble being entrained from the DNS results.[]{data-label="fig:ent_mech"}](figure24d.pdf "fig:"){height=".65in"} (a) (d) ![Proposed air entrainment mechanism schematics, a breaking wave (a), vortices oriented parallel to the free surface (b), and vortices oriented perpendicular to the free surface (c). (d-f) shows an example sequence of a single bubble being entrained from the DNS results.[]{data-label="fig:ent_mech"}](figure24b.png "fig:"){height=".65in"} ![Proposed air entrainment mechanism schematics, a breaking wave (a), vortices oriented parallel to the free surface (b), and vortices oriented perpendicular to the free surface (c). (d-f) shows an example sequence of a single bubble being entrained from the DNS results.[]{data-label="fig:ent_mech"}](figure24e.pdf "fig:"){height=".65in"} (b) (e) ![Proposed air entrainment mechanism schematics, a breaking wave (a), vortices oriented parallel to the free surface (b), and vortices oriented perpendicular to the free surface (c). (d-f) shows an example sequence of a single bubble being entrained from the DNS results.[]{data-label="fig:ent_mech"}](figure24c.png "fig:"){height=".65in"} ![Proposed air entrainment mechanism schematics, a breaking wave (a), vortices oriented parallel to the free surface (b), and vortices oriented perpendicular to the free surface (c). (d-f) shows an example sequence of a single bubble being entrained from the DNS results.[]{data-label="fig:ent_mech"}](figure24f.pdf "fig:"){height=".65in"} (c) (f) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The typical process of this entrainment regime begins with the interface being pulled into the flow beneath, creating a “neck” where the ligament attaches to the free surface (Figure \[fig:ent\_mech\] d). The neck continues to narrow until it breaks and an air bubble is released into the flow. In the numerical studies, we focus our attention on bubble generation and will not discuss the fate of the bubbles later on. We can examine turbulent air entrainment by considering the local turbulent eddies using the vortex identification scheme introduced by @Hunt1988 known as the Q-criterion. Any spatial point where the Eulerian norm of the vorticity tensor dominates that of the rate of strain is designated as being part of the vortical structure. Figure \[fig:all\_vorts\] shows the vortical structures in the vicinity of an entrainment event. The vortex core closest to the neck of the air ligament is identified in the figure. The low pressure center of the vortex has pulled in the interface and is narrowing the neck. [figure25.pdf]{} (75,48)[(-1,1)[20]{}]{} (50,40) The Q-criterion can be useful in identifying the vortex cores but is not a reliable method for establishing the size of the vortices and hence their local length scale. We employ the following 8 step procedure to identify and quantify the local scale of vortices that entrain air.\ **Step 1:** Identify an entrainment event.\ **Step 2:** Visually identify the vortex responsible for entrainment by setting the appropriate value for $Q_c$ (Figure \[fig:vortexQuantMethod\]a).\ **Step 3:** Define the centerline of the vortex by the line of minimum pressure along the length of the vortex (Figure \[fig:vortexQuantMethod\]b).\ **Step 4:** Create planes perpendicular to and along the spine (Figure \[fig:vortexQuantMethod\]c) and calculate the vorticity magnitude on the planes.\ **Step 5:** For each plane, identify the isoline for $\omega_c$.\ **Step 6:** For each plane calculate the circulation within the area designated by the $\omega_c$ isoline. Define $\Gamma$ as the average of circulation across all planes.\ **Step 7:** Find the maximum distance from the vortex center to points of the $\omega_c$ isoline for each plane ($r$). The average of $r$ across all planes as the average radius of the vortex $R$ which is also the local length scale of the vortex.\ **Step 8:** Define local Reynolds, Froude and Weber numbers as $Re_l=\frac{\Gamma}{\nu}$, $Fr_l=\frac{\Gamma}{\sqrt{gR^3}}$ and $We_l=\frac{\rho\Gamma^3}{\sigma R}$, respectively. \ These three dimensionless groups along with the distance of the vortex from the free surface are the independent variables of turbulent entrainment. Owing to the small value of surface tension, the typical local Weber numbers are very large (O($10^4$)) compared to the other three variables and will not be discussed further. We also non-dimensionalized the distance of the vortex from the surface $d$ using the radius of the vortex. As a result we have a three-dimensional parametric space $(d/r,Re_l,Fr_l)$ for all the entraining and non-entraining vortices. By placing these vortices on the parametric space, we seek to find the dominant factors separating the two types of vortices. We gathered data for 447 entraining and 450 non-entraining vortices for all computations at the two aforementioned Froude numbers. Figure \[fig:parameterSpace\]a shows the scatter plot of normalized distance against local Froude number for both entraining and non-entraining vortices. Most of the entraining vortices are clustered in the corner with $Fr_l<50$ and their population thins out as Froude number is increased. Non-entraining vortices however are spread much more uniformly with respect to the Froude number. For $d/r>6$ only a few entraining vortices exist whereas non-entraining vortices are numerous. The opposite is true for $d/r<6$ indicating that a vortex is not able to entrain air if it sits more than six times its own radius below the surface. This is irrespective of Froude number or the Reynolds number as can be seen in Figure \[fig:parameterSpace\]b. The scatter plot of Reynolds number against Froude number is depicted in \[fig:parameterSpace\]c. There is a significant overlap of entraining and non-entraining vortices and therefore the $(Re_l,Fr_l)$ pair cannot predict entrainment. It is worth noting that most of the entraining vortices collocate in the corner of the plot and tend to assume smaller values of $Re_l$. [figure27a.pdf]{} (15,70) [figure27b.pdf]{} (15,70) [figure27c.pdf]{} (20,70) Figure \[fig:pdf\]a depicts the probability density function of the normalized distance $d/r$. The entraining vortices are closer to the surface with a mean of 3.01 and standard deviation of 1.55. The non-entraining vortices have a mean equal to 9.85 with a standard deviation of 3.25. The two density functions cross at $(5.45,0.05)$ and do not show significant overlap. The value $d/r=5.45$ seems reasonable as the critical distance for entrainment. [figure28a.pdf]{} (78,70) [figure28b.pdf]{} (78,70) [figure28c.pdf]{} (78,70) Figure \[fig:pdf\]b shows the probability density function of the local Reynolds number $Re_l$. The two profiles cross at $Re_l\approx 200$ and there is significant overlap of the two distributions for Reynolds numbers smaller than this value. However, for Reynolds numbers larger than 200, the overlap is very small in comparison and the probability of finding entraining vortices drops dramatically, much faster than the non-entraining vortices. Figure \[fig:pdf\]c shows the probability density function of $Fr_l$. The curves coincide at $Fr_l\approx 37$ and there is significant overlap below and above this value. Both entraining and non-entraining vortices cluster towards the smaller Froude numbers however this is more pronounced in the case of the entraining vortices. Figure \[fig:zplusPDF\] depicts the probability distribution function for the wall-normal location of the entraining vortices. The deep water velocity profile has also been included for comparison. The mean is at $z^+\approx100$ and the standard deviation is about 83 meaning that the majority of the entraining vortices reside in buffer layer and log-law regions of the boundary layer. ![Wall-normal distance PDF for entraining vortices. Deep water mean velocity is given for comparison.[]{data-label="fig:zplusPDF"}](figure29.pdf){width="\linewidth"} CONCLUSIONS =========== In this research, the problem of the interaction of a turbulent boundary layer with a free surface is studied experimentally using a laboratory-scale device and a DNS simulation of a similar problem. The experimental device utilizes a stainless steel belt, driven by two powered vertically oriented rollers, as a surface piercing vertical wall of infinite length. This belt accelerates in under 0.7 seconds to constant speed $U$ in an effort to mimic the sudden passage of a flat-sided ship. Utilizing the full length and velocity scales of large naval ships, this device creates a temporally-evolving boundary layer analogous to the spatially-evolving boundary layer along the length of a ship, using the transformation $x=Ut$, where $x$ is distance from the leading edge and $t$ is time. Water surface profiles measured along lines perpendicular and parallel to the belt surface were recorded with a cinematic LIF system to study the generation of surface height fluctuations by the sub-surface turbulence. Entrained bubbles were measured using a high-speed camera setup which was able to measure and track bubbles down to a radius of $r = 0.5$ mm. To complement the experiments, DNS of the two-phase boundary layer problem were carried out where a section of the belt was considered. The boundary conditions of the computational domain are similar to that of the experiments. The DNS results allow access to the entire flow field that is otherwise inaccessible through experiments. From qualitative observation of the free surface profiles in the LIF movies with the light sheet parallel to the belt, it was found that surface features that resemble breaking water waves travel downstream, parallel to the belt surface. Two of these surface features are observed in detail qualitatively and it is hypothesized that they are one of the mechanisms through which air is entrained into the free surface boundary layer. The speed of these free surface features is measured parallel to the belt and it is found that they move about three times faster than similar features moving away from the belt. Also, it is found that the downstream speed of these features decays quickly when they are measured further away from the belt. The experimental free surface profiles are compared to the computational results and are found to agree qualitatively. Experimental measurements of bubbles are also reported. The bubble size spectrum is found to have a break in slope at around $r = 2.46$ mm with two characteristic linear regions. The break in slope suggests a Hinze-like scale that dominates the bubble radius spectrum. The experimental results are compared to the bubble radius spectrum from the DNS and they seem to agree qualitatively. The number of bubbles per depth increment is found to decrease with depth, in agreement with the DNS results. The mean bubble radius and mean bubble speed versus depth are found to t decrease slowly with depth. Three entrainment mechanisms, breaking waves, vertically oriented vortices, and horizontally oriented vortices, are studied in detail from the DNS results. It is found that horizontally oriented vortices account for $\approx 88 \%$ of the entrainment events in the DNS, while entrainment events by breaking waves and vertically oriented vortices account for $\approx 11 \%$ and $< 1 \%$ of the total entrainment events, respectively. The support of the Office of Naval Research under grant number N000141712081 (Program Managers: Ki-Han Kim and Thomas Fu) is gratefully acknowledged. Discussion ---------- **Discusser I:** **Discusser:** This is an excellent paper that presents an experimental and numerical study of air entrainment and surface fluctuations in a developing boundary layer. Besides the interest on a fundamental fluid mechanics problem, the work is significant in that provides further insight into entrainment through the boundary layer/free surface contact line and may help develop appropriate models for air entrainment. **Reply**: We thank for the discusser for his careful reading of our paper and his insightful questions and comments. **Discusser:** Detailed questions and comments: 1. **Comment/Question**: It would be desirable to add to the introduction some discussion on how the work in this paper and other contributions by the authors could help development of air entrainment models for bubbly wake applications. **Reply**: Thank you for pointing this out. We have added some references to the bibliography and made a few modifications along these lines to the beginning of the introduction. 2. **Comment/Question**: The discussion on air entrainment mechanisms is interesting. Can the authors really resolve Mesler entrainment, to the extent of 12 % of the bubbles? Bubbles resulting from impact are usually very small. **Reply**: This portion of the bubbles is mainly due to the larger cavities collapsing and entrapping large blobs of air which subsequently break up. We rarely see entrainment due to narrow columns of splashing water. **Discusser II:** We thank for the discusser for his careful reading of our paper and his insightful questions and comments. **Detailed comments and questions:** 1. **Comment/Question**: The authors mention these experiments should be carried out in salt water. Were these experiments done in fresh or salt water, unclear. **Reply**: All the experiments presented in this paper were carried out in filtered fresh water. The paper was modified to clarify this point. Once a complete set of bubble measurements is performed in fresh water, we hope to perform an identical set of measurements in salt water. 2. **Comment/Question**: The authors state: It is worth noting that most of the entraining vortices collocate in the corner of the plot and tend to assume smaller values of $Re_l$. However, from Fig 27b, it looks like at higher $Re$ ($ > 600$) almost no vortices have $d/r<6$. It may be worth commenting/discussing this. Is this due to the nature of higher $Re$ or could your grid resolution be playing into it? **Reply**: The absence of vortices in that area of the plot ($d/r < $ 6 ; $Re > $ 600) is an issue still under investigation. We are at the moment acquiring additional data to rule out the the possibility that this is due to a small dataset. The resolution of the grid is unlikely to be the issue since we keep the near-interface region at the finest refinement level at all times.

--- abstract: 'We continue the investigation of twisted homology theories in the context of dimension drop phenomena. This work unifies previous equivariant index calculations in twisted cyclic cohomology. We do this by proving the existence of the resolvent cocycle, a finitely summable analogue of the JLO cocycle, under weaker smoothness hypotheses and in the more general setting of ‘modular’ spectral triples. As an application we show that using our twisted resolvent cocycle, we can obtain a local index formula for the Podleś sphere. The resulting twisted cyclic cocycle has non-vanishing Hochschild class which is in dimension 2.' author: - | Adam Rennie, Roger Senior [^1]\ Mathematical Sciences Institute, Australian National University\ Acton, ACT, 0200, Australia\ title: The resolvent cocycle in twisted cyclic cohomology and a local index formula for the Podleś sphere --- =0.08in Introduction {#sec:intro} ============ This paper proves a residue index formula in noncommutative geometry for ‘modular spectral triples’, which are analogues of spectral triples with twisted traces. This is the appropriate setting for examples arising from $q$-deformations which typically experience ‘dimension drop’ in homology, [@H; @HK; @NT; @SW; @W]. The main results are as follows. 1\) We show that for finitely summable modular spectral triples the resolvent cocycle exists, is continuous and is an index cocycle under weaker smoothness conditions than have previously been used. In particular we do not need the pseudodifferential calculus to establish these facts, so that we are free to replace the usual pseudodifferential calculus by other schemes later, in order to obtain local index formulae. 2\) We show that modular spectral triples have a well-defined pairing with equivariant $K$-theory. In the finitely summable and weakly smooth case we show that this pairing can be computed using the resolvent cocycle, which defines a twisted cyclic cocycle. 3\) We apply the results of 1) and 2) to prove a local index formula for the Podleś sphere in twisted cyclic cohomology. This index formula puts the results of several authors into a common framework, [@H; @KW; @W]. In particular, the twisted Hochschild class of our residue cocycle is an explicit constant multiple of the fundamental Hochschild cocycle for the Podleś sphere, [@H; @KW], and our explicit index pairings can be compared to those in [@W]. The computations in 3) are similar to what was done in [@NT], however they used the twisting by the modular automorphism, rather than the inverse of the modular automorphism. While the summability is the same in both cases, the twisted Hochschild homology for the modular automorphism is trivial in dimension 2, while the inverse of the modular automorphism avoids the dimension drop. Thus the cocycle obtained in [@NT] is cohomologous to a $0$-cocycle, while ours is not. We also note that in [@NT] the starting point was the JLO cocycle in entire cyclic cohomology rather than the resolvent cocycle. The exposition is as follows. In Section \[sec:KMS-index\] we introduce the basic definitions for modular spectral triples, including smoothness and summability. We then show that a modular spectral triple defines an equivariant $KK$-class, and so gives us a well-posed $K$-theory valued index problem. The remainder of Section 2 demonstrates that together with a representative of an equivariant $K$-theory class, we obtain a well-posed numerical index problem. The aim of Section \[sec:local-index\] is then to show that these notions are compatible. We address the existence, continuity and index properties of the resolvent cocycle in Section \[sec:local-index\]. We begin by looking at our weak smoothness condition, and proving some basic results that follow from this assumption. Then we prove the existence and continuity of the resolvent cocycle, which originated in [@CPRS2], and show that it computes the numerical index. Finally we show, using results from [@KNR], that this numerical index is compatible with the $K$-theory valued index in a precise way. In section \[sec:pods-chern\] we show that the spectral triple introduced by [@DS] defines a 2-dimensional modular spectral triple, which is weakly smooth in our sense. Numerous results of [@KW; @NT; @SW; @W] are incorporated into this statement. We employ Neshveyev and Tuset’s modification of the pseudodifferential calculus to obtain a version of the local index formula for the Podleś sphere. Thus we see that with a suitable pseudodifferential calculus, our resolvent index formula can be extended to a full local index formula as in [@CPRS2; @CM; @Hig]. We conclude by computing some explicit index pairings, and as a corollary see that the degree two term in the residue index cocycle is not a coboundary. [**Acknowledgements.**]{} It is a pleasure to acknowledge the assistance of our colleagues Alan Carey, Ulrich Krähmer and Joe Várilly. Both authors were supported by the Australian Research Council. Modular spectral triples and equivariant $K$-theory {#sec:KMS-index} =================================================== We begin this section by defining modular spectral triples, a generalisation of semifinite spectral triples, [@BeF; @CP2; @CPRS2], allowing for twisted traces (weights) in place of traces. We then consider the index pairings defined by modular spectral triples. The strategy to study index pairings is almost the same as in [@CPRS2; @CPRS3]. Given a representative of an equivariant $K$-theory class for an algebra ${\mathcal{A}}$, we show that a modular spectral triple over ${\mathcal{A}}$ allows us to formulate a well-defined (semifinite) index problem. By following the strategy of [@CPRS2; @CPRS3], we find that the index can be computed by pairing a cocycle with the Chern character of the $K$-theory class. Modular spectral triples {#subsec:mod-specs} ------------------------ Let ${\mathcal{N}}$ be a semifinite von Neumann algebra acting on a Hilbert space ${\mathcal{H}}$, and fix a faithful normal semifinite weight $\phi$. We denote the modular automorphism group of $\phi$ by $\sigma^\phi_t$. Then as $\phi$ is ${\sigma}^\phi_t$ invariant, we see that for all $T\in{\rm dom}\,\phi\subset {\mathcal{N}}$ and $t\in{\mathbb{R}}$ $$\phi(T)=\phi({\sigma}^\phi_t(T)).$$ Suppose further that the modular group ${\sigma}^\phi_t$, which is inner since ${\mathcal{N}}$ is semifinite, is periodic, and let $\alpha$ be the (least) period of ${\sigma}^\phi_t$. Then $$\phi(T)=\frac{1}{\alpha}\int_0^\alpha \phi({\sigma}^\phi_t(T))dt =\phi\left(\frac{1}{\alpha}\int_0^\alpha {\sigma}^\phi_t(T)dt\right)=:(\phi\circ \Psi)(T),$$ where $\Psi:{\mathcal{N}}\to{\mathcal{M}}:={\mathcal{N}}^{{\sigma}^{\phi}}$ is the expectation onto the fixed point algebra ${\mathcal{M}}$ defined by the integral. Then the restriction of $\phi$ to ${\mathcal{M}}$ is a faithful normal trace. The restriction of $\phi$ to ${\mathcal{M}}$ is also semifinite if and only if $\phi$ is [*strictly*]{} semifinite, meaning that $\phi$ is the sum of normal positive linear functionals whose supports are mutually orthogonal, [@T p 105]. In everything that follows, we suppose that $\phi$ is strictly semifinite. Given a faithful normal semifinite trace $\tau$ on a von Neumann algebra ${\mathcal{N}}$, we define the ideal of $\tau$-compact operators ${\mathcal{K}}({\mathcal{N}},\tau)$ to be the norm closure of the ideal generated by the projections $p$ with finite trace, $\tau(p)<\infty$. Let ${\mathcal{N}}$ be a semifinite von Neumann algebra acting on a Hilbert space ${\mathcal{H}}$, and fix a faithful normal strictly semifinite weight $\phi$. Suppose further that the modular group ${\sigma}^\phi_t$ is periodic. Then we say that $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ is a [**unital modular spectral triple**]{} with respect to $({\mathcal{N}},\phi)$ if 0\) ${\mathcal{A}}$ is a separable unital $*$-subalgebra of ${\mathcal{N}}$ with norm closure $A$; 1\) ${\mathcal{A}}$ is invariant under ${\sigma}^\phi$, ${\mathcal{A}}$ consists of analytic vectors for ${\sigma}^\phi$, and ${\sigma}^\phi|_{A}$ is a strongly continuous action; 2\) ${\mathcal{D}}$ is a self-adjoint operator affiliated to the fixed point algebra ${\mathcal{M}}:={\mathcal{N}}^{{\sigma}^\phi}$; 3\) $[{\mathcal{D}},a]$ extends to a bounded operator in ${\mathcal{N}}$ for all $a\in{\mathcal{A}}$; 4\) $(1+{\mathcal{D}}^2)^{-1/2}\in {\mathcal{K}}({\mathcal{M}},\phi|_{\mathcal{M}})$. The triple is even if there exists $\gamma=\gamma^*\in{\mathcal{M}}$ with $\gamma^2=1$, $\gamma a=a\gamma$ for all $a\in{\mathcal{A}}$ and $\gamma{\mathcal{D}}+{\mathcal{D}}\gamma=0$. Otherwise the triple is odd. We say that the triple is finitely summable with spectral dimension $p\geq 1$ if $p$ is the least number such that $$\phi((1+{\mathcal{D}}^2)^{-s/2})<\infty\ \ \ \ \mbox{for all} \ \ \Re(s)>p.$$ Just as for ordinary spectral triples, there is a notion of smoothness and pseudodifferential operators for $QC^\infty$ modular spectral triples, just as in [@CPRS2; @CM], which we recall here. \[qck\] A modular spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ relative to $({\mathcal{N}},\phi)$ is $QC^k$ for $k\geq 1$ ($Q$ for quantum) if for all $a\in{\mathcal{A}}$ the operators $a$ and $[{\mathcal{D}},a]$ are in the domain of $\delta^k_1$, where $\delta_1(T)=[(1+{\mathcal{D}}^2)^{1/2},T]$ is the partial derivation on ${\mathcal{N}}$ defined by $(1+{\mathcal{D}}^2)^{1/2}$. We say that $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ is $QC^\infty$ if it is $QC^k$ for all $k\geq 1$. Equivalently, [@CPRS2 Proposition 6.5] and [@CM Lemma B.2], $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ is $QC^\infty$ if for all $a\in{\mathcal{A}}$ we have $a,\,[{\mathcal{D}},a]\in\bigcap_{k,l\geq 0}{\rm dom}L_1^k\circ R_1^l$, where $L,\,R$ are defined by $$L(T)=(1+{\mathcal{D}}^2)^{-1/2}[{\mathcal{D}}^2,T]\quad{\rm and}\quad R(T)=[{\mathcal{D}}^2,T](1+{\mathcal{D}}^2)^{-1/2}.$$ Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ be a modular spectral triple relative to $({\mathcal{N}},\phi)$. For $r\in{{\mathbb{R}}}$ $${\rm OP}^r=(1+{\mathcal{D}}^2)^{r/2}\left(\bigcap_{n\geq 0}{\rm dom}\,\delta_1^n\right).$$ If $T\in {\rm OP}^r$ we say that $T$ is a pseudodifferential operator and that the order of $T$ is (at most) $r$. The definition is actually symmetric, since for $r$ an integer (at least) we have by [@CPRS2 Lemma 6.2] $$\begin{aligned} {\rm OP}^r&=(1+{\mathcal{D}}^2)^{r/2}\left(\bigcap\mbox{dom}\,\delta_1^n\right) =(1+{\mathcal{D}}^2)^{r/2}\left(\bigcap\mbox{dom}\,\delta_1^n\right) (1+{\mathcal{D}}^2)^{-r/2}(1+{\mathcal{D}}^2)^{r/2}\\ &\subseteq\left(\bigcap\mbox{dom}\,\delta_1^n\right)(1+{\mathcal{D}}^2)^{r/2}.\end{aligned}$$ From this we easily see that ${\rm OP}^r\cdot {\rm OP}^s\subseteq {\rm OP}^{r+s}$. Finally, we note that if $b\in {\rm OP}^r$ for $r\geq 0$, then since $b=(1+{\mathcal{D}}^2)^{r/2}a$ for some $a\in {\rm OP}^0$, we get $[(1+{\mathcal{D}}^2)^{1/2},b]=(1+{\mathcal{D}}^2)^{r/2}[(1+{\mathcal{D}}^2)^{1/2},a]=(1+{\mathcal{D}}^2)^{r/2}\delta_1(a),$ so $[(1+{\mathcal{D}}^2)^{1/2},b]\in {\rm OP}^r$. [**Remarks**]{}: 1) An operator $T\in {\rm OP}^r$ if and only if $(1+{\mathcal{D}}^2)^{-r/2}T\in\bigcap_{n\geq 0}\mbox{dom}\,\delta_1^n$. Observe that operators of order at most zero are bounded. 2\) We will need a weaker notion of smoothness, introduced in Section 3, for modular spectral triples, as Definition \[qck\] is not satisfied for our main example, the Podleś sphere. [**Example.**]{} A semifinite spectral triple is a modular spectral triple with $\phi$ a semifinite normal trace (and so ${\mathcal{M}}={\mathcal{N}}$). [**Example.**]{} Given a circle action on a unital $C^*$-algebra $A$, every state on $A$ which is KMS for this circle action gives rise to a modular spectral triple of dimension 1. Explicit examples are the Cuntz algebra with its usual gauge action, [@CPR2], and the quantum group $SU_q(2)$ with its Haar state, [@CRT]. All these examples are $QC^\infty$ (or regular or smooth) when we use the algebra of analytic vectors ${\mathcal{A}}\subset A$ for the circle action. More examples arising from a topological version of the group-measure space construction are presented in [@CPPR]. [**Example.**]{} The only other unital example (known to the authors) is the Podleś sphere, which provides an example of a modular spectral triple of dimension 2. This was first presented in [@DS], and has been studied in numerous subsequent works by various authors. The paper [@W] provides a good summary. This example is not $QC^\infty$, but a replacement for the pseudodifferential calculus was developed in [@NT]. This example is our main motivation for weakening the $QC^\infty$ condition, and this example will be presented in detail in Section 4. [**Nonunital examples.**]{} We have chosen to work in the unital case for simplicity, but there are nonunital examples, [@CNNR; @CMR]. However, to simplify the discussion of the local index formula, we will restrict to the unital case. To handle the nonunital case in general, we would need to modify the definition of modular spectral triple in order to utilise (analogues of) the results of [@CGRS2], where the local index formula is proved in the nonunital case. Equivariant $KK$-theory and modular spectral triples. {#sec:spec-flow} ----------------------------------------------------- An odd modular spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ with respect to $({\mathcal{N}},\phi)$ defines an equivariant Kasparov module, and so a class $[{\mathcal{D}}]\in KK^{1,{\mathbb{T}}}(A,{\mathcal{K}}_{\mathcal{N}})$, where we recall that $A=\overline{{\mathcal{A}}}$. The construction of the Kasparov module associated to a modular spectral triple begins with the definition of a suitable ideal. We will deal explicitly with the odd case here, just stating the analogous results in the even case. \[defn\_j\_phi\] Given a modular spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},\phi)$, let [$$J_{\phi} := \{ SkT \colon S, \, T \in {\mathcal{N}}, \ k \in {\mathcal{K}}({{\mathcal M}}, \phi|_{{{\mathcal M}}}) \}$$]{} denote the norm closed two-sided ideal in ${\mathcal{N}}$ generated by ${\mathcal{K}}({{\mathcal M}}, \phi|_{{{\mathcal M}}})$. The ideal $J_{\phi}$ is a right Hilbert module over itself, and ${\mathcal{A}}$ acts on the left of $J_{\phi}$ by multiplication. The axioms of a modular spectral triple imply that $(1 + {\mathcal{D}}^{2})^{-1/2} \in J_{\phi}$. With a little effort we can show, as in [@KNR Theorem 4.1], that the pair $(J_\phi,{\mathcal{D}}(1+{\mathcal{D}}^2)^{-1/2})$ is a Kasparov module, except that the module $J_\phi$ may not be countably generated. To deal with this problem, we recall the following construction from [@KNR Theorem 5.3]. \[def:bee-phi\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},\phi)$ be a modular spectral triple, where we recall that ${\mathcal{A}}$ is separable. Write $F_{\mathcal{D}}:={\mathcal{D}}(1+{\mathcal{D}}^2)^{-1/2}$ and let $B_\phi$ be the smallest $C^*$-algebra in ${\mathcal{N}}$ containing the elements $$F_{\mathcal{D}}\,[F_{\mathcal{D}},a], \quad b\,[F_{\mathcal{D}},a], \quad F_{\mathcal{D}}\, b\,[F_{\mathcal{D}},a], \quad a\,\varphi({\mathcal{D}})$$ for all $a,b\in \mathcal{A}$ and $\varphi\in C_0({\mathbb{R}})$. Then $B_\phi$ is separable, and so $\sigma$-unital, and contained in $J_\phi$. \[prop:Kas-mod\] A modular spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},\phi)$ defines an equivariant $KK$-theory class $[{\mathcal{D}}] = [B_\phi, F_{{\mathcal{D}}}] \in KK^{1, {\mathbb{T}}}(A, B_\phi)$, where $F_{\mathcal{D}}:={\mathcal{D}}(1+{\mathcal{D}}^2)^{-1/2}$. A modular spectral triple is automatically a von Neumann spectral triple with respect to $J_\phi$ in the sense of [@KNR]. Then [@KNR Theorem 5.3] shows that $B_\phi$ is a countably generated right $C^*$ $B_\phi$-module, and that the pair $(B_\phi,F_{\mathcal{D}})$ is a Kasparov module. The equivariance is immediate. Having obtained an equivariant Kasparov module, and so a $KK$-class, the Kasparov product defines a $K_0^{{\mathbb{T}}}(B_\phi)$-valued index pairing between a modular spectral triple and equivariant $K$-theory. That is, [$$K_{1}^{{\mathbb{T}}}(A) \times KK^{1, {\mathbb{T}}}(A, B_{\phi}) \rightarrow K_{0}^{{\mathbb{T}}}(B_{\phi}).$$]{} See [@B Theorem 18.4.4] for example. We now seek an analytic formula to compute this index, and in Section \[sec:local-index\] we obtain such a formula, the resolvent index formula. The first step is the construction of a semifinite spectral triple which encodes the index pairing between a modular spectral triple and an equivariant $K$-theory class. This is necessary to obtain a well-defined numerical index problem. We now describe this procedure. Given a modular spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},\phi)$ and a class $[u]\in K_1^{\mathbb{T}}(A)$, there is a unitary $u\in M_n({\mathcal{A}})$ and a representation $V:{\mathbb{T}}\to M_n({\mathbb{C}})$ such that $u$ is $\sigma^\phi\otimes Ad\,V$ invariant, [@B; @CNNR]. In particular, if $n=1$ then $u$ is ${\sigma}^\phi$ invariant. We can diagonalise the representation $V_t=\oplus_{j=1}^n \lambda_j^{it}$, $\lambda_j\in[1,\infty)$, and in this basis it is clear that 1\) $u_{ij}$ transforms under $Ad\,V_t$ by $\lambda_i^{it}\lambda_j^{-it}$; 2\) $u_{ij}$ transforms under ${\sigma}_t^\phi$ by $\lambda_i^{-it}\lambda_j^{it}$; 3\) $V_t$ extends to an action of ${\mathbb{C}}$ which is not a $*$-action, but satisfies $V_z^*=V_{-\bar{z}}$. We define a positive functional $G:M_n({\mathbb{C}})\to {\mathbb{C}}$ by setting $$G(T)=\operatorname{Tr}(V_{-i}T),\ \ \ T\in M_n({\mathbb{C}}).$$ Then $G$ is a $KMS_{1}$ functional on $M_n({\mathbb{C}})$, [@BR], for the action $Ad\,V$, but is not a state as it is not normalised. Now consider the fixed point algebra ${\mathcal{M}}_n=(M_n({\mathcal{N}}))^{{\sigma}^\phi\otimes Ad\,V}$, which is the centralizer of the weight $\phi\otimes G$, [@T Proposition 4.3]. Then $\phi\otimes G$ restricts to a faithful normal semifinite trace on ${\mathcal{M}}$ and moreover $u \in {\mathcal{M}}_n$. The latter statement follows from the definition of $u$. The former follows since the strict semifiniteness of $\phi$ implies the strict semifiniteness of $\phi\otimes G$. \[lem:bob-the-builder\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},\phi)$ be a modular spectral triple which is finitely summable and $u\in M_n({\mathcal{A}})$ a $\sigma^\phi$ equivariant unitary, with associated representation $V:{\mathbb{T}}\to M_n({\mathbb{C}})$. Then $$(C^\infty(u),{\mathcal{H}}\otimes {\mathbb{C}}^n,{\mathcal{D}}\otimes {\rm Id}_n, {\mathcal{M}}_n,\phi\otimes G)$$ is a finitely summable semifinite spectral triple. Here $C^\infty(u)$ is the algebra of all $f(u)\in C^*(u)$ with $f$ a $C^\infty$ function on the spectrum of $u$. Let $B_{\phi\otimes G}\subseteq {\mathcal{K}}({\mathcal{M}}_n,\phi\otimes G)$ be defined as in Definition \[def:bee-phi\]. Then this semifinite spectral triple defines a Kasparov class in $KK^{1,{\mathbb{T}}}(C^*(u),B_{\phi\otimes G})$. The statement that we obtain a semifinite spectral triple follows from the construction, and that we get a Kasparov module follows from Proposition \[prop:Kas-mod\]. Thus given $[u,V]\in K_1^{\mathbb{T}}({\mathcal{A}})$, we apply [@KNR Theorem 6.9] to compute the spectral flow, [@Ph]. Let $i:B_{\phi\otimes G}\subset {\mathcal{K}}({\mathcal{M}}_n,\phi\otimes G)$ be the inclusion, and $i_*:K_0(B_{\phi\otimes G})\to K_0({\mathcal{K}}({\mathcal{M}}_n,\phi\otimes G))$. Then [@KNR Theorem 6.9] allows us to compute the spectral flow as [$$\label{eqn_sf} sf_{\phi\otimes G}({\mathcal{D}}{\otimes}{\rm Id}_n,\,u({\mathcal{D}}{\otimes}{\rm Id}_n)u^*) = (\phi\otimes G)_*(i_*([u]\otimes_{C^*(u)}[{\mathcal{D}}\otimes{\rm Id}_n])).$$]{} At this point, we have obtained an index problem which [*a priori*]{} depends on the representative $u$ of the equivariant $K$-theory class (through the use of $C^*(u)$). To show that we do indeed have a well-defined pairing with $K_1^{\mathbb{T}}({\mathcal{A}})$, we will show, via the resolvent index formula, that the index can be computed in terms of the Chern character of $u$, which is independent of the chosen representative of the class $[u]$. Finally, we show that the original index pairing between a modular spectral triple and equivariant $K$-theory can be described by the spectral flow above. The resolvent index formula in twisted cyclic cohomology {#sec:local-index} ======================================================== In this section we express the spectral flow from Equation in terms of the pairing between a twisted cyclic cocycle dependent only on the modular spectral triple and the Chern character of the equivariant unitary. In order to achieve this without invoking the $QC^\infty$ property, we make a technical improvement on the work of [@CPRS2] by using a weaker smoothness condition. This is necessary for our application, as the Podleś sphere modular spectral triple is not $QC^\infty$. Weakly $QC^\infty$ modular spectral triples {#subsec:weak-qcinfty} ------------------------------------------- We weaken the $QC^\infty$ condition with the aim of justifying a resolvent expansion, used in the proof of our index formulae, without recourse to the pseudodifferential calculus. There are two basic reasons for doing this. The first is that the example of the Podleś sphere shows that we do not always have the $QC^\infty$ property for modular spectral triples. The second reason is that, conceptually, the use of the pseudodifferential calculus to prove existence and continuity of the resolvent cocycle is overkill, requiring us to invoke much more smoothness than is necessary for the statment of existence and continuity. \[defn:weak-op\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ be a modular spectral triple relative to $({\mathcal{N}},\phi)$. For $T\in{\mathcal{N}}$ mapping the domain of ${\mathcal{D}}^2$ to itself, define $$\begin{aligned} WL(T):=(1+{\mathcal{D}}^2)^{-1}[{\mathcal{D}}^2,T]&=(1+{\mathcal{D}}^2)^{-1}T(1+{\mathcal{D}}^2)-T,\\ WR(T):=[{\mathcal{D}}^2,T](1+{\mathcal{D}}^2)^{-1}&=(1+{\mathcal{D}}^2)T(1+{\mathcal{D}}^2)^{-1}-T.\end{aligned}$$ We say that $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ is weakly $QC^\infty$ if $${\mathcal{A}}\subset {\rm OP}^0\subset {\mathcal{N}}\quad{\rm and}\quad [{\mathcal{D}},{\mathcal{A}}]\subset {{\rm w{\text -}OP}}^0:=\bigcap_{k,l\geq 0}\mbox{dom}(WL)^k(WR)^l\subset {\mathcal{N}}.$$ The analogous definition of weak $QC^k$ is awkward, since in Definition \[qck\], $QC^k$ is defined in terms of commutators with $|{\mathcal{D}}|$ or $(1+{\mathcal{D}}^2)^{1/2}$. We will leave aside these questions, and just work with weak $QC^\infty$. Also, $QC^\infty$ implies weak $QC^\infty$ by the boundedness of $(1+{\mathcal{D}}^2)^{-1/2}$. While we do not have a pseudodifferential calculus for a weakly $QC^\infty$ modular spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$, we may consider the weak pseudodifferential operators of order $s\in{\mathbb{R}}$ given by $${{\rm w{\text -}OP}}^s:=(1+{\mathcal{D}}^2)^{s/2}\left(\bigcap_{k,l}\,{\rm dom}\,WL^k\circ WR^l\right).$$ This definition is symmetric, in the sense that $${{\rm w{\text -}OP}}^s=\left(\bigcap_{k,l}\,{\rm dom}\,WL^k\circ WR^l\right)(1+{\mathcal{D}}^2)^{s/2},$$ since for all $s\in{\mathbb{R}}$, ${{\rm w{\text -}OP}}^s$ is preserved by $T\mapsto (1+{\mathcal{D}}^2)^{\pm s}T(1+{\mathcal{D}}^2)^{\mp s}$, by Lemma \[lem:bdd-conj\] below. Observe also that we have ${\rm OP}^s\subset {{\rm w{\text -}OP}}^s$. It follows from the definitions that if $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ is a weakly $QC^\infty$ modular spectral triple and $u\in M_n({\mathcal{A}})$ is an equivariant unitary, then the associated semifinite spectral triple $(C^\infty(u),{\mathcal{H}}\otimes{\mathbb{C}}^n,{\mathcal{D}}\otimes{\rm Id}_n, {\mathcal{M}}_n,\phi\otimes G)$ is also weakly $QC^\infty$. The next few lemmas record some basic properties of the maps $WL$ and $WR$. \[lem:bdd-conj\] Let ${\mathcal{D}}:{\rm dom}{\mathcal{D}}\subset {\mathcal{H}}\to {\mathcal{H}}$ be an unbounded self-ajoint operator. Then $T\in {\mathcal{B}}({\mathcal{H}})$ belongs to $$\bigcap_{k,l\geq 0}{\rm dom}(WL)^k(WR)^l$$ if and only if $(1+{\mathcal{D}}^2)^{s/2}T(1+{\mathcal{D}}^2)^{-s/2}$ extends to a bounded operator for all $s\in{\mathbb{R}}$. It follows from Definition \[defn:weak-op\] that $T\in \bigcap_{k,l\geq 0}\mbox{dom}(WL)^k(WR)^l$ if and only if $(1+{\mathcal{D}}^2)^{k}T(1+{\mathcal{D}}^2)^{-k}$ is a bounded operator for all $k\in{\mathbb{Z}}$. It is also immediate that if $(1+{\mathcal{D}}^2)^{s/2}T(1+{\mathcal{D}}^2)^{-s/2}$ is bounded for all $s\in{\mathbb{R}}$, then $T\in \bigcap_{k,l\geq 0}\mbox{dom}(WL)^k(WR)^l$. So let $0<s<1$, and recall that $$(1+{\mathcal{D}}^2)^{-s}=\frac{\sin(s\pi)}{\pi}\int_0^\infty \lambda^{-s}(1+\lambda+{\mathcal{D}}^2)^{-1}d\lambda.$$ Then for $T\in \bigcap_{k,l\geq 0}\mbox{dom}(WL)^k(WR)^l$ we have $$\begin{aligned} &(1+{\mathcal{D}}^2)^sT(1+{\mathcal{D}}^2)^{-s}\\ &=(1+{\mathcal{D}}^2)^sT\frac{\sin(s\pi)}{\pi}\int_0^\infty \lambda^{-s}(1+\lambda+{\mathcal{D}}^2)^{-1}d\lambda\\ &=(1+{\mathcal{D}}^2)^s\frac{\sin(s\pi)}{\pi}\int_0^\infty \lambda^{-s} \left((1+\lambda+{\mathcal{D}}^2)^{-1}T+(1+\lambda+{\mathcal{D}}^2)^{-1}[{\mathcal{D}}^2,T](1+\lambda+{\mathcal{D}}^2)^{-1}\right)d\lambda\\ &=(1+{\mathcal{D}}^2)^s\frac{\sin(s\pi)}{\pi}\int_0^\infty \lambda^{-s} (1+\lambda+{\mathcal{D}}^2)^{-1}\left(T+[{\mathcal{D}}^2,T](1+{\mathcal{D}}^2)^{-1}(1+{\mathcal{D}}^2)(1+\lambda+{\mathcal{D}}^2)^{-1}\right)d\lambda\\ &=T+(1+{\mathcal{D}}^2)^s\frac{\sin(s\pi)}{\pi}\int_0^\infty \lambda^{-s} (1+\lambda+{\mathcal{D}}^2)^{-1}[{\mathcal{D}}^2,T](1+{\mathcal{D}}^2)^{-1}(1+{\mathcal{D}}^2)(1+\lambda+{\mathcal{D}}^2)^{-1}d\lambda.\end{aligned}$$ An application of the functional calculus now shows that the integral is norm convergent, but in order to show that $(1+{\mathcal{D}}^2)^s$ times the integral is bounded, we must work a little harder. We write $$(1+{\mathcal{D}}^2)(1+\lambda+{\mathcal{D}}^2)^{-1}=1-\lambda(1+\lambda+{\mathcal{D}}^2)^{-1},$$ so that the integral can be written, with $B=[{\mathcal{D}}^2,T](1+{\mathcal{D}}^2)^{-1}$, as $$\begin{aligned} &\int_0^\infty \lambda^{-s} (1+\lambda+{\mathcal{D}}^2)^{-1}[{\mathcal{D}}^2,T](1+{\mathcal{D}}^2)^{-1}(1+{\mathcal{D}}^2)(1+\lambda+{\mathcal{D}}^2)^{-1}d\lambda\\ &=\int_0^\infty \lambda^{-s} (1+\lambda+{\mathcal{D}}^2)^{-1}B\,d\lambda-\int_0^\infty \lambda^{-s}\,\lambda (1+\lambda+{\mathcal{D}}^2)^{-1}B(1+\lambda+{\mathcal{D}}^2)^{-1}\,d\lambda.\end{aligned}$$ The first integral on the right hand side converges in norm to $\frac{\pi}{\sin(s\pi)}(1+{\mathcal{D}}^2)^{-s}B$. For the second integral on the right hand side, we suppose first that $B$ is self-adjoint. Then $$\lambda(1+\lambda+{\mathcal{D}}^2)^{-1}B(1+\lambda+{\mathcal{D}}^2)^{-1}\leq \Vert B\Vert\,\lambda(1+\lambda+{\mathcal{D}}^2)^{-2}\leq \Vert B\Vert\,(1+\lambda+{\mathcal{D}}^2)^{-1}.$$ Thus for $B$ self-adjoint, the second integral converges in norm to an operator which is bounded above by $\frac{\pi}{\sin(s\pi)}(1+{\mathcal{D}}^2)^{-s}\Vert B\Vert$. By decomposing $B$ into its real and imaginary parts, this is true for any bounded $B$. Thus for $0<s<1$, $(1+{\mathcal{D}}^2)^sT(1+{\mathcal{D}}^2)^{-s}$ is bounded, and a similar argument shows that $(1+{\mathcal{D}}^2)^{-s}T(1+{\mathcal{D}}^2)^{s}$ is bounded. In all the following, we define $R_s(\lambda):=(\lambda-(1+s^2+{\mathcal{D}}^2))^{-1}$ for $s\geq 0$ and $\lambda$ in the vertical line $$l:=\{a+iv: \ -\infty<v<\infty\}$$ for some fixed $0<a<1/2$. \[lem:bdd\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ be a weakly $QC^\infty$ modular spectral triple relative to $({\mathcal{N}},\phi)$. Then $R_s(\lambda)[{\mathcal{D}}^2,T]$ is uniformly bounded on the line $l$ independent of $s,\,\lambda$ for all $T\in{\mathcal{A}}\cup[{\mathcal{D}},{\mathcal{A}}]$. For all $T\in{\mathcal{A}}\cup[{\mathcal{D}},{\mathcal{A}}]$, the function $\lambda\mapsto R_s(\lambda)TR_s(\lambda)^{-1}$ is uniformly bounded and differentiable on the line $l$ with derivative $-R_s(\lambda)^2[{\mathcal{D}}^2,T]$ which vanishes as $\lambda\to a\pm i\infty$. First $R_s(\lambda)[{\mathcal{D}}^2,T]=R_s(\lambda)(1+{\mathcal{D}}^2)(1+{\mathcal{D}}^2)^{-1}[{\mathcal{D}}^2,T]$ and $R_s(\lambda)(1+{\mathcal{D}}^2)$ is uniformly bounded. Then $R_s(\lambda)TR_s(\lambda)^{-1}=R_s(\lambda)[{\mathcal{D}}^2,T]+T$ is uniformly bounded on $l$. For the differentiability, we form the difference quotients where $\epsilon$ is chosen so that $\lambda+\epsilon$ lies in a small ball centred on $\lambda=a+iv$ $$\begin{aligned} &R_s(\lambda+\epsilon)TR_s(\lambda+\epsilon)^{-1}-R_s(\lambda)TR_s(\lambda)^{-1}\\ &=(R_s(\lambda+\epsilon)-R_s(\lambda))TR_s(\lambda+\epsilon)^{-1} +R_s(\lambda)T(R_s(\lambda+\epsilon)-R_s(\lambda)^{-1})\\ &=-\epsilon R_s(\lambda+\epsilon)R_s(\lambda)TR_s(\lambda+\epsilon)^{-1} +\epsilon R_s(\lambda)T.\end{aligned}$$ Now the uniform boundedness of $R_s(\lambda)TR_s(\lambda)^{-1}$ and the boundedness of $R_s(\lambda)T$ show that after dividing by $\epsilon$, the norm limit as $\epsilon\to 0$ exists and is given by $$R_s(\lambda)T-R_s(\lambda)^2TR_s(\lambda)^{-1}=-R_s(\lambda)^2[{\mathcal{D}}^2,T].$$ This is not only bounded, but goes to zero as $|\lambda|\to \infty$ along the line $l=a+iv$. \[lemma\_bdd\_diff\_resolvent\] With $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ as above and $T\in{\mathcal{A}}\cup[{\mathcal{D}},{\mathcal{A}}]$, we have the formula $$R_s(\lambda)^nTR_s(\lambda)^{-n}=T+\sum_{j=1}^n(n-j+1)R_s(\lambda)^jT^{(j)}.$$ Induction and the easy formula $R_s(\lambda)TR_s(\lambda)^{-1}=T+R_s(\lambda)[{\mathcal{D}}^2,T]$. \[cr:deriv\] The function $\lambda\mapsto R_s(\lambda)^nTR_s(\lambda)^{-n}$ is norm differentiable for all $T\in{\mathcal{A}}\cup[{\mathcal{D}},{\mathcal{A}}]$. The derivative goes to zero in norm as $\lambda\to a\pm i\infty$ and is given by [$$\label{eqn_ddl_rtr} \frac{d}{d\lambda}R_s(\lambda)^nTR_s(\lambda)^{-n} =-R_s(\lambda)\sum_{j=1}^nj(n-j+1)R_s(\lambda)^jT^{(j)}.$$]{} We now prove the main technical result we require, which weakens the smoothness hypotheses of [@CPRS2 Lemma 7.2]. \[lemma\_Bs\_trace\] Let $({\mathcal{A}}, {\mathcal{H}}, {\mathcal{D}}, {\mathcal{N}}, \phi)$ be a weakly $QC^\infty$ modular spectral triple of dimension $p \geq 1$. Let $m$ be a non-negative integer and $j = 0, \dots, m$. 1. Let $A_j \in {{\rm w{\text -}OP}}^{k_j}$, $k_j \geq 0$, with the product $A_{0} A_{1} \cdots A_{m}$ being $\sigma^{\phi}_{t}$-invariant and affiliated to ${{\mathcal M}}={\mathcal{N}}^{\sigma^\phi}$. Then the map [$$r \mapsto B^{r}(s) = \frac{1}{2\pi i} \int_{l} \lambda^{-p/2-r}A_0 R_s(\lambda)A_1 R_s(\lambda) A_2 \cdots R_s(\lambda) A_m R_s(\lambda) d\lambda,$$]{} is an analytic function with values in $\mathrm{Dom}(\phi)$ for $r \in \{z \in {\mathbb{C}}:\ \Re(z)>|k|/2 - m,\ z\not\in{\mathbb{N}}-p/2\}$, where $|k| = k_0 + k_1 + \cdots + k_m$. For $\alpha > 0$, the function $s \mapsto s^{\alpha} \times \phi \big( |B^{r}(s)| \big)$ is integrable on $[0,\infty)$ when in addition we have $1+\alpha+|k|-2m<2\Re(r)$. 2. Define $\hat{R}_s(\lambda) = (\lambda-(1 + s^{2} + {\mathcal{D}}^{2} + sK))^{-1}$, for an operator $K = K^{\ast}$ with $\Vert K \Vert_{\infty} \leq \sqrt{2}$. For $a_j \in {\mathcal{A}}$, with $a_{0} a_{1} \cdots a_{m} \in {{\mathcal M}}$, and $r \in {\mathbb{C}}$, with $\Re(r)>0$, the operator [$$\tilde{B}^{r}(s) = \frac{1}{2\pi i}\int_l\lambda^{-p/2-r}a_0R_s(\lambda)[{\mathcal{D}}, a_1] R_s(\lambda)[{\mathcal{D}}, a_2]\cdots R_s(\lambda) [{\mathcal{D}}, a_m] \hat{R}_s(\lambda) d\lambda$$]{} is in $\mathrm{Dom}(\phi)$, and the function $s \mapsto s^{m} \times \phi \big( |\tilde{B}^{r}(s)| \big)$ is integrable on $[0,\infty)$ when $p < 1 + m$ and $1 < m + 2\Re(r)$. The restriction of $\phi$ to the fixed point algebra ${{\mathcal M}}:= {\mathcal{N}}^{\sigma^{\phi}}$ is a semifinite trace. By assumption, we have $(1 + {\mathcal{D}}^{2})^{-1/2} \in {{\mathcal M}}$, so $R_{s}(\lambda) \in {{\mathcal M}}$, and $A_{0} A_{1} \cdots A_{m}$ is affiliated to ${{\mathcal M}}$. Hence, the estimates in this proof will be done in the von Neumann algebra ${{\mathcal M}}$, and we denote the trace norm, with respect to $\phi$ on ${{\mathcal M}}$, by $\Vert \cdot \Vert_{1}$. To prove statement 1, the strategy is to use the fundamental theorem of calculus, at first just doing norm convergence of integrals and norm differentiability. We abbreviate $R := R_s(\lambda)$, fix $k_0,\dots,k_m$ as in the statement, and with $\Re(r)>0$ sufficiently large, we have for any integer $M>m$ [$$\begin{aligned} &\frac{1}{2\pi i} \int_l \lambda^{-p/2-r}A_0 R A_1 R \cdots R A_m R d\lambda \nonumber \\ & \qquad = \frac{1}{2\pi i} \int_l \lambda^{-p/2-r}A_0RA_1R^{-1}R^2A_2R^{-2}\cdots R^mA_mR^{-m}R^{m+1}d\lambda \nonumber \\ &\qquad = \frac{(-1)^{M-m}}{2\pi i(p/2+r-1)(p/2+r-2)\cdots(p/2+r-(M-m))}\times \nonumber \\ & \qquad\qquad \times \int_l \frac{d^{M-m}}{d\lambda^{M-m}} \left(\lambda^{-p/2-r+(M-m)}\right) A_0RA_1R^{-1}R^2A_2R^{-2}\cdots R^mA_mR^{-m}R^{m+1}d\lambda \nonumber \\ & \qquad = \frac{\Gamma(p/2 + r - (M-m))}{2\pi i \, \Gamma(p/2+r)} \int_l \lambda^{-p/2-r+(M-m)} \sum_{j=0}^{M-m} \times \nonumber \\ & \qquad \qquad \times {\left(\begin{array}{c}}M-m \\ j {\end{array}\right)}\frac{d^j}{d\lambda^j}\! \left(A_0RA_1R^{-1}R^2A_2R^{-2}\cdots R^mA_mR^{-m}\right)\!\!\frac{d^{M-m-j}}{d\lambda^{M-m-j}}(R^{m+1})d\lambda. \label{eqn_ch3_ftc_derivs}\end{aligned}$$]{} Iterating the derivative $\frac{d}{d\lambda}(R^{m+1}) = -(m+1)R^{m+2}$ yields [$$\frac{d^{M-m-j}}{d\lambda^{M-m-j}}(R^{m+1}) = (-1)^{M-m-j} \left( \prod_{n = m+1}^{M+1-j} n \right) R^{M+1-j}.$$]{} Now we consider [$$\frac{d^j}{d\lambda^j} \left(A_0 R A_1 R^{-1} R^2 A_2 R^{-2} \cdots R^m A_m R^{-m} \right).$$]{} We would like to apply Lemma \[lem:bdd\] to this term, however recall that each $A_{j} \in {{\rm w{\text -}OP}}^{k_{j}}$ and not ${{\rm w{\text -}OP}}^{0}$. So, we rewrite [$$\begin{aligned} &A_0 R A_1 R^{-1} R^2 A_2 R^{-2} \cdots R^m A_m R^{-m} \\ &\qquad = A_0 (1 + {\mathcal{D}}^{2})^{-k_{0}/2} \Big(R (1 + {\mathcal{D}}^{2})^{k_{0}/2} A_1 (1 + {\mathcal{D}}^{2})^{-(k_{0}+k_{1})/2} R^{-1} \Big) \times \\ & \qquad \qquad \times \Big( R^2 (1 + {\mathcal{D}}^{2})^{(k_{0}+k_{1})/2} A_2 (1 + {\mathcal{D}}^{2})^{-(k_{0}+k_{1}+k_{2})/2} R^{-2} \Big) \times \cdots \times \\ & \qquad \qquad \qquad \times \Big( R^m (1 + {\mathcal{D}}^{2})^{(|k|-k_{m})/2} A_m (1 + {\mathcal{D}}^{2})^{-|k|/2} R^{-m} \Big) (1 + {\mathcal{D}}^{2})^{|k|/2}.\end{aligned}$$]{} By definition we have $A_{j} (1 + {\mathcal{D}}^{2})^{-k_{j}/2} \in {{\rm w{\text -}OP}}^{0}$, so using Lemma \[lem:bdd-conj\], we now find that $(1 + {\mathcal{D}}^{2})^{s} A_{j} (1 + {\mathcal{D}}^{2})^{-s-k_{j}/2} \in {{\rm w{\text -}OP}}^{0}$ for all $s \in {\mathbb{R}}$. Hence, we define [$$A'_{j} := (1 + {\mathcal{D}}^{2})^{{\tfrac{1}{2}}\sum_{n = 0}^{j-1} k_{n}} \, A_{j} \, (1 + {\mathcal{D}}^{2})^{- {\tfrac{1}{2}}\sum_{n = 0}^{j} k_{n}} \in {{\rm w{\text -}OP}}^{0},$$]{} so that [$$A_0 R A_1 R^{-1} R^2 A_2 R^{-2} \cdots R^m A_m R^{-m} = A'_0 R A'_1 R^{-1} R^2 A'_2 R^{-2} \cdots R^m A'_m R^{-m} (1 + {\mathcal{D}}^{2})^{|k|/2}.$$]{} The purpose of introducing $A'_j$ is to move all the ${{\rm w{\text -}OP}}^{k_{j}}$ behaviour into the factor $(1 + {\mathcal{D}}^{2})^{|k|/2}$ on the right. We now invoke Corollary \[cr:deriv\], and find that each factor $R^{j} A'_{j} R^{-j}$ is norm differentiable with respect to $\lambda$. Indeed, by Equation we have [$$\label{eqn_ch3_diff_rn_bdd} \frac{d^n}{d\lambda^n} R^{j} A'_{j} R^{-j} = R^{n} B(\lambda)$$]{} for $n \geq 0$, and some operator $B(\lambda)$ uniformly bounded in $s, \, \lambda$. So, we apply the chain rule to [$$\begin{aligned} &\frac{d^j}{d\lambda^j} \left(A'_0 R A'_1 R^{-1} R^2 A'_2 R^{-2} \cdots R^m A'_m R^{-m} \right) (1 + {\mathcal{D}}^{2})^{|k|/2} \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \frac{d^j}{d\lambda^j} \left(A_0 R A_1 R^{-1} R^2 A_2 R^{-2} \cdots R^m A_m R^{-m} \right),\end{aligned}$$]{} and use Equation to compute the derivatives. Then Lemma \[lemma\_bdd\_diff\_resolvent\] allows us to move each resolvent $R^{n}$ arising from Equation to the left, which gives [$$\frac{d^j}{d\lambda^j} \left(A_0 R A_1 R^{-1} R^2 A_2 R^{-2} \cdots R^m A_m R^{-m} \right) = R^{j} B_{j}(A'_{0}, \dots, A'_{m}) (1 + {\mathcal{D}}^{2})^{|k|/2},$$]{} where $B_{j}(A'_{0}, \dots, A'_{m}) \in {{\rm w{\text -}OP}}^{0}$ is uniformly bounded in $s, \, \lambda$. We absorb the constants $(-1)^{M-m-j} \left( \prod_{n = m+1}^{M+1-j} n \right)$ into $B_{j}(A'_{0}, \dots, A'_{m})$, and apply the derivative computations to Equation , which yields [$$\begin{aligned} & \frac{1}{2\pi i} \int_l \lambda^{-p/2-r}A_0 R A_1 R \cdots R A_m R d\lambda \\ &\qquad \qquad = \frac{\Gamma(p/2 + r - (M-m))}{2\pi i \, \Gamma(p/2+r)} \int_l \lambda^{-p/2-r+(M-m)} \sum_{j=0}^{M-m} \times\\ &\qquad \qquad \qquad \quad \times {\left(\begin{array}{c}}M-m \\ j {\end{array}\right)}R^{j} B_{j}(A'_{0}, \dots, A'_{m}) (1 + {\mathcal{D}}^{2})^{|k|/2} R^{M+1-j} d\lambda\\ &\qquad \qquad = \frac{\Gamma(p/2 + r - (M-m))}{2\pi i \, \Gamma(p/2+r)} \int_l \lambda^{-p/2-r+(M-m)} \sum_{j=0}^{M-m} \times\\ &\qquad \qquad \qquad \quad \times {\left(\begin{array}{c}}M-m \\ j {\end{array}\right)}R^{j} B_{j}(A'_{0}, \dots, A'_{m}) R^{-j} (1 + {\mathcal{D}}^{2})^{|k|/2} R^{|k|/2} R^{M+1-|k|/2} d\lambda,\end{aligned}$$]{} where the square roots use the principal branch of $\log$. For each $j$, the operator $R^{j} B_{j}(A'_{0}, \dots, A'_{m}) R^{-j}$ is uniformly bounded in $s,\,\lambda$ by Lemma \[lemma\_bdd\_diff\_resolvent\] and the uniform boundedness of $B_{j}(A'_{0}, \dots, A'_{m})$. Also, the operator $(1 + {\mathcal{D}}^{2})^{|k|/2} R^{|k|/2}$ is uniformly bounded in $s, \, \lambda$, so we are left with estimating $R^{M+1-|k|/2}$. The trace estimate for the resolvent in [@CPRS2 Lemma 5.3] states that for $M$ large enough and all $\epsilon>0$, there is a constant $C_\epsilon>0$ such that $$\Vert R^{M+1-|k|/2}\Vert_1\leq C_\epsilon ((1/2+s^2-a)^2+v^2)^{-(M+1-|k|/2)/2+(p/4+\epsilon)}. \label{eq:cee-eps}$$ This estimate, and the uniform boundedness of each $R^{j} B_{j}(A'_{0}, \dots, A'_{m}) R^{-j}$, implies that [$$\frac{1}{2\pi i}\int_l \lambda^{-p/2-r}A_0RA_1R\cdots RA_mRd\lambda \in \mathrm{Dom}(\phi)$$]{} for $|k|-2m + \epsilon<2\Re(r)$, for all $\epsilon>0$. We may apply this estimate only when $r \neq (M-j)-p/2$ as the prefactor [$$\frac{\Gamma(p/2 + r - (M-m))}{2\pi i \, \Gamma(p/2+r)} = \frac{1}{2\pi i(p/2+r-1)(p/2+r-2)\cdots(p/2+r-(M-m))}$$]{} has a pole at these points. So now we estimate [$$\int_0^\infty s^\alpha\,\phi \left(\frac{1}{2\pi i}\int_l \lambda^{-p/2-r}A_0RA_1R\cdots RA_mRd\lambda\right)ds$$]{} in trace norm (recall that we regard $\phi$ as a trace on the fixed point algebra ${{\mathcal M}}$). The calculations above show that the trace norm is bounded by [$$\begin{aligned} & \frac{|\Gamma(p/2 + r - (M-m))|}{2\pi |\Gamma(p/2+r)|} \int_0^\infty s^\alpha\, \int_{-\infty}^\infty \sqrt{a^2+v^2}^{-p/2-\Re(r)+(M-m)} \times \\ & \qquad \qquad \times \sum_{j=0}^{M-m}\binom{M-m}{j} \Vert R^{j}B_{j}(A'_0,A'_1,\dots,A'_m)R^{-j}\Vert_\infty\,\Vert R^{M+1-|k|/2}\Vert_1 dvds\\ &\leq \sum_{j=0}^{M-m}\binom{M-m}{j} \frac{|\Gamma(p/2 + r - (M-m))|}{2\pi |\Gamma(p/2+r)|} \, C_{\epsilon}' \times\\ &\times\int_0^\infty s^\alpha\, \int_{-\infty}^\infty \sqrt{a^2+v^2}^{(M-m)-p/2-\Re(r)} \sqrt{(1/2+s^2-a)^2+v^2}^{|k|/2-M-1+(p+\epsilon)/2} dvds,\end{aligned}$$]{} where the constant $C_\epsilon'$ incorporates the constant from the estimate in Equation and the constant coming from $\Vert R^{j}B_j(A'_0,A'_1,\dots,A'_m)R^{-j}\Vert_\infty\leq C$. Now by [@CPRS2 Lemma 5.4], the double integral converges for [$$(\alpha+|k|-M)+(p+\epsilon-M)<1 \quad \text{and} \quad (\alpha+|k|)-2m+\epsilon-2\Re(r)<-1.$$]{} The first constraint can always be satisfied by taking $M$ sufficiently large. The second holds precisely when $\alpha+|k|+1-2m<2\Re(r)$, by choosing $\epsilon$ small enough. Statement 2 of the lemma is proved just as above, with the extra $\hat{R}_s(\lambda)$ just estimated in operator norm, using [@CPRS2 Lemma 5.1]: [$$\Vert \hat{R}_s(\lambda)\Vert_\infty\leq (v^2+(1+s^2-a-s\Vert K \Vert_\infty)^2)^{-1/2},$$]{} and the general integral estimate [@CPRS2 Lemma 5.4]. Existence of the resolvent cocycle for weakly $QC^\infty$ modular spectral triples {#subsec:resolvent} ---------------------------------------------------------------------------------- First we explicitly define the resolvent cocycle associated to a modular spectral triple, again just working in the odd case. The definitions in the even case can be deduced from [@CPRS3]. \[expectation\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},{\mathcal{N}},\phi)$ be a weakly $QC^\infty$ odd modular spectral triple of dimension $p \geq 1$. Let $N=[p/2]+1$ be the least integer strictly greater than $p/2$. Let $m$ be an odd integer, $1 \leq m \leq 2N-1$, and let $A_j \in {{\rm w{\text -}OP}}^{k_j}$, $j=0,\dots,m$, be operators whose product $A_{0} A_{1} \cdots A_{m}$ is $\sigma^{\phi}_{t}$-invariant and affiliated to ${{\mathcal M}}$. For $2 \Re(r) > (k_{0} + \dots + k_{m}) + 1 - m$, $r\not\in{\mathbb{N}}-p/2$, define [$$\langle A_0,\dots,A_m\rangle_{m,s,r} := \frac{1}{2\pi i} \phi \left( \int_l \lambda^{-p/2-r} A_0R_s(\lambda)A_1\cdots A_mR_s(\lambda)d\lambda\right).$$]{} The resolvent cocycle $(\Phi^r_m)_{m=1,3,\dots,2N-1}$ is defined to be [$$\Phi_m^r(a_0,a_1,\dots,a_m) := \frac{-2\,\sqrt{2\pi i}}{\Gamma((m+1)/2)} \int_0^\infty s^m\langle a_0,[{\mathcal{D}},a_1],\dots, [{\mathcal{D}},a_m]\rangle_{m,s,r}ds,$$]{} for $a_{i} \in {\mathcal{A}}$ satisfying $a_{0} a_{1} \cdots a_{m} \in {{\mathcal M}}$. We observe that $\Phi_m^r$ is finite for $\Re(r)>(1-m)/2$, by Lemma \[lemma\_Bs\_trace\]. In this subsection we show that for weakly smooth modular spectral triples, $(\Phi_m^r)_{m=1,\dots,2N-1}$ defines a twisted $b,B$ cocycle modulo functions holomorphic in the half-plane $r > (1-p)/2$. We start by presenting the $s$- and $\lambda$-tricks, which are the main tools needed to prove continuity of the resolvent cocycle. These tricks appeared in [@CPRS2; @CPRS3; @CPRS4] without appropriate justification for the convergence of the derivatives in trace norm. In [@CGRS2] the justification was given with the aid of the pseudodifferential calculus. Here we present a different proof using only the weak $QC^\infty$ hypothesis. Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ be a weakly $QC^\infty$ odd modular spectral triple relative to $({\mathcal{N}},\phi)$ of dimension $p\geq 1$. For any integers $m\geq 0, k\geq 1$ and operators $A_0,\dots,A_m$ with $A_j\in {{\rm w{\text -}OP}}^{k_j}$, and $2\Re(r)>k+2\sum k_j-2m$, $r\not\in {\mathbb{N}}-p/2$, we may choose $r$ with $\Re(r)$ sufficiently large such that $$\begin{aligned} k\int_0^\infty s^{k-1}\langle A_0,\dots,A_m\rangle_{m,s,r}ds= -2\sum_{j=0}^m\int_0^\infty s^{k+1}\langle A_0,\dots,A_j, 1, A_{j+1},\dots,A_m\rangle_{m+1,s,r}ds.\end{aligned}$$ The only thing that needs justification is the trace norm derivative formula $$\frac{d}{ds}\langle A_0,\dots,A_m\rangle_{m,s,r}= 2s\sum_{k=0}^m\langle A_0,\dots,A_k,1,A_{k+1},\dots,A_m\rangle_{m+1,s,r}$$ for suitable $m,s,r$ and weak pseudodifferential operators $A_j$. So start with the difference quotient leading to one of the terms on the right hand side. $$\begin{aligned} &\frac{1}{2\pi i}\int_l \lambda^{-p/2-r}A_0R\cdots RA_k \left(\frac{R_{s+\epsilon}-R_s}{\epsilon}\right)A_{k+1}R\cdots RA_mRd\lambda\\ &=(2s+\epsilon)\frac{1}{2\pi i}\int_l \lambda^{-p/2-r}A_0R\cdots RA_k R_{s+\epsilon}R_sA_{k+1}R\cdots RA_mRd\lambda.\end{aligned}$$ Now repeat the trick of Lemma \[lemma\_Bs\_trace\], giving $$\begin{aligned} &=(2s+\epsilon)\frac{1}{2\pi i(p/2+r-1)(p/2+r-2)\cdots(p/2+r-(2M-1-m))}\times\\ &\times\int_l \lambda^{-p/2-r+(2M-1-m)} \sum_{j=0}^{2M-1-m}\binom{2M-1-m}{j}\\ &\times \frac{d^j}{d\lambda^j} \left(a_0Rda_1R^{-1}R^2da_2R^{-2}\cdots R^kA_kR^{-k}R_{s+\epsilon}R^{k+1}A_{k+1}R^{-k-1}\cdots R^mda_mR^{-m}\right)\times\\ &\times\frac{d^{2M-1-m-j}}{d\lambda^{2M-1-m-j}}(R^{m+1})d\lambda.\end{aligned}$$ Performing the derivatives yields a formula similar to that in the proof of Lemma \[lemma\_Bs\_trace\], but in place of the uniformly bounded $B_j$’s, we have uniformly bounded operators [*and*]{} one extra resolvent. Thus the same trace norm estimates apply and we see that the difference quotients converge in trace norm. Thus $\langle A_0,\dots,A_m\rangle_{m,s,r}$ is trace norm differentiable in $s$, and the derivative goes to zero as $\lambda\to a\pm i\infty$. The proof is completed by applying the fundamental theorem of calculus to $$\frac{d}{ds}\left(s^k\langle A_0,\dots,A_m\rangle_{m,s,r}\right). \qedhere$$ A completely analogous argument proves the $\lambda$-trick with our weak smoothness hypotheses. Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ be a weakly $QC^\infty$ odd modular spectral triple relative to $({\mathcal{N}},\phi)$ of dimension $p\geq 1$. For any integer $m\geq0$, operators $A_j\in {{\rm w{\text -}OP}}^{k_j}$, $j=0,\dots,m$, and $r$ such that $2\Re(r) > 2\sum k_j-2m$, $r\not\in {\mathbb{N}}-p/2$, we have $$-(p/2+r)\langle A_0,\dots,A_m\rangle_{m,s,r+1}=\sum_{k=0}^m\langle A_0,\dots,A_k, 1, A_{k+1},\dots,A_m\rangle_{m+1,s,r}.$$ \[hii\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ be a weakly $QC^\infty$ odd modular spectral triple relative to $({\mathcal{N}},\phi)$ of dimension $p\geq 1$. Let $m=1,3,\dots,2N-1$. Let ${\mathcal{A}}\otimes{\mathcal{A}}^{\otimes m}$ have the projective tensor product topology coming from the seminorms $a\mapsto \Vert WR^k\circ WL^l(a)\Vert_\infty +\Vert WR^k\circ WL^l([{\mathcal{D}},a])\Vert_\infty$ on ${\mathcal{A}}$, and restrict this topology to the subspace $({\mathcal{A}}\otimes{\mathcal{A}}^{\otimes m})^{\sigma^\phi}$ of $\sigma^\phi$ invariant tensors. (This can be called the weak $QC^\infty$-topology). Then the maps $$({\mathcal{A}}\otimes{\mathcal{A}}^{\otimes m})^{\sigma^\phi}\ni a_0\otimes\dots\otimes a_m\mapsto\big[r\mapsto\Phi_{m}^r(a_0,\dots,a_m)\big]\,,$$ are continuous multilinear maps from $ ({\mathcal{A}}\otimes{\mathcal{A}}^{\otimes m})^{\sigma^\phi}$ to the functions holomorphic in $\{z\in{\mathbb{C}}:\ \Re(z)>(1-m)/2,\,z\not\in{\mathbb{N}}-p/2\}$, with the topology of uniform convergence on compacta. Let us first fix $r\in {\mathbb{C}}$ with $\Re(r)>(1-m)/2$, and set $M=2N-1$. Lemma \[lemma\_Bs\_trace\] ensures that our functionals are finite for these values of $r$, and it is an exercise (see [@CPRS2 Lemma 7.4]) to show that these functionals are holomorphic there. Thus all that we need to do is to improve the estimates to prove continuity. We do this, following [@CPRS4 Proposition 5.18], using the $s$- and $\lambda$-tricks. We recall that we have defined $M=2N-1$. By applying successively the $s$- and $\lambda$-tricks (which commute) $(M-m)/2$ times each, we obtain $$\begin{aligned} \Phi_{m}^r(a_0,\dots,a_m)&= 2^{(M-m)/2}(M-n)!\prod_{l_1=1}^{(M-m)/2}\frac1{p/2+r-l_1}\prod_{l_2=1}^{(M-m)/2}\frac1{m+l_2}\nonumber\\ &\hspace{2cm}\times\sum_{|k|=M-m} \int_0^\infty s^M \langle a_0,1^{k_0},da_1,1^{k_1},\dots,da_m,1^{k_m}\rangle_{M,r-(M-m)/2,s}ds\,, \label{haa}\end{aligned}$$ where $1^{k_i}=1,1,\dots,1$ with $k_i$ entries. Since $M\leq p+1$, the poles associated to the prefactors are outside the region $\{z\in\mathbb C:\Re(z)>(1-m)/2\}$. Ignoring the prefactors, setting $n_i=k_i+1$ and $R:=R_{s,t}(\lambda)$, we need to deal with the integrals $$\int_0^\infty s^M\phi\Big( \gamma\int_l \lambda^{-p/2-r-(M-m)/2}a_0R^{n_0}da_1R^{n_1}\cdots da_mR^{n_m}d\lambda\Big)ds\,, \qquad |n|=M+1\,,$$ where $l$ is the vertical line $l=\{a+iv:v\in{\mathbb{R}}\}$ with $a=1/2$. To estimate the trace norm (using the trace given by restricting $\phi$ to the invariant subalgebra ${\mathcal{N}}^{\sigma^\phi}$) we first write $$\begin{aligned} &a_0R^{n_0}da_1R^{n_1}\cdots da_mR^{n_m}\\ &\quad=a_0(R^{n_0}da_1 R^{-n_0})(R^{n_0+n_1}da_2R^{-(n_0+n_1)})\cdots (R^{n_0+\cdots+n_{m-1}}da_mR^{-(n_0+\cdots+n_{m-1})}) R^{n_0+\cdots+n_m}.\end{aligned}$$ Then, using [@CPRS2 Lemma 5.2], and the fact that $ |n|=M+1$, for each $\epsilon>0$ we obtain $C_\epsilon>0$ such that $$\begin{aligned} &\|a_0R^{n_0}da_1R^{n_1}\cdots da_mR^{n_m}\|_1\\ &\leq \| a_0(R^{n_0}da_1 R^{-n_0})(R^{n_0+n_1}da_2R^{-(n_0+n_1)})\cdots (R^{n_0+\cdots+n_{m-1}}da_mR^{-(n_0+\cdots+n_{m-1})})\|_\infty \, \| R^{M+1}\|_1 \\ &\qquad\leq \|a_0R^{n_0}da_1R^{n_1}\cdots da_mR^{n_m}R^{-(M+1)}\|_\infty \,C_\epsilon\,((s^2+a^2)+v^2)^{-(M+1)/2+(p+\epsilon)/4}.\end{aligned}$$ The operator norm of the product yields a constant $C(a_0,a_1,\dots,a_m)$ depending on $a_0,a_1,\dots,a_m$, which varies continuously as the $a_j$ vary in a weak $QC^\infty$ continuous way. Integrating now shows that $$|\Phi^r_m(a_0,\dots,a_m)|\leq |f(r)|\,C_{\epsilon,M,m}C(a_0,a_1,\dots,a_m),$$ for a function $f$ continuous for $\Re(r)>(1-m)/2$, $r\not\in {\mathbb{N}}-p/2$ (coming from the prefactor and the integral) and some constant $C_{\epsilon,M,m}$. \[cocycle\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ be a weakly $QC^\infty$ odd modular spectral triple relative to $({\mathcal{N}},\phi)$ of dimension $p\geq 1$, and let $N=[p/2]+1$. The collection of functionals $\Phi^r=\{\Phi_m^r\}_{m=1}^{2N-1}$, $m$ odd, is such that $$(B^\sigma\Phi^r_{m+2}+b^\sigma\Phi^r_m)(a_0,\dots,a_{m+1})=0\ \ \ m=1,3,\dots,2N-3,\ \ \ (B^\sigma\Phi^r_1)(a_0)=0 \label{coc}$$ where the $a_i\in{\mathcal{A}}$, $\sigma=\sigma^\phi_i$ and $b^\sigma, B^\sigma$ are the twisted coboundary operators of cyclic cohomology. Moreover, there is a $\delta'$, $0<\delta'<1$ such that $b^\sigma\Phi_{2N-1}^r(a_0,\dots,a_{2N})$ is a holomorphic function of $r$ for $\Re(r)>-p/2+\delta'/2$. The proof is just as in [@CPRS2 Proposition 7.10], using the formula for the twisted coboundaries $b^\sigma,\,B^\sigma$, and the twisted tracial property of $\phi$, until we compute $$(b^\sigma\Phi^r_{2N-1})(a_0,\dots,a_{2N})=\int_0^\infty s^m\langle a_0,[{\mathcal{D}},a_1],\dots,[{\mathcal{D}}^2,a_j],\dots,[{\mathcal{D}},a_{m+1}] \rangle_{2N,r,s}ds. \label{eq:bdry}$$ Since ${\mathcal{A}}\subset {\rm OP}^0$ we have $[{\mathcal{D}}^2,{\mathcal{A}}]\subset {\rm OP}^1$, and then the proof is just as in [@CPRS2 Proposition 7.10]. We now specialise to the semifinite case so that we may relate the resolvent cocycle to the index problem (that is, to compute spectral flow). Proposition \[cocycle\] establishes that the resolvent cocycle is almost a cocycle, so we have the following theorem, proven just as in [@CPRS2]. \[thm:weak-index\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ be a weakly $QC^\infty$ odd semifinite spectral triple relative to $({\mathcal{N}},\tau)$ of dimension $p\geq 1$. Let $N = [p/2] + 1$ be the least positive integer strictly greater than $p/2$ and let $u \in {\mathcal{A}}$ be unitary. Then [$$sf_\tau({\mathcal{D}},u^*{\mathcal{D}}u) = \frac{1}{\sqrt{2\pi i}} \mathrm{Res}_{r = (1-p)/2} \left( \sum_{m=1,odd}^{2N-1}\Phi^r_m(Ch_m(u)) \right),$$]{} where $Ch_m(u)$ is defined to be $$Ch_m(u)=(-1)^{(m-1)/2}\,((m-1)/2)!\, u^*\otimes u\otimes\cdots\otimes u\qquad (m+1)\ \mbox{entries}.$$ This ‘resolvent index formula’ is proved as in [@CPRS2], where the differences for the weak $QC^{\infty}$ assumption are detailed above. [**Remark.**]{} In the even case we have a similar statement with $N=[(p+1)/2]$ and the sum runs over even integers from $m=0$ to $2N$; see [@CPRS3] for the $QC^\infty$ case and [@S] for the weakly $QC^\infty$ case. The resolvent index formula for modular spectral triples {#subsec:local-index} -------------------------------------------------------- Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ be a modular spectral triple relative to $({\mathcal{N}},\phi)$ with modular group $\sigma^\phi$, of spectral dimension $p\geq 1$, and weakly $QC^\infty$ so that $${\mathcal{A}}\subset {\rm OP}^0,\qquad [{\mathcal{D}},{\mathcal{A}}]\subset {{\rm w{\text -}OP}}^0.$$ Let $u\in M_n({\mathcal{A}})$ be unitary, $V:{\mathbb{T}}\to M_n({\mathbb{C}})$ a representation and suppose that $u$ is $\sigma^\phi\otimes Ad\,V$ invariant. Lemma \[lem:bob-the-builder\] constructs a semifinite spectral triple from $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ and $u$. The semifinite resolvent index formula, Theorem \[thm:weak-index\], then shows that the resolvent cocycle defined using the trace $\phi\otimes G$ is ‘almost’ a $b,\,B$ cocycle, and computes the spectral flow from ${\mathcal{D}}$ to $u{\mathcal{D}}u^*$. With $N=[p/2]+1$, we have $$sf_{\phi\otimes G}({\mathcal{D}}{\otimes}{\rm Id}_n, u({\mathcal{D}}{\otimes}{\rm Id}_n) u^*) =\frac{1}{\sqrt{2\pi i}}\operatorname{Res}_{r=(1-p)/2} \sum_{m=1,odd}^{2N-1}(\Phi_G)^r_m(Ch_m(u)), \label{eq:sf}$$ where $(\Phi_G)^r_m$ is the resolvent cocycle defined using the trace $\phi\otimes G$. In particular the sum on the right hand side of analytically continues to a deleted neighbourhood of $r=(1-p)/2$ with [*at worst*]{} a simple pole at $r=(1-p)/2$. We will compute the $G$ part of the trace, leaving us with a functional defined in terms of $\phi$. The Chern character of $u$ is defined to be the (infinite) sum $\oplus_j Ch_{2j+1}(u)\in HE_{2j+1}(M_N({\mathcal{A}}))$, the entire cyclic homology, with $$Ch_{2j+1}(u)=(-1)^j\,j!\, u^*\otimes u\otimes\cdots\otimes u^*\otimes u\qquad (2j+2)\ \mbox{entries}.$$ Now in [@W Lemma 4.1], Wagner has shown, in a slightly different context, that the map $$G_*:\oplus_jHE_{2j+1}(M_N({\mathcal{A}})^{{\sigma}\otimes Ad V})\to \oplus_jHE_{2j+1}^\sigma({\mathcal{A}})$$ to ${\sigma}$-twisted cyclic homology given on chains by $$G_*(T_0\otimes\cdots\otimes T_{2j+1}) =\sum_{i_0,i_1,\dots,i_{2j+2}}(V_{-i})_{i_{2j+2},i_0}(T_0)_{i_0,i_1}\otimes (T_1)_{i_1,i_2}\otimes\cdots \otimes (T_{2j+1})_{i_{2j+1},i_{2j+2}}$$ is an isomorphism. Now each equivariant unitary with class $[u]\in K_1^{\mathbb{T}}(A)$ is equivariant for its own representation of the circle. So it makes sense to regard the representation $V$ as part of the data, so $[u]=[u,V]$. We define $Ch_{2j+1}([u,V])\in HE_{2j+1}^\sigma({\mathcal{A}})$ by $$Ch_{2j+1}([u,V])=(-1)^j\,j!\,\sum_I (V_{-i})_{i_{2j+2},i_0}(u^*)_{i_0,i_1}\otimes (u)_{i_1,i_2}\otimes\cdots \otimes (u_{2j+1})_{i_{2j+1},i_{2j+2}}.$$ Then it is straightforward to check that this does indeed define an entire twisted cyclic cycle. Moreover it is immediate from the definitions that $$\begin{aligned} sf_{\phi\otimes G}({\mathcal{D}}{\otimes}{\rm Id}_n,u({\mathcal{D}}{\otimes}{\rm Id}_n) u^*) &=\frac{1}{\sqrt{2\pi i}}\operatorname{Res}_{r=(1-p)/2} \sum_{m=1,odd}^{2N-1}(\Phi_G)^r_m(Ch_m(u))\\ &=\frac{1}{\sqrt{2\pi i}}\operatorname{Res}_{r=(1-p)/2} \sum_{m=1,odd}^{2N-1}\Phi^r_m(Ch_m([u,V]))\end{aligned}$$ Here $\Phi^r_m$ is the resolvent cocycle given by the modular spectral triple. We now collect the results proved above into a statement describing the resolvent index formula for weakly smooth modular spectral triples. \[thm\_lif\_odd\] For a weakly $QC^\infty$ odd modular spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ relative to $({\mathcal{N}},\phi)$ of spectral dimension $p\geq 1$, and with $N=[p/2]+1$, the function valued cochain $(\Phi^r_m)_{m=1,\dots,2N-1}$ is a twisted cyclic cocycle modulo cochains with values in functions holomorphic in a half-plane containing $(1-p)/2$. Moreover, for $[u,V]\in K_1^{\mathbb{T}}({\mathcal{A}})$ with representative $u\in M_n({\mathcal{A}})$ we have [$$sf_{\phi\otimes G}({\mathcal{D}}{\otimes}{\rm Id}_n,u^*({\mathcal{D}}{\otimes}{\rm Id}_n) u) = \frac{1}{\sqrt{2\pi i}} \mathrm{Res}_{r = (1-p)/2} \left( \sum_{m=1,odd}^{2N-1}\Phi^r_m(Ch_m([u,V])) \right).$$]{} In particular, there is a well-defined map $$K_1^{\mathbb{T}}({\mathcal{A}})\mapsto {\mathbb{R}},\qquad [u,V]\mapsto sf_{\phi\otimes G}({\mathcal{D}}{\otimes}{\rm Id}_n,u^*({\mathcal{D}}{\otimes}{\rm Id}_n) u).$$ Though we have not proved it here, a similar result is true in the even case, see [@S]. \[thm\_lif\_even\] For a weakly $QC^\infty$ even modular spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},\gamma)$ relative to $({\mathcal{N}},\phi)$ of spectral dimension $p\geq 1$, and with $M=[(p+1)/2]$, the function valued cochain $(\Phi^r_m)_{m=0,\dots,2M}$ is a twisted cyclic cocycle modulo cochains with values in functions holomorphic in a half-plane containing $(1-p)/2$. Moreover, for $[P,V]\in K_0^{\mathbb{T}}({\mathcal{A}})$ with representative $P\in M_n({\mathcal{A}})$ and ${\mathcal{D}}_+=\frac{1}{4}(1-\gamma){\mathcal{D}}(1+\gamma)$ we have [$${\rm Index}_{\phi{\otimes}G}(P({\mathcal{D}}_+{\otimes}{\rm Id}_n)P)= \frac{1}{\sqrt{2\pi i}} \mathrm{Res}_{r = (1-p)/2} \left( \sum_{m=0,even}^{2M}\Phi^r_m(Ch_m([P,V])) \right).$$]{} In particular, there is a well-defined map $$K_0^{\mathbb{T}}({\mathcal{A}})\mapsto {\mathbb{R}},\qquad [P,V]\mapsto {\rm Index}_{\phi{\otimes}G}(P({\mathcal{D}}_+{\otimes}{\rm Id}_n)P).$$ [**Remark.**]{} The Chern character of an equivariant projection is $$Ch_0([P,V])={\rm Tr}(V_{-i}P),\quad Ch_{2k}([P,V])=(-1)^k\frac{(2k)!}{k!}\sum (V_{-i}(P-1/2))_{i_0i_1}\otimes P_{i_1i_2}\otimes\cdots\otimes P_{i_{2k}i_0}. \label{eq:even-chern}$$ Finally, the next two results relate the even index given by the resolvent index formula above back to the $K$-theory valued index pairing between the $KK$-class defined by the modular spectral triple and equivariant $K$-theory. Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}})$ be a modular spectral triple relative to $({\mathcal{N}},\phi)$. Let $J_\phi\subset {\mathcal{N}}$ be the ideal from Definition \[defn\_j\_phi\], and $J_\phi^\sim$ its unitisation. Let $E \in M_{k}(J_{\phi}^\sim)$ be a $\sigma^\phi {\otimes}Ad \, W$-invariant projection, for the associated representation $W \colon {\mathbb{T}}\to M_k({\mathbb{C}})$, so that $[E, W] \in K_{0}^{{\mathbb{T}}}(J_{\phi}^\sim)$. Define [$$\phi_{\ast}([E, W]) := (\phi {\otimes}G_{W})(E) \in [0, \infty],$$]{} where $G_{W}(T) = \operatorname{Tr}(W_{-i} T)$, for $T \in M_{k}({\mathbb{C}})$. Then $\phi_{\ast}$ is a well-defined map on the semigroup of Murray-von Neumann equivalence classes of equivariant projections in $J_\phi^\sim\otimes {\mathcal{K}}$, where ${\mathcal{K}}$ is the compact operators. The Grothendieck group of the sub-semigroup for which $\phi_*$ takes finite values is (isomorphic to) a subgroup of $K_{0}^{{\mathbb{T}}}(J_{\phi})$, and we call this the domain of $\phi_*$. Let $W_{1} \colon {\mathbb{T}}\to M_{n}({\mathbb{C}})$ and $W_{2} \colon {\mathbb{T}}\to M_{m}({\mathbb{C}})$ be representations. Let $E_{1} \in M_{n}(J_{\phi})$ denote a $\sigma {\otimes}Ad \, W_{1}$ projection, and let $E_{2} \in M_{m}(J_{\phi})$ denote a $\sigma^\phi {\otimes}Ad \, W_{2}$ projection. Suppose that $[E_{1}, W_{1}]$ and $[E_{2}, W_{2}]$ are equivariantly Murray-von Neumann equivalent ([@W Definition 3.1]), meaning there exists some $S \in M_{m \times n}(J_{\phi})$ such that [$$S^{\ast} S = E_{1}, \qquad S S^{\ast} = E_{2}, \qquad \text{and} \qquad W_{2, z} S = S W_{1, z} \quad \text{for all} \ \ z \in {\mathbb{T}}.$$]{} Then we compute [$$\begin{aligned} \phi_{\ast}([E_{1}, W_{1}]) &= (\phi {\otimes}G_{W_{1}})(E_{1}) = \phi \left( \operatorname{Tr}_{n}(W_{1, -i} E_{1}) \right) = \phi \left( \operatorname{Tr}_{n}(W_{1, -i} S^{\ast} S) \right) \\ &= \phi \left( \operatorname{Tr}_{n}(S W_{1, -i} S^{\ast}) \right).\end{aligned}$$]{} Now, by analytically continuing, $S W_{1, -i} = W_{2, -i} S$, so [$$\begin{aligned} \phi_{\ast}([E_{1}, W_{1}]) &= \phi \left( \operatorname{Tr}_{m}(W_{2, -i} S S^{\ast}) \right) = \phi \left( \operatorname{Tr}_{m}(W_{2, -i} E_{2}) \right) = \phi_{\ast}([E_{2}, W_{2}]).\end{aligned}$$]{} Using the universal property of the Grothendieck group, we see that the Grothendieck group of equivalence classes for which $\phi_*$ takes finite values may be regarded as a subgroup of $K_{0}^{{\mathbb{T}}}(J_{\phi})$. On this subgroup, $\phi_*$ is well-defined. \[thm\_ch3\_kk\_index\_resolvent\] Let $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},\gamma)$ be a weakly $QC^\infty$ even modular spectral triple relative to $({\mathcal{N}},\phi)$ of spectral dimension $p\geq 1$, and $[P,V]\in K_0^{\mathbb{T}}(A)$. Let $B_\phi\subset J_\phi$ be as in Definition \[def:bee-phi\], and let $i:B_\phi\to J_\phi$ be the inclusion. Then $i_*([P, V] {\otimes}_{A} [(B_\phi,F_{{\mathcal{D}}})]) \in K_{0}^{{\mathbb{T}}}(J_{\phi})$ is in the domain of $\phi_{\ast}$. Furthermore, [$$\phi_{\ast}(i_*([P, V] {\otimes}_{A} [(B_\phi,F_{{\mathcal{D}}})] )) = \operatorname{Res}_{r = (1-p)/2} \left( \sum_{m=0,even}^{2N}\Phi^{r}_{m}(Ch_{m}([P, V])) \right).$$]{} Given the modular spectral triple $({\mathcal{A}},{\mathcal{H}},{\mathcal{D}},\gamma,{\mathcal{N}},\phi)$, we define $[(B_\phi,F_{{\mathcal{D}}})] \in KK^{0, {\mathbb{T}}}(A, B_{\phi})$. Also, let $V \colon {\mathbb{T}}\to M_n({\mathbb{C}})$ be a representation and $P \in M_n({\mathcal{A}})$ a projection which is $\sigma^\phi \otimes Ad \,V$ invariant, so that we obtain a class $[P, V] \in K_{0}^{{\mathbb{T}}}(A)$. Define the projections [$$N_{\pm} := \ker(P ({\mathcal{D}}{\otimes}\mathrm{Id}_{n})^{\pm} P),$$]{} so that [$$\operatorname{\mathrm{Ind}}_{\phi {\otimes}G}(P ({\mathcal{D}}{\otimes}\mathrm{Id}_{n})^{+} P) = (\phi {\otimes}G)(N_{+}) - (\phi {\otimes}G)(N_{-}).$$]{} By the construction of the semifinite spectral triple $(C^\infty(P), {\mathcal{H}}{\otimes}{\mathbb{C}}^{n}, {\mathcal{D}}{\otimes}\mathrm{Id}_{n}, {{\mathcal M}}_{n}, \phi {\otimes}G)$, we have $N_{\pm} \in {\mathcal{K}}((M_{n}({\mathcal{N}}))^{\sigma^\phi {\otimes}Ad \, V}, \phi {\otimes}G)$, since the $N_{\pm}$ are kernel projections, and [$$\label{eqn_ch3_N_leq_pdp} N_{\pm} \leq (P + (P ({\mathcal{D}}{\otimes}\mathrm{Id}_{n}) P)^{2})^{-1}.$$]{} Also, the $\sigma^\phi {\otimes}Ad \, V$-invariance of $P$ implies the same invariance for $N_{\pm}$. We now want to show that we also have $N_{\pm} \in M_{n}(B_{\phi})$, so that they define classes in $K_{0}^{{\mathbb{T}}}(B_{\phi})$. We do this by proving that the operator $(P + (P ({\mathcal{D}}{\otimes}\mathrm{Id}_{n}) P)^{2})^{-1} \in M_{n}(B_{\phi})$, then applying Equation again to see that $N_{\pm} \in M_{n}(B_{\phi})$. For brevity let ${\mathcal{D}}_{n} := \mathrm{Id}_{n} {\otimes}{\mathcal{D}}$. Consider the operator $(P + (P{\mathcal{D}}_{n}P)^{2})^{-1} \colon P({\mathbb{C}}^{n} {\otimes}B_\phi) \rightarrow P({\mathbb{C}}^{n} {\otimes}B_\phi)$. The inverse exists because $P$ acts as the identity on $P({\mathbb{C}}^{n} {\otimes}B_\phi)$, and $(P {\mathcal{D}}_{n} P)^{2} \geq 0$. The adjointable endomorphisms on $P({\mathbb{C}}^{n} {\otimes}B_\phi)$ are $P M_{n}(M(B_\phi)) P$, where $M(B_\phi)$ is the multiplier algebra, while the compact operators are $P M_{n}(B_\phi) P$. A priori, we know only that $(P + (P{\mathcal{D}}_{n}P)^{2})^{-1}$ is bounded on $P({\mathbb{C}}^{n} {\otimes}B_\phi)$. To show the compactness of $(P + (P{\mathcal{D}}_{n}P)^{2})^{-1}$, we compute [$$\begin{aligned} (P + (P{\mathcal{D}}_{n}P)^{2})^{-1} &= (P + P[{\mathcal{D}}_{n}, P] {\mathcal{D}}_{n} P + P{\mathcal{D}}_{n}^{2}P)^{-1} \\ &= (P + P{\mathcal{D}}_{n}^{2}P)^{-1} + \Big[ (P + P[{\mathcal{D}}_{n}, P] [{\mathcal{D}}_{n}, P] P + P{\mathcal{D}}_{n}^{2}P)^{-1} - (P + P{\mathcal{D}}_{n}^{2}P)^{-1} \Big],\end{aligned}$$]{} where the last line follows from the observation [$$P[{\mathcal{D}}_{n}, P]P = P[{\mathcal{D}}_{n}, P^{2}]P = P(P[{\mathcal{D}}_{n}, P] + [{\mathcal{D}}_{n}, P]P)P = 2P[{\mathcal{D}}_{n}, P]P = 0,$$]{} so that $P[{\mathcal{D}}_{n}, P] [{\mathcal{D}}_{n}, P] P = P[{\mathcal{D}}_{n}, P] {\mathcal{D}}_{n} P$. Now, the algebraic result $\alpha^{-1} - \beta^{-1} = \beta^{-1} (\beta - \alpha) \alpha^{-1}$ yields [$$\begin{aligned} &(P + P[{\mathcal{D}}_{n}, P] [{\mathcal{D}}_{n}, P] P + P{\mathcal{D}}_{n}^{2}P)^{-1} - (P + P{\mathcal{D}}_{n}^{2}P)^{-1} \\ & \qquad \qquad \qquad = - (P + P{\mathcal{D}}_{n}^{2}P)^{-1} \Big( P[{\mathcal{D}}_{n}, P] [{\mathcal{D}}_{n}, P] P \Big) (P + P[{\mathcal{D}}_{n}, P] [{\mathcal{D}}_{n}, P] P + P{\mathcal{D}}_{n}^{2}P)^{-1}.\end{aligned}$$]{} Hence [$$(P + (P{\mathcal{D}}_{n}P)^{2})^{-1} = (P + P{\mathcal{D}}_{n}^{2}P)^{-1} B(P),$$]{} where $B(P)$ is a bounded operator, given by [$$B(P) = 1 - (P[{\mathcal{D}}_{n}, P] [{\mathcal{D}}_{n}, P] P) (P + P[{\mathcal{D}}_{n}, P] [{\mathcal{D}}_{n}, P] P + P{\mathcal{D}}_{n}^{2}P)^{-1}.$$]{} Now consider $(1 + {\mathcal{D}}_{n})^{-1} \colon {\mathbb{C}}^{n} {\otimes}B_\phi \rightarrow {\mathbb{C}}^{n} {\otimes}B_\phi$. Then we have [$$\label{eqn_ch3_parametrix} (P + P{\mathcal{D}}_{n}^{2}P) \, P (1 + {\mathcal{D}}_{n})^{-1} P = P + P [{\mathcal{D}}_{n}^{2}, P] (1 + {\mathcal{D}}_{n}^{2})^{-1} P=P+PC(P)(1+{\mathcal{D}}_n^2)^{-1/2}P.$$]{} Here $C(P)$ is bounded since $P\in M_n({\mathcal{A}})\subset OP^0$ (where $OP^0$ is defined using ${\mathcal{D}}_n$). Now $(1 + {\mathcal{D}}^{2})^{-1/2} \in B_\phi$ by definition, so $(1 + {\mathcal{D}}_{n}^{2})^{-1/2} \in M_{n}(B_\phi)$. Hence [$$P [{\mathcal{D}}_{n}^{2}, P] (1 + {\mathcal{D}}_{n}^{2})^{-1} P \in P M_{n}(B_\phi) P,$$]{} so Equation now implies that [$$(P + P{\mathcal{D}}_{n}^{2}P)^{-1} \in P M_{n}(B_\phi) P.$$]{} We know $B_\phi$ is an ideal in the endomorphisms, so Equation now implies that $N_{\pm} \in M_{n}(B_\phi)$. By the $\sigma^\phi {\otimes}Ad \, V$-invariance of $N_{\pm}$, we have $[N_{\pm}, V] \in K_{0}^{{\mathbb{T}}}(B_\phi)$. Then [$$\label{eqn_ch3_phig_phiast} (\phi {\otimes}G)(N_{+}) - (\phi {\otimes}G)(N_{-}) = \phi_{\ast}\left(i_*\left( [N_{+}, V] - [N_{-}, V] \right)\right).$$]{} In order to compare Equation to the Kasparov product $[P, V] {\otimes}_{A} [(B_\phi, F_{{\mathcal{D}}})]$, we rewrite the classes $[N_{\pm}, V]$ as Kasparov modules. We have [$$[N_{+}, V] - [N_{-}, V] = \left[ \left( N_{+}({\mathbb{C}}^{n} {\otimes}B_\phi) \oplus N_{-}({\mathbb{C}}^{n} {\otimes}B_\phi), \, 0, \, {\left(\begin{array}{cc}}N_{+} & 0 \\ 0 & -N_{-} {\end{array}\right)}, \, V \oplus V \right) \right],$$]{} where $N_{+}({\mathbb{C}}^{n} {\otimes}B_\phi) \oplus N_{-}({\mathbb{C}}^{n} {\otimes}B_\phi)$ is the right Hilbert $B_\phi$-module, $0$ is the operator, ${\left(\begin{array}{cc}}N_{+} & 0 \\ 0 & -N_{-} {\end{array}\right)}$ is the grading and $V \oplus V$ is the ${\mathbb{T}}$-action giving the equivariance. Now the operator $P (\mathrm{Id}_{n} {\otimes}F_{{\mathcal{D}}})^{+} P$ gives an isomorphism from $(1 - N_{+})({\mathbb{C}}^{n} {\otimes}B_\phi)$ to $(1 - N_{-})({\mathbb{C}}^{n} {\otimes}B_\phi)$. Hence, the Kasparov module constructed from $(1 - N_{\pm})({\mathbb{C}}^{n} {\otimes}B_\phi)$ and $P (\mathrm{Id}_{n} {\otimes}F_{{\mathcal{D}}})^{+} P$ has trivial class. Consequently, [$$\begin{aligned} & \left[ \left( N_{+}({\mathbb{C}}^{n} {\otimes}B_\phi) \oplus N_{-}({\mathbb{C}}^{n} {\otimes}B_\phi), \, 0, \, {\left(\begin{array}{cc}}N_{+} & 0 \\ 0 & -N_{-} {\end{array}\right)}, \, V \oplus V \right) \right] \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \left[ \left( {\mathbb{C}}^{n} {\otimes}B_\phi, \, P (\mathrm{Id}_{n} {\otimes}F_{{\mathcal{D}}}) P, \, \mathrm{Id}_{n} {\otimes}\gamma, \ V \right) \right].\end{aligned}$$]{} Finally, observe that, see [@B] for example, we have an explicit representative of the Kasparov product [$$[P, V] {\otimes}_{A} [( B_\phi,F_{{\mathcal{D}}})] = \left[ \left( {\mathbb{C}}^{n} {\otimes}B_\phi, \, P (\mathrm{Id}_{n} {\otimes}F_{{\mathcal{D}}}) P, \, \mathrm{Id}_{n} {\otimes}\gamma, \ V \right) \right].$$]{} Reiterating the above results, we have proved that [$$\begin{aligned} \operatorname{\mathrm{Ind}}_{\phi {\otimes}G}(P ({\mathcal{D}}{\otimes}\mathrm{Id}_{n})^{+} P) &= (\phi {\otimes}G)(N_{+}) - (\phi {\otimes}G)(N_{-}) \\ &= \phi_{\ast}\left(i_*\left( [N_{+}, V] - [N_{-}, V] \right)\right) \\ &= \phi_{\ast}\left(i_*\left( \left[ \left( {\mathbb{C}}^{n} {\otimes}B_\phi, \, P (\mathrm{Id}_{n} {\otimes}F_{{\mathcal{D}}}) P, \, \mathrm{Id}_{n} {\otimes}\Gamma, \ V \right) \right] \right)\right) \\ &= \phi_{\ast}\left(i_*\left( [P, V] {\otimes}_{A} [(B_\phi,F_{{\mathcal{D}}})] \right)\right). \qedhere\end{aligned}$$]{} The local index formula for the Podleś sphere {#sec:pods-chern} ============================================= In this section we will explicitly compute a twisted $b,B$ cocycle for the (modular) spectral triple over the Podleś sphere first investigated in [@DS]. We do this by applying the modified pseudodifferential calculus of [@NT] to the twisted resolvent cocycle of the previous section. Having done this, we construct some equivariant projections for a circle action arising from the Haar state and compute the index pairing via a residue formula, yielding a local index formula. The modular spectral triple for the Podleś sphere {#subsec:pods-mod} ------------------------------------------------- We first recall (see [@KS]) that the quantum algebra ${\mathcal{A}}= {\mathcal{O}}(SU_{q}(2))$, for $q \in [0, 1]$, is generated by elements $a,\, b,\, c,\, d$ modulo the relations $$\begin{aligned} ab = qba, \ \ \ ac = qca, \ \ \ bd &= qdb, \ \ \ cd = qdc, \ \ \ bc = cb \\ ad = 1 + qbc, \ & \ \ da = 1 + q^{-1}bc \\ a^{\ast} = d, \ \ \ b^{\ast} = -qc, \ &\ \ c^{\ast} = -q^{-1}b, \ \ \ d^{\ast} = a.\end{aligned}$$ The Podleś sphere, which we denote by ${\mathcal{B}}$, is (isomorphic to) the unital $\ast$-subalgebra of ${\mathcal{O}}(SU_{q}(2))$ generated by $q^{-1}ab$, $-cd$ and $-q^{-1}bc$. Recall that for each $l \in {\tfrac{1}{2}}{\mathbb{N}}_0$, there is a unique (up to unitary equivalence) irreducible corepresentation $V_l$ of the coalgebra ${\mathcal{A}}$ of dimension $2l+1$, and that ${\mathcal{A}}$ is cosemisimple. That is, if we fix a vector space basis in each of the $V_l$ and denote by $t^l_{r,s} \in {\mathcal{A}}$ the corresponding matrix coefficients, then we have the following analogue of the Peter-Weyl theorem. \[thm\_rep\_basis\] Let $ I_{l} := \{ -l, -l+1, \ldots, l-1,l \} $. Then the set $\{ {t_{r, s}^{l}} \ | \ l \in {\tfrac{1}{2}}{\mathbb{N}}_{0}, \ r, s \in I_{l} \}$ is a vector space basis of ${\mathcal{A}}$. This will be referred to as the Peter-Weyl basis. With a suitable choice of basis in $V_{{\tfrac{1}{2}}}$, one has [$$\begin{aligned} a &= {t_{-{\tfrac{1}{2}}, -{\tfrac{1}{2}}}^{{\tfrac{1}{2}}}}, & b &= {t_{-{\tfrac{1}{2}}, {\tfrac{1}{2}}}^{{\tfrac{1}{2}}}}, & c &= {t_{{\tfrac{1}{2}}, -{\tfrac{1}{2}}}^{{\tfrac{1}{2}}}}, & d &= {t_{{\tfrac{1}{2}}, {\tfrac{1}{2}}}^{{\tfrac{1}{2}}}}.\end{aligned}$$]{} The expressions for the Peter-Weyl basis elements as linear combinations of the polynomial basis elements can be found in [@KS Section 4.2.4]. The algebra ${\mathcal{A}}$ has a useful direct sum decomposition. For $m, \, n \in {\mathbb{Z}}$ where $m - n$ is even, define [$${\mathcal{A}}[m, n] := \mathrm{span} \{ a^{\frac{1}{2}(m+n)}b^{k + \frac{1}{2}(m-n)}c^{k}, \ b^{k + \frac{1}{2}(m-n)}c^{k}d^{-\frac{1}{2}(m+n)} \colon k + \min\{ 0, \, \tfrac{1}{2}(m-n) \} \in {\mathbb{N}}_{0} \},$$]{} and for $m - n$ odd, let ${\mathcal{A}}[m, n] := \{ 0 \}$. Then [$${\mathcal{A}}= \bigoplus_{m, n \in {\mathbb{Z}}} {\mathcal{A}}[m, n], \qquad \text{and} \qquad {\mathcal{A}}[m_{1}, n_{1}] \cdot {\mathcal{A}}[m_{2}, n_{2}] \subseteq {\mathcal{A}}[m_{1}+m_{2}, n_{1}+n_{2}].$$]{} With this notation, we have ${\mathcal{B}}= \bigoplus_{m \in {\mathbb{Z}}} {\mathcal{A}}[m, 0]$. Let $h$ be the Haar state on the universal $C^{\ast}$-completion of the $\ast$-algebra ${\mathcal{A}}$, whose value on the Peter-Weyl basis is $h({t_{r, s}^{l}}) = \delta_{l, 0}$. Define an automorphism $\vartheta$ on ${\mathcal{A}}$ by [$$\vartheta(a) = q^{2} a, \quad \vartheta(b) = b, \quad \vartheta(c) = c, \quad \vartheta(d) = q^{-2} d.$$]{} Then $\vartheta$ is the modular automorphism for the Haar state, in the sense that $h(\alpha \beta) = h(\vartheta(\beta) \alpha)$ for all $\alpha, \beta \in {\mathcal{A}}$. For all $n \in {\mathbb{Z}}$ define [$${\mathcal{H}}_{n} := L^{2} \left( {\rm span} \left\{ {t_{r, \frac{n}{2}}^{l}} \colon l \in \tfrac{n}{2} + {\mathbb{N}}_{0}, r \in I_{l} \right\}, h \right).$$]{} The left action of the dual Hopf algebra to ${\mathcal{A}}$ provides the unbounded operators ${\partial}_e \colon {\mathcal{H}}_{n} \rightarrow {\mathcal{H}}_{n+2}$ and ${\partial}_f \colon {\mathcal{H}}_{n} \rightarrow {\mathcal{H}}_{n-2}$ given by [$$\begin{aligned} {\partial}_e({t_{r, s}^{l}}) &= \sqrt{\left[ l + {\tfrac{1}{2}}\right]_{q}^{2} - \left[ s+ {\tfrac{1}{2}}\right]_{q}^{2} } \,\,{t_{r, s+1}^{l}}, & {\partial}_f({t_{r, s}^{l}}) &= \sqrt{\left[ l + {\tfrac{1}{2}}\right]_{q}^{2} - \left[ s- {\tfrac{1}{2}}\right]_{q}^{2} } \,\, {t_{r, s-1}^{l}}\end{aligned}$$]{} where our definition of the $q$-number $[a]_{q}$ is [$$[a]_{q} := \frac{q^{-a} - q^{a}}{q^{-1} - q}=Q(q^{-a} - q^{a}) \qquad \text{for any} \ a \in {\mathbb{C}},$$]{} and we abbreviated $Q := (q^{-1} - q)^{-1} \in (0, \infty)$. Finally, we define an unbounded linear operator $\Delta_{R}$ on ${\mathcal{A}}\subset \bigoplus {\mathcal{H}}_{n}$ by [$$\Delta_{R}({t_{r, s}^{l}}) := q^{2r} {t_{r, s}^{l}}.$$]{} Define the Hilbert space ${\mathcal{H}}:= {\mathcal{H}}_{1} \oplus {\mathcal{H}}_{-1}$, and represent ${\mathcal{B}}$ on ${\mathcal{H}}$ by left multiplication. The Hilbert space ${\mathcal{H}}$ is graded by $\gamma := {\left(\begin{array}{cc}}1 & 0 \\ 0 & -1 {\end{array}\right)}$. Define the weight $\Psi_R$ on ${\mathcal{B}}({\mathcal{H}})$ by $\Psi_R(T):={\rm Trace}(\Delta_R^{-1/2}T\Delta_R^{-1/2})$. Finally, on a suitable domain in ${\mathcal{H}}$, define the self-adjoint operator ${\mathcal{D}}:= {\left(\begin{array}{cc}}0 & {\partial}_{e} \\ {\partial}_{f} & 0 {\end{array}\right)}$. In fact $({\mathcal{B}},{\mathcal{H}},{\mathcal{D}},\gamma)$ defines an honest spectral triple, [@DS], (i.e. a modular spectral triple with von Neumann algebra $B({\mathcal{H}})$ and weight given by the operator trace) which is $\epsilon$-summable for all $\epsilon>0$. \[lemma\_podles\_wop\_zero\] The data $({\mathcal{B}},{\mathcal{H}},{\mathcal{D}},{\mathcal{B}}({\mathcal{H}}),\Psi_R)$ defines a weakly $QC^\infty$ even modular spectral triple, which is finitely summable with spectral dimension 2. We first show that the data produces a modular spectral triple. Certainly ${\mathcal{B}}$ is a separable $\ast$-subalgebra of ${\mathcal{B}}({\mathcal{H}})$, and by construction the modular automorphism group of $\Psi_{R}$ is $\vartheta^{-1}_{t}$, and ${\mathcal{B}}$ consists of analytic vectors for $\vartheta^{-1}_{t}$. Also, the commutators $\left[ {\mathcal{D}}, \beta \right]$ extend to bounded operators for all $\beta \in {\mathcal{B}}$, given by $$\label{eqn_podles_com} d\beta := \left[ {\mathcal{D}}, \beta \right] = {\left(\begin{array}{cc}}0 & q^{-\frac{1}{2}} {\partial}_{e}(\beta) \\ q^{\frac{1}{2}} {\partial}_{f}(\beta) & 0 {\end{array}\right)}.$$ We also observe that $\gamma = \gamma^{\ast}$ and $\gamma^{2} = I$, and by construction $\gamma {\mathcal{D}}+ {\mathcal{D}}\gamma = 0$. Now, for $T \in {\mathcal{B}}({\mathcal{H}})$ set $T^{+} = (1 + \gamma) T (1 + \gamma) /4$ and $T^{-} = (1 - \gamma) T (1 - \gamma) /4$. From the definition of the operator trace, and using the normalised Peter-Weyl basis $\xi^{l}_{r, j} := {t_{r, j}^{l}} / \Vert {t_{r, j}^{l}} \Vert$, we find for $T \geq 0$ that [$$\Psi_{R}(T) = \sum_{l, r} q^{-2r} \left( {\langle}\xi^{l}_{r, 1/2}, T^{+} \xi^{l}_{r, 1/2} {\rangle}+ {\langle}\xi^{l}_{r, -1/2}, T^{-} \xi^{l}_{r, -1/2} {\rangle}\right).$$]{} We first observe from the above formula that the finite rank operators are in the domain of $\Psi_{R}$, so $\Psi_{R}$ is semifinite. Next, we see that $\Psi_{R}$ is a sum of vector states with orthogonal support, as the Peter-Weyl basis is orthogonal. Hence $\Psi_{R}$ is strictly semifinite. The Peter-Weyl basis elements can be used to construct a common eigenbasis for ${\mathcal{D}}$ and $\Delta_{R}$ on ${\mathcal{H}}$, so the spectral projections of ${\mathcal{D}}$ and $\Delta_{R}$ commute. We conclude that ${\mathcal{D}}$ is affiliated to the fixed point algebra ${{\mathcal M}}:= {\mathcal{B}}({\mathcal{H}})^{\vartheta^{-1}}$. All that remains to be proved is that $(1 + {\mathcal{D}}^{2})^{-1/2} \in {\mathcal{K}}({{\mathcal M}}, \Psi_{R}|_{{{\mathcal M}}})$. To establish this, we observe that ${\mathcal{D}}^{2}$ has the following spectral projections [$${\mathcal{P}}_{l} {\left(\begin{array}{c}}{t_{r, 1/2}^{k}} \\ 0 {\end{array}\right)}:= \delta_{l, k} {\left(\begin{array}{c}}{t_{r, 1/2}^{k}} \\ 0 {\end{array}\right)}, \qquad \qquad {\mathcal{P}}_{l} {\left(\begin{array}{c}}0 \\ {t_{r, -1/2}^{k}} {\end{array}\right)}:= \delta_{l, k} {\left(\begin{array}{c}}0 \\ {t_{r, -1/2}^{k}} {\end{array}\right)},$$]{} for $l = 1/2, \, 3/2, \, \ldots$, which correspond to the eigenvalues $[l + {\tfrac{1}{2}}]_{q}^{2}$. Now, $\Psi_{R}({\mathcal{P}}_{l}) = \sum_{r = -l}^{l} q^{-2r} = [2l+1]_{q}$, and the sum $\sum_{l = \frac{1}{2}, \frac{3}{2}, \ldots} (1 + [l + {\tfrac{1}{2}}]_{q}^{2})^{-1/2} < \infty$ implies that [$$(1 + {\mathcal{D}}^{2})^{-1/2} = \sum_{l = \frac{1}{2}, \frac{3}{2}, \ldots} (1 + [l + {\tfrac{1}{2}}]_{q}^{2})^{-1/2} {\mathcal{P}}_{l}$$]{} is norm convergent. Hence $(1 + {\mathcal{D}}^{2})^{-1/2} \in {\mathcal{K}}({{\mathcal M}}, \Psi_{R}|_{{{\mathcal M}}})$, and so $({\mathcal{B}},{\mathcal{H}},{\mathcal{D}},\gamma,{\mathcal{B}}({\mathcal{H}}),\Psi_{R})$ is a modular spectral triple. The spectral dimension is shown to be 2 in [@KW]. We now prove that ${\mathcal{B}}\subset \mathrm{OP}^{0}, \, [{\mathcal{D}}, {\mathcal{B}}] \subset {{\rm w{\text -}OP}}^{0}$, so that the modular spectral triple is weakly $QC^\infty$. The first statement is proved in [@NT Proposition 3.2]. To prove the second statement we show that for all $\beta \in {\mathcal{B}}$ and $z \in {\mathbb{C}}$, the operators $(1+{\mathcal{D}}^{2})^{-z} [{\mathcal{D}}, \beta] (1+{\mathcal{D}}^{2})^{z} \in {\mathcal{B}}({\mathcal{H}})$, as per Lemma \[lem:bdd-conj\]. We begin by observing that ${\mathcal{D}}^{2}$ has eigenbasis given by [$$\begin{aligned} {\mathcal{D}}^{2} {\left(\begin{array}{c}}t^{l}_{r, \frac{1}{2}} \\ 0 {\end{array}\right)}&= [l + \tfrac{1}{2}]^{2} {\left(\begin{array}{c}}t^{l}_{r, \tfrac{1}{2}} \\ 0 {\end{array}\right)}, & {\mathcal{D}}^{2} {\left(\begin{array}{c}}0 \\ t^{l}_{r, -\frac{1}{2}} {\end{array}\right)}&= [l + \tfrac{1}{2}]^{2} {\left(\begin{array}{c}}0 \\ t^{l}_{r, -\tfrac{1}{2}} {\end{array}\right)}.\end{aligned}$$]{} Now we consider $\beta \in {\mathcal{B}}$ to be of the form $t^{p}_{r,0}$ (as finite linear combinations of these elements span ${\mathcal{B}}$). Then the commutator $[{\mathcal{D}}, t^{p}_{r, 0}] = \left(\begin{array}{cc} 0 & \kappa_{1} t^{p}_{r, 1} \\ \kappa_{2} t^{p}_{r, -1} & 0 \end{array}\right)$ for some coefficients $\kappa_{1}, \kappa_{2}$. We expand the product $t^{p}_{r, 0} t^{l}_{s, \frac{1}{2}}$ using the Clebsch-Gordan coefficients (see [@DLSSV; @KS]), giving [$$(1+{\mathcal{D}}^{2})^{-z} t^{p}_{r, 0} (1+{\mathcal{D}}^{2})^{z} {\left(\begin{array}{c}}t^{l}_{s, \frac{1}{2}} \\ 0 {\end{array}\right)}= (1 + [l + \tfrac{1}{2}]_{q}^{2})^{z} \sum_{k = |l-p|}^{l+p} (1 + [k + \tfrac{1}{2}]_{q}^{2})^{-z} c_{s, r}^{p, l, k} {\left(\begin{array}{c}}t^{k}_{s+r, \frac{1}{2}} \\ 0 {\end{array}\right)}$$]{} where $c_{s, r}^{p, l, k}$ is some product of Clebsch-Gordan coefficients that will be subsumed later. The norm of $(1+{\mathcal{D}}^{2})^{-z} t^{p}_{r, 0} (1+{\mathcal{D}}^{2})^{z} \left(\begin{array}{c} t^{l}_{s, \frac{1}{2}} \\ 0 \end{array}\right)$ can be computed using the orthogonality of the Peter-Weyl basis, so $$\begin{aligned} \left\Vert (1+{\mathcal{D}}^{2})^{-z} t^{p}_{r, 0} (1+{\mathcal{D}}^{2})^{z} \left(\begin{array}{c} t^{l}_{s, \frac{1}{2}} \\ 0 \end{array}\right) \right\Vert^{2} &= \sum_{k = |l-p|}^{l+p} \left( \frac{1 + [l + \tfrac{1}{2}]^{2}}{1 + [k + \tfrac{1}{2}]^{2}} \right)^{2\Re(z)} \left| c_{s, r}^{p, l, k} \right|^{2} \left\Vert \left(\begin{array}{c} t^{k}_{s+r, \frac{1}{2}} \\ 0 \end{array}\right) \right\Vert^{2}.\end{aligned}$$ Let $M_{l, p} := \max_{|l-p| \leq k \leq l+p} \{ \left( (1 + [l + \tfrac{1}{2}]^{2})/(1 + [k + \tfrac{1}{2}]^{2}) \right)^{2\Re(z)} \}$, then $$\begin{aligned} \left\Vert (1+{\mathcal{D}}^{2})^{-z} t^{p}_{r, 0} (1+{\mathcal{D}}^{2})^{z} \left(\begin{array}{c} t^{l}_{s, \frac{1}{2}} \\ 0 \end{array}\right) \right\Vert^{2} &\leq M_{l, p} \sum_{k = |l-p|}^{l+p} \left| c_{s, r}^{p, l, k} \right|^{2} \left\Vert \left(\begin{array}{c} t^{k}_{s+r, \frac{1}{2}} \\ 0 \end{array}\right) \right\Vert^{2} = M_{l, p} \left\Vert t^{p}_{r, 0} \left(\begin{array}{c} t^{l}_{s, \frac{1}{2}} \\ 0 \end{array}\right) \right\Vert^{2}.\end{aligned}$$ It remains to show that there exist finite $M_{p}$ such that $M_{l, p} \leq M_{p}$ for all $l \geq 0$. Let $\varepsilon_{k} = Q(1 - q^{2k})$ so that $[k]_{q} = q^{-k}\varepsilon_{k}$. Then for all $l \geq p + {\tfrac{1}{2}}$, $$\frac{1 + [l + \tfrac{1}{2}]^{2}}{1 + [|l-p| + \tfrac{1}{2}]^{2}} = \frac{\varepsilon_{l + \frac{1}{2}}^{2} + q^{2l+1}}{q^{2p}\varepsilon_{l -p + \frac{1}{2}}^{2} + q^{2l+1}}, \quad \implies \quad \frac{\varepsilon_{1}^{2}}{1 + q^{2p}Q^{2}} \leq \frac{1 + [l + \tfrac{1}{2}]^{2}}{1 + [|l-p| + \tfrac{1}{2}]^{2}} \leq \frac{Q^{2} + 1}{q^{2p} \varepsilon_{1}^{2}}.$$ It follows that the operator $(1+{\mathcal{D}}^{2})^{-1} t^{p}_{r, 0} (1+{\mathcal{D}}^{2})$ is bounded on the set of vectors of the form $\left(\begin{array}{c} t^{l}_{s, \frac{1}{2}} \\ 0 \end{array}\right)$. The same calculation can be performed for the vectors $\left(\begin{array}{c} 0 \\ t^{l}_{s, -\frac{1}{2}} \end{array}\right)$, and again for the operators $\left(\begin{array}{cc} 0 & t^{p}_{r, 1} \\ 0 & 0 \end{array}\right)$ and $\left(\begin{array}{cc} 0 & 0 \\ t^{p}_{r, -1} & 0 \end{array}\right)$, completing the proof. The residue cocycle for the Podleś sphere {#subsec:head-aches} ----------------------------------------- Lemma \[lemma\_podles\_wop\_zero\] shows that the modular spectral triple $({\mathcal{B}}, {\mathcal{H}}, {\mathcal{D}})$ satisfies the hypotheses of Theorem \[thm\_lif\_even\]. Hence we can employ the resolvent cocycle to compute index pairings with equivariant $K$-theory, or at least those classes which can be represented as projections over ${\mathcal{B}}$. As $\Delta_{R}$ implements the modular automorphism $\vartheta$, then it follows that the weight $\Psi_{R}$ is $\vartheta^{-1}$-twisted. The resolvent cocycle, which we denote by, $( \phi_{m}^{r})_{m=0,2}$ therefore lives is $\vartheta^{-1}$-twisted cohomology. To simplify the computation of the resolvent cocycle, we would like to have a version of the pseudodifferential calculus. A simple replacement for the pseudodifferential calculus for this example was presented in [@NT]. \[lemma\_podles\_pdc\] Define $\chi := {\left(\begin{array}{cc}}q^{-1} & 0 \\ 0 & q {\end{array}\right)}$ on $\mathcal{H}_{1} \oplus \mathcal{H}_{-1}$. For any $\beta \in {\mathcal{B}}$ there exists an analytic function $z \mapsto M(z) \in {{\rm w{\text -}OP}}^0\subset {\mathcal{B}}({\mathcal{H}})$ with at most linear growth on vertical strips such that $$|D|^{-z} d\beta = d\beta \chi^{z} |D|^{-z} + M(z) |D|^{-z-1} = \chi^{-z} d\beta |D|^{-z} + M(z) |D|^{-z-1}.$$ We can now use this pseudodifferential calculus to simplify the computation of the resolvent cocycle, $(\phi_{0}^{r},\, \phi_{2}^{r})$, and arrive at a twisted version of the local index formula. The first simplification we make is to discard the $1$ from the resolvent, replacing $R_{s}(\lambda) = (\lambda - (1 + s^{2} + {\mathcal{D}}^{2}))^{-1}$ with $R_{s}(\lambda) = (\lambda - (s^{2} + {\mathcal{D}}^{2}))^{-1}$. This is possible because ${\mathcal{D}}$ is invertible in this example, so we can employ the method of [@CPRS4 Section 5.3], in particular [[@CPRS4 Proposition 5.20]]{}. (The transgression cochain defined there is well defined for weakly $QC^\infty$ modular spectral triples since ${\mathcal{D}}\in {\rm OP}^1$, by essentially the same arguments as we employed for the resolvent cocycle). Removing the $1$ from the resolvents modifies the resolvent cocycle by coboundaries and cochains holomorphic at $r = -1/2$. Before proceeding, we recall the detailed summability properties of the spectral triple $({\mathcal{B}}, {\mathcal{H}}, {\mathcal{D}})$ computed in [@KW]. \[lemma\_podles\_basic\_trace\] The function $r \mapsto \operatorname{Tr}(\Delta_{R}^{-1} {\tfrac{1}{2}}(1 \pm \gamma) |{\mathcal{D}}|^{-3-2r})$ has a meromorphic continuation to the complex plane which is holomorphic for $\Re(r) > -1/2$, and has a simple pole at $r = -1/2$. Furthermore, for all $\beta \in {\mathcal{B}}$ we have the following equality [$$\mathrm{Res}_{r = -1/2} \operatorname{Tr}(\Delta_{R}^{-1} {\tfrac{1}{2}}(1 \pm \gamma) \beta |{\mathcal{D}}|^{-3-2r}) = \frac{(q^{-1}-q)}{2 \ln q^{-1}} \varepsilon(\beta)$$]{} where $\varepsilon$ is the counit of ${\mathcal{A}}$ restricted to ${\mathcal{B}}$ satisfying $\varepsilon({t_{i, 0}^{l}}) = \delta_{i, 0}$. The degree zero component $\phi^r_0$ of the resolvent cocycle is computed from the definition using the Cauchy formula and [@CPRS3]. This yields the formula, for $a_0\in{\mathcal{B}}$, $$\phi_{0}^{r}(a_{0}) = 2 \int_{0}^{\infty} \operatorname{Tr}\left(\Delta_{R}^{-1} \gamma \frac{1}{2\pi i} \int_{l} \lambda^{-1-r} a_{0} R_{s}(\lambda) d\lambda \right) ds= \frac{\Gamma(\frac{1}{2})\Gamma(r+\frac{1}{2})}{\Gamma(r+1)} \operatorname{Tr}\left(\Delta_{R}^{-1} \gamma a_{0} |{\mathcal{D}}|^{-2r-1} \right).$$ Since $\operatorname{Tr}\left(\Delta_{R}^{-1} \gamma |{\mathcal{D}}|^{-2r-1} \right) = 0$ for all sufficiently large $r \in {\mathbb{R}}$, then taking the trivial continuation to the whole real line gives $\phi_{0}^{r}(I) = 0$ for all $r \in {\mathbb{R}}$. This is the only evaluation of $\phi_{0}^{r}$ needed to compute the index pairing later on. We can compute $\mathrm{Res}_{r=-1/2}\,\phi^r_0$ explicitly, but as the calculation is quite lengthy and we do not require this full computation for computing the index pairing, we just quote the result; see [@S] for full details. The functional $\phi_0:=\mathrm{Res}_{r=-1/2}\,\phi^r_0$ is supported on the span of the powers $(bc)^k$, $k=0,1,2\dots$. We have seen that $\phi_0(I)=0$. For the remaining values we have [$$\mathrm{Res}_{r = -\frac{1}{2}} \phi_{0}^{r}(bc) = \frac{1}{2}\left( 1 - \frac{\gamma}{\ln q^{-1}}\right) - qQ$$]{} where $\gamma$ is Euler’s constant. For $k=0,1,2,\dots$ and with $h$ the Haar state [$$\mathrm{Res}_{r = -\frac{1}{2}} \phi_{0}^{r}((bc)^{k+2}) = \frac{(-1)^{k+1} q^{k+1}}{1 - q^{2k+2}} = \frac{(-1)^{k+1}}{q^{-k-1} - q^{k+1}} = -\frac{h((bc)^{k})}{q^{-1} - q}.$$]{} We now compute the degree two term $\phi^r_2$ of the resolvent cocycle starting with the definition, $$\phi_{2}^{r}(a_{0}, a_{1}, a_{2}) = 4 \int_{0}^{\infty} s^{2} \operatorname{Tr}\left(\Delta_{R}^{-1} \gamma \frac{1}{2\pi i}\int_{l} \lambda^{-1-r} a_{0} R_{s}(\lambda) da_{1} R_{s}(\lambda) da_{2} R_{s}(\lambda) d\lambda \right) ds.$$ We proceed by employing the pseudodifferential calculus described in Lemma \[lemma\_podles\_pdc\] in order to rewrite the expression $a_{0} R_{s}(\lambda) da_{1} R_{s}(\lambda) da_{2} R_{s}(\lambda)$ by moving all the resolvents to the right. From Lemma \[lemma\_podles\_pdc\], for each $\beta \in {\mathcal{B}}$ there exist bounded operators $M_{1}, \, M_{2}$ such that $$\begin{aligned} (\lambda - s^{2} - {\mathcal{D}}^{2}) d\beta &= d\beta (x\lambda - s^{2}- \chi^{-2} {\mathcal{D}}^{2}) + M_{1}|{\mathcal{D}}| ,\\ &(\lambda - s^{2} - \chi^{-2} {\mathcal{D}}^{2})d\beta = d\beta (\lambda - s^{2} - {\mathcal{D}}^{2}) + M_{2}|{\mathcal{D}}|.\end{aligned}$$ This gives the formulae $$\begin{aligned} R_{s}(\lambda) d\beta &= d\beta (\lambda - s^{2} - \chi^{-2} {\mathcal{D}}^{2})^{-1} - R_{s}(\lambda)M_{1} |{\mathcal{D}}| (\lambda - s^{2} - \chi^{-2} {\mathcal{D}}^{2})^{-1} \nonumber\\ d\beta R_{s}(\lambda) &= (\lambda - s^{2} - \chi^{-2} {\mathcal{D}}^{2})^{-1} d\beta + (\lambda - s^{2} - \chi^{-2} {\mathcal{D}}^{2})^{-1}M_{2} |{\mathcal{D}}| R_{s}(\lambda). \label{eq:commute-chi}\end{aligned}$$ Observe that the operators $R_{s}(\lambda)M_{1} |{\mathcal{D}}| (\lambda - s^{2} - \chi^{-2} {\mathcal{D}}^{2})^{-1}$ and $(\lambda - s^{2} - \chi^{-2} {\mathcal{D}}^{2})^{-1}M_{2} |{\mathcal{D}}| R_{s}(\lambda)$ are in ${{\rm w{\text -}OP}}^{-3}$ by Lemma \[lemma\_podles\_pdc\]. Using this observation, and Equation , we can move all the resolvents to the right, and in doing so we only introduce errors which are functions holomorphic at $r=-1/2$. More precisely, for any $a_0,\,a_1,\,a_2\in{\mathcal{B}}$, we obtain the formula $$\begin{aligned} \phi_{2}^{r}(a_{0}, a_{1}, a_{2}) &= 4 \int_{0}^{\infty} s^{2} \operatorname{Tr}\left(\Delta_{R}^{-1} \gamma a_{0} da_{1} da_{2} \frac{1}{2\pi i}\int_{l} \lambda^{-1-r} R_{s}(\lambda) (\lambda - s^{2} - \chi^{-2} {\mathcal{D}}^{2})^{-1} R_{s}(\lambda) d\lambda \right) ds \\ &= 4 \int_{0}^{\infty} s^{2} \operatorname{Tr}\left(\Delta_{R}^{-1} \gamma a_{0} da_{1} da_{2} \frac{1}{2\pi i}\int_{l} \lambda^{-1-r} (\lambda - s^{2} - \chi^{-2} {\mathcal{D}}^{2})^{-1} R_{s}(\lambda)^2 d\lambda \right) ds\end{aligned}$$ modulo functions holomorphic at $r = -1/2$. The integral [$$\label{eqn_cauchy_integral} \int_{l} \lambda^{-1-r} (\lambda - s^{2} - \chi^{-2} {\mathcal{D}}^{2})^{-1} R_{s}(\lambda)^{2} d\lambda$$]{} is evaluated on the spectra of the operators $(\lambda - s^{2} - \chi^{-2} {\mathcal{D}}^{2})^{-1}$ and $R_{s}(\lambda)^{2}$. We want to use the Cauchy integral formula, however because there are two poles to consider, $\lambda = s^{2} + {\mathcal{D}}^{2}$ and $\lambda = s^{2} + \chi^{-2} {\mathcal{D}}^{2}$, we outline the process. First note that $\chi$ and ${\mathcal{D}}^{2}$ are commuting operators with discrete spectra, and they can be simultaneously diagonalised with respect to the direct sum ${\mathcal{H}}= {\mathcal{H}}_{1} \oplus {\mathcal{H}}_{-1}$. Indeed, the integrand in Equation has the eigenbasis [$$\label{eqn_ch4_int_basis} \left\{ {\left(\begin{array}{c}}{t_{i, 1/2}^{l}} \\ 0 {\end{array}\right)}, \ {\left(\begin{array}{c}}0 \\ {t_{i, -1/2}^{l}} {\end{array}\right)}\colon l - {\tfrac{1}{2}}\in {\mathbb{N}}_{0}, i \in \{ -l, -l+1, \ldots, l \} \right\},$$]{} on which $\chi^{-2}$ simply acts via multiplication by the scalar $q^{\pm 2} \neq 1$. We specialise to the eigenbasis in ${\mathcal{H}}_{1}$, where $\chi^{-2}$ acts via multiplication by $q^{2}$. The argument we now present can be applied analogously to the remaining eigenbasis elements. On each eigenvector, the integral in Equation reduces to a scalar integral over $\lambda$, where we may apply the usual Cauchy integral formula. The integrand of this scalar integral has two poles; on the eigenbasis elements in ${\mathcal{H}}_{1}$ described in Equation these poles are $\lambda_{1} = s^{2} + q^{2} [l + {\tfrac{1}{2}}]_{q}^{2}$ and $\lambda_{2} = s^{2} + [l + {\tfrac{1}{2}}]_{q}^{2}$, with $\lambda_{1} < \lambda_{2}$. The contour of integration $l$ is a vertical line to the left of the spectrum for all $s\geq 0$. (8,5)(-5,-2.25) (0,-2)[(0,1)[4]{}]{} (-2,0)[(1,0)[8]{}]{} (0.5,-2)[(0,1)[4]{}]{} (0.5,-2)(0.3,0)[15]{}[(1,0)[0.2]{}]{} (0.5,2)(0.3,0)[15]{}[(1,0)[0.2]{}]{} (0.36, 1)[$\blacktriangle$]{} (0.36, -1)[$\blacktriangle$]{} (3.95, -2.1)[$\blacktriangleleft$]{} (3.95, 1.9)[$\blacktriangleright$]{} (1.5, 0) (2.5, 0) (0.25, 0.1)[$a$]{} (0.6, -1.5)[$l$]{} (1.38, 0.2)[$\lambda_{1}$]{} (2.38, 0.2)[$\lambda_{2}$]{} In order to apply the Cauchy integral formula, we modify the contour $l$ by adding a vertical line $l' = \{ a' + iv \colon \lambda_{1} < a' < \lambda_{2}, \ \ v \in {\mathbb{R}}\}$ between the poles $\lambda_{1}$ and $\lambda_{2}$. We integrate along this line in both directions, allowing us to split the integral into two parts. We denote by $\Gamma_1$ the contour obtained by going up along $l$ and down along $l'$ , and denote by $\Gamma_2$ the remaining integration along $l'$. Lemma \[lemma\_Bs\_trace\] shows that the horizontal dashed integrals go to zero. (8,5)(-5,-2.25) (0,-2)[(0,1)[4]{}]{} (-2,0)[(1,0)[8]{}]{} (0.5,-2)[(0,1)[4]{}]{} (2.1,-2)[(0,1)[4]{}]{} (0.5,-2)(0.3,0)[15]{}[(1,0)[0.2]{}]{} (0.5,2)(0.3,0)[15]{}[(1,0)[0.2]{}]{} (0.36, 1)[$\blacktriangle$]{} (0.36, -1)[$\blacktriangle$]{} (1.96, 1.2)[$\blacktriangle$]{} (1.96, -0.8)[$\blacktriangle$]{} (1.96, 0.8)[$\blacktriangledown$]{} (1.96, -1.2)[$\blacktriangledown$]{} (3.95, -2.1)[$\blacktriangleleft$]{} (3.95, 1.9)[$\blacktriangleright$]{} (1.2, -2.1)[$\blacktriangleleft$]{} (1.2, 1.9)[$\blacktriangleright$]{} (1.5, 0) (2.5, 0) (0.25, 0.1)[$a$]{} (0.6, -1.5)[$l$]{} (2.2, -1.5)[$l'$]{} (1.38, 0.2)[$\lambda_{1}$]{} (2.38, 0.2)[$\lambda_{2}$]{} (1.1, 1)[$\Gamma_{1}$]{} (3, 1)[$\Gamma_{2}$]{} Define [$$f_{1}(\lambda) := \lambda^{-1-r} R_{s}(\lambda) R_{s}(\lambda), \qquad \qquad f_{2}(\lambda) := \lambda^{-1-r} (\lambda - s^{2} - \chi^{-2} {\mathcal{D}}^{2})^{-1}.$$]{} By construction, the function $f_{1}$ is holomorphic on the domain defined by the contour $\Gamma_{1}$, while $f_{2}$ is holomorphic on the domain defined by $\Gamma_{2}$. Therefore, we may apply the (scalar) Cauchy integral formula for each contour $\Gamma_{1}$ and $\Gamma_{2}$, so we write [$$\int_{l} \lambda^{-1-r} (\lambda - s^{2} - \chi^{-2} {\mathcal{D}}^{2})^{-1} R_{s}(\lambda)^{2} d\lambda = \int_{\Gamma_{1}} \frac{f_{1}(\lambda)}{ (\lambda - s^{2} - \chi^{-2} {\mathcal{D}}^{2})} d\lambda + \int_{\Gamma_{2}} \frac{f_{2}(\lambda)}{(\lambda - s^{2} - {\mathcal{D}}^{2})^{2}} d\lambda$$]{} This yields [$$\begin{aligned} &\frac{1}{2\pi i} \int_{l} \lambda^{-1-r} (\lambda - s^{2} - \chi^{-2} {\mathcal{D}}^{2})^{-1} R_{s}(\lambda)^{2} d\lambda = f_{1}(s^{2}+\chi^{-2}{\mathcal{D}}^{2}) + f_{2}'(s^{2}+{\mathcal{D}}^{2}) \\ &\qquad\qquad= (s^{2}+\chi^{-2}{\mathcal{D}}^{2})^{-1-r}(\chi^{-2} - 1)^{-2} {\mathcal{D}}^{-4} -(1+r)(s^{2}+{\mathcal{D}}^{2})^{-2-r}(1 - \chi^{-2})^{-1}{\mathcal{D}}^{-2} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-(s^{2}+{\mathcal{D}}^{2})^{-1-r}(1 - \chi^{-2})^{-2} {\mathcal{D}}^{-4}.\end{aligned}$$]{} Inserting the result of the Cauchy integral into our previous formula for $\phi_2^r$, and evaluating the $s$-integrals (see for example [[@CPRS3 Lemma 5.9]]{}) yields [$$\begin{aligned} \phi_{2}^{r}(a_{0}, a_{1}, a_{2}) &= \frac{\sqrt{\pi} \Gamma\left(r - \frac{1}{2} \right)}{\Gamma(r+1)} \operatorname{Tr}\bigg(\Delta_{R}^{-1} \gamma a_{0} da_{1} da_{2} \chi^{2r-1}|{\mathcal{D}}|^{-2r+1} (\chi^{-2} - 1)^{-2} {\mathcal{D}}^{-4} \bigg) \\ & \quad - \frac{\sqrt{\pi} \Gamma\left(r + \frac{1}{2} \right)}{\Gamma(r+1)} \operatorname{Tr}\bigg(\Delta_{R}^{-1} \gamma a_{0} da_{1} da_{2} |{\mathcal{D}}|^{-2r-1} (1 - \chi^{-2})^{-1}{\mathcal{D}}^{-2} \bigg) \\ & \quad - \frac{\sqrt{\pi} \Gamma\left(r - \frac{1}{2} \right)}{\Gamma(r+1)} \operatorname{Tr}\bigg(\Delta_{R}^{-1} \gamma a_{0} da_{1} da_{2} |{\mathcal{D}}|^{-2r+1} (1 - \chi^{-2})^{-2} {\mathcal{D}}^{-4} \bigg),\end{aligned}$$]{} modulo functions holomorphic at $r=-1/2$. Writing $\Gamma(r + {\tfrac{1}{2}}) = (r - {\tfrac{1}{2}}) \Gamma(r-{\tfrac{1}{2}})$ and collecting terms, $\phi_{2}^{r}(a_{0}, a_{1}, a_{2}) $ is given by [$$\begin{aligned} & \frac{\sqrt{\pi} \Gamma\left(r - \frac{1}{2} \right)}{\Gamma(r+1)} \operatorname{Tr}\bigg(\Delta_{R}^{-1} \gamma a_{0} da_{1} da_{2} |{\mathcal{D}}|^{-2r-3} (1 - \chi^{-2})^{-2} \Big( \chi^{2r-1} - (r - {\tfrac{1}{2}}) (1 - \chi^{-2}) - 1 \Big) \bigg) .\end{aligned}$$]{} Observe that [$$\begin{aligned} \Gamma\left(r - \tfrac{1}{2} \right) & \Big( \chi^{2r-1} - (r - {\tfrac{1}{2}}) (1 - \chi^{-2}) - 1 \Big) = \Gamma\left(r - \tfrac{1}{2} \right) \Big( \chi^{-2} (\chi^{2r+1} - 1) - (r + {\tfrac{1}{2}}) (1 - \chi^{-2}) \Big) \nonumber\\ &= (r + {\tfrac{1}{2}}) \Gamma\left(r - \tfrac{1}{2} \right) \Big( \chi^{-2} (1 + \ln \chi^{2}) - 1 \Big) + \Gamma\left(r - \tfrac{1}{2} \right) \chi^{-2} \sum_{n = 2}^{\infty} \frac{(\ln \chi^{2})^{n}}{n!} (r + {\tfrac{1}{2}})^{n}. \label{eq:gamma}\end{aligned}$$]{} Now, $\Gamma\left(r - \tfrac{1}{2} \right)$ has a simple pole at $r = -1/2$, so the function in Equation is holomorphic at $r = -1/2$ with constant term $1 - \chi^{-2} (1 + \ln \chi^{2})$. Therefore [$$\begin{aligned} \phi_2(a_0,a_1,a_2):=\mathrm{Res}_{r = -1/2} \phi_{2}^{r}(a_{0}, a_{1}, a_{2}) &= \mathrm{Res}_{r = -1/2} \operatorname{Tr}(\Delta_{R}^{-1} \gamma a_{0} da_{1} da_{2} |{\mathcal{D}}|^{-2r-3} C )\end{aligned}$$]{} where $C = ( 1 - \chi^{-2} (1 + \ln \chi^{2}) ) (1 - \chi^{-2})^{-2} = ( \chi^{2} - 1 - \ln \chi^{2} ) (\chi - \chi^{-1})^{-2}$ is a constant diagonal matrix. Finally, Equation yields [$$a_{0} da_{1} da_{2} = {\left(\begin{array}{cc}}a_{0} {\partial}_{e}(a_{1}) {\partial}_{f}(a_{2}) & 0 \\ 0 & a_{0} {\partial}_{f}(a_{1}) {\partial}_{e}(a_{2}) {\end{array}\right)},$$]{} and so invoking Lemma \[lemma\_podles\_basic\_trace\] gives the formula [$$\begin{aligned} &\phi_{2}(a_{0}, a_{1}, a_{2}) \label{eqn_phi_two} \\ &= \frac{1}{2 (q^{-1} - q) \ln q^{-1}} \Big( ( q^{-2} - 1 - \ln q^{-2}) \varepsilon(a_{0} {\partial}_{e}(a_{1}) {\partial}_{f}(a_{2})) - ( q^{2} - 1 - \ln q^{2}) \varepsilon(a_{0} {\partial}_{f}(a_{1}) {\partial}_{e}(a_{2})) \Big). \nonumber\end{aligned}$$]{} Some equivariant projections and their Chern characters {#subsec:projns} ------------------------------------------------------- Our aim is to construct representatives in the equivariant $K$-theory $K_{0}^{{\mathbb{T}}}({\mathcal{B}})$ for the action of the modular automorphism group $\Psi_{R}$, which is given by $\sigma^{\Psi_{R}}_{t}=\vartheta_{t}^{-1}$. These representatives will take the form of projections $p \in M_{N \times N}({\mathcal{B}})$ together with a representation $V \colon {\mathbb{T}}\rightarrow M_{N \times N}({\mathbb{C}})$ such that $p$ is $\vartheta^{-1} {\otimes}\mathrm{Ad}(V)$-invariant. See [@W] for similar constructions. For $n \in \frac{1}{2} {\mathbb{Z}}$, define [$$T_{n}^{l} := \left( \begin{array}{c} {t_{l, n}^{l}} \\ {t_{l-1, n}^{l}} \\ \vdots \\ {t_{-l, n}^{l}} \end{array} \right), \qquad \text{and} \qquad P_{n} := T_{n}^{|n|} T_{n}^{|n|\ast}.$$]{} More explicitly, [$$P_{n} = \left( \begin{array}{cccc} {t_{|n|, n}^{|n|}}{t_{|n|, n}^{|n|\ast}} & {t_{|n|, n}^{|n|}}{t_{|n|-1, n}^{|n|\ast}} & \cdots & {t_{|n|, n}^{|n|}}{t_{-|n|, n}^{|n|\ast}} \\ {t_{|n|-1, n}^{|n|}}{t_{|n|, n}^{|n|\ast}} & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \vdots \\ {t_{-|n|, n}^{|n|}}{t_{|n|, n}^{|n|\ast}} & \cdots & \cdots & {t_{-|n|, n}^{|n|}}{t_{-|n|, n}^{|n|\ast}} \end{array} \right), \qquad (P_{n})_{r, s} = {t_{|n|-r+1, n}^{|n|}}{t_{|n|-s+1, n}^{|n|\ast}}.$$]{} By construction, $P_{n}^{\ast} = P_{n}$ and $P_{n} \in M_{(2|n|+1) \times (2|n|+1)}({\mathcal{B}})$. Furthermore, [$$T_{n}^{l\ast}T_{n}^{l} = \sum_{p = -l}^{l} {t_{p, n}^{l\ast}} {t_{p, n}^{l}} ={\rm Id}_n$$]{} and hence $P_{n}^{2} = P_{n}$, so $P_{n}$ is a projection. Now define $\lambda_{j} = q^{-2j+2} \in [1, \infty)$ for $j \in \{ 1, 2, \ldots, 2|n|+1 \}$, and define $V_n:{\mathbb{T}}\to M_{2|n|+1}({\mathbb{C}})$ by [$$V_{n}(t) := \left( \begin{array}{cccc} \lambda_{1}^{it} & 0 & \cdots & 0 \\ 0 & \lambda_{2}^{it} & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & \lambda_{2|n|+1}^{it} \end{array} \right).$$]{} While we have defined $V_n$ to be a real action on $M_{(2|n|+1) \times (2|n|+1)}({\mathbb{C}})$, the action is periodic and so induces a circle action. Observe that [$$\sigma_{t}((P_{n})_{r, s}) = q^{-2it(|n|-r+1 - (|n|-s+1))} {t_{|n|-r+1, n}^{|n|}}{t_{|n|-s+1, n}^{|n|\ast}} = q^{2it(r-s)} (P_{n})_{r, s}$$]{} and [$$\mathrm{Ad}(V_{n}(t))(P_{n})_{r, s} = (V_{n}(t) P_{n} V_{n}^{-1}(t))_{r, s} = \lambda_{r}^{it} (P_{n})_{r, s} (\lambda_{s}^{it})^{-1} = q^{2it(s-r)} (P_{n})_{r, s}.$$]{} So $P_{n}$ is $\vartheta^{-1}{\otimes}\mathrm{Ad}(V_{n})$-invariant and we define the weight $G \colon M_{(2|n|+1) \times (2|n|+1)}({\mathbb{C}}) \rightarrow {\mathbb{C}}$ by [$$G(X) := \operatorname{Tr}(V_{n,-i} X)$$]{} for $X \in M_{(2|n|+1) \times (2|n|+1)}({\mathbb{C}})$, and $(V_{n,-i})_{k, m} = \delta_{k, m} q^{-2k+2}$. We have demonstrated that $P_{n}$ is an equivariant projection for the circle action represented by $V_n$, and therefore defines a class in $K_{0}^{{\mathbb{T}}}({\mathcal{B}})$. We now write down the Chern character of this representative, Equation . \[lemma\_chern\_pn\] The Chern character of $[P_{n},V_n]$ is [$$\begin{aligned} Ch_{0}([P_{n}, V_n]) &= q^{2(n - |n|)} I, \\ Ch_{2}([P_{n}, V_n]) &= \!-2 \sum_{k_{0}, k_{1}, k_{2}=0}^{2|n|} \!\!\!q^{-2k_{0}}\left( {t_{|n|-k_{0}, n}^{|n|}}{t_{|n|-k_{1}, n}^{|n|\ast}}-\frac{1}{2}\delta_{k_0,k_1}\right) {\otimes}{t_{|n|-k_{1}, n}^{|n|}}{t_{|n|-k_{2}, n}^{|n|\ast}} {\otimes}{t_{|n|-k_{2}, n}^{|n|}}{t_{|n|-k_{0}, n}^{|n|\ast}}.\end{aligned}$$]{} Using Equation we have [$$Ch_{0}([P_{n}, V_n]) = \sum_{k_{0}, k_{1}=1}^{2|n|+1} (V_{n,-i})_{k_{1}, k_{0}} (P_{n})_{k_{0}, k_{1}} = \sum_{k_{0}, k_{1}=1}^{2|n|+1} \delta_{k_{1}, k_{0}} q^{-2k_{0}+2} {t_{|n|-k_{0}+1, n}^{|n|}}{t_{|n|-k_{1}+1, n}^{|n|\ast}}.$$]{} Now we apply the formulae for adjoints $({t_{i, j}^{l}})^* = (-q)^{j-i} {t_{-i, -j}^{l}}$ and ${t_{i, j}^{l}} = (-q)^{j-i} ({t_{-i, -j}^{l}})^*$, along with the unitary relations for the Peter-Weyl basis elements, [@KS Proposition 16, Chapter 4], to obtain [$$\begin{aligned} Ch_{0}([P_{n}, V_n]) &= \sum_{k=0}^{2|n|} q^{-2k} q^{2(n - |n| + k)} {t_{k-|n|, -n}^{|n|\ast}}{t_{k-|n|, -n}^{|n|}} = q^{2(n - |n|)} I.\end{aligned}$$]{} Finally, $$\begin{aligned} Ch_{2}([P_{n}, V_n]) &= -\frac{2!}{1!} \sum_{k_{0}, k_{1}, k_{2}, k_{3}=1}^{2|n|+1} (V_{n,-i})_{k_{3}, k_{0}} (P_{n} - \tfrac{1}{2})_{k_{0}, k_{1}} {\otimes}(P_{n})_{k_{1}, k_{2}} {\otimes}(P_{n})_{k_{2}, k_{3}} \\ &= -2 \sum_{k_{0}, k_{1}, k_{2}=1}^{2|n|+1} (V_{n,-i})_{k_{0}, k_{0}} (P_{n} - \tfrac{1}{2})_{k_{0}, k_{1}} {\otimes}(P_{n})_{k_{1}, k_{2}} {\otimes}(P_{n})_{k_{2}, k_{0}} \\ &= -2 \!\!\!\!\!\!\sum_{k_{0}, k_{1}, k_{2}=0}^{2|n|} \!\!\!\!\!q^{-2k_{0}}\!\! \left( {t_{|n|-k_{0}, n}^{|n|}}{t_{|n|-k_{1}, n}^{|n|\ast}} - \tfrac{1}{2} \delta_{k_{0}, k_{1}} \right)\! {\otimes}{t_{|n|-k_{1}, n}^{|n|}}{t_{|n|-k_{2}, n}^{|n|\ast}} \!{\otimes}{t_{|n|-k_{2}, n}^{|n|}}{t_{|n|-k_{0}, n}^{|n|\ast}}. \qed\end{aligned}$$ The index pairing ----------------- The resolvent index formula established in Section \[subsec:local-index\] proves that the index pairing defined by the modular spectral triple $({\mathcal{B}}, {\mathcal{H}}, {\mathcal{D}}, {\mathcal{B}}({\mathcal{H}}), \Psi_{R})$ and the equivariant $K$-theory class defined by the projection $P_{n}$ is given by the formula [$$\mathrm{Ind}_{\Psi_{R} {\otimes}G}(P_{n} ({\mathcal{D}}{\otimes}\mathrm{Id}_{2|n|+1})^{+} P_{n}) = \phi_{2}(Ch_{2}([P_{n}, V_n])) + \phi_{0}(Ch_{0}([P_{n}, V_n])) .$$]{} Now that we have explicit formulae for the cocycle $(\phi_0,\phi_2)$ and the cycle $Ch_{\ast}([P_{n}, V_n])$, the computation is straightforward. \[prop\_phi\_p\_two\] The evaluation of $\phi_{2}$ on $Ch_{2}([P_{n}, V_n])$ is [$$\phi_{2}(Ch_{2}([P_{n}, V_n])) = q^{-2|n|} [2n]_{q}.$$]{} Recalling the formula for $\phi_{2}$ from Equation and the expression for $Ch_{2}([P_{n}, V_n])$ from Lemma \[lemma\_chern\_pn\], we compute [$$\begin{aligned} & \varepsilon\left(\left( {t_{|n|-k_{0}, n}^{|n|}}{t_{|n|-k_{1}, n}^{|n|\ast}} - \tfrac{1}{2} \delta_{k_{0}, k_{1}} \right) {\partial}_{e}({t_{|n|-k_{1}, n}^{|n|}}{t_{|n|-k_{2}, n}^{|n|\ast}}) {\partial}_{f}({t_{|n|-k_{2}, n}^{|n|}}{t_{|n|-k_{0}, n}^{|n|\ast}}) \right) \\ & \qquad = \left( \delta_{|n|-k_{0}, n} \delta_{|n|-k_{1}, n} - \tfrac{1}{2} \delta_{k_{0}, k_{1}} \right) \varepsilon({\partial}_{e}({t_{|n|-k_{1}, n}^{|n|}}{t_{|n|-k_{2}, n}^{|n|\ast}})) \varepsilon({\partial}_{f}({t_{|n|-k_{2}, n}^{|n|}}{t_{|n|-k_{0}, n}^{|n|\ast}})) \\ & \qquad = \delta_{k_{0}, k_{1}} \left( \delta_{k_{0}, |n|-n} - \tfrac{1}{2} \right) \varepsilon({\partial}_{e}({t_{|n|-k_{1}, n}^{|n|}}{t_{|n|-k_{2}, n}^{|n|\ast}})) \varepsilon({\partial}_{f}({t_{|n|-k_{2}, n}^{|n|}}{t_{|n|-k_{1}, n}^{|n|\ast}})).\end{aligned}$$]{} We observe that this expression is zero for the case $n = 0$, because ${\partial}_{e}(I) = {\partial}_{f}(I) = 0$. So for the remainder we consider only nonzero $n$. Observe that ${t_{|n|-k_{2}, n}^{|n|}}{t_{|n|-k_{1}, n}^{|n|\ast}} = ({t_{|n|-k_{1}, n}^{|n|}}{t_{|n|-k_{2}, n}^{|n|\ast}})^{\ast}$. Now we use the property that $(g \triangleright \alpha)^{\ast} = S(g)^{\ast} \triangleright \alpha^{\ast}$ for all $g \in {\mathcal{U}}_{q}(su_{2})$ and $\alpha \in {\mathcal{A}}$, so that $\varepsilon({\partial}_{e}(\alpha^{\ast})) = -q \varepsilon({\partial}_{f}(\alpha))$ and $\varepsilon({\partial}_{f}(\alpha^{\ast})) = -q^{-1} \varepsilon({\partial}_{e}(\alpha))$. Then [$$\begin{aligned} & \varepsilon\left(\left( {t_{|n|-k_{0}, n}^{|n|}}{t_{|n|-k_{1}, n}^{|n|\ast}} - \tfrac{1}{2} \delta_{k_{0}, k_{1}} \right) {\partial}_{e}({t_{|n|-k_{1}, n}^{|n|}}{t_{|n|-k_{2}, n}^{|n|\ast}}) {\partial}_{f}({t_{|n|-k_{2}, n}^{|n|}}{t_{|n|-k_{0}, n}^{|n|\ast}}) \right) \\ & \qquad = -q^{-1} \delta_{k_{0}, k_{1}} \left( \delta_{k_{0}, |n|-n} - \tfrac{1}{2} \right) \varepsilon({\partial}_{e}({t_{|n|-k_{1}, n}^{|n|}}{t_{|n|-k_{2}, n}^{|n|\ast}}))^{2},\end{aligned}$$]{} and similarly [$$\begin{aligned} & \varepsilon\left(\left( {t_{|n|-k_{0}, n}^{|n|}}{t_{|n|-k_{1}, n}^{|n|\ast}} - \tfrac{1}{2} \delta_{k_{0}, k_{1}} \right) {\partial}_{f}({t_{|n|-k_{1}, n}^{|n|}}{t_{|n|-k_{2}, n}^{|n|\ast}}) {\partial}_{e}({t_{|n|-k_{2}, n}^{|n|}}{t_{|n|-k_{0}, n}^{|n|\ast}}) \right) \\ & \qquad = -q^{-1} \delta_{k_{0}, k_{1}} \left( \delta_{k_{0}, |n|-n} - \tfrac{1}{2} \right) \varepsilon({\partial}_{e}({t_{|n|-k_{2}, n}^{|n|}}{t_{|n|-k_{1}, n}^{|n|\ast}}))^{2}.\end{aligned}$$]{} Using the twisted derivation property of ${\partial}_{e}$ on ${\mathcal{A}}$, we find, for $r, s \in \{ 0, \ldots, 2|n|\}$, [$$\begin{aligned} \varepsilon({\partial}_{e}({t_{|n|-r, n}^{|n|}}{t_{|n|-s, n}^{|n|\ast}}))^{2} &= \varepsilon({\partial}_{e}({t_{|n|-r, n}^{|n|}}) {\partial}_{k}({t_{|n|-s, n}^{|n|\ast}}) + {\partial}_{k}^{-1}({t_{|n|-r, n}^{|n|}}) {\partial}_{e}({t_{|n|-s, n}^{|n|\ast}}))^{2} \\ &= q^{-2n} \left( \varepsilon({\partial}_{e}({t_{|n|-r, n}^{|n|}})) \delta_{s, |n|-n} + \delta_{r, |n|-n} \varepsilon({\partial}_{e}({t_{|n|-s, n}^{|n|\ast}})) \right)^{2} \\ &= q^{-2n} \left( \varepsilon({\partial}_{e}({t_{|n|-r, n}^{|n|}})) \delta_{s, |n|-n} - q\delta_{r, |n|-n} \varepsilon({\partial}_{f}({t_{|n|-s, n}^{|n|}})) \right)^{2} \\ &= q^{-2n} \left( \varepsilon(\kappa^{|n|}_{n+1} {t_{|n|-r, n+1}^{|n|}}) \delta_{s, |n|-n} - q\delta_{r, |n|-n} \varepsilon(\kappa^{|n|}_{n} {t_{|n|-s, n-1}^{|n|}}) \right)^{2} \\ &= q^{-2n} \left( \kappa^{|n|}_{n+1} \delta_{r, |n|-n-1} \delta_{s, |n|-n} - q\kappa^{|n|}_{n} \delta_{r, |n|-n} \delta_{s, |n|-n+1} \right)^{2} \\ &= q^{-2n} \left( (\kappa^{|n|}_{n+1})^{2} \delta_{r, |n|-n-1} \delta_{s, |n|-n} + q^{2} (\kappa^{|n|}_{n})^{2} \delta_{r, |n|-n} \delta_{s, |n|-n+1} \right)\end{aligned}$$]{} where $\kappa^{l}_{j} = ([l+j]_{q}[l-j+1]_{q})^{1/2}$. Combining these results with the formula for $\mathrm{Res}_{r = -\frac{1}{2}}\phi_{2}^{r}$ and the expression for $Ch_{2}([P_{n}, V])$ gives [$$\begin{aligned} &\phi_{2}(Ch_{2}([P_{n}, V_n])) = \frac{-2 }{2 (q^{-1} - q) \ln q^{-1}} \sum_{k_{0}, k_{1}, k_{2}=0}^{2|n|} q^{-2k_{0}} (-q^{-1} \delta_{k_{0}, k_{1}} ( \delta_{k_{0}, |n|-n} - \tfrac{1}{2})) \\ & \qquad \times \Big( ( q^{-2} - 1 - \ln q^{-2}) \varepsilon({\partial}_{e}({t_{|n|-k_{1}, n}^{|n|}}{t_{|n|-k_{2}, n}^{|n|\ast}}))^{2} - ( q^{2} - 1 - \ln q^{2}) \varepsilon({\partial}_{e}({t_{|n|-k_{2}, n}^{|n|}}{t_{|n|-k_{1}, n}^{|n|\ast}}))^{2} \Big) \\ &= \frac{q^{-1} }{(q^{-1} - q) \ln q^{-1}} \sum_{k_{1}, k_{2}=0}^{2|n|} q^{-2k_{1}} ( \delta_{k_{1}, |n|-n} - \tfrac{1}{2}) q^{-2n} \\ & \qquad \times \Big( ( q^{-2} - 1 - \ln q^{-2}) \left( (\kappa^{|n|}_{n+1})^{2} \delta_{k_{1}, |n|-n-1} \delta_{k_{2}, |n|-n} + q^{2} (\kappa^{|n|}_{n})^{2} \delta_{k_{1}, |n|-n} \delta_{k_{2}, |n|-n+1} \right) \\ & \qquad \qquad - ( q^{2} - 1 - \ln q^{2}) \left( (\kappa^{|n|}_{n+1})^{2} \delta_{k_{2}, |n|-n-1} \delta_{k_{1}, |n|-n} + q^{2} (\kappa^{|n|}_{n})^{2} \delta_{k_{2}, |n|-n} \delta_{k_{1}, |n|-n+1} \right) \Big).\end{aligned}$$]{} Using $( \delta_{k_{1}, |n|-n} - \tfrac{1}{2}) \delta_{k_{1}, |n|-n} = \tfrac{1}{2} \delta_{k_{1}, |n|-n}$ and $( \delta_{k_{1}, |n|-n} - \tfrac{1}{2}) \delta_{k_{1}, |n|-n \pm 1} = -\tfrac{1}{2} \delta_{k_{1}, |n|-n \pm 1}$ yields [$$\begin{aligned} &\mathrm{Res}_{r = -\frac{1}{2}}\phi_{2}^{r}(Ch_{2}([P_{n}, V_n])) = \frac{q^{-1} }{(q^{-1} - q) \ln q^{-1}} \sum_{k_{1}, k_{2}=0}^{2|n|} \tfrac{1}{2} q^{-2k_{1}-2n} \\ & \qquad \times \Big( ( q^{-2} - 1 - \ln q^{-2}) \left( -(\kappa^{|n|}_{n+1})^{2} \delta_{k_{1}, |n|-n-1} \delta_{k_{2}, |n|-n} + q^{2} (\kappa^{|n|}_{n})^{2} \delta_{k_{1}, |n|-n} \delta_{k_{2}, |n|-n+1} \right) \\ & \qquad \qquad - ( q^{2} - 1 - \ln q^{2}) \left( (\kappa^{|n|}_{n+1})^{2} \delta_{k_{1}, |n|-n} \delta_{k_{2}, |n|-n-1} - q^{2} (\kappa^{|n|}_{n})^{2} \delta_{k_{1}, |n|-n+1} \delta_{k_{2}, |n|-n} \right) \Big).\end{aligned}$$]{} We can reduce the different summations over $k_{1}$ and $k_{2}$ down to two distinct sums, either [$$\sum_{k = 0}^{2|n|} \delta_{k, |n|-n - 1} = \delta_{n, -|n|}, \qquad \qquad \text{or} \qquad \qquad \sum_{k = 0}^{2|n|} \delta_{k, |n|-n + 1} = \delta_{n, |n|}.$$]{} Hence [$$\begin{aligned} &\phi_{2}(Ch_{2}([P_{n}, V])) = \frac{q^{-1} }{2(q^{-1} - q) \ln q^{-1}} \\ & \qquad \times \Big( ( q^{-2} - 1 - \ln q^{-2}) \left( -(\kappa^{|n|}_{n+1})^{2} \delta_{n, -|n|} q^{-2(|n|-n-1)-2n} + q^{2} (\kappa^{|n|}_{n})^{2} \delta_{n, |n|} q^{-2(|n|-n)-2n} \right) \\ & \qquad \qquad - ( q^{2} - 1 - \ln q^{2}) \left( (\kappa^{|n|}_{n+1})^{2} \delta_{n, -|n|} q^{-2(|n|-n)-2n} - q^{2} (\kappa^{|n|}_{n})^{2} \delta_{n, |n|} q^{-2(|n|-n+1)-2n} \right) \Big) \\ &= \frac{q^{-1} }{2(q^{-1} - q) \ln q^{-1}} \\ & \qquad \times \Big( ( q^{-2} - 1 - \ln q^{-2}) \left( -(\kappa^{|n|}_{1-|n|})^{2} \delta_{n, -|n|} q^{-2|n|+2} + q^{2} (\kappa^{|n|}_{|n|})^{2} \delta_{n, |n|} q^{-2|n|} \right) \\ & \qquad \qquad - ( q^{2} - 1 - \ln q^{2}) \left( (\kappa^{|n|}_{1-|n|})^{2} \delta_{n, -|n|} q^{-2|n|} - q^{2} (\kappa^{|n|}_{|n|})^{2} \delta_{n, |n|} q^{-2|n|-2} \right) \Big).\end{aligned}$$]{} Observe that $(\kappa^{|n|}_{1-|n|})^{2} = (\kappa^{|n|}_{|n|})^{2} = [2|n|]_{q}$ as $[1]_{q} = 1$, and so [$$\begin{aligned} &\phi_{2}(Ch_{2}([P_{n}, V_n])) = \frac{q^{-1} }{2(q^{-1} - q) \ln q^{-1}} [2|n|]_{q} q^{-2|n|} \\ & \qquad \times \Big( ( q^{-2} - 1 - \ln q^{-2}) q^{2} (\delta_{n, |n|} - \delta_{n, -|n|}) - ( q^{2} - 1 - \ln q^{2}) (\delta_{n, -|n|} - \delta_{n, |n|}) \Big) \\ &= \frac{q^{-2|n|-1} [2|n|]_{q} }{2(q^{-1} - q) \ln q^{-1}} (\delta_{n, |n|} - \delta_{n, -|n|}) \Big( ( q^{-2} - 1 - \ln q^{-2}) q^{2} + ( q^{2} - 1 - \ln q^{2}) \Big) \\ &= \frac{q^{-2|n|-1} [2|n|]_{q} }{2(q^{-1} - q) \ln q^{-1}} (\delta_{n, |n|} - \delta_{n, -|n|}) \Big( - q^{2}\ln q^{-2} - \ln q^{2} \Big) \\ &= \frac{q^{-2|n|-1} [2|n|]_{q} }{2(q^{-1} - q) \ln q^{-1}} (\delta_{n, |n|} - \delta_{n, -|n|}) (1 - q^{2}) \ln q^{-2} \\ &= q^{-2|n|} [2|n|]_{q} (\delta_{n, |n|} - \delta_{n, -|n|}).\end{aligned}$$]{} Considering $n \neq 0$, then $(\delta_{n, |n|} - \delta_{n, -|n|}) = \mathrm{sgn}(n)$ and $\mathrm{sgn}(n)[2|n|]_{q} = [2n]_{q}$. As $[0]_{q} = 0$, then for all $n \in \frac{1}{2} {\mathbb{Z}}$ we have [$$\phi_{2}(Ch_{2}([P_{n}, V_n])) = q^{-2|n|} [2n]_{q}. \qedhere$$]{} We can now write down the index pairing and compute the classical limit as $q \rightarrow 1$. For $N \in {\mathbb{Z}}$, the index pairing of the modular spectral triple $({\mathcal{B}}, {\mathcal{H}}, {\mathcal{D}}, {\mathcal{B}}({\mathcal{H}}), \Psi_{R})$ with the equivariant projections $P_{N/2}$ is [$$\mathrm{Ind}(P_{N/2} ({\mathcal{D}}{\otimes}{\rm Id}_{|N| + 1})^{+} P_{N/2}) = q^{-|N|} [N]_{q}.$$]{} The classical limit of the index as $q \rightarrow 1$ is [$$\lim_{q \rightarrow 1} \mathrm{Ind}(P_{N/2} ({\mathcal{D}}{\otimes}{\rm Id}_{|N| + 1})^{+} P_{N/2}) = N.$$]{} First, the degree zero contribution is $\phi_{0}(Ch_{0}([P_{N/2}, V_{N/2}])) = 0$. This follows from $\phi_{0}(I) = 0$, and from Lemma \[lemma\_chern\_pn\], which gives $Ch_{0}([P_{N/2}, V_{N/2}]) = q^{(N - |N|)} I$. Thus the index pairing is computed just with the degree 2 part, which comes from Proposition \[prop\_phi\_p\_two\]. To compute the classical limit of the index we recall that $\lim_{q \rightarrow 1} [N]_{q} = N$ (see for example [@KS]). =0.0in [999999]{} M-T. Benameur, T. Fack, [*Type II noncommutative geometry. I. Dixmier trace in von Neumann algebras*]{}, Adv. Math. [**199**]{} (2006), 29–87. B. Blackadar, [$K$-Theory for Operator Algebras]{}, 2nd ed., MSRI, 1998. O. Bratteli, D. Robinson, [*Operator Algebras and Quantum Statistical Mechanics 2*]{}, Springer-Verlag, 2nd Ed, 1987. A. L. Carey, V. Gayral, A. Rennie, F. Sukochev, [*Index theory for locally compact noncommutative spaces*]{}, arXiv:1107.0805 A. Carey, M. Marcolli, A. Rennie, [*Modular Index Invariants of Mumford Curves*]{}, to appear in ‘Noncommutative Geometry, Arithmetic, and Related Topics’, editor Caterina Consani, John Hopkins University Press. A. Carey, S. Neshveyev, R. Nest, A. Rennie, [*Twisted cyclic theory, equivariant $KK$-theory and KMS states*]{}, J. reine ange Math., [**650**]{} (2011), 161–191. A. L. Carey, J. Phillips, [*Unbounded Fredholm modules and spectral flow*]{}, Can. J. Math [**50**]{} (1998), 673–718. A. L. Carey, J. Phillips, [*Spectral Flow in $\theta$-summable Fredholm modules, eta invariants and the JLO cocycle*]{}, K-Theory [**31**]{} (2004), 135–194. A. L. Carey, J. Phillips, I. Putnam, A. Rennie, [*Families of type $III$ KMS states on a class of $C^*$-algebras containing $O_n$ and $Q_{\mathbb{N}}$.*]{} J. Funct. Anal., [**260**]{}, no. 6 (2011), 1637–1681. A. L. Carey, J. Phillips, A. Rennie, [*Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebras*]{}, J. $K$-Theory, [**6**]{}, no. 2 (2010), 339–380. A. L. Carey, J. Phillips, A. Rennie, F. Sukochev, [*The local index formula in semifinite von Neumann algebras I: Spectral Flow*]{}, Adv. Math. [**202**]{} (2006), 451–516. A. L. Carey, J. Phillips, A. Rennie, F. Sukochev, [*The local index formula in semifinite von Neumann algebras II: The Even Case*]{}, Adv. Math. [**202**]{} (2006), 517–554. A. L. Carey, J. Phillips, A. Rennie, F. Sukochev, [*The Chern character of semifinite spectral triples*]{}, J. Noncommut. Geom., [**2**]{}, no. 2 (2008), 253–283. A. L. Carey, A. Rennie, K. Tong, [*Spectral flow invariants and twisted cyclic theory for the Haar state on $SU_q(2)$*]{}, J. Geom. Phys., [**59**]{} (2009), 1431–1452. A. Connes, [*Noncommutative Geometry*]{}, Academic Press, 1994. A. Connes, H. Moscovici, [*The local index formula in noncommutative geometry*]{}, Geom. Funct. Anal. [**5**]{} (1995), 174–243. L. Dabrowski, G. Landi, A. Sitarz, W. van Suijlekom, J. Várilly [*The Dirac operator on $SU_{q}(2)$*]{}, Commun. Math. Phys. [**259**]{} (2005), 729–759. L. Dabrowski, A. Sitarz, [*Dirac operator on the standard Podleś sphere*]{}, Banach Centre Publ. [**61**]{}, 49–58. J. M. Gracia-Bondía, J. C. Varilly, H. Figueroa, Elements of Non-commutative Geometry, Birkhauser, Boston, 2001. T. Hadfield, [*Twisted cyclic homology of all Podleś quantum spheres*]{}, J. Geom. Phys. [**57**]{} (no. 2) (2007), 339–351. T. Hadfield, U. Krähmer, [*Twisted homology of quantum SL(2) - Part II*]{}, J. $K$-Theory [**6**]{} (2010), no. 1, 69–98. N. Higson, [*The Local Index Formula in Noncommutative Geometry*]{}, Contemporary Developments in Algebraic $K$-Theory, ictp Lecture Notes, no 15, (2003), 444–536. J. Kaad, R. Nest, A. Rennie, [*$KK$-theory and spectral flow in von Neumann algebras*]{}, arXiv math.OA /0701326. G. G. Kasparov, [*The operator $K$-functor and extensions of $C^*$-algebras*]{}, Math. USSR. Izv. [**16**]{} (1981), 513–572. U. Krähmer, E. Wagner, [*A residue formula for the fundamental Hochschild class on the Podleś sphere*]{}, arXiv:1008.1830, to appear in J. $K$-Theory. D. Kucerovsky, [*The $KK$-product of unbounded modules*]{}, $K$-Theory **11** (1997), 17–34. J. Kustermans, G. Murphy, L. Tuset, [*Differential calculi over quantum groups and twisted cyclic cocycles*]{}, J. Geom. Phys., [**44**]{} (2003), 570–594. A. Klimyk, K. Schmüdgen, [*Quantum Groups and Their Representations*]{}, Springer, 1997. S. Neshveyev, L.Tuset, [*A local index formula for the quantum sphere*]{}, Comm. Math. Phys. [**254**]{} (2005), 323–341. J. Phillips, *Self-Adjoint Fredholm Operators and Spectral Flow*, Canad. Math. Bull., [**39**]{}(1996), 460–467. K. Schmüdgen, E. Wagner, [*Dirac operator and a twisted cyclic cocycle on the standard Podleś quantum sphere*]{}, J. Reine Angew. Math. [**574**]{} (2004), 219–235. R. Senior, [*Modular Spectral Triples and KMS States in Noncommutative Geometry*]{}, PhD thesis, ANU, 2011. M. Takesaki, [*Theory of Operator Algebras II*]{}, Encyclopedia of Mathematics, 125, Springer-Verlag, 2003. E. Wagner [*On the noncommutative spin geometry of the standard Podleś sphere and index computations*]{}, J. Geom. Phys. [**59**]{} (2009), 998–1016. [^1]: email: `adam.rennie@anu.edu.au`, `roger.senior@anu.edu.au`

--- abstract: 'In this paper, we introduce a discrete variant of the meta-learning framework. Meta-learning aims at exploiting prior experience and data to improve performance on future tasks. By now, there exist numerous formulations for meta-learning in the continuous domain. Notably, the Model-Agnostic Meta-Learning (MAML) formulation views each task as a continuous optimization problem and based on prior data learns a suitable initialization that can be adapted to new, unseen tasks after a few simple gradient updates. Motivated by this terminology, we propose a novel meta-learning framework in the discrete domain where each task is equivalent to maximizing a set function under a cardinality constraint. Our approach aims at using prior data, i.e., previously visited tasks, to train a proper initial solution set that can be quickly adapted to a new task at a relatively low computational cost. This approach leads to (i) a personalized solution for each individual task, and (ii) significantly reduced computational cost at test time compared to the case where the solution is fully optimized once the new task is revealed. The training procedure is performed by solving a challenging discrete optimization problem for which we present deterministic and randomized algorithms. In the case where the tasks are monotone and submodular, we show strong theoretical guarantees for our proposed methods even though the training objective may not be submodular. We also demonstrate the effectiveness of our framework on two real-world problem instances where we observe that our methods lead to a significant reduction in computational complexity in solving the new tasks while incurring a small performance loss compared to when the tasks are fully optimized.' author: - 'Arman Adibi[^1] ,Aryan Mokhtari[^2] ,Hamed Hassani[^3]' bibliography: - 'ref.bib' - 'ref\_2.bib' title: ' Submodular Meta-Learning ' --- Introduction ============ Many applications in artificial intelligence necessitate exploiting prior data and experience to enhance quality and efficiency on new tasks. This is often manifested through a set of tasks given in the training phase from which we can learn a model or representation that can be later used for new unseen tasks in the test phase. In this regard, meta-learning aims at exploiting the data from the available tasks to learn model parameters or representation that can be later used to perform well on new unseen tasks, in particular, when we have access to limited data and computational power at the test time [@thrun2012learning; @schmidhuber1992learning; @bengio1990learning; @vilalta2002perspective]. By now, there are several formulations for meta-learning, but perhaps one of the most successful ones is the Model-Agnostic Meta-Learning (MAML) framework proposed in [@finn2017model]. In MAML, we aim to train the model parameters such that applying a few steps of gradient-based updates with a small number of samples from a new task would perform well on that task. MAML can also be viewed as a way to provide a proper initialization, from which performance on a new task can be optimized after a few gradient-based updates. Alas, this scheme only applies to settings in which the decision variable belongs to a *continuous* domain and can be adjusted using gradient-based methods at the test time. Our goal is to extend the methodology of MAML to the *discrete* setting. We consider a setting in which our decision variable is a discrete set, and our goal is to come up with a good initial set that can be quickly adjusted to perform well over a wide range of new tasks. In particular, we focus on submodular maximization to represent the tasks which is an essential class of discrete optimization. There are numerous applications where the submodular meta-learning framework can be applied to find a personalized solution for each task while significantly reducing the computation load. In general, most recommendation tasks can be cast as an instance of this setting[@gabillon2013adaptive; @el2009turning; @yue2011linear]. Consider the task of recommending a set of items, e.g., products, locations, ads, to a set of users. One approach for solving such a problem is to find the subset of items that have the highest score over all the previously-visited users and recommend that subset to a new user. Indeed, this approach leads to a reasonable performance at test time; however, it does not provide a user-specific solution for a new user. Another approach is to find the whole subset at the test time when the new user arrives. In contrast to the previous approach, this scheme leads to a user-specific solution, but at the cost of running a computationally expensive algorithm to select all the elements at the test time. In our meta-learning framework, the process of selecting set items to be recommended to a new user is done in two parts: In the first part, a set of items is selected offline according to prior experience. These items are the most popular items to the previously-visited users (depending on the context). In the second part, which happens at the test time, a set of items that is *personalized* to the coming user is selected. These are items that are computed specifically according to the features of the coming user. In this manner, the computation for each coming user would be reduced to the selection of the second part, which typically constitutes a small portion of the final set of recommended items. The first part can be done offline with a lower frequency. For instance, in a real recommender system, the first part can be computed once every hour, and the second part can be computed specifically for each coming user (or for a class of similar users). While we have mentioned recommendation (or more generally facility location) as a specific example, it is easy to see that this framework can be easily used to reduce computation in other notable applications of submodular optimization. **Contributions.** Our contributions are threefold: - We propose a novel discrete meta-learning framework where each task is equivalent to maximizing a set function under some cardinality constraint. Our framework aims at using prior data, i.e., previously visited tasks, to train a proper initial solution set that can be quickly adapted to a new task at a low computational cost to obtain a task-specific solution. - We present computationally efficient deterministic and randomized meta-greedy algorithms to solve the resulting meta-learning problem. When the tasks are monotone and submodular, we prove that the solution obtained by the deterministic algorithm is at least $0.53$-optimal, and the solution of the randomized algorithm is $(1-1/e-o(1))$-optimal in expectation, where the $o(1)$ term vanishes by the size of the solution. These guarantees are obtained by introducing new techniques, despite that the meta-learning objective is *not* submodular. - We study the performance of our proposed meta-learning framework and algorithms for movie recommendation and ride-sharing problems. Our experiments illustrate that the solution of our proposed meta-learning scheme, which chooses a large portion of the solution in the training phase and a small portion adaptively at test time, is very close to the solution obtained by choosing the entire solution at the test time when a new task is revealed. Related work ------------ **Continuous Meta-Learning.** Meta-learning has gained considerable attention recently mainly due to its success in few shot learning [@vinyals2016matching; @DBLP:conf/iclr/RaviL17; @snell2017prototypical; @wang2019few] as well as reinforcement learning [@DBLP:journals/corr/DuanSCBSA16; @wang2016learning; @DBLP:conf/iclr/SongGYCPT20; @fallah2020provably]. One of the most successful forms of meta-learning is the gradient-based *Model Agnostic Meta-learning* (MAML) approach[@finn2017model]. MAML aims at learning an initialization that can be adapted to a new task after performing one (or a few) gradient-based update(s); see, e.g., [@fallah2019convergence]. This problem can be written as $$\min_{w \in W} \mathbb{E}_{a\sim P}[f_a(w-\nabla f_a(w))],$$ where $W \subseteq \mathbb{R}^d$ is the feasible set and $l$ is the probability distribution over tasks. The previous works on MAML including [@nichol2018first; @finn2018probabilistic; @DBLP:conf/iclr/GrantFLDG18; @yoon2018bayesian; @DBLP:conf/iclr/AntoniouES19; @rajeswaran2019meta; @fallah2019convergence; @collins2020distribution] consider the case where $W$ is a continuous space. In fact none of these works can be applied to the case where the feasible parameter space is discrete. In this paper, we aim to close this gap and extend the terminology of MAML to discrete settings. **Submodular Maximization.** Submodular functions have become key concepts in numerous applications such as data summarization [@lin2011class; @wei2013using; @kirchhoff2014submodularity; @mirzasoleiman2016distributed], viral marketing [@kempe2003maximizing], sensor placement [@krause2008near], dictionary learning [@das2011submodular], and influence maximization [@kempe2003maximizing]. It is well-known that for maximizing a monotone and submodular function under the cardinality constraint, the greedy algorithm provides a $(1-1/e)$-optimal solution [@krause2014submodular; @nemhauser1978best; @wolsey1982analysis]. There has been significant effort to improve the scalability and efficiency of the greedy algorithm using lazy, stochastic, and distributed methods [@mirzasoleiman2015lazier; @karimi2017stochastic; @barbosa2015power; @mirrokni2015randomized; @kumar2015fast; @hassani2017gradient; @mokhtari2020stochastic; @karbasi2019stochastic; @balkanski2019exponential]. However, our framework is fundamentally different and complementary to these approaches as it proposes a new approach to use data at training time to improve performance at new tasks. Indeed, all the aforementioned techniques can be readily used to further speed-up our algorithms. Optimization of related submodular tasks has been a well-studied problem with works on structured prediction [@lin2012learning], submodular bandits [@yue2011linear; @zhang2019online], online submodular optimization [@jegelka2011online; @streeter2009online; @golovin2014online; @chen2018projection], and public-private data summarization [@mirzasoleiman2016fast]. However, unlike our work, these approaches are not concerned with train-test phases for optimization. Another recently-developed methodology to reduce computation is the two-stage submodular optimization framework [@balkanski2016learning; @mitrovic2018data; @stan2017probabilistic], which aims at summarizing the ground set to a reasonably small set that can be used at test time. The main difference of our framework with the two-stage approaches is that we allow for *personalization*: A small subset of items that can be found at test time specific to the task at hand. This leads to a completely new problem formulation, and consequently, new algorithms. Problem Statement: Discrete Meta-Learning {#sect:prob-statement} ========================================= **Setup.** We consider a family of tasks $\mathcal{T}=\{\mathcal{T}_i\}_{i\in \mathcal{I}}$, where the set $\mathcal{I}$ could be of infinite size. Each task $\mathcal{T}_i$ is represented via a set function $f_i:2^{V} \to {\mathbb{R}}_+$ that measures the reward of a set $S \subseteq V $ for the $i$-th task, and performing the task $\mathcal{T}_i$ would mean to maximize the function $f_i$ subject to a given constraint. For instance, in a recommender system where we aim to recommend a subset of the items to the users, the set $\mathcal{I}$ denotes the set of all the possible users and selecting which items to recommend to a user $i \in \mathcal{I}$ is viewed as the task $\mathcal{T}_i$. Moreover, the function $f_i$ encodes the users satisfaction, i.e., $f_i(S)$ quantifies how suitable the set of items $S$ is for user $i$. Taking a statistical perspective, we assume that the tasks $\mathcal{T}_i$ occur according to a possibly unknown probability distribution $i \sim p$. In this paper, we focus on the case where the functions $f_i$ are monotone and submodular set functions and each task $\mathcal{T}_i$ amounts to maximizing $f_i$ under the $k$-cardinality constraint. That is, the task $\mathcal{T}_i$ is to select a subset $S \subseteq V$ of size $k$ such that the value of $f_i(S)$ is maximized. Submodularity of $f_i$ means that for any $A,B \subseteq V$ the following inequality holds $f_i(A) + f_i(B) \geq f_i(A \cup B) + f_i(A \cap B)$. Furthermore, $f_i$ is called monotone if for any $A \subseteq B$ we have $f_i(A) \leq f_i(B)$. **Training and test tasks.** We assume access to a collection of *training* tasks $\{\mathcal{T}_i\}_{i=1}^m$. These are the tasks that we have already experienced, i.e., they correspond to the users that we have already seen. Formally, this means that for each training task $\mathcal{T}_i$, we assume knowledge of the corresponding function $f_i$. In our formulation, each of the training tasks is assumed to be generated i.i.d. according to the distribution $p$. Indeed, eventually we aim to optimize performance at *test* time, i.e., obtain the best performance for new and unseen tasks generated independently from the distribution $p$. For instance, in our recommendation setting, test tasks correspond to new users that will arrive in the future. Our goal is to use the training tasks to reduce the computation load at test time. **Two extremes of computation.** Let us use $\mathcal{T}_{\rm test}$ (and $f_{\rm{test}}$) to denote the task (and its corresponding set function) that we aim to learn at test time. Ideally, if we have sufficient computational power, then we should directly optimize $f_{\rm{test}}$ by solving the following problem $$\label{test_problem} \max_{S\in V, |S|\leq k}\ f_{\rm{test}}(S).$$ We denote the optimal solution of by $S_{\rm test}^*$. For instance, we can use the greedy procedure to solve  which leads to a $(1-1/e)$-optimal solution using $\mathcal{O}(kn)$ evaluations of $f_{\rm test}$, and through $k$ passes over the ground set. However, the available computational power and time in the test phase is often limited, either because we need to make quick decisions to respond to new users or since we need to save energy. For instance, in real-world advertising or recommendation systems, both these requirements are crucial: many users arrive within each hour which means fast optimization is crucial (especially if $n,k$ are large), and also, reducing computation load would lead to huge energy savings in the long run. In such cases, Problem  should be solved approximately with less computation. An alternative to reduce computation at test time is to solve the problem associated with the expected reward over all possible tasks in the training phase (when we have sufficient computation time), i.e., $$\label{expected_max} \max_{S\in V, |S|\leq k} \ \mathbb{E}_{i \sim p}\; [f_i(S)].$$ We denote the optimal solution of by $S_{\rm exp}^*$. The rationale behind this approach is that the optimal solution to this problem would generalize well over an unseen task if the new task is also drawn according to the probability distribution $p$. In other words, the solution of should perform well for the problem in that we aim to solve at the test time, assuming that $f_{\rm test}$ is sampled according to $p$. In this way, we do not need any extra computation at the test time. However, in this case, the solution that we obtain would not be the best possible solution for the task that we observe at the test time, i.e., $S_{\rm test}^*$ is not equal to $S_{\rm exp}^*$. Note that we often do not have access to the underlying probability distribution $p$, and we only have access to a large set of realizations of tasks in the training phase. As a result, instead of solving , we settle for maximizing the sample average function $$\label{empirical_max} \max_{S\in V, |S|\leq k} \ \frac{1}{m}\sum_{i=1}^m f_i(S),$$ where $m$ is the number of available tasks in the training phase. Problems and can be considered as two different extreme cases. In the first option, by solving , we avoid any pre-processing in the training phase, and we obtain the best possible guarantee for the new task, but at the cost of performing computationally expensive operations (e.g., full greedy) at the test time. In the second approach, by solving in the training phase, we obtain a solution that possibly performs reasonably without any computation at the test phase, but the quality of the solution may not be as good as the first option. In summary, there exists a trade-off between the required computational cost at the test time and the performance guarantee on the unseen task. Hence, a fundamental question that arises is what would be the best scheme at the training phase assuming that at test time we have some limited computational power. For instance, in the monotone submodular case, assume that instead of running the greedy algorithm for $k$ rounds, which has a complexity of $\mathcal{O}(kn)$, we can only afford to run $\alpha k$ rounds of greedy at test time, which has a complexity of $\mathcal{O}(\alpha nk)$, where $\alpha\in(0,1)$ is small. In this case, a natural solution would be to find an appropriate set of $(1-\alpha)k$ elements in the training phase, and add the remaining $\alpha k$ elements at test time when a new task arrives. This discussion also applies to any other greedy method (e.g., lazy or stochastic greedy). We now formally state this problem. [0.21]{} {width="1.0\linewidth" height="1\linewidth"} [0.21]{} {width="1.0\linewidth" height="1\linewidth"} [0.21]{} {width="1.0\linewidth" height="1\linewidth"} [0.21]{} {width="1.0\linewidth" height="1\linewidth"} **Discrete Meta-Learning.** As we discussed so far, when computational power is limited at test time, it makes sense to divide the process of choosing the best decision between training and test phases. To be more specific, in the training phase, we choose a subset of elements from the ground set that would perform over the training tasks, and then select (or optimize) the remaining elements at the test time *specifically* with respect to the task at hand. To state this problem, consider ${S_{\rm{tr}}}\subseteq V$ with cardinality $|{S_{\rm{tr}}}|=l$, where $l<k$, as the initial set that we aim to find at the training phase, and the set $S_i$ that we add to the initial set ${S_{\rm{tr}}}$ at test time (See Figure \[fig:toy\] for an illustration). Hence, the problem of interest can be written as $$\label{eq:ML_submodular} \max_{{S_{\rm{tr}}}\in V, |{S_{\rm{tr}}}|\leq l} \ \mathbb{E}_{i \sim p} \Big[\max_{S_i\in V, |S_i|\leq k-l} f_i({S_{\rm{tr}}}\cup S_i)\Big],$$ Note that the critical decision variable that we need to find is ${S_{\rm{tr}}}$ which is the best initial subset of size $l$ overall all possible choices of task when a best subset of size $k-l$ is added to that. In fact, if we define $f_i'({S_{\rm{tr}}}):=\max_{S_i\in V, |S_i|\leq k-l} f_i({S_{\rm{tr}}}\cup S_i)$, then we can rewrite the problem in as $$\max_{{S_{\rm{tr}}}\in V, |{S_{\rm{tr}}}|\leq l} \ \mathbb{E}_{i \sim p}\; \left[ f_i'({S_{\rm{tr}}}) \right].$$ As described previously, we often do not have access to the underlying probability distribution $p$ of the tasks, and we instead have access to a large number of sampled tasked that are drawn independently according to $p$. Hence, instead of solving , we solve its sample average approximation given by $$\label{eq:ML_submodular_sample_avg} \max_{{S_{\rm{tr}}}\in V, |{S_{\rm{tr}}}|\leq l} \ \frac{1}{m}\sum_{i=1}^m\; \left[\max_{S_i\in V, |S_i|\leq k-l} f_i({S_{\rm{tr}}}\cup S_i)\right]\ = \ \max_{{S_{\rm{tr}}}\in V, |{S_{\rm{tr}}}|\leq l} \ \frac{1}{m}\sum_{i=1}^m\; \left[ f_i'({S_{\rm{tr}}}) \right],$$ where $m$ is the number of tasks in the training set which are sampled according to $p$. Even though the functions $f_i$ are submodular, $f'_i$ *is not submodular* or $k$-submodular [@ohsaka2015monotone] (see Appendix \[sec:counter\_example\] for specific counter examples). Hence, Problem  is not a submodular maximization problem. In the next section, we present algorithms for solving Problem  with provable guarantees. We finally note that Problem  will be solved at *training* time to find the solution ${S_{\rm{tr}}}$ of size $l$. This solution is then *completed at test time*, by, e.g., running $k-l$ further rounds of greedy on the new task, to obtain a task-specific solution of size $k$. Submodular Cross-Learning ------------------------- \[pr: cross-learning submodular\] Let $f_i:2^{\mathcal{X}}\xrightarrow{}{\mathbb{R}}_+$ for $i\in [m-1]$, be a monotone submodular function over ground set ${\mathcal{X}}$. We want to find $S_m$ with size at most $l$ whose subests with average size of at most $k$ maximize the sum of $f_i$ for $i\in [m-1]$, more formally: $$\max_{S_m\subseteq {\mathcal{X}}, \mid S_m \mid\leq l}\;\sum_{i=1}^{m-1}\; \max_{S_i\subseteq S_m,\; \frac{1}{m-1}\sum\limits_{i=1}^{m-1} \mid S_i\mid \leq k}\; f_i(S_i)$$ \[eq:cross-learning submodular\] $$\begin{aligned} {2} &\operatorname*{maximize}&&\sum_{i=1}^{m-1}\; \; f_i(S_i) \\ &\operatorname*{subject\,\, to \quad}&& {S_m\subseteq {\mathcal{X}}, \mid S_m \mid\leq l} \\ & && {S_i\subseteq S_m,\; \frac{1}{m-1}\sum\limits_{i=1}^{m-1} \mid S_i\mid \leq k}\end{aligned}$$ \[eq:basic-formulation\] Algorithms for Discrete Submodular Meta-Learning {#sec:algorithms_ML} ================================================ Solving Problem requires finding a set ${S_{\rm{tr}}}$ for the outer maximization and sets $\{S_i\}_{i=1}^m$ for the inner maximization. In this section, we describe our proposed greedy-type algorithms to select the elements ${S_{\rm{tr}}}$ and $\{S_i\}_{i=1}^m$. As we deal with $m+1$ sets, the order in which the sets ${S_{\rm{tr}}}$ and $\{S_i\}_{i=1}^m$ are updated becomes crucial, i.e., it is not clear which of the sets ${S_{\rm{tr}}}$ or $S_i$’s should be preferably updated in each round and how can the functions $f_i$ be incorporated in finding the right order, which is the main challenge in designing greedy methods to solve . We design greedy procedures with both deterministic and randomized orders and provide strong guarantees for their solutions. Deterministic Algorithms ------------------------ In this section, we first describe Algorithms \[alg: discrete meta-Greedy\] and \[alg: Reverse discrete meta-Greedy\] which use specific orderings to solve Problem . Based on these two, we then design Algorithm \[alg: main discrete meta-Greedy\] as our *main deterministic* algorithm. Throughout this section, we use $\Delta_i(e|S)=f_i(S\cup\{e\})-f_i(S)$ to denote the marginal gain of adding an element $e$ to set $S$ for function $f_i$. In brief, Algorithm \[alg: discrete meta-Greedy\] first fills ${S_{\rm{tr}}}$ greedily up to completion and then it constructs each of the $S_i$’s greedily on the top of ${S_{\rm{tr}}}$. Specifically, starting from the empty set initialization for ${S_{\rm{tr}}}$ and $S_i$’s, Algorithm \[alg: discrete meta-Greedy\] constructs in its first phase the set ${S_{\rm{tr}}}$ in $l$ rounds, by adding one element per round, where the next element in each round is chosen according to $ e^* = \operatorname*{arg\,max}_{e \in V} \sum_{i=1}^m f_i({S_{\rm{tr}}}\cup \{e\}) - f_{i}({S_{\rm{tr}}})$. Once ${S_{\rm{tr}}}$ is completed, in the second phase, each of the sets $S_i$ is constructed in parallel by running the greedy algorithm on $f_i$. That is, each $S_i$ is updated in $k-l$ rounds where in each round an element with maximum marginal on $f_i$ is added to $S_i$ based on $ e^*_i = \operatorname*{arg\,max}_{e \in V} f_i({S_{\rm{tr}}}\cup S_i \cup\{e\}) - f_i({S_{\rm{tr}}}\cup S_i)$. **Initialize** ${S_{\rm{tr}}}=\{S_i\}_{i=1}^m=\emptyset$ Find $e^{*}\!\!\! =\!\! \operatorname*{arg\,max}_{ e\in V}\sum\limits_{i=1}^{m}\Delta_i(e|{S_{\rm{tr}}}) $ ${S_{\rm{tr}}}\xleftarrow{} {S_{\rm{tr}}}\cup\{e^{*}\}$ **end for** Find $e_i^{*} \!=\!\operatorname*{arg\,max}_{ e\in V} \Delta_i(e|{S_{\rm{tr}}}\!\cup\! S_i)$ $S_i\xleftarrow{} S_i\cup\{e_i^{*}\}$ **end for** **end for** Return ${S_{\rm{tr}}}$ and $S_i$ **Initialize** ${S_{\rm{tr}}}=\{S_i\}_{i=1}^m=\emptyset$ Find $e_i^{*} = \operatorname*{arg\,max}_{ e\in V} \Delta_i(e|S_i)$ $S_i\xleftarrow{} S_i\cup\{e_i^{*}\}$ **end for** **end for** Find $ e^{*} =\!\operatorname*{arg\,max}_{ e\in V} \!\sum\limits_{i=1}^{m}\!\Delta_{i}(e|{S_{\rm{tr}}}\cup S_i)$ ${S_{\rm{tr}}}\xleftarrow{} {S_{\rm{tr}}}\cup\{e^{*}\}$ **end for** Return ${S_{\rm{tr}}}$ and $S_i$ **Initialize** the sets ${S_{\rm{tr}}}$ and $\{S_i\}_{i=1}^m$ to the empty set. Find $e^{*} = \operatorname*{arg\,max}_{ e\in V} \sum_{i=1}^{m}f_i({S_{\rm{tr}}}\cup\{e\})-f_i({S_{\rm{tr}}})$ ${S_{\rm{tr}}}\xleftarrow{} {S_{\rm{tr}}}\cup\{e^{*}\}$ **end for** Find $e_i^{*} = \operatorname*{arg\,max}_{ e\in V} f_i({S_{\rm{tr}}}\cup S_i\cup\{e\})-f_i({S_{\rm{tr}}}\cup S_i)$ $S_i\xleftarrow{} S_i\cup\{e_i^{*}\}$ **end for** **end for** return ${S_{\rm{tr}}}$ and $S_i$ Algorithm \[alg: Reverse discrete meta-Greedy\] uses the opposite ordering of Algorithm \[alg: discrete meta-Greedy\]. Initializing with all sets to be empty, in the first phase it constructs the sets $S_i$ using the greedy procedure on $f_i$, i.e., each $S_i$ is updated in parallel in $k-l$ rounds, where in each round the element $e_i^{*}$ defined as $e_i^{*} = \operatorname*{arg\,max}_{ e\in V} f_i({S_{\rm{tr}}}\cup S_i\cup\{e\})-f_i({S_{\rm{tr}}}\cup S_i)$ is added to $S_i$. In the second phase, the set ${S_{\rm{tr}}}$ is formed greedily in $l$ rounds, and in each round the element $e^{*} $ defined as $e^{*} = \operatorname*{arg\,max}_{ e\in V} \sum_{i=1}^{m}f_i({S_{\rm{tr}}}\cup\{e\}\cup S_i)-f_i({S_{\rm{tr}}}\cup S_i)$ is added. **Initialize** the sets ${S_{\rm{tr}}}$ and $\{S_i\}_{i=1}^m$ to the empty set. Find $e_i^{*} = \operatorname*{arg\,max}_{ e\in V} f_i(S_i\cup\{e\})-f_i(S_i)$ $S_i\xleftarrow{} S_i\cup\{e_i^{*}\}$ **end for** **end for** Find $e^{*} = \operatorname*{arg\,max}_{ e\in V} \sum_{i=1}^{m}f_i({S_{\rm{tr}}}\cup\{e\}\cup S_i)-f_i({S_{\rm{tr}}}\cup S_i)$ ${S_{\rm{tr}}}\xleftarrow{} {S_{\rm{tr}}}\cup\{e^{*}\}$ **end for** return ${S_{\rm{tr}}}$ and $S_i$ While the solutions obtained by Algorithms \[alg: discrete meta-Greedy\] and \[alg: Reverse discrete meta-Greedy\] are guaranteed to be near-optimal, it turns out that they can be complementary with respect to each other. Our *main* deterministic algorithm, called `Meta-Greedy`, runs both Algorithms \[alg: discrete meta-Greedy\] and \[alg: Reverse discrete meta-Greedy\] and chooses as output the solution, among the two, that leads to a higher objective value in . To be more specific, if we consider ${S_{\rm{tr}}}^{(1)}, \{S_i^{(1)}\}_{i=1}^m$ as the outputs of Algorithm \[alg: discrete meta-Greedy\] and ${S_{\rm{tr}}}^{(2)}, \{S_i^{(2)}\}_{i=1}^m$ as the outputs of Algorithm \[alg: Reverse discrete meta-Greedy\], then `Meta-Greedy` compares the values of $ \sum_{i=1}^{m}f_i({S_{\rm{tr}}}^{(1)}\cup S_i^{(1)})$ and $ \sum_{i=1}^{m}f_i({S_{\rm{tr}}}^{(2)}\cup S_i^{(2)})$ and chooses the solution set that has the higher objective function value. Note that as we described earlier, the main output of this procedure should be the set ${S_{\rm{tr}}}$ of size $l$. Hence, the output of `Meta-Greedy` is either ${S_{\rm{tr}}}^{(1)}$ or ${S_{\rm{tr}}}^{(2)}$ and the sets $ \{S_i^{(1)}\}_{i=1}^m$ and $ \{S_i^{(2)}\}_{i=1}^m$ are only evaluated for the purpose of comparing objective function values. Next, we explain why our $\texttt{Meta-Greedy}$ method can outperform both Algorithms \[alg: discrete meta-Greedy\] and \[alg: Reverse discrete meta-Greedy\]. This will be done by providing the theoretical guarantees for these methods and consequently explaining why Algorithms \[alg: discrete meta-Greedy\] and \[alg: Reverse discrete meta-Greedy\] are complementary. Run Algorithms \[alg: discrete meta-Greedy\] and \[alg: Reverse discrete meta-Greedy\] and obtain respective solution sets ${S_{\rm{tr}}}^{(1)}, \{S_i^{(1)}\}_{i=1}^m$ and ${S_{\rm{tr}}}^{(2)}, \{S_i^{(2)}\}_{i=1}^m$. Compute the objective value $ \sum_{i=1}^{m}f_i({S_{\rm{tr}}}\cup S_i)$ for both solution sets. **Theoretical guarantees.** We begin with the analysis of Algorithm \[alg: discrete meta-Greedy\]. The following proposition relates the overall performance of Algorithm \[alg: discrete meta-Greedy\] to its performance after phase 1 and shows that the output of the algorithm is at least $1/2$-optimal. We use OPT for the optimal value of Problem . \[alg1lemma\] Let ${S_{\rm{tr}}}^{(1)}, \{S_i^{(1)}\}_{i=1}^m$ be the output of Algorithm \[alg: discrete meta-Greedy\], and define $\beta$ as $\beta :=\frac{1}{m} \sum_{i=1}^m f_i({S_{\rm{tr}}}^{(1)})$. If the set functions $f_i$ are monotone and submodular, then $$\frac{1}{m} \sum_{i=1}^m f_i ({S_{\rm{tr}}}^{(1)} \cup S_i^{(1)}) \geq \max \Bigl\{ \beta \,, \, (1-1/e)({\rm{OPT}} - 2 \beta) + \beta \Bigr \}.$$ Consequently, the solution obtained by Algorithm \[alg: discrete meta-Greedy\] is at least $1/2$-optimal for any value of $\beta$. Check Appendix \[sec:proof\_prop\_1\]. The proof of this proposition is relegated to the appendix. The key step in the proof is to relate the progress made in phase 1 to the gap to OPT. This is indeed challenging as phase 1 only involves updates on the outer maximization of . In this regard, we prove a novel technical lemma that can be generally applicable to any mini-max submodular problem. The guarantee given in Proposition \[alg1lemma\] is minimized when $\beta = \text{OPT}/2$. If $\beta$ is small (e.g., $\beta = 0$) or if $\beta$ is large (e.g. if $\beta = (1-1/e)\text{OPT}$) then the guarantee becomes tight (e.g. $(1-1/e)\text{OPT})$. This is indeed expected from the greedy nature of the two phases of Algorithm \[alg: discrete meta-Greedy\]. What is non-trivial about the result of Proposition \[alg1lemma\] is that it provides a strong guarantee for any value of $\beta$, and not just cases that $\beta$ is small or large. Similarly, we can provide near-optimality guarantees for Algorithm \[alg: Reverse discrete meta-Greedy\]. \[alg2lemma\] Let ${S_{\rm{tr}}}^{(2)}, \{S_i^{(2)}\}_{i=1}^m$ be the output of Algorithm \[alg: Reverse discrete meta-Greedy\], and define $\gamma$ as $\gamma := \frac{1}{m}\sum_{i=1}^m f_i(S_i^{(2)})$. If the set functions $f_i$ are monotone and submodular, then $$\frac{1}{m} \sum_{i=1}^m f_i ({S_{\rm{tr}}}^{(2)} \cup S_i^{(2)}) \geq \max \Bigl\{ \gamma \,, \, (1-1/e)({\rm{OPT}} - 2\gamma) + \gamma \Bigr \}.$$ Consequently, the solution obtained by Algorithm \[alg: Reverse discrete meta-Greedy\] is at least $1/2$-optimal for any value of $\gamma$. Check Appendix \[sec:proof\_prop\_2\]. Similarly, we can show that $\gamma=\text{OPT}/2$ leads to (the worst) guarantee $1/2$-OPT, while for large and small values of $\gamma$ the bound in Proposition \[alg2lemma\] approaches the optimal approximation $(1-1/e)\text{OPT}$. We note that the values $\beta$ in Proposition \[alg1lemma\] (Algorithm \[alg: discrete meta-Greedy\]) and $\gamma$ in Proposition \[alg2lemma\] (Algorithm \[alg: Reverse discrete meta-Greedy\]) represent two different extremes. The value $\beta$ represents the significance of the role of ${S_{\rm{tr}}}$ in solving Problem , and $\gamma$ represents how significant the role of the sets $\{S_i\}_{i=1}^m$ can be. Even though the worst-case guarantees of Propositions \[alg1lemma\] and \[alg2lemma\] are obtained when $\beta, \gamma = \text{OPT}/2$, a coupled analysis of the algorithms show that in this case at least one of the algorithms should output a solution which is strictly better than $1/2$-optimal. In other words, the outcomes of Algorithms \[alg: discrete meta-Greedy\] and \[alg: Reverse discrete meta-Greedy\] are dependent to one another, and the best performance is achieved when the maximum of the two is considered. This justifies why our main algorithm $\texttt{Meta-Greedy}$ can perform strictly better than each of the Algorithms \[alg: discrete meta-Greedy\] and \[alg: Reverse discrete meta-Greedy\]. Using a coupled analysis of the outcome of Algorithms \[alg: discrete meta-Greedy\] and \[alg: Reverse discrete meta-Greedy\], we can bound the performance of $\texttt{Meta-Greedy}$ for different values of $\beta$ and $\gamma$ (see the proof of Theorem \[thm:meta-greedy\] in the appendix). In particular, we can show that the output of $\texttt{Meta-Greedy}$ is at least $0.53$-optimal. The proof of the following theorem carefully analyzes the interplay between the role of the inner and outer maximization problems in . We emphasize that the proof introduces new techniques applicable to other types of minimax submodular problems. \[thm:meta-greedy\] Consider the $\texttt{Meta-Greedy}$ algorithm outlined in Algorithm \[alg: main discrete meta-Greedy\]. If the functions $f_i$ are monotone and submodular, then we have $$\label{bound-meta-greedy} \max \Bigl\{ \frac{1}{m}\sum_{i=1}^m f_i ({S_{\rm{tr}}}^{(1)} \cup S_i^{(1)}) \, , \, \frac{1}{m} \sum_{i=1}^m f_i ({S_{\rm{tr}}}^{(2)} \cup S_i^{(2)}) \Bigr\} \geq 0.53 \times {\rm{OPT}}.$$ Check Appendix \[sec:proof\_of\_main\_thm\]. Note that for all the results in Propositions \[alg1lemma\] and \[alg2lemma\] as well as Theorem \[thm:meta-greedy\], for given output sets ${S_{\rm{tr}}}$ or $\{S_i\}_{i=1}^m$, the value of $\frac{1}{m} \sum_{i=1}^m f_i({S_{\rm{tr}}}\cup S_i)$ is a lower bound for the objective function value of Problem evaluated at the output set ${S_{\rm{tr}}}$. To be more precise, the accurate measure for evaluating the quality of the output set ${S_{\rm{tr}}}$ is $\frac{1}{m}\sum_{i=1}^m\; \left[\max_{S_i\in V, |S_i|\leq k-l} f_i({S_{\rm{tr}}}\cup S_i)\right]$ which is indeed larger than $\frac{1}{m} \sum_{i=1}^m f_i({S_{\rm{tr}}}\cup S_i)$. Hence, all the guarantees that have obtained in the statements above (as well as Theorem \[thm: randomized meta\] below) would directly translate into the same guarantees when we evaluate the objective in on the set ${S_{\rm{tr}}}$. Randomized Algorithm -------------------- In this section, we consider greedy procedures in which the decision to alternate between the set ${S_{\rm{tr}}}$ (the outer maximization) and the sets $\{S_i\}_{i=1}^m$ (the inner maximization) is done based on a randomized scheme. The `Randomized meta-Greedy` procedure, outlined in Algorithm \[alg: Randomized meta-Greedy\], provides a specific randomized order. In each round, with probability $l/k$ we choose to perform a greedy update on ${S_{\rm{tr}}}$, and with probability $1-l/k$ we choose to perform a greedy update on *all* the $S_i$’s, $i=1, \cdots,m$. This procedure continues until either ${S_{\rm{tr}}}$ or $\{S_i\}_{i=1}^m$ hit their corresponding carnality constraint, in which case we continue to update the other set(s) greedily until they also become full. **Initialize** the sets ${S_{\rm{tr}}}$ and $\{S_i\}_{i=1}^m$ to the empty set. $e^{*}_i \xleftarrow{} \operatorname*{arg\,max}_{ e\in V} f_i({S_{\rm{tr}}}\cup S_i\cup\{e\})-f_i({S_{\rm{tr}}}\cup S_i)$ $e^{*}_{tr} \xleftarrow{} \operatorname*{arg\,max}_{ e\in V} \sum_{i=1}^{m} f_i({S_{\rm{tr}}}\cup S_i\cup\{e\})-f_i({S_{\rm{tr}}}\cup S_i)$ **w.p. $\frac{l}{k}$:** ${S_{\rm{tr}}}={S_{\rm{tr}}}\cup\{e^{*}_{tr}\}$ **w.p. $\frac{k-l}{k}$:** $S_i=S_i\cup\{e_i^{*}\}$, $\forall i = 1, \cdots, m$ **end** If ${S_{\rm{tr}}}$ or $S_i$’s have not yet reached their cardinality limit then fill them greedily until their limit is reached Return ${S_{\rm{tr}}}$ and $S_i$ The randomized update of Algorithm \[alg: Randomized meta-Greedy\] is designed to optimally connect the expected increase the objective value at each round with the gap to OPT (as shown in the proof of Theorem \[thm: randomized meta\]). Hence, the `Randomized meta-Greedy` procedure is able to achieve in expectation a guarantee close to the tight value $(1-1/e)\text{OPT}$. However, due to the randomized nature of the algorithm, the sets ${S_{\rm{tr}}}$ or $S_i$ might hit their carnality constraint earlier than expected. Analyzing the function value at this “stopping time” is another technical challenge that we resolve in the following theorem to obtain a guarantee that becomes slightly worse than $(1-1/e)\text{OPT}$ depending on the values of $k-l$ and $l$. \[thm: randomized meta\] Let the (random) sets ${S_{\rm{tr}}}$, $\{S_i\}_{i=1}^m$ be the output of Algorithm \[alg: Randomized meta-Greedy\]. If the functions $f_i$ are monotone and submodular, then $$\mathbb{E}\Bigl[\frac{1}{m}\sum_{i=1}^m f_i ({S_{\rm{tr}}}\cup S_i) \Bigr] \geq \left(1 - \frac{1}{e} - b \right) \rm{OPT},$$ where $b \to 0$ as $k-l$ and $l$ grow. More precisely, letting $c = \max\{\frac{1}{k-l}, \frac{1}{l}\},$ we have $b = c + (\exp(3\sqrt{c\log1/c}) - 1)/e = {\mathcal{O}}(\sqrt{c \log 1/c}).$ Check Appendix \[sec:proof\_of\_main\_random\_theorem\]. All presented algorithms are designed for the training phase and their output is the set ${S_{\rm{tr}}}$ with size $l$. The sets $\{S_i\}_{i=1}^m$ are only computed for algorithmic purposes. Given a new task at the test phase, the remaining $k-l$ task-specific elements will be added to ${S_{\rm{tr}}}$ using for instance greedy updates that require a total complexity of $O((k-l)n)$ in function evaluations. Also, the training complexity of the proposed algorithms is $O(kmn)$, however, certain phases can be implemented in parallel. Simulation Results {#sec:Simulation} ================== We provide two experimental setups to evaluate the performance of our proposed algorithms and compare with other baselines. Each setup involves a different set of tasks which are represented as submodular maximization problems subject to the $k$-cardinality constraint. In our experiments, we consider the following algorithms: **Meta-Greedy** (Algorithm \[alg: main discrete meta-Greedy\]), **Randomized Meta-Greedy** (Algorithm \[alg: Randomized meta-Greedy\]), **Greedy-Train** (which chooses all the $k$ elements during the training phase–see and the discussion therein), **Greedy-Test** (which chooses all the $k$ elements during the test phase–see and the discussion therein), and **Random** (which chooses a random set of $k$ elements). In the following, we briefly explain the data and tasks and refer the reader to the supplementary materials for more details. We described the Meta submodular learning problem and algorithms in previous sections. In this section, we provide two applications: - Ride Share Optimization - Movie Recommendation We compare the performance of proposed algorithms on them by showing the comparison in figure\[fig:simulation-ride\]. [0.48]{} {width="1\linewidth"} [0.48]{} {width="1\linewidth"} [0.48]{} {width="1\linewidth"} [0.48]{} {width="1\linewidth"} [0.48]{} {width="1.1\linewidth"} [0.48]{} {width="1.1\linewidth"} **Ride Share Optimization.** We will formalize and solve a facility location problem on the Uber dataset [@uber_2019]. Our experiments were run on the portion of data corresponding to Uber pick-ups in Manhattan in the period of September 2014. This portion consists of $\sim 10^6$ data points each represented as a triplet $(latitude, longitude, DateTime)$. A customer and a driver are specified through their locations on the map. We use $u=(x_u,y_u)$ for a customer a and $r=(x_r, y_r)$ for a driver. We define the “convenience score” of a (customer, driver) pair as $c(u,r)=2-\frac{2}{1+e^{-200 d(u,r)}}$, where $d(u,r)$ denotes the Manhattan distance [@mitrovic2018data]. Given a specific time $a$, we define a time slot $T_a$ and picking inside the data set 10 points in half an hour prior to time $a$, and for each point we further pick 10 points in its 1 km neighborhood, which makes a total of 100 points (locations) on the map. A task ${\mathcal{T}}_i$ takes place at a corresponding time $a_i$, and by defining the set of locations $T_{a_i}$ as above, we let $f_i$ be a monotone submodular function defined over a set $S$ of driver locations as $f_i(S)=\sum_{u\in T_{a_i}}\max_{r\in S}c(u,r)$. We pick 100,000 locations at random from the September 2014 Uber pick-up locations as a ground set. For training we form $m=50$ tasks by picking for each task a random time in the *first* week of Sept. 2014. We test on $m=50$ new tasks formed similarly from the *second* week of Sept. 2014 and report in the figures the average performance obtained at test tasks. Figures \[fig:simulation-1-rideshare\] and \[fig:simulation-2-rideshare\] show the performance of our proposed algorithms against the baselines mentioned above. Figure shows the performance of all algorithms when we fix $k=20$, and vary $l$ from 5 to 18. Larger $l$ means less computation at test time (since we need to further choose $k-l$ elements at test). However, we see that even for large values of $l$ (e.g. $l=16$), the performance of Meta-Greedy is still quite close to the ideal performance of Greedy-Test. Putting this together with the fact that the performance of Greedy-Train is not so good, we can conclude that adding a few personalized elements at test time significantly boosts performance to be even close to the ideal. In Figure , we compare the performance of all the algorithms when $k$ changes from 5 to 30, and $l$ is $80\%$ of $k$ ($l = \lfloor 0.8 k \rfloor$). As we can see, even when we just learn $20\%$ of the set in test time, the performance of Meta-greedy is close to Test-Greedy. Also, when $k-l$ increases, Random-Meta-Greedy performs better than Meta-Greedy. This is in compliance with the results of Theorems \[thm:meta-greedy\], \[thm: randomized meta\]. **Movie Recommendation.** In this application, we use the Movielens dataset [@harper2015movielens] which consists of $10^6$ ratings (from 1 to 5) by $6041$ users for $4000$ movies. We pick the 2000 most rated movies, and 200 users who rated the highest number of movies (similar to [@stan2017probabilistic]). We partitioned the 200 users into 100 users for the training phase and 100 other users for the test phase. Each movie can belong to one of 18 genres. For each genre $t$ we let $G_t$ be the set of all movies with in genre $t$. For each user $i$, we let $R_i$ be the set of all movie rated by the user, and for each movie $v \in R_i$ the corresponding rating is denoted by $r_i(v)$. Furthermore, for user $i$ we define $ f_i(S)=\sum_{t=1}^{18} w_{i,t}.\max_{v\in R_i\cap G_t \cap S} r_i(v)$ which is the weighted average over maximum rate that user $i$ gives to movies from each genre and $w_{i,t}$ is proportion of movies in genre $t$ which is rated by user $i$ out of all the rating he provides. A task ${\mathcal{T}}_i$ involves 5 users $i_1, \cdots, i_5$ and the function assigned to the task is the average of $f_{i_1}, \cdots, f_{i_5}$. We formed $m=50$ training tasks from the users in the training phase, and $m=50$ test tasks from the users in the test phase. Figure (resp. \[fig:simulation-2-movrecom\]) has been obtained in a similar format as Figure \[fig:simulation-1-rideshare\] (resp. Figure \[fig:simulation-2-rideshare\]). We observe a very similar pattern as in the ride share experiments. ![Comparison of two-stage framework and submodular meta-learning framework []{data-label="fig:twostage"}](rideshare_twostage.eps){width="8cm"} Comparison with Two-stage Submodular Optimization {#sec:two_stage} ================================================= Two-stage submodular optimization is another way to deal with limited computational power in test time. In this framework, at training time, a reduced ground set is learned which will be used as a ground set at test time. This procedure will reduce the computational time in test time. More formally, the two-stage submodular optimization framework aims to solve the following problem. Let $f_i:2^{\mathcal{X}}\xrightarrow{}{\mathbb{R}}_+$ for $i\in [m]$, be a monotone submodular function over ground set $V$. The goal is to find $S$ with size at most $q$ whose subests of size $k$ maximize the sum of $f_i$ for $i\in [m]$: $$\label{eq:two-stage submodular} \max_{S\subseteq {\mathcal{X}}, \mid S \mid\leq q}\; \frac{1}{m}\sum_{i=1}^{m}\; \max_{S_i\subseteq S, \mid S_i\mid \leq k}\; f_i(S_i)$$ Once the set $S$ is found, it will be used in the test phase (e.g., by running full greedy on $S$ as the reduced ground set) to find $k$ elements for a new task. This framework uses ${\mathcal{O}}(qk)$ function evaluations for each new test task; however, it poorly personalizes to a test task because the set $S$ has been optimized only for the tasks at the training time. This intuition is indeed consistent with our experimental findings reported below. We further remark that the two-stage framework requires very high computational power in training. For example, the Replacement-Greedy algorithm [@stan2017probabilistic] requires computational complexity ${\mathcal{O}}(qkmn)$ (which is a factor $q$ larger than the complexity of the algorithms in this paper). As a result of this issue, we were not able to run the state-of-the-art two-stage algorithms to solve in the setting considered in our main simulation results (presented in Section \[sec:Simulation\]). e.g., for ground set of size $n = 10^5$ our two-stage implementation would take a very long time. We have considered the ride-sharing application discussed in Section \[sec:Simulation\] and let $n=500$ (ground set size), $m=50$ (number of tasks), and $k$ changing from 5 to 30 (cardinality constraint) while $l=80\%k$ (portion that will fill in the submodular meta-learning during training), and $q=100$ (size of reduced ground set for two-stage framework). For solving the two-stage problem we have used the Replacement-Greedy algorithm introduced in [@stan2017probabilistic]. We choose these parameters based on the following two facts: 1. Because of the high computational cost of the Replacement Greedy algorithm in training for the ride-sharing application, we chose $n$ to be 500. 2. We provide a fair comparison in terms of computational power at test time, which means both Meta-Greedy (our algorithm) and Replacement-Greedy have exactly *the same computational cost* at test time. Formally, $n(k-l)=qk$. we report the result for the above setting in the Figure \[fig:twostage\]. A few comments are in order: (i) The two stage implementation reduces the ground set of size $n = 500$ to $q = 100$. When $k$ is small, some of the popular elements found at training time would be good enough to warrant a good performance at test time. However, when $k$ increases, the role of personalizing becomes more apparent. As we see, the performance of Replacement-Greedy does not improve much when we increase $k$ and it is close to the performance of Greedy-Train (which chooses all the $k$ elements during the training phase–see and the discussion therein). However, since Meta-Greedy does (a small) task-specific optimization at test time, its performance becomes much better. We emphasize again that, in order to be fair, the comparison in Figure \[fig:twostage\] has been obtained using *the same* computational power allowed at test time for both meta-learning and two-stage approaches. Conclusion and Future Work ========================== In this paper, we extended the notion of Model-Agnostic Meta-Learning (MAML) to discrete optimization and in particular to submodular maximization. We proposed a novel formulation in which we aim to find an initial solution set that can be quickly adapted to a new task at a relatively low computational cost. In our meta-learning framework, the process of selecting set items is done in two parts: In the first part, a set of items are selected offline according to prior experience and data. In the second part, which happens at test time, a set of elements that is personalized to the new revealed task is selected. For the proposed problem, we introduced a deterministic variant of the greedy algorithm which obtains a solution that is at least $0.53$-optimal, when the tasks are monotone and submodular. We further presented a randomized algorithm that improves this result and obtains $(1-1/e-o(1))$-approximation in expectation. We also studied the performance of our proposed meta-learning framework and algorithms for two real-world applications: movie recommendation and ride-sharing problems. Our numerical results indicate the advantage of our proposed scheme with respect to traditional learning procedures as well as methods based on two-stage submodular optimization. There are numerous open directions that can be investigated along the lines of discrete meta-learning and user-specific adaptation for discrete problems (indeed, this work can be considered as a first step). Examples include extending the results to a more general setting when the tasks are (approximately) submodular but non-monotone, considering the case that the tasks at training and test times are drawn according to two different probability distributions (possibly with bounded distance), and desinig Further, exploring meta-learning continuous extensions and Acknowledgements {#acknowledgements .unnumbered} ================ The research of Arman Adibi and Hamed Hassani is supported by NSF award CPS-1837253, NSF CAREER award CIF 1943064, and Air Force Office of Scientific Research Young Investigator Program (AFOSR-YIP) under award FA9550-20-1-0111. Appendix {#appendix .unnumbered} ======== Proof of Proposition \[alg1lemma\] {#sec:proof_prop_1} ================================== First, we prove lemma which will help us to show . \[lemma:greedy procedure\] Consider the monotone submodular function $f$ where and the greedy procedure for maximizing it starting from $S^{'}$ and in $t^{th}$ step adding the element with maximum marginal gain $e^{(t)}$ to $S^{(t-1)}$ which means $e^{(t)}=\operatorname*{arg\,max}\limits_e f(S^{(t-1)}\cup e \cup S^{'})-f(S^{(t-1)}\cup S^{'})$, then the following holds for every set $T$ and $S^{''}$ where $S^{'}\subseteq T$ and $\mid T\mid=k$. $$\label{eq: eq1 pf lemma:greedy} f(S{''}\cup T ) -f( S^{(k)}\cup S{''})\leq f( S^{(k)}\cup S^{'})- f(S^{'})$$ To show define $J^{(t)}$ iteratively as follows. Start from $J^{(0)}=T$ and let $D^{(t)}=J^{(t-1)}\setminus S^{(t-1)}$ and define $o^{(t)}$ in the following way: 1. If $e^{(t)}\in D^{(t)}$, then let $o^{(t)}=e^{(t)}$. 2. Otherwise, if $e^{(t)}\notin D^{(t)}$, let $o^{(t)}$ be one of the elements of $D^{t}$ chosen uniformly at random. Define $J^{(t)}:=J^{(t-1)}\cup e^{(t)}\setminus o^{(t)}$.\ (TS\^[”]{}) (TJ\^[(1)]{}) … (TJ\^[(t)]{})\ (S\^[(0)]{}S\^[’]{}) (S\^[(1)]{} S\^[’]{}) … (S\^[(t)]{}S\^[’]{}) \ then we can write the following inequalities: $$\begin{aligned} f(S^{(t)}\cup S^{'})-f(S^{(t-1)}\cup S^{'}) &= f(S^{(t-1)}\cup e^{(t)}\cup S^{'})-f(S^{(t-1)}\cup S^{'}) \\\label{eq: chain1 l1:greedy}&\geq f(S^{(t-1)}\cup S^{'}\cup o_i^{(t)})-f(S^{(t-1)}\cup S^{'}) \\\label{eq: chain2 l1:greedy}&\geq f(J^{(t-1)}\cup S^{'})-f(J^{(t-1)}\cup S^{'}\setminus o^{(t)})\\&\geq f(J^{(t-1)}\cup S^{''})-f(J^{(t-1)}\cup S^{''}\setminus o^{(t)}) \\&\geq f(J^{(t-1)}\cup S^{''})-f(J^{(t-1)}\cup S^{''}\setminus o^{(t)}) \nonumber\\ &\quad-f(S^{''}\cup J^{(t-1)}\cup e^{(t)}\setminus o^{(t)})+f(S^{''}\cup J^{(t-1)}\setminus o^{(t)})\label{eq: chain3 l1:greedy}\\&= f(J^{(t-1)}\cup S^{''})-f(J^{(t-1)}\cup S^{''}\setminus o^{(t)}) \nonumber\\ &\quad-f(S^{''}\cup J^{(t)})+f(S^{''}\cup J^{(t-1)}\setminus o^{(t)}) \\&=f(J^{(t-1)}\cup S^{''})-f(S^{''}\cup J^{(t)})\end{aligned}$$ where follows from $f(S^{(t-1)}\cup e^{(t)} \cup S^{'})-f(S^{(t-1)}\cup S^{'})=\max\limits_e f(S^{(t-1)}\cup e \cup S^{'})-f(S^{(t-1)}\cup S^{'})\geq f(S^{(t-1)}\cup o^{(t)} \cup S^{'})-f(S^{(t-1)}\cup S^{'})$ and follows from submodularity since in each step $S^{(t-1)}\subseteq J^{(t-1)}$ and $o^{(t)}\not\in S^{(t-1)}$. Finally, equation follows from the fact that $-f(J^{(t-1)}\cup e^{(t)}\cup S^{''}\setminus o^{(t)})+f(J^{(t-1)}\cup S^{''}\setminus o^{(t)})\leq 0$. Then, by summing over the above inequality we get the following inequality: $$\begin{aligned} f(S^{(t)}\cup S^{'})-f(S^{(0)}\cup S^{'})&=\sum_{i=0}^{t}f(S^{(i)}\cup S^{'})-f(S^{(i-1)}\cup S^{'}) \\&\geq \sum_{i=0}^{t} f(S^{''}\cup J^{(i-1)})-f(S^{''}\cup J^{(i)}) \\&=f(S^{''}\cup J^{(0)})-f(S^{''}\cup J^{(t)}) \\&\geq f(S^{''}\cup T)-f(S^{''}\cup J^{(t)})\end{aligned}$$ Note because of the definition of $J^{(t)}$ after k step $J^{(k)}=S^{(k)}$; therefore, we can conclude that: $$\begin{aligned} f(S^{(k)}\cup S^{'})-f( S^{'})&\geq f(S^{''}\cup T)-f(S^{''}\cup J^{(t)})\notag\\&=f(S^{''}\cup T)-f(S^{''}\cup S^{(k)})\end{aligned}$$ . Let ${S_{\rm{tr}}}$, $\{S_i\}_{i=1}^{m}$ be the output of Algorithm \[alg: discrete meta-Greedy\] and ${S_{\rm{tr}}}^{*}$, $\{S_i^{*}\}_{i=1}^{m}$ be the optimal solution for problem . We first show that the output of Algorithm \[alg: discrete meta-Greedy\] in phase 1 satisfies the following inequality: $$\label{eq: eq1 pf l1} \sum_{i=1}^m f_i({S_{\rm{tr}}}^{*} \cup S_i^{*}) - \sum_{i=1}^m f_i( {S_{\rm{tr}}})\leq \sum_{i=1}^m f_i({S_{\rm{tr}}}\cup S_i^{*})$$ To show let $e^{(t)}$ be the ${t}^{th}$ element of greedy procedure in phase 1, and ${S_{\rm{tr}}}^{(t)}$ be the $t^{th}$ set in this procedure, where $e^{(t)}=\operatorname*{arg\,max}\limits_e \sum\limits_{i=1}^{m} f_i({S_{\rm{tr}}}^{(t-1)}\cup e)-f_i({S_{\rm{tr}}}^{(t-1)})$. let $J^{(0)}={S_{\rm{tr}}}^{*}$ and define $J^{(t)}$ iteratively as follows. Let $D^{(t)}=J^{(t-1)}\setminus {S_{\rm{tr}}}^{(t-1)}$ and define $o^{(t)}$ in the following way: 1. If $e^{(t)}\in D^{(t)}$, then let $o^{(t)}=e^{(t)}$. 2. Otherwise, if $e^{(t)}\notin D^{(t)}$, let $o^{(t)}$ be one of the elements of $D^{t}$ chosen uniformly at random. Define $J^{(t)}:=J^{(t-1)}\cup e^{(t)}\setminus o^{(t)}$. We show this procedure in the following chain.\ ([S\_[[tr]{}]{}]{}\^[\*]{},{S\_i\^[\*]{}}\_[i=1]{}\^[m]{}) (J\^[(1)]{},{S\_i\^[\*]{}}\_[i=1]{}\^[m]{}) … (J\^[(l)]{},{S\_i\^[\*]{}}\_[i=1]{}\^[m]{})\ ([S\_[[tr]{}]{}]{}=,{S\_i\^0}\_[i=1]{}\^[m]{}=) ([S\_[[tr]{}]{}]{}\^[(1)]{},{}\_[i=1]{}\^[m]{}) … ([S\_[[tr]{}]{}]{}\^[(l)]{},{}\_[i=1]{}\^[m]{}) \ then we can write the following inequalities: $$\begin{aligned} \sum_{i=1}^m f_i({S_{\rm{tr}}}^{(t)})-f_i({S_{\rm{tr}}}^{(t-1)}) &= \sum_{i=1}^m f_i({S_{\rm{tr}}}^{(t-1)}\cup e^{(t)})-f_i({S_{\rm{tr}}}^{(t-1)}) \\\label{eq: chain1 l1}&\geq \sum_{i=1}^m f_i({S_{\rm{tr}}}^{(t-1)}\cup o_i^{(t)})-f_i({S_{\rm{tr}}}^{(t-1)}) \\\label{eq: chain2 l1}&\geq \sum_{i=1}^m f_i(S_{i}^{*}\cup J^{(t-1)})-f_i(S_{i}^{*}\cup J^{(t-1)}\setminus o^{(t)}) \\&\geq \sum_{i=1}^m f_i(S_i^{*}\cup J^{(t-1)})-f_i(S_i^{*}\cup J^{(t-1)}\setminus o_i^{(t)}) \nonumber\\ &\quad+\sum_{i=1}^m -f_i(S_{i}^{*}\cup J^{(t)})+f_i(S_{i}^{*}\cup J^{(t-1)}\setminus o_i^{(t)})\label{eq: chain3 l1} \\&=\sum_{i=1}^m f_i(S_{i}^{*}\cup J^{(t-1)})-f_i(S_{i}^{*}\cup J^{(t)})\end{aligned}$$ where follows from definition of $e^{(t)}$ and the greedy procedure and follows from submodularity since in each step ${S_{\rm{tr}}}^{(t-1)}\subseteq J^{(t-1)}$ and $o^{(t)}\not\in {S_{\rm{tr}}}^{(t-1)}$ and finally, equation follows from the fact that $-f_i(J^{(t)}\cup S_i^{*})+f_i(J^{(t-1)}\cup S_i^{*}\setminus o^{(t)})\leq 0$. Then, by summing over $t$ from 0 to $l$ we get the following inequality: $$\begin{aligned} \sum_{i=1}^m f_i({S_{\rm{tr}}})=\sum_{i=1}^m f_i({S_{\rm{tr}}}^{(l)})-f_i({S_{\rm{tr}}}^{(0)})&=\sum_{i=1}^m\sum_{t=0}^{l}f_i({S_{\rm{tr}}}^{(t)})-f_i({S_{\rm{tr}}}^{(t-1)}) \\&\geq \sum_{i=1}^m\sum_{t=0}^{l}f_i(S_{i}^{*}\cup J^{(t-1)})-f_i(S_{i}^{*}\cup J^{(t)}) \\&=\sum_{i=1}^m f_i(S_{i}^{*}\cup J^{(0)})-f_i(S_{i}^{*}\cup J^{(l)}) \\&=\sum_{i=1}^m f_i(S_{i}^{*}\cup {S_{\rm{tr}}}^{*})-f_i(S_{i}^{*}\cup {S_{\rm{tr}}})\end{aligned}$$ where the last equality comes from the process of defining $J$. Because, we only change one element by adding element found in greedy process and removing one element from the optimal set in each step and the size of $J^{(t)}$ is $l$ in each step; therefore, after $l$ step $J^{(l)}={S_{\rm{tr}}}$. By rearranging the terms and summing over $i$ the claim in follows.\ \ Second, for the phase 2 of the algorithm \[alg: discrete meta-Greedy\] we can use the usual analysis of greedy[@krause2014submodular] for set $S_i$: $$\begin{aligned} \sum_{i=1}^m f_i({S_{\rm{tr}}}\cup S_i)-f_i( {S_{\rm{tr}}})&\geq (1-\frac{1}{e})(\sum_{i=1}^m f_i({S_{\rm{tr}}}\cup S_i^{opt})-f_i({S_{\rm{tr}}})) \label{eq: eq3 pf l1 1} \\&\geq (1-\frac{1}{e})(\sum_{i=1}^m f_i({S_{\rm{tr}}}\cup S_i^{*})-f_i({S_{\rm{tr}}})) \label{eq: eq3 pf l1 2}\\&\geq (1-\frac{1}{e})(\sum_{i=1}^m f_i({S_{\rm{tr}}}^{*}\cup S_i^{*})-2f_i({S_{\rm{tr}}})) \label{eq: eq3 pf l1 3}\end{aligned}$$ where $S_i^{opt}=\operatorname*{arg\,max}\limits_{|S_i|\leq k-l}f_i({S_{\rm{tr}}}\cup S_i)$ in the equation . Equation follows from usual greedy analysis, equation follows from definition of ${S_{\rm{tr}}}^{opt}$, and equation  follows from equation . Now divide both sides of by $1/m$ and regroup the terms to obtain $$\label{eq:result_alg_1_part_1} \frac{1}{m} \sum_{i=1}^m f_i({S_{\rm{tr}}}\cup S_i) \geq \left(1-\frac{1}{e}\right)({\rm{OPT}}-2\beta)+\beta,$$ where $\beta := \frac{1}{m}\sum_{i=1}^m f_i({S_{\rm{tr}}}).$ Finally, since $S_i\subseteq S_i\cup {S_{\rm{tr}}}$ by monotonicity [[$f_i(S_i\cup {S_{\rm{tr}}})\geq f_i({S_{\rm{tr}}})$]{}]{}. Then, combing this observation with the result in implies $$\frac{1}{m}\sum_{i=1}^m f_i ({S_{\rm{tr}}}\cup S_i) \geq \max \biggl\{ \beta \,, \, (1-1/e)({\rm{OPT}} - 2\beta) + \beta \biggr \}.$$ Proof of Proposition \[alg2lemma\] {#sec:proof_prop_2} ================================== Let ${S_{\rm{tr}}}$, $\{S_i\}_{i=1}^{m}$ be the output of Algorithm \[alg: Reverse discrete meta-Greedy\] and ${S_{\rm{tr}}}^{*}$, $\{S_i^{*}\}_{i=1}^{m}$ be the optimal solution for problem . We first show the following about the output of algorithm \[alg: Reverse discrete meta-Greedy\], phase 1. $$\label{eq: eq1 pf l2} \sum_{i=1}^m f_i({S_{\rm{tr}}}^{*} \cup S_i^{*}) - \sum_{i=1}^m f_i( S_i)\leq \sum_{i=1}^m f_i({S_{\rm{tr}}}^{*} \cup S_i)$$ to show consider the following: let $e_i^{(t)}=\operatorname*{arg\,max}\limits_e f_i(S_i^{(t-1)}\cup e)-f_i(S_i^{(t-1)})$. let $J^{(0)}_i=S_i^{*}$ and define $J^{(t)}_i$ iteratively as follows. Let $D^{t}_{i}=J^{(t-1)}_i\setminus S^{(t-1)}_i$ and define $o_i^{(t)}$ in the following way: 1. If $e_i^{(t)}\in D^{t}_{i}$, then $o^{(t)}=e_i^{(t)}$; 2. Otherwise, if $e_i^{(t)}\notin D^{t}_{i}$, let $o_i^{(t)}$ be one of the elements of $D^{t}_{i}$ chosen uniformly at random; Define $J^{(t)}_i:=J^{(t-1)}_i\cup e_i^{(t)}\setminus o_i^{(t)}$.\ ([S\_[[tr]{}]{}]{}\^[\*]{},{S\_i\^[\*]{}}\_[i=1]{}\^[m]{}) ([S\_[[tr]{}]{}]{}\^[\*]{},{J\_i\^[(1)]{}}\_[i=1]{}\^[m]{}) … ([S\_[[tr]{}]{}]{}\^[\*]{},{J\_i\^[(k-l)]{}}\_[i=1]{}\^[m]{})\ ([S\_[[tr]{}]{}]{}=,{S\_i\^0}\_[i=1]{}\^[m]{}=) (,{S\_i\^[(1)]{}}\_[i=1]{}\^[m]{}) … (,{S\_i\^[(k-l)]{}}\_[i=1]{}\^[m]{}) \ then we can write the following inequalities: $$\begin{aligned} f_i(S_i^{(t)})-f_i(S_i^{(t-1)}) &= f_i(S_i^{(t-1)}\cup e_i^{(t)})-f_i(S_i^{(t-1)}) \\\label{eq: chain1}&\geq f_i(S_i^{(t-1)}\cup o_i^{(t)})-f_i(S_i^{(t-1)}) \\\label{eq: chain2}&\geq f_i({S_{\rm{tr}}}^{*}\cup J_i^{(t-1)})-f_i({S_{\rm{tr}}}^{*}\cup J_i^{(t-1)}\setminus o_i^{(t)}) \\&\geq f_i({S_{\rm{tr}}}^{*}\cup J_i^{(t-1)})-f_i({S_{\rm{tr}}}^{*}\cup J_i^{(t-1)}\setminus o_i^{(t)}) \nonumber\\ &\quad-f_i({S_{\rm{tr}}}^{*}\cup J_i^{(t)})+f_i({S_{\rm{tr}}}^{*}\cup J_i^{(t-1)}\setminus o_i^{(t)})\label{eq: chain3} \\&=f_i({S_{\rm{tr}}}^{*}\cup J_i^{(t-1)})-f_i({S_{\rm{tr}}}^{*}\cup J_i^{(t)})\end{aligned}$$ where follows from definition of $e_i^{(t)}$ and the greedy procedure and follows from the submodularity since in each step $S_i^{(t-1)}\subseteq J_i^{(t-1)}$ and $o_i^{(t)}\not\in S_i^{(t-1)}$ and finally, equation follows from the fact that $-f_i({S_{\rm{tr}}}^{*}\cup J_i^{(t)})+f_i({S_{\rm{tr}}}^{*}\cup J_i^{(t-1)}\setminus o_i^{(t)})\leq 0$ because of monotonicity. Then, by summing over $t$ from 0 to $k-l$ we get the following inequality: $$\begin{aligned} f_i(S_i)=f_i(S_i^{(k-l)})-f_i(S_i^{(0)})&=\sum_{t=0}^{k-l}f_i(S_i^{(t)})-f_i(S_i^{(t-1)}) \\&\geq \sum_{t=0}^{k-l}f_i({S_{\rm{tr}}}^{*}\cup J_i^{(t-1)})-f_i({S_{\rm{tr}}}^{*}\cup J_i^{(t)}) \\&=f_i({S_{\rm{tr}}}^{*}\cup J_i^{(0)})-f_i({S_{\rm{tr}}}^{*}\cup J_i^{(k-l)}) \\&=f_i({S_{\rm{tr}}}^{*}\cup S_i^{*})-f_i({S_{\rm{tr}}}^{*}\cup S_i)\end{aligned}$$ where the last equality comes from the process of defining $J_i^{(k-l)}$; since, the size of $J_i^{(t)}$ is $k-l$ in each step and after $k-l$ step $J_i^{(k-l)}=S_i$. Then, by rearranging and summing over $i$ we can obtain .\ \ Second, for phase 2 of algorithm \[alg: Reverse discrete meta-Greedy\] we can use the usual analysis of greedy[@krause2014submodular] for set ${S_{\rm{tr}}}$ : $$\begin{aligned} \sum_{i=1}^m f_i({S_{\rm{tr}}}\cup S_i)-f_i( S_i)&\geq (1-\frac{1}{e})(\sum_{i=1}^m f_i({S_{\rm{tr}}}^{opt}\cup S_i)-f_i(S_i)) \label{eq: eq3 pf l2 1} \\&\geq (1-\frac{1}{e})(\sum_{i=1}^m f_i({S_{\rm{tr}}}^{*}\cup S_i)-f_i(S_i)) \label{eq: eq3 pf l2 2}\\&\geq (1-\frac{1}{e})(\sum_{i=1}^m f_i({S_{\rm{tr}}}^{*}\cup S_i^{*})-2f_i(S_i)) \label{eq: eq3 pf l2 3}\end{aligned}$$ where ${S_{\rm{tr}}}^{opt}=\operatorname*{arg\,max}\limits_{|{S_{\rm{tr}}}|\leq l}\sum\limits_{i=1}^m f_i({S_{\rm{tr}}}\cup S_i)$ in equation . Equation follows from the usual greedy analysis, equation follows from the definition of ${S_{\rm{tr}}}^{opt}$, and equation follows the from equation . Now divide both sides of by $1/m$ and regroup the terms to obtain $$\label{eq:result_alg_2_part_1} \frac{1}{m} \sum_{i=1}^m f_i({S_{\rm{tr}}}\cup S_i) \geq \left(1-\frac{1}{e}\right)({\rm{OPT}}-2\gamma)+\gamma,$$ where $\gamma := \frac{1}{m}\sum_{i=1}^m f_i(S_i).$ Finally, since $S_i\subseteq S_i\cup {S_{\rm{tr}}}$ by monotonicity $f_i(S_i\cup {S_{\rm{tr}}})\geq f_i(S_i)$. Then, by combing this result wit we obtain $$\frac{1}{m}\sum_{i=1}^m f_i ({S_{\rm{tr}}}\cup S_i) \geq \max \biggl\{ \gamma \,, \, (1-1/e)({\rm{OPT}} - 2\gamma) + \gamma \biggr \}.$$ The following shows the ratio of lower bound to optimum (a similar plot can be obtained for the lower bound of Proposition \[alg1lemma\] when $\gamma$ is replaced with $\beta$.). As we observe, in the worst case, the approximation factor is $0.5$. ![y-axis: The lower bound of Proposition \[alg2lemma\] divided by OPT, x-axis: $\gamma/\text{OPT}$.[]{data-label="fig:bound"}](boundplot.eps){width="8cm"} Proof of Theorem \[thm:meta-greedy\] {#sec:proof_of_main_thm} ==================================== Let $\theta_2=\frac{1}{m}\sum_{i=1}^m f_i ({S_{\rm{tr}}}^{(2)} \cup S_i^{(2)})$. Since ${S_{\rm{tr}}}^{(2)}$ found greedily given $\{S_i\}_{i=1}^{m}$ we can write: $$\label{eq: thm1 pf eq1} \theta_2-\gamma\geq ({\rm{OPT}}-\gamma)(1-\frac{1}{e})\geq( \frac{1}{m} \sum_{i=1}^{m} f_i(S^{'} \cup S_i^{(2)})-\gamma)(1-\frac{1}{e})$$ for every $\mid S^{'}\mid\leq l$. Also, we can write $$\begin{aligned} {\rm{OPT}}-\gamma&= \frac{1}{m}\sum_{i=1}^{m}f_i({S_{\rm{tr}}}^{*}\cup S_i^{*})-f_i(S_i^{(2)}) \label{eq:tm1 pf align 1} \\&\leq \frac{1}{m}\sum_{i=1}^{m}f_i({S_{\rm{tr}}}^{*}\cup S_i^{(2)} \cup S_i^{*})-f_i(S_i^{(2)}) \label{eq:tm1 pf align 2} \\&= \frac{1}{m}\sum_{i=1}^{m}f_i({S_{\rm{tr}}}^{*}\cup S_i^{(2)} \cup S_i^{*})+f_i({S_{\rm{tr}}}^{*} \cup S_i^{(2)})-f_i({S_{\rm{tr}}}^{*} \cup S_i^{(2)})-f_i(S_i^{(2)}) \label{eq:tm1 pf align 3} \\&\leq \frac{1}{m}\sum_{i=1}^{m}f_i({S_{\rm{tr}}}^{*}\cup S_i^{(2)} \cup S_i^{*})-f_i({S_{\rm{tr}}}^{*} \cup S_i^{(2)})+\frac{\theta_2-\gamma}{1-1/e} \label{eq:tm1 pf align 4}\\&\leq \frac{1}{m}\sum_{i=1}^{m}f_i( S_i^{(2)} \cup S_i^{*})-f_i( S_i^{(2)})+\frac{\theta_2-\gamma}{1-1/e} \label{eq:tm1 pf align 5}\end{aligned}$$ where comes from , and comes from submodularity. We thus obtain $$\label{eq: thm l1 1} {\rm{OPT}} -\frac{\theta_2-\gamma}{1-1/e}-\gamma \leq \frac{1}{m}\sum_{i=1}^{m}f_i( S_i^{(2)} \cup S_i^{*})-f_i( S_i^{(2)})$$ Also we can write for any set $S'$ such that $\mid S^{'}\mid\leq l$: $$\begin{aligned} \label{eq:tm1 pf align2 1} \frac{1}{m}\sum_{i=1}^{m}f_i(S^{'}\cup S_i^{*})-f_i(S^{'}) &\geq \frac{1}{m}\sum_{i=1}^{m}f_i(S^{'}\cup S_i^{*}\cup S_i)-f_i(S^{'}\cup S_i)\\ \label{eq:tm1 pf align2 2} &\geq \frac{1}{m}\sum_{i=1}^{m}f_i(S^{'}\cup S_i^{*}\cup S_i)-f_i(S_i)+f_i(S_i)-f_i(S^{'}\cup S_i)\\ \label{eq:tm1 pf align2 3} &\geq \frac{1}{m} \sum_{i=1}^{m}f_i( S_i^{*}\cup S_i)-f_i(S_i)+f_i(S_i)-f_i(S^{'}\cup S_i)\\ \label{eq:tm1 pf align2 4} &\geq {\rm{OPT}} -\frac{\theta_2-\gamma}{1-1/e}-\gamma+ \frac{1}{m} \sum_{i=1}^{m}f_i(S_i)-f_i(S^{'}\cup S_i)\\ \label{eq:tm1 pf align2 5} &\geq {\rm{OPT}} -2\frac{\theta_2-\gamma}{1-1/e}-\gamma\end{aligned}$$ where follows from submodularity, follows from monotonicity, and follows from , and follows from . This results the following for any set $S'$ such that $|S^{'}|\leq l$: $$\label{eq: thm l1 2} \frac{1}{m} \sum_{i=1}^{m}f_i(S^{'}\cup S_i^{*})-f_i(S^{'}) \geq {\rm{OPT}} -2\frac{\theta_2-\gamma}{1-1/e}-\gamma$$ Now, from we can find a new bound for the performance of algorithm \[alg: main discrete meta-Greedy\]. From we can write: $$\label{eq: thm1 pf eq2} \frac{1}{m} \sum_{i=1}^{m}f_i({S_{\rm{tr}}}^{(1)}\cup S_i^{*})-f_i({S_{\rm{tr}}}^{(1)}) \geq {\rm{OPT}} -2\frac{\theta_2-\gamma}{1-1/e}-\gamma$$ Also, since in Algorithm 1 the set $S_i^{(1)}$ is constructed greedily on the top of ${S_{\rm{tr}}}^{(1)}$, we have: $$\begin{aligned} \label{eq: thm1 pf eq3} \frac{1}{m} \sum_{i=1}^{m} f_i({S_{\rm{tr}}}^{(1)} \cup S_i^{(1)})-\beta &\geq( \frac{1}{m} \sum_{i=1}^{m} f_i({S_{\rm{tr}}}^{(1)} \cup S_i^{*})-\beta)(1-\frac{1}{e})\\ &\geq ( {\rm{OPT}} -2\frac{\theta_2-\gamma}{1-1/e}-\gamma)(1-\frac{1}{e}),\label{eq: thm1 pf eq3}\end{aligned}$$ where follows from . We thus obtain: $$\begin{aligned} \label{eq: thm1 pf eq4} \frac{1}{m} \sum_{i=1}^{m} f_i({S_{\rm{tr}}}^{(1)} \cup S_i^{(1)}) \geq ( {\rm{OPT}} -2\frac{\theta_2-\gamma}{1-1/e}-\gamma)(1-\frac{1}{e})+\beta\end{aligned}$$ Using the same procedure as above, by defining $\theta_1= \frac{1}{m} \sum_{i=1}^{m} f_i({S_{\rm{tr}}}^{(1)} \cup S_i^{(1)})$, we can prove: $$\begin{aligned} \label{eq: thm1 pf eq5} \frac{1}{m} \sum_{i=1}^{m} f_i({S_{\rm{tr}}}^{(2)} \cup S_i^{(2)}) \geq ( {\rm{OPT}} -2\frac{\theta_1-\gamma}{1-1/e}-\gamma)(1-\frac{1}{e})+\beta\end{aligned}$$ which results in the following lower bound: $$\begin{aligned} &\max \biggl\{ \frac{1}{m} \sum_{i=1}^m f_i ({S_{\rm{tr}}}^{(1)} \cup S_i^{(1)}) \, , \, \frac{1}{m}\sum_{i=1}^m f_i ({S_{\rm{tr}}}^{(2)} \cup S_i^{(2)}) \biggr\} \nonumber\\ & \geq \max \biggl\{\theta_1, \theta_2, (1-1/e)({\rm{OPT}} \!-\!\gamma) + \beta -2 (\theta_2-\gamma) ,(1-1/e)({\rm{OPT}} \!-\!\beta) + \gamma -2 (\theta_1-\beta) \biggr\}.\label{eq:thm p6}\end{aligned}$$ Finally, given and , the factor $0.53$ is obtained as a result of the following procedure. Let $\beta$ and $\gamma$ given as $\beta := \frac{1}{m} \sum_{i=1}^m f_i({S_{\rm{tr}}}^{(1)})$ and $\gamma := \frac{1}{m} \sum_{i=1}^m f_i(S_i^{(2)})$. Then the left-hand-side term in is lower bounded by: \[eq: thm1 opt\] $$\begin{aligned} {4} \min\limits_{\theta_1, \theta_2} & \,\, \max \biggl\{\theta_1, \theta_2, (1-1/e)({\rm{OPT}} \!-\!\gamma) + \beta -2 (\theta_2-\gamma) , (1-1/e)({\rm{OPT}} \!-\!\beta) + \gamma -2 (\theta_1-\beta) \biggr\} \\ & \operatorname*{subject\,\, to \quad}\theta_1 \geq \max\{\beta, (1-1/e)({\rm{OPT}} - 2\beta) + \beta\} \\ & \!\qquad \quad \quad \quad \quad \theta_2 \geq \max\{\gamma, (1-1/e)({\rm{OPT}} - 2\gamma) +\gamma\} \end{aligned}$$ Note that the constraints hold due to the results of Proposition 1 and 2. In particular, the above bound is always larger than $ 0.53 \times \rm{OPT} $ for any value of $\beta$ and $\gamma$. Proof of Theorem \[thm: randomized meta\] {#sec:proof_of_main_random_theorem} ========================================= Consider round $t$ in which $\mid {S_{\rm{tr}}}\mid<l$ and $\mid S_i \mid<k-l$ the expected gain of the algorithm with probability $\frac{l}{k}$ is the maximum gain from adding an element $e^{*}=\operatorname*{arg\,max}\limits_e \sum\limits_{i=1}^{m}f_i({S_{\rm{tr}}}^{t}\cup{e}\cup S_i^{t})-f_i({S_{\rm{tr}}}^t\cup S_i^t)$ or with probability $\frac{k-l}{k}$ the gain is $\sum\limits_{i=1}^{m}\max_{e_i}f_i({S_{\rm{tr}}}^{t}\cup{e_i} \cup S_i^{t})-f_i({S_{\rm{tr}}}^t\cup S_i^t)$ which can be written as follows. $$\begin{aligned} \mathbb{E}[&\sum\limits_{i=1}^{m}f_i({S_{\rm{tr}}}^{t+1}\cup S_i^{t+1})-f_i({S_{\rm{tr}}}^t\cup S_i^t)|{S_{\rm{tr}}}^{t},S_i^t]\nonumber\\=&\frac{l}{k}\max_e\sum\limits_{i=1}^{m}f_i({S_{\rm{tr}}}^{t}\cup{e}\cup S_i^{t})-f_i({S_{\rm{tr}}}^t\cup S_i^t)+\frac{k-l}{k}\sum\limits_{i=1}^{m}\max_{e_i}f_i({S_{\rm{tr}}}^{t}\cup{e_i} \cup S_i^{t})-f_i({S_{\rm{tr}}}^t\cup S_i^t)\label{eq: rand alg pf eq begin}\end{aligned}$$ assuming ${S_{\rm{tr}}}^* , S_i^{*}$ is optimal solution, we can also write: $$\begin{aligned} \frac{1}{k}\sum\limits_{i=1}^{m}f_i({S_{\rm{tr}}}^{*}\cup S_i^{*})-f_i({S_{\rm{tr}}}^t\cup S_i^t) &\leq \frac{1}{k}\sum\limits_{i=1}^{m}f_i({S_{\rm{tr}}}^{*}\cup S_i^{*}\cup {S_{\rm{tr}}}^t\cup S_i^t)-f_i({S_{\rm{tr}}}^t\cup S_i^t) \label{eq: rand alg pf eq 1} \\&\leq \frac{1}{k}\sum_{e\in {S_{\rm{tr}}}^{*}\setminus {S_{\rm{tr}}}^t}\sum\limits_{i=1}^{m}f_i({e} \cup {S_{\rm{tr}}}^t\cup S_i^t)-f_i( {S_{\rm{tr}}}^t\cup S_i^t) \nonumber\\&+\frac{1}{k}\sum\limits_{i=1}^{m}\sum_{e\in S_i^{*}\setminus S_i^t}f_i({e} \cup {S_{\rm{tr}}}^t\cup S_i^t)-f_i( {S_{\rm{tr}}}^t\cup S_i^t) \label{eq: rand alg pf eq 2} \\&\leq \frac{l}{k}\max_e\sum\limits_{i=1}^{m}f_i({S_{\rm{tr}}}^{t}\cup{e}\cup S_i^{t})-f_i({S_{\rm{tr}}}^t\cup S_i^t) \nonumber\\&+\frac{k-l}{k}\sum\limits_{i=1}^{m}\max_{e_i}f_i({S_{\rm{tr}}}^{t}\cup{e_i} \cup S_i^{t})-f_i({S_{\rm{tr}}}^t\cup S_i^t) \label{eq: rand alg pf eq 3}\end{aligned}$$ where follows from monotonicity, and follows from submodularity. Then, from and we conclude that: $$\begin{aligned} \mathbb{E}[&\sum\limits_{i=1}^{m}f_i({S_{\rm{tr}}}^{t+1}\cup S_i^{t+1})-f_i({S_{\rm{tr}}}^t\cup S_i^t)|{S_{\rm{tr}}}^{t},S_i^t]\leq\frac{1}{k}\sum\limits_{i=1}^{m}f_i({S_{\rm{tr}}}^{*}\cup S_i^{*})-f_i({S_{\rm{tr}}}^t\cup S_i^t) \label{eq: rand alg pf eq final}\end{aligned}$$ In other words, the expected improvement in the objective (left-hand side of ) is at least $1/k$ times the gap of the current objective value to OPT (i.e. right-hand side of ). Note that is only valid when $\mid {S_{\rm{tr}}}\mid<l$ and $\mid S_i \mid<k-l$. Hence, by defining the stopping time $\tau$ as first time that either $\mid {S_{\rm{tr}}}\mid=l$ or $\mid S_i \mid=k-l$, and a telescopic usages of the bounds in , we obtain the following bound: $$\begin{aligned} \mathbb{E} \left[ \frac{1}{m}\sum\limits_{i=1}^{m}f_i({S_{\rm{tr}}}^{\tau}\cup S_i^{\tau}) \right]\nonumber\geq \text{OPT} \times \mathbb{E} \left[\left(1-\left(1-\frac{1}{k}\right)^\tau\right)\right]\end{aligned}$$ The following theorem finds an upper bound on $\mathbb{E}[(1-\frac{1}{k})^\tau]$ which finishes the proof. \[thm: tau analyze randomized meta convergence\] If stopping time $\tau$ is first time that either $\mid {S_{\rm{tr}}}\mid=l$ or $\mid S_i \mid=k-l$ then $\mathbb{E}[(1-\frac{1}{k})^\tau]\leq c+\exp(-1+\sqrt{3c.log(\frac{k}{c}}))$ where $c=\frac{1}{\min\{l,k-l\}}$. let $u_1, u_2, \cdots$ be $i.i.d$ random variables with distribution $u_i \sim \text{Bernoulli}(1 - l/k)$, i.e. $p(u_i = 1) = (k-l)/k$. The stopping time $\tau$ is the first time that $\sum_{i=1}^{\tau}u_i=k-l$ or $\tau - \sum_{i=1}^{\tau}u_i = l$. Let us define $X_r=\sum_{i=1}^{r}u_i$. As a result, $\tau=r$ if and only if either $X_r=k-l$ or $X_r=r-l$. Assume without loss of generality $k-l \leq l$ which means $\max\{\sqrt{1/(k-l)},\sqrt{1/l}\}=\sqrt{1/(k-l)}$. Then, we can write: $$\begin{aligned} \nonumber \mathbb{E}[(1-\frac{1}{k})^\tau]&\leq \sum_{r=1}^{k-1}(1-\frac{1}{k})^{r}[p(X_r=k-l)+p(X_r=r-l)]\\&\leq\sum_{r=1}^{k-1}\text{exp}(\frac{-r}{k})[p(X_r=k-l)+p(X_r=r-l)] \label{eq_partition}\end{aligned}$$ We focus on the first term in and show that it is less than $e^{-1 + c}/2$. The second term can be shown similarly to have the same upper bound. By using Sterling and Chernoff bounds we can find upper bound on $p(X_r=k-l)$ as follows: $$\begin{aligned} p(X_r=k-l)&\leq \frac{1}{\sqrt{r}}\text{exp}(-\frac{(k-r)^2(k-l)}{3kr})\\&\leq \frac{1}{\sqrt{r}}\text{exp}(-\frac{(k-r)^2(k-l)}{3k^2})\end{aligned}$$ As a result: $$\begin{aligned} \sum_{r=1}^{k-1} \text{exp}(\frac{-r}{k})[p(X_r=k-l)]&\leq \sum_{r=1}^{k-1} \frac{1}{\sqrt{r}}\text{exp}(\frac{-r}{k})\text{exp}(-\frac{(k-r)^2(k-l)}{3k^2})\\&\leq \sum_{i=0}^{ 2\sqrt{k-l} -1 }\frac{1}{\sqrt{k(1-\frac{i}{2\sqrt{k-l}})}}\text{exp}(\frac{-i^2}{12})\text{exp}(-1+\frac{i}{2\sqrt{k-l}})\\&\leq \text{exp}(-1+\frac{1}{\sqrt{k-l}})\sum_{i=0}^{2\sqrt{k-l}-1}\frac{1}{\sqrt{k(1-\frac{i}{2\sqrt{k-l}})}}\text{exp}(\frac{-i^2}{12})\text{exp}(\frac{i-1}{2\sqrt{k-l}})\end{aligned}$$ where one can show using Gaussian integrals that $\sum\limits_{i=0}^{2\sqrt{k-l}-1}\frac{1}{\sqrt{k(1-\frac{i}{2\sqrt{k-l}})}}\text{exp}(\frac{-i^2}{12})\text{exp}(\frac{i-1}{2\sqrt{k-l}})$ is always less than 0.5[@bullen1998dictionary]. Therefore, $\sum\limits_{r=1}^{k-1} \text{exp}(\frac{-r}{k})[p(X_r=k-l)]\leq \frac 12 \text{exp}(-1+\frac{1}{\sqrt{k-l}})$ Similarly, we can use the same argument to prove $\sum\limits_{r=1}^{k-1} \text{exp}(\frac{-r}{k})[p(X_r=r-l)]\leq \frac12 \text{exp}(-1+\frac{1}{\sqrt{k-l}})$, which finishes the proof. Furthermore, we define $\tau^{'}=r$ when $r$ is the first time that $X_r=r-l$ and $\tau^{''}=r$ when $r$ is the first time that $X_r=k-l$. Also, let $c=\frac{1}{\min\{l,k-l\}}$ as it was defined in the lemma. By this definition, $\tau=\min\{\tau^{''},\tau^{'}\}$ and we can write the following about the probabilities of $\tau^{'}$ and $\tau^{''}$: $$p(\tau^{'}=r)={r-1 \choose l-1}(\frac{k-l}{k})^ {r-l}(\frac{l}{k})^{l}$$ $$p(\tau^{''}=r)={r-1 \choose k-l-1}(\frac{l}{k})^ {r-k+l}(\frac{k-l}{k})^{k-l}$$ then, based on the definition of $\tau^{'}$ and $\tau^{''}$ we have the following properties for $\tau^{'}$ and $\tau^{''}$: - if $r<k-l$ then $p(\tau^{''}=r)=0$. - if $r<l$ then $p(\tau^{'}=r)=0$. - if $r>k$ then $p(\tau^{'}\leq \tau^{''}|\tau^{'}=r)=0$. - if $r<k$ then $p(\tau^{'}\leq \tau^{''}|\tau^{'}=r)=1$. - if $r<k$ then $p(\tau^{'}\geq \tau^{''}|\tau^{''}=r)=1$ - if $r>k$ then $p(\tau^{'}\geq \tau^{''}|\tau^{''}=r)=0$. - $p(\tau^{''}=r|\tau^{'}\geq \tau^{''})=p(\tau=r|\tau^{'}\geq \tau^{''})$. - $p(\tau^{'}=r|\tau^{'}\leq \tau^{''})=p(\tau=r|\tau^{'}\leq \tau^{''})$. Moreover using Bayes rule we can write: - $$p(\tau^{'}=r|\tau^{'}\leq \tau^{''})=\frac{p(\tau^{'}\leq \tau^{''}|\tau^{'}=r)p(\tau^{'}=r)}{p(\tau^{'}\leq \tau^{''})}=\frac{\mathbbm{1}(r\leq k)p(\tau^{'}=r)}{p(\tau^{'}\leq \tau^{''})}.$$ - $$p(\tau^{''}=r|\tau^{'}\geq \tau^{''})=\frac{\mathbbm{1}(r\leq k)p(\tau^{''}=r)}{p(\tau^{''}\leq \tau^{'})}.$$ Let $\bar{X_r}=r-X_r$ we can write $\bar{X_r}=\sum_{i=1}^{r}v_i$ where $v_1,v_2,v_3,\dots$ are $i.i.d$ random variable with distribution $v_i\sim \text{Bernoulli}(l/k)$. Then, we can write the following using Chernoff bound: $$\begin{aligned} p(\tau^{'}=r) &\leq p(X_r= r-l) \\&\leq p(\bar{X_r}\geq l) \\&\leq p(\bar{X_r}\geq r(\frac{l}{k})-(k-r)\frac{l}{k}) \\&\leq \exp{\left(-\frac{(k-r)^2(\frac{l}{k})^2}{3r(\frac{l}{k})}\right)} \\&=\exp{\left(-\frac{(k-r)^2(l)}{3rk}\right)}\end{aligned}$$ Similarly: $$\begin{aligned} p(\tau^{''}=r) &\leq p(X_r= k-l) \\&\leq p({X_r}\geq k-l) \\&\leq p({X_r}\geq r(1-\frac{l}{k})-(k-r)(1-\frac{l}{k})) \\&\leq \exp{\left(-\frac{(k-r)^2(1-\frac{l}{k})^2}{3r(1-\frac{l}{k})}\right)} \\&\leq \exp{\left(-\frac{(k-r)^2(k-l)}{3rk}\right)} \\&\leq \exp{\left(-\frac{(k-r)^2}{3rkc}\right)}\end{aligned}$$ then we can write the $\mathbb{E}[(1-\frac{1}{k})^\tau]$ as follows: $$\begin{aligned} \label{eq:pf expectation} \mathbb{E}[(1-\frac{1}{k})^\tau]&=\sum_{r=1}^{k}(1-\frac{1}{k})^rp(\tau=r)\leq(1-\frac{1}{k})^{k-\alpha \sqrt{c}}+\sum_{r=1}^{k-\alpha \sqrt{c}}(1-\frac{1}{k})^rp(\tau=r)\end{aligned}$$ Our goal is to find proper bound for . we focus on the second term in - and try to find proper bound for it. $$\begin{aligned} \sum_{r=1}^{k-\alpha \sqrt{c}}&(1-\frac{1}{k})^rp(\tau=r) \label{eq:pf thm2 2nd term eq1}\\&=\sum_{r=1}^{k-\alpha \sqrt{c}}(1-\frac{1}{k})^r(p(\tau^{'}=r|\tau^{'}< \tau^{''})p(\tau^{'}<\tau^{''})+ p(\tau^{''}=r|\tau^{'}\geq \tau^{''})p(\tau^{'}\geq\tau^{''})) \label{eq:pf thm2 2nd term eq2}\\&=\sum_{r=1}^{k-\alpha \sqrt{c}}(1-\frac{1}{k})^r(p(\tau^{'}=r)+ p(\tau^{''}=r)) \label{eq:pf thm2 2nd term eq3}\\&=\sum_{r=l}^{k-\alpha \sqrt{c}}(1-\frac{1}{k})^rp(\tau^{'}=r)+\sum_{r=k-l}^{k-\alpha \sqrt{c}}(1-\frac{1}{k})^r p(\tau^{''}=r) \label{eq:pf thm2 2nd term eq4}\\& \leq \sum_{r=l}^{k-\alpha \sqrt{c}}\exp{\left(-\frac{(k-r)^2}{3rkc}\right)} + \sum_{r=k-l}^{k-\alpha \sqrt{c}} \exp{\left(-\frac{(k-r)^2}{3rkc}\right)} \label{eq:pf thm2 2nd term eq5}\\& \leq (k-l)\exp{\left(-\frac{(k-(k-\alpha \sqrt{c}))^2}{3k^2c}\right)} + l \exp{\left(-\frac{(k-(k-\alpha \sqrt{c}))^2l}{3k^2}\right)} \label{eq:pf thm2 2nd term eq6}\\& \leq (k-l)\exp{\left(-\frac{(\alpha \sqrt{c})^2}{3k^2c}\right)} + l \exp{\left(-\frac{(\alpha \sqrt{c})^2l}{3k^2}\right)} \label{eq:pf thm2 2nd term eq7}\end{aligned}$$ where follows from law of total probability, follows from bayes rule, follows from Chernoff bound, follows from the fact that $r<k$. Let $\alpha=3\sqrt{\log(\frac{1}{c})}.k$. As result, we have: $$\begin{aligned} \sum_{r=1}^{k-\alpha \sqrt{c}}&(1-\frac{1}{k})^rp(\tau=r)\leq (k-l)c^{3} + l c^{3cl} \end{aligned}$$ Assume without loss of generality $k-l\leq l$ and $k-l\geq 2$. As a result, $c=\frac{1}{k-l}$. we want to show that $(k-l)c^{3} + l c^{3cl}=c^2+lc^{3cl}\leq c$. To show this, we show the following equivalent inequality : $$\begin{aligned} l (k-l)^{-3cl}\leq c(1-c)=\frac{k-l-1}{(k-l)^2}\end{aligned}$$ This holds since $k-l\geq 2$ we have $ \frac{l}{(k-l)^3} (k-l)^{-3(cl-1)}\leq \frac{l}{(k-l)^3} 2^{-3(\frac{l}{k-l}-1)}\leq \frac{l}{(k-l)^3\frac{l}{k-l}}=\frac{1}{(k-l)^2}\leq \frac{k-l-1}{(k-l)^2}$. Moreover, we can bound the first term in as follows: $$\begin{aligned} (1-\frac{1}{k})^{k-\alpha \sqrt{c}}\leq \exp(-1+3\sqrt{c.\log(\frac{1}{c}}))\end{aligned}$$ summing up we can find the following bound for $\mathbb{E}[(1-\frac{1}{k})^\tau]$ which finishes the proof. $$\begin{aligned} \mathbb{E}[(1-\frac{1}{k})^\tau]&\leq c+\exp(-1+3\sqrt{c.log(\frac{1}{c}}))\end{aligned}$$ Continuous Algorithm for Submodular Meta-Learning ================================================= In this section, we provide method that accomplish $1-1/e$ approximation ratio for problem \[eq:ML\_submodular\_sample\_avg\] . In high level, we use the continuous optimization method and dependent rounding technique in [@balkanski2016learning] to obtain a solution.\ \ **New Ground Set:** Similar to [@balkanski2016learning], we define new ground set of size $nm+n$ which has the orginal ground set elements and element for every $(element, function)$ pair ${\mathcal{X}}^{'}=V \cup \{a_{i,j}\}_{i\in[n],j\in[m]}$. $$\label{eq:cont reformulation} g(S)=\sum_{j=1}^{m} f_j(\{a_i: a_{i,j}\in S\})$$ \ \ **Continuous problem:** let us associate with each element $a_{i}$ a variable $x_i \in [0,1]$ and for each element $a_{i,j}$ a variable $x_{i,j} \in [0,1]$; then, we define the $G(x)$ as in : $$\label{eq:cont obj1} G(x)= \mathbb{E}_{s \sim D(x)}g(S)$$ where $D(x)$ is the following distribution: 1. $a_i \in S \sim D(x)$ for each i independently with probability $x_i$ 2. $a_{i,j} \in S \sim D(x)$ for each i and for each j independently with probability $\frac{x_{i,j}}{x_i}$ if $a_i \in S$ and with probability 0 otherwise. we can write the continuous version of the problem as note that the difference between this problem and two-stage submodular problem shows itself in . \[eq: cont\] $$\begin{aligned} {4} & \max_{\mathcal{S}}&& G(x) \\ &\operatorname*{subject\,\, to \quad}&& x_i\in [0,1]&&&\forall i\in[n]\\ & && x_{i,j}\in [0,1]&&&\forall i\in[n]\forall j\in[m]\\ & && x_{i} \leq x_{i,j} \;\;\;&&&\forall j\in[m]\label{eq: dif two-stage meta sub}\\ & && \sum_{j} x_{i,j} \leq k \;\;\;\\ & && \sum_{i} x_{i} \leq l \;\;\;\end{aligned}$$ \[lemma: cont\] $G_{ME}=G$ for every x where $$\label{eq:cont obj} G_{ME}(x)= \mathbb{E}_{s \sim x}g(S)$$ [@balkanski2016learning] then we can reformulate the problem as: \[eq: cont2\] $$\begin{aligned} {4} & \max_{\mathcal{S}}&& G_{ME}(x) \\ &\operatorname*{subject\,\, to \quad}&& x_i\in [0,1]&&&\forall i\in[n]\\ & && x_{i,j}\in [0,1]&&&\forall i\in[n]\forall j\in[m]\\ & && x_{i} \leq x_{i,j} \;\;\;&&&\forall j\in[m]\\ & && \sum_{j} x_{i,j} \leq k \;\;\;\\ & && \sum_{i} x_{i} \leq l \;\;\;\end{aligned}$$ Lemma \[lemma: cont\] reduces the problem to submodular maximization over solvable polytope [@calinescu2011maximizing] which can be solve using continuous greedy method and the solution achieves $(1- \frac{1}{e})$ approximation ratio. But, this algorithm is very slow and can not be use in applications with large amount of data. Generalization of Submodular Meta-Learning ------------------------------------------ Generalization of submodular maximization has been studied before as $k$-submodularity . The $k$-submodularity studies function of $k$ subset of ground set which are disjoint. And function is submodular in each set[@ohsaka2015monotone]. However, in submodular meta-learning framework, sets can overlap and we don’t have disjoint restriction on the sets. Therefore, we need to come up with more general notion that can be generalize submodular meta-learning. We introduce the notion of orthant-submodularity and propose two algorithm to solve this problem. ### General Submodular Meta-Learning Formulation A inherent structure of submodular meta-learning involves the m subsets of ground set which lead us to generalize to more general case in which we are dealing with general function of m subsets of ground set. In order to pursue this goal we need to define following definitions. **Set of m-tuples.** Let ${\mathcal{X}}$ be finite set of elements of size n, and ${\mathcal{Y}}:=2^{\mathcal{X}}$ be the all the subset of ${\mathcal{X}}$, we call ${\mathcal{Y}}^m:=\{(S_1,S_2,\dots,S_m)|\;S_i\in{\mathcal{Y}}\;\}$ set of all m-tuples. **Partial Order on m-tuples.** For two element ${\mathcal{S}}\, , \,{\mathcal{S}}^{'}\in{\mathcal{Y}}^m$ we call ${\mathcal{S}}\leq {\mathcal{S}}^{'}$ iff for every $i$ in $\{1,2,\dots, m\}$, $S_i\subseteq S_i^{'}$. **Addition on m-tuples.** For ${\mathcal{S}},{\mathcal{S}}^{'}\in {\mathcal{Y}}^m$ and $e\in {\mathcal{X}}$ we denote ${\mathcal{S}}\xrightarrow{\{j,e\}}{\mathcal{S}}^{'}$ whenever for every $i$ in $\{1,2,\dots, m\}$ except $j$, formally ${S^{'}_i}=S_i$ and $S^{'}_j=S_j\cup\{e\}$. **Addition and Subtraction on m-tuples.** For ${\mathcal{S}},{\mathcal{S}}^{'}\in {\mathcal{Y}}^m$ and $e\in {\mathcal{X}}$ we denote ${\mathcal{S}}\xrightarrow[\{k,d\}]{\{j,e\}}{\mathcal{S}}^{'}$ whenever for every $i$ in $\{1,2,\dots, m\}$ except $j,k$ ${S^{'}_i}=S_i$ and $S^{'}_j=S_j\cup\{e\}$, $S^{'}_k=S_j\setminus\{d\}$. **Discrete Derivative of Function on m-tuples.** function $f:{\mathcal{Y}}^m\xrightarrow{}{\mathbb{R}}_+$ is function that map every element of ${\mathcal{Y}}^m=\{(S_1,S_2,\dots,S_m)|\;S_i\in{\mathcal{Y}}\;\}$ to a positive number. Define the discrete derivative of $f$ at ${\mathcal{S}}\in {\mathcal{Y}}^m , e\in{\mathcal{X}}$ as follows: $$\Delta_{i,e}f(S):=f({\mathcal{S}}^{'})-f({\mathcal{S}})$$ where ${\mathcal{S}}\xrightarrow{\{j,e\}}{\mathcal{S}}^{'}$. **Orthant-Submodularity.** Let ${\mathcal{X}}$ be finite set of elements, and ${\mathcal{Y}}=2^{\mathcal{X}}$ be the all the subset of ${\mathcal{X}}$, function $f:{\mathcal{Y}}^m\xrightarrow{}{\mathbb{R}}_+$ is function that map every element of ${\mathcal{Y}}^m=\{(S_1,S_2,\dots,S_m)|\;S_i\in{\mathcal{Y}}\;\}$ to a positive number. $f$ is **Orthant-submodular** if for every ${\mathcal{S}}$,${\mathcal{S}}^{'}\in {\mathcal{Y}}^m $ ,where ${\mathcal{S}}\leq {\mathcal{S}}^{'}$: $$\Delta_{i,e}f({\mathcal{S}})\geq \Delta_{i,e}f({\mathcal{S}}^{'})$$ **Monotonicity.** Let ${\mathcal{X}}$ be finite set of elements, and ${\mathcal{Y}}=2^{\mathcal{X}}$ be the all the subset of ${\mathcal{X}}$, function $f:{\mathcal{Y}}^m\xrightarrow{}{\mathbb{R}}_+$ is function that map every element of ${\mathcal{Y}}^m=\{(S_1,S_2,\dots,S_m)|\;S_i\in{\mathcal{Y}}\;\}$ to a positive number. $f$ is **monotone** iff for every ${\mathcal{S}}$,${\mathcal{S}}^{'}\in {\mathcal{Y}}^m $ ,where ${\mathcal{S}}\leq {\mathcal{S}}^{'}$: $$f({\mathcal{S}}^{'})\geq f({\mathcal{S}})$$ \[pr: monotone orthant submodular\] Consider the function $f:{\mathcal{Y}}^m\xrightarrow{}{\mathbb{R}}_+$ which is monotone and orthant-submodular. **Monotone orthant-submodular maximization with individual size constraints** is the problem of maximizing $f({\mathcal{S}})$ over m-tuples, ${\mathcal{S}}\in {\mathcal{Y}}^m$ subject to individual size constraint on every $S_i$ , formally: \[eq:monotone orthant submodular\] $$\begin{aligned} {3} & \max_{\mathcal{S}}&& f({\mathcal{S}}) \label{eq:monotone orthant submodular obj}\\ &\operatorname*{subject\,\, to \quad}&& {\mathcal{S}}\in {\mathcal{Y}}^m\\ & && |S_i| \leq k_i \;\;\;\forall i\in[m]\end{aligned}$$ \[co: equivalent of two-stage submodular\] let $f_i:{\mathcal{Y}}\xrightarrow{}{\mathbb{R}}_+$ be monotone submodular function for $i\in[m-1]$ let ${\mathcal{S}}\in {\mathcal{Y}}^m$ be ${\mathcal{S}}=(S_1,S_2,...,S_m)$ define $f:{\mathcal{Y}}^m\xrightarrow{}{\mathbb{R}}_+$ as written in \[eq: equivalent of two-stage submodular function\]: $$\label{eq: equivalent of two-stage submodular function} f({\mathcal{S}}):=\sum_{i=1}^{m-1} f_i(S_i\cap S_m)$$ then the function $f$ is monotone orthant-submodular. Then consider the problem which is monotone orthant-submodular maximization with individual size constraints as in \[eq: equivalent of few shot submodular optimization\] this problem has same optimal solution as problem \[pr: two-stage submodular\]. \[eq: equivalent of two-stage submodular optimization\] $$\begin{aligned} {4} & \max_{\mathcal{S}}&& f({\mathcal{S}}) \label{eq: equivalent of two-stage submodular optimization obj}\\ &\operatorname*{subject\,\, to \quad}&& {\mathcal{S}}\in {\mathcal{Y}}^m\\ & && |S_i| \leq k \;\;\;\forall i\in[m-1]\\ & && |S_m| \leq l \;\;\;\end{aligned}$$ As we saw both problem \[pr: Few-Shot Submodular optimization\] and \[pr: two-stage submodular\] are special case of monotone orthant-submodular maximization with individual size constraints. Therefore, the generalization of every meta submodular task can be formulated as maximization of monotone orthant-submodular maximization with individual size constraints. ### Algorithms for General Submodular Meta-Learning {#sec:algorithms} ### Deterministic Algorithms In this section we will describe two greedy based algorithms that solve the problem \[pr: monotone orthant submodular\]. Note that in each algorithm we use the similar technique as in [@ohsaka2015monotone].\ In algorithm 1, we begin with $\emptyset$ for every set $S_i$ and in each step $t$ we add the pair $e$ to set $S_i$ in which $\{i,e\}$ maximize the marginal gain of $f$ and choosing $\{i,e\}$ does not violate cardinality constraints. **Input:** a monotone orthant-submodular $f:{\mathcal{Y}}^m\xrightarrow{}{\mathbb{R}}_+$ and size constraints $k_i$ $\forall i \in [m]$\ **Output:** ${\mathcal{S}}\in {\mathcal{Y}}^m$, where $|S_i|=k_i$ $\forall i \in [m]$ Initialize $K=\sum^{m}_{i=1}k_i$ , ${\mathcal{S}}\xleftarrow[]{}\emptyset$ ${\mathcal{I}}\xleftarrow[]{}\{i \in [m]\;|\;|S_i|< k_i\}$ $V_i\xleftarrow{}{\mathcal{X}}\setminus S_i$ for $i \in {\mathcal{I}}$ $\{i,e\}\xleftarrow{} \operatorname*{arg\,max}_{ i \in {\mathcal{I}}, e\in V_i} \Delta_{i,e} f({\mathcal{S}})$ ${\mathcal{S}}\xrightarrow{\{i,e\}}{\mathcal{S}}$ **end for** return ${\mathcal{S}}$ \[thm: main M meta convergence\] Algorithm \[alg: M meta-Greedy\] returns $\frac{1}{2}$-approximation in ${\mathcal{O}}(Kmn)$ evaluation of $f$. ### Fast Randomized Lazy Algorithm Algorithm \[alg: M meta-Greedy\] has a high computational complexity; therefore, inspired by the idea of [@lazygreedy] we construct randomized algorithm which has lower computational complexity. The only difference is instead of choosing any element in $V_i$ we substitute a random subset of it $R_i$. This change significantly reduce the computational cost. \[thm: Fast M meta-Greedy\] Algorithm \[alg: Fast M meta-Greedy\] returns $\frac{1}{2}$-approximation with probablity $1-\delta$ in ${\mathcal{O}}(n\,m\;logK\;log\frac{K}{\delta})$ evaluation of $f$. **Input:** a monotone orthant-submodular $f:{\mathcal{Y}}^m\xrightarrow{}{\mathbb{R}}_+$ and size constraints $k_i$ $\forall i \in [m]$\ **Output:** ${\mathcal{S}}\in {\mathcal{Y}}^m$, where $|S_i|=k_i$ $\forall i \in [m]$ Initialize $K=\sum^{m}_{i=1}k_i$ , ${\mathcal{S}}\xleftarrow[]{}\emptyset$ ${\mathcal{I}}\xleftarrow[]{}\{i \in [m]\;|\;|S_i|< k_i\}$ $V_i\xleftarrow{}{\mathcal{X}}\setminus S_i$ for $i \in {\mathcal{I}}$ $R_i\xleftarrow{}$a random subset of size $ \min\{\frac{n-|S_i|}{k_i-|S_i|}log\frac{K}{\delta},n\}$ sampled uniformly from $ V_i $ for $ i \in {\mathcal{I}}$. $\{i,e\} \xleftarrow{} \operatorname*{arg\,max}_{ i \in {\mathcal{I}}, e\in R_i} \Delta_{i,e} f({\mathcal{S}})$ ${\mathcal{S}}\xrightarrow{\{i,e\}}{\mathcal{S}}$ **end for** return ${\mathcal{S}}$ ### Special Example \[co: equivalent of few shot submodular\] let $f_i:{\mathcal{Y}}\xrightarrow{}{\mathbb{R}}_+$ be monotone submodular function for $i\in[m-1]$ let ${\mathcal{S}}\in {\mathcal{Y}}^m$ be ${\mathcal{S}}=(S_1,S_2,...,S_m)$ define $f:{\mathcal{Y}}^m\xrightarrow{}{\mathbb{R}}_+$ as written in : $$\label{eq: equivalent of few shot submodular function} f({\mathcal{S}}):=\sum_{i=1}^{m-1} f_i(S_i\cup S_m)$$ then the function $f$ is monotone orthant-submodular. Consider the problem which is a monotone orthant-submodular maximization with individual size constraints for $f$ as in , this problem has same optimal solution as problem . \[eq: equivalent of few shot submodular optimization\] $$\begin{aligned} {4} & \max_{\mathcal{S}}&& f({\mathcal{S}}) \label{eq: equivalent of few shot submodular optimization obj}\\ &\operatorname*{subject\,\, to \quad}&& {\mathcal{S}}\in {\mathcal{Y}}^m\\ & && |S_i| \leq k-l \;\;\;\forall i\in[m-1]\\ & && |S_m| \leq l \;\;\;\end{aligned}$$ Using Corollary \[co: equivalent of few shot submodular\] we can use algorithms for maximization of general orthant-submodular with individual size constraint, algorithm and , for discrete meta learning problem, problem . In order to do that first note that for objective function $f({\mathcal{S}}):=\sum_{i=1}^{m-1} f_i(S_i\cup S_m)$, the step 5 of algorithm can be simplified as follows. For every i and e we have: $$\Delta_{m,e} f({\mathcal{S}})=\sum_{i=1}^{m-1} f_i(S_i\cup S_m\cup{e})-f_i(S_i\cup S_m)\geq \Delta_{i,e} f({\mathcal{S}})=\ f_i(S_i\cup{e}\cup S_m)-f_i( S_i\cup{e}\cup S_m)$$ which means in algorithm step 5, until the $S_m$ is not hit the cardinality constraint non of the other sets have chosen. This results simplified algorithm for discrete meta learning problem. Since, algorithm . ### Convergence Analysis ### Proof of corollary \[co: equivalent of few shot submodular\] We want to proof first that the function in equation \[eq: equivalent of few shot submodular function\] is monotone. Suppose ${\mathcal{S}}\leq {\mathcal{S}}^{'}$ we want to show $f({\mathcal{S}})\leq f({\mathcal{S}}^{'})$. we Know that every $f_i$ is monotone which means for every $j\in[m-1]$ we have $S_j\cup S_m\subseteq S_j^{'}\cup S_m^{'}$; therefore, we can conclude $f_j(S_j\cup S_m)\leq f_j(S_j^{'}\cup S_m^{'})$ which results $f({\mathcal{S}})\leq f({\mathcal{S}}^{'})$. Secondly, we want to show $\Delta_{i,e}f(S^{'})\leq \Delta_{i,e}f(S)$. let us first consider the case that $i\not = m$ then $\Delta_{i,e}f({\mathcal{S}})=f_i((\{e\}\cup S_i)\cup S_m)-f_i( S_i\cup S_m)\geq f_i((\{e\}\cup {S^{'}_i})\cup S_m^{'})-f_i( {S^{'}_i}\cup S_m^{'})=\Delta_{i,e}f({\mathcal{S}}^{'})$ follows from submodularity of $f_i$. If $i=m$ then $\Delta_{i,e}f({\mathcal{S}})=\sum_{t=1}^{m-1}f_t( S_t\cup S_m\cup \{e\})-\sum_{t=1}^{m-1}f_t( S_t\cup S_m)\geq\sum_{t=1}^{m-1}f_t( S_t^{'}\cup S_m^{'}\cup \{e\})-\sum_{t=1}^{m-1}f_t( S_t^{'}\cup S_m^{'})=\Delta_{i,e}f({\mathcal{S}}^{'})$ follows from submodularity of $f_i$.\ For proving the equivalence of two problem. First note that every feasible solution for problem \[eq: equivalent of few shot submodular optimization\] is also feasible for the maximization problem in \[eq:ML\_submodular\_sample\_avg\]. let the ${\mathcal{S}}^{*}=(S^{*}_1,S^{*}_2,\dots,S^{*}_m)$ be a maximizer of function in \[eq: equivalent of few shot submodular function\] with maximum value $OPT_1=\sum_{i=1}^{m-1} f_i(S^{*}_i\cup S^{*}_m)$, and ${\mathcal{S}}^{'}=(S^{'}_1,S^{'}_2,\dots,S^{'}_m)$ be a maximizer of problem in \[eq:ML\_submodular\_sample\_avg\] with maximum value $OPT_2=\sum_{i=1}^{m-1} f_i({S^{'}_i}\cup S^{'}_m)$, then we can conclude $f({\mathcal{S}}^*)\geq f({\mathcal{S}}^{'})$ which means $OPT_2=\sum_{i=1}^{m-1} f_i(S^{*}_i\cup S^{*}_m)\geq \sum_{i=1}^{m-1} f_i({S^{'}_i}\cup S^{'}_m)=OPT_1$. By same argument $OPT_1\geq OPT_2$, which results $OPT_1=OPT_2$. ### Proof of corollary \[co: equivalent of two-stage submodular\] Let the ${\mathcal{S}}^{*}=(S^{*}_1,S^{*}_2,\dots,S^{*}_m)$ be a maximizer of function in \[eq: equivalent of two-stage submodular function\] with maximum value $OPT_1=\sum_{i=1}^{m-1} f_i(S^{*}_i\cap S^{*}_m)$, and ${\mathcal{S}}^{'}=(S^{'}_1,S^{'}_2,\dots,S^{'}_m)$ be a maximizer of problem \[eq:ML\_submodular\_sample\_avg\] with maximum value $OPT_2=\sum_{i=1}^{m-1} f_i({S^{'}_i})$ . We want to show ${\mathcal{S}}^{''}:=(S^{*}_1\cap S^{*}_m ,\, S^{*}_2\cap S^{*}_m ,\,\dots,\,S^{*}_{m-1}\cap S^{*}_m,S^{*}_m)$ is a optimal solution for problem \[eq:ML\_submodular\_sample\_avg\]. Firstly, ${\mathcal{S}}^{''}$ is feasible solution for problem \[eq:ML\_submodular\_sample\_avg\]. Secondly, $OPT_1=f({\mathcal{S}}^{*})=\sum_{i=1}^{m-1} f_i(S^{*}_i\cap S^{*}_m)=\sum_{i=1}^{m-1} f_i(S^{''}_i)$ , from optimality of $S^{'}$ for problem \[eq:ML\_submodular\_sample\_avg\], $OPT_2=\sum_{i=1}^{m-1} f_i({S^{'}_i})\geq \sum_{i=1}^{m-1} f_i(S^{''}_i)=OPT_1$. Finally, observe that ${\mathcal{S}}^{'}$ is a feasible solution for \[eq: equivalent of two-stage submodular optimization\] which means $OPT_1=\sum_{i=1}^{m-1} f_i(S^{''}_i)\geq \sum_{i=1}^{m-1} f_i({S^{'}_i})=OPT_2$ results in $OPT_1=OPT_2$ . ### Proof of theorem \[thm: main M meta convergence\] {#proof: thm1} **Incidence Matrix for m-tuples.** for m-tuple ${\mathcal{S}}$ is matrix $L({\mathcal{S}})\in {\mathbb{R}}^{n\times m}$ which define as: $$L({\mathcal{S}}):=\begin{bmatrix} \mathbbm{1}_{S_1} & \mathbbm{1}_{S_2} &\dots& \mathbbm{1}_{S_m} \end{bmatrix}$$ Let ${\mathcal{S}}^{(0)}=\emptyset,{\mathcal{S}}^{(1)},\dots,{\mathcal{S}}^{(t)},\dots,{\mathcal{S}}^{(K)}$ be a chain obtained by algorithm \[alg: M meta-Greedy\] and let $S^*$ be the optimal solution. Define the sequence ${\mathcal{J}}^{(0)}={\mathcal{S}}^{*},{\mathcal{J}}^{(1)},\dots,{\mathcal{J}}^{(K)}={\mathcal{S}}^{(K)}$ as follows. In step 5 of algorithm \[alg: M meta-Greedy\], we have defined $i^{(t)},e^{(t)}$ to be ..... We then define $D^{t}_{i}=J^{(t-1)}_i\setminus S^{(t-1)}_i$ and also define $o^{(t)}$ in the following way: (i) If $e^{(t)}\in D^{t}_{i^{(t)}}$, then $o^{(t)}=e^{(t)}$; (ii) Otherwise, if $e^{(t)}\notin D^{t}_{i^{(t)}}$, let $o^{(t)}$ be one of the elements of $D^{t}_{i^{(t)}}$ chosen uniformly at random. We also, define the set ${\mathcal{J}}^{(t-\frac{1}{2})}$ by deleting $o^{(t)}$ from ${\mathcal{J}}^{(t-1)}$, i.e. .... and ${\mathcal{J}}^{(t)}$ by adding $e^{(t)}$ to ${\mathcal{J}}^{(t-\frac{1}{2})}$, i.e. ...... Figure, explain how the sets.... are evolved when $t$ increases from .. to ... \^[(0)]{}= \^[(1)]{} … \^[(t)]{} … \^[(K)]{}\ \^[(0)]{}=\^\* \^[(1)]{} … \^[(t)]{} … \^[(K)]{} Note that by definition ${\mathcal{S}}^{(t-1)}\leq {\mathcal{J}}^{t-\frac{1}{2}}$. \[lemma: in proof thm1\] For every $t\in [K]$ the following inequality holds: $$\label{eq:lemma_arman} f({\mathcal{S}}^{(t)})-f({\mathcal{S}}^{(t-1)})\geq f({\mathcal{J}}^{(t-1)})-f({\mathcal{J}}^{(t)}).$$ Define $a^{(t)}=\Delta_{e^{(t)},i^{(t)}}f({\mathcal{J}}^{(t-\frac{1}{2})})$, and $a^{(t-\frac{1}{2})}=\Delta_{o^{(t)},i^{(t)}}f({\mathcal{J}}^{(t-\frac{1}{2})})$, and $b^{(t)}=\Delta_{e^{(t)},i^{(t)}}f({\mathcal{S}}^{(t-1)})$. We observe that $f({\mathcal{J}}^{(t-1)})-f({\mathcal{J}}^{(t)})=a^{(t-\frac{1}{2})}-a^{(t)}$. Using this notation, becomes $b^{(t)}\geq a^{(t-\frac{1}{2})}-a^{(t)}$. By monotonicity of $f$ it suffices to show $b^{(t)}\geq a^{(t-\frac{1}{2})}$. By step 5 of algorithm \[alg: M meta-Greedy\] (greedily choosing elements), we have $b^{(t)}=\Delta_{e^{(t)},i^{(t)}} f({\mathcal{S}}^{(t-1)})\geq \Delta_{o^{(t)},i^{(t)}} f({\mathcal{S}}^{(t-1)})$. Finally, by orthant-submodularity of ..., we obtain $\Delta_{o^{(t)},i^{(t)}} f({\mathcal{S}}^{(t-1)})\geq \Delta_{o^{(t)},i^{(t)}}f({\mathcal{J}}^{(t-\frac{1}{2})})=a^{(t-\frac{1}{2})}$ since ${\mathcal{S}}^{(t-1)}\leq {\mathcal{J}}^{t-\frac{1}{2}}$. Finally, by summing over the relations , we obtain: $$\begin{aligned} f_{final}-0= f({\mathcal{S}}^{(K)})-f({\mathcal{S}}^{(0)})&=\sum_{t=1}^{t=K} f({\mathcal{S}}^{(t)})-f({\mathcal{S}}^{(t-1)})\geq \sum_{t=1}^{t=K}f({\mathcal{J}}^{(t-1)})-f({\mathcal{J}}^{(t)}) \\&=f({\mathcal{J}}^{(K)})-f({\mathcal{J}}^{(0)})= f_{final}-OPT\end{aligned}$$ ### Proof of theorem \[thm: Fast M meta-Greedy\] We reruse the notation $i^{(t)},e^{(t)},{\mathcal{J}}^{(t)},{\mathcal{S}}^{(t)},D_i^t$ in section \[proof: thm1\]. Assume $R_i^{(t)}$ be the $R_i$ in $t^{th}$ iteration. Let us iteratively define ${\mathcal{J}}^{(0)}={\mathcal{S}}^{*},{\mathcal{J}}^{(1)},\dots,{\mathcal{J}}^{(m)}={\mathcal{S}}^{(K)}$ as follows. If $R_{i^{(t)}}^{t}\cap D_{i^{(t)}}^{t}$ is empty, then algorithm fails if not set $o^{(t)}=e^{(t)}$ if $e^{(t)}\in R_{i^{(t)}}^{t}\cap D_{i^{(t)}}^{t} $ or if $e^{(t)}\not\in R_{i^{(t)}}^{t}\cap D_{i^{(t)}}^{t} $ set $o^{(t)}$ a random element from $R_{i^{(t)}}^{t}\cap D_{i^{(t)}}^{t}$. Every other part of the proof follows as in section \[proof: thm1\]. We just need to find the probability that the algorithm fails. With probability $1-\delta$ , $R_{i^{(t)}}^{t}\cap D_{i^{(t)}}^{t}\not = \emptyset$ for every $t\in [K]$. If for any $t\in [K]$ $|R_{i^{(t)}}^{t}|=n$ then $Pr[R_{i^{(t)}}^{t}\cap D_{i^{(t)}}^{t}=\emptyset]=0$ $$\begin{aligned} Pr[R_{i^{(t)}}^{t}\cap D_{i^{(t)}}^{t}=\emptyset]&=(1-\frac{|S_{i^{(t)}}^{t}|}{|V_{i^{(t)}}^t|})^{(|R_{i^{(t)}}^{t}|)}=(1-\frac{k_{i^{(t)}}-|S_{i^{(t)}}^{t-1}|}{n-|S_{i^{(t)}}^{t-1}|})^{(|R_{i^{(t)}}^{t}|)}\\&\leq exp({-(\frac{k_{i^{(t)}}-|S_{i^{(t)}}^{t-1}|}{n-|S_{i^{(t)}}^{t-1}|}) (\frac{n-|S_{i^{(t)}}^{t-1}|}{k_{i^{(t)}}-|S_{i^{(t)}}^{t-1}|})log(\frac{K}{\delta})})=\frac{K}{\delta}\end{aligned}$$ union bound over $t\in [K]$ prove the lemma. Algorithm for Worst Case Setting {#sec: Distribution Agnostic Algorithm} -------------------------------- In the first section, we discussed the proper formulation for the case that we aim to choose an initial subset of the desired set at the training time and the remaining elements at the test time. In particular, we introduced the formulation that finds the best initial set that in expectation or on average would perform well when we later add the best subset to it. Indeed, this formulation leads to a good generalization result since we assume all tasks at the training time are drawn from a common distribution and the task that we observe at the test time is also drawn from that distribution. However, in some cases, either the tasks in the training phase are not necessarily drawn from the same distribution, or instead of average performance we care about the worst-case performance. In either case, instead of measuring the performance of the initial set in terms of the maximum value of the expected objective function, we aim to maximize the minimum objective function value over all given tasks. In other words, we aim to solve $$\label{eq:ML_submodular_minmax} \max_{{S_{\rm{tr}}}\in V, |{S_{\rm{tr}}}|\leq l} \ \min_{i=1,\dots, m}\; \max_{S_i\in V, |S_i|\leq k-l} f_i({S_{\rm{tr}}}\cup S_i) \ = \ \max_{{S_{\rm{tr}}}\in V, |{S_{\rm{tr}}}|\leq l} \ \min_{i=1,\dots, m}\; f_i'({S_{\rm{tr}}}).$$ Note that both problems in are equivalent since $f_i'({S_{\rm{tr}}}):=\max_{S_i\in V, |S_i|\leq k-l} f_i({S_{\rm{tr}}}\cup S_i)$. Later, in Section \[sec:algorithms\_ML\] we formally present efficient methods for solving this problem and characterize their theoretical guarantees. ### Equivalent of Worst Case Setting \[co: equivalent of MASML\] let $f_i:{\mathcal{Y}}\xrightarrow{}{\mathbb{R}}_+$ be monotone submodular function for $i\in[m]$ let ${\mathcal{S}}\in {\mathcal{Y}}^{m+1}$ be ${\mathcal{S}}=(S_1,S_2,...,{S_{\rm{tr}}})$ define $f:{\mathcal{Y}}^m\xrightarrow{}{\mathbb{R}}_+$ as written in : $$\label{eq: equivalent of MASML function} f({\mathcal{S}}):=\min_{i\in [m]} f_i(S_i\cup {S_{\rm{tr}}})$$ Then consider the problem which is a maximization of $f$ with individual size constraints as in \[eq: equivalent of MASML\], this problem has same optimal solution as problem . \[eq: equivalent of MASML\] $$\begin{aligned} {4} & \max_{\mathcal{S}}&& f({\mathcal{S}}) \label{eq: equivalent of MASML obj}\\ &\operatorname*{subject\,\, to \quad}&& {\mathcal{S}}\in {\mathcal{Y}}^{m+1}\\ & && |S_i| \leq k-l \;\;\;\forall i\in[m]\\ & && |{S_{\rm{tr}}}| \leq l \;\;\;\end{aligned}$$ ### Proof of corollary \[co: equivalent of MASML\] For proving the equivalence of two problem. First note that every feasible solution for problem in \[eq: equivalent of MASML\] is also feasible for the maximization problem in . let the ${\mathcal{S}}^{*}=(S^{*}_1,S^{*}_2,\dots,S^{*}_{tr})$ be a maximizer of function in \[eq: equivalent of MASML\] with maximum value $OPT_1=\min\limits_{i\in [m]} f_i(S^{*}_i\cup S^{*}_{tr})$, and ${\mathcal{S}}^{'}=(S^{'}_1,S^{'}_2,\dots,S^{'}_{tr})$ be a maximizer of problem with maximum value $OPT_2=\min\limits_{i\in [m]} f_i({S^{'}_i}\cup S^{'}_{tr})$, then we can conclude $OPT_1\geq \min\limits_{i\in [m]} f_i(S_i\cup S_m)$ for every ${\mathcal{S}}$ in feasible set specially ${\mathcal{S}}={\mathcal{S}}^{'}$ which results $OPT_1\geq OPT_2$. Similarly, $OPT_2=\min\limits_{i\in [m]}\max\limits_{S_i}f_i({S_{\rm{tr}}}^{*}\cup S_i)\geq\min\limits_{i\in [m]}\max\limits_{S_i}f_i({S_{\rm{tr}}}^{'}\cup S_i)\geq\min\limits_{i\in [m]}f_i({S_{\rm{tr}}}^{'}\cup {S^{'}_i})=OPT_1$. \[co: Hardness of MASML\] Finding any constant factor of problem in is NP-Hard. Because, even in the case of $m=1$ the problem is equivalent to robust submodular optimization which is NP-Hard to find any constant approximation factor. ### Bi-criteria Approximation Algorithm It is computationally hard to find the approximation solution for problem in \[eq: equivalent of MASML\]. Therefore, we try to solve the problem with Bi-criteria Approximation method. First, Using slack variable $c$ we can reformulate the \[eq: equivalent of MASML\] as \[eq MASML equal slack variable\]. \[eq MASML equal slack variable\]$$\begin{aligned} {4} & \max_{{\mathcal{S}},c} && c \label{eq: MASML equal slack variable obj}\\ &\operatorname*{subject\,\, to \quad}&& f_i(S_i\cup {S_{\rm{tr}}})\geq c\\ & && {\mathcal{S}}\in {\mathcal{Y}}^{m+1}\\ & && |S_i| \leq k-l &&&\forall i\in[m]\\ & && |{S_{\rm{tr}}}| \leq l \;\;\;\end{aligned}$$ Then, inspired by [@krause2008robust] we relaxed the cardinality constraints in \[eq MASML equal slack variable\] which results \[eq: Relaxed MASML2\]. \[eq: Relaxed MASML2\] $$\begin{aligned} {4} & \max_{{\mathcal{S}},c} && c \label{eq: Relaxed MASML2 obj}\\ &\operatorname*{subject\,\, to \quad}&& f_i(S_i\cup {S_{\rm{tr}}})\geq c \;\;\;&&&\forall i\in[m]\label{eq: Relaxed MASML2 const1}\\ & && {\mathcal{S}}\in {\mathcal{Y}}^{m+1}\label{eq: Relaxed MASML2 const2}\\ & && |S_i| \leq \alpha (k-l) &&&\forall i\in[m]\label{eq: Relaxed MASML2 const3}\\ & && |{S_{\rm{tr}}}| \leq \alpha l\label{eq: Relaxed MASML2 const4} \;\;\;\end{aligned}$$ Our goal is approximately solve \[eq: Relaxed MASML2\]. In order to do that we will use binary search procedure over c, by itertaively solving \[eq: Relaxation Binary Search Procedure\] and make the search interval smaller in each step. We can reformulate and relax the constraints in \[eq: Relaxed MASML2 const1\]-\[eq: Relaxed MASML2 const4\] as \[eq: Relaxation Binary Search Procedure const1\]-\[eq: Relaxation Binary Search Procedure const3\]. \[eq: Relaxation Binary Search Procedure\] $$\begin{aligned} {4} & {\mathcal{S}}_c^{*}=\operatorname*{arg\,min}_{{\mathcal{S}}} && \mid {\mathcal{S}}\mid = \sum_{i=1}^{m} \mid S_i \mid+\mid {S_{\rm{tr}}}\mid \label{eq: Relaxation Binary Search Procedure obj}\\ &\operatorname*{subject\,\, to \quad}&& F_c({\mathcal{S}}):=\frac{1}{m}\sum_{i=1}^{m}\min\{f_i(S_i\cup {S_{\rm{tr}}}),c\}= F_c({\mathcal{X}}^{m+1}) \;\;\label{eq: Relaxation Binary Search Procedure const1}\\ & && {\mathcal{S}}\in {\mathcal{Y}}^{m+1}\\ & && \frac{\mid S_i \mid}{\mid {\mathcal{S}}\mid} \leq \frac{(k-l)}{(k-l)(m)+l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \forall i\in[m]\label{eq: Relaxation Binary Search Procedure const2}\\ & && \frac{\mid {S_{\rm{tr}}}\mid}{\mid {\mathcal{S}}\mid} \leq \frac{l}{(k-l)(m)+l} \label{eq: Relaxation Binary Search Procedure const3}\end{aligned}$$ We named the optimal value of problem $Z=\min \mid{\mathcal{S}}^{*}_c\mid$. The bi-criteria approximation procedure start with $c_{min}=0$ , $c_{max}=\min_if_i({\mathcal{X}})$ and In each step solves optimization problem in \[eq: Relaxation Binary Search Procedure\] by algorithm \[alg: meta-Saturation Submodular algorithm \]and give us the solution ${\mathcal{S}}^{*}_{c}$ and update $c_{min}$ , $c_{max}$ as follows. - if $ \mid {\mathcal{S}}^{*}_{c} \mid :=\sum\limits_{i=1}^{m} \mid S_i^{c} \mid +\mid {S_{\rm{tr}}}^{c} \mid\; \geq \; \alpha (l+(m)(k-l))$ or the problem was infeasible then $c_{max}=c$ - if $ \mid {\mathcal{S}}^{*}_{c} \mid =\sum\limits_{i=1}^{m} \mid S_i^{c} \mid+\mid {S_{\rm{tr}}}^{c} \mid \; \leq \; \alpha (l+(m)(k-l))$ then $c_{min}=c$ \[eq: Relaxation Binary Search Procedure\] $$\begin{aligned} {4} & {\mathcal{S}}_c^{*}=\operatorname*{arg\,min}_{{\mathcal{S}}} && \mid {\mathcal{S}}\mid = \sum_{i=1}^{m} \mid S_i \mid +\mid {S_{\rm{tr}}}^{c} \mid \label{eq: Relaxation Binary Search Procedure obj}\\ &\operatorname*{subject\,\, to \quad}&& f_i(S_i\cup {S_{\rm{tr}}})\geq c \;\;&&&\;\; \forall i\in[m]\\ & && {\mathcal{S}}\in {\mathcal{Y}}^m\\ & && \frac{\mid S_i \mid}{\mid {\mathcal{S}}\mid} \leq \frac{k}{k(m)+l} &&&\;\; \forall i\in[m]\\ & && \frac{\mid {S_{\rm{tr}}}\mid}{\mid {\mathcal{S}}\mid} \leq \frac{l}{k(m)+l} &&&\;\; \end{aligned}$$ **Input:** a function $f:{\mathcal{Y}}^{m+1}\xrightarrow{}{\mathbb{R}}_+$ ,c , ${\mathcal{S}}^0=\emptyset$, $t=0$\ **Output:** ${\mathcal{S}}\in {\mathcal{Y}}^m$ ${\mathcal{I}}\xleftarrow[]{}\{i \in [m]\;|\; i\; $satisfy \[eq: Relaxation Binary Search Procedure const2\] and \[eq: Relaxation Binary Search Procedure const3\]$ \}$ ${\mathcal{I}}=\{tr\}$$V_i\xleftarrow{}{\mathcal{X}}\setminus S_i$ for $i \in {\mathcal{I}}$ $\{i^t,j^t\} \xleftarrow{} \operatorname*{arg\,max}_{ i \in {\mathcal{I}}, j\in V_i} \Delta_{i,j} f({\mathcal{S}}^t)$ ${\mathcal{S}}^t \xrightarrow{\{i^t,j^t\}}{\mathcal{S}}^{t+1}$, $\theta^{t}:=\;\frac{1}{\Delta_{i^{t},j^{t}}F_c({\mathcal{S}}^{t})}$ , $t \xleftarrow{}t+1$ **end While** return ${\mathcal{S}}$ In the following lemma we analyze the quality of solution for each run of the greedy algorithm. \[eq: model agnostic main lemma \] For integral valued $f_i$ and feasible c Algorithm \[alg: M meta-Greedy submodular partial cover (MGPC)\] finds m tuple ${\mathcal{S}}^{G}$ which $f_i({\mathcal{S}}^{G})\geq c$ for all $i$ and $\mid S_i^{G}\mid\leq \alpha \mid S_i^{*} \mid$ where ${\mathcal{S}}^{*}$ is optimal solution and $\alpha=1 +log(max_{e\in{\mathcal{X}},i\in[m]}\Delta_{i,e}f(\emptyset))$. **Sketch of the proof:**\ Our proof is similar to the proof in [@wolsey1982analysis]. We want to find upper bound on the greedy solution provided by Algorithm \[alg: M meta-Greedy submodular partial cover (MGPC)\] for optimization problem . To do that, we propose some related problems( , and ) that leads us to find a upper bound. First, we propose problem which is equivalent to . This equivalency will be proven in corollary . Then, we relaxed two constraints , and in and provide a upper bound for solution in . In particular, if $Z^L$ is the optimal value of ,then $Z^L\leq Z$. Afterward, using duality we argue that if $D^L$ is the optimal value of dual problem of we have $Z^L\geq D^L$. Finally, we show that if $Z^G$ is the value of for solution of algorithm  we have $D^L\geq Z^G(\alpha)^{-1}$ in lemma \[lemma: final in proof of MASMAl1\] and \[lemma: final in proof of MASMAl2\]. In nutshell, we proved lemma \[eq: model agnostic main lemma \] by showing: $$Z\geq Z^L\geq D^L\geq Z^G(\alpha)^{-1}$$ where $Z$ is the optimal value of , $Z^G$ is the value of for solution of algorithm , $Z^L$ is the optimal value of , and $D^L$ is the optimal value of dual problem of . \[col: MASML1\] We will propose the problem which is equivalent to and have same optimal value. For any m-tuple ${\mathcal{S}}$ which is feasible for let it is indicator matrix $z$ be a matrix in which for every $i\in [m]$ and $j \in [n]$, $z_{i,j}={\mathbf{1}}\{j\in S_i\}$ and let $f_i^c(S)=\min\{c,f_i(S)\}$. Then, $z$ is feasible for and the optimal value of two problem is same. (the argument is similar to proposition 2 in [@wolsey1982analysis] ). \[eq: Relaxation LP\] $$\begin{aligned} {4} & Z=\min_{z} && \sum_{i=1}^{m}\sum_{j=1}^{n} z_{i,j} \label{eq: LPRelaxation}\\ &\operatorname*{subject\,\, to \quad}&& \sum_{i=l,m}\sum_{j=1}^{n} \Delta_{i,j} f_{i}^{c} (S_{l}\cup {S_{\rm{tr}}})z_{i,j} \geq -f_i^c(S_{l}\cup {S_{\rm{tr}}})+f_i^c({\mathcal{X}})\notag \label{eq:Relaxation LP eq1}\\ & && \;\;\;\;\;\;\quad\;\forall t=0,\dots,T-1 \;,\; l\in [m] \;,\;\forall {\mathcal{S}}\in{\mathcal{Y}}^m\\ & && z_{i,j} \in \{0,1\}\\ & && \frac{\sum_{j=1}^{n} z_{i,j} }{\sum_{i=1}^{m}\sum_{j=1}^{n} z_{i,j} } \leq \frac{k}{k(m)+l} &&&\;\; \forall i\in[m]\label{eq:Relaxation LP eq3}\\ & && \frac{\sum_{j=1}^{n} z_{m,j} }{\sum_{i=1}^{m}\sum_{j=1}^{n} z_{i,j} } \leq \frac{l}{k(m)+l} &&&\label{eq:Relaxation LP eq4}\;\; \end{aligned}$$ The equivalency of and is direct result of the proposition 2 in [@wolsey1982analysis]. let $\theta^{t}$ be the output of algorithm \[alg: M meta-Greedy submodular partial cover (MGPC)\] for $t=0,1,\dots T$ where $T$ is number of iteration of algorithm \[alg: M meta-Greedy submodular partial cover (MGPC)\]. we will prove $\mid S_i^{G}\mid\leq \alpha \frac{l}{k(m)+l} \mid {\mathcal{S}}^{*} \mid=\alpha \mid {\mathcal{S}}_i^{*} \mid$. To show that let us define $Z^l$ as follows: \[eq: Relaxation LP2\] $$\begin{aligned} {4} & Z^{L}=\min_{z} && \sum_{i=1}^{m}\sum_{j=1}^{n} z_{i,j} \label{eq: LPRelaxation2}\\ &\operatorname*{subject\,\, to \quad}&& \sum_{i=l,m}\sum_{j=1}^{n} \Delta_{i,j}f_i^c({S_{\rm{tr}}}^t\cup {S_{\rm{tr}}}^t)z_{i,j} \geq -f_i^c({S_{\rm{tr}}}^t\cup {S_{\rm{tr}}}^t)+f_i^c({\mathcal{Y}}) \notag\\ \; \; & &&\;\; \forall t=0,\dots,T-1 \;,\; l\in [m]\\ & && z_{i,j} \geq 0\end{aligned}$$ This problem relaxed two constraints , and in and as consequence $Z^L\leq Z$. Then, Let $D^L$ be the optimal value of dual problem of . Then, by weak duality $D^L\leq Z^L$. \[lemma: final in proof of MASMAl1\] Let us call the final value of Algorithm \[alg: M meta-Greedy submodular partial cover (MGPC)\] $Z^G$, and let $\bar f_{i^t}^c(S):=\sum\limits_{i=1}^m f_{i}^c(S){\mathbf{1}}\{i=i^t\; or\; i^t=tr\} $ .Then, we can write $Z^G$ as follows. $$Z^G =\sum_{t=1}^{T}\theta^{t} (F_c({\mathcal{S}}^{t})-F_c({\mathcal{S}}^{t-1}))=\theta^1(\bar f_{i^1}^c({\mathcal{X}}))+\sum_{t=1}^{T}(\theta^{t}-\theta^{t-1})(\bar f_{i^t}^c({\mathcal{X}})-F_c({\mathcal{S}}^{t-1}))$$ First Inequality is a direct consequence of definition of $\theta^t $ and by rearranging and add and subtract $\bar f^c_i({\mathcal{X}})$ we get the second equality. \[lemma: final in proof of MASMAl2\] Let $D^L$ be optimal value of dual problem of . We can find following lower bound for $D^L$. $$D^L\geq(\theta^1(f_{i^1}^c({\mathcal{X}}))+ \sum_{t=1}^{T}(\theta^{t}-\theta^{t-1})(\bar f_{i^t}^c({\mathcal{X}})-F_c({\mathcal{S}}^{t-1})))(\alpha)^{-1}$$ First for every $j\in {\mathcal{X}}$ there exist $r\leq T$ that $(\Delta_{i^t,j}F_c({\mathcal{S}}^{r-1}))>0$ and $(\Delta_{i^t,j}F_c({\mathcal{S}}^{r}))=0$ . We want to prove $\alpha^{-1}\theta^*$ is dual feasible where $\theta^*=(\theta^1{\mathbf{1}}_{i^0},(\theta^2 -\theta^1){\mathbf{1}}_{i^1},\dots (\theta^T-\theta^{T-1}){\mathbf{1}}_{i^{T-1}})\in {\mathbb{R}}^{(m)\times T }$ (${\mathbf{1}}_{i}$ is vector of size $m$ which is 1 in the $i^{th}$ place or if $i=tr$ it is all ones) for every $j$ we have the following. $$(\theta^1(f_{i^1}^c({\mathcal{X}}))+ \sum_{t=1}^{T}(\theta^{t}-\theta^{t-1})(\Delta_{i^t,j}F_c({\mathcal{S}}^t))\leq (\max_{t=1,...,r} (\theta^{t})(\Delta_{i^t,j}F_c({\mathcal{S}}^t))) (\alpha)\leq \alpha$$ The first inequality follows from proposition 3 in [@wolsey1982analysis] and $(\theta^{t})(\Delta_{i^t,j}F_c({\mathcal{S}}^t))\leq1$ is consequence of definition of $(\theta^{t})$ and the fact that $i^t$ chosen greedily. Therefore, $\alpha^{-1}\theta^*$ is dual feasible and we can write the . $$\label{eq: dual LPR} (\theta^1(f_{i^1}^c({\mathcal{X}}))+ \sum_{t=1}^{T}(\theta^{t}-\theta^{t-1})(\bar f_{i^t}^c({\mathcal{X}})-F_c({\mathcal{S}}^{t-1}))) (\alpha^{-1})\leq D^L$$ Then, combing all we have: $$Z\geq Z^L\geq D^L\geq(\theta^1(f_{i^1}^c({\mathcal{X}}))+ \sum_{t=1}^{T}(\theta^{t}-\theta^{t-1})(\bar f_{i^t}^c({\mathcal{X}})-F_c({\mathcal{S}}^{t-1})))(\alpha)^{-1}=Z^G(\alpha)^{-1}$$ which resualts $\mid S_i^{G}\mid\leq \alpha \frac{l}{k(m)+l} Z=\alpha \mid {\mathcal{S}}_i^{*} \mid$. In algorithm \[alg: meta-Saturation Submodular algorithm \], we present the whole procedure for finding solution for worst case setting. **Input:** a function $f:{\mathcal{Y}}^m\xrightarrow{}{\mathbb{R}}_+$\ **Output:** ${\mathcal{S}}\in {\mathcal{Y}}^m$ $c\xleftarrow[]{}\frac{c_{max}+c_{min}}{2}$ $\Tilde{{\mathcal{S}}}=MGPC(F_c,c)$ $c_{max}\xleftarrow[]{}c$ $c_{min}\xleftarrow[]{}c$ , ${\mathcal{S}}_{best}\xleftarrow[]{}\Tilde{{\mathcal{S}}}$ **end While** return ${\mathcal{S}}_{best}$ The following theorem prove that the optimal value of worst case setting can be achieve by algorithm \[alg: meta-Saturation Submodular algorithm \] if we relax the conditions as follows. for any $k,l$ algorithm \[alg: meta-Saturation Submodular algorithm \] find m-tuple ${\mathcal{S}}^{'}$ where $$\min_i f_i(S_i^{'}\cup {S_{\rm{tr}}}^{'})\geq \max_{{\mathcal{S}}}\min_i f_i(S_i\cup {S_{\rm{tr}}})$$ and $\mid S_i^{'}\mid\leq \alpha (k-l)$ for $i\in[m]$ and $\mid {S_{\rm{tr}}}^{'}\mid\leq \alpha l$. In each step $\mid {\mathcal{S}}_{best}\mid \leq \alpha( l+ (m)(k-l))$ and $c^{*}\in [c_{min}, c_{max})$. Since, $f_i$ are integer, and $c_{max}-c_{min}< \frac{1}{m}$; we can conclude $\min\limits_i f_i(S_i^{'}\cup {S_{\rm{tr}}}^{'})\geq \max\limits_{{\mathcal{S}}}\min\limits_i f_i(S_i\cup {S_{\rm{tr}}})$. Simulation supplementary ------------------------ ### Ride Share Optimization In this application, we try to solve facility location problem on Uber dataset [@uber_2019]. One part of this dataset, that we work with consists of the September 2014 Uber pick-ups in Manhattan, which is around 1,000,000 data point, each one in the form of $(latitude, longitude, datatime)$. Our goal is to use the data from the past to find the better waiting location for drivers in the future. In particular, We want to find subset of location that minimizes the defined metric between each customers to his closest driver. To be more precise, given a customer $p=(x_p,y_p)$ and driver $r=(x_r, y_r)$ define a manhatan distance between these two point as $d(p,r)=|x_p-x_r|+|y_p-y_r|$. We use the same metric as [@mitrovic2018data], the “convenience score”, which is $c(p,r)=2-\frac{2}{1+e^{-200 d(p,r)}}$.Next, let us define the tasks and learning procedure more clearly. We use September 1 of 2014 as a training set and September 2 of 2014 as a test set. September 1 of 2014 consists of around 700 points and September 1 of 2014 consists of around 1200 points. We define a Time slot $T_a$ by selecting point $a$ and pick 10 data points in half an hour before point $a$ and for each one pick 10 points in 1 km neighborhood of that which makes a total of 100 points. Each task is ${\mathcal{T}}_a=\{\ f_a,T_a\}$ where $f_a$ is monotone submodular function defined for every driver waiting locations set $S$ as $f_a(S)=\sum_{p\in T_a}\max_{r\in S}c(p,r)$, and $T_a$ is corresponding time slot set. We pick 100,000 points at random from the September 2014 Uber pick-ups locations as ground set. For training we pick $m$ points at random from the first week $a_1,a_2,\dots, a_m$ and consider the tasks ${\mathcal{T}}_1, {\mathcal{T}}_2,\dots,{\mathcal{T}}_m$. We run the **Greedy Train**(running greedy at training on sum),**meta-Greedy**, and **Random meta-Greedy** algorithms on them which means we try to optimize $\sum_{i=1}^{m}f_{a_{i}}({S_{\rm{tr}}}\cup S_i)$ and find set $S$ with size $l$ which can be used in test time as initial set. In test time we test the performance of algorithms on the series of tasks ${\mathcal{T}}^{'}_{1}, {\mathcal{T}}^{'}_{2},\dots,{\mathcal{T}}^{'}_{h}$ and report the comparison of performance on average. First, we pick $m$ points at random from second week $a^{'}_{1}, a^{'}_{2},\dots,a^{'}_{m}$ and look at the performance of the **Greedy Train**,**Greedy Test**,**meta Greedy**, and **random meta Greedy** algorithms on this tasks. We report the normalized performance of algorithms in figure \[fig:simulation-ride\]. We can see if we run greedy for each task in test time independently, we achieve better performance, but in the price of computation time. ### Movie Recommendation In this application we use the data set of movies from movielens dataset [@fivethirtyeight_2019]. The data set consists of movie information and movie ratings between 1 to 5. We pick the subset of 2000 most rated movies and 200 user that rate the highest to these movies same as [@stan2017probabilistic]. Each movies can belong to 18 genres(Horror, Thriller, and etc.). We define a set of all movies in genre $t$ as $G_t$ and define $R_i$ be the set of all movie rated by user $i$, in which for each movie $v \in R_i$, $r_i(v)$ is a corresponding rating. Here we define a user based task which involves 5 users ratings and the related objective functions; formally ${\mathcal{T}}_j=\{ \{f_{j_i}\}_{i=1}^{5},\{R_{j_i}\}_{i=1}^{5}\}$ which $f_{j_i}$ is monotone submodular function and $R_{j_i}$ is set of movie ratings by user $j_i$. In particular, we define $f_i$ as in equation . $$\label{eq: mov recom obj2} f_i(S)=\sum_{t=1}^{18} w_{i,t}.\max_{v\in R_i\cap G_t \cap S} r_i(v)$$ which is the weighted average over maximum rate that user $i$ gives to movies from each genre and $w_{i,t}$ is proportion of movies in genre $t$ which is rated by user $i$ out of all the rating he provides. We divide 200 users to two parts the first 100 users for training, and the second 100 users for test. We show the results in figure \[fig:simulation-movrecom\]. As we can see the performance of meta-Greedy is really close to Greedy test while it is faster. For example, in simulation of Figure the average time for Greedy test algorithm is 4.33 sec per running, and meta-Greedy is 0.91 sec per running while the performance of meta-greedy is close to Greedy test. Counter-example for Submodularity of the Objective in {#sec:counter_example} ====================================================== In this section, we provide a counterexample for submodularity of the objective function in the equation . We consider a maximum coverage problem in which the function value is an area covered by a set of elements. We define the ground set $V=\{ABIJ,BCDI,ACDJ,IDEH\\,HEFG,BCEH\}$ which has shown in Figure \[fig:Counter Example\]. Each element is a rectangle, and a function value of that element is an area covered by that element. We refer to each element (rectangle) by it’s vertices. Let $AC=CD=DE=EF=1$, and $BC=0.75$. Also in we let $m=1$ and $k-l=1$ which means that we are considering a single set function $f$ defined as: $f(S)=\max_{e \in V} A(S\cup e)$, where $A(T)$ is a area of set $T$. Note that the area function $A$ is monotone and submodular, however as we will show below, the function $f$ is not submodular. To do so, we consider two sets $T_1=\emptyset$ and $T_2=\{ACDJ\}$ and add the element $IDEH$ to both sets and observe that $f$ does not satisfy the diminishing returns property. Let us first compute the function value at $T_1$ and $T_2$ as follows: $$f(T_1)=\max_{e \in V} A(e)=A(\{BCEH\})=1.5,$$ and $$f(T_2)=\max_{e \in V} A(T_2\cup e)=A(\{ACDJ,IDEH\})=1.75.$$ Similarly, we compute the function value at $T_1^{'}=T_1\cup \{IDEH\} $, and $T_2^{'}=T_2\cup \{IDEH\} $: $$f(T_1^{'})=\max_{e \in V} A(T_1^{'}\cup e)=A(\{IDEH,ACDJ\})=1.75,$$ and $$f(T_2^{'})=\max_{e \in V} A(T_2^{'}\cup e)=A(\{IDEH,ACDJ,EFGH\})=2.5.$$ We can now see that $T_1\subseteq T_2$, but $f(T_2^{'})-f(T_2)\not \leq f(T_1^{'})-f(T_1)$. Therefore, $f$ is not submodular. Also let us make a remark about $k$-submodularity which studies functions of $k$ subsets of the ground set that are disjoint sets. This class of functions is submodular in each orthant [@ohsaka2015monotone]. However, in the submodular meta-learning framework, sets can have overlap, and there is no restriction on the sets to be disjoint. Therefore, our framework is different from $k$-submodular maximization. Objective Value ------------------------ ----------------- -- Greedy-Train 0.896822 Greedy-Test 0.963639 Random 0.839599 Meta-Greedy 0.949555 Randomized Meta-Greedy 0.945020 Replacement Greedy 0.910977 : Comparison of two-stage framework and submodular meta-learning framework ### Two-Stage Submodular Meta-Learning Although, our submodular meta-learning has the ability of personalizing the solution in test time but it is still use the same ${S_{\rm{tr}}}$ for every task. But learning all the set ${S_{\rm{tr}}}\cup S_i$ is computationally expensive in the test time. Therefore, we need to find the way to personalize ${S_{\rm{tr}}}$ too without increasing the computational power. In order to that, we come up with following approach: Let assume similar to two-stage framework we learn reduced ground set in. training but instead of choosing the. whole set as a subset of this set instead we just choose ${S_{\rm{tr}}}$ as subset of this set and we let $S_i$ be personalized at the test time. Formally, we try to solve the following problem: \[eq:two-stage sub meta-learning\] $$\begin{aligned} {4} & \max_{S,\tilde{S_i}} && \sum_{i=1}^{m}\; \max_{\mid S_i\mid \leq k-l}\; f_i(S_i\cup \tilde{S_i}) \label{eq: LPRelaxation2}\\ &\operatorname*{subject\,\, to \quad}&&\tilde{S_i}\subseteq S\\ \; \; & &&\mid \tilde{S_i}\mid \leq l\;\;\\ & &&\mid S\mid \leq q\;\;\\ \end{aligned}$$ Note that the optimal value of this problem is higher than related two-stage submodular maximization and related meta-submodular maximization problem; since, if $q=l$ this problem reduced to meta-submodular problem and if $k=l$ this problem reduced to two-stage submodular maximization. ### Two-Stage Submodular Meta-Learning Algorithm For solving the problem \[eq:two-stage sub meta-learning\] the natural way that come to mind is to some how combining the replacement greedy algorithm and meta-Greedy algorithm but how we can do that? One way to doing that is to fill $S_i$ greedily and then using the replacement greedy to obtain $S,\tilde{S_i}$ then we can use $S$ as reduced ground set in test time. **Initialize** ${S_{\rm{tr}}}=\{S_i\}_{i=1}^m=\{\tilde{S}_i\}_{i=1}^m=\emptyset$ Find $e_i^{*} = \operatorname*{arg\,max}_{ e\in V} \Delta_i(e|S_i)$ $S_i\xleftarrow{} S_i\cup\{e_i^{*}\}$ **end for** **end for** Find $ e^{*} =\!\operatorname*{arg\,max}_{ e\in V} \!\sum\limits_{i=1}^{m}\!\nabla_{i}(e|\tilde{S_i} \cup S_i)$ $S\xleftarrow{} S\cup\{e^{*}\}$ $\tilde{S_i}\xleftarrow{} \tilde{S_i}\cup\{e^{*}\}\setminus \text{REP}_i(e^{*},\tilde{S_i})$ **end for** Return $\tilde{S_i}$, $S$ and $S_i$ In the above algorithm $\nabla_{i}(e|S)=\Delta_i(e|S)$ if $\mid S\mid\not=l $, and $\nabla_{i}(e|S)=\max\{0,\max_{x \in S}{\nabla_i{(e,x,S)}}\}$, and $\text{REP}_i(e,S)=\operatorname*{arg\,max}_{x \in S}{\nabla_i{(e,x,S)}}$ if $\mid S\mid=l $ and is $\emptyset$ if $\mid S\mid\not=l $. Finally, $\nabla_i{(e,x,S)}=f_i(e\cup S\setminus x)-f_i(S)$. ### Analysis of Two-Stage submodular Meta-Learning Algorithm First suppose $S_i$ and $\tilde{S_i}$ and $S$ is a output of algorithm \[alg: two-stage meta-Greedy\]; because, the first phase of algorithm \[alg: two-stage meta-Greedy\] is same as first phase of algorithm \[alg: Reverse discrete meta-Greedy\] we can have the same analysis as the one for which results: $$\label{eq: two-stage eq1 pf} \sum_{i=1}^m f_i(\tilde{S_i}^{*} \cup S_i^{*}) - \sum_{i=1}^m f_i( S_i)\leq \sum_{i=1}^m f_i(\tilde{S_i}^{*} \cup S_i)$$ Also by analysis of replacement greedy in [@stan2017probabilistic], we can obtain the following: $$\begin{aligned} \sum_{i=1}^m f_i(\tilde{S_i}\cup S_i)-f_i( S_i)&\geq \frac{1}{2}(1-\frac{1}{e^2})(\sum_{i=1}^m f_i(\tilde{S_i}^{opt}\cup S_i)-f_i(S_i)) \label{eq: eq3 pf two-stage} \\&\geq \frac{1}{2}(1-\frac{1}{e^2})(\sum_{i=1}^m f_i(\tilde{S_i}^{*}\cup S_i)-f_i(S_i)) \label{eq: eq3 pf two-stage }\\&\geq \frac{1}{2}(1-\frac{1}{e^2})(\sum_{i=1}^m f_i(\tilde{S_i}^{*}\cup S_i^{*})-2f_i(S_i)) \label{eq: eq3 pf two-stage}\end{aligned}$$ Also $S_i\subseteq S_i\cup \tilde{S_i}$ and monotonicity $\sum_{i=1}^m f_i(\tilde{S_i}\cup S_i)\geq \sum_{i=1}^m f_i( S_i)$. therefore, if we let $\theta=\sum_{i=1}^m f_i( S_i)$: $$\sum_{i=1}^m f_i(\tilde{S_i}\cup S_i)\geq \max\{ \theta,\frac{1}{2}(1-\frac{1}{e^2})(\text{OPT}-2\theta)+\theta\}$$ then we can write the as follows: $$\begin{aligned} \nonumber \mathbb{E}[(1-\frac{1}{k})^\tau]&=\mathbb{E}[(1-\frac{1}{k})^\tau|\tau^{''}>\tau^{'}]p(\tau^{''}>\tau^{'})+\mathbb{E}[(1-\frac{1}{k})^\tau|\tau^{''}<\tau^{'}]p(\tau^{''}<\tau^{'}) \\&=\mathbb{E}[(1-\frac{1}{k})^{\tau^{'}}|\tau^{''}>\tau^{'}]p(\tau^{''}>\tau^{'})+\mathbb{E}[(1-\frac{1}{k})^{\tau^{''}}|\tau^{''}<\tau^{'}]p(\tau^{''}<\tau^{'}) \\&=\sum_{r=1}^{k}(1-\frac{1}{k})^r(p(\tau^{'}=r)+p(\tau^{''}=r)) \\&=\sum_{r=l}^{k}(1-\frac{1}{k})^r({r-1 \choose l-1}(\frac{k-l}{k})^ {r-l}(\frac{l}{k})^{l}\nonumber\\&\quad+\sum_{r=k-l}^{k}{r-1 \choose k-l-1}(\frac{l}{k})^ {r-k+l}(\frac{k-l}{k})^{k-l})\\&\leq\sum_{r=l}^{k}\exp(\frac{-r}{k}){r-1 \choose l-1}(\frac{k-l}{k})^ {r-l}(\frac{l}{k})^{l}\nonumber\\&\quad+\sum_{r=k-l}^{k}\exp(\frac{-r}{k}){r-1 \choose k-l-1}(\frac{l}{k})^ {r-k+l}(\frac{k-l}{k})^{k-l}\end{aligned}$$ Then using moment generating function(MGF) of negative binomial(NB) distribution we have : $$\begin{aligned} \sum_{r=l}^{k}\exp(\frac{-r}{k}){r-1 \choose l-1}(\frac{k-l}{k})^ {r-l}(\frac{l}{k})^{l}&=\left(\frac{1-\frac{l}{k}}{1-\frac{l}{k}\exp({-\frac{1}{k}})}\right)^l \exp(-\frac{l}{k})\nonumber\\&- \sum_{r=k+1}^{\infty}\exp(\frac{-r}{k}){r-1 \choose l-1}(\frac{k-l}{k})^ {r-l}(\frac{l}{k})^{l}\nonumber\\&\leq \frac{k-l}{k}\exp\left(-\frac{l^2}{k}(1-\exp(-\frac{1}{k}))\right)\nonumber\\&\quad-\exp({-{I_{\frac{l}{k}}}(k+1-l,l+1)}) \\&\leq \frac{k-l}{k}\exp\left(\frac{l^2}{k}((\exp(-1)-1)\frac{1}{k})\right)\nonumber\\&\quad-\exp({-{I_{\frac{l}{k}}}(k+1-l,l+1)})\end{aligned}$$ where the first equality comes from definition of MGF and the first inequality comes from Jensen inequality and definition of CDF of NB distribution and $I_x$ is regularized incomplete beta function. The second inequality comes from the fact that $-1+\exp(-1/k)\leq (\exp(-1)-1)1/k$.\ similarly: $$\begin{aligned} \sum_{r=k-l}^{k}\exp(\frac{-r}{k}){r-1 \choose k-l-1}&(\frac{l}{k})^ {r-k+l}(\frac{k-l}{k})^{k-l}=\left(\frac{1-\frac{k-l}{k}}{1-\frac{k-l}{k}\exp({-\frac{1}{k}})}\right)^{k-l} \exp(-\frac{k-l}{k})\nonumber \\&- \sum_{r=k+1}^{\infty}\exp(\frac{-r}{k}){r-1 \choose k-l-1}(\frac{l}{k})^ {r-k+l}(\frac{k-l}{k})^{k-l}\nonumber\\&\leq \frac{l}{k}\exp\left(\frac{(k-l)^2}{k}(1-\exp(-\frac{1}{k}))\right) \\&-\exp({-{I_{\frac{k-l}{k}}}(l+1,k+1-l)}) \\&\leq \frac{l}{k}\exp\left(\frac{(k-l)^2}{k}((\exp(-1)-1)\frac{1}{k})\right)-\exp({-{I_{\frac{k-l}{k}}}(l+1,k+1-l)})\end{aligned}$$ summing up we can write: $$\begin{aligned} \mathbb{E}[(1-\frac{1}{k})^\tau]&\leq\frac{l}{k}\exp\left((\frac{k-l}{k})^2((\exp(-1)-1))\right)\notag\\&+(\frac{k-l}{k})\exp\left((\frac{l}{k})^2((\exp(-1)-1)\right)\notag\\&-\exp({-{I_{\frac{k-l}{k}}}(l+1,k+1-l)})\notag\\&-\exp({-{I_{\frac{l}{k}}}(k+1-l,l+1)})\leq \exp\left(((\exp(-1)-1))(1-c)^2\right).\end{aligned}$$ where c is $\max\{k-l/k,l/k\}$. [^1]: Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, USA. {aadibi@seas.upenn.edu}. [^2]: Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX, USA. {mokhtari@austin.utexas.edu}. [^3]: Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, USA.{hassani@seas.upenn.edu}.

‘=11 makefntext\#1[ to 3.2pt [-.9pt $^{{\ninerm\@thefnmark}}$]{}\#1]{} makefnmark[to 0pt[$^{\@thefnmark}$]{}]{} PS. @myheadings[mkbothgobbletwo oddhead[ ]{} oddfootevenhead[ ]{}evenfoot \#\#1\#\#1]{} 6.0in 8.6in -0.25truein 0.30truein 0.30truein =1.5pc **THE LAKE BAIKAL EXPERIMENT: SELECTED RESULTS** *$^a$ Institute for Nuclear Research, Moscow, Russia* $^b$ Irkutsk State University, Irkutsk, Russia $^c$ Institute of Nuclear Physics, MSU, Moscow, Russia $^d$ Nizhni Novgorod State Technical University, Nizhni Novgorod, Russia $^e$ St.Petersburg State Marine Technical University, St.Petersburg, Russia $^f$  Kurchatov Institute, Moscow, Russia $^g$ Joint Institute for Nuclear Research, Dubna, Russia $^h$ DESY-Zeuthen, Berlin/Zeuthen, Germany $^i$ KFKI, Budapest, Hungary presented by Zh.DZHILKIBAEV E-mail: djilkib@pcbai10.inr.ruhep.ru Detector ========= The deep underwater Cherenkov detector [*NT-200*]{}, the medium-term goal of the BAIKAL collaboration [@Project; @APP; @APP2], was put into operation at April 6th, 1998. [*NT-200*]{} is deployed in Lake Baikal, Siberia, from shore at a depth of . The detector comprises 192 optical modules (OM) at 8 vertical strings, see Fig.1. The OMs are grouped in pairs along the strings. They contain 37-cm diameter [*QUASAR*]{} PMTs which have been developed specially for our project [@Project; @APP; @OM2]. The two PMTs of a pair are switched in coincidence in order to suppress background from bioluminescence and PMT noise. A pair defines a [*channel*]{}. All OMs face downward, with the exception of the OMs of the second and eleventh layers, which look upward. The distance between downward oriented layers is 6.25m, the distance between layers facing to each other (layers 1/2 and 10/11) is 7.5m, the distance between back-to-back layers (2/3 and 11/12) is 5.0m. A [*muon-trigger*]{} is formed by the requirement of (with [*hit*]{} referring to a channel) within . $N$ is typically set to For such events, amplitude and time of all fired channels are digitized and sent to shore. A separate [*monopole trigger*]{} system searches for clusters of sequential hits in individual channels which are characteristic for the passage of slowly moving, bright objects like GUT monopoles. In April 1993, the first part of [*NT-200*]{}, the detector [*NT-36*]{} with 36 OMs at 3 strings, was put into operation and took data up to March 1995. A 72-OM array, [**]{}, run in . In 1996 it was replaced by the four-string array [*NT-96*]{}. [ *NT-144*]{}, a six-string array with 144 OMs, was taking data in . Analysis of experimental data taken with intermediate arrays, especially with [*NT-36*]{} and [*NT-96*]{}, proves the capability of the Baikal neutrino telescope to investigate various problems of neutrino and muon physics. Below we present results which illustrate the capability of the Baikal experiment to search for atmospheric muons and neutrinos, neutrinos induced by neutralino annihilation in the center of the Earth, magnetic monopoles and showers produced by high energy neutrinos. Atmospheric Muons ================= Muon angular distributions as well as depth dependence of the vertical flux obtained from data taken with [*NT-36*]{} have been presented earlier [@APP]. Another example which confirms the efficiency of track reconstruction procedure relates to the investigation of the shore “shadow” in muons with [*NT-96*]{}. The Baikal Neutrino Telescope is placed at a distance of 3.6 km to the nearby shore of the lake. The opposite shore is about 30 km away. This asymmetry opens the possibility to investigate the influence of the close shore to the azimuth distribution under large zenith angles, where reconstruction for the comparatively “thin” [*NT-96*]{} is most critical. A sharp decrease of the muon intensity at zenith angles of 70$^0$-90$^0$ is expected. The comparison of the experimental muon angular distribution with MC calculations gives us an estimation of the accuracy of the reconstruction error close to the horizontal direction. Indeed, the [*NT-96*]{} data show a pronounced dip of the muon flux in the direction of the shore and for zenith angles larger than 70$^0$ – in very good agreement with calculations which take into the effect of the shore. Atmospheric Neutrinos ===================== The main results have been obtained with the first small detector [*NT-36*]{} - investigation of atmospheric muon flux, searching for nearly vertically upward moving muons and searching for slowly moving GUT monopoles have been presented elsewhere [@APP; @FRST_vert; @GUT_monop]. Below we present selected results obtained with [*NT-96*]{}. Identification of nearly vertically upward moving muons ------------------------------------------------------- Different to the standard analysis [@APP], the method presented in this section relies on the application of a series of cuts which are tailored to the response of the telescope to nearly vertically upward moving muons [@FRST_vert; @INR_vert]. The cuts remove muon events far away from the opposite zenith as well as background events which are mostly due to pair and bremsstrahlung showers below the array and to naked downward moving atmospheric muons with zenith angles close to the horizon ($\theta>60^{\circ}$). The candidates identified by the cuts are afterwards fitted in order to determine the zenith angle. We included all events with $\ge$4 hits along at least one of all hit strings. To this sample, a series of 6 cuts is applied. Firstly, the time differences of hit channels along each individual string have to be compatible with a particle close to the opposite zenith (1). The event length should be large enough (2), the maximum recorded amplitude should not exceed a certain value (3) and the center of gravity of hit channels should not be close to the detector bottom (4). The latter two cuts reject efficiently brems showers from downward muons. Finally, also time differences of hits along [*different*]{} strings have to correspond to a nearly vertical muon (5) and the time difference between top and bottom hit in an event has to be larger than a minimum value (6). The effective area for muons moving close to opposite zenith and fulfilling all cuts exceeds $1000$ m$^2$. Within 70 days of effective data taking, $8.4 \cdot 10^7$ events with the muon trigger $N_{hit} \ge 4$ have been selected. Table1 summarizes the number of events from all 3 event samples (MC signal and background, and experiment) which survive the subsequent cuts. After applying all cuts, four events were selected as neutrino candidates, compared to 3.5 expected from MC. One of the four events has 19 hit channels on four strings and was selected as neutrino candidate by the standard analysis too. The zenith angular distribution of these four neutrino candidates is shown in the inner box of Fig.3. [||c|c|c|c|c|c|c||]{} after cut [N]{}$^o$ $\rightarrow$ & 1 & 2 & 3 & 4 & 5 & 6\ atm. $\nu$, MC & 11.2 & 5.5 & 4.9 & 4.1 & 3.8 & 3.5\ background, MC & 7106 & 56 & 41 & 16 & 1.1 & 0.2\ experiment & 8608 & 87 & 66 & 28 & 5 & 4\ Regarding the four detected events as being due to atmospheric neutrinos, one can derive an the upper limit on the flux of muons from the center of the Earth due to annihilation of neutralinos - the favored candidate for cold dark matter. The limits on the excess muon flux obtained with underground experiments [@Bak; @MACRO; @Kam] and [*NT-96*]{} are shown in Table 2. The limits obtained with [*NT-96*]{} are 4–7 times worse then the best underground limits since the data collecting time of [*NT-96*]{} was only $\approx 70$ days. \[limit\] -------------------- ------------- -------------- ------------- -------------- Zenith [*NT-96*]{} [*Baksan*]{} [*MACRO*]{} [*Kam-de*]{} angles $>10GeV$ $>1GeV$ $>1.5GeV$ $>3GeV$ $\geq 150^{\circ}$ $11.0$ $2.1$ $2.67$ $4.0$ $\geq 155^{\circ}$ $9.3 $ $3.2$ $2.14$ $4.8$ $\geq 160^{\circ}$ $ 5.9-7.7 $ $2.4$ $1.72$ $3.4$ $\geq 165^{\circ}$ $4.8$ $1.6$ $1.44$ $3.3$ -------------------- ------------- -------------- ------------- -------------- This result, however, illustrates the capability of underwater experiments with respect to the search for muons due to neutralino annihilation in the center of the Earth. Selection of neutrino events over a large solid angle ----------------------------------------------------- The signature of neutrino induced events is a muon crossing the detector from below. With the flux of downward muons exceeding that of upward muons from atmospheric neutrino interactions by about 6 orders of magnitude, a careful reconstruction is of prime importance. In contrast to first stages of the detector ([*NT-36*]{} [@FRST_vert]), [*NT-96*]{} can be considered as a real neutrino telescope for a wide region in zenith angle $\theta$. After the reconstruction of all events with $\ge$ 9 hits at $\ge$ 3 strings (trigger[*9/3*]{}), quality cuts have been applied in order to reject fake events. Furthermore, in order to guarantee a minimum lever arm for track fitting, events with a projection of the most distant channels on the track ($Z_{dist}$) less than 35 meters have been rejected. Due to the small transversal dimensions of [*NT-96*]{}, this cut excludes zenith angles close to the horizon. The efficiency of the procedure has been tested with a sample of $ 1.8 \cdot 10^6$ MC-generated atmospheric muons, and with MC-generated upward muons due to atmospheric neutrinos. It turns out that the signal to noise ratio is $ > 1$ for this sample. The reconstructed angular distribution of events taken with [*NT-96*]{} in April/September 1996 – after all cuts – is shown in Fig.3. From 70 days of [**]{} data, 12 neutrino candidates have been found. Nine of them have been fully reconstructed. Three nearly upward vertical tracks (see subsection 3.1) hit only 2 strings and give a clear zenith angle but ambiguities in the azimuth angle – similar to the two events from [*NT-36*]{} [@APP]. This is in good agreement with MC expectations. Search for Fast Monopoles ($\beta > 0.75$) ========================================== Fast bare monopoles with unit magnetic Dirac charge and velocities greater than the Cherenkov threshold in water ($\beta = v/c > 0.75$) are promising survey objects for underwater neutrino telescopes. For a given velocity $\beta$ the monopole Cherenkov radiation exceeds that of a relativistic muon by a factor $(gn/e)^2=8.3\cdot10^3$ ($n=1.33$ - index of refraction for water) [@Fr; @DA]. Therefore fast monopoles with $\beta \ge 0.8$ can be detected up to distances $55$ m $\div$ $85$ m which corresponds to effective areas of (1–3)$\cdot 10^4$ m$^2$. The natural way for fast monopole detection is based on the selection of events with high multiplicity of hits. In order to reduce the background from downward atmospheric muons we restrict ourself to monopoles coming from the lower hemisphere. Two independent approaches have been used for selection of upward monopole candidates from the 70 days of [*NT-96*]{} data. The first one is similar to the method which was applied to upward moving muons (see subsection 3.1), with an additional cut $N_{hit}>25$ on the hit multiplicity. The second one cuts on the value of space-time correlation, followed by a cut $N_{hit}>35$ on the hit multiplicity. The upper limits on the monopole flux obtained with the two different methods coincide within errors. The same type of analysis was applied to the data taken during $0.42$ years lifetime with the neutrino telescope [*NT-36*]{} [@INR]. The combined $90\%$ C.L. upper limit obtained by the Baikal experiment for an isotropic flux of bare fast magnetic monopoles is shown in Fig.4, together with the best limits from underground experiments Soudan2, KGF, MACRO and Ohya [@Oh; @MA; @KGF; @Sou] in Fig.4. Search for Very High Energy Electron Neutrinos ============================================== In this section we present very preliminary results with the aim to illustrate the capability of the Baikal Neutrino Telescope to search for extraterrestrial high energy neutrinos from AGNs, GRBs and other sources. The idea used here to search for high energy electron neutrinos ($E_{\nu} > 100$ TeV) is to detect the Cherenkov light emitted by the electromagnetic and (or) hadronic particle cascade produced at the neutrino interaction vertex in the sensitive volume of the neutrino telescope. Earlier this idea has been used by DUMAND [@DUMAND] and to obtain upper limits on the diffuse flux of high energy neutrinos. In order to reduce the background from downward moving atmospheric muons we restrict ourself to cascades produced in a sensitive volume below the detector (see Fig.5) and cause high multiplicity events in detector. The trigger conditions for event selection are the same as those which were used for fast monopole detection (see sec.4). The effective volumes of [*NT-96*]{} averaged over neutrino directions for detection of cascades ith energy $E_{sh}$ are presented in Fig.6. The curves marked as “DOWN”, “UP” and “TOTAL” correspond to effective volumes averaged over lower and upper hemisphere and over all directions. Also effective volumes of detectors SPS (DUMAND) and AMANDA-A are presented in Fig.6. After analysis of 70 days of [*NT-96*]{} data no evidence for any neutrino-induced cascades is found. The limit to the $\tilde{\nu_e}$ flux at the W resonance energy ---------------------------------------------------------------- Although the neutrino-electron interactions can generally be neglected with respect to neutrino-nucleon interactions due to the small electron mass, the resonance cross section of $\tilde{\nu_{e}}e$ interaction at 6.3 PeV is larger than the $\nu N$ cross section at any energy up to $10^{21}$ eV. The resonant cross section at 6.3 PeV for $\tilde{\nu_e}e$ scattering with a hadronic cascade in the final state: $$\tilde{\nu_e} + e \rightarrow W^- \rightarrow hadrons$$ is $3.41 \times 10^{-31}$cm$^2$ [@Gandi]. The cross section averaged over the energy range $$\Delta E=(M_w+2\Gamma_w)^2/2m_e - (M_w-2\Gamma_w)^2/2m_e,$$ $$M_w=80.22 GeV, \, \, \, \Gamma_w=2.08 GeV$$ is $\bar{\sigma}=1.12 \times 10^{-31}$cm$^2$. Eq.3 is used to calculate the upper limit on the diffuse flux of $\tilde{\nu_e}$: $$\frac{dF_{\tilde{\nu}}}{dE_{\tilde{\nu}}} \leq \frac{2.3}{\frac{10}{18}\rho N_A \bar{\sigma} T \Omega_{eff} V_{eff}\Delta E }.$$ Here T is the detector livetime (70 days), $\Omega_{eff}$ and $ V_{eff}$ are the average effective solid angle and volume of the detector respectively. The 90% CL limit at the W resonance energy is: $$\frac{dF_{\tilde{\nu}}}{dE_{\tilde{\nu}}} \leq 3.7 \times 10^{-18} cm^{-2}s^{-1}sr^{-1}GeV^{-1}.$$ This limit lies between limits obtained by SPS ($1.1 \times 10^{-18}$ cm$^{-2}$s$^{-1}$sr$^{-1}$GeV$^{-1}$) and EAS-TOP ($7.6 \times 10^{-18}$cm$^{-2}$s$^{-1}$sr$^{-1}$GeV$^{-1}$). The limit to the $\nu_e + \tilde{\nu_e}$ flux --------------------------------------------- For setting a limit to the $\nu_e + \tilde{\nu_e}$ flux we have used the cross sections for $\nu_e$($\tilde{\nu_e}$) CC-interactions with nucleons [@Gandi] $$\nu_e(\tilde{\nu_e}) + N \stackrel{CC}{\rightarrow} e^-(e^+) + hadrons$$ when all neutrino energy is transferred to the cascade. The energy dependence of neutrino absorption in the Earth has been taken into account. Assuming the $F(E_{\nu})dE=A \delta (E_{\nu}-E)dE$ behavior of the differential neutrino flux the 90% CL limit has been obtained within the $10^{13} \div 6 \times 10^{15}$eV, see Fig.7. To compare [*NT-96*]{} limit with those obtained by SPS and EAS-TOP [@EAS] we assume that a possible signal of 2.3 events originate in the energy interval from $10^5$ to $10^6$ GeV with an $E^{-2}$ differential spectrum of neutrinos. This limit as well as limits obtained by other groups are shown in Fig.8. Also, the resulting neutrino fluxes from a number of different models \[22\] as well as backgrounds from atmospheric neutrinos \[23\] are shown in Fig.8. Conclusions and Outlook ======================= The results obtained with intermediate detector stages show the capability of Baikal Neutrino Telescope to search for the wide variety of phenomena in neutrino astrophysics, cosmic ray physics and particle physics. The first atmospheric neutrinos have been identified. Also muon spectra have been measured, and limits on the fluxes of magnetic monopoles as well as of neutrinos from WIMP annihilation in the center of the Earth have been derived. In the following years, [*NT-200*]{} will be operated as a neutrino telescope with an effective area between 1000 and 5000 m$^2$, depending on the energy and will investigate atmospheric neutrino spectra above 10 GeV. [*NT-200*]{} can be used to search for neutrinos from WIMP annihilation and for magnetic monopoles. It will also be a unique environmental laboratory to study water processes in Lake Baikal. Apart from its own goals, [*NT-200*]{} is regarded to be a prototype for the development a telescope of next generation with an effective area of 50,000 to 100,000 m$^2$. The basic design of such a detector is under discussion at present. [*This work was supported by the Russian Ministry of Research,the German Ministry of Education and Research and the Russian Fund of Fundamental Research ( grants* ]{} , , , and ). [99]{} I.A.Sokalski and Ch.Spiering (eds.) [*The Baikal Neutrino Telescope NT-200, BAIKAL 92-03*]{} (1992) I.A.Belolaptikov [*et al.*]{}, [*Astroparticle Physics*]{} 7 (1997) 263. I.A.Belolaptikov [*et al.*]{}, [*astro-ph/9903341*]{} (1999) (accepted for publ. in [*Astropart. Phys.*]{}). R.I.Bagduev [*et al.*]{} [*Preprint DESY*]{} 98-091, (1998) L.B.Bezrukov [*et al.*]{}, [*Proc. of the 2nd Workshop on the Dark Side of the Universe*]{}, 221 (Rome, 1995) ([astro-ph/9601161]{}) L.B.Bezrukov [*et al.*]{}, [*Proc. of the 2nd Workshop on the Dark Side of the Universe*]{}, 221 (Rome, 1995) ([astro-ph/9601160]{}) V.A.Balkanov [*et al.*]{} 1998 [*Preprint INR 0972/98*]{} (in russian) M.M.Boliev [*et al.*]{} [*Nucl.Phys.*]{} (Proc. Suppl.) [**48**]{} (1996) 83 T.Montaruli [*et al.*]{} [*Proc. 25-th ICRC*]{} Durban–South Africa, vol.7, (1997) 185 M.Mori [*et al.*]{} [*Phys. Rev.*]{} [**D48**]{} (1993) 5505 I.M.Frank 1988 [*Vavilov-Cherenkov Radiation*]{} (Moscow: Nauka) 192 (in russian) D.A.Kirzhnits and V.V.Losjakov [*Pis’ma Zh. Eksp.Theor.Fz.*]{} [**42**]{} (1985) 226 V.A.Balkanov [*et al.*]{} [Preprint INR]{} (Moscow: INR) (1998) (in russian) S.Orito [*et al.*]{} [*Phys. Rev. Lett.*]{} [**66**]{} (1992) 1951. M.Ambrosio [*et al.*]{} [*MACRO Preprint*]{} MACRO/PUB 98/3 (1998) H.Adarkar [*et al.*]{} [*Proc. 21st ICRC. Adelaide.*]{} (1990) 95 J.L.Thorn [*et al.*]{} [*Phys. Rev.*]{} [**D46**]{} (1992) 4846 J.W.Bolesta [*et al.*]{} [*Proc. 25-th ICRC*]{} Durban–South Africa, vol.7, (1997) 29 R.Porrata [*et al.*]{} [*Proc. 25-th ICRC*]{} Durban–South Africa, vol.7, (1997) 9 R.Gandhi [*et al.*]{} [*Astroparticle Physics*]{} [**5**]{} (1996) 81 M.Aglietta [*et al.*]{} [*Phys. Lett.*]{} [**B333**]{} (1994) 555 R.J.Protheroe [*e-preprint astro-ph/9809144*]{} (1998) P.Lipari [*Astropart. Phys.*]{} [**1**]{} (1993) 195

--- abstract: 'Deep images taken with the Wide Field Channel of the Advanced Camera for Surveys on board the *Hubble Space Telescope* provide the basis for study the resolved stellar population of the M81 companion dwarf irregular galaxy Holmberg [ix]{}. Based on color-magnitude diagrams the stellar population toward Holmberg [ix]{} contains numerous stars with ages of $\la\,$200 Myr as well as older red giant stars. By charting the spatial distribution of the red giant stars and considering their inferred metallicities, we concluded that most of these older stars are associated with M81 or its tidal debris. At least 20% of the stellar mass in Holmberg [ix]{}was produced in the last $\sim\,$200 Myr, giving it the youngest stellar populations if any nearby galaxy. The location of Holmberg [ix]{}, its high gas content, and its youthful stellar population suggests that it is a tidal dwarf galaxy, perhaps formed during the last close passage of M82 around M81.' author: - 'E. Sabbi, J.S. Gallagher, L.J.  Smith, D.F. de Mello, & M. Mountain' title: 'HOLMBERG [IX]{}: THE NEAREST YOUNG GALAXY' --- Introduction {#intro} ============ @zwicky56 presented the possibility that new stellar systems might form in the tidal debris around interacting galaxies. This concept has since received observational support from the detection of condensations of gas and newly formed stars in the tidal tails of interacting galaxies [e.g., @schwiezer78]. In some cases these structures have internal kinematics that are indicative of gravitational binding and they thus can be considered as dwarf galaxies in formation [e.g., @hibbard94; @duc00; @weilbacher00; @mundell04]. The long term fate of these tidal dwarf galaxies (TDGs), however, remains unclear, although theory [e.g., @bournaud06] supports the possibility that TDGs separate from other tidal debris and become relatively long-lived dwarfs. TDGs form in dynamically cool tidal tails and may favor gas rich regions [e.g., @wetzstein07]. In these cases the TDG stellar populations would consist of a mix of pre-existing stars from tidally disrupted materials combined with a prominent “new generation” of young stars produced as gas collects in the TDG. Young TDGs may stand out due to their large young stellar content; however, no TDG candidates have been found in the Local Group (LG). On the other hand the major post-formation interactions in the LG appear to have been associated with M31 and are themselves quite old, so any TDG could be difficult to distinguish and may not have survived. However if the conventional wisdom concerning a lack of dark matter in galaxy disks holds, then TDGs should contain little dark matter. Few if any nearby dwarf galaxies are candidates for satisfying this condition [@hunter00]. In contrast to the Local Group, the nearby [3.6 Mpc, @freedman94] M81 galaxy group hosts a major ongoing interaction involving M81, M82, and NGC 3077. This interaction produced extensive H[i]{} arms that link the three galaxies [e.g., @gottesman75; @hulst79; @appleton81; @yun94] and which are potential sites for TDG formation [e.g., @boyce01]. In particular H[i]{} knots with optical counterparts have been identified within the tidal bridges, the most prominent to the east/north-east of M81 being Holmberg [ix]{}, a dwarf galaxy close to M81 catalogued by @Holmberg1969, and Arp’s Loop, a region located along the bridge between M81 and M82 discovered by @Arp1965. We have begun a study of young stellar populations in the M81 galaxy group with the overall aim of elucidating star formation modes in tidal debris. Combining multi-wavelength observations of the stellar features in the H[i]{} tidal bridge connecting M81 and M82, we identified eight young star-forming regions in Arp’s Loop. The FUV luminosities of these objects are modest and typical of small clusters and of associations of O and B stars. We suggested that these young stars recently formed in gas that was tidally stripped from M81 and M82 but is not part of a gravitationally bound galaxy [@demello08]. The stellar content of Holmberg [ix]{} was recently investigated by @makarova02. While their *HST* Wide Field Planetary Camera 2 (WFPC2) observations of Holmberg [ix]{} were not deep enough to unambiguously rule out the presence of old stars, they noted “...no clear signs of an RGB \[red giant branch\] in the color magnitude diagram of this object.” This conclusion was reached despite overall optical properties that have led to Holmberg [ix]{} being considered a dwarf irregular galaxy by most investigators on the basis of its structure and kinematics. In this [*Letter*]{} we analyze deep archival *Hubble Space Telescope* (*HST*) images of Holmberg [ix]{} obtained with the Wide Field Channel (WFC) of the Advanced Camera for Surveys (ACS). The data are $\sim\,$2.5 mag deeper than those obtained by @makarova02, and readily detect stars along the upper RGB, allowing us to set strong limits on the presence of old stars in Holmberg [ix]{}. We show that this galaxy contains a predominantly young stellar population, consistent with it being a recently formed TDG, as suggested by @makarova02. Observations {#obs} ============ We retrieved deep broadband F555W and F814W ACS/WFC images from the Multimission Archive at STScI (MAST) of the dwarf galaxy Holmberg [ix]{} (P.I. E.D. Skillman, GO-10605). The dataset consists of eight 1192 s dithered exposures in both the F555W and F814W filters. The data were processed through the standard Space Telescope Science Institute ACS calibration pipeline CALACS and, for each filter, all the exposures were co-added using the MULTIDRIZZLE package [@koekemoer02]. For each filter the total exposure time is 4768 s and the images cover an area of $200\arcsec \times 200\arcsec$ corresponding, at a distance for Holmberg [ix]{} of 3.6 Mpc, to $3.5\times 3.5$ kpc. The color-combined image of the data is presented in Plate 1. The photometric reduction has been performed with the DAOPHOT package within the IRAF[^1] environment. Stars were independently detected in each filter using the DAOFIND routine, with a detection threshold set at $4\sigma$ above the local background level. Their fluxes were measured by aperture photometry using an aperture size of $0.15\arcsec$. We then performed PSF-fitting photometry to refine the photometric measurements of the individual sources. To take into account the spatial variations in the core width and shape of the PSF [@krist03; @sirianni05], we computed a spatially-variable PSF using $\sim\,$180 isolated and moderately bright stars, uniformly distributed over the entire region. We transformed the instrumental magnitudes to the *HST* VEGAMAG system by converting the individual stellar magnitudes to an aperture radius of $0.5\arcsec$ and applying the zero points listed in @sirianni05. We applied selection criteria to our catalogs based on the shape of the objects, with the aim of distinguishing bona-fide single stars from extended, blended or spurious objects. To accomplish this, we considered the DAOPHOT $\chi^2$ and sharpness parameters: $\chi^2$ gives the ratio of the observed pixel-to-pixel scatter in the fit residuals to the expected scatter calculated from a predictive model based on the measured detector features, while sharpness sets the intrinsic angular size of the objects. Only objects with $\chi^2<3$ and $-0.6<\mathrm{sharpness}< 0.6$ in the F555W filter and $\chi^2<4$ and $-0.5<\mathrm{sharpness}< 0.5$ in the F814W filter have been retained. We found these values to be the best for rejecting spurious and extended objects, without also eliminating the bright stars. By inspecting the rejected objects, we recognized several candidate star clusters (i.e., fairly round but extended objects) and background galaxies. The final catalog contains 23,182 stars. RESULTS ======= Figure \[f:cmdA\] shows the $m_{\rm F814W}$ versus $m_{\rm F555W}-m_{\rm F814W}$ color-magnitude diagram (CMD) of Holmberg [ix]{}. The young stellar population is represented by well-defined blue and red plumes. The blue plume is located at $m_{\rm F555W}-m_{\rm F814W}\simeq 0.0$ with the brightest stars at $m_{\rm F814W}\simeq 20.5$ and is composed of upper main sequence (MS) stars as well as stars along the hot edge of the core helium burning blue loop phase. The quality of the photometry is such that the gap between the MS and the blue edge of the blue loop phase is clearly visible. The red plume is at $1.2\le m_{\rm F555W}-m_{\rm F814W}\le 2.0$ and extends between $20.0\le m_{\rm F814W}<24.0$. It is populated by red supergiants (RSGs) at the brighter magnitudes that define the red side of the blue loop, and by asymptotic giant branch (AGB) stars at fainter luminosities. The rich concentration of stars fainter than $m_{\rm F814W}>24.0$, with colors redder than $m_{\rm F555W}-m_{\rm F814W}\ge1$ corresponds to low-mass, old stars (age $\ge 1$ Gyr) in the RGB evolutionary phase. The region of Holmberg [ix]{} then contains stars covering a considerable range in age. An interesting feature in the CMD is the lack of stars with $m_\mathrm{F814W}\le 24.0$ and colors in the range of $1\le m_\mathrm{F555W}-m_\mathrm{F814W}\le 1.5$ (marked as the “Gap” in Fig. 1). This shortage in stellar density at the base of the red plume indicates a prolonged quiescent star formation occurred about $\le\,$1 Gyr ago. Evidently vigorous star formation started again recently to produce the large young population seen along the blue MS and both sides of the blue loop. Spectroscopic observations of H[ii]{} regions in Holmberg [ix]{} indicate a metallicity for the ionized gas of $Z\simeq 0.0076$ [@makarova02]. Therefore we used $Z=0.008$ Padova isochrones [@salasnich00] to derive the ages of the two stellar components identified in the Holmberg [ix]{} region (Fig. 2). From the structure of the RGB, the older stellar population covers a likely range in age of $\sim\,$1 to 12 Gyr, as well as having a span in metallicity that reaches to relatively high values. A second major episode of star formation evidently started $\simeq\!200$ Myr in the past, when M81 and M82 experienced the their nearest approach [as derived from dynamical simulations by @yu99], suggesting that this event could be the origin of Holmberg [ix]{} as a gravitationally bound entity. As can be seen from the presence of H[ii]{} regions, active star formation continues into the present epoch. We next consider whether the RGB and young stars are both part of Holmberg [ix]{}. This is accomplished by comparing spatial distributions of the older and younger stellar populations visible in the ACS image. RGB stars with $26.0<m_\mathrm{F814W}<24.0$ and $1.0<m_\mathrm{F555W}-m_\mathrm{F814W}<3.0$ trace the old stellar population. MS and blue loop stars, the blue plume, with $26.0<m_\mathrm{F814W}<24.0$ and $-0.3<m_\mathrm{F555W}-m_\mathrm{F814W}<0.4$ chart the spatial distribution of the younger stellar populations. Inspection of Figure \[mappe\] (*top panels*) reveals that the blue plume and RGB stars have very different spatial distributions. Blue plume stars (*top left panel*) are strongly concentrated in the center of Holmberg [ix]{}, as normally observed in a star forming dwarf galaxy. The density of RGB stars (*top right panel*) peaks at the left bottom corner of the image (the position closest to the M81 galaxy), and rapidly decreases toward the image center. This is not the typical spatial distribution for the old stellar population in a dwarf galaxy, which is usually symmetrically distributed around the center of the dwarf. This effect is highlighted in Figure \[mappe\] (*lower panels*), which presents variations in stellar counts in a grid along the diagonal across the image extending from pixel (0;0) to pixel (4500;4500). Each grid element has dimensions $\simeq 150 \times150$ pixels$^2$, which corresponds to $\sim\!130\times130$ pc$^2$, at the distance of Holmberg [ix]{}. We performed a similar analysis on Arp’s Loop [see Fig. 8 in @demello08], but found that the spatial distributions of the young and old stars are closely correlated. Experiments with more than 10$^6$ artificial stars following the procedure described in @sabbi07 tested the quality of our photometry and the completeness of our data. These experiments demonstrate that the differences in the spatial distributions of blue plume and RGB stars is due neither to crowding nor incompleteness in the central region of Holmberg [ix]{}. At $m_\mathrm{F814W}\le 25.7$ and $m_\mathrm{F555W} - m_\mathrm{F814W}= 1.5$, the photometry remains complete at the 90% level (Fig. \[f:cmdA\]). This conclusion can be visually confirmed from the inspection of Plate 1, where a multitude of background galaxies is easily distinguishable even through the center of Holmberg [ix]{} where we have the highest stellar density. The majority of RGB stars have spatial distributions consistent with them being an extended component of M81. This also would explain the presence of a relatively metal-rich RGB stellar population in a dwarf irregular galaxy, which normally would have a more vertical RGB structure with blue colors associated with low metallicity older stars. These results reinforce the findings of @makarova02. Holmberg [ix]{} is impressively dominated by stars with ages of $\la\,$200 Myr, but is it free of any traces of an older stellar populations? If Holmberg [ix]{} is a TDG formed in the recent M81–M82–NGC 3077 interaction, then its older stars should have come from one of the interacting systems while its young stars formed on site. DISCUSSION ========== The lack of an obvious concentration of old stars associated with Holmberg [ix]{} is consistent with its being a TDG that formed out of a mixture of gas and stars from the disks of interacting systems in the M81 group. Since this material comes from the outer parts of galaxies, we expect it to initially have had a high gas-to-star mass ratio and low star formation rate. Alternatively Holmberg [ix]{} could be an old dwarf galaxy whose evolutionary path was modified through interactions. In particular the flattening in RGB star counts (Fig. \[mappe\]—right panel) may indicate that while the M81 halo dominates the lower left corner of the observed region, the RGB stars detected in the right upper could be associated with Holmberg [ix]{}. To test this hypothesis we assumed that [*all*]{} the RGB stars detected in the right upper corner (e.g., between $\sim\,$2,000 and $\sim\,$4,000 pixels) are part of the Holmberg [ix]{} old stellar population. In this region we identified 447 RGB stars between $24.3<m_\mathrm{F814W}<25.7$ and $1.0<m_\mathrm{ F555W}-m_\mathrm{F814W}<2.6$. If, as usually observed in dwarf galaxies, the old stellar population is uniformly distributed, then up to $\sim\!2000$ RGB stars in the CMD could belong to Holmberg [ix]{}. This is a conservative upper limit since we assumed no contamination from M81 in selecting our Holmberg [ix]{} RGB sample. We applied the synthetic CMD method of @tosi91 to make a first assessment of the total numbers of young and old stars formed in Holmberg [ix]{}. Under the assumption of a continuous SF between 1 and 12 Gyr ago, with a Salpeter mass function, the presence of 2000 RGB stars in the observed magnitude range requires a star formation rate of $\leq 5.5\times 10^{-4}\, M_\odot$ yr$^{-1}$. The total mass in older stars for this model is $M_\star\leq 6\times 10^6\, M_\odot$. Similarly we counted 1237 blue loop stars between $19.8<m_\mathrm{F814W}<24.5$ and $-0.3<m_\mathrm{F555W}-m_\mathrm{F814W}<2.7$. Assuming that the galaxy has constantly formed stars in the last 200 Myr, we derive a recent $\mathrm{SFR}\approx 8.1\times10^{-3}\, M_\odot$ yr$^{-1}$, a factor 15 times higher than the lifetime average assuming that all of the observed younger stars formed in Holmberg [ix]{}. The corresponding young stellar mass is $M_\star\simeq 1.6\times 10^6\,M_\odot$. While we cannot exclude that Holmberg [ix]{} is an old dwarf galaxy, it then would have an extreme ratio of young-to-old stars. *No matter what its origin, Holmberg [ix]{} has the youngest mean stellar population age of any nearby galaxy*. Similarly it also has an unusually high ratio of gas-to-stellar mass. We find M(H[i]{}) $=3.3\times 10^8\, M_\odot$ from @swaters02 [on-line Table A.1], which implies a remarkable M(H[i]{})/M$_\star > 40$. Although dwarf galaxies with this degree of gas richness are known, it is extraordinary to find such an object apparently in close proximity to a giant spiral. Given the current data the most viable possibility is that Holmberg [ix]{} is a young TDG assembled from gravitational collapse of gas and stars stripped off during the interaction of M81 with M82. It is likely that the majority of the old stars in the Holmberg [ix]{} region belong to the extended halo of M81 or is tidal debris not associated with Holmberg [ix]{}. This model receives further support from the location of Holmberg [ix]{} along one of the main H[i]{} tidal arms, close to M81. Further tests of this model can be made by checking for significant dark matter in Holmberg [ix]{} and through more detailed studies of the age distribution of its stellar populations. CONCLUSIONS =========== We analyzed archival *HST*/WFC/ACS images of the region around the dwarf galaxy Holmberg [ix]{}. The resulting CMD clearly shows that this galaxy experienced an intense episode of star formation in the last $\sim\,$200 Myr. Although a prominent RGB is present in this field, the spatial distribution of the stars, and in most cases their relatively high metallicities as judged from colors, are consistent with a projected large old stellar population contribution from the halo or disk of M81. However, a slight excess of RGB stars in the region of Holmberg [ix]{} opens the possibility that a low mass, $M_\star\leq 6\times 10^6\, M_\odot$, old stellar component could be present in this dwarf galaxy. The best proof that Holmberg [ix]{} not a TDG would come from evidence for a dark matter halo. Whatever its origin, Holmberg [ix]{} is a low-density stellar system, with most of its baryonic mass in the form of gas. Whether it remains gravitationally bound will depend on a variety of factors, including the fate of the gas and presence or absence of dark matter. If Holmberg [ix]{} dissolves near its present location, then in a few hundred million years its stars should begin forming tidal streams similar to those observed in M31 [@ibata05]. While we cannot strictly exclude that Holmberg [ix]{} is an old dwarf galaxy, the high ratio of gas to stellar mass would then be highly peculiar for a system located near a giant spiral. On the other hand all of the observed properties of Holmberg [ix]{} can be understood in the context of a TDG that formed in tidal debris $\sim\,$200 Myr in the past. In particular the gap in star formation activity at $\approx\,$1 Gyr is consistent with this galaxy having become an active star forming and likely gravitationally bound entity about 200 Myr ago, near the time of the closest M81–M82 approach, as is its location along a tidal arm. We thank Evan Skillman for having proposed to obtain these ACS WFC observations ands the STScI Hubble Heritage team for making Figure 4. We thank Monica Tosi and Francesca Annibali for usefull discussions and suggestions. E.S. was founded by STScI GO grant GO-1208. JSG appreciates research support for this project from the University of Wisconsin Graduate School. DFdM was founded by STScI grant-44185. Appleton, P. N., Davies, R. D., & Stephenson, R. J. 1981, , 195, 327 Arp, H. 1965, Science, 148, 363 Bournaud, F., & Duc, P.-A. 2006, , 456, 481 Boyce, P. J., et al. 2001, , 560, L127 de Mello, D. F., Smith, L. J., Sabbi, E., Gallagher, J. S., Mountain, M., & Harbeck, D. R. 2008, AJ (accepted), astro-ph/0711.2685 Duc, P.-A., Brinks, E., Springel, V., Pichardo, B., Weilbacher, P., & Mirabel, I. F. 2000, , 120, 1238 Fagotto, F., Bressan, A., Bertelli, G., & Chiosi, C. 1994a, A&AS, 104, 365 Fagotto, F., Bressan, A., Bertelli, G., & Chiosi, C. 1994b, A&AS, 105, 29 Freedman, W. L., et al. 1994, , 427, 628 Grebel, E. K. 2005, Stellar Astrophysics with the World’s Largest Telescopes, 752, 161 Gottesman, S. T., & Weliachew, L. 1975, , 195, 23 Hibbard, J. E., Guhathakurta, P., van Gorkom, J. H., & Schweizer, F. 1994, , 107, 67 Holmberg, E. 1969, A study of physical groups of galaxies. Arkiv för Astronomii, Vol. 5, No. 20, pp. 305–343 Hunter, D. A., Hunsberger, S. D., & Roye, E. W. 2000, , 542, 137 Ibata, R., Chapman, S., Ferguson, A. M. N., Lewis, G., Irwin, M., & Tanvir, N. 2005, , 634, 237 Koekemoer, A. M., Fruchter, A. S., Hook, R., & Hack, W. 2002, in Hubble after the Installation of the ACS and the NICMOS Cooling System, ed.  S. Arribas, A. Koekemoer, & B. Whitmore (Baltimore: STScI), 337 Krist, J. E. 2003, ACS Inst. Sci. Rep. 2003-06 (Baltimore: STScI) Makarova, L. N., Karachentsev, I. D., Grebel, E. K., & Barsunova, O. Y. 2002, , 384, 72 Mundell, C. G., James, P. A., Loiseau, N., Schinnerer, E., & Forbes, D. A. 2004, , 614, 648 Ryan-Weber, E. V., Meuerer, G. R., Freeman, K. C., Putman, M. E., & Webster, R. L. 2004, , 127, 1431 Sabbi, E., Sirianni, M., Nota, A., Tosi, M., Gallagher, J., Meixner, M., Oey, M. S., Walterbos, R., Pasquali, A., Smith, L. J., & Angeretti, L. 2007, , 133, 44 Salasnich, B., Girardi, L., Weiss, A., & Chiosi, C. 2000, A&A, 361, 1023 Schwiezer, G. 1978, AN, 58, 151 Sirianni, M., et al. 2005, PASP, 117, 1049 Sirianni, M., Nota, A., De Marchi, G., Leitherer, C., & Clampin, M. 2002, ApJ, 579, 275 Sirianni, M., Nota, A., Leitherer, C., De Marchi, G., & Clampin, M. 2000, ApJ, 533, 203 Swaters, R. A., & Balcells, M. 2002, A&A, 390, 863 Tosi, M., Greggio, L., Marconi, G., & Focardi, P. 1991, , 102, 951 van der Hulst, J. M. 1979, A&A, 75, 97 Weilbacher, P. M., Duc, P.-A., Fritze v. Alvensleben, U., Martin, P., & Fricke, K. J. 2000, , 358, 819 Wetzstein, M., Naab, T., & Burkert, A. 2007, , 375, 805 Yun, M. S., Ho, P. T. P., & Lo, K. Y. 1994, Nature, 372, 530 Yun, M. S. 1999, IAU Symp., 186, 81 Zwicky, F. 1956, Ergebnisse der exakten Naturwissenschaften, 29, 344 [^1]: IRAF is distributed by the National Optical Astronomy Observatory, which is operated by AURA, Inc., under cooperative agreement with the National Science Foundation.

--- abstract: 'The complete electroweak radiative ${\cal O}{(\alpha)}$ corrections to the Drell-Yan process at large invariant dimuon mass have been studied. All formulas for the cross sections and kinematical restrictions are presented in explicit form, for the simplification of calculation and coding the $\theta$– and $\delta$–functions are actively used. The FORTRAN code READY for the numerical analysis in the high energy region corresponding to the future experiments at the CERN Large Hadron Collider has been constructed. To simulate the detector acceptance we used the standard CMS detector cuts. The radiative corrections are found to become large at high dimuon mass $M$, the complete corrections at “bare” setup change the dimuon mass distribution up to $\sim -5.6\%\ (-23.2\%;\ -35.3\%) $ at the LHC energy and $M=1\ (3;\ 5) \mbox{TeV}$.' author: - 'Vladimir A. Zykunov' title: ' Electroweak corrections to the Drell-Yan process in the high dimuon mass range' --- Introduction ============ More than twenty years the Standard Model (SM) keeps for oneself the status of consistent and experimentally confirmed theory since the experimental data of past and present accelerators (LEP, SLC and Tevatron) have shown no significant deviation from the SM predictions up to energy scales of several hundred GeV. However, New Physics (NP) models: various left-right symmetric models, extended gauge theories including grand unification theories, models of composite gauge bosons [@34], some extra dimension scenarios [@extra-dim], extra neutral gauge bosons [@extra-bos] and fermion compositeness models [@89] predict various deviations out of the SM predictions and, therefore, their testing in the new energy scale (thousands GeV) is one of the main tasks of modern physics. The forthcoming experiments at the collider LHC provides such possibility and probably will shed the light on this important problem in the immediate future. The experimental investigation of the continuum for the Drell-Yan production of a dilepton pair, i.e. data on the cross section and the forward-backward asymmetry of the reaction $$pp \rightarrow \gamma,Z \rightarrow \mu^+\mu^-X \label{1}$$ at large invariant mass of a dimuon pair (see [@cmsnote] and references therein) is considered to be one of the powerful tool in the experiments at the LHC from the NP exploration standpoint. The studies of the NP effects are impossible without the exact knowledge of the SM predictions including higher order QCD and ElectroWeak radiative Corrections (EWC). The EWC to the reaction (\[1\]) are studied well (see papers on pure QED corrections [@MosShuSor], and the QED and electroweak corrections in the Z-peak region and above in [@DY2002] and numerous papers cited there). The EWC result contains so-called Double Sudakov Logarithms (DSL) [@sud-log], i.e. the expressions which are growing with the scale of energy, and thus giving one of the main effect in the region of large invariant dimuon mass. By now extensive studies have been done in this area. For instance, the weak Sudakov expansion for general four-fermion processes has been studied in detail (see, for example, [@DENPOZ] and the recent paper [@ARX05] along with the extensive list of references therein). Obviously, the collinear logarithms of QED radiative corrections can compete with the DSL in the investigated region. This important issue has been studied at the one-loop level in [@DY2002], where both the QED and weak corrections have been calculated for $M \leq$ 2 TeV, but it has yet remained unsolved in the region of $M>$ 2 TeV (with the exception of short numerical estimation of EWC to $pp \rightarrow e^+e^-X$ by program ZGRAD [@ZGRAD] in [@Baur2006], see Fig.7 there). Other important contributions in the investigated reaction at high invariant masses are the higher-order corrections (two-loop electroweak logarithms, at least), which also have been studied in the works [@ARX05], [@Denner:2006jr] (see also the numerous papers cited therein). Weak boson emission contribution has been recently calculated in [@Baur2006] and the contribution of higher-order corrections due to multiple photon emission has been computed in [@CarCal], these contributions are beyond the presented calculations. Thus, for the future experiments at LHC aimed at the searches of NP in the reaction (\[1\]) it is urgent to know exactly the SM predictions, including the radiative background, i.e. the processes, which are experimentally indistinguishable from (\[1\]). The important task is the insertion of this background into the LHC Monte Carlo generators and they should be both accurate and fast. For them to be fast it is necessary to have a set of compact formulas for the EWC. They are obtained in our previous paper [@YAFDY] using the Asymptotic Approach (AA) and improved in subsequent paper [@PRD]. To speed up the bremsstrahlung calculation and increase its accuracy it is very important to have all formulas for the cross sections and kinematical restrictions in explicit form appropriate for adaptive multidimensional integration. We obtained such form here and reached the high speed and good accuracy of calculation using the $\theta$– and $\delta$–functions apparatus. Notations and the cross section with the Born kinematics ======================================================== Our notations are the following (see Fig.1,a): $p_1 (p_2)$ is the 4-momentum of the first (second) quark or antiquark with the flavor $q$ and mass $m_q$;$k_1 (k_2)$ is the 4-momentum of the final muon $\mu^+ (\mu^-)$ with the mass $m$; $q=k_1+k_2$ is the 4-momentum of the $i$-boson with the mass $m_i$ ($i=\gamma, Z$); $P_{A(B)}$ is the 4-momentum of initial nucleon $A(B)$. We use the standard set of Mandelstam invariants for the partonic elastic scattering $s,\ t,\ u$: $$s=(p_1+p_2)^2,\ t=(p_1-k_1)^2,\ u=(k_1-p_2)^2,$$ and the invariant  $S=(P_A+P_B)^2$ for hadron scattering. The invariant mass of dimuon is $M=\sqrt{q^2}$. For a start let us present the convolution formula for the total hadronic (H) cross section, where we used such abbreviations and indices: “Born” (index 0), V-contribution: (indices: BSE for boson self energies, HV for “heavy” vertices, “$\gamma \gamma$” for the IR-finite part of $\gamma \gamma$-boxes, “$\gamma Z$” for the IR-finite part of $\gamma Z$-boxes, “$ZZ$” for the $ZZ$-boxes, “$WW$” for the $WW$-boxes, “$fin$” for the sum of “Light” Vertices (LV), the IR-part of $\gamma\gamma$ and $\gamma Z$–boxes and emission of “soft” photon, with the energy less than $\omega$). The “$fin$”-part is IR-finite and described by Born kinematics. Also we used common index for V-contribution $V=0,\mbox{BSE},\mbox{HV},b,fin$ and special index for boxes $b=\gamma\gamma,\gamma Z,ZZ,WW$. Thus, the hadronic cross section looks like $$\begin{aligned} \sigma_{V}^H = \frac{1}{3} \int\limits_{0}^{1}dx_1 \int\limits_{0}^{1}dx_2 \int\limits_{-S}^{0}dt \sum\limits_{q=u,d,...} &&[ f_q^A(x_1,Q^2)f_{\bar q}^B(x_2,Q^2) \hat \sigma_{V}^{q\bar q}(t) + f_{\bar q}^A(x_1,Q^2)f_{q}^B(x_2,Q^2) \hat \sigma_{V}^{\bar q q}(t)] \theta(t+\hat s) \theta_M \hat \theta_{D}, \label{xsfin}\end{aligned}$$ here the $f_q^H(x,Q^2)$ is the probability at energy scale $Q^2$ of constituent $q$ with the fraction $x$ of the hadron’s momentum in hadron $H$ finding, $\hat s=x_1 x_2 S$, the $\theta$-function under integral sign determined by the kinematics of parton reaction, the factor $$\theta_M=\theta(\hat s-M_1^2)\theta(M_2^2-\hat s)$$ provides the integration in interval of invariant mass $M_1 \leq M \leq M_2$ and the factor $$\theta_{D}= \theta(\zeta^*-\cos\theta)\theta(\zeta^*+\cos\theta) \theta(\zeta^*-\cos\alpha)\theta(\zeta^*+\cos\alpha) \theta(p_T(\mu^+)-p_T^{min})\theta(p_T(\mu^-)-p_T^{min}) \label{tetad}$$ cuts the region of integration according detector geometry, the parameters $\zeta^*$ and $p_T^{min}$ will be discussed below. The expressions for the angles ($\theta\ (\alpha)$ is the scattering angle of the muon with the 4-momenta $k_1\ (k_2)$ in the center mass system of hadrons) and energies (also in the centre of hadron mass system) can be obtained as special situation of radiation case (“radiative” invariants $v,\ z,\ u_1,\ z_1$=0) from the formulas (\[uie\]) presented below. For transverse components the expressions take place: $p_T(\mu^+)={k_1}_0 \sin\theta,\ p_T(\mu^-)={k_2}_0 \sin\alpha$. Let us enumerate all quark cross sections in (\[xsfin\]) using agreement $\sigma(t) \equiv d\sigma/{dt}$. The Born cross section looks like $$\begin{aligned} \sigma_{0}^{q\bar q}(t) = \frac{2\pi\alpha^2}{s^2} \sum\limits_{i,j=\gamma,Z} D^{i}{D^{j}}^* (b_+^{i,j}t^2+b_-^{i,j}u^2),\end{aligned}$$ where the non-radiative boson propagators look like $$D^{j} =\frac{1}{s-m_j^2+im_j\Gamma_j},$$ $\Gamma_j$ is the $j$-boson width, $$b_{\pm}^{n,k}= {\lambda_q}^{n,k}_+{\lambda_l}^{n,k}_+ \pm {\lambda_q}^{n,k}_-{\lambda_l}^{n,k}_- \label{bplusminus}$$ and the combinations of coupling constants for $f$-fermion with $i$- (or $j$-) boson have the form $${\lambda_f}^{i,j}_+=v^i_fv^j_f+a^i_fa^j_f,\ {\lambda_f}^{i,j}_-=v^i_fa^j_f+a^i_fv^j_f, \label{lamb}$$ where $$v^{\gamma}_f=-Q_f,\ a^{\gamma}_f=0,\ v^Z_f=\frac{I_f^3-2s_W^2Q_f} {2s_Wc_W},\ a^Z_f=\frac{I_f^3}{2s_Wc_W},$$ $Q_f$ is the electric charge of fermion $f$, $I_f^3$ is the third component of the weak isospin of fermion $f$, and $s_W\ (c_W)$ is the sine(cosine) of the weak mixing angle: $s_W=\sqrt{1-c_W^2}$, $c_W= m_W/m_Z$. The BSE-part is $$\begin{aligned} \sigma^{q \bar q}_{\rm BSE}(t)=-\frac{4\alpha^2\pi}{s^2} \bigl[ && \sum\limits_{i,j=\gamma,Z} \Pi_S^i D^{i} {D^{j}}^* \sum\limits_{\chi=+,-} {\lambda_q}^{i,j}_{\chi} {\lambda_l}^{i,j}_{\chi} B_{\chi} + \nonumber \\&& + \Pi_S^{\gamma Z} D^{Z} \sum\limits_{i=\gamma,Z} {D^{j}}^* \sum\limits_{\chi=+,-} ( {\lambda_q}^{\gamma,j}_{\chi} {\lambda_l}^{Z,j}_{\chi} + {\lambda_q}^{Z,j}_{\chi} {\lambda_l}^{\gamma,j}_{\chi}) B_{\chi} \bigr].\end{aligned}$$ Here $\Pi_S^{\gamma,Z,\gamma Z}$ are connected with the renormalized photon–, Z– and $\gamma$Z–self energies [@BSH86; @Hollik] as $$\Pi_S^{\gamma}=\frac{\hat\Sigma^{\gamma}}{s},\ \Pi_S^{Z}=\frac{\hat\Sigma^{Z}}{s-m_Z^2},\ \Pi_S^{\gamma Z}=\frac{\hat\Sigma^{\gamma Z}}{s}.$$ The HV-part looks like $$\begin{aligned} \sigma^{q \bar q}_{\rm HV}(t)= \frac{4 \pi \alpha^2}{s^2} {\rm Re} \sum_{i,j=\gamma,Z} D^{i} {D^{j}}^* \sum_{\chi=+,-} ({\lambda_q^{\rm F}}^{i,j}_{\chi} {\lambda_l}^{i,j}_{\chi} + {\lambda_q}^{i,j}_{\chi} {\lambda_l^{\rm F}}^{i,j}_{\chi}) B_{\chi},\end{aligned}$$ where the form factors ${\lambda_f^{\rm F}}^{i,j}_{\pm}$ are explained in [@YAFDY]. The boxes can be presented as $$\begin{aligned} \sigma^{q \bar q}_{ b}(t)= \frac{2 \alpha^3}{s^2} \sum_{k=\gamma,Z} {D^{k}}^* (\delta^{b,k}(t,u,b_+,b_-)-\delta^{b,k}(u,t,b_-,b_+)),\end{aligned}$$ where functions $\delta^{b,k}(t,u,b_+,b_-)$ and all prescriptions for them can be found in [@YAFDY; @PRD]. The “$fin$”-part (the result of infrared singularity cancellation of $\gamma\gamma, \gamma Z, \mbox{LV}$ and “soft” bremsstrahlung) is $$\begin{aligned} \sigma_{fin}^{q\bar q}(t) = && \frac{\alpha}{\pi} \delta_{fin}^{q\bar q} \sigma_{0}^{q\bar q}(t), \nonumber \\[0.3cm] \displaystyle \delta_{fin}^{q\bar q} =&& J_0 {\log}\frac{2\omega}{\sqrt{s}} + Q_l^2 (\frac 32 {\log}\frac{s}{m^2}-2+\frac{\pi^2}{3}) + Q_q^2 (\frac 32 {\log}\frac{s}{m_q^2}-2+\frac{\pi^2}{3}) \nonumber \\[0.3cm] \displaystyle &&-Q_qQ_l ({\log}\frac{s^2}{tu}{\log}\frac{t}{u}+\frac{\pi^2}{3} +{\log}^2\frac{t}{u}+4{\mbox Li}_2\frac{-t}{u}), \nonumber \\[0.3cm] \displaystyle &&J_0=2\bigl(Q_q^2\bigl({\log}\frac{s}{m_q^2}-1\bigr)-2Q_qQ_l{\log}\frac{t}{u} +Q_l^2\bigl({\log}\frac{s}{m^2}-1\bigr)\bigr), \label{soft}\end{aligned}$$ where $\mbox{Li}_2$ denotes the Spence dilogarithm. Let us note that correction $\delta_{fin,{\rm FSR}}^{q\bar q}$ is well known and presented, for example, in paper [@baur], and the correction $\delta_{fin,{\rm ISR}}^{q\bar q}$ can be found in the following way $$\delta_{fin,{\rm ISR}}^{q\bar q} = \delta_{fin,{\rm FSR}}^{q\bar q} (m\rightarrow m_q, Q_l\rightarrow Q_q).$$ To find the cross section for ${\bar q q}$-case, it is necessary to change $t \leftrightarrow u$ in the Born part and $Q_qQ_l \rightarrow -Q_qQ_l$ in “$fin$”-part. The “hat” in formula (\[xsfin\]) means only $s \rightarrow \hat s$. “Hard” photons ============== Let us present the hadronic cross section induced by bremsstrahlung (Fig.1,i-Fig.1,l). Introducing the total phase space of reaction as $$\begin{aligned} I_{\Omega}^6[A]= \int\limits_{0}^{1}dx_1 \int\limits_{0}^{1}dx_2 \int \!\!\!\! \int\limits_{~\Omega} \!\!\!\! \int \!\!\!\! \int dt dv dz du_1 \frac{1}{\pi\sqrt{R_{u_1}}} \theta(\hat R_{u_1}) \theta_M^R \hat \theta_{D}^R \ A, \label{i6}\end{aligned}$$ where $z=2k_1p,\ v=2k_2p,\ z_1=2p_1p,\ u_1=2p_2p$ and $p$ is the 4-momentum of a real bremsstrahlung photon. The factors $\theta_M^R$ and $\theta_{D}^R$ look in “radiative” case slightly different in comparison with “non-radiative” ones: $$\theta_M^R=\theta(\hat s-z-v-M_1^2)\theta(M_2^2-\hat s+z+v),$$ and for $\theta_{D}^R$ we use “non-radiative”expression $\theta_{D}$ (\[tetad\]), we should change only the angles and energies: $$\begin{aligned} &&\cos\theta=1+\frac{t}{{k_1}_0 x_1 \sqrt{S}},\ \cos\alpha=1+\frac{u+z-u_1}{{k_2}_0 x_1 \sqrt{S}}, \nonumber \\[0.3cm] \displaystyle &&{k_1}_0=-\frac{1}{2\sqrt{S}} \bigl(\frac{t}{x_1}+\frac{u}{x_2}\bigr),\ {k_2}_0= \frac{1}{2\sqrt{S}} \bigl( \frac{s+t-z_1}{x_1}+\frac{s+u-u_1}{x_2} \bigr), \label{uie}\end{aligned}$$ where $u=v-\hat s-t$ and $z_1=z-u_1+v$. The physical region $\Omega$ is determined by $\theta(R_{u_1})$, where $R_{u_1}$ ($R_{u_1}$ is the Gram determinant multiplied by constant factor) is described by: $$\begin{aligned} R_{u_1}&=& -A_{u_1} u_1^2 - 2 B_{u_1} u_1 - C_{u_1}, \nonumber \\[0.3cm] \displaystyle A_{u_1}&=& -4m^2s + (s-v)^2, \nonumber \\[0.3cm] \displaystyle B_{u_1}&=& v[ m^2 (3 s - v) + (s - v) (m_q^2 - s - t + v) ]+ z[ m^2(s - v) - m_q^2(s + v) + st + v(s+t-v) ], \nonumber \\[0.3cm] \displaystyle C_{u_1}&=& z^2[ (m^4 + m_q^4 - 2 m^2(m_q^2 + t - v) - 2 m_q^2 ( t + v) + (t - v)^2 ] + \nonumber \\[0.3cm] \displaystyle &+& 2zv [ m^4 + m_q^4 + m_q^2(s - 2 t) - m^2 (2 m_q^2 + s + 2 t - 2 v) + (t - v)(s + t - v) ] + \nonumber \\[0.3cm] \displaystyle &+& v^2 [ m^4 - 2 m^2 (m_q^2 + s + t - v) + (m_q^2 - s - t + v)^2 ].\end{aligned}$$ Then the total bremsstrahlung cross section have form: $$\begin{aligned} \sigma_{R}^H =&& \frac{\alpha^3}{3} I_{\Omega}^6 [ \ \hat s^{-2} \!\!\! \sum\limits_{\chi=+,-} \ \sum\limits_{q=u,d,...} \sum\limits_{i,j=\gamma,Z} {\lambda_q}^{i,j}_{\chi} {\lambda_l}^{i,j}_{\chi} \times \nonumber \\[0.3cm] \displaystyle && \bigl( [ f_q^A(x_1,Q^2)f_{\bar q}^B(x_2,Q^2) + \chi f_{\bar q}^A(x_1,Q^2)f_{q}^B(x_2,Q^2) ] [ Q_l^2 {R_l}_{\chi}^{q\bar q} D^{i}{D^{j}}^* +Q_q^2 {R_{qk}}_{\chi}^{q\bar q} \Pi^{i}{\Pi^{j}}^* ] \nonumber \\[0.3cm] \displaystyle && \left. + [ f_q^A(x_1,Q^2)f_{\bar q}^B(x_2,Q^2) - \chi f_{\bar q}^A(x_1,Q^2)f_{q}^B(x_2,Q^2) ] Q_lQ_q {R_{int}}_{\chi}^{q\bar q} \frac{\Pi^{i}{D^{j}}^* + D^{i}{\Pi^{j}}^*}{2} \bigr)\right|_{s \rightarrow \hat s} ]. \label{xshard}\end{aligned}$$ Indices $l,\ qk$ and $int$ mean the origin of emmited photon: lepton, quark and lepton-quark interference, i.e. Final State Radiation (FSR), Initial State Radiation (ISR) and their INTerference (INT), correspondingly. The “radiative” boson propagators look like $$\Pi^{j} =\frac{1}{s-z-v-m_j^2+im_j\Gamma_j}$$ and the expressions $R$ can be found in Appendix A. Dissecting the region of integration with the help of function $$\theta_{\omega} = \theta(\frac{v+z}{2\sqrt{s}}-\omega),$$ we divide the cross section (\[xshard\]) in two parts: first one is corresponding to “soft” photons with the energy less then $\omega$ (it goes to IR singularity cancellation in formula (\[soft\]) of Sect.I) and the second one is corresponding to “hard” photons with the energy more then $\omega$. We realize the numerical integration of (\[i6\]) (and, certainly, of (\[xsfin\])) by Monte Carlo routine based on the VEGAS algorithm [@VEGAS]. Discussion of numerical results =============================== First, we want to demonstrate the independence the results on unphysical parameters: “soft”-“hard” photon separator $\omega$ (Fig.\[fig:2\]) and the quark mass (Fig.\[fig:3\]). In these Figs. we can see the relative corrections $$\delta^C = {\sigma^H_C}/{\sigma^H_0}$$ to cross sections integrated over interval of invariant dimuon mass 1 TeV $\leq M \leq$ 14 TeV and assuming $\zeta^*=1$ and $p_T^{min}=0$. Fig.\[fig:2\] shows the $\omega$-independence for FSR (left picture), ISR (middle picture) and INT-part (right picture) separately in wide range of $\omega$: $10^{-2}$ GeV $\leq~\omega~\leq$ 10 GeV. We can see also the obvious property for sums the SOFT and the HARD parts: $|\mbox{FSR}| > |\mbox{ISR}| > |\mbox{INT}|$; all of them are negative. For the decision of the quark mass singularity problem we used the $\rm \overline{MS}$ scheme [@MSbar] and the procedure of linearization which is well-grounded in [@SANC]. After all manipulations the part of cross section which must be subtracted to be free from the quark mass dependence has the form (here we used abbreviations Q.S.=QUARK SING. and $q(x) = q(x,Q^2) \equiv f_q(x,Q^2)$) $$\begin{aligned} \sigma_{Q.S.}^H = \frac{1}{3} \int\limits_{0}^{1}dx_1 \int\limits_{0}^{1}dx_2 \int\limits_{0}^{1}dz \int\limits_{-S}^{0}dt \sum\limits_{q=u,d,...} && \left[ \left( q(x_1)\Delta \bar q(x_2,z) \theta(z-x_2) +\Delta q(x_1,z)\bar q(x_2) \theta(z-x_1) ) \hat \sigma_{0}^{q\bar q}(t) \right.\right. + \nonumber \\[0.3cm] \displaystyle && \left.\left. + (q \leftrightarrow \bar q) \right) \right] \theta(t+\hat s) \theta_M \hat \theta_{D}, \label{quarksing}\end{aligned}$$ where $$\begin{aligned} \Delta q(x,z) = \frac{\alpha}{2\pi}Q_q^2 \left[ \frac{1}{z}q(\frac{x}{z},M_{sc}^2)-q(x,M_{sc}^2)\right] \frac{1+z^2}{1-z} \left( \log\frac{M_{sc}^2}{m_q^2}-2\log(1-z)-1\right) \label{deq}\end{aligned}$$ and $M_{sc}$ is the factorization scale [@MSbar]. Fig.\[fig:3\] shows the $m_q$-independence for ISR part of cross section (the asterisks on plot are the points where the calculation was made, they are connected by straight lines). We can see that in the range of rather big values (10 – 100) of ratio $m_q/m_u$ the difference (SOFT+HARD)-(QUARK SING.) is constant (i.e. independent on $m_q$). In the region of small $m_q$ this property is slightly broken. The reason is simple: at small parameter of mass the calculation of mass singularity cross section demands of more time (it is better to say – more iterations in integration). In Fig.\[fig:3\] all of points are obtained with the same number of iterations, so in the region of small $m_q$ the result for HARD ISR part has not so good accuracy, in actual fact this part is slightly more. Surely increasing the accuracy (and simultaneously the running time of code) we provide the exact cancellation of $m_q$, this obvious graph of less importance than Fig.\[fig:3\] and we do not present it here. In the following using FORTRAN program READY [^1] (READY is “Radiative corrEctions to lArge invariant mass Drell-Yan process”) the scale of electroweak radiative corrections and their effect on the observables of the Drell-Yan processes for future CMS experiments will be discussed. In Fig.\[xs0\] and Fig.\[RC\] it is shown the differential Born cross section and the relative corrections to it $$\delta_M^C = \frac{d\sigma^H_C}{dM}/\frac{d\sigma^H_0}{dM}$$ as a functions of $M$. The pure weak (total electroweak) corrections are presented in left (right) picture of Fig.\[RC\]. The translation from total to the differential cross sections realized with the help of trick presented in Appendix B. We used the following set of prescriptions: - investigated reaction is (1) with the energy of LHC $\sqrt{S}=14\ \mbox{TeV}$, - the set of SM input electroweak parameters: $\alpha=1/137.03599911$,  $m_Z=91.1876\ \mbox{GeV}$, $m_W=80.37399\ \mbox{GeV}$,  $\Gamma_Z=2.4924\ \mbox{GeV}$, $\Gamma_W=2.0836\ \mbox{GeV}$,  $m_H=115\ \mbox{GeV}$, - muon mass $m_\mu=0.105658369\ \mbox{GeV}$, masses of fermions for loop contributions to the BSE: $m_e=0.51099892\ \mbox{keV}$, $m_\tau=1.77699\ \mbox{GeV}$,  $m_u=0.06983\ \mbox{GeV}$, $m_c=1.2\ \mbox{GeV}$, $m_t=174\ \mbox{GeV}$, $m_d=0.06984\ \mbox{GeV}$, $m_s=0.15\ \mbox{GeV}$, $m_b=4.6\ \mbox{GeV}$ (the light quark masses provide $\Delta \alpha_{had}^{(5)}(m_Z^2)$=0.0276), - 5 active flavors of quarks in proton, their masses as regulators of the collinear singularity $m_q=10\times m_u$, - non-diagonal elements of CKM matrix = 0, diagonal ones = 1, - “soft”-“hard” photon separator $\omega=10$ GeV, - the MRST2004QED set of unpolarized parton distribution functions [@MRST] (with the choice $Q=M_{sc}=m_Z$), - we impose the experimental restriction conditions on the detected lepton angle $-\zeta^* \leq \zeta \leq \zeta^*$ and on the rapidity $|y(l)|\leq y(l)^*$, see (\[tetad\]); for CMS detector the cut values of $\zeta^*$ and $y(l)^*$ are determined as $$y(l)^* = - {\log}\ \tan \frac{\theta^*}{2} = 2.5,\ \zeta^* = \cos\theta^* \approx 0.986614, \label{restr}$$ also we used the second standard CMS restriction $p_T(l) \geq 20\ \mbox{GeV}$, - here we used so-called “bare” setup for muons identification requirements (no smearing, no recombination of muon and photon). Let us discuss briefly the effects of EWC induced by different contributions (in region $M=1~\mbox{TeV}$). The BSE-contribution is positive and $\sim 0.12$, it is usual effect of BSE. The HV-part gives positive contribution $\sim 0.07$ in spite of the negative sigh of DSL ($-\log^2({m_{Z(W)}^2/s})=-l^2_{Z(W),s}$) in diagrams Fig.1,b and Fig.1,c with the Z and W as additional virtual particle. Analysis shows that the Single Sudakov Logs (SSL=$l_{Z(W),s}$) of diagrams Fig.1,d and Fig.1,e play the very important role in the region of TeV’s M. To determine that we can compare, for example, the coefficients at the functions $\Lambda_2(k^2,M_W)$ (it contains DSL and SSL) and $\Lambda_3(k^2,M_W)$ (it contains only SSL) in formulas (6.8)-(6.12) from [@BSH86], the second one is sometimes much more than first one (up to 9 times), whereas $|\mbox{DSL/SSL}|=|\mbox{SSL}|\approx$4.79 at $M=1\ \mbox{TeV}$. The combined effect of all HV becomes positive. Then, the WW boxes are uniquely negative and they are the dominant contributions. Let us explain it by the example of WW-diagram and $\gamma$-exchange Born diagram interference: extracting this part of cross section (denote it $\sigma^{u\bar u+\bar u u}_{WW \times \gamma}$) and retaining only u-type of quark contributions and leading power of Sudakov logs (there is no SSL in WW-boxes). Then (see formula (40) of [@PRD]) $$\sigma^{u\bar u+\bar u u}_{WW \times \gamma} \sim u \bar u \cdot \delta^{WW,\gamma}(t,u,b_+,b_-) + \bar u u \cdot \delta^{WW,\gamma}(u,t,b_+,b_-),$$ here $u (\bar u) \equiv f_{u(\bar u)}^p(x_{1(2)})$. In that way we can see the fact: the terms $u\bar u$ and $\bar u u$ contain the same $b$ and different invariants $t$ and $u$ as arguments of $\delta^{WW,\gamma}$. Further, using $b_+^{WW,\gamma} = -2{(v^{WW})}^2 Q_q$ and $b_-^{WW,\gamma}=0$ (see (\[bplusminus\]), (\[lamb\]), and $v^{ij}=v^iv^j+a^ia^j$) we can make sure that $$\sigma^{u\bar u+\bar u u}_{WW \times \gamma} \sim -2{(v^{WW})}^2 Q_u [u \bar u \cdot t^2 l^2_{W,t} + \bar u u\cdot u^2 l^2_{W,u} ] < 0.$$ Corresponding contribution of d-type of quarks also less than zero and looks like $$\sigma^{d\bar d+\bar d d}_{WW \times \gamma} \sim 2{(v^{WW})}^2 Q_d [d \bar d \cdot t^2 l^2_{W,t} + \bar d d\cdot u^2 l^2_{W,u} ] < 0.$$ The same situation takes place also for ${WW \times Z}$-case. At last, ZZ-, ISR-, INT- parts are small enough to give determinant effect (ISR gives $\sim -0.019$, INT gives $\sim -0.008$, ZZ is $\sim 0.0003$) and FSR-part is negative and $\sim -0.071$, so the total effect of EWC is found to be negative $\sim -0.056$. Conclusions =========== The complete electroweak radiative ${\cal O}{(\alpha)}$ corrections to the Drell-Yan process at large invariant dimuon mass have been studied. For the shortening of code running time (keeping an enough accuracy) we simplify the calculation as much as possible (using AA and generalized functions). Using FORTRAN code READY the numerical analysis is performed in the high energy region corresponding to the future experiments at the CERN Large Hadron Collider. To simulate the detector acceptance we used the standard CMS detector cuts. The radiative corrections are found to become large at high dimuon mass $M$, the complete corrections at “bare” setup change the dimuon mass distribution up to $\sim -5.6\%\ (-23.2\%;\ -35.3\%) $ at the LHC energy and $M=1\ (3;\ 5)\ \mbox{TeV}$. Some issues have became beyond the scope of the presented paper (the detailed numerical analysis of process $pp \rightarrow e^+e^-X$ and other interesting observables: total inclusive cross section, forward-backward asymmetries; al last, “calo” results – taking into consideration also smearing and recombination). All that will be the subjects of future investigation but, first of all, due the importance and complexity of investigated problem, we should cross-check with the results of other groups (programs SANC [@SANC], ZGRAD [@DY2002; @ZGRAD], ...) to make sure that our result is correct. Author will be grateful to all interested groups for giving a chance to compare the results in that stage. Acknowledgments =============== I am grateful to A. Arbuzov, D. Bardin, S. Bondarenko, I. Golutvin, A. Ilyichev, E. Kuraev, A. Lanyov, V. Mossolov, S. Shmatov, N. Shumeiko, D. Wackeroth for the stimulating discussions. [99]{} M. Cveti[$\rm \check c$]{} and S. Godfrey, hep-ph/9504216; T.G. Rizzo, Proceedings of the 1996 DPF/DPB Summer Study on New Directions for High Energy Physics Snowmass96, Snowmass, CO, 25 June - 12 July (1996); J.L. Hewett and T.G. Rizzo, Phys. Rept. 183 (1989) 193. N. Arkani-Hamed et.al, Phys. Lett. B [**429**]{}, 263 (1998) \[arXiv:hep-ph/9803315\]; I. Antoniadis et.al, Phys. Lett B [**436**]{}, 257 (1998) \[arXiv:hep-ph/9804398\]; L. Randall and R. Sundrum, Phys. Rev. Lett. [**83**]{}, 3370 (1999) \[arXiv:hep-ph/9905221\],  Phys. Rev. Lett. [**83**]{}, 4690 (1999) \[arXiv:hep-th/9906064\]; C. Kokorelis, Nucl. Phys. [**677**]{} (2004) 115 \[arXiv:hep-th/0207234\] A. Leike, Phys. Rep. [**317**]{}, 143 (1999), \[arXiv:hep-ph/9805494\]; T.G. Rizzo, arXiv:hep-ph/9612440 D. Bourilkov, arXiv:hep-ph/0305125; CERN-CMS-NOTE-2006-085 (2006) I. Belotelov et.al, CERN-CMS-NOTE-2006-123 (2006) V. Mosolov, N. Shumeiko, Nucl.Phus. B [**186**]{}, 397 (1981); A. Soroko, N. Shumeiko, Yad. Fiz [**52**]{}, 514 (1990) U. Baur et al., Phys. Rev. D [**65**]{}: 033007, (2002) \[arXiv:hep-ph/0108274\] V. Sudakov, Sov. Phys. JETP [**3**]{}, 65 (1956) A. Denner and S. Pozzorini, Eur. Phys. J. C [**18**]{} (2001) 461 \[arXiv:hep-ph/0010201\];  Eur. Phys. J. C [**21**]{} (2001) 63 \[arXiv:hep-ph/0104127\] B. Jantzen, J.H. Kuhn, A.A. Penin, and V.A. Smirnov, TTP05-17, PSI-PR-05-04 \[arXiv:hep-ph/0509157\] `http://ubhex.physics.buffalo.edu/~baur/zgrad2.tar.gz` U. Baur, Phys. Rev. D [**75**]{}, 013005 (2007) \[arXiv:hep-ph/0611241\] A. Denner, B. Jantzen and S. Pozzorini, Nucl. Phys.  B [**761**]{} (2007) 1 \[arXiv:hep-ph/0608326\] C.M. Carloni Calame [*et al.*]{} JHEP [**0505**]{}, 019 (2005) V. Zykunov, Yad. Fiz. [**69**]{}, 1557 (2006) (Engl. vers.: Phys. of Atom. Nucl. [**69**]{}, 1522 (2006)) V. Zykunov, Phys. Rev. D [**75**]{}, 073019 (2007) \[arXiv:hep-ph/0509315\] M. Böhm, H. Spiesberger, W. Hollik, Fortschr. Phys. [**34**]{}, 687 (1986) W. Hollik, Fortschr. Phys. [**38**]{}, 165 (1990) U. Baur [*et al.*]{}, Phys. Rev. D [**57**]{}, 199 (1998) G. Peter Lepage, J. Comput.Phys. [**27**]{}, 192 (1978) W. Bardeen [*et al.*]{}, Phys. Rev. D [**18**]{}, 3998 (1978) A. Andonov, A. Arbuzov, D. Bardin et al., Comput. Phys. Commun. [**174**]{}, 481 (2006) \[arXiv:hep-ph/0411186\]; SANC project website: `http://sanc.jinr.ru, http://pcphsanc.cern.ch` A.D. Martin [*et al.*]{} Eur.Phys.J. C [**39**]{}, 155 (2005) \[arXiv:hep-ph/0411040\] Expressions for the $R$ ======================= The expressions for the $R$ have such form: for lepton emission (see Fig.1,i and Fig.1,j) $$\begin{aligned} {R_l}_+^{q\bar q}=&& 2 s -2 \frac{m^2}{z^2} (2 u_1^2 + 2 u_1 s + 4 u_1 t - 4 u_1 v + s^2 + 2 s t -2 s v + 2 t^2 - 4 t v + 2 v^2) \nonumber \\[0.3cm] \displaystyle &&- 2 \frac{s}{zv} ( - u_1^2 - u_1 s - 2 u_1 t - s^2 - 2 s t - 2 t^2) - \frac{s}{z} (4 u_1 + 4 s + 6 t - 3 v) \nonumber \\[0.3cm] \displaystyle &&- 2 \frac{m^2}{v^2} (s^2 + 2 s t + 2 t^2) - \frac{1}{v} s ( - z + 2 u_1 + 2 s + 2 t), \nonumber \\[0.3cm] \displaystyle {R_l}_-^{q\bar q}=&& 2 \frac{m^2}{z^2} s (2 u_1 + s + 2 t - 2 v) + 2 \frac{s^2}{zv} ( - u_1 - s - 2 t) + \frac{s}{z} (4 s + 2 t - v) \nonumber \\[0.3cm] \displaystyle && + 2 \frac{m^2}{v^2} s (s + 2 t) + \frac{s}{v} ( - z + 2 u_1 + 2 s + 2 t) - 2 s,\end{aligned}$$ for quark emission (see Fig.1,k and Fig.1,l) $$\begin{aligned} {R_{qk}}_+^{q\bar q}=&& 2 ( z - s + v) - 2 \frac{m_q^2}{z_1^2} (z^2 + 2 z t + s^2 + 2 s t - 2 s v + 2 t^2 - 2 t v + v^2) \nonumber \\[0.3cm] \displaystyle && - 2 \frac{s}{z_1 u_1} ( - s^2 - 2 s t + s v - 2 t^2 + 2 t v - v^2) \nonumber \\[0.3cm] \displaystyle && - \frac{1}{z_1} (z^2 - z s + 2 z t + 2 s^2 + 2 s t - s v + 4 t^2 - 2 t v + v^2) \nonumber \\[0.3cm] \displaystyle && - 2 \frac{m_q^2}{u_1^2} (z^2 - 2 z s - 2 z t + 2 z v + s^2 + 2 s t - 2 s v + 2 t^2 - 2 t v + v^2) \nonumber \\[0.3cm] \displaystyle && - \frac{1}{u_1} (z^2 - 3 z s - 2 z t + 2 z v + 4 s^2 + 6 s t - 5 s v + 4 t^2 - 6 t v + 3 v^2), \nonumber \\[0.3cm] \displaystyle {R_{qk}}_-^{q\bar q}=&& 2 \frac{m_q^2}{z_1^2} ( - z^2 - 2 z t + s^2 + 2 s t - 2 s v - 2 t v + v^2) + 2 \frac{ s^2}{z_1 u_1} ( - s - 2 t + v) \nonumber \\[0.3cm] \displaystyle && + \frac{1}{z_1} ( - z^2 + z s - 2 z t + 2 s^2 + 6 s t - 3 s v - 2 t v + v^2) \nonumber \\[0.3cm] \displaystyle && + 2 \frac{ m_q^2}{u_1^2} (z^2 - 2 z s - 2 z t + 2 z v + s^2 + 2 s t - 2 s v - 2 t v + v^2) \nonumber \\[0.3cm] \displaystyle && + \frac{1}{u_1} (z^2 - 3 z s - 2 z t + 2 z v + 4 s^2 + 6 s t - 5 s v - 2 t v + v^2),\end{aligned}$$ for lepton-quark interference $$\begin{aligned} {R_{int}}_+^{q\bar q}=&& 2 (z - u_1 - s - 4 t + 3 v) + \frac{t}{z z_1} (2 s^2 + 4 s t - 2 s v + 4 t^2 - 2 t v + v^2) \nonumber \\[0.3cm] \displaystyle && +\frac{1}{zu_1} (2 s^3 + 6 s^2 t - 6 s^2 v + 8 s t^2 - 14 s t v + 7 s v^2 + 4 t^3 - 10 t^2 v + 9 t v^2 - 3 v^3) \nonumber \\[0.3cm] \displaystyle && + \frac{2}{z} (u_1 s - u_1 v + s^2 + 2 s t - 3 s v - 3 t v + 2 v^2) \nonumber \\[0.3cm] \displaystyle && + \frac{1}{z_1v} (z^2 s + z^2 t + 2 z s t + 2 z t^2 + 2 s^3 + 6 s^2 t + 8 s t^2 + 4 t^3) + \frac{1}{z_1} (z t - 2 s^2 - 4 s t + s v + t v) \nonumber \\[0.3cm] \displaystyle && + \frac{t}{v u_1} (z^2 - 2 z s - 2 z t + 2 s^2 + 4 s t + 4 t^2) + \frac{2}{v} ( - z t - u_1 s - s^2 - 2 s t) \nonumber \\[0.3cm] \displaystyle && + \frac{1}{u_1} ( - z^2 + 3 z s + 5 z t - 3 z v - 4 s^2 - 12 s t + 8 s v - 12 t^2 + 13 t v - 5 v^2), \nonumber \\[0.3cm] \displaystyle {R_{int}}_-^{q\bar q}=&& 2 (-z + 2 s - v) - \frac{t}{z z_1} (2 s^2 + 4 s t - 2 s v - 2 t v + v^2) \nonumber \\[0.3cm] \displaystyle && - \frac{1}{z u_1} (2 s^3 + 6 s^2 t - 6 s^2 v + 4 s t^2 - 10 s t v + 5 s v^2 - 2 t^2 v + 3 t v^2 - v^3) \nonumber \\[0.3cm] \displaystyle && - \frac{2s}{z} (s - v) - \frac{1}{z_1 v} ( - z^2 s - z^2 t - 2 z s t - 2 z t^2 + 2 s^3 + 6 s^2 t + 4 s t^2) \nonumber \\[0.3cm] \displaystyle && - \frac{1}{z_1} ( - z t - 2 s^2 - 4 s t + s v - 4 t^2 + t v) - \frac{t}{v u_1} (z^2 - 2 z s - 2 z t + 2 s^2 + 4 s t) \nonumber \\[0.3cm] \displaystyle && - \frac{2s}{v} (z - s) - \frac{1}{u_1} ( - z^2 + 3 z s + 5 z t - 3 z v - 4 s^2 - 12 s t + 8 s v - 4 t^2 + 7 t v - 3 v^2).\end{aligned}$$ Translation from total to the differential cross section ======================================================== To translate the total non-radiative (\[xsfin\]) (or radiative (\[xshard\])) cross section to the differential one we should just differentiate it on variable $M$ using obvious rule $$\frac{d\sigma}{dM}=-\left.{\frac{d\sigma}{dM_1}}\right|_{M_1=M}.$$ After that and taking into consideration the formula $$\frac{d\theta(x)}{dx}=\delta(x)$$ we can significantly simplify the form of cross section, as we are in a position to integrate analytically over one of variable (here we choose $x_2$) in such a way $$\int_a^b f(x)\delta(x-z) dx= f(z)\theta(z-a)\theta(b-z).$$ Finally, we get very simple recipe for translation from total to the differential cross section: so, for radiative case to pass $$\sigma^H_{hard} \rightarrow \frac{d\sigma^H_{hard}}{dM}$$ we should in formula (\[xshard\]) 1. do not integrate over $x_2$ and omit $\theta(\hat s-z-v-M_1^2)$, 2. substitute $x_2 \rightarrow {(M^2+z+v)}{(Sx_1)^{-1}}$ (or $\hat s \rightarrow M^2+z+v$), 3. multiply by factor ${2M}{(Sx_1)^{-1}}\theta(Sx_1-M^2-z-v)$. For nonradiative case the translation steps are the same but “radiative” invariants should equal to zeros: $v = z = 0$ since in that case $p \rightarrow 0$. [^1]: FORTRAN code READY is available by contacting to author via e-mail

--- abstract: 'Most of the existing work on automatic facial expression analysis focuses on discrete emotion recognition, or facial action unit detection. However, facial expressions do not always fall neatly into pre-defined semantic categories. Also, the similarity between expressions measured in the action unit space need not correspond to how humans perceive expression similarity. Different from previous work, our goal is to describe facial expressions in a continuous fashion using a compact embedding space that mimics human visual preferences. To achieve this goal, we collect a large-scale faces-in-the-wild dataset with human annotations in the form: Expressions A and B are visually more similar when compared to expression C, and use this dataset to train a neural network that produces a compact (16-dimensional) expression embedding. We experimentally demonstrate that the learned embedding can be successfully used for various applications such as expression retrieval, photo album summarization, and emotion recognition. We also show that the embedding learned using the proposed dataset performs better than several other embeddings learned using existing emotion or action unit datasets.' author: - | Raviteja Vemulapalli\ Google AI\ [ravitejavemu@google.com]{} - | Aseem Agarwala\ Google AI\ [aseemaa@google.com]{} title: A Compact Embedding for Facial Expression Similarity --- Introduction ============ Automatic facial expression analysis has received significant attention from the computer vision community due to its numerous applications such as emotion prediction, expression retrieval (Figure \[fig:example\]), photo album summarization, candid portrait selection [@Candid], etc. Most of the existing work [@EmotionSurvey; @AUSurvey] focuses on recognizing discrete emotions or action units defined by the Facial Action Coding System (FACS) [@FACS]. However, facial expressions do not always fit neatly into semantic boxes, and there could be significant variations in the expression within the same semantic category. For example, smiles can come in many subtle variations, from shy smiles, to nervous smiles, to laughter. Also, not every human-recognizable facial expression has a name. In general, the space of facial expressions can be viewed as a continuous, multi-dimensional space. In this work, we focus on learning a compact, language-free, subject-independent, and continuous expression embedding space that mimics human visual preferences. If humans consider two expressions to be visually more similar when compared to a third one, then the distance between these two expressions in the embedding space should be smaller than their distances from the third expression. To learn such an embedding we collect a new dataset, referred to as the Facial Expression Comparison (FEC) dataset, that consists of around 500K expression triplets generated using 156K face images, along with annotations that specify which two expressions in each triplet are most similar to each other. To the best of our knowledge, this is the first large-scale face dataset with expression comparison annotations. This dataset can be downloaded from <https://ai.google/tools/datasets/google-facial-expression/>. ![Expression retrieval results for embeddings learned using the proposed dataset (top) and an existing emotion classification dataset (bottom).[]{data-label="fig:example"}](images/kissy_face_comparison.jpg){width="45.00000%" height="20.00000%"} We show that a compact (16-dimensional) expression embedding space can be learned by training a deep network with the proposed FEC dataset using triplet loss [@FaceNet]. Based on the distances in the learned embedding space, we are able to predict the most similar pair in a triplet with an accuracy of 81.8% when evaluated on a held-out validation set. The accuracy of median human rater is 87.5% on this validation set, and the accuracy of random selection is 33.3%. We also show that the embedding learned using the FEC dataset performs better than several other embeddings learned using existing emotion or action unit datasets. We experimentally demonstrate that the expression embedding learned using the FEC dataset can be successfully used for various applications such as expression retrieval, photo album summarization, and emotion recognition. Our contributions ----------------- - We introduce the FEC dataset, which is the first large-scale face dataset with expression comparison annotations. This dataset is now available to public. - We experimentally demonstrate that a 16-dimensional expression embedding learned by training a deep neural network with the FEC dataset can be successfully used for several expression-based applications. - We show that the embedding learned using the FEC dataset performs better than several other embeddings learned using existing emotion or action unit datasets. Related work ============ Most of the existing research in the area of automatic facial expression analysis focuses on the following three topics: *(i) Categorical model:* Assigning discrete emotion category labels, *(ii) FACS model*: Detecting the presence/absence (and the strength) of various action units defined by FACS [@FACS], and *(iii) Dimensional model*: Describing emotions using two or three dimensional models such as valence-arousal [@ValAro], pleasure-arousal-dominance [@PAD], etc. Summarizing the vast amount of existing research on these topics is beyond the scope of this paper and we refer the readers to [@Aff-wild; @EmotionSurvey; @AUSurvey] for recent surveys on these topics.\ **Expression datasets:** Several facial expression datasets have been created in the past that consist of face images labeled with discrete emotion categories [@Emotionnet; @afew-7; @Afew; @Sfew; @Fer2013; @Multipie; @Raf-db; @CK+; @AffectNet; @Fer-Wild; @MMI; @ExpW; @Oulu-casia], facial action units [@Emotionnet; @CK+; @Disfa; @AM-FED; @MMI], and strengths of valence and arousal [@Deap; @Aff-wild; @afew-va; @AffectNet; @Reloca]. While these datasets played a significant role in the advancement of automatic facial expression analysis in terms of emotion recognition, action unit detection and valence-arousal estimation, they are not the best fit for learning a compact expression embedding space that mimics human visual preferences.\ **Expression embedding:** A neural network was trained in [@TripletEmbedding] using an emotion classification dataset and category label-based triplet loss [@FaceNet] to produce a 128-dimensional embedding, which was combined with an LSTM-based network for animating three basic expressions. Emotion labels do not provide information about within-class variations and hence a network trained with label-based triplets may not encode fine-grained expression information. The proposed FEC dataset addresses this issue by including expression comparison annotations for within-class triplets. A self-supervised approach was proposed in [@EmbeddingBMVC18] to learn a 256-dimensional facial attribute embedding by watching videos, and the learned embedding was used for multiple tasks such as head pose estimation, facial landmarks prediction, and emotion recognition by training an additional classification or regression layer using labeled training data. However, as reported in [@EmbeddingBMVC18], its performance is worse than existing approaches on these tasks. Different from [@EmbeddingBMVC18], we follow a fully-supervised approach for learning a compact (16-dimensional) expression embedding.\ **Triplet loss-based representation learning:** Several existing works have used triplet-based loss functions for learning image representations. While majority of them use category label-based triplets [@HTL; @TCLoss; @PersonReId; @DRDL; @FaceNet; @LiftingLoss; @AngularLoss; @BinaryEmb], some existing works [@ImgSimJMLR; @DeepRanking] have focused on learning fine-grained representations. While [@DeepRanking] used a similarity measure computed using several existing feature representations to generate groundtruth annotations for the triplets, [@ImgSimJMLR] used text-image relevance based on Google image search to annotate the triplets. Different from these approaches, we use human raters to annotate the triplets. Also, none of these works focus on facial expressions. {width="\textwidth" height="\textwidth"} {width="\textwidth" height="\textwidth"} {width="\textwidth" height="\textwidth"} {width="\textwidth" height="\textwidth"} {width="\textwidth" height="\textwidth"} {width="\textwidth" height="\textwidth"} {width="\textwidth" height="\textwidth"} {width="\textwidth" height="\textwidth"} {width="\textwidth" height="\textwidth"} Facial expression comparison dataset {#sec::motivation} ==================================== In this section, we introduce the FEC dataset, which is a large-scale faces-in-the-wild dataset with expression comparison annotations provided by human raters. To the best of our knowledge, there is no such publicly-available expression comparison dataset. Most of the existing expression datasets are either annotated with emotion labels, or facial action units, or strengths of valence and arousal. One may think that we could generate comparison annotations for the existing datasets using the available emotion or action unit labels. However, there are several issues with such an approach: - Emotion labels do not provide information about within-class variations and hence we cannot generate comparison annotations within a class. For example, while all the expressions in Figure \[fig:triplets\](a) fall into the *Happiness* category, the left and middle expressions are visually more similar when compared to the right expression. Such within-class comparisons are important to learn a fine-grained expression representation. - Due to within-class variations and between-class similarities, two expressions from the same category need not be visually more similar when compared to an expression from a different category. For example, while the middle and right expressions in Figure \[fig:triplets\](b) belong to the *Surprise* category, the middle expression is visually more similar to the left expression which belongs to the *Anger* category. - It is difficult to predict the visual similarity relationships between expressions from three different emotion categories by using labels. For example, while the three expressions in Figure \[fig:triplets\](c) belong to three different categories, the left and middle expressions are visually more similar when compared to the right expression. Such comparisons are useful for learning an embedding that can model long-range visual similarity relationships between different categories. - It is unclear how the difference in the strengths of action units between two expressions could be converted into a distance function that mimics visual preferences. {width="\textwidth"} Dataset {#sec::dataset} ------- Each sample in the FEC dataset consists of a face image triplet $(I_1, I_2, I_3)$ along with a label $L \in \{1, 2, 3\}$ that indicates which two images in the triplet form the most similar pair in terms of facial expression. For example, $L=1$ means $I_2$ and $I_3$ are visually more similar when compared to $I_1$. Note that these triplets do not have a notion of anchor, and each triplet provides two annotations: $I_2$ is closer to $I_3$ than $I_1$, and $I_3$ is closer to $I_2$ than $I_1$. This is different from the commonly-used triplet annotation [@FaceNet; @LMNN] that consists of an anchor, a positive image and a negative image. Also, in this dataset, an image A can be (relatively) closer to another image B in one triplet and (relatively) farther from the same image B in another triplet. This is different from the triplets generated using category labels [@FaceNet], in which any two images will either form a similar pair or a dissimilar pair in all the triplets they appear in. -- --------------- ----------- ----------- ------------- --------- -- One-class Two-class Three-class All Strong 115,544 124,665 117,540 357,749 Strong + Weak 137,266 138,034 132,435 407,735 All 152,674 150,234 146,235 449,143 Strong 13,046 14,607 13,941 41,594 Strong + Weak 15,411 15,908 15,404 46,723 All 17,059 17,107 16,894 51,060 Strong 128,590 139,272 131,481 399,343 Strong + Weak 152,677 153,942 147,839 454,458 All 169,733 167,341 163,129 500,203 -- --------------- ----------- ----------- ------------- --------- -- The triplets in the FEC dataset were generated by sampling images from a partially-labeled [^1] internal face dataset in which each face image has one or more of the following emotion labels [@emotions-pnas; @emotions-cognitive]: *Amusement, Anger, Awe, Boredom, Concentration, Confusion, Contemplation, Contempt, Contentment, Desire, Disappointment, Disgust, Distress, Doubt, Ecstasy, Elation, Embarrassment, Fear, Interest, Love, Neutral, Pain, Pride, Realization, Relief, Sadness, Shame, Surprise, Sympathy, and Triumph*. To reduce the effect of category-bias, we sampled the images such that all these categories are (roughly) equally represented in the triplet dataset. Each triplet was annotated by six human raters, and the raters were instructed to focus only on expressions ignoring other factors such as identity, gender, ethnicity, pose and age. A total of 40 raters participated in the process, each annotating a subset of the entire dataset. Based on the existing emotion labels, each triplet in this dataset can be categorized into one of the following types [^2]: - *One-class triplets*: All the three images share a category label, see Figure \[fig:triplets\](a). These triplets are useful for learning a fine-grained expression representation. - *Two-class triplets*: Only two images share a category label and the third image belongs to a different category, see Figure \[fig:triplets\](b). As explained in Section \[sec::motivation\], images sharing a category label need not form the (visually) most similar pair in these triplets. - *Three-class triplets*: None of the images share a common category label, see Figure \[fig:triplets\](c). These triplets are useful for learning long-range visual similarity relationships between different categories. One-class triplets are relatively the most difficult ones since the expressions could be very close to each other, and two-class triplets are relatively the easiest ones since the images sharing a label could potentially be different from the remaining image (though not always). While there are other possible types of triplets based on other label combinations (for example, $I_1$, $I_2$ sharing a label, and $I_2$, $I_3$ sharing another label), we prioritized the above three types while collecting the dataset as the other types could be confusing for the raters. Extending the dataset to include the other types will be considered in the future. Table \[tab:dataset\] shows the number of triplets in this dataset along with the number of faces used to generate the triplets. The dataset is further divided into training (90%) and test (10%) sets, and we encourage the users of this dataset to use the training set for training their algorithms and the test set to validate them.\ **Annotation agreement:** Each triplet in this dataset was annotated by six raters. For a triplet, we say that the raters *agree strongly* if at least two-thirds of them voted for the maximum-voted label, and *agree weakly* if there is a unique maximum-voted label and half of the raters voted for it. The number of such triplets for each type are shown in Table \[tab:dataset\]. Raters agree strongly for about 80% of the triplets suggesting that humans have a well-defined notion of visual expression similarity. Facial expression embedding network {#sec:network} =================================== In the recent past, the performance of face recognition systems has improved significantly [@MegaFaceLeaderboard; @MSCeleb1MLeaderboard; @MegaFace1; @MegaFace2] in part due to the availability of large-scale (several million data samples) training datasets such as MS-Celeb-1M [@MSCeleb1M], MegaFace [@MegaFace2], SFC [@DeepFace] and Google-Face [@FaceNet]. Neural networks trained on these large-scale datasets see images with significant variations along different dimensions such as lighting, pose, age, gender, ethnicity, etc. during training. Compared to these large-scale face datasets, our facial expression comparison dataset is significantly smaller (just 130K training faces). Hence, in order to leverage the power of a large training set, we build our facial expression embedding network using the pre-trained FaceNet proposed in [@FaceNet], see Figure \[fig:network\]. We use the NN2 version of pre-trained FaceNet [@FaceNet] up to the inception (4e) block [^3] whose output is a $7 \times 7$ feature map with 1024 channels. This feature map is processed by a DenseNet which consists of a $1\times 1$ convolution layer (512 filters) followed by a Dense block [@DenseNet] with 5 layers and growth rate of 64. The output of DenseNet is passed to a $7\times 7$ average pooling layer followed by a fully connected (FC) layer with 512 hidden units and an embedding layer (a linear FC layer + $\ell_2$ normalization layer). Batch normalization [@BatchNorm] and ReLu6 [@Relu6] activation function are used in the DenseNet and the first FC layer. We also use dropout for regularization. The input to our network is an aligned (rotated to undo roll transformation and scaled to maintain an inter-ocular distance of 55 pixels) $224\times 224$ face image $I$, and the output is a $d$-dimensional embedding $e_{I}$ of unit $\ell_2$ norm. Triplet loss function --------------------- For training the embedding network using the proposed FEC dataset, we use a triplet loss function that encourages the distance between the two images that form the most similar pair to be smaller than the distances of these two images from the third image. For a triplet $(I_1, I_2, I_3)$ with the most similar pair $(I_1, I_2)$, the loss function is given by $$\begin{aligned} l(I_1, I_2, I_3) &= max(0, \|e_{I_1} - e_{I_2} \|_2^2 - \|e_{I_1} - e_{I_3} \|_2^2 + \delta)\\ &+ max(0, \|e_{I_1} - e_{I_2} \|_2^2 - \|e_{I_2} - e_{I_3} \|_2^2 + \delta), \end{aligned}$$ where $\delta$ is a small margin. Experiments =========== In this section, we demonstrate the usefulness of the expression embedding learned from the proposed FEC dataset for various applications such as expression retrieval, photo album summarization, and emotion classification. In all our experiments, we only use the triplets with strong rater agreement for both training and evaluation. We also tried using the triplets with weak rater agreement for training, but the results did not improve (see Section \[sec::comparison\]). In the rest of the paper, we refer to the proposed expression embedding network trained on the proposed FEC dataset as *FECNet*. Comparative approaches ---------------------- Most of the existing large-scale expression datasets focus on the task of classification. One can train a classification network with such a dataset, and use the output of the final or penultimate layer as an expression embedding. Here, we train two networks: *AFFNet-CL* for emotion recognition using the AffectNet dataset [@AffectNet], and *FACSNet-CL* for facial action unit detection using the DISFA dataset [@Disfa]. AffectNet is a large-scale faces-in-the-wild dataset manually labeled with eight emotion categories. This dataset has around 288K training and 4K validation images. DISFA is a widely-used spontaneous facial actions dataset manually labeled with the presence/absence of 12 action units [^4]. This dataset has around 260K images, out of which 212K images are used for training and 48K images are used for validation. We create four expression embeddings using these two classification networks: - *AFFNet-CL-P* and *AFFNet-CL-F*: Penultimate and final layer outputs of AFFNet-CL. - *FACSNet-CL-P* and *FACSNet-CL-F*: Penultimate and final layer outputs of FACSNet-CL. Another way to learn an embedding using a classification dataset is to train an embedding network with a category label-based triplet loss similar to [@FaceNet]. So, we also train an embedding network (referred to as *AFFNet-TL*) on AffectNet dataset using triplet loss. For a fair comparison, the input and architecture of all the networks are chosen to be same as FECNet (Figure \[fig:network\]) except that the embedding layer is replaced by a softmax classifier for AFFNet-CL and separate binary classifiers for FACSNet-CL. Training and validation ----------------------- We define *triplet prediction accuracy* as the percentage of triplets for which the distance (in the embedding space) between the visually most similar pair is less than the distances of these two images from the third. As for the validation measure during training, we use triplet prediction accuracy on the FEC test set for FECNet, (following [@Fabnet]) average area under ROC curve (AUC-ROC) on the AffectNet validation set for AFFNet-CL and AFFNet-TL [^5], and (following [@FAUECCV18; @AUDandAlign]) average F1-score on the DISFA validation set for FACSNet-CL. For all the networks, the parameters of the FaceNet layers were kept fixed and the newly-added DenseNet and FC layers were trained starting from Xavier initialization [@Xavier] using Adam optimizer [@Adam] with a learning rate of $5\times10^{-4}$ and dropout of 0.5. FECNet was trained on the FEC training set with mini-batches of 90 samples (30 triplets from each of the triplet types) for 50K iterations, AFFNet-CL and AFFNet-TL were trained on the AffectNet training set with mini-batches of 128 samples (16 samples from each of the eight emotion categories) for 10K iterations, and FACSNet-CL was trained on the DISFA training set with mini-batches of 130 samples (at least 10 positive samples for each action unit and 10 samples with no action units) for 20K iterations. For training FECNet, the value of margin $\delta$ was set to 0.1 for one-class triplets, and 0.2 for two-class and three-class triplets. For training AFFNet-TL, the loss margin was set to 0.2 and the embedding dimensionality was set to 16. All the hyper-parameter values were chosen based on the corresponding validation measures. Average human performance ------------------------- To estimate how good humans are at identifying the most similar pair in a triplet, we computed the triplet prediction accuracy values for individual raters based on how often they agree with the maximum-voted label. Figure \[fig:human-accuracy\] shows the accuracy values for all the 30 raters who contributed to the FEC test set annotations. The mean and median values are 86.2% and 87.5%. Dimensionality of the FECNet embedding -------------------------------------- While we want to represent facial expressions in a continuous fashion using an embedding, it is unclear how many dimensions should be used for the embedding space. To answer this question, we trained FECNet for different values of the output dimensionality. Figure \[fig:dimensions\] shows how the triplet prediction accuracy on the FEC test set varies with the dimensionality of the embedding space. The accuracy increases till 16 dimensions and drops slightly after that. Based on these results, we choose 16-dimensions to represent the expression embedding space (referred to as FECNet-16d). Figure \[fig:dimensions\] also shows the median rater accuracy. Using 16 dimensions, the proposed FECNet is able to achieve an accuracy of 81.8%, which is fairly close to the median rater accuracy (87.5%). Note that the triplet prediction accuracy of random choice is 33.3%. -------------- ---------- ---------- -------- $\ell_1$ $\ell_2$ Cosine FACSNet-CL-F 47.1 47.1 40.7 FACSNet-CL-P 45.3 44.2 48.3 AFFNet-CL-F 49.0 47.7 49.0 AFFNet-CL-P 52.4 51.6 53.3 AFFNet-TL - 49.6 - FECNet-16d - 81.8 - -------------- ---------- ---------- -------- : Triplet prediction accuracy on the FEC test set.[]{data-label="tab:affnet-embeddings"} Comparison of different embeddings {#sec::comparison} ---------------------------------- Table \[tab:affnet-embeddings\] shows the triplet prediction accuracy of various embeddings on the FEC test set using different distance functions. Among all the AFFNet and FACSNet embeddings, the combination of AFFNet-CL-P and cosine distance gives the best accuracy, and hence, we use this combination for comparison with FECNet-16d in the rest of the experiments. It is worth noting that the proposed FECNet-16d (81.8%) performs significantly better than the best competing approach (AFFNet-CL-P + Cosine distance; 53.3%). We also trained FECNet-16d by adding the triplets with weak rater agreement to the training set, but the test accuracy dropped from 81.8% to 80.5%. Triplet type AFFNet-CL-P FECNet-16d Median rater -------------- ------------- ------------ -------------- One-class 49.2 77.1 85.3 Two-class 59.8 85.1 89.3 Three-class 50.4 82.6 87.2 All triplets 53.3 81.8 87.5 : Triplet prediction accuracy for different types of triplets in the FEC test set.[]{data-label="fig:triplet-type-performance"} Performance for different triplet types --------------------------------------- Table \[fig:triplet-type-performance\] shows the triplet prediction accuracy of median rater, FECNet-16d and AFFNet-CL-P for each triplet type in the FEC test set. As expected, the performance is best (85.1%) for two-class triplets, which are relatively the easiest ones, and is lowest (77.1%) for one-class triplets, which are relatively the most difficult ones. Visualization of the FECNet embedding space ------------------------------------------- ![2D visualization of the FECNet-16d embedding space using t-SNE [@tSNE].[]{data-label="fig:tSNE"}](images/tSNE.png) Figure \[fig:tSNE\] shows a 2D t-SNE [@tSNE] visualization of the learned FECNet-16d embedding space using the AffectNet validation set. The amount of overlap between two categories in this figure roughly tells us about the extent of visual similarity between them. For example, fear and surprise have a high overlap indicating that they could be confused easily, and both of them have a very low overlap with contempt indicating that they are visually very distinct from contempt. Also, the spread of a category in this figure tells us about the visual diversity within that category. For example, happiness category maps to three distinct regions indicating that there are three visually distinct modes within this category. See Figure \[fig:tsne\_faces\] for a visualization of the face images that fall into different regions in Figure \[fig:tSNE\]. Album BO CB DT GB HC JL JC KM LJ LS --------------------------- ----- ----- ----- ----- ------ ----- ----- ------ ----- ----- FECNet-16d vs AFFNet-CL-P 5-2 9-1 5-1 9-0 10-0 9-0 7-1 10-0 1-4 1-6 Applications ------------ ### Image retrieval We can perform expression-based image retrieval by using nearest neighbor search in the expression embedding space. To compare the retrieval performance of FECNet-16d and AFFNet-CL-P embeddings, we use a query set consisting of 25 face images and a database (CelebA dataset [@CelebA]) consisting of 200K face images. For each query, we retrieved $N$ nearest neighbors ($N$ varied from 1 to 10) using both the embeddings and ranked the $2N$ retrieved images based on how close they are to the query as judged by ten human raters. Since ranking all $2N$ images at once is difficult for human raters, we asked them to rank two images at a time. In each pairwise ranking, the winner and looser get a score of +1 and -1, respectively. If it is a tie, i.e., the two images get equal number of rater votes, then both of them get a score of zero. We obtained such pairwise ranking scores for all pairs and converted them into a global ranking based on the overall scores. For numerical evaluation, we use *rank-difference* metric, defined as the average difference in the ranks of images retrieved by AFFNet-CL-P and FECNet-16d embeddings, respectively, divided by the number of retrieved images $N$. Positive value of this rank-difference metric indicates that FECNet-16d embedding is better than AFFNet-CL-P embedding. The lowest value for this metric is $-1$, corresponding to the case when all the AFFNet-CL-P retrieval results are ranked lower than all the FECNet-16d retrieval results, and the highest value is $+1$, corresponding to the case when all the FECNet-16d retrieval results are ranked lower than all the AFFNet-CL-P retrieval results. Figure \[fig:rank-difference\] shows the rank-difference metric for different values of $N$. Positive value of the metric for all values of $N$ clearly indicates that the proposed FECNet-16d embedding produces better matches compared to the AFFNet-CL-P embedding. Figure \[fig:retrieval\_results\] shows the top-5 retrieved images for some of the queries. The overall results of the proposed FECNet-16d embedding are clearly better than the results of AFFNet-CL-P embedding. Specifically, the FECNet-16d embedding pays attention to finer details such as teeth-not-visible (first query), eyes-closed (second and third queries) and looking straight (fourth query). See Figures \[fig:retrieval-first\] to \[fig:retrieval-last\] for additional retrieval results. ### Photo album summarization In this task, we are interested in summarizing the diverse expression content present in a given photo album using a fixed number of images. Expression embedding can be used for this task by combining it with a clustering algorithm. For evaluation, we created ten photo albums (100-200 images in each album) by downloading images of ten celebrities using Google image search. For each album, we ran hierarchical agglomerative clustering [^6] (10 clusters) with FECNet-16d and AFFNet-CL-P embeddings, and used the images that are closest to cluster centers for generating the summaries. We showed these two summaries to ten human raters and asked them which one is better. The raters were also allowed to choose the *difficult-to-decide* option. Table \[tab:clustering\_results\] shows the number of votes received by both the embeddings for all the albums. Humans prefer the summaries generated by the proposed FECNet-16d embedding for eight out of ten albums. Figures \[fig:summary1\] and \[fig:summary2\] show the summary results for all the albums. We can see that the expression content is more diverse in the summaries produced by the FECNet-16d embedding for most of the albums. Approach Neutral Happiness Sadness Surprise Fear Disgust Anger Contempt Average ------------------------------- ---------- ----------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- AFFNet-CL 84.6 **96.5** **90.7** **88.5** **90.2** **85.2** **88.3** **85.0** **88.6** AFFNet-TL **85.9** 96.0 89.2 **88.5** 89.6 83.7 87.9 82.6 87.9 FECNet-16d + K-NN 83.3 94.9 78.0 83.0 84.5 79.3 78.7 81.2 82.9 AlexNet [@AffectNet] - - - - - - - - 82.0 VGG-Face descriptor [@Fabnet] 75.9 92.2 80.5 81.4 82.3 81.4 81.2 77.1 81.5 FAb-Net [@Fabnet] 72.3 90.4 70.9 78.6 77.8 72.5 76.4 72.2 76.4 {width="\textwidth" height="50.00000%"} \[fig:retrieval\_results\] ### Emotion classification The proposed FECNet-16d embedding can be used for emotion classification by combining it with K-Nearest Neighbor (K-NN) classifier. Figure \[fig:classification\] shows the average AUC-ROC of the FECNet-16d embedding on the AffectNet validation set as a function of the number of neighbors used. The performance increases up to 200 neighbors and then remains stable. Table \[tab:affnet-classification\] compares the classification performance of the FECNet-16d embedding (using 200 neighbors) with other approaches. Note that AFFNet-CL and AFFNet-TL have the same architecture as FECNet-16d and are specifically trained for classification using AffectNet training data. Hence, as expected, they perform a bit better than FECNet-16d. However, despite not being trained for classification, the FECNet-16d embedding outperforms AlexNet and VGG-Face based classifiers, demonstrating that it is well-suited for classification. Conclusions and Future Work =========================== In this work, we presented the first large-scale facial expression comparison dataset annotated by human raters, and learned a compact (16-dimensional) facial expression embedding using this dataset with triplet loss. The embedding learned using this dataset performs better than various other embeddings learned using existing emotion and action units datasets. We experimentally demonstrated the usefulness of the proposed embedding for various applications such as expression retrieval, photo album summarization, and emotion classification. Another interesting application of the FECNet embedding is hard-negative mining for expression classification. Since FECNet is trained using human visual preferences, negative samples that are close to the positive samples in the FECNet embedding space can be considered as hard negatives while training a classification model. We plan to explore this further in our future work. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Gautam Prasad, Ting Liu, Brendan Jou, Alan Cowen, Florian Schroff and Hartwig Adam from Google for their support and suggestions during the data collection process.  \[fig:tsne\_faces\] {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} \[fig:summary1\] {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} {width="8cm" height="3.5cm"} \[fig:summary2\] {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} \[fig:retrieval-first\] {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} {width="85.00000%" height="85.00000%"} \[fig:retrieval-last\] [^1]: The images in this dataset are not exhaustively labeled, i.e., an image may not have all the labels that are applicable to it. [^2]: The images in the dataset (from which we sampled the faces) were not exhaustively labeled, and hence, a triplet classified as a two/three-class triplet based on the existing labels may not be be a two/three-class triplet if the images had been exhaustively labeled. [^3]: We also experimented with features from inception 4d, 5a and 5b blocks, and features from 4e block performed the best. [^4]: The frames with action unit intensities greater than 2 are treated as positives and the remaining are treated as negatives. [^5]: Nearest neighbor classifier with 800 neighbors is used. [^6]: Cosine distance and maximum linkage were used.

--- author: - 'F. M. Brunbauer,' - 'C. Chatterjee,' - 'G. Cicala,' - 'A. Cicuttin,' - 'P. Ciliberti,' - 'M. L. Crespo,' - 'D. D‘Ago,' - 'S. Dalla Torre,' - 'S. Dasgupta,' - 'M. Gregori,' - 'T. Ligonzo,' - 'S. Levorato,' - 'M. Lisowska,' - 'G. Menon,' - 'F. Tessarotto,' - 'L. Ropelewski,' - 'Triloki,' - 'A. Valentini,' - 'L. Velardi,' - 'Y. X. Zhao' title: 'Nanodiamond photocathodes for MPGD-based single photon detectors at future EIC' --- Introduction {#sec:intro} ============ The future Electron Ion Collider (EIC) [@EIC] is the facility dedicated to understanding quantum chromodynamics (QCD), including the elusive non-perturbative effects and the answer to key questions, pending since long. Among them: the origin of nucleon mass and spin and the properties of dense gluon systems. The experimental activity at EIC requires efficient hadron Particle IDentification (PID) in a wide momentum range, including the challenging scope of hadron PID at high momenta, namely larger than $6$-$8~GeV/c$. A gaseous Ring Imaging CHerenkov (RICH) is the only possible choice for this specific task. The number of Cherenkov photons generated in a light radiator is limited. In spectrometer setups, these number of photons is recovered by using long radiators. The compact design of the experimental setup at the EIC collider imposes limitations on the radiator length, requiring a dedicated strategy. In the far UltraViolet (UV) spectral region ($\sim120$ nm), the number of generated Cherenkov photon increases, according to the Frank-Tamm distribution [@Frank:1937fk]. This suggests the detection of photons in the very far UV range. The standard fused-silica windows are opaque for wavelengths below 165 nm. Therefore, a windowless RICH is a potential option. The approach also points to the use of gaseous photon detectors operated with the radiator gas itself [@windowless-RICH]. The MicroPattern Gaseous Detector (MPGD)-based Photon Detectors (PD) have recently been demonstrated as effective devices [@PM18] for the detection of single photon in Cherenkov imaging counters. These PDs are composed of a hybrid structure, where two layers of THick GEM (THGEM) multipliers [@thgem] are followed by a MICRO-MEsh GAseous Structure (MICROMEGAS) [@mm] stage; the top layer of the first THGEM is coated with a reflective CsI PhotoCathode (PC). CsI PC is, so far, the only feasible option for gaseous detectors thanks to its relatively high work function that makes it more robust than other ones commonly used in vacuum-based detectors. CsI has high Quantum Efficiency (QE) in the far UV spectral region. In spite of its successful application [@RD26], it presents problematic aspects. It is hygroscopic: the absorbed water vapour splits the CsI molecule, causing a degradation in QE [@NIMA_695_2012_279]. Therefore, the handling of CsI PC is a very delicate operation. QE degradation also appears after intense ion bombardment, when the integrated charge is $1~mC/cm^{2}$ [@NIMA_574_2007_28] or larger. In gaseous detectors, ion bombardment of the cathode is by the ion avalanche produced in the multiplication process. The fraction of ions reaching the cathode depends on the detector architecture. In recent years, MPGD schemes with enhanced ion blocking capability have been developed [@PM18; @ibf-blocking]. The search for an alternative UV sensitive photocathode overcoming these limitations is therefore an important goal for the research and development (R&D) program for the experiments at the EIC. In the present article, we present the preliminary results on NanoDiamond (ND) and Hydrogenated-NanoDiamond (H-ND) coated THGEM detectors. They also include preliminary results about QE robustness with respect to ion bombardment. Nanodiamond based PC as an alternative of CsI PC ================================================ The high QE value of CsI photocathode makes it the mostly used photoconverter for the UV sensitive devices. This high QE value is related to its low electron affinity ($0.1~eV$) and wide band gap ($6.2~eV$)  [@JAP_77_1995_2138]. The ND particles have a comparable band gap of $5.5~eV$ and low electron affinity of $0.35$-$0.50~eV$. H-NDs exhibit chemical inertness and radiation hardness. ND hydrogenation lowers the electron affinity to -$1.27~eV$. The Negative Electron Affinity (NEA) allows an efficient escape into vacuum of the generated photoelectrons without an energy barrier at the surface [@NDRep-1]. A novel ND hydrogenation procedure, developed in Bari [@NDRep-1; @NDreport], provides high and stable QE. A comparison of CsI and ND QE can be extracted from literature  [@NDRep-1; @NIMA_502_2003_76]. The R&D activity ================ THGEM characterization ---------------------- The initial phase of our R&D studies consisted in coating five THGEMs with ND and H-ND powder. THGEMs are robust gaseous electron multipliers based on GEM principle scaling the geometrical parameters. It is obtained via standard PCB drilling and etching processes. The 35 $\mu$m copper layer is coated with $\approx$5 $\mu$m of Ni, followed by 200 nm Au. The THGEMs used for our studies have an active area of $30\times30~mm^{2}$ with a hole diameter of 0.4 mm, a pitch of 0.8 mm and a thickness of 470 $\mu$m. THGEMs with different rim i.e. the clearance ring around the hole edge have been used: $\le5\mu$m (no rim), $\sim10~\mu$m and $\sim20~\mu$m. Each THGEM is characterized in the setup schematized in figure  \[fig:Schematic\_of\_Detector\_setup\]. A plane of drift wires above it and a segmented readout anode plane, both properly biased, provide the drift and induction field respectively. The detector is operated with various gas mixtures, all including Ar. The electrons from $^{55}Fe$ converted by Ar are collected and multiplied in the hole region of the THGEM. The electron avalanche generated in the multiplication process, while drifting towards the anode, induces the detected signal. ![Top-left panel: The schematic of our detector assembly. Bottom-left panel: The detector is illuminated with an ${}^{55}Fe$ X-ray source . Top-right panel: A typical ${}^{55}Fe$ X-ray spectrum obtained in $Ar-CO_{2}~(70\%-30\%)$ gas mixture when the applied voltages at drift, top and bottom of THGEM are -2520 V, -1720 V and -500 V respectively, while the anode is at ground. Bottom-right panel: shows the gain dependence of the THGEM versus the applied voltage.[]{data-label="fig:Schematic_of_Detector_setup"}](Schematic_of_Detector_setup.jpg){width="\textwidth"} All THGEMs used for our studies have been characterized using different gas mixtures at INFN Trieste before applying coating procedures: the goal is to perform comparative studies after coating them with UV sensitive films. A typical ${}^{55}Fe$ X-ray spectrum obtained in $Ar-CO_{2}~(70\%-30\%)$ gas mixture is shown in figure \[fig:Schematic\_of\_Detector\_setup\], top-right panel. The bottom right panel of figure \[fig:Schematic\_of\_Detector\_setup\] shows the gain dependence of THGEM versus the voltages applied between the two faces. Coating procedure ----------------- ![ (A) Au\_PCB of 1 inch diameter substrate used for the QE measurement. (B) Uncoated THGEM of active area 30 mm$\times$30 mm. (C) Half uncoated and half coated THGEM, mounted into the test chamber and zoomed view of the both coated (D) and uncoated (E) part. (F) test chamber with readout pad where the THGEMs are tested. (G) The test chamber after installation of a THGEM, illuminated by an ${}^{55}Fe$ X-ray source.[]{data-label="fig:Detector_Images"}](Detector_Images.png){width="\textwidth"} THGEMs have been coated with raw ND, namely as-received powder or with H-ND. ND powder with an average grain size of 250 nm produced by Diamonds & Tools srl has been employed. The coating is covering either the whole surface of one of the THGEM faces, or half of it. The standard procedure of hydrogenation of ND powder photocathodes is performed by using the MicroWave Plasma Enhanced Chemical Vapor Deposition (MWPECVD) technique. For the treatment in microwave (mw) H2 plasma, 30 mg of ND powder was placed in a tungsten boat (overall length 32 mm, trough 12 mm long × 5 mm wide × 1 mm deep, Agar Scientific Ltd) positioned on a heatable substrate holder of an ASTeX-type reactor evacuated to a base pressure better than $7\times{10}^{-7}$ mbar. The powder was heated to $650^{0}$C using an external radiative heater (via a Proportional-Integral-Derivative feedback control system), then H2 gas was flowed in the chamber at 200 sccm, the pressure and the mw power were maintained at 50 mbar and 1250 W, respectively. The heating due to the mw power increases further the temperature of the powders up to $1138^{0}$C as determined by a dual wavelength ($\lambda$1 = 2.1 $\mu m$ and $\lambda$2 =2.4 $\mu m$) infrared pyrometer (Williamson Pro 9240). After 1 h of H2 plasma exposure, the hydrogenated powder were cooled to room temperature under high vacuum.This procedure can not be used for THGEMs which are made by fiberglass, which does not tolerate temperatures above $180^{0}$C. This limitation is overcome by the novel and low-cost technique developed at INFN Bari  [@coating-I; @coating-II]. The H-ND is obtained by treating the as-received powder in $H_{2}$ microwave plasma for one hour before deposition. The ND and H-ND powder were separately dispersed in deionized water and sonicated for 30 minutes by a Bandelin Sonoplus HD2070 system. Then, the emulsion was sprayed on the THGEM at $120 ^{0}$C or slightly higher temperature. Four THGEMs with different geometrical characteristics have been coated as listed below: -  0 $\mu m$ rim - ND half coated -  0 $\mu m$ rim - H-ND half coated - 10 $\mu m$ rim - H-ND full coated - 20 $\mu m$ rim - ND half coated A fifth THGEM with 10 $\mu m$ rim was coated with a reflective CsI film by thermal evaporation technique at INFN Bari. Images of the coated substrates and the setup for the characterization are provided in Fig. [\[fig:Detector\_Images\]]{}. ![The gain of the THGEM with 10  $\mu$m rim measured before and after CsI coating are compared. Left panel: Gain versus applied voltage across the THGEM electrodes. Right panel: gain evolution versus time.[]{data-label="fig:CsI_Comp"}](EG_CsI_Comp.png "fig:"){width=".56\linewidth"}![The gain of the THGEM with 10  $\mu$m rim measured before and after CsI coating are compared. Left panel: Gain versus applied voltage across the THGEM electrodes. Right panel: gain evolution versus time.[]{data-label="fig:CsI_Comp"}](10um_id2_CsI.png "fig:"){width=".44\linewidth"} Post-coating THGEM characterization ----------------------------------- The characterization of THGEMs coated with CsI, ND and H-ND provides interesting indications. The gain in the coated part tends to be larger than the gain for the uncoated part in all the three cases. However, the increase for the coated part depends on the coating materials as well as on rim size. The THGEM with $\sim10~\mu$m rim size, coated with a reflective CsI showed a 20% gain increment in comparison to the uncoated ones as shown in Fig.  \[fig:CsI\_Comp\]. A tentative explanation of the observed gain increase is the lower rate of charging-up of the free dielectric surface. The surface resistivity is decreased when the coating is present due to the resistivity of the coating film. The left panel of Fig.  \[fig:X\_Ray\_Spectra\], shows the amplitude distribution for both uncoated and ND coated parts of the THGEM with $\sim20~\mu$m rim size. The voltages applied to drift, top and bottom of the THGEM electrodes are 3510 V, 2110 V and 750 V respectively. The gain of the ND coated part is $\approx $ 2 times higher compared to the one of the uncoated part. ![ Left panel: ${}^{55}Fe$ X-ray spectra obtained with a 20 $\mu$m rim THGEM half-coated with ND powder in $Ar-CO_{2} (70\%-30\%)$ gas mixture. The voltages applied to drift, top and bottom of the THGEM electrodes are 3510 V, 2110 V and 750 V respectively, while the anode is kept at ground. Right panel: ${}^{55}Fe$ X-ray spectra obtained with a 0 $\mu$m rim uncoated THGEM. The same measurement after coating the same THGEM with a H-ND emulsion prepared 17 months earlier in a $Ar-CH_{4}~(50\%-50\%)$ gas mixture[]{data-label="fig:X_Ray_Spectra"}](THGEM_ND_Comp_new.pdf "fig:"){width=".485\linewidth"} ![ Left panel: ${}^{55}Fe$ X-ray spectra obtained with a 20 $\mu$m rim THGEM half-coated with ND powder in $Ar-CO_{2} (70\%-30\%)$ gas mixture. The voltages applied to drift, top and bottom of the THGEM electrodes are 3510 V, 2110 V and 750 V respectively, while the anode is kept at ground. Right panel: ${}^{55}Fe$ X-ray spectra obtained with a 0 $\mu$m rim uncoated THGEM. The same measurement after coating the same THGEM with a H-ND emulsion prepared 17 months earlier in a $Ar-CH_{4}~(50\%-50\%)$ gas mixture[]{data-label="fig:X_Ray_Spectra"}](BeforeAfterCoating_new.pdf "fig:"){width=".48\linewidth"} In case of a ND coated THGEM with no rim the gain of the coated part is larger by a factor of $\sim$1.4 as shown in Fig.\[fig:Nor\_Gain\_LR\] left panel. The gain is maximum when the X-ray source starts illuminating and it decreases gradually by ${\sim}$ 20% in about 500 minutes. This effect is observed both for the ND coated and uncoated THGEM parts as illustrated in Fig.\[fig:Nor\_Gain\_LR\], right panel. The tentative explanation of the gain increase is the same one already proposed for the gain increase observed with CsI coating. The increase is higher when the open dielectric surface is larger, namely for the $\sim20~\mu$m rim THGEM. ![Evolution versus time of the effective gain behavior of a THGEM with 0 $\mu$m rim, half-coated with ND. Gain versus time (left panel); the source is moved to the coated region at 700 minutes. The same data normalized to the maximum gain measured in the coating region versus time, where t=0 is when the illumination of a region starts (right panel).[]{data-label="fig:Nor_Gain_LR"}](Gain_Evaluation_Uncoated_ND_coated_THGEM_Corrected.pdf "fig:"){width=".48\linewidth"}![Evolution versus time of the effective gain behavior of a THGEM with 0 $\mu$m rim, half-coated with ND. Gain versus time (left panel); the source is moved to the coated region at 700 minutes. The same data normalized to the maximum gain measured in the coating region versus time, where t=0 is when the illumination of a region starts (right panel).[]{data-label="fig:Nor_Gain_LR"}](Nor_Gain_Evaluation_Corrected.pdf "fig:"){width=".48\linewidth"} The H-ND coated THGEMs with 0 $\mu m$ and 10 $\mu m$ rim show a lower electrical stability as compared to the uncoated THGEMs and cannot be operated at nominal voltage. In order to study this unexpected behaviour a second exercise was performed using a new THGEM with no rim fully coated with H-ND and $Ar-CH_{4}~(50\%-50\%)$ gas mixture. Immediately after the coating the THGEM could not be operated at the nominal voltage. A heat treatment in an electric oven at ${120}~^{0}C$ for 24 hours allowed to perform the characterization. The right panel of figure  \[fig:X\_Ray\_Spectra\] shows the signal amplitude distributions measured before and after the H-ND coating followed by the heat treatment. No evidence for increase of gain is observed. This observation suggests that the electrical instability present before the heat treatment can be related to water molecules present at the H-ND surface. A consistent picture is emerging in spite of the initially unexpected results obtained characterising the coated THGEMs. The consolidation of this picture requires further investigation. Ageing test of H-ND PC ---------------------- QE measurements can be performed with the available setup only in case of small-size samples. Therefore, the same coating procedure used for the THGEM samples has been applied to disk substrates, one inch diameter. The disk material and surface preparation are the same of the THGEMs, even if no hole structure is present (Fig. \[fig:Detector\_Images\] A). The H-ND coating is with an emulsion of ND powder hydrogenated 17 months earlier. ![Quantum efficiency as a function of wavelength for fresh and various charge accumulations ($0.263mC{/}cm^{2}, ~2.895mC{/}cm^{2}, ~5.527mC{/}cm^{2} and ~8.159mC{/}cm^{2}$) due to ion bombardment on H-ND coated Au\_PCB substrate.[]{data-label="fig:Ageing_New"}](X_ray_Ageing_Corrected.pdf){width="\textwidth"} The H-ND coated PCB disc was irradiated with a mini X-ray source in $Ar-CO_{2} (70\%-30\%)$ gas mixture. For the QE measurement a McPherson VUV monochromator (model 234/302) was employed. The RD-51 ASSET system [@asset] at CERN has been used. In this setup both the QE measurement and ion bombardment can be performed in the same system without exposure to air. Ions are generated by a gaseous multiplier metallic grid set at 5 mm from the disc surface and they impinge on the sample surface. The QE of an H-ND coated disc was measured before irradiation in a wavelength range from 130 nm to 180 nm with a scan step of 5 nm. The sample was then moved to the X-ray irradiation chamber using an automated manipulator under vacuum ($\approx {1\times{10}^{-7} mbar}$). The QE has been measured again after controlled doses of accumulated charge and the measurements are reported in Fig. \[fig:Ageing\_New\]. The QE before irradiation and after an accumulated charge of $0.263~ mC{/}cm^{2}$ are similar in whole wavelength range. This strongly supports the hypothesis that H-ND photocathode are more robust than CsI once respect to ion bombardment. In fact, in case of a CsI photocathode, a 25% drop in QE is observed for an irradiation of $1.0~ mC{/}cm^{2}$ [@NIMA_574_2007_28]. For the H-ND coated sample, a decrease in the QE of 42$\%$ and 74$\%$ was observed as charge accumulation reached the values of $2.895~ mC{/}cm^{2}$ and $5.527 ~mC{/}cm^{2}$, respectively. Interestingly, we did not observe any further degradation in the QE for an accumulated charge of $8.159~ mC{/}cm^{2}$. This is the first preliminary irradiation ageing study of H-ND photocathodes ever performed. Conclusion ========== THGEM samples coated with different types of photosensitive layers (CsI, ND and H-ND) have been studied. An increase in the gain response for the CsI and ND coated THGEMs was observed compared to the uncoated ones. The electrical instability of the H-ND coated THGEM, initially observed, is overcome by a heat treatment. No gain enhancement is observed. The X-ray irradiation study on H-ND photocathodes performed by us for the first time indicates that H-ND is more robust than CsI. We can conclude that H-ND photocathode material is a promising alternative to CsI based photocathodes for all applications in the far VUV domain requiring high robustness. Acknowledgment ============== This R&D activity is partially supported by - EU Horizon 2020 research and innovation programme, STRONG-2020 project, under grant agreement No 824093; - the Program Detector Generic R&D for an Electron Ion Collider by Brookhaven National Laboratory, in association with Jefferson Lab and the DOE Office of Nuclear Physics. [99]{} A. Accardi et al., *Electron-Ion Collider: The next QCD frontier*, Eur. Phys. J. A **52** (2016) 268. I. Frank and I. Tamm, *“Coherent visible radiation of fast electrons passing through matter,”* Compt. Rend. Acad. Sci. URSS **14**, **no.3**, (1937) 109-114. M. Blatnik et al., *Performance of a Quintuple-GEM Based RICH Detector Prototype*, IEEE NS **62** (2015) 3256. J. Agarwala et al., *The MPGD-based photon detectors for the upgrade of COMPASS RICH-1 and beyond*, Nucl. Instr. and Meth. A **936** (2019) 416. L. Periale et al., *Detection of the primary scintillation light from dense Ar, and Xe with novel photosensitive gaseous detectors*, Nucl. Instr. and Meth. A **478** (2002) 377; P. Jeanneret, *Time Projection Chambers and detection of neutrinos, PhD thesis*, Neuchatel University, 2001; P.S. Barbeau et al, *Toward coherent neutrino detection using low-background micropattern gas detectors* IEEE **NS-50** (2003) 1285; R. Chechik et al, *Thick GEM-like hole multipliers: properties and possible applications*, Nucl. Instr. and Meth. A **535** (2004) 303. Y. Giomataris et al., *MICROMEGAS: a high-granularity positionsensitive gaseous detector for high particle-flux environments*, Nucl. Instr. and Meth. A **376** (1996) 29. The RD26 Collaboration, *RD26 status reports: CERN/DRDC* **93-36** (1993), CERN/DRDC **94-49** (194), CERN/DRDC (1996). Triloki n, B.Dutta,B.K.Singh, *Influence of humidity on the photoemission properties and surface morphology of cesium iodide photocathode*, Nucl. Instr. and Meth. A, **695** (2012) 279. H.Hoedlmoser et al., *Long term performance and ageing of CsI photocathodes for the ALICE/HMPID detector*, Nucl. Instr. and Meth. A, **574** (2007) 28. A. Bondar et al., *Study of ion feedback in multi-GEM structures*, Nucl. Instr. and Meth. A **496** (2003) 325; A. Breskin et al., *Sealed GEM photomultiplier with a CsI photocathode: ion feedback and ageing*, Nucl. Instr. and Meth. A **478** (2002) 225; J.F.C.A. Veloso et al., *A proposed new microstructure for gas radiation detectors: The microhole and strip plate*, Rev. Sc. Instr. **71** (2000) 2371; A.V. Lyashenko et al., *Further progress in ion back-flow reduction with patterned gaseous hole - multipliers*,JINST **2** (2007) P08004;A.V. Lyashenko et al., *Efficient ion blocking in gaseous detectors and its application to gas-avalanche photomultipliers sensitive in the visible-light range*, Nucl. Instr. and Meth. A **598** (2009) 116;M. Alexeev et al., *Ion backflow in thick GEM-based detectors of single photons*, JINST **8** (2013) P01021. M. Bari et al., *RHIP, a Radio-controlled High-Voltage Insulated Picoammeter and its usage in studying ion backflow in MPGD-based photon detectors*; PoS **MPGD2017** 068; M.M. Aggarwal et al, (ALICE TPC Collaboration), *Particle identification studies with a full-size 4-GEM prototype for the ALICE TPC upgrade*, Nucl. Instr. and Meth. A **903** (2018) 215. A.Buzulutskov, A.Breskin, R.Chechik, *Field enhancement of the photoelectric and secondary electron emission from CsI* ,Journal of Applied Physics **77** (1995) 2138. L. Velardi, A. Valentini, G. Cicala, *Highly efficient and stable UV photocathode based on nanodiamond particles*, Appl. Phys. Lett. **108** (2016) 083503-1-5. A. Valentini, D. Melisi, G. De Pascali, G. Cicala, L. Velardi, A. Massaro, *High-efficiency nanodiamond-based ultraviolet photocathodes*, 30-03-2017 Patent n. **WO 2017/051318 A9**; International Patent n. **PCT/IB2016/055616** of September 21, 2016; National Patent Italia - n. 102015000053374 del 21 Settembre 2015, Istituto Nazionale di Fisica Nucleare e Consiglio Nazionale delle Ricerche. F. Piuz, *Ring imaging Cherenkov systems based on gaseous photo-detectors: trends and limits around particle accelerators*, Nucl. Instr. and Meth. A , **502** (2003) 76. G. Cicala, A. Massaro, L. Velardi, G. S. Senesi, A. Valentini, *Self-Assembled Pillar-Like Structures in Nanodiamond Layers by Pulsed Spray Technique*, ACS Appl. Mater. Interfaces **6** (2014) 21101. L. Velardi, A. Valentini, G. Cicala, *UV photocathodes based on nanodiamond particles: effect of carbon hybridization on the efficiency*, Diam. Relat. Mater. **76** (2017) 1-8. $https://indico.cern.ch/event/872501/contributions/3726017/attachments/1985809/3308869/Marta\_Lisowska\_-\_RD51\_Mini\_Week\_-\_Asset\_photocathode\_characterisation\_device.pdf$

--- abstract: 'We apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the Analytic Hierarchy Process in multi-criteria problems of rating alternatives from pairwise comparison data. In this framework, we first consider the minimax Chebyshev approximation of pairwise comparison matrices by consistent matrices in the logarithmic scale. Recasting this approximation problem as a problem of tropical pseudo-quadratic programming, we then write out a closed-form solution to it. This solution might be either a unique score vector (up to a positive factor) or a set of different score vectors. To handle the problem when the solution is not unique, we develop tropical optimization techniques of maximizing and minimizing the Hilbert seminorm to find those vectors from the solution set that are the most and least differentiating between the alternatives with the highest and lowest scores, and thus are well representative of the entire solution set.' author: - Nikolai Krivulin - 'Serge[ĭ]{} Sergeev' title: | Tropical implementation of\ the Analytical Hierarchy Process decision method --- tropical optimization ,max-algebra ,pairwise comparison ,log-Chebyshev approximation ,Analytical Hierarchy Process ,Hilbert distance 90B50 ,15A80 ,90C47 ,41A50 ,15B48 Introduction {#s:intro} ============ Tropical (idempotent) mathematics, which deals with the theory and applications of algebraic systems with idempotent operations [@Golan2003Semirings; @Heidergott2006Maxplus; @Mceneaney2006Maxplus; @Butkovic2010Maxlinear; @Maclagan2015Introduction], is widely used as a coherent analytical framework to solve problems in engineering, operations research and computer science. Tropical optimization presents an important research domain in this area, focused on optimization problems that are formulated and solved in the tropical mathematics setting. Methods and techniques of tropical optimization are applied to solve many well-known and new optimization problems in various fields, including decision making [@Elsner2004Maxalgebra; @Elsner2010Maxalgebra; @Gursoy2013Analytic; @Tran2013Pairwise; @Gavalec2015Decision]. The traditional Analytic Hierarchy Process (AHP) method [@Saaty1977Scaling; @Saaty1990Analytic; @Saaty2013Onthemeasurement] consists of two principal levels of pairwise comparisons: the upper level, where the relative importance of criteria is estimated, and the lower level, where the relative quality of choices is evaluated with respect to each criterion. The final decision is made by combining the rates of all choices computed on the lower level, and the weights of all criteria on the higher level. The rates of choices with respect to each criterion are found through the rank-one approximation of pairwise comparison matrices, typically by using the principal (Perron) eigenvector methods [@Saaty1977Scaling; @Saaty1990Analytic; @Saaty1984Comparison; @Saaty2013Onthemeasurement], and, sometimes, by other techniques, including the least squares or the logarithmic least squares methods [@Saaty1984Comparison; @Chu1998Ontheoptimal; @Saaty1984Comparison; @Barzilai1997Deriving; @Farkas2003Consistency]. The weights of criteria can be evaluated in the same manner from pairwise comparisons or obtained in a different way. More specifically, assume that there are $m$ criteria and $n$ choices. Given a pairwise comparison matrix $A_{k}=(a_{ij}^{(k)})$ of order $n$, and a weight $w_{k}$ for each criterion $k$, the vector of the priorities of all choices $x=(x_{i})$ is calculated as $$x = \sum_{k=1}^{m}w_{k}y_{k}, \label{e:AHP}$$ where $y_{k}$ is the vector of rates, obtained from $A_{k}$. Since the first appearance of the AHP decision approach, several new implementations of AHP have been developed using various mathematical techniques, including fuzzy AHP [@Laarhoven1983Fuzzy; @Kubler20163Sateoftheart], interval AHP [@Ahn2017Analytic], and some others. Specifically, in [@Gursoy2013Analytic] a variant of AHP based on tropical mathematics is proposed, using the matrix approximation in terms of the minimization of the maximum relative error [@Elsner2004Maxalgebra; @Elsner2010Maxalgebra]. Based on the observation of [@Elsner2010Maxalgebra] that this minimum is attained by any tropical subeigenvector, the paper [@Gursoy2013Analytic] suggests to seek a common subeigenvector of pairwise comparison matrices $A^{(k)}$ for all criteria $k$ (giving a number of conditions for existence of that common subeigenvector) or, alternatively, to seek a Pareto optimal solution. In this paper, we develop a new theoretical and computational framework for the implementation of AHP, based on the tropical optimization techniques proposed in [@Krivulin2015Rating; @Krivulin2016Using]. The new AHP method, which we offer and investigate below, aims to find a rank-one matrix that should be the closest, in the sense of the maximum of weighted log-Chebyshev distances, to all the pairwise comparison matrices corresponding to different criteria. As we argue in Section \[s:AHP-intro\] the vector of priorities $x=(x_{i})$ is a solution of the following optimization problem $$\begin{aligned} & \min_{x} && \max_{i,j=1}^{n}\max_{k=1}^{m}(w_{k}a_{ij}^{(k)})x_{j}/x_{i}. \end{aligned} \label{P-minxwkaijkxjxi}$$ The new approach is a modification of the direct weighted sum calculation in the basic AHP scheme . In this modification, the weights of criteria are incorporated into the objective function of the optimization problem solved at the lower level of AHP. We thus minimize a max-linear combination of lower level objective functions (representing the consistency of judgment with respect to various criteria), in which they are multiplied by their proper weights taken from the higher level of AHP. The solution obtained as a result of the tropical AHP method is, in general, non-unique. However, inconsistency and ambiguity of determining the pairwise preferences are inherent to the process of forming the pairwise comparison matrices, and therefore it seems quite natural that the method ends up with a set of solution vectors rather than with just one vector as in the ordinary AHP. To make the non-unique result tractable and useful for practice, we focus on two kinds of solutions that can be considered, in some sense, as the best and worst solutions. The solution set is characterized by vectors that are the most and least differentiating between the choices with the highest and lowest priorities. See Section \[s:AHP-intro\] and problem formulations , . Our next goal is then to describe the sets of most and least differentiating vectors in the framework of tropical mathematics, using some basic facts about the tropical linear algebra and the tropical pseudoquadratic optimization given in Section \[s:elements\]. Such description is obtained in Subsections \[ss:maximization\] and \[ss:minimization\], and its geometric sense is then illustrated in Subsection \[ss:geometry\]. Note that the solution set of is a tropical convex cone: a subset of nonnegative orthant, which is closed under multiplication by a scalar factor and componentwise maximum of two vectors. The problem of minimizing the Hilbert semidistance of a point from a tropical convex cone or, more generally, from an idempotent semimodule, was considered in [@Cohen2004Duality] and [@Akian2011Best], where it was shown to be in a close relation with the tropical Hahn-Banach theorem. Below we will consider a special case of this problem where a simple algebraic description of the whole solution set will be available: see Subsection \[ss:minimization\]. The problem of finding the maximum of Hilbert seminorm over the tropical column span of normal matrices[^1] – but without describing the whole solution set – was solved in [@Delapuente2013Ontropical]. A complete solution of this problem that does not assume the normality or any other property of the matrix will be given in Subsection \[ss:maximization\]. Examples of application of our tropical AHP framework are given in Section \[s:AHP\], where we consider two multi-criteria decision problems from [@Saaty1977Scaling]. This is followed by a discussion of the differences between our approach and the traditional implementation of AHP, and some possibilities for further research. The main contribution of this paper is a new tropical framework for implementation of AHP. This implementation is based on complete description of the most and least differentiating vectors. In mathematical terms, this description results from a complete closed-form solution of two optimization problems over the tropical column span of a nonnegative matrix. We refine these solutions and provide a new geometric interpretation, which consists in maximization and minimization of the Hilbert seminorm over the tropical column span of a nonnegative matrix. Let us mention that the new tropical implementation of AHP presented here has been outlined in our short conference paper [@Krivulin2017Tropicaloptimization]. That conference paper also presents an application of our new scheme to one of the examples described in Section 5 of the present paper. It also contains the formulation of some of the basic facts on which the new method is based, but without any proofs. Minimax approximation based AHP {#s:AHP-intro} =============================== In this section, we describe a new approach to develop an AHP decision scheme that is based on the rank-one log-Chebyshev approximation of pairwise comparison matrices. We also call it the tropical implementation of AHP since the tropical linearity is essential for closed-form description of solutions at each step. Log-Chebyshev approximation of pairwise comparison matrices ----------------------------------------------------------- Consider the problem of evaluating the rates of $n$ choices from pairwise comparisons of these choices. The outcome of these comparisons is described by a square symmetrically reciprocal matrix $A=(a_{ij})$, where $a_{ij}$ specifies the relative priority of choice $i$ over $j$, and satisfies the condition $a_{ij}=1/a_{ji}>0$ for all $i,j$. The pairwise comparison matrix $A$ is called consistent if its entries are transitive, that is, if they satisfy the equality $a_{ij}=a_{ik}a_{kj}$ for all $i,j,k$. For each consistent matrix $A$, there is a positive vector $x=(x_{i})$ whose elements completely determine the entries of $A$ by the relation $a_{ij}=x_{i}/x_{j}$, which, in particular, means that $A$ is a matrix of unit rank. Provided that the matrix $A$ is consistent, its corresponding vector $x$, which can be readily obtained from $A$, directly represents, up to a positive factor, the individual preferences of choices in question. Since the pairwise comparison matrices, encountered in practice, are generally inconsistent, the solution usually involves approximating these matrices by consistent matrices. The approximation with the principal (Perron) eigenvector as well as the least squares or the logarithmic least squares approximation are often used as solution approaches. Consider another approach [@Krivulin2015Rating; @Krivulin2016Using], which is based on the approximation of a pairwise comparison matrix $A=(a_{ij})$ by a consistent matrix $X=(x_{ij})$ in the log-Chebyshev sense, where the approximation error is measured with the Chebyshev metric on the logarithmic scale. Taking into account that both matrices $A$ and $X$ have positive entries, and that the logarithmic function (to the base more than one) is monotonically increasing, the error can be written as $$\max_{i,j=1}^{n}|\log a_{ij}-\log x_{ij}| = \log\max_{i,j=1}^{n}\max\{a_{ij}/x_{ij},x_{ij}/a_{ij}\}.$$ Observing that the minimization of the logarithm is equivalent to the minimization of its argument, and that $a_{ij}=1/a_{ji}$ and $x_{ij}=x_{i}/x_{j}$, we replace the last logarithm by $\max_{i,j=1}^{n}\max\{a_{ij}/x_{ij},x_{ij}/a_{ij}\}=\max_{i,j=1}^{n}a_{ij}x_{j}/x_{i}$. This reduces solving the approximation problem to solving, with respect to the unknown vector of priorities $x=(x_{i})$, the optimization problem $$\min_{x}\ (\max_{i,j=1}^{n}a_{ij}x_{j}/x_{i}). \label{P-minxaijxjxi}$$ Note that problem is equivalent to that arising in the approximation by minimizing the maximum relative error in [@Elsner2004Maxalgebra; @Elsner2010Maxalgebra]. Weighted approximation under several criteria --------------------------------------------- Suppose the priorities of choices are evaluated based on pairwise comparisons according to $m$ criteria, each having a given weight. For each criterion $k$, we denote the pairwise comparison matrix by $A_{k}=(a_{ij}^{(k)})$ and the positive weight by $w_{k}$. To determine the priority vector $x=(x_{i})$, we minimize the maximum of the functions $\max_{i,j=1}^{n}a_{ij}^{(k)}x_{j}/x_{i}$, taken with the weights $w_{k}$ for all $k$. That is, we pose the following problem: $$\min_{x}\ (\max_{k=1}^{m}w_{k}(\max_{i,j=1}^{n}a_{ij}^{(k)}x_{j}/x_{i})) = \min_{x}\ (\max_{i,j=1}^{n}\max_{k=1}^{m}(w_{k}a_{ij}^{(k)})x_{j}/x_{i}). \label{P-weighted-approx}$$ Introducing the matrix $B=(b_{ij})$ with the entries $$b_{ij} = \max_{k=1}^{m}w_{k}a_{ij}^{(k)}, \label{E-bij-maxwkaijk}$$ we can reduce to a problem in the form of , with $B$ instead of $A$. The solution of can be considered as a modification of the basic AHP scheme, in which the log-Chebyshev approximation is used instead of the principal eigenvector method, and the weights of criteria are incorporated into the lower-level evaluation of choices. Most and least differentiating priority vectors {#ss:mostandleast} ----------------------------------------------- In general, two priority vectors that solve problem or cannot necesarily be obtained from one another by means of multiplication by a positive factor. That is, solution of  or can be essentially non-unique, in general. Below, we develop an approach in which the entire solution is “represented” by two vectors, which can be considered, in some sense, as the best and worst solutions. Assume that problem or has a set $S$ of solutions $x=(x_{i})$ rather than a unique one (up to a scalar factor multiplication). Since the main purpose of evaluating priorities is to differentiate between choices, we find the solutions that are the most and least differentiating between the choices with the highest and lowest priorities. The calculation of the most and least differentiating vectors involves determining the exact bounds for the contrast ratio $$(\max_{i=1}^{n}x_{i})/(\min_{j=1}^{n}x_{j}) = (\max_{i=1}^{n}x_{i})(\max_{j=1}^{n}(1/x_{j}))$$ aiming to find the vectors $x$, which solve the problem of the Hilbert (span, range) seminorm maximization $$\max_{x\in S}\ (\max_{i=1}^{n}x_{i})(\max_{j=1}^{n}(1/x_{j})), \label{P-maxxmaxximax1xj}$$ and the problem of the Hilbert seminorm minimization $$\min_{x\in S}\ (\max_{i=1}^{n}x_{i})(\max_{j=1}^{n}(1/x_{j})). \label{P-minxmaxximax1xj}$$ The Hilbert seminorm of $x$ in the logarithmic scale is actually defined as $\log ((\max_{i=1}^{n}x_{i})(\max_{j=1}^{n}(1/x_{j})))$, but, for the sake of optimization, the logarithm can be omitted, since it is a monotone function. In subsequent sections, we will treat problems , and in terms of tropical mathematics, and give direct and explicit solutions, which are ready for immediate computation. Tropical linear algebra and tropical pseudo-quadratic programming {#s:elements} ================================================================= We start with a brief overview of basic definitions and notation of tropical (idempotent) algebra to provide a formal framework for describing tropical optimization techniques, used below in the development of tropical implementation of AHP. Further details on tropical mathematics can be found, e.g., in [@Golan2003Semirings; @Heidergott2006Maxplus; @Mceneaney2006Maxplus; @Butkovic2010Maxlinear; @Maclagan2015Introduction]. Tropical linear algebra {#ss:maxalg} ----------------------- We consider the set of non-negative reals $R_{+}$ equipped with two operations: addition $\oplus$, defined as $\max$, and multiplication $\otimes$, defined as the usual multiplication, with the neutral elements: zero $0$ and one $1$. Addition $\oplus$ is idempotent since $x\oplus x=\max(x,x)=x$ for each $x\in R_{+}$, multiplication distributes over addition $\oplus$ and is invertible, providing each $x>0$ with the inverse $x^{-1}$ such that $x\otimes x^{-1}=xx^{-1}=1$. The idempotent algebraic system $(R_{+},0,1,\oplus,\otimes)$ is commonly referred to as the tropical algebra or max-algebra, and denoted by $R_{\max}$. In the tropical algebra, both addition and multiplication are monotone in their arguments, which means that the inequality $x\leq y$ implies the inequalities $x\oplus z\leq y\oplus z$ and $x\otimes z\leq y\otimes z$ for any $x,y,z\in R_{+}$. Moreover, the inequality $x\oplus y\leq z$ is equivalent to the system of inequalities $x\leq z$ and $y\leq z$. The inversion is antitone in the sense that if $x\leq y$ for some $x,y>0$, then $x^{-1}\geq y^{-1}$. Since the multiplication $\otimes$ defined in $R_{\max}$ coincides with the standard arithmetic multiplication, the power notation $x^{p}$ has the usual interpretation for all $x>0$ and rational $p$. In what follows, the multiplication sign $\otimes$ is omitted for the sake of brevity. The scalar tropical algebra is routinely extended to the set of non-negative matrices over $R_{+}$ with the matrix operations defined by the conventional rules, where the scalar addition and multiplication are replaced by the operations $\oplus$ and $\otimes$. This is referred to as the tropical linear algebra. As usual, a matrix with all zero entries is called the zero matrix. The multiplicative conjugate transpose (or simply the conjugate transpose) of a nonzero $(m\times n)$-matrix $A=(a_{ij})$ is the $(n\times m)$-matrix $A^{-}=(a_{ij}^{-})$ with the entries $a_{ij}^{-}=a_{ji}^{-1}$ if $a_{ji}\ne0$, and $a_{ij}^{-}=0$ otherwise. Any matrix that consists of one column is a column vector. The column vector with all zero entries is the zero vector $0$. The column vector with all entries equal to $1$ is denoted by $1$. The conjugate transpose of a nonzero column vector $x=(x_{i})$ is the row vector $x^{-}=(x_{i}^{-})$, where $x_{i}^{-}=x_{i}^{-1}$ if $x_{i}\ne0$, and $x_{i}^{-}=0$ otherwise. The monotone properties of the scalar operations $\oplus$ and $\otimes$ are readily carried over to the matrix and vector operations, where the relations are understood entrywise. Specifically, for all positive matrices $A$ and $B$ such that $A\leq B$, the conjugate transposition satisfies $A^{-}\geq B^{-}$. An $m$-vector $b$ is linearly dependent on $m$-vectors $a_{1},\ldots,a_{n}$ if there are non-negative numbers $x_{1},\ldots,x_{n}$ such that $b=x_{1}a_{1}\oplus\cdots\oplus x_{n}a_{n}$. Specifically, a vector $b$ is collinear with a vector $a$, if $b=xa$ for some scalar $x$. A positive matrix $A$ is of rank $1$ if and only if $A=xy^{T}$, where $x$ and $y$ are positive column vectors. A matrix $A$ that satisfies the condition $A^{-}=A$ is called symmetrically reciprocal (or simply reciprocal). A reciprocal matrix $A$ is of rank $1$ if and only if $A=xx^{-}$, where $x$ is a positive column vector. For any square matrix $A$ and integer $p>0$, the tropical (or max-algebraic) power notation is routinely defined by the inductive rule $A^{p}=A^{p-1}A$, where $A^{0}=I$ is the usual identity matrix. The tropical (max-algebraic) spectral radius of an $(n\times n)$-matrix $A=(a_{ij})$ is computed as the maximum cycle geometric mean of the matrix entries, which is given by $$\lambda = \bigoplus_{1\leq k\leq n}\bigoplus_{1\leq i_{1},\ldots,i_{k}\leq n}(a_{i_{1}i_{2}}a_{i_{2}i_{3}}\cdots a_{i_{k}i_{1}})^{1/k} = \mathop\mathrm{tr}A \oplus\cdots\oplus \mathop\mathrm{tr}\nolimits^{1/n}(A^{n}). \label{E-lambda-ai1i2ai2i3aiki1}$$ We also use the function, which maps the matrix $A$ onto the scalar $$\mathop\mathrm{Tr}(A) = \bigoplus_{m=1}^{n}\mathop\mathrm{tr}A^{m} = \mathop\mathrm{tr}A \oplus\cdots\oplus \mathop\mathrm{tr}A^{n},$$ and note that if $\lambda>0$ then the inequality $\mathop\mathrm{Tr}(\lambda^{-1}A)\leq1$ holds. Provided that $\mathop\mathrm{Tr}(A)\leq1$, the asterate operator (the Kleene star) yields the matrix $$A^{\ast} = \bigoplus_{m=0}^{n-1}A^{m} = I\oplus A\oplus\cdots\oplus A^{n-1}.$$ Finally, we consider the problem to find positive vectors $x$ that solve the inequality $$Ax\leq x. \label{I-Axx}$$ The next result obtained in [@Krivulin2015Extremal] offers a complete solution to this inequality (see also [@Butkovic2010Maxlinear] and references therein). \[L-Axx\] For any square matrix $A$, the following statements hold: 1. If $\mathop\mathrm{Tr}(A)\leq1$, then all positive solutions to are given by $x=A^{\ast}u$, where $u$ is any positive vector. 2. If $\mathop\mathrm{Tr}(A)>1$, then there is no positive solution. Below, we use the algebraic preliminaries introduced above to describe tropical optimization problems and their solutions. Tropical pseudo-quadratic programming {#ss:pseudoquad} ------------------------------------- In this section, we consider optimization problems, which are formulated and solved in the tropical algebra setting, to provide the basis for our tropical implementation of AHP. First, assume that, given a non-negative $(n\times n)$-matrix $A$, we need to find positive $n$-vectors $x$ that solve the problem $$\begin{aligned} & \min_{x} && x^{-}Ax. \end{aligned} \label{P-minxxAx}$$ A complete, direct solution to the problem was obtained in [@Krivulin2015Extremal] (see also [@Butkovic2010Maxlinear] and references therein). \[L-minxxAx\] Let $A$ be a matrix with tropical spectral radius $\lambda>0$. Then, the optimal value in problem is equal to $\lambda$, and all positive solutions are given by $$x = (\lambda^{-1}A)^{\ast}u, \quad u>0.$$ We now suppose that $A_{1},\ldots,A_{m}$ are given non-negative $(n\times n)$-matrices, and $w_{1},\ldots,w_{m}$ are given positive numbers. The problem is to find positive $n$-vectors $x$ that attain the minimum in $$\begin{aligned} & \min_{x} && \bigoplus_{k=1}^{m}w_{k}x^{-}A_{k}x. \end{aligned} \label{P-minxwkxAkx}$$ As a direct consequence of the previous result, we have the following solution [@Krivulin2016Using]. \[C-minxwkxAkx\] Let $A_{1},\ldots,A_{m}$ be non-negative matrices and $w_{1},\ldots,w_{m}$ be positive numbers such that the matrix $B=w_{m}A_{m}\oplus\cdots\oplus w_{m}A_{m}$ has the tropical spectral radius $\mu>0$. Then, the minimum value in is equal to $\mu$, and all positive solutions are given by $$x = (\mu^{-1}B)^{\ast}u, \quad u>0.$$ Furthermore, given a matrix and two vectors, we examine two problems, which take the form of an unconstrained maximization and a constrained minimization problems. Let $A=(a_{j})$ be a non-negative $(m\times n)$-matrix with columns $a_{j}=(a_{ij})$, and $p=(p_{i})$ be an $m$-vector and $q=(q_{j})$ an $n$-vector. Consider the problem to find positive vectors $x=(x_{j})$ that attain the maximum $$\begin{aligned} & \max_{x} && q^{-}x(Ax)^{-}p. \end{aligned} \label{P-maxxqxAxp}$$ The next result obtained in [@Krivulin2016Maximization; @Krivulin2017Algebraic] offers a complete solution to problem under fairly general conditions. \[L-maxxqxAxp\] Let $A$ be a positive matrix, $p$ be a nonzero vector, $q$ be a positive vector, and $\Delta=q^{-}A^{-}p$. Let $A_{lk}$ be the matrix obtained from $A$ by keeping the entry $a_{lk}$ for some indices $l$ and $k$, and replacing the other entries by zero. Then, the optimal value in problem is equal to $\Delta$, and all positive solutions are given by the conditions $$x = (I\oplus A_{lk}^{-}A)u, \quad u>0,$$ for all indices $k$ and $l$ defined by the condition $$k = \arg\max_{j=1}^{m}q_{j}^{-1}a_{j}^{-}p, \qquad l = \arg\max_{i=1}^{n}a_{ik}^{-1}p_{i}.$$ Finally, suppose that we need to find positive solutions of the constrained minimization problem $$\begin{aligned} & \min_{x} && q^{-}xx^{-}p, \\ &&& Ax \leq x. \end{aligned} \label{P-minxqxxp-Axx}$$ A complete solution was obtained in [@Krivulin2017Tropical], and it can be described as follows. \[L-minxqxxp-Axx\] Let $A$ be a matrix such that $\mathop\mathrm{Tr}(A)\leq1$, $p$ be a nonzero vector, $q$ be a positive vector, and $\delta=q^{-}A^{\ast}p$. Then, the optimal value in problem is equal to $\delta$, and all regular solutions are given by $$x = (\delta^{-1}pq^{-}\oplus A)^{\ast}u, \quad u>0.$$ Maximizing and minimizing the Hilbert seminorm {#s:math} ============================================== In this section, we further simplify and refine the solutions of problems and to make them more appropriate for use in the tropical AHP below. We give a new more simple proof for the statement of Proposition \[L-maxxqxAxp\] for the maximization problem, and then obtain a compact closed-form solution of the special cases of maximization problem and minimization problem . The geometric sense of these problems and their solutions is then illustrated in Subsection \[ss:geometry\]. Maximization problem {#ss:maximization} -------------------- We start with the derivation of a new more compact solution to the maximization problem given by . First note that, without loss of generality, we can consider the vector $p$ to be positive. If $p$ has zero components, then these components can be eliminated together with the corresponding rows of the matrix $A$, which yields an equivalent problem in the form of with a positive vector $p$. The next statement establishes the maximum value of the objective function in problem and describes all vectors $x$ that yield this minimum. The derivation of the upper bound for the objective function is taken from [@Krivulin2016Maximization] and included in the proof below for the sake of completeness. \[T-maxxqxAxp\] Let $A$ be a positive matrix, $p$ be a nonzero vector, $q$ be a positive vector, and $\Delta=q^{-}A^{-}p$. Then, the optimal value in problem is equal to $\Delta$, and all positive solutions are given by $$\bigoplus_{j=1}^{n}a_{lj}x_{j} = a_{lk}x_{k} \label{E-aljxjalkxk}$$ for all indices $k$ and $l$ defined by the condition $$q_{k}^{-1}a_{lk}^{-1}p_{l} = \Delta. \label{E-qkalkplqAp}$$ We take the obvious inequality $xx^{-}\geq I$, which is valid for all positive vectors $x$. Multiplying it from the left by $A$, we obtain the inequality $Axx^{-}\geq A$. Since both sides of this inequality are matrices with positive entries, we further have $(Axx^{-})^{-}\leq A^{-}$ by conjugate transposing. The latter inequality is the same as $x(Ax)^{-}\leq A^{-}$, which we multiply by $q^{-}$ on the left and by $p$ on the right to obtain $q^{-}x(Ax)^{-}p\leq q^{-}A^{-}p=\Delta$. Thus $q^{-}A^{-}p$ is an upper bound for $q^{-}x(Ax)^{-}p$. Let us show that there exists $x$ such that $q^{-}x(Ax)^{-}p\geq\Delta=q^{-}A^{-}p$. We define indices $k$ and $l$, and take a vector $x$ according to the conditions $$q^{-}A^{-}p = \bigoplus_{i=1}^{n}\bigoplus_{j=1}^{m}q_{i}^{-1}a_{ji}^{-1}p_{j} = q_{k}^{-1}a_{lk}^{-1}p_{l}, \qquad \bigoplus_{j=1}^{n}a_{lj}x_{j} = a_{lk}x_{k}.$$ With this vector $x$, we obtain $$q^{-}x(Ax)^{-}p \geq q_{k}^{-1}x_{k}(a_{lk}x_{k})^{-1}p_{l} = q_{k}^{-1}a_{lk}^{-1}p_{l} = q^{-}A^{-}p.$$ With the opposite inequality, we have $q^{-}x(Ax)^{-}p=q^{-}A^{-}p$, which means that $q^{-}A^{-}p$ is a strict upper bound, and thus the maximum in problem . Let us now take an arbitrary solution $x$ of , and then verify that $x$ satisfies under condition . First note, that, for such $x$, we have $q^{-}x(Ax)^{-}p=q^{-}A^{-}p$. Let indices $s$ and $t$ be defined by the conditions $$q^{-}x=q_{s}^{-1}x_s, \qquad (Ax)^{-}p=(Ax)^{-1}_{t}p_{t}.$$ Then, we can write the following chain of equalities and inequalities: $$q^{-}x(Ax)^{-}p = q_{s}^{-1}x_{s}(Ax)^{-1}_{t}p_{t} \leq q_{s}^{-1}x_{s}a_{ts}^{-1}x_{s}^{-1}p_{t} = q_{s}^{-1}a_{ts}^{-1}p_{t} \leq q^{-}A^{-}p.$$ However, since $q^{-}A^{-}p=q^{-}x(Ax)^{-}p$, both inequalities in this chain turn into equalities. As a result, we have $$q_{s}^{-1}a_{ts}^{-1}p_{t} = q^{-}A^{-}p, \qquad (Ax)_{t} = a_{ts}x_{s},$$ which means that and are satisfied with $k=s$ and $l=t$ . The entrywise positivity of the matrix $A$ is important for the above proof, particularly in inverting the inequality $Axx^{-}\geq A$. Indeed, if we take $A=I$ then $Axx^{-}\geq A$ does not imply $(Axx^{-})^{-}\leq A^{-}$, where $A^{-}=A=I$. If we define $A^{-}$ as a matrix with $+\infty$ entries then $q^{-}A^{-}p$ may become $+\infty$: a trivial bound, which is never attained. We now examine a special case of problem that arises in the tropical implementation of AHP and as the problem of Hilbert seminorm maximization. We set $q^{-}=1^{T}A$ and $p=1$ in , and consider the problem $$\begin{aligned} & \max_{x} && 1^{T}Ax(Ax)^{-}1. \end{aligned} \label{P-maxx1AxAx1}$$ A complete solution to this problem is formulated as follows. \[C-maxx1AxAx1\] Let $A$ be a positive matrix, and $\Delta=1^{T}AA^{-}1$. Let $A_{lk}$ be the matrix obtained from $A$ by keeping the entry $a_{lk}$ for some indices $l$ and $k$, and replacing the other entries by zero. Then, the optimal value in problem is equal to $\Delta$, and all positive solutions are given by $$x = (I\oplus A_{lk}^{-}A)u, \qquad u>0,$$ for all indices $k$ and $l$ defined by the condition $$1^{T}a_{k}a_{lk}^{-1} = \Delta.$$ We apply Theorem \[T-maxxqxAxp\] with $q^{-}=1^{T}A$ and $p=1$ to represent the maximum in the problem as $q^{-}A^{-}p=1^{T}AA^{-}1$, and the left-hand side of the condition at as $q_{k}^{-1}a_{lk}^{-1}p_{l}=1^{T}a_{k}a_{lk}^{-1}$. Furthermore, we note that the equality does not include the vectors $p$ and $q$, and thus remains unchanged. To represent the set of solutions in a compact vector form, we multiply both sides of the equality by $a_{lk}^{-1}$. Furthermore, we introduce a positive $n$-vector of parameters $u=(u_{j})$ and rewrite this equality in a parametric form using the scalar equalities $$x_{k} = \bigoplus_{j=1}^{n}a_{lk}^{-1}a_{lj}u_{j}, \qquad x_{i} = u_{i}, \qquad i\ne k.$$ We denote by $A_{lk}$ the matrix obtained from $A$ by setting all entries other than $a_{lk}$ to zero. With this matrix, we represent the scalar equalities in the vector form $$x = (I\oplus A_{lk}^{-}A)u, \qquad u>0,$$ which completes the proof. Minimization problem {#ss:minimization} -------------------- We now consider a constrained minimization problem that we use in the tropical implementation of AHP, and solve it by reducing to problem . Suppose that, given a matrix $A$ with spectral radius $\lambda>0$, the problem is to find positive vectors $x$ that yield the minimum $$\begin{aligned} & \min_{x} && 1^{T}xx^{-}1, \\ &&& x = (\lambda^{-1}A)^{\ast}u, \quad u>0. \end{aligned} \label{P-minx1xx1-xlambdaAastu}$$ The next result offers a complete solution to the problem. \[C-minx1xx1-xlambdaAastu\] Let $A$ be a matrix with spectral radius $\lambda>0$, and $\delta=1^{T}(\lambda^{-1}A)^{\ast}1$. Then, the optimal value in problem is equal to $\delta$, and all positive solutions are given by $$x = (\delta^{-1}11^{T}\oplus\lambda^{-1}A)^{\ast}u, \qquad u>0.$$ We consider the equality $x=(\lambda^{-1}A)^{\ast}u$, and note that by Lemma \[L-Axx\], this equality means that $x$ is determined by the inequality $\lambda^{-1}Ax\leq x$. Observing that $\mathop\mathrm{Tr}(\lambda^{-1}A)\leq1$, we apply Proposition \[L-minxqxxp-Axx\], where $A$ is replaced by $\lambda^{-1}A$ and both $q$ and $p$ by $1$, and thus complete the proof. Geometric interpretation {#ss:geometry} ------------------------ Problems and consist in maximizing and minimizing $$\label{e:Hilb-ratio} 1^{T}xx^{-}1 = (\max_{i=1}^{n}x_{i})(\max_{j=1}^{n}(1/x_{j})),$$ where $x$ belongs to the set $\{Au\colon u\in R_{+}^{n}\}$, $A$ is an $n\times n$ nonnegative matrix (and, more specifically, a Kleene star). That set will be referred to as the tropical column span of $A$ and denoted by $\operatorname{span}(A)$. The logarithm of ratio is known as the Hilbert seminorm or range seminorm [@Butkovic2010Maxlinear], or as the Hilbert semidistance between $x$ and $1$ [@Cohen2004Duality]. Therefore, problems and consist in finding the maximum and minimum of the Hilbert seminorm of vectors in the tropical column span, $\operatorname{span}(A)$, or, in other words, finding the maximum and minimum of the Hilbert semidistance between $x$ and $1$. We now give two three-dimensional examples, which illustrate the geometry of the optimization problems under consideration. \[E-1AuAu1-left\] We start with the following matrix: $$A = A^{\ast} = \begin{pmatrix} 1 & 3/4 & 1/2 \\ 4/3 & 1 & 2/3 \\ 2/3 & 1/2 & 1 \end{pmatrix}.$$ The problem of minimizing the Hilbert seminorm over $\operatorname{span}(A)$ is posed as follows: $$\begin{aligned} & \min_{x} && 1^{T}xx^{-}1, \\ &&& x=Au, \quad u>0, \end{aligned}$$ and it is the same as . We solve this problem by applying Corollary \[C-minx1xx1-xlambdaAastu\]. We observe that the matrix $A$ has the spectral radius $\lambda=1$, and hence $\lambda^{-1}A=A$. The optimal value of this problem is equal to $$\delta = 1^{T}A1 = \max_{i,j=1}^{n}a_{ij} = 4/3.$$ To find the solution set, we successively compute $$\delta^{-1}11^{T}\oplus A = \begin{pmatrix} 1 & 3/4 & 3/4 \\ 4/3 & 1 & 3/4 \\ 3/4 & 3/4 & 1 \end{pmatrix}, \quad (\delta^{-1}11^{T}\oplus A)^{\ast} = \begin{pmatrix} 1 & 3/4 & 3/4 \\ 4/3 & 1 & 1 \\ 1 & 3/4 & 1 \end{pmatrix}.$$ As the first two columns of the last matrix are proportional to one another, the solution set can be written as $$x = \begin{pmatrix} 1 & 3/4 \\ 4/3 & 1 \\ 1 & 1 \end{pmatrix}u, \quad u>0.$$ The section of this solution set by the plane $\{x\colon x_{3}=1\}$ is the segment between $(1, 4/3)$ and $(3/4, 1)$: see the thick blue segment on Figure \[F-1AuAu1\]. (1.5,1.5) – (1.5,2.5) – (2.5,4.5) – (4.5,4.5) – (4.5,2.5) – (2.5,1.5) – (1.5,1.5); (2,2) – (2,2.5) – (2.5,3.16) – (3.16,3.16) – (3.16,2.5) – (2.5,2) – (2,2); (1.5,1.83) – (3.5,4.5); (2,2.5) – (2.5,3.16); (1.5,1.83) circle \[radius=0.05\]; (3.5,4.5) circle \[radius=0.05\]; (0,0.5) – (5, 0.5); (0.5,0) – (0.5,5); at (0.3,0.2) [$0$]{}; (1,0.5)–(1,0.6); (1.5,0.5)– (1.5,0.7); at (1.5,0.1) [$\frac{1}{2}$]{}; (2,0.5) – (2,0.6); (2.5,0.5) – (2.5,0.7); at (2.5,0.2) [$1$]{}; (3,0.5)– (3,0.6); (3.5,0.5) – (3.5,0.7); at (3.5,0.1) [$\frac{3}{2}$]{}; (4,0.5) – (4,0.6); (4.5,0.5) – (4.5,0.7); at (4.5,0.2) [$2$]{}; (0.5,1) – (0.6,1); (0.5,1.5) – (0.7,1.5); at (0.3,1.5) [$\frac{1}{2}$]{}; (0.5,2) – (0.6,2); (0.5,2.5) – (0.7,2.5); at (0.3,2.5) [$1$]{}; (0.5,3) – (0.6,3); (0.5,3.5) – (0.7,3.5); at (0.3,3.5) [$\frac{3}{2}$]{}; (0.5,4) – (0.6,4); (0.5,4.5) – (0.7,4.5); at (0.3,4.5) [$2$]{}; Let us now consider the problem of maximizing the Hilbert seminorm . To find the optimal value by Corollary \[C-maxx1AxAx1\], we calculate $$A^{-} = \begin{pmatrix} 1 & 3/4 & 3/2 \\ 4/3 & 1 & 2 \\ 2 & 3/2 & 1 \end{pmatrix}, \qquad \Delta = 1^{T}AA^{-}1 = 2.$$ Furthermore, we calculate $$1^{T}a_{1} = 4/3, \qquad 1^{T}a_{2} = 1^{T}a_{3} = 1,$$ and then observe that the condition $1^{T}a_{k}a_{lk}^{-1}=\Delta$ is satisfied if either $k=2$ and $l=3$, or $k=3$ and $l=1$. Let us take $k=2$ and $l=3$. Then, we calculate $$A_{32}^{-} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{pmatrix}, \quad (I\oplus A_{32}^{-}A) = \begin{pmatrix} 1 & 0 & 0 \\ 4/3 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}$$ and consider the following solution of : $$x = \begin{pmatrix} 1 & 0 & 0 \\ 4/3 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}u, \quad u>0.$$ The corresponding vector in $\operatorname{span}(A)$ is $$Ax = \begin{pmatrix} 1 & 3/4 & 3/2 \\ 4/3 & 1 & 2 \\ 2/3 & 1/2 & 1 \end{pmatrix}u.$$ Since all columns of this matrix are proportional to the third column, we take this column to represent the solution. The section of the solution set by the plane $x_{3}=1$ is just one point $(3/2, 2)$. It is not difficult to verify in the similar way that with $k=3$ and $l=1$, we have the solution $$Ax = \begin{pmatrix} 1/2 \\ 2/3 \\ 1 \end{pmatrix}v, \quad v>0.$$ which intersects the plane $x_{3}=1$ in the point $(1/2, 2/3)$. Both solutions are shown on Figure \[F-1AuAu1\] (thick red dots). \[E-1AuAu1-right\] Consider problems and with the matrix $$A = A^{\ast} = \begin{pmatrix} 1 & 3/4 & 1/2 \\ 3/4 & 1 & 1/2 \\ 1/2 & 1/2 & 1 \end{pmatrix}.$$ To solve problem of minimizing the Hilbert seminorm, we note that $\lambda=1$. Next, we find $$\delta = 1^{T}A1 = 1,$$ which is attained on the ray of points whose all coordinates are equal to each other. The section of this ray by the plane $x_{3}=1$ coincides with the point $(1,1)$ (the thick blue dot on Figure \[F-1AuAu2\]). (1.5,1.5) – (1.5,2.5) – (2.5,4.5) – (4.5,4.5) – (4.5,2.5) – (2.5,1.5) – (1.5,1.5); (1.5,1.5) – (1.5,1.83) – (3.5,4.5) – (4.5,4.5)– (4.5,3.5) – (1.83,1.5) – (1.5,1.5); (1.5,1.5) – (1.5,1.83) – (3.5,4.5) – (4.5,4.5)– (4.5,3.5) – (1.83,1.5) – (1.5,1.5); (2.5,2.5) circle \[radius=0.05\]; (1.83,1.5) – (1.5,1.5) – (1.5,1.83); (3.5,4.5) – (4.5,4.5) – (4.5,3.5); (0,0.5) – (5, 0.5); (0.5,0) – (0.5,5); at (0.3,0.2) [$0$]{}; (1,0.5)–(1,0.6); (1.5,0.5)– (1.5,0.7); at (1.5,0.1) [$\frac{1}{2}$]{}; (2,0.5) – (2,0.6); (2.5,0.5) – (2.5,0.7); at (2.5,0.2) [$1$]{}; (3,0.5)– (3,0.6); (3.5,0.5) – (3.5,0.7); at (3.5,0.1) [$\frac{3}{2}$]{}; (4,0.5) – (4,0.6); (4.5,0.5) – (4.5,0.7); at (4.5,0.2) [$2$]{}; (0.5,1) – (0.6,1); (0.5,1.5) – (0.7,1.5); at (0.3,1.5) [$\frac{1}{2}$]{}; (0.5,2) – (0.6,2); (0.5,2.5) – (0.7,2.5); at (0.3,2.5) [$1$]{}; (0.5,3) – (0.6,3); (0.5,3.5) – (0.7,3.5); at (0.3,3.5) [$\frac{3}{2}$]{}; (0.5,4) – (0.6,4); (0.5,4.5) – (0.7,4.5); at (0.3,4.5) [$2$]{}; Let us examine problem of maximizing the Hilbert seminorm. First, we calculate $$A^{-} = \begin{pmatrix} 1 & 4/3 & 2 \\ 4/3 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix}, \qquad \delta = 1^{T}AA^{-}1 = 2.$$ Let us observe here that since all entries of $A$ are not greater than one, the entrywise logarithm of that matrix is a normal matrix and hence the results of [@Delapuente2013Ontropical] apply to it. According to [@Delapuente2013Ontropical], $\delta$ is the greatest entry of $A^{-}$, which is the same as $1^{T}A^{-}1$. However, this is also clear from our computation since $1^{T}A=1^{T}$ and hence $1^{T}AA^{-}1=1^{T}A^{-}1$ in this case. Next, we have $$1^{T}a_{1} = 1^{T}a_{2} = 1^{T}a_{3} = 1,$$ and then conclude that the condition $1^{T}a_{k}a_{lk}^{-1}=\Delta$ is satisfied at four $(k,l)$ pairs: $(3,1)$, $(3,2)$, $(1,3)$ and $(2,3)$. For $k=3$ and $l=1$, we have to calculate $$A_{13}^{-} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 2 & 0 & 0 \end{pmatrix}, \quad (I\oplus A_{13}^{-}A) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 3/2 & 1 \end{pmatrix}.$$ and then we consider the following solutions of : $$x = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 3/2 & 1 \end{pmatrix}u, \quad u>0.$$ The corresponding vector in $\operatorname{span}(A)$ is calculated as $$Ax = \begin{pmatrix} 1 & 3/4 & 1/2 \\ 1 & 1 & 1/2 \\ 2 & 3/2 & 1 \end{pmatrix}u.$$ The first and third column of the last matrix are proportional to each other, hence solution subset corresponding to $(k,l)=(3,1)$ is $$\left\{ \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 2/3 \\ 1 & 1 \end{pmatrix}v \colon v>0\right\}.$$ The intersection of this subset with $\{x\colon x_3=1\}$ is the segment with ends $(1/2, 1/2)$ and $(1/2, 2/3)$. Assume that $(k,l)=(3,2)$. In this case, we have $$A_{23}^{-} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 2 & 0 \end{pmatrix}, \quad (I\oplus A_{23}^{-}A) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 3/2 & 2 & 1 \end{pmatrix}.$$ The solution of problem is written in the form $$x = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 3/2 & 1 \end{pmatrix}u, \quad u>0.$$ The solution yields the vector $$Ax = \begin{pmatrix} 1 & 1 & 1/2 \\ 3/4 & 1 & 1/2 \\ 3/2 & 2 & 1 \end{pmatrix}u.$$ The second and third column of the matrix are proportional to each other, hence solution subset corresponding to $(k,l)=(3,2)$ is $$\left\{ \begin{pmatrix} 1/2 & 2/3 \\ 1/2 & 1/2 \\ 1 & 1 \end{pmatrix}v \colon v>0 \right\}.$$ The intersection of this subset with the plane $\{x\colon x_{3}=1\}$ is the segment with the ends $(1/2, 1/2)$ and $(2/3, 1/2)$. We can similarly find solutions for $(k,l)=(1,3)$ and $(k,l)=(2,3)$. Solutions for all four pairs $(k,l)$ are shown on Figure \[F-1AuAu2\] (thick red lines). Tropical implementation of AHP {#s:AHP} ============================== We are now in a position to describe our tropical implementation of AHP. Let us consider a multi-criteria decision problem to rate $n$ alternatives (choices) from pairwise comparisons with respect to $m$ criteria. Suppose that $C$ is an $(m\times m)$-matrix of pairwise comparisons of the criteria, and $A_{1},\ldots,A_{m}$ are $(n\times n)$-matrices of pairwise comparisons of the alternatives for every criterion. Given the above matrices, the problem consists in finding $n$-vectors $x$ of scores (rates, priorities) of the alternatives. We propose a decision procedure that involves the following steps: (i) log-Chebyshev approximation of the pairwise comparison matrix of criteria to find the vector of weights in a parametric form; (ii) simultaneous weighted minimax approximation of the pairwise comparison matrices of choices according to each criterion to obtain the vectors of priorities for choices; (iii) solution of the optimization problems of maximizing and minimizing the Hilbert seminorm to derive the vectors, which most and least differentiate between the choices with the highest and lowest priorities. At the first step, we need to evaluate the relative importance of the criteria by solving problem with the matrix $C$. In terms of the tropical linear algebra, problem takes the form of . Therefore, we can apply Lemma \[L-minxxAx\] to obtain the vector of weights in the parametric form $$w = (\lambda^{-1}C)^{\ast}v,$$ where $\lambda$ is the tropical spectral radius of $C$, and $v$ is any positive vector. The next step is the evaluation of the priorities of alternatives, which involves the solution of problem with the matrix $B$ defined by , where the vector $w$ is the weight vector obtained at the first step. After translation into the tropical linear algebra language, we have problem . Corollary \[C-minxwkxAkx\] offers the solution to the problem in the form $$x = (\mu^{-1}B)^{\ast}u, \qquad B = \bigoplus_{k=1}^{m}w_{k}A_{k}, \label{e:xB}$$ where $\mu$ is the spectral radius of the matrix $B$, and $u$ is any positive vector. If the obtained solution $x=Su$, where $S=(\mu^{-1}B)^{\ast}$ or $S$ is a submatrix of linearly independent columns of $(\mu^{-1}B)^{\ast}$, is not unique (up to a positive factor), we need to solve problems and to determine the most and least differentiating priority vectors. In the tropical linear algebra setting, problem reduces to where $A$ is replaced by $S$, and $x$ by $u$. Application of Corollary \[C-maxx1AxAx1\] to solve the last problem requires calculating $\Delta=1^{T}SS^{-}1$, and yields the solution $$u = S(I\oplus S_{lk}^{-}S)v, \qquad v>0,$$ where the indices $k$ and $l$ satisfy the condition $$1^{T}s_{k}s_{lk}^{-1} = \Delta.$$ The most differentiating priority vectors are then given by $$x_{1} = S(I\oplus S_{lk}^{-}S)v, \qquad v>0.$$ Problem takes the form of with $A$ replaced by $B$ and $\lambda$ by $\mu$. The solution is given by Corollary \[C-minx1xx1-xlambdaAastu\], and involves calculating $\delta=1^{T}(\mu^{-1}B)^{\ast}1$, which is used to obtain the least differentiating vector of priorities $$x_{2} = (\delta^{-1}11^{T}\oplus\mu^{-1}B)^{\ast}u, \qquad u>0.$$ We now present two examples, which will illustrate the computational technique involved in the tropical implementation of AHP described above. In the first examples, the matrix $C$ has an essentially unique weight vector $w$ associated with it, which is then used to form the matrix $B$ as in . In the second example, the weight vector associated with $C$ is not unique and represented in a parametric form. In this case, we combine the choice of the appropriate weight vector with the solution of the optimization problems to find the most and least differentiating vectors on the next step of the procedure. Vacation site selection example {#S-VSSE} ------------------------------- Consider an example from [@Saaty1977Scaling], where a plan for vacation is to be selected. The places considered are $\mathbf{S}$: short trips from Philadelphia (i.e., New York, Washington, Atlantic City, New Hope, etc.), $\mathbf{Q}$: Quebec, $\mathbf{D}$: Denver, $\mathbf{C}$: California. The problem is to evaluate the places with respect to the following criteria: (1) cost of the trip from Philadelphia, (2) sight-seeing opportunities, (3) entertainment (doing things), (4) way of travel, (5) eating places. The comparison matrix of criteria for places is given by $$C = \begin{pmatrix} 1 & 1/5 & 1/5 & 1 & 1/3 \\ 5 & 1 & 1/5 & 1/5 & 1 \\ 5 & 5 & 1 & 1/5 & 1 \\ 1 & 5 & 5 & 1 & 5 \\ 3 & 1 & 1 & 1/5 & 1 \end{pmatrix}.$$ The pairwise comparison matrices of vacation sites with respect to the criteria are defined as follows: $$\begin{gathered} A_{1} = \begin{pmatrix} 1 & 3 & 7 & 9 \\ 1/3 & 1 & 6 & 7 \\ 1/7 & 1/6 & 1 & 3 \\ 1/9 & 1/7 & 1/3 & 1 \end{pmatrix}, \qquad A_{2} = \begin{pmatrix} 1 & 1/5 & 1/6 & 1/4 \\ 5 & 1 & 2 & 4 \\ 6 & 1/2 & 1 & 6 \\ 4 & 1/4 & 1/6 & 1 \end{pmatrix}, \\ A_{3} = \begin{pmatrix} 1 & 7 & 7 & 1/2 \\ 1/7 & 1 & 1 & 1/7 \\ 1/7 & 1 & 1 & 1/7 \\ 2 & 7 & 7 & 1 \end{pmatrix}, \qquad A_{4} = \begin{pmatrix} 1 & 4 & 1/4 & 1/3 \\ 1/4 & 1 & 1/2 & 3 \\ 4 & 2 & 1 & 3 \\ 3 & 1/3 & 1/3 & 1 \end{pmatrix}, \\ A_{5} = \begin{pmatrix} 1 & 1 & 7 & 4 \\ 1 & 1 & 6 & 3 \\ 1/7 & 1/6 & 1 & 1/4 \\ 1/4 & 1/3 & 4 & 1 \end{pmatrix}.\end{gathered}$$ To solve the problem, we first evaluate the priorities of criteria. We take the pairwise comparison matrix $C$, and find its tropical spectral radius (its maximum cycle geometric mean). Using , we obtain $$\lambda = (c_{14}c_{43}c_{32}c_{21})^{1/4} = 5^{3/4} \approx 3.3437.$$ Furthermore, we consider the matrix $$\lambda^{-1}C = \begin{pmatrix} 1/\lambda & 1/5\lambda & 1/5\lambda & 1/\lambda & 1/3\lambda \\ 5/\lambda & 1/\lambda & 1/5\lambda & 1/5\lambda & 1/\lambda \\ 5/\lambda & 5/\lambda & 1/\lambda & 1/5\lambda & 1/\lambda \\ 1/\lambda & 5/\lambda & 5/\lambda & 1/\lambda & 5/\lambda \\ 3/\lambda & 1/\lambda & 1/\lambda & 1/5\lambda & 1/\lambda \end{pmatrix},$$ and calculate its powers to obtain the Kleene star matrix $$\begin{gathered} (\lambda^{-1}C)^{\ast} = I\oplus\lambda^{-1}C\oplus\lambda^{-2}C^{2}\oplus\lambda^{-3}C^{3}\oplus\lambda^{-4}C^{4} \\ = \begin{pmatrix} 1 & \lambda/5 & 5/\lambda^{2} & 1/\lambda & 5/\lambda^{2} \\ 5/\lambda & 1 & \lambda/5 & 5/\lambda^{2} & \lambda/5 \\ \lambda^{2}/5 & 5/\lambda & 1 & \lambda/5 & 1 \\ \lambda & \lambda^{2}/5 & 5/\lambda & 1 & 5/\lambda \\ 3/\lambda & 3/5 & 3\lambda/25 & 3/\lambda^{2} & 3\lambda/25 \end{pmatrix}.\end{gathered}$$ The columns of the Kleene matrix generate the set of all weight vectors of criteria. Since all columns of this matrix are collinear, any one of them can serve as the weight vector. We take the first column, and use its elements as coefficients to combine the matrices $A_{1},\ldots,A_{5}$ into one matrix $$B = A_{1} \oplus 5\lambda^{-1} A_{2} \oplus %25\lambda^{-2} 5^{-1}\lambda^{2} A_{3} \oplus \lambda A_{4} \oplus 3\lambda^{-1} A_{5} \\ = \begin{pmatrix} \lambda & 7\lambda^{2}/5 & 7\lambda^{2}/5 & 9 \\ 25/\lambda & \lambda & 6 & 3\lambda \\ 4\lambda & 2\lambda & \lambda & 3\lambda \\ 3\lambda & 7\lambda^{2}/5 & 7\lambda^{2}/5 & \lambda \end{pmatrix}.$$ We now apply Corollary \[C-minxwkxAkx\] to find all priority vectors that correspond to the matrix $B$. Evaluation of the tropical spectral radius (the maximum cycle mean) of $B$ yields $$\mu = (b_{13}b_{31})^{1/2} %= %140^{1/2}5^{1/8} = 2\cdot5\cdot7^{1/2}/\lambda^{1/2} = 2\cdot5^{5/8}7^{1/2} \approx 14.4689.$$ Furthermore, we calculate powers of the matrix $$\mu^{-1}B = \begin{pmatrix} \lambda/\mu & 7\lambda^{2}/5\mu & 7\lambda^{2}/5\mu & 9/\mu \\ 25/\lambda\mu & \lambda/\mu & 6/\mu & 3\lambda/\mu \\ 4\lambda/\mu & 2\lambda/\mu & \lambda/\mu & 3\lambda/\mu \\ 3\lambda/\mu & 7\lambda^{2}/5\mu & 7\lambda^{2}/5\mu & \lambda/\mu \end{pmatrix},$$ and combine them to construct the matrix $$(\mu^{-1}B)^{\ast} = I\oplus\mu^{-1}B\oplus\mu^{-2}B^{2}\oplus\mu^{-3}B^{3} = \begin{pmatrix} 1 & \mu/4\lambda & \mu/4\lambda & 3/4 \\ 3\lambda/\mu & 1 & 3/4 & 3\lambda/\mu \\ 4\lambda/\mu & 1 & 1 & 3\lambda/\mu \\ 1 & \mu/4\lambda & \mu/4\lambda & 1 \end{pmatrix}$$ whose columns generate all priority vectors for alternatives. Observing that the first column is collinear with the third, one of them, say the third column, can be removed from the set of generators. Thus, we represent a complete solution as the set of vectors $$x = Su, \quad S = \begin{pmatrix} 1 & \mu/4\lambda & 3/4 \\ 3\lambda/\mu & 1 & 3\lambda/\mu \\ 4\lambda/\mu & 1 & 3\lambda/\mu \\ 1 & \mu/4\lambda & 1 \end{pmatrix}, \quad u>0.$$ To find solutions that most differentiate alternatives with the highest and lowest priorities, we apply Corollary \[C-maxx1AxAx1\]. We start with the calculation $$\begin{aligned} 1^{T}s_{1} &= 1, & 1^{T}s_{2} &= \mu/4\lambda, & 1^{T}s_{3} &= 1, \\ s_{1}^{-}1 &= \mu/3\lambda, & s_{2}^{-}1 &= 1, & s_{3}^{-}1 &= \mu/3\lambda,\end{aligned}$$ and then obtain $$\Delta %= %1^{T}SS^{-}1 = 1^{T}s_{1}s_{1}^{-}1 \oplus 1^{T}s_{2}s_{2}^{-}1 \oplus 1^{T}s_{3}s_{3}^{-}1 = \mu/3\lambda = 2\cdot3^{-1}5^{-1/8}7^{1/2} \approx 1.4424.$$ The condition $1^{T}s_{k}s_{lk}^{-1}=\Delta$ holds if we take the following $(k,l)$ pairs: $(1,2)$, $(3,2)$ and $(3,3)$. First, assume that $k=1$ and $l=2$. We form the matrices $$S_{21} = \begin{pmatrix} 0 & 0 & 0 \\ 3\lambda/\mu & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad S_{21}^{-}S = \begin{pmatrix} 1 & \mu/3\lambda & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix},$$ and then derive the matrix, which generates the most differentiating priority vectors, $$S(I\oplus S_{21}^{-}S) = \begin{pmatrix} 1 & \mu/3\lambda & 1 \\ 3\lambda/\mu & 1 & 3\lambda/\mu \\ 4\lambda/\mu & 4/3 & 4\lambda/\mu \\ 1 & \mu/3\lambda & 1 \end{pmatrix}.$$ Since all columns in the last matrix are collinear to each other, we take one of them, say the first, to write one of the most differentiating solutions as $$x_{1}^{\prime} = \begin{pmatrix} 1 \\ 3\lambda/\mu \\ 4\lambda/\mu \\ 1 \end{pmatrix} u, \quad \lambda = 5^{3/4}, \quad \mu = 2\cdot5^{5/8}7^{1/2}, \quad u>0.$$ Specifically, by setting $u=1$, we have $x_{1}^{\prime}\approx(1.0000, 0.6933, 0.9244, 1.0000)^{T}$. This vector specifies the order of choices as $\mathbf{C}\equiv\mathbf{S}\succ\mathbf{D}\succ\mathbf{Q}$. Next, we examine the case where $k=3$ and $l=2$. In a similar way, we obtain the vector $$x_{1}^{\prime\prime} = \begin{pmatrix} 3/4 \\ 3\lambda/\mu \\ 3\lambda/\mu \\ 1 \end{pmatrix} u, \quad \lambda = 5^{3/4}, \quad \mu = 2\cdot5^{5/8}7^{1/2}, \quad u>0,$$ which suggests another most differentiating solution. If $u=1$, then $x_{1}^{\prime\prime}\approx(0.7500, 0.6933, 0.6933, 1.0000)^{T}$, which puts the choices in the order $\mathbf{C}\succ\mathbf{S}\succ\mathbf{D}\equiv\mathbf{Q}$. It is not difficult to verify that the case with $k=3$ and $l=3$ introduces no other solutions than those already found. We now turn to an application of Corollary \[C-minx1xx1-xlambdaAastu\] to derive the least differentiating vector of priorities; as it will turn out, in this example it is essentially unique. First, we calculate $$\delta = 1^{T}(\mu^{-1}B)^{\ast}1 = \mu/4\lambda = 2^{-1}5^{-1/8}7^{1/2} \approx 1.0818,$$ and then construct the matrix $$\delta^{-1}11^{T} \oplus \mu^{-1}B = \begin{pmatrix} 1/\delta & \delta & \delta & 1/\delta \\ 1/\delta & 1/\delta & 1/\delta & 1/\delta \\ 1/\delta & 1/\delta & 1/\delta & 1/\delta \\ 1/\delta & \delta & \delta & 1/\delta \end{pmatrix}.$$ The least differentiating priority vectors are generated by the columns of the Kleene star matrix $$\begin{gathered} (\delta^{-1}11^{T}\oplus\mu^{-1}B)^{\ast} \\ = I \oplus (\delta^{-1}11^{T}\oplus\mu^{-1}B) \oplus (\delta^{-1}11^{T}\oplus\mu^{-1}B)^{2} \oplus (\delta^{-1}11^{T}\oplus\mu^{-1}B)^{3} \\ = \begin{pmatrix} 1 & \delta & \delta & 1 \\ 1/\delta & 1 & 1 & 1/\delta \\ 1/\delta & 1 & 1 & 1/\delta \\ 1 & \delta & \delta & 1 \end{pmatrix}.\end{gathered}$$ Observing that all columns in the matrix obtained are collinear, we take one of them, say the first, to write the least differentiating solutions as $$x_{2} = \begin{pmatrix} 1 \\ 1/\delta \\ 1/\delta \\ 1 \end{pmatrix}u, \quad u>0.$$ Setting $u=1$, we have $x_{2}\approx(1, 0.9244, 0.9244, 1)^{T}$. This vector arranges the alternatives in the order $\mathbf{C}\equiv\mathbf{S}\succ\mathbf{D}\equiv\mathbf{Q}$. As one can see, all solutions indicate the highest score of the fourth choice (California). The score assigned to the first choice (short trip) is the same or lower. The third choice (Denver) has the same or higher score, than the second choice (Quebec), and both of them always have a lower score than the first. Combining both the most and least differentiating solutions yields the order of choices $\mathbf{C}\succeq\mathbf{S}\succ\mathbf{D}\succeq\mathbf{Q}$. Note that the results obtained above with the tropical implementation of AHP are quite different from those offered by the classical AHP method. Specifically, the order of choices, found in [@Saaty1977Scaling], is $\mathbf{S}\succ\mathbf{D}\succ\mathbf{C}\succ\mathbf{Q}$. School selection example {#S-SSE} ------------------------ As another example, we investigate a problem in [@Saaty1977Scaling; @Saaty1990Analytic] to rank three high schools $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$, according to the following characteristics (criteria): (1) learning, (2) friends, (3) school life, (4) vocational training, (5) college preparation, (6) music classes. The results of pairwise comparison of criteria are given by the matrix $$C = \begin{pmatrix} 1 & 4 & 3 & 1 & 3 & 4 \\ 1/4 & 1 & 7 & 3 & 1/5 & 1 \\ 1/3 & 1/7 & 1 & 1/5 & 1/5 & 1/6 \\ 1 & 1/3 & 5 & 1 & 1 & 1/3 \\ 1/3 & 5 & 5 & 1 & 1 & 3 \\ 1/4 & 1 & 6 & 3 & 1/3 & 1 \end{pmatrix}.$$ The matrices of pairwise comparison of schools for each criterion take the following forms: $$\begin{gathered} A_{1} = \begin{pmatrix} 1 & 1/3 & 1/2 \\ 3 & 1 & 3 \\ 2 & 1/3 & 1 \end{pmatrix}, \quad A_{2} = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}, \quad A_{3} = \begin{pmatrix} 1 & 5 & 1 \\ 1/5 & 1 & 1/5 \\ 1 & 5 & 1 \end{pmatrix}, \\ A_{4} = \begin{pmatrix} 1 & 9 & 7 \\ 1/9 & 1 & 1/5 \\ 1/7 & 5 & 1 \end{pmatrix}, \quad A_{5} = \begin{pmatrix} 1 & 1/2 & 1 \\ 2 & 1 & 2 \\ 1 & 1/2 & 1 \end{pmatrix}, \quad A_{6} = \begin{pmatrix} 1 & 6 & 4 \\ 1/6 & 1 & 1/3 \\ 1/4 & 3 & 1 \end{pmatrix}.\end{gathered}$$ The solution of the problem involves evaluation of the priority vectors that most and least differentiate the schools with the highest and lowest priorities. To find the most differentiating solution, we first obtain a parametric description of all weight vectors from the pairwise comparison matrix of criteria. The components of the weight vector are used to form a weighted sum of comparison matrices of schools for each criteria. Next, the priority vectors for schools are evaluated based on the sum of matrices with parameterized weights. Finally, those priority vectors, which most and least differentiate the schools with the highest and lowest scores, are taken as the most and leats differentiating solutions to the problem. As the first step, we need to obtain the priority vectors for criteria, which specify the weights of the criteria. We evaluate the spectral radius of the matrix $C$ by using to write $$\lambda = (c_{15}c_{52}c_{24}c_{41})^{1/4} %= %(45)^{1/4} = 3^{1/2}5^{1/4} \approx 2.5900.$$ We calculate the first five powers of the matrix $$\lambda^{-1}C = \begin{pmatrix} 1/\lambda & 4/\lambda & 3/\lambda & 1/\lambda & 3/\lambda & 4/\lambda \\ 1/4\lambda & 1/\lambda & 7/\lambda & 3/\lambda & 1/5\lambda & 1/\lambda \\ 1/3\lambda & 1/7\lambda & 1/\lambda & 1/5\lambda & 1/5\lambda & 1/6\lambda \\ 1/\lambda & 1/3\lambda & 5/\lambda & 1/\lambda & 1/\lambda & 1/3\lambda \\ 1/3\lambda & 5/\lambda & 5/\lambda & 1/\lambda & 1/\lambda & 3/\lambda \\ 1/4\lambda & 1/\lambda & 6/\lambda & 3/\lambda & 1/3\lambda & 1/\lambda \end{pmatrix},$$ and then combine these powers to derive the Kleene star matrix $$\begin{gathered} (\lambda^{-1}C)^{\ast} = I\oplus\lambda^{-1}C\oplus\lambda^{-2}C^{2}\oplus\lambda^{-3}C^{3}\oplus\lambda^{-4}C^{4}\oplus\lambda^{-5}C^{5} \\ = \begin{pmatrix} 1 & \lambda^{2}/3 & 7\lambda/3 & \lambda & 3/\lambda & 4/\lambda \\ 3/\lambda^{2} & 1 & 7/\lambda & 3/\lambda & \lambda/5 & 4\lambda/15 \\ 1/3\lambda & \lambda/9 & 1 & 1/3 & 1/\lambda^{2} & 4/3\lambda^{2} \\ 1/\lambda & \lambda/3 & 7/3 & 1 & 3/\lambda^{2} & 4/\lambda^{2} \\ \lambda/3 & 5/\lambda & 7\lambda^{2}/9 & \lambda^{2}/3 & 1 & 4/3 \\ 3/\lambda^{2} & 1 & 7/\lambda & 3/\lambda & \lambda/5 & 1 \end{pmatrix}.\end{gathered}$$ Note that the first, second, fourth and fifth columns in the matrix $(\lambda^{-1}C)^{\ast}$ are collinear, and thus all of them but one, say the fourth, can be omitted. We combine the fourth column together with the third multiplied by $3/7$ and the sixth multiplied by $\lambda^{2}/4$ to obtain the generating matrix, and introduce the vector of parameters $v=(v_{1},v_{2},v_{3})^{T}>0$ to represent the weight vector in parametric form as $$w = \begin{pmatrix} \lambda & \lambda & \lambda \\ 3/\lambda & 3/\lambda & 3/\lambda \\ 3/7 & 1/3 & 1/3 \\ 1 & 1 & 1 \\ \lambda^{2}/3 & \lambda^{2}/3 & \lambda^{2}/3 \\ 3/\lambda & 3/\lambda & \lambda^{2}/4 \end{pmatrix} v, \quad v>0.$$ We now use the components of the vector $w$ as weights to combine the matrices $A_{1},\ldots,A_{6}$ into the matrix $$\begin{gathered} B = \lambda(v_{1}\oplus v_{2}\oplus v_{3}) A_{1} \oplus 3\lambda^{-1}(v_{1}\oplus v_{2}\oplus v_{3}) A_{2} \oplus (3\cdot7^{-1}v_{1}\oplus3^{-1}v_{2}\oplus3^{-1}v_{3}) A_{3} \\ \oplus (v_{1}\oplus v_{2}\oplus v_{3}) A_{4} \oplus 3^{-1}\lambda^{2}(v_{1}\oplus v_{2}\oplus v_{3}) A_{5} \oplus (3\lambda^{-1}v_{1} \oplus 3\lambda^{-1}v_{2} \oplus 4^{-1}\lambda^{2}v_{3}) A_{6} \\ = \begin{pmatrix} \lambda(v_{1}\oplus v_{2}\oplus v_{3}) & 9(v_{1}\oplus v_{2})\oplus3\lambda^{2}v_{3}/2 & 7(v_{1}\oplus v_{2}\oplus v_{3}) \\ 3\lambda(v_{1}\oplus v_{2}\oplus v_{3}) & \lambda(v_{1}\oplus v_{2}\oplus v_{3}) & 3\lambda(v_{1}\oplus v_{2}\oplus v_{3}) \\ 2\lambda(v_{1}\oplus v_{2}\oplus v_{3}) & 5(v_{1}\oplus v_{2})\oplus3\lambda^{2}v_{3}/4 & \lambda(v_{1}\oplus v_{2}\oplus v_{3}) \end{pmatrix}.\end{gathered}$$ Observing that the parameters $v_{1}$ and $v_{2}$ occur in all entries of the matrix in the form of the sum $v_{1}\oplus v_{2}$, we change the variables by replacing this sum by $v_{1}$ and $v_{3}$ by $v_{2}$ to rewrite the matrix in the more simple form $$B = \begin{pmatrix} \lambda(v_{1}\oplus v_{2}) & 9v_{1}\oplus3\lambda^{2}v_{2}/2 & 7(v_{1}\oplus v_{2}) \\ 3\lambda(v_{1}\oplus v_{2}) & \lambda(v_{1}\oplus v_{2}) & 3\lambda(v_{1}\oplus v_{2}) \\ 2\lambda(v_{1}\oplus v_{2}) & 5v_{1}\oplus3\lambda^{2}v_{2}/4 & \lambda(v_{1}\oplus v_{2}) \end{pmatrix}.$$ Furthermore, we take the matrix $B$ to derive all solutions by using Corollary \[C-minxwkxAkx\]. Evaluation of the spectral radius of the matrix $B$ yields $$\mu = (b_{12}b_{21})^{1/2} = (3\lambda(v_{1}\oplus v_{2})(9v_{1}\oplus3\lambda^{2}v_{2}/2))^{1/2}.$$ We consider the matrix $\mu^{-1}B$ and calculate the Kleene star matrix $$\begin{gathered} (\mu^{-1}B)^{\ast} = I\oplus\mu^{-1}B\oplus\mu^{-2}B^{2} \\ = \begin{pmatrix} 1 & \mu/3\lambda(v_{1}\oplus v_{2}) & 1 \\ 3\lambda(v_{1}\oplus v_{2})/\mu & 1 & 3\lambda(v_{1}\oplus v_{2})/\mu \\ 2\lambda(v_{1}\oplus v_{2})/\mu & 2/3 & 1 \end{pmatrix}.\end{gathered}$$ As the first two columns of the obtained matrix are collinear, we take one of them, say the second, to write the solution in the form $$x = Su, \quad S = \begin{pmatrix} \mu/3\lambda(v_{1}\oplus v_{2}) & 1 \\ 1 & 3\lambda(v_{1}\oplus v_{2})/\mu \\ 2/3 & 1 \end{pmatrix}, \quad u>0.$$ Since the solution is not unique up to a positive factor, we need to find the vectors, which most and least differentiate between the alternatives with the highest and lowest priorities. We begin with the application of Corollary \[C-maxx1AxAx1\] to obtain the most differentiating solutions of the problem. First, we note that $\mu=(3\lambda(v_{1}\oplus v_{2})(9v_{1}\oplus3\lambda^{2}v_{2}/2))^{1/2}>3\lambda(v_{1}\oplus v_{2})$, and calculate $$\begin{aligned} 1^{T}s_{1} &= \mu/3\lambda(v_{1}\oplus v_{2}), & 1^{T}s_{2} &= 1, \\ s_{1}^{-}1 &= 3/2, & s_{2}^{-}1 &= \mu/3\lambda(v_{1}\oplus v_{2}).\end{aligned}$$ Next, we have to find vectors $v$ that maximize $$\Delta_{v} %= %1^{T}SS^{-}1 = 1^{T}s_{1}s_{1}^{-}1 \oplus 1^{T}s_{2}s_{2}^{-}1 = \mu/2\lambda(v_{1}\oplus v_{2}) = \left(\frac{3(9v_{1}\oplus3\lambda^{2}v_{2}/2)}{4\lambda(v_{1}\oplus v_{2})}\right)^{1/2}.$$ Observing that $9<3\lambda^{2}/2$, we see that the maximum of $\Delta_{v}$ is attained if and only if $v_{2}\geq v_{1}$, and equal to $\Delta=(9\lambda/8)^{1/2}$. In this case, we have $$\mu = (9\lambda^{3}/2))^{1/2}v_{2},$$ whereas the matrix $S$ becomes $$S = \begin{pmatrix} (\lambda/2)^{1/2} & 1 \\ 1 & (2/\lambda)^{1/2} \\ 2/3 & 1 \end{pmatrix}.$$ The condition $1^{T}s_{k}s_{lk}^{-1}=\Delta$ holds if we take $k=1$ and $l=3$. According to Corollary \[C-maxx1AxAx1\], we construct the matrices $$S_{31} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \\ 2/3 & 0 \end{pmatrix}, \qquad S_{31}^{-}S = \begin{pmatrix} 1 & 3/2 \\ 0 & 0 \end{pmatrix},$$ and then calculate the generating matrix $$S(I\oplus S_{31}^{-}S) = \begin{pmatrix} (\lambda/2)^{1/2} & 3(\lambda/2)^{1/2}/2 \\ 1 & 3/2 \\ 2/3 & 1 \end{pmatrix}.$$ Since both columns of the generating matrix are collinear, we take the first column to write the solution $$x_{1} = \begin{pmatrix} (\lambda/2)^{1/2} \\ 1 \\ 2/3 \end{pmatrix} u, \quad \lambda = 3^{1/2}5^{1/4} \approx 2.5900, \quad u>0.$$ With $u=(\lambda/2)^{-1/2}$, we have the vector of ratings $x_{1}\approx(1,0.8787,0.5858)^{T}$, which gives the order $\mathbf{A}\succ\mathbf{B}\succ\mathbf{C}$. Let us derive the least differentiating solution by using Corollary \[C-minx1xx1-xlambdaAastu\]. We take the matrix $\mu^{-1}B$ and calculate $$\delta_{v} = 1^{T}(\mu^{-1}B)^{\ast}1 = \mu/3\lambda(v_{1}\oplus v_{2}) = \left(\dfrac{9v_{1}\oplus3\lambda^{2}v_{2}/2}{3\lambda(v_{1}\oplus v_{2})}\right)^{1/2}.$$ To find the minimum of $\delta_{v}$ with respect to $v$, consider two cases. First, assume that $v_{1}\leq v_{2}$. Observing that $3\lambda^{2}/2>9$, we have $$\delta_{v} = 2^{-1/2}\lambda^{1/2} \approx 1.1380.$$ If $v_{1}>v_{2}$, we obtain the lower bound $$\delta_{v} = (3/\lambda \oplus \lambda v_{2}/2v_{1})^{1/2} \geq 3^{1/2}\lambda^{-1/2} = 3^{1/4}5^{-1/8} \approx 1.0762.$$ This bound is achieved if $v_{2}\leq6v_{1}/\lambda^{2}$, and thus is the minimum $\delta$ under consideration. Finally, under the condition $v_{2}\leq6v_{1}/\lambda^{2}<v_{1}$, we have $$B = v_{1} \begin{pmatrix} \lambda & 9 & 7 \\ 3\lambda & \lambda & 3\lambda \\ 2\lambda & 5 & \lambda \end{pmatrix}, \quad \mu = 3^{3/2}\lambda^{1/2}v_{1}, \quad \delta = 3^{1/2}\lambda^{-1/2} = 9v_{1}/\mu.$$ The matrix $\mu^{-1}B$ takes the form $$\mu^{-1}B = \begin{pmatrix} \lambda^{1/2}/3^{3/2} & 3^{1/2}/\lambda^{1/2} & 7/3^{3/2}\lambda^{1/2} \\ \lambda^{1/2}/3^{1/2} & \lambda^{1/2}/3^{3/2} & \lambda^{1/2}/3^{1/2} \\ 2\lambda^{1/2}/3^{3/2} & 5/3^{3/2}\lambda^{1/2} & \lambda^{1/2}/3^{3/2} \end{pmatrix} = \begin{pmatrix} 1/3\delta & \delta & 7\delta/9 \\ 1/\delta & 1/3\delta & 1/\delta \\ 2/3\delta & 5\delta/9 & 1/3\delta \end{pmatrix}.$$ Furthermore, we construct the matrix $$\delta^{-1}11^{T} \oplus \mu^{-1}B = \delta^{-1}11^{T} \oplus \begin{pmatrix} 1/3\delta & \delta & 7\delta/9 \\ 1/\delta & 1/3\delta & 1/\delta \\ 2/3\delta & 5\delta/9 & 1/3\delta \end{pmatrix} = \begin{pmatrix} 1/\delta & \delta & 1/\delta \\ 1/\delta & 1/\delta & 1/\delta \\ 1/\delta & 1/\delta & 1/\delta \end{pmatrix},$$ and then find the matrix $$\begin{gathered} (\delta^{-1}11^{T}\oplus\mu^{-1}B)^{\ast} = I \oplus (\delta^{-1}11^{T}\oplus\mu^{-1}B) \oplus (\delta^{-1}11^{T}\oplus\mu^{-1}B)^{2} \\ = \begin{pmatrix} 1 & \delta & 1 \\ 1/\delta & 1 & 1/\delta \\ 1/\delta & 1 & 1 \end{pmatrix}.\end{gathered}$$ Since the first two columns in the matrix are collinear, we take the first one to write a solution, which least differentiate the alternatives, as $$x_{2}^{\prime} = \begin{pmatrix} 1 \\ 1/\delta \\ 1/\delta \end{pmatrix} u, \quad \delta = 3^{1/4}5^{-1/8}, \quad u>0.$$ With $u=1$, we have the vector $x_{2}^{\prime}\approx(1,0.9292,0.9292)^{T}$, which produces the order $\mathbf{A}\succ\mathbf{C}\equiv\mathbf{B}$. The third column in the matrix presents another solution $$x_{2}^{\prime\prime} = \begin{pmatrix} 1 \\ 1/\delta \\ 1 \end{pmatrix} u, \quad \delta = 3^{1/4}5^{-1/8}, \quad u>0,$$ which for $u=1$ becomes $x_{2}^{\prime\prime}\approx(1,0.9292,1)^{T}$, and thus defines the order $\mathbf{A}\equiv\mathbf{C}\succ\mathbf{B}$. By combining all least differentiating solutions, we put the schools in the order $\mathbf{A}\succeq\mathbf{C}\succeq\mathbf{B}$. Note that this result, as well as the most differentiating solution, which produces the order $\mathbf{A}\succ\mathbf{B}\succ\mathbf{C}$, significantly differ from that obtained by the traditional AHP in [@Saaty1977Scaling; @Saaty1990Analytic] and given by $\mathbf{B}\succ\mathbf{A}\succ\mathbf{C}$. Conclusion and Discussion ========================= In the paper, we have developed a new approach to solve multi-criteria decision problems of ranking the priorities of choices from pairwise comparison judgments. The approach mainly follows the general AHP methodology, but offers a new analytical and computational framework based on tropical optimization to solve the problems in a different way. The new approach offers an exact direct solution to the problems in analytical form, and may have the potential to complement and supplement other AHP solutions. The main differences between the proposed and traditional approaches are as follows. First, to approximate pairwise comparison matrices by consistent matrices, the new AHP applies rank-one matrix approximation in the log-Chebyshev sense instead of the approximation in Frobenius (or spectral) norm in the traditional AHP. Using the log-Chebyshev approximation yields the solution in the form of the tropical subeigenvectors of pairwise comparison matrices rather than the usual Perron vector of these matrices, provided by the traditional AHP. Note that the log-Chebyshev approximation is equivalent to minimizing the maximum relative error over the matrix entries. Therefore, this approximation technique seems to be quite reasonable to handle pairwise comparison matrices that consists of reciprocal entries with their values covering a wide range of magnitude. Furthermore, given the weights of criteria, the new AHP finds final priorities of choices by solving one optimization problem of the minimax weighted log-Chebyshev approximation rather than by obtaining separate solutions to the Frobenius approximation problems for each criteria and calculating the weighted sum of these solutions. The proposed minimax solution incorporates the weights into the objective function of the optimization problem, which provides a more general and comprehensive solution technique than that based on the direct calculation of the weighted sum. Specifically, this technique may result in a set of different solution vectors instead of a single solution in the traditional AHP, and thereby enhances the decision-making capabilities by extending the range of effective choice. To simplify the analysis and interpretation of non-unique solutions, the whole set of priority vectors is characterized by the vectors, which most and least differentiate between alternatives with the highest and lowest priorities. The most and least differentiating vectors are found by maximizing and minimizing the Hilbert (span, range) seminorm of priority vectors in the solution set. A key feature of the new approach is its close connection with tropical optimization, which results in a strong possibility to formulate all decision-making procedures as tropical optimization problems, and then to solve these problems directly using results available in the area of tropical optimization. In contrast to the traditional AHP, which involves numerical algorithms to calculate priority vectors, the application of tropical optimization yields analytical solutions, which describe all priority vectors in a compact vector form, ready for both formal analysis and immediate computations. Let us now consider the examples presented in the paper to discuss the difference between the outcomes of the traditional AHP and the tropical AHP, which we are suggesting here. Matrix $B$, from which the final vector of priorities results, is computed as the max-linear combination of the matrices $A_{i}$ for all criteria multiplied by the corresponding weight. In the vacation site selection example, all but two entries of this matrix come from $A_{3}$ (entertainment) and $A_{4}$ (way of travel), so all other criteria are not so important. Note that $A_{3}$ clearly ranks California better than short trips and both of them much better than the other two alternatives. The key entries of $A_{3}$ are those equal to $7$ and they “survive” (multiplied by some factors) in $B$ and in $(\mu^{-1}B)^{*}$. This is the main reason why the ranking of $A_{3}$ is also the final ranking, although the preference of California over Denver and Quebec is not so overwhelming as in $A_{3}$, due to some admixture from $A_{4}$ and other matrices. In contrast to this result, the traditional AHP ranks California third. In the school selection example, six entries of $B$ come from $A_{1}$ (learning) and three entries come from $A_{4}$ (vocational training) with some help of $A_{6}$ (music classes). Friends, school life and college preparation are completely ruled out. This seems reasonable, since school life is unimportant, making friends is the same in all the schools and the ranking of $A_{5}$ (college preparation) is similar to that of learning but less important and less distinguishing between the schools. The second school is the champion in learning, but the first school is much better in vocational training and music lessons, which, in the end, makes it the winner, albeit with a small margin even for the most differentiating vector and a possibility of being on the par with the third school for one of the least differentiating vectors. The outcome of the traditional AHP is the same as if we judged the schools first with respect to learning and college preparation (which puts the second school first) and only then with respect to vocational training and music lessons (which decides between the remaining two schools). Contemplating these examples and thinking of a general case, we see that the tropical AHP picks the highest entries of the matrices $A_{i}$ (with the corresponding weights) resulting in matrix $B$ on which the final comparison is based. Unlike the traditional AHP, the unimportant criteria and the criteria, which distinguish between the alternatives too weakly, are dispensed with. Also, a criterion $i$, which rates one alternative higher, can win over any number of other criteria which rate another alternative higher, if $A_{i}$ has high enough entries. After matrix $B$ is formed, we deal with a solution set to a Chebyshev approximation problem, in which we minimize the largest deviation between the logarithms of entries of $B$ and the logarithms of entries of a rank-one comparison matrix. From this solution set, we pick the solutions that most and least differentiate between the alternatives, and this allows us to see (unlike the vector produced by the traditional AHP) a whole set of reasonable priority vectors and rankings and, in particular, to observe when the judgments based on different criteria are in conflict with each other, and there is no clear winner among the alternatives. The school selection example shows that the new approach may lead to a set of weight vectors that represents the weights of criteria in parametric form as the whole tropical column span of the matrix $(\lambda^{-1}C)^{*}$. This involves the derivation of the most and least differentiating priority vectors based on solving parameterized maximization and optimization problems, which may be a difficult task. Another alternative would be to find the sets of the most, least and average (fair) differentiating vectors from the tropical column span of $(\lambda^{-1}C)^{*}$, but it is not clear whether this would necessarily lead to the most, least, and fair differentiating priority vectors in the end. Finally, we remark that it is possible to combine the traditional and tropical AHP approaches, for instance, by using the usual Perron eigenvector as the vector of criteria weights or by consistently using the geometric column barycenter of Kleene stars on both levels of the traditional AHP. Acknowledgements ================ This work was supported by the Russian Foundation of Basic Research (RFBR) \[grant number 18-010-00723\] and the Engineering and Physical Sciences Research Council (EPSRC) \[grant number EP/P019676/1\]. The authors are grateful to the referees for valuable criticism of the initial versions of this paper, and, in particular, for suggesting to apply the tropical AHP to Saaty’s school selection example. [30]{} natexlab\#1[\#1]{}\[2\][\#2]{} , , volume of **, , , . , , , , Princeton series in applied mathematics, , , . , , Systems and control: foundations and applications, , , . , , Springer Monographs in Mathematics, , , . , , , volume of **, , , . , , , () . , , , () . , , , , () . , , () . , , , , volume of **, , , . , , () . , , , , edition, . , , () . , , , () . , , () . , , () . , , , , () . , , , () . , , , , , , () . , , () . , , in: , , , , (Eds.), , , , pp. . , , in: , , (Eds.), , , , , pp. . , , , , () . , , , , , () . , , () . , , , in: , , , (Eds.), , , , pp. . . , , () . , , () . , , () . , , () . [^1]: A square matrix is tropically normal if it has the diagonal entries equal to and the off-diagonal entries less than or equal to the tropical unit [@Butkovic2010Maxlinear].

--- abstract: 'Trimerization-polarization domains in ferroelectric hexagonal YMnO$_3$ were resolved in all three spatial dimensions by piezoresponse force microscopy. Their topology is dominated by electrostatic effects with a range of 100 unit cells and reflects the unusual electrostatic origin of the spontaneous polarization. The response of the domains to locally applied electric fields explains difficulties in transferring YMnO$_3$ into a single-domain state. Our results demonstrate that the wealth of non-displacive mechanisms driving ferroelectricity that emerged from the research on multiferroics are a rich source of alternative types of domains and domain-switching phenomena.' author: - Tobias Jungk$^1$ - Ákos Hoffmann$^1$ - Manfred Fiebig$^2$ - Elisabeth Soergel$^1$ title: 'Electrostatic topology of ferroelectric domains in YMnO$_3$' --- In materials with a coexistence of magnetic and ferroelectric order, called multiferroics,[@khomskii09; @eerenstein06; @fiebig05a] the most prominent type of ferroelectricity, i.e., ferroelectricity of the displacive type found in perovskites like BaTiO$_3$, is usually avoided.[@hill00] Therefore, multiferroics research lead to the awareness of a wealth of alternative mechanisms driving the emergence of a spontaneous polarization. This includes ferroelectricity from electron lone pairs, charge order, helical spin structures, electrostatic effects, and more.[@khomskii09; @aken04] The unconventional origin of the spontaneous polarization in these systems should also affect the basic properties of a ferroelectric such as the distribution and switching behavior of its domains. However, in spite of the implications of this aspect for technological applications it has not attracted much attention thus far. Very recently, the ferroelectric domain structure of hexagonal YMnO$_3$ was investigated.[@choi10] Hexagonal manganites are textbook multiferroics in which the spontaneous polarization is induced by electrostatic instead of displacive effects. Choi et al.found that perpendicular to the spontaneous polarization intriguing intersections of as many as six ferroelectric domains are common to YMnO$_3$ and related this to the atomic displacement at the domain wall. In this letter we show that in addition (or alternatively) the unusual domain topology is a direct consequence of the electrostatic nature of the ferroelectric state in YMnO$_3$. This is derived from piezoresponse force microscopy (PFM) measurements showing that in spite of the anisotropic crystal structure kaleidoscopic intersections of domain walls are present in [*all three*]{} spatial dimensions. At the intersections electrostatic repulsion leads to a relative displacement of polarization and trimerization domains that are otherwise rigidly coupled. PFM tip-poling experiments confirm that the electrostatic discontinuity at the domain walls controls the spontaneous polarization within a range of about 100 unit cells. In hexagonal YMnO$_3$ the Mn$^{3+}$ ions are found in a rare fivefold coordination with the O$^{2-}$ ions. Planes of MnO$_5^{3+}$ bipyramids are interspaced with planes of Y$^{3+}$ ions along the hexagonal $z$ axis. Ferroelectricity emerges in two steps.[@lonkai04; @fennie05] At 1270 K tilting of the MnO$_5^{3+}$ polyhedra and corrugation of the Y$^{3+}$ layers occurs. At 920 K additional displacement of the MnO$_5^{3+}$ polyhedra induces a spontaneous polarization $P_z$ with a saturation value of 5.6 $\mu$C/cm$^2$. The transitions are driven by electrostatic and geometric effects, rather than by the usual changes in chemical bonding associated with displacive ferroelectric phase transitions as in perovskite oxides. This mechanism permits the coexistence of magnetism and ferroelectricity so that the compound becomes multiferroic at cryogenic temperatures.[@aken04] As shown in Fig. \[fig1\] unit-cell tripling at 1270 K leads to three trimerization domains ($\alpha$, $\beta$, $\gamma$). The polarization along $+z$ or $-z$ emerging at 920 K leads to a total of six domains ($\alpha^{\pm}$, $\beta^{\pm}$, $\gamma^{\pm}$). The trimerization domains are translation domains for which the identification as $\alpha$, $\beta$, or $\gamma$ is ambiguous. In contrast, the polarization domains are $180^{\circ}$ domains with a unique assignment of $+$ and $-$. From a variety of experiments as-grown ferroelectric domains in YMnO$_3$ are known to possess an extension of $\lesssim 1$ $\mu$m.[@safrankova67; @oleinik75; @fiebig02; @neacsu09] but a detailed study of their topology has only been presented by Choi et al. YMnO$_3$ samples were flux-grown $z$-oriented platelets with a lateral extension of a few mm and a thickness in the order of 100 $\mu$m.[@kim00] Chemical-mechanical polishing with a silica slurry was applied to the $z$ face and to $x$- and $y$-oriented bars cut from one platelet. The distribution of ferroelectric domains was measured by PFM which allows us to probe all three crystallographic directions non-invasively with an impressive sensitivity and high spatial resolution.[@jungk08; @jungk09] Figure \[fig2\] shows the $x$, $y$, and $z$ face[**s**]{} of as-grown YMnO$_3$ crystals from the same batch under ambient conditions. As expected, all faces reveal domains of $\lesssim 1$ $\mu$m with two grey levels corresponding to domains with $+P_z$ (bright) or $-P_z$ (dark). However, the topology of the ferroelectric domains is striking. On the $z$ face of the crystal the same kaleidoscopic domain structure with meeting points of six domains as in Ref.  is obtained. In Ref.  the intersections were assigned to $\alpha^+$, $\beta^-$, $\gamma^+$, $\alpha^-$, $\beta^+$, $\gamma^-$ (anti-) vortices of domains with a rigid clamping of trimerization and polarization domain walls. The clamping was attributed to the microscopic structure at the domain wall: The coexistence of trimerization and polarization walls minimizes the displacement of the Y$^{3+}$ position with respect to the paraelectric phase which might be energetically favorable. This argument referred to domain walls in the $xy$ plane so that kaleidoscopic intersections are expected in this plane only. However, Fig. \[fig2\] reveals that in spite of the highly anisotropic uniaxial structure of the YMnO$_3$ crystal the distribution of the domains is almost isotropic with only a slight elongation along $z$. In [*all three*]{} spatial dimensions the peculiar ferroelectric domain topology with meeting points of six domains is observed. We therefore conclude that in addition (or alternative) to mechanisms rooting in the microscopic structure of the YMnO$_3$ unit cell a more general mechanism that is less sensitive to the microscopic anisotropy must be responsible for the domain structure in Fig. \[fig2\]. For elucidating this issue we conducted high-resolution PFM scans of the domain vortex region. The images in Fig. \[fig3\] reveal that the six trimerization-polarization domains do not really meet in one point. In the majority of cases three equally polarized domains approach one another up to a distance of about 30 nm on the $z$ face and of about 100 nm on the $x$ and $y$ faces. Hence, at the center of the corresponding domain vortex the spontaneous polarization is $+P_z$ or $-P_z$ instead of approaching zero. In a minority of cases two domains of equal polarization are connected via a thin bridge but separated from the third, equally polarized domain. On the $x$ and $y$ faces this scenario is met an order of magnitude less often than the former one. For the $z$ face the occurrence of the second scenario is not clear. As in the case of Fig. \[fig2\] the local microscopy of the domain walls cannot be responsible for the topology of domains observed here. Domain walls in ferroelectrics are at best a few unit cells wide while separations in the order of 100 unit cells are observed in Fig. \[fig3\]. This rather indicates that Coulomb interactions determine the domain structure at the vortex. The Coulomb force is a central force, thus acting in all three spatial dimensions and it is a long-range force that can act across many unit cells. In order to understand the relation of the trimerization and polarization domain walls to the electrostatic repulsion and vortex structure of the domains we subjected the YMnO$_3$ crystal in Fig. \[fig4\] to an electric poling field. A DC voltage of 50 V was applied via the SFM tip along the $z$ axis while scanning a square of $5\times 5$ $\mu$m$^2$. In contrast to Ref.  the voltage was applied non-invasively and the same region was imaged before and after poling. For the domain vortex we have to distinguish between the two scenarios depicted in Figs. \[fig4\](a) and \[fig4\](b): (1) an $\alpha^+$, $\alpha^-$, $\beta^+$, $\beta^-$, $\gamma^+$, $\gamma^-$ and (2) an $\alpha^+$, $\beta^-$, $\gamma^+$, $\alpha^-$, $\beta^+$, $\gamma^-$ sequence of domains.[@footnote] Scenario (1) involves two types of boundaries: polarization-trimerization and polarization-only walls. This may occur if the trimerization walls are so stable that each trimerization domain splits into two polarization domains. Scenario (2) involves polarization-trimerization walls only. This will occur if the creation of additional trimerization domains costs less energy than the creation of a polarization-only wall. Scenario (2) was proposed in Ref.  but not explicitly confirmed there because trimerization and polarization domains were observed in separate experiments, i.e., by force and electron microscopy, respectively. According to the scan shown in Fig. \[fig4\](d) [*all*]{} walls respond to the electric field, thus indicating that scenario (2) as depicted in Fig. \[fig4\](b) holds. This result was confirmed by annealing experiments in which a sample was heated to 1150 K for three hours. After re-cooling all the domain walls, including the trimerization walls, were found to have moved. We therefore conclude that in YMnO$_3$ trimerization and polarization walls are rigidly coupled except in the vortex region where the correlation becomes “flexible” with a relative spatial displacement of trimerization and polarization walls as shown in Fig. \[fig3\](c). Note that in contrast to displacive ferroelectrics it is not possible to convert the entire YMnO$_3$ sample into a single-domain state. Figure \[fig4\](d) shows that the presence of remanent interstitial domains polarized oppositely to the applied electric field are enforced by the electrostatic discontinuity at the domain wall. This nicely reflects the electrostatic nature of the ferroelectric state and explains why in former experiments evidence for a multi-domain state was still obtained when the coercive field was exceeded by an order of magnitude.[@fiebig02] Quantitative analysis of Fig. \[fig4\](d) reveals a width of $60 \pm 10$ nm for the interstitial domain which matches the electrostatic reach found in Fig. \[fig3\]. Our observations thus reveal the following scenario. Growth-induced trimerization leads to intersections where one $\alpha$, one $\beta$, and one $\gamma$ domain meet. The electrostatic discontinuity associated to a trimerization wall controls the ferroelectric polarization up to a distance of about 100 unit cells. It enforces a reversal of polarization whenever a trimerization wall is crossed. The simplest arrangement of domains satisfying these requirements involve kaleidoscopic intersection of six domains as shown in Fig. \[fig4\](b) and Ref. . At the intersection the electrostatic repulsion between equally polarized domains is solved by a decoupling of trimerization and polarization domains near the center of the domain vortex. In summary the unusual kaleidoscopic topology of ferroelectric trimerization-polarization domains in multiferroic hexagonal YMnO$_3$ was shown to be a consequence of electrostatic effects which reflects the unusual electrostatic origin of the spontaneous polarization in this compound. The discontinuity at the domain walls leads to locally polarizing fields that determine the spontaneous polarization within a range of 100 unit cells in all three spatial dimensions. At the intersection of domains this leads to a flexible coupling of trimerization and polarization domains with correlated, but non-overlapping domain walls. Away from the intersection trimerization and polarization walls coincide with domains maintaining a width of at least 100 unit cells. Even beyond saturation fields a remanent ferroelectric multi-domain structure is inherent to the sample which explains former difficulties to transform YMnO$_3$ into a single-domain state. Our results show that the wealth of non-displacive mechanisms driving ferroelectricity that emerged from the present intense research activities on multiferroics are a potential source of alternative domain and polarization phenomena with basic-research as well as technological implications. The authors thank the Deutsche Telekom AG and the DFG for financial support and J. F. Scott for useful advice about multidimensional order parameters. [99]{} W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature [**442**]{}, 759 (2006). D. Khomskii, Physics [**2**]{}, 20 (2009). M. Fiebig, J. Phys. D [**38**]{}, 123 (2005). N. A. Hill, J. Phys. Chem. B [**104**]{}, 6694 (2000). B. B. van Aken, T. T. M. Palstra, A. Filippetti, and N. A. Spaldin, Nat. Mater. [**3**]{}, 164 (2004). T. Choi, Y. Horibe, H. T. Yi, Y. J. Choi, W. Wu, and S.-W. Cheong, Nat. Mater. [**9**]{}, 253 (2010); M. Mostovoy, Nat. Mater. [**9**]{}, 188 (2010). T. Lonkai, D. G. Tomuta, U. Amann, J. Ihringer, R. W. A. Hendrikx, D. M. Többens, and J. A. Mydosh, Phys. Rev. B [**69**]{}, 134108 (2004). C. J. Fennie and K. M. Rabe, Phys. Rev. B [**72**]{}, 100103(R) (2005). M. Šafránková, J. Fousek, and S. A. Kižaev, Czech. J. Phys. B [**17**]{}, 559 (1967). A. S. Oleinik and V. A. Bokov, Sov. Phys. Solid State [**17**]{}, 560 (1975). C. C. Neacsu, B. B. van Aken, M. Fiebig, and M. B. Raschke, Phys. Rev. B [**79**]{}, 100107(R) (2009). M. Fiebig, D. Fr[ö]{}hlich, Th. Lottermoser, and M. Maat, Phys. Rev. B [**66**]{}, 144102 (2002). S. H. Kim, S. H. Lee, T. H. Kim, T. Zyung, Y. H. Jeong, and M. S. Jang, Cryst. Res. Technol. [**35**]{}, 19 (2000). T. Jungk, Á. Hoffmann, and E. Soergel, New J. Phys. [**10**]{}, 013019 (2008). T. Jungk, Á. Hoffmann, and E. Soergel, New. J. Phys. [**11**]{}, 033029 (2009). Scenarios like (1) $\alpha^-$, $\alpha^+$, $\beta^-$, $\beta^+$, $\gamma^-$, $\gamma^+$ or (2) $\alpha^+$, $\gamma^-$, $\beta^+$, $\alpha^-$, $\gamma^+$, $\beta^-$ are equivalent. ![\[fig1\] (Color) (a) YMnO$_3$ crystal at $>1270$ K with the unit cell (small diamond) and three choices for the trimerization at 1270 K (large diamonds). (b) Ferroelectric crystal with tripled unit cell. Legend — Mn$^{3+}$ ions: $z$ position in the unit cell; Y$^{3+}$ ions: local symmetry.](fig1) ![\[fig2\] PFM images ($50\times 50$ $\mu$m$^2$) of the $x$, $y$ and $z$ faces of as-grown YMnO$_3$ crystals from the same batch. Bright and dark areas correspond to ferroelectric domains with $+P_z$ and $-P_z$, respectively.](fig2)  ![\[fig4\] (Color) Structure of trimerization and polarization domains and their response to an electric field. (a, b) Scenarios 1 and 2 as discussed in the text. $\alpha$, $\beta$, $\gamma$ and dotted black lines denote the distribution of domains in the trimerized phase at $>920$ K. $\alpha^{\pm}$, $\beta^{\pm}$, $\gamma^{\pm}$ and straight red lines denote the distribution of domains in the ferroelectric phase at $<920$ K. (c, d) PFM image ($10\times 10$ $\mu$m$^2$) of a $z$ faced YMnO$_3$ crystal (c) before and (d) after tip-voltage poling.](fig4)

--- abstract: | Following Haken [@Ha] and Casson-Gordon [@CG], it was shown in [@Sc] that given a reducing sphere or ${\mbox{$\partial$}}$-reducing disk $S$ in a Heegaard split manifold $M$ in which every sphere separates, the Heegaard surface $T$ can be isotoped so that it intersects $S$ in a single circle. Here we show that when this is achieved by two different positionings of $T$, one can be moved to the other by a sequence of - isotopies of $T$ rel $S$ - pushing a stabilizing pair of $T$ through $S$ and - eyegelass twists of $T$. This last move is inspired by one of Powell’s proposed generators for the Goeritz group [@Po]. address: - | - Microsoft Station Q\ University of California\ Santa Barbara, CA 93106-6105\ and\ Mathematics Department\ University of California\ Santa Barbara, CA 93106-3080 USA - | - Martin Scharlemann\ Mathematics Department\ University of California\ Santa Barbara, CA 93106-3080 USA author: - Michael Freedman - Martin Scharlemann title: 'Uniqueness in Haken’s Theorem' --- It is a classic theorem of Haken [@Ha] that any Heegaard splitting $M = A \cup_T B$ of a closed orientable reducible $3$-manifold $M$ is reducible; that is, there is an essential sphere in the manifold that intersects $T$ in a single circle. Casson-Gordon [@CG Lemma 1.1] refined and generalized the theorem, showing that it applies also to essential disks, when $M$ has boundary and, more specifically, if $E$ is a disjoint union of essential disks and $2$-spheres in $M$ then there is a similar family $E^*$, obtained from $E$ by ambient $1$-surgery and isotopy, so that each component of $E^*$ intersects $T$ in a single circle. (We say $E^*$ is a $T$-reducing system in $M$.) It is now known [@Sc] that in fact we may take $E^* = E$ so long as every sphere in $M$ separates. Here we consider a naturally related uniqueness question: Suppose $E_0$ and $E_1$ are each $T$-reducing systems in $M$, and the systems $E_0$, $E_1$ are isotopic rel ${\mbox{$\partial$}}$ in $M$. Is $E_0$ isotopic to $E_1$ [*via $T$-reducing systems*]{}? Counterexamples spring to mind, even when $M$ is irreducible and each $E_i$ is simply a single disk. [**Example:**]{} Suppose $E_0$ is a ${\mbox{$\partial$}}$-reducing disk for $M$ which intersects $T$ in a single circle, so it is a ${\mbox{$\partial$}}$-reducing disk for $T$. Suppose $T$ is stabilized, with stabilizing disks $D_A \subset A$, $D_B \subset B$ and both disks are disjoint from $E_0$. A regular neighborhood of $D_A \cup D_B$ is a ball ${\mbox{$\beta$}}$ which $T$ intersects in a standard genus $1$ summand. We call such a pair $({\mbox{$\beta$}}, T)$ a [*standard bubble*]{}. We can imagine ${\mbox{$\beta$}}$ as a small ball, the regular neighborhood of a point in the destabilized Heegaard surface $T'$. Now isotope ${\mbox{$\beta$}}$ along a path in $T'$ which passes once through $E_0$, and let $E_1$ be the result of pushing $E_0$ by the resulting ambient isotopy of $M$. Then $E_0$ and $E_1$ are isotopic, but they can’t be isotopic via ${\mbox{$\partial$}}$-reducing disks for $T$, since the circles $T \cap E_i, i = 0, 1$ are not isotopic in $T$. [**Example:**]{} More subtly, suppose $D_A$, $D_B$ are disjoint essential disks in $A$ and $B$ respectively, and $\gamma$ is a path in $T$ connecting their boundaries. The complex $D_A \cup \gamma \cup D_B$ is called an [*eyeglass*]{} for $T$ ([@FS Definition 2.1]). Associated to such an eyeglass is an isotopy of $T$ in $M$ back to itself (with support near the eyeglass) called an [*eyeglass twist*]{}. It is illustrated in Figure \[fig:eyeglass1\]. Suppose $E_0$ is a reducing disk for $T$ and the circle $E_0 \cap T$ essentially intersects the bridge $\gamma$ of the eyeglass. Then the disk $E_1$ obtained by pushing $E_0$ along by the resulting ambient isotopy of $M$ cannot be isotopic via ${\mbox{$\partial$}}$-reducing disks, again since the circles $T \cap E_i, i = 0, 1$ are not isotopic in $T$. $D_A$ at 110 130 $\gamma$ at 155 133 $D_B$ at 200 135 ![Eyeglass twist[]{data-label="fig:eyeglass1"}](eyeglass1 "fig:") Our goal is to show that the two operations just described are essentially the only two obstacles to uniqueness in general, so long as every 2-spheres separates. The extra condition is needed only to invoke [@Sc]. Background and results ====================== Suppose $M = A \cup_T B$ is a Heegaard splitting of a compact orientable $3$-manifold. \[defin:reducer\] A disjoint collection of reducing spheres and ${\mbox{$\partial$}}$-reducing disks for $M$ is an $M$-reducing system; each element is called an $M$-reducer. For $M = A \cup_T B$ a Heegaard splitting, say an $M$-reducing system (resp $M$-reducer) is a $T$-reducing system (resp $T$-reducer) if each component intersects $T$ in a single circle. A disk $M$-reducer whose boundary lies in ${\mbox{$\partial$}}_- A$ (resp ${\mbox{$\partial$}}_- B$) is an $A$-disk (resp $B$-disk). Given an $M$-reducing system $E$, a positioning of $T$ so that $E$ is a $T$-reducing system is called a [*solution*]{} to the $M$-reducing system. Suppose $(E, {\mbox{$\partial$}}E) \subset (M, {\mbox{$\partial$}}M)$ is a $T$-reducing system in $M$. Let $S \subset M$ be a reducing sphere for $T$ that is disjoint from $E$ and cuts off a genus $1$ stabilizing summand of $T$ inside a ball. Let $\gamma$ be an arc in $T$ with one end at a component ${\mbox{$\overline{E}$}}$ of $E$, the other end at $S \cap T$, and $\gamma$ is otherwise disjoint from both $E$ and $S$. Alter ${\mbox{$\overline{E}$}}$ by tube summing it to $S$ along a neighborhood of $\gamma$ and call the result ${\mbox{$\overline{E}$}}'$. Replace ${\mbox{$\overline{E}$}}$ by ${\mbox{$\overline{E}$}}'$ in $E$ and call the result $E'$. $E'$ is obtained by a [*bubble move*]{} on $E$ along $\gamma$ with bubble $S$. We can think of $S$ as a ‘bubble’ that passes through $E$ to create $E'$. Note that $E$ and $E'$ are properly isotopic in $M$. Let $(D_A, {\mbox{$\partial$}}D_A) \subset (A, {\mbox{$\partial$}}A)$ and $(D_B, {\mbox{$\partial$}}D_B) \subset (B, {\mbox{$\partial$}}B)$ be disjoint essential disks that are also disjoint from $E$. Let $\gamma$ be an arc in $T$ transverse to $E$, with one end at ${\mbox{$\partial$}}D_A$, other end at ${\mbox{$\partial$}}D_B$, and otherwise disjoint from $D_A \cup D_B$. Perform an eyeglass twist on $T$ using the eyeglass $D_A \cup \gamma \cup D_B$. The eyeglass twist returns $T$ to itself, but may alter $E$. The image $E'$ of $E$ is said to be obtained from $E$ by an eyeglass twist. Note that if $E$ is disjoint from the eyeglass then $E = E'$; in any case $E$ and $E'$ are properly isotopic in $M$. Suppose $E_0$ and $E_1$ are each a $T$-reducing system in $M$. An isotopy $E_s, 0 \leq s \leq 1$ from $E_0$ to $E_1$ in $M$ is an equivalence (and $E_0, E_1$ are equivalent) if $E_s$ is a $T$-reducing system for all $s$. $E_0$ and $E_1$ are [*congruent*]{} if a sequence of equivalences, bubble moves and eyeglass twists carries $E_0$ to $E_1$. We intend to show: \[thm:main\] Suppose every $2$-sphere in $M$ separates. If $E_0, E_1$ are $T$-reducing systems that are properly isotopic in $M$, then $E_0$ and $ E_1$ are congruent. In conjunction with [@Sc], this means that, when $M$ contains no $S^1 \times S^2$ summand, any $M$-reducing system is isotopic in $M$ to a $T$-reducing system that is unique up to congruence. For the purposes of the proof we will assume that ${\mbox{$\partial$}}M$ contains no sphere components; these add a small amount of complexity, which we leave for the reader to resolve. [**Example:**]{} Theorem \[thm:main\] is obvious for reducing spheres $E_0, E_1$ that intersect $T$ in disjoint circles: Then each component of $E_0 \cap E_1$ is a circle lying in either $A$ or $B$. An innermost one in $E_0 \cap A$, say, cuts off also a subdisk of $E_1 \cap A$. Since $A$ is irreducible, the latter disk can be isotoped to the former in $A$. Eventually such isotopies make $E_0$ and $E_1$ disjoint, so they are parallel in $M$. The splitting that $T$ induces inside of the collar between them is simply a sum of stabilizing pairs, per Waldhausen ([@Wa], [@R]), which can be passed through $E_0$ bubble by bubble until the spheres are equivalent. A similar argument applies if $E_0, E_1$ are ${\mbox{$\partial$}}$-reducing disks. So the interest focuses on cases in which $E_0 \cap E_1 \cap T \neq \emptyset$. Sweepouts, spines, and labels for the graphic ============================================= Here we briefly review the classical sweep-out technology on $M = A \cup_T B$. $A$ (and also $B$) is a compression-body, which can be viewed (dually to the original definition in [@Bo]) as a compact connected orientable $3$-manifold obtained from a (possibly disconnected) surface $\times I$ by attaching $1$-handles to surface $\times \{1\}$. The boundary of $A$ is the disjoint union of ${\mbox{$\partial$}}_- A =$ surface $\times \{0\}$ and a connected surface denoted ${\mbox{$\partial$}}_+ A$. From its construction we see that $A$ deformation retracts to the union of ${\mbox{$\partial$}}_- A$ and the cores of the $1$-handles, where the latter are extended down through ${\mbox{$\partial$}}_- A \times I$ via the product structure. More generally, a [*spine*]{} ${\mbox{$\Sigma$}}$ of $A$ is the union of ${\mbox{$\partial$}}_- A$ and a certain type of graph in $A$: all valence $1$ vertices in the graph lie on ${\mbox{$\partial$}}_- A$, and $A$ deformation retracts to ${\mbox{$\Sigma$}}$; indeed $A - {\mbox{$\Sigma$}}\cong {\mbox{$\partial$}}_+ A \times (0, 1]$. (Sometimes we will not distinguish between ${\mbox{$\Sigma$}}$ and a thin regular neighborhood of ${\mbox{$\Sigma$}}$.) $A$ has many spines, but in an argument that goes back to Whitehead [@Wh] (who was concerned with spheres, not disks) one can change one spine to any other by a sequence of “edge-slides”, in which one edge is slid over others and along ${\mbox{$\partial$}}_- A$ [@ST Section 1]. A properly embedded annulus in $A$ is [*spanning*]{} if its two boundary components lie, one each, in ${\mbox{$\partial$}}_+ A$ and ${\mbox{$\partial$}}_- A$. Let $E_A \subset A$ be a properly embedded disjoint collection of spanning annuli and disks that compress ${\mbox{$\partial$}}_+A$ in $A$. Essentially the same argument as in [@ST Section 1] shows that there is a spine ${\mbox{$\Sigma$}}$ for $A$ with the properties: - Each disk in $E_A$ intersects ${\mbox{$\Sigma$}}$ in a single meridian of an edge. - Each annulus in $E_A$ intersects ${\mbox{$\Sigma$}}$ only in ${\mbox{$\partial$}}_- A$. Moreover, given $E_A$, one can choose the parameterization $A - {\mbox{$\Sigma$}}\cong {\mbox{$\partial$}}_+ A \times (0, 1]$ so that the half-open annuli $E_A - {\mbox{$\Sigma$}}$ are parameterized as $(E_A \cap {\mbox{$\partial$}}_+ A) \times (0, 1]$. We will say that such a spine and parameterization [*comports with $E_A$*]{}. Note that, via Hatcher’s work [@Ha1] ,[@Ha2], the exact parameterization involves no choice, in the sense that its space of parameters is contractible. Combining these ideas, if $E'_A \subset A$ is another such collection, then one can move from a spine (and associated parameterization) that comports with $E_A$ to one that comports with $E'_A$ via a sequence of edge slides. Now we export all these ideas to the setting at hand: a Heegaard split $M = A \cup_T B$ and two $T$-reducing systems $E_0$ and $E_1$ that are isotopic in $M$. Each $E_i$ intersects each compression-body $A$ (resp $B$) in a collection of spanning annuli and essential disks $E_{i, A} = E_i \cap A$ (resp $E_{i, B} = E_i \cap B$). Choose spines ${\mbox{$\Sigma$}}_{i, A} \subset A$ (resp $ {\mbox{$\Sigma$}}_{i, B}\subset B$) so that each comports with $E_{i, A}$ (resp $E_{i, B}$). For each $i = 0, 1$ combine the comporting parameterizations in each compression-body, to parameterize the entire complement of the spines in $M$ as $T \times (0, 1)$, picking the convention that the spine of $A$ is the limit of $T \times \{t\}$ as $t \to 0$. Then the complement of the spines in $M$ is swept-out by copies of $T$ in such a way that each copy of $T$ intersects each component of $E_i$ in a single circle. Denote the copy $T \times \{t\}$ in such a sweepout by $T_t$. The core argument will mirror that of [@FS Section 4], with the isotopy $E_s, 0 \leq s \leq 1$ from $E_0$ to $E_1$ replacing what was there a sweepout of $S^3$ by level $2$-spheres. In addition we use $s$ to simultaneously parameterize a movie of the sequence of edge slides on the spines that take ${\mbox{$\Sigma$}}_{0, A} \cup {\mbox{$\Sigma$}}_{0, B}$ to ${\mbox{$\Sigma$}}_{1, A} \cup {\mbox{$\Sigma$}}_{1, B}$. Together, this sweep-out and the isotopy $E_s$ (together with edge slides on the spines) produce a “graphic" ${\mbox{$\Gamma$}}$ in the $(t, s)$-square $I \times I$. The graphic consists of open regions $R_i$ where $E_s$ and $T_t$ intersect transversely, edges or “walls" where the two have a tangency, and cusp points where two types of tangencies cancel. As argued in [@RS] only domain walls corresponding to saddle tangencies need to be tracked. Cusps and tangencies of index 2 or 0 can be erased as they amount only to births/deaths of inessential simple closed curves of intersection in $E_s \cap T_t$. The most interesting event which occurs are transverse crossings of saddle walls; at this point two independent saddle tangencies occur. Label a region of the graphic as follows: - Ignore circles in $T_t \cap E_s$ that are inessential in $T_t$, - Label the region $A$ if there is a circle $a$ of $T_t \cap E_s$ so that either - $a$ is innermost in $E_s$ among essential circles in $T_t \cap E_s$, and the disk in $E_s$ that it bounds lies in $A$ or - $a$ is parallel in $E_s$ to a boundary component, and the collar between them lies in $A$ - Label a region $B$ if there are [*no*]{} such circles $a$ as above, but there is [*at least one circle*]{} $T_t \cap E_s$ that is innermost in $E_s$ among essential circles in $T_t \cap E_s$ and the disk in $E_s$ that it bounds lies in $B$. Note that the definition of the labeling breaks symmetry: A collar of ${\mbox{$\partial$}}A$ is counted as if it were an innermost disk but a collar of ${\mbox{$\partial$}}B$ is not; and a region in which there are essential disks in both $A$ and $B$ is labeled $A$. In the figures illustrating our argument we will distinguish between the compression-bodies $A$ and $B$ by color: pinkish (nominally red) will denote $A$, while azure (nominally blue) will denote $B$. This distinction will color regions of $E$ cut out by $T$ alternately red and blue. For example, consider Figure \[fig:hidden\]. The bi-colored horizontal plane shows part of a component of $E$. Two parts of $T$ are shown - a large-diameter vertical annulus, separating the visible part of $E$ into a blue unbounded region and a red ‘pair of pants’; and - an inverted-U-shaped annulus that separates a blue $1$-handle of $B$ (the ‘blue tube’) from the part of $A$ that contains the red pair of pants. Figure \[fig:hidden\] shows the blue tube bounded by part of $T$ being lowered through the reducer $E$. In so doing the $(s, t)$ parameter designating the two surfaces passes from one region of the graphic to another. An astute viewer will notice a gray area at the top of the blue tube reflecting ambiguity on what might lie there: Is the top of the tube just a disk, or does a chimney filled with blue ascend through it? This could be an important question, as we will discuss shortly. We will also use the red-blue coloring scheme on the graphic: Regions that are labeled $A$ will be colored red; those labeled $B$ will be colored blue. Skip ahead to Figure \[fig:graphicsketch\] to see how the coloring scheme might then appear in $I \times I$ containing the graphic. The idea of the proof of Theorem \[thm:main\] can already be discerned in this figure: Ultimately we will walk around the outside boundary of the big red region and observe that every step corresponds to some combination of an isotopy, a bubble move or an eyeglass twist. Return now to the ongoing argument: The first labeling rule above – ignore circles of intersection that are inessential in $T_t$ – raises a [*caveat*]{}: When we say that an essential circle $a$ bounds a disk in $A$ (similarly for $B$), what is technically meant is that there is a planar surface $P \subset A$ whose boundary consists of $a$ and, possibly, a collection of circles that are inessential in $T$. $E$ at 30 120 $T$ at 195 150 $T$ at 185 100 ![Label change or not?[]{data-label="fig:hidden"}](hidden "fig:") As a consequence, crossing from one region of the graphic to another may change the label from $B$ to $A$ in surprising ways, as shown in Figure \[fig:hidden\]. As the saddle point in $T$ passes down through $E$, the label will change from $B$ to $A$ if the grey disk at the top is inessential in $T$. If the grey disk is essential in $T$, so $T$ ascends beyond it, the label remains $B$. The difference between the two situations cannot be determined just by examining the behavior of $E \cap T$. Labels around the boundary of the graphic ========================================= In thinking about the labeling scheme, consider first the situation near $s = 0, 1$. Since the parameterization $T \times (0, 1)$ in each case comports with $E$, any component of $ E_i$ is swept out by a single circle. One side of the circle is a disk or annulus lying entirely in $A$. Thus all regions near $s = 0, 1$ are labeled $A$. Also, near $t = 0$, $T_t$ is near a spine of $A$, which must intersect each component of $E_s$, since $B$ is incompressible and ${\mbox{$\partial$}}_- B$ does not compress in $B$. Such an intersection point with the spine (or possibly a component of ${\mbox{$\partial$}}E_s$ in ${\mbox{$\partial$}}_- A$) means that near $t=0$, $S_s \cap T_t$ will cut out from $E_s$ a small disk in $A$ (or a thin annulus near a component of ${\mbox{$\partial$}}E_s$ at ${\mbox{$\partial$}}_- A$). Thus the regions adjacent to $t=0$ are again all labeled $A$. The labeling of regions near $t=1$ is more subtle and contains a warm-up for the general case. Because $T$ is near a spine of $B$, each circle of intersection with $E_s$ either bounds a disk in $B$ or is parallel in $B$ to a boundary component, as we have just noted. Consider first the three simple cases that can arise when $E = E_s$ is a single component: - Suppose $E$ is a reducing sphere for $M$. Since each component of $E \cap T$ bounds a small disk in $B$, a region will be labeled $A$ if and only if there is a single circle of intersection, in other words, if and only if $E$ is a reducing sphere for $T$. - If $E$ is a ${\mbox{$\partial$}}$-reducing disk whose boundary lies in ${\mbox{$\partial$}}_- B$ then $T$ intersects $E$ in at least a circle parallel to ${\mbox{$\partial$}}E$ and the collar lies in $B$. But if there is any other circle of intersection, the small disk it bounds lies in $B$, so there are no disks in $A$. Again, a region will be labeled $A$ if and only if $E$ is a ${\mbox{$\partial$}}$-reducing disk for $T$. - Suppose $E$ is a ${\mbox{$\partial$}}$-reducing disk for $M$ whose boundary lies on ${\mbox{$\partial$}}_- A$. Since ${\mbox{$\partial$}}_- A$ is incompressible in $A$, not all of $E$ can lie in $A$ so there is at least one circle of intersection. There is exactly one circle if and only if that circle is ${\mbox{$\partial$}}$-parallel in $A$ and so labels the region $A$. So, once again, the region is labeled $A$ if and only if $E$ is a ${\mbox{$\partial$}}$-reducing disk for $T$. Hence we have (see Figure \[fig:graphicsketch\]): Suppose $E$ has a single component and the regions adjacent to the side $t = 1$ are all labeled $A$. Then $E_0$ and $E_1$ are equivalent $T$-reducers. When $E$ has many components, the argument is more complicated, since our labeling scheme assigns the label $A$ if just one component of $E_s$ is a $T$-reducer. So at this point we make a crucial inductive assumption: \[ass:induct\] Theorem \[thm:main\] is true in all cases for which the genus of the splitting surface is less than the genus of $T$. Note that Theorem \[thm:main\] is more or less obvious when $genus(T) = 1$. Suppose in a region of the graphic a component of $E = E_s$ is a $T$-reducer $\overline{E}$. Then there is a natural way of isotoping $T$ rel $\overline{E}$ to a solution for all of $E$: Reduce or ${\mbox{$\partial$}}$-reduce $(M, T)$ along $\overline{E}$ to obtain a new Heegaard split manifold $M' = A' \cup_{T'} B'$ (disconnected if $E$ is separating). Each component of $T'$ has genus less than $genus(T)$. By [@Sc], $T'$ can be isotoped in $M'$ so that the family $E - \overline{E}$ in $M'$ is $T'$-reducing, and by Assumption \[ass:induct\] this solution for $(M', T')$ is well-defined up to congruence. This solution, together with ${\mbox{$\overline{E}$}}$, constitutes a natural solution to $(M, T)$. Call it the [*solution generated by $\overline{E}$*]{}. \[lemma:generate\] Suppose $\overline{E}'$ is another $T$-reducer in $E$. Then the solution generated by $\overline{E}'$ is congruent to that generated by $\overline{E}$. If $E$ is a $T$-reducing system, then it is the solution generated by any of its members. The solution obtained by compressing (or ${\mbox{$\partial$}}$-compressing) along both $\overline{E}'$ and $\overline{E}$ is congruent to that generated by either, per Assumption \[ass:induct\]. In view of Lemma \[lemma:generate\] we can simply call such a solution in the region [*internally generated*]{} without naming the component of $E$ that generates it. \[lemma:pairgenerate\] Suppose two regions of the graphic, adjacent along an edge of each, have internally generated solutions. Then these solutions are congruent. Let $\overline{E}$ and $\overline{E}'$ be generators in adjacent regions $R, R'$ respectively. If $\overline{E} = \overline{E}'$ congruence follows by definition, so we assume $\overline{E} \neq \overline{E}'$. The edge between regions $R$ and $R'$ represents $E$ passing through a single saddle tangency with $T$, a point that may lie on $\overline{E}$ or $ \overline{E}'$ but not both. Thus at least one of the two is a generator in both regions, from which congruence of solutions follows by Lemma \[lemma:generate\]. Return now to the setting for Theorem \[thm:main\] and we have: \[prop:side1\] Suppose the regions adjacent to the side $t = 1$ are all labeled $A$. Then $E_0$ and $E_1$ are congruent. The label $A$ implies that each region adjacent to the side $t=1$ has a self-generated solution. Lemma \[lemma:pairgenerate\] ensures that the congruence class of the solution doesn’t change as we move along the side $t=1$ from $E_0$ to $E_1$. A forbidden labeling around a vertex {#sect:forbidden} ==================================== Focus now how labels behave around a vertex in the interior of the graphic $\Gamma$. Such a vertex corresponds to a position of $T = T_t$ in which it has two simultaneous tangency points with $E = E_s$. The non-trivial cases arise when both points of tangency lie on a single component ${\mbox{$\overline{E}$}}\subset E$. If ${\mbox{$\overline{E}$}}$ is a disk, a simple combinatorial argument shows that there are 15 possible configurations of these tangency points, shown in Figure \[fig:vertexresolved\]. The same diagram can be used when ${\mbox{$\overline{E}$}}$ is a sphere, but far fewer panels are needed because of the extra symmetry this introduces. For example, panels 10, 11 and 12 are the same in a sphere, as are 13 and 14. We will proceed assuming ${\mbox{$\overline{E}$}}$ is a disk; if it is a sphere, just delete an open disk near a point in $A$, converting it to an $A$-disk, and apply the arguments there. There are typically many more circles in ${\mbox{$\overline{E}$}}\cap T$ than are shown in the panels of Figure \[fig:vertexresolved\]; these only show the components containing tangency points. The two tangency points will be denoted $\rho = \rho_{\pm}$; the 4 quadrants near it correspond to the 4 ways of resolving the tangencies, each by perturbing $T$ slightly near $\rho$. $1$ at 85 195 $2$ at 160 195 $3$ at 235 195 $4$ at 305 195 $5$ at 380 195 $6$ at 85 105 $7$ at 160 105 $8$ at 235 105 $9$ at 305 105 $10$ at 380 105 $11$ at 85 15 $12$ at 160 15 $13$ at 230 15 $14$ at 305 15 $15$ at 385 15 ![At a vertex in the graphic $\Gamma$ []{data-label="fig:vertexresolved"}](vertexresolved "fig:") The picture in $T$ can be more complicated than these panels suggest. For example, panel 15 might look like Figure \[fig:resolved\] in $T$. (0.75, -1.5) – (4.75,-1.5); (2.75, -3.5) – (2.75,0.5); (1,-2.4) ellipse (0.08 and 0.25); (2.3,-2.65) arc (-90:90:0.08 and 0.25); (2.3,-2.15) arc (90:270:0.08 and 0.25); (1,-2.15) to \[out=25,in=155\] (2.3,-2.15); (1,-2.65) to \[out=-25,in=205\] (2.3,-2.65); (1.3,-2.05) arc (90:-90:0.1 and 0.35); (1.3,-2.05) arc (90:270:0.1 and 0.35); (1.85,-2.4) arc (0:-180:0.2 and 0.1); (1.8,-2.46) arc (0:180:0.15 and 0.07); (3.2,-0.6) ellipse (0.08 and 0.25); (3.2,-0.35) to \[out=25,in=155\] (4.5,-0.35); (4.5,-0.35) arc (90:-90:0.08 and 0.25); (4.5,-0.35) arc (90:270:0.08 and 0.25); (3.2,-0.85) to \[out=-25,in=205\] (4.5,-0.85); (4.06,-0.6) arc (0:-180:0.2 and 0.1); (4.01,-0.66) arc (0:180:0.15 and 0.07); (4.22,-0.25) arc (90:-90:0.1 and 0.35); (4.22,-0.25) arc (90:270:0.1 and 0.35); (3.2,-2.4) ellipse (0.08 and 0.25); (3.2,-2.15) to \[out=25,in=155\] (4.5, -2.15); (4.5,-2.15) arc (90:-90:0.08 and 0.25); (4.5,-2.15) arc (90:270:0.08 and 0.25); (3.2,-2.65) to \[out=-25,in=205\] (4.5,-2.65); (4.06,-2.4) arc (0:-180:0.2 and 0.1); (4.01,-2.46) arc (0:180:0.15 and 0.07); (3.85, -2) arc (90:-90:0.07 and 0.195); (3.85, -2) arc (90:270:0.07 and 0.195); (3.85, -2.5) arc (90:-90:0.05 and 0.155); (3.85, -2.5) arc (90:270:0.05 and 0.155); (1,-0.6) ellipse (0.08 and 0.25); (1,-0.35) to \[out=25,in=155\] (2.3,-0.35); (2.3,-0.35) arc (90:-90:0.08 and 0.25); (2.3,-0.35) arc (90:270:0.08 and 0.25); (1,-0.85) to \[out=-25,in=205\] (2.3,-0.85); (1.86,-0.6) arc (0:-180:0.2 and 0.1); (1.81,-0.66) arc (0:180:0.15 and 0.07); (1.66,-0.63) ellipse (0.3 and 0.2); (1.26, -0.25) arc (90:270:0.1 and 0.35); (1.26, -0.25) .. controls (2.35,-0.3) and (2.35,-0.95) .. (1.26, -0.95); \[prop:diagonals\] No vertex in the graphic is surrounded by labeling pattern $\arraycolsep = 2.0pt \begin{array}{c|c} A & B \\ \hline B & A \end{array}$. The simply connected components of ${\mbox{$\overline{E}$}}- T$ that are shown in Figure \[fig:vertexresolved\] will each become a disk in some resolution of the tangency points; if such a disk contains no other essential circles of $T$ and is essential in $A$ we will call the component an $A$-leaf component and the disk it becomes an $A$-leaf. (The terminology is explained in the next section.) A component of ${\mbox{$\overline{E}$}}- T$ shown in the diagram is incident either to one of $\rho_{\pm}$ or to both. \[lemma:1tangency\] If an $A$-leaf component is incident to only one of $\rho_{\pm}$, then the labeling around the vertex is not $\arraycolsep = 2.0pt \begin{array}{c|c} A & B \\ \hline B & A \end{array}$ Resolve the single tangent point so that the component becomes an $A$-leaf. Either resolution at the other tangent point (these corresponding to two adjacent quadrants in $\Gamma$) leaves the $A$-leaf intact, so these two adjacent quadrants both get labeled $A$. \[lemma:2tangency\] At a vertex in $\Gamma$ with surrounding labels $\arraycolsep = 2.0pt \begin{array}{c|c} A & B \\ \hline B & A \end{array}$ the two $A$ labels cannot both come from $A$-leaf components. Following Lemma \[lemma:1tangency\] each $A$-label must come from an $A$-leaf component that is incident to both $\rho_{\pm}$. This eliminates panels 1 through 9. Moreover, the two $A$-leaf components arise from different resolutions on each tangency point, since they are diagonally opposite. Only panels 11 and 15 have two $2$-vertex components, but in panel 11 they are adjacent and so they can’t both lie in $A$. In panel 15, a resolution of the tangencies that turns an $A$-leaf component into an $A$-leaf, when reversed, only gives disk components that contain points in $B$. This would seem to prove Proposition \[prop:diagonals\], until we remember that $A$-labels may arise in another way, as shown in Figure \[fig:hidden\]. For example in panel 4, the annulus component of ${\mbox{$\overline{E}$}}- T$ that is shown might lie in $A$, and the interior pair of circles might bound parallel disks in $B$, but when the pair is resolved into a single circle, it is inesential in $T$. Call such a component of $E - T$ in a panel an $A$-annulus. \[lemma:annulus\] At a vertex in $\Gamma$ with surrounding labels $\arraycolsep = 2.0pt \begin{array}{c|c} A & B \\ \hline B & A \end{array}$, neither $A$ label can come from an $A$-annulus. Suppose one of the quadrants gets its $A$-label via an $A$-annulus, as described. Such a component could arise in panels 4, 5, 6, 7, 8 and 9. In order to have the given labeling the opposite resolution at both $\rho_{\pm}$ should again generate an $A$-label. The label can’t come from the same $A$-annulus since its inner boundary is no longer adjacent to an inessential disk. Thus the $A$-label must come from an $A$-leaf component and, by observation, each $A$-leaf component is incident to only one of $\rho_{\pm}$. This contradicts Lemma \[lemma:1tangency\] A final way in which $A$-labels might arise is via ‘hidden components’. Remember that the panels only show components of $T \cap E$ that are incident to the tangency points. Imagine a circle $c$ of $E \cap T$ bounding a disk that contains the pair of components shown in panel 1. If both of $\rho_{\pm}$ resolve as in Figure \[fig:hidden\], the resulting disk bounded by $c$ could generate a label $A$. The hidden component here is the ‘pair of pants’ bounded by $c$ and the two components in the panel; it is hidden because $c$ does not appear in the panel. But it is easy to see that hidden pairs of pants (which could arise in panels 1, 2, and 3) can’t possibly give rise to the labeling $\arraycolsep = 2.0pt \begin{array}{c|c} A & B \\ \hline B & A \end{array}$. Hidden annuli require more thought. Suppose a circle component $c$ of $E \cap T$ cobounds an annulus with a component $X$ from one of the panels. It may be possible to resolve the tangency points in $X$ so that the end of the annulus at $X$ bounds an inessential circle, so it might in this way be possible for $c$ to generate a label $A$. By the argument of Lemma \[lemma:1tangency\] such a hidden annulus can be part of a labeling $\begin{array}{c|c} A & B \\ \hline B & A \end{array}$ only if the end at $X$ is incident to both tangency points $\rho_{\pm}$. This immediately rules out panels 1 through 9 as well as 11 and 14. The end at $X$ must also have the property that the opposite resolution at both $\rho_{\pm}$ will still give rise to an $A$-disk, and the new $A$-disk must be incident to both $\rho_{\pm}$, by Lemma \[lemma:1tangency\]. This eliminates panels 12 and 13. Panel 10 won’t work: each leaf component shown has points in $B$, since the hidden annulus lies in $A$. Finally, these requirements can be fulfilled in panel 15 only if the middle sector lies in $A$ and the two other sectors are inessential in $B$. But in that case, there could be no label $B$ in any quadrant. A technical note: our labeling convention assigns the label $A$ also if one of the regions in ${\mbox{$\overline{E}$}}- T$ is a collar of the boundary in an $A$-disk. The argument in this case is identical to that given above for the case in which there is a hidden circle that completely surrounds the figure in each panel. From weakly reducing to reducing {#sect:weakreduce} ================================ In [@CG] Casson-Gordon introduced the notion of a weakly reducible Heegaard splitting, rejuvenating Heegaard theory. They showed that if there are disjoint essential disks in $A$ and $B$, then simultaneous compression on a maximal family of such disjoint disks in $A$ and $B$ will produce either a reducing sphere for $T$ or an incompressible surface or both. In considering uniqueness, the choice of a ’maximal family’ is problematic, since such a family is far from unique. In this section we avoid this problem of choice, by deriving from the entire pattern of circles $T \cap E$ in $E$ a recipe to move from a weakly reducing system for $T$ to a full $T$-reducing system, in a series of steps that is well-defined up to congruence. Suppose $E$ is an $M$-reducing system for $M = A \cup_T B$. $E$ will be called a [*weak solution*]{} if, among the components of $E - T$, there are ${\mbox{$\partial$}}$-reducing disks for both $A$ and $B$. Continuing under Assumption \[ass:induct\], we will describe a natural algorithm that transforms a weak solution $E$ into a $T$-reducing system, an algorithm that is well-defined up to congruence. Denote by ${\mbox{$\mathcal {D}_A$}}$ (resp ${\mbox{$\mathcal{D}_B$}}$) the collection of all disk components of $E$ cut off by $T$ that lie in $A$ (resp $B$). Figure \[fig:Eview\] illustrates the idea in a $B$-disk component of $E$. ${\mbox{$\mathcal {D}_A$}}$ at 110 35 ${\mbox{$\mathcal{D}_B$}}$ at 175 150 ![The view in $E$[]{data-label="fig:Eview"}](Eview "fig:") Consider the surface $T_c \subset M$ obtained by compressing $T$ along $\mathcal{D}_A \cup \mathcal{D}_B $. $T_c$ divides $M$ into two (possibly disconnected) 3-manifolds $M_A$ and $M_B$. Imagine thickening $T_c$ by expanding it into a bi-collar as shown in Figure \[fig:MAMB\]. This would induce Heegaard splitting surfaces $T_A \subset M_A$, obtained from the original $T$ by compressing only along $\mathcal{D}_A$ and then pushing towards the $A$-side of the bicollar. The symmetric construction gives a Heegaard splitting surface $T_B$ in $M_B$, ${\mbox{$\mathcal {D}_A$}}$ at 120 115 $T$ at 135 200 ${\mbox{$\mathcal{D}_B$}}$ at 70 140 $A$ at 30 115 $B$ at 150 115 $M_A$ at 200 140 $T_A$ at 200 215 $M_B$ at 285 110 $T_B$ at 285 200 $T_c$ at 245 200 ![The view in $M$, and a mental image[]{data-label="fig:MAMB"}](MAMB "fig:") Both $T_A$ and $T_B$ have lower genus than $T$, so our inductive Assumption \[ass:induct\] applies. In particular, given any $M_A$-reducing system of disks and spheres, $T_A$ can be isotoped, uniquely up to congruence, so that the system becomes $T_A$ reducing (and similarly for $(M_B, T_B)$). Such an isotopy of $T_A$ can be described [@Sc] as a sequence of handle-slides of and over the handles whose cocores are the ${\mbox{$\mathcal {D}_A$}}$ disks. But these same handle-slides could have been done on and over the handles as they actually lie on $T_c$, avoiding (by general position) the attaching disks for the handles on the other side, those with cocores the disks ${\mbox{$\mathcal{D}_B$}}$. In thinking of this as an isotopy of the original Heegaard surface $T$, the exact trajectory which the handle-slides follow across $T_c$ in order to avoid the disks ${\mbox{$\mathcal{D}_B$}}$ is, for our purposes, unimportant: one choice can be moved to another by eyeglass twists of $T$. The symmetric argument applies to $M_A$. The upshot is: \[prop:handleslide\] Suppose $E_A \subset M_A$ and $E_B \subset M_B$ are embedded collections of ${\mbox{$\partial$}}$-reducing disks whose boundaries lie on $T_c \subset M$. Then there is an isotopy of $T$, keeping $T_c$ setwise fixed, to a position in which the boundary of each disk remains unchanged in $T_c$ and the interior of each disk is disjoint from $T$. The isotopy is well-defined up to congruence. Consider a component $P$ of $E - T$ which is next to innermost, i.e. all but one of its boundary component is an innermost circle in $E \cap T$, and so each bounds a disk in ${\mbox{$\mathcal {D}_A$}}$ (or each bounds a disk in ${\mbox{$\mathcal{D}_B$}}$). Then the exceptional boundary component ${\mbox{$\partial$}}_0 P$ lies in $T_c$ and bounds a disk $D_P$ in $M_B$ (or $M_A$), through which the $1$-handles dual to ${\mbox{$\mathcal {D}_A$}}$ or ${\mbox{$\mathcal{D}_B$}}$ may pass. The algorithm is then: 1. Apply Proposition \[prop:handleslide\] to the collection of all such components $P$ of $E - T$, isotoping $T$ without changing $T_c$ so that afterwards the interior of each disk $D_P$ is disjoint from $T$. 2. Add each such disk $D_p$ to ${\mbox{$\mathcal{D}_B$}}$ or ${\mbox{$\mathcal {D}_A$}}$ as appropriate, compressing $T_c$ to $T'_c$ 3. Repeat the process until at least one component $\overline{E}$ of $E$ is a $T$-reducer. 4. The output is the solution generated by $\overline{E}$. It will be important for its application that the algorithm is robust: a minimal change in input information will result in the same output. To understand more fully how the algorithm operates, we can describe it schematically. The pattern of circles $T \cap E$ in $E$ defines a tree in each component ${\mbox{$\overline{E}$}}$ of $E$, with a vertex for each component of $E - T$ and an edge connecting two components if there is a single circle of $E \cap T$ between them. The tree has a natural base or root when ${\mbox{$\overline{E}$}}$ is a disk, namely the component of ${\mbox{$\overline{E}$}}- T$ containing the boundary. Let $Y$ denote the forest that is the whole collection of trees. The innermost disks of $E - T$ can be thought of as leaves in the forest $Y$. One measure of the complexity of each tree is the diameter of the tree, when ${\mbox{$\overline{E}$}}$ is a sphere, or the height of the tree when ${\mbox{$\overline{E}$}}$ is a disk. (Tree height is the edge-distance from the root of the tree to the most distant leaf. See Figure \[fig:Tree1\]). $0$ at 550 60 $4$ at 315 180 $2$ at 360 220 ![Tree height is 4[]{data-label="fig:Tree1"}](Tree1 "fig:") The $B$-leaves of $Y$ correspond to $\mathcal{D}_B$, and $A$-leaves to $\mathcal{D}_A$. The [*branch-structure*]{} $Y_c$ of $Y$ is obtained from $Y$ by removing all leaves; alternatively, it is the forest determined by the circles $T_c \cap E$ in $E$. $Y_c$ at 400 60 ![Branch structure from $T_c \cap E$[]{data-label="fig:Branch"}](Branch "fig:") The leaves of the branch structure correspond to the “second-innermost" circles in the algorithm described above or, in terms of the original forest, they are the outermost forks. Applying the algorithm described above replaces the original leaves with new leaves, corresponding to what were originally outermost forks. Since we have no control over how the $1$-handles of $T_A \subset M_A$ and $T_B \subset M_B$ intersect the non-disk components of $E - T_c$, leaves may also sprout from every other vertex in $Y_c$. But one iteration of the algorithm described will decrease the height (or diameter) of each tree. This is shown schematically in Figure \[fig:Step2b\], where new leaves sprout in non-disk components of $E - T_c$. at 350 80 at 350 60 at 350 40 ![De/refoliation of $B$-leaves; height is now 3[]{data-label="fig:Step2b"}](Step2b "fig:") Since $genus(T_A) < genus (T)$, the inductive hypothesis says that the new $A$-leaves of $Y'$ (the ones contributed by what were previously outermost forks) are well-defined in $T_A$ up to congruence, so similarly well-defined in $T$. Add them to $\mathcal{D}_A$, compressing $T_c$ into $A$ and effectively ${\mbox{$\partial$}}$-reducing $T_A \subset M_A$. See Figure \[fig:MakeY2\], Call the augmented collection $\mathcal{D'}_A$. ${\mbox{$\mathcal {D}_A$}}$ at 120 115 $T$ at 135 200 ${\mbox{$\mathcal {D}_A$}}'$ at 360 120 $A$ at 30 115 $B$ at 150 115 ![New $A$-leaves added to $\mathcal{D}_A$ []{data-label="fig:MakeY2"}](MakeY2) A similar argument applies in $M_B$, resulting in new surfaces $T'_A, T'_B$ and $T'_c$, the latter dividing $M$ now into $M'_A, M'_B$. Continue with the algorithm until the height or diameter in some component $\overline{E}$ is $1$. (We pause to note the last step). $T$ now divides $\overline{E}$ into a planar surface in $B$, say, (which is incident to ${\mbox{$\partial$}}\overline{E}$ if $\overline{E}$ is a disk) and a collection of disks in $A$, all of them lying in a submanifold of $M$ with a lower genus Heegaard splitting surface. Once again apply the Strong Haken Theorem [@Sc] together with the inductive hypothesis on this lower genus splitting to isotope $T$ so that $\overline{E}$ is a solution. This completes the algorithm. Appendages and convergence ========================== Section \[sect:weakreduce\] described an algorithm which proceeds by well-defined iteration from a weak solution of an $M$-reducing system $E$ into a $T$-reducing system for $M = A \cup_T B$, at each step augmenting the number of disjoint weakly reducing disks. \[defin:converge\] Two weak solutions for $E$ [*converge*]{} if the algorithm results in congruent $T$-reducing systems. The term is meant to convey that, after perhaps some iterations in the algorithm, the two weak solutions may become indistinguishable, even before they each become full $T$-reducing systems. An important example of convergent weak solutions begins with this easy corollary of Theorem \[thm:main\]: \[cor:appendage\] With hypotheses as in Theorem \[thm:main\], suppose that $\mathcal{D} \subset M$ is a collection of ${\mbox{$\partial$}}$-reducing disks for $T$ that is disjoint from the two systems $ E_i$. Then the sequence of eyeglass twists and bubble moves creating the congruence may be taken to be disjoint from $\mathcal{D}$. The Corollary follows from Assumption \[ass:induct\], which allows us to apply Theorem \[thm:main\] to the lower-genus Heegaard split manifold $(M', T')$ obtained from $(M, T)$ by ${\mbox{$\partial$}}$-reducing along the disks $\mathcal{D}$. Suppose $E$ is an $M$-reducing system whose intersection with $T$ is a weak solution, and ${\mbox{$\mathcal {D}_A$}}\subset M_A$ and ${\mbox{$\mathcal{D}_B$}}\subset M_B$ are the leaves, as described above. \[prop:appendage\] Suppose $(D, {\mbox{$\partial$}}D) \subset (M_B, {\mbox{$\partial$}}M_B)$ (resp $(M_A, {\mbox{$\partial$}}M_A)$ is a properly embedded essential disk that is disjoint from $E$. Then the solution to $E$ given by the algorithm is unaffected by adding $D$ to ${\mbox{$\mathcal{D}_B$}}$ (resp ${\mbox{$\mathcal {D}_A$}}$) at the start. We will show that in both $M_A$ and $M_B$ the algorithm is unaffected by the addition of $D$ to ${\mbox{$\mathcal{D}_B$}}$. (Of course if $D$ is parallel in $T$ to an element in ${\mbox{$\mathcal{D}_B$}}$ there is nothing to show.) This follows from Corollary \[cor:appendage\] for $M_B$, since $D$ can be regarded as a ${\mbox{$\partial$}}$-reducing disk in $M_B$. In $M_A$ the addition of $D$ changes the status of its dual $1$-handle from being part of the boundary of $M_A$ to being a $1$-handle in the splitting of $M_A$ by $T_A$. This is a profound change, but the original algorithm describes passing $1$-handles past ${\mbox{$\partial$}}D$, i. e. over the new $1$-handle, and this can still be done. Since $D$ is disjoint from $E$ the algorithm never requires the new $1$-handle to move. And so the algorithm can proceed step after step, never noticing the new $1$-handle, until a solution is achieved. Because of its inactivity in the proof, a disk $D$ as in Proposition \[prop:appendage\] is called an [*appendage disk*]{}. \[lemma:redundant\] Suppose there are multiple components in a weakly reducing system $E$, and the system $E'$ obtained by removing one of them remains weakly reducing. Then the solution provided by the algorithm is unaffected by the removal. (That is, the two weak solutions converge.) The case is in which $E$ consists of only two components $E = {\mbox{$\overline{E}$}}_{\pm}$ is definitive; the general case is no harder. At the beginning of the algorithm on $E$, $M_A$ and $M_B$ are defined by compressing $T$ along the disk components of $E - T$. Now remove ${\mbox{$\overline{E}$}}_-$ and note that the algorithm applied to ${\mbox{$\overline{E}$}}_+$ would call for compressing only along the disk components of ${\mbox{$\overline{E}$}}_+ - T$. But the outcome of that algorithm is unaffected by further compressing by disk components on ${\mbox{$\overline{E}$}}_-$, by Proposition \[prop:appendage\]. So, at the initial stage, there is no difference between the eventual solutions. Just continue in this manner, using how the algorithm behaves on the ‘virtual’ componoent ${\mbox{$\overline{E}$}}_-$ to present extra disks to be included as appendages (under Proposition \[prop:appendage\]) as the algorithm is applied to ${\mbox{$\overline{E}$}}_+$ alone. Eventually the parallel algorithms stop, when one of ${\mbox{$\overline{E}$}}_{\pm}$ becomes a $T$-reducer. If it stops because ${\mbox{$\overline{E}$}}_{+}$ is a $T$-reducer, then we have shown that the solutions on $E$ and ${\mbox{$\overline{E}$}}_+$ result in the same solution, as required. If it stops because ${\mbox{$\overline{E}$}}_{-}$ is a $T$-reducer, then the algorithm for $E$ declares that a solution consists of ${\mbox{$\overline{E}$}}_{-}$, together with [*any*]{} solution for ${\mbox{$\overline{E}$}}_{+}$ in the manifold obtained by reducing (or ${\mbox{$\partial$}}$-reducing) $(M, T)$ along ${\mbox{$\overline{E}$}}_{-}$. An example of such a solution is given by the output of the algorithm further played out on ${\mbox{$\overline{E}$}}_+$. Suppose an edge in the graphic lies between a region labeled $A$ and a region labeled $B$. The edge indicates a saddle tangency of $E$ with $T$. Let $\overline{E}$ be the component of $E$ that contains the saddle tangency. Let ${\mbox{$\overline{E}$}}_A, {\mbox{$\overline{E}$}}_B$ be slight push-offs of ${\mbox{$\overline{E}$}}$ corresponding to the regions labeled $A$ and $B$ respectively. The [*edge weak solution*]{} is the weak solution obtained by deleting ${\mbox{$\overline{E}$}}$ from $E$ and replacing with ${\mbox{$\overline{E}$}}_A \cup {\mbox{$\overline{E}$}}_B$. \[prop:edgesoltn\] An edge weak solution converges with a weak solution (if any) determined by either adjacent region. Suppose there is a weak solution for the adjacent region labeled $A$ (resp $B$). That weak solution is obtained from the edge weak solution by just deleting ${\mbox{$\overline{E}$}}_B$ (resp ${\mbox{$\overline{E}$}}_A$). The result then follows from Lemma \[lemma:redundant\]. Return to the graphic ===================== Return now to the proof of Theorem \[thm:main\] by examining the graphic more closely, inspired by [@FS Subsection 4.5] and adopting similar conventions. An edge in the graphic that lies between a region labeled $A$ and a region labeled $B$ will be called a [*border edge*]{}. Following Section \[sect:forbidden\], any vertex in the graphic that is incident to a border edge is incident to exactly two border edges (or to the boundary of the graphic). Thus the collection of border edges constitute a properly embedded $1$-manifold in the graphic that separates $A$ regions from $B$ regions. We have shown earlier that three sides of the graphic ($s = 0, 1$ and $t = 0$) are adjacent to $A$-regions. Since the union of the three sides is connected, there is a single component $\mathcal{A}$ of the complement of the border edges that contains all three sides in its boundary. $\mathcal{A}$ consists entirely of regions labeled $A$. See Figure \[fig:graphicsketch\]. We focus on the $1$-manifold component $C$ of ${\mbox{$\partial$}}\mathcal{A}$ that contains the three sides $s = 0, 1$ and $t = 0$. If $C$ contains the fourth side $t=1$ then per Proposition \[prop:side1\] we are done, so our interest will focus on the arc in $C$ whose ends are at the corners $s \in \{0, 1\}, t = 1$ of the graphic, or more specifically, the border arcs that $C$ contains. (See Figure \[fig:graphicsketch\]). at 45 20 at 70 05 at 90 30 at 180 100 at 115 60 at 115 80 at 170 120 ![Graphic labels: red = A, blue = B[]{data-label="fig:graphicsketch"}](graphicsketch "fig:") It follows from Proposition \[prop:edgesoltn\] that the lowest border edge (i. e. minimal $s$) on the lowest border arc in $C$ generates the solution $E_0$ and the highest border edge on the highest border arc of $C$ generates the solution $E_1$. If we can show that the weak solutions given by successive border edges in $C$ always generate congruent solutions, then we will have shown that the solutions $E_0$ and $E_1$ are congruent, as required. So we examine how passing through a vertex of the graphic that lies on a border arc affects the weak solutions given by the incident edges. We will show the following, from which Theorem \[thm:main\] then follows. \[prop:vertex\] At any vertex in a border arc, the weak solutions given by the incident border edges converge. There is an important feature distinguishing between the two diagonals in a labeling diagram around a vertex in $\Gamma$. Put $T$ in the position determined by the vertex of $\Gamma$, so that $E = E_s$ and $T = T_t$ are tangent at two points $\rho = \rho_{\pm}$. We assume that $\rho_{\pm}$ lie on the same component ${\mbox{$\overline{E}$}}$ of $E$; if they lie on different components of $E$ the proof is easier. Let ${\mbox{$\overline{E}$}}_{\pm}$ be slight push-offs of ${\mbox{$\overline{E}$}}$ to each of its sides. Then the disks ${\mbox{$\overline{E}$}}_{\pm}$ correspond to positionings of ${\mbox{$\overline{E}$}}$ that lie in diagonally opposite quadrants of the graphic, since in moving from one to the other, the resolution of each of the tangencies at $\rho_{\pm}$ is changed. The curves of ${\mbox{$\overline{E}$}}_{+} \cap T$ and ${\mbox{$\overline{E}$}}_{-} \cap T$ are visibly disjoint in $T$, since the disks ${\mbox{$\overline{E}$}}_{\pm}$ are disjoint in $M$. Call this the [*aligned*]{} diagonal. (The other diagonal was called the [*dangerous diagonal*]{} in [@FS]. In Figure \[fig:resolved\] the antidiagonal is aligned and the main diagonal is dangerous.) The argument will be symmetric in $A$ and $B$ and also indifferent to symmetries of the quadrants about the vertex, so, following Proposition \[prop:diagonals\], there are just two cases to consider, corresponding to the labelings: $\arraycolsep = 2.0pt \begin{array}{c|c} A & A \\ \hline A & B \end{array}$ and $\arraycolsep = 2.0pt \begin{array}{c|c} A & A \\ \hline B & B \end{array}$. [**Case 1:**]{} The labelings around the vertex are $\arraycolsep = 2.0pt \begin{array}{c|c} A & A \\ \hline A & B \end{array}$; and the antidiagonal $\arraycolsep = 2.0pt \begin{array}{c|c} & \bullet \\ \hline \bullet & \end{array}$ is aligned. In this case, replace the component ${\mbox{$\overline{E}$}}$ in $E$ by three parallel components, namely, the two components ${\mbox{$\overline{E}$}}_{\pm}$ representing the antidiagonal, and a component ${\mbox{$\overline{E}$}}_B$ representing the quadrant labeled $B$. Call the resulting system $E^+$. Deleting exactly ${\mbox{$\overline{E}$}}_+$ from $E^+$ gives the weakly reducing system for one boundary edge and deleting exactly ${\mbox{$\overline{E}$}}_-$ gives the weakly reducing system for the other boundary edge. Now apply Lemma \[lemma:redundant\]: both of these converge with the system $E^+$. [**Case 2:**]{} The labelings around the vertex are $\arraycolsep = 2.0pt \begin{array}{c|c} A & A \\ \hline A & B \end{array}$; and the main diagonal $\arraycolsep = 2.0pt \begin{array}{c|c} \bullet & \\ \hline & \bullet \end{array}$ is aligned. In a similar fashion, replace ${\mbox{$\overline{E}$}}$ in $E$ by three parallel components: ${\mbox{$\overline{E}$}}_+$ representing the upper left quadrant, ${\mbox{$\overline{E}$}}_-$ representing the lower right quadrant and a component ${\mbox{$\overline{E}$}}^{12}$ representing the upper right quadrant. Call the resulting system $E^{12}$. Deleting exactly ${\mbox{$\overline{E}$}}_+$ from $E^{12}$ gives the weakly reducing system for the right boundary edge; deleting exactly ${\mbox{$\overline{E}$}}_{12}$ gives a weakly reducing system we call here the [*diagonal system*]{}. By Lemma \[lemma:redundant\] the two systems converge. Now repeat the argument using the system $E^{21}$ obtained by replacing ${\mbox{$\overline{E}$}}^{12}$ with a component ${\mbox{$\overline{E}$}}^{21}$ representing the lower left quadrant. The argument shows that the diagonal system also converges to the weakly reducing system for the lower boundary edge. Therefore the weak solutions representing the two boundary edges converge to each other. [**Case 3:**]{} The labelings around the vertex are $\arraycolsep = 2.0pt \begin{array}{c|c} A & A \\ \hline B & B \end{array}$. In this case we may as well assume the main diagonal is aligned. Then a minor variant of the argument for Case 2 suffices. [5]{} F. Bonahon, Cobordism of automorphisms of surfaces. [*Ann. Sci. École Norm. Sup.* ]{} [**16**]{} (1983) 237?270. A. Casson and C. McA. Gordon, [*Reducing Heegaard splittings*]{}, Topology and its applications, [**27**]{} (1987), 275-283. M. Freedman and M. Scharlemann, [*Powell moves and the Goeritz group*]{}, arXiv:1804.05909. W. Haken, [*Some results on surfaces in 3-manifolds*]{}, Studies in Modern Topology, Math. Assoc. Am., Prentice Hall, 1968, 34-98. A. Hatcher, Homeomorphisms of sufficiently large $P^2$-irreducible $3$-manifolds, [*Topology*]{} [**11**]{} (1976) 343-347. A.. Hatcher, A proof of the Smale Conjecture, [*Annals of Mathematics*]{}, [**117**]{} (1983) 553–607. J. Powell, Homeomorphisms of $S^{3}$ leaving a Heegaard surface invariant, [*Trans. Amer. Math. Soc.*]{} [**257**]{} (1980) 193–216. Y. Rieck, A proof of Waldhausen’s uniqueness of splittings of S3 (after Rubinstein and Scharlemann). Workshop on Heegaard Splittings, 277–284, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007. H. Rubinstein, Hyam and M.  Scharlemann, Comparing Heegaard splittings of non-Haken 3-manifolds. [*Topology* ]{} [**35**]{} (1996), 1005–1026. M. Scharlemann, A Strong Haken Theorem, Arxiv 2003.08523. M. Scharlemann and A. Thompson, [*Heegaard splittings of (surface) $\times$ I are standard*]{}, Math. Ann., [**295**]{} (1993), 549-564. J. H. C. Whitehead, On equivalent sets of elements in a free group, [*Ann. of Math.*]{} [**37**]{} (1936), 782–800. F. Waldhausen, Heegaard-Zerlegungen der 3-Sphäre. [*Topology*]{} [**7**]{} (1968) 195-203.

--- abstract: | We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite difference discretization of the time-fractional derivatives, and discuss its stability and error estimate. Extensive numerical experiments for one and two-dimension problems confirm the convergence rates of the theoretical results.\ [**Keywords**]{}: multi-term time-fractional diffusion equation, finite element method, error estimate, semidiscrete scheme, Caputo derivative address: - 'Department of Mathematics, University of California, Riverside, 900 University Ave., Riverside, CA 92521, USA (bangti.jin@gmail.com)' - | Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA\ (lazarov@math.tamu.edu) - 'Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153, Japan (ykliu@ms.u-tokyo.ac.jp)' - | Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA\ (zzhou@math.tamu.edu) author: - Bangti Jin - Raytcho Lazarov - Yikan Liu - Zhi Zhou bibliography: - 'frac.bib' date: 'started August, 2013; today is ' title: 'The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation' --- \#1[H\^[\#1]{}()]{} \#1\#2[\#1 \_[H\^[\#2]{}()]{}]{} introduction {#sec:intro} ============ We consider the following initial/boundary value problem for a multi-term time fractional diffusion equation in $u(x,t)$: $$\begin{aligned} {3}\label{eqn:goveq} \PD u-\Delta u&= f,&&\quad \text{in } \Omega&&\quad T \ge t > 0,\notag\\ u&=0,&&\quad\text{on}\ \partial\Omega&&\quad T \ge t > 0,\\ u(0)&=v,&&\quad\text{in }\Omega,&&\notag\end{aligned}$$ where $\Omega$ denotes a bounded convex polygonal domain in $\mathbb R^d\,(d=1,2,3)$ with a boundary $\partial\Omega$, $f$ is the source term, and the initial data $v$ is a given function on $\Omega$ and $T>0$ is a fixed value. Here the differential operator $\PD$ is defined by $$\PD=\partial_t^{{\alpha}}+\sum_{i=1}^m b_i \partial_t^{{\alpha}_i},$$ where $0< {\alpha}_m \le ...\le {\alpha}_1<{\alpha}<1$ are the orders of the fractional derivatives, $b_i > 0$, $i=1,2,...,m$, with the left-sided Caputo fractional derivative $\partial_t^\beta u$ being defined by (cf. [@KilbasSrivastavaTrujillo:2006 pp.91]) $$\label{eqn:McT} \partial_t^\beta u(t)= \frac{1}{\Gamma(1-\beta)} \int_0^t(t-\tau)^{-\beta}\frac{d}{d\tau}u(\tau) d\tau,$$ where $\Gamma(\cdot)$ denotes the Gamma function. In the case of $m=0$, the model reduces to its single-term counterpart $$\label{eqn:single} \partial_t^\alpha u -\Delta u = f\quad \mbox{ in } \Omega\times(0,T].$$ This model has been studied extensively from different aspects due to its extraordinary capability of modeling anomalous diffusion phenomena in highly heterogeneous aquifers and complex viscoelastic materials [@AdamsGelhar:1992; @Nigmatulin:1986]. It is the fractional analogue of the classical diffusion equation: with $\alpha=1$, it recovers the latter, and thus inherits some of its analytical properties. However, it differs considerably from the latter in the sense that, due to the presence of the nonlocal fractional derivative term, it has limited smoothing property in space and slow asymptotic decay in time [@SakamotoYamamoto:2011], which in turn also impacts related numerical analysis [@JinLazarovZhou:2013] and inverse problems [@JinRundell:2012; @SakamotoYamamoto:2011]. The model was developed to improve the modeling accuracy of the single-term model for describing anomalous diffusion. For example, in [@SchumerBensonMeerschaertBaeumer:2003], a two-term fractional-order diffusion model was proposed for the total concentration in solute transport, in order to distinguish explicitly the mobile and immobile status of the solute using fractional dynamics. The kinetic equation with two fractional derivatives of different orders appears also quite naturally when describing subdiffusive motion in velocity fields [@MetzlerKlafterSokolov:1998]; see also [@KellyMcGoughMeerschaert:2008] for discussions on the model for wave-type phenomena. There are very few mathematical studies on the model . Luchko [@Luchko:2011] established a maximum principle for problem , and constructed a generalized solution for the case $f\equiv0$ using the multinomial Mittag-Leffler function. Jiang et al [@JiangLiuTurnerBurrage:2012b] derived analytical solutions for the diffusion equation with fractional derivatives in both time and space. Li and Yamamoto [@LiYamamoto:2013] established existence, uniqueness, and the Hölder regularity of the solution using a fixed point argument for problem with variable coefficients $\{b_i\}$. Very recently, Li et al [@LiLiuYamamoto:2013] showed the uniqueness and continuous dependence of the solution on the initial value $v$ and the source term $f$, by exploiting refined properties of the multinomial Mittag-Leffler function. The applications of the model motivate the design and analysis of numerical schemes that have optimal (with respect to data regularity) convergence rates. Such schemes are especially valuable for problems where the solution has low regularity. The case $m=0$, i.e., the single-term model , has been extensively studied, and stability and error estimates were provided; see [@LinXu:2007; @ZhangSunWu:2011] for the finite difference method, [@LiXu:2009; @ZayernouriKardiadakis:2014] for the spectral method, [@McLeanThomee:2010; @Mustapha:2011; @MustaphaMcLean:2013; @JinLazarovZhou:2013; @JinLazarovPasciakZhou:2013; @JinLazarovPasciakZhou:2013a] for the finite element method, and [@BrunnerLingYamamoto:2010; @FuChenYang:2013] for meshfree methods based on radial basis functions, to name a few. In particular, in [@JinLazarovPasciakZhou:2013a; @JinLazarovPasciakZhou:2013; @JinLazarovZhou:2013], the authors established almost optimal error estimates with respect to the regularity of the initial data $v$ and the right hand side $f$ for a semidiscrete Galerkin scheme. These studies include the interesting case of very weak data, i.e., $ v \in \dH {q}$ and $f\in L^\infty(0,T;\dH q)$ for $-1 < q \le 0$. Numerical methods for the general multi-term case for an ordinary differential equation were considered in [@Katsikadelis:2009; @ElSayedElKallaZiada:2010]. In [@ZhaoXiaoXu:2013], a scheme based on the finite element method in space and a specialized finite difference method in time was proposed for , and error estimates were derived. We also refer to [@LiuMeerschaert:2013] for a numerical scheme based on a fractional predictor-corrector method for the multi-term time fractional wave-diffusion equation. The error analysis in these works is done under the assumption that the solution is sufficiently smooth and therefore it excludes the case of low regularity solutions. This is the main goal of the present study. However, the derivation of optimal with respect to the regularity error estimates requires additional analysis of the properties of problem , e.g., stability, asymptotic behavior for $t \to 0^+$. Relevant results of this type have recently been obtained in [@LiLiuYamamoto:2013], which, however, are not enough for the analysis of the semidiscrete Galerkin scheme, and hence in Section \[sec:prelim\], we make the necessary extensions. Now we describe the semidiscrete Galerkin scheme. Let ${\{\mathcal{T}_h\}}_{0<h<1}$ be a family of shape regular and quasi-uniform partitions of the domain $\Omega$ into $d$-simplexes, called finite elements, with a maximum diameter $h$. The approximate solution $u_h$ is sought in the finite element space $X_h$ of continuous piecewise linear functions over the triangulation $\mathcal{T}_h $ $$X_h =\left\{\chi\in H^1_0(\Omega): \ \chi ~~\mbox{is a linear function over} ~~\tau, \,\,\,\,\forall \tau \in \mathcal{T}_h\right\}.$$ The semidiscrete Galerkin FEM for problem is: find $ u_h (t)\in X_h$ such that $$\label{eqn:fem} {( \PD u_{h},\chi)}+ a(u_h,\chi)= {(f, \chi)}, \quad \forall \chi\in X_h,\ T \ge t >0, \quad u_h(0)=v_h,$$ where $a(u,w)=(\nabla u, \nabla w) ~~ \text{for}\ u, \, w\in H_0^1(\Omega)$, and $v_h \in X_h$ is an approximation of the initial data $v$. The choice of $v_h$ will depend on the smoothness of the initial data $v$. We shall study the convergence of the semidiscrete scheme for the case of initial data $ v \in \dH q$, $-1<q\leq 2$, and right hand side $f\in L^\infty(0,T;\dH q)$, $-1<q<1$. The case of nonsmooth data, i.e., $-1<q\leq 0$, is very common in inverse problems and optimal control [@JinRundell:2012; @SakamotoYamamoto:2011]; see also [@XieZou:2005; @JinLu:2012; @CasasClasonKunisch:2013; @CasasZuazua:2013] for the parabolic counterpart. The goal of this work is to develop a numerical scheme based on the finite element approximation for the model , and provide a complete error analysis. We derive error estimates optimal with respect to the data regularity for the semidiscrete scheme, and a convergence rate $O(h^2+\tau^{2 -\alpha})$ for the fully discrete scheme in case of a smooth solution. Specifically, our essential contributions are as follows. First, we obtain an improved regularity result for the inhomogeneous problem, by allowing less regular source term, cf. Theorem \[thm:regeps2\]. This is achieved by first establishing a new result, i.e., the complete monotonicity of the multinomial Mittag-Leffler function, cf. Lemma \[lem:MMLcm\]. Second, we derive nearly optimal error estimates for a semidiscrete Galerkin scheme for both homogeneous and inhomogeneous problems, cf. Theorems \[thm:SG-smooth\]-\[thm:gal:l2\], which cover both smooth and nonsmooth data. Third, we develop a fully discrete scheme based on a finite difference method in time, and establish its stability and error estimates, cf. Theorem \[thm:estfull\]. We note that the derived error estimate for the fully discrete scheme holds only for smooth solutions. The rest of the paper is organized as follows. In Section \[sec:prelim\], we recall the solution theory for the model for both homogeneous and inhomogeneous problems, using properties of the multinomial Mittag-Leffler function. The readers not interested in the analysis may proceed directly to Section \[sec:galerkin\]. Almost optimal error estimates for their Galerkin finite element approximations are given in Section \[sec:galerkin\]. Then a fully discrete scheme based on a finite difference approximation of the Caputo fractional derivatives is given in Section \[sec:fulldis\], and an error analysis is also provided. Finally, extensive numerical experiments are presented to illustrate the accuracy and efficiency of the Galerkin scheme, and to verify the convergence theory. Throughout, we denote by $C$ a generic constant, which may differ at different occurrences, but always independent of the mesh size $h$ and time step size $\tau$. Solution theory {#sec:prelim} =============== In this part, we recall the solution theory for problem . We shall describe the solution representation using the multinomial Mittag-Leffler function, and derive optimal solution regularity for the homogeneous and inhomogeneous problems. Multinomial Mittag-leffler function {#ssec:MultiML} ----------------------------------- First we recall the multinomial Mittag-Leffler function, introduced in [@HadidLuchko:1996]. For $0<\beta<2$, $0<\beta_i<1$ and $z_i\in \mathbb C$, $i=1,...,m$, the multinomial Mittag-Leffler function $E_{(\beta_1,...,\beta_m),\beta} (z_1,...,z_m)$ is defined by $$ E_{(\beta_1,...,\beta_m),\beta}(z_1,...,z_m)=\sum_{k=0}^{\infty} \sum_{\substack{l_1+...+l_m=k\\l_1\ge0,...,l_m\ge0}} (k;l_1,...,l_m) \frac{\prod_{i=1}^m z_i^{l_i}}{\Gamma(\beta+\Sigma_{i=1}^m \beta_i l_i)},$$ where the notation $(k;l_1,...,l_m)$ denotes the multinomial coefficient, i.e., $$(k;l_1,...,l_m)=\frac{k!}{l_1!...l_m!}\quad \mbox{with } k = \sum_{i=1}^ml_i.$$ It generalizes the exponential function $e^z$: with $m=1$ and $\beta=\beta_1=1$, it reproduces the exponential function $e^z$. It appears in the solution representation of problem , cf. below. We shall need the following two important lemmas on the function $E_{(\beta_1,...,\beta_m),\beta}(z_1,...,z_m)$, recently obtained in [@LiLiuYamamoto:2013 Section 2.1]. \[lem:MLbound\] Let $0<\beta<2$, $0<\beta_i<1$, $\beta_1>\max\{\beta_2,...,\beta_m\}$ and $\frac{\beta_1\pi}{2}<\mu<\beta_1\pi$. Assume that there is $K >0$ such that $-K\le z_i<0$, $i=2,...,m$. Then there exists a constant $C=C(\beta_1,...,\beta_m,\beta,K,\mu)>0$ such that $$ E_{(\beta_1,...,\beta_m),\beta}(z_1,...,z_m)\le \frac{C}{1+|z_1|}, \quad \quad\quad \mu\leq|\mathrm{arg}(z_1)|\leq \pi.$$ \[lem:multiMLprop\] Let $0<\beta<2$, $0<\beta_i<1$ and $z_i \in \mathbb C$, $i=1,...,m$. Then we have $$\frac{1}{\Gamma(\beta_0)}+ \sum_{i=1}^{m}z_iE_{(\beta_1,...,\beta_m),{\beta_0+\beta_i}}(z_1,...,z_m) =E_{(\beta_1,...,\beta_m),\beta_0}(z_1,...,z_m).$$ Solution Representation {#ssec:represent} ----------------------- For $s\ge-1$, we denote by $\dH s\subset H^{-1}(\Omega)$ the Hilbert space induced by the norm: $$\|v\|_{\dH s}^2=\sum_{j=1}^{\infty}{\lambda}_j^s \langle v,{\varphi}_j \rangle^2$$ with $\{{\lambda}_j\}_{j=1}^\infty$ and $\{{\varphi}_j\}_{j=1}^\infty$ being respectively the eigenvalues and the $L^2(\Omega)$-orthonormal eigenfunctions of the Laplace operator $-\Delta$ on the domain $\Omega$ with a homogeneous Dirichlet boundary condition. Then $\{{\varphi}_j\}_{j=1}^\infty$ and $\{{\lambda}_j^{1/2} {\varphi}_j\}_{j=1}^\infty$, form an orthonormal basis in $L^2(\Omega)$ and $H^{-1}(\Omega)$, respectively. Further, $\|v\|_{\dH 0}=\|v\|_{L^2(\Omega)}=(v,v)^{1/2}$ is the norm in $L^2(\Omega)$ and $\|v\|_{\dH {-1}} = \|v\|_{H^{-1}(\Omega)}$ is the norm in $H^{-1}(\Omega)$. It is easy to verify that $\|v\|_{\dH 1}= \|\nabla v\|_{L^2(\Omega)}$ is also the norm in $H_0^1(\Omega)$ and $\|v\|_{\dH 2}=\|\Delta v\|_{L^2(\Omega)}$ is equivalent to the norm in $H^2(\Omega)\cap H^1_0(\Omega)$ [@Thomee:2006 Lemma 3.1]. Note that $\dH s$, $s\ge -1$ form a Hilbert scale of interpolation spaces. Hence, we denote $\|\cdot\|_{H^s(\Omega)}$ to be the norm on the interpolation scale between $H^1_0(\Omega)$ and $L^2(\Omega)$ for $s\in [0,1]$ and $\|\cdot\|_{H^{s}(\Omega)}$ to be the norm on the interpolation scale between $L^2(\Omega)$ and $H^{-1}(\Omega)$ for $s\in [-1,0]$. Then, $\| \cdot \|_{H^s(\Omega)}$ and $\|\cdot\|_{\dH s}$ are equivalent for $s\in [-1,1]$. Further, for a Banach space $B$, we define the space $$L^r(0,T;B) = \{u(t)\in B \mbox{ for a.e. } t\in (0,T) \mbox{ and } \|u\|_{L^r(0,T;B)}<\infty\},$$ for any $r\geq 1$, and the norm $\|\cdot\|_{L^r(0,T;B)}$ is defined by $$\|u\|_{L^r(0,T;B)} = \left\{\begin{aligned}\left(\int_0^T\|u(t)\|_B^rdt\right)^{1/r}, &\quad r\in [1,\infty),\\ \displaystyle {\mathrm{esssup}_{t\in(0,T)}}\|u(t)\|_B, &\quad r= \infty. \end{aligned}\right.$$ Upon denoting $\vecal=({\alpha},{\alpha}-{\alpha}_1,...,{\alpha}-{\alpha}_m)$, we introduce the following solution operator $$\label{eqn:opE} E(t)v=\sum_{j=1}^{\infty} \left(1-{\lambda}_j t^{{\alpha}} E_{\vecal,1+{\alpha}}(-{\lambda}_jt^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m})\right) (v,{\varphi}_j){\varphi}_j.$$ This operator is motivated by a separation of variable [@LuchkoGorenflo:1999; @Luchko:2011]. Then for problem with a homogeneous right hand side, i.e., $f\equiv0$, we have $u(x,t)=E(t)v$. However, the representation is not always very convenient for analyzing its smoothing property. We derive an alternative representation of the solution operator $E$ using Lemma \[lem:multiMLprop\]: $$\label{eqn:opE2} \begin{split} E(t)v = &\sum_{j=1}^{\infty} E_{\vecal,1} (-{\lambda}_jt^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m}) (v,{\varphi}_j){\varphi}_j\\ &\ \ + \sum_{i=1}^m b_it^{{\alpha}-{\alpha}_i} \sum_{j=1}^{\infty} E_{\vecal,1+{\alpha}-{\alpha}_i} (-{\lambda}_jt^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m}) (v,{\varphi}_j){\varphi}_j. \end{split}$$ Besides, we define the following operator $\bar{E}$ for $\chi\in L^2(\Omega)$ by $$\label{eqn:Ebar} \bar E(t)\chi=\sum_{j=1}^{\infty} t^{{\alpha}-1} E_{\vecal,{\alpha}}(-{\lambda}_jt^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m}) (\chi,{\varphi}_j){\varphi}_j.$$ The operators $E(t) $ and $ {\bar E}(t) $ can be used to represent the solution $u$ of as: $$\label{eqn:solrep} u(t)=E(t)v + \int_0^t {\bar E}(t-s) f(s) ds.$$ The operator $\bar{E}$ has the following smoothing property. \[lem:barE\] For any $t>0$ and $\chi\in \dH q$, $q\in(-1,2]$, there holds for $0\le p-q \le 2$ $$\|\bar E(t) \chi \|_{\dH p} \le Ct^{-1+{\alpha}(1+(q-p)/2)}\|\chi\|_{\dH q}.$$ The definition of the operator $\bar{E}$ in and Lemma \[lem:MLbound\] yield $$\begin{split} \|\bar E(t) \chi \|_{\dH p}^2 &=t^{-2+(2+q-p){\alpha}}\sum_{j=1}^{\infty} ({\lambda}_j t^{{\alpha}})^{p-q} | E_{\vecal,{\alpha}}(-{\lambda}_j t^{\alpha},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m})|^2 {\lambda}_j^q |(\chi,{\varphi}_j)|^2\\ &\le C t^{-2+(2+q-p){\alpha}}\sum_{j=1}^{\infty} \frac{({\lambda}_j t^{{\alpha}})^{p-q}}{(1+{\lambda}_jt^{{\alpha}})^2} {\lambda}_j^q |(\chi,{\varphi}_j)|^2\\ &\le C t^{-2+(2+q-p){\alpha}}\sum_{j=1}^{\infty} {\lambda}_j^q |(\chi,{\varphi}_j)|^2 \le C t^{-2+(2+q-p){\alpha}}\|\chi\|_{\dH q}, \end{split}$$ where the last line follows by the inequality $\sup_{j\in \mathbb{N}} \frac{ ({\lambda}_j t^{\alpha})^{p-q}}{(1+{\lambda}_j t^{\alpha})^2}\le C$, for $0\leq p-q\leq 2$. Solution regularity ------------------- First we recall known regularity results. In [@LiYamamoto:2013], Li and Yamamoto showed that in the case of variable coefficients $\{b_i(x)\}$, there exists a unique mild solution $u\in C((0,T]; \dH{\gamma})\cap C([0,T];L^2(\Omega))$ and $u\in C([0,T];\dH {\gamma}) \cap L^{\infty}(0,T;\dH 2)$ when $ v \in L^2(\Omega)$, $f=0$ and $v =0$, $f\in L^{\infty}(0,T]; L^2(\Omega))$, respectively, with $\gamma\in[0,1)$. These results were recently refined in [@LiLiuYamamoto:2013] for the case of constant coefficients, i.e., problem . In particular, it was shown that for $v\in \dH q$, $0\leq q \leq 1$, and $f=0$, $u\in L^{1/(1-q/2)}(0,T;H^2(\Omega)\cap H_0^1(\Omega))$; and for $v=0$ and $f\in L^r(0,T; \dH q)$, $0\leq q\leq 2$, $r\geq1$, $u\in L^r(0,T;\dH {q+2-\gamma})$ for some $\gamma\in(0,1]$. Here we follow the approach in [@LiLiuYamamoto:2013], and extend these results to a slightly more general setting of $v\in \dH q$, $-1<q\leq 2$, and $f\in L^2(0,T;\dH q)$, $-1<q\le 1$. The nonsmooth case, i.e., $-1<q\leq 0$, arises commonly in related inverse problems and optimal control problems. We shall derive the solution regularity to the homogeneous problem, i.e., $f\equiv 0$, and the inhomogeneous problem, i.e., $v\equiv 0$, separately. These results will be essential for the error analysis of the space semidiscrete Galerkin scheme in Section \[sec:galerkin\]. First we consider the homogeneous problem with initial data $v\in \dH q$, $-1<q\leq 2$. \[thm:homogreg\] Let $u(t)=E(t)v$ be the solution to problem with $f\equiv 0$ and $v\in \dH q$, $q\in(-1,2]$. Then there holds $$ \|\PD^\ell u(t) \|_{\dH p} \le Ct^{-{\alpha}(\ell+(p-q)/2)}\|v\|_{\dH q},\quad t>0,$$ where for $\ell=0$, $ 0 \le p-q\le 2$ and for $\ell=1$, $-2 \le p-q \le 0$. We show that represents indeed the weak solution to problem with $f\equiv 0$ and further it satisfies the desired estimate. We first discuss the case $\ell=0$. By Lemma \[lem:MLbound\] and we have for $0\le p-q \le 2$ $$\begin{split} \| E(t)v \|_{\dH p}^2 & = \displaystyle \sum_{j=1}^{\infty}\lambda_j^p \Big (E_{\vecal,1} (-{\lambda}_jt^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m}) \Big .\\ & \Big . \quad\quad +\sum_{i=1}^mb_it^{{\alpha}-{\alpha}_i} E_{\vecal,1+{\alpha}-{\alpha}_i} (-{\lambda}_jt^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m}) \Big )^2 (v,{\varphi}_j)^2\\ &\le C t^{-(p-q){\alpha}}\sum_{j=1}^{\infty}\frac{({\lambda}_jt^{{\alpha}})^{p-q}}{(1+{\lambda}_jt^{\alpha})^2} {\lambda}_j^q |(v,{\varphi}_j)|^2\le C t^{-(p-q){\alpha}} \| v \|_{\dH q}^2, \end{split}$$ where the last line follows from the inequality $\sup_{j\in\mathbb N}\frac{ ({\lambda}_jt^{\alpha})^{p-q} }{(1+{\lambda}_jt^{\alpha})^2}\le C$ for $0\le p-q\le 2$. The estimate for the case $\ell=1$ follows from the identity $\|\PD E(t)v \|_{\dH p} = \| E(t)v \|_{\dH {p+2}}$. It remains to show that satisfies also the initial condition in the sense that $\lim_{t\rightarrow 0^+} \| E(t)v-v \|_{\dH q}=0$. By identity and Lemma \[lem:MLbound\], we deduce $$\begin{split} \| E(t)v-v \|_{\dH q}^2 &=\sum_{j=1}^{\infty} {\lambda}_j^{2} t^{2{\alpha}} \bigg|E_{\vecal,1+{\alpha}}(-{\lambda}_jt^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m})\bigg|^2{\lambda}_j^q| (v,{\varphi}_j)|^2\\ & \leq C\|v\|_{\dH q}^2 <\infty. \end{split}$$ Using Lemma \[lem:multiMLprop\], we rewrite the term ${\lambda}_j t^{{\alpha}}E_{\vecal,1+{\alpha}} (-{\lambda}_jt^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m})$ as $$\begin{aligned} {\lambda}_j t^{{\alpha}} E_{\vecal,1+{\alpha}}&(-{\lambda}_jt^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m})\\ = & (1-E_{\vecal,1} (-{\lambda}_jt^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m})) \\ & \ \ - \sum_{i=1}^m b_it^{{\alpha}-{\alpha}_i}E_{\vecal,1+{\alpha}-{\alpha}_i} (-{\lambda}_jt^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m}). \end{aligned}$$ Upon noting the identity $\lim_{t\to 0^+}(1-E_{\vecal,1} (-{\lambda}_jt^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},..., -b_mt^{{\alpha}-{\alpha}_m}))=0$, and the boundedness of $E_{\vecal,1+{\alpha}-{\alpha}_i}(-{\lambda}_jt^{{\alpha}}, -b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m})$ from Lemma \[lem:MLbound\], we deduce that for all $j$ $$\lim_{t\to0^+}{\lambda}_j t^{{\alpha}} E_{\vecal,1+{\alpha}}(-{\lambda}_jt^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m})=0.$$ Hence, the desired assertion follows by Lebesgue’s dominated convergence theorem. Now we turn to the inhomogeneous problem with a nonsmooth right hand side, i.e., $f\in L^{\infty}(0,T;\dH q)$, $-1<q\leq 1$, and a zero initial data $v\equiv0$. \[thm:inhomogreg\] For $f\in L^{\infty}(0,T;\dH q)$, $-1<q\le1$, and $v\equiv 0$, the solution $u$ belongs to $L^{\infty}(0,T;\dH {q+2-{\epsilon}})$ for any $\epsilon>0$ and satisfies $$\label{eq:regeps} \|u(\cdot,t)\|_{\dH {q+2-\epsilon}} \leq C\epsilon^{-1}t^{{\epsilon}{\alpha}/2}\|f\|_{L^\infty(0,t;\dH q)}.$$ Hence, it is a solution to problem with a homogeneous initial data $v=0$. By construction, it satisfies the governing equation. By Lemma \[lem:barE\], we have $$\begin{aligned} \|u(\cdot,t)\|_{\dH {q+2-{\epsilon}}} & = \|\int_0^t \bar{E}(t-s)f(s)ds\|_{\dH {q+2-{\epsilon}}}\\ & \leq \int_0^t \|\bar{E}(t-s)f(s)\|_{\dH {q+2-{\epsilon}}} ds \\ & \leq C\int_0^t (t-s)^{{\epsilon}{\alpha}/2-1} \|f(s)\|_{\dH q}ds\\ &\leq C\epsilon^{-1}t^{{\epsilon}{\alpha}/2}\|f\|_{L^\infty(0,t;\dH q)} \end{aligned}$$ which shows the desired estimate. Further, it satisfies the initial condition $u(0)=0$, i.e., for any ${\epsilon}>0$, $\lim_{t\to 0^+}\|u(\cdot,t) \|_{\dH {q+2-{\epsilon}}}=0$, and thus it is indeed a solution of . Next we extend Theorem \[thm:inhomogreg\] to allow less regular right hand sides $f\in L^2(0,T;\dH q)$, $-1<q\le 1$. Then the function $u(x,t)$ satisfies also the differential equation as an element in the space $L^2(0,T;\dot H^{q+2}(\Omega))$. However, it may not satisfy the homogeneous initial condition $u(x,0)=0$. In Remark \[weakest\_sol\] below, we argue that the weakest class of source term that produces a legitimate weak solution of is $f \in L^r(0,T;\dH q)$ with $r>1/{\alpha}$ and $-1 < q \le 1$. Obviously, for $1/2<{\alpha}<1$, it does give a solution $u(x,t) \in L^2(0,T;\dH {q+2})$. To this end, we introduce the shorthand notation $$\bar E_{\vecal}^j(t) = t^{{\alpha}-1} E_{\vecal,{\alpha}}(-{\lambda}_jt^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m}).$$ The function $\bar E_{\vecal}^j(t)$ is completely monotone; see Appendix \[app:MMLcm\] for the technical proof. \[lem:MMLcm\] The function $ \bar E_{\vecal}^j(t)$ for $j\in \mathbb{N}$ has the following properties: $$\begin{aligned} \bar E_{\vecal}^j(t)\ \mbox { is completely monotone} \quad \mbox{and}\quad \int_0^T | \bar E_{\vecal}^j(t)|\,dt<\frac{1}{{\lambda}_j}. $$ \[thm:regeps2\] For $f\in L^2(0,T;\dH q)$, $-1<q\le1$, the representation belongs to $L^{2}(0,T;\dH {q+2})$ and satisfies the [*a priori*]{} estimate $$\label{eq:regeps2} \|u\|_{L^2(0,t;\dH {q+2})} + \| \PD u \|_{L^2(0,t;\dH {q})} \leq C\|f\|_{L^2(0,t;\dH q)}.$$ By Young’s inequality for the convolution $\|k\ast f\|_{L^p}\le\|k\|_{L_1}\|f\|_{L^p}$, $k\in L^1$, $f\in L^p$, $p\ge 1$, and Lemma \[lem:MMLcm\], we deduce $$\begin{aligned} \left\|\int_0^t \bar E_{\vecal}^n (t-\tau)f_n(\tau)\,d\tau\right\|_{L^2(0,T)}^2&\le \left(\int_0^T|\bar E_{\vecal}^n (t)|\,dt\right)^2 \left(\int_0^T |f_n(t)|^2\,dt\right) \le \frac{1}{{\lambda}_n^2} \int_0^T |f_n(t)|^2\,dt. \end{aligned}$$ Hence, $$\begin{aligned} \| u\|_{L^2(0,T;\dot H^{q+2}(\Omega))}^2&\le \sum_{n=1}^{\infty}{\lambda}_n^{q+2}\left\|\int_0^t \bar E_{\vecal}^n (t-\tau)f_n(\tau)\,d\tau\right\|_{L^2(0,T)}^2\\ &\le \sum_{n=1}^{\infty}\lambda_n^q\int_0^T |f_n(t)|^2\,dt= \|f\|_{L^2(0,T;\dH q)}^2. \end{aligned}$$ The estimate on $\| \PD u \|_{L^2(0,t;\dH {q})}$ follows analogously. This completes the proof. \[weakest\_sol\] The condition $f\in L^\infty(0,T;\dH q)$ in Theorem \[thm:inhomogreg\] can be weakened to $f\in L^r(0,T;\dH q)$ with $r>1/\alpha$. This follows from Lemma \[lem:barE\] and Hölder’s inequality with $r'$, $1/r' + 1/r=1$ $$\begin{aligned} \|u(\cdot,t)\|_{\dH q} & \leq \int_0^t \|\bar{E}(t-s)f(s)\|_{\dH {q}} ds \leq C\int_0^t (t-s)^{\alpha-1} \|f(s)\|_{\dH q}ds \\ & \leq C\left(\frac{t^{1+r'(\alpha-1)}}{1+r'(\alpha-1)}\right)^{1/r'}\|f\|_{L^r(0,t;\dot H^q(\Omega))}, \end{aligned}$$ where $1+r'(\alpha-1)>0$ by the condition $r>1/\alpha$. It follows from this that the initial condition $u(\cdot,0)=0$ holds in the following sense: $\lim_{t\to0^+}\|u(\cdot,t)\|_{\dH q}=0$. Hence for any $\alpha\in (1/2,1)$ the representation remains a legitimate solution under the weaker condition $f\in L^2(0,T;\dH q)$. Error Estimates for Semidiscrete Galerkin Scheme {#sec:galerkin} ================================================ Now we derive and analyze a space semidiscrete Galerkin finite element scheme. First we describe the semidiscrete scheme, and then derive almost optimal error estimates for the homogeneous and inhomogeneous problems separately. In the analysis we essentially use the technique developed in [@JinLazarovZhou:2013] and improved in [@JinLazarovPasciakZhou:2013; @JinLazarovPasciakZhou:2013a]. Semidiscrete scheme ------------------- To describe the scheme, we need the $L^2(\Omega)$ projection $P_h:L^2(\Omega)\to X_h$ and Ritz projection $R_h:H^1_0(\Omega)\to X_h$, respectively, defined by $$\begin{aligned} (P_h \psi,\chi) & =(\psi,\chi) \quad\forall \chi\in X_h,\\ (\nabla R_h \psi,\nabla\chi) & =(\nabla \psi,\nabla\chi) \quad \forall \chi\in X_h. \end{aligned}$$ The operators $R_h$ and $P_h$ satisfy the following approximation property. \[lem:prh-bound\] For any $\psi\in \dH q$, $q=1,2$, the operator $R_h$ satisfies: $$\begin{aligned} \|R_h \psi-\psi\|_{L^2(\Omega)}+h\|\nabla(R_h \psi-\psi)\|_{L^2(\Omega)}\le Ch^q\| \psi\|_{\dot H^q(\Omega)}.\end{aligned}$$ Further, for $s\in [0,1]$ we have $$\begin{aligned} \|(I-P_h)\psi \|_{H^s(\Omega)} &\le Ch^{2-s} \|\psi\|_{\dH 2}\quad \forall \psi\in H^2(\Omega)\cap H^1_0(\Omega),\\ \|(I-P_h)\psi \|_{H^s(\Omega)} &\le Ch^{1-s} \|\psi\|_{\dH 1}\quad \forall \psi\in H^1_0(\Omega). \end{aligned}$$ Now we can describe the semidiscrete Galerkin scheme. Upon introducing the discrete Laplacian $\Delta_h: X_h\to X_h$ defined by $$-(\Delta_h\psi,\chi)=(\nabla\psi,\nabla\chi)\quad\forall\psi,\,\chi\in X_h,$$ and $f_h= P_h f$, we may write the spatially discrete problem as $$\label{eqn:fem-operator} \PD u_{h}(t)-\Delta_h u_h(t) =f_h(t) \for t\ge0 \quad \mbox{with} \quad u_h(0)=v_h,$$ where $v_h\in X_h$ is an approximation to the initial data $v$. Next we give a solution representation of using the eigenvalues and eigenfunctions $\{{\lambda}^h_j\}_{j=1}^{N}$ and $\{{\varphi}_j^h\}_{j=1}^{N}$ of the discrete Laplacian $-\Delta_h$. First we introduce the operators $E_h$ and $\bar{E}_h$, the discrete analogues of and , for $t>0 $, defined respectively by $$\label{eqn:Eh} \begin{split} E_h(t)v_h&=\sum_{j=1}^{N}E_{\vecal,1} (-{\lambda}_j^ht^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m}) (v,{\varphi}_j^h){\varphi}_j^h\\ & + \sum_{i=1}^m b_it^{{\alpha}-{\alpha}_i} \sum_{j=1}^{N} E_{\vecal,1+{\alpha}-{\alpha}_i} (-{\lambda}_j^ht^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m}) (v,{\varphi}_j^h){\varphi}_j^h,\\ \end{split}$$ and $$\label{eqn:Ehbar} \Etilh(t) f_h = \sum_{j=1}^N t^{{\alpha}-1} E_{\vecal,{\alpha}} (-{\lambda}_j^ht^{{\alpha}},-b_1t^{{\alpha}-{\alpha}_1},...,-b_mt^{{\alpha}-{\alpha}_m})\,(f_h,{\varphi}^h_j) \, {\varphi}_j^h.$$ Then the solution $u_h$ of the discrete problem can be expressed by: $$\label{eqn:Duhamelh} u_h(x,t)= E_h(t) v_h + \int_0^t \Etilh(t-s) f_h(s)\,ds.$$ On the finite element space $X_h$, we introduce the discrete norm $\tribar \cdot\tribar_{\dH p}$ defined by $$\tribar \psi\tribar_{\dH p}^2 = \sum_{j=1}^N({\lambda}_j^h)^p(\psi,{\varphi}_j^h)^2\quad \psi\in X_h.$$ The norm $\tribar \cdot\tribar_{\dH p}$ is well defined for all real $p$. Clearly, $\tribar \psi\tribar_{\dH 1}=\|\psi\|_{\dH 1}$ and $\tribar \psi\tribar_{\dH 0}=\|\psi\|_{L^2(\Omega)}$ for any $\psi\in X_h$. Further, the following inverse inequality holds [@JinLazarovZhou:2013]: if the mesh $\mathcal{T}_h$ is quasi-uniform, then for any $l>s$ $$\label{eqn:inverse} \tribar \psi\tribar_{\dH l}\le Ch^{s-l}\tribar \psi\tribar_{\dH {s}}\quad \forall \psi\in X_h.$$ \[lem:regh\] Assume that the mesh $\mathcal{T}_h$ is quasi-uniform. Then for any $v_h \in X_h$ the function $u_h(t)=E_h(t)v_h$ satisfies $$\tribar \PD^\ell u_h(t) \tribar_{\dH p} \le Ct^{ -{\alpha}(\ell + (p-q)/2)}\tribar v_h\tribar_{\dH q}, \quad t >0,$$ where for $\ell=0$, $0\leq p-q\leq 2$ and for $\ell=1$, $p \le q \le p+2$. Upon noting $\tribar \PD E_h(t)v_h \tribar_{\dH p}=\tribar E_h(t)v_h \tribar_{\dH {p+2}}$, it suffices to show the case $\ell=0$. Using the representation and Lemma \[lem:MLbound\], we have for $0\leq p-q\leq 2$ $$\begin{split} \tribar E_h(t)v_h \tribar_{\dH p}^2 &\le C \sum_{j=1}^{N}\frac{({\lambda}_j^h)^p}{(1+{\lambda}_j^h t^{\alpha})^2} |(v_h,{\varphi}_j^h)|^2\\ &\le C \sup_{1 \le j\le N}\frac{ ({\lambda}_j^h t^{\alpha})^{p-q}}{(1+{\lambda}_j^h t^{\alpha})^2} t^{-(p-q){\alpha}}\sum_{j=1}^N\frac{({\lambda}_j^h t^{{\alpha}})^{p-q}}{(1+{\lambda}_j^h t^{\alpha})^2}({\lambda}_j^h)^q |(v_h,{\varphi}_j^h)|^2\\ &\le C t^{-(p-q){\alpha}} \tribar v_h \tribar_{\dH q}^2, \end{split}$$ where the last inequality follows from $\sup_{1 \le j\le N}\frac{ ({\lambda}_j^h t^{\alpha})^{p-q}}{(1+{\lambda}_j^h t^{\alpha})^2}\le C$ for $0\leq p-q\leq 2$. The next result is a discrete analogue to Lemma \[lem:barE\]. \[lem:reg-f\] Let $\Etilh$ be defined by and $ \chi \in X_h$. Then for all $t >0$ $$\tribar \Etilh(t) \chi \tribar_{\dH p} \le \left \{ \begin{array}{ll} Ct^{ -1 + {\alpha}(1 + (q -p)/2)}\tribar \chi \tribar_{\dH q}, & \quad 0\le p-q\le 2, \\[1.3ex] Ct^{ -1 + \alpha }\tribar \chi \tribar_{\dH q}, & \quad p< q. \end{array} \right.$$ The proof for the case $0\le p-q\le2$ is similar to Lemma \[lem:barE\]. The other assertion follows from the fact that $\{\lambda_j^h\}_{j=1}^N$ are bounded from zero independent of $h$. Error estimates for the homogeneous problem ------------------------------------------- To derive error estimates, first we consider the case of smooth initial data, i.e., $v \in \dH 2$. To this end, we split the error $u_h(t)-u(t)$ into two terms: $$u_h-u= (u_h-R_hu)+(R_hu-u):={\vartheta}+ \varrho.$$ By Lemma \[lem:prh-bound\] and Theorem \[thm:homogreg\], we have for any $t>0$ $$\label{eqn:rho-bound} \| \varrho(t) \|_{L^2(\Omega)} + h \|\nabla\varrho(t)\|_{L^2(\Omega)}\le Ch^2 t^{-(1-q/2){\alpha}}\|v\|_{\dH q} \quad v\in \dH q, q=1,2.$$ So it suffices to get proper estimates for ${\vartheta}(t) $, which is given below. \[lem:vth-smooth\] The function ${\vartheta}(t):=u_h(t)-R_hu(t)$ satisfies for $p=0,1$ $$\|{\vartheta}(t)\|_{\dH p} \leq Ch^{2-p}\|v\|_{\dH 2} .$$ Using the identity $\Delta_hR_h=P_h\Delta$, we note that ${\vartheta}$ satisfies $$\PD {\vartheta}(t) -\Delta_h{\vartheta}(t) =-P_h \PD \varrho(t) \for t > 0,$$ with ${\vartheta}(0)=0$. By the representation , $${\vartheta}(t)=-\int_0^t \Etilh(t-s) P_h \PD \varrho(s)\,ds.$$ Then by Lemmas \[lem:reg-f\] and \[lem:prh-bound\], and Theorem \[thm:homogreg\] we have for $p=0,1$ $$\begin{split} \|{\vartheta}(t)\|_{\dH p} &\le \int_0^t \| \Etilh(t-s) P_h \PD \varrho(s) \|_{\dH p} \,ds \\ & \le C \int_0^t (t-s)^{(1-p/2){\alpha}-1} \|\PD \varrho(s) \|_{L^2(\Omega)}\,ds \\ & \le C h^{2-p}\int_0^t (t-s)^{(1-p/2){\alpha}-1} \|\PD u(s) \|_{\dH {2-p}}\,ds \\ & \le C h^{2-p} \int_0^t (t-s)^{(1-p/2){\alpha}-1} s^{-(1-p/2){\alpha}} \,ds \|v\|_{\dH 2} \le Ch^{2-p}\|v\|_{\dH 2}, \end{split}$$ which is the desired result. Using , Lemma \[lem:vth-smooth\] and the triangle inequality, we arrive at our first estimate, which is formulated in the following Theorem: \[thm:SG-smooth\] Let $v\in \dH 2$ and $f\equiv 0$, and $u$ and $u_h$ be the solutions of and with $v_h=R_hv$, respectively. Then $$\|u_h(t)- u(t)\|_{L^2(\Omega)} + h\|\nabla (u_h(t) - u(t))\|_{L^2(\Omega)} \le Ch^2 \|v\|_{\dH 2}.$$ Now we turn to the nonsmooth case, i.e., $v\in \dH q$ with $-1< q \leq 1$. Since the Ritz projection $R_h$ is not well-defined for nonsmooth data, we use instead the $L^2 (\Omega)$-projection $v_h=P_hv$ and split the error $u_h-u$ into: $$\label{eqn:splitnonsmooth} u_h-u=(u_h-P_hu)+(P_hu-u):={{\widetilde \vartheta}}+ {{\widetilde \varrho}}.$$ By Lemma \[lem:prh-bound\] and Theorem \[thm:homogreg\] we have for $-1\leq q\leq 1$ $$ \| {{\widetilde \varrho}}(t) \|_{L^2(\Omega)} + h \|\nabla{{\widetilde \varrho}}(t) \|_{L^2(\Omega)} \leq Ch^2\|u(t)\|_{\dH 2} \leq Ch^2 t^{-{\alpha}(1-q/2)}\|v\|_{\dH q}. $$ Thus, we only need to estimate the term ${{\widetilde \vartheta}}(t)$, which is stated in the following lemma. \[lem:vtht\] Let ${{\widetilde \vartheta}}(t)=u_h(t)-P_hu(t)$. Then for $p=0,1$, $-1< q\le1$, there holds (with $\ell_h=|\ln h|$) $$\|{{\widetilde \vartheta}}(t)\|_{\dH p}\leq Ch^{\min(q,0)+2-p}\ell_h t^{-{\alpha}\left(1-\max(q/2,0)\right)} \| v \|_{\dH q}.$$ Obviously, $ P_h \PD {{\widetilde \varrho}}= \PD P_h(P_hu-u)=0$ and using the identity $\Delta_hR_h=P_h\Delta$, we get the following problem for ${{\widetilde \vartheta}}$: $$\label{eqn:vtht2} \PD {{\widetilde \vartheta}}(t) -\Delta_h {{\widetilde \vartheta}}(t) = - \Delta_h (R_h u - P_h u)(t), \quad t>0, \quad {{\widetilde \vartheta}}(0)=0.$$ Using , ${{\widetilde \vartheta}}(t)$ can be represented by $$\label{eqn:vtht} {{\widetilde \vartheta}}(t) = - \int_0^t\Etilh(t-s)\Delta_h(R_hu-P_hu)(s)\,ds.$$ Let $A=\Etilh(t-s)\Delta_h(R_hu-P_hu)(s)$. Then by Lemma \[lem:regh\], there holds for $p=0,1$: $$\begin{split} \| A \|_{\dH p} &\le C (t-s)^{{\epsilon}{\alpha}/2-1}\tribar \Delta_h(R_hu-P_hu)(s) \tribar_{\dH {p-2+{\epsilon}}} \\ &\le C (t-s)^{{\epsilon}{\alpha}/2-1}\tribar(R_hu-P_hu)(s) \tribar_{\dH {p+{\epsilon}}}.\\ \end{split}$$ Then by , Theorem \[thm:homogreg\], Lemma \[lem:prh-bound\] we have for $p=0,1$ and $-1\le q\le 1$ $$\begin{split} \| A \|_{\dH p} &\le Ch^{\min(q,0)+2-p-{\epsilon}}(t-s)^{{\epsilon}{\alpha}/2-1}\| u(s) \|_{\dH {\min(q,0)+2}}\\ &\le Ch^{\min(q,0)+2-p-{\epsilon}}(t-s)^{{\epsilon}{\alpha}/2-1} s^{-\left(1-\max(q/2,0)\right){\alpha}}\| v \|_{\dH {q}}. \end{split}$$ Then plugging the estimate into yields $$\begin{split} \| {{\widetilde \vartheta}}\|_{\dH p} &\le Ch^{\min(q,0)+2-p-{\epsilon}} \int_0^t (t-s)^{{\epsilon}{\alpha}/2-1} s^{-\left(1-\max(q/2,0)\right){\alpha}} \, ds \| v \|_{\dH {q}}\\ &\le C\epsilon^{-1}h^{\min(q,0)+2-p-{\epsilon}} t^{-{\alpha}\left(1-\max(q/2,0)\right)}\| v \|_{\dH q}. \end{split}$$ Now with the choice ${\epsilon}=1/\ell_h$, we obtain the desired estimate. Now the triangle inequality yields an error estimate for nonsmooth initial data. \[thm:SG-nonsmooth\] Let $f\equiv 0$, $u$ and $u_h$ be the solutions of with $v\in \dH q$, $-1<q\le 1$, and with $v_h=P_hv$, respectively. Then with $\ell_h=|\ln h|$, there holds $$\| u_h(t) - u(t) \|_{L^2(\Omega)} + h \|\nabla(u_h(t) - u(t))\|_{L^2(\Omega)} \le Ch^{\min(q,0)+2} \, \ell_h \,t^{-{\alpha}(1-\max(q/2,0))}\|v\|_{\dH q}.$$ Error estimates for the inhomogeneous problem --------------------------------------------- Now we derive error estimates for the semidiscrete Galerkin approximations of the inhomogeneous problem with $f\in L^{\infty}(0,T;\dH q)$, $-1<q\leq 0$, and $v\equiv0$, in both $L^2$ and $L^\infty$-norm in time. To this end, we appeal again to the splitting . By Theorem \[thm:inhomogreg\] and Lemma \[lem:prh-bound\], the following estimate holds for ${{\widetilde \varrho}}$: $$\| {{\widetilde \varrho}}(t) \|_{L^2(\Omega)} + h \|\nabla{{\widetilde \varrho}}(t)\|_{L^2(\Omega)} \le Ch^{2+q-\epsilon} \|u(t)\|_{\dH {2+q-\epsilon}} \le C\epsilon^{-1}h^{2+q-\epsilon}\|f\|_{L^\infty(0,t;\dH q)}.$$ Now the choice $\ell_h= |\ln h|,\, \epsilon=1/\ell_h$, yields $$\label{eqn:Ph-bound} \| {{\widetilde \varrho}}(t) \|_{L^2(\Omega)} + h \|\nabla{{\widetilde \varrho}}(t)\|_{L^2(\Omega)}\le C\ell_hh^{2+q} \|f\|_{L^\infty(0,t;\dH q)}.$$ Thus, it suffices to bound the term ${{\widetilde \vartheta}}$; see the lemma below. \[lem:vtht-f\] Let ${{\widetilde \vartheta}}(t)$ be defined by , and $f\in L^\infty(0,T;\dH q)$, $-1<q\leq 0$. Then with $\ell_h=|\ln h|$, there holds $$\|{{\widetilde \vartheta}}(t)\|_{L^2(\Omega)}+h\|\nabla{{\widetilde \vartheta}}(t)\|_{L^2(\Omega)}\leq Ch^{2+q}\ell_h^2\|f\|_{L^\infty(0,t;\dH q)}.$$ By and Lemma \[lem:reg-f\], we deduce that for $p=0,1$ $$\begin{aligned} \|{{\widetilde \vartheta}}(t)\|_{\dH p} &\leq \int_0^t \|\Etilh(t-s)\Delta_h(R_h u-P_hu)(s)\|_{\dH p} ds\\ &\leq C\int_0^t (t-s)^{\epsilon{\alpha}/2 - 1} \tribar\Delta_h(R_h u -P_hu)(s)\tribar_{\dH {p-2+\epsilon}}ds\\ & \leq C\int_0^t(t-s)^{\epsilon{\alpha}/2-1} \tribar R_h u(s) -P_hu(s)\tribar_{\dH {p+\epsilon}}ds. \end{aligned}$$ Further, using and Lemma \[lem:prh-bound\], we deduce for $p=0,1$ $$\begin{aligned} \|{{\widetilde \vartheta}}(t)\|_{\dH p} & \leq C h^{-\epsilon} \int_0^t(t-s)^{\epsilon{\alpha}/2-1} \| R_h u(s) -P_hu(s)\|_{\dH p}ds \\ & \leq C h^{2+q-p-2\epsilon} \int_0^t(t-s)^{\epsilon{\alpha}/2-1} \| u(s) \|_{\dH {2+q-\epsilon}}ds. \end{aligned}$$ Now by and the choice ${\epsilon}=1/\ell_h$ we get for $p=0,1$: $$\begin{aligned} \|{{\widetilde \vartheta}}(t)\|_{\dH p} & \leq C \epsilon^{-1} h^{2+q-p-2\epsilon} \|f\|_{L^\infty(0,t;\dH q)} \int_0^t(t-s)^{\epsilon{\alpha}/2-1}t^{\epsilon \alpha/2}ds\\ &\leq C \epsilon^{-2} h^{2+q-p-2\epsilon}\|f\|_{L^\infty(0,t;\dH q)} \le Ch^{2+q-p}\ell_h^2\|f\|_{L^\infty(0,t;\dH q)}. \end{aligned}$$ This completes the proof of the lemma. An inspection of the proof of Lemma \[lem:vtht-f\] indicates that for $0<q <1$, one can get rid of one factor $\ell_h$. Now we can state an error estimate in $L^\infty$-norm in time. \[thm:gal:linf\] Let $v\equiv0$, $f\in L^\infty(0,T;\dH q)$, $-1<q\leq0$, and $u$ and $u_h$ be the solutions of and with $f_h=P_hf$, respectively. Then with $\ell_h =| \ln h|$ and $t > 0$, there holds $$\| u_h(t) - u(t) \|_{L^2(\Omega)} + h\|\nabla(u_h(t) - u(t))\|_{L^2(\Omega)} \le Ch^{2+q} \ell_h^{2} \|f\|_{L^\infty(0,t;\dH q)}.$$ Last, we derive an error estimate in $L^2$-norm in time. To this end, we need a discrete analogue of Theorem \[thm:regeps2\], which follows from the identical proof. \[lem:reg-d-l2\] Let $u_h$ be the solution of with $v_h=0$. Then for arbitrary $p>-1$ $$\int_0^T\normh{ \PD u_h(t)}{p}^2 +\normh{ u_h(t) }{p+2}^2 \, dt \le \int_0^T \normh {f_h(t)}{p}^2 dt.$$ \[thm:gal:l2\] Let $v\equiv 0$, $f\in L^\infty(0,T;\dH q)$, $-1<q\leq 0$, and $u$ and $u_h$ be the solutions of and with $f_h=P_hf$, respectively. Then $$\| u_h - u \|_{L^2(0,T;L^2(\Omega))} + h\|\nabla(u_h- u)\|_{L^2(0,T;L^2(\Omega))} \le Ch^{2+q} \|f\|_{L^2(0,T;\dH q)}.$$ We use the splitting . By Theorem \[thm:regeps2\] and Lemma \[lem:prh-bound\] $$\begin{aligned} \| {{\widetilde \varrho}}\|_{L^2(0,T;L^2(\Omega))} + h \|\nabla {{\widetilde \varrho}}\|_{L^2(0,T;L^2(\Omega))} &\le Ch^{2+q} \|u\|_{L^2(0,T;\dH {2+q})}\\ & \le Ch^{2+q} \|f\|_{L^2(0,T;\dH q)}. \end{aligned}$$ By , and Lemmas \[lem:reg-d-l2\] and \[lem:prh-bound\], we have for $p=0,\,1$: $$\begin{split} \int_0^T \|{{\widetilde \vartheta}}(t)\|^2_{\dH p} dt & \le C\int_0^T \normh{\Delta_h (R_h u - P_h u)(t) }{p-2}^2 dt \\ & \le C\int_0^T \normh{(R_h u - P_h u)(t)}{p}^2 dt\\ & \le C h^{4+2q-2p} \| u(t) \|_{L^2(0,T;\dH {2+q})}^2\\ & \le C h^{4+2q-2p}\| f(t) \|_{L^2(0,T;\dH q)}^2. \end{split}$$ Combing the preceding two estimates yields the desired assertion. A Fully Discrete Scheme {#sec:fulldis} ======================= Now we describe a fully discrete scheme for problem based on the finite difference method introduced in [@LinXu:2007]. To discretize the time-fractional derivatives, we divide the interval $[0,T]$ uniformly with a time step size $\tau=T/K$, $K\in\mathbb{N}$. We use the following discretization: $$\label{eqn:difference} \begin{aligned} \frac{\pa^{\alpha}u(x,t_{n+1})}{\partial t^{\alpha}} &= \frac{1}{\Gamma(1-{\alpha})} \sum_{j=0}^{n} \int_{t_j}^{t_{j+1}}(t_{n+1}-s)^{-{\alpha}}\frac{\partial u(x,s)}{\partial s}\,ds \\ &= \frac{1}{\Gamma(1-{\alpha})} \sum_{j=0}^{n} \frac{u(x,t_{j+1})-u(x,t_j)}{\tau} \int_{t_j}^{t_{j+1}}(t_{n+1}-s)^{-{\alpha}}\,ds +r_{{\alpha},\tau}^{n+1}\\ &= \frac{1}{\Gamma(2-{\alpha})} \sum_{j=0}^{n} d_{{\alpha},j} \frac{u(x,t_{n+1-j})-u(x,t_{n-j})}{\tau^{\alpha}}+ r_{{\alpha},\tau}^{n+1}, \end{aligned}$$ where $d_{{\alpha},j}=(j+1)^{1-{\alpha}}-j^{1-{\alpha}}$ with $j=0,1,2,...,n$ and $r_{{\alpha},\tau}^{n+1}$ denotes the local truncation error, which is given by $$\label{eqn:residue} \begin{split} | r_{{\alpha},\tau}^{n+1} | \le C \max_{0\le t \le T} |u_{tt}(x,t)| \sum_{j=1}^{n} \int_{t_j}^{t_{j+1}}\frac{2s-t_j-t_{j+1}}{({t_{n+1}-s})^{{\alpha}}}\,ds +O(\tau^2). \end{split}$$ Lin and Xu [@LinXu:2007 Lemma 3.1] showed that the truncation error $r_{\alpha,\tau}^{n+1}$ can be bounded by $$\label{eqn:residueerror} | r_{{\alpha},\tau}^{n+1} | \le C \max_{0\le t \le T} \left |u_{tt}(x,t) \right | \tau^{2-{\alpha}}.$$ Then the multi-term fractional derivative $\PD u(t)$ at $t=t_{n+1}$ in can be discretized by $$\label{eqn:difference2} \PD u(t_{n+1}) = P_\tau (\bar \pa_t) u(t_{n+1})+R_\tau^{n+1},$$ where the discrete differential operator $P_\tau(\bar\pa_t)$ is defined by $$\label{eqn:discdiff} {\small P_\tau (\bar \pa_t) u(t_{n+1}) := \frac{1}{\Gamma(2-{\alpha})} \sum_{j=0}^{n} P_j \frac{u(x,t_{n+1-j})-u(x,t_{n-j})}{\tau^{\alpha}},}$$ where the coefficients $\{P_j\}$ are defined by $$P_j=d_{{\alpha},j} + \sum_{i=1}^m\frac{\Gamma(2-{\alpha})b_id_{{\alpha}_i,j}\tau^{{\alpha}-{\alpha}_i}}{\Gamma(2-{\alpha}_i)}, \quad j\in \mathbb{N}.$$ Then by the local truncation error $R_\tau^{n+1}$ of the approximation $P_\tau(\bar\partial_t)u(t_{n+1})$ is bounded by $$\label{eqn:residue2} \begin{split} | R_\tau^{n+1} | \le C \max_{0\le t \le T} \left |u_{tt}(x,t) \right | \left(\tau^{2-{\alpha}} + \sum_{i=1}^m b_i \tau^{2-{\alpha}_i}\right)\le C \tau^{2-{\alpha}}\max_{0\le t \le T} \left |u_{tt}(x,t) \right |. \end{split}$$ By the monotonicity and convergence of $\{d_{{\alpha},j}\}$ [@LinXu:2007 equation (13)], we know that $$\label{eqn:Pmono} P_0 > P_1 >...>0 \quad \text{and} \quad P_j \rightarrow 0 \quad \text{for} \quad j \rightarrow \infty.$$ Now we arrive at the following fully discrete scheme: find $U^{n+1}\in X_h$ such that $$\label{eqn:fully1} (P_\tau(\bar \pa_t)U^{n+1},\chi) + (\nabla U^{n+1},\nabla \chi) = (F^{n+1}, \chi)\quad \forall \chi\in X_h,$$ where $F^{n+1}=f(x,t_{n+1})$. Upon setting $\gamma=\Gamma(2-{\alpha})\tau^{{\alpha}}$, the fully discrete scheme is equivalent to finding $U^{n+1}\in X_h$ such that for all $\chi \in X_h$ $$\label{eqn:fully} P_0(U^{n+1},\chi) + \gamma(\nabla U^{n+1}, \nabla \chi) =\sum_{j=0}^{n-1} (P_j-P_{j+1})(U^{n-j},\chi)+P_n (U^0 ,\chi) + \gamma(F^{n+1},\chi).$$ The next result gives the stability of the fully discrete scheme. \[lem:fullystab\] The fully discrete scheme is unconditionally stable, i.e., for all $n\in \mathbb{N}$ $$\label{eqn:fullystab} \| U^n \|_{L^2(\Omega)} \le \| U^0 \|_{L^2(\Omega)}+ c\max_{1\le j\le n} \| F^{j} \|_{L^2(\Omega)}.$$ where the constant $c$ depends only on ${\alpha}$ and $T$. The case $n=1$ is trivial. Then the proof proceeds by mathematical induction. By noting the monotone decreasing property of the sequence $\{P_n\}$ from and choosing $\chi=U^{n+1}$ in , we deduce $$\begin{split} P_0 \| U^{n+1}\|_{L^2(\Omega)} &\le \sum_{j=0}^{n-1} (P_j-P_{j+1}) \| U^{n-j} \|_{L^2(\Omega)} + P_n \| U^0 \|_{L^2(\Omega)} + \gamma \| F^{n+1} \|_{L^2(\Omega)} \\ &\le \sum_{j=0}^{n-1} (P_j-P_{j+1}) \| U^{n-j} \|_{L^2(\Omega)} + P_n \| U^0 \|_{L^2(\Omega)} + \gamma \max_{1\le j\le n+1} \| F^{j} \|_{L^2(\Omega)} \\ &\le P_0 \| U^0 \|_{L^2(\Omega)}+ \left(c(P_0-P_n)+\gamma\right)\max_{1\le j\le n+1} \| F^{j} \|_{L^2(\Omega)} \\ \end{split}$$ Using the monotonicity of $\{P_n\}$ again gives $$c(P_0-P_n)+\gamma \le cP_0-(c P_{N} - \gamma).$$ It suffices to choose a constant $c$ such that $c P_N - \gamma>0$. By taking $\tau=T/N$, we get $$\begin{split} P_N &= (N+1)^{1-{\alpha}}-N^{1-{\alpha}}=((T+\tau)^{1-{\alpha}}-T^{1-{\alpha}})\tau^{{\alpha}-1} \leq (1-{\alpha})T^{-{\alpha}}\tau^{{\alpha}} \end{split}$$ upon noting the concavity of the function $g(\tau)=(T+\tau)^{1-\alpha}$. Then by choosing $c=\Gamma(2-{\alpha})T^{\alpha}/(1-{\alpha})$ we obtain $$\begin{split} P_0 \| U^{n+1}\|_{L^2(\Omega)} \le P_0 \| U^0 \|_{L^2(\Omega)}+ cP_0\max_{1\le j\le n+1} \| F^{j} \|_{L^2(\Omega)}. \\ \end{split}$$ The desired result follows by dividing both sides by $P_0$. Next we state an error estimate for the fully discrete scheme. In order to analyze the temporal discretization error, we assume the solution is sufficiently smooth. \[thm:estfull\] Let the solution $u$ be sufficiently smooth, and $\{U^n\}\subset X_h$ be the solution of the fully discrete scheme with $U^0$ such that $$\| U^0-v\|_{L^2(\Omega)} \le Ch^2\| v\|_{\dH 2}.$$ Then there holds $$\begin{split} \| u(t_n)-U^n\|_{L^2(\Omega)} \le C \bigg(h^2 (\|v\|_{H^2(\Omega)} +\| f \|_{L^\infty(0,T;L^2(\Omega))} & + \max_{0< t\le t_n} \| u_t \|_{\dH 2} )\\ & +\tau^{2-{\alpha}} \max_{0< t\le t_n}\| u_{tt}(t)\|_{L^2(\Omega)} \bigg). \end{split}$$ We split the error $e^n=u(t_n)-U^n$ into $$e^n = (u(t_n)-R_h u(t_n))+(R_h u(t_n)- U^n)=: \varrho^n+\theta^n.$$ The term $\varrho^n$ can be bounded by $$\| u(t_n)-R_h u(t_n) \|_{L^2(\Omega)}\le Ch^2\| u(t_n)\|_{\dH 2} \le Ch^2(\|v\|_{H^2(\Omega)}+\| f \|_{L^\infty(0,T;L^2(\Omega))}).$$ It suffices to bound the term $\theta^n$. By comparing and , we have the error equation $$\label{eqn:erroreq} (P_\tau(\bar \pa_t)\theta^n,\chi) + (\nabla\theta^n,\nabla\chi)=(\omega^n,\chi),$$ where the right hand side $\omega^n$ is given by $$\omega^n = R_h P_\tau(\bar \pa_t)u(t_n)-\PD u(t_n)= -P_\tau(\pa_t)\varrho(t_n)-R_\tau^n :=\omega_1^n+\omega_2^n,$$ where the truncation error $R_\tau^n$ is defined in . Using the identity $$\varrho(x,t_{j+1})-\varrho(x,t_{j})=\int_{t_{j}}^{t_{j+1}} \varrho_t(x,t) \,dt,$$ we can bound the term $\omega_1^n$ by $$\begin{split} \| \omega_1^n\|_{L^2(\Omega)} &\le C \bigg|\hspace{-0.6mm}\bigg| \sum_{j=0}^{n-1} \frac{\varrho(t_{j+1})-\varrho(t_j)}{\tau} \int_{t_j}^{t_{j+1}}(t_{n}-s)^{-{\alpha}}+\sum_{i=1}^m b_i (t_{n}-s)^{-{\alpha}_i}\,ds\bigg|\hspace{-0.6mm}\bigg|_{L^2(\Omega)}\\ &\leq C\sum_{j=0}^{n-1} \tau^{-1} \int^{t_{j+1}}_{t_{j}} \| \varrho_t(t) \|_{L^2(\Omega)} \,dt\int_{t_j}^{t_{j+1}}(t_{n}-s)^{-{\alpha}}+\sum_{i=1}^m b_i (t_{n}-s)^{-{\alpha}_i}\,ds\\ &\leq C h^2\max_{0< t\le t_n} \| u_t \|_{\dH 2} \left( \int_0^{t_n}(t_{n}-s)^{-{\alpha}}+\sum_{i=1}^m b_i (t_{n}-s)^{-{\alpha}_i}\,ds\right)\\ &\le C h^2\max_{0< t\le t_n} \| u_t \|_{\dH 2}. \end{split}$$ Meanwhile, the second term $\omega_2^n$ can be bounded using . Then by the stability from Lemma \[lem:fullystab\] for the error equation , we obtain $$\begin{aligned} \| \theta^n \|_{L^2(\Omega)} & \le C\bigg( \| \theta^0 \|_{L^2(\Omega)} + \max_{1\le j\le n} \| \omega_1^j \|_{L^2(\Omega)} + \max_{1\le j\le n} \| \omega_2^j \|_{L^2(\Omega)}\bigg)\\ & \le C \bigg(h^2\| v \|_{\dH 2} + h^2 \max_{0< t\le t_n} \| u_t \|_{\dH 2} + \tau^{2-{\alpha}} \max_{0< t\le t_n} \|u_{tt}(t)\|_{L^2(\Omega)} \bigg). \end{aligned}$$ The error estimate in Theorem \[thm:estfull\] holds only if the solution $u$ is sufficiently smooth. There seems no known error estimate expressed in terms of the initial data (and right hand side) only for fully discrete schemes for nonsmooth initial data even for the single-term time-fractional diffusion equation with a Caputo fractional derivative. Numerical Experiments {#sec:numeric} ===================== In this part we present one- and two-dimensional numerical experiments to verify the error estimates in Sections \[sec:galerkin\] and \[sec:fulldis\]. We shall discuss the cases of a homogeneous problem and an inhomogeneous problem separately. The case of a smooth solution ----------------------------- Here we consider the following one-dimensional problem on the unit interval $\Omega=(0,1)$ with $0<\beta<{\alpha}<1$ $$\label{1Dnum} \begin{split} \partial_t^{\alpha}u(x,t) + \partial_t^\beta u(x,t) -\partial_{xx}^2 u(x,t)&=f, \quad 0< x <1 \quad 0\le t\le T,\\ u(0,t)=u(1,t)&=0, \quad 0\le t \le T, \\ u(x,0)&=v(x), \quad 0\le x \le 1. \end{split}$$ In order to verify the estimate in Theorem \[thm:estfull\], we first check the case that the solution $u$ is sufficiently smooth. To this end, we set initial data $v$ to $v(x)=x(1-x)$ and the source term $f$ to $f(x,t)=(2t^{2-{\alpha}}/{\Gamma(3-{\alpha})}+2t^{2-\beta} /{\Gamma(3-\beta)})(-x^2+x)+2(1+t^2)$. Then the exact solution $u$ is given by $u(x,t)= (1+t^2)(-x^2+x)$, which is very smooth. In our computation, we divide the unit interval $\Omega$ into $M$ equally spaced subintervals, with a mesh size $h = 1/N$. Similarly, we fix the time step size at $\tau=1/K$. Here we choose $N$ large enough so that the space discretization error is negligible, and the time discretization error dominates. We measure the accuracy of the numerical approximation $U^n$ by the normalized $L^2$ error $\| U^n-u(t_n) \|_{L^2(\Omega)}/ \| v \|_{L^2(\Omega)}$. In Table \[tab:smooth-sol-time\], we show the temporal convergence rates, indicated in the column `rate` (the number in bracket is the theoretical rate), for three different $\alpha$ values, which fully confirm the theoretical result, cf. also Fig. \[fig:time\_error\] for the plot of the convergence rates. ${\alpha}$ $\tau$ $1/10$ $1/20$ $1/40$ $1/80$ $1/160$ rate ----------------- ------------ --------- --------- --------- --------- --------- ------------------------- ${\alpha}=0.25$ $L^2$-norm 5.58e-4 1.73e-4 5.25e-5 1.51e-5 3.90e-6 $\approx 1.81$ ($1.75$) ${\alpha}=0.5$ $L^2$-norm 1.45e-3 5.11e-4 1.78e-4 6.17e-5 2.08e-5 $\approx 1.55$ ($1.50$) ${\alpha}=0.95$ $L^2$-norm 7.92e-3 3.79e-3 1.82e-3 8.73e-4 4.20e-4 $\approx 1.06$ ($1.05$) : Numerical results for the case with a smooth solution at $t=1$ with $\beta=0.2$ and ${\alpha}=0.25, 0.5, 0.95$, discretized on a uniform mesh with $h= 2^{-10}$ and $\tau =0.2\times2^{-k}$.[]{data-label="tab:smooth-sol-time"} Homogeneous problems -------------------- In this part we present numerical results to illustrate the spatial convergence rates in Section \[sec:galerkin\]. We performed numerical tests on the following three different initial data: - Smooth data: $v(x)=\sin(2\pi x)$ which belongs to $H^2(\Omega)\cap H^1_0(\Omega)$. - Nonsmooth data: $v(x)=\chi_{(0,1/2]}$ which lies in the space $\dH {{\epsilon}}$ for any ${\epsilon}\in [0,1/2)$. - Very weak data: $v(x)=\delta_{1/2}(x)$, a Dirac $\delta_{1/2}(x)$-function concentrated at $x=1/2$, which belongs to the space $\dH {- {\epsilon}}$ for any ${\epsilon}\in (1/2,1]$. In order to check the convergence rate of the semidiscrete scheme, we discretize the fractional derivatives with a small time step $\tau$ so that the temporal discretization error is negligible. In view of the possibly singular behavior as $t \rightarrow 0$, we set the time step $\tau$ to $\tau=t/(5\times 10^4)$, with $t$ being the terminal time. For each example, we measure the error $e(t)=u(t)-u_h(t)$ by the normalized errors $\| e(t) \|_{L^2(\Omega)}/ \| v \|_{L^2(\Omega)}$ and $\| \nabla e(t) \|_{L^2(\Omega)}/ \| v \|_{L^2(\Omega)}$. The normalization enables us to observe the behavior of the error with respect to time in case of nonsmooth initial data. ### Numerical results for example (2a): smooth initial data The numerical results show $O(h^2)$ and $O(h)$ convergence rates for the $L^2$- and $H^1$-norms of the error, respectively, for all three different $\alpha$ values, cf. Fig. \[fig:spaceerror:smooth\]. As the value of $\alpha$ increases from $0.25$ to $0.95$, the error at $t=1$ decreases accordingly, which resembles that for the single-term time-fractional diffusion equation [@JinLazarovZhou:2013]. ### Numerical results for example (2b): nonsmooth initial data For nonsmooth initial data, we are particularly interested in errors for $t$ close to zero, and thus we also present the errors at $t=0.01$ and $t=0.001$; see Table \[tab:nonsmooth1\_initial\]. The numerical results fully confirm the theoretically predicted rates for nonsmooth initial data. Further, in Table \[tab:check\_singular\] we show the $L^2$-norm of the error for fixed $h = 2^{-6}$ and $t\rightarrow 0$. We observe that the error deteriorates as $t\rightarrow0$. Upon noting $v\in \dH{1/2-{\epsilon}}$, it follows from Theorem \[thm:SG-nonsmooth\] that the error grows like $O(t^{-3{\alpha}/4})$, which agrees well with the results in Table \[tab:check\_singular\]. $t$ $k$ $3$ $4$ $5$ $6$ $7$ rate ----------- ------------ --------- --------- --------- --------- --------- ------------------------- $t=1$ $L^2$-norm 1.86e-3 4.64e-4 1.16e-4 2.87e-5 6.88e-6 $\approx 2.02$ ($2.00$) $H^1$-norm 4.89e-2 2.44e-2 1.22e-2 6.07e-3 2.96e-3 $\approx 1.01$ ($1.00$) $t=0.01$ $L^2$-norm 8.04e-3 2.00e-3 5.01e-4 1.24e-4 2.98e-5 $\approx 2.03$ ($2.00$) $H^1$-norm 2.31e-2 1.16e-1 5.79e-2 2.88e-2 1.40e-2 $\approx 1.01$ ($1.00$) $t=0.001$ $L^2$-norm 1.65e-2 4.14e-3 1.03e-3 2.56e-4 6.18e-4 $\approx 2.01$ ($2.00$) $H^1$-norm 5.15e-1 2.58e-1 1.29e-2 6.41e-2 3.13e-2 $\approx 1.01$ ($1.00$) : Numerical results for the nonsmooth case (2b) with ${\alpha}=0.5$ and $\beta=0.2$ at $t=1, 0.01, 0.001$, discretized on a uniform mesh with $h = 2^{-k}$ and $\tau =t/(5\times10^4)$. []{data-label="tab:nonsmooth1_initial"} $t$ 1e-3 1e-4 1e-5 1e-6 1e-7 1e-8 rate ---------- --------- --------- --------- --------- --------- --------- ------------------------- Case(2b) 2.56e-4 5.39e-4 1.15e-3 2.91e-3 6.77e-3 1.55e-2 $\approx-0.37 (-0.37) $ : $L^2$-error with ${\alpha}=0.5$ and $h=2^{-6}$ for $t \to 0$ for nonsmooth initial data (2b).[]{data-label="tab:check_singular"} ### Numerical results for example (2c): very weak initial data The numerical results show a superconvergence with a rate of $O(h^2)$ in the $L^2$-norm and $O(h)$ in the $H^1$-norm, cf. Fig. \[fig:weak\](a). This is attributed to the fact that in one-dimension the solution with the Dirac $\delta$-function as the initial data is smooth from both sides of the support point and the finite element spaces $X_h$ have good approximation property. When the singularity point $x=1/2$ is not aligned with the grid, Fig. \[fig:weak\](b) indicates an $O(h^{3/2})$ and $O(h^{1/2})$ convergence rate for the $L^2$- and $H^1$-norm of the error, respectively, which agrees with our theory. Inhomogeneous problems ---------------------- Now we consider the inhomogeneous problem with $v\equiv0$ on the unit interval $\Omega=(0,1)$ and test the following two examples: 1. Nonsmooth data: $f(x,t)=(\chi_{[1/2,1]}(t)+1)\chi_{[0,{1/2}]}(x)$. The jump at $x=1/2$ leads to $f(t,\cdot) \notin \dot H^1(\Omega)$; nonetheless, for any $\epsilon >0$, $f \in L^\infty(0,T;\dH {{1/2}-\epsilon})$. 2. Very weak data: $f(x,t)=(\chi_{[1/2,1]}(t)+1)\delta_{1/2}(x)$ where $f$ involves a Dirac $\delta_{1/2}(x)$-function concentrated at $x=0.5$. ### Numerical results for example (3a) Since the errors are bounded independently of the time, cf. Theorem \[thm:gal:linf\], we only present the errors in $L^\infty$ in time, i.e., $\|e(t)\|_{L^2(\Omega)}$ and $\| \nabla e(t) \|_{L^2(\Omega)}$. In Table \[tab:nonsmooth1\_right\], we present the $L^2$- and $H^1$-error at $t=1$, $0.01$, and $0.001$. The numerical results agree well with our theoretical predictions, i.e., $O(h^2)$ and $O(h)$ convergence rates for the $L^2$- and $H^1$-norms of the error, respectively. $t$ $k$ $3$ $4$ $5$ $6$ $7$ rate ----------- ------------ --------- --------- --------- --------- --------- ------------------------- $t=1$ $L^2$-norm 1.76e-3 4.40e-4 1.10e-4 2.71e-5 6.53e-6 $\approx 2.01$ ($2.00$) $H^1$-norm 4.72e-2 2.36e-2 1.18e-2 5.86e-3 2.86e-3 $\approx 1.01$ ($1.00$) $t=0.01$ $L^2$-norm 6.34e-4 1.59e-4 3.96e-5 9.82e-6 2.38e-6 $\approx 2.01$ ($2.00$) $H^1$-norm 1.89e-2 9.46e-3 4.72e-3 2.35e-3 1.15e-3 $\approx 1.01$ ($1.00$) $t=0.001$ $L^2$-norm 4.55e-4 1.15e-4 2.88e-5 1.15e-6 1.73e-6 $\approx 2.02$ ($2.00$) $H^1$-norm 1.45e-2 7.31e-3 3.66e-3 1.82e-3 8.88e-4 $\approx 1.01$ ($1.00$) : Numerical results for example (3a) with ${\alpha}=0.5$ and $\beta=0.2$ at $t=1, 0.01, 0.001$, discretized on a uniform mesh $h = 2^{-k}$ and $\tau=t/(5\times 10^{4})$. []{data-label="tab:nonsmooth1_right"} ### Numerical results for example (3b) In Table \[tab:weak\_right\_G\] we show convergence rates at three different times, i.e., $t=1$, $0.01$, and $0.001$. Here the mesh size $h$ is chosen to be $h=1/(2^k+1)$, and thus the support of the Dirac $\delta$-function does not align with the grid. The results indicate an $O(h^{1/2})$ and $O(h^{3/2})$ convergence rate for the $H^1$- and $L^2$-norm of the error, respectively, which agrees well with the theoretical prediction. If the Dirac $\delta$-function is supported at a grid point, both $L^2$- and $H^1$-norms of the error exhibit a superconvergence of one half order, cf. Table \[tab:weak\_right\_d\]. This, however, theoretically remains to be established. $t$ $k$ $3$ $4$ $5$ $6$ $7$ rate ----------- ------------ --------- --------- --------- --------- --------- ------------------------- $t=0.1$ $L^2$-norm 1.02e-2 4.01e-3 1.49e-3 5.35e-4 1.82e-4 $\approx 1.49$ ($1.50$) $H^1$-norm 3.24e-1 2.35e-1 1.65e-1 1.11e-1 6.94e-2 $\approx 0.50$ ($0.50$) $t=0.01$ $L^2$-norm 4.66e-3 1.91e-3 7.29e-4 2.64e-4 9.02e-5 $\approx 1.45$ ($1.50$) $H^1$-norm 1.54e-1 1.14e-1 8.16e-2 5.54e-2 3.47e-2 $\approx 0.55$ ($0.50$) $t=0.001$ $L^2$-norm 4.30e-3 1.83e-3 7.12e-4 2.61e-4 8.97e-5 $\approx 1.45$ ($1.50$) $H^1$-norm 1.47e-1 1.11e-1 8.05e-2 5.50e-2 3.45e-2 $\approx 0.55$ ($0.50$) : Numerical results for example (3b) with ${\alpha}=0.5$ and $\beta=0.2$ at $t=0.1, 0.01, 0.001$, discretized on a uniform mesh $h = 1/(2^{k}+1)$ and $\tau=t/(5\times 10^{4})$. []{data-label="tab:weak_right_d"} $t$ $k$ $3$ $4$ $5$ $6$ $7$ rate ----------- ------------ --------- --------- --------- --------- --------- ------------------------- $t=1$ $L^2$-norm 5.35e-4 1.34e-4 3.35e-5 8.31e-6 2.01e-6 $\approx 2.01$ ($1.50$) $H^1$-norm 1.49e-2 7.48e-3 3.74e-3 1.86e-3 9.07e-4 $\approx 1.01$ ($0.50$) $t=0.01$ $L^2$-norm 6.67e-4 1.67e-4 4.17e-5 1.04e-5 2.52e-6 $\approx 2.03$ ($1.50$) $H^1$-norm 2.56e-2 1.29e-2 6.44e-3 3.20e-3 1.56e-3 $\approx 1.02$ ($0.50$) $t=0.001$ $L^2$-norm 8.19e-4 2.08e-4 5.22e-5 1.30e-5 3.19e-6 $\approx 2.02$ ($1.50$) $H^1$-norm 3.96e-2 2.00e-2 1.00e-3 4.98e-3 2.45e-3 $\approx 1.01$ ($0.50$) : Numerical results for example (3b) with ${\alpha}=0.5$ and $\beta=0.2$ at $t=1, 0.01, 0.001$, discretized on a uniform mesh with $h= 2^{-k}$ and $\tau=t/(5\times 10^{4})$. []{data-label="tab:weak_right_G"} Examples in two-dimension ------------------------- In this part, we present three two-dimensional examples on the unit square $\Omega=(0,1)^2$. 1. Nonsmooth initial data: $v=\chi_{(0,1/2)\times(0,1)}$ and $f\equiv0$. 2. Very weak initial data: $v=\delta_\Gamma$ with $\Gamma$ being the boundary of the square $[1/4,3/4]^2$ and $\langle \delta_\Gamma,\phi\rangle=\int_\Gamma\phi(s)\,ds$. By Hölder’s inequality and the continuity of the trace operator from $\dot H^{{1/2}+\epsilon}(\Omega)$ to $L^2(\Gamma)$ [@AdamsFournier:2003], we deduce $\delta_\Gamma \in H^{-1/2-\epsilon}(\Omega)$. 3. Nonsmooth right hand side: $f(x,t)=(\chi_{[1/20,1/10]}(t)+1)\chi_{(0,1/2)\times(0,1)}(x)$ and $v\equiv0$. To discretize the problem, we divide each direction into $N=2^k$ equally spaced subintervals, with a mesh size $h=1/N$ so that the domain $[0,1]^2$ is divided into $N^2$ small squares. We get a symmetric mesh by connecting the diagonal of each small square. The numerical results for example (4a) are shown in Table \[tab:nonsmooth2D\], which agree well with Theorem \[thm:SG-nonsmooth\], with a rate $O(h^2)$ and $O(h)$, respectively, for the $L^2$- and $H^1$-norm of the error. Interestingly, for example (4b), both the $L^2$-norm and $H^1$-norm of the error exhibit super-convergence, cf. Table \[tab:weak2D\_G\]. The numerical results for example (4c) confirm the theoretical results; see Table \[tab:2Dnonsmooth\_right\]. The solution profiles for examples (4b) and (4c) at $t=0.1$ are shown in Fig. \[fig:2D\_solution\], from which the nonsmooth region of the solution can be clearly observed. $t$ $k$ $3$ $4$ $5$ $6$ $7$ rate ----------- ------------ --------- --------- --------- --------- --------- ------------------------- $t=0.1$ $L^2$-norm 5.25e-3 1.35e-3 3.38e-4 8.24e-5 1.98e-5 $\approx 2.06$ ($2.00$) $H^1$-norm 9.10e-2 4.53e-2 2.25e-2 1.09e-2 4.99e-3 $\approx 1.04$ ($1.00$) $t=0.01$ $L^2$-norm 1.25e-2 3.23e-3 8.09e-4 1.97e-4 4.65e-5 $\approx 2.05$ ($2.00$) $H^1$-norm 2.18e-1 1.08e-1 5.35e-2 2.62e-2 1.27e-2 $\approx 1.05$ ($1.00$) $t=0.001$ $L^2$-norm 3.02e-2 7.84e-3 1.97e-3 4.81e-4 1.16e-4 $\approx 2.03$ ($2.00$) $H^1$-norm 5.30e-1 2.64e-1 1.31e-1 6.38e-2 3.14e-2 $\approx 1.04$ ($1.00$) : Numerical results for (4a) with ${\alpha}=0.5$ and $\beta=0.2$ at $t=0.1, 0.01, 0.001$, discretized on a uniform mesh, $h = 2^{-k}$ and $\tau=t/10^{4}$. []{data-label="tab:nonsmooth2D"} $t$ $k$ $3$ $4$ $5$ $6$ $7$ rate ----------- ------------ --------- --------- --------- --------- --------- ------------------------- $t=0.1$ $L^2$-norm 1.18e-2 3.18e-3 8.41e-4 2.18e-4 5.41e-5 $\approx 1.92$ ($1.50$) $H^1$-norm 2.25e-1 1.13e-1 6.60e-2 3.40e-2 1.66e-2 $\approx 0.92$ ($0.50$) $t=0.01$ $L^2$-norm 2.82e-2 7.62e-3 2.28e-3 5.26e-4 1.25e-4 $\approx 1.95$ ($2.00$) $H^1$-norm 5.66e-1 3.09e-1 1.65e-1 8.52e-2 4.19e-2 $\approx 0.94$ ($1.00$) $t=0.001$ $L^2$-norm 6.65e-2 1.83e-3 4.98e-3 1.33e-3 3.30e-4 $\approx 1.91$ ($2.00$) $H^1$-norm 1.66e0 8.93e-1 4.75e-1 2.43e-1 1.21e-1 $\approx 0.95$ ($1.00$) : Numerical results for example (4b) with ${\alpha}=0.5$ and $\beta=0.2$ at $t=0.1, 0.01, 0.001$ for a uniform mesh with $h = 2^{-k}$ and $\tau=t/10^{4}$. []{data-label="tab:weak2D_G"} $t$ $k$ $3$ $4$ $5$ $6$ $7$ rate ----------- ------------ --------- --------- --------- --------- --------- ------------------------- $t=0.1$ $L^2$-norm 2.28e-3 5.86e-4 1.47e-4 3.58e-5 7.91e-6 $\approx 2.07$ ($2.00$) $H^1$-norm 3.97e-2 1.97e-2 9.77e-3 4.76e-3 2.13e-3 $\approx 1.06$ ($1.00$) $t=0.01$ $L^2$-norm 1.06e-3 2.73e-4 6.86e-5 1.67e-6 3.70e-6 $\approx 2.06$ ($2.00$) $H^1$-norm 1.85e-2 9.18e-3 4.56e-3 2.22e-3 9.94e-3 $\approx 1.06$ ($1.00$) $t=0.001$ $L^2$-norm 8.66e-4 2.28e-4 5.75e-5 1.40e-6 3.11e-6 $\approx 2.04$ ($2.00$) $H^1$-norm 1.56e-2 7.82e-3 3.88e-3 1.90e-3 8.47e-4 $\approx 1.05$ ($1.00$) : Numerical results for example (4c) with ${\alpha}=0.5$ and $\beta=0.2$ at $t=0.1, 0.01, 0.001$ for a uniform mesh with $h= 2^{-k}$ and $\tau=t/10^{4}$. []{data-label="tab:2Dnonsmooth_right"} Concluding remarks ================== In this work, we have developed a simple numerical scheme based on the Galerkin finite element method for a multi-term time fractional diffusion equation which involves multiple Caputo fractional derivatives in time. A complete error analysis of the space semidiscrete Galerkin scheme is provided. The theory covers the practically very important case of nonsmooth initial data and right hand side. The analysis relies essentially on some new regularity results of the multi-term time fractional diffusion equation. Further, we have developed a fully discrete scheme based on a finite difference discretization of the Caputo fractional derivatives. The stability and error estimate of the fully discrete scheme were established, provided that the solution is smooth. The extensive numerical experiments in one- and two-dimension fully confirmed our convergence analysis: the empirical convergence rates agree well with the theoretical predictions for both smooth and nonsmooth data. Acknowledgements {#acknowledgements .unnumbered} ================ The research of B. Jin has been supported by US NSF Grant DMS-1319052, R. Lazarov was supported in parts by US NSF Grant DMS-1016525 and also by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST), and Y. Liu was supported by the Program for Leading Graduate Schools, MEXT, Japan. Proof of Lemma \[lem:MMLcm\] {#app:MMLcm} ============================ First, we define an auxiliary function $v_j(t)$ by $$\hat v_j(z)=\mathcal{L}(v_j) = \frac{z^{{\alpha}-1}+\sum_{k=1}^m b_k z^{{\alpha}_k-1}}{z^{\alpha}+\sum_{k=1}^m b_k z^{{\alpha}_k}+{\lambda}_j}.$$ Now by the property of the Laplace transform $f(0^+)=\lim_{z\to \infty} z \widehat{f}(z)$, we obtain $v_j(0+)=1$. The function $\bar E_{\vecal}^j(t)$ is the inverse Laplace integral of $\widehat {\bar E_{\vecal}^j} =(z^{\alpha}+\sum_{k=1}^m b_k z^{{\alpha}_k}+{\lambda}_j)^{-1}$, i.e. $$\label{L1} \bar E_{\vecal}^j(t)=\frac{1}{2\pi\mathrm{i}} \int_{Br}e^{zt}\frac{1}{{z^{\alpha}+\sum_{k=1}^m b_k z^{{\alpha}_k}+{\lambda}_j}}\,dz,$$ where $Br=\{z;\ \text{Re}~~z=\sigma,\ \sigma>0\}$ is the Bromwich path. The function $\widehat{\bar E_{\vecal}^j}(z)$ has a branch point $0$, so we cut off the negative part of the real axis. Note that the function $z^{\alpha}+\sum_{k=1}^m b_k z^{{\alpha}_k}+{\lambda}_j$ has no zero in the main sheet of the Riemann surface including its boundaries on the cut. Indeed, if $z=\varrho e^{i\theta}$, with $\rho>0$, $\theta\in(-\pi,\pi)$, then $$\Im\left\{z^{\alpha}+\sum_{k=1}^m b_k z^{{\alpha}_k}+{\lambda}_j\right\}=\varrho^{\alpha}\sin{\alpha}\theta+\sum_{k=1}^m b_k \varrho^{{\alpha}_k}\sin{\alpha}_k\theta\neq 0,\ \ \forall \theta\neq 0,$$ since $\sin \alpha \theta$ and $\sin\alpha_k\theta$ have the same sign for any $\theta\in(-\pi,\pi)$ and $b_k>0$. Hence, $ \bar E_{\vecal}^j(t)$ can be found by bending the Bromwich path into the Hankel path $Ha({\epsilon})$, which starts from $-\infty$ along the lower side of the negative real axis, encircles the disc $|s|={\epsilon}$ counterclockwise and ends at $-\infty$ along the upper side of the negative real axis. Then by taking ${\epsilon}\to 0$, we obtain $$ \bar E_{\vecal}^j(t)= \int_0^\infty e^{-rz}K_n(r)\,dr,$$ where $$ K_n(r)=-\frac{1}{\pi}\Im\left\{\left.\frac{1} {z^{\alpha}+\sum_{k=1}^m b_k z^{{\alpha}_k}+{\lambda}_j}\right|_{z=r e^{i\pi}}\right\}.$$ It is easy to check $$K_n(r)=\frac{1}{\pi} \frac{r^{{\alpha}}\sin{\alpha}\pi+\sum_{k=1}^m b_kr^{{\alpha}_k}\sin{\alpha}_k\pi} {(r^{{\alpha}}\cos{\alpha}\pi+\sum_{k=1}^m b_kr^{{\alpha}_k}\cos{\alpha}_k\pi+{\lambda}_j)^2 +(r^{{\alpha}}\sin{\alpha}\pi+\sum_{k=1}^m b_kr^{{\alpha}_k}\sin{\alpha}_k\pi)^2} $$ which is greater than zero for all $r>0$. Therefore, $\bar E_{\vecal}^j(t)$ is completely monotone. A similar argument shows that $v_j(t)$ is also completely monotone. Consequently, $$\int_0^T |\bar E_{\vecal}^j(t)|\,dt=\int_0^T \bar E_{\vecal}^j(t)\,dt= -\frac{1}{{\lambda}_j}\int_0^T v'_j(t)\,dt=\frac{1}{{\lambda}_n}(1-v_j(T))<\frac{1}{{\lambda}_n},$$ which concludes the proof of the lemma.

--- author: - 'Colin M. Hardy' - 'Philip W. Livermore' - Jitse Niesen bibliography: - 'allrefs.bib' title: Constraints on the magnetic field within a stratified outer core --- Mounting evidence from both seismology and experiments on core composition suggests the existence of a layer of stably stratified fluid at the top of Earth’s outer core. In this work we examine the structure of the geomagnetic field within such a layer, building on the important but little known work of [@malkus1979dynamo]. We assume (i) an idealised magnetostrophic spherical model of the geodynamo neglecting inertia, viscosity and the solid inner core, and (ii) a strongly stratified layer of constant depth immediately below the outer boundary within which there is no spherically radial flow. Due to the restricted dynamics, Malkus showed that the geomagnetic field must obey certain a condition which is a refined and more restrictive version of the well known condition of @Taylor_63 which holds on an infinite set of azimuthal rings within the stratified layer. By adopting a spectral representation with truncation $N$ in each direction, we show that this infinite class collapses to a discrete set of $O(N^2)$ Malkus constraints. Although fewer than the $N^3$ degrees of freedom of the magnetic field, their nonlinear nature makes finding a magnetic field that obeys such constraints, here termed a [*Malkus state*]{}, a challenging task. Nevertheless, such Malkus states when constrained further by geomagnetic observations have the potential to probe the interior of the core. By focusing on a particular class of magnetic fields for which the Malkus constraints are linear, we describe a constructive method that turns any purely-poloidal field into an exact Malkus state by adding a suitable toroidal field. We consider poloidal fields following a prescribed smooth profile within the core that match a degree-13 observation-derived model of the magnetic field in epoch 2015 or a degree-10 model of the 10000-yr time averaged magnetic field. Despite the restrictions of the Malkus constraints, a significant number of degrees of freedom remain for the unknown toroidal field and we seek extremal examples. The Malkus state with the least toroidal energy has in both cases a strong azimuthal toroidal field, about double the magnitude of that observed from the poloidal field at the core-mantle boundary. For the 2015 field for a layer of depth 300 km, we estimate a root mean squared azimuthal toroidal field of $3$ mT with a pointwise maximum of 8 mT occurring at a depth of about 70 km. Introduction ============ The question of whether or not Earth’s liquid outer core contains a stratified layer just below its outer boundary has long been debated [@whaler1980does; @Braginsky_67; @braginsky1987waves; @hardy2019stably; @gubbins2007geomagnetic]. A stratified layer may result from the pooling of buoyant elements released from the freezing of the solid inner core [@braginsky2006formation; @bouffard2019chemical], diffusion from the mantle above [@jeanloz1990nature; @buffett_seagle_2010] or sub-adiabatic thermal effects [@Pozzo_etal_2012]. Within a strongly stratified layer, the dynamics would be very different to the remainder of the convecting core because spherical radial motion would be suppressed [@braginsky1999dynamics; @davies2015constraints; @cox2019penetration]. In terms of using observations of the changing internal geomagnetic field as a window on the dynamics within the core, the existence of a stratified layer is crucial because motion confined to the stratified layer such as waves may have a pronounced geomagnetic signature, which may be falsely interpreted as emanating from the large-scale dynamo process ongoing beneath. Observational constraints on the stratified layer are largely from seismology, where analysis of a specific ‘SmKS’ class of waves has revealed a localised decrease in wave velocities in the outermost $100-300 \text{km}$ of the core [@helffrich2013causes; @lay1990stably; @helffrich2010outer], suggesting that the outermost part of the core has a different density and/or elasticity than the rest of the core. However, this evidence is far from conclusive because not all studies agree that a stratified layer is necessary to explain seismic measurements [@irving2018seismically], and there are inherent uncertainties due to the remoteness of the core [@alexandrakis_eaton_2010]. So far, observational geomagnetism has offered equivocal evidence for stratified layers. Time dependent observational models can be explained by simple core flow structures on the core-mantle boundary (CMB) which have either no layer [@Holme_2015; @Amit_2014] (upwelling at the CMB is permitted), or a strongly stratified layer (in which all radial motion is suppressed), [@Lesur_etal_2015]. A complementary approach to understanding the observational signature of a stratified layer is by numerical simulation of a stratified geodynamo model [@nakagawa2011effect]. Models of outer core dynamics have demonstrated that dynamo action can be sensitive to variations in the assumed background state of a fully convective outer core, and that the presence of stably stratified layers can significantly alter the dynamics and morphology of the resultant magnetic field [@glane2018enhanced; @christensen2018geodynamo; @olson2018outer]. Hence comparisons between the magnetic fields from stratified models with the geomagnetic field can be used to infer compatibility with the presence of a stratified layer. This has been used to constrain the possible thickness of a stratified layer such that it is consistent with geomagnetic observations. [@yan2018sensitivity] find that unstratified dynamo simulations significantly underpredict the octupolar component of the geomagnetic field. Their model endorses the presence of a thin stably stratified layer, as the resultant magnetic field can be rendered Earth-like by the inclusion of 60-130 km layer. However, the results are rather sensitive to both the strength of stratification and layer depth, with a thicker layer of 350 km resulting in an incompatible octupole field. Similarly [@olson2017dynamo] find that stratified model results compare favorably with the time-averaged geomagnetic field for partial stratification in a thin layer of less than 400 km, but unfavorable for stratification in a thick 1000 km layer beneath the CMB. Additionally, in terms of dynamics, [@Braginsky_93; @Buffett_2014] show that MAC (Magnetic, buoyancy (Archimedean) and Coriolis forces) waves in the [*hidden ocean*]{} at the top of the core provide a mechanism for the 60 year period oscillations detected in the geomagnetic field [@roberts200760]. The model of [@buffett2016evidence] suggests that MAC waves underneath the CMB are also able to account for a significant part of the fluctuations in length of day (LOD) [@gross2001combined; @holme2005geomagnetic] through explaining the dipole variation, but are contingent on the existence of a stratified layer at the top of the core with a thickness of at least 100 km. However, not all stratified dynamo model results champion this scenario for the Earth. It has been found that the inclusion of a thin stable layer in numerical models can act to destablise the dynamo, through generating a thermal wind which creates a different differential rotation pattern in the core [@stanley2008effects]. Additionally many distinctive features of the geomagnetic field are not reproduced, as strong stratification leads to the disappearance of reverse flux patches and suppression of all non-axisymmetric magnetic field components [@mound2019regional; @christensen2008models]. One reason why there is no clear message from existing geodynamo models is perhaps that they all have been run in parameter regimes very far from Earth’s core [@Roberts_Aurnou_2011]. Two important parameters, the Ekman and Rossby numbers, quantify the ratio of rotational to viscous forces $E \sim 10^{-15}$ and the ratio of inertial to viscous forces $R_o \sim 10^{-7}$ respectively [@Christensen_2015]. These parameters being so small causes difficulties when attempting to numerically simulate the geodynamo because they lead to small spatial and temporal scales that need to be resolved in any direct numerical simulation, but are extremely computationally expensive to do so. Despite this challenge, numerical models have been used with great success to simulate aspects of the geodynamo, reproducing features such as torsional oscillations [@Wicht_Christensen_2010] that are consistent with observational models [@gillet2010fast], geomagnetic jerks [@aubert2019geomagnetic] and allowing predictions of the Earth’s magnetic field strength [@christensen2009energy]. Recent simulations have been able to probe more Earth-like parameter regimes than previously possible, achieving very low Ekman numbers of $E = 10^{-7} - 10^{-8}$ [@schaeffer2017turbulent; @aubert2019approaching]. However despite this progress, these simulations remain in parameter regimes vastly different to that of the Earth [@Christensen_2015], posing the inescapable question of how representative of the Earth they really are, as force balances can still vary significantly between the simulation regime and the correct regime of the Earth [@wicht2019advances], with the ability to simultaneously reproduce Earth-like field morphology and reversal frequency still beyond current capabilities [@christensen2010conditions]. The assessments conducted by [@sprain2019assessment] highlight that present geodynamo models able unable to satisfactorily reproduce all aspects of Earth’s long term field behaviour. In this paper we consider the approach proposed by [@Taylor_63], based on the assumption that the inertia-free and viscosity-free asymptotic limit is more faithful to Earth’s dynamo than adopting numerically-expedient but nevertheless inflated parameter values. This amounts to setting the values of $R_o$ and $E$ to zero, which simplifies the governing equations significantly, enabling numerical solutions at less computational expense and importantly for us, analytic progress to be made. The resulting dimensionless magnetostrophic regime then involves an exact balance between the Coriolis force, pressure, buoyancy and the Lorentz force associated with the magnetic field ${{\bf B}}$ itself: $${\bm{\hat{z}}} \times {\bm{u}} = -{{\bm{\nabla}}}p + F_B{\bm{\hat r}} + ({{{\boldsymbol \nabla}}\times}{\bm{B}}) \times {\bm{B}}, \label{eqn:magneto}$$ where $F_B$ is a buoyancy term that acts in the unit radial direction ${\bm{\hat r}}$ [@Fearn_98]. Throughout this paper we consider the magnetostrophic balance of \[eqn:magneto\]. @Taylor_63 showed that, as a consequence of this magnetostrophic balance, the magnetic field must obey at all times $t$ the well-known condition $$T(s,t) \equiv \int_{C(s)} (({{{\boldsymbol \nabla}}\times}{{\bf B}}) \times {{\bf B}})_\phi ~ s \text{d}\phi \text{d}z =0,\label{eqn:Taylor}$$ for any geostrophic cylinder $C(s)$ of radius $s$, aligned with the rotation axis, where $(s,\phi,z)$ are cylindrical coordinates. This constraint applies in the general case for fluids independent of stratification. It was first shown by [@malkus1979dynamo] how can be refined within a stratified layer of constant depth, which in the limit of zero radial flow leads to a more strict constraint. This constraint now applies on every axisymmetric ring coaxial with the rotation axis that lies within the layer and is known as the [*[Malkus constraint]{}*]{} $$M(s,z,t) \equiv \int_0^{2\pi} (({{\bm{\nabla}}}\times {\bm{B}}) \times {\bm{B}})_{\phi} ~ \text{d}\phi =0,$$ for any $s$ and $z$ within the layer. Magnetic fields that satisfy the Taylor or Malkus constraints respectively are termed Taylor or Malkus states. The associated timescale over which the dominant force balance described by the magnetostrophic equations evolves is $\sim 10^4$ years. However observations show changes in the geomagnetic field on much shorter timescale of years to decades [@jackson2015geomagnetic]. This vast discrepancy in timescales motivates distinguishing between the slowly evolving background state and perturbations from it and considering these two features separately. The theoretically predicted magnetostrophic timescale, represented by Taylor or Malkus states, describes the slow evolution of the magnetic field, and may explain dynamics such as geomagnetic reversals and also the longstanding dominance of the axially symmetric dipolar component of the field. Although rapid dynamics such as MHD waves occur on a much shorter timescale, they cannot be considered in isolation as their structure depends critically upon the background state that they perturb. Thus although insightful models of perturbations can be based upon simple states [e.g. @Malkus_67], ultimately a close fit to the observed geomagnetic field requires accurate knowledge of the background state. It is the search for such a state that is explored in this paper. Dynamical models of a non-stratified background state, produced by evolving the magnetic field subject to Taylor’s constraint, have appeared very recently [@Wu_Roberts_2015; @roberts2018magnetostrophic; @li2018taylor] and are currently restricted to axisymmetry, although the model of [@li2018taylor] can be simply extended to a three dimensional system. These models can additionally be used to probe the effect of incorporating inertia driven torsional waves within this framework [@roberts2014modified]. In this paper we adopt a different strategy and explore the use of both the Taylor and Malkus constraints as a tool for analytically constraining instantaneous structures of the magnetic field throughout Earth’s core. This method ignores any dynamics and asks simply whether we can find a set of magnetic fields which satisfy the necessary constraints: Taylor’s constraint in the interior and Malkus’s constraint in the stratified layer, which will provide plausible background geomagnetic states. However, constructing Malkus states is a non-trivial task. Firstly we need to establish whether such fields can even exist, and if so how numerous they are, before we are able to construct examples of Malkus states. Since we are geophysically motivated, we also wish to determine whether such fields can be compatible with geomagnetic observations. Our task is a challenging one: even finding magnetic fields that exactly satisfy the comparatively simple case of Taylor’s constraint has proven to be difficult in the 55 years since the seminal paper of @Taylor_63, although notable progress has been made in axisymmetry [@Hollerbach_Ierley_91; @Soward_Jones_83] and in 3D [@Jault_Cardin_99] subject to imposing a specific symmetry. Recently, significant progress has been made in this regard by presenting a more general understanding of the mathematical structure of Taylor’s constraint in three dimensions [@livermore2008structure]. This method was implemented by [@livermore2009construction] to construct simple, large scale magnetic fields compatible with geomagnetic observations. It is this which provides the foundation for the work presented here. The remainder of this paper is structured as follows. In section 2 we present a new, more general derivation of the condition required to be satisfied with a stratified layer of fluid, which under an idealised limit reduces to what is known as Malkus’ constraint. In section 3 we summarise the method for discretising and constructing a Taylor state before extending this to Malkus states in section 4. In section 5 we prove that an arbitrary poloidal field can be transformed into a Malkus state through the addition of an appropriate toroidal field and show how this is a useful approach due to the resultant equations being linear. In section 6 we present our results for an Earth like magnetic field satisfying all relevant constraints, within the linear framework. In section 7 we discuss these results with regard to Earth’s internal field, specifically our estimate of toroidal field strength, before concluding in section 8. Derivation of Malkus’ constraint {#sec:Malk_derivation} ================================ Within stably stratified fluids radial flows are suppressed, hence in the limit of strong stratification radial fluid velocities are negligibly small [@braginsky1999dynamics; @davies2015constraints]. We proceed within this idealistic limit and require that $u_r=0$ within a region of stratified fluid that is a volume of revolution: we represent the proposed stratified layer within Earth’s core as a spherically symmetric layer of constant depth. We assume further that the system is in magnetostrophic balance; that is, rapidly rotating with negligible inertia and viscosity. The resulting constraint was first derived by [@malkus1979dynamo], however, here we present an alternative and more straightforward derivation courtesy of Dominique Jault (personal communication). We use the condition for incompressible flow that ${{\bm{\nabla}}}\cdot {\bm{u}} = 0$ and the standard toroidal poloidal decomposition within spherical coordinates $(r,\theta,\phi)$. From the condition that there is no spherically-radial component of velocity then ${\bm{u}}$ must be purely toroidal and hence can be written as $${\bm{u}} = {{\bm{\nabla}}}\times (\mathcal{T}(r,\theta,\phi){\bm{\hat{r}}})= \frac{1}{r\sin\theta} {\dfrac{\partial \mathcal{T}}{\partial \phi}} {\bm{\hat{\theta}}} - \frac{1}{r} {\dfrac{\partial \mathcal{T}}{\partial \theta}} {\bm{\hat{\phi}}}.$$ Therefore the cylindrically-radial velocity, written in spherical coordinates, is$$u_s=\sin\theta u_r+\cos\theta u_\theta = \frac{\cos\theta}{r\sin\theta} {\dfrac{\partial \mathcal{T}}{\partial \phi}}$$ and so $$\int_0^{2\pi} u_s ~ \text{d}\phi = \frac{\cos\theta}{r\sin\theta} \int_0^{2\pi} {\dfrac{\partial \mathcal{T}}{\partial \phi}} ~ \text{d}\phi = 0.$$ Now, since ${\bm{\hat{\phi}}} \cdot ({\hat{\bf z}}\times {\bm{u}}) = u_s $ then, from the azimuthal component of the magnetostrophic \[eqn:magneto\] we have $$u_s = -{\dfrac{\partial p}{\partial \phi}} +(({{\bm{\nabla}}}\times {\bm{B}}) \times {\bm{B}})_{\phi}.$$ Integrating this around any circle in a plane orthogonal to ${\bm{\hat{r}}}$ centred on the rotation axis, (as illustrated by the red rings in \[fig:constraint\_surfaces\]), and using the single-valued nature of pressure, gives Malkus’ constraint, $$\underbrace{\int_0^{2\pi}u_s ~ \text{d}\phi}_{=0} = -\underbrace{\int_0^{2\pi}{\dfrac{\partial p}{\partial \phi}} ~ \text{d}\phi}_{=0} +\int_0^{2\pi} (({{\bm{\nabla}}}\times {\bm{B}}) \times {\bm{B}})_{\phi} ~d\phi = 0, \nonumber$$ or equivalently requiring that the Malkus integral $M$ is zero: $$M(s,z,t) \equiv \int_0^{2\pi} (({{\bm{\nabla}}}\times {\bm{B}}) \times {\bm{B}})_{\phi} ~ \text{d}\phi =0. \label{eqn:Malkcon}$$ We are also able to generalise this constraint from considering the idealistic limit of requiring $u_r=0$ within the stratified fluid to the more general situation of permitting $u_r \neq 0$, where we express the Malkus integral in terms of the radial flow. Now, the flow ${{\bf u}}$ is no longer purely toroidal and hence $$\begin{aligned} M(s,z,t) =\int_0^{2\pi} u_s ~ \text{d}\phi = \int_0^{2\pi} u_\theta \cos\theta \text{d}\phi + \int_0^{2\pi} u_r \sin\theta ~ \text{d}\phi. \end{aligned}$$ We now use the condition for incompressible flow that ${{\bm{\nabla}}}\cdot {\bm{u}} = 0,$ $$0 = {{\bm{\nabla}}}\cdot {{\bf u}}= \frac{1}{r^2}{\dfrac{\partial (r^2u_r)}{\partial r}} +\frac{1}{r\sin\theta}{\dfrac{\partial (u_\theta \sin\theta)}{\partial \theta}} + \frac{1}{r\sin\theta} {\dfrac{\partial u_\phi}{\partial \phi}},$$ $$\Rightarrow \int_0^{2\pi} \left(\frac{\sin\theta}{r}{\dfrac{\partial (r^2u_r)}{\partial r}} + {\dfrac{\partial (u_\theta \sin\theta)}{\partial \theta}} \right) \text{d}\phi = - \int_0^{2\pi} {\dfrac{\partial u_\phi}{\partial \phi}} \text{d}\phi = 0.$$ Now integrating over $[0, \theta]$ we find $$\int_0^{2\pi}u_\theta \text{d}\phi = \frac{1}{\sin\theta}\int_0^\theta \frac{\sin\theta'}{r} \int_0^{2\pi}{\dfrac{\partial (r^2u_r)}{\partial r}} \text{d}\phi \text{d}\theta' = - \frac{1}{r\sin\theta}\int_0^\theta \sin \theta' {\dfrac{\partial }{\partial r}}\left( r^2 \int_0^{2\pi} u_r d\phi\right) ~ \text{d}\theta'$$ $$\Rightarrow M = -\frac{1}{r\tan\theta} \int_0^\theta {\dfrac{\partial }{\partial r}} \left(r^2\int_0^{2\pi} u_r \sin\theta' \text{d}\phi \right) \text{d}\theta' + \int_0^{2\pi} u_r \sin\theta ~ \text{d}\phi.$$ In the above derivation, no assumption has been made about stratification and this equation holds as an identity in the magnetostrophic regime independent of stratification. In the case considered by Malkus, $M=0$ is recovered in the limit of $u_r \rightarrow 0$. It is clear that Malkus’ constraint is similar to Taylor’s constraint except now not only does the azimuthal component of the Lorentz force need to have zero average over fluid cylinders, it needs to be zero for the infinite set of constant-$z$ slices of these cylinders (here termed [*Malkus rings*]{}, see figure \[fig:constraint\_surfaces\]) that lie within the stratified region. In terms of the flow, the increased restriction of the Malkus constraint arises because it requires zero azimuthally-averaged $u_s$ at any given value of $z$, whereas Taylor’s constraint requires only that the cylindrically averaged $u_s$ vanishes and allows outward flow at a given height to be compensated by inward flow at another. We note that all Malkus states are Taylor states, but the converse is not true. [.48]{} ![ Geometry of constraint surfaces \[fig:constraint\_surfaces\]](Tay_Malk_domain_dash.png "fig:"){width="55.00000%"} [.48]{} ![ Geometry of constraint surfaces \[fig:constraint\_surfaces\]](Tay_Malk_cylinders2_rings_neartop_label.png "fig:"){width="98.00000%"} Geometry and representation of a stratified magnetostrophic model {#sec:fulldom} ================================================================= The physical motivation for applying Malkus’ constraint arises from seeking to represent a realistic model for the magnetic field in the proposed stratified layer within Earth’s outer core. Hence we compute solutions for the magnetic field in the Earth-like configuration illustrated in \[fig:fulldomain\], consisting of a spherical region in which Taylor’s constraint applies, representing the convective region of Earth’s core, surrounded by a spherical shell in which Malkus’ constraint applies, representing the stratified layer immediately beneath the CMB. Our method allows a free choice of inner radius $r_{SL}$, so in order to agree with the bulk of seismic evidence [@helffrich2010outer; @helffrich2013causes; @lay1990stably], the value $r_{SL}=0.9R$ is chosen for the majority of our solutions, where $R$ is the full radius of the core (3845 km). However due to the uncertainty which exists for the thickness of Earth’s stratified layer [@Kaneshima_2017], we also probe how sensitive our results are to layer thickness, considering $r_{SL}=0.85R$ and $r_{SL}=0.95R$ as well. The Earth’s inner core is neglected throughout, since incorporating it would lead to additional intricacies due to the cylindrical nature of Taylor’s constraint which leads to a distinction between regions inside and outside the tangent cylinder [@Livermore_Hollerbach_2012; @livermore2008structure]. Since the focus here is on the outermost reaches of the core, we avoid such complications. The method used to construct the total solution for the magnetic field throughout Earth’s core that is consistent with the Taylor and Malkus constraints is sequential. Firstly, we use a regular representation of the form shown in \[eqn:torpolexpan\] to construct a Malkus state in the stratified layer. Secondly, we construct a Taylor state which matches to the Malkus state at $r=r_{SL}$; overall the magnetic field is continuous but may have discontinuous derivatives on $r=r_{SL}$. We note that any flow driven by this magnetic field through the magnetostrophic balance may also be discontinuous at $r=r_{SL}$ because in general $u_r \neq 0$ in the inner region but $u_r = 0$ is assumed in the stratified region. Considerations of such dynamics lie outside the scope of the present study focussed only on the magnetic constraints, but imposing continuity of $u_r$ for example would clearly require additional constraints. As a pedogogical exercise we also construct some Malkus states within a fully stratified sphere ($r_{SL} = 0$), as detailed in \[sec:Apa\_both\]. Without the complications of matching to a Taylor state, the equations take a simpler form and we present some first examples in \[sec:Ap\_sol\_simp\]. Dynamically, sustenance of a magnetic field within a fully stratified sphere is of course ruled out by the theory of [@Busse_75a], which provides a strictly positive lower bound for the radial flow as a condition on the existence of a dynamo. Nonetheless it can be insightful to first consider the full sphere case, as it facilitates the consideration of fundamental principles of the magnetic field and Malkus constraint structure, and allows direct comparisons to be made with similar full sphere Taylor states. In what follows we represent a magnetic field by a sum of toroidal and poloidal modes with specific coefficients $$\label{eqn:Brep} {\bm{B}} = \sum_{l=1}^{L_{max}} \sum_{m=-l}^{l} \sum_{n=1}^{N_{max}} a_{l,n}^m {{\bm{\mathcal{T}}} }_{l,n}^m + b_{l,n}^m {{\bm{\mathcal{S}}}}_{l,n}^m$$ where ${\bm{\mathcal{T}}}_{l,n}^m={{{\boldsymbol \nabla}}\times}(T_{l,n}(r) Y_l^m {\hat{\bf r}})$, ${\bm{\mathcal{S}}}_{l,n}^m={{{\boldsymbol \nabla}}\times}{{{\boldsymbol \nabla}}\times}(S_{l,n}(r) Y_l^m {\hat{\bf r}})$, $N_{max}$ is the radial truncation of the poloidal and toroidal field. In the above, $Y_l^m$ is a spherical harmonic of degree $l$ and order $m$, normalised to unity by its squared integral over solid angle. Positive or negative values of $m$ indicate respectively a $\cos m\phi$ or $\sin m\phi$ dependence in azimuth. The scalar functions ${{T} }_{l,n}^m$ and ${{S} }_{l,n}^m$, $n\ge 1$, are respectively chosen to be the functions $\chi_{l,n}$ and $\psi_{l,n}$ composed of Jacobi polynomials [@li2010optimal; @Li_etal_2011]. They are orthogonal, and obey regularity conditions at the origin and the electrically insulating boundary condition at $r=R$ $$\frac{d \mathcal{S}_l^m}{dr} + l \mathcal{S}_l^m/R = \mathcal{T}_l^m = 0. \label{eqn:bc}$$ We note that this description is convenient but incomplete when used within the spherical shell, for which the magnetic field does not need to obey regularity at the origin. For simplicity, we nevertheless use this representation in both layers, although restricting the domain of the radial representation to $[0,r_{SL}]$ for the inner region. Discretisation of the Taylor constraint {#sec:Tay_disc} ======================================= Since the Malkus constraint forms a more restrictive constraint which encompasses the Taylor constraint it is useful for us to first summarise the structure of the Taylor constraint in a full sphere. The integral given in \[eqn:Taylor\], which Taylor’s constraint requires to be zero, is known as the Taylor integral. Although applied on an infinite set of surfaces, [@livermore2008structure] showed that Taylor’s constraint reduces to a finite number of constraint equations for a suitably truncated magnetic field expansion $$\mathcal{S}_l^m(r)=r^{l+1}\sum_{j=0}^{N_{max}}c_j r^{2j} ~~~~ \text{and} ~~~~ \mathcal{T}_l^m(r)=r^{l+1}\sum_{j=0}^{N_{max}}d_j r^{2j}, \label{eqn:torpolexpan}$$ which is an expanded version of for some $c_j$ and $d_j$. The Taylor integral itself then collapses to a polynomial of finite degree which depends upon $s^2$ [@lewis1990physical] and the coefficients $a_{l,n}^m, b_{l,n}^m$, and takes the form $$\label{eqn:Taypoly} T(s) = s^2\sqrt{R-s^2}Q_{D_{T}}(s^2)=0,$$ for some polynomial $Q_{D_{T}}$ of maximum degree $D_T$. Taylor’s constraint is now equivalent to enforcing that the coefficients of all powers of $s$ in the polynomial $Q_{D_{T}}$ equal zero, as this ensures $T(s)$ vanishes identically on every geostrophic contour. This reduces the infinite number of constraints to a finite number of simultaneous, coupled, quadratic, homogeneous equations. This reduction is vital as it gives a procedure for enforcing Taylor’s constraint in general, and allows the implementation of a method to construct magnetic fields which exactly satisfy this constraint, known as Taylor states, as demonstrated by [@livermore2009construction]. In the next section we see how, with some relatively simple alterations this procedure can be extended to the construction of exact Malkus states. Malkus states {#sec:Malk_state} ============= This section outlines some general properties of the mathematical structure of Malkus’ constraints and provides the methodology for constructing the first known Malkus states; we also address the questions of existence and uniqueness of solutions and the dimension of the resultant solution space. Along similar lines as we showed for Taylor’s constraints in \[sec:Tay\_disc\], on adopting the representation the Malkus integral reduces to a multinomial in $s^2$ and $z$ [@lewis1990physical] and we require $$M(s,z) = Q_{D_{M}}(s^2,z) = 0$$ for some finite degree multinomial $Q_{D_{M}}$ in $s$ and $z$. Note that the Taylor integral is simply a z-integrated form of $Q_{D_{M}}$. Equating every multinomial term in $Q_{D_{M}}(s^2,z)$ to zero results in a finite set of constraints that are nonlinear in the coefficients $a_{l,n}^m$ and $b_{l,n}^m$. The number of constraints can be quantified for a given truncation following a similar approach as that employed by [@livermore2008structure] for Taylor’s constraint, by tracking the greatest exponent of the dimension of length. This analysis is conducted in \[sec:enum\_con\] and results in the number of Malkus constraints given by $$\label{eq:Malk_numcon} C_M= {C_T}^2+3C_T+2,$$ where the number of Taylor constraints for an equivalent magnetic field is $C_T = L_{max} + 2N_{max} - 2$ (after the single degeneracy due to the electrically insulating boundary condition is removed) [@livermore2008structure]. Therefore we find that as expected the Malkus’ constraints are more numerous than Taylor’s constraints. It is significant to notice that $C_M \gg C_T$ and in particular for high degree/resolution systems $C_M \approx {C_T}^2$. In order to satisfy these constraints, the magnetic field has $2L_{max}N_{max}(L_{max}+2)$ degrees of freedom (this being the number of unknown spectral coefficients within the truncation of $(L_{max}, N_{max})$. In axisymmetry the number of degrees of freedom reduces to $2N_{max}L_{max}$. If we truncate the magnetic field quasi uniformly as $N= \mathcal{O}(L_{max}) \approx \mathcal{O}(N_{max})$, then we observe that at high $N$ the number of constraints ($O(N^2)$ Malkus constraints; $O(N)$ Taylor constraints) is exceeded by the number of degrees of freedom of $N^3$. A simple argument based on linear algebra suggests that many solutions exist at high $N$, however this may be misleading because the constraints are nonlinear and it is not obvious *a priori* whether any solutions exist, or if they do, how numerous they might be. We consider a simple example in \[sec:Apb\], which shows the structure of constraint equations that arise. The example highlights that degeneracy of the constraint equations plays a far more significant role for the Malkus constraints compared with the Taylor constraints, which only have a single weak degeneracy due to the electrically insulating boundary condition [@livermore2008structure]. However, due to the complex nature of these nonlinear equations, at present we have no theory to predict which constraints will be degenerate and hence the total number of independent constraints. Because of the apparent uncertainty of the existence of Malkus states, it is instructive to identify whether imposing strong symmetry is useful to identify very simple examples. Owing to symmetries inherent in the spherical harmonics, many classes of simple Taylor states exist, as outlined by [@livermore2009construction]: for example any field that is either symmetric or anti-symmetric with respect to a rotation of $\pi$ radians about the $x$-axis is a Taylor state. Due to the absence of averaging in $z$, such symmetric magnetic fields do not automatically satisfy the Malkus constraints. However some simple classes of field are guaranteed to be Malkus states, such as single spherical harmonic modes, axisymmetric purely toroidal or poloidal fields since the integrand itself $(({{{\boldsymbol \nabla}}\times}{{\bf B}}) \times {{\bf B}})_\phi$ is zero. Also equatorially symmetric purely toroidal or poloidal fields comprising either only cosine or only sine dependence in azimuth are Malkus states as the resultant integrand is anti-symmetric with respect to a rotation of $\pi$ radians and hence the azimuthal average over $[0,2\pi]$ causes the Malkus integral to vanish. Finding a Malkus state ====================== Owing to the nonlinear albeit finite nature of the Malkus constraints, it is far from obvious whether any solutions exist beyond those of the simple structure explored above. In the next section, we demonstrate the existence of a class of solutions with arbitrarily complex lateral structure. A special class of Malkus states {#sec:theo} -------------------------------- Here we demonstrate that within the class of magnetic fields that all contain a known poloidal component (but whose toroidal component is unknown) then there exists systems where all the Malkus constraints are linear in the unknown spectral parameters. A formal statement of this fact is given in the theorem given below. Any arbitrary, prescribed, polynomial poloidal field can be transformed into a Malkus state through the addition of an appropriate polynomial toroidal field. \[proof:linear\] We prove below that by considering an arbitrary, prescribed, truncated polynomial poloidal field, the addition of a specific choice of toroidal modes renders the Malkus constraints linear in the unknown toroidal coefficients. By taking a sufficient number of such modes such that the degrees of freedom exceed the number of constraints, it follows that for the general case (barring specific degenerate cases) by solving the linear system the resultant magnetic field is a Malkus state. To show this, because the Malkus constraint is quadratic in the magnetic field, we introduce the concept of a magnetic field interaction. In general there are three possible field interactions within the Malkus integral, toroidal-toroidal, poloidal-poloidal and toroidal-poloidal, respectively $$M = \sum_{l_1,l_2}^{L_{max}} \sum_{m}^{L_{max}} \left( [{\bm{T}}_{l_l}^m, {\bm{T}}_{l_2}^m] + [{\bm{S}}_{l_l}^m, {\bm{S}}_{l_2}^m] + [{\bm{T}}_{l_l}^m, {\bm{S}}_{l_2}^m] \right)$$ where $$\begin{aligned} \label{eq:tortor} [{\bm{T}}_{l_l}^m, {\bm{T}}_{l_2}^m] &= \int_0^{2\pi} \frac{l_1(l_1+1)\mathcal{T}_{l_l}^m \mathcal{T}_{l_2}^m}{r^3 \sin\theta}\left({Y}_{l_l}^m{\dfrac{\partial {Y}_{l_2}^m}{\partial \phi}}\right) s ~ \text{d}\phi +sc, \\ [{\bm{S}}_{l_l}^m, {\bm{S}}_{l_2}^m] &= \int_0^{2\pi} \frac{l_1(l_1+1)\mathcal{S}_{l_l}^m (\frac{\text{d}^2}{\text{d}r^2}-l_2(l_2+1)/r^2)\mathcal{S}_{l_2}^m}{r^3 \sin\theta}\left({Y}_{l_l}^m{\dfrac{\partial {Y}_{l_2}^m}{\partial \phi}}\right) s ~ \text{d}\phi +sc,\nonumber \\ [{\bm{T}}_{l_l}^m, {\bm{S}}_{l_2}^m] &= \int_0^{2\pi} \frac{1}{r^3}\left( l_1(l_1+1){T}_{l_l}^m \frac{\text{d}\mathcal{S}_{l_2}^m}{\text{d}r} Y_{l_1}^m {\dfrac{\partial {Y}_{l_2}^m}{\partial \theta}}\right. \left.- l_2(l_2+1)\mathcal{S}_{l_2}^m \frac{\text{d}T_{l_1}^m}{\text{d}r} Y_{l_2}^m {\dfrac{\partial {Y}_{l_1}^m}{\partial \theta}} \right) s ~ \text{d}\phi, \nonumber \end{aligned}$$ where $sc$ is the symmetric counterpart given by interchanging the vector harmonics and hence the positions of $l_1$ and $l_2$ [@livermore2008structure]. Note that there is no poloidal-toroidal interaction since the curl of a poloidal vector is toroidal and (${\bm{\mathcal{T}_1}} \times {\bm{\mathcal{T}_2}})_\phi = 0,$ for any two toroidal vectors ${\bm{\mathcal{T}_1}}$ and ${\bm{\mathcal{T}_2}}$. For the situation we consider of a given poloidal field, then the only non-linearity within the unspecified coefficients arises from the toroidal-toroidal interactions, which results in quadratic dependence, just as for the general case with unprescribed poloidal field. However, by restricting attention to toroidal fields that result in no toroidal-toroidal interaction, the unknown toroidal coefficients appear only in a linear form through the toroidal-poloidal interactions. Axisymmetric modes are the simplest set of toroidal modes which are non-self-interacting, however there are too few of them (within the truncation) to solve the resulting linear system which is over-constrained (see \[fig:new\_con\_dof\]). Therefore we require additional non-axisymmetric toroidal modes, which we choose such that the total set of toroidal modes remains non-self-interacting. This is achieved by exploiting the previously noted observations that any single harmonic is a Malkus state and that the set of equatorially symmetric toroidal modes $T_l^l$ is a Malkus state (and therefore has no self-interaction). Owing additionally to azimuthal symmetry, the modes $${T_1^0}, T_2^0, \cdots, T_1^{-1}, T_{1}^1,T_2^{-2}, T_{2}^2, \dots,$$ that is, the modes $T^m_l$ with $m = 0$ or $m = \pm l$, have no self-interactions. Each harmonic mode may be expanded in radial modes up to the truncation $N_{max}$. The non-interacting nature of the modes may be confirmed from \[eq:tortor\]. The addition of these nonaxisymmetric modes increases the number of degrees of freedom from the axisymmetric case by a factor of three such that it is now larger than the number of constraints (which are now all linear). This can be shown in general since for a toroidal field truncated at $L_1, N_1$ and a poloidal field truncated at $L_2, N_2$ the number of Taylor constraints is equal to half of the maximum degree of the polynomial in $s$, (i.e. $C_T = \frac{1}{2}(L_1+L_2+2N_1+2N_2) - 2$) [@livermore2008structure] and the maximum number of Malkus constraints we have shown is given in terms of $C_T$ by \[eq:Malk\_numcon\]. This results in a situation where if the poloidal field is fixed at a chosen resolution then for a toroidal field truncated quasi uniformly as $N= \mathcal{O}(L_{max}) \approx \mathcal{O}(N_{max})$ then we can see that the number of Malkus constraints scales as $\frac{9}{4} N^2$, which importantly, grows slower than the number of degrees of freedom for the non-axisymmetric linear system which scales as $3 N^2$. Hence it is guaranteed that at a sufficiently large resolution toroidal field representation then there will be more degrees of freedom than constraints. Therefore, barring degenerate cases, Malkus states exist. Compared with the case of a purely axisymmetric toroidal field, the number (but not the specific form) of linear constraints remains unaltered by the addition of these extra non-axisymmetric modes. It is worth noting that the depth of the stratified layer does not enter into above derivation. The magnetic field solution in fact satisfies the Malkus constraints everywhere within its region of definition: in our case, this is the full sphere $0 \le r \le R$. provides a specific example of the number of constraints given a poloidal field of degree $13$. It demonstrates two important things. Firstly, that due to degeneracy (for which we have no explanation) the independent linear constraints (red triangles) are much fewer than the full set of linear constraints (red squares). Secondly, that the number of degrees of freedom exceed the number of independent constraints at $L_{max}=N_{max} \geq 10$ if we consider the non-axisymmetric toroidal basis (blue circles) but is not exceeded at any truncation if we adopt the axisymmetric toroidal basis (blue stars). In particular, taking a non-axisymmetric toroidal field with truncation $L_{max}=N_{max}=13$ gives an infinite set of Malkus states. We note that the above deviation is based upon a polynomial representation, which is sufficient for our purposes here. However, we know that any continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function, and hence it can be extended to include an arbitrary magnetic field structure by expressing the relevant scalars in a polynomial basis of suitably large truncation. We need to match the Malkus state (physically defined within the stratified layer) to a Taylor state in the region beneath. One way of proceeding is to simply evaluate the Malkus state beneath the stratified layer (where it also satisfies Taylor constraint); however this effectively imposes additional constraints on the inner region and is overly restrictive. Instead, we impose the same profile of poloidal field and expand the toroidal component of the Taylor state in the same set of spherical harmonic modes as used for the Malkus state. Such a choice also renders the Taylor constraints linear in the unknown toroidal coefficients. ![This graph compares the number of constraints to degrees of freedom (DOF) as a function of toroidal field spherical harmonic resolution with $L_{max}=M_{max}=N_{max}$, given a fixed poloidal field of $L_{max}=M_{max}=13$. This illustrates that for the non-axisymmetric linear system we construct then the number of degrees of freedom (red) exceeds the number of independent constraints (red triangles) for a toroidal field of resolution $L_{max}=N_{max} \geq 10$. \[fig:new\_con\_dof\]](new_dof_vs_constraints5.pdf){width="70.00000%"} Further geophysical constraints ------------------------------- In order to construct a Malkus state according to the above procedure, we need to completely specify the poloidal field. Following [@livermore2009construction], we downwards continue observation-derived models inside the core $r \le R$ by assuming a profile for each poloidal harmonic of degree $l$ that minimises the Ohmic dissipation within the modelled core $$\label{eqn:polprofile} (2l+3)r^{l+1} - (2l+1)r^{l+3}.$$ We adopt two choices of observation-derived model. First, we use the CHAOS-6 model [@Finlay_etal_2016] at epoch 2015 evaluated to degree 13, the maximum obtainable from geomagnetic observations without significant interference due to crust magnetism [@Kono2015Geointro]. Second, we use the time-averaged field over the last 10000 years from the CALS10k.2 model [@constable2016persistent], which although is defined to degree 10 it has power concentrated mostly at degrees 1–4 because of strong regularisation of sparsely-observed ancient magnetic field structures. Recalling that the magnetostrophic state that we seek is defined over millenial timescales, this longer average provides on the one hand a better approximation to the background state, but on the other a much lower resolution. Even within these geomagnetically consistent Malkus states, there are nevertheless multiple degrees of freedom remaining. This raises the question of which of the multiple possible solutions are most realistic for the Earth, and motivates us to incorporate additional conditions to distinguish ‘Earth-like’ solutions. We determine specific solutions through optimising the toroidal field $\bf T$ through either its Ohmic dissipation or its energy, respectively $$Q = \frac{\eta}{\mu_0}\int_V ({{{\boldsymbol \nabla}}\times}{\bf T})^2 dV, \qquad \mathcal{E} = \frac{1}{2\mu_0} \int_V {\bf T}^2 dV,$$ where $\eta \approx 1$ m$^2$s$^{-1}$ is magnetic diffusivity and $\mu_0=4\pi \times 10^{-7}~\text{NA}^{-2}$ is the permeability of free space. Both of these target functions are quadratic in the magnetic field, and so seeking a minimal value subject to the now linear constraints is straightfoward. In our sequential method to find a matched Malkus-Taylor state, we first optimise the Malkus state, and then subsequently find an optimal matching Taylor state. Of the dissipation mechanisms in the core: Ohmic, thermal and viscous, the Ohmic losses are believed to dominate. On these grounds, the most efficient arrangement of the geomagnetic field would be such that Ohmic dissipation $Q$ is minimised. It is worth noting that in general our procedure is not guaranteed to provide the Malkus state field with least dissipation, but only an approximation to it, since we effectively separately optimise for the poloidal and toroidal component with least dissipation. In terms of finding a Malkus state with minimum toroidal field energy, this is useful in allowing us to determine the weakest toroidal field which is required in order to transform the imposed poloidal field into a Malkus state. In \[sec:Ap\_sol\_simp\] we compare the method of finding the weakest toroidal field required to make a Malkus state, between using only selected toroidal modes, and all toroidal modes (resulting in a nonlinear system). For low truncation, minimisation of the toroidal energy subject to these nonlinear constraints is computationally solvable, and the two approaches produce comparable results. This suggests that estimates for the lower bound of Earth’s toroidal field strength obtained using our linearised approach will not differ greatly from related full non-linear optimisation (that is computationally infeasible). An Earth-like example {#sec:highres} ===================== We now present some visualisations of the specific class of Malkus states discussed above with minimal toroidal field energy for which the system of equations which enforce the constraints is linear. The geometry assumed here is as illustrated in \[fig:fulldomain\], with a Malkus state in the stratified layer in the region $0.9R < r \leq R$, matching to an inner Taylor state. We shall show the adjustment of the imposed poloidal field structure to a Malkus state by the required additive toroidal field. The strength of this toroidal field will be shown by contour plots of its azimuthal component. We note that the radial component of the magnetic field is defined everywhere by the imposed poloidal field, with the smooth degree 2 radial profile defined in \[eqn:polprofile\]. Magnetic field at 2015 {#sec:Tay_Malk_sol} ---------------------- We begin by showing in \[fig:CMB\_field\_basemap\_cor\_both\] both the radial and azimuthal structure, $B_r$ and $B_\phi$, of the CHAOS-6 model at epoch 2015 on the CMB, $r=R$. Of note is that at the truncation to degree 13, the azimuthal component is about half as strong as the radial component. [.47]{} ![Magnetic field at the CMB based on the poloidal field from CHAOS-6 at epoch 2015. Visualised using the Mollweide projection and centred on the Greenwich meridian. \[fig:CMB\_field\_basemap\_cor\_both\]](CMB_field_basemap_rot__phi_new2_a_r.pdf "fig:"){width="90.00000%"} [.47]{} ![Magnetic field at the CMB based on the poloidal field from CHAOS-6 at epoch 2015. Visualised using the Mollweide projection and centred on the Greenwich meridian. \[fig:CMB\_field\_basemap\_cor\_both\]](CMB_field_basemap_rot__phi_new2_a_phi.pdf "fig:"){width="90.00000%"} summarises the strength of toroidal field (in terms of its azimuthal root mean squared value over solid angle) as a function of radius, for different toroidal truncations $L_{max} = N_{max}$ (shown in different colours). The toroidal field is required to be four orders of magnitude stronger in the stratified layer in order to satisfy the more restrictive Malkus constraints, compared with the inner region in which the weaker Taylor constraint applies, and adopts a profile that is converged by degree 13. The strong toroidal field throughout the stratified layer occurs despite the electrically insulating boundary condition at the outer boundary that requires the toroidal field to vanish. Within the stratified layer, the azimuthal toroidal field strength attains a maximum rms value of 2.5 mT at a radius of about $0.98R$ or a depth of about 70 km, about double the observed value at the CMB, and locally exceeds the imposed azimuthal poloidal magnetic field (of rms $0.28$ mT at this radius). ![\[fig:Tor\_Pol\_field\_radial\_profile\] The root mean squared azimuthal field strength (defined over solid angle) as a function of radius, comparing the strengths of the poloidal field (red) and toroidal field (blue, green, magneta and cyan) for toroidal fields with maximum spherical harmonic degree, order and radial resolution, 13 – 16 respectively. The poloidal field is the degree 13 field of minimum Ohmic dissipation compatible with the CHAOS-6 model at epoch 2015 [@Finlay_etal_2016]. ](Toroidal_strength_radial_profile.pdf){width="70.00000%"} shows $B_\phi$ for both the total field and the toroidal component in isolation, using a toroidal truncation of 13 (corresponding to the blue line in \[fig:Tor\_Pol\_field\_radial\_profile\].) The top row shows the structure at the radius of maximum rms toroidal field ($r=0.98R$), demonstrating that the additive toroidal field component (of maximum 8 mT) dominates the total azimuthal field. The bottom row shows a comparable figure at $r=0.7R$, in the inner region where only Taylor’s constraint applies. Plotted on the same scale, the required additive toroidal component is tiny compared with the imposed poloidal field. This highlights again that the Malkus constraint is much more restrictive than the Taylor constraint. [.47]{} ![\[fig:fullsolution\] Minimal toroidal-energy solution (a,c) shown by the azimuthal component, of a Malkus state ($0.9R < r \leq R$) and Taylor state $r \le 0.9R$, compared with the total azimuthal component (b,d). Figures (a,b) show the field at a radius of $r=0.98R$, close to where the maximum rms azimuthal toroidal field occurs, while (c,d) show the inner region at $r=0.7R$.](Malkus_Toroidal_2_a2_r098_2.pdf "fig:"){width=".85\linewidth"} [.47]{} ![\[fig:fullsolution\] Minimal toroidal-energy solution (a,c) shown by the azimuthal component, of a Malkus state ($0.9R < r \leq R$) and Taylor state $r \le 0.9R$, compared with the total azimuthal component (b,d). Figures (a,b) show the field at a radius of $r=0.98R$, close to where the maximum rms azimuthal toroidal field occurs, while (c,d) show the inner region at $r=0.7R$.](Malkus_Total_2_a2_r098_2.pdf "fig:"){width=".85\linewidth"} [.47]{} ![\[fig:fullsolution\] Minimal toroidal-energy solution (a,c) shown by the azimuthal component, of a Malkus state ($0.9R < r \leq R$) and Taylor state $r \le 0.9R$, compared with the total azimuthal component (b,d). Figures (a,b) show the field at a radius of $r=0.98R$, close to where the maximum rms azimuthal toroidal field occurs, while (c,d) show the inner region at $r=0.7R$.](tor07_2.pdf "fig:"){width=".85\linewidth"} [.47]{} ![\[fig:fullsolution\] Minimal toroidal-energy solution (a,c) shown by the azimuthal component, of a Malkus state ($0.9R < r \leq R$) and Taylor state $r \le 0.9R$, compared with the total azimuthal component (b,d). Figures (a,b) show the field at a radius of $r=0.98R$, close to where the maximum rms azimuthal toroidal field occurs, while (c,d) show the inner region at $r=0.7R$.](tot07_2.pdf "fig:"){width=".85\linewidth"} For comparison, \[Tay\_fullsphere\] shows an equivalent solution to \[fig:fullsolution\](a,b) but in the absence of stratification (where the magnetic field satisfies only Taylor’s constraint). The toroidal contribution to the azimuthal field is very weak (note the colourbar range is reduced from that of \[fig:fullsolution\](a,b) from 8 to 0.04 mT) and is of very large scale. This further highlights the weakness of the Taylor constraints compared with the Malkus constraints. [.47]{} ![Azimuthal field for an unstratified comparative case, for which the magnetic field satisfies only Taylor’s constraint. \[Tay\_fullsphere\]](Tay_tor098.pdf "fig:"){width="85.00000%"} [.47]{} ![Azimuthal field for an unstratified comparative case, for which the magnetic field satisfies only Taylor’s constraint. \[Tay\_fullsphere\]](Tay_tot098.pdf "fig:"){width=".85\linewidth"} Time averaged field over the past ten millenia ---------------------------------------------- Here we show results for a poloidal field that is derived from the 10000-year time averaged field from the CALS10k.2 model [@constable2016persistent]. The model is only available up to spherical harmonic degree 10, hence we adopt a truncation of $L_{max} = N_{max} = 10$ for the toroidal field. Due to the absence of small-scale features in the field (caused by regularisation) the maximum value of $B_r$ is reduced to about $1/2$ of the comparable value from the degree-13 CHAOS-6 model from epoch 2015, and similarly the azimuthal field to about $1/6$ of its value. We note that over a long enough time span, the magnetic field is generally assumed to average to an axial dipole: a field configuration that is both a Malkus state and one in which the azimuthal component vanishes. Thus a small azimuthal component is consistent with such an assumption. [.47]{} ![Magnetic field at the CMB based on the 10000-year time average field from CALS10k.2 \[fig:CALS\_CMB\_field\_basemap\_cor\_both\]](CMB_field_basemap_r_2_a.pdf "fig:"){width="90.00000%"} [.47]{} ![Magnetic field at the CMB based on the 10000-year time average field from CALS10k.2 \[fig:CALS\_CMB\_field\_basemap\_cor\_both\]](CMB_field_basemap_r_2_a_phi.pdf "fig:"){width="90.00000%"} Contours of the azimuthal field within the stratified layer (at $r=0.97R$) are shown in \[fig:CALS\_Min\_total\_r095\_phi\_rot\_both\], which is approximately the radius at which the maximum rms azimuthal toroidal field occurs. As before, the toroidal field dominates the azimuthal component, and its rms (1.66 mT) is about double that on the CMB (0.085 mT). Although its maximum absolute value is about 3 mT, less than the 8 mT found in the 2015 example above, this is consistent with the overall reduction in structure of the imposed poloidal field. [.47]{} ![The azimuthal component of the Malkus state magnetic field within the stratified layer at a radius of $r=0.97R$, approximately the radius of maximum rms azimuthal toroidal field. \[fig:CALS\_Min\_total\_r095\_phi\_rot\_both\]](CALSMalkus_Toroidal_2_a2_r097_2.pdf "fig:"){width="90.00000%"} [.47]{} ![The azimuthal component of the Malkus state magnetic field within the stratified layer at a radius of $r=0.97R$, approximately the radius of maximum rms azimuthal toroidal field. \[fig:CALS\_Min\_total\_r095\_phi\_rot\_both\]](CALSMalkus_Total_2_a2_r097_2.pdf "fig:"){width="90.00000%"} Discussion ========== Estimates of the internal magnetic field strength {#sec:tor_min} ------------------------------------------------- Estimating the magnetic field strength inside the core is challenging, because observations made on Earth’s surface, using a potential-field extrapolation, only constrain the poloidal magnetic structure down to the CMB and not beneath, but also even this structure is visible only to about spherical harmonic degree 13. Furthermore, within the framework of such an extrapolation, the toroidal field is zero on the CMB. Estimating the field within the core beyond these surface values requires insight from numerical models or observations of physical mechanisms that are sensitive to the interior field. Based on numerical models, [@Zhang_Fearn_93] suggest that a criterion for stability of the geomagnetic field is a toroidal field no more than 10 times that of the poloidal field, resulting an approximate upper bound of 5 mT. In a more recent and conflicting study, [@sreenivasan2017damping] suggest that the mean toroidal field is approximately 10 mT or higher, since this intensity is required for the slow magnetostrophic waves present to be able to originate from small-scale motions in the core. Observation-based studies based on electric field measurements [@shimizu1998observational] suggest a toroidal field strength at the CMB of anywhere in the range of 1-100 times that of the poloidal magnetic field there (i.e. up to about 100 mT). [@Buffett_2010] calculated a core averaged field strength of 2.5 mT from measurements of tidal dissipation; @gubbins2007geomagnetic estimated the toroidal field strength of 1 mT as compatible with patches of reversed magnetic flux. Lastly, the magnetic signature of both torsional and Rossby waves have led to respective estimates of at least 2 mT for $B_s$ within the core and therefore an RMS strength of $4$ mT assuming isotropy [@gillet2010fast], and an RMS estimate of $B_\phi$ of 12 mT [@Hori_etal_2015]. The strong toroidal field within our 2015 models of up to 8 mT (and rms $B_\phi$ of $2.5$ mT) within the stratified layer (at radius $r=0.98R$ or a depth of about 70 km) is in agreement with the majority of these estimates. This maximum value is notably about 8 times stronger than the observed radial field on the CMB. In both the 2015 and the 10000-yr averaged model, the rms toroidal field within the stratified layer was about double the radial field on the CMB. Interestingly, the azimuthal component of our solution within the inner unstratified region is about 100 times weaker, demonstrating the extent to which Malkus’ constraint is far more restrictive than Taylor’s constraint. Limitations of our model ------------------------ Our model does not produce a formal lower bound on the azimuthal component of a magnetic field that (a) satisfies both the Malkus and Taylor constraints in their relevant regions along with (b) constraints on the radial field at the CMB. Instead, our results give only an upper bound on the lower bound [e.g. @jackson2011ohmic] because we have made a variety of simplifying assumptions, the most notable of which are (i) we have restricted ourselves to a subspace of Malkus states for which the constraints are linear (ii) we have imposed the entire poloidal profile and (iii) we have used a regular basis set for all magnetic fields even within the stratified layer when this is not strictly necessary. However, we show for the example considered in \[sec:Ap\_sol\_simp\] that in this case assumption (i) does not have a significant impact and our estimate is close to the full nonlinear lower bound. It may be that the other assumptions also do not cause our azimuthal field estimates to deviate significantly from the actual lower bound. Leaving aside the minimum toroidal field suggested by our model, our analysis allows two statements to be made on the weakness of the Malkus constraints, and the ability of magnetic structures assumed on the CMB to probe the magnetic structure within the stratified layer. Firstly, our method can find a toroidal field that converts any poloidal field into a Malkus state within a stratified layer of any depth. This means that we cannot use consistency of observation-derived models of the radial field with the Malkus constraints as a discriminant to test the probe the existence (and depth) of a stratified layer: all such models are consistent. Second, even if a stratified layer is assumed, the lateral radial magnetic field structure at the bottom of the layer is unconstrained by its structure at the top because we can find a Malkus state assuming any poloidal profile within the layer. Thus using considerations of the Malkus constraints, models of the surface magnetic field, such as CHAOS-6, cannot be downwards-continued further than the CMB into a stratified layer beneath. Model robustness ---------------- There remains much uncertainty over the depth of any stably stratified layer at the top of the Earth’s core [@hardy2019stably]. Hence it is natural to consider how our results may change if the layer were to be of a different thickness to the $10\%$ of core radius used, as such we also calculated minimum toroidal-energy solutions matched to CHAOS-6 in epoch 2015 for layer thicknesses of $5\%$ and $15\%$. We find very little dependence of the field strengths internal to the layer on the depth of the layer itself, with our root mean square azimuthal field taking peak values of 2.7, 2.5 and 2.4 mT for thicknesses of $5\%$, $10\%$ and $15\%$ respectively. The resolution of poloidal field also impacts significantly our optimal solutions. This has already been identified in the comparison between the degree-13 2015 model, and the degree-10 10,000-yr time-averaged model, that respectively resulted in rms azimuthal field estimates of 2.5 and 1.2 mT. Interestingly, for very long time-averaging windows the magnetic field is widely supposed to converge to an axial dipole, and assuming a simple poloidal profile is itself an exact Malkus state, with zero azimuthal field strength. We can further test the effect of resolution by considering maximum poloidal degrees of 6 and 10 for the 2015 model to compare with our solution at degree 13. We find that our estimates for the root mean square azimuthal field (taken over their peak spherical surface) were 1.6 and 2.2 mT respectively. In all these calculations, the spherical harmonic degree representing the toroidal field was taken high enough to ensure convergence. Thus stronger toroidal fields are apparently needed to convert more complex poloidal fields into a Malkus state. This has important implications for the Earth, for which we only know the degree of the poloidal field to about $13$ due to crustal magnetism. Our estimates of the azimuthal field strength would likely increase if a full representation of the poloidal field were known. Ohmic dissipation ----------------- Our method can be readily amended to minimise the toroidal Ohmic dissipation, rather than the toroidal energy. In so doing, we provide a new estimate of the lower bound of Ohmic dissipation within the core. Such lower bounds are useful geophysically as they are linked to the rate of entropy increase within the core, which has direct implications for: core evolution, the sustainability of the geodynamo, the age of the inner core and the heat flow into the mantle [@jackson2011ohmic]. The poloidal field with maximum spherical harmonic degree 13 that we use, based on CHAOS-6 [@Finlay_etal_2016] and the minimum Ohmic dissipation radial profile [@Book_Backus_etal_96] has by itself an Ohmic dissipation of 0.2 GW. [@Jackson_Livermore_2009] showed that by adding additional constraints for the magnetic field, a formal lower bound on the dissipation could be raised to 10 GW, and even higher to 100 GW with the addition of further assumptions about the geomagnetic spectrum. This latter bound is close to typical estimates of 1 - 15 TW [@Jackson_Livermore_2009; @jackson2011ohmic]. The addition of extra conditions derived from the assumed dynamical balance, namely Taylor constraints, were considered by @jackson2011ohmic by adopting a very specific magnetic field representation. These constraints alone raised their estimate of the lower bound from 0.2 to 10 GW, that is, by a factor of 20. In view of the much stronger Malkus constraints (compared to the Taylor constraints), we briefly investigate their impact here. We follow our methodology and find an additive toroidal field of minimal dissipation (rather than energy) that renders the total field a Malkus state. The dissipation is altered from $0.2$ to $0.7$ GW. That this increase is rather small (only a small factor of about 3) is rather disappointing, but is not in contradiction to our other results. It is generally true that the Malkus constraints are more restrictive than the Taylor constraint, but this comparison can only be made when the same representation is used for both. The method of @jackson2011ohmic assumed a highly restrictive form, so that in fact their Taylor states were apparently actually more tightly constrained than our Malkus states and thus produced a higher estimate of the lower bound. Despite our low estimate here, additional considerations of Malkus constraint may increase the highest estimates of @Jackson_Livermore_2009 well into the geophysically interesting regime. Further extensions ------------------ The Malkus states we have computed, which match to field observations, provide a plausible background state at the top of the core. It may be interesting for future work to investigate how waves thought to exist within such a stratified layer [@Buffett_2014] may behave when considered as perturbations from such a background state, and whether they remain valid suggestions for explaining secular variation in the geomagnetic field. Similarly, combining our analysis with constraints on $B_s$ from torsional wave models [@gillet2010fast] may be insightful, and would combine aspects of both long and short-term dynamics. It is worth noting though, that we have investigated only static Malkus states without consideration of dynamics: we do not require the magnetic field to be either steady or stable, both of which would apply additional important conditions. An obvious extension to this work then is to investigate the fluid flows which are generated by the Lorentz force associated with these fields. This would then allow a consideration of how such flows would modify the field through the induction equation. These dynamics are however, are still relatively poorly known even for the much simpler problem of Taylor states. Recent progress by [@hardy2018determination] now allows a full calculation of the flow driven by a Taylor state. A general way to discover stationary and stable Taylor states comparable with geomagnetic observations is still out of reach, and currently the only way to find a stable Taylor state is by time-stepping [e.g. @li2018taylor]. The well established test used to determine whether the appropriate magnetostrophic force balance is achieved within numerical dynamo simulations is ‘Taylorisation’, which represents a normalised measure of the magnitude of the Taylor integral \[eqn:Taylor\] and hence the departure from the geophysically relevant, magnetostrophic regime [e.g. @Takahashi_etal_2005]. We propose an analogous quantity termed ‘Malkusisation’ defined in the same way, in terms of the Malkus integral: $$\text{Malkusization} = \frac{|\int_0^{2\pi} ([{{{\boldsymbol \nabla}}\times}{{\bf B}}] \times {{\bf B}})_\phi d\phi|}{ \int_0^{2\pi} | ([{{{\boldsymbol \nabla}}\times}{{\bf B}}] \times {{\bf B}})_\phi | d\phi }$$ This quantity is expected to be very small within a stratified layer adjacent to a magnetostrophic dynamo, provided stratification is sufficiently strong. The recently developed dynamo simulations of [@olson2018outer; @stanley2008effects] which incorporate the presence of a stratified layer can utilise the computation of this quantity to access the simulation regime. Finally, we note that the appropriate description of a stratified layer may in fact need to be more complex than a single uniform layer that we assume. Numerical simulations of core flow with heterogeneous CMB heat flux by [@mound2019regional] find that localised subadiabatic regions that are stratified are possible amid the remaining actively convecting liquid. If indeed local rather than global stratification is the more appropriate model for the Earth’s outermost core then the condition of requiring an exact Malkus state would not apply, and the constraints on the magnetic field would be weakened by the existence of regions of non-zero radial flow. Conclusion ========== In this paper we have shown how to construct magnetic fields that are consistent with geomagnetic observations, a strongly stratified layer and the exact magnetostrophic balance thought to exist within Earth’s core. To do this, we derived the Malkus constraints that must be satisfied by such a magnetic field, whose structure gives insight into the nature of the Earth’s magnetic field immediately beneath the CMB, where a layer of stratified fluid may be present. For a fixed magnetic field resolution, although the Malkus constraints are more numerous than the Taylor constraints, many solutions compatible with geomagnetic observations still exist. By making further assumptions about the field structure, we estimate that the toroidal field within the stratified layer is about 8 mT, significantly stronger than the 1 mT of the radial field inferred from degree-13 observations. Acknowledgements ================ This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) Centre for Doctoral Training in Fluid Dynamics at the University of Leeds under Grant No. EP/L01615X/1. P.W.L. acknowledges partial support from NERC grant NE/G014043/1. The authors would also like to thank Dominique Jault and Emmanuel Dormy for helpful discussions, as well as the Leeds Deep Earth group for useful comments. Figures were produced using matplotlib [@Hunter_2007]. Full sphere Malkus states {#sec:Apa_both} ========================= Simple Example {#sec:Apb} -------------- Here we consider a simple example of an axisymmetric magnetic field in a full sphere of radius $R$, consisting of four modes: a toroidal $l=1$, $n=1$ mode, a toroidal $l=1$, $n=2$ mode, a poloidal $l=1$, $n=1$ mode and a poloidal $l=1$, $n=2$ mode, each of which has an unspecified corresponding coefficient $\alpha_{l,n}$ and $\beta_{l,n}$ for toroidal and poloidal modes respectively. Through this we demonstrate the form of the linear constraints which arise from Malkus’ constraint in this case. It is significant to note the vital role of degeneracy within these constraints in permitting a solution. Through computing the Malkus integral and enforcing that this is zero for all values of $s$ and $z$ by requiring that the coefficients of all powers of $s$ and $z$ vanish we obtain a series of simultaneous equations: $$\left(-\frac{11}{8}\beta_{1,2}+2\beta_{1,1}\right)\alpha_{1,2}-\frac{253}{140}\left(\frac{77}{69}\beta_{1,2}+\frac{56}{759}\beta_{1,1}\right)\alpha_{1,1}=0,$$ $$\left(\frac{319}{84}\beta_{1,2}-\frac{10}{3}\beta_{1,1}\right)\alpha_{1,2}-\frac{253}{70}\beta_{1,2}\alpha_{1,1}=0,$$ $$\left(-\frac{165}{56}\beta_{1,2}+\beta_{1,1}\right)\alpha_{1,2}-\frac{253}{140}\beta_{1,2}\alpha_{1,1}=0,$$ $$\left(\frac{319}{84}\beta_{1,2}-\frac{10}{3}\beta_{1,1}\right)\alpha_{1,2}-\frac{253}{70}\beta_{1,2}\alpha_{1,1}=0,$$ $$\left(-\frac{165}{28}\beta_{1,2}+2\beta_{1,1}\right)\alpha_{1,2}-\frac{253}{70}\beta_{1,2}\alpha_{1,1}=0,$$ $$\left(-\frac{165}{56}\beta_{1,2}+\beta_{1,1}\right)\alpha_{1,2}-\frac{253}{140}\beta_{1,2}\alpha_{1,1}=0.$$ Although there are 6 equations here, there are only two independent conditions: $$\alpha_{1,1}\beta_{1,2}+\frac{5}{2}\alpha_{1,2}\beta_{1,2}=0, ~~~~ \text{and} ~~~~ \alpha_{1,2}\beta_{1,1}+\frac{11}{7}\alpha_{1,2}\beta_{1,2}=0.$$ If both $\beta_{1,2}$ and $\alpha_{1,2}$ are nonzero, then these become linear constraints. Hence, in this case we can see that there are 4 degrees of freedom, 6 constraint equations but only 2 independent constraints. This means that while on first inspection the system appears to be overconstrained with no solution, there are in fact multiple Malkus state solutions, with the solution space being spanned by two degrees of freedom ($\beta_{1,2}, \alpha_{1,2}$) with the other coefficients determined in terms of these by the relationships: $$\alpha_{1,1}=-\frac{5}{2}\alpha_{1,2} ~~~~ \text{and} ~~~~ \beta_{1,1}=-\frac{11}{7}\beta_{1,2}.$$ Despite the significant degenercy in the Malkus constraints, they are notably more restrictive than the Taylor constraints for this truncation of $L_{max}=1$, $N_{max}=2$, for which the Taylor integral is identically zero and so provides no restriction. Solution of a low-resolution system {#sec:Ap_sol_simp} ----------------------------------- We now present the first known non-trivial solution of a Malkus state. Here we consider a full sphere magnetic field truncated at $L_{max}=3, ~ N_{max}=3, ~ M_{max}=3$, and impose a minimum Ohmic dissipation poloidal profile that matches the CHAOS-6 model (to degree 3) at $r=R$.We seek a toroidal field using all spherical harmonic modes within the truncation $L_{max}=3,~N_{max}=3$ (described by 45 degrees of freedom) that when added to this poloidal field satisfies the Malkus constraints. Of the 72 nonlinear constraint equations, only 42 are independent. Thus the number of degrees of freedom exceed the number of independent constraints, although since the constraints are nonlinear it is not immediate that a solution exists. However, using the computer algebra software Maple, we find the solution that minimises toroidal field strength as well as satisfying all the constraints, which is visualised in \[fig:fixpol\_nonlin\_all\]. We cannot generalise this procedure to higher resolutions because of the numerical difficulty in finding optimal solutions in such a nonlinear problem. [.4]{} ![\[fig:fixpol\_nonlin\_all\] Malkus state with $L_{max}=3, ~ N_{max}=3, ~ M_{max}=3$. ](r09.pdf "fig:"){width="100.00000%"} [0.4]{} ![\[fig:fixpol\_nonlin\_all\] Malkus state with $L_{max}=3, ~ N_{max}=3, ~ M_{max}=3$. ](phi_r09.pdf "fig:"){width="100.00000%"} [.4]{} ![\[fig:fixpol\_nonlin\_all\] Malkus state with $L_{max}=3, ~ N_{max}=3, ~ M_{max}=3$. ](r08.pdf "fig:"){width="100.00000%"} [0.4]{} ![\[fig:fixpol\_nonlin\_all\] Malkus state with $L_{max}=3, ~ N_{max}=3, ~ M_{max}=3$. ](phi_r08.pdf "fig:"){width="100.00000%"} For comparison we also compute the solution using the method described in \[sec:theo\], which owing to the specific choice of toroidal spherical harmonic modes results in a linear system. The qualitative similarity between these solutions is important in giving an insight into how important it is that we make the (necessary) choice for our higher resolution Earth-like solutions, of only including modes that result in a linear system. Quantitatively this holds too, with rms values of $B_\phi$ of 0.21 and 0.23 mT for the non-linear and linear solutions respectively. Hence we suggest that the estimates for the lower bound of Earth’s toroidal field strength we have calculated would not be significantly different were it possible to solve the full non-linear system. ![Linear solution for $B_\phi$ at $r=0.9 R$, using the method outlined in \[sec:theo\] and used for the Earth-like solutions](phi_r09_linear_2.pdf){width="40.00000%"} Enumeration of constraints {#sec:enum_con} ========================== In order to determine the number of Malkus constraints, we calculate the maximum possible exponent in dimension of length within the Malkus integral. Since each constraint equation arises from ensuring a coefficient of a different exponent vanishes, enumerating all possibilities gives the maximum number of constraints. There are three possible non-zero interactions whose sum comprise the Malkus integral, toroidal-toroidal, toroidal-poloidal and poloidal-poloidal as defined in \[eq:tortor\]. Since the poloidal field definition contains two curls whereas the toroidal field only one, then this extra derivative reduces the maximum exponent by one for interations involving a poloidal field as opposed to a toroidal one. This means that the maximal case is determined by the toroidal-toroidal interaction, $[{\bm{\mathcal{T}_1}},{\bm{\mathcal{T}}}_2]$. Since the Malkus integrand is identical to the Taylor integrand, we observe that the maximum radial exponent in the Malkus integrand $(({{{\boldsymbol \nabla}}\times}{\bm{\mathcal{T}_1}})\times {\bm{\mathcal{T}_2}})_\phi$ is $2L_{max}+4N_{max}-1$, as derived by [@livermore2008structure]. This is then reduced by two due to the property that the interaction of two toroidal harmonics that have identical spherical harmonic degrees and orders is zero [@livermore2008structure]. This requires that one of the two modes has an $L_{max}$ of at least one smaller than the other, hence resulting in a maximum possible degree in $r$ of $2L_{max}+4N_{max}-3$. Now under a transform in coordinate systems we note that $r^n$ in spherical coordinates can be expressed as $s^jz^k$ in cylindrical coordinates, where $n=j+k$. Since only even values of $j$ are present this results in $L_{max}+2N_{max}-2 = C_T$ non-trivial constraint equations in this dimension. There is no such restriction on $k$, which can take all values up to the maximum of $2L_{max}+4N_{max}-3 = 2C_T+1$. Each one of the constraints arises from a coefficient of a term with a different combination of exponents in $s$ and $z$, explicitly, these terms have the following form: $$\begin{aligned} &(A_{C_T,0}z^0 + A_{C_T,1}z) s^{2C_T} +(A_{C_T-1,0}z^0+A_{C_T-1,1}z+A_{C_T-1,2}z^2+A_{C_T-1,3}z^3)s^{2(C_T-1)}\nonumber \\&+(A_{C_T-2,0}z^0+\dots+A_{C_T-2,5}z^5)s^{2(C_T-2)} +\dots \nonumber\\& +(A_{1,0}+\dots+A_{1,2C_T-1}z^{2C_T-1})s^2+(A_{0,0}+\dots+A_{0,2C_T+1}z^{2C_T+1}).\end{aligned}$$ Hence from the summation of the total number of these terms for every combination of $j$ and $k$, with $j$ even, such that $j+k \leq 2C_T+1$ we have the following expression for the maximum number of Malkus constraints, $$C_M = 2 \sum_{n=0}^{C_T} (n+1) = (C_T+1)(C_T+2) = C_T^2 + 3C_T + 2.$$

--- abstract: 'Automatic fake news detection is a challenging problem in deception detection, and it has tremendous real-world political and social impacts. However, statistical approaches to combating fake news has been dramatically limited by the lack of labeled benchmark datasets. In this paper, we present <span style="font-variant:small-caps;">liar</span>: a new, publicly available dataset for fake news detection. We collected a decade-long, 12.8K manually labeled short statements in various contexts from <span style="font-variant:small-caps;">PolitiFact.com</span>, which provides detailed analysis report and links to source documents for each case. This dataset can be used for fact-checking research as well. Notably, this new dataset is an order of magnitude larger than previously largest public fake news datasets of similar type. Empirically, we investigate automatic fake news detection based on surface-level linguistic patterns. We have designed a novel, hybrid convolutional neural network to integrate meta-data with text. We show that this hybrid approach can improve a text-only deep learning model.' author: - | William Yang Wang\ Department of Computer Science\ University of California, Santa Barbara\ Santa Barbara, CA 93106 USA\ [william@cs.ucsb.edu]{} bibliography: - 'all.bib' title: | “Liar, Liar Pants on Fire”:\ A New Benchmark Dataset for Fake News Detection --- Introduction ============ In this past election cycle for the 45th President of the United States, the world has witnessed a growing epidemic of fake news. The plague of fake news not only poses serious threats to the integrity of journalism, but has also created turmoils in the political world. The worst real-world impact is that fake news seems to create real-life fears: last year, a man carried an AR-15 rifle and walked in a Washington DC Pizzeria, because he recently read online that “this pizzeria was harboring young children as sex slaves as part of a child-abuse ring led by Hillary Clinton”[^1]. The man was later arrested by police, and he was charged for firing an assault rifle in the restaurant [@nyt2016]. The broadly-related problem of deception detection [@Mihalcea:2009:LDE:1667583.1667679] is not new to the natural language processing community. A relatively early study by Ott et al.  focuses on detecting deceptive review opinions in sentiment analysis, using a crowdsourcing approach to create training data for the positive class, and then combine with truthful opinions from TripAdvisor. Recent studies have also proposed stylometric [@feng2012syntactic], semi-supervised learning [@haideceptive], and linguistic approaches [@perez2015experiments] to detect deceptive text on crowdsourced datasets. Even though crowdsourcing is an important approach to create labeled training data, there is a mismatch between training and testing. When testing on real-world review datasets, the results could be suboptimal since the positive training data was created in a completely different, simulated platform. The problem of fake news detection is more challenging than detecting deceptive reviews, since the political language on TV interviews, posts on Facebook and Twitters are mostly short statements. However, the lack of manually labeled fake news dataset is still a bottleneck for advancing computational-intensive, broad-coverage models in this direction. Vlachos and Riedel  are the first to release a public fake news detection and fact-checking dataset, but it only includes 221 statements, which does not permit machine learning based assessments. To address these issues, we introduce the <span style="font-variant:small-caps;">liar</span> dataset, which includes 12,836 short statements labeled for truthfulness, subject, context/venue, speaker, state, party, and prior history. With such volume and a time span of a decade, <span style="font-variant:small-caps;">liar</span> is an order of magnitude larger than the currently available resources [@vlachos2014fact; @ferreira2016emergent] of similiar type. Additionally, in contrast to crowdsourced datasets, the instances in <span style="font-variant:small-caps;">liar</span> are collected in a grounded, more natural context, such as political debate, TV ads, Facebook posts, tweets, interview, news release, etc. In each case, the labeler provides a lengthy analysis report to ground each judgment, and the links to all supporting documents are also provided. Empirically, we have evaluated several popular learning based methods on this dataset. The baselines include logistic regression, support vector machines, long short-term memory networks [@hochreiter1997long], and a convolutional neural network model [@kim:2014:EMNLP2014]. We further introduce a neural network architecture to integrate text and meta-data. Our experiment suggests that this approach improves the performance of a strong text-only convolutional neural networks baseline. <span style="font-variant:small-caps;">liar</span>: a New Benchmark Dataset =========================================================================== The major resources for deceptive detection of reviews are crowdsourced datasets [@ott2011finding; @perez2015experiments]. They are very useful datasets to study deception detection, but the positive training data are collected from a simulated environment. More importantly, these datasets are not suitable for fake statements detection, since the fake news on TVs and social media are much shorter than customer reviews. Vlachos and Riedel  are the first to construct fake news and fact-checking datasets. They obtained 221 statements from <span style="font-variant:small-caps;">Channel 4</span>[^2] and <span style="font-variant:small-caps;">PolitiFact.com</span>[^3], a Pulitzer Prize-winning website. In particular, PolitiFact covers a wide-range of political topics, and they provide detailed judgments with fine-grained labels. Recently, Ferreira and Vlachos  have released the Emergent dataset, which includes 300 labeled rumors from PolitiFact. However, with less than a thousand samples, it is impractical to use these datasets as benchmarks for developing and evaluating machine learning algorithms for fake news detection. Therefore, it is of crucial significance to introduce a larger dataset to facilitate the development of computational approaches to fake news detection and automatic fact-checking. Dataset Statistics -------------------------------- -------- Training set size 10,269 Validation set size 1,284 Testing set size 1,283 Avg. statement length (tokens) 17.9 Top-3 Speaker Affiliations Democrats 4,150 Republicans 5,687 None (e.g., FB posts) 2,185 : The <span style="font-variant:small-caps;">liar</span> dataset statistics.[]{data-label="tab:stats"} [We show some random snippets from our dataset in Figure \[fig:example\]. The <span style="font-variant:small-caps;">liar</span> dataset[^4] includes 12.8K human labeled short statements from  <span style="font-variant:small-caps;">PolitiFact.com</span>’s API[^5], and each statement is evaluated by a <span style="font-variant:small-caps;">PolitiFact.com</span> editor for its truthfulness. After initial analysis, we found duplicate labels, and merged the full-flop, half-flip, no-flip labels into false, half-true, true labels respectively. We consider six fine-grained labels for the truthfulness ratings: *pants-fire, false, barely-true, half-true, mostly-true, and true*. The distribution of labels in the <span style="font-variant:small-caps;">liar</span> dataset is relatively well-balanced: except for 1,050 pants-fire cases, the instances for all other labels range from 2,063 to 2,638. We randomly sampled 200 instances to examine the accompanied lengthy analysis reports and rulings. Not that fact-checking is not a classic labeling task in NLP. The verdict requires extensive training in journalism for finding relevant evidence. Therefore, for second-stage verifications, we went through a randomly sampled subset of the analysis reports to check if we agreed with the reporters’ analysis. The agreement rate measured by Cohen’s kappa was 0.82. We show the corpus statistics in Table \[tab:stats\]. The statement dates are primarily from 2007-2016.]{} The speakers in the <span style="font-variant:small-caps;">liar</span> dataset include a mix of democrats and republicans, as well as a significant amount of posts from online social media. We include a rich set of meta-data for each speaker—in addition to party affiliations, current job, home state, and credit history are also provided. In particular, the credit history includes the historical counts of inaccurate statements for each speaker. For example, Mitt Romney has a credit history vector $h=\{19,32,34,58,33\}$, which corresponds to his counts of “pants on fire”, “false”, “barely true”, “half true”, “mostly true” for historical statements. Since this vector also includes the count for the current statement, it is important to subtract the current label from the credit history when using this meta data vector in prediction experiments. These statements are sampled from various of contexts/venues, and the top categories include *news releases, TV/radio interviews, campaign speeches, TV ads, tweets, debates, Facebook posts, etc*. To ensure a broad coverage of the topics, there is also a diverse set of subjects discussed by the speakers. The top-10 most discussed subjects in the dataset are *economy, health-care, taxes, federal-budget, education, jobs, state-budget, candidates-biography, elections, and immigration*. ![The proposed hybrid Convolutional Neural Networks framework for integrating text and meta-data.[]{data-label="fig:cnn"}](framework.png){width="1\linewidth"} Automatic Fake News Detection ============================= One of the most obvious applications of our dataset is to facilitate the development of machine learning models for automatic fake news detection. In this task, we frame this as a 6-way multiclass text classification problem. And the research questions are: - Based on surface-level linguistic realizations only, how well can machine learning algorithms classify a short statement into a fine-grained category of fakeness? - Can we design a deep neural network architecture to integrate speaker related meta-data with text to enhance the performance of fake news detection? Since convolutional neural networks architectures (CNNs) [@collobert2011natural; @kim:2014:EMNLP2014; @zhang2015character] have obtained the state-of-the-art results on many text classification datasets, we build our neural networks model based on a recently proposed CNN model [@kim:2014:EMNLP2014]. Figure \[fig:cnn\] shows the overview of our hybrid convolutional neural network for integrating text and meta-data. We randomly initialize a matrix of embedding vectors to encode the metadata embeddings. We use a convolutional layer to capture the dependency between the meta-data vector(s). Then, a standard max-pooling operation is performed on the latent space, followed by a bi-directional LSTM layer. We then concatenate the max-pooled text representations with the meta-data representation from the bi-directional LSTM, and feed them to fully connected layer with a softmax activation function to generate the final prediction. <span style="font-variant:small-caps;">liar</span>: Benchmark Evaluation ======================================================================== In this section, we first describe the experimental setup, and the baselines. Then, we present the empirical results and compare various models. Experimental Settings --------------------- We used five baselines: a majority baseline, a regularized logistic regression classifier (LR), a support vector machine classifier (SVM) [@crammer2001algorithmic], a bi-directional long short-term memory networks model (Bi-LSTMs) [@hochreiter1997long; @graves2005framewise], and a convolutional neural network model (CNNs) [@kim:2014:EMNLP2014]. For LR and SVM, we used the <span style="font-variant:small-caps;">LibShortText</span> toolkit[^6], which was shown to provide very strong performances on short text classification problems [@Wang:2015:EMNLP]. For Bi-LSTMs and CNNs, we used TensorFlow for the implementation. We used pre-trained 300-dimensional word2vec embeddings from Google News [@mikolov2013efficient] to warm-start the text embeddings. We strictly tuned all the hyperparameters on the validation dataset. The best filter sizes for the CNN model was (2,3,4). In all cases, each size has 128 filters. The dropout keep probabilities was optimized to 0.8, while no $L_2$ penalty was imposed. The batch size for stochastic gradient descent optimization was set to 64, and the learning process involves 10 passes over the training data for text model. For the hybrid model, we use 3 and 8 as filter sizes, and the number of filters was set to 10. We considered 0.5 and 0.8 as dropout probabilities. The hybrid model requires 5 training epochs. We used grid search to tune the hyperparameters for LR and SVM models. We chose accuracy as the evaluation metric, since we found that the accuracy results from various models were equivalent to f-measures on this balanced dataset. Models Valid. Test ---------------------- ----------- ----------- Majority 0.204 0.208 SVMs 0.258 0.255 Logistic Regress0ion 0.257 0.247 Bi-LSTMs 0.223 0.233 CNNs 0.260 0.270 Hybrid CNNs Text + Subject 0.263 0.235 Text + Speaker **0.277** 0.248 Text + Job 0.270 0.258 Text + State 0.246 0.256 Text + Party 0.259 0.248 Text + Context 0.251 0.243 Text + History 0.246 0.241 Text + All 0.247 **0.274** : The evaluation results on the <span style="font-variant:small-caps;">liar</span> dataset. The top section: text-only models. The bottom: text + meta-data hybrid models.[]{data-label="tab:eval"} Results ------- We outline our empirical results in Table \[tab:eval\]. First, we compare various models using text features only. We see that the majority baseline on this dataset gives about 0.204 and 0.208 accuracy on the validation and test sets respectively. Standard text classifier such as SVMs and LR models obtained significant improvements. Due to overfitting, the Bi-LSTMs did not perform well. The CNNs outperformed all models, resulting in an accuracy of 0.270 on the heldout test set. We compare the predictions from the CNN model with SVMs via a two-tailed paired t-test, and CNN was significantly better ($p<.0001$). When considering all meta-data and text, the model achieved the best result on the test data. Conclusion ========== We introduced <span style="font-variant:small-caps;">liar</span>, a new dataset for automatic fake news detection. Compared to prior datasets, <span style="font-variant:small-caps;">liar</span> is an order of a magnitude larger, which enables the development of statistical and computational approaches to fake news detection. <span style="font-variant:small-caps;">liar</span>’s authentic, real-world short statements from various contexts with diverse speakers also make the research on developing broad-coverage fake news detector possible. We show that when combining meta-data with text, significant improvements can be achieved for fine-grained fake news detection. Given the detailed analysis report and links to source documents in this dataset, it is also possible to explore the task of automatic fact-checking over knowledge base in the future. Our corpus can also be used for stance classification, argument mining, topic modeling, rumor detection, and political NLP research. [^1]: http://www.nytimes.com/2016/12/05/business/media/comet-ping-pong-pizza-shooting-fake-news-consequences.html [^2]: http://blogs.channel4.com/factcheck/ [^3]: http://www.politifact.com/ [^4]: <https://www.cs.ucsb.edu/~william/data/liar_dataset.zip> [^5]: <http://static.politifact.com/api/v2apidoc.html> [^6]: https://www.csie.ntu.edu.tw/\~cjlin/libshorttext/

--- abstract: 'The Josephson ring modulator (JRM) is a device, based on Josephson tunnel junctions, capable of performing non-degenerate mixing in the microwave regime without losses. The generic scattering matrix of the device is calculated by solving coupled quantum Langevin equations. Its form shows that the device can achieve quantum-limited noise performance both as an amplifier and a mixer. Fundamental limitations on simultaneous optimization of performance metrics like gain, bandwidth and dynamic range (including the effect of pump depletion) are discussed. We also present three possible integrations of the JRM as the active medium in a different electromagnetic environment. The resulting circuits, named Josephson parametric converters (JPC), are discussed in detail, and experimental data on their dynamic range are found to be in good agreement with theoretical predictions. We also discuss future prospects and requisite optimization of JPC as a preamplifier for qubit readout applications.' author: - Baleegh Abdo - Archana Kamal - Michel Devoret title: 'Non-degenerate, three-wave mixing with the Josephson ring modulator' --- The photon energy of microwave radiation in the band from $4-8$ GHz ($\sim$ $8-4$ cm wavelength) is approximately $10^{5}$ smaller than that of the visible light. Yet, at a temperature $10^{4}$ smaller than room temperature, now routinely achievable with a dilution refrigerator, it is now possible to resolve the energy of single microwave photons [@DSnature]. There are three advantages of single photon microwave electronics when compared with quantum optics. First, signal shapes at carrier frequencies of a few GHz with a relative bandwidth of few percent can be controlled with much greater relative precision than their equivalent at a few hundreds of THz. This is partly due to the fact that microwave generators have more short term stability than lasers, but also because microwave components are mechanically very stable, particularly when cooled, compared with traditional optical components. Second, in single photon microwave electronics, the on-chip circuitry can be well in the lumped element regime, and spatial mode structure can be controlled more thoroughly and more reliably than in the optical domain. Finally, there exists a simple, robust non-dissipative component, the Josephson tunnel junction (JJ), whose non-linearity can be ultra-strong even at the single photon level [@MDphysik]. Many quantum signal processing functions have been realized using JJs, both digital and analog, and this short review will not attempt to describe all of them. We will focus on analog Josephson devices pumped with a microwave tone. They recently led to microwave amplifiers working at the single photon level [@CastellanosNat; @JPCnature]. These novel devices have taken the work pioneered by B. Yurke at Bell labs 25 years ago [@YurkePRL; @ParamYurkePRA; @MovshovichPRL] to the point where actual experiments can be performed using Josephson amplifiers as the first link in the chain of measurement [@TeufelNatTech; @QuantumJumps; @QubitJPC]. In this paper, we address one particular subclass of analog signal processing devices based on Josephson tunnel junction, namely those performing non-degenerate three-wave mixing. Examples are Josephson circuits based on the Josephson ring modulator [@JPCnaturePhys; @Jamp] which we will describe below. The Hamiltonian of such a device is of the form $$\begin{aligned} H_{0} & = & \frac{1}{2}\left( \frac{P_{X}^{2}}{\mathcal{M}_{X}} +\frac{P_{Y}^{2}}{\mathcal{M}_{Y}} +\frac{P_{Z}^{2}}{\mathcal{M}_{Z}}\right) \nonumber\\ & & + \frac{1}{2}\left( \mathcal{K}_{X}X^{2}+\mathcal{K}_{Y}Y^{2} +\mathcal{K}_{Z}Z^{2}\right) +KXYZ, \label{Hamiltonian1}$$ where ($X$, $Y$, $Z$) and ($P_{X}$, $P_{Y}$, $P_{Z}$) are the generalized position and momentum variables for the three independent oscillators, $\mathcal{M}_{X,Y,Z}$ and $\mathcal{K}_{X,Y,Z}$ represent the “mass" and “spring constant" of the relevant oscillator (see table I), and $K$ is the three-wave mixing constant which governs the non-linearity of the system. We will discuss later how such simple minimal non-linear term can arise. The classical equation of motions for the standing waves in such a device are symmetric and are given by: $$\begin{aligned} \overset{\cdot\cdot}{X}+\gamma_{a}\overset{\cdot}{X}+\omega_{a}^{2} X+K^{\prime}YZ & =x\left( t\right) \cos\omega_{a}t, \label{non-degenerate1}\\ \overset{\cdot\cdot}{Y}+\gamma_{b}\overset{\cdot}{Y}+\omega_{b}^{2} Y+K^{\prime}XZ & =y\left( t\right) \cos\omega_{b}t, \label{non-degenerate2}\\ \overset{\cdot\cdot}{Z}+\gamma_{c}\overset{\cdot}{Z}+\omega_{c}^{2}Z+K^{\prime}XY & =z\left( t\right) \cos\omega_{c}t, \label{non-degenerate3}$$ where $K^{\prime}=K/\mathcal{M}$ (we assume, for simplicity, equal masses $\mathcal{M}_{X,Y,Z}=\mathcal{M}$) and $\omega_{a,b,c}$ $=$ $\sqrt {\mathcal{K}_{X,Y,Z}/\mathcal{M}}$ are the angular resonant frequencies of the three coordinates satisfying $$\omega_{a}<\omega_{b}\,<\omega_{c}=\omega_{a}+\omega_{b}.$$ ------------------------------------------------- $X $ $P$ $\mathcal{M}$ $\mathcal{K}$ -------- -------- --------------- --------------- $\Phi$ $Q$ $C$ $L^{-1} $ $Q$ $\Phi$ $L$ $C^{-1} $ ------------------------------------------------- \[Table1\] We also suppose the oscillators are well in the underdamped regime $$\begin{aligned} \gamma_{a} & \ll\omega_{a},\\ \gamma_{b} & \ll\omega_{b},\\ \gamma_{c} & \ll\omega_{c},\end{aligned}$$ a sufficient but not strictly necessary hypothesis, which has the principal merit of keeping the problem analytically soluble under the conditions of interest. It is worth noting that the system is non-degenerate both spatially and temporally. On the other hand, it is important to suppose that the envelope functions $x(t)$, $y\left( t\right) $ and $z\left( t\right) $ of the drive signals are supposed to be slow compared to the respective drive frequencies $\omega_{b}-\omega_{a}\gg\gamma_{a}+\gamma_{b}$. The equations (\[non-degenerate1\]-\[non-degenerate3\]) must be contrasted with that of a degenerate three-wave mixing device for which two cases are possible. In the first case, where the $Y$ and $Z$ degrees of freedom have merged into a single oscillator, the Hamiltonian has a non-linear term of the form $KXZ^{2}$ and the equations read: $$\begin{aligned} \overset{\cdot\cdot}{X}+\gamma_{a}\overset{\cdot}{X}+\omega_{a}^{2}X+K^{\prime}Z^{2} & =x\left( t\right) \cos\omega_{a}t,\\ \overset{\cdot\cdot}{Z}+\gamma_{c}\overset{\cdot}{Z}+\omega_{c}^{2}Z+2K^{\prime}ZX & =z\left( t\right) \cos\omega_{c}t.\end{aligned}$$ This is the case of electromechanical resonators [@TeufelNature] in which one of the capacitance plates of a microwave oscillator ($Z$) is itself the mass of a mechanical resonator ($X$). There $\omega_{c}\gg\omega_{a}$, and pumping the microwave oscillator in the vicinity of $\omega_{c}-\omega_{a}$ leads to cooling of the mechanical oscillator provided $\gamma_{c}\gg \gamma_{a}$. In the second case, it is the $X$ and the $Y$ degrees of freedom that merge into a single oscillator, leading to a non-linear term in the Hamiltonian of the form $KX^{2}Z$. The equations then read $$\begin{aligned} \overset{\cdot\cdot}{X}+\gamma_{a}\overset{\cdot}{X}+\omega_{a}^{2}X+2K^{\prime}XZ & =x\left( t\right) \cos\omega_{a}t, \label{degenerate 2}\\ \overset{\cdot\cdot}{Z}+\gamma_{c}\overset{\cdot}{Z}+\omega_{c}^{2}Z+K^{\prime}X^{2} & =z\left( t\right) \cos\omega_{c}t \label{degenerate 3}$$ and we have now $$\omega_{c}=2\omega_{a}. \label{degenerate 1}$$ This case is implemented in Josephson circuits as a dcSQUID whose flux is driven by a microwave oscillating signal at twice the plasma frequency of the SQUID [@Yamamoto]. When $z\left( t\right) =z_{d}\gg K^{\prime}X^{2}$ (so-called stiff“ or non-depleted” pump condition), the system of equations (\[degenerate 2\],\[degenerate 3\]) reduces to the parametrically driven oscillator equation $$\overset{\cdot\cdot}{X}+\gamma_{a}\overset{\cdot}{X}+\omega_{a}^{2} \left[1+\frac{K^{\prime}z_{d}}{\gamma_{c}\omega_{c}}\sin\left( \omega_{c}t\right)\right] X=x\left( t\right) \cos\omega_{a}t. \label{parametric oscillator}$$ Note that there is, in addition to the parametric drive on the left hand side, a small perturbing drive signal $x\left( t\right) \cos\omega_{a}t$ on the right hand side. The theory of the degenerate parametric amplifier starts with this latter equation, the term $\frac{K^{\prime}z_{d}}{\gamma_{c}\omega_{c}}\sin\left( \omega_{c}t\right) $ corresponding to the pump and $x\left( t\right) \cos\omega_{a}t$ corresponding to the input signal. The output signal is obtained from a combination of the loss term $\gamma_{a}\dot{X}$ and the input signal. In the context of Josephson devices, another route to the effective parametric oscillator of equation (\[parametric oscillator\]) can be obtained by a driven, Duffing-type oscillator [@VijayJBAreview; @JBA]. This system (Josephson bifurcation amplifier) has only one spatial mode and quartic non-linearity, $$\overset{\cdot\cdot}{X}+\gamma_{a}\overset{\cdot}{X}+\omega_{a}^{2}X-\lambda X^{3}=\left[ z_{d}+x\left( t\right) \right] \cos\omega_{d}t. \label{parametric oscillator2}$$ Driven by a strong tone $z_{d}\cos\omega_{d}t$ in the vicinity of the bifurcation occurring at $$\begin{aligned} \omega_{d} & =\omega_{a}-\frac{\sqrt{3}}{2}\gamma_{a},\\ z_{d} & =\frac{128}{27}\sqrt{\frac{\gamma_{a}^{3}\omega_{a}}{3\lambda}},\end{aligned}$$ it will lead to an equation of the form (\[parametric oscillator\]) for small deviations around the steady-state solution. It will, therefore, amplify the small drive modulation signal $x\left( t\right) $ of equation (\[parametric oscillator2\]) \[\]. Similar amplifying effects can be found in pumped superconducting microwave resonators without Josephson junctions [@ImBaleegh; @TholenAPL; @Zmuidzinas]. In the following section, we will treat Eqs. (\[non-degenerate1\]-\[non-degenerate3\]) using input-output theory [@LinearScatteringNote] and obtain the quantum-mechanical scattering matrix of the signal and idler amplitudes in the stiff-pump approximation. This allows us to find the photon gain of the device in its photon amplifier mode as a function of the pump amplitude, and the corresponding reduction of bandwidth. We then discuss the implementation of the device using a ring of four Josephson junctions flux-biased at half-quantum in Sec. II. It is the non-dissipative analogue of the semiconductor diode ring modulator [@Pozar]. In Sec. III, we treat the finite amplitude of signals and establish useful relations between the dynamic range, gain and bandwidth. In Sec. IV we introduce the Josephson parametric converter (JPC) as an example of a non-degenerate, three-wave mixing device operating at the quantum limit. We present three different realizations schemes for the JPC and point out their practical advantages and limitations. In Sec. V we present experimental results for different JPC devices and compare the data with the maximum bounds predicted by theory. We follow this with a discussion, in Sec. VI, of general requirements for an amplifier to meet the needs of qubit readout and how the maximum input power of the device can be increased by two orders of magnitude beyond typical values achieved nowadays. We conclude with a brief summary of our results in Sec. VII. Input-output treatment of a generic non-degenerate, three-wave mixing device ============================================================================ The three oscillators of Eqs. (\[non-degenerate1\]-\[non-degenerate3\]) correspond to three quantum LC oscillators coupled by a non-linear, trilinear mutual inductance, whose mechanism we will discuss in the next section. They are fed by transmission lines which carry excitations both into and out of the oscillators, as shown on Fig. \[three\_osc\_fig\]. The Hamiltonian of the system is (leaving out the transmission lines for the moment), ![General non-degenerate three-wave mixing device consisting of three LC oscillators coupled by a non-linear medium, giving a trilinear term in the Hamiltonian of the form $K\Phi_{a}\Phi_{b}\Phi_{c}$ where the fluxes $\Phi_{a,b,c}$ are those of the inductors. Each oscillator is fed by a transmission line with characteristic impedance $R_{a,b,c}$.[]{data-label="three_osc_fig"}](three_oscillators.pdf "fig:"){width="\columnwidth"}\ $$\begin{aligned} \frac{H_{0}}{\hbar} & =\omega_{a}a^{\dag}a+\omega_{b}b^{\dag}b+\omega_{c}c^{\dag}c\nonumber\\ & +g_{3}\left( a+a^{\dag}\right) \left( b+b^{\dag}\right) \left(c+c^{\dag}\right) ,\end{aligned}$$ where $a$, $b$ and $c$ are the annihilation operators associated with each of the three degrees of freedom. Their associated angular frequencies are given in terms of the inductances and capacitances as $$\omega_{a,b,c}=\frac{1}{\sqrt{L_{a,b,c}C_{a,b,c}}}.$$ The bosonic operators of different modes (a, b, c) commute with each other and those associated with the same mode satisfy the usual commutation relations of the form $$\left[ a,a^{\dag}\right] =1.$$ The link between the mode amplitude such as $X$, which represents the flux through the inductance of the oscillator, and a quantum operator such as $a$ can be written as, $$X=X^{ZPF}\left( a+a^{\dag}\right),$$ where “ZPF" stands for “zero-point fluctuations" and $$\begin{aligned} X^{ZPF} & =\sqrt{\frac{\hbar Z_{a}}{2}},\\ Z_{a} & =\sqrt{\frac{L_{a}}{C_{a}}},\end{aligned}$$ the last equation defining the impedance of the oscillator, equal to the modulus of the impedance on resonance of either the inductance or the capacitance. The link between $K$ and $g_{3}$ is therefore $$\hbar g_{3}=KX^{ZPF}Y^{ZPF}Z^{ZPF}.$$ We now work in the framework of Rotating Wave Approximation (RWA), in which we only keep terms commuting with the total photon number $$\frac{H_{0}^{\mathrm{RWA}}}{\hbar} =\omega_{a}a^{\dag}a+\omega_{b}b^{\dag}b+\omega_{c}c^{\dag}c+g_{3}\left( a^{\dag}b^{\dag}c+abc^{\dag}\right).$$ Treating in RWA the coupling of each oscillator with a transmission line carrying waves in and out of the oscillator (see Appendix for complements of the next 6 equations), one arrives at three coupled quantum Langevin equations for $a\left(t\right)$, $b\left(t\right)$ and $c\left( t\right)$: $$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}a & = -i\omega_{a}a-ig_{3}b^{\dag}c-\frac{\gamma_{a}}{2}a+\sqrt{\gamma_{a}}\tilde{a}^{\mathrm{in}}\left(t\right), \nonumber\\ \frac{\mathrm{d}}{\mathrm{d}t}b & = -i\omega_{b}b-ig_{3}a^{\dag}c-\frac{\gamma_{b}}{2}b+\sqrt{\gamma_{b}}\tilde{b}^{\mathrm{in}}\left(t\right), \nonumber\\ \frac{\mathrm{d}}{\mathrm{d}t}c & =-i\omega_{c}c-ig_{3}ab-\frac{\gamma_{c}}{2}c+\sqrt{\gamma_{c}}\tilde{c}^{\mathrm{in}}\left(t\right), \label{threeAmpEqs}\end{aligned}$$ In these equations, the second term in the right hand side corresponds to the non-linear term producing photon conversion. The third term says that photons introduced in one resonator leave with a rate $$\gamma_{a,b,c}=\omega_{a,b,c}\frac{Z_{a,b,c}}{R_{a,b,c}},$$ with the resistances $R_{a,b,c}$ denoting the characteristic impedances of the transmission lines. Finally, in the fourth term of the Langevin equations, the input fields such as $\tilde{a}^{\mathrm{in}}\left(t\right)$ correspond to the negative frequency component of the drive terms in the classical equations. They obey the relation $$\tilde{a}^{\mathrm{in}}(t)=\frac{1}{\sqrt{2\pi}}\int_{0}^{+\infty}a^{\mathrm{in}}\left[ \omega\right] e^{-i\omega t}\mathrm{d}\omega,$$ where $a^{\mathrm{in}}[\omega]$ are the usual field operators obeying the commutation relations $$\left[a^{\mathrm{in}}\left[ \omega\right] ,a^{\mathrm{in}}\left[\omega^{\prime}\right] \right] =\mathrm{sgn}\left(\frac{\omega-\omega^{\prime}}{2}\right) \delta\left(\omega+\omega^{\prime}\right)$$ in which $\omega$ denotes a frequency that can be either positive or negative. The transmission lines thus both damp and drive the oscillators. The incoming field operator treats the drive signals and the Nyquist equilibrium noise of the reservoir on the same footing. Photon spectral densities $\mathcal{N}^{\mathrm{in}}[\omega]$ of the incoming fields, introduced by relations of the form $$\left\langle \left\{a^{\mathrm{in}}\left[\omega\right], a^{\mathrm{in}}\left[\omega^{\prime}\right]\right\}\right\rangle =2\mathcal{N}_{a}^{\mathrm{in}}\left[ \frac{\omega-\omega^{\prime}}{2}\right]\delta\left(\omega+\omega^{\prime}\right), \label{Na_in_first}$$ have the value $$\begin{aligned} \mathcal{N}_{a}^{\mathrm{in}}\left[ \omega\right] & = \frac{\mathrm{sgn}\left( \omega\right) }{2}\coth\left( \frac{\hbar\omega}{2k_{B}T}\right) \nonumber\\ & + 2\pi P_{a}^{\mathrm{in}}\left[ \delta\left( \omega-\omega_{1}\right)+\delta\left( \omega+\omega_{1}\right) \right], \label{Na_in_sec}$$ where $P_{a}^{\mathrm{in}}$ is the photon flux of the incoming drive signal at angular frequency $\omega_{1}$ (in units of photons per unit time) and $T$ is the temperature of the electromagnetic excitations of the line. Note that the dimensionless function $\mathcal{N}_{a}^{\mathrm{in}}\left[ \omega\right] $ is defined for both positive and negative frequencies. It is symmetric $\mathcal{N}_{a}^{\mathrm{in}}\left[ \omega\right] =\mathcal{N}_{a}^{\mathrm{in}}\left[ -\omega\right] $ and its value at frequency $\left\vert \omega\right\vert $ represents the average number of photons per unit time per unit bandwidth in the incoming signal, which in the high temperature limit is $k_{B}T/\left( \hbar\left\vert \omega\right\vert \right) $. It includes the $\frac{1}{2}$ contribution of zero-point quantum noise. It is worth insisting that we treat the non-linear coupling strength as a perturbation compared with the influence of the reservoirs, treated themselves as a perturbation compared with the Hamiltonian of the oscillators: $$g_{3}\ll\gamma_{a},\gamma_{b}<\gamma_{c}\ll\omega_{a},\omega_{b}<\omega_{c}=\omega_{a}+\omega_{b}.$$ In general, only one strong drive tone is applied to one of the resonators and is called the “pump". Two cases must then be distinguished at this stage, as shown in Fig. \[frequencies\]: Case 1 (amplification and frequency conversion with photon gain): the pump tone is applied to the $c$ resonator. The device is usually used as an amplifier [@JPCnature; @Jamp]. It can also be used as a two-mode squeezer [@JPCSqueezer]. Case 2 (noiseless frequency conversion without photon gain): the pump tone is applied to either the $a$ or $b$ resonator [@BSconv]. The device is useful as a noiseless up- and down-converter and can perform dynamical cooling of the lowest energy oscillator, transferring its spurious excitations to the highest frequency one, which is more easily void of any excitations and plays the role of a cold source. \[h\] ![Characteristic frequency landscape of non-degenerate three-wave mixing devices. Three separate oscillators have resonant frequencies $\omega_{a}<\omega_{b}<\omega_{c}=\omega_{a}+\omega_{b}$. They are fed by transmission lines, giving them a full linewidth at half-maximum $\gamma_{a}$, $\gamma_{b}$ and $\gamma_{c}$ respectively. The non-linear coupling strength, expressed in photon amplitude language, is much smaller than these linewidths. The device can be pumped at $\omega_{c}$ and operates then as a phase-preserving amplifier with photon gain for frequencies $\omega_{a}$ and $\omega_{b}$ (top), or it can be pumped at one of the two lower frequencies $\omega_{a}$ or $\omega_{b}$ and operates then as a noiseless frequency converter or dynamical cooler, upconverting signals into oscillator at $\omega_{c}$ (bottom). In this figure, the spectral density of weak signal corresponds to thin arrows whereas the spectral densities of pump signals corresponds to thick arrows.[]{data-label="frequencies"}](converter-frequencies-positive.pdf "fig:"){width="\columnwidth"}\ Photon gain (case 1) -------------------- We will first suppose that the pump is “stiff", namely $$\begin{aligned} \left\vert \left\langle \tilde{c}^{\mathrm{in}}\right\rangle \right\vert ^{2} & \gg1\\ \gamma_{c} & \gg\gamma_{a},\gamma_{b}$$ This means that the pump tone will not be easily depleted despite the fact that its photons are converted into the signal and idler photons at $\omega_{a}$ and $\omega_{b}$. For solving the quantum Langevin equations, we replace the pumped oscillator annihilation operator $c$ by its average value in the coherent state produced by the pump as $$c\left( t\right) \rightarrow\left\langle c\left( t\right) \right\rangle =\sqrt{\bar{n}_{c}}e^{-i\left( \omega_{c}t+\phi\right) }.$$ The Langevin equations can then be transformed into the linear equations (see equation (\[IOT\]) of Appendix) $$\begin{gathered} \left[ \begin{array} [c]{cc}O_{a}^{+} & ig_{b}^{a}e^{-i\omega_{c}t}\\ -ig_{a}^{b\ast}e^{+i\omega_{c}t} & O_{b}^{+\ast}\end{array} \right] \left[ \begin{array} [c]{c}\tilde{a}^{\mathrm{out}}\\ \tilde{b}^{\mathrm{out}\dagger}\end{array} \right] =\\ -\left[ \begin{array} [c]{cc}O_{a}^{-} & ig_{b}^{a}e^{-i\omega_{c}t}\\ -ig_{a}^{b\ast}e^{+i\omega_{c}t} & O_{b}^{-\ast}\end{array} \right] \left[ \begin{array} [c]{c}\tilde{a}^{\mathrm{in}}\\ \tilde{b}^{\mathrm{in}\dagger}\end{array} \right] ,\end{gathered}$$ where $$\begin{aligned} O_{a,b}^{\pm} & =\frac{\mathrm{d}}{\mathrm{d}t}+i(\omega_{a,b}\mp i\Gamma_{a,b}),\\ \Gamma_{a,b} & =\frac{\gamma_{a,b}}{2},\\ g_{b,a}^{a,b} & =g_{3}\sqrt{\bar{n}_{c}}e^{-i\phi}\sqrt{\frac{\Gamma_{a,b}}{\Gamma_{b,a}}}.\end{aligned}$$ After a Fourier transform, we obtain in the frequency domain, a simpler relation $$\begin{gathered} \left[ \begin{array} [c]{cc}h_{a}\left[ \omega_{1}\right] & +ig_{b}^{a}\\ -ig_{a}^{b\ast} & h_{b}^{\ast}\left[ \omega_{2}\right] \end{array} \right] \left[ \begin{array} [c]{c}a^{\mathrm{out}}\left[ +\omega_{1}\right] \\ b^{\mathrm{out}}\left[ -\omega_{2}\right] \end{array} \right] =\label{in-out-amp5}\\ \left[ \begin{array} [c]{cc}h_{a}^{\ast}\left[ \omega_{1}\right] & -ig_{b}^{a}\\ +ig_{a}^{b\ast} & h_{b}\left[ \omega_{2}\right] \end{array} \right] \left[ \begin{array} [c]{c}a^{\mathrm{in}}\left[ +\omega_{1}\right] \\ b^{\mathrm{in}}\left[ -\omega_{2}\right] \end{array} \right] ,\end{gathered}$$ where $$h_{a,b}\left[ \omega\right] =-i\omega+i(\omega_{a,b}-i\Gamma_{a,b})$$ and the signal and idler angular frequencies $\omega_{1}$ and $\omega_{2}$ are both positive, satisfying the relationship $$\omega_{1}+\omega_{2}=\omega_{c}.$$ The scattering matrix of the device for small signals is defined by $$\left[ \begin{array} [c]{c}a^{\mathrm{out}}\left[ +\omega_{1}\right] \\ b^{\mathrm{out}}\left[ -\omega_{2}\right] \end{array} \right] =\left[ \begin{array} [c]{cc}r_{aa} & s_{ab}\\ s_{ba} & r_{bb}\end{array} \right] \left[ \begin{array} [c]{c}a^{\mathrm{in}}\left[ +\omega_{1}\right] \\ b^{\mathrm{in}}\left[ -\omega_{2}\right] \end{array} \right] . \label{RedS}$$ It can be computed from Eq. (\[in-out-amp5\]) and one finds $$\begin{aligned} r_{aa} & =\frac{\chi_{a}^{-1\ast}\chi_{b}^{-1\ast}+\left\vert \rho \right\vert ^{2}}{\chi_{a}^{-1}\chi_{b}^{-1\ast}-\left\vert \rho\right\vert ^{2}},\\ r_{bb} & =\frac{\chi_{a}^{-1}\chi_{b}^{-1}+\left\vert \rho\right\vert ^{2}}{\chi_{a}^{-1}\chi_{b}^{-1\ast}-\left\vert \rho\right\vert ^{2}},\\ s_{ab} & =\frac{-2i\rho}{\chi_{a}^{-1}\chi_{b}^{-1\ast}-\left\vert \rho\right\vert ^{2}},\\ s_{ba} & =\frac{2i\rho^{\ast}}{\chi_{a}^{-1}\chi_{b}^{-1\ast}-\left\vert \rho\right\vert ^{2}},\end{aligned}$$ where the $\chi$’s are the bare response functions of modes *a* and *b* (whose inverses depend linearly on the signal frequency) $$\begin{aligned} \chi_{a}^{-1} & =1-i\frac{\omega_{1}-\omega_{a}}{\Gamma_{a}},\\ \chi_{b}^{-1} & =1-i\frac{\omega_{2}-\omega_{b}}{\Gamma_{b}},\end{aligned}$$ and $\rho$ is the dimensionless pump amplitude $$\rho=\frac{g_{3}\sqrt{\bar{n}_{c}}e^{-i\phi}}{\sqrt{\Gamma_{a}\Gamma_{b}}}. \label{rho}$$ Note that the matrix in Eq. (\[RedS\]) has unity determinant and the property $$\begin{aligned} \left\vert r_{aa}\right\vert ^{2}-\left\vert s_{ab}\right\vert ^{2} & =1,\\ \left\vert r_{bb}\right\vert ^{2}-\left\vert s_{ba}\right\vert ^{2} & =1.\end{aligned}$$ For zero frequency detuning, i.e. $\chi_{a}^{-1}=\chi_{b}^{-1}=1$, the scattering matrix displays a very simple form $$\left[ \begin{array} [c]{cc}\cosh\tau_{0} & -ie^{-i\phi}\sinh\tau_{0}\\ +ie^{+i\phi}\sinh\tau_{0} & \cosh\tau_{0}\end{array} \right] , \label{hyperbolic}$$ where $\tanh(\tau_{0}/2)=|\rho|$. The zero frequency detuning power gain $G_{0}$ is given by $$G_{0}=\left( \cosh\tau_{0}\right) ^{2}=\left( \frac{1+\left\vert \rho\right\vert ^{2}}{1-\left\vert \rho\right\vert ^{2}}\right) ^{2}. \label{gain-formula}$$ For non-zero detuning, the scattering matrix acquires extra phase factors but the minimal scattering matrix for a quantum-limited phase-preserving amplifier represented in Fig. \[minimal-amp\] still describes the device. \[h\] ![An amplifier reaching the quantum limit must have a minimal scattering matrix, with the signal in port $a$ being reflected with amplitude gain $G^{1/2}$ while the signal in port $b$ is phase-conjugated and transmitted to port $a$ with amplitude gain $(G-1)^{1/2}$. This can be realized in case 1 of Fig. \[frequencies\].[]{data-label="minimal-amp"}](phase_preserving_amplifier.pdf "fig:"){width="\columnwidth"}\ The gain $G_{0}$ diverges as $\left\vert \rho\right\vert \rightarrow1^{-}$, i.e. when the photon number $\bar{n}_{c}$ in the pump resonator reaches the critical number given by $$\bar{n}_{c}^{po}=\frac{\Gamma_{a}\Gamma_{b}}{\left\vert g_{3}\right\vert ^{2}}, \label{onset-param-osc}$$ a result that is common to all forms of parametric amplification. Increasing the pump power beyond the critical power yielding $\bar{n}_{c}^{po}$ leads to the parametric oscillation regime. This phenomenon is beyond the scope of our simple analysis and cannot be described by our starting equations, since higher order non-linearities of the system need to be precisely modelled if the saturation of the oscillation is to be accounted for. Introducing the detuning $$\Delta\omega=\omega_{1}-\omega_{a}=\omega_{b}-\omega_{2},$$ we can give a useful expression for the gain as a function of frequency as $$G\left( \Delta\omega\right) \underset{\left\vert \rho\right\vert \rightarrow1^{-}}{=}\frac{G_{0}}{1+\left( \frac{\Delta\omega}{\gamma G_{0}^{-1/2}}\right) ^{2}},$$ which shows that in the limit of large gain, the response of the amplifier for both the signal and idler port is Lorentzian with a bandwidth given by $$B=2\gamma G_{0}^{-1/2}=\frac{2\gamma_{a}\gamma_{b}G_{0}^{-1/2}}{\gamma _{a}+\gamma_{b}}. \label{B}$$ The product of the maximal amplitude gain times the bandwidth is thus constant and is given by the harmonic average of the oscillator bandwidths. Another interesting prediction of the scattering matrix is the two-mode squeezing function of the device demonstrated in Ref. . Conversion without photon gain (case 2) --------------------------------------- The case of conversion without photon gain can be treated along the same line as in the previous subsection, where scattering takes place between *c* and *a* or *c* and *b* modes. Without loss of generality we assume that the pump is applied to the intermediate frequency resonance. In this case the scattering matrix reads $$\left[ \begin{array} [c]{c}a^{\mathrm{out}}\left[ +\omega_{1}\right] \\ c^{\mathrm{out}}\left[ +\omega_{3}\right] \end{array} \right] =\left[ \begin{array} [c]{cc}r_{aa} & t_{ac}\\ t_{ca} & r_{cc}\end{array} \right] \left[ \begin{array} [c]{c}a^{\mathrm{in}}\left[ +\omega_{1}\right] \\ c^{\mathrm{in}}\left[ +\omega_{3}\right] \end{array} \right] ,$$ where $$\begin{aligned} r_{aa} & =\frac{\chi_{a}^{-1\ast}\chi_{c}^{-1}-\left\vert \rho^{\prime }\right\vert ^{2}}{\chi_{a}^{-1}\chi_{c}^{-1}+\left\vert \rho^{\prime }\right\vert ^{2}},\nonumber\\ r_{cc} & =\frac{\chi_{a}^{-1}\chi_{c}^{-1\ast}-\left\vert \rho^{\prime }\right\vert ^{2}}{\chi_{a}^{-1}\chi_{c}^{-1}+\left\vert \rho^{\prime }\right\vert ^{2}},\nonumber\\ t_{ac} & =\frac{2i\rho^{\prime}}{\chi_{a}^{-1}\chi_{c}^{-1}+\left\vert \rho^{\prime}\right\vert ^{2}},\nonumber\\ t_{ca} & =\frac{2i\rho^{\prime\ast}}{\chi_{a}^{-1}\chi_{c}^{-1}+\left\vert \rho^{\prime}\right\vert ^{2}},\nonumber\\ & \label{SparamsConvYSeries}$$ and $$\begin{aligned} \chi_{c}^{-1} & =1-i\frac{\omega_{3}-\omega_{c}}{\Gamma_{c}},\\ \rho^{\prime} & =\frac{g_{3}\sqrt{\bar{n}_{b}}e^{-i\phi}}{\sqrt{\Gamma _{a}\Gamma_{c}}}.\end{aligned}$$ The reduced pump strength $\rho^{\prime}$ plays the same role here as $\rho$ in the photon amplification case. Note that the scattering matrix is now unitary (conservation of total number of photons) and satisfies the following relations: $$\begin{aligned} \left\vert r_{aa}\right\vert ^{2}+\left\vert t_{ac}\right\vert ^{2} & =1,\\ \left\vert r_{cc}\right\vert ^{2}+\left\vert t_{ca}\right\vert ^{2} & =1.\end{aligned}$$ For zero frequency detuning, i.e. $\chi_{a}^{-1}=\chi_{c}^{-1}=1$, the scattering matrix can be written as $$\left[ \begin{array} [c]{cc}\cos\tau_{0} & e^{-i\phi}\sin\tau_{0}\\ e^{i\phi}\sin\tau_{0} & \cos\tau_{0}\end{array} \right] ,$$ which corresponds to replacing the parameter $\tau_{0}$ by $i\tau_{0}$ or $\left\vert \rho\right\vert $ by $i\left\vert \rho\right\vert $ in the scattering matrix (\[hyperbolic\]). A scattering representation of the two-port device in conversion mode is shown in Fig. \[convflow\]. In this mode the device operates as a beam splitter, the only difference being that the photons in different arms have different frequencies [@BSconv]. Full conversion $\left( \sin\tau_{0}=1\right) $ is obtained on resonance when the pump power reaches the critical value. However, here, the critical value can be traversed without violating the validity of the equations. Full photon conversion is desirable in dynamical cooling: in that case, the higher frequency resonator will be emptied of photons, and the lower frequency resonator can be cooled to its ground state by pumping the intermediate frequency resonator (see lower panel of Fig. \[frequencies\]). \[h\] ![Signal flow graph for a three-wave mixing device operating in conversion without photon gain, realized in case 2 of Fig. \[frequencies\]. The incoming signal in port $a$ ($b$) is reflected with amplitude $r$ and transmitted with up-conversion (down-conversion) to port $b$ (a) with amplitude $(1-r^{2})^{1/2}$. []{data-label="convflow"}](ConvFlow.pdf "fig:"){width="\columnwidth"}\ Added Noise ----------- The number of output photons generated per mode in the amplification (case 1) is given by $$\mathcal{N}_{a,b}^{\mathrm{out}}=\left\vert r\right\vert ^{2}\mathcal{N}_{a,b}^{\mathrm{in}}+\left\vert s\right\vert ^{2}\mathcal{N}_{b,a}^{\mathrm{in}}, \label{Nab_out_ampl}$$ where $\mathcal{N}^{\mathrm{in}}$ is the input photon spectral density given by Eq. (\[Na\_in\_sec\]) and we assume that there is no cross-correlations between the input fields $a^{\mathrm{in}}$ and $b^{\mathrm{in}}$. Assuming that the three-wave mixing device is in thermal equilibrium at temperature $T\ll\hbar\omega_{1,2}/k_{B}$ and that the dominant noise entering the system at each port is zero-point fluctuations $\hbar\omega_{1,2}/2$ ($\mathcal{N}^{\mathrm{in}}=1/2$), then in the limit of high gain $\left\vert r\right\vert \gg1$, the number of noise equivalent photons effectively feeding the system is $$\mathcal{N}_{eq}^{\mathrm{in}}=\mathcal{N}^{\mathrm{out}}/\left\vert r\right\vert ^{2}\simeq1. \label{N_in_eff_refl}$$ This means that the number of noise equivalent photons added by the device to the input is given by $\mathcal{N}^{\mathrm{add}}=\mathcal{N}_{eq}^{\mathrm{in}}-\mathcal{N}^{\mathrm{in}}=1/2$. Hence, when operated as a non-degenerate amplifier with $G_{0}\gg1$, the device adds noise which is equivalent to at least half a photon at the signal frequency to the input, in agreement with Caves theorem [@Caves]. In contrast, in the conversion mode of operation, assuming that there is no correlation between the input fields, the number of generated output photons per mode reads $$\mathcal{N}_{a,b}^{\mathrm{out}}=\left\vert r\right\vert ^{2}\mathcal{N}_{a,b}^{\mathrm{in}}+\left\vert t\right\vert ^{2}\mathcal{N}_{b,a}^{\mathrm{in}}.$$ Therefore, in pure conversion where $\left\vert r\right\vert =0$ and $\left\vert t\right\vert =1$, when referring the noise back to the input, one gets noise equivalent photons $$\mathcal{N}_{eq}^{\mathrm{in}}=\mathcal{N}^{\mathrm{out}}/\left\vert t\right\vert ^{2}=1/2. \label{N_in_eff_pure}$$ This means that, as a converter, the device is not required to add noise to the input since $\mathcal{N}_{eq}^{\mathrm{in}}=\mathcal{N}^{\mathrm{in}}$. Three-wave mixing using JRM =========================== The Josephson ring modulator is a device consisting of four Josephson junctions, each with critical current $I_{0}=\frac{\hbar}{2eL_{J}}$ forming a ring threaded by a flux $\Phi=\Phi_{0}/2$ where $\Phi_{0}$ is the flux quantum (see Fig. \[Josephson\_ring\_modulator\]). The device has the symmetry of a Wheatstone bridge. \[h\] ![Three-wave mixing element (see ellipse marked $K$ in Fig. \[three\_osc\_fig\]) consisting of a loop of four nominally identical Josephson junctions threaded by a flux in the vicinity of half a flux quantum. Mutual inductances, not shown here, couple this circuit to inductances $L_{a}$, $L_{b}$ and $L_{c}$ of Fig. \[three\_osc\_fig\] via the inductances $L_{X}$, $L_{Y}$ and $L_{Z}$ respectively, which are much larger than the junction inductance $L_{J}$. The three currents $I_{X}$, $I_{Y}$ and $I_{Z}$ correspond to the three orthogonal modes of the structure.[]{data-label="Josephson_ring_modulator"}](ring-modulator-currents.pdf "fig:"){width="\columnwidth"}\ There are thus three orthogonal electrical modes coupled to the junctions, corresponding to the currents $I_{X}$, $I_{Y}$ and $I_{Z}$ flowing in three external inductances $L_{X}$, $L_{Y}$ and $L_{Z}$ that are much larger than the junction inductance $L_{J}=\varphi_{0}^{2}E_{J}^{-1}$, where $\varphi _{0}=\hbar/2e$ is the reduced flux quantum. Each junction $j\in\left\{ \alpha,\beta,\gamma,\delta\right\} $ is traversed by a current $I_{j}$ and at the working point (i.e. $\Phi=\Phi_{0}/2$) its energy is, keeping terms up to order four in $I_{j}$, given by $$E_{j}=\frac{1}{2}L_{J}^{eff}I_{j}^{2}-\frac{1}{24}\frac{L_{J}^{eff}}{I_{0}^{\prime2}}I_{j}^{4},$$ where $L_{J}^{eff}=\sqrt{2}L_{J}$ and $I_{0}^{\prime}=I_{0}/\sqrt{2}$. The currents in the junctions are expressed by $$\begin{aligned} I_{\alpha} & =\frac{-I_{X}-I_{Y}}{2}+\frac{I_{Z}}{4}+I_{\Phi},\\ I_{\beta} & =\frac{+I_{X}-I_{Y}}{2}-\frac{I_{Z}}{4}+I_{\Phi},\\ I_{\gamma} & =\frac{+I_{X}+I_{Y}}{2}+\frac{I_{Z}}{4}+I_{\Phi},\\ I_{\delta} & =\frac{-I_{X}+I_{Y}}{2}-\frac{I_{Z}}{4}+I_{\Phi},\end{aligned}$$ where $I_{\Phi}$ is the supercurrent induced in the ring by the externally applied flux $\Phi$. The total energy of the ring is, keeping terms up to third order in the currents [@HuardProc], $$E_{ring}=\frac{1}{2}L_{J}^{eff}\left( I_{X}^{2}+I_{Y}^{2}+\frac{1}{4}I_{Z}^{2}\right) -\frac{1}{4}\frac{L_{J}^{eff}I_{\Phi}}{I_{0}^{\prime2}}I_{X}I_{Y}I_{Z}.$$ We can express the currents as $$I_{X,Y,Z}=\frac{\Phi_{a,b,c}}{L_{a,b,c}}\frac{M_{a,b,c}}{L_{X,Y,Z}}=\frac {\Phi_{a,b,c}}{L_{a,b,c}^{eff}},$$ where $M_{a,b,c}$ are the mutual inductances between $L_{X,Y,Z}$ and the oscillator inductances $L_{a,b,c}$. The non-linear coefficient in the energy is, therefore, $$K=\frac{\left( L_{J}^{eff}\right) ^{2}}{4\varphi_{0}}\frac{1}{L_{a}^{eff}L_{b}^{eff}L_{c}^{eff}},$$ and we finally arrive at the result $$g_{3}^{2}=\frac{p_{a}p_{b}p_{c}\omega_{a}\omega_{b}\omega_{c}}{\omega _{J}^{eff}}. \label{eqng3}$$ Here the participation ratios are defined as $$p_{a,b,c}=\frac{L_{J}^{eff}}{L_{a,b,c}^{eff}},$$ and, at $\Phi=\Phi_{0}/2$, $$\omega_{J}^{eff}=\frac{128}{\sqrt{2}}\frac{E_{J}}{\hbar}.$$ The participation ratios are linked to the maximal number of photons in each resonator, defined as those corresponding to an oscillation amplitude reaching a current of $I_{0}$ in each junction of the ring modulator, $$p_{a,b,c}\bar{n}_{a,b,c}^{\max}=\frac{E_{J}^{a,b,c}}{\hbar\omega_{a,b,c}}, \label{p-min}$$ where the $E_{J}^{a,b,c}$ are of order $E_{J}$ with factors accounting for the different participation of modes $X,$ $Y$ and $Z$ in the current of each junction. Equations (\[eqng3\]) and (\[p-min\]) are valid for all types of coupling between the Josephson ring modulator and signal/pump oscillators, which can be realized in practice by inductance sharing rather than by the mutual inductances discussed here. Equation (\[p-min\]) can also be rewritten in terms of the maximum circulating power in cavities *a* and *b* as $$P_{\mathrm{cav}}^{\mathrm{\max}}=\frac{\gamma_{a,b}}{p_{a,b}}\frac{E_{J}}{\sqrt{2}} \label{Pmaxcav}$$ where we substituted $E_{J}/\sqrt{2}$ as an upper bound for $E_{J}^{a,b}$. The maximum number of photons in equation (\[p-min\]) determine the maximum signal input power handled by the device $$P_{a,b}^{\max}=\frac{1}{G}\gamma_{a,b}\hbar\omega_{a,b}\bar{n}_{a,b}^{\max}. \label{P-Max}$$ We can now combine the notion of maximum power in resonator $c$ compatible with weak non-linearity with that of a critical power for the onset of parametric oscillation given by Eq. (\[onset-param-osc\]): $$\bar{n}_{c}^{\max}=\frac{E_{J}^{c}}{p_{c}\hbar\omega_{c}}>\bar{n}_{c}^{\mathrm{po}}=\frac{\Gamma_{1}\Gamma_{2}}{g_{3}^{2}},$$ arriving at the important relation $$p_{a}p_{b}Q_{a}Q_{b}>\Xi, \label{PQ-product}$$ where $\Xi$ is a number of order unity depending on the exact implementation of the coupling between the ring modulator and the oscillators. The quality factors of the resonators obey the well-known relation $$Q_{a,b}=\frac{\omega_{a,b}}{\gamma_{a,b}}.$$ Another maximum limit on the gain of the amplifier is set by the saturation of the device due to amplified zero-point fluctuations present at the input given by $$G_{\mathrm{ZPF}}^{\max}=\frac{E_{J}}{\sqrt{2}p_{a,b}}\frac{2}{\hbar \omega_{a,b}}. \label{Gmaxzpf}$$ Eqs. (\[p-min\]), (\[P-Max\]) and (\[PQ-product\]) show that it is not possible to maximize simultaneously gain, bandwidth and dynamic range. \[c\][|c|c|]{} &\ $\omega_{a,b}/2\pi$ & 1 - 16\ $Q_{a,b}$ & 50 - 500\ $Z_{a,b} $ & 10 - 150 $\Omega$\ $\gamma_{c} $ & 0.5 - 10\ $I_{0} $ & 0.5 - 10 $\mu$\ $E_{J} $ & 10 - 230\ $p_{a,b,c} $ & 0.01 - 0.5\ $g_{3}/2\pi$ & 0.1 - 15\ $\bar{n}^{\;\mathrm{max}}_{a,b,c} $ & $20 - 10^{4}$\ \[Table2\] In table II we enlist general bounds on the characteristic parameters of the three-wave mixing device, which are feasible with superconducting microwave circuits and standard Al-AlOx-Al junction fabrication technology. A few comments regarding the values listed in the table are in order. The frequency ranges of resonators *a* and *b* is mainly set by the center frequency of the system whose signal one needs to amplify or process. It is also important that these frequencies are very small compared to the plasma frequency of the Josephson junction. The total quality factor range listed in the table $\left(50-500\right)$ is suitable for practical devices. Quality factors in excess of $500$ can be easily achieved with superconducting resonators but, as seen from Eq. (\[B\]), higher the quality factor, smaller the dynamical bandwidth of the device. Quality factors lower than $50$ on the other hand are not recommended either for a variety of reasons. For example, in the limit of very low $Q$ the pump softens (becomes less stiff), and the dynamic range decreases as more quantum noise will be admitted by the device bandwidth and amplified “unintentionally" by the junctions. The characteristic impedance of the resonators $Z_{a,b}$ is set by microwave engineering considerations as discussed in Sec. IV but, in general, this value varies around 50 $\Omega$. The rate $\gamma_{c}$ at which pump photons leave the circuit varies from one circuit design to the other as discussed in Sec. IV and is limited by $\omega_{c}$. This parameter also affects the maximum input power performance of the device as explained in Sec. III. As to the values of $I_{0}$, on the one hand it is beneficial to work with large Josephson junctions in order to increase the processing capability of the device; on the other hand a critical current larger than 10 $\mu$A adds complexity to the microwave design of the resonators and makes the fabrication process of the Josephson junction more involved. This might even require switching to a different fabrication process such as Nb-AlOx-Nb trilayer junctions [@Trilayer] or nanobridges [@Nanobridges]. The other parameters listed in the table, namely $p_{a,b,c}$, $E_{J}$, $g_{3}$, $\bar{n}_{a,b,c}^{\max}$, their values depend, to a large extent, on the device parameters already discussed. Limitation of dynamic range due to pump depletion ================================================= In the last two sections, we were using results obtained by solving only the first two of the equations of motion Eqs. (\[threeAmpEqs\]) under the restriction of the stiff pump approximation. In this section, we extend our analysis and include the third equation describing the dynamics of the pump to calculate the pump depletion and its effect on the dynamic range of the device. For this purpose, we consider the average value of the third equation of motion for field $c$ $$\frac{\mathrm{d}}{\mathrm{d}t}\left\langle c\right\rangle =-i\omega _{c}\left\langle c\right\rangle -ig_{3}\left\langle ab\right\rangle -\frac{\gamma_{c}}{2}\left\langle c\right\rangle +\sqrt{\gamma_{c}}\left\langle \tilde{c}^{\mathrm{in}}\left( t\right) \right\rangle .$$ In steady state and using RWA we obtain $$ig_{3}\left\langle ab\right\rangle +\frac{\gamma_{c}}{2}\left\langle c\left( t\right) \right\rangle =\sqrt{\gamma_{c}}\left\langle \tilde{c}^{\mathrm{in}}\left( t\right) \right\rangle . \label{cin_c_relation}$$ In the limit of vanishing input, the cross-correlation term $\left\langle ab\right\rangle $ is negligible and, therefore, $$\left\langle c\left( t\right) \right\rangle =\frac{2}{\sqrt{\gamma_{c}}}\left\langle \tilde{c}^{\mathrm{in}}\left( t\right) \right\rangle .$$ The average number of photons in the $c$ resonator in this case is, thus, $$\underset{\left\langle ab\right\rangle \rightarrow0}{\lim}\bar{n}_{c}=\frac {4}{\gamma_{c}}\left\vert \left\langle \tilde{c}^{\mathrm{in}}\left( t\right) \right\rangle \right\vert ^{2}. \label{c_sq_c_in_sq}$$ We now establish a self-consistent equation for $\bar{n}_{c}$, taking into account input signals of finite amplitude. We first evaluate the value of $\left\langle a\left( t\right) b\left( t\right) \right\rangle $ in the frame rotating with the pump phase, $$\begin{aligned} & \left\langle a\left( t\right) b\left( t\right) \right\rangle \nonumber\\ & =\frac{1}{2\pi}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\left\langle a\left[ \omega\right] b\left[ \omega^{\prime}\right] \right\rangle e^{-i\left( \omega+\omega^{\prime}\right) t}\mathrm{d}\omega\mathrm{d}\omega^{\prime}.\end{aligned}$$ Using the field relations (see Appendix) $$\begin{aligned} \sqrt{\gamma_{a}}a\left[ \omega\right] & =\tilde{a}^{\mathrm{in}}\left[ \omega\right] +\tilde{a}^{\mathrm{out}}\left[ \omega\right] ,\\ \sqrt{\gamma_{b}}b\left[ \omega\right] & =\tilde{b}^{\mathrm{in}}\left[ \omega\right] +\tilde{b}^{\mathrm{out}}\left[ \omega\right]\end{aligned}$$ and the input-output relations given by Eq. (\[RedS\]), we obtain (transforming back into the time domain) $$-ig_{3}\left\langle a\left( t\right) b\left( t\right) \right\rangle =-\frac{\gamma_{eff}\left( G\right) }{2}\left\langle c\left( t\right) \right\rangle ,$$ where, in the limit of large gains $G\gg1$, $$\gamma_{eff}\left( G\right) =\frac{1}{2\pi}\frac{\gamma_{c}}{4\overline {n}_{c}^{\mathrm{in}}}\int_{0}^{+\infty}\mathrm{d}\omega\left( \mathcal{N}_{a}^{\mathrm{in}}\left[ \omega\right] +\mathcal{N}_{b}^{\mathrm{in}}\left[ \omega\right] \right) G\left( \Delta\omega\right) \label{gamma_eff}$$ denotes an effective decay rate of pump photons due to generation of entangled signal and idler photons. This last relation expresses, in another form, the Manley-Rowe relations [@ManleyProc] that establish the equality between the number of created signal photons by the amplifier to the number of destroyed pump photons. It shows that even in the absence of any deterministic signal applied to the oscillator $a$ or $b$, pump photons are used to amplify zero-point fluctuations. Therefore, the pump tone always encounters a dissipative load even when no signals are injected into the device. For a continuous wave (CW) input power sent at the center frequency of the $a$ or $b$ oscillator, or both, we have $$\gamma_{eff}\left( G_{0},P^{\mathrm{in}}\right) =\frac{\gamma_{c}}{4\overline{n}_{c}^{\mathrm{in}}}G_{0}P^{\mathrm{in}}, \label{gamma_eff1}$$ where $P^{\mathrm{in}}=P_{a}^{\mathrm{in}}+P_{b}^{\mathrm{in}}$ is given in units of photon number per unit time and, in steady state, $$\overline{n}_{c}\left( G_{0},P^{\mathrm{in}}\right) =\frac{4\gamma_{c}}{\left( \gamma_{c}+\gamma_{eff}\left( G_{0},P^{\mathrm{in}}\right) \right) ^{2}}\overline{n}_{c}^{\mathrm{in}}. \label{nc_ncin}$$ As a finite input power is applied to the signal oscillators, oscillator $c$ depopulates and, keeping the pump power constant, we get $$\begin{aligned} \frac{\overline{n}_{c}\left( G_{0},P^{\mathrm{in}}\right) }{\overline{n}_{c}\left( G_{0},P^{\mathrm{in}}=0\right) } & =\frac{1}{\left( 1+\frac{G_{0}P^{\mathrm{in}}}{4\overline{n}_{c}^{\mathrm{in}}}\right) ^{2}}\\ & \simeq1-\frac{G_{0}P^{\mathrm{in}}}{2\overline{n}_{c}^{\mathrm{in}}}.\end{aligned}$$ On the other hand, from Eqs. (\[rho\]) and (\[gain-formula\]), the left hand side is given by $$\frac{\overline{n}_{c}\left( G_{0},P^{\mathrm{in}}\right) }{\overline{n}_{c}\left( G_{0},P^{\mathrm{in}}=0\right) }=\frac{\frac{\sqrt{G}-1}{\sqrt {G}+1}}{\frac{\sqrt{G_{0}}-1}{\sqrt{G_{0}}+1}},$$ where $G$ denotes the gain in the presence of $P_{\mathrm{in}}$. In the large gain limit, if we fix the maximum decrease of gain due to pump depletion to be $$\frac{G}{G_{0}}>1-\varepsilon$$ with $\varepsilon\ll1$, then we obtain $$\frac{P^{\mathrm{in}}}{2\overline{n}_{c}^{\mathrm{in}}}<\varepsilon G_{0}^{-3/2}, \label{pumpDepletion1}$$ which can also be rewritten as $$\frac{2\overline{n}_{c}^{\mathrm{in}}}{G_{0}P^{\mathrm{in}}}>\varepsilon ^{-1}\sqrt{G_{0}}.$$ This relation shows that the ratio of the power of the pump tone to that of the signal at the output of the amplifier must always be much larger than the amplitude gain, in order for the linearity of the amplifier not to be compromised by pump depletion effects. In Fig. \[drthyfig\] we plot a calculated response of the signal output power $P_{\mathrm{out}}$ versus the signal input power $P_{\mathrm{in}}$ for a typical three-wave mixing device. The device parameters employed in the calculation and listed in the figure caption are practical values yielding a maximum input power, which is limited by the effect of pump depletion. The different blue curves are obtained by solving Eq. (\[nc\_ncin\]) for $G$ and using the input-output relation $P_{\mathrm{out}}=GP_{\mathrm{in}}$, where $P_{\mathrm{in}}$ expressed in units of power is taken as the independent variable and $G_{0}$ is treated as a parameter. Note that in solving Eq. (\[nc\_ncin\]), equations (\[gamma\_eff1\]), (\[c\_sq\_c\_in\_sq\]), (\[rho\]) and (\[gain-formula\]) are used. When drawn on logarithmic scale, the device gain translates into a vertical offset (arrow indicating $G_{0}$) off the $P_{\mathrm{out}}=P_{\mathrm{in}}$ line, indicated in red. The dashed black vertical line corresponds to a signal input power of 1 photon at the signal frequency per inverse dynamical bandwidth of the device at $G_{0}=20$ dB. The dashed green line corresponds to the maximum gain set by the amplified zero-point fluctuations given by Eq. (\[Gmaxzpf\]), while the cyan line corresponds to the maximum circulating power in the cavity given by Eq. (\[Pmaxcav\]). \[h\] ![(Color online). A calculated response of the signal output power $P_{\mathrm{out}}$ versus the signal input power $P_{\mathrm{in}}$ of a typical three-wave mixing device which exhibits a pump depletion effect. The different blue curves correspond to different $G_{0}$ setpoints. The definition of the other lines in the figure is given in the text. The parameters used in the calculation are: $\omega_{a}/2\pi=7\operatorname{GHz}$, $\omega_{b}/2\pi=8\operatorname{GHz}$, $\omega_{c}/2\pi=15\operatorname{GHz}$, $\gamma_{a}/2\pi=\gamma_{b}/2\pi=50\operatorname{MHz}$, $\gamma_{c}/2\pi=0.6\operatorname{GHz}$, $Q_{a}=140$, $Q_{b}=160,$ $p_{a}=p_{b}=0.03$, $p_{c}=0.02,$ $I_{0}=1\operatorname{\mu A}$, $E_{J}^{a,b}=E_{J}/\sqrt {2}=16.3\operatorname{K}$, $P_{\mathrm{1ph}}=-128$ dBm, $G_{\mathrm{ZPF}}^{\max}=35$ dB, $P_{\mathrm{cav}}^{\mathrm{\max}}=P_{b}^{\mathrm{\max}}=-86$ dBm and $g_{3}/2\pi=0.7\operatorname{MHz}$.[]{data-label="drthyfig"}](DRthyfig.pdf "fig:"){width="\columnwidth"}\ Furthermore, the maximum bound $P^{\max}$ indicated by the solid magenta line corresponds to $P_{\mathrm{out}}^{\max}=G_{0}P_{\mathrm{in}}^{\max}$, where $P_{\mathrm{in}}^{\max}=P_{b}^{\max}/G_{0}^{3/2}$ and $P_{b}^{\max }=P_{\mathrm{cav}}^{\mathrm{\max}}$. As can be seen in the figure the predicted power scaling due to pump depletion effect, expressed in relation (\[pumpDepletion1\]), follows the calculated response quite well. Finally, it is straightforward to see that the usable region in the parameter space of the device with respect to gain, bandwidth and maximum input power lies within the boundaries of the fictitious triangle ABC indicated in the figure which is formed by the intersection of the magenta, black and red lines. The Josephson parametric converter ================================== We discuss here three different realizations of the Josephson parametric converter (JPC), which constitutes a fully non-degenerate three-wave mixing device capable of amplification and conversion as discussed in the previous sections. The three schemes differ in the resonator circuit design and the coupling between the feedline and the resonator. Microstrip Resonator JPC (MRJ) ------------------------------ \[h\] ![Circuit model of the Microstrip Resonator JPC (MRJ).[]{data-label="MJcirc"}](MJCirc.pdf "fig:"){width="\columnwidth"}\ The MRJ comprises two superconducting microstrip resonators which intersect at a JRM at the center as shown in the circuit model of the device in Fig. \[MJcirc\]. The resonance frequencies of the MRJ are determined by the lengths of the microstrips $l_{a}\simeq\lambda_{a}/2$ and $l_{b}\simeq \lambda_{b}/2$ and the Josephson inductance of the JRM, where $\lambda_{a}$, $\lambda_{b}$ are the wavelengths of the fundamental resonances at $\omega _{a}$ and $\omega_{b}$. It is worth mentioning that in addition to the differential modes *a* and *b*, this configuration of two coupled resonators also supports a common (even) mode. The angular frequency $\omega_{e}$ at which this even mode resonates lies between $\left( \omega_{b}+\omega_{a}\right) /2$ and $\omega_{b}$ (where $\omega_{b}>\omega_{a}$). The characteristic impedance of the resonators in the MRJ model is designed to be 50 $\Omega$ to ensure optimal coupling to the feedlines. Figure \[MJphoto\] exhibits an optical image of a typical MRJ device. The resonators are usually made of Al or Nb over sapphire or high-resistivity silicon and are coupled to the (transmission-line) feedlines using gap capacitors. The main role of these coupling capacitors is to set the external quality factor of the resonators. For a large bandwidth device operating in the $6-10$ GHz band, the external $Q$ of the resonators is typically in the range $60-100$. In all JPC designs discussed here the total $Q$ essentially coincides with the external $Q$, since the internal losses of the resonators are less than $10^{-4}$. Signals at $\omega_{1}$ and $\omega_{2}$, which lie within the bandwidths of resonators *a* and *b*, are fed into the JPC through the delta port of a $180$ degree hybrid, whereas the pump drive applied at $\omega_{3}=\omega_{1}+\omega_{2}$, for amplification, is a non-resonant tone and is injected into the device through the sigma port of the hybrid (Fig. \[MJcirc\]). The main advantage of the MRJ is that it is easy to design and fabricate. On the other hand, the main disadvantages are: (1) the area of the device can be relatively large depending on the frequencies of interest, (2) the characteristic impedance of the device is limited to around 50 $\Omega$, (3) the pump can be less stiff than the designs discussed below. The latter is due to the fact that the transmission-line resonators support higher resonance modes such as $2\omega_{a}$ and $2\omega_{b}$ with finite $Q$, which can be relatively close to the pump angular frequency $\omega_{3}$. \[h\] ![(Color online). Optical microscope image of a microstrip resonator JPC (MRJ). The resonators denoted *a* and *b* are half-wave microstrip resonators which intersect at a JRM. A zoomed-in view of the JRM, which consists of four Josephson junctions arranged in Wheatstone bridge configuration, is shown on the right. The MRJ is coupled to 50 $\Omega$ feedlines via gap capacitors.[]{data-label="MJphoto"}](MJfig.pdf "fig:"){width="\columnwidth"}\ Compact Resonator JPC (CRJ) --------------------------- \[h\] ![Circuit model of the Compact Resonator JPC (CRJ).[]{data-label="CJcirc"}](CJCirc.pdf "fig:"){width="\columnwidth"}\ In order to mitigate some of the drawbacks of the MRJ, we developed a new JPC design based on compact resonators known as CRJ. The circuit model of the CRJ, shown in Fig. \[CJcirc\], consists of four equal capacitors denoted as $C$ and two pairs of linear inductors connected in series with the JRM whose total inductance is $L_{a}$ and $L_{b}$ respectively. Using symmetry considerations one can verify that this circuit has three eigenmodes. Two differential eigenmodes which resonate at bare angular frequencies $\omega_{a}=1/\sqrt{\left( L_{a}+L_{J}^{eff}\right) C}$, $\omega_{b}=1/\sqrt{\left( L_{b}+L_{J}^{eff}\right) C}$, where $L_{J}^{eff}$ is the equivalent Josephson inductance of the JRM biased at half a flux quantum, and an even eigenmode which resonates at a lower bare angular frequency $\omega_{e}=1/\sqrt{\left( L_{a}+L_{b}+L_{J}^{eff}\right) C}$. Figure \[cjphoto\] shows an optical image of a typical compact JPC. The resonators of the device are made of Nb deposited over sapphire substrate. They are fabricated using a standard photolithography step and RIE etching. The JRM at the center of the device is made of Aluminum. It is fabricated using e-beam lithography, and angle shadow evaporation. As can be seen in the figure, the capacitance elements (including the coupling capacitors) of the device are implemented using interdigitated capacitors, whereas the inductive elements are realized using long narrow superconducting lines. Unlike the microstrip resonator JPC, the compact resonator JPC does not have higher harmonic resonances. The next closest resonance of this structure resides above $4\omega_{a}$, therefore the pump applied at $\omega_{a}+\omega_{b}$ can be considered stiff to a very good approximation. Other advantages of this realization are: (1) small size, with dimensions much smaller than the wavelengths corresponding to the resonance frequencies, (2) no requirement of a definite ground plane, unlike the MRJ, (3) greater flexibility in engineering the characteristic impedance of the resonators higher or lower than 50 $\Omega$, (4) higher internal quality factor resonators than the microstrip design. On the other hand, the main disadvantages of this design are: (1) the narrow lines and the interdigitated capacitors (as well as the lines connecting them) have parasitic capacitances and parasitic inductances associated with them, therefore scaling these devices to match a certain frequency or certain characteristic impedance requires using a microwave simulation tool, (2) there is a limit to how big the capacitance can be using the interdigitated configuration (values above 0.5 pF is difficult to achieve), therefore engineering characteristic impedances below 30 $\Omega$ is not quite feasible with this design. ![(Color online). Optical microscope image of a compact resonator JPC (CRJ). The device consists of four equal interdigitated capacitors denoted $C$ and two inductive elements denoted $L_{a}$ and $L_{b}$ which are realized using narrow superconducting lines of different lengths. The JRM of the device resides at the intersection of the two lines. An optical image of the JRM is shown in the inset. The CRJ is coupled to $50\operatorname{\Omega }$ microstrip feedlines via interdigitated capacitors denoted $C_{c}$.[]{data-label="cjphoto"}](CJ3fig.pdf "fig:"){width="\columnwidth"}\ Shunted JPC (SJ) ---------------- ![Circuit model of the Shunted JPC (SJ).[]{data-label="LJcirc"}](LJCirc.pdf "fig:"){width="\columnwidth"}\ In this subsection we discuss a third promising design called the capacitively and inductively shunted JPC (SJ) which is still a work in progress in our lab. In this version of the JPC, the capacitive elements are parallel plate capacitors and the inductive elements are mainly Josephson junctions. A schematic circuit model of the SJ is drawn in Fig. \[LJcirc\]. It is straightforward to show that the SJ model has two differential eigenmodes with angular resonance frequencies $\omega_{a}=1/\sqrt{L_{J}^{^{\prime}}C_{a}}$, $\omega_{b}=1/\sqrt{L_{J}^{^{\prime}}C_{b}}$, where $L_{J}^{^{\prime}}$ corresponds to the equivalent inductance of the JRM shunted by linear inductors [@Roch], as shown in Fig. \[LJcirc\]. The main purpose of these shunting inductors is to eliminate the hysteretic flux response of the JRM and extend the frequency tunability of the device beyond the bandwidth limit of the resonators. Such frequency tunability is achieved by varying the flux threading the loop which, in turn, varies $L_{J}^{^{\prime}}$. Note that the addition of these shunting inductors can be employed in other realizations of the JPC also, such as the MRJ, as shown in Ref. [@Roch] and the CRJ. It is important to emphasize, however, that the main difference between the SJ and the CRJ or MRJ schemes is that the shunted JRM in the SJ design is the only inductive element in the circuit that forms an integral part of the resonators *a* and *b*. Thus, the larger lumped capacitors employed in the SJ design play a crucial role in keeping the resonance frequencies of the device below $10$ GHz. Similar to the Josephson bifurcation amplifier (JBA) implementation [@VijayJBAreview], the plate capacitors in the SJ design can be made of Nb electrodes separated by a thin SiN dielectric layer. Using plate capacitors in this realization has two advantages: (1) the plate capacitors can be made very large, i.e. their capacitance can vary in the range $1-40$ pF, (2) they are easy to design as their capacitance scales linearly with the electrode area. Furthermore, due to the lumped nature of the capacitive and inductive elements in the SJ design and the fact that the capacitors can be large, the SJ has three important advantages over the previous designs: (1) the characteristic impedance of the resonators can be of the order of a few ohms, which yields an improved coupling between the resonators and the JRM, (2) due to the impedance mismatch between the characteristic impedance of the resonators and the 50 $\Omega$ feedlines, the coupling capacitors are unnecessary to achieve low external Q and the feedlines can be connected directly to the resonators, (3) the maximum input power of the amplifier can be increased by increasing the critical current of the JRM junctions while keeping the resonance frequencies fixed by enlarging the capacitors. Experimental results ==================== The set of JPC parameters which can be directly measured in an experiment are: the angular resonance frequencies of the resonators *a* and *b* $\omega_{a}$, $\omega_{b}$, the inverse of residence times of photons at resonance $\gamma_{a}$, $\gamma_{b}$, the participation ratios $p_{a}$, $p_{b}$, the maximum input power which the device can handle with no applied pump tone $P_{a}^{\max}$, $P_{b}^{\max}$, and the maximum measured gain at vanishing input power $G_{0}^{\max}$. One way to find $p_{a}$, $p_{b}$ is by measuring $\omega_{a}$, $\omega_{b}$ as a function of applied magnetic flux threading the JRM loop. To establish this relation, we model the resonators near resonance as an LC oscillator with effective inductance $L_{a,b}$ and effective capacitance $C_{a,b}$. In this model, the bare angular resonance frequencies of the device (with the junctions) $\omega_{a}$, $\omega_{b}$, can be written as $$\omega_{a,b}\left( \varphi\right) =\frac{1}{\sqrt{C_{a,b}\left( L_{a,b}+L_{J}\left( \varphi\right) \right) }},$$ where $L_{J}\left( \varphi\right) $ is the effective Josephson inductance of the JRM given by $$L_{J}\left( \varphi\right) =\frac{L_{J}}{\cos\left( \frac{\varphi}{4}\right) }$$ with $\varphi=2\pi\Phi/\Phi_{0}$. By calculating the derivative of $\omega_{a,b}\left( \varphi\right) $ with respect to the reduced flux $\varphi$, one gets $$\begin{aligned} \frac{1}{\omega_{a,b}}\frac{\mathrm{d}\omega_{a,b}}{\mathrm{d}\varphi} & =-\frac{1}{8}\tan\left( \frac{\varphi}{4}\right) \frac{L_{J}\left( \varphi\right) }{\left( L_{a,b}+L_{J}\left( \varphi\right) \right) },\\ & =-\frac{1}{8}\tan\left( \frac{\varphi}{4}\right) p_{a,b}(\varphi).\end{aligned}$$ Hence, at the device working point $\Phi=\Phi_{0}/2$ ($\varphi=\pi$), $p_{a,b}$ reads $$p_{a,b}=-8\left. \left( \frac{1}{\omega_{a,b}}\frac{\mathrm{d}\omega_{a,b}}{\mathrm{d}\varphi}\right) \right\vert _{\varphi=\pi}.$$ Furthermore, using Eq. (\[p-min\]) and the measured values $P_{a}^{\max}$, $P_{b}^{\max}$, one can infer the Josephson energy $E_{J}^{a,b}$ which is available for amplification $$E_{J}^{a,b}=p_{a,b}\frac{P_{a,b}^{\max}}{\gamma_{a,b}}.$$ It is important to mention that, in our experiments, we find that this value is lower by about one order of magnitude than the Josephson energy of the junctions at the working point $E_{J}=I_{0}\varphi_{0}/\sqrt{2}$, where $I_{0}$ is evaluated using dc resistance measurement of the junctions. Using Eqs. (\[rho\]) and (\[gain-formula\]) for the case of maximum gain $G_{0}^{\max}$ yields $$g_{3}^{2}\overline{n}_{c,\rho\rightarrow1}=\frac{\gamma_{a}\gamma_{b}}{4}\frac{\sqrt{G_{0}^{\max}}-1}{\sqrt{G_{0}^{\max}}+1}, \label{g3_sq_times_n_c}$$ which in the limit of high gains gives an upper bound on the product $g_{3}^{2}\overline{n}_{c,\rho\rightarrow1}$ $$g_{3}^{2}\overline{n}_{c,\rho\rightarrow1}\leq\frac{\gamma_{a}\gamma_{b}}{4}.$$ Here $\overline{n}_{c,\rho\rightarrow1}$ is the number of pump photons in the device at $G_{0}^{\max}$. \[h\] ![(Color online). Output power $P_{\mathrm{out}}$ measurement of a CRJ amplifier (device A) as a function of input power $P_{\mathrm{in}}$ measured at $\omega_{b}$. The data curves plotted in blue correspond to different $G_{0}$ setpoints obtained for different pump powers. The red line corresponds to $0$ dB (unity gain) where $P_{\mathrm{out}}=P_{\mathrm{in}}$. The dashed black vertical line indicates the input power of $1$ photon at the signal frequency per inverse dynamical bandwidth of the device at $G_{0}=20$ dB. The top horizontal line labelled $P_{\mathrm{cav}}^{\max}$ corresponds to the maximum circulating power in the resonator cavity given by Eq. (\[Pmaxcav\]). The green line corresponds to an upper limit on the device gain set by the saturation of the amplifier due to zero-point fluctuations given by Eq. (\[Gmaxzpf\]). The dashed magenta line is a theoretical prediction for $P_{\mathrm{out}}^{\max}$, which corresponds to the maximum circulating power in the device given by Eq. (\[DRstiff\]). The measured and calculated parameters of this device (A) are listed in table III.[]{data-label="drcjfig"}](DRCJfig.pdf "fig:"){width="\columnwidth"}\ In Figs. (\[drcjfig\]), (\[drpsfig\]), (\[drchfig\]) we plot on logarithmic scale the output power $P_{\mathrm{out}}$ of three different JPCs with different characteristics as a function of input power $P_{\mathrm{in}}$. For simplicity, we refer to the three devices as A, B and C respectively. The parameters of the three devices are listed in table III. The data curves plotted in blue are measured at resonance and satisfy the relation $$P_{\mathrm{out}}=G\left( P_{\mathrm{in}},G_{0}\right) P_{\mathrm{in}},$$ where $G\left( P_{\mathrm{in}},G_{0}\right) $ is the amplifier gain. This depends on $P_{\mathrm{in}}$ and $G_{0}$, the device gain for $P_{\mathrm{in}}=0$ which is set by the applied pump power. In this measurement, we apply a fixed pump power and vary $P_{\mathrm{in}}$ treating $G_{0}$ as a parameter. In log units, the device gain translates into a vertical offset from the $0$ dB baseline (red line) which corresponds to $P_{\mathrm{out}}=P_{\mathrm{in}}$. As expected, the devices maintain an almost constant gain $G_{0}$ as a function of $P_{\mathrm{in}}$ before they saturate and their gain drops for elevated input powers. However, as can be seen in Figs. (\[drcjfig\]), (\[drpsfig\]), (\[drchfig\]), the three devices exhibit qualitatively different behaviors in the vicinity of their maximum input power, which correspond to different saturation mechanisms taking place in the device as will be discussed shortly. Note that the order in which the different results are presented in this section does not depend on the specific implementation of the device (see Sec. IV) but rather on the saturation mechanism involved in each case. In Fig. \[drcjfig\], device A exhibits almost a plateau in $P_{\mathrm{out}}$ as it reaches its maximum input power for different $G_{0}$ setpoints. This result can be explained by assuming a stiff pump for which Eq. (\[P-Max\]) applies. By employing $P_{b}^{\max}$, measured with no applied pump tone, we plot the dashed magenta line labelled $P^{\max}$ which corresponds to $$P_{\mathrm{out}}^{\max}=P_{b}^{\max}. \label{DRstiff}$$ The dashed black vertical line indicates the input power of $1$ photon at the signal frequency per inverse dynamical bandwidth of the device at $G_{0}=20$ dB. In practice, as we discuss in Sec. VI, the usable region in the parameter space of the device with respect to gain, bandwidth and maximum input power lies within the boundaries of the fictitious triangle formed by the magenta, red and black lines. ![(Color online). Output power $P_{\mathrm{out}}$ measurement of a MRJ amplifier (device B) as a function of input power $P_{\mathrm{in}}$ measured at $\omega_{a}$. The data curves plotted in blue correspond to different $G_{0}$ setpoints obtained for different pump powers. The red line corresponds to $0$ dB (unity gain) where $P_{\mathrm{out}}=P_{\mathrm{in}}$. The dashed black vertical line indicates the input power of $1$ photon at the signal frequency per inverse dynamical bandwidth of the device at $G_{0}=20$ dB. The top horizontal line labelled $P_{\mathrm{cav}}^{\max}$ corresponds to the maximum circulating power in the resonator cavity given by Eq. (\[Pmaxcav\]). The green line corresponds to an upper limit on the device gain set by the saturation of the amplifier due to zero-point fluctuations given by Eq. (\[Gmaxzpf\]). The solid magenta line is a theoretical prediction for $P_{\mathrm{in}}^{\max}$ of the device and the corresponding $P_{\mathrm{out}}^{\max}$ due to pump depletion effect given by Eq. (\[DRpumpDep\]). The measured and calculated parameters of this device (B) are listed in table III.[]{data-label="drpsfig"}](DRPSfig.pdf "fig:"){width="\columnwidth"}\ Furthermore, in Figs. (\[drcjfig\]), (\[drpsfig\]), (\[drchfig\]) we plot two fundamental limits on the maximum gain $G_{\mathrm{ZPF}}^{\max}$ (green line) which corresponds to saturation of the device due to amplified zero-point fluctuations and the maximum circulating power $P_{\mathrm{cav}}^{\mathrm{\max}}$ (cyan line), given by Eq. (\[Gmaxzpf\]) and Eq. (\[Pmaxcav\]) respectively. The fact that these lines lie considerably above the experimental data in Figs. (\[drcjfig\]), (\[drpsfig\]), (\[drchfig\]), suggests that the energy threshold, at which nonlinear effects in these devices become significant, is much lower than the Josephson energy of the junctions, i.e. $E_{J}^{a,b}\ll E_{J}$. In contrast to Fig. \[drcjfig\], the data curves shown in Fig. \[drpsfig\] for device B, exhibit a gradual decrease in the gain in the vicinity of the maximum input power which can be explained in terms of pump depletion effect discussed in Sec. III. The maximum bound $P^{\max}$ indicated by the solid magenta line corresponds to $P_{\mathrm{out}}^{\max}=G_{0}P_{\mathrm{in}}^{\max}$, where in this case $P_{\mathrm{in}}^{\max}$ satisfies the inequality \[pumpDepletion1\] and is given by $$P_{\mathrm{in}}^{\max}=\frac{P_{a}^{\max}}{G_{0}^{3/2}}. \label{DRpumpDep}$$ ![(Color online). Output power $P_{\mathrm{out}}$ measurement of another CRJ amplifier (device C, with different parameters from device A) as a function of input power $P_{\mathrm{in}}$ measured at $\omega_{b}$. The data curves plotted in blue corresponds to different $G_{0}$ setpoints obtained for different pump powers. The data curves of this device exhibit abrupt drop in the gain in the vicinity of the maximum input powers which suggests that the device enters an unstable regime at elevated input powers. The red line corresponds to $0$ dB (unity gain) where $P_{\mathrm{out}}=P_{\mathrm{in}}$. The dashed black vertical line indicates the input power of $1$ photon at the signal frequency per inverse dynamical bandwidth of the device at $G_{0}=20$ dB. The top horizontal line labelled $P_{\mathrm{cav}}^{\max}$ corresponds to the maximum circulating power in the resonator cavity given by Eq. (\[Pmaxcav\]). The green line correspond to an upper limit on the device gain set by the saturation of the amplifier due to zero-point fluctuations given by Eq. (\[Gmaxzpf\]). The solid magenta line is a theoretical prediction for $P_{\mathrm{in}}^{\max}$ of the device and the corresponding $P_{\mathrm{out}}^{\max}$ due to pump depletion effect given by Eq. (\[DRpumpDep\]). The measured and calculated parameters of this device (C) are listed in table III.[]{data-label="drchfig"}](DRChfig.pdf "fig:"){width="\columnwidth"}\ On the other hand the data curves shown in Fig. \[drchfig\] for device C exhibit an abrupt drop in the device gain in the vicinity of $P_{\mathrm{in}}^{\max}$ of the device, which indicates that the device enters an unstable regime at elevated input powers. As can be seen in this case the solid magenta line — which satisfies $P_{\mathrm{out}}^{\max}=G_{0}P_{\mathrm{in}}^{\max}$, where $P_{\mathrm{in}}^{\max}$ is given by Eq. (\[DRpumpDep\]) — lies above the experimental data. This suggests that the maximum input power in this sample, which displays a steeper power scaling than Eq. (\[DRpumpDep\]), is mainly limited by nonlinear effects arising from higher order terms in the Hamiltonian of the system and cannot be attributed to a pump depletion effect alone. It is worthwhile noting that a similar power scaling for the maximum input power has been observed as well for an MRJ amplifier in Ref. . To understand which properties are responsible for the different gain behaviors exhibited by devices A, B and C, we point out a few important distinctions in their design (respective parameters are listed in table III). The data in Fig. \[drcjfig\] (device A) and Fig. \[drchfig\] (device C) is measured on JPC devices realized using the CRJ configuration which yields, in general, a stiff pump response as explained in Sec. IV (B). However, the main two differences between devices A and C are: (1) device A has a narrower bandwidth as compared to C (70 MHz vs. 142 MHz) and (2) the JRM junctions in A have a smaller $I_{0}$ compared to those in C (2 $\mu$A vs. 4 $\mu$A). The relatively large bandwidth of device C leads to a larger dynamical bandwidth 14 MHz at $G_{0}=16$ dB, as opposed to 10 MHz achieved in device A for the same gain, and also yields (with the larger $I_{0}$ of device C) higher $P_{a,b}^{\max}$ values. However, the large bandwidth translates into a lower $pQ$ product for C as compared to A, thus making it more susceptible to parametric oscillation (at high gains or high input powers) as implied by inequality (\[PQ-product\]). Device B, on the other hand, exhibits a pump depletion effect as shown in Fig. \[drpsfig\]. This can be attributed to its MRJ configuration, which, in general, exhibits a less stiff pump response than the CRJ, due to the presence of high order modes as explained in Sec. IV (A). Furthermore, as opposed to the MRJ amplifier in Ref. with an idler frequency of 6.4 GHz, device B has a higher idler frequency of 15 GHz which leads to a higher $pQ$ product.  $\backslash$        -------------------------------------------------- ---------- ------------- ----------         $\omega_{a}/2\pi$   $6.576$ $8.436$ $7.051$ $\omega_{b}/2\pi$   $6.873$ $15.087$ $7.673$ $\omega_{3}/2\pi$   $13.449$ $23.523$ $14.724$ $\gamma_{a}/2\pi$   $69$ $116$ $79$ $\gamma_{b}/2\pi$   $71$ $250\pm25$ $142$ $Q_{a},Q_{b}$ $94,96$ $73,60$ $89,54$ $p_{a},p_{b}$ $0.02$ $0.03,0.05$ $0.03$ $p_{a}p_{b}Q_{a}Q_{b}$ $8.1$ $6.6$ $4.3$ $I_{0}$ $(\mu A)$ $2$ $3$ $4$ $P_{\;\mathrm{cav}}^{\;\mathrm{max}}$   $-82$ $-77$ $-76$ $P_{a,b}^{\;\mathrm{max}}$   $-97$ $-89$ $-87$ $P_{\;\mathrm{1ph}}$   $-127$ $-125$ $-123$ $G_{\;\mathrm{ZPF}}^{\;\mathrm{max}}$   $38$ $39$ $40$ $G_{0}^{\;\mathrm{max}}$   $22$ $20$ $16$ $g_{3}\bar{n}_{c,\rho\rightarrow1}^{1/2}/2\pi$   $33$ $77$ $45$ : Parameters of JPCs A, B and C. Precision is last significant digit unless indicated otherwise. \[Table3\] Requirements for qubit readout ============================== One of the leading architectures which is used to manipulate and readout the state of superconducting qubits such as transmons and fluxoniums [@TransmonThy; @fluxonium] is circuit Quantum Electrodynamics (cQED). In such a system a quantum non-demolition measurement of the qubit state can be performed using dispersive readout in which the frequencies of the qubit and the cavity are detuned. As a result, the qubit and the cavity interact via exchanging virtual microwave photons [@RSL1] and the qubit state gets encoded in the output microwave field of the cavity. However, since the energy of microwave photons is very small, the detection of single photons is difficult especially considering the fact that state-of-the-art cryogenic amplifiers (i.e. high electron mobility transistor (HEMT) [@HEMT]) following the cQED setup add noise to the input signal, equivalent to about $20-40$ photons at the signal frequency. Therefore, adding a quantum-limited amplifier in series between the cQED sample and the HEMT amplifier can substantially decrease the noise temperature of the system and enable real-time tracking of the qubit state [@QuantumJumps; @QubitJPC]. The desired requirements of a Josephson parametric amplifier for such high-fidelity qubit readout can be summarized as follows: - A center frequency in the range $5-12$ GHz which is widely used in readout cavities of superconducting qubits. - A large power gain on the order of $20$ dB in order to beat the noise of the following amplifier, i.e. the HEMT. - A minimum added noise to the signal, equivalent to a half input photon at the signal frequency $T_{\mathrm{N}}=\hbar\omega_{a,b}/2k_{B}$ when operated in the phase preserving mode [@Caves]. - A large dynamical bandwidth of the order of $10$ MHz, which corresponds to a signal processing time of less than $100$ ns and matches the bandwidth of most readout cavities. - A maximum input power of a few photons per inverse dynamical bandwidth of the device at the highest gain. Such requirement is essential in quantum non-demolition readout schemes which employ of the order of a photon on average [@fluxonium]. - A tunable bandwidth of more than 100 MHz so that the center frequency of the amplifier can match the frequency of the readout signal. Recently, Roch *et al.* [@Roch] have achieved a tunable bandwidth of more than 500 MHz in a MRJ device by shunting the Josephson junctions of the JRM with linear inductors realized using superconducting wires \[see Fig. \[LJcirc\]\]. Similar results were obtained by our group in a MRJ device by utilizing large Josephson junctions instead of superconducting wires [@QubitJPC]. - Minimal out-of-band back-action to avoid qubit relaxation. In table IV we enlist the parameters achievable with a JPC which show its viability as a low-noise preamplifier for qubit measurements.       --------------------------- ------------ --------------------- $\omega_{a,b}/2\pi$ 5 - 12  6.4 & 8.1   $T_{\;\mathrm{N}}$ $@$ 8  0.2  0.4  $G$ $20$  $21$  $B$ 10  11    100  60  $P_{\;\mathrm{max}}$ $1$ photon 3 photons $@$ 20 dB OB back-action Negligible None measurable : Preamplifier requirements and JPC merits achieved to date (OB=out-of-band). \[Table4\] The last question which we would like to address in this paper is whether there exists a set of technologically feasible parameters for which the JPC can be optimized with respect to dynamic range while maintaining a gain of $20$ dB and a reasonable dynamical bandwidth of more than 2 MHz. In order to provide a quantitative answer we choose a signal frequency of 12 GHz, which is a good choice for readout frequency for qubits as it is higher than most qubit frequencies. We set an ambitious goal for the processing capability of the JPC of about $100$ input photons at the signal frequency with gain $20$ dB. ![(Color online). An optimized JPC response drawn in blue for large maximum input power in excess of $100$ input photons at $12$ $\operatorname{GHz}$ per inverse dynamical bandwidth of the device at $20$ dB. The definition of the other lines shown in the figure is similar to Fig. (\[drcjfig\]). The parameters used in the calculation are: $\omega_{a}/2\pi=11\operatorname{GHz}$, $\omega_{b}/2\pi=12\operatorname{GHz}$, $\omega_{c}/2\pi=23\operatorname{GHz}$, $Z_{a}=36\operatorname{\Omega }$, $Z_{b}=33\operatorname{\Omega }$, $L_{a}=0.51$ n$\operatorname{H}$, $L_{b}=0.42$ n$\operatorname{H}$, $C_{a}=C_{b}=0.4\operatorname{pF}$, $C_{C_{a}}=31$ f$\operatorname{F}$, $C_{C_{b}}=28$ f$\operatorname{F}$, $\gamma_{a}/2\pi=\gamma_{b}/2\pi=44.2\operatorname{MHz}$, $\gamma_{c}/2\pi=3\operatorname{GHz}$, $Q_{a}=249$, $Q_{b}=271$, $Q_{c}=8$, $p_{a}=0.028$, $p_{b}=0.034$, $p_{c}=0.02$, $I_{0}=30\operatorname{\mu A}$, $E_{J}/\sqrt{2}=490\operatorname{K}$, $E_{J}^{a,b}=49\operatorname{K}$, $P_{\mathrm{1ph}}=-126$ dBm, $G_{\mathrm{ZPF}}^{\max}=48$ dB, $P_{\mathrm{cav}}^{\mathrm{\max}}=P_{b}^{\mathrm{\max}}=-86.3$ dBm and $g_{3}/2\pi =0.6\operatorname{MHz}$. []{data-label="DRprospectsfig"}](DRprospectsThyfig.pdf "fig:"){width="\columnwidth"}\ To that end, we choose to perform the optimization process for the CRJ configuration which inherently yields large $\gamma_{c}$ values and also allows variation of the external quality factor of the resonators more easily than the SJ scheme. We also choose a resonance frequency for mode *a* of $\omega_{a}/2\pi=$ 11 GHz, a relatively high critical current $I_{0}=$ 30 $\mu$A and limit ourselves to capacitance values below or equal to 0.4 pF. The advantage of working with large $I_{0}$ for the purpose of large dynamic range is that it increases $E_{J}$ and lowers $g_{3}$. However, such large $I_{0}$ yields very low $L_{J}^{eff}=\sqrt{2}\varphi_{0}/I_{0}=15$ pH which requires coupling to relatively low impedance resonators while maintaining participation ratios of a few percent. The next challenge in the optimization is to increase the $pQ$ product of the device which promotes high gains by increasing the quality factor of the resonators. Nevertheless, care must be taken not to increase the quality factors beyond what is strictly necessary for two reasons (i) high Q resonators limit the dynamical bandwidth of the device as can be seen in Eq. (\[B\]), (ii) high Q resonators increase the pump depletion effect and, in turn, lower the dynamic range of the device. In Fig. \[DRprospectsfig\] we plot the calculated response of such an optimized JPC which takes into account the above considerations and limitations. As can be seen in the figure the optimized device exhibits, for the chosen set of parameters, a maximum input power of about $100$ photons at the signal frequency per inverse dynamical bandwidth of the device $B/2\pi =$ 4.4 MHz at $20$ dB of gain. The device parameters which are used in the calculation are listed in the figure caption. It is important to note that in the calculation of the expected response, which is indicated by the blue curves for different values of $G_{0}$, we assumed an available Josephson energy $10$ times smaller than $E_{J}/\sqrt{2}$ of the junctions, in agreement with experimental conditions. Finally, we verify that the set of parameters of the optimized device satisfy the inequalities $\bar{n}_{c}^{\max}=3.7\cdot10^{3}>\bar{n}_{c}^{\mathrm{po}}=1.3\cdot10^{3}> \bar{n}_{c}^{20\mathrm{dB}}=10^{3}$. Conclusion ========== We have addressed in this paper a new type of quantum signal processing device based on Josephson tunnel junctions. In contrast with the devices based on SQUIDS and driven non-linear Josephson oscillators, it performs a fully non-degenerate three-wave mixing in which the modes of the signal, pump and idler are separate both spatially and temporally. The heart of the device consists of a ring modulator constructed from four Josephson junctions arranged in a loop. Both quantum-limited amplification and noiseless frequency conversion are possible with this device, and the characteristics of these analog signal processing operations are entirely calculable analytically. We have established the limitations preventing the simultaneous maximization of photon number gain, bandwidth and dynamic range. Nevertheless, we have shown that a device satisfying all the requirements of superconducting qubit readout is realizable with present day technology. Discussions with Flavius Schackert, Michael Hatridge, Nicolas Bergeal, Benjamin Huard and Ananda Roy are gratefully acknowledged. The assistance of Michael Power and Luigi Frunzio in the fabrication process is highly appreciated. This work was supported by Yale University, NSF, IARPA, ARO and College de France. Appendix: Quantum signals propagating along a transmission line and input-output formalism {#appendix Quantum signals .unnumbered} ========================================================================================== This appendix treats quantum-mechanically the damping of a circuit by a resistance modelled as a semi-infinite transmission line, as shown in Fig. \[Nyquist-model\]. It borrows heavily from the book by Gardiner and Zoller [@QuantumNoise] but uses slightly different notations that are adapted to the specificities of our Josephson circuits. We first describe an infinite transmission line extending from $x=-\infty$ to $x=+\infty$. Later, we will cut the line at $x=0$ and replace the left portion by two terminals of the circuit. ![The damping of a circuit by a resistance $R$ can take place in a parallel or series way, depending on whether the resistance is placed across a branch or in series with it. The Nyquist model represents the resistance by a transmission line with characteristic impedance $Z_{c}=R$. The noise source associated with the resistance (fluctuation-dissipation theorem) is a parallel current source in the parallel case and a series voltage source in the series case. The noise source is replaced in the Nyquist model by incoming thermal radiation whose amplitude $A^{\mathrm{in}}$ is the square root of the power flux of the radiation ($A^{\mathrm{in}}$ should not be associated to a vector potential and is rather like the square root of the length of the Poynting vector).[]{data-label="Nyquist-model"}](nyquist_model_parallel-series.pdf "fig:"){width="\columnwidth"}\ Infinite transmission line {#infinite-transmission-line .unnumbered} -------------------------- The capacitance and inductance per unit length of the line are $C_{\ell}$ and $L_{\ell}$, respectively. The equations obeyed by the current $I$ along and the voltage $V$ across the line are $$\begin{aligned} -\frac{\partial}{\partial x}V\left( x,t\right) & =L_{\ell}\frac{\partial }{\partial t}I\left( x,t\right) ,\label{propagation_eq_1}\\ -\frac{\partial}{\partial x}I\left( x,t\right) & =C_{\ell}\frac{\partial }{\partial t}V\left( x,t\right) , \label{propagation_eq_2}$$ in which, for the moment, we treat the fields classically. The characteristic impedance and propagation velocity are given by $$\begin{aligned} Z_{c} & =\sqrt{\frac{L_{\ell}}{C_{\ell}}},\\ v_{p} & =\sqrt{\frac{1}{L_{\ell}C_{\ell}}}.\end{aligned}$$ In order to solve Eqs. (\[propagation\_eq\_1\]) and (\[propagation\_eq\_2\]), we introduce two new fields: the left-moving and right-moving wave amplitudes, $$\begin{aligned} A^{\rightarrow}\left( x,t\right) & =\frac{1}{2}\left[ \frac{1}{\sqrt{Z_{c}}}V\left( x,t\right) +\sqrt{Z_{c}}I\left( x,t\right) \right] ,\\ A^{\leftarrow}\left( x,t\right) & =\frac{1}{2}\left[ \frac{1}{\sqrt {Z_{c}}}V\left( x,t\right) -\sqrt{Z_{c}}I\left( x,t\right) \right] ,\end{aligned}$$ which have the advantage of treating currents and voltage on the same footing (note that these amplitudes are not directly related to the vector potential). The dimension of these fields is \[watt\]$^{1/2}$ and they are normalized such that the total power $P$ traversing, in the forward direction, a section of the line at position $x$ and time $t$ is given by $$P\left( x,t\right) =\left[ A^{\rightarrow}\left( x,t\right) \right] ^{2}-\left[ A^{\leftarrow}\left( x,t\right) \right] ^{2}. \label{poynting}$$ The quantity $P$ here plays the role of the Poynting vector in full 3D electrodynamics. Each of the terms at the right hand side of the last equation is thus the separate contribution of the corresponding wave to the total power flow. When solving Eqs. (\[propagation\_eq\_1\]-\[propagation\_eq\_2\]), we find $$\frac{\partial}{\partial x}A^{\rightleftarrows}\left( x,t\right) =\mp \frac{1}{v_{p}}\frac{\partial}{\partial t}A^{\rightleftarrows}\left( x,t\right) . \label{Eq._of_motion}$$ This relation means that $A^{\rightleftarrows}$ does not depend separately on $x$ or $t$ but a combination of both and thus: $$\begin{aligned} A^{\rightarrow}\left( x,t\right) & =A^{\rightarrow}\left( x=0,t-\frac {x}{v_{p}}\right) =A^{\rightarrow}\left( x-v_{p}t,t=0\right) ,\nonumber\\ A^{\leftarrow}\left( x,t\right) & =A^{\leftarrow}\left( x=0,t+\frac {x}{v_{p}}\right) =A^{\leftarrow}\left( x+v_{p}t,t=0\right) .\nonumber\\ &\end{aligned}$$ The properties of the wave amplitude can be summarized by writing $$\begin{aligned} A^{\rightleftarrows}\left( x,t\right) & =A_{0}^{\rightleftarrows}\left( \tau\right) ,\\ \tau & =t+\frac{\varepsilon^{\rightleftarrows}}{v_{p}}x,\\ \varepsilon^{\rightleftarrows} & =\mp1.\end{aligned}$$ Note that the detailed definition of the retardation $\tau$ depends on the wave direction. We now turn to the energy density $U\left( x,t\right) $, related to $P$ by the local energy conservation law $$\frac{\partial U}{\partial t}=-\frac{\partial P}{\partial x}.$$ Combining Eqs. (\[poynting\]) and (\[Eq.\_of\_motion\]), we get $$\begin{aligned} & \frac{\partial U\left( x,t\right) }{\partial t}\nonumber\\ & =\frac{2}{v_{p}}\left[ A^{\rightarrow}\left( x,t\right) \frac{\partial }{\partial t}A^{\rightarrow}\left( x,t\right) +A^{\leftarrow}\left( x,t\right) \frac{\partial}{\partial t}A^{\rightarrow}\left( x,t\right) \right] ,\nonumber\\ & =\frac{1}{v_{p}}\frac{\partial}{\partial t}\left\{ \left[ A^{\rightarrow }\left( x,t\right) \right] ^{2}+\left[ A^{\leftarrow}\left( x,t\right) \right] ^{2}\right\} .\end{aligned}$$ The total energy of the line at time $t$ is, thus [@SumofEMenergy], $$H=\frac{1}{v_{p}}\int_{-\infty}^{+\infty}\left\{ \left[ A^{\rightarrow }\left( x,t\right) \right] ^{2}+\left[ A^{\leftarrow}\left( x,t\right) \right] ^{2}\right\} \mathrm{d}x. \label{HinA}$$ When $H$ in Eq. (\[HinA\]) is considered as a functional of dynamical field variables $A^{\rightarrow}$ and $A^{\leftarrow}$, the equation of motion Eq. (\[Eq.\_of\_motion\]) can be recovered from Hamilton’s equation of motion as $$\frac{\partial}{\partial t}A^{\rightleftarrows}\left( x,t\right) =-\left\{ H,A^{\rightleftarrows}\left( x,t\right) \right\} _{P.B.},$$ on imposing the Poisson bracket $$\begin{aligned} & \left\{ A^{\rightleftarrows}\left( x_{1},t_{1}\right) ,A^{\rightleftarrows}\left( x_{2},t_{2}\right) \right\} _{P.B.} =\frac{1}{2}\frac{\partial}{\partial\left( \tau_{1}-\tau_{2}\right) }\delta\left( \tau_{1}-\tau_{2}\right). \label{PBinA2}$$ Therefore, from the classical-quantum correspondence involving the replacement of Poisson brackets by commutators, we find that the quantum operator version $\hat{A}^{\rightleftarrows}$ of the fields satisfy the commutation relation $$\left[ \hat{A}^{\rightleftarrows}\left( x_{1},t_{1}\right) ,\hat {A}^{\rightleftarrows}\left( x_{2},t_{2}\right) \right] =\frac{i\hbar}{2}\frac{\partial}{\partial\left( \tau_{1}-\tau_{2}\right) }\delta\left( \tau_{1}-\tau_{2}\right) ,$$ which is analogous to the commutation relation between the electric and magnetic field in 3-D quantum electrodynamics. Note that the fields are Hermitian at this stage. Introducing the Fourier transform, $$\hat{A}^{\rightleftarrows}\left[ \omega\right] =\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\hat{A}^{\rightleftarrows}\left( x=0,\tau\right) e^{i\omega\tau}\mathrm{d}\tau,$$ where the Fourier components (which are now non-hermitian operators) satisfy $$\hat{A}^{\rightleftarrows}\left[ \omega\right] ^{\dag}=A^{\rightleftarrows }\left[ -\omega\right] ,$$ we can also write the Hamiltonian as $$\sum_{\sigma=\rightleftarrows}\int_{-\infty}^{+\infty}\hat{A}^{\sigma}\left[ \omega\right] \hat{A}^{\sigma}\left[ -\omega\right] \mathrm{d}\omega.$$ The field operators in the frequency domain satisfy $$\left[ \hat{A}^{\rightleftarrows}\left[ \omega_{1}\right] ,\hat {A}^{\rightleftarrows}\left[ \omega_{2}\right] \right] =\frac{\hbar}{4}\left( \omega_{1}-\omega_{2}\right) \delta\left( \omega_{1}+\omega _{2}\right) .$$ We now introduce the usual quantum field annihilation operators $$\begin{aligned} a^{\rightarrow}\left[ \omega\right] & =\frac{\hat{A}^{\rightarrow}\left[ \omega\right] }{\sqrt{\hbar\left\vert \omega\right\vert /2}}=a^{\rightarrow }\left[ -\omega\right] ^{\dagger},\\ a^{\leftarrow}\left[ \omega\right] & =\frac{\hat{A}^{\leftarrow}\left[ \omega\right] }{\sqrt{\hbar\left\vert \omega\right\vert /2}}=a^{\leftarrow }\left[ -\omega\right] ^{\dagger}.\end{aligned}$$ They satisfy the commutation relations $$\left[ a^{\rightleftarrows}\left[ \omega_{1}\right] ,a^{\rightleftarrows }\left[ \omega_{2}\right] \right] =\mathrm{sgn}\left( \frac{\omega _{1}-\omega_{2}}{2}\right) \delta\left( \omega_{1}+\omega_{2}\right) .\label{bosonic_com}$$ It is useful to note that since $$a^{\rightleftarrows}\left[ \omega\right] =a^{\rightleftarrows}\left[ -\omega\right] ^{\dag},$$ Eq. (\[bosonic\_com\]) exhaustively describes all possible commutator cases. In the thermal state of the line, at arbitrary temperature (including $T=0$), $$\left\langle a^{\rightleftarrows}\left[ \omega_{1}\right] a^{\rightleftarrows}\left[ \omega_{2}\right] \right\rangle =S_{a^{\rightleftarrows}a^{\rightleftarrows}}\left[ \frac{\omega_{1}-\omega_{2}}{2}\right] \delta\left( \omega_{1}+\omega_{2}\right) ,$$ where $$S_{a^{\rightleftarrows}a^{\rightleftarrows}}\left[ \omega\right] =\mathrm{sgn}\left( \omega\right) N_{T}\left( \omega\right) .$$ When $\omega$ is strictly positive $N_{T}\left( \omega\right) $ is the number of available photons per unit bandwidth per unit time travelling on the line in a given direction around frequency $\omega$ $$\begin{aligned} N_{T}\left( \omega\right) & =\frac{1}{\exp\left( \frac{\hbar\omega}{k_{B}T}\right) -1}\\ & =\frac{1}{2}\left[ \coth\left( \frac{\hbar\omega}{2k_{B}T}\right) -1\right] .\end{aligned}$$ Negative frequencies $\omega$ correspond to the possibility of emitting photons into the line $$N_{T}\left( -\left\vert \omega\right\vert \right) =-N_{T}\left( \left\vert \omega\right\vert \right) -1.$$ The Bose-Einstein expression $N_{T}\left( \omega\right) $ is expected from the Hamiltonian of the line, which reads, with the $a$ operators, $$H=\frac{\hbar}{2}\sum_{\sigma=\rightleftarrows}\int_{-\infty}^{+\infty }\left\vert \omega\right\vert a^{\sigma}\left[ \omega\right] a^{\sigma }\left[ -\omega\right] \mathrm{d}\omega.$$ We can now give the expression for the anticommutator of the fields $$\begin{aligned} & \left\langle \left\{ a^{\rightleftarrows}\left[ \omega_{1}\right] ,a^{\rightleftarrows}\left[ \omega_{2}\right] \right\} \right\rangle _{T}=2\mathcal{N}_{T}\left[ \frac{\omega_{1}-\omega_{2}}{2}\right] \delta\left( \omega_{1}+\omega_{2}\right) \nonumber\\ & =\mathrm{sgn}\left( \frac{\omega_{1}-\omega_{2}}{2}\right) \coth\left( \frac{\hbar\left( \omega_{1}-\omega_{2}\right) }{4k_{B}T}\right) \delta\left( \omega_{1}+\omega_{2}\right) . \label{bos-anticom6}$$ Equation (\[Na\_in\_first\]) with no external drive is identical to Eq. (\[bos-anticom6\]) $$\begin{aligned} \mathcal{N}_{T}\left[ \omega\right] & =\frac{\mathrm{sgn}\left( \omega\right) }{2}\coth\left( \frac{\hbar\omega}{2k_{B}T}\right) \\ & =\mathrm{sgn}\left( \omega\right) \left[ N_{T}\left( \left\vert \omega\right\vert \right) +\frac{1}{2}\right] .\end{aligned}$$ We now introduce the forward-propagating and backward-propagating voltage and current amplitudes obeying $$\begin{aligned} V^{\rightarrow}\left( x,t\right) & =\sqrt{Z_{c}}A^{\rightarrow}\left( x,t\right) ,\\ V^{\leftarrow}\left( x,t\right) & =\sqrt{Z_{c}}A^{\leftarrow}\left( x,t\right) ,\end{aligned}$$$$\begin{aligned} I^{\rightarrow}\left( x,t\right) & =V^{\rightarrow}\left( x,t\right) /Z_{c},\\ I^{\leftarrow}\left( x,t\right) & =V^{\leftarrow}\left( x,t\right) /Z_{c}.\end{aligned}$$ Quantum-mechanically, the voltage and current amplitudes become hermitian operators $$\begin{aligned} V^{\rightleftarrows}\left( x,t\right) & \rightarrow\hat{V}^{\rightleftarrows}\left( x,t\right) ,\\ I^{\rightleftarrows}\left( x,t\right) & \rightarrow\hat{I}^{\rightleftarrows}\left( x,t\right) .\end{aligned}$$ These operators, in turn, can be expressed in terms of field annihilation operators as $$\begin{aligned} \hat{V}^{\rightleftarrows}\left( x,t\right) & =\sqrt{\frac{\hbar Z_{c}}{4\pi}}\int_{-\infty}^{+\infty}\mathrm{d}\omega\sqrt{\left\vert \omega\right\vert }\hat{a}^{\rightleftarrows}\left[ \omega\right] e^{-i\omega\left( t\,\mp\,x/v_{p}\right) },\\ \hat{I}^{\rightleftarrows}\left( x,t\right) & =\sqrt{\frac{\hbar}{4\pi Z_{c}}}\int_{-\infty}^{+\infty}\mathrm{d}\omega\sqrt{\left\vert \omega \right\vert }\hat{a}^{\rightleftarrows}\left[ \omega\right] e^{-i\omega \left( t\,\mp\,x/v_{p}\right) }.\end{aligned}$$ All physical operators can be deduced from these primary expressions. For instance, the transmission line charge operator, describing the charge in the line brought from one end to the position $x$, is $$\hat{Q}^{\rightleftarrows}\left( x,t\right) =i\sqrt{\frac{\hbar}{4\pi Z_{c}}}\int_{-\infty}^{+\infty}\frac{\mathrm{d}\omega\sqrt{\left\vert \omega\right\vert }}{\omega}\hat{a}^{\rightleftarrows}\left[ \omega\right] e^{-i\omega\left( t\,\mp\,x/v_{p}\right) }.$$ Nyquist model of resistance: semi-infinite transmission line {#nyquist-model-of-resistance-semi-infinite-transmission-line .unnumbered} ------------------------------------------------------------ We now are in a position to deal with the semi-infinite line extending from $x=0$ to $x=\infty$, whose terminals at $x=0$ models a resistance $R=Z_{c}$ \[see Fig. \[Nyquist-model\]\]. In that half-line, the left- and right-moving propagating waves are no longer independent. We will now refer to the wave amplitude $A^{\leftarrow}\left( x=0,t\right) $ as $A^{\mathrm{in}}\left( t\right) $ and $A^{\rightarrow}\left( x=0,t\right) $ as $A^{\mathrm{out}}\left( t\right) $. The quantum-mechanical voltage across the terminal of the resistance and the current flowing into it satisfy the operator relations $$\begin{aligned} \hat{V}\left( t\right) & =\hat{V}^{\mathrm{out}}\left( t\right) +\hat {V}^{\mathrm{in}}\left( t\right) ,\\ \hat{I}\left( t\right) & =\hat{I}^{\mathrm{out}}\left( t\right) -\hat {I}^{\mathrm{in}}\left( t\right) .\end{aligned}$$ These relations can be seen either as continuity equations at the interface between the damped circuit and the resistance/line, or as boundary conditions linking the semi-infinite line quantum fields $\hat{A}^{\mathrm{in}}\left( t\right) $ and $\hat{A}^{\mathrm{out}}\left( t\right) $. From the transmission line relations, $$\hat{V}^{\mathrm{out},\mathrm{in}}\left( t\right) =R\hat{I}^{\mathrm{out},\mathrm{in}}\left( t\right) ,$$ we obtain $$\begin{aligned} \hat{I}\left( t\right) & =\frac{1}{R}\hat{V}\left( t\right) -2\hat {I}^{\mathrm{in}}\left( t\right) ,\\ & =\frac{1}{R}\hat{V}\left( t\right) -\frac{2}{\sqrt{R}}\hat{A}^{\mathrm{in}}\left( t\right) .\end{aligned}$$ For a dissipationless circuit with Hamiltonian $H_{bare}\left( \hat{\Phi },\hat{Q}\right) $, where $\hat{\Phi}$ is the generalized flux of the node electrically connected to the transmission line, and $\hat{Q}$ its canonically conjugate operator (top panel of Fig. \[Nyquist-model\]), we can write the Langevin equation, $$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\hat{Q} & =\frac{i}{\hbar}\left[ H_{bare},\hat{Q}\right] -\hat{I},\nonumber\\ & =\frac{i}{\hbar}\left[ H_{bare},\hat{Q}\right] -\frac{\mathrm{d}}{R\mathrm{d}t}\hat{\Phi}+\frac{2}{\sqrt{R}}\hat{A}^{\mathrm{in}}\left( t\right) . \label{Langevin-example}$$ The latter equation is just a particular case of the more general quantum Langevin equation giving the time evolution of any operator $\hat{Y}$ of a system with Hamiltonian $H_{bare}$, which is coupled to the semi-infinite transmission line by an Hamiltonian term proportional to another system operator $\hat{X}$, $$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\hat{Y} & =\frac{i}{\hbar}\left[ H_{bare},\hat{Y}\right] \nonumber\\ & +\frac{1}{2i\hbar}\left\{ \left[ \hat{X},\hat{Y}\right] ,2R^{\zeta /2}\hat{A}^{\mathrm{in}}\left( t\right) -R^{\zeta}\frac{\mathrm{d}}{\mathrm{d}t}\hat{X}\right\} .\nonumber\\ & \label{general-QLE1}$$ The value of $\zeta$ in Eq. (\[general-QLE1\]) depends on whether the damping is parallel“ ($\zeta=-1$) or series” type ($\zeta=+1$) \[see Fig. \[Nyquist-model\]\]. In the parallel case, the greater the line impedance the smaller the damping, whereas in the series case the situation is reversed. Equation (\[general-QLE1\]) should be supplemented by $$\left[ \hat{A}^{\mathrm{in}}\left( t_{1}\right) ,\hat{A}^{\mathrm{in}}\left( t_{2}\right) \right] =\frac{i\hbar}{2}\frac{\partial}{\partial\left( t_{1}-t_{2}\right) }\delta\left( t_{1}-t_{2}\right)$$ and $$\hat{A}^{\mathrm{out}}\left( t\right) =\zeta\left[ \hat{A}^{\mathrm{in}}\left( t\right) -R^{\zeta/2}\frac{\mathrm{d}}{\mathrm{d}t}\hat{X}\right] .$$ It follows from the last three equations that the output fields have the same commutation relation as the input fields $$\left[ \hat{A}^{\mathrm{out}}\left( t_{1}\right) ,\hat{A}^{\mathrm{out}}\left( t_{2}\right) \right] =\frac{i\hbar}{2}\frac{\partial}{\partial\left( t_{1}-t_{2}\right) }\delta\left( t_{1}-t_{2}\right) .$$ Quantum Langevin equation in the RWA approximation {#quantum-langevin-equation-in-the-rwa-approximation .unnumbered} -------------------------------------------------- We now consider an approximate form of the input-output formalism which is valid when the system degree of freedom consists of an oscillator with very low damping, and for which all the frequencies of interest will lie in a narrow range around the oscillator frequency $\omega_{a}$. We start from Eq. (\[Langevin-example\]) and use $$\begin{aligned} \hat{\Phi} & =\Phi^{ZPF}\left( a+a^{\dag}\right) ,\\ \hat{Q} & =Q^{ZPF}\frac{\left( a-a^{\dag}\right) }{i},\end{aligned}$$ where $\Phi^{ZPF}=\sqrt{\hbar Z_{a}/2}$ and $Q^{ZPF}=\sqrt{\hbar/2Z_{a}}$. We then obtain, neglecting the effect of driving terms oscillating at twice the resonance frequency, $$\frac{\mathrm{d}}{\mathrm{d}t}a=\frac{i}{\hbar}\left[ H_{bare},a\right] -\omega_{a}\frac{Z_{a}}{2R}a+\sqrt{\frac{2Z_{a}}{\hbar R}}\tilde {A}^{\mathrm{in}}\left( t\right)$$ with $$\tilde{A}^{\mathrm{in}}(t)=\int_{0}^{\infty}\hat{A}^{\mathrm{in}}[\omega]e^{-i\omega t}\mathrm{d}\omega.$$ The field amplitude $\tilde{A}^{\mathrm{in}}(t)$ is non-hermitian and contains only the negative frequency component of $A^{\mathrm{in}}(t)$. For signals in a narrow band of frequencies around the resonance frequency, we can make the substitution $$\sqrt{\frac{2}{\hbar\omega_{a}}}\tilde{A}^{\mathrm{in}}\left( t\right) \rightarrow\tilde{a}^{\mathrm{in}}\left( t\right) ,$$ where $$\tilde{a}^{\mathrm{in}}(t)=\int_{0}^{\infty}a^{\mathrm{in}}[\omega]e^{-i\omega t}\mathrm{d}\omega.$$ The input field operator $a^{\mathrm{in}}[\omega]$ is identical to $a^{\leftarrow}[\omega]$ of the infinite line. We finally arrive at the RWA quantum Langevin equation, also referred to in the quantum optics literature as the quantum Langevin equation in the Markov approximation $$\frac{\mathrm{d}}{\mathrm{d}t}a=\frac{i}{\hbar}\left[ H_{bare},a\right] -\frac{\gamma_{a}}{2}a+\sqrt{\gamma_{a}}\tilde{a}^{\mathrm{in}}\left( t\right) ,$$ where $$\left[ \tilde{a}^{\mathrm{in}}\left( t\right) ,\tilde{a}^{\mathrm{in}}\left( t^{\prime}\right) ^{\dagger}\right] =\delta\left( t-t^{\prime }\right) .$$ For any oscillator, the input output relationship is obtained from $$\sqrt{\gamma_{a}}a\left( t\right) =\tilde{a}^{\mathrm{in}}\left( t\right) -\zeta\tilde{a}^{\mathrm{out}}\left( t\right) . \label{IOT}$$ It is worth noting that although $a^{\mathrm{in}}$ and $a^{\mathrm{out}}$ play the role of $a^{\leftarrow}$ and $a^{\rightarrow}$ in Eq. (\[bosonic\_com\]), only the average values of the moments of $a^{\mathrm{in}}$ can be imposed, $a^{\mathrm{out}}$ being a slave" of the dynamics of $a^{\mathrm{in}}$, as processed by the oscillator. [99]{} D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M. Gambetta, A. Blais, L. Frunzio, J. Majer, B. R. Johnson, M. H. Devoret, S. M. Girvin and R. J. Schoelkopf, Nature **445**, 515 (2007). M. H. Devoret, S. M. Girvin and R. J. Schoelkopf, Annalen der Physik **16**, 767 (2007). M. A. Castellanos-Beltran, K. D. Irwin, G. C. Hilton, L. R Vale, and K. W. Lehnert, Nature Phys. **4**, 928 (2008). N. Bergeal, F. Schackhert, M. Metcalfe, R. Vijay, V. E. Manucharyan, L. Frunzio, D.E. Prober, R. J. Schoelkopf, S. M. Girvin and M. H. Devoret, Nature **465**, 64 (2010). B. Yurke, P. G. Kaminsky, R. E. Miller, E. A. Whittaker, A. D. Smith, A. H. Silver and R. W. Simon, Phys. Rev. Lett. **60**, 764 (1988). B. Yurke, L. R. Corruccini, P. G. Kaminsky, L. W. Rupp, A. D. Smith, A. H. Silver, R. W. Simon, and E. A. Whittaker, Phys. Rev. A **39**, 2519 (1989). R. Movshovich, B. Yurke, P. G. Kaminsky, A. D. Smith, A. H. Silver, R. W. Simon, and M. V. Schneider, Phys. Rev. Lett. **65**, 1419 (1990). J. D. Teufel, T. Donner, M. A. Castellanos-Beltran, J. M. Harlow and K. W. Lehnert, Nature Nanotechnology **4**, 820 (2009). R. Vijay, D. H. Slichter, and I. Siddiqi, Phys. Rev. Lett. **106**, 110502 (2011). M. Hatridge, S. Shankar, M. Mirrahimi, F. Schackert, K. Geerlings, T. Brecht, K. Sliwa, B. Abdo, L. Frunzio, S. Girvin, R. Schoelkopf, M. Devoret, submitted. N. Bergeal, R. Vijay, V. E. Manucharyan, I. Siddiqi, R. J. Schoelkopf, S. M. Girvin and M. H. Devoret, Nature Physics **6**, 296 (2010). B. Abdo, F. Schackert, M. Hatridge, C. Rigetti and M. H. Devoret, Appl. Phys. Lett. **99**, 162506 (2011). J. D. Teufel, T. Donner, Li, Dale, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert and R. W. Simmonds, Nature **475**, 359 (2011). T. Yamamoto, K. Inomata, M. Watanabe, K. Matsuba, T. Miyazaki, W. D. Oliver, Y. Nakamura and J. S. Tsai, Appl. Phys. Lett. **93**, 042510 (2008). R. Vijay, M. H. Devoret and I. Siddiqi, Rev. Sci. Instrum. **80**, 111101 (2009). I. Siddiqi, R. Vijay, F. Pierre, C. M. Wilson, M. Metcalfe, C. Rigetti, L. Frunzio, and M. H. Devoret, Phys. Rev. Lett. **93**, 207002 (2004). M. Hatridge, R. Vijay, D. H. Slichter, J. Clarke and I. Siddiqi, Phys. Rev. B **83**, 134501 (2011). B. Abdo, E. Segev, O. Shtempluck, and E. Buks, Appl. Phys. Lett. **88**, 022508 (2006). E. A. Tholen, A. Ergul, E. M. Doherty, F. M. Weber, F. Gregis and D. B. Haviland, Appl. Phys. Lett. **90**, 253509 (2007). B. H. Eom, P. K. Day, H. G. LeDuc, J. Zmuidzinas, Nature Physics **8**, 623 (2012). Treatment of a linear dissipative environment by scattering formalism, usually employed in quantum optics. See appendix for details. D. M. Pozar, *Microwave Engineering*, (Wiley, 2005), Third edition. E. Flurin, N. Roch, F. Mallet, M. H. Devoret, B. Huard, arXiv:1204.0732v1. B. Abdo, K. Sliwa, F. Schackert, N. Bergeal, M. Hatridge, L. Frunzio and M. H. Devoret, manuscript in preparation. N. Bergeal, F. Schackert, L. Frunzio and M. H. Devoret, Phys. Rev. Lett. **108**, 123902 (2012). C. M. Caves, Phys. Rev. D **26**, 1817 (1982). B. Huard, N. Bergeal and M. H. Devoret, Proceedings of the International School of Physics Enrico Fermi CLXXI 151 (2009). R. E. Miller, W. H. Mallison, A. W. Kleinsasser, K. A. Delin and E. M. Macedo, Appl. Phys. Lett. **63**, 1423 (1993). R. Vijay, E. M. Levenson-Falk, D. H. Slichter, and I. Siddiqi, Appl. Phys. Lett. **96**, 223112 (2010). J. M. Manley and R. H. Rowe, Proceedings of the IRE 904 (1956). N. Roch, E. Flurin, F. Nguyen, P. Morfin, P. Campagne-Ibarcq, M. H. Devoret, B. Huard, Phys. Rev. Lett. **108**, 147701 (2012). J. Koch, T. M. Yu, J. Gambetta, A. Houck, D. Schuster, J. Majer, A. Blais, M. H. Devoret, S. M. Girvin and R. J. Schoelkopf, Phys. Rev. A **76**, 042319 (2007). V. E. Manucharyan, J. Koch, L. I. Glazman and M. H. Devoret, Science **326**, 113 (2009). A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S Huang, J. Majer, S. Kumar, S. M. Girvin, R. J. Schoelkopf, Nature **431**, 162 (2004). R. F. Bradley, Nucl. Phys. B (Proc. Suppl.) **72**, 137 (1999). C. W. Gardiner and P. Zoller, *Quantum Noise* (Springer, 2004), Third edition. Its value can also be obtained as the sum of the electrostatic and magnetic energy in the line $$\frac{1}{2}\int_{-\infty}^{+\infty}\left\{ C_{\ell}\left[ V\left( x,t\right) \right] ^{2}+L_{\ell}\left[ I\left( x,t\right) \right] ^{2}\right\} dx.$$

--- abstract: 'Phase matching has been studied for the Grover algorithm as a way of enhancing the efficiency of the quantum search. Recently Li and Li found that a particular form of phase matching yields, with a single Grover operation, a success probability greater than 25/27 for finding the equal-amplitude superposition of marked states when the fraction of the marked states stored in a database state is greater than 1/3. Although this single operation eliminates the oscillations of the success probability that occur with multiple Grover operations, the latter oscillations reappear with multiple iterations of Li and Li’s phase matching. In this paper we introduce a multi-phase matching subject to a certain matching rule by which we can obtain a multiple Grover operation that with only a few iterations yields a success probability that is almost constant and unity over a wide range of the fraction of marked items. As an example we show that a multi-phase operation with six iterations yields a success probability between 99.8% and 100% for a fraction of marked states of 1/10 or larger.' author: - 'F.M. Toyama' - 'W. van Dijk' - 'Y. Nogami' - 'M. Tabuchi' - 'Y. Kimura' title: 'Multi-phase matching in the Grover algorithm' --- Introduction {#sec:intro} ============ The quantum search algorithm introduced by Grover [@grover96; @grover97a; @grover97b; @grover01] constitutes a major advance in quantum computing. It enables us to find a marked state stored in a database state consisting of $N$ unordered basis states in only ${\cal O}(\sqrt{N})$ Grover operations. A number of modifications and generalizations of the original Grover search algorithm have been proposed [@grover98; @long99; @long01; @collins02; @tulsi06; @li07; @younes07]. In particular, phase matching methods in the Grover algorithm have been extensively examined [@long99; @long01; @li07]. The outcome of the search algorithm is characterized in terms of $P(\lambda)$, the probability of obtaining an equal-amplitude superposition of the marked states where $\lambda$ is the ratio of the marked states to all the states stored in the original database state. Recently, Li and Li [@li07] proposed a new phase matching for the Grover algorithm and they obtained an improved success probability $P(\lambda)$ over a wide range of the ratio $\lambda$. They introduced the set of the Grover operators (details are described in Eqs. (\[eq:1\]) and (\[eq:2\])): $U = I-(1-e^{i\alpha})\sum_{l=0}^{M-1}|t_l\rangle\langle t_l|$ and $V = Ie^{i\beta}+(1-e^{i\beta})|0^{\otimes n}\rangle \langle 0^{\otimes n}|$. The phase factor $e^{i\beta}$ in the first term of the operator $V$ was first introduced in Ref. [@li07]. In the new phase matching the number of phases is the same as the usual one but the form of the phase shift operator $V$ is different. Li and Li found the remarkable result that *a single Grover operation* of the new phase matching yields $P(\lambda)>25/27$ for $1/3\leq\lambda\leq 1$. This is significant in the sense that with *only one Grover operation* the efficiency of the Grover algorithm is substantially improved in the range of values of $\lambda$ where the efficiency of the original algorithm deteriorates. This phase matching has another interesting aspect that was not explicitly pointed out by Li and Li [@li07]. For a given values of $\lambda$ in the range $1/4\leq\lambda \leq 1$, one Grover operation with the phases $\alpha=-\beta=\arccos(1-1/2\lambda)$ yields exactly $P=1$. \[See Eq. (\[eq:12a\]) in the following.\] This results was obtained earlier by Chi and Kim [@chi97] who considered a modified Grover operator of arbitrary phase. The special case of $\lambda=1/2$ yields $\alpha=-\beta=\pm\pi/2$, which are the phases found in Ref. [@li07]. This aspect of the phase matching is also significant because it implies that one can always find the equal-amplitude superposition of the marked states by only one Grover operation when $\lambda$ is greater than 1/4 by tuning the phases $\alpha$ and $\beta$ appropriately for the given $\lambda$. Conditions for a success probability of unity have been studied by previous authors. See, for example, Refs. [@hoyer00; @long01a]. It should be pointed out, however, that the so-called new phase matching of Ref. [@li07] is equivalent to the original phase matching of Long *et al.* [@long99]. When the second operator is defined as $V'=e^{-i\beta}V$, it becomes the phase-matching operator of Long *et al.* The only difference between the two is that the overall state is multiplied by a phase factor and so the amplitudes of the components are different, but the probabilities are the same. Thus the remarkable result of Li and Li can also be seen to follow from the operator of Long *et al.* Analytically the formulation by Li and Li is somewhat more transparent and hence we use it throughout this paper, except in the Appendix where we explicitly show the equivalence of the two formulations by calculating the probability profile. Thus a number of aspects of the Grover algorithm with phase matching, already alluded to, are of particular interest and they form the objectives of this study. We focus on high success probabilities with as few iterations as possible in order to enhance the efficiency of the quantum search. We emphasize the following three objectives: (1) the elucidation of features of the phase-matched Grover operations with a small number of iterations that yield success probabilities $P(\lambda)$ close to one over a wide range of values of $\lambda$, (2) given a value of $\lambda$ the determination of the phase-matched Grover operator(s) that results in $P(\lambda)=1$ exactly, and (3) the elucidation of the features of the phase-matched Grover operators that allow us to obtain $P(\lambda)=1$ for very small values of $\lambda$. In this paper we explore the search algorithm with these objectives in mind using the advantages of a few multiple Grover operations with phase matching. It is well known that a multiple application of the original Grover operation gives rise to intensive oscillations of $P$ as a function of $\lambda$ and such oscillations deteriorate the efficiency of the algorithm. This undesirable feature remains even in the new phase matching of Li and Li, as we will illustrate. We show that if we introduce a *multi-phase* matching subject to a certain matching rule, we can obtain a multiple Grover operation that yields a success probability almost constant and unity over a wide range of $\lambda$, e.g., $0.1 \leq \lambda\leq 1$. This is also significant in the sense that when $\lambda$ is greater than a small minimum value we can always find the superposition of the marked states with high degree of certainty without (re)tuning the phases. In the next section we set up the algorithm of the multi-phase matching in the framework of the phase matching of Li and Li [@li07] and analyze the efficiency of the algorithm by considering a single matched phase and a two-stage multi-phase matching. We also obtain an exemplar of a good probability profile for a six-stage multi-phase matched operator. In Sec. \[k\_iterations\] we consider the success probability for small $\lambda$ by using the Grover operations with a phase other than $\pi$. We summarize our results in Sec. \[summary\]. Multi-phase matching in the framework of the new phase matching {#multiphase} =============================================================== The new phase matching in the Grover algorithm proposed by Li and Li [@li07] is defined with the two operators, $$\label{eq:1} U=I-(1-e^{i\alpha})\sum_{l=0}^{M-1}|t_l\rangle\langle t_l|$$ $$\label{eq:2} V=Ie^{i\beta}+(1-e^{i\beta})|0^{\otimes n}\rangle\langle 0^{\otimes n}|.$$ where $|0^{\otimes n}\rangle$ is the $n$-qubits initial state, $M$ is the number of target (marked) states stored in an unstructured database state, and the $|t_l\rangle$ denote the target or marked states. The database state is given as $|\phi\rangle = H^{\otimes n}|0^{\otimes n}\rangle$, where $H$ is the Walsh-Hadamard transformation. The state $|\phi\rangle$ is an equally-weighted superposition of the $N=2^n$ basis states, $|w_l\rangle, \ l = 0,\dots,N-1$. The fraction $\lambda$ of the target states is defined as $\lambda = M/N$. The $U$ and $V$ of Eqs. (\[eq:1\]) and (\[eq:2\]) are both unitary as was shown in Ref. [@li07]. With $\alpha=\beta=\pi$, $U$ and $V$ reduce to the Grover operators of the original algorithm. As we mentioned in Sec. \[sec:intro\], Li and Li showed explicitly that *a single Grover operation* of the new phase matching $(H^{\otimes n}VH^{\otimes n})UH^{\otimes n}|0^{\otimes n}\rangle$ with $\alpha = -\beta=\pi/2$ yields a success probability $P(\lambda)>25/27$ for $1/3\leq\lambda\leq 1$. We introduce a multi-phase matching within the framework of the new phase matching. We rewrite the database state $|\phi\rangle = H^{\otimes n}|0^{\otimes n}\rangle = N^{-1/2}\sum_{l=0}^{N-1} |\omega_l\rangle$ in terms of $\lambda$ as $$\begin{aligned} \label{eq:3} |\phi\rangle = \frac{1}{\sqrt{N}}\sum_{l=0}^{N-1}|\omega_l\rangle & = & \sqrt{\frac{N-M}{N}}|R\rangle + \sqrt{\frac{M}{N}}|T\rangle \nonumber \\ & = & \sqrt{1-\lambda}|R\rangle + \sqrt{\lambda} |T\rangle,\end{aligned}$$ where $$\label{eq:4} |R\rangle = \frac{1}{\sqrt{N-M}}\sum_{l=0}^{N-M-1}|r_l\rangle, \ \ |T\rangle = \frac{1}{\sqrt M}\sum_{l=0}^{M-1}|t_l\rangle.$$ The state $|T\rangle$ is the uniform superposition of the marked states and $|R\rangle$ is that of the remaining states $|r_l\rangle$. They are both normalized to unity and orthogonal to each other. In the following, for convenience, we work in the two-dimensional space defined by the basis $\{|R\rangle, |T\rangle\}$. The two-dimensional representations of $U$ and $H^{\otimes n}VH^{\otimes n} = Ie^{i\beta} + (1-e^{i\beta})|\phi\rangle\langle\phi|$ are $$\label{eq:5} U:\begin{pmatrix} 1 & 0 \\ 0 & e^{i\alpha} \end{pmatrix}, \ \ \ H^{\otimes n}VH^{\otimes n} : \begin{pmatrix} (1-e^{i\beta})(1-\lambda)+e^{i\beta} & (1-e^{i\beta}) \sqrt{\lambda(1-\lambda)} \\ (1-e^{i\beta})\sqrt{\lambda(1-\lambda)} & (1-e^{i\beta})\lambda + e^{i\beta} \end{pmatrix}.$$ We write the multiple Grover operation with the multiple phases $\alpha_j$ and $\beta_j$ $(j=1,\dots,k)$ as $$\label{eq:7} \begin{pmatrix} u_k \\ d_k \end{pmatrix} = G(\alpha_k,\beta_k)G(\alpha_{k-1},\beta_{k-1})\cdots G(\alpha_1,\beta_1) \begin{pmatrix} \sqrt{1-\lambda} \\ \sqrt{\lambda} \end{pmatrix},$$ where one Grover operation $G(\alpha_j,\beta_j)$ $(j=1,\dots,k)$ in this representation is $$\label{eq:8} G(\alpha_j,\beta_j)= \begin{pmatrix} (1-e^{i\beta_j})(1-\lambda) + e^{i\beta_j} & (e^{i\alpha_j} - e^{i(\alpha_j+\beta_j)})\sqrt{\lambda(1-\lambda)} \\ (1-e^{i\beta_j})\sqrt{\lambda(1-\lambda)} & (e^{i\alpha_j} - e^{i(\alpha_j+\beta_j)})\lambda + e^{i(\alpha_j+\beta_j)} \end{pmatrix}.$$ The success probability of finding the superposition of target states is given by $P_k(\lambda)\equiv|d_k|^2$. We now consider the one- and two-pair-phase cases before increasing the phase-matching to six different pairs of phases in order to obtain $P(\lambda)$ nearly equal to unity over a large range of values of $\lambda$. In other words, we discuss the $k=1$ and the $k=2$ cases in detail first, and then proceed to the numerical results of the $k=6$ case. Multi-phase matching with one pair of phases {#one} -------------------------------------------- When $k=1$, Eq. (\[eq:7\]) reduces to $$\label{eq:10} \begin{pmatrix}u_1 \\ d_1 \end{pmatrix} = G_1(\alpha,\beta) \begin{pmatrix}\sqrt{1-\lambda} \\ \sqrt{\lambda} \end{pmatrix}.$$ Since we first focus on cases of complete success $P=|d_1|^2 = 1 - |u_1|^2 = 1$, we can equivalently consider the condition $u_1 = 0$. In general $$\label{eq:11} u_1 = \sqrt{1-\lambda} \, [1-\lambda+e^{i\beta}\lambda + (e^{i\alpha} -e^{i(\alpha+\beta)})\lambda \, ].$$ The condition that $u_1$ be zero leads to $$\begin{aligned} \label{eq:12} \frac{1}{\lambda} & = & 1 -\cos\alpha -\cos\beta - \cos(\alpha+\beta) \nonumber \\ && \ \ + i[\sin{(\alpha+\beta)}-\sin\alpha-\sin\beta].\end{aligned}$$ The fact that $\lambda$ must be real implies that (1) $\beta = -\alpha$, (2) either $\alpha$ or $\beta$ are zero, or (3) both $\alpha$ and $\beta$ are zero. When $\beta =0$ in the operator $V$ of Eq. (\[eq:2\]), the operator is the identity and the overall effect of operator $U$ of Eq. (\[eq:1\]) by itself would cause the phase of the marked states to be changed, but the probabilities of marked and unmarked states would remain the same. When $\alpha=0$, then $U=I$ and $G=H^{\otimes n} VH^{\otimes n}$. The initial state $|\phi\rangle$ is an eigenvector of $G$ with eigenvalue 1. Thus $G$ does not cause any evolution in $|\phi\rangle$. The success probability is $P=\lambda$, which is the success probability of the classical algorithm. As no quantum improvement to the search algorithm is achieved, we eliminate the case of $\alpha=0$ and any nonzero $\beta$ from the solutions of Eq. (\[eq:12\]). Thus only the solution $\beta=-\alpha$ is meaningful, and yields, as mentioned in Sec. \[sec:intro\] and in Ref. [@chi97], $P=1$ when $$\label{eq:12a} \alpha = -\beta = \arccos(1-1/2\lambda).$$ Since $\lambda$ lies between zero and one, the range of $\alpha$ is $\pi/3 \leq \alpha \leq \pi$. The boundary point of this range $\alpha=\pi/3$ occurs when $P(\lambda=1)=1$, and similarly $\alpha=\pi$ when $P(\lambda=1/4)=1$. We can express $P(\lambda)$ as a function of $\lambda$ depending on the parameter $\alpha$, $$\begin{aligned} \label{eq:13} P(\lambda) & = & 1 - |u_1|^2 \nonumber \\ & = & \lambda \, [5-4 \, \cos\alpha - 4 \, (1-\cos\alpha) \, (2-\cos\alpha) \, \lambda \nonumber \\ && \ \ \ \ +4 \, (1-\cos\alpha)^2\lambda^2].\end{aligned}$$ For $\alpha = \pi/2$ the equation reduces to Eq. (14) of Li and Li [@li07] . Since Eq. (\[eq:13\]) is cubic in $\lambda$ we expect a local maximum and a local minimum in the range $0<\lambda\leq 1$ at $\lambda_\mathrm{max}$ and $\lambda_\mathrm{min}$ respectively, where $$\label{eq:14} \lambda_\mathrm{max}=\frac{1}{2 \, (1-\cos\alpha)}, \ \ \lambda_\mathrm{min} = \frac{5-4\cos\alpha}{6 \, (1-\cos\alpha)}.$$ Furthermore the extrema are $$\label{eq:15} P(\lambda_\mathrm{max}) = 1, \ \ P(\lambda_\mathrm{min}) = \frac{(1+\cos\alpha)(5-4\cos\alpha)^2}{27 \, (1-\cos\alpha)}.$$ We illustrate different cases in Fig. \[fig:1\]. It is evident that the $\alpha=\pi/2$ case, which is the one used by Li and Li [@li07], gives the optimal profile for the success probability. Optimal here could be defined as the largest average $P$ over the range of $\lambda$, or the largest range of $\lambda$ over which $P\geq 25/27$. Multi-phase matching with two pairs of phases {#two} --------------------------------------------- We now consider Eq. (\[eq:7\]) for $k=2$ and we again concentrate on the upper component of the vector $(u_2,d_2)^T$. The general expression for it is too lengthy to give here, but again we demand that for an arbitrary value of $\lambda$ the imaginary part is zero to obtain the matching relationship for the phases. Apart from a factor of $\sqrt{1-\lambda}$ the expression of $\mathrm{Im} \, u_2$ contains a term in $\lambda$ and another in $\lambda^2$. Demanding that the coefficients of each power of $\lambda$ vanishes gives us two equations involving $\alpha_1$, $\alpha_2$, $\beta_1$, and $\beta_2$. Solving for $\beta_1$ and $\beta_2$ in terms of $\alpha_1$ and $\alpha_2$ we obtain the following four solutions: $$\label{eq:16} \left. \begin{array}{l} \{\beta_1=-\alpha_1, \ \beta_2 = 0\} \nonumber \\ \{\beta_1=-\alpha_2, \ \beta_2 = -\alpha_1\} \nonumber \\ \{\beta_1 = \beta_2 = 0\} \nonumber \\ \{\beta_1 = 0, \ \beta_2 = -(\alpha_1 + \alpha_2)\}. \end{array} \right.$$ Since one of $\beta_1$ and $\beta_2$ is zero for the first and last solution, the operation is then reduced to one iteration, and for the third solution the two iterations would not change the probabilities of the marked and unmarked states. Thus the only solution that gives new information is the one where $\beta_1=-\alpha_2$ and $\beta_2=-\alpha_1$. (The fact that $\mathrm{Im} \, u_2 = 0$ is a necessary, but not a sufficient, condition for this solution.) After obtaining the matched phases for which $\mathrm{Im} \, u_2 =0$, we set $\mathrm{Re} \, u_2 = 0$ to solve for the values of $\lambda$ which gives $P=1$. The expression for $u_2$ is then real and can be written as $$\begin{aligned} \label{eq:17} u_2 & = & \{1+ 2\, [(1-\cos\alpha_1)(-2+\cos\alpha_2) -\sin\alpha_1\sin\alpha_2]\, \lambda \nonumber \\ & & + 4\, (1-\cos\alpha_1) (1-\cos\alpha_2) \, \lambda^2 \} \sqrt{1-\lambda}.\end{aligned}$$ The factor multiplying $\sqrt{1-\lambda}$ is quadratic in $\lambda$ and hence it can vanish for two values of $\lambda$. Thus we can ask ourselves the questions, suppose two values of $\lambda$ between zero and one are given at which $P(\lambda)=1$, what are the corresponding values of $\alpha$ and what limits are there on the possible values of $\lambda$ that satisfy $P(\lambda)=1$? If $\lambda_1$ and $\lambda_2$ are the roots of the equation $$\label{eq:17a} u_2(\lambda)/\sqrt{1-\lambda}=0,$$ then $\cos\alpha_1$ and $\cos\alpha_2$ satisfy the equations $$\begin{aligned} \label{eq:18} && 8\lambda_1\lambda_2\cos^3\alpha_2 + [4(\lambda_1+\lambda_2) (1-\lambda_1-\lambda_2) - 8\lambda_1\lambda_2]\cos^2\alpha_2 + [8(\lambda_1+\lambda_2)^2 - 12(\lambda_1+\lambda_2)-8\lambda_1\lambda_2 + 4]\cos\alpha_2 \nonumber \\ && \hspace{1.7in} - 4(\lambda_1+\lambda_2)^2 + 8(\lambda_1+\lambda_2) -5 + 8\lambda_1\lambda_2 = 0,\end{aligned}$$ $$\label{eq:19} \cos\alpha_1 = 1 - \frac{1}{4(1-\cos\alpha_2)\lambda_1\lambda_2}.$$ In order to have a sense of the values of $\alpha_1$ and $\alpha_2$ that are valid, we have minimally the condition that the discriminant of Eq. (\[eq:17a\]) (quadratic in $\lambda$) should be nonnegative to avoid complex values of $\lambda$. In Fig. [\[fig:3a\]]{} we plot the discriminant as a surface $z= D(\alpha_1,\alpha_2)$; the intersection of the surface with the $xy$ plane gives the boundary of the non-allowed $\alpha_1$ and $\alpha_2$ values. Given $\lambda_1$ and $\lambda_2$ one can solve Eq. (\[eq:18\]) for $\cos\alpha_2$ and using it we obtain $\cos\alpha_1$ from the second equation. Only those solutions that yield real angles $\alpha_1$ and $\alpha_2$ are meaningful for the unitary operators. The minimum value of $\lambda$ for which $P=1$ occurs when $\alpha_1=\alpha_2 = \pi$. In that case $\lambda= (3-\sqrt{5})/8 =0.09549$. It can be shown that varying $\alpha_1$ or $\alpha_2$ by a small amount away from $\pi$ always leads to an increase in the $\lambda$ which corresponds to the smaller of the two values of $\lambda$. When we let $\alpha_{1,2}=\pi+\epsilon_{1,2}$ we obtain a change in the smaller $\lambda$ of $$\label{eq:20} \Delta\lambda = \frac{1}{160}\left[\left(2\sqrt{5}\epsilon_1 +\frac{5-3\sqrt{5}}{\sqrt{2\sqrt{5}}}\epsilon_2\right)^2+(22-8\sqrt{5}) \epsilon_2^2\right],$$ which is positive regardless of the signs of $\epsilon_{1,2}$. The larger $\lambda$ can increase or decrease with changes in the phases(s). We obtain a particular example using the procedure described above. We search through combinations of $\lambda_{1,2}$ and find that $\lambda_1 = 2/5$ and $\lambda_2=4/5$ give good results. In this case $\alpha_1=1.00889485$ and $\alpha_2=2.30794928$. We find local minima of $P(\lambda)$ at $\lambda = 0.5767$ and $\lambda = 0.9433$ at which $P = 0.9936$ and 0.9966, respectively. The corresponding graph of the success probability as a function of $\lambda$ obtained with the two-stage multi-phase operator is shown in Fig. \[fig:3\] and compared with double iterations of the Grover operation and that of Li and Li [@li07]. It would be interesting to examine a classical counterpart of $P(\lambda)$. The probability of failing to find one of $M$ marked objects out of $N$ objects is $(N-M)/N = 1-\lambda$. The probability of failing twice in a row is $$(1-\lambda)\left(\frac{N-1-M}{N-1}\right)=(1-\lambda) \left(1-\frac{\lambda}{1-1/N}\right).$$ The probability of failing $k$ times in a row is $$\begin{aligned} && (1-\lambda)\left(1-\frac{\lambda}{1-1/N}\right)\cdots \left(1-\frac{\lambda}{1-(k-1)/N}\right) \nonumber \\ &&=\prod_{n=1}^k \left[1-\lambda\left(1-\frac{n-1}{N}\right)^{-1}\right]. \nonumber\end{aligned}$$ Thus the probability of finding at least one of the $M$ items in $k$ successive attempts is $$\label{eq:class} P_\mathrm{classical}(\lambda) = 1 - \prod_{n=1}^k\left[1-\lambda\left(1-\frac{n-1}{N}\right)^{-1}\right].$$ If $k\ll N$, this probability is approximately $P_\mathrm{classical}(\lambda)\approx 1-(1-\lambda)^k$, which we interpret as the classical counterpart of $P(\lambda)$. This probability with $k=2$ is also plotted in Fig. \[fig:3\]. Multi-phase matching with six pairs of phases {#six} --------------------------------------------- We show that if we match the multi-phase $\alpha_j$ and $\beta_j$ $(j=1,\dots,k)$ with $k=6$ in accordance with a certain matching rule (best fit), we can obtain a multiple Grover operation that yields $P(\lambda)\approx 1$ in a wide range of $\lambda$. We found this best solution for six Grover iterations by a nonlinear fitting to the ideal probability curve $P(\lambda)=1$ for $0<\lambda\leq 1$. The phases $\alpha_j$ and $\beta_j$ found in this way are given in the left side of Table \[table:1\]. $~~j~~$ $\beta_j$ $\lambda_j^{(P(\lambda_j)=1)}$ $P(\lambda_j^{(\mathrm{local~min})})$ --------- ------------- --------------- -------------------------------- --------------------------------------- 1 1.20560132   $-\alpha_6$  0.10777 0.9980 2 1.29806396 $-\alpha_5$ 0.23793 0.9993 3 1.31701508 $-\alpha_4$ 0.41889 0.9996 4 1.33356767 $-\alpha_3$ 0.62393 0.9997 5 0.47289426 $-\alpha_2$ 0.81366 0.9997 6 1.66668634 $-\alpha_1$ 0.94483 0.9995 : Phase parameters for the six-parameter multi-phase matching, and results for local maxima ($P(\lambda)=1$) and local minima of $P(\lambda)$.[]{data-label="table:1"} It is remarkable that $\alpha_j$ and $\beta_j$ are matched to each other such that $\alpha_j=-\beta_{6-j+1}$. The signs of $\alpha_j$ and $\beta_j$ are opposite to each other, which is consistent with the case of the new phase matching of Ref. [@li07], i.e., the $k=1$ case with $\alpha_1=-\beta_1 = \pi/2$. The matching rule $\alpha_j = -\beta_{k-j+1}$ between the multi-phases $\alpha_j$ and $\beta_j$ holds for any $k$ in the best solution obtained by the nonlinear fitting to the ideal probability curve $P(\lambda)=1$ for $0 < \lambda \leq 1$, although we omit to show cases other than those for which $k=1$, 2, and 6. Fig. \[fig:4\] shows the success probabilities obtained by six Grover operations with the multi-phase matching of Eq. (\[eq:7\]). The inset of Fig. \[fig:4\] shows that there are six values of $\lambda_i$ at which $P(\lambda_i)=1$ exactly. They are given in the right side of Table \[table:1\] along with local minimum values of the function $P(\lambda)$ which occur between 0.1 and 1. We studied the $k=5$ case in the same way and obtained a graph similar to Fig. \[fig:4\] with $P(\lambda)=1$ for *five* values of $\lambda$ other than unity. The local minima of $P(\lambda)$ are lower and the minimum value of $\lambda$ for which $P(\lambda)=1$ is slightly larger than in the $k=6$ case. The matching rule $\alpha_j=-\beta_{k-j+1}$ is also satisfied for the $k=5$ case as it was for $k=1$, 2, and 6 cases. We are confident that for any $k>1$ this matching rule for the best fit holds so that in general one finds $k$ values of $\lambda$ for which $P(\lambda)=1$ and the smallest $\lambda$ for which $P(\lambda)=1$ decreases as $k$ increases. Returning to the six-stage multiple phase operation, we define $P_j(\lambda)$ $(j=1,\dots,6)$ as the success probability curves after $j$ steps of the six-stage multi-phase operation. As seen in the Figs. \[fig:4\] and \[fig:5\], $P_6(\lambda)\approx 1$ is achieved for $0.1\leq\lambda\leq 1$ in the sixth Grover operation. This is significant in the sense that if $\lambda$ is greater than 0.1, we can always find the superposition of marked states by just six Grover operations. In contrast to the shape of the curve for $P_6(\lambda)$, Fig. \[fig:5\] shows that each $P_j(\lambda)$ for $j=1,\dots,5$ depends strongly on $\lambda$ and is far from the desired success probability $P_6(\lambda)$. The curves do not monotonically approach the desired success probability $P_6(\lambda)$ when $\lambda > 0.05$. In particular, $P_5(\lambda)$ is quite different from the desired probability $P_6(\lambda)$. However, in the final (sixth) step the desired probability $P_6(\lambda)$ is obtained. This is in contrast to the fixed-point iteration schemes studied in Refs. [@grover05; @tulsi06; @younes07]. Figure \[fig:6\] shows the success probabilities obtained by six Grover operations with the single phase matching with $\alpha_j=-\beta_j=\pi/2$ $(j=1,\dots,6)$ , where we showed only $P_1(\lambda)$, $P_3(\lambda)$ and $P_6(\lambda)$. The $P_1(\lambda)$ is the success probability of the new phase matching obtained by Li and Li [@li07]. As stressed in Ref. [@li07], the success probability is substantially improved in $\lambda>1/3$ by a single Grover operation, compared with that of the original Grover algorithm indicated by the yellow line, where the probability is plotted for optimal iteration times indicated by $k$. However, $P_3(\lambda)$ and $P_6(\lambda)$ obtained by multiple Grover operations with the single phase matching show intensive oscillations with $\lambda$. As we have shown, such undesirable oscillations can be eliminated by the multi-phase matching subject to the matching rule $\alpha_j=-\beta_{6-j+1}$. Here we should note that the nonlinear fitting is not unique. The phases $\alpha_j$ and $\beta_j$ given in Table \[table:1\] were obtained by minimizing the function $\sum_i\chi_i^2$ where the $\chi_i$ are the differences at $\lambda=\lambda_i$ of the ideal probability $P(\lambda)=1$ and the probability function $P_6(\lambda)$. If we take, for example, a function such as $\sum_i|\chi_i|$ we obtain another solution. Although this solution gives almost the same $P_6(\lambda)$, the probability curve is shifted slightly toward larger values of $\lambda$, so that the local extrema are also slightly moved to the right. Since we emphasize obtaining $P(\lambda)\approx 1$ over as wide a range of $\lambda$ as possible we adopted the solution that uses the $\chi^2_i$ for the fitting. Iteration of Grover’s operation with phase other than $\pi$ {#k_iterations} =========================================================== In this section we consider the repeated application of Grover’s original operation generalized to have a phase other than $\pi$. We focus in particular on cases with small $\lambda$ for which the success probability with the multi-phase matching is small, and determine the conditions that yield success probabilities close to unity. Consider a single Grover operation with matched phase, Eq. (\[eq:8\]), but with $\beta=-\alpha$, $$\label{eq:a1} G_1 = \begin{pmatrix} (1-e^{-i\alpha})(1-\lambda)+ e^{-i\alpha} & (e^{i\alpha}-1)\sqrt{\lambda(1-\lambda)} \\ (1-e^{-i\alpha})\sqrt{\lambda(1-\lambda)} & (e^{i\alpha}-1)\lambda+1 \end{pmatrix}.$$ Note that $\det G_1 = 1$. We obtain eigenvalues $\sigma$ of the matrix $G_1$ by solving $$\label{eq:a2} f(\sigma)=\det(G_1-\sigma I) = 0.$$ The characteristic function $f(\sigma)$ is $$\label{eq:a3} f(\sigma) = \sigma^2 + 2[-1+(1-\cos\alpha)\lambda]\sigma + 1.$$ The equation $f(\sigma)=0$ yields solutions $$\label{eq:a4} \sigma = 1 - (1-\cos\alpha)\lambda \pm i\sqrt{(1-\cos\alpha)\lambda [2-(1-\cos\alpha)\lambda]}$$ We define $x$ as $$\label{eq:a4a} x =(1-\cos\alpha)\lambda,$$ so that the eigenvalues can be written as $$\label{eq:a5} \sigma = e^{\pm i\phi}, \ \ \phi = \arctan\left(\frac{\sqrt{x(2-x)}}{1-x}\right).$$ We choose the definition of the arc tangent so that as $x$ varies from 0 to 2, $\phi$ goes from 0 to $\pi$. We can rewrite the function $f(\sigma)$ as $$\label{eq:a6} f(\sigma) = \sigma^2 - 2\sigma\cos\phi +1.$$ By the Cayley-Hamilton theorem [@pipes58 page 91] $f(G_1) =0$, so that we obtain the identity $$\label{eq:a7} G_1^2 = 2G_1\cos\phi -1.$$ This means that $G_1$ iterated any number of times can be written as a linear expression of $G_1$. In fact for $k$ iterations it can be shown by induction [@sprung93] that $$\label{eq:a8} G_1^k = \frac{1}{\sin\phi}\left[G_1\sin(k\phi) - \sin((k-1)\phi)\right].$$ Consider now the $k$ iterations of the Grover operation, so that $$\label{eq:a9} \begin{pmatrix} u_k \\ d_k \end{pmatrix} = G_1^k \begin{pmatrix} \sqrt{1-\lambda} \\ \sqrt{\lambda} \end{pmatrix}.$$ This yields $$\label{eq:a10} \begin{pmatrix} u_k \\ d_k \end{pmatrix} = \frac{1}{\sin\phi} \left[ \sin{(k\phi)} \begin{pmatrix} (1-e^{-i\alpha})(1-\lambda)+ e^{-i\alpha} & (e^{i\alpha}-1)\sqrt{\lambda(1-\lambda)} \\ (1-e^{-i\alpha})\sqrt{\lambda(1-\lambda)} & (e^{i\alpha}-1)\lambda+1 \end{pmatrix} - \sin{((k-1)\phi)} \ I \right] \begin{pmatrix} \sqrt{1-\lambda} \\ \sqrt{\lambda} \end{pmatrix}.$$ Thus the expression for $u_k$ is $$\label{eq:a11} u_k = \frac{\sqrt{1-\lambda}}{\sin\phi}\left\{\sin{(k\phi)}(1-2x) -\sin{((k-1)\phi)} \right\}.$$ We require $u_k=0$ so that $P=1$. A trivial solution is $\lambda=1$. We also note that $\phi=0$ yields $x=-1/(2k)$. Since $x$ must be positive $\sin{\phi}\neq 0$. Thus we need to solve only $$\label{eq:a12} \sin{(k\phi)}(1-2x)-\sin{((k-1)\phi)} = 0.$$ The solutions are values of $x=(1-\cos{\alpha})\lambda$ for which $P=1$. Thus we have $P=1$ for combinations of $\alpha$ and $\lambda$. For instance, when $\alpha=\pi$, then $\lambda = x/2$. In Fig. \[fig:ap1\], we display the $P(\lambda)$ curves for six ($k=6$) iterations when $\alpha$ has different values. For large $k$ we can estimate the smallest value of $\lambda$ for which $P$ is unity. We rewrite Eq. (\[eq:a12\]) so that $$\label{eq:a13} \tan(k\phi) = \frac{\sin\phi}{\cos\phi-1+2x}.$$ The value of $x$ which is the solution occurs for the $x$ coordinate of the point of intersection of the curves represented by the left side and the right side of Eq. (\[eq:a13\]). The curve on the right is a smoothly decreasing positive function starting at infinity when $x=0$ and asymptotically approaching the positive $x$ axis. The curve on the left starts at zero and increases to positive infinity when $k\phi= \pi/2$. When $k$ is large this occurs for small values of $\phi$ or small values of $x$. Thus using the condition $\phi\approx \pi/(2k)$, we obtain $$\label{eq:a14} \arctan\frac{\sqrt{x(2-x)}}{1-x} {\mathrm{\raisebox{-0.6ex}{$\stackrel{\textstyle <}{\sim}$}}}\frac{\pi}{2k} \ \ \mathrm{or} \ \ \frac{\sqrt{x(2-x)}}{1-x} {\mathrm{\raisebox{-0.6ex}{$\stackrel{\textstyle <}{\sim}$}}}\frac{\pi}{2k}.$$ This leads to the approximation of the smallest value of $x$ for which $P$ is one as $x_\mathrm{min} {\mathrm{\raisebox{-0.6ex}{$\stackrel{\textstyle <}{\sim}$}}}\pi^2/(8k^2)$; for $\alpha = \pi$ (the Grover case) $\lambda_\mathrm{min}= x_\mathrm{min}/2$. This approximation leads to $\lambda_\mathrm{min} = 0.017$ for $k=6$ and $\alpha=\pi$, whereas the exact solution of Eq. (\[eq:a13\]) gives $0.014$. As $k$ gets larger the approximation improves further. The Grover algorithm as a special case -------------------------------------- We recover the Grover algorithm starting with Eq. (\[eq:a11\]) and setting $\alpha=\pi$ or $x=2\lambda$. Then it follows from Eq. (\[eq:a5\]) that $\sin\phi = \sqrt{4\lambda(1-\lambda)}$ and $\cos\phi = (1-2\lambda)$. The $u_k$ of Eq. (\[eq:a11\]) can be reduced to $$\label{eq:a15} u_k=-\sqrt{\lambda}\sin(k\phi) + \sqrt{1-\lambda}\cos(k\phi).$$ Define $\sin\theta = \sqrt{\lambda}$. Then $$\label{eq:a16} u_k=\cos(k\phi+\theta).$$ We can show that $\phi = 2\theta$, so that $$\label{eq:a17} u_k = \cos[(2k+1)\theta] = \cos[(2k+1)\arcsin(\sqrt{\lambda})].$$ This is Eq. (6) of Ref. [@li07]. Furthermore $$\label{eq:a18} P=1-u_k^2 = \sin^2[(2k+1)\arcsin(\sqrt{\lambda})].$$ Thus each iteration effectively rotates the state through an angle of $\theta/2 = \arcsin(\sqrt{\lambda})/2$. We can use this to estimate the number of iterations that are required to obtain $P(\lambda) =1$. That occurs when the argument of the sine function in Eq. (\[eq:a18\]) is $\pi/2$, i.e., $$\label{eq:a19} k=\frac{1}{2}\left(\frac{\pi}{2\theta}-1\right).$$ For small $\theta$ (or large $k$) $$\label{eq:a20} k \approx\frac{\pi}{4\theta} \approx \left[\frac{\pi}{4\theta}\right] = \mathrm{~integer~value~of~} \frac{\pi}{4\theta} \approx \left[ \frac{\pi}{4}\frac{1}{\sqrt{\lambda}}\right].$$ Thus after approximately $\pi/(4\sqrt{\lambda})$ iterations one has certainty of having found the superposition of marked states. Classically the number of search operations to have this certainty is on the average approximately $1/(2\lambda)=N/(2M)$ for $N$ much larger than $M$. By the same reasoning we find $P=0$ with twice as many quantum iterations. Thus by continuing to iterate indefinitely we can end with any probability of success. However, if we iterate close to the number that gives 100% probability of success we have a good approximation to a successful search. Effect of phase $\mathbf{\alpha}$ {#general_phase} --------------------------------- For a general value of $x=(1-\cos\alpha)\lambda$, Eq. (\[eq:a11\]) can be written as $$\label{eq:a21} u_k = A\cos(k\phi+\theta),$$ where $$\label{eq:a22} A=\sqrt{\frac{2(1-\lambda)}{2-x}}, \ \ \ \theta = \arctan\sqrt{\frac{x}{2-x}}.$$ Since $P(\lambda) = 1 - u_k^2 = 1 - A^2\cos^2(k\phi+\theta)$, the minimum of $P(\lambda)$ is $$\label{eq:a23} P_\mathrm{min}(\lambda) = 1 - A^2 = \frac{\lambda(1+\cos\alpha)} {2-(1-\cos\alpha)\lambda}.$$ In Fig. \[fig:ap2\] it is seen that $P_\mathrm{min}(\lambda)$ has at most a linear rise as $\lambda$ increases from zero to one. Summary ======= We have proposed a multi-phase matching for the Grover search algorithm, which is an extension of the new phase matching proposed in Ref. [@li07]. The multi-phase matching is characterized by multiple Grover operations with two kinds of multi-phases $\alpha_j$ and $\beta_j$ $(j=1,\dots,k)$. We showed that if we match $\alpha_j$ and $\beta_j$ in accordance with the rule $\alpha_j = -\beta_{k-j+1}$ for a given $k$ we can obtain an optimal solution for $\alpha_j$, $\beta_j$ that gives a success probability curve such that it is almost constant and unity in a wide range of the fraction of marked states. As an example we presented an optimal solution obtained for $k=6$. The solution yields the desired success probability $P = 1$ to within 0.2% for the fraction of the marked states greater than 0.1. This is significant in the sense that when the fraction of marked states is greater than 0.1, we can always with a high degree of confidence find a uniform superposition of the marked states by repeating the Grover operation just six times. To clarify the mechanism of the multi-phase matching we studied in detail the one- and two-iteration cases. We showed that it is possible to obtain $P=1$ exactly for a particular fraction $\lambda$ by tuning the phases. This can be generalized to having $k$ values of $\lambda$ for which $P(\lambda)=1$ when we go to a $k$-iteration scheme. One can obtain $P=1$ for a given very small $\lambda$ by using the original Grover algorithm or the phase-matched version of it. In this case usually a specified large number of iterations is required. Further study is needed to obtain an efficient algorithm for extremely small $\lambda$. This work was supported by the Japan Society for the Promotion of Sciences and the Natural Sciences and Engineering Research Council of Canada. Equivalence of two phase-matching schemes ========================================= Li and Li [@li07] claim to have generalized the Long phase-matching algorithm in order to produce a higher success probability. In actual fact the phase matching of Li and Li and that of Long *et al.* [@long99] result in the same success probability. We show that in the following. Instead of Eqs. (\[eq:1\]) and (\[eq:2\]), Long *et al.* work with the operators $$\label{eq:aa1} U=I-(1-e^{i\theta})\sum_{l=0}^{M-1}|t_l\rangle\langle t_l|$$ $$\label{eq:aa2} V=I-(1-e^{i\phi})|0^{\otimes n}\rangle\langle 0^{\otimes n}|.$$ These unitary transformations lead to the Grover operator (in the notation of this paper) $G(\theta,\phi)$, where $$\label{eq:aa3} G = \begin{pmatrix} 1-(1-e^{i\phi})(1-\lambda) & -(1 - e^{i\phi})e^{i\theta}\sqrt{\lambda(1-\lambda)} \\ -(1-e^{i\phi})\sqrt{\lambda(1-\lambda)} & [1-(1 - e^{i\phi})\lambda]e^{i\theta} \end{pmatrix}.$$ For one operation we calculate the final state $$\label{eq:aa4} \begin{pmatrix}u \\ d \end{pmatrix} = G(\theta,\phi) \begin{pmatrix}\sqrt{1-\lambda} \\ \sqrt{\lambda} \end{pmatrix}$$ with $$\label{eq:aa5} u=\sqrt{1-\lambda}[1-(1-e^{i\phi})(1-\lambda) - (1-e^{i\phi}) e^{i\theta}\lambda].$$ Setting $u=0$ we obtain (in addition to $\lambda=1$) the solution $$\label{eq:aa6} \phi=\theta, \ \ \ \lambda= \frac{1}{2}\frac{\cos\theta +1}{\sin^2\theta}.$$ Note that the signs of $\phi$ and $\theta$ are the same, unlike the opposite signs of the matched phases of Li and Li, i.e., $\beta=-\alpha$. In order that $0 < \lambda\leq 1$ with this phase matching, $\theta$ varies from $\pi/3$ to $\pi$. For $P(\lambda) = 1-|u|^2$, we obtain the expression of Eq. (\[eq:13\]) with $\alpha$ replaced by $\theta$. Thus the impressive result by Li and Li of a single phase-matched Grover operation can also be obtained with the earlier-proposed operation of Long *et al.* However, the formulation of Li and Li results in $\mathrm{Im} \, u = 0$ when $\beta = -\alpha$, whereas $\mathrm{Im} \, u \neq 0$ when $\phi=\theta$ for the operator of Long *et al.* It should be noted however that the remarkable single-operation result was first reported by Li and Li [@li07]. Although the probabilities are the same the amplitudes are not, and Li and Li’s formulation gives a more straightforward derivation of the probabilities. (See Sec. IIA.) One can relate the two formulations by suggesting that instead of the operator acting on $(\sqrt{1-\lambda},\sqrt{\lambda})^T$ initially, in the case of Long *et al.* it operates on this state multiplied by a phase factor. [17]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , in ** (, , ), pp. . , ****, (). , ****, (). , ****, (). , ****, (). , , , , ****, (). , , , , , , , , , , , ****, (). , ****, (). , , , ****, (). , ****, (). , (). , (). , ****, (). , ****, (). , ****, (). , ** (, , ), ed. , , , ****, ().

--- abstract: 'In this paper, we study the entanglement entropy of a single interval on a cylinder in two-dimensional $T\overline{T}$-deformed conformal field theory. For such case, the (Rényi) entanglement entropy takes a universal form in a CFT. We compute the correction due to the deformation up to the leading order of the deformation parameter in the framework of the conformal perturbation theory. We find that the correction to the entanglement entropy is nonvanishing in the finite temperature case, while it is vanishing in the finite size case. For the deformed holographic large $c$ CFT, which is proposed to be dual to a AdS$_3$ gravity in a finite region, we find the agreement with the holographic entanglement entropy via the Ryu-Takayanagi formula. Moreover, we compute the leading order correction to the Rényi entropy, and discuss its holographic picture as well.' author: - 'Bin Chen$^{1,2,3}$, Lin Chen$^{1}$ and Peng-xiang Hao$^{1}$[^1]' title: '**Entanglement Entropy in $T\overline{T}$-Deformed CFT**' --- *$^{1}$Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, No.5 Yiheyuan Rd, Beijing 100871, P.R. China*\ *$^{1}$Center for High Energy Physics, Peking University, No.5 Yiheyuan Rd, Beijing 100871, P. R. China*\ *$^{3}$Collaborative Innovation Center of Quantum Matter, No.5 Yiheyuan Rd, Beijing 100871, P. R. China*\ Introduction ============ The integrable quantum field theory allows us to understand the non-perturbative aspects of the quantum field theory. In a remarkable paper by Zamolodchikov[@ttcft], the operator $T_{zz}T_{\bar z \bar z}-T^2_{z\bar z}$ of a two-dimensional(2D) quantum field theory(QFT) was studied and its expectation value had been shown to have an analytic form. Such deformation is now called $T\overline{T}$-deformation. The $T\overline{T}$ deformation has some interesting properties, as shown in various studies including the spectrum and the S-matrix[@ttcft2]. In particular, an integrable QFT deformed by such operator was found to be still integrable [@ttcft2; @ttcft3]. In [@ttcft3], it was shown that the deformation of the theory of 24 free scalars leads to the Nambu-Goto action. In [@cardytt], Cardy explained the solvability of the deformation by considering it as a stochastic process. For other studies on the $T\overline{T}$ deformation of a field theory, see [@ttatlargec; @ttinclosedform; @ttpartition]. The $T\overline{T}$ deformation of a 2D conformal field theory(CFT) is of particular interest. To be more precise the $T\overline{T}$ deformed CFT form a one-parameter family of the theories $\mathcal{T}^{(\mu)}$ parametrized by $\mu\geq 0$. The original CFT sits on $\mu=0$. Moving infinitesimally from $\mathcal{T}^{(\mu)}$ to $\mathcal{T}^{(\mu+\delta \mu)}$ is achieved by adding a term $$\begin{aligned} \label{ttdeform} \delta \mu \int d^2x \left(T^{(\mu)}\overline{T}^{(\mu)}-\Theta^{(\mu)2}\right)\end{aligned}$$ to the action of $\mathcal{T}^{(\mu)}$, where $$\begin{aligned} T^{(\mu)}=-2\pi T_{zz}^{(\mu)},\;\;\overline{T}^{(\mu)}=-2\pi T_{\bar{z}\bar{z}}^{(\mu)},\;\;\Theta^{(\mu)}=2\pi T_{z\bar{z}}^{(\mu)}\end{aligned}$$ are the stress tensor of $\mathcal{T}^{(\mu)}$. In this case, the spectrum could be determined explicitly. Considering the deformed CFT on a cylinder of circumference $L$, the spectrum is E\_n(,L)L=(1- ), \[spectrum\] where $ \tmu=\frac{\mu}{4\pi L^2}$ is a dimensionless quantity and M\_n=\_n+|\_n-, J\_n=\_n-|\_n are the conformal dimensions and the spins of the primary operators in the undeformed CFT. As the spectrum could be imaginary for a fixed $\tmu$, the theory should have a UV cutoff. It is certainly an interesting problem to find a UV completion of such deformation. On the other hand, the $T\overline{T}$-deformation opens a new window to study the AdS/CFT correspondence. It is a double-trace deformation, and could change the boundary condition of the AdS gravity. For a $T\overline{T}$-deformed holographic CFT, McGough, Mezei, and Verlinde [@verlinde] proposed that the dual AdS$_3$ gravity should be defined in a finite region, with the asymptotic boundary being at a finite radial position. More precisely if a CFT i.e. $\mathcal{T}^{(0)}$ has a gravity dual, then the theory $\mathcal{T}^{(\mu)}$ is dual to the original gravitational theory with the new boundary at $r=r_c$. With our convention, the relation between $\mu$ and $r_c$ is $$\begin{aligned} \label{muandrc} \mu=\frac{6R^4}{\pi c r_c^2},\end{aligned}$$ where $R$ is the AdS radius, and $c$ is the central charge of the original CFT. This new correspondence has been checked from various points of view. First of all, the spectrum (\[spectrum\]) is reproduced by considering the quasi-local energy of a BTZ black hole of mass $M_n$ and angular momentum $J_n$ in a spatial region $r<r_c$. Secondly, the superluminal propagation of the perturbation of the stress tensor[@Cardy:2015xaa] can be understood holographically by the metric perturbations preserving Dirichlet boundary condition on the surface $r=r_c$[@Marolf:2012dr]. Moreover, the exact RG equation could be understood holographically as well[@verlinde]. More on the holographic interpretation of the $T\overline{T}$ deformation can be found in [@kutasov1; @kutasov2; @dubovsky1; @dubovsky2; @bihdtott; @ttandcf; @cutoffads; @commentsontt; @ttingenerald][^2]. In this paper, we would like to study the entanglement entropy in the $T\overline{T}$-deformed conformal field theory. In particular we pay special attention to the entanglement entropy in the deformed holographic CFT and investigate its implication in the AdS/CFT correspondence. In a holographic CFT, the entanglement entropy could be captured by the area of the minimal surface via the Ryu-Takayanagi(RT) formula [@rtformula; @Ryu:2006ef]. When considering the new duality proposed in [@verlinde], it seems that the RT-formula still holds. We would like to use the entanglement entropy to test their proposal. More concretely we are going to compute the entanglement entropy of a single interval on a cylinder in the $T\overline{T}$-deformed CFT by using the conformal perturbation method. We will investigate two cases: the one at a finite temperature and the other one with a finite size. We find that in the finite temperature case, there is indeed nonvanishing correction from the deformation, while in the finite size case, the correction is vanishing. We discuss the holographic entanglement entropy via the RT formula and find the consistent picture. Moreover we compute analytically the leading order correction to the Rényi entropy, and discuss its holographic picture. We show that for the AdS$_3$ gravity with a cutoff surface, the on-shell action includes a cutoff-dependent term, which corresponds to the leading order correction due to the $T\overline{T}$-deformation in the partition function in the CFT. The remaining parts of the paper are organized as follows. In section 2, we compute perturbatively the single-interval (Renyi) entanglement entropy on a cylinder in the deformed CFT. In section 3, we compute the holographic entanglement entropy of a single interval in the BTZ black hole and global AdS$_3$ with a finite radius cutoff, and compare with the field theory results. In section 4, we show that the on-shell action of the gravitational configuration in a cut-off restrained region could be dual to the CFT partition function with the leading order correction under the $T\overline{T}$-deformation. We end with discussions in section 5. In the appendix, we collect some technical details. While this papar was in preparation, closely related studies were presented in [@eeandtt]. The authors in [@eeandtt] considered the entanglement entropy for an entangling surface consisting of two antipodal points on a sphere. Entanglement entropy in $T\overline{T}$-deformed CFT ==================================================== Let us consider a $T\overline{T}$-deformed CFT living on some manifold $\mathcal{M}$. And we are interested in the entanglement entropy of some subsystem $A \in \mathcal{M}$. The entanglement entropy is given by $$\begin{aligned} S(A)=\lim_{n\to1}S_n(A),\;\;\;\;S_n(A)=\frac{1}{1-n}\log\frac{Z_n(A)}{Z^n},\end{aligned}$$ where $Z$ is the partition function on $\mathcal{M}$, $Z_n(A)$ is the partition function on the manifold $\mathcal{M}^n(A)$ which is obtained by gluing $n$ copies of $\mathcal{M}$ together along $A$. The precise definition of $\mathcal{M}^n$ and more details about the above formulas can be found in [@cardy]. In this work, we only consider the small $\mu$ case, i.e. $\mu\to0$. According to (\[ttdeform\]) the action of the deformed CFT can be written as $$\begin{aligned} S=S_{CFT}+\mu \int_{\mathcal{M}}(T\overline{T}-\Theta^2),\end{aligned}$$ where $T,\overline{T}$ and $\Theta$ are the quantities of the original CFT. Now we have $$\begin{aligned} \frac{Z_n(A)}{Z^n}=\frac{\int_{\mathcal{M}^n} e^{-S_{CFT}-\mu \int_{\mathcal{M}^n}(T\overline{T}-\Theta^2)}}{\left[\int_{\mathcal{M}} e^{-S_{CFT}-\mu \int_{\mathcal{M}}(T\overline{T}-\Theta^2)}\right]^n}.\end{aligned}$$ Since $\mu$ is small, we further expand in terms of $\mu$ and get $$\begin{aligned} \frac{Z_n(A)}{Z^n}=\frac{\int_{\mathcal{M}^n} e^{-S_{CFT}}\left(1-\mu\int_{\mathcal{M}^n}(T\overline{T}-\Theta^2)+O(\mu^2)\right)}{\left[\int_{\mathcal{M}} e^{-S_{CFT}}\left(1-\mu\int_{\mathcal{M}}(T\overline{T}-\Theta^2)+O(\mu^2)\right)\right]^n}.\end{aligned}$$ We know that in a CFT which is defined on a flat manifold, any correlation function with $T^\mu_\mu$ insertion is zero, i.e. $\left<T^\mu_\mu\dots\right>=0$. Later we will always consider the case $\mathcal{M}$ is a cylinder. Thus $$\begin{aligned} \int_{\mathcal{M}} e^{-S_{CFT}}\Theta^2&\sim&\left<\Theta^2\right>_{\mathcal{M}}=0,\\ \int_{\mathcal{M}^n} e^{-S_{CFT}}\Theta^2&\sim&\left<\Theta^2\sigma\right>_{\mathcal{M}}=0,\end{aligned}$$ with $\sigma$ being the operator inducing the field identification such that the adjacent replicas are pasted along $A$. After some simple algebra, we get $$\begin{aligned} \frac{Z_n(A)}{Z^n}=\left(\frac{\int_{\mathcal{M}^n} e^{-S_{CFT}}}{\left[\int_{\mathcal{M}} e^{-S_{CFT}}\right]^n}\right)\left(1-\mu\int_{\mathcal{M}^n}\left<T\overline{T}\right>_{\mathcal{M}^n}+n\mu\int_{\mathcal{M}}\left<T\overline{T}\right>_{\mathcal{M}}+O(\mu^2)\right).\end{aligned}$$ Notice that $\left<T\overline{T}\right>_{\mathcal{M}^n}$ is only a function defined on $\mathcal{M}^n$. Actually we have $$\begin{aligned} \int_{\mathcal{M}^n}\left<T\overline{T}\right>_{\mathcal{M}^n}=n\int_{\mathcal{M}}\left<T\overline{T}\right>_{\mathcal{M}^n},\end{aligned}$$ from which we get $$\begin{aligned} \frac{Z_n(A)}{Z^n}=\left(\frac{\int_{\mathcal{M}^n} e^{-S_{CFT}}}{\left[\int_{\mathcal{M}} e^{-S_{CFT}}\right]^n}\right)\left(1-n\mu\int_{\mathcal{M}}\left[\left<T\overline{T}\right>_{\mathcal{M}^n}-\left<T\overline{T}\right>_{\mathcal{M}}\right]+O(\mu^2)\right).\end{aligned}$$ Then we can read the leading order correction to $S_n(A)$ $$\begin{aligned} \label{deltasn} \delta S_n(A)=\frac{-n\mu}{1-n}\int_{\mathcal{M}}\left[\left<T\overline{T}\right>_{\mathcal{M}^n}-\left<T\overline{T}\right>_{\mathcal{M}}\right].\end{aligned}$$ Taking the $n\to1$ limit, we have the leading order correction to $S(A)$. In the following, let us consider two concrete cases where $\delta S(A)$ can be calculated. Finite temperature {#finitet} ------------------ The first case is a $2D$ deformed CFT at a finite temperature $1/\beta$. The spatial direction is not compactified and the manifold $\mathcal{M}$ on which the theory is defined is an infinitely long cylinder with circumference $\beta$. We introduce complex coordinate $w=x+i\tau$ and $\bar{w}=x-i\tau$ on the cylinder $\mathcal{M}$, where $x\in(-\infty,\infty)$ and $\tau\in(0,\beta)$ with the identification $\tau\sim\tau+\beta$. The subsystem $A$ is chosen to be a single interval of length $l$ which will be parallel to the axis of the cylinder. The endpoints of $A$ are put at $(w,\bar{w})=(0,0)$ and $(w,\bar{w})=(l,l)$. Consider the transformation $$\begin{aligned} w\to z=e^{\frac{2\pi w}{\beta}},\end{aligned}$$ which maps the cylinder to a plane $\mathcal{C}$. The stress tensor obeys the well-known transformation law $$\begin{aligned} \label{sttransf} T(w)=\left(\frac{dz}{dw}\right)^2T(z)+\frac{c}{12}\{z,w\},\end{aligned}$$ where {z,w}=(z\^z\^-z\^[2]{})/z\^[2]{} is the Schwarzian derivative. There is a similar relation for $\overline{T}$. Using (\[sttransf\]) and $\left<T(z)\right>_{\mathcal{C}}=0$, we find $$\begin{aligned} \label{tt1} \nonumber \left<T\overline{T}(w,\bar{w})\right>_{\mathcal{M}}&=&\left(\frac{c}{12}\right)^2\{z,w\}\{\bar{z},\bar{w}\}\\ &=&\left(\frac{c}{12}\right)^2\left(\frac{2\pi^2}{\beta^2}\right)^2.\end{aligned}$$ To obtain $\left<T\overline{T}(w,\bar{w})\right>_{\mathcal{M}^n}$, one should consider the following maps. The first map is $$\begin{aligned} w\to w^\prime=e^{\frac{2\pi w}{\beta}},\end{aligned}$$ which maps each sheet of $\mathcal{M}^n$ to a plane $\mathcal{C}$. The interval $A$ on the cylinder $\mathcal{M}$ is mapped to an interval $A^\prime$ on the plane $\mathcal{C}$ whose endpoints become $(w^\prime,\bar{w}^\prime)=(1,1)$ and $(w^\prime,\bar{w}^\prime)=(e^{\frac{2\pi l}{\beta}},e^{\frac{2\pi l}{\beta}})$. After this map $\mathcal{M}^n$ becomes a manifold $\mathcal{C}^n$ which is obtained by gluing $n$ copies of the plane $\mathcal{C}$ together along $A^\prime$. The next map is $$\begin{aligned} w^\prime\to z=\left(\frac{w^\prime-1}{w^\prime-e^{\frac{2\pi l}{\beta}}}\right)^{\frac{1}{n}},\end{aligned}$$ which maps $\mathcal{C}^n$ to a plane $\mathcal{C}$. More about this map can be found in section 3 of [@cardy]. Combining these two maps, we find a map $w\to z$ relating $\mathcal{M}^n$ to the plane $\mathcal{C}$. Once again using (\[sttransf\]) and $\left<T(z)\right>_{\mathcal{C}}=0$, we find $$\begin{aligned} \left<T\overline{T}(w,\bar{w})\right>_{\mathcal{M}^n}=\left(\frac{c}{12}\right)^2\{z,w\}\{\bar{z},\bar{w}\}.\end{aligned}$$ In order to read the entanglement entropy, we only need the information under the limit of $n\to 1$. Expanding $\{z,w\}$ and $\{\bar{z},\bar{w}\}$ near $n=1$, we have $$\begin{aligned} \{z,w\}&=&-\frac{2\pi^2}{\beta^2}+(n-1)\frac{4\pi^2\left(1-e^{\frac{2\pi l}{\beta}}\right)^2e^{\frac{4\pi w}{\beta}}}{\beta^2\left(e^{\frac{2\pi w}{\beta}}-e^{\frac{2\pi l}{\beta}}\right)^2\left(e^{\frac{2\pi w}{\beta}}-1\right)^2}+O((n-1)^2),\\ \{\bar{z},\bar{w}\}&=&-\frac{2\pi^2}{\beta^2}+(n-1)\frac{4\pi^2\left(1-e^{\frac{2\pi l}{\beta}}\right)^2e^{\frac{4\pi \bar{w}}{\beta}}}{\beta^2\left(e^{\frac{2\pi \bar{w}}{\beta}}-e^{\frac{2\pi l}{\beta}}\right)^2\left(e^{\frac{2\pi \bar{w}}{\beta}}-1\right)^2}+O((n-1)^2).\end{aligned}$$ Then we find $$\begin{aligned} \label{ttn} \nonumber \left<T\overline{T}(w,\bar{w})\right>_{\mathcal{M}^n}&=&\left(\frac{c}{12}\right)^2\left[\left(\frac{2\pi^2}{\beta^2}\right)^2+(n-1)\left(-\frac{2\pi^2}{\beta^2}\right)\left(\frac{4\pi^2\left(1-e^{\frac{2\pi l}{\beta}}\right)^2e^{\frac{4\pi w}{\beta}}}{\beta^2\left(e^{\frac{2\pi w}{\beta}}-e^{\frac{2\pi l}{\beta}}\right)^2\left(e^{\frac{2\pi w}{\beta}}-1\right)^2}+h.c.\right)\right]\\ &\;&+O((n-1)^2).\end{aligned}$$ Plugging (\[tt1\]) and (\[ttn\]) into (\[deltasn\]), then taking the $n\to 1$ limit, we get $$\begin{aligned} \delta S(A)=-\mu\left(\frac{c}{12}\right)^2\frac{8\pi^4}{\beta^4}\left(1-e^{\frac{2\pi l}{\beta}}\right)^2\int_{\mathcal{M}}\left[\frac{e^{\frac{4\pi w}{\beta}}}{\left(e^{\frac{2\pi w}{\beta}}-e^{\frac{2\pi l}{\beta}}\right)^2\left(e^{\frac{2\pi w}{\beta}}-1\right)^2}+h.c.\right].\end{aligned}$$ The integration is a little bit tricky and the details can be found in the appendix \[integral\]. In the end, we obtain $$\begin{aligned} \delta S(A)=\frac{-\mu\pi ^4 c^2 l \coth \left(\frac{\pi l}{\beta }\right)}{9 \beta ^3}. \label{mucorrection}\end{aligned}$$ In the “low temperature" limit, $\beta\gg l$, the correction to the entanglement entropy (\[mucorrection\]) is $\frac{-\pi ^3 c^2 \mu }{9 \beta ^2}$. In the “high temperature" limit, $\beta\ll l$, the correction is $\frac{-\pi ^4 c^2 l \mu }{9 \beta ^3}$. Actually, at high enough energy, the deformed theory cannot be taken as a local field theory and the above discussion breaks down. Moreover in order to compare with the bulk dual, we have to take the large $c$ limit carefully. It turns out that we should keep $\mu c$ finite in the large $c$ limit[@ttatlargec; @cutoffads]. Under this limit, the correction of the entanglement entropy is proportional to $c$, which could be compared with the semi-classical action of the gravity. Recall that the entanglement entropy of $A$ in a CFT with the same setup is $$\begin{aligned} S_0(A)=\frac{c}{3} \log \left(\frac{\beta }{\pi \epsilon_0 }\sinh \left(\frac{\pi l}{\beta }\right)\right),\end{aligned}$$ with $\epsilon_0$ the CFT cutoff. So to the leading order in $\mu$, we have $$\begin{aligned} \label{sat} \nonumber S(A)&=&S_0(A)+\delta S(A). $$ It is remarkable that although our perturbative computation is to the leading order of $\mu$ and seems work for any temperature, the parameter $\mu$ is of dimension of length square. In the finite temperature case, there is a dimensionless quantity \_=, which cannot be large. In terms of $\tmu_\b$, the change of the entanglement entropy is S(A)=(). In fact, the leading order correction to the R$\acute{\text{e}}$nyi entropy can also be worked out. The computation of it is more tedious, and the details can be found in the appendix \[renyit\]. The final result is $$\begin{aligned} \label{renyientropyt} \nonumber \delta S_n(A)&=&-\frac{\pi ^4 c^2 l \mu (n+1) \coth \left(\frac{\pi l}{\beta }\right)}{18 \beta ^3 n}+\frac{\pi c^2 \mu (n-1) (n+1)^2}{576 n^3 \epsilon ^2}\\ \nonumber &\;&-\frac{\pi ^3 c^2 \mu (n-1) (n+1)^2 \left(\cosh \left(\frac{2 \pi l}{\beta }\right)-7\right) \text{csch}^2\left(\frac{\pi l}{\beta }\right)}{864 \beta ^2 n^3}\\ &\;&+\frac{\pi ^3 c^2 \mu (n-1) (n+1)^2 \coth ^2\left(\frac{\pi l}{\beta }\right) \log \left(\frac{\beta \sinh \left(\frac{\pi l}{\beta }\right)}{2 \pi \epsilon }\right)}{36 \beta ^2 n^3}.\end{aligned}$$ When $n=1$, only the first term survives, and it gives the leading order correction (\[mucorrection\]) to the entanglement entropy. The second term diverges as $1/\epsilon^2$ and does not depend on $\beta$ and $l$. The third term does not depend on the cutoff $\epsilon$ and can have a finite contribution when $n\neq 1$. The last term has the form $\#\log \big(\frac{\beta \sinh \left(\frac{\pi l}{\beta }\right)}{2 \pi \epsilon }\big)$, recalling that $\log \big(\frac{\beta \sinh \left(\frac{\pi l}{\beta }\right)}{2 \pi \epsilon }\big)$ is the original entanglement entropy. Finite size {#finitel} ----------- Another simple case is a $2D$ deformed CFT at zero temperature but with a finite size $L$. The spatial direction is now compactified, while the time direction is non-compact so the manifold $\mathcal{M}$ is still an infinitely long cylinder with circumference $L$. We introduce complex coordinate $w=x+i\tau$ and $\bar{w}=x-i\tau$ on the cylinder $\mathcal{M}$, where $\tau\in(-\infty,\infty)$ and $x\in(0,L)$ with the identification $x\sim x+L$. The subsystem $A$ is chosen to be a single interval of length $l<L $ which will be vertical to the axis of the cylinder. The endpoints of $A$ are put at $(w,\bar{w})=(0,0)$ and $(w,\bar{w})=(l,l)$. The computation procedure is similar to the finite temperature case. Using the map $$\begin{aligned} w\to w^\prime=\tan\left(\frac{\pi w}{L}\right),\end{aligned}$$ which can also map the cylinder to a plane $\mathcal{C}$, we obtain $$\begin{aligned} \label{tt1l} \nonumber \left<T\overline{T}(w,\bar{w})\right>_{\mathcal{M}}&=&\left(\frac{c}{12}\right)^2\{w^\prime,w\}\{\bar{w}^\prime,\bar{w}\}\\ &=&\left(\frac{c}{12}\right)^2\left(\frac{2\pi^2}{L^2}\right)^2.\end{aligned}$$ The interval $A$ on the cylinder $\mathcal{M}$ is mapped to an interval $A^\prime$ on the plane $\mathcal{C}$ whose endpoints become $(w^\prime,\bar{w}^\prime)=(0,0)$ and $(w^\prime,\bar{w}^\prime)=(\tan \left(\frac{\pi l}{L}\right),\tan \left(\frac{\pi l}{L}\right))$. Combining with the map $$\begin{aligned} w^\prime\to z=\left(\frac{w^\prime}{w^\prime-\tan \left(\frac{\pi l}{L}\right)}\right)^{\frac{1}{n}}\end{aligned}$$ yields a map $w\to z$ which relates $\mathcal{M}^n$ to the plane $\mathcal{C}$. Then we have $$\begin{aligned} \left<T\overline{T}(w,\bar{w})\right>_{\mathcal{M}^n}=\left(\frac{c}{12}\right)^2\{z,w\}\{\bar{z},\bar{w}\}.\end{aligned}$$ Expanding $\{z,w\}$ and $\{\bar{z},\bar{w}\}$ near $n=1$, we find $$\begin{aligned} \nonumber \{z,w\}&=&\frac{2 \pi ^2}{L^2}+(n-1)\frac{\pi ^2 \sin ^2\left(\frac{\pi l}{L}\right) \csc ^2\left(\frac{\pi w}{L}\right) \csc ^2\left(\frac{\pi (l-w)}{L}\right)}{L^2}+O((n-1)^2),\\ \{\bar{z},\bar{w}\}&=&\frac{2 \pi ^2}{L^2}+(n-1)\frac{\pi ^2 \sin ^2\left(\frac{\pi l}{L}\right) \csc ^2\left(\frac{\pi \bar{w}}{L}\right) \csc ^2\left(\frac{\pi (l-\bar{w})}{L}\right)}{L^2}+O((n-1)^2).\end{aligned}$$ Then $$\begin{aligned} \label{ttnl} \nonumber \left<T\overline{T}(w,\bar{w})\right>_{\mathcal{M}^n}&=&\left(\frac{c}{12}\right)^2\left[\left(\frac{2 \pi ^2}{L^2}\right)^2+(n-1)\frac{2 \pi ^2}{L^2}\left(\frac{\pi ^2 \sin ^2\left(\frac{\pi l}{L}\right) \csc ^2\left(\frac{\pi w}{L}\right) \csc ^2\left(\frac{\pi (l-w)}{L}\right)}{L^2}+h.c.\right)\right]\\ &\;& +O((n-1)^2).\end{aligned}$$ Plugging (\[tt1l\]) and (\[ttnl\]) into (\[deltasn\]), then taking the $n\to 1$ limit, we get $$\begin{aligned} \delta S(A)=\frac{\mu \pi ^4 c^2 \sin ^2\left(\frac{\pi l}{L}\right)}{72 L^4}\int_{\mathcal{M}}\left[\csc ^2\left(\frac{\pi w}{L}\right) \csc ^2\left(\frac{\pi (l-w)}{L}\right)+h.c.\right].\end{aligned}$$ The integral involved is $$\begin{aligned} \int_{\mathcal{M}}\csc ^2\left(\frac{\pi w}{L}\right) \csc ^2\left(\frac{\pi (l-w)}{L}\right)=\int_{-\infty}^{\infty}d\tau \int_0^Ldx\csc ^2\left(\frac{\pi (x+i\tau))}{L}\right) \csc ^2\left(\frac{\pi (l-(x+i\tau)))}{L}\right).\;\;\;\;\end{aligned}$$ We first do the $x$ integral. Fortunately the primitive function can be found, which is $$\begin{aligned} \nonumber &\;&-\frac{8 i L e^{\frac{2 i \pi l}{L}} \left(1+e^{\frac{2 i \pi l}{L}}\right) \left(\log \left(1-e^{\frac{2 i \pi (x+i \tau )}{L}}\right)-\log \left(1-e^{\frac{2 i \pi (-l+i \tau +x)}{L}}\right)\right)}{\pi \left(-1+e^{\frac{2 i \pi l}{L}}\right)^3}\\ &\;&+\frac{8 i L e^{\frac{2 \pi (\tau +i l)}{L}} \left(2 e^{\frac{2 \pi (\tau +i l)}{L}}-e^{\frac{2 i \pi (l+x)}{L}}-e^{\frac{2 i \pi x}{L}}\right)}{\pi \left(-1+e^{\frac{2 i \pi l}{L}}\right)^2 \left(-e^{\frac{2 \pi \tau }{L}}+e^{\frac{2 i \pi x}{L}}\right) \left(e^{\frac{2 \pi (\tau +i l)}{L}}-e^{\frac{2 i \pi x}{L}}\right)}.\end{aligned}$$ The term on the second line has no contribution since plugging $x=0$ or $x=L$ into it gives the same result. The term on the first line has no contribution as well since the two $\log$ terms always cancel each other. This is very different from the finite temperature case. Thus we learn that $$\begin{aligned} \int_{\mathcal{M}}\csc ^2\left(\frac{\pi w}{L}\right) \csc ^2\left(\frac{\pi (l-w)}{L}\right)=0,\end{aligned}$$ which means $$\begin{aligned} \delta S(A)=0.\end{aligned}$$ So to the leading order of $\mu$, the entanglement entropy of $A$ is still $$\begin{aligned} \label{sal} S(A)=\frac{c}{3} \log \left(\frac{L }{\pi \epsilon_0 }\sin \left(\frac{\pi l}{L}\right)\right),\end{aligned}$$ with $\epsilon_0$ the CFT cutoff. On the contrary, the leading order correction to the R$\acute{\text{e}}$nyi entropy in this case is not vanishing. The computation is similar to the finite-temperature case, and the details can be found in the appendix \[renyil\]. And the result is $$\begin{aligned} \delta S_n(A)&=&\frac{\pi c^2 \mu (n-1) (n+1)^2}{576 n^3 \epsilon ^2}\nn\\ &\;&-\frac{\pi ^3 c^2 \mu (n-1) (n+1)^2 \left(11 \cos \left(\frac{2 \pi l}{L}\right)+19\right) \csc ^2\left(\frac{\pi l}{L}\right)}{864 L^2 n^3}\nn\\ &\;&+\frac{\pi ^3 c^2 \mu (n-1) (n+1)^2 \cot ^2\left(\frac{\pi l}{L}\right) \log \left(\frac{L \sin \left(\frac{\pi l}{L}\right)}{2 \pi \epsilon }\right)}{36 L^2 n^3}.\end{aligned}$$ When $n=1$, it is vanishing as we expect. Let us compare it with (\[renyientropyt\]): the quadratic divergent terms ($1/\epsilon^2$) are the same, which is independent of the finite temperature or finite size; their logarithmic terms are the same under the identification $L \leftrightarrow i\b$. The main difference between them is that $\delta S_n(A)$ in the finite $T$ case has an additional term $$\begin{aligned} -\frac{\pi ^4 c^2 l \mu (n+1) \coth \left(\frac{\pi l}{\beta }\right)}{18 \beta ^3 n},\end{aligned}$$ which is nonzero when $n=1$. Gravity dual ============ The AdS/CFT correspondence[@Maldacena:1997re] states that the gravitational theory living in the bulk is dual to a CFT living on the asymptotic boundary of the AdS spacetime. Especially the Ryu-Takayanagi formula [@rtformula; @Ryu:2006ef] relates the entanglement entropy in the CFT with the area of the corresponding minimal surface in the gravitational theory. The holographic entanglement entropy could be understood as a generalized gravitational entropy[@proofhee]. For the AdS$_3$/CFT$_2$ correspondence, it has been found that after imposing appropriate asymptotic boundary condition[@Brown:1986nw], the AdS$_3$ gravity could be dual to a 2D CFT with central charge[@Strominger:1997eq] c=. The authours of [@verlinde] proposed that under $T\overline{T}$ deformation the bulk dual gravitational theory should be defined by moving the asymptotic boundary inwards with the radius being at r\_c\^2=. Here $R$ is the AdS radius and $c$ is the central charge of the dual CFT. In this case, we expect that the holographic entanglement entropy is still given by the RT-formula, $$\begin{aligned} \label{rtformu} S(A)=\frac{\text{Area}\;\text{of}\;\gamma_A}{4G},\end{aligned}$$ where $\gamma_A$ is the minimal surface in the bulk whose boundary is given by $\partial A$. In the cases we are considering $A$ is an interval, and $\gamma_A$ is the geodesic whose endpoints coincide with $A$’s. BTZ black hole {#btzblackhole} -------------- A $2d$ CFT at high temperature is dual to a BTZ black hole. According to [@verlinde] the $T\overline{T}$ deformed CFT at high temperature is naturally dual to a BTZ black hole with a radial cutoff. The metric of Euclidean BTZ black hole is $$\begin{aligned} \label{btz} ds^2=\frac{r^2-r_+^2}{R^2}dt^2+\frac{R^2}{r^2-r_+^2}dr^2+r^2dx^2,\end{aligned}$$ with $R$ the AdS radius, $r_+$ the position of the horizon, $t$ compactified as $t\sim t+\beta$. $\beta=\frac{2\pi R^2}{r_+}$ is the temperature of the black hole and the corresponding CFT. Originally the boundary is located at $r\to \infty$. Now we move the boundary inwards to $r=r_c$ where the $T\overline{T}$ deformed CFT lives. On this new boundary, the metric is $$\begin{aligned} \nonumber ds^2_b&=&\frac{r_c^2-r_+^2}{R^2}dt^2+r_c^2dx^2\\ &=&\frac{r_c^2-r_+^2}{R^2}\left(dt^2+\frac{R^2r_c^2}{r_c^2-r_+^2}dx^2\right).\end{aligned}$$ The black hole temperature is still $\beta$, which means the temperature of the $T\overline{T}$ deformed CFT is also $\beta$. So $t$ is the physical time of the deformed CFT, whose physical metric shall be $$\begin{aligned} \label{pmetric} ds_p^2=dt^2+\frac{R^2r_c^2}{r_c^2-r_+^2}dx^2.\end{aligned}$$ At some time $t_0$, we put the endpoints of the subsystem $A$ at $(t,x)=(t_0,0)$ and $(t,x)=(t_0,\delta x)$. According to (\[pmetric\]), the length of $A$ is $$\begin{aligned} \label{lengthl} l=\frac{\delta xRr_c}{\sqrt{r_c^2-r_+^2}}.\end{aligned}$$ What we are left to do is to find the geodesic distance $\lambda$ between $(r,t,x)=(r_c,t_0,0)$ and $(r,t,x)=(r_c,t_0,\delta x)$. To achieve this we define the new coordinates $$\begin{aligned} r=r_+\cosh \rho,\;\;t=\frac{ R^2\theta}{r_+},\;\;x=\frac{R \tau}{r_+},\end{aligned}$$ following which the metric (\[btz\]) becomes $$\begin{aligned} ds^2=R^2\left(\sinh^2 \rho d\theta^2+d\rho^2+\cosh^2 \rho d\tau^2\right),\end{aligned}$$ which is the Euclidean version of global $AdS_3$ metric. The endpoints become $(\rho,\theta,\tau)=(\rho_c,\theta_0,0)$ and $(\rho,\theta,\tau)=(\rho_c,\theta_0,\frac{ r_+\delta x}{R})$ with $$\begin{aligned} \label{rcandrho} r_c=r_+\cosh \rho_c.\end{aligned}$$ Now the geodesic distance $\lambda$ can be easily found: $$\begin{aligned} \cosh\left(\frac{\lambda}{R}\right)=1+2\cosh^2 \rho_c \sinh^2 \frac{r_+\delta x}{2R}.\end{aligned}$$ Plugging (\[rcandrho\]) and (\[lengthl\]) into it, we find $$\begin{aligned} \cosh\left(\frac{\lambda}{R}\right)=1+2\left(\frac{r_c}{r_+}\right)^2\sinh^2\left(\frac{\pi l}{\beta}\sqrt{1-\left(\frac{r_+}{r_c}\right)^2}\right).\end{aligned}$$ When $r_c\gg r_+$, we can expand $\lambda$ in terms of $r_+/r_c$ and obtain $$\begin{aligned} \label{lambdat} \frac{\lambda}{4G}=\frac{R}{2G}\log\left(\frac{\beta r_c\sinh\left(\frac{\pi l}{\beta}\right)}{\pi R^2}+\frac{\pi R^2}{\beta r_c\sinh\left(\frac{\pi l}{\beta}\right)}-\frac{2\pi^2 R^2 l \cosh\left(\frac{\pi l}{\beta}\right)}{\beta^2 r_c}+O\left(\left(\frac{r_+}{r_c}\right)^2\right)\right),\end{aligned}$$ where we have used $\beta=\frac{2\pi R^2}{r_+}$ to replace $r_+$ by $\frac{2\pi R^2}{\beta}$. If we consider the “high temperature" case $\beta <l$, the second term in the parenthesis can be naturally ignored since it is much smaller than the third term. On the other hand since $r_c$ is very large, we can treat the third term as a small quantity compared with the first term. This leads to &=&()-\ &=&()-() after considering the relations $c=3R/2G$ and (\[muandrc\]). Now as the cutoff boundary is at $r_c$ so the corresponding cutoff in the field theory is $\epsilon=R^2/r_c$, then we find the perfect match with the field theory result. Global AdS ---------- A $2d$ CFT at zero temperature with the spatial direction compactified lives on the asympotic boundary of the global $AdS_3$. So the $T\overline{T}$ deformed CFT which we considered in section (\[finitel\]) is dual to the global $AdS_3$ with a radial cutoff. The metric of global $AdS_3$ is $$\begin{aligned} ds^2=R^2\left(-\cosh^2\rho dt^2+d\rho^2+\sinh^2 \rho d\phi^2\right),\end{aligned}$$ with $\phi$ compactified as $\phi\sim \phi+2\pi$. We put the boundary at $\rho=\rho_c$, on which the $T\overline{T}$ deformed CFT lives. At some time $t_0$, the endpoints of the subsystem $A$ are put at $(t,\phi)=(t_0,0)$ and $(t,\phi)=(t_0, \delta \phi)$. Suppose that the total length of the quantum system is $L$, then the length of $A$ is given by $$\begin{aligned} \label{landphi} l=\frac{\delta \phi L}{2\pi}.\end{aligned}$$ The geodesic distance $\lambda$ between $(\rho,t,\phi)=(\rho_c,t_0,0)$ and $(\rho,t,\phi)=(\rho_c,t_0,\delta \phi)$ is given by $$\begin{aligned} \cosh \left(\frac{\lambda }{R}\right)=1+2\sinh^2\rho_c \sin^2\left(\frac{\pi l}{L}\right),\end{aligned}$$ where we have used (\[landphi\]) to replace $\delta \phi$ by $2\pi l/L$. When $\rho_c\gg1\leftrightarrow \sinh \rho_c\gg1$, we can expand $\lambda$ in terms of $1/ \sinh \rho_c$ and obtain $$\begin{aligned} \label{lambdal} \frac{\lambda}{4G}=\frac{R}{2G}\log\left(2\sinh \rho_c \sin\left(\frac{\pi l}{L}\right)+\frac{1}{2\sinh \rho_c \sin\left(\frac{\pi l}{L}\right)}+O\left(\left(\frac{1}{\sinh\rho_c}\right)^2\right)\right).\end{aligned}$$ Now there is no other correction except that the cutoff surface is moved inward. This fact is in accordance with the fact that there is no correction to the entanglement entropy from the $T\overline{T}$ deformation in the finite size CFT in the leading order of $\mu$. After the careful calculations on the bulk side, we notice that the main difference between these two cases lies on the difference between (\[lengthl\]) and (\[landphi\]). (\[lengthl\]) says that $\delta x$ depends on the cutoff $r_c$ when $r_+\neq0$ (i.e. $1/\beta\neq0$), while (\[landphi\]) shows that $\delta \phi$ does not depend on the cutoff. In the finite temperature case the leading order correction comes actually from the $r_c$ dependence of $\delta x$. More general holographic picture ================================ In the above discussion on the holographic entanglement entropy, we actually assumed the RT prescription. This expectation turns out to be good. However, for the single-interval Rényi entropy, we need to consider the backreaction of the twist operator[@proofhee; @Dong:2016fnf]. In the following, we try to argue that the holographic picture is still true for general configurations, using the method developed in [@Skenderis:1999nb; @Hung:2011nu; @Faulkner:2013yia; @Barrella:2013wja]. We start from the $T\overline{T}$-deformed CFT defined on the boundary metric $$ds^2=g_{ab}dx^adx^b.$$ It is dual to the gravitational theory living on a compact sub-region of AdS. The metric of the bulk configuration could be $$ds^2=\frac{dr^2}{r^2}+r^2g_{ab}dx^adx^b.$$ We have set $R_{AdS}=1$. The Poincare coordinate is recovered by setting $\xi=1/r$. In the Fefferman-Graham gauge, the metric is expanded as $$ds^2=\frac{d\rho^2}{4\rho^2}+\frac{g_{ab}}{\rho}dx^adx^b.$$ Having fixed the leading order $g_{(0)}$, the metric above is characterized by the stress tensor of the classical Liouville field[@Krasnov:2000zq]. In other words, the classical gravitational solution is characterized by the stress tensor, which is determined by the conformal weights and the accessory parameters in particular. In general, it is hard to find the explicit form of the metric. In the following discussion we denote the $x^a$ in Poincare coordinate as $z,\bar{z}$, while the $x^a$ in FG coordinate is denoted as $w, \bar{w}$. For the $T\overline{T}$-deformed holographic CFT, there is a certain regulator surface at a fixed radial position. We choose the regulator surface in the Poincare coordinate, so that the induced metric of the surface coincides with the one in CFT. The regulator surface is located at $$\xi_c\approx be^{\phi},$$ where $\phi$ is the classical Liouville field, relating to the Weyl factor, and $$b^2=\frac{\mu c}{24\pi}.$$ There is a coordinate transformation between the FG coordinate and the Poincare coordinate[@Krasnov:2001cu] $$\xi=\frac{\rho^{1/2}e^{-\phi}}{1+\rho e^{-2\phi}a^2},$$ where $a=\partial \phi$. The semi-classical action of the gravitational theory is $$I=I_{EH}+I_{GH}+I_{CT},$$ including the Einstein-Hilbert term plus a negative cosmological constant, the Gibbons-Hawking term and the counter term. The counter term cancels the power-law divergence in the bulk integral and the boundary integral. More concretely, the on-shell Einstein-Hilbert action reduces to $$I_{EH}=-\frac{c}{96\pi}\int dzd\bar{z}\xi_c^{-2},$$ $$\xi_c^{-2}=a^4 b^2 e^{-2 \phi }+2 a^2-\frac{e^{2 \phi }}{b^2}.$$ The Gibbons-Hawking term and the counter term give $$I_{GH}+I_{CT}=-\frac{c}{96\pi}\int dzd\bar{z}(\frac{e^{2 \phi }}{b^2}+8\partial\bar{\partial}\phi).$$ The final on-shell action is $$I=-\frac{c}{96\pi}\int dzd\bar{z}(2a^2+a^4 b^2 e^{-2 \phi }-8\partial\bar{\partial}\phi).$$ Note that as $b\rightarrow 0$, the action above is just the Liouville action, and the changes in the choice of cut-off surface is sub-leading in $b$. Note also that there is an ambiguity in the choice of the counter term, so the linear order change in the bulk action does matter, leaving the other potential terms depending on certain regularization prescription. To go further, we turn to calculate the above action in the FG gauge with a proper regulator surface. It turns out that the last part vanishes and the first part becomes $$I_{1}=-n\frac{c}{96\pi}\int dwd\bar{w} 4\sqrt{T_L\bar{T}_L}.$$ The result above can be understood as follows $$-\frac{c}{96\pi}\int dzd\bar{z} 2a_z^2=-\frac{c}{96\pi}\int dwd\bar{w}e^{-2\phi}e^{2\phi} 2a_w^2=-n\frac{c}{96\pi}\int dwd\bar{w} 4\sqrt{ T_L\bar{T}_L}.$$ The $T_L $ is the Liouville stress tensor, related to the vacuum expectation value of the CFT stress tensor by $${\left\langle}T {\right\rangle}_{CFT}=-\frac{1}{2l_p}T_L.$$ The action above can be used to get the HRE when certain conformal transformation has been made[@Hung:2011nu], and $l_p=8\pi G$. The remaining part, which is associated to the regulator surface and gives the correction to the HRE, can be calculated by $$I_{2}=-\frac{c}{96\pi}\int dzd\bar{z}a^4 b^2 e^{-2 \phi }=-\frac{cb^2}{96\pi}\int dwd\bar{w} a^4_w e^{4\phi} e^{-2\phi}e^{-2\phi}=-n\frac{cb^2}{96\pi}\int dwd\bar{w}4 T_L\bar{T}_L.$$ Considering the fact that $$c=\frac{12\pi}{l_p}=\frac{3}{2G},$$ we find that the integrand from field theory side is $$-\frac{cb^2}{96\pi}4T_L\bar{T}_L=-\frac{\mu c^2}{576\pi^2}T_L\bar{T}_L.$$ Recall the involved partition function $Z_n$ from the conformal perturbation theory, where the linear term in $\mu$ is just $$-\mu{\left\langle}T\bar{T} {\right\rangle}_{CFT} =-\mu{\left\langle}T{\right\rangle}_{CFT} {\left\langle}\bar{T} {\right\rangle}_{CFT}=-\frac{\mu}{4l_p^2}T_L \bar{T}_L =-\frac{\mu c^2}{576\pi^2}T_L\bar{T}_L.$$ Thus at the linear level, the QFT partition function calculated by the conformal perturbation theory matches with the gravitational result. Note that the discussion may apply to the more general cases than the single-interval Rényi entropy. For example, for the two-interval case[@Barrella:2013wja; @Chen:2013kpa] and the single-interval on a torus case[@Barrella:2013wja; @Chen:2014unl], the leading order correction in $\mu$ to the Rényi entropy should match with the holographic computation as well. Discussion ========== In this papar we have calculated the entanglement entropy of a single interval on a cylinder in the $T\overline{T}$-deformed CFT. We find that the leading order correction to the entanglement entropy is nonzero in the finite temperature case while it is vanishing in the finite size case. In the dual bulk side it is expected naively that moving inwards will certainly change the geodesic distances which means the leading order correction should be nonzero in both cases. However in the finite size case, the change of the boundary could actually be taken into account by a different cutoff. On the contrary, in the high temperature case, such a change do modify the geodesic distance. Our study supports the conjecture proposed in [@verlinde]. Unlike the work done in [@eeandtt], our field theory results are only valid when $\mu\to 0$. To obtain the finite $\mu$ results, we need to know the partition function of the theory $\mathcal{T}^{(\mu)}$ on $\mathcal{M}$ and $\mathcal{M}^n$, which is a much harder job. It would be definitely interesting to study this issue. On the gravity side, the discussion in the present work relies also on the condition that $\mu$ is very small. In the finite $\mu$ case, it is not clear if the RT prescription can be applied naively. To determine whether the RT formula is still valid or not, one should go to the non-perturbative level. It is interesting to consider the full version of this duality, saying arbitrary geometry and finite deformation. The holographic entanglement entropy in the standard $AdS/CFT$ correspondence has brought us many new understandings of the holographic duality. The new duality proposed in [@verlinde] is fascinating, and provides a new window to study various problems in the $AdS/CFT$ holography , like holographic entanglement entropy, bulk reconstruction, holographic complexity etc.. We wish to address these issues in the future. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to Han Liu for the participation at the early stage of the project. We would like to thank Yan-jun Liu for valuable discussions. The work is supported in part by NSFC Grant No. 11275010, No. 11325522, No. 11335012 and No. 11735001. The Integral {#integral} ============ In this appendix, we present the details of the integration in section (\[finitet\]). In order to work out the integral $$\begin{aligned} \int_{\mathcal{M}}\frac{e^{\frac{4\pi w}{\beta}}}{\left(e^{\frac{2\pi w}{\beta}}-e^{\frac{2\pi l}{\beta}}\right)^2\left(e^{\frac{2\pi w}{\beta}}-1\right)^2}=\int_{-\infty}^{\infty} dx\int_0^\beta d\tau\frac{e^{\frac{4\pi (x+i\tau)}{\beta}}}{\left(e^{\frac{2\pi (x+i\tau)}{\beta}}-e^{\frac{2\pi l}{\beta}}\right)^2\left(e^{\frac{2\pi (x+i\tau)}{\beta}}-1\right)^2},\end{aligned}$$ we first do the $\tau$ integral. Luckily the primitive function can be found to be $$\begin{aligned} \nonumber \frac{i \beta \left(\frac{e^{\frac{2 \pi l}{\beta }}-1}{-1+e^{\frac{2 \pi (x+i \tau )}{\beta }}}+\frac{e^{\frac{2 \pi l}{\beta }}-e^{\frac{4 \pi l}{\beta }}}{e^{\frac{2 \pi l}{\beta }}-e^{\frac{2 \pi (x+i \tau )}{\beta }}}-\left(e^{\frac{2 \pi l}{\beta }}+1\right) \left(\log \left(e^{\frac{2 \pi (x+i \tau )}{\beta }}-1\right)-\log \left(e^{\frac{2 \pi (x+i \tau )}{\beta }}-e^{\frac{2 \pi l}{\beta }}\right)\right)\right)}{2 \pi \left(e^{\frac{2 \pi l}{\beta }}-1\right)^3}.\;\;\end{aligned}$$ Plugging $\tau=0$ or $\tau=\beta$ into the term $$\begin{aligned} \frac{e^{\frac{2 \pi l}{\beta }}-1}{-1+e^{\frac{2 \pi (x+i \tau )}{\beta }}}+\frac{e^{\frac{2 \pi l}{\beta }}-e^{\frac{4 \pi l}{\beta }}}{e^{\frac{2 \pi l}{\beta }}-e^{\frac{2 \pi (x+i \tau )}{\beta }}}\end{aligned}$$ gives the same result, so this term has no contribution. The term we shall analyze carefully is $$\begin{aligned} -\left(e^{\frac{2 \pi l}{\beta }}+1\right) \left(\log \left(e^{\frac{2 \pi (x+i \tau )}{\beta }}-1\right)-\log \left(e^{\frac{2 \pi (x+i \tau )}{\beta }}-e^{\frac{2 \pi l}{\beta }}\right)\right).\end{aligned}$$ Let us first focus on $$\begin{aligned} \log \left(e^{\frac{2 \pi (x+i \tau )}{\beta }}-1\right).\end{aligned}$$ Fixing $x$, when $\tau$ runs from $0$ to $\beta$, $e^{\frac{2 \pi (x+i \tau )}{\beta }}$ runs around the origin once with circular orbit of radius $e^{\frac{2 \pi x}{\beta }}$. If the radius $e^{\frac{2 \pi x}{\beta }}>1$, $e^{\frac{2 \pi (x+i \tau )}{\beta }}$ will run around $1$ once, which means that $\log \left(e^{\frac{2 \pi (x+i \tau )}{\beta }}-1\right)$ will contribute $2\pi i$. It is demonstrated explicitly in Fig.\[fig1\]. So we have $$\begin{aligned} \nonumber \log \left(e^{\frac{2 \pi (x+i \tau )}{\beta }}-1\right)\Big|_{\tau=0}^{\tau=\beta}&=&\left\{ \begin{aligned} 0, \;\;\;\;x<0 \leftrightarrow e^{\frac{2 \pi x}{\beta }}<1\\ 2\pi i, \;\;\;\;x>0 \leftrightarrow e^{\frac{2 \pi x}{\beta }}>1, \end{aligned} \right.\\\end{aligned}$$ For the other logarithmic function, the discussion is similar: $$\begin{aligned} \nonumber \log \left(e^{\frac{2 \pi (x+i \tau )}{\beta }}-e^{\frac{2 \pi l}{\beta }}\right)\Big|_{\tau=0}^{\tau=\beta}&=&\left\{ \begin{aligned} 0, \;\;\;\;x<l \leftrightarrow e^{\frac{2 \pi x}{\beta }}<e^{\frac{2 \pi l}{\beta }}\\ 2\pi i, \;\;\;\;x>l \leftrightarrow e^{\frac{2 \pi x}{\beta }}>e^{\frac{2 \pi l}{\beta }}. \end{aligned} \right.\end{aligned}$$ Consequently, we have $$\begin{aligned} \label{logterm} g(x)\equiv\log \left(e^{\frac{2 \pi (x+i \tau )}{\beta }}-1\right)-\log \left(e^{\frac{2 \pi (x+i \tau )}{\beta }}-e^{\frac{2 \pi l}{\beta }}\right)\Big|_{\tau=0}^{\tau=\beta}&=&\left\{ \begin{aligned} 0&,& \;\;\;\;x<0 \\ 2\pi i&,& \;\;\;\;0<x<l\\ 0&,& \;\;\;\;l<x. \end{aligned} \right.\end{aligned}$$ The value of $g(x)$ can be easily seen from the contour in Fig.\[fig2\]. With these in hand, the integral turns out to be $$\begin{aligned} \int_{\mathcal{M}}\frac{e^{\frac{4\pi w}{\beta}}}{\left(e^{\frac{2\pi w}{\beta}}-e^{\frac{2\pi l}{\beta}}\right)^2\left(e^{\frac{2\pi w}{\beta}}-1\right)^2}=\frac{\beta l \left(e^{\frac{2 \pi l}{\beta }}+1\right)}{\left(e^{\frac{2 \pi l}{\beta }}-1\right)^3},\end{aligned}$$ and $\delta S(A)$ is simplified to $$\begin{aligned} \delta S(A)=-\frac{\mu\pi ^4 c^2 l \coth \left(\frac{\pi l}{\beta }\right)}{9 \beta ^3}.\end{aligned}$$ Correction of the R$\acute{\text{e}}$nyi entropy ================================================ Finite temperature {#renyit} ------------------ The leading order correction to $S_n(A)$ is given by (\[deltasn\]). We already have $\left<T\overline{T}(w,\bar{w})\right>_{\mathcal{M}}$ and $\left<T\overline{T}(w,\bar{w})\right>_{\mathcal{M}^n}$. Previously we expand $\left<T\overline{T}(w,\bar{w})\right>_{\mathcal{M}^n}$ near $n=1$. Now we need its exact form, which is given by $$\begin{aligned} \left<T\overline{T}(w,\bar{w})\right>_{\mathcal{M}^n}=\left(\frac{c}{12}\right)^2\{z,w\}\{\bar{z},\bar{w}\},\end{aligned}$$ where the Schwarzian derivatives $\{z,w\}$ and $\{\bar{z},\bar{w}\}$ are determined by the map $$\begin{aligned} w\to z=\left(\frac{e^{\frac{2\pi w}{\beta}}-1}{e^{\frac{2\pi w}{\beta}}-e^{\frac{2\pi l}{\beta}}}\right)^{\frac{1}{n}}.\end{aligned}$$ Then $I\equiv\left<T\overline{T}\right>_{\mathcal{M}^n}-\left<T\overline{T}\right>_{\mathcal{M}}$ can be obtained. Using $w=x+i \tau$ and $\bar{w}=x-i \tau$, we arrive at $$\begin{aligned} \nonumber \delta S_n(A)&=&\frac{-n\mu}{1-n}\int_{\mathcal{M}}I\\ &=&\frac{-n\mu}{1-n}\int_{-\infty}^{\infty} dx\int_0^{\beta} d\tau\;I(x,\tau).\end{aligned}$$ We first do the $\tau$ integral, and the primitive function of $I(x,\tau)$ can be found. Let us call it $\mathcal{I}(x,\tau)$. As before there are two kinds of terms in $\mathcal{I}(x,\tau)$: the terms with $\log$ and the terms without $\log$. The analyses of them are similar with those in appendix \[integral\]. After some carefull analyses we can express the integral as $$\begin{aligned} \label{integralint} \int_{-\infty}^{\infty} dx\int_0^{\beta} d\tau\;I(x,\tau)=\frac{\pi ^4 c^2}{36 \beta ^4}\left(\int_{-\infty}^0dx F(x)-\int_0^{\infty}dxF(x)+\int_{-\infty}^ldxG(x)-\int_l^{\infty}dx G(x)\right),\end{aligned}$$ where $$\begin{aligned} F(x)&=& \frac{i \beta \text{C}_1(x) \left(n^2-1\right) (-2\pi i)}{2 \pi n^4 \left(e^{\frac{2 \pi l}{\beta }}-1\right) \left(e^{\frac{4 \pi x}{\beta }}-1\right)^3 \left(e^{\frac{2 \pi l}{\beta }}-e^{\frac{4 \pi x}{\beta }}\right)^3},\\ G(x)&=&\frac{-i \beta \text{C}_2(x) \left(n^2-1\right) (-2\pi i)}{2 \pi n^4 \left(e^{\frac{2 \pi l}{\beta }}-1\right) \left(e^{\frac{2 \pi l}{\beta }}-e^{\frac{4 \pi x}{\beta }}\right)^3 \left(e^{\frac{4 \pi l}{\beta }}-e^{\frac{4 \pi x}{\beta }}\right)^3}.\end{aligned}$$ Here $\text{C}_1, \text{C}_2$ are two functions of $x$, and their expressions are so long that we would not like to show them here. Notice that $F(x)$ has poles at $x=0,\;l/2$, and $G(x)$ has poles at $x=l/2,\;l$. With a cutoff $\epsilon$, the integral becomes $$\begin{aligned} \frac{\pi ^4 c^2}{36 \beta ^4}\left(\int_{-\infty}^{-\epsilon}dx F(x)-\int_{\epsilon}^{\frac{l}{2}-\epsilon}dxF(x)-\int_{\frac{l}{2}+\epsilon}^{\infty}dxF(x)+\int_{-\infty}^{\frac{l}{2}-\epsilon}dx G(x)+\int_{\frac{l}{2}+\epsilon}^{l-\epsilon}dxG(x)-\int_{l+\epsilon}^{\infty}dxG(x)\right).\;\;\;\end{aligned}$$ The primitive functions of $F(x)$ and $G(x)$ can also be found, which are denoted by $\mathcal{F}(x),\;\mathcal{G}(x)$. Now the term in the bracket becomes $$\begin{aligned} \nonumber &\;&\mathcal{F}(-\epsilon)+\mathcal{F}(\epsilon)+\mathcal{F}(\frac{l}{2}+\epsilon)-\mathcal{F}(\frac{l}{2}-\epsilon)-\mathcal{F}(\infty)-\mathcal{F}(-\infty)\\ &\;&+\mathcal{G}(l+\epsilon)+\mathcal{G}(l-\epsilon)+\mathcal{G}(\frac{l}{2}-\epsilon)-\mathcal{G}(\frac{l}{2}+\epsilon)-\mathcal{G}(\infty)-\mathcal{G}(-\infty).\end{aligned}$$ We find that $$\begin{aligned} \mathcal{F}(\infty)+\mathcal{G}(\infty)&=&0,\\ \mathcal{F}(-\infty)+\mathcal{G}(-\infty)&=&0,\\ \mathcal{F}(\frac{l}{2}+\epsilon)-\mathcal{F}(\frac{l}{2}-\epsilon)&=&O(\epsilon),\\ \mathcal{G}(\frac{l}{2}-\epsilon)-\mathcal{G}(\frac{l}{2}+\epsilon)&=&O(\epsilon).\end{aligned}$$ And $$\begin{aligned} \nonumber \mathcal{F}(-\epsilon)+\mathcal{F}(\epsilon)+\mathcal{G}(l+\epsilon)+\mathcal{G}(l-\epsilon)&=&-\frac{2 \beta l \left(n^2-1\right) \coth \left(\frac{\pi l}{\beta }\right)}{n^2}+\frac{\beta ^4 \left(n^2-1\right)^2}{16 \pi ^3 n^4 \epsilon ^2}\\ \nonumber &\;&-\frac{\beta ^2 \left(n^2-1\right)^2 \left(\cosh \left(\frac{2 \pi l}{\beta }\right)-7\right) \text{csch}^2\left(\frac{\pi l}{\beta }\right)}{24 \pi n^4}\\ \nonumber &\;&+\frac{\beta ^2 \left(n^2-1\right)^2 \coth ^2\left(\frac{\pi l}{\beta }\right) \log \left(\frac{\beta \sinh \left(\frac{\pi l}{\beta }\right)}{2 (\pi \epsilon )}\right)}{\pi n^4}\\ &\;&+O(\epsilon^2).\end{aligned}$$ Multiplying the prefactor back, we finally get $$\begin{aligned} \nonumber \delta S_n(A) &=&-\frac{\pi ^4 c^2 l \mu (n+1) \coth \left(\frac{\pi l}{\beta }\right)}{18 \beta ^3 n}+\frac{\pi c^2 \mu (n-1) (n+1)^2}{576 n^3 \epsilon ^2}\\ \nonumber &\;&-\frac{\pi ^3 c^2 \mu (n-1) (n+1)^2 \left(\cosh \left(\frac{2 \pi l}{\beta }\right)-7\right) \text{csch}^2\left(\frac{\pi l}{\beta }\right)}{864 \beta ^2 n^3}\\ \nonumber &\;&+\frac{\pi ^3 c^2 \mu (n-1) (n+1)^2 \coth ^2\left(\frac{\pi l}{\beta }\right) \log \left(\frac{\beta \sinh \left(\frac{\pi l}{\beta }\right)}{2 \pi \epsilon }\right)}{36 \beta ^2 n^3}\\ &\;&+O(\epsilon).\end{aligned}$$ Finite size {#renyil} ----------- The discussion in this case is similar to the one in Appendix \[renyit\]. Now we have $$\begin{aligned} \left<T\overline{T}(w,\bar{w})\right>_{\mathcal{M}^n}=\left(\frac{c}{12}\right)^2\{z,w\}\{\bar{z},\bar{w}\}\end{aligned}$$ with the Schwarzian derivatives $\{z,w\}$ and $\{\bar{z},\bar{w}\}$ determined by the map $$\begin{aligned} w\to z=\left(\frac{\tan\left(\frac{\pi w}{L}\right)}{\tan\left(\frac{\pi w}{L}\right)-\tan \left(\frac{\pi l}{L}\right)}\right)^{\frac{1}{n}}.\end{aligned}$$ Defining $I\equiv\left<T\overline{T}\right>_{\mathcal{M}^n}-\left<T\overline{T}\right>_{\mathcal{M}}$ and using $w=x+i \tau,\;\bar{w}=x-i \tau$, we get $$\begin{aligned} \nonumber \delta S_n(A)&=&\frac{-n\mu}{1-n}\int_{\mathcal{M}}I\\ &=&\frac{-n\mu}{1-n}\int_{-\infty}^{\infty} d\tau \int_0^{L} dx\;I(x,\tau).\end{aligned}$$ Now we first do the $x$ integral, and the primitive function of $I(x,\tau)$ can be found which is denoted as $\mathcal{I}(x,\tau)$. After some efforts we can express the integral as $$\begin{aligned} \label{integralinl} \int_{-\infty}^{\infty} d\tau\int_0^{L} dx\;I(x,\tau)=\frac{\pi ^4 c^2}{2304 L^4}\left(\int_{-\infty}^0d\tau F(\tau)-\int_0^{\infty}d\tau F(\tau)+\int_{-\infty}^0 d\tau G(\tau)-\int_0^{\infty}d\tau G(\tau)\right),\;\end{aligned}$$ where $$\begin{aligned} F(\tau)&=& \frac{64 \text{D}_1(\tau) L \left(n^2-1\right)}{n^4 \left(-1+e^{\frac{2 i \pi l}{L}}\right) \left(e^{\frac{4 \pi \tau }{L}}-1\right)^3 \left(-e^{\frac{4 \pi \tau }{L}}+e^{\frac{2 i \pi l}{L}}\right)^3},\\ G(\tau)&=&\frac{64 \text{D}_2(\tau) L \left(n^2-1\right)}{n^4 \left(-1+e^{\frac{2 i \pi l}{L}}\right) \left(e^{\frac{4 \pi \tau }{L}}-1\right)^3 \left(-1+e^{\frac{4 \pi \tau +2 i \pi l}{L}}\right)^3}.\end{aligned}$$ Here $\text{D}_1, \text{D}_2$ are two functions of $\tau$. We should notice that the form of the integral (\[integralinl\]) is slightly different from the one of (\[integralint\]), i.e. the $G$ integral is changed from $\int_{-\infty}^l-\int_l^{\infty}$ to $\int_{-\infty}^0-\int_0^{\infty}$. This difference is significant. Now $F(\tau)$ only has a pole at $\tau=0$, so does $G(\tau)$. With a cutoff $\epsilon$, the integral becomes $$\begin{aligned} \frac{\pi ^4 c^2}{2304 L^4}\left(\int_{-\infty}^{-\epsilon}d\tau F(\tau)-\int_{\epsilon}^{\infty}d\tau F(\tau)+\int_{-\infty}^{-\epsilon} d\tau G(\tau)-\int_{\epsilon}^{\infty}d\tau G(\tau)\right).\;\;\;\end{aligned}$$ The primitive functions of $F(\tau)$ and $G(\tau)$ can be found out, which are denoted as $\mathcal{F}(\tau),\;\mathcal{G}(\tau)$. Now the term in the bracket becomes $$\begin{aligned} \nonumber &\;&\mathcal{F}(\epsilon)+\mathcal{F}(-\epsilon)-\mathcal{F}(\infty)-\mathcal{F}(-\infty)+\mathcal{G}(\epsilon)+\mathcal{G}(-\epsilon)-\mathcal{G}(\infty)-\mathcal{G}(-\infty).\end{aligned}$$ There are $\log$ terms in $\mathcal{F}(\tau)$ and $\mathcal{G}(\tau)$, so we should deal with them very carefully. After figuring out everything carefully, we finally get $$\begin{aligned} \nonumber \delta S_n(A)&=&\frac{\pi c^2 \mu (n-1) (n+1)^2}{576 n^3 \epsilon ^2}\\ \nonumber &\;&-\frac{\pi ^3 c^2 \mu (n-1) (n+1)^2 \left(11 \cos \left(\frac{2 \pi l}{L}\right)+19\right) \csc ^2\left(\frac{\pi l}{L}\right)}{864 L^2 n^3}\\ \nonumber &\;&+\frac{\pi ^3 c^2 \mu (n-1) (n+1)^2 \cot ^2\left(\frac{\pi l}{L}\right) \log \left(\frac{L \sin \left(\frac{\pi l}{L}\right)}{2 \pi \epsilon }\right)}{36 L^2 n^3}\\ &\;&+O(\epsilon^2).\end{aligned}$$ A. B. Zamolodchikov, “Expectation value of composite field T anti-T in two-dimensional quantum field theory," arXiv:hep-th/0401146 \[hep-th\]. F. A. Smirnov and A. B. Zamolodchikov, “On space of integrable quantum field theories," *Nucl. Phys.* [**B915**]{} (2017) 363-383, arXiv:1608.05499 \[hep-th\]. A. Cavaglià, S. Negro, I. M. Szécsényi and R. Tateo, “$T \bar{T}$-deformed 2D Quantum Field Theories," *JHEP* [**10**]{} (2016) 112, arXiv:1608.05534 \[hep-th\]. J. Cardy, “The TT deformation of quantum field theory as a stochastic process," arXiv:1801.06895 \[hep-th\]. O. Aharony and T. Vaknin, “The TT\* deformation at large central charge," arXiv:1803.00100 \[hep-th\]. G. Bonelli, N. Doroud and M. Zhu, “$T\bar{T}$-deformations in closed form," arXiv:1804.10967 \[hep-th\]. S. Datta and Y. Jiang, “$T\bar{T}$ deformed partition functions," arXiv:1806.07426 \[hep-th\]. L. McGough, M. Mezei and H. Verlinde, “Moving the CFT into the bulk with $ T\overline{T} $," *JHEP* [**04**]{} (2018) 010, arXiv:1611.03470 \[hep-th\]. J. Cardy, “Quantum Quenches to a Critical Point in One Dimension: some further results,” J. Stat. Mech.  [**1602**]{}, no. 2, 023103 (2016) doi:10.1088/1742-5468/2016/02/023103 \[arXiv:1507.07266 \[cond-mat.stat-mech\]\]. D. Marolf and M. Rangamani, “Causality and the AdS Dirichlet problem,” JHEP [**1204**]{}, 035 (2012) doi:10.1007/JHEP04(2012)035 \[arXiv:1201.1233 \[hep-th\]\]. A. Giveon, N. Itzhaki and D. Kutasov, “$T\bar{T}$ and LST,” *JHEP* [**07**]{} (2017) 122, arXiv:1701.05576 \[hep-th\]. S. Chakraborty, A. Giveon, N. Itzhaki and D. Kutasov, “Entanglement Beyond AdS,” arXiv:1805.06286 \[hep-th\]. S. Dubovsky, V. Gorbenko and M. Mirbabayi, “Asymptotic fragility, near $AdS_2$ holography and $T\bar{T}$,” *JHEP* [**09**]{} (2017) 136, arXiv:1706.06604 \[hep-th\]. S. Dubovsky, V. Gorbenko and G. Hernndez-Chifflet, “$T\bar{T}$ Partition Function from Topological Gravity,” arXiv:1805.07386 \[hep-th\]. V. Shyam, “Background independent holographic dual to $T\bar{T}$ deformed CFT with large central charge in 2 dimensions,” *JHEP* [**10**]{} (2017) 108, arXiv:1707.08118 \[hep-th\]. G. Giribet, “$T\bar{T}$-deformations, AdS/CFT and correlation functions,” *JHEP* [**02**]{} (2018) 114, arXiv:1711.02716 \[hep-th\]. P. Kraus, J. Liu and D. Marolf, “Cutoff $AdS_3$ versus the $T\bar{T}$ deformation," arXiv:1801.02714 \[hep-th\]. W. Cottrell and A. Hashimoto, “Comments on $T\bar{T}$ double trace deformations and boundary conditions,” arXiv:1801.09708 \[hep-th\]. M. Taylor, “TT deformations in general dimensions,” arXiv:1805.10287 \[hep-th\]. M. Guica, “An integrable Lorentz-breaking deformation of two-dimensional CFTs,” arXiv:1710.08415 \[hep-th\]. A. Bzowski and M. Guica, “The holographic interpretation of $J \bar T$-deformed CFTs,” arXiv:1803.09753 \[hep-th\]. S. Chakraborty, A. Giveon and D. Kutasov, “$J\bar{T}$ deformed $CFT_2$ and String Theory,” arXiv:1806.09667 \[hep-th\]. L. Apolo and W. Song, “Strings on warped AdS$_3$ via $T\bar{J}$ deformations,” arXiv:1806.10127 \[hep-th\]. S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT," *Phys. Rev. Lett.* [**96**]{} (2006) 181602, arXiv:hep-th/0603001 \[hep-th\]. S. Ryu and T. Takayanagi, “Aspects of Holographic Entanglement Entropy,” JHEP [**0608**]{}, 045 (2006) doi:10.1088/1126-6708/2006/08/045 \[hep-th/0605073\]. W. Donnelly and V. Shyam, “Entanglement entropy and $T \overline{T}$ deformation,” arXiv:1806.07444 \[hep-th\]. P. Calabrese and J. Cardy, “Entanglement entropy and conformal field theory," *J. Phys.* [**A42**]{} (2009) 504005, arXiv:0905.4013 \[cond-mat.stat-mech\]. A. Lewkowycz and J. Maldacena, “Generalized gravitational entropy," *JHEP* [**08**]{} (2013) 090, arXiv:1304.4926 \[hep-th\]. J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J. Theor. Phys.  [**38**]{}, 1113 (1999) \[Adv. Theor. Math. Phys.  [**2**]{}, 231 (1998)\] \[hep-th/9711200\]. J. D. Brown and M. Henneaux, “Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity,” Commun. Math. Phys.  [**104**]{}, 207 (1986). A. Strominger, “Black hole entropy from near horizon microstates,” JHEP [**9802**]{}, 009 (1998) \[hep-th/9712251\]. X. Dong, “The Gravity Dual of Renyi Entropy,” Nature Commun.  [**7**]{}, 12472 (2016) doi:10.1038/ncomms12472 \[arXiv:1601.06788 \[hep-th\]\]. K. Skenderis and S. N. Solodukhin, “Quantum effective action from the AdS / CFT correspondence,” Phys. Lett. B [**472**]{}, 316 (2000) doi:10.1016/S0370-2693(99)01467-7 \[hep-th/9910023\]. L. Y. Hung, R. C. Myers, M. Smolkin and A. Yale, “Holographic Calculations of Renyi Entropy,” JHEP [**1112**]{}, 047 (2011) doi:10.1007/JHEP12(2011)047 \[arXiv:1110.1084 \[hep-th\]\]. T. Faulkner, “The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT,” arXiv:1303.7221 \[hep-th\]. T. Barrella, X. Dong, S. A. Hartnoll and V. L. Martin, “Holographic entanglement beyond classical gravity,” JHEP [**1309**]{}, 109 (2013) doi:10.1007/JHEP09(2013)109 \[arXiv:1306.4682 \[hep-th\]\]. K. Krasnov, “Holography and Riemann surfaces,” Adv. Theor. Math. Phys.  [**4**]{}, 929 (2000) \[hep-th/0005106\]. K. Krasnov, “On holomorphic factorization in asymptotically AdS 3-D gravity,” Class. Quant. Grav.  [**20**]{}, 4015 (2003) doi:10.1088/0264-9381/20/18/311 \[hep-th/0109198\]. B. Chen and J. J. Zhang, “On short interval expansion of Rényi entropy,” JHEP [**1311**]{}, 164 (2013) doi:10.1007/JHEP11(2013)164 \[arXiv:1309.5453 \[hep-th\]\].\ B. Chen, J. Long and J. j. Zhang, “Holographic Rényi entropy for CFT with W symmetry,” JHEP [**1404**]{}, 041 (2014) \[arXiv:1312.5510 \[hep-th\]\]. B. Chen and J. q. Wu, “Single interval Renyi entropy at low temperature,” JHEP [**1408**]{}, 032 (2014) doi:10.1007/JHEP08(2014)032 \[arXiv:1405.6254 \[hep-th\]\].\ B. Chen and J. q. Wu, “Holographic calculation for large interval Rényi entropy at high temperature,” Phys. Rev. D [**92**]{}, no. 10, 106001 (2015) doi:10.1103/PhysRevD.92.106001 \[arXiv:1506.03206 \[hep-th\]\]. [^1]: bchen01@pku.edu.cn, linchen91@pku.edu.cn, pxhao@pku.edu.cn [^2]: There is another interesting generalization of the $T\overline{T}$ deformation, the so-called $T\overline{J}$-deformation[@Guica:2017lia], which breaks the Lorentz symmetry but is still solvable. For other relevant studies on this kind of deformation, see [@Bzowski:2018pcy; @Chakraborty:2018vja; @Apolo:2018qpq].

--- abstract: 'We investigate radio-frequency (rf) reflectometry in a tunable carbon nanotube double quantum dot coupled to a resonant circuit. By measuring the in-phase and quadrature components of the reflected rf signal, we are able to determine the complex admittance of the double quantum dot as a function of the energies of the single-electron states. The measurements are found to be in good agreement with a theoretical model of the device in the incoherent limit. Besides being of fundamental interest, our results present an important step forward towards non-invasive charge and spin state readout in carbon nanotube quantum dots.' author: - 'S.J. Chorley' - 'J. Wabnig' - 'Z.V. Penfold-Fitch' - 'K.D. Petersson' - 'J. Frake' - 'C.G. Smith' - 'M.R. Buitelaar' title: Measuring the complex admittance of a carbon nanotube double quantum dot --- An important requirement in any quantum information processing scheme is fast manipulation and readout of the quantum system in which the quantum information is encoded. This requires an understanding of the response of the quantum system at finite frequencies which, in the case of an electronic device, involves an understanding of its complex admittance [@Buttiker1; @Buttiker2]. Of particular interest in the context of quantum information processing are double quantum dots which are widely used to define charge and spin qubits [@Loss]. However, while double quantum dots have been investigated in detail over the last decade, experiments to measure and analyze their complex admittance have not yet been performed and this topic has only recently been addressed theoretically [@Cottet]. The admittance of quantum dots at finite frequencies is non-trivial as exemplified by recent experiments on single quantum dots [@Gabelli; @Delbecq]. The physics is even richer for double quantum dots as internal charge dynamics, i.e. charge transfer between the quantum dots, has to be taken into account. However, the dependence of the admittance on the internal charge dynamics also provides a route towards charge and spin state readout [@Petersson]. In this work we present a detailed experimental study of the complex admittance of a carbon nanotube double quantum dot which is measured using rf reflectometry techniques. The measurements are compared with a theoretical model of the device where we use a density matrix approach to calculate the double quantum dot admittance. The good quantitative agreement between the experimental and theoretical results allows us to determine the effective conductance and susceptance of the double dot as a function of the energies of the single-electron states. Our measurements thus present a first quantitative analysis of the complex admittance of a double quantum dot. The demonstrated technique also provides the basis for a simple and fast detection scheme for charge and spin state readout in carbon nanotubes - a material with considerable potential for spin-based quantum information processing [@Kuemmeth; @Galland; @Churchill1; @Churchill2; @Jespersen; @Chorley] - without the need for a separate charge detector [@Biercuk]. ![\[Fig1\]**(a)** (color online) Schematic of the carbon nanotube double quantum dot and resonant circuit. The device is connected to a 50 $\Omega$ transmission line for the rf reflectometry measurements. A dc signal is applied via a bias-tee. **(b)** Measured amplitude and phase response for the resonant circuit relative to the transmission line background.](Fig1.eps){width="85mm"} The device we consider is a carbon nanotube grown by chemical vapour deposition on degenerately doped Si terminated by 300 nm SiO$_2$, see Fig. 1(a). The nanotube is contacted by Au source and drain electrodes which form the outer tunnel barriers of the quantum dots. A capacitively coupled top gate, separated from the nanotube by $\sim$ 3 nm AlO$_x$, is used to define a tunable coupling between the dots while two plunger gates vary the energies of the dots. The nanotube device is embedded in a resonant circuit consisting of a parasitic capacitance $C=2.5$ pF and on-chip inductor $L=180$ nH [@capacitance]. The circuit has a resonance frequency $f_0 \sim 236$ MHz and loaded quality factor $Q \sim 1/Z_0 \sqrt{L/C} \sim 5.4$, where $Z_0 = 50$ $\Omega$ is the characteristic impedance of the transmission lines, see Fig. 1(b). We note that higher quality factors ($Q \sim 30$) were readily obtained on nanotube devices grown on undoped Si and quartz substrates. A highly doped Si substrate is used here because of its convenience as a back gate. ![\[Fig2\]**(a)** Schematic of the carbon nanotube double quantum dot device. The rf signal is applied to the right electrode. **(b,c)** Demodulated response of the resonant circuit as a function of $V_L$ and $V_R$. In each plot the background signal measured inside a stable charge region is subtracted. The top gate voltage is set to $V_t = 0$ V. The back gate voltages are $V_{bg} = 5$ V and $V_{bg} = -0.18$ V in (b) and (c), respectively. **(d)** Measured rf signal (right) and dc current (left) for a triple point pair in the presence of an applied bias $V_{sd} = 1$ mV. The full scale of the dc current represents 150 pA.](Fig2.eps){width="85mm"} The nanotube device is characterized by dc transport measurements and rf reflectometry in a dilution refrigerator with a base temperature $T \sim 40$ mK. The dc signal is applied via a bias-tee while the rf excitation is directed to the source electrode of the double quantum dot through the coupled port of a directional coupler, connected to the sample holder via a stainless-steel semi-rigid coaxial line, see Fig. 1(a). The reflected signal is sent back via a cryogenic preamplifier which is thermally anchored at 4 K, followed by room temperature amplification and demodulation by mixing with the reference signal. We drive the resonator at its resonant frequency with an amplitude at the double dot $V_{RF} \sim 10$ $\mu$eV. Depending on the energies of the quantum dots, the oscillating potential on the source electrode may induce charge transfer (tunneling) between the electrodes and the quantum dots and/or redistribution of charge between the two dots. The resulting oscillating current will generally have both out-of-phase and in-phase components with respect to the driving voltage which give rise to a resonance frequency shift and damping of the resonator. The response of the double quantum dot is thus characterized by a complex admittance which can be deduced by measuring the phase and amplitude of the reflected rf signal. We first determine the various energy scales of the double quantum dot by measuring the dc and rf response as a function of the side-gate voltages $V_L$ and $V_R$ which modulate the left and right quantum dot energies, respectively, see Fig. 2. For simplicity in-phase and out-of-phase components of the rf response are shown together here (i.e. the demodulated signal is sensitive to both amplitude and phase). The nanotube device is fully tunable by the top $V_t$ and back gate $V_{bg}$ voltages. The stability diagram of Fig. 2(b), for example, illustrates the situation where the two quantum dots are fully decoupled: no dc current could be detected but a strong rf response is observed at charge transitions of the right quantum dot. For more positive back gate voltages, the stability diagram evolves into a honeycomb pattern that is characteristic of the double quantum dot investigated here. At finite bias $V_{sd} = 1$ mV, bias triangles are observed at the triple points in both the dc and rf response, see Fig. 2(d), which allows us to extract the characteristic energy scales of the double quantum dot. We obtain charging energies $U_L \sim 6.5$ meV and $U_R \sim 5$ meV for the left and right quantum dots, respectively, and an inter-dot electrostatic coupling energy $U'=0.6$ meV. The analysis also allows us to deduce the various geometric capacitances of the device [@Wiel]. We did not observe any obvious four-fold periodicity in the stability diagrams, an indication that the orbital degeneracy of the nanotubes is broken [@Liang; @Buitelaar]. To determine the complex admittance of the nanotube device we measured the in-phase and quadrature components of the reflected signal for the double quantum dot which allows us to extract the amplitude and phase information, see Fig. 3(a) and (b). The observed phase shifts $\Delta \Phi$, relative to the phase measured inside a stable charge region, are strongest at the $(n,m)-(n,m+1)$ charge transition, with a signal of about half the strength observed at the internal $(n+1,m)-(n,m+1)$ charge transition. The amplitude shifts on the other hand are concentrated around the triple points. Using standard circuit analysis [@Pozar], the measured phase signal at resonance can be related to a change in capacitance as $\Delta \Phi/\Delta C \sim 2Q/C$, where $C = 2.5$ pF for the circuit considered here. At the $(n,m)-(n,m+1)$ charge transition, for example, the measured phase shift of $\sim 0.18$ degrees implies $\Delta C = 0.74$ fF. Note that this is several orders of magnitude larger than the geometric capacitances of carbon nanotube quantum dots which are typically in the aF range. The amplitude modulation is related to the double dot resistance $R$ via $\Delta |\Gamma| /|\Gamma| = 2Q^2Z_0/R$. The measured damping $\Delta |\Gamma| /|\Gamma| \sim 0.2 \%$ at the triple points therefore implies $R \sim 1.5$ M$\Omega$ which is in agreement with dc conductance measurements. In the following we compare the experimental results with a theoretical model of the device, the full results of which are shown in Fig. 3(c) and (d) which show the real ($R^{-1}_{eff}$) and imaginary ($C_{eff}$) part of the admittance as a function of the energies of the quantum dots’ single-electron states. We model the the admittance measured at the source electrode using a master equation approach similar to Ref. [@Cottet]. We take into account all relevant many body states, i.e. for Fig. 3(c) and (d) the empty state and all one and two electron states. For a given set of gate voltages $V_{L}$ , $V_{R}$ we obtain the steady state density matrix at zero bias. We then calculate the current flowing into the source electrode as a linear response to a periodic driving of the source potential. This enables us to deduce the double quantum dot admittance at the source electrode. We assume that the driving frequency is too small to induce transitions between quantum dot eigenstates and we therefore treat the perturbation as adiabatic. The current into the source has two contributions: the particle current due to tunneling from the source to the left dot and the displacement current induced on the source capacitance by the tunneling induced redistribution of charges on the quantum dot. We include spin relaxation as well as phonon assisted tunneling between the left and the right dot. Overall, we obtain good agreement with the experimental results as demonstrated by Fig. 3. ![\[Fig3\]**(a)** Measured phase shift of the carbon nanotube double quantum dot device.**(b)** Measured amplitude response. **(c)** Calculated effective capacitance and phase shift of the double quantum dot. The parameters used for the model calculations are $T=80$ mK. $U'=0.6$ meV, $t=40$ $\mu$eV, $\omega_0 = 1.2$ GHz, $\gamma_L = 0.15 \gamma_R$. **(d)** Calculated conductance and damping for the same parameters as used in (c).](Fig3.eps){width="85mm"} For a physical understanding of the results (and the model) it is instructive to consider several limiting cases. Let us first consider transitions between $(n,m)$ and $(n,m+1)$ charge states such that direct charge transfer between the two electrodes (a dc current) and internal transitions of the double quantum dot can be neglected. An oscillating potential $V_{RF}(t)$ on the right electrode modulates the energy difference $\delta\epsilon (t)$ between the right quantum dot states and the Fermi energy of the lead as $\delta\epsilon (t) = \epsilon_0 -e \alpha V_{RF}(t)$, where $\epsilon_0$ is an offset, tunable with the plunger gates. The constant $e\alpha$ converts between voltage and energy and its value depends on the various geometric capacitances of the device [@Wiel]. As a results of the oscillating potential, charge moves back and forth between the right electrode and right quantum dot. Depending on the ratio of the tunnel rate $\gamma_R$ and angular driving frequency $\omega_0$, the induced current has both in-phase and out-of-phase components with respect to the voltage which can be expressed as a complex admittance $Y(\omega_0)= R_{eff}^{-1} + j \omega_0 C_{eff}$. In the incoherent limit, i.e. for $h \gamma_R \ll k_B T$ the terms are given by [@Gabelli]: $$\label{Reff} R_{eff}=\frac{4 k_B T}{e^2 \alpha^2 \gamma_R} \left(1 + \frac{\gamma_R^2}{\omega^2_0}\right)$$ $$\label{Ceff} C_{eff}=\frac{e^2 \alpha^2}{4 k_B T}\frac{1}{1+\frac{\omega^2_0}{\gamma_R^2}}$$ As expressed by Eqs. (1) and (2), both the conductance and capacitance are dependent on the ratio of the tunnel coupling and angular driving frequency. In the transparent limit ($\gamma \gg \omega_0$), the effective capacitance can be approximated by $C_{eff} \approx e^2 \alpha^2 / 4 k_B T$. The conductance has a vanishing contribution, i.e. $R^{-1}_{eff}\rightarrow 0$, in both the transparent and opaque limit. This can be understood intuitively as for very large $\gamma$ electrons will tunnel on the quantum dot as soon as this is energetically possible and no energy is dissipated in the process. In the opaque limit, tunneling occurs out of equilibrium. However, as the probability of a tunnel event on the timescale of the driving frequency vanishes for weak coupling, no energy is dissipated either. Damping is therefore strongest in the intermediate regime where $\gamma \sim \omega$, as recently observed for single-electron tunneling devices coupled to electrical [@Persson; @Ferguson] and mechanical [@Jun] resonators. We can compare these predictions with the experimental data obtained on the carbon nanotube double quantum dot. Using $\alpha \approx 0.35$, as determined from dc transport experiments, we obtain quantitative agreement between the experiment ($\Delta \Phi \sim 0.18$ degrees) and the change in capacitance predicted by Eq. 2 for the $(n,m)$-$(n,m+1)$ transition, if we assume an effective electron temperature $T \sim 80$ mK [@limit]. This is somewhat larger than the base temperature of the dilution fridge, most likely due to heat loading of the device by the coaxial cables. The estimate of the effective capacitance at the $(n,m)$-$(n+1,m)$ transition, i.e. due to charge transfer between the left lead and left dot also follows Eq. 2. However in this case, the prefactor is much smaller $\alpha \sim 0.05$ reflecting the weak coupling between the left quantum dot and the right electrode at which the rf signal is applied. Since the expected phase shift $\propto \alpha^2$ the response along this line is too weak to be detected. We observe a very weak amplitude modulation along the $(n,m)$-$(n,m+1)$ charge transition, $\Delta |\Gamma| /|\Gamma| \sim 0.05 \%$, most clearly observed in the top half of Fig. 3(b). This is consistent with damping due to out-of-equilibrium tunneling and in agreement with the theoretical calculations of the effective conductance shown in Fig. 3(d). The fact that the signal is rather weak implies that $\gamma_R \gtrsim \omega_0$ in our device. Stronger damping is observed at the triple points where the behavior of the device is similar to that of the conventional rf single-electron transistor [@Schoelkopf]. Of particular interest is the phase signal along the polarization line which reflects the movement of electrons between the quantum dots, i.e. transitions between the $(n+1,m)$ and $(n,m+1)$ charge states. In this case, the amplitude and width of the signal is not set by temperature but the tunnel coupling $t$. More precisely, it has been predicted [@Cottet] that, $C_{eff} = e^2 \beta^2/ 4t$, where here $\beta \sim 0.6$ converts between $V_{RF}$ and detuning $\epsilon_L - \epsilon_R$. The predictions can be directly verified with our experiments. The tunnel coupling can be deduced from the stability diagram [@Graber], yielding $t \sim 40$ $\mu$eV, and the estimate for $C_{eff} \approx 0.46$ fF therefore has no free parameters. This result is in excellent agreement with the experimental data of Fig. 3(a) where we measure a phase shift of $\sim 0.11$ degrees, roughly a factor $\sim 2$ smaller than that observed at the $(n,m)-(n+1,m)$ transition, as also seen in the model calculation of Fig. 3(c). We note that we did not observe spin blockade in dc transport experiments on the carbon nanotube double quantum dot studied here and the effect is therefore not taken into account in the analysis. Nevertheless, spin blockade has been observed previously in carbon nanotubes [@Churchill1; @Churchill2; @Chorley]. In the spin blockade regime, transitions between a (1,1) and (0,2) charge state directly depend on whether the (1,1) state is a singlet or triplet. Since the (0,2) charge state is a singlet by virtue of the Pauli principle the (1,1) triplet state will be a blocked state which is reflected in the admittance [@Cottet]. Measurements of the admittance of a carbon nanotube double quantum dot as demonstrated here can therefore also be used for spin state readout [@Petersson]. In conclusion, we have measured the complex admittance of a carbon nanotube double quantum using rf reflectometry. The results are in quantitative agreement with a theoretical model of the device of which several limiting cases are discussed in detail. The demonstrated technique is of particular interest as a tool for fast and sensitive charge and spin state readout of carbon nanotube quantum dots. A further interesting possibility is to extend the measurement to the coherent limit where the nanotubes are more strongly coupled to the leads. This should allow the observation of a quantized charge relaxation resistance [@Gabelli] and possible deviations thereof in long nanotubes for which interactions (Luttinger liquid physics) become important [@Martin; @Burke]. We thank David Cobden and Jiang Wei for the carbon nanotube growth and Andrew Ferguson for useful discussion. This work was supported by the Newton Trust and the Royal Society (M.R.B.). [10]{} M. Büttiker, A. Prêtre, and H. Thomas, Phys. Rev. Lett. **70**, 4114 (1993). A. Prêtre, H. Thomas, and M. Büttiker, Phys. Rev. B **54**, 8130 (1996). D. Loss and D.P. DiVincenzo, Phys. Rev. A **57**, 120 (1998). A. Cottet, C. Mora, and T. Kontos, Phys. Rev. B **83**, 121311 (2011). J. Gabelli, G. Fève, J.-M. Berroer, B. Plaçais, A. Cavanna, B. Etienne, Y. Jin, and D.C. Glattli, Science **313**, 499 (2006). M.R. Delbecq, V. Schmitt, F.D. Parmentier, N. Roch, J.J. Viennot, G. Fève, B. Huard, C. Mora, A. Cottet, and T. Kontos, arXiv:1108.4371. K.D. Petersson, C.G. Smith, D. Anderson, P. Atkinson, G.A.C. Jones, and D.A. Ritchie, Nano Lett. **10**, 2789 (2010). F. Kuemmeth, S. Ilani, D.C. Ralph, and P.L. McEuen, Nature (London) **452**, 448 (2008). C. Galland and A. Imamoǧlu, Phys. Rev. Lett. **101**, 157404 (2008). H.O.H. Churchill, A.J. Bestwick, J.W. Harlow, F. Kuemmeth, D. Marcos, C.H. Stwertka, S.K. Watson, and C.M. Marcus, Nature Phys. **5**, 321 (2009). H.O.H. Churchill, F. Kuemmeth, J.W. Harlow, A.J. Bestwick, E.I. Rashba, K. Flensberg, C.H. Stwertka, T. Taychatanapat, S.K. Watson, and C.M. Marcus, Phys. Rev. Lett. **102**, 166802 (2009). T.S. Jespersen, K. Grove-Rasmussen, J. Paaske, K. Muraki, T. Fujisawa, J. Nyg[å]{}rd, and K. Flensberg, Nature Phys. **7**, 348 (2011). S.J. Chorley, G. Giavaras, J. Wabnig, G.A.C. Jones, C.G. Smith, G.A.D. Briggs, and M.R. Buitelaar, Phys. Rev. Lett. **106**, 206801 (2011). M.J. Biercuk, D.J. Reilly, T.M. Buehler, V.C. Chan, J.M. Chow, R.G. Clark, and C.M. Marcus, Phys. Rev. B **73**, 201402(R) (2006). The main component of the capacitance is the parallel plate capacitor formed by the bond pad and Si back gate. W.G. van der Wiel, S. De Franceschi, J.M. Elzerman, T. Fujisawa, S. Tarucha, and L.P. Kouwenhoven, Rev. Mod. Phys. **75**, 1 (2002). W.J. Liang, M. Bockrath, and H. Park, Phys. Rev. Lett. **88**, 126801 (2002). M.R. Buitelaar, A. Bachtold, T. Nussbaumer, M. Iqbal and C. Schönenberger, Phys. Rev. Lett. **88**, 156801 (2002). D.M. Pozar, *Microwave Engineering*, 3rd ed. (Wiley, New York, 2005). F. Persson, C.M. Wilson, M. Sandberg, G. Johansson, and P. Delsing, Nano Lett. **10**, 953 (2010). C. Ciccarelli and A.J. Ferguson, arXiv:1108.3463. Z. Jun, M. Brink, P.L. McEuen, Nano Lett. **8**, 2399 (2008). For this temperature $k_B T / h \gamma_R \sim 2$. R.J. Schoelkopf, P. Wahlgren, A.A. Kozhevnikov, P. Delsing, and D.E. Prober, Science **280**, 1238 (1998). M.R. Gräber, W.A. Coish, C. Hoffmann, M. Weiss, J. Furer, S. Oberholzer, D. Loss, and C. Schönenberger, Phys. Rev. B **74**, 075427 (2006). Y. Hamamoto, T. Jonckheere, T. Kato, and T. Martin, Phys. Rev. B **81**, 153305 (2010). P.J. Burke, IEEE T. Nanotechnol. **1**, 129 (2002).

--- abstract: 'The fuzzy quantification model $\mathcal{F}^{A}$ has been identified as one of the best behaved quantification models in several revisions of the field of fuzzy quantification. This model is, to our knowledge, the unique one fulfilling the strict *Determiner Fuzzification Scheme* axiomatic framework that does not induce the standard min and max operators. The main contribution of this paper is the proof of a convergence result that links this quantification model with the Zadeh’s model when the size of the input sets tends to infinite. The convergence proof is, in any case, more general than the convergence to the Zadeh’s model, being applicable to any quantitative quantifier. In addition, recent revisions papers have presented some doubts about the existence of suitable computational implementations to evaluate the $\mathcal{F}^{A}$ model in practical applications. In order to prove that this model is not only a theoretical approach, we show exact algorithmic solutions for the most common linguistic quantifiers as well as an approximate implementation by means of Monte Carlo. Additionally, we will also give a general overview of the main properties fulfilled by the $\mathcal{F}^{A}$ model, as a single compendium integrating the whole set of properties fulfilled by it has not been previously published.' author: - 'Félix Díaz-Hermida, Marcos Matabuena, Juan C. Vidal[^1]' bibliography: - 'biblio.bib' title: 'The $\mathcal{F}^{A}$ Quantifier Fuzzification Mechanism: analysis of convergence and efficient implementations.' --- Fuzzy quantification, theory of generalized quantifiers, quantifier fuzzification mechanisms, Zadeh’s quantification model. Introduction ============ great range of models have been proposed for the evaluation of fuzzy quantified sentences, being [@Barro02; @Delgado00; @Delgado14; @Sanchez16; @DiazHermida00; @DiazHermida02-FuzzySets; @DiazHermida04IPMU; @DiazHermida10Arxiv; @DiazHermida04-IEEE; @Dubois85; @DVORAK2014; @Glockner03-Generalized; @Glockner06Libro; @Liu98; @Ming2006; @Ralescu95; @Yager83; @Yager88; @Yager2016; @Zadeh83] only an example. Several revision papers have also been published, being [@Delgado14] possibly the one that makes a more exhaustive comparison. Other revision works is worth to mention are [@Barro02; @Delgado00; @Glockner06Libro; @DiazHermida06Tesis; @Yager2016]. There also exists an specific paper [@DiazHermida17-FuzzySets], comparing the models following the quantification framework presented in [@Glockner06Libro]. Moreover, fuzzy quantifiers have been used in a wide range of applications like fuzzy control, temporal reasoning, fuzzy databases, information retrieval, multi-criteria decision making, data fusion, natural language generation, etc. In [@Delgado14] a list of the main applications of fuzzy quantifiers is presented. This paper is devoted to present some relevant new results of the $\mathcal{F}^{A}$ *quantification model* proposed in [@DiazHermida04-IEEE; @DiazHermida04IPMU]. The main result we will present is a convergence proof that, in the particular case of proportional quantifiers, assures the convergence of the $\mathcal{F}^{A}$ model to the *Zadeh’s quantification model* [@Zadeh83] when the intersection of fuzzy sets is modelled by means of the probabilistic *tnorm* operator. Moreover, although several revisions of the fuzzy quantification field [@Delgado14; @Yager2016; @DiazHermida17-FuzzySets] have presented this model as one of the best fuzzy quantification models available, some doubts persist about the possibility of efficiently implementing it [@Delgado14; @Yager2016]. For this reason, we will also provide efficient computational implementations of the $\mathcal{F}^{A}$ quantification model for the most common linguistic quantifiers as well as the explanation of how to extend these implementations to other types of linguistic quantifiers. The definition of the $\mathcal{F}^{A}$ quantification model follows the Glöckner’s approximation to fuzzy quantification [@Glockner06Libro] instead of the common one based on type I, type II quantified expressions proposed by Zadeh [@Zadeh83]. Glöckner’s approximation generalizes the concept of generalized classic quantifier [@Barwise81] (second order predicates or set relationships) to the fuzzy case as fuzzy relationships between fuzzy sets. Following this idea he recasts the problem of evaluating fuzzy quantified expressions as a problem of searching for adequate mechanisms to transform semi-fuzzy quantifiers (specification means) into fuzzy quantifiers (operational means). The author denominates these transformation mechanisms *Quantifier Fuzzification Mechanism* (*QFMs*). The followed approach also generalizes the *Theory of Generalized Quantifiers* (*TGQ*), that deals with the analysis and modelling of the phenomena of quantification in natural language, [@Barwise81] to the fuzzy case. In his proposal the author also defined a rigorous axiomatic framework to ensure the good behavior of QFMs. Models fulfilling this strict framework are denominated *Determiner Fuzzification Schemes* (*DFSs*) and they comply with a broad set of properties that guarantee a good behavior from a linguistic and fuzzy point of view. In [@Sanchez16] or [@Glockner06Libro] can be consulted a comparison between Zadeh’s and Glöckner’s approaches. The $\mathcal{F}^{A}$ model is a QFM which fulfills the strict axiomatic framework proposed by Glöckner, which makes it a DFS. It is to our knowledge the unique *non-standard DFS* (a DFS non inducing the standard max and min operators). The structure of this paper is the following. First, we will introduce the fuzzy quantification framework proposed by Glöckner. Second, we will present two alternative definitions of the $\mathcal{F}^{A}$ QFM, one based on the use of fuzzy connectives and the other based on a probabilistic interpretation of fuzzy sets. After that, we will give a brief review of the main properties fulfilled by the $\mathcal{F}^{A}$ QFM. The following section is dedicated to introduce the convergence results of the $\mathcal{F}^{A}$ QFM, that as a particular case includes the convergence to the Zadeh’s model. However, the rate of convergence is too slow to be of utility in most applications, but thanks to this result is argued that we can expect very good approximations of the $\mathcal{F}^{A}$ QFM by means of Monte Carlo. The final section is devoted to present exact and approximate algorithmic implementations. The fuzzy quantification framework ================================== In the specification of the fuzzy quantification framework [@Glockner06Libro], the author rewrote the problem of defining fuzzy quantification models as the problem of looking for adequate means to convert *semi-fuzzy quantifiers* (i.e., mechanisms adequate to specify the meaning of linguistic quantifiers) into *fuzzy quantifiers* (i.e., operational means adequate to apply semi-fuzzy quantifiers to fuzzy inputs). In this framework, fuzzy quantifiers are just a generalization of crisp quantifiers to fuzzy sets. We will show below the definition of classic quantifiers conforming to TGQ. A two valued (generalized) quantifier on a base set $E\neq\varnothing$ is a mapping $Q:\mathcal{P}\left( E\right) ^{n}\longrightarrow\mathbf{2}$, where $n\in\mathbb{N}$ is the arity (number of arguments) of $Q$, $\mathbf{2}=\left\{ 0,1\right\} $ denotes the set of crisp truth values, and $\mathcal{P}\left( E\right) $ is the powerset of $E$. Below we show two examples of classic quantifiers: $$\begin{aligned} & \mathbf{all}\left( Y_{1},Y_{2}\right) = Y_{1}\subseteq Y_{2}\\ & \mathbf{at\_least\_60\%}\left( Y_{1},Y_{2}\right) \qquad = \left\{ \begin{array} [c]{cc}\frac{\left\vert Y_{1}\cap Y_{2}\right\vert }{\left\vert Y_{1}\right\vert }\geq0.60 & Y_{1}\neq\varnothing\\ 1 & Y_{1}=\varnothing \end{array} \right. .\end{aligned}$$ From here on, we will denote $\left\vert E\right\vert =m$. A fuzzy quantifier assigns a fuzzy value to each possible choice of $X_{1},\ldots,X_{n}\in\widetilde{\mathcal{P}}\left( E\right) $, where by $\widetilde{\mathcal{P}}\left( E\right) $ we are denoting the fuzzy powerset of $E$. [@Glockner06Libro definition 2.6] An n-ary fuzzy quantifier $\widetilde{Q}$ on a base set $E\neq\varnothing$ is a mapping $\widetilde {Q}:\widetilde{\mathcal{P}}\left( E\right) ^{n}\longrightarrow \mathbf{I=}\left[ 0,1\right] $. The next example shows a possible definition of the fuzzy quantifier $\widetilde{\mathbf{all}}:\widetilde{\mathcal{P}}\left( E\right) ^{2}\longrightarrow\mathbf{I}$: $$\widetilde{\mathbf{all}}\left( X_{1},X_{2}\right) =\inf\left\{ \max\left( 1-\mu_{X_{1}}\left( e\right) ,\mu_{X_{2}}\left( e\right) \right) :e\in E\right\}$$ where by $\mu_{X}\left( e\right) $ we are denoting the membership function of $X\in\widetilde{\mathcal{P}}\left( E\right) $. Previous definition of the fuzzy quantifier $\widetilde{\mathbf{all}}$ seems plausible. However, the reader could think of in other possible plausible expressions to model $\widetilde{\mathbf{all}}$ by simply changing the *tconorm* operator *max* by other *tconorm* operator. For other quantifiers, like *‘at least sixty percent’*, the problem of establishing adequate models is far from obvious. In the search of possible solutions for defining fuzzy quantifiers, in [@Glockner06Libro] the concept of semi-fuzzy quantifier was introduced to work as a ‘middle point’ between classic and fuzzy quantifiers. Semi-fuzzy quantifiers are close but more powerful than the Zadeh’s concept of linguistic quantifiers [@Zadeh83]. Semi-fuzzy quantifiers only accept crisp arguments, as classic quantifiers, but they have a fuzzy output, as in the case of fuzzy quantifiers. Semi-fuzzy quantifiers are adequate to capture the semantics of linguistic quantified expressions. [@Glockner06Libro definition 2.8] An n-ary semi-fuzzy quantifier $Q$ on a base set $E\neq\varnothing$ is a mapping $Q:\mathcal{P}\left( E\right) ^{n}\longrightarrow\mathbf{I}$. $Q$ assigns a gradual result to each pair of crisp sets $\left( Y_{1},\ldots,Y_{n}\right) $. Some examples of semi-fuzzy quantifiers are: $$\begin{aligned} & \mathbf{about\_10}\left( Y_{1},Y_{2}\right) = T_{6,8,12,14}\left(\left\vert Y_{1}\cap Y_{2}\right\vert \right)\\ & \mathbf{about\_60\%\_or\_more}\left( Y_{1},Y_{2}\right) \qquad = \left\{ \begin{array} [c]{cc}S_{0.4,0.6}\left( \frac{\left\vert Y_{1}\cap Y_{2}\right\vert }{\left\vert Y_{1}\right\vert }\right) & X_{1}\neq\varnothing\\ 1 & X_{1}=\varnothing \end{array} \right. \end{aligned}$$ where $T_{6,8,12,14}\left( x\right) $ and $S_{0.4,0.6}\left( x\right) $ represent the ordinary trapezoidal[^2] and $S$ fuzzy numbers[^3]. We generally will denominate the fuzzy numbers used in the definition of the semi-fuzzy quantifiers as ‘support functions of the semi-fuzzy quantifiers’. Although the semantics of semi-fuzzy quantifiers is intuitive, they do not permit to evaluate fuzzy quantified expressions. In [@Glockner06Libro], the author proposes to use an additional mechanism to transform semi-fuzzy quantifiers into fuzzy quantifiers. This mechanism allows to map semi-fuzzy quantifiers into fuzzy quantifiers: [@Glockner06Libro definition 2.10] A quantifier fuzzification mechanism (*QFM*) $\mathcal{F}$ assigns to each semi-fuzzy quantifier $Q:\mathcal{P}\left( E\right) ^{n}\rightarrow\mathbf{I}$ a corresponding fuzzy quantifier $\mathcal{F}\left( Q\right) :\widetilde{\mathcal{P}}\left( E\right) ^{n}\rightarrow\mathbf{I}$ of the same arity $n\in\mathbb{N}$ and on the same base set $E$. The *QFM* $\mathcal{F}^{A}$\[SubSectionModeloFA\] ================================================= In this section we present the finite *QFM* $\mathcal{F}^{A}$ [@DiazHermida04IPMU; @DiazHermida04-IEEE; @DiazHermida06Tesis; @DiazHermida10Arxiv; @DiazHermida17-FuzzySets]. The $\mathcal{F}^{A}$ *QFM* can be defined using two different strategies. The first definition uses the equipotence concept [@Bandler80] and remains purely on the use of fuzzy operators. The second is based on a probabilistic interpretation of fuzzy sets. Both definitions are equivalent. Following [@Bandler80] the equipotence between a crisp set $Y$ and a fuzzy set $X$ can be defined as: $$Eq\left( Y,X\right) \\ = \wedge_{e\in E}\left( \mu_{X}\left( e\right) \rightarrow\mu_{Y}\left( e\right) \right) \wedge\left( \mu_{Y}\left( e\right) \rightarrow\mu_{X}\left( e\right) \right). \label{Eq_equipotence_1}$$The concept of equipotence is basically a measure of equality between fuzzy sets. Let us consider the product tnorm ($\wedge\left( x_{1},x_{2}\right) =x_{1}\cdot x_{2}$) and the Lukasiewicz implication ($\rightarrow\left( x_{1},x_{2}\right) =\min\left( 1,1-x_{1}+x_{2}\right) $). In previous expression, if $e\in Y$ then $\mu_{Y}\left( e\right) =1$ and if $e\notin Y$ then $\mu_{Y}\left( e\right) =0$. Then $$\left( \mu_{X}\left( e\right) \rightarrow\mu_{Y}\left( e\right) \right) \wedge\left( \mu_{Y}\left( e\right) \rightarrow\mu_{X}\left( e\right) \right) \\ =\left\{ \begin{array} [c]{ccc}\mu_{X}\left( e\right) & : & e\in Y\\ 1-\mu_{X}\left( e\right) & : & e\notin Y \end{array} \right. \label{Eq_equipotence_2}.$$ Then from (\[Eq\_equipotence\_1\]) and (\[Eq\_equipotence\_2\]) $$\begin{aligned} Eq\left( Y,X\right) & ={\prod\limits_{e\in Y}}\mu_{X}\left( e\right) {\prod\limits_{e\in E\backslash Y}}\left( 1-\mu_{X}\left( e\right) \right).\end{aligned}$$ Using the equipotence concept, the $\mathcal{F}^{A}$ model can be defined as: [ ]{}Let $Q:\mathcal{P}\left( E\right) ^{n}\rightarrow\mathbf{I}$ be a semi-fuzzy quantifier, $E$ finite. The *QFM* $\mathcal{F}^{A}$ is defined as: $$\mathcal{F}^{A}(X_{1}, \ldots, X_{n}) = \bigvee\limits_{Y_{1} \in \mathcal{P}(E)} \ldots \bigvee\limits_{Y_{n} \in \mathcal{P}(E)} \\ Eq(Y_1,X_1) \wedge \ldots \wedge Eq(Y_n,X_n) \wedge Q(Y_1, \ldots, Y_n)$$ where $\vee$ is the Lukasiewicz tconorm ($\vee\left( x_{1},x_{2}\right) =\min\left( x_{1}+x_{2},1\right) $), and $\wedge$ is the product tnorm ($\wedge\left( x_{1},x_{2}\right) =x_{1}\cdot x_{2}$). Now, we will present an alternative definition based on a probabilistic interpretation of fuzzy sets. The semantic interpretation of fuzzy sets based on likelihood functions [@Mabuchi92; @Thomas95; @Tursken2000Fundamentals; @Dubois2000Fundamentals] simply interprets vagueness in the data as a consequence of making a random experiment in which a set of individuals are asked about the fulfillment of a certain property. For example, let us consider $h\in\mathbb{R}$. We can define the degree of fulfillment of the statement *the value of height* $h$* is tall* as: $$\begin{aligned} \mu\left( \text{\textquotedblleft}h\text{ is }tall\text{\textquotedblright}\right) & =\Pr\left( \text{\textquotedblleft}h\text{ is considered }tall\text{\textquotedblright}\right) =\frac{\left\vert v\in V:C\left( v,\text{\textquotedblleft}h\text{ is considered }tall\text{\textquotedblright}\right) =1\right\vert }{\left\vert V\right\vert }$$ where $V$ is a set of voters and $C\left( v,\text{\textquotedblleft}h\text{ is considered }tall\text{\textquotedblright}\right) $ denotes the answer of the voter $v$ to the question $h$ is considered $tall$. We can apply the same idea to compute the probability that a crisp set $Y\in\mathcal{P}\left( E\right) $ is a representative of a fuzzy set $X\in\widetilde{\mathcal{P}}\left( E\right) $ when we suppose the base set $E$ finite and that the probabilities of the different elements are independent. The intuition is to measure the probability that only the elements in $Y$ belongs to $X$: \[DefInterpretProbConj\]Let $X\in\widetilde{\mathcal{P}}\left( E\right) $ be a fuzzy set, $E$ finite. The probability of the crisp set $Y\in \mathcal{P}\left( E\right) $ being a representative of the fuzzy set $X\in\widetilde{\mathcal{P}}\left( E\right) $ is defined as $$\begin{aligned} \Pr\left( representative_{X} = Y\right) & = m_{X}\left(Y\right) = {\prod\limits_{e\in Y}}\mu_{X}\left( e\right) {\prod\limits_{e\in E\backslash Y}}\left( 1-\mu_{X}\left( e\right) \right)\end{aligned}$$ We would like to point out that in the previous definition the probability points are the subsets of $E$. In this way the $\sigma$-algebra on which the probability is defined is $\mathcal{P}\left( E\right) $. Using expression \[DefInterpretProbConj\] the definition of the *QFM* $\mathcal{F}^{A}$ is easily made: [@DiazHermida04IPMU page. 1359]. Let $Q:\mathcal{P}\left( E\right) ^{n}\rightarrow\mathbf{I}$ be a semi-fuzzy quantifier, $E$ finite. The *QFM* $\mathcal{F}^{A}$ is defined as $$\mathcal{F}^{A}\left( Q\right) \left( X_{1},\ldots,X_{n}\right) = \sum_{Y_{1}\in\mathcal{P}\left( E\right) }\ldots\sum_{Y_{n}\in \mathcal{P}\left( E\right)} \\ m_{X_{1}}\left( Y_{1}\right) \ldots m_{X_{n}}\left( Y_{n}\right) Q\left( Y_{1},\ldots,Y_{n}\right) \label{ModeloVerosimilitudes}$$ for all $X_{1},\ldots,X_{n}\in\widetilde{\mathcal{P}}\left( E\right) $. In expression \[ModeloVerosimilitudes\] we are assuming that the probability of being $Y_{i}$ a representative of the fuzzy set $X_{i}$ is independent of the probability of being $Y_{j}$ a representative of the fuzzy set $X_{j}$ for $i\neq j$. $\mathcal{F}^{A}\left( Q\right) \left( X_{1},\ldots ,X_{n}\right) $ can then be interpreted as the average opinion of voters[^4]. The following example shows the application of the *QFM* $\mathcal{F}^{A}$: \[EjemVerosimilitudes\]Let us consider the sentence: $$\text{\textquotedblleft Nearly all big houses are expensive\textquotedblright}$$ where the semi-fuzzy quantifier $Q=$ **** *‘nearly all’*, and the fuzzy sets *‘big houses’* and *‘expensive’* take the following values: $$\begin{aligned} \mathbf{big}\text{ }\mathbf{houses} & =\left\{ 0.8/e_{1},0.9/e_{2},1/e_{3},0.2/e_{4}\right\}\\\mathbf{expensive} & =\left\{ 1/e_{1},0.8/e_{2},0.3/e_{3},0.1/e_{4}\right\} \\ Q\left( X_{1},X_{2}\right) & =\left\{ \begin{array} [c]{cc}\max\left\{ 2\left( \frac{\left\vert X_{1}\cap X_{2}\right\vert }{\left\vert X_{1}\right\vert }\right) -1,0\right\} & X_{1}\neq\varnothing\\ 1 & X_{1}=\varnothing \end{array} .\right.\end{aligned}$$ We compute the probabilities of the representatives of the fuzzy sets *‘big houses’* and *‘expensive’*: $$\begin{aligned} & m_{\mathbf{big}\text{ }\mathbf{houses}}\left( \varnothing\right) =\left( 1-0.8\right) \left( 1-0.9\right) \left( 1-1\right) \left(1-0.2\right) =0, \\ & \ldots \\ & m_{\mathbf{expensive}}\left( \left\{ e_{1},e_{2},e_{3},e_{4}\right\} \right) = 0.8\cdot0.9\cdot1\cdot0.2=0.144, \\ & m_{\mathbf{expensive}}\left( \varnothing\right) =\left( 1-1\right) \left( 1-0.8\right) \left( 1-0.3\right) \left( 1-0.1\right) =0,\\ & m_{\mathbf{expensive}}\left( \left\{ e_{1}\right\} \right) = 1\cdot\left( 1-0.8\right) \left( 1-0.3\right) \left( 1-0.1\right) =0.126,\\ & \ldots \\ & m_{\mathbf{expensive}}\left( \left\{ e_{1},e_{2},e_{3},e_{4}\right\} \right) =0.8\cdot0.9\cdot1\cdot0.2=0.144.\end{aligned}$$ And using expression (\[ModeloVerosimilitudes\]): $$\begin{aligned} & \mathcal{F}^{A}\left( Q\right) \left( \mathbf{big}\text{ }\mathbf{houses},\mathbf{expensive}\right) =\sum_{Y_{1}\in\mathcal{P}\left( E\right) }\sum_{Y_{2}\in\mathcal{P}\left( E\right) }m_{X_{1}}\left( Y_{1}\right) m_{X_{2}}\left( Y_{2}\right) Q\left( Y_{1},Y_{2}\right) =0.346.\end{aligned}$$ The DFS axiomatic framework =========================== We will present now the definition of the *Determiner fuzzification scheme (DFS*) axiomatic framework [@Glockner06Libro]. It is impossible in this paper to explain in full detail the DFS axiomatic framework, as in [@Glockner06Libro] the author needed chapters three and four to present it in adequate detail. We will limit us to introduce the framework, referring the reader to the previous reference for further study. A *QFM* $\mathcal{F}$ is called a determiner fuzzification scheme (DFS) if the conditions listed in $TABLE$ $I$ are satisfied for all semi-fuzzy quantifiers $Q:\mathcal{P}\left( E\right) ^{n}\longrightarrow\mathbf{I}$. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- **Name** **Condition** **Reference** ------------------------------ ----------------------------------------------------------------------------------------------------------------------- --------------- Correct generalization $\mathcal{U}\left( \mathcal{F}\left( Q\right) (Z-1) \right) =Q$if $n\leq1$ Projection quantifiers $\mathcal{F}\left( Q\right) =\widetilde{\pi_{e}}$if $Q=\pi_{e}$ for some $e\in E$ (Z-2) Dualisation $\mathcal{F}\left( Q\square\right) =\mathcal{F}\left( Q\right) \widetilde{\square}$$n>0$ (Z-3) Internal joins $\mathcal{F}\left( Q\cup\right) =\mathcal{F}\left( (Z-4) Q\right) \widetilde{\cup}$$n>0$ Preservation of monotonicity If $Q$ is nonincreasing in the $n$-th arg, then $\mathcal{F}\left( Q\right) $ is nonincreasing in $n$-th arg, $n>0$ (Z-5) Functional application $\mathcal{F}\left( Q\circ\underset{i=1}{\overset (Z-6) {n}{\times}}\widehat{f_{i}}\right) =\mathcal{F}\left( Q\right) \circ\underset{i=1}{\overset{n}{\times}}\widehat{\mathcal{F}}\left( f_{i}\right) $ where $f_{1},\ldots,f_{n}:E^{\prime}\rightarrow E,E^{\prime}\neq\varnothing$ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- In the following section, we will present the main properties of the models fulfilling the framework in relation to the $\mathcal{F}^{A}$ QFM, and some others that are not consequence of the DFS axiomatic framework but which are important in order to adequately characterize the behavior of QFMs. Analysis of the behavior of the $\mathcal{F}^{A}$ QFM ===================================================== In this section we will give a general overview of the main properties of the $\mathcal{F}^{A}$ QFM referring the different publications where the proofs and extended explanations can be found. Main properties of the $\mathcal{F}^{A}$ QFM derived from the DFS framework\[PropertiesDerivedDFS\] --------------------------------------------------------------------------------------------------- As we have advanced, the $\mathcal{F}^{A}$ QFM fulfills the DFS axiomatic framework. This fact guarantees that it also fulfills all the adequacy properties the framework guarantees. We will present now the main properties derived from it in relation with the behavior of the $\mathcal{F}^{A}$ QFM. ### Correct generalization (P1) This property is possibly the most important property derived from the DFS framework. *Correct generalization* requires that the behavior of a fuzzy quantifier $\mathcal{F}\left( Q\right) $ when we apply it to crisp arguments would be equal to the application of the semi-fuzzy quantifier $Q$ over the same crisp arguments. That is, for all the crisp subsets $Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) $, then it holds that $\mathcal{F}\left( Q\right) \left( Y_{1},\ldots,Y_{n}\right) =Q\left( Y_{1},\ldots,Y_{n}\right) $. For example, given the crisp sets $\mathbf{big}$ $\mathbf{houses},\mathbf{expensive}\in\mathcal{P}\left( E\right) $, this property guarantees that: $$\mathcal{F}\left(\mathbf{some}\right) \left( \mathbf{big}\text{ }\mathbf{houses},\mathbf{expensive}\right) = \\ \mathbf{some}\left(\mathbf{big}\text{ }\mathbf{houses},\mathbf{expensive}\right)$$ The proof of this property for the $\mathcal{F}^{A}$ QFM can be found in [@DiazHermida06Tesis page 291],[@DiazHermida10Arxiv page 31] for the unary case, as in conjunction with the other axioms of the DFSs is enough to assure the fulfillment of the property in the general case. ### Quantitativity (P2) In TGQ, a quantifier is *quantitative* if it does not depend on any particular property fulfilled by the elements. Most common examples of quantifiers we can find in the literature are quantitative (e.g., *‘many’, ‘about 10’*). *Non-quantitative quantifiers* involve the reference to particular elements of the base set (e.g., *‘Spain’* in a set of countries). A *QFM* $\mathcal{F}$ retains the *quantitativity property* if quantitative semi-fuzzy quantifiers are converted into quantitative fuzzy quantifiers by the application of $\mathcal{F}.$ The fulfillment of the quantitativity property for the $\mathcal{F}^{A}$ QFM is a consequence of the fulfillment of the DFS framework. ### Projection quantifier (P3) The *Axiom Z-2* of the DFS framework establishes that the *projection crisp quantifier* $\pi_{e}\left( Y\right) $ (which returns $1$ if $e\in Y$ and $0$ in other case) is transformed into the *fuzzy projection quantifier* $\widetilde{\pi_{e}}\left( X\right) $ (which returns $\mu_{X}\left( e\right) $). The proof of this property for the $\mathcal{F}^{A}$ QFM can be found in [@DiazHermida06Tesis page 272], [@DiazHermida10Arxiv page 31]. ### Induced propositional logic (P4) In [@Glockner06Libro] a mechanism was proposed to embed crisp logical functions ($\lnot\left( x\right) $, $\wedge\left( x_{1},x_{2}\right) $, $\vee\left( x_{1},x_{2}\right) $, $\rightarrow\left( x_{1},x_{2}\right) $) into semi-fuzzy quantifiers. For example, the ‘and’ function can be embedded into a semi-fuzzy quantifier $Q_{\wedge}:\mathcal{P}\left( \left\{ e_{1},e_{2}\right\} \right) \rightarrow\left\{ 0,1\right\} $ such that $Q_{\wedge}\left( \varnothing\right) =Q_{\wedge}\left( \left\{ e_{1}\right\} \right) =Q_{\wedge}\left( \left\{ e_{2}\right\} \right) =0$ and $Q_{\wedge}\left( \left\{ e_{1},e_{2}\right\} \right) =1$. The property of *induced propositional logic* assures that crisp logical functions are transformed into acceptable fuzzy logical functions. In the case of the $\mathcal{F}^{A}$ QFM the induced propositional functions are respectively the *strong negation*, the probabilistic *tnorm*, the probabilistic *tconorm* and the *Rechenbach* ** fuzzy implication. The proof of this property for the $\mathcal{F}^{A}$ QFM can be found in [@DiazHermida06Tesis page 265], [DiazHermida10Arxiv]{}. ### External negation (P5) We will say that a QFM fulfills the *external negation property* if $\mathcal{F}\left( \widetilde{\lnot}Q\right) $ is equivalent to $\widetilde{\lnot}\mathcal{F}\left( Q\right) $. In words*,* equivalence of expressions *it is false that at least 60% of the good students are good athletes *and *less than 40% of the good students are good athletes *is assured. Here, $\widetilde{\lnot}$ is assumed to be the induced negation of the QFM (the strong negation for the $\mathcal{F}^{A}$ QFM). The proof of this property for the $\mathcal{F}^{A}$ QFM can be found in [@DiazHermida06Tesis page 273], [@DiazHermida10Arxiv page 32]. ### Internal negation (P6) The *internal negation or antonym* of a semi-fuzzy quantifier $Q:\mathcal{P}\left( E\right) ^{n}\longrightarrow\mathbf{I}$ is defined as $Q\mathbf{\lnot}\left( Y_{1},\ldots,Y_{n}\right) =Q\left( Y_{1},\ldots,\mathbf{\lnot}Y_{n}\right) $. For example, *‘all’* is the antonym of *‘no’* as $\mathbf{no}\left( Y_{1},Y_{2}\right) =\mathbf{all}\left( Y_{1},\lnot Y_{2}\right) =\mathbf{all}\lnot\left( Y_{1},Y_{2}\right) $. Fulfillment of the *internal negation property* assures that internal negation transformations are translated to the fuzzy case. The proof of this property for the $\mathcal{F}^{A}$ QFM can be found in [@DiazHermida06Tesis page 273], [@DiazHermida10Arxiv page 32]. ### Dualisation (P7) The *dualisation property* is a consequence of the fulfillment of the external negation and internal negation properties. In conjunction, these negation properties assure the maintenance of the equivalences in the ‘Aristotelian square’ [@Gamut84]. It forms part of the DFS framework (*Z-3 axiom*), being the dual of a semi-fuzzy quantifier $Q:\mathcal{P}\left( E\right) ^{n}\longrightarrow\mathbf{I}$ defined as $Q\widetilde {\square}\left( Y_{1},\ldots,Y_{n}\right) =\widetilde{\lnot}Q\left( Y_{1},\ldots,\mathbf{\lnot}Y_{n}\right) $ and equivalently in the fuzzy case. As an example, the equivalence of $\mathcal{F}\left( \mathbf{no}\right) \left( \mathbf{big}\text{ }\mathbf{houses},\widetilde{\lnot}\mathbf{expensive}\right) $ and $\mathcal{F}\left( \mathbf{all}\right) \left( \mathbf{big}\text{ }\mathbf{houses},\mathbf{expensive}\right) $ is assured; or in words, *no big house is not expensive* and *all big houses are expensive* are equivalent. ** The proof of this property for the $\mathcal{F}^{A}$ QFM can be found in [@DiazHermida06Tesis page 275], [@DiazHermida10Arxiv page 33]. ### Union/intersection of arguments P8 The properties of *union and intersection of arguments* guarantee the compliance with some transformations to construct new quantifiers using unions and intersections of arguments. Being $Q:\mathcal{P}\left( E\right) ^{n+1}\longrightarrow\mathbf{I}$ an $n+1$-ary semi-fuzzy quantifier, $Q\cup$ is defined as $Q\cup\left( Y_{1},\ldots,Y_{n+1}\right) =Q\left( Y_{1},\ldots,Y_{n}\cup Y_{n+1}\right) $ and equivalently in the fuzzy case. *Z-4 axiom* specifies this property for the union of quantifiers, as the property is also fulfilled for the intersection of arguments as a consequence of the DFS axiomatic framework. One particular example of the consequences of fulfilling these properties is that the equivalence between absolute unary and binary quantifiers is assured, guaranteeing that we obtain the same result when we evaluate *around 5 big houses are expensive* and *there are around 5 houses that are big and expensive,* where the evaluation of the first quantified expression is computed by means of an absolute binary quantifier and the evaluation of the second expression is computed by applying the corresponding absolute unary quantifier to the intersection of ‘big houses’ and ‘expensive houses’ computed by means of the induced *tnorm.* In combination with the internal and external negation properties, they allow the preservation of the boolean argument structure that can be expressed in natural language when none of the boolean variables $X_{i}$ occurs more than once [@Glockner06Libro section 3.6]. The proof of these properties for the $\mathcal{F}^{A}$ QFM can be found in [@DiazHermida06Tesis page 275], [@DiazHermida10Arxiv page 33]. **** ### Coherence with standard quantifiers P9 By *standard quantifiers* we mean the classical quantifiers $\exists,\forall$ and their binary versions $\mathbf{some}$ and $\mathbf{all}$. Every QFM fulfilling the DFS axiomatic framework guarantees that the fuzzy version of these classical quantifiers is the expected. For example, the $\mathcal{F}^{A}$ QFM fulfills (where $\widetilde{\vee},\widetilde{\wedge },\widetilde{\rightarrow}$ are the logical operators induced by the $\mathcal{F}^{A}$ ** model): $$\mathcal{F}\left( \exists\right) \left( X\right) =\sup\left\{\overset{m}{\underset{i=1}{\widetilde{\vee}}}\mu_{X}\left( a_{i}\right): \right.\\ \left. {} A=\left\{ a_{1},\ldots,a_{m}\right\} \in\mathcal{P}\left( E\right),a_{i}\neq a_{j}\text{ if }i\neq j\right. \bigg\}$$ $$\mathcal{F}\left( \mathbf{all}\right) \left( X_{1},X_{2}\right) = \inf\left\{ \overset{m}{\underset{i=1}{\widetilde{\wedge}}}\mu_{X_{1}} \left( a_{i}\right) \widetilde{\rightarrow}\mu_{X_{2}}\left( a_{i}\right) : \right.\\ \left. {} A=\left\{a_{1},\ldots,a_{m}\right\} \in\mathcal{P}\left( E\right), a_{i}\neq a_{j}\text{ if }i\neq j\right. \bigg\}$$ This property is a consequence of being the $\mathcal{F}^{A}$ QFM a DFS. ### Monotonocity in arguments P10 In [@Glockner06Libro] different definitions to assure the preservation of monotonicity relationships were included. The property of *monotonicity in arguments*, which forms part of the DFS axiomatic framework (*axiom Z5*) assures that monotonic behaviors in arguments are translated from the semi-fuzzy to the fuzzy case. For example, for the binary semi-fuzzy quantifier *‘most’,* that is increasing in its second argument (e.g. *most politics are rich*), the fulfillment of this property guarantees that its fuzzy version will also be increasing in its second argument. The DFS framework also guarantees the maintenance of ‘local’ monotonicity properties [Glockner06Libro]{}. The proof of these properties for the $\mathcal{F}^{A}$ QFM can be found in [@DiazHermida06Tesis page 282], [DiazHermida10Arxiv]{}. ### Monotonicity between quantifiers P11 The DFS axiomatic framework also guarantees the preservation of *monotonicity relationships between quantifiers.* For example, *‘between 4 and 6’* is more specific than *‘between 2 and 8’*. Thanks to this property, monotonicity relationships between semi-fuzzy quantifiers are preserved between fuzzy quantifiers. The fulfillment of the property of monotonicity in quantifiers is a consequence of the DFS axiomatic framework. ### Crisp argument insertion P12 The operator of *crisp argument insertion,* applied to a semi-fuzzy quantifier $Q:\mathcal{P}\left( E\right) ^{n}\rightarrow\mathbf{I}$, allows to construct a new quantifier $Q:\mathcal{P}\left( E\right) ^{n-1}\rightarrow\mathbf{I}$ by means of the restriction of $Q$ by a crisp set $A$. More explicitly, the crisp argument insertion $Q\lhd A$ is defined as $Q\lhd A\left( Y_{1},\ldots,Y_{n-1}\right) =Q\left( Y_{1},\ldots,Y_{n-1},A\right) $. A QFM preserves this property if $\mathcal{F}\left( Q\lhd A\right) =\mathcal{F}\left( Q\right) \lhd A$; that is, the crisp argument insertion commutes for semi-fuzzy and fuzzy quantifiers. Crisp argument insertion allows to model the ‘adjectival restriction’ of natural language in the crisp case. The fulfillment of this property by the $\mathcal{F}^{A}$ QFM is also a consequence of the DFS axiomatic framework. Some relevant properties considered in the QFM framework but not derived from the DFS axioms \[PropertiesAdditionalDFS\] ------------------------------------------------------------------------------------------------------------------------ In [@Glockner06Libro chapter six] it can be found the definition of some additional adequacy properties for characterizing QFMs. These properties were not included in the DFS framework in some cases, for not being compatible with it, and in other cases, in order to not excessively constraint the set of theoretical models fulfilling the DFS framework. We will present now the most relevant ones: ### Continuity in arguments P13 The property of *continuity in arguments* ** assures the continuity of the models with respect to the input sets. It is fundamental to guarantee that small variations in the inputs do not cause jumps in the outputs. The $\mathcal{F}^{A}$ QFM is a finite DFS and it is continuous. The proof of this property for the $\mathcal{F}^{A}$ QFM can be found in [@DiazHermida06Tesis page 293], [@DiazHermida10Arxiv page 48]. ### Continuity in quantifiers P14 The property of *continuity in quantifiers* ** assures the continuity of the QFMs with respect to small variations in the quantifiers. The proof of this property for the $\mathcal{F}^{A}$ QFM can be found in [@DiazHermida06Tesis page 297], [@DiazHermida10Arxiv page 48]. ### Propagation of fuzziness P15 *Propagation of fuzziness properties* assure that fuzzier inputs (understood as fuzzier input sets) and fuzzier quantifiers produce fuzzier outputs[^5]. This property is not fulfilled by the $\mathcal{F}^{A}$ QFM because it is not fulfilled by the induced product *tnorm* and the induced probabilistic sum *tconorm* of the model. An extensive analysis of the fulfillment of this property by the main QFMs that can be found in the literature is presented in [@DiazHermida17-FuzzySets]. ### Fuzzy argument insertion P16 The property of *fuzzy argument insertion* is the fuzzy counterpart of the crisp argument insertion. To our knowledge, this property has only been proved for the DFSs $\mathcal{M}_{CX}$ [@Glockner06Libro definition 7.56] and $\mathcal{F}^{A}$ [@DiazHermida06Tesis page 292],[@DiazHermida10Arxiv page 48]. Additional properties fulfilled by the $\mathcal{F}^{A}$ QFM not included in the QFM framework ---------------------------------------------------------------------------------------------- In this section we summarize three other properties fulfilled by the $\mathcal{F}^{A}$ QFM that do not form part of the ones considered in the QFM framework by Glöckner [@Glockner06Libro]. We will explain these properties in some more detail as they are not commonly considered in the bibliography about fuzzy quantification. In [@DiazHermida17-FuzzySets] these properties were used, in combination with other criteria, to present a comparison of the behavior of different QFMs thinking in their convenience for practical applications. ### Property of averaging for the identity quantifier\[SubSubSubPropMedia\] The fulfillment of this property by a QFM $\mathcal{F}$ assures that when we apply the model to the unary semi-fuzzy quantifier $\mathbf{identity}\left( Y\right) =\frac{\left\vert Y\right\vert }{\left\vert E\right\vert },Y\in\mathcal{P}\left( E\right) $ we obtain the average of the membership grades. For the ‘identity’ semi-fuzzy quantifier the addition of one element increases the result in $\frac{1}{m}$. We could expect that a QFM $\mathcal{F}$ would translate this linearity relationship into the fuzzy case. The QFM $\mathcal{F}^{A}$ fulfills the property of averaging for the identity quantifier that assures: $$\mathcal{F}^{A}\left( \mathbf{identity}\right) \left( X\right) =\frac {1}{m}\sum_{j=1}^{m}\mu_{X}\left( e_{j}\right)$$ The proofs can be found in [@DiazHermida06Tesis page 298] or in [@DiazHermida10Arxiv page 50]. ### Property of the probabilistic interpretation of quantifiers\[SubSubSubPropRecubProbab\] Let us suppose we use a set of semi-fuzzy quantifiers (*at most about 20%*, *between 20% and 80%*, *at least about 80%*) to split the quantification universe. We will say that a set of semi-fuzzy quantifiers $Q_{1},\ldots,Q_{r}:\mathcal{P}^{n}\left( E\right) \rightarrow\mathbf{I}$ forms a *quantified Ruspini partition* of the quantification universe if for all $Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) $ it holds that $$Q_{1}\left( Y_{1},\ldots,Y_{n}\right) +\ldots+Q_{r}\left( Y_{1},\ldots,Y_{n}\right) =1$$ The QFM $\mathcal{F}^{A}$ translates this relationship to the fuzzy case. Forming $Q_{1},\ldots,Q_{r}:\mathcal{P}\left( E\right) ^{n}\rightarrow \mathbf{I}$ a quantified Ruspini partition it is fulfilled:$$\mathcal{F}^{A}\left( Q_{1}\right) \left( X_{1},\ldots,X_{n}\right) +\ldots+\mathcal{F}^{A}\left( Q_{r}\right) \left( X_{1},\ldots ,X_{n}\right) =1$$ This property is very interesting because it will permit to interpret the result of evaluating a fuzzy quantified expression as a probability distributed over the labels related to the quantifiers. Proofs can be found in [@DiazHermida06Tesis page 298] or in [@DiazHermida10Arxiv page 52]. ### Fine distinction between objects\[SectionRankingGeneration\] This property is particularly useful for the application of fuzzy quanfiers in ranking problems. Let us consider a set of objects $o_{1},\ldots,o_{N}$ for which the fulfillment of a set of criteria $p_{1},\ldots,p_{m}$ is represented by means of a fuzzy set $X^{o_{i}}=\left\{ \mu_{X^{i}}\left( p_{1}\right) /p_{1},\ldots,\mu_{X^{i}}\left( p_{m}\right) /p_{m}\right\} $, where $\mu_{X^{i}}\left( p_{j}\right) /p_{j}$ indicates the fulfillment of the criteria $p_{j}$ by the object $o_{i}$. Generally, we also have a set of weights $W=\left\{ \mu_{W}\left( p_{1}\right) /p_{1},\ldots,\mu_{W}\left( p_{m}\right) /p_{m}\right\} $ to indicate the relative relevance of the different criteria $p_{1},\ldots,p_{m}$. Using fuzzy quantification, a ranking can be constructed assigning to each object a weight computed by means of an unary proportional quantified expression $r^{o_{i}}=\widetilde{Q}\left( X^{o_{i}}\right) $ (in the case that a vector of weights is not involved) or a binary proportional quantified expression $r^{o_{i}}=\widetilde{Q}\left( W,X^{o_{i}}\right) $ (in the case that a vector of weights $W$ is used to indicate the relative importance of each criteria). In this way, computing $r^{o_{i}}$ for each $i=1,\ldots,N$, we can sort the objects of the collection with respect to the linguistic expression *‘how* $\widetilde{Q}$’ criteria are fulfilled (e.g., for $\widetilde{Q}=\mathbf{many}$, *‘how many’*). In order to guarantee a sufficient discriminative power, even small variations in the inputs should produce some effect in the outputs. In [@DiazHermida17-FuzzySets section 5.6] it was proposed to analyze the behavior of QFMs with respect to the following semi-fuzzy quantifiers defined by means of increasing fuzzy numbers: Let $h\left( x\right) :\left[ 0,1\right] \rightarrow\mathbf{I}$ be an strictly increasing continuous mapping; i.e., $h\left( x\right) >h\left( y\right) $ for every $x>y$. We define the unary and binary semi-fuzzy quantifiers $Q_{h}:\mathcal{P}\left( E\right) \rightarrow\mathbf{I}$ and $Q_{h}:\mathcal{P}\left( E\right) ^{2}\rightarrow\mathbf{I}$ as $$\begin{aligned} Q_{h}\left( Y\right) & =h\left( \left\vert Y\right\vert \right) ,Y\in\mathcal{P}\left( E\right) \\ Q_{h}\left( Y_{1},Y_{2}\right) & =\left\{ \begin{array} [c]{cc}h\left( \frac{\left\vert Y_{1}\cap Y_{2}\right\vert }{\left\vert Y_{1}\right\vert }\right) & Y_{1}\neq\varnothing\\ 1 & Y_{1}=\varnothing \end{array} \right.\end{aligned}$$ And then, to require to a QFM $\mathcal{F}$ the maintanace of the strictly increasing relationships in the fuzzy case. That is, that any increase in the fulfillment of a criteria will increase $\mathcal{F}\left( Q_{h}\right) $ in the unary case, and that any increase in the fulfillment of a criteria associated with a strictly positive weight will increase $\mathcal{F}\left( Q_{h}\right) $ in the binary case. The $\mathcal{F}^{A}$ DFS fulfills this property as can be found in [@DiazHermida17-FuzzySets section 5.6]. Limit case approximation of the $\mathcal{F}^{A}$ QFM ===================================================== In this section we will prove that in the general case of semi-fuzzy quantifiers defined by means of continuous proportional fuzzy numbers (i.e., ‘unary proportional’, ‘binary proportional’, ‘comparative proportional’, etc.) the $\mathcal{F}^{A}$ QFM can be approximated by simply evaluating the fuzzy number that supports the quantifier over a function which depends on the average of the different boolean combinations of the input sets (more details below). As an additional result, in the specific case of unary and binary proportional linguistic quantifiers, the $\mathcal{F}^{A}$ QFM converges to the Zadeh’s model when the intersection of the inputs sets is computed with the *probabilistic tconorm* for binary proportional quantifiers. Before proceeding, we will make a brief summary of the ideas of the proof in order to facilitate its understanding. In the proof, we will start introducing some previous results which guarantee that quantitative quantifiers can be expressed by means of a function of the cardinalities of the boolean combinations of the input sets. This will allow us to develop a general proof, valid for each quantitative quantifier defined by means of a proportional fuzzy number. After that, we will use the fact that in the definition of the $\mathcal{F}^{A}$ QFM we are interpreting membership degrees $\mu_{X}(e_{i})$ as probabilities, and that independence is fulfilled for $\mu_{X}(e_{i}),\mu_{X}(e_{j}), i \neq j$. In this case, a fuzzy set $X=\{a_1/e_1, \dots,a_{m}/e_{m}\}$ will induce an specific probability distribution over the function of the possible cardinalities $0, \dots,m$ of the set. In other words, as we are interested in the number of elements of $X$ fulfilling the property, each possible cardinality $i$ will have a probability value measuring the probability that exactly ‘$i$’ elements fulfill the property. We will see that this probability follows a poisson binomial distribution. Moreover, we will also prove that the projections of the probability function $f(i_1,\dots,i_{K})$ induced by the $\mathcal{F}^{A}$ QFM for n-ary quantifiers follow poisson binomial distributions. In that case, the probability parameters of the $j$ projection will be determined by the $j$-th boolean combination used in the specification of the semi-fuzzy quantifier. When $m$ tends to infinite, the fuzzy set $X=\{a_1/e_1, \dots, a_{m}/e_{m}\}$ will induce a sequence $B_1,B_2,\dots$ of poisson binomial distributions on $0, \dots, m$. But we will see that the variance of $Z_{i}=B_{i}/m$ will tend to 0. As in the definition of proportional quantifiers we use fuzzy numbers defined over $[0,1]$ instead of $\{0,\dots,m\}$, when we normalize the probability distribution $f(i_1,\dots,i_{K})$ to $[0,1]^n$ we will obtain a probability distribution whose projections are poisson binomial distributions such that their average converge in probability to the average of the membership degrees of the fuzzy set ‘induced’ by the boolean combination, and their variance tend to $0$. Then, as each marginal distribution converges in probability to a constant, by the theorem of the continuous mapping the joint distribution converges in probability to a constant. In practice, this implies that the probability distribution will be more and more concentrated around the average of the boolean combinations as the size of the input sets tends to infinite. As a consequence, we could simply evaluate $\mathcal{F}^{A}$ computing the value of the proportonal fuzzy number used in the definition of $Q$ over the average of the boolean combinations. After this summary we will present the proof in full detail. The next theorem establishes that, in the finite case, quantitative semi-fuzzy quantifiers can be expressed by means of a function of the cardinalities of the boolean combinations of the input sets. \[TeoremaCuantitativo\][@Glockner06Libro Theorem 11.32, chapter 11] A semi-fuzzy quantifier $Q:\mathcal{P}\left( E\right) ^{n}\longrightarrow \mathbf{I}$ on a finite base set $E\neq\varnothing$ is quantitative if and only if $Q$ can be computed from the cardinalities of its arguments and their Boolean combinations, i.e. there exist Boolean expressions $\Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) ,\ldots,\Phi_{K}\left( Y_{1},\ldots,Y_{n}\right) $ for some $K\in\mathbb{N}$, and a mapping $q:\left\{ 0,\ldots,m\right\} ^{K}\longrightarrow\mathbf{I}$ such that$$Q\left( Y_{1},\ldots,Y_{n}\right) = \\ q\left( \left\vert \Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) \right\vert ,\ldots,\left\vert \Phi_{K}\left( Y_{1},\ldots,Y_{n}\right) \right\vert \right) \label{EqTeoremaCuantitativo}$$ for all $Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) $. We will also introduce the following notation for denoting the boolean combinations: Let be $l_{1},\ldots,l_{n}\in\left\{ 0,1\right\} $, we define $\Phi _{l_{1},\ldots,l_{n}}\left( Y_{1},\ldots,Y_{n}\right) $ as:$$\Phi_{l_{1},\ldots,l_{n}}\left( Y_{1},\ldots,Y_{n}\right) =Y_{1}^{\left( l_{1}\right) }\cap\ldots\cap Y_{n}^{\left( l_{n}\right) }$$ where$$Y^{\left( l\right) }=\left\{ \begin{tabular} [c]{lll}$Y$ & $:$ & $l=1$\\ $\lnot Y$ & $:$ & $l=0$\end{tabular} \ \right.$$ Let us remember we are denoting $\left\vert E\right\vert =m$. Then, in the finite case, quantitative semi-fuzzy quantifiers can be expressed by means of a function $q:\left\{ 0,\ldots,m\right\} ^{K}\longrightarrow\mathbf{I}$ depending only of the cardinalities of the boolean combinations of $Y_{1},\ldots,Y_{n}$. For example, proportional binary semi-fuzzy quantifiers can be defined by means of the boolean combinations $\Phi_{1}\left( Y_{1},Y_{2}\right) =Y_{1}\cap Y_{2}$ and $\Phi_{2}\left( Y_{1},\overline{Y_{2}}\right) =Y_{1}\cap \overline{Y_{2}}$. Let $Q:\mathcal{P}\left( E\right) ^{n}\longrightarrow\mathbf{I}$ be a quantitative semi-fuzzy quantifier $Q:\mathcal{P}\left( E\right) ^{n}\longrightarrow\mathbf{I}$ on a finite base set $E\neq\varnothing$, and let us suppose it can be expressed following expression \[EqTeoremaCuantitativo\] for some set $\Phi_{1}\left( Y_{1},\ldots ,Y_{n}\right) ,\ldots,\Phi_{K}\left( Y_{1},\ldots,Y_{n}\right) $ of boolean combinations and some $q:\left\{ 0,\ldots,m\right\} ^{K}\longrightarrow \mathbf{I}$. For convenience, we will define $q^{\prime}:\left[ 0,1\right] ^{K}\longrightarrow\mathbf{I}$ such that:$$q^{\prime}\left( \frac{\left\vert \Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) \right\vert }{m},\ldots,\frac{\left\vert \Phi_{K}\left( Y_{1},\ldots ,Y_{n}\right) \right\vert }{m}\right) = \\ q\left( \left\vert \Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) \right\vert ,\ldots,\left\vert \Phi_{K}\left( Y_{1},\ldots,Y_{n}\right) \right\vert \right)$$ $q^{\prime}:\left[ 0,1\right] ^{K}\longrightarrow\mathbf{I}$ simply normalizes $q$ in the interval of proportions $\left[ 0,1\right] ^{K}$[^6]. We introduce now the definition of the *poisson binomial distribution*. Let us consider a sequence of $m$ *independent bernoulli trials* $\mathbf{B}=P_{1},\ldots,P_{m}$ that are not necessarily identically distributed. Let be $p_{1},\ldots,p_{m}$ the corresponding probabilities of the independent bernouilli trials. The probability function of the poisson binomial distribution is: $$\Pr^{\mathbf{B}}\left( K=k\right) =\sum_{A\in F_{k}}{\displaystyle\prod\limits_{i\in A}} p_{i}{\displaystyle\prod\limits_{j\in A^{c}}} \left( 1-p_{j}\right)$$ where $F_{k}$ is the set of all subsets of $k$ integers that can be selected from $\left\{ 1,2,3,\ldots,m\right\} $. We now introduce a notation for representing the poisson bernoulli succession $\mathbf{B}=P_{1},\ldots,P_{m}$ with probabilities $p_{1},\ldots,p_{m}$ by means of a fuzzy set: Let be $\mathbf{B}=P_{1},\ldots,P_{m}$ a poisson bernoulli succession with probabilities $p_{1},\ldots,p_{m}$. We will denote by $X^{\mathbf{B}}\in\widetilde{\mathcal{P}}\left( E\right) $ the fuzzy set defined in the following way:$$\mu_{X^{\mathbf{B}}}\left( e_{i}\right) =p_{i}$$ Under the probabilistic interpretation of the $\mathcal{F}^{A}$ QFM, a crisp set $Y\in\mathcal{P}\left( E\right) $ can be interpreted as a realization of a poisson bernoulli succession $\mathbf{B}=P_{1},\ldots,P_{m}$ with probabilities $p_{1},\ldots,p_{m}$ such that $\chi_{Y}\left( e_{i}\right) =P_{i}$[^7]. In this sense,$$\begin{aligned} \overset{\mathbf{B}}{\Pr}\left( Y\right) & ={\displaystyle\prod_{i|e_{i}\in Y}} p_{i}{\displaystyle\prod_{j|e_{j}\notin Y}} \left( 1-p_{j}\right) ={\displaystyle\prod_{i|e_{i}\in Y}} \left( P_{i}=1\right) {\displaystyle\prod_{j|e_{j}\notin Y}} \left( P_{j}=0\right) =m_{X^{\mathbf{B}}}\left( Y\right).\end{aligned}$$ Now, we will compute the projection of the probability function used in the definition of the $\mathcal{F}^{A}$ QFM for the cardinalities of each possible boolean combination associated to a quantitative semi-fuzzy quantifier $Q:\mathcal{P}\left( E\right) ^{n}\longrightarrow\mathbf{I}$. As $Q$ is quantitative, by theorem \[TeoremaCuantitativo\] it can be defined by means of a function $q:\left\{ 0,\ldots,\left\vert E\right\vert \right\} ^{K}\longrightarrow\mathbf{I}$ depending on the cardinalities of the boolean combinations of the input sets ($\left\vert \Phi_{1}\left( Y_{1},\ldots ,Y_{n}\right) \right\vert ,\ldots,\left\vert \Phi_{K}\left( Y_{1},\ldots,Y_{n}\right) \right\vert $). Then:$$\begin{aligned} \mathcal{F}^{A}\left( Q\right) \left( X_{1},\ldots,X_{n}\right) & =\sum_{Y_{1}\in\mathcal{P}\left( E\right) }\ldots\sum_{Y_{n}\in\mathcal{P}\left( E\right) }m_{X_{1}}\left( Y_{1}\right) \ldots m_{X_{n}}\left( Y_{n}\right) Q\left( Y_{1},\ldots,Y_{n}\right) \\ & =\sum_{\substack{\left( i_{1},\ldots,i_{K}\right) \in \\ \left\{ 0,\ldots,m\right\}^{K}}} \sum_{\substack{Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) \; | \\ \left\vert \Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) \right\vert =i_{1}\wedge\\\ldots\\\left\vert \Phi_{k}\left( Y_{1},\ldots,Y_{n}\right) \right\vert =i_{K}}} \; m_{X_{1}}\left( Y_{1}\right) \ldots m_{X_{n}}\left( Y_{n}\right) \times\\ & \qquad q\left( \left\vert \Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) \right\vert ,\ldots,\left\vert \Phi_{K}\left( Y_{1},\ldots,Y_{n}\right) \right\vert \right) \\ & =\sum_{\substack{\left( i_{1},\ldots,i_{K}\right) \in \\ \left\{ 0,\ldots,m\right\} ^{K}}} q\left( i_{1},\ldots,i_{K}\right) \sum_{\mathclap{\substack{Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) \; |\\\left\vert \Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) \right\vert =i_{1}\wedge\\\ldots\\\left\vert \Phi _{k}\left( Y_{1},\ldots,Y_{n}\right) \right\vert =i_{K}}}} m_{X_{1}}\left(Y_{1}\right) \ldots m_{X_{n}}\left( Y_{n}\right).\end{aligned}$$ Let us denote by$$f\left( i_{1},\ldots,i_{K}\right) = \sum_{\mathclap{\substack{Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) \; |\\\left\vert \Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) \right\vert =i_{1}\wedge\\\ldots\\\left\vert \Phi _{k}\left( Y_{1},\ldots,Y_{n}\right) \right\vert =i_{K}}}} \; m_{X_{1}}\left( Y_{1}\right) \ldots m_{X_{n}}\left( Y_{n}\right) \label{EqProbabilityMx}$$where $f\left( i_{1},\ldots,i_{K}\right) $ is a probability function. Take into account that $m_{X_{1}}\left( Y_{1}\right) \ldots m_{X_{n}}\left( Y_{n}\right) $ define a probility over $\left( Y_{1},\ldots,Y_{n}\right) \in\mathcal{P}\left( E\right) ^{n}$, and $f\left( i_{1},\ldots ,i_{K}\right) $ simply distributes the probabilities of $\left( Y_{1},\ldots,Y_{n}\right) \in\mathcal{P}\left( E\right) ^{n}$ over the cardinalities of the $K$ boolean combinations. Let $f\left( i_{1},\ldots,i_{K}\right) $ be the probability distribution that is obtained when we compute the probability induced by the $X_{1},\ldots,X_{n}\in\mathcal{P}\left( E\right) ^{n}$ fuzzy sets over the cardinalities of the boolean combinations $\left\vert \Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) \right\vert ,\ldots,\left\vert \Phi_{K}\left( Y_{1},\ldots,Y_{n}\right) \right\vert $ following equation \[EqProbabilityMx\]. The probability projection $j$ of $f\left( i_{1},\ldots,i_{K}\right) $ will follow a poisson binomial distribution of parameters:$$\begin{aligned} p_{1}^{j} & =\mu_{X_{1}^{\left( l_{j,1}\right) }\widetilde{\cap}\ldots\widetilde{\cap}X_{n}^{\left( l_{j,n}\right) }}\left( e_{1}\right) \\ & \ldots\\ p_{m}^{j} & =\mu_{X_{1}^{\left( l_{j,1}\right) }\widetilde{\cap}\ldots\widetilde{\cap}X_{n}^{\left( l_{j,n}\right) }}\left( e_{m}\right)\end{aligned}$$ where $X_{1}^{\left( l_{j,1}\right) }\widetilde{\cap}\ldots\widetilde{\cap }X_{n}^{\left( l_{j,n}\right) }=\Phi_{j}\left( X_{1},\ldots,X_{n}\right) $ is the j-th boolean combination. We will only give an intuitive idea of this result. In appendix \[AnnexProof\] an analytical proof can be consulted.By assumption, the $\mathcal{F}^{A}$ QFM is interpreting membership grades of the input sets as probabilities, and considering that the independence assumption is always fulfilled between different elements and sets. The probability projection $j$ of $f\left( i_{1},\ldots,i_{K}\right) $ simply denotes the probability of the different cardinalities of one of these boolean combinations. But the probability of an element $e_{s}$ of pertaining to the boolean combination $\Phi_{j}\left( Y_{1},\ldots,Y_{n}\right) $ is just the probability of $e_{s}$ pertaining to every fuzzy set $X_{r}^{\left( l_{r}\right) }$ such that $l_{r}=1$ and non pertaining to every fuzzy set $X_{r}^{\left( l_{r}\right) }$ set such that $l_{r}=0$. As this is fulfilled for every $e\in E$, the cardinality of the boolean combination follows a poisson binomial distribution with the indicated parameters. *\[LimitCase\]*Let $Q:\mathcal{P}\left( E\right) ^{n}\longrightarrow\mathbf{I}$ be a semi-fuzzy quantitative quantifier on a finite base set $E\neq\varnothing$, $\Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) ,\ldots,\Phi_{K}\left( Y_{1},\ldots,Y_{n}\right) $, $K\in\mathbb{N}$ boolean combinations, and $q:\left\{ 0,\ldots,m\right\} ^{K}\longrightarrow \mathbf{I}$ the corresponding function for which:$$\begin{aligned} Q\left( Y_{1},\ldots,Y_{n}\right) & = q\left( \left\vert \Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) \right\vert ,\ldots,\left\vert \Phi_{K}\left( Y_{1},\ldots,Y_{n}\right) \right\vert \right) \\ & = q^{\prime}\left( \frac{\left\vert \Phi_{1}\left( Y_{1},\ldots ,Y_{n}\right) \right\vert }{m},\ldots,\frac{\left\vert \Phi_{K}\left( Y_{1},\ldots,Y_{n}\right) \right\vert }{m}\right)\end{aligned}$$ If $q^{\prime}:\left[ 0,1\right] ^{K}\longrightarrow\mathbf{I}$ is continuous around $$\left( \frac{\sum_{i=1}^{m}\mu_{\Phi_{1}\left( X_{1},\ldots,X_{n}\right) }}{m},\ldots,\frac{\sum_{i=1}^{m}\mu_{\Phi_{K}\left( X_{1},\ldots ,X_{n}\right) }}{m}\right)$$ then the following result will be fulfilled when the size of $E$ tend to infinite: $$\begin{aligned} \lim_{\left\vert E\right\vert \rightarrow\infty}\mathcal{F}^{A}\left( Q\right) \left( X_{1},\ldots,X_{n}\right) & = q^{\prime}\left( \frac {\sum_{i=1}^{m}\mu_{\Phi_{1}\left( X_{1},\ldots,X_{n}\right) }}{m},\ldots,\frac{\sum_{i=1}^{m}\mu_{\Phi_{K}\left( X_{1},\ldots,X_{n}\right) }}{m}\right)\end{aligned}$$ Before proving proposition \[LimitCase\], we would like to make some appointments about the applicability of the result. In general, we always could find a $q^{\prime}$ continuous around $\left( \frac{\sum_{i=1}^{m}\mu_{\Phi_{1}\left( X_{1},\ldots,X_{n}\right) }}{m},\ldots,\frac{\sum _{i=1}^{m}\mu_{\Phi_{K}\left( X_{1},\ldots,X_{n}\right) }}{m}\right) $ such that previous result would be applicable. But in choosing a ‘proportional expression’ for $q^{\prime}$, we are indicating that the types of fuzzy quantifiers in which we are mainly interested are ‘proportional quantifiers’. In practical applications, support functions associated to proportional quantifiers are generally defined by means of ‘smooth’ fuzzy numbers over $\left[ 0,1\right] $, which guarantees a good approximation when the size of the referential set is sufficiently large. Let $f\left( i_{1},\ldots,i_{K}\right) $ be the probability distribution that is obtained when we compute the probability induced by the $X_{1},\ldots,X_{n}\in\mathcal{P}\left( E\right) ^{n}$ fuzzy sets over the cardinalities of the boolean combinations $\left\vert \Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) \right\vert ,\ldots,\left\vert \Phi_{K}\left( Y_{1},\ldots,Y_{n}\right) \right\vert $. We know that the probability projection $f^{j}\left( i_{s}\right) $ follows a poisson binomial distribution of parameters $$\begin{aligned} p_{1}^{j} & =\mu_{X_{1}^{\left( l_{j,1}\right) }\widetilde{\cap}\ldots\widetilde{\cap}X_{n}^{\left( l_{j,n}\right) }}\left( e_{1}\right) \\ & \ldots\\ p_{m}^{j} & =\mu_{X_{1}^{\left( l_{j,1}\right) }\widetilde{\cap}\ldots\widetilde{\cap}X_{n}^{\left( l_{j,n}\right) }}\left( e_{m}\right)\end{aligned}$$ Moreover,$$\begin{aligned} \mathcal{F}^{A}\left( Q\right) \left( X_{1},\ldots,X_{n}\right) & =\sum_{\substack{\left( i_{1},\ldots,i_{k}\right) \in \\ \left\{ 0,\ldots,m\right\}^{K}}} q\left(i_{1},\ldots,i_{K}\right) \sum_{\mathclap{\substack{Y_{1},\ldots,Y_{n} \in\mathcal{P}\left( E\right) \; | \\\left\vert \Phi_{1}\left( Y_{1} ,\ldots,Y_{n}\right) \right\vert =i_{1}\wedge\\\ldots\\\left\vert \Phi_{k}\left( Y_{1},\ldots,Y_{n}\right) \right\vert =i_{K}}}} \; m_{X_{1}}\left(Y_{1}\right) \ldots m_{X_{n}}\left( Y_{n}\right) \\ & =\sum_{\substack{\left( i_{1},\ldots,i_{k}\right) \in \\ \left\{ 0,\ldots,m\right\}^{K}}} q^{\prime}\left( \frac{i_{1}}{m},\ldots,\frac{i_{K}}{m}\right) f\left( i_{1},\ldots,i_{K}\right)\end{aligned}$$ Let $f^{\prime}:\left[ 0,1\right] ^{K}\longrightarrow\mathbf{I}$ be probability distribution defined by:$$f^{\prime}\left( s_{1},\ldots,s_{K}\right) =f\left( m\times i_{1},\ldots,m\times i_{K}\right)$$ that normalizes $f$ in the interval $\left[ 0,1\right] ^{K}$. Then,$$\mathcal{F}^{A}\left( Q\right) \left( X_{1},\ldots,X_{n}\right) \\ =\sum_{\left( i_{1},\ldots,i_{K}\right) \in m^{K}}q^{\prime}\left( \frac{i_{1}}{m},\ldots,\frac{i_{K}}{m}\right) f^{\prime}\left( \frac{i_{1}}{m},\ldots,\frac{i_{K}}{m}\right)$$ As we are normalizing $f$ by $m$, the corresponding $f^{j\prime}\left( i_{s}\right) $ projection of $f^{\prime}$ will follow a probability distribution such that:$$\begin{aligned} average\left( f^{j\prime}\right) & =\frac{average\left( f^{j}\right) }{m}=\frac{\sum_{i=1}^{m}p_{i}^{j}}{m}\\ var\left( f^{j\prime}\right) & =\frac{1}{m^{2}}var\left( f^{j}\right) =\frac{1}{m^{2}}\sum_{i=1}^{m}p_{i}^{j}\left( 1-p_{i}^{j}\right)\end{aligned}$$ but when $m\longrightarrow\mathbf{\infty}$ the variance tends to $0$. And as the variance tends to $0$, $f^{j\prime}\overset{p}{\longrightarrow }\frac{\sum_{i=1}^{m}p_{i}^{j}}{m}$, and as $q^{\prime}\left( s_{1},\ldots,s_{K}\right) $ is continuous around $\left( \frac{\sum_{i=1}^{m}p_{i}^{1}}{m},\ldots,\frac{\sum_{i=1}^{m}p_{i}^{K}}{m}\right) $, by continuous mapping theorem [^8]: $$\begin{aligned} & \lim_{m\rightarrow\infty}\sum_{\left( i_{1},\ldots,i_{k}\right) \in m^{k}}q^{\prime}\left( \frac{i_{1}}{m},\ldots,\frac{i_{K}}{m}\right) f\left( \frac{i_{1}}{m},\ldots,\frac{i_{K}}{m}\right) \nonumber \overset{p}{\longrightarrow}q^{\prime}\left( \frac{\sum_{i=1}^{m}p_{i}^{1}}{m},\ldots,\frac{\sum_{i=1}^{m}p_{i}^{K}}{m}\right) \label{EqApproximation}\\ & =q^{\prime}\left( \frac{\sum_{i=1}^{m}\mu_{\Phi_{1}\left( X_{1},\ldots,X_{n}\right) }}{m},\ldots,\frac{\sum_{i=1}^{m}\mu_{\Phi_{K}\left( X_{1},\ldots,X_{n}\right) }}{m}\right). \nonumber\end{aligned}$$ This result guarantees that the $\mathcal{F}^{A}$ QFM converges to the Zadeh’s model for unary proportional and binary proportional quantifiers when the size of the referential set tends to infinite and the intersection is modelled by means of the *product tnorm* in the proportional case, as these quantifiers basically depend on[^9]:$$\begin{array} [c]{ccc}q:\frac{\left\vert Y\right\vert }{m}\longrightarrow\mathbf{I} & \text{:} & \text{unary quantifiers}\\ q:\left( \frac{\left\vert Y_{1}\cap Y_{2}\right\vert }{m},\frac{\left\vert Y_{1}\cap\overline{Y_{2}}\right\vert }{m}\right) ,\longrightarrow\mathbf{I} & \text{:} & \text{binary quantifiers}\end{array}$$ As we introduced below, the normalization by $m$ is coherent with proportional linguistic quantifiers, that are generally defined by means of ‘smooth’ fuzzy numbers in $\left[ 0,1\right] $. In these situations, the result guarantees that the probability of the projections of $f^{\prime}\left( s_{1},\ldots,s_{s}\right) $ will concentrate around the average of the projections as we increase the size of the referential set. As a consequence, if the variation of the fuzzy number that supports the linguistic quantifier is small around this average, we could expect a good approximation of the $\mathcal{F}^{A}$ QFM using \[EqApproximation\] when the size of the referential set tends to infinite. Quality of the convergence and Monte Carlo approximation of the $\mathcal{F}^{A}$ QFM ===================================================================================== In section \[ComputationalAlgorithms\] we will present some computational exact implementations of the $\mathcal{F}^{A}$ QFM for evaluating the most common linguistic quantifiers. We advance that the complexity of the exact implementation of the $\mathcal{F}^{A}$ QFM is $O\left( m^{2}\right) $ for unary quantifiers, $O\left( m^{3}\right) $ for binary proportional quantifiers and $O\left( m^{r+1}\right) $ in the general case, being $r$ the number of boolean combinations that are necessary for the definition of the semi-fuzzy quantifier. For some applications, and specifically for quantifiers depending on a high value of $r$, this complexity could be too high for applying the model to big fuzzy sets. One consequence of the result of the previous section is that the $\mathcal{F}^{A}$ QFM can be approximated in linear time for fuzzy sets containing a sufficiently large number of elements. But we do not know if the proposed approximation is sufficiently accurate for problems where the exact implementation could not be applied due to its computational demands. We will make now a deeper analysis about the applicability of the results of previous section for approximating the $\mathcal{F}^{A}$ QFM, connecting them with a proposal to use a Monte Carlo simulation. Let us consider the following example: \[ExampleConvergence\]Let us consider a fuzzy set $X=\left\{ 0.5/e_{1},\ldots,0.5/e_{m}\right\} $. In this situation, the probability distribution subjacent to the $\mathcal{F}^{A}$ QFM is a binomial distribution with parameters $\left( m,0.5\right) $. Let us consider a trapezoidal function $T_{0.5,0.6,\infty,\infty}\left( x\right) $ and the unary semi-fuzzy quantifier defined as $Q\left( Y\right) =T_{0.5,0.6,\infty ,\infty}\left( \left\vert Y\right\vert \right) $. The following table compares the result of the application of the $\mathcal{F}^{A}$ QFMwith its approximation by means of the Zadeh’s model: $m,X$ $F^{A}\left( X\right) $ $f_{Q}\left(\overline{X}\right) $ ---------------------------------------------------------------------- --------------------------- ------------------------------------ $50,X=\left\{ \underset{50}{\underbrace{0.5,\ldots,0.5}}\right\}$ $0.260$ $0$ $100,X=\left\{ \underset{100}{\underbrace{0.5,\ldots,0.5}}\right\}$ $0.195$ $0$ $500,X=\left\{ \underset{500}{\underbrace{0.5,\ldots,0.5}}\right\}$ $0.089$ $0$ Previous example proves that, even for a large fuzzy set containing 500 elements, the error of the approximation is not negligable for a semi-fuzzy quantifier defined by means of a fuzzy number that seems very plausible from a practical viewpoint. Moreover, the error will be greater for a semi-fuzzy quantifier definfed by means of a fuzzy number with a higher slope. We will now introduce a theorem applicable to the poisson binomial distribution [@Degroot88 page 263]. Central limit theorem applied to Bernoulli variables. Let $X_{1},\ldots,X_{m}$ be independent random variables, each $X_{i}$ following a Bernoulli distribution with parameter $p_{i}$. Moreover, let us suppose that the infinite sum $\sum_{i=1}^{\infty}p_{i}\left( 1-p_{i}\right) $ is divergent and let $Y_{m}$ be$$Y_{m}=\frac{\sum_{i=1}^{m}X_{i}-\sum_{i=1}^{n}p_{i}}{\left( \sum_{i=1}^{m}p_{i}q_{i}\right) ^{1/2}}.$$ Then $$\lim_{n\rightarrow\infty}\Pr\left( Y_{m}\leq x\right) =\Phi\left( x\right)$$ where $\Phi\left( x\right) $ is the standard normal distribution function. In practical situations, this result allow us to approximate a poisson binomial distribution by a normal distribution when the variance of the distribution is high (take into account that we are interpreting the cardinality of a fuzzy set as a poisson binomial distribution). Cases of a low variance for poisson binomial distributions with a high number of parameters will be associated to situations in which most parameters are really close to 0 or 1[^10]. In these cases, the approximation by means of the normal distribution will be poor, but the probability distribution will be extremely concentrated around the average, which will guarantee an even better approximation by means of Montecarlo. Let us consider again a fuzzy set $X=\left\{ 0.5/e_{1},\ldots,0.5/e_{m}\right\} $ whose underlying probability distribution following the $\mathcal{F}^{A}$ QFM interpretation is a binomial distribution with parameters $\left( m,0.5\right) $. We will compute the confidence intervals for the $0.95$ and $0.99$ probability mass approximating the underlying probability of the $\mathcal{F}^{A}$ QFM by means of a normal distribution. The following table shows the confidence intervals for the underlying probability distribution of a fuzzy set $X=\left\{ 0.5/e_{1},\ldots ,0.5/e_{m}\right\} $, following the $\mathcal{F}^{A}$ QFM interpretation: $m$ $\overline{X}$ $\frac{X}{m},0.95$ $\frac{X}{m},0.99$ --------- ---------------- ------------------------------ ---------------------------- $50$ $25$ $\left( 0.36,0.64\right) $ $\left( 0.32,0.68\right)$ $100$ $50$ $\left( 0.40,0.60\right) $ $\left( 0.37.0.63\right)$ $1000$ $500$ $\left( 0.47,0.53\right) $ $\left( 0.45,0.54\right)$ $10000$ $5000$ $\left( 0.49.0.51\right) $ $\left( 0.49,0.51\right)$ Previous example shows that the probability distribution is really concentrated around the average for medium size fuzzy sets. In previous example, we have chosen the binomial distribution of parameter $0.5$ as it is the highest variance distribution in the family of poisson binomial distributions. Take into account that for a poisson binomial distribution $\mathbf{B}$, $var\left( \mathbf{B}\right) =\sum_{i=1}^{m}p_{i}\left( 1-p_{i}\right) $, and that the maximum of $p_{i}\left( 1-p_{i}\right) \,$is obtained for $p_{i}=0.5$. The idea of the Monte Carlo simulation is simply to generate, for each $X_{i}$, a random binary vector using a Bernoulli trial of probability $\mu_{X_{i}}\left( j\right) $ for each $e_{j}$. Previous example indicates that the $f^{j\prime}\left( i_{s}\right) $ projections of $f^{\prime}$ would be very concentrated around the average when the size of the referential set contains a large number of elements, which will allow to expect a really good approximation of the $\mathcal{F}^{A}$ QFM by means of a Monte Carlo simulation. Moreover, a Monte Carlo simulation can be easily parallelized. In section \[MonteCarloApproximation\] the algorithm for unary quantifiers is presented. The extension to higher arity quantifiers is trivial. Efficient implementation of the $\mathcal{F}^{A}$ model\[ComputationalAlgorithms\] ================================================================================== For quantititative quantiers is possible to develop polynomial algorithms for the $\mathcal{F}^{A}$ DFS. Let us remember that the class of quantitative quantifiers is composed of the semi-fuzzy quantifiers that are invariant under automorphims [@Glockner06Libro section 4.13], and that they can be expressed as a function of the cardinalities of their arguments and their boolean combinations. The class of quantitative quantifiers include the most interesting ones for applications, and in particular the common *absolute, proportional and comparative* quantifiers. Quantitative unary quantifiers ------------------------------ Let $Q:\mathcal{P}\left( E\right) \rightarrow\mathbf{I}$ be an unary semi-fuzzy quantifier defined over a referential set $E^{m}=\left\{ e_{1},\ldots,e_{m}\right\} $. Quantitative unary semi-fuzzy quantifiers can always be expressed by means of a function $q:\left\{ 0,\ldots,\left\vert E\right\vert \right\} \rightarrow\mathbf{I}$ (theorem \[TeoremaCuantitativo\]); that is, a function that goes from cardinality values in $\mathbf{I}$. In this way, there exists $q$ such that $q\left( j\right) =Q\left( Y_{j}\right) $ where $Y_{j}\in\mathcal{P}\left( E\right) $ is an arbitrary set of cardinality $j$ ($\left\vert Y_{j}\right\vert =j$). Let $X\in\mathcal{P}\left( E\right) $ be a fuzzy set. Then, $$\begin{aligned} \mathcal{F}^{A}\left( Q\right) \left( X\right) & =\sum_{Y \in \mathcal{P}\left( E\right) }m_{X}\left( Y\right) Q\left( Y\right) \\ & =\sum_{\substack{Y \in \mathcal{P}\left( E\right) \\ | \; \left\vert Y\right\vert =0}} m_{X}\left( Y\right) Q\left( Y\right) + \ldots + \sum_{\substack{Y\in\mathcal{P}\left(E\right) \\ | \; \left\vert Y\right\vert =m}} m_{X}\left( Y\right) Q\left( Y\right) \\ & =\sum_{\substack{Y \in \mathcal{P}\left( E\right) \\ | \; \left\vert Y\right\vert =0}} m_{X}\left( Y\right) q\left( 0\right) + \ldots + \sum_{\substack{Y\in\mathcal{P}\left(E\right) \\ | \; \left\vert Y\right\vert =m}} m_{X}\left( Y\right) q\left( m\right) \\ & =\sum_{j=0}^{m}\Pr\left( card_{X}=j\right) q\left( j\right)\end{aligned}$$ The algorithm we will present uses the fact that it is possible to compute the probability $\Pr_{E^{m}}\left( card_{X}=j\right) ,$ $j=0,\ldots,m$ for a referential set $E^{m}$ of $m$ elements using the probabilities $\Pr_{E^{m-1}}\left( card_{X^{E^{m-1}}}=j\right) ,j=0,\ldots,m-1$ where $E^{m-1}=\left\{ e_{1},\ldots,e_{m-1}\right\} $ and $X^{E^{m-1}}$ is the projection of $X$ over $E^{m-1}$ (that is, the fuzzy set $X$ without the element $e_{m}$). In this way, it is easy to develop a recursive function for computing the probabilities of the cardinalities in $E^{m}$. In the case of a referential set of one element ($E^{1}=\left\{ e_{1}\right\} $) the probabilities of the cardinalities of a fuzzy set $X\in\mathcal{P}\left( E^{1}\right) $ are simply:$$\begin{aligned} \Pr\left( card_{X}=0\right) & = m_{X}\left(\varnothing\right) =1-\mu_{X}\left( e_{1}\right)\\ \Pr\left( card_{X}=1\right) & = m_{X}\left(\left\{ e_{1}\right\} \right) =\mu_{X}\left( e_{1}\right)\end{aligned}$$ Let us suppose now a referential set of $m+1$ elements ($E^{m+1}=\left\{ e_{1},\ldots,e_{m+1}\right\} $), let $X\in\widetilde{\mathcal{P}}\left( E^{m+1}\right) $ be a fuzzy set on $E^{m+1}$, $E^{m}=\left\{ e_{1},\ldots,e_{m}\right\} $ and $X^{E^{m}}\in\widetilde{\mathcal{P}}\left( E^{m}\right) $ the projection of $X$ in $E^{m}$; that is, $\mu_{X^{E^{m}}}\left( e_{j}\right) =\mu_{X}\left( e_{j}\right) ,1\leq j\leq m$. Moreover, let us suppose we know the probabilities of the cardinalities associated to $X^{E^{m}}$ ($\Pr\left( card_{X^{E^{m}}}=0\right) ,\ldots ,\Pr\left( card_{X^{E^{m}}}=m\right) $). Now, we will compute the probabilities of $X$ using the probabilities of the cardinalities on $X^{E^{m}}$: **Case 1:** $\Pr\left( card_{X}=0\right) $$$\begin{aligned} \Pr\left( card_{X}=0\right) & = \sum_{\substack{Y\in\mathcal{P}\left( E^{m+1} \right) | \; \left\vert Y\right\vert =0}} m_{X}\left( Y\right) \\ & = m_{X}\left(\varnothing\right) \\ & = \left( 1-\mu_{X}\left( e_{1}\right) \right) \ldots\left( 1-\mu _{X}\left( e_{m}\right) \right) \left( 1-\mu_{X}\left( e_{m+1}\right) \right) \\ & = m_{X^{E^{m}}}\left( \varnothing\right) \left( 1-\mu_{X}\left(e_{m+1}\right) \right) \\ & = \Pr\left( card_{X^{E^{m}}}=0\right) \left( 1-\mu_{X}\left(e_{m+1}\right) \right)\end{aligned}$$ **Case 2:** $\Pr\left( card_{X}=m+1\right) $$$\begin{aligned} \Pr\left( card_{X}=m+1\right) & =\sum_{\substack{Y\in\mathcal{P}\left(E^{m+1}\right) | \; \left\vert Y\right\vert =m+1}} m_{X}\left( Y\right) \text{\hspace{18mm}} \\ & = m_{X}\left( E^{m+1}\right) \\ & = \mu_{X}\left( e_{1}\right) \ldots\mu_{X}\left( e_{m}\right) \mu_{X}\left( e_{m+1}\right) \\ & = m_{X^{E^{m}}}\left( E^{m}\right) \mu_{X}\left( e_{m+1}\right) \\ & = \Pr\left( card_{X^{E^{m}}}=m\right) \mu_{X}\left( e_{m+1}\right)\end{aligned}$$ **Case 3:** ** **$\Pr\left( card_{X}=j\right) ,0<j<m+1$$$\begin{aligned} \Pr\left( card_{X}=j\right) & =\sum_{\substack{Y\in\mathcal{P}\left( E^{m+1}\right) \\ | \; \left\vert Y\right\vert =j}} m_{X}\left( Y\right) \\ & = \sum_{\substack{Y\in\mathcal{P}\left( E^{m+1}\right) \\ | \; \left\vert Y\right\vert =j\wedge e_{m+1}\notin Y}} m_{X}\left( Y\right) + \sum_{\substack{Y\in\mathcal{P}\left(E^{m+1}\right) \\ | \; \left\vert Y\right\vert =j\wedge e_{m+1}\in Y}} m_{X}\left( Y\right) \\ & = \sum_{\substack{Y\in\mathcal{P}\left( E^{m}\right) \\ | \; \left\vert Y\right\vert =j}} m_{X^{E^{m}}}\left( Y\right) \left( 1-\mu_{X}\left( e_{m+1}\right) \right) + \ldots + \sum_{\substack{Y\in\mathcal{P}\left( E^{m}\right) \\ | \; \left\vert Y\right\vert =j-1}} m_{X}\left( Y\right) \mu_{X}\left( e_{m+1}\right) \\ & = \Pr\left( card_{X^{E^{m}}}=j\right) \left( 1-\mu_{X}\left( e_{m+1}\right) \right) +\Pr\left( card_{X^{E^{m}}}=j-1\right) \mu _{X}\left( e_{m+1}\right)\end{aligned}$$ Previous computations are summarized in expression \[EqAlgoritmoUnarioFA\_1\]. In algorithm \[alg:AlgoritmoUnarioFA\], the code for evaluating $\mathcal{F}^{A}\left( Q\right) \left( X\right) $ is presented. Complexity of the algorithm is $O\left( n^{2}\right) $.$$\begin{aligned} & \Pr\left( card_{X}=j\right) \label{EqAlgoritmoUnarioFA_1} =\left\{ \begin{array} [c]{lll}\Pr\left( card_{X^{E^{m}}}=0\right) \left( 1-\mu_{X}\left( e_{m+1}\right) \right) & : & j=0\\\Pr\left( card_{X^{E^{m}}}=j\right) \left( 1-\mu_{X}\left( e_{m+1}\right) \right) & &\\ \; +\Pr\left( card_{X^{E^{m}}}=j-1\right) \mu_{X}\left( e_{m+1}\right) & : & 1\leq j\leq m\\ \Pr\left( card_{X^{E^{m}}}=m\right) \mu_{X}\left( e_{m+1}\right) & : & j=m+1 \end{array} \right. \nonumber$$ Conservative binary quantifiers\[ConservativeBinaryQuantifiers\] ---------------------------------------------------------------- In this section we will present the algorithm for evaluating conservative binary quantifiers [@Keenan97VanBenthem], which includes proportional quantitative quantifiers as a particular case. The strategy we are going to detail can be easily generalized for implementing other kinds of quantitative quantifiers. A semi-fuzzy conservative quantitative quantifier $Q\left( Y_{1},Y_{2}\right) $ depends on the cardinalities of $\left\vert Y_{1}\right\vert $ and $\left\vert Y_{1}\cap Y_{2}\right\vert $; that is, there exists a function $q:\left\{ 0,\ldots,m\right\} ^{2}\rightarrow\mathbf{I}$ such that:$$Q\left( Y_{1},Y_{2}\right) =q\left( \left\vert Y_{1}\right\vert ,\left\vert Y_{1}\cap Y_{2}\right\vert \right)$$ for all $Y_{1},Y_{2}\in\mathcal{P}\left( E\right) $. Let $X_{1},X_{2}\in\mathcal{P}\left( E\right) $ be two fuzzy sets. By$$\begin{aligned} & \Pr\left( card_{X_{1},X_{1}\cap X_{2}}=\left( j,k\right) \right) =\sum_{\substack{Y_{1}\in\mathcal{P}\left( E\right) \; | \\ Y_{2}\in\mathcal{P} \left( E\right)}} \sum_{\substack{\left\vert Y_{1}\right\vert =j \; \wedge \\ \left\vert Y_{1}\cap Y_{2}\right\vert =k}} m_{X_{1}}\left( Y_{1}\right) m_{X_{2}}\left(Y_{2}\right) ,0\leq k,j\leq m\end{aligned}$$ we will denote the probability of choosing a pair of representatives $Y_{1},Y_{2}\in\mathcal{P}\left( E\right) $ of $X_{1},X_{2}\in \widetilde{\mathcal{P}}\left( E\right) $ such that$\left\vert Y_{1}\right\vert =j$ and $\left\vert Y_{1}\cap Y_{2}\right\vert =k$. It should be noted that for $k>j$ $\Pr\left( card_{X_{1},X_{1}\cap X_{2}}=\left( j,k\right) \right) =0$. Let $X_{1},X_{2}\in\widetilde{\mathcal{P}}\left( E^{m}\right) $ be two fuzzy sets over $E^{m}=\left\{ e_{1},\ldots,e_{m}\right\} $. And let us suppose we know the probabilities:$$\Pr\left( card_{X_{1},X_{1}\cap X_{2}}=\left( j,k\right) \right)$$ for all $j,k$ such that $0\leq j,k\leq m$. Let us suppose now we add an element $e_{m+1}$ to the referential. That is, the new referential set is $E^{m+1}=\left\{ e_{1},\ldots,e_{m+1}\right\} $. And let $X_{1}^{\prime },X_{2}^{\prime}\in\widetilde{\mathcal{P}}\left( E^{m+1}\right) $ be two fuzzy sets in $E^{m+1}$ resulting of adding $e_{m+1}$. That is, $\left( X_{1}^{\prime}\right) ^{E^{m}}=X_{1},\left( X_{2}^{\prime}\right) ^{E^{m}}=X_{2}$; where by $\left( {}\right) ^{E^{m}}$ we are denoting the projections of $X_{1}^{\prime},X_{2}^{\prime}$ over $E^{m}$. By definition of the $\mathcal{F}^{A}$ DFS, belongniness of $e_{m+1}$ to the set $X_{i}^{\prime},i=1,2$ is an event of probability $\mu_{X_{_{i}}^{\prime}}\left( e_{m+1}\right) $ and this probability is independent of the belongniness of other elements. Then, if the cardinality of $X_{1},\,X_{2}$ were $\left( j,k\right) $ and it would happen that $e_{m+1}\in X_{1}^{\prime}$ and $e_{m+1}\in X_{2}^{\prime}$ then the cardinality of $X_{1}^{\prime}\,X_{2}^{\prime}$ would be $\left( j+1,k+1\right) $. Let $0\leq j,k\leq m$ be arbitray indexes and let us consider the probability $\Pr(card_{X_{1},X_{1}\cap X_{2}}$ $=\left( j,k\right) )$. When we include the element $e_{m+1}$ the probability of $e_{m+1}$ contributes to the probability $\Pr\left( card_{X_{1}^{\prime},X_{1}^{\prime}\cap X_{2}^{\prime }}=\left( j,k\right) \right) $ with: $$\begin{aligned} & \left( 1-\mu_{X_{1}^{\prime}}\left( e_{m+1}\right) \right) \left( 1-\mu_{X_{2}^{\prime}}\left( e_{m+1}\right) \right) \Pr\left( card_{X_{1},X_{1}\cap X_{2}}=\left( j,k\right) \right) + \left( 1-\mu_{X_{1}^{\prime}}\left( e_{m+1}\right) \right) \mu _{X_{2}^{\prime}}\left( e_{m+1}\right) \Pr\left( card_{X_{1},X_{1}\cap X_{2}}=\left( j,k\right) \right) \\ & =\left( 1-\mu_{X_{1}^{\prime}}\left( e_{m+1}\right) \right) \Pr\left( card_{X_{1},X_{1}\cap X_{2}}=\left( j,k\right) \right)\end{aligned}$$ that is, if we know that the cardinality $card_{X_{1},X_{1}\cap X_{2}}$ is $\left( j,k\right) $ then the cardinality $card_{X_{1}^{\prime},X_{1}^{\prime}\cap X_{2}^{\prime}}$ would be $\left( j,k\right) $ with probability $\left( 1-\mu_{X_{1}^{\prime}}\left( e_{m+1}\right) \right) $. It should be noted that if $e_{m+1}\notin X_{1}^{\prime}$ and $e_{m+1}\in X_{2}^{\prime}$ then $e_{m+1}\notin X_{1}^{\prime}\cap X_{2}^{\prime}$. Similarly, the contribution to $\Pr\left( card_{X_{1}^{\prime},X_{1}^{\prime }\cap X_{2}^{\prime}}=\left( j+1,k\right) \right) $ will be:$$\mu_{X_{1}^{\prime}}\left( e_{m+1}\right) \left( 1-\mu_{X_{2}^{\prime}}\left( e_{m+1}\right) \right) \Pr\left( card_{X_{1},X_{1}\cap X_{2}}=\left( j,k\right) \right)$$ that is, as the cardinality $card_{X_{1},X_{1}\cap X_{2}}$ is $\left( j,k\right) $ then the cardinality $card_{X_{1}^{\prime},X_{1}^{\prime}\cap X_{2}^{\prime}}$ will be $\left( j+1,k\right) $ with probability $\mu _{X_{1}^{\prime}}\left( e_{m+1}\right) \left( 1-\mu_{X_{2}^{\prime}}\left( e_{m+1}\right) \right) $. And the contribution to the probability $\Pr\left( card_{X_{1}^{\prime},X_{1}^{\prime}\cap X_{2}^{\prime}}=\left( j+1,k+1\right) \right) $ will be:$$\mu_{X_{1}^{\prime}}\left( e_{m+1}\right) \mu_{X_{2}^{\prime}}\left( e_{m+1}\right) \Pr\left( card_{X_{1},X_{1}\cap X_{2}}=\left( j,k\right) \right)$$ that is, as the cardinality $card_{X_{1},X_{1}\cap X_{2}}$ is $\left( j,k\right) $ then the cardinality $card_{X_{1}^{\prime},X_{1}^{\prime}\cap X_{2}^{\prime}}$ will be $\left( j+1,k+1\right) $ with probability $\mu_{X_{1}^{\prime}}\left( e_{m+1}\right) \mu_{X_{2}^{\prime}}e_{m+1}$. Using previous expressions a polynomial algorithm can be developed to evaluate conservative semi-fuzzy quantifiers (table \[alg:AlgoritmoConservativoFA\]). Complexity of the algorithm is $O\left( n^{3}\right) $. It is not difficult to generalize the strategy we have presented to other quantifiers. For example, let us consider the case of a ternary comparative quantifier (e.g., *the number of brilliant investors that earn high salaries is about twice the number of brilliant investors that earn low salaries*). For this example, the semi-fuzzy quantifier will follow the expression $Q\left( Y_{1},Y_{2},Y_{3}\right) =q\left( \left\vert Y_{1}\cap Y_{2}\right\vert ,\left\vert Y_{1}\cap Y_{3}\right\vert \right) $ where $q:\left\{ 0,\ldots,m\right\} ^{2}\rightarrow\mathbf{I}$ is the fuzzy number we use to model *‘about twice’.* As this quantifier only depends on two boolean combinations, a binary probability matrix will be enough to compute and update the probabilities of the cardinalities. In this way, the complexity of the resulting algorithm will be again $O\left( n^{3}\right) $. In the general case, if a quantitative semi-fuzzy quantifier $Q:\mathcal{P}\left( E\right) ^{n}\longrightarrow\mathbf{I}$ depens on $r$ boolean combinations its complexity will be $O\left( m^{r+1}\right) $, one iteration to go through the input vectors and $r$ to go through the probability matrix of cardinalities. Monte Carlo Approximation\[MonteCarloApproximation\] ---------------------------------------------------- Monte Carlo simulation permits the approximation of the $\mathcal{F}^{A}$ QFM when efficiency restrictions do not permit the use of the exact implementations presented in previous sections. The idea of the Monte Carlo simulation is simply to generate, for each $X_{i}$, a random binary vector using a Bernoulli trial of probability $\mu_{X_{i}}\left( j\right) $ for each $e_{j}$. The code for the Monte Carlo implementation of the $\mathcal{F}^{A}$ model for an unary quantifier can be seen in table \[alg:MonteCarloUnarioFA\]. The extension to higher arity quantifiers is trivial, by simply generating a random binary vector for each $X_{i}$. The Monte Carlo approximation can be parallelized by simply dividing the number of simulations between different processors. Conclusions =========== In this paper we have presented several relevant results about the $\mathcal{F}^{A}$ QFM. First, we summarized some of the most relevant properties fulfilled by this model, in order to give a comprehensive and integrative summary of its behavior. After that, we introduced a convergence result that guarantees that, in the limit case, the model converges to the Zadeh’s model for semi-fuzzy quantifiers defined by means of proportional continuous fuzzy numbers. Moreover, this result is more general than the specific convergence to the Zadeh’s model, being applicable to every proportional quantitative quantifier. For sufficiently big fuzzy sets, this will allow to approximate the $\mathcal{F}^{A}$ QFM in linear time. However, the rate of convergence could be too slow to make this approximation useful in most applications. For this reason, we also provided the exact computational implementation for some of the most common quantifiers (unary and proportional quantitative quantifiers), introducing a scheme that can be easily extended to other types of quantifiers. Complexity of the exact implementation is $O\left( m^{r+1}\right) $, being $r$ the number of boolean combinations that are involved in the definition of the semi-fuzzy quantifier. Finally, the convergence result has a strong implication. The underlying probability of the $\mathcal{F}^{A}$ QFM will concentrate around the average of the boolean combinations necessary to define the semi-fuzzy quantifier, as we increase the number of elements of the input fuzzy sets. This property was used to propose a Monte Carlo approximation of the $\mathcal{F}^{A}$ QFM that can be used when the complexity of the exact implementation is too elevate to compute an exact solution. Acknowledgment {#acknowledgment .unnumbered} ============== This work has received financial support from the Consellería de Cultura, Educación e Ordenación Universitaria (accreditation 2016-2019, ED431G/08 and reference competitive group 2019-2021, ED431C 2018/29) and the European Regional Development Fund (ERDF) and is also supported by the Spanish Ministry of Economy and Competitiveness under the project TIN2015-73566-JIN. Appendix: Computation of the probability of the projections of the boolean combinations\[AnnexProof\] ===================================================================================================== In this section we will develop an analyticial proof to show that the probability projection $j$ of $f\left( i_{1},\ldots,i_{K}\right) $ follows a binomial poisson distribution. For developing the proof, we will need to introduce the complete definition of some of the axioms of the DFS framework and of its derived properties. The next definition allows the construction of a new semi-fuzzy quantifier that simply permutes the arguments in the input: \[**Argument permutations**\]\[ArgPerm\][Glockner06Libro]{} Let $Q:\mathcal{P}\left( E\right) ^{n}\rightarrow \mathbf{I}$ be a semi-fuzzy quantifier and $\beta:\left\{ 1,\ldots,n\right\} \rightarrow\left\{ 1,\ldots,n\right\} $ a permutation. By $Q\beta :\mathcal{P}\left( E\right) ^{n}\rightarrow\mathbf{I}$ we denote the semi-fuzzy quantifier defined by:$$Q\beta\left( Y_{1},\ldots,Y_{n}\right) =Q\left( Y_{\beta\left( 1\right) },\ldots,Y_{\beta\left( n\right) }\right)$$ for all $Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) $. In the case of fuzzy quantifiers $\widetilde{Q}\beta:\widetilde{\mathcal{P}}\left( E\right) ^{n}\rightarrow\mathbf{I}$ is defined analogously. The next definition will allow us to rewrite permutations as a combination of transpositions: \[**Trasposition**\][@Glockner06Libro Definition 4.14]\[DefPropTraspArg\_1\] For all $n\in\mathbb{N}$ $(n>0$) and $i,j\in\left\{ 1,\ldots,n\right\} $, the transposition $\tau_{i,j}:\left\{ 1,\ldots ,n\right\} \rightarrow\left\{ 1,\ldots,n\right\} $ is defined as:$$\tau_{i,j}\left( k\right) =\left\{ \begin{tabular} [c]{lll}$i$ & $:$ & $k=j$\\ $j$ & $:$ & $k=i$\\ $k$ & $:$ & $k\neq j\wedge k\neq i$\end{tabular} \ \right.$$ for all $k\in\left\{ 1,\ldots,n\right\} $. Moreover, by $\tau_{i}$ we will denote the transposition $\tau_{i,n}$ (that interchanges positions $i$ and $n$). It should be noted that $\tau_{i,j}=\tau_{i}\circ\tau_{j}\circ\tau_{i}$. We also can apply \[DefPropTraspArg\_1\] to fuzzy and semi-fuzzy quantifiers: \[**Argument transpositions**\]Let $Q:\mathcal{P}\left( E\right) ^{n}\rightarrow\mathbf{I}$ be a semi-fuzzy quantifier, $n>0$. By $Q\tau _{i}:\mathcal{P}\left( E\right) ^{n}\rightarrow\mathbf{I}$ we denote the semi-fuzzy quantifier defined by: $$Q\tau_{i}\left( Y_{1},\ldots,Y_{i-1},Y_{i},Y_{i+1},\ldots,Y_{n}\right) \\ =Q\left( Y_{1},\ldots,Y_{i-1},Y_{n},Y_{i+1},\ldots,Y_{i}\right)$$ for all $Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) $. In the case of semi-fuzzy quantifiers $\widetilde{Q}\tau_{i}:\widetilde{\mathcal{P}}\left( E\right) ^{n}\rightarrow\mathbf{I}$ is defined analogously. The DFS axiomatic framework guarantees the adequate generalization of argument transpositions: [@Glockner06Libro Theorem 4.16]\[DefPropTrasArg\]** **Every DFS $\mathcal{F}$ is compatible with argument transpositions, i.e. for every semi-fuzzy quantifier $Q:\mathcal{P}\left( E\right) ^{n}$ $\rightarrow \mathbf{I}$ $i\in\left\{ 1,\ldots,n\right\} $,$$\mathcal{F}\left( Q\tau_{i}\right) =\mathcal{F}\left( Q\right) \tau_{i}$$ Note that as permutations can be expressed as compositions of transpositions, every DFS $\mathcal{F}$ also conmutes with permutations. We will also need to introduce the full definition of external and internal negation. \[**External negation**\][@Glockner06Libro Definition 3.8] The external negation of a semi-fuzzy quantifier $Q:\mathcal{P}\left( E\right) ^{n}\rightarrow\mathbf{I}$ is defined by $$\left( \widetilde{\lnot}Q\right) \left( Y_{1},\ldots,Y_{n}\right) =\widetilde{\lnot}\left( Q\left( Y_{1},\ldots,Y_{n}\right) \right)$$ for all $Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) $. The definition of $\widetilde{\lnot}\widetilde{Q}:\widetilde{\mathcal{P}}\left( E\right) \rightarrow\mathbf{I}$ in the case of fuzzy quantifiers $\widetilde {Q}:\widetilde{\mathcal{P}}\left( E\right) \rightarrow\mathbf{I}$ is analogous[^11]. The next theorem expresses that every DFS correctly generalizes the external negation property: [@Glockner06Libro Theorem 4.20]\[DefPropNegExterna\]** **Every DFS $\mathcal{F}$ is compatible with the formation of negations. Hence if $Q:\mathcal{P}\left( E\right) ^{n}\rightarrow\mathbf{I}$ is a semi-fuzzy quantifier then $\mathcal{F}\left( \widetilde{\lnot}Q\right) =\widetilde {\lnot}\mathcal{F}\left( Q\right) $. The internal negation or antonym of a semi-fuzzy quantifier is defined as: \[**Internal negation/antonym**\][@Glockner06Libro Definition 3.9] Let a semi-fuzzy quantifier $Q:\mathcal{P}\left( E\right) ^{n}\rightarrow \mathbf{I}$ of arity $n>0$ be given. The internal negation $Q\lnot :\mathcal{P}\left( E\right) ^{n}\rightarrow\mathbf{I}$ of $Q$ is defined by $$Q\lnot\left( Y_{1},\ldots,Y_{n}\right) =Q\left( Y_{1},\ldots,\lnot Y_{n}\right)$$ for all $Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) $. The internal negation $\widetilde{Q}\widetilde{\lnot}:\widetilde{\mathcal{P}}\left( E\right) ^{n}\rightarrow\mathbf{I}$ of a fuzzy quantifier $\widetilde {Q}:\widetilde{\mathcal{P}}\left( E\right) ^{n}\rightarrow\mathbf{I}$ is defined analogously, based on the given fuzzy complement $\widetilde{\lnot}$. The next theorem expresses that every DFS correctly generalizes the internal negation property: [@Glockner06Libro Theorem 4.19]\[DefPropNegInterna\] **** Every DFS $\mathcal{F}$ is compatible with the negation of quantifiers. Hence if $Q:\widetilde{\mathcal{P}}\left( E\right) ^{n}\rightarrow\mathbf{I}$ is a semi-fuzzy quantifier, then $\mathcal{F}\left( Q\lnot\right) =\mathcal{F}\left( Q\right) \widetilde{\lnot}$. We will now show the necessary definitions to establish the compatibility with unions and intersections of quantifiers: \[**Union quantifier**\][@Glockner06Libro Definition 3.12] Let a semi-fuzzy quantifier $Q:\mathcal{P}\left( E\right) ^{n}\rightarrow \mathbf{I}$ of arity $n>0$ be given. We define the fuzzy quantifier $Q\cup:\mathcal{P}\left( E\right) ^{n+1}\rightarrow\mathbf{I}$ as $$Q\cup\left( Y_{1},\ldots,Y_{n},Y_{n+1}\right) =Q\left( Y_{1},\ldots ,Y_{n-1},Y_{n}\cup Y_{n+1}\right)$$ for all $Y_{1},\ldots,Y_{n+1}\in\mathcal{P}\left( E\right) $. In the case of fuzzy quantifiers $\widetilde{Q}\widetilde{\cup}$ is defined analogously, based on a fuzzy definition of $\widetilde{\cup}$. Analogously, the definition of the intersection of quantifiers is: \[**Intersection quantifier**\]Let $Q:\mathcal{P}\left( E\right) ^{n}\rightarrow\mathbf{I}$ a semi-fuzzy quantifier, $n>0$, be given. We define the semi-fuzzy quantifier $Q\cap:\mathcal{P}\left( E\right) ^{n+1}\rightarrow\mathbf{I}$ as$$Q\cap\left( Y_{1},\ldots,Y_{n},Y_{n+1}\right) =Q\left( Y_{1},\ldots ,Y_{n-1},Y_{n}\cap Y_{n+1}\right)$$ for all $Y_{1},\ldots,Y_{n+1}\in\mathcal{P}\left( E\right) $. In the case of fuzzy quantifiers $\widetilde{Q}\widetilde{\cap}$ is defined analogously, based on a fuzzy definition of $\widetilde{\cap}$. [@Glockner06Libro sections 3.9 and 4.9]** **Let $Q:\mathcal{P}\left( E\right) ^{n}\rightarrow\mathbf{I}$ be a semi-fuzzy quantifier, $n>0$. Every DFS $\mathcal{F}$ is compatible with the union an intersection of arguments$$\begin{aligned} \mathcal{F}\left( Q\cup\right) & =\mathcal{F}\left( Q\right) \widetilde{\cup}\\ \mathcal{F}\left( Q\cap\right) & =\mathcal{F}\left( Q\right) \widetilde{\cap}$$ Previous properties guarantee that arbitrary boolean combinations conmute between fuzzy and semi-fuzzy quantifiers. This is one to the consequences of the DFS axiomatic framework and it will be fundamental to prove that the projection $j$ of $f\left( i_{1},\ldots,i_{K}\right) $ follows a binomial poisson distribution. \[ExampleBooleanCombination\]Let $Q:\mathcal{P}\left( E\right) \rightarrow\mathbf{I}$ be a semi-fuzzy quantifier. And let $\Phi\left( Y_{1},Y_{2},Y_{3}\right) =\lnot Y_{1}\cap\lnot Y_{2}\cap Y_{3}$ be a boolean combination of the crisp sets $Y_{1},Y_{2},Y_{3}\in\mathcal{P}\left( E\right) $, and $\Phi^{\prime}\left( X_{1},X_{2},X_{3}\right) =\widetilde{\lnot}X_{1}\widetilde{\cap}\widetilde{\lnot}X_{2}\widetilde{\cap }X_{3}$ be the analogous boolean combination of fuzzy sets $X_{1},X_{2},X_{3}\in\mathcal{P}\left( E\right) $ where $\widetilde{\lnot},\widetilde{\cap}$ are defined by means of the corresponding negation and tnorm induced by a particular DFS $\mathcal{F}$. Then[^12]$$\begin{aligned} \mathcal{F}\left( Q\circ\Phi\right) \left( X_{1},X_{2},X_{3}\right) & =\mathcal{F}\left( Q\cap\cap\tau_{1}\lnot\tau_{1}\tau_{2}\lnot\tau _{2}\right) \left( X_{1},X_{2},X_{3}\right) \\ & =\mathcal{F}\left( Q\right) \widetilde{\cap}\widetilde{\cap}\tau _{1}\widetilde{\lnot}\tau_{1}\tau_{2}\widetilde{\lnot}\tau_{2}\left( X_{1},X_{2},X_{3}\right) \\ & =\left( \mathcal{F}\left( Q\right) \circ\Phi^{\prime}\right) \left( X_{1},X_{2},X_{3}\right) \\ & =\mathcal{F}\left( Q\right) \left( \widetilde {\lnot}X_{1}\widetilde{\cap}\widetilde{\lnot}X_{2}\widetilde{\cap}X_{3}\right)\end{aligned}$$ Now, we will introduce some notation to specify that the cardinality of the input sets of a semi-fuzzy quantifiers is exactly of ‘$i$ elements’: We will denote by $q_{exactly}^{i}:\left\{ 0,\ldots,m\right\} \longrightarrow\left\{ 0,1\right\} $ the function defined by$$q_{exactly}^{i}\left( x\right) =\left\{ \begin{tabular} [c]{lll}$1$ & $:$ & $x=i$\\ $0$ & $:$ & $otherwise$\end{tabular} \ \right.$$ and by $Q_{exactly}^{i,n}:\mathcal{P}\left( E\right) ^{n}\longrightarrow \mathbf{I}$ the semi-fuzzy quantifier defined as:$$Q_{exactly}^{i,n}\left( Y_{1},\ldots,Y_{n}\right) =q_{exactly}^{i}\left( \left\vert Y_{1}\cap\ldots\cap Y_{n}\right\vert \right)$$ where with the superindex $n$ we are indicating the arity of the semi-fuzzy quantifier. \[PropEqBinFA\]Let $X^{\mathbf{B}}\in\widetilde{\mathcal{P}}\left( E\right) $ a fuzzy set where $\mathbf{B}=P_{1},\ldots,P_{m}$ is its corresponding poisson bernoulli succession with probabilities $p_{1},\ldots,p_{m}$. It is fulfilled:$$\Pr^{\mathbf{B}}\left( K=k\right) =\mathcal{F}^{A}\left( Q_{exactly}^{i,1}\right) \left( X^{\mathbf{B}}\right)$$ Simply:$$\begin{aligned} \mathcal{F}^{A}\left( Q_{exactly}^{i}\right) \left( X^{\mathbf{B}}\right) & \qquad =\sum_{Y\in\mathcal{P}\left( E\right) }m_{X^{\mathbf{B}}}\left( Y\right) Q_{exactly}^{i,1}\left( Y\right) \\ & \qquad =\sum_{Y\in\mathcal{P}\left( E\right) |\left\vert Y\right\vert =i}m_{X^{\mathbf{B}}}\left( Y\right) \\ & \qquad =\sum_{Y\in\mathcal{P}\left( E\right) |\left\vert Y\right\vert =i}{\displaystyle\prod\limits_{e\in Y}} \mu_{X^{\mathbf{B}}}\left( e\right) {\displaystyle\prod\limits_{e\in Y^{c}}} \left( 1-\mu_{X^{\mathbf{B}}}\left( e\right) \right) \\ & \qquad =\sum_{A\in F_{k}}{\displaystyle\prod\limits_{i\in A}} p_{i}{\displaystyle\prod\limits_{j\in A^{c}}} \left( 1-p_{j}\right) \\ & \qquad =\Pr^{\mathbf{B}}\left( K=k\right)\end{aligned}$$ And before proceeding to the main proof of this section, we need to introduce the following lemma: \[LemaBooleanCombinations\]Let $X_{1},\ldots,X_{n}\in\widetilde {\mathcal{P}}\left( E\right) $ and $\Phi_{l_{1},\ldots,l_{n}}\left( X_{1},\ldots,X_{n}\right) =X_{1}^{\left( l_{1}\right) }\widetilde{\cap }\ldots\widetilde{\cap}X_{n}^{\left( l_{n}\right) }$[^13] a boolean combination of $X_{1},\ldots,X_{n}$, then it is fulfilled:$$\begin{aligned} \mathcal{F}^{A}\left( Q_{exactly}^{i_{j},1}\right) \left( X_{1}^{\left( l_{1}\right) }\widetilde{\cap}\ldots\widetilde{\cap}X_{n}^{\left( l_{n}\right) }\right) & =\sum_{Y\in\mathcal{P}\left( E\right) \; | \; \left\vert Y\right\vert =j}m_{X_{1}^{\left( l_{1}\right) }\widetilde{\cap}\ldots\widetilde{\cap}X_{n}^{\left( l_{n}\right) }}\left( Y\right) \\ & =\sum_{\substack{Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) \; | \\ \left\vert Y_{1} \cap\ldots\cap Y_{n}\right\vert =j}} m_{X_{1}^{\left( l_{1}\right) }}\left( Y_{1}\right) \ldots m_{X_{n}^{\left( l_{n}\right) }}\left( Y_{n}\right) \\ & =\mathcal{F}^{A}\left( Q_{exactly}^{i_{j},1}\circ\Phi_{l_{1},\ldots,l_{n}}\right) \left( X_{1},\ldots,X_{n}\right)\end{aligned}$$ Being $\mathcal{F}^{A}$ a DFS, we have seen it conmutes with boolean combinations. Then:$$\begin{aligned} \mathcal{F}^{A}\left( Q_{exactly}^{i_{j},1}\right) \left( X_{1}^{\left( l_{1}\right) }\widetilde{\cap}\ldots\widetilde{\cap}X_{n}^{\left( l_{n}\right) }\right) & =\sum_{Y\in\mathcal{P}\left( E\right) } m_{X_{1}^{\left( l_{1}\right) }\widetilde{\cap}\ldots\widetilde{\cap}X_{n}^{\left( l_{n}\right) }}\left( Y\right) Q_{exactly}^{j,1}\left( Y\right) \\ & =\sum_{Y\in\mathcal{P}\left( E\right) \; | \; \left\vert Y\right\vert =j}m_{X_{1}^{\left( l_{1}\right) }\widetilde{\cap}\ldots\widetilde{\cap }X_{n}^{\left( l_{n}\right) }}\left( Y\right) \\ & =\mathcal{F}^{A}\left( Q_{exactly}^{j,1}\right) \left( \Phi _{l_{1},\ldots,l_{n}}\left( X_{1},\ldots,X_{n}\right) \right) \\ & =\mathcal{F}^{A}\left( Q_{exactly}^{j,1}\circ\Phi_{l_{1},\ldots,l_{n}}\right) \left( X_{1},\ldots,X_{n}\right)\end{aligned}$$ and let $j_{1},\ldots,j_{s}\in\left\{ 1,\ldots,n\right\} $ the ordered set of indexes in $l_{1},\ldots,l_{n}$ such that $l_{j_{s}}=0$ (i.e., the ones that complement the input argument). Example \[ExampleBooleanCombination\] showed that $\Phi_{l_{1},\ldots,l_{n}}$ is of the form $\underset{n}{\cap\ldots\cap}\tau_{j_{1}}\lnot\tau_{j_{1}}\tau_{j_{2}}\lnot\tau_{j_{2}}\ldots\tau_{j_{s}}\lnot\tau_{j_{s}}$. That is:$$\Phi_{l_{1},\ldots,l_{n}}\left( Y_{1},\ldots,Y_{n}\right) \\ =\underset{n}{\cap\ldots\cap}\tau_{j_{1}}\lnot\tau_{j_{1}}\tau_{j_{2}}\lnot\tau_{j_{2}}\ldots\tau_{j_{s}}\lnot\tau_{j_{s}}\left( Y_{1},\ldots,Y_{n}\right)$$ then $$\begin{aligned} \mathcal{F}^{A}\left( Q_{exactly}^{i_{j},1}\right) \left( X_{1}^{\left( l_{1}\right) }\widetilde{\cap}\ldots\widetilde{\cap}X_{n}^{\left( l_{n}\right) }\right) & =\ldots\\ & =\mathcal{F}^{A}\left( Q_{exactly}^{j,1}\underset{n}{\cap\ldots\cap}\tau_{j_{1}}\lnot\tau_{j_{1}}\tau_{j_{2}}\lnot\tau_{j_{2}}\ldots\tau_{j_{s}}\lnot\tau_{j_{s}}\right) \\ & \qquad \left( X_{1},\ldots,X_{n}\right) \\ & =\mathcal{F}^{A}\left( Q_{exactly}^{j,1}\underset{n}{\cap\ldots\cap }\right) \left( X_{1}^{\left( l_{1}\right) },\ldots,X_{n}^{\left( l_{n}\right) }\right) \\ & =\sum_{Y_{1}\in\mathcal{P}\left( E\right) }\ldots\sum_{Y_{n}\in\mathcal{P}\left( E\right) }m_{X_{1}^{\left( l_{1}\right) }}\left( Y_{1}\right) \ldots m_{X_{n}^{\left( l_{n}\right) }}\left( Y_{n}\right) \\ & \qquad \left( Q_{exactly}^{j,1}\underset{n}{\cap\ldots\cap}\right) \left( Y_{1},\ldots,Y_{n}\right) \\ & =\sum_{\substack{Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) \; | \\ \left\vert Y_{1}\cap\ldots\cap Y_{n}\right\vert =j}} m_{X_{1}^{\left( l_{1}\right) }}\left( Y_{1}\right) \ldots m_{X_{n}^{\left( l_{n}\right) }}\left( Y_{n}\right)\end{aligned}$$ Now, we will compute the projection of the probability function for quantitative quantifiers. By theorem \[TeoremaCuantitativo\] every semi-fuzzy quantifier $Q:\mathcal{P}\left( E\right) ^{n}\longrightarrow\mathbf{I}$ can be defined by means of a function $q:\left\{ 0,\ldots,\left\vert E\right\vert \right\} ^{K}\longrightarrow\mathbf{I}$ depending on the cardinalities of the boolean combinations of the input sets ($\left\vert \Phi_{1}\left( Y_{1},\ldots ,Y_{n}\right) \right\vert ,\ldots,\left\vert \Phi_{K}\left( Y_{1},\ldots,Y_{n}\right) \right\vert $). Then: $$\begin{aligned} \mathcal{F}^{A}\left( Q\right) \left( X_{1},\ldots,X_{n}\right) & =\sum_{Y_{1}\in\mathcal{P}\left( E\right) }\ldots\sum_{Y_{n}\in\mathcal{P}\left( E\right) }m_{X_{1}}\left( Y_{1}\right) \ldots m_{X_{n}}\left( Y_{n}\right) Q\left( Y_{1},\ldots,Y_{n}\right) \\ & =\sum_{\left( i_{1},\ldots,i_{K}\right) \in m^{K}}\sum_{\substack{Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) \; | \\\left\vert \Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) \right\vert =i_{1}\wedge\\\ldots\\\left\vert \Phi_{k}\left( Y_{1},\ldots,Y_{n}\right) \right\vert =i_{K}}}m_{X_{1}}\left( Y_{1}\right) \ldots m_{X_{n}}\left( Y_{n}\right) q\left( \left\vert \Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) \right\vert ,\ldots,\left\vert \Phi_{K}\left( Y_{1},\ldots,Y_{n}\right) \right\vert \right) \\ & =\sum_{\left( i_{1},\ldots,i_{K}\right) \in m^{K}} q\left( i_{1},\ldots,i_{K}\right) \sum_{\mathclap{\substack{Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) |\\\left\vert \Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) \right\vert =i_{1}\wedge\\\ldots\\\left\vert \Phi_{K}\left( Y_{1},\ldots,Y_{n}\right) \right\vert =i_{K}}}} \; m_{X_{1}}\left( Y_{1}\right) \ldots m_{X_{n}}\left( Y_{n}\right)\end{aligned}$$ Let us denote by$$f\left( i_{1},\ldots,i_{K}\right) =\sum_{\substack{Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) \; |\\\left\vert \Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) \right\vert =i_{1}\wedge\\\ldots\\\left\vert \Phi _{K}\left( Y_{1},\ldots,Y_{n}\right) \right\vert =i_{K}}}m_{X_{1}}\left( Y_{1}\right) \ldots m_{X_{n}}\left( Y_{n}\right) \label{EqProbabilityMx2}$$ $f\left( i_{1},\ldots,i_{K}\right) $ is a probability function. Take into account that $m_{X_{1}}\left( Y_{1}\right) \ldots m_{X_{n}}\left( Y_{n}\right) $ define a probability over $\left( Y_{1},\ldots,Y_{n}\right) \in\mathcal{P}\left( E\right) ^{n}$, and $f\left( i_{1},\ldots ,i_{K}\right) $ simply distributes the probabilities on $\left( Y_{1},\ldots,Y_{n}\right) \in\mathcal{P}\left( E\right) ^{n}$ over the cardinalities of the $K$ boolean combinations. Let $f\left( i_{1},\ldots,i_{K}\right) $ be the probability distribution that is obtained when we compute the probability induced by $X_{1},\ldots,X_{n}\in\mathcal{P}\left( E\right) ^{n}$ fuzzy sets over the cardinalities of the boolean combinations $\left\vert \Phi_{1}\left( Y_{1},\ldots,Y_{n}\right) \right\vert ,\ldots,\left\vert \Phi_{K}\left( Y_{1},\ldots,Y_{n}\right) \right\vert $. The probability projection $j$ of $f\left( i_{1},\ldots,i_{K}\right) $ will follow a binomial poisson distribution of parameters$$\begin{aligned} p_{1}^{j} & =\mu_{X_{1}^{\left( l_{j,1}\right) }\widetilde{\cap}\ldots\widetilde{\cap}X_{n}^{\left( l_{j,n}\right) }}\left( e_{1}\right) \\ & \ldots\\ p_{m}^{j} & =\mu_{X_{1}^{\left( l_{j,1}\right) }\widetilde{\cap}\ldots\widetilde{\cap}X_{n}^{\left( l_{j,n}\right) }}\left( e_{m}\right)\end{aligned}$$ where $X_{1}^{\left( l_{j,1}\right) }\widetilde{\cap}\ldots\widetilde{\cap }X_{n}^{\left( l_{j,n}\right) }=\Phi_{j}\left( X_{1},\ldots,X_{n}\right) $ Let us consider the projection $j$ of $f\left( i_{1},\ldots,i_{K}\right) $. Using the same ideas than in the proof of lemma \[LemaBooleanCombinations\]:$$\begin{aligned} f^{j}\left( i_{j}\right) & =\sum_{i_{1},\ldots,i_{j-1},i_{j+1},\ldots,i_{K}}f\left( i_{1},\ldots,i_{j},\ldots,i_{K}\right) \\ & =\sum_{\substack{Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) |\\\left\vert \Phi_{j}\left( Y_{1},\ldots,Y_{n}\right) \right\vert =i_{j}}}m_{X_{1}}\left( Y_{1}\right) \ldots m_{X_{n}}\left( Y_{n}\right) \\ & =\sum_{Y_{1},\ldots,Y_{n}\in\mathcal{P}\left( E\right) }m_{X_{1}}\left( Y_{1}\right) \ldots m_{X_{n}}\left( Y_{n}\right) \\ & \qquad \left( Q_{exactly}^{i_{j},1}\circ\Phi_{j}\right) \left( Y_{1},\ldots,Y_{n}\right) \\ & =\mathcal{F}^{A} \left( Q_{exactly}^{i_{j},1}\circ\Phi_{j}\right) \left( X_{1},\ldots,X_{n}\right) \\ & =\sum_{Y\in\mathcal{P}\left( E\right) |\left\vert Y\right\vert =j}m_{X_{1}^{\left( l_{1}\right) }\widetilde{\cap}\ldots\widetilde{\cap }X_{n}^{\left( l_{n}\right) }}\left( Y\right)\end{aligned}$$ but this is a binomial poisson bernoulli succession with distribution $\mathbf{B}=P_{1},\ldots,P_{m}$ with probabilities (proposition \[PropEqBinFA\])$$\begin{aligned} p_{1} & =\mu_{X_{1}^{\left( l_{1}\right) }\widetilde{\cap}\ldots \widetilde{\cap}X_{n}^{\left( l_{n}\right) }}\left( e_{1}\right) \\ & \ldots\\ p_{m} & =\mu_{X_{1}^{\left( l_{1}\right) }\widetilde{\cap}\ldots \widetilde{\cap}X_{n}^{\left( l_{n}\right) }}\left( e_{m}\right)\end{aligned}$$ [^1]: F. Díaz-Hermida, M. Matabuena, and J.C. Vidal are with the Centro Singular de Investigación en Tecnoloxías da Información (CiTIUS), Universidade de Santiago de Compostela, 15782, Santiago de Compostela, SPAIN. [^2]: Function $T_{a,b,c,d}$ is defined as $$T_{a,b,c,d}\left( x\right) =\left\{ \begin{array} [c]{cc}0 & x\leq a\\ \frac{x-a}{b-a} & a<x\leq b\\ 1 & b<x\leq c\\ 1-\frac{x-c}{d-c} & c<x\leq d\\ 0 & d<x \end{array} \right.$$ [^3]: Function $S_{\alpha,\gamma}$ is defined as $$S_{\alpha,\gamma}\left( x\right) =\left\{ \begin{tabular} [c]{ll}$0$ & $x<\alpha$\\ $2\left( \frac{\left( x-\alpha\right) }{\left( \gamma-\alpha\right) }\right) ^{2}$ & $\alpha<x\leq\frac{\alpha+\gamma}{2}$\\ $1-2\left( \frac{\left( x-\gamma\right) }{\left( \gamma-\alpha\right) }\right) ^{2}$ & $\frac{\alpha+\gamma}{2}<x\leq\gamma$\\ $1$ & $\gamma<x$\end{tabular} \right.$$ [^4]: We would like to point out that the probabilistic interpretation of the *QFM* $\mathcal{F}^{A}$ holds some relationships with a similar probabilistic interpretation of the Zadeh’s model. Let $X\in\widetilde{\mathcal{P}}\left( E\right) $ be a fuzzy set representing the linguistic concept  *‘big houses’*, and let us suppose we want to select an element $e\in E$. Let us assume we have the same probability of selecting each element, and that $\mu_{e}\left( X\right) $ represents the probability that the element $e$ fulfills the property of being a big house. Let $fq:\left[ 0,1\right] \rightarrow\mathbf{I}$ be a function representing a proportional unary linguistic quantifier (e.g. *‘most’*). Then, the Zadeh’s model just applies the linguistic quantifier to the average probability of selecting an element fulfilling the property of being a ‘big house’: $$fq\left( Avg\left( \Pr\left( e\_is\_big|e\_is\_selected\right) \right) \right) =fq\left( \frac{1}{m}\sum_{e\in E}\mu_{e}\left( X\right) \right)$$ . In contrast, the $\mathcal{F}^{A}$ QFM computes the probability of every possible combination in which the elements of $E$ can fulfill the property of ‘being a big house’. After computing the probability of each combination, we compute the average of applying the support function of the quantifier $fq$ to the possible combinations. [^5]: Let be $\preceq_{c}$ a partial order in $\mathbf{I}\times\mathbf{I}$ defined as [@Glockner06Libro section 5.2 and 6.3]:$$x\preceq_{c}y\Leftrightarrow y\leq x\leq\frac{1}{2}\text{ or }\frac{1}{2}\leq x\leq y$$ for $x,y\in\mathbf{I}$. A fuzzy set $X_{1}$ is at least as fuzzy as a fuzzy set $X_{2}$ if for each $e\in E$, $\mu_{X_{1}}\left( e\right) \preceq_{c}\mu_{X_{2}}$; that is, membership degrees of $X_{1}$ are closer to $0.5$ than membership degrees of $X_{2}$. In the case of fuzzy quantifiers a similar definition is applied. [^6]: For simplicity of the notation, we will use $\left[ 0,1\right] ^{K}$ instead of $\left\{ 0,\frac{1}{m},\ldots,\frac{m-1}{m},1\right\}^{K} $. [^7]: By $\chi_{Y}\left( e_{i}\right) $ we are representing the characteristic function of $Y$; that is: $\chi_{Y}\left( e_{i}\right) =1$ if $e_{i}\in Y$ and $0$ otherwise. [^8]: Take into account that, as the variance tends to $0$, by the *Chebyshev inequality* we always could find an interval around $average\left( f^{j\prime}\right) $ as small and containing a probability mass as high as desired for any $j$. This will allow to put as much probability around $\left( \frac{\sum_{i=1}^{m}p_{i}^{1}}{m},\ldots,\frac{\sum_{i=1}^{m}p_{i}^{k}}{m}\right) $ as we wanted, where $q^{\prime}$ is continuous by hypothesis. [^9]: Take into account that for proportional quantifiers $\frac{\left\vert Y_{1}\cap Y_{2}\right\vert }{\left\vert Y_{1}\right\vert }=\frac{\left\vert Y_{1}\cap Y_{2}\right\vert }{m}/\left( \frac{\left\vert Y_{1}\cap Y_{2}\right\vert }{m}+\frac{\left\vert Y_{1}\cap\overline{Y_{2}}\right\vert }{m}\right) $. In this case, the $\mathcal{F}^{A}$ QFM will converge to $f_{Q}\left( \frac{\sum_{e\in E}\mu_{X_{1}}\left( e\right) \mu_{X_{2}}\left( e\right) }{\sum_{e\in E}\mu_{X_{1}}\left( e\right) }\right) $. [^10]: For many quantifiers, results of the $\mathcal{F}^{A}$ QFM and of the Zadeh’s model will be extremely close even for small fuzzy sets. There are two main reasons for that. Once is that the variance of the probability projections associated to the different boolean combinations was very low and as a consequence, that the probability distributions would be very concentrated around the average. The other situation is that the fuzzy number used in the definition of the semi-fuzzy quantifier was approximately linear in the area in which much of the probability is concentrated. In this situation, the symmetry of the normal distribution (to which the poisson binomial distribution converges) will cause that the result of the evaluation will be really close to the result of the Zadeh’s model. [^11]: The reasonable choice of the fuzzy negation $\widetilde {\lnot}:\mathbf{I}\rightarrow\mathbf{I}$ is the induced negation of the QFM. [^12]: Notation used in the example can result very confusing. For this reason, we will present below the full detail $Q\cap\cap\tau_{1}\lnot\tau_{1}\tau_{2}\lnot\tau_{2}$, explicitily detaling the application of the different transformations to the semi-fuzzy quantifier: $$\begin{aligned} \left( Q\cap\cap\tau_{1}\lnot\tau_{1}\tau_{2}\lnot\tau_{2}\right) & =\left( f^{\prime}:\left( Y_{1}^{\prime},Y_{2}^{\prime}\right) \rightarrow Q\left( Y_{1}^{\prime}\cap Y_{2}^{\prime}\right) \right) \cap\tau_{1}\lnot\tau_{1}\tau_{2}\lnot\tau_{2}\\ & =\left( f^{\prime\prime}:\left( Y_{1}^{\prime\prime},Y_{2}^{\prime\prime },Y_{3}^{\prime\prime}\right) \rightarrow f^{\prime}\left( Y_{1}^{\prime\prime},Y_{2}^{\prime\prime}\cap Y_{3}^{\prime\prime}\right) \right) \tau_{1}\lnot\tau_{1}\tau_{2}\lnot\tau_{2}\\ & =\left( f^{\prime\prime}:\left( Y_{1}^{\prime\prime},Y_{2}^{\prime\prime },Y_{3}^{\prime\prime}\right) \rightarrow Q\left( Y_{1}^{\prime\prime}\cap Y_{2}^{\prime\prime}\cap Y_{3}^{\prime\prime}\right) \right) \tau_{1}\lnot\tau_{1}\tau_{2}\lnot\tau_{2}\\ & =\left( f^{\prime\prime\prime}:\left( Y_{1}^{\prime\prime\prime},Y_{2}^{\prime\prime\prime},Y_{3}^{\prime\prime\prime}\right) \rightarrow f^{\prime\prime}\left( Y_{3}^{\prime\prime\prime},Y_{2}^{\prime\prime\prime },Y_{1}^{\prime\prime\prime}\right) \right) \lnot\tau_{1}\tau_{2}\lnot \tau_{2}\\ & =\left( f^{\prime\prime\prime}:\left( Y_{1}^{\prime\prime\prime},Y_{2}^{\prime\prime\prime},Y_{3}^{\prime\prime\prime}\right) \rightarrow Q\left( Y_{3}^{\prime\prime\prime}\cap Y_{2}^{\prime\prime\prime}\cap Y_{1}^{\prime\prime\prime}\right) \right) \lnot\tau_{1}\tau_{2}\lnot\tau _{2}\\ & =\left( f^{\prime\prime\prime\prime}:\left( Y_{1}^{\prime\prime \prime\prime},Y_{2}^{\prime\prime\prime\prime},Y_{3}^{\prime\prime\prime \prime}\right) \rightarrow f^{\prime\prime\prime}\left( Y_{1}^{\prime \prime\prime\prime},Y_{2}^{\prime\prime\prime\prime},\lnot Y_{3}^{\prime \prime\prime\prime}\right) \right) \tau_{1}\tau_{2}\lnot\tau_{2}\\ & =\left( f^{\prime\prime\prime\prime}:\left( Y_{1}^{\prime\prime \prime\prime},Y_{2}^{\prime\prime\prime\prime},Y_{3}^{\prime\prime\prime \prime}\right) \rightarrow Q\left( \lnot Y_{3}^{\prime\prime\prime\prime }\cap Y_{2}^{\prime\prime\prime\prime}\cap Y_{1}^{\prime\prime\prime\prime }\right) \right) \tau_{1}\tau_{2}\lnot\tau_{2}\\ & =\left( f^{\prime\prime\prime\prime\prime}:\left( Y_{1}^{\prime \prime\prime\prime\prime},Y_{2}^{\prime\prime\prime\prime\prime},Y_{3}^{\prime\prime\prime\prime\prime}\right) \rightarrow f^{\prime\prime \prime\prime}\left( Y_{3}^{\prime\prime\prime\prime\prime},Y_{2}^{\prime\prime\prime\prime\prime},Y_{1}^{\prime\prime\prime\prime\prime }\right) \right) \tau_{2}\lnot\tau_{2}\\ & =\left( f^{\prime\prime\prime\prime\prime}:\left( Y_{1}^{\prime \prime\prime\prime\prime},Y_{2}^{\prime\prime\prime\prime\prime},Y_{3}^{\prime\prime\prime\prime\prime}\right) \rightarrow Q\left( \lnot Y_{1}^{\prime\prime\prime\prime\prime}\cap Y_{2}^{\prime\prime\prime \prime\prime}\cap Y_{3}^{\prime\prime\prime\prime\prime}\right) \right) \tau_{2}\lnot\tau_{2}\\ & =\left( f^{\prime\prime\prime\prime\prime\prime}:\left( Y_{1}^{\prime\prime\prime\prime\prime\prime},Y_{2}^{\prime\prime\prime\prime \prime\prime},Y_{3}^{\prime\prime\prime\prime\prime\prime}\right) \rightarrow f^{\prime\prime\prime\prime\prime}:\left( Y_{1}^{\prime\prime\prime \prime\prime\prime},Y_{3}^{\prime\prime\prime\prime\prime\prime},Y_{2}^{\prime\prime\prime\prime\prime\prime}\right) \right) \lnot\tau_{2}\\ & =\left( f^{\prime\prime\prime\prime\prime\prime}:\left( Y_{1}^{\prime\prime\prime\prime\prime\prime},Y_{2}^{\prime\prime\prime\prime \prime\prime},Y_{3}^{\prime\prime\prime\prime\prime\prime}\right) \rightarrow Q\left( \lnot Y_{1}^{\prime\prime\prime\prime\prime\prime}\cap Y_{3}^{\prime\prime\prime\prime\prime\prime}\cap Y_{2}^{\prime\prime\prime \prime\prime\prime}\right) \right) \lnot\tau_{2}\\ & =\left( f^{\prime\prime\prime\prime\prime\prime\prime}:\left( Y_{1}^{\prime\prime\prime\prime\prime\prime\prime},Y_{2}^{\prime\prime \prime\prime\prime\prime\prime},Y_{3}^{\prime\prime\prime\prime\prime \prime\prime}\right) \rightarrow f^{\prime\prime\prime\prime\prime\prime }\left( Y_{1}^{\prime\prime\prime\prime\prime\prime\prime},Y_{2}^{\prime\prime\prime\prime\prime\prime\prime},\lnot Y_{3}^{\prime\prime \prime\prime\prime\prime\prime}\right) \right) \tau_{2}\\ & =\left( f^{\prime\prime\prime\prime\prime\prime\prime}:\left( Y_{1}^{\prime\prime\prime\prime\prime\prime\prime},Y_{2}^{\prime\prime \prime\prime\prime\prime\prime},Y_{3}^{\prime\prime\prime\prime\prime \prime\prime}\right) \rightarrow Q\left( \lnot Y_{1}^{\prime\prime \prime\prime\prime\prime\prime}\cap\lnot Y_{3}^{\prime\prime\prime\prime \prime\prime\prime}\cap Y_{2}^{\prime\prime\prime\prime\prime\prime\prime }\right) \right) \tau_{2}\\ & =\left( f^{\prime\prime\prime\prime\prime\prime\prime\prime}:\left( Y_{1}^{\prime\prime\prime\prime\prime\prime\prime\prime},Y_{2}^{\prime \prime\prime\prime\prime\prime\prime\prime},Y_{3}^{\prime\prime\prime \prime\prime\prime\prime\prime}\right) \rightarrow f^{\prime\prime \prime\prime\prime\prime\prime}\left( Y_{1}^{\prime\prime\prime\prime \prime\prime\prime\prime},Y_{3}^{\prime\prime\prime\prime\prime\prime \prime\prime},Y_{2}^{\prime\prime\prime\prime\prime\prime\prime\prime}\right) \right) \\ & =\left( f^{\prime\prime\prime\prime\prime\prime\prime\prime}:\left( Y_{1}^{\prime\prime\prime\prime\prime\prime\prime\prime},Y_{2}^{\prime \prime\prime\prime\prime\prime\prime\prime},Y_{3}^{\prime\prime\prime \prime\prime\prime\prime\prime}\right) \rightarrow Q\left( \lnot Y_{1}^{\prime\prime\prime\prime\prime\prime\prime\prime}\cap\lnot Y_{2}^{\prime\prime\prime\prime\prime\prime\prime\prime}\cap Y_{3}^{\prime\prime\prime\prime\prime\prime\prime\prime}\right) \right)\end{aligned}$$ [^13]: Following notation in \[TeoremaCuantitativo\], by $X_{1}^{\left( l_{1}\right) }$ we are denoting the set $X_{1}$ in case $l_{1}=1$ and $\widetilde{\lnot}X_{1}$ in case $l_{1}=0$.

--- abstract: | This paper deals with the following quasilinear Keller-Segel-Navier-Stokes system modeling coral fertilization $$\left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c)-nm,\quad x\in \Omega, t>0,\\ c_t+u\cdot\nabla c=\Delta c-c+m,\quad x\in \Omega, t>0,\\ m_t+u\cdot\nabla m=\Delta m-nm,\quad x\in \Omega, t>0,\\ u_t+\kappa(u \cdot \nabla)u+\nabla P=\Delta u+(n+m)\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in \Omega, t>0\\ \end{array}\right.\eqno(*)$$ under no-flux boundary conditions in a bounded domain $\Omega\subset \mathbb{R}^3$ with smooth boundary, where $\phi\in W^{2,\infty} (\Omega)$. Here $S(x,n,c)$ denotes the rotational effect which satisfies $|S(x,n,c)|\leq S_0 (c)(1 + n)^{-\alpha}$ with $\alpha\geq0$ and some nonnegative nondecreasing function $S_0$. Based on a new weighted estimate and some carefully analysis, if $\alpha>0$, then for any $\kappa\in\mathbb{R},$ system $(*)$ possesses a global weak solution for which there exists $T > 0$ such that $(n,c,m , u)$ is smooth in $\Omega\times( T ,\infty)$. Furthermore, for any $p>1,$ this solution is uniformly bounded in with respect to the norm in $L^p(\Omega)\times L^\infty(\Omega) \times L^\infty(\Omega)\times L^2 (\Omega; \mathbb{R}^3)$. Building on this boundedness property and some other analysis, it can finally even be proved that in the large time limit, any such solution approaches the spatially homogeneous equilibrium $(\hat{n},\hat{m},\hat{m},0)$ in an appropriate sense, where $\hat{n}=\frac{1}{|\Omega|}\{\int_{\Omega}n_0-\int_{\Omega}m_0\}_{+}$ and $\hat{m}=\frac{1}{|\Omega|}\{\int_{\Omega}m_0 -\int_{\Omega}n_0\}_{+}$. author: - | Jiashan Zheng[^1]\ School of Mathematics and Statistics Science,\ Ludong University, Yantai 264025, P.R.China\ title: '**A new result for the global existence (and boundedness), regularity and stabilization of a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization** ' --- [***Key words:***]{} Navier-Stokes system; Keller-Segel model; Global existence; Large time behavior; Tensor-valued sensitivity [***2010 Mathematics Subject Classification***]{}:  35K55, 35Q92, 35Q35, 92C17 Introduction ============ This work is concerned with the following chemotaxis-fluid system modelling coral fertilization: $$\left\{\begin{array}{ll} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c)-nm,\quad x\in \Omega, t>0,\\ c_t+u\cdot\nabla c=\Delta c-c+m,\quad x\in \Omega, t>0,\\ m_t+u\cdot\nabla m=\Delta m-nm,\quad x\in \Omega, t>0,\\ u_t+\kappa(u \cdot \nabla)u+\nabla P=\Delta u+(n+m)\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in \Omega, t>0\\ \disp{(\nabla n-nS(x, n, c))\cdot\nu=\nabla c\cdot\nu=\nabla m\cdot\nu=0,u=0,}\quad x\in \partial\Omega, t>0,\\ \disp{n(x,0)=n_0(x),c(x,0)=c_0(x),m(x,0)=m_0(x),u(x,0)=u_0(x),}\quad x\in \Omega,\\ \end{array}\right.\label{334451.1fghyuisda}$$ where $\Omega\subset \mathbb{R}^3$ is a bounded domain with smooth boundary $\partial\Omega$ and the matrix-valued function $S(x,n,c)$ indicates the rotational effect which satisfies $$\label{x1.73142vghf48rtgyhu} S\in C^2(\bar{\Omega}\times[0,\infty)^2;\mathbb{R}^{3\times3})$$ and $$\label{x1.73142vghf48gg}|S(x,n,c)|\leq (1+n)^{-\alpha}S(c)~~ \mbox{for all}~~ (x,n,c)\in\Omega\times[0,\infty)^2~~\mbox{with}~~ S(c)~~ \mbox{nondecreasing on}~~ [0,\infty)$$ and $\alpha\geq0.$ As described in [@Kiselevdd793; @Kiselevsssdd793; @EspejoEspejojainidd793; @EspejojjEspejojainidd793], problems of this type arise in the modeling of the phenomenon of coral broadcast spawning, where the sperm $n$ chemotactically moves toward the higher concentration of the chemical $c$ released by the egg $m$, while the egg $m$ is merely affected by random diffusion, fluid transport and degradation upon contact with the sperm (see also [@Lidfff00]). Here $\kappa,u,P$ and $\phi$ denote, respectively, the strength of nonlinear fluid convection, the velocity field, the associated pressure of the fluid and the potential of the gravitational field. We further note that the sensitivity tensor $S(x,n,c)$ may take values that are matrices possibly containing nontrivial off-diagonal entries, which reflects that the chemotactic migration may not necessarily be oriented along the gradient of the chemical signal, but may rather involve rotational flux components (see [@Xusddeddff345511215; @XueXuejjainidd793] for the detailed model derivation). Chemotaxis is the directed movement of the cells as a response to gradients of the concentration of the chemical signal substance in their environment, where the chemical signal substance may be produced or consumed by cells themselves (see e.g. Hillen and Painter [@Hillen] and [@Bellomo1216]). The classical chemotaxis system was introduced in 1970 by Keller and Segel ([@Keller2710]), which is called Keller-Segel system. Since then, the Keller-Segel model has attracted more and more attention, and also has been constantly modified by various authors to characterize more biological phenomena (see Cieślak and Stinner [@Cie791], Cieślak and Winkler [@Cie72], Ishida et al. [@Ishida], Painter and Hillen [@Painter55677], Hillen and Painter [@Hillen], Wang et al. [@Wang76; @Wang79], Winkler et al. [@Cie72; @Horstmann791; @Winkler72; @Winkler792; @Winkler21215; @Winkler2233444ssdff51215; @Winkler793], Zheng [@Zheng00] and references therein for detailed results). For related works in this direction, we mention that a corresponding quasilinear version ( see e.g. [@Tao794; @Winkler72; @Zhengssdefr23; @Zheng00; @Zhengsddfffsdddssddddkkllssssssssdefr23]), the logistic damping or the signal consumed by the cells, has been deeply investigated by Cieślak and Stinner [@Cie791; @Cie201712791], Tao and Winkler [@Tao794; @Winkler79; @Winkler72], and Zheng et al. [@Zheng00; @Zhengssssdefr23; @Zhengssdefr23; @Zhengssssssdefr23]. In various situations, however, the interaction of chemotactic movement of the gametes and the surrounding fluid is not negligible (see Tuval et al. [@Tuval1215]). In 2005, Tuval et al. ([@Tuval1215]) proposed the following prototypical [**signal consuming**]{} model (with tensor-valued sensitivity): $$\left\{\begin{array}{ll} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot( nS(x,n,c)\nabla c),\quad x\in \Omega, t>0,\\ c_t+u\cdot\nabla c=\Delta c-nf(c),\quad x\in \Omega, t>0,\\ u_t+\kappa (u\cdot\nabla)u+\nabla P=\Delta u+n\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in \Omega, t>0,\\ \end{array}\right.\label{1.1hhjffggjddssggtyy}$$ where $f(c)$ denotes the [**consumption**]{} rate of the oxygen by the cells. Here $S$ is a tensor-valued function or a scalar function which is the same as . The model describes the interaction of oxygen-taxis bacteria with a surrounding incompressible viscous fluid in which the oxygen is dissolved. After this, assume that the chemotactic sensitivity $S(x, n, c):=S(c)$ is a scalar function. This kind of models have been studied by many researchers by making use of energy-type functionals (see e.g. Chae et. al. [@Chaexdd12176], Duan et. al. [@Duan12186], Liu and Lorz [@Liu1215; @Lorz1215], Tao and Winkler [@Tao41215; @Winkler31215; @Winkler61215; @Winkler51215], Zhang and Zheng [@Zhang12176] and references therein). In fact, if $S(x, n, c):=S(c)$, Winkler ([@Winkler31215] and [@Winkler61215]) proved that in two-dimensional space admits a unique global classical solution which stabilizes to the spatially homogeneous equilibrium $(\frac{1}{|\Omega|}\int_{\Omega}n_0 , 0, 0)$ in the large time limit. While in three-dimensional setting, he (see [@Winkler51215]) also showed that there exists a globally defined weak solution to . Experiment [@Xusddeddff345511215] show that the chemotactic movement could be not directly along the signal gradient, but with a rotation, so that, the the corresponding chemotaxis-fluid system with tensor-valued sensitivity loses entropy-like functional structure, which gives rise to considerable mathematical difficulties during the process of analysis. The global solvability of corresponding initial value problem for chemotaxis-fluid system with tensor-valued sensitivity have been deeply investigated by Cao, Lankeit [@CaoCaoLiitffg11], Ishida [@Ishida1215], Wang et al. [@Wang11215; @Wang21215] and Winkler [@Winkler11215]. If $-nf(c)$ in the $c$-equation is replaced by $-c+n$, and the $u$-equation is a (Navier-)Stokes equation, then becomes the following chemotaxis-(Navier-)Stokes system in the context of signal [**produced**]{} other than consumed by cells (see [@Winkler444ssdff51215; @Wddffang11215; @Wang21215; @Wangss21215; @Zhenddddgssddsddfff00; @Kegssddsddfff00]) $$\left\{\begin{array}{ll} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot( n S(x,n,c)\cdot\nabla c),\quad x\in \Omega, t>0,\\ c_t+u\cdot\nabla c=\Delta c-c+n,\quad x\in \Omega, t>0,\\ u_t+\kappa (u\cdot\nabla)u+\nabla P=\Delta u+n\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in \Omega, t>0. \\ \end{array}\right.\label{1sdfdffgggggsxdcfffggvgb.1}$$ Due to the presence of the tensor-valued sensitivity $S(x,n,c)$ as well as the strongly nonlinear term $(u\cdot\nabla)u$ and lower regularity for $n$, the analysis of with tensor-valued sensitivity began to flourish (see [@Winkler444ssdff51215; @Wddffang11215; @Wang21215; @Wangss21215; @Zhenddddgssddsddfff00; @Kegssddsddfff00]). In fact, the global boundedness of classical solutions to the Stokes-version ($\kappa=0$ in the third equation of system ) of system with the tensor-valued $S$ satisfying $|S(x,n,c)|\leq C_S(1+n)^{-\alpha}$ with some $C_S > 0$ and $\alpha > 0$ which implies that the effect of chemotaxis is weakened when the cell density increases has been proved for any $\alpha > 0$ in two dimensions (see Wang and Xiang [@Wang21215]) and for $\alpha >\frac{1}{2}$ in three dimensions (see Wang and Xiang [@Wangss21215]). Then Wang-Winkler-Xiang ([@Wddffang11215]) further shows that when $\alpha > 0$ and $\Omega\subset R^2$ is a bounded [**convex**]{} domain with smooth boundary, system possesses a global-in-time classical and bounded solution. Recently, Zheng ([@Zhenddsdddddgssddsddfff00]) extends the results of [@Wddffang11215] to the general bounded domain by some new entropy-energy estimates. More recently, by using new entropy-energy estimates, Zheng and Ke ([@Kegssddsddfff00]) presented the existence of global and weak solutions for the system under the assumption that $S$ satisfies and $$|S(x,n,c)|\leq (1+n)^{-\alpha}~~ \mbox{for all}~~ (x,n,c)\in\Omega\times[0,\infty)^2$$ with $\alpha > \frac{1}{3}$, which, in light of the known results for the fluid-free system (see Horstmann and Winkler [@Horstmann791] and Bellomo et al. [@Bellomo1216] ), is an optimal restriction on $\alpha$. For more works about the chemotaxis-(Navier-)Stokes models , we mention that a corresponding quasilinear version or the logistic damping has been deeply investigated by Zheng [@Zhengsdsd6], Wang and Liu [@Liuddfffff], Tao and Winkler [@Tao41215], Wang et. al. [@Wang21215; @Wangss21215]. Other variants of the model has been used in the mathematical study of coral broad- cast spawning. In fact, Kiselev and Ryzhik ([@Kiselevdd793] and [@Kiselevsssdd793]) introduced the following Keller-Segel type system to model coral fertilization: $$\left\{\begin{array}{ll} \rho_t+u\cdot\nabla\rho=\Delta \rho-\chi\nabla\cdot( \rho\nabla c)-\varepsilon\rho^q, \quad \\ \disp{ 0=\Delta c +\rho,}\quad\\ \end{array}\right.\label{722dff344101.dddddgghggghhff2ffggffggx16677}$$ where $\rho,u,\chi$ and $-\varepsilon \rho^q$, respectively, denote the density of egg (sperm) gametes, the smooth divergence free sea fluid velocity as well as the positive chemotactic sensitivity constant and the reaction (fertilization) phenomenon. In fact, under suitable conditions, the global-in-time existence of the solution to is presented by Kiselev and Ryzhik in [@Kiselevdd793]. Moreover, they proved that the total mass $m_0(t) =\int_{R^2}\rho(x,t)dx$ approaches a positive constant whose lower bound is $C(\chi,\rho_0 ,u)$ as $t\rightarrow\infty$ when $q>2$. In the critical case of $N = q = 2$, a corresponding weaker but yet relevant effect within finite time intervals is detected (see [@Kiselevsssdd793]). In order to analyze a further refinement of the model which explicitly distinguishes between sperms and eggs, Espejo and Winkler ([@EspejojjEspejojainidd793]) have recently considered the Navier-Stokes version of : $$\left\{\begin{array}{ll} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(n\nabla c)-nm,\quad x\in \Omega, t>0,\\ c_t+u\cdot\nabla c=\Delta c-c+m,\quad x\in \Omega, t>0,\\ m_t+u\cdot\nabla m=\Delta m-nm,\quad x\in \Omega, t>0,\\ u_t+\kappa(u \cdot \nabla)u+\nabla P=\Delta u+(n+m)\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in \Omega, t>0\\ \end{array}\right.\label{1.dddddffdffg1}$$ in a bounded domain $\Omega\subset \mathbb{R}^2$. If $N=2,$ Espejo and Winkler ([@EspejojjEspejojainidd793]) established the global existence of classical solutions to the associated initial-boundary value problem , which tend towards a spatially homogeneous equilibrium in the large time limit. Furthermore, if $S(x, n, c)$ satisfying and with $\alpha \geq \frac{1}{3}$ or $\alpha \geq0$ and the initial data satisfy a certain smallness condition, Li-Pang-Wang ([@Lidfff00]) proved the same result for the three-dimensional Stokes ($\kappa=0$ in the fourth equation of ) version of system . From [@Lidfff00], we know that $\alpha\geq\frac{1}{3}$ is enough to warrant the boundedness of solutions to system for any large data (see Li-Pang-Wang [@Lidfff00]). We should point that the core step of [@Lidfff00] is to establish the estimates of the functional $$\|n(\cdot,t) \|_{L^2(\Omega)}^2+\|\nabla c(\cdot,t) \|_{L^2(\Omega)}^2+\|u(\cdot,t) \|_{W^{1,2}(\Omega)}^2,$$ which strongly relies on $\alpha\geq\frac{1}{3} $ and $\kappa=0$ (see the proof of Lemma 3.1 of [@Lidfff00]). To the best of our knowledge, it is yet unclear whether for $\alpha<\frac{1}{3}$ or $\kappa\neq0$, the solutions of exist (or even bounded) or not. Recently, relying on the functional $$\left\{ \begin{array}{rl} \disp{\int_{\Omega} n_{\varepsilon}^{4\alpha+\frac{2}{3}}+\int_{\Omega} |\nabla c_{\varepsilon}|^2+\int_{\Omega} | {u_{\varepsilon}}|^2~~~\mbox{if}~~\alpha\neq\frac{1}{12},}\\ \disp{\int_{\Omega} n_{\varepsilon}\ln n_{\varepsilon}+\int_{\Omega} |\nabla c_{\varepsilon}|^2+\int_{\Omega} | {u_{\varepsilon}}|^2~~~~\mbox{if}~~\alpha=\frac{1}{12},} \end{array} \right.$$ we ([@Zhenssssssdffssdddddddgssddsddfff00]) presented the existence of global [**weak**]{} solutions for the system under the assumption that $S$ satisfies and with $\alpha > 0$. However, the existence of global (stronger than the result of [@Zhenssssssdffssdddddddgssddsddfff00]) [**weak**]{} solutions is still open. In this paper, by using a new weighted estimate (see Lemma \[lemmaghjffggssddgghhmk4563025xxhjklojjkkk\]), we try to obtain enough regularity and compactness properties (see Lemmas \[lemmaghjffggssddgghhmk4563025xxhjklojjkkk\], \[ssdddlemmaghjffggssddgghhmk4563025xxhjklojjkkk\], and \[lemmddaghjsffggggsddgghhmk4563025xxhjklojjkkk\]), then show that system possesses a globally defined [**weak**]{} solution, which improves the result of [@Zhenssssssdffssdddddddgssddsddfff00]. Therefore, collecting the above results, it is meaningful to analyze the following question: Whether or not the assumption of $\alpha$ is optimal? Can we further relax the restriction on $\alpha$, say, to $\alpha > 0 $? Moreover, can we consider the regularity of global solution for system ? Inspired by the above works, the first result of paper is to prove the existence of global (and bounded) solution for any $\alpha > 0.$ Moreover, we also show that the corresponding solutions converge to a spatially homogeneous equilibrium exponentially as $t \rightarrow\infty$ as well. Throughout this paper, we assume that $$\phi\in W^{2,\infty}(\Omega) \label{x1.73142vghf481}$$ and the initial data $(n_0, c_0, u_0)$ fulfills $$\label{ccvvx1.731426677gg} \left\{ \begin{array}{ll} \displaystyle{n_0\in C(\bar{\Omega})~~~~ \mbox{with}~~ n_0\geq0 ~~\mbox{and}~~n_0\not\equiv0},\\ \displaystyle{c_0\in W^{1,\infty}(\Omega)~~\mbox{with}~~c_0\geq0~~\mbox{in}~~\bar{\Omega},}\\ \displaystyle{m_0\in C(\bar{\Omega})~~~~ \mbox{with}~~ m_0\geq0 ~~\mbox{and}~~m_0\not\equiv0},\\ \displaystyle{u_0\in D(A^\gamma_{r})~~\mbox{for~~ some}~~\gamma\in ( \frac{3}{4}, 1)~~\mbox{and any}~~ {r}\in (1,\infty),}\\ \end{array} \right.$$ where $A_{r}$ denotes the Stokes operator with domain $D(A_{r}) := W^{2,{r}}(\Omega)\cap W^{1,{r}}_0(\Omega) \cap L^{r}_{\sigma}(\Omega)$, and $L^{r}_{\sigma}(\Omega) := \{\varphi\in L^{r}(\Omega)|\nabla\cdot\varphi = 0\}$ for ${r}\in(1,\infty)$ ([@Sohr]). In the context of these assumptions, the first of our main results can be read as follows. \[theorem3\] Let $\Omega \subset \mathbb{R}^3$ be a bounded domain with smooth boundary. Suppose that the assumptions – and – hold. If $$\label{x1.73142vghf48}\alpha>0,$$ then for any $\kappa\in \mathbb{R}$, there exist $$\left\{\begin{array}{ll} n\in L^{\infty}_{loc}([0,\infty),L^p(\Omega))\cap L^{2}_{loc}([0,\infty),W^{1,2}(\Omega))~~~\mbox{for any}~~p>1,\\ c\in L^{\infty}(\Omega\times(0,\infty))\cap L^{2}_{loc}([0,\infty),W^{2,2}(\Omega))\cap L^{4}_{loc}([0,\infty),W^{1,4}(\Omega)),\\ m\in L^{\infty}(\Omega\times(0,\infty))\cap L^{2}_{loc}([0,\infty),W^{2,2}(\Omega))\cap L^{4}_{loc}([0,\infty),W^{1,4}(\Omega)),\\ u\in L^{2}_{loc}([0,\infty),W^{1,2}_{0,\sigma}(\Omega))\cap L^{\infty}_{loc}([0,\infty),L^{2}(\Omega)),\\ \end{array}\right.\label{1.1ddfghyuisdsdddda}$$ such that $(n,c,m,u)$ is a global weak solution of the problem in the natural sense as specified in [@Zhenssssssdffssdddddddgssddsddfff00]. Moreover, if $\kappa=0$, the problem possesses at least one global classical solution $(n, c,m, u, P)$. Moreover, this solution is bounded in $\Omega\times(0,\infty)$ in the sense that $$\|n(\cdot, t)\|_{L^\infty(\Omega)}+\|c(\cdot, t)\|_{W^{1,\infty}(\Omega)}+\|m(\cdot, t)\|_{W^{1,\infty}(\Omega)}+\| u(\cdot, t)\|_{L^{\infty}(\Omega)}\leq C~~ \mbox{for all}~~ t>0. \label{1.163072xggttyyu}$$ \(i) Theorem \[theorem3\] indicates that $\alpha > 0$ and $\kappa=0$ is enough to ensure the global existence and uniform boundedness of solution of the three-dimensional Keller-Segel-Stokes system , which improves the result obtained in [@Lidfff00], therein $\alpha\geq \frac{1}{3}$ is required. (ii)This result also improves the result of our recent paper ([@Zhenssssssdffssdddddddgssddsddfff00]), where the more [**weak**]{} solution than our result was obtained by using different method (see [@Zhenssssssdffssdddddddgssddsddfff00]). We can secondly prove that in fact any such [**weak**]{} solution $(n,c,m,u)$ becomes smooth ultimately, and that it approaches the unique spatially homogeneous steady state $(\hat{n},\hat{m},\hat{m},0)$, where $\hat{n}=\frac{1}{|\Omega|}\{\int_{\Omega}n_0-\int_{\Omega}m_0\}_{+}$ and $\hat{m}=\frac{1}{|\Omega|}\{\int_{\Omega}m_0 -\int_{\Omega}n_0\}_{+}$. \[thaaaeorem3\] Under the assumptions of Theorem \[theorem3\], then there are $T > 0$ and $\iota\in (0,1)$ such that the solution $(n,c,m,u)$ given by Theorem \[theorem3\] satisfies $$n,c,m\in C^{2+\iota,1+\frac{\iota}{2}}(\bar{\Omega}\times[T,\infty)),~~~u\in C^{2+\iota,1+\frac{\iota}{2}}(\bar{\Omega}\times[T,\infty);\mathbb{R}^3).$$ Moreover, $$n(\cdot,t)\rightarrow \hat{n},~c(\cdot,t)\rightarrow \hat{m} ~~\mbox{as well as } ~~~m(\cdot,t)\rightarrow \hat{m}~~\mbox{and}~~~u(\cdot,t)\rightarrow0 ~~\mbox{in}~~~L^\infty(\Omega),$$ where $\hat{n}=\frac{1}{|\Omega|}\{\int_{\Omega}n_0-\int_{\Omega}m_0\}_{+}$ and $\hat{m}=\frac{1}{|\Omega|}\{\int_{\Omega}m_0 -\int_{\Omega}n_0\}_{+}$. Theorem \[theorem3\] indicates that if $\alpha > 0$, then for arbitrarily large initial data and for any $\kappa\in \mathbb{R}$, this problem admits at least one global weak solution for which there exists $T > 0$ such that $(n,c,m, u)$ is smooth in $\Omega\times ( T ,\infty)$. Moreover, it is asserted that such solutions are shown to approach a spatially homogeneous equilibrium in the large time limit, which improves the result obtained in [@Lidfff00], therein $\kappa=0$ is required. [**Mathematical challenges for the regularity and stabilization of the solution for system .**]{} System incorporates fluid and rotational flux, which involves more complex cross-diffusion mechanisms and brings about many considerable mathematical difficulties. Firstly, even when posed without any external influence, that is, $n=c=m\equiv0,$ the corresponding Navier-Stokes system does not admit a satisfactory solution theory up to now (see Leray [@LerayLerayer792] and Sohr [@Sohr], Wiegner [@Wiegnerdd79]). As far as we know that the question of global solvability in classes of suitably regular functions yet remains open except in cases when the initial data are appropriately small (see e.g. Wiegner [@Wiegnerdd79]). Moreover, the tensor-valued sensitivity functions result in new mathematical difficulties, mainly linked to the fact that a chemotaxis system with such rotational fluxes thereby loses an energy-like structure (see e.g. [@Xusddeddff345511215]). In [@Lidfff00] and [@EspejojjEspejojainidd793], relying on globally [**bounded**]{} for the solution, Espejo-Winkler ([@EspejojjEspejojainidd793]) Li-Pang-Wang ([@Lidfff00]) proved that all these solutions of problem are shown to approach a spatially homogeneous equilibrium in the large time limit when $N=2$ or $N=3$ and $\kappa=0,$ respectively. As already mentioned in the above, in the case $N = 3$, it is not only unknown whether the incompressible Navier-Stokes equations possess global smooth solutions for arbitrarily large smooth initial data (see e.g. Wiegner [@Wiegnerdd79] and Sohr [@Sohr]). Therefore, when $\kappa\not=0$ and $N=3$, we can not use the idea of [@Lidfff00] and [@EspejojjEspejojainidd793] to discuss the large time behavior to problem , since, the globally [**bounded**]{} for the solutions are needed in [@Lidfff00] and [@EspejojjEspejojainidd793]. In order to derive these theorems, in Section 2, we introduce the regularized system of , establish some basic estimates of the solutions and recall a local existence result. In Section 3, a key step of the proof of our main results is to establish a bound for $n_\varepsilon(\cdot, t)$ in $L^{{p}} (\Omega)$ for any $p>1$. The approach is based on the weighted estimate of $\int_{\Omega}n_\varepsilon^{{p}}g(c_\varepsilon)$ with some weight function $g(c_\varepsilon)$ which is uniformly bounded both from above and below by positive constants. Here $n_\varepsilon$ and $c_\varepsilon$ are components of the solutions to below. On the basis of the previously established estimates and the compactness properties thereby implied, we shall pass to the limit along an adequate sequence of numbers $\varepsilon = \varepsilon_j\searrow0$ and thereby verify Theorem \[theorem3\]. Using the basic relaxation properties expressed in and , Section 4 is devoted to showing the large time behavior of global solutions to obtained in the above section. To this end, thanks to the decay property of $m_\varepsilon(\cdot,t)-\hat{m}+n_\varepsilon(\cdot,t) -\hat{n}$ formulated in Lemmas \[ssdddlemmddddaddffffdfffgg4sssdddd5630\] and \[11aaalemdfghkkmaddffffdfffgg4sssdddd5630\], this actually entails a certain eventual regularity and decay of $u_\varepsilon$ also in the present situation, where $\hat{n}=\frac{1}{|\Omega|}\{\int_{\Omega}n_0-\int_{\Omega}m_0\}_{+}$ and $\hat{m}=\frac{1}{|\Omega|}\{\int_{\Omega}m_0 -\int_{\Omega}n_0\}_{+}$. Using these bounds (see Lemmas \[ssdddlemmddddaddffffdfffgg4sssdddd5630\]–\[aaalemmaddffffsddddfffgg4sssdddd5630\]), based on maximal Sobolev regularity in the Stokes evolution system as well as inhomogeneous linear heat equations and the standard Schauder theory, we then prove eventual Hölder regularity and smoothness of solution $(n_\varepsilon,c_\varepsilon,m_\varepsilon,u_\varepsilon)$ (see Lemmas \[lemma45630hhuujjuuyy\]–\[lemma45630hhuujjsdfffggguuyy\]). For convergence as $t\rightarrow\infty$, we draw upon uniform Hölder bounds and smoothness for solution $(n,c,m,u)$ (see Lemmas \[lemma45630223\]–\[sssslemma45ssddddff630hhuujjsdfffggguuyy\]). Finally, applying an Ehrling-type lemma, we can prove any such solution approaches the spatially homogeneous equilibrium by using the above convergence properties (see Lemma \[lemma4dd5630hhuujjuuyy\]). Preliminaries ============= As mentioned in the introduction, the chemotactic sensitivity $S$ in the first equation in and the nonlinear convective term $\kappa(u \cdot \nabla)u$ in the Navier-Stokes subsystem of bring about a great challenge to the study of system . To deal with these difficulties, according to the ideas in [@Winkler51215] (see also [@Winklerssdd51215; @Kegssddsddfff00; @Winklerpejoevsssdd793]), we first consider the approximate problems given by $$\left\{\begin{array}{ll} n_{\varepsilon t}+u_{\varepsilon}\cdot\nabla n_{\varepsilon}=\Delta n_{\varepsilon}-\nabla\cdot(n_{\varepsilon}S_\varepsilon(x, n_{\varepsilon}, c_{\varepsilon})\nabla c_{\varepsilon})-n_{\varepsilon}m_{\varepsilon},\quad x\in \Omega, t>0,\\ c_{\varepsilon t}+u_{\varepsilon}\cdot\nabla c_{\varepsilon}=\Delta c_{\varepsilon}-c_{\varepsilon}+m_{\varepsilon},\quad x\in \Omega, t>0,\\ m_{\varepsilon t}+u_{\varepsilon}\cdot\nabla m_{\varepsilon}=\Delta m_{\varepsilon}-n_{\varepsilon}m_{\varepsilon},\quad x\in \Omega, t>0,\\ u_{\varepsilon t}+\nabla P_{\varepsilon}=\Delta u_{\varepsilon}-\kappa (Y_{\varepsilon}u_{\varepsilon} \cdot \nabla)u_{\varepsilon}+(n_{\varepsilon}+m_{\varepsilon})\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u_{\varepsilon}=0,\quad x\in \Omega, t>0,\\ \disp{\nabla n_{\varepsilon}\cdot\nu=\nabla c_{\varepsilon}\cdot\nu=\nabla m_{\varepsilon}\cdot\nu=0,u_{\varepsilon}=0,\quad x\in \partial\Omega, t>0,}\\ \disp{n_{\varepsilon}(x,0)=n_0(x),c_{\varepsilon}(x,0)=c_0(x),m_{\varepsilon}(x,0)=m_0(x),u_{\varepsilon}(x,0)=u_0(x)},\quad x\in \Omega,\\ \end{array}\right.\label{1.1fghyuisda}$$ where $$\begin{array}{ll} S_\varepsilon (x, n, c) = \rho_\varepsilon(x)\chi_\varepsilon (n)S(x, n, c),~~ x\in\bar{\Omega},~~n\geq0,~~c\geq0 \end{array}\label{3.10gghhjuuloollyuigghhhyy}$$ and $$\begin{array}{ll} Y_\varepsilon w := (1 + \varepsilon A)^{-1}w ~~~~\mbox{for all}~~ w\in L^2_{\sigma}(\Omega) \end{array}\label{aasddffgg1.1fghyuisda}$$ is the standard Yosida approximation. Here $(\rho_{\varepsilon} )_{\varepsilon\in(0,1)} \in C^\infty_0 (\Omega)$ and $(\chi_{\varepsilon} )_{\varepsilon\in(0,1)} \in C^\infty_0 ([0,\infty))$ are a family of functions which satisfy $$0\leq\rho_\varepsilon \leq 1~~~\mbox{in}~~\Omega,~~\rho_\varepsilon \nearrow1~~~~\mbox{in}~~\Omega~~\mbox{as}~~\varepsilon\searrow0$$ and $$0\leq\chi_\varepsilon \leq 1~~~\mbox{in}~~[0,\infty),~~\chi_\varepsilon \nearrow1~~\mbox{in}~~[0,\infty)~~\mbox{as}~~\varepsilon\searrow0.$$ Without essential difficulty, the local existence of approximate solutions to can be easily proved according to the corresponding procedure in Lemma 2.1 of [@Winkler51215] (see also [@Winkler11215] and Lemma 2.1 of [@Painter55677]). Therefore, we give the following lemma without proof. \[lemma70\] Assume that $\varepsilon\in(0,1).$ Then there exist $T_{max,\varepsilon}\in (0,\infty]$ and a classical solution $(n_\varepsilon, c_\varepsilon, m_\varepsilon,u_\varepsilon, P_\varepsilon)$ of in $\Omega\times(0, T_{max,\varepsilon})$ such that $$\left\{\begin{array}{ll} n_\varepsilon\in C^0(\bar{\Omega}\times[0,T_{max,\varepsilon}))\cap C^{2,1}(\bar{\Omega}\times(0,T_{max,\varepsilon})),\\ c_\varepsilon\in C^0(\bar{\Omega}\times[0,T_{max,\varepsilon}))\cap C^{2,1}(\bar{\Omega}\times(0,T_{max,\varepsilon})),\\ m_\varepsilon\in C^0(\bar{\Omega}\times[0,T_{max,\varepsilon}))\cap C^{2,1}(\bar{\Omega}\times(0,T_{max,\varepsilon})),\\ u_\varepsilon\in C^0(\bar{\Omega}\times[0,T_{max,\varepsilon}))\cap C^{2,1}(\bar{\Omega}\times(0,T_{max,\varepsilon})),\\ P_\varepsilon\in C^{1,0}(\bar{\Omega}\times(0,T_{max,\varepsilon})),\\ \end{array}\right.\label{1.1ddfghyuisda}$$ classically solving in $\Omega\times[0,T_{max,\varepsilon})$. Moreover, $n_\varepsilon,c_\varepsilon$ and $m_\varepsilon$ are nonnegative in $\Omega\times(0, T_{max,\varepsilon})$, and $$\|n_\varepsilon(\cdot, t)\|_{L^\infty(\Omega)}+\|c_\varepsilon(\cdot, t)\|_{W^{1,\infty}(\Omega)}+\|m_\varepsilon(\cdot, t)\|_{W^{1,\infty}(\Omega)}+\|A^\gamma u_\varepsilon(\cdot, t)\|_{L^{2}(\Omega)}\rightarrow\infty~~ \mbox{as}~~ t\nearrow T_{max,\varepsilon}, \label{1.163072x}$$ where $\gamma$ is given by . ([@Horstmann791; @Winkler792; @Zhengddfggghjjkk1])\[llssdrffmmggnnccvvccvvkkkkgghhkkllvvlemma45630\] The Stokes operator $A$ denotes the realization of the Stokes operator under homogeneous Dirichlet boundary conditions in the solenoidal subspace $L^2_{\sigma}(\Omega)$ of $L^2(\Omega)$. Let $\mathcal{P}: L^p (\Omega)\rightarrow L^p_{\sigma}(\Omega)$ stand for the Helmholtz projection in $L^p (\Omega).$ Then there exist positive constants $\kappa_i(i=1,\ldots,3)$ such that $$\| e^{-tA}\mathcal{P}\varphi\|_{L^p(\Omega)} \leq \kappa_1(\Omega)t^{-\gamma}\|\varphi\|_{L^q(\Omega)} ~~~\mbox{for all}~~~ t > 0~~~\mbox{and any}~~~\varphi\in L^{q}(\Omega) \label{1.1ddfghssdddyddffssddduisda}$$ as well as $$\| e^{-tA}\mathcal{P}\nabla\cdot\varphi\|_{L^p(\Omega)} \leq \kappa_2(\Omega)t^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{q}-\frac{1}{p})}\|\varphi\|_{L^{q}(\Omega)} ~~~\mbox{for all}~~~ t > 0~~~\mbox{and any}~~~\varphi\in L^{q}(\Omega) \label{1.1ddfghssdddyssddduisda}$$ and $$\| e^{-tA}\mathcal{P}\varphi\|_{L^p(\Omega)} \leq \kappa_3(\Omega)\|\varphi\|_{L^p(\Omega)} ~~~\mbox{for all}~~~ t > 0~~~\mbox{and any}~~~\varphi\in L^{p}(\Omega). \label{1.1ddddfghhfghssdddyssddduisda}$$ Invoking the divergence free of the fluid and the homogeneous Neumann boundary conditions on $n_\varepsilon, m_\varepsilon$ and $c_\varepsilon$, we can establish the following basic estimates by using the maximum principle to the second and third equations. The proof of this lemma is very similar to that of Lemmas 2.2 and 2.6 of [@Tao41215] (see also Lemma 3.2 of [@Wangssddss21215]), so we omit its proof here \[fvfgsdfggfflemma45\] There exists $\lambda > 0$ such that the solution of satisfies $$\frac{d}{dt}\int_{\Omega}c_{\varepsilon}(\cdot,t)\leq0,~~~\mbox{and}~~~\frac{d}{dt}\int_{\Omega}m_{\varepsilon}(\cdot,t)\leq0~~\mbox{for all}~~ t\in(0, T_{max,\varepsilon}), \label{ddfgczhhhh2.5ghjjjssddju48cfg924ghyuji}$$ $$\|c_{\varepsilon}(\cdot,t)\|_{L^\infty(\Omega)}+\|m_{\varepsilon} (\cdot,t)\|_{L^\infty(\Omega)}\leq \lambda~~\mbox{for all}~~ t\in(0, T_{max,\varepsilon}), \label{ddfgczhhhh2.5ghju48cfg924ghyuji}$$ $$\int_{\Omega}{n_{\varepsilon} }-\int_{\Omega}{m_{\varepsilon}}= \int_{\Omega}{n_0}-\int_{\Omega}{m_0}~~\mbox{for all}~~ t\in(0, T_{max,\varepsilon}), \label{sssddfgczhhhh2.5ghju48cfg924ghyuji}$$ $$\|c_{\varepsilon}(\cdot,t)\|_{L^2(\Omega)}^2+2\int_0^{t}\int_{\Omega}{|\nabla c_{\varepsilon}|^{2}}\leq \lambda~~\mbox{for all}~~ t\in(0, T_{max,\varepsilon}), \label{ddczhjjjj2.5ghju48cfg9ssdd24}$$ $$\int_0^{t}\int_{\Omega}{n_{\varepsilon}m_{\varepsilon}}dxds\leq \lambda~~\mbox{for all}~~ t\in(0, T_{max,\varepsilon}) \label{ddczhjjjj2.5ghxxccju48cfg9ssdd24}$$ as well as $$\frac{1}{2}\int_0^{t}\int_{\Omega}{|\nabla m_{\varepsilon}|^2}dxds\leq \frac{1}{2}\int_{\Omega}{m_0^2}~~\mbox{for all}~~ t\in(0, T_{max,\varepsilon}) \label{ddczhjjjj2.5ghxxssddccju48cfg9ssdd24}$$ and $$\int_0^{t}\int_{\Omega}{|\nabla c_{\varepsilon}|^{2}}\leq \lambda~~\mbox{for all}~~ t\in(0, T_{max,\varepsilon}). \label{ddczhjjjj2.5ghju48cfgffff924}$$ For simplicity, here and hereafter, we take the notations $$C_S:=\sup_{0\leq s\leq \|c_{0}\|_{L^\infty(\Omega)}}S(s) \label{hnjmssddaqwswddaassffssff3.10deerfgghhjdddfgggjjuuloollgghhhyhh}$$ by using and . A-priori estimates ==================== In this section we want to ensure that the time-local solutions obtained in Lemma \[lemma70\] are in fact global solutions. To this end, for any $p>1,$ we firstly obtain boundedness of $n_\varepsilon$ in $L^p (\Omega)$ under the assumption that $\alpha>0.$ Inspired by the weighted estimate argument developed in [@Winkler2233444ssdff51215] (see also [@Winkler61215; @Winklerpejoevsssdd793]), we shall invoke a weight function $g(c_\varepsilon)$ which is uniformly bounded from above and below by positive constants. Before deriving the uniform of $L^p$ norm of $n_\varepsilon$, let us first recalling the well-known facts for $g(c_\varepsilon)$. \[fvfgfflemma45\] Let $$g(s) = e^{\beta s^2},~~~\mbox{for any}~~s\in(0,\|c_{0}\|_{L^\infty(\Omega)}],$$ where $$\beta=\frac{1}{8\|c_0\|_{L^\infty(\Omega)}^2} \label{czfvgbdfgg2.5ddffghddffsddffhjuyddfffdddddddggfffuccvviihjj}$$ and $C_S$ is given by . Then for any $s\in(0,\|c_{0}\|_{L^\infty(\Omega)}]$, $$\begin{array}{rl} 1\leq g(s)\leq\mu_0:=e^{\frac{1}{8}} \end{array} \label{czfvgb2.5ddffghsddffhjuyddfffdddddddggfffuccvviihjj}$$ and $$g'(s)\leq \mu_1:= \frac{1}{4\|c_0\|_{L^\infty}}e^{\frac{1}{8}}. \label{1111czfvgb2.ghhjkl5ghsddffhjuyddfffudddfccvviihjj}$$ Obviously, holds. On the other hand, a direct computation shows $$\begin{array}{rl}g'(s)=&2\beta se^{\beta s^2}.\\ \end{array} \label{czfvgb2.5ghsddffhjuyddfffddfffuccvviihjj}$$ This combined with the fact that $\beta=\frac{1}{8\|c_0\|_{L^\infty(\Omega)}^2}$ implies . \[lemmaghjffggssddgghhmk4563025xxhjklojjkkk\] Let $\alpha>0$. Then for any $p>1$, there exists $C>0$ such that the solution of satisfies $$\begin{array}{rl} &\disp{\int_{\Omega} n_{\varepsilon} ^p\leq C~~~\mbox{for all}~~ t\in (0, T_{max,\varepsilon}).}\\ \end{array} \label{czfvgb2.5ghhjuyuccvviihjj}$$ Firstly, we define a functional $$L(n_{\varepsilon} ,c_{\varepsilon})=\frac{1}{p}\int_{\Omega}n_{\varepsilon}^pg(c_{\varepsilon}),$$ where $p>\max\{1,2\alpha\}$, $g(c_{\varepsilon})=e^{\beta c ^2}$ and $\beta$ is the same as . Using the first two equations in , we find: $$\begin{array}{rl} &\disp{\frac{d}{dt}L(n_{\varepsilon} ,c_{\varepsilon})}\\ =&\disp{\int_{\Omega}n_{\varepsilon} ^{p-1}n_{\varepsilon t} g(c_{\varepsilon})+\frac{1}{p}\int_{\Omega}n_{\varepsilon} ^pg'(c_{\varepsilon})c_{\varepsilon t}}\\ =&\disp{\int_{\Omega}n_{\varepsilon} ^{p-1}g(c_{\varepsilon})(\Delta n_{\varepsilon} -\nabla\cdot(n_{\varepsilon} S_{\varepsilon} (x, n_{\varepsilon} , c_{\varepsilon})\nabla c_{\varepsilon}) -u_{\varepsilon} \cdot\nabla n_{\varepsilon} -n_{\varepsilon} m _{\varepsilon})}\\ &\disp{+\frac{1}{p}\int_{\Omega}n_{\varepsilon} ^pg'(c_{\varepsilon})(\Delta c_{\varepsilon}-c_{\varepsilon} +m_{\varepsilon} -u \cdot\nabla c_{\varepsilon})}\\ =&\disp{-\frac{1}{p} \int_{\Omega}n_{\varepsilon} ^pg'(c_{\varepsilon})u_{\varepsilon} \cdot\nabla c_{\varepsilon}-\int_{\Omega}n_{\varepsilon} ^{p-1}g(c_{\varepsilon})u_{\varepsilon} \cdot\nabla n_{\varepsilon} }\\ &\disp{-\frac{1}{p}\int_{\Omega}n_{\varepsilon} ^pg'(c_{\varepsilon})c_{\varepsilon}-\int_{\Omega}n_{\varepsilon} ^pg(c_{\varepsilon})m_{\varepsilon} +\frac{1}{p}\int_{\Omega}n_{\varepsilon} ^pg'(c_{\varepsilon})m_{\varepsilon} }\\ &+\disp{\int_{\Omega}n_{\varepsilon} ^{p-1}g(c_{\varepsilon})\Delta n_{\varepsilon} -\int_{\Omega}n_{\varepsilon} ^{p-1}g(c_{\varepsilon})\nabla\cdot(n_{\varepsilon} S_{\varepsilon}(x, n_{\varepsilon}, c_{\varepsilon})\nabla c_{\varepsilon}) }\\ &\disp{+\frac{1}{p}\int_{\Omega}n_{\varepsilon} ^pg'(c_{\varepsilon})\Delta c_{\varepsilon} }\\ =:&\disp{\sum_{i=1}^8I_i~~~\mbox{for all}~~ t\in (0, T_{max,\varepsilon}).} \end{array} \label{55hhjjcffghhhjkkllz2dddd.5}$$ In the following, we will estimate the right-hand sides of one by one. To this end, firstly, applying the elementary calculus identity $$\frac{1}{p}n_{\varepsilon} ^pg'(c_{\varepsilon})\nabla c_{\varepsilon} +n_{\varepsilon} ^{p-1}g(c_{\varepsilon})\nabla n_{\varepsilon} =\frac{1}{p}\nabla(n_{\varepsilon} ^pg(c_{\varepsilon}))$$ and the fact that $$\nabla\cdot u_{\varepsilon}=0,\quad x\in \Omega, t>0,\\$$ we once more integrate by parts to find that $$I_1+I_2=- \frac{1}{p}\int_{\Omega}n_{\varepsilon} ^pg'(c_{\varepsilon})u_{\varepsilon} \cdot\nabla c_{\varepsilon}-\int_{\Omega}n_{\varepsilon} ^{p-1}g(c_{\varepsilon})u_{\varepsilon} \cdot\nabla n_{\varepsilon} =0 \label{55hhjjcfddfffddghhhjkkllz2dddd.5}$$ by using $u_{\varepsilon} =0, x\in \partial\Omega, t>0$. Next, we derive from the non-negativity of $g',g$ (see and ), $c_{\varepsilon} ,m_{\varepsilon} $ and $n_{\varepsilon} $ that $$I_3+I_4=-\frac{1}{p}\int_{\Omega}n_{\varepsilon} ^pg'(c_{\varepsilon})c_{\varepsilon}-\int_{\Omega}n_{\varepsilon} ^pg(c_{\varepsilon})m_{\varepsilon} \leq0, \label{55hhjjcfddfffddghhhjkkssdddsssllz2dddd.5}$$ which combined with and yields $$\begin{array}{rl} &\disp{\frac{d}{dt}L(n_{\varepsilon} ,c_{\varepsilon})}\\ \leq&\disp{\int_{\Omega}n_{\varepsilon} ^{p-1}g(c_{\varepsilon})\Delta n_{\varepsilon} -\int_{\Omega}n_{\varepsilon} ^{p-1}g(c_{\varepsilon})\nabla\cdot(n_{\varepsilon} S_{\varepsilon}(x, n_{\varepsilon}, c_{\varepsilon})\nabla c_{\varepsilon})+\frac{1}{p}\int_{\Omega}n_{\varepsilon} ^pg'(c_{\varepsilon})\Delta c_{\varepsilon} }\\ &+\disp{\frac{1}{p}\int_{\Omega}n_{\varepsilon} ^pg'(c_{\varepsilon})m_{\varepsilon} .}\\ \end{array} \label{55hhjjcffghhhjkddffkllz2dddd.5}$$ Now we proceed to estimate the fourth term on the right-hand side herein by using and to find that $$\begin{array}{rl} \disp{\frac{1}{p}\int_{\Omega}n_{\varepsilon} ^pg'(c_{\varepsilon})m_{\varepsilon} }\leq&\disp{ \frac{1}{p}\mu_1\lambda\int_{\Omega}n_{\varepsilon} ^p,}\\ \end{array} \label{55hhjjcffghhhjkdddddssdddffkllz2dddd.5}$$ where $\mu_1$ is the same as . Now we estimate the term $\int_{\Omega}n_{\varepsilon} ^{p-1}g(c_{\varepsilon})\Delta n_{\varepsilon} $ and $\frac{1}{p}\int_{\Omega}n_{\varepsilon} ^pg'(c_{\varepsilon})\Delta c_{\varepsilon} $ in the right hand side of . In fact, we once more integrate by parts to see that $$\begin{array}{rl} &\disp{\int_{\Omega}n_{\varepsilon} ^{p-1}g(c_{\varepsilon})\Delta n_{\varepsilon} +\frac{1}{p}\int_{\Omega}n_{\varepsilon} ^pg'(c_{\varepsilon})\Delta c_{\varepsilon} }\\ =&\disp{-(p-1)\int_{\Omega}n_{\varepsilon} ^{p-2}g(c_{\varepsilon}) |\nabla n_{\varepsilon} |^2 - \int_{\Omega}n_{\varepsilon} ^{p-1}g'(c_{\varepsilon}) \nabla n_{\varepsilon} \cdot \nabla c_{\varepsilon} }\\ &\disp{-\frac{1}{p}\int_{\Omega}n_{\varepsilon} ^pg''(c_{\varepsilon})|\nabla c_{\varepsilon} |^2-\int_{\Omega}n_{\varepsilon} ^{p-1}g'(c_{\varepsilon})\nabla c_{\varepsilon} \cdot \nabla n_{\varepsilon} }\\ \leq&\disp{-(p-1)\int_{\Omega}n_{\varepsilon} ^{p-2}g(c_{\varepsilon}) |\nabla n_{\varepsilon} |^2 -\frac{1}{p}\int_{\Omega}n_{\varepsilon} ^pg''(c_{\varepsilon})|\nabla c_{\varepsilon} |^2 }\\ &\disp{+2\int_{\Omega}n_{\varepsilon} ^{p-1}g'(c_{\varepsilon})|\nabla n_{\varepsilon} || \nabla c_{\varepsilon} |}\\ \leq&\disp{-\frac{3(p-1)}{4}\int_{\Omega}n_{\varepsilon} ^{p-2}g(c_{\varepsilon}) |\nabla n_{\varepsilon} |^2 -\frac{1}{p}\int_{\Omega}n_{\varepsilon} ^pg''(c_{\varepsilon})|\nabla c_{\varepsilon} |^2 }\\ &\disp{+\frac{4}{p-1}\int_{\Omega}n_{\varepsilon} ^p\frac{g'(c_{\varepsilon})^2}{g(c_{\varepsilon})}| \nabla c_{\varepsilon} |^2}\\ \end{array} \label{55hhjjcffghdrgddffffhjkhhjkddffkllz2dssdsdddsdddddd.5}$$ by using the Young inequality. Next, recall , we can estimate second term on the right-hand side of as follows: $$\begin{array}{rl} &\disp{-\int_{\Omega}n_{\varepsilon} ^{p-1}g(c_{\varepsilon})\nabla\cdot(n_{\varepsilon} S_{\varepsilon}(x, n_{\varepsilon}, c_{\varepsilon})\nabla c_{\varepsilon})}\\ =&\disp{(p-1)\int_{\Omega}n_{\varepsilon} ^{p-1}g(c_{\varepsilon})S (x, n_{\varepsilon} , c_{\varepsilon})\nabla c_{\varepsilon} \cdot\nabla n_{\varepsilon} +\int_{\Omega}n_{\varepsilon} ^pg'(c_{\varepsilon})S (x, n_{\varepsilon} , c_{\varepsilon})\nabla c_{\varepsilon} \cdot\nabla c_{\varepsilon} }\\ \leq&\disp{(p-1)C_S\int_{\Omega}n_{\varepsilon} ^{p-1-\alpha}g(c_{\varepsilon})|\nabla c_{\varepsilon} ||\nabla n_\varepsilon |+C_S\int_{\Omega}n_\varepsilon ^{p-\alpha}g'(c_{\varepsilon})|\nabla c_{\varepsilon} |^2}\\ \leq&\disp{\frac{p-1}{4}\int_{\Omega}n_{\varepsilon} ^{p-2} g(c_{\varepsilon})|\nabla n_{\varepsilon} |^2+ C_S^2(p-1)\int_{\Omega}n_\varepsilon ^{p-2\alpha} g(c_{\varepsilon})|\nabla c_{\varepsilon} |^2}\\ &\disp{+C_S\int_{\Omega}n_\varepsilon ^{p-\alpha}g'(c_{\varepsilon})|\nabla c_{\varepsilon} |^2,}\\ \end{array} \label{55hhjjcffghhhjkdssdddffkllz2dddd.5}$$ where in the last inequality, we have used the Young inequality. Now, collecting –, we may have $$\begin{array}{rl} &\disp{\frac{d}{dt}L(n_{\varepsilon} ,c_{\varepsilon})+\frac{p-1}{2}\int_{\Omega}n_{\varepsilon} ^{p-2}g(c_{\varepsilon}) |\nabla n_{\varepsilon} |^2 }\\ &+\disp{\int_{\Omega}n_{\varepsilon} ^p\left[\frac{1}{p}g''(c_{\varepsilon})-\frac{4}{p-1}\frac{g'(c_{\varepsilon})^2}{g(c_{\varepsilon})}- C_S^2(p-1)n_{\varepsilon} ^{-2\alpha}g(c_{\varepsilon})-C_Sg'(c_{\varepsilon})n_{\varepsilon} ^{-\alpha}\right]|\nabla c_{\varepsilon} |^2 }\\ \leq&\disp{\frac{1}{p}\mu_1\lambda\int_{\Omega}n_{\varepsilon} ^p,}\\ \end{array} \label{55hhjjcffghhhjkddffkllz2ddddsddll.5}$$ whence returning to the definition of $g(c_{\varepsilon})$ we conclude that $$\begin{array}{rl} &\disp{\frac{d}{dt}L(n_{\varepsilon} ,c_{\varepsilon})+\frac{p-1}{2}\int_{\Omega}n_{\varepsilon} ^{p-2}g(c_{\varepsilon}) |\nabla n_{\varepsilon} |^2 }\\ \leq&\disp{-\int_{\Omega}n_{\varepsilon} ^pe^{\beta c_{\varepsilon} ^2}\left[\frac{1}{p}(4\beta^2 c_{\varepsilon} ^2+2\beta)-\frac{16}{p-1}\beta^2 c_{\varepsilon} ^2- C_S^2(p-1)n_{\varepsilon} ^{-2\alpha}-2C_S\beta c_{\varepsilon} n_{\varepsilon} ^{-\alpha}\right]|\nabla c_{\varepsilon} |^2}\\ &\disp{ +\frac{1}{p}\mu_1\lambda\int_{\Omega}n_{\varepsilon} ^p}\\ \leq&\disp{-\int_{\Omega}n_{\varepsilon} ^pe^{\beta c_{\varepsilon} ^2}\left[\frac{2\beta}{p}-\frac{16}{p-1}\beta^2 c_{\varepsilon} ^2- C_S^2(p-1)n_{\varepsilon} ^{-2\alpha}-2C_S\beta c_{\varepsilon} n_{\varepsilon} ^{-\alpha}\right]|\nabla c_{\varepsilon} |^2}\\ &\disp{ +\frac{1}{p}\mu_1\lambda\int_{\Omega}n_{\varepsilon} ^p}\\ \leq&\disp{-\int_{\Omega}n_{\varepsilon} ^pe^{\beta c_{\varepsilon} ^2}\left[\frac{2\beta}{p}-\frac{16}{p-1}\beta^2 \|c_0\|_{L^\infty(\Omega)}^2- C_S^2(p-1)n_{\varepsilon} ^{-2\alpha}-2C_S\beta \|c_0\|_{L^\infty(\Omega)} n_{\varepsilon} ^{-\alpha}\right]|\nabla c_{\varepsilon} |^2}\\ &\disp{ +\frac{1}{p}\mu_1\lambda\int_{\Omega}n_{\varepsilon} ^p}\\ \leq&\disp{-\int_{\Omega}n_{\varepsilon} ^pe^{\beta c_{\varepsilon} ^2}\left[\frac{2\beta}{p}-\frac{8}{p}\beta^2 \|c_0\|_{L^\infty(\Omega)}^2- C_S^2pn_{\varepsilon} ^{-2\alpha}-2C_S\beta \|c_0\|_{L^\infty(\Omega)} n_{\varepsilon} ^{-\alpha}\right]|\nabla c_{\varepsilon} |^2}\\ &\disp{ +\frac{1}{p}\mu_1\lambda\int_{\Omega}n_{\varepsilon} ^p}\\ \end{array} \label{55hhjjcffghhhjkddffkllz2ddddsddll.ssddd5}$$ by using $p>\max\{1,2\alpha\}.$ In view of $\beta=\frac{1}{8\|c_0\|_{L^\infty(\Omega)}^2}$, thus, implies that $$\begin{array}{rl} &\disp{\frac{d}{dt}L(n_{\varepsilon} ,c_{\varepsilon})+\frac{p-1}{2}\int_{\Omega}n_{\varepsilon} ^{p-2}g(c_{\varepsilon}) |\nabla n_{\varepsilon} |^2 }\\ \leq&\disp{-\int_{\Omega}n_{\varepsilon} ^pe^{\beta c ^2}\left[\frac{\beta}{p}- C_S^2pn_{\varepsilon} ^{-2\alpha}-2C_S\beta \|c_0\|_{L^\infty(\Omega)} n_{\varepsilon} ^{-\alpha}\right]|\nabla c_{\varepsilon} |^2 +\frac{1}{p}\mu_1\lambda\int_{\Omega}n_{\varepsilon} ^p.}\\ \end{array} \label{55hhjjcffghhhjkddffkllz2ddddsddll.ssdddssd5}$$ On the other hand, due to $\alpha>0,$ we may have $$\lim_{s\rightarrow+\infty}[ C_S^2ps^{-2\alpha}+2C_S\beta \|c_0\|_{L^\infty(\Omega)} s^{-\alpha}]=0,$$ so that, there exists $\eta_0>0$, such that for any $s>\eta_0$, $$[C_S^2ps^{-2\alpha}+2C_S\beta \|c_0\|_{L^\infty(\Omega)} s^{-\alpha}]<\frac{\beta}{2p}.$$ Therefore, by some basic calculation, we derive from that $$\begin{array}{rl} &\disp{\int_{\Omega}n_{\varepsilon} ^pe^{\beta c_{\varepsilon} ^2}\left[C_S^2pn_{\varepsilon} ^{-2\alpha}+2C_S\beta \|c_0\|_{L^\infty(\Omega)} n_{\varepsilon} ^{-\alpha}\right]|\nabla c_{\varepsilon} |^2 }\\ \leq&\disp{\int_{n_{\varepsilon} >\eta_0}n_{\varepsilon} ^pe^{\beta c_{\varepsilon} ^2}\left[C_S^2pn_{\varepsilon} ^{-2\alpha}+2C_S\beta \|c_0\|_{L^\infty(\Omega)} n_{\varepsilon} ^{-\alpha}\right]|\nabla c_{\varepsilon} |^2}\\ &\disp{+\int_{n_{\varepsilon} \leq\eta_0}n_{\varepsilon} ^pe^{\beta c_{\varepsilon} ^2}\left[C_S^2pn_{\varepsilon} ^{-2\alpha}+2C_S\beta \|c_0\|_{L^\infty(\Omega)} n ^{-\alpha}\right]|\nabla c_{\varepsilon} |^2 }\\ \leq&\disp{\int_{n_{\varepsilon} >\eta_0}n_{\varepsilon} ^pe^{\beta c_{\varepsilon} ^2}\frac{\beta}{2p}|\nabla c_{\varepsilon} |^2+\int_{n_{\varepsilon} \leq\eta_0}n_{\varepsilon} ^pe^{\beta c_{\varepsilon} ^2}\left[C_S^2pn_{\varepsilon} ^{-2\alpha}+2C_S\beta \|c_0\|_{L^\infty(\Omega)} n ^{-\alpha}\right]|\nabla c_{\varepsilon} |^2 }\\ \leq&\disp{\int_{\Omega}n_{\varepsilon} ^pe^{\beta c_{\varepsilon} ^2}\frac{\beta}{2p}|\nabla c_{\varepsilon} |^2+\int_{n_{\varepsilon} \leq\eta_0}e^{\beta c_{\varepsilon} ^2}\left[C_S^2pn_\varepsilon ^{p-2\alpha}+2C_S\beta \|c_0\|_{L^\infty(\Omega)} n_{\varepsilon} ^{p-\alpha}\right]|\nabla c_{\varepsilon} |^2 }\\ \leq&\disp{\int_{\Omega}n_{\varepsilon} ^pe^{\beta c_{\varepsilon} ^2}\frac{\beta}{2p}|\nabla c_{\varepsilon} |^2+\gamma_0\int_{n_{\varepsilon} \leq\eta_0}|\nabla c_{\varepsilon} |^2 }\\ \leq&\disp{\int_{\Omega}n_{\varepsilon} ^pe^{\beta c_{\varepsilon} ^2}\frac{\beta}{2p}|\nabla c_{\varepsilon} |^2+\gamma_0\int_{\Omega}|\nabla c_{\varepsilon} |^2 }\\ \end{array} \label{55hhjjcffghhhjkddfssddfkllz2ddddsddll.sssddsdddssd5}$$ with $$\gamma_0=e^{\beta \|c_0\|_{L^\infty(\Omega)}^2}\left[C_S^2p\eta_0^{p-2\alpha}+2C_S\beta \|c_0\|_{L^\infty(\Omega)} \eta_0^{p-\alpha}\right]$$ by using and $p>\max\{1,2\alpha\}$. Substituting into , we have $$\begin{array}{rl} &\disp{\frac{d}{dt}L(n_{\varepsilon} ,c_{\varepsilon})+\frac{p-1}{2}\int_{\Omega}n_{\varepsilon} ^{p-2}g(c_{\varepsilon}) |\nabla n_{\varepsilon} |^2 +\int_{\Omega}n_{\varepsilon} ^pe^{\beta c_{\varepsilon} ^2}\frac{\beta}{2p}|\nabla c_{\varepsilon} |^2}\\ \leq&\disp{\gamma_0\int_{\Omega}|\nabla c_{\varepsilon} |^2 +\frac{1}{p}\mu_1\lambda\int_{\Omega}n_{\varepsilon} ^p.}\\ \end{array} \label{55hhjjcffghhhjkddffkllz2ddddsddll.sssddsdddssd5}$$ Now, according to , we therefore obtain on using the Gagliardo-Nirenberg inequality that $$\begin{array}{rl} \disp \int_{\Omega}n_{\varepsilon} ^{p+\frac{1}{3}}=&\disp{ \|n_{\varepsilon} ^{\frac{p}{2}}\|_{L^{\frac{2(3p+1)}{3p}}(\Omega)}^{\frac{2(3p+1)}{3p}}}\\ \leq&\disp{ C_1[\|\nabla n_{\varepsilon} ^{\frac{p}{2}}\|_{L^{\frac{2}{p}}(\Omega)}^{\frac{2(3p-2)}{3p-1}}\| n_{\varepsilon} ^{\frac{p}{2}}\|_{L^{\frac{2}{p}}(\Omega)}^{\frac{2(3p+1)}{3p}-\frac{2(3p-2)}{3p-1}}+ \|n_{\varepsilon} ^{\frac{p}{2}}\|_{L^{\frac{2}{p}}(\Omega)}^{\frac{2(3p+1)}{3p}}]}\\ \leq&\disp{ \frac{(p-1)}{4}\frac{4}{p^2}\|n_{\varepsilon} ^{\frac{p}{2}}\|_{L^{2}(\Omega)}^2+C_2}\\ =&\disp{ \frac{(p-1)}{4}\int_{\Omega}n_{\varepsilon} ^{p-2} |\nabla n_{\varepsilon} |^2 +C_2}\\ \leq&\disp{ \frac{(p-1)}{4}\int_{\Omega}n_{\varepsilon} ^{p-2}g(c_{\varepsilon}) |\nabla n_{\varepsilon}|^2 +C_2}\\ \end{array} \label{55hhjjcffghhhjssddkdssddddfsdddfkllz2dddd.5}$$ for some positive constants $C_1$ and $C_2$, where in the last inequality, we have used . Collecting and , we have $$\begin{array}{rl} &\disp{\frac{d}{dt}L(n_{\varepsilon} ,c_{\varepsilon})+\frac{p-1}{4}\int_{\Omega}n_{\varepsilon} ^{p-2}g(c_{\varepsilon}) |\nabla n_{\varepsilon} |^2 +\int_{\Omega}n_{\varepsilon} ^pe^{\beta c_{\varepsilon} ^2}\frac{\beta}{2p}|\nabla c_{\varepsilon} |^2+\int_{\Omega}n_\varepsilon ^{p+\frac{1}{3}}}\\ \leq&\disp{\gamma_0\int_{\Omega}|\nabla c_{\varepsilon} |^2 +\frac{1}{p}\mu_1\lambda\int_{\Omega}n_{\varepsilon} ^p+C_2}\\ \leq&\disp{\gamma_0\int_{\Omega}|\nabla c_{\varepsilon} |^2 +\frac{1}{2}\int_{\Omega}n_{\varepsilon} ^{p+\frac{1}{3}}+C_3}\\ \end{array} \label{55hhjjcffghhhjkddffkllz2ddddsddll.ssdddssddsdddssdddsd5}$$ by using the Young inequality. Therefore, $$\begin{array}{rl} &\disp{\frac{d}{dt}L(n_{\varepsilon} ,c_{\varepsilon})+\frac{p-1}{4}\int_{\Omega}n_{\varepsilon} ^{p-2}g(c_{\varepsilon}) |\nabla n_{\varepsilon} |^2 +\int_{\Omega}n_{\varepsilon} ^pe^{\beta c_{\varepsilon} ^2}\frac{\beta}{2p}|\nabla c_{\varepsilon} |^2+\frac{1}{2}\int_{\Omega}n_{\varepsilon} ^{p+\frac{1}{3}}}\\ \leq&\disp{\gamma_0\int_{\Omega}|\nabla c_{\varepsilon} |^2 +C_3.}\\ \end{array} \label{55hhjjcffghhhjkddffddfsdddfkllz2dddd.5}$$ To track the time evolution of $c_\varepsilon $, testing the second equation in by $c_\varepsilon$ and using $\nabla\cdot u_\varepsilon =0$ and yields that for some positive constant $C_4$ such that $$\begin{array}{rl} \disp\frac{1}{{2}}\disp\frac{d}{dt}\|{c_{\varepsilon} }\|^{{{2}}}_{L^{{2}}(\Omega)}+ \int_{\Omega} |\nabla c_{\varepsilon} |^2=&\disp{-\int_{\Omega} c_{\varepsilon} ^2+\int_{\Omega}m_{\varepsilon} c_{\varepsilon} }\\ \leq&\disp{-\frac{1}{2}\int_{\Omega} c_{\varepsilon} ^2+C_4~~\mbox{for all}~~ t\in(0, T_{max,\varepsilon}),}\\ \end{array} \label{hhxxcdfvvjjcz2.5}$$ wereafter integrating the above inequality in time yields $$\begin{array}{rl} &\disp{\int_{t}^{t+1}\int_{\Omega} |\nabla {c_{\varepsilon} }|^2\leq C_5~~\mbox{for all}~~ t>0}\\ \end{array} \label{bnmbncz2.5ghhjuddfghhdddddffggyhjkklluivvbnnihjj}$$ for some $C_5> 0$ by an integration. This yields to $$\begin{array}{rl} &\disp{\int_{t}^{t+\tau}\left[\gamma_0\int_{\Omega}| \nabla c_{\varepsilon} |^2+C_3\right]\leq C_6~~\mbox{for all}~~ t\in(0, T_{max,\varepsilon}-\tau)}\\ \end{array} \label{bnmbncz2.5ghhjuddfghhdddddffggyhjkkllddffguivvbnnihjj}$$ with $$\tau:=\min\{1,\frac{1}{6}T_{max,\varepsilon}\}. \label{cz2.5ghju48cfg924vbhu}$$ Finally, in conjunction with Lemma 2.3 of [@Wddffang11215] (see also [@Zhenddsdddddgssddsddfff00]) and establish . In a straightforward manner, the estimates gained above can be seen to imply the following $\varepsilon$-independent estimates, which plays an important role in proving Theorem \[theorem3\]. \[ssdddlemmaghjffggssddgghhmk4563025xxhjklojjkkk\] Let $\alpha>0$. Then there exists $C>0$ such that the solution of satisfies $$\begin{array}{rl} &\disp{\int_{\Omega} \left(|\nabla c_{\varepsilon}| ^{2}+|\nabla m_{\varepsilon}| ^{2}\right)\leq C~~~\mbox{for all}~~ t\in (0, T_{max,\varepsilon}).}\\ \end{array} \label{czfvgb2.5ghhjuyuccvviihjj}$$ Moreover, for $t\in(0, T_{max,\varepsilon}-\tau)$, it holds that $$\begin{array}{rl} &\disp{\int_{t}^{t+\tau}\int_{\Omega} \left[ |\nabla {c_{\varepsilon} }|^4+ |\nabla {m_{\varepsilon}}|^4+|\nabla {u_{\varepsilon} }|^2+| {u_{\varepsilon} }|^{\frac{10}{3}}\right]\leq C,}\\ \end{array} \label{bnmbncz2.5ghhjuyuivvbnnihjj}$$ where $\tau$ is the same as . We multiply the second equation in by $-\Delta c_{\varepsilon} $ and integrate by parts to see that $$\begin{array}{rl} &\disp\frac{1}{{2}}\disp\frac{d}{dt}\|\nabla{c_{\varepsilon} }\|^{{{2}}}_{L^{{2}}(\Omega)}+ \int_{\Omega} |\Delta c_{\varepsilon} |^2+ \int_{\Omega} | \nabla c_{\varepsilon} |^2 \\ =&\disp{-\int_{\Omega} m_{\varepsilon} \Delta c_{\varepsilon} +\int_{\Omega} (u_{\varepsilon} \cdot\nabla c_{\varepsilon})\Delta c_{\varepsilon} } \\ =&\disp{-\int_{\Omega} m_{\varepsilon} \Delta c_{\varepsilon}-\int_{\Omega}\nabla c_{\varepsilon} \nabla (u_{\varepsilon} \cdot\nabla c_{\varepsilon})} \\ =&\disp{-\int_{\Omega} m_{\varepsilon} \Delta c_{\varepsilon}-\int_{\Omega}\nabla c_{\varepsilon} \nabla (\nabla u_{\varepsilon} \cdot\nabla c_{\varepsilon}),} \end{array} \label{hhxxcsssdfvvjjczddfdddfff2.5}$$ where we have used the fact that $$\disp{\int_{\Omega}\nabla c_{\varepsilon} \cdot(D^2 c_{\varepsilon} \cdot u_{\varepsilon} ) =\frac{1}{2}\int_{\Omega} u_{\varepsilon} \cdot\nabla|\nabla c_{\varepsilon} |^2=0 ~~\mbox{for all}~~ t\in(0,T_{max,\varepsilon}).}$$ On the other hand, by the Young inequality and , $$\begin{array}{rl} \disp-\int_{\Omega} m_{\varepsilon}\Delta c_{\varepsilon}\leq&\disp{\int_{\Omega} m^2_{\varepsilon}+\frac{1}{4}\int_{\Omega}|\Delta c_{\varepsilon}|^2 }\\ \leq&\disp{|\Omega|\| m_0\|^2_{L^\infty(\Omega)}+\frac{1}{4}\int_{\Omega}|\Delta c_{\varepsilon}|^2 ~~\mbox{for all}~~ t\in(0, T_{max,\varepsilon}).}\\ \end{array} \label{ssdddaassshhxxcdfvvjjcz2.5}$$ In the last summand in , we use the Cauchy-Schwarz inequality to obtain $$\begin{array}{rl} \disp-\int_{\Omega}\nabla c_{\varepsilon} \nabla (\nabla u_{\varepsilon} \cdot\nabla c_{\varepsilon}) \leq&\disp{\|\nabla u_{\varepsilon} \|_{L^{2}(\Omega)}\|\nabla c_{\varepsilon} \|_{L^{4}(\Omega)}^2~~\mbox{for all}~~ t\in(0,T_{max,\varepsilon}).} \end{array} \label{hhxxcsssdfvvjjcddssdddffzddfdddfff2.5}$$ Now thanks to and in view of the Gagliardo-Nirenberg inequality, we can find $C_1> 0$ and $C_2> 0$ fulfilling integrate by parts to find that $$\begin{array}{rl} \disp \|\nabla c_{\varepsilon} \|_{L^{4}(\Omega)}^2\leq&\disp{C_{1}\|\Delta c_{\varepsilon} \|_{L^{2}(\Omega)}\|c_{\varepsilon} \|_{L^{\infty}(\Omega)}+C_{1}\|c_{\varepsilon} \|_{L^{\infty}(\Omega)}^4}\\ \leq&\disp{C_{2}\|\Delta c_{\varepsilon} \|_{L^{2}(\Omega)}+C_{2} ~~\mbox{for all}~~ t\in(0,T_{max,\varepsilon}).}\\ \end{array} \label{hhxxcsssdfvvjjcddfffddffzddfdddfff2.5}$$ This, together with the Young inequality, yields $$\begin{array}{rl} &\disp-\int_{\Omega}\nabla c_{\varepsilon} \nabla (\nabla u_{\varepsilon} \cdot\nabla c_{\varepsilon}) \\ \leq&\disp{\|\nabla u_{\varepsilon} \|_{L^{2}(\Omega)}[C_{2}\|\Delta c_{\varepsilon} \|_{L^{2}(\Omega)}+C_{2}]} \\ \leq&\disp{C_{2}^2\|\nabla u_{\varepsilon} \|_{L^{2}(\Omega)}^2 +\frac{1}{4}\|\Delta c_{\varepsilon} \|_{L^{2}(\Omega)}^2+C_3~~\mbox{for all}~~ t\in(0,T_{max,\varepsilon}).} \end{array} \label{hhxxcsssdfvvjjcddffzddfdddfff2.5}$$ Inserting and into , we have $$\begin{array}{rl} &\disp\frac{1}{{2}}\disp\frac{d}{dt}\|\nabla{c_{\varepsilon} }\|^{{{2}}}_{L^{{2}}(\Omega)}+\frac{1}{2} \int_{\Omega} |\Delta c_{\varepsilon} |^2+ \int_{\Omega} | \nabla c_{\varepsilon} |^2 \\ \leq&\disp{C_{2}^2\|\nabla u_{\varepsilon} \|_{L^{2}(\Omega)}^2+C_4.} \end{array} \label{hhxxcsssdfvvjjczddfssdddddfff2.5}$$ Now, multiplying the third equation of by $u_\varepsilon$, integrating by parts and using $\nabla\cdot u_{\varepsilon}=0$ $$\begin{array}{rl} \disp{\frac{1}{2}\frac{d}{dt}\int_{\Omega}{|u_{\varepsilon}|^2}+\int_{\Omega}{|\nabla u_{\varepsilon}|^2}}= &\disp{ \int_{\Omega}(n_{\varepsilon}+m_{\varepsilon})u_{\varepsilon}\cdot\nabla \phi~~\mbox{for all}~~ t\in(0, T_{max,\varepsilon}).}\\ \end{array} \label{ddddfgcz2.5ghju48cfg924ghyuji}$$ Here we use the Hölder inequality, the Young inequality and the continuity of the embedding $W^{1,2}(\Omega)\hookrightarrow L^6(\Omega)$ and to find $C_{5} $ and $C_{6}> 0$ such that $$\begin{array}{rl} \disp\int_{\Omega}(n_{\varepsilon}+m_{\varepsilon})u_{\varepsilon}\cdot\nabla \phi\leq&\disp{\|\nabla \phi\|_{L^\infty(\Omega)}\| n_{\varepsilon} \|_{L^{\frac{6}{5}}(\Omega)}\| u_{\varepsilon}\|_{L^{6}(\Omega)}+\|\nabla \phi\|_{L^\infty(\Omega)}\| m_{\varepsilon} \|_{L^{\frac{6}{5}}(\Omega)}\| u_{\varepsilon}\|_{L^{6}(\Omega)}}\\ \leq&\disp{C_{5}\|\nabla \phi\|_{L^\infty(\Omega)}(\| n_{\varepsilon} \|_{L^{\frac{6}{5}}(\Omega)}+\|m_{\varepsilon} \|_{L^{\frac{6}{5}}(\Omega)})\|\nabla u_{\varepsilon}\|_{L^{2}(\Omega)}}\\ \leq&\disp{\frac{1}{2}\|\nabla u_{\varepsilon}\|_{L^{2}(\Omega)}^2+C_{6}~~\mbox{for all}~~ t\in(0, T_{max,\varepsilon})}\\ \end{array} \label{dddddddddfgcz2.5ghju48cfg924ghyuji}$$ by using and . Inserting into and integrating in time to see that $$\begin{array}{rl} &\disp{\int_{t}^{t+\tau}\int_{\Omega} |\nabla {u_{\varepsilon}}|^2\leq C_7~~\mbox{for all}~~ t\in(0,T_{max,\varepsilon}-\tau)}\\ \end{array} \label{bnmbncz2.5ghhjuddfghssddhdddddffggyhjkklluivvbnnihjj}$$ and $$\begin{array}{rl} &\disp{\int_{\Omega} u_{\varepsilon}^{2}\leq C_7~~~\mbox{for all}~~ t\in (0, T_{max,\varepsilon}),}\\ \end{array} \label{czfvgb2.5ghhjussddyuccvviihjj}$$ where we use that once more employing the Gagliardo-Nirenberg inequality, the Hölder inequality and the Young inequality we can find $C_8> 0$ and $C_9> 0$ satisfying $$\begin{array}{rl} \disp\int_{t}^{t+\tau}\disp\int_{\Omega} |u_{\varepsilon}|^{\frac{10}{3}} =&\disp{\int_{t}^{t+\tau}\| {u_{\varepsilon}}\|^{{\frac{10}{3}}}_{L^{\frac{10}{3}}(\Omega)}}\\ \leq&\disp{C_8\int_{t}^{t+\tau}\left(\| \nabla{u_{\varepsilon}}\|^{2}_{L^{2}(\Omega)}\|{u_{\varepsilon}}\|^{{\frac{4}{3}}}_{L^{2}(\Omega)}+ \|{u_{\varepsilon}}\|^{{\frac{10}{3}}}_{L^{2}(\Omega)}\right)}\\ \leq&\disp{C_9~~\mbox{for all}~~ t\in(0,T_{max,\varepsilon}-\tau).}\\ \end{array} \label{5555bnmbncz2ddfssdddvgffghhbhh.htt678hyuiihjj}$$ Next, combining , and rearranging shows that $$\begin{array}{rl} &\disp{\int_{t}^{t+\tau}\int_{\Omega} \left(|\Delta {c_{\varepsilon}}|^2+|\nabla {c_{\varepsilon}}|^4\right)\leq C_{10}~~\mbox{for all}~~ t\in(0,T_{max,\varepsilon}-\tau)}\\ \end{array} \label{bnmbncz2.5ghhjuddfghssddhddddddddffggyhjkklluivvbnnihjj}$$ and $$\begin{array}{rl} &\disp{\int_{\Omega} |\nabla {c_{\varepsilon}}|^{2}\leq C_{10}~~~\mbox{for all}~~ t\in (0, T_{max,\varepsilon})}\\ \end{array} \label{czfvgb2.5ghhjussddyuccvvissddihjj}$$ by using . Testing the third equation in by $-\Delta m_{\varepsilon} $ and integrating by parts and using and , one can finally derive $$\begin{array}{rl} &\disp{\int_{t}^{t+\tau}\int_{\Omega} \left(|\Delta {m_{\varepsilon}}|^2+|\nabla {m_{\varepsilon}}|^4\right)\leq C_{11}~~\mbox{for all}~~ t\in(0,T_{max,\varepsilon}-\tau)}\\ \end{array} \label{bnmbncz2.5ghhjuddfghsddfgggsddhddddddddffggyhjkklluivvbnnihjj}$$ and $$\begin{array}{rl} &\disp{\int_{\Omega} |\nabla {m_{\varepsilon}}|^{2}\leq C_{11}~~~\mbox{for all}~~ t\in (0, T_{max,\varepsilon}).}\\ \end{array} \label{czfvgb2.5ghhjussddyuccvsddddvissddihjj}$$ With Lemmas \[fvfgsdfggfflemma45\] and \[lemmaghjffggssddgghhmk4563025xxhjklojjkkk\]–\[ssdddlemmaghjffggssddgghhmk4563025xxhjklojjkkk\] at hand, we can proceed to show that our approximate solutions are actually global in time. \[lemma45630hhuujj\] For any $\varepsilon > 0,$ then one can find $C > 0$ such that the solutions of fulfill Firstly, under the assumption that $T_{max,\varepsilon}< \infty$, for any $\varepsilon > 0$, Lemmas \[fvfgsdfggfflemma45\]–\[ssdddlemmaghjffggssddgghhmk4563025xxhjklojjkkk\] would provide us with $C_1 > 0$ such that $$\begin{array}{rl} &\disp{\int_{\Omega} n_{\varepsilon} ^p\leq C_1~~~\mbox{for all}~~ t\in (0, T_{max,\varepsilon})~~~\mbox{for all}~~p>4,}\\ \end{array} \label{czfvgb2sss.5ghhjuyuccvviihjj}$$ $$\begin{array}{rl} &\disp{\|c_{\varepsilon} (\cdot,t)\|_{L^\infty(\Omega)}\leq C_1~~~\mbox{for all}~~ t\in (0, T_{max,\varepsilon})}\\ \end{array} \label{ssssczfvgb2sss.aass5ghhjuyuccvviihjj}$$ as well as $$\begin{array}{rl} &\disp{\|m_{\varepsilon} (\cdot,t)\|_{L^\infty(\Omega)}\leq C_1~~~\mbox{for all}~~ t\in (0, T_{max,\varepsilon})}\\ \end{array} \label{czfvgb2sss.aass5ghhjuyuccvviihjj}$$ and $$\begin{array}{rl} &\disp{\int_{t}^{t+\tau}\int_{\Omega} \left[ |\nabla {c_{\varepsilon} }|^4+ |\nabla {m}|^4\right]\leq C_1.}\\ \end{array} \label{bnmbncz2.5ghssffhjuyuivvbnnihjj}$$ Then, aided by the $L^2$-estimate for $\nabla u_{\varepsilon}$ (from a testing argument), we can obtain that $$\begin{array}{rl} \|A^\gamma u_{\varepsilon} (\cdot, t)\|_{L^2(\Omega)}\leq&\disp{C_{2}~~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon})}\\ \end{array} \label{cz2.571hhhhh51csdddcvvhddfccvvhjjjkkhhggjjllll}$$ with some $C_2= C_2(\varepsilon) > 0$. Thus, the continuous embedding $D(A^\gamma)\hookrightarrow L^\infty(\Omega)$ implies the $L^\infty$ -estimate for $u_{\varepsilon}.$ Therefore, employing the same arguments as in the proof of Lemma 3.2 in [@Zhenssssssdffssdddddddgssddsddfff00] (see also [@Kegssddsddfff00; @Zhengsddfffsdddssddddkkllssssssssdefr23; @Winkler11215; @Winkler51215]), and taking advantage of –, we conclude the estimates $$\|n_{\varepsilon} (\cdot,t)\|_{L^\infty(\Omega)} \leq C_3 ~~\mbox{for all}~~ t\in(0,\infty) \label{zjscz2.5297x9630111kk}$$ as well as $$\|c_{\varepsilon} (\cdot,t)\|_{W^{1,\infty}(\Omega)} \leq C_3 ~~~~\mbox{for all}~~ t\in(0,\infty) \label{zjscz2.5297x9630111kkhh}$$ and $$\|m_{\varepsilon} (\cdot,t)\|_{W^{1,\infty}(\Omega)} \leq C_3~~~~\mbox{for all}~~ t\in(0,\infty) \label{sssszjscz2.5297x9630111kkhh}$$ and some positive constant $C_3.$ In view of –, we apply Lemma \[lemma70\] to reach a contradiction. Further a-priori estimates -------------------------- With the help of Lemma \[lemmaghjffggssddgghhmk4563025xxhjklojjkkk\] and the Gagliardo–Nirenberg inequality, one can derive the following Lemma: \[lemmddaghjsffggggsddgghhmk4563025xxhjklojjkkk\] Let $\alpha>0$. Then for each $T\in(0, T_{max,\varepsilon})$, there exists $C>0$ independent of $\varepsilon$ such that the solution of satisfies $$\begin{array}{rl} &\disp{\int_{0}^T\int_{\Omega}|\nabla n_{\varepsilon}|^{2}\leq C(T+1).}\\ \end{array} \label{bnmbncz2.ffghh5ghhjuyuivvbnnihjj}$$ Multiply the first equation in $\dref{1.1fghyuisda}$ by $ n_{\varepsilon}$ and using $\nabla\cdot u_\varepsilon=0$, we derive $$\begin{array}{rl} &\disp{\frac{1}{{2}}\frac{d}{dt}\|{ n_{\varepsilon} }\|^{{{{2}}}}_{L^{{{2}}}(\Omega)}+ \int_{\Omega} |\nabla n_{\varepsilon}|^2}\\ =&\disp{- \int_{\Omega} n_{\varepsilon}\nabla\cdot(n_{\varepsilon}S_\varepsilon(x, n_{\varepsilon}, c_{\varepsilon})\cdot\nabla c_{\varepsilon})-\int_{\Omega}n_{\varepsilon}^{2}m_{\varepsilon}}\\ \leq&\disp{ \int_{\Omega} n_{\varepsilon}|S_\varepsilon(x, n_{\varepsilon}, c_{\varepsilon})||\nabla n_{\varepsilon}||\nabla c_{\varepsilon}|~~\mbox{for all}~~ t\in(0, T_{max,\varepsilon}).} \end{array} \label{55hhjjcffgjjjkkkkhhhjkklddfgggffgglffghhhz2.5}$$ Recalling and using $\alpha\geq0$, from Young inequality again, we derive from Lemma \[lemmaghjffggssddgghhmk4563025xxhjklojjkkk\] that $$\begin{array}{rl} &\disp\int_{\Omega} n_{\varepsilon} |S_\varepsilon(x, n_{\varepsilon}, c_{\varepsilon})||\nabla n_{\varepsilon}||\nabla c_{\varepsilon}|\\ \leq&\disp{C_S\int_{\Omega}n_{\varepsilon} |\nabla n_{\varepsilon}||\nabla c_{\varepsilon}|}\\ \leq&\disp{\frac{1}{2}\int_{\Omega} |\nabla n_{\varepsilon}|^2+\frac{C_S^2}{2}\int_{\Omega}n_{\varepsilon}^{2} |\nabla c_{\varepsilon}|^2}\\ \leq&\disp{\frac{1}{2}\int_{\Omega} |\nabla n_{\varepsilon}|^2+\frac{1}{4} \int_{\Omega}n_{\varepsilon}^{4}+\frac{C_S^4}{4} \int_{\Omega}|\nabla c_{\varepsilon}|^4}\\ \leq&\disp{\frac{1}{2}\int_{\Omega} |\nabla n_{\varepsilon}|^2+\frac{C_S^4}{4} \int_{\Omega}|\nabla c_{\varepsilon}|^4+C_1~~\mbox{for all}~~ t\in(0, T_{max,\varepsilon}),} \end{array} \label{55hhjjcffghhhjkkllfffghggggghgghjjjhfghhhz2.5}$$ which combined with implies that $$\begin{array}{rl} &\disp{\frac{1}{{2}}\frac{d}{dt}\|{ n_{\varepsilon} }\|^{{{{2}}}}_{L^{{{2}}}(\Omega)}+ \frac{1}{2}\int_{\Omega} |\nabla n_{\varepsilon}|^2\leq\frac{C_S^4}{4} \int_{\Omega}|\nabla c_{\varepsilon}|^4+C_1~~\mbox{for all}~~ t\in(0, T_{max,\varepsilon}),} \end{array} \label{55hhjjcffghhhjkklddfgggffgglffghhhz2.5}$$ so that, gather and , one can get . Passing to the limit: The proof of Theorem \[theorem3\] ======================================================= With the help of a priori estimates, in this subsection, by means of a standard extraction procedure we can now derive the following lemma which actually contains our main existence result (Theorem \[theorem3\]) already. \[lemma45hyuuuj630223\] Assume that $\alpha>0$. Then for any $\kappa\in\mathbb{R}$, there exists $(\varepsilon_j)_{j\in \mathbb{N}}\subset (0, 1)$ such that $\varepsilon_j\rightarrow 0$ as $j\rightarrow\infty$ and that $$n_\varepsilon\rightarrow n ~~\mbox{a.e.}~~\mbox{in}~~\Omega\times(0,\infty)~~\mbox{and in}~~ L_{loc}^{2}(\bar{\Omega}\times[0,\infty))\label{zjscz2.5297x963ddfgh0ddfggg6662222tt3}$$ $$n_\varepsilon\rightharpoonup n ~~~\mbox{weak star in}~~ L^{\infty}_{loc}([0,\infty),L^p(\Omega))~~~\mbox{for any}~~p>1,\label{zjsczssdd2.5297x963ddfgh0ddfggg6662222tt3}$$ $$\nabla n_\varepsilon\rightharpoonup \nabla n ~~\mbox{in}~~ L_{loc}^{2}(\bar{\Omega}\times[0,\infty)),\label{zjscz2.5297x963ddfgh0ddgghjjfggg6662222tt3}$$ $$c_\varepsilon\rightarrow c ~~\mbox{in}~~ L^{2}_{loc}(\bar{\Omega}\times[0,\infty))~~\mbox{and}~~\mbox{a.e.}~~\mbox{in}~~\Omega\times(0,\infty), \label{zjscz2.fgghh5297x963ddfgh0ddfggg6662222tt3}$$ $$m_\varepsilon\rightarrow m ~~\mbox{in}~~ L^{2}_{loc}(\bar{\Omega}\times[0,\infty))~~\mbox{and}~~\mbox{a.e.}~~\mbox{in}~~\Omega\times(0,\infty), \label{zjscz2.fgghh5297x96ddddd3ddfgh0ddfggg6662222tt3}$$ $$\nabla c_\varepsilon\rightharpoonup \nabla c ~~\mbox{in}~~ L^{4}_{loc}(\bar{\Omega}\times[0,\infty)), \label{zjscz2.fgghh5297x963ddfgh0dddddfggg6662222tt3}$$ $$\nabla m_\varepsilon\rightharpoonup \nabla m ~~\mbox{in}~~ L^{4}_{loc}(\bar{\Omega}\times[0,\infty)), \label{zjscz2.fgghh5297x963ddfgh0dddddfggg6662222tt3}$$ $$u_\varepsilon\rightarrow u~~\mbox{in}~~ L_{loc}^2(\bar{\Omega}\times[0,\infty))~~\mbox{and}~~\mbox{a.e.}~~\mbox{in}~~\Omega\times(0,\infty), \label{zjscz2.5297x96302222t666t4}$$ $$\nabla c_\varepsilon\rightharpoonup \nabla c~~\begin{array}{ll} \mbox{in}~~ L_{loc}^{2}(\bar{\Omega}\times[0,\infty)) \end{array}\label{1.1ddfgghhhge666ccdf2345ddvbnmklllhyuisda}$$ as well as $$\nabla m_\varepsilon\rightharpoonup \nabla m~~\begin{array}{ll} \mbox{in}~~ L_{loc}^{2}(\bar{\Omega}\times[0,\infty)) \end{array}\label{1.1dddddfgghhhge666ccdf2345ddvbnmklllhyuisda}$$ and $$\nabla u_\varepsilon\rightharpoonup \nabla u ~~\mbox{ in}~~L^{2}_{loc}(\bar{\Omega}\times[0,\infty);\mathbb{R}^{3}) \label{zjscz2.5297x96366602222tt4455}$$ some quadruple $(n, c,m, u)$ which is a global weak solution of in the natural sense as specified in [@Zhenssssssdffssdddddddgssddsddfff00]. Firstly, applying the discussion in Section 3, under the assumptions of Theorem \[theorem3\], for each $T > 0$, we can find $\varepsilon$-independent constant $C(T)$ such that $$\|n_\varepsilon(\cdot,t)\|_{L^p(\Omega)}+ \|c_\varepsilon(\cdot,t)\|_{W^{1,2}(\Omega)}+ \|m_\varepsilon(\cdot,t)\|_{W^{1,2}(\Omega)}+\|u_\varepsilon(\cdot,t)\|_{L^2(\Omega)} \leq C(T) ~~\mbox{for all}~~ t\in(0,T)~~\mbox{and}~~p>1 \label{zjscz2.ssddd5297x9630111kk}$$ as well as $$\int_{0}^T\int_{\Omega}\left(|\nabla n_\varepsilon|^2+|\nabla c_\varepsilon|^4+|\nabla m_\varepsilon|^4+ |\Delta c_\varepsilon|^2+ |\Delta m_\varepsilon|^2\right)\leq C(T) ~~\mbox{for all}~~ t\in(0, T) \label{fvgbhzjscz2.5sss297x96302222tt4455hyuhii}$$ and $$\int_{0}^T\int_{\Omega}|\nabla u_\varepsilon|^2\leq C(T) ~~\mbox{for all}~~ t\in(0, T) . \label{fvgbhzjscz2.5297x96302222tt4455hyuhii}$$ Now, choosing $\varphi\in W^{1,2} (\Omega)$ as a test function in the first equation in and using , we have $$\begin{array}{rl} &\disp\left|\int_{\Omega}(n_{\varepsilon,t})\varphi\right|\\ =&\disp{\left|\int_{\Omega}\left[\Delta n_{\varepsilon}-\nabla\cdot(n_{\varepsilon} S_\varepsilon(x, n_{\varepsilon}, c_{\varepsilon})\nabla c_{\varepsilon})-u_{\varepsilon}\cdot\nabla n_{\varepsilon}-n_{\varepsilon}m_{\varepsilon}\right]\varphi\right|} \\ =&\disp{\left|\int_\Omega \left[-\nabla n_{\varepsilon}\cdot\nabla\varphi+n_{\varepsilon} S_\varepsilon(x, n_{\varepsilon}, c_{\varepsilon})\nabla c_{\varepsilon}\cdot\nabla\varphi+ n_{\varepsilon}u_{\varepsilon}\cdot\nabla \varphi- n_{\varepsilon}m_{\varepsilon} \varphi\right]\right|}\\ \leq&\disp{\left\{\|\nabla n_{\varepsilon}\|_{L^{2}(\Omega)}+ \|n_{\varepsilon}S_\varepsilon(x, n_{\varepsilon}, c_{\varepsilon})\nabla c_{\varepsilon}\|_{L^{2}(\Omega)}+ \|n_{\varepsilon}u_{\varepsilon}\|_{L^{2}(\Omega)}+\|n_{\varepsilon}m_{\varepsilon}\|_{L^{2}(\Omega)} \right\}}\\ &\times\disp{\|\varphi\|_{W^{1,2}(\Omega)}}\\ \end{array} \label{gbhncvbmdcfvgcz2.5ghju48}$$ for all $t>0$. Along with and , further implies that $$\begin{array}{rl} &\disp\int_0^T\|\partial_tn_\varepsilon(\cdot,t)\|_{({W^{1,2}(\Omega)})^*}^{2}dt \\ \leq&\disp{\int_0^T\left\{\|\nabla n_{\varepsilon}\|_{L^{2}(\Omega)}+ \|n_{\varepsilon} S_\varepsilon(x, n_{\varepsilon}, c_{\varepsilon})\nabla c_{\varepsilon}\|_{L^{2}(\Omega)}+ \|n_{\varepsilon}u_{\varepsilon}\|_{L^{2}(\Omega)} \right\}}^{2}dt \\ \leq&\disp{C_1\int_0^T\left\{\|\nabla n_{\varepsilon}\|_{L^{2}(\Omega)}^{2}+ \|n_{\varepsilon}\|_{L^4(\Omega)}^4 \|\nabla c_{\varepsilon}\|_{L^{4}(\Omega)}^{4}\right\}dt}\\ &\disp{+C_1\int_0^T\left\{\|n_{\varepsilon}u_{\varepsilon}\|_{L^{2}(\Omega)}^{2}+ \|m_{\varepsilon}\|^{2}_{L^{\infty}(\Omega)}\|n_{\varepsilon}\|_{L^{2}(\Omega)}^{2} \right\}}dt\\ \leq&\disp{C_2\int_0^T\left\{\|\nabla n_{\varepsilon}\|_{L^{2}(\Omega)}^{2}+ \|\nabla c_{\varepsilon}\|_{L^{4}(\Omega)}^{4}\right\}dt}\\ &\disp{+C_2\int_0^T\left\{\|n_{\varepsilon}u_{\varepsilon}\|_{L^{2}(\Omega)}^{2}+ 1 \right\}}dt,\\ \end{array} \label{gbhncvbmdcfvgczffghhh2.5ghju48}$$ where $C_1$ and $C_2$ are positive constants independent of $\varepsilon$. Now, due to the Hölder inequality, we have $$\begin{array}{rl} &\disp\int_{0}^T\int_{\Omega}|n_{\varepsilon}u_\varepsilon|^{2} \\ \leq&\disp{\int_{0}^T\|n_{\varepsilon}\|^{2}_{L^3(\Omega)} \|u_\varepsilon\|^{2}_{L^6(\Omega)}}\\ \leq&\disp{C_3\int_{0}^T \|\nabla u_\varepsilon\|^{2}_{L^2(\Omega)}}\\ \leq&\disp{C_4(T+1)~~\mbox{for all}~ T > 0}\\ \end{array} \label{gbhncvbmdcfvgsddddczffghhh2.5ghju48}$$ by using and . Inserting into and applying and , we can obtain for some positive constant $C_1(T)$ such that $$\|n_{\varepsilon t}\|_{L^2(0,T;(W^{1,2}(\Omega))^*)}\leq C_1(T) \label{fvgbhzjsczsssd2.5297x9630222ssdd2tt4455hyuhii}$$ by using and Lemma \[lemma45630hhuujj\]. In a similar way, one can derive $$\|c_{\varepsilon t}\|_{L^2(0,T;(W^{1,2}(\Omega))^*)}\leq C_2(T) \label{111fvgbhzjsczsssd2.5297x9630ssddd2222tt4455hyuhii}$$ and $$\|m_{\varepsilon t}\|_{L^2(0,T;(W^{1,2}(\Omega))^*)}\leq C_2(T) \label{fvgbhzjsczsssd2.5297x9630ssddd2222tt4455hyuhii}$$ with some $C_2(T) > 0.$ Next, for any given $\varphi\in C^{\infty}_{0,\sigma} (\Omega;\mathbb{R}^3)$, we infer from the fourth equation in that $$\begin{array}{rl} \disp\left|\int_{\Omega}\partial_{t}u_{\varepsilon}(\cdot,t)\varphi\right|=&\disp{\left|-\int_\Omega \nabla u_{\varepsilon}\cdot\nabla\varphi-\kappa\int_\Omega (Y_{\varepsilon}u_{\varepsilon}\otimes u_{\varepsilon})\cdot\nabla\varphi+\int_\Omega (n_{\varepsilon}+m_{\varepsilon})\nabla \phi\cdot\varphi\right| ~~\mbox{for all}~~ t>0.} \end{array}$$ Now, by virtue of , Lemma \[lemmaghjffggssddgghhmk4563025xxhjklojjkkk\] and Lemma \[fvfgsdfggfflemma45\], we thus infer that there exist positive constants $C_{3},C_{4}$ and $C_{5}$ such that $$\begin{array}{rl} &\disp\int_0^T\|\partial_{t}u_{\varepsilon}(\cdot,t)\|_{(W^{1,2}_{0,\sigma}(\Omega))^*}^{2}dt\\ \leq&\disp{C_{3}\left(\int_0^T\int_\Omega|\nabla u_{\varepsilon}|^{2}+\int_0^T\int_\Omega |Y_{\varepsilon}u_{\varepsilon}\otimes u_{\varepsilon}|^{2}+\int_0^T\int_\Omega n_\varepsilon^{2}+\int_0^T\int_\Omega m_\varepsilon^{2}\right)}\\ \leq&\disp{C_{4}\left(\int_0^T\int_\Omega|\nabla u_{\varepsilon}|^{2}+\int_0^T\int_\Omega |u_\varepsilon|^{2}+\int_0^T\int_\Omega n_{\varepsilon}^{2}+T\right)}\\ \leq&\disp{C_{5}(T+1) ~~\mbox{for all}~~ T>0.} \end{array} \label{fvgbhzjsczsssssddd2.5297x9630sssssdddssss2222tt4455hyuhii}$$ Here we have used the fact that $$\|Y_{\varepsilon}v\|_{L^2(\Omega)}\leq \|v\|_{L^2(\Omega)}~~~\mbox{for all}~~ v\in L^2_{\sigma}(\Omega).$$ Combining estimates –, we conclude from Aubin-Lions lemma (see e.g. [@Simon]) that $(u_\varepsilon)_{\varepsilon\in (0,1)}$ is relatively compact in $L^2_{loc} (\Omega\times[0,\infty);\mathbb{R}^3)$ and $(n_\varepsilon)_{\varepsilon\in (0,1)}$, $(c_\varepsilon)_{\varepsilon\in (0,1)}$, $(m_\varepsilon)_{\varepsilon\in (0,1)}$ are relatively compact in $L^2_{loc} (\Omega\times[0,\infty)).$ Therefore, in conjunction with – and standard compactness arguments, we can thus find a sequence $(\varepsilon_j)_{j\in N} \subset (0,1)$ such that $\varepsilon_j\searrow0$ as $j \rightarrow\infty$, and such that $$n_\varepsilon\rightarrow n ~~\mbox{in}~~ L^2_{loc}(\bar{\Omega}\times[0,\infty))~~~~\mbox{and}~~ n_\varepsilon\rightarrow n~~~a.e. ~~~\mbox{in}~~~\Omega\times(0,\infty), \label{zjscz2.5297sssssssx963sss0222222ee}$$ $$n_\varepsilon \rightharpoonup n ~~\mbox{weak star in}~~ L^{\infty}_{loc}([0,\infty),L^p(\Omega))~~~~\mbox{for any}~~ p>1, \label{zjscz2.5297ssdsssssssx963sss0222222ee}$$ $$\nabla n_\varepsilon\rightharpoonup \nabla n ~~\mbox{in}~~ L^2_{loc}(\bar{\Omega}\times[0,\infty)), \label{zjscz2.5297ssssssssssx963sss0222222ee}$$ $$c_\varepsilon\rightarrow c ~~\mbox{in}~~ L^2_{loc}(\bar{\Omega}\times[0,\infty))~~~~\mbox{and}~~ c_\varepsilon\rightarrow c~~~a.e. ~~~\mbox{in}~~~\Omega\times(0,\infty), \label{zjscz2.5297sssssssssx9630222222ee}$$ $$\nabla c_\varepsilon\rightharpoonup \nabla c ~~\mbox{in}~~ L^4_{loc}(\bar{\Omega}\times[0,\infty)) \label{zjscz2.5297ssssssdddsssssssx963sss0222222ee}$$ $$\Delta c_\varepsilon\rightharpoonup \Delta c ~~\mbox{in}~~ L^2_{loc}(\bar{\Omega}\times[0,\infty)) \label{zjscz2.5297sddffssddsssssdddsssssssx963sss0222222ee}$$ $$m_\varepsilon\rightarrow m ~~\mbox{in}~~ L^2_{loc}(\bar{\Omega}\times[0,\infty))~~~~\mbox{and}~~ m_\varepsilon\rightarrow m~~~a.e. ~~~\mbox{in}~~~\Omega\times(0,\infty), \label{zjscz2.5297sssssssssx9630222222ee}$$ $$\nabla m_\varepsilon\rightharpoonup \nabla m ~~\mbox{in}~~ L^4_{loc}(\bar{\Omega}\times[0,\infty)) \label{zjscz2.5297sssssssddssssssssx963sss0222222ee}$$ $$\Delta m_\varepsilon\rightharpoonup \Delta m ~~\mbox{in}~~ L^2_{loc}(\bar{\Omega}\times[0,\infty)) \label{zjscz2.5297sssssdddsssddssssssssx963sss0222222ee}$$ as well as $$u_\varepsilon\rightarrow u ~~\mbox{in}~~ L^2_{loc}(\bar{\Omega}\times[0,\infty))~~~~\mbox{and}~~ u_\varepsilon\rightarrow u~~~a.e. ~~~\mbox{in}~~~\Omega\times(0,\infty) \label{zjscz2.5297sssfffssssssx9630222222ee}$$ and $$\nabla u_\varepsilon\rightharpoonup \nabla u ~~\mbox{in}~~ L^2_{loc}(\bar{\Omega}\times[0,\infty)). \label{zjscz2.5297ssssssssssssssssx963sss0222222ee}$$ for some limit function $(n,c,m,u).$ On the other hand, according to the bounds provided by Lemma \[fvfgsdfggfflemma45\] and Lemmas \[lemmaghjffggssddgghhmk4563025xxhjklojjkkk\]–\[ssdddlemmaghjffggssddgghhmk4563025xxhjklojjkkk\], this readily yields that, for any $\varepsilon\in(0,1)$, $$m_{\varepsilon}-c_{\varepsilon}-u_{\varepsilon}\cdot\nabla c_{\varepsilon}~~~\mbox{is bounded in}~~~L^{\frac{5}{3}} (\Omega\times(0, T)), \label{zjsdffggghhcz2.5297sssssdddsssddssssssssx963sss0222222ee}$$ where we have used the fact that $$\begin{array}{ll} &\disp\int_0^T\int_{\Omega}\left[|u_{\varepsilon}\cdot\nabla c_{\varepsilon}|^{\frac{5}{3}} \right]\\ \leq&\disp C_6\left[\int_{0}^T\disp\int_{\Omega} |u_{\varepsilon}|^{\frac{10}{3}}\right]^{\frac{1}{2}}\left[\int_{0}^T\disp\int_{\Omega} |\nabla c_\varepsilon|^{\frac{10}{3}}\right]^{\frac{1}{2}}\\ \leq&\disp C_7\left[\int_{0}^T\disp\int_{\Omega} |u_{\varepsilon}|^{\frac{10}{3}}\right]^{\frac{1}{2}} \left[\int_{0}^T\disp\int_{\Omega} |\nabla c_\varepsilon|^{4}\right]^{\frac{5}{12}}\\ \leq &C_8(T+1) \end{array}$$ by using Lemma \[ssdddlemmaghjffggssddgghhmk4563025xxhjklojjkkk\]. Therefore, in light of , regularity estimates for the second equation of (see e.g. [@Ladyzenskajaggk7101]) ensure that $(c_{\varepsilon})_{\varepsilon\in(0,1)}$ is bounded in $L^{\frac{5}{3}} ((0, T); W^{2,\frac{5}{3}}(\Omega))$. Hence, by virtue of , we derive form the Aubin–Lions lemma that $(c_{\varepsilon})_{\varepsilon\in(0,1)}$ relatively compacts in $L^{\frac{5}{3}} ((0, T); W^{1,\frac{5}{3}}(\Omega))$. Thus, we can choose an appropriate subsequence that is still written as $(\varepsilon_j )_{j\in \mathbb{N}}$ such that $\nabla c_{\varepsilon_j} \rightarrow z_1$ in $L^{\frac{5}{3}} (\Omega\times(0, T))$ for all $T\in(0, \infty)$ and some $z_1\in L^{\frac{5}{3}} (\Omega\times(0, T))$ as $j\rightarrow\infty$. Therefore, by , we can also derive that $\nabla c_{\varepsilon_j} \rightarrow z_1$ a.e. in $\Omega\times(0, \infty)$ as $j \rightarrow\infty$. In view of and the Egorov theorem, we conclude that $z_1=\nabla c$ and hence $$\nabla c_\varepsilon\rightarrow \nabla c ~~~a.e. ~~~\mbox{in}~~~\Omega\times(0,\infty). \label{zjscz2.5297sssssssssssssx9630222222ee}$$ This combined with , as well as and implies that $$n_\varepsilon S_\varepsilon(x, n_{\varepsilon}, c_{\varepsilon})\nabla c_\varepsilon\rightharpoonup nS(x, n, c)\nabla c ~\mbox{in}~ L^{2}(\Omega\times(0,T))~\mbox{as}~\varepsilon = \varepsilon_j\searrow 0~\mbox{for each}~ T\in(0,\infty) \label{1.1ddddfddffttyygghhyujiikkkffghhhifgghhhgffddgge6bhhjh66ccdf2345ddvbnmklddfggllhyuisda}$$ by using the Egorov theorem. Next we shall prove that $(n,c,m,u)$ is a weak solution of problem . To this end, testing the first equation in by $\varphi\in C^\infty_0(\Omega\times [0,\infty))$, we obtain $$\begin{array}{rl}\label{eqx4ss5xx12112ccgghh} \disp{-\int_0^{\infty}\int_{\Omega}n_\varepsilon\varphi_t-\int_{\Omega}n_0\varphi(\cdot,0) }=&\disp{- \int_0^\infty\int_{\Omega}\nabla n_\varepsilon\cdot\nabla\varphi+\int_0^\infty\int_{\Omega}n_\varepsilon S_\varepsilon(x,n_\varepsilon,c_\varepsilon)\nabla c_\varepsilon\cdot\nabla\varphi}\\ &+\disp{\int_0^\infty\int_{\Omega}n_\varepsilon u_\varepsilon\cdot\nabla\varphi-\int_0^{\infty}\int_{\Omega}n_\varepsilon m_\varepsilon\varphi}\\ \end{array}$$ for all $\varepsilon\in (0,1)$. Then – and the dominated convergence theorem enables us to conclude $$\begin{array}{rl}\label{eqx45xx12112ccgghh} \disp{-\int_0^{\infty}\int_{\Omega}n\varphi_t-\int_{\Omega}n_0\varphi(\cdot,0) }=&\disp{- \int_0^\infty\int_{\Omega}\nabla n\cdot\nabla\varphi+\int_0^\infty\int_{\Omega}n S(x,n,c)\nabla c\cdot\nabla\varphi}\\ &+\disp{\int_0^\infty\int_{\Omega}nu\cdot\nabla\varphi-\int_0^\infty\int_{\Omega}nm\varphi}\\ \end{array}$$ by a limit procedure. Next, multiplying the second equation and the third equation in by $\varphi\in C^\infty_0(\Omega\times [0,\infty))$, we derive from a limit procedure that $$\begin{array}{rl}\label{222eqx45xx12112ccgghhjj} \disp{-\int_0^{\infty}\int_{\Omega}c\varphi_t-\int_{\Omega}c_0\varphi(\cdot,0) }=&\disp{- \int_0^\infty\int_{\Omega}\nabla c\cdot\nabla\varphi-\int_0^\infty\int_{\Omega}c\varphi+\int_0^\infty\int_{\Omega}m\varphi+ \int_0^\infty\int_{\Omega}cu\cdot\nabla\varphi}\\ \end{array}$$ and $$\begin{array}{rl}\label{111eqx45fffffxx12112ccgghhjj} \disp{-\int_0^{\infty}\int_{\Omega}m\varphi_t-\int_{\Omega}m_0\varphi(\cdot,0) }=&\disp{- \int_0^\infty\int_{\Omega}\nabla m\cdot\nabla\varphi-\int_0^\infty\int_{\Omega}nm\varphi+ \int_0^\infty\int_{\Omega}mu\cdot\nabla\varphi}\\ \end{array}$$ in a completed similar manner (see [@Zhenssssssdffssdddddddgssddsddfff00] for details). Then testing the fourth equation of by $\varphi\in C_0^{\infty} (\bar{\Omega}\times[0, T);\mathbb{R}^3)$, we obtain $$\begin{array}{rl}\label{eqx45xx12ddd112ccgghhjjgghh} \disp{-\int_0^{\infty}\int_{\Omega}u\varphi_t-\int_{\Omega}u_0\varphi(\cdot,0)+ \kappa \int_0^T\int_{\Omega} u\otimes u\cdot\nabla\varphi}=&\disp{- \int_0^\infty\int_{\Omega}\nabla u\cdot\nabla\varphi- \int_0^\infty\int_{\Omega}(n+m)\nabla\phi\cdot\varphi}\\ \end{array}$$ by using Lemma \[lemma45630hhuujj\] and a limit procedure (see [@Zhenssssssdffssdddddddgssddsddfff00] for details). This means that $(n,c,m,u)$ is a weak solution of , in the natural sense as specified in [@Zhenssssssdffssdddddddgssddsddfff00]. Moreover, if in addition we assume that $\kappa\in\mathbb{R}$, then our solutions will actually be bounded and smooth and hence classical. In fact, by applying the standard parabolic regularity and the classical Schauder estimates for the Stokes evolution, we will show that it is sufficiently regular so as to be a classical solution. \[lemmassddddff45630223\] Let $(n,c,m,u)$ be a weak solution of . Assume that $\alpha>0$ and $\kappa=0$. Then $(n,c,m,u)$ solves in the classical sense in $\Omega\times (0,\infty).$ Moreover, this solution is bounded in $\Omega\times(0,\infty)$ in the sense that $$\|n(\cdot, t)\|_{L^\infty(\Omega)}+\|c(\cdot, t)\|_{W^{1,\infty}(\Omega)}+\|m(\cdot, t)\|_{W^{1,\infty}(\Omega)}+\|A^\gamma u(\cdot, t)\|_{L^2(\Omega)}\leq C~~ \mbox{for all}~~ t>0. \label{1.163072xggttsdddyyu}$$ In what follows, let $C, C_i$ denote some different constants, and if no special explanation, they depend at most on $\Omega, \phi, m, n_0, c_0$ and $u_0$. [**Step 1. The boundedness of $\|A^\gamma u (\cdot, t)\|_{L^2(\Omega)}$ and $\| u (\cdot, t)\|_{L^{\infty}(\Omega)}$ for all $t\in (0, T_{max,\varepsilon})$**]{} On the basis of the variation-of-constants formula for the projected version of the third equation in , we derive that $$u (\cdot, t) = e^{-tA}u_0 +\int_0^te^{-(t-\tau)A} \mathcal{P}((n (\cdot,\tau)+m (\cdot,\tau))\nabla\phi)d\tau~~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon}).$$ On the other hand, in view of Lemma \[lemmaghjffggssddgghhmk4563025xxhjklojjkkk\] as well as and , $$\|h (\cdot,t)\|_{L^{{4+2\alpha}}(\Omega)}\leq C ~~~\mbox{for all}~~ t\in(0,T_{max,\varepsilon})$$ with $h :=\mathcal{P}((n (\cdot,\tau)+m (\cdot,\tau)\nabla\phi)$. Therefore, according to standard smoothing properties of the Stokes semigroup we see that there exist $C_1 , C_2> 0$ and $\lambda_1 > 0$ such that $$\begin{array}{rl} \|A^\gamma u (\cdot, t)\|_{L^2(\Omega)}\leq&\disp{\|A^\gamma e^{-tA}u_0\|_{L^2(\Omega)} +\int_0^t\|A^\gamma e^{-(t-\tau)A}h (\cdot,\tau)d\tau\|_{L^2(\Omega)}d\tau}\\ \leq&\disp{\|A^\gamma u_0\|_{L^2(\Omega)} +C_{1}\int_0^t(t-\tau)^{-\gamma-\frac{3}{2}(\frac{1}{{4+2\alpha}}-\frac{1}{2})}e^{-\lambda_1(t-\tau)}\|h (\cdot,\tau)\|_{L^{{4+2\alpha}}(\Omega)}d\tau}\\ \leq&\disp{C_{2}~~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon})}\\ \end{array} \label{cz2.571hhhhh51ccvvhddfccvvhjjjkkhhggjjllll}$$ with $\gamma\in ( \frac{3}{4}, 1),$ where in the last inequality, we have used the fact that $$\begin{array}{rl}\disp\int_{0}^t(t-\tau)^{-\gamma-\frac{3}{2}(\frac{1}{{4+2\alpha}}-\frac{1}{2})}e^{-\lambda_1(t-\tau)}ds \leq&\disp{\int_{0}^{\infty}\sigma^{-\gamma-\frac{3}{2}(\frac{1}{{4+2\alpha}}-\frac{1}{2})} e^{-\lambda_1\sigma}d\sigma<+\infty}\\ \end{array}$$ by using $-\gamma-\frac{3}{2}(\frac{1}{{4+2\alpha}}-\frac{1}{2})>-1.$ implies to $$\begin{array}{rl} \|u (\cdot, t)\|_{L^\infty(\Omega)}\leq \sigma_{0}~~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon})\\ \end{array} \label{cz2ddfgjjj.5jkkcvvvhjkfffffkhhgll}$$ by using the fact that $D(A^\gamma)$ is continuously embedded into $L^\infty(\Omega)$ (by $\gamma>\frac{3}{4}$). [**Step 2. The boundedness of $\|\nabla c (\cdot, t)\|_{L^4(\Omega)}$ for all $t\in (0, T_{max,\varepsilon})$**]{} Now, multiply the second equation in $\dref{334451.1fghyuisda}$ by $-\Delta c $, in view of , and , we derive from and the Young inequality that $$\begin{array}{rl} \|\nabla c (\cdot, t)\|_{L^2(\Omega)}\leq \sigma_{1}~~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon}).\\ \end{array} \label{cz2ddfgjjj.5jddfghkkcvvvhjkfffffkhhgll}$$ Considering the fact that $\nabla c \cdot\nabla\Delta c = \frac{1}{2}\Delta |\nabla c |^2-|D^2c |^2$, by a straightforward computation using the second equation in and several integrations by parts, we find that $$\begin{array}{rl} &\disp{\frac{1}{{4}}\frac{d}{dt} \|\nabla c \|^{{{4}}}_{L^{{4}}(\Omega)}} \\ = &\disp{\int_{\Omega} |\nabla c |^{2}\nabla c \cdot\nabla(\Delta c -c +m -u \cdot\nabla c )} \\ =&\disp{\frac{1}{{2}}\int_{\Omega} |\nabla c |^{2}\Delta |\nabla c |^2 -\int_{\Omega} |\nabla c |^{2}|D^2 c |^2-\int_{\Omega} |\nabla c |^{4}} \\ &+\disp{\int_\Omega m \nabla\cdot( |\nabla c |^{2}\nabla c ) +\int_\Omega (u \cdot\nabla c )\nabla\cdot( |\nabla c |^{2}\nabla c )} \\ =&\disp{-\frac{1}{{2}}\int_{\Omega} \left|\nabla |\nabla c |^{2}\right|^2-\int_{\Omega} |\nabla c |^{4} +\frac{1}{{2}}\int_{\partial\Omega} |\nabla c |^{2}\frac{\partial |\nabla c |^{2}}{\partial\nu}}\\ &-\disp{\int_{\Omega} |\nabla c |^{2}|D^2 c |^2 +\int_\Omega m |\nabla c |^{2}\Delta c +\int_\Omega m \nabla c \cdot\nabla( |\nabla c |^{2})} \\ &+\disp{\int_\Omega (u \cdot\nabla c ) |\nabla c |^{2}\Delta c +\int_\Omega (u \cdot\nabla c )\nabla c \cdot\nabla( |\nabla c |^{2})} \\ \end{array} \label{cz2.5gdfhjjkhju48156}$$ for all $t\in(0,T_{max,\varepsilon})$. On the other hand, since Lemma 2.2 of [@Zhengssssssdefr23], we derive from and the Young inequality that $$\begin{array}{rl} \|\nabla {c }\|_{L^{4}(\Omega)}^4\leq&\displaystyle{\kappa_0\||\nabla{c }|D^2{c } \|_{L^2(\Omega)}^{\frac{2}{3}} \|{c }\|_{L^\infty(\Omega)}^{\frac{8}{3}}+\kappa_0\|{c }\|_{L^\infty(\Omega)}^4} \\ \leq&\displaystyle{\frac{1}{4(1+16\sigma_0^2)}\int_{\Omega} |\nabla c |^{2}|D^2 c |^2+\kappa_1,}\\ \end{array} \label{cz2.563022222ikossdddpl255}$$ where $\sigma_0$ is the same as and $\kappa_0 $ and $\kappa_{1}$ are some positive constants. Thanks to the pointwise inequality $|\Delta c | \leq\sqrt{3}|D^2c |$, along with as well as and this implies that $$\begin{array}{rl} &\disp\int_\Omega m |\nabla c |^{2}\Delta c \\ \leq&\disp{\sqrt{3} \int_\Omega m |\nabla c |^{2}|D^2c |} \\ \leq&\disp{\frac{1}{8}\int_\Omega |\nabla c |^{2}|D^2c |^2+6\|m \|_{L^\infty(\Omega)}^2\int_\Omega |\nabla c |^{2}} \\ \leq&\disp{\frac{1}{8}\int_\Omega |\nabla c |^{2}|D^2c |^2+6\lambda^2\sigma_{1}^2} \\ \leq&\disp{\frac{1}{8}\int_\Omega |\nabla c |^{2}|D^2c |^2+C_3~~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon})} \\ \end{array} \label{cz2.5ghju48hjuikl1}$$ and $$\begin{array}{rl} &\disp\int_\Omega (u \cdot\nabla c ) |\nabla c |^{2}\Delta c \\ \leq&\disp{\sqrt{3}\int_\Omega |u \cdot\nabla c | |\nabla c |^{2}|D^2c |} \\ \leq&\disp{\frac{1}{16}\int_\Omega |\nabla c |^{2}|D^2c |^2 +12\int_\Omega |u \cdot\nabla c |^2 |\nabla c |^{2}} \\ \leq&\disp{\frac{1}{16}\int_\Omega |\nabla c |^{2}|D^2c |^2 +12\|u \|^2_{L^\infty(\Omega)}\int_\Omega |\nabla c |^{4}~~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon})} \end{array} \label{cz2.5ghju48hjuikl451}$$ by using and the Young inequality. Now, inserting into , this shows that $$\begin{array}{rl} &\disp\int_\Omega (u \cdot\nabla c ) |\nabla c |^{2}\Delta c \\ \leq&\disp{\frac{1}{16}\int_\Omega |\nabla c |^{2}|D^2c |^2 +12\|u \|^2_{L^\infty(\Omega)}\times[\frac{1}{4(1+16\sigma_0^2)}\int_{\Omega} |\nabla c |^{2}|D^2 c |^2+\kappa_1]}\\ \leq&\disp{\frac{1}{4}\int_\Omega |\nabla c |^{2}|D^2c |^2+C_4~~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon})} \end{array} \label{3344cz2.5ghju48hssddjuikl451}$$ by using . Again, from the Young inequality, as well as and , we have $$\begin{array}{rl} &\disp\int_\Omega m \nabla c \cdot\nabla( |\nabla c |^{2}) \\ \leq &\disp{\frac{1}{8}\int_{\Omega} \left|\nabla |\nabla c |^{2}\right|^2+2 \lambda^2\sigma_{1}^2} \\ \leq &\disp{\frac{1}{8}\int_{\Omega}\left|\nabla |\nabla c |^{2}\right|^2+C_5} \end{array} \label{cz2.5ghju4ghvvvbbbjuk81}$$ and $$\begin{array}{rl} &\disp\int_\Omega (u \cdot\nabla c )\nabla c \cdot\nabla( |\nabla c |^{2}) \\ \leq &\disp{\frac{1}{16}\int_{\Omega}\left|\nabla |\nabla c |^{2}\right|^2+4\int_\Omega |u \cdot\nabla c |^2 |\nabla c |^{2}} \\ \leq &\disp{\frac{1}{16}\int_{\Omega}\left|\nabla |\nabla c |^{2}\right|^2 +4 \|u \|^2_{L^\infty(\Omega)}\times[\frac{1}{4(1+16\sigma_0^2)}\int_{\Omega} |\nabla c |^{2}|D^2 c |^2+\kappa_1]}\\ \leq &\disp{\frac{1}{16}\int_{\Omega}\left|\nabla |\nabla c |^{2}\right|^2 +\frac{1}{16}\int_{\Omega} |\nabla c |^{2}|D^2 c |^2+C_6.} \end{array} \label{cz2.5ghju4ccvvvghjuk81}$$ Given the the boundedness of $\|\nabla c \|_{L^2(\Omega)}^2$ (see ), it is well-known that (cf. [@Ishida; @Tao41215; @Zhengsdsd6]) the boundary trace embedding implies that $$\label{com-est-4} \begin{split} \int_{\partial\Omega} |\nabla c |^{2}\frac{\partial}{\partial \nu} |\nabla c |^2&\leq \frac{1}{16}\int_\Omega |\nabla |\nabla c |^2|^2+C_7 \Bigr(\int_{\Omega} |\nabla c |^2\Bigr)^2\\ &\leq \frac{1}{16}\int_\Omega |\nabla |\nabla c |^2|^2+C_{8}. \end{split}$$ Now, together with , –, we can derive that, for some positive constant $C_9$, $$\begin{array}{rl} \disp{\frac{1}{{4}}\frac{d}{dt} \|\nabla c \|^{{{4}}}_{L^{{4}}(\Omega)}+\frac{3}{{4}}\int_{\Omega} \left|\nabla |\nabla c |^{2}\right|^2 +\frac{1}{{2}}\int_{\Omega} |\nabla c |^{2}|D^2 c |^2}\leq&\disp{C_{9}~~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon}),}\\ \end{array} \label{cz2.sss5gdfhjjkhjsdfggu48156}$$ which combined with yields to $$\begin{array}{rl} \disp{\frac{1}{{4}}\frac{d}{dt} \|\nabla c \|^{{{4}}}_{L^{{4}}(\Omega)}+ C_{10}\|\nabla c \|^{{{4}}}_{L^{{4}}(\Omega)}}\leq&\disp{C_{11}~~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon}).}\\ \end{array} \label{cz2.5gdfhjjkhjsdfggu48156}$$ This implies $$\begin{array}{rl} \|\nabla c (\cdot, t)\|_{L^4(\Omega)}\leq C_{12}~~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon})\\ \end{array} \label{cz2ddfgjjj.5jddfghkkcvkkkllvvhjkfffffkhhgll}$$ by integration. [**Step 3. The boundedness of $\|\nabla m (\cdot, t)\|_{L^{4}(\Omega)}$ for all $t\in (0, T_{max,\varepsilon})$**]{} An application of the variation of constants formula for $c $ leads to $$\begin{array}{rl} &\disp{\|\nabla m (\cdot, t)\|_{L^{4}(\Omega)}}\\ \leq&\disp{\|\nabla e^{t(\Delta-1)} m_0\|_{L^{4}(\Omega)}+ \int_{0}^t\|\nabla e^{(t-s)(\Delta-1)}(m (s)-n (s)m (s))\|_{L^{4}(\Omega)}ds}\\ &\disp{+\int_{0}^t\|\nabla e^{(t-s)(\Delta-1)}\nabla \cdot(u (s) m (s))\|_{L^{4}(\Omega)}ds.}\\ \end{array} \label{44444zjccfgghhhfgbhjcvvvbscz2.5297x96301ku}$$ To estimate the terms on the right of , in light of and , applying the $L^p$-$L^q$ estimates associated heat semigroup, for some positive constant $\lambda_1$ such that $$\begin{array}{rl} \|\nabla e^{t(\Delta-1)} m_0\|_{L^{4}(\Omega)}\leq &\disp{C_{13}~~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon})}\\ \end{array} \label{zjsssddddccffgbhjcghhhjjjvvvbscz2.5297x96301ku}$$ as well as $$\begin{array}{rl} &\disp{\int_{0}^t\|\nabla e^{(t-s)(\Delta-1)}(m (s)-n (s)m (s))\|_{L^{4}(\Omega)}ds}\\ \leq&\disp{C_{14}\int_{0}^t[1+(t-s)^{-\frac{1}{2}-\frac{3}{2}(\frac{1}{4}-\frac{1}{4})}] e^{-\lambda_1(t-s)}(\|n (s)\|_{L^{4}(\Omega)}+\|m (s)\|_{L^{\infty}(\Omega)})ds}\\ \leq&\disp{C_{15}~~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon})}\\ \end{array} \label{zjccffgbhjcvvvbscz2.5297x96301ku}$$ and $$\begin{array}{rl} &\disp{\int_{0}^t\|\nabla e^{(t-s)(\Delta-1)}\nabla \cdot(u (s) c (s))\|_{L^{4}(\Omega)}ds}\\ \leq&\disp{C_{16}\int_{0}^t\|(-\Delta+1)^\iota e^{(t-s)(\Delta-1)}\nabla \cdot(u (s) c (s))\|_{L^{4}(\Omega)}ds}\\ \leq&\disp{C_{17}\int_{0}^t(t-s)^{-\iota-\frac{1}{2}-\tilde{\kappa}} e^{-\lambda_1(t-s)}\|u (s) c (s)\|_{L^{4}(\Omega)}ds}\\ \leq&\disp{C_{18}\int_{0}^t(t-s)^{-\iota-\frac{1}{2}-\tilde{\kappa}} e^{-\lambda_1(t-s)}\|u (s)\|_{L^{\infty}(\Omega)}\| c (s)\|_{L^{\infty}(\Omega)}ds}\\ \leq&\disp{C_{19}~~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon}),}\\ \end{array} \label{zjccffgbhjcvdgghhhhdfgghhvvbscz2.5297x96301ku}$$ where $\iota=\frac{13}{28},\tilde{\kappa}=\frac{1}{56}$. Inserting – into , one has $$\begin{array}{rl} \|\nabla m (\cdot, t)\|_{L^4(\Omega)}\leq \sigma_{2}~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon}).\\ \end{array} \label{cz2ddfgjjj.5jssddddddfghkkcvkkkllvvhjkfffffkhhgll}$$ [**Step 4. The boundedness of $\|c (\cdot, t)\|_{W^{1,\infty}(\Omega)}$ and $\|m (\cdot, t)\|_{W^{1,\infty}(\Omega)}$ for all $t\in (\tau, T_{max,\varepsilon})$ with $\tau\in(0,T_{max,\varepsilon})$**]{} Choosing $\theta\in(\frac{1}{2}+\frac{3}{8},1),$ then the domain of the fractional power $D((-\Delta + 1)^\theta)\hookrightarrow W^{1,\infty}(\Omega)$ (see e.g. [@Horstmann791; @Winkler792]). Thus, in light of $\alpha>0$, using the Hölder inequality and the $L^p$-$L^q$ estimates associated heat semigroup, $$\begin{array}{rl} &\| c (\cdot, t)\|_{W^{1,\infty}(\Omega)}\\ \leq&\disp{C_{20}\|(-\Delta+1)^\theta c (\cdot, t)\|_{L^{4}(\Omega)}}\\ \leq&\disp{C_{21}t^{-\theta}e^{-\lambda_1 t}\|c_0\|_{L^{4}(\Omega)}+C_{21}\int_{0}^t(t-s)^{-\theta}e^{-\lambda_1(t-s)} \|(m -u \cdot \nabla c )(s)\|_{L^{4}(\Omega)}ds}\\ \leq&\disp{C_{22}+C_{22}\int_{0}^t(t-s)^{-\theta}e^{-\mu(t-s)}[\|m (s)\|_{L^{\infty}(\Omega)}+\|c (s)\|_{L^{\infty}(\Omega)}+\|u (s)\|_{L^\infty(\Omega)} \|\nabla c (s)\|_{L^{4}(\Omega)}]ds}\\ \leq&\disp{C_{23}~~ \mbox{for all}~~ t\in(\tau,T_{max,\varepsilon})}\\ \end{array} \label{zjccffgbhjcvvvbscz2.5297x96301ku}$$ and $$\begin{array}{rl} &\| m (\cdot, t)\|_{W^{1,\infty}(\Omega)}\\ \leq&\disp{C_{24}\|(-\Delta+1)^\theta m (\cdot, t)\|_{L^{4}(\Omega)}}\\ \leq&\disp{C_{25}t^{-\theta}e^{-\lambda_1 t}\|m_0\|_{L^{4}(\Omega)}+C_{25}\int_{0}^t(t-s)^{-\theta}e^{-\lambda_1(t-s)} \|(m -m n -u \cdot \nabla m )(s)\|_{L^{4}(\Omega)}ds}\\ \leq&\disp{C_{26}+C_{26}\int_{0}^t(t-s)^{-\theta}e^{-\mu(t-s)}[\|m (s)\|_{L^{\infty}(\Omega)}+\|n (s)\|_{L^{4}(\Omega)}+\|u (s)\|_{L^\infty(\Omega)} \|\nabla m (s)\|_{L^{4}(\Omega)}]ds}\\ \leq&\disp{C_{27}~~ \mbox{for all}~~ t\in(\tau,T_{max,\varepsilon})}\\ \end{array} \label{zjccffgbhjcvbscz97x96u}$$ with $\tau\in(0,T_{max,\varepsilon})$, where we have used , , , as well as the Hölder inequality and $$\int_{0}^t(t-s)^{-\theta}e^{-\lambda_1(t-s)}\leq \int_{0}^{\infty}\sigma^{-\theta}e^{-\lambda_1\sigma}d\sigma<+\infty.$$ [**Step 5. The boundedness of $\|c (\cdot, t)\|_{W^{1,\infty}(\Omega)}$ and $\|n (\cdot, t)\|_{L^{\infty}(\Omega)}$ for all $t\in (0, T_{max,\varepsilon})$**]{} Recalling Lemma \[lemma70\], and , we infer that $$\begin{array}{rl} \|\nabla c (\cdot, t)\|_{L^{{\infty}}(\Omega)}\leq \kappa_{1} ~~ \mbox{for all}~~~ t\in(0,T_{max,\varepsilon}) \\ \end{array} \label{cz2.5g5ddfgggggghh56789hhjui78jj90099}$$ and $$\begin{array}{rl} \|\nabla m (\cdot, t)\|_{L^{{\infty}}(\Omega)}\leq \kappa_{2} ~~ \mbox{for all}~~~ t\in(0,T_{max,\varepsilon}). \\ \end{array} \label{cz2.5g5ddfggggggdffgghh56789hhjui78jj90099}$$ [**Step 6. The boundedness of $\|n (\cdot, t)\|_{L^{\infty}(\Omega)}$ for all $t\in (\tau, T_{max,\varepsilon})$ with $\tau\in(0,T_{max,\varepsilon})$**]{} Fix $T\in (0, T_{max,\varepsilon})$. Let $M(T):=\sup_{t\in(0,T)}\|n (\cdot,t)\|_{L^\infty(\Omega)}$ and $\tilde{h} :=n S (x, n , c )\nabla c +u $. Then by , and , there exists $C_{28} > 0$ such that $$\begin{array}{rl} \|\tilde{h} (\cdot, t)\|_{L^{4}(\Omega)}\leq&\disp{C_{28}~~ t\in(0,T_{max,\varepsilon}),}\\ \end{array} \label{cz2ddff.57151ccvhhjjjkkkuuifghhhivhccvvhjjjkkhhggjjllll}$$ where we have used and the boundedness of $\|c (\cdot, t)\|_{W^{1,\infty}(\Omega)}$ for all $t\in (\tau, T_{max,\varepsilon})$ with $\tau\in(0,T_{max,\varepsilon})$. Hence, due to the fact that $\nabla\cdot u =0$, again, by means of an associate variation-of-constants formula for $n $, we can derive $$n (t)=e^{(t-t_0)\Delta}n (\cdot,t_0)-\int_{t_0}^{t}e^{(t-s)\Delta}\nabla\cdot(n (\cdot,s)\tilde{h} (\cdot,s)) ds-\int_{t_0}^{t}e^{(t-s)\Delta}(n (\cdot,s)m (\cdot,s)) ds,~~ t\in(t_0, T), \label{sss5555fghbnmcz2.5ghjjjkkklu48cfg924ghyuji}$$ where $t_0 := (t-1)_{+}$. As the last summand in is non-positive by the maximum principle, we can thus estimate $$\|n (t)\|_{L^\infty(\Omega)}\leq \|e^{(t-t_0)\Delta}n (\cdot,t_0)\|_{L^\infty(\Omega)}+\int_{t_0}^{t}\| e^{(t-s)\Delta}\nabla\cdot(n (\cdot,s)\tilde{h} (\cdot,s))\|_{L^\infty(\Omega)} ds,~~ t\in(t_0, T). \label{5555fghbnmcz2.5ghjjjkkklu48cfg924ghyuji}$$ If $t\in(0,1]$, by virtue of the maximum principle, we derive that $$\begin{array}{rl} \|e^{(t-t_0)\Delta}n (\cdot,t_0)\|_{L^{\infty}(\Omega)}\leq &\disp{\|n_0\|_{L^{\infty}(\Omega)},}\\ \end{array} \label{zjccffgbhjffghhjcghhhjjjvvvbscz2.5297x96301ku}$$ while if $t > 1$ then with the help of the $L^p$-$L^q$ estimates for the Neumann heat semigroup and , we conclude that $$\begin{array}{rl} \|e^{(t-t_0)\Delta}n (\cdot,t_0)\|_{L^{\infty}(\Omega)}\leq &\disp{C_{29}(t-t_0)^{-\frac{3}{2}}\|n (\cdot,t_0)\|_{L^{1}(\Omega)}\leq C_{30}.}\\ \end{array} \label{zjccffgbhjffghhjcghghjkjjhhjjjvvvbscz2.5297x96301ku}$$ Finally, we fix an arbitrary $\frac{7}{2}\in(3,4)$ and then once more invoke known smoothing properties of the Stokes semigroup and the Hölder inequality to find $C_4 > 0$ such that $$\begin{array}{rl} &\disp\int_{t_0}^t\| e^{(t-s)\Delta}\nabla\cdot(n (\cdot,s)\tilde{h} (\cdot,s)\|_{L^\infty(\Omega)}ds\\ \leq&\disp C_{31}\int_{t_0}^t(t-s)^{-\frac{1}{2}-\frac{3}{7}}\|n (\cdot,s)\tilde{h} (\cdot,s)\|_{L^p(\Omega)}ds\\ \leq&\disp C_{32}\int_{t_0}^t(t-s)^{-\frac{1}{2}-\frac{3}{7}}\| n (\cdot,s)\|_{L^{28}(\Omega)}\|\tilde{h} (\cdot,s)\|_{L^{4}(\Omega)}ds\\ \leq&\disp C_{33}\int_{t_0}^t(t-s)^{-\frac{1}{2}-\frac{3}{7}}\| u (\cdot,s)\|_{L^{\infty}(\Omega)}^\frac{27}{28}\| u (\cdot,s)\||_{L^1(\Omega)}^{\frac{1}{28}}\|\tilde{h} (\cdot,s)\|_{L^{4}(\Omega)}ds\\ \leq&\disp C_{34}M^b(T)~~\mbox{for all}~~ t\in(0, T),\\ \end{array} \label{ccvbccvvbbnnndffghhjjvcvvbccfbbnfgbghjjccmmllffvvggcvvvvbbjjkkdffzjscz2.5297x9630xxy}$$ In combination with – and using the definition of $M(T)$ we obtain $C_{35}> 0$ such that $$\begin{array}{rl} &\disp M(T)\leq C_{35}+C_{35}M^{\frac{27}{28}}(T)~~\mbox{for all}~~ T\in(0, T_{max,\varepsilon}).\\ \end{array} \label{ccvbccvvbbnnndffghhjjvcvvfghhhbccfbbnfgbghjjccmmllffvvggcvvvvbbjjkkdffzjscz2.5297x9630xxy}$$ Hence, with some basic calculation, in light of $T\in (0, T_{max,\varepsilon})$ was arbitrary, one can get $$\begin{array}{rl} \|n (\cdot, t)\|_{L^{\infty}(\Omega)}\leq&\disp{C_{36}~~ \mbox{for all}~~ t\in(0,T_{max,\varepsilon}).}\\ \end{array} \label{cz2.57ghhhh151ccvhhjjjkkkffgghhuuiivhccvvhjjjkkhhggjjllll}$$ Finally, by virtue of Lemma \[lemma70\] and , –, , the local solution can be extend to the global-in-time solutions. Employing almost exactly the same arguments as in the proof of Lemma 3.1 in [@LiLiLiLisssdffssdddddddgssddsddfff00] (see also [@Zhenddsdddddgssddsddfff00]), and taking advantage of , we conclude the regularity theories for the Stokes semigroup and the Hölder estimate for local solutions of parabolic equations, we can obtain weak solution $(n,c,m,u)$ is a classical solution. The most important consequence of Lemmas \[lemma45hyuuuj630223\]–\[lemmassddddff45630223\] is the following: [**Proof of Theorem \[theorem3\]**]{}: The theorem \[theorem3\] is part of the statement proven by Lemmas \[lemma45hyuuuj630223\]–\[lemmassddddff45630223\]. Eventual smoothness and asymptotics ----------------------------------- Given the preliminary lemma collected in the above, in this subsection, we now establish the claimed asymptotic behavior of the solutions to under $\alpha>0$. Before going further, we list the following lemma, which will be used to derive the convergence properties of solution with respect to the norm in $L^2 (\Omega)$. \[fhhghfbglemma4563025xxhjklojjkkkgyhuissddff\] (Lemma 4.6 of [@EspejojjEspejojainidd793]) Let $\lambda > 0, C > 0$, and suppose that $y\in C^1 ([0,\infty))$ and $h\in C^0 ([0,\infty))$ are nonnegative functions satisfying $y'(t)+\lambda y(t)\leq h(t)$ for some $\lambda > 0$ and all $t > 0$. Then if $\int_0^\infty h(s)ds \leq C,$ we have $\lim_{t\rightarrow\infty}y(t)=0$. To begin with, let us collect some basic solution properties which essentially have already been used in [@EspejojjEspejojainidd793]. \[lemmadsssddffffdfffgg4dddd5630\] The global solution of satisfies $$\begin{array}{rl} \disp\disp\int_0^\infty\int_\Omega \left(n_{\varepsilon}m_{\varepsilon}+|\nabla m_{\varepsilon}|^2\right)<&\disp{+\infty.}\\ \end{array} \label{hhxxcdfvvsssjjdfffssddcz2.5}$$ These properties are immediate consequences of and . As an immediate consequence, we obtain the following which will firstly serve as a fundament for our proof of stabilization in the first and third solution components. \[lemmaddffffdfffgg4sssdddd5630\] Under the assumptions of Lemma \[lemma45hyuuuj630223\], for any $\eta> 0$, there are $T > 0$ and $\varepsilon_0>0$ such that for any $t > T$ and such that for any $\varepsilon\in(0,\varepsilon_0 )$ $$0\leq\frac{1}{|\Omega|}\int_{\Omega}m_\varepsilon(\cdot,t)-\hat{m}<\eta~~~\mbox{for any}~~~t>T \label{11111hhxxcdfvhhhvssssssfftggggsssjjghjjsdggggdddfffddffssddcssdz2.5}$$ and $$0\leq\frac{1}{|\Omega|}\int_{\Omega}n_\varepsilon(\cdot,t)-\hat{n}<\eta~~~\mbox{for any}~~~t>T, \label{11111hhxxcdfvhhhvsssssssssfggjjsdggggdddfffddffddfggssddcssdz2.5}$$ where $$\hat{m}=\left\{\frac{1}{|\Omega|}\int_{\Omega}m_{0}-\frac{1}{|\Omega|}\int_{\Omega}n_{0}\right\}_{+} \label{1111hhxxcdfvhhhvsddfffgssjjdfffsfffsddcsssz2.5}$$ and $$\hat{n}=\left\{\frac{1}{|\Omega|}\int_{\Omega}n_{0}-\frac{1}{|\Omega|}\int_{\Omega}m_{0}\right\}_{+}. \label{1111hhxxcddffdfvhhhvsddfffgssdfffjjdfffssddcsssz2.5}$$ Pursuing a strategy demonstrated in lemma 4.2 of [@Winkler61215], we start by noting that as a first consequence of Lemma \[lemmadsssddffffdfffgg4dddd5630\] we know that $$\begin{array}{rl} \disp\disp\int_{t-1}^t\int_\Omega \left(n_\varepsilon m_\varepsilon+|\nabla m_\varepsilon|^2\right)\rightarrow&\disp{0~~~\mbox{as}~~t\rightarrow\infty.}\\ \end{array} \label{1111hhxxcdfvhhhvsssjjdfffssddcsssz2.5}$$ Next, in view of , by using the Hölder inequality and the Poincaré inequality, for some positive constant $K$, $$\begin{array}{rl} \disp\disp\int_{t-1}^t\int_\Omega n_\varepsilon m_\varepsilon=&\disp{\int_{t-1}^t\int_\Omega n_\varepsilon(m_{\varepsilon}-\bar{m})+\int_{t-1}^t\bar{m}\int_\Omega n_{\varepsilon}}\\ \geq&\disp{-\int_{t-1}^t\|n_{\varepsilon}\|_{L^2(\Omega)}\|m_{\varepsilon}-\bar{m}\|_{L^2(\Omega)}+\int_{t-1}^t\bar{m}\int_{\Omega}n_\varepsilon(x,s)dxds}\\ \geq&\disp{-K\int_{t-1}^t\|\nabla m_{\varepsilon}\|_{L^2(\Omega)}+\frac{1}{|\Omega|}\int_{t-1}^t\left[\int_{\Omega}m_\varepsilon(x,s)dx\int_{\Omega}n_{\varepsilon}(x,s)dx\right]ds}\\ \geq&\disp{-K\left(\int_{t-1}^t\|\nabla m_{\varepsilon}\|_{L^2(\Omega)}^2\right)^{\frac{1}{2}}+\frac{1}{|\Omega|}\int_{t-1}^t\left[\int_{\Omega}m_\varepsilon(x,s)dx\int_{\Omega}n_{\varepsilon}(x,s)dx\right]ds.}\\ \end{array} \label{11112222hhxxcdfvssdhhhvsssjjdfffssddcsssz2.5}$$ Inserting into , we obtain $$\begin{array}{rl} \disp\int_{t-1}^t\left[\int_{\Omega}m_\varepsilon(x,s)dx\int_{\Omega}n_{\varepsilon}(x,s)dx\right]ds\rightarrow0~~~\mbox{as}~~t\rightarrow\infty. \end{array} \label{11111111111hhxxcdfvssdhhhvsssjjdfffssddcsssz2.5}$$ Now if $\int_{\Omega}n_ 0-\int_{\Omega}m_ 0 \geq 0,$ warrants that $\int_{\Omega}n_{\varepsilon}-\int_{\Omega}m_{\varepsilon} \geq 0$, which along with implies that $$\begin{array}{rl} \disp\int_{t-1}^t\left(\int_{\Omega}m_\varepsilon(x,s)dx\right)^2ds\rightarrow0~~~\mbox{as}~~t\rightarrow\infty. \end{array} \label{hhxxcdfvssdhhhvssssssjjdfffssddcsssz2.5}$$ Noticing that $\int_{\Omega}m_\varepsilon(s) \geq \int_{\Omega}m_\varepsilon(t)$ for all $t\geq s,$ we have $$0\leq \left(\int_{\Omega}m_\varepsilon(x,t)dx\right)^2 \leq \int_{t-1}^t\left(\int_{\Omega}m_\varepsilon(x,s)dx\right)^2ds\rightarrow0~~~\mbox{as}~~t\rightarrow\infty,$$ where we invoke to obtain $$\int_{\Omega}n_{\varepsilon}(\cdot,t)\rightarrow \int_{\Omega}n_{0}-\int_{\Omega}m_{0}~~\mbox{as}~~t\rightarrow\infty.$$ By very similar argument, one can see that $\int_{\Omega}n_{\varepsilon}\rightarrow 0$ and $\int_{\Omega}m_{\varepsilon}\rightarrow \int_{\Omega}m_{0}-\int_{\Omega}n_{0}$ as $t\rightarrow\infty$ in the case of $\int_{\Omega}n_0 - \int_{\Omega}m_0 < 0$. This readily establishes and . \[ssdddlemmddddaddffffdfffgg4sssdddd5630\] Under the assumptions of Lemma \[lemma45hyuuuj630223\], for any $\eta > 0$, there are $T > 0$ and $\varepsilon_0>0$ such that for any $t > T$ and such that for any $\varepsilon\in(0,\varepsilon_0 )$ $$\int_{t}^{t+1}\|m_\varepsilon(\cdot,t)-\hat{m}\|_{L^\infty(\Omega)}<\eta \label{11111hhxxcdfvhhhvsssssssssjjghjjsdggggdddfffddffssddcssdz2.5}$$ and $$\|m_\varepsilon(\cdot,t)-\hat{m}\|_{L^p(\Omega)}< \eta, \label{11111hhxxcdfvhhhvsssdddjjkddffkksssssssscccjjghjjsdggggdddfffddffssddcssdz2.5}$$ where $\hat{m}$ is give by . Firstly, since Lemma \[lemma45630hhuujj\] asserts the existence of $\kappa_1$ such that $$\int_t^{t+1}\|\nabla m_\varepsilon (\cdot,t)\|_{L^{4}(\Omega)}^4 \leq \kappa_1 \label{1111hhxxcdfvhhhvsssssssssjjdfffddffssddcssdz2.ssdd5}$$ and since implies that $$\frac{1}{|\Omega|}\int_{\Omega}m_\varepsilon(\cdot,t)\geq \left\{\frac{1}{|\Omega|}\int_{\Omega}m_{0}-\frac{1}{|\Omega|}\int_{\Omega}n_{0}\right\}_{+}, \label{11111hhxxcdfvhssdddhhvsssssssssjjghjjsdggggdddfffddffssddcssdz2.5}$$ thus, by we infer from the interpolation inequality and the Hölder inequality that $$\label{fvgbccvvhnjmkfhhhhhgbdffrhnkkkn6291} \begin{array}{rl} \disp &\int_t^{t+1}\|m_\varepsilon-\hat{m}\|_{L^\infty(\Omega)}ds\\ \leq&{ C(\disp\int_t^{t+1}(\|\nabla m_\varepsilon\|_{L^{4}(\Omega)}^{\frac{12}{13}}\|m_\varepsilon-\hat{m}\|_{L^1(\Omega)}^{\frac{1}{13}}+\|m_\varepsilon-\hat{m}\|_{L^1(\Omega)})ds}\\ \leq&{ C\disp\int_t^{t+1}\|\nabla m_\varepsilon\|_{L^{4}(\Omega)}^{4}ds)^{\frac{3}{13}}(\disp\int_t^{t+1}\|m_\varepsilon-\hat{m}\|_{L^1(\Omega)}ds)^{\frac{1}{13}}+C\disp\int_t^{t+1}\|m_\varepsilon-\hat{m}\|_{L^1(\Omega)})ds}\\ \leq&{ C\disp\int_t^{t+1}\|\nabla m_\varepsilon\|_{L^{4}(\Omega)}^{4}ds)^{\frac{3}{13}}(\sup_{t>0}\|m_\varepsilon(\cdot,t)-\hat{m}\|_{L^1(\Omega)})^{\frac{1}{13}}+C\disp\sup_{t>0}\|m_\varepsilon(\cdot,t)-\hat{m}\|_{L^1(\Omega)})}\\ \rightarrow&{0~~~\mbox{as}~~t\rightarrow+\infty,}\\ \end{array}$$ which immediately implies . Here we have used the fact that $$\|m_\varepsilon-\hat{m}\|_{L^1(\Omega)}=\int_{\Omega}[m_\varepsilon(\cdot,t)-\hat{m}]= |\Omega|\left[\frac{1}{|\Omega|}\int_{\Omega}m_\varepsilon(\cdot,t)-\hat{m}\right]\rightarrow0~~~\mbox{as}~~t\rightarrow+\infty$$ by using . Next, for any $p>1$, in view of Lemma \[fvfgsdfggfflemma45\], we derive from the the interpolation and the Hölder inequality that $$\label{fvgbccvsddfgddffvhnsdfffjmkfhhhhhgbdffrhnkkkn6291} \begin{array}{rl} \disp &\|m_{\varepsilon}-\hat{m}\|_{L^p(\Omega)}\\ \leq&{ \disp\|m_{\varepsilon}-\hat{m}\|_{L^{\infty}(\Omega)}^{\frac{p-1}{p}}\|m_{\varepsilon}-\hat{m}\|_{L^1(\Omega)}^{\frac{1}{p}}}\\ \rightarrow&{0~~~\mbox{as}~~t\rightarrow+\infty,}\\ \end{array}$$ which yields directly. \[sedddlemmaddffffdfffgg4sssdddd5630\] Under the assumptions of Lemma \[lemma45hyuuuj630223\], for any $\eta > 0$, there are $T > 0$ and $\varepsilon_0>0$ such that for any $t > T$ and such that for any $\varepsilon\in(0,\varepsilon_0 )$ $$\|c_\varepsilon(\cdot,t)-\hat{m}\|_{L^2(\Omega)}<\eta \label{11111hhxxcdfvhhhvsssssssssjjsdgkkkgggdddfffddffssddcssdz2.5}$$ and $$\begin{array}{rl} \disp\disp\int_t^{t+1}\int_\Omega |\nabla c_\varepsilon|^2&<\disp{\eta.}\\ \end{array} \label{hhxxcdfvhhhvsssssddffsssjjssdfffssddcssdz2.5}$$ where $\hat{m}$ is give by . Firstly, by means of the testing procedure, we may derive from the Young inequality that $$\begin{array}{rl} &\disp{\frac{1}{2}\frac{d}{dt}\|c_\varepsilon-\hat{m}\|^{{2}}_{L^{{2}}(\Omega)}} \\ =&\disp{ \int_{\Omega}(c_\varepsilon-\hat{m})[\Delta c_\varepsilon-u_\varepsilon\cdot\nabla c_\varepsilon-(c_\varepsilon-\hat{m})+(m_\varepsilon-\bar{m})]}\\ =&\disp{ \int_{\Omega}(c_\varepsilon-\hat{m})(\Delta c_\varepsilon-u_\varepsilon\cdot\nabla c_\varepsilon)-\int_{\Omega}(c_\varepsilon-\hat{m})^2+\int_{\Omega}(c_\varepsilon-\hat{m})(m_\varepsilon-\hat{m})}\\ \leq&\disp{- \int_{\Omega}|\nabla c_\varepsilon|^2-\int_{\Omega}(c_\varepsilon-\hat{m})^2+\frac{1}{2}\int_{\Omega}(m_\varepsilon-\hat{m})^2}\\ \leq&\disp{-\int_{\Omega}(c_\varepsilon-\hat{m})^2+\frac{1}{2}\int_{\Omega}(m_\varepsilon-\hat{m})^2~~\mbox{for all}~~ t>0, }\\ \end{array} \label{wwwwwcz2.511ssssdfggsddffggg4ddfggg114}$$ where we have used the fact that $\nabla\cdot u_\varepsilon = 0$ and $u_\varepsilon |_{\partial\Omega} = 0$. On the other hand, the bounds from \[ssdddlemmddddaddffffdfffgg4sssdddd5630\] entails $$\begin{array}{rl} \disp\int_{t}^{t+1}\int_{\Omega}(m_\varepsilon-\hat{m})^2ds\rightarrow0~~~\mbox{as}~~t\rightarrow\infty. \end{array} \label{hhxxcdfvssdhhhvssssssjjdfffssddcsssddssz2.5}$$ This together with and Lemma \[fhhghfbglemma4563025xxhjklojjkkkgyhuissddff\] imply and . \[11aaalemdfghkkmaddffffdfffgg4sssdddd5630\] Under the assumptions of Lemma \[lemma45hyuuuj630223\], for any $p > 1$ and $\eta>0$, there are $T > 0$ and $\varepsilon_0>0$ such that for any $t > T$ and such that for any $\varepsilon\in(0,\varepsilon_0 )$ $$\|n_\varepsilon(\cdot,t)-\hat{n}\|_{L^p(\Omega)}< \eta, \label{11111hhxxcdfvhhhvssssssssscccjjghjjsdggggdddfffddffssddcssdz2.5}$$ where $\hat{n}$ is given by . Firstly, for any $p>1$, by Lemma \[lemmaghjffggssddgghhmk4563025xxhjklojjkkk\], there exist positive constants $\alpha_1$ and $q>p$ such that $$\begin{array}{rl} &\disp{\int_{\Omega} n_\varepsilon ^{q}(x,t)\leq \alpha_1~~~\mbox{for all}~~ t>0.}\\ \end{array} \label{czfvgb2.5ghhjussdyuccvviihjj}$$ By the interpolation and the Hölder inequality, we have $$\label{fvgbccvsddfgddffvhnjmkfhhhhhgbdffrhnkkkn6291} \begin{array}{rl} \disp \|n_{\varepsilon}-\hat{n}\|_{L^p(\Omega)} \leq&{ \disp\|n_{\varepsilon}-\hat{n}\|_{L^{q}(\Omega)}^{\frac{q(p-1)}{p(q-1)}}\|n_{\varepsilon}-\hat{n}\|_{L^1(\Omega)}^{\frac{q-p}{p(q-1)}}}\\ \rightarrow&{0~~~\mbox{as}~~t\rightarrow+\infty}\\ \end{array}$$ by using . From we readily derive and thereby completes the proof. The stabilization property implied by Lemmas \[ssdddlemmddddaddffffdfffgg4sssdddd5630\] and \[11aaalemdfghkkmaddffffdfffgg4sssdddd5630\] can now be turned into a preliminary statement on decay of $u_\varepsilon$ by making use of Lemma \[fhhghfbglemma4563025xxhjklojjkkkgyhuissddff\] and the standard testing procedures. \[11aaalemmaddffffdsddfffffgg4sssdddd5630\] Under the assumptions of Lemma \[lemma45hyuuuj630223\], for any $\eta > 0$, there are $T > 0$ and $\varepsilon_0>0$ such that for any $t > T$ and such that for any $\varepsilon\in(0,\varepsilon_0 )$ $$\|u_\varepsilon(\cdot,t)\|_{L^2(\Omega)}<\eta, \label{11111hhxxcdfvhhhvssssssssscccjjghjjsdgggddddgdddfffddffssddcssdz2.5}$$ $$\int_{t}^{t+1}\|\nabla u_\varepsilon\|_{L^2(\Omega)}^2dx<\eta \label{11111hhxxcdfvhhhvsssssddfffssddffsscccjjghjjsdggggdddfffddffssddcssdz2.5}$$ as well as $$\int_{t}^{t+1}\|u_\varepsilon\|_{L^q(\Omega)}^2dx<\eta \label{11111hhxxcdfvhhhvsssddffssddfffssddffsscccjjghjjsdggggdddfffddffssddcssdz2.5}$$ and $$\int_{t}^{t+1}\|u_\varepsilon\|_{L^q(\Omega)}dx<\eta \label{11111hhxxcdfvhhhvddfffsssddffssddfffssddffsscccjjghjjsdggggdddfffddffssddcssdz2.5}$$ for any $q\in[1,6)$. From the fourth equation in we obtain the associated Navier-Stokes energy inequality in the form $$\begin{array}{rl} &\disp{\frac{1}{2}\frac{d}{dt}\|u_{\varepsilon}\|^{{2}}_{L^{{2}}(\Omega)}} \\ =&\disp{- \int_{\Omega}|\nabla u_{\varepsilon}|^2+\int_{\Omega}(n_{\varepsilon}+m_{\varepsilon})\nabla\phi\cdot u_{\varepsilon}-\int_{\Omega}\nabla P_{\varepsilon}\cdot u_{\varepsilon}}\\ =&\disp{- \int_{\Omega}|\nabla u_{\varepsilon}|^2+\int_{\Omega}(n_{\varepsilon}-\hat{n}+m_{\varepsilon}-\hat{m})\nabla\phi\cdot u_{\varepsilon}}\\ \leq&\disp{- \int_{\Omega}|\nabla u_{\varepsilon}|^2+K_1\left(\int_{\Omega}(n_{\varepsilon}-\hat{n}+m_{\varepsilon}-\hat{m})^2\right)^{\frac{1}{2}}\left(\int_{\Omega} |u_{\varepsilon}|^2\right)^{\frac{1}{2}}}\\ \leq&\disp{- \int_{\Omega}|\nabla u_{\varepsilon}|^2+K_1\left(\int_{\Omega}|n_{\varepsilon}-\hat{n}|^2+\int_{\Omega}|m_{\varepsilon}-\hat{m}|^2\right)^{\frac{1}{2}}\left(\int_{\Omega} |u_{\varepsilon}|^2\right)^{\frac{1}{2}},}\\ \end{array} \label{cddddz2.51kkk1ssssdfssddjjkkkggsddffggg4ddfggg114}$$ where we have used the fact that $\hat{n}=-\hat{m}$ as well as $\nabla\cdot u_{\varepsilon} = 0$ and $u_{\varepsilon} |_{\partial\Omega} = 0$. Due to the Poincaré inequality again, we have $$\eta_0\int_{\Omega} |u_{\varepsilon}|^2\leq \int_{\Omega}|\nabla u_{\varepsilon}|^2,$$ therefore, collecting and , we derive from that $$\begin{array}{rl} \disp\disp \lim_{t\rightarrow+\infty}\int_{\Omega}|u_{\varepsilon}(x,t)|^2dx&=\disp{0}\\ \end{array} \label{hhxxcdfvhhhvsssssssssjjdfffddffssddcssdz2.5}$$ and $$\int_{t}^{t+1}\|\nabla u_{\varepsilon}\|_{L^2(\Omega)}^2dx\rightarrow 0 ~~~\mbox{as}~~~t\rightarrow\infty \label{11111hhxxcdfvhhhvddddsssssddfffssddffsscccjjghjjsdggggdddfffddffssddcssdz2.5}$$ and thereby proves –. Finally, we make use of the embedding $W^{1,2}(\Omega)\hookrightarrow L^q (\Omega)$ (for any $q\in[1,6)$) and the Young inequality to find that and hold. Using thge decay property of $m_\varepsilon(\cdot,t)-\hat{m}+n_\varepsilon(\cdot,t) -\hat{n}$ (see Lemmas \[ssdddlemmddddaddffffdfffgg4sssdddd5630\] and \[11aaalemdfghkkmaddffffdfffgg4sssdddd5630\]), by means of a contraction mapping argument, we may derive a certain eventual regularity and decay of $u_\varepsilon$ in $L^p(\Omega)$ with some $p\geq6$. \[aaalemmaddffffsddddfffgg4sssdddd5630\] For any $p \in[6,\infty)$ and $\eta>0$, there are $T > 0$ and $\varepsilon_0>0$ such that for any $t > T$ and such that for any $\varepsilon\in(0,\varepsilon_0 )$ $$\|u_{\varepsilon}(\cdot,s)\|_{L^p(\Omega)}<\eta~~~\mbox{for any}~~~ s\in[t,t + 1]. \label{11111hhxxcdfvhhhvssssssssscccjjghjjsdgggddddgdddffddfffddffssddcssdz2.5}$$ We let $p\geq 6$ and choose $q \in (3,6)$ which is close to $6$ (e.g. $q=\frac{6p}{p+2}$) such that $$\begin{array}{rl} \disp{\frac{1}{2}-\frac{3}{2p}-3(\frac{1}{q}-\frac{1}{p})} >\disp{0.}\\ \end{array} \label{hhxxcdfvhhhvsssjjdfffssssddssdddcz2.5}$$ Now, we define $\gamma_0=\frac{3}{2}(\frac{1}{q}-\frac{1}{p})$. Next, in view of and using $p\geq6$, $q\in(3,6)$, we have $$-2\gamma_0-\frac{1}{2}-\frac{3}{2p}>-1~~~\mbox{and}~~ -\frac{1}{2}-\frac{3}{2p}>-1.$$ Therefore, $$2\kappa_2\int_0^3s^{-2\gamma_0-\frac{1}{2}-\frac{3}{2p}}ds<+\infty~~~\mbox{and}~~ 2\kappa_2\int_0^3s^{-\frac{1}{2}-\frac{3}{2p}}ds<+\infty, \label{2222dddddf1.1ddfghssdddyddffssddduisda}$$ where $\kappa_2$ is give by Lemma \[llssdrffmmggnnccvvccvvkkkkgghhkkllvvlemma45630\]. Thus for any $\eta>0$, we any choose $\eta_0\in(0,\eta)$ small enough such that $$\eta_0<\min\{\frac{1}{2\kappa_2\int_0^3s^{-2\gamma_0-\frac{1}{2}-\frac{3}{2p}}ds},\frac{1}{2\kappa_2\int_0^3s^{-\frac{1}{2}-\frac{3}{2p}}ds}\}. \label{2222dddddf1.1ddfghssdddysddddddffssddduisda}$$ On the other hand, by , Lemmas \[ssdddlemmddddaddffffdfffgg4sssdddd5630\]–\[11aaalemmaddffffdsddfffffgg4sssdddd5630\], we then pick $T_0$ and $\varepsilon_0>0$ such that for any $t > T$ and such that for any $\varepsilon\in(0,\varepsilon_0 )$ $$\int_{t}^{t+1}\|u_\varepsilon\|_{L^q(\Omega)}dx<\frac{\eta_0}{3\kappa_1}~~~ \mbox{and}~~\|m_\varepsilon(\cdot,t)-\hat{m}+n_\varepsilon(\cdot,t)-\hat{n}\|_{L^p(\Omega)}<\frac{\eta_0}{3^{\gamma_0+1}\kappa_3\|\nabla\phi\|_{L^\infty(\Omega)}}, \label{11111hhxxcdfvhhhvddfssdddffsssddffssddfffssddffsscccjjghjjsdggggdddfffddffssddcssdz2.5}$$ where $\kappa_1$ and $\kappa_3$ are given by Lemma \[llssdrffmmggnnccvvccvvkkkkgghhkkllvvlemma45630\]. In view of , for any $t_1>T_0$ and $\varepsilon\in (0,\varepsilon_{t_1})$, one can find $\tilde{t}_0\in(t_1,t_1+1)$ such that $$\|u_\varepsilon(\cdot,\tilde{t}_0)\|_{L^q(\Omega)}dx<\frac{\eta_0}{3\kappa_1}. \label{11111hhxxcdfvhhhvdssddddfssdddffsssddffssddfffssddffsscccjjghjjsdggggdddfffddffssddcssdz2.5}$$ Now, we define $$T=T_0+2.$$ In the following, we will prove that holds for any $t>T.$ To this end, for the above $\tilde{t}_0,p,\gamma_0$ and $\eta_0$, we let $$X=\{v:\Omega\times(\tilde{t}_0,\tilde{t}_0+3)\rightarrow \mathbb{R};\sup_{s\in(0,3)}s^{\gamma_0}\|v(\tilde{t}_0+s)\|_{L^p(\Omega)}\leq\eta_0\}. \label{cz2.571hhhhh51lllllccvvddfgghddfccvvhjjjkkhhggjjlsdddlll}$$ Then we consider the mapping $\varphi: X\rightarrow \mathbb{R}$ defined by $$\varphi(v) = e^{-tA}u_{\varepsilon}(\tilde{t}_0) +\int_{\tilde{t}_0}^te^{-(t-\tau)A} \mathcal{P}\left[-\kappa\nabla\cdot(Y_\varepsilon v\otimes v) (\tau)+(n_\varepsilon (\tau)+m_\varepsilon (\tau))\nabla\phi\right]d\tau.$$ Now, we will show that $\varphi$ is a contraction on $X$. In fact, in view of Lemma \[llssdrffmmggnnccvvccvvkkkkgghhkkllvvlemma45630\], for any $s>1$ and for any such $v$ we may derive from the Hölder inequality that $$\begin{array}{rl} &\|\varphi(v) (\cdot, t)\|_{L^p(\Omega)}\\ \leq&\disp{\kappa_1(t-\tilde{t}_0)^{-\gamma_0}\|u_\varepsilon(\tilde{t}_0)\|_{L^q(\Omega)} + \kappa_2\int_{\tilde{t}_0}^t(t-\tau)^{-\frac{1}{2}-\frac{3}{2}(\frac{2}{p}-\frac{1}{p})}\|v\oplus v\|_{L^{\frac{p}{2}}(\Omega)}d\tau}\\ &\disp{+\kappa_3\|\nabla\phi\|_{L^\infty(\Omega)}\int_{\tilde{t}_0}^t\|m_\varepsilon(\cdot,\tau)-\hat{m}+n_\varepsilon(\cdot,\tau)-\hat{n}\|_{L^p(\Omega)}d\tau}\\ \leq&\disp{\kappa_1(t-\tilde{t}_0)^{-\gamma_0}\|u_\varepsilon(\tilde{t}_0)\|_{L^q(\Omega)} + \kappa_2\int_{\tilde{t}_0}^t(t-\tau)^{-\frac{1}{2}-\frac{3}{2}(\frac{2}{p}-\frac{1}{p})}\|v\|_{L^{p}(\Omega)}^2d\tau}\\ &\disp{+\kappa_3\|\nabla\phi\|_{L^\infty(\Omega)}\int_{\tilde{t}_0}^{\tilde{t}_0+3}\|m_\varepsilon(\cdot,\tau)-\hat{m}+n_\varepsilon(\cdot,\tau) -\hat{n}\|_{L^p(\Omega)}d\tau.}\\ \end{array} \label{cz2.571hhhhh51lllllccvvhddfccvvhjjjkkhhggjjlsdddlll}$$ Therefore, in light of , as well as and , we see that for every $t\in (\tilde{t}_0 ,\tilde{t}_0 + 3)$ and every $v\in X$ $$\begin{array}{rl} &(t-\tilde{t}_0)^{\gamma_0}\|\varphi(v) (\cdot, t)\|_{L^p(\Omega)}\\ \leq&\disp{\kappa_1\|u_{\varepsilon}(\tilde{t}_0)\|_{L^q(\Omega)} + \kappa_2(t-\tilde{t}_0)^{\gamma_0}\int_{\tilde{t}_0}^t(t-\tau)^{-\frac{1}{2}-\frac{3}{2}(\frac{2}{p}-\frac{1}{p})}\|v\oplus v\|_{L^{\frac{p}{2}}(\Omega)}d\tau}\\ &\disp{+\kappa_3(t-\tilde{t}_0)^{\gamma_0}\|\nabla\phi\|_{L^\infty(\Omega)}\int_{\tilde{t}_0}^{\tilde{t}_0+3}\|m_\varepsilon(\cdot,\tau)-\hat{m}+ n_\varepsilon(\cdot,\tau)-\hat{n}\|_{L^p(\Omega)}d\tau}\\ \leq&\disp{\kappa_1\|u_{\varepsilon}(\tilde{t}_0)\|_{L^q(\Omega)} + \kappa_2\eta_0^23^{\gamma_0}\int_{0}^3(t-\tau)^{-\frac{1}{2}-\frac{3}{2p}}\tau^{-2\gamma_0}d\tau}\\ &\disp{+\kappa_33^{\gamma_0}\|\nabla\phi\|_{L^\infty(\Omega)}\frac{\eta_0}{3^{\gamma_0+1}\kappa_3\|\nabla\phi\|_{L^\infty(\Omega)}}}\\ <&\disp{\eta_0.}\\ \end{array} \label{cz2.571hhhhh51lllllccvvhddfccvvhjjjkkhhggjjllll}$$ from which it readily follows that $\varphi(X)\subset X$. Likewise, for $v\in X$ and $w\in X$ we can use Lemma \[llssdrffmmggnnccvvccvvkkkkgghhkkllvvlemma45630\] to find that $$\begin{array}{rl} &\|\varphi(v) (\cdot, t)-\varphi(w)(\cdot, t) \|_{L^p(\Omega)}\\ \leq&\disp{ \kappa_2\int_{\tilde{t}_0}^t(t-\tau)^{-\frac{1}{2}-\frac{3}{2p}}\|v\oplus v-w\oplus w\|_{L^{\frac{p}{2}}(\Omega)}d\tau}\\ =&\disp{ \kappa_2\int_{\tilde{t}_0}^t(t-\tau)^{-\frac{1}{2}-\frac{3}{2p}}\|v\oplus (v-w)+(v-w)\oplus w\|_{L^{\frac{p}{2}}(\Omega)}d\tau}\\ \leq&\disp{ \kappa_2\int_{\tilde{t}_0}^t(t-\tau)^{-\frac{1}{2}-\frac{3}{2p}}(\|v\|_{L^{p}}+\|w\|_{L^{p}})\|v-w\|_{L^{p}(\Omega)}d\tau}\\ \leq&\disp{2 \kappa_2\eta_0\int_{0}^3\tau^{-\frac{1}{2}-\frac{3}{2p}}d\tau\|v-w\|_{L^{\infty}((0,3);L^{p}(\Omega)}.}\\ \end{array} \label{234cz2.571hhhhh51lllllccvvhsdddddfccvvhjjjkkhhggjjlsdddlll}$$ On the other hand, implies that $$2 \kappa_2\eta_0\int_{0}^3\tau^{-\frac{1}{2}-\frac{3}{2p}}d\tau<1,$$ whence shows that $\varphi$ acts as a contraction on $X$ and hence possesses a unique fixed point on $X$, which, in view of the definition of $\varphi$, must coincide with the unique weak solution $u_\varepsilon$ of fourth equation of on $(\tilde{t}_0 ,\tilde{t}_0+3)$ (see e.g. Thm. V.2.5.1 of [@Sohr]). Now, by , we also derive that $$\|u_\varepsilon(\cdot,t)\|_{L^p(\Omega)}dx<\eta_0,~~\mbox{for all}~~t\in(\tilde{t}_0+1 ,\tilde{t}_0+3), \label{11111hhxxcdfvhhhssddvdssddddfssdddffsssddffssddfffssddffsscccjjghjjsdggggdddfffddffssddcssdz2.5}$$ from $(\tilde{t}_0+1 ,\tilde{t}_0+3)\supset(t_1+2,t_1+3)$ we readily derive . In the following lemmas, we next plan to prove Hölder regularity of the components of a solution on intervals of the form $(T_0 ,T_0 + 1)$ for $T_0> 0$ by using the maximal Sobolev regularity. To this end, we introduce the following cut-off functions, which will play a key role in deriving higher order regularity for solution of problem . \[aaalemmaddffffdssfffgg4sssdddd5630\] Let $\xi: \mathbb{R} \rightarrow [0,1]$ be a smooth, monotone function, satisfying $\xi\equiv 0$ on $(-\infty,0]$ and $\xi\equiv 0$ on $(1,\infty)$ and for any $t_0\in \mathbb{R}$ we let $\xi_{t 0}:= \xi (t -t_0 )$. Due to the above cut-off function, it follows from maximal Sobolev regularity that the solution $(n_\varepsilon, c_\varepsilon, m_\varepsilon,u_\varepsilon)$ even satisfies estimates in appropriate Hölder spaces: \[lemma45630hhuujjuuyy\] Let $\alpha>0$. Then one can find $\mu\in(0, 1)$ and $T,\varepsilon_0,C>0$ such that for all $\varepsilon\in(0,\varepsilon_0)$ $$\|u_\varepsilon(\cdot,t)\|_{C^{1+\mu,\frac{\mu}{2}}(\bar{\Omega}\times[t,t+1])} \leq C ~~\mbox{for all}~~ t>T, \label{zjscz2.5297x9630111kkhhffrroojj}$$ $$\|c_\varepsilon(\cdot,t)\|_{C^{1+\mu,\frac{\mu}{2}}(\bar{\Omega}\times[t,t+1])} \leq C ~~\mbox{for all}~~ t>T \label{zjscz2.5297x9630111kkhhiioo}$$ as well as $$\|m_\varepsilon(\cdot,t)\|_{C^{1+\mu,\frac{\mu}{2}}(\bar{\Omega}\times[t,t+1])} \leq C ~~\mbox{for all}~~ t>T \label{ddhhhzjscz2.5297x9630111kkhhiioo}$$ and $$\|n_\varepsilon(\cdot,t)\|_{C^{1+\mu,\frac{\mu}{2}}(\bar{\Omega}\times[t,t+1])} \leq C ~~\mbox{for all}~~ t>T. \label{zjscz2.5297x96dfgg30111kkhhffrroojj}$$ Firstly, for any $T_0 > 0$ let $\xi:=\xi_{T_0}$ and $v:=\xi u_\varepsilon.$ $$\left\{\begin{array}{ll} v_t-\Delta v=h~~~\mbox{in}~~\Omega\times (T_0,\infty),\\ \nabla\cdot v=0,~~~\mbox{in}~~\Omega\times (T_0,\infty),\\ \disp{v(x,T_0)=0},\quad x\in \Omega,\\ \disp{v=0},\quad \mbox{on}~~ \partial\Omega\times(T_0,\infty),\\ \end{array}\right.$$ where $$h=-\kappa(Y_\varepsilon u_\varepsilon \cdot \nabla)v+\nabla (\xi P_\varepsilon)+\xi(n_\varepsilon+m_\varepsilon)\nabla \phi+\xi' u_\varepsilon.$$ To estimate the inhomogeneity $h$ herein, we first note that the known maximal Sobolev regularity estimate for the Stokes semigroup ([@Gigass12176]) yields a constant $k_1 > 0$ such that $$\begin{array}{rl} &\disp\int_{T_0}^{T_0+2}\|v_t\|_{L^s(\Omega)}^s+\int_{T_0}^{T_0+2}\|D^2v\|_{L^s(\Omega)}^s\\ \leq&\disp{k_1\int_{T_0}^{T_0+2}\|\mathcal{P}(\xi Y_\varepsilon u_\varepsilon\cdot)u_\varepsilon\|_{L^s(\Omega)}^s+k_1\int_{T_0}^{T_0+2} \left(\|\mathcal{P}\xi(m_\varepsilon(\cdot,\tau)-\hat{m}+n_\varepsilon(\cdot,\tau)-\hat{n})\nabla\phi\|_{L^s(\Omega)}^s\right)}\\ &\disp{+k_1\int_{T_0}^{T_0+2}\|\mathcal{P}\xi' u_\varepsilon\|_{L^s(\Omega)}^s.}\\ \end{array} \label{cz2.571hhhhh51lllllccvvhsdddddfccvvhjjjkkhhggjjlsdddlll}$$ From the boundedness of the Helmholtz projection in $L^s$-spaces and the Hölder inequality we derive from Lemma \[aaalemmaddffffsddddfffgg4sssdddd5630\] that there exist positive constant $k_2 $ and $k_3$ such that for any $ T_0 > T$ $$\begin{array}{rl} &\disp k_1\int_{T_0}^{T_0+2}\|\mathcal{P}(Y_\varepsilon u_\varepsilon\cdot)v\|_{L^s(\Omega)}^s\\ \leq&\disp{k_2\int_{T_0}^{T_0+2}\left(\|Y_\varepsilon u_\varepsilon\|_{L^{l'}(\Omega)}^s\|\nabla v\|_{L^{l}(\Omega)}^s\right) }\\ \leq&\disp{k_2\int_{T_0}^{T_0+2}\left(\| u_\varepsilon\|_{L^{l'}(\Omega)}^s\|\nabla v\|_{L^{l}(\Omega)}^s\right) }\\ \leq&\disp{k_3\int_{T_0}^{T_0+2}\|\nabla v\|_{L^{l}(\Omega)}^s~~~~~\mbox{for any}~~\varepsilon\in(0,\varepsilon_{T_0}),}\\ \end{array} \label{cz2.57ssdd1hhhhh51lllllccvvhsddddddfdfccvvhjjjkkhhggjjlsdddlll}$$ where $l>2s$ and $\frac{1}{l}+\frac{1}{l'}=1.$ Thanks to the Gagliardo-Nirenberg inequality, from Lemma \[aaalemmaddffffsddddfffgg4sssdddd5630\] again, we can estimate the right of by following: $$\begin{array}{rl} &\disp k_3\int_{T_0}^{T_0+2}\|\nabla v\|_{L^{l}(\Omega)}^s\\ \leq&\disp{k_4\int_{T_0}^{T_0+2}\left(\|D^2 v\|_{L^s(\Omega)}^{as}\| v\|_{L^{r_0}(\Omega)}^{(1-a)s}\right) }\\ \leq&\disp{k_4\int_{T_0}^{T_0+2}\left(\|D^2 v\|_{L^s(\Omega)}^{as}\| u_\varepsilon\|_{L^{r_0}(\Omega)}^{(1-a)s}\right) }\\ \leq&\disp{k_5\int_{T_0}^{T_0+2}\|D^2 v\|_{L^s(\Omega)}^{as}~~~~~\mbox{for any}~~\varepsilon\in(0,\varepsilon_{T_0}),}\\ \end{array} \label{cz2.57ssdd1hhhhh51lllllccvvhsddddddfdfccvvhjddffjjkkhhgdddgjjlsdddlll}$$ where $a\in(0,1)$ satisfies $$\frac{1}{l}-\frac{1}{3}=a(\frac{1}{s}-\frac{2}{3})+(1-a)\frac{1}{r_0}.$$ Therefore, inserting into and applying the Young inequality, we find $k_6> 0$ such that for all $t_0 > T$ $$\begin{array}{rl} \disp k_1\int_{T_0}^{T_0+2}\|\mathcal{P}(Y_\varepsilon u_\varepsilon\cdot)v\|_{L^s(\Omega)}^s \leq&\disp{\frac{1}{2}\int_{T_0}^{T_0+2}\|D^2 v\|_{L^s(\Omega)}^{s}+k_6~~~~~\mbox{for any}~~\varepsilon\in(0,\varepsilon_{T_0}).}\\ \end{array} \label{cz2.57ssdd1hhhhh51lllllkkllccvvhsddddddfdfccvvhjjjkkhhggjjlsdddlll}$$ Moreover, we derive from Definition \[aaalemmaddffffdssfffgg4sssdddd5630\] and Lemmas \[ssdddlemmddddaddffffdfffgg4sssdddd5630\], \[11aaalemmaddffffdsddfffffgg4sssdddd5630\] and \[aaalemmaddffffsddddfffgg4sssdddd5630\], there is $k_7> 0$ such that $$\begin{array}{rl} &\disp k_1\int_{T_0}^{T_0+2}\|\mathcal{P}\xi(m_\varepsilon(\cdot,\tau)-\hat{m}+n_\varepsilon(\cdot,\tau)-\hat{n})\nabla\phi\|_{L^s(\Omega)}^s +k_1\int_{T_0}^{T_0+2}\|\mathcal{P}\xi' u_\varepsilon\|_{L^s(\Omega)}^s\\ \leq&\disp{ k_7,}\\ \end{array} \label{cz2.571hhhhh51lllllccvvhsdddddfccvvhjjjkkhhgssdgjjlsdddlll}$$ so that invoking and we can estimate $$\begin{array}{rl} \disp\int_{T_0}^{T_0+2}\|v_t\|_{L^s(\Omega)}^s+\int_{T_0}^{T_0+2}\|D^2v\|_{L^s(\Omega)}^s \leq&\disp{k_8.}\\ \end{array} \label{cz2.571hhhhh51lllllccddfffvvhsdddddfccvvhjjjkkhhggjjlsdddlll}$$ Therefore, by the definition of $\xi$, for any $s > 1$, there exist positive constants $C $ and $T$ such that for any $t > T$ there is $\varepsilon_0 > 0$ satisfying that for any $\varepsilon\in (0,\varepsilon_0)$ $$\|u_\varepsilon(\cdot,t)\|_{L^{s}((t,t+1);W^{2,s}(\Omega))} +\|u_{\varepsilon t}(\cdot,t)\|_{L^{s}(\Omega\times(t,t+1))}\leq C ~~\mbox{for all}~~ t\geq T, \label{zjscz2.5297x9dddjkkkkkd630111kkhhffrddroojj}$$ which in view of a known embedding result ([@AmannAmannmo1216]) implies that for all $ t_0 > 0$, we can find $\theta_1\in (0, 1)$ and $C_{10}$ such that $$\|u_\varepsilon(\cdot,t)\|_{C^{1+\theta_1,\theta_1}(\bar{\Omega}\times[t,t+1])} \leq C_{10} ~~\mbox{for all}~~ t> T. \label{zjscz2.5297x9ddddfggdd630111kkhhffrfffroojj}$$ Likewise, again using the maximal Sobolev regularity estimates and the Gagliardo-Nirenberg inequality, we can claim that – hold by applying Lemmas \[fvfgsdfggfflemma45\], \[lemmaghjffggssddgghhmk4563025xxhjklojjkkk\] and \[aaalemmaddffffsddddfffgg4sssdddd5630\]. Straightforward applications of standard Schauder estimates for the Stokes evolution equation and the heat equation, respectively, finally yield eventual smoothness of the solution $(n_\varepsilon, c_\varepsilon, m_\varepsilon,u_\varepsilon)$. \[lemma45630hhuujjsdfffggguuyy\] Let $\alpha>0$. Then one can find $\mu\in(0, 1)$ and $T>0$ such that for some $C > 0$ $$\|u_\varepsilon(\cdot,t)\|_{C^{2+\mu,1+\frac{\mu}{2}}(\bar{\Omega}\times[t,t+1];\mathbb{R}^3)} \leq C ~~\mbox{for all}~~ t>T, \label{222zjscz2.5297x9630111kkhhffrroojj}$$ $$\|c_\varepsilon(\cdot,t)\|_{C^{2+\mu,1+\frac{\mu}{2}}(\bar{\Omega}\times[t,t+1])} \leq C ~~\mbox{for all}~~ t>T \label{222zjscz2.5297x9630111kkhhiioo}$$ as well as $$\|m_\varepsilon(\cdot,t)\|_{C^{2+\mu,1+\frac{\mu}{2}}(\bar{\Omega}\times[t,t+1])} \leq C ~~\mbox{for all}~~ t>T \label{222ddhhhzjscz2.5297x9630111kkhhiioo}$$ and $$\|n_\varepsilon(\cdot,t)\|_{C^{2+\mu,1+\frac{\mu}{2}}(\bar{\Omega}\times[t,t+1])} \leq C ~~\mbox{for all}~~ t>T. \label{222zjscz2.5297x96dfgg30111kkhhffrroojj}$$ We first combine Lemma \[lemma45630hhuujjuuyy\] to infer the existence of $\alpha_1\in(0,1)$, $T_1 > 0$ and $C_1 > 0$ such that for all $t>T_1$, $$\begin{array}{rl} &\|u_\varepsilon\cdot \nabla c_\varepsilon\|_{C^{\alpha_1,\frac{\alpha_1}{2}}(\bar{\Omega}\times[t,t+1])}+\|u_\varepsilon\cdot \nabla m_\varepsilon\|_{C^{\alpha_1,\frac{\alpha_1}{2}}(\bar{\Omega}\times[t,t+1])}\\ &+\|n_\varepsilon m_\varepsilon\|_{C^{\alpha_1,\frac{\alpha_1}{2}}(\bar{\Omega}\times[t,t+1])}+\| m_\varepsilon\|_{C^{\alpha_1,\frac{\alpha_1}{2}}(\bar{\Omega}\times[t,t+1])}\\ \leq &C_1. \end{array}\label{222zjscz2.52ssdd97x9630111ffggkkhhffrroojj}$$ Standard parabolic Schauder estimates applied to the second and third equation in ([@Ladyzenskajaggk7101]) thus provide $C_2 > 0$ fulfilling $$\|c_\varepsilon\|_{C^{2+\alpha_1,1+\frac{\alpha_1}{2}}(\bar{\Omega}\times[t,t+1])}+\| m_\varepsilon\|_{C^{2+\alpha_1,1+\frac{\alpha_1}{2}}(\bar{\Omega}\times[t,t+1])}\leq C_2~~~\mbox{for all}~~ t > T_1 + 1. \label{222zjscz2.52ssdd97x9630111kddffkhhffrroojj}$$ According to Lemma \[lemma45630hhuujjuuyy\], it is possible to fix $\alpha_2\in (0,1)$, $T_2> 0$ and $C_3> 0$ such that $$\|n_\varepsilon\|_{C^{1+\alpha_2,\frac{\alpha_2}{2}}(\bar{\Omega}\times[t,t+1])}+\| u_\varepsilon\|_{C^{1+\alpha_2,\frac{\alpha_2}{2}}(\bar{\Omega}\times[t,t+1];\mathbb{R}^3)}\leq C_3~~~\mbox{for all}~~ t > T_2. \label{222zjscz2.52ssdd97x9630111kddffkhhffrssddroojj}$$ We next set $T:= T_2+1$ and let $t_0 > T$ be given. Then with $\xi_{t_0}$ taken from Definition \[aaalemmaddffffdssfffgg4sssdddd5630\], we again use that $v(\cdot,t) := \xi_{t_0}u_\varepsilon(\cdot,t), (x\in\Omega,t> t_0-1)$, is a solution of $$\left\{\begin{array}{ll} v_t-\Delta v=h~~~~x\in\Omega, t>t_0-1,\\ \disp{v(x,t_0-1)=0},\quad x\in \Omega,\\ \end{array}\right. \label{222zjscz2.52ssdd9ddff7x9630111kddffkhhffrssddroojj}$$ where $$h_\varepsilon=-\kappa(Y_\varepsilon u_\varepsilon \cdot \nabla)v+\nabla (\xi P_\varepsilon)+\xi(n_\varepsilon+m_\varepsilon)\nabla \phi+\xi' u_\varepsilon.$$ Now from and the smoothness of $\xi$ we readily obtain $\alpha_3\in (0,1)$ and $C_4> 0$ fulfilling $$\|h_\varepsilon\|_{C^{\alpha_3,\frac{\alpha_3}{2}}(\bar{\Omega}\times[t_0-1,t_0+1];\mathbb{R}^3)}\leq C_4, \label{222zjscz2.52ssdd97x9dddd630111kddffkhhffrssddroojj}$$ so that regularity estimates from Schauder theory for the Stokes evolution equation ([@SolonnikovSolonnikov1216]) ensure that possesses a classical solution $\bar{v}\in {C^{2+\alpha_3,1+\frac{\alpha_3}{2}}(\bar{\Omega}\times[t_0-1,t_0+1])}$ satisfying $$\|\bar{v}\|_{C^{2+\alpha_3,1+\frac{\alpha_3}{2}}(\bar{\Omega}\times[t_0-1,t_0+1];\mathbb{R}^3)}\leq C_5 \label{222zjscz2.52ssdd97x9dddd63011ssdd1kddffkhhffrssddroojj}$$ with some $C_5 > 0$ which is independent of $t_0$. This combined with the uniqueness property of , one can prove $$\|u_\varepsilon\|_{C^{2+\alpha_3,1+\frac{\alpha_3}{2}}(\bar{\Omega}\times[t,t+1];\mathbb{R}^3)}\leq C_6. \label{222zjscz2.52ssdd9ddff7x9dddd63011ssdd1kddffkhhffrssddroojj}$$ Again relying on Lemma \[lemma45630hhuujjuuyy\], this in turn warrants that for some $\alpha_4\in(0,1), T_4> 0$ and $C_7> 0$ such that for all $t>T_4$ $$\|\nabla\cdot(n_{\varepsilon}S_\varepsilon(x, n_{\varepsilon}, c_{\varepsilon})\nabla c_{\varepsilon})\|_{C^{\alpha_4,\frac{\alpha_4}{2}}(\bar{\Omega}\times[t,t+1])}+\|u_{\varepsilon}\cdot\nabla n_{\varepsilon}\|_{C^{\alpha_4,\frac{\alpha_4}{2}}(\bar{\Omega}\times[t,t+1])}+ \|n_{\varepsilon}m_{\varepsilon}\|_{C^{\alpha_4,\frac{\alpha_4}{2}}(\bar{\Omega}\times[t,t+1])}\leq C_7, \label{222zjscz2.52ssdd9ddff7x9ddkklldd63011ssdd1kddffkhhffrssddroojj}$$ which along with the Schauder theory says establishes $$\|n_\varepsilon\|_{C^{2+\alpha_4,1+\frac{\alpha_4}{2}}(\bar{\Omega}\times[t,t+1])}\leq C_8. \label{222zjscz2.52ssdd9ddff7x9dsddddd63011ssdd1kddffkhhffrssddroojj}$$ Finally, choose $T=\max\{T_1,T_1,T_2,T_3,T_4\}$ and $\mu=\min\{\alpha_1,\alpha_2,\alpha_3,\alpha_4\}$, then , , imply –. Having found uniform Hölder bounds on $n_\varepsilon, c_\varepsilon, m_\varepsilon$ and $u_\varepsilon$ for $\varepsilon> 0$ in the previous three lemmas (see Lemmas \[lemma45630hhuujjuuyy\] and \[lemma45630hhuujjsdfffggguuyy\]), also $n, c,m$ and $u$ share this regularity and these bounds. \[lemma45630223\] Assume that $\alpha>0$. There exist $\theta\in (0,1)$ as well as $T_0 > 0$, $(\varepsilon_j)_{j\in \mathbb{N}}\subset (0, 1)$ of the sequence from Lemma \[lemma45hyuuuj630223\] such that for any $t>T_0 $ $$\left\{\begin{array}{ll} n\in C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[t,t+1]),\\ c\in C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[t,t+1]),\\ m\in C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[t,t+1]),\\ u\in C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[t,t+1];\mathbb{R}^3),\\ \end{array}\right.\label{1.ffhhh1hhhjjkdffggdfghyuisda}$$ that $\varepsilon_j\searrow 0$ as $j\rightarrow\infty$ and $$\left\{\begin{array}{ll} n_\varepsilon\rightarrow n~~\in C^{1+\theta,\frac{\theta}{2}}(\bar{\Omega}\times[t,t+1]),\\ c_\varepsilon\rightarrow c~~\in C^{1+\theta,\frac{\theta}{2}}(\bar{\Omega}\times[t,t+1]),\\ m_\varepsilon\rightarrow m~~\in C^{1+\theta,\frac{\theta}{2}}(\bar{\Omega}\times[t,t+1]),\\ u_\varepsilon\rightarrow u~~\in C^{1+\theta,\frac{\theta}{2}}(\bar{\Omega}\times[t,t+1];\mathbb{R}^3)\\ \end{array}\right. and\label{1.ffgghhhhh1dffggdfghyuisda}$$ as $\varepsilon=\varepsilon_j\searrow 0$. Moreover, there is $C > 0$ such that $$\|c(\cdot,t)\|_{C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[t,t+1])}+ \|m\cdot,t)\|_{C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[t,t+1])}\leq C ~~\mbox{for all}~~ t>T_0 \label{222zjscz2.5297x9630111kkhhfsddddfrroojj}$$ as well as $$\|n(\cdot,t)\|_{C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[t,t+1])}+\|u(\cdot,t)\|_{C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[t,t+1]);\mathbb{R}^3)}\leq C ~~\mbox{for all}~~ t>T_0. \label{sss222zjscz2.5297x9sdddd630111kkhhfsddddfrroojj}$$ In conjunction with Lemmas \[lemma45630hhuujjsdfffggguuyy\] and \[lemma45hyuuuj630223\] and the standard compactness arguments (see [@Simon]), we can thus find a sequence $(\varepsilon_j)_{j\in \mathbb{N}}\subset (0, 1)$ such that $\varepsilon_j\searrow 0$ as $j\rightarrow\infty$, and such that – hold. The proof of Lemma \[lemma45630223\] is completed. \[sssslemma45ssddddff630hhuujjsdfffggguuyy\] Let $\alpha>0$. Then one can find $\theta\in(0, 1)$ and $T>0$ such that $$\|c(\cdot,t)\|_{C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[T,\infty))} \leq C \label{222zjscffgggz2.5fff297x9630111kkhhffrroojj}$$ as well as $$\|u(\cdot,t)\|_{C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[T,\infty);\mathbb{R}^3)} \leq C \label{222zjscffgggz2.5sdddff297x9630111kkhhffrroojj}$$ and $$\|n(\cdot,t)\|_{C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[T,\infty))} + \|m(\cdot,t)\|_{C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[T,\infty))} \leq C. \label{222zjscz2ggggg.5297x9630111kkhhiioo}$$ Let $g := -\xi c+m\xi -\xi u\cdot\nabla c+c\xi'$, where $\xi:=\xi_{T_0}$ is given by [Definition]{} \[aaalemmaddffffdssfffgg4sssdddd5630\] and $T_0$ is same as the previous lemmas. Then we consider the following problem $$\left\{\begin{array}{ll} \tilde{c}_t-\Delta \tilde{c}=g~~~~x\in\Omega, t>T_0,\\ \disp{\tilde{c}(T_0)=0},\quad x\in \Omega,\\ \disp{\frac{\partial\tilde{c}}{\partial \nu}=0},\quad x\in \partial\Omega,\\ \end{array}\right. \label{222zjscz2.52ssdd9dssddddff7x9630111kddffkhhffssdddrssddroojj}$$ In view of Lemma \[lemma45630223\] and [Definition]{} \[aaalemmaddffffdssfffgg4sssdddd5630\], we drive that $$g~~\mbox{is bounded in}~~ C^{\theta} (\bar{\Omega}\times(T, \infty)),$$ so that, regularity estimates from Schauder theory for the parabolic equation (see e.g. III.5.1 of [@Ladyzenskajaggk7101]) ensure that problem admits a unique solution $\tilde{c}\in C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[T_0+1,\infty)).$ This combined with the property of $\xi$ implies that $$c\in C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[T_0+1,\infty)). \label{222zjscz2.52ssdd9dssddddff7x9630111kddffkhhffssddddffdrssddroojj}$$ Applying the same argument one can derive the third equation of that $$m\in C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[T_0+1,\infty)). \label{222zjscz2.52ssdd9dssddddff7x9630111kddffkhhffssdddrssddsdfgghhroojj}$$ Finally, employing almost exactly the same arguments as in the proof of Lemma \[lemma45630hhuujjsdfffggguuyy\] (the minor necessary changes are left as an easy exercise to the reader), and taking advantage of , we conclude that $$u\in C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[T_0+1,\infty);\mathbb{R}^3) \label{222zjscz2.52ssdd9dssddddff7x9630111kddddffffkhhffssdddrssddsdfgghhroojj}$$ and $$n\in C^{2+\theta,1+\frac{\theta}{2}}(\bar{\Omega}\times[T_0+1,\infty)), \label{222zjscz2.52ssdd9dssddddff7x9630111kddddffffkhhffssdddrssddssddffdfgghhroojj}$$ whence combining the result of with completes the proof. On the basis of the eventual uniform continuity properties implied by the estimates in this section (see Lemma \[sssslemma45ssddddff630hhuujjsdfffggguuyy\]), by using the interpolation inequality, we can now turn the weak stabilization properties of $n,c,m$ and $u$ from Lemmas \[ssdddlemmddddaddffffdfffgg4sssdddd5630\]–\[11aaalemmaddffffdsddfffffgg4sssdddd5630\] into convergence with regard to the norm in $L^\infty(\Omega)$. \[lemma4dd5630hhuujjuuyy\] Let $\alpha>0$. The solution $(n,c,m,u)$ of constructed in Lemma \[lemma45hyuuuj630223\] satisfies $$n(\cdot,t)\rightarrow \hat{n}, m(\cdot,t)\rightarrow \hat{m}~~\mbox{as well as } ~~~c(\cdot,t)\rightarrow \hat{m}~~\mbox{and}~~~u(\cdot,t)\rightarrow0 ~~\mbox{in}~~~L^\infty(\Omega),$$ where $\hat{n}=\frac{1}{|\Omega|}\{\int_{\Omega}n_0-\int_{\Omega}m_0\}_{+}$ and $\hat{m}=\frac{1}{|\Omega|}\{\int_{\Omega}m_0 -\int_{\Omega}n_0\}_{+}$. Firsly, due to Lemmas \[ssdddlemmddddaddffffdfffgg4sssdddd5630\]–\[11aaalemmaddffffdsddfffffgg4sssdddd5630\], we derive from Lemma \[sssslemma45ssddddff630hhuujjsdfffggguuyy\] that $$n(t)\rightarrow \hat{n}, c(t)\rightarrow \hat{m}, m(t)\rightarrow \hat{m}~~\mbox{and}~~u(t)\rightarrow0~~~\mbox{in}~~~L^2(\Omega), \label{aahhxxcdfvhhhvssssssssdsssjjdfffddffssllllddcssdz2.ssdd5}$$ where $\hat{m}$ and $\hat{n}$ are given by and , respectively. Next, due to Lemma \[lemma45630223\], one can obtain there exist positive constants $\kappa_1$ and $T$ such that for all $t>T$ $$\|n (\cdot,t)\|_{C^{2+\theta}(\bar{\Omega})} +\|c (\cdot,t)\|_{C^{2+\theta}(\bar{\Omega})}+\|m (\cdot,t)\|_{C^{2+\theta}(\bar{\Omega})}+\|u (\cdot,t)\|_{C^{2+\theta}(\bar{\Omega})} \leq \kappa_1. \label{aahhxxcdfvhhhvssssssssdsssjjdfffddffssddcssdz2.ssdd5}$$ Therefore, for any $\eta > 0$, we may use the compactness of the first of the embeddings $C^{2+\theta}(\bar{\Omega}) \hookrightarrow\hookrightarrow L^\infty (\Omega)\hookrightarrow L^2 (\Omega)$ to fix, through an associated Ehrling lemma, a constant $\kappa_{2} > 0$ such that $$\|n(\cdot,t)-\hat{n}\|_{L^{\infty}(\Omega)}\leq\frac{\eta}{2\kappa_{1} }\|n(\cdot,t)-\hat{n}\|_{C^{2+\theta}(\bar{\Omega})}+\kappa_{2}\|n(\cdot,t)-\hat{n}\|_{L^{2}(\Omega)} \label{233ddxcvbbggdddddddfghhdfgcz2vv.5ghju4ss8cfg9ddsddddffff24ssdddghddfgggyddfggusdffji}$$ $$\|c(\cdot,t)-\hat{m}\|_{L^{\infty}(\Omega)}\leq\frac{\eta}{2\kappa_{1} }\|c(\cdot,t)-\hat{m}\|_{C^{2+\theta}(\bar{\Omega})}+\kappa_{2}\|c(\cdot,t)-\hat{m}\|_{L^{2}(\Omega)} \label{ghhddxcvbbggdddddddfghhdfgcz2vv.5ghju4ss8cfg9ddsddddffff24ssdddghddfgggyddfggusdffji}$$ as well as $$\|m(\cdot,t)-\hat{m}\|_{L^{\infty}(\Omega)}\leq\frac{\eta}{2\kappa_{1} }\|m(\cdot,t)-\hat{m}\|_{C^{2+\theta}(\bar{\Omega})}+\kappa_{2}\|m(\cdot,t)-\hat{m}\|_{L^{2}(\Omega)} \label{klllddxcvbbggdddddddfghhdfgcz2vv.5ghju4ss8cfg9ddsddddffff24ssdddghddfgggyddfggusdffji}$$ and $$\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\leq\frac{\eta}{2\kappa_{1} }\|u(\cdot,t)\|_{C^{2+\theta}(\bar{\Omega})}+\kappa_{2}\|u(\cdot,t)\|_{L^{2}(\Omega)}. \label{dllkoooopdxcvbbggdddddddfghhdfgcz2vv.5ghju4ss8cfg9ddsddddffff24ssdddghddfgggyddfggusdffji}$$ Now due to , we may choose $t_0 > \max\{1,T\}$ large enough such that for all $t>t_0$, $$\|n(\cdot,t)-\hat{n}\|_{L^2(\Omega)}+\|c(\cdot,t)-\hat{m}\|_{L^2(\Omega)}+\|m(\cdot,t)-\hat{m}\|_{L^2(\Omega)}+\|u(\cdot,t)\|_{L^2(\Omega)}<\frac{\eta}{2\kappa_{2}}. \label{234ddxcvbbggdddddddfghhdfgcz2vv.5ghju4ss8cfg9ddsddddffff24ghddfgggyddfggusdffji}$$ Combined with –, this shows that in fact $$\begin{array}{rl}\|n(\cdot,t)-\hat{n}\|_{L^{\infty}(\Omega)}\leq&\disp{\frac{\eta}{2\kappa_{1} }\|n(\cdot,t)-\hat{n}\|_{C^{2+\theta}(\bar{\Omega})}+\kappa_{2}\|n(\cdot,t)-\hat{n}\|_{L^{2}(\Omega)}}\\ <&\disp{\frac{\eta}{2\kappa_{1} }\kappa_{1}+\kappa_{2}\frac{\eta}{2\kappa_{2} }}\\ =&\eta~~~\mbox{for all}~~ t > t _0,\\ \end{array}$$ $$\begin{array}{rl}\|c(\cdot,t)-\hat{m}\|_{L^{\infty}(\Omega)}\leq&\disp{\frac{\eta}{2\kappa_{1} }\|c(\cdot,t)-\hat{m}\|_{C^{2+\theta}(\bar{\Omega})}+\kappa_{2}\|c(\cdot,t)-\hat{m}\|_{L^{2}(\Omega)}}\\ <&\disp{\frac{\eta}{2\kappa_{1} }\kappa_{1}+\kappa_{2}\frac{\eta}{2\kappa_{2} }}\\ =&\eta~~~\mbox{for all}~~ t > t _0\\ \end{array}$$ as well as $$\begin{array}{rl}\|m(\cdot,t)-\hat{m}\|_{L^{\infty}(\Omega)}\leq&\disp{\frac{\eta}{2\kappa_{1} }\|m(\cdot,t)-\hat{m}\|_{C^{2+\theta}(\bar{\Omega})}+\kappa_{2}\|m(\cdot,t)-\hat{m}\|_{L^{2}(\Omega)}}\\ <&\disp{\frac{\eta}{2\kappa_{1} }\kappa_{1}+\kappa_{2}\frac{\eta}{2\kappa_{2} }}\\ =&\eta~~~\mbox{for all}~~ t > t _0\\ \end{array}$$ and $$\begin{array}{rl}\|u(\cdot,t)\|_{L^{\infty}(\Omega)}\leq&\disp{\frac{\eta}{2\kappa_{1} }\|u(\cdot,t)\|_{C^{2+\theta}(\bar{\Omega})}+\kappa_{2}\|u(\cdot,t)\|_{L^{2}(\Omega)}}\\ <&\disp{\frac{\eta}{2\kappa_{1} }\kappa_{1}+\kappa_{2}\frac{\eta}{2\kappa_{2} }}\\ =&\eta~~~\mbox{for all}~~ t > t _0,\\ \end{array}$$ which together with the fact that $\eta> 0$ was arbitrary implies the claimed estimates. In order to prove Theorem \[thaaaeorem3\], we now only have to collect the results prepared during this section: [**Proof of Theorem \[thaaaeorem3\].**]{} Combining Lemmas \[lemma45630223\]–\[lemma4dd5630hhuujjuuyy\] this convergence statement results immediately. [**Acknowledgement**]{}: This work is partially supported by the National Natural Science Foundation of China (No. 11601215), Shandong Provincial Science Foundation for Outstanding Youth (No. ZR2018JL005) and Project funded by China Postdoctoral Science Foundation (No. 2019M650927, 2019T120168). [00]{} H. Amann, *Compact embeddings of vector-valued Sobolev and Besov spaces*, Glasnik Mat., 35(55)(2000), 161–177. N. Bellomo, A. Belloquid, Y. Tao, M. Winkler, *Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues*, Math. Models Methods Appl. Sci., 25(9)(2015), 1663–1763. X. Cao, J. Lankeit, *Global classical small-Data solutions for a three-dimensional Chemotaxis Navier-Stokes System involving Matrix-Valued Sensitivities*, Cal. Var. Partial Differential Equations, 55(4)(2016), 107-123. M. Chae, K. Kang, J. Lee, *Global Existence and temporal decay in Keller–Segel models coupled to fluid equations*, Comm. Part. Diff. Eqns., 39(2014), 1205–1235. T. Cieślak, C. Stinner, *Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions,* J. Diff. Eqns., 252(2012), 5832–5851. T. Cieślak, C. Stinner, *New critical exponents in a fully parabolic quasilinear Keller–Segel system and applications to volume filling models*, J. Diff. Eqns., 258(2015), 2080–2113. T. Cieślak, M. Winkler, *Finite-time blow-up in a quasilinear system of chemotaxis,* Nonlinearity, 21(2008), 1057–1076. R. Duan, A. Lorz, P. A. Markowich, *Global solutions to the coupled chemotaxis-fluid equations*, Comm. Part. Diff. Eqns., 35 (2010), 1635–1673. E. E. Espejo, T. Suzuki, *Reaction enhancement by chemotaxis*, Nonlinear Anal. Real World Appl., 35(2017), 102–131. E. E. Espejo, M. Winkler, *Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier-Stokes system modeling coral fertilization*, Nonlinearity, 31(2018), 1227-1259. Y. Giga, H. Sohr, *Abstract $L^p$ estimate for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains*, J. Funct. Anal., 102(1991), 72–94. T. Hillen, K. Painter, *A user’s guide to PDE models for chemotaxis,* J. Math. Biol., 58(2009), 183–217. D. Horstmann, M. Winkler, *Boundedness vs. blow-up in a chemotaxis system*, J. Diff. Eqns, 215(2005), 52–107. S. Ishida, *Global existence and boundedness for chemotaxis–Navier–Stokes system with position-dependent sensitivity in $2d$ bounded domains*, Discrete Contin. Dyn. Syst. Ser. A, 32(2015), 3463–3482. S. Ishida, K. Seki and T, Yokota, *Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains*, J. Diff. Eqns., 256 (2014), 2993–3010. Y. Ke, J. Zheng, *An optimal result for global existence in a three-dimensional Keller–Segel–Navier–Stokes system involving tensor-valued sensitivity with saturation*, Calculus of Variations and Partial Diff. Eqns., https://doi.org/10.1007/s00526-019-1568-2. E. Keller, L. Segel, *Model for chemotaxis*, J. Theor. Biol., 30(1970), 225–234. A. Kiselev, L. Ryzhik, *Biomixing by chemotaxis and efficiency of biological reactions: the critical reaction case*, J. Math. Phys., 53(11):115609, 9, 2012. A. Kiselev and L. Ryzhik, *Biomixing by chemotaxis and enhancement of biological reactions*, Comm. Partial Diff. Eqns.,37(2012), 298–318. O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Uraĺceva, *Linear and quasi-Linear Equations of Parabolic Type*, Amer. Math. Soc. Transl., Providence, RI, 1968. J. Leray, *Sur le mouvement dun liquide visqueus amplissant lespace*, Acta Math., 63(1934), 193–248. J. Li, P. Pang, Y. Wang, *Global boundedness and decay property of a three-dimensional Keller–Segel–Stokes system modeling coral fertilization*, Nonlinearity, In press. T. Li, A. Suen, M. Winkler and C. Xue, *Gobal small-data solutions in a chemotaxis system with rotation*, Math. Mod. Meth. Appl. Sci., 25(2015), 721–747. J. Liu, Y. Wang, *Boundedness and decay property in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation*, J. Diff. Eqns., 261(2)(2016), 967–999. J.-G. Liu, A. Lorz, *A coupled chemotaxis–fluid model: global existence*, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28(5)(2011), 643–652. A. Lorz, *Coupled chemotaxis fluid equations*, Math. Models Methods Appl. Sci., 20(2010), 987–1004. K. Painter, T. Hillen, *Volume-filling and quorum-sensing in models for chemosensitive movement*, Can. Appl. Math. Q. 10(2002), 501–543. J. Simon, *Compact sets in the space $L^{p}(O, T;B)$*, Annali di Matematica Pura ed Applicata, 146(1)(1986), 65–96. H. Sohr, *The Navier–Stokes equations, An elementary functional analytic approach*, Birkhäuser Verlag, Basel (2001). V. A. Solonnikov, *Schauder estimates for the evolutionary generalized Stokes problem, Non- linear equations and spectral theory*, Amer. Math. Soc. Transl. Ser. 2, vol. 220, Amer. Math. Soc., Providence, RI, 2007, pp. 165–200. Y. Tao, M. Winkler, *Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity*, J. Diff. Eqns., 252(2012), 692–715. Y. Tao, M. Winkler, *Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis–fluid system*, Z. Angew. Math. Phys., 66(2015), 2555–2573. I. Tuval, L. Cisneros, C. Dombrowski, et al., *Bacterial swimming and oxygen transport near contact lines*, Proc. Natl. Acad. Sci. USA, 102(2005), 2277–2282. L. Wang, C. Mu, P. Zheng, *On a quasilinear parabolic–elliptic chemotaxis system with logistic source*, J. Diff. Eqns., 256(2014), 1847–1872. L. Wang, Y. Li, C. Mu, *Boundedness in a parabolic–parabolic quasilinear chemotaxis system with logistic source*, Discrete Contin. Dyn. Syst. Ser. A., 34(2014), 789–802. Y. Wang, X. Cao, *Global classical solutions of a $3d$ chemotaxis–Stokes system with rotation*, Discrete Contin. Dyn. Syst. Ser. B, 20(2015), 3235–3254. Y. Wang, *Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity*, Math. Models Methods Appl. Sci., (27)(14)(2017), 2745–2780. Y. Wang, M. Winkler, Z. Xiang, *Global classical solutions in a two-dimensional chemotaxis–Navier–Stokes system with subcritical sensitivity*, Annali della Scuola Normale Superiore di Pisa–Classe di Scienze. XVIII, (2018), 2036–2145. Y. Wang, Z. Xiang, *Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation*, J. Diff. Eqns., 259(2015), 7578–7609. Y. Wang, Z. Xiang, *Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: the 3D case*, J. Differ. Eqns. 261(2016), 4944–4973. M. Wiegner, *The Navier-Stokes equations〞a neverending challenge*? Jahresber. Deutsch. Math.-Verein., 101(1)(1999), 1–25. M. Winkler, *Does a volume-filling effect always prevent chemotactic collapse*, Math. Methods Appl. Sci., 33(2010), 12–24. M. Winkler, *Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model*, J. Diff. Eqns., 248(2010), 2889–2905. M. Winkler, *Absence of collapse in a parabolic chemotaxis system with signal-dependent sensiticity*, Math. Nachr., 283(11)(2010), 1664–1673. M. Winkler, *Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source*, Comm. Partial Diff. Eqns., 35(2010), 1516–1537. M. Winkler, *Global large-data solutions in a chemotaxis–(Navier–)Stokes system modeling cellular swimming in fluid drops*, Comm. Partial Diff. Eqns., 37(2012), 319–351. M. Winkler, *Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system*, J. Math. Pures Appl., 100(2013), 748–767. M. Winkler, *Stabilization in a two-dimensional chemotaxis–Navier–Stokes system*, Arch. Ration. Mech. Anal., 211(2014), 455–487. M. Winkler, *Boundedness and large time behavior in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion and general sensitivity*, Calculus of Variations and Partial Diff. Eqns., (54)(2015), 3789–3828. M. Winkler, *Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system*, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33(5)(2016), 1329—1352. M. Winkler, *How far do oxytaxis-driven forces influence regularity in the Navier-Stokes system*? Trans. Am. Math. Soc., 369(2017),3067–125. M. Winkler, *Does fluid interaction affect regularity in the three-dimensional Keller-Segel System with saturated sensitivity?*, J. Math. Fluid Mechanics, 20(4)(2018), 1889–1909. M. Winkler, *A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization*, J. Func. Anal., 276(5)(2019), 1339–1401. M. Winkler, K. C. Djie, *Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect*, Nonlinear Anal. TMA., 72(2010), 1044–1064. C. Xue, *Macroscopic equations for bacterial chemotaxis: integration of detailed biochemistry of cell signaling*, J. Math. Biol., 70(2015), 1–44. C. Xue, H. G. Othmer, *Multiscale models of taxis-driven patterning in bacterial populations*, SIAM J. Appl. Math., 70(1)(2009), 133–167. Q. Zhang, X. Zheng, *Global well-posedness for the two-dimensional incompressible chemotaxis–Navier–Stokes equations*, SIAM J. Math. Anal., 46(2014), 3078–3105. J. Zheng, *Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source*, J. Diff. Eqns., 259(1)(2015), 120–140. J. Zheng, *Boundedness in a three-dimensional chemotaxis–fluid system involving tensor-valued sensitivity with saturation*, J. Math. Anal. Appl., 442(1)(2016), 353–375. J. Zheng, *A note on boundedness of solutions to a higher-dimensional quasi-linear chemotaxis system with logistic source*, Zeitsc. Angew. Mathe. Mech., 97(4)(2017), 414–421. J. Zheng, *Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion*, J. Diff. Eqns., 263(5)(2017), 2606–2629. J. Zheng, *An optimal result for global classical and bounded solutions in a two-dimensional Keller-Segel-Navier-Stokes system with sensitivity*, arXiv:1903.01033, 2019. J. Zheng, *Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization*, arXiv: 1905.08647, 2019. J. Zheng, *An optimal result for global existence and boundedness in a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion*, J. Diff. Eqns., (4)(267)(2019), 2385–2415. J. Zheng, Y. Ke, *Large time behavior of solutions to a fully parabolic chemotaxis–haptotaxis model in $N$ dimensions*, J. Diff. Eqns., 266(2019), 1969–2018. J. Zheng, Y. Wang, *Boundedness and decay behavior in a higher-dimensional quasilinear chemotaxis system with nonlinear logistic source*, Compu. Math. Appl., 72(10)(2016), 2604–2619. J. Zheng, Y. Wang, *A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant*, Discrete Contin. Dyn. Syst. Ser. B, 22(2)(2017), 669–686. [^1]: Corresponding author. E-mail address: zhengjiashan2008@163.com (J.Zheng)

0.6cm \#1 \#1\#2 UCB-PTH-12/07\ 1.3cm [**The Static Quantum Multiverse**]{} 0.7cm [Yasunori Nomura]{} 0.4cm [*Berkeley Center for Theoretical Physics, Department of Physics,\ University of California, Berkeley, CA 94720, USA*]{} 0.1cm [*Theoretical Physics Group, Lawrence Berkeley National Laboratory, CA 94720, USA*]{} 0.8cm Introduction {#sec:intro} ============ The goal of fundamental physics is to find a prescription in which (potentially) testable predictions can be made and, through it, to learn how nature works at the most fundamental level. In the present way physics is formulated, this can be done in three steps: - [*“Theory”*]{} — We must specify the fundamental structure of the theory, which consists of the following two parts: - [*Kinematics*]{} — We must understand what is a “state” which somehow represents the status of a physical system. We must also understand how it is related to the observed reality. For example, in conventional quantum mechanics a state is a ray in Hilbert space, which is related to reality through the Born rule, while in classical mechanics a state is a point in classical phase space (so is directly observable). - [*Dynamics*]{} — We must know a (set of) fundamental law(s) the states obey. In quantum mechanics it is the Schrödinger equation, $i \frac{d}{dt} \ket{\Psi(t)} = H \ket{\Psi(t)}$, while in classical mechanics it is the Newton equation, $m\, \ddot{\bf x}(t) = {\bf F}$. - [*“System”*]{} — We then need to specify a system we consider, which again consists of two parts: - [*Kinematics*]{} — We need to know the kinematical structure of the system. In quantum mechanics this corresponds to specifying the Hilbert space, which is characterized by its dimension and operators acting on its elements. In classical mechanics, it is given by the dimension of the phase space. - [*Dynamics*]{} — We also need to specify dynamics of the system. In the examples in (i-2), we need to give the forms of $H$ and ${\bf F}$, respectively. - [*“Selection Conditions”*]{} — Even if (i) and (ii) are known, we still need to provide “selection conditions” on a state. Usually, they are given in the form of boundary conditions, for example as the knowledge one already has, e.g. $\ket{\Psi(0)}$ and $\{ {\bf x}(0), \dot{\bf x}(0) \}$, before making predictions on something unknown, e.g. $\ket{\Psi(t)}$ and $\{ {\bf x}(t), \dot{\bf x}(t) \}$ for $t > 0$. To understand the ultimate structure of nature, we would want to do the above in the context of cosmology, and see whether the resulting predictions are consistent with what we observe. In this respect, physics of eternal inflation—which occurs under rather general circumstances [@Guth:1982pn]—has caused tremendous confusions in recent years. A major problem has been the so-called measure problem:  even if we know the initial state and its subsequent evolution, we cannot define (even probabilistic) predictions unambiguously.[^1] This occurs because in eternal inflation anything that can happen will happen infinitely many times, so it apparently leads to arbitrariness in predictions, associated with how these infinities are regularized [@Guth:2000ka]. Such an arbitrariness would prevent us from making well-defined predictions, so it seemed that to define the theory we needed to specify the exact way of regulating spacetime, where the infinities occur. This would be quite uncomfortable, since then the theory requires a specification of a (ad hoc) regularization prescription [*beyond the basic principles of quantum mechanics and relativity*]{}. Recently, a framework that addresses this problem has been proposed in Refs. [@Nomura:2011dt; @Nomura:2011rb], which allows for an intrinsically quantum mechanical treatment of the eternally inflating multiverse (see Ref. [@Nomura-review] for a review directed to a wide audience). In this framework, physics is described in [*a fixed reference (local Lorentz) frame*]{} associated with a fixed reference point $p$, with spacetime existing only within its (stretched) apparent horizon. An essential point is that the principles of quantum mechanics constrain the space of states ${\cal H}_{\rm QG}$ [@Nomura:2011rb] in such a way that the problem of infinity does not arise. Namely, the correct identification of (ii-1) avoids the problem, without changing (i) from that of usual unitary quantum mechanics. A state representing the multiverse $\ket{\Psi(t)}$ “evolves” deterministically and unitarily in ${\cal H}_{\rm QG}$, following the laws of quantum mechanics: $i \frac{d}{dt} \ket{\Psi(t)} = H \ket{\Psi(t)}$. Here, $t$ is an auxiliary parameter introduced to describe the “evolution” of the state, and need not be directly related to physical time we observe. Once the state $\ket{\Psi(t)}$ is known, physical predictions can be obtained through the (extended) Born rule [@Nomura:2011dt; @Nomura:2011rb] without suffering from an infinity or ambiguity. This framework makes it possible that once a boundary condition on the state, e.g. $\ket{\Psi(t)}$ at some $t = t_0$, is given (element (iii)) and the explicit form of $H$ acting on ${\cal H}_{\rm QG}$ is understood, e.g. by studying string theory (element (ii-2)), then unambiguous predictions are obtained for any physical questions one asks. While the framework does not achieve all of (i)–(iii), it does eliminate the ambiguity associated with the measure problem and provides a setting in which the remaining issues can be discussed. In this paper we consider the issue of (iii) in the quantum mechanical framework of the multiverse described above. We take the following hypothesis: $$\ovalbox{Hypothesis I: The laws of quantum mechanics are not violated.} \label{eq:hypo-1}$$ This—in particular the fact that the evolution of a quantum state is deterministic and unitary—implies that the multiverse state exists all the way from $t = -\infty$ to $+\infty$. Namely, the multiverse does not have a beginning or end. (For recent discussions on the beginning of the eternally inflating multiverse, see, e.g., Refs. [@Mithani:2012ii].) There are three potential issues in this picture: - [*Uniqueness*]{} — What is the selection condition imposed on the multiverse state, on which physical predictions will depend? In particular, what is the principle determining it? - [*Well-definedness*]{} — The (extended) Born rule formula in general involves $t$ integrals, which would now run from $-\infty$ to $+\infty$. Will this give well-defined probabilities? - [*Consistency*]{} — Are the resulting predictions consistent with observation? In particular, are they consistent with the observed arrow of time, even if there is no beginning or end? In this paper we argue that consistency with observation excludes the possibility that the selection condition is determined purely in ${\cal H}_{\rm QG}$, without referring to an operator algebra. In particular, this excludes the possibility that the multiverse is in the maximally mixed state in ${\cal H}_{\rm QG}$. We then propose that the multiverse state must satisfy the following simple criterion: $$\ovalbox{Hypothesis II: Physical predictions do not depend on the reference frame one chooses.} \label{eq:hypo-2}$$ We show that this requirement leads to the condition $$\frac{d}{dt}\ket{\Psi(t)} = 0 \qquad\Leftrightarrow\qquad H \ket{\Psi(t)} = 0, \label{eq:static}$$ where we have taken $t$ to be the proper time at $p$; namely, we find that [*the multiverse state must be static!*]{} We will argue that despite its naive appearance, this does not contradict observation, including the fact that we observe that time flows in a definite direction. It simply gives constraints on the structure of $H$, on which we will allow for making arbitrary assumptions, given that its explicit form is not available under current theoretical technology. We will also argue that the hypothesis leads to unique and well-defined predictions for any physical questions, once one knows the explicit form of $H$ (element (ii-2) listed at the beginning). Specifically, any physical question can be phrased in the form: given what we know $A$ about a state, what is the probability for it to be consistent also with $B$? And the relevant probability is given by $$P(B|A) = \frac{\bra{\Psi} {\cal O}_{A \cap B} \ket{\Psi}} {\bra{\Psi} {\cal O}_A \ket{\Psi}}, \label{eq:prob-final}$$ where $\ket{\Psi} \equiv \ket{\Psi(0)}$, and ${\cal O}_X$ is the operator projecting onto states consistent with condition $X$. There are two comments. First, given Hypothesis I, Hypothesis II arises as a consequence of general covariance (and its suitable extension to the quantum regime) if we assume that the multiverse is in a zero-eigenvalue eigenstate of global energy and boost operators. This condition, therefore, provides another, more technical way of stating Hypothesis II. Second, without knowledge of the ultimate structure of $H$ in quantum gravity, the scenario presented here is not the only option available within the framework of Refs. [@Nomura:2011dt; @Nomura:2011rb], although it seems to be the most natural possibility. For example, one might imagine that the multiverse has a “beginning,” and evolves only thereafter. (This violates both Hypotheses I and II.) The framework of Refs. [@Nomura:2011dt; @Nomura:2011rb] itself may still be applied in such a case. The organization of this paper is as follows. In the next section, we review the framework of the quantum multiverse given in Refs. [@Nomura:2011dt; @Nomura:2011rb], and discuss the issue of selection conditions in that context. In Section \[sec:obs\], we reconsider what the arrow of time is. We emphasize that the observed flow of time does not necessarily mean that the state is actually evolving. In Section \[sec:max-mixed\], we explore the possibility that the selection condition is expressed in ${\cal H}_{\rm QG}$ without referring to any quantum operator. We find that this forces the multiverse to be in the maximally mixed state in ${\cal H}_{\rm QG}$, which is observationally excluded. In Section \[sec:stat\], we present our main scenario in which the multiverse state is determined by the two hypotheses described above. We find this implies that the multiverse state must be static, and discuss how it can be realized in the cosmological context. We also see that the scenario arises as a consequence of quantum mechanics and general covariance if we assume that the multiverse is in a zero-eigenvalue eigenstate of global energy and boost operators. In Section \[sec:consistency\], we discuss the consistency of the scenario with observation, specifically the observed arrow of time. In Section \[sec:discuss\], we provide our final discussions. We draw a close analogy of the present scenario with the case of the hydrogen atom, underscoring the intrinsically quantum nature of the scenario. Framework—the Quantum Multiverse {#sec:framework} ================================ In this section we review the framework of Refs. [@Nomura:2011dt; @Nomura:2011rb], describing the quantum multiverse. We also discuss the issue of selection conditions in making predictions within this framework. The Hilbert space {#subsec:Hilbert} ----------------- The framework is based on the principles of quantum mechanics. In particular, we formulate it using Hamiltonian (canonical) quantum mechanics, although the equivalent Lagrangian (path integral) formulation should also be possible. We take the Schrödinger picture throughout. Recall that to do Hamiltonian quantum mechanics, we need to fix all gauge redundancies. Since these redundancies include coordinate transformations in a theory with gravity, states must be defined [*as viewed from a fixed (local Lorentz) reference frame*]{} associated with a fixed reference point $p$. Moreover, to avoid violation of the principles of quantum mechanics, they must represent only spacetime regions within the (stretched) apparent horizons of $p$, as suggested first in the study of black hole physics [@Susskind:1993if]. Together with the states associated with spacetime singularities, these states form the Hilbert space for quantum gravity ${\cal H}_{\rm QG}$. The construction of ${\cal H}_{\rm QG}$ can proceed analogously to the usual Fock space construction in quantum field theory. For a set of fixed semi-classical geometries ${\cal M} = \{ {\cal M}_i \}$ having the same apparent horizon $\partial {\cal M}$, the Hilbert space is given by $${\cal H}_{\cal M} = {\cal H}_{{\cal M}, {\rm bulk}} \otimes {\cal H}_{{\cal M}, {\rm horizon}}, \label{eq:ST-H_M}$$ where ${\cal H}_{{\cal M}, {\rm bulk}}$ and ${\cal H}_{{\cal M}, {\rm horizon}}$ represent Hilbert space factors associated with the degrees of freedom inside and on the horizon $\partial {\cal M}$. The dimensions of these factors are both $\exp({\cal A}_{\partial {\cal M}}/4)$, where ${\cal A}_{\partial {\cal M}}$ is the area of the horizon in Planck units: $${\rm dim}\,{\cal H}_{\cal M} = {\rm dim}\,{\cal H}_{{\cal M}, {\rm bulk}} \times {\rm dim}\,{\cal H}_{{\cal M}, {\rm horizon}} = \exp\left(\frac{{\cal A}_{\partial {\cal M}}}{2}\right), \label{eq:H_M-dimension}$$ consistently with the holographic principle [@'tHooft:1993gx]. The full Hilbert space for dynamical spacetime is then given by the direct sum of the Hilbert spaces for different ${\cal M}$’s $${\cal H} = \bigoplus_{\cal M} {\cal H}_{\cal M}. \label{eq:ST-H}$$ In addition, the complete Hilbert space for quantum gravity must contain “intrinsically quantum mechanical” states, associated with spacetime singularities [@Nomura:2011rb]: $${\cal H}_{\rm QG} = {\cal H} \oplus {\cal H}_{\rm sing}, \label{eq:QG-H}$$ where ${\cal H}_{\rm sing}$ represents the Hilbert space for the singularity states. The evolution of the multiverse state $\ket{\Psi(t)}$, which represents the entire multiverse, is deterministic and unitary in ${\cal H}_{\rm QG}$, but not in ${\cal H}_{\cal M}$ or ${\cal H}$. The dimension of the complete Hilbert space ${\cal H}_{\rm QG}$ is infinite, as the dimensions of Hilbert subspaces associated with stable Minkowski space and spacetime singularities are infinite: $${\rm dim}\,{\cal H}_{\rm Minkowski} = \infty, \qquad {\rm dim}\,{\cal H}_{\rm sing} = \infty. \label{eq:dim-inf}$$ This implies, by the second law of thermodynamics, that a [*generic*]{} multiverse state in ${\cal H}_{\rm QG}$ will evolve at large $t$ into a superposition of terms corresponding to supersymmetric Minkowski space or spacetime singularity: $$\ket{\Psi(t)} \,\,\stackrel{t \rightarrow \infty}{\longrightarrow}\,\, \sum_i a_i(t) \ket{\mbox{supersymmetric Minkowski space $i$}} \,+\, \sum_j b_j(t) \ket{\mbox{singularity state $j$}}, \label{eq:asympt}$$ where we have assumed that the only absolutely stable Minkowski vacua are supersymmetric ones, as suggested by the string landscape picture [@Bousso:2000xa]. Note that an infinite number of states exist only in a Hilbert subspace associated with a spacetime singularity or a Minkowski space [*in which the area of the apparent horizon diverges ${\cal A}_{\partial {\cal M}} = \infty$*]{}. In particular, the number of states associated with a fixed Friedmann-Robertson-Walker (FRW) time in a Minkowski bubble is finite for any finite energy density $\rho$, since the area of the apparent horizon is given by ${\cal A}_{\partial {\cal M}} = 3/2\rho$ (with $\rho$ in Planck units) [@Bousso:2002ju-FRW], so that ${\cal A}_{\partial {\cal M}} < \infty$ for $\rho > 0$. The (extended) Born rule {#subsec:Born} ------------------------ For a given multiverse state $\ket{\Psi(t)}$, physical predictions can be obtained following the rules of quantum mechanics. An important point is that the “time” parameter $t$ here is simply an auxiliary parameter introduced to describe the “evolution” of the state. The physical information is only in [*correlations*]{} between events; specifically, time evolution of a physical quantity $X$ is nothing more than a correlation between $X$ and a quantity that can play the role of time, such as the location of the hands of a clock or the average temperature of the cosmic microwave background in our universe. A particularly useful choice for $t$ is the proper time at $p$, which we will assume for the rest of the paper. Any physical question can then be phrased as: given what we know $A$ about a state, what is the probability for that state to be consistent also with condition $B$? In the context of the multiverse, this probability is given by [@Nomura:2011dt] $$P(B|A) = \frac{\int\!dt \bra{\Psi(0)} U(0,t)\, {\cal O}_{A \cap B}\, U(t,0) \ket{\Psi(0)}} {\int\!dt \bra{\Psi(0)} U(0,t)\, {\cal O}_A\, U(t,0) \ket{\Psi(0)}}, \label{eq:prob}$$ where $U(t_1,t_2) = e^{-iH(t_1-t_2)}$ is the “time evolution” operator with $H$ being the Hamiltonian of the entire system for a fixed “time” parameterization $t$ (here the proper time at $p$), and ${\cal O}_X$ is the operator projecting onto states consistent with condition $X$. Note that since we have already fixed a reference frame, conditions $A$ and $B$ in general must involve specifications of ranges of location and velocity in which a physical object must be with respect to the reference point $p$. As we will discuss in more detail in Section \[sec:obs\], the formula in Eq. (\[eq:prob\]) can be used to answer any physical questions including those about dynamical evolution of a system, despite the fact that conditions $A$ and $B$ both act at the same moment $t$. We therefore base all our discussions on Eq. (\[eq:prob\]) in this paper. (For a different formula that can be used more easily in many practical contexts, see Ref. [@Nomura:2011rb].) The $t$ integrals in the equation run over the entire region under consideration. Suppose, for example, that we know the universe/multiverse is in a particular, e.g. eternally inflating, state $\ket{\Psi(0)}$ at $t = 0$, and want to predict what happens in $t > 0$. In this case, the integrals must be taken from $t = 0$ to $\infty$, since condition $A$ may be satisfied at any value of $t > 0$ in some component of $\ket{\Psi(t)}$. Note that despite the integrals running to $\infty$ the resulting probability is well-defined, because Eq. (\[eq:asympt\]) prohibits an event from occurring infinitely many times with a finite probability, which would cause divergences. The issue of selection conditions {#subsec:s-c} --------------------------------- What kind of predictions does the framework described above allow us to make? While the framework addresses the issues of infinity and the ambiguity associated with it (i.e. the measure problem as defined here), it is certainly not complete. In particular, ... - [*“Unspecified System”*]{} — We did not identify the system [ *explicitly*]{}. Specifically, the complete theory of quantum gravity is not known, so that we do not know the form of $H$, especially the part acting on the horizon degrees of freedom. This particular issue can be bypassed if we focus only on questions addressed at the semi-classical level. Even then, however, current technology does not give us the explicit form of $H$, e.g. the structure of the string landscape. - [*“Selection Conditions”*]{} — Predictions in general depend on the selection condition we impose on $\ket{\Psi(t)}$ (even if we know $H$ explicitly). For example, in the situation considered at the end of the previous subsection, they depend on the initial condition $\ket{\Psi(0)}$. These limitations may still allow us to make certain predictions, possibly with some assumption on the dynamics of the system. First of all, if we are interested in a system localized in a small region compared with the horizon scale, then we can make predictions on the evolution of the system (i.e. correlation with a physical quantity that plays the role of time) using prior information about the system—indeed, one can show that Eq. (\[eq:prob\]) is reduced to the standard Born rule in such a case. Second, if we are interested in quantities whose distributions in $H$ are reasonably inferred in an anthropically allowed range, then we can predict the probability distribution of these quantities seen by a typical observer, under the assumption that the selection condition provides a statistically uniform prior [@Weinberg:1987dv]. This is, for example, the case if we are interested in the probability distribution of the cosmological constant one observes [@Larsen:2011mi]. However, if we want to answer general “multiversal” questions, e.g.  if we want to predict the probability distribution of the structure of the low-energy Lagrangian found by an intellectual observer in the multiverse, then we would need to address both (a) and (b) above. (What the intellectual observer means can be specified explicitly by condition $A$.) For (a), one could hope that future progress, e.g.  in string theory, might provide us (at least the relevant information on) the form of $H$ in ${\cal H}_{\rm QG}$. But what about (b)? There are at least three aspects which make this problem substantial: - One might speculate that a physical theory only allows for relating a given initial state to another final state, which is indeed the case in conventional Newtonian and quantum mechanics. In the present context, this implies that to make general predictions, we need to know the state $\ket{\Psi(t)}$ explicitly for some $t$. This is, however, impossible to do observationally! Quantum mechanics does not allow us to know the exact state [*including us, the observer*]{}. Moreover, $\ket{\Psi(t)}$ is the quantum state for the whole multiverse, so it in general contains terms representing different semi-classical universes than what we live in. - General predictions in the multiverse, therefore, will be possible only if we have a [*theoretical*]{} input on the selection condition of $\ket{\Psi(t)}$. Suppose it takes the form of a specific “initial condition,” $\ket{\Psi(0)}$. Then, the predictions depend on $\ket{\Psi(0)}$, so that, unless we have a separate theory of the initial condition, the uniqueness of (even statistical) predictions will be lost. - Imagine that there is, indeed, a theory of the initial condition giving a particular state $\ket{\Psi(0)}$, and that the framework described in Sections \[subsec:Hilbert\] and \[subsec:Born\] applies only to $t > 0$. In this case, the laws of quantum mechanics, especially deterministic and unitary evolution of the state, is violated at $t = 0$. While this is possible, it would be more comfortable if fundamental principles, such as those of quantum mechanics, do not have an “exception” like this. In the rest of the paper, we will address the problem of selection conditions, i.e. issue (b), from the viewpoint of extrapolating the principles of quantum mechanics to the maximum extent possible. By postulating a certain simple criterion, and requiring consistency with observation, we will arrive at the picture that the multiverse state must, in fact, be static. This provides a strong selection of the possible states. The observed flow of time arises from the structures of $H$ in ${\cal H}_{QG}$, and not because of a $t$ dependence of $\ket{\Psi(t)}$. The Observational “Data” {#sec:obs} ======================== Any selection condition imposed on the multiverse state must not lead to results inconsistent with observation, if it is to do with nature. The basic observational fact in our universe is that we see time flow in a definite direction, and predictions of a theory must not contradict it. As we will see, this seemingly weak requirement, in fact, provides a powerful tool to determine the selection condition. Here, we carefully consider what the observed flow of time actually means in the context of the quantum multiverse. What is the arrow of time? {#subsec:arrow-time} -------------------------- What does the fact that we see time flow really mean? At the most elementary level, it just means that the memory state of my (or your) brain is consistent with the hypothesis that it is generated by an environment whose coarse-grained entropy evolves from lower to higher values. The point is that the states consistent with such a hypothesis are very special ones among all the possible states the brain can take. What the fundamental theory must explain is why my brain is in one of these highly exceptional states. ![Suppose you know that there are a half of a chair and of a room in the first half of the scene (the upper picture). In a regular ordered world, you expect the second half of the scene contains the other half of the chair and the room, possibly with some other things (the lower left picture). On the other hand, the number of such states is much smaller than that of states in which the second half contains random, disordered configurations (the lower right picture).[]{data-label="fig:chair"}](chair.epsi){width="16cm"} To illustrate the basic idea further, let us consider a more corporeal example of a chair in a room. Suppose you are looking at only a half of the scene and find a half of a chair and of a room there; see the upper picture in Fig. \[fig:chair\]. What would you expect to be in the other half? In the ordered world we live in, we expect to see the other half of the chair and the room, possibly with some other things such as a painting on the wall, as depicted in the lower left picture in Fig. \[fig:chair\]. However, any such configurations are extremely rare among all the possible configurations physically allowed and consistent with the first half of the scene. The vast majority of these general configurations correspond to the ones in which the other half of the scene is completely disordered, as depicted in the lower right picture in Fig. \[fig:chair\]. The arrow of time refers to the fact that we always find ordered configurations (as in the lower left picture) rather than disordered ones (as in the lower right picture) in any similar situation, i.e. not only for a chair in a room but also for other objects. Such ordered configurations can be naturally expected if the entire system is evolved from a state having a much lower coarse-grained entropy; otherwise, we would expect disordered ones since the number of states corresponding to disordered configurations is much larger than that corresponding to ordered ones. In the context of the multiverse, the fact that we live in our universe and see the arrow of time tells us two things: - A typical observer among all the “conscious” observers in the multiverse (including fluke, Boltzmann brain observers [@Dyson:2002pf]) must live in a universe consistent with our current knowledge, i.e.  a universe whose low energy physics is described by the standard model of particle physics and cosmology. - When we ask any conditional probability $P(B|A)$ within our universe, i.e. when precondition $A$ is chosen such that it selects a situation in our universe (e.g. my brain state), the answer should be dominated by one that arises from a low coarse-grained entropy state through evolution. These two are the only things we definitely know from observation about the structure of the multiverse; for example, the arrow of time may not exist in other universes, i.e. the probabilities may be dominated by disordered configurations in those universes. What we must require is that the theory must (at least) be compatible with these two conditions. The above discussion shows that the following two statements are literally [*equivalent*]{} as concepts: “An observer sees the arrow of time” and “There is no Boltzmann brain problem.” This is consistent with the picture presented recently by Bousso [@Bousso:2011aa], who analyzed the arrow of time in the context of the evolving multiverse in the landscape. Historically, the argument like the one here was first used to exclude the possibility that our universe, which has a positive cosmological constant, is absolutely stable [@Dyson:2002pf]. It was also argued in Ref. [@Nomura:2011rb] that it excludes the possibility that the multiverse is a closed, finite system if it has a [*generic*]{} initial condition in ${\cal H}_{\rm QG}$. This possibility, however, is allowed if the selection condition imposed on the entire multiverse state is special, as is the case in the scenario considered in this paper. In summary, a selection condition imposed on the multiverse state must be such that the resulting probabilities are consistent with conditions (A) and (B) listed above. In particular, this leads to the following corollary: - Any selection condition on $\ket{\Psi(t)}$ that leads to an (almost) equal probability for all the possible states in ${\cal H}_{\rm QG}$ corresponding to our universe is observationally excluded. This is because such a scenario would lead to the probabilities being dominated by disordered configurations in our universe, contradicting observation. This condition will play an important role in rejecting a possible selection condition in Section \[sec:max-mixed\]. Is the multiverse really evolving? {#subsec:evolve} ---------------------------------- The consideration given above also illuminates the following question:  is the multiverse really evolving? The answer is: it need not. In order to be consistent with the observed arrow of time, it is only necessary that the probabilities [*in our universe*]{} are dominated by configurations that are consistent with the hypothesis that the system has evolved from a lower coarse-grained entropy state. This, however, does not necessarily mean that the multiverse state $\ket{\Psi(t)}$ is actually evolving in $t$. It simply says that the probabilities obtained from $\ket{\Psi(t)}$ should be consistent with the hypothesis that our universe has evolved from a lower entropy state. One might think that we actually “witnessed” that the state evolved as we came into being and grew. The interpretation of this fact, however, needs care—all we know is that our memory states are such that they are [*consistent with*]{} those obtained by interacting with environments that evolve from lower to higher entropy states. Similarly, we usually consider that our universe has evolved from the early big-bang, but all we really know is that the current state of the universe is consistent with the hypothesis that it has evolved from a lower entropy, big-bang state. As we have seen in the previous subsection, what these observations are really telling us is that in our universe different parts of physical configurations are correlated in certain (very) special ways. They do not mean that the multiverse state $\ket{\Psi(t)}$ [*must*]{} be evolving. The question of whether a physical system is viewed as evolving or not, therefore, can be determined by asking questions about a “current” configuration, i.e. configuration at a fixed value of $t$. If the configuration is consistent with the hypothesis that the system has evolved from a lower entropy state, then we [*interpret*]{} it as the system evolving—it is not necessary that the state itself is actually changing with $t$. To do such a determination, it is enough to use the formula of Eq. (\[eq:prob\]), in which conditions $A$ and $B$ act at the same moment. In fact, in quantum mechanics, when we obtain information about a system we do that indirectly by observing imprints in the environment left by the system [@q-Darwinism], so this is almost exactly what we do in reality when we study the “history” of a system. Summarizing, the observed flow of time does not require that the multiverse state is actually changing with $t$. It simply requires that the resulting probabilities satisfy the two conditions described in the previous subsection: (A) and (B). The probability formula in which conditions $A$ and $B$ both act at the same moment can be used to answer any physical questions, including those about a system that we interpret as dynamically evolving. Selection Conditions and Operators {#sec:max-mixed} ================================== We now start exploring possible selection conditions that can be imposed on the multiverse state. As stated in the introduction, we consider that the laws of quantum mechanics are not violated (Hypothesis I), which forces the multiverse state to exist for all values of $t$: from $-\infty$ to $+\infty$. This implies that, once a selection condition is given at a particular moment, which we take as $t = 0$, then the state is uniquely determined by solving the Schrödinger equation both forward and backward in $t$. In this section, we ask the following question: can the selection condition be given in Hilbert space ${\cal H}_{\rm QG}$ without referring to any quantum operator? If this is possible, then it would imply that the form of the selection condition, written purely in terms of quantum states, must be basis independent, since we cannot specify a basis without knowledge of operators and how they act on elements in the Hilbert space. (Note that Hilbert space itself does not contain any physical information except for its dimension, i.e. any complex Hilbert spaces having the same dimension are identical with each other.) We will see that there is only one possible selection condition satisfying this criterion, and that it is observationally excluded. We will therefore learn that the expression for the selection condition in ${\cal H}_{\rm QG}$ must involve some information about the quantum operators. The selection condition without an operator {#subsec:state-MM} ------------------------------------------- Suppose that the multiverse is in a pure state, and that the selection condition at $t = 0$ is given by $$\ket{\Psi(0)} = \sum_i c_i \ket{\alpha_i}, \label{eq:bc-pure}$$ where $\ket{\alpha_i}$ represents a complete, orthonormal basis for the elements in ${\cal H}_{\rm QG}$, and $c_i$ are fixed coefficients characterizing the selection condition. Can the expression in Eq. (\[eq:bc-pure\])—[*including the values of $c_i$*]{}—be basis independent? Consider that we perform an arbitrary basis change $$\ket{\alpha_i} = \sum_j U_{ij} \ket{\alpha'_j}, \label{eq:basis-change}$$ where $U_{ij}$ is an arbitrary unitary matrix. In the new basis, the expression in Eq. (\[eq:bc-pure\]) is written as $\ket{\Psi(0)} = \sum_i c'_i \ket{\alpha'_i}$, where the new coefficients $c'_i$ are given by $c'_i = \sum_j c_j U_{ji}$. In order for the form of the selection condition to be basis independent, we need to have $$c_i = c'_i = \sum_j c_j U_{ji} \label{eq:pure-indep}$$ [*for an arbitrary $U_{ij}$*]{}. This condition cannot be satisfied unless $c_i = 0$ for all $i$. Therefore, it is not possible to write a selection condition without referring to any quantum operator if the multiverse state is pure. Suppose now that the multiverse is in an intrinsically mixed state, which takes the form $$\rho(0) = \sum_{i,j} d_{ij} \ket{\alpha_i} \bra{\alpha_j} \label{eq:bc-mixed}$$ at $t = 0$, where $d_{ij}$ is a positive semi-definite Hermitian matrix. The basis change in Eq. (\[eq:basis-change\]) then leads to $\rho(0) = \sum_i d'_{ij} \ket{\alpha'_i} \bra{\alpha'_j}$, where the new coefficients are given by $d'_{ij} = \sum_{k,l} U_{ik} d_{kl} U^*_{jl}$. In order for the selection condition to be basis independent, we must have $$d_{ij} = d'_{ij} = \sum_{k,l} U_{ik} d_{kl} U^*_{jl} \label{eq:mixed-indep}$$ for an arbitrary $U_{ij}$. This has the unique solution (up to the overall coefficient): $$d_{ij} \propto \delta_{ij}. \label{eq:d-indep}$$ We thus find that the requirement is satisfied if the multiverse state is specified by $$\rho(0) \propto \sum_i \ket{\alpha_i} \bra{\alpha_i}, \label{eq:rho-MM}$$ namely if the multiverse is in the maximally mixed state in ${\cal H}_{\rm QG}$ at $t = 0$. Can the multiverse be in the maximally mixed state? {#subsec:max-ig} --------------------------------------------------- Once the selection condition is given by Eq. (\[eq:rho-MM\]), the multiverse state $\rho(t)$ for arbitrary $t$ can be obtained using the evolution equation $$\rho(t) = U(t,0)\, \rho(0)\, U(0,t). \label{eq:mixed-evol}$$ Since $\rho(0)$ is proportional to the unit matrix in ${\cal H}_{\rm QG}$, however, this gives $$\rho(t) = \rho(0), \label{eq:mixed-0}$$ i.e. the multiverse is in the maximally mixed state at all times. Equations (\[eq:rho-MM\]) and (\[eq:mixed-0\]) imply that all the possible states in ${\cal H}_{\rm QG}$ corresponding to our universe are equally probable. This is exactly the possibility that is observationally excluded by corollary ($\ast$) in Section \[subsec:arrow-time\]. Since we have arrived at this conclusion only by assuming that the selection condition is written without referring to a quantum operator in ${\cal H}_{\rm QG}$, we learn that the condition must in fact involve a quantum operator. The significance of this result lies in the fact that in quantum mechanics, operators are the objects that contain information about the system—the condition imposed on the multiverse state must reflect the structure of the system. The Static Quantum Multiverse {#sec:stat} ============================= What operators can be used in the condition imposed on the multiverse state? Since the multiverse contains many universes in which low energy physical laws differ, they cannot be “vacuum specific” operators. In this section, we identify candidate operators—those generating reference frame changes and that generating evolution. We then impose the requirement that physical predictions are independent of a reference frame one chooses to describe the multiverse (Hypothesis II in the introduction). We will see that this implies that the multiverse state is independent of $t$, i.e. it must be static. As discussed in Section \[subsec:evolve\], this does not necessarily contradict observation. (The consistency with the observed flow of time will be discussed further in Section \[sec:consistency\].) We will also see that with Hypothesis I, Hypothesis II can be viewed as a consequence of requiring that the multiverse is in an eigenstate of global energy and boost operators with zero eigenvalues. Reference frame changes {#subsec:ref-change} ----------------------- Recall that in the framework of Refs. [@Nomura:2011dt; @Nomura:2011rb], quantum states allowing for spacetime interpretation, i.e. elements of ${\cal H} \subset {\cal H}_{\rm QG}$, represent only the spacetime regions inside and on the (stretched) apparent horizons as viewed from a fixed reference frame associated with a fixed reference point $p$. What happens if we change the reference frame? Consider a state representing a configuration in de Sitter space. If we perform a spatial translation, which is equivalent to shifting the location of $p$, then it will necessarily mix the degrees of freedom inside and on the horizon because the state is defined only in the restricted spacetime region. This is precisely the phenomenon we call the observer dependence of the horizon: (some of) the degrees of freedom associated with internal space for one observer are described as those associated with the horizon by another. Next, consider a state which will later form a black hole, with $p$ staying outside of the black hole horizon. Such a state will not contain the spacetime region inside the black hole horizon because it will be outside $p$’s horizon. Now, imagine that we change the reference frame by performing a boost at an early time so that $p$ will be inside the black hole horizon at late times. In this new frame, the state at late times [*does*]{} contain the spacetime region inside the black hole horizon, although now it does [*not*]{} contain Hawking radiation quanta escaping to the future null infinity, which were included in the state before performing the reference frame change. This is exactly the phenomenon of black hole complementarity [@Susskind:1993if]. The present framework, therefore, allows us to understand the two phenomena described above in a unified manner as special cases of general reference frame changes [@Nomura:2011rb]; in particular, the concept of spacetime depends on the reference frame. As any symmetry transformation, reference frame changes must be represented by unitary transformations acting on Hilbert space ${\cal H}_{\rm QG}$. What is the set of generators representing these transformations, and what is the algebra they satisfy? In the limit $G_N \rightarrow 0$, the set of transformations associated with the reference frame changes and a shift of the origin of $t$ (time translation) is reduced to the standard Poincaré transformations, which is analogous to the fact that the standard Poincaré group is reduced to the Galilean group in the limit $c \rightarrow \infty$ [@Nomura:2011rb]. Here, $G_N$ and $c$ are Newton’s constant and the speed of light, respectively. In the case of the reduction associated with $c \rightarrow \infty$, the structure of infinitesimal transformations changes. This is seen clearly in the Poincaré algebra: $$\begin{array}{c} [ J_{[ij]}, J_{[kl]} ] = i \left( \delta_{ik} J_{[jl]} - \delta_{il} J_{[jk]} - \delta_{jk} J_{[il]} + \delta_{jl} J_{[ik]} \right), \\[10pt] [ J_{[ij]}, K_k ] = i \left( \delta_{ik} K_j - \delta_{jk} K_i \right), \qquad [ K_i, K_j ] = - \frac{i}{c^2} J_{[ij]}, \\[10pt] [ J_{[ij]}, P_k ] = i \left( \delta_{ik} P_j - \delta_{jk} P_i \right), \qquad [ K_i, P_j ] = \frac{i}{c^2} \delta_{ij} H, \qquad [ P_i, P_j ] = 0, \\[10pt] [ J_{[ij]}, H ] = [ P_i, H ] = [ H, H ] = 0, \qquad [ K_i, H ] = i P_i, \end{array} \label{eq:Poincare-alg}$$ where $J_{[ij]}$, $K_i$, and $P_i$ are the generators of spatial rotations, boosts, and spatial translations, respectively, and we have exhibited $c$ explicitly. This algebra is reduced to a different algebra, i.e. that of the Galilean group, as $c \rightarrow \infty$: $$\begin{array}{c} [ J_{[ij]}, J_{[kl]} ] = i \left( \delta_{ik} J_{[jl]} - \delta_{il} J_{[jk]} - \delta_{jk} J_{[il]} + \delta_{jl} J_{[ik]} \right), \\[10pt] [ J_{[ij]}, K_k ] = i \left( \delta_{ik} K_j - \delta_{jk} K_i \right), \qquad [ K_i, K_j ] = 0, \\[10pt] [ J_{[ij]}, P_k ] = i \left( \delta_{ik} P_j - \delta_{jk} P_i \right), \qquad [ K_i, P_j ] = i \delta_{ij} M, \qquad [ P_i, P_j ] = 0, \\[10pt] [ J_{[ij]}, H ] = [ P_i, H ] = [ H, H ] = 0, \qquad [ K_i, H ] = i P_i, \end{array} \label{eq:Galilean-alg}$$ where we have rescaled $H \rightarrow c^2 M + H$ to allow for the possibility that the original $H$ has a constant piece that goes as $c^2$. Can the algebra corresponding to the reference frame changes and time translation have extra terms beyond Eq. (\[eq:Poincare-alg\]) that disappears in the limit $G_N \rightarrow 0$? One can immediately see that it cannot. The generators of the reference frame changes consist of $J_{[ij]}$, $K_i$, and $P_i$, while that of time translation is $H$. Taking natural units, the mass dimensions of these generators are $[J_{[ij]}] = [K_i] = 0$ and $[P_i] = [H] = 1$, while that of Newton’s constant is $[G_N] = -d+2$, where $d$ is the number of spacetime dimensions. It is then easy to find that for $d \geq 4$, where gravity is dynamical, there is no term one can add to the commutators in Eq. (\[eq:Poincare-alg\]) that is linear in generators and has a positive integer power of $G_N$.[^2] The algebra for the reference frame changes and time translation, therefore, is the same as that of the Poincaré transformations in Eq. (\[eq:Poincare-alg\]). The effect of nonzero $G_N$ appears as the reduction of the Hilbert space, but not in the transformation generators of the Poincaré group. Selecting the multiverse state {#subsec:selec} ------------------------------ Let us now require that predictions do not depend on the reference frame one chooses to describe the multiverse (Hypothesis II). Physically, this implies that there is neither absolute center nor the frame of absolute rest in the multiverse. Formally, our requirement can be stated as follows. Suppose we want to make physical predictions using projection operators ${\cal O}_X$, e.g. $X = A$, $A \cap B$, and so on. The relevant matrix elements are then $\bra{\Psi(t)} {\cal O}_X \ket{\Psi(t)}$. Now, consider a multiverse state as viewed from a different reference frame:  $\ket{\Psi'(t)} = S \ket{\Psi(t)}$, where $S$ is the unitary operator representing the corresponding reference frame change. Our requirement is then $$\bra{\Psi(t)} {\cal O}_X \ket{\Psi(t)} = \bra{\Psi'(t)} {\cal O}_X \ket{\Psi'(t)} \label{eq:req}$$ [*for arbitrary $S$ and ${\cal O}_X$*]{}. Note that the operator in the right-hand side is [*not*]{} ${\cal O}'_X = S {\cal O}_X S^\dagger$, but the same ${\cal O}_X$ as in the left-hand side. This equation, therefore, has a nontrivial physical content, imposing constraints on the multiverse state. (If we had ${\cal O}'_X$ in the right-hand side, then the equation would simply represent a basis change, and thus would be trivial.) In order to satisfy Eq. (\[eq:req\]), the multiverse state must satisfy $S \ket{\Psi(t)} \propto \ket{\Psi(t)}$, so that it must be a simultaneous eigenstate of operators $J_{[ij]}$, $K_i$ and $P_i$.[^3] One can then easily see from Eq. (\[eq:Poincare-alg\]) that this requires that the multiverse state is also an eigenstate of $H$, and that the eigenvalues under $J_{[ij]}$, $K_i$, $P_i$, and $H$ are all zero. The fact that the multiverse state is an eigenstate of $H$ with zero eigenvalue means that $$\frac{d}{dt}\ket{\Psi(t)} = 0, \label{eq:d-dt_Psi}$$ i.e. the multiverse state is static! We can therefore write it simply as $\ket{\Psi} \equiv \ket{\Psi(t)} = \ket{\Psi(0)}$. The conditions coming from Hypothesis II can then be summarized as $$J_{[ij]} \ket{\Psi} = K_i \ket{\Psi} = P_i \ket{\Psi} = H \ket{\Psi} = 0. \label{eq:Psi-cond}$$ This provides selection conditions for the multiverse state. In fact, given Hypothesis I, the conditions in Eq. (\[eq:Psi-cond\]) follow from a standard procedure of quantizing a system with redundancies [@Dirac-book], if we assume that the multiverse state is invariant under the action of global energy and boost operators. In this procedure, any gauge redundancy, including general coordinate transformations, appears as a supplementary condition imposed on quantum states, which eliminates unphysical degrees of freedom from the states. Starting from a consistent, general covariant quantum theory of gravity (which is presumably string theory), the states are subject to a huge number of supplementary conditions, some of which will be used to reduce the number of degrees of freedom from that implied by local field theory to that suggested by the holographic principle, as in Eq. (\[eq:H\_M-dimension\]). (This implies that the number of constraints is much larger than that of the standard constraints associated with classical general coordinate transformations [@DeWitt:1967yk].) In this bigger (more redundant) picture, the framework of Refs. [@Nomura:2011dt; @Nomura:2011rb] corresponds to the scheme in which all the gauge redundancies are explicitly fixed, except for the ones associated with the reference frame changes. These residual redundancies, i.e. those of the reference frame changes, must then have their own supplementary conditions imposed on the states living in ${\cal H}_{\rm QG}$. To illustrate this in a simple example, let us consider a spacetime that admits rectilinear coordinates $x_i$ in a constant $t$ hypersurface. In terms of Hamiltonian and momentum densities, ${\cal H}(x)$ and ${\cal P}_i(x)$, the Hilbert space ${\cal H}_{\rm QG}$ then corresponds to the space of states in which the constraints of the form $$\begin{aligned} && \int\! x_i x_j {\cal H}(x)\, d^3x \ket{\Psi} = \int\! x_i x_j x_k {\cal H}(x)\, d^3x \ket{\Psi} = \cdots \nonumber\\ && \qquad = \int\! x_i x_j {\cal P}_k (x)\, d^3x \ket{\Psi} = \int\! x_i x_j x_k {\cal P}_l(x)\, d^3x \ket{\Psi} = \cdots = 0, \label{eq:const-1}\end{aligned}$$ as well as those associated with holography and complementarity, are [*already*]{} imposed; namely, the states in ${\cal H}_{\rm QG}$ satisfy these constraints by construction. On the other hand, the constraints of the form $$\int\! {\cal H}(x)\, d^3x \ket{\Psi} = \int\! x_i {\cal H}(x)\, d^3x \ket{\Psi} = \int\! {\cal P}_i (x)\, d^3x \ket{\Psi} = \int\! x_i {\cal P}_j(x)\, d^3x \ket{\Psi} = 0 \label{eq:const-2}$$ are [*not*]{} imposed to obtain ${\cal H}_{\rm QG}$, so they must still be imposed on the states in ${\cal H}_{\rm QG}$. Now, the generators of time translation and the reference frame changes are given by $$\begin{array}{c} H = \int\! {\cal H}(x)\, d^3x + \epsilon, \qquad P_i = \int\! {\cal P}_i(x)\, d^3x + p_i, \\[10pt] K_i = \int\! x_i {\cal H}(x)\, d^3x + k_i, \qquad J_{[ij]} = \int\! (x_i {\cal P}_j(x) - x_j {\cal P}_i(x)) d^3x + j_{[ij]}, \end{array}$$ where we have included global energy $\epsilon$ and momentum $p_i$ operators (and the corresponding quantities in $K_i$ and $J_{[ij]}$) that represent possible contributions from surface terms. Such terms can indeed arise in asymptotically Minkowski space, and play the role of what we consider the total energy and momentum of the system [@Arnowitt:1962hi]. Note that it is the effect of global energy $\epsilon$ that allows for any evolution of states in $t$ in quantum gravity, because $$\ket{\Psi(t_1)} = e^{-iH(t_1-t_2)} \ket{\Psi(t_2)} = e^{-i\epsilon (t_1-t_2)} \ket{\Psi(t_2)}, \label{eq:gen-evol}$$ so unless $\ket{\Psi(t)}$ is a superposition of terms that give different values of $\epsilon$, the state is stationary. In this picture, our Hypothesis II corresponds to the assumption that the multiverse is an eigenstate of $\epsilon$ and $k_i$ with vanishing eigenvalues: $$\epsilon \ket{\Psi} = k_i \ket{\Psi} = 0, \label{eq:zero-ep}$$ in which case we immediately see that $\ket{\Psi}$ also has zero eigenvalues under $p_i$ and $j_{[ij]}$, and that Eq. (\[eq:Psi-cond\]) follows from the constraints in Eq. (\[eq:const-2\]) (and vice versa). An important point is that for a state in ${\cal H}_{\rm QG}$, the surface terms reside on the (stretched) apparent horizon, so that Eq. (\[eq:zero-ep\]) is the assumption about the structure of the theory on this surface. This is in the intrinsically quantum gravitational regime, over which we currently do not have good theoretical control. The selection of possible multiverse states, therefore, is boiled down to solving the infinite-dimensional matrix equations in Eq. (\[eq:Psi-cond\]). Here, we assume that there is no other selection condition, i.e.  Eq. (\[eq:Psi-cond\]) is enough to fully select the system. (We assume that other supplementary conditions, e.g. those associated with standard gauge symmetries, are already taken care of. Also, since all the redundancies associated with gravity other than those corresponding to the reference frame changes are supposed to be fixed in the present framework [@Nomura:2011dt], there are no more conditions arising from considerations of gravity.) We look for solutions to Eq. (\[eq:Psi-cond\]) of the form $$\ket{\Psi} = \sum_i c_i \ket{\alpha_i}, \qquad \sum_i |c_i|^2 < \infty, \label{eq:sol-cond}$$ where $\ket{\alpha_i}$ represents a complete, orthonormal basis in ${\cal H}_{\rm QG}$, so that the sums of $i$ run to infinity; see Eq. (\[eq:dim-inf\]). The normalizability condition here is imposed for the following (usual) reason. Suppose there are normalizable solutions $\ket{\Psi_I}$ ($I = 1,\cdots,N$) satisfying Eq. (\[eq:sol-cond\]), as well as non-normalizable solutions $\ket{\Psi_I}$ ($I = N+1,\cdots,K$). The non-normalizable solutions will have coefficients which strongly diverge as the dimensions of corresponding Hilbert subspaces ${\cal H}_{\cal M}$ become large. This is because the process transforming an element of ${\cal H}_{\cal M}$ to that of ${\cal H}_{\cal M'}$ with ${\rm dim}\,{\cal H}_{\cal M'} < {\rm dim}\,{\cal H}_{\cal M}$ becomes highly suppressed as ${\rm dim}\,{\cal H}_{\cal M}$ gets large (because of Eq. (\[eq:dim-inf\])). Let us now imagine regulating the sums of $i$ as $\sum_i \rightarrow \sum_{i=1}^{n}$, in which case we can normalize all the solutions so that $\inner{\Psi_I}{\Psi_J} = \delta_{IJ}$ for $I,J = 1,\cdots,K$. We can then consider state $\rho = \sum_{I,J=1}^{K} d_{IJ} \ket{\Psi_I} \bra{\Psi_J}$ with arbitrary finite positive semi-definite Hermitian matrix $d_{IJ}$, and calculate probabilities arising from $\rho$ using a projection operator that selects (a finite number of) configurations compatible with some condition $X$: ${\cal O}_X = \sum_{i \in X} \ket{\alpha_i} \bra{\alpha_i}$. The resulting probabilities are the same as those arising from $\rho' = \sum_{I,J=1}^{N} d_{IJ} \ket{\Psi_I} \bra{\Psi_J}$, i.e. the state obtained by eliminating all the non-normalizable solutions from $\rho$, up to terms disappearing for $n \rightarrow \infty$. Therefore, the non-normalizable solutions can all be dropped from physical considerations. The Hilbert space relevant for the multiverse ${\cal H}_{\rm Multiverse}$, then, is spanned by the normalizable solutions to Eq. (\[eq:Psi-cond\]), and so is much smaller than ${\cal H}_{\rm QG}$: $${\cal H}_{\rm Multiverse} \subset {\cal H}_{\rm QG}, \qquad {\rm dim}\,{\cal H}_{\rm Multiverse} \ll {\rm dim}\,{\cal H}_{\rm QG}. \label{eq:H_multiverse}$$ We note that this situation is analogous to usual quantum mechanical systems, e.g. a hydrogen atom. In the hydrogen atom, the state factor corresponding to a radial wavefunction $c(r)$ can be written as $\ket{\psi} = \int_0^\infty\!dr\, c(r) \ket{r}$. The only states relevant to physics of the hydrogen atom are those satisfying the normalizability condition $\int_0^\infty\!dr\, |c(r)|^2 < \infty$ in the Hilbert space spanned by $\ket{r}$. The other, non-normalizable solutions (which behave as $\ln c(r) \sim r$ at large $r$) are irrelevant. The situation in the quantum multiverse is similar. The non-normalizable solutions have infinitely strong supports in supersymmetric Minkowski vacua or singularity worlds, which have infinite-dimensional Hilbert spaces. These solutions, therefore, are irrelevant in making predictions in a “realistic world,” i.e. in a universe that has nonzero free energy. The only relevant states are those that are normalizable in the Hilbert space of quantum gravity, ${\cal H}_{\rm QG}$. For a schematic drawing of this analogy, see Fig. \[fig:analogy\]. ![A schematic depiction of the analogy between the hydrogen atom and the quantum multiverse. In the case of the hydrogen atom, the only relevant states are those that satisfy the Schrödinger equation and are normalizable in the Hilbert space spanned by $\ket{r}$ (solid line); the non-normalizable modes are irrelevant (dashed line). In the quantum multiverse, the relevant states are those that satisfy Eq. (\[eq:Psi-cond\]) and are normalizable in Hilbert space ${\cal H}_{\rm QG}$ (solid line); the non-normalizable modes, which have diverging coefficients for supersymmetric Minkowski or singularity states, are irrelevant (dashed line).[]{data-label="fig:analogy"}](analogy.epsi){width="17cm"} The static multiverse states in [${\cal H}_{\rm QG}$]{} {#subsec:static} ------------------------------------------------------- We now discuss how our conditions Eqs. (\[eq:d-dt\_Psi\], \[eq:sol-cond\]) can be compatible with Eq. (\[eq:asympt\]), which says that a generic multiverse state in ${\cal H}_{\rm QG}$ will evolve into a superposition of supersymmetric Minkowski and singularity states as $t \rightarrow \infty$. In order for Eq. (\[eq:d-dt\_Psi\]) to be satisfied, the coefficients $c_i$ of all the terms in $\ket{\Psi(t)} = \ket{\Psi}$ must be constant when expanded in components $\ket{\alpha_i}$. In a basis in which $\ket{\alpha_i}$ in ${\cal H}$ have well-defined semi-classical configurations, the evolution operator $\exp(-iHt)$ (and thus $H$ as well) is not diagonal. Therefore, the processes in Eq. (\[eq:asympt\]) will occur for [*generic*]{} $\ket{\Psi(t)}$, but they must exactly be canceled by some “inverse processes” in $\ket{\Psi}$. In particular, in order for the normalization condition in Eq. (\[eq:sol-cond\]) to be satisfied, this must occur before the state is dissipated into infinite-dimensional Hilbert space. Let us consider a physical configuration in a Minkowski universe in which there is a bubble wall surrounding us, which, however, is contracting toward us rather than expanding away. Such a configuration, which is exactly the time reversal of a usual expanding bubble configuration, is physically allowed, as the fundamental equation of the theory is symmetric under $t \rightarrow -t$. Usually, we do not consider this kind of configuration as it is only an exponentially small subset of all the configurations allowed by the theory; in particular, there is only an exponentially small probability for forming such a configuration starting from a generic, e.g. thermal, state. We are, however, now considering very special states, i.e. the states that satisfy Eqs. (\[eq:d-dt\_Psi\], \[eq:sol-cond\]), and in these states such configurations could balance the “loss” of semi-classically unstable states in Eq. (\[eq:asympt\]). For example, the entire multiverse state is so “fine-tuned” that a reheating that occurs in a Minkowski universe produces exactly the configuration that puts the system back to (a superposition of) states in unstable vacua. Similar processes must also occur for singularities. Note that since these processes are exponentially suppressed under normal circumstances, they are invisible in the usual semi-classical analysis. The states given by Eq. (\[eq:sol-cond\]) are the ones in which all these and other processes are balanced.[^4] Since the inverse processes are unlikely to occur at the zero density, these states will explore only a finite-dimensional portion of Minkowski vacua (see the discussion at the end of Section \[subsec:Hilbert\]). The number of independent states, therefore, may well be finite:  ${\rm dim}\,{\cal H}_{\rm Multiverse} < \infty$, which we will assume to be the case. Note that the sizes of various elements in $H$ represented as a matrix acting on ${\cal H}_{\rm QG}$ differ significantly; in fact, they are expected to differ exponentially, or even double-exponentially, as some of the processes are highly suppressed. This implies that the values of $|c_i|$’s in Eq. (\[eq:sol-cond\]) will also vary significantly. The resulting states $\ket{\Psi}$, therefore, are not excluded by corollary ($\ast$) in Section \[subsec:arrow-time\]. The structure of $\ket{\Psi}$, and its consistency with observation, will be discussed further in Section \[sec:consistency\]. Predictions in the static quantum multiverse {#subsec:pred} -------------------------------------------- The number of independent normalizable solutions to Eq. (\[eq:Psi-cond\]) will depend on the structure of the multiverse, i.e. issue (a) in Section \[subsec:s-c\]. In particular, the existence of a solution requires $H$ to take a certain special form (so that it has at least one normalizable, zero-eigenvalue eigenvector), which we assume to be the case. Suppose there are $N$ such solutions $\ket{\Psi_I}$ ($I = 1,\cdots,N = {\rm dim}\,{\cal H}_{\rm Multiverse} < \infty$). How can the physical predictions be made? If $N = 1$, the multiverse state is simply $\ket{\Psi} \equiv \ket{\Psi_1}$. The probabilities are then given by the generalized Born rule, Eq. (\[eq:prob\]), but now without the $t$ integrals. (They simply give a constant factor $\int_{-\infty}^{+\infty} dt$, which cancels between the numerator and denominator.) The final formula is given by Eq. (\[eq:prob-final\]), which we reproduce here: $$P(B|A) = \frac{\bra{\Psi} {\cal O}_{A \cap B} \ket{\Psi}} {\bra{\Psi} {\cal O}_A \ket{\Psi}}. \nonumber$$ As discussed in Section \[subsec:evolve\], this formula can be used to answer any physical questions, including those about a system that we view as dynamically evolving. In the case that $N > 1$, any multiverse states of the form $\ket{\Psi} = \sum_{I=1}^N c_I \ket{\Psi_I}$ or $\rho = \sum_{I,J=1}^{N} d_{IJ} \ket{\Psi_I} \bra{\Psi_J}$ are allowed. In the absence of more information (or selection conditions), it is natural to assume that the multiverse is in the maximally mixed state $$\rho = \frac{1}{N} \sum_{I=1}^N \ket{\Psi_I} \bra{\Psi_I}, \label{eq:multiverse-mixed}$$ where we have taken $\ket{\Psi_I}$’s to be orthonormal. This state is invariant under the basis change $\ket{\Psi_I} \rightarrow U_{IJ} \ket{\Psi_J}$, and is reduced to $\ket{\Psi} = \ket{\Psi_1}$ for $N = 1$. The probabilities are given by the mixed-state version of Eq. (\[eq:prob-final\]): $$P(B|A) = \frac{{\rm Tr}\left[ \rho\, {\cal O}_{A \cap B} \right]} {{\rm Tr}\left[ \rho\, {\cal O}_A \right]}. \label{eq:prob-final-mixed}$$ Note that Eq. (\[eq:multiverse-mixed\]), i.e. the maximally mixed state in ${\cal H}_{\rm Multiverse}$, is different from Eq. (\[eq:rho-MM\]), i.e. the maximally mixed state in ${\cal H}_{\rm QG}$, in which the sum runs over all the possible states in ${\cal H}_{\rm QG}$ including the ones that do not satisfy Eq. (\[eq:Psi-cond\]). The state in Eq. (\[eq:multiverse-mixed\]), therefore, is not excluded by corollary ($\ast$) in Section \[subsec:arrow-time\]. Consistency with Observation {#sec:consistency} ============================ In this section we discuss the consistency of the present scenario with observation, specifically the observed arrow of time. Our approach here will be to allow for making assumptions on the structures of $H$ and ${\cal H}_{\rm QG}$ (unless they are inconsistent with what we already know about string theory), and to see if the scenario is consistent. We do not claim that all of these assumptions are absolutely necessary—our purpose here is to argue that, despite its naive appearance, the scenario is not excluded by observation. More detailed analysis/modeling of the landscape will be left for future work. The structure of [${\cal H}_{\rm QG}$]{} {#subsec:structure} ---------------------------------------- Solutions to Eq. (\[eq:Psi-cond\]) depend on the structure of ${\cal H}_{\rm QG}$ as well as the form of $H$ (and other operators). Here we assume that ${\cal H}_{\rm QG}$ contains only “cosmologically relevant” states. The minimally required set of ${\cal H}_{\cal M}$’s that must be included in ${\cal H}$, i.e. in the right-hand side of Eq. (\[eq:ST-H\]), will then be those of FRW universes corresponding to all the possible vacua in the theory (and their straightforward generalizations, e.g. those of FRW universes with black holes). Not all spacetime must be contained in ${\cal H}$; for example, ${\cal H}$ need not contain a stable anti-de Sitter space without a singularity, which might only be a mathematical idealization because it does not arise through dynamical evolution in the FRW universes. For each vacuum $I$ of the theory, the number of states associated with an FRW universe in $I$ is estimated as $${\cal N}_I \,\,= \sum_{n=\exp({\cal A}_{I,{\rm min}}/2)}^{\exp({\cal A}_{I,{\rm max}}/2)} \!\!\!\! n \,\,\simeq\,\, \frac{1}{2}\, e^{{\cal A}_{I,{\rm max}}}, \label{eq:N_I}$$ where ${\cal A}_{I,{\rm min}}$ and ${\cal A}_{I,{\rm max}}$ are the minimum and maximum areas of the apparent horizon in this universe, and we have used ${\cal A}_{I,{\rm max}} \gg {\cal A}_{I,{\rm min}}$ in the last equation. While possible deformations of the apparent horizon, e.g.  by the existence of black holes, can have corrections to the explicit expression, we expect that the above estimate gives a qualitatively correct result: $\ln {\cal N}_I \approx O({\cal A}_{I,{\rm max}})$. The area ${\cal A}_{I,{\rm max}}$ is given by the inverse of the absolute value of the vacuum energy density (in Planck units) ${\cal A}_{I,{\rm max}} \sim 1/|\rho_{\Lambda,I}|$, since in a de Sitter universe the apparent horizon approaches the event horizon at late times, while in an anti-de Sitter universe it has the maximum area when $p$ hits the singularity at $t \sim 1/|\rho_{\Lambda,I}|^{1/2}$. We therefore find $$\ln {\cal N}_I \sim \frac{1}{|\rho_{\Lambda,I}|}. \label{eq:ln-N_I}$$ This implies that the number of states associated with a vacuum with $\rho_{\Lambda,I} \neq 0$ is finite. The arrow of time in the static multiverse {#subsec:arrow} ------------------------------------------ We now consider a solution to the equation $H \ket{\Psi} = 0$, a part of the conditions in Eq. (\[eq:Psi-cond\]). We can view this equation as requiring that $\ket{\Psi}$ is in a stationary state in ${\cal H}_{\rm QG}$. (In fact, the equation is stronger than that, since the eigenvalue of $H$ must be zero.) In particular, it implies that the probability current creating states in vacuum $I$ must be balanced with that destroying those for each $I$ (in fact, each state in $I$). At the semi-classical level, this condition is impossible to satisfy for terminal vacua. As discussed in Section \[subsec:static\], however, our state is special, obtained after solving the “quantization condition” $H \ket{\Psi} = 0$, so that it can also be satisfied for these vacua. Let us now consider vacuum $J$ that can support any observer, either an ordinary observer or a Boltzmann brain. We will argue that the arrow of time is predicted if the following three conditions are met for all possible $J$’s: - Transitions to states in $J$ from those in other vacua are mainly through the states having low coarse-grained entropies in $J$, i.e.  elements of ${\cal H}_{\cal M}$ with $\ln {\rm dim}\,{\cal H}_{\cal M} \ll {\cal A}_{J,{\rm max}}$. - Subsequent evolution in vacuum $J$ produces ordinary observers with probability $\epsilon_J$, which may be suppressed exponentially but not double-exponentially. - The rate of producing Boltzmann brains $\Gamma_{{\rm BB},J}$ in vacuum $J$, which is double-exponentially suppressed (see, e.g. [@Bousso:2008hz]), is smaller than the decay rate $\Gamma_J$ of the vacuum itself. Namely, if the structure of $H$ is such that it satisfies all these conditions, then the scenario is compatible with observation. (The “transitions” and “evolution” here, of course, refer to the apparent ones in $\ket{\Psi}$, which is in itself static.) To see this, let us consider the distribution of the size of the coefficients $|c^J_i|$ of various terms in $\ket{\Psi}$ corresponding to the states in vacuum $J$, $\ket{\alpha^J_i}$. For this purpose, we define the quantity $P^J_\tau$ corresponding to the probability for a universe to be at FRW time $t_{\rm FRW}$ between $\tau$ and $\tau+d\tau$: $$P^J_\tau\, d\tau = \sum_{i | \tau < t_{\rm FRW} < \tau+d\tau}\!\!\!\! |c^J_i|^2,$$ where $t_{\rm FRW}$ should be specified by physical configurations in $\ket{\alpha^J_i}$. The distribution of $P^J_\tau$ then follows from the definition of $\Gamma_J$: $$P^J_\tau = P^J_0 e^{-\Gamma_J\, \tau}, \label{eq:PJtau}$$ where we have assumed that the transitions to states in $J$ occur at $\tau = 0$ either through Coleman-De Luccia [@Coleman:1980aw] or Hawking-Moss [@Hawking:1981fz] processes (or their inverses), although our conclusion is insensitive to this assumption. Note that in these cases it is indeed natural to expect that states just after the transitions are the ones having low coarse-grained entropies, i.e. in ${\cal H}_{\cal M}$ with $\ln {\rm dim}\,{\cal H}_{\cal M} \ll {\cal A}_{J,{\rm max}}$, because both the start and end points of the Coleman-De Luccia tunneling in field space are away from local minima (if the false vacuum has a positive vacuum energy), and the Hawking-Moss transition is a thermal process occurring through the field climbing up the potential barrier [@Weinberg:2006pc]. Now, the definitions of $\epsilon_J$ and $\Gamma_{{\rm BB},J}$ in (II) and (III) above imply that if we compute the probability of $\ket{\Psi}$ containing ordinary observers (OO) or Boltzmann brains (BB) in vacuum $J$ using the corresponding projection operators ${\cal O}_{{\rm OO},J}$ and ${\cal O}_{{\rm BB},J}$, then we obtain $$\begin{aligned} && \bra{\Psi} {\cal O}_{{\rm OO},J} \ket{\Psi} \sim \epsilon_J P^J_0, \\ && \bra{\Psi} {\cal O}_{{\rm BB},J} \ket{\Psi} \sim \Gamma_{{\rm BB},J} \int\! P^J_\tau\, d\tau = \frac{\Gamma_{{\rm BB},J}}{\Gamma_J} P^J_0. \label{eq:OO-BB}\end{aligned}$$ Here, the projection operators select observers in a specific range of location and velocity with respect to $p$, although the results do not depend on the chosen location or velocity because of Eq. (\[eq:Psi-cond\]). Under conditions (II) and (III), this gives $$\frac{\bra{\Psi} {\cal O}_{{\rm BB},J} \ket{\Psi}}{\bra{\Psi} {\cal O}_{{\rm OO},J} \ket{\Psi}} \sim \frac{\Gamma_{{\rm BB},J}}{\epsilon_J \Gamma_J} \lll 1, \label{eq:BB-OO-ratio}$$ where we have used the fact that $\Gamma_{{\rm BB},J}$ is double-exponentially suppressed while $\epsilon_J$ is not. (In fact, we only need $\epsilon_J > \Gamma_{{\rm BB},J}/\Gamma_J$ to obtain this result, so $\epsilon_J$ may be double-exponentially suppressed.) We therefore find that the overwhelming majority of observers are indeed ordinary observers, and thus perceive time’s arrow (as discussed in Section \[subsec:max-ig\]). Perhaps not surprisingly, the conditions described above are similar to the ones obtained in Ref. [@Bousso:2011aa] in the context of the evolving multiverse, despite the fact that the overall physical pictures are rather different. One distinct feature of the present scenario in this respect is that since there is no “initial vacuum,” the absolute nonexistence of Boltzmann brains in such a vacuum ($\Gamma_{{\rm BB},\ast} = 0$ in the notation of Ref. [@Bousso:2011aa]) need not be imposed. In any case, as discussed in Ref. [@Bousso:2011aa], the conditions described above, in particular (I), are likely to be satisfied in the string landscape. It is, therefore, quite promising that the scenario discussed in this paper is indeed consistent with observation in the realistic string theory setup. Discussions {#sec:discuss} =========== In this paper we have studied the multiverse in the quantum mechanical framework recently proposed in Refs. [@Nomura:2011dt; @Nomura:2011rb]. By requiring that the laws of quantum mechanics are not violated (Hypothesis I) and that physical predictions do not depend on the reference frame one chooses to described the multiverse (Hypothesis II), we have found that the multiverse state must be static; in particular, the multiverse does not have a beginning or end. Despite its naive appearance, the scenario does not contradict observation, including the fact that we observe that time flows in a definite direction. [*The arrow of time is simply an emergent phenomenon that is occurring in the branch (terms) corresponding to our universe in the static multiverse state*]{}—the terms that would be obtained by evolving the system from lower entropy states have much larger coefficients than the terms that cannot. The scenario is summarized by the selection conditions in Eq. (\[eq:Psi-cond\]), imposed on the states in ${\cal H}_{\rm QG}$. With these conditions, any multiversal questions can be answered using the Born rule, Eq. (\[eq:prob-final\]) or (\[eq:prob-final-mixed\]), [*without any additional input*]{}, once the explicit form of the operators such as $H$ is known. This scenario, therefore, provides a completion of the framework of the quantum multiverse in Refs. [@Nomura:2011dt; @Nomura:2011rb]. The supplementary condition of the form $H \ket{\Psi} = 0$ has certainly been considered before—indeed, this is nothing but the well-known Wheeler-DeWitt equation [@DeWitt:1967yk]. The scenario presented here, however, differs from standard applications of this equation in several important ways: - The redundancies associated with gravity are much larger than what are usually imagined. In particular, they reduce the Hilbert space in such a way that it contains only the spacetime region within the reference point’s (stretched) apparent horizon [@Nomura:2011dt; @Nomura:2011rb]. This is important to avoid ambiguities associated with eternally inflating spacetime. The ultimate origin of these large redundancies will, presumably, be string theory. - We apply the supplementary conditions corresponding to the whole set of time translation and reference frame changes with zero global charges, even if the universe is not closed. Since spacetime is defined only within the apparent horizon, this requires the assumption on the structure of the theory on this surface, which is intrinsically quantum mechanical. Note that it is this assumption that is responsible for the static nature of the multiverse state, which in turn excludes the possibility for the multiverse to have a beginning or end. - We analyze the consequences of the supplementary conditions at the microscopic level. This selects very special states that are not visible in the analysis at the semi-classical level. In fact, normalizable solutions to the conditions correspond to the states in which the processes of Eq. (\[eq:asympt\]) are balanced with the inverse processes, which put the system back from terminal vacua to unstable vacua. It is quite satisfying that such simple requirements as Hypotheses I and II lead to a consistent and predictive scheme for the entire multiverse. Finally, it is instructive to draw a close analogy between the situation in the quantum multiverse described here and that in the standard, hydrogen atom. As is well known, the hydrogen atom cannot be correctly described using classical mechanics. Any orbit of the electron is unstable with respect to the emission of synchrotron radiation. Even if we artificially ignore the emission, the electron can orbit the nucleus at an arbitrary radius, unable to explain the discrete spectral lines. The solution to these problems is intrinsically quantum mechanical, i.e. quantum mechanics is responsible for the very existence of the hydrogen atom, not just providing a correction to the classical picture. The situation in the quantum multiverse is similar. At the semi-classical level, the multiverse is unstable to the decay to terminal states, such as supersymmetric Minkowski vacua and singularities. Even if we artificially ignore the process of vacuum decays, it would lead to phenomena such as Poincaré recurrence, contradicting observation (the dominance of Boltzmann brains). The picture presented here says that the solution to these problems is [*intrinsically quantum mechanical*]{}—one cannot see it in the usual semi-classical analysis. The multiverse state is very special: a normalizable state satisfying the “quantization conditions” of Eq. (\[eq:Psi-cond\]), as in the case of the hydrogen atom. In the case of the hydrogen atom, these conditions make the dimension of Hilbert space from continuous infinity $\psi(r, \theta, \varphi)$ to countable infinity $(n,l,m)$. In the quantum multiverse, they will presumably make it from countable infinity to finite: ${\rm dim}\,{\cal H}_{\rm QG} \rightarrow {\rm dim}\,{\cal H}_{\rm Multiverse}$. After all, quantum mechanics treats the multiverse very similarly to the hydrogen atom. Our job is then to figure out the precise structure of the multiverse, a system which we are a part of. Hopefully, further progress in string theory will serve this purpose. Acknowledgments {#acknowledgments .unnumbered} =============== I would like to thank Alan Guth and Grant Larsen for useful conversations. This work was supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, of the US Department of Energy under Contract DE-AC02-05CH11231, and in part by the National Science Foundation under grant PHY-0855653. [99]{} Y. Nomura, JHEP [**11**]{}, 063 (2011) \[arXiv:1104.2324 \[hep-th\]\]. Y. Nomura, arXiv:1110.4630 \[hep-th\]. A. H. Guth and E. J. Weinberg, Nucl. Phys. B [**212**]{}, 321 (1983); A. Vilenkin, Phys. Rev. D [**27**]{}, 2848 (1983); A. D. Linde, Phys. Lett. B [**175**]{}, 395 (1986); Mod. Phys. Lett. A [**1**]{}, 81 (1986). For reviews, see e.g.A. H. Guth, Phys. Rept. [**333**]{}, 555 (2000) \[arXiv:astro-ph/0002156\]; A. Vilenkin, J. Phys. A [**40**]{}, 6777 (2007) \[arXiv:hep-th/0609193\]; S. Winitzki, Lect. Notes Phys. [**738**]{}, 157 (2008) \[arXiv:gr-qc/0612164\]; A. Linde, Lect. Notes Phys. [**738**]{}, 1 (2008) \[arXiv:0705.0164 \[hep-th\]\]. Y. Nomura, AstRv.  [**7**]{}, 36 (2012) \[arXiv:1205.2675 \[hep-th\]\]. A. Mithani and A. Vilenkin, arXiv:1204.4658 \[hep-th\]; L. Susskind, arXiv:1204.5385 \[hep-th\]; arXiv:1205.0589 \[hep-th\]. L. Susskind, L. Thorlacius and J. Uglum, Phys. Rev.  D [**48**]{}, 3743 (1993) \[arXiv:hep-th/9306069\]; L. Susskind, Phys. Rev. Lett.  [**71**]{}, 2367 (1993) \[arXiv:hep-th/9307168\]; C. R. Stephens, G. ’t Hooft and B. F. Whiting, Class. Quant. Grav.  [**11**]{}, 621 (1994) \[arXiv:gr-qc/9310006\]. G. ’t Hooft, arXiv:gr-qc/9310026; L. Susskind, J. Math. Phys.  [**36**]{}, 6377 (1995) \[arXiv:hep-th/9409089\]; R. Bousso, JHEP [**07**]{}, 004 (1999) \[arXiv:hep-th/9905177\]. R. Bousso and J. Polchinski, JHEP [**06**]{}, 006 (2000) \[arXiv:hep-th/0004134\]; S. Kachru, R. Kallosh, A. Linde and S. P. Trivedi, Phys. Rev. D [**68**]{}, 046005 (2003) \[arXiv:hep-th/0301240\]; L. Susskind, arXiv:hep-th/0302219; M. R. Douglas, JHEP [**05**]{}, 046 (2003) \[arXiv:hep-th/0303194\]. See, e.g., R. Bousso, Rev. Mod. Phys.  [**74**]{}, 825 (2002) \[hep-th/0203101\], Section VII A. S. Weinberg, Phys. Rev. Lett. [**59**]{}, 2607 (1987). G. Larsen, Y. Nomura and H. L. L. Roberts, Phys. Rev. D [**84**]{}, 123512 (2011) \[arXiv:1107.3556 \[hep-th\]\]. L. Dyson, M. Kleban and L. Susskind, JHEP [**10**]{}, 011 (2002) \[arXiv:hep-th/0208013\]. R. Bousso, arXiv:1112.3341 \[hep-th\]. H. Ollivier, D. Poulin and W. H. Zurek, Phys. Rev. Lett. [**93**]{}, 220401 (2004) \[arXiv:quant-ph/0307229\]; R. Blume-Kohout and W. H. Zurek, Phys. Rev.  A [**73**]{}, 062310 (2006) \[arXiv:quant-ph/0505031\]. P. A. M. Dirac, [*Lectures on Quantum Mechanics*]{}, (Belfer Graduate School of Science, Yeshiva University, New York, 1964). B. S. DeWitt, Phys. Rev.  [**160**]{}, 1113 (1967). R. Arnowitt, S. Deser and C. W. Misner, in [*Gravitation: An Introduction to Current Research*]{}, ed. L. Witten (John Wiley & Sons, New York, 1962) \[gr-qc/0405109\]. R. Bousso, B. Freivogel and I-S. Yang, Phys. Rev. D [**79**]{}, 063513 (2009) \[arXiv:0808.3770 \[hep-th\]\]; A. De Simone, A. H. Guth, A. Linde, M. Noorbala, M. P. Salem and A. Vilenkin, Phys. Rev. D [**82**]{}, 063520 (2010) \[arXiv:0808.3778 \[hep-th\]\]. S. R. Coleman and F. De Luccia, Phys. Rev. D [**21**]{}, 3305 (1980). S. W. Hawking and I. G. Moss, Phys. Lett. B [**110**]{}, 35 (1982). E. J. Weinberg, Phys. Rev. Lett.  [**98**]{}, 251303 (2007) \[hep-th/0612146\]. [^1]: There are several varying, though related, definitions of the measure problem in literature. In this paper we adopt the definition as stated here. [^2]: For $d=3$, one can add terms $\varDelta [J, K_i] = i\gamma G_N P_i$ and $\varDelta [K_1,K_2] = -i\gamma G_N H$, where $J \equiv J_{[12]}$ and $\gamma$ is a real constant, without violating Jacobi identities. The significance of this is not clear. [^3]: It is, in principle, possible that the predictions are reference frame independent because the multiverse is in an intrinsically mixed state that satisfies $S \rho(t) S^\dagger = \rho(t)$ at all $t$ but each component $\ket{\psi_i(t)}$ in $\rho(t)$ is [*not*]{} a simultaneous eigenstate of all the $S$’s. This is, however, the case only if $\rho(t)$ is the maximally mixed state in ${\cal H}_{\rm QG}$ (because of Schur’s lemma), which is observationally excluded as we saw in Section \[subsec:max-ig\]. We must therefore require that each pure-state component leads to reference-frame independent predictions even if the multiverse is in a mixed state. [^4]: There is also the possibility that some (or all) of the states given by Eq. (\[eq:sol-cond\]) do not contain any Minkowski or singularity components. This does not affect any of our discussions below.

--- author: - | HOLGER STARK $\,$\ Institut für Theoretische und Angewandte Physik, Universität Stuttgart, $\enspace$\ Pfaffenwaldring 57, 70550 Stuttgart, Germany $\,$\ $\enspace$ title: | $\enspace$\ **Radiative Transfer Theory and Diffusion of Light in $\,$\ Nematic Liquid Crystals $\enspace$** --- In nematic liquid crystals light is strongly scattered from director fluctuations. We are interested in the limit where the incoming light wave is scattered many times. Then, the light transport can be described by a diffusion equation for the energy density of light with diffusion constants $D_{\|}$ and $D_{\perp}$, respectively, parallel and perpendicular to the director. We start from a radiative transfer theory, connect the diffusion constants to the dynamic structure factor of director fluctuations, and shortly discuss our results. Temporal correlations of the diffusing light probe the dynamics of director modes on much shorter time scales than single light scattering experiments. To account for the decaying temporal correlations, one has to add an absorption term to the diffusion equation, which we also link to the dynamic structure factor. $\enspace$\ Nematic liquid crystals; light scattering; radiative transfer theory; diffusion\ $\enspace$\ **1. INTRODUCTION** {#introduction .unnumbered} =================== More then one decade ago the discovery of coherent backscattering or weak localization of light in colloidal suspensions \[1\] initiated a tremendous research activity in multiple light scattering \[2\]. Researchers were attracted by the possibility to achieve the equivalent to the Anderson localization of electrons in disordered solids \[3\]. So far strong localization of light waves has not been observed. In the theoretical description of coherent backscattering the diffusion limit for multiply scattered light is employed \[4\]. Photons are considered as random walkers with a scattering mean free path $l$, measuring the length they travel between two scattering events, and a diffusion constant $D = c l^{\ast} / 3$, which involves the transport mean free path $l^{\ast} = l / \langle 1 - \cos\vartheta_{s} \rangle$. It stands for the path length beyond which the direction of propagation of a photon is randomized. The angular brackets denote an average over all possible scattering events and $\vartheta_{s}$ is the scattering angle. In 1987 Wolf and Maret discovered that diffusing light could be used for spectroscopy \[5\], which was later called Diffusing Wave Spectroscopy (DWS) by Pine [*et al.*]{} \[6\]. This was an important step forward, since so far turbid systems could not be investigated with conventional dynamic light scattering. In DWS temporal correlations of the detected intensities decay much faster than in single scattering since phase shifts of electric field waves from many scattering events are added up. DWS, therefore, detects dynamic phenomena at much shorter time scales then normal dynamic light scattering \[7\]. Research on diffusing light and DWS has focused on isotropic media, like colloidal suspensions \[7, 8\], emulsions \[9\], and foams \[10\]. Recently, diffusing photons were used for the imaging of objects \[11, 12\] which found its application in medicine \[13\]. In magnetic fields a photonic Hall effect was discovered \[14\]. Interest in diffusing light in nematic liquid crystals was again initiated by the observation of coherent backscattering first in multidomain samples \[15\] and then in a perfectly aligned nematic state \[16\]. In treating multiply scattered light in the nematic phase, one has to deal with the anisotropy in light propagation and a different scattering mechanism. Light is not scattered from local objects like particles in colloidal suspensions but rather from long-range correlated director fluctuations \[17, 18\]. The theory for diffusing light in nematic liquid crystals was developed independently by Stark and Lubensky \[19, 20, 21\] and Tiggelen, Maynard, and Heiderich \[22, 23\]. For a review see ref. \[24\]. Its final content can be summarized in an anisotropic diffusion equation with absorption for the electric field autocorrelation function $W({\mathbfit{R}},T,t) = \langle {\mathbfit{E}}({\mathbfit{R}},T+t/2) \cdot {\boldsymbol{\varepsilon}}_{0} {\mathbfit{E}}({\mathbfit{R}},T-t/2) \rangle$: $$\left[ \frac{\partial}{\partial T} - D_{\|} \nabla^{2}_{\|} - D_{\perp} \mathbf{\nabla}^{2}_{\perp} + \mu(t) \right] W({\mathbfit{R}},T,t) = 0 \enspace,$$ \ where $D_{\|}$ and $D_{\perp}$ denote, respectively, the diffusion constants parallel and perpendicular to the nematic director. The dynamic absorption coefficient $\mu(t)$, measured in DWS experiments, governs the temporal decay of the autocorrelation function \[$\mu(t=0) =0$\]. The mentioned theories link the parameters $D_{\|}$, $D_{\perp}$, and $\mu(t)$ to the structure factor of the director fluctuations. The absorption coefficient $\mu(t)$ appears as an angular average over all director modes. Careful experiments by Kao [*et al.*]{} demonstrated the anisotropic diffusion and the application of DWS \[21, 25\]. Their results were in excellent agreement with theory. The developed theories used a Green function approach to arrive at the diffusion equation. In the present article we present a derivation without the extended formalism of Green functions but which nevertheless contains the main ideas. To do so we start from a radiative transfer theory which was first introduced as early as 1905 by Schuster to describe the transport of light through the atmosphere \[26\]. The radiative transfer theory is basically equivalent to a Boltzmann equation for the energy density of light \[27, 28\]. Any coherent superposition of electric field waves are omitted. For anisotropic media it needs some refinements which we present in section 3. The advantage of the radiative transfer theory is that it only requires some intuitive understanding. It can, however, be derived directly from Maxwell’s theory. For nematic liquid crystals this was done first by Romanov and Shalaginov \[29\]. The article intends to stimulate further research on diffusing light in liquid crystals. There exists a wealth of material which scatters light very strongly, namely porous media filled with nematics \[30\], polymer dispersed liquid crystals \[31\], focal conic textures in cholesterics \[32\], and the Blue Phase III. Also applications to lyotropic and polymeric liquid crystals and to liquid crystalline colloids could be of interest. Section 2 first summarizes light propagation in a homogeneous nematic phase emphasizing the differences to isotropic systems. Then we review single light scattering from director fluctuations and introduce the dynamic structure factor. Section 3 presents the transport equation for radiative transfer. An approximation for the diffusion constants is derived and the formula for the dynamic absorption coefficient is developed. The main features of both quantities are discussed. **2. LIGHT PROPAGATION AND SINGLE SCATTERING IN NEMATIC LIQUID CRYSTALS** {#light-propagation-and-single-scattering-in-nematic-liquid-crystals .unnumbered} ========================================================================= There are two necessary ingredients to treat multiple light scattering in a turbid medium: the propagation of a plane wave between scattering events in an effectively homogeneous system and the single scattering itself. We start with the first point. **A. Light Propagation** {#a.-light-propagation .unnumbered} ------------------------ A homogeneous nematic with the equilibrium Frank director ${\mathbfit{n}}_{0}$ represents a uniaxial and therefore birefringent material. It posseses the dielectric tensor \[17, 18\] $$\label{1.1} {\boldsymbol{\varepsilon}}_{0} = \varepsilon_{\perp} \mathbf{1} + \Delta \varepsilon {\mathbfit{n}}_{0} \otimes {\mathbfit{n}}_{0} \enspace ,$$\ where $\varepsilon_{\|}$ and $\varepsilon_{\perp}$ denote the dielectric constants for electric fields, respectively, parallel and perpendicular to the director, and where $\Delta \varepsilon = \varepsilon_{\|} - \varepsilon_{\perp}$ stands for the dielectric anisotropy. There exist two characteristic light modes which we describe by plane waves for the electric field: ${\mathbfit{E}}({\mathbfit{r}},t) = E^{\alpha} {\mathbfit{e}}_{\alpha}(\hat{{\mathbfit{k}}}) \, \exp[i(-\omega t + {\mathbfit{k}} \cdot {\mathbfit{r}} )]$. They are characterized by the polarization vector ${\mathbfit{e}}_{\alpha}(\hat{{\mathbfit{k}}})$ and the refractive index $n_{\alpha}(\hat{{\mathbfit{k}}}) = ck/\omega$ \[33\]. The ordinary light wave ($\alpha = 2$) behaves as in an isotropic medium. It has a constant index of refraction $n_{2} = \sqrt{\varepsilon_{\perp}}$ and the polarization vector ${\mathbfit{e}}_{2}(\hat{{\mathbfit{k}}})$ is perpendicular to both ${\mathbfit{n}}_{0}$ and ${\mathbfit{k}}$. For the extraordinary mode ($\alpha =1$), however, $n_{1}(\hat{{\mathbfit{k}}})$ depends on the angle $\vartheta$ between ${\mathbfit{k}}$ and ${\mathbfit{n}}_{0}$: $$\label{1.2} \frac{1}{n_{1}^{2}(\hat{{\mathbfit{k}}})} = \frac{\sin^{2}\vartheta}{\varepsilon_{\|}} + \frac{\cos^{2}\vartheta}{\varepsilon_{\perp}} \enspace.$$ \ The vector ${\mathbfit{e}}_{1}(\hat{{\mathbfit{k}}})$ lies in the plane defined by ${\mathbfit{n}}_{0}$ and ${\mathbfit{k}}$ but is not perpendicular to the wave vector ${\mathbfit{k}}$. It makes sense to introduce the polarization vector ${\mathbfit{d}}^{\alpha}(\hat{{\mathbfit{k}}}) = {\mathbfit{\varepsilon}}_{0} {\mathbfit{e}}_{\alpha}(\hat{{\mathbfit{k}}})$ for the electric displacement field. Then, the biorthogonality relation ${\mathbfit{d}}^{\alpha} \cdot {\mathbfit{e}}_{\beta} = \delta^{\alpha}_{\beta}$ holds \[20, 34\]. The normalization of the polarization vectors is chosen such that the energy density ${\mathbfit{E}}\cdot {\boldsymbol{\varepsilon}}_{0} {\mathbfit{E}} /4\pi$ of a light mode, averaged over on time period, becomes $W^{\alpha} = |E^{\alpha}|^{2} / 8\pi$. Besides the phase velocity ${\mathbfit{v}}_{p \alpha} = c \hat{{\mathbfit{k}}}/ n_{\alpha}$ there exists the group velocity ${\mathbfit{v}}_{g\alpha} = {\boldsymbol{\nabla }}_{{\mathbfit{k}}} \omega_{\alpha}({\mathbfit{k}})$, where $\omega_{\alpha}({\mathbfit{k}}) = c k / n_{\alpha}(\hat{{\mathbfit{k}}})$ stands for the dispersion relation. Electromagnetic energy is transported along the direction of ${\mathbfit{v}}_{g\alpha}$. For ordinary light ${\mathbfit{v}}_{g2} = {\mathbfit{v}}_{p2}$. For extraordinary light the group velocity $$\label{1.3} {\mathbfit{v}}_{g1} = c n_{1}(\hat{{\mathbfit{k}}}) \, \left( \frac{\cos \vartheta}{\varepsilon_{\perp}} {\mathbfit{n}}_{0} + \frac{\sin \vartheta}{\varepsilon_{\|}} \hat{{\mathbfit{u}}}_{1} \right)$$ \ differs in direction and magnitude from the phase velocity. We introduced the unit vector $\hat{{\mathbfit{u}}}_{1}$ perpendicular to both ${\mathbfit{n}}_{0}$ and ${\mathbfit{e}}_{2}(\hat{{\mathbfit{k}}})$. In a medium without absorption the energy transport is described by the Poynting vector $c({\mathbfit{E}} \times {\mathbfit{H}}) / 4\pi$. It can be rewritten so that for each light mode it equals the energy density times the group velocity: ${\mathbfit{S}}^{\alpha} = W^{\alpha} {\mathbfit{v}}_{g\alpha}$ \[33\]. **B. Single Scattering** {#b.-single-scattering .unnumbered} ------------------------ Single light scattering from thermally activated director modes has been well understood for a long time \[17, 18\]. The fluctuating part of the director, $\delta {\mathbfit{n}}({\mathbfit{r}},t)$, induces fluctuations in the dielectric tensor, $\delta {\boldsymbol{\varepsilon}} = \Delta \varepsilon [\delta {\mathbfit{n}} \otimes {\mathbfit{n}}_{0} + {\mathbfit{n}}_{0} \otimes \delta {\mathbfit{n}}]$, which scatter light. The typical scattering experiment involves incoming light with wave vector $ {\mathbfit{k}}^{\alpha} = \omega n_{\alpha} \hat{{\mathbfit{k}}} /c$ and with polarization ${\mathbfit{e}}_{\alpha}(\hat{{\mathbfit{k}}})$ which is partially scattered into light with wave vector ${\mathbfit{q}}^{\beta} = \omega n_{\beta} \hat{{\mathbfit{q}}} /c$ and polarization ${\mathbfit{e}}_{\beta}(\hat{{\mathbfit{q}}})$. In the weak-scattering approximation, the temporal autocorrelation function of the scattered electric field is proportional to the dynamic structure factor $$\label{1.4} B_{\alpha \beta}(\hat{{\mathbfit{k}}}, \hat{{\mathbfit{q}}}, t) = \frac{\omega^{4}}{c^{4}} \langle \, \delta \varepsilon_{\alpha \beta}({\mathbfit{q}}_{s},t) \, \delta \varepsilon_{\alpha \beta}^{\ast}({\mathbfit{q}}_{s},0) \, \rangle \enspace ,$$ \ where ${\mathbfit{q}}_{s} = {\mathbfit{q}}^{\beta} - {\mathbfit{k}}^{\alpha}$ denotes the scattering vector and $\delta \varepsilon_{\alpha \beta}({\mathbfit{q}}_{s},t) = $${\mathbfit{e}}_{\beta}(\hat{{\mathbfit{q}}}) \cdot \delta {\boldsymbol{\varepsilon}}({\mathbfit{q}}_{s},t) {\mathbfit{e}}_{\alpha}(\hat{{\mathbfit{k}}}) $ stands for the projection of $\delta {\boldsymbol{\varepsilon}}$ on the polarization vectors. In its final form we obtain for the structure factor: $$\label{1.5} B_{\alpha \beta}(\hat{{\mathbfit{k}}}, \hat{{\mathbfit{q}}}, t) = (\Delta \varepsilon)^{2} \frac{\omega^{4}}{c^{4}} \sum_{\delta =1}^{2} N(\alpha,\beta,\delta) \frac{k_{\mathrm{B}} T}{K_{\delta}({\mathbfit{q}}_{s})} \, \exp\left[-\frac{K_{\delta}({\mathbfit{q}}_{s})}{\eta_{\delta}({\mathbfit{q}}_{s})} t \right] \enspace.$$ \ The term $k_{\mathrm{B}} T / K_{\delta}({\mathbfit{q}}_{s})$ represents the thermally activated director modes. It involves the elastic coefficient $K_{\delta}({\mathbfit{q}}_{s}) = K_{\delta} q^{2}_{\perp} + K_{3} q^{2}_{\|} + \Delta \chi H^{2}$ with Frank constants $K_{i}$, magnetic field $H$, magnetic anisotropy $\Delta \chi$, and the components of ${\mathbfit{q}}_{s} = ({\mathbfit{q}}_{\perp} , q_{\|})$. The exponential factor reflects the diffusive nature of the director modes. The relaxation frequency depends on the viscosity $\eta_{\delta}({\mathbfit{q}}_{s}) = \gamma - \mu({\mathbfit{q}}_{s})$, where the rotational viscosity $\gamma$ plays the important role. Finally, $N(\alpha,\beta,\delta)$ stands for a geometry factor which forbids ordinary-to-ordinary scattering and forward scattering along the director. From the structure factor we derive the scattering mean free path $l_{\alpha}(\hat{{\mathbfit{k}}})$ \[20, 22, 35\]: $$\label{1.6} \frac{1}{l_{\alpha}(\hat{{\mathbfit{k}}})} = n_{\alpha}(\hat{{\mathbfit{k}}}) \sum_{\beta} \int \frac{d\Omega_{{\mathbfit{q}}}}{(4\pi)^{2}} B_{\alpha \beta}(\hat{{\mathbfit{k}}}, \hat{{\mathbfit{q}}}, 0) n_{\beta}^{3}(\hat{{\mathbfit{q}}}) \enspace,$$ \ which depends on the polarization $\alpha$ and the direction $\hat{{\mathbfit{k}}}$ of light. In the photon picture it gives the average path length a photon travels between two scattering events. For light intensity it gives the distance after which the initial intensity $I_{0}$ has decayed to $I_{0}/e$. However, multiple scattering events are totally neglected in the last interpretation. For a careful interpretation of the scattering mean free path in connection with $\hat{{\mathbfit{k}}}$ and ${\mathbfit{v}}_{g\alpha}$ see ref.. The mean free path is discussed in refs.. We only note, that $l_{1}(\hat{{\mathbfit{k}}})$ goes to zero when $H \rightarrow 0$ since the structure factor for extraordinary-to-extraordinary scattering diverges for ${\mathbfit{q}}_{s} \rightarrow \mathbf{0}$. Finally, we make some comments about the symmetry of the structure factor. In isotropic systems $B(\hat{{\mathbfit{k}}} \cdot \hat{{\mathbfit{q}}})$ only depends on the scattering angle $\vartheta_{s}$ via $\hat{{\mathbfit{k}}} \cdot \hat{{\mathbfit{q}}} = \cos \vartheta_{s}$. Therefore, if expanded into spherical harmonics $Y_{l'm'}(\hat{{\mathbfit{k}}})$ and $Y_{lm}(\hat{{\mathbfit{q}}})$, it is fully diagonal: $\langle l'm' | B(\hat{{\mathbfit{k}}} \cdot \hat{{\mathbfit{q}}}) | lm \rangle \propto \delta_{ll'} \delta_{mm'}$. The structure factor for the director modes just posseses the rotational symmetry around the equilibrium director ${\mathbfit{n}}_{0}$. It can be written as a function of the relative azimuthal angle $\varphi = \varphi_{{\mathbfit{q}}} - \varphi_{{\mathbfit{k}}}$ between $\hat{{\mathbfit{q}}}$ and $\hat{{\mathbfit{k}}}$. Using the additional symmetry $B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},0) = B_{\alpha \beta}(-\hat{{\mathbfit{k}}},-\hat{{\mathbfit{q}}},0)$ the following expansion holds: $$\label{1.7} B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},0) = \sum_{m \ge 0} B_{\alpha \beta}^{m}(\vartheta_{{\mathbfit{k}}},\vartheta_{{\mathbfit{q}}}) \cos[m(\varphi_{{\mathbfit{q}}} - \varphi_{{\mathbfit{k}}})] \enspace,$$ \ where $\vartheta_{{\mathbfit{k}}}$ and $\vartheta_{{\mathbfit{q}}}$ are polar angles of $\hat{{\mathbfit{k}}}$ and $\hat{{\mathbfit{q}}}$ with respect to ${\mathbfit{n}}_{0}$. It expresses the fact that $\langle l'm'| B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},0)| lm \rangle \propto \delta_{mm'}$ is diagonal in the index $m$ but not in $l$. **3. RADIATIVE TRANSFER THEORY AND DIFFUSION APPROXIMATION** {#radiative-transfer-theory-and-diffusion-approximation .unnumbered} ============================================================= In the following we deal with the temporal auto correlation function of the electric field $$\label{2.1} n^{3}_{\alpha}(\hat{{\mathbfit{k}}}) W_{\hat{{\mathbfit{k}}}}^{\alpha}({\mathbfit{R}},T,t) = \langle \, E^{\alpha}_{\hat{{\mathbfit{k}}}}({\mathbfit{R}},T-t/2) \, E^{\alpha\ast}_{\hat{{\mathbfit{k}}}}({\mathbfit{R}},T+t/2) \, \rangle \enspace.$$ \ For $t=0$, it stands for the energy density of a light wave at time $T$ and space point ${\mathbfit{R}}$ travelling into direction $\hat{{\mathbfit{k}}}$ with polarization $\alpha$. The light frequency $\omega$ is omitted. We pulled out a factor $n^{3}_{\alpha}(\hat{{\mathbfit{k}}})$. It is proportional to the number $N_{\alpha}(\omega,\hat{{\mathbfit{k}}}) d \omega d\Omega_{{\mathbfit{k}}}$ of photon states for a given polarization $\alpha$, frequency $\omega$, and direction $\hat{{\mathbfit{k}}}$, since $N_{\alpha}(\omega,\hat{{\mathbfit{k}}}) d \omega d\Omega_{{\mathbfit{k}}} \propto k^{2} d k d\Omega_{{\mathbfit{k}}} = n^{3}_{\alpha}(\hat{{\mathbfit{k}}})\omega^{2} d\omega d\Omega_{{\mathbfit{k}}} / c^{3}$. The equation of radiative transfer theory formally corresponds to a Boltzmann equation balancing all the changes in $W_{\hat{{\mathbfit{k}}}}^{\alpha}({\mathbfit{R}},T,t)$: $$\begin{aligned} \label{2.2} \lefteqn{\left(\frac{\partial}{\partial T} + {\mathbfit{v}}_{g\alpha} \cdot {\boldsymbol{\nabla}} + \frac{1}{l_{\alpha}}\frac{c}{n_{\alpha}}\right) W_{\hat{{\mathbfit{k}}}}^{\alpha}({\mathbfit{R}},T,t) \qquad \qquad} \nonumber \\ & & = c \sum_{\beta} \int \frac{d \Omega_{{\mathbfit{q}}}}{(4\pi)^{2}} B_{\alpha\beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},t) n^{3}_{\beta}(\hat{{\mathbfit{q}}}) W_{\hat{{\mathbfit{q}}}}^{\beta}({\mathbfit{R}},T,t) + S^{\alpha}_{\hat{{\mathbfit{k}}}}({\mathbfit{R}},T) \enspace.\end{aligned}$$ \ The first term gives temporal variations of $W_{\hat{{\mathbfit{k}}}}^{\alpha}({\mathbfit{R}},T,t)$ due to e.g. time dependent light sources. The second term involves the divergence of the Poynting vector ${\mathbfit{v}}_{g\alpha} W_{\hat{{\mathbfit{k}}}}^{\alpha}({\mathbfit{R}},T,t)$. The correlation function $W_{\hat{{\mathbfit{k}}}}^{\alpha}({\mathbfit{R}},T,t)$ changes when there is a net flow of energy ($t=0$) or correlation ($t \ne 0$) out of the volume element around ${\mathbfit{R}}$. The third and fourth terms describe losses and gains due to scattering. The scattering mean free path $l_{\alpha}(\hat{{\mathbfit{k}}})$ has been already introduced in Eq. (\[1.6\]). Finally, $S^{\alpha}_{\hat{{\mathbfit{k}}}}({\mathbfit{R}},T)$ indicates a source term. The transport equation (\[2.2\]) is intuitively understandable. Compared to standard textbooks on multiple light scattering \[27, 28\] it contains two generalizations. First, it is valid for general anisotropic random media. Secondly, for $t \ne 0$, it describes the transport of electric field correlations \[11, 36\]. The transport equation can be derived from first principles, i. e. starting from Maxwell’s theory. It follows in a straightforward way from the Bethe-Salpeter equation for the averaged “two-particle” Green function \[19, 20, 22, 29\]. Its validity is restricted to length and time scales much longer than the wavelength and the time period of light. **A. Diffusion Approximation** {#a.-diffusion-approximation .unnumbered} ------------------------------ To derive the diffusion approximation from Eq. (\[2.2\]) we set $t=0$ and neglect any source terms. We first study the equilibrium solution of Eq. (\[2.2\]), where $W_{\hat{{\mathbfit{k}}}}^{\alpha}({\mathbfit{R}},T)$ equals a constant $W_{0}$. As a result formula (\[1.6\]) for the scattering mean free path $l_{\alpha}(\hat{{\mathbfit{k}}})$ is reproduced. The energy density $ n^{3}_{\alpha}(\hat{{\mathbfit{k}}}) W_{0}$ still depends on the direction $\hat{{\mathbfit{k}}}$ and polarization $\alpha$ of light, due to the equipartition of the light energy on all available photon states. So far, this still needs an experimental confirmation. The diffusion approximation follows when the $\hat{{\mathbfit{k}}}$ dependence of $W_{\hat{{\mathbfit{k}}}}^{\alpha}({\mathbfit{R}},T)$ deviates only slightly from the equilibrium angular distribution \[27, 28\]. Let us therefore look at an angular expansion of $W_{\hat{{\mathbfit{k}}}}^{\alpha}({\mathbfit{R}},T)$: $$\begin{aligned} W_{\hat{{\mathbfit{k}}}}^{\alpha}({\mathbfit{R}},T) & = & \frac{1}{8\pi} W_{0}({\mathbfit{R}},T) + \frac{3}{4\pi} \frac{1}{c \overline{n^{3}_{\alpha}}} \, n_{\alpha} \hat{{\mathbfit{k}}} \cdot {\mathbfit{J}}^{\alpha}({\mathbfit{R}},T) \nonumber\\ \label{2.3} & & + \sum_{l>1,m} W^{\alpha}_{lm}({\mathbfit{R}},T) \widetilde{Y}_{lm}^{\alpha}(\hat{{\mathbfit{k}}}) \enspace,\end{aligned}$$ \ where we introduced the total energy density $$\label{2.4} \overline{n^{3}} W_{0}({\mathbfit{R}},T) = \sum_{\alpha} \int d\Omega_{{\mathbfit{k}}} n^{3}_{\alpha}(\hat{{\mathbfit{k}}}) W^{\alpha}_{\hat{{\mathbfit{k}}}}({\mathbfit{R}},T)$$ \ as $W_{0}$ times an angular average over the cubes of both refractive indices: $$\label{2.5} \overline{n^{3}} = (\overline{n_{1}^{3}} + \overline{n_{2}^{3}}) / 2 = \int d\Omega_{{\mathbfit{k}}} [n^{3}_{1}(\hat{{\mathbfit{k}}}) + n^{3}_{2}(\hat{{\mathbfit{k}}})] / 8\pi = ( \varepsilon_{\perp}^{1/2} \varepsilon_{\|} + \varepsilon_{\perp}^{3/2} )/2 \enspace.$$ \ Our goal is to establish a diffusion equation for $W_{0}({\mathbfit{R}},T)$. We also defined the total energy density current ${\mathbfit{J}}^{\alpha}({\mathbfit{R}},T)$ of light with polarization $\alpha$, $$\label{2.6} {\mathbfit{J}}^{\alpha}({\mathbfit{R}},T) = \int d\Omega_{{\mathbfit{k}}} n^{3}_{\alpha} {\mathbfit{v}}_{g\alpha} W^{\alpha}_{\hat{{\mathbfit{k}}}}({\mathbfit{R}},T) \enspace.$$ \ A few comments are necessary to understand Eq. (\[2.3\]). In isotropic systems, the second term on the right-hand side just corresponds to an expansion into the components of $\hat{{\mathbfit{k}}} = (\sin \vartheta \cos \varphi, \sin \vartheta \sin \varphi, \cos \vartheta)$, which basically stand for $l=1$ spherical harmonics. Here, for uniaxial systems, it is useful to choose basis functions with $n_{\alpha}^{3}(\hat{{\mathbfit{k}}})$ as a weight function: $$\label{2.7} \int d\cos\vartheta d\varphi \, n_{\alpha}^{3}(\hat{{\mathbfit{k}}}) \widetilde{Y}_{lm}^{\alpha}(\hat{{\mathbfit{k}}}) \widetilde{Y}_{l'm'}^{\alpha}(\hat{{\mathbfit{k}}}) = \overline{n^{3}_{\alpha}} \delta_{ll'} \delta_{mm'} \enspace.$$ \ We will see below that this choice establishes an approximation scheme for the diffusion constants of light. For ordinary waves we still obtain the conventional spherical harmonics. For extraordinary waves we introduce a new coordinate $C = n_{1}(\hat{{\mathbfit{k}}}) \cos\vartheta / n_{2}$, which is equivalent to ${\mathbfit{n}}_{0} \cdot \hat{{\mathbfit{k}}} = \cos\vartheta$ since it also ranges from $-1$ to $1$. With this choice the weight function becomes a constant: $$\label{2.8} \int d \cos \vartheta n^{3}_{1}(\hat{{\mathbfit{k}}}) \ldots = \overline{n^{3}_{1}} \int dC \ldots$$ \ Hence, the basis functions $\widetilde{Y}_{lm}^{1}(\hat{{\mathbfit{k}}})$ for $\alpha =1$ simply follow from spherical harmonics when $\cos \vartheta$ is replaced by $C$. With the abbreviation $C=\cos \vartheta$ for ordinary light, the generalized spherical harmonics are the same for $\alpha=1$ and 2. For $l=1$ in real representation, they read: $$\widetilde{Y}^{\alpha}_{10}(\hat{{\mathbfit{k}}}) = \sqrt{\frac{3}{4\pi}} C \quad , \quad \frac{\widetilde{Y}^{\alpha}_{11}(\hat{{\mathbfit{k}}}) \pm \widetilde{Y}^{\alpha}_{1-1}(\hat{{\mathbfit{k}}}) }{2} = \sqrt{\frac{3}{4\pi}} \sqrt{1-C^{2}} \left\{ \begin{array}{c} \cos\varphi \\ \sin{\varphi} \end{array} \right.$$ \ In expansion (\[2.3\]) we have already used them explicitly in the second term. We also note that $n_{1}(\hat{{\mathbfit{k}}}) \sin \vartheta / n_{2} = \sqrt{\varepsilon_{\|\rule[-1.5mm]{0mm}{2mm}} / \varepsilon_{\perp}} \sqrt{1-C^{2}}$. We are ready to extract the diffusion approximation from the transport equation (\[2.2\]). Multiplying Eq. (\[2.2\]) by $n_{\alpha}^{3}$ and summing over all directions of $\hat{{\mathbfit{k}}}$ and the two polarizations leads to the continuity equation for the energy density $$\label{2.9} \frac{\partial }{\partial T} \overline{n^{3}} W_{0} + {\boldsymbol{\nabla}} \cdot {\mathbfit{J}} = 0 \enspace.$$ \ The vector ${\mathbfit{J}} = {\mathbfit{J}}^{1} + {\mathbfit{J}}^{2}$ denotes the total energy density current. A second equation, Fick’s law, relates ${\mathbfit{J}}$ to the gradient of the energy density: $$\label{2.10} {\mathbfit{J}} = - {\mathbfit{D}} {\boldsymbol{\nabla}} \overline{n^{3}}W_{0} \enspace.$$ \ It will be derived below. We introduced the diffusion tensor $$\label{2.11} {\mathbfit{D}} = D_{\perp} \mathbf{1} + (D_{\|} - D_{\perp}) {\mathbfit{n}}_{0} \otimes {\mathbfit{n}}_{0}$$ \ with its two independent light diffusion constants $D_{\|}$ and $D_{\perp}$, respectively, parallel and perpendicular to the director ${\mathbfit{n}}_{0}$. Eliminating the current ${\mathbfit{J}}$ finally gives the diffusion equation for $W_{0}({\mathbfit{R}},T)$, $$\label{2.12} \left( \frac{\partial}{\partial T} - D_{\|} \nabla_{\|}^{2} - D_{\perp} {\boldsymbol{\nabla}}_{\perp}^{2} \right) W_{0}({\mathbfit{R}},T) = 0 \enspace,$$ \ where ${\boldsymbol{\nabla}} = ({\boldsymbol{\nabla}}_{\perp} , \nabla_{\|\rule[-1.5mm]{0mm}{2mm}} )$. To obtain Fick’s law (\[2.10\]), we first look at components parallel to the director ${\mathbfit{n}}_{0}$. We project the transport equation (\[2.2\]) on $ \widetilde{Y}^{\alpha}_{10}(\hat{{\mathbfit{k}}}) \propto C_{{\mathbfit{k}}}$ and arrive at a set of equations which couple the energy density $W_{0}$ and the components $J^{1}_{\|}$ and $J^{2}_{\|}$: $$\label{2.13} \frac{(4\pi)^{3}}{18} \frac{c}{n_{2}^{2}} \nabla_{\|} W_{0} \left( \begin{array}{c} 1 \\ 1 \end{array} \right) + \left( \begin{array}{cc} {\cal B}_{11}^{\|} & {\cal B}_{12}^{\|} \\ {\cal B}_{12}^{\|} & {\cal B}_{22}^{\|} \end{array} \right) \, \left( \begin{array}{c} J^{1}_{\|} \\ J^{2}_{\|} \end{array} \right) + \dots = \mathbf{0} \enspace.$$ \ The quantitites ${\cal B}_{\alpha \beta}^{\|}$ are extended matrix elements of $B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}})$: $$\begin{aligned} {\cal B}_{11}^{\|} & = & \int_{\hat{{\mathbfit{k}}}} \int_{\hat{{\mathbfit{q}}}} \, [(C_{{\mathbfit{k}}}^{2} - C_{{\mathbfit{k}}}C_{{\mathbfit{q}}}) B_{11}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}}) + C_{{\mathbfit{k}}}^{2} \frac{\varepsilon_{\perp}}{\varepsilon_{\|}} B_{12}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}})] \nonumber \\ \label{2.14} {\cal B}_{22}^{\|} & = & \int_{\hat{{\mathbfit{k}}}} \int_{\hat{{\mathbfit{q}}}} \, [(C_{{\mathbfit{k}}}^{2} - C_{{\mathbfit{k}}}C_{{\mathbfit{q}}}) B_{22}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}}) + C_{{\mathbfit{k}}}^{2} \frac{\varepsilon_{\perp}}{\varepsilon_{\|}} B_{21}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}})] \\ {\cal B}_{12}^{\|} & = & - \int_{\hat{{\mathbfit{k}}}} \int_{\hat{{\mathbfit{q}}}} C_{{\mathbfit{k}}} C_{{\mathbfit{q}}} B_{12}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}}) \nonumber\end{aligned}$$ \ where we used the abbreviation $\int_{\hat{{\mathbfit{k}}}} = \int dC_{{\mathbfit{k}}} d \varphi_{{\mathbfit{k}}} $. In Eq. (\[2.13\]) we neglected terms proportional to $\partial J^{\alpha}_{\|} / \partial T$. We also assumed that further contributions containing $W^{\alpha}_{l0}$ with $l>1$ are small and discuss this approximation below. Solving for the two currents $J_{\|}^{1}$ and $J_{\|}^{2}$ leads to the parallel component of Eq. (\[2.10\]): $$\label{2.15} J_{\|} = J_{\|}^{1} + J_{\|}^{2} = - D_{\|} \nabla_{\|} \overline{n^{3}} W_{0}$$ \ with the diffusion constant $$\label{2.16} D_{\|} = \frac{(4\pi)^{3}}{18} \, \frac{c}{n_{2}^{2} \overline{n^{3}}} \, \frac{{\cal B}_{11}^{\|} + {\cal B}_{22}^{\|} -2 {\cal B}_{12}^{\|}}{ {\cal B}_{11}^{\|} {\cal B}_{22}^{\|} - ({\cal B}_{12}^{\|})^{2} } \enspace.$$ \ The perpendicular component of Eq. (\[2.10\]) follows in an analogous way after projecting the transport equation (\[2.2\]) on $[\widetilde{Y}_{11}^{\alpha}(\hat{{\mathbfit{k}}}) + \widetilde{Y}_{1-1}^{\alpha}(\hat{{\mathbfit{k}}}) ] \propto$$\sqrt{1-C_{{\mathbfit{k}}}^{2}} \cos \varphi_{{\mathbfit{k}}}$: $$\label{2.17} J_{\perp} = J_{\perp}^{1} + J_{\perp}^{2} = - D_{\perp} {\boldsymbol{\nabla}}_{\perp} \overline{n^{3}} W_{0}$$ with $$\label{2.18} D_{\perp} = \frac{(4\pi)^{3}}{18} \, \frac{c}{n_{2}^{2} \overline{n^{3}}} \, \frac{{\cal B}_{11}^{\perp} + {\cal B}_{22}^{\perp} \varepsilon_{\perp}/ \varepsilon_{\|} - 2 {\cal B}_{12}^{\perp} \sqrt{\varepsilon_{\perp}/\varepsilon_{\|}}}{ {\cal B}_{11}^{\perp} {\cal B}_{22}^{\perp} - ({\cal B}_{12}^{\perp})^{2} } \enspace.$$ \ The matrix elements ${\cal B}_{\alpha \beta}^{\perp}$ are defined as in Eqs. (\[2.14\]) but with $C$ replaced by $\sqrt{1-C^{2}} \cos \varphi$. The expressions (\[2.16\]) and (\[2.18\]) for the diffusion constants are approximate formulas since in Eq. (\[2.13\]) we have neglected terms containing $W^{\alpha}_{l0}$ with $l>1$. In ref.we showed that corrections of $D_{\|}$ and $D_{\perp}$ resulting from l=3 spherical harmonics are essentially 1% or smaller. One could ask for the reason. The additional coefficients $W^{\alpha}_{l0}$ in Eq. (\[2.13\]) with the non-diagonal matrix elements $\langle 10 | B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},0)|l0 \rangle$ as prefactors have to be determined by projecting the transport equation on higher generalized spherical harmonics ($l>1$). Because of the choice of our basis functions the resulting equations only couple $W^{\alpha}_{l0}$ to $J_{\|}^{\beta}$. The coefficient $W_{0}$ does not appear. That means $W^{\alpha}_{l0}$ ($l > 1$) depends in a complicated way but directly on $J_{\|}^{\beta}$. As a result the matrix elements ${\cal B}^{\|}_{\alpha \beta}$ are renormalized. However, the renormalization is small if the non-diagonal elements $\langle 10 | B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},0)|l0\rangle$ are small. For symmetry reasons it is clear that $D_{\|}$ and $D_{\perp}$ involve, respectively, $m=0$ or $|m| = 1$ spherical harmonics only. In isotropic systems higher spherical harmonics do not contribute at all, since $\langle 10 | B(\hat{{\mathbfit{k}}} \cdot \hat{{\mathbfit{q}}},0)|l0\rangle \propto \delta_{l1}$, and $D \propto \langle 1 - \cos \vartheta_{s} \rangle^{-1}$ follows. An extensive discussion of the diffusion constants and their dependence on Frank constants $K_{i}$, dielectric anisotropy $\Delta \varepsilon$, and applied magnetic field $H$ is given in refs. and by Tiggelen [*et al.***]{} \[22\]. We just summarize the important results. For a typical material, 5CB, we find $D_{\|} = 1.43 \times 10^{9} \mathrm{cm}^{2}/\mathrm{s}$ and $D_{\perp} = 0.98 \times 10^{9} \mathrm{cm}^{2}/\mathrm{s}$ with a ratio of $D_{\|}/D_{\perp} = 1.45$ in excellent agreement with experiment and numerical simulations \[21, 25\]. The values were calculated in the limit of $H \rightarrow 0$, which demonstrates that the divergence of the structure factor for ${\mathbfit{q}}_{s} \rightarrow \mathbf{0}$ does not affect the diffusion constants. If one introduces transport mean free paths via $ l^{\ast}_{\|/\perp} = 3 n_{2} D_{\|/\perp} / c $, one arrives at $l_{\|} = 2.2 \mathrm{mm}$ and $l_{\perp} = 1.5 \mathrm{mm}$ for 5CB. We stress that it is not obvious how to define transport mean free paths in anisotropic turbid media. The calculated values give an orientation only. In the case of $\Delta \varepsilon = 0$ and the one-constant approximation $K_{i} = K$ there is still a remaining anisotropy of $D_{\|} / D_{\perp} = 1.06$ due to the inherent anisotropy in the nematic structure factor. For $\Delta \varepsilon < 0$ there exists a point with $D_{\|} = D_{\perp}$ beyond which the ratio $D_{\|} / D_{\perp}$ is smaller than 1. This behavior should be observable in discotic nematics. **B. Dynamic Absorption** {#b.-dynamic-absorption .unnumbered} ------------------------- Now, we set $t \ne 0$ and look at the transport of electric field correlations. In doing so, we restrict ourselves to times $t$ much smaller than typical director relaxation times $\tau = \gamma / (K q_{s}^{2})$. For wave numbers $q_{s}$ of light they cover the range $\tau = 10 - 100 \,\mu \mathrm{s}$. We rewrite the dynamical structure factor, $$\label{2.19} B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},t) = B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},0) + [B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},t) - B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},0) ] \enspace,$$ \ where the second part on the right-hand side assumes the form $$\label{2.20} B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},t) - B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},0) = - (\Delta \varepsilon)^{2} k_{\mathrm{B}} T \frac{\omega^{4}}{c^{4}} \sum_{\delta =1}^{2} \frac{N(\alpha,\beta,\delta)}{\eta_{\delta} ({\mathbfit{q}}_{s})} \, t$$ \ after an expansion of the exponential factor in $B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},t)$. All the coefficients in an angular expansion of $W_{\hat{{\mathbfit{k}}}}^{\alpha}({\mathbfit{R}},T,t)$ now carry the relative time $t$ as a further argument. The important quantity is the total autocorrelation function $$\label{2.21} \overline{n^{3}} W_{0}({\mathbfit{R}},T,t) = \sum_{\alpha} \int d\Omega_{{\mathbfit{k}}} n^{3}_{\alpha}(\hat{{\mathbfit{k}}}) W^{\alpha}_{\hat{{\mathbfit{k}}}}({\mathbfit{R}},T,t) \enspace.$$ \ In repeating the derivation of the continuity equation (\[2.9\]) we have to add a dynamic absorption term $\mu(t) \overline{n^{3}} W_{0}({\mathbfit{R}},T,t)$: $$\label{2.22} \left[ \frac{\partial }{\partial T} + \mu(t) \right] \overline{n^{3}} W_{0}({\mathbfit{R}},T,t) + {\boldsymbol{\nabla}} \cdot {\mathbfit{J}}({\mathbfit{R}},T,t) = 0 \enspace,$$ \ which means that temporal correlations are not conserved quantities because they decay to zero. The dynamic absorption coefficient $\mu(t)$ follows generally from an angular average over all dynamic modes of a system: $$\mu(t) = \frac{c}{8 \pi \overline{n^{3}}} \sum_{\alpha \beta} \int \! \int \frac{d \Omega_{{\mathbfit{k}}} d \Omega_{{\mathbfit{q}}}}{(4\pi)^{2}} n^{3}_{\alpha}(\hat{{\mathbfit{k}}}) [B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},0) - B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},t) ] n^{3}_{\beta}(\hat{{\mathbfit{q}}}) \enspace.$$ \ Further terms in Eq. (\[2.22\]) containing coefficients $W^{\alpha}_{lm}$ are small, due to the assumption that $ B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},0) \gg B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},0) - B_{\alpha \beta}(\hat{{\mathbfit{k}}},\hat{{\mathbfit{q}}},t)$, and can be neglected. The same applies to Fick’s law in Eq. (\[2.10\]), so that we arrive at the diffusion equation with absorption which we already formulated in the introduction. Using Eq. (\[2.20\]) for director modes we calculate $$\mu(t) = \mu_{0} t \quad \mathrm{with} \quad \mu_{0} = \frac{2k_{B}T}{9\pi}\, \frac{\omega^{4}}{c^{3}} \, \frac{(\Delta \varepsilon)^{2}}{\sqrt{\varepsilon_{\perp}}} \, \frac{\widetilde{\mu}}{\gamma} \enspace,$$ \ where the numerical factor $\widetilde{\mu}$ stands for a dimensionless angular average over the geometrical factor $N(\alpha,\beta,\delta)$ and the viscosities $\eta_{\delta}({\mathbfit{q}}_{s}) / \gamma$. The factor $\widetilde{\mu}$ is of the order of 1 in thermotropic nematics but always larger than 1 \[20\]. Note that $\mu(t)$ only depends on viscosities and not at all on elastic properties. They cancel because they determine both static light scattering and hydrodynamics of the director modes. For 5CB, $\gamma = 0.81\,\mathrm{P}$ which agrees very well with the experimentally determined value of $\gamma / \widetilde{\mu} = 0.60 \pm 0.20\,\mathrm{P}$ \[21, 25\]. In experiments one fits the autocorrelation function with the help of a numerical or exact solution of the diffusion equation under the appropriate boundary conditions \[7\]. Down to the experimental resolution of $4 \times 10^{-8} \, \mathrm{s}$ no deviation of the director dynamics from the Leslie-Erickson theory was observed. It should show up in a different temporal power law of $\mu(t)$. It would be interesting to study systems with higher viscosities like polymer liquid crystals and to look for such a deviation. The Brownian motion of colloidal particles in a fluid e.g. clearly does not show a simple diffusive behavior on very short time scales \[7, 37\]. **Acknowledgements** {#acknowledgements .unnumbered} -------------------- I thank Bart van Tiggelen, Roger Maynard, Georg Maret, Michael Heckmeier and Martin Čopič for stimulating discussions. Tom Lubensky, Arjun Yodh, Ming Kao and Kristen Jester contributed considerably to develop my understanding of diffusing light in nematics. **References** {#references .unnumbered} -------------- \[10\]= Y. Kuga and A. Ishimaru, [*J. Opt. Soc. Am. A*]{}, [**1**]{}, 831 (1984); M. P.\ van Albada and A. Lagendijk, [*Phys. Rev. Lett.*]{}, [ **55**]{}, 2692 (1985);\ P.-E. Wolf and G. Maret, [*Phys. Rev. Lett.*]{}, [**55**]{}, 2696 (1985).\ ,\ edited by P. Sheng (World Scientific, Singapore, 1990).\ P. W. Anderson, [*Phys. Rev.*]{}, [**109**]{}, 1492 (1958).\ A. A. Golubentsev, [*Sov. Phys. JETP*]{}, [**86**]{}, 26 (1984); E. Akkermans\ and R. Maynard, [*J. Physique Lett.*]{}, [**46**]{}, L-1045 (1985); E. Akker-\ mans, P. E. Wolf, and R. Maynard, [*Phys. Rev. Lett.*]{}, [**56**]{}, 1471 (1986).\ G. Maret and P. E. Wolf, [*Z. Phys. B*]{}, [**65**]{}, 409 (1987).\ D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, [*Phys.*]{}\ , [**60**]{}, 1134 (1988).\ D. A. Weitz and D. J. Pine, in [*Dynamic Light Scattering*]{}, edited by\ W. Brown (Oxford University Press, Oxford, 1992), pp. 652–720.\ X. L. Wu [*et al.*]{}, [*J. Opt. Soc. Am. B*]{}, [**7**]{}, 15 (1990).\ H. Gang, A. H. Krall, and D. A. Weitz, [*Phys. Rev. Lett.*]{}, [ **73**]{}, 3435\ (1994).\ \[10\] D. J. Durian, D. A. Weitz, and D. J. Pine, [*Science*]{}, [**252**]{}, 617 (1994).\ \[11\] D. A. Boas, L. J. Campbell, and A. G. Yodh, [*Phys. Rev. Lett.*]{}, [**75**]{},\ 1855 (1995).\ \[12\] M. Heckmeier and G. Maret, [ *Europhys. Lett.*]{}, [**34**]{}, 257 (1996); M.\ Heckmeier, S. E. Skipetrov, G. Maret, and R. Maynard, [ *J. Opt. Soc.*]{}\ , [**14**]{}, 185 (1997).\ \[13\] A. G. Yodh and B. Chance, [*Physics Today*]{}, [**48**]{}, 34 (1995).\ \[14\] B. A. van Tiggelen, [*Phys. Rev. Lett.*]{}, [**75**]{}, 422 (1995); G. L. J. A. Rik-\ ken and B. A. van Tiggelen, [*Nature (London)*]{}, [**381**]{}, 54 (1996).\ \[15\] D. V. Vlasov, L. A. Zubkov, N. V. Orekhova, and V. P. Romanov,\ , [**48**]{}, 91 (1988).\ \[16\] H. K. M. Vithana, L. Asfaw, and D. L. Johnson, [*Phys. Rev. Lett.*]{},\ , 3561 (1993).\ $\!\!$\[17\] P. G. de Gennes and J. Prost, 2nd ed. [*The Physics of Liquid Crys-*]{}\ (Clarendon Press, Oxford, 1993).\ \[18\] S. Chandrasekhar, [*Liquid Crystals*]{}, 2nd ed. (Cambridge University\ Press, Cambridge, 1992).\ \[19\] H. Stark and T. C. Lubensky, [*Phys. Rev. Lett.*]{}, [**77**]{}, 2229 (1996).\ \[20\] H. Stark and T. C. Lubensky, [*Phys. Rev. E*]{}, [**55**]{}, 514 (1997).\ \[21\] H. Stark, M. H. Kao, K. A. Jester, T. C. Lubensky, A. G. Yodh, and\ P. J. Collings, [*J. Opt. Soc. Am. A*]{}, [**14**]{}, 156 (1997).\ \[22\] B. A. van Tiggelen, R. Maynard, and A. Heiderich, [*Phys. Rev. Lett.*]{},\ , 639 (1996); B. A. van Tiggelen, A. Heiderich, and R. Maynard,\ , 205 (1997).\ \[23\] A. Heiderich, R. Maynard, and B. A. van Tiggelen, [ *J. Phys. (France)*]{}\ , [**7**]{}, 765 (1997).\ \[24\] B. A. van Tiggelen and H. Stark, [*Int. J. Mod. Phys. B*]{} (to be pub-\ lished).\ \[25\] M. H. Kao, K. Jester, A. G. Yodh, and P. J. Collings, [ *Phys. Rev.*]{}\ , [**77**]{}, 2233 (1996).\ \[26\] A. Schuster, [*Astrophys. J.*]{}, [**21**]{}, 1 (1905).\ \[27\] A. Ishimaru, [*Wave Propagation in Random Media*]{}, Vols. 1 and 2,\ (Academic, New York, 1978).\ \[28\] H. C. van de Hulst, [*Multiple Light Scattering*]{}, Vols. 1 and 2, (Acade-\ mic, New York, 1980).\ \[29\] V. P. Romanov and A. N. Shalaginov, [*Opt. Spectrosc. (USSR)*]{}, [**64**]{},\ 774 (1988).\ \[30\] T. Bellini and N. A. Clark, in [*Liquid Crystals in Complex Geome-*]{}\ , edited by G. P. Crawford and S. Žumer (Tayler & Francis, Lon-\ don, 1996), Chap. 19, p. 381.\ \[31\] P. S. Drzaic, [*Liquid Crystal Dispersions*]{} (World Scientific, Singapore,\ 1995).\ \[32\] D.-K. Yang, L.-C. Chien, and Y. K. Fung, in [*Liquid Crystals in Com-*]{}\ , edited by G. P. Crawford and S. Žumer (Tayler &\ Francis, London, 1996), Chap. 5, p. 103.\ \[33\] L. D. Landau and E. M. Lifschitz, [*Electrodynamics of Continuous*]{}\ , Vol. 8 of [*Course of Theoretical Physics*]{}, first englisch ed.\ (Pergamon Press, Oxford, 1960).\ \[34\] M. Lax and D. F. Nelson, [*Phys. Rev. B*]{}, [**4**]{}, 3694 (1971).\ \[35\] D. Langevin and M.-A. Bouchiat, [*J. Phys. (Paris) Colloq.*]{}, [**36**]{}, C1-\ 197 (1975); A. Y. Val’kov and V. P. Romanov, [*Sov. Phys. JETP*]{} [**63**]{},\ 737 (1986).\ \[36\] Dougherty [*et al.*]{}, [*J. Quant. Spectrosc. Radiat. Transfer*]{}, [**52**]{}, 713\ (1994).\ \[37\] M. H. Kao, A. G. Yodh, and D. J. Pine, [*Phys. Rev. Lett.*]{}, [**70**]{}, 242\ (1993).

--- abstract: 'The influence of compressibility on the stability of the scaling regimes of the passive scalar advected by a Gaussian velocity field with finite correlation time is investigated by the field theoretic renormalization group within two-loop approximation. The influence of compressibility on the scaling regimes is discussed as a function of the exponents $\varepsilon$ and $\eta$, where $\varepsilon$ characterizes the energy spectrum of the velocity field in the inertial range $E\propto k^{1-2\varepsilon}$, and $\eta$ is related to the correlation time at the wave number $k$ which is scaled as $k^{-2+\eta}$. The restrictions given by nonzero compressibility on the regions with stable infrared fixed points which correspond to the stable infrared scaling regimes are discussed in detail. A special attention is paid to the case of so-called frozen velocity field, when the velocity correlator is time independent. In this case, explicit inequalities which must be fulfilled in the plane $\varepsilon-\eta$ are determined within two-loop approximation. The existence of a “critical” value $\alpha_c$ of the parameter of compressibility $\alpha$ at which one of the two-loop conditions is canceled as a result of the competition between compressible and incompressible terms is discussed. Brief general analysis of the stability of the scaling regime of the model with finite correlations in time of the velocity field within two-loop approximation is also given.' address: - '$^{1}$ Institute of Experimental Physics, Slovak Academy of Sciences, Watsonova 47, 040 01, Košice, Slovakia' - '$^{2}$ Department of Mathematics Faculty of Civil Engineering, Technical University Vysokoškolská 4, 040 01 Košice, Slovakia' author: - 'M Hnatič$^{1,2}$, E Jurčišinová$^{1}$[^1], M Jurčišin$^{1}$[^2] and M Repašan$^{1}$' title: 'Compressible advection of a passive scalar: Two-loop scaling regimes' --- Introduction {#sec1} ============ One of the main problems in the modern theory of fully developed turbulence is to verify the validity of the basic principles of Kolmogorov-Obukhov (KO) phenomenological theory and their consequences within the framework of a microscopic model [@MonYag75; @Frisch95]. On the other hand, recent experimental, numerical and theoretical studies signify the existence of deviations from the well-known Kolmogorov scaling behavior. The scaling behavior of the velocity fluctuations with exponents, which values are different from Kolmogorov ones, is known as anomalous and is associated with intermittency phenomenon [@Frisch95]. Even thought the understanding of the intermittency and anomalous scaling within the theoretical description of the fluid turbulence on basis of the “first principles”, i.e., on the stochastic Navier-Stokes equation, still remains an open problem, considerable progress has been achieved in the studies of the simplified model systems which share some important properties of the real turbulence. The crucial role in these studies is played by models of advected passive scalar field [@Obu49]. Maybe the most known model of this type is a simple model of a passive scalar quantity advected by a random Gaussian velocity field, white in time and self-similar in space, the so-called Kraichnan’s rapid-change model [@Kra68]. It was shown by both natural and numerical experimental investigations that the deviations from the predictions of the classical KO phenomenological theory is even more strongly displayed for a passively advected scalar field than for the velocity field itself (see, e.g., [@FaGaVe01] and references cited therein). At the same time, the problem of passive advection is much more easier to be consider from theoretical point of view. There, for the first time, the anomalous scaling was established on the basis of a microscopic model [@Kraichnan94], and corresponding anomalous exponents was calculated within controlled approximations (see review [@FaGaVe01] and references therein). In paper [@AdAnVa98+] the field theoretic renormalization group (RG) and operator-product expansion (OPE) were used in the systematic investigation of the rapid-change model. It was shown that within the field theoretic approach the anomalous scaling is related to the very existence of so-called “dangerous” composite operators with negative critical dimensions in OPE (see, e.g., [@Vasiliev; @AdAnVa99] for details). Important advantages of the RG approach are its universality and calculational efficiency: a regular systematic perturbation expansion for the anomalous exponents was constructed, similar to the well-known $\epsilon$-expansion in the theory of phase transitions. Afterwards, various generalized descendants of the Kraichnan model, namely, models with inclusion of large and small scale anisotropy [@AdAnHnNo00], compressibility [@AdAn98] and finite correlation time of the velocity field [@Antonov99; @Antonov00] were studied by the field theoretic approach. General conclusion is: the anomalous scaling, which is the most important feature of the Kraichnan rapid change model, remains valid for all generalized models. In paper [@Antonov99] the problem of a passive scalar advected by the Gaussian self-similar velocity field with finite correlation time [@all2] was studied by the field theoretic RG method. There, the systematic study of the possible scaling regimes and anomalous behavior was present at one-loop level. The two-loop corrections to the anomalous exponents were obtained in [@AdAnHo02]. In paper [@Antonov00] the influence of compressibility on the problem studied in [@Antonov99] was analyzed. In what follows, we shall continue with the investigation of this model from the point of view of the influence of compressibility on the stability of the scaling regimes within two-loop approximation. It can lead to sufficient restrictions of the parameter space where the stable fixed points can exist. This, as we shall see rather complicated task, is the first nontrivial step on the way to understand the influence of the compressibility of the system on the two-loop corrections to anomalous dimensions of the measurable quantities [@AdAnHo02]. Description of the model {#sec2} ======================== We consider the advection of a passive scalar field $\theta \equiv \theta(x)\equiv \theta(t, {\bf x})$ which is described by the stochastic equation $$\partial_t \theta + v_i \partial_i \theta=\nu_0 \Delta \theta + f^{\theta},\label{scalar1}$$ where $\partial_t \equiv \partial/\partial t$, $\partial_i \equiv \partial/\partial x_i$, $\nu_0$ is the coefficient of molecular diffusivity (hereafter all parameters with a subscript $0$ denote bare parameters of unrenormalized theory; see below), $\Delta \equiv \partial^2$ is the Laplace operator, and $f^{\theta} \equiv f^{\theta}(x)$ is a Gaussian random noise with zero mean and correlation function $$\langle f^{\theta}(x) f^{\theta}(x^{\prime})\rangle = \delta(t-t^{\prime})C({\bf r}/L), \,\,\, {\bf r}={\bf x}-{\bf x^{\prime}},\label{correlator}$$ where parentheses $\langle...\rangle$ hereafter denote average over corresponding statistical ensemble. The noise maintains the steady-state of the system but the concrete form of the correlator is not essential. The only condition which must be fulfilled by the function $C({\bf r}/L)$ is that it must decrease rapidly for $r\equiv |{\bf r}| \gg L$, where $L$ denotes an integral scale related to the stirring. The velocity field ${\bf v(x)}$ obeys a Gaussian distribution with zero mean and correlator [@Antonov00] $$\begin{aligned} \fl \langle v_i(x) v_j(x^{\prime}) \rangle &=& D^v_{ij}(x,x^{\prime})\label{corv} \\ &=&\int \frac{d\omega d^d k}{(2\pi)^{d+1}} \left(P_{ij}({\bf k}) + \alpha Q_{ij}({\bf k}) \right) \tilde{D}^v(\omega,k) \exp[-i\omega(t-t^{\prime})+i{\bf k}({\bf x}-{\bf x^{\prime}})],\nonumber\end{aligned}$$ where $k=|{\bf k}|$ is the wave number, $\omega$ is frequency, $d$ is the dimensionality of the ${\bf x}$ space. In what follows, we shall work with compressible velocity field which is demonstrated by the form of the tensor structure of the correlator (\[corv\]), namely, it consists of two parts: the standard transverse projector $P_{ij}({\bf k})=\delta_{ij}-k_ik_j/k^2$, and the longitudinal projector $Q_{ij}({\bf k})=k_i k_j/k^2$ which is related to compressibility. The parameter $\alpha \geq 0$ is a free parameter. The value $\alpha=0$ corresponds to the divergence-free (incompressible) advecting velocity field. The function $\tilde{D}^v$ is chosen as follows [@Antonov99; @Antonov00] $$\tilde{D}^v(\omega, k) = \frac{g_0 \nu_0^3 k^{4-d-2\varepsilon-\eta}}{(i\omega+u_0 \nu_0 k^{2-\eta})(-i\omega+u_0 \nu_0 k^{2-\eta})}.\label{corrvelo}$$ The correlator (\[corrvelo\]) is related to the energy spectrum via the frequency integral $$E(k)\simeq k^{d-1} \int d\omega \tilde{D}^v(\omega, k) \simeq \frac{g_0 \nu_0^2}{u_0} k^{1-2\varepsilon}.$$ It means that the coupling constant $g_0$ (more precisely $g_0/u_0$ [@Antonov00]) and the exponent $\varepsilon$ describe the equal-time velocity correlator or, equivalently, energy spectrum. On the other hand, the constant $u_0$ and the second exponent $\eta$ are related to the frequency $\omega \simeq u_0 \nu_0 k^{2-\eta}$ which characterizes the mode $k$. Thus, in our notation, the value $\varepsilon=4/3$ corresponds to the well-known Kolmogorov “five-thirds law” for the spatial statistics of velocity field, and $\eta=4/3$ corresponds to the Kolmogorov frequency. Simple dimensional analysis shows that the charges $g_0$ and $u_0$ are related to the characteristic ultraviolet (UV) momentum scale $\Lambda$ (of the order of inverse Kolmogorov length) by $$g_0\simeq \Lambda^{2\varepsilon + \eta},\,\,\, u_0\simeq \Lambda^{\eta}.$$ In the end of this section, let us briefly discuss two important limits of the considered model (\[corv\]), (\[corrvelo\]) (see also [@Antonov99; @Antonov00]). First of them is so-called rapid-change model limit when $u_0\rightarrow \infty$ and $g_0^{\prime}\equiv g_0/u_0^2=$ const, $$\tilde{D}^v(\omega, k)\rightarrow g_0^{\prime} \nu_0 k^{-d-2\varepsilon + \eta},$$ and the second one is so-called quenched (time-independent or frozen) velocity field limit which is defined by $u_0\rightarrow 0$ and $g_0^{\prime\prime}\equiv g_0/u_0=$ const, $$\tilde{D}^v(\omega, k)\rightarrow g_0^{\prime\prime} \nu_0^2 k^{-d+2-2\varepsilon} \pi \delta(\omega).$$ Here the velocity correlator is independent of time in the $t$ representation. Field Theoretic Formulation of the Model {#sec3} ======================================== The stochastic problem (\[scalar1\])-(\[corv\]) is equivalent to the field theoretic model of the set of fields $\Phi \equiv \{\theta, \theta^{\prime}, {\bf v}\}$ (see, e.g., [@Vasiliev; @ZinnJustin]) with action functional $$\begin{aligned} \hspace{-1cm} S(\Phi)=&-&\frac{1}{2} \int dt_1\,d^d{\bf x_1}\,dt_2\,d^d{\bf x_2} \,\,v_i(t_1,{\bf x_1}) [D^{v}_{ij}(t_1,{\bf x_1};t_2,{\bf x_2})]^{-1} v_j(t_2,{\bf x_2}) \nonumber \\ &+& \int dt\,d^d{\bf x}\,\, \theta^{\prime}\left[-\partial_t \theta - v_i\partial_i\theta+\nu_0\triangle\theta \right], \label{action1}\end{aligned}$$ where, in what follows, unimportant term related to the noise (\[correlator\]) is omitted, $\theta^{\prime}$ is an auxiliary scalar field, and summations are implied over the vector indices. The second line in (\[action1\]) represent the Martin-Siggia-Rose action for the stochastic problem (\[scalar1\]) at fixed velocity field ${\bf v}$, and the first line describes the Gaussian averaging over ${\bf v}$ defined by the correlator $D^v$ in (\[corv\]) and (\[corrvelo\]). Standardly, the formulation through the action functional (\[action1\]) replaces the statistical averages of random quantities in the stochastic problem (\[scalar1\])-(\[corv\]) with equivalent functional averages with weight $\exp S(\Phi)$. Generating functionals of total Green functions G(A) and connected Green functions W(A) are then defined by the functional integral $$G(A)=e^{W(A)}=\int {\cal D}\Phi \,\, e^{S(\Phi) + A\Phi},\label{green}$$ where $A(x)=\{A^{\theta},A^{\theta^{\prime}},{\bf A^{v}}\}$ represents a set of arbitrary sources for the set of fields $\Phi$, ${\cal D}\Phi \equiv {\cal D}\theta{\cal D}\theta^{\prime}{\cal D}{\bf v}$ denotes the measure of functional integration, and linear form $A\Phi$ is defined as $$A\Phi= \int d\,x [A^{\theta}(x)\theta(x)+A^{\theta^{\prime}}(x)\theta^{\prime}(x) + A_i^{v}(x) v_i(x)].\label{form}$$ epsf epsf Action (\[action1\]) is given in a form convenient for a realization of the field theoretic perturbation analysis with the standard Feynman diagrammatic technique. The matrix of bare propagators is obtained from the quadratic part of the action. The wave-number-frequency representation of, in what follows, important propagators are: a) the bare propagator $\langle\theta \theta^{\prime}\rangle_0$ defined as $$\langle\theta \theta^{\prime}\rangle_0=\langle\theta^{\prime} \theta\rangle^*_0=\frac{1}{-i\omega+\nu_0 k^2},$$ and b) the bare propagator for the velocity field $\langle v v\rangle_0$ given directly by (\[corrvelo\]), namely $$\langle v_i v_j\rangle_0 = \left(P_{ij}({\bf k}) + \alpha Q_{ij}({\bf k}) \right) D^v(\omega, k).$$ Their graphical representation is present in figure \[propagators\]. The triple (interaction) vertex $-\theta^{\prime} v_j\partial_j \theta = \theta^{\prime} v_j V_j \theta $ is present in figure \[vertex\], where momentum ${\bf k}$ is flowing into the vertex via the scalar field $\theta$. Renormalization and RG analysis {#sec4} =============================== The model under consideration is logarithmic at $\varepsilon=\eta=0$ (the coupling constants $g_0$, and $u_0$ are dimensionless), therefore the UV divergences in the correlation functions have the form of the poles in $\varepsilon, \eta$, and their linear combinations. The crucial role in the renormalization of the model is played by the total canonical dimension of an arbitrary one-particle irreducible correlation (Green) function $\Gamma=\langle \Phi \cdots \Phi \rangle_{1-ir}$. It plays the role of the formal index of the UV divergence and it is given as follows [@Vasiliev; @AdAnVa99] $$d_{\Gamma}=d^k_{\Gamma}+2 d^{\omega}_{\Gamma}=d+2-N_{\Phi} d_{\Phi},$$ where $N_{\Phi}=\{N_{\theta},N_{\theta^{\prime}},N_{{\bf v}}\}$ are the numbers of corresponding fields entering into the function $\Gamma$, $d^k_{\Gamma}$ and $d^{\omega}_{\Gamma}$ are the canonical momentum dimension and the canonical frequency dimension of the function $\Gamma$, respectively, and summation over all types of fields is implied. In what follows, we shall use the definitions of the canonical dimensions of the fields $\Phi$ as they are given in [@Antonov99; @Antonov00]. It is well-known that superficial UV divergences, whose removal requires counterterms, can be presented only in those Green functions $\Gamma$ for which the total canonical index $d_{\Gamma}$ is non-negative integer. From the dimensional analysis of the model (see, e.g., [@Antonov99; @Antonov00]), we conclude that for any $d$, superficial UV divergences can exist only in the 1-irreducible functions $\langle \theta^{\prime} \theta \rangle_{1-ir}$ and $\langle \theta^{\prime} \theta {\bf v}\rangle_{1-ir}$. To remove them one needs to include into the action functional the counterterm of the form $\theta^{\prime} \triangle \theta$ and $\theta^{\prime} v_i \partial_i \theta$. Their inclusion is manifested by the multiplicative renormalization of the bare parameters $g_0, u_0$, and $\nu_0$, and the velocity field ${\bf v}$ in the action functional (\[action1\]): $$\nu_0=\nu Z_{\nu},\,\,\, g_0=g \mu^{2\varepsilon+\eta} Z_g,\,\,\,u_0=u\mu^{\eta} Z_u,\,\,\, {\bf v}\rightarrow Z_v {\bf v}. \label{zetka}$$ Here, the dimensionless parameters $g,u$,and $\nu$ are the renormalized counterparts of the corresponding bare ones, $\mu$ is the renormalization mass (a scale setting parameter), and $Z_i=Z_i(g,u,\alpha), i=\nu,g,u,v$ are renormalization constants. The renormalized action functional has the following form $$\begin{aligned} \fl S(\Phi)=&-&\frac{1}{2} \int dt_1\,d^d{\bf x_1}\,dt_2\,d^d{\bf x_2} v_i(t_1,{\bf x_1}) [D_{ij}^v(t_1,{\bf x_1};t_2,{\bf x_2})]^{-1} v_j(t_2,{\bf x_2}) \label{actionRen} \\ &+& \int dt\,d^d{\bf x}\,\, \theta^{\prime}\left[-\partial_t \theta - Z_2 v_i\partial_i\theta+\nu Z_1 \triangle\theta \right],\nonumber\end{aligned}$$ where the correlator $D_{ij}^v$ is written in renormalized parameters (in wave-number-frequency representation) $$\tilde{D}_{ij}^v(\omega, k) = \frac{\left(P_{ij}({\bf k}) + \alpha Q_{ij}({\bf k}) \right) g \nu^3 \mu^{2\varepsilon+\eta} k^{4-d-2\varepsilon-\eta}}{(i\omega+u \nu \mu^{\eta} k^{2-\eta})(-i\omega+u \nu \mu^{\eta} k^{2-\eta})}.\label{corrveloRen}$$ By comparison of the renormalized action (\[actionRen\]) with definitions of the renormalization constants $Z_i$, $i=g,u,\nu$ (\[zetka\]) we are coming to the relations among them: $$Z_{\nu}=Z_1,\,\,\,Z_u=Z_1^{-1},\,\,\,Z_g=Z_2^2 Z_1^{-3},\,\,\, Z_v=Z_2. \label{zetka1}$$ The second and the third relations are consequences of the absence of the renormalization of the term with $D^v$ in renormalized action (\[actionRen\]). The issue of interest are especially multiplicatively renormalizable equal-time two-point quantities $G(r)$ (see, e.g., [@Antonov00]). The example of such quantity are the equal-time structure functions $$S_{n}(r)\equiv\langle[\theta(t,{\bf x})-\theta(t,{\bf x'})]^{n}\rangle \label{struc}$$ in the inertial range, specified by the inequalities $l\sim 1/\Lambda <<r<<L=1/m$ ($l$ is an internal length). The infrared (IR) scaling behavior of the function $G(r)$ (for $r/l\gg 1$ and any fixed $r/L$) $$G(r)\simeq \nu_0^{d^{\omega}_G} l^{-d_G} (r/l)^{-\Delta_G} R(r/L) \label{frscaling}$$ is related to the existence of IR stable fixed points of the RG equations (see next section). In (\[frscaling\]) $d^{\omega}_G$ and $d_G$ are corresponding canonical dimensions of the function $G$, $R(r/L)$ is so-called scaling function which cannot be determined by RG equation (see, e.g., [@Vasiliev]), and $\Delta_G$ is the critical dimension defined as $$\Delta_G=d_G^k+\Delta_{\omega} d_G^{\omega} + \gamma_G^*.$$ Here $\gamma_G^*$ is the fixed point value of the anomalous dimension $\gamma_G\equiv \mu \partial_{\mu} \ln Z_G$, where $Z_G$ is renormalization constant of multiplicatively renormalizable quantity $G$, i.e., $G=Z_G G^R$ [@Antonov00], and $\Delta_{\omega}=2-\gamma_{\nu}^*$ is the critical dimension of frequency with $\gamma_{\nu}=\gamma_1$ which is defined further in the text (for more details see, e.g., [@Antonov99; @Antonov00]). On the other hand, the small $r/L$ behavior of the scaling function $R(r/L)$ can be studied using the Wilson OPE [@Vasiliev]. It shows that, in the limit $r/L\to 0$, the function $R(r/L)$ have the following asymptotic form $$R(r/L) = \sum_{F} C_{F}(r/L)\, (r/L)^{\Delta_F}, \label{ope}$$ where $C_{F}$ are coefficients regular in $r/L$. In general, the summation is implied over certain renormalized composite operators $F$ with critical dimensions $\Delta_F$. In present paper we shall study only the first stage of the RG analysis, namely, the influence of compressibility of the velocity field on the stability of possible scaling regimes of the model. The influence of compressibility on the anomalous scaling (the second stage of the RG analysis) will be studied in the subsequent paper. In what follows we shall work with two-loop approximation. But the calculation of higher-order corrections is more difficult in the models with turbulent velocity field with finite correlation time than in the cases with $\delta$-correlations in time. It is related to the fact that the diagrams for the finite correlated case involve two different dispersion laws, namely, $\omega \propto k^2$ for the scalar field and $\omega \propto k^{2-\eta}$ for the velocity field which complicates situation even in the one-loop approximation [@Antonov99; @Antonov00]. But, as was discussed in [@Antonov99; @Antonov00; @AdAnHo02], this difficulty can be avoided by the calculation of all renormalization constants in an arbitrary specific choice of the exponents $\varepsilon$ and $\eta$ that guarantees UV finiteness of the Feynman diagrams. From the calculational point of view the most suitable choice is to put $\eta=0$ and leave $\varepsilon$ arbitrary. Thus, the knowledge of the renormalization constants for the special choice $\eta=0$ is sufficient to obtain all important quantities as the $\gamma$-functions, $\beta$-functions, coordinates of fixed points, and the critical dimensions. But such possibility is not automatic in general. In the model under consideration, it is the consequence of an analysis which shows that in the minimal subtraction (MS) scheme, which is used in what follows, all needed anomalous dimensions are independent of the exponents $\varepsilon$ and $\eta$ in the two-loop approximation. But in the three-loop approximation this dependence can simply appear [@AdAnHo02]. Now let us continue with renormalization of the model. The relation $S(\theta,\theta^{\prime},{\bf v}, e_0)=S^R(\theta,\theta^{\prime},{\bf v}, e, \mu)$, where $e_0$ stands for the complete set of bare parameters and $e$ stands for renormalized one, leads to the relation $W(A, e_0)=W^R(A, e, \mu)$ for the generating functional of connected Green functions. By application of the operator $\tilde{\cal{D}}_{\mu}\equiv\mu \partial_{\mu}$ at fixed $e_0$ on both sides of the latest equation one obtains the basic RG differential equation $${\cal{D}}_{RG} W^R(A,e,\mu)=0, \label{RGE}$$ where ${\cal{D}}_{RG}$ represents operation $\tilde{\cal{D}}_{\mu}$ written in the renormalized variables. Its explicit form is $${\cal{D}}_{RG} = {\cal{D}}_{\mu} + \beta_g(g,u)\partial_g+\beta_u(g,u)\partial_u-\gamma_{\nu}(g,u){\cal{D}}_{\nu},\label{RGoper}$$ where we standardly denote ${\cal{D}}_x\equiv x\partial_x$ for any variable $x$, and the RG functions (the $\beta$ and $\gamma$ functions) are given by well-known definitions and, in our case, using the relations (\[zetka1\]) for renormalization constants, they have the following form $$\begin{aligned} \gamma_{i}&\equiv& \tilde{\cal{D}}_{\mu} \ln Z_{i},\,\,\, i=1,2 \label{gammanu}\\ \beta_g&\equiv&\mu \partial_{\mu} g =g (-2\varepsilon-\eta+3\gamma_1-2\gamma_2), \label{betag}\\ \beta_u&\equiv&\mu \partial_{\mu} u = u(-\eta+\gamma_1).\label{betau}\end{aligned}$$ The renormalization constants $Z_1$, and $Z_2$ are determined by the requirement that one-particle irreducible Green functions $\langle \theta^{\prime} \theta \rangle_{1-ir}$ and $\langle \theta^{\prime} \theta {\bf v}\rangle_{1-ir}$ must be UV finite when are written in renormalized variables. In our case, it means that they have no singularities in the limit $\varepsilon, \eta\rightarrow0$. The one-particle irreducible Green function $\langle \theta^{\prime} \theta\rangle_{1-ir}$ is related to the self-energy operator $\Sigma_{\theta^{\prime}\theta}$ by the Dyson equation $$\langle \theta^{\prime}\theta \rangle_{1-ir}=-i\omega+\nu_0 p^2 - \Sigma_{\theta^{\prime}\theta}(\omega, p),\label{Dyson}$$ where the self-energy operator $\Sigma_{\theta^{\prime}\theta}$ is represented by the corresponding one-particle irreducible diagrams. In the two loop approximation, it is defined by the diagrams which are shown in figure \[fig3\]. epsf =8.5cm On the other hand, the renormalized function $\langle \theta^{\prime} \theta {\bf v}\rangle_{1-ir}$ is defined as $$\langle\theta^{\prime} \theta v_i\rangle_{1-ir}= Z_2 V_i + {\cal V}_i,\label{vrcholl}$$ where the function ${\cal V}_i$ is defined by diagrams of figure \[fig4\] (in two-loop approximation). epsf Thus, $Z_{1}$, and $Z_2$ are found from the requirement that the UV divergences are canceled in (\[Dyson\]), and (\[vrcholl\]) after substitution $\nu_0=\nu Z_{\nu}=\nu Z_{1}$. This determines $Z_{1}$, and $Z_2$ up to an UV finite contribution, which are fixed by the choice of the renormalization scheme. In the MS scheme all renormalization constants have the form: 1 + [*poles in $\varepsilon,\eta$ and their linear combinations*]{}. As was already mentioned, in our calculations we can put $\eta=0$. This possibility essentially simplifies the evaluations of all quantities [@Antonov99; @Antonov00; @AdAnHo02]. The analytical expressions for one-loop diagrams in figure \[fig3\] and figure \[fig4\] (in the MS scheme) have the following form $$\begin{aligned} G_{1p}&=&-\frac{S_d}{(2\pi)^d}\frac{g \nu p^2}{4 u (1+u)^2}\frac{(1 + u)(d - 1 + \alpha) - 2\alpha}{d} \left(\frac{\mu}{m}\right)^{2\varepsilon}\frac{1}{\varepsilon},\\ G_{1v}&=& i \frac{S_d}{(2\pi)^d} \frac{g p_j}{4 u (1+u)^2}\frac{\alpha}{d} \left(\frac{\mu}{m}\right)^{2\varepsilon}\frac{1}{\varepsilon},\end{aligned}$$ where $G_{1p}$ is result for the one-loop diagram in figure \[fig3\], and $G_{1v}$ is result for the one-loop diagram in figure \[fig4\]. Here, $S_d=2 \pi^{d/2}/\Gamma(d/2)$ denotes the $d$-dimensional sphere. The two-loop expressions for the diagrams in figure \[fig3\] and figure \[fig4\] are rather huge, therefore we shall not present their explicit form separately but rather we present complete expressions for renormalization constants $Z_1$, and $Z_2$ which have the following structure $$Z_i=\frac{g}{\varepsilon} A_i + \frac{g^2}{\varepsilon}\left(\frac{1}{\varepsilon}B_i+C_i\right),\,\,\,i=1,2.\label{ZZZ}$$ Now using the definition of the anomalous dimensions $\gamma_{1,2}$ in (\[gammanu\]) we obtain $$\begin{aligned} \gamma_1&\equiv&\mu \partial_{\mu} \ln Z_1 = -2 (\bar{g} A_1 + 2 \bar{g}^2 C_1),\label{g1}\\ \gamma_2&\equiv&\mu \partial_{\mu} \ln Z_2 = -2 (\bar{g} A_2 + 2 \bar{g}^2 C_2), \label{g2}\end{aligned}$$ where we denote $\bar{g}=g S_d/(2\pi)^d$. The one-loop contributions $A_1$ and $A_2$ in (\[g1\]) and (\[g2\]) are defined as follows $$\begin{aligned} A_1&=& -\frac{1}{4 u (1+u)^2}\frac{(1 + u)(d - 1 + \alpha) - 2\alpha}{d}, \label{aa1}\\ A_2&=&\frac{\alpha}{4 d u (1+u)^2},\label{aa2}\end{aligned}$$ and the two-loop contributions $C_1$ and $C_2$ have the form $$\begin{aligned} C_1&=&\frac{1}{16 d^2 u^2 (1+u)^3}\left(C_{10} +\alpha C_{11} + \alpha^2 C_{12} \right), \label{cc1}\\ C_2&=&\frac{1}{32 d^3 u^2 (1+u)^6}\left(\alpha C_{21} + \alpha^2 C_{22}\right),\label{cc2}\end{aligned}$$ where $$\begin{aligned} &&\hspace{-2.3cm} C_{10}=\frac{(d-1)(d+u) H_2}{(d+2)(1+u)^2},\\ &&\hspace{-2.3cm} C_{11}=\frac{d-1}{1+u}-\frac{u (d-1)(2 + u)(2 (u-2) u + d (2 + 3 u))}{d(1+u)^3} H_0 \nonumber \\ && \hspace{-1.3cm}+\frac{(d-1) (4 (-2 + u) u + d^2 (-2 + 3 u^2 (2 + u)) + 2 d (2 + u (5 - 5 u + u^3)))}{d^2(1+u)^3} H_1,\\ &&\hspace{-2.3cm} C_{12}=\frac{3u-1}{(1+u)^2} + \frac{u H_0}{d(1+u)^4}\Big(2 d^2 u (1+u)^2 -(u-3) (u-1) u (2 + u)\nonumber \\ &&\hspace{2.5cm} + d (2 + u) (1 + u (-2 + (u-6) u)) \Big) \nonumber \\ && \hspace{-1.3cm} +\frac{H_1}{d^2(1+u)^4} \Big( 2 (u-3) (u-1) u - 2 d^3 u^2 (1 + u)^2 \nonumber \\ && \hspace{1.3cm} - d^2 (-1 + u (5 + u (2 + u) (3 + (u-6) u))) \nonumber \\ && \hspace{1.3cm} + d (-2 + u (1 + u (22 + (u-4) u (2 + u))))\Big), \\ &&\hspace{-2.3cm} C_{21}= -(d-1) d (1 + u)^2 -(d-1) d u (3 + 5 u + 2 u^2) H_0 \nonumber \\ && \hspace{-1.3cm}+ (d-1) (1 + 2 u) (2 + d u (2 + u)) H_1 + \frac{2 (d-1) d (u-1)}{d+2}H_2, \\ &&\hspace{-2.3cm} C_{22}= -4 d (1 + u)\nonumber \\ && \hspace{-1.3cm} + u (-2 (2 + u) + d (5 - 2 u (1 + u) + d (-1 + 2 u (1 + u)))) H_0 \nonumber \\ && \hspace{-1.3cm} + \frac{2 (2 (1 + u) - d^3 u^2 (1 + u)^2 + d (-3 + u + 5 u^2 + u^3) + d^2 (1 - u + u^3 + u^4))}{d (1 + u)}H_1\nonumber \\ && \hspace{-1.3cm} + \frac{2 d ((u-1)^2 + d (1 + 2 u - u^2))}{(2 + d) (1 + u)}H_2,\end{aligned}$$ where we have used the following notation $$H_i={_2F_1}\left[1,1;i+\frac{d}{2};\frac{1}{(1+u)^2}\right],\,\,\, i=0,1,2$$ for the corresponding hypergeometric function ${_2F_1}[a,b;c;z]=1+\frac{a\, b}{c\cdot1}z+\frac{a(a+1)b(b+1)}{c(c+1)\cdot1\cdot2}z^{2}+\ldots$. The functions $B_i, i=1,2$ which are introduced in (\[ZZZ\]) are not important in what follows, therefore we shall not define them explicitly. Fixed points and scaling regimes {#sec5} ================================ Possible scaling regimes of a renormalizable model are directly given by the IR stable fixed points of the corresponding system of RG equations [@Vasiliev; @ZinnJustin]. The coordinates of the fixed point of the RG equations are defined by $\beta$-functions, namely, by requirement of their vanishing. In our model the coordinates $g_*, u_*$ of the fixed points are found from the system of two equations $$\beta_g(g_*,u_*)=\beta_u(g_*,u_*)=0.$$ The beta functions $\beta_g$ and $\beta_u$ are defined in (\[betag\]) and (\[betau\]). The IR asymptotic behavior is governed by the IR stable fixed point which is given by the positive eigenvalues of the matrix $\Omega$ of the first derivatives: $$\Omega_{ij}=\left(\begin{array}{cc}\partial \beta_g/\partial g & \partial \beta_g/\partial u \\ \partial \beta_u/\partial g & \partial \beta_u/\partial u \end{array} \right).$$ The influence of compressibility on the scaling regimes of the present model in one-loop approximation was investigated in [@Antonov00]. We are interested in the answer on the following question: how can the two-loop approximation change the picture of the scaling regimes discussed in [@Antonov00]? In what follows, we shall try to study possible scaling regimes in detail. First of all, we shall investigate the rapid-change limit: $u\rightarrow\infty$. In this regime, it is necessary to make transformation to new variables, namely, $w\equiv1/u$, and $g^{\prime}\equiv g/u^2$, with the corresponding changes in the $\beta$ functions: $$\begin{aligned} \beta_{g^{\prime}}&=&g^{\prime} (-2\varepsilon+\eta+\gamma_1-2\gamma_2), \label{betag1}\\ \beta_w &=& w(\eta-\gamma_1).\label{betau1}\end{aligned}$$ It is well-known that in the rapid change model the higher than one-loop corrections to the self-energy operator are equal to zero. On the other hand, the renormalization of the velocity field is absent at all as a consequence of the fact that $Z_2=1$ at all orders of the perturbation theory. It can be also seen directly by the corresponding manipulations with our $\gamma$-functions (\[g1\]) and (\[g2\]). Therefore, we are coming to the one-loop results of [@Antonov00] (in the rapid-change model limit), namely $$\gamma_1=\bar{g}^{\prime}\frac{d-1+\alpha}{2d},\,\,\,\gamma_2=0, \label{gamma10}$$ where again $\bar{g}^{\prime}=g^{\prime} S_d/(2\pi)^d$. In this regime we have two fixed points denoted as FPI and FPII in [@Antonov99; @Antonov00]. The first of them is trivial one $$\mathrm{FPI}:\,\,\,\, w_*=g_*^{\prime}=0,$$ with $\gamma_{1}^*=0$, and diagonal matrix $\Omega$ with eigenvalues (diagonal elements) $$\Omega_1=\eta,\,\,\,\,\,\Omega_2=\eta-2\varepsilon.$$ Thus, this fixed point is IR stable when $\eta>0$, and, at the same time, $\eta>2\varepsilon$. The second point is defined as $$\mathrm{FPII}:\,\,\,\,w_*=0,\,\,\, \bar{g}_*^{\prime}=\frac{2d}{d-1+\alpha}(2\varepsilon-\eta),$$ with exact one loop result $\gamma_{1}^*=2\varepsilon-\eta$. The corresponding $\Omega$ matrix is triangular with diagonal elements (eigenvalues): $$\Omega_1=2(\eta-\varepsilon),\,\,\,\,\Omega_2=2\varepsilon-\eta.$$ It means that this kind of the fixed point is IR stable when $\eta<2\varepsilon$ together with $\eta>\varepsilon$. The second special case of the present model is so-called “frozen regime” with the frozen velocity field. It is obtained from our model in the limit $u\rightarrow0$. To consider this transition, it is again appropriate to change the variable $g$ to the new variable $g^{\prime\prime} \equiv g/u$ [@Antonov99]. Then the $\beta$ functions are transform to the following ones: $$\begin{aligned} \beta_{g^{\prime\prime}}&=&g^{\prime\prime} (-2\varepsilon+2\gamma_1-2\gamma_2), \label{betag2}\\ \beta_u &=& u(-\eta+\gamma_1),\label{betau2}\end{aligned}$$ with unchanged $\beta$ function for parameter $u$. In this notation, the anomalous dimensions $\gamma_{1,2}$ have the form $$\begin{aligned} \gamma_1&=& -2 (\bar{g}^{\prime\prime} A_1^{\prime\prime} + 2 \bar{g}^{\prime\prime2} C_1^{\prime\prime}),\label{g1u0}\\ \gamma_2&=& -2 (\bar{g}^{\prime\prime} A_2^{\prime\prime} + 2 \bar{g}^{\prime\prime2} C_2^{\prime\prime}), \label{g2u0}\end{aligned}$$ where, as obvious, $\bar{g}^{\prime\prime}=g^{\prime\prime} S_d/(2\pi)^d$, and the one-loop contributions are now given as $$\begin{aligned} A_1^{\prime\prime}&=& -\frac{d - 1 - \alpha}{4d}, \\ A_2^{\prime\prime}&=&\frac{\alpha}{4d},\end{aligned}$$ and the two-loop contributions $C_1^{\prime\prime}$ and $C_2^{\prime\prime}$ are now $$\begin{aligned} C_1^{\prime\prime}&=&\frac{1}{16 d^2}\left(C_{10}^{\prime\prime} +\alpha C_{11}^{\prime\prime} + \alpha^2 C_{12}^{\prime\prime} \right), \\ C_2^{\prime\prime}&=&\frac{1}{32 d^3}\left(\alpha C_{21}^{\prime\prime} + \alpha^2 C_{22}^{\prime\prime}\right),\end{aligned}$$ with $$\begin{aligned} &&\hspace{-2.3cm} C_{10}^{\prime\prime}=\frac{(d-1)d }{(d+2)}H_{02}=d-1,\\ &&\hspace{-2.3cm} C_{11}^{\prime\prime}=(d-1)\left(1 -\frac{2(d - 2 )}{d} H_{01}\right)=1-d,\\ &&\hspace{-2.3cm} C_{12}^{\prime\prime}=-1+\frac{d - 2}{d}H_{01}=0, \\ &&\hspace{-2.3cm} C_{21}^{\prime\prime}= 2(d-1)\left( H_{01} - \frac{d}{d+2}H_{02}\right)=\frac{4(d-1)}{d-2}, \\ &&\hspace{-2.3cm} C_{22}^{\prime\prime}= \frac{2 (d-1)(d-2)}{d }H_{01}+ \frac{2 d (1 + d)}{(2 + d)}H_{02}=4d,\end{aligned}$$ where we denote $$H_{0i}={_2F_1}\left[1,1;i+\frac{d}{2};1\right]=\frac{d-2+2i}{d-4+2i},\,\,\, i\geq1.$$ The system of $\beta$ functions (\[betag2\]) and (\[betau2\]) exhibits two fixed points, denoted as FPIII and FPIV in [@Antonov99]. They are related to the corresponding two scaling regimes. One of them is trivial, $$\mathrm{FPIII}:\,\,\,\, u_*=g_*^{\prime\prime}=0,$$ with $\gamma_{1}^*=\gamma_2^*=0$. The eigenvalues of the corresponding matrix $\Omega$, which is diagonal in this case, are $$\Omega_1=-2\varepsilon,\,\,\,\,\Omega_2=-\eta.$$ Thus, this regime is IR stable only if both parameters $\varepsilon$, and $\eta$ are negative simultaneously. The second, non-trivial, point is $$\mathrm{FPIV}:\,\,\,\, u_*=0,\,\,\,\, \bar{g}_*^{\prime\prime}=-\frac{\varepsilon}{2 (A_1^{\prime\prime}-A_2^{\prime\prime})}-\frac{C_1^{\prime\prime}-C_2^{\prime\prime}}{2 (A_1^{\prime\prime}-A_2^{\prime\prime})^3} \varepsilon^2,$$ with exact one-loop relation $\gamma_1^*=\gamma_2^*+ \varepsilon$. After substitution of the corresponding quantities one obtains the following expression for the coordinates of the fixed point $$\hspace{-1.8cm}u_*=0,\,\,\,\, \bar{g}_*^{\prime\prime}=\frac{2 d \varepsilon}{d-1}\left\{1+\frac{\varepsilon}{(d-1)^2} \left[(d-1)\left(1-\alpha\left(1+\frac{2}{d(d-2)}\right)\right)-2\alpha^2\right]\right\}.$$ The eigenvalues of the matrix $\Omega$ (taken at the fixed point) are $$\Omega_1=2\varepsilon \left(1- \frac{C_1^{\prime\prime}-C_2^{\prime\prime}}{(A_1^{\prime\prime}-A_2^{\prime\prime})^2}\varepsilon\right),\,\,\, \Omega_2=\varepsilon-\eta+\gamma_2^*.$$ After corresponding substitutions one has $$\begin{aligned} &&\hspace{-2cm}\Omega_1=2 \varepsilon\left\{1-\frac{\varepsilon}{(d-1)^2} \left[(d-1)\left(1-\alpha\left(1+\frac{2}{d(d-2)}\right)\right)-2\alpha^2\right]\right\}, \\ && \hspace{-2cm}\Omega_2=\varepsilon-\eta + \frac{\alpha\, \varepsilon}{d-1}\left[-1+ \varepsilon \frac{2 \alpha^2 (d-2) d - (d-1)(d^2-2)-\alpha (2+(d-3) d^2)}{(d(d-1)^2(d-2))} \right].\end{aligned}$$ The conditions $\bar{g}_*^{\prime\prime}>0, \Omega_{1}>0$, and $\Omega_2>0$ for the IR stable fixed point lead to the restrictions on the values of the parameters $\varepsilon$ and $\eta$. First, suppose that $\varepsilon<0$. Then from the conditions $\bar{g}_*^{\prime\prime}>0$, and $\Omega_{1}>0$ one has the following restrictions which must be fulfilled simultaneously $$1+\varepsilon D<0,\,\,\,1-\varepsilon D<0,\label{cond1}$$ but they cannot be fulfilled at the same time. Thus, our first condition is $\varepsilon>0$. In (\[cond1\]) $D$ is given as $$D=\frac{1}{(d-1)^2} \left[(d-1)\left(1-\alpha\left(1+\frac{2}{d(d-2)}\right)\right)-2\alpha^2\right].$$ To have $\bar{g}_*^{\prime\prime}>0$, and $\Omega_{1}>0$ together with $\varepsilon>0$, the following inequalities must be held $$-1< \varepsilon D < 1, \label{cond2}$$ which restricts the value of $\varepsilon$ as a function of the parameter $\alpha$, and the dimension of the space $d$. In the incompressible case ($\alpha=0$) the condition (\[cond2\]) is reduced into the simple inequality $$\varepsilon<d-1.$$ In the general case, for each value of $d$, there exists a “critical” value of $\alpha$ in which $D=0$. We denote it as $\alpha_c$. In this situation $\varepsilon$ can be arbitrary, i.e., the condition (\[cond2\]) is fulfilled automatically. The value of $\alpha_c$ is defined as follows $$\hspace{-1.5cm}\alpha_c=\frac{2-4 d+3 d^2-d^3+(4-16 d-4 d^2+36 d^3-23 d^4+2 d^5+d^6)^{1/2}}{4 d (d-2)}.$$ For example, for $d=3$ its value is $\alpha_c=\frac{\sqrt{61}-5}{6}\simeq0.468$. Therefore, in the compressible model, the situation is a little bit more complicated as a result of a competition between incompressible and compressible terms within two-loop approximation which leads to the existence of $\alpha_c$. How does it work? The answer is the following. If we continuously increase the value of the parameter $\alpha$, the region of stability of the fixed point defined by the inequalities (\[cond2\]) increases too. This restriction vanishes completely when $\alpha$ reaches the “critical” value $\alpha_c$. In this rather specific situation the two-loop influence on the region of stability of fixed point defined by condition (\[cond2\]) is exactly zero. Then, if the value of parameter $\alpha$ increases further, the condition (\[cond2\]) appears again, and restriction on $\varepsilon$ becomes stronger when $\alpha$ tends, in principle, to infinity. In this limit $\varepsilon \rightarrow 0$. On the other hand, it must be stressed that in our model only relatively small values of $\alpha$ are allowed ($\alpha \ll 1$). It corresponds to small fluctuations of the density $\rho$ in the system which is supposed in our investigation. In other words, it is supposed that the stochastic component of the velocity field of the fluid is much smaller than the velocity of the sound in the system (the Mach number $Ma\ll 1$). The last condition on the stability of the IR fixed point is found from the requirement to have $\Omega_2>0$. It reads $$\hspace{-1cm}\eta< \varepsilon + \frac{\alpha\, \varepsilon}{d-1}\left[-1+ \varepsilon \frac{2 \alpha^2 (d-2) d - (d-1)(d^2-2)-\alpha (2+(d-3) d^2)}{(d(d-1)^2(d-2))} \right].$$ In the incompressible case it is reduced into the simple condition $$\eta<\varepsilon,$$ which is held at each order of the perturbation theory. In the end, let us consider the most interesting scaling regime with finite value of the fixed point for variable $u$. The coordinates of the fixed point is now defined by the requirement of vanishing of the $\beta$ functions which are given in (\[betag\]) and (\[betau\]). The fixed point value for $\bar{g}=g S_d/(2\pi)^d$ is given as $$\mathrm{FPV}:\,\,\,\, \bar{g}_*=-\frac{\varepsilon}{2(A_{1}-A_{2})}- \frac{C_{1}-C_{2}}{2 (A_{1}-A_{2})^3} \varepsilon^2,$$ where the functions $A_1, A_2, C_1$, and $C_2$ are given in (\[aa1\])-(\[cc2\]), and where the parameter $u$ is taken at its fixed point value $u_*$ which is given implicitly by the equation $$-\eta+\gamma_1^*(u_*)=0.$$ Using the exact relations $$\gamma_1^*=\eta,\,\,\,\,\,\gamma_2^*=\eta-\varepsilon$$ the expression for the fixed point value of $\bar{g}$ can be rewritten as a series (expansion) of the parameter $\eta$ or a linear combination of $\eta$ and $\varepsilon$. For example, in [@Antonov00], where the problem was analyzed in one-loop approximation, it was expressed as a function of $2\varepsilon-\eta$ (in our notation). In the framework of one-loop approximation it allows one to have linear dependence of $\bar{g}_*$ on the fixed point value of the parameter $u$. Together with another choice of the linear combination of $\eta$ and $\varepsilon$, namely $\eta-\varepsilon$ it leads also to the simple expression for the fixed point value of $u$. Thus, the coordinates of the fixed point in one-loop approximation are [@Antonov00] $$\bar{g}_*=\frac{2 d (1+u_*)}{d-1+\alpha}(2\varepsilon-\eta),\,\,\,\,\,\, u_*=-1+\frac{\alpha}{d-1+\alpha}\frac{\eta-2\varepsilon}{\eta-\varepsilon}.$$ It allows, together with the requirement of the positive eigenvalues of the corresponding matrix of the first derivatives $\Omega$, to find simple conditions for the IR stable fixed point. They are defined by inequalities $\varepsilon>0, \varepsilon>\eta$, and $\eta>\varepsilon\frac{d-1-\alpha}{d-1}$ [@Antonov00]. The situation is essentially more complicated when we are working in two-loop approximation. It is given by the fact that now we have nonlinear dependence of $\bar{g}$ on the parameters $\eta$ and $\varepsilon$, and the expression for the fixed point value of $u$ is now given only implicitly in rather complicated expression containing hypergeometric functions. Another complication, which defends to analyze the problem in general, is related to the fact that contrary to the incompressible case when one has additional condition, namely $\eta=\varepsilon$, no such condition exists in compressible case under consideration. As a result, the analysis of the IR stability of the general case of the present model have to be done individually for concrete situation. It is rather cumbersome and it will be done in the subsequent work. In what follows, let us only give the general analysis of the most interesting case when one suppose the relation $\eta=\varepsilon$. In this situation from the definition of the $\beta$ functions given in (\[betag\]) and (\[betau\]) one obtains the condition $$\gamma_2^*=0.$$ Thus, in this case, the coordinates of the fixed points are given as $$\begin{aligned} &&\bar{g}_*=-\frac{\varepsilon}{2 A_{1}}- \frac{C_{1}}{2 A_{1}^3} \varepsilon^2, \\ && A_2(u_*) + 2 \bar{g}_* C_2(u_*)=0, \label{impu}\end{aligned}$$ but even in this situation the fixed point value of $u$ is defined by complicated implicit equation (\[impu\]) and its exhausted analysis must be discussed separately. Conclusions {#sec6} =========== We have studied the influence of compressibility on the possible IR scaling regimes of the model of a passive scalar advected by a Gaussian velocity field with finite time correlations by means of the field theoretic RG technique. The possible scaling regimes are directly connected to the existence of IR stable fixed points of the RG equations. The dependence of the fixed points on the parameter of compressibility and their IR stability is discussed. The most attention is paid to the frozen limit of the model where inequalities which define the stable IR scaling regimes are found analytically. The existence of a “critical” value $\alpha_c$ of the parameter of compressibility $\alpha$ at which one of the two-loop conditions is canceled as a result of the competition between compressible and incompressible terms is discussed in detail. The main conclusion is that for the small value of parameter $\alpha$ the region of stability is not restricted considerably. It is also shown that the most general case with finite time correlations of the velocity field is more complicated within two-loop approximation and have to be consider separately once more. The work was supported in part by VEGA grant 6193 of Slovak Academy of Sciences, by Science and Technology Assistance Agency under contract No. APVT-51-027904. References {#references .unnumbered} ========== [10]{} Monin A S, Jaglom A M 1975 [*Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 2*]{} (Cambridge: MIT Press) Frisch U 1995 [*Turbulence: the legacy of A. N. Kolmogorov*]{} (Cambridge: Cambridge University Press) Obukhov A M 1949 [*Izv. Akad. Nauk SSSR, Geogr. Geofiz.*]{} [**13**]{} 58 Kraichnan R H 1968 [*Phys. Fluids*]{} [**11**]{} 945 Falkovich G, Gawȩdzki K, Vergassola M 2001 [*Rev. Mod. Phys.*]{} [**73**]{} 913 Kraichnan R H 1994 [*Phys. Rev. Lett.*]{} [**72**]{} 1016 Adzhemyan L Ts, Antonov N V and Vasil’ev A N 1998 [*Phys. Rev.*]{} E [**58**]{} 1823 Vasil’ev A N 1998 [*Quantum-Field Renormalization Group in the Theory of Critical Phenomena and Stochastic Dynamics*]{} (St. Petersburg: St. Petersburg Institute of Nuclear Physics) \[in Russian; English translation: Gordon & Breach, 2004\] Adzhemyan L Ts, Antonov N V and Vasil’ev A N 1999 [*The Field Theoretic Renormalization Group in Fully Developed Turbulence*]{} (London: Gordon $\&$ Breach) Adzhemyan L Ts, Antonov N V, Hnatich M and Novikov S V 2000 [*Phys. Rev.*]{} E [**63**]{} 016309 Adzhemyan L Ts and Antonov N V 1998 [*Phys. Rev.*]{} E [**58**]{} 7381 Antonov N V 1999 [*Phys. Rev.*]{} E [**60**]{} 6691 Antonov N V 2000 [*Physica*]{} D [**144**]{} 370 Antonov N V 2000 [*Zap. Nauchn. Semin. POMI*]{} [**269**]{} 79 Shraiman B I and Siggia E D 1994 [*Phys. Rev.*]{} E [**49**]{} 2912 Shraiman B I and Siggia E D 1995 [*C. R. Acad. Sci., Ser. IIb: Mec., Phys., Chim., Astron.*]{} [**321**]{} 279 Shraiman B I and Siggia E D 1996 [*Phys. Rev. Lett.*]{} [**77**]{} 2463 Adzhemyan L Ts, Antonov N V and Honkonen J 2002 [*Phys. Rev.*]{} E [**66**]{} 036313 Zinn-Justin J 1989 [*Quantum Field Theory and Critical Phenomena*]{} (Oxford: Clarendon) Adzhemyan L Ts, Antonov N V, Barinov V A, Kabrits Yu S and Vasil’ev A N 2001 [*Phys. Rev.*]{} E [**64**]{} 056306 Adzhemyan L Ts, Antonov N V, Barinov V A, Kabrits Yu S and Vasil’ev A N 2001 [*Phys. Rev.*]{} E [**63**]{} 025303(R) [^1]: Present address: Laboratory of Information Technologies, JINR, 141 980 Dubna, Russia [^2]: Present address: Laboratory of Theoretical Physics, JINR, 141 980 Dubna, Russia

--- bibliography: - 'library.bib' --- [**** ]{}\ Vina Ayumi^1^, L.M. Rasdi Rere^1,2^, Mohamad Ivan Fanany^1^, Aniati Murni Arymurthy^1^,\ **[1]{} Machine Learning and Computer Vision Laboratory,\ Faculty of Computer Science, Universitas Indonesia\ **[2]{} Computer System Laboratory, STMIK Jakarta STI&K\ vina.ayumi@ui.ac.id**** Abstract {#abstract .unnumbered} ======== Convolutional neural network (CNN) is one of the most prominent architectures and algorithm in Deep Learning. It shows a remarkable improvement in the recognition and classification of objects. This method has also been proven to be very effective in a variety of computer vision and machine learning problems. As in other deep learning, however, training the CNN is interesting yet challenging. Recently, some metaheuristic algorithms have been used to optimize CNN using Genetic Algorithm, Particle Swarm Optimization, Simulated Annealing and Harmony Search. In this paper, another type of metaheuristic algorithms with different strategy has been proposed, i.e. Microcanonical Annealing to optimize Convolutional Neural Network. The performance of the proposed method is tested using the MNIST and CIFAR-10 datasets. Although experiment results of MNIST dataset indicate the increase in computation time (1.02x - 1.38x), nevertheless this proposed method can considerably enhance the performance of the original CNN (up to 4.60%). On the CIFAR10 dataset, currently, state of the art is 96.53% using fractional pooling, while this proposed method achieves 99.14%. [***Keywords—*** Metaheuristic, Microcanonical Annealing, Convolutional Neural Network, MNIST, CIFAR10]{} Introduction {#sec1} ============ Essentially, Deep learning (DL) is motivated by the artificial intelligent (AI) research, where the objective is to replicate the human brain capability, i.e. to observe, learn, analyze and make a decision, particularly for complex problems [@Naja]. DL is about learning the representation of a hierarchical feature, and it contains a variety of methods, such as neural network, hierarchical of probabilistic models, and supervised as well as unsupervised learning algorithms.[@Liangpei]. The current good reputation of DL is due to the decrease in the price of computer hardware, improvement in the computational processing capabilities, and advanced research in the Machine Learning and Signal Processing [@Deng]. In general, DL models can be classified into discriminative, generative, and hybrid models[@Deng]. Recurrent neural network (RNN), deep neural networks (DNN), and convolutional neural networks (CNN) are some examples of Discriminative models. Examples of generative models are deep Boltzmann machine (DBM), regularized autoencoder, and deep belief network (DBN). In the case of the hybrid model, it refers to a combination of generative and discriminative models. An example of such hybrid model is a pre-trained deep CNN using DBN, where it can improve the performance of deep CNN better than if it uses only random initialization. Among all of these DL techniques, this paper focuses on CNN. Although the good reputation of DL for solving any learning problem is known, how to train it is challenging. The successful proposal to optimize this technique using layered-wise pre-training was proposed by Hinton and Salakhutdinov [@Hinton]. Some other methods are Hessian-free optimization suggested by Marten [@Martens], and Krylov Subspace Descent by Vinyal et al. [@Vinyal] Recently, some of the metaheuristic algorithms have been used to optimize DL, especially CNN. Some papers[@You][@Oul][@Rasdi][@Gustavo] [@Laode] report that these methods can improve the accuracy of CNN. Metaheuristic is a powerful method to solve difficult optimization problems, and it has been used in almost all research area of engineering, science, and even industrial application [@Yang]. In general, this method works with three main objectives, i.e. solving big problems, solving the problem faster, and finding robust algorithms [@Talbi]. Besides, they are not difficult to be designed, flexible, and relatively easy to be applied. Almost all metaheuristics algorithms inspired by nature, which is based on several principles of phenomena in physics, biology, and ethology. Some examples of biology phenomena are Differential Evolution (DE), Evolution Strategy (ES), Genetic Algorithm (GA). Phenomena of physics are Threshold Accepting method (TA), Microcanonical Annealing (MA), Simulated Annealing (SA), and Ethology phenomena are Ant Colony Optimization (ACO), Firefly Algorithm (FA), Particle Swarm Optimization (PSO)[@Boussaid]. Another metaheuristic phenomenon is inspired by music, such as Harmony Search algorithm [@Lee]. Classifications of metaheuristic can also be based on single-solution based metaheuristic (S-metaheuristic) and population-based metaheuristic (P-metaheuristic). Examples of S-metaheuristic are SA, TA, MA, Guided Local Search, and Tabu Search. In the case of P-metaheuristic, it can be divided into Swarm Intelligent (SI) and Evolutionary Computation (EC). Examples of SI are FA, PSO, ACO, Bee Colony Optimization and examples of EC are GA, ES, DE [@Boussaid]. Of the various types of the metaheuristic algorithm, in this paper we use the MA, with the consideration that the S-Metaheuristic is simple to implement on DL, and to the best of our knowledge, has never been used for optimizing CNN. The Macrocanonic algorithm is the variant of Simulated Annealing. Uses an adaptation of the Metropolis algorithm, the conventional SA algorithm aims to bring a system to equilibrium at decreasing temperatures [@Stephen]. On the other hand, MA based on Creutz’s microcanonical simulation technique, where the system’s evolution is controlled by its internal energy, not by its temperature. The advantages Creutz algorithm over the Metropolis algorithm is since it does not require the generation of quality random numbers or the evaluation of transcendental functions, thus allowing much faster implementation. Experiments on the Creutz method indicate that, it can be programmed to run an order of magnitude faster than the conventional Metropolis method for discrete systems [@Bhanot]. A further significant advantage is that microcanonical simulation does not require high-quality random numbers. The organization of this paper is as follows: Section 1 is an introduction; Section 2 provides an overview of Microcanonical Annealing; Section 3 describes the method convolutional neural network; Section 4 presents the proposed method; Section 5 gives the results of the experiment, and lastly, Section 6 presents the conclusion of this paper. Microcanonical Annealing ======================== Microcanonical Annealing (MA) corresponds to a variant of simulated annealing (SA). This technique is based on the Creutz algorithm, known as “demon” algorithm or microcanonical Monte Carlo simulation. In which the algorithm tolerates attainment of the equilibrium of thermodynamic in an isolated system, where in this condition, total energy of the system $E_p$ is constant [@Boussaid]. Total energy is the sum of kinetic energy $E_k$ and potential energy $E_p$ of the system, as the equation (2) follow: $$\label{eq1} E_{total} = E_k + E_p$$ In case of minimum optimization problem, potential energy $E_p$ is the objective function to be minimized, and the kinetic energy is used as temperature in SA, that is forced to remain positive [@Boussaid]. When the change of energy is negative $(- \Delta E)$, while it increases the kinetic energy $(E_k \leftarrow E_k - \Delta E )$, this new states is accepted. Otherwise, it is accepted when $- \Delta E < E_k$, and the energy obtained in the form of potential energy is cut off from the kinetic energy. So that the total energy remains constant. The standard algorithm for MA is shown in **Algorithm 1**.[@Boussaid]. Randomly, select an initial solution $x$ Initialize the kinetic energy $E_k$ Convolutional neural network ============================ One variant of the standard multilayer perceptron (MLP) is CNN. Its capability in reducing the dimension of data, extracting the feature sequentially, and classifying in one structure of network are distinguished advantages of this method, especially, for pattern recognition compared with the conventional approaches [@Bengio]. The classical CNN by LeCun et al [@LeCun] is an extension of traditional MLP based on three ideas: local receive fields, weights sharing, and spatial/temporal sub-sampling. There are two types of processing layers, which are convolution layers and sub-sampling layers. As demonstrated in Fig.1, the processing layers contain three convolution layers C1, C3, and C5, combined in between with two sub-sampling layers S2 and S4, and output layer F6. These convolution and sub-sampling layers are arranged into planes called features maps. In convolution layer, each neuron is locally linked to a small input region (local receptive field) in the preceding layer. All neurons with similar feature maps obtain data from different input regions until the whole plane input is skimmed, but the similar weights are used together (weights sharing). The feature maps are spatially down-sampled in sub-sampling layer, in which the map size is reduced by a factor 2. For instance, the feature map in layer C3 of size 10x10 is sub-sampled to a conforming feature map of size 5x5 in the subsequent layer S4. The last layer is F6 that is the process of classification [@LeCun]. Basically, a convolution layer is correlated with some feature maps, the size of the kernel, and connections to the previous layer. Each feature map is the result of a sum of convolution from the maps of the previous layer, by their corresponding kernel and a linear filter. Furthermore, a bias term is added to the map then and applying it to a non-linear function. The k-th feature map $M_{ij}^k$ with the weights $W^k$ and bias $b_k$ is obtained using the $\tanh$ function as follow: $$M_{ij}^k=\tanh((W^k \times x)_{ij} + b_k)$$ The purpose of a sub-sampling layer is the spatially invariant reached by reducing the feature maps resolution, where each feature map is pooled relating to one of the feature map of the previous layer where each map feature is collected relating to one of the maps of the features of the previous layer. Where $a_i^{n \times n} $ are the inputs, $ \beta $ is a scalar of trainable, and $ b $ is bias of trainable, the sub-sampling function,is given by the following equation: $$a_j=\tanh\left(\beta\sum_{N\times N}{a_i^{n \times n} + b}\right)$$ After several convolutions and sub-samplings, the last structure is a classification layer. This layer works as an input for a series of fully connected layers that will execute the classification task. In this layer, each output neuron is assigned to one class label, and in the case of CIFAR10 or MNIST data set, this layer contains ten neurons corresponding to their classes. Design of proposed methods ========================== In this proposed method, the algorithm of MA is used to train CNN to find the condition of best accuracy, as well as to minimize estimated error and indicator of network complexity. This objective can be realized by computing the loss function of vector solution or the standard error on the training set. The following is the loss function used in this paper: $$f= \frac {1}{2} \left({\frac{\sum_{i=N}^{N}{(x - y)^2}}{N}}\right)^{0.5}$$ where the expected output is $x$, the real output is $y$, and some of the training samples are $N$. The two situations are used in this method for termination criterion. The first is when the maximum iteration has been reached and the second is when the loss function is less than a certain constant. Both conditions mean that the most optimal state has been achieved. The architecture of this proposed method is i-6c-2s-12c-2s, where the number of C1 is 6, and C3 is 12. The size of kernel for all convolution layer is 5x5, and the scale of sub-sampling is 2. This architecture is a simple CNN structure (LeNet-5), not a complex structure like AlexNet[@Alexnet], SPP[@KHe], and GoogLeNet[@Szegedy]. In this paper, these architecture is designed for MNIST dataset. Technically in these proposed methods, CNN will compute the values of bias and weight. These values ($x$) are used to calculate the loss function $f(x)$. The values of $x$ are used as a vector of solution in MA, which will be optimized, by adding a value of $\Delta x$ randomly. Meanwhile, $f(x)$ is used as a potential energy $E_k$ in MA. In this proposed method, $\Delta x$ is one of the important parameters. The value of accuracy will be improved significantly by providing an appropriate value of the $\Delta x$ parameter. As an example of one epoch, if $ \Delta x = 0.001 \times rand$, then the maximum accuracy is 87.60%, in which this value is 5.21% greater than the original CNN (82.39%). However, if $ \Delta x = 0.0001 \times rand$, its accuracy is 85.45% and its is only 3.06% greater than the original CNN. Another important parameter of the proposed method is the size of neighborhood. For example in one epoch, if neighborhood is 5, 10 or 20, and then the accuracy values are respectively 85.74%, 87.52%, or 88.06%. While the computing time are respectively 98.06 seconds, 99.18 seconds and 111.80 seconds. Furthermore, this solution vector is updated based on MA algorithm. In case of termination criterion has been reached, all of biases and weights for all layers on the system will be updated. Experiment and results ====================== In this paper, there are two categories of experiments conducted, based on the dataset. The first experiment was using MNIST dataset, and the second experiment using CIFAR10 dataset. Some of the examples image for MNIST dataset are shown in Figure 2 and for CIFAR10 dataset are shown in Figure 3. ![Examples of some image from MNIST data-set[]{data-label="fig:my_label"}](MNIST5) Experiment using MNIST data set ------------------------------- The experiment for MNIST data set was implemented in MATLAB-R2011a, windows 10, on a PC with processor Intel Core i7-4500u, and 8 GB RAM running memory, with five experiments for each epoch. The original program of this experiment is DeepLearn Toolbox from Palm[@Palm]. In this research, the program of CNN is modified with the algorithm of MA. In all experiment, the size of neighborhood was set to 10, maximum of iteration (maxit) = 10, as well as kinetic energy = 100. We also set the parameter of CNN i.e., the learning rate ($\alpha = 1$) and the batch size (100). On the MNIST dataset, all of the experiment results of the proposed methods are compared with the experiment result from the original CNN. The results of CNN and CNN based on MA is summarized in Table 1, for accuracy (A1, A2) and computation time (T1, T2), as well as Figure 4 for Error and Figure 5 for computation time. In case of 100 epochs, as is shown in Figure 6 and 7, the accuracy of original CNN is 98.65% and the accuracy of CNN by MA is 98.75%. The computation time of both methods are 10731 seconds and 17090s seconds respectively. [c c c c c]{} & &\ & & & &\ 1 & 82.39 & 91.75 & 86.99 $\pm$ 0.53 & 109.99 $\pm$ 10.47\ 2 & 89.06 & 193.39 & 91.33 $\pm$ 0.43 & 203.04 $\pm$ 0.83\ 3 & 91.13 & 297.31 & 93.14 $\pm$ 0.31 & 302.84 $\pm$ 1.74\ 4 & 92.33 & 379.44 & 94.48 $\pm$ 0.15 & 402.42 $\pm$ 0.57\ 5 & 93.11 & 479.04 & 95.11 $\pm$ 0.28 & 514.52 $\pm$ 7.54\ 6 & 93.67 & 576.38 & 95.72 $\pm$ 0.12 & 612.07 $\pm$ 2.33\ 7 & 94.25 & 676.57 & 95.99 $\pm$ 0.22 & 781.32 $\pm$ 23.57\ 8 & 94.77 & 768.24 & 96.26 $\pm$ 0.26 & 1062.11 $\pm$ 6.79\ 9 & 95.37 & 855.85 & 96.49 $\pm$ 0.27 & 1144.01 $\pm$ 79.84\ 10 & 95.45 & 954.54 & 96.89 $\pm$ 0.15 & 1274.00 $\pm$ 47.69\ 100 & 98.65 & 10731 & 98.75 & 17090\ In general, the experiments conducted for MNIST data set shown that the proposed methods are better than the original CNN, for any given epoch. As an example for the second epoch, the accuracy of original CNN is 89.06%, while for CNNMA is 91.33%. Accuracy improvement of the proposed method, compared to original CNN, varies of each epoch, with a range of values between 1.12% (CNNMA, 9 epoch) up to 4.60% (CNNMA, 1 epoch). The computation time for the proposed method, compared to the original CNN, is in the range of $1.02\times$ (CNNMA, three epochs : 302.84/297.31) up to $1.38\times$ (CNNMA, eight epochs: 1062.11/768.24). Experiment using CIFAR10 dataset -------------------------------- The experiment of CIFAR10 dataset was conducted in MATLAB-R2014a, Ubuntu 14.04 LTS 64, on a PC with Processor Intel Core i7-5820K, Four GPU GTX Titan X, Memory DDR2 RAM 64.00 GB, Hard disk 240 GB. The original program is MatConvNet from [@vedaldi]. In this paper, the program was modified with MA algorithm. The results can be seen in Fig. 8 for top-1 error and top-5 error. The proposed method has proven very effective on CIFAR10 dataset with an accuracy of 99.6%, for the last epoch in the top-1 error. In Table II different results from state of the art approaches are listed as a comparison. Another work proposed fine-tuning CNN using metaheuristic algorithm, harmony search (HS) [@Rosa] also compared in Table II. Method Accuracy (%) ---------------------------- -------------- **CNN-MA \[ours\]** **99,14** CNN-HS [@Rosa] 72.28 Fractional Pooling [@BG] 96.53 Large ALL-CNN [@Jost] 95.59 Spatially Sparse CNN [@BG] 95,53 LSUV [@Dmytro] 94.16 CNN [@matconvnet] 80.46 : State of the art CIFAR10 dataset[]{data-label="tab:example"} Conclusion ========== This paper proposed a type of metaheuristic called Microcanonical Annealing algorithm to optimize the Convolutional Neural Network. Experimental result using MNIST and CIFAR-10 dataset demonstrated that although MA requires more computational time, the accuracy is reasonably better than the standard CNN without metaheuristic. This paper shows that on MNIST dataset, Microcanonical Annealing can improve the accuracy of Convolutional Neural Network, for all variations of epoch up to 4.60%. The results obtained for CIFAR10 dataset, with an accuracy of 99.14% (top-1 error), indicates that the proposed method is able to compete on the current state of the art approaches (96.53%), in the field of image classification. For the future study, fining the proper MA parameters need to be investigated. Furthermore application of this proposed method using the other benchmark data set need to be explored, such as MMI and CKP facial expression data set, as well as ORI and ImageNet. For future research, we will investigate further on the computation time comparison between Microcanonical Annealing to the Simulated Annealing. We also need to examine further the accuracy on the CIFAR-10 dataset using other GPU-based deep learning frameworks such as Torch, Theano, Tensorflow, and Keras with more number of iterations.

--- abstract: | This paper is devoted to study the dynamics of gravitational collapse in the Misner and Sharp formalism. We take non-viscous heat conducting charged anisotropic fluid as a collapsing matter with cylindrical symmetry. The dynamical equations are derived and coupled with the transport equation for heat flux obtained from the M$\ddot{u}$ller-Israel-Stewart causal thermodynamic theory. We discuss the role of anisotropy, electric charge and radial heat flux over the dynamics of the collapse with the help of coupled equation.\ author: - | M. Sharif [^1] and G. Abbas [^2]\ Department of Mathematics, University of the Punjab,\ Quaid-e-Azam Campus, Lahore-54590, Pakistan. title: '**Dynamics of Non-adiabatic Charged Cylindrical Gravitational Collapse**' --- [**Keywords:**]{} Gravitational collapse; Electromagnetic Field; Dynamical and transport equations.\ [**PACS:**]{} 04.20.Cv; 04.20.Dw Introduction ============ One of the most important problems in the gravitation theory and relativistic astrophysics is to understand the end state of a continual gravitational collapse. A massive star undergoes to gravitational collapse at the end of its life cycle. This happens when all the internal nuclear forces fail to supply the sufficiently high pressure to counter-balance gravity. The compact objects such as white dwarfs, neutron stars and black hole are the results of possible stages of the collapsing astronomical objects. In white dwarfs and neutron stars, gravity is neutralized by electron and neutron degeneracy pressure respectively and black hole is a complete collapsed object. Oppenheimer and Snyder (1939) are the pioneers who studied gravitational collapse of an adiabatically flowing dust. This was idealized problem because dust is unrealistic matter and one cannot ignore the effects of pressure on the formation of spacetime singularity. A more analytic analysis was made by Misner and Sharp (1964) with perfect fluid in the inner region of a star. They formulated the dynamical equations governing adiabatic relativistic collapse. In both cases, vacuum was taken in the exterior region of a star. The concept of non-vacuum exterior of a star was introduced by Vaidya (1951) for the radiating fluid in the interior region of the star. Goswami (2007) formulated a more realistic collapsing model by taking the radiating dust matter in the interior of a star. He remarked that bounce in the collapse is due to the dissipation. Debnath et al. (2005) explored gravitational collapse of the non-adiabatic fluid by assuming quasi-spherical Szekeres spacetime in the interior and plane symmetric Vaidya solution in the exterior region. By using the local conservation of momentum, they studied the thermodynamical behavior of the collapsing matter. An extensive literature survey (Herrera et al. 2004 and Mitra 2006)predicts that gravitational collapse is highly dissipative process. This indicates that the effects of the dissipation must be included in the study of collapse for its better understanding. Herrera and Santos (2004) explored dynamical description of gravitational collapse by using Misner and Sharp’s formulation. Matter under consideration was distributed with spherically symmetric and energy loss in the form of heat flow and radiation. Chan (2001) studied the realistic model of radiating star which undergoes dissipation in the form of radial heat flow and shear viscosity. Herrera et al. (2009) also formulated the dynamical equations by including dissipation in the form of heat flow, radiation, shear and bulk viscosity and then coupled with causal transport equations with spherical symmetry. Herrera (2006) discussed the inertia of heat and its role in the dynamics of dissipative collapse with outgoing radial heat flux by using spherical symmetry. Most of the work available in spherical symmetry is due its simplest symmetry. To generalize the geometry of the star, people worked on gravitational collapse using the non-spherical symmetry. The existence of cylindrical and plane gravitational waves provides strong motivation in this regard. Herrera et al. (2005) formulated the set of equations with regularity and matching conditions for the static cylindrically symmetric distribution of matter. Sharif and Ahmad (2007) studied cylindrically symmetric gravitational collapse of two perfect fluids using the high speed approximation scheme. They investigated the emission of gravitational radiations from cylindrically symmetric gravitational collapse. Nolan (2002) investigated naked singularities in the cylinderical gravitational collapse of counter rotating dust shell. Di Prisco et al. (2009) discussed the shear free cylindrical gravitational collapse by using junction conditions. Nakao et al. (2009) studied gravitational collapse of a hollow cylinder composed of dust. Recently, Sharif and Rehmat (2010) discussed the dynamics of viscous dissipative plane symmetric gravitational collapse. The behavior of electromagnetic field in gravitational field has been the subject of interest for many people. Thorne (1965) developed the concept of cylindrical energy and investigated that a strong magnetic field along the symmetry axis may halt the cylindrical collapse of a finite cylinder before it reached to singularity. In recent papers (Sharif and Abbas 2009, 2010a, 2010b), we have studied the effects of the electromagnetic field on the gravitational collapse by taking the homogenous, non-homogeneous and spherical model. Di Prisco et al. (2007) derived the dynamical equations for the spherically symmetric collapse by including electromagnetic field. This work has been extended by Sharif and Siddiqa (2011) for the charged plane symmetric gravitational collapse. Also, Sharif and Fatima (2011) discussed dynamics of adiabatic charged viscous cylindrical gravitational collapse. This paper is aimed to study the dynamics of non-adiabatic charged cylindrically symmetric gravitational collapse to see the effects of charge and heat flux on the process of collapse. The plan of the paper is the following. In the next section, we describe the gravitational source and the Einstein-Maxwell field equations. Section **3** is devoted to matching conditions. We formulate the dynamical equations in section **4** and the derivation of the transport equation and their coupling with the dynamical equations are presented in section **5**. The last section contains the conclusion of the paper. Interior Matter Distribution and the Field Equations ==================================================== We take non-static cylindrically symmetric as an interior metric in the co-moving coordinates in the form $$\begin{aligned} \label{1} &&ds^{2}_{-}=-A^{2}dt^{2}+B^{2}dr^{2}+C^{2}(d{\theta}^{2}+dz^{2}),\\ &&-\infty<t<\infty,\quad 0\leqslant {r} <\infty,\quad 0\leqslant\theta\leqslant{2}\pi,\quad -\infty<z<+\infty\nonumber\end{aligned}$$ where $A$, $B$ and $C$ are functions of $t$ and $r$. Matter under consideration is anisotropic fluid which undergoes dissipation in the form of heat flux. The energy-momentum tensor for such a fluid dissipating only at diffusion approximation, i.e., $\epsilon=0$ is defined as (Herrera 2006)) $$\label{2} T_{\alpha\beta}=(\mu+P_{\bot})V_{\alpha}V_{\beta}+P_{\bot}g_{\alpha\beta} +(P_{r}-P_{\bot})\chi_{\alpha} \chi_{\beta}+V_{\alpha}q_{\beta}+V_{\beta}q_{\alpha},$$ where $\mu,~P_{r},~P_{\bot},~q_{\alpha},~V_{\alpha}$ and $\chi_{\alpha}$ are the energy density, the radial pressure, the tangential pressure, heat flux, the four-velocity of the fluid and the unit four-vector along the radial direction respectively. For the metric (1), the four-vector velocity, heat flux and unit four-vector along the radial direction are given by $$\label{3} V^{\alpha}=A^{-1}\delta^{\alpha}_{0},\quad \chi^{\alpha}=B^{-1}\delta^{\alpha}_{1}, \quad q^{\alpha}=B^{-1}q{\delta}^{{\alpha}}_{1},$$ which satisfy $$\begin{aligned} \label{3} V^{\alpha}V_{\alpha}=-1,\quad \chi^{\alpha}\chi_{\alpha}=1,\quad \chi^{\alpha}V_{\alpha}=0, \quad q^{\alpha}V_{\alpha}=0.\end{aligned}$$ We can write the electromagnetic energy-momentum tensor in the form $$\label{3} T^{(em)}_{\alpha\beta}=\frac{1}{4\pi}\left(F_{\alpha}^{\gamma}F_{\beta\gamma} -\frac{1}{4}F^{\gamma\delta}F_{\gamma\delta}g_{\alpha\beta}\right).$$ The Maxwell equations are given by $$\begin{aligned} \label{4} F_{\alpha\beta}&=&\phi_{\beta,\alpha}-\phi_{\alpha,\beta},\\\label{5} {F^{\alpha\beta}}_{;\beta}&=&4\pi J^{\alpha},\end{aligned}$$ where $F_{\alpha\beta}$ is the Maxwell field tensor, $\phi_{\alpha}$ is the four potential and $J_{\alpha}$ is the four current. Since the charge is at rest with respect to the co-moving coordinate system, thus the magnetic field is zero. Consequently, the four potential and the four current will become $$\label{6} \phi_{\alpha}=\phi{\delta^{0}_{\alpha}},\quad J^{\alpha}=\sigma V^{\alpha},$$ where $\phi=\phi(t,r)$ is an arbitrary function and $\sigma=\sigma(t,r)$ is the charge density. For the interior spacetime, using Eq.(\[6\]), the Maxwell field equations take the following form $$\begin{aligned} \label{8} \phi''-\left(\frac{A'}{A}+\frac{B'}{B}-2\frac{C'}{C}\right){\phi'}&=&{4 \pi }{\sigma}AB^{2}, \\\label{9} {\dot{\phi}}'-\left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}-2\frac{\dot{C}}{C}\right){\phi'}&=&0,\end{aligned}$$ where dot and prime represent the partial derivatives with respect to $t$ and $r$ respectively. Integration of Eq.(\[8\]) implies that $$\label{10} \phi'=\frac{2 sAB}{C^{2}},$$ where $s\left(r\right)=2{\pi}{\int^{r}_{0}}\sigma BC^{2}dr$ is the total charge distributed per unit length of the cylinder and is the consequence of law of conservation of charge, $J^\mu_{; \mu}=0$. Obviously Eq.(\[9\]) is identically satisfied by Eq.(\[10\]). The Einstein field equations, $G_{\alpha\beta}=8\pi(T_{\alpha\beta}+T^{(em)}_{\alpha\beta})$, for the metric (1) can be written as $$\begin{aligned} \label{12} 8{\pi}(T_{00}+T^{(em)}_{00})&=&8{\pi}{\mu}A^{2}+\frac{4s^{2} {A^{2}}}{{C^{4}}}\nonumber\\ &=&\frac{\dot{C}}{C}\left(2\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right) +\left(\frac{A}{B}\right)^{2}\left(-2\frac{C''}{C} +2\frac{B'C'}{BC}-(\frac{C'}{C})^2\right),\nonumber\\\\ \label{13} 8{\pi}(T_{01}+T^{(em)}_{01})&=&8{\pi}q=\frac{2}{AB} \left(\frac{\dot{C'}}{C}-\frac{\dot{B}C'}{BC}-\frac{\dot{C}A'}{CA}\right),\\ \label{14} 8{\pi}(T_{11}+T^{(em)}_{11})&=&8{\pi}P_{r}B^{2} -\frac{4s^{2}B^{2}}{C^{4}}\nonumber\\ &=&-\left(\frac{B}{A}\right)^{2}\left(2\frac{\ddot{C}}{C}+\left(\frac{\dot{C}}{C}\right)^{2} -2\frac{\dot{A}\dot{C}}{AC}\right)+\left(\frac{C'}{C}\right)^{2}+2\frac{A'C'}{AC},\nonumber\\\end{aligned}$$ $$\begin{aligned} \label{15} 8{\pi}(T_{22}+E_{22})&=&8{\pi}P_{\bot}C^{2} +\frac{4s^{2}}{C^{2}}\nonumber\\ &=&-\left(\frac{C}{A}\right)^{2}\left(\frac{\ddot{B}} {B}+\frac{\ddot{C}}{C}-\frac{\dot{A}}{A}\left(\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right) +\frac{\dot{B}\dot{C}}{BC}\right)\nonumber\\ &+&\left(\frac{C}{B}\right)^{2}\left(\frac{A''} {A}+\frac{C''}{C}-\frac{A'}{A}\left(\frac{B'}{B}-\frac{C'}{C}\right)-\frac{B'C'} {BC}\right).\end{aligned}$$ The C-energy for the cylindrically symmetric spacetime is defined by (Thorne 1965) $$\begin{aligned} \label{16} E=\frac{1}{8}(1-l^{-2}\nabla^{a}\tilde{r}\nabla_{a}\tilde{r}),\end{aligned}$$ The circumference radius $\rho$, specific length $l$ and areal radius $\tilde{r}$ can be defined as $$\begin{aligned} \label{2.1.16} \rho^{2}=\xi_{(1)a}\xi^{a}_{(1)},\quad l^{2}=\xi_{(2)a}\xi^{a}_{(2)},\quad \tilde{r}={\rho}l,\end{aligned}$$ where $\xi_{(1)}=\frac{\partial}{\partial{\theta}},~\xi_{(2)}=\frac{\partial}{\partial{z}}$ are Killing vectors and $E$ represents the gravitational energy per unit specific length of the cylinder. The specific energy of the cylinder (Poisson 2004) analogous to Misner and Sharp energy for the spherical symmetry in the interior region with the contribution of electromagnetic field can be written as follows $$\begin{aligned} \label{17} E'=\frac{l}{8}+\frac{C}{2}\left(\frac{\dot{C}^{2}}{A^{2}} -\frac{C'^{2}}{B^{2}}\right)+\frac{s^{2}}{2C}.\end{aligned}$$ We would like to mention here that this energy is also analogous to Taub’s mass function in the plane symmetric spacetime (Sharif and Rehmat 2010 ). Junction Conditions =================== In this section, we assume that the $3D$ timelike boundary surface ${\Sigma}$ splits the two $4D$ cylindrically symmetric spacetimes $V^+$ and $V^-$. The metric which describes the internal region $V^-$ is given by Eq.(\[1\]) while for the representation of exterior region $V^+$, a metric in the retarded time coordinate is considered. If $M(\nu)$ and $Q(\nu)$ are mass and charge, respectively, in retarded time then the corresponding cylindrically symmetric spacetime given by (Chao-Guang 1995) will take the form $$\label{1j} ds^{2}_{+}=-(\frac{-2M(\nu)}{R}+\frac{Q^{2}(\nu)}{R^{2}})d\nu^{2}-2dRd\nu+R^{2}(d\theta^2+dz^2),$$ where $\nu$ is the retarded time coordinate. We can write the induced metric for the hypersurface $\Sigma$ in the following form $$\label{2j} (ds^{2})_{\Sigma}=-{d{\tau}}^2+y^{2}(\tau)(d\theta^{2}+dz^{2}),$$ where $\xi^{i}\equiv(\tau,\phi,z)~(i=0,2,3)$ represent the intrinsic coordinates of $\Sigma$. The Darmois junction conditions (Darmois, 1927) can be stated as follows: - The continuity of the first fundamental form over the hypersurface $\Sigma$ i.e., $$\label{3j} (ds^{2})_{\Sigma}=(ds^{2}_{-})_{\Sigma}=(ds^{2}_{+})_{\Sigma}.$$ - The continuity of the second fundamental form over the hypersurface $\Sigma$ $$\label{4j} [K_{ij}]=K^{+}_{ij}-K^{-}_{ij}=0.$$ Here, $K^{\pm}_{ij}$ is the extrinsic curvature given by $$\label{5j} K^{\pm}_{ij}=-n^{\pm}_{\sigma}(\frac{{\partial}^2\chi^{\sigma}_{\pm}} {{\partial}{\xi}^i{\partial}{\xi}^j}+{\Gamma}^{\sigma}_{{\mu}{\nu}} \frac{{{\partial}\chi^{\mu}_{\pm}}{{\partial}\chi^{\nu}_{\pm}}} {{\partial}{\xi}^i{\partial}{\xi}^j}),\quad({\sigma}, {\mu},{\nu}=0,1,2,3).$$ where $n^{\pm}_{\sigma}$ are the components of outward unit normal to the hypersurface in the coordinates $\chi^{{\pm}\mu}$. We can write the equations of hypersurface as follows $$\begin{aligned} \label{6j} h_{-}(t,r)&=&r-r_{\Sigma}=0,\\ h_{+}(\nu,R)&=&R-R_{\Sigma}(\nu)=0,\label{7j}\end{aligned}$$ where $r_{\Sigma}$ is a constant. Using above equations, we have the interior and exterior spacetimes on $\Sigma$ as follows $$\begin{aligned} \label{8j} (ds^{2}_{-})_{\Sigma}&=&-A^{2}(t,r_{\Sigma})dt^{2}+C^{2}(t,r_{\Sigma})(d\theta^{2}+dR_{\Sigma}(\nu){2}).\\ \label{9j} (ds^{2}_{+})_{\Sigma}&=&-[(\frac{-2M(\nu)}{R_{\Sigma}(\nu)}+\frac{Q^2(\nu)}{R_{\Sigma}(\nu)^2}) +\frac{2dR_{\Sigma}(\nu)}{d\nu}]d\nu^{2}\nonumber\\ &+&R^{2}_{\Sigma}(d\theta^2+dz^2).\end{aligned}$$ The continuity of the first fundamental form gives $$\begin{aligned} \label{10j} R_{\Sigma}(\nu)&=&C(t,r_{\Sigma}),\\ \label{11j} \frac{dt}{d\tau}&=&\frac{1}{A},\\ \label{12j} \frac{d\nu}{d\tau}&=&[(\frac{-2M(\nu)}{R_{\Sigma}}+\frac{Q^2(\nu)}{R_{\Sigma}^2}) +\frac{2dR_{\Sigma}}{d\nu}]^{\frac{-1}{2}}.\end{aligned}$$ Now we consider the second fundamental form over $\Sigma$. For this purpose, we need the outward unit normals to $\Sigma$ using Eqs.(\[6j\]) and (\[7j\]) $$\begin{aligned} \label{13j} n^{-}_{a}&=&B(0,1,0,0), \\\label{14j} n^{+}_{a}&=& \left(\frac{-2M(\nu)}{R_{\Sigma}}+\frac{Q^2(\nu)}{R_{\Sigma}^2} +\frac{2dR_{\Sigma}}{d\nu}\right)^{\frac{-1}{2}}(-\frac{d{R_{\Sigma}}}{d\nu},1,0,0).\end{aligned}$$ The non-zero components of the extrinsic curvature $K^{\pm}_{ij}$ are $$\begin{aligned} \label{15j} K^{-}_{00}&=&-\left(\frac{A'}{AB}\right)_{\Sigma},\\ \label{41} \label{16j} K_{00}^{+}&=&[\frac{d^{2}\nu}{d\tau^{2}}(\frac{d\nu}{d\tau})^{-1}- (\frac{M}{R^2}-\frac{Q^2}{R^3})(\frac{d\nu}{d\tau})]_{\Sigma}.\\ \label{17j} K^{-}_{22}&=& K^{-}_{33}=\left(\frac{CC'}{B}\right)_{\Sigma},\\ \label{18j} K_{22}^{+}&=&[R\frac{dR}{d\tau}+(\frac{Q^2}{R}-2M) \frac{d\nu}{d\tau}]_{\Sigma}=K_{33}^{+}.\end{aligned}$$ The continuity of the extrinsic curvature components yields $$\begin{aligned} \label{19j} [\frac{d^{2}\nu}{d\tau^{2}}(\frac{d\nu}{d\tau})^{-1}- (\frac{M}{R^2}-\frac{Q^2}{R^3})(\frac{d\nu}{d\tau})]_{\Sigma} =-\left(\frac{A'}{AB}\right)_{\Sigma}\\\label{20j} [R\frac{dR}{d\tau}+(\frac{Q^2}{R}-2M) \frac{d\nu}{d\tau}]_{\Sigma}=\left(\frac{CC'}{B}\right)_{\Sigma}\end{aligned}$$ Using Eqs.(\[10j\])-(\[12j\]), (\[12\]) and (\[13\]) in Eqs.(\[19j\]) and (\[20j\]), it follows that $$\begin{aligned} {E'-M}\overset{\Sigma}{=}\frac{l}{8} \Leftrightarrow s\overset{\Sigma}{=}Q, \\\ q\overset{\Sigma}{=}P_{r}-\frac{3s^2}{2C^4}.\end{aligned}$$ The first equation indicates that the difference between two masses is equal to $\frac{l}{8}$ as shown in the adiabatic case (Sharif and Fatima 2011). This is due to the least unsatisfactory definition of Thorne C-energy (Thorne 1965). The second equation describes a relation between heat flux, radial pressure and charge over the hypersurface ${\Sigma}$. It is obvious from this equation that for uncharged radiating fluid, radial pressure and heat flux are equal over the boundary of the collapsing cylinder. The Dynamical Equations ======================= Here we derive the dynamical equations for non-adiabatic charged anisotropic fluid. The energy-momentum conservation, $(T^{\alpha\beta}+T^{{(em)}{\alpha\beta}})_{;\beta}=0$, implies that $$\begin{aligned} \label{18} \left(T^{\alpha\beta}+{T^{(em)}}^{\alpha\beta}\right)_{;\beta}V_{\alpha}=&-&\frac{\dot{\mu}}{A} -\frac{\dot{B}}{AB}(\mu+P_{r})-\frac{2\dot{C}}{AC}(\mu+P_{\bot})\nonumber\\ &-&\frac{2q}{B}(\frac{A'}{A}+\frac{C'}{C})-\frac{q'}{B} =0\end{aligned}$$ and $$\begin{aligned} \label{19} \left(T^{\alpha\beta}+{T^{(em)}}^{\alpha\beta}\right)_{;\beta}\chi_{a} &=&\frac{1}{B}P_{r}'+\frac{A'}{AB}(\mu+P_{r}) +\frac{\dot{q}}{A}+\frac{2q}{A}(\frac{\dot{B}}{B}+\frac{\dot{C}}{C})\nonumber\\ &+&\frac{2C'}{BC}(P_{r}-P_{\bot}) -\frac{ss'}{\pi BC^{4}}=0\end{aligned}$$ Following Misner and Sharp formalism (Misner and Sharp 1964), we discuss the dynamics of the collapsing system. We introduce proper time derivative as well as the proper radial derivative constructed from the circumference radius of a cylinder inside $\Sigma$ as (Sharif and Fatima 2011) $$\label{20} D_{T}=\frac{1}{A}\frac{\partial}{\partial{t}},\quad D_{R}=\frac{1}{R'}\frac{\partial}{\partial{r}},\quad R=C.$$ The fluid velocity in the case of collapse can be defined as $$\label{23} U=D_{T}(R)=D_{T}(C)<0.$$ Consequently, we can write Eq.(\[17\]) as $$\label{24} \tilde{E}=\frac{C'}{B}=\left[U^{2}+\frac{s^{2}}{C^{2}}-\frac{2}{C} \left(E'-\frac{1}{8}\right)\right]^{1/2}.$$ Using Eqs.(\[13\]), (\[14\]), (\[17\]) and (\[20\]), the time rate of change of the C-energy turns out to be $$\label{26} D_{T}E'=-4{\pi}R^{2}\left[\left(P_{r}-\frac{1}{32{\pi}R^{2}}\right)U + q \tilde{E}\right]+\frac{3s^{2}U}{2R^{2}}.$$ This equation represents the variation of total energy inside the collapsing cylinder. Since $U<0$, the first term on the right hand side of this equation will increase the energy of the system provided that the factor within the round brackets is positive. The second term in the square brackets due to negative sign describes the outflow of energy in the form of radiation during the collapse. For the collapsing cylinder containing the same species of the charges, the third term will decrease the energy of the system as $\frac{3s^2}{2R^2}$ plays the role of Coulomb repulsive force and $U<0$. Similarly, using Eqs.(\[12\]), (\[13\]), (\[17\]) and (\[20\]), we obtain $$\label{27} D_{R}E'=4{\pi}R^{2}({\mu}+q\frac{U}{\tilde{E}})+\frac{l}{8} +\frac{s}{R}D_{R}s+\frac{3s^{2}}{2R^{2}}.$$ This equation gives the variation of energy between the adjacent cylindrical surfaces inside the matter distribution. The first term is the energy density of the fluid element along with heat flux contribution. Since $U<0$, the heat flux factor decreases the energy of the system during the collapse of cylinder. The term $\frac{l}{8}$ comes from the definition of C-energy and the remaining terms are due to the electromagnetic field. Using Eqs.(\[14\]), (\[17\]), (\[23\]) and (\[24\]), we can obtain the acceleration $D_{T}U$ of a collapsing matter inside $\Sigma$ $$\label{28} D_{T}U=-\frac{1}{R^{2}}\left(E'-\frac{l}{8}\right) -4\pi{R}P_{r}+\frac{\tilde{E}A'}{AB} +\frac{5s^{2}}{2R^{3}}.$$ Inserting the value of $\frac{A'}{A}$ from Eq.(\[28\]) into Eq.(\[19\]), it follows that $$\begin{aligned} \label{29} (\mu+P_{r})D_{T}U=&-&(\mu+P_{r}) \left[\frac{1}{R^{2}}(E'-\frac{l}{8})+4\pi P_{r}R -\frac{5s^{2}}{2R^{3}}\right]\nonumber\\ &-&\tilde{E}\left[D_{T}q+\frac{4qU}{R}+2qG\right]\nonumber\\ &-&\tilde{E}^2\left[D_{R}P_{r} +2(P_{r}-P_{\bot} )\frac{1}{R}-\frac{s}{{\pi}R^{4}}D_{R}s\right],\end{aligned}$$ where $G=\frac{1}{A}(\frac{\dot{B}}{B}-\frac{\dot{C}}{C})\neq0$ for simplicity. Now the complete dynamics of the system is described by Eq.(\[29\]). The system will evolve radially outward or inward according to $D_{T}U<0$ or $D_{T}U>0$. Thus the terms in Eq.(\[29\]) contributing negatively, favors the collapse while the other contribution prevents the collapse. If both of these cancel each other then there will be a hydrostatic equilibrium. Since the left hand side of Eq.(\[29\]) represents force, so the factor $\mu+P_{r}$ refers to an inertial mass density independent of charge and heat flux contributions. The first and third terms on the right hand side represents the gravitational force. The second term represents the heat flux contribution which seems to leave the system (due to negative sign) through the outward radially directed streamlines. Being in the same direction of pressure, it supports the pressure and would prevent the collapse. The term $(\mu+P_{r})[\frac{1}{R^{2}}(E'-\frac{l}{8})+4\pi P_{r}R -\frac{5s^{2}}{2R^{3}}]$ represents the gravitational force. The factor within the first square brackets shows the effects of specific length and the electric charge on the active gravitational mass term $(\mu+P_{r})$. The third term has three main contributions, i.e., the first is the pressure gradient which is negative, the second is the local anisotropy of the fluid which will be negative for $P_{r}<P_{\bot}$ and the third is the electromagnetic field term. For an isotropic pressure, the second contribution will be vanished. Further, following Di Prisco et al. (2007), it can be found that the third term contributes negatively for $\frac{s}{R}>D_{R}s $. Thus the third square brackets, under these conditions with negative sign, contributes positively by reducing an attractive behavior of force appearing on left hand side of the equation. Since the attractive force is decreased, so the third term prevents the gravitational collapse of the cylinder. The Transport Equation ====================== The transport equation predicts the processes of mass, heat and momentum transfer during the dynamics of a realistic matter. The transport equation for heat flux derived from the M$\ddot{u}$ller-Israel-Stewart causal thermodynamic theory is given by (Herrera 2006) $$\label{30} \tau h^{ab}V^{c}q_{b;c}+q^{a}=-\kappa h^{ab}(T_{,b}+a_{b}T)-\frac{1}{2}\kappa T^2(\frac{\tau V^{b}}{\kappa T^2})_{;b}q^{a},$$ where $h^{ab}=g^{ab}+V^{a}V^{b}$ is the projection tensor, $\kappa$ denotes thermal conductivity, $T$ is temperature, $\tau$ stands for relaxation time which is the time taken by a perturbed system to return into an equilibrium state and $a_{b}T$ is the Tolman inertial term. Due to symmetry of the spacetime, the transport equation reduces to the following form $$\label{31} \tau \dot{q}=-\frac{1}{2}\kappa qT^2(\frac{\tau}{\kappa T^2})^{\cdot}-\frac{1}{2}\tau q(\frac{\dot{B}}{B}+2\frac{\dot{C}}{C}) -\frac{\kappa}{B}(TA)'-qA.$$ Using Eqs.(\[20\]) and (\[23\]) in this equation, it follows that $$\begin{aligned} \label{32} D_{T}q=&-&\frac{\kappa T^2q}{2\tau}D_{T}(\frac{\tau }{\kappa T^2})-q[\frac{3U}{2 R}+{G}+\frac{1}{\tau}]-\frac{\kappa \tilde{E}}{\tau }D_{R}T-\frac{\kappa T}{\tau\tilde{ E}}\nonumber\\ &\times&D_{T}U-\frac{\kappa T}{\tau \tilde{E}}[E'+\frac{l}{8}+4\pi P_{r}R^3-\frac{5s^{2}}{2R}]\frac{1}{R^2}.\end{aligned}$$ In order to understand the effects of heat flux or dissipation on collapsing process, we couple Eq.(\[32\]) with dynamical Eq.(\[29\]). Thus the replacement of Eq.(\[32\]) in Eq.(\[29\]), yields $$\begin{aligned} \label{33} (\mu+P_{r})(1-\alpha)D_{T}U&=& (1-\alpha)F_{grav}+F_{hyd}+\frac{\kappa \tilde{E}^2}{\tau} D_{R}T\nonumber\\&+&\tilde{E}[\frac{\kappa T^2q}{2\tau}D_{T}(\frac{\tau}{\kappa T^2})]-\tilde{E}q(\frac{5U}{2{R}} +G -\frac{1}{\tau}) ,\end{aligned}$$ where $F_{grav},~F_{hyd}$ and $\alpha$ are given by the following equations $$\begin{aligned} \label{34} F_{grav}&=&-(\mu+P_{r})[E'-\frac{C}{8}+4\pi P_{r}{R}^3-\frac{5s^2}{2{R}}]\frac{1}{{R}^2},\\\label{35} F_{hyd}&=&-\tilde{E}^2[D_{R}(P_{r})+\frac{2}{R}(P_{r}-P_{\perp}) -\frac{\mu_{0}^{2}sD_{R}s}{4\pi R^4}], \\\label{36} \alpha&=&\frac{\kappa T}{\tau}(\mu+P_{r})^{-1}.\end{aligned}$$ For the physical interpretation of Eq.(\[33\]), it can be observed that the left hand side of this equation being the product of inertial mass density $(\mu+P_{r}) (1-\alpha)$ and acceleration, $D_{T}U$ represents the Newtonian force. Thus we can write $F=(\mu+P_{r})(1-\alpha)D_{T}U$. It is clear that when $\alpha\rightarrow1$, then $F\rightarrow0$, which means that there is no inertial force and matter would experience the gravitational attraction which causes the collapse. For $0<\alpha<1$, the inertial mass density goes on decreasing while $1<\alpha$ indicates the increase of inertial mass density. Of course, by the equivalence principle, there would occur decrease and increase in the gravitational mass. In this way, one can explicitly distinguish the expanding and collapsing mechanism during the dynamics of dissipative system. Also, Eq.(\[33\]) implies that gravitational force is affected by the same factor but hydrodynamical force is independent of this factor. Further, as long as $(\mu+P_{r})(1-\alpha)D_{T}U<0$, the total Newtonian force of the system remains directed downward which is the indication for the gravitational collapse and the converse is true for the expansion. If there is a continuous change in $\alpha$ from a value greater than one to less than one and vice versa, then there is a transition phase in the system and bouncing would occur. This phenomenon causes the loss of energy from the system and hence the collapsing cylinder with non-adiabatic source leads to the emission of the gravitational radiations. On the basis of this fact, the exterior of the collapsing cylinder is radiation zone which is completely described by a spacetime in radiation coordinates like in the present case. Outlook ======= This paper deals with the effects of the charge and heat conduction on the dynamics of cylindrical anisotropic fluid collapse. We have extended the recent work of Sharif and Fatima (2011) to non-adiabatic case for the transportation process of heat flux during the dynamics of realistic matter collapse. For this purpose, the non-viscous heat conducting anisotropic fluid with cylindrical symmetry has been taken as the source of gravitation in the presence of electromagnetic field. Using the Misner and Sharp formalism, the dynamical equations are derived. We have found that during the collapse of non-viscous heat conducting charged anisotropic fluid, the radial heat flux and electric charge causes to reduce the energy of the system. For $q>0,~E'=\frac{C'}{B}>0$, Eq.(\[27\]) yields that the second term in the first square brackets is negative which indicates that heat is emitting from the system. Thus the external region of such a collapsing system, being non-vacuum, is defined by a cylindrical geometry in the radiation (single null) coordinate. This prediction is analogous to the Vaidya (1951) for spherically symmetric case. The time evolution of the system, given by dynamical equation (\[29\]), indicates that the inertial mass density is independent of heat flux and electric charge. The left hand side of this equation corresponds to the Newtonian force of the system which is decreased by the heat flux. Since the collapse of a star is an irreversible process. The transport process of such non-equilibrium objects and connection between their dynamics and thermodynamics are important for the better understanding of the problem. Thus using the M$\ddot{u}$ller-Israel-Stewart causal thermodynamic theory, the transport equation for the dissipative fluid has been formulated and coupled to the dynamical equation. The coupled equation helps to determine the influence of the heat flux over the dynamics of collapsing cylinder. It has been found that in the coupled dynamical Eq.(\[33\]), the inertial as well as gravitational masses are influenced by the factor $(1-\alpha)$. The role of $\alpha$ during the dynamics of system can be explained as follows: For $\alpha$ tends to one, we get zero mass density. For $0<\alpha<1$, the inertial and the gravitational mass density are decreased while for $\alpha>1$, the gravitational force term becomes negative. This is the case for the reversal of collapse. The conditions on $\alpha$ have been investigated for the bouncing behavior of the system. [**Acknowledgment**]{} We would like to thank the Higher Education Commission, Islamabad, Pakistan for its financial support through the [*Indigenous Ph.D. 5000 Fellowship Program Batch-IV*]{}. [40]{} Chao-Guang, H.: Acta Phys. Sin. **4**, 617(1995)\ Chan, R.: Astron. Astrophys. **368**, 325(2001)\ Darmois, G.: Memorial des Sciences Mathematiques (Gautheir-Villars, Paris, 1927) Fasc. 25\ Debnath, U., Nath, S., Chakraborty, S.: Gen. Relativ. Grav. **37**, 215(2005)\ Di Prisco, A., Herrera, L., Denmat, G.Le., MacCallum, M.A.H., Santos, N.O.: Phys. Rev. **D76**, 064017(2007)\ Di Prisco, A., Herrera, L., Denmat, G.Le., MacCallum, M.A.H., Santos, N.O.: Phys. Rev. **D80**, 064031(2009)\ Goswami, R.: *Gravitational Collapse of Dustlike Matter with Heat Flux*, ArXiv: 0707.1122.\ Hayward, S.A.: Class. Quantum Grav.: **17**, 1749(2000)\ Herrera, L., Di Prisco, A., Martin, J. Ospino, J. Santos, N.O., Troconis, O.: Phys. Rev. **D69**, 084026(2004)\ Herrera, L., Santos, N.O.: Phys. Rev. **D70**, 084004(2004)\ Herrera, L., Di Prisco, A., Fuenmayor, E., Troconis, O.: Int. J. Mod. Phys. **D18**, 129(2009)\ Herrera, L.: Int. J. Mod. Phys. **D15**, 2197(2006)\ Herrera, L., Denmat, G.Le., Marcilhacy, G., Santos, N.O.: Int. J. Mod. Phys. **D14**, 657(2005)\ Misner, C.W., Sharp, D.: Phys. Rev. **136**, B571(1964)\ Mitra, A.: Phys. Rev. **D74**, 024010(2006)\ Nolan, B.C.: Phys. Rev. **D65**, 104006(2002)\ Nakao, K., Harada, T., Kurita, Y., Morisawa, Y.: Prog. Theor. Phys. **122**, 521(2009)\ Oppenheimer, J.R., Snyder, H.: Phys. Rev. **56**, 455(1939)\ Poisson, E.: *A Relaivistic’s Toolkit* (Cambrridge University Press, 2004)\ Sharif, M., Abbas, G.: Mod. Phys. Lett. **A24**, 2551(2009)\ Sharif, M., Abbas, G.: J. Korean Phys. Society **56**, 529(2010)\ Sharif, M., Abbas, G.: Astrophys. Space Sci. **327**, 285(2010)\ Sharif, M., Siddiqa, A.: Gen. Relativ. Grav. **43**, 37(2011)\ Sharif, M., Fatima, S.: Gen. Relativ. Grav. **43**, 127(2011)\ Sharif, M., Ahmad, Z.: Gen. Relativ. Grav. **39**, 1331(2007)\ Sharif, M., Rehmat, Z.: Gen. Relativ. Grav. **42**, 1795(2010)\ Thorne, K.S.: Phys. Rev. **138**(1965)B251; ibid. **139**, B244(1965)\ Vaidya, P.C.: Proc. Indian Acad. Sci. **A33**, 264(1951) [^1]: msharif.math@pu.edu.pk [^2]: abbasg91@yahoo.com

--- abstract: 'The purpose of this study is to investigate observational features of Brans-Dicke wormholes in a case if they exist in our Universe. The energy flux from accretion onto a Brans-Dicke wormhole and the so-called “maximum impact parameter” are studied (the last one might allow to observe light sources through a wormhole throat). The computed values were compared with the corresponding ones for GR-wormholes and Schwarzschild black holes. We shown that Brans-Dicke wormholes are quasi-Schwarzschild objects and should differ from GR wormholes by about one order of magnitude in the accretion energy flux.' author: - 'S.O. Alexeyev' - 'K.A. Rannu' - 'D.V. Gareeva' title: 'Possible observation sequences of Brans-Dicke wormholes' --- Introduction ============ Different types of wormholes were extensively studied recently, especially in the framework of extended gravitational models. A lot of attention was paid to wormholes with massless scalar fields [@ellis; @bronn2], in multidimentional theories [@clement1; @clement2; @clement3; @bhawal; @dotti], in brane-world models [@anch; @bronn3; @cam; @lobo1], in semiclassical gravitation [@garat], and for different equations of state addressing the Dark Matter (or energy) issues [@sushkov]. An important question naturally arises: would it be possible to distinguish between wormholes in different gravitational models using observational data available in the near future with better precision? There are several methods of compact objects’ study nowadays. Novikov and Shatsky [@shat1; @shat2; @shat3] have extracted distinctive features of gravitational lensing of light passing through the wormhole throat. It this work we focus on the possible observations of accretion disks around Brans-Dicke wormholes. According to the latest results, there are gas accretion disks or gas clouds around almost all galactic nuclei at distances from $0.1$ to $10^3$ pc [@urry]. It is possible to evaluate the mass of the object through the gas flux in the clouds. The accretion speed allows to establish the existence of a surface and hence to determine the object’s nature because black holes and wormholes do not have any surface in the opposite to neutron stars that have the one. In the articles by T. Harko, Z. Kovacs and F.S.N. Lobo [@lobo2; @lobo3] it was shown that fluxes of accretion onto different types of wormholes in General Relativity (GR) should differ by orders of magnitude. This approach therefore sounds meaningful. Our work is devoted to analogous considerations for Brans-Dicke wormholes and studies some of their topological aspects as well. We consider a stationary model of accretion disk [@lobo3]. Isotropic coordinates are used, as in the original Brans-Dicke formalism. All the results are written down in Planck units $c = G = \hbar = 1$. Brans-Dicke wormholes ===================== Brans-Dicke theory is a scalar-tensor gravitational model that leads to GR when the coupling constant $| \omega | \to \infty$. The action is: $$\begin{gathered} S = \frac{1}{16\pi} \int d^4x \ \sqrt{-g} \ (\phi R + \frac{\omega}{\phi} \ g^{\mu\nu} \phi_{,\mu} \phi_{,\nu} + L_{matter}),\end{gathered}$$ where $R$ is the Ricci scalar, $\phi$ is a scalar field, $g_{\mu\nu}$ is the metric tensor and $L_{matter}$ is the contribution of matter fields. The corresponding field equations are $$\begin{gathered} \label{eq:02} \begin{split} R_{\mu\nu} - \frac{1}{2} \ g_{\mu\nu} R &= \frac{8\pi}{\phi} \ T_{\mu\nu} + \frac{\omega}{\phi^2} \left(\phi_{,\mu} \phi_{,\nu} - \frac{1}{2} \ g_{\mu\nu} \phi^{,\sigma} \phi_{,\sigma} \right) \\ & + \ \frac{1}{\phi}(\nabla_{\mu} \nabla_{\nu} \phi - g_{\mu\nu} \ \Box \phi), \\ \Box \phi &= \frac{8\pi T}{2 \omega + 3}, \end{split}\end{gathered}$$ where $T = T^{\mu}_{\mu}$, $T_{\mu\nu}$ is the stress-energy tensor and $\Box$ is d’Alembert operator. There is a set of energy conditions in GR [@hock1] that imposes limits on the stress-energy tensor $T_{\mu\nu}$ [@caroll]. To form a wormhole, one needs a matter that breaks the null energy condition but still remains stable. As usual, Jordan-Brans-Dicke theory allows gravity to influence matter via the space-time metric tensor. But the matter itself can change the metric both directly and via the additional scalar field. Thus, the gravitational constant $G$ depends on the scalar field which is variable in space and time. In the theory just the scalar field of Brans-Dicke plays the role of matter. This model does not restrict other kinds of matter or dust to be added of course, it just can not be completely matterless. As the Einstein tensor breaks the null energy condition due to its definition, the right part of expression (\[eq:02\]) also breaks that condition. There are four static spherically-symmetric solutions in Brans-Dicke theory [@brans] but only the first and the fourth ones are independent. Scalar fields in Brans-Dicke model should satisfy the equation [@bhadra] $$\begin{gathered} \label{eq:03} \phi = \phi_0 \left(1 + \frac{1}{\omega + 2} \ \frac{M}{r} \right)\end{gathered}$$ up to first order in $1/r$. $M$ is the asymptotic mass of the wormhole at infinity. The forth solution breaks this condition so we consider only the first Brans-Dicke class of solutions with the metric $$\begin{gathered} \label{eq:04} \begin{split} ds^2 &= -\left( \cfrac{1 - 1/x}{1 + 1/x} \right)^{2l} dt^2 + \left( 1 + \cfrac{1}{x} \right)^4 \left( \frac{1 - 1/x}{1 + 1/x} \right)^{n} \\ &\quad \times \ (d \rho^2 + \rho^2 d \Omega^2), \\ \phi &= \phi_0 \left( \cfrac{1 - 1/x}{1 + 1/x} \right)^{p}. \end{split}\end{gathered}$$ Here $x = \rho/B$, $\quad l = 1 / \lambda$, $\quad n = (\lambda - C - 1) / \lambda$, $\quad p = C / \lambda$, $\rho$ is the isotropic radial coordinate, $\lambda, \ B, \ C$ and $\omega$ are the constants related by [@brans]: $$\begin{gathered} \label{eq:05} \lambda = \sqrt{\cfrac{2\omega+3}{2\omega+4}}, \quad B = \cfrac{M}{2 \phi_0} \ \sqrt{\cfrac{2 \omega + 4}{2 \omega + 3}}, \quad C = - \ \frac{1}{\omega + 2}.\end{gathered}$$ At infinity, the metric asymptotically approaches the Schwarzschild solution $$\begin{gathered} ds^2 = \left( \cfrac{1 - \cfrac{r_g}{4 \rho}}{1 + \cfrac{r_g}{4 \rho}} \right)^2 dt^2 - \left( 1 - \cfrac{r_g}{4 \rho} \right)^4 \left( d {\rho}^2 + {\rho}^2 d \Omega^2 \right)\end{gathered}$$ if $\omega \to \pm \ \infty$. The case $\omega \to + \ \infty$ corresponds to black holes and the one $\omega \to - \ \infty$ leads to wormholes. Thus $\lambda \to 1$, $C \to 0$ and $B \to M/2$ at the infinity. Numerical values for the expressions in Brans-Dicke theory depend on the value of the coupling constant $\omega$. So it should be possible to fix $\left| \omega \right|$ from observational data. As it was shown by Agnese and La Camera, the discussed wormhole is traversable if $\omega < - \ 2$ [@agness]. Thus the scalar field itself plays the role of the required exotic matter. Using Cassini experiments on PPN measuring, it is easy to find that $\left| \omega \right| > 50000$ and only such values are considered. Flux and topology. Results ========================== By solving geodesic equations, it is straightforward to establish that energy, angular momentum and angular velocity for particles on Keplerian orbits in the equatorial region of the accretion disk are: $$\begin{gathered} \label{eq:06} \begin{split} \tilde{E} &= \left(\cfrac{x - \lambda}{x + \lambda} \right)^l \sqrt{\cfrac{x^2 + \lambda^2 - 2x \ (C + 1)}{x^2 + \lambda^2 - 2x \ (C + 2)}}, \\ \tilde{L} &= \sqrt{\cfrac{2}{x}} \ B \ \cfrac{x^2 - \lambda^2}{\sqrt{x^2 + \lambda^2 - 2x \ (C + 2)}} \left(\cfrac{x + \lambda}{x - \lambda} \right)^{l+p}, \\ \Omega &= \cfrac{x}{B} \ \cfrac{1}{x^2 - \lambda^2} \ \sqrt{\cfrac{2x}{x^2 + \lambda^2 - 2x \ (C + 1)}} \left(\cfrac{x - \lambda}{x + \lambda} \right)^{p + 2l}. \end{split}\end{gathered}$$ The flux of energy emitted from the disk surface during the accretion to the Brans-Dicke wormhole can be calculated numerically after substituting (\[eq:06\]) into the expression [@pagethorn]: $$\begin{gathered} \label{eq:07} F(r) = - \ \frac{\dot{M_0}}{4 \pi \sqrt{-g}} \ \frac{\Omega_{,r}}{(\tilde{E} - \Omega \tilde{L})^2} \int \limits_{r_{ms}}^{r} (\tilde{E} - \Omega \tilde{L}) \ \tilde{L}_{,r} \ dr,\end{gathered}$$ where $r_{ms}$ is the marginally stable orbit, and $\dot{M_0}$ is the accretion speed $\cfrac{dM}{dt}$. On Fig. \[fig1\]–\[fig4\] we compare the obtained values of angular velocity and moment, energy and energy flux of the accretion disk particles with the corresponding magnitudes for wormholes in GR and Schwarzschild black holes. $$\epsfxsize=8cm % \epsfysize=5cm \epsfbox{fig1.eps}$$ $$\epsfxsize=8cm % \epsfysize=5cm \epsfbox{fig2.eps}$$ $$\epsfxsize=8cm % \epsfysize=4cm \epsfbox{fig3.eps}$$ $$\epsfxsize=8cm % \epsfysize=5cm \epsfbox{fig4.eps}$$ $$\epsfxsize=8cm % \epsfysize=4cm \epsfbox{fig5.eps}$$ The wormhole throat is the surface with the minimal possible area surrounding the entry into another universe. The isotropic radial coordinate $\rho$ on the throat surface is given by $$\begin{gathered} \label{eq:08} \rho_0 = \cfrac{\sqrt{2} B}{2} \left( \cfrac{2 \ | \omega + 1 | \pm \sqrt{- 8 - 6 \omega}}{\sqrt{(2 \omega + 3) (\omega + 2)}} \right).\end{gathered}$$ If $\left| \omega \right| > 50000$ the throat radius in arbitrary coordinates is $r_0 = 2M$ (Fig. \[fig5\]) with a high precision. Hence, it coincides with the Schwarzschild gravitational radius of the black hole with corresponding mass. The maximum impact parameter $h_{max}$ that allows to observe light sources from the other universe [@shat1] for Brans-Dicke wormholes in almost the full range of $\omega$ is $h_{max} = 3 \sqrt{3} M\approx 5.18 M$ (Fig. \[fig6\]). This expected result confirms the fact that the observable Brans-Dicke wormholes must be quasi-Schwarzschild (for them $\omega \to - \ \infty$ by definition). The marginally stable orbit in the considered model was found numerically and has a value $r_{ms} \approx 5M$. One can find the stress-energy tensor for the Brans-Dicke scalar field from: $$\begin{gathered} \label{eq:09} \begin{split} T_{\mu\nu} &= \frac{\omega}{\varphi^2} \left(\varphi_{,\mu} \varphi_{,\nu} - \frac{1}{2} \ g_{\mu\nu} \varphi^{,\sigma} \varphi_{,\sigma} \right) \\ &+ \frac{1}{\varphi} \ (\nabla_{\mu} \nabla_{\nu} \varphi - g_{\mu\nu} \ \Box \varphi). \end{split}\end{gathered}$$ Therefore $$\begin{gathered} \label{eq:10} \begin{split} T_{rr} &= \cfrac{\omega}{2} \ C^2 l^2 \left(\cfrac{2}{\rho x \left(1 - 1/x^2 \right)} \right)^2 \left( \cfrac{1 - 1/x}{1 + 1/x} \right)^{2 C l} \\ &+ \ \cfrac{1}{\rho^2} \left( \cfrac{1 - 1/x}{1 + 1/x} \right)^{C l} \left\{ 1 - \left( 1 + \cfrac{1}{x} \right)^4 \right. \\ &\times \left. \left( \cfrac{1 - 1/x}{1 + 1/x} \right)^{2 \left( 1 - C l - l \right)} \right\} \\ & \times \left[ \left( - \cfrac{2Cl}{x \left(1 - 1/x^2 \right)} - 1 \right)^2 - \cfrac{4Cl}{x \left( 1 - 1/x^2 \right)^2} - 1 \right]. \end{split} \end{gathered}$$ The study of this expression can reveal new properties of the scalar field and will be the subject or further researches. $$\epsfxsize=8cm % \epsfysize=4cm \epsfbox{fig6.eps}$$ Conclusions =========== We have calculated the flux from accretion onto the Brans-Dicke wormhole. It is important to underline that the observer at infinity sees only the integral flux. The distribution of the flux of energy emitted form the accretion disk is almost Gauss one, so its integral flux is proportional to the maximum of the energy one. These maxima are just the values to be compared. The maximum energy for the accretion onto spherically-symmetric wormholes in GR is one order of magnitude larger that the one for the accretion onto the Brans-Dicke wormhole or on a Schwarzschild black hole. As shown here, for the last two types the values or the energy maximum are almost the same. As allowed values for $\omega$ are already restricted, measuring of the energy flux allows to test the considered model and can help to distinguish between different types of compact objects in future. The throat radius and maximum impact parameter for a Brans-Dicke wormhole were found. We show that these values do not differ from the ones associated with a quasi-Schwarzschild wormhole. According to the Birkhoff theorem the Schwarzschild metric is the most common spherically-symmetric one in a curved space-time. This solution by itself describes the black hole and does not lead to such objects as wormholes, but the studied wormholes tend to the Schwarzschild metric at the infinity. So it is possible to claim that Brans-Dicke wormholes are asymptotically Schwarzschild. This fact allows to search for them basing on future observational data with more accuracy. Acknowledgements ================ Authors would like to thank Prof. Aurelien Barrau and Dr. Alexander Shatsky for the useful discussions on the subject of this work. The work was supported by the Federal Agency on Science and Innovations of Russian Federation via State Contract No. 02.740.11.0575. [99]{} H.G. Ellis, J. Math. Phys. [**14**]{}, 104 (1973) K.A. Bronnikov, Acta Phys. Polon. B [**4**]{}, 251 (1973) G. Clement, Gen. Rel. Grav. [**16**]{}, 131 (1984) G. Clement, Gen. Rel. Grav. [**16**]{}, 477 (1984) G. Clement, Gen. Rel. Grav. [**16**]{}, 491 (1984) B. Bhawal, S. Kar, Phys. Rev. D [**46**]{}, 2464 (1992) G. Dotti, J. Oliva, R. Troncoso, Phys. Rev. D [**75**]{}, 024002 (2007) L.A. Anchordoqui, S.E. PerezBergliaffa, Phys. Rev. D [**62**]{}, 067502 (2000) K.A. Bronnikov, S.-W. Kim, Phys. Rev. D [**67**]{}, 064027 (2003) M. La Camera, Phys.Lett. B [**573**]{}, 27 (2003) F.S.N. Lobo, Phys. Rev. D [**75**]{}, 064027 (2007) R. Garattini, F.S.N. Lobo, Class. Quant. Grav. [**24**]{}, 2401 (2007) S. Sushkov, Phys. Rev. D [**71**]{}, 043520 (2005) A.A. Shatsky, Phys.Usp. [**52**]{}, 811 (2009) A.A. Shatsky, I.D. Novikov, N.S. Kardashev, Phys.Usp. [**51**]{}, 457 (2008) I.D. Novikov, N.S. Kardashev, A.A. Shatsky, Phys.Usp. [**50**]{}, 965 (2007) C.M. Urry, P. Padovani, Publ. Astron. Soc. of the Pacific [**107**]{}, 803 (1995) T. Harko, Z. Kovacs, F.S.N. Lobo, Phys. Rev. D [**78**]{}, 084005 (2008) T. Harko, Z. Kovacs, F.S.N. Lobo, Phys. Rev. D [**79**]{}, 064001 (2009) S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, (Cambridge, Cambridge University Press: 1973) S.M. Carroll, M. Hoffman, M. Trodden, Phys. Rev. D [**68**]{}, 023509 (2003) C.H. Brans, R.H. Dicke, Phys. Rev. [**124**]{}, 925 (1961) A. Bhadra, K. Sarkar, Mod. Phys. Lett. A [**20**]{}, 1831 (2005) A.G. Agnese, M. La Camera, Phys. Rev. D [**51**]{}, 2011 (1995) D.N. Page, K.S. Thorn, Astrophys. J. [**191**]{}, 499 (1974)

--- abstract: 'We address the problem of estimating steady-state quantities associated to systems of stochastic chemical kinetics. In most cases of interest these systems are analytically intractable, and one has to resort to computational methods to estimate stationary values of cost functions. In this work we consider a previously introduced variance reduction method and present an algorithm for its application in the context of stochastic chemical kinetics. Using two numerical examples, we test the efficiency of the method for the calculation of steady-state parametric sensitivities and evaluate its performance in comparison to other estimation methods.' author: - 'Andreas Milias-Argeitis [^1]' - 'John Lygeros [^2]' - 'Mustafa Khammash [^3]' title: 'Variance reduction for steady-state simulation and sensitivity analysis of stochastic chemical systems' --- Introduction ============ Knowledge of steady-state quantities related to an ergodic stochastic chemical system can provide useful insights into its properties. Moreover, steady-state values of cost functions are often easier to estimate with good accuracy compared to stationary distributions. When the system propensities are affine in the state, mean values of polynomial functions of the system state can be computed analytically, as the system of moments is closed. However, when non-polynomial functions are considered, or the system propensities are not affine, analytic calculations are no longer possible and the only solution left is simulation. While moment closure methods [@singh11] can be used to provide good approximations to system moments over a finite time interval, they commonly tend to diverge from the true solution over time, thus resulting in biased steady-state values. The solution presented in Ref. [@ruess11] works only when polynomial functions of the state are considered, and generalization to arbitrary functions is still very difficult. The Finite State Projection algorithm [@munsky06] can be alternatively employed to provide moment estimates with guaranteed accuracy bounds, however the number of states required to attain a certain accuracy makes the method applicable to small problems. On the other hand, stochastic simulation [@gillespie07] can always provide estimates for the stationary mean of any function of the state, however these estimates are inevitably noisy. Brute-force noise reduction can only be achieved at an increased computational cost, either by simulating longer trajectories or by running many trajectories in parallel. Another possibility for reducing the noise in the estimated quantities is the application of a variance reduction technique [@asmussen], provided the added computational cost of the reduced variance estimator is significantly smaller than the gain in computer time. In this work we present the application of such a variance reduction technique [@henderson1; @henderson2] to systems of stochastic chemical kinetics. The idea is based on so-called *shadow functions* and originated in the queueing systems simulation literature, a field where the range of analytically tractable systems in that field is much larger. We demonstrate how the same idea can be applied to steady-state simulation of stochastic chemical systems. We further test the capabilities of the reduced-variance estimators by performing parametric sensitivity calculations for two systems governed by nonlinear propensity functions. The paper is organized as follows: in Sections II and III we define the steady-state estimation problem and define the naïve and shadow function estimators. In Sections IV and V we present one possible implementation of the variance reduction technique to stochastic chemical kinetics and its applicability to steady-state sensitivity analysis. The numerical examples in Section VI serve to demonstrate the effectiveness of the shadow function method in practice and assess its computational cost in comparison to naïve estimation. The conclusions of our study and some future research directions are finally summarized in Section VII. Problem statement ================= Setup ----- Assume an irreducible positive recurrent Markov chain $X=\{X(t):t\geq 0\}$ on a countable space $\mathcal{S}$. In the case of stochastic chemical kinetics, $\mathcal{S}\subseteq\mathbb{Z}^N_{\geq 0}$, where $N$ is the number of chemical species in the system. The chain moves according to a finite set of available transitions $\{\zeta_r\in \mathcal{S}\}_{r=1}^R$, with a corresponding set of propensity functions $\{\lambda_r:\mathcal{S}\to\mathbb{R}\}_{r=1}^R$. The infinitesimal generator of $X$ is the operator $Q$ satisfying $$\label{generator} (Qf)(x) = \sum_{r=1}^R\lambda_r(x)(f(x+\zeta_r)-f(x))$$ for all $f:\mathcal{S}\to\mathbb{R}$ such that $|Qf(x)|<\infty~\forall x\in \mathcal{S}$. The discreteness of $\mathcal{S}$ allows us to enumerate its elements and think of $Q$ as an infinite matrix $Q=(Q_{ij}),~i,j\in\mathbb{N}$. Similarly, any function $f$ on $\mathcal{S}$ can be thought of as an infinite column vector, and distributions on $\mathcal{S}$ can be defined as infinite row vectors. Steady-state estimators ----------------------- Let $f:\mathcal{S}\to\mathbb{R}$ be a $\pi$-integrable cost function associated with $X$. The ergodic theorem for Markov chains [@norris98] ascertains that for any initial condition $$\lim_{t\to\infty}\frac{1}{t}\int_0^t f(X(s))\,ds=\pi f:=\alpha~~\mbox{almost surely,}$$ where $\pi$ is the unique invariant distribution of the system and $\alpha$ the steady-state mean value of $f$. Since the analytic calculation of $\alpha$ is possible only in very special cases, its estimation from simulation is usually the only possibility. The most straightforward estimator of $\alpha$ is $$\alpha_1(t)=\frac{1}{t}\int_0^t f(X(s))\,ds,$$ which is also strongly consistent [@asmussen]. Under some further general conditions on $X$, and $f$, we also know that $$\label{CLT} \sqrt{t}(\alpha_1(t)-\alpha)\Rightarrow \sigma_1\mathcal{N}(0,1),$$ as $t\to\infty$, where $\Rightarrow$ denotes weak convergence, $\mathcal{N}(0,1)$ is the standard normal distribution and $\sigma^2_1$ is called the *time average variance constant (TAVC) for $\alpha_1(t)$* [@asmussen]. The TAVC can be expressed in terms of the integrated autocovariance function of the process $(f_c(X(t)):t\geq 0)$, where $f_c(x):=f(x)-\alpha$, according to the formula [@asmussen]: $$\sigma_1^2=2\int_0^\infty\mathbb{E}_{\pi}[f_c(X(0))f_c(X(s))]\,ds.$$ An alternative expression for $\sigma^2$ can be derived from the functional Central Limit Theorem for continuous-time Markov chains [@bhattacharya]: $$\label{cltvar} \sigma_1^2=-2\langle Qg,g\rangle=-2\int g(x)\cdot Qg(x)\,d\pi(x),$$ where $g$ is a solution to the so-called *Poisson’s equation*[@asmussen2] (Note that solutions to the Poisson equation are unique up to an additive constant, i.e. if $g$ is a solution, then $g'=g+c,~c\in\mathbb{R}$ is also a solution [@asmussen2]): $$\label{poissonexact} Qg=-f_c.$$ A more general class of estimators for $\alpha$ has the form $$\label{a2}\alpha_2(t)=\frac{1}{t}\int_0^t (f+h)(X(s))\,ds,$$ where $h:\mathcal{S}\to\mathbb{R}$ is chosen such that $t^{-1}\int_0^t h(X(s))\,ds\to 0$ almost surely for all $x\in \mathcal{S}$ [@henderson1]. The function $h$ offers an extra degree of freedom in the design of the estimator, which can be exploited to achieve variance reduction. In other words, $h$ can be chosen such that the TAVC of $\alpha_2$, denoted by $\sigma^2_2$, is smaller than $\sigma_1^2$. The obvious choice $h^{opt}=\alpha-f$ is of course intractable, however it suggests that a function $h$ with a zero steady-state mean that is approximately equal to $\alpha-f$ could also achieve variance reduction. Such functions would result in a process $h(X(\cdot))$ that behaves almost antithetically from $f(X(\cdot))$, thus making the variance of $(f+h)(X(\cdot))$ smaller than that of $f(X(\cdot))$ alone. In the steady-state simulation literature, a function $h:\mathcal{S}\to\mathbb{R}$ that satisfies $\pi h=0$ is called a *shadow function* [@henderson2]. The problem then becomes the selection of an appropriate shadow function $h$, so that $\sigma_2^2=c\cdot\sigma_1^2$, with $c<1$. From we see that a reduction of variance by a factor $c$ implies that the variance of $\alpha_2(T)$ is equal to the variance of $\alpha_1(T/c)$. Assuming that the computational cost of both estimators is dominated by the cost of simulating the process $X$, $c^{-1}$ can be used as an indicator of the efficiency of $\alpha_2$ relative to $\alpha_1$. The basic idea of the shadow function method of Ref. [@henderson2], outlined in the next section, is to obtain such an $h$ by using analytical information from a second Markov chain that approximates the original one and is mathematically tractable. A second alternative solution of more general applicability will be described after presenting the method in more detail. The shadow function method: main idea ===================================== The basic idea to the shadow function method is to consider candidate functions of the form $$h=Qg,$$ where $Q$ is the generator matrix of the Markov chain and $g$ is any $\pi$-integrable function (so that the ergodic theorem holds for it as well). In this case, and under the assumption that $\pi(Qg)=(\pi Q)g$ (that holds under some not-too-stringent conditions on $g$ [@henderson2]), $Qg$ becomes a shadow function. We are then naturally led to consider the solution of the Poisson equation , which could provide us with the appropriate function $g$. Solving is of course not possible, since the state space is countable and $\alpha$ is unknown. However, we can look for so-called *surrogate functions* that approximate this solution to build a better estimator. Following the analysis from [@henderson2], we consider another Markov chain $\tilde{X}$ evolving on a countable space $\tilde{\mathcal{S}}$, with stationary distribution $\tilde{\pi}$ and generator $\tilde{Q}$. We also assume a map $r:\mathcal{S}\to\tilde{\mathcal{S}}$ (not necessarily one-to-one) and a function $\tilde{f}$ that is somehow closely related to the original cost function $f$. If $\tilde{f}$ is $\tilde{\pi}$-integrable, we further assume that we can compute the solution to the Poisson equation $$\label{poissonappr} \tilde{Q}\tilde{g}=\tilde{\pi}\tilde{f}-\tilde{f},$$ through which we arrive at a surrogate function $$\label{newg}g(x)=\tilde{g}(r(x))~\forall x\in \mathcal{S}.$$ Summing up, the approach outlined above is based on the fact that if $\tilde{X}$ is: 1) a relatively good approximation of $X$ and 2) tractable analytically, then we can derive a surrogate function $g$ and an estimator $\alpha_2(t)$ which is better than the original estimator in terms of TAVC (assuming that the extra calculation time needed for $\alpha_2$ is not significant). Practical implementation of the shadow function method in chemical kinetics =========================================================================== The shadow function method was originally developed for steady-state simulation of queueing systems, for which a wide range of known and tractable approximations exists. The solution of the approximating Poisson equation can thus be calculated explicitly in many cases, and the application of the method is straightforward. This is not the case for stochastic chemical kinetic systems, where explicit solutions are very hard or impossible to calculate. One thus has to resort to different types of approximation schemes, outlined below. State-space truncation ---------------------- The Markov chains we are interested in satisfy the following properties: 1. They have a finite number of bounded increments over each finite time interval 2. Each state leads to a finite number of states (i.e. for every $i$, $Q(i,j)\neq 0$ for finitely many $j$’s) For such chains, an obvious idea for obtaining an approximating process is to consider a chain evolving on a finite truncation of $\mathcal{S}$ (i.e. consider $\tilde{\mathcal{S}}$ to be a finite subset of $\mathcal{S}$). Actually, under quite weak assumptions and careful definition of $\tilde{Q}$, one can show that the invariant distribution of $\tilde{X}$ on $\tilde{\mathcal{S}}$ approaches that of $X$ as the truncation size grows [@tweedie98]. This of course implies that $\tilde{\pi}\tilde{f}$ also approaches $\pi f$. In this case, the function $r$ between the two state spaces can be intuitively defined to map every $s\in \mathcal{S}\bigcap\tilde{\mathcal{S}}$ to itself, and every $s\in \mathcal{S}\setminus\tilde{\mathcal{S}}$ to some $\tilde{s}\in\tilde{\mathcal{S}}$ (which may vary with $s$). In this way, $\tilde{f}=f|_{\tilde{\mathcal{S}}}$. In order to arrive at a good approximation with this approach, one first has to study a few simulations of $X$, to determine a finite set that contains a good amount of its invariant mass and then perform the necessary calculation of the solution to the Poisson equation on $\tilde{\mathcal{S}}$. The size of this set is determined in practice as a trade-off between tractability and approximation accuracy. However, the applicability of this approach is in general very limited due to the fact that the required truncations grow exponentially with the system dimension. Another problem is that the approximation $\tilde{g}$ of $g$ (the solution to the original intractable Poisson equation) will be very poor for states $s\in \mathcal{S}\setminus\tilde{\mathcal{S}}$, because of the form of $r$, which projects are states outside $\tilde{\mathcal{S}}$ back into the set. This implies that significant variance reduction will be hard to achieve (and in some cases variance may even increase), if the chain sample paths exit $\tilde{\mathcal{S}}$ too frequently during simulation. Approximating solutions of the Poisson equation ----------------------------------------------- Instead of searching for an approximating Markov process, one may try to approximate the solution of directly, to arrive at a suitable shadow function $h$. This approach is also followed in Ref. [@meyn07], where the discrete-time steady-state simulation problem is considered. Given a set of functions $\{\psi_i:\mathcal{S}\to\mathbb{R},~i=1,\dots,n\}$ [^4], one can define $$\label{ghat} \hat{g}=\sum_{i=1}^n\theta_i\psi_i=\psi\cdot\theta,$$ where $\psi=\begin{bmatrix}\psi_1\dots\psi_n\end{bmatrix}$ and $\theta\in\mathbb{R}^{n\times 1}$ is a vector of weights. In principle one could then try to calculate the value of $\theta$ that minimizes the TAVC of $\alpha_2$. Using and , this variance constant turns out to be (see Appendix \[app\_A\]) $$\label{theta_opt} \sigma_2^2=\sigma_1^2-2\left[\langle f_c,\psi\theta\rangle -\langle Q(\psi\theta),g\rangle +\langle Q(\psi\theta),\psi\theta\rangle\right],$$ where $g$ solves . Thus, minimizing the TAVC of $\eqref{a2}$ requires knowledge of $g$, which is unavailable. We thus have to resort to heuristic methods for obtaining a suboptimal estimate of $\theta$, for example by determining the value of $\theta$ that minimizes $$L(\theta):=\int(Q(\psi\theta)+f_c)^2\,d\mu(x)$$ for some suitable measure $\mu$. This is a linear least squares regression problem, which can be solved approximately by generating a set of training data $(f_c(x_1),Q\psi(x_1)),\dots,(f_c(x_m),Q\psi(x_m))$, $x_1,\dots,x_m$, with weights $\mu(x_1),\dots,\mu(x_m)$. If we define the finite-sample version of $\psi$ by $$\Psi:=\begin{bmatrix}\psi_1(x_1)&\psi_2(x_1)&\dots&\psi_n(x_1)\\\psi_1(x_2)&\psi_2(x_2)&\dots&\psi_n(x_2)\\ \dots&\dots&\dots&\dots\\ \psi_1(x_m)&\psi_2(x_m)&\dots& \psi_n(x_m)\end{bmatrix}$$ and similarly set $$F_c:=\begin{bmatrix}f_c(x_1)&f_c(x_2)&\dots&f_c(x_m)\end{bmatrix}^T,$$ we can then calculate the matrix $\Psi_Q\in\mathbb{R}^{m\times n}$ corresponding to $Q\psi$ by using the explicitly known form of the Markov chain generator and finally obtain $$\theta^*=(\Psi_Q^TM\Psi_Q)^{-1}\Psi_Q^TMF_c,$$ where $M=diag(\mu(x_1),\dots,\mu(x_m))$, as the (weighted) least squares minimizer of $L(\theta)$. Variance reduction algorithm using a shadow function ---------------------------------------------------- Putting together all the elements presented above, we summarize below the basic steps of the variance reduction algorithm implemented in this work: - Simulate a long path of the process $X$ using any preferred version of the stochastic simulation algorithm [@gillespie07]. - Obtain a rough estimate of $\alpha$ from the simulated trajectory using $\alpha_1$. - Pick a set of functions $\psi_i,~i=1,\dots,n$ and approximate the solution $g$ to the Poisson equation by $\hat{g}=\psi\cdot\theta^*$ using the approach outlined above. - Evaluate $h=Qg$ along the simulated sample path. - Refine the estimate of $\alpha$ using $\alpha_2$. - Verify that variance reduction has been achieved. The last step is necessary to ensure that the variance has not actually increased due to the use of a suboptimal weight vector $\theta$, and it can be carried out quite straightforwardly using the method of batch means [@asmussen] and the simulated trajectory from Step 1. In all cases we have tested, Steps 2-6 do not contribute more than a few seconds to the computational cost of this algorithm, which implies that the main computational bottleneck still lies at Step 1. ### Implementation issues {#implementation} The estimate of $\theta^*$ obtained by weighted least squares is clearly suboptimal, however it may still yield a reduced-variance estimator. The choice of the weighting measure $\mu$ in the optimization problem above is completely free, and one could in principle try to optimize over both $\mu$ and $\theta$ for a given problem. In practice however, such an approach would increase computational cost of the reduced-variance estimator and possibly eliminate the benefit of variance reduction. To maintain estimator efficiency, one should thus consider a single (or a few) “generic” choices for $\mu$, and preferably re-use the points generated at Step 1. A reasonable choice of weighting measure would be $\pi$ itself. The training set for regression would then consist of all distinct points visited by the process over the course of simulation in Step 1 (possibly after discarding the burn-in period), weighted according to the empirical distribution of the process. A more coarse approximation of $\pi$ would be to use the same sample with all weights being equal. Yet another possibility consists of sampling from a uniform grid that is centered on the area containing the bulk of the invariant mass of the chain. This area can also be crudely determined from the sample of Step 1. All these approaches can achieve variance reduction, however the optimal choice remains problem-dependent. Given that the calculation of least squares estimates can be carried out very efficiently using linear algebraic techniques, it is highly advisable to test several alternatives for the problem at hand. In Ch.11 of Ref. [@meyn07], the problem of selecting an optimal $\theta$ is overcome by introducing a least-squares temporal difference learning (LSTD) algorithm for the approximation of the value of $\theta$ that minimizes the variance of $\alpha_2$ in the context of discrete-time chains. The same algorithm could in principle be applied to continuous-time chains using the embedded discrete-time Markov chain and carrying out the necessary modifications to the original algorithm, based on the results of Ref. [@hordijk]. While this solution is theoretically justified, it requires setting up and running an LSTD estimator in parallel with the simulated chain that will asymptotically converge to the optimal value of $\theta$. Depending on the convergence properties of this estimator, the overall efficiency of the variance reduction scheme may be smaller than the efficiency achieved by using a sub-optimal value for $\theta$, especially when several approximating functions $\psi_i$ are considered. Another degree of freedom in the design of shadow function estimators is the choice of the approximating set $\{\psi_i,i=1,\dots,n\}$. Here, the probabilistic interpretation of Poisson’s equation may assist the selection of approximating functions by providing some useful intuition: Assuming $f$ is $\pi$-integrable and $X$ ergodic, it holds that [@asmussen2; @makowski02] $$g(x)=\mathbb{E}_x\left[\int_0^{\tau(x_0)}f_c(X_s)\,ds\right],$$ where $\tau(x_0)$ is the hitting time of some state $x_0$ (changing $x_0$ simply shifts $g(x)$ by a constant) and $\mathbb{E}_x$ denotes expectation given $X(0)=x$. From this equation one may infer some general properties of $g$ (e.g. monotonicity, oscillatory behavior etc.) based on the form of the propensity functions. The same formula can be used to provide some crude simulation-based estimates of $g(x)$, which can be also helpful for the selection of $\{\psi_i\}$. Finally, a Lyapunov-type analysis can be employed to infer the asymptotic behavior of $g$ [@glynn96]. Steady-state parameter sensitivity ================================== Chemical reaction systems typically depend on several kinetic parameters, and the calculation of the output sensitivity with respect to these parameters is an essential step in the analysis of a given model. While there are several powerful parameter sensitivity methods available today [@sheppard12; @anderson12], they are mostly appropriate for transient sensitivity analysis, as the variance of their estimates tends to grow with the simulation length. Indeed, it can be shown that the variance of sensitivity methods based on the so-called likelihood ratio [@glynn90] or the Girsanov transformation [@plyasunov07] grows linearly with time. On the other hand, the variance of estimators based on finite parametric perturbations can be shown to remain bounded under mild conditions on the propensity functions, provided the underlying process is ergodic. However, the stationary variance can be still quite large, which makes necessary the use of a variance reduction method, such as the one presented here. Besides providing reduced-variance estimates of various steady-state functions of the chain, the shadow function estimator can be also employed for sensitivity analysis using a finite difference scheme [@asmussen] and the Common Random Numbers (CRN) estimator [@rathinam10]. More analytically, assuming that the propensity functions of $X$ are of the form $\lambda(x,p)$, where $p$ is a parameter of interest, the finite difference method aims to characterize the sensitivity of the steady-state value of a given function $f$ to a small finite perturbation of $\delta$ of $p$ around a nominal value $p_0$. If $\delta$ is small enough, we expect that $(\alpha(p_0+\delta)-\alpha(p_0))/\delta$ will be approximately equal to $\partial\alpha/\partial p$. Finite difference-based sensitivity analysis using shadow functions can be simply carried out by generating process trajectories for the nominal and perturbed parameter values, and estimating $\partial\alpha/\partial p$ by $(\alpha_2(p_0+\delta)-\alpha_2(p_0))/\delta$. As shown in Ref. [@rathinam10], use of the same random number stream for the generation of both the nominal and perturbed trajectories can result in great variance decrease compared to using independent streams. Numerical Examples ================== To demonstrate the efficiency of shadow function estimators, we next present two applications of the method to steady-state sensitivity estimation. We compare our finite difference scheme that uses common random numbers and the shadow function estimator to the method of Coupled Finite Differences (CFD) [@anderson12], which frequently outperforms finite-difference estimators based on common random numbers and the Random Time Change representation [@anderson12; @rathinam10]. All numerical examples were generated using custom-written Matlab scripts running on a 3.4 Ghz quad-core PC with 8 GB of RAM. Stochastic focusing ------------------- As a first example, we consider the stochastic focusing model of [@paulsson00], where an input signaling molecule $S$ inhibits the production of another molecule $R$. Stochastic focusing arises due to the presence of stochastic fluctuations in $S$, that make the mean value of $R$ more sensitive to changes $S$ than predicted by the deterministic model of the system. The same system is treated in Ref. [@warren12] using a more sophisticated method based on trajectory reweighting. The system reactions are given below: $$\emptyset\xrightarrow{k_s}S\xrightarrow{k_d}\emptyset,~\emptyset\xrightarrow{k(S)}R\xrightarrow{1}\emptyset,$$ where $k(S)=k_r/(S+K_m)$. The parameters used are $k_d=100$, $k_r=900$ and $K_m=0.9$, while $k_s$ is varied between 200 and 900 to study the effect of varying $\alpha_S:=\mathbb{E}_{\pi}[S]$ on $\alpha_R:=\mathbb{E}_{\pi}[R]$. More specifically (and similarly to Ref. [@warren12]), we want to calculate the gain $$g=\frac{\partial\mbox{ln}(\alpha_R)}{\partial\mbox{ln}(\alpha_S)}=\frac{k_s}{\alpha_R}\frac{\partial\alpha_R}{\partial k_s}.$$ To this end we estimate $\partial\alpha_R/\partial k_s$ using finite differences with $\delta=2\cdot 10^{-2}k_s$ at several points between $k_s=200$ and $k_s=900$. Figure \[SF\_gain\] shows the calculated confidence intervals for $|g|$ obtained by the Common Random Number (CRN) estimator, the CRN estimator in conjunction with a shadow function and the CFD method. For each value of $k_s$, a simulated sample path of length $T=8000$ time units (t.u.) was used to generate 19 batches of length 400 t.u. each, while the first 400 t.u. were discarded as burn-in. Shadow functions consisted of linear combinations of all monomials in two variables up to order three (that is, $\psi_i=S^j\cdot R^k$, with $0<j+k\leq 3$), together with [^5] $\log(S+2)$. This set of $\psi_i$’s was selected manually and is definitely not the “optimal” choice. The training set used for regression consisted of all unique points visited by the process sample paths after a burn-in period. Two alternative weighting schemes were tested for each value of $k_s$: according to the first, all points were assigned equal weight ($M=I$), while in the second one the points were weighted according to the empirical distribution of the process, calculated using the simulated sample paths ($M\approx diag(\pi))$. Both schemes lead to variance reduction, and calculation of $\theta^*$ in each case can be performed very fast ($\sim 0.15$ sec), given the small number of training points ($\sim 2000$). Post-processing of the trajectories for the evaluation of the shadow function over the different batches takes another 5 sec of CPU time. On the other hand, SSA simulation takes on average 40 sec, which demonstrates that the overhead associated with the shadow function usage is relatively small, while the computational savings in the estimation of $\alpha_R$ are significant, as Table \[VR\_aR\] demonstrates. Finally, a CFD simulation of the same length requires 220 sec of CPU time on average, while achieving a smaller magnitude of variance reduction. ![Absolute value of steady-state gain from $\alpha_S$ to $\alpha_R$, estimated with the finite difference method. Shown are 95% confidence intervals obtained with the method of batch means [@asmussen]. Green: CRN estimator. Blue: CFD estimator. Red: CRN combined with shadow functions.[]{data-label="SF_gain"}](SF_sens_CFD){width="\columnwidth"} $k_s/k_d$ 2 2.5 3 3.5 4 4.5 5 5.5 6 7 9 ----------------------------------- ------ ------ ----- ----- ------ ----- ------ ------ ----- ------ ------ CFD: $(\sigma_1^2/\sigma_2^2)$ 5.0 5.3 4.0 5.6 21.0 6.5 16.7 25.3 6.9 22.2 16.6 CRN+SF: $(\sigma_1^2/\sigma_2^2)$ 7464 2051 968 735 444 379 232 475 386 108 129 : Variance reduction in the estimation of $g$[]{data-label="VR_aR"} Before we leave this example, we should point out that application of the shadow function method to just the birth-and-death process of $S$ results in tremendous variance reduction for $f(S)=S^n$ and $n\leq 3$. As an example, Table \[poisson\] shows the confidence intervals of uncentered moment estimates obtained with and without a shadow function for $k_s=500$, $k_d=100$, using a simulated trajectory of $T=5000$ t.u. and 10 batches. We attribute this phenomenon to the fact that the chosen set of functions $\psi_i$ can approximate the true solution to the Poisson equation very closely for $n\leq 3$. Of course, it is known that $S$ has a Poisson stationary distribution, which makes the use of a moment estimator pointless in this case. However, this interesting observation provides some heuristic justification for using polynomial approximating functions $\psi_i$. Moment True value $\alpha_1$ C.I. $\alpha_2$ C.I. ----------------------- ------------ -------------------- --------------------------- $\mathbb{E}_\pi[S]$ 5 $5.0012\pm0.008$ $5\pm7.8\cdot 10^{-15}$ $\mathbb{E}_\pi[S^2]$ 30 $30.0241\pm0.0721$ $30\pm1\cdot 10^{-12}$ $\mathbb{E}_\pi[S^3]$ 205 $205.42\pm1.219$ $205\pm3.3\cdot 10^{-11}$ $\mathbb{E}_\pi[S^4]$ 1555 $1561.5\pm18.88$ $1555.5\pm1.05$ : Variance reduction for a Poisson stationary distribution[]{data-label="poisson"} A six-dimensional system ------------------------ Our second example is a system consisting of two interacting genes, $A$ and $B$. The product of gene $A$ forms homodimers, which repress the expression of $A$, as well as heterodimers with the product of $B$, which repress the expression of $B$. The system species are listed in Table \[species\], while Table \[reactions\] displays the reaction scheme and propensities of our model. Note that several parameters are assumed to be the same for the two genes for simplicity. For the same reason, the gene states are omitted from the model. Name Symbol ------------------- ----------- Gene A mRNA $m_A$ Protein A monomer $p_A$ Protein A dimer $p_{A_2}$ Gene B mRNA $m_B$ Protein B momomer $p_{B_2}$ A-B dimer $p_{AB}$ : Molecular species of the two-gene system[]{data-label="species"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Reactions Propensities ---------------------------------------------------------------------------------- -------------------------------------------------------------------------------------- $\emptyset\xrightarrow{\lambda_1}m_A\xrightarrow{\lambda_2}\emptyset$ $\lambda_1=k_{r}\displaystyle\frac{\phi^4}{\phi^4+p_{A_2}^4}$, $\lambda_2=k_{dr}m_A$ $\emptyset\xrightarrow{\lambda_3}p_A\xrightarrow{\lambda_4}\emptyset$ $\lambda_3=k_{p}m_A$, $\lambda_4=k_{dp}p_A$ $p_A+p_A{\displaystyle $\lambda_5=k_{1}p_A(p_A-1)$, $\lambda_6=k_2p_{A_2}$ \mathrel{\longrightleftharpoons^{\lambda_5\mathstrut}_{\lambda_6}}}p_{A_2}$ $\emptyset\xrightarrow{\lambda_7}m_B\xrightarrow{\lambda_8}\emptyset$ $\lambda_1=k_{r}\displaystyle\frac{\phi^2}{\phi^2+p_{AB}^2}$, $\lambda_2=k_{dr}m_B$ $\emptyset\xrightarrow{\lambda_9}p_B\xrightarrow{\lambda_{10}}\emptyset$ $\lambda_9=k_{p}m_A$, $\lambda_10=k_{dp}p_A$ $p_A+p_B{\displaystyle $\lambda_{11}=k_3p_Ap_B$, $\lambda_{12}=k_4p_{AB}$ \mathrel{\longrightleftharpoons^{\lambda_{11}\mathstrut}_{\lambda_{12}}}}p_{AB}$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : Reactions and propensities[]{data-label="reactions"} The system comprises six molecular species interacting through twelve reactions [^6]. Our goal is to estimate the sensitivity of the steady-state mean of $p_{AB}$ (the second repressor dimer), denoted by $\alpha_{AB}$, to small variations of each of the system parameters. Once more, we compare the behavior of the CRN steady-state estimator with and without a shadow function to the performance of the CFD method. Shadow functions for this system consisted of linear combinations of all monomials of state pairs up to order two. Since the number of unique points visited by this six-dimensional process during simulation was (expectedly) too large to be handled with the least squares method, 10000 points sampled uniformly at random from this set were used in the regression step. For the finite difference method we perturbed each parameter $p$ by $\delta=10^{-2}\cdot p$ and estimated the 95% confidence intervals of each sensitivity estimate using batch means with 24 batches of length 4000 time units each (with an additional 4000 t.u. for burn-in). Prior to parameter perturbations, the estimate of $\alpha_{AB}$ was calculated for $$\begin{aligned} p_0&=&\begin{bmatrix}k_r&\phi&k_{dr}&k_p&k_{dp}&k_1&k_2&k_3&k_4\end{bmatrix}\\ &=&\begin{bmatrix}1&60&0.1&1&0.5&0.02&0.08&0.02&0.1\end{bmatrix}.\end{aligned}$$ The estimates and their corresponding variances were: $\alpha_1=64.46$, $\sigma_1^2=2.68$, $\alpha_2=65.31$ and $\sigma_2^2=3.5\cdot 10^{-3}$. The results of the sensitivity analysis are summarized in Table \[sens\]. CRN sensitivity estimates and their associated confidence intervals not accurate enough to provide useful information. On the contrary, using a shadow function results in great improvements, as now the relative magnitudes and signs of the various sensitivity coefficients can be meaningfully compared to each other. Normalized sensitivity coefficient CRN 95% CI CFD 95% CI CRN+SF 95% CI ---------------------------------------------------------- ---------------------- -------------------- ------------------- $\alpha_2^{-1}\cdot\partial\alpha_{AB}/\partial{k_r}$ $2.461\pm 1.339$ $1.048\pm0.104$ $0.982\pm0.034$ $\alpha_2^{-1}\cdot\partial\alpha_{AB}/\partial{\phi}$ $0.025\pm 0.020$ $0.013\pm0.002$ $0.013\pm0.0005$ $\alpha_2^{-1}\cdot\partial\alpha_{AB}/\partial{k_{dr}}$ $1.853\pm 11.222$ $-10.246\pm 1.453$ $-9.777\pm 0.232$ $\alpha_2^{-1}\cdot\partial\alpha_{AB}/\partial{k_{p}}$ $1.894\pm 1.236$ $1.020\pm0.122$ $0.967\pm0.025$ $\alpha_2^{-1}\cdot\partial\alpha_{AB}/\partial{k_{dp}}$ $0.513\pm 2.049$ $-1.876\pm0.286$ $-1.888\pm0.058$ $\alpha_2^{-1}\cdot\partial\alpha_{AB}/\partial{k_1}$ $137.769\pm 123.344$ $18.437\pm8.181$ $22.374\pm3.383$ $\alpha_2^{-1}\cdot\partial\alpha_{AB}/\partial{k_2}$ $21.675\pm 16.660$ $-3.053\pm1.417$ $-2.785\pm0.333$ $\alpha_2^{-1}\cdot\partial\alpha_{AB}/\partial{k_3}$ $87.001\pm 62.754$ $27.001\pm4.43$ $23.351\pm1.704$ $\alpha_2^{-1}\cdot\partial\alpha_{AB}/\partial{k_4}$ $8.003\pm 16.060$ $-4.818\pm0.756$ $-4.597\pm0.328$ : Sensitivity estimates and associated confidence intervals for the two-gene system. Note that a reduction of a confidence interval by a factor $r$ requires a variance reduction by a factor $r^2$, which can be achieved by running simulations $r^2$ times longer.[]{data-label="sens"} The variance reduction method remains quite efficient computationally in this case as well: SSA simulation of a $10^5$ t.u. trajectory takes about 17 sec of CPU time, while calculation of $\theta$ requires 1 sec and post-processing of the sample path another 3 sec. At the same time, a CFD simulation of the same length requires 95 sec of CPU time on average, while failing to achieve a comparable level of variance reduction. Discussion ========== We demonstrated the applicability of the powerful shadow function method to the problem of steady-state simulation of stochastic chemical kinetics. Our results suggest that a significant increase in the efficiency of a steady-state estimator is possible by only a small increase in its computational cost. The method can be applied to the steady-state estimation of practically any function of the process, and can thus provide improved estimates of high order (cross-)moments, as well as estimates of stationary probabilities for subsets of the process state space, by using set indicators as cost functions. The magnitude of variance reduction achieved by the shadow function method allows also the efficient and precise computation of steady-state parameter sensitivities using the finite difference method. The comparison of the efficiency of this approach for providing steady-state sensitivity estimates with the one presented in [@warren12] is the topic of our ongoing work. It would also be instructive to assess the relative strengths and weaknesses of the LSTD approximation algorithm for optimizing the shadow function [@meyn07] and test its scalability with system size and number of approximating functions (note that only one-dimensional examples are treated in [@meyn07]). The proposed workflow for arriving at a useful shadow function can be improved at several points, by drawing from the large literature on function approximation techniques, in order to enlarge its range of applicability and its accuracy. However, even a crude approach such as the one presented above seems to be sufficient for systems of practical interest. TAVC for shadow function estimator {#app_A} ================================== From and , $\sigma_2^2=-2\langle Qg_2,g_2\rangle$, where $g_2$ solves the Poisson equation $Qg_2=-f_c-Q(\psi\theta)$. This implies that $g_2=g_1-\psi\theta$, where $g_1$ is the solution of the Poisson equation $Qg_1=-f_c$. The variance of the shadow function estimator thus becomes $$\begin{aligned} \sigma_2^2&=-2\langle -f_c-Q(\psi\theta),g_1-\psi\theta\rangle\\ &= -2\left[\langle -f_c,g_1\rangle+\langle f_c,\psi\theta\rangle -\langle Q(\psi\theta),g_1\rangle +\langle Q(\psi\theta),\psi\theta\rangle\right]\\ &= \sigma_1^2-2\left[\langle f_c,\psi\theta\rangle -\langle Q(\psi\theta),g_1\rangle +\langle Q(\psi\theta),\psi\theta\rangle\right].\end{aligned}$$ [10]{} An efficient finite difference method for parameter sensitivities of continuous time markov chains. , 5 (2012), 2237–2258. . Springer, 2003. , vol. 57. Springer, 2007. On the functional central limit theorem and the law of the iterated logarithm for markov processes. , 2 (1982), 185–201. Stochastic simulation of chemical kinetics. (2007), 35–55. Likelihood ratio gradient estimation for stochastic systems. , 10 (1990), 75–84. . , 2 (1996), 916–931. . PhD thesis, Dept. of Operations Research, Stanford University, 1997. Approximating martingales for variance reduction in markov process simulation. , 2 (2002), 253–271. Discrete time methods for simulating continuous time markov chains. (1976), 772–788. The poisson equation for countable markov chains: probabilistic methods and interpretations. In [*Handbook of Markov decision processes*]{}. Springer, 2002, pp. 269–303. . Cambridge University Press, 2007. The finite state projection algorithm for the solution of the chemical master equation. (2006), 044104. . Cambridge University Press, 1998. Stochastic focusing: Fluctuation-enhanced sensitivity of intracellular regulation. , 13 (2000), 7148–7153. Efficient stochastic sensitivity analysis of discrete event systems. , 2 (2007), 724–738. Efficient computation of parameter sensitivities of discrete stochastic chemical reaction networks. , 3 (2010). Moment estimation for chemically reacting systems by extended kalman filtering. , 16 (2011), 165102–165102. A pathwise derivative approach to the computation of parameter sensitivities in discrete stochastic chemical systems. (2012), 034115. Approximate moment dynamics for chemically reacting systems. , 2 (2011), 414–418. Truncation approximations of invariant measures for [M]{}arkov chains. (1998), 517–536. Steady-state parameter sensitivity in stochastic modeling via trajectory reweighting. (2012), 104106. [^1]: Department of Biosystems Science and Engineering, ETH Zurich, 4058 Basel, Switzerland; andreas.milias@bsse.ethz.ch [^2]: Automatic Control Lab, ETH Zurich, Physikstrasse 3, 8092, Zurich, Switzerland; lygeros@control.ee.ethz.ch [^3]: Department of Biosystems Science and Engineering, ETH Zurich, 4058 Basel, Switzerland; mustafa.khammash@bsse.ethz.ch [^4]: Candidate functions $\psi_i$ must satisfy a boundedness condition derived from a Foster-Lyapunov inequality. For more details, see Ref. [@glynn96] or Ch.8 of Ref. [@meyn07]. In the sequel we will assume that all the functions considered satisfy this property. [^5]: This is an example where the probabilistic interpretation of the Poisson equation given in subsection \[implementation\] can provide useful intuition for the selection of approximating functions. In the case at hand, $f=R$, so $g$ (the solution to the Poisson equation) is expected to grow only very slowly with $S$, as the production rate of $R$ tends to zero as $S\to\infty$. [^6]: Note that all examples presented in Ref. [@warren12] consist of two-species systems and no more than four reactions

--- abstract: 'Nontrivial collective behavior may emerge from the interactive dynamics of many oscillatory units. Chimera states are chaotic patterns of spatially localized coherent and incoherent oscillations. The recently-introduced notion of a weak chimera gives a rigorously testable characterization of chimera states for finite-dimensional phase oscillator networks. In this paper we give some persistence results for dynamically invariant sets under perturbations and apply them to coupled populations of phase oscillators with generalized coupling. In contrast to the weak chimeras with nonpositive maximal Lyapunov exponents constructed so far, we show that weak chimeras that are chaotic can exist in the limit of vanishing coupling between coupled populations of phase oscillators. We present numerical evidence that positive Lyapunov exponents can persist for a positive measure set of this inter-population coupling strength.' address: 'Centre for Systems, Dynamics and Control and Department of Mathematics, University of Exeter, Exeter EX4 4QF, UK' author: - Christian Bick and Peter Ashwin bibliography: - 'citations\_18\_9.bib' title: Chaotic Weak Chimeras and their Persistence in Coupled Populations of Phase Oscillators --- Introduction ============ The emergence of collective behavior is a fascinating effect of the interaction of oscillatory units [@Strogatz2000; @Pikovsky2003; @Strogatz2004] and coupled phase oscillators [@Ashwin1992] serve as paradigmatic mathematical models to study these dynamical states in many system of interest ranging from technology to neuroscience [@Acebron2005; @Tchistiakov1996; @Ashwin2015]. In addition to global synchronization, the emergence of locally synchronized coherence-incoherence patterns—commonly known as chimera states [@Kuramoto2002; @Abrams2004]—has received a lot of attention in recent years [@Panaggio2015a]. These are particularly of interest where the patterns have broken symmetry with respect to the networks. Similar states have been shown to exist in real-world experiments [@Tinsley2012; @Hagerstrom2012; @Martens2013] and may be exploited for applications [@Bick2014a]. While chimera states are typically studied at or near the limit of infinitely many oscillators as stationary patterns of the phase density distribution [@Omel'chenko2013], they have generally only been described phenomenologically. Ashwin and Burylko recently introduced a testable definition of a chimera state in the context of finite networks of indistinguishable phase oscillators—a weak chimera [@Ashwin2014a]. Weak chimeras are defined for oscillators where the phases $\vphi_k\in \Tor=\R/2\pi \Z$ evolve according to $$\frac{\ud\vphi_k}{\ud t} = \omega - \frac{1}{n} \sum_{j=1}^n H_{kj} g(\vphi_k-\vphi_j) \label{eq:COsc}$$ in terms of partial frequency synchronization on trajectories; here $H_{kj}$ gives the network topology (respecting a permutation symmetry that acts transitively on the indices of the oscillators) and $g(\phi)$ is the generalized coupling (phase interaction) function. Such weak chimeras cannot appear in fully symmetric globally coupled phase oscillator networks or in any system with three or fewer oscillators. For coupling functions with more than one Fourier mode, there are examples of weak chimeras in systems of four oscillators that are relative periodic orbits, relative quasiperiodic orbits for six oscillators, and weak chimeras of heteroclinic type in a system of ten oscillators [@Ashwin2014a]. Note that coupling functions with multiple Fourier modes are not necessary for the occurrence of attracting weak chimeras: they can be found even for Kuramoto–Sakaguchi coupling $g(\phi)=\sin(\phi+\alpha)$ in such a system with only four oscillators [@Panaggio2015b]. However, the weak chimeras in [@Ashwin2014a] fail to capture one important dynamical feature expected of chimera states in higher-dimensional phase oscillator network: namely that they are chaotic. Moreover, the definition assumes existence of limiting frequency differences, which may not be the case for general trajectories even for a chaotic invariant set possessing a natural measure. Numerical investigations indicate that the chimeras in certain nonlocally coupled rings of oscillators may exhibit positive maximal Lyapunov exponents [@Wolfrum2011a]—while for attractive coupling, they may appear only as transients for typical initial conditions [@Wolfrum2011b]. By contrast, weak chimeras constructed in [@Ashwin2014a; @Panaggio2015b] have Lyapunov exponents that are presumably nonpositive, and can be attractors or repellers. A natural question is whether it is possible find “chaotic” weak chimeras in relatively small finite networks of indistinguishable oscillators—are there weak chimeras whose dynamics have positive maximal Lyapunov exponents for typical orbits? In this paper we construct systems that exhibit such chaotic weak chimera states. It is already known that positive maximal Lyapunov exponents may arise in a fully symmetric phase oscillator system if the coupling function is chosen appropriately [@Bick2011]. We use this to show the existence of weak chimeras for weakly coupled clusters of oscillators with positive Lyapunov exponents in the limit of vanishing coupling. Then we show numerically that the positive Lyapunov exponents persist for nonvanishing coupling. Consequently, the study of chaotic weak chimeras sheds some light on the dynamics of regular chimera states; indeed one might conjecture that “typical” weak chimeras will be chaotic in all but the smallest and simplest systems. The paper is organized as follows. In Section \[sec:Persistence\] we discuss some results on persistence of invariant sets for general nonautonomously perturbed dynamical systems and their consequences for weakly coupled product systems. In Section \[sec:Prelims\] we review and make a simple generalization of the definition of a weak chimera state to cases where the frequency difference may not exist on certain trajectories, but may only have upper and lower bounds. In Section \[sec:CWC\] we apply the results of Section \[sec:Persistence\] to prove the existence of weak chimeras for systems that are weakly coupled populations of phase oscillators. We numerically investigate the maximal Lyapunov exponents of these weak chimeras in Section \[sec:Numerics\] and observe that the maximal Lyapunov exponent stays positive for a large (presumably positive measure) set of coupling strengths between clusters. Finally, we give some concluding remarks. Invariant sets and their persistence under perturbations {#sec:Persistence} ======================================================== In this section we give a general result for the persistence of absorbing sets under general nonautonomous bounded perturbations. Applied to weakly coupled systems, this yields a persistence result for dynamically invariant sets. We will use this result in the following section to show the existence of weak chimeras. The assumptions we make are sufficient to prove the results but not necessary as far as we can determine. Some notation we will use throughout this manuscript. Let $\R$ denote the field of real numbers and $\Rn$ then $\maxdim$-dimensional vector space over $\R$. If $x,y\in\Rn$ then we denote their scalar product by $x\cdot y$. If $F:\Rn\to\R^m$ then let $F'$ denote the (total) derivative and for $x:\R\to\Rn, t\mapsto x(t)$ write $\dot x=\frac{\ud x}{\ud t}$. A (time-dependent) smooth vector field $f:\Rn\times\R\to\Rn$ defines an ordinary differential equation $$\label{eq:ODE} \dot x = f(x, t).$$ If the system is autonomous, that is, $f(x, t)=f(x)$, then for $x\in\Rn$ the associated $\alpha$ and $\omega$-limit sets are closed and invariant with respect to the dynamics. Persistence of absorbing regions -------------------------------- A set $R\subset\Rn$ with compact smooth boundary is *forward absorbing* for  if there is a differentiable function $W:\Rn\to\R$ such that $\partial R = \set{x\in\Rn}{W(x)=0}$ and $W'(x)\cdot f(x, t)<0$ for all $x\in\partial R$, $t\in\R$. A set is *backward absorbing* if it is forward absorbing for $-f$. In other words, an absorbing set has a boundary given by the zeros of some differentiable function and the vector field $f(x, t)$ points either inward or outward everywhere on this boundary. Clearly, if  is autonomous and $R$ is a forward or backward absorbing set then $R$ contains an invariant set of the dynamics. We now show persistence of absorbing regions for an autonomous system $$\begin{aligned} \label{eq:aut} \dot{x}&=f(x)\end{aligned}$$ subject to bounded nonautonomous perturbations. More precisely, consider $$\begin{aligned} \label{eq:nonaut} \dot{x}&=f(x)+\e g(x,t)\end{aligned}$$ with $\e\geq 0$ and $g:\Rn\times\R\to\Rn$ smooth with $\norm{g(x,t)}<M<\infty$ for all $t>0$ and $x\in\Rn$. \[lem:AbsorbingRegion\] Suppose that $R$ is forward absorbing for $\e=0$. Then there is an $\e_0>0$ such that whenever $0\leq\e<\e_0$ the set $R$ is forward absorbing for the dynamics of . We just have to check that $W'(x)\cdot (f(x)+\e g(x,t))<0$ for all $x\in \partial R$ and $t>0$. Since $W'(x)\cdot f(x)$ is negative on the compact surface $\partial R$ it has a lower bound, i.e., there is a $\xi>0$ such that $W'(x)\cdot f(x)<-\xi$ for all $x\in \partial R$. Thus we have for $x\in\partial R$ $$W'(x)\cdot(f(x)+\e g(x,t)) \leq -\xi + \e \max\set{W'(x)\cdot g(x,t)}{x\in \partial R, t\in\R}$$ so if we pick $$\e_0= \frac{\xi}{\max \tset{M\norm{W'(x)}}{x\in \partial R}}$$ then for each $\e<\e_0$ and $t>0$ we have $W'(x)\cdot(f(x)+\e g(x,t))<0$ as required. Let $A$ be a compact invariant set for the dynamics of . (a) A nonnegative differentiable function $V:\Rn\to\R$ is a *Lyapunov function for $A$* if $V(x) = 0$ for all $x\in A$ and $\dot{V}(x) = V'(x)\cdot f(x)<0$ for $x\not\in A$. (b) If $V$ is a Lyapunov function for $A$ then we call $A$ *sufficiently stable*. (c) If $V$ is a Lyapunov function for $A$ for $-f$ then we call $A$ *sufficiently unstable*. One can check that sufficient stability of a set implies asymptotic stability, though the converse only holds under additional assumptions; see for example [@Michel2015]. We now show that Lyapunov functions give absorbing regions arbitrarily close to a compact invariant set $A$ of the unperturbed dynamics as long as the perturbation is sufficiently small. We write $B_{\delta}(A)$ to denote a $\delta$-neighborhood of $A$. \[thm:NonautPerturb\] Suppose that $A$ is a compact invariant sufficiently stable (or unstable) set for the unperturbed dynamics  and $\norm{g(x,t)}<M$ for all $x\in\Rn$ and $t>0$. For any $\delta>0$ there is a compact set $R$ with $$A\subset R \subset B_{\delta}(A)$$ and an $\e_0$ such that whenever $0\leq\e<\e_0$ the set $R$ is an absorbing region for the dynamics of the perturbed system . Sufficient persistence implies that there is a Lyapunov function $V$. For $\eta>0$ define $$R_\eta:= \set{x\in \Rn}{V(x)\leq\eta}.$$ For given $\delta>0$ choose $\eta>0$ such that $A\subset R_{\eta} \subset B_{\delta}(A)$. Since $V$ is a Lyapunov function the set $R=R_{\eta}$ is an absorbing region for the dynamics of . Lemma \[lem:AbsorbingRegion\] now implies that there is an $\e_0$ such that $R$ is also absorbing for the dynamics of . Persistence of invariant sets in product systems ------------------------------------------------ Nonautonomous perturbations may arise from weak coupling of two dynamical systems. ### Weak forcing respecting an invariant set Let $x\in\Rno$, $y\in\Rnt$ and consider the product system $$\label{eq:coupledforced} \begin{aligned} \dot{x}&=f(x)+ \e g(x,y)\\ \dot{y}&=h(x,y). \end{aligned}$$ Suppose that $V\subset\Rnt$ is a compact set such that $\Rno\times V$ is dynamically invariant for all $\e\geq 0$. \[thm:PersistenceFactor\] Let $A\subset\Rno$ be a compact and sufficiently stable (or sufficiently unstable) set for $\dot{x}=f(x)$ and suppose $\norm{g(x,y)}\leq M<\infty$ for all $(x,y)$. For any $\delta>0$ there exists an $\e_0>0$ such that whenever $0\leq\e<\e_0$ there is a dynamically invariant set $A_{\e} \subset B_\delta(A)\times V$ of the perturbed system . For given $\delta >0$, Proposition \[thm:NonautPerturb\] yields an $\e_0$ and $R\subset B_\delta(A)$ such that $R$ is absorbing for all $0\leq\e<\e_0$. Suppose that $A$ is sufficiently stable. The $\omega$-limit set $A_{\e}=\omega(x, y)$ of $(x,y)\in R\times V$ is dynamically invariant and we have $D\subset B_\delta(A)\times V$ since $R$ is absorbing. If $A$ is sufficiently unstable take the $\alpha$-limit set instead. ### Weakly coupled product systems We now present a similar result for weakly coupled systems, $$\label{eq:coupled} \begin{aligned} \dot{x}&=f_1(x)+\e g_1(x,y)\\ \dot{y}&=f_2(y)+\e g_2(x,y) \end{aligned}$$ with $x\in\Rno$, $y\in\Rnt$ and $f_\ell,g_\ell$ smooth such that $\norm{g_\ell}<M<\infty$, $\ell=1, 2$. We refer to $\dot{x}=f_1(x)$ and $\dot{y}=f_2(y)$ as the *uncoupled factors* of . \[thm:Products\] Suppose that the uncoupled factors of have sufficiently stable attractors $A_1$ and $A_2$. Let $A=A_1\times A_2$. Then for any $\delta>0$ there exists an $\e_0>0$ such that for all $0\leq \e<\e_0$ there is an invariant set $A_\e\subset B_\delta(A)$ for the dynamics of . The same holds if $A_1$ and $A_2$ are sufficiently unstable. We repeat the same argument as for Theorem \[thm:NonautPerturb\] for the Lyapunov function $V(x,y):=V_1(x)+V_2(y)$ to show that for any $\delta>0$ there is an $\e_0$ and an absorbing region $R$ with $$A_1\times A_2 \subset R \subset B_{\delta}(A_1\times A_2).$$ for the dynamics of for all $0\leq\e<\e_0$. Since Theorems \[thm:PersistenceFactor\] and \[thm:Products\] are local results, they clearly generalize the case where the assumptions are satisfied on a sufficiently large neighborhood of the set of interest. In fact, the existence of absorbing regions $R_1\supset A_1$, $R_2\supset A_2$ suffices to show the existence of an invariant set $A_{\e}\subset R_1\times R_2$ for sufficiently small $\e$. By contrast, a Lyapunov function allows one to construct invariant sets arbitrarily close to the product $A_1\times A_2$. ### Dynamics on the invariant sets Note that the dynamics on $A_\e$ described in Theorems \[thm:PersistenceFactor\] and \[thm:Products\] may be qualitatively very different than the dynamics on $A$. In fact, even if one assumes appropriate contraction and expansion properties such that $A$ is uniformly normally hyperbolic and $A_\e$ is the invariant set that arises when the vector field is perturbed, one cannot expect that the dynamics on $A$ and $A_\e$ are conjugate [@Fenichel1972]. Persistence relates to the upper semicontinuity of attractors. In fact, many attractors of dynamical systems will be persistent in the sense of upper semicontinuity of the attracting set to a wide class of perturbations that may include nonautonomous perturbations. A more precise statement of this requires framing assumptions on the properties of the attractor and of the class of perturbations. For example, [@Afraimovich1998] give some conditions that ensure persistence of attractors to nonautonomous perturbations, with applications to synchronization problems, while semicontinuity of attractors to discretization is clearly an important topic in the study of numerical approximations of dynamical systems [@Stuart1994]. Weak chimeras for coupled phase oscillators {#sec:Prelims} =========================================== Frequency synchronization and weak chimeras ------------------------------------------- We now consider the dynamics of $\maxdim\in\N$ phase oscillators where oscillator $k$ is characterized by its state $\vphi_k\in\Tor$. Rather than restricting to systems of the form , we first consider a more general setting. Let $F=(F_1, \dots, F_\maxdim)$ be a smooth vector field on the torus $\Torn$. The evolution of the $k$th oscillator is given by $$\label{eq:Dyn} \dot\vphi_k = F_k(\vphi_1,\ldots,\vphi_n).$$ ### Average angular frequency intervals and weak chimeras For any $\vphi^0\in\Torn$ and $T>0$ let us define the [*average angular frequency of oscillator $k$ on the time interval*]{} $[0, T]$ by $$\langle F_k \rangle_{T, \vphi^0} := \frac{1}{T}\int_{0}^{T} F_k(\vphi(t))\ud t,$$ where $\vphi(t)$ is the solution of  with initial condition $\vphi(0)=\vphi^0$. More generally, for $A\subset \Torn$ a compact and forward invariant set under the flow determined by $F$ we define $$\begin{aligned} \Oml_k(F, A) &:= \inf_{\vphi\in A}\liminf_{T\to\infty}\langle F_k \rangle_{T, \vphi},\\ \Omu_k(F, A) &:= \sup_{\vphi\in A}\limsup_{T\to\infty}\langle F_k \rangle_{T, \vphi}\end{aligned}$$ to describe the minimal and maximal average frequency on $A$. Similarly, define the angular frequency differences $$\begin{aligned} \OmlD{k;j}(F, A) & := \inf_{\vphi\in A}\liminf_{T\to\infty} \big(\langle F_k \rangle_{T, \vphi}-\langle F_j \rangle_{T, \vphi}\big),\\ \OmuD{k;j}(F, A) & := \sup_{\vphi\in A}\limsup_{T\to\infty}\big(\langle F_k \rangle_{T, \vphi}-\langle F_j \rangle_{T, \vphi}\big)\end{aligned}$$ where we use the subscript $\text{df}(k;j)$ to indicate the frequency difference between oscillators $k$ and $j$. If $F$ describes the dynamics of weakly coupled phase oscillators then $\Oml_k(F, A), \Omu_k(F, A)$ converge under fairly weak assumptions on smoothness of the dynamics [@Karabacak2009]. There are various alternative ways to express the interval of angular frequency differences, for example one can use a continuous version of [@Jenkinson2006 Prop. 2.1] to write $$\begin{aligned} \OmlD{k;j}(F, A) & = \inf_{\mu\in M_A}\int \langle F_k \rangle_{T, \vphi}-\langle F_j \rangle_{T, \vphi}\, d\mu,\\ \OmuD{k;j}(F, A) & = \sup_{\mu\in M_A}\int \langle F_k \rangle_{T, \vphi}-\langle F_j \rangle_{T, \vphi}\, d\mu.\end{aligned}$$ where $M_A$ is the set of ergodic probability measures invariant under the flow generated by $F$ that are supported on $A$. The *average angular frequency interval* of the $k$th oscillator on $A$ is given by $$\label{eq:Freqs} \Omk(F, A) := \left[\Oml_k(F, A), \Omu_k(F, A)\right]\subset\R$$ and the *average angular frequency difference interval* between oscillators $k, j$ is given by $$\label{eq:FreqDiffs} \OmD{k;j}(F, A) := \left[\OmlD{k;j}(F, A), \OmuD{k;j}(F, A)\right]\subset\R.$$ We say the oscillators $k, j$ are *frequency synchronized* on $A$ if $$\OmD{k;j}(F, A) = \sset{0}.$$ Note that even if $\Omk(F, A)$ and $\Omj(F, A)$ are intervals with interior, it is possible that oscillators $k, j$ are frequency synchronized. We now define weak chimeras by frequency synchronization—this is a generalization of the definition in [@Ashwin2014a]. A compact, connected, chain-recurrent forward-invariant set $A$ is a *weak chimera* if for there are distinct oscillators $k, j, l$ such that $$\begin{aligned} \OmD{k;j}(F, A) &= \sset{0},\\ \Om_l(F, A)\cap\Omk(F, A)&=\emptyset.\end{aligned}$$ As in [@Ashwin2014a] we assume minimal conditions on $A$ that are satisfied if it is an $\omega$-limit set for the dynamics. The definition in [@Ashwin2014a] can be seen a special case of this definition in the case that there is convergence of the average frequency differences: $\OmD{k;j}(F, A)=\OmD{k;j}(F, A)$. ### Average angular frequencies under perturbations What is the effect of a bounded perturbation on the average frequencies of oscillators with dynamics given by ? More precisely, if $\e>0$, $M \geq 0$, $Y_k:\R\to\R$ is smooth with $\abs{Y_k(t)}\leq M<\infty$ for $k\in\sset{1, \dotsc, \maxdim}$ and all $t>0$, we are interested in the average angular frequencies of the dynamics determined by the vector field $$\begin{aligned} \label{eq:Feps} \dot{\vphi}_k = F^{(\e)}_k(\vphi, t) = F_k(\vphi) + \e Y_k(t).\end{aligned}$$ We now give a (very) approximate bound for the average angular frequencies for dynamics with respect to the perturbed vector field $F^{(\e)}$. For $A\subset\Torn$ define the interval $$\label{eq:Bound} N_k(F, A) := \Big[\inf_{x\in A}F_k(x), \sup_{x\in A}F_k(x)\Big].$$ Note that if $A$ is compact, so is $N_k(F, A)$. \[lem:CloseCoarseBound\] Let $A$ be a compact dynamically invariant set for  with $\e=0$ and let $\delta\geq 0$, $\e_0>0$ be given. Then there is an $\eta\geq 0$ such that for any $0\leq\e<\e_0$, if there is a compact dynamically invariant set $A_\e\subset B_\delta(A)$ for  then $$\begin{aligned} \Omk(F^{(\e)}, A_\e)&\subset B_{\eta}(N_k(F, B_\delta(A))).\end{aligned}$$ The statement follows from explicit integral estimates; we give here the upper bound—the lower bound is obtained analogously. If $\vphi(t)$ is a solution of  with $\vphi(0)=\vphi^0\in A_\e$ set $D=\set{\vphi(t)}{t\geq 0}$. For any $T>0$ we have $$\begin{aligned} \langle F^{(\e)}\rangle_{T, \vphi^0} &= \frac{1}{T}\int_{0}^{T}F^{(\e)}_k(\vphi(t))\ud t = \frac{1}{T}\int_{0}^{T}F_k(\vphi(t)) + \e Y_k(t)\ud t\\ & \leq \sup_{\psi\in D} {F_k(\psi)} + \e M \\ & \leq \sup_{\psi\in {B_\delta(A)}} {F_k(\psi)} + \e_0 M.\end{aligned}$$ Thus, for $\eta=\e_0 M$ we have $$\Omu_k(F^{(\e)}, A_\e)=\sup_{\vphi^0\in A_\e}\limsup_{T\to\infty}\langle F^{(\e)}\rangle_{T, \vphi^0}\leq \sup_{\psi\in {B_\delta(A)}} {F_k(\psi)} + \eta$$ which proves the assertion. In particular, Lemma \[lem:CloseCoarseBound\] implies that $\Omega_k(F,A)\subset N_k(F,A)$. While Lemma \[lem:CloseCoarseBound\] suffices for our purposes, using continuity of $F$ one can prove a stronger statement: for any $\eta>0$ there exists a $\delta>0$ and $\e_0>0$ such that for any compact and invariant $A_\e\subset B_\delta(A)$ for  with $0\leq\e<\e_0$ we have $\Omk(F^{(\e)}, A_\e)\subset B_{\eta}(N_k(F, A))$. Moreover, if one assumes ergodicity and the existence of suitable invariant measures then the time averages may be replaced by spatial averages. If in addition the measure deforms nicely under the perturbation of the vector field then we can obtain better approximation of the frequencies of the perturbed system than the coarse bound given in Lemma \[lem:CloseCoarseBound\]. Networks of globally coupled phase oscillators ---------------------------------------------- The notion of frequency synchronization and weak chimeras applies to systems of identical and diffusively coupled phase oscillators. We define $\Th:\Torn\times\Torn\to\Rn$ by $$\label{eq:Theta} \Th_k(\vphi, \psi) := \frac{1}{\maxdim}\sum_{j=1}^{\maxdim}g(\vphi_k-\psi_j).$$ where $g:\Tor\to\R$ is the $2\pi$-periodic coupling (phase interaction) function. Now consider the system of $n$ globally coupled identical phase oscillators $$\label{eq:OscVF} \dot\vphi_k = F_k(\vphi) = \omega+\Th_k(\vphi, \vphi)$$ with $\Th$ defined as in . We may assume $\omega=0$ without loss of generality. The system  is $\Sn\times\Tor$-equivariant [@Ashwin1992]; the group $\Sn$ of permutations acts by permuting the indices of the oscillators and the continuous symmetry $\Tor$ acts by shifting the phase of all oscillators. As a consequence, both the diagonal $$\begin{aligned} \Dn &= \set{(\vphi_1, \dotsc, \vphi_\maxdim)\in\Torn}{\vphi_1 = \dotsb = \vphi_\maxdim}\\ \intertext{and the open~\emph{canonical invariant region}} \label{eq:CIR} \C &:= \set{(\vphi_1, \dotsc, \vphi_\maxdim)}{\vphi_1 < \dotsb < \vphi_n < \vphi_1+2\pi},\end{aligned}$$ are dynamically invariant with respect to the dynamics of . The latter is bounded by codimension one invariant subspaces corresponding to $(n-1)$-cluster states that all intersect at $\Dn$. The preservation of phase ordering implies that weak chimeras cannot exist for  for any $\maxdim$ even for our more general definition of a weak chimera [@Ashwin2014a]. For some specific solutions, the average angular frequencies are easy to calculate: (a) If $A=\sset{\vphi^*}$ is a fixed point of  (relative to the continuous group action) then we have $\Omk(F, A) = \tsset{\frac{1}{\maxdim}\sum_jg(\vphi^*_n-\vphi^*_j)}$. (b) If $A$ is an arbitrary (relative) periodic orbit $\vphi(t)$ with period $P>0$ then $\Omk(A) = \tsset{\frac{1}{P}\int_0^P F_k(\vphi(t))\ud t}$. (c) We have $\Omk(F, \Dn) = \sset{g(0)}$ where $\Dn$ is either a continuum of fixed points (if $g(0)=0$) or the periodic orbit $\vphi(t)=(g(0)t, \dotsc, g(0)t)$. In particular, all oscillators are frequency synchronized. The following two observations become important in the next section; they assert that changing the coupling function $g$ locally at zero affects the angular frequencies on $\Dn$ but not the dynamics of  on compact $A\subset\C$. First, Lemma \[lem:CloseCoarseBound\] gives information about the average angular frequencies of almost fully (phase) synchronized oscillators. Write $Z^{(\e)}=F+\e Y$ for a small perturbation of $F$ as in  and set $D=\set{\vphi(t)}{t\geq 0}$ for a solution $\vphi$ for the flow of $Z^{(\e)}$ that stays close to $\Dn$. We have $\Omk(F, \Dn) = N_k(F, \Dn)= \sset{g(0)}$ and for $\e_0>0$ there exists $\eta>0$ such that $$\label{eq:omgdn} \Omk(Z^{(\e)}, D) \subset [g(0)-\eta, g(0)+\eta]$$ for all $0\leq\e<\e_0$. In particular, if $D \subset \Dn$ then $\eta$ does not depend on $F$. Second, the following lemma implies that the dynamics on the canonical invariant region are independent of $g$ in a neighborhood of zero. To highlight the dependence of $F$ on $g$, we write $F^{(g)}$ for the remainder of this section. \[lem:Coupling\] Let $A\subset \C$, with $\C$ as in , be compact for the coupled phase oscillator system  with coupling function $g$. Then there exists a closed interval $I\subset (0, 2\pi)$ such $F^{(g)}|_A = F^{(\hat{g})}|_A$ for all coupling functions $\gh$ with $\gh|_{I} = g|_{I}$. Since $A\subset\C$ note that $\abs{\vphi_j-\vphi_k}>0$ for $k\neq j$ and $(\vphi_1, \dotsc, \vphi_\maxdim)\in A$. Set $J = \bigcup_{j\neq k}\set{\vphi_k-\vphi_j}{\vphi\in A} \subset \Tor\sm\sset{0}$ and we may write $J\subset (0, 2\pi)$. Since $A$ is compact, $I = [\inf J, \sup J]$ is the desired compact interval. Now $F^{(g)}|_A$ only depends on the values $g$ takes on $I$ and the result follows. Persistence of weak chimeras {#sec:CWC} ============================ We now apply the results of Section \[sec:Persistence\] to networks with weakly coupled populations of phase oscillators, where each population is as introduced in the previous section. Persistence for weakly symmetrically coupled populations -------------------------------------------------------- The weakly coupled product of the dynamics  with itself which defines a dynamical system with $\vphi = (\vphi_{1}, \vphi_{2})\in\Torn\times\Torn=\Tornn$ where $\vphi_{\ell}=(\vphi_{\ell,1}, \dotsc, \vphi_{\ell,\maxdim})$. More explicitly, the dynamics in the case of weak coupling are given by $$\dot{\vphi}=F^{(\e,g)}(\vphi)$$ where $$\label{eq:DynP} \begin{aligned} \dot\vphi_{1,k} &= F^{(\e,g)}_{1,k}(\vphi) := F_k(\vphi_{1}) + \e \Th_k(\vphi_{1},\vphi_{2}) ,\\ \dot\vphi_{2,k} &= F^{(\e,g)}_{2,k}(\vphi) := F_k(\vphi_{2}) + \e \Th_k(\vphi_{2},\vphi_{1}) , \end{aligned}$$ for $k=1, \dotsc, \maxdim$ where $F_k$ is given by and $\Theta$ by . If $A_\e\subset\Tornn$ is compact and dynamically invariant for  we let $\Omega_{\ell,k}(F^{(\e,g)}, A_\e)$ denote the angular frequency intervals for oscillator $(\ell, k)$ with phase $\vphi_{\ell,k}$, $\ell\in\sset{1, 2}$, $k\in\sset{1, \dotsc, \maxdim}$. Moreover, for $D\subset\Tornn$ write $$N_{\ell, k}(F^{(\e, g)}, D) = \Big[\inf_{\vphi\in D}F^{(\e, g)}_{\ell, k}(\vphi), \sup_{\vphi\in D}F^{(\e, g)}_{\ell, k}(\vphi)\Big]$$ as in  and we have $\Omega_{\ell,k}(F^{(\e,g)}, A_\e)\subset N_{\ell, k}(F^{(\e, g)}, A_\e)$ if $A_\e$ is dynamically invariant. Observe that for $\e=0$ the system decouples into two identical groups of $n$ oscillators—both of which with nontrivial dynamics . Note that, in addition to the continuous $\Tor$ symmetry, is equivariant with respect to the action of a symmetry group $\Sn\wr \Sk{2}$ where $\wr$ is the wreath product [@Dionne1996]. That is, $\Sn\wr\Sk{2}= (\Sn)^2\times_s \Sk{2}$ where the $\Sn$ permute the oscillators within the each group of $n$ oscillators and the $\Sk{2}$ permutes the two groups. Observe that this is only a semidirect product $\times_s$ as the two sets of permutations do not necessarily commute. This group acts transitively on the oscillators: the oscillators are indistinguishable. Both $\Torn\times\Dn\subset\Tornn$ and $\Dn\times\Torn\subset\Tornn$ are dynamically invariant for any $\e\geq 0$ as fixed point subspaces of the action of $\Sn\wr\Sk{2}$. ### Persistence of weak chimeras {#persistence-of-weak-chimeras} We now state the main result of this section; the notation $\Ai$ suggests that this invariant set corresponds to the cluster of incoherent oscillators for the chimera while the remaining oscillators are coherent in the sense that they are fully phase-synchronized. \[thm:CWC\] Suppose that $g$ is a coupling function such that $\Ai\subset\C$ is a compact, forward invariant, and sufficiently stable (or unstable) set for the dynamics of . For any $\delta>0$ with $\overline{B_\delta(\Ai)}\subset\C$ there exist a smooth coupling function $\hat{g}$ and an $\e_0>0$ such that the weakly coupled product system  for $F^{(\e,\hat{g})}$ has a weak chimera $A_\e$ with $A_\e \subset \Dn\times B_\delta(\Ai)$ for all $0\leq\e<\e_0$. Let $\delta>0$ be given and let us assume $\Ai$ is sufficiently stable. Compactness and continuity imply that $M := \max_{k\in\sset{1, \dotsc, \maxdim}}\max_{\vphi\in\Tornn}\abs{\Th_k(\vphi)}<\infty$. Define $$N(g) := \bigcup_{k=1}^{\maxdim} N_{2,k}(F^{(0, g)}, \Dn \times B_\delta(\Ai)),$$ assume that $\e_0$ is fixed (we will determine the exact value below) and set $\eta=\e_0M$. If $B_{\eta}(\sset{g(0)})\cap B_{\eta}(N(g)) = \emptyset$ then we choose $\gh=g$, otherwise we choose $\gh$ by Lemma \[lem:Coupling\] such that $F^{(0, \gh)}|_{B_\delta(\Ai)^2}=F^{(0, g)}|_{B_\delta(\Ai)^2}$—implying $N(g)=N(\gh)$—and $\gh(0)$ sufficiently large so that $B_{\eta}(\sset{\gh(0)})\cap B_{\eta}(N(\gh)) = \emptyset$. In particular, for this choice of $\gh$ the set $\Ai$ is still forward invariant, and sufficiently stable in $B_\delta(\Ai)$ for . A similar argument applies if $\Ai$ is sufficiently unstable. Applying Theorem \[thm:PersistenceFactor\] (note that $\Delta_n$ is compact in the topology of $\Tor^n$) yields an $\e_0>0$—take this as the unspecified value of $\e_0$ above—and compact sets $A_\e\subset\Dn\times B_\delta(\Ai)$ for all $0\leq\e<\e_0$ such that $A_\e$ is dynamically invariant for the flow defined by $F^{(\e, g)}$. Moreover, $A_\e$ can be assumed to be connected and chain-recurrent buy taking a subset if necessary. It remains to be shown that any $A_\e$ is a weak chimera. For $\e=0$ the dynamics are uncoupled with $\Omega_{1,k}(F^{(0,\gh)}) = N_{1,k}(F^{(0, \gh)}, \Dn \times \Ai) = \sset{\gh(0)}$ for $k\in\sset{1, \dotsc, \maxdim}$. The weak coupling for $\e>0$ in  may be seen as a bounded nonautonomous perturbation to each factor, that is, $\dot\vphi_{\ell}=F(\vphi_{\ell})+\e Y(\vphi_{\ell}, t)$ with $\norm{Y}\leq M$. By Lemma \[lem:CloseCoarseBound\] we have (with $\eta$ as above) $$\begin{aligned} \Om_{1,k}(F^{(\e,\gh)}, A_\e) &\subset B_\eta(N_{1,k}(F^{(0, \gh)}, \Dn \times B_\delta(\Ai))) = B_\eta(\sset{\gh(0)}),\\ \Om_{2,k}(F^{(\e,\gh)}, A_\e) &\subset B_\eta(N_{2,k}(F^{(0, \gh)}, \Dn \times B_\delta(\Ai))) \subset B_\eta(N(\gh))\end{aligned}$$ for all $k\in\sset{1, \dotsc, \maxdim}$ and any $0\leq \e<\e_0$. For $\vphi\in A_\e$ we have $\vphi_{1, 1}=\dotsb=\vphi_{1, \maxdim}$ and hence $\Om_{1,1}(F^{(\e,g)}, A_\e)=\dotsb=\Om_{1,\maxdim}(F^{(\e,g)}, A_\e)$. Since $B_\eta(\sset{\gh(0)})\cap B_\eta(N(\gh))=\emptyset$ by the choice of $\gh$, we have $\Om_{1,k}(F^{(\e,g)}, A_\e)\cap \Om_{2,k}(F^{(\e,g)}, A_\e)=\emptyset$ for all $k\in\sset{1, \dotsc, \maxdim}$. Therefore any $A_\e$ is a weak chimera for  with the coupling function $\gh$. We remark that the results of Theorem \[lem:Coupling\] can be generalized in a straightforward way to $m\geq 2$ populations of $n$ coupled phase oscillators with dynamics given by $$\label{eq:DynPnm} \dot\vphi_{\ell,k} = F_k(\vphi_{\ell}) + \e\sum_{r\neq \ell}\Th_k(\vphi_{\ell}, \vphi_r)$$ for phases $\vphi_{\ell,k}$ with $k=1, \dotsc, \maxdim$ and $\ell=1, \dotsc, m$. This more general system has symmetry $\Sn\wr \Sk{m}$ which acts transitively on the oscillators. ### Stability The stability of a weak chimeras $A_\e$ of Theorem \[thm:CWC\] as a subset of $\Dn\times\Torn$ depends on the stability properties of $\Ai\subset\Torn$. Introduction of coordinates $\psi_{\ell,k}=\vphi_{\ell,k}-\vphi_{1,k}$ eliminates the phase shift symmetry. In fact $\psi_{1,k} = 0$ and thus the reduced system is a dynamical system on $\Torn$. If $\Ai$ is sufficiently stable so is $A_\e$ in the reduced system; if $\Ai$ is sufficiently unstable in the unperturbed system so is $A_\e$. Of course, if $\Ai$ is sufficiently stable we obtain a sufficiently unstable set by reversing time and vice versa. Transversal stability of the invariant set $\Dn\times\Torn\subset\Tornn$ is determined by the sign of $g'(0)$ [@Ashwin1992]. More precisely, $\Dn\times\Torn\subset\Tornn$ is asymptotically stable if $g'(0)<0$ and asymptotically unstable if $g'(0)>0$. Through local perturbation of the coupling function, one can get the desired transversal stability properties. \[cor:StabilitySwap\] Let $A_\e\subset\Dn\times\Torn$ be a weak chimera of  with coupling function $g$ such that $A_\e$ is asymptotically stable in $\Dn\times\Torn$. Then there exists a coupling function such that $A_\e$ is asymptotically stable in $\Tornn$. An application of Lemma \[lem:Coupling\]; choose a coupling function $\gh$ such that $F^{(0,\gh)}|_{\Dn\times\overline{B_\delta(\Ai)}}=F^{(0,g)}|_{\Dn\times\overline{B_\delta(\Ai)}}$ and $\gh'(0)<0$. Consequently, the weak chimera states constructed above can be asymptotically stable, asymptotically unstable, or of saddle type. They are asymptotically stable (unstable) if $\Ai$ is sufficiently stable (unstable) and $\Dn\times\Torn$ is transversally stable (unstable) and of saddle type if if $\Ai$ is sufficiently stable and $\Dn\times\Torn$ is transversally unstable and vice versa. ### Chaotic weak chimeras: dynamics on the invariant set Suitable coupling functions may now give rise to chaotic weak chimera in the limit of vanishing coupling. So far, we have not made any assumptions on the dynamics on $\Ai$; they may be chaotic. If $g$ is chosen such that the dynamics of on $\Ai$ has positive Lyapunov exponents then so will the dynamics of  on $A_0=\Dn\times\Ai$. Therefore it is reasonable to assume that the positive Lyapunov exponents will persist for a set of $\e>0$ of positive measure, that is $A_\e$ are chaotic weak chimeras for $\e$ sufficiently small. \[rem:PersistenceChaos\] To rigorously show that positive Lyapunov exponents persist for nonzero $\e>0$ one would have to make further assumptions that guarantee there is a suitable invariant measure for $\e=0$ that deforms nicely for sufficiently small $\e>0$; see also Section \[sec:Persistence\]. We give explicit examples of coupling functions that give rise to chaotic dynamics on $A_0$ and numerical evidence that the persisting weak chimeras $A_\e$ are chaotic for nonzero coupling in Section \[sec:Numerics\]. Persistence for coupling breaking symmetry {#sec:CWCnosym} ------------------------------------------ So far we have considered weak coupling between populations that preserved the symmetry of the system for any choice of $\e$. Using the notation otherwise as for and assume that $Y_1, Y_2: \Tornn\to \Rn$ are Lipshitz continuous. The system $$\label{eq:DynPns} \begin{aligned} \dot\vphi_{1,k} &= F^{(\e,g)}_{1,k}(\vphi) = F_k(\vphi_1) + \e Y_{1,k}(\vphi),\\ \dot\vphi_{2,k} &= F^{(\e,g)}_{2,k}(\vphi) = F_k(\vphi_2) + \e Y_{2,k}(\vphi), \end{aligned}$$ for $k=1, \dotsc, \maxdim$ is a weakly coupled product system. While  is $(\Sn\wr\Sk{2})\times\Tor$ equivariant for $\e=0$, for a generic choice of $Y_1, Y_2$ the symmetry is broken whenever $\e>0$[^1]. However, the weak chimera may persist as the next result shows. \[thm:CWCns\] Suppose that $g$ is a coupling function such that $\Ai\subset\C$ and $\Ac = \Dn\subset\Torn$ are compact, forward invariant and sufficiently stable (or sufficiently unstable) sets for the dynamics of  for $F^{(0,g)}$. For any $\delta>0$ there exist a smooth coupling function $\gh$ and $\e_0>0$ such that the weakly coupled product system  has a sufficiently stable (unstable) weak chimera $A$ with $A \subset B_\delta(\Ac\times\Ai)$ for $F^{(\e,\gh)}$ and any $0\leq\e<\e_0$. Persistence of invariant sets $A_\e$ in each factor follows from Theorem \[thm:Products\]. The same argument as in the proof of Theorem \[thm:CWC\] shows that there is a smooth coupling function such that $A_\e$ is a weak chimera since Lemma \[lem:Coupling\] allows us to modify the coupling function in an open neighborhood of zero. While Theorem \[thm:CWCns\] only gives existence of sufficiently stable (or sufficiently unstable) weak chimeras for systems  there may be others of saddle type. Note also that there are coupling functions for which there is a Lyapunov function for the fully synchronized solution [@VanHemmen1993]. Examples of chaotic weak chimeras {#sec:Numerics} ================================= In this section we explicitly construct coupling functions that give rise to chaotic weak chimeras for  with $\e=0$. Moreover, we demonstrate numerically that these chaotic weak chimeras persist for $\e>0$. While we use the methods developed in the previous section to construct the chaotic weak chimeras, we do not check rigorously whether the assumptions are satisfied. A chaotic weak chimera of saddle type ------------------------------------- Consider the dynamics of  for $\maxdim=4$ oscillators. Suppose that the coupling function is given by the truncated Fourier series $$\label{eq:gchaos} {g}(\phi) = \sum_{r=0}^{4} c_r \cos (r\phi+\xi_r)$$ with $c_1 = -2$, $c_2 = -2$, $c_3 = -1$, and $c_4 = -0.88$. For $\xi_1 = \eta_1$, $\xi_2 = -\eta_1$, $\xi_3 = \eta_1+\eta_2$, and $\xi_4 = \eta_1+\eta_2$ with $\eta_1=0.11151$, $\eta_2=0.05586$ the function ${g}$ gives rise to a chaotic attractor $\Ai\subset\C$ with positive maximal Lyapunov exponents [@Bick2011]. ![\[fig:N4CWCSaddle\] A transient started near a chaotic weak chimera of saddle type for two populations of $\maxdim=4$ oscillators with coupling function  for $\eta_1=0.1104$, $\eta_2=0.057511$ and $\e=0.2$. The phase evolution of each oscillator in a co-rotating frame at the speed of oscillator $1$ is shown using a periodic grey scale ($\vphi_{\ell,k}(t)=0$, in black $\vphi_{\ell,k}(t)=\pi$ in white). Middle and bottom panels show the order parameters $R_\ell(t)$ of the populations $\vphi_{\ell}$, $\ell=1,2$, and the instantaneous frequencies $\dot\vphi_{\ell,k}(t)$. Note that for $t\lessapprox 70$ the populations are clearly are not frequency synchronized with chaotic oscillations—evidence of a chaotic weak chimera. After the trajectory leaves the vicinity of the saddle, it converges to a homogeneous chaotic state.](img/gCWCn4Transient.pdf) ![\[fig:N4CWCSaddleScan\] Positive maximal Lyapunov exponents persist for $\e>0$ for two populations of $\maxdim=4$ oscillators  with coupling function  for $\eta_1=0.1104$, $\eta_2=0.057511$. We approximated the maximal Lyapunov exponent (black dots) by integrating  from a random initial condition (sampled uniformly on $\Dn\times\Torn$) for $T=7000$ time units. The shaded regions show the intervals $[\min_{k,t}\dot\vphi_{\ell,k}(t), \max_{k,t}\dot\vphi_{\ell,k}(t)]$ for $\ell=1$ (dark grey) and $\ell=2$ (light grey)—where these do not overlap, there is no frequency synchronization between the two populations and hence a weak chimera. ](img/gLyapScanCinf.pdf) For this choice of coupling function and suitable $\e$, the product system  gives rise to a chaotic weak chimera of saddle type. While these chaotic weak chimeras are attracting in $\Dn\times\Torn$, they are transversally unstable in $\Tornn$ since ${g}'(0)>0$. Figure \[fig:N4CWCSaddle\] shows a trajectory that is initialized slightly off the invariant set. The population order parameter $R_\ell(t) = \tabs{\frac{1}{\maxdim}\sum_{j=1}^\maxdim \exp(i\vphi_{\ell,j})}$, $\ell=1, 2$, characterizes synchronization in the system is equal to one for the synchronized population and fluctuating chaotically for the incoherent population. Numerical simulations show that these chaotic weak chimeras persist for $\e>0$. We calculated the maximal Lyapunov exponent $\lmax$ by numerically integrating the variational equations for [^2]. As shown in Figure \[fig:N4CWCSaddleScan\], chaotic weak chimeras apparently exist for most values $\e\lessapprox 0.3$. An attracting chaotic weak chimera ---------------------------------- As indicated in Corollary \[cor:StabilitySwap\] the chaotic weak chimeras of saddle type can be made attracting in $\Tornn$ with a suitable local perturbation to the coupling function $g$. Define $$\Bm(x) := \begin{cases}\exp\fleft(-\frac{1}{1-x^2}\right)&\text{if} -1<x<1,\\ 0& \text{otherwise}\end{cases}$$ and let $a\in\R$, $b \in (0, \pi)$, $c\in \Tor$ be parameters. Consider the “bump function” $\Bm_{abc}(\phi) = a \Bm\big(\frac{\phi}{b}-c\big)$ with $\phi$ taken modulo $2\pi$ with values in $(-\pi, \pi]$. Thus, $\Bm_{abc}(\phi)$ is a $2\pi$-periodic $\Cinf$ function. Define $$\label{eq:ghat} \gh := {g} + \Bm_{abc}.$$ Let $D$ be an open set with $\Ai\subset D\subset\C$. Now choose $a, b, c$ such that $F^{(\e, g)}|_D=F^{(\e, \gh)}|_D$ and $\gh'(0)<0$. The dynamics of  with coupling function $\gh$ give rise to attracting chaotic weak chimeras. Figure \[fig:CinftyN4CWC\](<span style="font-variant:small-caps;">a</span>) shows a single trajectory for $\e=0.2$ that is initialized slightly off the invariant set. \ Attracting chaotic weak chimeras also appear for larger population sizes. With the same coupling function $\gh$ as for $\maxdim=4$ oscillators we also find chaotic weak chimeras for $\maxdim\in\sset{5, 7}$; cf. [@Bick2011]. A trajectory for $\maxdim=7$ oscillator in each population is depicted in Figure \[fig:CinftyN4CWC\](<span style="font-variant:small-caps;">b</span>). Other coupling functions giving chaotic weak chimeras {#subsec:trigpoly} ----------------------------------------------------- \ \ The appearance of attracting chaotic weak chimeras is not limited to the perturbed coupling function constructed above; these typically have infinitely many nontrivial coefficients in their Fourier expansion. Attracting chaotic weak chimeras can also be found for trigonometric polynomial coupling functions with only finitely many nontrivial Fourier modes. Consider the dynamics of  with two populations of $\maxdim=4$ phase oscillators and coupling function $$g(\phi) = \sum_{r=1}^{L} c_r \cos (r\phi)+ s_r\sin(r\phi)$$ with $L=11$ and Fourier coefficients as specified in Table \[tab:FourierCoeafficients\] in Appendix \[app:TrigPoly\]. This coupling function is analytic and the dynamics show that there is an attracting chaotic weak chimera for a range of $\e$; cf. Figure \[fig:ComegaN4CWC\](<span style="font-variant:small-caps;">a</span>). Our simulations indicate that apart from the chaotic weak chimera there are other stable attractors in the system. Further numerical investigation, using the fixed initial conditions on the chaotic weak chimera in Figure \[fig:ComegaN4CWC\](<span style="font-variant:small-caps;">a</span>), is summarized in Figure \[fig:ComegaN4CWC\](<span style="font-variant:small-caps;">b</span>). Positive maximal Lyapunov exponents can arise for a range of positive coupling values $\e\lessapprox 0.15$. It is possible that adiabatic continuation—that is, using a point on the attractor for one parameter value as an initial condition for a nearby parameter value—may give more detailed insights into how the chaotic weak chimeras develop and bifurcate as $\e$ is varied. Discussion {#sec:discuss} ========== In Section \[sec:CWC\] we prove a general existence result for weak chimeras in coupled phase oscillator systems. Chimeras are constructed in “modular” networks that are weakly coupled but the numerical investigations indicate that they exist even beyond the weak coupling limit. As dynamically invariant sets, the weak chimeras are not transient—“persistent” chimera states in systems with generalized coupling were recently observed numerically [@Suda2015]. But the existence of weak chimeras of saddle type induces transient dynamics for initial conditions close to the saddle. While our results are stated for the symmetric case that each population has the same number of oscillators, it is straightforward to generalize the construction to asymmetric systems with populations of differing sizes, say of $n_1$ and $n_2$ oscillators. If the oscillators belonging to the coherent region are not only frequency synchronized but also phase locked (their phase difference stays constant over time) the minimal number of oscillators to get similar chaotic weak chimera dynamics is $n_1+n_2=5$ since the dynamics are effectively three dimensional; however in this case the oscillators will not longer be indistinguishable. The weak chimeras of saddle type (i.e., with both stable and unstable directions) are certainly of interest as analogues to the states studied in [@Wolfrum2011b]. Further existence results of such chaotic weak chimeras may be possible with the methods mentioned in Section \[sec:Persistence\]: normally hyperbolic sets persist under small perturbations. Moreover, for suitable choice of parameters, such chimera saddles could have connecting orbits, thus leading to transitions from one chimera state to the next one. Constructions of such connections could be seen as a form of control of spatially localized dynamics similar to chimera control [@Bick2014a]. The relationship between the chaotic weak chimeras we constructed here and “classical” chimera states in multiple populations needs to be clarified further: in [@Abrams2004] the coupling function only contains a single nontrivial Fourier mode and the coupling strength between the populations is almost as strong as within a population. Moreover, “large” populations are considered in the continuum limit of infinitely many oscillators where the dynamics reduce to mean field equations [@Abrams2004; @Panaggio2015a]. By contrast, the chaotic weak chimeras here arise in weakly coupled populations of four to seven oscillators with coupling given by functions with four or more nontrivial Fourier components. Continuation of these solutions in a suitable parameter space may shed some light on whether there these solutions are directly related. It is worth noting that the chaotic weak chimeras constructed here in such small networks also exhibit chaotic fluctuations of the order parameter [@Bick2011] similar what is observed for large ensembles of interacting nonsmooth oscillators [@Pazo2014]. Clearly, our results for small populations extend to more general (limit cycle) oscillators whose phase reduction is given by the coupling functions of our construction; see [@Kori2008] for approaches to design general oscillator systems with a desired phase reduction. In fact, we expect chaotic dynamics to be more common for general oscillator networks that are not necessarily weakly coupled (as long as phases can still be defined, see remarks in [@Bick2015d]) as amplitude variations facilitate chaotic dynamics in globally coupled identical oscillators [@Nakagawa1993]. In summary, the notion of a weak chimera yields a mathematically precise definition of chimera states for finite dimensional systems. Here we show that weak chimeras which mimic the dynamical behavior of regular chimeras can be constructed explicitly. We anticipate that a similar study of weak chimeras may give further insight into how chimeras arise in finite-dimensional coupled phase oscillator networks. Acknowledgements {#acknowledgements .unnumbered} ================ CB and PA would like to thank Mike Field and Tiago Pereira for helpful discussions. The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007–2013) under REA grant agreement n^o^ 626111 (CB). A trigonometric polynomial coupling function {#app:TrigPoly} ============================================ Table \[tab:FourierCoeafficients\] gives coefficients for a trigonometric polynomial coupling function that gives an attracting chaotic weak chimera for the system discussed in Section \[subsec:trigpoly\]. $$\begin{aligned} c_0 &= -0.48239 & s_0 &= 0.0538\\ c_1 &= -0.48239 & s_1 &= -0.05766\\ c_2 &= -0.23244 & s_2 &= 0.03754\\ c_3 &= -0.20325 & s_3 &= 0.0313\\ c_4 &= 0.01322 & s_4 &= -0.00626\\ c_5 &= 0.01261 & s_5 &= -0.0074\\ c_6 &= 0.01191 & s_6 &= -0.00849\\ c_7 &= 0.01111 & s_7 &= -0.00951\\ c_8 &= 0.01023 & s_8 &= -0.01045\\ c_9 &= 0.00927 & s_9 &= -0.01131\\ c_{10} &= 0.00823 & s_{10} &= -0.01209\end{aligned}$$ [^1]: Note that in this case the oscillators may not be indistinguishable anymore. [^2]: Integration for $T=7000$ time units was carried out in MATLAB using the standard adaptive Runge–Kutta scheme with relative and absolute error tolerances of $10^{-9}$ and $10^{-11}$ respectively.

--- abstract: 'After inflation, a period of preheating may have produced a stochastic background of high frequency gravitational waves (GWs) that would persist until today. The nature of the inflaton’s coupling to Standard Model or other fields is unknown, so it is useful to ask what features such fields may typically have, and how these affect predictions for the GW’s produced. Here we consider the inflaton to be coupled to a light scalar field, and show that even a very small quartic self-interaction term will reduce the amplitude of the GW spectrum. For self-coupling $\lambda_{\chi} \gtrsim g^2$, where $g^2$ is the inflaton-scalar coupling, the peak energy density goes as $\Omega_{\rm gw}^{(\lambda_{\chi})} / \Omega_{\rm gw}^{(\lambda_{\chi}=0)} \sim (g^2/\lambda_{\chi})^{2}$. A consequence is that if the universe reheats through an inflaton-Higgs coupling then the spectrum would be suppressed but the dynamics would be sensitive to the Higgs potential near the energy scale of inflation.' author: - 'Jeffrey M. Hyde' bibliography: - 'refs.bib' title: 'Sensitivity of gravitational waves from preheating to a scalar field’s interactions' --- Introduction {#sec:intro} ============ Inflation leaves the universe cold and nearly empty of particles, so there needs to be a reheating mechanism for energy transfer between inflaton and Standard Model fields in order to create the thermalized particles that existed before Big Bang Nucleosynthesis began. This is typically modeled by a small, direct coupling between inflaton and another field. The first discussions of reheating [@Linde:1981mu; @Albrecht:1982mp; @Dolgov:1982th; @Abbott:1982hn; @Traschen:1990sw; @Kofman:1994rk; @Kofman:1997yn] studied a perturbative calculation of inflaton decay into the coupled field, with energy gradually transferred to matter fields. (Also see the reviews [@Allahverdi:2010xz; @Amin:2014eta].) However, inflaton decay occurs in the context of large, coherent field oscillations and nonperturbative effects should also be taken into account [@Shtanov:1994ce; @Kofman:1994rk; @Kofman:1997yn; @Greene:1997fu]. Typically, the inflaton $\phi$ is considered to be coupled to a field $\chi$ by an interaction $\frac{1}{2}g^2\phi^2\chi^2$, which is $\chi$’s only potential energy term. As the inflaton oscillates about the bottom of its potential after inflation, the phenomenon of parametric resonance leads to some modes of the decay product $\chi$ being excited at an exponential rate. This effect, which may occur briefly at the beginning of a longer period of reheating, is called preheating. (Most of the work on this subject has been in the context of direct couplings between inflaton and matter fields; see [@Alexander:2014bsa] for a scenario that does not require this.) Preheating in these models can produce gravitational waves [@Khlebnikov:1997di; @Easther:2006gt; @Easther:2006vd; @Dufaux:2007pt; @GarciaBellido:2007af; @Dufaux:2008dn], since the exponential amplification of certain modes leads to a large contribution to anisotropic stress, which sources tensor perturbations. Predictions for the resulting spectrum are around $h^2\Omega_{\rm gw} \sim 10^{-10}$ and $f \sim 10^4$ to $10^6$ Hz today for massive or $\lambda\phi^4$ inflation or could be as low as $10^2$ to $10^3$ Hz for hybrid inflation models. Some work [@GarciaBellido:2007af; @GarciaBellido:2008ab; @Dufaux:2010cf; @Enqvist:2012im; @Enqvist:2012tc; @Enqvist:2014tta; @Figueroa:2014aya] has addressed this problem in the context of various models that relate to processes that are more specific. These find a wide range of possibilities. For example, [@Enqvist:2012im] found that decay into fermions after inflation could produce $\Omega_{\rm gw} \sim 10^{-12}$ to $10^{-18}$, $f \sim 10^{9}$ to $10^{10}$ Hz today, depending on the parameters. These tend to fall outside the range of current, planned or proposed gravitational wave experiments such as Advanced LIGO and VIRGO, KAGRA, Einstein Telescope, eLISA, DECIGO or BBO (for an exception, see [@Dufaux:2010cf]). Roughly speaking, these are most sensitive to frequencies around $10^{-3}$ to $10^3$ Hz and signal strength corresponding to $h^2\Omega_{\rm gw} \sim 10^{-5}$ to possibly $10^{-14}$. (See [@Moore:2014lga; @Thrane:2013oya] or the review [@Riles:2012yw].[^1]) LIGO and VIRGO have jointly placed upper bounds on a stochastic gravitational wave background on the order of $\Omega_{\rm gw} \sim 5 \times 10^{-6}$ around $10^2$ Hz [@Aasi:2014zwg]. Gravitational wave detection at MHz frequencies has also been considered [@Akutsu:2008qv; @PhysRevD.77.022002; @Goryachev:2014yra]. It has not been a major focus, though, since comparatively reliable astrophysical sources (e.g. neutron star mergers) are not expected in this frequency range. This motivates the study of how robust are the predictions for the gravitational wave spectrum from preheating. We would expect that a realistic preheating process in the early universe would include couplings of the decay product to other fields, as well as possible self-interactions. It will be useful to know whether these can significantly affect the observability of such a process.[^2] Specifically, it would be interesting to answer the question “Given a model of preheating with some self-interaction strength, how does one estimate the overall gravitational wave production?" This is analogous to the discussion in [@Giblin:2014dea], which estimates the maximum energy density in gravitational waves that could be produced by a cosmological process such as preheating. Previous work has shown that for self-couplings $\lambda_{\chi} \sim \mathcal{O}(10^{-2}) \gg g^2$, where $g^2$ is the coupling between the inflaton and scalar, parametric resonance can be significantly affected [@Prokopec:1996rr; @Allahverdi:1996xc]. However, there has been little discussion of gravitational wave production in this scenario.[^3] Therefore it is difficult to give a thorough answer to the above question based on the existing literature. This also means that it is unclear how general a gravitational wave prediction is when it ignores interactions of the decay products. In this work, we begin to address this by studying the development and termination of parametric resonance and the production of gravitational waves in the context of $\lambda\phi^4$ chaotic inflation coupled to a self-interacting light scalar field. We verify by lattice simulation that the resonance terminates early for self-coupling $\lambda_{\chi} \gtrsim g^2$, demonstrating the condition $\rho_{\chi}^{\rm final} \sim g^2/\lambda_{\chi}$ mentioned in footnote 19 of [@Prokopec:1996rr] (their $g$ is our $g^2$), and show that this leads to significant suppression of gravitational wave production. The resonance terminates early because the self-interaction term allows more efficient rescattering of particles out of the resonant mode, and this can be characterized by a condition comparing the energy density associated with the self-interaction to the inflaton-scalar interaction energy. The early termination of the resonance means that there is less energy in the light scalar’s fluctuations, which directly source gravitational waves. Therefore, gravitational wave production is reduced. For $\lambda_{\chi} \gtrsim \lambda_{\chi}^{\ast} = g^2$, the energy density goes as $\Omega_{\rm gw}^{(\lambda_{\chi})} \sim (g^2/\lambda_{\chi})^2 \, \Omega_{\rm gw}^{(\lambda_{\chi}=0)}$. In [Sec. \[sec:generality\]]{} we show that this result is robust to changes in initial conditions, and that the same scaling occurs in massive ($m^2\phi^2$) inflation. Although this suggests generality to inflationary models that are quadratic or quartic about the minimum, we point out that an important goal of future work is to understand the effect of realistic interactions on other models that have predicted gravitational wave spectra. As an application of this result, one could imagine the universe reheating by a coupling between the Higgs and inflaton, and we argue in [Sec. \[sec:discussion\]]{} that such a scenario would likely produce no observable gravitational radiation. This is due to the size of the Higgs self-coupling, despite its eventual running to zero in the Standard Model. However, we point out that even a resonance too brief to produce observable gravitational waves could be relevant for the issue of vacuum stability. Finally, if the inflaton preheats a scalar field with an extremely small self-coupling, then the gravitational wave spectrum could directly measure this potential. Model {#sec:model} ===== Representing the universe by a spatially flat Friedmann-Robertson-Walker metric, we will describe gravitational waves as transverse and traceless perturbations to this metric, specifically as $h_{ij}$ such that $$\begin{aligned} \label{eq:metric} ds^2 = a^2(\eta)\left( -d\eta^2 + (\delta_{ij} + h_{ij})dx^idx^j \right)\end{aligned}$$ with $\partial_ih_{ij} = 0$ and $h_{ii} = 0$. We will take the inflaton to be a real scalar field, $\phi(t,\vec{x})$, and consider it to be coupled to a massless real scalar field $\chi(t,\vec{x})$, with potential given by $$\begin{aligned} \label{eq:potential} V = \frac{1}{4}\lambda\phi^4 + \frac{1}{4}\lambda_{\chi}\chi^4 + \frac{1}{2}g^2\phi^2\chi^2\end{aligned}$$ Here we have chosen to study $\lambda\phi^4$ chaotic inflation, and this requires some justification since standard slow-roll inflation with this potential is inconsistent with Cosmic Microwave Background (CMB) observations [@Ade:2015oja]. Much literature on gravitational waves from preheating takes the potential as $\frac{1}{4}\lambda\phi^4$, in particular the thorough numerical study [@Dufaux:2007pt], whose model corresponds to ours with the choice $\lambda_{\chi}=0$. We expect the qualitative nature of our results to be relevant to a broad range of inflationary scenarios (this will be discussed further in [Sec. \[sec:discussion\]]{}), and it will be useful to refer to specific previous results in order to understand the production of gravitational waves. We are also studying the behavior of a “light" scalar field, and so we neglect a $\chi$ mass term in comparison with the effective $\chi$ mass that comes from the interaction term $\frac{1}{2}g^2\phi^2\chi^2$. Comparing these terms using the amplitude of the $\phi$ oscillations shows that this is roughly equivalent to requiring the $\chi$ mass to be $m_{\chi} \ll \sqrt{g^2/\lambda} \times 10^{12}$ GeV. Here the inflaton self-coupling is set by the amplitude of the scalar power spectrum of the CMB as $\lambda = 10^{-13}$. The unknown coupling $g^2$ must be small, but we will also take it to be larger than $\lambda$; in terms of the resonance parameter $q \equiv g^2/\lambda$ this means $1 \ll q \ll \lambda^{-1}$; here we will examine the range $10 \lesssim q \lesssim 2000$, which contains most of the region with the largest gravitational wave production [@Dufaux:2007pt]. We will see that this peaks around $q \approx 100-200$ and falls off slightly as $q$ gets larger or smaller (see [Fig. \[fig:q\_dependence\]]{}), although there are examples with smaller $q$ that do not exactly follow this trend [@Dufaux:2007pt]. We consider the light scalar’s self-interaction in the range $\lambda < \lambda_{\chi} < 1$. We study the dynamics in this model beginning at the end of inflation, $t_0 \equiv 0$, once the comoving horizon $(aH)^{-1}$ begins to expand, with the inflaton as a homogeneous field given everywhere by $\phi_0 = 0.342 \, M_{\rm Pl}$.[^4] The field $\chi$ is a light “spectator" field during inflation, and at the end of inflation each $\chi$ mode is in the de Sitter vacuum state. As shown in previous work [@Khlebnikov:1996mc; @Polarski:1995jg], as the inflaton decays the quantum state quickly approaches a semiclassical regime with large occupation numbers, and the evolution here is equivalent to the classical evolution of an initial classical distribution that gives $$\begin{aligned} \langle |\chi_k(0)|^2\rangle = 1/(2\lambda^{3/2}\phi_0^3\omega_k), \, \, \, \, \, \, \, \, \dot{\chi}_k(0) = \left(i\omega_k + H(0)\right)\chi_k(0)\end{aligned}$$ at the beginning of reheating.[^5] The dynamics considered here occurs on sub-horizon scales.[^6] Since $\phi$ is homogeneous, the equations of motion for these fields in a spatially flat Friedmann-Robertson-Walker (FRW) background are $$\begin{aligned} \ddot{\phi} + 3H\dot{\phi} + \lambda\phi^3 & = 0 \label{eq:phi_eom} \\ \square \chi + 3H\dot{\chi} + \lambda_{\chi}\chi^3 + g^2\phi^2\chi & = 0 \label{eq:chi_eom}\end{aligned}$$ where $H \equiv \dot{a}/a$ is the Hubble parameter, whose value is related to the total energy density $\rho$ by the Friedmann equation $$\begin{aligned} H^2 = \frac{8\pi G}{3} \rho.\end{aligned}$$ In order to study the behavior of $\phi$ and $\chi$ that follows from the above, we will express $\chi$ in terms of modes $\chi_k$[^7]: $$\begin{aligned} \chi(t,\vec{x}) = \frac{1}{(2\pi)^{3/2}}\int d^3k \left( a_k \chi_k(t) e^{-i\vec{k}\cdot\vec{x}} + a_k^{\dagger} \chi_k^{\ast}(t) e^{i\vec{k}\cdot\vec{x}} \right).\end{aligned}$$ The amplitude of $\phi$ is still very large at the end of inflation, $\lambda_{\chi}\chi^2 \ll g^2\phi^2$, and [Eq. (\[eq:chi\_eom\])]{} is approximately linear in $\chi$. We can then use the mode equation $$\begin{aligned} \label{eq:linear_chi_eom} \ddot{\chi}_k + 3 H \dot{\chi}_k + \left( \frac{k^2}{a^2} + g^2\phi^2 \right)\chi_k = 0\end{aligned}$$ to study the beginning of the reheating process. It will be convenient to introduce conformally rescaled fields $\overline{\phi} \equiv a\phi/\phi_0$, $\overline{\chi} \equiv a\chi/\phi_0$, and time $d\eta \equiv dt / a$ and define a dimensionless time parameter and wave number $$\begin{aligned} \label{eq:x_def} \tau \equiv \sqrt{\lambda}\phi_0 \eta, \, \, \, \, \, \kappa \equiv k / (\sqrt{\lambda}\phi_0).\end{aligned}$$ Following e.g. [@Greene:1997fu; @Dufaux:2007pt], we study the field spectrum in terms of a comoving number density for the field $\chi$, $$\begin{aligned} n_{\kappa} = \frac{1}{2}\left( \omega_{\kappa} |\overline{\chi}_{\kappa}|^2 + \frac{1}{\omega_{\kappa}}|\overline{\chi}_{\kappa}^{ \, \, '}|^2 \right),\end{aligned}$$ and comoving energy density $\rho_{\kappa} = \omega_{\kappa} n_{\kappa}$, where $\omega_{\kappa} = \sqrt{\kappa^2 + m_{\rm eff}^2} = \sqrt{\kappa^2 + q\overline{\phi}^{\, 2} + 3(\lambda_{\chi}/\lambda)\overline{\chi}^{\, 2}}$. Preheating in This Model {#sec:preheating} ======================== We begin by briefly outlining some results from previous studies of preheating, beginning with the case $\lambda_{\chi}=0$ (see [@Greene:1997fu] and references therein). We then use these to develop an approximate relation that quantifies the end of preheating and that will be useful in the gravitational wave calculation. After the end of inflation, $\phi$ oscillates in its potential with period $T \approx 7.416$ (in terms of the dimensionless time parameter $\tau$) [@Greene:1997fu] while the modes $\chi_{\kappa}$ can be excited by the phenomenon of parametric resonance. This process is typically described in terms of the resonance parameter $q = g^2/\lambda$. In general, certain modes $\kappa$ will be excited as $\chi_{\kappa} \propto \exp(\mu_{\kappa} \tau)$. The exponential growth factor $\mu_{\kappa}$ will vary with $\kappa$, giving rise to resonance “bands" characterized by some central $\kappa$ and width $\Delta \kappa$. We will consider the case of “broad resonance" where $q \gg 1$ (as compared with “narrow resonance" when $q < 1$). In this case the spectrum of resonantly excited modes takes the form of a broad peak whose location and width are approximately characterized by $$\begin{aligned} \kappa_{\ast}, \, \Delta \kappa \, \, \, \sim \, \, \, q^{1/4}.\end{aligned}$$ For a particular value of $q$, the maximum growth exponent $\mu_{\rm max} \equiv {\rm max}\{ \mu_{\kappa} \}$ is [@Greene:1997fu] $$\begin{aligned} \mu_{\rm max} & = \frac{1}{\pi} \ln \left( \sqrt{1 + \exp\left[-\pi \kappa^2\sqrt{2/q}\right]} + \exp\left[-\pi \kappa^2/\sqrt{2q}\right] \right)\end{aligned}$$ and the resonance is efficient when $\kappa^2 \leq \sqrt{q/(2\pi^2)}$. Numerically we find that typical resonant momenta are $\kappa_{\ast}\sim 1$, so $\mu_{\rm max} \sim (3/2\pi)\exp(-\pi\sqrt{2/q})$ which is $\mathcal{O}(10^{-1})$ for the range of $q$ we consider. Number density $n_{\chi} \equiv \int d^3\kappa \, n_{\chi \, \kappa}$ increases in steps, twice per $\phi$ oscillation – every time the inflaton passes through $\phi=0$ and $\chi$’s effective mass-squared $m_{\chi}^2 = g^2\phi^2$ goes to zero, a burst of $\chi$ particles are created. The exponential amplification of some $\chi_{\kappa}$ derived from [Eq. (\[eq:linear\_chi\_eom\])]{} is a solution for small $\chi$ (approximately zero) and homogeneous $\phi$, when the mode equation for $\chi_{\kappa}$ is linear. As this process evolves, this will become a worse approximation and the problem will become fully nonlinear. Therefore, [Eq. (\[eq:linear\_chi\_eom\])]{} is only useful for understanding the beginning of the reheating process, and in general it is the coupled equations of motion [Eq. (\[eq:phi\_eom\])]{} and [Eq. (\[eq:chi\_eom\])]{} that must be solved. [0.5]{} ![\[fig:preheat\_120\] Evolution of preheating for $q = 120$. (a) Energy density $\rho_{\chi}$ as a function of time for $\lambda_{\chi}=0$. (b) Energy density $\rho_{\chi}$ as a function of time for $\lambda_{\chi}=10^{-8}$. (c) Energy density of $\phi$ and $\chi$, as well as energy density in the interaction term, for $\lambda_{\chi}=0$. (d) Same as (c), but for $\lambda_{\chi} = 10^{-8}$. The spatially averaged quantity $q\langle \frac{1}{4}\lambda_{\chi}\chi^4\rangle$ is also shown. (e) The spectrum in $\chi$ at several times of interest, for $\lambda_{\chi}=0$. The solid line corresponds to approximately the time when the exponential growth ends. (f) Same as (e), for $\lambda_{\chi}=10^{-8}$.](fig_energychi_q120_lx0.png "fig:"){width="\textwidth"} [0.5]{} ![\[fig:preheat\_120\] Evolution of preheating for $q = 120$. (a) Energy density $\rho_{\chi}$ as a function of time for $\lambda_{\chi}=0$. (b) Energy density $\rho_{\chi}$ as a function of time for $\lambda_{\chi}=10^{-8}$. (c) Energy density of $\phi$ and $\chi$, as well as energy density in the interaction term, for $\lambda_{\chi}=0$. (d) Same as (c), but for $\lambda_{\chi} = 10^{-8}$. The spatially averaged quantity $q\langle \frac{1}{4}\lambda_{\chi}\chi^4\rangle$ is also shown. (e) The spectrum in $\chi$ at several times of interest, for $\lambda_{\chi}=0$. The solid line corresponds to approximately the time when the exponential growth ends. (f) Same as (e), for $\lambda_{\chi}=10^{-8}$.](fig_energychi_q120_lx8.png "fig:"){width="\textwidth"} [0.5]{} ![\[fig:preheat\_120\] Evolution of preheating for $q = 120$. (a) Energy density $\rho_{\chi}$ as a function of time for $\lambda_{\chi}=0$. (b) Energy density $\rho_{\chi}$ as a function of time for $\lambda_{\chi}=10^{-8}$. (c) Energy density of $\phi$ and $\chi$, as well as energy density in the interaction term, for $\lambda_{\chi}=0$. (d) Same as (c), but for $\lambda_{\chi} = 10^{-8}$. The spatially averaged quantity $q\langle \frac{1}{4}\lambda_{\chi}\chi^4\rangle$ is also shown. (e) The spectrum in $\chi$ at several times of interest, for $\lambda_{\chi}=0$. The solid line corresponds to approximately the time when the exponential growth ends. (f) Same as (e), for $\lambda_{\chi}=10^{-8}$.](fig_energy_q120_lx0.png "fig:"){width="\textwidth"} [0.5]{} ![\[fig:preheat\_120\] Evolution of preheating for $q = 120$. (a) Energy density $\rho_{\chi}$ as a function of time for $\lambda_{\chi}=0$. (b) Energy density $\rho_{\chi}$ as a function of time for $\lambda_{\chi}=10^{-8}$. (c) Energy density of $\phi$ and $\chi$, as well as energy density in the interaction term, for $\lambda_{\chi}=0$. (d) Same as (c), but for $\lambda_{\chi} = 10^{-8}$. The spatially averaged quantity $q\langle \frac{1}{4}\lambda_{\chi}\chi^4\rangle$ is also shown. (e) The spectrum in $\chi$ at several times of interest, for $\lambda_{\chi}=0$. The solid line corresponds to approximately the time when the exponential growth ends. (f) Same as (e), for $\lambda_{\chi}=10^{-8}$.](fig_energy_q120_lx8.png "fig:"){width="\textwidth"} [0.5]{} ![\[fig:preheat\_120\] Evolution of preheating for $q = 120$. (a) Energy density $\rho_{\chi}$ as a function of time for $\lambda_{\chi}=0$. (b) Energy density $\rho_{\chi}$ as a function of time for $\lambda_{\chi}=10^{-8}$. (c) Energy density of $\phi$ and $\chi$, as well as energy density in the interaction term, for $\lambda_{\chi}=0$. (d) Same as (c), but for $\lambda_{\chi} = 10^{-8}$. The spatially averaged quantity $q\langle \frac{1}{4}\lambda_{\chi}\chi^4\rangle$ is also shown. (e) The spectrum in $\chi$ at several times of interest, for $\lambda_{\chi}=0$. The solid line corresponds to approximately the time when the exponential growth ends. (f) Same as (e), for $\lambda_{\chi}=10^{-8}$.](fig_energy_spectrum_q120_lx0.png "fig:"){width="\textwidth"} [0.5]{} ![\[fig:preheat\_120\] Evolution of preheating for $q = 120$. (a) Energy density $\rho_{\chi}$ as a function of time for $\lambda_{\chi}=0$. (b) Energy density $\rho_{\chi}$ as a function of time for $\lambda_{\chi}=10^{-8}$. (c) Energy density of $\phi$ and $\chi$, as well as energy density in the interaction term, for $\lambda_{\chi}=0$. (d) Same as (c), but for $\lambda_{\chi} = 10^{-8}$. The spatially averaged quantity $q\langle \frac{1}{4}\lambda_{\chi}\chi^4\rangle$ is also shown. (e) The spectrum in $\chi$ at several times of interest, for $\lambda_{\chi}=0$. The solid line corresponds to approximately the time when the exponential growth ends. (f) Same as (e), for $\lambda_{\chi}=10^{-8}$.](fig_energy_spectrum_q120_lx8.png "fig:"){width="\textwidth"} These can be studied by lattice simulation, and we have used the C++ code LATTICEEASY [@Felder:2000hq] in order to simulate the evolution of these interacting scalar fields in an expanding universe. [Fig. \[fig:preheat\_120\]]{} shows results for $q = 120$. This is a useful example since [@Dufaux:2007pt] presents detailed results for preheating and gravitational wave production for $q=120$ in the absence of a self-coupling. [Fig. \[fig:egychi\_120\_0\]]{} shows the spatially-averaged energy density $\rho_{\chi} \equiv \langle \frac{1}{2}\dot{\chi}^2 + \frac{1}{2a^2}(\partial_j\chi)^2 + \frac{1}{4}\lambda_{\chi}\chi^4 \rangle$ as a function of time. [Fig. \[fig:egydens\_120\_0\]]{} shows for $\lambda_{\chi}=0$ the sum of the spatially-averaged energy densities $\rho_{\phi} \equiv \langle \frac{1}{2}\dot{\phi}^2 + \frac{1}{2a^2}(\partial_j\phi)^2 + \frac{1}{4}\lambda\phi^4 \rangle$ and $\rho_{\chi}$, as well as the energy density only in the interaction term, $\rho_{\rm int} \equiv \langle \frac{1}{2}g^2\phi^2\chi^2 \rangle$. [Fig. \[fig:spectrum\_120\_0\]]{} shows the spectrum in $\chi$ for the same choice of parameters. The spectrum is shown at several times, and the solid line corresponds to approximately the time when the exponential growth ends. We can understand how preheating progresses by observing that the transfer of energy between $\chi$ and $\phi$, and among different modes $\chi_{\kappa}$ and $\phi_{\kappa}$, occurs in the following distinct stages. First, oscillations of the homogeneous $\phi$ excite modes of $\chi$ centered around some $\kappa = \kappa_{\ast}$, and the initially small inhomogeneities of $\chi$ become large. There is some backreaction onto $\phi$, whereby the $\frac{g^2}{2}\phi^2\chi^2$ interaction term broadly excites modes $\phi_{\kappa}$ up through $\approx 2 \kappa_{\ast}$, and inhomogeneities in $\phi$ begin to grow. The second stage occurs once $q^{1/2}\rho_{\rm int} \approx \rho_{\phi}+\rho_{\chi}$. This is a useful, approximate numerical result, that is essentially the same as Eq. 6 in [@Khlebnikov:1996zt]. Then $\chi_{\kappa_{\ast}}$ efficiently rescatters, i.e. interacts with other modes, and its exponential growth ends. The total energy in $\chi$ continues to grow a bit until $\rho_{\chi}\approx\rho_{\phi}$. This is evident in [Fig. \[fig:egychi\_120\_0\]]{}. Large field inhomogeneities break up and the spectrum broadens towards larger $k$. This broad spectrum where energy density becomes approximately evenly distributed among modes is evident in [Fig. \[fig:spectrum\_120\_0\]]{}. This figure indicates the spectrum at the time when the exponential growth ends with a solid curve. Spectra before this time are indicated by dashed curves, and spectra after this time are indicated by dotted curves. This stage is discussed and examples of field configurations are shown in [@Felder:2006cc]. Some work has also examined the final, so-called “turbulent thermalization" stage in detail [@Micha:2002ey; @Micha:2004bv]. We now consider the case of nonzero $\lambda_{\chi}$. This has been studied to some extent in [@Prokopec:1996rr; @Allahverdi:1996xc; @Frolov:2008hy], and here we find results consistent with theirs. [Fig. \[fig:egychi\_120\_8\]]{} shows $\rho_{\chi}$ as a function of time. The resonance ends earlier in comparison with the $\lambda_{\chi}=0$ situation of [Fig. \[fig:egychi\_120\_0\]]{}. [Fig. \[fig:egydens\_120\_8\]]{} shows $\rho_{\phi}+\rho_{\chi}$, $\rho_{\rm int}$ and $\langle \frac{1}{4}\lambda_{\chi}\chi^4 \rangle$ for $\lambda_{\chi} = 10^{-8}$. [Fig. \[fig:spectrum\_120\_8\]]{} shows the spectrum in $\chi$ at several times of interest, and the solid line again corresponds to approximately the time when the exponential growth ends. Here the end of this stage still corresponds to a large mixing between modes, but in this case it is the quartic self-interaction that is significant. In general, we find from numerical simulation that when $\lambda_{\chi}$ becomes significantly larger than $g^2$, the resonance terminates earlier than for the $\lambda_{\chi}=0$ case, i.e. for any $\lambda_{\chi} > \lambda_{\chi}^{\ast} \sim g^2$. In terms of energy transfer, when $q^{1/2}\left\langle \frac{1}{4}\lambda_{\chi}\chi^4 \right\rangle \approx \rho_{\rm int}$, the resonance ends. This is analogous to the condition we described for $\lambda_{\chi}=0$, and will be useful. Depending on the size of $\lambda_{\chi}$, this may occur before or after the relation $q^{1/2}\rho_{\rm int} \approx \rho_{\phi}+\rho_{\chi}$ becomes true. To summarize, the resonant stage of preheating ends by the following condition: $$\begin{aligned} \left( \rho_{\phi} + \rho_{\chi} \right) \approx q^{1/2} \rho_{\rm int} & \, \, \, \, \, \, \, \, \, \, \, {\rm for \, \, }\lambda_{\chi}<\lambda_{\chi}^{\ast}, \label{eq:end_1}\\ \rho_{\rm int} \approx q^{1/2}\left\langle \frac{1}{4}\lambda_{\chi}\chi^4 \right\rangle & \, \, \, \, \, \, \, \, \, \, \, {\rm for \, \, }\lambda_{\chi}>\lambda_{\chi}^{\ast}. \label{eq:end_2}\end{aligned}$$ The powers of $1/2$ are approximate – when comparing the size of the oscillating energy densities, as in [Fig. \[fig:egydens\_120\_0\]]{} and [Fig. \[fig:egydens\_120\_8\]]{} for example, there is some ambiguity in determining exactly what the value of the energy is when the resonance ends. We can estimate the value $\lambda_{\chi}^{\ast}$ where the condition [Eq. (\[eq:end\_2\])]{} becomes more important than [Eq. (\[eq:end\_1\])]{} in terms of an energy argument. For small enough $\lambda_{\chi}$, we will have $\langle \frac{1}{4}\lambda_{\chi}\chi^4\rangle \ll \rho_{\phi}+\rho_{\chi}$, so the self-interaction will not play a role in ending the resonance. This will no longer be true once $$\begin{aligned} q^{1/2}\left\langle \frac{1}{4}\lambda_{\chi}\chi^4\right\rangle \sim \rho_{\phi}+\rho_{\chi}.\end{aligned}$$ This can be related to the value of $\chi$ when the resonance ends by observing that, around this critical value $\lambda_{\chi}^{\ast}$ where behavior transitions from [Eq. (\[eq:end\_1\])]{} to [Eq. (\[eq:end\_2\])]{}, we will also have $$\begin{aligned} \rho_{\phi} + \rho_{\chi} \sim q^{1/2}\left\langle \frac{1}{2}g^2\phi_{\rm end}^2\chi_{\rm end}^2 \right\rangle\end{aligned}$$ so that $$\begin{aligned} \label{eq:end} \frac{1}{4}\lambda_{\chi}\langle\chi_{\rm end}^4\rangle \sim \frac{1}{2}g^2\langle\phi_{\rm end}^2\chi_{\rm end}^2\rangle\end{aligned}$$ For $( \langle \phi^2\chi^2 \rangle / \langle \chi^4 \rangle )_{\rm end} \sim \mathcal{O}(1)$ this means that $$\begin{aligned} \label{eq:lambda_star} \lambda_{\chi}^{\ast} \sim g^2.\end{aligned}$$ This agrees with numerical results showing that the maximum energy density begins to decrease dramatically with increasing $\lambda_{\chi}$ around this value. For example, $q=120$ will give $\lambda_{\chi}^{\ast} \sim 120\times 10^{-13} \sim 10^{-11}$. We check this by defining for each $\lambda_{\chi}$ the quantity $\rho_{\chi}^{\rm max}$ as the time average over several oscillations once $\rho_{\chi}$ has stopped increasing with time. [Fig. \[fig:compare\_gw\_120\]]{} shows that around $\lambda_{\chi}^{\ast} \approx 10^{-11}$, $\rho_{\chi}^{\rm max}$ begins to decrease as $\lambda_{\chi}^{-1}$. We now seek to quantify the effect that this has on gravitational wave production. Gravitational Wave Spectrum {#sec:gw_spectrum} =========================== The metric perturbation $h_{ij}$ defined in [Eq. (\[eq:metric\])]{} can be rescaled as $\overline{h}_{ij} \equiv a h_{ij}$. Neglecting a term that goes as $a''/a \sim (aH)^2$ [@Dufaux:2007pt], the equation of motion is $$\begin{aligned} \label{eq:h_eom} \overline{h}_{ij}'' - \nabla^2\overline{h}_{ij} & = 16\pi G a^3 \Pi_{ij}^{\rm TT}\end{aligned}$$ where $G$ is Newton’s constant and $\Pi_{ij}^{\rm TT}$ is the transverse traceless projection of the anisotropic stress: $$\begin{aligned} \label{eq:stress} \Pi_{ij} = a^{-2}\left( T_{ij} -\langle p \rangle g_{ij} \right).\end{aligned}$$ The second term in [Eq. (\[eq:stress\])]{} will be neglected since $g_{ij}$ is the sum of a homogeneous, isotropic part whose transverse traceless projection is zero, and a perturbation that is higher order in $G$. The Fourier Transform of [Eq. (\[eq:h\_eom\])]{} is $$\begin{aligned} \overline{h}_{ij}''(\vec{k}) + k^2\overline{h}_{ij}(\vec{k}) & = 16\pi G a^3 \Pi_{ij}^{\rm TT}(\vec{k})\end{aligned}$$ We consider $\Pi_{ij}^{\rm TT}$ to be a source acting continuously during the time interval $\eta_0 < \eta < \eta_f$, solve [Eq. (\[eq:h\_eom\])]{} using Green’s functions, and use this solution to find the energy density of the tensor perturbation. As shown in [@Dufaux:2007pt], the result of this procedure is $$\begin{aligned} \label{eq:gw_spec} \frac{d \rho_{\rm gw}}{d \ln k}(\eta > \eta_f) & = \frac{S_k}{a^4(\eta)}\end{aligned}$$ where $S_k$ is defined by $$\begin{aligned} \label{eq:sk_def} S_k & = \frac{4\pi G k^3}{V} \int d\Omega \sum_{i,j}\left( \left|\int_{\eta_i}^{\eta_f}d\eta'\cos(k\eta')a(\eta')T_{ij}^{\rm TT}(\eta',\vec{k})\right|^2 + \left|\int_{\eta_i}^{\eta_f}d\eta'\sin(k\eta')a(\eta')T_{ij}^{\rm TT}(\eta',\vec{k})\right|^2 \right)\end{aligned}$$ where $V$ is the volume of the box considered and $\int d\Omega$ is an integral over directions in $k$ space.[^8] $S_k$ only depends on the dynamics occurring during gravitational wave generation, and the TT part of the energy-momentum tensor is defined in terms of projection operators by $$\begin{aligned} T_{ij}^{\rm TT}(\eta,\vec{k}) & = \left( P_{il}(\hat{k})P_{jm}(\hat{k}) - \frac{1}{2}P_{ij}(\hat{k})P_{lm}(\hat{k}) \right) T_{lm}(\eta,\vec{k}) \\ P_{ij}(\hat{k}) & = \delta_{ij} - \hat{k}_i\hat{k}_j\end{aligned}$$ We obtain the spectrum of gravitational waves numerically using the LATTICEEASY code mentioned above, modified to in order to compute [Eq. (\[eq:gw\_spec\])]{} as described above. We will give results in terms of $\Omega_{\rm gw} = \rho_{\rm gw} / \rho_{\rm total}$, at the “time of production" defined as approximately the time when energy in gravitational waves stops increasing noticeably. This is very well approximated by the value at the end of the simulation at $\tau=250$, and denote with a subscript “p" quantities evaluated at this time. The relation between the results we give and their present values depends somewhat on the equation of state throughout reheating, but previous works have established that in $\lambda\phi^4$ preheating, the equation of state very rapidly becomes that of radiation, so that the energy density in gravitational waves will be [@Dufaux:2007pt] $$\begin{aligned} \label{eq:gw_amp_today} h^2\Omega_{\rm gw} & = \left(\frac{S_k}{a^4\rho}\right)_{p}\left(\frac{g_{0}}{g_{\ast}}\right)^{1/3}h^2\Omega_{\rm rad} \nonumber \\ & = \left( 9.3 \times 10^{-6} \right)\left(\Omega_{\rm gw}\right)_{p}\end{aligned}$$ where $h^2\Omega_{\rm rad} = 4.3\times 10^{-5}$, $g_{\ast}/g_0 \approx 100$. Similarly, frequencies today are related to comoving wave numbers at the time of preheating by $$\begin{aligned} \label{eq:gw_freq_today} f = \left(\frac{k}{a\rho^{1/4}}\right)_{p}4\times 10^{10} \, {\rm Hz} \, \sim \, \kappa \times 10^{7} \, {\rm Hz}\end{aligned}$$ where in the last step we have taken $(a^4\rho)_p \sim \lambda\phi_0^4$ (see e.g. [Fig. \[fig:egychi\_120\_0\]]{}; we begin with $\rho_{\chi} \approx 0$ and $\rho_{\phi} \approx 2\times \frac{1}{4}\lambda\phi_0^4$ and throughout the simulation the quantity $a^4(\rho_{\phi} + \rho_{\chi}) \approx$ constant). [0.5]{} ![\[fig:gw\_spectra\] Peak of gravitational wave energy density spectrum, defined in [Eq. (\[eq:gw\_spec\])]{}, as a fraction of total energy density at end of preheating stage. (a) Spectrum for $q=120$ and two choices of self-coupling $\lambda_{\chi}$. (b) Amplitude of peak of GW spectrum, and final average value for $\rho_{\chi}$ after preheating ends, for $q=120$ and as a function of $\lambda_{\chi}$. These quantities are presented as fractions of their value in the $\lambda_{\chi}=0$ case. For comparison, dashed curves are also shown for the scaling behavior [Eq. (\[eq:rho\_scaling\])]{} and [Eq. (\[eq:gw\_scaling\])]{}. (c) Amplitude of peak of GW spectrum, $\Omega_{\rm gw}^{\ast}$, as a function of $\lambda_{\chi}$ for $q=12$ and $q=1200$, compared with [Eq. (\[eq:gw\_scaling\])]{}. (d) Value of $\Omega_{\rm gw}$ as a function of the resonance parameter, $q$, for $\lambda_{\chi}=0$ and $\lambda_{\chi}=10^{-7}$, and the prediction [Eq. (\[eq:gw\_scaling\])]{} applied to the latter case.](fig_gwavespec_0and9.png "fig:"){width="\textwidth"} [0.5]{} ![\[fig:gw\_spectra\] Peak of gravitational wave energy density spectrum, defined in [Eq. (\[eq:gw\_spec\])]{}, as a fraction of total energy density at end of preheating stage. (a) Spectrum for $q=120$ and two choices of self-coupling $\lambda_{\chi}$. (b) Amplitude of peak of GW spectrum, and final average value for $\rho_{\chi}$ after preheating ends, for $q=120$ and as a function of $\lambda_{\chi}$. These quantities are presented as fractions of their value in the $\lambda_{\chi}=0$ case. For comparison, dashed curves are also shown for the scaling behavior [Eq. (\[eq:rho\_scaling\])]{} and [Eq. (\[eq:gw\_scaling\])]{}. (c) Amplitude of peak of GW spectrum, $\Omega_{\rm gw}^{\ast}$, as a function of $\lambda_{\chi}$ for $q=12$ and $q=1200$, compared with [Eq. (\[eq:gw\_scaling\])]{}. (d) Value of $\Omega_{\rm gw}$ as a function of the resonance parameter, $q$, for $\lambda_{\chi}=0$ and $\lambda_{\chi}=10^{-7}$, and the prediction [Eq. (\[eq:gw\_scaling\])]{} applied to the latter case.](fig_compare_gw_produced.png "fig:"){width="\textwidth"} [0.5]{} ![\[fig:gw\_spectra\] Peak of gravitational wave energy density spectrum, defined in [Eq. (\[eq:gw\_spec\])]{}, as a fraction of total energy density at end of preheating stage. (a) Spectrum for $q=120$ and two choices of self-coupling $\lambda_{\chi}$. (b) Amplitude of peak of GW spectrum, and final average value for $\rho_{\chi}$ after preheating ends, for $q=120$ and as a function of $\lambda_{\chi}$. These quantities are presented as fractions of their value in the $\lambda_{\chi}=0$ case. For comparison, dashed curves are also shown for the scaling behavior [Eq. (\[eq:rho\_scaling\])]{} and [Eq. (\[eq:gw\_scaling\])]{}. (c) Amplitude of peak of GW spectrum, $\Omega_{\rm gw}^{\ast}$, as a function of $\lambda_{\chi}$ for $q=12$ and $q=1200$, compared with [Eq. (\[eq:gw\_scaling\])]{}. (d) Value of $\Omega_{\rm gw}$ as a function of the resonance parameter, $q$, for $\lambda_{\chi}=0$ and $\lambda_{\chi}=10^{-7}$, and the prediction [Eq. (\[eq:gw\_scaling\])]{} applied to the latter case.](fig_compare_gw_12and1200.png "fig:"){width="\textwidth"} [0.5]{} ![\[fig:gw\_spectra\] Peak of gravitational wave energy density spectrum, defined in [Eq. (\[eq:gw\_spec\])]{}, as a fraction of total energy density at end of preheating stage. (a) Spectrum for $q=120$ and two choices of self-coupling $\lambda_{\chi}$. (b) Amplitude of peak of GW spectrum, and final average value for $\rho_{\chi}$ after preheating ends, for $q=120$ and as a function of $\lambda_{\chi}$. These quantities are presented as fractions of their value in the $\lambda_{\chi}=0$ case. For comparison, dashed curves are also shown for the scaling behavior [Eq. (\[eq:rho\_scaling\])]{} and [Eq. (\[eq:gw\_scaling\])]{}. (c) Amplitude of peak of GW spectrum, $\Omega_{\rm gw}^{\ast}$, as a function of $\lambda_{\chi}$ for $q=12$ and $q=1200$, compared with [Eq. (\[eq:gw\_scaling\])]{}. (d) Value of $\Omega_{\rm gw}$ as a function of the resonance parameter, $q$, for $\lambda_{\chi}=0$ and $\lambda_{\chi}=10^{-7}$, and the prediction [Eq. (\[eq:gw\_scaling\])]{} applied to the latter case.](fig_gwmax_vs_q.png "fig:"){width="\textwidth"} [Fig. \[fig:gwavespec\_0and9\_120\]]{} shows the spectrum obtained in the case $q=120$, for $\lambda_{\chi} = 0$ and $\lambda_{\chi} = 10^{-9}$. The decrease in the energy produced in gravitational waves is evident from this, and [Fig. \[fig:compare\_gw\_120\]]{} shows how this depends on $\lambda_{\chi}$, as a fraction of the peak energy density when $\lambda_{\chi} = 0$. The solid lines in [Fig. \[fig:compare\_gw\_12and1200\]]{} show $\Omega_{\rm gw}$ for the cases $q=12$ and $q=1200$. Evidently the effect of $\lambda_{\chi}$ is to end the resonance early and suppress gravitational wave production. Once preheating ends, the additional contribution of inhomogeneities to the gravitational wave spectrum is negligible [@Dufaux:2007pt]. To estimate how this effect depends on the model parameters $q$ and $\lambda_{\chi}$, we note that $\Omega_{\rm gw} \sim (T_{ij}^{\rm TT})^2 \sim (\partial_i \chi)^4$. The energy density $\Omega_{\rm gw}$ is dominated by the most recently produced part of the spectrum before the resonance ends (this is particularly clear in Fig. 8 of [@Dufaux:2007pt]), so for the purposes of this estimate we will ask how the maximum amplitude of $\chi$ depends on $q$ and $\lambda_{\chi}$. We have seen that $\chi$ grows until the condition [Eq. (\[eq:end\_2\])]{}, $\frac{1}{2}q\lambda\langle\phi^2\chi^2\rangle \sim \frac{1}{4}\lambda_{\chi}\langle\chi^4\rangle$, is satisfied. (Also, comparison of [Fig. \[fig:egychi\_120\_0\]]{} with [Fig. \[fig:egychi\_120\_8\]]{} shows this since $\rho_{\chi} \sim (\partial_{i}\chi)^2$.) This suggests a parametric scaling $$\begin{aligned} \label{eq:rho_scaling} \chi_{\rm end}^2 \propto q \lambda / \lambda_{\chi} = g^2 / \lambda_{\chi}\end{aligned}$$ Then the expectation that $\Omega_{\rm gw} \sim (\chi_{\rm end}^2)^2$ becomes $$\begin{aligned} \label{eq:gw_scaling} \Omega_{\rm gw} \propto \left(g^2 / \lambda_{\chi}\right)^2.\end{aligned}$$ Our numerical results confirm this relation as shown in [Fig. \[fig:compare\_gw\_120\]]{} and [Fig. \[fig:compare\_gw\_12and1200\]]{}. For $\lambda_{\chi} < \lambda_{\chi}^{\ast}$, the peak energy in gravitational waves decreases only very slightly with increasing $\lambda_{\chi}$, as the self-interaction term plays a small role in mixing modes and damping inhomogeneities. Once $\lambda_{\chi} > \lambda_{\chi}^{\ast}$, the energy density in gravitational waves scales in the manner given by [Eq. (\[eq:gw\_scaling\])]{}. For $\lambda_{\chi} \sim 10^{-2}$, we see that $\rho_{\chi}$ and $\Omega_{\rm gw}$ no longer decrease significantly with increasing $\lambda_{\chi}$. This is simply because the unstable resonance never begins, and the quartic self-interaction can no longer dramatically decrease $\Omega_{\rm gw}$ by ending the resonance earlier. [Fig. \[fig:q\_dependence\]]{} shows how the value of $\Omega_{\rm gw}$ at the time of production depends on the resonance parameter, $q$, for both $\lambda_{\chi}=0$ and $\lambda_{\chi}=10^{-7}$. In the latter case, we also show the prediction of the scaling relation [Eq. (\[eq:gw\_scaling\])]{}. Generality {#sec:generality} ========== So far, we have examined results in the context of $\lambda\phi^4$ chaotic inflation, with the self-coupling $\lambda$ and the initial condition of the inflaton field identical to a previous work that thoroughly investigated the dynamics of gravitational wave production during preheating [@Dufaux:2007pt]. This allows the results of the previous sections to be directly compared with that work. However, observational data indicates that the $\lambda\phi^4$ chaotic potential is not favored [@Ade:2015oja], so an important question is the generality of the results we have quoted above. In this section we will address this question in two ways, before pointing out interesting directions for future work. We will consider massive ($m^2\phi^2$) inflation, another standard example in which preheating is studied, and we will also consider a range of initial conditions for $\phi$ within both the $\lambda\phi^4$ and $m^2\phi^2$ cases. Specifically, this means that we will begin the numerical situation – corresponding to the end of inflation, with the inflaton’s energy about evenly split between kinetic and potential – with the inflaton field at various lower points on its potential than in the original case. Here, we are not primarily concerned with representing a complete model of inflation, but rather are studying how preheating and gravitational wave production proceed within a potential that is quadratic or quartic about the minimum, without regard to the model’s behavior at higher (inflationary) field values. In this spirit, we also study the $m^2\phi^2$ case with a few choices of $m_{\phi}$. It is worth pointing out that not all inflationary models end with oscillations of the field responsible for inflation about its zero; see for example the Abelian Higgs and Higgs-dilaton models [@Dufaux:2010cf; @GarciaBellido:2011de]. ![\[fig:gwavespec\_phi4\_newic\] Peak of gravitational wave energy density spectrum, defined in [Eq. (\[eq:gw\_spec\])]{}, as a fraction of total energy density at end of preheating stage. Spectrum for $q=120$ and two choices of self-coupling $\lambda_{\chi}$. The curves labeled $\phi(0) = \phi_0$ are identical to those shown in [Fig. \[fig:gwavespec\_0and9\_120\]]{}, corresponding to the original initial condition for the inflaton field. The curves labeled $\phi(0) = \phi_0/10$ correspond to starting the inflaton a factor of 10 lower on the potential, as described in the text. The magnitude of the gravitational wave spectrum is changed, but the effect of turning on $\lambda_{\chi}$ is the same.](fig_gwavespec_phi4_newic.png){width="50.00000%"} For every situation we have tried, the same approximate scaling behavior of reduced gravitational wave production with increased self-interaction $\lambda_{\chi}$ holds. In particular, we display some typical results in [Fig. \[fig:gwavespec\_phi4\_newic\]]{} and [Fig. \[fig:gwcomp\_gen\]]{}. For the case of $\lambda\phi^4$ inflation, with $q = 120$ and $\phi(0) = \phi_0 \equiv 0.342 \, M_{\rm Pl}$, we plot the gravitational wave spectrum in [Fig. \[fig:gwavespec\_phi4\_newic\]]{} and the scaling behavior with $\lambda_{\chi}$ in [Fig. \[fig:compare\_gw\_phi4\_newic\]]{}. These results were presented in [Sec. \[sec:gw\_spectrum\]]{}, and they are provided again for direct comparison with alternative scenarios. We label this choice of parameters as $\phi^4-{\rm I}$. We also show results for $q = 120$ and $\phi(0) = \phi_0 / 10$, referred to as $\phi^4-{\rm II}$, as well as $q = 120$ and $\phi(0) = \phi_0 / 100$, referred to as $\phi^4-{\rm III}$. [0.5]{} ![\[fig:gwcomp\_gen\] Peak of gravitational wave energy density spectrum, defined in [Eq. (\[eq:gw\_spec\])]{}, as a fraction of total energy density at end of preheating stage and normalized to the value when $\lambda_{\chi}=0$. Here we show several typical examples where the initial condition and/or parameters of the model are varied, as described in the text. As in [Sec. \[sec:gw\_spectrum\]]{}, there is some value $\lambda_{\chi}^{\ast}$ above which the peak of the gravitational wave spectrum decreases as $\lambda_{\chi}^{-2}$. Once $\lambda_{\chi}$ is large enough, the preheating resonance never starts and there is no further suppression with increasing $\lambda_{\chi}$, an effect also seen in [Sec. \[sec:gw\_spectrum\]]{}. (a) Varying initial conditions for $\lambda\phi^4$ inflaton potential. (b) Varying initial conditions and mass parameter for $m^2\phi^2$ inflaton potential. For ease of comparison, this result is given as a function of $\lambda_{\chi} / \lambda_{\ast}$, where $\lambda_{\ast}^{\rm I} = 10^{-6}$, $\lambda_{\ast}^{\rm II} = 10^{-9}$, $\lambda_{\ast}^{\rm III} = 10^{-7}$.](fig_compare_gw_phi4_newic.png "fig:"){width="\textwidth"} [0.5]{} ![\[fig:gwcomp\_gen\] Peak of gravitational wave energy density spectrum, defined in [Eq. (\[eq:gw\_spec\])]{}, as a fraction of total energy density at end of preheating stage and normalized to the value when $\lambda_{\chi}=0$. Here we show several typical examples where the initial condition and/or parameters of the model are varied, as described in the text. As in [Sec. \[sec:gw\_spectrum\]]{}, there is some value $\lambda_{\chi}^{\ast}$ above which the peak of the gravitational wave spectrum decreases as $\lambda_{\chi}^{-2}$. Once $\lambda_{\chi}$ is large enough, the preheating resonance never starts and there is no further suppression with increasing $\lambda_{\chi}$, an effect also seen in [Sec. \[sec:gw\_spectrum\]]{}. (a) Varying initial conditions for $\lambda\phi^4$ inflaton potential. (b) Varying initial conditions and mass parameter for $m^2\phi^2$ inflaton potential. For ease of comparison, this result is given as a function of $\lambda_{\chi} / \lambda_{\ast}$, where $\lambda_{\ast}^{\rm I} = 10^{-6}$, $\lambda_{\ast}^{\rm II} = 10^{-9}$, $\lambda_{\ast}^{\rm III} = 10^{-7}$.](fig_compare_gw_m2phi2_newic.png "fig:"){width="\textwidth"} [Fig. \[fig:gwavespec\_phi4\_newic\]]{} compares the gravitational wave spectra of the $\phi^4-{\rm I}$ and $\phi^4-{\rm II}$ parameter choices, for both $\lambda_{\chi} = 0$ and $\lambda_{\chi} = 10^{-9}$. In both cases, evidently, there is a significant reduction in gravitational wave production that accompanies an increase in $\lambda_{\chi}$, despite the difference in overall amplitude of the spectrum. In [Fig. \[fig:compare\_gw\_phi4\_newic\]]{}, we show how this reduction depends on $\lambda_{\chi}$ for each of the parameter choices $\phi^4-{\rm I}$, $\phi^4-{\rm II}$, $\phi^4-{\rm III}$. We find the same scaling behavior as before: there is a $\lambda_{\chi}^{\ast}$ above which gravitational wave production is suppressed by a factor of $(g^2 / \lambda_{\chi} )^2$. In the case of massive inflation, we replace [Eq. (\[eq:potential\])]{} with the potential $$\begin{aligned} \label{eq:m2p2_potential} V = \frac{1}{2}m_{\phi}^2\phi^2 + \frac{1}{4}\lambda_{\chi}\chi^4 + \frac{1}{2}g^2\phi^2\chi^2\end{aligned}$$ i.e. the light field $\chi$ has the same potential and interactions with the inflaton as it did previously, but the inflaton potential is quadratic rather than quartic. In this case we find that, as above, there is some $\lambda_{\chi}^{\ast}$ such that for $\lambda_{\chi} > \lambda_{\chi}^{\ast}$, gravitational wave production tends to be suppressed by $\lambda_{\chi}^{-2}$. We again plot three typical examples. We refer to $q \equiv g^2\phi(0)^2 / 4m_{\phi}^2 = 60$, $\phi(0) = 0.1 \, M_{\rm Pl}$, $m_{\phi} = 10^{-6}M_{\rm Pl}$ as $\phi^2-{\rm I}$. We refer to $q = 15$, $\phi(0) = 0.01$, $m_{\phi} = 10^{-9}$ as $\phi^2-{\rm II}$. We refer to $q = 15$, $\phi(0) = 0.001$, $m_{\phi} = 10^{-9}$ as $\phi^2-{\rm III}$. [Fig. \[fig:compare\_gw\_m2phi2\_newic\]]{} shows how the gravitational wave spectra in these cases scale with $\lambda_{\chi}$. For ease of comparison with the scaling relation $\lambda_{\chi}^{-2}$ we plot the results as a function of $\lambda_{\chi}/\lambda_{\chi}^{\ast}$, where $\lambda_{\chi}^{\ast} = 10^{-6}, \, 10^{-9}, \, 10^{-7}$ for $\phi^2-{\rm I}$, $\phi^2-{\rm II}$, $\phi^2-{\rm III}$ respectively. As before, $\lambda_{\chi}^{-2}$ fits well (until $\lambda_{\chi}$ becomes large enough that preheating no longer starts, so that increasing $\lambda_{\chi}$ won’t further decrease the gravitational wave production). The numerical computations involved make it impractical to check here every imaginable situation of interest to verify this relation. We have shown that gravitational wave production from preheating in potentials with minimum at zero can be extremely sensitive to the value of the light field’s self-coupling term, and that result is not exclusive to one particular model or choice of parameters. Therefore, an important goal of future work will be to fully characterize this effect in other realistic models, and better understand the implications for observability. Discussion and Conclusions {#sec:discussion} ========================== In this work we have studied the effect of a nonzero self-interaction on gravitational wave production during preheating of a scalar field. Previous work has considered the dynamics of preheating for a light, self-interacting scalar, as well as gravitational wave production by preheating of a non-self-interacting scalar. This work is an extension of these results, and in particular shows that the spectrum of gravitational waves that survive until today is very sensitive to the light scalar’s self-interaction. Our main result within the $\lambda\phi^4$ model is that for self-coupling $\lambda_{\chi} \gtrsim g^2$, the preheating resonance is terminated early, and the gravitational wave spectrum is significantly reduced: $$\begin{aligned} \label{eq:result1} \Omega_{\rm gw} \approx \left(\frac{g^2}{\lambda_{\chi}}\right)^2 \Omega_{\rm gw}^{(\lambda_{\chi}=0)} \, \, \, \, \, \, \, \, {\rm for} \, \, \, \, \, \, \, \, \lambda_{\chi} \gtrsim g^2.\end{aligned}$$ We have also begun to address the question of generality of this result, as discussed in [Sec. \[sec:generality\]]{}. For various choices of the inflaton’s initial condition in the $\lambda\phi^4$ model, we have seen that [Eq. (\[eq:result1\])]{} holds. Additionally, for an $m^2\phi^2$ inflationary potential, the result that the gravitational waves are suppressed as $\lambda_{\chi}^{-2}$ is shown, for several parameter choices. While this suggests generality to inflation models with potentials quadratic or quartic about a minimum at zero, an important question for future work is to study the effect of the light field’s interactions in other preheating models that have been shown to predict gravitational waves. As our work shows, predictions that neglect such interactions - even if they are extremely small - may not necessarily be accurate. It is easy to imagine that in a realistic preheating scenario, decay products will have their own self-interactions or further interactions with other fields, that will end the resonance early. Recently, another paper studied the effect of interactions of $\chi$ with further light degrees of freedom, as well as self-interactions in the context of a curvaton decaying to Higgs [@Lerner:2015uca]. Although the model is not identical to ours, it also found that self-interactions can be important in terminating the resonance early. Furthermore, they found that interaction with the additional light scalars, as characterized by the contribution to a thermal term, has the ability to significantly affect the resonance and either end it early or prevent it from occurring at all. They did not consider gravitational wave production, but following the argument given here in [Sec. \[sec:gw\_spectrum\]]{} it is reasonable to expect that this early termination of the resonance can further reduce any production of gravitational waves. Analyses of other scenarios have shown that preheating can be sensitive to nonlinear interaction terms of decay products [@Enqvist:2014tta], or other nonperturbative effects motivated by new physics above the TeV scale [@Kusenko:2008zm; @Kusenko:2009cv; @Chiba:2009zu; @Zhou:2013tsa; @Zhou:2015yfa]. Another interesting goal for future work would be to incorporate the effects of interactions such as those studied in this paper into a more general framework for obtaining order-of-magnitude estimates of gravitational wave production, as in [@Giblin:2014dea]. Although current constraints on MHz gravitational wave backgrounds are not sensitive to these processes [@Akutsu:2008qv], this could be very useful in evaluating the potential for observability in future experiments. One interesting possibility is that reheating occurred through an inflaton-to-Higgs coupling, since the Higgs is a natural candidate to couple to beyond-Standard Model fields [@Patt:2006fw; @Bhattacharya:2014gva; @Kamada:2014ufa; @Gross:2015bea]. The running of the Higgs self-coupling is sensitive to any new physics that comes in at high energies, but it has not been directly measured and will be difficult to measure at the LHC. One might hope that since $\lambda_{\rm H}$ runs from 0.13 at the weak scale to zero around $10^{10}$ GeV in the Standard Model [@Chetyrkin:2012rz], the condition $\lambda_{\rm H} \ll 1$ could be satisfied. This would avoid enormous damping of the preheating resonance, and thereby provide a possible cosmological probe of $\lambda_{\rm H}$. The self-coupling remains $\mathcal{O}\left( 10^{-1} \right)$ up to $\sim 10^8$ GeV, though, which suggests that there will not be significant (or any) preheating resonance. However, above this scale the self-coupling decreases and the effective potential reaches a maximum (in the Standard Model – small changes in input parameters or new physics beyond the Standard Model can significantly affect this; see e.g. [@Lebedev:2012zw; @Branchina:2013jra; @Branchina:2014usa; @Branchina:2014rva]). The condition [Eq. (\[eq:end\_2\])]{} suggests that a more relevant condition than the self-coupling may be the magnitude of the Higgs potential. The configuration of $\chi$ at the end of inflation (initial configuration for this problem) is certainly sensitive to the potential at large field values, as it corresponds to approximately $\chi_{\rm rms} \sim 10^{12}$ GeV $\sim H_{\rm inf}$.[^9] If one takes [Eq. (\[eq:end\_2\])]{} to apply as the condition for whether parametric resonance does or does not occur, then the result could be a resonance pushing Higgs oscillations toward the vacuum instability region.[^10] New physics that prevents $\lambda_{\rm H}$ from becoming negative would likely be more than sufficient to prevent a resonance from occuring. These rough estimates also ignore the possibilities of a different running of $\lambda_{\rm H}$ from the new inflaton coupling, as well as thermal effects. We leave the resolution of these questions to future work. I am grateful to Jessica Cook, Emanuela Dimastrogiovanni, Francis Duplessis, Damien Easson, Andrew Long and especially Tanmay Vachaspati for valuable discussions and comments on a draft of this paper. I also thank Michael Landry for bringing the references on MHz gravitational wave detection to my attention, and I thank the referee for useful suggestions that have improved this work. This work was supported by the U.S. Department of Energy at ASU. [^1]: Note that some results are given in terms of $h^2\Omega_{\rm gw}$, others in terms of $\Omega_{\rm gw}$ and still others in terms of strain $h$, which is distinct from today’s Hubble constant in units of 100 km/s/Mpc that appears in $h^2\Omega_{\rm gw}$. Consistent comparison of experimental sensitivities is discussed in [@Moore:2014lga]. [^2]: While this paper was in preparation, another work [@Lerner:2015uca] appeared that addresses some of these questions. We will discuss it in [Sec. \[sec:discussion\]]{}. [^3]: A study of gravitational waves in M-flation preheating [@Ashoorioon:2013oha] mentions that a self-interaction can suppress the resonance, but does not quantify this in a way that allows comparison with [@Prokopec:1996rr]. [^4]: This particular point along the inflaton’s phase space trajectory is identical to that of [@Dufaux:2007pt]. This choice is further addressed in [Sec. \[sec:generality\]]{}. [^5]: $\chi_k$ and $\omega_k$ are defined below. The specific implementation for initial field conditions of [@Khlebnikov:1996mc] is as described in the documentation for LATTICEEASY code, available at http://www.felderbooks.com/latticeeasy/. [^6]: For the typical example $q=120$, numerical results show that preheating begins at about $H=1.1 \times 10^{-9} \, M_{\rm Pl}$ and $a=5.5$ (for $a=1$ at the beginning of the simulation) and the mode $k_{\ast} \approx \sqrt{\lambda}\phi_0$ is excited. Then at formation the wavelength of these perturbations is a fraction $R_{\ast} / R_{\rm horizon} = (a \, k_{\ast}^{-1}) H \sim 10^{-2}$ of the horizon size. Since inflation has ended, the comoving horizon $(aH)^{-1}$ is increasing, so $aH$ is decreasing and the modes excited later will be an even smaller fraction of the horizon size. [^7]: Here we always use the Fourier Transform convention $f(\vec{x})=(2\pi)^{-3/2}\int d^3k \, f(\vec{k}) \, \exp(i\vec{k}\cdot\vec{x})$. [^8]: Our physical results are independent of box size, as we use a numerical Fourier Transform that takes this into account. This is described in the LATTICEEASY documentation. [^9]: The behavior of the Higgs after inflation, when there is no coupling to the inflaton, is discussed in [@Enqvist:2013kaa]. [^10]: There has been much work on the Higgs and vacuum stability, including discussion of the reheat temperature; see e.g. [@Espinosa:2007qp; @Kobakhidze:2013tn; @Degrassi:2012ry] and references therein.

--- abstract: 'In this paper we investigate metric properties of the groups $\Gamma_d(q)$ whose Cayley graphs are the Diestel-Leader graphs $DL_d(q)$ with respect to a given generating set $S_{d,q}$. These groups provide a geometric generalization of the family of lamplighter groups, whose Cayley graphs with respect to a certain generating set are the Diestel-Leader graphs $DL_2(q)$. Bartholdi, Neuhauser and Woess in [@BNW] show that for $d \geq 3$, $\Gamma_d(q)$ is of type $F_{d-1}$ but not $F_d$. We show below that these groups have dead end elements of arbitrary depth with respect to the generating set $S_{d,q}$, as well as infinitely many cone types and hence no regular language of geodesics. These results are proven using a combinatorial formula to compute the word length of group elements with respect to $S_{d,q}$ which is also proven in the paper and relies on the geometry of the Diestel-Leader graphs.' address: - 'Department of Mathematics, Trinity College, Hartford, CT 06106' - 'Department of Mathematics, Bowdoin College, Brunswick, ME 04011' author: - Melanie Stein - Jennifer Taback bibliography: - 'refs.bib' date: - - title: 'Metric properties of Diestel-Leader groups' --- Introduction ============ We investigate the metric properties of a family of groups whose Cayley graphs with respect to a carefully chosen generating set are the Diestel-Leader graphs ${DL_d(q)}$. These graphs are subsets of a product of $d$ infinite trees of valence $q+1$. We call these groups [*Diestel-Leader groups*]{} and denote them ${\Gamma_d(q)}$. More general Diestel-Leader graphs were introduced in [@DL] as a potential answer to the question “Is any connected, locally finite, vertex transitive graph quasi-isometric to the Cayley graph of a finitely generated group?" It was first shown in [@EFW] that $DL_2(m,n)$, the Diestel-Leader graph which is a subset of a product of two trees of valence $m+1$ and $n+1$ respectively, is not quasi-isometric to the Cayley graph of any finitely generated group when $m \neq n$. It is proven in [@BNW] that Diestel-Leader graphs which are subsets of the product of any number of trees of differing valence are not Cayley graphs of finitely generated groups. It is well known that the Cayley graph of the wreath product $L_n={\mathbb Z}_n \wr {\mathbb Z}$, often called the [*lamplighter group*]{}, with respect to the generating set $\{ t,ta, ta^2, \dots, ta^{n-1}\}$ (where $a$ is the generator of ${\mathbb Z}_n$ and $t$ generates ${\mathbb Z}$) is the Diestel-Leader graph $DL_2(n)$. This graph is a subset of the product of two trees of constant valence $n+1$. The groups we study below provide a geometric generalization of the family of lamplighter groups, as their Cayley graphs generalize the geometry of the lamplighter groups, that is, their Cayley graphs with respect to a natural generating set $S_{d,q}$ are the “larger" Diestel-Leader graphs $DL_d(q)$, which are subsets of the product of $d$ trees of constant valence $q+1$, and are defined explicitly in Section \[sec:DLgraphs\] below. Bartholdi, Neuhauser and Woess in [@BNW] present a construction of a group which we denote ${\Gamma_d(q)}$, a generating set $S_{d,q}$ and an identification with the graph $DL_d(q)$ which they prove to be the Cayley graph $\Gamma({\Gamma_d(q)},S_{d,q})$. Moreover, they provide a simple metric criterion for when their construction holds, namely either $d=2$, $d=3$ or if $d \geq 4$ and $q = p_1^{e_1}p_2^{e_2} \cdots p_r^{e_r}$ is the prime power decomposition of $q$, then $p_i > d-1$ for all $i$. They show that the groups ${\Gamma_d(q)}$ are type $F_{d-1}$ but not $F_d$ when $d \geq 3$, hence not automatic. We note that there are still open cases where it is not known whether $DL_d(q)$ is the Cayley graph of a finitely generated group; the smallest open case is $DL_4(2)$. Random walks in the Cayley graph $\Gamma({\Gamma_d(q)},S_{d,q})$ are studied in [@BNW], and a presentation for the group is given explicitly. For example, when $d=3$ Bartholdi, Neuhauser and Woess obtain the presentation $$\Gamma_3(m) \cong \langle a,s,t | a^m=1, \ [a,a^t] = 1, \ [s,t] = 1, \ a^s = aa^t \rangle.$$ When $m=p$ is prime, it is shown in [@CT] that $\Gamma_3(p)$ is a cocompact lattice in $Sol_5 \left({\mathbb F}_p((t)) \right)$, and that its Dehn function is quadratic. The Dehn function of $\Gamma_3(m)$ is studied for any $m$ in [@KR] where it is shown to be at most quartic. It was mentioned to the authors by Kevin Wortman that arguments analogous to those of Gromov in [@G] imply that the Dehn function of ${\Gamma_d(q)}$ is quadratic regardless of the values of $d \geq 3$ and $q$. When the relation $a^m=1$ is removed from the presentation above, one obtains Baumslag’s metabelian group $\Gamma$ which, in contrast to $\Gamma_3(m)$, has exponential Dehn function [@KR]. Baumslag defined this group to provide the first example of a finitely presented group with an abelian normal subgroup of infinite rank. It is noted in [@BNW] that ${\Gamma_d(q)}$ is in most cases an automata group, hence a self-similar group. Metric properties of self-similar groups are in general not well understood. In this paper we seek to answer many of the standard geometric group-theoretic questions related to metric properties of groups and their Cayley graphs for these Diestel-Leader groups ${\Gamma_d(q)}$. Such properties often rely on the ability to compute word length of elements within the group; we begin by proving that a particular combinatorial formula yields the word length of elements of ${\Gamma_d(q)}$ with respect to the generating set $S_{d,q}$. This formula relies on the symmetry present in the Diestel-Leader graph, and we subsequently use it to prove that ${\Gamma_d(q)}$ has dead end elements of arbitrary depth with respect to $S_{d,q}$. This generalizes a result of Cleary and Riley [@CR1; @CR2] which proves that $\Gamma_3(2)$ with respect to a generating set similar to $S_{3,2}$ has dead end elements of arbitrary depth, the first example of a finitely presented group with this property. The word length formula is used in later sections to show that ${\Gamma_d(q)}$ has infinitely many cone types, and hence no regular language of geodesics with respect to $S_{d,q}$. Definitions and Background on Diestel-Leader graphs {#sec:DLgraphs} =================================================== To define $DL_d(q)$, let $T$ be a homogeneous, locally finite, connected tree in which the degree of each vertex is $q+1$. This tree has an orientation such that each vertex $v$ has a unique predecessor $v^-$ and $q$ successors $w_1,w_2, \cdots ,w_q$ such that $w_i^-=v$ for $1 \leq i \leq q$. The transitive closure of the set of relationships of the form $v^- < v$ induces the partial order $\preccurlyeq$. In this partial order, any two vertices $v, w \in T$ have a greatest common ancestor $v\curlywedge w$. Choose a basepoint $o \in T$, and define a height function $h(v)= d(v, o \curlywedge v)-d(o, o\curlywedge v)$, where $d(x,y)$ denotes the number of edges on the unique path in $T$ from $x$ to $y$. With this definition, note that $h(v^-) = h(v) - 1$. Let $T_1$, $T_2, \cdots ,T_d$ denote $d$ copies the tree $T$, with basepoints $o_i$ and height functions $h_i$ for $1 \leq i \leq d$. The Diestel-Leader graph $DL_d(q)$ is the graph whose vertex set $V_d(q)$ is the set of $d$-tuples $(x_1, x_2, \dots , x_d)$ where $x_i$ is a vertex of $T_i$ for each $i$, and $h_1(x_1)+ \cdots +h_d(x_d)=0$. Two vertices $x=(x_1, \dots, x_d)$ and $y=(y_1, \dots, y_d)$ are connected by an edge if and only if there are two indices $i$ and $j$, with $i \neq j$, such that $x_i$ and $y_i$ are connected by an edge in $T_i$, $x_j$ and $y_j$ are connected by an edge in $T_j$, and $x_k=y_k$ for $k \neq i,j$. There is a projection $\Pi: V_d(q) \rightarrow ({\mathbb Z}^2)^d$ given by $$\Pi(x)=\Pi(x_1, x_2, \dots, x_d)={\left((m_1, l_1), (m_2, l_2), \dots, (m_d, l_d)\right)}$$ where $m_i=d(o_i,o_i\curlywedge x_i)$ and $l_i=d(x_i,o_i\curlywedge x_i)$. In particular, $0 \leq m_i$ and $0 \leq l_i$ for all $i$. Note that in $T_i$, the shortest path from $o_i$ to $x_i$ has length $m_i+l_i$, and recall that $h_i(x_i)=l_i-m_i$. The defining conditions of the Diestel-Leader graph ensure that $\sum_{i=1}^d l_i-m_i = 0$. In [@BNW] it is shown that these graphs are Cayley graphs of certain matrix groups, when a simple metric condition is satisfied. Specifically, let $\mathcal{L}_q$ be a commutative ring of order $q$ with multiplicative unit 1, and suppose $\mathcal{L}_q$ contains distinct elements $l_1, \dots, l_{d-1}$ such that if $d\geq 3$, their pairwise differences are invertible. Define a ring of polynomials in the formal variables $t$ and $(t+l_i)^{-1}$ for $1 \leq i \leq d-1$ with finitely many nonzero coefficients lying in ${\mathcal L}_q$: $${\mathcal R}_d({\mathcal L}_q) = {\mathcal L}_q[t,(t+l_1)^{-1},(t+l_2)^{-1}, \cdots ,(t+l_{d-1})^{-1}].$$ It is proven in [@BNW] that the group $\Gamma_d(\mathcal{L}_q)$ (which we denote by ${\Gamma_d(q)}$) of affine matrices of the form $$\left( \begin{array}{cc} (t+l_1)^{k_1} \cdots (t+l_{d-1})^{k_{d-1}} & P \\ 0 & 1 \end{array} \right), \text{ with } k_1,k_2, \cdots ,k_{d-1} \in {\mathbb Z}\text{ and }P \in {\mathcal R}_d({\mathcal L}_q)$$ has Cayley graph ${DL_d(q)}$ with respect to the generating set $S_{d,q}$ consisting of the matrices $$\left( \begin{array}{cc} t+l_i & b \\ 0 & 1 \end{array} \right)^{\pm 1}, \text{ with } b \in {\mathcal L}_q, \ i \in \{1,2, \cdots ,d-1\} \text{ and }$$ $$\left( \begin{array}{cc} (t+l_i)(t+l_j)^{-1} & -b(t+l_j)^{-1} \\ 0 & 1 \end{array} \right), \text{ with } b \in {\mathcal L}_q, \ i,j \in \{1,2, \cdots ,d-1\}, \ i \neq j.$$ This construction holds for any value of $q$ when $d=2$ or $d=3$, and when $d \geq 4$ and $q = p_1^{e_1}p_2^{e_2} \cdots p_r^{e_r}$ is the prime power decomposition of $q$, we require that $p_i > d-1$ for all $i$. We refer the reader to [@BNW] for more details on this construction and the identification between the group and the Cayley graph $DL_d(q)$. In exploring the metric properties of the groups ${\Gamma_d(q)}$, and hence the Cayley graphs $DL_d(q)$, one often needs to keep track of [*edge types*]{} along a path in $DL_d(q)$ rather than the specific generators which label the edges along the path. Given any vertex $x=(x_1, x_2, \dots, x_d)$, by an edge of type ${\textbf e_i}-{\textbf e_j}$ emanating from vertex $x$ we mean an edge with one endpoint at $x$ and the other at $y=(y_1, y_2, \dots , y_d)$ where $y_k=x_k$ for $k \notin \{i,j\}$, $y_i^-=x_i$, and $y_j=x_j^-$. Note that $h_i(y_i)=h_i(x_i)+1$ and $h_j(y_j)=h_j(x_j)-1$. There are exactly $q$ possible choices for $y_i$, so there are $q$ distinct edges of type ${\textbf e_i}- {\textbf e_j}$ emanating from $x$. Since the vertices of $DL_d(q)$ are identified with the elements of ${\Gamma_d(q)}$, we abuse notation and consider the projection map $\Pi$ to be a map from the group ${\Gamma_d(q)}$ to $({\mathbb Z}_2)^d$, and write $$\Pi(g) =\Pi(x=(x_1, x_2, \dots, x_d)) ={\left((m_1(g),l_1(g)),(m_2(g),l_2(g)), \cdots ,(m_d(g),l_d(g))\right)}$$ when $g \in {\Gamma_d(q)}$ is identified with the vertex $x$ in ${DL_d(q)}$. We remark that the basepoint vertex $o=(o_1, \ldots, l_d)$ in $DL_d(q)$ is identified with the identity element in ${\Gamma_d(q)}$. Computing Word length in ${\Gamma_d(q)}$ with respect to $S_{d,q}$ {#sec:wordlength} ================================================================== Let $S_{d,q}$ be the generating set for $\Gamma_d(q)$ so that the Cayley graph $\Gamma({\Gamma_d(q)},S_{d,q})$ is $DL_d(q)$. We will show that the word length of an element with respect to $S_{d,q}$ depends only on $\Pi(g)$, and not on $g$ itself. In the course of establishing the formula for word length, it is often sufficient to keep track of the edge types along a path, rather than the edge labels themselves. Given a vertex $v \in DL_d(q)$, we have defined edges of type ${\textbf e_i}- {\textbf e_j}$ emanating from $v$. By a path of type $\alpha_1 \alpha _2 \cdots \alpha_r$ starting at $v$, where $\alpha_k=({\textbf e_{i_k}}-{\textbf e_{j_k}})^{p_k}$, with $p_k\geq 0$ for each $k$, we mean a path beginning at $v$ which follows $p_1$ edges of type ${\textbf e_{i_1}}-{\textbf e_{j_1}}$, then $p_2$ edges of type ${\textbf e_{i_2}}-{\textbf e_{j_2}}$, and so on. We begin by defining a function $f$ from $\Gamma_d(q)$ to the natural numbers, a candidate for the word length function $l: {\Gamma_d(q)}\rightarrow \mathbb{N}$ for elements of ${\Gamma_d(q)}$ with respect to the generating set $S_{d,q}$. \[thm:wordlength\] Let $g \in \Gamma_d(q)$, with $\Pi(g)= {\left((m_1(g),l_1(g)),(m_2(g),l_2(g)), \cdots ,(m_d(g),l_d(g))\right)}$, and $\sigma$ in $\Sigma_d$, the symmetric group on $d$ letters. Define - $A_{\sigma(d)}(g)= \sum_{j=1}^d m_{\sigma(j)}(g)$ and $A_{\sigma(i)}(g)=\sum_{j=2}^{i}m_{\sigma(j)}(g)+\sum_{k=i}^{d-1}l_{\sigma(k)}(g)$ for $2 \leq i \leq d-1$. - $f_{\sigma}(g,i)=m_{\sigma(1)}(g)+l_{\sigma(d)}(g)+ A_{\sigma(i)}(g)$ for $2 \leq i \leq d$. - $f_{\sigma}(g)= \max_{2 \leq i \leq d} f_{\sigma}(g,i)$. - $f(g)= \min_{\sigma \in \Sigma_d}f_{\sigma}(g)$. In order to establish that the function $f$ defined above is the word length function, we use the following general lemma. \[lemma:length\] Given a group $G$ with generating set $S$, let $l: G \rightarrow {\mathbb N}$ be the word length with respect to $S$. If $f:G \rightarrow {\mathbb N}$ is another function satisfying 1. $f(g)=0$ if and only if $g$ is the identity element, 2. For every $g \in G$, $l(g) \geq f(g)$, 3. For every $g \in G$, there exists some $s \in S$ with $f(gs)=f(g)-1$, then $l(g)=f(g)$ for every $g \in G$. Let $g \in G$, and suppose $f(g)=n$. Then by property (3) there exist $s_1, s_2, \dots, s_n \in S$ satisfying $f(gs_1s_2 \cdots s_n)=0$. By property (1), $g=s_n^{-1} \cdots s_2^{-1}s_1^{-1}$, so $l(g) \leq f(g)$. Hence by property (2) we have $l(g)=f(g)$. Clearly, for the function $f$ defined in Definition \[thm:wordlength\] we have $f(g)=0$ if and only if $g$ is the identity element. The other two properties of the function $f$ will be verified in Propositions \[prop2\] and \[prop:decrease\] below. It then follows from Lemma \[lemma:length\] that the function $f$ defined in Definition \[thm:wordlength\] is the word length function for ${\Gamma_d(q)}$ with respect to the generating set $S_{d,q}$. \[prop2\] Let $g \in \Gamma_d(q)$ with $\Pi(g)= {\left((m_1(g),l_1(g)),(m_2(g),l_2(g)), \cdots ,(m_d(g),l_d(g))\right)}$, let $f(g)$ be as in Definition \[thm:wordlength\] and let $l(g)$ be the word length of $g$ with respect to the generating set $S_{d,q}$. Then $l(g) \geq f(g)$. Let $\gamma$ be a path of length $n$ in $DL_d(q)$ from $o$ to the vertex $x$ identified with $g$, thus $\gamma$ corresponds naturally to a word $a_1a_2a_3 \cdots a_n$ with $a_i \in S_{d,q}$ for $1 \leq i \leq n$. We will show that for some choice of $\sigma \in \Sigma_d$ we have $n \geq f_{\sigma}(g,i)$ for every $2 \leq i \leq d$. It follows that $n \geq f_{\sigma}(g) \geq f(g)$, and thus $l(g) \geq f(g)$. We begin by choosing the permutation $\sigma \in \Sigma_d$. Along the path $\gamma$ from $o$ to $x$, there must be points where the $k^{th}$ coordinate is $y_k=o_k \curlywedge x_k $ for $1 \leq k \leq d$. Let $v^1$ be the first such point, so $v^1_{i_1}=y_{i_1}$ for some $i_1$ with $1 \leq i_1 \leq d$. By the definition of $v^1$, we know that $v^1_k \curlywedge x_k =y_k$ for $k \neq i_1$. Thus, on the portion on the path from $v^1$ to $x$, there must be points where the $k^{th}$ coordinate is $y_k=o_k \curlywedge x_k $ for each $1 \leq k \leq d$, $k \neq i_1$. Let $v^2$ be the first such point, so $v^2_{i_2}=y_{i_2}$ for some $i_2$ with $1 \leq i_2 \leq d$, $i_2 \neq i_1$. Continuing in this manner, we define points $v^1,v^2, \dots, v^d$, each with a distinct associated coordinate $i_1, i_2, \dots i_d$ such that the $i_k^{th}$ coordinate of $v^k$ is $y_{i_k}$. Let $\sigma \in \Sigma_d$ be the unique permutation defined by $\sigma(k)=i_k$ for $1 \leq k \leq d$. First we consider the point $v^j$, for $2 \leq j \leq d-1$, and suppose that the prefix $a_1 \dots a_r$ corresponds to the subpath of $\gamma$ starting at $o$ and ending at $v^j$. Then for every $p$ with $1 \leq p \leq j$ the path $a_1 \dots a_r$ must contain at least $m_{\sigma(p)}(g)$ edges of type ${\textbf e_t}-{\textbf e_{\sigma(p)}}$, where $t \neq {\sigma(p)}$ may vary by edge, so $r \geq \sum_{p=1}^{j}m_{\sigma(p)}(g)$. However, for every $p$ with $j \leq p \leq d$, the path $a_{r+1} \dots a_n$ must contain at least $l_{\sigma(p)}(g)$ edges of type ${\textbf e_{\sigma(p)}}-{\textbf e_t}$, where $t \neq {\sigma(p)}$ may vary by edge, so $n-r \geq \sum_{p=j}^{d}l_{\sigma(p)}(g)$. Thus, $$\begin{aligned} n=r+(n-r) &\geq \sum_{p=1}^{j}m_{\sigma(p)}(g) + \sum_{p=j}^{d}l_{\sigma(p)}(g)\\ &=m_{\sigma(1)}(g)+A_{\sigma(j)}(g) +l_{\sigma(d)}(g)\\ &=f_{\sigma}(g,j)\end{aligned}$$ for every $2 \leq j \leq d-1$. For the case $j=d$, we use a slightly different argument. In this case, let $a_1 \dots a_r$ be the path from $o$ to $v^1$, and let $a_{r+1} \dots a_s$ be the path from $v^1$ to $v^d$. Then the path $a_1 \dots a_r$ must contain at least $m_{\sigma(1)}(g)$ edges of type ${\textbf e_t}-{\textbf e_{\sigma(1)}}$, so $r \geq m_{\sigma(1)}(g)$. Similarly, the path $a_{s+1} \dots a_n$ must contain at least $l_{\sigma(d)}(g)$ edges of type ${\textbf e_{\sigma(d)}}-{\textbf e_t}$, so $n-s \geq l_{\sigma(d)}(g)$. For each $p \neq 1$, $y_{\sigma(p)} \curlywedge v^1_{\sigma(p)} = y_{\sigma(p)}$, so for each such $p$, there must by at least $h_{\sigma(p)}(v^1_{\sigma(p)})- h_{\sigma(p)}( y_{\sigma(p)})$ letters corresponding to generators of type ${\textbf e_t}-{\textbf e_{\sigma(p)}}$ for various choices of $t$ in the word $a_{r+1} \cdots a_s$. Thus, $s-r \geq \sum_{p=2}^d h_{\sigma(p)}(v^1_{\sigma(p)})- h_{\sigma(p)}(y_{\sigma(p)})$. Now since $\sum_{p=1}^d h_{\sigma(p)}(v_{\sigma(p)}^1) = 0$ and $h_{\sigma(1)}(v^1_{\sigma(1)})=-m_{\sigma(1)}(g)$, we must have $\sum_{p=2}^d h_{\sigma(p)}(v^1_{\sigma(p)}) = -h_{\sigma(1)}(v^1_{\sigma(1)})=m_{\sigma(1)}(g)$. Furthermore, $h_{\sigma(p)}(y_{\sigma(p)})=-m_{\sigma(p)}(g)$ for every $2 \leq p \leq d$. Hence, $$\begin{aligned} s-r &\geq \sum_{p=2}^d \left( h_{\sigma(p)}(v^1_{\sigma(p)})- h_{\sigma(p)}(y_{\sigma(p)}) \right) \\ &= m_{\sigma(1)}(g)- \sum_{p=2}^d h_{\sigma(p)}(y_{\sigma(p)})\\ &= \sum_{p=1}^d m_{\sigma(p)}(g) = A_{\sigma(d)}(g). \end{aligned}$$ Thus we have $$\begin{aligned} n &= r + (s-r) + (n-s) \\ &\geq m_{\sigma(1)}(g)+ A_{\sigma(d)}(g)+l_{\sigma(d)}(g)\\ &=f_{\sigma}(g,d). \end{aligned}$$ Hence, we have shown that $n \geq f_{\sigma}(g,j)$ for every $2 \leq j \leq d$, as desired. To complete the argument, we must prove that $f$ satisfies the third and final property of Lemma \[lemma:length\]. In doing so, it is often necessary to keep track of which values of $l_{\chi(i)}(g)$ in $\Pi(g)$ are zero for a given permutation $\chi$, so we first prove several preliminary lemmas. \[lemma:index-set\] Let $\chi \in \Sigma_d$ be any permutation and $g \in {\Gamma_d(q)}$ any nontrivial element. 1. If $l_{\chi(d)}(g)=0$, let $n$ be the maximal value of $j$ with $1 \leq j \leq d-1$ so that $l_{\chi(j)}(g) \neq 0$. Then $$\max_{2 \leq i \leq d} A_{\chi(i)}(g) = \max_{2 \leq i \leq n, i=d} A_{\chi(i)}(g)$$ 2. If $l_{\chi(1)}(g)=0$, let $k$ be the minimum value of $j$ with $2 \leq j \leq d$ so that $l_{\chi(j)}(g) \neq 0$. Then $$\max_{2 \leq i \leq d} A_{\chi(i)}(g) = \max_{k \leq i \leq d} A_{\chi(i)}(g).$$ Since $g$ is nontrivial, the values of $n$ and $k$ defined in the statement above both exist. The proof of (1) follows from the fact that if $l_{\chi(n+1)}(g) = l_{\chi(n+2)}(g) = \cdots = l_{\chi(d-1)}(g) = 0$, then for $n+1 \leq i \leq d-1$ we have $A_{\chi(i)}(g) \leq A_{\chi(d)}(g)$. Similarly, to prove (2), if $l_{\chi(2)}(g)= \cdots = l_{\chi(k-1)}(g)=0$ for $2 \leq i \leq k-1$ we have $A_{\chi(i)}(g) \leq A_{\chi(k)}(g)$. \[prop:main\] Fix $g \in {\Gamma_d(q)}$. Let $\sigma \in \Sigma_d$ with $l_{\sigma(1)}(g) = 0$. Let $\tau \in \Sigma_d$ be defined by $\tau(i) = \sigma(i+1)$ for $1 \leq i <d$ and $\tau(d) = \sigma(1)$. Then $f_{\sigma}(g) \geq f_{\tau}(g)$. First note that for $2 \leq i \leq d-2$ we have $$A_{\sigma(i+1)}(g) = m_{\sigma(2)}(g) + \cdots + m_{\sigma(i+1)}(g) + l_{\sigma(i+1)}(g) + \cdots + l_{\sigma(d-1)}(g)$$ and $$\begin{aligned} {A_{\tau(i)}}(g) &= m_{\tau(2)}(g) + \cdots + m_{\tau(i)}(g) + l_{\tau(i)}(g) + \cdots l_{\tau(d-1)}(g) \\ &= m_{\sigma(3)}(g) + \cdots + m_{\sigma(i+1)}(g) + l_{\sigma(i+1)}(g) + \cdots + l_{\sigma(d)}(g). \end{aligned}$$ Hence for $2 \leq i \leq d-2$ we have $$A_{\sigma(i+1)}(g) = {A_{\tau(i)}}(g) +m_{\sigma(2)}(g)-l_{\sigma(d)}(g).$$ The lemma is clearly true if $g$ is the identity element, so we may assume for the remainder of the proof that $g$ is nontrivial. Using the definition of $k$ given in Lemma \[lemma:index-set\] we have $\max_{2 \leq i \leq d} A_{\sigma(i)}(g) = \max_{k \leq i \leq d} A_{\sigma(i)}(g)$, we may assume that $f_{\sigma}(g) = f_{\sigma}(g,i)$ for $k \leq i \leq d$, that is, $f_{\sigma}(g) = m_{\sigma(1)}(g) + l_{\sigma(d)}(g) + {A_{\sigma(i)}}(g)$ for some $i$ with $k \leq i \leq d$. We must show for each $j$ with $2 \leq j \leq d$ that $f_{\sigma}(g,i) \geq f_{\tau}(g,j)$. From this it follows that $f_{\sigma}(g) \geq f_{\tau}(g)$ as desired. We consider three subcases, as follows. **Case 1.** Suppose that $2 \leq j \leq d-2$. Using the formula above, we see that $$\begin{aligned} f_{\sigma}(g,i)&= m_{\sigma(1)}(g) + l_{\sigma(d)}(g) + A_{\sigma(i)}(g)\\ &\geq m_{\sigma(1)}(g) + l_{\sigma(d)}(g) + A_{\sigma(j+1)}(g) \\ &= m_{\sigma(1)}(g) + l_{\sigma(d)}(g) + A_{\tau(j)}(g) + m_{\sigma(2)}(g) - l_{\sigma(d)}(g) \\ & \geq m_{\sigma(2)}(g) + A_{\tau(j)}(g) = m_{\tau(1)}(g) + l_{\tau(d)}(g) + A_{\tau(j)}(g)\\ & = f_{\tau}(g,j)\end{aligned}$$ where the first inequality holds because $f_{\sigma}(g)=f_{\sigma}(g,i)$ and thus $A_{\sigma(i)}(g) \geq A_{\sigma(j)}(g)$ for $i \neq j$, and the penultimate equality holds because $l_{\tau(d)}(g) = l_{\sigma(1)}(g) =0$. **Case 2.** Suppose that $j=d$. Then, $$\begin{aligned} f_{\sigma}(g,i) &= m_{\sigma(1)}(g) + l_{\sigma(d)}(g) + A_{\sigma(i)} (g)\\ &\geq m_{\sigma(1)}(g) + l_{\sigma(d)}(g) + A_{\sigma(k)}(g) \\ &= m_{\sigma(1)}(g) + m_{\sigma(2)}(g) + \cdots + m_{\sigma(k)}(g) + l_{\sigma(k)}(g) + l_{\sigma(k+1)}(g) + \cdots + l_{\sigma(d)}(g) \\ &\geq m_{\sigma(2)}(g) + 0 + l_{\sigma(k)}(g) + l_{\sigma(k+1)}(g) + \cdots + l_{\sigma(d)}(g) \\ &= m_{\tau(1)}(g) + l_{\tau(d)}(g) + \sum_{r=1}^d m_{\tau(r)(g)} = f_{\tau}(g,d) \end{aligned}$$ where the last line relies on the fact thats that $l_{\tau(d)}(g) = l_{\sigma(1)}(g) = 0$ and by our choice of $k$, $$\sum_{r=1}^d m_{\tau(r)}(g) = \sum_{r=1}^d m_{\sigma(r)}(g) = \sum_{r=1}^d l_{\sigma(r)}(g) = \sum_{r=k}^d l_{\sigma(r)}(g).$$ **Case 3.** When $j=d-1$ we differentiate between $2 \leq i \leq d-1$ and $i=d$. Recall that $f_{\sigma}(g) = f_{\sigma}(g,i)$. First let $2 \leq i \leq d-1$ and recall that $l_{\tau(d)}(g) = l_{\sigma(1)}(g) = 0$ by assumption. In this case, $$\begin{aligned} f_{\tau}(g,d-1) &= m_{\tau(1)}(g) + m_{\tau(2)}(g) + \cdots + m_{\tau(d-1)}(g) + l_{\tau(d-1)}(g) + l_{\tau(d)}(g) \\ &= m_{\sigma(2)}(g) + \cdots + m_{\sigma(d)}(g) + l_{\sigma(d)}(g).\end{aligned}$$ Additionally, we are assuming that $A_{\sigma(i)} \geq A_{\sigma(d)}$. Writing out this inequality and canceling identical terms from both sides of the inequality yields $$l_{\sigma(i)}(g) + l_{\sigma(i+1)}(g) + \cdots l_{\sigma(d-1)}(g) \geq m_{\sigma(1)}(g) + m_{\sigma(i+1)}(g) + m_{\sigma(i+2)}(g)+ \cdots + m_{\sigma(d)}(g).$$ Hence, $$\begin{aligned} f_{\sigma}(g,i) &= m_{\sigma(1)}(g) + m_{\sigma(2)}(g) + \cdots + m_{\sigma(i)}(g) + l_{\sigma(i)}(g) + \cdots + l_{\sigma(d)}(g)\\ &\geq \left(m_{\sigma(1)}(g)+ \cdots + m_{\sigma(i)}(g)\right) + \left(m_{\sigma(1)}(g) + m_{\sigma(i+1)}(g) + m_{\sigma(i+2)}(g)+ \cdots + m_{\sigma(d)}(g)\right) + l_{\sigma(d)} (g) \\ &\geq m_{\sigma(2)}(g) + \cdots + m_{\sigma(d)}(g) + l_{\sigma(d)}(g) \\ &= m_{\tau(1)}(g) + m_{\tau(2)}(g) + \cdots + m_{\tau(d-1)}(g) + l_{\tau(d-1)}(g)+l_{\tau(d)}(g) = f_{\tau}(g,d-1).\end{aligned}$$ Now assume that $i=d$. In this case, $$\begin{aligned} f_{\sigma}(g,d) &= m_{\sigma(1)}(g) + l_{\sigma(d)}(g) + \sum_{r=1}^d m_{\sigma(i)}(g) \\ &\geq m_{\sigma(2)}(g) + \cdots +m_{\sigma(d)}(g) + l_{\sigma(d)}(g)\\ &= m_{\tau(1)}(g) + m_{\tau(2)}(g) + \cdots + m_{\tau(d-1)}(g) + l_{\tau(d-1)}(g) \\ &= m_{\tau(1)}(g) + m_{\tau(2)}(g) + \cdots + m_{\tau(d-1)}(g) + l_{\tau(d-1)}(g) + l_{\tau(d)}(g) = f_{\tau}(g,d-1) .\end{aligned}$$ If $g \in \Gamma_d(q)$ is nontrivial, let $\Theta_g = \{ \sigma \in \Sigma_d| f(g) = f_{\sigma}(g) \}$, and let $\Theta'_g = \{\sigma \in \Theta_g | l_{\sigma(1)}(g) \neq 0 \}$. Then Lemma \[prop:main\] has the following corollary. \[cor:main\] If $g \in {\Gamma_d(q)}$ is not the identity element, then $\Theta_g'$ is not empty, that is, there exists $\sigma \in \Sigma_d$ such that $f(g) = f_{\sigma}(g)$ and $l_{\sigma(1)}(g) \neq 0$. Suppose that $\chi \in \Theta_{g}$, and $l_{\chi(1)}(g) =0$. Let $k$ be defined as in Lemma \[lemma:index-set\]. Applying Lemma \[prop:main\] $k-1$ times, we obtain the corollary. \[lemma:twozeros\] Let $g \in {\Gamma_d(q)}$ and $\sigma \in \Sigma_d$ with $m_{\sigma(d)}(g)=l_{\sigma(d)}(g)=0$. Define $\tau \in \Sigma_d$ such that $\tau(i)=\sigma(i-1)$ for $i \geq 2$, and $\tau(1)=\sigma(d)$. Then $f_{\sigma}(g)=f_{\tau}(g)$. One directly verifies, using arguments as in the previous lemma, that $f_{\sigma}(g,i)=f_{\tau}(g, i+1)$ for $2 \leq i \leq d-1$, and $f_{\sigma}(g,d)=f_{\tau}(g,2)$. We immediately obtain the following corollary. \[cor:twozeroes\] If $g \in {\Gamma_d(q)}$ is not the identity element, there exists $\sigma \in \Theta_g$ such that either $l_{\sigma(d)}(g)\neq 0$ or $ m_{\sigma(d)}(g)\neq 0$. The next proposition uses the above lemmas and corollaries to prove that the function $f$ satisfies the third property of Lemma \[lemma:length\]. \[prop:decrease\] Let $g \in \Gamma_d(q)$ be a nontrivial group element, and let $f(g)$ be as in Definition \[thm:wordlength\]. Then there exists $s \in S_{d,q}$ with $f(gs)=f(g)-1$. If $g \in S_{d,q}$, that is, $g$ is a generator of $\Gamma_d(q)$, then it is easy to see that $f(g) = 1$ and choosing $s = g^{-1}$, that $f(gs) = 0$ and the condition of the proposition is satisfied. From now on we assume that $g \notin S_{d,q}$ and hence for any $s \in S_{d,q}$ we know that $gs$ is nontrivial. Let $x$ be the vertex in $DL_d(q)$ identified with $g$; recall that we write $\Pi(g)$ for $\Pi(x)$. **Case 1.** There exists $\sigma \in \Theta_g$ with $l_{\sigma(d)}(g) \neq 0$. If in addition $l_{\sigma(1)}(g) > 0$, or $l_{\sigma(1)}(g)=m_{\sigma(1)}(g)=0$, choose $s$ to be any generator corresponding to an edge of type ${\textbf e_{\sigma(1)}}-{\textbf e_{\sigma(d)}}$. If $l_{\sigma(1)}(g) = 0$ and $m_{\sigma(1)}(g) >0$, let $w$ be the vertex in $T_{\sigma(1)}$ adjacent to $x_{\sigma(1)}$ on the unique shortest path from $o_{\sigma(1)}$ to $x_{\sigma(1)}$. Choose $s$ to be any generator of type ${\textbf e_{\sigma(1)}}-{\textbf e_{\sigma(d)}}$ so that if $z$ is the vertex in $DL_d(q)$ identified with $gs$, then $z_{\sigma(1)} \neq w$. Then we have: 1. $(m_{\sigma(d)}(gs),l_{\sigma(d)}(gs)) = (m_{\sigma(d)}(g),l_{\sigma(d)}(g)-1)$, 2. $(m_{\sigma(1)}(gs),l_{\sigma(1)}(gs)) = (m_{\sigma(1)}(g),l_{\sigma(1)}(g)+1)$, and 3. $(m_{\sigma(i)}(gs),l_{\sigma(i)}(gs)) = (m_{\sigma(i)}(g),l_{\sigma(i)}(g))$ for $i \neq 1,d$. But this implies that $A_{\sigma(i)}(gs)=A_{\sigma(i)}(g)$ for every $2 \leq i \leq d$, and hence $$\begin{aligned} f_{\sigma}(gs)&=m_{\sigma(1)}(gs)+l_{\sigma(d)}(gs)+ \max_{2 \leq i \leq d} A_{\sigma(i)}(gs) \\ &= m_{\sigma(1)}(g)+(l_{\sigma(d)}(g)-1)+ \max_{2 \leq i \leq d} A_{\sigma(i)}(g)\\ &=f_{\sigma}(g)-1. \end{aligned}$$ First we note that the inequality $f(g)-1 \geq f(gs)$ is fairly easy to verify, since $f(g) - 1 = f_{\sigma}(g) - 1 = f_{\sigma}(gs) \geq f_{\tau}(gs)$ for any $\tau \in \Theta_{gs}$. Hence $f(g) - 1 \geq f(gs)$. Now for any $\tau \in \Sigma_d$, $m_{\tau (i)}(gs) = m_{\tau (i)}(g)$ for every $1 \leq i \leq d$, $l_{\tau (i)}(gs) \neq l_{\tau (i)}(g)$ for only two choices of $i$, and in addition, for one of these values, $l_{\tau (i)}(gs) = l_{\tau (i)}(g)-1$ and for the other, $l_{\tau (i)}(gs) = l_{\tau (i)}(g)+1$. Since for any value of $i$, $l_{\tau (i)}(g)$ (respectively $l_{\tau (i)}(gs)$) appears at most in the formula for $f_{\tau}(g,i)$ (respectively $f_{\tau}(gs,i)$) this implies that $f_{\tau}(g)-1 \leq f_{\tau}(gs)$. Thus for $\tau \in \Theta_{gs}$, $f(gs)=f_{\tau}(gs) \geq f_{\tau}(g)-1 \geq f(g)-1$, so $f(gs) \geq f(g)-1$ as well. Hence, $f(gs)= f(g)-1$, as desired. **Case 2.** For every $\chi \in \Theta_g$, we assume that $l_{\chi(d)}(g)=0$. Applying Corollary \[cor:twozeroes\] we may choose $\sigma \in \Theta_g$ so that $m_{\sigma(d)}(g) \neq 0$. Let $w$ be the vertex in $T_{\sigma(d)}$ adjacent to $x_{\sigma(d)}$ on the unique shortest path from $o_{\sigma(d)}$ to $x_{\sigma(d)}$, and let $n$ be as defined in Lemma \[lemma:index-set\]. Choose the generator $s \in S_{d,q}$ of type ${\textbf e_{\sigma(d)}}-{\textbf e_{\sigma(n)}}$ so that if $z$ is the vertex in $DL_d(q)$ identified with $gs$, then $z_{\sigma(d)}=w$. Then we have, for the pair $\sigma$ and $s$: 1. $(m_{\sigma(d)}(gs),l_{\sigma(d)}(gs)) = (m_{\sigma(d)}(g)-1,l_{\sigma(d)}(g))$, where we note that $l_{\sigma(d)}(gs) = l_{\sigma(d)}(g) = 0$, 2. $(m_{\sigma(n)}(gs),l_{\sigma(n)}(gs)) = (m_{\sigma(n)}(g),l_{\sigma(n)}(g)-1)$, and 3. $(m_{\sigma(i)}(gs),l_{\sigma(i)}(gs)) = (m_{\sigma(i)}(g),l_{\sigma(i)}(g))$ for $i \neq n,d$. For the above choice of $\sigma$ and $s$, we claim that $f_{\sigma}(gs) = f_{\sigma}(g)-1$. Applying Lemma \[lemma:index-set\] to $g$ we see that $$f_{\sigma}(g) = m_{\sigma(1)}(g) + l_{\sigma(d)}(g) + \max_{2 \leq i \leq n, i=d} A_{\sigma(i)}(g).$$ As the only index for which $l_{\sigma(i)}(gs) \neq l_{\sigma(i)}(g)$ is $i=n$, we again see that for $j>n$ we have $l_{\sigma(j)}(gs) =0$. Applying Lemma \[lemma:index-set\] to $gs$, we see that $$f_{\sigma}(gs) = m_{\sigma(1)}(gs) + l_{\sigma(d)}(gs) + \max_{2 \leq i \leq n, i=d} A_{\sigma(i)}(gs).$$ It follows from the definition of $s$ that $A_{\sigma(d)}(gs) = A_{\sigma(d)}(g)-1$. Similarly, for $2 \leq i \leq n$ we have $A_{\sigma(i)}(gs) = A_{\sigma(i)}(g)-1$ since neither expression contains $m_{\sigma(d)}$ and both contain $l_{\sigma(n)}$. Hence, $$\max_{2 \leq i \leq n, i=d} A_{\sigma(i)}(gs) = \max_{2 \leq i \leq n, i=d} \left( A_{\sigma(i)}(g)-1 \right) = \left( \max_{2 \leq i \leq n, i=d} A_{\sigma(i)}(g) \right) - 1.$$ Combining the above reasoning, we see that $$\begin{aligned} f_{\sigma}(gs) &= m_{\sigma(1)}(gs) + l_{\sigma(d)}(gs) + \max_{2 \leq i \leq n, i=d} A_{\sigma(i)}(gs) \\ &= m_{\sigma(1)}(g) + l_{\sigma(d)}(g) + \max_{2 \leq i \leq n, i=d} A_{\sigma(i)}(g) -1 = f_{\sigma}(g) -1. \end{aligned}$$ Since $f_{\sigma}(g) = f(g)$ and $f(gs) \leq f_{\sigma}(gs)$ it follows immediately from the fact that $f_{\sigma}(gs)= f_{\sigma}(g) -1$ that $f(gs) \leq f(g)-1$. To complete the proof of Proposition \[prop:decrease\] we must show that $f(gs) \geq f(g)-1$. First note that for any $\chi \in \Sigma_d$ it follows from the definition of $f$ that $f_{\chi}(gs) \geq f_{\chi}(g)-3$. We now show that this inequality can be improved slightly for $\tau \in \Theta'_{gs}$; for such $\tau$ we will show that $f_{\tau}(gs) \geq f_{\tau}(g)-2$. Suppose to the contrary that $\tau \in \Theta '_{gs}$ and $f_{\tau}(gs)=f_{\tau}(g)-3$. This can happen in only one way, namely all three of the following conditions must be met: 1. $\tau(1) = \sigma(d)$, 2. $\tau(d) = \sigma(n)$, and 3. $\max_{2 \leq i \leq n, i=d}A_{\tau(i)}(gs) = A_{\tau(d)}(gs)$. Now $\tau \in \Theta '_{gs}$ implies that $l_{\tau(1)}(gs) \neq 0$, but the first condition above requires that $l_{\tau(1)}(gs) = l_{\sigma(d)}(gs) = l_{\sigma(d)}(g) = 0$, a contradiction. Thus, for all $\tau \in \Theta '_{gs}$, we must have $f_{\tau}(gs) \geq f_{\tau}(g)-2$. It follows from Corollary \[cor:main\] that $\Theta'_{gs}$ is not empty, so we may choose $\chi \in \Theta'_{gs}$. If $\chi \notin \Theta_g$, then $f_{\chi}(g) > f_{\sigma}(g)$. Thus we have $f(gs) = f_{\chi}(gs) \geq f_{\chi}(g)-2 > f_{\sigma}(g)-2$, and hence $f(gs) \geq f_{\sigma}(g)-1=f(g)-1$, as desired. If, on the other hand, $\chi \in \Theta _g$, we make the following claim. [**Claim.**]{} If $\chi \in \Theta _g$, there exists $\tau \in \Theta '_{gs}$ with $f_{\tau}(gs) \geq f_{\tau}(g)-1$. Proposition \[prop:decrease\] follows immediately from the Claim, as follows. Let $\tau$ be as in the Claim, so that $f_{\tau}(gs) \geq f_{\tau}(g)-1$. Then $f(gs) = f_{\tau}(gs) \geq f_{\tau}(g)-1 \geq f(g)-1$, and hence $f(gs) \geq f(g) -1$, as desired. To prove the claim, if $f_{\chi}(gs) \geq f_{\chi}(g)-1$ then simply let $\tau = \chi$. If $f_{\chi}(gs) = f_{\chi}(g)-2$, we use $\chi$ to construct $\tau \in \Theta'_{gs}$ so that $f_{\tau}(gs) \geq f_{\tau}(g) -1$, as follows. There exist distinct $u,v \in \{1,2, \cdots ,d\}$ so that $\chi(u) = \sigma(d)$ and $\chi(v) = \sigma(n)$. We now show that $1 < u < v < d$. To see that $1<u$, observe that $l_{\sigma(d)}(gs) = 0$ but $l_{\chi(1)}(gs) \neq 0$ since $\chi \in \Theta'_{gs}$, hence $\sigma(d) \neq \chi(1)$, that is, $u \neq 1$. To see that $v<d$, observe that $l_{\sigma(n)}(g) =l_{\chi(v)}(g) \neq 0$. Recall that since $\chi \in \Theta _g$, $l_{\chi(d)}(g) = 0$, and hence $v \neq d$. Finally, we must show that $u<v$. Since $\chi(1) \neq \sigma(d)$, $m_{\chi(1)}(gs) = m_{\chi(1)}(g)$. Also, since $\chi(v) \neq \chi(d)$, $l_{\chi(d)}(gs) = l_{\chi(d)}(g)$. Thus, in order for $f_{\chi}(gs) = f_{\chi}(g)-2$ it must be the case that $\max_{2 \leq i \leq d}A_{\chi(i)}(g)-\max_{2 \leq i \leq d}A_{\chi(i)}(gs)=2$. The only way this can happen is if $\max_{2 \leq i \leq d}A_{\chi(i)}(g)$ is realized by an expression which contains both $m_{\sigma(d)}(g)$ and $l_{\sigma(n)}(g)$, that is, both $m_{\chi(u)}(g)$ and $l_{\chi(v)}(g)$. By the construction of the terms $A_{\chi(i)}(g)$ we must have $u<v$ for this to occur, for if $u>v$ and $l_{\chi(v)}(g)$ was a term in the expression which realized $\max_{2 \leq i \leq d}A_{\chi(i)}(g)$, this expression would also contain $l_{\chi(u)}(g)$, not $m_{\chi(u)}(g)$ as required. Thus $u<v$. We now construct $\tau \in \Theta'_{gs}$ which satisfies $f_{\tau}(gs) \geq f_{\tau}(g)-1$. Let $u$ and $v$ be as above, and define - For $i<u$ let $\tau(i) = \chi(i)$. - For $u \leq i < v$ let $\tau(i) = \chi(i+1)$. - For $i=v$ let $\tau(v) = \chi(u)$. - For $i>v$ let $\tau(i) = \chi(i)$. We first show that $\tau \in \Theta'_{gs}$. Since $\chi(1) = \tau(1)$ and $\chi(d) = \tau(d)$ we claim that for any $i$ with $2 \leq i \leq d$ we have $A_{\tau(i)}(gs) \leq \max_{2 \leq j \leq d} A_{\chi(j)}(gs)$. This is clearly true when $i=d$, since $A_{\tau(d)}(gs) = A_{\chi(d)}(gs)$. We consider four remaining cases, and abuse our notation by writing $m_{\chi(i)}$, $l_{\chi(i)}$, and $A_{\chi(i)}$ instead of $m_{\chi(i)}(gs)$, $l_{\chi(i)}(gs)$, and $A_{\chi(i)}(gs)$, respectively. 1. Let $i<u$. Then $$\begin{aligned} A_{\chi(i)} &= m_{\chi(2)} + \cdots + m_{\chi(i)} + l_{\chi(i)} + \cdots + l_{\chi(u)} + \cdots + l_{\chi(v)} + \cdots + l_{\chi(d-1)} \\ & \text{simply rearranging the terms yields}\\ &= m_{\chi(2)} + \cdots + m_{\chi(i)} + l_{\chi(i)} + \cdots + l_{\chi(u-1)} + l_{\chi(u+1)} + \cdots + l_{\chi(v)} + l_{\chi(u)} + l_{\chi(v+1)} + \cdots \\ & \ \ \ + l_{\chi(d-1)} \\ & \text{changing to the equivalent indices for } \tau \text{ yields}\\ &=m_{\tau(2)} + \cdots + m_{\tau(i)} + l_{\tau(i)} + \cdots + l_{\tau(d-1)} \\ &= A_{\tau(i)}.\end{aligned}$$ 2. If $u \leq i <v $ then $$\begin{aligned} A_{\tau(i)} &= m_{\tau(2)} + \cdots + m_{\tau(u)} + \cdots + m_{\tau(i)} + l_{\tau(i)} + \cdots + l_{\tau(d-1)} \\ &= m_{\chi(2)} + \cdots + m_{\chi(u-1)} + m_{\chi(u+1)} + \cdots + m_{\chi(i+1)} + l_{\chi(i+1)} + \cdots + l_{\chi(v)} + l_{\chi(u)} \\ & \ \ \ + l_{\chi(v+1)} + \cdots + l_{\chi(d-1)} \\ & \text{adding in the "missing" term }m_{\chi(u)}=m_{\sigma(d)}>0 \text{ and omitting }l_{\chi(u)}=0 \text{ yields the inequality}\\ &< m_{\chi(2)} + \cdots + + m_{\chi(u-1)} + m_{\chi(u)} + m_{\chi(u+1)} + \cdots + m_{\chi(i+1)} + l_{\chi(i+1)} + \cdots \\ & \ \ \ + l_{\chi(v)} + l_{\chi(v+1)} + \cdots + l_{\chi(d-1)} \\ &= A_{\chi(i+1)}.\end{aligned}$$ 3. If $i=v$ then recall that $\tau(v) = \chi(u)=\sigma(d)$. Note that $\tau(v-1) = \chi(v)$ by the definition of $\tau$. Then $$\begin{aligned} A_{\tau(v)} &= m_{\tau(2)} + \cdots + m_{\tau(u-1)} + m_{\tau(u)} + \cdots + m_{\tau(v-1)} + m_{\tau(v)} + l_{\tau(v)} + \cdots + l_{\tau(d-1)} \\ &= m_{\chi(2)} + \cdots + m_{\chi(u-1)}+ m_{\chi(u+1)}+ \cdots + m_{\chi(v)} + m_{\chi(u)} + l_{\chi(u)} + l_{\chi(v+1)} + \cdots + l_{\chi(d-1)} \\ & \text{rearranging the existing terms, omitting }l_{\chi(u)}=0 \text{ and adding in the term } \\ & l_{\chi(v)} \text{ yields the inequality}\\ &\leq m_{\chi(2)} + \cdots + m_{\chi(u-1)} + m_{\chi(u)} + m_{\chi(v)} + l_{\chi(v)} + \cdots + l_{\chi(d-1)} = A_{\chi(v)}.\end{aligned}$$ 4. Let $i > v$. Then $$\begin{aligned} A_{\chi(i)} &= m_{\chi(2)} + \cdots + m_{\chi(u)} + \cdots + m_{\chi(v)} + \cdots + m_{\chi(i)} + l_{\chi(i)} + \cdots + l_{\chi(d-1)} \\ & \text{rearranging the existing terms yields }\\ &= m_{\chi(2)} + \cdots + m_{\chi(u-1)} + m_{\chi(u+1)} + \cdots + m_{\chi(v)} + m_{\chi(u)} + m_{\chi(v+1)} + \cdots + m_{\chi(i)} \\ & \ \ \ + l_{\chi(i)} + \cdots + l_{\chi(d-1)} \\ &= A_{\tau(i)}.\end{aligned}$$ Combining these cases we see that for all $2 \leq i \leq d$ we have $A_{\tau(i)}(gs) \leq \max_{2 \leq j \leq d} A_{\chi(j)}(gs)$. Thus $f_{\tau}(gs) \leq f_{\chi}(gs) = f(gs)$ and hence $f(gs) = f_{\tau}(gs)$, that is, $\tau \in \Theta_{gs}$. Now since $\chi \in \Theta _{gs}'$, this implies that $l_{\chi(1)}(gs) \neq 0$. But $\tau(1)=\chi(1)$, so $l_{\tau(1)}(gs) \neq 0$ as well, and hence $\tau \in \Theta_{gs}'$. Finally, it remains to show that $f_{\tau}(gs) = f_{\tau}(g)-1$. Recall from the definition of $\tau$ that $\tau(v) = \chi(u) = \sigma(d)$ and $\tau(v-1) = \chi(v) = \sigma(n)$. Additionally, recall from the choice of $s$ that 1. $(m_{\sigma(d)}(gs),l_{\sigma(d)}(gs)) = (m_{\sigma(d)}(g)-1,l_{\sigma(d)}(g))$, 2. $(m_{\sigma(n)}(gs),l_{\sigma(n)}(gs)) = (m_{\sigma(n)}(g),l_{\sigma(n)}(g)-1)$, 3. $(m_{\sigma(i)}(gs),l_{\sigma(i)}(gs)) = (m_{\sigma(i)}(g),l_{\sigma(i)}(g))$ for $i \neq n,d$. Comparing $A_{\tau(i)}(gs)$ and $A_{\tau(i)}(g)$ for all possible values of $i$ shows that for all $i$, $A_{\tau(i)}(gs) = A_{\tau(i)}(g)-1$. From the definition of $\tau$ we see that $$m_{\tau(1)}(g) = m_{\chi(1)}(g) = m_{\chi(1)}(gs) = m_{\tau(1)}(gs)$$ and $$l_{\tau(d)}(g) = l_{\chi(d)}(g) = l_{\chi(d)}(gs) = l_{\tau(d)}(gs).$$ Thus $f_{\tau}(gs) = f_{\tau}(g)-1$, which concludes the proof of the Claim, and hence Proposition \[prop:decrease\]. Comparing word length in ${\Gamma_d(q)}$ and distance in the product of trees ============================================================================= As the Diestel-Leader graph $DL_d(q)$ is a subset of the product of $d$ trees of valence $q+1$, it is natural to compare the word metric on the Cayley graph $DL_d(q)$ to the product metric on the product of trees. This product metric assigns every edge length one, and simply counts edges in each tree between the coordinates corresponding to two different group elements. It is a straightforward consequence of the word length formula that these two metrics are quasi-isometric. In Corollary \[cor:treedist\] below we extend the word length function $f$ to compute the the distance in the word metric (with respect to the generating set $S_d(q)$) between arbitrary group elements. We conclude with a corollary which constructs a family of quasi-geodesic paths from the vertex corresponding to the identity to that corresponding to any group element. \[thm:treedist\] Let $l(g)$ denote the word length of $g \in {\Gamma_d(q)}$ with respect to the generating set $S_{d,q}$ and $d_T(g)$ the distance in the product metric on the product of trees between $g$ in $DL_d(q)$ and $\epsilon$, the fixed basepoint corresponding to the identity in ${\Gamma_d(q)}$. Then $$\frac{1}{2} d_T(g) \leq l(g) \leq 2 d_T(g)$$ that is, the word length is quasi-isometric to the distance from the identity in the product metric on the product of trees. Let $\Pi(g) = {\left((m_1, l_1), (m_2, l_2), \dots, (m_d, l_d)\right)}$. It follows that $d_T(g) = \sum_{i=1}^d m_i + l_i = 2 \sum_{i=1}^d m_i$. Using the word length formula from Section \[sec:wordlength\], we see that for some $\sigma \in \Sigma(d)$, $$l(g)=f_{\sigma}(g)= \left(m_{\sigma(1)} + l_{\sigma(d)}\right) + \max_{2 \leq i \leq d} A_{\sigma(i)} \leq \left(\sum_{i=1}^d m_i + \sum_{i=1}^d l_i \right) + \sum_{i=1}^d m_i + \sum_{i=1}^d l_i = 2 d_T(g).$$ To obtain a lower bound, note that $$\begin{aligned} l(g)= \min _{\sigma \in \Sigma_d} f_{\sigma}(g) &= \min_{\sigma \in \Sigma_d} (m_{\sigma(1)} + l_{\sigma(d)}+ \max _{2 \leq i \leq d} A_{\sigma(i)}(g)) \\ &\geq \min_{\sigma \in \Sigma_d} ( \max _{2 \leq i \leq d} A_{\sigma(i)}(g))\end{aligned}$$ But for every $\sigma\in \Sigma_d$, $\max _{2 \leq i\leq d} A_{\sigma(i)}(g) \geq A_{\sigma(d)}(g)= \sum_{i=1}^d m_i$, so $$l(g) \geq \sum_{i=1}^d m_i = \frac{1}{2} d_T(g).$$ Combining these inequalities proves the theorem. The first corollary to Theorem \[thm:treedist\] requires us to extend the techniques of Section \[sec:wordlength\] in order to compute the distance in the word metric between arbitrary group elements. \[cor:treedist\] Let $g,h \in {\Gamma_d(q)}$ and let $d_T(g,h)$ denote the distance between the two vertices in $DL_d(q)$ corresponding to $g$ and $h$ with respect to the product metric on the product of trees. Then $$\frac{1}{2} d_T(g,h) \leq l(g^{-1}h) \leq 2 d_T(g,h).$$ In Section \[sec:wordlength\] we show that $l(g) = f(g)$ for the function $f$ defined there. The calculation of the value of $f(g)$ depends only on the coordinates of $\Pi(g)=((m_1(g),l_1(g)), \ldots, (m_d(g),l_d(g)))$. Recall that if $g$ corresponds to the vertex $(g_1, \ldots, g_d)$ in $DL_d(q)$,then for $1 \leq i \leq d$, $$(m_i(g),l_i(g))= (d_{T_i}(o_i,o_i \curlywedge g_i),d_{T_i}(g_i,o_i \curlywedge g_i)),$$ where $(o_1, \ldots, o_d)$ is the vertex in $DL_d(q)$ corresponding to the identity element of ${\Gamma_d(q)}$. Define an analogous relative projection function $\Pi_h(g)= ((m_{h,1}(g), l_{h,1}(g)), \ldots, (m_{h,d}(g), l_{h,d}(g)))$, where for $1 \leq i \leq d$, $$(m_{h,i}(g), l_{h,i}(g))=(d_{T_i}(h_i,h_i \curlywedge g_i),d_{T_i}(g_i,h_i \curlywedge g_i)).$$ Now define $f_h(g)$ as in Section \[sec:wordlength\], replacing $\Pi(g)$ with $\Pi_h(g)$. Since the proof that $l(g) = f(g)$ is strictly combinatorial, the arguments in Section \[sec:wordlength\] then imply that $f_h(g)$ computes the word length of $g^{-1}h$ with respect to the generating set $S_{d,q}$, and the Corollary follows directly from Theorem \[thm:treedist\]. The component of the word length function which computes the maximum of the quantities $A_{\sigma(i)}$ over $\sigma \in \Sigma_d$ presents a combinatorial obstruction to writing down a family of geodesic paths representing elements of ${\Gamma_d(q)}$. The symmetry present in the Diestel-Leader graphs gives rise to a natural family of paths described by edge labels, with the property that any path with these edge labels is a quasi-geodesic path in the Cayley graph $DL_d(q)$. While it is often not difficult to write down a family of quasi-geodesic paths in a Cayley graph, the paths we describe are very natural paths to traverse and the construction is valid when the trees are permuted, capturing the symmetry of the Diestel-Leader graphs. Let $g \in {\Gamma_d(q)}$ have projection $\Pi(g) = {\left((m_1, l_1), (m_2, l_2), \dots, (m_d, l_d)\right)}$. Consider the sequence of edge labels $$({\textbf e_d}-{\textbf e_1})^{m_1} ({\textbf e_d}-{\textbf e_2})^{m_2} \cdots ({\textbf e_d}-{\textbf e_{d-1}})^{m_{d-1}}({\textbf e_1}-{\textbf e_d})^{l_1} ({\textbf e_2}-{\textbf e_d})^{l_2} \cdots ({\textbf e_{d-1}}-{\textbf e_d})^{l_{d-1}}({\textbf e_1}-{\textbf e_d})^{\alpha}({\textbf e_d}-{\textbf e_1})^{l_d}$$ where $\alpha = m_d+(m_1 + \cdots + m_{d-1})-(l_1 + \cdots + l_{d-1}) = m_d + (\sum_{i=1}^d m_i - m_d) - (\sum_{i=1}^d m_i - l_d) = l_d$. We claim there is such a path $\zeta_g$ from the basepoint $o$ to the point identified with $g$ in $DL_d(q)$; in general, there are many possible choices of path with the above edge labels. Moreover, this construction holds under permutation of the trees $T_1,T_2, \cdots T_d$. \[cor:quasigeo\] Let $g \in {\Gamma_d(q)}$ and $\zeta_g$ any path from $\epsilon$ to $\gamma$ with edge labels as listed above. The $\zeta_g$ is a quasi-geodesic path. The corollary follows from combining Theorem \[thm:treedist\] and Corollary \[cor:treedist\] and checking that for any two points $h_1$ and $h_2$ along $\zeta_g$ the distance between them along the path $\zeta_g$ is coarsely equivalent to the distance between them in the product metric on the product of trees. Dead end elements ================= An element in a group $G$ with finite generating set $S$ which corresponds to a vertex $x \in \Gamma(G,S)$ is a [*dead end element*]{} if no geodesic ray in $\Gamma(G,S)$ from can be extended past $x$ and remain geodesic. Intuitively, the [*depth*]{} of the dead end element $g$ is the length of the shortest path in $\Gamma(G,S)$ from $g$ to any point in the complement of the ball of radius $l(g)$. Both the existence of dead end elements and their depth are dependent on generating set; in [@RW] an example is given of a finitely generated group which has dead end elements of finite depth with respect to one generating set, and unbounded depth with respect to another. Theorem \[thm:deadend\] below generalizes the main result of [@CR1; @CR2], namely that $\Gamma_3(2)$ has dead end elements of arbitrary depth with respect to a generating set similar to $S_{3,2}$. An element $g$ in a finitely generated group $G$ is a [*dead end element with respect to a finite generating set $S$*]{} for $G$ if $l(g)=n$ and $l(gs) \leq n$ for all generators $s$ in $S \cup S^{-1}$, where $l(g)$ denotes the word length of $g \in G$ with respect to $S$. A dead end element $g$ in a finitely generated group $G$ with respect to a finite generating set $S$ has [*depth $k$*]{} if $k$ is the largest integer with the following property. If the word length of $g$ is $n$, then $l(g s_1 s_2 \ldots s_r) \leq n$ for $1 \leq r < k$ and all choices of generators $s_i \in S \cup S^{-1}$. The goal of this section is to prove the following theorem. \[thm:deadend\] The group ${\Gamma_d(q)}$ has dead end elements of arbitrary depth with respect to the generating set $S_{d,q}$. The outline of the proof of Theorem \[thm:deadend\] mimics the outline of the proof in [@CR1; @CR2] showing that $\Gamma_3(2)$ has dead end elements of infinite depth with respect to a generating set similar to $S_{3,2}$. However, the details of the proofs are quite different. In [@CR1; @CR2] the lamplighter model of an element of ${\Gamma_d(q)}$ is used to compute word length and analogous lemmas to those below. This model extends the well-known lamplighter model of an element in $L_n = {\mathbb Z}_n \wr {\mathbb Z}$ (due to Jim Cannon) in which a group element of $L_n$ is visualized using a bi-infinite string of multi-state light bulbs placed at integer points along a number line along with a “lamplighter." Then $g \in L_n$ corresponds to a finite collection of illuminated bulbs and an integral position of the lamplighter. However, in $\Gamma_3(2)$ the “lampstand" (analogous to ${\mathbb Z}$ for $L_n$) consists of three bi-infinite rays, and the illuminated bulbs are obtained using a series of relations derived from Pascal’s triangle modulo 2, and the “lamplighter" moves over a ${\mathbb Z}\times {\mathbb Z}$ grid. A precise extension of this model to describe elements of ${\Gamma_d(q)}$ for $d >3$ seems ambiguous. The proofs given below rely instead on the geometry of the Diestel-Leader graphs, and their inherent symmetry. Begin by defining, for any $n \in {\mathbb Z}^+$, the set $$H_n = \{g \in {\Gamma_d(q)}~ | ~\Pi(g) = {\left((m_1(g),l_1(g)),(m_2(g),l_2(g)), \cdots ,(m_d(g),l_d(g))\right)}$$ $$\text{ with } 0 \leq m_i(g) \leq n \text{ and } 0 \leq l_i(g) \leq m_i(g)+n \text{ for all } 1 \leq i \leq d \}.$$ In the two lemmas below, we show that the word length of any point in $H_n$ with respect to $S_{d,q}$ is bounded, and a set of vertices in $H_n$ at maximal distance from the identity is described. Proofs of both lemmas follow easily from the wordlength formula proven in Section \[sec:wordlength\]. \[lemma:Hn\] If $g \in H_n$ then $l(g) \leq (d+2)n$. Let $g \in H_n$ with $\Pi(g) = {\left((m_1, l_1), (m_2, l_2), \dots, (m_d, l_d)\right)}$. Choose $\sigma \in \Sigma_d$ so that $l_{\sigma(1)} \geq l_{\sigma(2)} \geq \cdots \geq l_{\sigma(d)}$. We claim that $m_{\sigma(1)}+ A_{\sigma(i)}(g)+l_{\sigma(d)} \leq (d+2)n$ for every $2 \leq i \leq d$, hence $f_{\sigma}(g) \leq (d+2)n$. It then follows from the word length formula that $l(g) \leq f_{\sigma}(g) \leq (d+2)n$. Choose $k$ so that $l_{\sigma(k)} > n$, but $l_{\sigma(k+1)} \leq n$, and $k=0$ if $l_{\sigma(i)} \leq n$ for every $i$. Since $$\sum_{i=1}^d l_{\sigma(i)}(g) = \sum_{i=1}^d m_{\sigma(i)}(g) \leq dn,$$ it follows that $k < d$. Furthermore, we claim that $l_{\sigma(i)} + \cdots l_{\sigma(d)} \leq (d-i+1)n$ for $1 \leq i \leq d$. This is clear if $i \geq k+1$, since then each term in the sum is less than $n$. But if $1 \leq i \leq k+1$, then $l_{\sigma(1)}+ \cdots l_{\sigma(i-1)} \geq (i-1)n$, so $l_{\sigma(i)}+ \cdots l_{\sigma(d)}\leq dn-(-i+1)n=(d-i+1)n$. For $2 \leq j \leq d-1$, we see that $m_{\sigma(1)}+ A_{\sigma(j)}(g)+l_{\sigma(d)}=\sum_{i=1}^j m_{\sigma(i)}+ \sum_{i=j}^{d}l_{\sigma(i)}\leq (j)n+(d-j+1)n=(d+1)n$. But $A_\sigma(d)(g)= \sum_{i=1}^d m _{\sigma(d)} \leq dn$, and hence $m_{\sigma(1)}+ A_{\sigma(d)}(g)+l_{\sigma(d)} \leq (d+2)n$. Thus, $m_{\sigma(1)}+ A_{\sigma(i)}(g)+l_{\sigma(d)} \leq (d+2)n$ for every $2 \leq i \leq d$, as claimed, and the lemma follows. The next lemma follows immediately from the word length formula of Section \[sec:wordlength\]. \[lemma:5n\] If $g_n \in H_n$ and $\Pi(g_n) = \left( (n,n)(n,n) \cdots (n,n) \right)$ then $l(g) = (d+2)n$. The proof of Theorem \[thm:deadend\] follows easily from Lemmas \[lemma:Hn\] and \[lemma:5n\]. [*Proof of Theorem \[thm:deadend\].*]{} Let $g_n \in H_n$ be any element with $\Pi(g_n) = ((n,n)(n,n) \cdots (n,n))$. In Lemma \[lemma:5n\] it is shown that $l(g)=(d+2)n$. It follows immediately from Lemma \[lemma:Hn\] that $g_n$ is a dead end element, as all vertices adjacent to $g_n$ lie in $H_n$. To see that the depth of $g_n$ is at least $n$, note that the length of a path from $g_n$ to a point outside $H_n$ must contain a subpath of at least $n$ edges. Thus the depth of $g_n$ is at least $n$ and we conclude that ${\Gamma_d(q)}$ has dead end elements of arbitrary depth with respect to the generating set $S_{d,q}$. Cone types and geodesic languages ================================= We now prove that ${\Gamma_d(q)}$ has no regular language of geodesics with respect to the generating set $S_{d,q}$, that is, there is no collection of geodesic representatives for elements of ${\Gamma_d(q)}$ which is accepted by a finite state automata. The existence of a regular language of geodesics for a finitely generated group $G$ is equivalent to the finiteness of the set of cone types of $G$. It is a well known theorem in computer science that a language is regular if and only if it has finitely many distinct left quotients. In the case of a geodesic language, the left quotients are exactly the cone types. We prove that ${\Gamma_d(q)}$ has infinitely many cone types with respect the generating set $S_{d,q}$, and it follows that ${\Gamma_d(q)}$ has no regular language of geodesics with respect to $S_{d,q}$. We begin by defining the [*cone*]{} and the [*cone type*]{} of an element $g \in G$, where $G$ is a group with finite generating set $S$. Cannon defined the cone type of an element $w \in G$ to be the set of geodesic extensions of $w$ in the Cayley graph $\Gamma(G,S)$. [@cannoncone] A path $p$ is [*outbound*]{} if $d(1,p(t))$ is a strictly increasing function of $t$. For a given $g\in G$, the [*cone*]{} at $g$, denoted $C'(g)$ is the set of all outbound paths starting at $g$. Define the [*cone type*]{} of $g$, denoted $C(g)$, to be $g^{-1}C'(g)$. This definition applies both in the discrete setting of the group and in the one-dimensional metric space which is the Cayley graph. A subtlety is that if the presentation for $G$ includes odd length relators, then the cone type of an element in the Cayley graph may include paths which end at the middle of an edge. If the presentation for $G$ consists entirely of even length relators, then every cone type viewed in the Cayley graph consists entirely of full edge paths. We refer the reader to [@NS] for a more detailed discussion of cone types. \[thm:conetypes\] The group ${\Gamma_d(q)}$ has infinitely many cone types with respect to the generating set $S_{d,q}$. The following corollary is an immediate consequence of Theorem \[thm:conetypes\]. \[cor:languages\] The group ${\Gamma_d(q)}$ has no regular language of geodesics with respect to the generating set $S_{d,q}$. We begin with a lemma stating sufficient but not necessary conditions on $\sigma \in \Sigma_d$ which ensure that $f(g) = f_{\sigma}(g)$; this lemma will be extremely useful in the the proof of Theorem \[thm:conetypes\] as realizing when $f(g)=f_{\sigma}(g)$ for a particular $g \in {\Gamma_d(q)}$ and $\sigma \in \Sigma_d$ can be quite difficult. Recall that we identify $g \in {\Gamma_d(q)}$ with the vertex $x \in DL_d(q)$ corresponding to it, and abuse notation by writing $\Pi(g)$ for $\Pi(x)$. \[lemma:max-m\] Let $g \in {\Gamma_d(q)}$ have projection $\Pi(g) = {\left((m_1(g),l_1(g)),(m_2(g),l_2(g)), \cdots ,(m_d(g),l_d(g))\right)}$. If $\sigma \in \Sigma_d$ satisfies 1. $\min_{\tau \in \Sigma_d} m_{\tau(1)}(g) + l_{\tau(d)}(g) = m_{\sigma(1)}(g) + l_{\sigma(d)}(g) $, and 2. $\max_{2 \leq i \leq d} A_{\sigma(i)}(g) = A_{\sigma(d)}(g)$ then $f(g) = f_{\sigma}(g)$. Let $\sigma$ be as in the statement of the lemma, and $\tau$ any element of $\Sigma_d$. It is always true that $\max_{2 \leq i \leq d} A_{\tau(i)}(g) \geq A_{\tau(d)}(g)$, and that $m_{\tau(1)}(g) + l_{\tau(d)}(g) \geq m_{\sigma(1)}(g) + l_{\sigma(d)}(g)$ by the choice of $\sigma$. Hence $$\begin{aligned} f_{\tau}(g) &=m_{\tau(1)}(g) + l_{\tau(d)}(g) + \max_{2 \leq i \leq d} A_{\tau(i)}(g)\\ & \geq m_{\sigma(1)}(g) + l_{\sigma(d)}(g) + A_{\tau(d)}(g) \\ & =m_{\sigma(1)}(g) + l_{\sigma(d)}(g) + A_{\sigma(d)}(g)\\ & = f_{\sigma}(g). \end{aligned}$$ Thus by the definition of $f(g)$ we must have $f(g) = f_{\sigma}(g)$. To prove Theorem \[thm:conetypes\] we define a sequence of elements {$g_n\}$ so that there is a geodesic path of length $n$ from $g_n$ terminating at a dead end element, and so that no shorter geodesic path from $g_n$ reaches any other dead end element of the group. Thus each $g_n$ lies in a different cone type, and the theorem follows. [*Proof of Theorem \[thm:conetypes\].*]{} Let $g_n$ for $n \in {\mathbb Z}^+$ be any element with projection $$\Pi(g_n) = \left( (2n,3n),(3n,4n),(4n,5n),(5n,6n), \cdots ,((d-1)n,dn),(dn,3n),(2n,n)\right).$$ We first show that $f(g_n) = f_{\epsilon}(g_n)$ where $\epsilon$ is the identity permutation, and specifically that $f(g_n) = 3n + \sum_{i=1}^d m_i(g_n)$. First note that $\min_{\tau \in \Sigma_d} m_{\tau(1)}(g_n) + l_{\tau(d)}(g_n) = 3n = m_{\epsilon(1)}(g_n) + l_{\epsilon(d)}(g_n) $. Second, consider $A_{\epsilon(d)}(g_n) = 2n+ \sum_{j=2}^{d} jn=4n+ \sum_{j=3}^{d} jn$ and compare this value to $A_{\epsilon(i)}(g_n)$ for $i \neq d$. When $2 \leq i < d-1$, $$\begin{aligned} A_{\epsilon(i)}(g_n) &= [m_2(g_n) + m_3(g_n) + \cdots + m_i(g_n)] + [l_i(g_n) + \cdots + l_{d-1}(g_n)] \\ &= [3n + 4n + \cdots (i+1)n] + [(i+2)n + \cdots + dn + 3n] \\ &= 3n+ \sum_{j=3}^{d} jn < 4n + \sum_{j=3}^{d} jn = A_{\epsilon(d)}(g_n).\end{aligned}$$ When $i=d-1$ we see that $A_{d-1}(g_n) = 3n+ \sum_{j=3}^{d} jn < 4n + \sum_{j=3}^{d} jn = A_{\epsilon(d)}(g_n).$ It then follows from Lemma \[lemma:max-m\] that $f(g_n) = f_{\epsilon}(g_n) = 3n + \sum_{i=1}^d m_i(g_n)$. We note for later use that $A_{\epsilon(d)}(g_n) - A_{\epsilon(i)}(g_n) = n$ when $i \neq d$. Let $h_n$ be any point connected to $g_n$ by a path of length at most $n$ in $DL_d(q)$. Then $h_n$ has projection $$\begin{aligned} \Pi(h_n) &= ( (2n,3n-r_1),(3n,4n-r_2),(4n,5n-r_3),(5n,6n-r_4), \cdots \\ & ((d-1)n,dn-r_{d-2}),(dn,3n-r_{d-1}),(2n,n-r_d) )\end{aligned}$$ where the $r_i$ satisfy: 1. $\sum_{i=1}^d r_i = 0$, and 2. the sum of the positive $r_i$ is at most $n$; hence the sum of the negative $r_i$ is at least $-n$. We first calculate $f(h_n)$, again using Lemma \[lemma:max-m\]. Note that for any $\tau \in \Sigma_d$, $$\min_{\tau \in \Sigma_d} m_{\tau(1)}(h_n) + l_{\tau(d)}(h_n) = 3n-r_d = m_{\epsilon(1)}(h_n) + l_{\epsilon(d)}(h_n)$$ and that $2n \leq 3n-r_d \leq 4n$. Moreover, $A_{\epsilon(d)}(h_n) = A_{\epsilon(d)}(g_n)$. We compare $A_{\epsilon(i)}(g_n)$ and $A_{\epsilon(i)}(h_n)$ for $i \neq d$, and see that $$\begin{aligned} A_{\epsilon(i)}(h_n) &= 3n + 4n + \cdots + (i+1)n + (i+2)n-r_{i} + (i+3)n-r_{i+1} + \cdots dn - r_{d-2} + 3n-r_{d-1} \\ &=A_{\epsilon(i)}(g_n) - (r_{i} + \cdots + r_{d-1}) \leq A_{\epsilon(i)}(g_n) + n.\end{aligned}$$ Above we saw that $A_{\epsilon(d)}(g_n) - A_{\epsilon(i)}(g_n) =n$ for $2 \leq i<d$. Combining this with the above inequality yields $$A_{\epsilon(i)}(h_n) \leq A_{\epsilon(i)}(g_n) + n = A_{\epsilon(d)}(g_n) -n +n = A_{\epsilon(d)}(g_n) = A_{\epsilon(d)}(h_n)$$ and hence $\max_{2 \leq i \leq d} A_{\epsilon(i)}(h_n) = A_{\epsilon(d)}(h_n)$. Lemma \[lemma:max-m\] then implies that $$f(h_n) = f_{\epsilon}(h_n) = 3n-r_d + A_{\epsilon(d)}(h_n).$$ Now choose $h_n$ to be a point of the above form which is connected to $g_n$ by a path of length at most $n-1$. We show that $h_n$ is not a dead end element by exhibiting a generator $s$ so that $f(h_ns) = f(h_n) + 1$. Let $s \in S_{d,q}$ be a generator corresponding to an edge of type ${\textbf e_d}-{\textbf e_1}$ emanating from $h_n$, so that $$\begin{aligned} \Pi(h_ns) &= ( (2n,3n-r_1-1),(3n,4n-r_2),(4n,5n-r_3),(5n,6n-r_4), \cdots \\ & \cdots ,((d-1)n,dn-r_{d-2}),(dn,3n-r_{d-1}),(2n,n-r_d+1) )\end{aligned}$$ As the ordered pairs in the projection are unchanged between $\Pi(h_n)$ and $\Pi(h_ns)$ except in the second coordinate of the first and last ordered pairs, it is still the case that $\max_{2 \leq i \leq d} A_{\epsilon(i)}(h_ns) = A_{\epsilon(d)}(h_ns)$. Note as well that $$\min_{\tau \in \Sigma_d} m_{\tau(1)}(h_ns) + l_{\tau(d)}(h_ns) = 3n-r_d+1 = m_{\epsilon(1)}(h_ns) + l_{\epsilon(d)}(h_ns).$$ The maximum value of $3n-r_d+1$ is $4n$; it may be possible to achieve a value of $4n$ using another permutation in $\Sigma_d$, but if $3n-r_d+1 =4n$, the value of $m_{\tau(1)}(h_ns) + l_{\tau(d)}(h_ns)$ can never be less than $4n$ with any non-identity permutation. Thus we can achieve the minimum value of this quantity using $\epsilon$. Lemma \[lemma:max-m\] now implies that $f(h_ns) = f_{\epsilon}(h_ns) = 3n-r_d+1 + A_{\epsilon(d)}(h_ns) = 3n-r_d+1 +A_{\epsilon(d)}(h_n) = f(h_n) + 1$ and thus $h_n$ is not a dead end element in ${\Gamma_d(q)}$ with respect to the generating set $S_{d,q}$. We now show that there is a geodesic path of length $n$ from $g_n$ which terminates at a dead end element which we denote $g_{n,n}$. Namely, consider any path of length $n$ originating at $g_n$ with the property that the $i$-th point on the path, denoted $g_{n,i}$, has projection $$\Pi(g_{n,i}) = ( (2n,3n-i),(3n,4n),(4n,5n),(5n,6n), \cdots,((d-1)n,dn),(dn,3n),(2n,n+i))$$ for $1 \leq i \leq n$. Letting $r_1 = i$, $r_d = -i$ and $r_j = 0$ for $1 < j < d$, the above argument implies that $$f(g_{n,i}) = f_{\epsilon}(g_{n,i}) = 3n-r_d +A_{\epsilon(d)}(g_{n,i}) = 3n-r_d + A_{\epsilon(d)}(g_{n})=f(g_n) - r_d = f(g_n) + i$$ and hence this path is geodesic. We now show that the endpoint $g_{n,n}$ of this path, which has projection $$\Pi(g_{n,n}) = \left( (2n,2n),(3n,4n),(4n,5n),(5n,6n), \cdots ,((d-1)n,dn),(dn,3n),(2n,2n)\right)$$ is a dead end element in ${\Gamma_d(q)}$ with respect to the generating set $S_{d,q}$. We know that $f(g_{n,n}) = 4n + A_{\epsilon(d)}(g_{n,n})$. Let $s \in S_{d,q}$ be any generator so that $g_{n,n}s \neq g_{n,n-1}$. We must show that $f(g_{n,n}s) \leq f(g_{n,n})$. Since $l_{i}(g_{n,n}) > 0$ for all $i$, there must be indices $j \neq k$ so that 1. $(m_j(g_{n,n}s),l_j(g_{n,n}s)) = (m_j(g_{n,n}),l_j(g_{n,n})+1)$, 2. $(m_k(g_{n,n}s),l_k(g_{n,n}s)) = (m_k(g_{n,n}),l_k(g_{n,n})-1)$, and 3. $(m_r(g_{n,n}s),l_r(g_{n,n}s)) = (m_r(g_{n,n}),l_r(g_{n,n}))$ for $r \neq j,k$. Case 1: $k=d$. Using the identity permutation $\epsilon$, note that $$\min_{\tau \in \Sigma_d} m_{\tau(1)}(g_{n,n}s) + l_{\tau(d)}(g_{n,n}s) = 4n-1 = m_{\epsilon(1)}(g_{n,n}s) + l_{\epsilon(d)}(g_{n,n}s).$$ It may now be the case that $A_{\epsilon(i)}(g_{n,n}s) = A_{\epsilon(i)}(g_{n,n})+1$ for some $i$; however, it is always true that for $2 \leq i \leq d-1$ we have $A_{\epsilon(i)}(g_{n,n}s) \leq A_{\epsilon(i)}(g_{n,n})+1$. Since $A_{\epsilon(d)}(g_{n,n})-A_{\epsilon(i)}(g_{n,n})=n$, we see that for $2 \leq i \leq d-1$ $$A_{\epsilon(i)}(g_{n,n}s) \leq A_{\epsilon(i)}(g_{n,n})+1 \leq A_{\epsilon(i)}(g_{n,n})+n = A_{\epsilon(d)}(g_{n,n}) = A_{\epsilon(d)}(g_{n,n}s).$$ It then follows from Lemma \[lemma:max-m\] that $f(g_{n,n}s) = f_{\epsilon}(g_{n,n}s) = 4n-1 +A_{\epsilon(d)}(g_{n,n}s)$. Since $A_{\epsilon(d)}(g_{n,n}s) =A_{\epsilon(d)}(g_{n,n})$ we see that $f(g_{n,n}s) =f(g_{n,n})-1$. Case 2: $k=1$. Replacing $\epsilon$ with the permutation $\sigma = (1 \ d) \in \Sigma_d$, the argument in Case 1 shows that $f(g_{n,n}s) =f_{\sigma}(g_{n,n}s)=f(g_{n,n})-1$. Case 3: $2 \leq k \leq d-1$ and $j \neq d$. First note that $$\min_{\tau \in \Sigma_d} m_{\tau(1)}(g_{n,n}s) + l_{\tau(d)}(g_{n,n}s) = 4n = m_{\epsilon(1)}(g_{n,n}s) + l_{\epsilon(d)}(g_{n,n}s)$$ and that $A_{\epsilon(d)}(g_{n,n}s) =A_{\epsilon(d)}(g_{n,n}).$ As in the above cases, for $2 \leq i \leq d-1$ we have $A_{\sigma(i)}(g_{n,n}s) \leq A_{\epsilon(i)}(g_{n,n})+1$ and the same reasoning yields $A_{\epsilon(i)}(g_{n,n}s) \leq A_{\epsilon(d)}(g_{n,n}s).$ Together this shows that $f(g_{n,n}s) = f_{\epsilon}(g_{n,n}s) = 4n +A_{\epsilon(d)}(g_{n,n}s)$. Since $A_{\epsilon(d)}(g_{n,n}s) =A_{\epsilon(d)}(g_{n,n})$ we see that $f(g_{n,n}s) =f(g_{n,n})$. Case 4: $2 \leq k \leq d-1$ and $j = d$. Replacing $\epsilon$ with the permutation $\sigma = (1 \ d) \in \Sigma_d$, the argument in Case 3 shows that $f(g_{n,n}s)=f_{\sigma}(g_{n,n}s) =f(g_{n,n})$. Combining the above four cases shows that $f(g_{n,n}s) \leq f(g_{n,n})$ for all $s \in S_{d,q}$ and hence $g_{n,n}$ is a dead end element in ${\Gamma_d(q)}$ with respect to this generating set. Thus, there is a geodesic path of length $n$ from $g_n$ which terminates at a dead end element of ${\Gamma_d(q)}$, and no shorter path from $g_n$ reaches a dead end element. Hence each $g_n$ lies in a distinct cone type, and the theorem follows.

--- author: - 'Arpita Ghosh and Somenath Chakrabarty$^\dagger$' date: 'Received: / Revised version: date' title: A Theoretical Study of the Magnetically Deformed Inner Crust Matter of Magnetars --- [leer.eps]{} gsave 72 31 moveto 72 342 lineto 601 342 lineto 601 31 lineto 72 31 lineto showpage grestore Introduction ============ From the observational evidence of a few strongly magnetized neutron stars, which are supposed to be the sources of anomalous X-rays and soft gamma rays, also called magnetars [@R1; @R2; @R3; @R4], the study of the effect of strong magnetic field on dense neutron star matter, including the crustal matter, both outer crust and inner crust regions of such compact stellar objects have gotten a new dimension. These exotic objects are also called anomalous X-ray pulsars (AXP) and soft gamma repeaters (SGR). The outer crust of a typical neutron star in general, is mainly composed of dense crystalline metallic iron [@hpy]. The density of the upper edge of such metallic crystalline matter is $\sim 10^6-10^7$gm cm$^{-3}$, whereas the bottom edge is $\sim 10^{11}$gm cm$^{-3}$, the matter consists of nuclei (also some highly neutron rich nuclei), surrounded by cylindrically deformed distribution of electron gas, makes the system electrically charge neutral and drifted out free neutrons, if the density of matter is $\geq$ neutron drift value. Since the density is high eneough, it is therefore absolutely impossible to investigate the properties of such matter in material science laboratories, even at zero magnetic field. The observed surface magnetic field of the magnetars is $\sim 10^{15}$G, which is again too high to achieve in the terrestrial laboratories. Also, it is quite possible that the interior field of such exotic objects can go up to $\sim 10^{18}$G (which can be shown theoretically by virial theorem). If the magnetic field at the interior is really so high, then most of the physical and chemical properties of dense neutron matter should change significantly from the conventional neutron star (radio pulsar) scenario (see the recent article by one of the co-authors [@R5] for necessary references). In [@R5], the matter in the outer crust region of strongly magnetized neutron stars have been studied using Thomas-Fermi approximation, in which the WS cells are assumed to be spherical in nature and arranged in a regular manner to form dense crystal of metallic iron ($\sim$ BCC-type). Even though the magnetic field strength at the crustal region is slightly higher than $10^{15}$G, it must change significantly most of the properties of dense matter, both in the outer crust and the inner crust regions of the magnetars [@R5; @R6]. It is believed that strong magnetic field can cause a structural deformation of the metallic atoms (see e.g., [@OST]) present in the inner crust region of a neutron star. The spherically symmetric structure of the atoms are destroyed and become cigar shape with the elongated axis along the direction of strong magnetic field. The atoms may even become almost an one dimensional string like object, i.e., needle shape, if the magnetic field strength is extremely high. In presence of ultra-strong magnetic field of strength $>4.4\times 10^{13}$G, the application of TF model for spherically symmetric WS cells is not a valid approximation [@LIMOD]. However, one can use TF model for sylindrically deformed WS cells because of ultra-strong magnetic field [@OST; @BKS; @XUE1; @XUE2]. In presence of ultra-strong magnetic field ($\gg 4.4\times 10^{13}$G), the WS cells get magnetically deformed and become ellipsoidal in nature. In this article, for the sake of simplicity, we shall assume a cylindrical type deformation of the atoms in the inner crust region and use cylindrical coordinate system with azimuthal symmetry. In reality, to investigate the cigar like deformed atoms in presence of strong magnetic field one has to use prolate spheroidal coordinate system [@RR6]. In future we shall present the problem related to structural deformation of atoms in a strong quantizing magnetic field using such coordinate system [@RR6]. In the present article we shall study the properties of inner crust matter composed of magnetically deformed metallic atoms. In section 2 we have developed the basic formalism and discuss the numerical results, whereas in the last section we have given the conclusions and the future prospect of this work. Basic formalism =============== The width of the outer crust of a typical neutron star is $\sim 0.3$ km, the density of matter, which is assumed to be a dense crystalline structure of metallic iron is $\sim 10^6-10^{11}$gm cm$^{-3}$, ranging from the upper edge to the bottom edge [@SHA] respectively. In one of our previous work appeared i1n [@R5], the properties of outer crust region has been studied. To investigate the properties of such dense exotic crystalline matter of metallic iron, we have replaced the outer crust matter by a regular array of spherically symmetric WS cells, with positively charged nuclei at the centre surrounded by a non-uniform electron gas. Whereas in the inner crust region, of width $\sim 0.5-0.7$km and density $\sim 10^{11}-10^{14}$gm cm$^{-3}$, we assume that since the magnetic field is high enough compared to outer crust region, the electron distribution around each nucleus (iron and also some neutron rich nuclei) gets deformed (the numerical values of widths and density ranges for various regions inside a neutron star strongly depends on the type of equation of state considered). They become cigar shape. In [@R6] we have studied the equation of states of inner crust matter with spherically symmetric electron distribution around each nucleus inside WS cells in presence of strong magnetic field. In this article for the sake of simplicity, we assume cylindrical type distribution of electron gas around each nucleus with the axis of each cylinder is along the direction of magnetic field and further assume azimuthal symmetry for all the cylindrically deformed WS cells. In this article, although we have considered magnetically deformed WS cells for electron distribution, the nuclei at the centre of these cells are assumed to be spherical in nature. We have also assumed that the magnetic field is not so strong to populate proton Landau levels and the magnetic (dipole) energy of neutron sector is negligibly small compared to the kinetic energy of these particles. To investigate various physical properties of deformed electron distribution in the inner crust matter, we start with the Poisson’s equation, given by $$\nabla^2\phi=4\pi en_e \theta(\vec{r}-\vec{r_n})-\frac{4\pi Ze}{V_n}\theta(\vec{r_n}-\vec{r})$$ where $V_n=4\pi r_n^3/3=4\pi r_0^3A/3$, is the nuclear volume, $r_0=1.12$fm and $A$ is the mass number of the nucleus [@XUE1; @XUE2]. In the cylindrical coordonate system, if one assumes the magnetic deformation of atomic nuclei in this region, the $\theta$-function associated with the contribution of protons within the nucleus has to be replaced by $\theta(\vec{r_n}- \vec{r})\theta(z_n-z)$, where in eqn.(1), $Z$ is the atomic number of the nucleus, $r_n$ is the nuclear radius, assumed to be spherical in shape, whereas in the above text, $r_n$ and $z_n$ given in the arguments of the $\theta$-functions are respectively the radial and axial dimensions of the cylidrically deformed nucleus. However, in this article we have not assumed magnetic deformation of the nuclei. Here $\phi$ is the electrostatic field, $e$ is the electron charge and $n_e$ is the electron density, which because of assumed non-uniformity within the WS cells, is a function of positional coordinates $(r,z)$. In our study, we shall consider the variation of $\phi(r,z)$ within the cylindrical distribution of the electrons surrounding the positively charged nucleus. Therefore, in the Poisson’s equation the nuclear part as shown in eqn.(1) will not contribute. Now in the cylindrical coordinate with circular symmetry, the above equation reduces to $$\frac{\partial^2\phi}{\partial r^2}+\frac{1}{r}\frac{\partial\phi}{\partial r}+\frac{\partial^2\phi}{\partial z^2}=4\pi en_e$$ It is well known that in presence of strong quantizing magnetic field, the number density of degenerate electron gas is given by $$n_e=\frac{eB}{2\pi^2}\sum_{\nu=0}^{\nu_{max}}(2-\delta_{\nu 0})p_F$$ where B is the constant external magnetic field, assumed to be acting along z-direction and is $>B_c ^{(e)}$, where $B_c ^{(e)}$ is the typical strength of magnetic field at and above which, in the relativistic scenario the Landau levels for the electrons are populated. For the sake of convenience, throughout this article we shall use natural units, i.e., $\hbar=c=1$. The critical strength is then given by $B_c=m_e ^2/\mid e\mid$ [@R5], where $m_e$ is the electron rest mass and $\mid e\mid$ is the magnitude of electron charge. This critical strength may be obtained by equating the cyclotron quantum with the rest mass energy for electrons. We further assume that the matter is at zero temperature. In eqn.(3), $p_F$ is the electron Fermi momentum, $\nu$ is the Landau quantum number, with $\nu_{max}$, the upper limit of $\nu$. The upper limit will be finite at zero temperature and infinity for finite temperature. The factor $(2-\delta_{\nu 0})$ takes care of singly degenerate $\nu=0$ state and doubly degenerate all other states with $\nu\neq 0$. To study the properties of inner crust matter with deformed WS cells, we make Thomas-Fermi approximation, which is semi-classical version of Hartree approximation [@R7]. In this model, the well known Thomas-Fermi condition is given by $$\mu_e= (p_F ^2+m_e ^2+2\nu eB)^{1/2}-e\phi={\rm{constant}}$$ where $\mu_e$ is the electron chemical potential, which is assumed to be constant throughout the WS cell. Hence one can express the Fermi momentum for electrons in the following form: $$p_F=[(\mu_e+e\phi)^2-m_e ^2-2\nu eB]^{1/2}$$ Since the electrostatic potential $\phi\equiv \phi(r,z)$, the Fermi momentum $p_F$ for the electron is also a function of positional coordinates $(r,z)$ within the cell. In principle one should use this exact expression for electron Fermi momentum in the equation for electron density (eqn.(3)) which in turn appearing on the right hand side of the cylindrical form of Poisson’s equation (eqn.(2)). However, with this exact expression for $p_F$, since the equation becomes non-linear in $\phi(r,z)$, it is absolutely impossible to proceed further analytically, even a single effective step. From the very beginning, therefore, one has to use some numerical technique to solve the Poisson’s equation. Of course, with the numerical method within the limitation of the algorithm followed, more exact results can be obtained. However, in numerical computation of $\phi(r,z)$, we only get a set of numbers, but the beauty of this model will be completely destroyed and a lot of interesting physics associated with the intermediate results of this problem will be totally lost. Therefore, to get an approximate analytical solution for $\phi(r,z)$, as a first approximation, we set the upper limit for Landau quantum number $\nu_{max}=0$ and also neglect the rest mass of electrons, i.e., we put $m_e=0$ (since in the inner crust region the density of electron gas is high enough, the electron Fermi momentum will also be large compared to electron rest mass, therefore we expect that without appreciable error one can neglect electron mass in the above expression) in the expression for Fermi momentum $p_F$ (eqn.(5)). The approximation $\nu_{max}=0$ is actually valid if the magnetic field is extremely high ($\sim 10^{15}$G), perhaps is a valid approximation at the inner crust region for magnetars. However, to investigate some of the properties of dense electron gas within the cylindrically deformed WS cells to somewhat exact manner, later in this article, we shall use this approximate solution for $\phi(r,z)$ only, but do not use the first approximation $\nu_{max}=0$ and $m_e=0$ to evaluate mathematical expressions for various physical quantities of the dense electron gas. However. later in this article we shall show that the upper limit $\nu_{max}$ for the electron Landau quantum number is also a function of $(r,z)$. Therefore to evaluate various physical quantities analytically in this region, first, one has to obtain an approximate solution for the Poisson’s equation (with the values of $\nu_{max}=0$ and $m_e=0$ in the first approximation). To achieve our objectives, we use first the approximate form of electron Fermi momentum obtained from the assumption as mentioned above and its mathematical form is given by $$p_F \approx \mu_e+e\phi$$ Next on substituting $$\mu_e+e\phi(r,z)=\psi(r,z),$$ the cylindrical form of Poisson’s equation reduces to $$\frac{\partial^2\psi}{\partial r^2}+\frac{1}{r}\frac{\partial\psi}{\partial r}+\frac{\partial^2\psi}{\partial z^2}=\lambda^2\psi$$ where $\lambda^2=2e^3 B/\pi$. From eqn.(8), it is quite obvious that under this approximation, the Poisson’s equation reduces to a linear partial differential equation. To solve this equation analytically we use the method of separation of variables, given by $$\psi(r,z)=R(r)L(z)$$ Substituting $\psi(r,z)$ from eqn.(9) in eqn.(8) and introducing a constant $\xi$, we get $$\begin{aligned} &&\frac{d^2 R}{dr^2}+\frac{1}{r}\frac{dR}{dr}+\xi^2 R=0\\ &&\frac{d^2 L}{dz^2}-(\xi^2+\lambda^2)L=0\end{aligned}$$ where $\xi$ is some real constant, independent of r and z but may change with the magnetic field strength and with the mass number and the atomic number of the type of elements present in this region. The solutions of eqns.(10) and (11) are well known. For eqn.(10), the solution is an ordinary Bessel function of order zero with the argument $\xi r$, whereas for eqn.(11), it is an exponentially decaying function of $z$. In the language of mathematics, the solution for $\psi(r,z)$ is then given by $$\psi(r,z)=CJ_0(\xi r)exp\left [\pm(\xi^2+\lambda^2)^{1/2}z\right ]$$ where $+$ and $-$ signs are for $z<0$ or $>0$ respectively. We consider a convenient form of cylindrical coordinate system, such that $z=0$ plane is at the middle of the finite size cylinder. In that case we have to take positive sign for the upper half of the cylinder and negative sign for the lower half. Here $C$ is a constant (again may change with the magnetic field strength and with the nuclear properties of the elements present in the inner crust region) and $J_0(\xi r)$ is the Bessel function of order zero. Now on the nuclear surface, at the centre of the WS cells, $\phi=Z e/r_n $ [@R9], where $Z$ is the atomic number and $r_n=r_0 A^{1/3}$ is the nuclear radius, $r_0=1.12$fm and $A$ is the mass number. We consider eqn.(12) at various point on the nuclear surface. To evaluate the parameters $C$ and $\xi$ numerically, we do the following: put $r=\alpha r_n$ and $z=\beta r_n$ at a particular point on the nuclear surface and also consider $r=\beta r_n$ and $z=\alpha r_n$ for another point, with $\alpha^2+\beta^2=1$. Then we have $$\psi(\alpha r_n,\beta r_n)=CJ_0(\alpha r_n\xi)\exp \left [-(\xi^2+\lambda^2)^{1/2}\beta r_n \right ]=\mu_e+\frac{Ze^2}{r_n}$$ and $$\psi(\beta r_n,\alpha r_n)=CJ_0(\beta r_n\xi)\exp \left [-(\xi^2+\lambda^2)^{1/2}\alpha r_n \right ]=\mu_e+\frac{Ze^2}{r_n}$$ From eqn.(14) we have $$C=\frac{1}{J_0(\beta r_n\xi)\exp \left [-(\xi^2+\lambda^2)^{1/2}\alpha r_n\right ]}\left [\mu_e+\frac{Ze^2}{r_n}\right ]$$ Combining eqns.(13)-(14) we get $$J_0(\alpha r_n\xi)\exp \left [-(\xi^2+\lambda^2)^{1/2}\beta r_n \right ]=J_0(\beta r_n\xi)\exp \left [-(\xi^2+\lambda^2)^{1/2}\alpha r_n \right ]$$ Hence we can write $$\exp\left [-(\xi^2 +\lambda^2)^{1/2}r_n (\beta-\alpha)\right ]=\frac{J_0(\beta r_n\xi)}{J_0(\alpha r_n\xi)}$$ This is a highly transcendental equation for $\xi$. However, it is possible to evaluate $\xi$ numerically from this equation for a given magnetic field strength and for a given type of element, e.g., for metallic iron. To obtain $\xi$ numerically we express eqn.(17) in the following convenient form $$\xi^2+\lambda^2=\frac{1}{r_n^2(\beta-\alpha)^2}\left [\ln \left \{\frac{J_0(\beta r_n\xi)}{J_0(\alpha r_n\xi)}\right\} \right ]^2$$ To evaluate numerical values for $\xi$, we now put (a) $r$ and $z$ values of the equator and poles of the nucleus, (b) $r=r_n/4$, (c) $r=r_n/2$ and (d) $r=3r_n/4$, for four different positions on the nuclear surface and obtain four different sets of $\xi$ as a function of magnetic field strength. For all these cases the $z$-coordinates are obtained from the relation $z=(r_n^2-r^2)^{1/2}$. Because of the symmetry about $z=0$ plane, the choice of negative values for $\alpha$ and $\beta$ will give identical results. In fig.(1) we have plotted $\xi$ (in MeV) as a function of magnetic field strength $B$, expressed in terms of critical magnetic field strength $B_c^{(e)}$, for the specific values of $r$ and $z$ as indicated above by (a), (b), (c) and (d). For all these cases, the variations are insensitive for the low and moderate values of magnetic field strengths. This figure shows that beyond field strength $10^{17}$G, $\xi$ increases sharply for all the cases. It is also obvious from fig.(1) that the parameter $\xi$ not only varies with the strength of magnetic field but also more strongly depends on the positional coordinates on the nuclear surface, giving one of the boundary conditionss. In fig.(2) we have plotted the same kind of variations for the parameter $C$ with the magnetic field strength, expressed in the same unit as in fig.(1) and also considering the same positions on the nuclear surface as considered for fig.(1). For low and moderate magnetic field values it is also almost constant, then it falls abruptly beyond $10^{17}$G and finally saturates to some constant values. Since beyond $10^{17}$G, electrons within the cells occupy their zeroth or very low lying Landau levels, the quantum mechanical effect of magnetic field becomes extremely important in this region and as a consequence both $\xi$ and $C$ change significantly beyond this magnetic field value. Since the magnetic field is not ultra strong to distort atomic nuclei, we have therefore considered spherical nuclei of radii $r_0A^{1/3}$, at the $z$ and $r$ symmetric position inside the cylinder. From the solutions of eqns.(13) and (14), we have noticed that the functional form of the solution given by eqn.(12) does not change, but the unknown parameters $C$ and $\xi$ can have a large number of roots. As a result the final solution of Poisson’s equation becomes degenerate (which has been shown in figs.(1) and (2)). As an hypothetical case, instead of spherically symmetric nuclei, we have replaced them by cylindrically deformed nuclei with axially symmetric nucleon distribution. The smaller cylinder (deformed nucleus) is coaxial with the bigger one and with identical $z=0$ plane. For the surface potential we have used eqn.(52) of Appendix A. In eqn.(52) we have replaced $r_{max}$ and $z_{max}$ by $r_n$ and $z_n$, the maximum values of $r$ and $z$ for the deformed nuclei. Here $r_1$ and $z_1$ are the surface values of $r$ and $z$ coordinates at any arbitrary point on the nuclear surface. To make the nucleon distributions within the nuclei axially symmetric, we put $\theta_1=0$. Of course, in such a geometrical configuration one of the geometrical parameter, e.g., either $r_n$ or $z_n$ has to be chosen arbitrarily, and the other one can be expressed in terms of the known one. Further, to choose one of the parameters, we assume that the nucleus is incompressible, as a consequence, the density will not change even if the geometrical configuration has changed. Then we have the simple relation $r_n^2z_n=3A/4r_0^3$. In this expression, if we choose one of the unknown arbitrarily (with the numerical value very close to the nuclear radius), the other can also be known. However, we have noticed that in this case the degree of degeneracy is even more than the spherically symmetric case. Along the curved surface for the nuclei, for $r_1=r_n$, we can have along the positive direction of $z$-axis, any value of $z$ from $0$ to $z_n$. Whereas on the plane faces, we have $z=z_n$ and $r_1$ can have any value from $0$ to $r_n$. All these points are on the cylindrically deformed nuclear surface and the potential on the surface is given by eqn.(52). Hence we may conclude that a non-degenerate solution can only be obtained, if and only if the electron distribution is spherically symmetric and the nucleus is also spherical with a common centre (concentric spheres). Which we have studied previously [@R5]. However, we expect that if both the electron distribution and the shape of the nucleus at the centre are ellipsoidal in nature with the major axis (which is common for both the ellipsoids) along the magnetic field and two other axes (again common for both of them) are symmetric, the problem of degenerate solutions for electric potential inside WS cells will be removed. At present we are persuing this analysis. We believe that with this type of geometrical configuration, the electric field at the surface of the WS cell will also vanish, which is non-zero in the cylindrical case as discussed below.. Although the WS cells are overall charge neutral, at any point $(r_{max},z)$ on the curved surface and at any point $(r,z_{max})$ on the plane faces the potential $\psi(r,z)$ can not be a constant, here $r_{max}$ and $z_{max}$ are transverse and half longitudinal dimensions. Which further means that the electric field at the surface (both radial and longitudinal components) can not be zero. This is a purely geometrical effect and such a deformed charge distribution exhibit quadrupole moment. The modified form of electro-static potential at any point on the curved face of the cylindrically deformed WS cell is given by $$\psi(s,\theta)=CJ_0(\xi r_{max})\exp({-\Lambda s \sin \theta}),$$ whereas on the plane faces, this potential is given by $$\psi(s,\theta)=CJ_0(\xi s \cos \theta)\exp({-\Lambda s \sin \theta})$$ where $s=(r_{max}^2+z^2)^{1/2}$ for the curved face and $=(r^2+z_{max}^2)^{1/2}$ for the plane faces, and $\Lambda=(\lambda^2+\xi^2)^{1/2}$ is a constant. Here the variable $\theta$ is introduced to obtain the variation of $z$ on the curved surface and also $r$ on the plane faces. Therefore this $\theta$ variable is not the conventional $\theta$ coordinate used in cylindrical system. From eqns.(18) and (19), it is obvious that the potential can not be constant on the surfaces. The corresponding electric field on the curved surface is given by $$\begin{aligned} \vec E&=&\frac{\partial \psi}{\partial s}\hat{e_s}+\frac{1}{s}\frac{\partial \psi}{\partial \theta}\hat{e_\theta}\\ &=&-2C\exp({-\Lambda (s^2-r_{max}^2)^{1/2}})(J_0(\xi r_{max})\Lambda \hat{e_z}+J_1(\xi r_{max})\xi \hat{e_r})\end{aligned}$$ Similarly the electric field at the plane faces is given by $$\begin{aligned} \vec E&=&\frac{\partial \psi}{\partial s}\hat{e_s}+\frac{1}{s}\frac{\partial \psi}{\partial \theta}\hat{e_\theta}\\ &=&-2C\exp({-\Lambda z_{max}})(J_0(\xi (s^2-z_{max}^2)^{1/2})\Lambda \hat{e_z}+J_1(\xi (s^2-z_{max}^2)^{1/2})\xi \hat{e_r})\end{aligned}$$ Which are obviously non vanishing on the surface of WS cells. However, neighboring cylindrical WS cells will interact electro-magnetically because of the non-zero values of electrostatic fields at the surfaces. Due to some kind of electro-magnetic induction there will be charge polarization on the surfaces of cylindrically deformed WS cells. As a result a number of charged (induced by the nearest neighbor WS cells) WS cells will form a bundle of charge neutral WS cylinders, instead of a single cylindrical cell [@RU1]. Again we expect that with ellipsoidal coordinate system, the non-zero electric field problem at the WS cell surface will also be solved. Now from the Thomas-Fermi condition, we have $$p_F=[\psi^2(r,z)-m_{\nu}^2]^{1/2}$$ where $m_{\nu}=(m_e^2+2\nu eB)^{1/2}$. With this exact expression for $p_F$, the number density for electron gas can be expressed as $$n_e(r,z)=\frac{eB}{2\pi^2}\sum_{\nu=0}^{\nu_{max}}(2-\delta_{\nu 0})[\psi^2(r,z)-m_{\nu}^2]^{1/2}$$ This is obviously more exact than eqn.(3), where the value of Fermi momentum $p_F$ is taken from eqn.(6). Further, this expression shows that the electron density is a function of both $r$ and $z$ within the WS cells. Which actually justifies the assumption that the electron distribution inside each WS cell around the fixed nucleus is non-uniform. Now from the non-negative nature of $p_F ^2$, we have $$\nu_{max}=\frac{\psi^2(r,z)-m_e ^2}{2eB}=\nu_{max}(r,z)$$ The upper limit of Landau quantum number $\nu_{max}$ will therefore also depend on the positional coordinates $r$ and $z$ within and on the curved surface and the plane faces of the cylinder. In our model, the upper limit of Landau quantum number is zero at the deformed WS cell surface. Had the electron distribution at the crustal region been homogeneous, like electron gas in a highly conducting metal, with a background of positively charged nuclei at rest, the value of $\nu_{max}$ would have remain same at all points and for all the electrons depending on the density of electron gas and the strength of magnetic field. However, in our model, the electrons are distributed in an axially symmetric cylinder around a symmetrically placed positively charged nuclei of spherical in shape. The electrostatic potential changes with $r$ and $z$ accordingly following eqns.(9) and (12). From the nature of such variation, we found that $\nu_{max}$ is maximum near the nuclear surface and vanishes at the WS surfaces (both curved surface and the plane faces). Therefore replacing both $r$ and $z$ by their maximum values $r_{max}$ and $z_{max}$ respectively, we have $$\nu_{max}(r_{max},z_{max})=0$$ Further, from the overall charge neutrality of the WS cells we can write $$Z=2\pi e \int_{r_n}^{r_{max}} \int_{r_n}^{z_{max}} n_e(r,z)r dr dz=~~{\rm{constant}}$$ These two equations (eqns.(28) and (28)) are solved numerically to obtain $r_{max}$ and $z_{max}$ for various values of magnetic field strength for the inner crust matter with metallic iron only. Knowing both $r_{max}$ and $z_{max}$, if we solve the equations $\nu_{max}(r,z_{max})=0$ and $\nu_{max}(r_{max},z)=0$ numerically for $r$ (ranges from $0$ to $r_{max}$) and $z$ (ranges from $0$ to $z_{max}$), which are on the surface of the cylinder, one can generate the whole surface of the cylinder, including the plane faces. In fig.(3) we have plotted electron number density obtained from eqn.(26), in terms of normal nuclear density multiplied by $10^4$, as a function of radial distance from nuclear surface to the WS cell boundary for the magnetic field strengths $10^1$, $5\times 10^2$, $5\times 10^3$, $10^4$ and $5\times 10^4$ times $B_c^{(e)}$, indicated by the curves $a$, $b$, $c$, $d$ and $e$ respectively (In all the plots, we have used the values of $C$ and $\xi$ for the boundary condition on the equator and poles of the spherically symmetric nucleus. Although theoretically speaking the electrostatic potential inside the WS cells are degenerate with respect to the boundary condition on the nuclear surface, in practice, we have noticed from the numerical calculation that the variation with other points as the boundary is not so appreciable). These curves show that the electron number density is maximum near the nuclear surface and minimum near the WS cell boundary $r_{max}$. This figure also shows that electron number density increases with the strength of magnetic field. In fig.(4) we have plotted the same quantity as in fig.(3) but against the axial distance $z$ from the nuclear surface to the WS cell boundary (plane faces) indicated by $z_{max}$. In this case also the variations are exactly same, qualitatively and quantitatively, as in fig.(3). The little qualitative difference is because of different types of functional dependencies. In fig.(5) we have plotted the upper limit of Landau quantum number $\nu_{max}$ as a function of radial coordinate from the nuclear surface to the WS cell boundary. In this figure the upper curve is for $10\times B_c^{(e)}$, the middle one is for $10^2\times B_c^{(e)}$ and the lower one is for $500\times B_c^{(e)}$. The value of $\nu_{max}$ decreases with the increase in magnetic field strengths. It has been observed that beyond $500 \times B_c^{(e)}$, $\nu_{max}$ becomes identically zero throughout the WS cell. Further, the value of $\nu_{max}$ for $B\leq 500 \times B_c^{(e)}$ is largest near the nuclear surface and exactly zero near the cell boundary. In fig.(6) we have plotted the same kind of variation of $\nu_{max}$, but against the axial distance from nuclear surface to the cell boundary. The qualitative and the quantitative variations are exactly identical with fig.(5). These two figures show that the electrons are completely spin polarized in the direction opposite to the external magnetic field $\vec B$ at the cell boundary, including the plane faces, even for $B\leq 500 \times B_c^{(e)}$, but beyond this value they are polarized at all the points within the cells. Although the variations along radial and axial directions are shown in these two figures, we expect that such polarized picture of electron gas exists throughout the cylindrically deformed WS cell surface, including the two plane faces. To explain the phenomenon of electron spin polarization within the WS cells, one has to solve Dirac equation in presence of strong magnetic field. It is then trivial to show that the Landau quantum number $\nu$ is given by: $2\nu =2n+1+\alpha$, where $n=0,1,2,.....$ is the principal quantum number and $\alpha=\pm 1$ are the eigen-values of spin operator $\sigma_z$ corresponding to up and down spin states of electrons respectively. Hence it is quite obvious that all the Landau levels with $\nu \neq 0$ are doubly degenerate (with two possible combinations of $n$ and $\alpha$), whereas zeroth order Landau level is singly degenerate with only possible combination is $\nu=0$ is $n=0$ and $\alpha=-1$. Which actually mean that in the zeroth Landau level spins of all the electrons are aligned in the direction opposite to the external magnetic field. This is a kind of electron spin polarization under the influence of ultra-strong magnetic field, occurring at the surface region of WS cells, and also throughout the cells beyond some magnetic field strength. This type of spin polarization will not be observed if electrons satisfy schr${\ddot{\rm{o}}}$dinger equation, even if we introduce electron spin by hand. Therefore, the spin polarization is a purely relativistic effect. In fig.(7) we have shown the variation of $z_{max}$ with the strength of magnetic field $B$. This figure shows that the variation is almost insensitive for low and moderate magnetic field strengths but decreases almost abruptly beyond $10^{16}$G, when most of the electrons occupy their zeroth Landau level, and finally tends to saturate to a constant value $\sim 10$fm. The variation of $r_{max}$ with the strength of magnetic field is shown in fig.(8). The nature of variation is more or less same as that of $z_{max}$. However, we have noticed that for extremely large field strength, $r_{max}\longrightarrow 0$, instead of saturation. This is a remarkable difference from its longitudinal counter part. It actually shows that in presence of extremely strong magnetic field the cylindrically deformed WS cells become more and more thin in the transverse direction. We therefore conclude that with the increase in magnetic field strength the radial contraction will be enormous compared to the axial one. From figs.(7)-(8) we have noticed that the variations are most significant beyond $B=10^{16}$G. The reason is again because the electrons occupy only their zeroth Landau level in presence of such strong magnetic field, at which the quantum mechanical effect of the magnetic field dominates. In these two figures we have taken $Z=26$ and $A=56$, which are the atomic number and mass number respectively for the deformed iron atoms. Next we calculate the different kinds of energies associated with the electron gas within WS cells. The cell averaged kinetic energy density of the electron gas is given by $$\begin{aligned} \epsilon_k=\frac{2}{V}\frac{eB}{2\pi^2}\int d^3r&&\sum_{\nu=0}^{\nu_{max}} (2-\delta_{\nu 0}) \nonumber \\ && \int_0^{p_F(r,z)} dp_z [(p_z^2+m_\nu^2)^{1/2}-m_e]\end{aligned}$$ where in the cylindrical coordinate system with azimuthal symmetry $d^3r=2\pi rdr dz$, with the limits $r_n \leq r \leq r_{max}$ and $r_n \leq z \leq z_{max}$ and $V=\pi r_{max}^2 2z_{max}$, the volume of each cell. The $p_z$ integral is trivial, which gives an analytical expression for the local kinetic energy density i.e., at a particular point $(r,z)$ within the WS cell. The factor $2$ in the expression for $V$ is for $z$-symmetry about $z=0$ plane. In fig.(9) we have plotted the variation of local kinetic energy density as a function of radial distance $r$ from the nuclear surface to the cell boundary for various magnetic field values, keeping $z=0$. This figure shows that the kinetic energy density increases with the increase in magnetic field strength. We have indicated the curves by $a$, $b$, $c$, $d$ and $e$ for the magnetic field strengths $10B_c^{(e)}$, $5\times 10^2B_c^{(e)}$, $5\times 10^3B_c^{(e)}$, $10^4B_c^{(e)}$ and $5\times 10^4B_c^{(e)}$ respectively. In fig.(10) we have shown the same kind of variations along axial distance from the nuclear surface to one of the plane faces of the WS cell. The variation with magnetic field strength is again almost identical with fig.(9). Similar to the variation of electron number density within the cell (fig.(3) and fig.(4)), both from fig.(9) and fig.(10) it may be concluded that the local kinetic energy density for electron gas is maximum near the nuclear surface and minimum at the cell boundary. Similarly, the cell averaged electron-nucleus interaction energy per unit volume is given by $$E_{en}=-\frac{2}{V} Ze^2 \int d^3r \frac{n_e(r,z)}{(r^2+z^2)^{1/2} }$$ Analogous to the local kinetic energy density, here also one can obtain the local interaction energy per unit volume, $E_{en}(r,z)$ at a particular point within the cell. In fig.(11), the variation of the magnitude of electron-nucleus interaction energy per unit volume with the radial distance is plotted. In this figure we have indicated the curves by $a$, $b$, $c$, $d$ and $e$ for the magnetic field strengths $10B_c^{(e)}$, $5\times 10^2B_c^{(e)}$, $5\times 10^3B_c^{(e)}$, $10^4B_c^{(e)}$ and $5\times 10^4B_c^{(e)}$ respectively. This figure shows that the magnitude of electron-nucleus local interaction energy increases with the increase in magnetic field strength. Which means that with the increase in magnetic field strength the electrons become more strongly bound by the Lorentz force of the form $e\vec v\times\vec B$. In fig.(12) we have shown the same kind of variation along z-axis. Both the qualitative and the quantitative nature of variations with the strength of magnetic field are same as that of fig.(11). It is also obvious from figs.(9)-(12) that for a given magnetic field strength the kinetic energy density and the magnitude of electron-nucleus interaction energy per unit volume at a particular point, either along axial direction or in the radial direction, inside WS cells are of the same order of magnitude. We do expect that the same type of variations will be obtained at all the points inside the WS cells. In fig.(13) we have shown the variation of cell averaged kinetic energy of electron gas (solid curve) and the magnitude of cell averaged electron-nucleus interaction energy (dashed curve) with the magnetic field strength. This figure shows that both the quantities are increasing function of magnetic field strength. However, for low and moderate field values, the variations are not very much sensitive, particularly for the kinetic energy part of electron gas. Next we consider the cell averaged electron-electron direct interaction energy density, given by $$E_{ee}^{dir}=\frac{1}{V}e^2\int d^3r n_e(r,z)\int d^3r^\prime n_e(r^\prime,z^\prime) \frac{1}{[(\vec r -\vec r^\prime)^2 +(z-z^\prime)^2]^{1/2}}$$ Assuming $\vec r$ as the reference axis, we have $(\vec r-\vec r^\prime)^2 =r^2+{r^\prime}^2 -2rr^\prime\cos \theta$, where $\theta$ is the angle between $\vec r$ and $\vec r^\prime$, assumed to be on the same plane. Then $d^3r^\prime=r^\prime dr^\prime d\theta dz^\prime$, with $0\leq \theta \leq 2\pi$. In this case the $z$ integral has to be broken into two parts, one with limit $-z_{max} \leq z \leq \overline z$ and the other one with the limit $\overline z \leq z \leq +z_{max}$. The value of $\overline z$ is not easy to evaluate in the region between $z$-axis and $r$-axis. With $\theta$ symmetry, for the sake of simplicity we put $\overline z=r_n$ and expect that the error will be nominal. Now to obtain electron-electron direct interaction energy one has to evaluate the five dimensional integral as shown in eqn.(32). None of them can be obtained analytically, hence it is necessary to follow some numerical methods. Even the $\theta$ integral can not be obtained analytically. One can express the $\theta$ integral in the form of an elliptical integral of first kind, given by $$I_{\cal{EL}}(r,r^\prime,z,z^\prime)=\int_0^{\pi/2} d\theta \frac{1}{(1-K\cos^2\theta)^{1/2}}$$ where $K=4rr^\prime/[(r+r^\prime)^2+(z-z^\prime)^2]$. The direct part is then given by $$E_{ee}^{dir}=\frac{4}{2V}e^2\int d^3r n_e(r,z)\int r^\prime dr^\prime dz n_e(r^\prime,z^\prime) \frac{1}{[(r + r^\prime)^2 +(z-z^\prime)^2]^{1/2}} I_{\cal{EL}}(r,r^\prime,z,z^\prime)$$ where the factor $4$ is coming from the angular integral over $\theta$ from $0$ to $2\pi$. Let us now consider the elliptic integral (eqn.(33)) on $z=0$-plane. In this case both $z$ and $z^\prime$ are zero and the factor $K=1$. Then it can be shown very easily that the integral given by eqn.(33) will diverge at the lower limit. To avoid this unphysical infinity we put a lower cut-off (infrared cut-off) $\delta$, which will now the lower limit for $\theta$. The physical meaning of non-zero lower limit for the $\theta$-integral is that the two electrons under consideration can not be at zero distance from each other on the arc of a circle whose centre is same as that of the nucleus. The infrared cut-off $\delta$ which is a measure of minimum angular distance between two neighboring electrons must necessarily depends on the minimum possible linear distance between them and also on the average radial distance from the centre of the nucleus. From a very elementary geometrical construction it can be shown that $$\delta=\frac{s}{r}$$ where $s$ is the arc length, or the distance between two neighboring electrons on the circular arc. Since $s$ is infinitesimal in nature, we can approximate it by a straight line of length $s$, which is the length of the cord connecting two points occupied by two neighboring electrons. Since $s$ is the minimum possible linear distance between two electrons, we can express it in terms of electron density near those points, given by $$s\sim n_e^{-1/3}$$ The cell averaged electron-electron direct energy has been obtained by evaluating the multi-dimensional integrals numerically. The electron-electron exchange energy corresponding to the $i$th electron in the cell is given by $$E_{ee}^{(ex)}= -\frac{e^2}{2}\sum_j \int d^3r d^3r^\prime \frac{1}{ [(\vec r-\vec {r^\prime})^2+(z-z^\prime)^2]^{1/2}} \bar\psi_i(\vec r,z)\bar\psi_j(\vec {r^\prime},z^\prime) \psi_j(\vec r,z)\psi_i(\vec {r^\prime},z^\prime)$$ where $\psi_i(r,z)$ is the spinor wave function of Dirac equation in cylindrical coordinate in presence of strong quantizing magnetic field, and $\bar \psi({\vec r},z)= \psi^\dagger({\vec r},z)\gamma_0$, the adjoint of the spinor and $\gamma_0$ is the zeroth part of the Dirac gamma matrices $\gamma_\mu$ in cylindrical coordinate system. We have evaluated the cell averaged exchange energy using Dirac spinors in cylindrical coordinate. An elaborate discussion is given in the Appendix. The kinetic pressure of non-uniform electron gas within the WS cell is given by $$P(r,z)=\frac{eB}{2\pi^2}\sum_{\nu=0}^{\nu_{max}(r,z)}(2-\delta_{\nu 0})\int_0^{p_F(r,z)} \frac{p_z^2dp_z}{(p_z^2+m_\nu^2)^{1/2}}$$ The $p_z$ integral is very easy to evaluate analytically and is given by $$\begin{aligned} P(r,z)&=&\frac{eB}{4\pi^2}\sum_{\nu=0}^{\nu_{max}(r.z)}\big [p_F(p_F^2+m_\nu^2)^{1/2}\nonumber \\ &-& m_\nu^2 \ln\left \{ \frac{p_F+(p_F^2+m\nu^2)^{1/2}}{m_\nu}\right \}\big ]\end{aligned}$$ which gives the local pressure at $(r,z)$. This equation shows that the electron kinetic pressure also changes from point to point within the WS cells. In fig.(14) we have shown the variation of kinetic pressure for the non-uniform electron gas with the radial distance within the cell. Curves $a$ and $b$ are for $B=10\times B_c^{(e)}$ and $B=500\times B_c^{(e)}$, whereas upper curve and lower curve as indicated by $c$ (almost identical) are for $B=5000\times B_c^{(e)}$ and $B=10000\times B_c^{(e)}$ respectively. In fig.(15) the same kind of variations are shown along z-axis. In this figure the curves for $B=5000\times B_c^{(e)}$ and $B=10000\times B_c^{(e)}$ are almost identical and indicated by single thick curve $c$. For $B=10\times B_c^{(e)}$ and $B=500\times B_c^{(e)}$ the curves are indicated by $a$ and $b$ respectively. These two figures show that the kinetic pressure is maximum near the nuclear surface and zero at the cell boundary. The variation with magnetic field strength shows that the non-uniform electron gas becomes softer for higher magnetic field within the cells. In fig.(16) we have shown the variation of cell averaged electron-electron direct interaction energy (solid curve) and the corresponding kinetic pressure (dashed curve) with the magnetic field strength. Both the quantities are monotonically increasing function of magnetic field strength. Since in TOV equation for neutron stars, in the equation of state the cell averaged electron gas kinetic pressure is used, hence we can conclude that since electron gas becomes harder in presence of strong magnetic field in this particular region, the width of inner crust region will increase with the increase in magnetic field strength. In fig.(17) we have shown the variation of cell averaged electron-electron exchange interaction energy, assuming the extreme case when electrons occupy only their zeroth Landau level (dashed curve) and the most general one, when electrons can have all possible Landau levels (solid curve). The figure shows that for $\nu\neq 0$, the exchange energy is oscillatory for low and moderate values of magnetic field strength. This oscillatory phenomenon is identical with the observed De Haas-van Alphen oscillation observed in Landau diamagnetism. The exchange energy becomes extremely small when the electron Fermi momentum suddenly becomes zero for some value of magnetic field strength and then rises sharply upto a certain magnitude of magnetic field strength and finally decreases to zero, when the magnetic energy dominates over the matter part. In the case of $\nu=0$, the nature of variation is almost identical, except the oscillatory nature at low and moderate magnetic field region. For the sake of completeness, we have obtained the classical form of electron-electron Coulomb potential energy and the corresponding electron-nucleus interaction energy within a WS cell. The electron-electron interaction energy is given by (an elaborate discussion is given in the Appendix) $$\begin{aligned} E_{ee}&=&\frac{Z^2 e^2}{r_{max}^4 z_{max}^2}\int_0^\infty \frac{dk}{k^5} [r_{max}J_1(k r_{max}) - r_nJ_1(k r_n)]^2 \nonumber \\ && [k(z_{max}-r_n) - (1-\exp \{-k(z_{max} - r_n)\})]\end{aligned}$$ and the corresponding electron-nucleus interaction part is given by $$E_{en}=-\frac{Z^2 e^2}{r_{max}^2 z_{max}}I$$ where $$\begin{aligned} I&=&\frac{1}{2}\left[(z_{max}^2 + r_{max}^2)^{1/2}z_{max} + r_{max}^2 \log \left(\frac{(z_{max}^2 + r_{max}^2)^{1/2} + z_{max}}{r_{max}}\right)\right] \nonumber \\ &-&\frac{1}{2}\left[(r_n^2 + r_{max}^2)^{1/2}r_n + r_{max}^2 \log \left(\frac{(r_n^2 + r_{max}^2)^{1/2} + r_n}{r_{max}}\right)\right] \nonumber \\ &-&\frac{1}{2}\left[(z_{max}^2 + r_n^2)^{1/2}z_{max} + r_n^2 \log \left(\frac{(z_{max}^2 + r_n^2)^{1/2} + z_{max}}{r_n}\right)\right] \nonumber \\ &+&\frac{1}{2}[2^{1/2}r_n^2 + r_n^2 \log (2^{1/2}+1)] \nonumber \\ \end{aligned}$$ The detail derivations for electron-nucleus Coulomb interaction energy is also given in the Appendix. In fig.(18) we have shown the variation of electron-electron Coulomb energy density (solid curve) and the corresponding magnitude of electron-nucleus classical interaction energy density (dashed curve) with the strength of magnetic field. Both the curves show that the classical interaction energies are also insensitive to the strength of magnetic field in the low and moderate regions, and both of them are affected significantly beyond $10^{16}$G. The reason is again that the electrons occupy only their zero-th Landau level. However, unlike the quantum mechanical cases, here at high magnetic field region both of them become extremely small. Moreover the overall magnitude does not change by an order of magnitude within the range of magnetic field considered. Finally to show that the cylindrically deformed WS cell structure of the inner crust matter of strongly magnetized neutron star (magnetar) is energetically favorable over the spherical structure, in fig.(19) we have compared the energy per electron for these two possible type of WS cell structures for various values of magnetic field strength. In this figure solid curve indicated by sph is for spherical cell structure and the dashed one indicated by cyl is for the cylindrically deformed WS cells. The energy per electron plotted along left y-axis is for the spherical case whereas for the cylindrical case the same quantity is plotted along right y-axis. This figure shows that for cylindrical case the total energy per electron is about one order of magnitude less than that of spherical case. Further beyond $B\sim 10^{17}$G, energy per electron for the cylindrical case becomes several orders of magnitude less than the spherical case (see also [@RU2]). Conclusions =========== In this article we have investigated various physical properties of non-uniform electron gas assuming cylindrically deformed atoms of metals, in particular the metallic iron at the inner crust of a strongly magnetized neutron star. Because of extremely strong surface magnetic field of magnetars, we have assumed a cylindrical type deformation of the atoms, which are subsequently replaced by WS cells with the same kind of geometrical structure. The longitudinal axis of all the cylinders are along the direction of magnetic lines of forces. The curved surfaces of these cylinders are therefore parallel to the boundary surface of the neutron stars in the region far away from the magnetic poles, where the magnetic lines of forces emerge almost perpendicularly with the surface (polar cap). We have studied various physical quantities for the dense electron gas in the inner crust region. We have investigated the variations with magnetic field strength for cell averaged quantities and also the spatial variations for constant magnetic field. We have noticed that the transverse dimension of a cylindrically deformed WS cell becomes extremely thin in presence of ultra-strong magnetic field. In this work we have also compared the total energy per electron with the spherical case and found that the cylindrically deformed WS structure in the inner crust region is energetically more favorable over the spherical case in presence of ultra-strong magnetic field. Although we have considered cylindrical type deformation for the iron atoms in this region, it is expected that the atoms become cigar shape in presence of strong magnetic field. The present investigation, assuming cylindrically deformed atoms is therefore an approximate model calculation. In our future study, the properties of neutron star inner crust matter with cigar shape atoms in the metallic crystal in presence of strong magnetic field will be investigated. Appendix ======== A. Coulomb energy of electron gas: 1\. Electron-nucleus Coulomb interaction energy: $$E_{en}=-Ze \int_v \frac{dq}{s}$$ Where the integral is over the whole volume and $s=(r^2+z^2)^{1/2}$. The charge element within an elementary volume $dv$ is given by $$dq=Ze \frac{dv}{v}=Ze \frac{r dr dz}{2r_{max}^2 z_{max}}$$ Hence $$E_{en}=-\frac{Z^2 e^2}{2 r_{max}^2 z_{max}} \int_{r_n}^{r_{max}} r dr \int_{r_n}^{z_{max}} \frac{1}{(r^2+z^2)^{1/2}}$$ The double integral can very easily be evaluated and finally we get eqn.(40) as given in the text. 2\. Electron-electron Coulomb interaction energy: $$E_{ee}= \frac{1}{2} \int \int \frac{dq_1 dq_2}{s}$$ Where $s=[(z_1-z_2)^2 + (\vec r_1-\vec r_2)^2]^{1/2}$. In this case the charge elements $dq_1$ and $dq_2$ are at $(r_1,\theta_1,z_1)$ and $(r_2,\theta_2,z_2)$ respectively. The factor $1/2$ is to take care of double counting. From a very simple geometrical construction it can very easily be shown that $$s=\left[(z_1-z_2)^2 + {(r_1^2+r_2^2-2r_1r_2 \cos (\theta_1-\theta_2))}\right]^{1/2}$$ Substituting for $dq_1$ and $dq_2$, with the definition as given in eqn.(42), we have $$\begin{aligned} E_{ee}&=&\frac{1}{2} \frac{Z^2 e^2}{4\pi ^2 r_{max}^4z_{max}^2} \int_{r_n}^{r_{max}}r_1 dr_1 \int_{r_n}^{r_{max}}r_2 dr_2 \int_{r_n}^{z_{max}}dz_1 \int_{r_n}^{z_{max}}dz_2 \int_0^{2\pi} d\theta_1 \int_0^{2\pi} d\theta_2 \nonumber \\ && \frac{1}{\left[(z_1-z_2)^2 + {(r_1^2+r_2^2-2r_1r_2 \cos (\theta_1-\theta_2))}\right]^{1/2}}\end{aligned}$$ To evaluate the integrals, we use the identity $$\begin{aligned} \frac{1}{s}&=&\frac{1}{\left[(z_1-z_2)^2 + {(r_1^2+r_2^2-2r_1r_2 \cos (\theta_1-\theta_2))}\right]^{1/2}}\nonumber \\ &=&\frac{1}{\pi} \sum_{m=-\infty} ^{+\infty} \exp \{im(\theta_1-\theta_2)\} \int_0^\infty J_m(kr_1)J_m(kr_2) \exp \{-k(z_>-z_<)\} dk\end{aligned}$$ To obtain electron-electron Coulomb energy, we first evaluate the potential at $(r_1,\theta_1,z_1)$ due to a charge element $dq (r_2,\theta_2,z_2)$. This is given by $$\phi(r_1,\theta_1,z_1)= \int \frac{dq(r_2,\theta_2,z_2)}{s}$$ Using the identity as defined above (eqn.(48)) and integrating over $z_2$, within the range $r_n\leq z_2\leq z_{max}$, we get $$\begin{aligned} \phi(r_1,\theta_1,z_1)&=&\frac{Ze}{2\pi r_{max}^2 z_{max}} \sum_{m=-\infty}^{+\infty} \int_0^{+\infty} \frac{dk}{k} \int_{r_n}^{r_{max}}r_2 dr_2 \int_0^{2\pi} d\theta_2 J_m(kr_1)J_m(kr_2) \nonumber \\ && (2 - \exp \{-k(z_1-r_n)\} - \exp\{-k(z_{max}-z_1)\}) \exp \{im(\theta_1-\theta_2)\}\end{aligned}$$ Integral over $\theta_2$ gives $2\pi\delta_{m0}$, which means only $m=0$ term of the series in the above identity (eqn.(48)). Hence, we have $$\begin{aligned} \phi(r_1,\theta_1,z_1)&=& \frac{Ze}{r_{max}^2 z_{max}} \int_0^\infty \frac{dk}{k} \int_{r_n}^{r_{max}} r_2 dr_2 J_0(kr_1) J_0(kr_2)\nonumber \\ && \left(2-\exp \{-k(z_1-r_n)\}-\exp\{-k(z_{max}-z_1)\}\right)\end{aligned}$$ Further, to evaluate the $r_2$ integral within the range $r_n\leq r_2\leq r_{max}$, we use the relation $$\begin{aligned} \int_0^x r dr J_m(r)=\frac{x \Gamma (\frac{m+2}{2})}{\Gamma(m/2)} \sum_{l=0}^\infty \frac{(\mid m\mid +2l+1)\Gamma(\frac{m}{2}+l)}{\Gamma(\frac{m+4}{2}+l)} J_{\mid m\mid +2l+1}(x)\end{aligned}$$ Since $m=0$, we have in the denominator of the above expression $\Gamma(m/2)=\infty$. Whereas the numerator of the first term of $l$ series in the above relation (i.e., for $l=0$ term) also contains a $\Gamma(m/2)=\infty$ term. These two terms will cancel each other and we get non-zero contribution for $l=0$ only with $m=0$. To evaluate $r_2$ integral we decompose it in the following form as given below $$\int_{r_n}^{r_{max}}.......dr_2=\int_0^{r_{max}}......dr_2 - \int_0^{r_n}.....dr_2$$ Then using eqn.(52) with $l=0$ and $m=0$, it is possible to evaluate $r_2$ integral analytically. Then we have the Coulomb potential due to the charge element $dq(r_2,\theta_2,z_2)$ $$\begin{aligned} \phi(r_1,z_1)&=&\frac{Ze}{r_{max}^2 z_{max}}\int_0^\infty \frac{dk}{k^2}\left[r_{max} J_1(kr_{max})-r_n J_1(kr_n)\right] \nonumber \\ && J_0(kr_1)\left[2-\exp\{-k(z-r_n)\}-\exp\{-k(z_{max}-z_1)\}\right]\end{aligned}$$ Hence it is also possible to evaluate the components of electric field, $E_z=-\partial\phi/\partial z_1$ along z-direction and $E_r=-\partial\phi/\partial r_1$ along the radial direction. It is trivial to show that these components are non-zero on the WS cell surface, including the plane faces. The Coulomb energy is then given by $$E_{ee}=\frac{1}{2} \frac{Ze}{2\pi r_{max}^2 z_{max}}\int \phi(r_1,z_1,\theta_1) dq(r_1,z_1,\theta_1)$$ The $\theta_1$ integral will give $2\pi$, the $z_1$ integral is very easy to evaluate and for $r_1$ integral we use the Bessel integral formula as given in eqn.(52). Then we have the electron-electron Coulomb interaction energy $$\begin{aligned} E_{ee}=\frac{Z^2 e^2}{r_{max}^4 z_{max}^2}\int_0^\infty \frac{dk}{k^5}\left[r_{max} J_1(kr_{max})-r_n J_1(kr_n)\right]^2 \left[k(z_{max}-r_n)-(1-\exp\{-k(z_{max}-r_n)\})\right]\end{aligned}$$ B. Electron-electron exchange energy: In the cylindrical coordinate system with the external magnetic field along z-axis, which is also the symmetry axis of the cylinder and considering the gauge $\vec A(\vec r)\equiv B(-y/2,x/2,0)$, we have the Dirac equation corresponding to upper component $\phi_\lambda(\vec r)$: $$\left [E^2-m^2+2\lambda k+\frac{\partial^2}{\partial \rho^2}+\frac{1}{\rho}\frac{\partial} {\partial \rho} + \frac{\partial^2}{\partial z^2}-k^2 \rho^2\right]\phi_\lambda(\vec r)=0$$ where $k=eB/2$ and $\sigma_z \phi(\vec r)=\lambda \phi$ with $\lambda=\pm 1$, eigen-values for spin-up and spin-down states respectively. Defining $\beta_\lambda=E^2-m^2+2\lambda k$, the above equation can be written in the following form. $$\left [\beta_\lambda +\frac{\partial^2}{\partial \rho^2}+\frac{1}{\rho}\frac{\partial}{\partial \rho}+\frac{\partial^2}{\partial z^2}-k^2\rho^2\right]\phi_\lambda(\rho,z)=0$$ writing the solution for the upper component in the separable form: $\phi_\lambda(\rho,z)=f_\lambda (\rho)\exp (ip_zz)$, we have from the above equation $$\left [\beta_\lambda +\frac{\partial^2}{\partial \rho^2}+\frac{1}{\rho}\frac{\partial}{\partial \rho}-k^2\rho^2\right ]f_\lambda(\rho)=0$$ where $\beta_\lambda$ is replaced by $\beta_\lambda-p_z^2$. The solutions are given by $$f_\lambda(\rho)=\exp\left (-\frac{t}{2}\right )g_\lambda(t)$$ where $t=\eta^2=k\rho^2$. Substituting $f_\lambda (\rho)$, we have $$g_\lambda=\frac{N}{L}^{1/2}\exp(ip_zz)\exp\left (-\frac{t}{2}\right )L_n(t)$$ where $L$ is the linear dimension along z-axis, $L_n(t)$ is the well known Laguerre polynomial, $$\mid N\mid = \left [\frac{k}{2E}\left (1+\frac{m}{E}\right )\right ]^{1/2}$$ is the normalization constant. Considering the effect of spin into account, we have $$g_{\lambda=+1}=L_n\left (\frac{1}{2} eB\rho^2\right )\left (\begin{array}{c} 1\\0\end{array} \right )$$ and $$g_{\lambda=-1}=L_n\left (\frac{1}{2} eB\rho^2\right )\left (\begin{array}{c} 0\\1\end{array} \right )$$ are the up-spin and down-spin states respectively for the upper component, with the corresponding energy eigen-values $E_+=(p_z^2+m^2+2neB)^{1/2}$ and $E_+=(p_z^2+m^2+2(n+1)eB)^{1/2}$. The spin-up and spin-down spinor states are then given by $$\psi^\uparrow (\rho,z)=\left [\frac{k}{2E}\left (1+\frac{m}{E}\right )\right ]^{1/2}\frac{\exp(ip_zz)\exp\left (-\frac{t}{2}\right )}{{L}^{1/2}}\left (\begin{array}{c} L_n\\ 0\\ p_z L_n/(E+m)\\ -2ik^{1/2}t^{1/2}/(E+m) L_n^\prime \end{array}\right)$$ and $$\psi^\downarrow (\rho,z)=\left [\frac{k}{2E}\left (1+\frac{m}{E}\right )\right ]^{1/2}\frac{\exp(ip_zz)\exp\left (-\frac{t}{2}\right )}{{L}^{1/2}}\left (\begin{array}{c} 0\\ L_n\\ -2ik^{1/2}t^{1/2}/(E+m)(L_n^\prime-L_n)\\ -p_z L_n/(E+m) \end{array}\right)$$ Substituting for up-spin state, we have $$\begin{aligned} \overline\psi_i^\uparrow(\vec r)\psi_i^\uparrow(\vec r^\prime)&=&\frac{k}{2\pi L}\left (1+\frac{m}{E}\right)\exp\{-ip_z(z-z^\prime)-\frac{1}{2}(t+t^\prime)\}\nonumber \\ && \left \{\left (1-\frac{p_z^2}{(E+m)^2}\right)L_n(t)L_n(t^\prime) - \frac{4k(tt^\prime)^{1/2}}{(E+m)^2}L_n^\prime(t)L_n^\prime(t^\prime)\right \}\end{aligned}$$ Similarly, $$\begin{aligned} \overline\psi_i^\downarrow(\vec r^\prime)\psi_i^\downarrow(\vec r)&=&\frac{k}{2\pi L}\left (1+\frac{m}{E}\right)\exp\left \{-ip_z(z-z^\prime)-\frac{1}{2}(t+t^\prime)\right \}\nonumber \\ && \left [\left (1-\frac{p_z^2}{(E+m)^2}\right)L_n(t)L_n(t^\prime) - \frac{4k(tt^\prime)^{1/2}}{(E+m)^2} \left\{(L_n^\prime(t^\prime)-L_n(t^\prime))(L_n^\prime(t)-L_n(t))\right\} \right ]\end{aligned}$$ Identical expressions can also be obtained for $j$th type particles, on which there is a sum. To obtain the exchange energy, we replace the sum over $j$ by the integral over momentum $p_z$ of $j$th particle, i.e., $\sum_j\rightarrow L\int dp_z^\prime$. Assuming $\vec r$ as the reference axis, we can write $\int d^3x d^3x^\prime ...=2\pi \int\rho d\rho dz \rho^\prime d\rho^\prime dz^\prime d\theta^\prime ...$. Replacing the variables $z\prime$ and $p_z^\prime$ by two new variables, $\overline z$ and $P_z$, where $\overline z=z^\prime-z$ and $P_z=p_z-p_z^\prime$, we have the $\overline z$ integral $$I=\int_{-\infty}^{+\infty}\frac{\exp (i\overline zP_z)}{(\overline z^2+X^2)^{1/2}} d\overline z =2K_0(XP_z)$$ where $X=\mid \vec \rho-\vec \rho^\prime \mid$ and $$K_0(x)=\int_0^\infty \frac{\cos (xt)}{(t^2+1)^{1/2}}dt$$ is the modified Bessel function of order zero. The z-integral simply gives $L$. The integral over $P_z^\prime$ can be obtained analytically using the following integrals (eqns.(71)-(72)): $$I=\int_0^{2p_F}K_0(P_z\mid X \mid)dP_z=\frac{1}{\mid X \mid}\int_0^\alpha K_0(w)dw$$ where $\alpha=2p_F\mid X\mid$ and $p_F$ is the electron Fermi momentum. The value of the integral on the right hand side of the above equation is obtained from standard mathematical hand book [@mhb] and is given by $$\begin{aligned} \int_0^x K_0(t)dt&=& -(\gamma+\log x/2)~x \sum_{k=0}^\infty \frac{(x/2)^{2k}}{(k!)^2(2k+1)} \nonumber \\ &+& x\sum_{k=0}^\infty \frac{(x/2)^{2k}}{(k!)^2(2k+1)^2}+ x\sum_{k=1}^\infty \frac{(x/2)^{2k}}{(k!)^2(2k+1)}\times \nonumber \\ && (1+1/2+...+1/k) = xI(x)\end{aligned}$$ where $\gamma$ (Euler’s constant) = $0.5772156649$. The rest three integrals, over $\rho$, $\rho^\prime$ and $\theta$ are evaluated numerically. The Laguerre polynomials $L_n(x)$ and their derivatives $L_n^\prime(x)$ are obtained numerically from the recursion relations and derivative formulas from the reference as cited above and finally obtained the exchange energy as a function of magnetic field strength as defined in eqn.(36). R.C. Duncan and C. Thompson, Astrophys. J. Lett. [**[392]{}**]{}, L9 (1992); C. Thompson and R.C. Duncan, Astrophys. J. [**[408]{}**]{}, 194 (1993); C. Thompson and R.C. Duncan, MNRAS [**[275]{}**]{}, 255 (1995); C. Thompson and R.C. Duncan, Astrophys. J. [**[473]{}**]{}, 322 (1996). P.M. Woods et. al., Astrophys. J. Lett. [**[519]{}**]{}, L139 (1999); C. Kouveliotou, et. al., Nature [**[391]{}**]{}, 235 (1999). K. Hurley, et. al., Astrophys. Jour. [**[442]{}**]{}, L111 (1999). S. Mereghetti and L. Stella, Astrophys. Jour. [**[442]{}**]{}, L17 (1999); J. van Paradihs, R.E. Taam and E.P.J. van den Heuvel, Astron. Astrophys. [**[299]{}**]{}, L41 (1995); S. Mereghetti, astro-ph/99111252; see also A. Reisenegger, astro-ph/01003010; see also S. Mereghetti, arXiv:0904.4880v1, for current status on the observational aspects of magnetars. see the review work by P.Haensel, A.Y.Potekhin and D.G.Yakovlev,Neutron Stars 1: Equation of State and Structure, Springer, Vol. 326 (2007). Nandini Nag, Sutapa Ghosh and Somenath Chakrabarty, Ann. of Phys., [**[324]{}**]{}, 499 (2009). Nandini Nag, Sutapa Ghosh and Somenath Chakrabarty, Euro. Phys. Jour. [**[A 45]{}**]{}, 99 (2010). Jon Eilif Skjervold and Erlend Ostgaard, Phys. Scr., [**[29]{}**]{}, 484 (1984). D. Lai, Rev. Mod. phys. [**[73]{}**]{}, 629 (2001). B. K. Shivamoggi and P. Mulser, EPL, [**[22]{}**]{}, 657 (1993). M. Rotondo, R. Ruffini and S. -S. Xue, arXiv:0903.4095. M. Rotondo, J. A. Rueda, R. Ruffini and S. -S. Xue, arXiv:0911.4622. Arpita Ghosh and Somenath Chakrabarty (in preparation). S.L. Shapiro and S.A. Teukolsky, Black Holes, White Dwarfs and Neutron Stars, John Wiley and Sons, New York, (1983). E.H. Lieb and B. Simon, Phys. Rev. Lett. [**[31]{}**]{}, 681 (1973); E.H. Lieb, J.P. Solovej and J. Yngvason, Phys. Rev. Lett. [**[69]{}**]{}, 749 (1992); E.H. Lieb, Bull. Amer. Math. Soc., [**[22]{}**]{}, 1 (1990); R. Ruffini, “Exploring the Universe”, a Festschrift in honour of Riccardo Giacconi, Advance Series in Astrophysics and Cosmology, World Scientific, Eds. H. Gursky, R. Rufini and L. Stella, vol. [**[13]{}**]{}, 383 (2000); Int. Jour. of Mod. Phys. [**[5]{}**]{}, 507 (1996). M. Ruderman, Phys. Rev. Lett. [**[27]{}**]{}, 1306 (1971). Hand Book of Mathematical Functions, Ed. M. Abramowitz and I.A. Stegun, Dover publication, INC., New York, 1970. R. O. Mueller, A. R. P. Rau and Larry Spruch, Phys. Rev. Lett. [**[26]{}**]{}, 1136 (1971).

--- abstract: 'We prove a two-term Weyl-type asymptotic law, with error term $O(\tfrac{1}{n})$, for the eigenvalues of the operator $\psi(-\Delta)$ in an interval, with zero exterior condition, for complete Bernstein functions $\psi$ such that $\xi \psi''(\xi)$ converges to infinity as $\xi \to \infty$. This extends previous results obtained by the authors for the fractional Laplace operator ($\psi(\xi) = \xi^{\alpha/2}$) and for the Klein–Gordon square root operator ($\psi(\xi) = (1 + \xi^2)^{1/2} - 1$). The formula for the eigenvalues in $(-a, a)$ is of the form $\lambda_n = \psi(\mu_n^2) + O(\tfrac{1}{n})$, where $\mu_n$ is the solution of $\mu_n = \tfrac{n \pi}{2 a} - \tfrac{1}{a} {\vartheta}(\mu_n)$, and ${\vartheta}(\mu) \in [0, \tfrac{\pi}{2})$ is given as an integral involving $\psi$.' address: | Department of Pure and Applied Mathematics\ Faculty of Fundamental Problems of Technology\ Wroc[ł]{}aw University of Technology\ ul. Wybrze[ż]{}e Wyspia[ń]{}skiego 27, 50-370 Wroc[ł]{}aw, Poland author: - 'Kamil Kaleta, Mateusz Kwa[ś]{}nicki, Jacek Ma[ł]{}ecki' title: 'Asymptotic estimate of eigenvalues of pseudo-differential operators in an interval' --- [^1] Introduction and statement of the results ========================================= This is the final one in the series of articles where asymptotic formulae for eigenvalues of certain pseudo-differential operators in the interval are studied. The fractional Laplace operator $(-\Delta)^{\alpha/2}$ was considered in [@bib:kkms10] for $\alpha = 1$ and in [@bib:k12] for general $\alpha \in (0, 2)$, while in [@bib:kkm13] the case of the Klein–Gordon square-root operator was solved ($\Delta$ dentotes the second derivative operator, the Laplace operator in dimension one). In the present article we extend the above results to operators $\psi(-\Delta)$, where $\psi$ is an arbitrary complete Bernstein function such that $\xi \psi'(\xi)$ converges to infinity as $\xi \to \infty$. Let $\lambda_n$ denote the nondecreasing sequence of eigenvalues of $\psi(-\Delta)$ in an interval $D = (-a, a)$, with zero condition in the complement of $D$. Furthermore, for $\mu > 0$ define [ ]{} We note that ${\vartheta}_\mu \in [0, \tfrac{\pi}{2})$ and $\tfrac{d}{d \mu} {\vartheta}_\mu = O(\tfrac{1}{\mu})$ as $\mu \to \infty$. Finally, let $\mu_n$ be a solution of [ ]{} We remark that the solution is unique for $n$ large enough, and [ ]{} The following is the main result of the present article. \[th:main\] If $\psi$ is a complete Bernstein function and $\lim_{\xi \to \infty} \xi \psi'(\xi) = \infty$, then [ ]{} In many cases, $\mu_n$ can be approximated with more explicit expressions, at the price of a weaker estimate of the error term. We provide two examples. Let $\psi(\xi) = \xi^{\alpha/2} + \xi^{\beta/2}$, where $0 < \beta < \alpha \le 2$. Then (see Example \[ex:sum\]) [ ]{} and consequently [ ]{} If $\psi$ is regularly varying at infinity with index $\tfrac{\alpha}{2} \in (0, 1]$, then one has $\lim_{\mu \to \infty} {\vartheta}_\mu = \tfrac{(2 - \alpha) \pi}{8}$ (see ). Therefore, [ ]{} and, using Karamata’s monotone density theorem, one easily finds that [ ]{} The *moderate growth condition* $\lim_{\xi \to \infty} \xi \psi'(\xi) = \infty$ is satisfied by all regularly varying functions with positive index. It is *not* satisfied by a slowly varying complete Bernstein function $\psi(\xi) = \log(1 + \xi)$. There are, however, slowly varying functions which do satisfy the moderate growth condition, for example, [ ]{} which is asymptotically equal to $\tfrac{1}{2} (\log \xi)^2$ as $\xi \to \infty$. Numerical simulations for $\psi(\xi) = \xi^{\alpha/2}$ (using the results of [@bib:dkk15]) strongly suggest that the error in  is in fact of order $O(\tfrac{1}{n^2})$. There is no numerical evidence for more general functions $\psi$. We believe that at least when $\psi(\xi) = \sqrt{\xi}$, one can use a method applied in a somewhat similar problem in [@bib:d70] to obtain a version of  with an additional term in the asymptotic expansion and an improved bound of the error term, but this is far beyond the scope of the present article. We point out that relatively little is known about $\lambda_n$. Most results, including all listed below, cover also higher-dimensionsinal domains, but provide significantly less detailed information. Extension of Theorem \[th:main\] for higher-dimensional domains seems out of reach with the present methods. Best known estimates of $\lambda_n$, proved in [@bib:cs05], are given in terms of the corresponding eigenvalues $\lambda_n^\Delta$ of the Laplace operator $-\Delta$, namely [ ]{} a more direct statement for the case of an interval is given in  below. First term of the asymptotic expansion of $\lambda_n$, namely $\lambda_n \sim \psi(\lambda_n^\Delta)$, is given in many cases in [@bib:bg59]. This result follows by a Tauberian theorem from the asymptotic expression for the *trace* $\sum_{n = 1}^\infty e^{-t \lambda_n}$ as $t \to 0^+$. Second term of the asymptotic expansion of the trace has been found in [@bib:bk08; @bib:bks09] for $(-\Delta)^{\alpha/2}$, in [@bib:bmn14; @bib:ps14] for $(-\Delta + 1)^{\alpha/2} - 1$, and finally in [@bib:bs15] for a rather general class of isotropic Lévy processes with unimodal Lévy measure, satisfying some mild regularity conditions. Tauberian theory is, however, insufficient to obtain a result similar to Theorem \[th:main\] from the two-term expansion of the trace. To the knowledge of the authors, no results of this kind are known for domains other than intervals, with the only exception of the well-studied classical situation of the Laplace operator $-\Delta$. The only related result, proved in [@bib:fg14], provides a two-term asymptotic expansion of Cesàro means $\tfrac{1}{N} \sum_{n = 1}^N \lambda_n$ for $(-\Delta)^{\alpha/2}$ using the methods of semi-classical analysis. The proof of Theorem \[th:main\] is based on the explicit expression for the generalised eigenfunctions of the operator $\psi(-\Delta)$ in the half-line, found in [@bib:kkms10] for $(-\Delta)^{1/2}$, and in [@bib:k11; @bib:kmr13] for $\psi(-\Delta)$ for a general complete Bernstein function $\psi$. The asymptotic expression  for $(-\Delta)^{\alpha/2}$ simplifies to [ ]{} because ${\vartheta}_\mu = \tfrac{(2 - \alpha) \pi}{8}$. As mentioned above, this was proved for $\alpha = 1$ in [@bib:kkms10], with constant $1$ in the asymptotic notation $O(\tfrac{1}{n})$, and for general $\alpha \in (0, 2)$ in [@bib:k12], with a rather big constant in the term $O(\tfrac{1}{n})$. A very careful estimate of [@bib:kkm13] yielded a version of  uniform in $a > 0$ for the operator $(-\Delta + 1)^{1/2} - 1$. In the present article we do not pay attention to the constant in the asymptotic term $O(\tfrac{1}{n})$. All our estimates are, however, explicit, and so it is theoretically possible to trace the dependence of this constant on $a$ and $\psi$. To facilitate the reading of the article, we sketch the main idea of the proof. The generalised eigenfunction of $\psi(-\Delta)$ in the half-line $(0, \infty)$ corresponding to the eigenvalue $\psi(\mu^2)$ is given by an explicit formula $F_\mu(x) = \sin(\mu x + {\vartheta}_\mu) - G_\mu(x)$, where $G_\mu$ is the Laplace transform of a certain non-negative measure (here ‘generalised’ essentially means ‘not square integrable’). We construct approximation $\tilde{{\varphi}}_n$ to eigenfunctions of $\psi(-\Delta)$ in $(-a, a)$ by interpolating between $F_\mu(a + x)$ near $-a$ and $\pm F_\mu(a - x)$ near $a$. In order that the sine terms agree, we need to set $\mu = \mu_n$ defined in . Due to non-locality of $\psi(-\Delta)$, $\tilde{{\varphi}}_n$ is not an eigenfunction; nevertheless, we show that the $L^2(D)$ distance of $\psi(-\Delta) \tilde{{\varphi}}_n$ and $\mu_n \tilde{{\varphi}}_n$ does not exceed $O(\tfrac{1}{n})$ (Lemma \[lem:approxnorm0\]). This is sufficient to prove that there is some eigenvalue $\lambda_{k(n)}$ within the $O(\tfrac{1}{n})$ range from $\psi(\mu_n^2)$. Using the assumption that $\xi \psi'(\xi)$ diverges to infinity as $\xi \to \infty$, one easily finds that the numbers $k(n)$ are distinct for sufficiently large $n$. It remains to estimate the number of eigenvalues $\lambda_j$ not counted as $\lambda_{k(n)}$: this turns out to follow from an estimate of the trace (Lemma \[lem:trace\]). We conjecture that  holds for arbitrary complete Bernstein functions, without the moderate growth condition $\lim_{\xi \to \infty} \xi \psi'(\xi) = \infty$. Note that, however, if this growth condition is not satisfied (for example, when $\psi(\xi) = \log(1 + \xi)$) and $a$ is large enough, then one cannot expect that the numbers $k(n)$ are distinct. Therefore, an extension of Theorem \[th:main\] to general complete Bernstein function would require a completely different approach. It is also natural to expect that  holds for more general functions $\psi$, for example, for all Bernstein functions $\psi$ satisfying the growth condition. However, no expressions for the generalised eigenfunctions $F_\mu$ are known unless $\psi$ is a complete Bernstein function, and so our approach cannot currently be used in this case. The method described above has been designed in [@bib:kkms10] and sucessfully used in [@bib:k12] and [@bib:kkm13]. The core of the argument remains the same in the present article. Nevertheless, proving Theorem \[th:main\] in this generality requires rather non-obvious estimates of ${\vartheta}_\mu$, $\tfrac{d}{d \mu} {\vartheta}_\mu$ and $G_\mu(x)$, as well as many other modfications; for example, the trace estimate in the final part of the proof needed some improvements. The remaining part of the article is divided into two sections. In Preliminaries, we recall definitions and basic properties of complete Bernstein functions (Section \[sec:precbf\]) and the operator $\psi(-\Delta)$ in full space ${\mathbf{R}}$ and in (bounded or unbounded) intervals (Sections \[sec:preop\]–\[sec:preopdom\]). We also recall known properties of the eigenvalues $\lambda_n$ (Section \[sec:prelambda\]) and the generalised eigenfunctions $F_\mu(x)$ (Section \[sec:prefmu\]). Finally, we prove the necessary estimates of ${\vartheta}_\mu$ (Section \[sec:prethetmu\]) and $G_\mu(x)$ (Section \[sec:pregmu\]). The proof of Theorem \[th:main\] is given in Section \[sec:proofs\], which is divided into five parts. Pointwise estimates for $\psi(-\Delta)$ (Section \[sec:prpoint\]) are taken from [@bib:kkm13]. Construction of approximations to eigenfunctions $\tilde{{\varphi}}_n$ is followed by a technical lemma, which asserts that $\tilde{{\varphi}}_n$ is in the domain of $\psi(-\Delta)$ in the interval (Section \[sec:prconstr\]). An estimate for $\psi(-\Delta) \tilde{{\varphi}}_n$ (Section \[sec:prest\]) follow then easily from the results given in Preliminaries. This is used to find estimates for the eigenvalues $\lambda_{k(n)}$ (Section \[sec:prlambda\]). We conclude the proof by showing that $k(n) = n$ for $n$ large enough (Section \[sec:prtrace\]). As a side-result, we obtain some properties of the eigenfunctions, listed in the final Section \[subsec:prop\]. Preliminaries {#sec:pre} ============= All functions considered below are Borel measurable. For $p \in [1, \infty)$ and an open set $D {\subseteq}{\mathbf{R}}$, the Lebesgue space $L^p(D)$ is the set of functions $f$ on $D$ such that $\|f\|_{L^p(D)} = (\int_D |f(x)|^p dx)^{1/p}$ is finite, and $f \in L^\infty(D)$ if and only if the essential supremum $\|f\|_{L^\infty(D)}$ of $|f(x)|$ over $x \in D$ is finite. The space of smooth functions with compact support contained in $D$ is denoted by $C_c^\infty(D)$. By $C_0(D)$ we denote the space of continuous functions in ${\mathbf{R}}$ which are equal to $0$ in ${\mathbf{R}}\setminus D$ and which satisfy the condition $\lim_{x \to \pm \infty} f(x) = 0$. The Fourier transform of a function $f \in L^2({\mathbf{R}})$ is denoted by ${\mathcal{F}}f$. If $f \in L^2({\mathbf{R}}) \cap L^1({\mathbf{R}})$, then ${\mathcal{F}}f(\xi) = \int_{-\infty}^\infty f(x) e^{-i \xi x} dx$. The Laplace transform of a function $f$ is denoted by ${\mathcal{L}}f$, ${\mathcal{L}}f(\xi) = \int_0^\infty f(x) e^{-\xi x} dx$. Symbols $x$, $y$, $z$ are used for spatial variables, while $\xi$, $\eta$, $\mu$ typically correspond to ‘Fourier space’ variables. We sometimes use standard asymptotic notation: we write $f(n) = O(g(n))$ if $\limsup_{n \to \infty} |f(n) / g(n)| < \infty$, and $f(n) = o(g(n))$ if $\lim_{n \to \infty} |f(n) / g(n)| = 0$. Complete Bernstein functions {#sec:precbf} ---------------------------- In this section we recall several classical definitions. A function $f(x)$ on $(0, \infty)$ is said to be *completely monotone* if $(-1)^n f^{(n)}(x) \ge 0$ for all $x > 0$ and $n = 0, 1, 2, \dots$ By Bernstein’s theorem ([@bib:ssv10 Theorem 1.4]), $f$ is completely monotone if and only if it is the Laplace transform of a (possibly infinite) Radon measure on $[0, \infty)$. If $f$ is nonnegative on $(0, \infty)$ and $f'$ is completely monotone, then $f$ is said to be a *Bernstein function*. By Bernstein’s theorem, Bernstein functions have the representation [ ]{} for some $c, \tilde{c} \ge 0$ and a Radon measure $M$ such that $\int_{(0, \infty)} \min(z, 1) M(dz) < \infty$. The above formula extends to complex $x$ such that $\operatorname{Re}x \ge 0$, and defines a continuous function holomorphic in the region $\operatorname{Re}x > 0$. If the measure $M$ in  is absolutely continuous with respect to the Lebesgue measure, and the density function is completely monotone, then $f$ is said to be a *complete Bernstein function*. One easily verifies that in this case [ ]{} for some $c, \tilde{c} \ge 0$ and a Radon measure $m$ such that $\int_{(0, \infty)} \min(1/z, 1/z^2) m(dz) < \infty$. The above formula defines a holomorphic extension of $f$ in the region ${\mathbf{C}}\setminus (-\infty, 0]$. Bernstein and complete Bernstein functions appear in a number of different areas of mathematics. For more information on these objects, we refer the reader to [@bib:ssv10]. We will need the following technical result, proved in the Appendix. \[lem:cbfm\] Let $f$ is a complete Bernstein function with representation . Let $g$ be a holomorphic function in $\{w \in {\mathbf{C}}: |\operatorname{Arg}w| < C_1 \}$ (with $0 < C_1 < \tfrac{\pi}{2}$) such that $g(x)$ is real for $x > 0$, and let $h$ be a continuous function on $(0, \infty)$. Denote [ ]{} and suppose that [ ]{} for $x > 0$. Then [ ]{} Following [@bib:k11; @bib:kmr13], if $\psi$ is a non-constant complete Bernstein function such that $\psi(0) = 0$, and $\mu > 0$, we denote [ ]{} for $\xi \in {\mathbf{C}}\setminus ((-\infty, 0] \cup \{\mu^2\})$, and $\psi_\mu(\mu^2) = \psi(\mu^2) / (\mu^2 \psi'(\mu^2))$. We also let [ ]{} for $\xi \in {\mathbf{C}}$ with $\operatorname{Re}\xi > 0$. Then $\psi_\mu$ is a complete Bernstein function, $\psi^\dagger$ extends to a complete Bernstein function, and we have the Wiener–Hopf identity $\psi^\dagger(\xi) \psi^\dagger(-\xi) = \psi(-\xi^2)$ for $\xi \in {\mathbf{C}}\setminus {\mathbf{R}}$, see, for example, [@bib:k11 Proposition 2.19 and Lemma 3.8]. Finally, we denote $\psi^\dagger_\mu = (\psi_\mu)^\dagger$. In principle we could extend the definition of $\psi_\mu$ to general non-constant complete Bernstein functions $\psi$, so that $\psi_\mu(\xi) = (1 - \xi^2 / \mu^2) / (1 - (\psi(\xi) - \psi(0)) / (\psi(\mu^2) - \psi(0)))$. All results proved below hold true with this definition. However, to keep the notation simpler, we will typically assume that $\psi(0) = 0$. For brevity, we also denote $\psi(\infty) = \lim_{\xi \to \infty} \psi(\xi) \in [0, \infty]$. Unless otherwise stated, in what follows we assume that $\psi(\xi)$ is a non-constant complete Bernstein function which satisfies $\psi(0) = 0$, that is, $\tilde{c} = 0$ in representation  for $\psi$. Analytical definition of the operator $\psi(-\Delta)$ in ${\mathbf{R}}$ and in intervals {#sec:preop} ---------------------------------------------------------------------------------------- In this section it is enough to assume that $\psi$ is an increasing, nonnegative function on $[0, \infty)$, which satisfies [ ]{} for all $\xi \ge \eta \ge 0$ and some $C, \alpha \ge 1$. When $\psi$ is a complete Bernstein function, then  holds with $\alpha = 1$ and $C = 1$, because $\psi(\xi + \eta) \le \psi(\xi) + \psi(\eta) \le \psi(\xi) + C (1 + \eta)$. The operator $A = \psi(-\Delta)$ is an unbounded, non-local, self-adjoint operator on $L^2({\mathbf{R}})$, defined as follows. The domain ${\mathcal{D}}(A)$ of $A$ consists of functions $f \in L^2({\mathbf{R}})$ such that $(1 + \psi(\xi^2)) {\mathcal{F}}f(\xi)$ is square integrable. Clearly, ${\mathcal{D}}(A)$ contains $C_c^\infty({\mathbf{R}})$. For $f \in {\mathcal{D}}(A)$, [ ]{} In other words, $A$ is a *Fourier multiplier* with symbol $\psi(\xi^2)$. This explains the notation $A = \psi(-\Delta)$: the second derivative operator $\Delta$ is a Fourier multiplier with symbol $-\xi^2$. Furthermore, by Plancherel’s theorem, $A$ is positive-definite. Let ${\mathcal{D}}({\mathcal{E}})$ denote the space of $f \in L^2({\mathbf{R}})$ such that $(1 + \psi(\xi^2))^{1/2} {\mathcal{F}}f(\xi)$ is square integrable. For $f, g \in {\mathcal{D}}({\mathcal{E}})$ the quadratic form ${\mathcal{E}}(f, g)$ associated to $A$ is defined by [ ]{} The inner product ${\mathcal{E}}_1(f, g) = {\langle f, g \rangle} + {\mathcal{E}}(f, g)$ makes ${\mathcal{D}}({\mathcal{E}})$ into a Hilbert space. If $f \in {\mathcal{D}}(A)$, then ${\mathcal{E}}(f, g) = {\langle A f, g \rangle}$, and ${\mathcal{D}}(A)$ is a dense subset of the Hilbert space ${\mathcal{D}}({\mathcal{E}})$. Let $D$ be an open subset of ${\mathbf{R}}$. The following definition states that the operator $A_D$ is the *Friedrichs extension* (or the *minimal self-adjoint extension*) of the restriction of $A$ to $C_c^\infty(D)$. \[def:h\] The domain ${\mathcal{D}}({\mathcal{E}}_D)$ of the form ${\mathcal{E}}_D$ is the closure of $C_c^\infty(D)$ in the Hilbert space ${\mathcal{D}}({\mathcal{E}})$, and ${\mathcal{E}}_D(f, g) = {\mathcal{E}}(f, g)$ for $f, g \in {\mathcal{D}}({\mathcal{E}}_D)$. The operator $A_D$ is associated to the form ${\mathcal{E}}_D$: $f \in {\mathcal{D}}({\mathcal{E}}_D)$ is in the domain ${\mathcal{D}}(A_D)$ of $A_D$ if and only if there is a function $A_D f \in L^2(D)$ such that ${\mathcal{E}}(f, g) = {\langle A_D f, g \rangle}$ for $g \in {\mathcal{D}}({\mathcal{E}}_D)$ (or, equivalently, for $g \in C_c^\infty(D)$). The following result is well-known in the context of general Dirichlet forms and generators of Lévy processes, see [@bib:fot10; @bib:s99] for more general results in this direction. For completeness, we provide a short proof. \[prop:domain\] If $D$ is a bounded interval, then $f \in {\mathcal{D}}({\mathcal{E}}_D)$ if and only if $f \in {\mathcal{D}}({\mathcal{E}})$ and $f = 0$ almost everywhere in ${\mathbf{R}}\setminus D$. By definition, if $f \in {\mathcal{D}}({\mathcal{E}}_D)$, then $f \in {\mathcal{D}}({\mathcal{E}})$ and $f = 0$ almost everywhere in ${\mathbf{R}}\setminus D$. Let $f \in {\mathcal{D}}({\mathcal{E}})$ and $f = 0$ almost everywhere in ${\mathbf{R}}\setminus D$. The result follows from the following claim: there is a sequence $f_n \in C_c^\infty(D)$ such that [ ]{} converges to $0$ as $n \to \infty$. Let $h_n \in C_c^\infty({\mathbf{R}}^D)$ be an approximation to the identity such that $h_n(x) = n h(n x)$, $h(x) \ge 0$, $\int_{\mathbf{R}}h(x) dx = 1$ and $h(x) = 0$ for $x \notin (-1, 1)$. Note that $h_n$ is zero outside $(-\tfrac{1}{n}, \tfrac{1}{n})$. Let [ ]{} where $(x - b_n) / a_n$ maps the $\tfrac{2}{n}$-neighbourhood of $I$ into $I$, with $a_n \ge 1$, $\lim_{n \to \infty} a_n = 1$ and $\lim_{n \to \infty} b_n = 0$. Observe that $f_n \in C_c^\infty(D)$ and [ ]{} Since $f, g \in L^1({\mathbf{R}})$, ${\mathcal{F}}f$ and ${\mathcal{F}}h$ are continuous. Furthermore, ${\mathcal{F}}h(0) = 1$ and $|{\mathcal{F}}h(\xi)| \le 1$ for $\xi \in {\mathbf{R}}$. It follows that ${\mathcal{F}}f_n$ converges pointwise to ${\mathcal{F}}f$, and for $n$ large enough, [ ]{} for all $\xi \in {\mathbf{R}}$. Hence, if $u(\xi) = (1 + \psi(\xi^2)) |{\mathcal{F}}f(\xi)|^2$, then for $n$ large enough, [ ]{} for all $\xi$. By the assumption, $u(\xi)$ is integrable. Therefore, the family of functions $(1 + \psi(\xi^2)) |{\mathcal{F}}f_n(\xi) - {\mathcal{F}}f(\xi)|^2$ is tight and uniformly integrable. By the Vitali’s convergence theorem, ${\mathcal{E}}_1(f_n - f, f_n - f)$ converges to $0$ as $n \to \infty$, as desired. We remark that the above result in general fails to be true for arbitrary open sets $D$. It is, in particular, not true when $D = {\mathbf{R}}\setminus \{0\}$ and $\psi(\xi) = \xi^{\alpha/2}$ with $\alpha \in (1, 2]$. Markov semigroup generated by $A$ {#sec:preopfull} --------------------------------- From now on, $\psi$ is a complete Bernstein function. The operator $-A = -\psi(-\Delta)$ generates a strongly continuous semigroup of self-adjoint contractions [ ]{} where $t \ge 0$. Note that $T(0)$ is the identity operator, $T(t)$ is the Fourier multiplier with symbol $\exp(-t \psi(\xi^2))$, and [ ]{} For $t > 0$, the operator $T(t)$ is a convolution operator: for all $f \in L^2({\mathbf{R}})$, [ ]{} where $T(t; dx)$ is a sub-probability measure with characteristic function $\exp(-t \psi(\xi^2))$ and total mass $e^{-t \psi(0)}$. Furthermore, [ ]{} where $T(t; x) = T(t; -x)$ is a decreasing function of $x > 0$ (see [@bib:ssv10]). Hence, $T(t)$ is a *Markov operator*, and formula  defines a contraction on every $L^p({\mathbf{R}})$ ($p \in [1, \infty]$), and also on $C_0({\mathbf{R}})$. In each of these Banach spaces, the generator of the semigroup $T(t)$ is defined in a similar way as in ; for example, [ ]{} Since the above definitions of $A f$ are consistent on the intersections of domains with limits in different function spaces: $L^p({\mathbf{R}})$ for $p \in [1, \infty]$ or $C_0({\mathbf{R}})$, we abuse the notation and use the same symbol $-A$ for the generator of the semigroup $T(t)$ in any of these spaces. Observe that $C_c^\infty({\mathbf{R}})$ is contained in ${\mathcal{D}}(A, L^p({\mathbf{R}}))$ ($p \in [1, \infty]$) and in ${\mathcal{D}}(A; C_0({\mathbf{R}}))$, and it is the core of $A$ in each of these Banach spaces except $L^\infty({\mathbf{R}})$ (see [@bib:a04; @bib:j01]). Whenever we write ${\mathcal{D}}(A)$, we mean ${\mathcal{D}}(A; L^2({\mathbf{R}}))$. If $\psi(\xi)$ has the representation given in , then for $f \in C_c^\infty({\mathbf{R}})$ we have [ ]{} where by the subordination formula, [ ]{} and ‘${\operatorname{pv}\!\!\int}$’ denotes the Cauchy principal value integral: [ ]{} see, for example, [@bib:k11 Proposition 2.14]. Markov semigroup generated by $A_D$ {#sec:preopdom} ----------------------------------- Let $D$ be a (possibly unbounded) interval. The operator $-A_D$ generates a strongly continuous semigroup of operators [ ]{} The operators $T_D(t)$ are given by [ ]{} where [ ]{} It is known that $0 \le T_D(t; x, y) \le T(t, x - y)$, and we let $T_D(t; x, y) = 0$ whenever $x \notin D$ or $y \notin D$. Hence, $T_D(t)$ form a contraction semigroup on each of the spaces $L^p(D)$ ($p \in [1, \infty]$), and if $\psi$ is unbounded, then also on $C_0(D)$ (see [@bib:c85; @bib:k11; @bib:s99]). The generator of each of these semigroups is again denoted by $-A_D$, and it acts on an appropriate domain ${\mathcal{D}}(A_D; L^p)$ or ${\mathcal{D}}(A_D; C_0)$. Spectral theory for $A_{(-a, a)}$ {#sec:prelambda} --------------------------------- Suppose that $D$ is a bounded interval and that $\exp(-2 t \psi(\xi^2))$ is integrable for some $t > 0$. Then $T_D(t; x, y)$ is a Hilbert–Schmidt kernel, and so $T_D(t)$ is a compact operator on $L^2(D)$. Hence, there is a complete orthonormal set of eigenfunctions ${\varphi}_n \in L^2(D)$ of $T_D(t)$. By strong continuity and the semigroup property, the eigenfunctions do not depend on $t > 0$, and the corresponding eigenvalues have the form $e^{-t \lambda_n}$ for all $t > 0$, where the sequence $\lambda_n$ is nondecreasing and converges to $\infty$. By translation invariance, with no loss of generality we may assume that $D = (-a, a)$. By symmetry, $T_D(t; x, y) = T_D(t; -x, -y)$, and hence the spaces of odd and even $L^2(D)$ functions are invariant under the action of $T_D(t)$. Therefore, we may assume that every ${\varphi}_n$ is either an odd or an even function. The *ground state eigenvalue* $\lambda_1$ is positive and simple (unless $\psi$ is constant), and the corresponding *ground state eigenfunction* has constant sign in $D$; we choose it to be positive in $D$. The functions ${\varphi}_n$ are also the eigenfunctions of $A_D$ (because $-A_D$ is the generator of the semigroup $T_D(t)$), and $\lambda_n$ are the corresponding eigenvalues. No closed-form expression for $\lambda_n$ and ${\varphi}_n$ is available, except when $\psi(\xi) = c \xi + \tilde{c}$. By a general result of [@bib:bg59] (see Theorem 2.3 therein), $\lambda_n \sim \psi((\tfrac{n \pi}{2 a})^2)$ as $n \to \infty$ (the original statement includes only the case when $\psi(\xi) \sim \xi^{\alpha/2}$ for some $\alpha \in (0, 2)$, but it can be easily extended to more general $\psi$). Best known general estimates of $\lambda_n$ are found in [@bib:cs05 Theorem 4.4], where it is proved that: [ ]{} Note that the upper bound in  follows relatively easily from the *operator monotonicity* of $\psi$: the form associated to $A_D$ is bounded above by the form of $\psi(-\Delta_D)$, and the eigenvalues of the latter are equal to $\psi((\tfrac{n \pi}{2 a})^2)$. The proof of the lower bound is more intricate. Spectral theory for $A_{(0, \infty)}$ {#sec:prefmu} ------------------------------------- The spectrum of $A_D$ for an unbounded interval $D$ is continuous. When $D = {\mathbf{R}}$, then $A_D = A$ takes diagonal form in the Fourier space, and $e^{i \xi x}$ ($\xi \in {\mathbf{R}}$) are the $L^\infty$ eigenfunctions of $A$. Similar eigenfunction expansion was obtained for the half-line using an appropriate version of the Wiener–Hopf method in [@bib:k11; @bib:kmr13]. Due to translation invariance and symmetry, it suffices to consider $D = (0, \infty)$. \[def:halfline\] Suppose that $\psi$ is a non-constant complete Bernstein function such that $\psi(0) = 0$. For $x, \mu > 0$, let [ ]{} where ${\vartheta}_\mu \in [0, \tfrac{\pi}{2})$ and $G_\mu$ is a completely monotone function on $(0, \infty)$. More precisely, [ ]{} (as in ), and $G_\mu$ is the Laplace transform of a measure $\gamma_\mu$, [ ]{} with [ ]{} for $\mu, \xi, x > 0$. Equivalently, $F_\mu(x)$ is defined by its Laplace transform: for $\xi \in {\mathbf{C}}$ with $\operatorname{Re}\xi > 0$, [ ]{} see [@bib:kmr13 Theorem 1.3] and [@bib:k11 Theorem 1.1]. We have the short-hand expressions [ ]{} again see [@bib:k11 Remark 4.12] (note the typo in formula (4.14) therein) and [@bib:kmr13 formulae (2.17)–(2.19)]. The expressions for $\gamma_\mu(d\xi)$ given above are slightly different than in [@bib:k11; @bib:kmr13], so we provide a short justification. By Lemma \[lem:cbfm\] and the identity [ ]{} we have [ ]{} The expression for ${\mathcal{L}}F_\mu(\xi)$ and the Wiener–Hopf identity $\psi^\dagger_\mu(\xi) \psi^\dagger_\mu(-\xi) = \psi_\mu(-\xi^2)$ give [ ]{} as desired; here we used Lemma \[lem:cbfm\] again. We extend the definition of $F_\mu$ and $G_\mu$ to ${\mathbf{R}}$ so that $F_\mu(x) = G_\mu(x) = 0$ for $x \le 0$. The functions $F_\mu$ ($\mu > 0$) are $L^\infty$ eigenfunctions of $A_D$ and play a similar role for $A_D$ as the Fourier kernel $e^{i \xi x}$ ($\xi \in {\mathbf{R}}$) for $A$. This is formally stated in the following result. \[th:halfline\] The functions $F_\mu$ are $L^\infty$ eigenfunctions of $A_{(0, \infty)}$; the corresponding eigenvalues are $\psi(\mu^2)$. The operator $A_{(0, \infty)}$ takes a diagonal form under the integral transform with kernel $F_\mu(x)$. More precisely, let [ ]{} for $f \in L^2((0, \infty)) \cap L^1((0, \infty))$. Then $(\tfrac{2}{\pi})^{1/2} \Pi$ extends to a unitary mapping on $L^2((0, \infty))$, such that for $f \in L^2((0, \infty))$, [ ]{} and if $f \in {\mathcal{D}}(A_{(0, \infty)})$, then [ ]{} In this article we only use the first part of the above result, namely, that $F_\mu$ are the $L^\infty((0, \infty))$ eigenfunctions of $A_{(0, \infty)}$. We remark that a similar eigenfunction expansion is available for $D = {\mathbf{R}}\setminus \{0\}$, see [@bib:jk15; @bib:k12a], and there are no other known explicit expressions for the eigenfunctions of $A_D$ unless $D = {\mathbf{R}}$ or $\psi(\xi) = c \xi + \tilde{c}$. Estimates of ${\vartheta}_\mu$ {#sec:prethetmu} ------------------------------ Recall that according to , Definition \[def:halfline\] and [@bib:k11 Proposition 4.16], [ ]{} We remark that if $\psi$ is regularly varying at infinity with index $\alpha \in (0, 2]$, then, by dominated convergence, [ ]{} see [@bib:k11 Example 6.1]. By [@bib:k11 Proposition 4.17], dominated convergence can be used to differentiate the right-hand side of  in $\mu > 0$ under the integral sign. This yields [ ]{} for all $\mu > 0$. In this section we prove two properties of ${\vartheta}_\mu$ that are needed in the remaining part of the article. First, we find estimates of ${\vartheta}_\mu$ that imply that the lower limits of ${\vartheta}_\mu$ as $\mu \to 0^+$ or $\mu \to \infty$ do not exceed $\tfrac{3 \pi}{8}$ (Lemma \[lem:thetmulim\]). Next, a simple estimate of $\tfrac{d}{d \mu} {\vartheta}_\mu$ is found (Lemma \[lem:dthetmuest\]). By [@bib:kmr13 Proposition 4.3], we have the following general estimate of ${\vartheta}_\mu$: [ ]{} for all $\mu > 0$. Furthermore, the supremum in the upper bound is always not greater than $2$. If $\psi$ is a Thorin–Bernstein function (see [@bib:ssv10]), then one easily checks that the supremum is in fact not greater than $1$, and therefore ${\vartheta}_\mu \le \tfrac{\pi}{4}$. Below we find more refined bounds for ${\vartheta}_\mu$. By [@bib:kmr13 Proposition 4.5], [ ]{} with [ ]{} (note that the factor $\tfrac{1}{\pi}$ is missing in the lower bound in the original statement). By the same argument as in the proof of the lower bound of [@bib:kmr13 Proposition 4.5] (using the lower bound for $\psi_\lambda(\lambda^2 \zeta^2)$ and the upper bound for $\psi_\lambda(\lambda^2 / \zeta^2)$), one easily shows that, with the same $P$ and $Q$, [ ]{} One can also verify that this bound is always at least as good as the upper bound of , with equality when $P + Q = 1$. The following technical result states that $P + Q \le 1$. This in fact follows indirectly from the proof of [@bib:kmr13 Proposition 4.5] (note that the right-hand side of  is not well-defined when $P + Q > 1$), but we choose to give a simple, direct argument. \[lem:cbfest\] If $\psi$ is a non-constant complete Bernstein function, then [ ]{} The lemma is equivalent to the inequality [ ]{} Assuming $\psi$ has the representation , we need to prove [ ]{} This follows by simple integration from the following bounds: $0 \le c \tilde{c}$, [ ]{} and [ ]{} the last two inequalities are easily proved by direct calculations. \[lem:increasing\] The left-hand side of  is decreasing in $P \in [0, 1 - Q]$. The right-hand side of  is increasing in $P \in [0, 1 - Q]$. Let $P = 1 - \tfrac{Q}{s + Q} = \tfrac{s}{s + Q}$, $s \in [0, 1 - Q]$. Note that $P$ increases with increasing $s$, and the left-hand side of  is equal to [ ]{} Since $\arcsin^2 \sqrt{s}$ is convex, the above expression is increasing in $s$. In a similar way, with $P = 1 - \tfrac{Q}{1 - s} = \tfrac{1 - s - Q}{1 - s}$, $s \in [0, 1 - Q]$, the right-hand side of  is equal to [ ]{} which is again an increasing function of $s$, but now $P$ decreases with increasing $s$. Substituting $P = 0$, we obtain immediately the following elegant result. \[cor:inversebound\] If $\psi$ is a non-constant complete Bernstein function such that $\psi(0) = 0$, then [ ]{} \[lem:thetmulim\] If $\psi$ is a non-constant complete Bernstein function such that $\psi(0) = 0$, then [ ]{} If $\psi$ is unbounded, then also [ ]{} Suppose that $\liminf_{\mu \to 0^+} {\vartheta}_\mu > \tfrac{3 \pi}{8}$. Then there are $\mu_0 > 0$ and $q \in (0, 1)$ such that ${\vartheta}_\mu \ge \tfrac{\pi}{2} - \tfrac{q \pi}{8}$ for $\mu \in (0, \mu_0)$. By Corollary \[cor:inversebound\], [ ]{} for $\mu \in (0, \mu_0)$, and hence [ ]{} for $\mu \in (0, \mu_0)$. If $\alpha$ denotes the right-hand side, then $\alpha > 1$. By integration (see [@bib:jk15 Lemma 2.2]), we have $\psi'(\mu^2) / \psi'(\mu_0^2) \ge (\mu_0^2 / \mu^2)^\alpha$ for all $\mu \in (0, \mu_0)$, which contradicts integrability of $\psi'$ at $0$. This proves the first statement of the lemma. In a similar manner, if $\liminf_{\mu \to \infty} {\vartheta}_\mu > \tfrac{3 \pi}{8}$, then there are $\mu_0 > 0$ and $q \in (0, 1)$ such that ${\vartheta}_\mu \ge \tfrac{\pi}{2} - \tfrac{q \pi}{8}$ for $\mu \in (\mu_0, \infty)$. Again this implies [ ]{} for $\mu \in (\mu_0, \infty)$. If $\alpha$ denotes the right-hand side, then $\alpha > 1$, and by integration, $\psi'(\mu^2) / \psi'(\mu_0^2) \le (\mu_0^2 / \mu^2)^\alpha$ for all $\mu \in (\mu_0, \infty)$. This implies integrability of $\psi'$ at $\infty$. We conjecture that the above lemma holds with $\tfrac{3 \pi}{8}$ replaced with $\tfrac{\pi}{4}$. An example of a complete Bernstein function $\psi$ for which the set of partial limits of ${\vartheta}_\mu$ as $\mu \to 0^+$ is equal to $[0, \tfrac{\pi}{2}]$ is given in [@bib:kmr13 Section 7.5]. \[lem:dthetmuest\] If $\psi$ is a non-constant complete Bernstein function such that $\psi(0) = 0$, then for all $\mu > 0$, [ ]{} By  and the Cauchy’s mean value theorem, for some $\xi_z \in (\mu^2 z^2, \mu^2)$ and $\xi_{1/z} \in (\mu^2, \mu^2 / z^2)$ (where $z \in (0, 1)$), [ ]{} By , $0 \le -\xi \psi''(\xi) \le 2 \psi'(\xi)$ and $0 \le \xi^2 \psi^{(3)}(\xi) \le 6 \psi'(\xi)$. Hence, [ ]{} Furthermore, $\xi_z \psi''(\xi_z) / \psi'(\xi_z) - \xi_{1/z} \psi''(\xi_{1/z}) / \psi'(\xi_{1/z}) \in [-2, 2]$. It follows that [ ]{} Recall that $\xi_{1/z} / \xi_z \le z^{-4}$. Hence, [ ]{} Since $-\log z \le \tfrac{1}{z} - 1$, we have [ ]{} We conjecture that in fact $-\tfrac{1}{\mu} < \tfrac{d}{d \mu} {\vartheta}_\mu \le \tfrac{1}{2 \mu}$. We close this section with the following simple example. \[ex:sum\] Let $\psi(\xi) = \xi^{\alpha/2} + \xi^{\beta/2}$, where $0 < \beta < \alpha \le 2$. By a short calculation, [ ]{} If we denote $w = (1 - z^\alpha) / (1 - z^\beta)$, then [ ]{} As in the last equality of , we obtain [ ]{} Clearly, the integrand is nonnegative. Since $\log(1 + s) \le s$, $z^{\alpha - \beta} \ge z^2$ and $w \ge 1$, [ ]{} Therefore, [ ]{} Estimates of $F_\mu(x)$ {#sec:pregmu} ----------------------- In the remaining part of the article we will need the following simple estimate of ${\mathcal{L}}F_\mu$ and a more refined estimate of $G_\mu$. \[lem:laplaceflambdaest\] If $\psi$ is a non-constant complete Bernstein function such that $\psi(0) = 0$, then for all $\mu > 0$ and $\xi$ such that $\operatorname{Re}\xi > 0$, [ ]{} Recall that $(\mu^2 + \xi^2) {\mathcal{L}}F_\mu(\xi) = \mu (\psi_\mu(\mu^2))^{-1/2} \psi^\dagger_\mu(\xi)$ is a complete Bernstein function of $\xi$, and hence by [@bib:k11 Proposition 2.21(c)] and [@bib:kmr13 Corollary 5.1], [ ]{} for all $\xi$ such that $\operatorname{Re}\xi > 0$. \[lem:glambdaest\] If $\psi$ is a non-constant complete Bernstein function such that $\psi(0) = 0$, then for all $\mu, x > 0$ such that $\mu x \ne 1$, [ ]{} In particular, if $\psi$ is unbounded, then [ ]{} Recall that $\psi^\dagger_\mu(\xi) \ge \psi^\dagger_\mu(0) = \psi_\mu(0) = 1$. Hence, [ ]{} After a substitution $\xi = \sqrt{s}$ it follows that [ ]{} Since $x \sqrt{s} \, e^{-x \sqrt{s}} \le 2 / (1 + x^2 s)$, we have [ ]{} for the last inequality note that the integral converges to the integral term in the representation  for the complete Bernstein function $\psi_\mu$, and we have $\psi_\mu(0) = 1$ (therefore the inequality becomes equality if $\psi_\mu$ contains no linear term, that is, if $\psi$ is unbounded). To prove the first statement, it remains to use the definition of $\psi_\mu$. The other statement of the lemma follows from the first one by the inequality $\xi \psi'(\xi) \le \psi(\xi)$. Proofs {#sec:proofs} ====== Throughout this section we implicitly assume that $\psi$ is a non-constant complete Bernstein function such that $\psi(0) = 0$, that is, $\tilde{c} = 0$ in the representation  for $\psi$. By $c$ and $\nu$ we denote the constant and the measure in the representation  for $\psi$. Finally, we let $D = (-a, a)$ for some $a > 0$. Pointwise estimates for the operator $A$ {#sec:prpoint} ---------------------------------------- Recall that $A = \psi(-d^2 / dx^2)$, and for $f \in C_c^\infty({\mathbf{R}})$ we have, as in , [ ]{} We denote the right-hand side by ${\mathcal{A}}f(x)$ (with a calligraphic letter ${\mathcal{A}}$) whenever the integral converges and, if $c > 0$, $f''$ is well-defined. The following estimates of ${\mathcal{A}}f(x)$ are proved in [@bib:kkm13] in the special case $\psi(\xi) = (\xi + 1)^{1/2} - 1$, but their proofs rely only on the symmetry, unimodality and positivity of the kernel function $\nu$. Note that in [@bib:kkm13] the notation ${\mathcal{A}}_0$ is used for ${\mathcal{A}}$. \[lem:genest1\] Let $x \in {\mathbf{R}}$, $b > 0$, and let $g$ have an absolutely continuous derivative in $(x - b, x + b)$. Then [ ]{} As in [@bib:kkm13; @bib:kkms10; @bib:k12], for $b > 0$ we define an auxiliary function: [ ]{} Note that $q$ is $C^1$, $q'$ is absolutely continuous, $0 \le q''(x) \le 1 / b^2$ (for $x \in {\mathbf{R}}\setminus \{-b, 0, b\}$), the distributional derivative $q^{(3)}$ is a finite signed measure, and $q(x) + q(-x) = 1$. \[lem:genest\] Let $b > 0$, let $f \in L^1({\mathbf{R}})$, and suppose that the second derivative $f''(x)$ exists for $x \in [-b, b]$ and it is continuous in $[-b, b]$. Define [ ]{} Let $q(x)$ be given by , and define $g(x) = q(x) f(x)$. For $x \in (-\infty, 0)$, we have [ ]{} More precisely, for $x \in (-\infty, -b]$ we have [ ]{} and for $x \in (-b, 0)$, [ ]{} Approximate eigenfunctions {#sec:prconstr} -------------------------- Recall that $D = (-a, a)$. Following [@bib:kkm13; @bib:kkms10; @bib:k12], for $n \ge 1$, let $\tilde{\mu}_n$ be the largest solution of [ ]{} with ${\vartheta}_\mu$ defined in ; this agrees with the definition of $\mu_n$ in , but we choose to use the notation $\tilde{\mu}_n$, so that all approximations are clearly distinguished from true values by the presence of a tilde. Although we are interested in large $n$ only, note that by Lemma \[lem:thetmulim\], the equation $a \mu + {\vartheta}_\mu = \tfrac{n \pi}{2}$ has a solution for all $n \ge 1$, and every such solution satisfies [ ]{} We remark that  may fail to have a unique solution for $n = 1$ (for example, when $a = 1$ and $\psi(\xi) = \xi / (10^4 + \xi) + \xi / 10^7$). Nevertheless, if $n \ge 3$ and $\mu \ge \tfrac{(n - 1) \pi}{2 a} = \tfrac{\pi}{a}$, then, by Lemma \[lem:dthetmuest\], [ ]{} and so the solution $\tilde{\mu}_n$ is in fact unique. We let [ ]{} In order to show that $\tilde{\lambda}_n$ is close to some eigenvalue of $A_D$, we construct an approximate eigenfunction $\tilde{{\varphi}}_n$ of $A_D$, using the eigenfunctions $F_{\tilde{\mu}_n}(a - x)$, $F_{\tilde{\mu}_n}(a + x)$ for the one-sided problems corresponding to $A_{(-\infty, a)}$ and $A_{(-a, \infty)}$. As in [@bib:kkm13; @bib:kkms10; @bib:k12], we define [ ]{} with the auxiliary function $q$ defined by . Here $x \in {\mathbf{R}}$, but we have $\tilde{{\varphi}}_n(x) = 0$ for $x \notin D$, so that $\tilde{{\varphi}}_n$ is equal to zero in the complement of $D$. Clearly, $\tilde{{\varphi}}_n$ is continuously differentiable in $D$, $\tilde{{\varphi}}_n'$ is absolutely continuous in $D$, $\tilde{{\varphi}}_n''$ exists in $D \setminus \{-b, b\}$, and $\tilde{{\varphi}}_n''$ is locally bounded in $D$. Note that $\tilde{\lambda}_n$ depends on $a$ and $n$, while $\tilde{{\varphi}}_n(x)$ depends also on $b$. We could fix $b$ in order to optimise the constants (in many cases $b = \tfrac{1}{3} a$ seems to be a reasonable choice), but since we do not track the exact value of the constants, we will simply indicate their dependence on $b$. Note also that $\tilde{{\varphi}}_n$ is not normed in $L^2(D)$, its norm is approximately equal to $\sqrt{a}$ (see Lemma \[lem:norm0\]). The notation introduced above is kept throughout this section. The following result is intuitively clear, although its formal proof is rather long and technical. \[lem:domain\] We have $\tilde{{\varphi}}_n \in {\mathcal{D}}(A_D)$ and $A_D \tilde{{\varphi}}_n(x) = {\mathcal{A}}\tilde{{\varphi}}_n(x)$ for almost all $x \in D$. For brevity, in this proof we write $\tilde{\mu} = \tilde{\mu}_n$ and $\tilde{{\varphi}} = \tilde{{\varphi}}_n$. The domain of $A_D$ is described in Definition \[def:h\]: we need to prove that $\tilde{{\varphi}} \in {\mathcal{D}}({\mathcal{E}})$ and that ${\langle \tilde{{\varphi}}, {\mathcal{A}}g \rangle} = {\langle {\mathcal{A}}\tilde{{\varphi}}, g \rangle}$ for all $g \in C_c^\infty(D)$. We first verify the latter condition. Note that ${\mathcal{A}}\tilde{{\varphi}}(x)$ is well-defined for all $x \in D \setminus \{-b, b\}$, since $\tilde{{\varphi}}$ is smooth in $D \setminus \{-b, b\}$ and bounded on ${\mathbf{R}}$. Let $g \in C_c^\infty(D)$. Since $\tilde{{\varphi}}'$ is absolutely continuous in $(-a, a)$, integration by parts gives [ ]{} Furthermore, by the definition of ${\mathcal{A}}$ (see ), [ ]{} We claim that the double integral exists. Then, by Fubini, it is equal to $0$, and so ${\langle \tilde{{\varphi}}, {\mathcal{A}}g \rangle} = {\langle {\mathcal{A}}\tilde{{\varphi}}, g \rangle}$, as desired. Denote the integrand by $I(x, z) \nu(z)$, and let ${\varepsilon}= \tfrac{1}{3} \operatorname{dist}(\operatorname{supp}g, {\mathbf{R}}\setminus D)$, so that $\operatorname{supp}g {\subseteq}(-a + 3 {\varepsilon}, a - 3 {\varepsilon})$. When $z \ge {\varepsilon}$, then $|I(x, z)| \le 4 \|\tilde{{\varphi}}\|_{L^\infty({\mathbf{R}})} \|g\|_{L^\infty({\mathbf{R}})}$. Suppose that $z \in (0, {\varepsilon})$. If $x \notin (-a + 2 {\varepsilon}, a - 2 {\varepsilon})$, then $I(x, z) = 0$. Otherwise, by first-order Taylor’s expansion of $I(x, z)$ around $z = 0$ (note that $I(x, 0) = \tfrac{\partial}{\partial z} I(x, 0) = 0$) with the remainder in the integral form, we obtain that [ ]{} (recall that $\tilde{{\varphi}}''$ is bounded in $(-a + {\varepsilon}, a - {\varepsilon})$). We conclude that $|I(x, z) \nu(z)| \le C_1(\tilde{{\varphi}}, g) \min(1, z^2) \nu(z)$, which implies joint integrability of $I(x, z) \nu(z)$. Our claim is proved. It remains to verify that $\tilde{{\varphi}} \in {\mathcal{D}}({\mathcal{E}})$, that is, $(1 + \psi(\xi^2)) |{\mathcal{F}}\tilde{{\varphi}}(\xi)|^2$ is integrable. Let $f(x) = q(a - x) F_{\tilde{\mu}}(x)$, so that $\tilde{{\varphi}}(x) = f(a + x) - (-1)^n f(a - x)$ (see ). It suffices to prove integrability of $(1 + \psi(\xi^2)) |{\mathcal{F}}f(\xi)|^2$. Fix ${\varepsilon}> 0$ and let $\tilde{q}(x) = q(a - x) e^{{\varepsilon}x}$. Since the distributional derivatives $q$, $q'$ and $q''$ are integrable functions, and the third distributional derivative of $q(x)$ is a finite signed measure on ${\mathbf{R}}$, the function $\tilde{q}(x)$ has the same property. Therefore, ${\mathcal{F}}q(\xi)$ and ${\mathcal{F}}q^{(3)}(\xi) = -i \xi^3 {\mathcal{F}}q(\xi)$ are bounded functions, and so $|{\mathcal{F}}\tilde{q}(\xi)| \le C_2({\varepsilon}, a, b) / (1 + |\xi|)^3$. The Fourier transform of $e^{-{\varepsilon}x} F_{\tilde{\mu}}(x)$ is equal to ${\mathcal{L}}F_{\tilde{\mu}}({\varepsilon}+ i \xi)$, and the Fourier transform of $f(x) = q(a - x) F_{\tilde{\mu}}(x) = \tilde{q}(x) e^{-{\varepsilon}x} F_{\tilde{\mu}}(x)$ is given by the convolution [ ]{} Suppose that $\xi > 0$. To estimate $|{\mathcal{F}}f(\xi)|$, we write [ ]{} By Lemma \[lem:laplaceflambdaest\], we have [ ]{} (for the second inequality observe that the expression under the square root is bounded by a constant when $s \le 2 \tilde{\mu}$ and by $\psi'(\tilde{\mu}^2) (1 + s^2) / (\psi(s^2) - \psi(\tilde{\mu}^2))$ when $s > 2 \tilde{\mu}$). The right-hand side decreases with $s > 0$. Hence, [ ]{} in the last inequality we used the fact that $4 \psi(\xi^2 / 4) \ge \psi(\xi^2)$ and that the integral is bounded by $1$. The estimate of the other integral in  is simpler: $|{\mathcal{L}}F_{\tilde{\mu}}({\varepsilon}+ i s)| \le C_4({\varepsilon}, \tilde{\mu})$ for all $s \in {\mathbf{R}}$, and hence [ ]{} Therefore, for $\xi > 0$, [ ]{} Since ${\mathcal{F}}f(-\xi) = \overline{{\mathcal{F}}f(\xi)}$, the above estimate extends to all $\xi \in {\mathbf{R}}$. We conclude that for all $\xi \in {\mathbf{R}}$, [ ]{} and the right-hand side is integrable because $(1 + |\xi|)^{-2} (1 + \psi(\xi^2))$ is bounded. Estimates for approximate eigenfunctions {#sec:prest} ---------------------------------------- Following [@bib:kkm13], we introduce the following notation: [ ]{} We recall two fundamental estimates, which were proved in [@bib:kkm13] for $\psi(\xi) = (\xi + 1)^{1/2} - 1$, but their proofs work for general non-constant complete Bernstein functions $\psi$ such that $\psi(0) = 0$. One minor change is required in the proof of Lemma \[lem:approx0\]: an extra term $M_2 c$ appears when Lemma \[lem:genest\] is applied (as compared to the application of [@bib:kkm13 Proposition 4.2] in the proof of [@bib:kkm13 Lemma 4.2]). This extra term is absorbed into $M_2 \nu_0(b)$. Also, note two typos in the first displayed formula in the original statement of [@bib:kkm13 Lemma 4.2]: the norm in the left-hand side should not be squared, and the term $\tilde{\lambda}_n G_{\tilde{\mu}_n,b}(a)$ is missing in the right-hand side. (These typos did not appear in the the other displayed formula in the original statement, which was the one used later in the proof of the main result.) \[lem:approx0\] We have [ ]{} More precisely, we have [ ]{} \[lem:norm0\] We have [ ]{} More precisely, [ ]{} \[lem:approxnorm0\] If $\psi$ is unbounded, then for $n \ge 2$, [ ]{} and [ ]{} By [@bib:k11 Lemma 4.21], [ ]{} Furthermore, by complete monotonicity, [ ]{} so that [ ]{} By Lemma \[lem:glambdaest\], for $\mu \ge \tilde{\mu}_2$, [ ]{} Finally, $\tilde{\mu}_n \ge \tfrac{(n - 1) \pi}{2 a} \ge \tfrac{n \pi}{4 a}$ for $n \ge 2$. The result follows from Lemmas \[lem:approx0\] and \[lem:norm0\]. Distance to nearest eigenvalue {#sec:prlambda} ------------------------------ Let $\sigma(A_D)$ denote the spectrum of $A_D$. Recall that the spectrum of $A_D$ is purely discrete (see Subsection \[sec:prelambda\]), and the eigenvalues of $A_D$ are denoted by $\lambda_n$. The following result was given in [@bib:kkm13] for $\psi(\xi) = (\xi + 1)^{1/2} - 1$ only, but the proof extends to arbitrary self-adjoint operators $A_D$ that preserve the spaces of even and odd functions. \[lem:dist0\] We have [ ]{} In fact, if $A_D^{{\mathrm{even}}}$ and $A_D^{{\mathrm{odd}}}$ are the restrictions of $A_D$ to the (invariant) subspaces of $L^2(D)$ consisting of even and odd functions, respectively, then  holds with $\sigma(A_D)$ replaced by $\sigma(A_D^{{\mathrm{even}}})$ when $n$ is odd, and by $\sigma(A_D^{{\mathrm{odd}}})$ when $n$ is even. The following result is an immediate consequence of Lemmas \[lem:approxnorm0\] and \[lem:dist0\]. \[cor:dist0\] If $\psi$ is unbounded, for all $n \ge 7$ there is a positive integer $k(n)$ such that [ ]{} \[lem:separation\] Suppose that $\lim_{\xi \to \infty} \xi \psi'(\xi) = \infty$. For $n$ larger than some (integer) constant $C(a, b, \psi)$ the numbers $k(n)$ are distinct. Moreover, for any ${\varepsilon}> 0$, for $n$ larger than some (integer) constant $C(a, b, \psi, {\varepsilon})$, [ ]{} Let ${\varepsilon}\in (0, \tfrac{\pi}{4 a})$. For some $\xi_n \in (\tilde{\mu}_n, \tilde{\mu}_n + {\varepsilon})$, [ ]{} Since $\xi_n \le \tfrac{n \pi}{2 a} + {\varepsilon}\le \tfrac{n \pi}{a}$, it follows that [ ]{} Since $\xi_n \ge \tfrac{(n - 1) \pi}{2 a}$, we have $\lim_{n \to \infty} \xi_n^2 \psi'(\xi_n^2) = \infty$, and so, by Corollary \[cor:dist0\], for $n$ greater than some constant $C(a, b, \psi, {\varepsilon})$, [ ]{} Since $\psi$ is concave, [ ]{} Finally, $\tilde{\lambda}_n = \psi(\tilde{\mu}_n^2)$. This proves . Observe that, by Lemma \[lem:dthetmuest\], [ ]{} so that $\tilde{\mu}_{n+1} - \tilde{\mu}_n \ge \tfrac{\pi}{2 a} (1 + \tfrac{6}{(n - 1) \pi})^{-1} \ge \tfrac{\pi}{4 a}$ for $n \ge 3$. The first statement of the lemma follows hence from  with ${\varepsilon}= \tfrac{\pi}{8 a}$. \[lem:cscor\] Suppose that $\lim_{\xi \to \infty} \xi \psi'(\xi) = \infty$. Then $k(n) \ge n$ for infinitely many $n$. By Lemma \[lem:separation\], [ ]{} for $n$ large enough. On the other hand, by , [ ]{} for all $n \ge 1$. Finally, by Lemma \[lem:thetmulim\] and Lemma \[lem:dthetmuest\], ${\vartheta}_{\tilde{\mu}_n} < \tfrac{3 \pi}{8} + \tfrac{\pi}{16}$ for infinitely many $n$, and hence [ ]{} for infinitely many $n$. Trace estimate {#sec:prtrace} -------------- Recall that the kernel functions of the operators $\exp(-t A)$ and $\exp(-t A_D)$ are denoted by $T(t; x - y)$ and $T_D(t; x, y)$, respectively. Furthermore, $0 \le T_D(t; x, y) \le T(t; x - y)$ for all $t > 0$ and $x, y \in D = (-a, a)$, and the Fourier transform of $T(t; x)$ is $\exp(-t \psi(\xi^2))$. In order to estimate the number of eigenvalues $\lambda_n$ not counted as $\lambda_{k(n)}$ for $n$ large enough, we use the trace estimate method, applied previously in in [@bib:kkms10 Section 9], [@bib:k12 Section 5] and [@bib:kkm13 Step 4 of the proof of Lemma 4.4], see also [@bib:bk08; @bib:k98]. \[lem:trace\] Suppose that $\lim_{\xi \to \infty} \xi \psi'(\xi) = \infty$. For $n$ greater than some constant $C(a, b, \psi)$ we have $k(n) = n$. Let ${\varepsilon}= \tfrac{\pi}{6 a}$ and let $N$ be the constant $C(a, b, \psi, {\varepsilon})$ in Lemma \[lem:separation\]. Define $J = \{ k(n) : n > N \}$ and let $J' = \{j \ge 1 : j \notin J\}$. We claim that it suffices to show that $|J'| \le N$. Indeed, there is $n_0 > N$ such that $k(n_0) = 1 + \max J'$, and $k(n)$ is strictly increasing for $n > N$. It follows that $k(n) = k(n_0) + n - n_0$ for $n \ge n_0$. If $|J'| \le N$, then $k(n_0) = |J'| + (n_0 - N) \le n_0$, so that $k(n) \le n$ for $n \ge n_0$. Since $k(n) \ge n$ infinitely many times by Lemma \[lem:cscor\], necessarily $k(n) = n$ for $n \ge n_0$, as desired. Let $t > 0$. By the assumption, $\psi(\xi) \ge \tfrac{1}{t} \log \xi - C(t)$ for some constant $C(t)$, and therefore $\exp(-t \psi(\xi^2))$ is integrable. Therefore, $T(t; x)$ is bounded in $x \in {\mathbf{R}}$. In particular, $T_D(t; x, \cdot)$ is in $L^2(D)$, and so, by Parseval’s identity, [ ]{} On the other hand, by Plancherel’s identity, [ ]{} It follows that for all $t > 0$, [ ]{} Observe that [ ]{} Denote $\xi_n = n \pi / (2 a) + {\varepsilon}= (n + \tfrac{1}{3}) \pi / (2 a)$. Since $e^{-t \psi(z)}$ is concave in $z > 0$, [ ]{} Hence, [ ]{} (the second inequality is a consequence of $\tfrac{3 n + 2}{6 n + 5} + \tfrac{3(n + 1) + 3}{6(n + 1) + 5} \le 1$, while the last one follows from $\lambda_{k(n)} \le \psi((\tilde{\mu}_n + {\varepsilon})^2) \le \psi(\xi_n^2)$ for $n > N$). By , [ ]{} Passing to a limit as $t \to 0^+$, we obtain [ ]{} This shows that $|J'| \le N$, as desired. By Lemma \[lem:trace\], $k(n) = n$ for $n$ large enough. Hence, by Corollary \[cor:dist0\], [ ]{} Properties of eigenfunctions {#subsec:prop} ---------------------------- As in the previous articles [@bib:kkm13; @bib:kkms10; @bib:k12], the intermediate results in the proof of Theorem \[th:main\] provide some approximation results for the eigenfunctions. The details of the argument differ slightly from that of [@bib:kkm13; @bib:kkms10; @bib:k12], so we sketch the proofs. \[prop:l2approx1\] Suppose that $\lim_{\xi \to \infty} \xi \psi'(\xi) = \infty$. With the appropriate choice of the signs of ${\varphi}_n$ and with [ ]{} we have $\beta_n = \sqrt{a} + O(\tfrac{1}{n})$ as $n \to \infty$, and [ ]{} By Lemma \[lem:approxnorm0\], indeed $\beta_n = \sqrt{a} + O(\tfrac{1}{n})$. Let $\alpha_{n,j} = \langle\tilde{{\varphi}}_n, {\varphi}_j\rangle_{L^2(D)}$, so that $\tilde{{\varphi}}_n = \sum_{j = 1}^\infty \alpha_{n,j} {\varphi}_j$ in $L^2(D)$. We choose the sign of ${\varphi}_n$ so that $\alpha_{n,n} \ge 0$. We have [ ]{} As in the proof of Lemma \[lem:separation\], for $n$ larger than some constant, if $j \ne n$ and ${\varepsilon}= \tfrac{\pi}{8 a}$, then [ ]{} Therefore, [ ]{} again by Lemma \[lem:approxnorm0\]. \[prop:l2approx2\] Suppose that $\lim_{\xi \to \infty} \xi \psi'(\xi) = \infty$. With the appropriate choice of the signs of ${\varphi}_n$ and with [ ]{} we have [ ]{} Clearly, [ ]{} The middle summand is $O(1 / ((\tfrac{n \pi}{2 a})^2 \psi'((\tfrac{n \pi}{2 a})^2)))$, while the last one is $O(\tfrac{1}{n})$. Finally, by the definition  of $\tilde{{\varphi}}_n$ and the properties of $q(x)$ and $F_\mu(x)$, [ ]{} Since $G_\mu(0) = \cos {\vartheta}_\lambda \le 1$ and ${\mathcal{L}}G_\mu(0) = I_\mu \le \tfrac{1}{\mu}$ (see ), we have [ ]{} \[prop:loo\] Suppose that if $\xi_2 > \xi_1 > 1$, then [ ]{} for some $M, {\varepsilon}> 0$. Suppose in addition that [ ]{} Then ${\varphi}_n(x)$ are bounded uniformly in $n \ge 1$ and $x \in (-a, a)$. Condition  is known under various names, including *weak lower scaling condition* and *subregularity*; such a function $\psi$ is also said to have positive *lower Matuszewska index*. We remark that although  does not imply , examples of complete Bernstein functions which satisfy , but not , are rather artificial. Observe that $\xi \psi'(\xi)$ diverges to $\infty$ as $\xi \to \infty$, and therefore main results of the present article apply. Furthermore, by , we have $T(t, 0) \le C_1(\psi) \sqrt{\psi^{-1}(1/t)}$ for $t \le 1$, see, for example, [@bib:bgr13 Theorem 21]. We have [ ]{} Since $|{\varphi}_n(x)| \le 2$, the latter term in the right-hand side does not exceed $\tfrac{2}{\beta_n} e^{\lambda_n t}$. For the former one, observe that $|T_D(t) f(x)| \le \|T_D(t, x, \cdot)\|_{L^2(D)} \|f\|_{L^2(D)}$, $T_D(t, x, y) \le T(t, x - y)$, and, by Plancherel’s theorem, [ ]{} Finally, $T(2 t, 0) \le C_1(\psi) \sqrt{\psi^{-1}(1/(2t))} \le C_1(\psi) \sqrt{\psi^{-1}(1/t)}$ when $t \le 1$. Therefore, with $t = \tfrac{1}{\lambda_n}$, [ ]{} In the right-hand side, $\beta_n = O(1)$, $\psi^{-1}(\lambda_n) \le (\tfrac{n \pi}{2 a})^2$ (by ), and, by Lemma \[lem:approxnorm0\], [ ]{} Appendix {#appendix .unnumbered} ======== Let $x > 0$, $0 < {\varepsilon}< \tfrac{1}{2} C_1$ and $y > 0$, and denote for simplicity $\xi = -e^{-i {\varepsilon}} x$. By the representation  of the complete Bernstein function $f$ and Fubini, we have [ ]{} (an estimate which allows us to use Fubini is shown below). Our goal is to provide estimates for the integrands and find their pointwise limits as ${\varepsilon}\to 0^+$ in order to apply dominated convergence. For the first integral in the right-hand side of , we simply use $|\xi g(-\xi) h(x)| \le x G(x) H(x)$, integrability of $x G(x) H(x)$ and $\operatorname{Im}(\xi g(-\xi)) \to 0$ as ${\varepsilon}\to 0^+$. By dominated convergence, the limit as ${\varepsilon}\to 0^+$ of the first integral in the right-hand side of  is zero. Similarly, $|g(-\xi) h(\xi)| \le G(x) H(x)$, $G(x) H(x)$ is integrable and $\operatorname{Im}(g(-\xi)) \to 0$ as ${\varepsilon}\to 0^+$, and so also the second integral in the right-hand side of  converges to zero as ${\varepsilon}\to 0^+$. To estimate the last integral in the right-hand side of , we consider separately two cases. When $x \le \tfrac{y}{2}$ or $x \ge 2 y$, we have [ ]{} so that by dominated convergence, [ ]{} When $\tfrac{y}{2} < x < 2 y$, we need a more careful estimate. Observe that [ ]{} The estimate for $g$ and Cauchy’s integral formula for $g'$ easily give [ ]{} in $\{z \in {\mathbf{C}}: |\operatorname{Arg}z| < \tfrac{1}{2} C_1, \, y/2 \le |z| \le 2 y\}$, with $C_4 = 4 C_1^{-1}$. By the mean value theorem, [ ]{} when $\tfrac{y}{2} \le x \le 2 y$, and therefore, by dominated convergence, [ ]{} Finally, if $P_t(s)$ and $Q_t(s)$ denote the (classical) Poisson and conjugate Poisson kernels for the half-plane, then [ ]{} Clearly, $P_{y \sin {\varepsilon}}(x - y \cos {\varepsilon}) {\mathbf{1}}_{(y/2, 2y)}(x) dx$ converges weakly to $\delta_y(x)$, and therefore [ ]{} Furthermore, $|t Q_t(s)| \le \tfrac{1}{\pi}$ and $t Q_t(s) \to 0$ as $t \to 0^+$, and hence, by dominated convergence, [ ]{} We have thus proved that [ ]{} Due to estimates , and , as well as the integrability condition on $m$, indeed we could use Fubini in . The same estimates allow us to use dominated convergence in the limit as ${\varepsilon}\to 0^+$. We conclude that [ ]{} This proves the first equality in . The other one follows by replacing the pair $g(z)$, $h(x)$ with $1$ and $g(x) h(x)$. [99]{} D. Applebaum, *L[é]{}vy Processes and Stochastic Calculus*. Cambridge University Press, Cambridge, 2004. R. Ba[ñ]{}uelos, T. Kulczycki, *Trace estimates for stable processes*. Probab. Theory Relat. Fields 142(3–4) (2008) 313–338. R. Ba[ñ]{}uelos, T. Kulczycki, *On the traces of symmetric stable processes on Lipschitz domains*. J. Funct. Anal. 257(10) (2009): 3329–3352. R. Ba[ñ]{}uelos, J. B. Mijena, E. Nane, *Two-term trace estimates for relativistic stable processes*. J. Math. Anal. Appl. 410(2) (2014): 837–846. R. M. Blumenthal, R. K. Getoor, *The asymptotic distribution of the eigenvalues for a class of Markov operators*. Pacific J. Math. 9(2) (1959) 399–408. K. Bogdan, T. Grzywny, M. Ryznar, *Density and tails of unimodal convolution semigroups*. J. Funct. Anal. 266(6) (2014): 3543–3571. K. Bogdan, B. Siudeja, *Trace estimates for unimodal Lévy processes*. Preprint, 2015, arXiv:1503.09126v1. Z. Q. Chen, R. Song, *Two sided eigenvalue estimates for subordinate Brownian motion in bounded domains*. J. Funct. Anal. 226 (2005) 90–113. K. L. Chung, *Doubly-Feller process with multiplicative functional*. In: *Seminar on Stochastic Processes, 1985* (Gainesville, Fla., 1985), Progr. Probab. Statist. 12, Birkhäuser, Boston, Boston, MA., 1986, 63–78. A. M. J. Davis, *Waves in the presence of an infinite dock with gap*. J. Inst. Math. Appl. 6 (1970): 141–156. B. Dyda, A. Kuznetsov, and M. Kwa[ś]{}nicki. *Eigenvalues of the fractional Laplace operator in the unit ball*. In preparation, 2015. R. Frank, L. Geisinger, *Refined semiclassical asymptotics for fractional powers of the Laplace operator*. J. Reine Angew. Math., to appear. M. Fukushima, Y. Osima, M. Takeda, *Dirichlet Forms and Symmetric Markov Processes*. De Gruyter Studies in Mathematics 19, de Gruyter, Berlin, Boston, 2010. N. Jacob, *Pseudo Differential Operators and Markov Processes*, Vol. 1. Imperial College Press, London, 2001 T. Juszczyszyn, M. Kwaśnicki, *Hitting times of points for symmetric Lévy processes with completely monotone jumps*. Electron. J. Probab. 20 (2015): 48:1–24. K. Kaleta, M. Kwaśnicki, J. Ma[ł]{}ecki, *One-dimensional quasi-relativistic particle in the box*. Rev. Math. Phys. 25(8) (2013) 1350014. T. Kulczycki, *Intrinsic ultracontractivity for symmetric stable processes*. Bull. Polish Acad. Sci. Math. 46(3) (1998) 325–334. T. Kulczycki, M. Kwa[ś]{}nicki, J. Ma[ł]{}ecki, A. St[ó]{}s, *Spectral Properties of the Cauchy Process on Half-line and Interval*. Proc. London Math. Soc. 101(2) (2010) 589–622. M. Kwa[ś]{}nicki, *Spectral analysis of subordinate Brownian motions on the half-line*. Studia Math. 206(3) (2011) 211–271. M. Kwa[ś]{}nicki, *Eigenvalues of the fractional Laplace operator in the interval*. J. Funct. Anal. 262 (2012) 2379–2402. M. Kwa[ś]{}nicki, *Spectral theory for one-dimensional symmetric Lévy processes killed upon hitting the origin*. Electron. J. Probab. 17 (2012), no. 83, p. 1–29. M. Kwa[ś]{}nicki, J. Ma[ł]{}ecki, M. Ryznar, *First passage times for subordinate Brownian motions*. Stoch. Proc. Appl 123 (2013) 1820–1850. H. Park, R. Song, *Trace Estimates for Relativistic Stable Processes*. Potential Anal. 41(4) (2014): 1273–1291. K. Sato, *L[é]{}vy Processes and Infinitely Divisible Distributions*. Cambridge Univ. Press, Cambridge, 1999. R. Schilling, R. Song, Z. Vondraček, *Bernstein Functions: Theory and Applications*. De Gruyter, Studies in Math. 37, Berlin, Boston, 2012. [^1]: Work supported by NCN grant no. 2011/03/D/ST1/00311

--- abstract: | This work introduces a new inversion formula for analytical functions. It is simple, generally applicable and straightforward to use both in hand calculations and for symbolic machine processing. It is easier to apply than the traditional Lagrange-Bürmann formula since no taking limits is required. This formula is important for inverting functions in physical and mathematical problems.[^1] Keywords -------- inversion of functions, Taylor series, Lagrange-Bürmann inversion formula, reversion of series Mathematical Classification --------------------------- Mathematics Subject Classification 2010: 11A25, 40E99, 32H02 author: - | Henrik Stenlund[^2]\ Visilab Signal Technologies Oy, Finland date: 'July 27, 2010' title: Inversion Formula --- Introduction ============ General ------- The inversion of an analytic function $f(z)$ with $z, u\in C$ $$f(z)=u$$ is defined as $$z=g(u)$$ There is no general simple method known to determine $g(u)$ unless the variable $z$ can be readily solved from $f(z)$. Lagrange [@Lagrange1770] was the first to find a useful series expansion. Bürmann [@Burmann1799] and [@Hindenburg1798] generalized it to the Lagrange-Bürmann formula. Good [@Good1960] extended the Lagrange-Bürmann formula to multiple variables. His formula is known as the Lagrange-Good formula and Hofbauer [@Hofbauer1979] supplied the proof. A number of investigations has been published over the Lagrange-Bürmann formula for various applications, like Zhao [@Zhao2004] and Merlini et al. [@Merlini2006]. Sokal [@Sokal2009] recently introduced a new generalization of the Lagrange-Bürmann formula. We first express the Lagrange-Bürmann inversion formula which is the present standard method for calculating the inverse. The new inversion formula is derived next. The Lagrange-Bürmann Inversion Formula -------------------------------------- Lagrange [@Lagrange1770] and Bürmann [@Burmann1799] introduced an inversion formula for a function $f(z)$ of a complex variable $z$. $$f(z) = u$$ with $f$ being analytic at some point $z_0$ and the first derivative at $z_0$, is required to be nonzero. $$[{\frac{df(z)}{dz}}]_{z_{0}}\neq{0}$$ $f(z)$ has a value $u_0$ at $z_0$. The inverse function is $g(u)$ $$z=g(u)=g(f(z))$$ The Lagrange inversion formula or the Lagrange-Bürmann formula is a Taylor series as follows. $$z=z_{0}+ \sum^{\infty}_{n=1}\frac{(u-u_{0})^{n}}{n!}[\lim_{z\rightarrow{z_0}}[\frac{d^{n-1}}{dz^{n-1}}(\frac{z-z_0}{f(z)-u_0})^{n}]] \label{eqn10}$$ Proof of this formula can be found in [@Lagrange1770] and [@Burmann1799]. Taking limits in terms in equation (\[eqn10\]) usually requires lengthy calculations and a repeated use of L’Hospital’s rule to get rid of the singularity. All terms belonging to a certain coefficient need to be kept together to determine the limit properly. This may be a very laborious task in hand calculations. The Inversion Formula ===================== Using the annotation of the preceding chapter, let $$u=f(z) \ z,u \in C$$ and $f(z)$ be analytic over the interior of a circle $$r=\left|z-z_0\right|$$ Let the inverse function $g(u)$ be analytic over the interior of a circle $R_0$ at $u_0$ $$R_0=\left|u-u_0\right| \label{eqn15}$$ We have a Taylor series $$z=z_{0}+ \sum^{\infty}_{n=1}\frac{(u-u_{0})^{n}}{n!}[\frac{d^{n}}{du^{n}}g(u)]_{u_{0}} \label{eqn20}$$ This series converges over the circle $R_1$ (as in equation (\[eqn15\])). The equation (\[eqn20\]) is very difficult to be used any further as such. Higher derivatives of $g(u)$ are requested and to get them, one would need the $g(u)$. We can use derivatives of $f(z)$ instead of $g(u)$. In order to circumvent the generation of progressively complicated terms, we proceed as follows. Differentiate equation (\[eqn30\]) below. $$z=g(u) \ \label{eqn30}$$ to obtain $$\frac{d}{dz}z=1=(\frac{d}{du}g)(\frac{d}{dz}u)=(\frac{d}{du}g(u))(\frac{d}{dz}f(z)) \ \label{eqn40}$$ and solve it as $$\frac{d}{du}g(u)=\frac{1}{(\frac{d}{dz}f(z))} \ \label{eqn50}$$ Differentiate (\[eqn50\]) further and solve it for $$\frac{d^2}{du^2}g(u)=\frac{1}{(\frac{d}{dz}f(z))}\cdot[{\frac{d}{dz}\frac{1}{(\frac{d}{dz}f(z))}}] \ \label{eqn60}$$ In the same manner the n’th derivative would be solved as $$\frac{d^n}{du^n}g(u)=\frac{1}{(\frac{d}{dz}f(z))}\cdot[{\frac{d}{dz}\frac{1}{(\frac{d}{dz}f(z))}}\cdot[{\frac{d}{dz}\frac{1}{(\frac{d}{dz}f(z))}}\cdot\cdot\cdot[{\frac{d}{dz}\frac{1}{(\frac{d}{dz}f(z))}}\cdot[{\frac{d}{dz}\frac{1}{(\frac{d}{dz}f(z))}}]]]] \ \label{eqn70}$$ having $n-1$ derivatives acting on the right side in addition to the bracketed derivatives acting on $f(z)$ alone. We can rearrange the brackets yielding $$\frac{d^n}{du^n}g(u)=[\frac{1}{\frac{d}{dz}f(z)}\cdot{\frac{d}{dz}]^{n-1}\frac{1}{(\frac{d}{dz}f(z))}} \ \label{eqn80}$$ The multiplying factor is a differential operator acting on all terms to the right containing any dependence on $z$. Placing this result to equation (\[eqn20\]) yields the simplified inversion formula $$z=z_{0}+ \sum^{\infty}_{n=1}\frac{(u-u_{0})^{n}}{n!}[[\frac{1}{\frac{d}{dz}f(z)}\cdot{\frac{d}{dz}]^{n-1}\frac{1}{(\frac{d}{dz}f(z))}}]_{u_{0}} \label{eqn90}$$ The necessary, but not sufficient, condition for the new inversion formula to converge is that the first derivative of $f(z)$ must be nonzero at $z_0$. The radius of convergence $R_1$ must be evaluated for each resulting series. If a singularity would appear at $z_0$, a translation to a nearby point should be made. Conclusions =========== The equation (\[eqn90\]) represents a simple alternative to the Lagrange-Bürmann formula (equation (\[eqn10\])). The Lagrange-Bürmann formula requires taking limits and repeated use of L’Hospital’s rule to remove the singularity. The new formula requires only elementary differentiation and evaluation at $z_0$. Comparison of coefficients in each term between the two formulas is not possible since the expansions are based on polynomials of $u$. A special case appears when $u_0=0$ making the expansions powers of $u$. This leads to equalities but not directly. One has to approach the limit $(z\rightarrow0)$ in equation (\[eqn10\]) finally reaching terms identical with equation (\[eqn90\]). Working in the opposite way is not possible. In spite of its simplicity, this inversion formula can be applied generally. It can be used for inversion of functions and polynomials and for reversion of series. It is valid also for real variables. It is useful for estimating the behavior of the inverse function at some point with a few beginning terms. The radius of convergence needs to be studied for each new series. [10]{} <span style="font-variant:small-caps;">Lagrange, J. L.</span>:*Nouvelle méthode pour résoudre des équations littérales par le moyen de séries.* Mém. Acad. Roy. des Sci. et Belles-Lettres de Berlin **24, (1770)** <span style="font-variant:small-caps;">Bürmann, H.</span>: *Essai de calcul fonctionnaire aux constantes ad-libitum*, Mem. Inst. Nat. Sci Arts. Sci. Math. Phys., **2 (1799)** pp. 13-17 <span style="font-variant:small-caps;">Good, I.J.</span>: *Generalization to several variables of Lagrange’s expansion, with applications to stochastic processes*, Proc. Cambridge Philos. Soc. 367-380, **56 (1960)** <span style="font-variant:small-caps;">Hofbauer, J.</span>: *A Short Proof of the Good-Lagrange Formula*, Discrete Mathematics, **25 (1979)** 135-139 <span style="font-variant:small-caps;">Zhao, W.</span>: *Recurrent Inversion Formulas*, **arXiv:Math/0305162v2** \[math.CV\] 19. Feb 2004 <span style="font-variant:small-caps;">Merlini, D. Sprugnoli, R. Verri, M. C.</span>: *Lagrange Inversion: When and How*, Acta Appl. Math., **(2006) 94** 233-249 <span style="font-variant:small-caps;">Sokal, A. D.</span>: *A Ridiculously Simple and Explicit Implicit Function Theorem*, **arXiv:0902.0069v1** \[math.CV\], 31 Jan 2009 <span style="font-variant:small-caps;">Hindenburg, C. F. ed.</span>: *Versuch einer vereinfachten Analysis, ein Auszug eines Auszuges von Herrn Bürmann*, Archiv der reinen und angewandten Mathematik, 2. Leipzig, Germany: Schäferischen Buchhandlung. pp. 495-499. **(1798)**, a summary to [@Burmann1799] [^1]: Visilab Report \#2010-07. Revision 1 written in LaTeXwith some sentences corrected, 2-1-2011. [^2]: The author is grateful to Visilab Signal Technologies for supporting this work.

--- abstract: | Our team won the second prize$^{3}$ of the Safe Aging with SPHERE Challenge organized by SPHERE, in conjunction with ECML-PKDD and Driven Data. The goal of the competition was to recognize activities performed by humans, using sensor data. This paper presents our solution. It is based on a rich pre-processing and state of the art machine learning methods. From the raw train data, we generate a synthetic train set with the same statistical characteristics as the test set. We then perform feature engineering. The machine learning modeling part is based on stacking weak learners through a grid searched XGBoost algorithm. Finally, we use post-processing to smooth our predictions over time.\ author: - 'Maxime Voisin$^{1}$, Leo Dreyfus-Schmidt$^{2}$, Pierre Gutierrez$^{2}$, Samuel Ronsin$^{2}$ and Marc Beillevaire$^{2}$[^1] [^2] [^3]' title: '**Dataiku’s Solution to SPHERE’s Activity Recognition Challenge** ' --- INTRODUCTION {#introduction .unnumbered} ============ SPHERE organized an activity recognition competition in conjunction with ECML-PKDD and DrivenData. The goal was to recognize activities - postures and movements - from sensor data collected from participants. Our solution reached second prize. Hopefully, this paper will contain sufficient information to reproduce our results, and we will try to make it transparent when our choices were time-driven rather than following proper scientific method. The paper is organized as follow : We first introduce the challenge’s goal and its data. Then, we describe our approach to create a train set suitable for the machine learning paradigm. The next sections deal respectively with the feature engineering, the machine learning models and the post-processing we used in the competition. CHALLENGE DESCRIPTION {#challenge-description .unnumbered} ===================== The target {#the-target .unnumbered} ---------- The goal of this challenge was to predict, on a second-by-second basis, a person’s activity based on sensor data. It was modeled as a multi-class classification problem. The target variable could take 20 different values, representing the individual’s activities, postures and transitions: ascend stairs, descend stairs, jump, walk with load, walk, bending, kneeling, lying, sitting, squatting, standing, stand-to-bend, kneel-to-stand, lie-to-sit, sit-to-lie, sit-to-stand, stand-to-kneel, stand-to-sit, bend-to-stand and turn. Several annotators have manually defined the ground truth for the target variable. For instance, if “jump” is given a value of 0.05 at a given second, this should be interpreted as meaning that on average the annotators marked 5% of that second as arising from jumping. Datasets {#datasets .unnumbered} -------- For this contest, the SPHERE team equipped a house with three sensor modalities. ### Accelerometers {#accelerometers .unnumbered} Participants wore a tri-axial accelerometer on their dominant wrist. The device wirelessly transmits the value of acceleration to several receivers positioned within the house. This device gives two valuable pieces of information. First, the value of acceleration, in three directions. Second, the signal power that was recorded by each receiver (in units of dBm) - this data will be informative for indoor localization. ### Cameras {#cameras .unnumbered} Three cameras were used in the living room, hallway and kitchen. Automatic detection of humans was performed. In order to preserve the anonymity of the participants, the raw video data are not shared. Instead, the coordinates of the 2D bounding box, 2D centre of mass, 3D bounding box and 3D centre of mass are provided. ### Environmental Sensors {#environmental-sensors .unnumbered} The values of passive (PIR) sensors positioned within the house are given.\ In order to generate the train data, 10 participants successively performed a script of daily-life actions in this house. Hence, the train data consists of 10 continuous sequences of monitoring. Each sequence was recorded on a second-by-second basis and lasts approximately 30 minutes.\ The test data was generated by 10 other participants who followed the same script of daily-life actions: these 10 test sequences of monitoring were also recorded on a second-by-second basis and have a similar duration. However, instead of supplying 10 continuous test sequences of 30 minutes of monitoring, the SPHERE team randomly split these 10 long sequences into 800 smaller subsequences. To do so, they iteratively sampled a subsequence duration and a number of seconds to drop between two subsequences. The subsequence duration was chosen to follow a uniform distribution between 10 and 30 seconds. The gap length follows a similar distribution. These subsequences were finally permuted so that it would be difficult to reconstitute the whole 30-minute test sequences. This was probably done to force the inference of test sequences to be independent of the daily-life action script. The competition entrants’ model would thus have to generalize to other scripts and participants, which would make them useful in real-life situations. Evaluation metric {#evaluation-metric .unnumbered} ----------------- Submissions to the competition are evaluated with the Brier score defined as: $$BS = \frac{1}{N} \sum_{n=1}^N \sum_{c=1}^C w_c(p_{n,c}-y_{n,c})^2$$ where $N$ is the number of test sequences, $C$ is the number of classes, $p_{n,c}$ is the predicted probability of instance $n$ being from class $c$, $y_{n,c}$ is the proportion of annotators that labeled instance $n$ as arising from class $c$, and $w_c$ is the weight for each class. Lower Brier scores indicate better performance, and optimal performance is achieved with a Brier score of 0. Class weights place more weight on the classes that are less frequent. PRE-PROCESSING {#pre-processing .unnumbered} ============== Changing the structure of the train and test sets {#changing-the-structure-of-the-train-and-test-sets .unnumbered} ------------------------------------------------- The first step was to change the structure of the train set to have its distribution follow that of the test set more closely. Therefore, we randomly split the 10 train sequences of 30 minutes into 800 smaller subsequences of 10 to 30 seconds, to follow the test set creation methodology. By doing this random splitting several times, with different random seeds, it is possible to generate several train sets. Then, we could follow a bagging approach: create one model per train set and average their predictions. This approach showed good results in cross-validation, but due to time constraints it was not part of our final model.\ The second operation was to optimize a hard target. In order to have an easier integration with existing Python machine learning libraries (such as scikit-learn and XGboost), we converted our probabilistic (soft) target into a hard target, by keeping for each line the target label with the highest probability. Cross-validation strategy {#cross-validation-strategy .unnumbered} ------------------------- A good cross-validation strategy is crucial to have a faithful estimation of the performance of our models on the leaderboard and to avoid overfitting. In the competition, train and test sets were generated using two distinct groups of participants. Since it was crucial for our model to generalize to unknown people, we split up the train data into: - a train subset: data generated by all individuals but number 6 and 10 - a validation subset: data generated by individuals 6 and 10\ We observed for each model a rather constant gap between our evaluation and the score obtained on the public leaderboard. So every improvement in our local cross-validation score led to a similar improvement on public leaderboard. Note that this cross-validation strategy might not be optimal. It might even cause overfitting if individuals 6 and 10 turn out to be more similar to the individuals involved in public leaderboard, than to those involved in the private one. Feature Engineering {#feature-engineering .unnumbered} =================== Initial features {#initial-features .unnumbered} ---------------- The raw train and test datasets contained 119 features. However, many of these features are highly correlated or are at a level of granularity too refined. For instance, each camera gives the x,y and z-coordinates of the individual’s centre of mass. But the camera records at 25 frames per second. For every second, we kept the mean, median, min, max and standard deviation of these 25 coordinate values. Hence we got 5 features that describe, for every second, the coordinate values of the individual’s centre of mass. Similarly, the accelerometers sample at 20 Hz. So, for a given second of monitoring, we keep the mean, median, min, max and standard deviation of the 5 acceleration values generated by the accelerometer. Basic feature engineering {#basic-feature-engineering .unnumbered} ------------------------- First, we extracted basic features: speeds, accelerations, derivatives of acceleration, second derivatives of acceleration and rotations. We noted that the accelerometer was fixed on the individual’s dominant wrist. Figure \[fig:acc\] clearly highlights two distributions of y-accelerations: one for right-handed individuals and another one for left-handed ones. To correct this bias, we multiplied accelerometer x and y data by -1 for left-handed individuals.\ ![The evolution of y-acceleration over time suggests that two distributions coexist in acceleration data[]{data-label="visina8"}](visualisation_accelerations.jpg){width="50.00000%"} \[fig:acc\] Lags and leads {#lags-and-leads .unnumbered} -------------- The previous features exploit the current value of sensors data. But they do not exploit the past or future sensors data. In order to take into account the time series component of the problem, we added lagged variables: values of existing features 1 to 10 seconds before. Note how important it is to make train and test sets look alike. In the test set, lagged values are always empty for the first line of each subsequence. Whereas in the train set, if we hadn’t split the 30-minute sequences into subsequences, lagged values would hardly be empty. We thus avoided a covariate shift on lagged variables.\ Experimentally, adding lags helped our model perform well, so we also added leads. That is we added new variables giving values of features 1 to 10 seconds after. One may wonder whether adding leads makes sense in real-life applications. Should we really wait a couple of seconds before sending help to an individual, in order to add leads to our model and make sure that the individual actually has a problem? Yet, given the context of the challenge, we decided to exploit this test set artifact.\ When looking at Random Forest or Gradient Boosting trees feature importance, we noted that lead variables were as important as lagged ones. Moreover and quite naturally, the importance of the 1 second lag/lead variables was greater than the importance of 2 second ones, etc. Enriching the data through stack transferring {#enriching-the-data-through-stack-transferring .unnumbered} --------------------------------------------- The room variable indicates the room where the individual is located. Intuitively, this variable should be very useful to predict activity: for instance, when someone is in the toilets, he is probably not jumping nor lying down. Unfortunately, this room variable is available in the train set, but missing in the test set. We here propose a technique, that we call *stack transferring*, to propagate this information. It consists in three steps. ### Step 1 - On the train set, update the room variable: replace its exact values by its out-of-fold predictions {#step-1---on-the-train-set-update-the-room-variable-replace-its-exact-values-by-its-out-of-fold-predictions .unnumbered} On the train set, we replace the exact values of the room variable by out-of-folds predictions of the room variable. In other words, use 9 folds of the train set - corresponding to 9 participants out of 10 - to predict the room variable on the remaining fold. By doing so 10 times, we can predict the room variable on all the train set. This generates out-of-folds predictions of the room variable on the train set. We can now update the room variable on the train set by dropping the exact values of the room variable and keeping its out-of-fold predictions. ### Step 2 - On the test set, predict the room variable {#step-2---on-the-test-set-predict-the-room-variable .unnumbered} The room variable being missing on the test set, we can add it as follows. In step 1, we have trained 10 models that predict room variable. We simply apply them to the test set and average these 10 predictions. Now, the room variable should be available on the test set. ### Step 3 - Use the out-of-folds predictions of the room variable to predict the activity variable {#step-3---use-the-out-of-folds-predictions-of-the-room-variable-to-predict-the-activity-variable .unnumbered} Now that we have updated the room variable on the train set, and that we have predictions of the room variable on the test set, we can add this variable to the model. It should improve the activity prediction. Notice that the individuals from train and test sets were asked to perform the same list of actions in the same order. Therefore, the room variable had the same distribution on train and test sets. This is a necessary condition for stack transferring to perform well.\ Eventually, feature engineering increased the number of variables from 119 to 2700. Table \[tab:feat\] shows the top 15 features by importance for a Random Forest trained on the engineered train set. 5 of them come from our feature engineering. Feature name Importance (%) ----------------------------------------------------- ---------------- median acceleration in x direction 0.5035 std deviation of acceleration in x direction 0.4622 max acceleration in x direction 0.4290 mean acceleration in x direction 0.4247 room variable is equal to living room 0.4119 lead 1 second of mean acceleration in x direction 0.3726 min y coordinate of 3D centre of mass 0.3676 lead 1 second of median acceleration in x direction 0.3463 min acceleration in x direction 0.3424 min x coordinate of 2D centre of mass 0.3262 std deviation of acceleration in x direction 0.3258 max PIR value of the receiver located upstairs 0.3251 lag 1 second of mean acceleration in x direction 0.3179 mean acceleration in y direction 0.3093 length in y direction of 3D bounding box 0.2991 : Top 15 most important features for our level-one random forest learners. Features colored in green come from feature engineering. \[tab:feat\] ACTIVITY-RECOGNITION MODELS {#activity-recognition-models .unnumbered} =========================== Individual models {#individual-models .unnumbered} ----------------- It is in general a good idea to start with a simple model that does not need much tuning - for instance a Random Forest - while doing feature engineering. They are easy to implement and able to handle large amounts of variables, so they give valuable feedback on the quality of our work. Feature engineering diminished our Random Forest’s error-rate from 22% to 16.4%, ranking us $15^{th}$ of the competition.\ When performance seemed to reach a plateau even when we were adding new features, we tried other models that require more tuning. We then went on for the machine learning blockbuster, XGBoost. We grid-searched its parameters - max depth, min child weight, column sample by tree, subsample - and derived the optimal number of estimators thanks to an early stopping on users 6 and 10. Optimizing XGBoost typically took one hour on our 12 cores computer, which was fast enough to explore a great number of feature combinations.\ The XGBoost classifier can optimize its predictions for a given loss function. This loss function can be chosen among several pre-implemented loss functions. But the metric of the challenge - Brier score - is not one of them. So, we chose a random pre-implemented loss function - logloss. It is not an optimal solution, because minimizing logloss should not necessarily lead to minimizing the Brier score. However, this already performed very well: our error rate reached 14.6% and ranked us top 5. We could then try to customize the XGBoost code to make it optimize the Brier score loss function instead of logloss. Customizing XGBoost {#customizing-xgboost .unnumbered} ------------------- Our goal was to make the XGBoost classifier optimize its predictions for the metric of the challenge - the Brier score. XGBoost provides a Python API to customize softmax loss functions, by defining their gradient and hessian. The first step was to define the softmax Brier score loss function: () = \_[n=1]{}\^N \_[c=1]{}\^C w\_c( \_[n,c]{}() - y\_[n,c]{})\^2 where $\mathbf{p}$ and $\sigma_{n,c_0}(\mathbf{p})$ are respectively equal to $$\mathbf{p}=\Big((p_{n,c})_{\substack{1\leq n\leq N \\ 1\leq c\leq C}}\Big), \sigma_{n,c_0}(\mathbf{p})=\frac{e^{p_{n,c_0}}}{\sum_{c}e^{p_{n,c}}}$$ We can then implement the loss gradient and hessian based on the following expressions. Notice that XGBoost does not work with the exact hessian but with its diagonal approximation. () = \_[n=1]{}\^N \_[n,c\_0]{}() & ( ) = \_[n=1]{}\^N & Unfortunately, the XGBoost Python API only allows this easy customization of loss function when the target is a hard target. In the SPHERE Challenge, the target is probabilistic. An easy way to deal with this issue was to convert the probabilistic (soft) target into a hard target, by keeping for every line the label that has the highest probability. This inevitably generated an approximation in the metric optimized by XGBoost. We managed to minimize this approximation by duplicating lines on our train dataset. For instance, for a given line, if label A has a probability of 0.7, label B of 0.1 and label C of 0.2, then we would create K new lines: $\floor*{0.7K}$ lines that would have label A as hard target, $\floor*{0.1K}$ line would have label B as hard target and $\floor*{0.2K}$ lines would have label C as hard target. By doing so, our XGBoost would optimize the following approximate softmax Brier score: \_[approx]{}() = \_[n=1]{}\^N \_[c=1]{}\^C w\_c(y\_[n,c]{}\^2- 2y\_[n,c]{} &&\ + ) which is quite close to the exact softmax Brier score: $$\mathcal{L}_{exact}(\mathbf{p}) = \frac{1}{N} \sum_{n=1}^N \sum_{c=1}^C w_c(y_{n,c}^2 -2y_{n,c}\sigma_{n,c}(\mathbf{p}) +\sigma_{n,c}(\mathbf{p}) ^2)$$ We refer to K as a resolution parameter, that governs the approximation in the Brier score metric made by XGBoost when dealing with a hard target instead of a probabilistic one. Higher values of K reduce this approximation. We have implemented this method with $K=10$, meaning that our train dataset consisted of 100,000 lines and 2700 features: but XGBoost training time was too long.\ Therefore, the only solution was to fork the C++ XGBoost source code to make it accept customized loss functions even when the target is probabilistic. This was much trickier, but unsurprisingly it gave slightly better results than the traditional XGBoost. Customized XGBoost decreased our CV score from 0.1817 to 0.1814. Stacking {#stacking .unnumbered} -------- Once we had trained 10 individual models - including linear regressions, Naive Bayes classifiers, Random Forests, extra-trees and XGBoost models -, we opted for ensemble learning methods. A grid-searched XGBoost combined the predictions of our individual models and leveraged their strengths. It turned out to be very efficient: it reduced our error rate to 12.9% and ranked us number 1 at that point. ![The stacking technique improved our CV results from 0.181 to 0.178 []{data-label="visina8"}](stacking2.jpg){width="50.00000%"} POSTPROCESSING {#postprocessing .unnumbered} ============== The previous approaches consider each prediction independently. However, it seems very unlikely that a person lying on a bed can be jumping the next second. This means that there are chances that transitions from one activity to another follow different probabilities. This mathematical property is known as the Markov chain property. A great way to take advantage of this underlying structure is to implement Hidden Markov Models. Smoothing predictions over time {#smoothing-predictions-over-time .unnumbered} ------------------------------- Yet, given the deadline, we did not have time to implement HMM models. We rather opted for a post-processing that smooths predictions over time. The idea is to make a weighted average between the activity predictions of a given second and the activity predictions of the last two seconds and of the future two seconds. We optimized the coefficients of this weighted average. Post-processing gave tremendous cross-validation results, with an error rate around 11%. However, we did not have time to submit it in our final model. CONCLUSION {#conclusion .unnumbered} ========== In this paper, we presented our solution to SPHERE Challenge as well as several techniques that may have worked if we had more time. Our final solution is based on a rich pre-processing and cutting-edge machine learning methods. After recreating a train set similar to the test set, we perform feature engineering. To our knowledge, what we call “stack transferring” - the idea of using predictions of a variable known in the train set but not in the test set as features - is new. The final model is based on the stacking of weak learners through a grid searched XGBoost algorithm.\ Our solution won the second prize of the challenge on the private leader-board, though we were ranked first on the public one. We hope that this work can modestly contribute to finding better way to detect old people fall for a quicker intervention. ACKNOWLEDGMENT {#acknowledgment .unnumbered} ============== We thank Dataiku for allocating time and servers for our team. We also thank DrivenData, ECML-PKDD and SPHERE for organizing and hosting this contest and supplying these valuable datasets.\ [99]{} Chen, Tianqi, and Carlos Guestrin. “Xgboost: A scalable tree boosting system.” arXiv preprint arXiv:1603.02754 (2016). Wolpert, David H. “Stacked generalization.” Neural networks 5.2 (1992): 241-259. Pan, Sinno Jialin, and Qiang Yang. “A survey on transfer learning.” IEEE Transactions on knowledge and data engineering 22.10 (2010): 1345-1359. Breiman, Leo. “Random forests.” Machine learning 45.1 (2001): 5-32. Pedregosa, Fabian, et al. “Scikit-learn: Machine learning in Python.” Journal of Machine Learning Research 12.Oct (2011): 2825-2830. Strobl, Carolin, et al. “Bias in random forest variable importance measures: Illustrations, sources and a solution.” BMC bioinformatics 8.1 (2007): 1. Chen, Tianqi, and Tong He. “xgboost: eXtreme Gradient Boosting.” R package version 0.4-2 (2015). [^1]: $^{1}$maxime.voisin@stanford.edu [^2]: $^{2}${leo.dreyfus-schmidt, pierre.gutierrez,samuel.ronsin, marc.beillevaire} @dataiku.com [^3]: $^{3}$Second Prize, but third place! The submission of a team ranked top 2 was deemed invalid

--- abstract: 'Investigation of the effect of the dynamical stage of heavy-ion collisions indicates that the increasing width of the initial isospin distributions is reflected by a significant modification of the isoscaling slope for the final isotopic distributions after de-excitation. For narrow initial distributions, the isoscaling slope assumes the limiting value of the two individual initial nuclei while for wide initial isotopic distributions the slope for hot fragments approaches the initial value. The isoscaling slopes for final cold fragments increase due to secondary emissions. The experimentally observed evolution of the isoscaling parameter in multifragmentation of hot quasiprojectiles at E$_{inc}$=50 AMeV, fragmentation of $^{86}$Kr projectiles at E$_{inc}$=25 AMeV and multifragmentation of target spectators at relativistic energies was reproduced by a simulation with the dynamical stage described using the appropriate model ( deep inelastic transfer and incomplete fusion at the Fermi energy domain and spectator-participant model at relativistic energies ) and the de-excitation stage described with the statistical multifragmentation model. In all cases the isoscaling behavior was reproduced by a proper description of the dynamical stage and no unambiguous signals of the decrease of the symmetry energy coefficient were observed.' author: - | Martin Veselsky\ \ Institute of Physics, Slovak Academy of Sciences,\ Dubravska cesta 9, Bratislava, Slovakia\ e-mail: fyzimarv@savba.sk title: Dynamical aspects of isotopic scaling --- Introduction {#introduction .unnumbered} ============ Multifragmentation studies in the recent years highlighted the importance of fragment yield ratios which can be used to extract thermodynamical observables of the fragmenting system such as temperature and chemical potential ( for a review of related methods see e.g. [@MVIsoTrnd] ). In the context of isotopic distributions, the fragment yield ratios represent the details of the distribution sensitive to the isospin degrees of freedom. Similar sensitivity can be explored globally by investigating the ratio of isotopic yields from two processes with different isospin asymmetry, essentially dividing the two isotopic distributions in a point-by-point fashion. When employing the macro-canonical formula for fragment yields, such a ratio will depend on N and Z as follows [@TsangIso] $$R_{21}(N,Z) = Y_{2}(N,Z)/Y_{1}(N,Z) = C \exp(\alpha N + \beta Z) \label{r21isots}$$ with $\alpha$ = $\Delta \mu_{n}$/T and $\beta$ = $\Delta \mu_{p}$/T, where $\Delta \mu_{n}$ and $\Delta \mu_{p}$ are the differences in the free neutron and proton chemical potentials, respectively, of the fragmenting systems. C is an overall normalization constant. Alternatively [@BotvIso] the N and Z dependence can be expressed as $$R_{21}(N,Z) = Y_{2}(N,Z)/Y_{1}(N,Z) = C \exp(\alpha^{\prime} A + \beta^{\prime} (N-Z) ) \label{r21isobt}$$ thus introducing the parameters which can be related to the isoscalar and isovector components of the free nucleon chemical potential since $\alpha^{\prime}$ = $\Delta (\mu_{n}+\mu_{p})$/2T and $\beta^{\prime}$ = $\Delta (\mu_{n}-\mu_{p})$/2T. An exponential scaling of $R_{21}$ with neutron and proton numbers was observed experimentally in multifragmentation data from collisions of high energy light particles with massive target nuclei [@BotvIso; @Lozhkin] and from collisions between mass symmetric projectiles and targets at intermediate energies [@TsangIso]. Such exponential behavior is called isotopic scaling or isoscaling [@TsangIso] ( the parameters $\alpha, \beta, \alpha^{\prime}, \beta^{\prime}$ being referred to as isoscaling parameters ). An isoscaling behavior was also reported in studies of heavy residues [@GSHRIso] and in fission data [@Fisiso]. The values of the isoscaling parameters were related by several authors to various physical quantities such as the symmetry energy [@TsangIso; @BotvIso], the level of isospin equilibration [@GSHRIso] and the values of transport coefficients [@Fisiso]. As demonstrated in the literature, isoscaling appears to be a global feature of nuclear reactions and multifragmentation data and the isoscaling parameters show sensitivity to both the dynamical and the thermodynamical properties of the hot source created in the early stage of the collision. It is of interest to clarify to which extent the isoscaling behavior is modified by the process of de-excitation in the late stage of the reaction and whether the dynamical and thermodynamical properties of the hot source can be disentangled. ![ Isoscaling plots after de-excitation by the SMM for nuclei with mass A = 50 and an excitation energy of 125 MeV. Upper and lower rows represent cold and hot partitions, respectively. Squares and thick solid lines - isoscaling plots and exponential fits for Z$\leq$6, scattered dots - all Z’s. The dynamical stage ( stars connected by thin solid lines ) was simulated by shifted Gaussians ( N/Z = 1.0, 1.3 ) with three different values of widths ( $\sigma_Z$ = 0.5, 2.5, 4.5 , see panels from left to right ). []{data-label="A50X125"}](isdnfg1.eps){width="10.0cm" height="7.0cm"} Effect of the de-excitation stage on isoscaling {#effect-of-the-de-excitation-stage-on-isoscaling .unnumbered} =============================================== To disentangle the dynamical and thermodynamical properties of the hot source in the experimental data and, specifically, to determine isoscaling properties after the dynamical stage, one can simulate the de-excitation process for various initial isotopic distributions produced in the dynamical stage. Such simulations allow to establish the relation between the isoscaling behavior prior to and after the de-excitation stage. In the present work the de-excitation stage is simulated using the code SMM [@SMM], representing the combination of the statistical multifragmentation model ( SMM ) for highly excited nuclei with evaporation/fission cascade at lower excitation energies. Simulations of the de-excitation stage with the SMM proved consistently better than sequential binary decay models, especially for the neutron-rich nuclides [@MVKrNi] and residues produced after the de-excitation of hot nuclei [@MVSnAl]. Some discrepancies were observed for the yields of a limited set of $\beta$-stable nuclei [@MVKrNi] close to the projectile, which were overestimated due to a low probability for the emission of complex fragments below multifragmentation threshold [@MVSnAl]. The effect of the de-excitation stage was investigated for initial ( dynamical ) isobaric distributions with three masses A = 25, 50, 100, thus covering the typical mass range where multifragmentation studies are commonly performed. In nuclear reactions, the de-excitation stage is preceded by the dynamical stage where hot nuclei are formed. An open question consists of understanding the isoscaling behavior after the dynamical stage and how it is modified by the effect of de-excitations. A possible way to answer such question is to generate the distributions of hot nuclei exhibiting isoscaling and to observe the effect of de-excitation. In order to generate the initial dynamical distributions exhibiting isoscaling one can explore the well known fact that after dividing two Gaussian distributions with equal width and shifted centers, an exponential is obtained and thus isoscaling is guaranteed. The logarithmic slope of such exponential ( commonly referred to as isoscaling parameter ) will thus be determined by the centers of the two Gaussian distributions, $x_1$ and $x_2$, and their common width, $\sigma$ $$\alpha = ( x_2 -x_1 )/\sigma^2 . \label{isogau}$$ Varying of the parameters of these Gaussian distributions allows one to vary the initial isoscaling behavior at the early dynamical stage. In the present work the positions of the centers are fixed and the Gaussian width is used to control isoscaling behavior. Such a choice reflects the situation occurring in damped nucleus-nucleus collisions where the initial isospin asymmetry is not changed dramatically while the width of distribution evolves quickly with damping of the initial kinetic energy. In the simulations presented here, isoscaling after the dynamical stage was simulated using initial isotopic distributions approximated by two Gaussians with shifted centers ( N/Z = 1.0, 1.3 ). Three different values of common Gaussian widths were used, $\sigma_Z$ = 0.5, 1.5, 3.5 for A = 25 and 0.5, 2.5, 4.5 for A = 50, 100 . For each mass, the yields of final products were simulated with good statistics for hot sources with five selected atomic numbers. For other elements, the yields of final products were estimated using polynomial interpolations. The calculation was carried out for two values of excitation energy, E$^{*}$ = 2.5 and 5.0 AMeV, the former being close to multifragmentation threshold while the latter corresponding to the region where multifragmentation is the main de-excitation mode. ![ Isoscaling plots after de-excitation by the SMM for nuclei with mass A=50 and an excitation energy of 250 MeV ( $\sigma_Z$ = 0.5, 2.5, 4.5 ). Symbols and lines as in Fig. \[A50X125\]. []{data-label="A50X250"}](isdnfg2.eps){width="10.0cm" height="7.0cm"} In Figs. \[A50X125\], \[A50X250\] are shown results for nuclei with mass A = 50 and excitation energies of 125 and 250 MeV, respectively. For the narrow initial Gaussian distributions ( left columns ) the isoscaling slopes appear to be governed by the intrinsic effect of de-excitation, while with increasing width ( middle and right columns ) the effect of initial distributions appears to take over. Secondary emission leads to a slight increase of the slope due to a lower temperature which, according to macro-canonical theory ( Eqs. (\[r21isots\]) and (\[r21isobt\]) ), enters into the denominator. For the widest initial distribution ( right columns ) the isoscaling plot in the hot partition appears to follow the initial isoscaling plot and the increase of the slope by secondary emission determines the final discrepancy. ![ Isoscaling plots after de-excitation by the SMM for nuclei with mass A = 25 and an excitation energy of 63 MeV ( $\sigma_Z$ = 0.5, 1.5, 3.5 ). Symbols and lines as in Fig. \[A50X125\]. []{data-label="A25X63"}](isdnfg3.eps){width="10.0cm" height="7.0cm"} ![ Isoscaling plots after de-excitation by the SMM for nuclei with mass A=25 and an excitation energy of 125 MeV ( $\sigma_Z$ = 0.5, 1.5, 3.5 ). Symbols and lines as in Fig. \[A50X125\]. []{data-label="A25X125"}](isdnfg4.eps){width="10.0cm" height="7.0cm"} Results for nuclei with mass A = 25 and excitation energies 63 and 125 MeV are shown, respectively, in Figs. \[A25X63\], \[A25X125\]. Analogous conclusions as for A = 50 can be made. However, the dominance of the initial isoscaling width an increase of the slope by secondary emissions is observed in both middle and right panels, indicating that such behavior takes over earlier as the initial width increases, as compared to the case of A=50. ![ Isoscaling plots after de-excitation by the SMM for nuclei with mass A = 100 and an excitation energy of 250 MeV ( $\sigma_Z$ = 0.5, 2.5, 4.5 ). Symbols and lines as in Fig. \[A50X125\]. []{data-label="A100X250"}](isdnfg5.eps){width="10.0cm" height="7.0cm"} ![ Isoscaling plots after de-excitation by the SMM for nuclei with mass A = 100 and an excitation energy of 500 MeV ( $\sigma_Z$ = 0.5, 2.5, 4.5 ). Symbols and lines as in Fig. \[A50X125\]. []{data-label="A100X500"}](isdnfg6.eps){width="10.0cm" height="7.0cm"} The case of A = 100 ( excitation energies 250 and 500 MeV ) is shown in Figs. \[A100X250\], \[A100X500\]. The behavior is analogous to the previous cases, but the isoscaling slope for hot fragments with Z$\leq$6 in the case of the widest initial distribution ( lower right panel ) appears to be larger than initial one. This is due to the fact that very isospin-asymmetric nuclei are produced and the symmetry energy of such light nuclei increases quickly and thus increasingly influencing the overall energy balance. However, for heavier fragments ( dots ) the isoscaling behavior appears to follow the initial distributions better. Such sensitivity to the symmetry energy predicted for light fragments originating from hot heavy nuclei can in principle provide a probe of the symmetry energy coefficient at the hot stage, if the reaction dynamics leads to wide initial distributions that can be reconstructed possibly via full calorimetry of the hot source or via reliable simulations of the initial stage. In general, the increasing width of initial isotopic distributions ( and the corresponding decrease of the initial isoscaling slope ) is reflected by significant modification of the final isoscaling slope after de-excitation. For narrow initial Gaussian distributions, the isoscaling slope assumes the limiting value fully determined by the details of the de-excitation stage. For wide initial Gaussian distributions, the isoscaling slope for hot fragments approaches the slope of initial isoscaling plots and it is thus fully determined by the initial stage. This correspondence is modified by secondary emission and the isoscaling slopes for final cold fragments are larger possibly due to a corresponding decrease of the temperature during secondary emissions. It is noteworthy that the width of initial Gaussian distributions induces a decrease of the isoscaling parameters comparable to the values, reported in the literature [@TsangIso; @BotvIso], and explained as an effect of a decreasing symmetry energy, according to liquid-drop based formula that relates the symmetry energy coefficient directly to the isoscaling parameter. However, the effect of the dynamical stage and specifically of the width of the initial distributions was not considered in the analysis and the estimates are based on simulation for individual initial nuclei, which appears to be an over-simplified approach. Investigation of the dynamical stage in selected reactions {#investigation-of-the-dynamical-stage-in-selected-reactions .unnumbered} ========================================================== The investigation presented in previous section suggests that the dynamical stage, leading to the evolution of a considerable width of the isospin distribution, plays an important role in determining the isoscaling behavior of final products. A detailed understanding of reaction dynamics is thus necessary to allow disentangling the properties of the hot multifragmentation source from the artifacts of the reaction dynamics. Three selected cases ( multifragmentation of hot quasiprojectiles at incident energy of 50 AMeV, fragmentation of a $^{86}$Kr beam at an incident energy of 25 AMeV and multifragmentation of target spectators at relativistic energies ) will be presented in this section in order to investigate the effects of reaction dynamics on isoscaling in few energy regions for hot nuclei with different masses. . Multifragmentation of projectiles with masses A $\sim$ 25 at Fermi energies {#multifragmentation-of-projectiles-with-masses-a-sim-25-at-fermi-energies .unnumbered} ---------------------------------------------------------------------------- Multifragmentation of hot quasiprojectiles with A$\sim$25 was studied in reactions $^{28}$Si+$^{124,112}$Sn at projectile energies of 30 and 50 AMeV [@SiSnNExch]. The observed fragment data [@SiSnNExch] provide full information ( with exception of emitted neutrons ) on the decay of thermally equilibrated hot quasi-projectiles with known masses ( A = 20 - 30 ), charges, velocities and excitation energies. A detailed investigation of the reaction mechanism [@SiSnNExch] allowed to establish a dominant reaction scenario. An excellent description of fragment observables was obtained using the deep-inelastic transfer ( DIT ) model [@DITTGSt] for the early stage of the collision and the statistical multifragmentation model ( SMM ) [@SMM] for the de-excitation stage. The DIT model describes well the dynamical properties of the reconstructed quasi-projectile such as its center of mass velocity, excitation energy and isospin-asymmetry. Fragment observables such as multiplicities, charge distributions and mean N/Z values for a given charge were also reproduced reasonably well [@SiSnNExch]. Thus the reaction mechanism can be considered well understood. The contribution from non-equilibrium processes such as pre-equilibrium emission was shown to be weak [@SiSnNExch]. According to successful DIT+SMM simulation, the number of emitted neutrons, not detected in the experiment, was between one and two per event and the underestimation of the excitation energy due to undetected neutrons can be estimated to be approximately 10–15 % in the whole range of excitation energies. The excitation energy dependence of the isoscaling slope, corrected for the 1/T temperature dependence, exhibits a turning-point at E$^{*}$=4 AMeV [@SiSnIso] which can be interpreted as a signal of the onset of separation into an isospin asymmetric dilute phase and an isospin symmetric dense phase. The onset of the chemical separation is correlated to the onset of the plateau in the caloric curve, thus signaling that chemical separation is accompanied by a latent heat. ![ Simulated isoscaling plots ( symbols ) and fits to experimental data ( lines ) from the statistical decay of hot quasi-projectiles in the reactions $^{28}$Si+$^{124,112}$Sn at incident energies of 50 AMeV. The upper left panel corresponds to inclusive data while the other panels correspond to the five excitation energy bins. []{data-label="Iso50"}](isdnfg7.eps){width="7.0cm" height="8.0cm"} In Fig. \[Iso50\] are presented simulated isoscaling data ( symbols ) from statistical decay of hot quasi-projectiles produced in the reactions $^{28}$Si+$^{124,112}$Sn at projectile energy 50 AMeV. The isoscaling plots are presented not only for the inclusive data ( upper left panel ) but also for five bins of excitation energy. The isoscaling slope in the simulations depends on the excitation energy almost identically as in the experimental data, represented by the solid lines. The DIT+SMM simulation fully reproduces experimental isoscaling behavior. The isoscaling parameters of the initial isospin distributions exhibit a similar trend as the final values, in agreement with the results of simulation presented in Figs. \[A25X63\], \[A25X125\]. The shift between simulated initial isotopic distributions is constant in all the excitation energy bins and the evolution is essentially determined by their widths that are smaller in the lowest excitation energy bins and then depend on excitation energy only weakly. Despite the overall success, the simulation does not allow one to extract unambiguously the values of the temperature corresponding to isospin trends of hot fragment distributions, due to the fact that the production of multiple fragments occurs mostly in secondary emissions represented by Fermi decay. The formalism of Fermi decay is analogous to the multifragmentation model with cold fragment partitions, which thus essentially duplicates the multifragmentation model with hot fragments used in the SMM. Thus the model does not provide an unambiguous equivalent to experimental double-isotope ratio or slope temperatures, used in [@SiSnIso; @SiSnNPA]. The duplicity of fragmentation stages in the calculation is consistent with the analogous success of the model of sequential binary decay for light nuclei, where the proper exploration of available phase space appears the most important requirement to successful models. Fragmentation of $^{86}$Kr beam with $^{124,112}$Sn targets at 25 AMeV {#fragmentation-of-86kr-beam-with-124112sn-targets-at-25-amev .unnumbered} ---------------------------------------------------------------------- The isoscaling phenomena are not restricted to relatively light fragments but can be observed also in heavy residue data, collected at forward angles. Yield ratios $ R_{21}(A,Z)$ of projectile residues from the reactions $^{86}$Kr+$^{124,112}$Sn at 25 AMeV [@KrSn_1] were investigated and isoscaling behavior was observed for each isotopic and isotonic chain. The isoscaling slopes are constant for residue mass range A = 25 – 60, corresponding to primary events with the maximum observed excitation energy of 2.2 AMeV. The slopes exhibited gradual decrease with increasing mass of the residues. Assuming that the fragmentation occurs at normal density, using C$_{sym}$ = 25 MeV [@BotvIso], the values of isoscaling parameters can be used to determine the values of $ \Delta( N/Z )_{qp} $ ( where qp means quasi-projectile ) as a function of the observed residue mass A and charge Z, thus demonstrating the evolution of the N/Z equilibration process in isospin-asymmetric collisions. The monotonic increase of $ \Delta( N/Z )_{qp} $ with excitation energy can be understood as a result of the mechanism of nucleon exchange. ![ Isoscaling plots for the reactions of $^{86}$Kr+$^{124,112}$Sn at an incident energy of 25 AMeV. Left panel - simulated data for final fragments, middle panel - experimental data [@GSKrSn], right panel - simulated data after dynamical stage. The lines represent exponential fits. []{data-label="IsoKrSn"}](isdnfg8.eps){width="14.0cm" height="5.0cm"} The left panel of Fig. \[IsoKrSn\] shows isoscaling plots corresponding to the simulations of the reactions of $^{86}$Kr (25AMeV) with $^{124,112}$Sn. The used simulation is the same as in [@MVKrSn] where it allowed to reproduce experimental cross sections for neutron-rich nuclides and residues from de-excitation of hot nuclei. Some discrepancies were observed in the yields of a limited set of $\beta$-stable nuclei close to the projectile, which were overestimated due to a low probability for the emission of complex fragments below multifragmentation threshold. As a comparison, in the middle panel experimental isoscaling plots are shown. For nuclei with Z=25-30 the simulation and experiment lead to a similar behavior with constant slopes and consistent values of the isoscaling parameters. For lower Z’s ( Z$\leq$24, not shown in the Fig. \[IsoKrSn\] ) the experimentally observed slopes become even larger than the calculated ones, thus eventually implying a physical phenomenon not included in the simulations, however this discrepancy can be caused by experimental limitations in measuring the yields of these elements in the tails of the isotopic distributions ( as it is discussed in [@MVKrSn] ) leading to lower widths of the isotopic distributions and thus larger apparent values of the isoscaling parameters. For heavier nuclei with N$>$44, the simulation leads to a reverse trend of the yield ratios toward unity, possibly signaling the onset of a reaction mechanism independent of the N/Z of the target, possibly quasi-elastic ( direct ) few-nucleon transfer taking place in very peripheral collisions. The experimental isoscaling behavior for these nuclei shows signs of a similar reverted trend, the transition is not as regular as in the simulation and the inclusion of the points from this region into the exponential fits ( lines ) leads to a decrease of the apparent isoscaling slopes. Such decrease of the slope of exponential ( “isoscaling” ) fits is shown by the lines in the left panel of Fig. \[IsoKrSn\], despite the very poor quality of such fits. Thus the evolution of the apparent exponential slopes in both experimental and simulated data suggest a mixing of two components: one component very sensitive to the N/Z of the target, possibly due to an intense nucleon exchange; a second component, insensitive to the N/Z of the target, possibly quasi-elastic few-nucleon exchange. This situation is demonstrated in the right panel of Fig. \[IsoKrSn\] where simulated isoscaling plots are shown for the dynamical stage prior to de-excitation. The isotopes with Z = 30 - 36 exhibit regular isoscaling behavior, except for a structure around N = 50 corresponding to elements close to the projectile charge, which can be identified with quasi-elastic processes. Despite minor effect on isoscaling plots, these points represent a significant portion of the reaction cross section and the corresponding wide excitation energy distribution leads to a mixing with the data for lighter elements and thus to a modification of their isoscaling behavior after de-excitation. The discrepancy of the final simulated and experimental isoscaling behavior can be possibly attributed to an underestimated probability for the emission of complex fragments below multifragmentation threshold in the SMM. ![ Evolution of the isoscaling parameter at the dynamical stage as a function of the centrality for the reactions $^{12}$C+$^{112,124}$Sn at relativistic energies. Solid line - results of simulations. Symbols - experimental data [@LeFevre]. []{data-label="IsoSnC"}](isdnfg9.eps){width="7.0cm" height="7.0cm"} Multifragmentation of target spectators at relativistic energies {#multifragmentation-of-target-spectators-at-relativistic-energies .unnumbered} ---------------------------------------------------------------- The dominant reaction mechanism at the relativistic energies is represented by the spectator-participant model where a hot region is formed in the participant zone ( zone of geometric overlap ) while the spectator regions are colder. These spectators can be warm enough to undergo multifragmentation. The isoscaling behavior in multifragmentation of target spectators was investigated in the literature [@Lozhkin; @LeFevre] and a dependence of the isoscaling parameters on the centrality of the collision was observed [@LeFevre]. The value of the isoscaling parameters was related to the symmetry energy [@TsangIso; @BotvIso] and the decrease of the symmetry energy coefficient was reported [@LeFevre]. However, based on the conclusions of the previous section, the effect of the reaction dynamics, specifically of the evolving width of the mass and charge distributions, can be considered as an alternative interpretation. The volume and thus the most probable mass $A_{TS}^{abr}$ of the target spectator can be estimated as a function of the impact parameter using the model of geometrical abrasion [@Gosset]. The number of nucleons in the spectator zone $A_{TS}$ can be estimated according to the binomial distribution $$P(A_{TS}) = (^{A_{T}}_{A_{TS}}) (\frac{A_{TS}^{abr}}{A_{T}})^{A_{TS}} (1-\frac{A_{TS}^{abr}}{A_{T}})^{A_{T}-A_{TS}} \label{a1binom }$$ where $A_{T}$ is the target mass number. The charge of the spectator $Z_{TS}$ can be determined as [@Fried] $$\label{ffried} P(Z_{TS}) = \frac{(^{Z_{T}}_{Z_{TS}})(^{N_{T}}_{N_{TS}})}{(^{A_{T}}_{A_{TS}})}$$ where $Z_{T}$ is the target mass number and $N_{TS}$ and $N_{T}$ are the neutron numbers of the target spectator and the target, respectively. The excitation energy of the target spectator can be estimated, according to [@Gaim], as proportional to the number of abraded nucleons with the proportionality factor 27 MeV, which was found to be consistent with experimental data [@27MeV]. Fig. \[IsoSnC\] shows the evolution of the isoscaling parameter after the dynamical stage ( solid line ), obtained using centers and widths of simulated fragment distributions from the abrasion calculation, as a function of centrality for the reactions $^{12}$C+$^{112,124}$Sn at relativistic energies. Symbols show the experimental points reported in [@LeFevre] for incident energies of 300 and 600 AMeV. The calculation is capable to reproduce the experimental trend without any assumptions about the decrease of the symmetry energy coefficient. The discrepancy at low centrality ( small excitation energy ) can be explained by the asymptotics of the experimental points ( after the de-excitation stage ) for the width approaching zero where the isoscaling parameter assumes a value determined by the intrinsic properties of the de-excitation. In central collisions the isoscaling parameter exhibits analogous saturation at a somewhat higher value ( by 10 - 15 % ) than the ones reported for the experimental data. The calculated widths of the target spectator isospin distributions at saturation are similar to the situation in the middle panel of Figs. \[A100X250\], \[A100X500\], where the initial ( dynamical ) isoscaling parameter reflects the initial value with remaining discrepancies due to secondary emissions. In the present case the excitation energy exceeds 11 AMeV and the effect of secondary emissions on the slope parameter in the test calculations was within statistical errors ( not exceeding 10 % ). Thus after taking secondary emissions into account the discrepancy will raise to the level of about 20 %, resulting in the corresponding decrease of the apparent symmetry energy coefficient from 25 MeV to about 20 MeV. Such a decrease is comparable with the uncertainty in the chosen initial values of the symmetry energy coefficient in the models of nuclear ground state properties ( with the value of 23 MeV being commonly used ). The decrease of the apparent symmetry energy coefficient can be further caused by other dynamical phenomena not included in the model such as emission at the pre-equilibrium stage ( leading to a decrease of the shift between the centroids and to an additional increase of the widths of the isotopic distributions ). The apparent value of the symmetry energy coefficient around 20 MeV can thus hardly be interpreted as a signal of a significant decrease of the nuclear symmetry energy. In any case it is much less significant than it was reported in [@LeFevre] ( up to factor of 6 after taking into account secondary emissions ), where the effect of the dynamical evolution of the initial isotopic distributions ( the width in particular ) was not considered. The model description of the dynamical evolution presented here thus allows one to reproduce the reported discrepancy, with the remaining discrepancies on the apparent symmetry energy coefficient being not significant enough to be declared as an unambiguous signal. Summary and conclusions {#summary-and-conclusions .unnumbered} ======================= Investigation of the effect of the dynamical stage of heavy-ion collisions established that the increasing width of the initial isotopic distributions induces a significant modification of the isoscaling slopes after the de-excitation stage. For narrow isotopic distributions, the isoscaling slope assumes the limiting value for two individual initial nuclei which is fully determined by the details of the de-excitation. For wide initial isotopic distributions, the isoscaling slope for hot fragments approaches the initial isoscaling slope and it is thus fully determined by the initial stage. This correspondence is modified by secondary emissions and the isoscaling slopes for final cold fragments are larger by an amount possibly corresponding to a lowering temperature during secondary emissions. The decrease of the isoscaling parameters, caused by the increase of the width of initial Gaussian distributions, is comparable in magnitude to the values, reported in the literature as an effect of the decrease of the symmetry energy. The experimentally observed evolution of the isoscaling parameter in the statistical decay of hot quasiprojectiles from the reactions $^{28}$Si+$^{124,112}$Sn at projectile energy 50 AMeV is reproduced by a simulation with the dynamical stage described by the deep inelastic transfer model and the de-excitation stage described using the statistical multifragmentation model. The evolution of the apparent isoscaling slopes in both experimental and simulated data for projectile residues from the reactions of $^{86}$Kr+$^{124,112}$Sn at incident energies of 25 AMeV suggests a mixing of two components, one sensitive to the N/Z of the target, possibly due to an intense nucleon exchange, and a second component due to a quasi-elastic few-nucleon exchange, almost insensitive to the N/Z of the target. The discrepancy between the final simulated and experimental isoscaling behavior can be possibly attributed to the absence of a mechanism for the emission of complex fragments below multifragmentation threshold in the SMM. The decrease of the isoscaling parameter in the multifragmentation of target spectators in central collisions was reproduced using the simulation using the spectator-participant model for the dynamical stage and the SMM model for the de-excitation stage. In all cases the isoscaling behavior was reproduced by a proper description of the dynamical stage and no unambiguous signals on the decrease of the symmetry energy coefficient were observed. The author acknowledges L. Tassan-Got for providing his DIT code and A.S. Botvina for providing his SMM code. This work was supported through grant of Slovak Scientific Grant Agency VEGA-2/5098/25. [00]{} M. Veselsky, Fiz. Elem. Chastits At. Yadra 36, 400 (2005); Physics of Part. and Nuclei 36, 213 (2005). M.B. Tsang et al., Phys. Rev. Lett. 2001. V.86. P.5023. A.S. Botvina et al., Phys. Rev. C. 2002. V.65. P.44610. O.V. Lozhkin, W. Trautmann, Phys. Rev. C. 1992. V.46. P.1996. G.A. Souliotis et al., Phys. Rev. C. 2003. V.68. P.24605. M. Veselsky, G.A. Souliotis, M. Jandel, Phys. Rev. C. 2004. V.69. P.44607. J.P. Bondorf et al., Phys. Rep. 1995. V.257. P.133. M. Veselsky, G.A. Souliotis, Nucl. Phys. A 765 (2006) 252. M. Veselsky et al., Nucl. Phys. A. 2003. V.724. P.431. M. Veselsky et al., Phys. Rev. C. 2000. V.62. P.064613. L. Tassan-Got, PhD Thesis, 1988, Orsay, France, IPNO-T-89-02, 1989; L. Tassan-Got, C. Stéfan, Nucl. Phys. A. 1991. V.524. P.121. M. Veselsky, G.A. Souliotis, S.J. Yennello, Phys. Rev. C. 2004. V.69. P.31603(R). M. Veselsky, S.J. Yennello, Nucl. Phys. A 749, 114c (2005). G.A. Souliotis et al., Phys. Lett. B. 2004. V.588. P.35. G.A. Souliotis et al., Phys. Rev. Lett. 91 (2003) 022701. M. Veselsky, G.A. Souliotis, arxiv.org:nucl-th/0607032. A. Le Fevre et al., Phys. Rev. Lett. 94, 162701 (2005). J. Gosset [*et al.*]{}, Phys. Rev. [**C 16**]{} (1977) 629. W.A. Friedman, Phys. Rev. [**C 27**]{} (1983) 569. J.-J. Gaimard, K.-H. Schmidt, Nucl. Phys. [**A 531**]{} (1991) 709. K.-H. Schmidt et al., Phys. Lett. [**B 300**]{} (1993) 313.

--- abstract: 'Approach-to-equilibrium molecular dynamics simulations have been used to study thermal transport in nanocrystalline graphene sheets. Nanostructured graphene has been created using an iterative process for grain growth from initial seeds with random crystallographic orientations. The resulting cells have been characterized by the grain size distribution based on the radius of gyration, by the number of atoms in each grain and by the number of atoms in the grain boundary. Introduction of nanograins with a radius of gyration of 1 nm has led to a significant reduction in the thermal conductivity to 3% of the value in single crystalline graphene. Analysis of the vibrational density of states has revealed a general reduction of the vibrational intensities and broadening of the peaks when nanograins are introduced which can be attributed to phonon scattering in the boundary layer. The thermal conductivity has been evaluated as a function of the grain size with increasing size up to 14 nm and it has been shown to follow an inverse rational function. The grain size dependent thermal conductivity could be approximated well by a function where transport is described by a connection in series of conducting elements and resistances (at boundaries).' author: - 'Konstanze R. Hahn' - Claudio Melis - Luciano Colombo title: 'Thermal transport in nanocrystalline graphene investigated by approach-to-equilibrium molecular dynamics simulations' --- This work is financially supported by the SNF grant with the project number P2ZHP2\_148667. Simulations have been conducted on the HPC resources of CINECA under the project ISCRA\_THETRASI. We also acknowledge financial support by MIUR under project PRIN 2010-2011 GRAF. CM additionally acknowledges Sardinian Regional Government for financial support (P.O.R. Sardegna ESF 2007-13). [29]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (). , , , , ****, (). , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , ****, (). , , , , ****, (). , ****, (). , , , , , , , , , , , ****, (). , , , , , , , , , , , ****, (). , , , , , , , ****, (). , ****, (). , , , ****, (). , , , , ****, (). , , , , ****, (). , ****, (). , ****, (). , , , **. , , , ****, (). , , , ****, (). , , , , , , , , , , , ****, (). , ****, (). , ****, (). , , , , , , ****, (). , , , , , ****, (). , , , ****, (). , , , , ****, ().

--- abstract: 'We study the shapes of pored membranes within the framework of the Helfrich theory under the constraints of fixed area and pore size. We show that the mean curvature term leads to a budding-like structure, while the Gaussian curvature term tends to flatten the membrane near the pore; this is corroborated by simulation. We propose a scheme to deduce the ratio of the Gaussian rigidity to the bending rigidity simply by observing the shape of the pored membrane. This ratio is usually difficult to measure experimentally. In addition, we briefly discuss the stability of a pore by relaxing the constraint of a fixed pore size and adding the line tension. Finally, the flattening effect due to the Gaussian curvature as found in studying pored membranes is extended to two-component membranes. We find that sufficiently high contrast between the components’ Gaussian rigidities leads to budding which is distinct from that due to the line tension.' author: - 'Zhenwei Yao, Rastko Sknepnek, Creighton K. Thomas and Monica Olvera de la Cruz' title: Shapes of pored membranes --- Introduction ============ The cell membrane is a complex bilayer sheet consisting of hundreds of lipid species embedded with numerous surface- and trans-membrane proteins.[@alberts2007molecular] Its main role is to separate the cell’s interior from its surroundings and to act as a conduit for exchanging matter and signaling between the cell and its environment. The cell membrane is a dynamic object whose conformational variations are associated with biological activities such as cell fission, fusion, and adsorption.[@Lip_Sack:1995] Most biological membranes exist in a liquid state where lipid molecules are rather strongly confined to the bilayer plane but can easily diffuse laterally within it. Fluidity allows the membrane to dynamically rearrange its local composition, quickly heal holes, and enable transmembrane transport besides allowing other metabolic functions. A substantial portion of the transport through the cell membrane takes place via pores.[@Biochemstry:book; @lipid_rafts:2009] The presence of a pore changes the topology of a membrane and can significantly influence its conformations and functions.[@talin:1998] Characterizing the conformations of closed membranes has been a subject of active research over the past four decades with numerous experimental[@Torus:1992; @Xavier:1994] and theoretical[@SMoMS; @OuYang:1989; @Lipowsky:1991] studies. Despite a high molecular complexity, when the length scale is large compared to the bilayer thickness and the energy scale is small compared to the typical intermolecular interactions, the cell membrane shape can be successfully described by a simple model proposed by Helfrich nearly forty years ago.[@Helfrich1973] In his seminal paper, Helfrich argued that the low-energy large-scale properties of a liquid membrane can be described in terms of a free energy that is a quadratic function of the two principal curvatures expressed in terms of their two invariants: the mean curvature and the Gaussian curvature. Within the framework of the Helfrich theory, various axisymmetric and non-axisymmetric shapes of closed membranes have been predicted.[@LCMembrane:1999; @Safran2003] In particular, the longstanding physiological puzzle about the biconcave shape typical of the red blood cell has been beautifully solved; the shape of the red blood cell shape has been understood as the conformation that minimizes the Helfrich free energy under a set of prescribed volume and area constraints.[@red_blood_cell:1976] Many predictions based on the Helfrich free energy have been observed experimentally. [@OuYang:1989] For example, the theoretical discovery of the thermal repulsion between membranes, that is to prevent sticking of cells, has been confirmed by small angle X-ray diffraction experiment.[@Helfrich1973; @X_ray:1978] There are various ways to form pores on membranes [*in vivo*]{} and [*in vitro*]{}. For example, pore-forming toxin proteins exist in a wide range of organisms including bacteria, fungi, plant and animal cells.[@Gilbert:2002] By binding at particular sites on a membrane, toxins can create pores via oligomerizing on the membrane surface. The pores created by toxin proteins are of limited sizes. For example, the maximum size of the pore formed by SecYEG on E. Coli is below $2.2-2.4\ \textrm{nm}$.[@pore_Sec] Recent studies have shown that larger pores can be created on a fluid membrane by detergents[@pore_exp:PNAS2001] or submembranous protein talin.[@talin:1998] Note that the size of the pore is controlled by tuning the talin concentration over an appropriate range.[@pore_exp:PNAS2001] The localization of talin mainly along the pore rim, as observed by fluorescent labelling, is likely responsible for stabilizing the pore. A recent experiment introduced a method to create pores of about $15\ \textrm{nm}$ in a lipid membrane.[@Dubois:2001] In a salt-free catanionic solution, charged pores are produced on membranes due to the partial segregation of the anionic surfactant in excess. In this case, the size of a pore can be controlled by tuning the relative amount of anionic and cationic surfactants and thus the charges on a pore. The size of a stable pore is determined by the competition of the line tension energy $\gamma R$ and the electrostatic self-energy $q^{2}/(\epsilon R)$, where $\gamma$ is the line tension, $R$ is the size of the pore, $q$ is the total charge on the pore and $\epsilon$ is the dielectric constant of the medium, such that $R\sim\sqrt{q^{2}/(\gamma\epsilon)}$. Both the increase of charge and the decrease of line tension can enlarge a pore on membrane. In this paper, we study how a pore modifies the morphology of a fluid membrane within the framework of the Helfrich theory. We discuss the equilibrium solutions of the Helfrich shape equation for fluid membranes with fixed area and pore size. In experiments the fixed pore size constraint can be realized by introducing stabilizing agents as discussed above. We find a budding structure in pored membranes, dictated by the mean curvature term in the Helfrich free energy. In studies of the conformation of closed single-component liquid membranes, the Gaussian curvature term in the Helfrich free energy can be omitted, as it is a constant that does not depend on the membrane’s shape. However, this is no longer the case if pores are present. We show that the Gaussian curvature term can significantly influence the shape of a pored membrane by imposing a local constraint on the shape of the membrane near the pore. The Gaussian curvature term tends to pull the membrane outside a pore to the plane where the pore loop lies and the membrane near the pore is flattened. This observation may lead to a simple method to fabricate polyhedral buckled membranes by manipulating the size and position of the pores. In addition, we propose a scheme to find the ratio of the Gaussian rigidity and the bending rigidity from the shape of a pored membrane. This ratio is usually difficult to measure experimentally.[@gaussian_rigidity:2012] The proposed scheme successfully passes the test on a pored membrane generated by Surface Evolver,[@brakke1992surface; @evolver] and is applied on an experimental case. Furthermore, we briefly discuss the stability of a pore on a membrane by relaxing the constraint of fixed pore size and adding the line tension. We find that a budding pore may be meta-stable with very shallow energy barrier and over a very narrow range of values of line tension. Therefore, stabilizing agents like talin proteins in the experiment of Ref.  are essential for a stable pore on fluid membranes. Finally, the flattening effect due to the Gaussian curvature as found in studying pored membranes is extended to two-component membranes. Multicomponent membranes can have a wide variety of morphologies, as has been recently discussed for both liquid[@Hu2011; @demers2012curvature] and polymerized membranes.[@Vernizzi2011; @sknepnek2011buckling; @Sknepnek2012] We find that the flattening effect due to the Gaussian curvature can induce budding in two-component membranes when there is sufficiently high contrast between the components’ Gaussian rigidities. This is recognized as a domain-induced budding, but via a mechanism that is distinct from the conventional line tension driven budding.[@domain:line_tension; @Kohyama2003] ![The mean curvature term in Eq. (\[bending\_energy\]) gives rise to a budding pore (right) instead of making a membrane spherical everywhere (left). The red line represents the opening of the membrane.\[why\_qiao\]](Figure_1) Model ===== The bending energy of a fluid membrane is modeled by the Helfrich free energy:[@Helfrich1973] $$\begin{aligned} E=\frac{1}{2}\kappa\int\left(2H\right){}^{2}dA+\kappa_{G}\int K_{G}dA,\label{bending_energy}\end{aligned}$$ where $\kappa$ ($\sim 10\ k_{B}\textrm{T}$)[@kBT:1992] and $\kappa_{G}$ are the bending rigidity and the Gaussian rigidity, respectively. The mean curvature $2H=1/R_{1}+1/R_{2}$ and the Gaussian curvature $K_{G}=1/\left(R_{1}R_{2}\right)$, where $R_{1}$ and $R_{2}$ are the radii of principal curvatures. For real membranes, $\kappa>0$ and $\kappa_{G}<0$.[@Webb:2005] Note that in Eq. (\[bending\_energy\]) we have assumed that the spontaneous curvature $H_{0}=0$, as is the case if there is no asymmetry with respect to the middle surface of the bilayer. The negative sign of the Gaussian rigidity indicates that it favors lower genus surfaces.[@Struik:1988] For example, without considering the mean curvature term, a spherical membrane is more stable than a toroidal membrane; the integrals of the Gaussian curvature for sphere and torus are $4\pi$ and zero, respectively. According to the Gauss-Bonnet theorem, the integral of the Gaussian curvature over a manifold $M$ is related to the integral of the geodesic curvature $k_{g}$ along the boundary of the manifold $\partial M$ by $$\int_{M}dAK_{G}=2\pi\chi\left(M\right)-\oint_{\partial M}k_{g}dl\label{GB}$$ where $\chi(M)$ is the Euler characteristic of the manifold $M$.[@Struik:1988] For a closed manifold $M$ without pores, the geodesic curvature term vanishes and the integral of the Gaussian curvature becomes a constant. Therefore, $\kappa_{G}$ plays no role for a topologically spherical membrane. However, $\kappa_{G}$ becomes important if a pore is introduced into a membrane to change its topology.[@Struik:1988] In fact, we find that even without considering the Gaussian curvature term in the Helfrich free energy, a pore on a membrane can induce an interesting budding structure. Results and discussion ====================== Single pore ----------- By exclusively considering the mean curvature term in the Helfrich free energy Eq. (\[bending\_energy\]), we analyze how the morphology of a topologically spherical membrane is influenced by a pore. Based on the intuition about closed membranes, one might guess that a punctured membrane would take a spherical shape everywhere except at the pore for minimizing the mean curvature term in the Helfrich free energy, as in Fig. \[why\_qiao\](a). Numerical experiments performed with Surface Evolver,[@brakke1992surface; @evolver] however, show that a budding pore appears, as in Fig. \[qiao\](a). It is thus natural to ask: Why does a budding of the pore appear? How does such a conformation minimize the integral of the squared mean curvature? To address these questions, we compare the energies of the two shapes in Fig. \[why\_qiao\](a, b). In order to minimize the integral of the squared mean curvature, the neck prefers to being a minimal surface with vanishing mean curvature. A catenoid is the only minimal surface with rotational symmetry.[@Nitsche] Consequently, the shape in Fig. \[qiao\](a) is, to first approximation, composed of a catenoid and part of a sphere (emphasized by the purple oval in Fig. \[qiao\](a)). The integrals of the squared mean curvature of the two shapes in Fig. \[why\_qiao\] are calculated as: $E_{a}=\frac{\pi}{2}\left(1+\cos\theta\right)$ and $E_{b}=\frac{\pi}{2}\left(1+\cos\theta'\right)$, where the angles $\theta$ and $\theta'$ are defined in Fig. \[why\_qiao\]. Note that the bending energy is independent of the radius of the sphere, as the integral of the squared mean curvature is scale invariant.[@SMoMS] Since $\theta'>\theta$, $E_{b}<E_{a}$, a budding pore is preferred. Theoretical model based on the boundary layer method shows that catenoidal necks between two asymptotically flat parallel membranes (a wormhole like structure, see Fig. \[double\_catenoid\]) are interacting like a gas of free particles with a hard core repulsion.[@Xavier:1994] The repulsion between necks comes from their overlap as they approach, which increases the bending energy of the system. It is analogous to the capillary interaction between particles floating or immersing on a liquid interface; their interaction originates from the overlap of the capillary deformations near particles.[@Particle_Interface] Considering that the budding pore structure is half of the wormhole like structure, we expect these budding pores also repel each other on the membrane as they approach.  ![The necks between two asymptotically flat parallel membranes repel each other as they approach; their overlap increases the bending energy of the system. \[double\_catenoid\] ](Figure_2){width="2.5in"} We further calculate the longitudinal size $L$ of a budding pore, as defined in Fig. \[why\_qiao\](b). We choose an x-y coordinate system such that the x-axis is along the solid red line in Fig. \[why\_qiao\](b) and the y-axis is along the symmetric axis of the membrane. The shape of the neck is characterized by $x\left(y\right)=r\ \cosh y$, where $r$ is the radius of the waist of the catenoid. By assuming that the boundary of the pore falls on the waist of the catenoid, we get the expression for the angle $\alpha$ between the x-axis and the tangent vector at the connecting circle of catenoid and sphere: $\cot\alpha=\sinh (L/r)$. On the other hand, a geometric argument leads to the relation between the radius $R$ of the sphere and the size of the pore $L$ as $R\sin\alpha=r\ \cosh (L/r)$. From these two expressions, we finally have $$R=r\ \cosh^2 (\frac{L}{r}),\label{eq:LR}$$ where $r$ is the radius of the pore. The dependence of the radius $R$ of sphere on the longitudinal size $L$ of the budding pore is plotted in Fig. \[LR\]. Measured in units of the radius of the pore, $L$ increases from $1.4$ to $1.8$ as $R$ increases from $5$ to $10$. The budding of a pore is more obvious in a bigger membrane. For the shape generated by Surface Evolver in Fig. \[qiao\](a), we measure $R=4.95$ and $L\approx 1.22$ which is close to our prediction $L=1.4$. The deviation comes from the assumption that the pore boundary falls on the waist of the catenoid, which is not precisely the case in Fig. \[qiao\](a). For very large values of $R$, from Eq. (\[eq:LR\]), the longitudinal size $L$ of the budding pore scales as $L\sim\frac{1}{2}\ln R$. The logarithm function comes from the exponential grow of the catenoidal neck from its waist. In real fluid membranes, the Gaussian rigidity can contribute more than $400\ \textrm{kJ}/\textrm{mol}$ in topological transformation of a membrane like creating a pore.[@gaussian_rigidity:2012] Theoretical microscopic models of monolayer fluid membranes show that $\kappa_G/\kappa\in\left[-1,0\right]$.[@templer1998gaussian] Therefore, the Gaussian rigidity can compete with the bending rigidity for influencing the shape of a pored membrane. In the following, we study this problem in the light of the Gauss-Bonnet theorem. ![Sphere radius, $R$, as a function of the longitudinal size $L$ of the budding pore when $|\kappa_G| \ll \kappa$, as given in Eq. (\[eq:LR\]). \[LR\] ](Figure_4) The Gauss-Bonnet theorem Eq. (\[GB\]) implies that the integral of the Gaussian curvature can be maximized by minimizing the line integral of the geodesic curvature, such that the bending energy is minimized as $\kappa_{G}$ is negative. Therefore, the Gaussian curvature term in the Helfrich free energy, which is an integral over the whole surface, essentially imposes a local constraint on the shape near the boundary, such that the integral of the geodesic curvature on the boundary is minimized. The geodesic curvature $k_{g}$ describes the deviation of a curve away from a geodesic, a generalization of a straight line in a plane. For example, the geodesic curvature of a big circle on a sphere is zero, since it corresponds to a straight line on spherical geometry. The geodesic curvature of a curve in a surface is defined in the following way. Consider a curve $\vec{x}(s)$ being parametrized by the arc length $s$, its curvature is $\vec{k}=\frac{d\hat{t}}{ds}$, where $\hat{t}=\frac{d\vec{x}}{ds}$ is the unit tangent vector of the curve. For a curve on a surface equipped with the coordinates $\left\{ \vec{e}_{u},\vec{e}_{v}\right\} $, the curvature $\vec{k}$ can be projected along the normal and tangent plane of the surface: $$\vec{k}=\frac{d\vec{t}}{ds}=\vec{k}_{n}+\vec{k}_{g},\label{eq:k}$$ where $\vec{k}_{n}=\left(\vec{k}\right){}_{\hat{n}}$ and $\vec{k}_{g}=\left(\vec{k}\right){}_{\textrm{TM}}$. $\vec{n}$ is the normal vector pointing *outward*; [*i.e.*]{}, along the direction of $\vec{e}_{u}\times\vec{e}_{v}$. TM represents the tangent plane. In the Gauss-Bonnet theorem, the sign of the geodesic curvature needs to be clarified. $k_{g}=\vec{k}_{g}\cdot\hat{u}$, where $\hat{u}=\hat{n}\times\hat{t}$.[@Struik:1988] The direction of $\hat{t}$ is chosen to be along the boundary of the pore such that the membrane stays on the left hand side of the boundary.[@Struik:1988] Under these conventions, the sign of the geodesic curvature is unambiguously determined. ![(a) The calculation of the geodesic curvature. (b) Possible shapes of a membrane near a circular pore which is represented by two dots.\[cal\_kg\] ](Figure_5){width="3in"} Using arguments based on the Gauss-Bonnet theorem, we show that the Gaussian curvature term in the Helfrich free energy tends to flatten the membrane near the pore. We first calculate the geodesic curvatures on the circular boundaries in the cut unit sphere as in Fig. \[cal\_kg\](a). For the upper bigger part of the cut sphere, the tangent vector on the boundary circle is clockwise seen from below, so the sphere is on the left hand side walking along the boundary circle. The other tangent vector $\hat{u}$ points upward, as shown in Fig. \[cal\_kg\](a), because the normal vector points outward. The curvature vector $\vec{k}$ of the boundary circle and the vector $\hat{u}$ makes an obtuse angle, so the geodesic curvature at any point on the boundary circle is negative $k_{g}=\vec{k}\cdot\hat{u}=-\frac{\sqrt{1-r^{2}}}{r}$, where $r$ is the radius of the boundary circle. A similar argument shows that the sign of the geodesic curvature at the boundary of the lower smaller part of the cut sphere in Fig. \[cal\_kg\](a) is positive. Fig. \[cal\_kg\](b) lists all the possible shapes around a symmetric circular pore of radius $r$ and the geodesic curvature for each case. The first shape has the minimum geodesic curvature, so it is preferred among other shapes. Therefore, the Gaussian curvature term in the Helfrich free energy tends to pull the membrane outside a pore to the plane where the pore loop lies. This conclusion also holds for multi-pored membranes. From the aspect of the Gauss-Bonnet theorem, the flattening effect of the Gaussian curvature term is disclosed. It also sheds light on the numerically generated flat surface in the vicinity of a pore on a membrane when the Gaussian rigidity is tuned to be negative.[@theory_of_PNAS:2005] We use Surface Evolver to generate the ground state shape of a pored membrane for exploring the flattening effect caused by the Gaussian curvature term in the Helfrich free energy. The Surface Evolver evolves a surface toward a local minimum energy shape by calculating the force on each vertex from the gradient of the total energy, which gives the direction of motion in the membrane’s configuration space.[@evolver] Therefore, the method to generate a ground state shape by Surface Evolver is distinct from that used in Ref. , where the equilibrium shapes are produced from solving the shape equation. The result is shown in Fig. \[qiao\] (b) for $\kappa=2$ and $\kappa_{G}=-1.5$. A comparison of Fig. \[qiao\](a) and (b) shows that the Gaussian rigidity does play a role in regulating the shape of a pored membrane. The mean curvature term prefers to form a neck while the Gaussian curvature term tends to flatten the membrane near the pore. A dark-field micrograph of an experiment on a liposome with a pore whose size (measured by the radius of the spherical body) is similar to that in Fig. \[qiao\](b) is shown in Fig. \[qiao\](c).[@talin:1998] The similarity of the shapes in Fig. \[qiao\](b, c) suggests that the experimental shape also results from the competition of the mean curvature and the Gaussian curvature terms. The shape of the pore, as the result of the competition of the mean curvature and the Gaussian curvature terms, encodes the information about the ratio $\kappa_{G}/\kappa$, as has been discussed in Refs.( ). Note that the absolute values of these rigidities cannot be derived from the shape, because the shape is determined only by their ratio. Here, we propose a scheme to determine the quantitative relation between the shape of the pore and the ratio $\kappa_{G}/\kappa$. Since the Gaussian curvature term flattens the membrane near a pore, we approximate the shape in Fig. \[qiao\](b) as a combination of a circular truncated cone (the section between the red line and the purple line) and a spherical crown. The whole shape is characterized by three parameters $r$, $A$, and $\theta$, where $r$ is the radius of the pore, $A$ is the area of the membrane, and $\theta$ is defined in Fig. \[qiao\](b), which is referred to as *the pore angle*. The pore angle reflects the flatness of the membrane near the pore. The total bending energy is $E_{b}\left(r,A,\theta;\kappa_{G}/\kappa\right) =\frac{1}{2}\kappa\int\left(2H\right){}^{2}dA+k_{G}2\pi\left(1+\cos\theta\right)$. The mean curvature for sphere is $2H=2/R$ and for cone $2H=\frac{\cos^{2}\delta}{z\sin\delta}$, where $2\delta$ is the cone angle and $z$ is the vertical distance to the tip of the cone. In $E_{b}\left(\theta,r,A;\kappa_{G}/\kappa\right)$, by specifying $r$, $A$ (as measured from a given shape) and $\kappa_{G}/\kappa$, we can find an optimal pore angle $\theta$ that minimizes the energy. We tune the ratio $\kappa_{G}/\kappa$ for fitting the optimal pore angle to the measured one. The ratio $\kappa_{G}/\kappa$ is thus found from a given shape. This scheme has its significance in application, considering that the ratio $\kappa_{G}/\kappa$ is usually very difficult to measure in experiment that only few results are available.[@gaussian_rigidity:2012] On the other hand, the scheme may be generalized to other systems, where the direct measurement of the elastic moduli is difficult, like for living materials.[@Fung_Biomechanics:1993; @cell_bubble]  where the fixed radius of the pore is defined to be unity. The membrane near the pore becomes more and more flat ($\theta$ decreases) with the increase of the absolute value of $\kappa_{G}/\kappa$. For a real fluid membrane, $\kappa_G/\kappa \in[-1,0]$, where more points are plotted. \[theta\_k\] ](Figure_6) We test the above method for finding the ratio $\kappa_{G}/\kappa$ of the shape in Fig. \[qiao\](b). The radius of the pore is defined as unity, so $R=2.3$ and the area is calculated as $62.8$. By varying the ratio $\kappa_{G}/\kappa$, we get different optimal pore angles, as shown in Fig. \[theta\_k\](a). It shows that the membrane near the pore becomes more and more flat ($\theta$ decreases) with stronger flattening effect by the Gaussian curvature term (the absolute value of $\kappa_{G}/\kappa$ increases). For fitting the optimal angle to the measured pore angle $47^{\circ}$, the ratio is required to be $\kappa_{G}/\kappa=-0.75$, which is exactly the one we use in Surface Evolver to generate the shape in Fig. \[qiao\](b). The validity of the scheme for obtaining the ratio $\kappa_{G}/\kappa$ is thus substantiated. Now we apply this scheme to the shape in Fig. \[qiao\](c) for identifying the ratio $\kappa_{G}/\kappa$ of the liposome used in the experiment of Ref. . From the experimental shape Fig. \[qiao\](c), we measure $R=2.25$, pore angle $\theta=55^{\circ}$ and calculate the area $A=66$. It is found that the observed pore angle can be fitted by using $\kappa_{G}/\kappa=-0.45$. Therefore, the value of the ratio $\kappa_{G}/\kappa$ of the liposome in the experiment of Ref.  is estimated as $-0.45$, which is of the same order as the experimentally-known values for typical liposomes.[@gaussian_rigidity:2012] ![The plot of energy versus the radius of the pore $r$ for membranes with budding (black curve) and flat (blue curve) pores. The two curves coincide at $r=\sqrt{2}/2 \approx 0.7$. It corresponds to a hemisphere beyond which the ansatz shape of a spherical cap plus catenoid does not apply. $\gamma=0.2049$. $\kappa=1,\ \kappa_G=0,\ \textrm{A}=\pi$. The radius of the pore $r$ is measured in the unit of the radius $r_0$ of the circular disk whose area is fixed in the evolution. The meta-stable pore has $r=0.43$, so the corresponding radius of the spherical cap is $R=0.50$, $L=0.17$, and $\theta=68$ degrees. \[line\_tension\] ](Figure_9){width="\columnwidth"} Stability of a pore with line tension ------------------------------------- Finally, we briefly discuss the consequences of relaxing the constraint of fixed pore size by introducing the line energy, $\gamma \oint_{\partial} dl$ for the pore. [@helfrich1974size] We explore the stability of a budding pore by working in the regime of $|\kappa_G|<<\kappa$ where a budding structure is expected to form. The pored membrane is assumed to take the shape of a spherical cap plus a catenoid, and the boundary of the pore is approximated as falling on the waist of the catenoid. The energy is thus obtained as $E=\frac{\pi\kappa}{4}(1+\cos\theta)+\gamma 2\pi r+\kappa_G 2\pi$, where $\theta$ is the pore angle and $r$ is the radius of the pore. The area of the pore membrane is fixed: $\textrm{A}=\pi r (2L+r\sinh(2L/r))+2\pi R^2(1+\cos\theta)$, where $L$ is the height of the pore and $R$ is the radius of the spherical cap (see Fig. \[why\_qiao\]). $L$ and $R$ are related by Eq. (\[eq:LR\]). For a given set of values for $\kappa,\ \kappa_G$ and $\textrm{A}$, the energy is a function of $r$ with the free parameter $\gamma$. Fig. \[line\_tension\] shows the plot of the energy versus $r$ for pored membranes with budding (black curve) and flat (blue curve) pores. The shape of a vesicle with flat pore is approximated as a spherical cap. [@helfrich1974size] Fig. \[line\_tension\] shows that for a specified value for the line tension the pore vanishes in both cases in the ground state. We notice that a budding pore has a meta-stable state at about $r=0.43$. However, this meta-stable state may be hard to see in an experiment, because the depth of the energy barrier ($\sim 0.01\kappa$) is very shallow and the range of values of the line tension where a meta-stable pore exists is very narrow: $0.19\lesssim r_0\gamma/\kappa \lesssim 0.22$, where $r_0$ is the radius of the circular disk as defined in the caption of Fig. \[line\_tension\]. We perform a series of simulations using Surface Evolver by adding the line energy to the pore. We were not able to observe a stable pore, *i.e.*, the pore either shrinks and closes up (for large values of line tension) or it fully opens and the membrane takes a form of a flat disk (for small line tension). While our numerical results cannot exclude the possibility of the existence of a stable pore within a certain parameter region, they suggest that even if such region exists, it is very narrow. Therefore, stabilizing agents like talin proteins in the experiment of Ref.  are essential for a stable pore on fluid membranes. Two-component membrane ---------------------- ![The schematic plot of budding on a two-component membrane. The red line represents the boundary of the two domains. $|\kappa_{G}^{1}|>|\kappa_{G}^{2}|$.\[two\_comp\] ](Figure_7){width="1.4in"} So far we have studied the effects of the mean curvature and the Gaussian curvature terms in the Helfrich free energy on the shape of pored membranes. It is interesting to extend the flattening effect due to the Gaussian curvature to two-component membranes where the components’ Gaussian rigidities are different. A pored membrane may be regarded as a limiting case of a two-component membrane, where one phase has vanishing bending and Gaussian rigidities. The effect of the inhomogeneity of the Gaussian rigidity in multicomponent membranes has been extensively discussed.[@Webb:2005; @allain2004fission; @idema2011analytical; @julicher1993domain; @Hu2011] Monte Carlo simulations show that a difference in the Gaussian rigidity of a two-component membrane can develop and stabilize multi-domain morphologies.[@julicher1993domain; @Hu2011; @allain2004fission] An explicit analytical expression for the shapes of axisymmetric closed membranes with multiple domains is derived in Ref. . However, the influence of the inhomogeneity of the Gaussian curvature on the local shape near the phase boundary was not explicitly discussed. In this subsection, we study how the same Gaussian-curvature effect that leads to the flattening near a pore can result in the onset of budding in a multicomponent membrane, if the Gaussian rigidities of the components are different. For simplicity, consider a two-component spherical membrane with Gaussian rigidities $\kappa_{G}^{(1)}$ and $\kappa_{G}^{(2)}$ for domain 1 and domain 2 of the sphere, respectively (see Fig. \[two\_comp\]). Suppose $\Delta\kappa_{G}=\kappa_{G}^{(2)}-\kappa_{G}^{(1)}>0$ without loss of generality. The integral of the Gaussian curvature over the whole surface is $\kappa_{G}^{(1)}\int_{1}K_{G}dA+\kappa_{G}^{(2)}\int_{2}K_{G}dA= 2\pi\left(\kappa_{G}^{(1)}+\kappa_{G}^{(2)}\right)- \Delta\kappa_{G}\oint_{2}k_{g}dl = 2\pi\left(\kappa_{G}^{(1)}+\kappa_{G}^{(2)}\right)+\Delta\kappa_{G}\oint_{1}k_{g}dl$, where the subscript numbers in the line integrals represent the boundary of the respective domains. The second and third expressions indicate that the geodesic curvature on the boundary of domain 2 (with larger Gaussian rigidity) prefers to increase and that on the boundary of domain 1 (with smaller Gaussian rigidity) prefers to decrease for lowering the Helfrich free energy. The effect is similar to imposing a “torque” rotating outward the original shape near the boundary loop (the dashes lines in Fig. \[two\_comp\]). ![The plot of the asphericity of a two-component membrane vs. the ratio of the two Gaussian rigidities in the two-component membrane with the 15% of the purple (domain 1) component. The budding becomes more obvious with the increase of the inhomogeneity of the membrane in Gaussian rigidity. \[kG12\] ](Figure_8) In order to confirm the proposed budding scenario, we performed a series of simulated annealing Monte Carlo simulations for a triangulated two-component membrane. Components were assigned to the vertices of the discrete mesh and liquid character of the membrane is ensured by using a dynamical triangulation; [*i.e.*]{}, we employed a Monte Carlo move in which an edge shared by two triangles was flipped to connect two vertices that were previously not connected.[@kazakov1985critical; @SMoMS] The discrete version of the mean curvature term in the Helfriech free energy was calculated following a prescription introduced by Gompper and Kroll,[@gompper1996random] while the Gaussian curvature term was treated according to Meyer, *et al*.[@meyer2002discrete] For a membrane with about $2\times10^{3}$ vertices typically $10^{5}$ Monte Carlo sweeps with a linear cooling protocol was sufficient to obtain low energy structures, with a sweep defined as an attempted move of each vertex followed by an attempted flip of each edge. The result is shown in Fig. \[kG12\]. In the simulation, $\kappa=2$, $\kappa_{G}^{(2)}=-0.5$ and $\kappa_{G}^{(1)}/\kappa_{G}^{(2)}$ increases from unity to 6. The deviation from a spherical shape is characterized by the asphericity $\frac{<(\Delta R)^{2}>}{<R>^{2}}=\frac{1}{N}\sum_{i=1}^{N}\frac{(R_{i}-<R>)^{2}}{<R>^{2}}$, where $R_{i}$ is the radial distance of vertex $i$ and $<R>=\frac{1}{N}\sum_{i=1}^{N}R_{i}$ is the mean radius.[@Lidmar03] With the increasing inhomogeneity in the Gaussian rigidity, the “torque” imposed on the phase boundary becomes stronger and the budding of the smaller component becomes more obvious as shown in Fig. \[kG12\]. This budding mechanism arising from an inhomogeneity in the Gaussian rigidity is distinct from the usual mechanism due to line tension. It sheds light on understanding shapes of multicomponent membranes and provides a novel method to control the shape of membranes. Conclusions =========== Our study of shapes of pored membranes of fixed area and pore size within the framework of the Helfrich theory shows that the presence of pores can be an important ingredient for generating various shapes of membranes. Several structures brought by pores have been disclosed, including the budding pores purely due to the mean curvature term and the flattening effect due to the Gaussian curvature term. The latter effect may be used to fabricate pore-controlled buckled membranes. Furthermore, we have proposed a method to extract the value of the Gaussian rigidity of a membrane simply from its shape. This scheme may be generalized to systems where the elastic moduli are difficult to measure, like in living materials. In addition, by relaxing the constraint of a fixed pore size and adding the line tension, we briefly discuss the stability of a pore and find that a budding pore may be meta-stable with very shallow energy barrier within a narrow range of line tension values. Finally, we extend the flattening effect due to the Gaussian curvature as found in studying pored membranes to two-component membranes. Theoretical analysis shows that sufficiently high contrast between the components’ Gaussian rigidities can lead to budding of a two-component membrane, which is substantiated by MC simulations. Numerical simulations were in part performed using the Northwestern University High Performance Computing Cluster Quest. ZY, MO and CT thank the financial support of the Air Force Office of Scientific Research (AFOSR) under Award No. FA9550-10-1-0167. RS and MO thank the financial support of the US Department of Energy Award DEFG02-08ER46539. @ifundefined [47]{} \[1\][\#1]{} \[1\] \[2\] \[3\] \[3\] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, P. Walter, D. Bray and J. D. Watson, *Molecular Biology of the Cell*, Garland Science, New York, 5th edn, 2007 R. Lipowsky and E. Sackmann, *Structure and dynamics of membranes*, Elsevier, 1995 D. L. Nelson and M. M. Cox, *Lehninger Principles of Biochemistry, 4th ed.*, W. H. Freeman, New York, 2005 D. Lingwood and K. Simons, *Science*, 2009, **327**, 46 A. Saitoh, K. Takiguchi, Y. Tanaka and H. Hotani, *Proceedings of the National Academy of Sciences*, 1998, **95**, 1026 B. Fourcade, M. Mutz and D. Bensimon, *Phys. Rev. Lett.*, 1992, **68**, 2551 X. Michalet and D. Bensimon, *Phys. Rev. Lett.*, 1994, **72**, 168 D. R. Nelson, T. Piran and S. Weinberg, *Statistical mechanics of membranes and surfaces*, World Scientific Singapore, 2nd edn, 2004 Z. C. Ou-Yang and W. Helfrich, *Phys. Rev. A*, 1989, **39**, 5280 R. Lipowsky, *Nature*, 1991, **349**, 475 W. Helfrich, *Zeitschrift Für Naturforschung C-a Journal Of Biosciences*, 1973, **C 28**, 693–703 Z. C. Ou-Yang, J. X. Liu and Y. Z. Xie, *Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases*, World Scientific Publishing Company, 1999 S. A. Safran, *Statistical Thermodynamics of Surfaces, Interfaces, and Membranes*, Westview Press, Colorado, 2003 H. J. Deuling and W. Helfrich, *Biophys. J.*, 1976, **16(8)**, 861 W. Helfrich, *Naturforsch A*, 1978, **33**, 305 R. J. C. Gilbert, *Cell. Mol. Life Sci.*, 2002, **59**, 832 F. Bonardi, E. Halza, M. Walko, F. Du Plessis, N. Nouwen, B. L. Feringa and A. J. M. Driessen, *Proceedings of the National Academy of Sciences*, 2011, **108**, 7775 F. Nomura, M. Nagata, T. Inaba, H. Hiramatsu, H. Hotani and K. Takiguchi, *Proceedings of the National Academy of Sciences*, 2001, **98**, 2340 M. Dubois, B. Deme, T. Gulik-Krzywicki, J. C. Dedieu, C. Vautrin, S. Desert, E. Perez and T. Zemb, *Nature*, 2001, **411**, 672–675 M. Hu, J. J. Briguglio and M. Deserno, *Biophys. J.*, 2012, **102**, 1403 K. Brakke, *Experimental Mathematics*, 1992, **1**, 141–165 K. Brakke, ttp://www.susqu.edu/brakke/evolver/evolver.html J. Hu, T. Weikl and R. Lipowsky, *Soft Matter*, 2011, **7**, 6092 M. F. Demers, R. Sknepnek and M. Olvera de la Cruz, *Phys. Rev. E*, 2012, **86**, 021504 G. Vernizzi, R. Sknepnek and M. Olvera de la Cruz, *Proceedings of the National Academy of Sciences*, 2011, **108**, 4292–4296 R. Sknepnek, G. Vernizzi and M. Olvera de la Cruz, *Soft Matter*, 2012, **8**, 636 R. Sknepnek and M. Olvera de la Cruz, *Phys. Rev. E*, 2012, **85**, 050501(R) R. Lipwsky, *Biophys. J.*, 1993, **64**, 1133 T. Kohyama, D. M. Kroll and G. Gompper, *Phys. Rev. E*, 2003, **68**, 061905 P. M[é]{}l[é]{}ard, J. Faucon, M. Mitov and P. Bothorel, *Europhys. Lett.*, 1992, **19**, 267 T. Baumgart, S. Das, W. W. Webb and J. T. Jenkins, *Biophys. J.*, 2005, **89**, 1067 D. J. Struik, *Lectures on Classical Differential Geometry*, Dover Publications, 2nd edn, 1988 J. C. C. Nitsche, *Lectures on Minimal Surfaces*, Cambridge University Press, 1989 P. A. Kralchevsky and K. Nagayama, *Particles at fluid interfaces and membranes*, Elsevier, 2001 R. Templer, B. Khoo and J. Seddon, *Langmuir*, 1998, **14**, 7427–7434 T. Umeda, Y. Suezaki, K. Takiguchi and H. Hotani, *Phys. Rev. E*, 2005, **71**, 011913 T. Idema and C. Storm, *The European Physical Journal E: Soft Matter and Biological Physics*, 2011, **34**, 1–10 S. Semrau, T. Idema, L. Holtzer, T. Schmidt and C. Storm, *Phys. Rev. Lett.*, 2006, **100**, 088101 Y. C. Fung, *Biomechanics: mechanical properties of living tissues*, Springer, 2nd edn, 1993 A. J. Crosby and J. J. McManus, *Physics Today*, 2011, **64**, 62 J. Allain, C. Storm, A. Roux, M. Amar and J. Joanny, *Phys. Rev. Lett.*, 2004, **93**, 158104 F. J[ü]{}licher and R. Lipowsky, *Phys. Rev. Lett.*, 1993, **70**, 2964–2967 V. A. Kazakov, I. Kostov and A. Migdal, *Phys. Lett. B*, 1985, **157**, 295–300 G. Gompper and D. Kroll, *J. Phys. I (France)*, 1996, **6**, 1305–1320 M. Meyer, M. Desbrun, P. Schr[ö]{}der and A. H. Barr, *Visualization and Mathematics*, 2002, **3**, 34–57 J. Lidmar, L. Mirny and D. R. Nelson, *Phys. Rev. E*, 2003, **68**, 051910 W. Helfrich, *Phys. Lett. A*, 1974, **50**, 115–116

--- author: - | Vladimir E. Manucharyan,$^{1}$ Jens Koch,$^{1}$ Leonid I. Glazman,$^{1}$ Michel H. Devoret$^{1}$\ \ \ title: - 'Fluxonium: single Cooper pair circuit free of charge offsets' - '**Supporting Material**' --- The promise of single Cooper pair quantum circuits based on tunnel junctions for metrology and quantum information applications is severely limited by the influence of offset charges – random, slowly drifting microscopic charges inherent to many solid-state systems. By shunting a small junction with the Josephson kinetic inductance of a series array of large capacitance tunnel junctions, thereby ensuring that all superconducting islands are connected to the circuit by at least one large junction, we have realized a new superconducting artificial atom which is totally insensitive to offset charges. Yet, its energy levels manifest the anharmonic structure associated with single Cooper pair effects, a useful component for solid state quantum computation. Electric charge can be manipulated at the level of a single charge quantum [@singlechargebook] in two types of superconducting circuits with different topologies. The minimal example of the first type of circuit is the Cooper pair box, which consists of an isolated superconducting electrode (island) connected to a superconducting reservoir on one side by a small tunnel junction, and on the other side by a gate capacitance in series with a voltage source. The dynamics of the island is described by two variables: the integer number of Cooper pairs occupying the island and its conjugate, the $2\pi $-cyclic superconducting phase difference between the island and the reservoir. The junction area must be chosen sufficiently small such that the electrostatic energy of the island due to an extra Cooper pair is larger than the Josephson energy of its coupling to the reservoir, thus confining fluctuations of the number of Cooper pairs below unity. Stated in electrical engineering language, one needs $Z_{J}\gtrsim R_{Q}$, where the junction reactive impedance $Z_{J}=(L_{J}/C_{J})^{1/2}$ is defined by the Josephson characteristic inductance $L_{J}$ and capacitance $C_{J}$ [@Josephson], and where the superconducting impedance quantum is given by $R_{Q}=\hbar /(2e)^{2}\approx 1~\mathrm{k\Omega }$, denoting Planck’s constant $\hbar $ and the charge quantum $e$. The second type of circuit is based on a superconducting loop connecting the two electrodes of a small junction with an inductance which exceeds $L_{J}$. The circuit conjugate variables are now the magnetic flux generated by the persistent current in the loop and the displacement charge on the plates of the small junction capacitance. When $Z_{J}\gtrsim R_{Q}$, the large loop inductance is submitted to quantum fluctuations of flux larger than the flux quantum $\Phi _{0}=2\pi \hbar /2e$, and therefore according to Heisenberg principle, the junction charge fluctuations are reduced below the value $2e$. In practice, the realization of both circuit types faces fundamental difficulties. Islands are exposed to random electric fields due to fluctuating charged impurities which are ubiquitous in most solid-state environments and whose compounded effect is described by a noisy offset charge. Although the fully developed charging effects were demonstrated for the Cooper pair box [@BouchiatCPB; @NakamuraCPB], it soon became clear that the low-frequency offset charge noise was a major source of decoherence for charge qubits derived from this device [NakamuraCPB,SaclayQuantronium,ChalmersCPB,MetcalfeQuantroniumPRB]{}. This state of affairs has prompted the development of alternative superconducting qubits based on large junctions with $Z_{J}\ll R_{Q}$, avoiding the single Cooper pair regime and the related charge offset problem [ChargeFreePhaseQubit,ChargeFreeFluxQubit,ChargeFreeTransmon]{}. On the other hand, implementing the island-free circuit, which is immune to charge offset noise, is another hard problem. This is because any finite-length wire with inductance $L$ always comes with self-capacitance $C$ which reduces the total charging energy of the circuit and therefore steers it away from the charging regime, unless $(L/C)^{1/2}\gg R_{Q}$. In fact, a purely electromagnetic inductance is incompatible with the single Cooper pair effects since $(L/C)^{1/2}$ is then bounded by the vacuum impedance $(\mu _{0}/\varepsilon _{0})^{1/2}\approx 377$ $\Omega <R_{Q}$, $\mu _{0}$ and $\varepsilon _{0}$ being vacuum permeability and permittivity [FeynmanVol2,FineStructureNote]{}. In this paper, we present experimental results on a novel single Cooper pair circuit based on a superconducting loop, which solves both the inductance and the offset charge noise problems. The small junction of our circuit is shunted by a series array of carefully chosen larger area tunnel junctions (Fig 1A-C). Here, all islands are connected to the rest of the circuit by at least one large junction so that quasistatic offset charges on all islands are screened. The large capacitances of the array junctions prevent phase slips within the array, and for excitations whose frequencies are below the junction plasma frequency, the array effectively behaves as an inductive wire. By choosing a sufficiently large number of array junctions it is possible to create an inductance exceeding that of the small junction. At low energies, the loop is effectively described by the loop flux $\hat{\Phi}$ and the small junction charge $\hat{Q}$, satisfying $[\hat{\Phi},\hat{Q}]=i\hbar $. To form a charge offset-free inductively shunted junction, four conditions involving the effective inductance $L_{JA}$ and capacitance $C_{JA}$ of the $\mathcal{N}$ array junctions are required: (i) $\mathcal{N}L_{JA}\gg L_{J}$, (ii) $e^{-8R_{Q}/Z_{JA}}<\varepsilon \ll 1$, (iii) $\mathcal{N}e^{-8R_{Q}/Z_{JA}}\ll e^{-8R_{Q}/Z_{J}}$, and (iv) $\mathcal{N<(}C_{JA}/C_{g})^{1/2}$. In the first relation (i), we simply estimate the total array inductance to be $\mathcal{N}L_{JA}$ and require that it exceeds the small junction inductance, allowing it to support the large flux fluctuations of the loop. The second relation (ii), where $Z_{JA}=(L_{JA}/C_{JA})^{1/2}$ is the array junction reactive impedance, dictates the minimum size of the array junctions necessary to reduce [JensTransmon]{} the uncontrolled offset charge on the islands of the circuit below the desired value of the order of $2e\times \varepsilon $. The third relation (iii) ensures that the inductive role of the array is not jeopardized by quantum phase slips [@GlazmanLarkinMatveev]. Specifically, the probability amplitude of a phase slip event within the array (l.h.s.) must be negligible compared to that in the small junction (r.h.s.). According to relation (iii) a fluxon tunnels in and out of the loop predominantly via the small junction, thus effectively erasing the discrete character of the array. Lastly, relation (iv) states that the inductance of the array is not shunted by the parasitic capacitances $C_{g}$ of array islands to ground. It is obtained by estimating the array parasitic resonance frequency to be $(L_{JA}\mathcal{N}\times C_{g}\mathcal{N})^{-1/2}$, and requiring that it is larger than the junction plasma frequency $(L_{JA}C_{JA})^{-1/2}$. Remarkably, it is the relation (iv) which, with present junction technology, most severely limits the maximum number of junction in the array and, thus, its maximum inductance. We have implemented the above array proposal and constructed a new superconducting artificial atom which we have nicknamed fluxonium. It contains $\mathcal{N}=43$ Al-AlOx-Al Josephson junctions [@MaterialsAndMethods] such that $Z_{JA}\simeq 0.5R_{Q}$ and a small junction with $Z_{J}\simeq 1.5R_{Q}$ [theEcfootnote]{}. The above four conditions being realized, the fluxonium can be modelled (Fig. 1D) as a small junction shunted by an inductance $L_{A}$ [@footnote2]. The three characteristic energies of this model, namely $E_{L}=(\Phi _{0}/2\pi )^{2}/L_{A}$, $E_{J}=(\Phi _{0}/2\pi )^{2}/L_{J}$ and $E_{C}=e^{2}/\left( 2C_{J}\right) $ have values corresponding to $0.52~\mathrm{GHz}$, $9.0~\mathrm{GHz}$ and $2.5~\mathrm{GHz}$, respectively. The additional $L_{R}C_{R}$ resonator, capacitively connected to the small junction (Fig. 1D), reads out the atomin a manner analogous to the dispersive measurement of cQED qubits [WallraffQED]{}. It is implemented by a quarter-wave superconducting coupled microstrip resonator (Fig. 1A) with quality factor $400$ due to capacitive coupling to the two $50~\mathrm{\Omega }$ measurement ports. The resonator frequency $\omega _{R}=(L_{R}C_{R})^{-1/2}\simeq 2\pi \times 8.17~\mathrm{GHz}$ is pulled by the reactance of the fluxonium circuit and is monitored by standard ultra low noise microwave reflection technique. The fluxonium reactance depends on its quantum state, an effect leading to a purely dispersive state measurement [@MaterialsAndMethods]. An externally imposed, static magnetic flux $\Phi _{\mathrm{ext}}$ threading the loop $\Phi _{0}$-periodically modulates the spacings of energy levels of our artificial atom. Introducing the operators $\hat{N}=\hat{Q}/2e$ and $\hat{\varphi}=2e\hat{\Phi}/\hbar $, describing the reduced charge on the junction capacitance and its conjugate reduced flux operator [@DevoretQFinEC], the Hamiltonian of the fluxonium coupled to its readout resonator can be written as $$\hat{H}=4E_{C}\hat{N}^{2}+\frac{1}{2}E_{L}\hat{\varphi}^{2}-E_{J}\cos \left( \hat{\varphi}-2\pi \Phi _{\mathrm{ext}}/\Phi _{0}\right) +g\hat{N}\left( \hat{a}+\hat{a}^{\dag }\right) +\hbar \omega _{R}\hat{a}^{\dag }\hat{a} \label{Ham}$$Here $\hat{a}$ is the photon annihilation operator for the resonator, $g$ is the atom-resonator coupling constant. The second term and the range of definition of $\hat{\varphi}$ and $\hat{N}$, whose eigenvalues are here both on the entire real axis, distinguishes the form of Hamiltonian (1) from that of the Cooper pair box in cQED experiments [@WallraffQED]. There are three important points to note concerning this Hamiltonian [@JensCPBL]: i) it is invariant under the transformation $\hat{N}\rightarrow \hat{N}+N_{\mathrm{offset}}\ $($N_{\mathrm{offset}}$ stands for offset charge value) hence the charge-free character of our device; ii) it differs from that of the transmon [@JensTransmon] since offset charge influence is screened for all states, not just for the low-lying states; iii) its second term, despite the fact that $E_{L}$ is the smallest of the fluxonium energies, has a non-perturbative influence on the full energy spectrum of this artificial atom, which presents strongly anharmonic transitions [@footnote3] (Fig. 1E). Our experiment probes these transitions by microwave spectroscopy, from which we infer the size of charge fluctuations. To characterize the fluxonium, we first measure the ground state resonator pull as a function of $\Phi _{\mathrm{ext}}$. The results (Fig. 2) show the expected $\Phi _{0}-$ periodicity as well as the avoided crossings of the resonator frequency and the ground to excited state transitions. This confirms that the entire $44$ junction loop is superconducting and that the resonator-atom system is in the strong coupling regime of cavity QED [RaymondRMP]{}. Next, we perform a two-tone spectroscopy measurement [SchusterSpectroscopy]{} at a fixed flux $\Phi _{\mathrm{ext}}=0.05\Phi _{0}$, during which, in addition to the fixed frequency readout tone, we probe the transition frequencies of the atom through a second, variable frequency spectroscopy tone. The resulting peaks (Fig. 3), correspond to the later-determined $0-1$, $0-2$, and $0-3$ transitions from the atom ground state. The peaks are well-fitted by Lorentzians and their power-dependent widths and heights are well-explained by the Bloch equations of precessing spin $1/2$ [@AbragamNMRbook] as shown in the insets of (Fig. 3). Extrapolating fitted linewidths to zero spectroscopy power, we obtain lower bound estimates of their decoherence time at $350$, $250$ and $80$ $\mathrm{ns}$ respectively. Our main result is the spectroscopic data collected as a function of both spectroscopy frequency and flux (Fig. 4A). Note that $\Phi _{\mathrm{ext}}$ variations span $20\%$ of $\Phi _{0}$ around $\Phi _{\mathrm{ext}}=0$ instead of the usual $1\%$ or less around $\Phi _{0}/2$ in flux qubit experiments [@ChargeFreeFluxQubit]. In (Fig. 4B) we compare the measured peak center frequencies with the prediction for the $0-1$, $0-2$, $0-3$ and the two-photon $0-4$ transitions obtained from numerical diagonalization of Hamiltonian (Eq. 2). Note that we are in effect fitting more than three flux dependent functions, i.e. the flux dependent transition frequencies,* *with only three a priori unknown energies $E_{C}$, $E_{L}$ and $E_{J}$ so the problem is severely overconstrained. The fit of the line (Fig. 4B) labeled SR (for array self-resonance) requires a minor extension of the model taking into account parasitic capacitances across the array [@MaterialsAndMethods]. Apart from introducing another resonator mode coupled to the atom, this extension by no means invalidates the inductive character of the array, at least as far as the $0-1$ and $0-2$ transition of the fluxonium are concerned. Even the perturbation of the $0-3$ and $0-4$ transition frequencies by this extra mode is less than $2\%$. Based on the excellent agreement between theory and experiment, we infer the wavefunctions of the first three energy levels, and plot their amplitudes both in charge (Fig. 4C) and flux (Fig. 4D) representations for $\Phi _{\mathrm{ext}}=0$. In the ground state, we find that the ratio of charge to flux fluctuations is $\Delta N/\Delta \varphi =0.56$, about $5$ times smaller than the fine structure constant allows for a conventional resonator. This confirms that the charge in our circuit is indeed localized at the single Cooper pair level ($\Delta N=0.53$, $\Delta \varphi =0.95$). The wavefunctions in flux representation (Fig. 4D) can be interpreted as simple superpositions of states in which the reduced flux $\varphi $ is localized in the wells of the Josephson cosine potential (fluxon states, hence the name fluxonium). The parity of fluxonium states, which forbids the $0-2$ transition at zero external flux, manifests itself explicitly by a remarkable “hole" in the corresponding spectroscopic line (Fig. 4A, inset). The allowed transition between the second and third level (data not shown) is particularly spectacular since it corresponds to motion of the total flux in the fluxonium loop by two whole flux quanta. This is to be contrasted with the $10\%$ of flux quantum or less flux motion involved in transitions of the flux and phase qubits [ChargeFreePhaseQubit,ChargeFreeFluxQubit]{}. Nevertheless, despite the large flux fluctuations of the system and the corresponding charge pinning, the circuit has complete immunity to offset charge variations: the data of (Fig. 4A) has been taken piecemeal in $72$ hours and no jumps or drifts have been observed during this period. We have thus demonstrated that an array of Josephson junctions with appropriately chosen parameters can perform two functions simultaneously: short-circuit the offset charge variations of a small junction and protect the strong non-linearity of its Josephson inductance from quantum fluctuations. The data shows that the array possesses a microwave inductance $10^{4}$ times larger than the geometric inductance of a wire of the same $20~\mathrm{\mu m}$ length. The reactance of such inductor is about $3R_{Q}\approx 20$ $\mathrm{k\Omega }$ at $10~\mathrm{GHz}$ while its resistance is less than $1~\Omega $. The spectrum of the fluxonium qubit suggests it is as anharmonic as the flux qubit but as insensitive to flux variations as the transmon qubit. Possible applications of this single Cooper pair charging effect immune to charge noise include the observation of fully developed macroscopic quantum coherent oscillations between fluxon states [@LeggetMQC], the search for a $\mathrm{\Lambda }$ or $\mathrm{V}$transition configurations for the shelving of quantum information [Shelving]{} in superconducting artificial atoms, topological protection of superconducting qubits [@KitaevTopProtQubit], and finally the long-sought quantum metrology of electrical current via Bloch oscillations [@AverinBO; @LikharevBO]. [99]{} H. Grabert, M. H. Devoret (Eds.), *Single Charge Tunneling: Coulomb Blockade Phenomena in Nanostructures* B. D. Josephson, *Rev. Mod. Phys* **36**,** **216 (1964). V. Bouchiat, D. Vion, P. Joyez, D. Esteve, M. H. Devoret, *Physica Scripta* **T76**, 165-170 (1998). Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, *Nature* **398**, 786 (1999). D. Vion et al., *Science* **296**, 866 (2002). K. Bladh, T. Duty, D. Gunnarsson P. Delsing, *New J. Phys.* **7,** 180 (2005). M. Metcalfe et al., *Phys. Rev. B* **76**, 174516 (2007). J. M. Martinis, S. Nam, J. Aumentado, C. Urbina, *Phys. Rev. Lett.* **89**, 117901 (2002). I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, J. E. Mooij, *Science* **299**, 1869 (2003). J. A. Schreier et al., *Phys. Rev. B* **77**, 180502(R) (2008). R. P. Feynman, R. B. Leighton, M. Sands, *The Feynman Lectures on Physics* (Addison-Wesley, 1964), vol. 2, chap. 23 (Springer, 1992). Note that the ratio $(\mu _{0}/\varepsilon _{0})^{1/2}/R_{Q}=16\pi \alpha $, where $\alpha \approx 1/137$ is the fine structure constant. J. Koch et al., *Phys. Rev. A* **76**, 042319 (2007). K. A. Matveev, A. I. Larkin, L. I. Glazman, *Phys. Rev. Lett.* **89**, 096802 (2002). Materials and methods are available as supporting material on Science online. Due to coupling of the fluxonium circuit to the readout resonator (Fig.1D) $C_{J}$ is renormalized to the value $C_{J}+C_{c}$, with corrections of order $C_c/C_R\simeq1\%$. Here we acknowledge experiments [@HavilandBO; @HavilandArrays] in which a small Josephson junction was DC-biased in series with a micron-scale disordered film resistance exceeding $R_K$ [HavilandBO]{}, or with an array of small-junction SQUIDs [@HavilandArrays] with zero-bias resistance tuned by magnetic field to exceed $R_K$. Both experiments aimed at protecting the small junction from the shunting effect of low impedance environment using highly dissipative biasing elements. While results of DC I-V measurements were consistent with single Cooper pair effects, they were distorted by Joule heating and other out-of-equilibrium effects in these biasing elements. We avoid problem of dissipation with an array employing the pure Josephson kinetic inductance. A. Wallraff et al., *Nature* **431**, 162 (2004). M. Devoret in *Quantum Fluctuations in Electrical Circuits*, S. Reynaud, E. Giacobiano, J. Zinn-Justin, eds. (Elsevier Science, 1997). J. Koch, V. Manucharyan, L. Glazman, M. Devoret, arXiv:0902.2980 (2009). Note that while the circuit in (Fig. 1D) might appear identical to the RF-SQUID circuit [@RF-SQUID], there is a drastic difference between the two when charging, Josephson and inductive energies are compared. In addition, RF-SQUID is addressed by coupling to its flux degree of freedom via mutual inductance, while the fluxonium is addressed by coupling to its charge degree of freedom via capacitance $C_{c}$. J. Raimond, M. Brune, S. Haroche, *Rev. Mod. Phys.* **73**, 565 (2001). D. Schuster et al., *Phys. Rev. Lett.* **94**, 123602 (2005). A. Abragam, *The principles of nuclear magnetism* (Oxford University Press, 1961). A. J. Leggett, *J. Phys.: Condens. Matter* **14,** R415 (2002) and references therein. W. Nagourney, J. Sandberg, H. Dehmelt, *Phys. Rev. Lett.* **56**, 2797 (1986). A. Kitaev, *arXiv:cond-mat/0609441* (2006). D. V. Averin, A. B. Zorin, K. K. Likharev, *Zh. Exp. Teor. Phys.* **88**, 692 (1985). K. K. Likharev, A. B. Zorin, *J. Low Temp. Phys.* **59**, 347 (1985). L. S. Kuzmin, D. B. Haviland, *Phys. Rev. Lett.* **67**, 2890 (1991). M. Watanabe, D. B. Haviland, *Phys. Rev. Lett.* **86**, 5120 (2001). J. R. Friedman, V. Patel, W. Chen, S. K. Tolpygo, J. E. Lukens, *Nature* **406**, 43 (2000). We acknowledge discussions with Markus Brink, Etienne Boaknin, Michael Metcalfe, R. Vijay, David Schuster, Leo DiCarlo, Luigi Frunzio, Robert Schoelkopf and Steven Girvin. This research was supported by the NSF under grants DMR-0754613, DMR-032-5580, the NSA through ARO Grant No. W911NF-05-01-0365, the Keck foundation, and Agence Nationale pour la Recherche under grant ANR07-CEXC-003. M.H.D. acknowledges partial support from College de France. Materials and Methods ===================== *Sample fabrication.* The device is made on a high-resistivity Si substrate, $300~\mathrm{\mu m}$ thick. Both Josephson junctions and the readout resonator are fabricated in a single step using e-beam lithography, double angle Al e-beam evaporation and lift-off techniques. The Al evaporation and oxidation is conducted in an e-gun evaporator at pressures less than $10^{-5}$ $\mathrm{Pa}$, AlOx grown in the environment of $680~\mathrm{Pa}$ of $15\%$ oxigen-in-argon mixture during $10$ minutes. The areas of the small junction and array junctions are designed to be $0.2\times 0.3~\mathrm{\mu m}^{2}$ and $0.25\times 2~\mathrm{\mu m}^{2}$, respectively. All $43$ array junctions are equally spaced at less than $200$ $\mathrm{nm}$ so that total length of the array is only $20~\mathrm{\mu m}$. The loop area of the array-small junction ring is $3\times 20~\mathrm{\mu m}^{2}$. *Sample mount.* The Si chip is glued using GE varnish to the copper base of a fully enclosing, custom-made microwave sample holder, shielding the sample from both residual RF, infrared and optical photons. The holder provides two well-matched transitions from the Anritsu K-connectors on the outside of the holder to the two microstrip lines made on a PCB inside the holder. The resonator’s on-chip launching pads, schematically indicated as two sections of a coaxial cable and marked $50~\mathrm{\Omega }$ in (Fig. 1A), are then wirebonded to the ends of the two microstip lines. *Cryogenic setup.* The experiment is performed in a dilution refrigerator with base temperature $10-20~\mathrm{mK}$. Both resonator and the qubit are differentially excited via the $\Delta $-port of a $180^{\circ}$ hybrid (Krytar, $2-18~\mathrm{GHz}$), whose two outputs are connected to the two ports of the sample holder. Incoming and outgoing signals are separated with a directional coupler (Krytar, $2-20~\mathrm{GHz}$). The incoming signal line is attenuated using $10$ and $20~\mathrm{dB}$ microwave attenuators (XMA) at all temperature stages of the refrigerator, to remove non-equilibrium noise. The output line is amplified at the $4~\mathrm{K}$ stage with a low-noise HEMT amplifier (Caltech, $1-12~\mathrm{GHz}$, $30~\mathrm{dB}$ gain). Two cryogenic isolators (Pamtech, $4-12$ $\mathrm{GHz}$, $15~\mathrm{dB}$) are placed between the amplifier and the sample, at the $800~\mathrm{mK}$ stage and at the base stage, again to remove non-equilibrium noise, especially that coming from the amplifier. Stainless steel SMA cables are used to connect between the different temperature stages. All components are thermally anchored to the proper refrigerator stages. A $\sim 1~\mathrm{cm}$ diameter custom made superconducting coil is glued to the sample holder, a few $\mathrm{mm}$ away from the chip, to provide perpendicular magnetic flux bias. The sample holder together with the coil is placed into a Cryoperm cylinder to shield it from stray quasistatic magnetic fields. *Room temperature measurement setup.* The readout resonator is excited with Agilent E8257D signal generator, the spectroscopy signal is generated using Agilent E8267D vector signal generator and Tektronix 520 AWG. Both signals are combined at room temperature and sent into the input line of the refrigerator. The reflected $\sim 8~\mathrm{GHz~}$readout signal from the refrigerator output line is amplified at room temperature with two Miteq amplifiers ($1-12~\mathrm{GHz}$, $30~\mathrm{dB}$ gain), mixed down with a local oscillator (a third Agilent E8257D) to an IF signal of $0-50~\mathrm{MHz}$, filtered and amplified with the IF amplifier (SRS SR445A), and finally digitized using $1~\mathrm{GS}/\mathrm{s}$ Agilent Acqiris digitizer. A software procedure then extracts the phase and the amplitude of the digitized wave. The experiment is typically repeated $10^{4}$ times to average the Gaussian noise to an acceptable level. Because the duration of each experiment is about $10$ microseconds, every averaged data point is taken in a fraction of a second. All microwave test equipment is phase locked using a Rb precision $10~\mathrm{MHz}$ reference (SRS FS725). The magnetic coil is biased in series with a resistor with Yokogawa 7751 voltage source. *Comments on the data.* The data in (Fig. 2) shows the digitized homodyne (zero IF) signal as a function of magnetic field, with the spectroscopy generator turned off. The data in (Fig. 3) shows the phase of the digitized heterodyne ($50$ $\mathrm{MHz}$ IF) signal, as a function of frequency of the spectroscopy generator. The data in (Fig. 4A) is taken in the pulsed regime, when the spectroscopy generator outputs a $6~\mathrm{\mu s}$ saturating pulse followed immediately by the $2~\mathrm{\mu s}$ readout pulse. This way we ensure that the sample is exposed to only one tone at a time, avoiding various spurious effects. The image presented in (Fig. 4A) contains $367\times4597$ data points. Supplementary Text ================== In our analysis of the fluxonium device, we use two simple models whose corresponding circuits are depicted in Figure \[fig:circuit\]: (A) the inductively shunted junction model, (B) the extended fluxonium model, describing the fluxonium coupled to a transmission-line resonator. ![Models for the fluxionium device. (A) Inductively shunted junction model. (B) Extended fluxionium model, including the capacitive coupling to the mode of a transmission-line resonator, and parasitic capacitances across the array. Numbers in cyan enumerate the nodes of the circuits.\[fig:circuit\]](circuit.eps){width="80.00000%"} *Inductively shunted junction model*. The simplest model of the fluxonium device is the inductively shunted junction model, see Fig. \[fig:circuit\](A). It neglects parasitic capacitances across the fluxonium’s Josephson junction array, and assumes that all internal degrees of freedom of the array are frozen out. In this limit, the fluxonium consists of a single Josephson junction with capacitance $C_J$ and Josephson energy $E_J$, shunted by a large inductance $L$. Quantization of this circuit [@DevoretQFinEC] is straightforward and leads to the Hamiltonian $$\hat{H}_0=4E_C \hat{N}^2 +\frac{1}{2}E_L\hat{\varphi}^2 -E_J\cos\left(\hat{\varphi}-2\pi\frac{\Phi_\mathrm{ext}}{\Phi_0}\right),$$ where the charge on the junction capacitance $\hat{N}$ (in units of $2e$) and the reduced flux $\hat{\varphi}$ are canonically conjugate variables, $[\hat{\varphi},\hat{N}]=i$. Structurally, this Hamiltonian is identical to the Hamiltonian describing one-junction flux qubits, and flux-biased phase qubits. However, the regime of large inductances relevant for the fluxionium differs from typical parameters in flux and phase qubits, and has been discussed in Ref. [@JensCPBL]. *Extended fluxonium model*. For a more complete modelling of the spectra obtained in the experiment, we take into account the coupling of the fluxonium device to a transmission-line resonator. In addition to this resonant mode, the experimental data shows another resonance coupling to the fluxonium device. Such additional resonances are expected when accounting for the parasitic capacitances of the Josephson junction array. The simplest effective model accurately describing the experimental data includes very few of these capacitances, and approximates the array by a combination of inductances and capacitances as shown in Fig. \[fig:circuit\](B). The Lagrangian describing this circuit can be written in the form $$\begin{aligned} \mathcal{L}=&\frac{C_J'}{2}\dot{\phi}_3 - \frac{1}{2L}(\phi_3-\Phi_\mathrm{ext})^2+E_J\cos\left( \frac{2\pi\phi_3}{\Phi_0} \right)+C'\dot\theta_2-\frac{1}{L'}\theta_2^2 +\frac{C_R'}{2}\dot\varphi_4^2-\frac{1}{2L_R}\varphi_4^2\\\nonumber &+\frac{C_c}{2}\dot\phi_3\dot\varphi_4 +\tilde{C}\dot\theta_2\dot\phi_3,\end{aligned}$$ where $C_J'=C_c/2+C_J+2C_1\lambda_1^2+C_2\lambda_2^2$, $\lambda_i=L_i/L$, $C'=9(C_1+2C_2)\lambda_1^2\lambda_2^2$, $L'=L/(9\lambda_1\lambda_2)$, $C_R'=C_R+C_c/2$, $\tilde{C}=6\lambda_1\lambda_2(C_1\lambda_1-C_2\lambda_2)$, and we have disposed of another resonant mode which does not couple to the fluxonium device. In terms of the original generalized flux $\phi_i$ at each node $i$, the relevant variables are $\phi_3$ (associated with the fluxonium subsystem), the resonator mode $\varphi_4=\phi_4-\phi_5$, and the additional resonant mode $\theta_2=-\frac{1}{6\lambda_1\lambda_2}(\phi_2-\phi_1-\lambda_2\phi_3)$. Employing canonical quantization of this circuit, we find the effective Hamiltonian $$\hat{H}=\hat{H}_0 + \sum_{j=1,2}\hbar\omega_j \hat{a}_j^\dag \hat{a}_j +\hbar\sum_{j=1,2}g_j\hat{N}(\hat{a}_j^\dag + \hat{a}_j),$$ describing the inductively shunted junction, $\hat{H}_0$, coupled to two resonant modes $j=1,2$ with coupling strengths $g_1$ and $g_2$, respectively. *Theory fits to experimental data*. Design and fabrication of the fluxonium system only allow for imprecise estimates of the system parameters. Thus, the comparison between experimental data and theoretical prediction requires the fitting of theory curves to determine the system parameters with more accuracy. The parameters at our disposal are: $E_J$, $E_C$, and $E_L$ (for both the inductively shunted junction and the extended fluxonium models). In addition, the extended fluxonium model takes the resonant mode frequencies and coupling strengths $\omega_{1,2}$ and $g_{1,2}$ as input. Fits are obtained by extracting the center frequencies from the experimental data and employing a least-squares fit algorithm. *Fit to inductively shunted junction model*. A simultaneous fit to the full flux-dependence of the 0–1 and 0–2 transitions around the zero-flux point fully determines the fluxonium parameters $E_C$, $E_J$, and $E_L$ (see Table \[tab\] for the obtained parameter values). A comparison between the resulting theory prediction of higher transitions can then be used as a consistency check. While the agreement for the 0–1 and 0–2 transitions is good, we find systematic deviations for higher levels. The reason for these deviations lies in the effect of the additional resonance on the 0–2 transition: the additional resonance leads to significant frequency shifts of the 0–2 transitions. Ignoring this effect leads to a systematic error in the estimation of the fluxonium parameters. *Fit to extended fluxonium model*. For best agreement, both resonator and additional resonant mode are taken into account. Using the full experimental data we obtain a fit for the extended fluxonium model, which shows excellent agreement with the data. The resulting parameter values are given in Table \[tab\]. ------------------ --------------------- -------------------------- inductively shunted extended fluxonium model junction model $E_C/h $ 2.39 2.47 $E_J/h $ 8.93 8.97 $E_L/h $ 0.52 0.52 $\omega_1/2\pi $ n.a. 8.18 $\omega_2/2\pi $ n.a. 10.78 $g_1/2\pi $ n.a. 0.135 $g_2/2\pi $ n.a. 0.324 ------------------ --------------------- -------------------------- : Fluxonium system parameters obtained from least-squares fits to the inductively shunted junction and the extended fluxonium model. All values are given in GHz. The coupling constants are expressed in terms of the coupling strength for the fluxonium 0–1 transition.\[tab\] [99]{} M. Devoret in *Quantum Fluctuations in Electrical Circuits*, S. Reynaud, E. Giacobiano, J. Zinn-Justin, eds. (Elsevier Science, 1997). J. Koch, V. Manucharyan, L. Glazman, M. Devoret, arXiv:0902.2980 (2009).

--- author: - 'Jacques Moret-Bailly [^1]' title: 'How the BAL quasars are quiet.' --- Pacs 42.65.Dr Stimulated Raman scattering, 98.54.Aj Quasars Introduction ============ Recent observations show two discrepancies in the standard interpretation of the spectra of quasars: i\) the relative Doppler or expansion frequency shift of spectral lines is strictly constant, while the observations show slight variations [@Webb]; the explanation by a variation of the fine structure constant is hardly credible; ii\) the concentration of iron in quasars whose spectrum is largely shifted requires a generation of this metal in old stars [@Hasinger], implying a too large increase of the age of the Universe. The standard model seems unable to explain that the loud quasars do not have broad lines. Twenty years ago, it seemed difficult to interpret various observed redshifts by the Doppler effect [@Reboul], but the lack of a credible alternative led to find solutions which do not convince all astrophysicists. In 1968, Giordmaine et al. [@Giordmaine] observed frequency shifts in experiments using high power ultrashort laser pulses; this observation is now common, used for analysis and called ”Impulsive Stimulated Raman Scattering” (ISRS)[@Weiner; @Dougherty; @Dhar]. The theory of ISRS shows that this effect has no intensity threshold [@Yan]. The intensities of the lasers make ISRS non-linear, so that the frequency shifts depend on these intensities; at low intensities, the effect becomes linear, so that it is similar to a Doppler effect, except for a small dispersion. This qualitative difference justifies a new name: “Coherent Raman Effect on time-Incoherent Light” (CREIL). In the next section, the properties of the CREIL are recalled. In the following sections, the influence of each of these properties on the standard model of quasars is given, solving very simply some problems, in particular the noisy quasars cannot show broad spectral lines. Properties of the “Coherent Raman Effect on Incoherent Light” (CREIL). ====================================================================== Any alternative to the Doppler effect in the vacuum requires a time-incoherence of the light: If, during a certain time, the received frequency of a coherent light is lower than the emitted frequency, the number of cycles, that is the number of wavelengths between the source and the receiver is increased; as the wavelength is an unit of length, the distance is increased, it is a Doppler effect. Thus, we understand that the length of the pulses of light is critical, the pulses of light must verify the condition set by G. L. Lamb when the length of the pulses is a condition for the observation of an optical effect: the “ultrashort light pulses”, must be “shorter than all relevant time constants” [@GLamb]. In an homogeneous gas, two relevant time constants must be longer than the pulses which make the usual incoherent light, that is longer than about 10 nanoseconds: i\) the collisional time $\tau$ must be longer than 10 ns, for instance $\tau=100ns$. For hydrogen atoms at 10000K, the density of atoms $N$ is $N\approx 1,7.10^{21}$ atomes/m$^3$; for H$_2$ at 100K, $N\approx 2,4.10^{22}$; these values are orders of magnitude because the mean diameter of the atoms depends on the criterion used to define a collision. This condition allows that all molecules on a wave surface remain in phase and scatter coherent waves which rebuild wave surfaces identical to the incident wave surfaces: no blur of the images. ii\) The period corresponding to the Raman transition must be larger than 10 ns, that is in the radiofrequencies range. As the period of the beats of the incident and scattered beams is much longer than the duration of the pulses, the beats do not appear, the incident and scattered beams interfere into a single frequency beam; the frequency shift is proportional to the ratio of the amplitudes and to the Raman frequency. Although the coherent scattering is much more powerful than an incoherent Raman scattering, the previous conditions make CREIL much weaker than ISRS which uses dense matter and infrared Raman frequencies; an other source of weakness is that the populations of the levels involved in the Raman transition are nearly the same because they are close, so that the opposed virtual transitions nearly cancel the shifts that they produce. Thus, it seems impossible to get in the labs long enough paths to observe the CREIL while the Universes provides long paths in low pressure gases. Previous papers [@Moret1; @Moret2; @Moret3; @Moret4] show that the cosmological redshift may be produced by CREIL if there are, in the average, some active molecules per litre in the intergalactic space; it is a very rough evaluation because a precise computation would require for each molecule present in the space, the knowledge of a lot of traces of Raman polarisation tensors. Suppose that a redshifting gas has absorption lines; the absorption by a line occurs from the redshifted frequency to the absolute frequency; if the redshift is large, the line is wide and weak. If the gas has many absorption lines, the lines are mixed, they cannot be observed. If ISRS and CREIL were simple Raman effects, the active molecules would be excited, and, without collisions, their de-excitations would be slow; ISRS and CREIL are parametric effects which mix the described Raman effect with an other which de-excite the molecules and blue-shift (an) other beam(s); in ISRS, the other beam is a laser beam, in CREIL, its role may be plaid by the thermal radiation which is amplified. In this last case, all frequencies are low, resonant. Distance (age) of the quasars ============================= The distance and the age of the quasars is deduced from the redshift of the sharp emission or absorption lines, which are the most redshifted lines. It was remarked that the neighbourhood of the most redshifted quasars contains a lot of matter, in particular molecular hydrogen detected by the 21 cm nuclear spin coupling line. It is reasonable to think that where the pressure is low, the molecular hydrogen is partly ionised by the UV into H$_2^+$. This gas cannot be detected by its absorption or emission lines because it has many relatively weak lines which are shifted during their absorption or emission by the CREIL; if the pressure is high enough to produce collisions which forbid the CREIL, these collisions destroy H$_2^+$ too quickly to allow its observation. The CREIL distorts the standard scale of distances deduced from the redshifts. This scale remains probably good in the intergalactic space, because there, the redshift is produced by expansion or by the CREIL in a nearly homogenous medium; but, where there is much molecular hydrogen, in particular in the neighbourhood of the quasars the distances must be strongly reduced. Absence of broad lines in the loud quasars ========================================== Suppose that a quasar has an accretion disk larger, but similar to Saturn’s disk, except maybe for an inner part which protects the remainder from burning. The friction of the stones on the halo of the quasar charges them, just as the drops of rain are charged in our clouds. While the lightnings discharge a volume of our clouds, the discharges of regions of the disk are limited to its plane, so that the electromagnetic field radiated by sheet-shaped currents propagate more in the direction of the axis of the disk. The disk is only partly and locally discharged by the lightnings; as the disk is inhomogeneous, the sign of its charges depend on the distance to the quasar; thus, the rotation of the disk produces circular currents which induce a magnetic field similar to the field introduced by de Kool and Begelman[@Kool]: near the disk, its orientation may change, so that it may have zero values, in particular if the disk has gaps. Raman transitions inside a Zeeman structure of the hydrogen atom require a principal quantum number larger than one, that is the atom must have been excited by a Lyman absorption; then the excited atom is active for CREIL were the magnetic field is not null; for atoms pumped by a Lyman $\alpha$ transition, neglecting the electronic spin, and supposing that the de-exciting Raman component is very strong, the computation of the CREIL requires only the trace of a single, simple Raman polarisability tensor. If the Lyman lines are redshifted while they are absorbed, they appear so wide and weak that they cannot be seen. If a line of sight makes large enough an angle with the axis of the quasar, it crosses places where the magnetic field is nearly zero, so that, as there is no CREIL in these places, the absorption of the Lyman lines is stable in the spectrum and the lines appear. The large width of the lines may be a consequence of a low CREIL (by a low magnetic field, or collisional decrease), or, as this process is close to the quasar, it may depend slightly on the line of sight. This description of the origin of the BAL applies to the BEL. Thus, the loudness, the presence of broad lines is not a property of the quasars, but depends on the direction of observation: BAL quasars, observed in the direction nearly perpendicular to the axis of the disk, cannot radiate much electromagnetic field in this direction so that they are ”quiet”. The Lyman forest ================= In the standard model, the Lyman forest corresponds to clouds, that is to variations of the density of hydrogen atoms. It may be the result of the absorption in a chemically homogenous medium and a space-variable redshifting power of the gas resulting of variations of the magnetic (or electric) field. But how could a space-variable magnetic field be generated in the far halo? Maybe by electric instabilities in the plasma, or it could be the magnetic part of an extremely low frequency electromagnetic field radiated by the quasar and propagating very slowly in the plasma, more probably by magnetised satellites of the quasar. In one of these hypotheses, the modulation of the absorption which structures the Lyman forest is self-amplifying: Follow the light; if, in a region, the magnetic field is very low, there is no redshift, the Lyman $\alpha$ line saturates, so that, in the following region there is no Lyman $\alpha$ pumping, thus no CREIL even if a magnetic field appears; a high magnetic field is necessary to shift the spectrum by a small number of remaining CREIL-active atoms ( hydrogen pumped by other Lyman lines, and other atoms). When the line is shifted of a fraction of its linewidth, the Lyman $\alpha$, the CREIL and the redshift become strong, the absorption is spread and seems low in the spectrum; a return to the saturation requires a low magnetic field. This discussion is valid where the forest is saturated, but its results remain true elsewhere: only small fluctuations of the field may induce the forest. Two effects explain that the density of lines in the high redshift region of the forest generally decreases: i) the density of hydrogen being higher and its illumination higher, the CREIL is larger, the spectrum is spread, so that the decrease is an artefact; ii) in this region, some small satellites may fall into the accretion disk. Along a line of sight, the frequencies are shifted near the magnetic satellites while the Lyman lines are written far from these satellites. When an eclipse is observed by a variation of the intensity of the quasar, if small variations of the redshift can be detected, the gap corresponding to the satellite in the Lyman forest may be found. It could help a study of the satellite and the quasar. High redshift quasars ===================== The high redshift quasars appear in regions of the Universe in which clouds of molecular hydrogen are detected by the nuclear spin coupling transition at 21 cm. They seem dusty too. If the pressure of molecular hydrogen allows a very long collisional time, the UV ionises a part of the hydrogen into H$_2^+$ which is active in CREIL, thus cannot be observed from its absorption or emission spectra. The CREIL increases the redshift of the quasar; it amplifies the thermal radiation, possibly up to 100K, so that this thermal radiation may be confused with radiation of hot dust. If a galaxy appears next to a quasar, and has a lower redshift, it may, however, be further than the quasar. Thus its line of sight may cross the neighbourhood of the quasar, so that absorptions of metal lines may have the same origin, appear with the same redshift. Conclusion ========== This draft intends only to show that taking the Coherent Raman Effect on time-Incoherent Light (CREIL) into account solves very easily, without any new physics or new matter, problems which appear in the study of the quasars: - the absence of BAL lines in the loud quasars is deduced from the hypothesis of a disk similar to Saturn’s; - the discrepancies in the observed spectra may be produced by the small dispersion of the CREIL; - the quasars are younger than they seem, so that they can contain metals made in old stars. The use of CREIL allows to find several explanations of a particular observation; the present qualitative study must be specified quantitatively to choose the best explanation, maybe the standard explanation: for instance the thermal radiation may be partly produced by hot dust, partly by CREIL. Having several possible solutions is better than having none ! The model of quasars which takes CREIL into account uses only commonly observed types of astrophysical objects and simple regular physics; with much less work, it seems to give interpretations of observations on the quasars better than the standard model; it does not require dark matter, but if this matter is necessary for the gravitational stability, it proposes partially ionised H$_2$. Logically, is it acceptable to neglect CREIL [*a priori*]{}? The present demonstration of the necessity to take CREIL into account in astrophysics weakens the two main proofs of the big bang, expansion and 2.7K radiation. Webb J., V. V. Flambaum, C. W. Churchill, M. J. Drinkwater, & J. D. Barrow, 1999, Phys. Rev. Lett, 82, 884 Hasinger G., S. Komossa and N. Schartel, 2002, Astrophys. J. Letters, 573, L77 Reboul, H.,1981, J. Astron. Astrophys. Suppl., 45 , 129 Giordmaine, J. A. , M. A. Duguay & J. W.Hansen, 1968, IEEE J. Quantum Electron., 4, 252 Weiner, A. M. , D. E. Leaird., G. P. Wiederrecht, & K. A. Nelson, 1990, Science 247, 1317 Dougherty, T. P., G. P. Wiederrecht, K. A. Nelson, M. H. Garrett, H. P. Jenssen & C. Warde, 1992, Science [**258,**]{}, 770 Dhar, L. , J. A. Rogers, & K. A. Nelson, 1994 Chem. Rev. [**94**]{}, 157 Yan, Y.-X. , E. B. Gamble Jr. & K. A. Nelson , 1985, J. Chem Phys., 83, 5391 Lamb, G. L. Jr., 1971, Rev. Mod. Phys., 43 , 99 Moret-Bailly J., 1998, Ann. Phys. Fr. , 23, C1-235 Moret-Bailly J., 1998, Quantum and Semiclassical Optics, 10, L35 Moret-Bailly J., 2001, J. Quantit. Spectr. & Radiative Ttransfer, 68, 575 Moret-Bailly J., 2001, arXiv:astro-ph//0103508 , //0110525 and //0203099 de Kool M., Begelman M. C., 1995, Astrophys. J.,455, 448. [^1]: Laboratoire de physique, Université de Bourgogne, BP 47870, F-21078 Dijon cedex, France. Email : Jacques.Moret-Bailly@u-bourgogne.fr

--- abstract: 'Using the derived gamma-ray burst $E_{\rm peak}$ and fluences from the complete BATSE 5B Spectral Catalog, we study the ensemble characteristics of the $E_{\rm peak}$–fluence relation for GRBs. This relation appears to be a physically meaningful and insightful fundamental discriminator between long and short bursts. We discuss the results of the lower limit test of the $E_{\rm peak}$–$E_{\rm iso}$ relations in the $E_{\rm peak}$–fluence plane for BATSE bursts with no observed redshift. Our results confirm the presence of two GRB classes as well as heavily suggesting two different GRB progenitor types.' author: - 'Adam Goldstein, Robert D. Preece and Michael S. Briggs' title: | A New Discriminator for Gamma Ray Burst Classification:\ The $E_{\rm peak}$ – Fluence Energy Ratio --- Introduction ============ Classification of Gamma-Ray Bursts (GRBs) is certainly a difficult task. Bursts are divided into long and short classes, based upon the bimodal duration histogram of bursts [@Kouveliotou] observed by the Burst And Transient Source Experiment (BATSE), which was on board the Compton Gamma Ray Observatory. One parameter, the split time of 2 seconds on the $t_{90}$ durations plot [@Koshut], was sufficient to classify bursts. Bursts from the two classes have another discriminator, spectral hardness as determined by the ratio of two broad energy channels. This hardness ratio, when used in conjunction with the duration, provides a means for classification, as shown by @Kouveliotou. In addition, another classification scheme uses the scatter plot of the fluence and duration fitted with two 2D Gaussians [@Balazs]. Some have indicated there are more than two clusters [@Mukherjee; @Horvath], while @Hakkila maintain that the reported third cluster is simply the result of a selection bias. Furthermore, significant overlap is present in the duration comparison of short and long bursts, complicating a clear distinction between two classes as has been discussed by the authors cited above. In any case, there are difficulties with all aforementioned classification measures in that the $t_{90}$ duration is somewhat subjective in the necessary selections of background regions, while a hardness ratio based upon counts is strongly detector dependent. The observation that some classically short bursts are extended in duration, when observed in an energy band (BATSE 20-50 keV) different from that of the natural BATSE 50-300 keV band, was reported by @Lazzati. The dedicated GRB mission, Swift [@Barthelmy], has introduced additional issues, while reinforcing the extended emission of short GRBs at lower energies. @Norris have introduced the time lag between broad energy channels (‘spectral lag’) as a classifier: ‘short’ GRBs have approximately zero lag, while the ‘long’ events have a lag that is significantly different from zero. Indeed, having a near-zero lag is the basis for claiming that some bursts observed by the Swift BAT that are significantly longer than 2 seconds belong in the ‘short’ class [@longShorts; @Zhang]. Following on the analysis of @NakarPiran on the Amati relation [@Amati02], @BandPreece investigated the implications of combinations of several observable GRB parameters, derived from an extensive data set of GRBs observed by BATSE. The BATSE data set used was a partial spectral catalog of the peak flux and fluence spectral parameters [@Mallozzi], complete up to the end of the BATSE 4B Catalog [@Paciesas]. The BATSE 5B Spectral Catalog has now been completed [@5BCatalog], which includes all BATSE bursts with sufficient counts in the spectral data that they could be analyzed. Based on this comprehensive data set, we present a new GRB classification measure that is as diagnostic as the $t_{90}$ duration for classification, but does not rely on the subjective choices required for the durations calculation [@Koshut]. Motivation ========== BATSE data has been used to study the $E_{\rm peak}$ distributions of GRBs [@Kaneko], and the time-integrated $E_{\rm peak}$ distribution for all bursts shows no evidence of discrimination between short and long bursts. Fluence hardness distributions [@Kouveliotou] show some evidence of bimodality [@Balazs], but there is only moderate significance with much overlap [@Nakar]. Using the BATSE 5B Spectral Catalog, we support previous observations on the distributions of $E_{\rm peak}$ and fluence. In addition, we split the distributions into long and short duration GRB distributions, following the $t_{90}$ classification of two seconds. The $E_{\rm peak}$ distribution for short bursts is completely overlapped by that of long bursts. Although the $E_{\rm peak}$ values for long bursts are centered around 150 keV, short burst $E_{\rm peak}$ values are shifted to higher energies around 300 keV. This is consistent with previous findings of @Paciesas2 and @Ghirlanda2. Approximately 65% of the fluence distribution for short bursts overlaps that of the fluence distribution for long bursts, with the position of the peak of the short GRB fluence distribution being an order of magnitude less than that for long GRBs. @LloydPetrosian have shown there to be a significant correlation between $E_{\rm peak}$ and the total fluence in gamma-rays, and it is desirable to investigate this correlation for both long and short GRBs. For this reason, we investigate the $E_{\rm peak}$– fluence distribution for all BATSE bursts with good spectral fits and devise a discriminator between long and short bursts based on the difference in correlation between $E_{\rm peak}$ and fluence for long and short bursts. A choice formulation for a discriminator is based on the hardness of a burst. @Kouveliotou used a hardness ratio based on the ratio of calculated fluence in different energy bands to compare to duration estimates. Instead, we propose to use a hardness measure represented by $E_{\rm peak}$/fluence. This so-called energy ratio is in units of area and should prove to be a good discriminator between long and short bursts if there is a strong correlation between $E_{\rm peak}$ and fluence. Observations ============ From the 5B spectral catalog (Goldstein, Preece & Mallozzi, in prep.), we extract bursts with a good model fit as determined by a 3-sigma confidence limit, with the time-integrated $E_{\rm peak}$ and fluence errors for each burst required to be no more than 40% of their respective fitted values. A total of 1121 long bursts and 168 short bursts, classified according to the classical $t_{90}$ cut of two seconds [@Kouveliotou], satisfied these criteria. We then calculated the $E_{\rm peak}$/fluence energy ratio for each of these bursts and plot a histogram of these values, as in Fig. 1. By using a standard nonlinear least-squares fitting algorithm, we fit a single lognormal function to the distribution with the resulting chi-square goodness-of-fit statistic 111 for 32 degrees of freedom. We then fit two lognormal functions to the distribution with the resulting chi-square statistic of 32 for 29 degrees of freedom. Since the two models are nested, we use Pearson’s chi-square test to show that the large change in chi-square per degree of freedom results in a chance probability of $5 \times 10^{-17}$ and that the two lognormals are statistically preferred with a high degree of significance . From this bimodal distribution an obvious distinction between long and short bursts emerges. In Fig. 2 we plot two histograms corresponding to long and short bursts as identified by their respective $t_{90}$ estimations to show that the bimodal distribution of the energy ratio is correlated to that of the duration distribution. A K-S test comparing the long and short burst distributions in Fig. 2 to the best fit lognormal functions in Fig. 1, however, finds the correlations to be statistically marginal with 2% and 0.6% respective probabilities that each distribution in Fig. 2 is drawn from their respective best fit lognormal functions in Fig. 1. The energy ratio distribution for short bursts is narrower compared to that of long bursts and is shifted to higher energies, resulting from the fact that short bursts are generally harder than long bursts. It appears the energy ratio values are a good discriminator between the classical definition of long and short bursts by merging two known discriminators into one quantity, and their relative overlap can be well estimated. Only 4% of long GRBs overlap the 1-sigma core of the short burst distribution, and 2% of short bursts overlap the the 1-sigma core of the long burst distribution. Similarly, the overlap for the 2-sigma cores is 11% and 23%, respectively. Comparatively, for our sample, the classical $t_{90}$ overlap of long bursts onto the 1-sigma (2-sigma) core of short bursts is 4% (23%), and and there is no overlap of short bursts onto either the 1-sigma or 2-sigma core of long bursts. The central value for the long burst energy ratio distribution is $\sim$0.06 while the central value for the short burst distribution is $\sim$1.5. @BandPreece showed that the Amati relation [@Amati02], $$E^{\rm rest}_{\rm peak} = C_{\rm A} \ \Biggl( \frac{E_{\rm iso}} {10^{52} \ \rm{erg}} \Biggr)^{\eta_{\rm A}}$$ and the Ghirlanda relation [@Ghirlanda], $$E^{\rm rest}_{\rm peak} = C_{\rm G} \ \Biggl( \frac{E_{\rm iso} \ f_{\rm B}} {10^{51} \ \rm{erg}} \Biggr)^{\eta_{\rm G}},$$ could be converted into a similar energy ratio $$\frac{E^{1/\eta_i}_{\rm peak, obs}}{S_\gamma} \propto F(z),$$ where the $C_{\rm i}$ in the previous equations are the respective normalization coefficients and $f_{\rm B}$ is the beaming fraction relevant for the Ghirlanda relation. Here, $S_\gamma$ is the fluence in gamma-rays, and $\eta_i$ are the best fit power law indices for the respective models. These energy ratios can be represented as functions of redshift, $F(z)$, and the upper limit of the ratios could be determined for any redshift. The energy ratio upper limit of the Amati relation, as well as the energy ratio upper limit of the Ghirlanda relation, can be projected into the $E_{\rm peak}$ – fluence plane where they become lower limits. Using the bursts described above, the $E_{\rm peak}$ values and fluences can be plotted in this plane. Fig. 3 shows the distribution of long bursts in the upper plot and the distribution of short bursts in the bottom plot. The lines denote the lower limits of the Amati and Ghirlanda relations in this plane. Note that the beaming fraction for the Ghirlanda relation, $f_{\rm B}$ = 1.0, which is related to the jet opening angle, $\theta$, by $f_{\rm B} = 1- \cos{\theta}$. This represents the energy radiated by the burst equalling $E_{\rm iso}$. It can easily be seen there are two separate distributions in the $E_{\rm peak}$ – $E_{\rm iso}$ plane. Long bursts appear to be clumped between the Amati and Ghirlanda limits, while the short bursts appear to distribute along the Ghirlanda lower limit. Conclusions =========== The energy ratio shows a clear distinction between two different types of GRBs. The fluence encodes the duration of the burst without deriving a subjective $t_{90}$ estimate, and $E_{\rm peak}$/fluence physically represents a ratio of the energy at which most of the gamma-rays are emitted to the total energy emitted in gamma-rays. This quantity effectively serves as a spectral hardness ratio and shows an increased hardness for short bursts compared to long bursts, consistent with @Kouveliotou. The distribution of short bursts is narrower, its 1-sigma width covering less than one order of magnitude in energy, while the long bursts 1-sigma width covers slightly more than an order of magnitude. This bimodal distribution heavily supports the original distinction between long and short bursts [@Kouveliotou] and suggests further investigation is desirable. The correlation between $E_{\rm peak}$ and the total fluence in gamma-rays is of particular interest, as the energy ratio removes the cosmological dependence for the energies involved since its value is merely proportional to the square of the luminosity distance. Because of this reason, the energy ratio could be considered physically superior to the duration classifier for GRBs. The low degree of correlation between the $t_{90}$ distribution and the energy ratio distribution may be attributed to the fact that the former is a an observed quantity in the observer’s frame, while the latter is a quantity that contains spectral information from the rest frame of the GRB. In addition, when comparing the difference in overlap between the energy ratio distributions and the $t_{90}$ distributions, the overlap for the $t_{90}$ distributions is marginally less pronounced, but gives little insight to the physical processes in the rest frame of the GRB. Note that there may be a truncation effect with the energy ratio associated with short bursts, due to the inability of BATSE to trigger on very low fluence events and bursts with $E_{\rm peak}$ outside the BATSE energy range. It is expected that Fermi/GBM will assist in discovering GRBs with $E_{\rm peak}$ values greater than 1 MeV, alleviating possible truncation effects due to the energy cutoff at the high end of the detector energy band. While a majority of BATSE bursts ($\sim$87 percent) fail the lower limit test for the Amati relation in the $E_{\rm peak}$ – fluence plane, very few BATSE bursts violate the lower limit for the Ghirlanda relation with $f_{\rm B}$ = 1.0. Especially intriguing is the fact that short GRBs fall close to the lower limit where the energy radiated is isotropic. Decreasing the beaming fraction shifts the lower limit to higher fluences, causing an increasing number of bursts to violate the Ghirlanda relation. In addition, only a few jet breaks have been discovered for short GRBs [@Soderberg], and @Watson caution those discoveries may be misleading due to the flaring activities of short GRB decay. Thus, the Ghirlanda relation, as well as the lack of reliable observed jet breaks for short bursts, suggests that short bursts release energy isotropically (or near isotropically), while longer bursts tend to have a much smaller beaming fraction and consequently have small opening angles. Clearly the Ghirlanda relation, if accurate, requires short bursts to release energy isotropically, as opposed to beamed radiation release in long bursts [@Frail; @Nakar]. In addition, if short GRBs are isotropic emitters, then virtually all short GRBs should be detectable within a given volume of space. Long GRBs, in general, have a small measured opening angle, and therefore only a small fraction are detectable [@Paradijs]. Since short GRBs are detected far less frequently than long GRBs [@Paciesas] this is indicative of a relatively rare and independent cause for short GRBs. Amati, L., et al. 2002, , 390, 81 Balázs, L. G., Bagoly, Z., Horváth, I., & M észáros, A., & Mészáros, P. 2003, , 401, 129 Band, D. L. & Preece, R. D. 2005, , 627, 319 Barthelmy, S. D., et al. 2005, Space Science Reviews, 120, 143 Frail, D. A. 2001, , 562, L55 Ghirlanda, G., Ghisellini, G., & Lazzati, D. 2004, , 616, 331 Ghirlanda, G., Nava, L., Ghisellini, G., Celotti, A., & Firmani, C. 2009, , 496, 585 Goldstein, A., Preece, R. D., & Mallozzi, R. S. 2010, The Complete BATSE GRB Spectral Catalog (http://gammaray.msfc.nasa.gov/$\sim$goldstein) Hakkila, J., Haglin, D. J., Roiger, R. J., Mallozzi, R. S., Pendleton, G. N., & Meegan, C. A. 2000, Gamma-ray Bursts, 5th Huntsville Symposium, 526, 33 Horváth, I. 1998, , 508, 757 Kaneko, Y., Preece, R. D., Briggs, M. S., Paciesas, W. S., Meegan, C. A., & Band, D. L. 2006, , 166, 298 Koshut, T. M., Paciesas, W. S., Kouveliotou, C., van Paradijs, J., Pendleton, G. N., Fishman, G. J., & Meegan, C. A. 1996, , 463, 570 Kouveliotou, C., Meegan, C. A., Fishman, G. J., Bhat, N. P., Briggs, M. S., Koshut, T. M., Paciesas, W. S., & Pendleton, G. N. 1993, , 413, L101 Lazzati, D., Ramirez-Ruiz, E., & Ghisellini, G. 2001, , 379, L39 Lloyd, N. M., Petrosian, V., & Mallozzi, R. S. 2000, , 534, 227 Mallozzi, R. S., Paciesas, W. S., Pendleton, G. N., Briggs, M. S., Preece, R. D., Meegan, C. A., & Fishman, G. J. 1995, AAS, 454, 597 Mukherjee, S., Feigelson, E. D., Jogesh Babu, G., Murtagh, F., Fraley, C., & Raftery, A. 1998, , 508, 314 Nakar, E. 2007, Physics Reports, 442, 166 Nakar, E., & Piran, T. 2005, MNRAS, 360, L73 Norris, J. P., & Bonnell, J. T. 2006, , 643, 266 Norris, J. P., Marani, G. F., & Bonnell, J. T. 2000, , 534, 248 Paciesas, W. S., et al. 1999, , 122, 465 Paciesas, W. S., Briggs, M. S., Preece, R. D., & Mallozzi, R. S. 2003, Gamma-ray Burst and Afterglow Astronomy 2001: A Workshop Celebrating the First Year of The HETE Mission, Conf. Proc, 662, 248 van Paradijs, J., Kouveliotou, C., Wijers, R. A. M. J. 2000, , 38, 379 Soderberg, A. M., Berger, E., Kasliwal, M., et al. 2006, , 650, 261 Watson, B., Hjorth, J., Jakobsson, P., Xu, D., Fynbo, J. P. U, Sollerman, J., Thöne, C. C., & Pedersen, K. 2006, , 454, L123 Zhang, Z. B., Xie, G. Z., Deng, J. G. 2006, , 373, 729 ![Histogram of the energy ratio distribution in the 20-2000 keV range for 1289 GRBs. The dotted line shows the best fit lognormal to the distribution, and the dashed lines show the fit of two lognormal functions. The chi-square goodness-of-fit statistic for one lognormal is 111 for 32 degrees of freedom, while for two lognormals is 32 for 29 degrees of freedom. Therefore, two lognormal distributions are statistically perferred, resulting in a bimodal distribution for the energy ratio. \[fig1\]](BATSE_EnRatio.eps) ![Histograms of the energy ratio distributions in the 20-2000 keV range for 1121 long bursts (white) and 168 short bursts (gray). There are clearly two distinct distributions, with long bursts centered around 0.6 and short bursts centered around 1.5. The solid curves are the best fit lognormal functions, and the dashed lines are the 1-sigma standard deviation of the distributions. \[fig2\]](eRatio.eps) ![Scatter plots of 1121 long bursts (top) and 168 short bursts (bottom) in the $E_{\rm peak}$–fluence plane. Also plotted are the lower limits of the Amati relation and Ghirlanda relation ($\rm{f}_B$=1.0). The dashed lines represent the 1-sigma errors about the lower limits. There is apparently a clear lower limit violation by most BATSE bursts for the Amati relation. Short bursts appear to cluster around the lower limit of the Ghirlanda relation. \[fig3\] ](epeakFluence.eps)

--- abstract: | CRTS J084133.15+200525.8 is an optically bright quasar at $z = 2.345$ that has shown extreme spectral variability over the past decade. Photometrically, the source had a visual magnitude of $V \sim 17.3$ between 2002 and 2008. Then, over the following five years, the source slowly brightened by approximately one magnitude, to $V \sim 16.2$. Only $\sim 1$ in 10,000 quasars show such extreme variability, as quantified by the extreme parameters derived for this quasar assuming a damped random walk model. A combination of archival and newly acquired spectra reveal the source to be an iron low-ionization broad absorption line (FeLoBAL) quasar with extreme changes in its absorption spectrum. Some absorption features completely disappear over the 9 years of optical spectra, while other features remain essentially unchanged. We report the first definitive redshift for this source, based on the detection of broad H$\alpha$ in a Keck/MOSFIRE spectrum. Absorption systems separated by several 1000 km s$^{-1}$ in velocity show coordinated weakening in the depths of their troughs as the continuum flux increases. We interpret the broad absorption line variability to be due to changes in photoionization, rather than due to motion of material along our line of sight. This source highlights one sort of rare transition object that astronomy will now be finding through dedicated time-domain surveys. author: - 'Daniel Stern, Matthew J. Graham, Nahum Arav, S. G. Djorgovski, Carter Chamberlain, Aaron J. Barth, Ciro Donalek, Andrew J. Drake, Eilat Glikman, Hyunsung D. Jun, Ashish A. Mahabal, Charles. C. Steidel' title: Extreme Variability in a Broad Absorption Line Quasar --- Introduction ============ For galaxies hosting active galactic nuclei (AGNs), time-domain surveys have long proven to be fertile avenues of research. Indeed, optical continuum variability was recognized as a common feature of quasars shortly after their initial discovery [@Matthews:63], and has since been exploited for purposes ranging from identifying quasars [, @vandenBergh:73], to determining black hole masses through reverberation mapping [, @Blandford:82; @Bentz:09], to studying the inner circumnuclear environment [, @Risaliti:02]. Recent efforts using wide-area, time-domain surveys have vastly extended this avenue of research by exploring the optical variability of extremely large samples of quasars, numbering in the tens to hundreds of thousands [, @MacLeod:12; @Graham:14]. Besides determining the light curve properties of typical quasars, such work has identified interesting new phenomenology such as candidate periodic light curves suggestive of sub-parsec binary super-massive black hole systems [, @DOrazio:15a; @DOrazio:15b; @Graham:15; @Graham:15b; @Jun:15b; @Liu:15], AGN undergoing major flaring suggestive of microlensing or explosive activity in the accretion disk such as superluminous supernovae, mergers, or tidal disruption events [, @Drake:11; @Lawrence:16 Graham , submitted], and changing look AGN with the abrupt appearance or disappearance of broad emission lines [, @LaMassa:15; @Gezari:17]. One topic where quasar variability has received particular attention has been the temporal characteristics of broad absorption line (BAL) quasars. Specifically, over the past few years, several teams have reported on multi-epoch spectroscopic observations of BAL quasars [, @Barlow:92; @Lundgren:07; @Gibson:08; @Gibson:10; @Capellupo:11; @Capellupo:12; @Capellupo:13; @FilizAk:12; @FilizAk:13; @FilizAk:14; @Vivek:12; @He:14; @He:15; @Joshi:14; @Wildy:14; @Wildy:15; @Grier:15; @Zhang:15]. While variability in BAL trough strengths is relatively common, large ($> 50\%$) changes in the absorption equivalent width is quite rare [, @Hall:11]. A primary question in BAL variability studies has been whether observed changes in BAL trough strengths are primarily due to changes in the ionization state of the outflowing wind [, @Wang:15], or whether they are due to high column density BAL clouds moving through our line of sight [, @McGraw:15]. For example, @FilizAk:13 present a detailed analysis of $\approx 650$ BAL troughs identified in 291 quasars observed by the Sloan Digital Sky Survey (SDSS), sampling rest-frame timescales between 1 and 3.7 years. They estimate that the average lifetime of a BAL trough is a few thousand years, and that the emergence/disappearance of BAL features are extremes of general BAL variability. @FilizAk:13 also report coordinated BAL variability across multiple troughs at different velocities. They argue that changes in the opacity of the shielding gas producing changes in the ionizing radiation incident on the BAL material are the most probable cause for such coordinated variability. @Grier:15 and @Wildy:15 reach similar conclusions based on the highly variable BAL lines seen in a spectroscopic monitoring campaigns. With variability seen on time-scales of just a few days, both authors conclude that the most likely cause of such rapid changes is the BAL gas responding to changes in the incident ionizing continuum. Leading to an alternative explanation of BAL variability, @Capellupo:11 [@Capellupo:12] report on an ongoing monitoring campaign of a sample of 24 BAL quasars at $1.2 < z < 2.9$ on timescales ranging from $\sim 4$ months to $\sim 8$ years. Studying the BAL feature, @Capellupo:11 found variability in 40% of their sample on month-long timescales, and in 65% of their sample on year-long timescales. They find that higher-velocity BALs are more likely to vary than lower-velocity BALs, and that weaker BALs are more likely to vary than stronger BALs. They suggest that the observations are best understood as the movement of clouds within 6 pc of the central engine across the line of sight. In a detailed study of the first observation of $\lambda \lambda 1118, 1128$ BAL variability in a quasar, @Capellupo:14 argue that the observations are best described by a BAL cloud at a distance of $\simlt 3.5$ pc moving across the line sight. The implied kinetic energy of the outflow would be $\sim 2\%$ of the quasar bolometric luminosity, which is sufficient to cause substantial feedback. Also supporting this interpretation that BAL variability is not dominated by photoionization, @He:14 report on 18 epochs of SDSS/BOSS spectroscopy of a BAL quasar at $z = 2.72$. They find only a weak correlation between the BAL variability and the continuum luminosity, suggesting that continuum changes are not driving changes in the BAL trough amplitudes. Here, we report on CRTS J084133.15+200525.8 (), an optically bright quasar that has shown extreme variability over the past decade (Figure \[fig:lc\]). The quasar transitioned from having a relatively stable visual magnitude of $V \sim 17.3$ between 2002 and 2008, to slowly brightening by a factor of $\sim 2.5$ over the course of 5 years and then plateauing at $V \sim 16.2$. As detailed below, a combination of archival and newly acquired spectroscopy reveal this source to be an iron low-ionization broad absorption line (FeLoBAL) quasar exhibiting extreme spectroscopic changes over the same time period, and the nature of these variations allow us to assess the likely cause of the BAL trough variability. Independent of our own work on , @Rafiee:16 recently reported on this same source as part of a sample of three FeLoBAL quasars that have shown significant spectroscopic variability over the past decade. Interestingly, all three show decreasing strength of their low-ionization iron absorption. The current paper has several additions relative to that work. Specifically, we provide new data on , including a new epoch of optical spectroscopy which demonstrates continued spectral changes, and a near-infrared spectrum which provides the first precise redshift for the quasar as well as an estimate of its black hole mass. Finally, @Rafiee:16 remain agnostic as to whether absorber transverse motion or ionization variability is the more likely cause of the changes in the absorption troughs of this source. In contrast, the additional epoch of Palomar spectroscopy presented here allows us to argue that ionization variability is the more likely cause of the extreme absorption variability seen in . Throughout this paper, we use Vega magnitudes unless otherwise indicated and we adopt the concordance cosmology, $\Omega_{\rm M} = 0.3$, $\Omega_\Lambda = 0.7$ and $H_0 = 70\, \kmsMpc$. Data and Results ================ Optical Light Curve ------------------- The Catalina Real-time Transient Survey[^1] [CRTS; @Drake:09] leverages the Catalina Sky Survey, designed to search for near-Earth objects, as a probe of the time-variable universe. CRTS has used three telescopes for much of the past decade, two in the northern hemisphere and one in Australia, to cover up to $\sim 2500\, {\rm deg}^2$ per night. The filterless observations are broadly calibrated to Johnson $V$ [for details, see @Drake:13] with a nominal depth of $V \sim 20$. The full CRTS data set contains time series for approximately 500 million sources.[^2] CRTS represents the best data set currently available with which to systematically study quasar variability with large samples over a decade-length timescale. In an analysis of characteristic timescales of 240,000 known spectroscopically confirmed objects using Slepian wavelet variance, Graham et al. (submitted) originally identified  as an extreme outlier in the plane defined by a linear trend (the Thiel-Sen statistic) and deviation from the median Slepian wavelet variance fit. In that analysis,  has a characteristic timescale $\tau = 109.9$ days, which is significantly larger than expected for a quasar of its magnitude, $\tau = 48.0 \pm 5.9$ days. If we instead characterize quasar light curves with a Gaussian process damped random walk model and only consider the subset of 79,749 quasars with at least 200 CRTS photometric measurements,  again stands out. The two parameters from this model are the amplitude, $\sigma$, and the characteristic timescale, $\tau$, of the damped random walk [, @Kelly:09]. We use a kernel density estimator to determine the distribution of sources in the $\sigma-\tau$ plane, and we find that  resides in an extreme location in this plane ($\log \Sigma = -7.8$, where $\Sigma$ is the density of sources in this plane). Only seven quasars stand out at this level or more from the population distribution, implying that only $\sim 1$ in 10,000 quasars show variability behavior as extreme as . Further inspection of the CRTS light curve of  (Figure \[fig:lc\]) also indicates that the variable behavior is different from the expected stochastic damped random walk model that describes most quasars, and instead appears more consistent with a state change. Further support for this interpretation comes from earlier photometry of  reported in @Rafiee:16 from the Palomar Sky Surveys (POSS-I, Palomar Quick V, and POSS-II), reaching back to the mid-1950s. @Rafiee:16 reports no evidence for a significant change in the optical brightness of  prior to 2000. SDSS imaged  on UT 2004 December 12 (MJD = 53351), which is prior to the brightening episode. The source was unresolved, and based on its unusual and red colors, SDSS targeted  for spectroscopic observations as a high-redshift quasar candidate. Optical Spectroscopy --------------------  was first observed spectroscopically by SDSS on UT 2005 December 1 [MJD = 53705; @Blanton:03] and was then re-observed by SDSS-III BOSS on UT 2011 January 4 [MJD = 55565; @Dawson:13]. The spectra, shown in Figure \[fig:optical\], show a source with many absorption features, making redshift identification challenging. Indeed, the SDSS data releases have reported a variety of redshifts for , always with warning flags, ranging from $z = 0.859$ (DR8; Warning = Many Outliers) to $z = 1.295$ (DR7; zStatus = Failed) to $z = 3.195$ (DR9; Warning = Negative Emission). Our visual inspection of the BOSS spectrum tentatively identified and blends in the region around 9400 Å, implying $z \sim 2.3$, consistent with both the visual inspection value of $z = 2.342$ in the SDSS DR12 quasar catalog [DR12Q; @Paris:14] and the results of our Keck infrared spectrum described in §2.4. We obtained additional optical spectroscopy of  using the Double Spectrograph on the Hale 200” Telescope at Palomar Observatory on UT 2014 April 22 (MJD = 56769). We obtained two 600 s exposures using the 1.0 slit in cloudy conditions. The data were reduced using standard procedures and relative spectrophotometric calibration was achieved using observations of standard stars obtained on the same night. Figure \[fig:optical\] presents the Palomar data, where we have scaled the spectra so that the long wavelength ($\simgt 5500$ Å) part of the spectra is of comparable flux density to the BOSS spectrum at the same wavelengths. The multi-epoch spectra show the extreme variability exhibited by , as well as multiple strong absorption features, characteristic of an FeLoBAL quasar. FeLoBALs are notoriously challenging targets for redshift identification [, @Becker:97; @Brunner:03]. We see dramatic changes across the full spectrum, particularly in the spectral region between redshifted Ly$\alpha$ and . Some features do not change across the near-decade timescale of the spectroscopy, such as the saturated absorption at 5150 Å. Other features completely disappear, such as absorption lines at $\approx$ 5450 and 8500 Å. There is an overall uncovering of blue continuum emission, with the flux around Ly$\alpha$ increasing by an order of magnitude over the $>8$ years spanned by the spectroscopy. In addition, while the continuum between \] and is extremely choppy in the 2005 spectrum, by 2014 it is smoother, which is more typical of normal quasar spectra. Imaging at Other Wavelengths ----------------------------  is a bright near- to mid-infrared source, well detected by both the Two Micron All Sky Survey [$K_s = 13.62 \pm 0.04$ — 2MASS; @Skrutskie:06] and the [*Wide-field Infrared Survey Explorer*]{} [$W3 = 9.58 \pm 0.06$ — ; @Wright:10]. With $W1 - W2 = 0.65$,  is slightly bluer than the mid-infrared AGN selection criteria of @Stern:12, which are $W1 - W2 \geq 0.8$ and $W2 \leq 15.05$. However, as shown in @Assef:13, the AGN selection color can be relaxed for brighter sources. There is little variability detected at longer wavelengths in this source. In AB magnitudes, the $z$-band magnitude recorded by SDSS was $z = 16.37 \pm 0.01$ on MJD 53351, closely matching the $z$-band magnitude of $Z = 16.34 \pm 0.01$ recorded by UKIRT Infrared Deep Sky Survey [UKIDSS; @Lawrence:07] on MJD 55141. In the near-infrared (in Vega magnitudes), 2MASS recorded $H = 14.41 \pm 0.05$ and $K_s = 13.62 \pm 0.04$ on MJD 51105, closely matching the UKIDSS values of $H = 14.32 \pm 0.02$ on MJD 54061 and $K = 13.57 \pm 0.02$ on both MJD 54061 and MJD 55238, where we have assumed a 2% floor on the UKIDSS photometric calibration [, @Hodgkin:09]. Similarly, the mid-infrared flux measured by [*WISE*]{} and [*NEOWISE*]{} [@Mainzer:14] varies by only $\sim 0.04$ mag, comparable to the typical uncertainty.  is not detected by [*ROSAT*]{}, nor was it (serendipitously) observed by either the [*Chandra X-Ray Observatory*]{} or [*XMM-Newton*]{}.  is also not detected by the Faint Images of the Radio Sky at Twenty cm survey [FIRST; @Becker:95], implying $S_{\rm 1.4~GHz} \simlt 1$ mJy ($5 \sigma$). Finally, as expected, observations by the [*Galaxy Evolution Explorer*]{} [[*GALEX*]{}; @Martin:05], which sample below the Lyman limit for $z = 2.35$, do not detect . Near-infrared Spectroscopy -------------------------- We obtained a $K$-band (1.95-2.39 $\mu$m) spectrum of  with the Multi-Object Spectrometer for InfraRed Exploration [MOSFIRE; @McLean:12; @Steidel:14] on UT 2014 May 5 (MJD=56782) in longslit mode. We obtained three dithered exposures of 180 s each through a 07 entrance slit under clear conditions with good seeing. The spectrum was reduced using a combination of the MOSFIRE data reduction pipeline (DRP) and custom routines [for details, see @Steidel:14]. Wavelength calibration was based on a combination of OH emission lines in the night sky and an internal Ne arc lamp. Flux calibration and telluric absorption removal was accomplished using spectra of an A0V star (Vega analog) observed at similar airmass. The final extracted spectrum (Figure \[fig:nearIR\]) shows strong continuum and a single broad emission line with a peak at 2.1956 $\mu$m, which we identify as H$\alpha$ at $z=2.3446$. The apparent asymmetry in the continuum straddling the line is well modeled by the @Boroson:92 template on the blue side of the line. We use the broad H$\alpha$ emission line to estimate the mass of the black hole in . First, we apply a multiplicative correction to the $K$-band spectrum to match the $K$-band photometry from the UKIDSS observations. We then approximate the uncertainties in the spectrum by considering the standard deviation of the spectrum outside the strong emission line. We model the H$\alpha$ spectral region as the sum of two broad Gaussian lines, a single narrow Gaussian, an iron template (which elevates the continuum on the blue side of the emission line), and a power-law continuum. The full-width at half-maximum (FWHM) of the broad H$\alpha$ emission is $6086 \pm 42\, \kms$, and the combined luminosity of the broad H$\alpha$ components is $L_{\rm H\alpha} = (5.56 \pm 0.05) \times 10^{45} \ergs$. Modeling the broad-band (3000Å to 7$\mu$m) spectral energy distribution of the quasar as a sum of a power-law continuum, two blackbody thermal components (500 and 1250 K, to model the rest-frame IR emission), and line emission from H$\alpha$ and as determined from the Keck spectrum, we derive $L_{5100} = (1.24 \pm 0.02) \times 10^{47} \ergs$. Following @Jun:15a, we derive $\log(M_{\rm BH}/M_\odot) = 10.36 \pm 0.16$ using the $L_{5100}$ estimator and $\log(M_{\rm BH}/M_\odot) = 10.29 \pm 0.17$ using the $L_{\rm H\alpha}$ estimator. We note that these statistical error bars underestimate the true uncertainty, both due to the non-simultaneity of the imaging and near-infrared spectroscopy and, more importantly, the systematic uncertainty in the virial scale factor, $f$, which is the typically the dominant source of uncertainty in black hole mass measurements; in this case, we adopt $f = 5.1 \pm 1.3$ from @Woo:13, as per @Jun:15a. For comparison, without access to any well-detected emission features, @Rafiee:16 simply adopted a black hole mass of $M_{\rm BH} = 6 \times 10^9\, M_\odot$ as a typical value. Adopting their value for the bolometric luminosity of , $L_{\rm bol} = (3.36 \pm 0.69) \times 10^{47}\, {\rm erg}\, {\rm s}^{-1}$ [based on the observed rest-frame 2900 Å flux density and a bolometric correction of ${\rm BC}_{2900} = 5 \pm 1$ from @Richards:06], we determine an Eddington ratio of $L_{\rm bol}/L_{\rm Edd} \sim 0.15$ (in comparison to their value of 0.45). Discussion ========== Figure \[fig:Fe\_Mg\] shows a comparison between the long wavelength portion of the three epochs of optical spectroscopy. In the spectral region beyond $\sim 7300$ Å there is much less blending of troughs from different ions, making the variability changes simpler to interpret. We identify two primary absorption systems. The first system, A, shows several troughs of UV absorption between 7500 and 8900 Å, as well as absorption at 9050 Å. The other system, B, shows absorption at 9300 Å. The two absorption systems are separated by 9000 , yet show coordinated reductions in the depth of their troughs as the quasar brightens. This is the expected behavior if the BAL spectral variability is driven by changes in the photoionization: as the ionizing continuum flux increases, the column densities of and decrease for all clouds along the line of sight. (Note that this expectation assumes that the ionizing continuum changes are correlated with the flux changes around 2500 Å). The scenario of clouds moving across our line of sight is hard pressed to explain both the coordinated changes of the trough depths as well as the observed trough weakening with increasing UV flux. A priori, there is no reason that troughs as widely separated in velocity as A and B would be correlated since they are different parcels of gas. Even more so, there is no reason in this scenario for changes in the trough depths to be correlated with flux changes. Therefore, we interpret the variability in the absorption troughs to be due to changes in photoionization, rather than motion of material into our line of sight. A follow-up paper will more carefully model the full multi-epoch spectroscopic data set, including additional spectroscopy from our continuing monitoring, with the goal of understanding the location and energetics of the outflow, and its impact on the host galaxy (Chamberlain , in preparation).  appears to be an FeLoBAL quasar in the process of transitioning to a more common low-ionization BAL (LoBAL) quasar, similar to FBQS J1408+3054 reported by @Hall:11. We note, however, that @Hall:11 interpreted the variability in that source as being related to structure in the BAL outflow moving out of our line of sight rather than being related to photoionzation changes.  highlights the sort of rare, extremely variable quasars that can be used to probe the physics of quasar outflows. We expect to find many more such examples with the new generation of wide-area, sensitive, high-cadence synoptic surveys. We were fortuitous in this case that multi-epoch archival spectroscopy was available for this source. In the future, it will be exciting to find similar major events in real time, allowing real-time multi-wavelength follow-up in order to more fully dissect the internal workings of AGN engines. We thank the anonymous referee for a prompt and helpful referee report. CRTS was supported by the NSF grants AST-1313422, AST-1413600, and AST-1518308. The work of DS and HJ was carried out at Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. DS also acknowledges support from NASA through ADAP award 12-ADAP12-0109. NA and CC acknowledge support from NSF through grant AST 1413319, and from NASA through STScI grants GO 11686 and GO 12022. Research by AJB was supported by NSF grant AST-1412693. EG acknowledges the generous support of the Cottrell College Award through the Research Corporation for Science Advancement. HJ is supported by an appointment to the NASA Postdoctoral Program at the Jet Propulsion Laboratory, administered by Universities Space Research Association under contract with NASA. The authors a grateful to the staff at the Palomar and Keck observatories, where some of the data presented here were obtained. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. , , , , , ©2017. All rights reserved. [63]{} natexlab\#1[\#1]{} Assef, R. J., Stern, D., Kochanek, C. S., [et al.]{} 2013, , 772, 26 Barlow, T. A., Junkkarinen, V. T., Burbidge, E. M., Weymann, R. J., Morris, S. L., & Korista, K. T. 1992, , 397, 81 Becker, R. H., Gregg, M. D., Hook, I. M., McMahon, R. G., White, R. L., & Helfand, D. J. 1997, , 479, 93 Becker, R. H., White, R. L., & Helfand, D. J. 1995, , 450, 559 Bentz, M. C., Walsh, J. L., Barth, A. J., [et al.]{} 2009, , 705, 199 Blandford, R. D. & McKee, C. F. 1982, , 255, 419 Blanton, M. R., Lin, H., Lupton, R. H., [et al.]{} 2003, , 125, 2276 Boroson, T. A. & Green, R. F. 1992, , 80, 109 Brunner, R. J., Hall, P. B., Djorgovski, S. G., [et al.]{} 2003, , 126, 53 Capellupo, D. M., Hamann, F., & Barlow, T. A. 2014, , 444, 1893 Capellupo, D. M., Hamann, F., Shields, J. C., Halpern, J. P., & Barlow, T. A. 2013, , 429, 1872 Capellupo, D. M., Hamann, F., Shields, J. C., [Rodríguez Hidalgo]{}, P., & Barlow, T. A. 2011, , 413, 908 —. 2012, , 422, 3249 Dawson, K. S., Schlegel, D. J., Ahn, C. P., [et al.]{} 2013, , 145, 10 , D. J., Haiman, Z., Duffell, P., Farris, B. D., & MacFadyen, A. I. 2015, , 452, 2540 , D. J., Haiman, Z., & Schiminovich, D. 2015, , 525, 351 Drake, A. J., Catelan, M., Djorgovski, S. G., [et al.]{} 2013, , 763, 32 Drake, A. J., Djorgovski, S. G., Mahabal, A., [et al.]{} 2009, , 696, 870 Drake, A. J., Graham, M. J., Djorgovski, S. G., [et al.]{} 2011, , 213, 9 , N., Brandt, W. N., Hall, P. B., [et al.]{} 2012, , 757, 114 —. 2013, , 777, 168 —. 2014, , 791, 88 Gezari, S., Hung, T., Cenko, S. B., [et al.]{} 2017, , 835, 144 Gibson, R. R., Brandt, W. N., Gallagher, S. C., Hewett, P. C., & Schneider, D. P. 2010, , 713, 220 Gibson, R. R., Brandt, W. N., Schneider, D. P., & Gallagher, S. C. 2008, , 675, 985 Graham, M. J., Djorgovski, S. G., Drake, A. J., [et al.]{} 2014, , 439, 703 Graham, M. J., Djorgovski, S. G., Stern, D., [et al.]{} 2015, , 518, 74 —. 2015, , 453, 1562 Grier, C. J., Hall, P. B., Brandt, W. N., [et al.]{} 2015, , 806, 111 Hall, P. B., Anosov, K., White, R. L., [et al.]{} 2011, , 411, 2653 He, Z., Bian, W., Ge, X., & Jiang, X. 2014, , 443, 2532 —. 2015, , 454, 3962 Hodgkin, S. T., Irwin, M. J., Hewett, P. C., & Warren, S. J. 2009, , 394, 675 Joshi, R., Chand, H., Srianand, R., & Majumdar, J. 2014, , 442, 862 Jun, H. D., Im, M., Lee, H. M., [et al.]{} 2015, , 806, 109 Jun, H. D., Stern, D., Graham, M. J., [et al.]{} 2015, , 814, L12 Kelly, B. C., Bechtold, J., & Siemiginowska, A. 2009, , 698, 895 LaMassa, S. M., Cales, S., Moran, E. C., [et al.]{} 2015, , 800, 144 Lawrence, A., Bruce, A. G., MacLeod, C., [et al.]{} 2016, , 463, 296 Lawrence, A., Warren, S. J., Almaini, O., [et al.]{} 2007, , 379, 1599 Liu, T., Gezari, S., Heinis, S., [et al.]{} 2015, , 803, 16 Lundgren, B. F., Wilhite, B. C., Brunner, R. J., [et al.]{} 2007, , 656, 73 Macleod, C. L., Ž. Ivezić, Sesar, B., [et al.]{} 2012, , 753, 106 Mainzer, A., Bauer, J., Grav, T., [et al.]{} 2014, , 784, 110 Martin, D. C., Schiminovick, D., Barlow, T. A., [et al.]{} 2005, , 619, L1 Matthews, T. A. & Sandage, A. R. 1963, , 138, 30 McGraw, S. M., Shields, J. C., Hamann, F. W., Capellupo, D. M., Gallagher, S. C., & Brandt, W. N. 2015, , 453, 1379 McLean, I. S., Steidel, C. C., Epps, H., [et al.]{} 2012, SPIE, 8446, 17 Pâris, I., Petitjean, P., Aubourg, E., [et al.]{} 2014, , 563, 54 Rafiee, A., Pirkola, P., Hall, P. B., Galati, N., Rogerson, J., & Ameri, A. 2016, , 459, 2472 Richards, G. T., Lacy, M., Storrie-Lombardi, L. J., [et al.]{} 2006, , 166, 470 Risaliti, G., Elvis, M., & Nicastro, F. 2002, , 571, 234 Sesar, B., Scott, S. J., Zeljko, I., Morgan, D. P., Becker, A. C., & Woźniak, P. 2011, , 142, 190 Skrutskie, M. F., Cutri, R. M., Stiening, R., [et al.]{} 2006, , 131, 1400 Steidel, C. C., Rudie, G. C., Strom, A. L., [et al.]{} 2014, , 795, 165 Stern, D., Assef, R. J., Benford, D. J., [et al.]{} 2012, , 753, 30 , S., Herbst, E., & Pritchet, C. 1973, , 78, 375 Vivek, M., Srianand, R., Petitjean, P., Noterdaeme, P., Mohan, V., Mahabal, A., & Kuriakose, V. C. 2012, , 423, 2879 Wang, T., Yang, C., Wang, H., & Ferland, G. 2015, , 814, 150 Wildy, C., Goad, M. R., & Allen, J. T. 2014, , 437, 1976 —. 2015, , 448, 2397 Woo, J., Schulze, A., Park, D., Kang, W., Kim, S. C., & Riechers, D. A. 2013, , 772, 49 Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., [et al.]{} 2010, , 140, 1868 Zhang, S., Zhou, H., Wang, T., [et al.]{} 2015, , 803, 58 [^1]: See [crts.caltech.edu/]{}. [^2]: See [http://catalinadata.org/]{}.

--- abstract: 'We study two methods for computing network features with topological underpinnings: the Rips and Dowker persistent homology diagrams. Our formulations work for general networks, which may be asymmetric and may have any real number as an edge weight. We study the sensitivity of Dowker persistence diagrams to asymmetry via numerous theoretical examples, including a family of highly asymmetric cycle networks that have interesting connections to the existing literature. In particular, we characterize the Dowker persistence diagrams arising from asymmetric cycle networks. We investigate the stability properties of both the Dowker and Rips persistence diagrams, and use these observations to run a classification task on a dataset comprising simulated hippocampal networks. Our theoretical and experimental results suggest that Dowker persistence diagrams are particularly suitable for studying asymmetric networks. As a stepping stone for our constructions, we prove a functorial generalization of a theorem of Dowker, after whom our constructions are named.' address: - 'Department of Mathematics, The Ohio State University. Phone: (614) 292-6805.' - 'Department of Mathematics and Department of Computer Science and Engineering, The Ohio State University. Phone: (614) 292-4975, Fax: (614) 292-1479.' author: - Samir Chowdhury and Facundo Mémoli bibliography: - 'biblio.bib' title: A functorial Dowker theorem and persistent homology of asymmetric networks --- Introduction ============ Networks are used throughout the sciences for representing the complex relations that exist between the objects of a dataset [@newman2003structure; @kleinberg-book]. Network data arises from applications in social science [@kumar2010structure; @kleinberg-book], commerce and economy [@elliott2014financial; @kleinberg-book; @acemoglu2015systemic], neuroscience [@sporns2011networks; @sporns2012discovering; @sporns2004motifs; @rubinov2010complex; @pessoa2014understanding], biology [@barabasi2004network; @huson2010phylogenetic], and defence [@masys2014networks], to name a few sources. Networks are often directed, in the sense that weights attached to edges do not satisfy any symmetry property, and this asymmetry often precludes the applicability of many standard methods for data analysis. Network analysis problems come in a wide range of flavors. One problem is in *exploratory data analysis*: given a network representing a dataset of societal, economic, or scientific value, the goal is to obtain insights that are meaningful to the interested party and can help uncover interesting phenomena. Another problem is *network classification*: given a “bag" of networks representing multiple instances of different phenomena, one wants to obtain a clustering which groups the networks together according to the different phenomena they represent. Because networks are often too complex to deal with directly, one typically extracts certain invariants of networks, and infers structural properties of the networks from properties of these invariants. While there are numerous such network invariants in the existing literature, there is growing interest in adopting a particular invariant arising from *persistent homology* [@frosini1992measuring; @robins1999towards; @edelsbrunner2002topological; @zomorodian2005computing], known as a *persistence diagram*, to the setting of networks. Persistence diagrams are used in the context of finite metric space or point cloud data to pick out *features of significance* while rejecting random noise [@edelsbrunner2002topological; @carlsson2009topology]. Since a network on $n$ nodes is regarded, in the most general setting, as an $n\times n$ matrix of real numbers, i.e. as a generalized metric space, it is conceivable that one should be able to describe persistence diagrams for networks as well. The motivation for computing persistence diagrams of networks is at least two-fold: (1) comparing persistence diagrams has been shown to be a viable method for *shape matching* applications [@frosini1992measuring; @frosini1999size; @collins2004barcode; @bot-stab; @carlsson2005persistence; @dgh-pers], analogous to the network classification problem described above, and (2) persistence diagrams have been successfully applied to feature detection, e.g. in detecting the structure of protein molecules (see [@krishnamoorthy2007topological; @xia2014persistent] and [@edelsbrunner2002topological §6]) and solid materials (see [@hiraoka2016hierarchical]) and might thus be a useful tool for exploratory analysis of network datasets. We point the reader to [@ghrist2008barcodes; @edelsbrunner2008persistent; @carlsson2009topology; @zigzag; @weinberger2011persistent; @burghelea2013topological; @dey2014computing] for surveys of persistent homology and its applications, and some recent extensions. Some extant approaches that obtain persistence diagrams from networks assume that the underlying network data actually satisfies metric properties [@lee2011computing; @khalid2014tracing]. A more general approach for obtaining persistence diagrams from networks is followed in [@horak2009persistent; @carstens2013persistent; @giusti2015clique; @petri2013topological], albeit with the restriction that the input data sets are required to be symmetric matrices. Our chief goal is to devise notions of persistent homology that are directly applicable to asymmetric networks in the most general sense, and are furthermore capable of absorbing structural information contained in the asymmetry. Contributions and an overview of our approach --------------------------------------------- \(N) at (0,0)[${\mathcal{N}}$]{}; (F) at (3,0)[${\mathcal}{F}$]{}; (D) at (6,0)[${\operatorname{Dgm}}$]{}; \(N) edge\[loop above, out=150, in=90, looseness=8, -&gt;\] node\[left\][${\mathfrak}{s}$]{} (N); (N) edge\[loop above, out=210, in=270, looseness=8, -&gt;\] node\[left\][${\mathfrak}{t}$]{} (N); \(N) edge\[-&gt;, bend left, looseness =1.5\] node\[above\][${\mathfrak}{R}$]{} (F); (N) edge\[-&gt;\] node\[above\][${{\mathfrak}{D}^{\operatorname{si}}}$]{} (F); (N) edge\[-&gt;, bend right, looseness=1.5\] node\[above\][${{\mathfrak}{D}^{\operatorname{so}}}$]{} (F); \(F) edge\[-&gt;\] node\[above\][$H_k$]{} (D); In this paper, we study two types of persistence diagrams: the *Rips* and *Dowker* diagrams. We define both invariants in the setting of asymmetric networks with real-valued weights, without assuming any metric properties at all (not symmetry and not even that the matrix representing the networks weights vanishes on the diagonal). As a key step in defining the Dowker persistence diagram, we first define two dual constructions, each of which can be referred to as a Dowker persistence diagram, and then prove a *functorial Dowker theorem* which implies that these two possible diagrams are equivalent. Following the line of work in [@dgh-pers], where stability of Rips persistence diagrams arising from finite metric spaces was first established, we formulate similar stability results for the Rips and Dowker persistence diagrams of a network. Through various examples, in particular a family of *cycle networks*, we espouse the idea that Dowker persistence diagrams are more appropriate than Rips persistence diagrams for studying asymmetric networks. We test our methods by solving a network classification problem on a database of simulated hippocampal networks. The first step in constructing a persistence diagram from a network is to construct a nested sequence of simplicial complexes, i.e. a simplicial filtration, which, in our work, will be the *Rips* or *Dowker* filtrations associated to a network. Rips and Dowker simplicial complexes and their associated filtrations are classically defined for metric spaces [@de2004topological; @ghrist-eat], and the generalization to networks that we use is a natural extension of the metric versions. After producing the simplicial filtrations, the standard framework of *persistent homology* takes over, and we obtain the Rips or Dowker persistence diagrams. Practitioners of persistent homology might recall that there are *two* Dowker complexes [@ghrist-eat p. 73], which we describe as the *source* and *sink* Dowker complexes. A subtle point to note here is that each of these Dowker complexes can be used to construct a persistence diagram. A folklore result in the literature about persistent homology of metric spaces, known as *Dowker duality*, is that the two persistence diagrams arising this way are equal [@chazal2014persistence Remark 4.8]. In this paper we prove a stronger result—a functorial Dowker theorem—from which the duality follows easily. Furthermore, the context of this result is strictly more general than that of metric spaces (see below for a more thorough description of the functorial version of Dowker’s theorem). Providing a construction of Rips and Dowker persistence diagrams is not enough: in order for these invariants to be useful in practice, one must verify that the diagrams are *stable*. In this context, stability means the following: the dissimilarity between two Rips (resp. Dowker) persistence diagrams obtained from two networks should be bounded above by a function of the dissimilarity between the two networks. To our knowledge, stability is not addressed in the existing literature on producing persistence diagrams from networks. In our work, we provide stability results for both the Rips and Dowker persistence diagrams (Propositions \[prop:rips-stab\] and \[prop:dowker-stab\]). One key ingredient in our proof of this result is a notion of *network distance* that follows previous work in [@clust-net; @nets-allerton; @nets-icassp]. This network distance is analogous to the Gromov-Hausdorff distance between metric spaces, which has previously been used to prove stability results for hierarchical clustering [@carlsson2008persistent; @clust-um] and Rips persistence diagrams obtained from finite metric spaces [@dgh-pers Theorem 3.1]. The Gromov-Hausdorff distance was later used in conjunction with the Algebraic Stability Theorem of [@chazal2009proximity] to provide alternative proofs of stability results for Rips and Dowker persistence diagrams arising from metric spaces [@chazal2014persistence]. Our proofs also involve this Algebraic Stability Theorem, but the novelty of our approach lies in a reformulation of the network distance (Proposition \[prop:dn-ko\]) that yields direct maps between two networks, thus passing naturally into the machinery of the Algebraic Stability Theorem (without having to define auxiliary constructions such as multivalued maps, as in [@chazal2014persistence]). A crucial issue that we point out in this paper is that even though we can construct both Rips and Dowker persistence diagrams out of asymmetric networks, Rips persistence diagrams appear to be *blind* to asymmetry, whereas Dowker persistence diagrams do exhibit sensitivity to asymmetry. In the case of Rips complexes, this purported insensitivity to asymmetry can be immediately seen from its definition. In the case of Dowker complexes, we argue about its sensitivity to asymmetry in two different ways. Firstly, we do so by explicitly computing Dowker persistence diagrams of multiple examples of asymmetric networks. In particular, we consider a family of highly asymmetric networks, the *cycle networks*, and by bulding upon results from [@adamaszek2015vietoris; @adamaszek2016nerve] we prove a complete characterization result for the Dowker persistence diagrams—across all dimensions—of any network belonging to this family. These networks constitute directed analogues of circles and may be *motifs* of interest in different applications related to network data analysis. More specifically, appearance of nontrivial 1-dimensional persistence in the Dowker persistence diagram of asymmetric network data may suggest the presence of directed cycles in the data. Some of our experimental results suggest that the Rips persistence diagrams of this family of networks are pathological, in the sense that they do not represent the signatures one would expect from the underlying dataset, which is a directed circle. Dowker persistence diagrams, on the other hand, are well-behaved in this respect in that they succeed at capturing relevant features. Secondly, we study the degree to which Dowker persistence diagrams are insensitive to changes (such as edge flips, or transposition) in the network structure. An overview of this thread of work is provided in Figure \[fig:overview\].\ #### Dowker’s theorem and a functorial generalization Let $X,Y$ be two totally ordered sets, and let $R\subseteq X\times Y$ be a nonempty relation. Then one can define two simplicial complexes $E_R$ and $F_R$ as follows. A finite subset ${\sigma}\subseteq X$ belongs to $E_R$ whenever there exists $y \in Y$ such that $(x,y) \in R$ for each $x\in {\sigma}$. Similarly a finite subset ${\tau}\subseteq Y$ belongs to $F_R$ whenever there exists $x\in X$ such that $(x,y) \in R$ for each $y\in {\tau}$. These constructions can be traced back to [@dowker1952homology], who proved the following result that we refer to as *Dowker’s theorem*: \[thm:dowker\] Let $X,Y$ be two totally ordered sets, let $R\subseteq X\times Y$ be a nonempty relation, and let $E_R, F_R$ be as above. Then for each $k \in {\mathbb{Z}}_+$, $$H_k(E_R) \cong H_k(F_R).$$ There is also a strong form of Dowker’s theorem that Björner proves via the classical *nerve theorem* [@bjorner-book Theorems 10.6, 10.9]: \[thm:dowker-strong\] Under the assumptions of Theorem \[thm:dowker\], we in fact have $|E_R| \simeq |F_R|$. The Functorial Dowker Theorem is the following generalization of the strong form of Dowker’s theorem: instead of a single nonempty relation $R \subseteq X\times Y$, consider any pair of nested, nonempty relations $R\subseteq R' \subseteq X\times Y$. Then there exist homotopy equivalences between the geometric realizations of the corresponding complexes that commute with the canonical inclusions, up to homotopy. We formalize this statement below. \[thm:dowker-functorial\] Let $X,Y$ be two totally ordered sets, let $R\subseteq R' \subseteq X\times Y$ be two nonempty relations, and let $E_R, F_R, E_{R'}, F_{R'}$ be their associated simplicial complexes. Then there exist homotopy equivalences $\Gamma_{|E_R|}:|F_R| {\rightarrow}|E_R|$ and $\Gamma_{|E_{R'}|}: |F_{R'}| {\rightarrow}|E_{R'}|$ such that the following diagram commutes up to homotopy: \(1) at (0,0)[$|F_R|$]{}; (2) at (3,0)[$|F_{R'}|$]{}; \(3) at (0,-2)[$|E_{R}|$]{}; (4) at (3,-2)[$|E_{R'}|$]{}; \(1) edge\[-&gt;\] node\[above\][$|\iota_E|$]{} (2); (3) edge\[-&gt;\] node\[above\][$|\iota_F|$]{}(4); (1) edge\[-&gt;\] node\[left\][$\Gamma_{|E_R|}$]{} node\[right\] [$\simeq$]{}(3); (2) edge\[-&gt;\] node\[right\][$\Gamma_{|E_{R'}|}$]{} node\[left\] [$\simeq$]{} (4); In other words, we have $|\iota_F|\circ \Gamma_{|E_R|} \simeq \Gamma_{|E_{R'}|} \circ |\iota_E|$, where $\iota_E,\iota_F$ are the canonical inclusions. From Theorem \[thm:dowker-functorial\] we automatically obtain Theorem \[thm:dowker-strong\] (the strong form of Dowker’s theorem) as an immediate corollary. The strong form does not appear in Dowker’s original paper [@dowker1952homology], but Björner has given a proof using the nerve theorem [@bjorner-book Theorems 10.6, 10.9]. Moreover, Björner writes in a remark following [@bjorner-book Theorem 10.9] that the nerve theorem and the strong form of Dowker’s theorem are equivalent, in the sense that one implies the other. We were not able to find an elementary proof of the strong form of Dowker’s theorem in the existing literature. However, such an elementary proof is provided by our proof of Theorem \[thm:dowker-functorial\] (given in Section \[sec:dowker-dual\]), which we obtained by extending ideas in Dowker’s original proof of Theorem \[thm:dowker\].[^1] Whereas the Functorial Dowker Theorem and our elementary proof are of independent interest, it has been suggested in [@chazal2014persistence Remark 4.8] that such a functorial version of Dowker’s theorem could also be proved using a functorial nerve theorem [@chazal2008towards Lemma 3.4]. Despite being an interesting possibility, we were not able to find a detailed proof of this claim in the literature. In addition, Björner’s remark regarding the equivalence between the nerve theorem and the strong form of Dowker’s theorem suggests the following question: \[q:f-nerve-f-dowker\] Are the Functorial Nerve Theorem (FNT) of [@chazal2008towards] and the Functorial Dowker Theorem (FDT, Theorem \[thm:dowker-functorial\]) equivalent? This question is of fundamental importance because the Nerve Theorem is a crucial tool in the applied topology literature and its functorial generalizations are equally important in persistent homology. In general, the answer is *no*, and moreover, one (of the FNT and FDT) is not stronger than the other. The FNT of [@chazal2008towards] is stated for paracompact spaces, which are more general than the simplicial complexes of the FDT. However, the FNT of [@chazal2008towards] is stated for spaces with *finitely-indexed* covers, so the associated nerve complexes are necessarily finite. All the complexes involved in the statement of the FDT are allowed to be infinite, so the FDT is more general than the FNT in this sense. To clarify these connections, we formulate a simplicial Functorial Nerve Theorem (Theorem \[thm:nerve-functorial-II\]) and prove it via a finite formulation of the FDT (Theorem \[thm:dowker-functorial-finite\]). In turn, we show that the simplicial FNT implies the finite FDT, thus proving the equivalence of these formulations (Theorem \[thm:dowker-nerve-eq\]). Dowker complexes are also known to researchers who use Q-analysis to study social networks [@johnson2013hypernetworks; @atkin1975mathematical; @atkin1972cohomology]. We perceive that viewing Dowker complexes through the modern lens of persistence will enrich the classical framework of Q-analysis by incorporating additional information about the *meaningfulness* of features, thus potentially opening new avenues in the social sciences. An announcement of part of our work has appeared in [@dowker-asilo]. Implementations --------------- Following work in [@curto2008cell; @dabaghian2012topological], we implement our methods in the setting of classifying simulated hippocampal networks. We simulate the activity pattern of hippocampal cells in an animal as it moves around arenas with a number of obstacles, and compile this data into a network which can be interpreted as the transition matrix for the time-reversal of a Markov process. The motivating idea is to ascertain whether, by just observing hippocampal activity and not using any higher reasoning ability, one might be able to determine the number of obstacles in the arena that the animal has just finished traversing. The results of computing Dowker persistence diagrams suggest that the hippocampal activity is indeed sufficient to accurately count the number of obstacles in each arena. Our datasets and software are available on <https://research.math.osu.edu/networks/Datasets.html> as part of the `PersNet` software package. Organization of the paper ------------------------- Notation used globally is defined directly below. §\[sec:background\] contains the necessary background on persistent homology. §\[sec:nets\] contains our formulations for networks, as well as some key ingredients of our stability results. §\[sec:rips\] contains details about the Rips persistence diagram. The first part of §\[sec:dowker\] contains details about the Dowker persistence diagram. §\[sec:dowker-dual\] contains the Functorial Dowker Theorem. The connection between the simplicial Functorial Nerve Theorem and the finite Functorial Dowker Theorem is detailed in §\[sec:dowker-nerve-equiv\]. In §\[sec:symmetry\] we show that Dowker complexes are sensitive to asymmetry. §\[sec:cycle\] contains a family of asymmetric networks, the *cycle networks*, and a full characterization of their Dowker persistence diagrams. In §\[sec:exp\] we provide details on an implementation of our methods. Finally, proofs of statements not contained in the main body of the paper are relegated to Appendix \[app:proofs\], whereas details about the characterization results for Dowker persistence diagrams of cycle networks are given in Appendix \[sec:cycle-addendum\]. Notation -------- We will write $\mathbb{K}$ to denote a field, which we will fix and use throughout the paper. We will write ${\mathbb{Z}}_+$ and ${\mathbb{R}}_+$ to denote the nonnegative integers and reals, respectively. The extended real numbers ${\mathbb{R}}\cup {\left\{\infty, -\infty\right\}}$ will be denoted $\overline{{\mathbb{R}}}$. The cardinality of a set $X$ will be denoted ${\operatorname{card}}(X)$. The collection of nonempty subsets of a set $X$ will be denoted ${\operatorname{pow}}(X)$. The natural numbers ${\left\{1,2,3,\ldots\right\}}$ will be denoted by ${\mathbb{N}}$. The dimension of a vector space $V$ will be denoted $\dim(V)$. The rank of a linear transformation $f$ will be denoted ${\operatorname{rank}}(f)$. An isomorphism between vector spaces $V$ and $W$ will be denoted $V\cong W$. A homotopy relation for two maps $f,g:X{\rightarrow}Y$ between topological spaces will be denoted $f\simeq g$. Occasionally we will need to take about multisets, i.e. sets where elements can have multiplicity greater than 1. We will use square bracket notation $[\ldots]$ to denote multisets. Identity maps will be denoted by the notation ${\operatorname{id}}_\bullet$. Given a simplicial complex ${\Sigma}$, we will often write $V({\Sigma})$ to denote the vertex set of ${\Sigma}$. We will write ${\operatorname{Bd}}({\sigma})$ to denote the boundary of a simplex ${\sigma}$. Background on persistent homology {#sec:background} ================================= We assume that the reader is familiar with terms and concepts related to simplicial homology, and refer to [@munkres-book] for details. Here we describe our choices of notation. Whenever we have a simplicial complex over a set $X$ and a $k$-simplex ${\left\{x_0,x_1,\ldots, x_k\right\}}$, $k\in {\mathbb{Z}}_+$, we will assume that the simplex is *oriented* by the ordering $x_0< x_1 < \ldots < x_k$. We will write $[x_0,x_1,\ldots,x_k]$ to denote the equivalence class of the even permutations of this chosen ordering, and $-[x_0,x_1,\ldots,x_k]$ to denote the equivalence class of the odd permutations of this ordering. Given a simplicial complex ${\Sigma}$, we will denote its geometric realization by $|{\Sigma}|$. The *weak topology* on $|{\Sigma}|$ is defined by requiring that a subset $A \subseteq |{\Sigma}|$ is closed if and only if $A \cap |{\sigma}|$ is closed in $|{\sigma}|$ for each ${\sigma}\in {\Sigma}$. A simplicial map $f: {\Sigma}{\rightarrow}\Xi$ between two simplicial complexes induces a map $|f|: |{\Sigma}| {\rightarrow}|\Xi|$ between the geometric realizations, defined as $|f|(\sum_{v\in {\Sigma}}a_v v):= \sum_{v\in {\Sigma}}a_v f(v)$. These induced maps satisfy the usual composition identity: given simpicial maps $f:{\Sigma}{\rightarrow}\Xi$ and $g:\Xi {\rightarrow}\Upsilon$, we have $|g\circ f| = |g| \circ |f|$. To see this, observe the following: $$\label{eq:htpy-func} |g\circ f|(\sum_{v\in {\Sigma}}a_v v) = \sum_{v\in {\Sigma}} a_vg(f(v)) = |g|(\sum_{v\in {\Sigma}}a_v f(v)) = |g|\circ|f|(\sum_{v\in {\Sigma}}a_v v).$$ A *filtration* of a simplicial complex ${\Sigma}$ (also called a *filtered simplicial complex*) is defined to be a nested sequence $\{{\Sigma}^{{\delta}}\subseteq {\Sigma}^{{\delta}'}\}_{{\delta}\leq {\delta}' \in {\mathbb{R}}}$ of simplicial complexes satisfying the condition that there exist ${\delta}_I,\, {\delta}_F \in {\mathbb{R}}$ such that ${\Sigma}^{{\delta}} = {\varnothing}$ for all ${\delta}\leq {\delta}_I$, and ${\Sigma}^{{\delta}} = {\Sigma}\text{ for all }{\delta}\geq {\delta}_F$. Fix a field ${\mathbb}{K}$. Given a finite simplicial complex ${\Sigma}$ and a dimension $k \in {\mathbb{Z}}_+$, we will denote a *$k$-chain* in ${\Sigma}$ as $\sum_ia_i{\sigma}_i$, where each $a_i \in \mathbb{K}$ and ${\sigma}_i \in {\Sigma}$. We write $C_k({\Sigma})$ or just $C_k$ to denote the $\mathbb{K}$-vector space of all $k$-chains. We will write ${\partial}_k$ to denote the associated *boundary map* ${\partial}_k : C_k {\rightarrow}C_{k-1}$: $${\partial}_k[x_0,\ldots,x_k]:=\sum_i(-1)^i[x_0,\ldots,\hat{x}_i,\ldots, x_k], \text{ where $\hat{x}_i$ denotes omission of $x_i$ from the sequence.}$$ We will write ${\mathcal{C}}=(C_k,{\partial}_k)_{k\in {\mathbb{Z}}_+}$ to denote a *chain complex*, i.e. a sequence of vector spaces with boundary maps such that ${\partial}_{k-1}\circ {\partial}_k =0$. Given a chain complex ${\mathcal{C}}$ and any $k\in {\mathbb{Z}}_+$, the *$k$-th homology of the chain complex ${\mathcal{C}}$* is denoted $H_k({\mathcal{C}}) :=\ker({\partial}_k)/{\operatorname{im}}({\partial}_{k+1})$. The *$k$-th Betti number* of ${\mathcal{C}}$ is denoted ${\beta}_k({\mathcal{C}})$. Given a simplicial map $f$ between simplicial complexes, we write $f_*$ to denote the induced chain map between the corresponding chain complexes [@munkres-book §1.12], and $(f_k)_{\#}$ to denote the linear map on $k$th homology vector spaces induced for each $k \in {\mathbb{Z}}_+$. The operations of passing from simplicial complexes and simplicial maps to chain complexes and induced chain maps, and then to homology vector spaces with induced linear maps, will be referred to as *passing to homology*. Recall the following useful fact, often referred to as *functoriality of homology* [@munkres-book Theorem 12.2]: given a composition $g\circ f$ of simplicial maps, we have $$(g_k\circ f_k)_\# = (g_k)_\#\circ (f_k)_\# \qquad\text{ for each } k\in {\mathbb{Z}}_+. \label{eq:functoriality}$$ A *persistence vector space* is defined to be a family of vector spaces $\{U^{\delta}{\xrightarrow}{\mu_{{\delta},{\delta}'}} U^{{\delta}'}\}_{{\delta}\leq {\delta}'\in {\mathbb{R}}}$ such that: (1) $\mu_{{\delta},{\delta}}$ is the identity for each ${\delta}\in {\mathbb{R}}$, and (2) $\mu_{{\delta},{\delta}''} = \mu_{{\delta}',{\delta}''}\circ \mu_{{\delta},{\delta}'}$ for each ${\delta}\leq {\delta}' \leq {\delta}'' \in {\mathbb{R}}$. The persistence vector spaces that we consider in this work also satisfy the following conditions: (1) $\dim(U^{\delta}) <\infty$ at each ${\delta}\in {\mathbb{R}}$, (2) there exist ${\delta}_I,\, {\delta}_F \in {\mathbb{R}}$ such that all maps $\mu_{{\delta},{\delta}'}$ are isomorphisms for ${\delta},{\delta}' \geq {\delta}_F$ and for ${\delta},{\delta}' \leq {\delta}_I$, and (3) there are only finitely many values of ${\delta}\in {\mathbb{R}}$ such that $U^{{\delta}-{\varepsilon}} \not\cong U^{{\delta}}$ for each ${\varepsilon}>0$. Here ${\delta}$ is referred to as a *resolution* parameter, and such a persistence vector space is described as being *${\mathbb{R}}$-indexed*. The collection of all such persistence vector spaces is denoted ${{\operatorname{\mathbf{PVec}}}}({\mathbb{R}})$. Observe that by fixing $k \in {\mathbb{Z}}_+$ and passing to the $k$th homology vector space at each step ${\Sigma}^{{\delta}}$ of a filtered simplicial complex $({\Sigma}^{{\delta}})_{{\delta}\in {\mathbb{R}}}$, the functoriality of homology gives us the $k$th persistence vector space associated to $({\Sigma}^{{\delta}})_{{\delta}\in {\mathbb{R}}}$, denoted $${\mathcal{H}}_k({\Sigma}) := \{H_k({\mathcal{C}}^{{\delta}}){\xrightarrow}{(\iota_{{\delta},{\delta}'})_\#} H_k({\mathcal{C}}^{{\delta}'})\}_{{\delta}\leq {\delta}' \in {\mathbb{R}}}.$$ The elements of ${{\operatorname{\mathbf{PVec}}}}({\mathbb{R}})$ contain only a finite number of vector spaces, up to isomorphism. By the classification results in [@carlsson2005persistence §5.2], it is possible to associate a full invariant, called a *persistence barcode* or *persistence diagram*, to each element of ${{\operatorname{\mathbf{PVec}}}}({\mathbb{R}})$. This barcode is a multiset of *persistence intervals*, and is represented as a set of lines over a single axis. The barcode of a persistence vector space ${\mathcal{V}}$ is denoted ${{\operatorname{\mathbf{Pers}}}}({\mathcal{V}})$. The intervals in ${{\operatorname{\mathbf{Pers}}}}({\mathcal{V}})$ can be represented as the *persistence diagram of ${\mathcal{V}}$*, which is as a multiset of points lying on or above the diagonal in $\overline{{\mathbb{R}}}^2$, counted with multiplicity. More specifically, $${\operatorname{Dgm}}({\mathcal{V}}):=\big[({\delta}_i,{\delta}_{j+1}) \in \overline{{\mathbb{R}}}^2 : [{\delta}_i,{\delta}_{j+1}) \in {{\operatorname{\mathbf{Pers}}}}({\mathcal{V}}) \big],$$ where the multiplicity of $({\delta}_i,{\delta}_{j+1})\in \overline{{\mathbb{R}}}^2$ is given by the multiplicity of $[{\delta}_i,{\delta}_{j+1}) \in {{\operatorname{\mathbf{Pers}}}}({\mathcal{V}})$. Persistence diagrams can be compared using the *bottleneck distance*, which we denote by ${d_{\operatorname{B}}}$. Details about this distance, as well as the other material related to persistent homology, can be found in [@chazal2012structure]. Numerous other formulations of the material presented above can be found in [@edelsbrunner2002topological; @zomorodian2005computing; @zigzag; @edelsbrunner2010computational; @edelsbrunner2014persistent; @bauer-isom; @ph-self]. \[rem:trivial-diag\] Whenever we describe a persistence diagram as being *trivial*, we mean that either it is empty, or it does not have any off-diagonal points. Interleaving distance and stability of persistence vector spaces. {#sec:background-int} ----------------------------------------------------------------- In what follows, we will consider ${\mathbb{R}}$-indexed persistence vector spaces ${{\operatorname{\mathbf{PVec}}}}({\mathbb{R}})$. Given ${\varepsilon}\geq 0$, two ${\mathbb{R}}$-indexed persistence vector spaces ${\mathcal{V}}=\{V^{\delta}{\xrightarrow}{\nu_{{\delta},{\delta}'}} V^{{\delta}'}\}_{{\delta}\leq {\delta}'}$ and ${\mathcal{U}}=\{U^{\delta}{\xrightarrow}{\mu_{{\delta},{\delta}'}} U^{{\delta}'}\}_{{\delta}\leq {\delta}'}$ are said to be *${\varepsilon}$-interleaved* [@chazal2009proximity; @bauer-isom] if there exist two families of linear maps $$\begin{aligned} \{{\varphi}_{{\delta},{\delta}+{\varepsilon}}&:V^{\delta}{\rightarrow}V^{{\delta}+ {\varepsilon}}\}_{{\delta}\in {\mathbb{R}}},\\ \{\psi_{{\delta},{\delta}+{\varepsilon}}&:U^{\delta}{\rightarrow}U^{{\delta}+ {\varepsilon}}\}_{{\delta}\in {\mathbb{R}}}\end{aligned}$$ such that the following diagrams commute for all ${\delta}' \geq {\delta}\in {\mathbb{R}}$: $$\begin{tikzcd}[column sep=large] V^{\delta}\arrow{r}{\nu_{{\delta},{\delta}'}} \arrow[swap]{dr}{{\varphi}_{\delta}} & V^{{\delta}'}\arrow{dr}{{\varphi}_{{\delta}'}} & {} & {} & V^{{\delta}+{\varepsilon}} \arrow{r}{\nu_{{\delta}+\eta,{\delta}'+\eta}} & V^{{\delta}'+{\varepsilon}} \\ {} & U^{{\delta}+{\varepsilon}} \arrow{r}{\mu_{{\delta}+\eta,{\delta}'+\eta}} & U^{{\delta}'+{\varepsilon}} & U^{{\delta}} \arrow{ur}{\psi_{\delta}} \arrow{r}{\mu_{{\delta},{\delta}'}} & U^{{\delta}'} \arrow[swap]{ur}{\psi_{{\delta}'}} & {} \end{tikzcd}$$ $$\begin{tikzcd}[column sep=large] V^{\delta}\arrow{rr}{\nu_{{\delta},{\delta}+2{\varepsilon}}} \arrow[swap]{dr}{{\varphi}_{\delta}} & {} & V^{{\delta}+2{\varepsilon}} & {}& V^{{\delta}+{\varepsilon}} \arrow{dr}{\psi_{{\delta}+{\varepsilon}}}\\ {} & U^{{\delta}+{\varepsilon}} \arrow[swap]{ur}{{\varphi}_{{\delta}+{\varepsilon}}} & {}& U^{{\delta}} \arrow{ur}{\psi_{{\delta}}} \arrow{rr}{\mu_{{\delta},{\delta}+2{\varepsilon}}} & {} & U^{{\delta}+2\eta} \end{tikzcd}$$ The purpose of introducing ${\varepsilon}$-interleavings is to define a pseudometric on the collection of persistence vector spaces. The *interleaving distance* between two ${\mathbb{R}}$-indexed persistence vector spaces ${\mathcal{V}},{\mathcal{U}}$ is given by: $${d_{\operatorname{I}}}({\mathcal{U}},{\mathcal{V}}) := \inf \{{\varepsilon}\geq 0 : \text{${\mathcal{U}}$ and ${\mathcal{V}}$ are ${\varepsilon}$-interleaved}\}.$$ One can verify that this definition induces a pseudometric on the collection of persistence vector spaces [@chazal2009proximity; @bauer-isom]. The interleaving distance can then be related to the bottleneck distance as follows: Let ${\mathcal{U}}, {\mathcal{V}}$ be two ${\mathbb{R}}$-indexed persistence vector spaces. Then, $${d_{\operatorname{B}}}({\operatorname{Dgm}}({\mathcal{U}}),{\operatorname{Dgm}}({\mathcal{V}}))\leq {d_{\operatorname{I}}}({\mathcal{U}},{\mathcal{V}}).$$ Stability results are at the core of persistent homology, beginning with the classical bottleneck stability result in [@bot-stab]. One of our key contributions is to use the Algebraic Stability Theorem stated above, along with Lemma §\[sec:nets\] stated below, to prove stability results for methods of computing persistent homology of a network. Before stating the following lemma, recall that two simplicial maps $f,g: {\Sigma}{\rightarrow}\Xi$ are *contiguous* if for any simplex ${\sigma}\in {\Sigma}$, $f({\sigma}) \cup g({\sigma})$ is a simplex of $\Xi$. Contiguous maps satisfy the following useful properties: \[prop:contigo-props\] Let $f,g: {\Sigma}{\rightarrow}\Xi$ be two contiguous simplicial maps. Then, 1. $|f|,|g|:|{\Sigma}| {\rightarrow}|\Xi|$ are homotopic [@spanier-book §3.5], and 2. The chain maps induced by $f$ and $g$ are chain homotopic, and as a result, the induced maps $f_\#$ and $g_\#$ for homology are equal [@munkres-book Theorem 12.5]. \[lem:stab\] Let ${\mathfrak}{F}, {\mathfrak}{G}$ be two filtered simplicial complexes written as $$\{{\mathfrak}{F}^{\delta}{\xrightarrow}{s_{{\delta},{\delta}'}} {\mathfrak}{F}^{{\delta}'}\}_{{\delta}'\geq {\delta}\in {\mathbb{R}}} \text{ and } \{{\mathfrak}{G}^{\delta}{\xrightarrow}{t_{{\delta},{\delta}'}} {\mathfrak}{G}^{{\delta}'}\}_{{\delta}'\geq {\delta}\in {\mathbb{R}}},$$ where $s_{{\delta},{\delta}'}$ and $t_{{\delta},{\delta}'}$ denote the natural inclusion maps. Suppose $\eta\geq 0$ is such that there exist families of simplicial maps ${\left\{{\varphi}_{\delta}:{\mathfrak}{F}^{\delta}{\rightarrow}{\mathfrak}{G}^{{\delta}+\eta}\right\}}_{{\delta}\in {\mathbb{R}}}$ and ${\left\{\psi_{\delta}:{\mathfrak}{G}^{\delta}{\rightarrow}{\mathfrak}{F}^{{\delta}+\eta}\right\}}_{{\delta}\in {\mathbb{R}}}$ such that the following are satisfied for any ${\delta}' \geq {\delta}$: 1. $t_{{\delta}+\eta,{\delta}'+\eta}\circ {\varphi}_{\delta}$ and ${\varphi}_{{\delta}'}\circ s_{{\delta},{\delta}'}$ are contiguous 2. $s_{{\delta}+\eta,{\delta}'+\eta}\circ \psi_{\delta}$ and $\psi_{{\delta}'}\circ t_{{\delta},{\delta}'}$ are contiguous 3. $\psi_{{\delta}+\eta}\circ {\varphi}_{\delta}$ and $s_{{\delta},{\delta}+2\eta}$ are contiguous 4. ${\varphi}_{{\delta}+\eta}\circ \psi_{\delta}$ and $t_{{\delta},{\delta}+2\eta}$ are contiguous. All the diagrams are as below: $$\begin{tikzcd}[column sep=large] {\mathfrak}{F}^{\delta}\arrow{r}{s_{{\delta},{\delta}'}} \arrow[swap]{dr}{{\varphi}_{\delta}} & {\mathfrak}{F}^{{\delta}'}\arrow{dr}{{\varphi}_{{\delta}'}} & {} & {} & {\mathfrak}{F}^{{\delta}+\eta} \arrow{r}{s_{{\delta}+\eta,{\delta}'+\eta}} & {\mathfrak}{F}^{{\delta}'+\eta} \\ {} & {\mathfrak}{G}^{{\delta}+\eta} \arrow{r}{t_{{\delta}+\eta,{\delta}'+\eta}} & {\mathfrak}{G}^{{\delta}'+\eta} & {\mathfrak}{G}^{{\delta}} \arrow{ur}{\psi_{\delta}} \arrow{r}{t_{{\delta},{\delta}'}} & {\mathfrak}{G}^{{\delta}'} \arrow[swap]{ur}{\psi_{{\delta}'}} & {} \end{tikzcd}$$ $$\begin{tikzcd}[column sep=large] {\mathfrak}{F}^{\delta}\arrow{rr}{s_{{\delta},{\delta}+2\eta}} \arrow[swap]{dr}{{\varphi}_{\delta}} & {} & {\mathfrak}{F}^{{\delta}+2\eta} & {} & {}& {\mathfrak}{F}^{{\delta}+\eta} \arrow{dr}{{\varphi}_{{\delta}+\eta}}\\ {} & {\mathfrak}{G}^{{\delta}+\eta} \arrow[swap]{ur}{\psi_{{\delta}+\eta}} & {} & {}& {\mathfrak}{G}^{{\delta}} \arrow{ur}{\psi_{\delta}} \arrow{rr}{t_{{\delta},{\delta}+2\eta}} & {} & {\mathfrak}{G}^{{\delta}+2\eta} \end{tikzcd}$$ For each $k\in {\mathbb{Z}}_+$, let ${\mathcal{H}}_k({\mathfrak}{F}), {\mathcal{H}}_k({\mathfrak}{G})$ denote the $k$-dimensional persistence vector spaces associated to ${\mathfrak}{F}$ and ${\mathfrak}{G}$. Then for each $k\in {\mathbb{Z}}_+$, $${d_{\operatorname{B}}}({\operatorname{Dgm}}_k({\mathcal{H}}_k({\mathfrak}{F})),{\operatorname{Dgm}}_k({\mathcal{H}}_k({\mathfrak}{G}))) \leq {d_{\operatorname{I}}}({\mathcal{H}}_k({\mathfrak}{F}),{\mathcal{H}}_k({\mathfrak}{G})) \leq \eta.$$ Background on networks and our network distance {#sec:nets} =============================================== A *network* is a pair $(X,{\omega}_X)$ where $X$ is a finite set and ${\omega}_X: X\times X {\rightarrow}{\mathbb{R}}$ is a *weight function*. Note that ${\omega}_X$ need not satisfy the triangle inequality, any symmetry condition, or even the requirement that ${\omega}_X(x,x) = 0$ for all $x\in X$. The weights are even allowed to be negative. The collection of all such networks is denoted ${\mathcal{N}}$. When comparing networks, one needs a way to correlate points in one network with points in the other. To see how this can be done, let $(X,{\omega}_X), (Y,{\omega}_Y) \in {\mathcal{N}}$. Let $R$ be any nonempty relation between $X$ and $Y$, i.e. a nonempty subset of $X \times Y$. The *distortion* of the relation $R$ is given by: $${\operatorname{dis}}(R):=\max_{(x,y),(x',y')\in R}|{\omega}_X(x,x')-{\omega}_Y(y,y')|.$$ A *correspondence between $X$ and $Y$* is a relation $R$ between $X$ and $Y$ such that $\pi_X(R)=X$ and $\pi_Y(R)=Y$, where $\pi_X:X\times Y {\rightarrow}X$ and $\pi_Y:X\times Y {\rightarrow}Y$ denote the natural projections. The collection of all correspondences between $X$ and $Y$ will be denoted ${\mathscr{R}}(X,Y)$. Following previous work in [@clust-net; @nets-allerton; @nets-icassp] the *network distance* ${d_{\mathcal{N}}}:{\mathcal{N}}\times {\mathcal{N}}{\rightarrow}{\mathbb{R}}_+$ is then defined as: $${d_{\mathcal{N}}}(X,Y):=\frac{1}{2}\min_{R\in{\mathscr{R}}}{\operatorname{dis}}(R).$$ It can be verified that ${d_{\mathcal{N}}}$ as defined above is a pseudometric, and that the networks at 0-distance can be completely characterized [@nets-allerton]. Next we wish to prove the reformulation in Proposition \[prop:dn-ko\]. First we define the distortion of a map between two networks. Given any $(X,{\omega}_X),(Y,{\omega}_Y)\in {\mathcal{N}}$ and a map ${\varphi}:(X,{\omega}_X) {\rightarrow}(Y,{\omega}_Y)$, the *distortion* of ${\varphi}$ is defined as: $${\operatorname{dis}}({\varphi}):= \max_{x,x'\in X}|{\omega}_X(x,x')-{\omega}_Y({\varphi}(x),{\varphi}(x'))|.$$ Next, given maps ${\varphi}:(X,{\omega}_X){\rightarrow}(Y,{\omega}_Y)$ and $\psi:(Y,{\omega}_Y){\rightarrow}(X,{\omega}_X)$, we define two *co-distortion* terms: $$\begin{aligned} C_{X,Y}({\varphi},\psi) &:= \max_{(x,y)\in X\times Y}|{\omega}_X(x,\psi(y)) - {\omega}_Y({\varphi}(x),y)|,\\ C_{Y,X}(\psi,{\varphi}) &:= \max_{(y,x)\in Y\times X}|{\omega}_Y(y,{\varphi}(x)) - {\omega}_X(\psi(y),x)|.\end{aligned}$$ \[prop:dn-ko\] Let $(X,{\omega}_X), (Y,{\omega}_Y)\in {\mathcal{N}}$. Then, $${d_{\mathcal{N}}}(X,Y) = \tfrac{1}{2}\min\{\max({\operatorname{dis}}({\varphi}),{\operatorname{dis}}(\psi),C_{X,Y}({\varphi},\psi), C_{Y,X}(\psi,{\varphi})) : {\varphi}:X {\rightarrow}Y, \psi:Y {\rightarrow}X \text{ any maps}\}.$$ Proposition \[prop:dn-ko\] is analogous to a result of Kalton and Ostrovskii [@kalton1997distances Theorem 2.1] where—instead of ${d_{\mathcal{N}}}$—one has the Gromov-Hausdorff distance between metric spaces. We remark that when restricted to the special case of networks that are also metric spaces, the network distance ${d_{\mathcal{N}}}$ agrees with the Gromov-Hausdorff distance. Details on the Gromov-Hausdorff distance can be found in [@burago]. An important remark is that in the Kalton-Ostrovskii formulation, there is only one co-distortion term. When Proposition \[prop:dn-ko\] is applied to metric spaces, the two co-distortion terms become equal by symmetry, and thus the Kalton-Ostrovskii formulation is recovered. But *a priori*, the lack of symmetry in the network setting requires us to consider both terms. In the following sections, we propose methods for computing persistent homology of networks, and prove that they are stable via Lemma \[lem:stab\]. Note that similar results, valid in the setting of metric spaces, have appeared in [@dgh-pers; @chazal2014persistence]. Whereas the proofs in [@chazal2014persistence] invoke an auxiliary construction of multivalued maps arising from correspondences, our proofs simply use the maps ${\varphi}, \psi$ arising directly from the reformulation of ${d_{\mathcal{N}}}$ (Proposition \[prop:dn-ko\]), thus streamlining the treatment. When studying the effect of asymmetry on persistent homology, it will be useful to consider the network transformations that we define next. \[defn:sym-trans\] Define the *max-symmetrization* map ${\mathfrak}{s}:{\mathcal{N}}{\rightarrow}{\mathcal{N}}$ by $(X,{\omega}_X)\mapsto (X,\widehat{{\omega}_X})$, where for any network $(X,{\omega}_X)$, we define $\widehat{{\omega}_X}:X\times X {\rightarrow}{\mathbb{R}}$ as follows: $$\widehat{{\omega}_X}(x,x'):= \max({\omega}_X(x,x'),{\omega}_X(x',x)), \text{ for } x,x'\in X.$$ Also define the *transposition* map ${\mathfrak}{t}:{\mathcal{N}}{\rightarrow}{\mathcal{N}}$ by $(X,{\omega}_X) \mapsto (X,{\omega}_X^\top)$, where for any $(X,{\omega}_X) \in {\mathcal{N}}$, we define ${\omega}_X^\top(x,x'):= {\omega}_X(x',x)$ for $x,x'\in X$. For convenience, we denote $X^\top:={\mathfrak}{t}(X)$ for any network $X$. We are now ready to formulate our two methods for computing persistent homology of networks. The Rips filtration is the “workhorse” of persistent homology of metric spaces so it is natural to consider its generalization to general asymmetric networks. The Rips filtration of a network {#sec:rips} ================================ Recall that for a metric space $(X,d_X)$, the *Rips complex* is defined for each ${\delta}\geq 0$ as follows: $${\mathfrak}{R}^{\delta}_X := {\left\{{\sigma}\in {\operatorname{pow}}(X): {\operatorname{diam}}({\sigma}) \leq {\delta}\right\}}, \text{ where } {\operatorname{diam}}({\sigma}) := \max_{x,x'\in {\sigma}}d_X(x,x').$$ Following this definition, we can define the Rips complex for a network $(X,{\omega}_X)$ as follows: $${\mathfrak}{R}^{\delta}_X:=\{{\sigma}\in {\operatorname{pow}}(X) : \max_{x,x'\in {\sigma}}{\omega}_X(x,x') \leq {\delta}\}.$$ To any network $(X,{\omega}_X)$, we may associate the *Rips filtration* $\{{\mathfrak}{R}^{\delta}_X{\hookrightarrow}{\mathfrak}{R}^{{\delta}'}_X\}_{{\delta}\leq {\delta}'}$. We denote the $k$-dimensional persistence vector space associated to this filtration by ${\mathcal{H}}_k^{{\mathfrak}{R}}(X)$, and the corresponding persistence diagram by ${\operatorname{Dgm}}_k^{{\mathfrak}{R}}(X)$. The Rips filtration is stable to small perturbations of the input data:9 \[prop:rips-stab\] Let $(X,{\omega}_X), (Y,{\omega}_Y) \in {\mathcal{N}}$. Then ${d_{\operatorname{B}}}({\operatorname{Dgm}}_k^{{\mathfrak}{R}}(X),{\operatorname{Dgm}}_k^{{\mathfrak}{R}}(Y)) \leq 2{d_{\mathcal{N}}}(X,Y).$ We omit the proof because it is similar to that of Proposition \[prop:dowker-stab\], which we will prove in detail. \[rem:rips-benefits\] The preceding proposition serves a dual purpose: (1) it shows that the Rips persistence diagram is robust to noise in input data, and (2) it shows that instead of computing the network distance between two networks, one can compute the bottleneck distance between their Rips persistence diagrams as a suitable proxy. The advantage to computing bottleneck distance is that it can be done in polynomial time (see [@efrat2001geometry]), whereas computing ${d_{\mathcal{N}}}$ is NP-hard in general. This follows from the fact that the problem of computing ${d_{\mathcal{N}}}$ includes the problem of computing the Gromov-Hausdorff distance between finite metric spaces, which is an NP-hard problem [@schmiedl]. We remark that the idea of computing Rips persistence diagrams to compare finite metric spaces first appeared in [@dgh-pers], and moreover, that Proposition \[prop:rips-stab\] is an extension of Theorem 3.1 in [@dgh-pers]. The Rips filtration in the setting of symmetric networks has been used in [@horak2009persistent; @carstens2013persistent; @giusti2015clique; @petri2013topological], albeit without addressing stability results. To our knowledge, Proposition \[prop:rips-stab\] is the first quantitative result justifying the constructions in these prior works. \[rem:rips-symm\] A critical weakness of the Rips complex construction is that it is not sensitive to asymmetry. To see this, recall the symmetrization map defined in Definition \[defn:sym-trans\], and let $(X,{\omega}_X) \in {\mathcal{N}}$. Now for any ${\sigma}\in {\operatorname{pow}}(X)$, we have $\max_{x,x' \in {\sigma}}{\omega}_X(x,x') = \max_{x,x'\in {\sigma}}\widehat{{\omega}_X}(x,x').$ It follows that for each ${\delta}\geq 0$, the Rips complexes of $(X,{\omega}_X)$ and $(X,\widehat{{\omega}_X})={\mathfrak}{s}(X,{\omega}_X)$ are equal, i.e. ${\mathfrak}{R} = {\mathfrak}{R} \circ {\mathfrak}{s}$. Thus the Rips persistence diagrams of the original and max-symmetrized networks are equal. The Dowker filtration of a network {#sec:dowker} ================================== Given $(X,\omega_X)\in \mathcal{N}$, and for any ${\delta}\in {\mathbb{R}}$, consider the following relation on $X$: $$R_{{\delta},X}:={\left\{(x,x') : {\omega}_X(x,x') \leq {\delta}\right\}}. \label{eq:relation}$$ Then $R_{{\delta},X} \subseteq X \times X$, and $R_{{\delta}_F,X} = X \times X$ for some sufficiently large ${\delta}_F$. Furthermore, for any ${\delta}' \geq {\delta}$, we have $R_{{\delta},X} \subseteq R_{{\delta}',X}$. Using $R_{{\delta},X}$, we build a simplicial complex ${{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$ as follows: $$\label{eq:d-sink} {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X}:={\left\{{\sigma}=[x_0,\ldots, x_n] : \text{ there exists } x'\in X \text{ such that } (x_i,x')\in R_{{\delta},X} \text{ for each } x_i\right\}}.$$ If ${\sigma}\in {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X}$, it is clear that any face of ${\sigma}$ also belongs to ${{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X}$. We call ${{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X}$ the *Dowker ${\delta}$-sink simplicial complex* associated to $X$, and refer to $x'$ as a *${\delta}$-sink* for ${\sigma}$ (where ${\sigma}$ and $x'$ should be clear from context). Since $R_{{\delta},X}$ is an increasing sequence of sets, it follows that ${{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X}$ is an increasing sequence of simplicial complexes. In particular, for ${\delta}'\geq {\delta}$, there is a natural inclusion map ${{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X} {\hookrightarrow}{{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}',X}$. We write ${{\mathfrak}{D}^{\operatorname{si}}}_X$ to denote the filtration $\{{{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X} {\hookrightarrow}{{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}',X}\}_{ {\delta}\leq {\delta}'}$ associated to $X$. We call this the *Dowker sink filtration on $X$*. We will denote the $k$-dimensional persistence diagram arising from this filtration by ${\operatorname{Dgm}}_k^{{\operatorname{si}}}(X)$. \(2) at (2,0)[$a$]{}; (3) at (3,2)[$b$]{}; (4) at (4,0)[$c$]{}; \(5) at (5,1)[ $ {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X}= \begin{cases} {\varnothing}&: {\delta}<-1 \\ \{[a]\} &:-1\leq {\delta}<0\\ \{[a],[b],[c]\} &: 0\leq {\delta}< 1\\ \{[a],[b],[c],[ab],[bc],[ac],[abc]\} &: {\delta}\geq 1 \end{cases} $ ]{}; \(50) at (5,1)[ $ {{\mathfrak}{D}^{\operatorname{so}}}_{{\delta},X}= \begin{cases} {\varnothing}&: {\delta}<-1 \\ \{[a]\} &:-1\leq {\delta}<0\\ \{[a],[b],[c]\} &: 0\leq {\delta}< 1\\ \{[a],[b],[c],[ab],[ac]\} &: 1\leq {\delta}< 2\\ \{[a],[b],[c],[ab],[bc],[ac],[abc]\} &: {\delta}\geq 2 \end{cases} $ ]{}; \(0) at (4,0); (1) at (6,0); \(2) edge \[loop left\] node\[left\][$-1$]{}(2); (3) edge \[loop right\] node\[right\][$0$]{}(3); (4) edge \[loop right\] node\[right\][$0$]{}(4); (2) edge \[bend left\] node\[above\][$1$]{} (3); (3) edge \[\] node\[below\][$1$]{} (2); (2) edge \[\] node\[above\][$2$]{} (4); (3) edge \[\] node\[below\][$2$]{} (4); (4) edge \[bend left\] node\[below\][$1$]{} (2); (4) edge \[bend right\] node\[below\][$2$]{} (3); Note that we can define a dual construction as follows: $$\label{eq:d-src} {{\mathfrak}{D}^{\operatorname{so}}}_{{\delta},X}:={\left\{{\sigma}=[x_0,\ldots, x_n] : \text{ there exists } x'\in X \text{ such that } (x',x_i)\in R_{{\delta},X} \text{ for each } x_i\right\}}.$$ We call ${{\mathfrak}{D}^{\operatorname{so}}}_{{\delta},X}$ the *Dowker ${\delta}$-source simplicial complex* associated to $X$. The filtration $\{{{\mathfrak}{D}^{\operatorname{so}}}_{{\delta},X} {\hookrightarrow}{{\mathfrak}{D}^{\operatorname{so}}}_{{\delta}',X}\}_{ {\delta}\leq {\delta}'}$ associated to $X$ is called the *Dowker source filtration*, denoted ${{\mathfrak}{D}^{\operatorname{so}}}_X$. We denote the $k$-dimensional persistence diagram arising from this filtration by ${\operatorname{Dgm}}_k^{{\operatorname{so}}}(X)$. Notice that any construction using ${{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X}$ can also be repeated using ${{\mathfrak}{D}^{\operatorname{so}}}_{{\delta},X}$, so we focus on the case of the sink complexes and restate results in terms of source complexes where necessary. In particular, we will prove in §\[sec:dowker-dual\] that $${\operatorname{Dgm}}_k^{{\operatorname{si}}}(X) = {\operatorname{Dgm}}_k^{{\operatorname{so}}}(X) \text{ for any } k\in {\mathbb{Z}}_+,$$ so it makes sense to talk about “the" Dowker diagram associated to $X$. The sink and source filtrations are not equal in general; this is illustrated in Figure \[fig:dowker-three-node\]. As in the case of the Rips filtration, both the Dowker sink and source filtrations are stable. \[prop:dowker-stab\] Let $(X,{\omega}_X), (Y,{\omega}_Y) \in {\mathcal{N}}$. Then ${d_{\operatorname{B}}}({\operatorname{Dgm}}_k^{\bullet}{(X)},{\operatorname{Dgm}}_k^{\bullet}(Y)) \leq 2{d_{\mathcal{N}}}(X,Y).$ Here ${\operatorname{Dgm}}^{\bullet}$ refers to either of ${\operatorname{Dgm}}^{{\operatorname{si}}}$ and ${\operatorname{Dgm}}^{{\operatorname{so}}}$. Both cases are similar, so we just prove the result for ${\operatorname{Dgm}}^{{\operatorname{si}}}$. Let $\eta=2{d_{\mathcal{N}}}(X,Y)$. Then by Proposition \[prop:dn-ko\], there exist maps ${\varphi}:X {\rightarrow}Y, \psi: Y {\rightarrow}X$ such that $$\max({\operatorname{dis}}({\varphi}),{\operatorname{dis}}(\psi), C_{X,Y}({\varphi},\psi),C_{Y,X}(\psi,{\varphi}))\leq \eta.$$ First we check that ${\varphi},\psi$ induce simplicial maps ${\varphi}_{\delta}:{{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X} {\rightarrow}{{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}+\eta,Y}$ and $\psi_{\delta}:{{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},Y} {\rightarrow}{{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}+\eta,Y}$ for each ${\delta}\in {\mathbb{R}}$. Let ${\delta}' \geq {\delta}\in {\mathbb{R}}$. Let ${\sigma}=[x_0,\ldots, x_n] \in {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X}$. Then there exists $x' \in X$ such that ${\omega}_X(x_i,x') \leq {\delta}$ for each $0\leq i \leq n$. Fix such an $x'$. Since ${\operatorname{dis}}({\varphi}) \leq \eta$, we have the following for each $i$: $$\vert {\omega}_X(x_i,x') - {\omega}_Y({\varphi}(x_i),{\varphi}(x'))\vert \leq \eta.$$ So ${\omega}_Y({\varphi}(x_i),{\varphi}(x')) \leq {\omega}_X(x_i,x') + \eta \leq {\delta}+ \eta$ for each $0\leq i\leq n$. Thus ${\varphi}_{{\delta}}({\sigma}):={\left\{{\varphi}(x_0),\ldots, {\varphi}(x_n)\right\}}$ is a simplex in ${{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}+ \eta,Y}$. Thus the map on simplices ${\varphi}_{\delta}$ induced by ${\varphi}$ is simplicial for each ${\delta}\in {\mathbb{R}}$. Similarly we can check that the map $\psi_{\delta}$ on simplices induced by $\psi$ is simplicial. Now to prove the result, it will suffice to check the contiguity conditions in the statement of Lemma \[lem:stab\]. Consider the following diagram: $$\begin{tikzcd}[column sep=large] {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X} \arrow{r}{s_{{\delta},{\delta}'}} \arrow[swap]{dr}{{\varphi}_{\delta}} & {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}',X}\arrow{dr}{{\varphi}_{{\delta}'}} & \\ &{{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}+\eta,Y} \arrow{r}{t_{{\delta}+\eta,{\delta}'+\eta}} & {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}'+\eta,Y} \end{tikzcd}$$ Here $s_{{\delta},{\delta}'}$ and $t_{{\delta}+\eta,{\delta}'+\eta}$ are the inclusion maps. We claim that $t_{{\delta}+\eta,{\delta}'+\eta} \circ {\varphi}_{\delta}$ and ${\varphi}_{{\delta}'} \circ s_{{\delta},{\delta}'}$ are contiguous simplicial maps. To see this, let ${\sigma}\in {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X}$. Since $s_{{\delta},{\delta}'}$ is just the inclusion, it follows that $t_{{\delta}+\eta,{\delta}'+\eta}({\varphi}_{\delta}({\sigma})) \cup {\varphi}_{{\delta}'}(s_{{\delta},{\delta}'}({\sigma}))={\varphi}_{\delta}({\sigma}),$ which is a simplex in ${{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}+\eta,Y}$ because ${\varphi}_{\delta}$ is simplicial, and hence a simplex in ${{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}'+\eta,Y}$ because the inclusion $t_{{\delta}+\eta,{\delta}'+\eta}$ is simplicial. Thus $t_{{\delta}+\eta,{\delta}'+\eta} \circ {\varphi}_{\delta}$ and ${\varphi}_{{\delta}'} \circ s_{{\delta},{\delta}'}$ are contiguous, and their induced linear maps for homology are equal. By a similar argument, one can show that $s_{{\delta}+\eta,{\delta}'+\eta}\circ \psi_{\delta}$ and $\psi_{{\delta}'}\circ t_{{\delta},{\delta}'}$ are contiguous simplicial maps as well. Next we check that the maps $\psi_{{\delta}+\eta}\circ {\varphi}_{\delta}$ and $s_{{\delta},{\delta}+2\eta}$ in the figure below are contiguous. $$\begin{tikzcd} {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X} \arrow{rr}{s_{{\delta},{\delta}+2\eta}} \arrow[swap]{dr}{{\varphi}_{\delta}} & {} & {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}+2\eta,X}\\ {} & {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}+\eta,Y} \arrow[swap]{ur}{\psi_{{\delta}+\eta}} & {} \end{tikzcd}$$ Let $x_i\in {\sigma}$. Note that for our fixed ${\sigma}= [x_0,\ldots, x_n]\in {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X}$ and $x'$, we have: $$\begin{aligned} |{\omega}_X(x_i,x')-{\omega}_X(\psi({\varphi}(x_i)),\psi({\varphi}(x')))|&\leq |{\omega}_X(x_i,x') - {\omega}_Y({\varphi}(x_i),{\varphi}(x'))|\\ &+ |{\omega}_Y({\varphi}(x_i),{\varphi}(x')) - {\omega}_X(\psi({\varphi}(x_i)),\psi({\varphi}(x')))| \\ &\leq 2\eta.\\ \text{Thus we obtain }\hfill {\omega}_X(\psi({\varphi}(x_i)),\psi({\varphi}(x')))&\leq {\omega}_X(x_i,x')+ 2\eta \leq {\delta}+2\eta.\end{aligned}$$ Since this holds for any $x_i \in {\sigma}$, it follows that $\psi_{{\delta}+\eta}({\varphi}_{\delta}({\sigma})) \in {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}+2\eta,X}$. We further claim that $${\tau}:={\sigma}\cup \psi_{{\delta}+\eta}({\varphi}_{\delta}({\sigma})) = {\left\{x_0,x_1,\ldots, x_n, \psi({\varphi}(x_0)),\ldots, \psi({\varphi}(x_n))\right\}}$$ is a simplex in ${{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}+2\eta,X}$. Let $0\leq i\leq n$. It suffices to show that ${\omega}_X(x_i,\psi({\varphi}(x')) \leq {\delta}+ 2\eta$. Notice that from the reformulation of ${d_{\mathcal{N}}}$ (Proposition \[prop:dn-ko\]), we have $$C_{X,Y}({\varphi},\psi) = \max_{(x,y)\in X\times Y}|{\omega}_X(x,\psi(y)) - {\omega}_Y({\varphi}(x),y)| \leq \eta .$$ Let $y = {\varphi}(x')$. Then $|{\omega}_X(x_i,\psi(y)) - {\omega}_Y({\varphi}(x_i),y)| \leq \eta$. In particular, $${\omega}_X(x_i,\psi({\varphi}(x'))) \leq {\omega}_Y({\varphi}(x_i),{\varphi}(x')) + \eta \leq {\omega}_X(x_i,x') + 2\eta \leq {\delta}+2\eta.$$ Since $0\leq i\leq n$ were arbitrary, it follows that ${\tau}\in {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}+2\eta,X}$. Thus $\psi_{{\delta}+\eta}\circ {\varphi}_{\delta}$ and $s_{{\delta},{\delta}+2\eta}$ are contiguous. Similarly, one can use the ${\operatorname{dis}}(\psi)$ and $C_{Y,X}(\psi,{\varphi})$ terms to show that $t_{{\delta},{\delta}+2\eta}$ and ${\varphi}_{{\delta}+\eta}\circ \psi_{\delta}$ are contiguous. The result now follows by an application of Lemma \[lem:stab\]. \[rem:dowker-benefits\] The preceding proposition shows that the Dowker persistence diagram is robust to noise in input data, and that the bottleneck distance between Dowker persistence diagrams arising from two networks can be used as a proxy for computing the actual network distance. Note the analogy with Remark \[rem:rips-benefits\]. Both the Dowker and Rips filtrations are valid methods for computing persistent homology of networks, by virtue of their stability results (Propositions \[prop:rips-stab\] and \[prop:dowker-stab\]). However, we present the Dowker filtration as an appropriate method for capturing directionality information in directed networks. In §\[sec:symmetry\] we discuss this particular feature of the Dowker filtration in full detail. In the setting of symmetric networks, the Dowker sink and source simplicial filtrations coincide, and so we automatically obtain ${\operatorname{Dgm}}_k^{{\operatorname{so}}}(X)={\operatorname{Dgm}}_k^{{\operatorname{si}}}(X)$ for any $k\in {\mathbb{Z}}_+$ and any $(X,{\omega}_X)\in {\mathcal{N}}$. When restricted to the setting of metric spaces, the Dowker complex resembles a construction called the witness complex [@de2004topological]. In particular, a version of the Dowker complex for metric spaces, constructed in terms of *landmarks* and *witnesses*, was discussed in [@chazal2014persistence], along with stability results. When restricted to the special networks that are pseudo-metric spaces, our definitions and results agree with those presented in [@chazal2014persistence]. The Functorial Dowker Theorem and equivalence of diagrams {#sec:dowker-dual} --------------------------------------------------------- Let $(X,{\omega}_X)\in {\mathcal{N}}$, and let ${\delta}\in {\mathbb{R}}$ be such that $R_{{\delta},X}$ is nonempty. By applying Dowker’s theorem (Theorem \[thm:dowker\]) to the setting $Y=X$, we have $H_k({{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X}) \cong H_k({{\mathfrak}{D}^{\operatorname{so}}}_{{\delta},X})$, for any $k \in {\mathbb{Z}}_+$. We still have this equality in the case where $R_{{\delta},X}$ is empty, because then ${{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X}$ and ${{\mathfrak}{D}^{\operatorname{so}}}_{{\delta},X}$ are both empty. Thus we obtain: \[cor:dowker\] Let $(X,{\omega}_X)\in{\mathcal{N}}$, ${\delta}\in {\mathbb{R}}$, and $k\in {\mathbb{Z}}_+$. Then, $$H_k({{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X}) \cong H_k({{\mathfrak}{D}^{\operatorname{so}}}_{{\delta},X}).$$ In the persistent setting, Theorem \[thm:dowker\] and Corollary \[cor:dowker\] suggest the following question: > *Given a network $(X,{\omega}_X)$ and a fixed dimension $k\in {\mathbb{Z}}_+$, are the persistence diagrams of the Dowker sink and source filtrations of $(X,{\omega}_X)$ necessarily equal?* In what follows, we provide a positive answer to the question above. Our strategy is to use the Functorial Dowker Theorem (Theorem \[thm:dowker-functorial\]), for which we will provide a complete proof below. The Functorial Dowker Theorem implies equality between sink and source persistence diagrams. \[cor:dowker-dual\] Let $(X,{\omega}_X)\in {\mathcal{N}}$, and let $k\in {\mathbb{Z}}_+$. Then, $${\operatorname{Dgm}}_k^{{\operatorname{si}}}(X)={\operatorname{Dgm}}_k^{{\operatorname{so}}}(X).$$ Thus we may call either of the diagrams above the *$k$-dimensional Dowker diagram of $X$*, denoted ${\operatorname{Dgm}}_k^{{\mathfrak}{D}}(X)$. Before proving the corollary, we state an ${\mathbb{R}}$-indexed variant of the Persistence Equivalence Theorem [@edelsbrunner2010computational]. This particular version follows from the *isometry theorem* [@bauer-isom], and we refer the reader to [@chazal2012structure Chapter 5] for an expanded presentation of this material. \[thm:pet\] Consider two persistence vector spaces ${\mathcal{U}}=\{U^{{\delta}} {\xrightarrow}{\mu_{{\delta},{\delta}'}} U^{{\delta}'}\}_{{\delta}\leq{\delta}' \in {\mathbb{R}}}$ and ${\mathcal{V}}=\{V^{{\delta}} {\xrightarrow}{\nu_{{\delta},{\delta}'}} V^{{\delta}'}\}_{{\delta}\leq {\delta}'\in {\mathbb{R}}}$ with connecting maps $f_{{\delta}}:U^{{\delta}}{\rightarrow}V^{{\delta}'}$. \(00) at (-3,0)[$\cdots$]{}; (1) at (0,0)[$V^{{\delta}}$]{}; (2) at (3,0)[$V^{{\delta}'}$]{}; (3) at (6,0)[$V^{{\delta}''}$]{}; (01) at (9,0)[$\cdots$]{}; \(02) at (-3,2)[$\cdots$]{}; (4) at (0,2)[$U^{{\delta}}$]{}; (5) at (3,2)[$U^{{\delta}'}$]{}; (6) at (6,2)[$U^{{\delta}''}$]{}; (03) at (9,2)[$\cdots$]{}; \(00) edge\[-&gt;\] (1); (3) edge\[-&gt;\] (01); (02) edge\[-&gt;\] (4); (6) edge\[-&gt;\] (03); \(1) edge\[-&gt;\] (2); (2) edge\[-&gt;\] (3); (4) edge\[-&gt;\] (5); (5) edge\[-&gt;\] (6); (1) edge\[&lt;-\] node\[right\][$f_{{\delta}}$]{} (4); (2) edge\[&lt;-\] node\[right\][$f_{{\delta}'}$]{} (5); (3) edge\[&lt;-\] node\[right\][$f_{{\delta}''}$]{} (6); If the $f_{{\delta}}$ are all isomorphisms and each square in the diagram above commutes, then: $${\operatorname{Dgm}}({\mathcal{U}}) = {\operatorname{Dgm}}({\mathcal{V}}).$$ Let ${\delta}\leq {\delta}' \in {\mathbb{R}}$, and consider the relations $R_{{\delta},X} \subseteq R_{{\delta}',X} \subseteq X\times X$. Suppose first that $R_{{\delta},X}$ and $R_{{\delta}',X}$ are both nonempty. By applying Theorem \[thm:dowker-functorial\], we obtain homotopy equivalences between the source and sink complexes that commute with the canonical inclusions up to homotopy. Passing to the $k$-th homology level, we obtain persistence vector spaces that satisfy the commutativity properties of Theorem \[thm:pet\]. The result follows from Theorem \[thm:pet\]. In the case where $R_{{\delta},X}$ and $R_{{\delta}',X}$ are both empty, there is nothing to show because all the associated complexes are empty. Suppose $R_{{\delta},X}$ is empty, and $R_{{\delta}',X}$ is nonempty. Then ${{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X}$ and ${{\mathfrak}{D}^{\operatorname{so}}}_{{\delta},X}$ are empty, so their inclusions into ${{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}',X}$ and ${{\mathfrak}{D}^{\operatorname{so}}}_{{\delta}',X}$ induce zero maps upon passing to homology. Thus the commutativity of Theorem \[thm:pet\] is satisfied, and the result follows by Theorem \[thm:pet\]. #### The proof of the Functorial Dowker Theorem It remains to prove Theorem \[thm:dowker-functorial\]. Because the proof involves numerous maps, we will adopt the notational convention of adding a subscript to a function to denote its codomain—e.g. we will write $f_B$ to denote a function with codomain $B$. First we recall the construction of a combinatorial barycentric subdivision (see [@dowker1952homology §2], [@lefschetz1942algebraic §4.7], [@barmak2011algebraic Appendix A]). \[def:subdivision\] For any simplicial complex ${\Sigma}$, one may construct a new simplicial complex ${\Sigma}^{(1)}$, called the *first barycentric subdivision*, as follows: $${\Sigma}^{(1)}:={\left\{[{\sigma}_1,{\sigma}_2,\ldots, {\sigma}_p] : {\sigma}_1 \subseteq {\sigma}_2 \subseteq \ldots \subseteq {\sigma}_p, \text{ each } {\sigma}_i \in {\Sigma}\right\}}.$$ Note that the vertices of ${\Sigma}^{(1)}$ are the simplices of ${\Sigma}$, and the simplices of ${\Sigma}^{(1)}$ are nested sequences of simplices of ${\Sigma}$. Furthermore, note that given any two simplicial complexes ${\Sigma}, \Xi$ and a simplicial map $f:{\Sigma}{\rightarrow}\Xi$, there is a natural simplicial map $f{^{(1)}}:{\Sigma}{^{(1)}}{\rightarrow}\Xi{^{(1)}}$ defined as: $$f{^{(1)}}([{\sigma}_1,\ldots,{\sigma}_p]):=[f({\sigma}_1),\ldots,f({\sigma}_p)], \qquad {\sigma}_1\subseteq {\sigma}_2\subseteq \ldots, {\sigma}_p, \text{ each } {\sigma}_i\in {\Sigma}.$$ To see that this is simplicial, note that $f({\sigma}_i) \subseteq f({\sigma}_j)$ whenever ${\sigma}_i\subseteq {\sigma}_j$. As a special case, observe that any inclusion map $\iota:{\Sigma}{\hookrightarrow}\Xi$ induces an inclusion map $\iota{^{(1)}}:{\Sigma}{^{(1)}}{\hookrightarrow}\Xi{^{(1)}}$. Given a simplex ${\sigma}=[x_0,\ldots, x_k]$ in a simplicial complex ${\Sigma}$, one defines the *barycenter* to be the point ${\mathcal}{B}({\sigma}):= \sum_{i=0}^k \tfrac{1}{k+1}x_i \in |{\Sigma}|$. Then the spaces $|{\Sigma}{^{(1)}}|$ and $|{\Sigma}|$ can be identified via a homeomorphism ${\mathcal}{E}_{|{\Sigma}|}:|{\Sigma}{^{(1)}}| {\rightarrow}|{\Sigma}|$ defined on vertices by ${\mathcal}{E}_{|{\Sigma}|}({\sigma}):= {\mathcal}{B}({\sigma})$ and extended linearly. Details on the preceding list of definitions can be found in [@munkres-book §2.14-15, 2.19], [@spanier-book §3.3-4], and also [@barmak2011algebraic Appendix A]. The next proposition follows from the discussions in these references, and is a simple restatement of [@barmak2011algebraic Proposition A.1.5]. We provide a proof in the appendix for completeness. \[prop:subdiv-identity\] Let ${\Sigma}$ be a simplicial complex, and let $\Phi: \Sigma{^{(1)}}{\rightarrow}\Sigma$ be a simplicial map such that $\Phi({\sigma}) \in {\sigma}$ for each ${\sigma}\in \Sigma$. Then $|\Phi| \simeq {\mathcal}{E}_{|\Sigma|}$. We now introduce some auxiliary constructions dating back to [@dowker1952homology] that use the setup stated in Theorem \[thm:dowker-functorial\]. For any nonempty relation $R\subseteq X\times Y$, one may define [@dowker1952homology §2] an associated map $\Phi_{E_R} : E_R{^{(1)}}{\rightarrow}E_R$ as follows: first define $\Phi_{E_R}$ on vertices of $E_R{^{(1)}}$ by $\Phi_{E_R}({\sigma})=s_{{\sigma}}$, where $s_{{\sigma}}$ is the least vertex of ${\sigma}$ with respect to the total order. Next, for any simplex $[{\sigma}_1,\ldots,{\sigma}_k]$ of $E_R{^{(1)}}$, where ${\sigma}_1\subseteq \ldots \subseteq {\sigma}_k$, we have $\Phi_{E_R}({\sigma}_i)=s_{{\sigma}_i} \in {\sigma}_k$ for all $1\leq i\leq k$. Thus $[\Phi_{E_R}({\sigma}_1),\ldots,\Phi_{E_R}({\sigma}_k)]=[s_{{\sigma}_1},s_{{\sigma}_2},\ldots,s_{{\sigma}_k}]$ is a face of ${\sigma}_k$, hence a simplex of ${\Sigma}$. This defines $\Phi_{E_R}$ as a simplicial map $E_R{^{(1)}}{\rightarrow}E_R$. This argument also shows that $\Phi_{E_R}$ is order-reversing: if ${\sigma}\subseteq {\sigma}'$, then $\Phi_{E_R}({\sigma}) \geq \Phi_{E_{R}}({\sigma}')$. \[rem:subdiv-id\] Applying Proposition \[prop:subdiv-identity\] to the setup above, one sees that $|\Phi_{E_R}| \simeq {\mathcal}{E}_{|E_R|}$. After passing to a second barycentric subdivision $E_R^{(2)}$ (obtained by taking a barycentric subdivision of $E_R{^{(1)}}$) and obtaining a map $\Phi_{E_R{^{(1)}}}:E_R^{(2)} {\rightarrow}E_R{^{(1)}}$, one also has $|\Phi_{E_R{^{(1)}}}| \simeq {\mathcal}{E}_{|E_R{^{(1)}}|}$. One can also define [@dowker1952homology §3] a simplicial map $\Psi_{F_R}: E_R{^{(1)}}{\rightarrow}F_R$ as follows. Given a vertex ${\sigma}=[x_0,\ldots, x_k] \in E_R{^{(1)}}$, one defines $\Psi_{F_R}({\sigma})=y_{\sigma}$, for some $y_{\sigma}\in Y$ such that $(x_i,y_{\sigma}) \in R$ for each $i$. To see why this vertex map is simplicial, let ${\sigma}{^{(1)}}= [{\sigma}_0,\ldots, {\sigma}_k]$ be a simplex in $E_R{^{(1)}}$. Let $x \in {\sigma}_0$. Then, because ${\sigma}_0 \subseteq {\sigma}_1 \subseteq \ldots \subseteq {\sigma}_k$, we automatically have that $(x,\Psi_{F_R}({\sigma}_i)) \in R$, for each $i=0,\ldots, k$. Thus $\Psi_{F_R}({\sigma}{^{(1)}})$ is a simplex in $F_R$. This definition involves a choice of $y_{\sigma}$ when writing $\Psi_{F_R}({\sigma}) = y_{\sigma}$, but all the maps resulting from such choices are contiguous [@dowker1952homology §3]. The preceding map induces a simplicial map $\Psi_{F_R{^{(1)}}}:E_R^{(2)} {\rightarrow}F_R{^{(1)}}$ as follows. Given a vertex ${\tau}{^{(1)}}= [{\tau}_0,\ldots, {\tau}_k] \in E_R^{(2)}$, i.e. a simplex in $E_R{^{(1)}}$, we define $\Psi_{F_R{^{(1)}}}({\tau}{^{(1)}}) := [\Psi_{F_R}({\tau}_0),\ldots, \Psi_{F_R}({\tau}_k)]$. Since $\Psi_{F_R}$ is simplicial, this is a simplex in $F_R$, i.e. a vertex in $F_R{^{(1)}}$. Thus we have a vertex map $\Psi_{F_R{^{(1)}}}:E_R^{(2)} {\rightarrow}F_R{^{(1)}}$. To check that this map is simplicial, let ${\tau}^{(2)} = [{\tau}{^{(1)}}_0,\ldots, {\tau}{^{(1)}}_p]$ be a simplex in $E_R^{(2)}$. Then ${\tau}{^{(1)}}_0 \subseteq {\tau}{^{(1)}}_1 \subseteq \ldots \subseteq {\tau}{^{(1)}}_p$, and because $\Psi_{F_R}$ is simplicial, we automatically have $$\Psi_{F_R}({\tau}{^{(1)}}_0) \subseteq \Psi_{F_R}({\tau}{^{(1)}}_1) \subseteq \ldots \subseteq \Psi_{F_R}({\tau}{^{(1)}}_p).$$ Thus $\Psi_{F_R{^{(1)}}}({\tau}^{(2)})$ is a simplex in $F_R{^{(1)}}$. We write $F_R^{(2)}$ to denote the barycentric subdivision of $F_R{^{(1)}}$, and obtain simplicial maps $\Phi_{F_R{^{(1)}}}:F_R^{(2)} {\rightarrow}F_R{^{(1)}}$, $\Phi_{F_R}:F_R^{(1)} {\rightarrow}F_R$, $\Psi_{E_R{^{(1)}}}:F_R^{(2)} {\rightarrow}E_R{^{(1)}}$, and $\Psi_{F_R}:E_R^{(1)} {\rightarrow}F_R$ as above. Consider the following diagram: \(1) at (0,0)[$F_R^{(2)}$]{}; (2) at (3,-1)[$F_R^{(1)}$]{}; (3) at (6,-2)[$F_R$]{}; (4) at (8,0)[$F_{R'}^{(2)}$]{}; (5) at (11,-1)[$F_{R'}^{(1)}$]{}; (6) at (14,-2)[$F_{R'}$]{}; (7) at (-1,-3)[$E_R^{(2)}$]{}; (8) at (2,-4)[$E_R{^{(1)}}$]{}; (9) at (5,-5)[$E_R$]{}; (10) at (7,-3)[$E_{R'}^{(2)}$]{}; (11) at (10,-4)[$E_{R'}{^{(1)}}$]{}; (12) at (13,-5)[$E_{R'}$]{}; \(1) edge\[-&gt;,violet!80,thick\] node\[right\] (2); (2) edge\[-&gt;,violet!80,thick\] node\[above\] (3); (1) edge\[-&gt;\] node\[left\] (4); (4) edge\[-&gt;\] node\[above right\][$\Phi_{F_{R'}{^{(1)}}}$]{} (5); (5) edge\[-&gt;\] node\[above right\][$\Phi_{F_{R'}}$]{} (6); (3) edge\[-&gt;\] node\[above\] (6); \(7) edge\[-&gt;,teal!80,thick\] node\[below\] (8); (8) edge\[-&gt;,teal!80,thick\] node\[below\] (9); (7) edge\[-&gt;,dashed\] node\[left\] (10); (10) edge\[-&gt;,dashed\] node\[above right\] (11); (11) edge\[-&gt;,dashed\] node\[above\] (12); (9) edge\[-&gt;\] node\[above\] (12); \(1) edge\[-&gt;,orange,thick\] node\[left,pos=0.2\] (8); (8) edge\[-&gt;,orange,thick\] node\[below right,pos=0.2\] (3); (4) edge\[-&gt;,dashed\] node\[left\] (11); (11) edge\[-&gt;,dashed\] node\[below right\] (6); \(7) edge\[-&gt;,NavyBlue!80,thick\] node\[above left,pos=0.2\] (2); (2) edge\[-&gt;,NavyBlue!80,thick\] node\[left,pos=0.2\] (9); (10) edge\[-&gt;,dashed\] node\[left\] (5); (5) edge\[-&gt;,dashed\] node\[below right\] (12); \(1) at (0,0)[$E_R^{(2)}$]{}; (2) at (2,-1)[$E_R^{(1)}$]{}; (3) at (4,-2)[$E_R$]{}; (4) at (6,0)[$E_{R'}^{(2)}$]{}; (5) at (8,-1)[$E_{R'}^{(1)}$]{}; (6) at (10,-2)[$E_{R'}$]{}; (7) at (3,1)[$F_R{^{(1)}}$]{}; (8) at (9,1)[$F_{R'}{^{(1)}}$]{}; (1) edge\[-&gt;\] node\[below left\] (2); (2) edge\[-&gt;\] node\[below left\] (3); (1) edge\[-&gt;,dashed\] node\[left\] (4); (4) edge\[-&gt;,dashed\] node\[left\] (5); (5) edge\[-&gt;,dashed\] node\[left\] (6); (3) edge\[-&gt;\] node\[above\][$\iota_E$]{} (6); (1) edge\[-&gt;\] node\[above\] (7); (7) edge\[-&gt;\] node\[right\] (3); (4) edge\[-&gt;,dashed\] node\[left\] (8); (8) edge\[-&gt;\] node\[right\] (6); (7) edge\[-&gt;\] node\[above\][$\iota_F$]{} (8); \(1) at (0,0)[$F_R^{(2)}$]{}; (2) at (2,-1)[$F_R^{(1)}$]{}; (3) at (4,-2)[$F_R$]{}; (4) at (6,0)[$F_{R'}^{(2)}$]{}; (5) at (8,-1)[$F_{R'}^{(1)}$]{}; (6) at (10,-2)[$F_{R'}$]{}; (7) at (1,-3)[$E_R{^{(1)}}$]{}; (8) at (7,-3)[$E_{R'}{^{(1)}}$]{}; (1) edge\[-&gt;\] node\[right\] (2); (2) edge\[-&gt;\] node\[above\] (3); (1) edge\[-&gt;\] node\[left\] (4); (4) edge\[-&gt;\] node\[above right\][$\Phi_{F_{R'}{^{(1)}}}$]{} (5); (5) edge\[-&gt;\] node\[above\][$\Phi_{F_{R'}}$]{} (6); (3) edge\[-&gt;\] node\[above\] (6); (1) edge\[-&gt;\] node\[left\] (7); (7) edge\[-&gt;\] node\[above left\] (3); (4) edge\[-&gt;,dashed\] node\[left\] (8); (8) edge\[-&gt;\] node\[below right\] (6); (7) edge\[-&gt;\] node\[above\] (8); We proceed by claiming contiguity of the following. \(1) at (0,0)[$E_R^{(2)}$]{}; (2) at (2,0)[$E_R^{(1)}$]{}; (3) at (4,0)[$E_R$]{}; (7) at (2,2)[$F_R{^{(1)}}$]{}; (1) edge\[-&gt;,teal\] node\[below\] (2); (2) edge\[-&gt;,teal\] node\[below\] (3); (1) edge\[-&gt;,NavyBlue\] node\[left\] (7); (7) edge\[-&gt;,NavyBlue\] node\[right\] (3); \(1) at (0,2)[$F_R^{(2)}$]{}; (2) at (2,2)[$F_R^{(1)}$]{}; (3) at (4,2)[$F_R$]{}; (7) at (2,0)[$E_R{^{(1)}}$]{}; (1) edge\[-&gt;,violet\] node\[below\] (2); (2) edge\[-&gt;,violet\] node\[below\] (3); (1) edge\[-&gt;,orange\] node\[below left\] (7); (7) edge\[-&gt;,orange\] node\[below right\] (3); \(1) at (0,2)[$F_R^{(2)}$]{}; (2) at (3,2)[$F_R^{(1)}$]{}; (7) at (3,0)[$E_R{^{(1)}}$]{}; (3) at (6,0)[$E_R$]{}; (1) edge\[-&gt;,violet\] node\[below\] (2); (2) edge\[-&gt;,NavyBlue\] node\[left\] (3); (1) edge\[-&gt;,orange\] node\[below left\] (7); (7) edge\[-&gt;,teal\] node\[above\] (3); \(1) at (0,0)[$E_R^{(2)}$]{}; (2) at (3,0)[$E_R^{(1)}$]{}; (7) at (3,2)[$F_R{^{(1)}}$]{}; (3) at (6,2)[$F_R$]{}; (1) edge\[-&gt;,teal\] node\[above\] (2); (2) edge\[-&gt;,orange\] node\[below right\] (3); (1) edge\[-&gt;,NavyBlue\] node\[above left\] (7); (7) edge\[-&gt;,violet\] node\[below\] (3); \[cl:dowker-contigo-items\] More specifically: 1. $\Phi_{E_R}\circ \Phi_{E_R{^{(1)}}}$ and $\Psi_{E_R}\circ \Psi_{F_R{^{(1)}}}$ are contiguous. \[item:dowker-contigo-1\] 2. $\Phi_{F_R}\circ \Phi_{F_R{^{(1)}}}$ and $\Psi_{F_R}\circ \Psi_{E_R{^{(1)}}}$ are contiguous. \[item:dowker-contigo-2\] 3. $\Psi_{E_R}\circ \Phi_{F_R{^{(1)}}}$ and $\Phi_{E_R}\circ \Psi_{E_R{^{(1)}}}$ are contiguous. \[item:dowker-contigo-3\] 4. $\Psi_{F_R}\circ \Phi_{E_R{^{(1)}}}$ and $\Phi_{F_R}\circ \Psi_{F_R{^{(1)}}}$ are contiguous. \[item:dowker-contigo-4\] Items (\[item:dowker-contigo-1\]) and (\[item:dowker-contigo-3\]) appear in the proof of Dowker’s theorem [@dowker1952homology Lemmas 5, 6], and it is easy to see that a symmetric argument shows Items (\[item:dowker-contigo-2\]) and (\[item:dowker-contigo-4\]). For completeness, we will verify these items in this paper, but defer this verification to the end of the proof. By passing to the geometric realization and applying Proposition \[prop:contigo-props\] and Remark \[rem:subdiv-id\], we obtain the following from Item (\[item:dowker-contigo-3\]) of Claim \[cl:dowker-contigo-items\]: $$\begin{aligned} |\Psi_{E_R}|\circ |\Phi_{F_R{^{(1)}}}| &\simeq |\Phi_{E_R}|\circ|\Psi_{E_R{^{(1)}}}|,\\ |\Psi_{E_R}|\circ {\mathcal}{E}_{|F_R{^{(1)}}|} &\simeq {\mathcal}{E}_{|E_R|}\circ|\Psi_{E_R{^{(1)}}}|, &&\text{(Remark \ref{rem:subdiv-id})}\\ {\mathcal}{E}{^{-1}}_{|E_R|} \circ |\Psi_{E_R}| \circ {\mathcal}{E}_{|F_R{^{(1)}}|} &\simeq |\Psi_{E_R{^{(1)}}}|. &&\text{(${\mathcal}{E}$ is a homeomorphism, hence invertible)}\end{aligned}$$ Replacing this term in the expression for Item (\[item:dowker-contigo-2\]) of Claim \[cl:dowker-contigo-items\], we obtain: $$\begin{aligned} |\Psi_{F_R}| \circ |\Psi_{E_R{^{(1)}}}| &\simeq |\Phi_{F_R}|\circ |\Phi_{F_R{^{(1)}}}| \simeq {\mathcal}{E}_{|F_R|}\circ {\mathcal}{E}_{|F_R{^{(1)}}|},\\ |\Psi_{F_R}| \circ {\mathcal}{E}{^{-1}}_{|E_R|} \circ |\Psi_{E_R}| \circ {\mathcal}{E}_{|F_R{^{(1)}}|} &\simeq {\mathcal}{E}_{|F_R|}\circ {\mathcal}{E}_{|F_R{^{(1)}}|},\\ |\Psi_{F_R}| \circ {\mathcal}{E}{^{-1}}_{|E_R|} \circ |\Psi_{E_R}|\circ {\mathcal}{E}{^{-1}}_{|F_R|} &\simeq {\operatorname{id}}_{|F_R|}.\end{aligned}$$ Similarly, we obtain the following from Item (\[item:dowker-contigo-4\]) of Claim \[cl:dowker-contigo-items\]: $$|\Psi_{F_R}|\circ |\Phi_{E_R{^{(1)}}}| \simeq |\Phi_{F_R}| \circ |\Psi_{F_R{^{(1)}}}|, \text{ so } {\mathcal}{E}{^{-1}}_{|F_R|} \circ |\Psi_{F_R}| \circ {\mathcal}{E}_{|E_R{^{(1)}}}| \simeq |\Psi_{F_R{^{(1)}}}|.$$ Replacing this term in the expression for Item (\[item:dowker-contigo-1\]) of Claim \[cl:dowker-contigo-items\], we obtain: $$\begin{aligned} |\Psi_{E_R}|\circ |\Psi_{F_R{^{(1)}}}| & \simeq|\Phi_{E_R}|\circ |\Phi_{E_R{^{(1)}}}| \simeq {\mathcal}{E}_{|E_R|}\circ {\mathcal}{E}_{|E_R{^{(1)}}|}, \\ |\Psi_{E_R}|\circ {\mathcal}{E}{^{-1}}_{|F_R|} \circ |\Psi_{F_R}| \circ {\mathcal}{E}_{|E_R{^{(1)}}}| &\simeq {\mathcal}{E}_{|E_R|}\circ {\mathcal}{E}_{|E_R{^{(1)}}|}\\ |\Psi_{E_R}|\circ {\mathcal}{E}{^{-1}}_{|F_R|} \circ |\Psi_{F_R}|\circ {\mathcal}{E}{^{-1}}_{|E_R|} &\simeq {\operatorname{id}}_{|E_R|}\end{aligned}$$ Define $\Gamma_{|E_R|}: |F_R| {\rightarrow}|E_R|$ by $\Gamma_{|E_R|} := |\Psi_{E_R}|\circ {\mathcal}{E}{^{-1}}_{|F_R|}$. Then $\Gamma_{|E_R|}$ is a homotopy equivalence, with inverse given by $|\Psi_{F_R}|\circ {\mathcal}{E}{^{-1}}_{|E_R|}$. This shows that $|F_R|\simeq |E_R|$, for any nonempty relation $R\subseteq X\times Y$. Next we need to show that $\Gamma_{|E_R|}$ commutes with the canonical inclusion. Consider the following diagram, where the $\iota_\bullet$ maps denote the respective canonical inclusions (cf. Definition \[def:subdivision\]): \(1) at (0,0)[$F_R{^{(1)}}$]{}; (2) at (3,0)[$F_{R'}{^{(1)}}$]{}; (3) at (0,-1.5)[$F_R$]{}; (4) at (3,-1.5)[$F_{R'}$]{}; (5) at (-2,-3)[$E_R$]{}; (6) at (5,-3)[$E_{R'}$]{}; (1) edge\[-&gt;\] node\[above\] (2); (3) edge\[-&gt;\] node\[above\] (4); (5) edge\[-&gt;\] node\[above\] (6); (1) edge\[-&gt;\] node\[right\] (3); (2) edge\[-&gt;\] node\[left\] (4); (1) edge\[-&gt;\] node\[left\] (5); (2) edge\[-&gt;\] node\[right\] (6); \[cl:dowker-func-1\] $\iota_E\circ \Psi_{E_R}$ and $\Psi_{E_{R'}}\circ \iota_{F{^{(1)}}}$ are contiguous. \[cl:dowker-func-2\] $\iota_F\circ \Phi_{F_R}$ and $\Phi_{F_{R'}}\circ \iota_{F{^{(1)}}}$ are contiguous. Suppose Claim \[cl:dowker-func-2\] is true. Then, upon passing to geometric realizations, we have: $$\begin{aligned} |\iota_F| \circ {\mathcal}{E}_{|F_R|} \simeq |\iota_F|\circ |\Phi_{F_R}| \simeq |\Phi_{F_{R'}}|\circ |\iota_{F{^{(1)}}}| &\simeq {\mathcal}{E}_{|F_{R'}|}\circ |\iota_{F{^{(1)}}}|,\\ {\mathcal}{E}{^{-1}}_{|F_{R'}|}\circ |\iota_F| \circ {\mathcal}{E}_{|F_R|} &\simeq |\iota_{F{^{(1)}}}|.\end{aligned}$$ Suppose also that Claim \[cl:dowker-func-1\] is true. Then we have: $$\begin{aligned} |\Psi_{E_{R'}}| \circ |\iota_{F{^{(1)}}}| &\simeq |\iota_E|\circ |\Psi_{E_R}|,\\ |\Psi_{E_{R'}}| \circ {\mathcal}{E}{^{-1}}_{|F_{R'}|}\circ |\iota_F| \circ {\mathcal}{E}_{|F_R|} &\simeq |\iota_E|\circ |\Psi_{E_R}|,\\ |\Psi_{E_{R'}}| \circ {\mathcal}{E}{^{-1}}_{|F_{R'}|}\circ |\iota_F| &\simeq |\iota_E|\circ |\Psi_{E_R}| \circ {\mathcal}{E}{^{-1}}_{|F_R|}, \text{ i.e. }\\ \Gamma_{|E_{R'}|}\circ |\iota_F| &\simeq |\iota_E| \circ \Gamma_{|E_R|}.\end{aligned}$$ This proves the theorem. It only remains to prove the various claims. In proving Claim \[cl:dowker-contigo-items\], we supply the proofs of Items (\[item:dowker-contigo-2\]) and (\[item:dowker-contigo-4\]). These arguments are adapted from [@dowker1952homology Lemmas 1, 5, and 6], where the proofs of Items (\[item:dowker-contigo-1\]) and (\[item:dowker-contigo-3\]) appeared. For Item (\[item:dowker-contigo-2\]), let ${\tau}^{(2)}=[{\tau}_0{^{(1)}},\ldots,{\tau}_k{^{(1)}}]$ be a simplex in $F_R^{(2)}$, where ${\tau}_0{^{(1)}}\subseteq \ldots \subseteq {\tau}_k{^{(1)}}$ is a chain of simplices in $F_R{^{(1)}}$. By the order-reversing property of the map $\Phi_{F_R{^{(1)}}}$, we have that $\Phi_{F_R{^{(1)}}}({\tau}_0{^{(1)}}) \supseteq \Phi_{F_R{^{(1)}}}({\tau}_i{^{(1)}})$ for each $i=0,\ldots, k$. Define $x:= \Psi_{E_R}(\Phi_{F_R{^{(1)}}}({\tau}_0{^{(1)}}))$. Then $(x,y) \in R$ for each $y \in \Phi_{F_R{^{(1)}}}({\tau}_0{^{(1)}})$. But we also have $(x,\Phi_{F_R}(\Phi_{F_R{^{(1)}}}({\tau}_i{^{(1)}}))) \in R$ for each $i=0,\ldots, k$, because $\Phi_{F_R}(\Phi_{F_R{^{(1)}}}({\tau}_i{^{(1)}})) \in \Phi_{F_R{^{(1)}}}({\tau}_i{^{(1)}}) \subseteq \Phi_{F_R}({\tau}_0{^{(1)}})$ for each $i=0,\ldots, k$. Next let $0\leq i \leq k$. For each ${\tau}\in {\tau}{^{(1)}}_i$, we have $\Psi_{E_R}({\tau}) \in \Psi_{E_R{^{(1)}}}({\tau}{^{(1)}}_i)$ (by the definition of $\Psi_{E_R{^{(1)}}}$). Because $\Phi_{F_R{^{(1)}}}({\tau}_0{^{(1)}}) \in {\tau}_0{^{(1)}}\subseteq {\tau}_i{^{(1)}}$, we then have $x = \Psi_{E_R}(\Phi_{F_R{^{(1)}}}({\tau}_0{^{(1)}})) \in \Psi_{E_R{^{(1)}}}({\tau}{^{(1)}}_i)$, which is a vertex of $E_R{^{(1)}}$ or alternatively a simplex of $E_R$. But then, by definition of $\Psi_{F_R}$, we have that $(x,\Psi_{F_R}(\Psi_{E_R{^{(1)}}}({\tau}_i{^{(1)}}))) \in R$. This holds for each $0\leq i \leq k$. Since ${\tau}^{(2)}$ was arbitrary, this shows that $\Phi_{F_R}\circ \Phi_{F_R{^{(1)}}}$ and $\Psi_{F_R}\circ \Psi_{E_R{^{(1)}}}$ are contiguous. For Item (\[item:dowker-contigo-4\]), let ${\sigma}^{(2)} = [{\sigma}{^{(1)}}_0,\ldots,{\sigma}{^{(1)}}_k]$ be a simplex in $E_R^{(2)}$. Let $0\leq i \leq k$. Then ${\sigma}{^{(1)}}_0 \subseteq \ldots \subseteq {\sigma}{^{(1)}}_k$, and $\Phi_{E_R{^{(1)}}}({\sigma}{^{(1)}}_i) \in {\sigma}{^{(1)}}_i \subseteq {\sigma}{^{(1)}}_k$. So $\Psi_{F_R}(\Phi_{E_R{^{(1)}}}({\sigma}_i{^{(1)}})) \in \Psi_{F_R{^{(1)}}}({\sigma}_k{^{(1)}})$. On the other hand, we have $\Psi_{F_R{^{(1)}}}({\sigma}_i{^{(1)}}) \subseteq \Psi_{F_R{^{(1)}}}({\sigma}_k{^{(1)}})$. Then $\Phi_{F_R}(\Psi_{F_R{^{(1)}}}({\sigma}_i{^{(1)}})) \in \Psi_{F_R{^{(1)}}}({\sigma}_i{^{(1)}}) \subseteq \Psi_{F_R{^{(1)}}}({\sigma}_k{^{(1)}})$. Since $i$ was arbitrary, this shows that $\Psi_{F_R}\circ \Phi_{E_R{^{(1)}}}$ and $\Phi_{F_R}\circ \Psi_{F_R{^{(1)}}}$ both map the vertices of ${\sigma}^{(2)}$ to the simplex $\Psi_{F_R{^{(1)}}}({\sigma}_k{^{(1)}})$, hence are contiguous. This concludes the proof of the claim. Let ${\tau}{^{(1)}}=[{\tau}_0,{\tau}_1,\ldots, {\tau}_k] \in F_R{^{(1)}}$, where ${\tau}_0 \subseteq {\tau}_1\subseteq \ldots \subseteq {\tau}_k$ is a chain of simplices in $F_R$. Then $\iota_{F{^{(1)}}}({\tau}{^{(1)}}) = {\tau}{^{(1)}}$, and $\Psi_{E_{R'}}({\tau}{^{(1)}})=[x_{{\tau}_0},\ldots, x_{{\tau}_k}]$, for some choice of $x_{{\tau}_i}$ terms. Also we have $\iota_E\circ \Psi_{E_{R}}({\tau}{^{(1)}}) = [x'_{{\tau}_0},\ldots, x'_{{\tau}_k}]$ for some other choice of $x'_{{\tau}_i}$ terms. For contiguity, we need to show that $$[x_{{\tau}_0},\ldots, x_{{\tau}_k}, x'_{{\tau}_0},\ldots, x'_{{\tau}_k}] \in E_{R'}.$$ But this is easy to see: letting $y \in {\tau}_0$, we have ${\left\{(x_{{\tau}_0},y),\ldots,(x_{{\tau}_k},y),(x'_{{\tau}_0},y),\ldots,(x'_{{\tau}_k},y)\right\}} \subseteq R$. Since ${\tau}{^{(1)}}$ was arbitrary, it follows that we have contiguity. Let ${\tau}{^{(1)}}=[{\tau}_0,{\tau}_1,\ldots, {\tau}_k] \in F_R{^{(1)}}$, where ${\tau}_0 \subseteq {\tau}_1\subseteq \ldots \subseteq {\tau}_k$ is a chain of simplices in $F_R$. Then $\Phi_{F_R}({\tau}_i) \in {\tau}_k$ for each $0\leq i \leq k$. Thus $\iota_F\circ \Phi_{F_R}({\tau}{^{(1)}})$ is a face of ${\tau}_k$. Similarly, $\Phi_{F_{R'}}\circ \iota_{F{^{(1)}}}({\tau}{^{(1)}})$ is also a face of ${\tau}_k$. Since ${\tau}{^{(1)}}$ was an arbitrary simplex of $F_R{^{(1)}}$, it follows that $\iota_F\circ \Phi_{F_R}$ and $\Phi_{F_{R'}}\circ \iota_{F{^{(1)}}}$ are contiguous. The equivalence between the finite FDT and the simplicial FNTs {#sec:dowker-nerve-equiv} -------------------------------------------------------------- In this section, we present our answer to Question \[q:f-nerve-f-dowker\]. We begin with a weaker formulation of Theorem \[thm:dowker-functorial\] and some simplicial Functorial Nerve Theorems. \[thm:dowker-functorial-finite\] Let $X,Y$ be two totally ordered sets, and without loss of generality, suppose $X$ is finite. Let $R\subseteq R' \subseteq X\times Y$ be two nonempty relations, and let $E_R, F_R, E_{R'}, F_{R'}$ be their associated simplicial complexes (as in Theorem \[thm:dowker-functorial\]). Then there exist homotopy equivalences $\Gamma_{|E_R|}:|F_R| {\rightarrow}|E_R|$ and $\Gamma_{|E_{R'}|}: |F_{R'}| {\rightarrow}|E_{R'}|$ that commute up to homotopy with the canonical inclusions. The finite FDT (Theorem \[thm:dowker-functorial-finite\]) is an immediate consequence of the general FDT (Theorem \[thm:dowker-functorial\]). Let ${\mathcal}{A} = {\left\{A_i\right\}}_{i\in I}$ be a family of nonempty sets indexed by $I$. The *nerve* of ${\mathcal}{A}$ is the simplicial complex ${\mathcal}{N}({\mathcal}{A}):= \{{\sigma}\in {\operatorname{pow}}(I) : {\sigma}\text{ is finite, nonempty, and } \cap_{i \in {\sigma}}A_i \neq {\varnothing}\}$. Let ${\Sigma}$ be a simplicial complex. Then a collection of subcomplexes ${\mathcal}{A}_{\Sigma}= \{{\Sigma}_i\}_{i\in I}$ is said to be a *cover of subcomplexes* for ${\Sigma}$ if ${\Sigma}= \cup_{i\in I}{\Sigma}_i$. Furthermore, ${\mathcal}{A}_{\Sigma}$ is said to be a *cover of simplices* if each ${\Sigma}_i \in {\mathcal}{A}_{\Sigma}$ has the property that ${\Sigma}_i = {\operatorname{pow}}(V({\Sigma}_i))$. In this case, each ${\Sigma}_i$ has precisely one top-dimensional simplex, consisting of the vertex set $V({\Sigma}_i)$. We present two *simplicial* formulations of the Functorial Nerve Theorem that turn out to be equivalent; the statements differ in that one is about covers of simplices and the other is about covers of subcomplexes. \[thm:nerve-functorial-I\] Let ${\Sigma}\subseteq {\Sigma}'$ be two simplicial complexes, and let ${\mathcal}{A}_{\Sigma}=\{{\Sigma}_i\}_{i\in I}$, ${\mathcal}{A}_{{\Sigma}'}=\{{\Sigma}'_i\}_{i\in I'}$ be finite covers of simplices for ${\Sigma}$ and ${\Sigma}'$ such that $I\subseteq I'$ and ${\Sigma}_i \subseteq {\Sigma}'_i$ for each $i \in I$. In particular, ${\operatorname{card}}(I') < \infty$. Suppose that for each finite subset ${\sigma}\subseteq I'$, the intersection $\cap_{i \in {\sigma}}{\Sigma}'_i$ is either empty or contractible (and likewise for $\cap_{i \in {\sigma}}{\Sigma}_i$). Then $|{\Sigma}| \simeq |{\mathcal}{N}({\mathcal}{A}_{\Sigma})|$ and $|{\Sigma}'| \simeq |{\mathcal}{N}({\mathcal}{A}_{{\Sigma}'})|$, via maps that commute up to homotopy with the canonical inclusions. \[thm:nerve-functorial-II\] The statement of Theorem \[thm:nerve-functorial-I\] holds even if ${\mathcal}{A}_{\Sigma}$ and ${\mathcal}{A}_{{\Sigma}'}$ are covers of subcomplexes. Explicitly, the statement is as follows. Let ${\Sigma}\subseteq {\Sigma}'$ be two simplicial complexes, and let ${\mathcal}{A}_{\Sigma}=\{{\Sigma}_i\}_{i\in I}$, ${\mathcal}{A}_{{\Sigma}'}=\{{\Sigma}'_i\}_{i\in I'}$ be finite covers of subcomplexes for ${\Sigma}$ and ${\Sigma}'$ such that $I\subseteq I'$ and ${\Sigma}_i \subseteq {\Sigma}'_i$ for each $i \in I$. In particular, ${\operatorname{card}}(I') < \infty$. Suppose that for each finite subset ${\sigma}\subseteq I'$, the intersection $\cap_{i \in {\sigma}}{\Sigma}'_i$ is either empty or contractible (and likewise for $\cap_{i \in {\sigma}}{\Sigma}_i$). Then $|{\Sigma}| \simeq |{\mathcal}{N}({\mathcal}{A}_{\Sigma})|$ and $|{\Sigma}'| \simeq |{\mathcal}{N}({\mathcal}{A}_{{\Sigma}'})|$, via maps that commute up to homotopy with the canonical inclusions. The following result summarizes our answer to Question \[q:f-nerve-f-dowker\]. \[thm:dowker-nerve-eq\] The finite FDT, the FNT I, and the FNT II are all equivalent. Moreover, all of these results are implied by the FDT, as below: (fdt) at (-2,0)[Theorem \[thm:dowker-functorial\]]{}; (ffdt) at (1,0)[Theorem \[thm:dowker-functorial-finite\]]{}; (sfnt1) at (3,1)[Theorem \[thm:nerve-functorial-I\]]{}; (sfnt2) at (5,0)[Theorem \[thm:nerve-functorial-II\]]{}; (fdt) – (ffdt); (ffdt) – (sfnt1); (sfnt1) – (sfnt2); (sfnt2) – (ffdt); We present the proof of Theorem \[thm:dowker-nerve-eq\] over the course of the next few subsections. By virtue of Theorem \[thm:dowker-nerve-eq\], we will write *simplicial FNT* to mean either of the FNT I or FNT II. Theorem \[thm:dowker-functorial-finite\] implies Theorem \[thm:nerve-functorial-I\] {#theorem-thmdowker-functorial-finite-implies-theorem-thmnerve-functorial-i .unnumbered} ----------------------------------------------------------------------------------- Let $V, V'$ denote the vertex sets of ${\Sigma},{\Sigma}'$, respectively. We define the relations $R\subseteq V\times I$ and $R' \subseteq V'\times I'$ as follows: $(v,i) \in R \iff v \in {\Sigma}_i$ and $(v',i') \in R' \iff v' \in {\Sigma}_i'.$ Then $R \subseteq R'$, the set $I'$ is finite by assumption, and so we are in the setting of the finite FDT (Theorem \[thm:dowker-functorial-finite\]) (perhaps invoking the Axiom of Choice to obtain the total order on $V'$). It suffices to show that $E_R = {\Sigma}$, $E_{R'} = {\Sigma}'$, $F_R = {\mathcal}{N}({\mathcal}{A}_{\Sigma})$, and $F_{R'} = {\mathcal}{N}({\mathcal}{A}_{{\Sigma}'})$, where $E_R, E_{R'}, F_R, F_{R'}$ are as defined in Theorem \[thm:dowker-functorial\]. First we claim the $E_R = {\Sigma}$. By the definitions of $R$ and $E_R$, we have $E_R = \{{\sigma}\subseteq V : \; \exists i \in I, \; (v,i) \in R \; \forall \; v\in {\sigma}\} = \{{\sigma}\subseteq V : \;\exists i \in I, \; v\in {\Sigma}_i \; \forall \; v\in {\sigma}\}.$ Let ${\sigma}\in E_R$, and let $i \in I$ be such that $v \in {\Sigma}_i$ for all $v \in {\sigma}$. Then ${\sigma}\subseteq V({\Sigma}_i)$, and since ${\Sigma}_i = {\operatorname{pow}}(V({\Sigma}_i))$ by the assumption about covers of simplices, we have ${\sigma}\in {\Sigma}_i \subseteq {\Sigma}$. Thus $E_R \subseteq {\Sigma}$. Conversely, let ${\sigma}\in {\Sigma}$. Then ${\sigma}\in {\Sigma}_i$ for some $i$. Thus for all $v \in {\sigma}$, we have $(v,i) \in R$. It follows that ${\sigma}\in E_R$. This shows $E_R = {\Sigma}$. The proof that $E_{R'} = {\Sigma}'$ is analogous. Next we claim that $F_R = {\mathcal}{N}({\mathcal}{A}_{\Sigma})$. By the definition of $F_R$, we have $F_R = \{{\tau}\subseteq I : \; \exists v \in V, \; (v,i) \in R \; \forall \; i\in {\tau}\}.$ Let ${\tau}\in F_R$, and let $v \in V$ be such that $(v,i) \in R$ for each $i \in {\tau}$. Then $\cap_{i \in {\tau}}{\Sigma}_i \neq {\varnothing}$, and so ${\tau}\in {\mathcal}{N}({\mathcal}{A}_{\Sigma})$. Conversely, let ${\tau}\in {\mathcal}{N}({\mathcal}{A}_{\Sigma})$. Then $\cap_{i \in {\tau}}{\Sigma}_i \neq {\varnothing}$, so there exists $v \in V$ such that $v \in {\Sigma}_i$ for each $i \in {\tau}$. Thus ${\sigma}\in F_R$. This shows $F_R = {\mathcal}{N}({\mathcal}{A}_{\Sigma})$. The case for $R'$ is analogous. An application of Theorem \[thm:dowker-functorial-finite\] now completes the proof. Theorem \[thm:nerve-functorial-II\] implies Theorem \[thm:dowker-functorial-finite\] {#theorem-thmnerve-functorial-ii-implies-theorem-thmdowker-functorial-finite .unnumbered} ------------------------------------------------------------------------------------ Let $X$ and $Y$ be two sets, and suppose $X$ is finite. Let $R\subseteq R' \subseteq X \times Y$ be two relations. Consider the simplicial complexes $E_R, F_R, E_{R'}, F_{R'}$ as defined in Theorem \[thm:dowker-functorial\]. Let $V_R:=V(E_R)$. For each $x \in V_R$, define $A_x:=\{{\tau}\in F_R : (x,y) \in R \text{ for all } y \in {\tau}\}$. Then $A_x$ is a subcomplex of $F_R$. Furthermore, $\cup_{x\in V_R}A_x = F_R$. To see this, let ${\tau}\in F_R$. Then there exists $x\in X$ such that $(x,y) \in R$ for all $y\in {\tau}$, and so ${\tau}\in A_x$. Let ${\mathcal}{A}:= \{A_x : x \in V_R\}$. We have seen that ${\mathcal}{A}$ is a cover of subcomplexes for $F_R$. It is finite because the indexing set $V_R$ is a subset of $X$, which is finite by assumption. Next we claim that ${\mathcal}{N}({\mathcal}{A}) = E_R$. Let ${\sigma}\in E_R$. Then there exists $y \in Y$ such that $(x,y) \in R$ for all $x\in {\sigma}$. Thus $\cap_{x\in {\sigma}}A_x \neq {\varnothing}$, and so ${\sigma}\in {\mathcal}{N}({\mathcal}{A})$. Conversely, let ${\sigma}\in {\mathcal}{N}({\mathcal}{A})$. Then $\cap_{x \in {\sigma}}A_x \neq {\varnothing}$, and so there exists $y \in Y$ such that $(x,y) \in R$ for all $x \in {\sigma}$. Thus ${\sigma}\in E_R$. Next we check that nonempty finite intersections of elements in ${\mathcal}{A}$ are contractible. Let ${\sigma}\in {\mathcal}{N}({\mathcal}{A}) = E_R$. Let $V_{\sigma}:= \cap_{x \in {\sigma}}V(A_x) \subseteq V(F_R)$. We claim that $\cap_{x \in {\sigma}}A_x = {\operatorname{pow}}(V_{\sigma})$, i.e. that the intersection is a full simplex in $F_R$, hence contractible. The inclusion $\cap_{x \in {\sigma}}A_x \subseteq {\operatorname{pow}}(V_{\sigma})$ is clear, so we show the reverse inclusion. Let ${\tau}\in {\operatorname{pow}}(V_{\sigma})$, and let $y \in {\tau}$. Then $y \in \cap_{x \in {\sigma}}A_x$, so $(x,y) \in R$ for each $x \in {\sigma}$. This holds for each $y \in {\tau}$, so it follows that ${\tau}\in \cap_{x\in {\sigma}}A_x$. Thus $\cap_{x \in {\sigma}}A_x = {\operatorname{pow}}(V_{\sigma})$. We remark that this also shows that ${\mathcal}{A}$ is a cover of simplices for $F_R$. Now for each $x \in V(E_{R'})$, define $A'_x:= \{{\tau}\in F_{R'} : (x,y) \in R' \text{ for all } y \in {\tau}\}$. Also define ${\mathcal}{A}':= \{A'_x : x \in V(E_{R'})\}$. The same argument shows that ${\mathcal}{A}'$ is a finite cover of subcomplexes (in particular, a cover of simplices) for $F_{R'}$ with all finite intersections either empty or contractible, and that $E_{R'} = {\mathcal}{N}({\mathcal}{A}')$. An application of Theorem \[thm:nerve-functorial-II\] now shows that $|E_R| \simeq |F_R|$ and $|E_{R'}| \simeq |F_{R'}|$, via maps that commute up to homotopy with the inclusions $|E_R| {\hookrightarrow}|E_{R'}|$ and $|F_R| {\hookrightarrow}|F_{R'}|$. Theorem \[thm:nerve-functorial-I\] implies Theorem \[thm:nerve-functorial-II\] {#theorem-thmnerve-functorial-i-implies-theorem-thmnerve-functorial-ii .unnumbered} ------------------------------------------------------------------------------ We lead with some remarks about the ideas involved in this proof. Theorem \[thm:nerve-functorial-II\] is a functorial statement in the sense that it is about an arbitrary inclusion ${\Sigma}\subseteq {\Sigma}'$. Restricting the statement to just ${\Sigma}$ would lead to a non-functorial statement. A proof of this non-functorial statement, via a non-functorial analogue of Theorem \[thm:nerve-functorial-I\], can be obtained using techniques presented in [@bjorner1985homotopy] (see also [@kozlov2007combinatorial Theorem 15.24]). We strengthen these techniques to our functorial setting and thus obtain a proof of Theorem \[thm:nerve-functorial-II\] via Theorem \[thm:nerve-functorial-I\]. We first present a lemma related to barycentric subdivisions and several lemmas about gluings and homotopy equivalences. These will be used in proving Theorem \[thm:nerve-functorial-II\]. Let ${\Sigma}$ be a simplicial complex, and let $\Delta$ be a subcomplex. Then $\Delta$ is an *induced subcomplex* if $\Delta = {\Sigma}\cap {\operatorname{pow}}(V(\Delta))$. \[lem:induced-cplx\] Let ${\Sigma}$ be a simplicial complex, and let $\Delta$ be a subcomplex. Then $\Delta{^{(1)}}$ is an *induced subcomplex* of ${\Sigma}{^{(1)}}$, i.e. $\Delta{^{(1)}}= {\Sigma}{^{(1)}}\cap {\operatorname{pow}}(V(\Delta{^{(1)}}))$. Let ${\sigma}$ be a simplex of $\Delta{^{(1)}}$. Then ${\sigma}$ belongs to ${\Sigma}{^{(1)}}$, and also to the full simplex ${\operatorname{pow}}(V(\Delta{^{(1)}}))$. Thus $\Delta{^{(1)}}\subseteq {\Sigma}{^{(1)}}\cap {\operatorname{pow}}(V(\Delta{^{(1)}}))$. Conversely, let ${\sigma}\in {\Sigma}{^{(1)}}\cap {\operatorname{pow}}(V(\Delta{^{(1)}}))$. Since ${\sigma}\in {\Sigma}{^{(1)}}$, we can write ${\sigma}= [{\tau}_0,\ldots, {\tau}_k]$, where ${\tau}_0 \subseteq \ldots \subseteq {\tau}_k$. Since ${\sigma}\in {\operatorname{pow}}(V(\Delta{^{(1)}}))$ and the vertices of $\Delta{^{(1)}}$ are simplices of $\Delta$, we also know that each ${\tau}_i$ is a simplex of $\Delta$. Thus ${\sigma}\in \Delta{^{(1)}}$. The equality follows. \[lem:carrier\] Let $X$ be a topological space, and let ${\Sigma}$ be a simplicial complex. Also let $f,g:X {\rightarrow}|{\Sigma}|$ be any two continuous maps such that $f(x), g(x)$ belong to the same simplex of $|{\Sigma}|$, for any $x \in X$. Then $f \simeq g$. \[lem:glue\] Let ${\Sigma}$ be a simplicial complex, and let $U \subseteq V({\Sigma})$. Suppose $|{\Sigma}\cap {\operatorname{pow}}(U)|$ is contractible. Then there exists a homotopy equivalence ${\varphi}: |{\Sigma}\cup {\operatorname{pow}}(U)| {\rightarrow}|{\Sigma}|$. The Gluing and Carrier Lemmas presented above are classical. We provide full details for the Gluing lemma inside the proof of the following functorial generalization of Lemma \[lem:glue\]. \[lem:glue-func\] Let ${\Sigma}\subseteq {\Sigma}'$ be two simplicial complexes. Also let $U \subseteq V({\Sigma})$ and $U' \subseteq V({\Sigma}')$ be such that $U \subseteq U'$. Suppose $|{\Sigma}\cap {\operatorname{pow}}(U)|$ and $|{\Sigma}' \cap {\operatorname{pow}}(U')|$ are contractible. Then, 1. There exists a homotopy equivalence ${\varphi}: |{\Sigma}\cup {\operatorname{pow}}(U)| {\rightarrow}|{\Sigma}|$ such that ${\varphi}(x)$ and ${\operatorname{id}}_{|{\Sigma}\cup {\operatorname{pow}}(U)|}(x)$ belong to the same simplex of $|{\Sigma}\cup {\operatorname{pow}}(U)|$ for each $x \in |{\Sigma}\cup {\operatorname{pow}}(U)|$. Furthermore, the homotopy inverse is given by the inclusion $\iota: |{\Sigma}| {\hookrightarrow}|{\Sigma}\cup {\operatorname{pow}}(U)|$. 2. Given a homotopy equivalence ${\varphi}: |{\Sigma}\cup {\operatorname{pow}}(U)| {\rightarrow}|{\Sigma}|$ as above, there exists a homotopy equivalence ${\varphi}': |{\Sigma}' \cup {\operatorname{pow}}(U')| {\rightarrow}|{\Sigma}'|$ such that ${\varphi}'|_{|{\Sigma}\cup {\operatorname{pow}}(U)|} = {\varphi}$, and ${\varphi}'(x)$ and ${\operatorname{id}}_{|{\Sigma}' \cup {\operatorname{pow}}(U')|}(x)$ belong to the same simplex of $|{\Sigma}'\cup {\operatorname{pow}}(U')|$ for each $x \in |{\Sigma}' \cup {\operatorname{pow}}(U')|$. Furthermore, the homotopy inverse is given by the inclusion $\iota': |{\Sigma}'| {\hookrightarrow}|{\Sigma}' \cup {\operatorname{pow}}(U')|$. The proof uses this fact: any continuous map of an $n$-sphere ${\mathbb{S}}^n$ into a contractible space $Y$ can be continuously extended to a mapping of the $(n+1)$-disk ${\mathbb{D}}^{n+1}$ into $Y$, where ${\mathbb{D}}^{n+1}$ has ${\mathbb{S}}^n$ as its boundary [@spanier-book p. 27]. First we define ${\varphi}$. On $|{\Sigma}|$, define ${\varphi}$ to be the identity. Next let ${\sigma}$ be a minimal simplex in $|{\operatorname{pow}}(U) \setminus {\Sigma}|$. By minimality, the boundary of ${\sigma}$ (denoted ${\operatorname{Bd}}({\sigma})$) belongs to $|{\Sigma}\cap {\operatorname{pow}}(U)|$, and $|{\Sigma}|$ in particular. Thus ${\varphi}$ is defined on ${\operatorname{Bd}}({\sigma})$, which is an $n$-sphere for some $n \geq 0$. Furthermore, ${\varphi}$ maps ${\operatorname{Bd}}({\sigma})$ into the contractible space $|{\Sigma}\cap {\operatorname{pow}}(U)|$. Then we use the aforementioned fact to extend ${\varphi}$ continuously to all of ${\sigma}$ so that ${\varphi}$ maps ${\sigma}$ into $|{\Sigma}\cap {\operatorname{pow}}(U)|$. Furthermore, both ${\operatorname{id}}_{|{\Sigma}\cup {\operatorname{pow}}(U)|}({\sigma}) = {\sigma}$ and ${\varphi}({\sigma})$ belong to the simplex $|{\operatorname{pow}}(U)|$. By iterating this procedure, we obtain a retraction ${\varphi}: |{\Sigma}\cup {\operatorname{pow}}(U)| {\rightarrow}|{\Sigma}|$ such that ${\varphi}(x)$ and $x$ belong to the same simplex in $|{\Sigma}\cup {\operatorname{pow}}(U)|$, for each $x \in |{\Sigma}\cup {\operatorname{pow}}(U)|$. Thus ${\varphi}$ is homotopic to ${\operatorname{id}}_{|{\Sigma}\cup {\operatorname{pow}}(U)|}$ by Lemma \[lem:carrier\]. Thus we have a homotopy equivalence: $${\operatorname{id}}_{|{\Sigma}|}= {\varphi}\circ \iota, \; \iota \circ {\varphi}\simeq {\operatorname{id}}_{|{\Sigma}\cup {\operatorname{pow}}(U)|} \tag{here $\iota:= \iota_{|{\Sigma}| {\hookrightarrow}|{\Sigma}\cup {\operatorname{pow}}(U)|}$} .$$ For the second part of the proof, suppose that a homotopy equivalence ${\varphi}: |{\Sigma}\cup {\operatorname{pow}}(U)| {\rightarrow}|{\Sigma}|$ as above is provided. We need to extend ${\varphi}$ to obtain ${\varphi}'$. Define ${\varphi}'$ to be equal to ${\varphi}$ on $|{\Sigma}\cup {\operatorname{pow}}(U)|$, and equal to the identity on $G:= |{\Sigma}'|\setminus |{\Sigma}\cup {\operatorname{pow}}(U)|$. Let ${\sigma}$ be a minimal simplex in $|{\operatorname{pow}}(U')| \setminus G$. Then by minimality, ${\operatorname{Bd}}({\sigma})$ belongs to $|{\Sigma}' \cap {\operatorname{pow}}(U')|$. As before, we have ${\varphi}'$ mapping ${\operatorname{Bd}}({\sigma})$ into the contractible space $|{\Sigma}' \cap {\operatorname{pow}}(U')|$, and we extend ${\varphi}'$ continuously to a map of ${\sigma}$ into $|{\Sigma}' \cap {\operatorname{pow}}(U')|$. Once again, ${\operatorname{id}}_{|{\Sigma}' \cup {\operatorname{pow}}(U')|}(x)$ and ${\varphi}'(x)$ belong to the same simplex $|{\operatorname{pow}}(U')|$, for all $x \in {\sigma}$. Iterating this procedure gives a continuous map ${\varphi}':|{\Sigma}' \cup {\operatorname{pow}}(U')| {\rightarrow}|{\Sigma}'|$. This map is not necessarily a retraction, because there may be a simplex ${\sigma}\in |{\Sigma}\cup {\operatorname{pow}}(U)| \cap |{\Sigma}'|$ on which ${\varphi}'$ is not the identity. However, it still holds that ${\varphi}'$ is continuous, and that $x, {\varphi}'(x)$ get mapped to the same simplex for each $x \in |{\Sigma}'\cup {\operatorname{pow}}(U')|$. Thus Lemma \[lem:carrier\] still applies to show that ${\varphi}'$ is homotopic to ${\operatorname{id}}_{|{\Sigma}' \cup {\operatorname{pow}}(U')|}$. We write $\iota'$ to denote the inclusion $\iota': |{\Sigma}'| {\hookrightarrow}|{\Sigma}' \cup {\operatorname{pow}}(U')|$. By the preceding work, we have $\iota'\circ {\varphi}' \simeq {\operatorname{id}}_{|{\Sigma}' \cup {\operatorname{pow}}(U')|}$. Next let $x \in |{\Sigma}'|$. Then either $x \in |{\Sigma}'| \cap |{\Sigma}\cup {\operatorname{pow}}(U)|$, or $x \in G$. In the first case, we know that ${\varphi}'(x) = {\varphi}(x)$ and ${\operatorname{id}}_{|{\Sigma}'|}(x) = {\operatorname{id}}_{|{\Sigma}\cup {\operatorname{pow}}(U)|}(x)$ belong to the same simplex of $|{\Sigma}\cup {\operatorname{pow}}(U)|$ by the assumption on ${\varphi}$. In the second case, we know that ${\varphi}'(x) = x = {\operatorname{id}}_{|{\Sigma}'|}(x)$. Thus for any $x \in |{\Sigma}'|$, we know that ${\varphi}'(x)$ and ${\operatorname{id}}_{|{\Sigma}'|}(x)$ belong to the same simplex in $|{\Sigma}' \cup {\operatorname{pow}}(U')|$. By Lemma \[lem:carrier\], we then have ${\varphi}'|_{|{\Sigma}'|} \simeq {\operatorname{id}}_{|{\Sigma}'|}$. Thus ${\varphi}'\circ \iota' \simeq {\operatorname{id}}_{|{\Sigma}'|}$. This shows that ${\varphi}'$ is the necessary homotopy equivalence. Now we present the proof of Theorem \[thm:nerve-functorial-II\]. **Notation.** Let $I$ be an ordered set. For any subset $J\subseteq I$, we write $(J)$ to denote the sequence $(j_1,j_2,j_3,\ldots)$, where the ordering is inherited from the ordering on $I$. The first step is to functorially deform ${\mathcal}{A}_{\Sigma}$ and ${\mathcal}{A}_{{\Sigma}'}$ into covers of simplices while still preserving all associated homotopy types. Then we will be able to apply Theorem \[thm:nerve-functorial-I\]. We can assume by Lemma \[lem:induced-cplx\] that each subcomplex ${\Sigma}_i$ is induced, and likewise for each ${\Sigma}'_i$. We start by fixing an enumeration $I' = \{l_1,l_2,\ldots\}$. Thus $I'$ becomes an ordered set. #### Passing to covers of simplices. {#passing-to-covers-of-simplices. .unnumbered} We now define some inductive constructions. In what follows, we will define complexes denoted ${\Sigma}^\bullet, {\Sigma}'^\bullet$ obtained by “filling in" ${\Sigma}$ and ${\Sigma}'$ while preserving homotopy equivalence, as well as covers of these larger complexes denoted ${\Sigma}_{\star,\bullet}, {\Sigma}'_{\star,\bullet}$. First define: $$\begin{aligned} {\Sigma}^{(l_1)} &:= {\begin{cases} {\Sigma}\cup {\operatorname{pow}}(V({\Sigma}_{l_1})) &: \text{ if }l_1 \in I\\ {\Sigma}&: \text{ otherwise}. \end{cases}}\\ {\Sigma}'^{(l_1)} &:= {\Sigma}' \cup {\operatorname{pow}}(V({\Sigma}'_{l_1})).\end{aligned}$$ Next, for all $i\in I$, define $$\begin{aligned} {\Sigma}_{i,(l_1)} &:= {\begin{cases} {\Sigma}_i \cup {\operatorname{pow}}(V({\Sigma}_i) \cap V({\Sigma}_{l_1}))&: \text{ if }l_1 \in I\\ {\Sigma}_i &: \text{ otherwise}. \end{cases}}\end{aligned}$$ And for all $i \in I'$, define $$\begin{aligned} {\Sigma}'_{i,(l_1)} &:= {\Sigma}'_i \cup {\operatorname{pow}}(V({\Sigma}'_i) \cap V({\Sigma}'_{l_1})).\end{aligned}$$ Now by induction, suppose ${\Sigma}^{(l_1,\ldots, l_n)}$ and ${\Sigma}_{i,(l_1,\ldots, l_n)}$ are defined for all $i \in I$. Also suppose ${\Sigma}'^{(l_1,\ldots, l_n)}$ and ${\Sigma}'_{i,(l_1,\ldots, l_n)}$ are defined for all $i \in I'$. Then we define: $$\begin{aligned} {\Sigma}^{(l_1,\ldots, l_n,l_{n+1})} &:= {\begin{cases} {\Sigma}^{(l_1,\ldots,l_n)} \cup {\operatorname{pow}}(V({\Sigma}_{l_{n+1},(l_1,\ldots,l_n)})) &: \text{ if }l_{n+1} \in I\\ {\Sigma}^{(l_1,\ldots,l_n)} &: \text{ otherwise}. \end{cases}}\\ {\Sigma}'^{(l_1,\ldots, l_n,l_{n+1})} &:= {\Sigma}'^{(l_1,\ldots,l_n)} \cup {\operatorname{pow}}(V({\Sigma}'_{l_{n+1},(l_1,\ldots,l_n)})).\end{aligned}$$ For all $i \in I$, we have $$\begin{aligned} {\Sigma}_{i,(l_1,l_2,\ldots, l_{n+1})} &:= {\begin{cases} {\Sigma}_{i,(l_1,l_2,\ldots, l_n)} \cup {\operatorname{pow}}(V({\Sigma}_{i,(l_1,l_2,\ldots, l_n)}) \cap V({\Sigma}_{l_{n+1},(l_1,l_2,\ldots, l_n)}))&: \text{ if }l_{n+1} \in I\\ {\Sigma}_{i,(l_1,l_2,\ldots, l_n)} &: \text{ otherwise}. \end{cases}}\end{aligned}$$ And for all $i \in I'$, we have $$\begin{aligned} {\Sigma}'_{i,(l_1,l_2,\ldots, l_{n+1})} &:= {\Sigma}'_{i,(l_1,l_2,\ldots, l_n)} \cup {\operatorname{pow}}(V({\Sigma}'_{i,(l_1,l_2,\ldots, l_n)}) \cap V({\Sigma}'_{l_{n+1},(l_1,l_2,\ldots, l_n)})).\end{aligned}$$ Finally, for any $n \leq {\operatorname{card}}(I')$, we define ${\mathcal}{A}_{{\Sigma},(l_1,\ldots, l_n)} :=\{{\Sigma}_{i,(l_1,\ldots, l_{n})} : i \in I\}$ and ${\mathcal}{A}_{{\Sigma}',(l_1,\ldots, l_n)} :=\{{\Sigma}'_{i,(l_1,\ldots, l_{n})} : i \in I'\}$. We will show that these are covers of ${\Sigma}^{(l_1,l_2,\ldots, l_n)}$ and ${\Sigma}'^{(l_1,l_2,\ldots, l_n)}$, respectively. The next step is to prove by induction that for any $n \leq {\operatorname{card}}(I')$, we have $|{\Sigma}|\simeq |{\Sigma}^{(l_1,l_2,\ldots, l_n)}|$ and $|{\Sigma}'|\simeq |{\Sigma}'^{(l_1,l_2,\ldots, l_n)}|$, that ${\mathcal}{N}({\mathcal}{A}_{\Sigma}) = {\mathcal}{N}({\mathcal}{A}_{{\Sigma},(l_1,l_2,\ldots, l_n)})$ and ${\mathcal}{N}({\mathcal}{A}_{{\Sigma}'}) = {\mathcal}{N}({\mathcal}{A}_{{\Sigma}',(l_1,l_2,\ldots, l_n)})$, and that nonempty finite intersections of the new covers ${\mathcal}{A}_{{\Sigma},(l_1,l_2,\ldots, l_n)}, {\mathcal}{A}_{{\Sigma}',(l_1,l_2,\ldots, l_n)}$ remain contractible. For the base case $n=0$, we have ${\Sigma}= {\Sigma}^{()}$, ${\Sigma}' = {\Sigma}'^{()}$. Thus the base case is true by assumption. We present the inductive step next. \[cl:nerve-cover-of-simplices\] For this claim, let $\bullet$ denote $l_1,\ldots, l_n$, where $0 < n < {\operatorname{card}}(I')$. Define $l:= l_{n+1}$. Suppose the following is true: 1. The collections ${\mathcal}{A}_{{\Sigma},(\bullet)}$ and ${\mathcal}{A}_{{\Sigma}',(\bullet)}$ are covers of ${\Sigma}^{(\bullet)}$ and ${\Sigma}'^{(\bullet)}$. 2. The nerves of the coverings are unchanged: ${\mathcal}{N}({\mathcal}{A}_{\Sigma}) = {\mathcal}{N}({\mathcal}{A}_{{\Sigma},(\bullet)})$ and ${\mathcal}{N}({\mathcal}{A}_{{\Sigma}'}) = {\mathcal}{N}({\mathcal}{A}_{{\Sigma}',(\bullet)})$. 3. Each of the subcomplexes ${\Sigma}_{i,(\bullet)}$, $i\in I$, and ${\Sigma}'_{j,(\bullet)}$, $j\in I'$ is induced in ${\Sigma}^{(\bullet)}$ and ${\Sigma}'^{(\bullet)}$, respectively. 4. Let ${\sigma}\subseteq I$. If $\cap_{i\in {\sigma}}{\Sigma}_{i,(\bullet)}$ is nonempty, then it is contractible. Similarly, let ${\tau}\subseteq I'$. If $\cap_{i\in {\tau}}{\Sigma}'_{i,(\bullet)}$ is nonempty, then it is contractible. 5. We have homotopy equivalences $|{\Sigma}|\simeq |{\Sigma}^{(\bullet)}|$ and $|{\Sigma}'|\simeq |{\Sigma}'^{(\bullet)}|$ via maps that commute with the canonical inclusions. Then the preceding statements are true for ${\Sigma}^{(\bullet,l)}$, ${\Sigma}'^{(\bullet,l)}$, ${\mathcal}{A}_{{\Sigma},(\bullet,l)}$, and ${\mathcal}{A}_{{\Sigma}',(\bullet,l)}$ as well. For the first claim, we have ${\Sigma}^{(\bullet, l)} = {\Sigma}^{(\bullet)} \cup {\operatorname{pow}}(V({\Sigma}_{l,(\bullet)})) \subseteq \cup_{i \in I}{\Sigma}_{i,(\bullet, l)}$. For the inclusion, we used the inductive assumption that ${\Sigma}^{(\bullet)} = \cup_{i \in I}{\Sigma}_{i,(\bullet)}$. Similarly, ${\Sigma}'^{(\bullet, l)} \subseteq \cup_{i \in I'} {\Sigma}'_{i,(\bullet,l)}$. For the second claim, let $i \in I$. Then $V({\Sigma}_{i,(l_1)}) = V({\Sigma}_i)$, and in particular, we have $V({\Sigma}_{i,(\bullet,l)}) = V({\Sigma}_{i,(\bullet)}) = V({\Sigma}_i)$. Next observe that for any ${\sigma}\subseteq I$, the intersection $$\cap_{i \in {\sigma}}{\Sigma}_i \neq {\varnothing}\iff \cap_{i \in {\sigma}}V({\Sigma}_i) \neq {\varnothing}\iff \cap_{i \in {\sigma}}V({\Sigma}_{i,(\bullet,l)}) \neq {\varnothing}\iff \cap_{i \in {\sigma}}{\Sigma}_{i,(\bullet,l)} \neq {\varnothing}.$$ Thus ${\mathcal}{N}({\mathcal}{A}_{\Sigma}) = {\mathcal}{N}({\mathcal}{A}_{{\Sigma},(\bullet)}) = {\mathcal}{N}({\mathcal}{A}_{{\Sigma},(\bullet,l)})$, and similarly ${\mathcal}{N}({\mathcal}{A}_{{\Sigma}'}) = {\mathcal}{N}({\mathcal}{A}_{{\Sigma}',(\bullet)}) = {\mathcal}{N}({\mathcal}{A}_{{\Sigma}',(\bullet,l)})$. For the third claim, again let $i \in I$. If $l \not\in I$, then ${\Sigma}_{i,(\bullet,l)} = {\Sigma}_{i,(\bullet)}$, so we are done by the inductive assumption. Suppose $l \in I$. Since ${\Sigma}_{i,(\bullet)}$ is induced by the inductive assumption, we have: $$\begin{aligned} {\Sigma}_{i,(\bullet,l)} &= {\Sigma}_{i,(\bullet)} \cup ({\operatorname{pow}}(V({\Sigma}_{i,(\bullet)})\cap V({\Sigma}_{l,(\bullet)})))\\ &= ({\Sigma}^{(\bullet)} \cap {\operatorname{pow}}(V({\Sigma}_{i,(\bullet)}))) \cup ({\operatorname{pow}}(V({\Sigma}_{i,(\bullet)})) \cap {\operatorname{pow}}(V({\Sigma}_{l,(\bullet)})))\\ & = ({\Sigma}^{(\bullet)} \cup {\operatorname{pow}}(V({\Sigma}_{l,(\bullet)}))) \cap {\operatorname{pow}}(V({\Sigma}_{i,(\bullet)}))\\ &= {\Sigma}^{(\bullet, l)} \cap {\operatorname{pow}}(V({\Sigma}_{i,(\bullet)})) = {\Sigma}^{(l)} \cap {\operatorname{pow}}(V({\Sigma}_{i,(\bullet,l)})).\end{aligned}$$ Thus ${\Sigma}_{i,(\bullet,l)}$ is induced. The same argument holds for the $I'$ case. For the fourth claim, let ${\sigma}\subseteq I$, and suppose $\cap_{i \in {\sigma}} {\Sigma}_{i,(\bullet,l)}$ is nonempty. By the previous claim, each ${\Sigma}_{i,(\bullet,l)}$ is induced. Thus we write: $$\begin{aligned} \cap_{i\in {\sigma}}{\Sigma}_{i,(\bullet,l)} &= {\Sigma}^{(\bullet,l)} \cap {\operatorname{pow}}(\cap_{i \in {\sigma}}V({\Sigma}_{i,(\bullet,l)})) \\ &= {\Sigma}^{(\bullet,l)} \cap {\operatorname{pow}}(\cap_{i \in {\sigma}}V({\Sigma}_{i,(\bullet)}))\\ &= ({\Sigma}^{(\bullet)} \cup {\operatorname{pow}}(V({\Sigma}_{l,(\bullet)}))) \cap {\operatorname{pow}}(\cap_{i \in {\sigma}}V({\Sigma}_{i,(\bullet)}))\\ &= (\cap_{i \in {\sigma}} ({\Sigma}^{(\bullet)} \cap {\operatorname{pow}}(V({\Sigma}_{i,(\bullet)}))) ) \cup {\operatorname{pow}}(\cap_{i \in {\sigma}}V({\Sigma}_{i,(\bullet)}) \cap V({\Sigma}_{l,(\bullet)}))\\ &= (\cap_{i \in {\sigma}} {\Sigma}_{i,(\bullet)} ) \cup {\operatorname{pow}}(\cap_{i \in {\sigma}}V({\Sigma}_{i,(\bullet)}) \cap V({\Sigma}_{l,(\bullet)})).\end{aligned}$$ For convenience, define $A:=(\cap_{i \in {\sigma}} {\Sigma}_{i,(\bullet)} )$ and $B:= {\operatorname{pow}}(\cap_{i \in {\sigma}}V({\Sigma}_{i,(\bullet)}) \cap V({\Sigma}_{l,(\bullet)}))$. Then $|A|$ is contractible by inductive assumption, and $|B|$ is a full simplex, hence contractible. Also, $A\cap B$ has the form $$\begin{aligned} &(\cap_{i \in {\sigma}} ({\Sigma}^{(\bullet)} \cap {\operatorname{pow}}(V({\Sigma}_{i,(\bullet)}))) ) \cap {\operatorname{pow}}(\cap_{i \in {\sigma}}V({\Sigma}_{i,(\bullet)}) \cap V({\Sigma}_{l,(\bullet)})) \\ =& {\Sigma}^{(\bullet)} \cap {\operatorname{pow}}(\cap_{i \in {\sigma}}V({\Sigma}_{i,(\bullet)}) \cap V({\Sigma}_{l,(\bullet)})) \\ =& \cap_{i \in {\sigma}}{\Sigma}_{i,(\bullet)} \cap {\Sigma}_{l,(\bullet)},\end{aligned}$$ and the latter is contractible by inductive assumption. Thus by Lemma \[lem:glue\], we have $|A\cup B|$ contractible. This proves the claim for the case ${\sigma}\subseteq I$. The case ${\tau}\subseteq I'$ is similar. Now we proceed to the final claim. Since ${\Sigma}_{l,(\bullet)}$ is induced, we have ${\Sigma}_{l,(\bullet)} = {\Sigma}^{(\bullet)} \cap {\operatorname{pow}}(V({\Sigma}_{l,(\bullet)}))$. By the contractibility assumption, we know that $|{\Sigma}_{l,(\bullet)}|$ is contractible. Also we know that $|{\Sigma}'_{l,(\bullet)}| = |{\Sigma}'^{(\bullet)} \cap {\operatorname{pow}}(V({\Sigma}'_{l,(\bullet)}))|$ is contractible. By assumption we also have $V({\Sigma}_{l,(\bullet)}) \subseteq V({\Sigma}'_{l,(\bullet)})$. Thus by Lemma \[lem:glue-func\], we obtain homotopy equivalences $\Phi_l : |{\Sigma}^{(\bullet,l)}| {\rightarrow}|{\Sigma}^{(\bullet)}|$ and $\Phi'_l : |{\Sigma}'^{(\bullet,l)}| {\rightarrow}|{\Sigma}'^{(\bullet)}|$ such that $\Phi'_l$ extends $\Phi_l$. Furthermore, the homotopy inverses of $\Phi_l$ and $\Phi'_l$ are just the inclusions $|{\Sigma}^{(\bullet)}| {\hookrightarrow}|{\Sigma}^{(\bullet,l)}|$ and $|{\Sigma}'^{(\bullet)}| {\hookrightarrow}|{\Sigma}'^{(\bullet,l)}|$. Now let $\iota: |{\Sigma}^{(\bullet)}| {\rightarrow}|{\Sigma}'^{(\bullet)}|$ and $\iota_l: |{\Sigma}^{(\bullet,l)}| {\rightarrow}|{\Sigma}'^{(\bullet,l)}|$ denote the canonical inclusions. We wish to show the equality $\Phi'_l\circ \iota_l = \iota\circ \Phi_l$. Let $x \in|{\Sigma}^{(\bullet,l)}|$. Because $\Phi'_l$ extends $\Phi_l$ (this is why we needed the *functorial* gluing lemma), we have $$\Phi'_l(\iota_l(x))) = \Phi'_l(x) = \Phi_l(x) = \iota(\Phi_l(x)).$$ Since $x \in|{\Sigma}^{(\bullet,l)}|$ was arbitrary, the equality follows immediately. By the inductive assumption, we already have homotopy equivalences $|{\Sigma}^{(\bullet)}| {\rightarrow}|{\Sigma}|$ and $|{\Sigma}'^{(\bullet)}| {\rightarrow}|{\Sigma}'|$ that commute with the canonical inclusions. Composing these maps with $\Phi_l$ and $\Phi'_l$ completes the proof of the claim. By the preceding work, we replace the subcomplexes ${\Sigma}_l, {\Sigma}'_l$ by full simplices of the form ${\Sigma}_{l,(\bullet,l)},{\Sigma}_{l,(\bullet,l)}'$. In this process, the nerves remain unchanged and the complexes ${\Sigma}, {\Sigma}'$ are replaced by homotopy equivalent complexes ${\Sigma}^{(\bullet,l)}, {\Sigma}'^{(\bullet,l)}$. Furthermore, this process is functorial—the homotopy equivalences commute with the canonical inclusions ${\Sigma}{\hookrightarrow}{\Sigma}^{(\bullet,l)}$ and ${\Sigma}' {\hookrightarrow}{\Sigma}'^{(\bullet,l)}$. Repeating the inductive process in Claim \[cl:nerve-cover-of-simplices\] for all the finitely many $l \in I$ yields a simplicial complex ${\Sigma}^{(I)}$ along with a cover of simplices ${\mathcal}{A}_{{\Sigma},(I)}$. We also perform the same procedure for all $l \in I' \setminus I$ (this does not affect ${\Sigma}^{(I)}$) to obtain a simplicial complex ${\Sigma}'^{(I')}$ along with a cover of simplices ${\mathcal}{A}_{{\Sigma}',(I')}$. Furthermore, ${\Sigma}^{(I)}$ and ${\Sigma}'^{(I)}$ are related to ${\Sigma}$ and ${\Sigma}'$ by a finite sequence of homotopy equivalences that commute with the canonical inclusions. Also, we have ${\mathcal}{N}({\mathcal}{A}_{{\Sigma}}) = {\mathcal}{N}({\mathcal}{A}_{{\Sigma},(I)})$ and ${\mathcal}{N}({\mathcal}{A}_{{\Sigma}'}) = {\mathcal}{N}({\mathcal}{A}_{{\Sigma}',(I')})$. Thus we obtain the following picture: (SI’) at (3,0)[$|{\Sigma}'^{(I')}|$]{}; (00) at (0,0)[$\cdots$]{}; (1) at (-3,0)[$|{\Sigma}'^{(l_1)}|$]{}; (2) at (-6,0)[$|{\Sigma}'|$]{}; (3) at (9,0)[$|{\mathcal}{N}({\mathcal}{A}_{{\Sigma}'})|$]{}; (01) at (6,0)[$|{\mathcal}{N}({\mathcal}{A}_{{\Sigma}',(I')})|$]{}; (SI) at (3,2)[$|{\Sigma}^{(I)}|$]{}; (02) at (0,2)[$\cdots$]{}; (4) at (-3,2)[$|{\Sigma}^{(l_1)}|$]{}; (5) at (-6,2)[$|{\Sigma}|$]{}; (6) at (9,2)[$|{\mathcal}{N}({\mathcal}{A}_{{\Sigma}})|$]{}; (03) at (6,2)[$|{\mathcal}{N}({\mathcal}{A}_{{\Sigma},(I)})|$]{}; (SI) edge\[-&gt;\]node\[above\][$\simeq$]{} (02); (SI’) edge\[-&gt;\]node\[above\][$\simeq$]{} (00); (00) edge\[-&gt;\] (1); (3) edge\[double distance=2pt\] (01); (02) edge\[-&gt;\] (4); (6) edge\[double distance=2pt\] (03); \(1) edge\[-&gt;\]node\[above\][$\simeq$]{} (2); (00) edge\[-&gt;\]node\[above\][$\simeq$]{} (1); (4) edge\[-&gt;\]node\[above\][$\simeq$]{} (5); (02) edge\[-&gt;\]node\[above\][$\simeq$]{} (4); (1) edge\[&lt;-\] node\[right\][$\iota_{(l_1)}$]{} (4); (2) edge\[&lt;-\] node\[right\][$\iota$]{} (5); (3) edge\[&lt;-\] node\[right\][$\iota_{{\mathcal}{N}}$]{} (6); (01) edge\[&lt;-\] node\[right\][$\iota_{{\mathcal}{N},(I)}$]{} (03); (SI) edge\[-&gt;\]node\[right\][$\iota_{(I)}$]{} (SI’); (SI) edge\[-&gt;\]node\[above\][$\simeq$]{} (03); (SI’) edge\[-&gt;\]node\[above\][$\simeq$]{} (01); By applying Theorem \[thm:nerve-functorial-I\] to the block consisting of $|{\Sigma}^{(I)}|$, $|{\Sigma}'^{(I')}|$, $|{\mathcal}{N}({\mathcal}{A}_{{\Sigma},I})|$ and $|{\mathcal}{N}({\mathcal}{A}_{{\Sigma}',I'})|$, we obtain a square that commutes up to homotopy. Then by composing the homotopy equivalences constructed above, we obtain a square consisting of $|{\Sigma}|$, $|{\Sigma}'|$, $|{\mathcal}{N}({\mathcal}{A}_{{\Sigma}})|$, and $|{\mathcal}{N}({\mathcal}{A}_{{\Sigma}'})|$ that commutes up to homotopy. Thus we obtain homotopy equivalences $|{\Sigma}| \simeq |{\mathcal}{N}({\mathcal}{A}_{\Sigma})|$ and $|{\Sigma}'| \simeq |{\mathcal}{N}({\mathcal}{A}_{{\Sigma}'})|$ via maps that commute up to homotopy with the canonical inclusions. Dowker persistence diagrams and asymmetry {#sec:symmetry} ========================================= From the very definition of the Rips complex at any given resolution, one can see that the Rips complex is blind to asymmetry in the input data (Remark \[rem:rips-symm\]). In this section, we argue that either of the Dowker source and sink complexes is sensitive to asymmetry. Thus when analyzing datasets containing asymmetric information, one may wish to use the Dowker filtration instead of the Rips filtration. In particular, this property suggests that the Dowker persistence diagram is a stronger invariant for directed networks than the Rips persistence diagram. In this section, we consider a family of examples, called *cycle networks*, for which the Dowker persistence diagrams capture meaningful structure, whereas the Rips persistence diagrams do not. We then probe the question “What happens to the Dowker or Rips persistence diagram of a network upon reversal of one (or more) edges?" Intuitively, if either of these persistence diagrams captures asymmetry, we would see a change in the diagram after applying this reversal operation to an edge. Cycle networks {#sec:cycle} -------------- For each $n\in {\mathbb{N}}$, let $(X_n,E_n,W_{E_n})$ denote the weighted graph with vertex set $X_n:={\left\{x_1,x_2,\ldots,x_n\right\}}$, edge set $E_n:={\left\{(x_1,x_2),(x_2,x_3),\ldots,(x_{n-1},x_n),(x_n,x_1)\right\}}$, and edge weights $W_{E_n}:E_n {\rightarrow}{\mathbb{R}}$ given by writing $W_{E_n}(e)=1$ for each $e\in E_n$. Next let ${\omega}_{G_n}:X_n\times X_n {\rightarrow}{\mathbb{R}}$ denote the shortest path distance induced on $X_n\times X_n$ by $W_{E_n}$. Then we write $G_n:=(X_n,{\omega}_{G_n})$ to denote the network with node set $X_n$ and weights given by ${\omega}_{G_n}$. Note that ${\omega}_{G_n}(x,x)=0$ for each $x\in X_n$. See Figure \[fig:cycle\] for some examples. We say that $G_n$ is the *cycle network of length n*. One can interpret cycle networks as being highly asymmetric, because for every consecutive pair of nodes $(x_i,x_{i+1})$ in a graph $G_n$, where $1\leq i\mod(n)\leq n$, we have ${\omega}_{G_n}(x_i,x_{i+1})=1$, whereas ${\omega}_{G_n}(x_{i+1},x_i)={\operatorname{diam}}(G_n) = n-1$, which is much larger than 1 when $n$ is large. To provide further evidence that Dowker persistence is sensitive to asymmetry, we computed both the Rips and Dowker persistence diagrams, in dimensions 0 and 1, of cycle networks $G_n$, for values of $n$ between 3 and 6. Computations were carried out using `Javaplex` in Matlab with ${\mathbb{Z}}_2$ coefficients. The results are presented in Figure \[fig:cycle\]. Based on our computations, we were able to conjecture and prove the result in Theorem \[thm:cycleH1\], which gives a precise characterization of the 1-dimensional Dowker persistence diagram of a cycle network $G_n$, for any $n$. Furthermore, the 1-dimensional Dowker persistence barcode for any $G_n$ contains only one persistent interval, which agrees with our intuition that there is only one nontrivial loop in $G_n$. On the other hand, for large $n$, the 1-dimensional Rips persistence barcodes contain more than one persistent interval. This can be seen in the Rips persistence barcode of $G_6$, presented in Figure \[fig:cycle\]. Moreover, for $n=3,4$, the 1-dimensional Rips persistence barcode does not contain any persistent interval at all. This suggests that Dowker persistence diagrams/barcodes are an appropriate method for analyzing cycle networks, and perhaps asymmetric networks in general. [0.3]{} \(1) at (0,1.5)[$x_1$]{}; (2) at (-1,0)[$x_2$]{}; (3) at (1,0)[$x_3$]{}; (4) at (0,-1); \(1) edge \[bend right\] node\[above,pos=0.5\][$1$]{} (2); (2) edge \[bend right\] node\[above,pos=0.5\][$1$]{} (3); (3) edge \[bend right\] node\[above,pos=0.5\][$1$]{} (1); [0.33]{} ![The first column contains illustrations of cycle networks $G_3,G_4,G_5$ and $G_6$. The second column contains the corresponding Dowker persistence barcodes, in dimensions 0 and 1. Note that the persistent intervals in the 1-dimensional barcodes agree with the result in Theorem \[thm:cycleH1\]. The third column contains the Rips persistence barcodes of each of the cycle networks. Note that for $n=3,4$, there are no persistent intervals in dimension 1. On the other hand, for $n=6$, there are two persistent intervals in dimension 1. []{data-label="fig:cycle"}](dowker-G3.png "fig:"){width="\linewidth"} [0.33]{} ![The first column contains illustrations of cycle networks $G_3,G_4,G_5$ and $G_6$. The second column contains the corresponding Dowker persistence barcodes, in dimensions 0 and 1. Note that the persistent intervals in the 1-dimensional barcodes agree with the result in Theorem \[thm:cycleH1\]. The third column contains the Rips persistence barcodes of each of the cycle networks. Note that for $n=3,4$, there are no persistent intervals in dimension 1. On the other hand, for $n=6$, there are two persistent intervals in dimension 1. []{data-label="fig:cycle"}](rips-G3.png "fig:"){width="\linewidth"} [0.3]{} \(1) at (0,1.5)[$x_1$]{}; (2) at (-1.5,0)[$x_2$]{}; (3) at (0,-1.5,0)[$x_3$]{}; (4) at (1.5,0)[$x_4$]{}; (5) at (0,-2); \(1) edge \[bend right\] node\[above,pos=0.5\][$1$]{} (2); (2) edge \[bend right\] node\[above,pos=0.5\][$1$]{} (3); (3) edge \[bend right\] node\[above,pos=0.5\][$1$]{} (4); (4) edge \[bend right\] node\[above,pos=0.5\][$1$]{} (1); [0.33]{} ![The first column contains illustrations of cycle networks $G_3,G_4,G_5$ and $G_6$. The second column contains the corresponding Dowker persistence barcodes, in dimensions 0 and 1. Note that the persistent intervals in the 1-dimensional barcodes agree with the result in Theorem \[thm:cycleH1\]. The third column contains the Rips persistence barcodes of each of the cycle networks. Note that for $n=3,4$, there are no persistent intervals in dimension 1. On the other hand, for $n=6$, there are two persistent intervals in dimension 1. []{data-label="fig:cycle"}](dowker-G4.png "fig:"){width="\linewidth"} [0.33]{} ![The first column contains illustrations of cycle networks $G_3,G_4,G_5$ and $G_6$. The second column contains the corresponding Dowker persistence barcodes, in dimensions 0 and 1. Note that the persistent intervals in the 1-dimensional barcodes agree with the result in Theorem \[thm:cycleH1\]. The third column contains the Rips persistence barcodes of each of the cycle networks. Note that for $n=3,4$, there are no persistent intervals in dimension 1. On the other hand, for $n=6$, there are two persistent intervals in dimension 1. []{data-label="fig:cycle"}](rips-G4.png "fig:"){width="\linewidth"} [0.3]{} \(1) at (0,1.5)[$x_1$]{}; (2) at (1.5,0)[$x_2$]{}; (3) at (1,-1.75)[$x_3$]{}; (4) at (-1,-1.75)[$x_4$]{}; (5) at (-1.5,0)[$x_5$]{}; (6) at (0,-2); \(1) edge node\[above,pos=0.5\][$1$]{} (2); (2) edge node\[left,pos=0.5\][$1$]{} (3); (3) edge node\[above,pos=0.5\][$1$]{} (4); (4) edge node\[right,pos=0.5\][$1$]{} (5); (5) edge node\[above,pos=0.5\][$1$]{} (1); [0.33]{} ![The first column contains illustrations of cycle networks $G_3,G_4,G_5$ and $G_6$. The second column contains the corresponding Dowker persistence barcodes, in dimensions 0 and 1. Note that the persistent intervals in the 1-dimensional barcodes agree with the result in Theorem \[thm:cycleH1\]. The third column contains the Rips persistence barcodes of each of the cycle networks. Note that for $n=3,4$, there are no persistent intervals in dimension 1. On the other hand, for $n=6$, there are two persistent intervals in dimension 1. []{data-label="fig:cycle"}](dowker-G5.png "fig:"){width="\linewidth"} [0.33]{} ![The first column contains illustrations of cycle networks $G_3,G_4,G_5$ and $G_6$. The second column contains the corresponding Dowker persistence barcodes, in dimensions 0 and 1. Note that the persistent intervals in the 1-dimensional barcodes agree with the result in Theorem \[thm:cycleH1\]. The third column contains the Rips persistence barcodes of each of the cycle networks. Note that for $n=3,4$, there are no persistent intervals in dimension 1. On the other hand, for $n=6$, there are two persistent intervals in dimension 1. []{data-label="fig:cycle"}](rips-G5.png "fig:"){width="\linewidth"} [0.3]{} \(1) at (-1,1.5)[$x_1$]{}; (2) at (1,1.5)[$x_2$]{}; (3) at (1.5,0)[$x_3$]{}; (4) at (1,-1.5)[$x_4$]{}; (5) at (-1,-1.5)[$x_5$]{}; (6) at (-1.5,0)[$x_6$]{}; (7) at (0,-2); \(1) edge node\[above,pos=0.5\][$1$]{} (2); (2) edge node\[left,pos=0.5\][$1$]{} (3); (3) edge node\[left,pos=0.5\][$1$]{} (4); (4) edge node\[above,pos=0.5\][$1$]{} (5); (5) edge node\[right,pos=0.5\][$1$]{} (6); (6) edge node\[right,pos=0.5\][$1$]{} (1); [0.33]{} ![The first column contains illustrations of cycle networks $G_3,G_4,G_5$ and $G_6$. The second column contains the corresponding Dowker persistence barcodes, in dimensions 0 and 1. Note that the persistent intervals in the 1-dimensional barcodes agree with the result in Theorem \[thm:cycleH1\]. The third column contains the Rips persistence barcodes of each of the cycle networks. Note that for $n=3,4$, there are no persistent intervals in dimension 1. On the other hand, for $n=6$, there are two persistent intervals in dimension 1. []{data-label="fig:cycle"}](dowker-G6.png "fig:"){width="\linewidth"} [0.33]{} ![The first column contains illustrations of cycle networks $G_3,G_4,G_5$ and $G_6$. The second column contains the corresponding Dowker persistence barcodes, in dimensions 0 and 1. Note that the persistent intervals in the 1-dimensional barcodes agree with the result in Theorem \[thm:cycleH1\]. The third column contains the Rips persistence barcodes of each of the cycle networks. Note that for $n=3,4$, there are no persistent intervals in dimension 1. On the other hand, for $n=6$, there are two persistent intervals in dimension 1. []{data-label="fig:cycle"}](rips-G6.png "fig:"){width="\linewidth"} **Notation.** In the remainder of this section, we will prove results involving Dowker sink complexes of the cycle networks $G_n$ and associated vector spaces at a range of resolutions ${\delta}$. For convenience, we will write ${{\mathfrak}{D}^{\operatorname{si}}}_{\delta}:= {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},G_n}$ (where $n$ will be fixed) and $C_k^{\delta}:= C_k({{\mathfrak}{D}^{\operatorname{si}}}_{\delta})$, the $k$-chain vector space associated to ${{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$ for each $k\in {\mathbb{Z}}_+$. For each $k\in {\mathbb{Z}}_+$, the boundary map from $C_k^{\delta}$ to $C_{k-1}^{\delta}$ will be denoted ${\partial}_k^{\delta}$. Whenever we write $x_i$ to denote a vertex of $G_n$, the subscript $i$ should be understood as $i{\ (\mbox{mod}\ n)}$. We write $e_i$ to denote the 1-simplex $[x_i,x_{i+1}]$ for each $1\leq i \leq n$, where $x_{n+1}$ is understood to be $x_1$. Given an element ${\gamma}\in \ker({\partial}_k^{\delta})\subseteq C_1^{\delta}$, we will write ${\langle}{\gamma}{\rangle}_{\delta}$ to denote its equivalence class in the quotient vector space $\ker({\partial}_k^{\delta})/{\operatorname{im}}({\partial}^{\delta}_k)$. We will refer to the operation of taking this quotient as *passing to homology*. The following theorem contains the characterization result for 1-dimensional Dowker persistence diagrams of cycle networks. \[thm:cycleH1\] Let $G_n=(X_n,{\omega}_{G_n})$ be a cycle network for some $n\in {\mathbb{N}}$, $n\geq 3$. Then we obtain: $${\operatorname{Dgm}}^{{\mathfrak}{D}}_1(G_n)={\left\{(1,{\left \lceil{n/2}\right \rceil })\in {\mathbb{R}}^2\right\}}.$$ Thus ${\operatorname{Dgm}}^{{\mathfrak}{D}}_1(G_n)$ consists of precisely the point $(1,{\left \lceil{n/2}\right \rceil })\in {\mathbb{R}}^2$ with multiplicity 1. The proof occurs in three stages: first we show that a 1-cycle appears at ${\delta}=1$, next we show that this 1-cycle does not become a boundary until ${\delta}={\left \lceil{n/2}\right \rceil }$, and finally that any other 1-cycle belongs to the same equivalence class upon passing to homology (this shows that the single point in the persistence diagram has multiplicity 1). Note that for ${\delta}<1$, there are no 1-simplices in ${{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$, and so $H_1({{\mathfrak}{D}^{\operatorname{si}}}_{\delta})$ is trivial. Suppose $1\leq {\delta}< 2$. \[cl:cycle-2simp\] There are no 2-simplices in ${{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$ for $1\leq {\delta}< 2$. To see this, let $x_i,x_j,x_k$ be any three distinct vertices in $X_n$. Assume towards a contradiction that there exists $x\in X_n$ such that $(x_i,x),(x_j,x),(x_k,x)\in R_{{\delta},X_n},$ where $R_{{\delta},X_n}$ is as given by Equation \[eq:relation\]. Thus ${\omega}_{G_n}(x_i,x)\in \{0,1\}$, so either $x=x_i$ or $x=x_{i+1}$. Similarly we get that $x=x_j$ or $x=x_{j+1}$, and that $x=x_k$ or $x=x_{k+1}$. But this is a contradiction, since $x_i,x_j,x_k$ are all distinct. By the claim, there are no 2-simplices in ${{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$, so ${\operatorname{im}}({\partial}^{\delta}_2)$ is trivial and the only 1-chains are linear combinations of $e_i$ terms. Next, we define: $$v_n:=e_1 + e_2 + \ldots + e_n=[x_1,x_2]+[x_2,x_3]+ \ldots + [x_n,x_1].$$ Note that $v_n\in C_1^{\delta}$ for all ${\delta}\geq 1$. One can further verify that ${\partial}^{\delta}_1(v_n) = 0$, for any ${\delta}\geq 1$. In other words, $v_n$ is a 1-cycle for any ${\delta}\geq 1$. \[cl:cycle-v\] Let $1\leq {\delta}< 2$. Then $v_n$ generates $\ker({\partial}^{\delta}_1)\subseteq C_1^{\delta}$. The only 1-simplices in ${{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$ are of the form $e_i$, for $1\leq i \leq n$. So it suffices to show that any linear combination of the $e_i$ terms is a multiple of $v_n$. Let $u = \sum_{i=1}^na_ie_i \in \ker({\partial}^{\delta}_1)$, for some $a_1,\ldots, a_n \in \mathbb{K}$. Then, $$\begin{aligned} 0={\partial}^{\delta}_1(u)=\sum_{i=1}^na_i{\partial}^{\delta}_1(e_i) &= \sum_{i=1}^na_i([x_{i+1}]-[x_i])\\ &=\sum_{i=1}^n(a_{i-1}-a_i)[x_i], &&\text{where $x_0$ is understood to be $x_n$.}\end{aligned}$$ Since all the $[x_i]$ are linearly independent, it follows that $a_1=a_2=\ldots=a_n$. Thus it follows that $u$ is a constant multiple of $v_n$. This proves the claim. By the two preceding claims, it follows that ${\left\{{\langle}v_n {\rangle}_{\delta}\right\}}$ is a basis for $H_1({{\mathfrak}{D}^{\operatorname{si}}}_{\delta})$, for ${\delta}\in [1,2)$. More specifically, ${\langle}v_n {\rangle}_{\delta}$ is a cycle that appears at ${\delta}=1$ and does not become a boundary until at least ${\delta}=2$, and any other cycle in $C_1^{\delta}$, for ${\delta}\in [1,2)$, is in the linear span of $v_n$. Next, suppose ${\delta}\geq 2$. Note that this allows the appearance of cycles that are not in the span of $v_n$. In the next claim, we show that upon passing to homology, the equivalence class of any such cycle coincides with that of $v_n$. This will show that there can be at most one nontrivial element in ${\operatorname{Dgm}}_1^{{\operatorname{si}}}(G_n)$. \[cl:cycle-decomp\] Let ${\delta}\geq 2$, and let $y = \sum_{i=1}^pa_i{\sigma}_i \in \ker({\partial}^{\delta}_1)$ for some $p\in {\mathbb{N}}$, some $a_1,\ldots,a_p \in \mathbb{K}$, and some ${\sigma}_1,\ldots, {\sigma}_p\in {{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$. Then there exists a choice of coefficients $(b_i)_{i=1}^n \in \mathbb{K}^n$ such that $z=\sum_{i=1}^nb_ie_i \in \ker({\partial}^{\delta}_1)$ and $y-z \in {\operatorname{im}}({\partial}^{\delta}_2)$. Moreover, we obtain ${\langle}y {\rangle}_{\delta}= {\langle}z {\rangle}_{\delta}= {\langle}v_n {\rangle}_{\delta}$ upon passing to homology. ![Given two points $x_j,x_k \in X_n$, we have either ${\omega}_{G_n}(x_j,x_k)\leq n/2$, or ${\omega}_{G_n}(x_k,x_j)< n/2$. To see this, note that ${\omega}_{G_n}(x,x')+{\omega}_{G_n}(x',x)=n$ for any $x\neq x'\in X_n$.[]{data-label="fig:dowker-cycle-prop"}](dowker-cycle-prop.jpg) ![Three possible locations for a ${\delta}$-sink $x$ of a simplex $[x_j,x_k]$, assuming that ${\omega}_{G_n}(x_j,x_k)\leq n/2$. For the figure on the left, note that ${\omega}_{G_n}(x_k,x)\geq n/2 \geq {\omega}_{G_n}(x_j,x_k)$. For the figure in the middle, note that ${\omega}_{G_n}(x_j,x)\geq {\omega}_{G_n}(x_j,x_k)$. Finally, for the figure on the right, where $x=x_k$, note that ${\omega}_{G_n}(x_j,x)={\omega}_{G_n}(x_j,x_k)$ and ${\omega}_{G_n}(x_k,x)=0$.[]{data-label="fig:dowker-cycle-prop-sink"}](dowker-cycle-prop-sink.png) To see this, fix ${\sigma}_i \in {{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$, and write ${\sigma}_i= [x_j,x_k]$ for some $1\leq j,k \leq n$. If $k=j+1$ (resp. $k=j-1$), then we already have ${\sigma}_i=e_j$ (resp. ${\sigma}_i=e_k$), so there is nothing more to show. Assume $k\not\in \{j+1,j-1\}$. Since ${\omega}_{G_n}(x_j,x_k) + {\omega}_{G_n}(x_k,x_j) = n$, we have two cases: (1) ${\omega}_{G_n}(x_j,x_k)\leq n/2$, or (2) ${\omega}_{G_n}(x_k,x_j)< n/2$. In the first case, we have $k=j+l $ for some integer $l\in [2,n/2]$ (all numbers are taken modulo $n$). In the second case, $j = k+l$ for some integer $l\in [2,n/2)$ (also modulo $n$). The situation is illustrated in Figure \[fig:dowker-cycle-prop\]. Both cases are similar, so we only prove the case ${\omega}_{G_n}(x_j,x_k) \leq n/2$. Recall that any ${\delta}$-sink $x\in X_n$ for $[x_j, x_k]$ satisfies $\max({\omega}_{G_n}(x_j,x), {\omega}_{G_n}(x_k,x)) \leq {\delta}$, by the ${\delta}$-sink condition (Equation \[eq:d-sink\]). Also note that such a ${\delta}$-sink $x$ satisfies $$\max({\omega}_{G_n}(x_j,x),{\omega}_{G_n}(x_k,x)) \geq {\omega}_{G_n}(x_j,x_k),$$ as can be seen from Figure \[fig:dowker-cycle-prop-sink\]. So whenever some $x\in X_n$ is a ${\delta}$-sink for $[x_j,x_k]$, we have $x_k$ as a valid ${\delta}$-sink for $[x_j,x_k]$. Since $[x_j,x_k] \in {{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$, it must have a ${\delta}$-sink $x\in X_n$. Thus $x_k$ is a valid ${\delta}$-sink for $[x_j,x_k]$. Next let $l\in [2,n/2]$ be an integer such that $k=j+l$ (modulo $n$). Notice that: $$0={\omega}_{G_n}(x_k,x_k)={\omega}_{G_n}(x_{j+l},x_k)< {\omega}_{G_n}(x_{j+l-1},x_k)< \ldots < {\omega}_{G_n}(x_{j+1},x_k)<{\omega}_{G_n}(x_j,x_k) \leq {\delta}.$$ Then observe that: $$[x_j,x_{j+1},x_k],[x_{j+1},x_{j+2},x_k],\ldots, [x_{k-2},x_{k-1},x_k] \in {{\mathfrak}{D}^{\operatorname{si}}}_{\delta},$$ since $x_k$ is a ${\delta}$-sink for all these 2-simplices. One can then verify the following: $$\begin{aligned} &{\partial}^{\delta}_2\left([x_j,x_{j+1},x_k]+[x_{j+1},x_{j+2},x_k]+\ldots+[x_{k-2},x_{k-1},x_k]\right)\\ &={\partial}_2^{\delta}\left(\sum_{q=0}^{k-j-2}[x_{j+q},x_{j+q+1},x_k]\right)\\ &=\sum_{q=0}^{k-j-2}[x_{j+q+1},x_k]-\sum_{q=0}^{k-j-2}[x_{j+q},x_k]+\sum_{q=0}^{k-j-2}[x_{j+q},x_{j+q+1}]\\ &=\sum_{q=0}^{k-j-2}[x_{j+q+1},x_k] - [x_j,x_k] - \sum_{q=0}^{k-j-3}[x_{j+q+1},x_k] +\sum_{q=0}^{k-j-2}[x_{j+q},x_{j+q+1}]\\ &=[x_j,x_{j+1}]+[x_{j+1},x_{j+2}]+\ldots+[x_{k-1},x_k]-[x_j,x_k]\\ &=e_j+e_{j+1} + \ldots + e_{k-1} -{\sigma}_i.\end{aligned}$$ Thus $a_i(e_j+e_{j+1}+\ldots+e_{k-1})-a_i{\sigma}_i \in{\operatorname{im}}({\partial}^{\delta}_2)$. Repeating this process for all ${\sigma}_i$, $i\in \{1,\ldots,p\}$, we may obtain the coefficients $(b_i)_{i=1}^n$ such that $\sum_{i=1}^pa_i{\sigma}_i - \sum_{i=1}^nb_ie_i \in {\operatorname{im}}({\partial}^{\delta}_2).$ Let $z=\sum_{i=1}^nb_ie_i$. Then $y-z \in {\operatorname{im}}({\partial}^{\delta}_2)$. Moreover, since ${\partial}^{\delta}_1\circ{\partial}^{\delta}_2 = 0$, it follows that ${\partial}^{\delta}_1(y)-{\partial}^{\delta}_1(z)=0$, so $z\in \ker({\partial}^{\delta}_1)$. Finally, note that an argument analogous to that of Claim \[cl:cycle-v\] shows that $b_1=b_2=\ldots=b_n$. Hence it follows that $z$ is a multiple of $v_n$. Thus ${\langle}z {\rangle}_{\delta}= {\langle}v_n {\rangle}_{\delta}$. This proves the claim. By Claims \[cl:cycle-v\] and \[cl:cycle-decomp\], it follows that $H_1({{\mathfrak}{D}^{\operatorname{si}}}_{\delta})$ is generated by ${\langle}v_n {\rangle}_{\delta}$ for all ${\delta}\geq 1$, so $\dim(H_1({{\mathfrak}{D}^{\operatorname{si}}}_{\delta}))\leq 1$ for all ${\delta}\geq 1$. It remains to show that ${\langle}v_n {\rangle}_{\delta}$ does not become trivial until ${\delta}={\left \lceil{n/2}\right \rceil }$. The cases $n=3,4$ can now be completed quickly, so we focus on these simpler situations first. For either of $n=3,4$, we have ${\left \lceil{n/2}\right \rceil }=2.$ Suppose ${\delta}=2$ and $n=3$. Then we have $[x_1,x_2,x_3]\in {{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$ because ${\operatorname{diam}}(G_n)=2$ and any of $x_1,x_2,x_3$ can be a $2$-sink for $[x_1,x_2,x_3]$. Then, $${\partial}_2^{\delta}([x_1,x_2,x_3])=[x_2,x_3]-[x_1,x_3]+[x_1,x_2]=e_1+e_2+e_3 = v_3.$$ Recall that by Claim \[cl:cycle-2simp\], $v_3\not\in {\operatorname{im}}({\partial}_2^{\delta})$ for any ${\delta}<2$. Thus by Claim \[cl:cycle-v\] and the preceding equation, $v_3$ generates $\ker({\partial}_1^{\delta})$ for $1\leq {\delta}<2$, and becomes a boundary for precisely ${\delta}\geq 2$. Thus ${\operatorname{Dgm}}_1^{{\operatorname{si}}}(G_3)={\left\{(1,2)\right\}}.$ Next, suppose ${\delta}=2$ and $n=4$. Then we have $[x_1,x_2,x_3],[x_1,x_3,x_4]\in {{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$ with $x_3, x_1$ as $2$-sinks, respectively. By a direct computation, we then have: $${\partial}_2^{\delta}([x_1,x_2,x_3]+[x_1,x_3,x_4])=e_1+e_2+e_3+e_4 = v_4.$$ By following the same argument as for the case $n=3$, we see that ${\operatorname{Dgm}}_1^{{\operatorname{si}}}(G_4)={\left\{(1,2)\right\}}.$ In the sequel, we assume that $n > 4$. Recall that it remains to show that ${\langle}v_n {\rangle}_{\delta}$ does not become trivial until ${\delta}={\left \lceil{n/2}\right \rceil }$, and that ${\langle}v_n {\rangle}_{\delta}= 0$ for all ${\delta}\geq {\left \lceil{n/2}\right \rceil }$. We have already shown that ${\langle}v_n {\rangle}_{\delta}$ is not trivial for ${\delta}\in [1,2)$. We proceed by defining the following: $${\gamma}_n:=[x_1,x_2,x_3] + [x_1,x_3,x_4] + \ldots + [x_1,x_{n-1},x_n] = \sum_{i=1}^{n-2}[x_1,x_{i+1},x_{i+2}].$$ For each ${\delta}\geq {\left \lceil{n/2}\right \rceil }$, we have ${\gamma}_n \in C_2^{\delta}$ and ${\partial}_2^{\delta}({\gamma}_n)=v_n$. In particular, ${\langle}v_n {\rangle}_{\delta}= 0$ for all such ${\delta}$. Let ${\delta}\geq {\left \lceil{n/2}\right \rceil }$. Notice that $${\omega}_{G_n}(x_{{\left \lceil{n/2}\right \rceil }+1},x_1) = n - {\omega}_{G_n}(x_1,x_{{\left \lceil{n/2}\right \rceil }+1}) = n- {\left \lceil{n/2}\right \rceil } \leq n/2 \leq {\left \lceil{n/2}\right \rceil } \leq {\delta},$$ so ${\omega}_{G_n}(x_i,x_1) \leq {\delta}$ for each $i \in \{{\left \lceil{n/2}\right \rceil }+1,{\left \lceil{n/2}\right \rceil }+2,\ldots, n\}$. Then for each $i\in \{{\left \lceil{n/2}\right \rceil }+1,{\left \lceil{n/2}\right \rceil }+2,\ldots, n-1\}$, we have $[x_i,x_{i+1},x_1] \in {{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$, with $x_1$ as a ${\delta}$-sink. Also notice that for each $i\in \{1,\ldots,{\left \lceil{n/2}\right \rceil }\}$, $${\omega}_{G_n}(x_i,x_{{\left \lceil{n/2}\right \rceil }+1})\leq{\omega}_{G_n}(x_1,x_{{\left \lceil{n/2}\right \rceil }+1})={\left \lceil{n/2}\right \rceil } \leq {\delta},$$ so ${\omega}_{G_n}(x_i,x_{{\left \lceil{n/2}\right \rceil }+1}) \leq {\delta}$. Thus for any $i\in \{2,\ldots,{\left \lceil{n/2}\right \rceil }\},$ we have $[x_1,x_i,x_{i+1}] \in {{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$, with $x_{{\left \lceil{n/2}\right \rceil }+1}$ as a ${\delta}$-sink. Combining the two preceding observations, we see that for any $i\in \{2,\ldots, n-2\}$, we have $[x_1,x_{i+1},x_{i+2}]\in {{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$. It follows that ${\gamma}_n \in C_2^{\delta}$. Next we observe the following: $$\begin{aligned} {\partial}_2^{\delta}({\gamma}_n)&={\partial}_2^{\delta}\left(\sum_{i=1}^{n-2}[x_1,x_{i+1},x_{i+2}]\right)\\ &=\sum_{i=1}^{n-2}[x_{i+1},x_{i+2}] - \sum_{i=1}^{n-2}[x_1,x_{i+2}]+ \sum_{i=1}^{n-2}[x_1,x_{i+1}]\\ &=\sum_{i=1}^{n-2}[x_{i+1},x_{i+2}] - \sum_{i=1}^{n-2}[x_1,x_{i+2}]+[x_1,x_2]+ \sum_{i=2}^{n-2}[x_1,x_{i+1}]\\ &=\sum_{i=1}^{n-2}[x_{i+1},x_{i+2}] + [x_1,x_2] - [x_1,x_n] = v_n.\end{aligned}$$ It follows that for any ${\delta}\geq {\left \lceil{n/2}\right \rceil }$, we have $v_n \in {\operatorname{im}}({\partial}_2^{\delta})$, and so ${\langle}v_n {\rangle}_{\delta}=0$ for each such ${\delta}$. ![Placement of $x_{{\left \lceil{n/2}\right \rceil }}$ and $x_{{\left \lceil{n/2}\right \rceil }-1}$, depending on whether $n$ is even or not.[]{data-label="fig:dowker-cycle-prop2"}](dowker-cycle-prop2.jpg) There does not exist ${\delta}\in [2,{\left \lceil{n/2}\right \rceil })$ such that ${\langle}v_n {\rangle}_{\delta}$ is trivial. Let $2\leq {\delta}< {\left \lceil{n/2}\right \rceil }$. As a first step, we wish to show that ${\gamma}_n \not\in C_2^{\delta}$. For this step, it suffices to show that the 2-simplex ${\sigma}:= [x_1,x_{{\left \lceil{n/2}\right \rceil }},x_{{\left \lceil{n/2}\right \rceil }+1}]$ does not belong to ${{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$. The placement of $x_{{\left \lceil{n/2}\right \rceil }}$ and $x_{{\left \lceil{n/2}\right \rceil }+1}$ is illustrated in Figure \[fig:dowker-cycle-prop2\]. By an argument similar to that used in Figure \[fig:dowker-cycle-prop-sink\], one can verify that there exists a ${\delta}$-sink for ${\sigma}$ if and only if at least one of $x_1,x_{{\left \lceil{n/2}\right \rceil }}, x_{{\left \lceil{n/2}\right \rceil }+1}$ is a ${\delta}$-sink for ${\sigma}$. But note the following: $$\begin{aligned} {\omega}_{G_n}(x_{{\left \lceil{n/2}\right \rceil }},x_1)= n-({\left \lceil{n/2}\right \rceil }-1) &= \begin{cases} n/2 + 1 &: n \text{ even}\\ {\left \lceil{n/2}\right \rceil } &: n\text{ odd} \end{cases}\\ &\geq {\left \lceil{n/2}\right \rceil } > {\delta},\end{aligned}$$ so $x_1$ cannot be a ${\delta}$-sink for ${\sigma}$. Similarly we note that ${\omega}_{G_n}(x_{{\left \lceil{n/2}\right \rceil }+1},x_{{\left \lceil{n/2}\right \rceil }}) = n > {\delta}$ and ${\omega}_{G_n}(x_1,x_{{\left \lceil{n/2}\right \rceil }+1})={\left \lceil{n/2}\right \rceil } > {\delta},$ so neither $x_{{\left \lceil{n/2}\right \rceil }}$ nor $x_{{\left \lceil{n/2}\right \rceil }+1}$ can be ${\delta}$-sinks for ${\sigma}$. Thus ${\sigma}\not \in {{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$, and so ${\gamma}_n \not\in C_2^{\delta}$. Suppose there exists ${\gamma}' \in C_2^{\delta}$ such that ${\partial}_2^{\delta}({\gamma}') = v_n$. Since $[x_1,x_2]$ is a summand of $v_n$, we must have $a_j[x_1,x_2,x_j]$ as a summand of ${\gamma}'$, for some coefficient $a_j$ and some $3\leq j \leq n$. First suppose that $x_j$ is a sink for $[x_1,x_2,x_j]$. We claim that ${\gamma}'$ is homologous to a chain containing $[x_1,x_2,x_3]$ as a summand. If $j=3$, then we are done, so suppose $j > 3$. Then we also know that $[x_1,x_2,x_3,x_j]$ is a 3-simplex in ${{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}}$. Let ${\gamma}''$ be the chain obtained from ${\gamma}'$ by replacing $[x_1,x_2,x_j]$ with $[x_1,x_2,x_3] - [x_2,x_3,x_j] + [x_1,x_3,x_j]$. Since ${\partial}_3^{\delta}([x_1,x_2,x_3,x_j]) = [x_2,x_3,x_j] - [x_1,x_3,x_j] + [x_1,x_2,x_j] -[x_1,x_2,x_3]$ and ${\partial}_2^{\delta}\circ {\partial}_3^{\delta}= 0$, we know that ${\partial}_2^{\delta}({\gamma}'') = {\partial}_2^{\delta}({\gamma}') = v_n$. Now ${\partial}_2^{\delta}([x_1,x_2,x_3])$ contributes an $[x_1,x_3]$ summand which does not appear in $v_n$, so it must be cancelled by some other terms in ${\gamma}'$ (resp. ${\gamma}''$). Thus there must exist another 2-simplex $[x_1,x_3,x_k]$ in ${{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$, where $k\neq 2$. But we can repeat the preceding argument to obtain a chain homologous to ${\gamma}'$ containing both $[x_1,x_2,x_3]$ and $[x_1,x_3,x_4]$ as summands. Proceeding in this way, we obtain a chain homologous to ${\gamma}'$ that contains $[x_1,x_{{\left \lceil{n/2}\right \rceil }},x_{{\left \lceil{n/2}\right \rceil }+1}]$ as a summand. But this is a contradiction to what we have shown previously, i.e. that $[x_1,x_{{\left \lceil{n/2}\right \rceil }},x_{{\left \lceil{n/2}\right \rceil }+1}]$ is not a simplex in ${{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$. In the case where $x_j$ is not a sink for $[x_1,x_2,x_j]$, we must have $x_2$ as a sink instead. Using similar reasoning as above, we can replace ${\gamma}'$ in this instance by a homologous chain containing $[x_n,x_1,x_2]$ as a summand. Since $[x_n,x_2]$ is not a summand of $v_n$, we can obtain another homologous chain containing $[x_{n-1},x_n,x_1]$ as a summand, then a homologous chain containing $[x_{n-1},x_n,x_1],[x_{n-2},x_{n-1},x_1]$ as summands, and so on until we again obtain a homologous chain containing $[x_1,x_{{\left \lceil{n/2}\right \rceil }},x_{{\left \lceil{n/2}\right \rceil }+1}]$ as a summand. Once again, this is a contradiction. This proves the claim. Thus we have shown that $v_n$ is a nontrivial cycle that appears at ${\delta}=1$, and becomes a boundary at exactly ${\delta}={\left \lceil{n/2}\right \rceil }$. Furthermore, we have shown that upon passing to homology, the equivalence classes of all cycles coincide with that of $v_n$. Thus there is only one off-diagonal point $(1,{\left \lceil{n/2}\right \rceil })$ on the 1-dimensional persistence diagram, which appears with multiplicity one. This concludes the proof. From our experimental results (see Figure \[fig:cycle\]), it appears that the 1-dimensional Rips persistence diagram of a cycle network does not admit a characterization as simple as that given by Theorem \[thm:cycleH1\] for the 1-dimensional Dowker persistence diagram. Moreover, the Rips complexes ${\mathfrak}{R}^{\delta}_{G_n}, {\delta}\in {\mathbb{R}}, n\in {\mathbb{N}}$ correspond to certain types of *independence complexes* that appear independently in the literature, and whose homotopy types remain open [@engstrom2009complexes Question 5.3]. On a related note, we point the reader to [@adamaszek2015vietoris] for a complete characterization of the homotopy types of Rips complexes of points on the circle (equipped with the restriction of the arc length metric). To elaborate on the connection to [@engstrom2009complexes], we write $H^k_n$ to denote the undirected graph with vertex set ${\left\{1,\ldots, n\right\}}$, and edges given by pairs $(i,j)$ where $1\leq i < j \leq n$ and either $j-i < k$ or $(n+i) - j < k$. Next we write ${\operatorname{Ind}}(H^k_n)$ to denote the *independence complex* of $H^k_n$, which is the simplicial complex consisting of subsets ${\sigma}\subseteq {\left\{1,2,\ldots, n\right\}}$ such that no two elements of ${\sigma}$ are connected by an edge in $H^k_n$. Then we have ${\operatorname{Ind}}(H^k_n) = {\mathfrak}{R}^{n-k}_{G_n}$ for each $k, n\in {\mathbb{N}}$ such that $k < n$. To gain intuition for this equality, fix a basepoint $1$, and consider the values of $j\in {\mathbb{N}}$ for which the simplex $[1,j]$ belongs to ${\operatorname{Ind}}(H^k_n)$ and to ${\mathfrak}{R}^{n-k}_{G_n}$, respectively. In either case, we have $k+1 \leq j \leq n-k+1$. Using the rotational symmetry of the points, one can then obtain the remaining 1-simplices. Rips complexes are determined by their 1-skeleton, so this suffices to construct ${\mathfrak}{R}^{n-k}_{G_n}$. Analogously, ${\operatorname{Ind}}(H^k_n)$ is determined by the edges in $H^k_n$, and hence also by its 1-skeleton. In [@engstrom2009complexes Question 5.3], the author writes that the homotopy type of ${\operatorname{Ind}}(H^k_n)$ is still unsolved. Characterizing the persistence diagrams ${\operatorname{Dgm}}_k^{{\mathfrak}{R}}(G_n)$ thus seems to be a useful future step, both in providing a computational suggestion for the homotopy type of ${\operatorname{Ind}}(H^k_n)$, and also in providing a valuable example in the study of persistence of directed networks. Theorem \[thm:cycleH1\] has the following implication for data analysis: nontrivial 1-dimensional homology in the Dowker persistence diagram of an asymmetric network suggests the presence of directed cycles in the underlying data. Of course, it is not necessarily true that nontrivial 1-dimensional persistence can occur *only* in the presence of a directed circle. Our motivation for studying cycle networks is that they constitute directed analogues of circles, and we were interested in seeing if the 1-dimensional Dowker persistence diagram would be able to capture this analogy. Theorem \[thm:cycleH1\] shows that this is indeed the case: we get a single nontrivial 1-dimensional persistence interval, which is what we would expect when computing the persistent homology of a circle in the metric space setting. We further studied the 2-dimensional Dowker persistence diagrams of cycle networks. Our computational examples, some of which are illustrated in Figure \[fig:cycle-2dim\], enabled us to conjecture: \[conj:dowker-dgm-2\] Let $n\in {\mathbb{N}}$, $n \geq 3$, and let $G_n$ be a cycle network. If $n$ is odd, then ${\operatorname{Dgm}}_2^{{\mathfrak}{D}}(G_n)$ is trivial. If $n$ is even, then ${\operatorname{Dgm}}_2^{{\mathfrak}{D}}(G_n) = [(\frac{n}{2},\frac{n}{2}+1)\in {\mathbb{R}}^2],$ and the multiplicity of this point is $\frac{n}{2}-1$. This computationally motivated conjecture is in fact true; moreover, we have a full characterization of the persistence diagram of a cycle network across all dimensions $k \in {\mathbb{Z}}_+$. This characterization relies on results in [@adamaszek2015vietoris] and [@adamaszek2016nerve], and is stated for even and odd dimensions below: [theorem]{}[thmdowkercyceven]{} \[thm:dowker-cyc-even\] Fix $n\in {\mathbb{N}}$, $n \geq 3$. If $l\in {\mathbb{N}}$ is such that $n$ is divisible by $(l+1)$, and $k:=\tfrac{nl}{l+1}$ is such that $0\leq k \leq n-2$, then ${\operatorname{Dgm}}^{{\mathfrak}{D}}_{2l}(G_n)$ consists of precisely the point $(\tfrac{nl}{l+1},\tfrac{nl}{l+1} + 1)$ with multiplicity $\tfrac{n}{l+1} -1$. If $l$ or $k$ do not satisfy the conditions above, then ${\operatorname{Dgm}}^{{\mathfrak}{D}}_{2l}(G_n)$ is trivial. As a special case, Theorem \[thm:dowker-cyc-even\] proves Conjecture \[conj:dowker-dgm-2\] by setting $l=1$. If $n$ is odd, then it is not divisible by $(l+1) = 2$, and so ${\operatorname{Dgm}}^{{\mathfrak}{D}}_2(G_n)$ is trivial. If $n$ is even, then it is divisible by $(l+1)=2$, and $\tfrac{nl}{l+1} = \tfrac{n}{2} \leq n-2$ because $n$ is at least 4. Thus ${\operatorname{Dgm}}^{{\mathfrak}{D}}_2(G_n)$ consists of the point $(\tfrac{n}{2},\tfrac{n}{2} + 1)$ with multiplicity $\tfrac{n}{2} - 1$. [theorem]{}[thmdowkercycodd]{}\[thm:dowker-cyc-odd\] Fix $n\in {\mathbb{N}}$, $n\geq 3$. Then for $l\in {\mathbb{N}}$, define $M_l:={\left\{m \in {\mathbb{N}}: \tfrac{nl}{l+1} < m < \tfrac{n(l+1)}{l+2}\right\}}$. If $M_l$ is empty, then ${\operatorname{Dgm}}^{{\mathfrak}{D}}_{2l+1}(G_n)$ is trivial. Otherwise, we have: $${\operatorname{Dgm}}^{{\mathfrak}{D}}_{2l+1}(G_n) = {\left\{\left(a_l,{\left \lceil{\tfrac{n(l+1)}{l+2}}\right \rceil }\right)\right\}},$$ where $a_l:=\min{\left\{m \in M_l\right\}}.$ We use set notation (instead of multisets) to mean that the multiplicity is 1. In particular, for $l=0$, we have $\tfrac{nl}{l+1} = 0$ and $\tfrac{n(l+1)}{l+2} = \tfrac{n}{2} \geq 3/2$, so $1 \in M_l$. Thus we have ${\operatorname{Dgm}}^{{\mathfrak}{D}}_1(G_n) = {\left\{\left(1,{\left \lceil{\tfrac{n}{2}}\right \rceil }\right)\right\}},$ and so Theorem \[thm:dowker-cyc-odd\] recovers Theorem \[thm:cycleH1\] as a special case. However, whereas the proof of Theorem \[thm:cycleH1\] is elementary and pedagogical (it relies on intuitive observations about the structure of a cycle network), the proofs of Theorems \[thm:dowker-cyc-even\] and \[thm:dowker-cyc-odd\] use sophisticated machinery developed across [@adamaszek2015vietoris] and [@adamaszek2016nerve]. We provide details for Theorem \[thm:cycleH1\] in the body of the paper, and relegate full details of Theorems \[thm:dowker-cyc-even\] and \[thm:dowker-cyc-odd\] to Appendix \[sec:cycle-addendum\]. [0.48]{} [0.48]{} [0.48]{} Sensitivity to network transformations -------------------------------------- We first make the following: Let $(X,{\omega}_X) \in {\mathcal{N}}$ be a network. For any $z,z'\in X$, define the *$(z,z')$-swap* of $(X,{\omega}_X)$ to be the network $S_X(z,z'):=(X^{z,z'},{\omega}_X^{z,z'})$ defined as follows: $$\begin{aligned} X^{z,z'}&:= X,\\ \text{For any $x,x'\in X^{z,z'}$,}\qquad {\omega}_X^{z,z'}(x,x')&:=\begin{cases} {\omega}_X(x',x) &: x=z,x'=z'\\ {\omega}_X(x',x) &: x'=z,x=z'\\ {\omega}_X(x,x') &: \text{otherwise.} \end{cases}\end{aligned}$$ We then pose the following question: > *Given a network $(X,{\omega}_X)$ and an $(x,x')$-swap $S_X(x,x')$ for some $x,x'\in X$, how do the Rips or Dowker persistence diagrams of $S_X(x,x')$ differ from those of $(X,{\omega}_X)$?* This situation is illustrated in Figure \[fig:3-node-networks\]. Example \[ex:3-node\] shows an example where the Dowker persistence diagram captures the variation in a network that occurs after a pair swap, whereas the Rips persistence diagram fails to capture this difference. Furthermore, Remark \[rem:swap-rips\] shows that Rips persistence diagrams always fail to do so. We also consider the extreme situation where all the directions of the edges of a network are reversed, i.e. the network obtained by applying the pair swap operation to each pair of nodes. We would intuitively expect that the persistence diagrams would not change. The following discussion shows that the Rips and Dowker persistence diagrams are invariant under taking the transpose of a network. \[prop:si-so\] Recall the transposition map ${\mathfrak}{t}$ and the shorthand notation $X^{\top}={\mathfrak}{t}(X)$ from Definition \[defn:sym-trans\]. Let $k\in {\mathbb{Z}}_+$. Then ${\operatorname{Dgm}}_k^{{\operatorname{si}}}(X)={\operatorname{Dgm}}_k^{{\operatorname{so}}}(X^\top),$ and therefore ${\operatorname{Dgm}}_k^{{\mathfrak}{D}}(X)={\operatorname{Dgm}}_k^{{\mathfrak}{D}}(X^\top)$ by Theorem \[thm:dowker-functorial\]. \[rem:swap-rips\] Let $(X,{\omega}_X)\in {\mathcal{N}}$, let $z,z'\in X$, and let ${\sigma}\in {\operatorname{pow}}(X)$. Then we have: $$\max_{x,x'\in {\sigma}}{\omega}_X(x,x') = \max_{x,x'\in {\sigma}}{\omega}_X^{z,z'}(x,x').$$ Using this observation, one can then repeat the arguments used in the proof of Proposition \[prop:si-so\] to show that: $${\operatorname{Dgm}}_k^{{\mathfrak}{R}}(X)={\operatorname{Dgm}}_k^{{\mathfrak}{R}}(S_X(z,z')),\text{ for each } k \in {\mathbb{Z}}_+.$$ This encodes the intuitive fact that Rips persistence diagrams are blind to pair swaps. Moreover, succesively applying the pair swap operation over all pairs produces the transpose of the original network, and so it follows that ${\operatorname{Dgm}}_k^{{\mathfrak}{R}}(X)={\operatorname{Dgm}}_k^{{\mathfrak}{R}}(X^\top)$. On the other hand, $k$-dimensional Dowker persistence diagrams are not necessarily invariant to pair swaps when $k\geq 1$. Indeed, Example \[ex:3-node\] below constructs a space $X$ for which there exist points $z,z'\in X$ such that $${\operatorname{Dgm}}_1^{{\mathfrak}{D}}(X)\neq{\operatorname{Dgm}}_1^{{\mathfrak}{D}}(S_X(z,z')).$$ However, 0-dimensional Dowker persistence diagrams are still invariant to pair swaps: \[prop:dowker0pair\] Let $(X,{\omega}_X)\in {\mathcal{N}}$, let $z,z'$ be any two points in $Z$, and let ${\sigma}\in {\operatorname{pow}}(X)$. Then we have: $${\operatorname{Dgm}}_0^{{\mathfrak}{D}}(X)={\operatorname{Dgm}}_0^{{\mathfrak}{D}}(S_X(z,z')).$$ \[ex:3-node\] Consider the three node dissimilarity networks $(X,{\omega}_X)$ and $(Y,{\omega}_Y)$ in Figure \[fig:3-node-networks\]. Note that $(Y,{\omega}_Y)$ coincides with $S_X(a,c)$. We present both the Dowker and Rips persistence barcodes obtained from these networks. Note that the Dowker persistence barcode is sensitive to the difference between $(X,{\omega}_X)$ and $(Y,{\omega}_Y)$, whereas the Rips barcode is blind to this difference. We refer the reader to §\[sec:exp\] for details on how we compute these barcodes. \(1) at (0,2.5)[$a$]{}; (2) at (-1.5,0)[$b$]{}; (3) at (1.5,0)[$c$]{}; at (0,-1)[$(X,{\omega}_X)$]{}; \(4) at (6.5,2.5)[$a$]{}; (5) at (5,0)[$b$]{}; (6) at (8,0)[$c$]{}; at (6.5,-1)[$(Y,{\omega}_Y)$]{}; \(1) edge \[loop above\] node\[above,pos=0.5\][$0$]{} (1); (2) edge \[loop left\] node\[above,pos=0.5\][$0$]{} (2); (3) edge \[loop right\] node\[above,pos=0.5\][$0$]{} (3); \(4) edge \[loop above\] node\[above,pos=0.5\][$0$]{} (4); (5) edge \[loop left\] node\[above,pos=0.5\][$0$]{} (5); (6) edge \[loop right\] node\[above,pos=0.5\][$0$]{} (6); \(1) edge \[bend left,in=180,out=0\] node\[above,pos=0.5\][$6$]{} (2); (2) edge \[bend left\] node\[above,pos=0.5\][$1$]{} (1); (1) edge \[bend left\] node\[above,pos=0.5\][$4$]{} (3); (3) edge \[bend left,out=0,in=180\] node\[above,pos=0.5\][$2$]{} (1); (2) edge \[ bend left\] node\[below,pos=0.5\][$5$]{} (3); (3) edge \[bend left\] node\[above,pos=0.5\][$3$]{} (2); \(4) edge \[bend left,in=180,out=0\] node\[above,pos=0.5\][$6$]{} (5); (5) edge \[bend left\] node\[above,pos=0.5\][$1$]{} (4); (4) edge \[bend left\] node\[above,pos=0.5\][$2$]{} (6); (6) edge \[bend left,in=180,out=0\] node\[above,pos=0.5\][$4$]{} (4); (5) edge \[ bend left\] node\[below,pos=0.5\][$5$]{} (6); (6) edge \[bend left\] node\[above,pos=0.5\][$3$]{} (5); [0.49]{} [0.49]{} [0.49]{} [0.49]{} To show how the Dowker complex is constructed, we also list the Dowker sink complexes of the networks in Figure \[fig:3-node-networks\], and also the corresponding homology dimensions across a range of resolutions. Note that when we write $[a,b](a)$, we mean that $a$ is a sink corresponding to the simplex $[a,b]$. $$\begin{aligned} {{\mathfrak}{D}^{\operatorname{si}}}_{0,X} = {\left\{[a],[b],[c]\right\}} && \dim(H_1({{\mathfrak}{D}^{\operatorname{si}}}_{0,X})) = 0\\ {{\mathfrak}{D}^{\operatorname{si}}}_{1,X} = {\left\{[a],[b],[c],[a,b](a)\right\}} && \dim(H_1({{\mathfrak}{D}^{\operatorname{si}}}_{1,X})) = 0\\ {{\mathfrak}{D}^{\operatorname{si}}}_{2,X} = {\left\{[a],[b],[c],[a,b](a),[a,c](a),[b,c](a),[a,b,c](a)\right\}} && \dim(H_1({{\mathfrak}{D}^{\operatorname{si}}}_{2,X})) = 0\\ {{\mathfrak}{D}^{\operatorname{si}}}_{3,X} = {\left\{[a],[b],[c],[a,b](a),[a,c](a),[b,c](a),[a,b,c](a)\right\}} && \dim(H_1({{\mathfrak}{D}^{\operatorname{si}}}_{3,X})) = 0\end{aligned}$$ $$\begin{aligned} {{\mathfrak}{D}^{\operatorname{si}}}_{0,Y} = {\left\{[a],[b],[c]\right\}} && \dim(H_1({{\mathfrak}{D}^{\operatorname{si}}}_{0,Y})) = 0\\ {{\mathfrak}{D}^{\operatorname{si}}}_{1,Y} = {\left\{[a],[b],[c],[a,b](a)\right\}} && \dim(H_1({{\mathfrak}{D}^{\operatorname{si}}}_{1,Y})) = 0\\ {{\mathfrak}{D}^{\operatorname{si}}}_{2,Y} = {\left\{[a],[b],[c],[a,b](a),[a,c](c)\right\}} && \dim(H_1({{\mathfrak}{D}^{\operatorname{si}}}_{2,Y})) = 0\\ {{\mathfrak}{D}^{\operatorname{si}}}_{3,Y} = {\left\{[a],[b],[c],[a,b](a),[a,c](c),[b,c](b)\right\}} && \dim(H_1({{\mathfrak}{D}^{\operatorname{si}}}_{3,Y})) = 1\\ {{\mathfrak}{D}^{\operatorname{si}}}_{4,Y} = {\left\{[a],[b],[c],[a,b](a),[a,c](a),[b,c](a),[a,b,c](a)\right\}} && \dim(H_1({{\mathfrak}{D}^{\operatorname{si}}}_{4,Y})) = 0\\ \end{aligned}$$ Note that for ${\delta}\in [3,4)$, $\dim(H_1({{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},Y})) = 1$, whereas $\dim(H_1({{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X}))=0$ for each ${\delta}\in {\mathbb{R}}$. Based on the discussion in Remark \[rem:swap-rips\], Proposition \[prop:dowker0pair\], and Example \[ex:3-node\], we conclude the following: **Moral:** *Unlike Rips persistence diagrams, Dowker persistence diagrams are truly sensitive to asymmetry.* We summarize some of these results: \[thm:sym-trans-summary\] Recall the symmetrization and transposition maps ${\mathfrak}{s}$ and ${\mathfrak}{t}$ from Definition \[defn:sym-trans\]. Then: 1. ${\mathfrak}{R}\circ {\mathfrak}{s} = {\mathfrak}{R}$, 2. ${\mathfrak}{D}^{{\operatorname{so}}}\circ {\mathfrak}{t} = {\mathfrak}{D}^{{\operatorname{si}}}$, and 3. ${\mathfrak}{D}^{{\operatorname{si}}}\circ {\mathfrak}{t} = {\mathfrak}{D}^{{\operatorname{so}}}$. Also, there exist $(X,{\omega}_X), (Y,{\omega}_Y) \in {\mathcal{N}}$ such that $({\mathfrak}{D}^{{\operatorname{si}}}\circ {\mathfrak}{s})(X) \neq {\mathfrak}{D}^{{\operatorname{si}}}(X)$, and $({\mathfrak}{D}^{{\operatorname{so}}}\circ {\mathfrak}{s})(Y) \neq {\mathfrak}{D}^{{\operatorname{so}}}(Y)$. These follow from Example \[ex:3-node\], Remark \[rem:rips-symm\], and Proposition \[prop:si-so\]. Implementation and an experiment on network classification {#sec:exp} ========================================================== In this section, we present the results of an experiment where we applied our methods to perform a classification task on a database of networks. All persistent homology computations were carried out using the `Javaplex` package for Matlab. A full description of Javaplex can be found in [@tausz2011javaplex]. We used $\mathbb{K}={\mathbb{Z}}_2$ as the field of coefficients for all our persistence computations. The dataset and software used for our computations are available as part of the `PersNet` software package on <https://research.math.osu.edu/networks/Datasets.html>. A version of our simulated hippocampal networks experiment has appeared in [@dowker-asilo]. All networks in the following experiment were normalized to have weights in the range $[0,1]$. For each network, we computed Dowker sink complexes at resolutions ${\delta}= 0.01,0.02,0.03,\ldots,1.00$. This filtration was then passed into Javaplex, which produced 0 and 1-dimensional Dowker persistence barcodes. Simulated hippocampal networks {#sec:exp-arenas} ------------------------------ In the neuroscience literature, it has been shown that as an animal explores a given *environment* or *arena*, specific “place cells" in the hippocampus show increased activity at specific spatial regions, called “place fields" [@o1971hippocampus]. Each place cell shows a *spike* in activity when the animal enters the place field linked to this place cell, accompanied by a drop in activity as the animal moves far away from this place field. To understand how the brain processes this data, a natural question to ask is the following: Is the time series data of the place cell activity, referred to as “spike trains", enough to detect the structure of the arena? Approaches based on homology [@curto2008cell] and persistent homology [@dabaghian2012topological] have shown positive results in this direction. In [@dabaghian2012topological], the authors simulated the trajectory of a rat in an arena containing “holes." A simplicial complex was then built as follows: whenever $n+1$ place cells with overlapping place fields fired together, an $n$-simplex was added. This yield a filtered simplicial complexed indexed by a time parameter. By computing persistence, it was then shown that the number of persistent bars in the 1-dimensional barcode of this filtered simplicial complex would accurately represent the number of holes in the arena. We repeated this experiment with the following change in methodology: we simulated the movement of an animal, and corresponding hippocampal activity, in arenas with a variety of obstacles. We then induced a directed network from each set of hippocampal activity data, and computed the associated 1-dimensional Dowker persistence diagrams. We were interested in seeing if the bottleneck distances between diagrams arising from similar arenas would differ significantly from the bottleneck distance between diagrams arising from different arenas. To further exemplify our methods, we repeated our analysis after computing the 1-dimensional Rips persistence diagrams from the hippocampal activity networks. In our experiment, there were five arenas. The first was a square of side length $L=10$, with four circular “holes" or “forbidden zones" of radius $0.2L$ that the trajectory could not intersect. The other four arenas were those obtained by removing the forbidden zones one at a time. In what follows, we refer to the arenas of each type as *4-hole, 3-hole, 2-hole, 1-hole,* and *0-hole arenas*. For each arena, a random-walk trajectory of 5000 steps was generated, where the animal could move along a square grid with 20 points in each direction. The grid was obtained as a discretization of the box $[0,L]\times [0,L]$, and each step had length $0.05L$. The animal could move in each direction with equal probability. If one or more of these moves took the animal outside the arena (a disallowed move), then the probabilities were redistributed uniformly among the allowed moves. Each trajectory was tested to ensure that it covered the entire arena, excluding the forbidden zones. Formally, we write the time steps as a set $T:={\left\{1,2,\ldots, 5000\right\}}$, and denote the trajectory as a map $\operatorname{traj}:T {\rightarrow}[0,L]^2$. For each of the five arenas, 20 trials were conducted, producing a total of 100 trials. For each trial $l_k$, an integer $n_k$ was chosen uniformly at random from the interval $[150,200]$. Then $n_k$ place fields of radius $0.05L$ were scattered uniformly at random inside the corresponding arena for each $l_k$. An illustration of the place field distribution is provided in Figure \[fig:arenas-rasters\]. A spike on a place field was recorded whenever the trajectory would intersect it. So for each $1\leq i\leq n_k$, the spiking pattern of cell $x_i$, corresponding to place field PF$_i$, was recorded via a function $r_i:T{\rightarrow}{\left\{0,1\right\}}$ given by: $$r_i(t)=\begin{cases} 1 &:\text{if } \operatorname{traj}(t)\text{ intersects } \text{PF}_i,\\ 0 &: \text{otherwise}\end{cases} \qquad\qquad t\in T.$$ The matrix corresponding to $r_i$ is called the *raster* of cell $x_i$. A sample raster is illustrated in Figure \[fig:arenas-rasters\]. For each trial $l_k$, the corresponding network $(X_k,{\omega}_{X_k})$ was constructed as follows: $X_k$ consisted of $n_k$ nodes representing place fields, and for each $1\leq i,j\leq n_k$, the weight ${\omega}_{X_k}(x_i,x_j)$ was given by: $$\begin{aligned} {\omega}_{X_k}(x_i,x_j) &:=1-\frac{N_{i,j}(5)}{\sum_{i=1}^{n_k}N_{i,j}(5)},\\ \text{ where } N_{i,j}(5)&={\operatorname{card}}\left({\left\{(s,t)\in T^2:t\in [2,5000], t-s\in [1,5], r_j(t)=1,r_i(s)=1\right\}}\right).\end{aligned}$$ In words, $N_{i,j}(5)$ counts the pairs of times $(s,t), s < t,$ such that cell $x_j$ spikes (at a time $t$) after cell $x_i$ spikes (at a time $s$), and the delay between the two spikes is fewer than 5 time steps. The idea is that if cell $x_j$ frequently fires within a short span of time after cell $x_i$ fires, then place fields PF$_i$ and PF$_j$ are likely to be in close proximity to each other. The column sum of the matrix corresponding to ${\omega}_{X_k}$ is normalized to 1, and so ${\omega}_{X_k}^\top$ can be interpreted as the transition matrix of a Markov process. [0.3]{} [0.3]{} [0.3]{} [0.3]{} [0.3]{} [0.3]{} Next, we computed the 1-dimensional Dowker persistence diagrams of each of the 100 networks. Note that ${\operatorname{Dgm}}_1^{{\mathfrak}{D}}({\omega}_X)={\operatorname{Dgm}}_1^{{\mathfrak}{D}}({\omega}_X^\top)$ by Proposition \[prop:si-so\], so we are actually obtaining the 1-dimensional Dowker persistence diagrams of transition matrices of Markov processes. We then computed a $100\times 100$ matrix consisting of the bottleneck distances between all the 1-dimensional persistence diagrams. The single linkage dendrogram generated from this bottleneck distance matrix is shown in Figure \[fig:dendro-dowker-arenas\]. The labels are in the format `env-<nh>-<nn>`, where `nh` is the number of holes in the arena/environment, and `nn` is the number of place fields. Note that with some exceptions, networks corresponding to the same arena are clustered together. We conclude that the Dowker persistence diagram succeeded in capturing the intrinsic differences between the five classes of networks arising from the five different arenas, even when the networks had different sizes. We then computed the Rips persistence diagrams of each network, and computed the $100\times 100$ bottleneck distance matrix associated to the collection of 1-dimensional diagrams. The single linkage dendrogram generated from this matrix is given in Figure \[fig:dendro-rips-arenas\]. Notice that the Rips dendrogram does not do a satisfactory job of classifying arenas correctly. We note that an alternative method of comparing the networks obtained from our simulations would have been to compute the pairwise network distances, and plot the results in a dendrogram. But ${d_{\mathcal{N}}}$ is NP-hard to compute—this follows from the fact that computing ${d_{\mathcal{N}}}$ includes the problem of computing Gromov-Hausdorff distance between finite metric spaces, which is NP-hard [@schmiedl]. So instead, we are computing the bottleneck distances between 1-dimensional Dowker persistence diagrams, as suggested by Remark \[rem:dowker-benefits\]. Discussion ========== We provided a complete description of the Rips and Dowker persistence diagrams of general networks. The stability results we have provided give quantitative guarantees on the robustness of these persistence diagrams. As a building block, we proved a functorial generalization of Dowker’s theorem, which also yields an independent proof of a folklore strengthening of Dowker’s theorem. We have provided numerous examples suggesting that Dowker persistence diagrams are an appropriate method for analyzing general asymmetric networks. For a particular class of such examples, the family of cycle networks, we have fully characterized their Dowker persistence diagrams in all dimensions. Finally, we have implemented our methods for a classification task on a database of networks, and provided interpretations for our results. We believe that the story of “persistent homology of asymmetric networks" has more aspects to be uncovered. Of particular interest to us is the analysis of alternative methods of producing simplicial complexes from asymmetric networks, for example, the *directed flag complex* construction of [@dlotko2016topological]. Yet another interesting extension to the non-metric framework has appeared in [@edelsbrunner2016topological], in the context of computing generalized Čech and Rips complexes for Bregman divergences. We remark that a persistent homology framework for the directed flag complex has been proposed by [@turner], but the computational aspects of this construction have not been addressed in the current literature. Another approach for computing persistence diagrams from asymmetric networks, which bypasses the construction of any simplicial complex and operates directly at the chain level is given in [@pph]. Some other interesting questions relate to cycle networks: for example, we would like to obtain a characterization of the Rips persistence diagrams of cycle networks for any dimension $k\geq 1$. Finally, it is important to devise more efficient implementations for the Dowker complexes we present here. It is likely that ideas from the literature on efficient construction of Čech complexes [@dantchev2012efficient; @edelsbrunner2016topological] will be helpful in this regard. #### **Acknowledgments.** This work was supported by NSF grants IIS-1422400 and CCF-1526513. We thank Pascal Wild and Zhengchao Wan for pointing out errors on an early preprint, and also Osman Okutan and Tim Porter for useful discussions. We are especially thankful to Henry Adams for numerous helpful observations and suggestions, especially regarding the material in Appendix \[sec:cycle-addendum\], and for suggesting the proof strategy for Theorem \[thm:cyc-cech-main\]. Proofs {#app:proofs} ====== The first inequality holds by the Algebraic Stability Theorem. For the second inequality, note that the contiguous simplicial maps in the diagrams above induce chain maps between the corresponding chain complexes, and these in turn induce equal linear maps at the level of homology vector spaces. To be more precise, first consider the maps $t_{{\delta}+\eta,{\delta}'+\eta}\circ {\varphi}_{\delta}$ and ${\varphi}_{{\delta}'}\circ s_{{\delta},{\delta}'}$. These simplicial maps induce linear maps $(t_{{\delta}+\eta,{\delta}'+\eta}\circ {\varphi}_{\delta})_\#, ({\varphi}_{{\delta}'}\circ s_{{\delta},{\delta}'})_\#: H_k({\mathfrak}{F}^{\delta}) {\rightarrow}H_k({\mathfrak}{G}^{{\delta}'+\eta})$. Because the simplicial maps are assumed to be contiguous, we have: $$(t_{{\delta}+\eta,{\delta}'+\eta}\circ {\varphi}_{\delta})_\# = ({\varphi}_{{\delta}'}\circ s_{{\delta},{\delta}'})_\#.$$ By invoking functoriality of homology, we then have: $$(t_{{\delta}+\eta,{\delta}'+\eta})_\# \circ ({\varphi}_{\delta})_\# = ({\varphi}_{{\delta}'})_\#\circ (s_{{\delta},{\delta}'})_\#.$$ Analogous results hold for the other pairs of contiguous maps. Thus we obtain commutative diagrams upon passing to homology, and so ${\mathcal{H}}_k({\mathfrak}{F}), {\mathcal{H}}_k({\mathfrak}{G})$ are $\eta$-interleaved for each $k\in {\mathbb{Z}}_+$. Thus we get: $${d_{\operatorname{I}}}({\mathcal{H}}_k({\mathfrak}{F}), {\mathcal{H}}_k({\mathfrak}{G}))\leq \eta. \qedhere$$ First we show that: $${d_{\mathcal{N}}}(X,Y) \geq \tfrac{1}{2}\inf\{\max({\operatorname{dis}}({\varphi}),{\operatorname{dis}}(\psi),C_{X,Y}({\varphi},\psi), C_{Y,X}(\psi,{\varphi})) : {\varphi}:X {\rightarrow}Y, \psi:Y {\rightarrow}X \text{ any maps}\}.$$ Let $\eta = {d_{\mathcal{N}}}(X,Y)$, and let $R$ be a correspondence such that ${\operatorname{dis}}(R) = 2\eta$. We can define maps ${\varphi}:X{\rightarrow}Y$ and $\psi:Y{\rightarrow}X$ as follows: for each $x\in X$, set ${\varphi}(x)=y$ for some $y$ such that $(x,y)\in R$. Similarly, for each $y\in Y$, set $\psi(y)=x$ for some $x$ such that $(x,y)\in R$. Thus for any $x \in X, y\in Y$, we obtain $|{\omega}_X(x,\psi(y)) - {\omega}_Y({\varphi}(x),y)| \leq 2\eta$ and $|{\omega}_X(\psi(y),x) - {\omega}_Y(y,{\varphi}(x))| \leq 2\eta$. So we have both $C_{X,Y}({\varphi},\psi) \leq 2\eta$ and $C_{Y,X}(\psi,{\varphi}) \leq 2\eta$. Also for any $x,x' \in X$, we have $(x,{\varphi}(x)),(x',{\varphi}(x')) \in R$. Thus we also have $$|{\omega}_X(x,x') - {\omega}_Y({\varphi}(x),{\varphi}(x'))| \leq 2\eta.$$ So ${\operatorname{dis}}({\varphi}) \leq 2\eta$ and similarly ${\operatorname{dis}}(\psi) \leq 2\eta$. This proves the “$\geq$" case. Next we wish to show: $${d_{\mathcal{N}}}(X,Y) \leq \tfrac{1}{2}\inf\{\max({\operatorname{dis}}({\varphi}),{\operatorname{dis}}(\psi),C_{X,Y}({\varphi},\psi), C_{Y,X}(\psi,{\varphi})) : {\varphi}:X {\rightarrow}Y, \psi:Y {\rightarrow}X \text{ any maps}\}.$$ Suppose ${\varphi}, \psi$ are given, and $\frac{1}{2}\max({\operatorname{dis}}({\varphi}),{\operatorname{dis}}(\psi),C_{X,Y}({\varphi},\psi),C_{Y,X}(\psi,{\varphi})) = \eta$. Let $R_X = {\left\{(x,{\varphi}(x) : x\in X\right\}}$ and let $R_Y = {\left\{(\psi(y),y) : y\in Y\right\}}$. Then $R = R_X \cup R_Y$ is a correspondence. We wish to show that for any $z = (a,b), z' = (a',b') \in R$, $$|{\omega}_X(a,a') - {\omega}_Y(b,b')| \leq 2\eta.$$ This will show that ${\operatorname{dis}}(R) \leq 2\eta$, and so ${d_{\mathcal{N}}}(X,Y) \leq \eta$. To see this, let $z,z' \in R$. Note that there are four cases: (1) $z,z' \in R_X$, (2) $z,z' \in R_Y$, (3) $z \in R_X, z' \in R_Y$, and (4) $z\in R_Y, z'\in R_X$. In the first two cases, the desired inequality follows because ${\operatorname{dis}}({\varphi}), {\operatorname{dis}}(\psi) \leq 2\eta$. The inequality follows in cases (3) and (4) because $C_{X,Y}({\varphi},\psi) \leq 2\eta$ and $C_{Y,X}(\psi,{\varphi}) \leq 2\eta$, respectively. Thus ${d_{\mathcal{N}}}(X,Y) \leq \eta$. It suffices to show that $\Phi$ is a simplicial approximation to ${\mathcal}{E}_{|\Sigma|}$, i.e. whenever ${\mathcal}{E}_{|\Sigma|}(x) \in |{{\sigma}}|$ for some vertex $x \in |{\Sigma}{^{(1)}}|$ and some simplex ${\sigma}\in |{\Sigma}|$, we also have $|\Phi|(x) \in |{{\sigma}}|$ [@spanier-book §3.4]. Here $|{\sigma}|$ denotes the *closed simplex* of ${\sigma}$; for any simplex ${\sigma}=[v_0,\ldots, v_k]$, this is the collection of formal convex combinations $\sum_{i=0}^ka_iv_i$ with $a_i \geq 0$ for each $0\leq i \leq k$ and $\sum_{i=0}^ka_i =1$. Let $x = \sum_{i=0}^ka_i{\sigma}_i$ be a vertex in $|\Sigma{^{(1)}}|$, with each $a_i > 0$. Then we have ${\mathcal}{E}_{|\Sigma|}(x) = \sum_{i=0}^ka_i{\mathcal}{B}({\sigma}_i) = \sum_{i=0}^ka_i\sum_{v\in {\sigma}_i}v/{{\operatorname{card}}({\sigma}_i)},$ a vertex in $|{\sigma}_k|$. Also we have $|\Phi|(x) = \sum_{i=0}^ka_i\Phi({\sigma}_i)$, a vertex in $|{{\sigma}_k}|$. Thus $\Phi$ is a simplicial approximation to ${\mathcal}{E}_{|\Sigma|}$, and so we have $|\Phi|\simeq {\mathcal}{E}_{|\Sigma|}$. Let ${\delta}\in {\mathbb{R}}$. We first claim that ${{\mathfrak}{D}^{\operatorname{si}}}_{\delta}(X) = {{\mathfrak}{D}^{\operatorname{so}}}_{\delta}(X^\top)$. Let ${\sigma}\in {{\mathfrak}{D}^{\operatorname{si}}}_{\delta}(X)$. Then there exists $x'$ such that ${\omega}_X(x,x')\leq {\delta}$ for any $x\in {\sigma}$. Thus ${\omega}_{X^\top}(x',x)\leq {\delta}$. So ${\sigma}\in {{\mathfrak}{D}^{\operatorname{so}}}_{\delta}(X^{\top})$. A similar argument shows the reverse containment. This proves our claim. Thus for ${\delta}\leq {\delta}' \leq {\delta}''$, we obtain the following diagram: $$\begin{tikzcd} {{\mathfrak}{D}^{\operatorname{si}}}_{\delta}(X) \arrow{r}\ar[-, double equal sign distance=3pt]{d} & {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}'}(X) \arrow{r}\ar[-, double equal sign distance=3pt]{d} & {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}''}(X) \arrow{r}\ar[-, double equal sign distance=3pt]{d} & \ldots \\ {{\mathfrak}{D}^{\operatorname{so}}}_{\delta}(X^\top) \arrow{r} & {{\mathfrak}{D}^{\operatorname{so}}}_{{\delta}'}(X^\top)\arrow{r} & {{\mathfrak}{D}^{\operatorname{so}}}_{{\delta}''}(X^\top)\arrow{r} & \ldots \end{tikzcd}$$ Since the maps ${{\mathfrak}{D}^{\operatorname{si}}}_{\delta}{\rightarrow}{{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}'}$, ${{\mathfrak}{D}^{\operatorname{so}}}_{\delta}{\rightarrow}{{\mathfrak}{D}^{\operatorname{so}}}_{{\delta}'}$ for ${\delta}'\geq {\delta}$ are all inclusion maps, it follows that the diagrams commute. Thus at the homology level, we obtain, via functoriality of homology, a commutative diagram of vector spaces where the intervening vertical maps are isomorphisms. By the Persistence Equivalence Theorem (\[thm:pet\]), the diagrams ${\operatorname{Dgm}}_k^{{\operatorname{si}}}(X)$ and ${\operatorname{Dgm}}_k^{{\operatorname{so}}}{(X^\top)}$ are equal. By invoking Corollary \[cor:dowker-dual\], we obtain ${\operatorname{Dgm}}_k^{{\mathfrak}{D}}(X)={\operatorname{Dgm}}_k^{{\mathfrak}{D}}(X^\top)$. Let ${\delta}\in {\mathbb{R}}$. For notational convenience, we write, for each $k\in {\mathbb{Z}}_+$, ---------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------ ${{\mathfrak}{D}^{\operatorname{si}}}_{\delta}:={{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X} $ $C_k^{\delta}:=C_k({{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},X})$ ${\partial}_k^{\delta}:={\partial}_k^{\delta}: C_k^{\delta}{\rightarrow}C_{k-1}^{\delta}$ ${{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},S}:={{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},S_X(z,z')}$ $C_k^{{\delta},S}:=C_k({{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},S_X(z,z')})$ ${\partial}_k^{{\delta},S}:={\partial}_k^{{\delta},S}:C_k^{{\delta},S} {\rightarrow}C_{k-1}^{{\delta},S}.$ ---------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------ First note that pair swaps do not affect the entry of 0-simplices into the Dowker filtration. More precisely, for any $x\in X$, we can unpack the definition of $R_{{\delta},X}$ (Equation \[eq:relation\]) to obtain: $$[x]\in {{\mathfrak}{D}^{\operatorname{si}}}_{\delta}\iff {\omega}_X(x,x)\leq {\delta}\iff {\omega}_X^{z,z'}(x,x)\leq {\delta}\iff [x]\in {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},S}.$$ Thus for any ${\delta}\in {\mathbb{R}}$, we have $C_0^{\delta}= C_0^{{\delta},S}$. Since all 0-chains are automatically 0-cycles, we have $\ker({\partial}_0^{\delta})=\ker({\partial}_0^{{\delta},S})$. Next we wish to show that ${\operatorname{im}}({\partial}_1^{{\delta}})={\operatorname{im}}({\partial}_1^{{\delta},S})$ for each ${\delta}\in {\mathbb{R}}$. Let ${\gamma}\in C_1^{\delta}$. We first need to show the forward inclusion, i.e. that ${\partial}_1^{\delta}({\gamma}) \in {\operatorname{im}}({\partial}_1^{{\delta},S})$. It suffices to show this for the case that ${\gamma}$ is a single 1-simplex $[x,x']\in {{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$; the case where ${\gamma}$ is a linear combination of 1-simplices will then follow by linearity. Let ${\gamma}=[x,x']\in {{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$ for $x,x'\in X$. Then we have the following possibilities: 1. $x'' \in X\setminus\{z,z'\}$ is a ${\delta}$-sink for $[x,x']$. 2. $z$ (or $z'$) is the only ${\delta}$-sink for $[x,x']$, and $x,x'\not\in {\left\{z,z'\right\}}$. 3. $z$ (or $z'$) is the only ${\delta}$-sink for $[x,x']$, and either $x$ or $x'$ belongs to ${\left\{z,z'\right\}}$. 4. $z$ (or $z'$) is the only ${\delta}$-sink for $[x,x']$, and both $x,x'$ belong to ${\left\{z,z'\right\}}$. In cases (1), (2), and (4), the $(z,z')$-pair swap has no effect on $[x,x']$, in the sense that we still have $[x,x']\in {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},S}$. So $[x']-[x]={\partial}_1^{\delta}({\gamma})={\partial}_1^{{\delta},S}({\gamma})\in {\operatorname{im}}({\partial}_1^{{\delta},S})$. Next consider case (3), and assume for notational convenience that $[x,x']=[z,x']$ and $z'$ is the only ${\delta}$-sink for $[z,x']$. By the definition of a ${\delta}$-sink, we have $\overline{{\omega}}_X(z,z')\leq {\delta}$ and $\overline{{\omega}}_X(x',z')\leq {\delta}$. Notice that we also have: $$[z,z'],[z',x']\in {{\mathfrak}{D}^{\operatorname{si}}}_{\delta}, \text{ with $z'$ as a ${\delta}$-sink}.$$ After the $(z,z')$-pair swap, we still have $\overline{{\omega}}_X^{z,z'}(x',z')\leq {\delta}$, but possibly $\overline{{\omega}}_X^{z,z'}(z,z')> {\delta}$. So it might be the case that $[z,x']\not\in {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},S}$. However, we now have: $$\begin{aligned} &[z',x']\in {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},S}, \text{ with $z'$ as a ${\delta}$-sink, and}\\ &[z,z'] \in {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},S}, \text{ with $z$ as a ${\delta}$-sink}.\end{aligned}$$ Then we have: $$\begin{aligned} {\partial}_1^{\delta}({\gamma})={\partial}_1^{\delta}([z,x'])=x'-z &=z'-z + x'-z'\\ &={\partial}_1^{{\delta}}([z,z'])+{\partial}_1^{{\delta}}([z',x'])\\ &={\partial}_1^{{\delta},S}([z,z'])+{\partial}_1^{{\delta},S}([z',x'])\in {\operatorname{im}}({\partial}_1^{{\delta},S}),\end{aligned}$$ where the last equality is defined because we have checked that $[z,z'],[z',x']\in {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},S}$. Thus ${\operatorname{im}}({\partial}_1^{{\delta}})\subseteq {\operatorname{im}}({\partial}_1^{{\delta},S})$, and the reverse inclusion follows by a similar argument. Since ${\delta}\in {\mathbb{R}}$ was arbitrary, this shows that ${\operatorname{im}}({\partial}_1^{{\delta}})= {\operatorname{im}}({\partial}_1^{{\delta},S})$ for each ${\delta}\in {\mathbb{R}}$. Previously we had $\ker({\partial}_0^{\delta})=\ker({\partial}_0^{{\delta},S})$ for each ${\delta}\in {\mathbb{R}}$. It then follows that $H_0({{\mathfrak}{D}^{\operatorname{si}}}_{\delta})=H_0({{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},S})$ for each ${\delta}\in {\mathbb{R}}$. Next let ${\delta}'\geq {\delta}\in {\mathbb{R}}$, and for any $k\in {\mathbb{Z}}_+$, let $f_k^{{\delta},{\delta}'}:C_k^{\delta}{\rightarrow}C_k^{{\delta}'}, g_k^{{\delta},{\delta}'}:C_k^{{\delta},S} {\rightarrow}C_k^{{\delta}',S}$ denote the chain maps induced by the inclusions ${{\mathfrak}{D}^{\operatorname{si}}}_{\delta}{\hookrightarrow}{{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}'}, {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},S} {\hookrightarrow}{{\mathfrak}{D}^{\operatorname{si}}}_{{\delta}',S}$. Since ${{\mathfrak}{D}^{\operatorname{si}}}_{\delta}$ and ${{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},S}$ have the same 0-simplices at each ${\delta}\in {\mathbb{R}}$, we know that $f_0^{{\delta},{\delta}'}\equiv g_0^{{\delta},{\delta}'}$. Let ${\gamma}\in \ker({\partial}_0^{\delta})=\ker({\partial}_0^{{\delta},S})$, and let ${\gamma}+{\operatorname{im}}({\partial}_1^{\delta}) \in H_0({{\mathfrak}{D}^{\operatorname{si}}}_{\delta})$. Then observe that $$\begin{aligned} (f_0^{{\delta},{\delta}'})_\#({\gamma}+ {\operatorname{im}}({\partial}_1^{\delta})) &=f_0^{{\delta},{\delta}'}({\gamma}) + {\operatorname{im}}({\partial}_1^{{\delta}'}) &&\text{($f_0^{{\delta},{\delta}'}$ is a chain map)} \\ &=g_0^{{\delta},{\delta}'}({\gamma}) + {\operatorname{im}}({\partial}_1^{{\delta}'}) &&\text{($f_0^{{\delta},{\delta}'}\equiv g_0^{{\delta},{\delta}'}$)}\\ &=g_0^{{\delta},{\delta}'}({\gamma}) + {\operatorname{im}}({\partial}_1^{{\delta}',S}) &&\text{(${\operatorname{im}}({\partial}_1^{{\delta}'})={\operatorname{im}}({\partial}_1^{{\delta}',S})$)}\\ &=(g_0^{{\delta},{\delta}'})_\#({\gamma}+ {\operatorname{im}}({\partial}_1^{{\delta},S})).&&\text{($g_0^{{\delta},{\delta}'}$ is a chain map)}\end{aligned}$$ Thus $(f_0^{{\delta},{\delta}'})_\#=(g_0^{{\delta},{\delta}'})_\#$ for each ${\delta}'\geq {\delta}\in {\mathbb{R}}$. Since we also have $H_0({{\mathfrak}{D}^{\operatorname{si}}}_{\delta})=H_0({{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},S})$ for each ${\delta}\in {\mathbb{R}}$, we can then apply the Persistence Equivalence Theorem (Theorem \[thm:pet\]) to conclude the proof. Higher dimensional Dowker persistence diagrams of cycle networks {#sec:cycle-addendum} ================================================================ The contents of this section rely on results in [@adamaszek2015vietoris] and [@adamaszek2016nerve]. We introduce some minimalistic versions of definitions from the referenced papers to use in this section. The reader should refer to these papers for the original definitions. Given a metric space $(M,d_M)$ and $m\in M$, we will write $\overline{B(m,{\varepsilon})}$ to denote a closed ${\varepsilon}$-ball centered at $m$, for any ${\varepsilon}> 0$. For a subset $X\subseteq M$ and some ${\varepsilon}>0$, the *Čech complex* of $X$ at resolution ${\varepsilon}$ is defined to be the following simplicial complex: $${\operatorname{\mathbf{\check{C}}}}(X,{\varepsilon}):={\left\{{\sigma}\subseteq X : \cap_{x\in {\sigma}}\overline{B(x,{\varepsilon})} \neq {\varnothing}\right\}}.$$ In the setting of metric spaces, the Čech complex coincides with the Dowker source and sink complexes. We will be interested in the special case where the underlying metric space is the circle. We write $S^1$ to denote the circle with unit circumference. Next, for any $n\in {\mathbb{N}}$, we write ${\mathbb}{X}_n:={\left\{0,\tfrac{1}{n},\tfrac{2}{n},\ldots, \tfrac{n-1}{n}\right\}}$ to denote the collection of $n$ equally spaced points on $S^1$ with the restriction of the arc length metric on $S^1$. Also let $G_n$ denote the $n$-node cycle network with vertex set ${\mathbb}{X}_n$ (in contrast with ${\mathbb}{X}_n$, here $G_n$ is equipped with the asymmetric weights defined in §\[sec:cycle\]). The connection between ${\mathbb}{X}_n$ and Dowker complexes of the cycle networks $G_n$ is highlighted by the following observation: \[prop:dowker-cech-cplx\] Let $n\in {\mathbb{N}}$. Then for any ${\delta}\in [0,1]$, we have ${\operatorname{\mathbf{\check{C}}}}({\mathbb}{X}_n,\frac{{\delta}}{2}) = {\mathfrak}{D}^{{\operatorname{si}}}_{n{\delta},G_n}.$ The scaling factor arises because $G_n$ has diameter $\sim n$, whereas ${\mathbb}{X}_n\subseteq S^1$ has diameter $\sim 1/2$. This proposition provides a pedagogical step which helps us transport results from the setting of [@adamaszek2015vietoris] and [@adamaszek2016nerve] to that of the current paper. For ${\delta}=0$, both the Čech and Dowker complexes consist of the $n$ vertices, and are equal. Similarly for ${\delta}=1$, both ${\operatorname{\mathbf{\check{C}}}}({\mathbb}{X}_n,1)$ and ${\mathfrak}{D}^{{\operatorname{si}}}_{n,G_n}$ are equal to the $(n-1)$-simplex. Now suppose ${\delta}\in (0,1)$. Let ${\sigma}\in {\mathfrak}{D}^{{\operatorname{si}}}_{n{\delta},G_n}$. Then ${\sigma}$ is of the form $[\tfrac{k}{n},\tfrac{k+1}{n},\ldots, \tfrac{{\left \lfloor{k+n{\delta}}\right \rfloor }}{n}]$ for some integer $0\leq k \leq n-1$, where the $n{\delta}$-sink is $\tfrac{{\left \lfloor{k+n{\delta}}\right \rfloor }}{n}$ and all the numerators are taken modulo $n$. We claim that ${\sigma}\in {\operatorname{\mathbf{\check{C}}}}({\mathbb}{X}_n,\tfrac{{\delta}}{2})$. To see this, observe that $d_{S^1}(\tfrac{k}{n},\tfrac{{\left \lfloor{k+n{\delta}}\right \rfloor }}{n}) \leq {\delta}$, and so $\overline{B(\tfrac{k}{n},\tfrac{{\delta}}{2})} \cap \overline{B(\tfrac{{\left \lfloor{k+n{\delta}}\right \rfloor }}{n},\tfrac{{\delta}}{2})} \neq {\varnothing}$. Then we have ${\sigma}\in \bigcap_{i=0}^{n{\delta}}\overline{B\left(\tfrac{{\left \lfloor{k+i}\right \rfloor }}{n},\tfrac{{\delta}}{2}\right)}$, and so ${\sigma}\in {\operatorname{\mathbf{\check{C}}}}({\mathbb}{X}_n,\tfrac{{\delta}}{2})$. Now let ${\sigma}\in {\operatorname{\mathbf{\check{C}}}}({\mathbb}{X}_n,\tfrac{{\delta}}{2})$. Then ${\sigma}$ is of the form $[\tfrac{k}{n},\tfrac{k+1}{n},\ldots, \tfrac{k+j}{n}]$ for some integer $0\leq k\leq n-1$, where $j$ is an integer such that $\tfrac{j}{n} \leq {\delta}$. In this case, we have ${\sigma}= {\mathbb}{X}_n \cap_{i=0}^j\overline{B\left(\tfrac{k+i}{n},{\delta}\right)}$. Then in $G_n$, after applying the scaling factor $n$, we have ${\sigma}\in {\mathfrak}{D}^{{\operatorname{si}}}_{n{\delta},G_n}$, with $\tfrac{k+j}{n}$ as an $n{\delta}$-sink in $G_n$. This shows equality of the two simplicial complexes. \[thm:cech-S1\] Fix $n\in {\mathbb{N}}$, and let $0\leq k \leq n-2$ be an integer. Then, $${\operatorname{\mathbf{\check{C}}}}({\mathbb}{X}_n,\tfrac{k}{2n})\simeq \begin{cases} \bigvee^{n-k-1}S^{2l} & \text{if } \tfrac{k}{n} = \tfrac{l}{l+1},\\ S^{2l+1} &\text{or if } \tfrac{l}{l+1} < \tfrac{k}{n} < \tfrac{l+1}{l+2}, \end{cases}$$ for some $l \in {\mathbb{Z}}_+$. Here $\bigvee$ denotes the wedge sum, and $\simeq$ denotes homotopy equivalence. Let $l \in {\mathbb{N}}$ be such that $(l+1)$ divides $n$ and $0\leq k\leq n-2$. Then ${\mathfrak}{D}^{{\operatorname{si}}}_{k,G_n} = {\operatorname{\mathbf{\check{C}}}}({\mathbb}{X}_n,\tfrac{k}{2n})$ has the homotopy type of a wedge sum of $(n-k-1)$ copies of $S^{2l}$, by Theorem \[thm:cech-S1\]. Here the equality follows from Proposition \[prop:dowker-cech-cplx\]. Notice that $n-k-1 = \tfrac{n}{l+1}-1$. Furthermore, by another application of Theorem \[thm:cech-S1\], it is always possible to choose ${\varepsilon}>0$ small enough so that ${\mathfrak}{D}^{{\operatorname{si}}}_{k-{\varepsilon},G_n} = {\operatorname{\mathbf{\check{C}}}}({\mathbb}{X}_n,\tfrac{k-{\varepsilon}}{2n})$ and ${\mathfrak}{D}^{{\operatorname{si}}}_{k+{\varepsilon},G_n} = {\operatorname{\mathbf{\check{C}}}}({\mathbb}{X}_n,\tfrac{k+{\varepsilon}}{2n})$ have the homotopy types of odd-dimensional spheres. Thus the inclusions ${\mathfrak}{D}^{{\operatorname{si}}}_{k-{\varepsilon},G_n} \subseteq {\mathfrak}{D}^{{\operatorname{si}}}_{k,G_n} \subseteq {\mathfrak}{D}^{{\operatorname{si}}}_{k+{\varepsilon},G_n}$ induce zero maps upon passing to homology. It follows that ${\operatorname{Dgm}}^{{\mathfrak}{D}}_{2l}(G_n)$ consists of the point $(\tfrac{nl}{l+1},\tfrac{nl}{l+1} + 1)$ with multiplicity $\tfrac{n}{l+1} -1$. If $l \in {\mathbb{N}}$ does not satisfy the condition described above, then there does not exist an integer $1\leq j \leq n-2$ such that $j/n = l/(l+1)$. So for each $1\leq j \leq n-2$, ${\mathfrak}{D}^{{\operatorname{si}}}_{j,G_n} = {\operatorname{\mathbf{\check{C}}}}({\mathbb}{X}_n,\tfrac{j}{2n})$ has the homotopy type of an odd-dimensional sphere by Theorem \[thm:cech-S1\], and thus does not contribute to ${\operatorname{Dgm}}^{{\mathfrak}{D}}_{2l}(G_n)$. If $l$ satisfies the condition but $k \geq n-1$, then ${\operatorname{\mathbf{\check{C}}}}({\mathbb}{X}_n,\tfrac{k}{2n})$ is just the $(n-1)$-simplex, hence contractible. Theorem \[thm:dowker-cyc-even\] gives a characterization of the even dimensional Dowker persistence diagrams of cycle networks. The most interesting case occurs when considering the 2-dimensional diagrams: we see that cycle networks of an even number of nodes have an interesting barcode, even if the bars are all short-lived. For dimensions 4, 6, 8, and beyond, there are fewer and fewer cycle networks with nontrivial barcodes (in the sense that only cycle networks with number of nodes equal to a multiple of 4, 6, 8, and so on have nontrivial barcodes). For a complete picture, it is necessary to look at odd-dimensional persistence diagrams. This is made possible by the next set of constructions. We have already recalled the definition of a Rips complex of a metric space. To facilitate the assessment of the connection to [@adamaszek2015vietoris], we temporarily adopt the notation ${\operatorname{\mathbf{VR}}}(X,{\varepsilon})$ to denote the Vietoris-Rips complex of a metric space $(X,d_X)$ at resolution ${\varepsilon}>0$, i.e. the simplicial complex ${\left\{{\sigma}\subseteq X : {\operatorname{diam}}({\sigma}) \leq {\varepsilon}\right\}}$. \[thm:vr-cech-cd\] Let $0< r < \tfrac{1}{2}$. Then there exists a map $T_r: {\operatorname{pow}}(S^1) {\rightarrow}{\operatorname{pow}}(S^1)$ and a map $\pi_r: S^1 {\rightarrow}S^1$ such that there is an induced homotopy equivalence $${\operatorname{\mathbf{VR}}}(T_r(X), \tfrac{2r}{1+2r}) {\xrightarrow}{\simeq} {\operatorname{\mathbf{\check{C}}}}(X,r).$$ Next suppose $X\subseteq S^1$ and let $0< r \leq r' < \tfrac{1}{2}$. Then there exists a map $\eta: S^1 {\rightarrow}S^1$ such that the following diagram commutes: $$\begin{tikzcd}[column sep=large] {\operatorname{\mathbf{VR}}}(T_r(X), \tfrac{2r}{1+2r}) \arrow{r}{\eta} \arrow{d}{\simeq}[swap]{\pi_r} & {\operatorname{\mathbf{VR}}}(T_{r'}(X), \tfrac{2r'}{1+2r'}) \arrow{d}{\simeq}[swap]{\pi_{r'}}\\ {\operatorname{\mathbf{\check{C}}}}(X,r) \arrow[hookrightarrow]{r}{\subseteq} & {\operatorname{\mathbf{\check{C}}}}(X,r') \end{tikzcd}$$ \[thm:cyc-cech-main\] Consider the setup of Theorem \[thm:vr-cech-cd\]. If ${\operatorname{\mathbf{\check{C}}}}(X,r)$ and ${\operatorname{\mathbf{\check{C}}}}(X,r')$ are homotopy equivalent, then the inclusion map between them is a homotopy equivalence. Before providing the proof, we show how it implies Theorem \[thm:dowker-cyc-odd\]. By Proposition \[prop:dowker-cech-cplx\] and Theorem \[thm:cech-S1\], we know that ${{\mathfrak}{D}^{\operatorname{si}}}_{k,G_n} = {\operatorname{\mathbf{\check{C}}}}({\mathbb}{X}_n,\tfrac{k}{2n}) \simeq S^1 $ for integers $0 < k < \tfrac{n}{2}$. Let $b \in {\mathbb{N}}$ be the greatest integer less than $n/2$. Then by Theorem \[thm:cyc-cech-main\], we know that each inclusion map in the following chain is a homotopy equivalence: $${{\mathfrak}{D}^{\operatorname{si}}}_{1,G_n} \subseteq \ldots \subseteq {{\mathfrak}{D}^{\operatorname{si}}}_{b,G_n} = {{\mathfrak}{D}^{\operatorname{si}}}_{{\left \lceil{n/2}\right \rceil }^-,G_n}.$$ It follows that ${\operatorname{Dgm}}^{{\mathfrak}{D}}_1(G_n) = {\left\{\left(1,{\left \lceil{\tfrac{n}{2}}\right \rceil }\right)\right\}}$. The notation in the last equality means that ${{\mathfrak}{D}^{\operatorname{si}}}_{b,G_n} = {{\mathfrak}{D}^{\operatorname{si}}}_{{\delta},G_n}$ for all ${\delta}\in [b,b+1)$, where $b+1 = {\left \lceil{n/2}\right \rceil }$. In the more general case, let $l \in {\mathbb{N}}$ and let $M_l$ be as in the statement of the result. Suppose first that $M_l$ is empty. Then by Proposition \[prop:dowker-cech-cplx\] and Theorem \[thm:cech-S1\], we know that ${{\mathfrak}{D}^{\operatorname{si}}}_{k,G_n}$ has the homotopy type of a wedge of even-dimensional spheres or an odd-dimensional sphere of dimension strictly different from $(2l+1)$, for any choice of integer $k$. Thus ${\operatorname{Dgm}}^{{\mathfrak}{D}}_{2l+1}(G_n)$ is trivial. Next suppose $M_l$ is nonempty. By another application of Proposition \[prop:dowker-cech-cplx\] and Theorem \[thm:cech-S1\], we know that ${{\mathfrak}{D}^{\operatorname{si}}}_{k,G_n} = {\operatorname{\mathbf{\check{C}}}}({\mathbb}{X}_n,\tfrac{k}{2n}) \simeq S^{2l+1} $ for integers $\tfrac{nl}{l+1} < k < \tfrac{n(l+1)}{l+2}$. Write $a_l:=\min{\left\{m\in M_l\right\}}$ and $b_l:=\max{\left\{m\in M_l\right\}}$. Then by Theorem \[thm:cyc-cech-main\], we know that each inclusion map in the following chain is a homotopy equivalence: $${{\mathfrak}{D}^{\operatorname{si}}}_{a_l,G_n} \subseteq \ldots \subseteq {{\mathfrak}{D}^{\operatorname{si}}}_{b_l,G_n} = {{\mathfrak}{D}^{\operatorname{si}}}_{{\left \lceil{n(l+1)/(l+2)}\right \rceil }^-,G_n}.$$ It follows that ${\operatorname{Dgm}}^{{\mathfrak}{D}}_{2l+1}(G_n) = {\left\{\left(a_l,{\left \lceil{\tfrac{n(l+1)}{l+2}}\right \rceil }\right)\right\}}$. It remains to provide a proof of Theorem \[thm:cyc-cech-main\]. For this, we need some additional machinery. #### Cyclic maps and winding fractions We introduce some more terms from [@adamaszek2015vietoris], but for efficiency, we try to minimize the scope of the definitions to only what is needed for our purpose. Recall that we write $S^1$ to denote the circle with unit circumference. So any $x\in S^1$ can be naturally identified with a point in $[0,1)$. We fix a choice of $0\in S^1$, and for any $x,x' \in S^1$, the length of a clockwise arc from $x$ to $x'$ is denoted by $\overrightarrow{d_{S^1}}(x,x')$. Then, for any finite subset $X\subseteq S^1$ and any $r \in (0,1/2)$, the *directed Vietoris-Rips graph* ${\overrightarrow{\operatorname{VR}}}(X,r)$ is defined to be the graph with vertex set $X$ and edge set $\{(x,x') : 0 < \overrightarrow{d_{S^1}}(x,x') < r\}$. Next, let $\overrightarrow{G}$ be a Vietoris-Rips graph such that the vertices are enumerated as $x_0,x_1,\ldots, x_{n-1}$, according to the *clockwise* order in which they appear. A *cyclic map* between $\overrightarrow{G}$ and a Vietoris-Rips graph $\overrightarrow{H}$ is a map of vertices $f$ such that for each edge $(x,x') \in \overrightarrow{G}$, we have either $f(x)=f(x')$, or $(f(x),f(x')) \in \overrightarrow{H}$, and $\sum_{i=0}^{n-1}\overrightarrow{d_{S^1}}(f(x_i),f(x_{i+1})) = 1$. Here $x_n:=x_0$. Next, the *winding fraction* of a Vietoris-Rips graph $\overrightarrow{G}$ with vertex set $V(\overrightarrow{G})$ is defined to be the infimum of numbers $\tfrac{k}{n}$ such that there is an order-preserving map $V(\overrightarrow{G}) {\rightarrow}{\mathbb{Z}}/n{\mathbb{Z}}$ such that each edge is mapped to a pair of numbers at most $k$ apart. A key property of the winding fraction, denoted ${\operatorname{wf}}$, is that if there is a cyclic map between Vietoris-Rips graphs $\overrightarrow{G} {\rightarrow}\overrightarrow{H}$, then ${\operatorname{wf}}(\overrightarrow{G}) \leq {\operatorname{wf}}(\overrightarrow{H})$. \[thm:vrcirc-main\] Let $X\subseteq S^1$ be a finite set and let $0 < r < \tfrac{1}{2}$. Then, $${\operatorname{\mathbf{VR}}}(X,r) \simeq \begin{cases} S^{2l+1} &: \tfrac{l}{2l+1} < {\operatorname{wf}}({\overrightarrow{\operatorname{VR}}}(X,r)) < \tfrac{l+1}{2l+3} \text{ for some } l\in {\mathbb{Z}}_+,\\ \bigvee^j S^{2l} &: {\operatorname{wf}}({\overrightarrow{\operatorname{VR}}}(X,r)) = \tfrac{l}{2l+1}, \text{ for some } j\in {\mathbb{N}}. \end{cases}$$ Next let $X' \subseteq S^1$ be another finite set, and let $r \leq r' < \tfrac{1}{2}$. Suppose $f:{\overrightarrow{\operatorname{VR}}}(X,r) {\rightarrow}{\overrightarrow{\operatorname{VR}}}(X',r')$ is a cyclic map between Vietoris-Rips graphs and $\tfrac{l}{2l+1} < {\operatorname{wf}}({\overrightarrow{\operatorname{VR}}}(X,r)) \leq {\operatorname{wf}}({\overrightarrow{\operatorname{VR}}}(X',r')) < \tfrac{l+1}{2l+3}$. Then $f$ induces a homotopy equivalence between ${\operatorname{\mathbf{VR}}}(X,r)$ and ${\operatorname{\mathbf{VR}}}(X',r')$. We now have the ingredients for a proof of Theorem \[thm:cyc-cech-main\]. Since the maps $\pi_r$ and $\pi_{r'}$ induce homotopy equivalences, it follows that $${\operatorname{\mathbf{VR}}}(T_r(X),\tfrac{2r}{1+2r}) \simeq {\operatorname{\mathbf{VR}}}(T_{r'}(X),\tfrac{2r'}{1+2r'}).$$ By the characterization result in Theorem \[thm:vrcirc-main\], we know that there exists $l \in {\mathbb{Z}}_+$ such that $$\tfrac{l}{2l+1} < {\operatorname{wf}}({\overrightarrow{\operatorname{VR}}}(T_r(X),\tfrac{2r}{1+2r})) \leq {\operatorname{wf}}({\overrightarrow{\operatorname{VR}}}(T_{r'}(X),\tfrac{2r'}{1+2r'})) < \tfrac{l+1}{2l+3}.$$ The map $\eta$ in Theorem \[thm:vr-cech-cd\] appears in [@adamaszek2015vietoris Proposition 9.5] through an explicit construction. Moreover, it is shown that $\eta$ induces a cyclic map ${\operatorname{wf}}({\overrightarrow{\operatorname{VR}}}(T_r(X),\tfrac{2r}{1+2r})) {\rightarrow}{\operatorname{wf}}({\overrightarrow{\operatorname{VR}}}(T_{r'}(X),\tfrac{2r'}{1+2r'}))$. Thus by Theorem \[thm:vrcirc-main\], $\eta$ induces a homotopy equivalence between ${\operatorname{\mathbf{VR}}}(T_r(X),\tfrac{2r}{1+2r})$ and ${\operatorname{\mathbf{VR}}}(T_{r'}(X),\tfrac{2r'}{1+2r'})$. Finally, the commutativity of the diagram in Theorem \[thm:vr-cech-cd\] shows that the inclusion ${\operatorname{\mathbf{\check{C}}}}(X,r) \subseteq {\operatorname{\mathbf{\check{C}}}}(X,r')$ induces a homotopy equivalence. The analogue of Theorem \[thm:cyc-cech-main\] for Čech complexes appears as Proposition 4.9 of [@adamaszek2015vietoris] for Vietoris–Rips complexes. We prove Theorem \[thm:cyc-cech-main\] by connecting Čech and Vietoris-Rips complexes using Proposition 9.5 of [@adamaszek2015vietoris]. However, as remarked in §9 of [@adamaszek2015vietoris], one could prove Theorem \[thm:cyc-cech-main\] directly using a parallel theory of winding fractions for Čech complexes. [^1]: A thread with ideas towards the proof of Theorem \[thm:dowker-strong\] was discussed in [@Nlab-dowker last accessed 4.24.2017], but the proposed strategy was incomplete. We have inserted an addendum in [@Nlab-dowker] proposing a complete proof with a slightly different construction.

--- abstract: 'The CMS Beam Conditions and Radiation Monitoring System, BRM, will support beam tuning, protect the CMS detector from adverse beam conditions, and measure the accumulated dose close to or inside all sub-detectors. It is composed of different sub-systems measuring either the particle flux near the beam pipe with time resolution between nano- and microseconds or the integrated dose over longer time intervals. This paper presents the Fast Beam Conditions Monitor, BCM1F, which is designed for fast flux monitoring measuring both beam halo and collision products. BCM1F is located inside the CMS pixel detector volume close to the beam-pipe. It uses sCVD diamond sensors and radiation hard front-end electronics, along with an analog optical readout of the signals. The commissioning of the system and its successful operation during the first beams of the LHC are described.' address: - 'Brandenburgische Technische Universität, 03046 Cottbus, Germany' - 'CERN, 1211 Geneva 23, Switzerland' - 'DESY, 15738 Zeuthen, Germany' - 'Rutgers University, 08854 Piscataway, NJ, USA' - 'Université de Genève, 1211 Geneva, Switzerland' - 'Canterbury University, 8041 Christchurch, New Zealand' - 'University of Wisconsin, Madison, WI 53706-1481, USA' author: - 'A. Bell' - 'E. Castro' - 'R. Hall-Wilton' - 'W. Lange' - 'W. Lohmann' - 'A. Macpherson' - 'M. Ohlerich' - 'N. Rodriguez' - 'V. Ryjov' - 'R.S. Schmidt' - 'R.L. Stone' title: Fast Beam Conditions Monitor BCM1F for the CMS Experiment --- LHC ,CMS ,beam conditions ,sCVD diamonds ,radiation hard sensors Introduction {#intro} ============ The CMS experiment [@CMS] at LHC [@LHC] will be situated in an unprecedentedly high radiation field. The LHC is designed to run with 362 MJ of stored energy in one beam and with proton intensities of more than 10$^{14}$ per beam. These beams will generate a continuous flux of halo particles near the beam-pipe and when colliding also interaction secondaries, predominantly at small polar angles. The collected dose will be largest for the innermost detectors, which are therefore designed with very high radiation tolerance. However, also short term losses of the beams may cause serious damage to detector elements, in particular to front-end electronics due to large ionisation. In addition, the innermost detectors need a sufficiently low occupancy level for successful data taking. To monitor the particle fluxes near the beam-pipe and the radiation level in the sub-detectors, beam conditions and radiation monitors, BRM [@CMS; @BCM1], are installed in the CMS detector. Several slow systems of BRM will be used to measure the accumulated dose near the volume of all sub-detectors. These measurements are necessary to understand potential longer term damage to detector elements. Particle flux monitors are installed close to the beam-pipe. Two such monitors measure the integral particle flux over half an orbit or over a bunch train using the signal current in polycrystalline small diamond sensors and integrate it over about 40 $\mu$s and 5 $\mu$s, respectively. These fluence measurements will assist beam tuning, indicate critical beam halo conditions for the inner detectors and initiate LHC beam aborts when conditions are such that detectors might be endangered. These systems are described in detail in Ref. [@abell]. Other LHC experiments also installed beam condition monitors using polycrystalline diamond sensors [@other_experiments]. The Fast Beam Conditions Monitor, hereafter referred to as BCM1F, will be sensitive to very fast changes of the beam conditions and provide diagnostics with a time resolution better than the time between bunch crossings, hence, for example, being able to flag problematic beam conditions resulting in bursts of beam loss over very short periods of time. Such beam losses are considered to be one of the principle damage scenarios for CMS detector components. In addition, it will store real-time data to allow post-mortem analyses in the case of beam accidents. BCM1F System Overview {#over} ===================== BCM1F uses single-crystal CVD[^1] diamond sensors, hereafter denoted as sCVD, for particle detection. Sensors made of sCVD are sufficiently fast to match the time resolution requirements, and small enough to be inserted into areas close to key detector components without adding substantial material or services. Four sCVD sensors, each with a volume of 5$\times$5$\times$0.5$\mm^3$, are positioned in a plane perpendicular to the beam-pipe on each side of the IP at a distance of about 1.8 m and at a radius of 4.5$\cm$ from the nominal beam position, as sketched in Figure \[fig:CMS\_detector\]. ![A sketch of the CMS detector. The positions of the BCM1F in front of the pixel detector and inside the tracker are indicated by the arrows. At both sides of the IP a carbon fibre structure around the beam-pipe, as shown in the upper part, supports the sensor modules.[]{data-label="fig:CMS_detector"}](CMS_BCM_carriages.eps){width="4in"} The position of the BCM1F is chosen to be optimal in terms of time separation between ingoing and outgoing particles from the IP. Relativistic particles need about 6$\ns$ to move between one of the detector planes and the IP. Hence, gated rate measurements of the BCM1F will allow to separate the fluxes from both beam halo of each direction and interactions products. Each sensor is connected to a radiation hard preamplifier. Its output signal is transmitted to the counting room over an analog optical link as shown schematically in Figure \[fig:readout\_scheme\]. ![ The readout scheme of BCM1F. The signal generated in the sensor is amplified and shaped in a JK16 preamplifier. The analog opto-hybrid drives a laser for analog signal transmission via a single-mode fibre. The signals are digitized and processed in the counting room. []{data-label="fig:readout_scheme"}](BCM1F_Readout_Sketch.eps){width="5.2in"} Since neither cooling nor slow control equipment are available at the mounting positions, the modules must be operated with low power dissipation and should work over long periods without a re-adjustment of the calibration parameters. At the back-end of the readout in the counting room signals are digitized and processed in a PC. Flash ADCs, scalers and multi-hit recording TDCs allow e.g. to monitor counts as a function of time over an orbit. Test-pulses are used to check the functionality of the system. Rates, multiplicities, timing and coincidence information are monitored and stored independently of the CMS data acquisition. sCVD Sensors {#sCVD} ============ Outstanding properties, such as a very low leakage current with negligible temperature dependence, a fast signal response and radiation hardness, make CVD diamond sensors attractive for the locations close to the interaction region. In previous experiments polycrystalline diamond sensors have been successfully used as beam conditions monitors [@CDF; @Babar] by measuring the currents created in the sensor by the crossing particles. However, integration over a certain time limits the time resolution of such devices. In addition, due to crystal defects the charge collection efficiency of polycrystalline CVD sensors is below 50% which may result in a signal-to-noise ratio not sufficient for the detection of minimum ionising particles, MIPs. Here single crystal CVD diamond sensors are used. They are characterised by nearly 100% charge collection efficiency and allow to count MIPs. They are operated as solid state ionisation chambers by applying high voltage to thin metal plates on both sides of a sensor to create an electric field in the bulk, as shown in Figure \[fig:readout\_scheme\]. Signals from crossing charged particles are created due to the drift of electrons and holes released in the bulk material. The sCVD sensors are of 5$\times$5$\mm^2$ area and 500 $\mu$m thickness. They have been manufactured by Element Six [@E6] after a few years of development and research in collaboration with the CERN RD42 project [@RD42]. A first application of an sCVD diamond sensor in a collider experiment was described in Ref. [@zeus]. ![ The schematic of the back-end readout. []{data-label="fig:backend"}](scheme_backend.eps){width="4in" height="6cm"} ![ The spectrum of signals from relativistic electrons of a $^{90}$Sr source taken with a fully assembled BCM1F module read out with a charge-integrating ADC. []{data-label="fig:flash_spectrum"}](characterisation_fullChain_s-w.eps){width="4in"} ![ The peak value of the signal spectrum as a function of the bias voltage. []{data-label="fig:signal_vs_voltage"}](signalYield_module10_s-w.eps){width="4in"} ![ The size of the analog pulse measured at the opto-receiver module output as a function of the amplitude of the test-pulse fed into the preamplifier. []{data-label="fig:linearity"}](linearity_largerTitle_s-w.eps){width="4in"} ![ Left side: The carbon fibre carrier structure of BCM1 with the four modules installed. Right side: Each module contains the sCVD sensor carrier (top left) and the preamplifier with the laser driver (top right), which are combined to a sandwich (middle) and then connected with cables and optical fibres and protected by an aluminum cage (bottom). []{data-label="fig:full_module"}](BCM1_subcomponents.eps){width="3.5in"} Sensor Tests {#test} ============ Before installation the sensors were tested in the laboratory. The leakage currents and signal response to electrons from a $^{90}$Sr source were measured for all diamond sensors before assembly. The leakage currents of the sensors are in the range of a few pico-Amperes. The signal amplitude, expressed as the average collected charge, is shown as a function of the bias voltage for both polarities in Figure \[fig:cvd\_characteristics\] for one of the sensors. The signal amplitude increases for increasing bias voltage up to about 120 V, corresponding to a field strength of about 0.25 V$\mu$m$^{-1}$, and is constant thereafter. The measurements for the other sensors show a very similar behavior. In a few cases slight differences in the signal size for different bias voltage polarities are observed. These sensors are operated with the polarity of the bias voltage giving the maximum signal yield. Two sample sensors were irradiated in a 60$\MeV$ proton beam at PSI up to a fluence of about 3$\times$10$^{14}$ protons per cm$^2$, corresponding to a fluence of 17.5$\times$10$^{14}$ MIPs per $\cm^2$ [@rad-hard_studies]. The signal amplitude obtained from electrons of a $^{90}$Sr source is shown in Figure \[fig:cvd\_characteristics\] for two sensors after irradiation. It drops to about 20% of the one measured with a non-irradiated sensor. Whereas in the non-irradiated sample the signal amplitude saturates already at about 0.25 V$\mu$m$^{-1}$, in the irradiated sample no saturation is seen up to 1 V$\mu$m$^{-1}$. Comparing these results to previous studies [@rad-hard_studies] of the performance of diamond sensors as a function of the fluence of 26$\MeV$ and 24$\GeV$ protons the hypothesis of enhanced damage at lower particle energies is supported. The fluence investigated here is approximately that expected at the location of the BCM1F detector over the baseline LHC program. Readout Electronics {#FE} =================== Each sensor is connected to a JK16 radiation hard amplifier ASIC [@FE1]. The chip is fabricated in a commercial 0.25 $\mu$m CMOS technology hardened by appropriate layout techniques. Each channel comprises a fast trans-impedance preamplifier with an active feedback loop and an amplifier-integrator stage with 20$\ns$ peaking time. An excellent noise performance is achieved by a careful adjustment of the feedback current through the gate voltage of the feedback FET. For a detector capacitance of 5 pF the measured noise amounts to about 700 electrons ENC in agreement with the specifications given in Ref. [@FE1]. The measured charge gain is 20$\mV$/fC. The analog signals are transmitted to the counting room using an analog optical chain [@optics1] developed for the CMS tracker. The preamplifier’s single-ended output is AC coupled to the custom-designed laser driver ASIC, which modulates the current of the edge-emitting laser diode. Single mode fibres from the pigtailed lasers are connected at the periphery of the tracker volume to an optical fan-in, which merges single fibres into a 12-fiber ribbon cable. In the counting room a corresponding ribbon connects directly to a 12-channel analog optical receiver card in a VME crate. ![ A snapshot of first signals in BCM1F from September 2008. The trigger was taken from the BPTX bunch pickup (yellow line). The blue and the green signals result from an analog sum of the BCM1F signals of each side. The shift of time between the two signals corresponds roughly to the time of flight of a relativistic particle between both sensor planes. []{data-label="fig:pulse_scope"}](bcm1f_scope.eps){width="4in"} ![ Distribution of the difference between arrival times of signals from sensors in different z-planes but equal azimuthal angles. []{data-label="fig:arrivel_time"}](time-lag_samePos_offsetCorrected_fit_s-w.eps){width="3.5in"} Minor modifications on the laser driver ASIC board were done to allow mounting in two opposite orientations of the laser diode, required by the minimal bending radius of the pigtail fibres. In contrast to the tracker application, for the BCM1F the gain and the laser diode bias current cannot be programmed via the foreseen I$^2$C interface. Hence, attention was paid to choose the input polarity and the laser bias setting to preserve the dynamic range of the receiver side. In addition, the impacts of heat dissipation and the expected radiation dose on the laser diode performance degradation were taken into account. To ensure a small package size a piggy-back architecture was used for interconnecting and mounting the sensor, the amplifier and the analog optical hybrid boards on their carriage. At the back-end side of the readout the optical signals are converted into electrical signals using an analog opto-receiver module. Its output signals are distributed by analog fan-outs to ADC inputs and to discriminators. A flash ADC performing 500 MS/s with 8 bit resolution, V1721 from CAEN, is used to digitise the signals. This module can be triggered internally or externally. It can read out in full data mode up to 45 consecutive beam orbits or a corresponding number of user definable time intervals. Data is written into a ring buffer and tagged with time stamps. It is read out via an on-board optical link and processed in a PC. The discriminated signals are counted in all channels with a V260 scaler from CAEN and used for on-line displays of hit rates. In addition, they are digitised with multi-hit capable TDCs V767 from CAEN with 20 bit dynamic range and 0.8 ns-LSB resolution. The TDCs and the scalers are read out via a VME-bridge. They will allow orbit-by-orbit counts to be obtained as a function of time for a detailed monitoring of beam halo and interaction products. Test-pulses are used to check the functionality of the system during operation. The amplitude of the test-pulse induces a signal similar to one MIP in the preamplifier. A schematic of the complete back-end is shown in Figure \[fig:backend\]. Performance of the System Before Installation {#performance} ============================================= The assembled front-end modules were tested before installation using a $^{90}$Sr source. Relativistic electrons crossing the sensor trigger a scintillation counter. An example of a spectrum recorded with a charge-integrating ADC is shown in Figure \[fig:flash\_spectrum\]. The distribution of the signal charge shows the expected Landau-shape. The signal is clearly separated from the pedestal peak. A signal-to-noise ratio of about 12 is estimated. The values for the other channels are very similar. These spectra were acquired for a range of increasing voltage applied across the sensor. Figure \[fig:signal\_vs\_voltage\] shows the most probable values of the pedestal-subtracted pulse height distributions measured as a function of the bias voltage. The maximum signal was reached, as expected, at an electric field of about 0.25 V$\mu$m$^{-1}$, corresponding to a bias voltage of 125 V for a sensor with 500 $\mu$m thickness. Above this voltage the signal is constant. The linearity of the response of the whole readout chain was investigated using test-pulses fed in a dedicated preamplifier input. The result is shown in Figure \[fig:linearity\]. For both polarities a linear response is found up to test-pulse amplitudes corresponding to approximately 5 MIP equivalents. For test-pulse amplitudes above this value the readout becomes non-linear and approaches saturation at about 10 MIPs. To test the proper functionality in the cooled tracker environment of CMS, the modules were operated in a climate chamber with five temperature cycles from $-$20 to $+$50 $^\circ$C. Leakage and supply currents as well as the test-pulse response were measured and found to be in the expected range. The stored results of these measurements will be compared with measurements of the same quantities taken during operation of the modules in CMS. Installation in CMS {#installation} =================== The components of the modules and the completed modules mounted on the carrier structure are shown in Figure \[fig:full\_module\]. The modules contain in addition to the BCM1F components also sensors from the current monitor BCM1L. Both systems are shielded with a double-cage structure. The inner cage is connected to the ground of the back-end readout. The outer cage is connected to the carbon-fibre support[^2]. The two cages are insulated from each other to mitigate frequency dependent pick-up effects on carbon fibre structures observed elsewhere [@johnson]. The eight BCM1F modules with their corresponding infrastructure were successfully installed and tested at the beginning of August 2008. First Measurements with LHC Beams {#measurements} ================================== When first beam circulated in the LHC at the beginning of September 2008, the BCM1F was operational and signals from beam-halo particles were recorded. One of the first beam-generated signals from BCM1F, observed on an oscilloscope, is shown in Figure \[fig:pulse\_scope\]. The readout of the BCM1F modules was triggered by a bunch pickup detector, BPTX [@bptx], indicating that a proton bunch crossed the CMS detector. A time window of about 500$\ns$ with respect to the trigger was recorded by the ADC. Figure \[fig:pulse\_height\] shows, as an example, the spectrum of signals from one of the detector modules. The signal size is obtained by integration of the signal pulse over time. The distributions for the other channels are very similar. The signal-to-noise ratios obtained from these distributions are also shown in Figure \[fig:pulse\_height\] for all channels[^3]. The values of the signal-to-noise ratio vary between 15 and 25 and are slightly better than the ones obtained in laboratory measurements before installation. Since the LHC was filled with beam in one direction only, the beam halo particles should follow this direction. To demonstrate the capabilities of BCM1F the signal arrivals times in the flash ADC are measured with respect to the BPTX trigger signal. The distribution of the difference between arrival times of signals from sensors in different z-planes but equal azimuthal positions is shown in Figure \[fig:arrivel\_time\]. From a Gaussian fit, a value of 12.4$\ns$ is obtained for the time difference, corresponding precisely to the expected time-of-flight of a relativistic particle between the two BCM1F planes on the $+$z and $-$z side. The variance of the Gaussian amounts to 1.8$\ns$, leading to an estimate of the single hit timing resolution of 1.3 ns. Conclusions ============ The BCM1F is a fully functional sub-detector of the BRM system of CMS and will be vital for monitoring beam conditions close to the beam-pipe inside CMS. It comprises 8 modules each containing an sCVD diamond sensor, a front-end ASIC and an optical analog signal transmission to scalers, flash ADCs and TDCs. The system is operated independently from the other CMS sub-detectors. System tests of each module in the laboratory show that performance matches requirements. Samples of sCVD sensors were exposed to a high intensity proton beam up to fluences expected at the location of BCM1F for the nominal LHC running. The signal amplitude measured for MIPs is reduced to 20% of the original one, approaching a level critically low for MIP counting. Using the ADC the size of the MIP amplitude will be monitored as a function of the LHC operation time allowing us to replace the sensors if necessary. BCM1F was successfully installed and was operational when LHC was filled with first beam. Data taken with beam show a slightely better signal-to-noise as reached in the laboratory tests. A measurement of the signal arrival times using a flash ADC indicates a promising single hit timing resolution of about 1.3$\ns$. This will allow the separation of incoming halo particles correlated to a certain bunch from interaction products created at the IP. Different readout-modes and time windows for data capture are programmable. Data can be written on a local disk and published to the CMS readout system. Local data preprocessing will deliver the shift-crew a detailed picture on the beam-halo count rates as a function of the time. Data of several orbits stored in circular memory will allow diagnostics just after a beam abort. The BCM1F detector is ready for data taking in the commissioning phase of the LHC. Acknowledgments =============== We thank J.P. Chatelain for his contributions to design and manufacture the mechanical structures. We would like to thank our colleagues in the BRM group for their advice and assistance. We are grateful to CMS Technical Coordination for assistance in launching and sustaining the project, particularly in its early phases. We also acknowledge the contribution of the CERN, CMS and DESY technical teams which helped bring it to a successful conclusion. We express our gratitude to PSI for allowing us to use the proton beam and in particular to K. Deiters for his collaboration there. The Rutgers University group is grateful for the support by NSF. R. Hall-Wilton is grateful for the support of the Israeli Technical Associates Program. [00]{} The CMS Collaboration, S. Chatrchyan et al. ,“The CMS experiment at the CERN LHC”, 2008 JINST 3S08004. L. Evans and P. Bryant (editors), “LHC Machine”, 2008 JINST 3S08001. L. Fernandez-Hernando et al., “Development of a CVD diamond Beam Condition Monitor for CMS at the Large Hadron Collider”, Nucl. Inst. and Meth. A552 (2005) 183, A. Macpherson, “Beam Condition Monitoring and radiation damage concerns of the experiment”, Proceedings LHC Project Workshop, Chamonix XV (2006) 198, D. Chong et al., “Validation of synthetic diamond for a Beam Condition Monitor for the Compact Muon Solenoid Experiment”, IEEE Trans. Nucl. Sci. 54 (2007) 182. A. J. Bell on the behalf of the BRM group, “Beam and Radiation Monitoring for CMS”, 2008 IEEE Nuclear Science Symposium Conference Record, pp 2322-2325, http://ieeexplore.ieee.org. V. Cidro et al., “The ATLAS Beam Conditions Monitor”, 2008 JINST 3P02004, The LHCb Collaboration, A. Augusto Alves Jr et al., “The LHCb Detector at the LHC”, 2008 JINST 3S08005. P. Dong et al., “Beam Condition Monitoring With Diamonds at CDF” IEEE Trans. Nucl. Sci. 55 (2008) 328. M. Brunisma et al., “CVD diamonds in the BaBar radiation monitoring system”, Nucl. Phys. B150 (2006) 164. Element Six Ltd., King’s Ride Park, Ascot, Berkshire SL5 8BP, UK. RD42 Status Report, http://cdsweb.cern.ch/record/1098155. W. Lange et al., “INVESTIGATION OF A SINGLE CRYSTAL DIAMOND SENSOR AND ITS APPLICATION IN BACKGROUND MEASUREMENTS FOR HERA”, talk given at the NoRHDia Workshop 2006, GSI Darmstadt, http://www-norhdia.gsi.de/index.php?mgid=6\#3rd. RD-42 Collaboration, CERN, “Development of Diamond Tracking Detectors for High Luminosity Experiments at the LHC”, CERN-LHCC-2006-10 (2006), W. de Boer et al., “Radiation Hardness of Diamond and Silicon Sensors Compared”, Physica Status Solidi A204 (2007) 3004. J. Kaplan and W. Dabrowski, “Fast CMOS Binary Front End for Silicon Strip Detectors at LHC Experiments”, IEEE Trans. Nucl. Sci. 52 (2005) 2713. J. Troska et al., “Optical readout and control systems for the CMS Tracker”, IEEE Trans. Nucl. Sci. 50 (2003) 1067, G. Cervelli et al., “A radiation tolerant linear laser driver array for optical transmission in the LHC experiments”, Proc. 7th Workshop on Electronics for LHC Experiments, CERN/LHCC/2001-034 (2001) pp 155-159. M. Johnson, “Grounding and Shielding Techniques for Large Scale Experiments”, Proceedings of the 8th Workshop for Electronics for the LHC (2002), B. Quinn, “Carbon Fibre Grounding Design in the D0 Run IIb Silicon Detector Design”, Nucl. Inst. and Meth. A511 (2003) 180. C. Ohm, “Phase and Intensity Monitoring of the Particle Beams at the ATLAS Experiment”, Linköping University, The Department of Physics, Chemistry and Biology, LITH-IFM-EX-07/1808-SE (2007), C. Ohm and T. Pauly, arxiv.org/pdf/0905.3648 (2009), T. Aumeyr, “Beam Phase and Intensity Monitoring for the Compact Muon Solenoid Experiment”, Vienna University of Technology (2008). [^1]: Chemical Vapor Deposition [^2]: The carbon fiber support is not connected to the CMS detector ground. [^3]: Channel one had a faulty cable at the ADC input at the time of this measurement.

--- abstract: 'The Chern-Simons forms for $\mathbb{R}$-linear connections on Lie algebroids are considered. A generalized Chern-Simons formula for such $\mathbb{R}$-linear connections is obtained. We it apply to define Chern character and secondary characteristic classes for $\mathbb{R}$-linear connections of Lie algebroids.' author: - Bogdan Balcerzak title: | Chern-Simons forms for $\mathbb{R}$-linear connections on\ Lie algebroids --- [^1][^2] Introduction ============ We observe that non-linear objects (forms, connections, mappings between modules of cross-sections of vector bundles, which are non–linear over a ring of smooth functions) have increasing meaning in problems of differential geometry. S. Evens, J. H. Lu and A. Weinstein considered especial non–linear connections of Lie algebroids called connections up to homotopy (see [@Evens-Lu-Weinstein]). Crainic and Fernandes [Crainic-up to homotopy]{}, [@Crainic-Fernandes-jets] introduce the Chern character for non–linear connections. They discuss non–linear forms on Lie algebroids with values in a super vector bundle as antisymmetric, multilinear maps over $\mathbb{R}$ (not necessarily multilinear over the ring of smooth functions), which have a local property. Every non–linear connection $\nabla $ establishes on non–linear forms the covariant derivative operator. If $\nabla $ is flat, the Chern character vanishes and the induced covariant derivative operator is the exterior derivative, and in classically way defines the cohomology space. Crainic and Fernandes introduced secondary characteristic classes for connections up to homotopy [@Crainic-up; @to; @homotopy], [@Crainic-Fernandes-jets]. We stay the question whenever these ideas refer to $\mathbb{R}$-linear forms and $\mathbb{R}$-linear connections – meaning as objects for which it is not supposed a local property. In the paper, using the generalized Stokes formula for $\mathbb{R}$-linear connections on Lie algebroids, we prove the Chern-Simons transgression formula without assumption locality for $\mathbb{R}$-linear connections. This is a helpful starting point to define characteristic classes for $\mathbb{R}$-linear connections on Lie algebroids. Some Crainic and Fernandes ideas we use to extend notions of Chern character and exotic (secondary) characteristic classes to $\mathbb{R}$-linear objects. Moreover, we found some explicit formulae for $\mathbb{R}$-linear Chern-Simons forms. In particular, we gain an direct formula of exotic (secondary) characteristic classes for an $\mathbb{R}$-linear connection as some trace $\mathbb{R}$-linear forms on a Lie algebroid. A *Lie algebroid* is a trip $\left( A,\rho _{A},[\![\bullet ,\bullet ]\!]_{A}\right) $, in which $A$ is a real vector bundle over a manifold $M$, $\rho _{A}:A\rightarrow TM$ (called an *anchor*) is a homomorphism of vector bundles, $\left( \Gamma \left( A\right) ,[\![\bullet ,\bullet ]\!]_{A}\right) $ is an $\mathbb{R}$-Lie algebra and the Leibniz identity$$\lbrack \![a,f\cdot b]\!]_{A}=f\cdot \lbrack \![a,b]\!]_{A}+\rho _{A}\left( a\right) \left( f\right) \cdot b\ \ \ \ \ \text{for all\ \ \ \ }a,b\in \Gamma \left( A\right) ,\ f\in \mathscr{C}^{\infty }\left( M\right)$$holds. Since the representation $\varrho :\mathscr{C}^{\infty }\left( M\right) \rightarrow \limfunc{End}\nolimits_{\mathscr{C}^{\infty }\left( M\right) }\left( \Gamma \left( A\right) \right) $, $\varrho \left( \nu \right) \left( a\right) =\nu \cdot a$, $\nu \in \mathscr{C}^{\infty }\left( M\right) $,$\ a\in \Gamma \left( A\right) $, is faithful ([@Herz], see also [@B-K-W-Primary]), the anchor induces a homomorphism of Lie algebras $\limfunc{Sec}\rho _{A}:\Gamma \left( A\right) \rightarrow \mathscr{X}\left( M\right) $, $a\mapsto \rho _{A}\circ a$. If $\rho _{A}$ is a constant rank (i.e. $\func{Im}\rho _{A}$ is a constant dimensional and completely integrable distribution), we say that $\left( A,\rho _{A},[\![\bullet ,\bullet ]\!]_{A}\right) $ is *regular*. A tangent bundle $TM$ to a manifold $M$ with the identity as an anchor and the bracket of vector fields is an elementary example of a Lie algebroid. For more about Lie algebroids and their properties we refer for example to [@Mackenzie], [Higgins-Mackenzie]{}, [@Kubarski-Lyon], [@Fernandes], [B-K-W-Primary]{}, [@Crainic-Fernandes-jets]. There are Lie functors from many geometric categories to the category of Lie algebroids (see a long list eg in [@Mackenzie], [@Kubarski-Lyon]). Especially meaning in the paper have algebroids of vector bundles. We recall that the module $\mathscr{CDO}\left( E\right) $ of sections of the Lie algebroid $\limfunc{A}\left( E\right) $ of a vector bundle $E$ is the space of all covariant differential operators in $E$, i.e. $\mathbb{R}$-linear operators $\ell :\Gamma \left( E\right) \rightarrow \Gamma \left( E\right) $ such that there exists exactly one $\widetilde{\ell }\in \mathscr{X}\left( M\right) $ with $\ell \left( f\zeta \right) =f\ell \left( \zeta \right) +\widetilde{\ell }\left( f\right) \zeta $ for all $f\in \mathscr{C}^{\infty }\left( M\right) $ and $\zeta \in \Gamma \left( E\right) $; see for example [@Teleman], [@Mackenzie], [@Kubarski-Lyon]. Let $\left( A,\rho _{A},[\![\bullet ,\bullet ]\!]_{A}\right) $ and $\left( B,\rho _{B},[\![\bullet ,\bullet ]\!]_{B}\right) $ be Lie algebroids over the same manifold $M$. A homomorphism $\nabla :A\rightarrow B$ of vector bundles is called an $A$-*connection* in $B$ if $\rho _{B}\circ \nabla =\rho _{A}$ (see [@B-K-W-Primary]). If an $A$-connection $\nabla $ in $B$ is a homomorphism of Lie algebroids ($\nabla $ preserves the Lie brackets) we say that $\nabla $ is *flat*. The notion of an $A$-connection in $B$ generalizes the known notions of connections (for example usual and partial covariant derivatives in vector bundles, a connection in principal bundles, a connection in extensions of Lie algebroids). In the case where $A=TM$ and $B=\limfunc{A}\left( E\right) $ is an algebroid of a vector bundle $E$, $TM$-connections in $\limfunc{A}\left( E\right) $ are one–to–one with covariant derivatives in $E$. For an arbitrary Lie algebroid $A$ and $B=\limfunc{A}\left( E\right) $ we have $A$-connections of $E$ considered in [@Mackenzie], [@Fernandes], [@Crainic-Fernandes-jets]. In case $B=\limfunc{A}\left( P\right) $ is a Lie algebroid of a principal bundle $P$, we get $A$-connections in $P$. In Poisson geometry an especially rule have connections acting from a Lie algebroid $T^{\ast }M$ associated to a given Poisson structure. In these examples a connection $\nabla $ considered as a mapping on modules of cross-sections is linear over $\mathscr{C}^{\infty }\left( M\right) $. By an $\mathbb{R}$*-linear connection *of $A$* *in $B$* *we called an $\mathbb{R}$-linear operator $\nabla :\Gamma \left( A\right) \rightarrow \Gamma \left( B\right) $* *such that $$\limfunc{Sec}\rho _{B}\circ \nabla =\limfunc{Sec}\rho _{A}\text{.}$$An $\mathbb{R}$-linear connection of $A$ in the Lie algebroid $A\left( E\right) $ is called the $\mathbb{R}$*-linear connection of* $A$*on the vector bundle* $E$. We call the map$$R^{\nabla }:\Gamma \left( A\right) \times \Gamma \left( A\right) \rightarrow \Gamma \left( B\right) ,\ \ R^{\nabla }\left( \alpha ,\beta \right) =[\![\nabla _{\alpha },\nabla _{\beta }]\!]_{B}-\nabla _{\lbrack \![\alpha ,\beta ]\!]_{A}}$$a *curvature* of $\nabla $. We see that $\nabla :\Gamma \left( A\right) \rightarrow \Gamma \left( B\right) $ is flat if $R^{\nabla }=0$. For every Lie algebroid $A$, the adjoint connection $\limfunc{ad}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( A\right) $, $\limfunc{ad}\left( a\right) =[\![a,\bullet ]\!]_{A}$ is an $\mathbb{R}$-linear connection of $A$ on $A$. The notion of an $\mathbb{R} $-linear connection includes so-called non-linear connections and connections up to homotopy on super-vector bundles ([@Crainic-up; @to; @homotopy], [@Crainic-Fernandes-jets], [@Evens-Lu-Weinstein]); such connections have a local property. Let $\left( A,\rho _{A},[\![\bullet ,\bullet ]\!]_{A}\right) $, $\left( B,\rho _{B},[\![\bullet ,\bullet ]\!]_{B}\right) $ be Lie algebroids over a manifold $M$. An $\mathbb{R}$-multilinear, antisymmetric map $$\omega :\underset{n}{\underbrace{\Gamma \left( A\right) \times \cdots \times \Gamma \left( A\right) }}\longrightarrow \Gamma \left( B\right)$$is called an $\mathbb{R}$*-linear* $n$-*form* on $A$ with values in $B$. The space of all such $\mathbb{R}$-linear $n$-forms will be denoted by $\mathcal{A}lt_{\mathbb{R}}^{n}\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right) $, and the space of $\mathbb{R}$-linear forms on $A$ with values in $B$ by$$\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right) =\bigoplus\limits_{k\geq 0}\mathcal{A}lt_{\mathbb{R}}^{k}\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right) ,$$where $\mathcal{A}lt_{\mathbb{R}}^{0}\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right) =\Gamma \left( B\right) $.[ ]{}Observe that if $\nabla :A\rightarrow B$ is an arbitrary $\mathbb{R}$-linear connection, then the curvature $R^{\nabla }$ is an element of $\mathcal{A}lt_{\mathbb{R}}^{2}\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right) $. We define the covariant differential operator$$d_{\mathbb{R}}^{\nabla }:\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right) \longrightarrow \mathcal{A}lt_{\mathbb{R}}^{\bullet +1}\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right)$$for $\mathbb{R}$-linear forms on $A$ with values in $B$ by the classical formula$$\begin{gathered} \left( d_{\mathbb{R}}^{\nabla }\eta \right) \left( a_{1},\ldots ,a_{n+1}\right) =\dsum\limits_{i=1}^{n+1}\left( -1\right) ^{i+1}\nabla _{a_{i}}\left( \eta \left( a_{1},\ldots \hat{\imath}\ldots ,a_{n+1}\right) \right) \\ +\dsum\limits_{i<j}\left( -1\right) ^{i+j}\eta \left( \lbrack \![a_{i},a_{j}]\!]_{A},a_{1},\ldots \hat{\imath}\ldots \hat{\jmath}\ldots ,a_{n+1}\right) .\end{gathered}$$$d_{\mathbb{R}}^{\nabla }$ is an antiderivation in $\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( E\right) \right) $ with respect to the product of $\mathbb{R}$-linear forms. A flat $\mathbb{R}$-linear connection $\nabla :\Gamma \left( A\right) \rightarrow \Gamma \left( B\right) $ induces, denoted by $H_{\nabla ,\mathbb{R}}^{\bullet }\left( A;B\right) $, the *Lie algebroid* $\mathbb{R}$*-cohomology space with coefficients in* $B$ as the cohomology space of the complex $\left( \mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( B\right) \right) ,d_{\mathbb{R}}^{\nabla }\right) $. The differential operator $d_{\mathbb{R}}^{\limfunc{Sec}\rho _{A}}$ induced by the anchor, i.e. by the flat[ ]{}$A$-connection in $TM$, will be denoted by $d_{A,\mathbb{R}}$. Since modules $\Gamma \left( M\times \mathbb{R}\right) $ and $\mathscr{C}^{\infty }\left( M\right) $ are isomorphic, it follows that $d_{A,\mathbb{R}} $ is an extension of the exterior derivative from the space $\Omega ^{\bullet }\left( A\right) $ of ($\mathscr{C}^{\infty }\left( M\right) $-linear) differential forms on $A$ to $\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) $. Let us recall that the cohomology space of the complex $\left( \Omega ^{\bullet }\left( A\right) ,d_{A}\right) $ where $\Omega ^{\bullet }\left( A\right) $$\mathscr{C}^{\infty }\left( M\right) $-linear $A$, $d_{A}=\left. d_{\mathbb{R}}^{\limfunc{Sec}\rho _{A}}\right\vert \Omega ^{\bullet }\left( A\right) :\Omega ^{\bullet }\left( A\right) \rightarrow \Omega ^{\bullet +1}\left( A\right) $, is called the *cohomology of Lie algebroid* and is denoted by $H^{\bullet }\left( A\right) $. Let $E$ be a vector bundle over $M$. Observe that $\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) $ is a left module over the algebra $\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) $ with the standard multiplication of forms. Moreover, $$\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) \otimes _{\mathscr{C}^{\infty }\left( M\right) }\Gamma \left( \func{End}E\right) \cong \mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right)$$as $\mathscr{C}^{\infty }\left( M\right) $-modules by the isomorphism defined in such a way that$$\omega \otimes \phi \longmapsto \omega \wedge \phi .$$ In the paper, we define the Chern-Simons forms for $\mathbb{R}$-linear connections on Lie algebroids. The generalized Chern-Simons formula is derived as a consequence of Stokes’ formula for $\mathbb{R}$-linear forms. The notion of the Chern classes of a vector bundle as cohomology classes of some $\mathbb{R}$-linear Chern-Simons forms is proposed. We show that such classes for a given $\mathbb{R}$-linear connection[ ]{}do not depend on the choice of the connection. In the paper, we discuss the wider then in [@Crainic-Fernandes-jets] for linear connections set of obstructions to the existence of a flat connection of a given Lie algebroid. Using ideas form papers Crainic and Fernandes, we introduce the secondary characteristic classes for arbitrary $\mathbb{R}$-linear connections of Lie algebroids in vector bundles. If an $\mathbb{R}$-linear $A$-connection $\nabla $ on a vector bundle $E$ is metrizable with respect to any metric $h$ in $E$ (i.e. $\nabla h=0$), the defined secondary characteristic classes vanishes. Therefore, secondary characteristic classes of $\nabla $ are obstructions to the existence of an invariant metric with respect to $\nabla $. In [@Crainic-Fernandes-jets] were considered connections up to homotopy (some non-linear connections with a local property). Here we examine all $\mathbb{R}$-linear connections. At the end of the last section we derive some comments on the Chern-Simons forms for $\mathbb{R}$-linear connections (in particular for Lie algebroids over odd dimensional manifolds). The Chern-Simons transgression forms on Lie algebroids and the Chern Character ============================================================================== Let $\left( A,\rho _{A},[\![\cdot ,\cdot ]\!]_{A}\right) $ be a Lie algebroid on a manifold $M$, $E$ a vector bundle over $M$, $k$ a natural number and $\func{pr}_{2}:\mathbb{R}^{k}\times M\rightarrow M$ a projection on the second factor. Consider an $\mathbb{R}$-linear connection $\nabla :\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ of $A$ on $E$. The standard fibrewise trace $\func{Tr}:\Gamma \left( \func{End}E\right) \rightarrow \mathscr{C}^{\infty }\left( M\right) $ on $\func{End}\left( E\right) $ induces a trace $$\func{Tr}_{\ast }:\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) \longrightarrow \mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right)$$such that $\func{Tr}_{\ast }\left( \omega \right) \left( a_{1},\ldots ,a_{n}\right) =\func{Tr}\left( \left( \omega \right) \left( a_{1},\ldots ,a_{n}\right) \right) $. Set (for $p\geq 1$)$$\func{ch}_{p}\left( \nabla \right) =\func{Tr}_{\ast }\left( R^{\nabla }\right) ^{p}\in \mathcal{A}lt_{\mathbb{R}}^{2p}\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right)$$where $\left( R^{\nabla }\right) ^{p}\in \mathcal{A}lt_{\mathbb{R}}^{2p}\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) $ is, for $a_{1},...,a_{2p}\in \Gamma \left( A\right) $, given by$$\left( R^{\nabla }\right) ^{p}\left( a_{1},...,a_{2p}\right) =\frac{1}{2^{p}}\sum\nolimits_{\tau \in S_{2p}}\func{sgn}\tau \cdot R_{a_{\tau \left( 1\right) },a_{\tau \left( 2\right) }}^{\nabla }\circ \cdots \circ R_{a_{\tau \left( 2p-1\right) },a_{\tau \left( 2p\right) }}^{\nabla }.$$The $2p$-form $\func{ch}_{p}\left( \nabla \right) $ is called the *Chern character form* associated to $\nabla $. \[comm\_Tr\_and\_diff\]$d_{A,\mathbb{R}}\circ \func{Tr}_{\ast }=\func{Tr}_{\ast }\circ d_{\mathbb{R}}^{\overline{\nabla }}$ where $\overline{\nabla }:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( \limfunc{End}E\right) $,* *$\overline{\nabla }_{a}=\left[ \nabla _{a},\bullet \right] $. First, recall that the space $\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) $ is isomorphic to$$\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) \otimes _{\mathscr{C}^{\infty }\left( M\right) }\Gamma \left( \func{End}E\right) .$$Let $\eta \in \mathcal{A}lt_{\mathbb{R}}^{n}\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) $, $\varphi \in \Gamma \left( \func{End}E\right) $. Then $\func{Tr}_{\ast }\left( \eta \otimes \varphi \right) =\eta \cdot \func{Tr}\varphi $. It is a simple matter to see that $d_{A,\mathbb{R}}\left( \func{Tr}\varphi \right) =\func{Tr}_{\ast }\left( d_{\mathbb{R}}^{\overline{\nabla }}\varphi \right) $. Therefore$$\begin{aligned} d_{A,\mathbb{R}}\func{Tr}_{\ast }\left( \eta \otimes \varphi \right) &=&d_{A,\mathbb{R}}\eta \cdot \func{Tr}\varphi +\left( -1\right) ^{n}\eta \wedge d_{A,\mathbb{R}}\left( \func{Tr}\varphi \right) \\ &=&d_{A,\mathbb{R}}\eta \cdot \func{Tr}\varphi +\left( -1\right) ^{n}\eta \wedge \func{Tr}_{\ast }\left( d_{\mathbb{R}}^{\overline{\nabla }}\varphi \right) \\ &=&\func{Tr}_{\ast }\left( d_{A,\mathbb{R}}\eta \otimes \varphi +\left( -1\right) ^{n}\eta \wedge d_{\mathbb{R}}^{\overline{\nabla }}\varphi \right) \\ &=&\func{Tr}_{\ast }\left( d_{\mathbb{R}}^{\overline{\nabla }}\left( \eta \otimes \varphi \right) \right) .\end{aligned}$$ $\mathscr{C}^{\infty }\left( \mathbb{R}\times M\right) $-modules $\Gamma \left( \func{pr}_{2}^{\ast }A\right) $ and $\mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) \otimes _{\mathscr{C}^{\infty }\left( M\right) }\Gamma \left( A\right) $ are isomorphic (see [Higgins-Mackenzie]{}) and this way the module of cross-sections of the inverse image$$\func{pr}_{2}^{\;\wedge }\hspace{-0.1cm}\left( A\right) =\left\{ \left( \gamma ,w\right) \in T\left( \mathbb{R}^{k}\times M\right) \times A:\left( \func{pr}_{2}\right) _{\ast }\gamma =\rho _{A}\left( w\right) \right\} \cong T\mathbb{R}^{k}\times A$$of $A$ by $\func{pr}_{2}$ is a $\mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) $-submodule of $$\mathscr{X}\left( \mathbb{R}^{k}\times M\right) \times \left( \mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) \otimes _{\mathscr{C}^{\infty }\left( M\right) }\Gamma \left( A\right) \right)$$($T\mathbb{R}^{k}\times A$ is the Cartesian product of Lie algebroids $T\mathbb{R}^{k}$ and $A$, see [@Kubarski-invariant]). We denote cross-sections $0\times a$, $\frac{\partial }{\partial t^{j}}\times 0$ of the vector bundle $T\mathbb{R}^{k}\times A$ briefly by $a$ and $\frac{\partial }{\partial t^{j}}$, respectively. Let $$\Delta ^{k}=\left\{ \left( t_{1},...,t_{k}\right) \in \mathbb{R}^{k};\;\;\;\forall i\;\;t_{i}\geq 0\,,\;\;\sum\nolimits_{i=1}^{k}t_{i}\leq 1\right\}$$be the *standard* $k$*-simplex* in $\mathbb{R}^{k}$. Additionally we set the *standard* $0$*-simplex* as $\Delta ^{0}=\left\{ 0\right\} $. Define$$\dint\nolimits_{\Delta ^{k}}:\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( T\mathbb{R}^{k}\times A\right) ;\mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) \right) \longrightarrow \mathcal{A}lt_{\mathbb{R}}^{\bullet -k}\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) ,$$$$\left( \int\nolimits_{\Delta ^{k}}\omega \right) \left( a_{1},...,a_{n-k}\right) =\int\nolimits_{\Delta ^{k}}\omega \left( \frac{\partial }{\partial t^{1}},...,\frac{\partial }{\partial t^{k}},a_{1},...,a_{n-k}\right) _{|\left( t_{1},...,t_{k},\bullet \right) }dt_{1}...dt_{k},$$$$\left( \int\nolimits_{\Delta ^{0}}\omega \right) \left( a_{1},...,a_{n}\right) =\iota _{0}^{\ast }\left( \omega \left( 0\times a_{1},...,0\times a_{n}\right) \right) ,\ \ \ \int\nolimits_{\Delta ^{0}}f=\iota _{0}^{\ast }f$$for all $n\geq 1$, $1\leq k\leq n$, $\omega \in \mathcal{A}lt_{\mathbb{R}}^{n}\left( \Gamma \left( T\mathbb{R}^{k}\times A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) $, $f\in \mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) $ and where $\iota _{0}:M\rightarrow \Delta ^{0}\times M$ is an inclusion defined by $\iota _{0}\left( x\right) =\left( 0,x\right) $. In view of the factorization property in $\mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) \otimes _{\mathscr{C}^{\infty }\left( M\right) }\Gamma \left( A\right) $, we conclude that for $\nabla $ there exists exactly one $\mathbb{R}$-linear connection $$\widetilde{\nabla }:\Gamma \left( T\mathbb{R}^{k}\times A\right) \longrightarrow \mathscr{CDO}\left( \func{pr}_{2}^{\;\ast }E\right)$$of $T\mathbb{R}^{k}\times A$ on $\func{pr}_{2}^{\;\ast }E$* *such that$$\left( \widetilde{\nabla }_{\left( X,\tsum_{i}r^{i}\otimes a^{i}\right) }\left( \nu \circ \func{pr}_{2}\right) \right) \left( t,\bullet \right) =\nabla _{\tsum_{i}r^{i}\left( t,\bullet \right) \cdot a^{i}}\left( \nu \right)$$for all $\left( X,\tsum_{i}r^{i}\otimes a^{i}\right) \in \mathscr{X}\left( \mathbb{R}^{k}\times M\right) \times \left( \mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) \otimes _{\mathscr{C}^{\infty }\left( M\right) }\Gamma \left( A\right) \right) $, $\nu \in \Gamma \left( E\right) $, $t=\left( t_{1},...,t_{k}\right) \in \mathbb{R}^{k} $. In particular, $\left( \widetilde{\nabla }_{\left( 0\times \left( \rho _{A}\circ a\right) ,1\otimes a\right) }\left( \nu \circ \func{pr}_{2}\right) \right) \left( t,\bullet \right) =\nabla _{a}\left( \nu \right) $, $a\in \Gamma \left( A\right) $.$\;$The connection$\;\widetilde{\nabla }$ is called the *lifting* of $\nabla $ to $T\mathbb{R}^{k}\times A$. Let $\nabla ^{0},\,\nabla ^{1},\ldots ,\nabla ^{k}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) \;$be $\mathbb{R}$-linear connections of a Lie algebroid $A$ on a vector bundle $E$ and $\widetilde{\nabla }^{0},\,\widetilde{\nabla }^{1},\ldots ,\,\widetilde{\nabla }^{k}:\Gamma \left( T\mathbb{R}^{k}\times A\right) \rightarrow \mathscr{CDO}\left( \func{pr}_{2}^{\;\ast }E\right) $ be their liftings to $T\mathbb{R}^{k}\times A$. Then there exists an $\mathbb{R}$-linear connection$$\nabla ^{\func{aff}_{k}}:\Gamma \left( T\mathbb{R}^{k}\times A\right) \longrightarrow \mathscr{CDO}\left( \func{pr}_{2}^{\;\ast }E\right) ,$$called the *affine combination of connections* $\nabla ^{0},\nabla ^{1},\ldots ,\nabla ^{k}$, given by $$\begin{aligned} &&\left( \nabla _{\,\,\,\left( X,\tsum_{i}r^{i}\otimes a^{i}\right) }^{\func{aff}_{k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( t,\bullet \right) \\ &=&\left( 1-\sum\nolimits_{i=1}^{k}t_{i}\right) \cdot \left( \nabla ^{0}\right) _{\tsum_{i}r^{i}\left( t,\bullet \right) \cdot a^{i}}\left( \nu \right) +\sum\nolimits_{i=1}^{k}t_{i}\cdot \left( \nabla ^{i}\right) _{\tsum_{i}r^{i}\left( t,\bullet \right) \cdot a^{i}}\left( \nu \right) .\end{aligned}$$For all $0<k\leq 2p$ we define an $\mathbb{R}$-linear form$$\limfunc{cs}\nolimits_{p}\left( \nabla ^{0},...,\nabla ^{k}\right) =\int\nolimits_{\Delta ^{k}}\func{ch}_{p}\left( \nabla ^{\func{aff}_{k}}\right) \in \mathcal{A}lt_{\mathbb{R}}^{2p-k}\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right)$$called the *Chern-Simons form* for $\left( \nabla ^{0},...,\nabla ^{k}\right) $ and additionally we put $cs_{p}\left( \nabla ^{0}\right) =\func{ch}_{p}\left( \nabla ^{0}\right) $. We have the following (useful) Stokes’ formula for $\mathbb{R}$-linear forms on $A$ (see [@Balcerzak-Stokes]) being a generalization of the one for tangent bundles given by R. Bott [@Bott]. For every natural number* *$k$,$$\int\nolimits_{\Delta ^{k}}\circ \,d_{T\mathbb{R}^{k}\times A,\mathbb{R}}+\left( -1\right) ^{k+1}d_{A,\mathbb{R}}\circ \int\nolimits_{\Delta ^{k}}=\dsum\nolimits_{j=0}^{k}\left( -1\right) ^{j}\int\nolimits_{\Delta ^{k-1}}\circ \,\left( d\sigma _{j}^{k-1}\times \func{id}_{A}\right) ^{\ast }, \label{Stokes}$$where $\sigma _{j}^{k}:\mathbb{R}^{k}\rightarrow \mathbb{R}^{k+1}$ for $0\leq j\leq k+1$ are functions defined by $\sigma _{0}^{0}\left( 0\right) =1 $, $\sigma _{1}^{0}\left( 0\right) =0$, and for $t=\left( t_{1},...,t_{k}\right) \in \mathbb{R}^{k}$ by$$\begin{aligned} \sigma _{0}^{k}\left( t\right) &=&\left( 1-\dsum\nolimits_{i=1}^{k}t_{i},t_{1},...,t_{k}\right) , \\ \sigma _{j}^{k}\left( t\right) &=&\left( t_{1},...,t_{j-1},0,t_{j},...,t_{k}\right) ,\;\;1\leq j\leq k+1,\end{aligned}$$and where $\left( \left( \int\nolimits_{\Delta ^{k-1}}\circ \,\left( d\sigma _{j}^{k-1}\times \func{id}_{A}\right) ^{\ast }\right) \omega \right) \left( a_{1},...,a_{n-k+1}\right) $ is, by definition, equal to$$\int\nolimits_{\Delta ^{k-1}}\omega \left( d\sigma _{j}^{k-1}\left( \frac{\partial }{\partial t^{1}}\right) ,...,d\sigma _{j}^{k-1}\left( \frac{\partial }{\partial t^{k-1}}\right) ,a_{1},...,a_{n-k+1}\right) _{|\left( t_{1},...,t_{k-1},\bullet \right) }\hspace{-0.3cm}dt_{1}...dt_{k-1}$$and $$\left( \left( \int\nolimits_{\Delta ^{0}}\circ \,\left( d\sigma _{j}^{0}\times \func{id}_{A}\right) ^{\ast }\right) \omega \right) \left( a_{1},...,a_{n}\right) =\left( \sigma _{j}^{0}\times \func{id}_{M}\circ \iota _{0}\right) ^{\ast }\left( \omega \left( a_{1},...,a_{n}\right) \right)$$if $k\geq 2$, $\omega \in \mathcal{A}lt_{\mathbb{R}}^{n}\left( \Gamma \left( T\mathbb{R}^{k}\times A\right) ;\mathscr{C}^{\infty }\left( \mathbb{R}^{k}\times M\right) \right) $, $a_{i}\in \Gamma \left( A\right) $, $j\in \left\{ 0,1\right\} $. The following lemma will be useful below in the proof of the Chern-Simons formula for $\mathbb{R}$-linear connections of Lie algebroids. \[lemma\_abc\]Let $a,\,b\in \Gamma \left( A\right) $, $\nu \in \Gamma \left( E\right) $, $t\in \mathbb{R}^{k-1}$, $0\leq j\leq k,$ $1\leq s\leq k, $ $1\leq z\leq k-1$. Denote here the affine combination $\nabla ^{\func{aff}_{k}}$ of$\ \nabla ^{0},\ldots ,\nabla ^{k}$ by $\nabla ^{0,...,k}$. Then - $\left( R_{a,b}^{\nabla ^{0,...,k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( \sigma _{j}^{k-1}\left( t\right) ,\bullet \right) =\left( R_{a,b}^{\nabla ^{0,...\widehat{j}...,k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( t,\bullet \right) ,$ - $\left( R_{\frac{\partial }{\partial \,\tilde{t}\,^{s}},a}^{\nabla ^{0,...,k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( \sigma _{j}^{k-1}\left( t\right) ,\bullet \right) $ is equal to $\left( R_{\frac{\partial }{\partial \,t^{s}},a}^{\nabla ^{0,...\widehat{j}...,k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( t,\bullet \right) $ if $1\leq s<j,$ and $\left( R_{\frac{\partial }{\partial \,t^{s-1}},a}^{\nabla ^{0,...\widehat{j}...,k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( t,\bullet \right) $ if $j\leq s\leq k$, and where $\tilde{t}\,^{i}$ are coordinates of the identity map of $\mathbb{R}^{k}$, - $\left( R_{d\sigma _{j}^{k-1}\left( \frac{\partial }{\partial t^{z}}\right) ,a}^{\nabla ^{0,...,k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( \sigma _{j}^{k-1}\left( t\right) ,\bullet \right) =\left( R_{\frac{\partial }{\partial t^{z}},a}^{\nabla ^{1...,k}}\left( \nu \circ \func{pr}_{2}\right) \right) \left( t,\bullet \right) $. Just calculations. *(The Chern-Simons formula for Lie algebroids and* $\mathbb{R}$*-linear connections)* Let $\left( A,\rho _{A},[\![\cdot ,\cdot ]\!]\right) $ be a Lie algebroid on a manifold $M$, $E$ a vector bundle over $M$, $k\in \mathbb{N}$, $\nabla ^{0},\,...,\,\nabla ^{k}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) \;$ $\mathbb{R}$-linear connections of $A$ on $E$. Then$$\left( -1\right) ^{k+1}\,d_{A,\mathbb{R}}\left( \limfunc{cs}\nolimits_{p}\left( \nabla ^{0},...,\nabla ^{k}\right) \right) =\sum\nolimits_{j=0}^{k}\left( -1\right) ^{j}\,\limfunc{cs}\nolimits_{p}\left( \nabla ^{0},...\widehat{\nabla ^{j}}...,\nabla ^{k}\right) \label{ChernSimonsformula}$$for all integer numbers $p$ such that $0<k\leq 2p$ and $d_{A,\mathbb{R}}\left( cs_{p}\left( \nabla ^{0}\right) \right) =0$. From Lemma \[comm\_Tr\_and\_diff\] and the Bianchi identity ($d_{\mathbb{R}}^{\overline{\nabla ^{j}}}\left( R^{\nabla ^{j}}\right) =0$) we deduce that forms $\func{ch}_{p}\left( \nabla ^{0}\right) $ and $\func{ch}_{p}\left( \nabla ^{\func{aff}_{k}}\right) $ are closed. Since these forms are closed, applying the Stokes formula (\[Stokes\]) we conclude that $$\left( -1\right) ^{k+1}\,d_{A,\mathbb{R}}\left( \limfunc{cs}\nolimits_{p}\left( \nabla ^{0},...,\nabla ^{k}\right) \right) =\sum\limits_{j=0}^{k}\left( -1\right) ^{j}\int\nolimits_{\Delta ^{k-1}}\left( d\sigma _{j}^{k-1}\times \func{id}_{A}\right) ^{\ast }\func{ch}_{p}\left( \nabla ^{\func{aff}_{k}}\right) .$$Let $a_{0},$...,$a_{2p-k}\in \Gamma \left( A\right) $. From the above$$\begin{gathered} \left( -1\right) ^{k+1}\,d_{A,\mathbb{R}}\left( \limfunc{cs}\nolimits_{p}\left( \nabla ^{0},...,\nabla ^{k}\right) \right) \left( a_{0},...,a_{2p-k}\right) \\ =\dsum\nolimits_{j=0}^{k}\left( -1\right) ^{j}\left( \int\nolimits_{\Delta ^{k-1}}\left( d\sigma _{j}^{k-1}\times \func{id}_{M}\right) ^{\ast }\func{ch}_{p}\left( \nabla ^{\func{aff}_{k}}\right) \right) \left( a_{0},...,a_{2p-k}\right) .\end{gathered}$$From the definition of $\left( R^{\nabla ^{\func{aff}_{k}}}\right) ^{p}$ and fact that $R_{\frac{\partial }{\partial \widetilde{t}^{i}},\frac{\partial }{\partial \widetilde{t}^{j}}}^{\nabla ^{\func{aff}_{k}}}=0$ (where $\left( \tilde{t}\,^{1},...,\tilde{t}\,^{k}\right) $ is the identity map on the manifold $\mathbb{R}^{k}$) we observe that the possible non-zero terms in the above sum are the form$$R_{d\sigma _{j}^{k-1}\left( \frac{\partial }{\partial t^{s}}\right) ,a}^{\nabla ^{\func{aff}_{k}}}\circ \cdots \circ R_{b,c}^{\nabla ^{\func{aff}_{k}}}\circ \cdots \circ R_{d,e}^{\nabla ^{\func{aff}_{k}}},\ \ \text{\ }a,b,c,d,e\in \Gamma \left( A\right) .$$Lemma \[lemma\_abc\] now yields that $\left( -1\right) ^{k+1}\,d_{A,\mathbb{R}}\left( \limfunc{cs}\nolimits_{p}\left( \nabla ^{0},...,\nabla ^{k}\right) \right) \left( a_{0},...,a_{2p-k}\right) $ is equal to$$\begin{gathered} \sum_{j=0}^{k}\left( -1\right) ^{j}\hspace{-0.1cm}\int\nolimits_{\Delta ^{k-1}}\hspace{-0.3cm}\func{ch}_{p}\left( \nabla ^{0,...\widehat{j}...,k}\right) \hspace{-0.1cm}\left. \left( \frac{\partial }{\partial t^{1}},...,\frac{\partial }{\partial t^{k-1}},a_{0\,},...,a_{2p-k}\right) \right\vert _{\left( t_{1},...,t_{k-1},\bullet \right) }\hspace{-0.3cm}dt_{1}\ldots dt_{k-1} \\ =\left( \sum_{j=0}^{k}\left( -1\right) ^{j}\,\limfunc{cs}\nolimits_{p}\left( \nabla _{0},...\widehat{\nabla ^{j}}...,\nabla _{k}\right) \right) \left( a_{0\,},...,a_{2p-k}\right) .\end{gathered}$$ *If* $\nabla ^{0},\nabla ^{1},\ldots ,\nabla ^{k}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) \;$*are* $\mathscr{C}^{\infty }\left( M\right) $*-linear connections, then* $\nabla ^{\func{aff}_{k}}$* is a* $\mathscr{C}^{\infty }\left( M\right) $*-linear connection. In this case, we obtain a formula due to property of Chern-Simons transgressions in [Crainic-Fernandes-jets]{} by M. Crainic and R. L. Fernandes.* *Let* $\nabla :\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ *be an* $\mathbb{R}$*-linear connection of* $A$* on a vector bundle* $E$*.* *The Chern character forms* $\func{ch}_{p}\left( \nabla \right) \in \mathcal{A}lt_{\mathbb{R}}^{2p}\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) $ *are closed and their cohomology classes*$$\func{ch}_{p}\left( A,E\right) =\left[ \func{ch}_{p}\left( \nabla \right) \right] \in H_{\rho _{A},\mathbb{R}}^{2p}\left( A;M\times \mathbb{R}\right) \emph{,}$$*do not depend on the choice of the connection* $\nabla $*. Indeed, let* $\nabla ^{0}$,$\nabla ^{1}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ *be* $\mathbb{R}$*-linear connections of* $A$* on* $E$*. According to (\[ChernSimonsformula\]), we have*$$\begin{aligned} d_{A,\mathbb{R}}\left( \limfunc{cs}\nolimits_{p}\left( \nabla ^{0},\nabla ^{1}\right) \right) &=&\limfunc{cs}\nolimits_{p}\left( \nabla ^{1}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ^{0}\right) \\ &=&\func{ch}_{p}\left( \nabla ^{1}\right) -\func{ch}_{p}\left( \nabla ^{0}\right) .\end{aligned}$$ In this way we have correctly defined the Chern character$$\func{ch}\left( A,E\right) \in H_{\rho _{A},\mathbb{R}}\left( A;M\times \mathbb{R}\right) .$$ *([@Crainic-up; @to; @homotopy], [@Crainic-Fernandes-jets]) In the particular case we can obtain the Chern character for a non-linear connection* $\nabla :\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $* of a Lie algebroid* $A$* on a vector bundle* $E$*, i.e. a local* $\mathbb{R}$*-linear connection* $\nabla :\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $*. In the space* $\Omega _{nl}\left( A\right) $*of non-linear differential forms on* $A$* (local* $\mathbb{R}$*-linear forms on* $A$*) we have the differential operator* $d_{nl}=d_{A,\mathbb{R}}|\Omega _{nl}^{\bullet }\left( A\right) :$* *$\Omega _{nl}^{\bullet }\left( A\right) \rightarrow \Omega _{nl}^{\bullet +1}\left( A\right) $. Secondary characteristic classes for $\mathbb{R}$-linear connections and some the Chern-Simons forms for a pair of connections ============================================================================================================================== Let $E$ be a vector bundle over $M$ with a metric $h$ and $\nabla :\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ be an $\mathbb{R}$-linear connection of a Lie algebroid $A$ on $E$. We define an $\mathbb{R}$-linear connection $\nabla ^{h}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ of $A$ on $E$ such that$$\left( \rho _{A}\circ a\right) \left( h\left( s,t\right) \right) =h\left( \nabla _{a}s,t\right) +h\left( s,\nabla _{a}^{h}t\right) ,\ \ \ a\in \Gamma \left( A\right) ,\ s,t\in \Gamma \left( E\right) .$$We can observe that$$R_{a,b}^{\nabla ^{h}}=-\left( R_{a,b}^{\nabla }\right) ^{\ast },\ \ \ \ \ \ a,b\in \Gamma \left( A\right) ,$$where $\left( R_{a,b}^{\nabla }\right) ^{\ast }$ is the adjoint map to $R_{a,b}^{\nabla }$ with respect to $h$. Therefore we obtain the following lemma. \[Lemma\_1\_about\_nabla\_h\]If $\nabla _{0}$, $\nabla _{1}$ are $\mathbb{R}$-linear connections of $A$ on $E$, then - $\limfunc{cs}\nolimits_{p}\left( \nabla _{0}^{h}\right) =\left( -1\right) ^{p}\limfunc{cs}\nolimits_{p}\left( \nabla _{0}\right) $, - $\limfunc{cs}\nolimits_{p}\left( \nabla _{0}^{h},\nabla _{1}^{h}\right) =\left( -1\right) ^{p}\limfunc{cs}\nolimits_{p}\left( \nabla _{0},\nabla _{1}\right) .$ From the Chern-Simons formula (\[ChernSimonsformula\]) and Lemma [Lemma\_1\_about\_nabla\_h]{} (a) we deduce that $$\begin{aligned} d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h}\right) &=&\limfunc{cs}\nolimits_{p}\left( \nabla \right) -\limfunc{cs}\nolimits_{p}\left( \nabla ^{h}\right) \\ &=&\limfunc{cs}\nolimits_{p}\left( \nabla \right) -\left( -1\right) ^{p}\limfunc{cs}\nolimits_{p}\left( \nabla \right) \\ &=&0,\end{aligned}$$because $\nabla $ is flat. In particular, we see that $\nabla ^{h}$ is also flat. The cohomology class $\left[ \limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h}\right) \right] \in H_{\rho _{A},\mathbb{R}}^{2p-1}\left( A\right) $ do not depend on the choice of metric $h$. Let $h_{1}$, $h_{2}$ be two metrics on $E$ and let $\nabla ^{M}$ be any $TM$-connection on $E$. Thus $\nabla _{o}=\nabla ^{M}\circ \rho _{A}$ is an $A$-connection on $E$ (i.e. a linear connection). The Chern-Simons formula ([ChernSimonsformula]{}) yields$$-d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{j}},\nabla _{o}^{h_{j}}\right) =\limfunc{cs}\nolimits_{p}\left( \nabla ^{h_{j}},\nabla _{o}^{h_{j}}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o}^{h_{j}}\right) +\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{j}}\right) \label{e1}$$and$$-d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o},\nabla _{o}^{h_{j}}\right) =\limfunc{cs}\nolimits_{p}\left( \nabla _{o},\nabla _{o}^{h_{j}}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o}^{h_{j}}\right) +\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o}\right) \label{e2}$$for $j\in \left\{ 1,2\right\} $. Lemma \[Lemma\_1\_about\_nabla\_h\] implies $\limfunc{cs}\nolimits_{p}\left( \nabla ^{h_{j}},\nabla _{o}^{h_{j}}\right) =\left( -1\right) ^{p}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o}\right) $. From this, (\[e1\]) and (\[e2\]) we get$$\begin{aligned} & \limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{1}}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{2}}\right) \\ & =d_{A,\mathbb{R}}\left( \limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{2}},\nabla _{o}^{h_{2}}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{1}},\nabla _{o}^{h_{1}}\right) \right) +\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o}^{h_{1}}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o}^{h_{2}}\right) \\ & =d_{A,\mathbb{R}}\left( \limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{2}},\nabla _{o}^{h_{2}}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{1}},\nabla _{o}^{h_{1}}\right) \right) +d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o},\nabla _{o}^{h_{1}}\right) \\ & -d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{o},\nabla _{o}^{h_{2}}\right) +\limfunc{cs}\nolimits_{p}\left( \nabla _{o},\nabla _{o}^{h_{1}}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla _{o},\nabla _{o}^{h_{2}}\right) .\end{aligned}$$Because of $\nabla _{o}$ is a linear connection, Proposition 1 from [Crainic-Fernandes-jets]{} yields $\limfunc{cs}\nolimits_{p}\left( \nabla _{o},\nabla _{o}^{h_{1}}\right) $$-\limfunc{cs}\nolimits_{p}\left( \nabla _{o},\nabla _{o}^{h_{2}}\right) $ is an exact form. In this way cohomology classes of $\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{1}}\right) $ and $\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h_{2}}\right) $ are both equal. *We call* $$\limfunc{u}\nolimits_{2p-1}\left( A,E\right) =\left[ \limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h}\right) \right] \in H_{\rho _{A},\mathbb{R}}^{2p-1}\left( A\right) ,\ \ \ p\in \left\{ 1,\ldots ,\limfunc{rank}E\right\} ,$$the secondary characteristic classes *of an* $\mathbb{R}$*-linear connection* $\nabla :\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $. If there exists in $E$ an invariant metric $h$ with respect to $\nabla $, then $\nabla ^{h}=\nabla $. Then classes $\limfunc{u}\nolimits_{2p-1}\left( A,E\right) $ are equal to zero. Hence these classes are obstructions to the existence of an invariant metric with respect to $\nabla $. We obtain the following theorem analogous to Proposition 2 in [Crainic-Fernandes-jets]{}. Let $\nabla $, $\nabla _{m}$ be $\mathbb{R}$-linear connections of $A$ on $E$ and $\nabla _{m}$ be additionally metric. - If $p$ is even, then $\limfunc{u}\nolimits_{2p-1}\left( A,E\right) =0$. - If $p$ is odd, then $\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{m}\right) $ is a closed form and$$\limfunc{u}\nolimits_{2p-1}\left( A,E\right) =\left[ 2\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{m}\right) \right] .$$ Let $\nabla _{m}$ be metric connection with respect to a metric $h$. On account of the Chern-Simons formula (\[ChernSimonsformula\]), we have$$-d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h},\nabla _{m}\right) =\limfunc{cs}\nolimits_{p}\left( \nabla ^{h},\nabla _{m}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{m}\right) +\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h}\right) .$$Now Lemma \[Lemma\_1\_about\_nabla\_h\] leads to $\limfunc{cs}\nolimits_{p}\left( \nabla ^{h},\nabla _{m}\right) =\left( -1\right) ^{p}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{m}\right) $, because $\nabla _{m}^{h}=\nabla _{m}$. It follows that$$\begin{aligned} \limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h}\right) &=&\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{m}\right) -\limfunc{cs}\nolimits_{p}\left( \nabla ^{h},\nabla _{m}\right) -d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h},\nabla _{m}\right) \\ &=&\left( 1+\left( -1\right) ^{p+1}\right) \limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla _{m}\right) -d_{A,\mathbb{R}}\limfunc{cs}\nolimits_{p}\left( \nabla ,\nabla ^{h},\nabla _{m}\right) ,\end{aligned}$$which completes the proof. For two $\mathbb{R}$-linear connections $\nabla ^{0}$,$\nabla ^{1}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ of $A$ on $E$ we define an $\mathbb{R}$-linear $1$-form$$\lambda =\nabla ^{1}-\nabla ^{0}\in \mathcal{A}lt_{\mathbb{R}}^{1}\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) .$$Let us observe that $$R^{\nabla ^{1}}=R^{\nabla ^{0}}+d^{\overline{\nabla }^{0}}\lambda +\left[ \lambda ,\lambda \right] , \label{curvature_and_pair}$$where $d^{\overline{\nabla }^{0}}$is the covariant derivative in $\mathcal{A}lt_{\mathbb{R}}^{\bullet }\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) $ determined by $\overline{\nabla }^{0}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( \limfunc{End}E\right) $, $\overline{\nabla }_{a}^{0}=\left[ \nabla _{a}^{0},\bullet \right] $ for all $a\in \Gamma \left( A\right) $, and $\left[ \lambda ,\lambda \right] \in \mathcal{A}lt_{\mathbb{R}}^{2}\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) $ is given by $\left[ \lambda ,\lambda \right] \left( a,b\right) =\left[ \lambda \left( a\right) ,\lambda \left( b\right) \right] $ for all $a,b\in \Gamma \left( A\right) $. *[@Balcerzak]* For two $\mathbb{R}$-linear connections $\nabla ^{0}$,$\nabla ^{1}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ the following properties hold: $$(R^{\nabla ^{\limfunc{aff}_{1}}})_{\frac{\partial }{\partial t},a}\left( \nu \circ \limfunc{pr}\nolimits_{2}\right) _{|\left( t,\bullet \right) }=\lambda \left( a\right) \left( \nu \right) , \label{properity_1}$$$$(R^{\nabla ^{\limfunc{aff}_{1}}})_{a,b}\left( \nu \circ \limfunc{pr}\nolimits_{2}\right) _{|\left( t,\bullet \right) }=\left( 1-t\right) \cdot R_{a,b}^{\nabla ^{0}}\left( \nu \right) +t\cdot R_{a,b}^{\nabla ^{1}}\left( \nu \right) +\left( t^{2}-t\right) \cdot \left[ \lambda ,\lambda \right] _{\left( a,b\right) }\left( \nu \right) \label{properity_2}$$for all $a,b\in \Gamma \left( A\right) $, $\nu \in \Gamma \left( E\right) $, $t\in \mathbb{R}$. #### **The Chern-Simons forms of the first and the second rank**   Let $\theta \in \mathcal{A}lt_{\mathbb{R}}^{1}\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) $, $\nabla :\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ be an $\mathbb{R}$-linear connection of $A$ on $E$. Therefore $\nabla +\theta $ is also an $\mathbb{R}$-linear $A$-connection on $E$, and $\limfunc{cs}\nolimits_{1}\left( \nabla ,\nabla +\theta \right) \in \mathcal{A}lt_{\mathbb{R}}^{1}\left( \Gamma \left( A\right) ;\mathscr{C}^{\infty }\left( M\right) \right) $ is given by $\limfunc{cs}\nolimits_{1}\left( \nabla ,\nabla +\theta \right) \left( a\right) =\limfunc{tr}\left( \theta \left( a\right) \right) $, $a\in \Gamma \left( A\right) $. Moreover, we conclude from (\[properity\_1\]), (\[properity\_2\]) and ([curvature\_and\_pair]{}) that$$\limfunc{tr}\left( R^{\nabla ^{\limfunc{aff}_{1}}}\right) ^{2}\left( \frac{\partial }{\partial t},\bullet \right) _{|\left( t,\bullet \right) }=2\limfunc{tr}\left( \theta \wedge R^{\nabla ^{0}}+t\cdot \theta \wedge d_{\mathbb{R}}^{\overline{\nabla ^{0}}}\theta +t^{2}\cdot \theta \wedge \theta \wedge \theta \right)$$for all $a_{1},a_{2},a_{3}\in \Gamma \left( A\right) $, $t\in \mathbb{R}$, hence$$\limfunc{cs}\nolimits_{2}\left( \nabla ,\nabla +\theta \right) =\limfunc{tr}\left( 2\theta \wedge R^{\nabla }+\theta \wedge d_{\mathbb{R}}^{\overline{\nabla }}\theta +\frac{2}{3}\theta \wedge \theta \wedge \theta \right) .$$If $\nabla $ and $\nabla +\theta $ are both flat, then $d_{\mathbb{R}}^{\overline{\nabla }}\theta =-\theta \wedge \theta $, which then yields $$\limfunc{cs}\nolimits_{2}\left( \nabla ,\nabla +\theta \right) =-~\frac{1}{3}\limfunc{tr}\left( \theta \wedge \theta \wedge \theta \right) .$$ For every manifold $M$ of an odd dimension $2m-1$, $\limfunc{cs}\nolimits_{m}\left( \nabla ,\nabla +\theta \right) $ is closed. In the case where $M$ is a $3$-dimensional manifold, $\limfunc{cs}\nolimits_{2}\left( \nabla ,\nabla +\theta \right) $ is closed and is given by the above formula; if additionally $\nabla $ is flat, we see that$$\limfunc{cs}\nolimits_{2}\left( \nabla ,\nabla +\theta \right) =\limfunc{tr}\left( \theta \wedge d_{\mathbb{R}}^{\overline{\nabla }}\theta +\frac{2}{3}\theta \wedge \theta \wedge \theta \right) . \label{formulaWZ}$$(\[formulaWZ\]) is a generalization of the known formula for tangent bundles of smooth, compact, oriented, three dimensional manifolds and standard connections (see for example [@Zhang]) to arbitrary rank three vector bundles and $\mathbb{R}$-linear connections. Moreover, we add (see [@Balcerzak]) that if both $\mathbb{R}$-linear connections $\nabla ^{0}$,$\nabla ^{1}:\Gamma \left( A\right) \rightarrow \mathscr{CDO}\left( E\right) $ of a Lie algebroid $A$ on a vector bundle $E$ are flat, then the Chern–Simons $\mathbb{R}$-linear form $\limfunc{cs}\nolimits_{p}\left( \nabla ^{0},\nabla ^{1}\right) $ is equal to $\left( -1\right) ^{p+1}\,\frac{\,p!\left( p-1\right) !\,}{\left( 2p-1\right) !}\func{Tr}_{\ast }\left( \lambda ^{2p-1}\right) $. In particular, for any flat $\mathbb{R}$-linear connection $\nabla $ of $A$ on $E$, $\nabla ^{h}$ is also flat and we conclude that the class $\limfunc{u}\nolimits_{2p-1}\left( A,E\right) $ is represented by the form $$\left( -1\right) ^{p+1}\,\frac{\,p!\left( p-1\right) !\,}{\left( 2p-1\right) !}\func{Tr}_{\ast }\left( \omega ^{2p-1}\right) ,$$where $\omega =\nabla ^{h}-\nabla \in \mathcal{A}lt_{\mathbb{R}}^{1}\left( \Gamma \left( A\right) ;\Gamma \left( \func{End}E\right) \right) $.   [99]{} <span style="font-variant:small-caps;">B. Balcerzak, J. Kubarski and W. Walas</span>,** ***Primary characteristic homomorphism of pairs of Lie algebroids and Mackenzie algebroid*, Banach Center Publ. **54** (2001)**,** 135–173. <span style="font-variant:small-caps;">B. Balcerzak</span>, *Modular classes of Lie algebroids homomorphisms as some the Chern-Simons forms*, Univ. Iagel. Acta Math. **47** (2009), 11–28. <span style="font-variant:small-caps;">B. Balcerzak</span>, *The Generalized Stokes theorem for* $\mathbb{R}$*-linear forms on Lie algebroids*, Accepted to Journal of Applied Analysis; available as preprint , 2011. <span style="font-variant:small-caps;">R. Bott</span>, *Lectures on characteristic classes and foliations,* Springer Lecture Notes in Math. 279, Springer, Berlin, 1972. <span style="font-variant:small-caps;">M. Crainic</span>, *Connections up to homotopy and characteristic classes*, Preprint arXiv, (2000). <span style="font-variant:small-caps;">M. Crainic and R. L. Fernandes</span>, *Secondary Characteristic Classes of Lie Algebroids*, In: Quantum Field Theory and Noncommutative Geometry, Lecture Notes in Phys. 662, pp. 157–176, Springer, Berlin, 2005. <span style="font-variant:small-caps;">S. Evens, J. H. Lu and A. Weinstein</span>, *Transverse measures, the modular class and a cohomology pairing for Lie algebroids*, Q. J. Math. **50 **(1999), 417–436. <span style="font-variant:small-caps;">R. L. Fernandes</span>, *Lie algebroids, holonomy and characteristic classes*, Adv. in Math. **170** (2002), 119–179. <span style="font-variant:small-caps;">J.-C. Herz</span>, Pseudo-algèbres de Lie, *C. R. Math. Acad. Sci. Paris* **263** (1953), I, 1935–1937, and II, 2289–2291. <span style="font-variant:small-caps;">Ph. J. Higgins and K. C. H. Mackenzie</span>, *Algebraic constructions in the category of Lie algebroids*, J. Algebra **129** (1990), 194–230. <span style="font-variant:small-caps;">J. Kubarski</span>, *The Chern-Weil homomorphism of regular Lie algebroids*, Publ. Dép. Math., Nouv. Sér., Univ. Claude Bernard, Lyon, 1991, 1–69. <span style="font-variant:small-caps;">J. Kubarski</span>, Invariant cohomology of regular Lie algebroids, in: *Analysis and Geometry in Foliated Manifolds* (Proceedings of the VII International Colloquium on Differential Geometry, Santiago de Compostella, Spain, 26–30 July 1994), pp. 137–151, World Sci. Publ., Singapore–New Yersey–London–Hong Kong, 1995. <span style="font-variant:small-caps;">K. C. H. Mackenzie</span>, *General Theory of Lie Groupoids and Lie Algebroids*, London Math. Soc. Lecture Note Ser. 213, Cambridge Univ. Press, 2005. <span style="font-variant:small-caps;">N. Teleman</span>, *A characteristic ring of a Lie algebra extension*, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (8) Mat. Appl., vol. **52** (1972),[** **]{}498–506 and 708–711. <span style="font-variant:small-caps;">W. Zhang</span>, *Lectures on Chern-Weil Theory and Witten Deformations*, Nankai Tracts Math., vol 4, World Sci. Publ., New Yersey–London–Singapore–Hong Kong, 2001.   [Bogdan Balcerzak]{} [Institute of Mathematics]{} [Technical University of Łódź]{} [Wólczańska 215]{} [90-924 Łódź, Poland]{} [*E-mail address*: bogdan.balcerzak@p.lodz.pl]{} [^1]: 2010 *Mathematics Subject Classification*: Primary 53C05; Secondary 58H05, 17B56. [^2]: *Key words and phrases*: Lie algebroid, connections, Lie algebroid cohomology, Chern-Simons forms.

**Modelling linguistic taxonomic dynamics** S[ø]{}ren Wichmann$^{1,2}$, Dietrich Stauffer$^3$, F. Welington S. Lima$^4$, and Christian Schulze$^3$ $^1$ Department of Linguistics, Max Planck Institute for Evolutionary Anthropology, Deutscher Platz 6, D-04103 Leipzig, Germany $^2$ Languages and Cultures of Indian America (TCIA), P.O. Box 9515, 2300 RA Leiden, The Netherlands $^3$ Institute for Theoretical Physics, Cologne University, D-50923 Köln, Euroland $^4$ Departamento de Física, Universidade Federal do Piauí, 57072-970 Teresina - PI, Brazil Abstract [This paper presents the results of the application of a bit-string model of languages (Schulze and Stauffer 2005) to problems of taxonomic patterns. The questions addressed include the following: (1) Which parameters are minimally needed for the development of a taxonomic dynamics leading to the type of distribution of language family sizes currently attested (as measured in the number of languages per family), which appears to be a power-law? (2) How may such a model be coupled with one of the dynamics of speaker populations leading to the type of language size seen today, which appears to follow a log-normal distribution?]{} Introduction ============ With few exceptions, such as Nettle (1999a,b), linguists have been little concerned with quantitative modeling and simulation, possibly due to the myriad of qualitative phenomena that scholars must analyze. An immense amount of structural differences exist not only from one language to the next, but also among different kinds of sociolinguistic situations. More recently, however, scholars belonging to the entirely different discipline of physics have taken an interest in simulating the aspect of historical sociolinguistics which concerns the competition among languages and have looked at how such competition may lead to various patterns of growth or extermination (see Schulze and Stauffer 2006 for a review). This interest among physicists for modeling language competition was triggered by Abrams and Strogatz (2003), who use differential equations to describe the vanishing of one language due to the dominance of another. Since then, a series of articles have appeared. For instance, Patriarca and Leppänen (2004) applied the Abrams-Strogatz model to a geographical situation where two languages, X and Y, may dominate in each their region, resulting in the survival of both, rather as in the original model where only one language will survive. Oliveira et al. (2006a,b) have looked at models in which speakers of small languages will tend to switch to geographically more widespread ones, to account for the fact that real geographical areas tend not to show equally-sized languages. Along the lines of the broader cultural model of Axelrod (1997), Teşileanu and Meyer-Ortmanns (2006) looked at the consequences of the possibility that a greater similarity among languages might further language shift. We build on some of this work, but so far the present paper is to our knowledge the first in this recent tradition to address the issue of taxonomic dynamics. Why simulate ============ Human languages have existed for at least about $10^4$ years, possible much longer. Only a few percent of this development is to some extent documented through writing, while another few percent may be inferred by comparative linguistic methods. Thus, we have no clues to aspects of the development of languages for 80% or more of their history other than what we might infer from abstract extrapolation or from simulations. Like the distant past, the future is also empirically impenetrable. Two aspects of simulations are important. First, we may hope to identify a minimal number of parameters that account for the present state of affairs seen as a result of a long development. Secondly, we may adjust these parameters to test the predictions that different models provide. It should be stressed that simulations are not necessarily suited to prove any particular model or to make predictions about what is in store for the languages of today; they can only represent tests of different models. Nor can the parameters identified be translated into direct explanatory factors for actual distributions. For instance, a simulation of language competition might restrict its parameters to, say, the relative size of languages and it might stipulate some simple mode of interaction, such as the tendency for speakers of smaller languages to shift to contiguous larger ones. Such a model might lead to a plausible picture of language distributions, perhaps even one resembling the current state of affairs. But this does not mean that this distribution is explained by language sizes and competition alone. The growth of a given language relates to socioeconomic, historical, geographical, ecological and many other circumstances. Since a primary aim of simulation is to reduce the set of parameters, it cannot and should not, however, take into account all relevant factors, but must remain an abstraction. The aim of the investigation ============================ Wichmann (2005) made some simple observations about the present-day quantitative distribution of language family sizes, as measured in numbers of languages per family, and about the distribution of language sizes, as measured in numbers of speakers per language (data drawn from *Ethnologue*). It was found that language family sizes approximate a so-called ‘power-law’, that is, a distribution described by the equation $y = ax^b$, which corresponds to a straight line on a log-log plot. Such distributions are frequent in both nature and the social world (cf. Newman 2005 for an excellent overview). The slope of the curve on the rank-by-size plot is described by the exponent $b$, which was found to be $-1.905$. (For a histogram of the number $n(S)$ of languages versus their size S this corresponds to another power-law with exponent $-1-1/1.905$, and if this histogram sums the raw numbers into bins whose size is proportional to the language size, then the exponent is $-1/1.905$.) When testing for the distribution of language sizes, however, no power-law emerged. The absence of a power-law distribution also comes out of studies by Novotny and Drozd (2000) and Sutherland (2003). (Gomes et al. 1999: 493) had earlier plotted the same data on a graph showing the cumulative size distribution, $n( > S)$, corresponding to the number of languages with a size greater than S. Cutting the curve up into different regions and describing each by a separate equation they then made the problematical claim of the existence of a “composite power-law”). The present paper takes up the challenge of Wichmann (2005: 139) to test, using computer simulations, what the expected past and future distributions of language family sizes and language sizes might look like. The question was raised whether the present distribution of language sizes might be characteristic of a stage of disequilibrium while the expected equilibrium might correspond to a power-law. Stauffer et al. (2006) supported the hypothesis of a disequilibrium. In the present paper we also report on language families. The bit-string model ==================== The model used is one eminently suited to computation. It is a variant of that of Schulze and Stauffer (2005), which operates with bit-strings of length L, where each bit has two values and where the total set of possible dialects has $2^L$ members. (A precursor to this kind of modelling is Wang and Minett 2005, which used strings of integers to simulate branching by the mutation and transfer of numbers.) Each bit may be interpreted as the presence or absence of some characteristic grammatical feature. Under this interpretation we might imagine that a number of diagnostic features were identified, the presence or absence of each of which would be sufficient to distinguish among the grammars of the world’s languages. This number corresponds to the length of the bit-string. An alternative model of language competition, also allowing for thousands of different languages, is that of de Oliveira et al (2006a,b). There, however, languages are characterized merely by consecutive numbers 1, 2, 3, ..., which is not suitable for simulating different taxonomic levels. In this model, language families would have to be determined by the history of language dynamics and their genealogical tree (Schulze and Stauffer 2006), and testing this approach is outside the scope of the present work. The other recent models of language competition to our knowledge allow only a relatively small number of languages and are, for this reason, also less suitable for taxonomy. We test two different variants of the model. In one, which we might call the “hierarchical” variant, the bit-string is divided into subsections corresponding to different taxonomic levels. Two languages are defined as belonging to the same family if their “family” parts of the bit-strings agree. In the other, “flat” variant, there is no such partitioning of the string. Instead, taxonomic levels are achieved by defined a certain threshold $k$ of differences among languages. Differences are measured by comparing two strings and noting the number of positions for which the two strings differ. If the difference is greater than $k$, the two languages are said to belong to different taxa. In both versions of the model we only operate with two taxonomic levels, but both could be extended to include more levels. In the following, each variant is described in more detail. The hierarchical variant ------------------------ This model achieves two taxonomic levels by partitioning the bit-string. The two levels may be conceptualized as corresponding to language families and languages within one family, respectively, but need not be translated exactly into these concepts (which are themselves not very well defined). The languages of individuals may be classified by comparing the bit-strings representing each individual. In the following we illustrate how the model works if we use a bit-string of length 64. People speaking the same language have to agree in all bits. In our implementation we have chosen to stipulate that people speaking languages belonging to the same family have to agree in the leftmost 19 bits. For example, 01101010011110101010-11101010010101101010010111010101010011100001 01101010011110101010-11101010010101101010010111010101010011101001 are two different (even if potentially closely related) languages, while 00101001101010100011-10100101011011010101110101010110101010100111 00101001101010100011-01011010001011000110010101110000110101001100 are two different languages belonging to the same family and 10100110100101011010-10010110101010101111010101101001110001010001 01011011010101101010-10100011110101011010101000110101010011101101 are two different languages belong to two different families. In these examples the dash “-” just indicates the boundary between the two segments of the string, analogously to the convention for phone numbers, which are structured much like our bit-string model. The choice of lengths of the whole string and its parts is of course arbitrary, and need not be 19 + 45 = 64. Nevertheless, various considerations led to single out certain lengths as more suitable than other. First, the model is computationally most effective for bit-string lengths which are powers of 2. Second, a shorter string is to be preferred to a longer one, all else being equal—again for computational reasons. Third, the string should not be so short that the sheer length imposes artificial constraints on the results. In earlier simulations the effect of different lengths ($L = 8$, 16, 32, and 64) were tested. Since it was found that the results were qualitatively similar for $L = 16$, 32, and 64, all values of $L$ higher than or equal to 16 would be equally suitable. Adding the criterion of minimal computation cost would single out $L = 16$ as preferable. However, we found that for this length a maximum number of languages was reached before a meaningfully interpretable distribution was found. (Unlike the 32 and 64 bit-string models and the real-life present-day distribution, see section 5 below, this did not lead to a power-law distribution, since power-laws require the absence of upper bounds. At a point where either all possible languages or all possible families are filled, the power-law distribution breaks down.) Instead, we have chosen a string with the larger length of 32 bits, of which the leading 10 bits define families and the remaining 22 define languages, yielding $2^{10} = 1024$ possible ‘families’ and $2^{22} = 4,194,304$ posssible ‘languages’). Using an ample $L$ also ensures that accidental ‘back mutation’, i.e. the phenomenon whereby, by chance, an identical bit-string occurs after some mutations–something which would not happen in real life–will occur so exceedingly rarely that its effects are completely negligible. (Even for $L = 8$ this situation occurs rarely, cf. Schulze and Stauffer 2006). Simulations using 64 bits, of which 19 bits are reserved for families and the remaining 45 bits for languages were also made (allowing for $2^19 = 524,288$ possible ‘families’ and $2^45 = 35,184,372,088,832$ possible ‘languages’). The results were qualitatively similar. Differentiation is simulated by setting the probability of the change in a bit to 0.0001 per iteration. An iteration is equivalent to a certain, average time step. After some time steps, a bit in either the family sub-string or the language bit-string will change, meaning the creation of a new entity at one of these levels. Given that there are fewer bits in the family bit-string, there is a smaller probability of a change in this part of the string per iteration, and there will therefore be a slower dynamics of families than of languages. In practice, with probability $0.0001 L$ at each iteration, one of the $L$ bits is selected randomly and then reverted, i.e. changed from 0 to 1 or from 1 to 0. In this process, analogously to biological mutations, all bit positions are equivalent and neither 0 nor 1 is in any way preferred. We neglect here for simplicity the diffusion of features from one language to the other used in other simulations involving this model. We assume a shift from small to large populations stipulating that at each iteration with probability $(1-x)^2$ or $(1-x^2)$ each individual gives up his/her old language and instead selects the language of one randomly selected individual of the whole population. Individuals get one child per iteration, and everybody dies with a Verhulst probability proportional to the current population size, something which takes into account factors such as limited food and space. We usually start with a population corresponding roughly to the equilibrium size determined by these Verhulst deaths, where everybody starts with a randomly selected language. After some time, one language may dominate and be spoken by more than 80 percent of the population. Stauffer et al. (2006a) list a complete Fortran program. The histograms of the number of languages spoken by a given number of people are smoothened by random multiplicative noise as in Stauffer et al. (2006b), which may correspond to external perturbations caused by migrations of individuals, intermarriage, changing political circumstances, and other non-systematic factors. The flat variant ---------------- This model is in all but one major respect similar to the hierarchical model. The difference is that taxonomic levels are achieved not by partitioning the string, but by stipulating that two languages which differ in more than one bit belong to different families. The size of each language family (i.e., the numbers of languages in each) is then measured by the number of languages that differ by just one bit from one reference language. We sum over all reference languages, and also over many samples, to get out final statistics. The definition allows one language to belong to different families, just as one person can belong to different friendship groups. Instead, one would get a clear separation into different families without such overlaps if we demand all languages within one family to be separated directly or indirectly by not more than one bit flip. But since we can move from each bit-string of 64 bits to every other possible bit-string through at most 64 such changes of single bits, this definition would mean that all possible languages form one huge family, which is not what we want. (Analogously, on a square lattice we can define a neighbourhood as the set of four nearest neighbours of a given site; then every lattice site belongs to several neighbourhoods. Alternatively, a cluster can be defined as the set of all sites connected directly or indirectly with a given site; then the whole lattice forms one large cluster. Neither definition leads to what we would like to have, which is non-overlapping clusters, corresponding to non-overlapping language families. A further disadvantage of the model is that its equilibrium is either dominance of one language spoken by most people, or fragmentation into numerous languages of about equal size; thus for dominance there is not much to analyze and for fragmentation nearly all languages could form one cluster, meaning that these more sophisticated definitions might not work better in equilibrium.) Results ======= Results for the application of the hierarchical model ----------------------------------------------------- The major results are shown in figs. 1-2 and 4-5. The interest of these are the shapes of the various curves, not the absolute numbers corresponding to each point. The mismatch between large numbers of families and small sizes of languages as compared to the real-world situation is due to the summation over iterations and could be normalized, but this would only serve presentational purposes. In fig. 1 it is shown how size histograms of families strongly depend on the temporal factor. At the initial stage of the simulation ($t = 1$) we see something close to a normal distribution (the rightmost curve in the diagram). At $t = 10$ the distribution forms a parabola (curve connecting x’s). This distribution is close to what the present-day *language* size distribution looks like (see fig. 6). At $t = 60$ (stars) a curve resembling the present-day distribution of *language family* sizes (fig. 3) is obtained, but it has a large hump on the right region of the curve. The real-life distribution also has a hump, but it is much smaller. At 300 iterations (squares) there is a discontinuous distribution with a number of small families and a leap up to a number of larger ones, which form a narrow normal distribution. Fig. 2a focuses on the range $20 \le t \le 150$, where the distribution most closely resembles the present-day one, and varies the population size $N$ to see the dependency on the graph on that variable. It appears that there is not much influence of $N$, provided $t$ is increased with increasing $N$. Moreover, fig. 2a suggests that the closest approximation to the present-day distribution is found around $10^2$ iterations. Statistically solid results for a long run of the 64-bit model in fig. 2b provide similar results. We now turn to the results for language sizes. Fig. 4 shows the sizes for the same number of iterations as in fig. 1. Since the simulations start with fragmentation, $t = 1$ represents a situation with many languages spoken by single speakers (single +). At $t = 10$ (x symbols) a curve roughly like a parabola and already strongly reminiscent of the present-day situation (fig. 6) has begun to form. At $t = 60$ (stars) this distribution is beginning to disrupt, as evidenced by the right tail. This situation further develops into one with many large languages and many small ones, with a large gap for language sizes in between, as shown by the curve for $t = 300$ (squares). Again we narrow in on the range, $20 \le t \le 60$, where the best approximation of the present situation (fig. 6) is found and vary the population size (fig. 5). For $t = 40$ and $N = 50,000$ the distribution closely approximates the present-day one. By comparing the curves for $t = 40$ in figs. 2a and 5 an interesting observation is obtained: at identical time steps the curve for language family sizes may approximate a power-law while the curve for language sizes does not, but rather something close to a parabola, as in real life. Wichmann (2005: 128) hypothesized that both curves should approximate a power-law, but the simulations rather suggest that this is only the case for language family sizes, at least given the model and the setting of parameters assumed here. The overall result, then, suggests that neither the present-day distribution of language family sizes nor that of language sizes are unexpected and that both may have been obtained for a long time and may continue to be obtained. Eventually a dominance of just one large language accompanied by other slightly different languages is possible, but this situation has not yet set in. Results for the application of the flat model --------------------------------------------- For investigating the distance among languages the ‘flat’ model is most useful because the distance among two languages belonging to two different families in the hierarchical model cannot easily be measured. (The hierarchical bit-strings representing languages in any two languages belong to two different families are not comparable since the positions no longer mean the same when one moves up one taxonomic level.) Thus we measured differences in simulations implementing the non-hierarchical model, i.e. the standard model of Schulze and Stauffer (2005, 2006) where all bits are equivalent. As in most of our previous studies, only short bit-strings of 8 or 16 bits were used, and the random multiplicative noise was omitted; for these studies we waited until a stationary state after about $10^3$ iterations was established. The distance measure used is the so-called ‘Hamming distances’, also investigated by Teşileanu and Meyer-Ortmanns (2006). The Hamming distance between two bit-strings is the number of bits which are different in a position-by-position comparison of the two strings. For example, the Hamming distance between 01001101 and 11000011 is four. As explained above, we define a language family in this model as a set of languages differing from a given reference language by not more then $k$ bits, in this case setting $k$ to one bit. The results of the simulations of bit-strings of lengths 8 and 16 are shown in fig. 7; as in fig. 2 above, the simulations represent states of non-equilibrium, i.e., they were stopped at some intermediate time and not let run until the distribution no longer changed apart from random fluctuations. These results are not very different from those shown in fig. 2 for the 64 bits string in the hierarchical model. More on Hamming distances ------------------------- The above results were obtained by stopping the simulations at a suitable time such that the results are closest to reality. In this section we report on the equilibrium properties for longer times where the distributions no longer change appreciably and where we will have either dominance of one language or fragmentation of the whole population into many different languages. Fig. 8 nicely shows the phase transition between dominance at low and fragmentation at high mutation rate $p$ per bit-string when we vary the mutation rate instead of fixing it to only 0.0001 mutations per bit and per iteration. For dominance, nearly everybody speaks one language, and most of the others speak a language differing in only one bit from this dominating language. Fragmentation happens for larger mutation rates; then all possible languages are represented about equally. We see in fig. 8 that dominance is characterized by a small average Hamming distance while for fragmentation the average Hamming distance is about 1/2 (here it is normalized by the length of the bit-string such that two random bit-strings have on average a distance 1/2.) This effect is already seen if one looks only at the two largest languages in the population, as done by Teşileanu and Meyer-Ortmanns (2006). For fragmentation, the distribution of Hamming distances between two pairs of speakers is roughly Gaussian (normal), shown by a parabola in the semi-logarithmic plot (stars in fig. 8). In the case of dominance, as observed for two lower mutation rates $p$ in fig. 9, the most probable Hamming distance is zero, and for higher distances the probability to observe them decays very rapidly. In these simulations we started with one language only and used the probability $1-x^2$ for the shift from small to large languages. We got qualitatively similar results when we started from a population fragmented into many languages, except that then the probability of a shift was set to $(1-x)^2$, to allow for a possible transition from fragmentation to dominance. Conclusion ========== The primary aim of our simulations was to capture, within one and the same model, how two different empirically observed distributions might arise, i.e. a roughly log-normal distribution of language sizes and an approximate power-law for the family sizes. With reasonable lengths of bit-strings, populations and observation times we could, indeed, find the two different behaviours in the same simulation. This suggests, contrary to the hypothesis of Wichmann (2005), that the present-day distribution of language family sizes in combination with that of language sizes may not be unexpected. In terms of simulation techniques the major contribution of the present paper has been the introduction of new models into the area of linguistic taxonomic dynamics, an area which, to our knowledge, has not previously been investigated by means of computer simulations. The best results were obtained in implementations of the hierarchical bitstring, a model which also has the advantage of being versatile and easy to implement. The investigations, however, also revealed some problems with the model. If for a fixed length of the bit-strings the population size N goes to infinity, then in the parameter region of fragmentation all possible languages will be spoken, and all possible families will exist, making taxonomy a mathematical triviality without connection to reality. Thus simulations of large but finite populations, as presented here, may be better than mathematically exact solutions for infinite populations. Moreover, we did find an effective power-law for the family size distribution, but that distribution decayed much faster with increasing number of languages than the real distribution, shown fig. 3. Thus future research should aim at also applying and testing other models, such as that of de Oliveira et al. (2006a,b), to problems of linguistic taxonomic dynamic. [**References**]{} Abrams, Daniel and Steven H. Strogatz. 2003. Modelling the dynamics of language death. *Nature* 424: 900. Axelrod, Robert. 1997. The dissimination of culture: a model with local convergence and global polarization. *The Journal of Conflict Resolution* 41: 203-226. *Ethnologue: Languages of the World* (14th edn. edited by Grimes, Barbara F. 2000, 15th edition edited by Raymond, G. Gordon 2005). Dallas, TX: Summer Institute of Linguistics. Gomes, Marcelo A. F., G. L. Vasconcelos, I. J. Tsang, and Ing Ren Tsang. 1999. Scaling relations for diversity of languages. *Physica A* 271: 489-495. Nettle, Daniel. 1999a. Linguistic diversity of the Americas can be reconciled with a recent colonization. *Proceedings of the National Academy of Sciences of the U.S.A.* 96: 3325-3329. Nettle, Daniel. 1999b. Using social impact theory to simulate language change. *Lingua* 108: 95-117. Newman, Mark E. J. 2005. Power laws, Pareto distributions and Zipf’s law. *Contemporary Physics* 46: 323-351. Novotny, Vojtech and Pavel Drozd. 2000. The size distribution of conspecific populations. *Proceedings of the Royal Society of London* B267: 947-952. Oliveira, Viviane M. de, Marcelo A. F. Gomes, and Ing Ren Tsang. 2006a. Theoretical model for the evolution of the linguistic diversity. *Physica A* 361: 361-370. Oliveira, Viviane M. de, Paulo R. A. Campos, Marcelo A. F. Gomes, and Ing Ren Tsang. 2006b. Bounded fitness landscapes and the evolution of the linguistic diversity, e-print physics 0510249 for *Physica A*. Patriarca, Marco and Teemu Leppänen. 2004. Modeling language competition. *Physica A* 338: 296-299. Schulze, Christian and Dietrich Stauffer. 2005. Monte Carlo simulation of the rise and fall of languages. *International Journal of Modern Physics C* 16: 781-787. Schulze, Christian and Dietrich Stauffer. 2006. Computer simulation of language competition by physicists. In: Chakrabarti, B. K., A. Chakraborti and A. Chatterjee (eds.), *Econophysics and Sociophysics: Trends and Perspectives*. Weinheim: WILEY-VCH Verlag; and: Recent developments in computer simulations of language competition, Computing in Science and Engineering 8 (May/June) 86-93. Stauffer, Dietrich, Suzana Moss de Oliveira, Paulo Murilo C. de Oliveira, Jorge S. Sá Martins. 2006a. *Biology, Sociology, Geology by Computational Physicists*. Amsterdam: Elsevier. Stauffer, Dietrich, Christian Schulze, F. Welington S. Lima, Søren Wichmann, and Sorin Solomon. 2006b. Non-equilibrium and irreversible simulation of competition among languages. *Physica A*. (In press). Sutherland, William J. 2003. Parallel extinction risk and global distribution of languages and species. *Nature* 423: 276-279. Tesşileanu, Tiberiu and Hildegard Meyer-Ortmanns. 2006. Competition and languages and their Hamming distance. arXiv:physics/0508229, *International Journal of Modern Physics C* 17: 259-278. Wang, William S. Y. and James W. Minett. 2005. The invasion of language: emergence, change and death. *Trends in Ecology and Evolution* 20.5: 263-296. Wichmann, S[ø]{}ren. 2005. On the power-law distribution of language family sizes. *Journal of Linguistics* 41: 117-131.

--- abstract: 'We introduce a new category called Quasi-Nash, unifying Nash manifolds and algebraic varieties. We define Schwartz functions, tempered functions and tempered distributions in this category. We show that properties that hold on affine spaces, Nash manifolds and algebraic varieties, also hold in this category.' address: 'Dept. of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel' author: - Boaz Elazar title: 'Schwartz Functions on Quasi-Nash Varieties' --- Introduction ============ Schwartz functions are named after Laurent Schwartz, who defined them in $\mathbb{R}^{n}$. In $\mathbb{R}^{n}$ they are usually defined as smooth functions which decay to zero with all their derivatives faster than the inverse of any polynomial when reaching infinity. We say $f$ is a Schwartz function on $\mathbb{R}$, for example, if for any $n,k\in\mathbb{N}\cup\left\{ 0\right\} $ we have $|x^{n}f^{\left(k\right)}|<\infty$ where $f^{\left(k\right)}$ is the $k$’th derivative of $f$. The space of all Schwartz functions on $\mathbb{R}^{n}$ will be denoted by $\mathcal{S}\left(\mathbb{R}^{n}\right)$. Schwartz functions on $\mathbb{R}^{n}$ have some nice properties such as: $\mathcal{S}\left(\mathbb{R}^{n}\right)$ is a Fréchet space, $\mathcal{S}\left(\mathbb{R}^{n}\right)$ is invariant under Fourier transform and every function in $\mathcal{S}\left(\mathbb{R}^{n}\right)$ is integrable. Later, in [\[]{}dC, AG[\]]{}, Schwartz functions were defined on Nash manifolds, which are smooth semi-algebraic varieties. For a Nash manifold $M$, $f$ is said to be Schwartz on $M$ if for any Nash differential operator $D$ we have $||Df||_{\infty}<\infty$. Nash differential operator on $M$ means an element of the algebra generated by multyplying by Nash functions and by deriving along Nash sections of the tangent bundle. Lately, in [\[]{}ES[\]]{} we defined Schwartz functions on real algebraic varieties which might have singularities, and showed how the affine algebraic varieties share the properties of Schwartz functions on Nash manifolds. We also showed that some of the results hold in the general case. In this paper we define a category such that both the Nash manifolds and the algebraic varieties are subcategories of the new one. Moreover, this category enables us to prove the rest of the claims about Schwartz functions on general algebraic varieties. We call this new category Quasi-Nash, or QN, where its affine objects correspond to semi-algebraic subsets of $\mathbb{R}^{n}$ and the morphisms are locally restrictions of Nash maps. A general variety is defined as a glueing of open affine varieties. This new category slightly extends the category of Nash varieties. The definitions of this category, and some usefull lemmas appear in section 3. The main results about Schwartz functions in this paper include: Isomorphic QN varieties $X_{1}\cong X_{2}$, imply an isomorphism of the Fréchet spaces $\mathcal{S}\left(X_{1}\right)\cong\mathcal{S}\left(X_{2}\right)$ where $\mathcal{S}\left(X_{i}\right)$ is the space of Schwartz functions on $X_{i}$ (Lemma \[lem-First-Result\]). The next Proposition deals with tempered functions. Informally, a tempered function on $\mathbb{R}^{n}$ is a smooth function bounded by a polynomial, and so does any of its derivatives. We define a tempered function on a QN variety later on. \[prop-elor\](Tempered partition of unity) Let $\{V_{i}\}_{i=1}^{m}$ be a finite open cover of a QN variety $X$. Then, there exist tempered functions $\{\beta_{i}\}_{i=1}^{m}$ on $X$, such that $supp(\beta_{i})\subset V_{i}$ and $\sum\limits _{i=1}^{m}\beta_{i}=1$ (Proposition \[prop-part-of-unity\]). \[thm-hardest\]For a QN variety $X$ and a closed subset $Z\subset X$, define $U:=X\setminus Z$ and $W_{Z}:=\{\phi\in\mathcal{S}(X)|\phi\text{ is flat on }Z\}$. Then $W_{Z}$ is a closed subspace of $\mathcal{S}(X)$ (and so it is a Fréchet space), and extension by zero $\mathcal{S}(U)\to W_{Z}$ is an isomorphism of Fréchet spaces, whose inverse is the restriction of functions (Theorem \[thm-general-char\]). It should be noted that the proof of Theorem \[thm-hardest\] was the hardest to prove. Usually, a function $f:\mathbb{R}^{n}\to\mathbb{R}$ is said to be flat at a point $p\in\mathbb{R}^{n}$ if $f$, and all its derivatives, vanish at $p$. We had to make clear what it means for a function on a singular variety to be flat on a singular point $p$. For subvarieties, of $\mathbb{R}^{n}$ we defined it to be a restriction of a smooth function on $\mathbb{R}^{n}$ which is flat at $p$. There are several other approaches to do so (see [\[]{}BMP2, F[\]]{} for example), but [\[]{}BMP2[\]]{} shows they are all equivalent in our case. Then, Theorem \[thm-hardest\] turns out to be a Whitney type extension problem. We proved it using subanalytical geometry results in [\[]{}BM1, BM2, BMP1, BMP2[\]]{}. Theorem \[thm-hardest\] also enables us to define Schwarz functions by a local condition rather then by the global ones we used. Instead of demanding a function that decays “fast at infinity”, we just have to demand a smooth function that is flat on the points “added at infinity” in some compactification process. Furthermore, Theorem \[thm-hardest\] gives us one more important result regarding tempered distributions. The space of tempered distributions is the space of linear continuous functionals on $\mathcal{S}\left(X\right)$. It is denoted by $\mathcal{S}^{*}\left(X\right)$. Theorem \[thm-hardest\] implies that for any open $U\subset X$, the restriction morphism of tempered distributions $\mathcal{S}^{*}(X)\to\mathcal{S}^{*}(U)$ is onto. This is not the case for general distributions. E.g. take the compactification of $\mathbb{R}$ into a circle. The distribution $e^{x}dx$ on $\mathbb{R}$ cannot be extended to the circle. \[cor-eimim\]Let $X$ be a QN variety. Then the assignment of the space of Schwartz functions (respectively tempered functions, tempered distributions) to any open $U\subset X$, together with the extension by zero $Ext_{U}^{V}$ from $U$ to any other open $V\supset U$ (restriction of functions, restrictions of functionals from $\mathcal{S}^{*}(V)$ to $\mathcal{S}^{*}(U)$), form a flabby cosheaf (sheaf, flabby sheaf) on $X$ (Corollaries \[cor-Sch-cosheaf\], \[cor-tempered-sheaf\], \[cor-Dist-sheaf\]). We would like to emphasize we proved Proposition \[prop-elor\], Theorem \[thm-hardest\], and Corollary \[cor-eimim\] also for non-affine varieties in this category, what we could not do in the algebraic category. Structure of this paper {#structure-of-this-paper .unnumbered} ----------------------- In **Section 2** we give preliminary definitions and results we use in this paper. Most of them concern with Nash manifolds and Schwartz functions on them, and basic properties of Fréchet spaces.\ In **Section 3** we define the new category and show some nice properties it has.\ In **Section 4** we define Schwartz functions, tempered functions and tempered distributions in this category, and prove some claims about them.\ In **Section 5** we show the co-sheaf structure of Schwartz functions, and the sheaf structures of tempered functions and distributions.\ In **Section 6** we define vector bundles over QN varieties and show some properties that hold on those bundles.\ Finally, in **Appendix A** we build some tools needed for tempered partition of unity. Conventions {#conventions .unnumbered} ----------- Throughout this paper, we use the restricted topology of semi algebraic sets over $\mathbb{R}^{n}$, unless otherwise stated. For the definition of restricted topology see \[AG-def-rest-top\]. We also use the convention that for two varieties of some kind $X\subset M$, we will denote by $I_{Sch}^{M}\left(X\right)$ the ideal of Schwartz functions on $M$ that vanish identically on $X$. We say a function $f$ is smooth over $M$ if $f\in C^{\infty}\left(M\right)$. Acknowledgments {#acknowledgments .unnumbered} --------------- I would like to extend my gratitude to my supervisor Prof. Dmitry Gourevitch, who taught me how to cross the barriers this work presented. A great thanks goes to Prof. Avraham Aizenbud, who suggested different approaches for defining our objects and dealing with the problems we started with. I would like to thank Prof. William A. Casselman, for his idea regarding the definition of Schwartz functions on closed subsets as restrictions of Schwartz functions from their neighborhoods. I thank Prof. Dmitry Novikov for patiently answering questions that arose along the way. Finally, I would like to thank Prof. Alessandro Tancredi for helping me understand the subtleties of the Nash varieties category. Preliminaries ============= We shall dedicate this section to definitions and results used in this paper. They will include a definition of a semi-algebraic set, and the algebraic Alexandrov compactification (2.1-2.3), Fréchet spaces (2.5-2.7), Nash manifolds and Schwartz functions on Nash manifolds (2.8-2.10), Schwartz and tempered functions over Nash manifolds (2.11-2.19). [\[]{}BCR[\]]{} A semi-algebraic subset of $\mathbb{R}^{n}$ is a subset of the form $\bigcup\limits _{i=1}^{n}\bigcap\limits _{j=1}^{m_{i}}\left\{ x\in\mathbb{R}^{n}|p_{i,j}\left(x\right)>0\text{ or }p_{ij}\left(x\right)=0\right\} $ where $p_{ij}\in\mathbb{R}[x_{1},...,x_{n}]$. An affine algebraic variety is called *complete* if regular function on it is bounded (cf. [\[]{}BCR 3.4.9 and 3.4.10[\]]{}). Thus, a closed embedding of a complete affine algebraic variety is compact in the Euclidean topology on $\mathbb{R}^{n}$. \[prop-Alexandrov\](Algebraic Alexandrov compactication [\[]{}BCR, Proposition 3.5.3[\]]{}). Let $X$ be an affine algebraic variety that is not complete, then there exists a pair $\left(\dot{X},i\right)$ such that: 1. $\dot{X}$ is a complete affine algebraic variety. 2. $i:X\to\dot{X}$ is an algebraic isomorphism from $X$ onto $i\left(X\right)$. 3. $\dot{X}\backslash i\left(X\right)$ consists of a single point. The chain rule for deriving composite functions can be extended to higher derivatives and higher dimensions. A relevant result is as follows: \[lem-Faa-di\][\[]{}CS, Theorem 2.1[\]]{} Let $x_{0}\in\mathbb{R}^{d}$, $V\subset\mathbb{R}^{d}$ be some open neighborhood of $x_{0}$ and $g:V\to\mathbb{R}^{m},\:g\in C^{k}\left(V,\mathbb{R}^{m}\right)$, for some $k\in\mathbb{N}$. Let $U\subset\mathbb{R}^{n}$ be some open neighborhood of $g\left(x_{0}\right)$ and $f:U\to\mathbb{R},\:f\in C^{k}\left(U\right)$. Assume $f$ is $k$-flat at $g(x_{0})$, i.e. its Taylor polynomial of degree $k$ at $g(x_{0})$ is zero. Then $f\circ g:g^{-1}\left(U\right)\to\mathbb{R}$ is $k$-flat at $x_{0}$ **Fréchet spaces** \[prop-Closed-Frechet-subspace\][\[]{}T, Chapter 10[\]]{}. A closed subspace of a Fréchet space is a Fréchet space (in the induced topology). \[thm-Banach-open-mapping\](Banach open mapping - [\[]{}T, Chapter 17, Corollary 1[\]]{}). A bijective continuous linear map from a Fréchet space to another Fréchet space is an isomorphism. $\:$ \[thm-Hahn-Banach\](Hahn-Banach - [\[]{}T, Chapter 18[\]]{}). Let $F$ be a Fréchet space, and $K\subset F$ a closed subspace. By Proposition \[prop-Closed-Frechet-subspace\] $K$ is a Fréchet space (with the induced topology). Define $F^{*}$ (respectively $K^{*}$) to be the space of continuous linear functionals on $F$ (on $K$). Then the restriction map $F^{*}\to K^{*}$ is onto. $\:$ **Nash manifolds** \[AG-def-rest-top\]A restricted topological space $M$ is a set $M$ equipped with a family of subsets of $M$, including $M$ and the empty set, called the set of open subsets of $M$, that is closed with respect to **finite** unions and finite intersections. Therefore, we will consider only finite open covers in restricted topology. $\:$ An **$\mathbb{R}$-space** is a pair $\left(M,\mathcal{O}_{M}\right)$ where $M$ is a restricted topological space and $\mathcal{O}_{M}$ a sheaf of $\mathbb{R}$-algebras over $M$ which is a subsheaf of the sheaf $C_{M}$ of all continuous real-valued functions on $M$. A continuous map $\varphi:\left(M,\mathcal{O}_{M}\right)\to\left(N,\mathcal{O}_{N}\right)$ is called a morphism of $\mathbb{R}$-spaces if for any open subset $U\subset N$ and any $f\in\mathcal{O}_{N}\left(U\right)$, we have $f\circ\varphi\left(\varphi^{-1}\left(U\right)\right)\in\mathcal{O}_{M}\left(\varphi^{-1}\left(U\right)\right)$. $\:$ \[AG-def-Nash-manif\](1) A Nash submanifold $M$ of $\mathbb{R}^{n}$ is a semi-algebraic subset of $\mathbb{R}^{n}$ which is a smooth submanifold. A Nash function on $M$ is a smooth semi-algebraic function. \(2) An affine Nash manifold is an $\mathbb{R}$-space which is isomorphic to an $\mathbb{R}$-space associated to a closed Nash submanifold of $\mathbb{R}^{n}$. \(3) A Nash manifold is an $\mathbb{R}$-space $\left(M,\mathcal{N}_{M}\right)$ with a sheaf of Nash functions, which has a finite open cover $\left(M_{i}\right)_{i=1}^{n}$ such that each $\mathbb{R}$-space $\left(M_{i},\mathcal{N}_{M}|_{M_{i}}\right)$ is an affine Nash manifold. $\:$ **Schwartz and tempered functions on Nash manifolds** \[AG-prop-sub-of-Nash-is-Nash\][\[]{}AG - Proposition 3.3.3[\]]{} (1) Any open (semi-algebraic) subset $U$ of an affine Nash manifold $M$ with the induced $\mathbb{R}$-space structure is an affine Nash manifold. \(2) Any open (semi-algebraic) subset $U$ of a Nash manifold $M$ with the induced $\mathbb{R}$-space structure is a Nash manifold. \(1) **Nash differential operator** on an affine Nash manifold is an element of the algebra with $1$ generated by multiplication by Nash functions and derivations along Nash sections of the tangent bundle (Nash vector fields). \(2) The space of **Schwartz functions on an affine Nash manifold $M$** is $\mathcal{S}\left(M\right):=\left\{ \phi\in C^{\infty}\left(M\right)|D\phi\text{ is bounded for any Nash differential operator}\right\} $. \(3) The topology on $\mathcal{S}\left(M\right)$ is defined by the semi norms $||\phi||{}_{D}:=\sup\limits _{x\in M}|D\phi\left(x\right)|$. \(4) Let $M$ be as in \[AG-def-Nash-manif\](3), and $\phi:\bigoplus\limits _{i=1}^{k}\mathcal{S}\left(M_{i}\right)\to C^{\infty}\left(M\right)$ defined by extension by zero and summing. Then $\mathcal{S}\left(M\right):=Im\left(\phi\right)$. $\:$ \[AG-prop-Sch-on-Nash-is-Fre\][\[]{}corollary of AG - Corollary 4.1.2[\]]{} Let $M$ be a Nash manifold. Then $\mathcal{S}\left(M\right)$ is a Fréchet space. [\[]{}AG - Definition 4.2.1 and Theorem 4.6.2[\]]{} A function $t:\mathbb{R}^{n}\to\mathbb{R}$ is called *tempered* if it is a smooth function such that for any $\alpha\in\left(\mathbb{N}\cup\left\{ 0\right\} \right)^{n}$ there exists a polynomial $p_{\alpha}\in\mathbb{R}\left[x_{1},...,x_{n}\right]$ such that $|\frac{\partial^{|\alpha|}t}{\partial^{\alpha}x}\left(x\right)|<p_{\alpha}\left(x\right)$ for any $x\in\mathbb{R}^{n}$. Let $M$ be an affine Nash manifold, and let $i:M\hookrightarrow\mathbb{R}^{n}$ be a closed embedding. A function $t:M\to\mathbb{R}$ is called a *tempered function* on $M$ if $i_{*}f:=f\circ i^{-1}$ is the restriction to $i(M)$ of a tempered function from $\mathbb{R}^{n}$. Denote the space of all tempered functions on $M$ by $\mathcal{T}\left(M\right)$. $\mathcal{T}\left(M\right)$ is a well defined space (independent of the embedding chosen). \[AG-prop-ts-is-s\][\[]{}corollary of AG - Proposition 4.2.1[\]]{} Let $M$ be a Nash manifold and $\alpha$ be a tempered function on $M$. Then $\alpha\mathcal{S}\left(M\right)\subset\mathcal{S}\left(M\right)$. \[AG-thm-red-to-cl-is-onto\][\[]{}corollary of AG - Theorem 4.6.1[\]]{} Let $M$ be a Nash manifold and $Z\hookrightarrow M$ be a closed Nash submanifold. The restriction $\mathcal{S}\left(M\right)\to\mathcal{S}\left(Z\right)$ is defined, continuous and onto. \[AG-prop-aff-temp-sheaf\][\[]{}AG - Proposition 5.1.3[\]]{} Let $M$ be a Nash manifold. The assignment of the space of tempered functions on $U$, to any open $U\subset M$, together with the usual restriction maps, define a sheaf of algebras on $M$. \[AG-thm-part-of-unity\][\[]{}AG - Theorem 5.2.1[\]]{} (Partition of unity for Nash manifold). Let $M$ be a Nash manifold, and let $\left(U_{i}\right)_{i=1}^{n}$ be a finite open cover. Then \(1) there exist tempered functions $\alpha_{1},...,\alpha_{n}$ on $M$ such that $supp\left(\alpha_{i}\right)\subset U_{i}$, $\sum\limits _{i=1}^{n}\alpha_{i}=1$. \(2) Moreover, we can choose $i$ in such a way that for any $\phi\in\mathcal{S}\left(M\right),\:\alpha_{i}\phi\in\mathcal{S}\left(U_{i}\right)$. $\:$ \[AG-thm-Char-Nash\][\[]{}AG - Theorem 5.4.1[\]]{} (Characterization of Schwartz functions on open subset) Let $M$ be a Nash manifold, $Z$ be a closed (semi-algebraic) subset and $U=M\backslash Z$. Let $W_{Z}$ be the closed subspace of $\mathcal{S}\left(M\right)$ defined by $W_{Z}:=\left\{ \phi\in\mathcal{S}\left(M\right)|\phi\text{ vanishes with all its derivatives on }Z\right\} $. Then restriction and extension by 0 give an isomorphism $\mathcal{S}\left(U\right)\cong W_{Z}$. Geometry ======== \[Def-NQN\] *“Naive” Quasi Nash category - NQN* - Let $X$ be a locally closed semi-algebraic subset of $\mathbb{R}^{n}$. A morphism in this category is a map from an NQN set $X$ to an NQN set $Y\subset\mathbb{R}^{m}$ which is a restriction of a Nash map on an open semi-algebraic neighborhood $U$ of $X$ to $\mathbb{R}^{m}$ such that $X$ is closed in $U$ and $X$ is mapped into $Y$. i,e, $\varphi:X\to Y$, $\varphi:=g|_{X}$ where $X\subset U$, and $g:U\to\mathbb{R}^{m}$ is Nash. \[NQN-equiv-property\]Let $X,\:Y$ be NQN, and let $\varphi:X\to Y$ a continuous map. Then $\varphi$ is an NQN morphism if and only if for any NQN function $f:Y\to\mathbb{R}$, $f\circ\varphi:X\to\mathbb{R}$ is an NQN function. (NQN function is an NQN map from an NQN set to $\mathbb{R}$) Let $\varphi$ be an NQN morphism. Take $f$ as required. As $f$ is a restriction of a Nash function on a neighborhood of $Y$, and $\varphi$ is a restriction of a Nash map on a neighborhood of $X$, we get a composition of Nash maps, which is Nash. Thus, $f\circ\varphi$ is an NQN function. Now let $\varphi$ pullback NQN functions to NQN functions. As $Y\subset\mathbb{R}^{m}$, $\varphi$ can be presented as $\varphi=\left(\varphi_{1},...,\varphi_{m}\right)$. For each $i$ take the Nash function $f_{i}=y_{i}$. This results in Nash functions on neighborhoods $U_{i}$ of $X$ where on each $U_{i}$: $f_{i}\circ\varphi=y_{i}\circ\varphi=\varphi_{i}$ for any $i$, what makes all $\varphi$’s coordinates NQN maps.\ Take the intersection $U=\bigcap\limits _{i=1}^{n}U_{i}$ of all those neighborhoods to get a neighborhood of $X$ where all of $\varphi$’s coordinates are NQN simultaneously and therefore $\varphi$ is an NQN map itself. Now let’s take a broader category - \[Def-Quasy-Nash\](1) Let $X$ be a locally closed semi-algebraic subset of $\mathbb{R}^{n}$. Define the **$\mathbb{R}$-space corresponding to $X$** to be the pair $\left(X,QN_{X}\right)$ where $QN_{X}$ is defined to be a sheaf as follows: For each open $U\subset X$, we say $f\in QN_{X}\left(U\right)$ if there exists a collection $\left\{ V_{i}\right\} _{i=1}^{m}$ of open semi-algebraic subsets of $\mathbb{R}^{n}$ and Nash functions $f_{i}$ from these sets, such that $U_{i}:=V_{i}\cap X$ is an open cover of $U$, for any $i,\:f|_{U_{i}}=f_{i}|_{U_{i}}$, and for any $i,j$ $f_{i}|_{U_{i}\cap U_{j}}=f_{j}|_{U_{i}\cap U_{j}}$. This is a sheafification of the NQN presheaf in definition \[Def-NQN\]. \(2) An **affine QN variety $X$** is defined to be an $\mathbb{R}$-space isomorphic to an $\mathbb{R}$-space obtained by sheafification from soN set $Y\subset\mathbb{R}^{n}$. I.e. $\left(X,O_{X}\right)$ is an affine QN variety if $\left(X,O_{X}\right)\cong\left(Y,O_{Y}\right)$ where $Y$ is NQN set, and $O_{Y}$ is the sheafification of the NQN functions on $Y$. \[Rem-NQN-Like\]Let $X$ be an affine QN variety. Then $X$ is QN isomorphic to a closed set in $\mathbb{R}^{n}$, which by abuse of notation is denoted by $X$ as well. Let $V\subset\mathbb{R}^{n}$ be some open semi-algebraic subset such that $U=V\cap X$ is an open subset of $X$. Then $V$ is Nash diffeomorphic to some closed affine Nash submanifold $V'$ in $\mathbb{R}^{N}$, by [\[]{}BCR - Theorems 8.4 Affine Nash manifolds form a full subcategory of affine QN varieties. Affine algebraic varieties form a subcategory, not a full subcategory. An affine Nash manifold is an $\mathbb{R}$-space isomorphic to an $\mathbb{R}$-space associated to a smooth closed semi-algebraic subset of $\mathbb{R}^{n}$. The sheaf on a Nash manifold is made of Nash functions - smooth semi-algebraic functions. In the QN category the sheaf is made of locally restrictions of Nash functions. Due to the Nash tubular neighborhood, the sheaves are the same on Nash manifolds. Let $\varphi:X\to Y$ be a QN morphism between the Nash manifolds $X,\:Y$. Take a Nash function $f$ on $Y$. So there exists a cover $\bigcup\limits _{i=1}^{n}X_{i}=X$ s.t. $g_{i}:=f\circ\varphi|_{X_{i}}$ is a Nash function. As Nash functions on the Nash manifold $X$ form a sheaf, $\exists g\in\mathcal{N}\left(X\right)$ s.t. $g|_{X_{i}}=g_{i}$. So for any QN map $\varphi$, and any $f\in\mathcal{N}\left(Y\right)$, we get $f\circ\varphi\in\mathcal{N}\left(X\right)$.\ Affine algebraic variety is isomorphic to an algebraic subset of $\mathbb{R}^{n}$. But as morphisms in this category are rational maps, and not any Nash map is such a map (e.g. $\sqrt{1+x^{2}}$), this is not a full subcategory. \[lem-affine-QN-equiv\]Let $X,\:Y$ be affine QN varieties and $\tilde{X}\subset\mathbb{R}^{n},\:\tilde{Y}\subset\mathbb{R}^{m}$ the corresponding closed QN sets. Let $\varphi:X\to Y$ be a continuous map and $\tilde{\varphi}:\tilde{X}\to\tilde{Y}$ the corresponding map. Then $\varphi$ is a QN morphism if and only if there exists an open cover $\bigcup\limits _{i=1}^{N}\tilde{X}_{i}=\tilde{X}$ such that $\tilde{\varphi}|_{\tilde{X}_{i}}$ is an NQN map for any $i$. As $\tilde{Y}\subset\mathbb{R}^{m}$, $\tilde{\varphi}$ can be presented as $\tilde{\varphi}=\left(\tilde{\varphi}_{1},...,\tilde{\varphi}_{m}\right)$. For each $j\in\left\{ 1,...,m\right\} $ take the Nash function $f_{j}=\tilde{y}_{j}$. When pulling it back, we get an open cover $\tilde{X}=\bigcup\limits _{k=1}^{N_{j}}\tilde{X}_{k}$ such that for any open subset $\tilde{X}_{k}$, the function $f_{j}\circ\tilde{\varphi}=\tilde{y}_{j}\circ\tilde{\varphi}=\tilde{\varphi}_{j}$ is a restriction of a Nash function, i.e. $\tilde{\varphi}_{j}|_{\tilde{X}_{k}}$ is an NQN map for any $k$. Now take the refinement of those $j$ covers of $\tilde{X}$, and denote it by $\tilde{X}=\bigcup\limits _{i=1}^{N}\tilde{X}_{i}$. As a restriction of a Nash function is a Nash function, we get that for any $i$, $\left(\tilde{\varphi}_{1},...,\tilde{\varphi}_{m}\right)|_{\tilde{X}_{i}}$ is an NQN map, what makes $\tilde{\varphi}|_{\tilde{X}_{i}}$ an NQN map. Now the other direction: Assume $\tilde{\varphi}|_{\tilde{X}_{i}}$ is an NQN map for any $i$ and let $f\in QN_{Y}\left(U\right)$ for an open $U\subset Y$. Those $f$ and $U$ correspond to some $\tilde{f}$ and $\tilde{U}$ respectively. This means there is an open cover $\tilde{U}=\bigcup\limits _{j=1}^{M}\tilde{U}_{j}$ and for each $j$, $\tilde{f}|_{\tilde{U}_{j}}$ is a restriction of a Nash function. The map $\tilde{\varphi}$ is continuous, so $\tilde{\varphi}|_{\tilde{X}_{i}\cap\tilde{\varphi}^{-1}\left(\tilde{U}_{j}\right)}:\tilde{X}_{i}\cap\tilde{\varphi}^{-1}\left(\tilde{U}_{j}\right)\to\tilde{U}$ is an NQN map, i.e. it is a restriction of a Nash map from an open neighborhod of $\tilde{X}_{i}\cap\tilde{\varphi}^{-1}\left(\tilde{U}_{j}\right)$. Therefore $\tilde{f}\circ\tilde{\varphi}|_{\tilde{X}_{i}\cap\tilde{\varphi}^{-1}\left(\tilde{U}_{j}\right)}$ is a restriction of a Nash function for any $j$ and any $i$. Thus, $f\circ\varphi\in QN_{X}\left(\varphi^{-1}\left(U\right)\right)$. \[def-gen-QN-variety\]A **QN variety** is an $\mathbb{R}$-space $\left(X,QN_{X}\right)$ where $X$ is a restricted topological space, which has a finite cover $\bigcup\limits _{i=1}^{m}X_{i}=X$ of open sets such that the $\mathbb{R}$-spaces $\left(X_{i},QN_{X}|_{X_{i}}\right)$ are affine QN varieties. \[lem-gen-QN-equiv\]Let $X,\:Y$ be general QN varieties. Let $\varphi:X\to Y$ be a continuous map. Then $\varphi$ is a QN morphism if and only if there exists an open affine cover $Y=\bigcup\limits _{i=1}^{M}Y_{i}$, and an open cover $\bigcup\limits _{j=1}^{N_{i}}X_{ij}=X_{i}:=\varphi^{-1}\left(Y_{i}\right)$, where each $X_{ij}$ corresponds to a locally closed semi-algebraic set $\tilde{X}_{ij}$ such that the induced map $\tilde{\varphi}|_{\tilde{X}_{ij}}$ is an NQN map to the closed semi-algebraic set $\tilde{Y}_{i}$ isomorphic to $Y_{i}$. $Y=\bigcup\limits _{i=1}^{M}Y_{i}$, and for each $i$ take the QN variety $X_{i}:=\varphi^{-1}\left(Y_{i}\right)$. Cover it by some open affine subvarieties $\bigcup\limits _{j=1}^{N_{i}}X^{ij}=X_{i}$. Now we have a restricted morphism $\varphi|_{X^{ij}}:X^{ij}\to Y_{i}$ of affine QN varieties and we can use Lemma \[lem-affine-QN-equiv\]. Refining the covers will yield the proof for the first direction. To prove the other direction we start with a function $f\in QN_{Y}\left(U\right)$ for an open $U\subset Y$. By remark \[Rem-NQN-Like\] each $X_{ij}$ corresponds to an affine QN variety, so by Lemma \[lem-affine-QN-equiv\] we have QN morphisms $\text{\ensuremath{\varphi}}|_{X_{ij}}:X_{ij}\to Y_{i}$. Thus, $f|_{Y_{i}\cap U}\in QN_{Y}\left(Y_{i}\cap U\right)$ is pulled back to $f\circ\varphi|_{X_{ij}\cap\varphi^{-1}\left(U\right)}\in QN_{X}\left(X_{ij}\cap\varphi^{-1}\left(U\right)\right)$ for any $i$, and $j$. As QN functions form a sheaf, we get that $f\circ\varphi|_{\varphi^{-1}\left(U\right)}\in QN_{X}\left(\varphi^{-1}\left(U\right)\right)$. Schwartz Functions, Tempered Functions and Tempered Distributions ================================================================= Naive Quasi-Nash ---------------- \[claim-Sch-from-open\]Let $X$ be a NQN set, and $U,\:V\subset\mathbb{R}^{n}$ be open sets containing $X$ as a closed subset. Then $\mathcal{S}\left(U\right)/I_{Sch}^{U}\left(X\right)\cong\mathcal{S}\left(V\right)/I_{Sch}^{V}\left(X\right)$. We start by showing $\mathcal{S}\left(U\right)/I_{Sch}^{U}\left(X\right)\cong\mathcal{S}\left(U\cap V\right)/I_{Sch}^{U\cap V}\left(X\right)$. By Theorem \[AG-thm-Char-Nash\], $\mathcal{S}\left(U\cap V\right)$ is isomorphic to a closed subspace of $\mathcal{S}\left(U\right)$. Thus, it is enough to check that ${\color{black}{\color{black}\mathcal{S}\left(X\right)}:=\mathcal{S}\left(U\right)/I_{Sch}^{U}\left(X\right)}$ and $\mathcal{S}\left(U\cap V\right)/I_{Sch}^{U\cap V}\left(X\right)$ are equal as sets, i.e. that a Schwartz function on $X$ is a restriction of a Schwartz function on $U$ if and only if it is a restriction of a Schwartz function on $U\cap V$. Let $f\in\mathcal{S}\left(U\cap V\right)/I_{Sch}^{U\cap V}\left(X\right)$. There exists $F\in\mathcal{S}\left(U\cap V\right)$ such that $F|_{X}=f$. By Theorem \[AG-thm-Char-Nash\], extending $F$ by zero to a function on $U$ (denote it by $\tilde{F}$) is a function in $\mathcal{S}\left(U\right)$. Then $f=\tilde{F}|_{X}$ and so $f\in\mathcal{S}\left(U\right)/I_{Sch}^{U}\left(X\right)$. For the other direction, let $f\in\mathcal{S}\left(U\right)/I_{Sch}^{U}\left(X\right)$. There exists $F\in\mathcal{S}\left(U\right)$ such that $F|_{X}=f$. Denote $U'=U\backslash X$. $\left\{ U',U\cap V\right\} $ form an open cover of $U$ and so, by Theorem \[AG-thm-part-of-unity\], there exist tempered functions $\alpha_{1},\:\alpha_{2}$ such that $supp\left(\alpha_{1}\right)\subset U\cap V,\:supp\left(\alpha_{2}\right)\subset U'$ and $\alpha_{1}+\alpha_{2}=1$ as a real valued function on $U$. Moreover, $\alpha_{1}$ and $\alpha_{2}$ can be chosen such that $\left(\alpha_{1}\cdot F\right)|_{U\cap V}\in\mathcal{S}\left(U\cap V\right)$. As $\alpha_{1}|_{X}=1$, it follows that $\left(\left(\alpha_{1}\cdot F\right)|_{U\cap V}\right)|_{X}=\left(\alpha_{1}\cdot F\right)|_{X}=F|_{X}=f$, and so $f\in\mathcal{S}\left(U\cap V\right)/I_{Sch}^{U\cap V}\left(X\right)$. As we can use the same proof exactly to show $$\mathcal{S}\left(V\right)/I_{Sch}^{V}\left(X\right)\cong\mathcal{S}\left(U\cap V\right)/I_{Sch}^{U\cap V}\left(X\right),$$ we end up having the result. \[claim-temp-from-open\]Let $X$ be a NQN set, and $U,\:V\subset\mathbb{R}^{n}$ be open sets containing $X$ as a closed subset. Then for any $\mathcal{T}\left(U\right)|_{X}=\mathcal{T}\left(V\right)|_{X}$. As $U,\:V$ are open Nash submnifolds (Preposition \[AG-prop-sub-of-Nash-is-Nash\]), as tempered functions form a sheaf on $U\cup V$ (by Proposition \[AG-prop-aff-temp-sheaf\]) and as we may use tempered partition of unity on $U\cup V$ (Theorem \[AG-thm-part-of-unity\]), the claim easily follows. *Let $X$ be an NQN set. A Schwartz function on $X$* is a restriction of a Schwartz function from an open semi-algebraic subset of $\mathbb{R}^{n}$ in which $X$ is closed, to $X$. Equivalently, we can define the space of Schwartz functions on $X$ as $\mathcal{S}\left(X\right):=\mathcal{S}\left(U\right)/I_{Sch}^{U}\left(X\right)$. *A tempered function on $X$* is a restriction of a tempered function from an open semi-algebraic subset of $\mathbb{R}^{n}$ in which $X$ is closed, to $X$. The space of tempered functions on $X$ is denoted by $\mathcal{T}\left(X\right)$. \[lem-Sch-NQN-is-Frechet\]Let $X$ be an NQN set. Then $\mathcal{S}\left(X\right)$ is a Fréchet space. $\mathcal{S}\left(U\right)$ is a Fréchet space (see [\[]{}Proposition \[AG-prop-sub-of-Nash-is-Nash\], Proposition \[AG-prop-Sch-on-Nash-is-Fre\][\]]{}) and as $I_{Sch}^{U}$ is a closed subset of $\mathcal{S}\left(U\right)$, we get that their quotient is a Fréchet space as well (by Proposition \[prop-Closed-Frechet-subspace\] and [\[]{}T - Proposition 7.9[\]]{}). \[lem-NQN-Frechet-iso\]Let $\varphi:X_{1}\to X_{2}$ be an NQN isomorphism, i.e. a bijective morphism whose inverse is also an NQN morphism. Then $\varphi^{*}|_{\mathcal{S}\left(X_{2}\right)}:\mathcal{S}\left(X_{2}\right)\to\mathcal{S}\left(X_{1}\right)$ is an isomorphism of Fréchet spaces. By definition we have an open semialgebraic neighborhood $U_{1}$ of $X_{1}$ and a Nash map $g_{1}:U_{1}\to\mathbb{R}^{n_{2}}$ such that $g_{1}|_{X}=\varphi$. Note that $U_{1}$ is an affine Nash manifold. Similarly to the construction of $U_{1}$ and $g_{1}$ above, we may construct an open $U_{2}\subset\mathbb{R}^{n_{2}}$ and a map $g_{2}:U_{2}\to\mathbb{R}^{n_{1}}$ such that $g_{2}|_{X_{2}}=\varphi{}^{-1}$. Note that $g_{2}\neq g_{1}^{-1}$: in general $g_{1}$ is not a bijection and $U_{1}\ncong U_{2}$. Consider the following diagram, where $\alpha$ is defined by $\alpha(x,y):=(x,y+g_{1}(x))$.  Clearly $U_{1}\times\{0\}$ is an affine Nash manifold isomorphic to $U_{1}$. Denote $\hat{U}_{1}:=\alpha(U_{1}\times\{0\})$, then $\alpha$ restricted to $U_{1}\times\{0\}$ is an isomorphism of the affine Nash manifolds $U_{1}\times\{0\}$ and $\hat{U}_{1}$ the inverse map is given by $\alpha^{-1}(x,y):=(x,y-g_{1}(x))$. Thus we have: $$\mathcal{S}(X_{1})\cong\mathcal{S}(U_{1})/I_{Sch}^{U}(X_{1})\cong\mathcal{S}(\hat{U}_{1})/I_{Sch}^{\hat{U}_{1}}(\alpha(X_{1}\times\{0\}))=\mathcal{S}(\hat{U}_{1})/I_{Sch}^{\hat{U}_{1}}((Id\times\varphi)(X_{1})),$$ where the first equivalence is by definition, the second is due the fact that $U_{1}\cong U_{1}\times\{0\}\cong\hat{U}_{1}$ and $\mathcal{S}(U_{1})\cong\mathcal{S}(U_{1}\times\{0\})\cong\mathcal{S}(\hat{U}_{1})$, and the third follows from the fact that $g_{1}|_{X_{1}}=\varphi$. As always $I_{Sch}^{U_{1}}(X_{1})$ is the ideal in $\mathcal{S}(U_{1})$ of Schwartz functions identically vanishing on $X_{1}$. As $\tilde{U_{1}}$ is closed in $U_{1}\times\mathbb{R}^{n_{2}}$ (as it is defined by Nash map on $U_{1}\times\mathbb{R}^{n_{2}}$), then by Theorem \[AG-thm-red-to-cl-is-onto\] we get that $$\mathcal{S}(\hat{U}_{1})/I_{Sch}^{\hat{U}_{1}}((Id\times\varphi)(X_{1}))\cong\mathcal{S}(U_{1}\times\mathbb{R}^{n_{2}})/I_{Sch}^{U_{1}\times\mathbb{R}^{n_{2}}}((Id\times\varphi)(X_{1})).$$ Applying claim \[claim-Sch-from-open\] for the open subset $U_{1}\times U_{2}\subset U_{1}\times\mathbb{R}^{n_{2}}$ we get that $$\mathcal{S}(U_{1}\times\mathbb{R}^{n_{2}})/I_{Sch}^{U_{1}\times\mathbb{R}^{n_{2}}}((Id\times\varphi)(X_{1}))\cong\mathcal{S}(U_{1}\times U_{2})/I_{Sch}^{U_{1}\times U_{2}}((Id\times\varphi)(X_{1})),$$ and thus we obtain $$\mathcal{S}(X_{1})\cong\mathcal{S}(U_{1}\times U_{2})/I_{Sch}^{U_{1}\times U_{2}}((Id\times\varphi)(X_{1})).$$ Repeating the above construction using the following diagram:  yields: $$\mathcal{S}(X_{2})\cong\mathcal{S}(U_{1}\times U_{2})/I_{Sch}^{U_{1}\times U_{2}}((\varphi^{-1}\times Id)(X_{2})).$$ Clearly $(Id\times\varphi)(X_{1})=(\varphi^{-1}\times Id)(X_{2})$, and so $\mathcal{S}(X_{1})\cong\mathcal{S}(X_{2})$. Note that the isomorphism constructed is in fact the pull back by $\varphi$ from $\mathcal{S}(X_{2})$ onto $\mathcal{S}(X_{1})$. This proves the lemma. A similar claim about tempered functions can be proved similarly, by replacing Theorem \[AG-thm-red-to-cl-is-onto\] with a similar claim about tempered functions ([\[]{}AG - 4.6.2[\]]{}). \[prop-sXt\]Let $X$ be an NQN set, $s\in\mathcal{S}\left(X\right)$, and $t\in\mathcal{T}\left(X\right)$. Then $t\cdot s\in\mathcal{S}\left(X\right)$. Take some $U\subset\mathbb{R}^{n}$ such that $X\subset U$ as a closed subset. Then there exist some Schwartz function $S\in\mathcal{S}\left(U\right)$ and a tempered function $T\in\mathcal{T}\left(U\right)$ such that $s=S|_{X}$ and $t=T|_{X}$. As $T\cdot S\in\mathcal{S}\left(U\right)$ (by Proposition \[AG-prop-ts-is-s\]), we get that $\left(T\cdot S\right)|_{X}\in\mathcal{S}\left(X\right)$. $\:$ Affine QN --------- Let $X,\:Y$ be affine QN varieties, and $\varphi:X\to Y$ a QN isomorphism. Let $\tilde{X},\:\tilde{Y}$ be closed NQN sets correponding to $X,\:Y$ respectively, and $\tilde{\varphi}$ the corresponding map. Then a Schwartz function on $\tilde{Y}$ is pulled-back by $\tilde{\varphi}$ to a Schwartz function on $\tilde{X}$. Let $f\in\mathcal{S}\left(\tilde{Y}\right)$. By Lemma \[lem-affine-QN-equiv\] there exist open covers $\bigcup\limits _{i=1}^{N}\tilde{X}_{i}=\tilde{X}$, $\bigcup\limits _{i=1}^{N}\tilde{Y}_{i}=\tilde{Y}$ such that $\tilde{\varphi}|_{\tilde{X}_{i}}:\tilde{X}_{i}\xrightarrow{\sim}\tilde{Y}_{i}$ are NQN isomorphisms. As Schwartz functions form a cosheaf, every Schwartz function on $\tilde{Y}$ is a sum of extensions of Schwartz functions on open subsets of $\tilde{Y}$. Let those subsets be $\tilde{Y}_{i}$. According to Lemma \[lem-NQN-Frechet-iso\], a Schwartz function $s_{i}\in\mathcal{S}\left(\tilde{Y}_{i}\right)$ is pulled back to $\varphi^{*}s_{i}\in\mathcal{S}\left(\tilde{X}_{i}\right)$, and thus, we get $\varphi^{*}s=\sum\limits _{i=1}^{n}Ext_{\tilde{X}_{i}}^{\tilde{X}}\left(\varphi^{*}s_{i}\right)\in\mathcal{S}\left(\tilde{X}\right)$. Together with the NQN isomorphism on each such subset, we get the result. This lemma enables us to use the following definition: Let $X$ be an affine QN variety. A function $s$ on $X$ is called a *Schwartz function on $X$* if it is a pullback of a Schwartz function from the closed NQN set corresponds to $X$. I.e. if $\varphi:X\xrightarrow{\sim}\tilde{X}\subset\mathbb{R}^{n}$ where $\tilde{X}$ is the closed semi-algebraic set QN isomorphic to $X$, and $\tilde{s}\in\mathcal{S}\left(\tilde{X}\right)$, then $\varphi^{*}\tilde{s}\in\mathcal{S}\left(X\right)$. A similar claim for tempered functions can be proven the same way. \[thm-Res To Closed\]Let $M$ be an affine QN variety, and let $X\subset M$ be a closed subset. Then the restriction of a Schwartz function from $M$ to $X$ defines an isomorphism $\mathcal{S}\left(X\right)=\mathcal{S}\left(M\right)/I_{Sch}^{M}\left(X\right)$ (with the quotient topology), where $I_{Sch}^{M}\left(X\right)$ is the ideal in $\mathcal{S}\left(M\right)$ of functions identically vanishing on $X$. Take the closed corresponding sets of $X$ and $M$, $\tilde{X}$ and $\tilde{M}$ correspondingly. As $\tilde{X}\subset\tilde{M}\subset U$ where $\tilde{M}$ is closed in some open $U\subset\mathbb{R}^{n}$, we get that $\mathcal{S}\left(M\right)/I_{Sch}^{M}\left(X\right)\cong\left(\mathcal{S}\left(U\right)/I_{Sch}^{U}\left(M\right)\right)/I_{Sch}^{M}\left(X\right)\cong\mathcal{S}\left(U\right)/I_{Sch}^{U}\left(X\right)\cong\mathcal{S}\left(X\right)$. Let $X$ be an affine QN variety corresponding to some closed subset $\tilde{X}\subset\mathbb{R}^{n}$. A function $f:\tilde{X}\to\mathbb{R}$ is called **flat** at $\tilde{p}\in\tilde{X}$ if there exists an open semi-algebraic neighborhood $U\subset\mathbb{R}^{n}$ of $\tilde{p}$ and a function $F\in C^{\infty}\left(U\right)$ such that $f|_{U\cap\tilde{X}}=F|_{U\cap\tilde{X}}$ and $F$’s Taylor series is identically zero at $\tilde{p}$. If $f$ is flat at all $\tilde{p}\in\tilde{Z}$ for some $\tilde{Z}\subset\tilde{X}$, $f$ is called **flat at $\tilde{Z}$**. Let $X$ and $Y$ be affine QN varieties, and $\varphi:X\to Y$ a QN isomorphism. Let $\tilde{X}\subset\mathbb{R}^{n}$ and $\tilde{Y}\subset\mathbb{R}^{m}$ be the corresponding closed subsets, and $\tilde{\varphi}:\tilde{X}\to\tilde{Y}$ the corresponding map. Let $f$ be a function on $\tilde{Y}$ that is flat at some $p\in\tilde{Y}$. Then $\tilde{\varphi}^{*}f$ is flat at $\tilde{\varphi}^{-1}\left(p\right)$. By definition there is some open neighborhood $U\subset\mathbb{R}^{m}$ of $p$ and a smooth function $F\in C^{\infty}\left(U\right)$ which is flat on $p$ and such that $f=F|_{\tilde{Y}\cap U}$. As $\tilde{\varphi}$ is an isomorphism, and by Lemma \[lem-affine-QN-equiv\], $\tilde{\varphi}^{-1}\left(p\right)$ has an open neighborhood $W\subset\mathbb{R}^{n}$ such that $\tilde{\varphi}|_{\tilde{X}\cap W}$ is a restriction of a Nash map $G:W\to\mathbb{R}^{m}$. Take an open subset $W'\subset W$ such that $p\in W'$ and $G\left(W'\right)\subset U$. By sheaf properties of Nash maps, $G|_{W'}$ is Nash as well. Thus, $F\circ G|_{W'}$ is a smooth function on $W'$, and by [\[]{}Lemma \[lem-Faa-di\][\]]{} it is flat on $\tilde{\varphi}^{-1}\left(p\right)$. We conclude that $F\circ G|_{\tilde{X}\cap W'}=F\circ\tilde{\varphi}|_{\tilde{X}\cap W'}=f\circ\tilde{\varphi}|_{\tilde{X}\cap W'}$, which means $\tilde{\varphi}^{*}f$ is flat at $\tilde{\varphi}^{-1}\left(p\right)$. Let $X$ be an affine QN variety. A function $f:X\to\mathbb{R}$ is called flat at $p\in X$ if the correponding map $\tilde{f}$ is flat at the corresponding point $\tilde{p}$. If $f$ is flat at all $p\in Z$ for some $Z\subset X$, $f$ is called **flat at $Z$**. \[lem-ext-by-0-affine\](Extension by zero - the affine case) Let $X$ be an affine QN variety, and let $V\subset X$ be an open subset. Then any $f\in\mathcal{S}\left(V\right)$ can be extended to a Schwartz function on $X$ which is flat on $X\backslash V$. First, let us take the closed set in $\mathbb{R}^{n}$ corresponding to $X$, and denote it, by abuse of notation, by $X$. By remark \[rem-NQN-Like-Sch-Tempered\], $f$ is a restriction of a Schwartz function $\hat{f}$ on some open neighborhood $\hat{V}\subset\mathbb{R}^{n}$. By Theorem \[AG-thm-Char-Nash\] $\hat{f}$ may be extended by zero to a Schwartz function on $\mathbb{R}^{n}$ which is flat on $\mathbb{R}^{n}\backslash\hat{V}$. Restricting this function to $X$ yields the desired result. In fact, for any open $V\subset X$, any restriction to $V$ of a Schwartz function on $X$ which is flat on $X\backslash V$, is a Schwartz function on $V$. In order to prove that, we need the following lemmas. The following lemmas and proposition are required for the proof of Theorem \[thm-affine-char\]: \[lem-subanalytic-ext\]Let $X$ be a QN variety corresponding to some compact set $\tilde{X}\subset\mathbb{R}^{n}$ closed in some $U\subset\mathbb{R}^{n}$. Let $Z\subset\tilde{X}$ be a closed subset. Define $V:=\tilde{X}\backslash Z$, $$W_{Z}:=\left\{ \phi:\tilde{X}\to\mathbb{R}|\exists\hat{\phi}\in C^{\infty}\left(U\right)\text{ such that }\hat{\phi}|_{\tilde{X}}=\phi\text{ and }\phi\text{ is flat at }Z\right\} ,$$ and $$\left(W_{Z}^{U}\right)^{comp}:=\left\{ \phi\in C^{\infty}\left(U\right)|\phi\text{ is compactly supported and is flat at }Z\right\} .$$ Then, for any $f\in W_{Z}$ there exists $\hat{f}\in\left(W_{Z}^{U}\right)^{comp}$ such that $\hat{f}|_{X}=f$. The proof of Lemma \[lem-subanalytic-ext\] is exactly the same as the proof of [\[]{}ES - Lemma 3.13[\]]{}, which deals with algebraic varieties. In the case of $Z=\left\{ p\right\} $, this extension is trivial. In the case $Z$ consists of more than one point, we need to use results on Whitney’s extension theorem. We used [\[]{}BM1[\]]{}, [\[]{}BM2[\]]{}, [\[]{}BMP1[\]]{}, dealing with subanalytic geometry, to prove this extension exists in the algebraic case in [\[]{}ES - Lemma 3.13 and Appendix A[\]]{}. As subanalytic geometry covers semi-algebraic sets as well, the proof is valid in our case with these minor changes:\ Semi-algebraic sets replace algebraic sets, affine QN varieties replace affine algebraic varieties, and open\\closed semi-algebraic sets replace Zariski open\\closed sets.\ Restricting a Schwartz function to an affine QN variety from an open neighborhood is given in Remark \[rem-NQN-Like-Sch-Tempered\], which replaces [\[]{}ES - Theorem 3.7[\]]{}.\ The Uniformization theorem - [\[]{}BM1, 5.1[\]]{} should be replaced by the more general Theorem [\[]{}BM1, Theorem 0.1[\]]{}. \[lem-Sch-are-smth-on-comp\]Let $X$ be an affine QN variety corresponding to some compact set $\tilde{X}\subset\mathbb{R}^{n}$. Let $U\subset\mathbb{R}^{n}$ such that $\tilde{X}$ is closed in $U$. Then $$\mathcal{S}\left(\tilde{X}\right)=\left\{ f:\tilde{X}\to\mathbb{R}|\exists\hat{f}\in C^{\infty}\left(U\right)\text{ such that }\hat{f}|_{\tilde{X}}=f\right\}$$ The inclusion $\subset$ is trivial as $\mathcal{S}\left(U\right)\subset C^{\infty}\left(U\right)$. For the other direction consider some $g:\tilde{X}\to\mathbb{R}$ and assume it extends to some $\hat{g}\in C^{\infty}\left(U\right)$ such that $\hat{g}|_{\tilde{X}}=g$. Let $\rho\in C^{\infty}\left(U\right)$ be a compactly supported function such that $\rho|_{\tilde{X}}=1$. Then $\rho\cdot\hat{g}$ is a smooth compactly supported function on $U$, so $\rho\cdot\hat{g}\in\mathcal{S}\left(U\right)$. Moreover, $\left(\rho\cdot\hat{g}\right)|_{\tilde{X}}=\hat{g}|_{\tilde{X}}=g$. Thus $g\in\mathcal{S}\left(\tilde{X}\right)$. \[prop-res-Wz-to-U-is-Sch\]Let $X$ be an affine QN variety corresponding to some closed $\tilde{X}$, and let $Z\subset\tilde{X}$ be some closed subset. Define $V:=\tilde{X}\backslash Z$ and $$W_{Z}:=\left\{ \phi\in\mathcal{S}\left(\tilde{X}\right)|\phi\text{ is flat on }Z\right\} .$$ Then restriction from $\tilde{X}$ to $V$ of a function in $W_{Z}$ is a Schwartz function on $V$, i.e. $Res_{V}^{\tilde{X}}\left(W_{Z}\right)\subset\mathcal{S}\left(V\right)$. The proof is divided into two parts. First we show the case where $X$ corresponds to a compact subset $\tilde{X}\subset\mathbb{R}^{n}$. Then, we deduce the general case. Assume $\tilde{X}$ is compact. Define $V^{\mathbb{R}^{n}}:=\mathbb{R}^{n}\backslash Z$ and $$W_{Z}^{\mathbb{R}^{n}}:=\left\{ \phi\in\mathcal{S}\left(\mathbb{R}^{n}\right)|\phi\text{ is flat on }Z\right\} .$$ As $Z$ is closed in $\mathbb{R}^{n}$, $V^{\mathbb{R}^{n}}$ is open. As $V=V^{\mathbb{R}^{n}}\cap X$, $V$ is closed in $V^{\mathbb{R}^{n}}$. The claim follows from the existence of these three maps: $$\xymatrix{ & W_{Z}^{\mathbb{R}^{n}}\ar@{->>}[dl]_{Res_{X}^{\mathbb{R}^{n}}}^{\left(1\right)}\ar[dr]_{\left(2\right)}^{Res_{U^{\mathbb{R}^{n}}}^{\mathbb{R}^{n}}}\\ W_{Z}\ar@{-->}[dr]_{Res_{U}^{X}} & & \mathcal{S}\left(U^{\mathbb{R}^{n}}\right)\ar[dl]_{\left(3\right)}^{Res_{U}^{U^{\mathbb{R}^{n}}}}\\ & \mathcal{S}\left(U\right) }$$ The existence of map (1) is clear. It is onto due to Lemma \[lem-subanalytic-ext\] and Lemma \[lem-Sch-are-smth-on-comp\]. Let $g\in W_{Z}^{\mathbb{R}^{n}}$. Then we get map (2) by Theorem \[AG-thm-Char-Nash\], $g|_{V^{\mathbb{R}^{n}}}\in\mathcal{S}\left(V^{\mathbb{R}^{n}}\right)$. Map (3) is obtained as for any $h\in\mathcal{S}\left(V^{\mathbb{R}^{n}}\right)$ we get $h|_{V}\in\mathcal{S}\left(V\right)$ by Remark \[rem-NQN-Like-Sch-Tempered\]. Now assume $\tilde{X}$ is not compact. But $\tilde{X}$ is closed in $\mathbb{R}^{n}$. By Proposition \[prop-Alexandrov\] we get an algebraic map $i:\mathbb{R}^{n}\to\dot{\mathbb{R}^{n}}$ where $\dot{\mathbb{R}^{n}}$ is an affine algebaic variety which is a one point compactification of $\mathbb{R}^{n}$, i.e. $\dot{\mathbb{R}^{n}}=i\left(\mathbb{R}^{n}\right)\cup\left\{ \infty\right\} $. We also get that $\mathbb{R}^{n}$ and $i\left(\mathbb{R}^{n}\right)$ are algebraically isomorphic, which means they are also QN isomorphic. Thus, $\tilde{X}$ is QN isomorphic to some locally closed subset $i\left(\tilde{X}\right)$ of the compact variety $\dot{\mathbb{R}^{n}}$. As $i\left(\tilde{X}\right)\cup\left\{ \infty\right\} =:\dot{\tilde{X}}$ is closed in $\dot{\mathbb{R}^{n}}$, it is compact. Now take some $f\in W_{Z}\subset\mathcal{S}\left(\tilde{X}\right)$, and get that $i_{*}f:=f\circ i^{-1}\in\mathcal{S}\left(i\left(\tilde{X}\right)\right)$. By Lemma \[lem-ext-by-0-affine\], as $i\left(\tilde{X}\right)$ is open in $\dot{\tilde{X}}$, there exist $\dot{f}\in\mathcal{S}\left(\dot{\tilde{X}}\right)$ such that $i_{*}f=\dot{f}|_{i\left(\tilde{X}\right)}$ . Now define $\dot{V}:=\dot{\tilde{X}}\backslash\left(i\left(Z\right)\cup\left\{ \infty\right\} \right)$. As $i$ is a QN isomorphism, $\dot{V}$ is open in $\dot{\tilde{X}}$. By the compact case, $Res_{\dot{V}}^{\dot{\tilde{X}}}\left(\dot{f}\right)\in\mathcal{S}\left(\dot{V}\right)$. Note that $\dot{V}$ is QN isomorphic to $V$ by $i^{-1}|_{\dot{V}}$ . Thus, $\left(i^{-1}|_{\dot{V}}\right)_{*}Res_{\dot{V}}^{\dot{\tilde{X}}}\left(\dot{f}\right)\in\mathcal{S}\left(V\right)$. Finally, $\left(i^{-1}|_{\dot{V}}\right)_{*}Res_{\dot{V}}^{\dot{\tilde{X}}}\left(\dot{f}\right)=\left(i^{-1}|_{\dot{V}}\right)_{*}\left(\left(i_{*}f\right)|_{\dot{V}}\right)=f|_{V}$ and thus $f|_{V}\in\mathcal{S}\left(V\right)$. \[thm-affine-char\](Characterization of Schwartz functions on open subset - the affine case) Let $X$ be an affine QN variety, and let $Z\subset X$ be some closed semi-algebraic subset. Define $V:=X\backslash Z$ and $W_{Z}:=\left\{ \phi\in\mathcal{S}\left(X\right)|\phi\text{ is flat on }Z\right\} $. Then extension by zero $Ext_{V}^{X}:\mathcal{S}\left(V\right)\to W_{Z}$ is an isomorphism of Fréchet spaces, whose inverse is $Res_{V}^{X}:W_{Z}\to\mathcal{S}\left(V\right)$. By \[prop-Closed-Frechet-subspace\], as $W_{Z}=\bigcap\limits _{z\in Z}\left\{ \phi\in\mathcal{S}\left(X\right)|\phi\text{ is flat on }z\right\} $ is a closed subspace of $\mathcal{S}\left(X\right)$, as an intersection of closed subets, it is a Fréchet space. Lemma \[lem-ext-by-0-affine\] shows that for any $f\in\mathcal{S}\left(V\right)$, we get $Ext_{V}^{X}\left(f\right)\in\mathcal{S}\left(X\right)$ and $Ext_{V}^{X}\left(f\right)$ is flat on $Z$, i.e. $Ext_{V}^{X}\left(\mathcal{S}\left(V\right)\right)\subset W_{Z}$. This extension is a continuous map. To show that, consider the closed set $\tilde{X}$ corresponding to $X$. Define $W:=\mathbb{R}^{n}\backslash Z$. This is an open semi-algebraic set, thus a Nash variety. $V=W\cap\tilde{X}$ so $V$ is closed in $W$. Take some closed embedding $W\hookrightarrow\mathbb{R}^{N}$. Then, by Remark \[rem-NQN-Like-Sch-Tempered\] $\mathcal{S}\left(V\right)\cong\mathcal{S}\left(W\right)/I_{Sch}^{W}\left(V\right)$. $W$ is open in $\mathbb{R}^{n}$ so by Theorem \[AG-thm-Char-Nash\], $Ext_{W}^{\mathbb{R}^{n}}$ is a closed embedding, and thus continuous, $\mathcal{S}\left(W\right)\hookrightarrow\mathcal{S}\left(\mathbb{R}^{n}\right)$. Therefore, the map $$\mathcal{S}\left(V\right)\cong\mathcal{S}\left(W\right)/I_{Sch}^{W}\left(V\right)\to\mathcal{S}\left(\mathbb{R}^{n}\right)/I_{Sch}^{\mathbb{R}^{n}}\left(\tilde{X}\right)=\mathcal{S}\left(\tilde{X}\right)$$ is continuous, i.e. $Ext_{V}^{\tilde{X}}$ is continuous, and $Ext_{V}^{X}$ is continuous as well. We saw in Proposition \[prop-res-Wz-to-U-is-Sch\] that $Res_{V}^{X}\left(W_{Z}\right)\subset\mathcal{S}\left(V\right)$. As $Res_{V}^{X}\circ Ext_{V}^{X}:\mathcal{S}\left(V\right)\to\mathcal{S}\left(V\right)$ is the identity operator by definition, and so is $Ext_{V}^{X}\circ Res_{V}^{X}:W_{Z}\to W_{Z}$, we get that $Ext_{V}^{X}$ is a continuous bijection. By Theorem \[thm-Banach-open-mapping\] this means $Ext_{V}^{X}$ is an isomorphism of Fréchet spaces. \[cor-Sch-flat-at-point\]Let X be an affine QN variety. A Schwartz function $f\in\mathcal{S}\left(X\right)$ is flat at $p\in X$ if and only if $f|_{X\backslash\left\{ p\right\} }\in\mathcal{S}\left(X\backslash\left\{ p\right\} \right)$. apply Theorem \[thm-affine-char\] to $Z=\left\{ p\right\} $. By the same argument for an arbitrary function $f\in C^{\infty}\left(X\right)$ (i.e. a function that is a restriction of a smooth function from an open set in which the set corresponding to $X$ is closed) and any $p\in X$, the following conditions are equivalent: \(1) $f$ is flat at $p$. \(2) There exists a smooth compactly supported function $\rho$ on $\mathbb{R}^{n}$, such that $\rho$ is identically 1 on some open neighborhood of $p$ and $\left(f\cdot\rho\right)|_{X\backslash\left\{ p\right\} }\in\mathcal{S}\left(X\backslash\left\{ p\right\} \right)$. Let $X$ be an affine QN variety. Define the **space of tempered distributions on $\boldsymbol{X}$** as the space of continuous linear functionals on $\mathcal{S}\left(X\right)$. Denote this space by $\mathcal{S}^{*}\left(X\right)$. $\:$ General QN ---------- *\[def-Schwartz-on-Gen-QN-Cov-C\]A Schwartz function on a (general) QN variety $X$ with cover $C$:*$C$ be an open affine QN cover of $X$ - i.e. $\bigcup\limits _{i=1}^{m}X_{i}=X$. Denote by $Func\left(X,\mathbb{R}\right)$ the space of all real valued functions on $X$. There is a natural map $\psi:\bigoplus\limits _{i=1}^{m}Func\left(X_{i},\mathbb{R}\right)\to Func\left(X,\mathbb{R}\right)$. Define the space of Schwartz functions on $X$ associated with the cover $C$ by $\mathcal{S}_{C}\left(X\right):=\psi\left(\bigoplus\limits _{i=1}^{m}\mathcal{S}\left(X_{i}\right)\right)$. \[lem-Schwartz-Gen-Frech\]Let $X$ be a QN variety, and let $C$ be an open affine QN cover of $X$. Then the space $\mathcal{S}_{C}\left(X\right)$ is a Fréchet space. First note that $\psi\left(\bigoplus\limits _{i=1}^{m}\mathcal{S}\left(X_{i}\right)\right)\cong\bigoplus\limits _{i=1}^{m}\mathcal{S}\left(X_{i}\right)/Ker\left(\psi|_{\bigoplus\limits _{i=1}^{m}\mathcal{S}\left(X_{i}\right)}\right)$ with the natural quotient topology. A direct sum of Fréchet spaces is a Fréchet space. The kernel of $\psi|_{\bigoplus\limits _{i=1}^{m}\mathcal{S}\left(X_{i}\right)}$ is a closed subspace, as $\bigoplus\limits _{i=1}^{m}s_{i}\in Ker\left(\psi|_{\bigoplus\limits _{i=1}^{m}\mathcal{S}\left(X_{i}\right)}\right)$ if and only if for any $x\in X$, $\sum\limits _{i\in J_{x}}s_{i}\left(x\right)=0$, where $J_{x}:=\left\{ 1\leq i\leq m|x\in X_{i}\right\} $, i.e. the kernel is given by infinitely many closed conditions. So by Proposition \[prop-Closed-Frechet-subspace\] and [\[]{}T - Proposition 7.9[\]]{}, the quotient is a Fréchet space as well. \[lem-Sch-cover-indep\]Let $X$ be a QN variety and let $C,\:D$ be two open QN covers of $X$. Then $\mathcal{S}_{C}\left(X\right)\cong\mathcal{S}_{D}\left(X\right)$ as Fréchet spaces. To prove this lemma, we will show the spaces have a continuous bijective map between them. Thus, by Theorem \[thm-Banach-open-mapping\] they are isomorphic as Fréchet spaces. We start with bijectiveness. Let the open affine cover $C$ be $X=\bigcup\limits _{i=1}^{m}Y_{i}$ and the open affine cover $D$ be $X=\bigcup\limits _{j=1}^{n}X_{j}$. By definition, $s\in\mathcal{S}_{D}\left(X\right)\Leftrightarrow s=\sum\limits _{j=1}^{m}Ext_{X_{j}}^{X}s_{j}$ where for each $j$, $s_{j}\in\mathcal{S}\left(X_{j}\right)$. Fix one such affine QN variety $X_{j}$. $X_{j}$ can be covered by the open QN cover $X_{j}=\bigcup\limits _{i=1}^{n}\left(Y_{i}\cap X_{j}\right)=:\bigcup\limits _{i=1}^{n}X_{ij}$. As $X_{j}$ is affine for any $j$, denote by $\tilde{X}_{j}\subset\mathbb{R}^{n_{j}}$ a closed set corresponding to $X_{j}$. Denote by $\tilde{X}_{ij}$ the open subsets of $\tilde{X}{}_{j}$ corresponding to $X_{ij}$. For any $j$, $s_{j}\in\mathcal{S}\left(\tilde{X}_{j}\right)$ is a restriction of a Schwartz function $S_{j}\in\mathcal{S}\left(U_{j}\right)$ to $\tilde{X}{}_{j}$, where $U_{j}\subset\mathbb{R}^{n_{j}}$ is an open set in which $\tilde{X}_{j}$ is closed. As $\tilde{X}{}_{ij}\subset\tilde{X}{}_{j}$ are open semi-algebraic, we may find open semi-algebraic subsets $U_{ij}\subset U_{j}$ such that $U_{ij}\cap\tilde{X}{}_{j}=\tilde{X}{}_{ij}$. As $U_{j}$ and the $U_{ij}$’s are Nash manifolds, we may add the open Nash set $W_{j}:=U_{j}\backslash\tilde{X}{}_{j}$ to get a Nash open cover of $U_{j}$, and use Theorem \[AG-thm-part-of-unity\](partition of unity) to get the Schwatrz functions $S_{ij}\in\mathcal{S}\left(U_{ij}\right),\:S_{W_{j}}\in\mathcal{S}\left(W_{j}\right)$, such that extending those functions by zero sum up to $S_{j}$, i.e. $\sum\limits _{i=1}^{n}Ext_{U_{ij}}^{U_{j}}S_{ij}+Ext_{W_{j}}^{U_{j}}S_{W_{j}}=S_{j}$. Thus, after restricting those functions to $\tilde{X}{}_{j}$, we can pull them back to get $\sum\limits _{i=1}^{n}Ext_{X_{ij}}^{X_{j}}s_{ij}=s_{j}$. So $s\in\mathcal{S}_{D}\left(X\right)\Leftrightarrow s=\sum\limits _{j=1}^{m}\sum\limits _{i=1}^{n}Ext_{X_{ij}}^{X_{j}}s_{ij}$. As both covers are finite, the sums may commute, and we get that $s\in\mathcal{S}_{D}\left(X\right)\Leftrightarrow s\in\mathcal{S}_{C}\left(X\right)$. To prove the map is continuous, let $\psi:\bigoplus\limits _{j=1}^{n}Func\left(X_{j},\mathbb{R}\right)\to Func\left(X,\mathbb{R}\right)$ be the natural map from Definition \[def-Schwartz-on-Gen-QN-Cov-C\]. We begin with the claim that $\mathcal{S}\left(X_{ij}\right)\rightarrow\mathcal{S}\left(X_{j}\right)$ is a continuous map. Take the closed sets $\tilde{X}_{j}\subset\mathbb{R}^{n_{j}}$, and $\tilde{X}_{ij}$’s as before. Each function $f_{ij}\in\mathcal{S}\left(\tilde{X}_{ij}\right)$ is a restriction of some $F_{ij}\in\mathcal{S}\left(U_{ij}\right)$, where $U_{ij}\subset\mathbb{R}^{n_{j}}$ is an open subset such that $U_{ij}\cap\tilde{X}_{j}=\tilde{X}_{ij}$. Denote $U_{j}:=\bigcup\limits _{i=1}^{n}U_{ij}$. By Theorem \[AG-thm-Char-Nash\], the extension by zero of Schwartz functions $Ext_{U_{ij}}^{U_{j}}F_{ij}=F_{j}\in\mathcal{S}\left(U_{j}\right)$ is a closed embedding, thus continuous. As $\tilde{X}_{j}\subset U_{j}$ is a closed subset, restricting the extension maps to $\tilde{X}_{j}$ shows that the extension by zero $\mathcal{S}\left(\tilde{X}_{ij}\right)\rightarrow\mathcal{S}\left(\tilde{X}_{j}\right)$ is continuous. Thus, by Theorem \[thm-Banach-open-mapping\], $\bigoplus\limits _{i}\mathcal{S}\left(X_{ij}\right)/Ker\left(\phi_{j}|_{\bigoplus\limits _{i}\mathcal{S}\left(X_{ij}\right)}\right)\cong\mathcal{S}\left(X_{j}\right)$, where $\phi_{j}:\bigoplus\limits _{i}Func\left(X_{ij},\mathbb{R}\right)\to Func\left(X_{j},\mathbb{R}\right)$ is the natural map from Definition \[def-Schwartz-on-Gen-QN-Cov-C\]. Thus, we get that $$\mathcal{S}_{C}\left(X\right)=\psi\left(\bigoplus\limits _{j}\mathcal{S}\left(X_{j}\right)\right)\cong\psi\left(\bigoplus\limits _{j}\phi_{j}\left(\bigoplus\limits _{i}\mathcal{S}\left(X_{ij}\right)\right)\right)\cong\phi\left(\bigoplus\limits _{j}\bigoplus\limits _{i}\mathcal{S}\left(X_{ij}\right)\right),$$ where $\phi:\bigoplus\limits _{i,j}Func\left(X_{ij},\mathbb{R}\right)\to Func\left(X,\mathbb{R}\right)$ is the natural map. The same can be done with the cover $D$ to get the result. In view of this lemma, we will denote the space of Schwartz functions on a QN variety $X$ just by $\mathcal{S}\left(X\right)$ without specifying the cover. \[lem-First-Result\]Let $\varphi:X\to Y$ be a QN isomorphism. Then $\varphi^{*}|_{\mathcal{S}\left(Y\right)}:\mathcal{S}\left(Y\right)\to\mathcal{S}\left(X\right)$ is an isomorphism of Fréchet spaces. Let $f\in\mathcal{S}\left(\tilde{Y}\right)$. By Lemma \[lem-gen-QN-equiv\] there exist open covers $\bigcup\limits _{i=1}^{N}\tilde{X}_{i}=\tilde{X}$, $\bigcup\limits _{i=1}^{N}\tilde{Y}_{i}=\tilde{Y}$ such that $\tilde{\varphi}|_{\tilde{X}_{i}}:\tilde{X}_{i}\xrightarrow{\sim}\tilde{Y}_{i}$ are NQN isomorphisms. As Schwartz functions form a cosheaf, every Schwartz function on $\tilde{Y}$ is a sum of extensions of Schwartz functions on open subsets of $\tilde{Y}$. Let those subsets be $\tilde{Y}_{i}$. According to Lemma \[lem-NQN-Frechet-iso\], a Schwartz function $s_{i}\in\mathcal{S}\left(\tilde{Y}_{i}\right)$ is pulled back to $\varphi^{*}s_{i}\in\mathcal{S}\left(\tilde{X}_{i}\right)$, and thus, we get $\varphi^{*}s=\sum\limits _{i=1}^{n}Ext_{\tilde{X}_{i}}^{\tilde{X}}\left(\varphi^{*}s_{i}\right)$. By definition and Lemma \[lem-Sch-cover-indep\] this means $\varphi^{*}s\in\mathcal{S}\left(\tilde{X}\right)$. Together with the NQN isomorphism on each such subset, we get the result. \[lem-Sch-Res-to-closed\]Let $X$ be a QN variety, and $Z\subset X$ be some semi-algebraic closed subset. Then $Res_{Z}^{X}\left(\mathcal{S}\left(X\right)\right)=\mathcal{S}\left(Z\right)$. Let $s\in\mathcal{S}\left(X\right)$, and let $X=\bigcup\limits _{i=1}^{m}X_{i}$ be some open affine QN cover, such that $s=\sum\limits _{i=1}^{m}Ext_{X_{i}}^{X}\left(s_{i}\right)$ for some $s_{i}\in\mathcal{S}\left(X_{i}\right)$. $Z\cap X_{i}$ is open in $Z$ and closed in $X_{i}$. By theorem \[thm-Res To Closed\] $s_{i}|_{Z\cap X_{i}}\in\mathcal{S}\left(Z\cap X_{i}\right)$, and thus $s|_{Z}=\sum\limits _{i=1}^{m}Ext_{Z\cap X_{i}}^{Z}\left(s|_{Z\cap X_{i}}\right)\in\mathcal{S}\left(Z\right)$. Now let $h\in\mathcal{S}\left(Z\right)$, and let $Z=\bigcup\limits _{i=1}^{m}\left(Z\cap X_{i}\right)$ where $X_{i}$ are as before. This is an open affine cover of $Z$ so $h=\sum\limits _{i=1}^{m}Ext_{Z\cap X_{i}}^{Z}\left(h_{i}\right)$ for some $h_{i}\in\mathcal{S}\left(Z\cap X_{i}\right)$. Each $Z\cap X_{i}$ is closed in $X_{i}$ so by theorem \[thm-Res To Closed\] there exist functions $H_{i}\in\mathcal{S}\left(X_{i}\right)$ such that $H_{i}|_{Z\cap X_{i}}=h_{i}$. Thus, $\sum\limits _{i=1}^{m}Ext_{X_{i}}^{X}\left(H_{i}\right)\in\mathcal{S}\left(X\right)$. \[lem-tempered-sheaf-affine\]Let $X$ be an affine QN variety. The assignment of the space of tempered functions to any open $V\subset X$, together with the restriction of functions, form a sheaf on $X$. First let us show that tempered functions restricted to an open subset remain tempered. Take a closed subset $\tilde{X}\subset\mathbb{R}^{n}$ corresponding to $X$. Tempered functions on $\tilde{X}$ are defined as restrictions of tempered functions on some open neighborhood $U$ of $\tilde{X}$. As tempered functions on $U$ form a sheaf, and by Remark \[rem-NQN-Like-Sch-Tempered\], we get that a restricted tempered function remains tempered. It is now clear the above forms a presheaf. The proof of the glueing property follows [\[]{}ES - Proposition 4.3[\]]{}. It uses again the definition of functions $f_{i}\in\mathcal{T}\left(V_{i}\right)$ on subsets $V_{i}$ of $\tilde{X}$ as restrictions of functions $\hat{f}_{i}\in\mathcal{T}\left(U_{i}\right)$ from neighborhoods $U_{i}$ of the $V_{i}$’s. Then, it uses tempered partition of unity on the $\hat{f}_{i}$’s to create function $\hat{f}$ on $\bigcup\limits _{i}U_{i}$ such that $\hat{f}|_{\tilde{X}}=f$. Finally, it proves that $\hat{f}|_{U_{i}}\in\mathcal{T}\left(U_{i}\right)$ for any $i$, in order to show $\hat{f}$ is tempered, what implies $f$ is tempered. \[lem-Tempered-eqiv\]Let $X$ be a QN variety, and let $t:X\to\mathbb{R}$ be some function. Then the following conditions are equivalent: \(1) There exists an open affine QN cover $X=\bigcup\limits _{i=1}^{k}X_{i}$ such that for any $1\leq i\leq k$, $t|_{X_{i}}\in\mathcal{T}\left(X_{i}\right)$. \(2) For any open affine QN cover $X=\bigcup\limits _{i=1}^{k}X_{i}$ and any $1\leq i\leq k$, $t|_{X_{i}}\in\mathcal{T}\left(X_{i}\right)$. Clearly (2) implies (1). For the other side assume there exist two open affine QN covers $X=\bigcup\limits _{i=1}^{k}X_{i}=\bigcup\limits _{j=k+1}^{l}X_{j}$ such that for any $k+1\leq j\leq l$, $t|_{X_{j}}\in\mathcal{T}\left(X_{j}\right)$. Fix some $1\leq i\leq k$. Note that $\left\{ X_{i}\cap X_{j}\right\} _{j=k+1}^{l}$ is an open cover of $X_{i}$. $t|_{X_{i}\cap X_{j}}$ is a restriction of the tempered function $t|_{X_{j}}$to the open subset $X_{i}\cap X_{j}\subset X_{i}$. By remark \[rem-NQN-Like-Sch-Tempered\] we get that $t|_{X_{j}}$ is a restriction to $X_{j}$ of a tempered function $T$ on an open neighborhood $U$ in which $X_{j}$ is closed. As tempered functions on Nash manifolds form a sheaf, take the open neighborhood $V\subset U$ of $X_{i}\cap X_{j}$ in which $X_{i}\cap X_{j}$ is closed, and get that $\hat{t}:=T|_{V}$ is a tempered function, and so, by remark \[rem-NQN-Like-Sch-Tempered\] again, we get that $t|_{X_{i}\cap X_{j}}=\hat{t}|_{X_{i}\cap X_{j}}\in\mathcal{T}\left(X_{i}\cap X_{j}\right)$. By \[lem-tempered-sheaf-affine\], these functions can be glued to a unique tempered function on $X_{i}$ as they form a sheaf. Thus, we get that $t|_{X_{i}}\in\mathcal{T}\left(X_{i}\right)$. \[def-temp-func\]Let $X$ be a QN variety. A real valued function $t:X\to\mathbb{R}$ is called ** if it satisfies the equivalent conditions of Lemma \[lem-Tempered-eqiv\]. Denote the space of all tempered functions on $X$ by $\mathcal{T}\left(X\right)$. \[lem-sXt-non-affine\]Let $X$ be a QN variety, $t\in\mathcal{T}\left(X\right)$ and $s\in\mathcal{S}\left(X\right)$. Then $t\cdot s\in\mathcal{S}\left(X\right)$. Let $X=\bigcup\limits _{i=1}^{k}X_{i}$ be some open affine QN cover such that $s=\sum\limits _{i=1}^{k}Ext_{X_{i}}^{X}\left(s_{i}\right)$ for some $s_{i}\in\mathcal{S}\left(X_{i}\right)$. Then $t|_{X_{i}}\in\mathcal{T}\left(X_{i}\right)$ and by proposition \[prop-sXt\] $t|_{X_{i}}\cdot s_{i}\in\mathcal{S}\left(X_{i}\right)$. Thus, $t\cdot s=\sum\limits _{i=1}^{k}Ext_{X_{i}}^{X}\left(s_{i}\cdot t|_{X_{i}}\right)\in\mathcal{S}\left(X\right)$. \[prop-part-of-unity\](tempered partition of unity) - Let $X$ be a QN variety, and let $\left\{ V_{i}\right\} _{i=1}^{m}$ be a finite open cover of $X$. Then: \(1) There exist tempered functions $\left\{ \alpha_{i}\right\} _{i=1}^{m}$ on $X$, such that $supp\left(\alpha_{i}\right)\subset V_{i}$ and $\sum\limits _{i=1}^{m}\alpha_{i}=1$. \(2) We can choose $\left\{ \alpha_{i}\right\} _{i=1}^{m}$ in such a way that for any $\varphi\in\mathcal{S}\left(X\right),\:\left(\alpha_{i}\varphi\right)|_{V_{i}}\in\mathcal{S}\left(V_{i}\right)$. The proof for the affine case is similar to that in [\[]{}ES - Proposition 3.11[\]]{}, with the corresponding claims here (e.g. Lemma \[lem-Sch-Res-to-closed\] replaces Theorem 3.7 in [\[]{}ES[\]]{}). The idea behind this proof is to take the corresponding set in the Nash manifold $\mathbb{R}^{n}$, and extend the $V_{i}$’s to some open semi-algebraic sets covering $\mathbb{R}^{n}$. By Theorem \[AG-thm-part-of-unity\] there is a tempered partition of unity on those extending sets, so we can reduce to our $V_{i}$’s to get the result. For the general case we use claims in the Appendix as follows: \(1) By definition of $X$, there exists an open affine cover $X=\bigcup\limits _{j=1}^{n}V_{j}$. Thus, for each $i$, there exists an open affine cover $V_{i}=\bigcup\limits _{j=1}^{n}V_{ij}$ where $V_{ij}:=V_{i}\cap V_{j}$. By Proposition \[prop-proper-ref\] there exists a cover of $X$ by $\bigcup\limits _{i,j}V_{ij}'=X$ where $V'_{ij}\subset\bar{V}_{ij}'\subset V_{ij}$. By Corollary \[cor-basis-Mf\], there exist a finite collection of continuous functions $G_{ijk}:V_{ij}\rightarrow\mathbb{R}$ and open sets $V'_{ijk}$ such that $V'_{ijk}=\left\{ x\in V_{ij}|G_{ijk}\left(x\right)\neq0\right\} ,\:G_{ijk}|_{V'_{ijk}}$ is positive and QN and $\bigcup\limits _{k}V'_{ijk}=V'_{ij}$. It gives a finite cover of $X$ which is a refinement of $U_{i}$. In order to have a unified system of indices we denote $V_{ijk}:=V_{ij}$. We re-index it to one index cover $V_{l}$. By the same re-indexation we get $G_{l}$ and $V'_{l}$ . Extend $G_{l}$ by zero to a function $\widetilde{G_{l}}$ on $X$. It is continuous. Denote $G=\left(\sum\widetilde{G_{l}}\right)/\left(2n\right)$ where $n$ is the number of values of the index $l$. Consider $G|_{V_{l}}$ . This is a strictly positive continuous semi-algebraic function on an affine QN variety. Lemma \[lem-majoration\] shows that continuous strictly positive semi-algebraic function on an affine QN variety can be bounded from below by a strictly positive QN function. Thus, $G|_{V_{l}}$ can be bounded from below by a strictly positive QN function $g'_{l}$. Denote $H_{l}:=G_{l}/g'_{l}$. Extending $H_{l}$ by zero outside $V_{l}$ to $X$ we obtain a collection of continuous semi-algebraic functions $F_{l}$. Note that $F_{l}$ is not smooth. It is easy to see that $X_{F_{l}}$ is a refinement of $V_{i}$. Now, let $\rho:\mathbb{R}\rightarrow[0,1]$ be a smooth function such that $$\rho\left((-\infty,0.1]\right)=\left\{ 0\right\} ,\:\rho\left([1,\infty)\right)=\left\{ 1\right\}$$ Denote $\beta_{l}:=\rho\circ F_{l}$ and $\gamma_{l}=\frac{\beta_{l}}{\sum\beta_{l}}$. It is easy to see that $\gamma_{l}$ are tempered. For every $l$ we choose $i\left(l\right)$ such that $X_{F_{l}}\subset U_{i\left(l\right)}$. Define $\alpha_{i}:=\sum\limits _{l|i\left(l\right)=i}\gamma_{l}$. It is easy to see that $\alpha_{i}$ is a tempered partition of unity. \(2) Take some $\varphi\in\mathcal{S}\left(X\right)$. By definition $\varphi=\sum\limits _{j=1}^{n}Ext_{X_{j}}^{X}\varphi_{j}$ where the $X_{j}$ ’s are the affine open cover of $X$ and $\varphi_{j}\in\mathcal{S}\left(X_{j}\right)$. By definition, $\alpha_{i}|_{X_{j}}\in\mathcal{T}\left(X_{j}\right)$, and by Lemma \[prop-sXt\] $\varphi_{j}\cdot\alpha_{i}|_{X_{j}}\in\mathcal{S}\left(X_{j}\right)$. By part (1), $supp\left(\alpha_{i}\right)\subset V_{i}$ so $supp\left(\varphi_{j}\cdot\alpha_{i}|_{X_{j}}\right)\subset V_{i}\cap X_{j}$ and $\varphi_{j}\cdot\alpha_{i}|_{X_{j}}$ is flat on $X_{j}\backslash V_{i}$. By Theorm \[thm-affine-char\], $\left(\varphi_{j}\cdot\alpha_{i}|_{X_{j}}\right)|_{V_{i}\cap X_{j}}\in\mathcal{S}\left(V_{i}\cap X_{j}\right)$. Note that $V_{i}\cap X_{j}$ is an open affine QN cover of $V_{i}$, so $$\sum\limits _{j=1}^{n}Ext_{V_{i}\cap X_{j}}^{V_{i}}\left(\varphi_{j}\cdot\alpha_{i}|_{X_{j}}\right)|_{V_{i}\cap X_{j}}\in\mathcal{S}\left(V_{i}\right).$$ Finally, by the definition of the $\varphi_{j}$’s we get that $\sum\limits _{j=1}^{n}Ext_{V_{i}\cap X_{j}}^{V_{i}}\left(\varphi_{j}\cdot\alpha_{i}|_{X_{j}}\right)|_{V_{i}\cap X_{j}}=\left(\varphi\cdot\alpha_{i}\right)|_{V_{i}}$. \[rem-temp-Sch-is-Sch\]Actually, what we proved in part (2) is that for any tempered function $\beta\in\mathcal{T}\left(X\right)$ whose support is a subset of some open subset $U$, i.e. $supp\left(\beta\right)\subset U,\:U\subset X$, and for any $\varphi\in\mathcal{S}\left(X\right)$ we get $\left(\beta\cdot\varphi\right)|_{U}\in\mathcal{S}\left(U\right)$. Let $X$ be a QN variety. A function $f:X\to\mathbb{R}$ is called **flat** at $p\in X$ if there exists an open affine QN subset $U\subset X$ such that $p\in U$ and $f|_{U}$ is flat at $p$. If $f$ is flat at all $p\in Z$ for some $Z\subset X$, $f$ is called **flat** at $Z$. \[prop-ext-by-0\](extension by zero for non affine varieties). Let $X$ be a QN variety, and $U$ an open subset of $X$. Then the extension by zero to $X$ of a Schwartz function on $U$ is a Schwartz function on $X$, which is flat at $X\backslash U$. Take an open affine QN cover $X=\bigcup\limits _{i=1}^{k}X_{i}$. Then $U=\bigcup\limits _{i=1}^{k}\left(U\cap X_{i}\right)$ is an open affine QN cover of $U$, what makes $U$ a QN variety as well. Take some $s\in\mathcal{S}\left(U\right)$. So $s=\sum\limits _{i=1}^{k}Ext_{U\cap X_{i}}^{U}\left(s_{i}\right)$ for some $s_{i}\in\mathcal{S}\left(U\cap X_{i}\right)$. The set $U_{i}:=U\cap X_{i}$ is open in $X_{i}$, so take the closed corresponding set $\tilde{X}$ in $\mathbb{R}^{n_{i}}$, and its corresponding open subset $\tilde{U}_{i}$. Then there exists an open semi-algebraic set $V_{i}\subset\mathbb{R}^{n}$ such that $\tilde{U}_{i}=V_{i}\cap\tilde{X}_{i}$. By remark \[rem-NQN-Like-Sch-Tempered\], $s_{i}$ corresponds to some $\tilde{s}_{i}=S_{i}|_{\tilde{U}_{i}}$where $S_{i}\in\mathcal{S}\left(V_{i}\right)$. By Theorem \[AG-thm-Char-Nash\], $S_{i}$ can be extended by zero to the whole of $\mathbb{R}^{n_{i}}$ and this extension is flat outside $V_{i}$, and in particular in $\tilde{X}_{i}\backslash\tilde{U}_{i}$. This extension can be reduced to $\tilde{X}_{i}$ to make a Schwartz function on $\tilde{X}_{i}$ which extends $\tilde{s}_{i}$ to $\tilde{X}_{i}$ by zero, and thus is flat on $\tilde{X}_{i}\backslash\tilde{U}_{i}$. Thus, $s_{i}$ can be extended to a Schwartz function on $X_{i}$ which is flat on $X_{i}\backslash U_{i}$ . So $Ext_{U}^{X}\left(s\right)=Ext_{U}^{X}\left(\sum\limits _{i=1}^{k}Ext_{U_{i}}^{U}\left(s_{i}\right)\right)=\sum\limits _{i=1}^{k}Ext_{U}^{X}\left(Ext_{U_{i}}^{U}\left(s_{i}\right)\right)=\sum\limits _{i=1}^{k}Ext_{X_{i}}^{X}\left(Ext_{U_{i}}^{X_{i}}\left(s_{i}\right)\right)$ which, by definition, is a Schwartz function on $X$ which is flat on $X\backslash U$. \[thm-general-char\](Characterization of Schwartz functions on open subset - the general case) Let $X$ be a QN variety, and let $Z\subset X$ be some closed subset. Define $U:=X\backslash Z$ and $W_{Z}:=\left\{ \phi\in\mathcal{S}\left(X\right)|\phi\text{ is flat on }Z\right\} $. Then $W_{Z}$ is a closed subspace of $\mathcal{S}\left(X\right)$, and thus it is a Fréchet space. Furthermore, extension by zero $Ext_{U}^{X}:\mathcal{S}\left(U\right)\to W_{Z}$ is an isomorphism of Fréchet spaces, whose inverse is $Res_{U}^{X}:W_{Z}\to\mathcal{S}\left(U\right)$. As for the first part, $W_{Z}=\bigcap\limits _{z\in Z}\left\{ \phi\in\mathcal{S}\left(X\right)|\phi\text{ is flat on }z\right\} $ is an intersection of closed sets, it is a closed subspace of $\mathcal{S}\left(X\right)$, and thus a Fréchet space. For the second part, by Proposition \[prop-ext-by-0\] the extension of a function in $\mathcal{S}\left(U\right)$ by zero to $X$ is a function in $\mathcal{S}\left(X\right)$ that is flat at $Z$, i.e. $Ext_{U}^{X}\left(\mathcal{S}\left(U\right)\right)\subset W_{Z}$. Furthermore, we will claim further on that $Ext_{U}^{X}$ is continuous. Before that, we will show the opposite direction - that $Res_{U}^{X}\left(W_{Z}\right)\subset\mathcal{S}\left(U\right)$. Let $f\in W_{Z}$. We want to show $f|_{U}=\sum\limits _{i=1}^{n}Ext_{U_{i}}^{U}s_{i}$ where $X=\bigcup\limits _{i=1}^{n}X_{i}$ is an open affine QN cover of $X$, $U_{i}:=U\cap X_{i}$, (this makes $U=\bigcup\limits _{i=1}^{n}U_{i}$ an open affine cover of $U$) and $s_{i}\in\mathcal{S}\left(U_{i}\right)$. Note that $s_{i}\neq f|_{U_{i}}$. As $f\in\mathcal{S}\left(X\right)$, and as $X=\bigcup\limits _{i=1}^{n}X_{i}$ is an affine QN open cover of $X$, we can use Proposition \[prop-part-of-unity\] to get tempered functions $\alpha_{i}$ such that $\alpha_{i}\cdot f\in\mathcal{S}\left(X_{i}\right)$. As $f\in W_{Z}$ is flat at any $x\in Z$, $\alpha_{i}\cdot f$ is flat at any $x\in Z_{i}:=Z\cap X_{i}$. Thus, $\alpha_{i}\cdot f\in W_{Z_{i}}$, and by Theorem \[thm-affine-char\] we get $\alpha_{i}\cdot f\in\mathcal{S}\left(U_{i}\right)$ where $U_{i}:=U\cap X_{i}$. $\bigcup\limits _{i=1}^{n}U_{i}$ is an open affine QN cover of $U$, so $\sum\limits _{i=1}^{n}Ext_{U_{i}}^{U}\left(\alpha_{i}\cdot f\right)\in\mathcal{S}\left(U\right)$. Furthermore, by the definition of the $\alpha$’s, $\sum\limits _{i=1}^{n}Ext_{U_{i}}^{U}\left(\alpha_{i}\cdot f\right)=f|_{U}$. By Theorem \[thm-affine-char\], for any $i=1,...,n$ we have the continuous map $Ext_{U_{i}}^{X_{i}}:\mathcal{S}\left(U_{i}\right)\rightarrow\mathcal{S}\left(X_{i}\right)$. As $n<\infty$, we get the continuous map $Ext_{U}^{X}:\bigoplus\limits _{i=1}^{n}\mathcal{S}\left(U_{i}\right)\rightarrow\bigoplus\limits _{i=1}^{n}\mathcal{S}\left(X_{i}\right)$. Recall that $\mathcal{S}\left(X\right):=\psi\left(\bigoplus\limits _{i=1}^{n}\mathcal{S}\left(X_{i}\right)\right)\cong\bigoplus\limits _{i=1}^{n}\mathcal{S}\left(X_{i}\right)/Ker\left(\psi|_{\bigoplus\limits _{i=1}^{n}\mathcal{S}\left(X_{i}\right)}\right)$ and get the continuous map $Ext_{U}^{X}:\mathcal{S}\left(U\right)\rightarrow\mathcal{S}\left(X\right)$. As this map is bijective, we get by the Theorem \[thm-Banach-open-mapping\] $Ext_{U}^{X}$ is an isomorphism of Fréchet spaces. Let $X$ be a QN variety. Define the **space of tempered distributions on $\boldsymbol{X}$** as the space of continuous linear functionals on $\mathcal{S}\left(X\right)$. Denote this space by $\mathcal{S}^{*}\left(X\right)$. \[thm-temp-dist-onto-general\]Let $X$ be a QN variety, and let $U\subset X$ be some semi-algebraic open subset. Then $Ext_{U}^{X}:\mathcal{S}\left(U\right)\hookrightarrow\mathcal{S}\left(X\right)$ is a closed embedding, and the restriction morphism $\mathcal{S}^{*}\left(X\right)\rightarrow\mathcal{S}^{*}\left(U\right)$ is onto. As $W_{Z}=\bigcap\limits _{z\in Z}\left\{ \phi\in\mathcal{S}\left(X\right)|\phi\text{ is flat on }z\right\} $ is an intersection of closed subsets, it is a closed subspace of $\mathcal{S}\left(X\right)$ and thus, by Theorem \[thm-general-char\] the first part is proved. The second part follows from the fact that $\mathcal{S}\left(X\right)$ is a Fréchet space and from Theorem \[thm-Hahn-Banach\]. **Sheaves and Cosheaves** ========================= The aim of this section is to prove that tempered functions and tempered distributions form sheaves and that Schwartz functions form a cosheaf. Unlike the algebraic case, this can be done to both the affine and the general case. The proofs for the affine cases, and for the general case of tempered functions, are the same as the those in the [\[]{}ES[\]]{} regarding algebraic varieties. Thus we give short sketches of the proofs together with relevant statements in this paper replacing statements in [\[]{}ES[\]]{}. First, let us recall Lemma \[lem-tempered-sheaf-affine\]: Let $X$ be an affine QN variety. The assignment of the space of tempered functions to any open $V\subset X$, together with the restriction of functions, form a sheaf on $X$. \[cor-tempered-sheaf\]Let $X$ be a QN variety. The assignment of the space of tempered functions to any open $U\subset X$, together with the restriction of functions, form a sheaf on $X$. The proof follows the proof of [\[]{}ES - Proposition 5.11[\]]{}. By the definition of tempered functions on QN varieties and by Lemma \[lem-tempered-sheaf-affine\], they form a presheaf. Using induction on the number of the covering open subsets, it is enough to show the following: Let $X$ be a QN variety and let $U_{1}\cup U_{2}=X$ be an open cover of $X$. Assume we are given $t_{i}\in\mathcal{T}\left(U_{i}\right)$ such that The existence of a function $t:U_{1}\cup U_{2}\to\mathbb{R}$ such that $t|_{U_{i}}=t_{i}$ is clear. We shall now show it is tempered. Consider some affine open cover $X=\bigcup\limits _{j=1}^{k}X_{j}$. Then $U_{i}=\bigcup\limits _{j=1}^{k}\left(U_{i}\cap X_{j}\right)$ is an affine open cover of $U_{i}$, and $U_{1}\cup U_{2}=\bigcup\limits _{j=1}^{k}\left(\left(U_{1}\cup U_{2}\right)\cap X_{j}\right)$ is an affine open cover of $U_{1}\cup U_{2}$. As $t_{i}\in\mathcal{T}\left(U_{i}\right)$, we get that $t_{i}|_{U_{i}\cap X_{j}}\in\mathcal{T}\left(U_{i}\cap X_{j}\right)$. As $\left(U_{1}\cup U_{2}\right)\cap X_{j}$ is affine, and $\bigcup\limits _{i=1}^{2}\left(U_{i}\cap X_{j}\right)$ is an affine open cover of it, and as Before dealing with cosheaves in the category of real vector spaces, let us recall their definition: First, define the category $Top(X)$ to be such that its objects are the open sets of $X$, and its morphisms are the inclusion maps. A *pre-cosheaf* $F$ on a topological space $X$ is a covariant functor from $Top(X)$ to the category of real vector spaces. A *cosheaf* on a topological space $X$ is a pre-cosheaf, such that for any open $V\subset X$ and any open cover $\{V_{i}\}_{i\in I}$ of $V$, the following sequence is exact: $$\bigoplus\limits _{\left(i,j\right)\in I^{2}}F\left(V_{i}\cap V_{j}\right)\xrightarrow{Ext_{1}}\bigoplus\limits _{i\in I}F\left(V_{i}\right)\xrightarrow{Ext_{2}}F\left(V\right)\xrightarrow{}0,$$ where the $k$-th coordinate of $Ext_{1}(\bigoplus\limits _{(i,j)\in I^{2}}\xi_{i,j})$ is $\sum\limits _{i\in I}Ext_{V_{k}\cap V_{i}}^{V_{k}}(\xi_{k,i}-\xi_{i,k})$, and $Ext_{2}(\bigoplus\limits _{i\in I}\xi_{i}):=\sum\limits _{i\in I}Ext_{V_{i}}^{V}(\xi_{i})$. When exactness will be proven in Proposition \[Schwartz-is-a-cosheaf\] below all calculations will be quickly reduced to finite subcovers. A cosheaf is *flabby* if for any two open subsets $U,V\subset X$ such that $V\subset U$, the morphism $Ext_{V}^{U}:F(V)\to F(U)$ is injective. \[lem-Sch-cosheaf-affine\]Let $X$ be an affine QN variety. The assignment of the space of Schwartz functions to any open $U\subset X$, together with the extension by zero $Ext_{U}^{V}$ from $U$ to any other open $V\supset U$, form a flabby cosheaf on X. The proof follows [\[]{}ES - Proposition 4.5[\]]{} - By extension by 0 (see Theorem \[thm-affine-char\]), $X$ is a pre-cosheaf. Now we shall prove the exactness: The $\bigoplus\limits _{j=1}^{l}F\left(U_{j}\right)\longrightarrow F\left(U\right)\longrightarrow0$ part follows immediately from partition of unity. The second part uses induction on the number of the covering sets. The base step uses the fact that the two functions sum to zero everywhere, and that their extensions are flat outside the sets’ intersection. Thus, by the characterization property given in Theorem \[thm-affine-char\], their restriction to the intersection is Schwartz. In the inductive step, we use partition of unity on $k+1$ sets, to create some new Schwartz functions on the first $k$ sets whose extensions sum to zero. We then define and prove the claim for the $k+1$’th function, using the fact that each of the $k$ functions is flat at each point they are solely defined . \[cor-Sch-cosheaf\]Let $X$ be a QN variety. The assignment of the space of Schwartz functions to any open $U\subset X$, together with the extension by zero $Ext_{U}^{V}$ from $U$ to any other open $V\supset U$, form a flabby cosheaf on X. By Proposition \[prop-ext-by-0\], the Schwartz functions form a pre-cosheaf on $X$. Let $\bigcup\limits _{i=1}^{k}X^{i}=X$ be a finite open cover such that for any $i$, $X^{i}$ is affine. Let $U\subset X$ be some open set, $\bigcup\limits _{j=1}^{l}U_{j}=U$ is some open cover, and let $s\in\mathcal{S}\left(U\right)$. Then for any $i$, $U^{i}:=U\cap X^{i}$ is an open subset of the affine $X^{i}$, and $\bigcup\limits _{j=1}^{l}U_{j}^{i}=U^{i}$ where $U_{j}^{i}:=U^{i}\cap U_{j}$ is an open cover of $U^{i}$. By definition, $s=\sum\limits _{i=1}^{k}Ext_{U^{i}}^{U}\left(s^{i}\right)$ where $s^{i}\in\mathcal{S}\left(U^{i}\right)$. By Lemma \[lem-Sch-cosheaf-affine\] we know that $s^{i}=\sum\limits _{j=1}^{l}Ext_{U_{j}^{i}}^{U^{i}}\left(s_{j}^{i}\right)$ where $s_{j}^{i}\in\mathcal{S}\left(U_{j}^{i}\right)$. So we get that $s=\sum\limits _{i=1}^{k}Ext_{U^{i}}^{U}\sum\limits _{j=1}^{l}Ext_{U_{j}^{i}}^{U^{i}}s_{j}^{i}$ and as the sums are finite, we may write $s=\sum\limits _{j=1}^{l}Ext_{U_{j}}^{U}\sum\limits _{i=1}^{k}Ext_{U_{j}^{i}}^{U_{j}}s_{j}^{i}=\sum\limits _{j=1}^{l}Ext_{U_{j}}^{U}s_{j}$ where $s_{j}:=\sum\limits _{i=1}^{k}Ext_{U_{j}^{i}}^{U_{j}}s_{j}^{i}$ and $s_{j}\in\mathcal{S}\left(U_{j}\right)$ by definition. This proves the part $$\bigoplus\limits _{j=1}^{l}F\left(U_{j}\right)\longrightarrow F\left(U\right)\longrightarrow0$$ of the cosheaf definition. Now let us have $\left\{ s_{j}\right\} \subset\mathcal{S}\left(U_{j}\right)$ such that $\sum\limits _{j=1}^{l}Ext_{U_{j}}^{U}s_{j}=0$. As $\bigcup\limits _{i=1}^{k}U^{i}=U$ is an open (affine) cover of $U$, we may use Proposition \[prop-part-of-unity\] to get some tempered functions $\left\{ \alpha_{i}\right\} _{i=1}^{k}$ on $U$, such that $supp\left(\alpha_{i}\right)\subset U^{i}$ and $\sum\limits _{i=1}^{k}\alpha_{i}=1$, and for any $\varphi\in\mathcal{S}\left(U\right),\:\left(\alpha_{i}\varphi\right)|_{U^{i}}\in\mathcal{S}\left(U^{i}\right)$. Now, $S_{j}:=Ext_{U_{j}}^{U}s_{j}\in\mathcal{S}\left(U\right)$ by Proposition \[prop-ext-by-0\], so $\left(\alpha_{i}\cdot S_{j}\right)|_{U^{i}}\in\mathcal{S}\left(U^{i}\right)$. We get an affine QN variety $U^{i}$ with $\sum\limits _{j=1}^{l}\left(\left(\alpha_{i}\cdot S_{j}\right)|_{U^{i}}\right)=0$. Notice that for each $j$, the function $\alpha_{i}\cdot S_{j}$ is flat on $U^{i}\backslash U_{j}$, so we may use Theorem \[thm-affine-char\] to get $\left(\alpha_{i}\cdot S_{j}\right)|_{U_{j}^{i}}\in\mathcal{S}\left(U_{j}^{i}\right)$. Thus, we have $\sum\limits _{j=1}^{l}Ext_{U_{j}^{i}}^{U^{i}}\left(\left(\alpha_{i}\cdot S_{j}\right)|_{U_{j}^{i}}\right)=0$ and by Lemma \[lem-Sch-cosheaf-affine\], we know there exists $s_{jm}^{i}\in\mathcal{S}\left(U_{j}^{i}\cap U_{m}^{i}\right)$ satisfying $\left(\alpha_{i}\cdot S_{j}\right)|_{U_{j}^{i}}=\sum\limits _{m<j}Ext_{U_{j}^{i}\cap U_{m}^{i}}^{U_{j}^{i}}\left(s_{jm}^{i}\right)-\sum\limits _{m>j}Ext_{U_{j}^{i}\cap U_{m}^{i}}^{U_{j}^{i}}\left(s_{mj}^{i}\right)$. Sum the extensions by zero to $U_{j}$ of both sides to get $$\sum\limits _{i=1}^{k}Ext_{U_{j}^{i}}^{U_{j}}\left(\left(\alpha_{i}\cdot S_{j}\right)|_{U_{j}^{i}}\right)=$$ $$\sum\limits _{i=1}^{k}Ext_{U_{j}^{i}}^{U_{j}}\left(\sum\limits _{m<j}Ext_{U_{j}^{i}\cap U_{m}^{i}}^{U_{j}^{i}}\left(s_{jm}^{i}\right)-\sum\limits _{m>j}Ext_{U_{j}^{i}\cap U_{m}^{i}}^{U_{j}^{i}}\left(s_{mj}^{i}\right)\right),$$ $\:$ but $\:$ $$\sum\limits _{i=1}^{k}Ext_{U_{j}^{i}}^{U_{j}}\left(\left(\alpha_{i}\cdot S_{j}\right)|_{U_{j}^{i}}\right)=S_{j}|_{U_{j}}=s_{j}$$ $\:$ and $\:$ $$\sum\limits _{i=1}^{k}Ext_{U_{j}^{i}}^{U_{j}}\left(\sum\limits _{m<j}Ext_{U_{j}^{i}\cap U_{m}^{i}}^{U_{j}^{i}}\left(s_{jm}^{i}\right)-\sum\limits _{m>j}Ext_{U_{j}^{i}\cap U_{m}^{i}}^{U_{j}^{i}}\left(s_{mj}^{i}\right)\right)=$$ $$\sum\limits _{m<j}Ext{}_{U_{j}\cap U_{m}}^{U_{j}}\sum\limits _{i=1}^{k}Ext_{U_{j}^{i}\cap U_{m}^{i}}^{U_{j}\cap U_{m}}\left(s_{jm}^{i}\right)-\sum\limits _{m>j}Ext{}_{U_{j}\cap U_{m}}^{U_{j}}\sum\limits _{i=1}^{k}Ext_{U_{j}^{i}\cap U_{m}^{i}}^{U_{j}\cap U_{m}}\left(s_{mj}^{i}\right)=$$ $$\sum\limits _{m<j}Ext{}_{U_{j}\cap U_{m}}^{U_{j}}\left(s_{jm}\right)-\sum\limits _{m>j}Ext{}_{U_{j}\cap U_{m}}^{U_{j}}\left(s_{mj}\right)$$ $\:$ where $s_{jm}\in\mathcal{S}\left(U_{j}\cap U_{m}\right)$ by definition. This sums up to $\:$ $$s_{j}=\sum\limits _{m<j}Ext{}_{U_{j}\cap U_{m}}^{U_{j}}\left(s_{jm}\right)-\sum\limits _{m>j}Ext{}_{U_{j}\cap U_{m}}^{U_{j}}\left(s_{mj}\right)$$ $\:$ what proves the part $\:$ $$\bigoplus\limits _{j>m}F\left(U_{j}\cap U_{m}\right)\longrightarrow\bigoplus\limits _{j=1}^{l}F\left(U_{j}\right)\longrightarrow F\left(U\right)$$ $\:$ of the cosheaf definition. \[lem-temp-dist-sheaf-affine\]Let $X$ be an affine QN variety. The assignment of the space of tempered distributions to any open $U\subset X$, together with restrictions of functionals from $\mathcal{S}^{*}\left(U\right)$ to $\mathcal{S}^{*}\left(V\right)$, for any other open $V\subset U$, form a flabby sheaf on $X$. The proof follows [\[]{}ES - Proposition 4.4[\]]{}. The proof uses extension by zero to show presheaf structure. To show uniqueness, we use partition of unity which enables us to write $s\in\mathcal{S}\left(U\right)$ as $s=\sum\limits _{i=1}^{k}\left(\beta_{i}\cdot s\right)$ where $supp\left(\beta_{i}\right)\subset U_{i}$. Then, as $\left(\beta_{i}\cdot s\right)|_{U_{i}}\in\mathcal{S}\left(U_{i}\right)$, and as the functionals $\xi,\:\zeta\in\mathcal{S}^{*}\left(U\right)$ agree on each subset and by the linearity of the functionals, we get $\xi\left(s\right)-\zeta\left(s\right)=\xi\left(\sum\limits _{i=1}^{k}\left(\beta_{i}\cdot s\right)\right)-\zeta\left(\sum\limits _{i=1}^{k}\left(\beta_{i}\cdot s\right)\right)=\sum\limits _{i=1}^{k}\left(\xi\left(\beta_{i}\cdot s\right)-\zeta\left(\beta_{i}\cdot s\right)\right)=0$. This means the uniqueness is achieved. The existence is proven by partition of unity on $U=\bigcup\limits _{i=1}^{k}U_{i}$, and defining a functional $\xi\in\mathcal{S}^{*}\left(U\right)$ in the following way: $\xi\left(s\right)=\xi\left(\sum\limits _{i=1}^{k}\left(\beta_{i}\cdot s\right)\right):=\sum\limits _{i=1}^{k}\xi_{i}\left(\beta_{i}\cdot s\right)$ where $\xi_{i}\in\mathcal{S}^{*}\left(U_{i}\right)$, $s\in\mathcal{S}\left(U\right)$ and $\beta_{i}$ are the tempered functions obtained by the partition of unity. For any $U_{\alpha}\subset U$ open, where $\alpha\in\left\{ 1,...,k\right\} $ and $s_{\alpha}\in\mathcal{S}\left(U_{\alpha}\right)$, we get $\left(\beta_{i}|_{U_{\alpha}}\cdot s_{\alpha}\right)|_{U_{\alpha}\cap U_{i}}\in\mathcal{S}\left(U_{\alpha}\cap U_{i}\right)$ by Lemma \[lem-tempered-sheaf-affine\] and Proposition \[lem-sXt-non-affine\]. As $\xi_{\alpha}|_{\mathcal{S}\left(U_{\alpha}\cap U_{i}\right)}=\xi_{i}|_{\mathcal{S}\left(U_{\alpha}\cap U_{i}\right)}$ , and $s_{\alpha}$ may be extended to a Schwartz function on $U$, we get $\xi_{\alpha}\left(s_{\alpha}\right)=\xi_{\alpha}\left(\sum\limits _{i=1}^{k}\left(\beta_{i}\cdot s_{\alpha}\right)\right)=\sum\limits _{i=1}^{k}\xi_{\alpha}\left(\beta_{i}\cdot s_{\alpha}\right)=\sum\limits _{i=1}^{k}\xi_{i}\left(\beta_{i}\cdot s_{\alpha}\right)=\xi\left(s_{\alpha}\right)$. This means that for any $\alpha,\:\xi|_{\mathcal{S}\left(U_{\alpha}\right)}=\xi_{\alpha}$ and the existence holds. \[cor-Dist-sheaf\]Let $X$ be a QN variety. The assignment of the space of tempered distributions to any open $U\subset X$, together with restrictions of functionals from $\mathcal{S}^{*}\left(U\right)$ to $\mathcal{S}^{*}\left(V\right)$, for any other open $V\subset U$, form a flabby sheaf on $X$. The claim that tempered distributions form a sheaf is dual to the claim that Scwartz functions form a cosheaf, which we proved in Corollary \[cor-Sch-cosheaf\]. Schwartz, Tempered and Tempered “Distributions” Over QN Vector Bundles ====================================================================== We begin this section with definitions of QN bundles and their sections. Most of the definitions and results in this chapter follow [\[]{}AG[\]]{} with light adjustments to our category. Let $\pi:X\to B$ be a morphism of QN varieties. It is called a **QN locally trivial fibration** with fiber $Z$ if the following holds: - $Z$ is a QN variety. - There exists a *finite* cover $B=\bigcup\limits _{i=1}^{n}U_{i}$ by open QN sets and QN isomorphisms $\nu_{i}:\pi^{-1}\left(U_{i}\right)\tilde{\to}U_{i}\times Z$ such that $\pi\circ\nu_{i}^{-1}$ is the natural projection. Let $X$ be a QN variety. A **QN vector bundle $E$ over $X$** is a QN locally trivial fibration with linear fiber and such that the trivialization maps $\nu_{i}$ are fiberwise linear. By abuse of notation, we use the same letters to denote bundles and their total spaces. $\:$ Let $X$ be a QN variety and $E$ a QN bundle over $X$. A **QN section** of $E$ is a section of $E$ which is a QN morphism. Now we can use the above definitions to define the more specific QN bundles relevant to our work. Let $X$ be a QN variety, and $E$ be a QN bundle over it. Let $X=\bigcup\limits _{i=1}^{k}X_{i}$ be an affine QN trivialization of $E$. A global section $s$ of $E$ over $X$ is called **tempered** if for any $i$, all the coordinate components of $s|_{X_{i}}$ are tempered functions. The space of global tempered sections of $E$ is denoted by $\mathcal{T}\left(X,E\right)$. As tempered functions on QN varieties form sheaves, the definition above does not depend on the cover. *A Schwartz section on a QN bundle $X$:* $E$ be a QN bundle over $X$. Let $\bigcup\limits _{i=1}^{m}X_{i}=X$ be affine QN trivialization of $E$. Denote by $Func\left(X,E,\mathbb{R}\right)$ There is a natural map $\psi:\bigoplus\limits _{i=1}^{m}\mathcal{S}\left(X_{i}\right)^{n}\to Func\left(X,E,\mathbb{R}\right)$. Define **the space of global Schwartz sections of** $E$ by $\mathcal{S}\left(X,E\right):=Im\psi$. We define the topology on the space of global Schwartz sections of $E$ by the quotient topology, i.e. by the isomorphism $\mathcal{S}\left(X,E\right)\cong\bigoplus\limits _{i=1}^{m}\mathcal{S}\left(X_{i}\right)^{n}/Ker\psi$. $\:$ The definition above does not depend on the cover. See the proof of Lemma \[lem-Sch-cover-indep\]. Now we will define the local sections: Let $X$ be a QN variety, and let $E$ be a QN bundle over it. We define the **cosheaf $\mathcal{S}_{X}^{E}$ of Schwartz sections of $E$** by $\mathcal{S}_{X}^{E}\left(U\right):=\mathcal{S}\left(U,E|_{U}\right)$. We define in a similar way the **sheaf $\mathcal{T}_{X}^{E}$ of tempered sections of $E$** by $\mathcal{T}_{X}^{E}\left(U\right):=\mathcal{T}\left(U,E|_{U}\right)$. Let $X$ be a QN variety. For an open semi-algebraic subset $U\subset X$, $\mathcal{S}_{X}^{E}|_{U}=\mathcal{S}_{U}^{E|_{U}}$, $\mathcal{T}_{X}^{E}|_{U}=\mathcal{T}_{U}^{E|_{U}}$ . The lemma holds by definitions. \[thm-bundle-char\](Characterization of Schwartz sections on open subset - the bundle case) Let $X$ be a QN variety, and let $Z\subset X$ be some closed subset. Define $U:=X\backslash Z$ and $W_{Z}:=\left\{ \phi\in\mathcal{S}\left(X,E\right)|\phi\text{ vanish with all its derivatives on }Z\right\} $. Then extension by zero $Ext_{U}^{X}:\mathcal{S}_{X}^{E}\left(U\right)\to W_{Z}$ is an isomorphism of Fréchet spaces, whose inverse is $Res_{U}^{X}:W_{Z}\to\mathcal{S}_{X}^{E}\left(U\right)$. This theorem follows from Theorem \[thm-general-char\] - characterization of Schwartz functions on open subset for general QN variety, and Proposition \[prop-part-of-unity\] - tempered partition of unity for general QN variety. Preperations For Partition Of Unity =================================== In order to prove partition of unity we will follow the proof of Theorem \[AG-thm-part-of-unity\] using some definitions and lemmas, lightly adapted to our case. We will start with some definitions, and then show there is a certain refinement for the cover we will need later on. \[def-Basicness\]1) Let $M$ be a QN variety and $F$ be a continuous semi-algebraic function on $M$. We denote $M_{F}:=\left\{ x\in M|F\left(x\right)\neq0\right\} $. 2\) Let $M$ be a QN variety. A continuous semi-algebraic function $F$ on $M$ is called **basic** if $F|_{M_{F}}$ is a positive QN function. 3\) A collection of continuous semi-algebraic functions $\left\{ F_{i}\right\} $ is called **basic collection** if every one of them is basic, and in every point of $M$ one of them is larger than 1. ([\[]{}S III.1.1[\]]{}) Let $r<\infty$. A $C^{r}$ Nash manifold is affine. Thus, by [\[]{}AG - A.2.4[\]]{}, a QN variety M can be continuously embedded in $\mathbb{R}^{n}$ by a semi-algebraic map, where M and its image are homeomorphic. \[cor-metric\]Let M be a QN manifold. Then there exists a semi-algebraic continuous metric $d:M\times M\rightarrow\mathbb{R}$. A cover $M=\bigcup\limits _{j=1}^{m}V_{j}$ is called a **proper refinement** of the cover $M=\bigcup\limits _{i=1}^{n}U_{i}$ if for any $j$ there exists $i$ such that $\bar{V_{j}}\subset U_{i}$. \[prop-proper-ref\]Let $M=\bigcup\limits _{i=1}^{n}U_{i}$ be a finite open (semi-algebraic) cover of an affine QN variety M. Then there exists a finite open (semi-algebraic) cover $M=\bigcup\limits _{j=1}^{m}V_{j}$ which is a proper refinement of $\left\{ U_{i}\right\} $. Let $d$ be the metric from corollary \[cor-metric\]. If a set $A$ is closed in the classical topology, then the distance $d\left(x,A\right):=\inf\limits _{y\in A}d\left(x,y\right)$ is strictly positive for all points $x$ outside $A$. Now define $F_{i}:M\rightarrow\mathbb{R}$ by $F_{i}\left(x\right)=d\left(x,M\backslash U_{i}\right)$. It is semi-algebraic by the Tarski-Seidenberg principle. Define $G=\left(\sum\limits _{i=1}^{n}F_{i}\right)/2n$ and $V_{i}=\left\{ x\in M|F_{i}\left(x\right)>G\left(x\right)\right\} $. It is easy to see that $V_{i}$ is a proper refinement of $U_{i}$. (finiteness) Let $X\subset\mathbb{R}^{n}$ be a semi-algebraic set. Then every open semi-algebraic subset of $X$ can be presented as a finite union of sets of the form $$\left\{ x\in X|p_{i}\left(x\right)>0,\:i=1,...,n\right\} ,$$ where $p_{i}$ are polynomials in n variables. \[cor-basis-Mf\]Let M be an QN variety. Then it has a basis of open sets of the form $M_{F}$ where F is a basic function. \[lem-majoration\] ([\[]{}AG - Lemma A.2.1 + Lemma 2.2.11[\]]{}) Let M be an QN variety. Then any continuous semi-algebraic function on it can be majorated by a QN function, and any continuous strictly positive semi-algebraic function on it can be bounded from below by a strictly positive QN function. [BMP1]{} F. Treves, Topological vector spaces, distributions and kernels, Academic Press (1967). A. Aizenbud, D. Gourevitch, Schwartz functions on Nash manifolds, International Mathematics Research Notices (2008), doi:10.1093/imrn/rnm155. M. Shiota, Nash Manifolds. Lecture Notes in Mathematics 1269 (1987) Bochnak, J; Coste, M; Roy, M-F: Real Algebraic Geometry (1998) G. M. Constantine, T. H. Savits, A Multivariante Faa Di Bruno Formula With Applications, Transactions of the American Mathematical Society , 384.2 (1996) 503-520. E. Bierstone, P. D. Milman, Semi-analytic and subanalytic sets, Publications mathematiques de l’IHES, 67 (1988) 5-42. E. Bierstone, P. D. Milman, Geometric and dierential properties of subanalytic sets, Annals of Mathe- matics, 147 (1998) 731-785. E. Bierstone, P. D. Milman, W. Pawlucki, Composite dierential functions, Duke Mathematics Journal, 83.3 (1996) 607-620. E. Bierstone, P. D. Milman, W. Pawlucki, Higher-order tangents and Feerman’s paper on Whitney’s extension problem, Annals of Mathematics, 164 (2006) 361-370. B. Elazar, A. Shaviv, Schwartz functions on real algebraic varieties, Canadian Journal of Mathematics, Vol. 69 No. 5, (to appear) DOI: CJM-2017-042-6/10.4153

--- abstract: 'A semiconductor microcavity embedding donor impurities and excited by a laser field is modelled. By including general decay and dephasing processes, and in particular cavity photon leakage, detailed simulations show that control over the spin dynamics is significally enhanced in high-quality-factor cavities, in which case picosecond laser pulses may produce spin-flip with high-fidelity final states.' author: - 'G. F. Quinteiro$^1$, P. Dmitruk$^1$, A. A. Aligia$^2$' title: 'Efficient spin control in high-quality-factor planar micro-cavities' --- Introduction ============ Photons and excitons (X) can be made to strongly interact in high-quality cavities containing a semiconductor quantum well, leading to a repetitive coherent exchange of energy between the two particles.[@Wei92] When the energy exchange ocurrs faster than the decay time of the individual components, a combined state, the exciton-polariton, is said to have formed. Exciton-polaritons show a variety of features, that motivate studies in multiple directions. Current interest in exciton-polariton research in 2D micro-cavities mainly focus on its liquid state and non-equilibrium Bose-Einstein condensation (BEC) covering many aspects of the problem[@BEC]. The interaction of polariton fields with impurities or defects has been studied for the case of a polariton fluid scattered by centers acting on the photonic component of the field[@Carusotto04], a polariton gas scattered by spin-independent disorder potential acting on the exciton degree of freedom[@Sav97], and the scattering of polaritons from spinless impurities acting on the excitonic component of the field[@Ste86]. However, to the best of our knowledge, there are no reported studies on the interaction of a polariton field with a single spin degree of freedom. Here we study the dynamics, including relaxation processes, of a diluted exciton/photon field interacting with a single impurity of spin $s=1/2$. The system is depicted in Fig. \[fig:system\]. It consists of a 2D photon cavity embedding a quantum well (QW), which contains few donor impurities [@Khitrova-1; @Yamamoto-01]. The whole system is assumed to be at low temperature and excited by a laser from outside. We show that the quantum control of a single spin is more efficient for high-quality-factor cavities. Thus, a spin-flip in a high-fidelity final state could be produced with a single laser pulse of a few picoseconds. Since typical decoherence times for impurity spins in semiconductors are in the $\mu$s time-scale, the system can act as a high speed quantum memory or qubit[@Qui08; @Chi05; @And11; @Krout04]. We believe that the present proposal and that for the implementation of two-qubit polariton-induced operations[@Quinteiro-01; @Puri12] suggest that a complete quantum-computing scalable architecture based on a solid-state system is possible using polaritons in 2D-microcavities. ![Pictorial representation of the system. Two Distributed Bragg Mirror (DBR) structures, placed at the sides of a quantum well (QW), confine photons injected from outside by a laser. The photons produce QW excitons that interact with the impurity spin localized at position $\mathbf R$.[]{data-label="fig:system"}](fig_system_v3.eps) Coupling with the environment ============================= Real systems cannot be entirely isolated from their environment. This is specially true for solid-state systems where several particles co-exist. When control is in mind, undesired interactions make the evolution unpredictable with the possibility of partial or total failure of the control operation. In our case, excitons, photons, and spins suffer from the coupling with the environment. For highly pure samples, low temperatures and low exciton density, the relevant decoherence processes for excitons are those causing spin flip of electrons and holes, with conversion between bright and dark excitons [@Rui-1; @Murayama-1]. In general hole spins loss coherence faster than electron spins; for instance, in CdTe the spin relaxation of electrons is 29ps[@Murayama-1], while that for holes is $<7$ps[@Akas04]. In addition, the annihilation of excitons must be also considered, with associated lifetime of hundred ps in GaInNAs/GaAs [@Lu-1]. While different processes, such as structural disorder[@Savona-1] are responsible for the loss of photon population and coherence, the main process is photon leakage off the cavity due to its finite Q-factor, which leads to a lifetime of the order of $\tau = 15$ps [@Cerna-01]. At extremely low concentration of impurities with densities $n_I\simeq10^{13}$ cm$^{-3}$, electrons bound to different donors are well localized and do not interact among them [@Fu-01]. The interaction with the nuclei is dominant (due to the strong confinement of the localized state). At temperatures $T < 10 K$ the transverse relaxation time T2$^*$ is a few ns for the electron bound to a donor, and the spin relaxation time is of the order of $\mu$s for donors in GaAs [@Fu-01]. Hamiltonian {#Sec:Model} =========== In what follows, we work in the Heisenberg picture; thus, time-dependent operators shall be everywhere understood. The free Hamiltonian reads $$\label{Eq_H0} H_0 = \sum_{\alpha \, \mathbf k} \varepsilon_{\mathbf k}\, \hat{b}_{\alpha \mathbf k}^\dag \hat{b}_{\alpha \mathbf k} + \sum_{\chi\, \mathbf k} \hbar \omega_{\mathbf k}\, \hat{c}_{\chi \mathbf k}^\dag \hat{c}_{\chi \mathbf k}~,$$ where the first and second terms correspond to excitons ($\hat{b}_{\alpha \mathbf k}^\dag/\hat{b}_{\alpha \mathbf k}$) and cavity photons ($\hat{c}_{\chi \mathbf k}^\dag/\hat{c}_{\chi \mathbf k}$). The QW quantum confinement splits the heavy- and light-hole electronic bands, forming excitons out of conduction-band electrons with total angular momentum $j_z=1/2$ and valence-band heavy holes with $j_z=3/2$. Bright ($j_z=1$) and dark ($j_z=2$) excitons are included: $\alpha=\{1,2,3,4\}=\{\uparrow \Uparrow, \downarrow \Uparrow, \uparrow \Downarrow, \downarrow \Downarrow\}$, where the single (double) arrow identifies an electron (hole) angular momentum. The respective dispersion relations for excitons and photons are $\varepsilon_{\mathbf k} = \varepsilon_0 + (\hbar \mathbf k)^2/2m^*$ and $ \omega_{\mathbf k} = c/n \, (\mathbf k^2+\mathbf k_z^2)^{1/2}$, where $ \mathbf k$ is the in-plane momentum; the momentum $\mathbf k_z$ in the growth direction is determined by parameters of the cavity, $n$ is the index of refraction, and $c$ the speed of light. The polarization of the photon is $\chi=\{1,2\}$. The ground state energy of the donor is set to zero. The system is excited by a classical laser field producing photons that propagate inside the cavity. Using the quasi-mode approximation (useful for high Q-factor cavities)[@quasimode], the cavity-laser interaction reads $$\label{Eq_HLC} H_{LC}=\hbar \sqrt{A} \sum_{\chi \, \mathbf k} \, {\cal{V}}_{\chi \mathbf k} (t) \, e^{i (\Omega_{\mathbf k} - \bar \Omega ) t}\, \hat{c}_{\chi \mathbf k} + H.c.~,$$ where ${\cal{V}}_{\chi \mathbf q} (t)$ is the coupling constant, $A=L^2$ is the area of the system, $\Omega_{\mathbf k}$ is the laser frequency and $\bar \Omega$ a constant adequately chosen to ease the numerical solution, see below. Cavity photons interact with excitons according to $$\begin{aligned} \label{} H_{L} &=& \sum_{ \chi \, \alpha \, \mathbf k } g_{\alpha \chi \mathbf k}(\omega_{\mathbf k}) \, \hat{c}_{\chi \mathbf k} \, \hat{b}_{\alpha \mathbf k}^\dag + H.c.\,,\end{aligned}$$ where $g_{\alpha \chi \mathbf k}(\omega_{\mathbf k}) = 0$ for $\alpha = 1,4$. The QW contains donor impuritites. The assumption is made that, at low temperature, each impurity has an electron bound to it that contributes a spin $s=1/2$; in addition, the concentration of donors is low enough to ensure that excitons/polaritons will interact only with one selected impurity, located at position $\mathbf R$, when the laser spot is small enough. [@foot1] Via Coulomb exchange, the electrons belonging to the exciton and the donor impurity interact through $ H_{XS} = H_{XS}^{(+)} + H. c.$ $$\begin{aligned} \label{Eq:HXS} H_{XS}^{(+)} \hspace{-1mm} &=& \hspace{-3mm} \sum_{ \scriptsize \begin{array}{cc} \mathbf k \, \mathbf k' \\ \end{array} } \frac{ j_{ \mathbf k \mathbf k'}}{A} \, e^{-i({\bf{k - k'}})\cdot \mathbf R} \nonumber \\ && \times \, \hat{ \mathbf s} \cdot \left[ \frac{\hbar}{2} \hspace{-2mm} \sum_{ \scriptsize \begin{array}{cc} \chi \, \chi' \, \eta \\ \end{array} } \hat{b}_{(\chi \eta) \mathbf k}^\dag \, \boldsymbol \sigma_{\chi \chi'} \, \hat{b}_{(\chi' \eta) \mathbf k'} \right]\end{aligned}$$ where the vector spin operator $\hat{\mathbf s} = (\hat{s}_x, \hat{s}_y, \hat{s}_z)$, $j_{ \mathbf k \mathbf k'} = j_0 [1 + a_I^{*\, 2} (\mathbf k- \mathbf k')^2]^{-1/2}$, and $a_I$ is a measure of the impurity electron localization[@Boiko10]. Here we adopted a more detailed notation for the spin $\alpha$ of the exciton: the electron (hole) spin has index $\chi$ ($\eta$), and $\boldsymbol \sigma$ is the vector of Pauli matrices.[@Qui08] X-X interaction is disregarded, because we will study the case of low exciton concentration, where $n_X a_B^{*\,2}/A < 1$, with $a_B^{*}$ the exciton Bohr radius. Method {#Sec:Method} ====== Different theoretical tools are employed to solve problems in exciton-polariton research. Heisenberg equations of motion describe the dynamics of mean values of either exciton/photon operators or polariton operators [@Carusotto04; @Cerna-1; @Sarchi-1]. It is also common the use of the Gross-Pitaevskii equation[@Ciuti-3; @Shelykh-1]. Other methods have also been used, such as the Hartree-Fock-Popov[@Sarchi-2]. We make use of the Heisenberg equations of motion (HEM) $ \hbar{d\langle \hat{{\cal O}} \rangle}/{dt} = i \langle\left[ H,\hat{{\cal O}} \right]\rangle $ for mean values ($ \langle \ldots \rangle$) of operators describing separately excitons, photons, and the impurity spin. This allows us to treat the cases of weak —where no polaritons exist— and strong coupling, as well as to include easily spin-flip processes that cause a polariton to dissociate into a dark exciton and a photon. In general, the HEM comprise a set of infinitely coupled equations, that can be ordered in a heirarchy, much as the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy of classical statistical mechanics. One must then set equations for products or correlation of increasing number of operators. In order to close the system of equations, a truncation of the hierarchy is necessary. We use the truncation scheme $\langle \hat{{\cal O}}_1 \hat{{\cal O}}_2 \rangle = \langle \hat{{\cal O}}_1 \rangle \langle \hat{{\cal O}}_2 \rangle$. It is important to note that the photon-exciton coupling is not affected by the truncation scheme; therefore, the formation of polaritons (strong-coupling regime) is accurately described. The system-bath coupling is properly introduced in the HEM by the formalism of Quantum Heisenberg-Langevin equations, which leads to additional terms in the equations: damping, Lamb shift, and stochastic force $\cal F$. A simpler way to deal with the environment is by introducing constants, taken from experiments or other theoretical works, directly in the HEM, taking into account the results of the detailed microscopic derivation. Here we follow the phenomenological procedure, by adding constants directly in the HEM [@Carusotto04; @Savasta96]. Photons are coupled to the radiation field at zero temperature outside of the cavity, resulting in the addition of a term $- \xi_q c_{\chi \mathbf q}$; the damping $\xi_q$ becomes very large when $q$ is such that the normal component of the field exceeds the critical angle separating low and high DBR reflectivity. For excitons we introduced a term $ - \beta_\alpha b_{\alpha \mathbf k}$, with a spin-dependent constant $\beta_\alpha$, accounting for radiative recombination and spin flip (no scattering is considered). For the impurity spin, a general constant $\gamma$ is used in all component, since no extenal magnetic field exists to distinguish among them. For long times, the spin relaxes, but does not vanish; thus, an equilibrium state is defined. We consider a circularly-polarized laser field that excites $\hat{b}_{2 0}^\dag$, and an impurity located at $\mathbf R = 0$. To simplify the calculations, we eliminate fast oscillations by moving to a rotating reference frame, with frequency $\bar \Omega$, setting $\langle \hat{{\cal O}} \rangle = e^{-i \bar \Omega t} \, \langle \hat{{\cal O}}' \rangle $, for ${\cal O} = c, b$. For the sake of simplicity, we hereafter denote the rotating frame version $ \langle \hat{{\cal O}}' \rangle$, simply as ${\cal O}$. The equations of motion read $$\begin{aligned} \label{Eq:HEM_Sz} \frac{ds_{z}}{dt} &=& \hspace{-2mm} -\gamma \bar s_{z} +\frac{\hbar}{A} \sum_{ \scriptsize \begin{array}{cc} \mathbf k \mathbf k' \\ \end{array} } j_{\mathbf k \mathbf k'}^{+} \left( s_{y}{\rho}_{1 \mathbf k, 2 \mathbf k'}^{x} +s_{x}{\rho}_{1 \mathbf k, 2 \mathbf k'}^{y} \right) \\ \label{Eq:HEM_Sx} \frac{ds_{x}}{dt} &=& \hspace{-2mm} -\gamma s_{x} -\frac{\hbar}{A} \sum_{ \scriptsize \begin{array}{cc} \mathbf k \mathbf k' \\ \end{array} } j_{\mathbf k \mathbf k'}^{+} \left( s_{y}{\rho}_{1 \mathbf k, 1 \mathbf k'}^{z} +s_{z}{\rho}_{1 \mathbf k, 2 \mathbf k'}^{y} \right) \\ \label{Eq:HEM_Sy} \frac{ds_{y}}{dt} &=& \hspace{-2mm} -\gamma s_{y} +\frac{\hbar}{A} \sum_{ \scriptsize \begin{array}{cc} \mathbf k \mathbf k' \\ \end{array} } j_{\mathbf k \mathbf k'}^{+} \left( s_{x}{\rho}_{1 \mathbf k, 1 \mathbf k'}^{z} -s_{z}{\rho}_{1 \mathbf k, 2 \mathbf k'}^{x} \right)\end{aligned}$$ where $ \bar s_{z} = s_{z}-s_{z \infty} $, ${\rho}_{n \mathbf k, n \mathbf k'}^{z} = ( {\rho}_{n \mathbf k, n \mathbf k'} - {\rho}_{n+1 \mathbf k, n+1 \mathbf k'})/2$, ${\rho}_{n \mathbf k, m \mathbf k'}^{x} = ({\rho}_{n \mathbf k, m \mathbf k'} + {\rho}_{m \mathbf k, n \mathbf k'})/2$ and ${\rho}_{n \mathbf k, m \mathbf k'}^{y} = i ( {\rho}_{n \mathbf k, m \mathbf k'} - {\rho}_{m \mathbf k, n \mathbf k'})/2$,[@quin-pierma05] with ${\rho}_{n \mathbf k, m \mathbf k'} = b_{n \mathbf k}^* b_{m \mathbf k'}$ and $j_{\mathbf k \mathbf k'}^{+} = j_{\mathbf k \mathbf k'} + j_{\mathbf k' \mathbf k}$. $$\begin{aligned} \label{Eq:b1q} {d {b}_{1 \mathbf q} \over dt} &=& - \left[ \beta_{1} + i \left( \frac{\varepsilon_{\mathbf q}}{\hbar} - \bar \Omega \right) \right] {b}_{1 \mathbf q} +\beta_{12} {b}_{2 \mathbf q} \\ && \hspace{-5mm} - \frac{i}{2 A}\sum_{\mathbf k} j_{\mathbf q \mathbf k}^{+} \left( s_- \, {b}_{2 \mathbf k} + s_z \, {b}_{1 \mathbf k} \right) \nonumber \\ \label{Eq:b2q} {d {b}_{2 \mathbf q} \over dt} &=& - \left[ \beta_{2} + i \left( \frac{\varepsilon_{\mathbf q}}{\hbar} - \bar \Omega \right) \right] {b}_{2 \mathbf q} + \beta_{12} {b}_{1 \mathbf q} \nonumber \\ && \hspace{-7mm} - \frac{i}{\hbar} \sum_\chi g_{2 \chi \mathbf q} \, {c}_{\chi \mathbf q} - \frac{i}{2 A} \sum_{\mathbf k } j_{\mathbf q \mathbf k}^{+} \left( s_+ {b}_{1 \mathbf k} - s_z {b}_{2 \mathbf k} \right) \vspace{3mm}\end{aligned}$$ where $s_{\pm} = s_x \pm i s_y$. Similar equations hold between ${b}_{2 \mathbf q} \leftrightarrow {b}_{3 \mathbf q}$ and ${b}_{1 \mathbf q} \leftrightarrow {b}_{4 \mathbf q}$. $$\begin{aligned} {d {c}_{\chi \mathbf q} \over dt} &=& - \left[ \xi_{q} + i ( \omega_{\mathbf q} - \bar \Omega ) \right] {c}_{\chi \mathbf q} - \frac{i}{\hbar} \sum_{\sigma } g_{\sigma \chi \mathbf q}^* \, {b}_{\sigma \mathbf q} \nonumber \\ && \hspace{-5mm} - i \sum_{\sigma \mathbf k} \sqrt{A} {\cal{V}}_{\sigma \mathbf k}^*(t) e^{- i (\Omega_k - \bar \Omega) t } \delta_{\sigma \chi} \delta_{\mathbf k\,\mathbf q} \,.\end{aligned}$$ Results ======= Numerical solution of the HEM is obtained using a 4th-order Runge-Kutta method in a 2D grid of $N \times N$ modes in momentum space. Basic units are $\{$meV, ps, nm$\}$, and we use data compatible with GaAs[@Berger]. The values of the different parameters are taken, in most cases, directly from experimental or theoretical work, only ${\cal{V}}_{0}$ and $j_{0} $ are adjusted using our calculations. When $g_{\alpha' \alpha \mathbf q}=0$, $b_{2 \mathbf q}(0) \neq 0$ and $s_z(0)=\hbar$, the system of equations becomes linear, and can be solved exactly. $j_0$ is then adjusted to yield a negative eigenvalue that matches the reported binding energy of excitons to donors (about $1$meV). The value so obtained for $\hbar^2 j_0/A \simeq 10^{-5}$ meV is in agreement with previous reports[@Chi05; @Quinteiro-01]. We fix the value of the coupling ${\cal{V}}_{0}$ by demanding that the total exciton density $n_X = \sum_{i \mathbf q} b^\dag_{i \mathbf q}b_{i \mathbf q}$ be low, i. e.: $r=n_X a_B^{*2}/A<1$, so that the X-X interaction can be neglected. We studied the evolution of spin components, exciton and photon populations when the system, represented by $N = 50$ modes (larger $N$s do not change the result significantly), is excited by a circularly polarized normal-incidence monochromatic laser-pulse ${\cal V}_{\sigma \mathbf k} = {\cal V}_0 \exp\{-(t-t_p)^2/w^2\}$. Cases with and without decoherence are considered. It is instructive to analyze first the (idealized) decoherence-free case —no plot presented. We find neither $b_{3\mathbf q}^*b_{3\mathbf q}$ nor $b_{4\mathbf q}^*b_{4\mathbf q}$ populations, while $s_z$ and $b_{1\mathbf q}^*b_{1\mathbf q}$ change little from their initial values. The small change in $s_z$ can be understood as follows: according to Eqs. (\[Eq:HEM\_Sz\]) $ds_z/dt \propto (\hbar j_0/A) b_{1\mathbf q}^* b_{2 \mathbf q}$ and to Eqs. (\[Eq:b1q\]) $d{b_{1 \mathbf q}}/dt \propto (\hbar j_0/A) b_{2 \mathbf q}$, that roughly yields $ds_z/dt \propto (\hbar j_0/A)^2 b_{2 \mathbf q}^* b_{2\mathbf q }$. This, compared to $ds_x/dt \propto (\hbar j_0/A) b_{2 \mathbf q}^* b_{2 \mathbf q}$, is a very small quantity given the choosen value of $\hbar j_0/A \simeq 10^{-5}$ ps$^{-1}$. On the contrary, the spin projection in the $xy$-plane can rotate several cycles depending on the temporal width and intensity of the pulse. As it is well known from quantum optics, once the laser is turned off, there is a remaining oscillating population of excitons and photons. For certain values of the pulse parameters, these populations are so small ($r \rightarrow 0$) that cannot produce important changes in the spin. When decoherence is included, there is conversion to dark states $b_{4\mathbf q}$, due to hole spin-flip, and to a lesser extent due to electron spin-flip. Because of the long life-time of these dark states, the fraction $r$ remains finite (though very small compared to its peak value). Fig. \[fig:S\_osc\] presents the results for a simulation with parameters $\{ \Omega_ 0 =2270.$ps$^{-1}, \bar \Omega=2301.2$ps$^{-1}, \varepsilon_0/\hbar=2301.2$ps$^{-1}, \xi_0=6.6\, 10^{-2}$ps$^{-1}, \beta_1=\beta_4=0, \beta_2=\beta_3=10^{-2}$ps$^{-1}, \beta_{12}= \beta_{34}= 3\, 10^{-2}$ps$^{-1}, \beta_{13}= \beta_{24}= 1$ps$^{-1} \}$. We find that if ${\cal V}_0 < 32$ps$^{-1}$nm$^{-1}$ then $r<1$ and the neglect of the X-X interaction is justified. Under this condition, we see that a single inversion $s_x \rightarrow -s_x$ can be realized in few picoseconds. Faster spin motion is observed when the laser intensity (and so the photon/exciton populations) increases. As predicted in the previous paragraph, during the whole evolution, the change in $s_z$ is very small, as seen in the lower panel of Fig. \[fig:S\_osc\]. In addition the population of dark excitons is also very small compared to that of bright excitons: with the definition $r_i = b_{i\mathbf q}^*b_{i\mathbf q} a_B^{*2}/A$, we obtain at $t=10$ps $\{r_1 \simeq 0.3, r_2 \simeq 1.5 \times 10^{-6}\}$ and at $t=20$ps $\{r_1 \simeq 6 \times 10^{-6}, r_2 \simeq 10^{-6}\}$. ![(color online) Evolution under the excitation by a laser pulse of width $w \simeq 4.5$ps and ${\cal V}_0=25$ps$^{-1}$nm$^{-1}$; the initial state is a spin having mean values $\{s_x=\sqrt{3}\hbar/4, s_y=0,s_z=-\hbar/4\}$. Upper Panel: Spin components $s_x$ (solid blue) and $s_y$ (red). Lower Panel: Spin component $s_z$.Inset: fraction $r=n_X a_B^{*2}/A$. []{data-label="fig:S_osc"}](fig_S_osc_tot.eps) Spin rotation for strong and weak coupling ------------------------------------------ The addition of decoherence allows us to address the regimes of strong and weak coupling, and in particular to study the effect that cavity losses have in the spin control. Weak coupling is characterized by $|\xi_0-\beta_2|>2g/\hbar$ (in our case $2g/\hbar \simeq 2.2$ ps$^{-1}$), and this regime can be simulated by increasing the photon losses of all modes (increasing $\xi_0$), which amounts to considering different cavities with varying quality factor Q. Two notes of caution: First, we have treated the laser-photon coupling in the quasimode approximation, valid for high-quality-factor cavities. Therefore, we will refrain from studying cases with large values of $\xi_0$. Second, the laser-photon coupling ${\cal V}_0$ is, in general, affected by changes in the photon losses $\xi_0$; however, we can envisage situations where one can increase $\xi_0$ without affecting ${\cal V}_0$, for example –but not exclusively– by reducing only the reflectivity of the left DBR in Fig. \[fig:system\]. Fig \[fig:S\_PhEscape\] shows the effect that the increase in photon leakage, at fixed laser field intensity, has on the rotation of the impurity spin. For simplicity other sources of decoherence are disregarded. For all simulations, we used one set of laser parameters {$w\simeq 4.5$ps, ${\cal V}_0=15$ps$^{-1}$nm$^{-1}$} for a gaussian pulse which produces, [*without*]{} photon loss, a rotation from the initial state $s_x=\sqrt{3}\hbar/4$ to the final state $s_x=-\sqrt{3}\hbar/4$ at $t=15$ps, i. e. a change in the angle $\Delta \theta = \pi$. Next we simulated situations of increasing $\xi_0$ and plotted $\Delta \theta (\xi_0)$. In addition, we plotted the maximum photon population acchieved during the pulse. We observe that for $\xi_0<2$ ps$^{-1}$ ($Q > 1500$) there is almost full rotation of $s_x$, and that for lower quality-factor cavities (high $\xi_0$) the spin changes little. The population of cavity photons and excitons follow this tendency. ![(color online) Degree of spin in-plane rotation $\Delta\theta$ (dashed blue) as a function of the cavity photon loss, and maximum cavity-photon population (solid red). Inset: Zoom-in of rotation angle $\Delta\theta$ for low $\xi_0$.[]{data-label="fig:S_PhEscape"}](fig_S_PhEscape_v5.eps) The effect of decoherence can be characterized with the fidelity $F$. If the final state we wish to obtain is the [*pure*]{} spin state $-1/2{\ensuremath{\,|{\uparrow}\rangle}}+\sqrt{3}/2{\ensuremath{\,|{\downarrow}\rangle}}$, having mean values $\{s_x = -\sqrt{3}\hbar/4, s_y = 0, s_z = \hbar/4\}$, the formula for the fidelty reduces to $F=(-1/2\hbar) (\sqrt{3}s_x-s_z-\hbar)$, see Jozsa[@Jozsa]. For the cases $\xi_0=3.5$ps$^{-1}$ and $\xi_0=20.5$ps$^{-1}$ the resulting fidelity is $F=0.9965$ and $F=0.607$, respectively. We interpret the enhanced rotation in high-Q cavities in the following way. For high Q, as seen in Fig.  \[fig:S\_PhEscape\], the photon density is larger, and a repetitive and longer interaction with excitons is possible. This leads concomitantly to the formation of polaritons, with the excitonic component causing impurity spin rotations. In contrast, for lower values of Q, photons tend to leave the cavity faster, and there is small conversion to excitons. As was mentioned before, it is perhaps easier to envisage a cavity, whose Q factor is lowered by degrading the left DBR in Fig. \[fig:system\]. Then, a naive picture tells us that the laser field produces photons inside the cavity at the same rate in the high and moderate Q cases. In the latter, photons are more prompt to leak out and produce less excitons. In addition, we can ask what the pulse width $w$ should be to ensure a full rotation of $s_x$, for different values of cavity loss (see Fig. \[fig:S\_width\]). As expected from the previous analysis, we see that one requires longer pulses to produce the rotation, but in contrast to what happened before, the fidelity is almost unchanged. We attribute the behavior of $F$ to the fact that the only source of decoherence is photon loss in these simulations and that the final state is forced (by changing $w$) to be the closest possible to the ideal state. ![(color online) Pulse width $w$ (solid blue) required to produce full rotation of the $s_x$ spin component, and corresponding fidelity $F$ (dashed red) as a function of cavity loss $\xi_0$. []{data-label="fig:S_width"}](fig_S_width_v2.eps) Finally, it is worth mentioning that we have used a conservative value for $j_0$. For example, Puri [*et al*]{}[@Puri12] reports a much higher value of $j_0$ for QDs replacing impurities. This would lead to even faster spin control, together with more efficient control on $s_z$. However, fs laser pulses have a broad frequency spectrum, and may excite several polariton modes. This may lead to destructive interference effects, which may reduce the effectiveness of a large $j_0$. Conclusions =========== We studied the optical control of single spins in micro-cavities accounting for all sources of decoherence. When the system is in the strong-coupling regime, the spin manipulation is most efficient and can be done by a few-picoseconds laser pulse. This suggests that single spins embedded in high Q-factor planar cavities can act as quantum memories and as qubits, with the optical excitation being the mechanism to control the state of the memory or to perform one-qubit operations for quantum computing. This optical control produces high-fidelity final states in very short time: a single operation can be performed $10^6$ faster than the typical decoherence time of the impurity spin qubit (compared to other proposals using for example ion traps[@Kirch09]) with a fidelity of $F>99.8\%$. We believe that the present proposal for one-qubit operations, together with a previous one for implementing two-qubit operations in the same system[@Quinteiro-01; @Puri12] show that polaritons in 2D-microcavities is a promising system for the implementation of solid-state quantum-computing scalable architectures. acknowledgments =============== G. F. Quinteiro thanks financial assistance by CONICET and ANPCyT (PICT2007-873), Argentina, and the Fulbright Commission, USA. A. A. Aligia thanks financial assistance by CONICET (PIP No. 11220080101821) and ANPCyT (PICT R1776). C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, Phys. Rev. Lett. [*69*]{}, 3314 (1992). F. Tassone, C. Piermarocchi, V. Savona, and A. Quattropani, P. Schwendimann Phys. Rev. B [**56**]{} 7554 (1997). L. Ferrier, E. Wertz, R. Johne, D. D. Solnyshkov, P. Senellart, I. Sagnes, A. Lemaitre, G. Malpuech, J. Bloch, Phys. Rev. Lett. [**106**]{}, 126401 (2011) Iacopo Carusotto, and Cristiano Ciuti, Phys. Rev. B [**72**]{} 125335 (2005). Iacopo Carusotto, Michiel Wouters, Cristiano Ciuti J Low Temp Phys (2007) [*148*]{}: 459–464. C. Ciuti, P. Schwendimann, B. Deveaud, and A. Quattropani, Phys. Rev. B [**62**]{}, R4825 (2000). Markus Muller and Joel Bleuse, Regis Andre Phys. Rev. B [**62**]{}, 16 886 (2000). S. Portolan, O. Di Stefano, S. Savasta, F. Rossi, and R. Girlanda Phys. Rev. B [**77**]{}, 035433 (2008). S. Savasta, G. Martino, R. Girlanda, Solid State Comm. [*111*]{}, 495 (1999) V. Savona, Z. Hradil, A. Quattropani, and P. Schwendimann, *Phys. Rev. B* [**49**]{}, 8774 (1994). Iacopo Carusotto and Cristiano Ciuti, Phys. Rev. Lett. [*93*]{}, 166401 (2004). V. Savona, C. Piermarocchi, and A. Quattropani, F. Tassone, P. Schwendimann Phys. Rev. Lett. [**78**]{}, 4470 (1997). T. Steiner, M. L. W. Thewalt, E. S. Koteles and J. P. Salerno, *Phys. Rev. B* **34**, 1006 (1986). B. Lavigne and R. T. Cox, *Phys. Rev. B* **43**, 12374 (1991). G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode-coupling semiconductor microcavities”, Rev. Mod. Phys. [*71*]{}(5), 1591-1639 (1999). Y. Yamamoto, F. Tassone, H. Cao, *Semiconductor Cavity Quantum Electrodynamics*, Springer Tracks in Modern Physics [**169**]{}, 2000. G. Chiappe, J. Fernandez-Rossier, E. Louis, and E. V. Anda, Phys. Rev. B [**72**]{}, 245311 (2005) G. F. Quinteiro, Phys. Rev. B [**77**]{}, 075301 (2008) J. A. Andrade, A. A. Aligia and G. F. Quinteiro, J. Phys.: Condens. Matter [**23**]{}, 215304 (2011). Miro Kroutvar, Yann Ducommun, Dominik Heiss, Max Bichler, Dieter Schuh, Gerhard Abstreiter & Jonathan J. Finley, Nature [**432**]{}, 81-84 (2004) G. F. Quinteiro, J. Fernandez-Rossier, and C. Piermarocchi, *Phys. Rev. Lett.* **97**, 097401 (2006). Shruti Puri, Na Young Kim, Yoshihisa Yamamoto, Phys. Rev. B [**85**]{}, 241403(R) (2012). P. Ramvall, N. Carlsson, P. Omling, L. Samuelson, W. Seifert, and Q. Wang Appl. Phys. Lett. [**70**]{} 243 (1997). I. -K. Oh, Jai Singh, Journal of Luminescence **85** (2000) 233-246. E. Tsitsishvili, R. v. Baltz, H. Kalt Phys. Rev B [**72**]{}, 155333 (2005). Igor Zutic, Jaroslav Fabian, S. Das Sarma Rev. Mod. Phys. [**76**]{} 323 (2004). A. Murayama, K. Seo, K. Nishibayashi, I. Souma, and Y. Oka, Applied physics letters, **88**, 261105 (2006). S. L. Lu, L. F. Bian, M. Uesugi, H. Nosho, A. Tackeuchi, and Z. C. Niu, Applied physics letters, **92**, 051908 (2008). C. W. Gardiner and M. J. Collett, *Phys. Rev. A* [**31**]{}, 3761 (1985). B. J. Dalton, Stephen M. Barnett and P. L. Knight, *Journal Modern Physics* [**46**]{}, 1315-1341 (1999). Hiroshi Ajiki, *J. Opt. B: Quantum Semiclass. Opt.* [**7**]{} (2005) 29-34. S. Akasaka, et al, Applied Physics Letters [**85**]{} 2083 (2004). Rui Shen, Hirofumi Mino, Tasuya Kimukawa, Shojiro Takeyama, Grzegrz Karczewski, Tomasz Wojtowicz, Jacek Kossut, Phisica E **22** (2004) 611-614. Vincenzo Savona, J. Phys.: Condens. matter **19** (2007) 295208. R. Cerna, D. Sarchi, T. K. Paraiso, G. Nardin, Y. Leger, M. Richard, B. Pietka, O. El Daif, F. Morier-Genoud, V. Savona, M. T. Portella-Oberli, B. Deveaud-Pledran, ’Optical manipulation of the wave function of quasiparticles in a solid’ (no reference available). Iacopo Carusotto and Cristiano Ciuti, arxiv:cond-mat/0504554v1 (2005). I. A. Shelykh, Y. G. Rubo, G. Malpuech, D. D. Solnyshkov, and A. Kavokin, *Phys. Rev. Lett.* **97** 066402 (2006). N. A. Gippius, I. A. Shelykh, D. D. Solnyshkov, S. S. Gavrilov, Yuri. G. Rubo, A. V. Kavokin, S. G. Tikhodeev, and G. Malpuech, *Phys. Rev. Lett.* **98** 236401 (2007). T. C. H. Liew and I. A. Shelykh, *Phys. Rev. B* **80** 161303(R) (2009). Kai-Mei C. Fu, C. Santori, C. Stanley, M. C. Holland, and Y. Yamamoto, Phys. Rev. Lett. **95** 187405 (2005). When quantum computing is in mind, two-qubit operations are performed by illuminating simultaneously two impurities. I. I. Boiko, Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. **13**, N 2. P. 214-220. R. Cerna, D. Sarchi, T. K. Paraïso, G. Nardin, Y. Leger, M. Richard, B. Pietka, O. El Daif, F. Morier-Genoud, V. Savona, M. T. Portella-Oberli, B. Deveaud-Plédran, *Phys. Rev. B* **80** 121309(R) (2009). D. Sarchi. and V. Savona, *Phys. Rev. B* **77**, 045304 (2008). D. Sarchi. and V. Savona, arXiv:0706.3719v1 \[cond-mat.mes-hall\]. S. Savasta and R. Girlanda, Phys. Rev. Lett. **77**, 4736 (1996). G. F. Quinteiro, C. Piermarocchi, Phys. Rev. B [**72**]{}, 045334 (2005) Lev. I. Berger, [*Semiconductor Materials*]{} (CRC Press, Inc. 1997). Richard Jozsa, Journal of Modern Optics **41**, 2315-2323 (1994). G. Kirchmair, J. Benhelm, F. Zahringer, R. Gerritsma, C. F. Roos, and R. Blatt, Phys. Rev. A **79** 020304 (2009). J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, Nature Physics **4**, 463 (2008)

--- abstract: 'As a first step toward a characterization of the limiting extremal process of branching Brownian motion, we proved in a recent work \[*Comm. Pure Appl. Math.* **64** (2011) 1647–1676\] that, in the limit of large time $t$, extremal particles descend with overwhelming probability from ancestors having split either within a distance of order 1 from time $0$, or within a distance of order 1 from time $t$. The result suggests that the extremal process of branching Brownian motion is a randomly shifted cluster point process. Here we put part of this picture on rigorous ground: we prove that the point process obtained by retaining only those extremal particles which are also maximal inside the clusters converges in the limit of large $t$ to a random shift of a Poisson point process with exponential density. The last section discusses the *Tidal Wave Conjecture* by Lalley and Sellke \[*Ann. Probab.* **15** (1987) 1052–1061\] on the full limiting extremal process and its relation to the work of Chauvin and Rouault \[*Math. Nachr.* **149** (1990) 41–59\] on branching Brownian motion with atypical displacement.' address: - | L.-P. Arguin\ Département de Mathématiques\ et Statistique\ Université de Montréal\ C.P. 6128, succ. Centre-ville\ Montréal, Québec\ Canada H3C 3J7\ - | A. Bovier\ N. Kistler\ Institut für Angewandte Mathematik\ Rheinische Friedrich-Wilhelms-Universität Bonn\ Endenicher Allee 60\ 53115 Bonn\ Germany\ \ author: - - - title: Poissonian statistics in the extremal process of branching Brownian motion --- , . . Introduction {#sec1} ============ Branching Brownian motion (BBM) is a continuous-time Markov branching process which plays an important role in the theory of partial differential equations [@aronsonweinberger; @aronsonweinbergertwo; @mckean], in particle physics [@munierpeschanski] and in the theory of disordered systems [@BovierKurkovaII; @derridaspohn]. It is also widely used in biology to model the genealogies of evolving populations, the spread of advantageous genes, etc., [@fisher; @kessleretal]. It is constructed as follows. Start with a standard Brownian motion (BM) (we will often refer to Brownian motions as “particles”), $x(t)$, starting at $0$. After an exponential random time, $T$, of mean 1, the BM splits into $k$ independent BMs, independent of $x$ and $T$, with probability $p_k$, where $\sum_{k=1}^\infty p_k = 1$, $\sum_{k=1}^\infty k p_k = 2$ and $K {\equiv}\sum_{k} k(k-1) p_k < \infty$. Each of these processes continues in the same way as first BM. Thus, after time $t>0$, there will be $n(t)$ BMs located at $x_1(t), \ldots, x_{n(t)}(t)$, with $n(t)$ being the random number of offspring generated up to that time \[note that ${{\mathbb{E}}}n(t)=e^t$\]. An interesting link between BBM and partial differential equations was observed by McKean [@mckean]: denote by $$\label{bbmrepr} u(t, x) {\equiv}{\mathbb{P}}\Bigl[ \max_{1\leq k \leq n(t)} x_k(t) \leq x \Bigr]$$ the law of the maximal displacement. Then, a renewal argument shows that $u(t,x)$ solves the Kolmogorov–Petrovsky–Piscounov or Fisher \[F-KPP\] equation, $$\begin{aligned} \label{kppequation} u_t &=& {1\over2} u_{xx} + \sum_{k=1}^\infty p_k u^k -u, \nonumber \\[-8pt] \\[-8pt] \nonumber u(0, x)&=& \cases{ 1, &\quad$\mbox{if } x\geq0,$\vspace*{2pt}\cr 0, &\quad$\mbox{if } x < 0.$} $$ The F-KPP equation admits traveling waves: there exists a unique solution satisfying $$\label{travellingone} u\bigl(t, m(t)+ x \bigr) \to\omega(x)\qquad \mbox{uniformly in } x \mbox{ as } t\to\infty,$$ with the centering term, the *front* of the wave, given by $$\label{centeringkpp} m(t) = \sqrt{2} t - {3 \over2 \sqrt{2}} \log t, $$ and ${\omega}(x)$ the unique (up to translation) distribution function which solves the ordinary differential equation $$\label{wavepde} \tfrac{1}{2} \omega_{xx} + \sqrt{2} \omega_x + \omega^2 - \omega= 0.$$ The leading order of the front has been established by Kolmogorov, Petrovsky and Piscounov [@kpp], whereas the logarithmic corrections have been obtained by Bramson [@bramson], using the probabilistic representation given above. The limiting law of the maximal displacement has been studied intensely. Let $$\label{eqnderiv} Z(t) {\equiv}\sum_{k=1}^{n(t)} \bigl( \sqrt{2}t -x_k(t) \bigr) \exp-\sqrt {2}\bigl( \sqrt{2}t -x_k(t) \bigr)$$ denote the so-called *derivative martingale*. Lalley and Sellke [@lalleysellke] proved that $Z(t)$ converges almost surely to a strictly positive random variable, $Z$, and established the integral representation $$\label{gumbellike} \omega(x) = {{\mathbb{E}}}\bigl[ {\mathrm{e}}^{- C Z {\mathrm{e}}^{-\sqrt{2}x} }\bigr],$$ with $C>0$ a constant. Thus the law of the maximum of BBM is a *random shift* of the Gumbel distribution. Moreover, it is known that $$\label{totheright} 1-\omega(x) \sim x {\mathrm{e}}^{-\sqrt{2} x},\qquad x\to+\infty,$$ where $\sim$ means that the ratio of the terms converges to a positive constant; see, for example, Bramson [@bramson] and Harris [@harris]. (There is emerging evidence that right-tails such as [(\[totheright\])]{}, manifestly different from those of the Gumbel, play an important role in a number of different fields, e.g., in models on spin glasses with logarithmic correlated potentials by Carpentier and Le Doussal [@carpentierledoussal], and Fyodorov and Bouchaud [@bouchaudfyodorov].) Contrary to the maximal displacement, very little is known on the full statistics of the extremal configurations (first-, second-, third-, etc., largest) in BBM. Such statistics are completely encoded in the extremal process, which is the random point measure associated to the collection of points shifted by the expectation of their maximum, that is, the point process $$\label{extremalprocess} \Xi(t) {\equiv}\sum_{i=1}^{n(t)} {\delta}_{\overline{x_i}(t)} ,\qquad \overline{x_i}(t) {\equiv}x_i(t)-m(t).$$ The key issue of interest is to characterize the limit of this process, as $t\uparrow\infty$. It can be shown that the limit of the point process exists using Bramson’s analysis [@bramsonmonograph] on the convergence of solutions of the KPP equations with appropriate initial conditions [@brunet; @kabluchko]. For given realization of the branching, the positions $\{x_i(t)\}_{i\leq n(t)}$ form a Gaussian process indexed by $i\in\{1, \ldots, n(t)\} {\equiv}\Sigma_t$ with correlations given by the *genealogical distance* $$Q_{ij}(t) {\equiv}\sup\{s\leq t\dvtx x_i(s)=x_j(s) \} \label{genealogical}$$ (the time to first branching of the common ancestor). The information about the correlation structure of any subsets of particles in BBM is encoded in their genealogical distance. This applies, in particular, to the subset of extremal particles, for which the following result was proved in [@abk]: with probability tending to 1, branching can happen only at “very early times,” smaller than $r_d$ with $r_d = O(1)$ as $t\to\infty$, or at times “very close” to the age of the system, namely greater than $t-r_g$ for $r_g = O(1)$ as $t \to\infty$. (The reason for this notation, in particular the use of the subscripts, will be explained below.) More precisely, denoting by $\Sigma_t(D) {\equiv}\{ i\in\Sigma_t\dvtx \overline{x_i}(t) \in D \}$ the set of particles in the subset $m(t)+D$, we have: \[genealogyabktheorem\] For any compact $D\subset{\mathbb{R}}$, $$\label{genealogyabkeqn} \lim_{r_d, r_g \to\infty} \sup_{t > 3 \max\{r_d, r_g \}} {\mathbb{P}}[ \exists i,j \in\Sigma_t(D)\dvtx Q_{ij}(t) \in(r_d, t-r_g) ] = 0.$$ Figure \[fig1\] presents a graphical representation of the genealogies of extremal particles of BBM. ![Genealogies of extremal particles.[]{data-label="fig1"}](809f01.eps) Theorem \[genealogyabktheorem\] gives insight into the limiting extremal process of BBM. In fact, it suggests the following picture, which holds with overwhelming probability in the limit when first $t\uparrow\infty$, and $r_d,r_g\rightarrow\infty$ after that. First, ancestries in the interval $[0, r_d]$ cannot be ruled out: this regime generates the derivative martingale appearing in the work of Lalley and Sellke [@lalleysellke]. Moreover, since the ancestors of the extremal particles evolved independently for most of the time (namely in the interval $[r_d,t-r_g]$), the extremal process must exhibit a structure similar to a Poisson process. Finally, since ancestors over the period $[t-r_g, t]$ also occur, it is natural to conjecture that *small grapes* of length at most $r_g = O(1)$, that is, clusters of particles with very recent common ancestor, appear at the end of the time-interval. (According to this picture, the subscript in $r_d$ refers to *derivative martingale*, while that in $r_g$ stands for *grape*.) It is the purpose of this work to make part of this picture rigorous. In Section \[mainresults\] we present our main result, which is proved in Section \[proofs\]. In Section \[secconvection\], we introduce a cluster point process, which we conjecture to correspond in the limit to the extremal process of BBM. We also discuss the cluster point process in relation to the work of Chauvin and Rouault [@chauvinrouault] on BBM conditioned to perform unusually large displacements, and in relation to the *Tidal Wave Conjecture* of Lalley and Sellke [@lalleysellke]. Detailed properties of this cluster point process will be the subject of a subsequent paper [@abkthree]. Main results {#mainresults} ============ Despite the rather clear image described above, a frontal attack on the extremal process appears to be difficult. This is in particular due to the fact that one has to take into account the self-similarity of BBM which is first and foremost detectable in the small clusters, an issue which remains rather elusive (see Section \[secconvection\] for more on this). On the other hand, the picture naturally suggests the existence of an underlying point process obtained from the extremal particles by a thinning procedure, which we describe next. Assume that the positions of particles at time $t$ are ordered in decreasing order: $$\overline{x}_1(t)\geq\overline{x}_2(t)\geq\cdots\geq\overline {x}_{n(t)}(t).$$ The inequalities will in fact be strict for almost all realizations of BBM for any deterministic time $t$. Define also $$\overline{Q}(t) =\{\overline{Q}_{ij}(t)\}_{i,j\leq n(t)}{\equiv}\{ t^{-1}Q_{ij}(t)\}_{i,j\leq n(t)} .$$ The pair $(\Xi(t), \overline{Q}(t))$ admits the following natural *thinning*. Since the matrix $\overline{Q}(t)$ is constructed from the branching of the BBM, the relation $\overline{Q}_{ij}(t)\geq q$ is transitive for any $q\geq0$: $$\label{eqnultra} \overline{Q}_{ij}(t)\geq q \quad\mbox{and} \quad \overline{Q}_{jk}(t)\geq q \quad\Longrightarrow\quad\overline{Q}_{ik}(t)\geq q.$$ In particular, for any $q>0$, this relation defines an equivalence relation on the set $\{1,\ldots,n(t)\}$. The corresponding equivalence classes are just the particles at time $t$ that had a common ancestor at a time later than $tq$. We want to select a representative of each class, namely the maximal particle within each class, and then consider the point process of these representatives. For any $q>0$, the of the process $(\Xi(t), \overline {Q}(t))$, denoted by $\Xi^{(q)}(t)$, is defined recursively as follows: $$\begin{aligned} \label{eqnthin} i_1&=&1; \nonumber \\[-8pt] \\[-8pt] \nonumber i_k&=&\min\{ j>i_{k-1}\dvtx \overline{Q}_{i_lj}(t)<q, \forall l\leq k-1\};\end{aligned}$$ and $$\Xi^{(q)}(t) {\equiv}\bigl(\Xi^{(q)}_{k}(t), k\in{\mathbb{N}}\bigr) {\equiv}\bigl(\overline {x}_{i_k}(t), k\in{\mathbb{N}}\bigr),$$ where it is understood that $\Xi^{(q)}_{k}(t)=0$ when an index $i_k$ in $\{1,\ldots,n(t)\}$ satisfying $\min\{ j>i_{k-1}\dvtx \overline{Q}_{i_lj}<q \ \forall l\leq k-1\}$ can no longer be found. The procedure selects the maximal position in each equivalence class defined from the relation . In addition, it is easily checked that the thinning map, $$\label{eqnthinmap} (\Xi(t),\overline{Q}(t))\mapsto\Xi^{(q)}(t),$$ considered at the level of realizations, is a continuous function on the space of pairs $(X,Q)$, where $X$ is a sequence of ordered positions and $Q$ is a symmetric matrix with entries in $[0,1]$, satisfying [(\[eqnultra\])]{} (when this space is equipped with the product topology in each coordinate of $X$ and $Q$). The thinning map can also be applied to $t$-dependent values of $q$. For example, take $q=q(t)=1-r_g/t$, where $r_g$ is fixed $t$. In this case, the thinning effectively retains those particles which are extremal within the class defined by a “very recent” common ancestor, which we refer to as *cluster-extrema*. Figure \[fig2\] presents a graphical representation of the set of such particles. ![Cluster-extrema.[]{data-label="fig2"}](809f02.eps) Our main result states that all such thinned processes converge to the same randomly shifted Poisson Point Process (PPP for short) with exponential density. \[convergencethinned\] For any $0<q<1$, the processes $\Xi^{(q)}(t)$ converge in law to the same limit, $\Xi ^0$. Also, $$\lim_{r_g \to\infty} \lim_{t\to\infty} \Xi^{(1-r_g/t)}(t)=\Xi^0.$$ Moreover, conditionally on $Z$, the limit of the derivative martingale [(\[eqnderiv\])]{}, $$\Xi^0=\operatorname{PPP}\bigl( C\cdot Z \cdot\sqrt{2}{\mathrm{e}}^{-\sqrt{2} x} \,{\mathrm{d}}x\bigr) ,$$ where $C>0$ is the constant appearing in [(\[totheright\])]{}. The point process $\Xi^{0}$ has a fundamental connection with the limiting extremal process of BBM. To see this, suppose for simplicity that the processes $(\Xi(t), \overline{Q}(t))$ induced by the law of BBM converge, as $t\uparrow\infty$, to a process, $(\Xi, \overline{Q})$. (The laws of these processes are in fact tight because the law of $\Xi (t)$ is itself tight; see, e.g., Corollary 2.3 in [@abk], and that $\overline{Q}_{ij}(t)\in[0,1]$ for any $i,j$. Convergence would evidently follow from a complete characterization of the extremal process.) It follows from Theorem \[genealogyabktheorem\] that $\overline{Q}_{ij}$ is either $0$ or $1$. This suggests: (1) to define a cluster of particles as the maximal set of particles such that $\overline{Q}_{ij}=1$ for all $i,j$ in the set; (2) to look at the process of the maxima of each cluster, denoted by, say, $\tilde{\Xi}^0$, defined as in [(\[eqnthin\])]{}, but where $i_k=\min \{ j>i_{k-1}\dvtx \overline{Q}_{i_lj}=0\ \forall l\leq k-1\}$. We claim that $\tilde{\Xi}^0$ is in fact the limit $\Xi^{0}$ of $\Xi ^{(q)}(t)$ in Theorem \[convergencethinned\]. Indeed, in view of the continuity of the thinning map [(\[eqnthinmap\])]{}, $\Xi^{(q)}(t)$ converges to the process, $\Xi^{(q)}$, constructed from $(\Xi, \overline{Q})$ for all $q$. But, for any $0<q<1$, the processes, $\Xi^{(q)}$, constructed from $(\Xi, \overline{Q})$ using [(\[eqnthin\])]{} are equal trivially to $\tilde{\Xi}^0$, since $\overline{Q}_{ij}$ is either $0$ or $1$. The claim then follows from Theorem \[convergencethinned\]. The point process describing the particles at the frontier of BBM in the limit of large times is thus formed by two “types” of particles: those coming from the randomly shifted PPP with exponential density, the cluster-extrema; and the second type of particles, those forming the clusters. Clearly, particles in the same cluster always lie on the left of the corresponding Poissonian particles, by the very definition of the cluster-extrema. It remains an open question to characterize the law of the clusters (see Section \[secconvection\] for some conjectures). We remark that, since $r_g = O(1)$ as $t\to\infty$, the thinned process $\Xi^{(1-r_g/t)}_t$ is obtained from the extremal one by removing only a small number of particles, those which have genealogical distance smaller than $t-r_g$ from the maximum in their class. It is rather surprising at first sight (but not quite when seen under the light of Theorem \[genealogyabktheorem\]) that such a point process converges, despite the high correlations among the branching Brownian particles, to a PPP with exponential density. Theorem \[convergencethinned\] also provides insights into a result by Bovier and Kurkova [@BovierKurkovaII], who addressed the weak limit of the Gibbs measure of BBM, the random probability measure on $\Sigma_t$ attaching weights $$\mathcal G_{\beta, t}(k) {\equiv}\frac{\exp(\beta x_k(t))}{\mathcal {Z}_t(\beta)},\qquad \mathcal{Z}_t(\beta) {\equiv}\sum_{j\in\Sigma_t} \exp(\beta x_j(t)),$$ where $\beta>0$ is the inverse of temperature. To see this, let us first recall the following. Consider the random set $(\xi_i, i \in{\mathbb{N}})$ where the $\xi$’s are generated according to a PPP with density $C Z \sqrt{2}{\mathrm{e}}^{-\sqrt{2}x} \,{\mathrm{d}}x$ on the real axis, $C$ and $Z$ as in Theorem \[convergencethinned\]. Construct then a new random set $(\rho_i, i\in {\mathbb{N}})$ where $\rho_i {\equiv}\exp({\beta}\xi_i)$. For $\beta> \sqrt{2}$, it is easily seen that $\mathcal N(\rho) {\equiv}\sum_j \rho_j < \infty$ almost surely, in which case the normalization $\hat{\rho}_i {\equiv}\rho _i/\mathcal N(\rho)$ is well defined, and the law of the normalized collection $(\hat{\rho}_i, i \in{\mathbb{N}})$ is the Poisson–Dirichlet distribution with parameter $m(\beta) = \sqrt{2}/\beta$, which we shall denote by $\operatorname{PD}(m(\beta))$. In a somewhat indirect way (by means of the so-called Ghirlanda–Guerra identities, which avoid the need to first identify the limiting extremal process) Bovier and Kurkova proved that, in the low temperature regime $\beta> \sqrt{2}$, the Gibbs measure $\mathcal G_{\beta, t}$ converges, in the limit of large times, to the $\operatorname{PD}(m(\beta ))$; together with our Theorem \[convergencethinned\], this naturally suggests that the Gibbs measure of BBM is concentrated, in fact, on the *cluster-extrema*. Finally, Theorem \[convergencethinned\] sheds light on a property of the extremal process of BBM that was conjectured by Brunet and Derrida [@derridabrunet]. They suggested that the statistics of the leading particles are invariant under superposition in the sense that the extremal process of two independent branching Brownian motions has the same law, up to a random shift, as the extremal process of a single one. This property at the level of the entire process is likely to involve specific features of the laws of the individual clusters. On the other hand, at the level of the thinned process, it is a straightforward consequence of Theorem \[convergencethinned\], since the law is Poisson with exponential density. \[corsuperposition\] Let $\Xi(t)$ and $\Xi'(t)$ be the extremal processes [(\[extremalprocess\])]{} of two independent branching Brownian motions. Denote by $Z$ and $Z'$ the pointwise limit of their respective derivative martingale. Then, for any $0<q<1$, the law of the $q$-thinning of $\Xi(t)+\Xi'(t)$ conditionally on $Z$ and $Z'$ converges to $$\operatorname{PPP}\bigl( C\cdot(Z+Z')\cdot\sqrt{2}{\mathrm{e}}^{-\sqrt{2} x} \,{\mathrm{d}}x\bigr) .$$ In particular, the thinned process of $\Xi(t)+\Xi'(t)$ has the same law in the limit as the thinned process $\Xi^0$ of a single branching Brownian motion, up to a random shift. As mentioned before, Theorem \[convergencethinned\] is a natural consequence of Theorem \[genealogyabktheorem\]. The main ingredient is the following lemma, which allows to compare thinning processes on a set of large probability. We use the notation $$\Xi^{(q)}(t)|_{(y,\infty)}{\equiv}\bigl\{\Xi^{(q)}_i(t)\dvtx \Xi^{(q)}_i(t)>y\bigr\} .$$ \[comparison\] For any $y\in{\mathbb{R}}$ and any ${\varepsilon}>0$, there exists $r_0=r_0(y,{\varepsilon})$ such that for $r_d,r_g>r_0$ and $t>3\max\{r_g,r_d\}$, on a set of probability $1-{\varepsilon}$, $$\Xi^{(q)}(t)|_{(y,\infty)}= \Xi^{(r_d/t)}_t|_{(y,\infty)},$$ for any $\frac{r_d}{t}<q<1-\frac{r_g}{t}$. Theorem \[convergencethinned\] is then proved by a standard Poisson convergence argument which exploits the weak correlations between the cluster-extrema in classes of the $\frac{r_d}{t}$-thinning. \[simplerpp\] With $C>0$ and $Z$ the limiting derivative martingale, conditionally on $Z$, $$\lim_{r_d \to\infty} \lim_{t\to\infty} \Xi^{(r_d/t)}(t) =\operatorname{PPP}\bigl( C Z \sqrt{2}{\mathrm{e}}^{-\sqrt{2} x} \,{\mathrm{d}}x\bigr) .\vspace*{-3pt}$$ Proofs ====== [Proof of Lemma \[comparison\]]{} Theorem \[genealogyabktheorem\] describes the genealogies of particles which fall into compact sets around the level of the maximum but for the proof of Lemma \[comparison\] we need a slight extension in order to cover the case of sets which are only bounded from below; more precisely, we claim that for $y\in{\mathbb{R}}$, $$\label{extension} \lim_{r_d, r_g \to\infty} \sup_{t > 3 \max\{r_d, r_g \}} {\mathbb{P}}[ \exists i,j \in\Sigma_t(y, \infty)\dvtx Q_{ij}(t) \in(r_d, t-r_g) ] = 0.$$ To see this, we recall the following estimate proved by Bramson [@bramson], Proposition 3: $$\label{boundmax} {\mathbb{P}}\Bigl[\max_{k\leq n(t)} \overline{x_k}(t) \geq Y \Bigr] \leq \kappa(Y+1)^2{\mathrm{e}}^{-\sqrt{2} Y},$$ which is valid for $t\geq2, 0< Y < \sqrt{t}$ and $\kappa>0$ a numerical constant. The bound [(\[boundmax\])]{} implies in particular that $$\label{newest} \lim_{Y \to\infty} \sup_{t\geq2} {\mathbb{P}}[ \sharp\Sigma_t(Y, \infty) >0 ] =0.$$ For $Y> y$, using the splitting $\Sigma_t(y, \infty)= \Sigma_t(y, Y) \cup \Sigma_t(Y, \infty)$, we have the bound $$\begin{aligned} && {\mathbb{P}}[ \exists i,j \in\Sigma_t(y, \infty)\dvtx Q_{ij}(t) \in(r_d, t-r_g) ] \nonumber \\[-9pt] \\[-9pt] \nonumber &&\qquad \leq{\mathbb{P}}[ \exists i,j \in\Sigma_t(y, Y)\dvtx Q_{ij}(t) \in(r_d, t-r_g) ] + {\mathbb{P}}[ \sharp\Sigma_t(Y, \infty) > 0].\end{aligned}$$ The first term on the right-hand side vanishes, by Theorem \[genealogyabktheorem\], in the limit $t\to\infty$ first and $r_d, r_g \to\infty$ next, whereas the second term vanishes, by [(\[newest\])]{}, in the limit $t\to \infty $ first and $Y\to\infty$ next: this proves [(\[extension\])]{}. Let us denote by $A_{t, r_d, r_g}(y, {\varepsilon})$ the event $$\{ \exists i, j \in\Sigma_t(y,\infty)\dvtx Q_{ij}(t) \in[r_d, t-r_g] \}.$$ By [(\[extension\])]{}, there exists $r_0 = r_0(y, {\varepsilon})$ such that, for $r_d, r_g > r_0$, ${\mathbb{P}}[ A_{t, r_d, r_g}^c ] > 1-{\varepsilon}$. By definition, on the event $A_{t, r_d, r_g}^c$, the following equivalence holds for any $\frac{r_d}{t} \leq q \leq1-\frac{r_g}{t}$: $$\overline{Q}_{ij}(t) < q \quad\Longleftrightarrow\quad\overline{Q}_{ij}(t) < \frac{r_d}{t}.$$ The assertion of the lemma is now a direct consequence of the definition of the thinning $\Xi^{(q)}(t)$ in [(\[eqnthin\])]{}. To prove Proposition \[simplerpp\], we need some control on the derivative martingale. \[squarederi\] Let $$Z^{(2)}(t) {\equiv}\sum_{k\leq n(t)} \bigl\{ \sqrt{2}t - x_k(t) \bigr\}^2 \exp\bigl[- 2 \sqrt{2}\bigl\{ \sqrt{2}t - x_k(t) \bigr\}\bigr].$$ For any given ${\varepsilon}>0$, $$\lim_{t\to\infty} {\mathbb{P}}\bigl[ Z^{(2)}(t) \geq{\varepsilon}\bigr] = 0.$$ First, by Bramson’s estimate [@bramson], we may find $Y = Y({\varepsilon})$ large enough, s.t. $$\label{derivo} {\mathbb{P}}\Bigl[ \max_k x_k(t) - m(t) > Y \Bigr] \leq(1+Y)^2 {\mathrm{e}}^{-\sqrt {2} Y} \leq{\varepsilon}/2.$$ Using this bound, and the Markov inequality, we get $$\begin{aligned} \label{squareone}\qquad && {\mathbb{P}}\bigl[ Z^{(2)}(t) \geq{\varepsilon}\bigr] \nonumber \\[-8pt] \\[-8pt] \nonumber &&\qquad\leq\frac{{\mathrm{e}}^{t}}{{\varepsilon}} {{\mathbb{E}}}\bigl[ \bigl\{\sqrt{2}t-x(t)\bigr\}^2 {\mathrm{e}}^{-2\sqrt{2} \{\sqrt {2}t-x(t)\}}; x(t) \leq m(t)+Y \bigr] +{\varepsilon}/2.\end{aligned}$$ The first term on the right-hand side is bounded from above by $$\begin{aligned} \label{squaretwo} && \frac{{\mathrm{e}}^{t}}{{\varepsilon}} \int_{( {3}/{(2\sqrt{2})}) \log t- Y}^{\infty} x^2 {\mathrm{e}}^{-2\sqrt{2} x} \exp\biggl\{-\frac{(\sqrt{2}t -x )^2}{2t}\biggr\} \frac{{\mathrm{d}}x}{\sqrt{2\pi t}} \nonumber\\ &&\qquad \leq\frac{1}{{\varepsilon}} \int_{ ({3}/{(2\sqrt{2})}) \log t- Y}^{\infty} x^2 {\mathrm{e}}^{-\sqrt{2} x} {\mathrm{e}}^{-x^2/2t} \frac{{\mathrm{d}}x}{\sqrt {2\pi t}} \nonumber\\ &&\qquad \leq\frac{\exp-\sqrt{2}({3}/{(2\sqrt{2})} \log t- Y)}{{\varepsilon}} \int_{ ({3}/{(2\sqrt{2})}) \log t- Y}^{\infty} x^2 {\mathrm{e}}^{-x^2/2t} \frac{{\mathrm{d}}x}{\sqrt{2\pi t}} \\ &&\qquad \leq\frac{\rho\cdot t^{-3/2}}{ {\varepsilon}} \int_0^{\infty} x^2 {\mathrm{e}}^{-x^2/2t} \frac{{\mathrm{d}}x}{\sqrt{2\pi t}} \nonumber\\ &&\qquad \leq\frac{\rho\cdot t^{-3/2}}{{\varepsilon}} t \to0 \qquad\mbox{as } t \to\infty.\nonumber\end{aligned}$$ This proves the lemma. [Proof of Proposition \[simplerpp\]]{} We will show the convergence of the Laplace functionals. For $\phi\dvtx {\mathbb{R}}\to{\mathbb{R}}_+$ measurable with compact support, we claim that $$\begin{aligned} \label{claimsimpler} && \lim_{r_d \to\infty} \lim_{t\to\infty} {{\mathbb{E}}}\biggl[ \exp- \int\phi ( x ) \Xi^{(r_d/t)}(t)({\mathrm{d}}x) \biggr] \nonumber \\[-8pt] \\[-8pt] \nonumber &&\qquad= {{\mathbb{E}}}\biggl[\exp{- C Z \int\bigl(1-{\mathrm{e}}^{-\phi(x)} \bigr) \sqrt{2}{\mathrm{e}}^{-\sqrt{2} x} \,{\mathrm{d}}x}\biggr],\end{aligned}$$ from which the proposition would evidently follow. We will prove [(\[claimsimpler\])]{} for simple step functions, that is, of the form $\phi(x) = \sum_{i=1}^N a_i 1_{A_i}$ for $a_i>0, i=1, \ldots , N$ and $A_i = [\underline{A_i}, \overline{A_i}], i=1\ldots N$ disjoint compact subsets. The extension to the general case of measurable $\phi$ follows by a standard monotone class argument. We will make use of the splitting $$\label{split} m(t) = \sqrt{2} r_d + m(t-r_d)+R_t$$ for some $R_t = o(1)$ as $t\uparrow\infty$. We also introduce, for $j \leq n(r_d)$, independent BBMs $\{x_{k}^{(j)}(t-r_d)\}_{k \leq n_j(t-r_d)}$, and use the abbreviation $$M_j(t-r_d) {\equiv}\max_{k\leq n_j(t-r_d)} x^{(j)}_{k}(t-r_d) - m(t-r_d).$$ Conditionally on everything that happened up to time $r_d$, the following equality holds in law due to the Markov property and the definition of the extrema in the $(r_d/t)$-thinning class: $$\label{equalitylaw} \Xi^{(r_d/t)}(t) \stackrel{(d)}{=} \bigl\{ x_j(r_d) -\sqrt{2} r_d + M_j(t-r_d)+R_t \bigr\}_{j=1\ldots n(r_d)}.$$ Since the $M_j$’s are i.i.d., with ${{\mathbb{E}}}_{M(t-r_d)}$ standing for expectation with respect to $M(t-r_d)$, $$\begin{aligned} \label{claimsimplerthree} &&{{\mathbb{E}}}\biggl[ \exp- \int\phi( x ) \Xi^{(r_d/t)}(t)({\mathrm{d}}x) \biggr] \nonumber \\[-8pt] \\[-8pt] \nonumber &&\qquad= {{\mathbb{E}}}\Biggl[ \prod_{j=1}^{n(r_d)} {{\mathbb{E}}}_{M(t-r_d)} \bigl[ {\mathrm{e}}^{-\phi ( x_{j}(r_d)-\sqrt{2} r_d + M(t-r_d) +R_t)}\bigr]\Biggr].\end{aligned}$$ As $t \to\infty$, the variable $M(t-r_d)$ converges weakly to $M$ with law $\omega$ by [(\[travellingone\])]{}. Hence $$\begin{aligned} \label{claimsimplerfour} &&\lim_{t\to\infty} {{\mathbb{E}}}\biggl[ \exp- \int\phi( x ) \Xi ^{(r_d/t)}(t)({\mathrm{d}}x) \biggr] \nonumber \\[-8pt] \\[-8pt] \nonumber &&\qquad = {{\mathbb{E}}}\Biggl[ \prod_{j=1}^{n(r_d)} {{\mathbb{E}}}_{M} \bigl[ {\mathrm{e}}^{-\phi( x_{j}(r_d)-\sqrt{2} r_d+ M )}\bigr]\Biggr].\end{aligned}$$ Define $y_j(r_d) {\equiv}\sqrt{2} r_d - x_j(r_d)$. We write $$\begin{aligned} {{\mathbb{E}}}_{M} \bigl[ {\mathrm{e}}^{-\phi( -y_j(r_d)+ M )}\bigr] &=& 1 - {{\mathbb{E}}}_{M} \bigl[1- {\mathrm{e}}^{-\phi( -y_j(r_d)+ M )}\bigr] \nonumber \\[-8pt] \\[-8pt] \nonumber & =:& 1 - F( -y_j(r_d)),\end{aligned}$$ and $$\label{claimsimplerfive} {(\ref{claimsimplerfour})} = {{\mathbb{E}}}\biggl[ \exp\biggl\{ \sum_{j \leq n(r_d)} \log[ 1 - F( -y_j(r_d)) ] \biggr\} \biggr]. $$ Observe that $$\min_{j\leq n(r_d)} y_j(r_d) \uparrow\infty\qquad \mbox{a.s.}$$ as $r_d\uparrow\infty$. This implies that $$\max_{j\leq n(r_d)} F( -y_j(r_d)) \downarrow0.$$ Using that $-x -x^2 < \log(1-x) < -x$ for $0<x<1/2$, for $r_d$ large enough, we obtain (up to a vanishing error) upper and lower bounds of the form $$\begin{aligned} \label{claimsimplersix} &&{{\mathbb{E}}}\biggl[ \exp\biggl\{ - \sum_{j \leq n(r_d)} F( -y_j(r_d)) \biggr\}\biggr]\nonumber\\ &&\qquad \geq{(\ref{claimsimplerfive})}\\ &&\qquad \geq {{\mathbb{E}}}\biggl[ \exp\biggl\{ - \sum_{j \leq n(r_d)} F( -y_j(r_d)) - F( -y_j(r_d))^2\biggr\} \biggr].\nonumber\end{aligned}$$ Since $\phi$ is chosen to be a simple step function, $$\label{claimsimplerseven} F( -y_j(r_d)) = \sum_{i=1}^N (1-{\mathrm{e}}^{-a_i}) \int_{A_i + y_j(r_d)} \,{\mathrm{d}}\omega.$$ Hence we can make use of the asymptotics [(\[totheright\])]{} to obtain $$\begin{aligned} \label{claimsimplereight} F( -y_j(r_d)) &\sim& \sum_{i=1}^N (1-{\mathrm{e}}^{-a_i}) C \bigl\{ \bigl(\underline{A_i}+y_j(r_d)\bigr) {\mathrm{e}}^{-\sqrt{2}(\underline{A_i}+y_j(r_d)) } \nonumber \\[-8pt] \\[-8pt] \nonumber &&\hspace*{48pt}\qquad{} - \bigl(\overline{A_i}+y_j(r_d)\bigr) {\mathrm{e}}^{-\sqrt{2}(\overline{A_i}+y_j(r_d)) }\bigr\},\end{aligned}$$ with $\sim$ meaning that the ratio of the left- and right-hand sides converges to 1, in the limit $r_d \to\infty$, ${\mathbb{P}}$-a.s. We regroup the terms on the right-hand side to get $$\begin{aligned} \label{claimsimplereight2} F( -y_j(r_d) ) &\sim& C y_j(r_d) {\mathrm{e}}^{-\sqrt{2} y_j(r_d)} \nonumber \\[-8pt] \\[-8pt] \nonumber &&{}\times\sum_{i=1}^N (1-{\mathrm{e}}^{-a_i}) \{ {\mathrm{e}}^{-\sqrt{2} \underline{A_i}} - {\mathrm{e}}^{-\sqrt{2} \overline{A_i}} \} + \mathcal R(y_j(r_d)),\end{aligned}$$ with $\mathcal R$ containing all the remaining terms; clearly, $$\label{firstrest} |\mathcal R(y_j(r_d))| \leq\rho\cdot{\mathrm{e}}^{-\sqrt{2} y_j(r_d)},$$ where $\rho$ depends on the $a_i$ and $A_i$, but not on $y_j(r_d)$. By the convergence of the derivative martingale as $r_d \uparrow\infty$ \[cf. [(\[eqnderiv\])]{}\], and the fact that, in the same limit, $$\sum_{j\leq n(r_d)} {\mathrm{e}}^{-\sqrt{2} y_j(r_d)} \to0,\vadjust{\goodbreak}$$ ${\mathbb{P}}$-almost surely, we get that $$\begin{aligned} \label{firstconv} \lim_{r_d \uparrow\infty} \sum_{j\leq n(r_d)} F\bigl( x_{j}(r_d)-\sqrt {2} r_d\bigr)&=& C Z \sum_{i=1}^N (1-{\mathrm{e}}^{-a_i}) \{ {\mathrm{e}}^{-\sqrt{2} \underline {A_i}} - {\mathrm{e}}^{-\sqrt{2} \overline{A_i}} \} \nonumber\hspace*{-35pt} \\[-8pt] \\[-8pt] \nonumber & =& C Z \int\bigl( 1- {\mathrm{e}}^{-\phi(x)}\bigr) \sqrt{2}{\mathrm{e}}^{-\sqrt{2} x} \,{\mathrm{d}}x,\hspace*{-35pt}\end{aligned}$$ ${\mathbb{P}}$-almost surely. This yields the correct asymptotics for the upper bound in \[claimsimplersix\]. The lower bound in [(\[claimsimplersix\])]{} involves exactly the same term as the left-hand side of [(\[firstconv\])]{}, and the additional term $$\label{secondconv} \sum_{j \leq n(r_d)} F\bigl( x_{j}(r_d)-\sqrt{2} r_d\bigr)^2.$$ It is straightforward to see that [(\[secondconv\])]{} converges to zero, as $r_d\uparrow\infty$. In fact, by the same argument as in [(\[claimsimplerseven\])]{}–[(\[firstrest\])]{}, one sees that $$|{(\ref{secondconv})}| = O\biggl( \sum_{j \leq n(r_d)} y_j(r_d)^2 {\mathrm{e}}^{-2\sqrt{2} y_j(r_d)}\biggr), \qquad r_d \uparrow\infty.$$ With the notation of Lemma \[squarederi\], $$\sum_{j \leq n(r_d)} y_j(r_d)^2 {\mathrm{e}}^{-2\sqrt{2} y_j(r_d)} = Z^{(2)}(r_d),$$ and this converges to zero in probability, by Lemma \[squarederi\]. Hence, in the limit of large $r_d$, the lower and upper bounds in [(\[claimsimplersix\])]{} coincide, which concludes the proof of the proposition. [Proof of Theorem \[convergencethinned\]]{} Let $\phi\dvtx {\mathbb{R}}\to{\mathbb{R}}_+$ be measurable, with compact support. We need to show that $$\begin{aligned} &&\lim_{t\to\infty} {{\mathbb{E}}}\biggl[ \exp-\int\phi(x) \Xi ^{(q)}(t)({\mathrm{d}}x)\biggr] \nonumber \\[-8pt] \\[-8pt] \nonumber &&\qquad={{\mathbb{E}}}\biggl[\exp{- C Z \int\bigl(1-{\mathrm{e}}^{-\phi(x)} \bigr) \sqrt{2}{\mathrm{e}}^{-\sqrt{2} x} \,{\mathrm{d}}x}\biggr] ,\end{aligned}$$ for any $\frac{r_d}{t}\leq q \leq{1-\frac{r_g}{t}}$. This is straightforward in view of Lemma \[comparison\] and Proposition \[simplerpp\] by taking $y$ smaller than the minimum of the support of $\phi$ and ${\varepsilon}$ arbitrarily small. Open problems {#secconvection} ============= On the extremal process of BBM ------------------------------ We consider the following cluster point process. Let $(\Omega', \mathcal F', P)$, $C>0$ be a probability space, and $Z\dvtx \Omega' \to{\mathbb{R}}_+$ with distribution as in Theorem \[convergencethinned\]. (Expectation w.r.t. $P$ will be denoted by $E$.) Conditionally on a realization of $Z$, let $(\eta_i; i \in{\mathbb{N}})$ be the position of particles generated according to a Poisson point process with density $$\label{density} C Z \bigl( -x {\mathrm{e}}^{-\sqrt{2} x} \bigr) \,{\mathrm{d}}x$$ on the negative axis. For each $i\in{\mathbb{N}}$, consider independent branching Brownian motions with drift $-\sqrt{2}$, that is, $\{ x_k^{(i)}(r) -\sqrt{2} r; k \leq n_i(r)\}$, issued on $(\Omega', \mathcal F', P)$. (“Time” is denoted here by $r$.) Remark that for given $i\in{\mathbb{N}}$, $$\label{driftingoff} \max_{k \leq n_i(r)} x^{(i)}_k(r)- \sqrt{2} r \to-\infty,$$ $P$-almost surely. The branching Brownian motions with drift are then superimposed on the Poissonian points, that is, the cluster point process is given by $$\qquad\Pi_r {\equiv}\{ \pi_{i, k}(r); i \in{\mathbb{N}}, k = 1\ldots n_i(r) \} , \qquad \pi_{i, k}(r) {\equiv}\eta_i + x_k^{(i)}(r)-\sqrt{2} r.$$ The existence of the large time limit of $\Pi_r$ is not straightforward. Due to [(\[driftingoff\])]{}, only those Poissonian points whose attached branching Brownian motion performs an unusually large displacement can contribute to the limiting object. It is thus not clear that one finds any Poissonian points at all which, together with their cluster of particles, achieve this feat. The fundamental observation here is that, in virtue of [(\[density\])]{}, the density of the Poissonian points on the negative axis grows (slightly faster than) exponentially when $x\to-\infty$. Together with the work of Chauvin and Rouault [@chauvinrouault] on branching Brownian motions conditioned to perform unusually large displacements, this observation can be exploited to rigorously establish the existence of the point process $\Pi_r$ in the limit of large times, as well as some of its statistical properties. We will report on this in a subsequent paper [@abkthree]. Here, we only put forward the following conjecture, which appears rather natural in the light of Theorem \[genealogyabktheorem\] and the results on the paths of extremal particles in BBM established in [@abk]: \[conjabk\] In the limit of large times, the distribution of the extremal process of BBM, $\Xi(t)$ and that of $\Pi _r$ coincide, that is, $$\label{equdistr} \lim_{t\to\infty} \Xi(t) \stackrel{(d)}{=} \lim_{r\to\infty} \Pi_r .$$ In particular, with $\phi\dvtx {\mathbb{R}}\to{\mathbb{R}}_+$ a measurable function with compact support, $$\begin{aligned} \label{laplacefull} && \lim_{t\to\infty} {{\mathbb{E}}}\biggl[\exp\biggl( -\sum_{k\leq n(t)} \phi \bigl(x_k(t)-m(t)\bigr) \biggr)\biggr] \nonumber \\[-8pt] \\[-8pt] \nonumber & &\qquad= \lim_{r\to\infty} E\biggl[ \exp- C Z \int_{-\infty}^0 \bigl( 1 - {\mathrm{e}}^{-\psi_r(x)} \bigr) \{- x {\mathrm{e}}^{- \sqrt{2} x} \} \,{\mathrm{d}}x \biggr],\end{aligned}$$ where $${\mathrm{e}}^{-\psi_r(x)} {\equiv}E\biggl[ \exp\biggl( - \sum_{k\leq n(r)} \phi \bigl(x + x_k(r) -\sqrt{2} r \bigr) \biggr)\biggr].\vspace*{-3pt}$$ We remark that densities of the form $-x \exp(-\sqrt{2} x) \,{\mathrm{d}}x$ on the negative axis have been conjectured to play an important role in the recent work by Brunet and Derrida [@derridabrunet], where the average size of the gaps between the $n$th- and ($n+1$)th-leading particle at the edge of BBM is numerically shown to behave as $$\frac{1}{n} - \frac{1}{n \log n}+\cdots,$$ (which is indeed “close” to the average size of the gaps in a PPP with density $-x {\mathrm{e}}^{-\sqrt{2} x} \,{\mathrm{d}}x$ on the negative axis). On a conjecture by Lalley and Sellke ------------------------------------ Conjecture \[conjabk\] is similar but fundamentally different from the *Tidal Wave Conjecture* formulated by Lalley and Sellke [@lalleysellke]. Lalley and Sellke suggested that the Poisson point process entering into the construction of $\Pi_r$ should have density $C Z {\mathrm{e}}^{-\sqrt{2} x} \,{\mathrm{d}}x$ conditionally on a realization of $Z$ where $C$ is some constant. However, this cannot be correct. We will show that such a point process does not exist in the limit $r\to\infty$: the density of the Poissonian component cannot compensate [(\[driftingoff\])]{} and all the particles are bound to drift off to $-\infty$. To formulate this precisely, consider the point process $$\label{lalleysellkepp} \widetilde\Pi_r{\equiv}\bigl( \tilde\eta_i + x^{(i)}_k(r)- \sqrt{2} r; i \in{\mathbb{N}}, k =1,\ldots, n_i(r) \bigr),$$ where the $\tilde\eta$’s are points of a PPP with density $C Z {\mathrm{e}}^{-\sqrt{2} x} \,{\mathrm{d}}x$, and the $x^{(i)}$’s independent BBMs. \[wrongdensity\] For given $y\in{\mathbb{R}}$, $$\label{driftoff} \lim_{r \to\infty} P\bigl[ \widetilde\Pi_r[y, \infty) \geq1 | Z \bigr] = 0.\vspace*{-3pt}$$ In order to prove Proposition \[wrongdensity\], we make use of the following bound established by Bramson: \[bramsonremarkablebound\] Let $y_0 < 0$ (strictly). There exists $r_0 = r_0(y_0)$ such that for $r\geq r_0$, $x \geq m(r)+1$ and $z{\equiv}x-m(r)$, $$\label{uppertight} P\Bigl[ \max_{k\leq n(r)} x_k(r) \geq x\Bigr] \leq\rho\cdot{\mathrm{e}}^r \int_{y_0}^0 \frac{{\mathrm{e}}^{-(x-y)^2/2r}}{\sqrt{2\pi r}} \bigl( 1- {\mathrm{e}}^{-2 (y-y_0) z/r}\bigr) \,{\mathrm{d}}y,$$ where $\rho>0$ is a numerical constant. Using this with $y_0 := -1$, we obtain the following corollary. (Here and below, denotes a numerical constant, not necessarily the same at different occurrences.) For $X>1$, and $r\geq r_o = r_o(-1)$, $$\begin{aligned} \label{uptightzero} &&P\Bigl[ \max_{k\leq n(t)} x_k(r) -m(r) \geq X \Bigr] \nonumber \\[-8pt] \\[-8pt] \nonumber &&\qquad\leq\rho\cdot X \cdot\exp\biggl(-\sqrt{2} X - \frac{X^2}{2r}+\frac{3 }{2\sqrt{2}}X \frac{\log r}{r}\biggr).\end{aligned}$$ According to Proposition \[bramsonremarkablebound\], for $X>1$, $$\begin{aligned} \label{uptightone} &&P\Bigl[ \max_{k\leq n(t)} x_k(r) -m(r) \geq X \Bigr] \nonumber \\[-8pt] \\[-8pt] \nonumber &&\qquad \leq\rho\cdot {\mathrm{e}}^r \int_{-1}^{0} \frac{{\mathrm{e}}^{-(X+m(r)-y)^2/2r}}{\sqrt{2\pi r}} \bigl( 1- {\mathrm{e}}^{-2 (y+1) X/r}\bigr) \,{\mathrm{d}}y.\end{aligned}$$ Since $y+1>0$ we have that $1-{\mathrm{e}}^{-2 (y+1) X/r} \leq2 (y+1)X/r$. Using this, the right-hand side of [(\[uptightone\])]{} is at most $$\label{uptightoneone} \rho\cdot\frac{ X {\mathrm{e}}^r}{r} \int_{-1}^{0} (y+1)\frac{{\mathrm{e}}^{-(X+m(r)-y)^2/2r}}{\sqrt{2\pi r}} \,{\mathrm{d}}y.$$ Expanding the square in the Gaussian density, [(\[uptightoneone\])]{} is at most $$\begin{aligned} \label{uptighttwo} && \rho\cdot X \cdot\exp\biggl(-\sqrt{2} X - \frac{X^2}{2r}+\frac{3 X \log r}{2\sqrt{2}r}\biggr)\nonumber\\ &&\quad{}\times \int_{-1}^{0} (y+1) \underbrace{{\mathrm{e}}^{Xy/r +\sqrt{2} y + y({3}/(2\sqrt{2})) (\log r)/r} {\mathrm{e}}^{-y^2/2r}}_{\leq1} \,{\mathrm{d}}y \\ &&\qquad \leq\rho\cdot X \cdot\exp\biggl(-\sqrt{2} X - \frac {X^2}{2r}+\frac{3 }{2\sqrt{2}}X \frac{\log r}{r}\biggr),\nonumber\end{aligned}$$ settling the proof of the corollary. [Proof of Proposition \[wrongdensity\]]{} In view of [(\[driftingoff\])]{}, it is plain that for any finite set $I\subset{\mathbb{N}}$ $$\max_{i \in I} \Bigl\{ \tilde\eta_i + \max_{k\leq n_i(r)} \bigl[x^{(i)}_k(r) -\sqrt{2} r \bigr] \Bigr\} \stackrel{r\uparrow\infty }{\longrightarrow} -\infty,$$ $P$-almost surely. But the number of Poissonian points $(\tilde\eta_i; i\in{\mathbb{N}})$ in the interval $[0, \infty)$ *is* finite, $P$-almost surely: this follows from the fact that the density $C Z {\mathrm{e}}^{-\sqrt{2}x} \,{\mathrm{d}}x$ is integrable on $x\in[0, \infty)$. Hence, Proposition \[wrongdensity\] will follow as soon as we prove that $$\label{driftofftwo} P\Bigl[ \exists_{i\in{\mathbb{N}}}\dvtx \tilde\eta_i + \max_{k\leq n_i(r)} \bigl\{ x^{(i)}_k(r)- \sqrt{2} r\bigr\} \geq y \mbox{ and } \tilde\eta _i \in(-\infty, 0) | Z \Bigr] \stackrel{r\uparrow\infty }{\longrightarrow} 0.\hspace*{-35pt}\vadjust{\goodbreak}$$ By the Markov inequality, and using that the BBMs superimposed on the Poissonian points are identically distributed, [(\[driftofftwo\])]{} is at most $$\label{driftoffthree} \int_{-\infty}^0 P\Bigl[ \max_{k\leq n(r)} \bigl\{ x_k(r)- \sqrt{2} r\bigr\} \geq y - x \Bigr] C Z {\mathrm{e}}^{-\sqrt{2} x} \,{\mathrm{d}}x.\vspace*{-1pt}$$ We rewrite this in terms of $M(r) {\equiv}\max_{k\leq n(r)} \{ x_k(r) - m(r) \} $: $$\begin{aligned} \label{driftofffour} \qquad{(\ref{driftoffthree})} & = &\int_{-\infty}^0 P\biggl[ M(r) \geq y - x + \frac{3}{2\sqrt{2}} \log r \biggr] C Z {\mathrm{e}}^{-\sqrt{2} x} \,{\mathrm{d}}x \nonumber \\[-10pt] \\[-10pt] \nonumber & = &\bigl(C Z {\mathrm{e}}^{-\sqrt{2} y}\bigr) \cdot\frac{1}{r^{3/2}}\int_{y+ {3}/{(2\sqrt{2})} \log r }^\infty P[ M(r) \geq X] {\mathrm{e}}^{\sqrt{2} X} \,{\mathrm{d}}X,\vspace*{-1pt}\end{aligned}$$ the last step by change of variable $y-x+\frac{3}{2\sqrt{2}} \log r \to X$. Let us abbreviate $\rho{\equiv}C Z {\mathrm{e}}^{-\sqrt{2} y}$. (Note that $y$ and $Z$ are fixed.) For $r$ large enough, $$y+ \frac{3}{2\sqrt{2}} \log r \geq1,\vspace*{-1pt}$$ hence we may use [(\[uptighttwo\])]{} to get that [(\[driftofffour\])]{} is at most $$\begin{aligned} \label{driftofffive}\qquad && \frac{\rho}{r^{3/2}} \int_{y+ ({3}/{(2\sqrt{2})}) \log r}^\infty X \exp\biggl( \frac{3 }{2\sqrt{2}}X \frac{\log r}{r}\biggr) {\mathrm{e}}^{- {X^2}/{(2r)}} \,{\mathrm{d}}X \nonumber \\[-9.5pt] \\[-9.5pt] \nonumber &&\qquad = \frac{\rho}{r^{3/2}} \underbrace{\exp\biggl( \frac{9}{16} \frac{(\log r)^2}{r} \biggr)}_{= 1+o(1), r\uparrow\infty} \int _y^\infty\biggl\{ Y+ \frac{3}{2\sqrt{2}} \log r \biggr\} {\mathrm{e}}^{-Y^2/2r} \,{\mathrm{d}}Y,\vspace*{-1pt}\end{aligned}$$ by change of variable $X-\frac{3}{2\sqrt{2}}\log r \to Y$. It thus remains to control the term $$\begin{aligned} \label{driftoffsix}\qquad && \frac{1}{r^{3/2}} \int_y^\infty\biggl\{ Y+ \frac{3}{2\sqrt{2}} \log r \biggr\} {\mathrm{e}}^{-Y^2/2r} \,{\mathrm{d}}Y \nonumber \\[-9.5pt] \\[-9.5pt] \nonumber &&\qquad = \frac{1}{r^{3/2}} \int_y^\infty Y {\mathrm{e}}^{-Y^2/2r} \,{\mathrm{d}}Y+ \frac {3}{2\sqrt{2}} \cdot\frac{\log r}{r^{3/2}} \int_y^\infty{\mathrm{e}}^{-Y^2/2r} \,{\mathrm{d}}Y.\vspace*{-1pt}\end{aligned}$$ As for the first term on the right-hand side of [(\[driftoffsix\])]{}: $$\label{driftoffseven}\qquad \frac{1}{r^{3/2}} \int_y^\infty Y {\mathrm{e}}^{-Y^2/2r} \,{\mathrm{d}}Y = \frac {1}{\sqrt {r}} \int_{y/\sqrt{r}}^\infty x {\mathrm{e}}^{-x^2/2} \,{\mathrm{d}}x \to0 \qquad\mbox{as } r\uparrow\infty.\vspace*{-1pt}$$ As for the second term on the right-hand side of [(\[driftoffsix\])]{}: $$\begin{aligned} \label{driftoffeight} &&\frac{3}{2\sqrt{2}} \cdot\frac{\log r}{r^{3/2}} \int_y^\infty{\mathrm{e}}^{-Y^2/2r} \,{\mathrm{d}}Y \nonumber \\[-10pt] \\[-10pt] \nonumber &&\qquad = \frac{3}{2\sqrt{2}} \cdot\frac{\log r}{r} \int _{y/\sqrt{r}}^\infty{\mathrm{e}}^{-x^2/2} \,{\mathrm{d}}x \to0\qquad \mbox{as } r\uparrow \infty.\vspace*{-1pt}\end{aligned}$$ This proves [(\[driftofftwo\])]{}, settling Proposition \[wrongdensity\]. The above computations also suggest that a point process which is obtained by superimposing independent BBMs with drift $-\sqrt{2}$ on a PPP with a certain density exists in the limit of large times if and only if such density is, up to a (possibly random) constant, $-x \exp(-\sqrt{2} x) \,{\mathrm{d}}x$ on the negative axis. In fact, a closer look at the above considerations reveals that the left-hand side of [(\[driftoffseven\])]{} is the leading order of the expected number of points (of the superimposed point process) which fall into the subset $[y, \infty)$. Choosing the density of the Poissonian component as in Conjecture \[conjabk\], [(\[driftoffseven\])]{} would then read $r^{-3/2} \int _y^\infty Y^2 {\mathrm{e}}^{-Y^2/2r} \,{\mathrm{d}}Y$, which indeed remains of order 1 in the limit $r\to\infty$. Note added in revision {#note-added-in-revision .unnumbered} ====================== There has been considerable activity concerning the extremal process of BBM after this paper was submitted for publication. Brunet and Derrida have shown in [@brunetderridatwo] that all statistical properties of the rightmost points can be extracted from the traveling wave solutions of the Fisher-KPP equation. The validity of Conjecture \[conjabk\] has been settled in a paper of ours [@abkthree], where it is proved that the extremal process of branching Brownian motion weakly converges in the limit of large times to a Poisson cluster process; shortly after that, Aidekon et al. [@aidekonetal] recovered the same results by means of “spine techniques.” The Poissonian structure of the extremal process can also be proved using the property of *superposability* as observed by Maillard [@maillard]. This property of the process was conjectured by Brunet and Derrida in [@brunetderridatwo] and proved in [@abkthree]. Acknowledgments {#acknowledgments .unnumbered} =============== The authors thank Éric Brunet and Zakhar Kabluchko for interesting discussions on the existence of the extremal process of branching Brownian motion. [25]{} , , (). . , (). . , (). . . (). . In . . , . (). . . (). . . (). . (). . . (). . (). . . (). . . (). . . (). . . (). . . (). . . (). . . (). . . (). . , , (). . . , (). . . (). . . (). . (). . . (). . .

--- author: - 'C. Lujan-Peschard,' - 'G. Pagliaroli,' - 'F. Vissani.' title: 'Spectrum of Supernova Neutrinos in Ultra-pure Scintillators' --- Introduction {#sec:intro} ============ A Core Collapse Supernova (SN) releases 99% of its total energy by emitting neutrinos of the six flavors. The capability to observe the electronic antineutrino component of this emission has already been proven by the detection of SN1987A neutrinos [@KII; @IMB; @Baksan]. The very large statistics that we will collect from the next galactic supernova will allow us to study the time dependence of the spectrum, specific features of the $\bar{\nu}_e$ luminosity and of its average energy [@SK; @IceCube]. The detection of the other neutrinos flavors, however, requires specific detectors and interactions, typically with smaller cross sections. This is true, in particular, for the non electronic component of the spectrum, that can be observed only through Neutral Current (NC) interactions. During the last years a new generation of ultra-pure liquid scintillators, Borexino (BRX) [@2009BorexinoColl] and KamLAND (KAM) [@2003KamlandColl], have been operated, obtaining excellent results thanks to the unprecedented low background levels reached and the new sensitivity in the very low energy range, below 1 MeV. They have a particularly good physics potential for the detection supernova NC channels, and quite remarkably, the Elastic Scattering (ES) of (anti)neutrinos on protons [@beato]. It has been argued that the high statistics from this reaction should suffice to constrain the spectra of the non electronic component for a SN emission, already with the existing detectors [@dasg]. In view of the importance of this conclusion, we would like to reconsider it in this work. The outline of this paper is as follows. First of all, we summarize the available information regarding SN neutrino detection in the existing ultra-pure scintillators, and calculate for each of them the total number of expected events as well as their spectral features. We consider the contributions of all neutrino interaction channels and obtain in this way the spectrum of events for a galactic supernova. In this way, we are in the position to evaluate which are the capabilities of the present generation of ultra-pure scintillators to identify and measure the different neutrino flavors. Emission from a Standard Core Collapse Supernova ================================================ The aim of this work is to discuss an important question: what we can really see with the existing ultra-pure scintillators and to which extent we can distinguish the different neutrino flavors. With this purpose in mind, we will use very conservative assumptions on the emission model. We suppose that the energy radiated in neutrinos is $\mathcal{E}=3\times 10^{53}$ erg, which is a typical theoretical value that does not contradict what is found in the most complete analyses of SN1987A events [@ll; @paglia]. We also assume that the energy is partitioned in equal amount among the six types of neutrinos, that should be true within a factor of 2 [@Keil:2002in]. In agreement with the recent studies, e.g., [@Tamborra:2012ac], we consider quasi-thermal neutrinos, each species being characterized by an average energy $\langle E_i\rangle$ and including a mild deviation from a thermal distribution described by the parameter $\alpha=3$ for all flavors. Thus, the neutrino fluence differential in the neutrino energy $E$ is $$\Phi_i=\frac{\mathcal{E}_i}{4\pi D^2}\times \frac{E^\alpha e^{-E/T_i}}{ T_i^{\alpha+2} \Gamma(\alpha+2)} \ \ \ i=\nu_e,\nu_\mu,\nu_\tau,\bar{\nu}_e,\bar{\nu}_\mu,\bar{\nu}_\tau$$ where the energy radiated in each specie is $\mathcal{E}_i=\mathcal{E} f_i$, with $f_i=1/6$ in the case of equipartition, and the ‘temperature’ is $T_i=\langle E_i\rangle/(\alpha+1)$. In particular, the neutrino and antineutrino fluences relevant to NC interactions $$\Phi^{\mbox{\tiny SN}}_\nu=\Phi_{\nu_e}+2 \Phi_{\nu_\mu} \mbox{ and } \Phi^{\mbox{\tiny SN}}_{\bar\nu}=\Phi_{\bar{\nu}_e}+2 \Phi_{\bar{\nu}_\mu}$$ since we suppose that the distribution of the 4 non-electronic species is identical. The average energies are fixed by the following considerations: consistent with the simulations in [@Tamborra:2012ac] and with the findings from SN1987A [@ll; @paglia], we set the electron antineutrino average energy to $\langle E_{\bar{\nu}_e}\rangle=$12 MeV. For the average energy of the non-electronic species, that cannot be seriously probed with SN1987A [@paglia], we suppose that the non-electronic temperature is 30% higher than the one of $\bar{\nu}_e$: $\langle E_x \rangle=15.6$ MeV, this is in the upper range of values, but still compatible with what is found in [@Keil:2002in]. For a comparison we will consider also the worst case in which the energies of the non electronic component is equal to the one of the $\bar{\nu}_e$, namely $\langle E_x \rangle=12$ MeV as showed in very recent simulation [@Mueller:2014rna]. We calculate the electron neutrino average energy by the condition that the proton (or electron) fraction of the iron core in the neutron star forming is 0.4: this gives $\langle E_{\nu_e}\rangle=9.5$ MeV. Note that the NC reactions are independent from neutrino oscillations, while for the CC interactions also considering only the standard oscillation scenario, the choice of the mass hierarchy has an important impact on the expectations. Thanks to the fact that $\theta_{13}$ is large (say, larger than about 1 degree) this means that, in normal mass hierarchy, the survival probability of electron neutrinos and antineutrinos are $|U_{e3}^2|$ and $|U_{e1}^2|$ respectively, whereas for inverted mass hierarchy, the two values become $|U_{e2}^2|$ and $|U_{e3}^2|$. Thus, the approximate numerical values that we will assume in the calculations are $P_{\bar\nu_e\to \bar\nu_e}$ $P_{\nu_e\to \nu_e}$ ---------- ------------------------------ ---------------------- Normal 0.7 0.0 Inverted 0.0 0.3 The value of $P_{\bar\nu_e\to \bar\nu_e}$ in the case of inverted hierachy means that what we measure as electronic antineutrinos in terrestrial detectors, are non-electronic antineutrinos at the emission in fact; thus, it has a particularly important impact on the interpretation of the data. We note also that these numerical values would imply that there are only little chances to probe the emission of electron neutrinos, which are, from the astrophysical point of view, the most important type of neutrinos emitted by a supernova. In the following we will consider only the case of the normal mass hierarchy for definiteness and adding a bit of theoretical bias; recall however that this hypothesis is immaterial for the discussion of the neutral current events. The total fluences expected to reach the Earth under these assumptions are shown in Fig.\[fig1\] for a Supernova exploding at $10$ kpc from us. {width="48.00000%"} {width="48.00000%"} Interaction Channels ==================== In the scintillators and at SN energies, we need consider the several interaction processes: #### **CC processes.** Those involving electronic antineutrinos are - Inverse Beta Decay (IBD), i.e. $\bar{\nu}_e+p\to n+e^+$; - $\bar{\nu}_e + ^{12}C \to ^{12}B + e^+$; while those involving electronic neutrinos are - Proton Knockout $^{12}$C$(\nu, p e^-) ^{11}$C; - $\nu_e + ^{12}C \to ^{12}N + e^-$. #### NC processes. We considered the following channels, - ES on protons, $\stackrel{(-)}\nu p\to \stackrel{(-)}\nu p$; - The 15.11 MeV de-excitation line of the $^{12}C$ nucleus, $\nu ^{12}C \to \nu ^{12}C^*$; - The Proton Knockout $\stackrel{(-)}\nu+ ^{12}C\to \stackrel{(-)}\nu+ ^{11}B$. Moreover, we consider the ES on electrons, that receives a contribution from both CC and NC. Let us discuss these reactions in detail. Detailed description of the cross sections {#detailed-description-of-the-cross-sections .unnumbered} ------------------------------------------ The [**IBD**]{}, i.e., $\bar{\nu}_e+p\to n+e^+$, represents the main signal not only in water Cherenkov and also in scintillator detectors. It produces a continuous spectrum due to the positrons energy release. The approximated kinematic of this reaction connects the neutrinos energy with the detected energy through $E_\nu=E_d+Q-m_e$ where $Q\simeq1.3$ MeV is the $Q$ value of the reaction and $m_e$ is the electron mass. For the calculations, the IBD cross section reported in [@strumiavissa] was used. The [**delayed neutron capture**]{} on a proton is characterized by a monochromatic $\gamma_{2.2 \text{ MeV}}$ emission. The coincidence in a typical time window of about 250 $\mu$s between the latter and the prompt signal from the $e^+$ gives a clear signature of an IBD event. This means that, in the time integrated events spectrum, there will be a very high peak around $2.2$ MeV that integrates the same number of events expected for IBD, reduced by the efficiency of the tag. The spectral shape of this peak is due to both the energy resolution of the detector and the quenching of the gamma ray energy in the scintillator. In this work, due to lack of information, we neglect the last effect and consider the optimistic case in which the width of this peak is only due to the energy resolution. The [**superallowed CC reactions**]{} $\nu_e+^{12}$C$\to e^- +^{12}$N and $\bar{\nu}_e+^{12}$C$\to e^+ +^{12}$B present physical thresholds of $E_{\nu_{e}}>\!17.3\,$ MeV and $E_{\bar{\nu}_{e}}\!>\!14.4\,$ MeV respectively. They are detectable through the prompt leptons $e^{-}$ ($e^{+}$), which give a continuous spectrum. Moreover the nucleus of both reactions in the final state, $^{12}$N and $^{12}$B, are unstable. The former will decay $\beta^+$ to $^{12}$C with a half life of $\sim 11$ ms. The latter will decay $\beta^-$ to $^{12}$C with a half life of $\sim 20$ ms [@N12]. The high energy positrons and electrons emitted in these beta decays can be observed, giving the possibility to tag these events. The cross sections used for the evaluation are those reported in [@Fukugita]. In the NC channels all neutrino flavors are involved potentially increasing the number of signal events detected. For the [**ES on protons**]{} channel the cross section in [@ahrens; @xxsec] was used, with a proton strangeness of $\eta=0.12$. However it is important to stress that the uncertainty on the number of events expected for this channel is not negligible due to the proton structure and amounts to about $20\%$ [@cyclotron]. To understand the spectral shape of this class of events it is necessary to model the quenching factor for protons in the scintillators; this accounts for the proton light output and depends on the liquid scintillator composition. A detailed description of this factor is given in the next section. The cross section for the [**superallowed NC reaction**]{} $\nu +^{12}$C $\to \nu+ ^{12}$C$^*$ followed by the emission of a monochromatic $\gamma$ at $15.11$ MeV is reasonably well known. It was measured in KARMEN [@Karmen], confirming the correctness of the calculations as reported in [@Fukugita] within an accuracy of 20%. Future measurements, most remarkably in OscSNS [@oscsns], claim the possibility of measuring more than 1,000 events in one year with a systematic estimated at 5% level or better. The prominent spectral feature of this channel can permit the identification of these events, as a sharp peak around $15$ MeV, standing out from the main signal due to IBD. The total cross section for [**NC proton knockout** ]{} $\nu+{}^{12}\mbox{C}\to \nu + \mbox{p}+^{11}$B has been calculated in [@yoshida], as a part of a network of reactions needed to describe the nucleosynthesis of light elements. However, the calculation of [@haxton] finds a cross section about 30% larger, which suggests an error of at least this order. The neutrino energy has to exceed a pretty high threshold, i.e. $E\!>\!\left[(M_B+m_p)^2-M_C^2\right]/(2 M_C)\!\simeq\!15.9$ MeV (where we use obvious symbols for the masses of the carbon nucleus, of the boron nucleus and of the proton). The initial neutrino energy (minus the activation energy, quantified by the threshold) is shared by the neutrino and the proton in the final state, $E+M_C\approx E'+T_p+M_B+m_p$ so that the maximum kinetic proton energy $T_p^{\mbox{\tiny max}}$ is obtained when the final state neutrino is almost at rest, $E'\approx 0$. The expression for the maximum of the proton kinetic energy is $$T_p^{\mbox{\tiny max}}=\left[(M^*-m_p)^2-M_B^2\right]/(2M^*),$$ with $M^*=\sqrt{M_C^2+2M_C E}$. In view of the smallness of this sample of events, we adopted a very simple procedure to describe the distribution in the kinetic energy of the proton $T_p$, namely, we resorted to the pure phase space, that gives ${d\sigma}/{dT_p}\propto {d\Phi}/{dT_p}\propto\sqrt{T_p}(M^*-m_p-m_B-T_p)^2$. We checked that the integral in the proton kinetic energy of this expression agrees at the level of few percent with the theoretical behaviour of the total cross sections as reported in [@yoshida].[^1] In the [**CC knockout proton**]{} reaction on $^{12}$C the outgoing kinetic energy is shared between the electron and the proton. The maximum kinetic proton energy $T_p^{\mbox{\tiny max}}$ is obtained when the final electron is at rest. Similarly to the previous case, this is given by $$T_p^{\mbox{\tiny max}}=\left[(M^*-m_p-m_e)^2-M_{C11}^2\right]/(2M^*-2m_e),$$ where again $M^*=\sqrt{M_C^2+2M_C E}$. In this case the phase space is ${d\sigma}/{dT_p} \propto {d\Phi}/{dT_p} \propto\sqrt{T_p}(M^*-m_p-m_{C11}-T_p)^2$. The theoretical cross sections reported in [@yoshida] and the value estimated from pure phase space agree at the level of $\sim 20$%. In the [**Elastic Scattering on electrons**]{} all the flavors participate, but the cross section is slightly different for the different flavors. The current best measurement of this interaction arises in a sample of 191 events [@ESe; @ESe2], and quotes 17% of total error. The error that we estimate in the standard model is instead absolutely negligible for our purposes. Numerical formulae {#numerical-formulae .unnumbered} ------------------ Let us conclude this section by giving two numerical formulas to evaluate easily the main neutral current cross sections:\ An easy-to-implement effective formula for the $\nu +^{12}C \to \nu+^{12}C^*$ cross section, that agrees with [@Fukugita] results at better than 1% in the region below 100 MeV, is (\_+\_[|]{})= (E-15.11 )\^2 10\^[p(E)]{}, with $p(E)=\sum_{n=0}^3 c_n \left(E/100\mbox{ MeV}\right)^n$ where $E$ is the incoming neutrino energy, and the numerical coefficients are $c_0=-0.146$, $c_1=-0.184$, $c_2=-0.884$, $c_3=+0.233$.\ A simple parametrization of the cross section for the ES scattering, $\nu p \to \nu p$, assuming that the proton strangeness is $\eta=0.12$, is simply (\_+\_[|]{})=G\_F\^2 E\^2 10\^[q(E)]{}, where $q(E)=-0.333-0.16 (E/100\mbox{ MeV})$. Description of the Ultra-pure Scintillating Detectors ===================================================== We consider the ultrapure liquid scintillators detectors that are running or under construction, namely the following three: Borexino (BRX) [@2009BorexinoColl] (0.3 kt of $C_9 H_{12}$) in Gran Sasso National Laboratory, Italy, KamLAND (KAM) in Kamioka Observatory, Japan [@2003KamlandColl] (1 kt of mixture of $C_{12}H_{20}$(80%) and $C_9H_{12}$(20%)) and SNO+ (0.8 kt of $C_6H_5C_{12}H_{25}$) currently under construction in the SNOLAB facility, located approximately 2 km underground in Sudbury, Ontario, Canada [@SNO]. We assume that the energy resolution for each detector is a Gaussian with an error described by $\sigma(E_d)=A\times \sqrt{E_d/\mbox{MeV}}$ and a different value of the constant $A$ for each detector. The trigger threshold in Borexino is as low as about 200 keV, reaching full efficiency at $E_d=250$ keV [@BorexPRL]. The overall light collection in Borexino is $\simeq$ 500 photoelectrons (p.e.)/MeV of deposited energy. The resolution is $\simeq$ 5% at 1 MeV (namely $A=50$ keV). The trigger efficiency in KamLAND currently reaches 100% at 350 keV. The energy resolution of the KamLAND detector can be expressed in terms of the deposited energy as $\sim\,6.9\%/\sqrt{E_d(\text{MeV})}$ (i.e., $A=69$ keV) [@dasg]. The energy threshold expected for SNO+ is the optimistic one of $200$ keV and the energy resolution is supposed to be the same as in Borexino. $a_1$ $a_2$ $a_3[\mbox{\small MeV$^{-1}$}]$ ------ ------- ----------- --------------------------------- BRX 0.624 $-0.175$ $-0.154$ KAM 0.581 $-0.0335$ $-0.207$ SNO+ 0.629 $ -0.286$ $-0.163$ : Constants appearing in the parametrized formula of the quenching function here adopted. \[const\] As we mentioned earlier when the detected particle is a proton, the visible energy is only a fraction of the kinetic energy $T_p$, as described by the ‘quenching function’. Each detector has its own quenching function, that depends on its chemical composition; for Borexino detector we consider the quenching function discussed in [@cyclotron], for KamLAND the one recently discussed in [@KamQuen] and finally for SNO+ the response to proton in LAB scintillator as measured in [@vonKrosigk:2013sa]. Following [@Madey] a simple parametrization of the quenching function is $$E_d=a_1[1-\exp(a_2+a_3\cdot T_p)]\cdot T_p \label{quench}$$ The values for the constants $a_1$, $a_2$ and $a_3$ that should be used for the different detectors are reported in Tab. \[const\] and the resulting functions are shown in Fig.\[fig3\]. At this point, we obtain the following important conclusion: > in ultrapure scintillators, the observation of protons from the NC elastic scattering reaction allows us to observe only the high energy part of the neutrino spectra. In fact, due to the thresholds and to the quenching, the protons below a minimum kinetic energy cannot be detected; this is $0.9$ MeV for SNO+, $1.8$ MeV for Kamland and $1.3$ MeV for Borexino. Thus, taking into account the kinematical relation between the proton kinetic energy and the one of incoming neutrinos, we find that the elastic scattering on protons is sensitive to neutrino energies above a threshold of $22$ MeV in the best situation of SNO+, it becomes 25 MeV for Borexino, and raises to $30$ MeV in the case of Kamland. In view of these considerations, one concludes that the exploration of the low energy of the spectrum via neutral currents is not possible with the existing ultrapure scientillators. Results ======= For each detection channel, we estimate the number of expected events and report them in Table \[tab\]. Moreover, we plot the energy distributions of the events, considering the specific features of the ultrapure scintillating detectors in Figure \[fig2\]. The IBD channel (red line) starts to dominate the global signal at $5$ MeV and reaches the maximum around $14$ MeV. The total number of interactions expected for a supernova located at $10$ kpc is of about $54$ events for Borexino, $257$ for Kamland and $176$ for SNO+. These results are reported in the first row of Table \[tab\]. The subsequent gamma from neutron capture gives the peak at 2.2 MeV, shown by a purple line. The efficiency of the neutron tag is (85$\pm$1)% in Borexino (see [@2010Borexino]), (78$\pm$2)% in KamLAND (see [@2003KamlandColl]) and also in SNO+. The condition for a successful IBD tag [@2010Borexino] is that no more than one interaction occurs during the time between the IBD interaction and the neutron capture inside a specific volume, namely $$R_{IBD}\times \Delta t \Delta V \rho \leq 1$$ where $R_{IBD}$ is the rate of IBD events per second and per unit mass, $\Delta t$ is the temporal window of the tag, that we assume to be $\Delta t=2\tau = 512 \mu s$, $\Delta V$ is the volume of a sphere with 1 meter of radius, $\rho$ is the density of the scintillator. This is related to the detector mass and to the distance of the supernova by $$R_{IBD}=\frac{256.5}{T}\cdot\left(\frac{10 \text{kpc}}{D}\right)^2\cdot \left(\frac{M}{1 \text{kton}} \right),$$ where $M$ is the mass of the detector, $D$ is the distance of the SN and $T$ is the duration of the emission. For example to allow the IBD tag in a detector with the density of Borexino and 1 kton of mass, considering that $50$% of the total emission is expected during the first second [@paglia], then the minimum distance of a SN is $D\geq 0.16$ kpc, that is not a severe limitation.[^2] Channel Color code Signal BRX KAM SNO+ ---------------------------------------------- ------------ ---------- ------------- --------------- --------------- $\bar{\nu}_e+p\to n+e^+$ red $e^+$ 54.1 (49.6) 256.5 (235.3) 175.8 (161.2) $n+p\to D +\gamma_{2.2 \text{ MeV}}$ purple $\gamma$ 46.0 (42.1) 200.1(183.5) 137.1 (125.8) $\nu +p\to \nu+p$ blue $p$ 12.7 (3.8) 29.0 (6.2) 74.9 (29.2) $\nu +^{12}C \to \nu+^{12}C^*$ orange $\gamma$ 4.7 (2.1) 15.0 (6.7) 12.3 (5.5) $\nu +e^-\to \nu+e^-$ green $e^-$ 4.4 (4.5) 14.8 (15.5) 12.0 (12.4) $\nu_e+^{12}C\to e^- +^{12}N$ magenta $e^-$ 2.0 (0.7) 6.4 (2.1) 5.3 (1.7) $\bar{\nu}_e+^{12}C\to e^+ +^{12}B$ black thin $e^+$ 1.2 (0.8) 3.7 (2.6) 3.0 (2.1) $\nu +^{12}C \rightarrow \nu + p + ^{11}B$ yellow $p$ 0.7 (0.2) 2.4 (0.6) 2.1 (0.6) $\nu_e +^{12}C \rightarrow e^- + p + ^{11}C$ red dashed $p$ 0.5 (0.1) 1.5 (0.3) 1.3 (0.2) Let us discuss now the NC elastic proton scattering. This channel dominates the low energy part of the spectrum, represented with the blue line in Figure \[fig2\], even if as discussed previously, this reaction probes only the high energy part of the supernova neutrinos. It is evident from Table \[tab\] and Fig. \[fig2\] that the event spectrum depends strongly on the quenching of the proton signal. The detector threshold used in the case of KamLAND is an optimistic value, since we assumed a threshold of $350$ keV, lower than the one that can be obtained with the current radioactivity level, i.e., $600$ keV [@Tolich]. If this higher threshold is assumed the ES with protons on KamLAND gives only $17$ signal events. It is important to remark that the number of events due to this channel is also very sensitive to the SN emission parameters; in fact, as discussed in the end of the previous section, this interaction is sensitive only to the high energy tail of the SN neutrinos spectra. In particular, the average energy of the different neutrino flavors have gradually been changing in recent years, moving toward lower mean values [@Janka:2012wk] and toward minor differences between the average energies of the different components [@Ott:2012mr; @Tamborra:2012ac]. For comparison we have considered the new paradigm of emission, where the average energy of non-electronic flavors is the same as of the electronic antineutrinos, namely $\langle E_x \rangle=12$ MeV [@Janka:2012wk; @Tamborra:2012ac] and have investigated the two different cases to outline the impact on this and the other NC process. As shown by the values in brackets in Table \[tab\], the expectations in this case are quite meager. The NC neutral current reaction $\nu +^{12}C \to \nu+ ^{12}C^*$ followed by the emission of a monochromatic $\gamma$ is shown in orange in Figure \[fig2\]. This channel does not require the low energy threshold and the efficiency for its detection is taken 100% for all the detectors considered. However, the possibility of a successful identification is affected by the quality of the energy resolution of the detector and by the effectiveness to tag the IBD signal. These events can be observed if the IBD events are identified through the correlated neutron capture signal, since they are expected to occur in the same energy region. With the assumed energy resolutions we have that this neutral current reaction can be observed in the energy range (14-16) MeV. For Borexino, the number of events due to the IBD signal in the same range is $5.7$; thus, more than the 50% of the total signal collected in this energy window is due to the IBD channel, while for KamLAND and SNO+ the IBD signal is 26.9 and 18.4, representing about 60% of the total one. In the case of Borexino, if the tagging efficiency is of 85% as assumed in the plot, we expect only 1 event due to IBD not identified, so the uncertainty on the $\gamma_{15.11 \text{MeV}}$ signal is reduced to 14%. The ESe involves all the flavors of neutrinos and we expect to collect about $4$ events for Borexino, $15 $ for KamLAND and $12 $ for SNO+. Their spectrum is reported with a dark green line in the spectra of Figure \[fig2\], and dominates in the energy region between the ES on protons and the IBD signals. As we mentioned the cross section for the different flavors are slightly different; the $\nu_e$ contribution produces half of the events. {width="45.00000%"} {width="45.00000%"}\ {width="45.00000%"} {width="45.00000%"}\ {width="45.00000%"} {width="45.00000%"} The rest of the detection channels have a low signal. All of them show a continuos spectrum, being from $e^{-}$, $e^{+}$ or protons. The two CC superallowed reactions are indicated by a magenta line for the $^{12}N$ final state nucleus and by a black thin line for the $^{12}B$ one. The decay products of the unstable nucleus are not considered in this plot. For both knockout channels, besides the high energy thresholds, the quenching has to be considered, so the total number of events collected for them is pretty small. The one due to NC is shown in yellow, while the one due to CC is shown with the dashed red line. Conclusions =========== In this paper, we have obtained and discussed the spectrum of supernova neutrino events in ultrapure scintillators for a supernova exploding at 10 kpc from the Earth. We have examined the capability to distinguish the various detection channels and we have quantified the uncertainties in this type of detectors. As discussed in the introduction, a major reason of specific interest for a future supernova is the possibility to observe neutral current interactions of neutrinos. We have investigated the three possible reactions of detection in ultrapure scintillators, namely: 1) the elastic scattering with protons, 2) the 15.11 MeV $\gamma$ de-excitation line, 3) the proton knockout channel. Our conclusions are as follows: The first reaction is characterized by the larger number of expected events in all the detectors; however the number of detectable events is strongly limited by the energy thresholds. The uncertainty on the total number of elastic scattering on protons, due to the proton structure, amounts to the 20%; moreover in the same energy region where this reaction can be observed there are also the indistinguishable events due to the elastic scattering with electrons and those due to NC and CC proton knockout. In other words, all we can observe is the total number of events collected in the energy region from the detector threshold to the threshold of the IBD signal, about $1.8$ MeV. In this detection window, we have found that a fraction of 8% (Borexino), the 7% (KamLAND), the 4% (SNO+) of the signal is due to the other channels and this uncertainty is small but irreducible. We have also seen that, in the case that the the energy of non-electronic neutrinos is low, the number of events due to this reaction is too small to permit the investigation of the $\nu_x$ spectrum at the level discussed in [@dasg]. The gamma line due to neutrino-induced $^{12}$C de-excitation is in principle easier, giving a signal at a high energies; its detection does not require the extreme performances at very low energies are not needed. However, in the same region of the spectrum where this line is visible we will have also positrons due to IBD reaction; thus, the efficiency to tag the concomitant neutron will be of crucial importance to identify cleanly a sample of this NC reaction. While this concern is not a severe issue for the type of detectors we have considered in this work, it is much more relevant for future scintillators with a much larger mass and limited performances at low energies. Finally, we have shown that the proton knockout will give a comparably small number of NC events. [99]{} K. Hirata, et al., *Observation of a Neutrino Burst from the Supernova SN 1987a*, *Phys. Rev. Lett.* [**58**]{} (1987) 1490. R. M. Bionta, et al., *Observation of a Neutrino Burst in Coincidence with Supernova SN 1987a in the Large Magellanic Cloud*, *Phys. Rev. Lett.* [**58**]{} (1987) 1494. E. N. Alekseev, et al., *Possible Detection of a Neutrino Signal on 23 February 1987 at the Baksan Underground Scintillation Telescope of the Institute of Nuclear Research*, *JETP Lett.* [**45**]{} (1987) 589. M. Ikeda, et al. \[Super-Kamiokande Collaboration\], *Search for Supernova Neutrino Bursts at Super-Kamiokande*, *Astrophys. J.*  [**669**]{} (2007) 519 \[arXiv:0706.2283 \[astro-ph\]\]. R. Abbasi, et al. \[IceCube Collaboration\], *IceCube Sensitivity for Low-Energy Neutrinos from Nearby Supernovae*, *Astron. Astrophys.* [**535**]{} (2011) A109 \[arXiv:1108.0171 \[astro-ph.HE\]\]. G. Alimonti, et al. \[Borexino Collaboration\], *The Borexino detector at the Laboratori Nazionali del Gran Sasso*, *Nucl. Instrum. Meth.* [**A 600**]{} (2009) 568 K. Eguchi, et al. \[KamLAND Collaboration\], *First results from KamLAND: Evidence for reactoranti-neutrino disappearance* *Phys. Rev. Lett.* [**90**]{} (2003) 021802 J. F. Beacom, W. M. Farr and P. Vogel, *Detection of supernova neutrinos by neutrino proton elastic scattering*, Phys. Rev. D [**66**]{} (2002) 033001 B. Dasgupta and J. .F. Beacom, *Reconstruction of supernova $\nu_\mu$, $\nu_\tau$, anti-$\nu_\mu$, and anti-$\nu_\tau$ neutrino spectra at scintillator detectors*, *Phys. Rev.* [**D 83**]{} (2011) 113006 T. J. Loredo and D. Q. Lamb, *Bayesian analysis of neutrinos observed from supernova SN-1987A*, *Phys. Rev.* [**D 65**]{} (2002) 063002 \[astro-ph/0107260\]. G. Pagliaroli, F. Vissani, M. L. Costantini and A. Ianni, *Improved analysis of SN1987A antineutrino events*, *Astropart. Phys.* [**31**]{} (2009) 163 \[arXiv:0810.0466 \[astro-ph\]\]. M. T. .Keil, G. G. Raffelt and H. -T. Janka, *Monte Carlo study of supernova neutrino spectra formation*, *Astrophys. J.* [**590**]{} (2003) 971 \[astro-ph/0208035\]. I. Tamborra, B. Muller, L. Hudepohl, H. -T. Janka and G. Raffelt, *High-resolution supernova neutrino spectra represented by a simple fit*, *Phys. Rev.* [**D 86**]{} (2012) 125031 \[arXiv:1211.3920 \[astro-ph.SR\]\]. B. Mueller and H. -T. .Janka, arXiv:1402.3415 \[astro-ph.SR\]. A. Strumia, F. Vissani, *Precise quasielastic neutrino/nucleon cross-section* *Phys. Lett.* [**B 564**]{} (2003) 42. F. Ajzenberg-Selove, *Energy levels of light nuclei A = 11-12*, *Nucl. Phys.* [**A 506**]{} (1990) 1 M. Fukugita, Y. Kohyama, K. Kubodera, *Neutrino Reaction Cross-Sections on C-12 Target* *Phys. Lett.* [**B 212**]{} (1988) 139. L.A. Ahrens, et al. \[BNL 734\] *Measurement of Neutrino - Proton and anti-neutrino - Proton Elastic Scattering*, *Phys. Rev.* [**D 35**]{} (1987) 785. G. Pagliaroli, C. Lujan-Peschard, M. Mitra and F. Vissani, *Using Low-Energy Neutrinos from Pion Decay at Rest to Probe the Proton Strangeness*, *Phys. Rev. Lett.* [**111**]{} (2013) 022001. B. E. Bodmann, et al. \[KARMEN. Collaboration\], *Neutrino interactions with carbon: Recent measurements and a new test of electron-neutrino, anti-muon-neutrino universality*, *Phys. Lett.* [**B 332**]{} (1994) 251. H. Ray \[OscSNS Collaboration\], *OscSNS: Precision Neutrino Measurements at the Spallation Neutron Source*, *J. Phys. Conf. Ser.* [**136**]{} (2008) 022029 \[arXiv:0810.3175 \[hep-ex\]\]. T. Yoshida, et al. *Neutrino-Nucleus Reaction Cross Sections for Light Element Synthesis in Supernova Explosions* *Astrophys. J.* [**686**]{} (2008) 448 A. Heger, E. Kolbe, W. C. Haxton, K. Langanke, G. Martinez-Pinedo and S. E. Woosley, *Neutrino nucleosynthesis*, *Phys. Lett.* [**B 606**]{} (2005) 258 L. B. Auerbach, et al., *Measurement of electron - neutrino - electron elastic scattering*, *Phys. Rev.* [**D 63**]{} (2001) 112001. R. C. Allen, et al., *Study of electron-neutrino electron elastic scattering at LAMPF* *Phys. Rev.* [**D 47**]{} (1992) 11. H. M. O’Keeffe, E. O’Sullivan, M. C. Chen, *Scintillation decay time and pulse shape discrimination in oxygenated and deoxygenated solutions of linear alkylbenzene for the SNO+ experiment*, *Nucl. Instrum. Meth.* [**A 640**]{} (2011) 119. C. Arpesella et al. \[Borexino Collaboration\], *Direct Measurement of the Be-7 Solar Neutrino Flux with 192 Days of Borexino Data*, *Phys. Rev. Lett.* [**101**]{} (2008) 091302. R. Madey, et al., *Nucl. Instrum. Meth.* [**151**]{} (1978) 445. S. Yoshida et al. *Light output response of KamLAND liquid scintillator for protons and 12 C nuclei*, *Nucl. Instrum. Meth.* [**A 622**]{} (2010) 574. B. von Krosigk, L. Neumann, R. Nolte, S. Rottger and K. Zuber, *Measurement of the proton light response of various LAB based scintillators and its implication for supernova neutrino detection via neutrino-proton scattering*, arXiv:1301.6403 \[astro-ph.IM\]. D. K. Nadyozhin, *The neutrino radiation for the hot neutron star formation and the envelope outburst problem*, *Astrophys. Space Sci.* [**53**]{} (1978) 131. H. A. Bethe, J. R. Wilson, *Revival of a stalled supernova shock by neutrino heating*, *Astrophys. J.* [**295**]{} (1985) 14. H. Duan, G. M. Fuller and Y. -Z. Qian, *Collective Neutrino Oscillations*, *Ann. Rev. Nucl. Part. Sci.* [**60**]{} (2010) 569 \[arXiv:1001.2799 \[hep-ph\]\]. G. Badino, et al., *The 90 ton liquid scintillator detector in the mont blanc laboratory*, *Nuovo Cimento* [**C 7**]{} (1984) 573. N. Yu. Agafonova, et al., *Study of the effect of neutrino oscillations on the supernova neutrino signal in the LVD detector*, *Astropart. Phys.* [**27**]{} (2007) 254. P. Antonioli, W. Fulgione, P. Galeotti and L. Panaro *Simulation of low-energy neutrino interactions in liquid scintillation counters*, *Nucl. Instrum. Meth.* [**A 309**]{} (1991) 569. T. Totani, K. Sato, H. E. Dalhed and J. R. Wilson, *Future detection of supernova neutrino burst and explosion mechanism*, *Astrophys. J.* [**496**]{} (1998) 216 \[astro-ph/9710203\]. G. Bellini, et al. \[Borexino Collaboration\], *Observation of Geo-Neutrinos*, *Phys. Lett.* [**B 687**]{} (2010) 299. K. Tolich \[KamLAND Collaboration\], *Supernova detection with KamLAND* *Nucl. Phys. Proc. Suppl.* [**221**]{} (2011) 355. H. -T. Janka, *Explosion Mechanisms of Core-Collapse Supernovae*, *Ann. Rev. Nucl. Part. Sci.* [**62**]{} (2012) 407 \[arXiv:1206.2503 \[astro-ph.SR\]\]. C. D. Ott, et al. *General-Relativistic Simulations of Three-Dimensional Core-Collapse Supernovae*, *Astrophys. J.* [**768**]{} (2013) 115 \[arXiv:1210.6674 \[astro-ph.HE\]\]. N. Smith, K.  H. Hinkle, and N. Ryde, *Red Supergiants as Potential Type IIn Supernova Progenitors: Spatially Resolved 4.6 $\mu$m CO Emission Around VY CMa and Betelgeuse*, *The Astronomical Journal* [**137**]{} (2009) 3558. [^1]: However, a word of caution is in order; while the above considerations on phase space are suggestive, they are just a reasonable way to explore of the consequences of this reaction in scintillator detectors: in fact, the distribution in $T_p$ of [@yoshida] is not available. Certainly, it would be better to have a true calculation of the distribution in $T_p$ of this reaction (or possibly its parameterization) along with an assessment of the theoretical error in the relevant energy range. [^2]: If instead, a similar detector but with a $50$ kton mass is considered the distance becomes $D\geq 1.13$ kpc, that includes several known potential Core-Collapse SNe as Betelgeuse and VY Canis Majoris [@bet].

--- abstract: 'This work investigates unsupervised learning of representations by maximizing mutual information between an input and the output of a deep neural network encoder. Importantly, we show that structure matters: incorporating knowledge about locality in the input into the objective can significantly improve a representation’s suitability for downstream tasks. We further control characteristics of the representation by matching to a prior distribution adversarially. Our method, which we call Deep InfoMax (DIM), outperforms a number of popular unsupervised learning methods and compares favorably with fully-supervised learning on several classification tasks in with some standard architectures. DIM opens new avenues for unsupervised learning of representations and is an important step towards flexible formulations of representation learning objectives for specific end-goals.' author: - | R Devon Hjelm\ MSR Montreal, MILA, UdeM, IVADO\ `devon.hjelm@microsoft.com`\ Alex Fedorov\ MRN, UNM\ Samuel Lavoie-Marchildon\ MILA, UdeM\ Karan Grewal\ U Toronto\ Phil Bachman\ MSR Montreal\ Adam Trischler\ MSR Montreal\ Yoshua Bengio\ MILA, UdeM, IVADO, CIFAR bibliography: - 'main.bib' title: Learning deep representations by mutual information estimation and maximization --- Introduction ============ Related Work ============ Deep InfoMax ============ Experiments =========== Conclusion ========== Appendix ========

--- abstract: 'The equilibrium properties of a minimal tiling model are investigated. The model has extensive ground state entropy, with each ground state having a quasiperiodic sequence of rows. It is found that the transition from the ground state to the high temperature disordered phase proceeds through a sequence of periodic arrangements of rows, in analogy with the commensurate-incommensurate transition. We show that the effective free energy of the model resembles the Frenkel-Kontorova Hamiltonian, but with temperature playing the role of the strength of the substrate potential, and with the competing lengths not explicitly present in the basic interactions.' author: - 'Nikolai Nikola, Daniel Hexner, and Dov Levine' title: 'Entropic commensurate-incommensurate transition ' --- Tilings provide a simple means to model systems with both simple and complex ground states. As statistical mechanical models, they include such systems as the Ising and Potts models, as well as other models with discrete spin variables[@fn1]. The class of tilings, however, is much larger[@tilings_and_patterns], and includes the various non-periodic Wang tilings[@fn2], the quasiperiodic Penrose tilings, the asymptotically isotropic Pinwheel tiling[@pinwheel], and the recently discovered generalizations of the Rudin-Shapiro sequence, which are neither periodic nor quasiperiodic[@Wolff]. The statistical mechanical behavior of tiling models is rich, and, as yet, largely uncharted. Apart from results which may be transcribed from discrete spin models, only a very small number of systems have been studied. First, a model based on the Amman set of 16 Wang tiles with quasiperiodic ground states was studied by several authors [@leuzzi_thermodynamics_2000; @rotman_finite-temperature_2011; @koch_modelling_2010]. It appears that this model undergoes a continuous phase transition from a disordered state to a quasiperiodic phase, and its non-equilibrium behavior was studied in the context of spin-glasses. A variation of the model allowing more complicated interactions and vacancies shows a first order transition[@aristoff_first_2011]. Hierarchical tilings[@miekisz_microscopic_1990] have also been studied, and a very recent model[@byington_hierarchical_2012] possesses limit-periodic ground states which undergo a series of phase transitions where motifs of ever larger scales order as the temperature is lowered. Finally, we note some recent studies [@sasa_pure_2012; @sasa_statistical_2012] on models with large number of degenerate disordered ground states aimed at studying glasses. In this paper we study the equilibrium behavior of a model based on the 13-tile Kari-Culik (KC) set [@culik_ii_aperiodic_1996; @kari_small_1996] of Wang tiles, both numerically and analytically. The KC set is the smallest known *aperiodic* set - it is the smallest set of tiles which can tile the plane, but not periodically. Allowed juxtapositions of tiles are enforced by matching rules, and these in turn induce a Hamiltonian: Every matching rule violation is penalized by a positive energy, while allowed matchings have zero energy. In what follows, we shall denote the energy cost of mismatching adjacent vertical or horizontal edges by $J_{x}$ or $J_{y}$, respectively. We shall argue that the equilibrium behavior of this system is analogous to the Frenkel-Kontorova (FK) model[@chaikin_principles_2000] of the commensurate-incommensurate (CI) transition. The FK model describes a chain of masses which are connected by springs and subjected to a periodic substrate potential. It exhibits rich behavior due to competition between these two interactions, each of which favors ordering with a different wavelength. However, in marked contrast with the FK model, where the favored length scales are present in the Hamiltonian, in the KC model they emerge spontaneously, from only nearest-neighbor interactions. Moreover, the role of substrate potential strength in the FK model is played by the temperature in the KC model. In this sense, the KC system exhibits an [*entropic CI*]{} transition. As indicated in Figure \[tiles\], the 13 KC tiles may be divided into two groups, which we will call types *A* and *B*. The markings on the edges of the tiles indicate the matching rules - abutting edges of adjacent tiles have the same markings in a perfect tiling. It is readily seen that in an undefected tiling, a given row can consist of *A* or *B* type tiles only, with no mixing, and thus, we may characterize a row as being *A-type* or *B-type.* The KC tiling differs from other tilings studied in that it is not generated by recursive substitution (inflation)[@tilings_and_patterns], and by the fact that it has a ground state degeneracy with extensive entropy*.* This should be contrasted with tilings created from a simple inflation rule where the degeneracy scales as a power of the system size[@koch_modelling_2010]. All the ground states, however, are characterized by a quasiperiodic arrangement of *A-*type and *B-*type ** rows. The finite temperature behavior of this model is striking - as the temperature is increased from zero, the rows, still identifiable as *A* or *B* type, order periodically with decreasing period. At high enough temperatures, the rows lose their *A* or *B* identity, and the system becomes disordered. ![The $13$ KC tiles. The seven upper tiles are type *A*, and the six lower tiles are type *B.* Note that $\frac{0}{3},\,0$ and $0'$ are considered different markings.[]{data-label="tiles"}](tiles2) The proof that there exists a zero-energy ground state is equivalent to showing that a perfect tiling exists. To do this[@Eigen], we note that there are two numbers which characterize the $n^{th}$ row in a perfect tiling - the “frequency” $\alpha_{n}$ and the number $q_{n}$ which are related through the mapping $$\alpha_{n+1}=q_{n}\,\alpha_{n}\label{eq:mapping}$$ where $$q_{n}=\begin{cases} 2 & \;\;\frac{1}{3}\leq\,\alpha_{n}<1\\ \frac{1}{3} & \;\;1\leq\,\alpha_{n}<2 \end{cases}\label{eq:q_of_m}$$ Note that we shall take *n* to increase in the *-y* direction. The reason for dividing the tile set into the *A* and *B* groups becomes apparent if we denote the markings of the *top*, *bottom*, *right*, and *left* edges of a tile by {*t,b,r,l*}, as shown in Figure \[basic\_tile\]. When needed, we shall indicate the position of the tile as a subscript; thus $t_{m,n}$ refers to the marking of the top edge of the tile centered at (*m,n*), *etc*. It is easily verified by inspecting Figure \[tiles\] that the markings on each tile satisfy the relation $$q_{n}\, t_{m,n}+l_{m,n}=r_{m,n}+b_{m,n}\label{eq:sym}$$ with a tile of type $A$ or *$B$* having $q_{n}=2\,$ or $\frac{1}{3}$, respectively. The mapping $\alpha_{n+1}=q_{n\,}\alpha_{n}$ has no periodic points, since $\alpha_{n+q}/\alpha_{n}=2^{p}/3^{q-p}\ne1$ for any positive integers $p$ and $q$; this implies that the resultant tilings are not periodic. The $\alpha_{n}$ are distributed densely, but not uniformly, in the range $\left(\frac{1}{3},2\right)$. To show that a perfect tiling exists, we must solve Equations \[eq:sym\] for all *m,n* with $q_{n}$ derived from Equations \[eq:mapping\] and \[eq:q\_of\_m\] whilst demanding that $b_{m,n}=t_{m,n+1}$ and $l_{m,n}=r_{m-1,n}$. It is easily verified that the markings $$\begin{aligned} t_{m,n} & = & \left\lfloor m\alpha_{n}\right\rfloor -\left\lfloor (m-1)\alpha_{n}\right\rfloor \label{eq:t&r}\\ r_{m,n} & = & q_{n}\left\lfloor m\alpha_{n}\right\rfloor -\left\lfloor mq_{n}\alpha_{n}\right\rfloor \nonumber \end{aligned}$$ constitutes a solution, where $\left\lfloor x\right\rfloor $ is the greatest integer less than or equal to $x$ (see Figure \[basic\_tile\]); this is one example of a perfect tiling. ![The marking scheme for the tile at (*m,n*). Note that *n* increases in the *-y* direction. []{data-label="basic_tile"}](tile) To facilitate later discussion, we shall employ a useful mapping. Let us define the variable $\phi_{n}=n\omega_{0}$ , where $\omega_{0}=log2/log\left(6\right)\approx0.3868$. Decomposing $\phi_{n}$ into its integer and fractional parts gives $$\phi_{n}=\lfloor n\omega_{0}\rfloor+\left\{ \phi_{n}\right\} \label{eq:Phi}$$ where $\left\{ \phi_{n}\right\} $, the fractional part of $\phi_{n}$, is defined through its relation to $\alpha_{n}$ by $$\left\{ \phi_{n}\right\} =\frac{log\left(\alpha_{n}\right)+log\left(3\right)}{log\left(2\right)+log\left(3\right)}\label{eq:theta}$$ This maps $\alpha_{n}\in[\frac{1}{3},2)$ onto $\left\{ \phi_{n}\right\} \in[0,1)$, and allows us to obtain[@futurepub] an explicit formula for $q_{n}$ : $$q_{n}=\left(\frac{1}{3}-2\right)\left(\left\lfloor \left(n+1\right)\omega_{0}\right\rfloor -\left\lfloor n\omega_{0}\right\rfloor \right)+2$$ Sequences of this type are called *Sturmian sequence*s, and are well known in the context of automatic sequences[@allouche_automatic_2003]. Here $\omega_{0}$ is irrational, and this gives a quasiperiodic sequence of the two “letters” $2$ and $\frac{1}{3}$, with the consequence that in the ground state the rows appear in a quasiperiodic sequence. ![An example of two different 2x2 patches with the same outer markings; these may be exchanged with no matching rule violations.[]{data-label="swaps"}](tiles_red) As noted above, this ground state is only one of many, and in fact, there is an extensive ground state entropy[@futurepub]. This may be inferred from Figure \[swaps\], where two patches with the same markings on their exterior are presented (there exist larger patches with the same property as well). This means that starting from some given ground state, we may obtain another by randomly exchanging the patches shown in Figure \[swaps\] (and any other pair of patches with the same exterior markings), provided, as we have verified[@futurepub], that they appear with a finite density in the ground state. This implies that almost all of the ground states are disordered in the sense that their patch entropy[@kurchan_levine] scales as the patch size for large enough patches. This notwithstanding, in all the ground states, the rows are arranged in a quasiperiodic sequence of *A-type* and *B-type,* and it is also true that $\alpha_{n}$ is unchanged for each row[@futurepub], which is relevant to what follows. Numerical studies of this model show that as the temperature is raised from 0, it goes through a series of phase transitions, where the *A-B* sequence of rows is periodic, and where the period decreases with increasing temperature. In analogy to the CI transition, we shall refer to these periodic phases as *commensurate phases.* In Figure \[period8\], we show a portion of a time averaged configuration from a $150\times150$ system (with $J_{x}=J_{y}$) at $T=0.304$ (in units of $J_{x}$), which was obtained by parallel tempering, where the color coding is as in Figure \[tiles\]. The *A-B* sequence is periodic with period 8, with 3 *B* rows per period. Characteristic defects are also present. At high enough temperature, of course, the rows lose their *A* or *B* character, and the system goes to a disordered phase. ![Finite temperature configuration averaged over time exhibiting 3/8 periodicity. The row structure is evident, with colors as in Figure \[tiles\]. Here $L=150$, $J_{x}=J_{y}$, and $T=0.304$.[]{data-label="period8"}](AvgConfig150x150_T0_304_period8CloseUp_color) ![$\Omega_{max}$ and $C_{V}$ as a function of temperature, for a $150\times150$ system with $J_{x}=J_{y}$. **.** []{data-label="omega_max"}](\string"Cv_omegaMax_150x150_improved\string".pdf) The different phase transitions are best traced by the winding number $\Omega=\frac{N_{B}}{N_{A}+N_{B}}$. Since at low *T* the rows are essentially pure type *A* or *B*, $\Omega$ ** essentially counts the fraction of type *B* rows in the system. ** For an ensemble of systems at any given temperature, $\Omega$ may take a variety of values, where the distribution has a well-defined maximum value, $\Omega_{max}$, which is typically close but not identical to the average $\langle\Omega\rangle$ [^1]. For large systems, both $\Omega_{max}$ and $\langle\Omega\rangle$ will equal $\omega_{0}\approx0.3868$ at $T=0$ and approach $\frac{6}{13}\sim0.46$ as $T\rightarrow\infty$. The first order nature of the transitions between the different commensurate phases can be verified by looking at the distribution of $\Omega$ near the transitions, which exhibits two well separated peaks, with one overtaking the other as the transition temperature is crossed [@futurepub]. This suggests that $\Omega_{max}(T)$ is a reliable indicator of these transitions. In Figure \[omega\_max\] we show curves of $\Omega_{max}$ and the specific heat $C_{V}$ *vs. T*, for a $150\times150$ system. At low temperature, the system is in its ground state, and $\Omega_{max}=\omega_{0}$ (to within $\frac{1}{L}$). As temperature is increased, $\Omega_{max}$ undergoes three jump discontinuities before becoming fully continuous as the rows are no longer homogeneous. While the step-wise behavior of $\Omega_{max}$ is a clear indication of the existence of the commensurate phases and the first order nature of the transitions between them, its value at the plateaus does not give the exact periodicities due to limited resolution and the existence of defects, and these then must be inferred by looking at the configurations themselves, as in Figure \[period8\]. The first order transitions are accompanied by small bumps in the specific heat at the same temperatures. In Figure \[omega\_max\], we identify phases with periods of $8$ and $13$ rows with the lower and middle plateaus, respectively. In our examinations of systems of sizes up to $L=300$, we have identified phases with $\Omega=\frac{1}{3},\frac{3}{8},\frac{5}{13},$ and $\frac{12}{31}$, depending on system size and values of the coupling constants. At these temperatures, we have verified numerically that the rows substantially maintain their pure *A* or *B* character. The broad peak in the specific heat at $T\sim.37$ attends the loss of purity in the rows. These results may be understood by an effective coarse-grained description of this system, appropriate for low temperatures, when the system may be considered as composed of *A* and *B* type rows[^2]. This description resembles the Frenkel-Kontorova model[@chaikin_principles_2000], and exhibits a competition between length scales. To see this, note that for the perfect ** tiling, the value of the frequency $\alpha_{n}$ for the $n^{th}$ row may be calculated from the tile markings: $\alpha_{n}=\lim_{L\rightarrow\infty}\frac{1}{L}\sum_{m=1}^{L}t_{mn}$. From this, using Equations \[eq:Phi\] and \[eq:theta\], we can compute $\phi_{n}$. Now although defects enter the rows at finite $T$, we may still define two frequencies, using the markings of the top and bottom of a row. The “top frequency” is defined as $\alpha_{n}^{T}=\frac{1}{L}\sum_{m=1}^{L}t_{mn}$ while the “bottom frequency” is given by $\alpha_{n}^{B}=\frac{1}{L}\sum_{m=1}^{L}b_{mn}$. For a perfect tiling, $\alpha_{n}^{B}=\alpha_{n+1}^{T}$, and therefore, from Equation \[eq:mapping\], $\alpha_{n}^{B}/\alpha_{n}^{T}=q_{n}$, but this will typically not be the case for defected tilings. This notwithstanding, the $\phi_{n}$ at finite *$T$* may be inferred from $\alpha_{n}^{T}$ using Equation \[eq:theta\] in the same manner as for the perfect tiling. These will be the variables used in our effective description. To construct an effective free energy for this model we assume that the dominant contributions come from the entropy of the rows, and the energy due to mismatches between the rows, each of which is characterized by its frequencies $\alpha^{T}$ and $\alpha^{B}$ . We argue that the energy cost associated with an imperfect interface between rows $n$ and $n+1$ goes as $LJ_{y}\left|\alpha_{n}^{B}-\alpha_{n+1}^{T}\right|$, where *L* is the length of the row. Clearly, this term is zero in the ground state, and numerical simulations bear out this functional form for low temperatures[@futurepub]. At this level of coarse graining, a row is characterized by its frequency $\alpha^{T}$ (or equivalently $\alpha^{B}$), and its entropy should be a function of this frequency which is extensive in *L,* so that we shall write the entropy of the $n^{th}$ row as $L\,\tilde{s}\left(\alpha_{n}^{T}\right)$. Taken together, we get that the free energy is given by $F/L=\sum_{n=1}^{L}J_{y}\left|q_{n}\alpha_{n}^{T}-\alpha_{n+1}^{T}\right|-T\tilde{s}\left(\alpha_{n}^{T}\right)$. It is convenient to express this in terms of the variables $\phi_{n}$ discussed above. The free energy is then of the form $$\frac{F}{L}=\sum_{m}J_{y}g\left(\phi_{m},\phi_{m+1}\right)-Ts\left(\phi_{m}\right)\label{eq:FreeEnergy-2}$$ where $g\left(\phi_{m},\phi_{m+1}\right)$ is a function that favors $\phi_{m+1}=\phi_{m}+\omega_{0}$, which holds identically in the ground state. The entropy $s\left(\phi_{n}\right)$ depends only on the fractional part $\left\{ \phi_{n}\right\} $, and thus it is a periodic function with period one. It is the competition between these two length scales which gives the novel behavior observed. Although it is tempting to expand $g\left(\phi_{n},\phi_{n+1}\right)$ to first order as $g\left(\phi_{n},\phi_{n+1}\right)\propto\left|\phi_{n+1}-\phi_{n}-\omega_{0}\right|$, we note that such an expression fails when both $\left\{ \phi_{n}\right\} $ and $\left\{ \phi_{n+1}\right\} $ are larger than $1-\omega_{0}$ , since this would imply two adjacent *B* rows, which carries a disproportionately large energy cost. The equilibrium configuration of the $\phi_{n}$ can be obtained by minimizing $F$. As in the FK model, the first term favors an incommensurate phase with a winding number $\omega_{0}$ while the entropy favors a commensurate configuration. The temperature $T$ plays the role of the strength of the periodic potential, so that commensurate phases are expected at high temperature while incommensurate phases are expected at low temperatures. The commensurate phases that are expected are those with a winding number close to $\omega_{0}$ such as $1/3,\,3/8,\,5/13$, *etc*. We have observed some of these phases in our numerical study, as seen in Figure \[omega\_max\], where their presence is indicated by the plateau values of $\Omega_{max}$. At still higher temperatures the segregation into type *A* and *B* rows breaks down, resulting in a disordered phase. It is interesting to speculate about the low *T* behavior of this system. It might be that only at $T=0$ an incommensurate phase appears, but it could be that such a phase, possibly with power-law correlations, is stable at finite *T.* These issues will be addressed in future work. We would like to thank P. Chaikin, J. Kurchan, T. Lubensky, F. Sausset, and G. Wolff for fruitful discussions. D.L. gratefully acknowledges support from Israel Science Foundation grant 1574/08 and US-Israel Binational Science Foundation grant 2008483. [10]{} As distinct, for example, from the Heisenberg or x-y models, which employ continuous spins. Wang tilings employ square tiles, and as such may be used to model systems on a square lattice. J. Allouche and J. Shallit. . Cambridge University Press, 2003. D. Aristoff and C. Radin. First order phase transition in a model of quasicrystals. , 44(25):255001, 2011. T. Byington and J. Socolar. Hierarchical freezing in a lattice model. , 108:045701, 2012. P. M. Chaikin and T. C. Lubensky. . Cambridge University Press, reprint edition, 2000. K. Culik [II]{}. An aperiodic set of 13 wang tiles. , 160:245–251, 1996. J. Eigen, S. Navarro and V. Prasad. An aperiodic tiling using a dynamical system and beatty sequences. , 54:207, 2007. B. Grünbaum and G. Shephard. . W.H. Freeman and Company, reprint edition, 1987. J. Kari. A small aperiodic set of wang tiles. , 160:259–264, 1996. H. Koch and C. Radin. Modelling quasicrystals at positive temperature. , 138(1):465–475, 2010. J. Kurchan and D. Levine. Order in glassy systems. , 44(3):035001, 2011. L. Leuzzi and G. Parisi. Thermodynamics of a tiling model. , 33(23):4215–4225, 2000. J. Miekisz. A microscopic model with quasicrystalline properties. , 58(5):1137–1149, 1990. N. Nikola, D. Hexner, and D. Levine. To be published. C. Radin. The pinwheel tilings of the plane. , 139(3):pp. 661–702, 1994. Z. Rotman and E. Eisenberg. Finite-temperature liquid-quasicrystal transition in a lattice model. , 83(1):011123, 2011. S.-i. Sasa. Pure glass in finite dimensions. , 2012. S.-i. Sasa. Statistical mechanics of glass transition in lattice molecule models. , 45(3):035002, 2012. G. Wolff and D. Levine. . [^1]: The distinction between $\Omega_{max}$and $\langle\Omega\rangle$ will vanish in the thermodynamic limit. [^2]: Although this is easiest to justify when $J_{x}>J_{y}$, our numerical study indicates that it has a large range of validity even when $J_{x}=J_{y}$.

--- abstract: 'In the brain, the structure of a network of neurons defines how these neurons implement the computations that underlie the mind and the behavior of animals and humans. Provided that we can describe the network of neurons as a graph, we can employ methods from graph theory to investigate its structure or use cellular automata to mathematically assess its function. Although, software for the analysis of graphs and cellular automata are widely available. Graph extraction from the image of networks of brain cells remains difficult. Nervous tissue is heterogeneous, and differences in anatomy may reflect relevant differences in function. Here we introduce a deep learning based toolbox to extracts graphs from images of brain tissue. This toolbox provides an easy-to-use framework allowing system neuroscientists to generate graphs based on images of brain tissue by combining methods from image processing, deep learning, and graph theory. The goals are to simplify the training and usage of deep learning methods for computer vision and facilitate its integration into graph extraction pipelines. In this way, the toolbox provides an alternative to the required laborious manual process of tracing, sorting and classifying. We expect to democratize the machine learning methods to a wider community of users beyond the computer vision experts and improve the time-efficiency of graph extraction from large brain image datasets, which may lead to further understanding of the human mind.' bibliography: - 'bibliography.bib' title: 'DeepTEGINN: Deep Learning Based Tools to Extract Graphs from Images of Neural Networks' --- neural network, deep learning, graph, segmentation, cellular automata, in-painting Introduction ============ One of the goals of systems neuroscience is to obtain a mechanistic model that describes and explains how network of neurons in the brain implements perception, thought, and behavior. Because structure implements function in biology, an unavoidable step towards these mechanistic models is to describe the structural connectivity of the network of neurons in the brain. Once a connectivity map (i.e., the description of the network) from a network of neurons is obtained, the it can be described as a graph or a cellular automata. This description may then be used as constraint to leverage methods from graph theory[@hassan1987review] to further analyze the network structure, or its functional complexity [@maccione2012multiscale], which could ultimately contribute in three ways. First, by moving further the understanding on how to use biological substrates for computing [@broersma2017computational; @aaser2017towards]. Second, by improving computing methods in AI by supplying biologically derived network structures that can be used as reservoir networks [@nichele2017deep]. Finally by offering an mathematical abstraction of the biological system that can be used to compare cell cultures with genetic diseases with the healthy ones [@sandvig2018neuroplasticity], providing ground for medical advancements. Nonetheless, obtaining these matrices of structural connections between the nodes of the nervous system is a challenging and cumbersome endeavour. Especially from microscopy images (see Figure \[fig:MEA\]. Although some methods have been recently developed to automatize the process, they still rely on basic image processing steps that demand substantial time to find suitable parameters and curate the results [@dirnberger2015nefi]. Furthermore, these automatic methods are not very robust, and as an effect the gold standard approach still is to trace these connections manually. Additionally, and unlike other biological substrates from which networks can be extracted, the brain is constituted of billions of neurons, with diversified morphology, and a few orders of magnitude higher number of connections (i.e., synapses) [@kandel2000principles]. This additional complexity implies that different nodes in a graph network should have different properties and represent different cell types or structures in the brain. Because this structural information is critical to understand the brain, it is of utmost importance that automatic tools take them into account, which is not currently available to the best of our knowledge. Furthermore, experimental constraints (e.g., multi-electrode arrays, patching pipettes) frequently obstruct the view of part of the image (black lines in Figure \[fig:MEA\]), consequently preventing the connection of nodes that otherwise would be connected. Although these gaps can be easily handled through human intervention (i.e., by estimating the edges that connect two nodes), no simple image processing method can properly handle this problem as the data in the obstructed area is missing. To cope with this problem, one must be able to reconstruct the missing data by inferring how it would look like based on the surrounding area and what is typically known about the morphology. Modern machine learning techniques that leverage the power of convolutional neural networks (ConvNets) may be used to automatize many of the steps mentioned before. In-painting algorithms can be used to estimate missing data caused by image obstruction, object detection algorithms can employed to locate and classify diverse structures in the brain, unsupervised segmentation algorithms can be leveraged to extract skeletonized versions of the image, just to name a few. This would allow for a comprehensive graph extraction from image in neuroscientific settings. The challenge regarding employing ConvNets is that due to its novelty and complexity, deep learning methods are not widely available to non-specialists in computer vision. Additionally, most of these methods require training, which by itself is generally poorly documented, preventing non-expert in computer vision from experimenting and benefiting from ConvNets in their neuroscientific research. Motivated by the foregoing shortcomings, we present a deep learning based toolbox to extract graphs from images of brain tissue. This toolbox is a framework constituted by an extensible library of methods that can be integrated into a computer vision pipeline. The library is based on a combination of standard image processing algorithms available in OpenCV[@Braski2000Opencv] and SKimage [@van2014scikit], and deep learning based methods for object detection, image/line segmentation, in-painting and style transfer implemented in Pytorch [@paszke2017automatic]. Additionally, the toolbox has a graphical user interface (GUI) that simplifies the steps of assembling the graph extraction pipeline, which includes the training of the supervised machine learning algorithms. The main contribution of this paper is to democratize deep learning based methods for computer vision to the neuroscientific community through a reusable, flexible and scalable tool. Through this toolbox, we hope to make deep learning methods more widely accessible to neuroscientists. ![Raw image example as acquired from the microscope. The black lines ending in circle are “blind-spots” created by the multi-electrode array.[]{data-label="fig:MEA"}](MEA1.jpg){width="\linewidth"} Image acquisition ================= The images were prepared as follows: Human cortical neural networks were differentiated and matured from iPSC-derived NSCs (ax0019, Axol bioscience), and fluorescently labelled using a two-color LIVE/DEAD viability/cytotoxicity kit (MP03224, Invitrogen). 0.8ul Ethidium homodimer-1 (2mM in DMSO/H2O 1:4) and 0,4ul Calcein AM (4mM in anhydrous DMSO) was diluted in 2ml PBS and applied to the neural networks for 15 minutes in 37C. The former produces an intense red fluorescence in dead or dying cells, while the latter produces an green fluorescence in live cells. The fluorescently labelled neural networks were imaged with a 10X objective using a automated EVOS 2 fluorescence microscope. Each multi-electrode array (MEA) cell culture chamber was briefly sterilized using ethanol, washed with water, UV-treated over-night, and hydrophilized by application of foetal bovine serum for 30-60 minutes at room temperature. The surface was subsequently double-coated using poly-L-ornithine (0,01%) and laminin. The appropriate neuronal cell culture media were heated to 37C and used to create a single-cell suspension, from which 100.000 cells were seeded directly onto the electrode area of each MEA in a dropwise manner. For some cultures, a feeder-layer of astrocytes (5000 per MEA) was first established, upon which 50 000 neuronal cells were seeded onto. The MEA neuronal cultures were kept in a standard humidified air incubator (5% CO2, 20%O2, 37C), and 50% of the media were changed every 2-3 days. Phase contrast images were acquired at various stages of neuronal differentiation and maturation on the MEAs using the laboratory light microscope Carl Zeiss Axiovert 25 with 5 and 10X objectives. The pipeline ============ The kernel of the toolbox is the graph extraction pipeline. It enables visualization, correction and analysis of the structures depicted in the input image. This pipeline, is constituted by an ordered sequence of methods, which will output a graph representation of the network from the input image. Additionally, some of the steps of the pipeline sequence require preparation (i.e. training). The obtained graph provides weights, edge lengths and node type, which should reflect anatomical structure. The default pipeline combines the following steps: pre-processing, structure detection, segmentation, thinning, graph extraction, graph pruning, and training. What follows is a high-level description of the steps. Pre-processing -------------- Pre-processing involves doing image transformations that allows the subsequent algorithms to perform more robustly. There is a set of image processing steps that may be employed interchangeably. Most of them are standard image transformations like color space change and filtering (including sharpening and blurring), widely available through OpenCV and SKimage libraries. Additionally, deep learning based algorithms were included, namely: style transfer [@gatys2015neural] and in-painting[@liu2018image]. These two methods rely on VGG16 networks pre-trained on ImageNet dataset[@simonyan2014very] and must be further fine-tuned to properly work with the dataset from the experiments (see training). The addition of in-painting(Figure \[fig:pipeline\] D) and style transfer (Figure \[fig:pipeline\] E) enables coping with data lost caused by obstruction of the field of view during experiments, and differences in imaging settings respectively (See black marks in Figure \[fig:MEA\]). {width="\textwidth"} Structure detection ------------------- In order to detect nodes that belong to different cell types, the object detection algorithm, yolov3 (You Only Look Once, version 3.0) [@redmon2018yolov3], was included in the pipeline. This method operates by detecting combinations of spatial features in the image, locating their position and area, and classifying them under a predefined category (namely: astrocytes, neurons and clusters of neurons; see Figure \[fig:pipeline\] F) with an explicit probability. This algorithm requires training, and substantial amount of data to be trained (see training). The center of these detected areas is used by later steps in the pipeline to discriminate synaptic nodes from cell body nodes. Segmentation ------------ We provide two interchangeable avenues for segmentation. Guided watershed [@osma2007improved] and W-Net [@xia2017w]. We noticed that depending on the characteristics of input image these algorithms perform the best. The main goal of this step is to separate the structures that compose the network from everything else and compose a mask. Thinning -------- The next step is to skeletonize the mask so no pixel in the mask has two or more neighbor pixels that belongs to the mask and are neighbor to each other (see Figure \[fig:pipeline\] H, yellow lines). To do that, we implemented the improved Zhang-Suen Thinning algorithm [@chen2012improved]. This method was chosen because it produced less artifacts in the intersection of lines (blobs and missing pixels). Graph extraction ---------------- Once obtained the skeletonized image we then detect the positions of nodes and the edges that connect them so we can create a graph. The graph is generated through the NetworkX [@hagberg2008exploring] library. To detect the nodes, we filter the image with series of 3x3 filters. Each filter represent one possible scenario for a node where the center pixel belongs to the filter and 1 or at least 3 other neighbours also belong to it and are not neighbours to each other. This guarantees that intersection nodes and end-of-the-line nodes are contemplated, but that points that belongs to lines are ignored. Edges between nodes are detected by the following steps. Firstly, the skeleton is segmented in edges by removing the node pixels from the skeletonized mask, each one of these edges has its own label automatically defined as 1 to the number of available edges. Secondly, these segmented edges are dilated. Thirdly, the edges that overlap with two nodes are added to the graph as a bidirectional edge. Finally, the overlapping segmented edge is removed from the set of possible edges. The process repeats until no edges are left (see Figure \[fig:pipeline\] H blue lines). If the Structure Detection step has been executed successfully, the nodes which are closest to the center of the regions of interest generated by the object detection algorithm will acquire the category of the identified object (eg., neuron, cluster). This type of representation of the network in graph is analogous to a connectivity estimation, which is highly relevant in neuroscience to infer functionality. Hence the graph extraction method can contribute to connectivity analysis as performed by Maccione et al. [@maccione2012multiscale] and Ullo et al. [@10.3389/fnana.2014.00137], but without the cumbersome and time-demanding step of extracting the connectivity map manually. Graph pruning ------------- As the last step, it is given to the user the opportunity to edit the graph extracted by removing or adding new edges, tracing new edges between them and assigning properties to each node. Training -------- In order to use the methods that depend on supervised learning, the user has to provide first a set of good examples so the algorithm can be properly trained. This process is usually poorly documented and the data format that the algorithm should receive is usually obscure. We make this step explicit, by declaring exactly what should be the data format, and providing a simple interface that the users can use to generate the training data themselves and train the model. Training in-painting -------------------- To train the in-painting[@liu2018image] network, we need a set of ground truth images, and a set of masks that would match in shape and size the typical artifact obstructing the original image. To generate this data, we segmented the dark areas of original raw image using a simple threshold. We dilated the segmented areas with a 5x5 circular kernel to include the edges of the artifact areas. We randomly cropped the mask image in images of 256x256 pixels. The images in which 1/4th of the area was occupied by the electrode mask were selected for the mask pool. Each mask was then copied 35 times and rotated cumulatively by 10 degrees until we had 36 versions of the same mask in all orientations. To extract the ground-truth images, we cropped patches of 256x256 pixels from the original image where no pixel in the patch overlapped with the cordinates of a pixel belonging to a mask. To expand the dataset, the selected patches were flipped and rotated 90, 180 and 270 degrees. Training object detection ------------------------- To train the Yolov3, we defined regions of interests (ROIs) by drawing a bounding boxes from a subset of patches (100 images, randomly picked). This ROIs are constrained by the vertical and horizontal coordinates of its centroid and its height and with as ratios of the original image. Each ROI is labeled as an instance of a class of objects. In our dataset, we defined three labeled structures: neurons, astrocytes and cluster of neurons. Only neurons and cluster of neurons were relevant for the graph extraction, thus only these two labels are displayed. The labeling of astrocytes was required to prevent falsely detecting astrocytes as neurons. Once training data is available, training follows by pointing the location of the data in the storage unit and running the training function. Because the amount of data was limited. The network was pre-trained with the COCO dataset[@lin2014microsoft] to learn and then fine-tuned and cross-validated using the labeled images. Our implementation of Yolov3 operates by predicting 3 boxes in 2 different scales. Thus, the tensor is N x N x\[2\*(4+1+3)\], where is for the 4 bounding box offsets, 1 for objectness prediction, and 3 is for the class predictions. Furthermore we chose 6 clusters in the k-means algorithm to establish our binding box priors. In our dataset the 6 clusters were (7x9), (15x16), (22x19), (31x32), (55 x 49), (89x91). The training progress was displayed on every set of epochs, which could defined by the user, and it could be interrupted at any time. The set of weights with the smaller error was highlighted to facilitate the use of the pipeline. Discussion ========== Many solutions exist to extract network graphs from images, including some generic flexible tools. But these tools assume that the network to be extracted is homogeneous (i.e., all the nodes are equal). This is a major problem in neuroscience because biological neurons form highly heterogeneous networks. In particular, the tools available cannot account for the difference between neurons and synapses as nodes. Additionally, they cannot account for differences between cell types (i.e., neurons vs. glia). This is a major source of error for describing a network. The abundance of false positives can lead to a description that is much bigger and dense, hence increasing the level of complexity which by itself increases the challenge of analysis. Our toolbox circumvent this problem by integrating machine learning methods into an easy to use pipeline to extract graphs from network of neurons. Because the algorithm detects objects by category, further developments may be implemented to extract sub-populations of neurons and include more cell types. One possible avenue is to develop a specific dataset for brain cell-type detection, which is currently unavailable in the best of our knowledge. A second major challenge in the path of automatizing graph extraction from images of cultured cells is that these images often come with major artifacts (e.g., electrodes, pipettes, objects that obstruct the view). We eliminate these artifacts by combining in-painting techniques and style transfer through deep learning methodologies. Although not perfect, we demonstrate that both techniques can provide qualitatively satisfactory results. Allowing to reconstruct a plausible network, despite the artifact. One further point of development could be to apply techniques to improve the resolution, as it may increase the performance of in-painting and style transfer techniques. We anticipate that this toolbox will enable neuroscientists to extract graphs from network of neurons in a more time-efficient way and consequently contribute in the pursue of the understanding of perception, intelligence and behavior. Acknowledgment {#acknowledgment .unnumbered} ============== This work was supported by Norwegian Research Council SOCRATES project (grant number 270961) and received internal support as a lighthouse project in Computer Vision from the Faculty of Technology, Art and Design (TKD) at Oslo Metropolitan University, Norway. Repository {#repository .unnumbered} ========== Code and example data can be found in the following repository: https://github.com/gmorenomello/deepteginn

--- abstract: 'We prove optimal lower bounds for the growth of the energy over balls of minimizers to the vectorial Allen-Cahn energy in two spatial dimensions, as the radius tends to infinity. In the case of radially symmetric solutions, we can prove a stronger result in all dimensions.' address: 'Department of Mathematics and Applied Mathematics, University of Crete.' author: - Christos Sourdis title: 'Optimal energy growth lower bounds for a class of solutions to the vectorial Allen-Cahn equation' --- Consider the semilinear elliptic system $$\label{eqEq} \Delta u=\nabla W(u)\ \ \textrm{in}\ \ \mathbb{R}^n,\ \ n\geq 1,$$ where $W:\mathbb{R}^m\to \mathbb{R}$, $m\geq 1$, is sufficiently smooth and nonnegative. This system has variational structure, and solutions in a smooth bounded domain $\Omega\subset \mathbb{R}^n$ are critical points of the energy $$E(v;\Omega)=\int_{\Omega}^{}\left\{\frac{1}{2}|\nabla v|^2+ W(v) \right\}dx$$ (subject to their own boundary conditions). A solution $u\in C^2(\mathbb{R}^n;\mathbb{R}^m)$ is called globally minimizing if $$E(u;\Omega)\leq E(u+\varphi;\Omega)$$ for every smooth bounded domain $\Omega\subset \mathbb{R}^n$ and for every $\varphi\in W^{1,2}_0(\Omega;\mathbb{R}^m)\cap L^\infty(\Omega;\mathbb{R}^m)$ (see also [@fuscoCPAA] and the references therein). In the scalar case, namely $m=1$, Modica [@modica] used the maximum principle to show that every bounded solution to (\[eqEq\]) satisfies the pointwise gradient bound $$\label{eqmodica} \frac{1}{2}|\nabla u|^2\leq W(u) \ \ \textrm{in}\ \ \mathbb{R}^n,$$ (see also [@cafamodica] and [@farinaFlat]). Using this, together with Pohozaev identities, it was shown in [@modicaProc] that the energies of such solutions satisfy the following monotonicity property: $$\label{eqmonotonicity} \frac{d}{dR}\left(\frac{1}{R^{n-1}}\int_{B(x_0,R)}^{}\left\{\frac{1}{2}|\nabla u|^2+ W\left(u\right) \right\}dx\right)\geq 0,\ \ R>0,\ x_0\in \mathbb{R}^n,$$ where $B(x_0,R)$ stands for the $n$-dimensional ball of radius $R$ that is centered at $x_0$. Combining the above two relations yields that, if $x_0\in \mathbb{R}^n$, the “potential” energy of each bounded nonconstant solution to the scalar problem satisfies the lower bound: $$\label{eqenergyLowerModi} \int_{B(x_0,R)}^{}W\left(u\right)dx\geq c R^{n-1},\ \ R>0,\ \ \textrm{for some}\ c>0.$$ In the scalar case, the most famous representative of this class of equations is the Allen-Cahn equation $$\label{eqallenSca} \Delta u=u^3-u\ \ \textrm{in}\ \ \mathbb{R}^n, \ \ \textrm{where}\ \ W(u)=\frac{(1-u^2)^2}{4},$$ which is used to model phase transitions (see [@farinaState] and the references therein). In the vectorial case, that is when $m\geq 2$, in the absence of the maximum principle, it is not true in general that the gradient bound (\[eqmodica\]) holds (see [@smyrnelis] for a counterexample). Nevertheless, it was shown by Alikakos [@alikakosBasicFacts] using a stress energy tensor (see also [@serfaty]), and earlier by Caffarelli and Lin [@caffareliLin] via Pohozaev identities, that the energy of every solution to (\[eqEq\]) (not necessarily bounded) satisfies the following weak monotonicity property: $$\label{eqmonotoniWeak} \frac{d}{dR}\left(\frac{1}{R^{n-2}}\int_{B(x_0,R)}^{}\left\{\frac{1}{2}|\nabla u|^2+ W\left(u\right) \right\}dx\right)\geq 0,\ \ R>0,\ x_0\in \mathbb{R}^n, \ n\geq 2.$$ In fact, as was observed in the former reference, if a solution $u$ satisfies Modica’s gradient bound (\[eqmodica\]), it follows that its energy satisfies the strong monotonicity property (\[eqmodica\]). Armed with (\[eqmonotoniWeak\]), and doing some more work in the case $n=2$ (see [@alikakosBasicFacts]), it is easy to show that, if $x_0\in \mathbb{R}^n$, the energy of each nonconstant solution to the system (\[eqEq\]) satisfies: $$\label{eqGrande} \int_{B(x_0,R)}^{}\left\{\frac{1}{2}|\nabla u|^2+ W\left(u\right) \right\}dx\geq \left\{\begin{array}{ll} c R^{n-2} & \textrm{if}\ n\geq 3, \\ & \\ c \ln R & \textrm{if}\ n=2, \end{array} \right.$$ for all $R>1$ and some $c>0$. The above results hold for arbitrary smooth and nonnegative $W$. If additionally $W$ vanishes at least at one point, it is easy to cook up a suitable competitor for the energy and show that bounded globally minimizing solutions satisfy $$\int_{B(x_0,R)}^{}\left\{\frac{1}{2}|\nabla u|^2+ W\left(u\right) \right\}dx\leq CR^{n-1},\ \ R>0,\ x_0\in \mathbb{R}^n,$$ for some $C>0$ (see for example [@ambrosioCabre Rem. 2.3]). The system (\[eqEq\]) with $W\geq 0$ vanishing at a finite number of global minima is used to model multi-phase transitions (see [@bronReih] and the references therein). In this case, the system (\[eqEq\]) is frequently referred to as the vectorial Allen-Cahn equation. Under appropriate assumptions (symmetries or non-degeneracy assumptions), it is possible to construct by variational methods “heteroclinic” solutions that “connect” the global minima of $W$ (see [@fuscoPreprint; @guiSchatz; @saez] and the references therein); the energy of these solutions over $B(x_0,R)$ is of order $R^{n-1}$ as $R\to \infty$. This observation implies that the estimate (\[eqGrande\]) is far from optimal for this class of $W$’s. On the other side, for the case of the Ginzburg-Landau system $$\Delta u=\left(|u|^2-1 \right)u,\ \ u:\mathbb{R}^2\to \mathbb{R}^2, \ \ \left(\textrm{here}\ W(u)=\frac{\left(1-|u|^2\right)^2}{4}\ \textrm{vanishes on}\ \mathbb{S}^1 \right),$$ there are globally minimizing solutions with energy over $B(x_0,R)$ of order $\ln R$ as $R\to \infty$ (see [@bethuel; @serfaty] and the references therein). In other words, the estimate (\[eqGrande\]) captures the optimal growth in the case of globally minimizing solutions to the Ginzburg-Landau system. In this note, we will establish the corresponding optimal lower bound in the case of the phase transition case when $n=2$. In fact, we will prove the analog of the lower bound (\[eqenergyLowerModi\]). As will be apparent, our proof does not work if $n\geq 3$. To the best of our knowledge, there is no related published result. Our approach combines ideas from two disciplines: - We adapt to this setting clearing-out arguments from the study of the Ginzburg-Landau system, see [@bethuel]. - We employ variational maximum principles for globally minimizing solutions that have been recently devised and used for the study of the vectorial Allen-Cahn equation in [@alikakosPreprint]. Our main result is the following. \[thmMine\] Assume that $W\in C^1(\mathbb{R}^m;\mathbb{R})$, $m\geq 1$, and that there exist finitely many $N\geq 1$ points $a_i\in \mathbb{R}^m$ such that $$\label{eqpoints} W(u)>0\ \ \textrm{in}\ \ \mathbb{R}^m\setminus \{a_1,\cdots, a_N \},$$ and there exists small $r_0>0$ such that the functions $$\label{eqmonot} r\mapsto W(a_i+r\nu),\ \ \textrm{where}\ \ \nu \in \mathbb{S}^1, \ \ \textrm{are strictly increasing\ for}\ r\in (0,r_0),\ \ i=1,\cdots,N.$$ Moreover, we assume that $$\label{eqinf}\liminf_{|u|\to \infty} W(u)>0.$$ If $u\in C^2(\mathbb{R}^2;\mathbb{R}^m)$ is a bounded, nonconstant, and globally minimizing solution to the elliptic system $$\label{eqEq2} \Delta u=\nabla W(u)\ \ \textrm{in}\ \ \mathbb{R}^2,$$ for any $x_0\in \mathbb{R}^2$, there exist constants $c_0, R_0>0$ such that $$\int_{B(x_0,R)}^{}W\left(u(x) \right) dx\geq c_0R\ \ \textrm{for}\ \ R\geq R_0.$$ Since the problem is translation invariant, without loss of generality, we may carry out the proof for $x_0=0$. Suppose, to the contrary, that there exists a bounded, nonconstant, and globally minimizing solution $u$ and radii $R_j\to \infty$ such that $$\label{eqcontra} \int_{B(0,R_j)}^{}W\left(u(x) \right) dx=o(R_j)\ \ \textrm{as}\ \ j\to \infty.$$ By the co-area formula (see for instance [@evans Ap. C]), the nonnegativity of $W$, and the mean value theorem, there exist $$s_j\in \left(\frac{R_j}{2},R_j\right)$$ such that $$\int_{\partial B(0,s_j)}^{}W\left(u(x) \right) dS(x)=o(1)\ \ \textrm{as}\ \ j\to \infty.$$ From this, as in the clearing-out lemma of [@bethuel], it follows that $$\label{eqbdry} \max_{|x|=s_j}W\left(u(x) \right)=o(1)\ \ \textrm{as}\ \ j\to \infty.$$ Indeed, if not, passing to a subsequence if necessary, there would exist $c_1>0$ and $x_j\in \partial B(0,s_j)$ such that $$W\left(u(x_j) \right)\geq c_1\ \ \textrm{for}\ \ j\geq 1.$$ On the other hand, since $u$ is bounded in $\mathbb{R}^2$, by standard interior elliptic regularity estimates (see [@evans; @Gilbarg-Trudinger]), the same is true for $\nabla u$. Hence, there exists $r_*>0$ such that $$W\left(u(x) \right)\geq \frac{c_1}{2},\ \ x\in B(x_j,r_*),\ \ \textrm{for}\ \ j\geq 1,$$ which implies that $$\int_{\partial B(0,s_j)}^{}W\left(u(x) \right) dS(x)\geq \frac{c_1}{2} \mathcal{H}^{1}\left\{B(x_j,r_*)\cap \partial B(0,s_j) \right\}\geq c_2\ \ \textrm{for}\ \ j\geq 1.$$ for some $c_2>0$. Clearly, the above relation contradicts (\[eqcontra\]). In view of (\[eqinf\]), relation (\[eqbdry\]) implies that there exist $i_j\in \{1,\cdots, N \}$ such that $$\max_{|x|=s_j}\left|u(x)-a_{i_j} \right|=o(1)\ \ \textrm{as}\ \ j\to \infty.$$ By virtue of (\[eqmonot\]), exploiting the fact that $u$ is a globally minimizing solution, we can apply a recent variational maximum principle from [@alikakosPreprint] to deduce that $$\max_{|x|\leq s_j}\left|u(x)-a_{i_j} \right|\leq \max_{|x|=s_j}\left|u(x)-a_{i_j} \right|\ \ \textrm{for} \ \ j\gg 1,$$ (so that the righthand side is smaller than $r_0/2$). The idea is that, if this is violated, one can construct a suitable function which agrees with $u$ on $\partial B(0,s_j)$ but with less energy, thus contradicting the minimality of $u$. The above two relations imply the existence of an $i_0\in \{1,\cdots,N\}$ such that $$\max_{|x|\leq s_j}\left|u(x)-a_{i_0} \right|=o(1)\ \ \textrm{as}\ \ j\to \infty.$$ Now, letting $j\to \infty$ in the above relation yields that $u\equiv a_{i_0}$ which contradicts our assumption that $u$ is nonconstant. Clearly, the monotonicity condition (\[eqmonot\]) is satisfied if the global minima are nondegenerate (the Hessian matrix $D^2W(a_i)$ is invertible for all $i=1,\cdot, N$). In dimensions $n\geq 2$, a Liouville type theorem of Fusco [@fuscoCPAA] tells us that, if $W$ is as in Theorem \[thmMine\], nonconstant global minimizing solutions to (\[eqEq\]) must enter any neighborhood of at least two of the global minima. Intuitively, this suggests that the optimal lower bound for the growth of the energy, that is kinetic (or interfacial) and potential, should be of order $R^{n-1}$ in all dimensions. In this regard, see [@cafa-cordoba; @modica] for the scalar case ($m=1$), provided that the global minima are nondegenerate. If we restrict our attention to radially symmetric solutions, we have a stronger result which follows at once from the following proposition which is of independent interest. Let $W\in C^{1}(\mathbb{R}^m;\mathbb{R})$, $m\geq 1$, possibly sign-changing. If $u\in C^2(\mathbb{R}^n;\mathbb{R}^m)$, $n\geq 2$, satisfies (\[eqEq\]), we have that $$\label{eq1} \frac{1}{2}\left|u'(R)\right|^2\leq W\left(u(R) \right)-W\left(u(0) \right),\ \ R>0,$$ and $$\label{eq2} \frac{d}{dR}\left(\frac{1}{R^n}\int_{B(0,R)}^{}\left\{\frac{n-2}{2}|\nabla u |^2+ n W(u)\right\} dx\right)\geq 0,\ \ R>0.$$ We know that $$u''+\frac{n-1}{r}u'-\nabla W(u)=0,\ \ r>0,\ \ u'(0)=0.$$ So, letting $$e(r)=\frac{1}{2}\left|u'(r)\right|^2-W\left(u(r) \right),\ \ r>0,$$ we find that $$\label{eq3} e'(r)=-\frac{n-1}{r}\left|u'(r)\right|^2,\ \ r>0.$$ Then, estimate (\[eq1\]) follows at once by integrating the above relation over $(0,R)$. By Pohozaev’s identity (the idea is to test the equation by $r u'(r)$, see for instance [@serfaty Ch. 5]), for $R>0$, we have that $$\frac{1}{R}\int_{B(0,R)}^{}\left\{\frac{n-2}{2}|\nabla u |^2+ n W(u)\right\}dx= \int_{\partial B(0,R)}^{}\left\{W\left(u(R) \right)-\frac{1}{2}\left|u'(R)\right|^2 \right\}dS=-\mathcal{H}^1\left\{\mathbb{S}^1\right\}R^{n-1}e(R).$$ Then, we can arrive at (\[eq2\]) by dividing both sides by $R^{n-1}$, differentiating, and using (\[eq3\]). Radial solutions to the Allen-Cahn equation (\[eqallenSca\]), decaying to zero in an oscillatory manner, as $r\to \infty$, have been constructed in [@guiRadial]. [50]{} , *Some basic facts on the system $\Delta u - W_u(u) = 0$*, Proc. Amer. Math. Soc. **139** (2011), 153-–162. , and [G. Fusco]{}, *A maximum principle for systems with variational structure and an application to standing waves*, arXiv:1311.1022 , and [X. Cabré]{}, *Entire solutions of semilinear elliptic equations in $\mathbb{R}^3$ and a conjecture of De Giorgi*, J. Amer. Math. Soc. **13** (2000), 725–-739. , [H. Brezis]{}, and [F. Hélein]{}, Ginzburg-Landau vortices, PNLDE **13**, Birkhäuser Boston, 1994. , and [F. Reitich]{}, *On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation*, Archive for Rational Mechanics and Analysis **124** (1993), 355–379. , [N. Garofalo]{}, and [F. Segála]{}, *A gradient bound for entire solutions of quasi–linear equations and its consequences*, Comm. Pure Appl. Math. **47** (1994), 1457–-1473. , and [A. Córdoba]{}, *Uniform convergence of a singular perturbation problem*, Comm. Pure Appl. Math. **48** (1995), 1-–12. , and [F. Lin]{}, *Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries*, Journal of AMS **21** (2008), 847–862. , Partial differential equations, Graduate studies in mathematics, American Mathematical Society **2** (1998). , and [E. Valdinoci]{}, *Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems*, Arch. Rational Mech. Anal. **195** (2010) 1025–-1058. , and [E. Valdinoci]{}, *The state of the art for a conjecture of De Giorgi and related problems*, in the book: “Recent progress on reaction-diffusion systems and viscosity solutions”, edited by H.Ishii, W.-Y.Lin and Y.Du, World Scientific, 2009. , *Equivariant entire solutions to the elliptic system $\Delta u - W_u(u) = 0$ for general $G$-invariant potentials*, Calc. Var. DOI 10.1007/s00526-013-0607-7 , *On some elementary properties of vector minimizers of the Allen-Cahn energy*, Comm. Pure. Appl. Analysis **13** (2014), 1045–1060. , and [N. S. Trudinger]{}, Elliptic partial differential equations of second order, second ed., Springer-Verlag, New York, 1983. , and [F. Zhou]{}, *Asymptotic behavior of oscillating radial solutions to certain nonlinear equations*, Methods Appl. Anal. **15** (2008), 285–296. , and [M. Schatzman]{}, *Symmetric quadruple phase transitions*, Ind. Univ. Math. J. **57** (2008), 781-–836. , *The gradient theory of phase transition and the minimal interface criterion*, Arch. Rational Mech. Anal. **98** (1987), 123–142. , *A gradient bound and a Liouville theorem for nonlinear Poisson equations*, Comm. Pure Appl. Math. **38** (1985), 679–684. , *Monotonicity of the energy for entire solutions of semilinear elliptic equations*, in Partial differential equations and the calculus of variations, Essays in honor of Ennio De Giorgi, Vol. 2, edited by F. Colombini, A. Marino, and L. Modica. Birkh¨auser, Boston, MA, 1989, 843-–850. , *Existence of a solution to a vector-valued Allen-Cahn equation with a three well potential*, Indiana Univ. Math. J. **58** (2009), 213–268. , and [S. Serfaty]{}, Vortices in the magnetic Ginzburg–-Landau model, Birkhäuser, Basel, 1997. , *Gradient estimates for semilinear elliptic systems and other related results*, arXiv:1401.4847

--- abstract: 'We present the design, fabrication, modeling and feedback control of an earthworm-inspired soft robot that crawls on flat surfaces by actively changing the frictional forces acting on its body. Earthworms are segmented and composed of repeating units called *metameres*. During crawling, muscles enable these metameres to interact with each other in order to generate peristaltic waves and retractable *setae* (bristles) produce variable traction. The proposed robot crawls by replicating these two mechanisms, employing pneumatically-powered soft actuators. Using the notion of controllable subspaces, we show that locomotion would be impossible for this robot in the absence of friction. Also, we present a method to generate feasible control inputs to achieve crawling, perform exhaustive numerical simulations of feedforward-controlled locomotion, and describe the synthesis and implementation of suitable real-time friction-based feedback controllers for crawling. The effectiveness of the proposed approach is demonstrated through analysis, simulations and locomotion experiments.' author: - 'Joey Z. Ge, Ariel A. Calderón, and Néstor O. Pérez-Arancibia[^1][^2]' bibliography: - 'REF.bib' title: '**An Earthworm-Inspired Soft Crawling Robot Controlled by Friction**' --- Introduction {#sec01} ============ Animal locomotion has long been a source of inspiration for robotic research. In particular, the study of limbless crawling has attracted significant attention during the past few years as the most effective method of traveling on unstructured terrains [@ref01; @ref02]. One of the most studied species that travel with a limbless gait is the *nightcrawler*, a type of earthworm (*Lumbricus terrestris*). A typical nightcrawler remains underground during the day and crawls above ground at night. As result of this behavior, they have evolved locomotive mechanisms that enable them to maneuver through their labyrinthine underground burrows and crawl over complex terrains. Specifically, nightcrawlers locomote by employing peristalsis, a motion pattern produced by the coordinated and repeated successive contraction and relaxation of the longitudinal and circular muscles embedded in the animals’ *metameres* (independent body segments). This periodic pattern can be thought of as a retrograde wave that travels along an earthworm’s body to propel it forward using friction-induced traction [@ref03; @ref04; @ref05]. In the case of nightcrawlers, traction is modulated employing microscopic bristle-like skin structures called *setae* [@ref06; @ref07]. Versatility, robustness and spatial efficiency make the nightcrawler’s peristaltic gait a very attractive natural model for robotic locomotion development. Numerous research projects have focused on creating robots that can replicate these earthworms’ peristalsis-based locomotion, adopting a variety of different actuation technologies, including *shape memory alloys* (SMAs) [@ref08; @ref09; @ref10], magnetic fluids [@ref11] and electric motors [@ref12; @ref13; @ref14]. Additionally, recent innovations in fabrication methods have enabled the development of biologically-inspired soft actuators, soft sensors and flexible electronics [@ref15; @ref16; @ref17]. An earthworm-inspired burrowing robot that incorporates these technologies is presented in [@ref18]. That artificial worm was designed to inspect and clean pipes, so its movements and functionalities are constrained to the interior of tubes with diameters in a limited predefined range. As expected, those prototypes are not capable of crawling on open surfaces, which is the problem addressed by the research presented in this paper. Here, we introduce a new soft robot capable of crawling on flat surfaces, whose basic conceptual design is inspired by the functionality of the *abstract notion* of a two-metamere earthworm. In this design, in order to produce the peristalsis-based retrograde waves required for crawling, a single central linear pneumatic actuator produces the deformations and forces that emulate the axial actions of metameres during earthworm locomotion. Two *extremal* pneumatic actuators produce and modulate the friction forces necessary to alternately anchor the robot’s *extremes* to the ground, which is the crucial action in the generation of friction-based crawling. The essential mechanism underlying most forms of terrestrial locomotion is friction. Drawing inspiration from nature, researchers have developed several different methods to exploit friction forces, including gecko-inspired adhesives [@ref19], microspine-based anchors [@ref20; @ref21] and anisotropic friction mechanisms [@ref22; @ref23]. In the context of soft robotics, [@ref24] presents a robot that employs materials with different coefficients of friction and a pair of unidirectional clutches to manipulate frictional forces to generate locomotion. In the robot presented here, each extremal actuator, made of silicone rubber, varies the friction coefficient between its surface of contact and the ground by expanding and contracting inside a hard 3D-printed smooth casing. This device is designed such that when the actuator is inflated, its silicone-rubber surface touches the ground, producing high friction. Conversely, when the actuator is deflated, its surface does not touch the ground and only the smooth edges of the casing make contact with the supporting surface, thus producing low friction. Friction is a nonlinear phenomenon, and consequently, the complete dynamics of the system presented here is both nonlinear and time-varying. However, by treating the forces generated by the central actuator and the friction forces as inputs, the robot’s dynamics can be described by a *linear time-invariant* (LTI) state-space model. This reduced-complexity model enables analysis of the system’s controllability and is instrumental in determining that locomotion is not feasible in the absence of friction. We explicitly show that the controllability subspace associated with the zero-friction case contains only states that define a constant position of the system’s center of mass with respect to the inertial frame of reference. Further analysis shows that if actuation and friction forces were to be chosen at will, the system would become fully controllable. This finding, despite being based on physically unattainable assumptions, indicates that there exists an infinite number of theoretically feasible gaits, and that biologically-inspired locomotion modes represent only a small set of what is possible to achieve with this framework. ![**(a) Metamere:** A metamere expands radially when its longitudinal muscles contract and expand longitudinally when its circular muscles contract. When a metamere is undergoing radial expansion, the 4 pairs of setae on its ventral and lateral surfaces will protrude and anchor it to the ground. **(b) Peristaltic crawling motion:** A *stride* is defined as a complete cycle of peristalsis and the *stride length* is the total distance advanced during one stride. The *protrusion time* is defined as the time span during which an earthworm moves forward within a stride. The head of the earthworm covers the stride length by the end of the protrusion time. Correspondingly, the *stance time* is the period during which the head of a earthworm remains anchored to the ground while the rest of the animal body recovers to the initial state. The sum of the protrusion time and stance time is the *stride period*. The thin dotted lines track the retrograde wave. **(c) Earthworm-inspired crawling robot:** The robot consists of two hard casings, a central actuator, a pair of front and rear actuators constrained by elastomeric o-rings, two machined steel plates, and pneumatic components. \[fig01\]](fig01.pdf){width="46.00000%"} The rest of the paper is organized as follows. Section \[sec02\] introduces the major concepts unique to earthworm-inspired locomotion. Section \[sec03\] presents a reduced-complexity dynamic model of the proposed robot and a set of locomotion simulations. Section \[sec04\] explains the design and fabrication processes of the soft-robotic components. Section \[sec05\] describes the locomotion planning and associated control strategy. Experimental results are presented and discussed in Section \[sec06\]. Lastly, Section \[sec07\] states the main conclusions of the presented research and provides directions for the future. Earthworm-Inspired Locomotion {#sec02} ============================= Earthworms belong to the phylum *annelida*, characterized by their segmented body structures. During locomotion, each ring-shaped segment (metamere) is actively reconfigured by the actions of layers of both longitudinal and circular muscle, as illustrated in Fig. \[fig01\]-(a). Internal sealed cavities in earthworms’ bodies, referred to as *coeloms*, are filled with incompressible fluid so that the volume of each metamere remains constant while reshaping and the structural integrity of the animal is continually preserved. Anatomical schemes of this type are known as hydrostatic skeletons. Also, the fluid inside each coelom is constrained within each metamere, partitioned by *septa* so that there is no movement of fluid across body segments [@ref05]. Such segmentation preserves, to some extent, the locomotion independence of each metamere, enhancing earthworms’ overall mobility [@ref03]. Thus, in order for an earthworm to locomote, the longitudinal and circular muscles of each segment contract alternately, causing each segment to shorten (expanding radially) and elongate (shrinking radially) according to the sequential pattern depicted in Fig. \[fig01\]-(b). Such motions from head to tail create the retrograde peristaltic gaits characteristic of worms. It can be proved mathematically that peristalsis-based crawling requires sufficient traction between anchoring metameres and the ground. In the case of *oligochaetas*, the subclass of *annelida* to which earthworms belong, traction is produced and modulated by retractable setae, as depicted in Fig. \[fig01\]-(a). Some species of earthworm are both geophagous (earth-eaters) and surface-feeders [@ref06]. That is the case of nightcrawlers, which emerge from their burrows and crawl on ground at night and remain underground during daytime [@ref25]. To transition and adapt to these two different surroundings, they switch between peristalsis-based crawling and burrowing locomotion modes. A worm-inspired burrowing soft robot was presented in [@ref18] and here we extend that work to the crawling case, which requires the active control of friction. This friction-based control strategy is loosely inspired by the morphology of nightcrawlers, which have evolved setae only on the ventral and lateral surfaces of each metamere to facilitate traction during surface crawling, as illustrated in Fig. \[fig01\]-(a). On the other hand, most purely geophagous earthworms have setae arranged in a ring around each body segment [@ref05; @ref07]. During crawling, setae protrude from radially expanding metameres (longitudinal muscle contraction) and anchor into the substratum to provide traction, thus preventing slipping while adjacent body segments contract or expand. Once a metamere’s circular muscle starts to contract, the longitudinal muscle relaxes and the setae retract from the ground to allow for the segment to slide forward. The basic crawling gait of the robot presented in this paper is loosely based on the nightcrawler’s crawling mechanism, depicted in Fig. \[fig01\]-(b). Following [@ref06], we define a *stride* as one cycle of peristalsis and describe the crawling kinematics as a function of four variables: *stride length*, *protrusion time*, *stance time* and *stride period* (illustrated in Fig. \[fig01\]-(b)). For simplicity, despite the fact that earthworms have numerous segments with staggered stride periods, we define these kinematic variables in relation to an earthworm’s first segment. A prototype of the proposed robot is shown in Fig. \[fig01\]-(c). This system can be thought of as a two-metamere crawling artificial worm composed of pneumatic soft actuators that emulate earthworms’ muscle structures as well as hard casings employed in friction regulation. The processes of locomotion modeling, robotic design, fabrication and controller development are discussed in the next sections. ![Reduced-complexity mass-spring-damper model of the robot in Fig. \[fig01\]-(c). A controllability analysis is carried out for two cases: without friction and with frictions $f_1$ and $f_2$ included as inputs. The values of $f_1$ and $f_2$ are positive when the associated vector forces act in the same direction as ${\boldsymbol{i}}$, and are negative, when the vector forces act in the opposite direction as ${\boldsymbol{i}}$. \[fig02\]](fig02.pdf){width="48.00000%"} Dynamic Modeling and Simulations {#sec03} ================================ Robot Dynamics and Controllability Analysis {#sec03a} ------------------------------------------- Several of the existing earthworm-inspired robots consist of three sections: a pair of posterior and anterior actuators that serve as artificial circular muscles, and an axial central actuator, which is the analogue of an earthworm’s longitudinal muscle [@ref10; @ref18; @ref26]. Limited by their configurations, those robots can only travel inside pipes with predetermined diameters. Thus, to locomote, a robot of that type replicates the peristaltic burrowing gaits of earthworms according to a scheme in which its anterior and posterior actuators alternately provide anchoring by pressure to the internal surface of a pipe, while its longitudinal actuator extends and contracts to generate displacements along the pipe’s axial axis. In this section, employing a reduced-complexity dynamic model, linear system theory and experimental data from [@ref18], we develop the analytical tools necessary to generate a conceptual design for an earthworm-inspired pneumatically-driven soft robot capable of crawling on flat surfaces. An abstraction of this system is shown in Fig. \[fig02\]. In this case, given its function and elastic characteristics, a longitudinal actuator is modeled as a massless elastic spring with stiffness constant $k$ and two forces with opposite directions and identical magnitudes $f_\textrm{a}$ (shown in Fig. \[fig02\]). The anterior and posterior actuators are modeled as two blocks with masses $m_1$ and $m_2$, capable of varying their friction coefficients with the ground in real time in order to modulate the resulting values of the friction forces $f_1$ and $f_2$. For a pneumatically-driven axial actuator of the type in [@ref18] and Fig. \[fig01\]-(c), the magnitude of the produced driving force can be estimated as $$\begin{aligned} f_{{\textrm{a}}}(t) = s_{{\textrm{a}}} p_{{\textrm{a}}}(t),\end{aligned}$$ where $p_{{\textrm{a}}}$ and $s_{{\textrm{a}}}$ are the instantaneous internal air pressure and constant cross-sectional area of the actuator, respectively. Note that in this model, in agreement with the experimental data presented in [@ref18], $k$ is considered to be constant. This approximation is sufficiently accurate for purposes of design, controllability analysis and controller synthesis. However, the true stiffness of the actuator is nonlinear, time-varying and depends on the air pressure inside the soft structure. Lastly, energy dissipation is modeled by a damper with a constant $c$ to be empirically identified. To address the problem of controllability, we first consider the frictionless case, in which the force magnitude $f_\textrm{a}$ is the sole input to the system. Thus, by defining $x_1$ and $x_2$ as the position variations of $m_1$ and $m_2$ with respect to an inertial frame of reference, and the corresponding speeds $v_1 = \dot{x}_1$ and $v_2 = \dot{x}_2$, we describe the system with the *single-input–multi-output* (SIMO) state-space realization $$\begin{aligned} \begin{split} \dot{x} (t) &= Ax(t)+B_0u_0(t), \\ y(t) &= Cx (t)+Du_0(t), \end{split} \label{eqn:eqlabel{2}}\end{aligned}$$ where $$\begin{aligned} A &= \left[ \begin{array}{cccc} ~0 & ~1 & ~0 & ~0 \\ -\frac{k}{m_1} & -\frac{c}{m_1} & ~\frac{k}{m_1} & ~\frac{c}{m_1} \\ ~0 & ~0 & ~0 & ~1 \\ ~\frac{k}{m_2} & ~\frac{c}{m_2} & -\frac{k}{m_2} & -\frac{c}{m_2} \\ \end{array} \right],~ B_0 = \left[\begin{array}{c} ~0 \\ -\frac{1}{m_1} \\ ~0 \\ ~\frac{1}{m_2} \end{array} \right],\\ C &= \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right] , ~D = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array} \right], ~x = y = \left[\begin{array}{c} x_1 \\ \dot{x}_1 \\ x_2 \\ \dot{x}_2 \end{array} \right], \label{eqn:eqlabel{3}}\end{aligned}$$ and $u_0 = f_{{\textrm{a}}}$. In this case, the controllability matrix $\mathcal{C}_{0} = \left[ \begin{array}{cccc} B_0 & AB_0 & A^2B_0 & A^3B_0 \end{array} \right]$ has rank $2$, thus the system is not controllable, meaning that there exists a set of states that cannot be reached from any possible initial state by the action of input signals [@ref27]. Associated with $\mathcal{C}_{0}$ is the controllable subspace, defined as $\mathcal{C}_{AB_0} = {\textrm{Image}} \left\{ \mathcal{C}_{0} \right\}$, which is equivalent to the set of reachable states from the initial condition $x(0) = 0$, $\mathcal{R}_{{\textrm{t}}}$ [@ref28]. It follows that $\mathcal{R}_{{\textrm{t}}} = \mathcal{C}_{AB_0} = {\textrm{Span}} \left\{ \chi_1, \chi_2\right\}$, where $$\begin{aligned} \chi_1 = \left[ \begin{array}{c} ~1 \\ ~0 \\ -\frac{m_1}{m_2} \\ ~0 \end{array} \right],~ \chi_2 = \left[ \begin{array}{c} ~0 \\ ~1 \\ ~0 \\ -\frac{m_1}{m_2} \end{array} \right].\end{aligned}$$ Therefore, every state in $\mathcal{C}_{AB_0}$ can be written as $\alpha_1\chi_1~+~\alpha_2 \chi_2$ for some $\alpha_1,\alpha_2 \in \mathbb{R}$, which implies that all the reachable positions for the masses take the form $\left\{ x_1 = \alpha_1, x_2 = -\alpha_1 \frac{m_1}{m_2} \right\}$. Thus, we conclude that for all possible inputs and initial state $x(0)=0$, the location of the system’s center of mass with respect to the inertial frame remains constant because the variation $$\begin{aligned} x_{{\textrm{CM}}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} = 0,\end{aligned}$$ for all $x \in \mathcal{R}_{{\textrm{t}}}$. The main implication of this analysis is that in the absence of friction, for a robot of the type in Fig. \[fig01\]-(c), locomotion is impossible. This result is consistent with generalizable physical intuition and biological observations [@ref29]. In the presence of friction, a simple model for the generation of traction forces in the system of Fig. \[fig02\] is $$\begin{aligned} f_i(t)~=~{\textrm{sign}}\left[\dot{x}_i(t) \right]\mu_i(t) m_i g,~\textrm{for}~i=1,2, \label{EQN05}\end{aligned}$$ where $\mu_i \in \mathbb{R}^+$ are kinetic friction coefficients [@ref30] and $g$ is the acceleration of gravity. Given this structure, the only possible way in which $f_1$ and $f_2$ can be modulated is by varying $\mu_1$ and $\mu_2$. From linear-systems-theory-based analysis it is not possible to determine if the system becomes fully controllable when the inputs are $\left\{f_{{\textrm{a}}},\mu_1,\mu_2 \right\}$. However, we explain the importance of friction for the control of locomotion by analyzing the simplified dynamics resulting from assuming unrestricted inputs $\left\{f_{{\textrm{a}}},f_1,f_2 \right\}$. This case can be modeled by the *multi-input–multi-output* (MIMO) state-space representation $\left\{ A,B_1,C,D\right\}$, where the new input-state matrix and input signal are given by $$\begin{aligned} B_1 = \left[ \begin{array}{ccc} ~0 & ~0 & ~0 \\ -\frac{1}{m_1} & -\frac{1}{m_1} & ~0 \\ ~0 & ~0 & ~0 \\ ~\frac{1}{m_2} & ~0 & -\frac{1}{m_2} \end{array} \right],~ u_1 = \left[ \begin{array}{c} f_{{\textrm{a}}} \\ f_1 \\ f_2 \end{array} \right].\end{aligned}$$ For this augmented state-space realization, the associated controllability matrix $\mathcal{C}_1=\left[ \begin{array}{cccc} B_1 & AB_1 & A^2B_1 & A^3B_1 \end{array} \right]$ has rank $4$, and therefore, the controllable subspace $\mathcal{C}_{AB_1}~=~\textrm{Image} \left\{ \mathcal{C}_1 \right\}$ spans $\mathbb{R}^4$. This analysis implies that if the input $u_1$ could be chosen without restriction, any desired state, and consequently, any position of the system’s center of mass could be reached in a finite amount of time. In reality, however, $u_1$ is highly restricted by actuator limitations and the nonlinear nature of friction. Despite of these restrictions, controlled locomotion can be achieved by varying the friction coefficients $\mu_1$ and $\mu_2$. This fact is explained using numerical examples in the next section. ![Discrete-time model used to implement numerical simulations. $\hat{G}(z)$ is the discretized version of $\left\{A, B_1, C, D\right\}$. Sign operators ensure opposing signs between velocities and frictional forces. In this case, both friction coefficients $\mu_i$, $i=1,2$, switch between $\underline{\mu_i} = 0.1$ and $\overline{\mu_i} = 1$. Zero initial conditions are set at the beginning of the simulations. \[fig03\]](fig03.pdf){width="48.00000%"} Locomotion Simulations {#sec03b} ---------------------- Through numerical simulations, we illustrate how the robot achieves locomotion with the use of feedforward-controlled time-varying friction. Here, a set of feasible control inputs is chosen via an exhaustive search and iteration process. As described in (\[EQN05\]), the values of the frictional forces $f_i$ are functions of kinematic coefficients of friction $\mu_i$ and the normal forces between the contact surfaces, $m_i g$ (assuming a perfectly flat supporting surface). In the proposed locomotion strategy, normal forces remain constant and friction is regulated by the active variation in real time of the friction coefficients. Specifically, the anterior and posterior actuators of the robot in Fig. \[fig02\] are designed and fabricated to switch their coefficients of friction between a small positive value, $\underline{\mu_i}$, and a larger positive value, $\overline{\mu_i}$, in order to produce friction forces with square-wave-signal shapes. This phenomenon is created by actively switching the surfaces of contact between the actuators and supporting ground. The magnitudes of friction coefficients depend on the materials of the surfaces in contact and range from $\sim \hspace{-0.4ex} 0.04$ for Teflon on steel to $\sim \hspace{-0.4ex} 0.8$ for rubber on concrete [@ref31]. According to experimental tests performed on the extremal friction-varying actuators of the robot in Fig. \[fig01\]-(c), the measured transition between $\underline{\mu_i}$ and $\overline{\mu_i}$ can be as fast as $0.4~\textrm{s}$, which enables the design and implementation of control strategies based on low-frequency *pulse width modulation* (PWM). For the simulations, we assume that these transitions are instantaneous. ![**(a)** Simulated displacements and instantaneous velocities of $m_1$ and $m_2$ when the frequencies of $f_\textrm{a}$, $f_1$ and $f_2$ are set to 1 Hz and $\phi = 0.4\pi~\textrm{rad}$ ($m_1 = m_2 = 0.2~\textrm{Kg},~k = 200~\textrm{N}\cdot \textrm{m}^{-1},~c=0$). **(b)** Simulated displacement of $m_1$ at 60 s across a variety of frequency combinations for $f_\textrm{a}$ (C. freq.) and $f_1$ (F. freq.). The forces $f_1$ and $f_2$ oscillate at the same frequency and $\phi$ is held at $0.4\pi~\textrm{rad}$ ($m_1 = m_2 = 0.2~\textrm{Kg},~k = 200~\textrm{N}\cdot \textrm{m}^{-1},~c=0$). **(c)** Relationship between displacement and phase difference $\phi$ when $f_\textrm{a}$, $f_1$ and $f_2$ are synchronized (at 1 Hz). It can be observed that the direction of locomotion can be reversed by controlling the phase difference between $f_\textrm{a}$ and $f_1$, $f_2$. Trial 1 and Trial 2 correspond to mass values of 0.1 Kg and 0.2 Kg, respectively ($k = 200~\textrm{N}\cdot \textrm{m}^{-1},~c=0$). Heavier masses correspond to higher frictions. These simulation results suggest that higher friction produces faster locomotion. \[fig04\]](fig04.pdf){width="44.00000%"} Thus, by combining the actuation model for the generation and control of friction forces with the system dynamics discussed in Subsection \[sec03a\], we implement numerical simulations aimed to study the dynamic behavior of the soft robot during surface crawling. This study is relevant for the search of feasible and, eventually, optimal locomotion patterns. The basic simulation scheme is shown in Fig. \[fig03\], where $\hat{G}(z)$ is the discretized version of $\hat{G}(s) = C \left(sI - A \right)^{-1}B_1 + D$, with the state-space representation $\left\{A_{{\textrm{D}}},{B_1}_{{\textrm{D}}},C_{{\textrm{D}}},D_{{\textrm{D}}} \right\}$, obtained with the *zero-order hold* (ZOH) method and employing a sampling frequency of $1~\hspace{-1.6ex}~{\textrm{KHz}}$. Consistently, the sequences $f_{{\textrm{a}}}[n]$, $f_1[n]$, $f_2[n]$, $x_1[n]$, $x_2[n]$, $v_1[n]$ and $v_2[n]$ are the discrete-time versions of the functions $f_{{\textrm{a}}}(t)$, $f_1(t)$, $f_2(t)$, $x_1(t)$, $x_2(t)$, $v_1(t)$ and $v_2(t)$. {width="100.00000%"} Assuming a periodic oscillation of the robot’s axial actuator, a sinusoidal signal, with amplitude and bias determined by the actuator’s minimum and maximum internal pressures, is chosen as input $f_{{\textrm{a}}}$. The extremum air-pressure values are estimated from the experimental data published in [@ref18]. The signals $f_1$ and $f_2$ are chosen to have square-wave shapes with amplitudes and biases given by the lower and upper bounds of the frictional forces associated with the lowest and highest friction coefficients, $\underline{\mu_i}$ and $\overline{\mu_i}$, respectively. For consistency with the experimental behaviors of the robot’s pneumatic actuators, the simulation inputs are limited to a frequency of $1~{\textrm{Hz}}$. Also, $f_1$ and $f_2$ are set to have the same frequency but set apart with a phase difference $\phi$ that can be varied between $0$ and $2\pi~{\textrm{rad}}$. Additionally, because kinetic friction always opposes an actuator’s motion, two sign operators are inserted in a feedback configuration, introduced to ensure opposing signs between velocities and frictional forces, as shown in Fig. \[fig03\]. A set of simulation results is presented in Fig. \[fig04\]. Here, for all the cases, we set $k = 200~{\textrm{N}} \cdot {\textrm{m}}^{-1}$, $c = 0 $, $\underline{\mu_1} = \underline{\mu_2} = 0.1$ and $\overline{\mu_1} = \overline{\mu_2} = 1$. Fig. \[fig04\]-(a) shows the displacements and velocities of the two mass-blocks ($m_1 = m_2 = 0.2~{\textrm{Kg}}$), when $f_{{\textrm{a}}}$, $f_1$ and $f_2$ oscillate at $1~{\textrm{Hz}}$ and $\phi = 0.4\pi~{\textrm{rad}}$. Both masses travel in an approximately linear motion at an average speed of $6.31~{\textrm{m}} \cdot {\textrm{min}}^{-1}$ ($10.52~{\textrm{cm}} \cdot {\textrm{s}}^{-1}$). Fig. \[fig04\]-(b) shows the total distance traveled by the robot in $60~{\textrm{s}}$ versus the frequencies of $f_\textrm{a}$ and $f_1$, $f_2$. All the simulations in this plot were run with a constant phase difference $\phi=0.4\pi~{\textrm{rad}}$ and $m_1 = m_2 = 0.2~{\textrm{Kg}}$. These simulation results suggest that for this specific selection of inputs, substantial locomotion can only be attained when the input frequencies are equal, with exception of a few frequency combinations. Also, in general, faster inputs generate faster locomotion. Fig. \[fig04\]-(c) shows the final position reached by the robot after $60~{\textrm{s}}$ across all $\phi$, when all the input frequencies are held at $1~{\textrm{Hz}}$, for two different choices of the pair $\left\{m_1,m_2 \right\}$. This plot suggests that, for this particular type of inputs, $\phi$ is critical for locomotion generation and direction reversal can be realized simply by varying the phase difference between friction inputs. Similar results have been observed in repeated simulations for friction forces with different amplitudes. These findings are limited to the specific set of inputs employed in the discussed cases, but nonetheless exemplify the challenges and potential of friction-controlled crawling. A more comprehensive study of input signals and control strategies to optimize locomotion is a matter of current and future research. {width="100.00000%"} Design and Fabrication {#sec04} ====================== The work presented in this paper extends that of [@ref18]. The earthworm-inspired soft robot introduced therein can only function constrained by the specific geometric configuration of pipes, employing a burrowing gait. Here, we develop the design, fabrication and control tools necessary to create an earthworm-inspired soft robot capable of crawling on flat surfaces. The key design innovation introduced in this work is the switching of friction forces by alternating the actuators’ surfaces of contact with the ground. To achieve such objective, we design the soft robot shown in Fig. \[fig01\]-(c), composed of a central longitudinal actuator, a pair of extremal longitudinal actuators, and a pair of hard casings that enclose the extremal actuators. In addition, a pair of soft modules, shown in Fig. \[fig05\]-(c), are employed to connect the central actuator with the extremal actuators. These two connecting modules are also enclosed within the hard casings. In the proposed robotic design, actuators are driven pneumatically. The central and extremal actuators are designed to emulate the earthworm’s longitudinal and circular muscles, respectively. All actuators are built to expand and contract axially as functions of their internal pressures, unlike those in [@ref18]. Both front and rear actuators are fixed to the upper interior surface of the hard casings and remain above the ground when deflated as the hard casings support the robot’s weight. When inflated, the front and rear actuators elongate and make contact with the surface. The hard casings provide low friction while the actuators yield high friction with the supporting surface. Thus, in this scheme, switching between high and low frictional force values is made possible by a simple inflation and deflation sequence. This actuation method is inspired by the traction variable mechanism employed by nightcrawlers, discussed in Section \[sec02\]. To see this, recall that, when crawling, their contracted longitudinal muscles (coupled with relaxed circular muscles) will cause a metamere to expand radially, pushing the setae into the ground to anchor and prevent backward slippage. Note that, even though the extremal actuators together with their casings are inspired by earthworm’s circular muscles and setae, the underlying working principles are significantly different. In addition, deformation of natural muscles is achieved through active contraction and passive elongation as opposed to the artificial actuators discussed here that elongate actively but contract passively. The methods and construction sequences employed to fabricate the soft robot are depicted in Fig. \[fig05\]. Fig. \[fig05\]-(a), Fig. \[fig05\]-(b) and Fig. \[fig05\]-(c) illustrate the fabrication processes of the front and rear actuators, central actuator and the connecting modules, respectively. Fig. \[fig05\]-(d) explains the steps leading to the final assembly of the robot. The parts fabricated and materials used to build this robot include 3D-printed *acrylonitrile butadiene styrene* (ABS) molds and casings, silicone elastomer (Ecoflex^^ 00-50, Smooth-On), butadiene rubber elastomeric o-rings, fiberglass sheets and pneumatic components. All actuators measure $35~{\textrm{mm}}$ in diameter, the central actuator measures $83~{\textrm{mm}}$ in length and the extremal actuators combined with the connecting modules measure $26~{\textrm{mm}}$ in height. The wall thickness of the soft components range between $2.5$ and $3~{\textrm{mm}}$. These dimensions were chosen based on the robot design in [@ref18], and were modified to accommodate off-the-shelf pneumatic components. To drive the system, an Elemental $\textrm{O}_2$ commercial air pump and a 12-V ROB-10398 vacuum pump are employed to inflate and deflate all actuators through a manifold (SMC VV3Q12). Three high speed solenoid valves (SMC VQ110-6M) and three Honeywell ASDX Series digital serial silicon pressure sensors provide regulation and measurement of each actuator’s internal pressure. Data acquisition and signal processing are performed with an AD/DA board (National Instruments PCI-6229) mounted on a target PC which communicates with a host PC via xPC Target 5.5 (P2013b). Locomotion Planning and Control {#sec05} =============================== In Section \[sec03b\], using simulations, we demonstrated that fast locomotion is contingent upon perfectly-shaped periodic driving and frictional forces, with perfectly-matched relatively high frequencies. These conditions are not realizable with pneumatically-powered soft actuators as those of the robot in Fig. \[fig01\]-(c) (discussed in Section \[sec04\]). Thus, replicating the high-speed simulated locomotion behaviors on the actual robot is, at this moment, not an attainable objective. However, we can implement bio-inspired locomotion strategies that are compatible with lower frequencies. It is easy to see from Fig. \[fig02\], that $m_1$ will remain stationary (anchored to the ground) and $m_2$ can slide forward as the central actuator inflates if $$\begin{aligned} |f_1|\geqslant|f_\textrm{a}|>|f_2| \label{eq07}. \end{aligned}$$ The signal $f_1$ corresponds to static friction while $f_2$ is considered to be kinetic friction. Similarly, $m_2$ will be anchored to the ground and $m_1$ will slide forward as the central actuator deflates if $$\begin{aligned} |f_2|\geqslant|f_\textrm{a}|>|f_1| \label{eq08}. \end{aligned}$$ Here, $f_2$ is a static friction force and $f_1$ is instead, considered to be kinetic friction. Thus, locomotion can be induced by actuating each actuator following a pattern such that the conditions defined in (\[eq07\]) and (\[eq08\]) are satisfied in an alternating sequence. In this way, a four-phase actuation sequence is designed to generate one complete stride for the robot as illustrated in Fig. \[fig06\]. Before implementing a locomotion sequence, an actuator characterization test is performed to determine a proper set of values for the robot’s stride length, stance time and protrusion time. [l\*[4]{}[c]{}r]{} **Phase** & **1** & **2** & **3** & **4**\ Rear Actuator & 1.2 & 1.2 & 1.2 & 0\ Central Actuator & 0 & 3 & 3 & 0\ Front Actuator & 0 & 0 & 1.2 & 1.2\ \[tab01\] ![Example of pressure-tracking experimental results. The continuous lines represent measurements and dashed lines represent references. These data were obtained employing the PID scheme in Fig. \[fig07\] to control the central (upper plot), frontal (middle plot) and rear (bottom plot) actuators, during locomotion. The *protrusion time* is $1.6~{\textrm{s}}$, the *stance time* is $2.4~{\textrm{s}}$ and the *stride period* is $4~{\textrm{s}}$. \[fig08\]](fig08.pdf){width="46.00000%"} To characterize each actuator, three *proportional-integral-derivative* (PID) controllers $\hat{K}_j,~j = 1, 2, 3$, depicted in Fig. \[fig07\], are implemented to regulate internal pressure. Both pumps are maintained at a constant flow rate and output pressure, and the response of each actuator is controlled by solenoid valves using PWM. The valves are normally closed, a state during which the manifold allows for the vacuum pump to deflate the actuators. The PWM duty cycle excites the valves to open and allows for each actuator to inflate individually. In this structure (Fig. \[fig07\]), the output of $\hat{K}_j$ is the duty cycle input to each valve. Every PID controller is tuned online in an exhaustive manner. The experimental characterization process follows the procedure introduced in [@ref18]. For the central actuator, a range of pressure values that produce substantial elongations without causing significant radial expansions is identified. For the front and rear actuators, the minimum pressure threshold for which firm contact between the actuators and supporting surface is established is chosen to be the reference pressure. Additionally, two 130-gram machined steel plates are fixed onto the top of both casings, as shown in Fig. \[fig01\]-(c), to increase frictional force and damp the vibration from the valves during actuation. Table \[tab01\] presents a set of reference pressures for individual actuators during the four phases described in Fig. \[fig06\]. Robot locomotion is achieved by controlling each actuator to track the reference pressure during each phase. In reference to the earthworm crawling kinematics described in Section \[sec02\], we define the protrusion time as the period during which the central actuator expands (phase 2). Similarly, the stance time is defined as the time duration after protrusion time during which the front actuator remains static horizontally and completes a cycle of inflation and deflation (phase 3 + phase 4 + phase 1). Protrusion time and stance time are prescribed in experiments. ![Photographic sequence showing the soft robot while crawling on a laboratory benchtop. Locomotion is achieved by tracking the actuators’ pressure references in Table \[tab01\]. In this case, a total distance of $52.4~{\textrm{cm}}$ is covered within $75~{\textrm{s}}$ at an average speed of $0.7~{\textrm{cm}}\cdot {\textrm{s}}^{-1}$. The complete set of locomotion experiments can be found in the supporting movie S1.mp4, also available at <http://www.uscamsl.com/resources/ROBIO2017/S1.mp4>. \[fig09\]](fig09.pdf){width="46.00000%"} To implement the described locomotion method, low-level PID controllers (Fig. \[fig07\]), tuned during the characterization process, are used to control the actions of each actuator. Experimental Results and Discussion {#sec06} =================================== Experiments were conducted to validate the locomotion sequence proposed in Section \[sec05\]. The first set of tests aims to optimize the crawling speed of the robot on a single uniform surface. The effect of different variables, including the duration of each phase and reference pressures for each actuator, is examined across a broad spectrum of values. Fig. \[fig08\] presents the pressure tracking signals of each actuator for the test that produced the fastest locomotion, in which the protrusion time and stance time were $1.6$ and $2.4~{\textrm{s}}$, respectively. A stride length of $2.79~{\textrm{cm}}$ and an average speed of $0.7~{\textrm{cm}} \cdot {\textrm{s}}^{-1}$ were observed and recorded, as shown in Fig. \[fig09\]. Since these experiments adopt a different actuation approach to that of the simulations in Section \[sec03\], the large differences between simulated speeds and experimental locomotion speeds are not surprising. As observed in Fig. \[fig08\], the front and rear actuators were able to track the reference pressures with minor overshoots. However, the central actuator was unable to deflate completely. Lower pressure references for the central actuator and longer protrusion times were found to produce better pressure tracking at the cost of overall locomotion speed. No obvious slippage was observed in any of the tests. The second set of tests was designed to validate the notion that the robot can travel on surfaces with different coefficients of friction. Using the same actuation sequence than that of Fig. \[fig09\], we proved that the robot can generate peristaltic locomotion on multiple surfaces, including a laboratory benchtop, plywood, *high-density polyethylene* (HDPE), aluminum and a foam pad. Furthermore, we showed that this robot is capable of traversing surfaces with different coefficients of friction by letting it crawl from a foam pad to an HDPE plate. The complete set of all the described tests can be found in the supporting movie S1.mp4, also available at <http://www.uscamsl.com/resources/ROBIO2017/S1.mp4>. The experiments presented in this section proved friction manipulation to be an effective way to generate peristaltic crawling in the proposed robot. During locomotion, pressure sensors provide feedback to regulate the elongation of each actuator, and therefore, displacement control was achieved indirectly. Direct displacement control can be implemented in the future by employing a motion-capture system or soft sensors. Also, note that actuator characterization in this case is performed empirically. An analytical model that can capture the nonlinear relationships between an actuator’s internal pressure and deformation is needed to improve the control strategy and optimize locomotion. Conclusion and Future Work {#sec07} ========================== We presented an earthworm-inspired soft crawling robot capable of locomoting on surfaces by manipulating friction. The robot consists of modular actuators and mechanisms that emulate the functionalities of an earthworm’s longitudinal and circular muscles as well as its bristle-like setae structures. We modeled the robot as a mass-spring-damper system and described its crawling dynamics with an LTI state-space representation. We proved mathematically that frictional forces can be employed as inputs that lead to system controllability. This finding was tested and validated through simulations. Experimentally, we demonstrated that the robot is capable of locomoting on surfaces with different coefficients of friction, emulating an earthworm’s peristaltic crawling. The modular structure of the robot makes it easily scalable, which leaves great potential for creating longer and more versatile robotic structures. Such complex modular systems will provide an ideal platform to develop and test novel decentralized control strategies. In this work, we empirically explored the feasibility of friction-controlled locomotion on flat surfaces. We anticipate that future research will further explore the proposed robotic concept, employing only soft materials and enabling steering and locomotion on uneven terrains. Additionally, the robot presented here is tethered to both the power source and feedback-control module. To achieve autonomy, novel sensing and wireless communication systems must be implemented. Also, portable sources of energy are required. Feasible options are electrolysis and combustion. These topics are a matter of future research. [^1]: This work was partially supported by the USC Viterbi School of Engineering through graduate fellowships to J. Z. Ge and A. A. Calderón, and a start-up fund to N. O. Pérez-Arancibia. Additional support was provided by the Chilean National Office of Scientific and Technological Research (CONICYT) through a graduate fellowship to A. A. Calderón. [^2]: The authors are with the Department of Aerospace and Mechanical Engineering, University of Southern California (USC), Los Angeles, CA 90089-1453, USA (e-mail: [zaoyuang@usc.edu]{}; [aacalder@usc.edu]{}; [perezara@usc.edu]{}).

--- abstract: 'We establish a large sieve inequality for power moduli in $\mathbb{Z}[i]$, extending earlier work by L. Zhao and the first-named author on the large sieve for power moduli for the classical case of moduli in $\mathbb{Z}$. Our method starts with a version of the large sieve for $\mathbb{R}^2$. We convert the resulting counting problem back into one for $\mathbb{Z}[i]$ which we then attack using Weyl differencing and Poisson summation.' address: - 'Stephan Baier, Ramakrishna Mission Vivekananda Educational Research Institute, Department of Mathematics, G. T. Road, PO Belur Math, Howrah, West Bengal 711202, India; email: email$_{-}$baier@yahoo.de' - 'Arpit Bansal, Jawaharlal Nehru University, School of Physical Sciences, New-Delhi 110067, India; email: apabansal@gmail.com' author: - 'Stephan Baier, Arpit Bansal' title: 'The large sieve with power moduli for $\mathbb{Z}[i]$' --- Introduction ============ The classical large sieve inequality with additive characters asserts that $$\sum\limits_{q\le Q} \sum\limits_{\substack{a=1\\ (a,q)=1}}^q \left| \sum\limits_{M<n\le M+N} a_n e\left(n\cdot \frac{a}{q}\right) \right|^2 \le \left(Q^2+N-1\right)\sum\limits_{M<n\le M+N} |a_n|^2,$$ where $Q,N\in \mathbb{N}$ and $M\in \mathbb{Z}$. This inequality has numerous applications in analytic number theory, in particular, in sieve theory and to questions regarding the distribution of arithmetic functions in arithmetic progressions. The large sieve with resticted sets of moduli $q$, in particular power moduli, was considered in a series of papers by Baier, Zhao and Halupczok (see [@Bai], [@BZ1], [@BZ2], [@Hal] and [@Zh1]), and these results turned out to be useful tools for applications (see [@BFKS] and [@BPS], for example). In the case of square moduli, it was first established by Zhao [@Zh1] that $$\label{firstls} \begin{split} & \sum\limits_{q\le Q} \sum\limits_{\substack{a=1\\ (a,q)=1}}^{q^2} \left| \sum\limits_{M<n\le M+N} a_n e\left(n\cdot \frac{a}{q^2}\right) \right|^2 \\ \ll_{\varepsilon} & (QN)^{\varepsilon}\left(Q^3+N\sqrt{Q}+\sqrt{N}Q^2\right)\sum\limits_{M<n\le M+N} |a_n|^2. \end{split}$$ This was improved in [@Bai], where the term $N\sqrt{Q}$ on the right-hand side of was replaced by $N$. A further improvement was obtained in [@BZ1], where with $N+\min\left\{N\sqrt{Q},\sqrt{N}Q^2\right\}$ in place of $N\sqrt{Q}+\sqrt{N}Q^2$ was established. In [@Zh1], Zhao conjectured that the bound $$\label{conjec} \sum\limits_{q\le Q} \sum\limits_{\substack{a=1\\ (a,q)=1}}^{q^k} \left| \sum\limits_{M<n\le M+N} a_n e\left(n\cdot \frac{a}{q^k}\right) \right|^2 \ll_{\varepsilon} (QN)^{\varepsilon}\left(N+Q^{k+1}\right)\sum\limits_{M<n\le M+N} |a_n|^2$$ should hold for $k$-th power moduli ($k\in \mathbb{N}$ arbitrary but fixed). This conjecture is still open for every $k\ge 2$. In the same paper [@Zh1], he established that $$\label{kls} \begin{split} & \sum\limits_{q\le Q} \sum\limits_{\substack{a=1\\ (a,q)=1}}^{q^k} \left| \sum\limits_{M<n\le M+N} a_n e\left(n\cdot \frac{a}{q^k}\right) \right|^2 \\ \ll_{\varepsilon} & (QN)^{\varepsilon}\left(Q^{k+1}+NQ^{1-1/\kappa}+N^{1-1/\kappa}Q^{1+k/\kappa} \right) \sum\limits_{M<n\le M+N} |a_n|^2, \end{split}$$ where $\kappa=2^{k-1}$, thus generalizing . Improvements of this result have been established in [@BZ2] and [@Hal]. The large sieve for additive characters was extended to number fields by Huxley. In the case of the number field $\mathbb{Q}(i)$ it takes the form $$\label{Huxley} \sum\limits_{\substack{q\in \mathbb{Z}[i]\setminus\{0\}\\ \mathcal{N}(q)\le Q}} \sum\limits_{\substack{r \bmod{q}\\ (r,q)=1}} \left|\sum\limits_{\substack{n\in \mathbb{Z}[i]\\ \mathcal{N}(n)\le N}} a_n \cdot e\left(\Re\left(\frac{nr}{q}\right)\right)\right|^2 \ll \left(Q^2+N\right)\sum\limits_{\substack{n\in \mathbb{Z}[i]\\ \mathcal{N}(n)\le N}} |a_n|^2.$$ Here as in the following $\mathcal{N}(q)$ denotes the norm of $q\in \mathbb{Z}[i]$, given by $$\mathcal{N}(q):=\Re(q)^2+\Im(q)^2.$$ The large sieve with square norm moduli for the number field $\mathbb{Q}(i)$ was investigated in [@Ba2], where an analogue of was established, namely the inequality $$\label{squarenorm} \begin{split} & \sum\limits_{\substack{q\in \mathbb{Z}[i]\setminus\{0\}\\ \mathcal{N}(q)\le Q^2\\ \mathcal{N}(q)=\Box}} \sum\limits_{\substack{r \bmod{q}\\ (r,q)=1}} \left|\sum\limits_{\substack{n\in \mathbb{Z}[i]\\ \mathcal{N}(n)\le N}} a_n \cdot e\left(\Re\left(\frac{nr}{q}\right)\right)\right|^2 \\ \ll & (QN)^{\varepsilon}\left(Q^3+Q^2\sqrt{N}+\sqrt{Q}N\right)\sum\limits_{\substack{n\in \mathbb{Z}[i]\\ \mathcal{N}(n)\le N}} |a_n|^2. \end{split}$$ In this paper, we go a step further and prove an analogue of for $\mathbb{Q}(i)$, i.e. a large sieve inequality with $k$-th power moduli for $\mathbb{Q}(i)$. Our approach will be more elegant than the previous one in [@Ba2], where the double large sieve and lattice point counting in $\mathbb{R}^2$ were used. Here our method starts with a version of the large sieve for $\mathbb{R}^2$. Then we convert the resulting counting problem back into one for $\mathbb{Z}[i]$ which can be attacked along similar lines as in [@Zh1] using Weyl differencing and Poisson summation. We begin with square moduli, for which we obtain the essentially same bound as for square norm moduli in . Then we generalize our method to $k$-th power moduli. Statement of main results ========================= We shall establish the following large sieve inequality for square moduli in $\mathbb{Z}[i]$. \[squaremodZi\] Let $Q,N\ge 1$ and $(a_n)_{n\in \mathbb{Z}[i]}$ be any sequence of complex numbers. Then $$\begin{split} & \sum\limits_{\substack{q\in \mathbb{Z}[i]\setminus\{0\}\\ \mathcal{N}(q)\le Q}} \sum\limits_{\substack{r \bmod{q^2}\\ (r,q)=1}} \left|\sum\limits_{\substack{n\in \mathbb{Z}[i]\\ \mathcal{N}(n)\le N}} a_n \cdot e\left(\Re\left(\frac{nr}{q^2}\right)\right)\right|^2\\ \ll & (QN)^{\varepsilon}\left(Q^3+Q^2\sqrt{N}+\sqrt{Q}N\right) \sum\limits_{\substack{n\in \mathbb{Z}[i]\\ \mathcal{N}(n)\le N}} |a_n|^2, \end{split}$$ where $\varepsilon$ is any positive constant, and the implied $\ll$-constant depends only on $\varepsilon$. Theorem \[squaremodZi\] will then be generalized to $k$-th power moduli, for which we establish the following. \[powermodZi\] Let $k\in \mathbb{N}$, $Q,N\ge 1$ and $(a_n)_{n\in \mathbb{Z}[i]}$ be any sequence of complex numbers. Set $\kappa:=2^{k-1}$. Then $$\begin{split} & \sum\limits_{\substack{q\in \mathbb{Z}[i]\setminus\{0\}\\ \mathcal{N}(q)\le Q}} \sum\limits_{\substack{r \bmod{q^k}\\ (r,q)=1}} \left|\sum\limits_{\substack{n\in \mathbb{Z}[i]\\ \mathcal{N}(n)\le N}} a_n \cdot e\left(\Re\left(\frac{nr}{q^k}\right)\right)\right|^2\\ \ll & (QN)^{\varepsilon}\left(Q^{k+1}+NQ^{1-1/\kappa}+N^{1-1/\kappa}Q^{1+k/\kappa} \right) \sum\limits_{\substack{n\in \mathbb{Z}[i]\\ \mathcal{N}(n)\le N}} |a_n|^2, \end{split}$$ where $\varepsilon$ is any positive constant, and the implied $\ll$-constant depends only on $k$ and $\varepsilon$. Large sieve for $\mathbb{R}^d$ ============================== We shall employ the following version of the large sieve for $\mathbb{R}^d$ (in fact, we shall need it for the case $d=2$ only). \[ls\] Let $R,N\in \mathbb{N}$, $N\ge 2$, $x_1,...,x_R\in \mathbb{R}^d$ and $(a_n)_{n\in \mathbb{Z}^d}$ be any $d$-fold sequence of complex numbers. Then $$\sum\limits_{i=1}^R \left|\sum\limits_{\substack{n\in \mathbb{Z}^d\\ ||n||_2\le N^{1/d}}} a_{n} \cdot e\left(n\cdot x_i\right)\right|^2 \ll KNZ,$$ where $$K:=\max\limits_{1\le i\le R} \sharp \left\{j\in \{1,...,R\} : \min\limits_{z\in \mathbb{Z}^d} ||x_j-x_i-z||_2\le \sqrt{d}N^{-1/d}\right\}$$ and $$\label{Zdef} Z:=\sum\limits_{\substack{n\in \mathbb{Z}^d\\ ||n||_2\le N^{1/d}}} |a_n|^2.$$ Here as in the following, $||s||_2$ denotes the Euclidean norm of $s\in \mathbb{R}^d$, given by $$||(s_1,s_2,..,s_d)||_2=\sqrt{s_1^2+s_2^2+\cdots + s_d^2}.$$ To prove Theorem \[ls\], we use the duality principle and the Poisson summation formula for $\mathbb{R}^d$. \[duality\] Let $C = [c_{mn}]$ be a finite matrix with complex entries. The following two statements are equivalent: 1. For any complex numbers $a_n$, we have $$\begin{aligned} \sum_m \mathrel \Big |\sum_n a_n c_{mn}\Big |^2 \leq \Delta \sum_n |a_n|^2.\end{aligned}$$ 2. For any complex numbers $b_m$, we have $$\begin{aligned} \sum_n \mathrel \Big |\sum_m b_m c_{mn}\Big |^2 \leq \Delta \sum_m |b_m|^2. \end{aligned}$$ \[poisson\] Let $f:\mathbb{R}^d \rightarrow \mathbb{C}$ be a smooth function of rapid decay and $\Lambda$ be a lattice of full rank in $\mathbb{R}^d$. Then $$\sum\limits_{y\in \Lambda} f(y) = \frac{1}{\mbox{\rm Vol}(\mathbb{R}^d/\Lambda)} \cdot \sum\limits_{x\in \Lambda'} \hat{f}(x),$$ where $\Lambda'$ is the dual lattice and $\hat{f}$ is the Fourier transform of $f$, defined as $$\hat{f}(x)=\int\limits_{\mathbb{R}^2} f(y)e\left(-x\cdot y\right)\ dy.$$ Here as in the following, by [*rapid decay*]{} we mean that the function $f:\mathbb{R}^d \rightarrow \mathbb{C}$ in question satisfies the bound $$f(y)\ll \left(1+||y||_2\right)^{-C}$$ for some $C>d$. By a linear change of variables, we immediately deduce the following more general version of the Poisson summation formula for shifted lattices from Proposition \[poisson\]. \[poissongen\] Let the conditions and notations of Proposition \[poisson\] be kept and assume that $B>0$ and $a\in \mathbb{R}^d$. Then $$\sum\limits_{y\in a+\Lambda} f\left(\frac{y}{B}\right) = \frac{B^d}{\mbox{\rm Vol}(\mathbb{R}^d/\Lambda)} \cdot \sum\limits_{x\in \Lambda'} e(a\cdot x)\hat{f}(Bx).$$ $ $\ [**Proof of Theorem \[ls\]:**]{} We first note that $$\label{KK} \begin{split} K = & \max\limits_{1\le i\le R} \sharp \left\{j\in \{1,...,R\} : \min\limits_{z\in \mathbb{Z}^d} ||x_j-x_i-z||_2\le \sqrt{d}N^{-1/d}\right\} \\ \geq & \max\limits_{1\le i\le R} \sharp \{j\in \{1,...,R\} : \min\limits_{z\in \mathbb{Z}^d} \max\limits_{1\le k\le d} |x_j^{(k)}-x_i^{(k)}-z^{(k)}| \le N^{-1/d}\}\\ = & \max\limits_{1\le i\le R} \sharp \{j\in \{1,...,R\} : \max\limits_{1\le k\le d} ||x_j^{(k)}-x_i^{(k)}-z^{(k)}|| \le N^{-1/d}\} =: K', \end{split}$$ where $||u||$ is the distance of $u\in \mathbb{R}$ to the nearest integer and we write $$x_i=\left(x_i^{(1)},...,x_i^{(d)}\right)\quad \mbox{and} \quad z=\left(z^{(1)},...,z^{(d)}\right)$$ for $i=1,...,R$. Now let $S = \{x_1, x_2, . . . , x_R\}$. Taking Proposition \[duality\], the duality principle, into account, it suffices to prove that $$\sum\limits_{\substack{n\in \mathbb{Z}^d\\ ||n||_2\le N^{1/d}}}\left|\sum\limits_{x\in S} b_x\cdot e(n\cdot x)\right|^2 \ll KN\sum\limits_{x\in S}|b_x|^2$$ for any complex numbers $b_x$. To this end, for $x=\left(x^{(1)},...,x^{(d)}\right)\in \mathbb{R}^d$, we define $$\phi(x) = \prod\limits_{k=1}^d \left(\frac{\sin\left(\pi x^{(k)}\right)}{2x^{(k)}}\right)^2$$ and note that $\phi(x)$ is non-negative and satisfies $\phi(x)\ge 1$ if $|x^{(k)}|\le 1/2$ for $k=1,...d$. Moreover, the Fourier transformation of $\phi(x)$ equals $$\hat{\phi}(s) = \left(\frac{\pi^2}{4}\right)^d\prod\limits_{k=1}^d\max\left\{1-\left|s^{(k)}\right|,\ 0\right\},$$ where we write $$s=\left(s^{(1)},...,s^{(d)}\right).$$ Hence, $$\begin{split} \sum\limits_{\substack{n\in \mathbb{Z}^d\\ ||n||_2\le N^{1/d}}} \left|\sum\limits_{x\in S} b_x\cdot e(n\cdot x)\right|^2 \le & \sum\limits_{n\in\mathbb{Z}^d}\phi\left(\frac{n}{2N^{1/d}}\right) \left|\sum\limits_{x\in S} b_x\cdot e(n\cdot x)\right|^2\\ = & \sum\limits_{x,x'\in S} b_x \overline{b_{x'}}\cdot V(x-x'), \end{split}$$ where $$V(y) = \sum\limits_{n\in \mathbb{Z}^d} \phi\left(\frac{n}{2N^{1/d}}\right)\cdot e(n\cdot y).$$ Using Proposition \[poissongen\], the Poisson summation formula, we transform $V(y)$ into $$\begin{split} V(y) = & 2^d N \cdot \sum\limits_{\alpha \in y+\mathbb{Z}^d} \tilde{\phi}\left(2N^{1/d} \alpha\right) = 2^d N \cdot \sum\limits_{\alpha\in -y+\mathbb{Z}^d} \hat{\phi}\left(2N^{1/d} \alpha \right)\\ = & \frac{\pi^{2d}}{2^d}\cdot N \cdot \prod\limits_{k=1}^d \max\left\{1-\left|2N^{1/d}y^{(k)}\right|,0\right\}, \end{split}$$ where $\tilde{\phi}$ is the inverse Fourier transform and $\hat{\phi}$ is the Fourier transform of $\phi$. Therefore, $$\begin{split} \sum\limits_{\substack{n\in \mathbb{Z}^d\\ ||n||_2\le N^{1/d}}}\left|\sum\limits_{x\in S} b_x\cdot e(n\cdot x)\right|^2 \le \frac{\pi^{2d}}{2^d}\cdot N \cdot \sum\limits_{\substack{x,x'\in S\\ ||x^{(i)}-{x'}^{(i)}||\le N^{-1/d} \mbox{\scriptsize\ for\ } i=1,...,d}} |b_x| |b_{x'}|. \end{split}$$ Now we observe that $$|b_x| |b_{x'}| \le \frac{1}{2}\cdot \left(|b_x|^2+|b_{x'}|^2\right).$$ It follows that $$\begin{split} \sum\limits_{\substack{n\in \mathbb{Z}^d\\ ||n||_2\le N^{1/d}}}\left|\sum\limits_{x\in S} b_x\cdot e(n\cdot x)\right|^2 \ll K'NZ\le KNZ, \end{split}$$ where we use and . This completes the proof. $\Box$ Conversion into a counting problem ================================== Now we return to the large sieve for $\mathbb{Q}(i)$. We begin with restricting the moduli $q$ to an arbitrary multiset $S$ of elements of $\mathbb{Z}[i]\setminus \{0\}$. We shall also restrict the norms of these moduli to dyadic intervals, which is for technical reasons. Thus, we are interested in estimating the quantity $$T:=\sum\limits_{\substack{q\in S\\ Q/2<\mathcal{N}(q)\le Q}} \sum\limits_{\substack{r \bmod{q}\\ (r,q)=1}} \left|\sum\limits_{\substack{n\in \mathbb{Z}[i]\\ \mathcal{N}(n)\le N}} a_n \cdot e\left(\Re\left(\frac{nr}{q}\right)\right)\right|^2.$$ We shall later confine ourselves to squares or, more generally, $k$-th powers. Our first step is to re-write $T$ in the form $$\label{T} T=\sum\limits_{\substack{q\in S\\ Q/2<\mathcal{N}(q)\le Q}} \sum\limits_{\substack{r \bmod{q}\\ (r,q)=1}} \left|\sum\limits_{\substack{n\in \mathbb{Z}[i]\\ \mathcal{N}(n)\le N}} a_n \cdot e\left(\left(\frac{xu+yv}{\mathcal{N}(q)},\frac{xv-yu}{\mathcal{N}(q)}\right)\cdot (s,t)\right)\right|^2,$$ where $$q=u+iv,\quad r=x+iy,\quad n=s+ti.$$ To bound $T$, we employ Theorem \[ls\] for the case $d=2$, which immediately gives us the following. \[preform\] For $T$ as defined in , we have the bound $$T\ll KNZ,$$ where $Z$ is defined as in and $$\begin{split} & K:=\max\limits_{r_1,q_1} \sharp \Bigg\{(r_2,q_2) : \\ & \min\limits_{z\in \mathbb{Z}^2} \left|\left|\left(\frac{x_2u_2+y_2v_2}{\mathcal{N}(q_2)}, \frac{x_2v_2-y_2u_2}{\mathcal{N}(q_2)}\right)- \left(\frac{x_1u_1+y_1v_1}{\mathcal{N}(q_1)},\frac{x_1v_1-y_1u_1}{\mathcal{N}(q_1)}\right)-z\right|\right|_2^2 \le 2N^{-1}\Bigg\} \end{split}$$ with the conventions that, for $j=1,2$, $$q_j\in S,\quad Q/2<\mathcal{N}(q_j)\le Q,$$ $\{r_j\}$ forms a system of representatives of reduced residue classes modulo $q_j$ and $$q_j=u_j+iv_j,\quad r_j=x_j+iy_j.$$ Thus, we have converted the problem into a counting problem. Switching back to $\mathbb{Z}[i]$ ================================= Now we observe that $$\frac{\overline{r_j}}{\overline{q_j}}=\frac{x_j-iy_j}{u_j-iv_j}=\frac{x_ju_j+y_jv_j}{\mathcal{N}(q_j)}+\frac{x_jv_j-y_ju_j}{\mathcal{N}(q_j)}i.$$ It follows that $$\begin{split} K= & \max\limits_{r_1,q_1} \sharp \Bigg\{(r_2,q_2) : \min\limits_{z\in \mathbb{Z}[i]} \left|\frac{\overline{r_2}}{\overline{q_2}}-\frac{\overline{r_1}}{\overline{q_1}}-z \right|^2 \le 2N^{-1}\Bigg\}\\ = & \max\limits_{\substack{q_1\in S\\ Q/2<\mathcal{N}(q_1)\le Q\\ (r_1,q_1)=1}} \sharp \Bigg\{(r_2,q_2) : q_2\in S,\ Q/2<\mathcal{N}(q_2)\le Q,\ (r_2,q_2)=1, \\ & \left|\frac{r_2}{q_2}-\frac{r_1}{q_1}\right|^2 \le 2N^{-1}\Bigg\}. \end{split}$$ Further, $$\left|\frac{r_2}{q_2}-\frac{r_1}{q_1}\right|_2^2 \le 2N^{-1} \Longleftrightarrow \mathcal{N}\left(r_1q_2-r_2q_1\right) \le 2N^{-1}\mathcal{N}(q_1)\mathcal{N}(q_2).$$ We deduce that $$\label{Ktransform} \begin{split} K\le & \max\limits_{\substack{q_1\in S\\ Q/2<\mathcal{N}(q_1)\le Q\\ (r_1,q_1)=1}} \ \sum\limits_{\substack{b\in \mathbb{Z}[i]\\ \mathcal{N}(b)\le 2N^{-1}\mathcal{N}(q_1)\mathcal{N}(q_2)}} \ \sum\limits_{\substack{q_2\in S,\\ Q/2<\mathcal{N}(q_2)\le Q\\ r_1q_2\equiv b \bmod{q_1}}} 1 \\ \ll & \max\limits_{\substack{q_1\in S\\ Q/2<\mathcal{N}(q_1)\le Q\\ (r_1,q_1)=1}} \ \sum\limits_{b\in \mathbb{Z}[i]} \Phi_1\left(\mathcal{N}\left(\frac{b\sqrt{N}}{q_1\sqrt{2Q}}\right)\right) \sum\limits_{\substack{q_2\in S,\\ r_1q_2\equiv b \bmod{q_1}}} \Phi_2\left(\mathcal{N}\left(\frac{q_2}{\sqrt{Q}}\right)\right)\\ = & \max\limits_{\substack{q_1\in S\\ Q/2<\mathcal{N}(q_1)\le Q\\ (r_1,q_1)=1}} \ \sum\limits_{q_2\in S} \Phi_2\left(\mathcal{N}\left(\frac{q_2}{\sqrt{Q}}\right)\right)\cdot \sum\limits_{b\equiv r_1q_2 \bmod{q_1}} \Phi_1\left(\mathcal{N}\left(\frac{b\sqrt{N}}{q_1\sqrt{2Q}}\right)\right), \end{split}$$ where, for $i=1,2$, $\Phi_{i} : \mathbb{R} \rightarrow \mathbb{R}^+$ are any smooth functions with rapid decay such that $\Phi_{i}(x)\gg 1$ if $|x|\le 1$. We shall fix $\Phi_{i}$ later suitably. Let $\Psi_{i}: \mathbb{C}\rightarrow \mathbb{R}^+$ be given by $$\Psi=\Phi \circ \mathcal{N} \quad \mbox{for } i=1,2.$$ Then the above inequality for $K$ turns into $$\label{superK} \begin{split} K\ll \max\limits_{\substack{q_1\in S\\ Q/2<\mathcal{N}(q_1)\le Q\\ (r_1,q_1)=1}} \ \sum\limits_{q_2\in S} \Psi_2\left(\frac{q_2}{\sqrt{Q}}\right)\cdot \sum\limits_{b\equiv r_1q_2 \bmod{q_1}} \Psi_1\left(\frac{b\sqrt{N}}{|q_1|\sqrt{2Q}}\right). \end{split}$$ Application of Poisson summation ================================ To transform the inner-most sum over $b$, we use Poisson summation again. The complex numbers $a\equiv 0 \bmod{q_1}$ form a square lattice $$\Lambda=\left\{x\binom{u_1}{v_1}+y\binom{-v_1}{u_1}: (x,y)\in \mathbb{Z}^2\right\}\subset \mathbb{R}^2$$ with volume $\mathcal{N}(q_1)$ when regarded as vectors in $\mathbb{R}^2$. The dual lattice turns out to be $$\Lambda'=\frac{1}{\mathcal{N}(q_1)}\cdot \Lambda,$$ which corresponds to the set $$\left\{\frac{a}{\mathcal{N}(q_1)} : a\equiv 0 \bmod{q_1}\right\}$$ in $\mathbb{C}$. Hence, Proposition \[poisson\] gives $$\label{afterpoisson} \begin{split} \sum\limits_{b\equiv r_1q_2 \bmod{q_1}} \Psi_1\left(\frac{b\sqrt{N}}{|q_1|\sqrt{2Q}}\right)= & \frac{2Q}{N}\cdot \sum\limits_{a\equiv 0\bmod{q_1}} e\left( \frac{\overrightarrow{a}\cdot \overrightarrow{r_1q_2}}{\mathcal{N}(q_1)}\right) \hat\Psi_1\left(\frac{a|q_1|\sqrt{2Q}}{\mathcal{N}(q_1)\sqrt{N}}\right)\\ = & \frac{2Q}{N}\cdot \sum\limits_{j\in \mathbb{Z}[i]} e\left(\overrightarrow{\frac{j}{\overline{q_1}}}\cdot \overrightarrow{r_1q_2}\right) \hat\Psi_1\left(\frac{j\sqrt{2Q}}{\sqrt{N}}\right), \end{split}$$ where for $x,z\in \mathbb{C}$, we write $$\overrightarrow{z}=\binom{\Re(z)}{\Im(z)} \quad \mbox{and} \quad \hat\Psi(x)=\int\limits_{\mathbb{C}} \Psi(y) e\left(-\overrightarrow{y}\cdot \overrightarrow{x}\right) \ dy_2dy_1$$ with $y_1:=\Re(y)$ and $y_2:=\Im(y)$. Combining and , and re-arranging summation, we deduce that $$\label{Kgen} K\ll \frac{Q}{N}\cdot\max\limits_{\substack{q_1\in S\\ Q/2<\mathcal{N}(q_1)\le Q\\ (r_1,q_1)=1}} \ \sum\limits_{j\in \mathbb{Z}[i]} \hat\Psi_1\left(\frac{j\sqrt{2Q}}{\sqrt{N}}\right) \cdot \sum\limits_{q_2\in S} \Psi_2\left(\frac{q_2}{\sqrt{Q}}\right)\cdot e\left(\overrightarrow{\frac{j}{\overline{q_1}}}\cdot \overrightarrow{r_1q_2}\right).$$ We observe that for $a,b\in \mathbb{C}$, $$\label{obs} e\left(\overrightarrow{a}\cdot \overrightarrow{b}\right)=e\left(\Re(\overline{a}b)\right).$$ Hence, upon a change of variables $j \rightarrow \overline{j}$, we arrive at $$\label{Kgen1} K\ll \frac{Q}{N}\cdot\max\limits_{\substack{q_1\in S\\ Q/2<\mathcal{N}(q_1)\le Q\\ (r_1,q_1)=1}} \ \sum\limits_{j\in \mathbb{Z}[i]} \hat\Psi_1\left(\frac{j\sqrt{2Q}}{\sqrt{N}}\right) \sum\limits_{q_2\in S} \Psi_2\left(\frac{q_2}{\sqrt{Q}}\right)\cdot e\left(\Re\left(\frac{jr_1}{q_1}\cdot q_2\right)\right).$$ The case of square moduli ========================= Now we restrict overselves to the case when $S$ is the set of non-zero squares in $\mathbb{Z}[i]$. We write $Q_0=\sqrt{Q}$ and replace $q_i$ by $q_i^2$ ($i=1,2$). Throughout the following, we assume that $Q_0>N^{1/4}$ for otherwise the desired result follows from upon extending the set of moduli to all non-zero Gaussian integers. We deduce from that $$\label{Knew} \begin{split} K\ll & \frac{Q_0^2}{N}\cdot \max\limits_{\substack{Q_0/\sqrt{2}<\mathcal{N}(q_1)\le Q_0\\ (r_1,q_1)=1}} \sum\limits_{j\in \mathbb{Z}[i]} \hat\Psi_1\left(\frac{\sqrt{2}jQ_0}{\sqrt{N}}\right) \cdot \sum\limits_{q_2\in \mathbb{Z}[i]} \Psi_2\left(\frac{q_2^2}{Q_0}\right)\cdot e\left(\Re\left(\frac{jr_1}{q_1^2}\cdot q_2^2 \right) \right)\\ \ll & \frac{Q_0^3}{N}+ \frac{Q_0^2}{N}\cdot \max\limits_{\substack{Q_0/\sqrt{2}<\mathcal{N}(q_1)\le Q_0\\ (r_1,q_1)=1}} \sum\limits_{j\in \mathbb{Z}[i]\setminus \{0\}} \hat\Psi_1\left(\frac{\sqrt{2}jQ_0}{\sqrt{N}}\right) S\left(q_1,r_1,j\right), \end{split}$$ where $$S\left(q_1,r_1,j\right):=\sum\limits_{q_2\in \mathbb{Z}[i]} \Psi_2\left(\frac{q_2^2}{Q_0}\right) \cdot e\left(\Re\left(\frac{jr_1}{q_1^2}\cdot q_2^2\right)\right).$$ Here we use the estimate $$\sum\limits_{q_2\in \mathbb{Z}[i]} \Psi_2\left(\frac{q_2^2}{Q_0}\right) \ll Q_0$$ to bound the contribution of $j=0$. Weyl differencing ----------------- Now we employ Weyl differencing in the setting of $\mathbb{Z}[i]$. Using the Cauchy-Schwarz inequality, we deduce that $$\label{afterCS} \begin{split} K\ll & \frac{Q_0^3}{N}+ \frac{Q_0^2}{N}\cdot \max\limits_{\substack{Q_0/\sqrt{2}<\mathcal{N}(q_1)\le Q_0\\ (r_1,q_1)=1}} \left(\sum\limits_{j\in \mathbb{Z}[i]} \hat\Psi_1\left(\frac{\sqrt{2}jQ_0}{\sqrt{N}}\right)\right)^{1/2}\times\\ & \left(\sum\limits_{j\in \mathbb{Z}[i]\setminus \{0\}} \hat\Psi_1\left(\frac{jQ_0}{\sqrt{N}}\right)\cdot \left| S\left(q_1,r_1,j\right) \right|^2\right)^{1/2}\\ \ll & \frac{Q_0^3}{N}+ \frac{Q_0}{\sqrt{N}}\cdot \max\limits_{\substack{Q_0/\sqrt{2}<\mathcal{N}(q_1)\le Q_0\\ (r_1,q_1)=1}} \left(\sum\limits_{j\in \mathbb{Z}[i]\setminus \{0\}} \hat\Psi_1\left(\frac{\sqrt{2}jQ_0}{\sqrt{N}}\right)\cdot \left| S\left(q_1,r_1,j\right) \right|^2\right)^{1/2}, \end{split}$$ where we use the estimate $$\sum\limits_{j\in \mathbb{Z}[i]} \hat\Psi_1\left(\frac{\sqrt{2}jQ_0}{\sqrt{N}}\right) \ll \frac{N}{Q_0^2}.$$ Multiplying out the square, we get $$\label{multi} \left| S\left(q_1,r_1,j\right) \right|^2 = \sum\limits_{q_2,q\in \mathbb{Z}[i]} \Psi_2\left(\frac{q_2^2}{Q_0}\right) \cdot \Psi_2\left(\frac{q^2}{Q_0}\right) \cdot e\left(\Re\left(\frac{jr_1}{q_1^2}\cdot \left(q_2^2-q^2\right) \right)\right).$$ We now set $$\alpha:=q_2-q$$ so that $$q_2^2-q^2=\alpha^2+2\alpha q \quad \mbox{and} \quad q_2=\alpha+q.$$ Then turns into $$\label{change} \begin{split} \left| S\left(q_1,r_1,j\right) \right|^2 = & \sum\limits_{\alpha\in \mathbb{Z}[i]} e\left(\Re\left(\frac{jr_1\alpha^2}{q_1^2}\right)\right) \times\\ & \sum\limits_{q\in \mathbb{Z}[i]} \Psi_2\left(\frac{q^2}{Q_0}\right) \cdot\Psi_2\left(\frac{(\alpha+q)^2}{Q_0}\right) \cdot e\left(\Re\left(\frac{2j\alpha r_1}{q_1^2}\cdot q\right)\right). \end{split}$$ Poisson summation ----------------- We shall apply Proposition \[poissongen\] with $d=2$ to transform the inner-most over $q$ on the right-hand side of . For $z=(z_1,z_2)\in \mathbb{R}^2$, we set $$g(z):=\Psi_2\left((z_1+iz_2)^2\right) \cdot\Psi_2\left(\left(\frac{\alpha}{\sqrt{Q_0}}+z_1+iz_2\right)^2\right).$$ Then using with $$a=q \quad \mbox{and} \quad b=\overline{\frac{2j\alpha r_1}{q_1^2}},$$ we deduce that $$\label{backtor2} \sum\limits_{q\in \mathbb{Z}[i]} \Psi_2\left(\frac{q^2}{Q_0}\right) \cdot\Psi_2\left(\frac{(\alpha+q)^2}{Q_0}\right) \cdot e\left(\Re\left(\frac{2j\alpha r_1}{q_1^2}\cdot q\right)\right) =\sum\limits_{x\in \mathbb{Z}^2} e\left(\overrightarrow{b}\cdot x\right)g\left(\frac{x}{\sqrt{Q_0}}\right).$$ Now applying Proposition \[poissongen\] with $B:=1/\sqrt{Q_0}$ and $f:=\tilde{g}$, the inverse Fourier transform of $g$, to the right-hand side of , we get $$\sum\limits_{x\in \mathbb{Z}^2} e\left(\overrightarrow{b}\cdot x\right)g\left(\frac{x}{\sqrt{Q_0}}\right) = Q_0\cdot \sum\limits_{y\in \overrightarrow{b}+\mathbb{Z}^2} \tilde{g}\left(\sqrt{Q_0}y\right) = Q_0\cdot \sum\limits_{y\in -\overrightarrow{b}+\mathbb{Z}^2} \hat{g}\left(\sqrt{Q_0}y\right).$$ It follows that $$\label{newpoiss} \left| S\left(q_1,r_1,j\right) \right|^2 = Q_0\cdot \sum\limits_{\alpha\in \mathbb{Z}[i]} e\left(\Re\left(\frac{jr_1\alpha^2}{q_1^2}\right)\right) \cdot \sum\limits_{\beta\in \mathbb{Z}[i]} \hat{g}\left(\sqrt{Q_0}\cdot \overrightarrow{\left(\beta-\overline{\frac{2j\alpha r_1}{q_1^2}}\right)}\right).$$ At this point, we specify our choice of $\Psi_2$ and compute the Fourier transform of $g$. We set $$\Phi_2(t):=\exp\left(-\frac{\pi}{2}\cdot \sqrt{|t|}\right)$$ so that $$\Psi_2(z)=\Phi_2(\mathcal{N}(z))=\exp\left(-\frac{\pi}{2}\cdot \sqrt{\mathcal{N}(z)}\right).$$ It follows that $$g(z)=\exp\left(-\frac{\pi}{2}\cdot \left(z_1^2+z_2^2+\left(\frac{\alpha_1}{\sqrt{Q_0}}+z_1\right)^2+ \left(\frac{\alpha_2}{\sqrt{Q_0}}+z_2\right)^2\right)\right),$$ where $\alpha_1:=\Re(\alpha)$ and $\alpha_2:=\Im(\alpha)$. Completing the squares, it follows that $$g(z)=\exp\left(-\frac{\pi}{4Q_0}\cdot \left(\alpha_1^2+\alpha_2^2\right)\right)\cdot \exp\left(-\pi\left(\left(z_1+\frac{\alpha_1}{2\sqrt{Q_0}}\right)^2+\left(z_2+\frac{\alpha_2}{2\sqrt{Q_0}}\right)^2\right)\right).$$ The Fourier transform of this function equals $$\label{gfourier} \begin{split} \hat{g}(z)= & \exp\left(-\frac{\pi}{4Q_0}\cdot \left(\alpha_1^2+\alpha_2^2\right)\right)\cdot e\left(-\frac{\alpha_1z_1+\alpha_2z_2}{2\sqrt{Q_0}}\right)\cdot \exp\left(-\pi \left(z_1^2+z_2^2\right)\right)\\ = &\exp\left(-\frac{\pi}{4Q_0}\cdot \mathcal{N}(\alpha)\right)\cdot e\left(-\frac{\Re(\overline{\alpha}(z_1+iz_2))}{2\sqrt{Q_0}}\right)\cdot \exp\left(-\pi \mathcal{N}(z_1+iz_2)\right). \end{split}$$ Plugging into , using the triangle inequality and bounding all terms of the form $e(\gamma)$ trivially by $|e(\gamma)|\le 1$, we get $$\label{finalafterps} \begin{split} \left| S\left(q_1,r_1,j\right) \right|^2\le & Q_0 \cdot \sum\limits_{\alpha\in \mathbb{Z}[i]} \exp\left(-\frac{\pi}{4Q_0}\cdot \mathcal{N}(\alpha)\right) \times\\ & \sum\limits_{\beta\in \mathbb{Z}[i]} \exp\left(-\pi Q_0 \mathcal{N}\left(\beta-\overline{\frac{2j \alpha r_1}{q_1^2}}\right)\right). \end{split}$$ Counting -------- The contributions of $\beta$’s with $$\mathcal{N}\left(\beta-\overline{\frac{2j \alpha r_1}{q_1^2}}\right) > Q_0^{\varepsilon-1}$$ and of $\alpha$’s with $$\mathcal{N}(\alpha)> Q_0^{1+\varepsilon}$$ to the right-hand side of are neglible. Therefore, it follows from that $$\label{squarebound} \begin{split} \left| S\left(q_1,r_1,j\right) \right|^2 \ll & \frac{1}{(Q_0N)^{2018}}+Q_0\sum\limits_{\mathcal{N}(\alpha)\le Q_0^{1+\varepsilon}} \sum\limits_{\substack{\beta\in \mathbb{Z}[i]\\ \mathcal{N}\left(\beta-\overline{2j \alpha r_1/q_1^2}\right)\le Q_0^{\varepsilon-1}}} 1\\ \ll & \frac{1}{(Q_0N)^{2018}}+Q_0\sum\limits_{\substack{\mathcal{N}(\alpha)\le Q_0^{1+\varepsilon}\\ ||2j\alpha r_1/q_1^2||\le Q_0^{\varepsilon-1/2}}} 1, \end{split}$$ where $||z||$ is the distance of $z\in \mathbb{C}$ to the nearest Gaussian integer. Now we want to bound the term in the maximum on the right-hand side of . To this end, we choose $\Psi_1$ in a suitable way so that $\hat\Psi_1$ decays exponentially. We set $$\Phi_1(t):=\exp\left(-\pi |t|\right)$$ Since $\Psi_1=\Phi_1\circ \mathcal{N}$, it follows that $$\label{psi1} \Psi_1(z)=\exp\left(-\pi \mathcal{N}(z)\right)=\hat{\Psi}_1(z).$$ Hence, using , we obtain $$\label{obvious} \sum\limits_{j\in \mathbb{Z}[i]\setminus \{0\}} \hat\Psi_1\left(\frac{\sqrt{2}jQ_0}{\sqrt{N}}\right)\cdot \left| S\left(q_1,r_1,j\right) \right|^2\\ \ll 1+Q_0\sum\limits_{0<\mathcal{N}(j)\le NQ_0^{\varepsilon-2}} \sum\limits_{\substack{\mathcal{N}(\alpha)\le Q_0^{1+\varepsilon}\\ ||2j\alpha r_1/q_1^2||\le Q_0^{\varepsilon-1/2}}} 1$$ upon noting that the contribution of $\mathcal{N}(j)>NQ_0^{\varepsilon-2}$ is negligible. The contribution of $\alpha=0$ to the right-hand side of is obviously bounded by $$\ll \frac{N}{Q_0^{1-\varepsilon}}.$$ Writing $d=2j\alpha$ and noting that the number of divisors of $d\in \mathbb{Z}[i]\setminus \{0\}$ in the Gaussian integers is bounded by $O\left(\mathcal{N}(d)^{\varepsilon}\right)$, we deduce that $$\label{super} \begin{split} \sum\limits_{j\in \mathbb{Z}[i]\setminus \{0\}} \hat\Psi_1\left(\frac{\sqrt{2}jQ_0}{\sqrt{N}}\right)\cdot \left| S\left(q_1,r_1,j\right) \right|^2 \ll & \frac{N}{Q_0^{1-\varepsilon}}+ N^{\varepsilon}Q_0\sum\limits_{\substack{\mathcal{N}(d)\le 4NQ_0^{2\varepsilon-1}\\ ||d r_1/q_1^2||\le Q_0^{\varepsilon-1/2}}} 1\\ \ll & \frac{N}{Q_0^{1-\varepsilon}}+ N^{\varepsilon}Q_0\sum\limits_{\substack{l\in \mathbb{Z}[i]\\ |l/q_1^2|\le Q_0^{\varepsilon-1/2}}} \sum\limits_{\substack{\mathcal{N}(d)\le 4NQ_0^{2\varepsilon-1}\\ d\equiv l\overline{r_1} \bmod{q_1^2}}} 1, \end{split}$$ where $\overline{r_1}$ is a multiplicative inverse of $r_1$ modulo $q_1^2$, i.e. $r_1\overline{r_1}\equiv 1 \bmod{q_1^2}$. The number of residue classes modulo $q_1^{2}$ is $\mathcal{N}(q_1^2)\le Q_0^2$, and hence $$\label{res} \sum\limits_{\substack{\mathcal{N}(d)\le 4NQ_0^{2\varepsilon-1}\\ d\equiv l\overline{r_1} \bmod{q_1^2}}} 1 \ll 1+\frac{N}{Q_0^{3-2\varepsilon}}.$$ Further, $$\label{lat} \sum\limits_{\substack{l\in \mathbb{Z}[i]\\ |l/q_1^2|\le Q_0^{\varepsilon-1/2}}} 1 \le \sum\limits_{\substack{l\in \mathbb{Z}[i]\\ |l|\le Q_0^{\varepsilon+1/2}}} 1 \ll Q_0^{1+2\varepsilon}.$$ Combining , and , we obtain $$\label{comb1} \sum\limits_{j\in \mathbb{Z}[i]\setminus \{0\}} \hat\Psi_1\left(\frac{\sqrt{2}jQ_0}{\sqrt{N}}\right)\cdot \left| S\left(q_1,r_1,j\right)\right|^2 \ll (Q_0N)^{4\varepsilon}\left( Q_0^{2}+\frac{N}{Q_0} \right),$$ and combining and , we arrive at $$\label{comb2} K\ll \frac{Q_0^3}{N}+(Q_0N)^{2\varepsilon}\left( \frac{Q_0^2}{N^{1/2}}+Q_0^{1/2} \right),$$ our final bound for $K$. Now the statement in Theorem \[squaremodZi\] follows immediately from Corollary \[preform\] and after dividing the moduli into dyadic intervals and replacing $Q_0$ by $Q$. The case of $k$-th power moduli =============================== Now we take $S$ as the set of non-zero $k$th-powers in $\mathbb{Z}[i]$. We write $Q_0=Q^{1/k}$ and replace $q_i$ by $q_i^k$ ($i=1,2$). Throughout the following, we assume that $Q_0>N^{1/(2k)}$ for otherwise the desired result follows from upon extending the set of moduli to all non-zero Gaussian integers. We deduce from that $$\label{generalKbound} \begin{split} K\ll & \frac{Q_0^k}{N}\cdot \max\limits_{\substack{Q_0/\sqrt[k]{2}<\mathcal{N}(q_1)\le Q_0\\ (r_1,q_1)=1}} \sum\limits_{j\in \mathbb{Z}[i]} \hat\Psi_1\left(\frac{\sqrt{2}jQ_0^{k/2}}{\sqrt{N}}\right) \times\\ & \sum\limits_{q_2\in \mathbb{Z}[i]} \Psi_2\left(\frac{q_2^k}{Q_0^{k/2}}\right)\cdot e\left(\Re\left(\frac{jr_1}{q_1^k}\cdot q_2^k \right) \right)\\ \ll & \frac{Q_0^{k+1}}{N}+ \frac{Q_0^k}{N}\cdot \max\limits_{\substack{Q_0/\sqrt[k]{2}<\mathcal{N}(q_1)\le Q_0\\ (r_1,q_1)=1}} \sum\limits_{j\in \mathbb{Z}[i]\textbackslash \{0\}} \hat\Psi_1\left(\frac{\sqrt{2}jQ_0^{k/2}}{\sqrt{N}}\right) \cdot \left| S_k\left(q_1,r_1,j\right)\right|, \end{split}$$ where $$S_k\left(q_1,r_1,j\right)=\sum\limits_{q_2\in \mathbb{Z}[i]} \Psi_2\left(\frac{q_2^k}{Q_0^{k/2}}\right) \cdot e\left(\Re\left(\frac{jr_1}{q_1^k}\cdot q_2^k \right) \right).$$ Here we use the estimate $$\sum\limits_{q_2\in \mathbb{Z}[i]} \Psi_2\left(\frac{q_2^k}{Q_0^{k/2}}\right) \ll Q_0$$ to bound the contribution of $j=0$. Weyl differencing ----------------- Multiplying out the square and setting $\alpha_1=q_2-q$, we obtain $$\begin{split} & \left|S_k\left(q_1,r_1,j\right) \right|^2 \\ = & \sum\limits_{q_2,q\in \mathbb{Z}[i]} \Psi_2\left(\frac{q_2^k}{Q_0^{k/2}}\right) \cdot \Psi_2\left(\frac{q^k}{Q_0^{k/2}}\right) \cdot e\left(\Re\left(\frac{jr_1}{q_1^k}\cdot(q_2^k-q^k) \right) \right),\\ = & \sum\limits_{\alpha_1,q\in \mathbb{Z}[i]} \Psi_2\left(\frac{q^k}{Q_0^{k/2}}\right) \cdot \Psi_2\left(\frac{(\alpha_1+q)^k}{Q_0^{k/2}}\right) \cdot e\left(\Re\left(\frac{jr_1}{q_1^k}\cdot\left((\alpha_1 + q)^k-q^k\right) \right) \right). \end{split}$$ We observe that the contribution of $\alpha_1$’s with $\mathcal{N}\left(\alpha_1\right)>Q_0^{1+\varepsilon}$ is negligible and write $$P_{k-1,\alpha_1}(q) = (\alpha_1 + q)^k - q^k = \binom{k}{1} \cdot \alpha_1 q^{k-1} + \binom{k}{2}\cdot \alpha_1^2q^{k-2} +\cdots + \binom{k}{k}\cdot \alpha_1^k,$$ thus obtaining $$\left|S_k\left(q_1,r_1,j\right) \right|^2 \ll \left|\sum\limits_{\mathcal{N}(\alpha_1)\le Q_0^{1+\varepsilon}} S_{k-1}\left(q_1,r_1,j,\alpha_1\right)\right|,$$ where $$S_{k-1}\left(q_1,r_1,j,\alpha_1\right):=\sum\limits_{q\in \mathbb{Z}[i]} \Psi_2\left(\frac{q^k}{Q_0^{k/2}}\right) \cdot \Psi_2\left(\frac{(\alpha_1+q)^k}{Q_0^{k/2}}\right) \cdot e\left(\Re\left(\frac{jr_1}{q_1^k}\cdot P_{k-1,\alpha_1}(q)\right) \right).$$ If $k > 2$, we apply the Cauchy-Schwarz inequality again to obtain $$\left|S_k\left(q_1,r_1,j\right) \right|^4 \ll Q_0^{1+ \varepsilon}\sum\limits_{\substack{\alpha_1 \in \mathbb{Z}[i] \\ \mathcal{N}(\alpha_1) \le Q_0^{1+\varepsilon}}} \left| S_{k-1}\left(q_1,r_1,j,\alpha_1\right) \right|^2.$$ Multiplying out the square, changing variables and truncating the resulting sums in a similar way as above, we now obtain $$\begin{split} & \left| S_{k-1}\left(q_1,r_1,j,\alpha_1\right) \right|^2\\ \ll & \Bigg|\sum\limits_{\substack{\alpha_2\in \mathbb{Z}[i]\\ \mathcal{N}(\alpha_2)\le Q_0^{1+\varepsilon}}} \sum\limits_{q \in \mathbb{Z}[i]} \Psi_2\left(\frac{q^k}{Q_0^{k/2}}\right) \cdot \Psi_2\left(\frac{(\alpha_1 + q)^k}{Q_0^{{k/2}}}\right) \Psi_2\left(\frac{(\alpha_2 + q)^k}{Q_0^{k/2}}\right)\times\\ & \Psi_2\left(\frac{(\alpha_1 + \alpha_2 + q)^k}{Q_0^{{k/2}}}\right)\cdot e\left(\Re\left(\frac{jr_1}{q_1^k}\cdot P_{k-2,\alpha_1,\alpha_2}(q)\right)\right)\Bigg|, \end{split}$$ where $$P_{k-2,\alpha_1,\alpha_2}(q) = k(k-1)\alpha_1 \alpha_2 q^{k-2} + \cdots$$ is a polynomial of degree $k-2$ in $q$. We continue this process of repeated use of Cauchy-Schwarz and differencing until we have reached a polynomial of degree 1. Eventually, after combining all inequalities obtained in this way, we get $$\label{again} \begin{split} & \left|S_k\left(q_1,r_1,j\right) \right|^{\kappa} \ll Q_0^{\kappa-k+ \varepsilon}\times\\ & \sum\limits_{\substack{\alpha \in \mathbb{Z}[i]^k\\ \mathcal{N}\left(\alpha_1\right),...,\mathcal{N}\left(\alpha_{k-1}\right)\le Q_0^{1+\varepsilon}}} \Bigg| \sum\limits_{q \in \mathbb{Z}[i]} \prod_{u\in \{0,1\}^{k-1}} \Psi_2\Bigg(\frac{(u\cdot \alpha+q)^k}{Q_0^{k/2}}\Bigg) \cdot e\left(\Re\left(\frac{jr_1}{q_1^k}\cdot P_{1,\alpha}(q)\right)\right)\Bigg|, \end{split}$$ where we write $\kappa=2^{k-1}$, $\alpha=\left(\alpha_1,...,\alpha_{k-1}\right)$ and $u=\left(u_1,...,u_k\right)$, $u\cdot \alpha$ is the standard inner product, and $P_{1,\alpha}(q)$ takes the form $$P_{1,\alpha}(q) = k! \alpha_1\cdots \alpha_{k-1}\cdot \left(q + \frac{1}{2}\cdot \left(\alpha_1+\cdots +\alpha_{k-1}\right)\right).$$ Poisson summation ----------------- Again, we shall apply Proposition \[poissongen\] with $d=2$ to transform the inner-most over $q$ on the right-hand side of . For $z=(z_1,z_2)\in \mathbb{R}^2$, we set $$g(z):=\prod_{u\in \{0,1\}^{k-1}} \Psi_2\left(\left(z_1+iz_2 + \frac{u\cdot \alpha}{\sqrt{Q_0}}\right)^k\right).$$ Then using with $$a=q \quad \mbox{and} \quad b=\overline{\frac{k! \alpha_1\cdots \alpha_{k-1}jr_1}{q_1^k}},$$ we deduce that $$\label{backtor3} \begin{split} & \Bigg| \sum\limits_{q\in \mathbb{Z}[i]} \prod_{u\in \{0,1\}^{k-1}} \Psi_2\Bigg(\frac{(u\cdot \alpha+q)^k}{Q_0^{k/2}}\Bigg) \cdot e\left(\Re\left(\frac{jr_1}{q_1^k}\cdot P_{1,\alpha}(q)\right)\right) \Bigg| \\ = & \Bigg|\sum\limits_{x\in \mathbb{Z}^2} e\left(\overrightarrow{b}\cdot x\right)g\left(\frac{x}{\sqrt{Q_0}}\right)\Bigg|. \end{split}$$ Now applying Proposition \[poissongen\] with $B:=1/\sqrt{Q_0}$ and $f:=\tilde{g}$, the inverse Fourier transform of $g$, to the right-hand side of , we get $$\sum\limits_{x\in \mathbb{Z}^2} e\left(\overrightarrow{b}\cdot x\right)g\left(\frac{x}{\sqrt{Q_0}}\right) = Q_0\cdot \sum\limits_{y\in \overrightarrow{b}+\mathbb{Z}^2} \tilde{g}\left(\sqrt{Q_0}y\right) = Q_0\cdot \sum\limits_{y\in -\overrightarrow{b}+\mathbb{Z}^2} \hat{g}\left(\sqrt{Q_0}y\right).$$ It follows that $$\label{newpoiss1} \begin{split} & \left|S_k\left(q_1,r_1,j\right) \right|^{\kappa} \ll Q_0^{\kappa-k+ \varepsilon}\times\\ & \sum\limits_{\substack{\alpha \in \mathbb{Z}[i]^k\\ \mathcal{N}\left(\alpha_1\right),...,\mathcal{N}\left(\alpha_{k-1}\right)\le Q_0^{1+\varepsilon}}} \sum\limits_{\beta\in \mathbb{Z}[i]} \hat{g}\left(\sqrt{Q_0}\cdot \overrightarrow{\left(\beta-\overline{\frac{k!\alpha_1\cdots \alpha_{k-1}j r_1}{q_1^k}}\right)}\right). \end{split}$$ Here we set $$\Phi_2(t):=\exp\left(-\frac{\pi}{\kappa}\cdot \sqrt[k]{|t|}\right)$$ so that $$\Psi_2(z)=\Phi_2(\mathcal{N}(z))=\exp\left(-\frac{\pi}{\kappa}\cdot \sqrt[k]{\mathcal{N}(z)}\right).$$ It follows that $$g(z)=\exp\left(-\frac{\pi}{\kappa}\cdot \sum\limits_{u\in \{0,1\}^{k-1}} \left( \left(z_1+\frac{u\cdot \alpha^{(1)}}{\sqrt{Q_0}}\right)^2+ \left(z_2 + \frac{u\cdot \alpha^{(2)}}{\sqrt{Q_0}} \right)^2\right)\right),$$ where $$\alpha^{(1)}:=\left(\alpha_1^{(1)},...,\alpha_{k-1}^{(1)}\right):=\left(\Re(\alpha_1),...,\Re(\alpha_{k-1})\right)$$ and $$\alpha^{(2)}:=\left(\alpha_1^{(2)},...,\alpha_{k-1}^{(2)}\right):= \left(\Im(\alpha_1),...,\Im(\alpha_{k-1})\right).$$ Completing the squares, it follows that $$g(z)=\exp\left(-\frac{\pi}{4Q_0}\cdot \sum\limits_{i=1}^2 \sum\limits_{v=1}^{k-1} \left(\alpha_k^{(i)}\right)^2\right)\cdot \exp\left(-\pi\cdot \sum\limits_{i=1}^2\left(z_i+\frac{\sum\limits_{v=1}^{k-1}\alpha_v^{(i)}}{2\sqrt{Q_0}}\right)^2\right).$$ The Fourier transform of this function satisfies $$\label{gfourier1} \begin{split} \hat{g}(z)= & \exp\left(-\frac{\pi}{4Q_0}\cdot \sum\limits_{i=1}^2 \sum\limits_{v=1}^{k-1} \left(\alpha_k^{(i)}\right)^2\right)\cdot e\left(-\frac{\sum\limits_{i=1}^2 z_i\sum\limits_{v=1}^{k-1}\alpha_v^{(i)}}{2\sqrt{Q_0}}\right) \cdot \exp\left(-\pi \left(z_1^2+z_2^2\right)\right)\\ \ll & \exp\left(-\pi \mathcal{N}(z_1+iz_2)\right). \end{split}$$ Plugging into , we get $$\label{finalafterps1} \begin{split} & \left|S_k\left(q_1,r_1,j\right) \right|^{\kappa} \ll Q_0^{\kappa-k+1+\varepsilon}\times\\ & \sum\limits_{\substack{\alpha \in \mathbb{Z}[i]^k\\ \mathcal{N}\left(\alpha_1\right),...,\mathcal{N}\left(\alpha_{k-1}\right)\le Q_0^{1+\varepsilon}}} \sum\limits_{\beta\in \mathbb{Z}[i]} \exp\left(-\pi Q_0\mathcal{N} \left(\beta-\overline{\frac{k!\alpha_1\cdots \alpha_{k-1}j r_1}{q_1^k}}\right)\right). \end{split}$$ Counting -------- Now we want to bound the term in the maximum in . We choose $\hat{\Psi}_1$ as in . Then $$\label{countbeg} \begin{split} & \sum\limits_{j\in \mathbb{Z}[i]\textbackslash \{0\}} \hat\Psi_1\left(\frac{\sqrt{2}jQ_0^{k/2}}{\sqrt{N}}\right) \cdot \left| S_k\left(q_1,r_1,j\right)\right| \ll 1+ \sum\limits_{\substack{j\in \mathbb{Z}[i]\textbackslash \{0\}\\ \mathcal{N}(j)\le NQ_0^{\varepsilon-k}}} \left| S_k\left(q_1,r_1,j\right)\right|\\ \ll & 1+ \left(\frac{N}{Q_0^{k-\varepsilon}}\right)^{1-1/\kappa} \left(\sum\limits_{\substack{j\in \mathbb{Z}[i]\textbackslash \{0\}\\ \mathcal{N}(j)\le NQ_0^{\varepsilon-k}}} \left| S_k\left(q_1,r_1,j\right)\right|^{\kappa}\right)^{1/\kappa}, \end{split}$$ where the second line follows from Hölder’s inequality. Using and taking into account that the contributions of $\beta$’s with $$\mathcal{N}\left(\beta-\overline{\frac{k!\alpha_1\cdots \alpha_{k-1}j r_1}{q_1^k}}\right) > Q_0^{\varepsilon-1}$$ is negligible, we deduce that $$\label{super1} \begin{split} & \sum\limits_{\substack{j\in \mathbb{Z}[i]\textbackslash \{0\}\\ \mathcal{N}(j)\le NQ_0^{\varepsilon-k}}} \left| S_k\left(q_1,r_1,j\right)\right|^{\kappa}\\ \ll & Q_0^{\kappa-k+1+ \varepsilon}\cdot \sum\limits_{\substack{j\in \mathbb{Z}[i]\textbackslash \{0\}\\ \mathcal{N}(j)\le NQ_0^{\varepsilon-k}}} \sum\limits_{\substack{\alpha \in \mathbb{Z}[i]^k\\ \mathcal{N}\left(\alpha_1\right),...,\mathcal{N}\left(\alpha_{k-1}\right)\le Q_0^{1+\varepsilon}}} \sum\limits_{\substack{\beta\in \mathbb{Z}[i] \\ \mathcal{N}\left(\beta-\overline{k!\alpha_1\cdots \alpha_{k-1}j r_1/q_1^k}\right)\le Q_0^{\varepsilon-1}}} 1\\ \ll & Q_0^{\kappa-k+1 \varepsilon}\cdot\sum\limits_{\substack{j\in \mathbb{Z}[i]\textbackslash \{0\}\\\mathcal{N}(j)\le NQ_0^{\varepsilon-k}}} \sum\limits_{\substack{\alpha \in \mathbb{Z}[i]^k\\ \mathcal{N}\left(\alpha_1\right),...,\mathcal{N}\left(\alpha_{k-1}\right)\le Q_0^{1+\varepsilon}\\ \left|\left|k!\alpha_1\cdots \alpha_{k-1}j r_1//q_1^k\right|\right|\le Q_0^{\varepsilon-1/2}}} 1\\ \ll & (NQ_0)^{(k+1)\varepsilon} \cdot Q_0^{\kappa-k+1}\cdot \Bigg(\frac{N}{Q_0^{2}}+ \sum\limits_{\substack{d\in \mathbb{Z}[i]\setminus\{0\}\\ \mathcal{N}(d)\le k!^2NQ_0^{k\varepsilon-1}\\ \left|\left|dr_1/q_1^k\right|\right|\le Q_0^{\varepsilon-1/2}}} 1\Bigg)\\ \ll & (NQ_0)^{(k+1)\varepsilon} \cdot \Big(NQ_0^{\kappa-k-1}+ Q_0^{\kappa-k+1}\cdot \sum\limits_{\substack{l\in \mathbb{Z}[i]\\ \left|l/q_1^k\right|\le Q_0^{\varepsilon-1/2}}} \sum\limits_{\substack{\mathcal{N}(d)\le k!^2NQ_0^{k\varepsilon-1}\\ d\equiv l\overline{r_1} \bmod{q_1^k}}} 1\Bigg), \end{split}$$ where we recall that $||z||$ is the distance of $z\in \mathbb{C}$ to the nearest Gaussian integer. In the above, we have set $d=k!\alpha_1\cdots \alpha_{k-1}j$ if $\alpha_1,...,\alpha_{k-1}\not=0$. The number of residue classes modulo $q_1^{k}$ is $\mathcal{N}(q_1^k)\le Q_0^k$, and hence $$\label{res1} \sum\limits_{\substack{\mathcal{N}(d)\le k!^2NQ_0^{k\varepsilon-1}\\ d\equiv l\overline{r_1} \bmod{q_1^k}}} 1 \ll 1+\frac{N}{Q_0^{k+1-k\varepsilon}}.$$ Further $$\label{lat1} \sum\limits_{\substack{l\in \mathbb{Z}[i]\\ |l/q_1^k|\le Q_0^{\varepsilon-1/2}}} 1 \le \sum\limits_{\substack{l\in \mathbb{Z}[i]\\ |l|\le Q_0^{\varepsilon+(k-1)/2}}} 1\ll Q_0^{k-1+2\varepsilon}.$$ Combining , and , we obtain $$\label{comb3} \sum\limits_{\substack{j\in \mathbb{Z}[i]\textbackslash \{0\}\\ \mathcal{N}(j)\le NQ_0^{\varepsilon-k}}} \left| S_k\left(q_1,r_1,j\right)\right|^{\kappa} \ll (Q_0N)^{(2k+3)\varepsilon}\left(Q_0^{\kappa}+ NQ_0^{\kappa-k-1}\right),$$ and combining , and , and changing $\varepsilon$ suitably, we arrive at $$\label{comb4} K\ll \frac{Q_0^k}{N}+(Q_0N)^{\varepsilon}\left(\frac{Q_0^{1+k/\kappa}}{N^{1/\kappa}}+Q_0^{1-1/\kappa}\right).$$ Now the statement in Theorem \[powermodZi\] follows immediately from Corollary \[preform\] and after dividing the moduli into dyadic intervals and replacing $Q_0$ by $Q$. [cccc]{} S. Baier, [*On the large sieve with sparse sets of moduli*]{}, J. Ramanujan Math. Soc. 21 (2006), no. 3, 279–295. S. Baier, [*The large sieve with square norm moduli in $Z[i]$*]{}, to appear in J. Theor. Nombr. Bordx., arXiv:1511.02470. S. Baier; L. Zhao, [*Large sieve inequality with characters for powerful moduli*]{}, Int. J. Number Theory 1 (2005), no. 2, 265–279. S. Baier; L. Zhao, [*An improvement for the large sieve for square moduli*]{}, J. Number Theory 128 (2008), no. 1, 154–174. W.D. Banks; F. Pappalardi; I.E. Shparlinski, [*On group structures realized by elliptic curves over arbitrary finite fields*]{}. Exp. Math. 21 (2012), no. 1, 11–25. J. Bourgain; K. Ford; S.V. Konyagin; I.E. Shparlinski, [*On the divisibility of Fermat quotients*]{}. Michigan Math. J. 59 (2010), no. 2, 313–328. K. Halupczok, [*A new bound for the large sieve inequality with power moduli*]{}, Int. J. Number Theory 8 (2012), no. 3, 689–695. G.H. Hardy; J.E. Littlewood; G. Pólya, [*Inequalities*]{}. Reprint of the 1952 edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988. 324 pp. M. Huxley, [*The large sieve inequality for algebraic number fields*]{}, Mathematika 15 (1968) 178-187. E.M. Stein; G. Weiss, [*Introduction to Fourier analyis on Euclidean spaces*]{}, Mir publishing house, Moscow, 1974. 336pp. L. Zhao, [*Large sieve inequality with characters to square moduli*]{}, Acta Arith. 112 (2004), no. 3, 297–308.

--- author: - 'B. Mosser' - 'K. Belkacem' - 'M.J. Goupil' - 'E.Michel' - 'Y. Elsworth' - 'C. Barban' - 'T. Kallinger' - 'S. Hekker' - 'J. De Ridder' - 'R. Samadi' - 'F. Baudin' - 'F.J.G. Pinheiro' - 'M. Auvergne' - 'A. Baglin' - 'C. Catala' bibliography: - 'biblio\_tu.bib' subtitle: an automated determination with CoRoT data title: 'The universal red-giant oscillation pattern' --- [The CoRoT and Kepler satellites have provided thousands of red-giant oscillation spectra. The analysis of these spectra requires efficient methods for identifying all eigenmode parameters.]{} [The assumption of new scaling laws allows us to construct a theoretical oscillation pattern. We then obtain a highly precise determination of the large separation by correlating the observed patterns with this reference. ]{}[We demonstrate that this pattern is universal and are able to unambiguously assign the eigenmode radial orders and angular degrees. This solves one of the current outstanding problems of asteroseismology hence allowing precise theoretical investigation of red-giant interiors.]{} = 1.5cm Introduction\[introduction\] ============================ Red giants are evolved stars that have depleted the hydrogen in their cores and are no longer able to generate energy from core-hydrogen burning. The physical processes taking place in their interiors are currently rather poorly understood. Observations with the space-borne mission CoRoT have revealed the oscillation pattern [@2009Natur.459..398D] of many of these stars, which is a crucial step on the route to probing their internal structure. Before the advent of the CoRoT data, complex oscillation patterns were explained by short-lived modes of oscillation . The new era of the space-borne missions CoRoT and *Kepler* has dramatically increased the amount and quality of the available asteroseismic data of red giants . The analysis of the oscillation eigenmodes now allows seismic inferences to be drawn about the internal structure. The identification of the angular degree and radial order of the eigenmodes represents a first and crucial step in an asteroseismic analysis. The values of the eigenfrequencies can be related to the order and degree of a mode with the commonly used asymptotic equation: $$\nu_{n,\ell} = \left[ n+{\ell\over 2} + \varepsilon \right] \, \Delta\nu - \dd_{0\ell} \label{tassoul}$$ where $\nu_{n,\ell}$ is the eigenfrequency of a mode with radial order $n$ and angular degree $\ell$; $\Delta\nu$ is the mean value of the large separation ($\dnumoy \simeq \nu_{n+1,\ell} - \nu_{n,\ell}$), and $\dd_{0\ell}$ is a second-order term, or small separation, dependent on the mode degree. This form, which is similar to the original expression developed for the Sun 30 years ago [@1980ApJS...43..469T], is useful for analysing the observations and performing the complete mode identification. It assumes $\dd_{00}$ equals 0. The parameter $\varepsilon$ comprises two parts: the offset due to the mode propagation in the upper-most layers of the star, and the second-order term of the asymptotic approximation which is sensitive to the gradient of sound speed in the stellar interior. In the absence of accurate determinations of the individual mode frequencies, the global seismic parameters used in the asymptotic expression above are important indicators of the physical parameters of the star. The large separation gives a measure of the mean stellar density; the small separation $\dd_{0\ell}$ describes the stratification of the central regions. Unfortunately, the methods currently used to determine the global oscillation parameters suffer from various sources of uncertainty . First, the stochastic excitation of the modes gives rise to variability in the amplitudes, resulting in an apparently irregular comb structure; second, the finite mode lifetime blurs the estimates of the eigenfrequencies; third, estimates are affected by the stellar noise and granulation signal superimposed on the oscillations . In fact, simulations have shown that the impact of realization noise on the measurement of the large separation $\dnumoy$, can be much larger than the background noise for red giants [@2010arXiv1008.2959H]. In an analysis of a sub-sample of Kepler red giants, [@huber2010] have shown the regularity of the oscillation spectra of such stars. In this paper, we show that all red giants have a regular pattern, as modelled recently by [@2010ApJ...721L.182M]. We propose a method which allows us to tag all the modes with their appropriate radial order and angular degree, regardless of the presence of the perturbing effects described above. Method\[method\] ================ The method to mitigate the effects of realization noise uses Eq. \[tassoul\] in a dimensionless form: $${\nu_{n,\ell} \over \dnumoy} = n+{\ell\over 2} + \varepsilon (\dnumoy) - d_{0\ell} (\dnumoy) . \label{tassoul_m}$$ In a departure from the previous practice, we have assumed that $\varepsilon$ obeys a scaling law $\varepsilon = A + B \log \dnumoy$, as derived from the observation of thousands of CoRoT targets and as observed by [@huber2010]. This is justified by the observation that scaling laws apparently govern *all* global asteroseismic parameters and is equivalent to assume that the underlying physics of $\varepsilon$ varies with the global stellar parameters. As the mixed nature of dipole modes ($\ell=1$) is more pronounced, we did not include them in the template, but only doublets corresponding to the eigenmodes with even degrees ($\nu_{n-1,2}$ and $\nu_{n,0}$), with equal amplitudes. As a first guess, we set the small separation $d_{02}$ at $- 0.14$ and then allowed it to vary with the value of $\dnumoy$ according to the same relationship as given for $\varepsilon$. For constructing the peaks of the template, we have also used the scaling laws of the Gaussian excess power derived by . We have assumed that the mode lifetime varies as $\dnumoy^{-1}$ and have used mode widths equal to about a few percent of $\dnumoy$. Finally, we stress that no background model is needed. ![CoRoT red giant power spectra stacked into an image after sorting on the large separation. One line corresponds to one star. [**a)**]{} The first sorting is based on the large separation before any correction. [**b)**]{} The blurred aspect disappears once the correction has been performed and reveals a clear comb-structure common to all red giants. The vertical lines at 23.2$\mu$Hz are the signature of the low-Earth orbit . \[deter\]](TU_avant.ps "fig:"){width="8.2cm"} ![CoRoT red giant power spectra stacked into an image after sorting on the large separation. One line corresponds to one star. [**a)**]{} The first sorting is based on the large separation before any correction. [**b)**]{} The blurred aspect disappears once the correction has been performed and reveals a clear comb-structure common to all red giants. The vertical lines at 23.2$\mu$Hz are the signature of the low-Earth orbit . \[deter\]](TU_apres.ps "fig:"){width="8.2cm"} ![The spectra presented in Fig. \[deter\]b have been rearranged in order to have the dimensionless frequency on the abscissa and the mean large separation on the ordinate. They show that the red-giant oscillation pattern is universal. The hyperbolic branch is here the signature of the low-Earth orbit. \[universal\]](TU_univer.ps){width="8.7cm"} The measurement of the large separation is performed in two steps. First, an initial-guess value $\dnug$ of the large separation is computed by an automated pipeline . This is used to form the initial synthetic template to correlate with the real spectrum. The best correlation between the observed and synthetic spectra provides then the corrected value of the large separation. The template was iteratively adjusted by varying its parameters to maximize the correlation. In Fig. \[deter\], we show the results obtained with all high signal-to-noise CoRoT data . In both cases the graphs show the spectra arranged in strips with the colour representing the strength of the signal, as was done by @2010PASP..122..131G. The spectra are sorted by increasing large separation with the smallest large separation at the top of the plots: in the upper plot we use the output from a conventional pipeline and in the lower one we use the corrected value. The remarkably regular structure within the oscillation spectra in the lower plot reveals the signature of comb-like structure of the asymptotic relationship in Eq. (\[tassoul\_m\]) already reported . Further, it validates the scaling law in $\varepsilon$ included in the reference template. The global agreement of all high signal-to-noise spectra of bright targets with the synthetic pattern (Fig. \[universal\]) shows that these oscillation patterns are homologous and that the red-giant oscillation pattern is universal. We further found that the template is significantly improved if it takes account of the linear dependence of the large separation in frequency, expressed by the degree-dependent gradient $\alpha_\ell = ({\mathrm{d}}\log \dnumoy / {\mathrm{d}}n)_\ell$: $${\nu_{n,\ell} \over \dnumoy} = n+{\ell\over 2} + \varepsilon (\dnumoy) - d_{0\ell} (\dnumoy) + {\alpha_\ell (\dnumoy) \over2} \left( n - {\numax \over \dnumoy} \right)^2 \label{tassoul_mod}$$ with $\numax$ the frequency of maximum oscillation amplitude. The corrected values of $\dnumoy$ are derived from this template. The values of the 12 free parameters that account for the variations in frequency of the parameter $\varepsilon$, of the small separations $d_{0\ell}$ and of the gradients $\alpha_\ell$ as derived from the fits to more than 6000 eigenmodes (Fig. \[identi\]) are given in Table \[fits\]. The value of $\varepsilon$, defined modulo 1, is fixed thanks to the extrapolation to the Solar case. As noted by and [@huber2010], the small separation $d_{01}$ is negative. All these fits are consistent with Kepler results. The average value of the correction from $\dnug$ to $\dnumoy$ is of the order of 2.5%. For the largest giants, with the smallest values of $\dnumoy$ and the smallest ratio $\numax / \dnumoy$, the correction can be as high as 6%. The absolute difference $|\dnug-\dnumoy|$ can be 10 times the estimate of the stellar noise contribution. At low frequency, with an observing run not much longer than the mode lifetimes [@baudin2010], the realization noise dominates the background noise and the mean accuracy of the determination of $\dnumoy$ is uniform, at about 0.015$\mu$Hz. We are aware that any bias in the $\varepsilon (\dnumoy)$ input relation will induce a bias in the results. We therefore took care to insure the iterative process to be unbiased. Analysis of synthetic data indicates that the precision gained with this method is of order 10 times better than that obtained with conventional methods [@2010arXiv1008.2959H]. --- --------------- ------------------------------ -------------------- ----------------- gradient of $\Delta\nu_\ell$ $A_\ell$ $B_\ell$ $\alpha_\ell$ 0 $\varepsilon$ $ 0.634 \pm 0.008$ $ 0.546 \pm 0.008$ $0.008\pm0.001$ 1 $d_{01}$ $-0.056 \pm 0.012$ $-0.002 \pm 0.010$ $0.003\pm0.002$ 2 $d_{02}$ $ 0.131 \pm 0.008$ $-0.033 \pm 0.009$ $0.005\pm0.001$ 3 $d_{03}$ $ 0.280 \pm 0.012$ $0$ $0.005\pm0.002$ --- --------------- ------------------------------ -------------------- ----------------- : Fits of the ridges, for $\dnumoy$ expressed in $\mu$Hz[]{data-label="fits"} ![Complete identification of the ridges, automatically derived from the eigenfrequencies extracted with a height-to-background ratio greater than 3. Each colour correspond to a different mode degree (radial modes in red, dipole modes in dark blue, $\ell=2$ modes in green, $\ell=3$ modes in light blue). The solid grey lines superimposed on the ridges indicate for each radial order the fits of $\varepsilon$ (with indication of the radial order $n$). The fits of $d_{01}$, $d_{02}$ and $d_{03}$ are superimposed on the respective ridges (respectively dash-dot, dot and dash lines for $\ell = $1, 2 and 3). The dark dashed lines, derived from the scaling law dealing with the oscillation excess power, delineate the region where the modes have noticeable amplitudes . \[identi\]](etap71.ps){width="8.8cm"} Discussion\[results\] ===================== This new method based on a simple hypothesis and an automated procedure removes any ambiguity on the identification of the modes (Fig. \[identi\]), despite the complexity induced by mixed modes. Mode identification is derived by looking at the closest ridge. In particular, we provide a straightforward determination of the mode radial orders, which were previously unknown. Radial eigenfrequencies are located at: $$\label{identin} \nu_{n,0} = \left[ n + \varepsilon (\dnumoy) \right] \; \dnumoy$$ Ridges were already shown in previous works. While [@2009Natur.459..398D] and looked at single stars separately, [@2010ApJ...713L.176B] and [@huber2010] used manual fine-tuning of the large separation to align the radial modes of a large sample of stars. However, the radial modes were identified in only one third of the spectra by [@huber2010], but they also showed the ridges with varying $\varepsilon$ in the folded and collapsed power spectrum. In most regions of the oscillation spectra we observe the presence of both radial and non-radial modes. Realization noise causes the height of the individual modes to show considerable variability, but on average, the ratio between the dipole and radial mode height is approximately independent of $\numax$. Although, at very low $\numax$ there is some reduction in the strength of the dipole mode. We also make clear that the larger spread of the ridges corresponding to dipole modes (Fig. \[identi\]) is due to the presence of many mixed modes, as already noticed . The universal pattern makes it easier to identify them opening up the possibility of exploring the conditions in the inner layers of the red giants. Despite their low amplitudes and the resulting poor signal, $\ell=3$ modes have been detected in Kepler data on red giants [@2010ApJ...713L.176B; @huber2010]. Our results represent the first such detection in CoRoT data. Their identification gives access to the fine structure of the oscillation spectra, as modes of different angular degree probe different depths within the star. Their detection and complete characterization will first be derived from the universal pattern, then the small differences to this pattern will be exploited to characterize in detail a given object [@2010arXiv1009.1024M]. More than 75% of the red-giant candidates with brighter magnitude than $m{_{\mathrm{R}}}=13$ observed with CoRoT show solar-like oscillations. In the remainder, we observe a large proportion of classical pulsators or of giants with a so large radius that the oscillations occur at a too low frequency for a positive detection. In a very limited number of cases at very low frequency, the possible confusion between radial and dipole modes is not clearly solved. This confusion increases toward dimmer targets with lower quality time series. Among the positive detection of bright stars, we did not observe any outliers when performing the correlation with the universal pattern. For this procedure to be effective, we require that the modes have a significant height-to-background ratio. Hence for all high signal-to-noise targets we are able to derive corrected values for the large frequency spacing. It is recognized that the majority of the red giants in the CoRoT field of view are in their post-flash helium-burning phase . In terms of stellar evolution, the demonstration of the universal regular pattern of red giants proves that these red giants have similar and homologous interior structures. On the other hand, despite the agreement of the fit in $\varepsilon$ with the Solar value, we have verified that the method does not work with subgiants or main-sequence stars . We explain this by the wide range of evolutionary phases pertaining outside the red clump, which certainly will cause a complex dependence of $\varepsilon$ with more parameters than just the large separation. This disagreement reinforces the homogenous properties of red-giant stars. The clearly observed pattern confirms that, on average, the linewidths of the modes are significantly smaller than the separation of the even mode pairs and hence makes short mode lifetimes unlikely [@2004SoPh..220..207S]. From the quasi-uniform width of the ridges (Fig. \[identi\]), we can see that the lifetimes of the modes increase significantly with decreasing large separation, contrary to [@huber2010]. It also suggests that very complex oscillation spectra previously observed may be an artefact of noise. ![Variation in stellar mass and radius as a function of the large separation $\dnumoy$. The vertical line corresponds to the error box in large separation and radius. \[masserayon\]](figure4_M_R.ps){width="8.8cm"} Finally, the better determination of the large separation $\dnumoy$ helps us to enhance the accuracy of the estimates of the stellar mass and radius as done in , with typical error-bars of 12 and 5% respectively, instead of 20 and 8% (Fig. \[masserayon\]). This accuracy, achieved without the need of stellar modeling [@kallinger2010], constitutes an important progress compared to the current photometric determination and demonstrates the power of asteroseismology. Conclusion ========== We have shown with CoRoT observations that the red-giant oscillation spectrum is very regular and can be described by its underlying universal pattern. This was modelled in parallel by [@2010ApJ...721L.182M]. As a consequence, the precise measures of the large separation and the scaling relation of the parameter $\varepsilon$ allow us to provide an unambiguous detection of the radial orders and angular degrees of the modes. Since the method is able to mitigate the realization noise, we consider it to give the most precise determination of the large separation available. It remains important to interpret the physical meaning of the scaling law for the term $\varepsilon$ in the Tassoul expression. We will have to disentangle the contributions of the surface and of the inner region. This will require an investigation of the term $\varepsilon$ in the context of the second-order corrections of the Tassoul development. Very long-duration observations with *Kepler* will help for this task. Despite the uniform aspect of the oscillation spectra, many differences invisible in the global approach are revealed by a detailed analysis of each individual spectrum, as, for example, the modulation with frequency of the large separation . Study of this variation will give access to the most accurate analysis of the red-giant interior structure. This summarizes the power of asteroseismology: first, the regular pattern provides the identification of the individual modes; second, the difference to this regular pattern unveils the detailed interior structure. We are confident that the shifts to the regular pattern will be explained by mass, age and/or metallicity effects. Similar analysis can be performed for oscillations in subgiants and solar-like stars . Due to the large variety of evolutionary stages among those stars, we expect the Tassoul parameter $\varepsilon$ to depend on more than the large separation. This work was supported by the Centre National d’Études Spatiales (CNES). It is based on observations with CoRoT. The research has made use of the Exo-Dat database, operated at LAM-OAMP, Marseille, France, on behalf of the CoRoT/Exoplanet program. KB acknowledges financial support through a postdoctoral fellowship from the Subside fédéral pour la recherche 2010, University of Liège. FJGP acknowledges FCT’s grant SFRH/BPD/37491/2007. YE and SH acknowledge financial support from the UK Science and Technology Facilities Council (STFC). The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement n$^\circ$227224 (PROSPERITY), as well as from the Research Council of K.U.Leuven grant agreement GOA/2008/04.

--- abstract: 'This paper provides a new proof of a theorem of Chandler-Wilde, Chonchaiya and Lindner that the spectra of a certain class of infinite, random, tridiagonal matrices contain the unit disc almost surely. It also obtains an analogous result for a more general class of random matrices whose spectra contain a hole around the origin. The presence of the hole forces substantial changes to the analysis.' author: - | S N Chandler-Wilde\ E B Davies date: December 2011 title: | Spectrum of a Feinberg-Zee\ Random Hopping Matrix --- Mathematics Subject Classification: 65F15, 15A18, 15A52, 47A10, 47A75, 47B80, 60H25. Key Words: spectrum, random matrix, hopping model, tridiagonal matrix, non-self-adjoint operator. Introduction {#intro} ============ Over the last fifteen years there have been many studies of the spectral properties of non-self-adjoint, random, tridiagonal matrices $A$, some of them cited in [@CCM; @FZ1; @FZ2; @HOZ]. It has become clear that if all of the off-diagonal entries $A_{i,j}$ with $i-j=\pm 1$ of the matrices concerned are positive, the almost sure limit as $N\to\infty$ of the spectra of random $N\times N$ matrices subject to periodic boundary conditions can be quite different from the spectral behaviour of the corresponding infinite random matrix, [@EBD0; @EBD1; @GK1; @GK2]. Indeed the limit in the first case can be the union of a small number of simple curves, while the second limit has a non-empty interior. Numerical calculations suggest that the situation is quite different if the off-diagonal entries have variable signs, but much less has been proved in this situation, which is the one that we consider here. In a recent paper, [@CCL], Chandler-Wilde, Chonchaiya and Lindner made important progress in determining the almost sure spectrum of a remarkably interesting class of non-self-adjoint, random, tridiagonal matrices introduced by Feinberg and Zee in [@FZ1], and sometimes called random hopping matrices, because the diagonal entries all vanish. Specifically they proved that, contrary to earlier conjectures, the infinite, tridiagonal matrix $$A_c= \left(\begin{array}{cccccc} \ddots&\ddots&&&&\\ \ddots&0&1&&&\\ &c_{n-1}&0&1&&\\ &&c_n&0&1&\\ &&&c_{n+1}&0&\ddots\\ &&&&\ddots&\ddots\\ \end{array}\right)$$ has spectrum that contains the unit disc almost surely, [@CCL]. The paper assumed that the entries $c_n$ are independent and identically distributed with values in $\{\pm 1\}$. In the present paper we assume that the entries $c_n$ are independent and identically distributed with values in $\{\pm {\sigma}\}$ for some fixed ${\sigma}\in (0,1]$. We assume that the probability $p$ that $c_n={\sigma}$ satisfies $0<p<1$; the corresponding probability measure on ${\Omega}_{\sigma}=\{\pm {\sigma}\}^{{\bf Z}}$ is denoted by $\mu$. The matrix $A_c$ is identified with the bounded operator acting in the natural manner on $\ell^2({{\bf Z}})$. In Lemma \[lemmaA\] we prove that $${{\rm Spec}}(A_c)\subseteq \{ {\lambda}: 1-{\sigma}\leq |{\lambda}|\leq 1+{\sigma}\}$$ by a perturbation argument. We also prove that $${{\rm Spec}}(A_c)\subseteq \{ x+iy:|x|+|y|\leq \sqrt{2(1+{\sigma}^2)}\}$$ by obtaining a bound on the numerical range of $A_c$. There are currently no general techniques for identifying the precise forms of holes in the spectra of non-self-adjoint operators, and we have not done so here, but numerical calculations are consistent with the hypothesis that it is the intersection, $H_{\sigma}$, of two elliptical regions as defined in (\[Hsigma\]); see the figures at the end of Section \[firstproof\]. Little is known about the part of the spectrum of $A_c$ outside the unit disc even in the case ${\sigma}=1$, but numerical studies suggest that the boundary of the spectrum has a self-similar fractal structure in that case; [@CCL; @HOZ]. The main result of [@CCL], that the spectrum contains the unit disc almost surely, is for the case that ${\sigma}=1$, when there is no hole in the spectrum. It depends upon the identification of a particular sequence $c\in {\Omega}_1$ such that the equation $A_cf={\lambda}f$ has a bounded solution $f$ for every ${\lambda}\in {{\bf C}}$ such that $|{\lambda}|<1$. Our Theorem \[maintheorem\] rederives the main result of [@CCL], in which ${\sigma}=1$, but depends on a certain operator identity introduced in the next section. Our main result, Theorem \[maintheorem3\], that the spectrum of $A_c$ contains that part of the unit disc which is not in $H_{\sigma}$, applies to all ${\sigma}\in (0,1)$. We give a second proof of this result in Theorem \[maintheorem6\], by combining some results for ${\sigma}=1$ with bounds on the Lyapunov exponents of certain transfer matrices. Both proofs depend, additionally, on results on the spectra of operators on $\ell^2({{\bf Z}})$ which have different periodic structures on the positive and negative half-axes. They also both depend on explicit spectral calculations which we are able to carry out for certain operators $A_c$ with $c$ having arbitrarily large period. Our main results, as just stated, concern the spectrum of the (bi-)infinite matrix $A_c$. In a shorter final section we spell out implications for the spectra of the corresponding semi-infinite and finite matrices, illustrating these observations with computations of the finite matrix spectra. In particular we show, by applying recent results of Lindner and Roch [@LR10], that, unlike $A_c$, the semi-infinite matrix has no hole in its spectrum for ${\sigma}\in (0,1)$, but contains the unit disc for all ${\sigma}\in (0,1]$. Let ${{\cal E}}_{\sigma}$ denote the set of all $c\in{\Omega}_{\sigma}$ that are pseudo-ergodic in the sense of [@EBD1]. Precisely, $c\in{{\cal E}}_{\sigma}$ if for every finite sequence $b:\{1,...,n\}\to\{\pm {\sigma}\}$ there exists $m\in{{\bf Z}}$ such that $b_r=c_{m+r}$ for all $r\in \{1,...,n\}$. Such sequences $c$ are easy to construct without any reference to probability theory. The following facts, proved in [@EBD1], and rederived in [@Li; @CWL] as an instance of the application of limit operator arguments, will be crucial in this paper. \[quasiergodic\] If $b,\, c \in{{\cal E}}_{\sigma}$ then ${{\rm Spec}}(A_b)={{\rm Spec}}(A_c)$. Let $S_{\sigma}$ denote this set, which is the main object of study in the paper. If $c\in{\Omega}_{\sigma}$, then $c\in{{\cal E}}_{\sigma}$ almost surely with respect to the measure $\mu$. Finally $$S_{\sigma}= \bigcup_{b\in{\Omega}_{\sigma}} {{\rm Spec}}( A_b ).$$ To describe a further result we establish, for $N\in {{\bf N}}$ and $\sigma\in (0,1]$ let $\pi_{N,\sigma}$ denote the union of ${{\rm Spec}}(A_c)$ over all $c\in \Omega_\sigma$ that are periodic with period $\leq N$. Let $$\label{eq:piinf} \pi_{\infty,{\sigma}} = \bigcup_{N\in{{\bf N}}} \pi_{N,{\sigma}}.$$ One obvious implication of the above proposition is that $$\label{eq:pidef} \pi_{\infty,{\sigma}} \subset S_\sigma.$$ As is well-known, the set $\pi_{N,\sigma}$ is the union of eigenvalues of $N\times N$ matrices. (Precisely, it is the union, over all sequences $c$ and all $|\alpha|=1$, of the eigenvalues of the matrix $A^{(N,\mathrm{per})}_{c,\alpha}$ defined in [(\[finitematrix2\])]{} below; see [(\[spectrumperiodic\])]{} and [@LOTS]. For another, equivalent characterisation see Lemma \[BIO\].) This simple observation is useful, in that it provides a method for computing what prove to be large subsets of $S_\sigma$, and will be one component in our arguments. An interesting question is whether $\pi_{\infty,{\sigma}}$ is dense in $S_{\sigma}$. We do not answer this question one way or the other, but our method of proof of Theorem \[maintheorem\], showing that the unit disc is a subset of $S_1$, as a by-product, and with some additional argument, leads to a proof that $\pi_{\infty,1}$ is dense in the unit disc (Theorem \[denseness\]). For the sake of simplicity we will, throughout the rest of the paper, omit the subscript ${\sigma}$ in our notations if ${\sigma}=1$. We use ${{\bf N}}$ and ${{\bf Z}}_+$, respectively, as our notations for the sets of positive and non-negative integers. An abstract theorem {#abstract} =================== In this section we present an abstract theorem that might be interesting in other contexts. It will be applied in Section \[main\]. Let $A$ be a bounded linear operator acting on the Hilbert space ${{\cal H}}$ and let ${{\cal H}}={{\cal H}}_e\oplus{{\cal H}}_o$ be an orthogonal decomposition of ${{\cal H}}$. \[interchange\] If $A({{\cal H}}_e)\subseteq {{\cal H}}_o$ and $A({{\cal H}}_o)\subseteq {{\cal H}}_e$ then ${{\cal H}}_e$ and ${{\cal H}}_o$ are invariant under the action of $A^2$. If $B$ is the restriction of $A^2$ to ${{\cal H}}_e$ and $M$ is the restriction of $A^2$ to ${{\cal H}}_o$ then $${{\rm Spec}}(A^2)\backslash \{0\}={{\rm Spec}}(B)\backslash \{0\}={{\rm Spec}}(M)\backslash \{0\}.\label{identity1}$$ If $A$ is invertible then $${{\rm Spec}}(A^2)={{\rm Spec}}(B)={{\rm Spec}}(M).\label{identity2}$$ The decomposition ${{\cal H}}={{\cal H}}_e\oplus{{\cal H}}_o$ allows one to write the operator $A$ in the form $$A={\left(\begin{array}{cc}0&X\\Y&0 \end{array}\right)}$$ where $X:{{\cal H}}_o\to{{\cal H}}_e$ and $Y:{{\cal H}}_e\to{{\cal H}}_o$. Therefore $$A^2={\left(\begin{array}{cc}XY&0\\0&YX \end{array}\right)}.\label{A2identity}$$ This implies that $B=XY$ and $M=YX$. The second identity in (\[identity1\]) follows by some simple algebra that holds for any pair of bounded operators $X$ and $Y$, and the first identity is a trivial consequence. If $A$ is invertible then (\[A2identity\]) implies that $B$ and $M$ are also invertible; therefore (\[identity2\]) is equivalent to (\[identity1\]). \[spectralequality\] Let ${{\cal H}}=\ell^2({{\bf Z}})$, let ${{\cal H}}_e$ be the closed subspace of sequences whose supports are contained in the set of even integers, and let ${{\cal H}}_o$ be the closed subspace of sequences whose supports are contained in the set of odd integers. Let $A$ be a bounded operator on ${{\cal H}}$ whose matrix satisfies $A_{r,s}= 0$ for all $r,\, s$ such that $|r-s|\not= 1$. Then $A({{\cal H}}_e)\subseteq {{\cal H}}_o$ and $A({{\cal H}}_o)\subseteq {{\cal H}}_e$. Moreover the identities $${{\rm Spec}}(A^2)={{\rm Spec}}(B)={{\rm Spec}}(M)$$ are valid in either of the following two cases. 1. $|A_{r,s}|=1$ for all $r,\, s$ such that $|r-s|=1$; 2. There exist constants ${\beta},\, {\gamma}$ such that $0<{\beta}<{\gamma}<\infty$ and $|A_{r,s}|\leq {\beta}$ if $r-s=1$ while $|A_{r,s}|\geq {\gamma}$ if $r-s=-1$. \ **Case 1.** An elementary calculation establishes that there exists a sequence $f:{{\bf Z}}\to{{\bf C}}$ such that $Af=0$, $|f_{2n}|=1$ for all $n$ and $f_{2n+1}=0$ for all $n$, so that $A$ and $B$ are not invertible viewed as operators on $\ell^\infty({{\bf Z}})$, and thus not invertible as operators on $\ell^2({{\bf Z}})$ (see e.g. [@RRS Theorem 2.5.2]). So $0\in{{\rm Spec}}(A)$ and $0\in{{\rm Spec}}(B)$. Similarly there exists a sequence $f:{{\bf Z}}\to{{\bf C}}$ such that $Af=0$, $|f_{2n+1}|=1$ for all $n$ and $f_{2n}=0$ for all $n$. Hence $0\in{{\rm Spec}}(M)$. The result follows by combining this with (\[identity1\]). **Case 2.** The operator $A_L$ associated with the matrix $$(A_L)_{r,s}=\begin{choices} A_{r,s}&\mbox{ if $r-s=-1$},\\ 0&\mbox{ otherwise}, \end{choices}$$ is invertible and satisfies ${\Vert}A_L^{-1}{\Vert}\leq {\gamma}^{-1}$. The operator $A_R=A-A_L$ satisfies ${\Vert}A_R{\Vert}\leq {\beta}$. Therefore $A$ is invertible with $${\Vert}A^{-1}{\Vert}={\Vert}A_L^{-1}(I+A_RA_L^{-1})^{-1}{\Vert}\leq \frac{{\gamma}^{-1}}{1-{\beta}/{\gamma}}=\frac{1}{{\gamma}-{\beta}}.$$ The proof is completed by applying [(\[identity2\])]{}. The case ${\sigma}=1$ {#main} ===================== The following lemma was noted in [@CCL]. \[symmetry\] If $c\in{\Omega}$ then ${{\rm Spec}}(A_c)$ is invariant with respect to both of the maps ${\lambda}\to\overline{{\lambda}}$ and ${\lambda}\to -{\lambda}$. If ${\lambda}\in S$ then $\overline{{\lambda}}$ and $i{\lambda}$ lie in $S$. Hence $S$ is invariant under the dihedral symmetry group $D_2$ generated by these two maps. The invariance of ${{\rm Spec}}(A_c)$ under complex conjugation follows directly from the fact that $A_c$ has real entries. If $D$ is the diagonal matrix with entries $D_{r,r}=(-i)^r$ for all $r\in{{\bf Z}}$ then $DA_cD^{-1}=iA_{-c}$, so $${{\rm Spec}}(A_c)=i{{\rm Spec}}(A_{-c}).\label{irotate}$$ Iterating this identity yields ${{\rm Spec}}(A_c)=-{{\rm Spec}}(A_{c})$. This proves the first part of the lemma. The second part follows once one observes that $c\in {{\cal E}}$ if and only if $-c\in{{\cal E}}$. The formulae in (\[iteration\]) are related to those in [@CCL Prop. 2.1], in a way that we will make explicit in Section \[sec:maps\]. However, nothing resembling the following lemma appears in [@CCL]. \[square\] Given $b\in{\Omega}$, let $c={\Gamma}_+(b)\in{\Omega}$ be the unique sequence satisfying $$c_0=1, \hspace{2em} c_{2n}+c_{2n+1}=0, \hspace{2em} c_{2n}c_{2n-1}=b_n,\label{iteration}$$ for all $n\in {{\bf Z}}$. Then $A_c^2$ is unitarily equivalent to $A_b\oplus M_b$ acting in $\ell^2({{\bf Z}})\oplus \ell^2({{\bf Z}})$, where $$(M_bf)_n=-f_{n-1}+(c_{2n+1}+c_{2n+2})f_n+f_{n+1}\label{Mbformula}$$ for all $f\in \ell^2({{\bf Z}})$. Moreover $${{\rm Spec}}(A_c^2) = {{\rm Spec}}(A_b)={{\rm Spec}}(M_b).$$ One may write $(A_cf)_n=c_nf_{n-1}+f_{n+1}$ for all $n\in {{\bf Z}}$, or equivalently $A_c=V_cR+L$ where $(Lf)_n=f_{n+1}$, $(Rf)_n=f_{n-1}$ and $(V_cf)_n=c_nf_n$ for all $f\in \ell^2({{\bf Z}})$. Therefore $$\begin{aligned} A_c^2&=& V_cRV_cR+LV_cR+V_cRL+L^2\\ &=& X_cR^2+Y_c+L^2\end{aligned}$$ where $X_c$ and $Y_c$ are the diagonal matrices with diagonal entries $$\begin{aligned} X_{c,n,n}&=&c_nc_{n-1}, \\ Y_{c,n,n}&=& c_n+c_{n+1}.\end{aligned}$$ The operator $A_c^2$ has two invariant subspaces $${{\cal H}}_e=\{f\in \ell^2({{\bf Z}}):f_{2n+1}=0 \mbox{ for all } n\in{{\bf Z}}\}$$ and ${{\cal H}}_o=\ell^2({{\bf Z}})\ominus {{\cal H}}_e$. After an obvious relabeling of the subscripts, the restriction of $A_c^2$ to ${{\cal H}}_e$ equals $A_b$ while the restriction of $A_c^2$ to ${{\cal H}}_o$ is equal to $M_b$, as defined in (\[Mbformula\]). The final statement of the lemma is now an application of Theorem \[spectralequality\], case 1. We will exploit extensively the formula ${{\rm Spec}}(A_c^2) = {{\rm Spec}}(A_b)$ which appears in the above lemma. The equation ${{\rm Spec}}(A_b)={{\rm Spec}}(M_b)$ will not play a role in our subsequent arguments, but makes an intriguing connection between spectra of rather different tridiagonal operators. Extending this connection slightly, for $b\in {\Omega}$ define $c={\Gamma}_+(b)$ and $\tilde M_b$ by $$(\tilde M_b f)_n = f_{n-1} + i^n(c_{2n+1}+c_{2n+2}) f_n + f_{n+1},$$ for all $f\in \ell^2({{\bf Z}})$. Then, arguing as we do above to show [(\[irotate\])]{}, we see that $${{\rm Spec}}(\tilde M_b) = i\,{{\rm Spec}}(M_b).$$ In particular, in the case $b\in {{\cal E}}$ when, by Lemma \[symmetry\], $i{{\rm Spec}}(M_b)=i{{\rm Spec}}(A_b) = {{\rm Spec}}(A_b)$, we see that $$S = {{\rm Spec}}(A_b) = {{\rm Spec}}(\tilde M_b).$$ Thus, in studying $S$, we are studying both the almost sure spectrum of the infinite hopping-sign matrix $A_b$ with respect to the measure $\mu$, and the almost sure spectrum, with respect to the same measure, of $\tilde M_b$, a discrete Schrödinger operator with a particular, complex random potential. In the next lemma we define the square root of any non-zero complex number to be the root whose argument lies in $(-\pi/2,\pi/2]$. \[squareroot\] If $b\in{\Omega}$ and $c={\Gamma}_+(b)$ then ${\lambda}\in{{\rm Spec}}(A_b)$ if and only if $\pm\sqrt{{\lambda}}$ both lie in ${{\rm Spec}}(A_c)$. If ${\lambda}\in S$ then $\pm \sqrt{{\lambda}}$ both lie in $S$. Lemma \[symmetry\] and Lemma \[square\] imply that the following statements are equivalent. ${\lambda}\in {{\rm Spec}}(A_b)$; ${\lambda}\in {{\rm Spec}}(A_c^2)=\left( {{\rm Spec}}(A_c)\right)^2$; either $\sqrt{{\lambda}}$ or $-\sqrt{{\lambda}}$ lies in ${{\rm Spec}}(A_c)$; $\pm\sqrt{{\lambda}}$ both lie in ${{\rm Spec}}(A_c)$. If ${\lambda}\in S$ and $b\in {{\cal E}}$ then ${\lambda}\in {{\rm Spec}}(A_b)$ by Proposition \[quasiergodic\]. Lemma \[square\] implies that $${\lambda}\in {{\rm Spec}}(A_c^2)=\left({{\rm Spec}}(A_c)\right)^2\subseteq S^2.$$ Therefore either $\sqrt{{\lambda}}$ or $-\sqrt{{\lambda}}$ lie in $S$. The proof is completed by applying Lemma \[symmetry\]. \[maintheorem\] The set $S$ contains $$\label{the_set} \bigcup_{n\in{{\bf Z}}_+,\; r\in \{0,...,2^{n+2}\}}\;{{\rm e}}^{\pi i r/2^{n+1}}\,[0,2^{1/2^n}] .$$ Hence $S$ contains the unit disc in ${{\bf C}}$. For $n=0$ the theorem states that $$[0,2]\times\{ 1,i,-1,-i\}\subset S.$$ This follows by combining Lemma \[symmetry\] with direct calculations of ${{\rm Spec}}(A_c)$ when $c_n=1$ for all $n\in{{\bf Z}}$ (in which case ${{\rm Spec}}(A_c)=[-2,2]$) and when $c_n=-1$ for all $n\in{{\bf Z}}$ (in which case ${{\rm Spec}}(A_c)=i[-2,2]$). For larger $n$ the first statement of the theorem follows by applying Lemma \[squareroot\] inductively. The second statement is now a consequence of the fact that the set [(\[the\_set\])]{} is dense in the unit disc. The maps ${\Gamma}_\pm$ {#sec:maps} ======================= A crucial role has been played in the proofs above by the nonlinear map $\Gamma_+$ on ${\Omega}$ introduced in Lemma \[square\], and this map will be key to the arguments that we make throughout this paper. And in fact a sequence which is almost a fixed point of $\Gamma_+$ (in a sense made precise below Lemma \[spaceinversion\]) is central to the proof of Theorem \[maintheorem\] in [@CCL], though the proof is quite different and no mapping $\Gamma_+$ appears in [@CCL]. The relationship between the above proof of Theorem \[maintheorem\] and that in [@CCL] is clarified to some extent by the following. Building on the definition of $\Gamma_+$ made above, let us define maps ${\Gamma}_\pm:{\Omega}\to{\Omega}$ by ${\Gamma}_\pm (b)=c$ where $$c_0=\pm 1, \hspace{2em} c_{2n}+c_{2n+1}=0, \hspace{2em} c_{2n}c_{2n-1}=b_n,\label{cfixed0}$$ for all $n\in{{\bf Z}}$. We also define the space inversion symmetry $b\to {\widehat}{b}$ by ${\widehat}{b}_n=b_{1-n}$ for all $n\in{{\bf Z}}$. \[spaceinversion\] If ${\Gamma}_\pm (b)=c$ then ${\Gamma}_\mp ({\widehat}{b})={\widehat}{c}$. In particular ${\Gamma}_\pm (c)=c$ if and only if ${\Gamma}_\mp ({\widehat}{c})={\widehat}{c}$. Each of the equations ${\Gamma}_\pm (c)=c$ has exactly one solution. Let $c={\Gamma}_+(b)$ and $d={\Gamma}_-({\widehat}{b})$. Then $d_0=-1$, $d_{2n}+d_{2n+1}=0$ and $d_{2n}d_{2n-1}={\widehat}{b}_n=b_{1-n}$ for all $n\in{{\bf Z}}$. Therefore ${\widehat}{d}_0=d_1=1$. Also $${\widehat}{d}_{2n+1}+{\widehat}{d}_{2n}=d_{1-(2n+1)}+d_{1-2n}=d_{-2n}+d_{1-2n}=0$$ and $${\widehat}{d}_{2n}{\widehat}{d}_{2n-1}=d_{1-2n}d_{1-(2n-1)}=d_{2(1-n)-1}d_{2(1-n)}={\widehat}{b}_{1-n}=b_n$$ for all $n\in{{\bf Z}}$. Therefore ${\widehat}{d}={\Gamma}_+(b)=c$ and $d={\widehat}{c}$. The proof that $c={\Gamma}_-(b)$ implies $d={\Gamma}_+({\widehat}{b})$ is similar. The other statements of the lemma follow immediately. This paper and [@CCL] use three different special sequences. The sequences $c_\pm$ are defined by ${\Gamma}_\pm (c_\pm)=c_\pm$. It follows directly from their definitions that $c_{+,0}=1$ and $c_{+,1}=-1$ while $c_{-,0}=-1$ and $c_{-,1}=1$. However $$c_{+,n}=c_{-,n}=c_{+,1-n}=c_{-,1-n}$$ for all $n\not= 0,\, 1$. The paper [@CCL] uses the sequence $c_e$ such that $c_{e,0}=c_{e,1}=1$, while $c_{e,n}=c_{\pm,n}$ for all other $n$. Because of the space inversion symmetry the use of $c_+$ or $c_-$ in any proof is really a matter of convenience. We now turn to the solution of the equation $A_c u={\lambda}u$ where $u:{{\bf Z}}\to{{\bf C}}$ is an arbitrary sequence. The eigenvalue equation is equivalent to the second order recurrence equation $$u_{n+1}+c_nu_{n-1}={\lambda}u_n.$$ \[uflip\] Suppose that $c\in{\Omega}$ and $\widehat{c}_n=c_{1-n}$ for all $n\in{{\bf Z}}$; that $u_{n+1}+c_nu_{n-1}={\lambda}u_n$ for some ${\lambda}\in{{\bf C}}$ and all $n\in{{\bf Z}}$ and $u_0=0$, $u_1=1$; and that $\widehat{u}_{n+1}+\widehat{c}_n\widehat{u}_{n-1}={\lambda}\widehat{u}_n$ for all $n\in{{\bf Z}}$ and $\widehat{u}_0=0$, $\widehat{u}_1=1$. Then $|\widehat{u}_{n}|=|u_{-n}|$ for all $n\in{{\bf Z}}$. In particular $u_n$ is bounded as $n\to\infty$ if and only if $\widehat{u}_n$ is bounded as $n\to -\infty$. If one puts $v_n=u_{-n}$ then $$\widehat{c}_{n+1}v_{n+1}+v_{n-1}=c_{-n}u_{-n-1}+u_{-n+1}={\lambda}u_{-n}={\lambda}v_n \label{growthlemma}$$ for all $n\in{{\bf Z}}$. Define $a:{{\bf Z}}\to\{\pm 1\}$ by $a_0=1$ and $a_n/a_{n-1}=\widehat{c}_n$ for all $n\in{{\bf Z}}$. If one now puts $w_n=a_n v_n$ for all $n\in{{\bf Z}}$ then (\[growthlemma\]) implies $$w_{n+1}+\widehat{c}_nw_{n-1}={\lambda}w_n$$ for all $n\in{{\bf Z}}$. Since $w_0=v_0=u_0=0$ it follows that there exist ${\gamma}$ such that $w_n={\gamma}\widehat{c}_n$ for all $n\geq 1$. But $|c_n|=1$ for all $n$, so one obtains $|{\gamma}|=1$ by evaluating this identity for $n=1$. Therefore $|\widehat{u}_n|=|w_n|=|v_n|=|u_{-n}|$ for all $n\geq 1$. \[uevenodd\] Let $u_+,\, u_-,\, u_e:{{\bf Z}}\to {{\bf C}}$ be the solutions of $u_{n+1}+c_nu_{n-1}={\lambda}u_n$ for all $n\in{{\bf Z}}$ subject to $u_0=0$ and $u_1=1$, if $c$ is put equal to $c_+,\, c_-,\, c_e$ respectively. Then $u_{+,n}=u_{e,n}$ for all $n\in{{\bf Z}}$. Moreover $|u_{-,n}|=| u_{e,n}|$ for all $n\in{{\bf Z}}$. The first statement is proved by an elementary computation. For the second we use $c_e=\widehat{c_e}$ and $c_-=\widehat{c_+}$. Lemma \[uflip\] now yields $$|u_{-,n}|=|u_{+,-n}|=|u_{e,-n}|=|u_{e,n}|$$ for all $n\in{{\bf Z}}$. The main step in the proof of Theorem \[maintheorem\] in [@CCL] is contained in the following proposition (we quote here the parts of [@CCL Proposition 2.1] which we use immediately or later in section \[secondproofsection\]). \[CCLprop\] Let $u_e$ be defined as in Corollary \[uevenodd\] and define $p_{i,j}\in {{\bf Z}}$ for $i,j\in{{\bf N}}$ by the formula $$u_{e,i}=\sum_{j=1}^i p_{i,j}{\lambda}^{j-1}$$ with $p_{i,j}=0$ if $j>i$. Let $Y$ denote the set of $(i,j)\in {{\bf N}}^2$ such that $p_{i,j}\neq 0$. Then $p_{i,j}\in\{0,1,-1\}$ for all $i,\, j$ and $(i,j)\in Y$ if and only if one of the following holds.\ (1) $i=j=1$;\ (2) $i$ and $j$ are both even and $(i/2,j/2)\in Y$;\ (3) $i$ and $j$ are both odd and $((i+1)/2,(j+1)/2)\in Y$;\ (4) $i$ and $j$ are both odd and $((i-1)/2,(j+1)/2)\in Y$. The following result is an immediate corollary of this proposition and Corollary \[uevenodd\], which together imply that $|u_{e,i}|\leq (1-|\lambda|)^{-1}$ for $i\in {{\bf Z}}$ and $|\lambda|<1$. [@CCL] \[CCLprop3\] As in Corollary \[uevenodd\], let $u_e:{{\bf Z}}\to {{\bf C}}$ be the solution of $u_{n+1}+c_nu_{n-1}={\lambda}u_n$ for all $n\in{{\bf Z}}$ subject to $u_0=0$ and $u_1=1$, with $c=c_e$. Then $u_e\in \ell^\infty({{\bf Z}})$ for $|{\lambda}|<1$, so that ${{\rm Spec}}(A_{c_e})$ contains the unit disc. Since Corollary \[uevenodd\] has shown that $u_{+}=u_{e}$, it is clear from Theorem \[CCLprop3\] that ${{\rm Spec}}(A_{c_+})$ also contains the unit disc. In fact this is precisely its spectrum. \[c+spectrum\] If $c_+$ is the unique solution of ${\Gamma}_+(c)=c$ then $${{\rm Spec}}(A_{c_+})=\{ z:|z|\leq 1\}.$$ It remains only to show that ${{\rm Spec}}(A_{c_+})\subset\{ z:|z|\leq 1\}$. If ${\lambda}\in {{\rm Spec}}(A_{c_+})$ then repeated applications of the first part of Lemma \[squareroot\] yield ${\lambda}^{2^n}\in {{\rm Spec}}(A_{c_+})$ for all $n\geq 1$. Since the spectrum is a bounded set, it follows that $|{\lambda}|\leq 1$. We will (rather arbitrarily) focus on the mapping $\Gamma_+$ rather than $\Gamma_-$ in the remainder of the paper. The following lemma, which shows that the set of periodic sequences is invariant under the action of ${\Gamma}_{+}$, will play a key role. \[lem\_periodic\] If $b\in \Omega$ is periodic with period $N$, i.e. $b_{n+N} = b_n$, $n\in{{\bf Z}}$, then $c={\Gamma}_{+}(b)$ is $4N$-periodic. Conversely, if $b\in\Omega$, $c={\Gamma}_{+}(b)$, and $c$ is $2N$-periodic for some $N\in{{\bf N}}$, then $b$ is $N$-periodic. First note that, if $c={\Gamma}_{+}(b)$ and one defines $\tilde c\in \Omega$ by $\tilde c_n=c_{2n}$, $n\in{{\bf Z}}$, then $$c={\Gamma}_{+}(b) \Leftrightarrow (\tilde c_0=1, \quad \tilde c_n = -b_n \tilde c_{n-1}, \; c_{2n+1} = - \tilde c_n, \; n\in{{\bf Z}}).\label{equiv}$$ Therefore $$\tilde c_{m+n} = \tilde c_m\,(-1)^n\prod_{j=1}^n b_{m+j}\label{equiv2}$$ for all $m\in{{\bf Z}}$ and $n\in{{\bf N}}$. If $b$ is $N$-periodic, then $$\tilde c_{m+2N} = \tilde c_m\,\prod_{j=1}^{2N} b_{m+j} = \tilde c_m\,\prod_{j=1}^{N} b_{m+j}^2 = \tilde c_m,$$ for all $m\in{{\bf Z}}$. Therefore $c$ is $4N$-periodic. Conversely, if $c={\Gamma}_{+}(b)$, for some $b\in \Omega$, and $c$ is $2N$-periodic for some $N\in{{\bf N}}$, then $\tilde c$ is $N$-periodic and, from [(\[equiv\])]{}, it follows that $b$ is $N$-periodic. To illustrate the above lemma, define $c^-,c^+ \in {\Omega}$ by $c^-_n = -1$, $c^+_n=1$, for $n\in{{\bf Z}}$, and define the sequences $c^{(m,+)}, c^{(m,-)}\in {\Omega}$, for $m=0,1,...$, by $$\label{periodic_sequences} c^{(0,\pm)} = c^\pm, \quad c^{(m,\pm)} = \Gamma_+(c^{(m-1,\pm)}), \quad m\in {{\bf N}}.$$ Then explicit calculations of the action of $\Gamma_+$ yield that $c^{(1,+)}={\Gamma}_{+}(c^+)$ is $4$-periodic (but not periodic with any smaller period), with $c^{(1,+)}_{-1}=c^{(1,+)}_0=1$, $c^{(1,+)}_1=c^{(1,+)}_2=-1$. On the other hand, $c^{(1,-)}={\Gamma}_{+}(c^-)$ is $2$-periodic (and so also $4$-periodic), with $c^{(1,-)}_n = (-1)^n$ for all $n\in{{\bf Z}}$. Both these calculations, of course, are consistent with the lemma, which implies that $c^{(m,\pm)}$ is $N$-periodic with $N=4^m$, so that, using the notation [(\[eq:piinf\])]{} (dropping ${\sigma}$ given that ${\sigma}=1$), $$\label{inclusion} {{\rm Spec}}(A_{c^{(m,\pm)}}) \subset \pi_{4^m}, \quad m=0,1,... .$$ Although we do not have an explicit formula for the sequences $c^{(m,\pm)}$, it is easy to compute ${{\rm Spec}}(A_{c^{(m,\pm)}})$. By Lemma \[squareroot\], if $c=\Gamma_+(b)$, then $$\label{speciter} {{\rm Spec}}(A_c) = \{\pm \sqrt{\lambda}: \lambda \in {{\rm Spec}}(A_b)\}.$$ The proof of Theorem \[maintheorem\] begins with the observation that ${{\rm Spec}}(A_{c^+})=[-2,2]$ and ${{\rm Spec}}(A_{c^-})=i[-2,2]$. Combining this observation with [(\[speciter\])]{} we easily prove by induction that $$\label{spec_cplus} {{\rm Spec}}(A_{c^{(m,+)}}) = \left\{r\,{{\rm e}}^{\pi i j/2^{m}}: 0\leq r \leq 2^{1/2^m}, \;j\in \{0,...,2^{m+1}-1\}\right\}$$ and $$\label{spec_cminus} {{\rm Spec}}(A_{c^{(m,-)}}) ={{\rm e}}^{\pi i/2^{m+1}}\,{{\rm Spec}}(A_{c^{(m,+)}}).$$ Combining equations [(\[inclusion\])]{}, [(\[spec\_cplus\])]{} and [(\[spec\_cminus\])]{}, we see that we have shown that $$\left\{r\,{{\rm e}}^{\pi i j/2^{m}}: 0\leq r \leq 2^{1/2^{m+1}}, \;j\in \{0,...,2^{m+2}-1\}\right\}\subset \pi_{4^m}, \quad m=0,1,... .$$ Thus we have shown the following modification of Theorem \[maintheorem\] which, of course, by [(\[eq:pidef\])]{}, has Theorem \[maintheorem\] as a corollary. \[denseness\] The set $\pi_\infty$ contains the set [(\[the\_set\])]{}, and so is dense in the unit disc in ${{\bf C}}$. We know ${{\rm Spec}}(A_{c^{(m,\pm)}})$ explicitly, but do not have explicit formulae for the sequences $c^{(m,\pm)}$. However we can show that $c^{(m,\pm)}$ converges pointwise to the sequence $c_+$, the unique fixed point of $\Gamma_+$, as $m\to \infty$. This is the content of the next two lemmas. We omit a proof of the first of these lemmas which is an easy consequence, by simple induction arguments, of the definition of ${\Gamma}_+$. \[maplem\] If $b\in\Omega$ and $c=\Gamma_+(b)$, then $c_0=c_{+,0}$ and $c_1 =c_{+,1}$. If, for some $N\in{{\bf N}}$, $b_m=c_{+,m}$ for $m = 1,...,N$, then also $c_m = c_{+,m}$ for $m = 2,...,2N+1$. If, for some $N\in{{\bf Z}}_+$, $b_{-m}=c_{+,-m}$ for $m = 0,1,...,N$, then $b_{-m}=c_{+,-m}$ for $m = 1,2,...,2N+2$. \[strong\_converg\] Let $b \in\Omega$, and define $c^{(n)}\in {\Omega}$ for $n\in {{\bf N}}$ by $c^{(1)}={\Gamma}_+(b)$ and $c^{(n+1)} = {\Gamma}_+(c^{(n)})$, $n\in{{\bf N}}$. Then, for $n\in{{\bf N}}$, $$c^{(n)}_m = c_{+,m}, \quad m = 2-2^n,3-2^n,...,2^n-1,$$ so that $c^{(n)}\to c_+$ pointwise and $A_{c^{(n)}}$ converges strongly to $A_{c_+}$ as $n\to\infty$. Further, $${{\rm Spec}}(A_{c^{(n)}}) \subset \{\lambda: |\lambda|\leq 2^{1/2^n}\}.$$ The first equation follows by induction from Lemma \[maplem\]. The second equation follows by induction from [(\[speciter\])]{} and the trivial bound that ${{\rm Spec}}(A_b)\subset \{\lambda:|\lambda|\leq 2\}$, which holds for all $b\in {\Omega}$. The mapping $\Gamma_{{\sigma},+}$ {#related} ================================= For the rest of the paper we consider operators $A_c$ for which the coefficients $c_n$ take values in $\{\pm {\sigma}\}$, where $0<{\sigma}\leq 1$; that is, in the notation we have introduced in the introduction, we assume that $c\in \Omega_{\sigma}$, for some ${\sigma}\in (0,1]$. The mapping $\Gamma_+$ that we have introduced continues to play an important role. We extend the mapping so that it operates on $\Omega_{{\sigma}^2}$, defining, for ${\sigma}\in (0,1]$, $\Gamma_{{\sigma},+}:\Omega_{{\sigma}^2}\to \Omega_{\sigma}$ by $$\label{def_Gam_sig_plus} \Gamma_{{\sigma},+}(c) = {\sigma}\Gamma_+({\sigma}^{-2}c).$$ In other words, for $b\in \Omega_{{\sigma}^2}$, $c=\Gamma_{{\sigma},+}(b)$ is the unique sequence in $\Omega_{\sigma}$ satisfying $$c_0={\sigma}, \hspace{2em} c_{2n}+c_{2n+1}=0, \hspace{2em} c_{2n}c_{2n-1}=b_n.\label{iteration2a}$$ Main properties of the mapping $\Gamma_{{\sigma},+}$ for our purposes are contained in the following extension of Lemma \[squareroot\]. We will need to refer to a number of circular annuli, and use in this lemma and subsequently the notation $$\label{annuli} { [ \hspace*{-0.15em} [ a,b ] \hspace*{-0.15em} ] } =\{ {\lambda}:a\leq |{\lambda}|\leq b\}.$$ \[lemmaB\]If $b\in{\Omega}_{{\sigma}^2}$ and $c={\Gamma}_{{\sigma},+}(b)\in {\Omega}_{\sigma}$, then $$(\,{\lambda}\in {{\rm Spec}}(A_b) \,)\Leftrightarrow (\, \pm \sqrt{{\lambda}} \in {{\rm Spec}}(A_c) \,).\label{sigmaroot0}$$ Hence $$(\,{\lambda}\in S_{{\sigma}^2}\,){\Rightarrow}(\, \pm \sqrt{{\lambda}} \in S_{\sigma}\,)\label{sigmaroot}$$ and $$(\, { [ \hspace*{-0.15em} [ a,b ] \hspace*{-0.15em} ] }\subseteq S_{{\sigma}^2}\,){\Rightarrow}( \, { [ \hspace*{-0.15em} [ a^{1/2},b^{1/2} ] \hspace*{-0.15em} ] }\subseteq S_{{\sigma}}\,).\label{sigroot2}$$ We modify the calculations in Section \[main\]. Lemma \[symmetry\] is valid as it stands. In Lemma \[square\] we assume that $b\in {\Omega}_{{\sigma}^2}$, and define $c\in {\Omega}_{\sigma}$ by $c={\Gamma}_{{\sigma},+}(b)$, and apply Case 2 of Theorem \[spectralequality\] in place of Case 1. This leads to the conclusion ${{\rm Spec}}(A_c^2)= {{\rm Spec}}(A_b)$ as in Lemma \[squareroot\]. (\[sigmaroot\]) and (\[sigroot2\]) follow by choosing $b\in{{\cal E}}_{{\sigma}^2}$ and using Proposition \[quasiergodic\]. Periodic and paired periodic operators {#periodic_ops} ====================================== To prove our main theorem we need results on operators $A_c$ on $\ell^2({{\bf Z}})$ that have one periodic structure for $n\geq 0$ and another for $n<0$ (which we term paired periodic operators). The essential spectrum of such an operator is the union of the essential spectra of the periodic operators involved, which may be calculated explicitly using their Bloch decompositions. There may also be substantial inessential spectrum, in particular, open subsets of the spectrum where $A_c-{\lambda}I$ is Fredholm but has non-zero index. These parts of the spectrum (and the corresponding values of the index) can be computed by application of general results for block Toeplitz operators, which have been developed to a high degree of sophistication; see [@B2; @BS; @BK; @BS2] and the references therein. We need only a small part of this theory, and it is easy to develop this from first principles. We do this in a short Lemma \[lem\_two\_periods\] below, inspired by earlier analysis in [@DS; @EBD0; @EBD1], and particularly [@EBD1 Theorem 12]. Both the proof of Lemma \[lem\_two\_periods\], and the effective application of this lemma to prove Theorem \[maintheorem3\], depend on the next two lemmas which describe properties of the spectra and eigenfunctions of periodic operators. We assume throughout this section that the parameter ${\sigma}\in (0,1)$. \[quadraticbounds\] Let $$\Phi(\tau,{\gamma})= \frac{{{\rm Re}}(\tau)^2}{(1+{\gamma})^2}+ \frac{{{\rm Im}}(\tau)^2}{(1-{\gamma})^2}\label{Phidef}$$ where $\tau\in{{\bf C}}$ and $-1< {\gamma}<1$. Then the quadratic equation $$z^2-\tau z+{\gamma}=0\label{zeq}$$ has a solution satisfying $|z|=1$ if and only if $\Phi=1$. If $\Phi<1$ then both solutions satisfy $|z|<1$. If $\Phi>1$ then one solution satisfies $|z|<1$ and the other satisfies $|z|>1$. For $\theta\in{{\bf R}}$, $z={{\rm e}}^{i\theta}$ is a solution of (\[zeq\]) if and only if $$\cos(\theta)=\frac{{{\rm Re}}(\tau)}{1+{\gamma}}, \hspace{2em} \sin(\theta)=\frac{{{\rm Im}}(\tau)}{1-{\gamma}},$$ so that [(\[zeq\])]{} has a solution satisfying $|z|=1$ if and only if $\Phi(\tau,\gamma)=1$. The set $U=\{\tau\in{{\bf C}}:\Phi<1\}$ is connected and contains the origin. Since the solutions of (\[zeq\]) depend continuously on $\tau$, and both solutions satisfy $|z|<1$ if $\tau=0$, it follows that both satisfy $|z|<1$ for all $\tau\in U$. The case $\Phi>1$ is similar. The following lemma is closely related to a similar result for the non-self-adjoint Anderson model in [@EBD1 Theorem 11]. \[BIO\] If $c\in {\Omega}_{\sigma}$ and ${\lambda}\in {{\bf C}}$ then the space of all solutions of $A_cf={\lambda}f$ is two-dimensional. If $c$ is periodic with period $p$ then the asymptotic behaviour as $n\to\pm\infty$ of the solutions is determined by the solutions $z_1,\, z_2$ of the polynomial $z^2-\tau({\lambda}) z+{\gamma}=0$, where $\tau({\lambda})$ is a monic polynomial in ${\lambda}$ with degree $p$, given by $\tau(\lambda) = {{\rm tr}}(T_p)$, where $T_p=X_p X_{p-1}\ldots X_1$ and $$X_n = {\left(\begin{array}{cc}0&1\\ -c_n&{\lambda}\end{array}\right)},$$ and ${\gamma}=\det(T_p)=\pm {\sigma}^p$. Ordering the two solutions so that $|z_1|\geq |z_2|$, there are three cases: 1. $\lambda$ lies in the closed set $$B_c=\{{\lambda}: |z_1|=1\mbox{ and } |z_2|={\sigma}^p\}.$$ This set is the spectrum of $A_c$, equivalently, the set of ${\lambda}$ for which $A_cf={\lambda}f$ has a bounded solution. 2. ${\lambda}$ lies in the open set $$I_c=\{{\lambda}: 1> |z_1|\geq|z_2|>{\sigma}^p\}.$$ This is the case if and only if all solutions of $A_cf={\lambda}f$ decay exponentially as $n\to +\infty$. 3. ${\lambda}$ lies in the open set $$O_c=\{{\lambda}: |z_1|>1\mbox{ and } |z_2|<{\sigma}^p\}.$$ This is the case if and only if there exists a solution of $A_cf={\lambda}f$ that decays exponentially as $n\to +\infty$ and grows exponentially as $n\to -\infty$, and another solution that decays exponentially as $n\to -\infty$ and grows exponentially as $n\to +\infty$. The sequence $f:{{\bf Z}}\to{{\bf C}}$ is a solution of $A_cf={\lambda}f$ if and only if $f_{n+1}+c_n f_{n-1}={\lambda}f_n$ for all $n\in{{\bf Z}}$. This recurrence relation can be rewritten in the form $$\begin{aligned} {\left(\begin{array}{c} f_n\\ f_{n+1} \end{array}\right)}&=&{\left(\begin{array}{cc}0&1\\ -c_n&{\lambda}\end{array}\right)}{\left(\begin{array}{c} f_{n-1}\\ f_n \end{array}\right)}\\ &=& X_n{\left(\begin{array}{c} f_{n-1}\\ f_n \end{array}\right)}\\ &=& T_n{\left(\begin{array}{c} f_{0}\\ f_1 \end{array}\right)}\end{aligned}$$ where $T_n=X_n X_{n-1}\ldots X_1$. If $c$ is periodic with period $p$, then the asymptotic behaviour of the two-dimensional space of eigenfunctions $f$ is determined by the magnitude of the eigenvalues $z_1,\, z_2$ of $T_p$. These are the solutions of the equation $z^2-\tau z+{\gamma}=0$ where $\tau={{\rm tr}}(T_p)$ and ${\gamma}=\det(T_p)$. A simple induction establishes that the $(i,j)$-th entry of $T_p$ is a polynomial in ${\lambda}$ with degree less than $p$ unless $i=j=2$ in which case it is a monic polynomial with degree $p$. Therefore $\tau$ is a monic polynomial in ${\lambda}$ with degree $p$. However $$\det(T_p)=\prod_{r=1}^p \det(X_r)= c_1 \dots c_p = \pm {\sigma}^p$$ does not depend on ${\lambda}$. The continuous dependence of the roots of a polynomial on its coefficients implies that $B_c$ is closed while $I_c$ and $O_c$ are open. An application of Lemma \[quadraticbounds\] now completes the proof. One sees, in particular, that $${{\rm Spec}}(A_c)= B_c=\{{\lambda}: \Phi(\tau,{\gamma})=1 \}.$$ Our next lemma enables us to determine the sets $I_c$ and $O_c$ for certain important periodic sequences $c$, and to determine the spectra of certain paired periodic operators. We continue with the assumptions and notation of Lemma \[BIO\]. \[findIcOc\] If $V$ is a connected component of ${{\bf C}}\backslash B_c$ then $V\subseteq I_c$ or $V\subseteq O_c$. If $V$ is unbounded then $V\subseteq O_c$, and if $0\in V$ then $V\subseteq I_c$. If ${{\bf C}}\backslash B_c$ has exactly two components then the bounded component equals $I_c$ and the unbounded component equals $O_c$. We first observe that $V$, $I_c$ and $O_c$ are all open sets and that their definitions imply directly that $I_c,\, O_c$ are disjoint. Therefore $V=(V\cap I_c)\cup (V\cap O_c)$, where the two intersections on the right-hand side are disjoint. Since $V$ is connected, it follows that $V=V\cap I_c$ or $V=V\cap O_c$ This completes the proof of the first statement. Lemma \[BIO\] case 1 implies that $$B_{\sigma}={{\rm Spec}}(A_c)\subseteq \{{\lambda}: |{\lambda}|\leq 1+{\sigma}\}.$$ Therefore ${{\bf C}}\backslash B_c$ has only one unbounded component $V$ and it contains $\{ {\lambda}: |{\lambda}|>1+{\sigma}\}$. To prove that $V\subseteq O_c$ it is sufficient by the first part of this proof to find a single point ${\lambda}\in V\cap O_c$. The fact that $\tau$ is a polynomial with degree $p$ implies that $|\tau({\lambda})|\to\infty$ as $|{\lambda}|\to\infty$. This implies that the solutions of $z^2-\tau({\lambda})z+{\gamma}=0$, where ${\gamma}=\pm {\sigma}^p$, are $z\sim \tau({\lambda})$ and $z\sim {\gamma}/\tau({\lambda})$ to leading order for all large enough $|{\lambda}|$. Therefore ${\lambda}\in O_c$ for all such ${\lambda}$. The proof is completed by proving that $0\in I_c$. For ${\lambda}=0$ one has $T_p=X_pX_{p-1}\ldots X_1$ where each $X_r$ is of the form ${\raisebox{0.25ex}{\scalebox{0.6}{${\left(\begin{array}{cc}0&1\\ \pm {\sigma}&0 \end{array}\right)}$}}}$. If $p=2m$ it follows that $T_p={\raisebox{0.25ex}{\scalebox{0.6}{${\left(\begin{array}{cc}\pm {\sigma}^m&0\\0&\pm {\sigma}^m \end{array}\right)}$}}}$. The fundamental equation must therefore take one of the forms $z^2-2{\sigma}^m z+{\sigma}^{2m}=0$, $z^2+2{\sigma}^m z+{\sigma}^{2m}=0$ or $z^2-{\sigma}^{2m}=0$. In each case both solutions have modulus ${\sigma}^{p/2}<1$. The same holds if $p=2m+1$. The final statement of the lemma follows from the following observations. There must be a component of ${{\bf C}}\setminus B_c$ that contains $0$ and there must be an unbounded component. The first part of the proof shows that these are distinct, and the extra hypothesis is that there are no other components. Our next task is to determine the sets $B_c,\, I_c$ and $O_c$ for certain particular periodic sequences. \[c1\] If $c_n={\sigma}$ for all $n\in{{\bf Z}}$ then ${{\rm Spec}}(A_c)$ is the ellipse $$\begin{aligned} {{\rm Spec}}(A_c)&=&\left\{ u+iv: \frac{u^2}{(1+{\sigma})^2}+\frac{v^2}{(1-{\sigma})^2}=1\right\}\label{c1Eucl}\\ &=&\left\{ \rho{{\rm e}}^{i\theta}: \rho=\frac{1-{\sigma}^2}{\sqrt{1+{\sigma}^2-2{\sigma}\cos(2\theta)}}\right\}.\label{c1polar}\end{aligned}$$ Moreover the interior $U$ of the ellipse equals $I_c$ and the exterior $V$ of the ellipse equals $O_c$. We have $p=1$ and $T_1={\raisebox{0.25ex}{\scalebox{0.6}{${\left(\begin{array}{cc}0&1\\ -{\sigma}&{\lambda}\end{array}\right)}$}}}$, so $\tau({\lambda})={\lambda}$ and ${\gamma}={\sigma}$. Using (\[Phidef\]) we deduce that ${{\rm Spec}}(A_c)$ is given by (\[c1Eucl\]). The proof is completed by using Lemma \[findIcOc\]. \[c2\] If $c_n=-{\sigma}$ for all $n\in{{\bf Z}}$ then ${{\rm Spec}}(A_c)$ is the ellipse $$\begin{aligned} {{\rm Spec}}(A_c)&=&\left\{ u+iv: \frac{u^2}{(1-{\sigma})^2}+\frac{v^2}{(1+{\sigma})^2}=1\right\}\label{c2Eucl}\\ &=&\left\{ \rho{{\rm e}}^{i\theta}: \rho=\frac{1-{\sigma}^2}{\sqrt{1+{\sigma}^2+2{\sigma}\cos(2\theta)}}\right\}.\label{c2polar}\end{aligned}$$ Moreover $I_c$ is the interior of the ellipse and $O_c$ is the exterior of the ellipse. We have $p=1$ and $T_1={\raisebox{0.25ex}{\scalebox{0.6}{${\left(\begin{array}{cc}0&1\\ {\sigma}&{\lambda}\end{array}\right)}$}}}$, so $\tau({\lambda})={\lambda}$ and ${\gamma}=-{\sigma}$. We omit the rest of proof, which is almost identical to that of Lemma \[c1\]. In the following lemma, starting from Lemmas \[c1\] and \[c2\], and making successive applications of Lemma \[lemmaB\], we compute the spectra of a family of periodic sequences, namely the sequences $c^\pm = {\sigma}c^{(n,\pm)}\in {\Omega}_{\sigma}$, defined by [(\[periodic\_sequences\])]{}. By Lemma \[lem\_periodic\], these sequences are periodic of period $\leq 4^n$. This next lemma applies for $0<{\sigma}<1$. The corresponding result for ${\sigma}=1$ is equations [(\[spec\_cplus\])]{} and [(\[spec\_cminus\])]{} above. \[lem\_sf\] Suppose $n\in {{\bf Z}}_+$ and $c^+ = {\sigma}c^{(n,+)}$, $c^- = {\sigma}c^{(n,-)}$. Then $$\label{spectral_formula} {{\rm Spec}}( A_{c^\pm}) = \{\,\rho{{\rm e}}^{{{\rm i}}\theta}: \rho = \rho_n^\pm(\theta,\sigma)\}$$ where $$\rho_0^+(\theta,\sigma) = \frac{1-\sigma^2}{\left(1+\sigma^2 - 2\sigma\cos 2\theta\right)^{1/2}}, \quad \rho_0^-(\theta,\sigma) = \frac{1-\sigma^2}{\left(1+\sigma^2 + 2\sigma\cos 2\theta\right)^{1/2}},$$ and, for $n\in {{\bf N}}$, $$\rho_n^\pm(\theta,\sigma) =\left(\rho_0^\pm(2^n\theta,\sigma^{2^n})\right)^{1/2^n} = \frac{\left(1-\sigma^{2^{n+1}}\right)^{1/2^n}}{\left(1+\sigma^{2^{n+1}} \mp 2\sigma^{2^n}\cos \left(2^{n+1}\theta\right)\right)^{1/2^{n+1}}}.$$ Moreover, $$I_{c^\pm} = \left\{\,\rho {{\rm e}}^{i\theta}: 0\leq \rho < \rho_n^\pm(\theta,\sigma) \right\}$$ and $$O_{c^\pm} = \left\{\,\rho {{\rm e}}^{i\theta}: \rho > \rho_n^\pm(\theta,\sigma) \right\}.$$ Our proof of [(\[spectral\_formula\])]{} is by induction. We note first that [(\[spectral\_formula\])]{} holds for $n=0$ by Lemmas \[c1\] and \[c2\]. Suppose now that [(\[spectral\_formula\])]{} holds for some $n\geq 0$ and all $0<{\sigma}<1$. Then $${{\rm Spec}}(A_{{\sigma}^2 c^{(n,\pm)}}) = \{\rho{{\rm e}}^{{{\rm i}}\theta}: \rho=\rho_n^\pm(\theta,\sigma^2)\} = \left\{\rho{{\rm e}}^{{{\rm i}}\theta}:\rho=\left( \rho_0^\pm(2^n\theta,\sigma^{2^{n+1}})\right)^{1/2^n}\right\}.$$ Further, since ${\sigma}c^{(n+1,\pm)} = {\sigma}\Gamma_+(c^{(n,+)})=\Gamma_{{\sigma},+}(\sigma^2 c^{(n,+)})$, it follows from Lemma \[lemmaB\] that $${{\rm Spec}}(A_{{\sigma}c^{(n+1,\pm)}}) = \left\{\pm \sqrt{\lambda}: \lambda\in {{\rm Spec}}(A_{{\sigma}^2 c^{(n,\pm)}})\right\}.$$ Combining these equations, we see that [(\[spectral\_formula\])]{} holds with $n$ replaced by $n+1$. Thus [(\[spectral\_formula\])]{} follows by induction. The formulae for $I_{c^\pm}$ and $O_{c^\pm}$ follow from [(\[spectral\_formula\])]{} and Lemma \[findIcOc\]. We remark that $\rho_n^-(\theta,\sigma) =\rho_n^+(\theta\pm\pi/2^{n+1},\sigma)$, so that the spectra of $A_{c^\pm}$ in the above lemma are related by $${{\rm Spec}}(A_{c^{+}}) = {{\rm e}}^{\pm i\pi/2^{n+1}}\, {{\rm Spec}}(A_{c^-}).$$ This is a symmetry which is surprising from an inspection of the sequences $c^\pm$, which need not even have the same period. (For example, as observed in Section \[sec:maps\], $c^+$ has period 4 and $c^-$ period 2 in the case $n=1$.) In principle, since $c^\pm$ is periodic, [(\[spectral\_formula\])]{} should be computable alternatively from the characterisation of the spectrum for general periodic sequences in Lemma \[BIO\]. As an example of this, for the sequence $c^-={\sigma}c^{(1,-)}$ which has period $2$, with $c^-_n = (-1)^n{\sigma}$, the transfer matrix $T_2$ is given by $$T_2=X_2X_1={\left(\begin{array}{cc}0&1\\ -{\sigma}&{\lambda}\end{array}\right)}{\left(\begin{array}{cc}0&1\\ {\sigma}&{\lambda}\end{array}\right)}={\left(\begin{array}{cc}{\sigma}&{\lambda}\\ {\sigma}{\lambda}&-{\sigma}+{\lambda}^2 \end{array}\right)}.$$ Applying Lemmas \[quadraticbounds\] and \[BIO\] with $\tau={\lambda}^2$ and ${\gamma}=-{\sigma}^2$, we find that ${{\rm Spec}}(A_c)$ is the set of all ${\lambda}=u+iv$ for which $$\frac{(u^2-v^2)^2}{(1-{\sigma}^2)^2}+ \frac{(2uv)^2}{(1+{\sigma}^2)^2}=1.$$ If one puts ${\lambda}=\rho {{\rm e}}^{i\theta}$, then this may be rewritten in the form [(\[spectral\_formula\])]{}. The main point of the above theory and calculations are to prove and prepare the use of the following result on operators $A_c$ that are paired periodic operators. To state this result let us introduce the notations $$E_{\sigma}=\left\{ x+iy:\frac{x^2}{(1+{\sigma})^2}+ \frac{y^2}{(1-{\sigma})^2}< 1\right\}\label{ell1}$$ and $$E_{-{\sigma}}=\left\{ x+iy:\frac{x^2}{(1-{\sigma})^2}+ \frac{y^2}{(1+{\sigma})^2}<1\right\}\label{ell2},$$ so that $E_{\sigma}$ and $E_{-{\sigma}}$ are the interiors of the ellipses introduced in Lemmas \[c1\] and \[c2\]. The following lemma is analogous to [@EBD1 Theorem 12], proved there for the non-self-adjoint Anderson model. \[lem\_two\_periods\] Suppose that $c\in {\Omega}_{\sigma}$ is periodic and $\tau\in \{{\sigma},-{\sigma}\}$, and define $c^*\in {\Omega}_{\sigma}$ by $c^*_n = c_n$, for $n\geq 0$, and $c^*_n=\tau$ for $n< 0$. Then $${{\rm Spec}}(A_{c^*}) \supset \overline{I_c}\setminus E_\tau.$$ Since ${{\rm Spec}}(A_{c^*})$ is closed it is enough to show that ${{\rm Spec}}(A_{c*}) \supset I_c\setminus \overline{E_\tau}$. So suppose that $\lambda \in I_c\setminus \overline{E_\tau}$. Then, by Lemmas \[c1\] and \[c2\] and Lemma \[BIO\], since ${\lambda}\not\in \overline{E_\tau}$, it follows that there exists a non-trivial solution $f$ of $A_{c^*}f={\lambda}f$ such that $f_n\to 0$ exponentially as $n\to -\infty$. Since ${\lambda}\in I_c$, again applying Lemma \[BIO\], it follows that this solution $f$ also decays exponentially as $n\to +\infty$. Thus ${\lambda}$ is an eigenvalue of $A_{c^*}$ so ${\lambda}\in {{\rm Spec}}(A_{c^*})$. First proof of the main theorem {#firstproof} =============================== This section is devoted to the proof of Theorem \[maintheorem3\], in which we assume that $0<{\sigma}<1$. \[lemmaA\] We have $${{\rm Spec}}(A_c)\subseteq { [ \hspace*{-0.15em} [ 1-{\sigma}, 1+{\sigma}] \hspace*{-0.15em} ] } \label{ring}$$ and $${{\rm Spec}}(A_c)\subseteq \{ x+iy:|x|+|y|\leq \sqrt{2(1+{\sigma}^2)}\},\label{numrange}$$ for every choice of $c\in{\Omega}_{\sigma}$. We regard $V_cR$ as a small perturbation of $L$ in the identity $A_c=V_cR+L$, noted in the proof of Lemma \[square\]. Since $L$ is a unitary operator with spectrum $\{z:|z|=1\}$, we have $${\Vert}(L-zI)^{-1}{\Vert}=\left| 1-|z|\, \right|^{-1}$$ for all $z$ not on the unit circle. The inclusion (\[ring\]) now follows from ${\Vert}V_cR{\Vert}={\sigma}$ by a perturbation argument; see [@LOTS Th. 9.2.13]. The inclusion (\[numrange\]) depends on an estimate of the numerical range of $A_c$. Following [@LOTS Section 9.3], $x+iy\in{{\rm Num}}(A_c)$ if there exists $f\in \ell^2({{\bf Z}})$ such that ${\Vert}f{\Vert}=1$ and $x+iy=\langle A_c f,f\rangle$. This implies that $$x=\frac{1}{2}\langle (A_c+A_c^\ast)f,f\rangle, \hspace{2em} y=-\frac{i}{2}\langle (A_c-A_c^\ast)f,f\rangle.$$ Therefore $$x+y=\frac{1}{2}\langle Bf,f\rangle$$ where $$B=(A_c+A_c^\ast)-i(A_c-A_c^\ast).$$ A simple calculation shows that $B_{m,n}=0$ unless $|m-n|=1$, while $$B_{n,n+1}=\overline{ B_{n+1,n}}=(1\pm {\sigma})-i(1\mp {\sigma}).$$ Therefore $|B_{n+1,n}|=|B_{n,n+1}|=\sqrt{2(1+{\sigma}^2)}$ for all $n\in{{\bf Z}}$ and $$x+y \leq \frac{1}{2}{\Vert}B{\Vert}\leq \sqrt{2(1+{\sigma}^2)}.$$ The other three steps in the proof of the bound for $|x|+|y|$ are similar. The statement of our main theorem refers to the open set $$H_{\sigma}=E_{\sigma}\cap E_{-{\sigma}},\label{Hsigma}$$ the intersection of the ellipses $E_{\sigma}$ and $E_{-{\sigma}}$. This set satisfies $${ [ \hspace*{-0.15em} [ 0,1-{\sigma}] \hspace*{-0.15em} ] } \subseteq \overline{H_{\sigma}}\subseteq { [ \hspace*{-0.15em} [ 0,r_{{\sigma}} ] \hspace*{-0.15em} ] } \label{Hsigsize}$$ where $$r_{{\sigma}}=\frac{ 1-{\sigma}^{2} }{ \sqrt{1+{\sigma}^{2}\,} }.\label{rndef}$$ \[maintheorem3\] If $0<{\sigma}<1$ then $$\{ {\lambda}:|{\lambda}|\leq 1\}\backslash H_{\sigma}\subseteq S_{\sigma}.$$ Note first that if $c^\pm$ and $\rho_n^\pm(\theta,{\sigma})$ are defined as in Lemma \[lem\_sf\], then $$\rho_n^\pm(\theta, {\sigma}) \geq \rho_{{\sigma},n} = \left(\frac{1-{\sigma}^{2^{n+1}}}{1+{\sigma}^{2^n}}\right)^{1/2^n},$$ for all $\theta\in {{\bf R}}$, so that $I_{c^\pm} \supset \{{\lambda}:|{\lambda}|< \rho_{{\sigma},n}\}$. Thus, defining $c^*\in {\Omega}_{\sigma}$ as in Lemma \[lem\_two\_periods\], with $c=c^+$ or $c^-$ and $\tau=\pm{\sigma}$, we see from Lemma \[lem\_two\_periods\] that $$\label{spectral_inc} {{\rm Spec}}(A_{c^*}) \supset \overline{I_{c}}\setminus E_\tau \supset \{{\lambda}:|{\lambda}| \leq \rho_{{\sigma},n}\}\setminus E_\tau.$$ Applying Proposition \[quasiergodic\], it follows that, for all $n\in {{\bf N}}$, $$S_{\sigma}\supset \{{\lambda}:|{\lambda}| \leq \rho_{{\sigma},n}\}\setminus H_\tau.$$ The theorem follows since $\sup_{n} \rho_{{\sigma},n} = 1$ and $S_{\sigma}$ is closed. Lemma \[lemmaA\] and Theorem \[maintheorem3\] together establish that there is a hole in $S_{\sigma}$ which is at least as big as $\{{\lambda}:|{\lambda}|<1-{\sigma}\}$ and which is no larger than $H_{\sigma}$. The numerical computations we have been able to carry out are consistent with a hypothesis that the hole is precisely the set $H_{\sigma}$, i.e. they are consistent with a hypothesis that ${{\rm Spec}}(A_c)\cap H_{\sigma}=\emptyset$ for every $c\in {\Omega}_{\sigma}$, and hence for every $c\in{{\cal E}}_{\sigma}$. It should be pointed out, however, that these numerical computations are only for instances where $c$ is periodic, for which we have a characterisation of the spectrum in Lemma \[BIO\]. Thus, strictly speaking, our calculations are evidence of the possibly weaker result that $\pi_{\infty,{\sigma}} \cap H_{\sigma}= \emptyset$; they become evidence that ${{\rm Spec}}(A_c)\cap H_{\sigma}=\emptyset$ with a hypothesis that $\pi_{\infty,{\sigma}}$ is dense in the part of $S_{\sigma}$ that is contained in the unit disc. This latter statement may or may not be true for ${\sigma}\in (0,1)$, but we have shown in Theorem \[denseness\] that it is true for ${\sigma}=1$. As an example of the numerical computations we have carried out, the right hand side of Figure \[fig:30pics1\] shows the union of ${{\rm Spec}}(A_c)$ over all periodic $c\in {\Omega}_{\sigma}$ for which the period $N\leq 12$. It is clear from this figure that $\pi_{12,{\sigma}}\cap H_{\sigma}= \emptyset$ for ${\sigma}=0.5$. We note that, rather than using the characterisation in Lemma \[BIO\], we use for these computations the standard Bloch-decomposition formula (e.g. [@LOTS]) that $$\label{spectrumperiodic} {{\rm Spec}}(A_c) = \bigcup_{|\alpha|=1} {{\rm Spec}}(A^{(N,\mathrm{per})}_{c,\alpha}),$$ where $A^{(N,\mathrm{per})}_{c,\alpha}$ is the $N\times N$ matrix defined in [(\[finitematrix2\])]{} below. It is not feasible to calculate $\pi_{N,{\sigma}}$, the union of all $2^{N-1}$ periodic spectra of period $N$, for very much larger values of $N$. In Figure \[Hsigplot\] we sample $\pi_{100,{\sigma}}$, for ${\sigma}=0.5$, plotting the union of the spectra of $10^5$ randomly chosen $N\times N$ matrices $A_{c,\alpha}^{(N,\mathrm{per})}$. By randomly chosen we mean here that, in each realisation, $N\in \{1,...,100\}$ is randomly chosen, with higher probabilities for the smaller matrix sizes, and then the vector $c=(c_1,...,c_N)$ is randomly chosen, with each $c_n=\pm {\sigma}$ independent and identically distributed with $\mathrm{Pr}(c_n={\sigma})=0.5$, and finally the phase factor $\alpha$ is randomly chosen from a uniform measure on the unit circle. We see in the figure that clearly, as they have to, the spectra are constrained to lie in the inclusion sets shown in Lemma \[lemmaA\]. We also note that all the spectra lie outside $H_{\sigma}$. Second proof of the main theorem {#secondproofsection} ================================ In Theorem \[maintheorem6\] of this section we provide a second proof that $$\{{\lambda}:|{\lambda}|\leq 1\}\backslash H_{\sigma}\subseteq S_{\sigma}\label{secondversion}$$ for all ${\sigma}\in (0,1)$. This proof has a lot in common with the previous one, but it reveals more about the asymptotic behaviour of the solutions of the second order recurrence relation for certain choices of $c\in{\Omega}_{\sigma}$. A key role in this section is played by the sequence $c_e\in {\Omega}_{\sigma}$ defined in Section \[sec:maps\], which sequence was central to the proof of Theorem \[maintheorem\] that appears in [@CCL] (see Theorem \[CCLprop3\] above for more details). We start with some calculations that do not depend on ${\sigma}$. Throughout this section $\widetilde{c}_n \in \{ \pm 1\}$ is defined for all $n\geq 1$ by the rules $\widetilde{c}_1=1$, $\widetilde{c}_{2n}=\widetilde{c}_{2n-1}\widetilde{c}_n$ and $\widetilde{c}_{2n}+\widetilde{c}_{2n+1}=0$. (In other words, $\widetilde{c}_n=c_{e,n}$, for $n\geq 1$.) The first few values are shown in Table \[tabledata2\]. We will obtain a bound on a transfer matrix $T_{m,{\lambda}}$ associated with this sequence and use this bound to prove Theorem \[maintheorem6\]. Let $u:{{\bf Z}}_+\to{{\bf C}}$ be the solution of $u_{n+1}={\lambda}u_n-\widetilde{c}_n u_{n-1}$ such that $u_0=0$ and $u_1=1$, so that $u$ is the restriction to ${{\bf Z}}_+$ of the bi-infinite sequence $u_e$ already studied in [@CCL] and discussed in Section \[sec:maps\]. We have observed already in Proposition \[CCLprop\] that $u_n$ is a polynomial of degree $n-1$ in ${\lambda}$ with integer coefficients for all $n\geq 2$. Similarly, if $v:{{\bf Z}}_+\to{{\bf C}}$ is the solution of $v_{n+1}={\lambda}v_n-\widetilde{c}_n v_{n-1}$ such that $v_0=1$ and $v_1=0$, it is easy to see that $v_n$ is a polynomial of degree $n-2$ with integer coefficients for all $n\geq 2$. $$\begin{array}{rrrr} n & \widetilde{c}_n & u_n & v_n \\ \hline 1&\hspace{2em}1&1& \,\,\,\,0 \\ 2&1&{\lambda}& -1 \\ 3&-1&{\lambda}^2-1& -{\lambda}\\ 4&-1&{\lambda}^3& -{\lambda}^2-1\\ 5&1&{\lambda}^4+{\lambda}^2-1&-{\lambda}^3 -2{\lambda}\\ 6&-1&{\lambda}^5-{\lambda}& -{\lambda}^4-{\lambda}^2 +1\\ 7&1&{\lambda}^6+{\lambda}^4-1& -{\lambda}^5 -2{\lambda}^3 -{\lambda}\\ 8&-1&{\lambda}^7& -{\lambda}^6 -{\lambda}^4 -1\\ 9&1&\hspace{2em}{\lambda}^8+{\lambda}^6+{\lambda}^4-1&\hspace{2em}-{\lambda}^7-2 {\lambda}^5-2{\lambda}^3-2{\lambda}\end{array}$$ One may check the computations of $u_m$ and $v_m$ in Table \[tabledata2\] by confirming the determinantal identity $u_mv_{m+1}-v_mu_{m+1}=\pm 1$ for all $m\geq 1$, the left hand side being a polynomial in ${\lambda}$. Here we are referring to a determinantal identity for the transfer matrix $T_{m,{\lambda}}$, defined by $$T_{m,{\lambda}}={\left(\begin{array}{cc} v_{m}&u_{m}\\ v_{m+1}&u_{m+1} \end{array}\right)},$$ which transfers the data of any solution of $x_{n+1}={\lambda}x_n-\widetilde{c}_n x_{n-1}$ from $\{ 0,1 \}$ to $\{ m,m +1 \}$ in the sense that $$T_{m,{\lambda}}{\left(\begin{array}{c}x_0\\ x_1 \end{array}\right)}={\left(\begin{array}{c} x_m\\x_{m+1} \end{array}\right)}.$$ It is easy to verify (see Lemma \[BIO\] above) that $$T_{m,{\lambda}}=X_mX_{m-1}\ldots X_1\label{TSprod}$$ where $$X_r={\left(\begin{array}{cc}0&1\\ -\widetilde{c}_r&{\lambda}\end{array}\right)}\label{Sdetvalue}$$ has determinant $\widetilde{c}_r\in \{\pm 1\}$ for every $r\geq 1$. The proof of Theorem \[maintheorem6\] we will give below was motivated by numerical evidence that $${{\rm tr}}(T_{2^r,{\lambda}}) ={\lambda}^{2^r}-2$$ holds for all $r\geq 1$ and all ${\lambda}\in{{\bf C}}$; see Table \[tabledata3\]. We prove this crucial identity in Lemma \[Ttau2\]. $$\begin{array}{rr} n & \hspace{2em}{{\rm tr}}(T_{n,{\lambda}})= u_{n+1}+ v_n \\ \hline 1&{\lambda}\\ 2&{\lambda}^2-2\\ 3& {\lambda}^3-{\lambda}\\ 4&{\lambda}^4-2\\ 5&{\lambda}^5-{\lambda}^3-3{\lambda}\\ 6&{\lambda}^6-{\lambda}^2\\ 7& {\lambda}^7-{\lambda}^5-2{\lambda}^3-{\lambda}\\ 8&{\lambda}^8-2 \end{array}$$ \[Tdet1\] Let $T$ be a $2\times 2$ matrix with determinant ${\delta}$ and trace $\tau$. If $\tau^2\not= 4{\delta}$ and ${\gamma}$ is the absolute value of the larger root of $z^2-\tau z+{\delta}=0$ then there exists a constant $b$ such that $${\Vert}T^r{\Vert}\leq b\,{\gamma}^r$$ for all $r\geq 1$. If $\tau^2 = 4{\delta}$ then for every ${\varepsilon}>0$ there exists a constant $b_{\varepsilon}$ such that $${\Vert}T^r{\Vert}\leq b_{\varepsilon}({\gamma}+{\varepsilon})^r\label{jordan}$$ for all $r\geq 1$. The eigenvalues $z_\pm$ of $T$ are the roots $z$ of $z^2-\tau z+{\delta}=0$. The condition $\tau^2\not= 4{\delta}$ implies that the eigenvalues are distinct, so $T$ is diagonalizable – there exists an invertible matrix $B$ such that $$T=B{\left(\begin{array}{cc} z_+&0\\0&z_- \end{array}\right)} B^{-1}.$$ Therefore $$\begin{aligned} {\Vert}T^r{\Vert}={\Vert}B{\left(\begin{array}{cc} z_+^r&0\\0&z_-^r \end{array}\right)} B^{-1}{\Vert}\leq {\Vert}B{\Vert}\, {\Vert}B^{-1}{\Vert}{\gamma}^r.\end{aligned}$$ The slightly worse bound (\[jordan\]) is obtained when $\tau^2= 4{\delta}$ because one has to use the Jordan canonical form for $T$. \[Tdet3\] The identity $\det(T_{2^n,{\lambda}}) =1$ holds for all $n\geq 1$ and all ${\lambda}\in{{\bf C}}$. If $m\in{{\bf N}}$ then (\[TSprod\]) and (\[Sdetvalue\]) imply $$\begin{aligned} \det(T_{2m,{\lambda}})&=& \prod_{r=1}^m \det(X_{2r}X_{2r-1})\\ &=&\prod_{r=1}^m (\widetilde{c}_{2r}\widetilde{c}_{2r-1})\\ &=&\prod_{r=1}^m \widetilde{c}_{r}\\ &=& \det(T_{m,{\lambda}}).\end{aligned}$$ It follows by induction that $$\det(T_{2^n,{\lambda}})=\det(T_{1,{\lambda}})=\widetilde{c}_1=1.$$ The following lemma depends on Proposition \[CCLprop\] above, abstracted from [@CCL], which notes properties of the integer coefficients $p_{i,j}$ of the polynomials $$u_i=\sum_{j=1}^i p_{i,j}{\lambda}^{j-1}.$$ \[Ttau1\] The polynomial $u_m$ is even for odd $m$ and odd for even $m$. Its leading term is ${\lambda}^{m-1}$. If $m=2^n$ and $n\geq 2$ then $$\begin{aligned} u_m&=& {\lambda}^{m-1},\label{ufortau1}\\ u_{m+1}&=&-1+{\lambda}^{m/2}\sum_{r=0}^{m/4} {\alpha}_r {\lambda}^{2r}\label{ufortau2}\end{aligned}$$ where ${\alpha}_r\in \{0,1,-1\}$ for all $r$. The statements in the first two sentences may be proved by induction, using the definition of $u_m$. We prove (\[ufortau1\]) and (\[ufortau2\]) for $m=2^n$ by induction in $n$, noting that both hold for $n\leq 3$; see Table \[tabledata2\]. As in Proposition \[CCLprop\], let $Y$ denote the set of $(i,j)\in {{\bf N}}^2$ such that $p_{i,j}\not= 0$. To prove (\[ufortau1\]) suppose that $(2^{n+1},j)\in Y$. Proposition \[CCLprop\] implies that $j$ is even and that $(2^n,j/2)\in Y$. The inductive hypothesis now implies that $j/2=2^n$, so $j=2^{n+1}$. To prove (\[ufortau2\]) suppose that $(2^{n+1}+1,j)\in Y$. Proposition \[CCLprop\] implies that $j$ is odd and either $(2^{n},(j+1)/2)\in Y$ or $(2^{n}+1,(j+1)/2)\in Y$. In the first case we have already proved that $(j+1)/2=2^n$, so $j=2^{n+1}-1$. In the second case the inductive hypothesis implies that $(j+1)/2 \geq 2^{n-1}+1$, so $j\geq 2^{n}+1$ or $(j+1)/2=1$, so $j=1$. We finally have to evaluate the constant coefficient ${\gamma}_m$ of $u_m$ when $m=2^n+1$. This may be done by considering the defining recurrence relation in the case ${\lambda}=0$, namely ${\gamma}_{r+1}=-\widetilde{c}_r{\gamma}_{r-1}$ subject to ${\gamma}_0=0$ and ${\gamma}_1=1$. This implies that $${\gamma}_{2m+1}=\prod_{r=1}^m \widetilde{c}_{2r}$$ for all $m\in{{\bf N}}$. Therefore ${\gamma}_5={\gamma}_3=-1$ and $$\begin{aligned} {\gamma}_{8m+1}&=&\prod_{r=1}^m\left(\widetilde{c}_{8r}\widetilde{c}_{8r-2}\widetilde{c}_{8r-4}\widetilde{c}_{8r-6}\right)\\ &=&\prod_{r=1}^m\left(\widetilde{c}_{4r}\widetilde{c}_{8r-1}\widetilde{c}_{8r-2}\widetilde{c}_{4r-2}\widetilde{c}_{8r-5}\widetilde{c}_{8r-6}\right)\\ &=&\prod_{r=1}^m\left(\widetilde{c}_{4r}\widetilde{c}_{4r-2}\right)\\ &=& {\gamma}_{4m+1}\end{aligned}$$ for all $m\in{{\bf N}}$. A simple induction now implies that ${\gamma}_{m}=-1$ for $m=2^n+1$ and all $n\geq 1$. \[Ttau2\] If $m=2^n$ and $n\geq 2$ then $$\tau={{\rm tr}}(T_{m,{\lambda}})=v_m+u_{m+1}={\lambda}^m-2\label{tauformula}$$ for all ${\lambda}\in{{\bf C}}$. The proof uses the identity $v_mu_{m+1}-u_mv_{m+1}=1$ of Lemma \[Tdet3\] together with the two identities proved in Lemma \[Ttau1\]. These are identities within the commutative ring ${{\bf Z}}({\lambda})$ of all polynomials with integer coefficents in the indeterminate quantity ${\lambda}$, but they imply similar identities in the commutative ring ${{\bf Z}}(\widehat{{\lambda}})$ of all polynomials with integer coefficients in an indeterminate quantity $\widehat{{\lambda}}$ that satisfies the identity $\widehat{{\lambda}}^{m-1}=0$. (Equivalently one may start by disregarding all terms in the identities that involve ${\lambda}^r$ with $r\geq m-1$.) The identities then simplify to $$\widehat{u}_{m+1} = -1+p,\hspace{2em} \widehat{v}_{m}\widehat{u}_{m+1}= 1,$$ where $$p(\widehat{{\lambda}})=\widehat{{\lambda}}^{m/2}\sum_{r=0}^{m/4} {\alpha}_r \widehat{{\lambda}}^{2r}$$ satisfies $p^{2}=0$ in ${{\bf Z}}(\widehat{{\lambda}})$. The second equation can be solved for $\widehat{v}_m$, yielding $$\widehat{v}_{m}=-1-p$$ and hence $\widehat{\tau}=-2$. Returning to the original variable ${\lambda}$ one deduces that $$\tau=-2+\sum_{r\geq m-1} {\beta}_r {\lambda}^r.$$ But (cf. Lemma \[Ttau1\]) it is easily shown that $v_n$ is an even polynomial of degree $n-2$ for odd $n$. Thus, and by Lemma \[Ttau1\], $\tau =v_m+u_{m+1}$ is an even polynomial of degree $m$ with leading coefficient $1$, so $\tau={\lambda}^m-2$. \[Tdet2\] Following the assumptions and notation of Lemma \[Tdet1\], suppose that ${\delta}=1$ and that there exist $m\in{{\bf Z}}_+$ and $\mu\in{{\bf C}}$ such that $\tau=\mu^m-2$. Then there exists a constant $b$ such that $${\Vert}T^r{\Vert}\leq b\, 4^r \max(|\mu|^{rm},1)$$ for all $r\geq 1$. \ **Case 1.** If $|\mu|\leq 1$ it suffices to obtain bounds on the solutions $z_\pm$ of $z^2-\tau z+1=0$ when $|\tau|\leq 3$. The solutions satisfy $$|z_\pm|= \left| \frac{\tau}{2}\pm \sqrt{\frac{\tau^2}{4} -1}\right| \leq\frac{3}{2}+\frac{\sqrt{13}}{2} <4.$$ Lemma \[Tdet1\] now implies that ${\Vert}T^r{\Vert}\leq b \, 4^r$ for all $r\geq 1$. **Case 2.** If $|\mu|> 1$ it suffices to obtain bounds on the solutions $z_\pm$ of $z^2-\tau z+1=0$ when $|\tau|\leq 3|\mu|^m$. The solutions satisfy $$|z_\pm|= \left| \frac{\tau}{2}\pm \sqrt{\frac{\tau^2}{4} -1}\right| <4|\mu|^m.$$ Lemma \[Tdet1\] now implies that ${\Vert}T^r{\Vert}\leq b \, 4^r|\mu|^{rm}$ for all $r\geq 1$. \[fillin\] Let $X_n$, $n\in{{\bf Z}}$, be a periodic sequence of $2\times 2$ matrices with period $m$ and let $T_r=X_rX_{r-1}\ldots X_1$ for all $r\geq 1$. If there exist constants $b_0,\, {\gamma}$ such that ${\Vert}(T_{m})^s{\Vert}\leq b_0 {\gamma}^{s}$ for all $s\geq 0$, then there exists a constant $b_2$ such that ${\Vert}T_r{\Vert}\leq b_2 {\gamma}^{r/m}$ for all $r\geq 1$. Every $r\in{{\bf Z}}_+$ may be written in the form $r=sm+v$ where $s\geq 0$ and $0\leq v<m$. Using the identity $T_{sm}=(T_m)^s$, one obtains $$\begin{aligned} {\Vert}T_r{\Vert}&=& {\Vert}X_rX_{r-1}\ldots X_{sm+1}T_{sm}{\Vert}\\ &=& {\Vert}X_vX_{v-1}\ldots X_1(T_{m})^s{\Vert}\\ &\leq & {\Vert}X_vX_{v-1}\ldots X_1{\Vert}b_0{\gamma}^{sm}\\ &\leq & b_0b_1 {\gamma}^{s}\\ &\leq & b_2 {\gamma}^{r/m},\end{aligned}$$ where $b_2=b_0b_1$ and $$b_1=\max_{0\leq v\leq m-1}\{ {\Vert}X_vX_{v-1}\ldots X_1{\Vert}\}.$$ \[maintheorem6\] One has $$\{{\lambda}:|{\lambda}|\leq 1\}\backslash H_{\sigma}\subseteq S_{\sigma}\label{strongform}$$ for all ${\sigma}\in (0,1)$. Given ${\sigma}\in (0,1)$ we put $m=2^d$ where $d\in {{\bf N}}$ is large enough to yield $${\sigma}^{1/2} < h=4^{-1/m}.\label{criticalmbound}$$ We use the identities $${\delta}=\det(T_{m,\mu})=1$$ and $$\tau={{\rm tr}}(T_{m,\mu})=\mu^{m}-2$$ proved in Lemmas \[Tdet3\] and \[Ttau2\] and valid for all $\mu\in{{\bf C}}$. Let $c\in{\Omega}_{\sigma}$ be the periodic sequence with period $m$ such that $c_n={\sigma}\widetilde{c}_n$ for all $1\leq n\leq m$. The main task is to prove that if $|{\lambda}|<h$ then all solutions $\xi:{{\bf Z}}\to{{\bf C}}$ of $$\xi_{n+1}={\lambda}\xi_n-c_n\xi_{n-1}\label{xiiterate}$$ decay exponentially as $n\to+\infty$. This will imply, by Lemma \[BIO\], and using the notations of that lemma, that $$I_c \supset \{{\lambda}:|{\lambda}| < h\}.$$ Arguing as in the proof of Theorem \[maintheorem3\], it will then follow from Lemma \[lem\_two\_periods\] and Proposition \[quasiergodic\] that $$S_{\sigma}\supset \{{\lambda}:|{\lambda}| \leq h\} \setminus H_{\sigma},$$ this holding for any $h=4^{-1/m}$ such that [(\[criticalmbound\])]{} holds and $m=2^d$, so that $$S_{\sigma}\supset \{{\lambda}:|{\lambda}| < 1\} \setminus H_{\sigma}.$$ Since $S_{\sigma}$ is closed, [(\[strongform\])]{} will follow. Thus it remains only to show that all solutions of [(\[xiiterate\])]{} decay exponentially at $+\infty$. To see that this holds, define $x_n={\sigma}^{-n/2}\xi_n$ and $\mu={\sigma}^{-1/2}{\lambda}$ so that (\[xiiterate\]) may be rewritten in the form $$x_{n+1}=\mu x_n-\widetilde{c}_nx_{n-1}$$ for $1\leq n\leq m$. Where $\theta=\max(1,|\mu|)$, Lemma \[Tdet2\] now yields $${\Vert}(T_{m,\mu})^r{\Vert}\leq b\, 4^r\theta^{rm}$$ for all $r\in {{\bf N}}$. Lemma \[fillin\] with ${\gamma}=4\theta^m$ implies $${\Vert}T_{r,\mu}{\Vert}\leq b\, 4^{r/m}\theta^r,$$ and hence $$|x_r|\leq b_3\, 4^{r/m}\theta^r,$$ again for all $r\in{{\bf N}}$. Hence, where $\phi = \max({\sigma}^{1/2},|{\lambda}|)$, $$|\xi_r|\leq b_3\, 4^{r/m}\theta^r{\sigma}^{r/2}=b_3 \left( \phi h^{-1}\right)^r$$ for all $r\in {{\bf N}}$. Since $0<\phi< h$, it follows that $\xi$ decays exponentially. Semi-infinite and finite matrices ================================= All our results so far have focused on calculations of the spectrum of the bi-infinite matrix $A_c$. In this final section we say something about the spectrum of the semi-infinite matrix $$A^+_c= \left(\begin{array}{cccc} 0&1&&\\ c_1&0&1&\\ & c_{2}&0&\ddots\\ &&\ddots&\ddots \end{array}\right)$$ in the case that $c=(c_1,c_2,...)\in \{\pm \sigma\}^{{\bf N}}$ is pseudo-ergodic (contains every finite sequence of $\pm \sigma$’s as a consecutive sequence). We also say something (though have mainly unanswered questions) about the finite $N\times N$ matrices $$\begin{aligned} A^{(N)}_c&=& \left(\begin{array}{ccccc} 0&1&&\\ c_1&0&1&\\ &c_{2}&0&\ddots\\ &&\ddots&\ddots& 1\\ &&& c_{N-1} & 0 \end{array}\right)\end{aligned}$$ and $$\begin{aligned} A^{(N,\mathrm{per})}_{c,\alpha}&=& \left(\begin{array}{ccccc} 0&1&&& \alpha c_N\\ c_1&0&1&\\ &c_{2}&0&\ddots\\ &&\ddots&\ddots& 1\\ \alpha^{-1} &&& c_{N-1} & 0 \end{array}\right).\label{finitematrix2}\end{aligned}$$ Here $A^{(N)}_c$ is tridiagonal, $A^{(N,\mathrm{per})}_c$ is tridiagonal except for “periodising” entries in row $1$ column $N$ and row $N$ column $1$ (in these entries we assume that $|\alpha|=1$), and each $c_j = \pm {\sigma}$: we have in mind particularly the random case where the $c_j$’s are independent and identically distributed random variables taking the values $\pm \sigma$. Our main result on the spectrum of $A_c$, proved in the previous sections, is that it contains the set $\{\lambda:|\lambda| \leq 1\}\setminus H_\sigma$. We suspect that $H_\sigma$ is a genuine hole in the spectrum for $0<\sigma<1$, i.e. that $H_\sigma\cap S_\sigma = \emptyset$. We have not shown this result but have shown in Lemma \[lemmaA\] the weaker result that $\{\lambda:|\lambda|< 1-\sigma\}\cap S_\sigma = \emptyset$. Our first result in this section is that this hole is not present in the spectrum of the semi-infinite matrix. The proof depends on recent results on semi-infinite pseudo-ergodic operators due to Lindner and Roch [@LR10], derived using characterisations of the index of Fredholm operators, whose matrix representations are banded semi-infinite matrices, in terms of so-called “plus indices” of limit operators, these characterisations derived using $K$-theory results for $C^*$-algebras in [@RRJ]. \[thm:specsemi\] Suppose $c\in \{\pm \sigma\}^{{{\bf N}}}$ is pseudo-ergodic. If $\sigma=1$ then ${{\rm Spec}}(A_c^+) = S$. For all $\sigma\in (0,1]$, $\{\lambda:|\lambda|\leq 1\}\subset {{\rm Spec}}( A_c^+)$. In the case that $\sigma=1$ it is shown in [@CCL2] that ${{\rm Spec}}(A_c^+)=S$. Thus, for $\sigma=1$, $$\{\lambda:|\lambda|\leq 1\}\subset S= {{\rm Spec}}(A_c^+)$$ follows from Theorem \[maintheorem\] (or [@CCL Theorem 2.3]). For all $\sigma\in (0,1]$ it follows from [@LR10 Theorem 2.1] that the essential spectrum of $A_c^+$, i.e. the set of $\lambda\in{{\bf C}}$ for which $A_c^+-\lambda I^+$ is not Fredholm (here $I^+$ is the identity operator on $\ell^2({{\bf N}})$), is the set $S_\sigma$. Thus and by Theorem \[maintheorem3\], $$(\{\lambda:|\lambda| \leq 1\}\setminus H_\sigma)\subset S_\sigma \subset {{\rm Spec}}(A_c^+).$$ It remains to show that $H_\sigma\subset {{\rm Spec}}(A_c^+)$. But, applying [@LR10 Theorem 2.4] (note that the set $E_-(U,W)$ in the notation of [@LR10 Theorem 2.4] is precisely the set $H_\sigma$ for this operator), it follows that, for $\lambda \in H_\sigma$, either $A_c^+-\lambda I^+$ is not Fredholm or $A_c^+-\lambda I^+$ is Fredholm with index 1: in either of these cases $\lambda\in {{\rm Spec}}(A_c^+)$. Our other result in this section is to say something about the spectra (sets of eigenvalues) of the finite matrices $A^{(N)}_c$ and $A^{(N,\mathrm{per})}_{c,\alpha}$. The notations $\pi_{N,{\sigma}}$ and $\pi_{\infty,{\sigma}}$ are as defined in and above equation [(\[eq:piinf\])]{} (and $\pi_N$ and $\pi_\infty$ are our abbreviations for $\sigma=1$). \[thmfinite\] If $0<{\sigma}\leq 1$, $|\alpha|=1$ and $c\in \{\pm{\sigma}\}^N$, then $${{\rm Spec}}(A^{(N,\mathrm{per})}_{c,\alpha}) \subset \pi_{N,{\sigma}}\subset \pi_{\infty,{\sigma}}\subset S_{\sigma}$$ while $${{\rm Spec}}(A^{(N)}_c)\subset \sqrt{{\sigma}} \pi_{2N+2}\subset \sqrt{{\sigma}} \pi_\infty\subset \sqrt{{\sigma}}S.$$ If $\lambda = x+{{\rm i}}y$ is an eigenvalue of $A^{(N,\mathrm{per})}_{c,\alpha}$ then $1-{\sigma}\leq |\lambda|\leq 1+{\sigma}$ and $|x|+|y|\leq \sqrt{2(1+{\sigma}^2)\,}$, while if $\lambda$ is an eigenvalue of $A^{(N)}_c$ then $|x|+|y|\leq 2\sqrt{{\sigma}}$. The first of these statements is clear from the definition of $\pi_{N,{\sigma}}$, [(\[spectrumperiodic\])]{}, and Proposition \[quasiergodic\] which gives that $\pi_{\infty,{\sigma}}\subset S_{\sigma}$. The second of these statements is shown for ${\sigma}=1$ in [@CCL2 Theorem 4.1]. The second statement follows for $0<{\sigma}<1$ by the observation that, where $d\in \{\pm 1\}^N$, $c={\sigma}d\in \{\pm {\sigma}\}^N$, and $D_N$ is the diagonal matrix with leading diagonal $(1, {\sigma}^{1/2},{\sigma},...,{\sigma}^{(N-1)/2})$, it holds that $ D_N^{-1} A^{(N)}_c D_N = \sqrt{{\sigma}}\,A^{(N)}_d. $ The last sentence then follows from Lemma \[lemmaA\]. Note that in the last sentence of the above theorem the condition $|x|+|y|\leq 2\sqrt{{\sigma}}$ implies both that $|\lambda|\leq 1+{\sigma}$ and that $|x|+|y|\leq \sqrt{2(1+{\sigma}^2)\,}$. In Figure \[fig:rand2000\_09025\] we plot the spectra of $A_c^{(N)}$ and $A^{(N,\mathrm{per})}_{c,\alpha}$ for $N=2000$ and $\alpha=1$ for a typical realisation with the entries $c\in \{\pm {\sigma}\}^N$ randomly chosen with the $c_j$ independently and identically distributed with $\mathrm{Pr}(c_j = {\sigma})=0.5$ and ${\sigma}= 0.9025$ so that $\sqrt{{\sigma}}=0.95$ (the several other realisations we have computed are very close in appearance to these plots). Theorem \[thmfinite\] tells us that ${{\rm Spec}}(A_c^{(N)})\subset 0.95 \pi_\infty \subset 0.95 S$ and that ${{\rm Spec}}(A^{(N,\mathrm{per})}_{c,\alpha})\subset S_{0.9025}$, and that if $\lambda=x+{{\rm i}}y$ is an eigenvalue of $A^{(N)}_c$ then $|x|+|y|\leq 1.9$, while if $\lambda$ is an eigenvalue of $A^{(N,\mathrm{per})}_{c,\alpha}$ then $0.075\leq |\lambda|\leq 1.9025$ and $|x|+|y|\leq \sqrt{2(1+{\sigma}^2)\,}\approx1.905$. It is clear from Figure \[fig:rand2000\_09025\] that Theorem \[thmfinite\] is only the beginning of the story. We observe in the figure a hole in the spectrum of $A^{(N,\mathrm{per})}_{c,\alpha}$, but it is a hole of radius approximately 0.6, not 0.075, with a large proportion of the eigenvalues positioned on the boundary of this hole, while outside the hole the spectra of $A^{(N,\mathrm{per})}_{c,\alpha}$ and $A^{(N)}_c$ appear near identical. The same qualitative behaviour is visible in Figure \[fig:rand2000\_05\], which is a similar plot except that ${\sigma}$ is reduced to 0.5 and we change the probability distribution, making it twice as likely that the entries of the vector $c$ are $-{\sigma}$ rather than ${\sigma}$. This change of probability distribution introduces an asymmetry, in particular an asymmetry in the hole in the spectrum (if we instead compute with $\mathrm{Pr}(c_j = \sigma)=1/2$ then typical realisations have spectra which are approximately invariant under the dihedral symmetry group $D_2$ of the square). Of course our methods, which are not probabilistic, have nothing to say about such asymmetries, indeed nothing, beyond Theorem \[thmfinite\], to say about the almost sure spectra of $A_c^{(N)}$ or $A^{(N,\mathrm{per})}_{c,\alpha}$ as $N\to\infty$. [99]{} A. Böttcher, Infinite matrices and projection methods, pp. 2-74 in “Lectures on Operator Theory and its Applications”, Fields Institute Monographs, ed. Peter Lancaster. Amer. Math. Soc. Publ., Providence, RI, 1995. A. Böttcher, Y. I. Karlovich, Carleson curves, Muckenhoupt weights, and Toeplitz operators, Basel, Birkhäuser, 1997. A. Böttcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, New York, Springer-Verlag, 1999. A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, 2nd Ed., New York, Springer-Verlag, 2006. S. N. Chandler-Wilde, R. Chonchaiya and M. Lindner, Eigenvalue problem meets Sierpinski triangle: computing the spectrum of a non-self-adjoint random operator, Oper. Matrices, to appear. Preprint at arXiv:1003.3946v3 S. N. Chandler-Wilde, R. Chonchaiya and M. Lindner, On the spectra and pseudospectra of a class of non-self-adjoint random matrices and operators, submitted for publication, 2011. Preprint at arXiv:1107.0177v1 S. N. Chandler-Wilde and M. Lindner, Limit Operators, Collective Compactness, and the Spectral Theory of Infinite Matrices, Memoirs AMS 210 (2011), No. 989. G. M. Cicuta, M. Contedini and L. Molinari: Non-Hermitian tridiagonal random matrices and returns to the origin of a random walk, J. Stat. Phys. 98 (2000), 685-699. E. B. Davies, Spectral properties of random non-self-adjoint matrices and operators, Proc. Roy. Soc. London A 457 (2001) 191-206. E. B. Davies, Spectral theory of pseudo-ergodic operators, Commun. Math. Phys. 216 (2001) 687-704. E. B. Davies, Linear Operators and Their Spectra, Camb. Univ. Press, Cambridge, 2007. E. B. Davies, B. Simon, Scattering theory for systems with different spatial asymptotics on the left and right, Commun. Math. Phys. 63 (1978) 277-301. J. Feinberg and A. Zee, Non-Hermitean localization and de-localization, Phys. Rev. E 59 (1999), 6433-6443. J. Feinberg and A. Zee, Spectral curves of non-Hermitean Hamiltonians, Nucl. Phys. B 552 (1999), 599-623. I. Y. Goldsheid, B. A. Khoruzhenko, Distribution of eigenvalues in non-Hermitian Anderson model, Phys. Rev. Lett. 80 (1998) 2897-2901. I. Y. Goldsheid, B. A. Khoruzhenko, Thouless formula for random non-Hermitian Jacobi matrices, Israel J. Math. 148 (2005) 331-346. D. E. Holz, H. Orland and A. Zee, On the remarkable spectrum of a non-Hermitian random matrix model, J. Phys. A, Math. and General, 36 (2003), 3385-3400. M. Lindner, Infinite Matrices and their Finite Sections: An Introduction to the Limit Operator Method, Birkhäuser, 2006. M. Lindner and S. Roch, Finite sections of random Jacobi operators, SIAM J. Numer. Anal., to appear. Preprint at arXiv:1011.0907v1 V. S. Rabinovich, S. Roch and J. Roe, Fredholm indices of band-dominated operators, Integral Equations Operator Theory 49 (2004), 221-–238. V. S. Rabinovich, S. Roch, and B. Silbermann, Limit Operators and Their Applications in Operator Theory, Birkhäuser, 2004.

$\mathbf{Salah\ Menouar}$[^1]$^*$, $\mathbf{Mustapha\ Maamache}^{1}$, $\mathbf{and\ Jeong\ Ryeol\ Choi}$[^2]$^\dagger$  $^{1}$*Laboratoire de Physique Quantique et Systèmes Dynamiques,* *  Département de Physique, Faculté des Sciences,* *  Université Ferhat Abbas de Sétif, Sétif 19000, Algeria* $\ \ ^{2}$*Division of Semiconductor and Display Engineering,*   *College of IT Engineering, Kyungpook National University,*    *1370 Sankguk-dong, Buk-gu, Daegu 702-701, Republic of Korea*    [Abstract ]{} The dynamics of time-dependent coupled oscillator model for the charged particle motion subjected to a time-dependent external magnetic field is investigated. We used canonical transformation approach for the classical treatment of the system, whereas unitary transformation approach is used when managing the system in the framework of quantum mechanics. For both approaches, the original system is transformed to a much more simple system that is the sum of two independent harmonic oscillators which have time-dependent frequencies. We therefore easily identified the wave functions in the transformed system with the help of invariant operator of the system. The full wave functions in the original system is derived from the inverse unitary transformation of the wave functions associated to the transformed system. **Keywords:*** charged particle motion; unitary transformation; canonical transformation; time-dependent coupled oscillator* **PACC numbers**: 0365G,  0365D, 4190 Introduction ============ The time-dependent harmonic oscillators have attracted considerable interest in the literature thanks to their usefulness in describing the dynamics of many physical systems. After the Bateman’s[@bate] proposition concerning the use of time-dependent harmonic oscillator model in describing dissipative systems, much attention was paid to quantum behavior of nonconservative and nonlinear systems. In the meantime, coupled oscillators have emerged to become powerful modeling tools and, consequently, are frequently used in modeling wide range of physical phenomena. With the progress of research, one may be interested in what would happen if two-dimensional harmonic oscillator is elaborated through the coupling of two additive potentials? As far as we know, dealing with such an issue was set thirty years ago by Kim et al. \[2-5\]. Abdalla demonstrated how to treat the time-dependent coupled oscillators in the context of quantum mechanics[@mscb]. The propagator for a time-dependent coupled and driven harmonic oscillators with time-varying frequencies and masses is investigated by Benamira [@benami] using path integral methods. Among various systems that can be modeled by time-dependent coupled oscillators, the dynamics of charged particle motion in the presence of time-varying magnetic fields has played an important role in condensed matter physics and plasma physics. There are plenty of applications for this system such as magnetoresistance[@bykov], the Aharonov-Bohm effect[wgv]{}, magnetic confinement devices for fusion plasmas[@varma], electromagnetic lenses with variable magnetic fields[@calvo], cyclotron resonance[@kenn], and entanglement of a two-qubit Heisenberg XY model[@sadi]. Though all of these problems are interesting, we can find their exact analytic solutions only for a few special cases due to their complex mathematical structures. The quantum properties of a free electron, which have a time-dependent effective mass under the influence of external magnetic field, are investigated in both the Landau and the symmetric gauges [@choi2; @smen]. Laroze and Rivera[@laroz] studied the dynamical behavior of electrons in the presence of a uniform time-dependent magnetic field and they presented the time evolution of the corresponding wave functions for the case that the initial state is a superposition of Landau levels. The propagators of a charged particle subjected to a time-dependent magnetic field are studied using the linear and the quadratic invariants [@abdella3]. Kim et al. \[2-5\] proposed a problem that what actually would take place if two harmonic oscillators are coupled so that the potential becomes $V(X_{1},X_{2})=\frac{1}{2}\left( c_{1}X_{1}^{2}+c_{2}X_{2}^{2}+c_{3}X_{1}X_{2}\right) $ where $c_{3}$ is a coupling constant. They studied the corresponding density matrix in order to establish the Wigner function. In this work, we are interested in the problem of Hamiltonian that involves the coupling term $X_{1}X_{2}$ in the presence of magnetic field. This system can be regarded as the generalization of the Hamiltonian model given in Refs. [@choi2] and [gel1]{}. Though the coupling of two or more oscillators is among the most basic concepts in dealing with gyroscopic motions, interactions, and complex structures, the related theory has been scarcely developed so far. This class of coupled harmonic oscillators can be used to describe numerous physical systems. Some of them are the Bogoliubov transformation model of superconductivity [@kim7], two-mode squeezed light [@cm], and the Lee model in quantum field theory [@ss]. One of the main focuses of research carried out by Zhang *et al*. in connection with time-dependent coupled oscillators including $X_{1}X_{2}$ term are some specific problems of time-dependent coupled electronic circuits[choi5,choi6]{}. We will use the invariant methods[@lewis3; @lej] in order to derive the exact wave functions for time-dependent coupled oscillators in a variable magnetic field. The invariant operator method in describing the quantum features of time-dependent harmonic oscillators is firstly introduced by Lewis[lewis3]{} and now became a very useful tool in developing quantum theory for the case where the Hamiltonian of the system is explicitly dependent on time. In Sec. 2, we formulate our problem by introducing a general time-dependent Hamiltonian describing the complicated motion of a charged particle in the presence of an arbitrary time-dependent magnetic field. Classical treatment of the system is presented in Sec. 3, on the basis of the canonical transformation method. Quantum analysis of the system is carried out in Sec. 4 using unitary transformation approach. The unitary transformation enables us to transform the original Hamiltonian (that is somewhat complicated) to that of a more simple system such as ordinary harmonic oscillator. We derive the quantum solutions of the system in Sec. 5 starting from the invariant operator associated to the transformed system described in Sec. 4. Finally, we give concluding remarks in the last section. Formulation of the problem ========================== For the dynamical system of our interest, the Hamiltonian has the form: $$H(X_{1},X_{2},t)=\frac{\Pi _{1}^{2}}{2m_{1}(t)}+\frac{\Pi _{2}^{2}}{2m_{2}(t)}+\frac{1}{2}\left( C_{1}(t)X_{1}^{2}+C_{2}(t)X_{2}^{2}+C_{3}(t)X_{1}X_{2}\right) ,$$ where $\ \Pi _{1}$ and $\Pi _{2}$ are the conjugate momenta. Note that $\ \Pi _{1}$ and $\Pi _{2}$ can be simplified by choosing an appropriate gauge. Actually, in the symmetric gauge with $\overrightarrow{A}\big(\frac{-B(t)}{2}X_{2},\frac{B(t)}{2}X_{1},0\big)$, they are given by$$\Pi _{1}=P_{1}-\frac{eB(t)}{2}X_{2}\ ,\ \Pi _{2}=P_{2}+\frac{eB(t)}{2}X_{1}.$$The parameters $m_{1}(t),$ $m_{2}(t)$, $C_{1}(t)$, $C_{2}(t)$, and $C_{3}(t)$ are arbitrary functions of time, $(X_{1},X_{2})$ are the pair of position variables, and $(P_{1}, P_{2})$ are the canonical conjugate momentum variables. The main difference of our study from that of Ref. [@laroz] is that we considered the coupling term $X_1 X_2$ in the Hamiltonian. Regarding the expressions of $\ \Pi _{1}$ and  $\Pi _{2}$, the Hamiltonian in Eq. (1) can be recasted into $$\begin{aligned} H(X_{1},X_{2},t) &=&\frac{P_{1}^{2}}{2m_{1}(t)}+\frac{P_{2}^{2}}{2m_{2}(t)}+\frac{1}{2}\left( c_{1}(t)X_{1}^{2}+c_{2}(t)X_{2}^{2}+c_{3}(t)X_{1}X_{2}\right) \notag \\ &&+\frac{1}{2}\left( \omega _{2c}(t)P_{2}X_{1}-\omega _{1c}(t)P_{1}X_{2}\right) ,\end{aligned}$$ where the new time-dependent functions $c_{1}(t)$, $c_{2}(t)$ and $c_{3}(t)$ are read $$c_{1}(t)=C_{1}(t)+m_{2}(t)\frac{\omega _{2c}^{2}(t)}{4},\text{\ \ }c_{2}(t)=C_{2}(t)+m_{1}(t)\frac{\omega _{1c}^{2}(t)}{4},\text{\ \ }c_{3}(t)=C_{3}(t),$$with the cyclotron frequencies $$\omega _{1c}(t)=\frac{eB(t)}{m_{1}(t)},\text{ \ \ \ \ }\omega _{2c}(t)=\frac{eB(t)}{m_{2}(t)}.$$ Classical treatment =================== The time-dependent canonical transformation approach is in fact very powerful in investigating the properties of dynamical systems described by a time-dependent Hamiltonian. In many cases, we can convert a given Hamiltonian into a simple and desired one by means of the canonical transformation. Therefore, in order to recast the solutions of this problem into a more soluble form, it is convenient to use the canonical transformation method. To simplify the Hamiltonian given in Eq. (3), let us transform the variables $(X_{1},X_{2},P_{1},P_{2})$  to the new variables $(x_{1},x_{2},p_{x_{1}},p_{x_{2}})$ such that $$x_{1}=\left( \frac{m_{1}(t)}{m_{2}(t)}\right) ^{1/4}X_{1},\text{ \ \ \ }x_{2}=\left( \frac{m_{2}(t)}{m_{1}(t)}\right) ^{1/4}X_{2},$$$$p_{x_{1}}=\left( \frac{m_{2}(t)}{m_{1}(t)}\right) ^{1/4}P_{1},\text{ \ \ \ }p_{x_{2}}=\left( \frac{m_{1}(t)}{m_{2}(t)}\right) ^{1/4}P_{2}.$$Replacing all of the canonical variables in Eq. (3) with the above ones, we have$$\begin{aligned} H(x_{1},x_{2},t) &=&\frac{1}{2m(t)}\left( p_{x_{1}}^{2}+p_{x_{2}}^{2}\right) +\frac{1}{2}\left( d_{1}(t)x_{1}^{2}+d_{2}(t)x_{2}^{2}+d_{3}(t)x_{1}x_{2}\right) \notag \\ &&+\frac{\omega _{c}(t)}{2}\left( x_{1}p_{x_{2}}-x_{2}p_{x_{1}}\right) ,\end{aligned}$$where $d_{1}-d_{3}$ are new time-dependent functions of the form$$\begin{aligned} d_{1}(t)&=&c_{1}(t)\left( \frac{m_{2}(t)}{m_{1}(t)}\right) ^{1/2}=\left( C_{1}(t)+\frac{1}{4}m_{2}(t)\omega _{2c}^{2}(t)\right) \left( \frac{m_{2}(t)}{m_{1}(t)}\right) ^{1/2},~~~ \\ d_2(t)&=&c_2(t)\left( \frac{m_{1}(t)}{m_{2}(t)}\right) ^{1/2}=\left( C_{2}(t)+\frac{1}{4}m_{1}(t)\omega _{1c}^{2}(t)\right) \left( \frac{m_{1}(t)}{m_{2}(t)}\right) ^{1/2},~~~\end{aligned}$$$$d_{3}(t)=c_{3}(t)=C_{3}(t),$$with the unique mass $m(t)=\left( m_{1}(t)m_{2}(t)\right) ^{1/2}$ and the cyclotron frequency $\omega _{c}(t)=\left( \omega _{1c}(t)\omega _{2c}(t)\right) ^{1/2}=eB(t)/m(t).$ To simplify the Hamiltonian of Eq. (8), we perform the following canonical transformation $$\begin{aligned} \binom{x_{1}}{x_{2}}&=&\left( \begin{array}{cc} \cos \phi (t) & \sin \phi (t) \\ -\sin \phi (t) & \cos \phi (t)\end{array}\right) \binom{q_{1}}{q_{2}}, \\ \binom{p_{x_{1}}}{p_{x_{2}}}&=&\left( \begin{array}{cc} \cos \phi (t) & \sin \phi (t) \\ -\sin \phi (t) & \cos \phi (t)\end{array}\right) \binom{p_{1}}{p_{2}},\end{aligned}$$where$$\phi (t)=-\frac{1}{2}\int \omega _{c}(t)dt.$$If $(q_{1},q_{2},p_{1},p_{2})$ are canonical coordinates, there should exist a new Hamiltonian $H(q_{1},q_{2},t)$ which is determined by only in terms of the Hamiltonian given in Eq. (8) with the aid of the linear transformation shown in Eqs. (12) and (13). The variables $\left( x_{1},x_{2},p_{x_{1}},p_{x_{2}}\right) $ and $(q_{1},q_{2},p_{1},p_{2})$ in two representations must satisfy the following relation [@gold]$$(p_{1}\dot{q}_{1}+p_{2}\dot{q}_{2}-H(q_{1},q_{2},t)=p_{x_{1}}\dot{x}_{1}+p_{x_{2}}\dot{x}_{2}-H(x_{1},x_{2},t)+\frac{\partial F_{1}}{\partial t},$$where $F_{1}$ is a time-dependent generating function in phase space, which should be determined afterwards. From the fundamental equations known in classical mechanics [@gold]$$\begin{aligned} p_{x_{1}} &=&\frac{\partial }{\partial x_{1}}F_{1}\left( x_{1},x_{2},p_{1},p_{2},t\right) ,~~~~q_{1}=\frac{\partial }{\partial p_{1}}F_{1}\left( x_{1},x_{2},p_{1},p_{2},t\right) , \\ p_{x_{2}} &=&\frac{\partial }{\partial x_{2}}F_{1}\left( x_{1},x_{2},p_{1},p_{2},t\right) ,~~~~q_{2}=\frac{\partial }{\partial p_{2}}F_{1}\left( x_{1},x_{2},p_{1},p_{2},t\right) ,\end{aligned}$$the generating function associated with the transformation is found to be$$F_{1}\left( x_{1},x_{2},p_{1},p_{2}t\right) =\left( p_{1}\cos \phi +p_{2}\sin \phi \right) x_{1}+\left( -p_{1}\sin \phi +p_{2}\cos \phi \right) x_{2},$$$$\frac{\partial F_{1}}{\partial t}=-\dot{\phi}(t)\left( x_{1}p_{x_{2}}-x_{2}p_{x_{1}}\right) =-\frac{\varpi _{c}(t)}{2}\left( x_{1}p_{x_{2}}-x_{2}p_{x_{1}}\right) .$$In terms of the new conjugate variables $(q_{1},q_{2},p_{1},p_{2}),$ the Hamiltonian of Eq. (8) becomes$$H(q_{1},q_{2},t)=\frac{1}{2m(t)}\left( p_{_{1}}^{2}+p_{_{2}}^{2}\right) +\frac{1}{2}\left( \lambda _{1}(t)q_{1}^{2}+\lambda _{2}(t)q_{2}^{2}+\lambda _{3}(t)q_{1}q_{2}\right) ,$$where$$\begin{aligned} & &\lambda _{1}(t)=d_{1}(t)\cos ^{2}\phi +d_{2}(t)\sin ^{2}\phi -d_{3}(t)\sin \phi \cos \phi \text{,} \\ & &\lambda _{2}(t)=d_{2}(t)\cos ^{2}\phi +d_{1}(t)\sin ^{2}\phi +d_{3}(t)\sin \phi \cos \phi \text{,} \\ & &\lambda _{3}(t)=2\left( d_{1}(t)-d_{2}(t)\right) \sin \phi \cos \phi +d_{3}(t)\left( \cos ^{2}\phi -\sin ^{2}\phi \right) \text{.}\end{aligned}$$To eliminate the coupling term $q_{1}q_{2},$ we now perform the following canonical transformation [@benami; @choi5; @choi6] $$\binom{q_{1}}{q_{2}}=\frac{1}{\sqrt{m(t)}}\left( \begin{array}{cc} \cos \frac{\theta (t)}{2} & \sin \frac{\theta (t)}{2} \\ -\sin \frac{\theta (t)}{2} & \cos \frac{\theta (t)}{2}\end{array}\right) \binom{Q_{1}}{Q_{2}},$$$$\binom{p_{1}}{p_{2}}=\sqrt{m(t)}\left( \begin{array}{cc} \cos \frac{\theta (t)}{2} & \sin \frac{\theta (t)}{2} \\ -\sin \frac{\theta (t)}{2} & \cos \frac{\theta (t)}{2}\end{array}\right) \binom{P_{1}}{P_{2}}-\left( \begin{array}{cc} \frac{\dot{m}(t)}{2} & 0 \\ 0 & \frac{\dot{m}(t)}{2}\end{array}\right) \binom{q_{1}}{q_{2}}.$$where $\theta(t)$ is an arbitrary function of time. Note that Eqs. (24) and (25) do not always represent the canonical transformation [@gold] between variables $\left( q_{i},p_{i}\right) [i=1,2]$ and $\left( Q_{i},P_{i}\right) $. If $\left( Q_{i},P_{i}\right) $ are canonical coordinates, there should exist a new Hamiltonian which is determined only by the Hamiltonian of Eq. (20) and the linear transformation given in Eqs. (24) and (25). The relation between variables $\left( q_{i},p_{i}\right) $ and $\left( Q_{i},P_{i}\right) $ in the two representations are [@gold] $$\sum_{i=1}^{2}P_{i}\dot{Q}_{i}-H_{Q}=\sum_{i=1}^{2}{}p_{i}\dot{q}_{i}-H_{q}+\frac{\partial F}{\partial t},$$where $F$ is an another time-dependent generating function in phase space. Using the basic equations $$p_{i}=\frac{\partial }{\partial q_{i}}F\left( q_{1},q_{2},P_{1},P_{2},t\right) ,~~~Q_{i}=\frac{\partial }{\partial P_{i}}F\left( q_{1},q_{2},P_{1},P_{2},t\right) ,$$where $i=1,2,$ we see that the generating function is given by$$\begin{aligned} F\left( q_{1},q_{2},P_{1},P_{2},t\right) &=&\sqrt{m(t)}\left( P_{1}\cos \frac{\theta (t)}{2}+P_{2}\sin \frac{\theta (t)}{2}\right) q_{1} \notag \\ &&+\sqrt{m(t)}\left( -P_{1}\sin \frac{\theta (t)}{2}+P_{2}\cos \frac{\theta (t)}{2}\right) q_{2} \notag \\ &&-\frac{1}{4}\dot{m}(t)\left( q_{1}^{2}+q_{2}^{2}\right) .\end{aligned}$$Then, in terms of the new conjugate variables $\left( Q_{i},P_{i}\right) $, the Hamiltonian can be represented in the form $$\begin{aligned} H_{Q}(Q_{1},Q_{2},t) &=&\frac{1}{2}\left( P_{1}^{2}+P_{2}^{2}\right) +\frac{1}{2}\Omega _{1}^{2}(t)Q_{1}^{2}+\frac{1}{2}\Omega _{2}^{2}(t)Q_{2}^{2} \notag \\ &&+\frac{\dot{\theta}(t)}{2}\left[ P_{1}Q_{2}-P_{2}Q_{1}\right] +\delta (t)Q_{1}Q_{2}.\end{aligned}$$Here, the time-dependent coefficients $\Omega _{1}(t),\Omega _{2}(t)$ and $\delta (t)$ are given by$$\Omega _{1}(t)=\left( \tilde{\omega}_{1}^{2}(t)\cos ^{2}\frac{\theta (t)}{2}+\tilde{\omega}_{2}^{2}(t)\sin ^{2}\frac{\theta (t)}{2}-\frac{\lambda _{3}(t)\sin \theta (t)}{m(t)}\right) ^{1/2},$$$$\Omega _{2}(t)=\left( \tilde{\omega}_{1}^{2}(t)\sin ^{2}\frac{\theta (t)}{2}+\tilde{\omega}_{2}^{2}(t)\cos ^{2}\frac{\theta (t)}{2}+\frac{\lambda _{3}(t)\sin \theta (t)}{m(t)}\right) ^{1/2},$$$$\delta (t)=\frac{1}{2}\left( \tilde{\omega}_{1}^{2}(t)-\tilde{\omega}_{2}^{2}(t)\right) \sin \theta (t)+\frac{\lambda _{3}(t)\cos \theta (t)}{m(t)},$$where$$\begin{aligned} \tilde{\omega}_{1}^{2}(t) &=&\frac{\lambda _{1}(t)}{m(t)}+\frac{1}{4}\left( \frac{\dot{m}^{2}(t)}{m^{2}(t)}-2\frac{\ddot{m}(t)}{m(t)}\right) , \\ \tilde{\omega}_{2}^{2}(t) &=&\frac{\lambda _{2}(t)}{m(t)}+\frac{1}{4}\left( \frac{\dot{m}^{2}(t)}{m^{2}(t)}-2\frac{\ddot{m}(t)}{m(t)}\right) .\end{aligned}$$ If we take the choice $\theta (t)=\mathrm{Const},$ the terms $P_{1}Q_{2}$ and $P_{2}Q_{1}$ in Eq. (29) are canceled out so that the Hamiltonian becomes$$H_{Q}(Q_{1},Q_{2},t)=\frac{1}{2}\left( P_{1}^{2}+P_{2}^{2}\right) +\frac{1}{2}\Omega _{1}^{2}(t)Q_{1}^{2}+\frac{1}{2}\Omega _{2}^{2}(t)Q_{2}^{2}+\delta (t)Q_{1}Q_{2}.$$ Notice that, with the above canonical transformation, the coupling $\delta (t)$ is a functional on the parameters of the original system. It is hence clear that the separation of variables in Eq. (35) requires that $\delta (t)=0$, i.e.$$\lambda _{3}(t)=\left( \tilde{\omega}_{2}^{2}(t)-\tilde{\omega}_{1}^{2}(t)\right) m(t)\tan \theta ,$$and consequently$$\tan \theta =\frac{\lambda _{3}(t)}{m(t)\left( \tilde{\omega}_{2}^{2}(t)-\tilde{\omega}_{1}^{2}(t)\right) }.$$ By taking into account Eq. (36), the Hamiltonian in Eq. (35) is rewritten as$$H_{Q}(Q_{1},Q_{2},t)=\frac{1}{2}\left( P_{1}^{2}+P_{2}^{2}\right) +\frac{1}{2}\Omega _{1}^{2}(t)Q_{1}^{2}+\frac{1}{2}\Omega _{2}^{2}(t)Q_{2}^{2}.$$Then, Eq.(38) represents the sum of two independent Hamiltonians of the simple harmonic oscillators with the time-dependent frequencies $\ \Omega _{1}(t)$ and $\Omega _{2}(t)$. Quantum treatment ================= The canonical transformations in classical mechanics, treated in the previous section, is the analogous of the unitary transformations in quantum mechanics. Now we are going to demonstrate this relationship between the two transformations and confirm how to obtain the quantum-mechanical Hamiltonian from the classical one. To manage the system in the context of quantum physics, we replace the canonical variables $\left( X_{1},X_{2}\right) $ in Eq. (3) by quantum operators $(\hat{X}_{1},\hat{X}_{2})$. Then the corresponding Hamiltonian has the form $$\begin{aligned} \hat{H}(\hat{X}_{1},\hat{X}_{2},t) &=&\frac{\hat{P}_{1}^{2}}{2m_{1}(t)}+\frac{\hat{P}_{2}^{2}}{2m_{2}(t)}+\frac{1}{2}\left( c_{1}(t)\hat{X}_{1}^{2}+c_{2}(t)\hat{X}_{2}^{2}+c_{3}(t)\hat{X}_{1}\hat{X}_{2}\right) \notag \\ &&+\frac{1}{2}\left( \omega _{2c}(t)\hat{P}_{2}\hat{X}_{1}-\omega _{1c}(t)\hat{P}_{1}\hat{X}_{2}\right) .\end{aligned}$$ In this quantum case, the pair of momentum operators are given by $(\hat{P}_{1}=-i\hbar \partial /\partial X_{1}$, $\hat{P}_{2}=-i\hbar \partial /\partial X_{2})$. The Schrödinger equation in the original system is $$i\hbar \frac{\partial }{\partial t}\Psi (X_{1},X_{2},t)=\hat{H}(\hat{X}_{1},\hat{X}_{2},t)\Psi (X_{1},X_{2},t).$$ To simplify the Hamiltonian in Eq. (39), we perform the unitary transformation such that $$\Psi (X_{1},X_{2},t)=\hat{U}_{1}(t)\psi (X_{1},X_{2},t),$$ where $\hat{U}_{1}(t)$ is a time-dependent unitary operator of the form$$\begin{aligned} \hat{U}_{1}(t) &=&\exp \frac{i}{2\hbar }\left[ (\hat{P}_{1}\hat{X}_{1}+\hat{X}_{1}\hat{P}_{1})\ln \left( \frac{m_{1}(t)}{m_{2}(t)}\right) ^{1/4}\right] \notag \\ &&\times\exp \frac{i}{2\hbar }\left[ (\hat{P}_{2}\hat{X}_{2}+\hat{X}_{2}\hat{P}_{2})\ln \left( \frac{m_{2}(t)}{m_{1}(t)}\right) ^{1/4}\right].\end{aligned}$$In this case, the Hamiltonian, Eq. (39), can be rewritten as$$\begin{aligned} & &\hat{H}_{1}(\hat{X}_{1},\hat{X}_{2},t) =\frac{1}{2m(t)}\left( \hat{P}_{1}^{2}+\hat{P}_{2}^{2}\right) \notag \\ & &~~~~~~~~~~~+\frac{1}{2}\left( d_{1}(t)\hat{X}_{1}^{2}+d_{2}(t)\hat{X}_{2}^{2}+d_{3}(t)\hat{X}_{1}\hat{X}_{2}\right) \notag \\ & &~~~~~~~~~~~+\frac{\omega _{c}(t)}{2}\left( \hat{P}_{2}\hat{X}_{1}-\hat{P}_{1}\hat{X}_{2}\right).\end{aligned}$$ It is easy to confirm that the commutation relations, $[\hat{L}_{Z}\text{ },\text{ }\hat{X}_{1}^{2}+\hat{X}_{2}^{2}]=0$ and $[\hat{L}_{z}\text{ },\text{ }\hat{P}_{1}^{2}+\hat{P}_{2}^{2}]=0$, are hold where $\hat{L}_{Z}$ is the angular momentum operator. This implies that there are common eigenfunctions between $\hat{L}_{Z}$ and $\hat{X}_{1}^{2}+\hat{X}_{2}^{2}$, and between $\hat{L}_{Z}$ and $\hat{P}_{1}^{2}+\hat{P}_{2}^{2}$. However, $\hat{L}_{Z}$ does not commutes with $\hat{X}_{1}\hat{X}_{2}$: $[\hat{L}_{Z}\text{ },\text{ }\hat{X}_{1}\hat{X}_{2}]\neq 0$, and consequently $[\hat{L}_{Z}\text{ },\text{ }\hat{H}]\neq 0$. If we regard that $\hat{L}_{Z}$ and $\hat{H}$ do not have the same eigenfunctions, it is not possible to simplify the Schrödinger equation $$i\hbar \frac{\partial }{\partial t}\psi (X_{1},X_{2},t)=\hat{H}_{1}(\hat{X}_{1},\hat{X}_{2},t)\psi (X_{1},X_{2},t),$$by decomposing it. However, we can overcome this difficulty through the transformation of the Hamiltonian of Eq. (39) into a simple form by introducing an appropriate unitary transformation operators. In the first step, we perform the following unitary transformation$$\psi (X_{1},X_{2},t)=\hat{U}_{2}(t)\varphi (X_{1},X_{2},t),$$where$$\begin{aligned} \hat{U}_{2}(t) &=&\exp \left( -\frac{i}{2\hbar }\left( \hat{P}_{2}\hat{X}_{1}-\hat{P}_{1}\hat{X}_{2}\right) \int \varpi _{c}(t)dt\right) \notag \\ &=&\exp \left( -\frac{i\hat{L}_{Z}}{2\hbar }\int \varpi _{c}(t)dt\right) .\end{aligned}$$Under this transformation, the Schrödinger equation (41) is mapped into$$i\hbar \frac{\partial }{\partial t}\varphi (X_{1},X_{2},t)=\hat{H}_{2}(\hat{X}_{1},\hat{X}_{2},t)\varphi (X_{1},X_{2},t),$$where the new Hamiltonian $\hat{H}_{2}(\hat{X}_{1},\hat{X}_{2},t)$ has the form$$\begin{aligned} &&\hat{H}_{2}(\hat{X}_{1},\hat{X}_{2},t)=\frac{1}{2m(t)}\left( \hat{P}_{1}^{2}+\hat{P}_{2}^{2}\right) \notag \\ &&~~~~~~~~~~+\frac{1}{2}\left( \lambda _{1}(t)\hat{X}_{1}^{2}+\lambda _{2}(t)\hat{X}_{2}^{2}+\lambda _{3}(t)\hat{X}_{1}\hat{X}_{2}\right) .\end{aligned}$$Now the term involving $\hat{L}_{Z}$ has disappeared in Eq. (48). This means that the magnetic field term is removed in the new frame rotating with the time-dependent phase $\phi (t)=-\frac{1}{2}\int \varpi _{c}(t)dt.$ To decouple the Hamiltonian of Eq. (48), we take another unitary transformation such that $$\varphi (X_{1},X_{2},t)=\hat{V}(t)\chi (X_{1},X_{2},t),$$where the unitary operator $\hat{V}(t)$ is given by$$\hat{V}(t)=\hat{V}_{1}(t)\hat{V}_{2}(t)\hat{V}_{3}(t),$$with $$\begin{aligned} \hat{V}_{1}(t) &=&\exp \frac{i}{2\hbar }\left[ (\hat{P}_{1}\hat{X}_{1}+\hat{X}_{1}\hat{P}_{1})\ln \sqrt{m(t)}\right] \notag \\ &&\times \exp \frac{i}{2\hbar }\left[ (\hat{P}_{2}\hat{X}_{2}+\hat{X}_{2}\hat{P}_{2})\ln \sqrt{m(t)}\right] , \\ \hat{V}_{2}(t) &=&\exp \left[ -\frac{i}{\hbar }\frac{\theta }{2}(\hat{P}_{2}\hat{X}_{1}-\hat{P}_{1}\hat{X}_{2})\right] , \\ \hat{V}_{3}(t) &=&\exp -\frac{i}{4\hbar }\dot{m}(t)\left( \hat{X}_{1}^{2}+\hat{X}_{2}^{2}\right) .\end{aligned}$$Some algebra with the substitution of Eqs. (48) and (49) into Eq.(47) yields a transformed Hamiltonian that represents the sum of two uncoupled simple harmonic oscillators having frequencies $\Omega _{1}(t)$ and $\Omega _{2}(t)$ and the unit mass: $$\begin{aligned} \hat{H}_{3}(\hat{X}_{1},\hat{X}_{2},t) &=&\hat{V}^{-1}(t)\hat{H}_{2}(\hat{X}_{1},\hat{X}_{2},t)\hat{V}(t)-i\hbar \hat{V}^{-1}(t)\frac{\partial }{\partial t}\hat{V}(t) \notag \\ &=&\frac{1}{2}\left( \hat{P}_{1}^{2}+\hat{P}_{2}^{2}\right) +\frac{1}{2}\Omega _{1}^{2}(t)\hat{X}_{1}^{2}+\frac{1}{2}\Omega _{2}^{2}(t)\hat{X}_{2}^{2}.\end{aligned}$$At this stage, it is possible to confirm that the classically transformed Hamiltonian given in Eq. (38) is right, since the above equation is consistent with it. Note that $\hat{U}_{1}(t)$ and$\ \hat{V}_{1}(t)$ given in Eqs. (42) and (51) are the squeeze operators, whereas $\hat{U}_{2}(t)$ and$\ \hat{V}_{2}(t)$ given in Eqs. (46) and (52) are the rotation operators characterized by the time-varying angles $\phi (t)$ and $\frac{\theta (t)}{2} $, respectively. Quantum solutions ================= It can be seen that there exists invariant for the harmonic oscillator with time-dependent mass and/or frequency[@lewis3]. In our case, the transformed system consists of the two independent harmonic oscillators which have time-dependent frequency. It is easy to verify, from Liouville-von Neumann equation for the invariant $\hat{I}$ $$\frac{d\hat{I}}{dt}=\frac{\partial \hat{I}}{\partial t}+\frac{1}{i\hbar }[\hat{I},\hat{H}_{3}]=0,$$that the invariant associated to the Hamiltonian of two-dimensional harmonic oscillator is given by$$\begin{aligned} \hat{I}(\hat{X}_{1},\hat{X}_{2},t) &=&\hat{I}(\hat{X}_{1},t)+\hat{I}(\hat{X}_{2},t) \notag \\ &=&\frac{1}{2}\left[ \left( \frac{\hat{X}_{1}}{\rho _{1}}\right) ^{2}+\left( \rho _{1}\overset{\cdot }{\hat{X}_{1}}-\dot{\rho}_{1}\hat{X}_{1}\right) ^{2}\right] \notag \\ &&+\frac{1}{2}\left[ \left( \frac{\hat{X}_{2}}{\rho _{2}}\right) ^{2}+\left( \rho _{2}\overset{\cdot }{\hat{X}_{2}}-\dot{\rho}_{2}\hat{X}_{2}\right) ^{2}\right] ,\end{aligned}$$where $\rho _{1}(t)$ and $\rho _{2}(t)$ are c-number quantities obeying the auxiliary equations $$\begin{aligned} \ddot{\rho}_{1}+\Omega _{1}^{2}(t)\rho _{1} &=&1/\rho _{1}^{3}, \\ \ddot{\rho}_{2}+\Omega _{2}^{2}(t)\rho _{2} &=&1/\rho _{2}^{3}.\end{aligned}$$ To guarantee the Hermiticity of Eq. (56) ($\hat{I}^{\dagger }=\hat{I}$), we choose only the real solutions of the above two equations. It is clear that $\hat{I}(\hat{X}_{1},\hat{X}_{2},t)$ satisfies the Liouville-Von Neumann equation. We now derive a complete orthonormal set of eigenfunctions $\xi _{n_{1}n_{2}}(X_{1},X_{2},t)$ of $\hat{I}(\hat{X}_{1},\hat{X}_{2},t)$ form the eigenvalue equation$$\hat{I}(\hat{X}_{1},\hat{X}_{2},t)\xi _{n_{1}n_{2}}(X_{1},X_{2},t)=\lambda _{n_{1}n_{2}}\xi _{n_{1}n_{2}}(X_{1},X_{2},t),$$where $\lambda _{n_{1}n_{2}}$ are time-[*in*]{}dependent eigenvalues. Through a straightforward evaluation after inserting Eq. (56) into the above equation, we get the eigenvalues and the eigenfunctions such that$$\lambda _{n_{1}n_{2}}=\hbar \left( n_{1}+\frac{1}{2}\right) +\hbar \left( n_{2}+\frac{1}{2}\right) ,$$$$\begin{aligned} &&\xi _{n_{1}n_{2}}(X_{1},X_{2},t)=\left[ \frac{1}{\pi \hbar n_{1}!n_{2}!2^{n_{1}+n_{2}}\rho _{1}\rho _{2}}\right] ^{1/2} \notag \\ &&~~~~~~~~~\times H_{n_{1}}\left( \frac{X_{1}}{\hbar ^{1/2}\rho _{1}}\right) H_{n_{2}}\left( \frac{X_{2}}{\hbar ^{1/2}\rho _{2}}\right) \notag \\ &&~~~~~~~~~\times \exp \left[ \frac{i}{2\hbar }\left( \frac{\dot{\rho}_{1}}{\rho _{1}}+\frac{i}{\rho _{1}^{2}}\right) X_{1}^{2}+\frac{i}{2\hbar }\left( \frac{\dot{\rho}_{2}}{\rho _{2}}+\frac{i}{\rho _{2}^{2}}\right) X_{2}^{2}\right] ,\end{aligned}$$where $H_{n_{1}}$ and $H_{n_{2}}$ are the usual Hermite polynomial of order $n_{1}$ and $n_{2}$ respectively. The solutions of the Schrödinger equation$$i\hbar \frac{\partial \chi _{n_{1}n_{2}}(X_{1},X_{2},t)}{\partial t}=\hat{H}_{3}(\hat{X}_{1},\hat{X}_{2},t)\chi _{n_{1}n_{2}}(X_{1},X_{2},t),$$can be written as$$\chi _{n_{1}n_{2}}(X_{1},X_{2},t)=e^{i\alpha _{n_{1}n_{2}}(t)}\xi _{n_{1}n_{2}}(X_{1},X_{2},t),$$where the phase functions $\alpha _{n_{1}n_{2}}(t)$ satisfy the equation$$\frac{\partial }{\partial t}\alpha _{n_{1}n_{2}}(t)=\frac{1}{\hbar }\left\langle \xi _{n_{1}n_{2}}(X_{1},X_{2},t)\right\vert \frac{\partial }{\partial t}-\hat{H}_{3}(\hat{X}_{1},\hat{X}_{2},t)\left\vert \xi _{n_{1}n_{2}}(X_{1},X_{2},t)\right\rangle .$$ According to Eqs. (61) and (63), the solutions $\chi_{n_{1}n_{2}} (X_{1},X_{2},t)$ of the Schrödinger equation (62), in the transformed system, becomes $$\begin{aligned} & &\chi _{n_{1}n_{2}}(X_{1},X_{2},t) = e^{i\alpha _{n_{1}n_{2}}(t)}\left[ \frac{1}{\pi \hbar n_{1}!n_{2}!2^{n_{1}+n_{2}}\rho _{1}\rho _{2}}\right] ^{1/2} \notag \\ &&~~~~~~~~~~~\times H_{n_{1}}\left( \frac{X_{1}}{\hbar ^{1/2}\rho _{1}}\right) H_{n_{2}}\left( \frac{X}{\hbar ^{1/2}\rho _{2}}\right) \notag \\ &&~~~~~~~~~~~\times \exp \left[ \frac{i}{2\hbar }\left( \frac{\dot{\rho}_{1}}{\rho _{1}}+\frac{i}{\rho _{1}^{2}}\right) X_{1}^{2}+\frac{i}{2\hbar }\left( \frac{\dot{\rho}_{2}}{\rho _{2}}+\frac{i}{\rho _{2}^{2}}\right) X_{2}^{2}\right] ,\end{aligned}$$where the time-dependent phase functions are given by $$\alpha _{n_{1}n_{2}}(t)=-\left( n_{1}+\frac{1}{2}\right) \int_{0}^{t}\frac{dt^{\prime }}{\rho _{1}^{2}(t^{\prime })}-\left( n_{2}+\frac{1}{2}\right) \int_{0}^{t}\frac{dt^{\prime }}{\rho _{2}^{2}(t^{\prime })}.$$ The relation between the wave functions, $\Psi _{n_{1}n_{2}}(X_{1},X_{2},t)$, in the original system described by the Hamiltonian of Eq. (3) and the wave functions $\chi _{n_{1}n_{2}}(X_{1},X_{2},t)$ in the transformed system is$$\begin{aligned} \Psi _{n_{1}n_{2}}(X_{1},X_{2},t) &=&\hat{U}_{1}(t)\hat{U}_{2}(t)\hat{V}(t)\chi _{n_{1}n_{2}}(X_{1},X_{2},t) \notag \\ &=&\hat{U}_{1}(t)\hat{U}_{2}(t)\hat{V}_{1}(t)\hat{V}_{2}(t)\hat{V}_{3}(t)\chi _{n_{1}n_{2}}(X_{1},X_{2},t).\end{aligned}$$Using Eqs. (42), (46), (50) and (65), we derive the full wave functions in the form$$\begin{aligned} &&\Psi _{n_{1}n_{2}}(X_{1},X_{2},t)=\left[ \frac{\sqrt{m_{1}m_{2}}}{\pi \hbar n_{1}!n_{2}!2^{n_{1}+n_{2}}\rho _{1}\rho _{2}}\right] ^{1/2} \notag \\ &&~~~~~~\times H_{n_{1}}\left( \frac{\sqrt{m_{1}}\cos \left( \phi +\theta /2\right) X_{1}-\sqrt{m_{2}}\sin \left( \phi +\theta /2\right) X_{2}}{\hbar ^{1/2}\rho _{1}}\right) \notag \\ &&~~~~~~\times H_{n_{2}}\left( \frac{\sqrt{m_{1}}\sin \left( \phi +\theta /2\right) X_{1}+\sqrt{m_{2}}\cos \left( \phi +\theta /2\right) X_{2}}{\hbar ^{1/2}\rho _{2}}\right) \notag \\ &&~~~~~~\times \exp \frac{im_{1}}{2\hbar }\left( \frac{\gamma }{2}+\frac{\beta }{2}+\left( \frac{\beta }{2}-\frac{\gamma }{2}\right) \sin \left( \theta +2\phi \right) \right) X_{1}^{2} \notag \\ &&~~~~~~\times \exp \frac{im_{2}}{2\hbar }\left( \frac{\gamma }{2}+\frac{\beta }{2}-\left( \frac{\beta }{2}-\frac{\gamma }{2}\right) \sin \left( \theta +2\phi \right) \right) X_{2}^{2} \notag \\ &&~~~~~~\times \exp \frac{i}{2\hbar }\sqrt{m_{1}m_{2}}\left( \left( \beta -\gamma \right) \cos \left( \theta +2\phi \right) \right) X_{1}X_{2} \notag \\ &&~~~~~~\times \exp i\left[ -\left( n_{1}+\frac{1}{2}\right) \int_{0}^{t}\frac{dt^{\prime }}{\rho _{1}^{2}(t^{\prime })}-\left( n_{2}+\frac{1}{2}\right) \int_{0}^{t}\frac{dt^{\prime }}{\rho _{2}^{2}(t^{\prime })}\right] ,\end{aligned}$$where the time-dependent coefficients $\gamma (t)$ and $\beta (t)$ are given as$$\gamma (t)=\left( \frac{\dot{\rho}_{1}}{\rho _{1}}+\frac{i}{\rho _{1}^{2}}-\frac{1}{2}\frac{d}{dt}\sqrt{m_{1}m_{2}}\right) ,$$$$\beta (t)=\left( \frac{\dot{\rho}_{2}}{\rho _{2}}+\frac{i}{\rho _{2}^{2}}-\frac{1}{2}\frac{d}{dt}\sqrt{m_{1}m_{2}}\right) .$$The full solutions in the original system, given in Eq. (68), are exact since we did not use approximation or perturbation methods. Though these solutions are somewhat complicated, they are very useful in predicting the quantum behavior of the system. A merit of such analytical solutions is that they can be employed in deriving the evolution of the probability distribution, regardless of the change of the system’s parameters. However, the numerical solutions in this field, such as the one obtained from FDTD (finite difference time domain) method[@fdtd], are somewhat inconvenient as inputs to further analyses, since one should recalculate the results whenever the parameters of the system changes. Using Eq. (68), one can easily take a complete description of the charged particle motion even when the parameters of the system vary from time to time provided that the classical solutions of Eqs. (57) and (58) are known. Conclusion ========== We investigated the quantal problem of the time-dependent coupled oscillator model associated to the charged particle motion in the presence of time-dependent magnetic field. Though the behavior of charged particle in magnetic field drew great concern in both quantum and classical view point, researches in this line are rather concentrated on static problems that can be modeled by time-*in*dependent harmonic oscillator. The system we treated in this work is however a more generalized one. It is summarized as follows: (i) We supposed that the effective mass of the charged particle varies explicitly with time under the influence of the time-dependent magnetic field. If electrons or holes in the condensed matter interact with environment or various excitations such as pressure, energy, temperature, and stress, their effective mass may naturally vary with time[@choi2]. Moreover, the random changes of the external field in the heterojunctions and solid solutions give rise to the variation of effective mass in accordance with the fluctuation of the composition in the system[@zsg]. (ii) We let the external magnetic field $B(t)$ be an *arbitrary* function of time. Therefore, the application of our theory is not confined in a special system that has a specific class of time-dependence for $B(t)$. In fact, we can apply it in wide range of practical systems with the flexible choice of the type of $B(t)$. (iii) Our system is further generalized by adding a coupling term $X_{1}X_{2}$ in the Hamiltonian. Through these generalization, the system became a somewhat complicated one that is described in terms of time-dependent Hamiltonian. Since the treatment of the original Hamiltonian system is not an easy task in this case, we transformed our system to that of a much more simplified one using two different techniques. In the first one, we carried out canonical transformations in order to simplify the problem relevant to the original classical Hamiltonian given in Eq. (1). After the transformation, the Hamiltonian reduced to a simple form associated to two uncoupled harmonic oscillators that each have time-dependent frequencies $\Omega _{1}(t)$ and $\Omega _{2}(t)$. In the second technique we used an alternative approach on the basis of the unitary transformation method. With the choice of unitary operators $\hat{U}_{1}(t)$, $\hat{U}_{2}(t)$ and $\hat{V}(t)$, the quantum Hamiltonian (39) has been transformed to an equally simple one as that of the canonical transformation previously performed, but within the realm of quantum mechanics. Since the Hamiltonian in the transformed system is very simple, we easily constructed dynamical invariant operator $\hat{I}(\hat{X}_1,\hat{X}_2,t)$ associated to the transformed system, as given in Eq. (55). The eigenstates $\xi _{n_{1}n_{2}}(X_{1},X_{2},t)$ of this invariant operator are represented in terms of the Hermite polynomial. The Schrödinger solutions $\chi _{n_{1}n_{2}}(X_{1},X_{2},t)$ in the transformed system are the same as $\xi _{n_{1}n_{2}}(X_{1},X_{2},t)$ except for the time-dependent phase factor $e^{i\alpha _{n_{1}n_{2}}(t)}$. From the inverse transformation of $\chi_{n_{1}n_{2}} (X_{1},X_{2},t)$ with the unitary operators, we derived the full wave functions (quantum solutions) in the original system \[see Eq. (68)\]. The quantum solutions are expressed in terms of $\rho_1$ and $\rho_2$ that are the two independent solutions of the classical equation of motion given in Eqs. (56) and (57), respectively. Even if we represented the quantum solutions in terms of the classical solutions associated with the *transformed system*, it is also possible to represent them in terms of the classical solutions associated with *original system*. The wave functions given in Eq. (68) can be used to investigate various quantum properties of the system such as the fluctuations of canonical variables, the evolution of quantum energy, and probability densities, even when the parameters of the system vary from time to time. This is the advantage of such analytical solutions over numerical solutions obtained, for example, using the FDTD method[@fdtd]. **Acknowledgements** The work of J. R. Choi was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (No. 2010-0016914). [99]{} Bateman H 1931 *Phys. Rev*. **38** 815. Kim Y S, Noz M E and Oh S H 1979 *Am. J. Phys*. **47** 892 Han D, Kim Y S and Noz M E 1989 *Phys. Lett*. *A*** 144** 111 Kim Y S and Wigner E P 1990 *Phys. Lett*. *A*** 147** 343 Han D, Kim Y S and Noz M E 1995 *J. Math. Phys.* **36** 3940 Abdalla M S 1996 *J. Phys A: Math. Gen.* **29** 1997 Benamira F and Ghechi L 2003 *Czechoslovac J. Phys*. **9** 717 Bykov A A, Gusev G M, Leite J R, Bakarov A K, Goran A V, Kudryashev V M and Toropov A I 2001 *Phys. Rev*. *B* **65** 035302 van der Wiel W G, Nazarov Yu V, De Franceschi S, Fujisawa T, Elzerman J M, Huizeling E W G M, Tarucha S and Kouwenhoven L P 2003 *Phys. Rev*. *B* **67** 033307 Varma R K 2003 *Phys. Rep*. **378** 301 Calvo M 2004 *Ultramicroscopy.* **99** 179 Kennedy T A, Wagner R, McCombe B and Tsui D 1975 *Phys. Rev. Lett*.** 35** 1031 Sadiek G, Lashin E I and Abdalla M S 2009 *Physica B*: *Condens. Matter.* **404** 1719 Choi J R 2003 *J. Phys: Condens. Matter.* **15** 823 S. Menouar, M. Maamache, and J. R. Choi, Gaussian wave packet for time-dependent harmonic oscillator model of a charged particle in a variable magnetic field, unpublished. Laroze D and Rivera R 2006 *Phys. Lett. A* **355** 348 Abdalla M S and Choi J R 2007 *Ann. Phys.* **322** 2795 Schrade G, Man’ko V I, Schleich W P and Glauber R J 1995 *Quantum Semiclass. Opt.* **7** 307 Han D, Kim Y S and Noz M E 1990 *Phys. Rev.* *A*** 41** 6233 Caves C M and Schumaker B L 1985 *Phys. Rev*. *A*** 31** 3068 Schweber S. S 1961 An Introduction to Relativistic Quantum Field Theory(Row-Peterson, Elmsford, New York) Zhang S, Choi J R, Um C I and Yeon K H 2001 *Phys. Lett*. *A* **289** 257 Zhang S, Choi J R, Um C I and Yeon K H 2002 *Phys. Lett*. *A* **294** 319 Lewis H R Jr 1968 *Math.Phys,* **9** 1976 Lewis H R Jr. and Riesenfeld W B 1969 *J. Math. Phys.* **10** 1458 Lewis H R Jr 1967 *Phys. Rev.Lett.***18** 510 Goldstein H 1980 Classical Mechanics (Addison-Wesley, Reading, MA) Sudiarta I W and Geldart D J W 2008 *Phys. Lett. A* **372** 3145 Gevorkian Z S and Lozovik Y E 1988 *J. Phys. C: Solid State Phys*. **21** 2181 [^1]: $^*$E-mail: menouar\_salah@yahoo.fr [^2]: $^\dagger$Corresponding author, E-mail: choiardor@hanmail.net

--- author: - '[^1]' title: 'Ginzburg-Landau Equations for Coexistent States of Superconductivity and Antiferromagnetism in $t-J$ Model ' --- Introduction ============ The discovery of the coexistence of antiferromagnetism (AF) and superconductivity (SC) in multilayer high-$T_c$ cuprates has stimulated wide interest. [@Kitaoka; @Mukuda] Antiferromagnetic superexchange interactions in high-$T_C$ cuprate superconductors, which are strongly correlated electron systems, are thought to be the origin of two ordered states; thus understanding the condition for coexistence may give insight into the mechanism of superconductivity. In single-layer and bilayer cuprates such as La- and Y-based compounds, it has been well known that AF is easily suppressed by a tiny amount of carrier doping.[@Keimer; @Sanna] On the contrary in multilayer systems (in this paper the term “multilayer” will refer to three or more layers in a unit cell) such as HgBa$_2$Ca$_4$Cu$_5$O$_{12+y}$, AF survives up to much higher doping rate and coexists with SC state. NMR measurements revealed that the coexistence was not due to a proximity effect but a genuine phase transition within a CuO$_2$ plane.[@Kitaoka; @Mukuda] Multilayer cuprates have flat CuO$_2$ planes with a perfect square lattice and are known to be free from disorder in contrast to La- and Y-based compounds. Combined with their high $T_C$ of more than 100K,[@Iyo] multilayer cuprates can be viewed as ideal systems to study the mechanism of high $T_C$. In this sense it is desirable to explore the nature of the coexistent state of AF and SC theoretically. Low-energy electronic states of high-$T_C$ cuprates are described by the $t-J$ model. [@Anderson; @Ogata; @Lee] In the case of single-layer and bilayer systems the AF order is easily destabilized by strong fluctuations due to low dimensionality. Assuming the absence of AF order, mean-field (MF) theories[@Kotliar; @Suzumura] based on the slave-boson (SB) scheme[@Zou] to treat the condition of no double occupancy and the gauge theory,[@Nagaosa; @Lee] which takes into account the low-energy fluctuations around mean-fields, capture many important properties of single-layer and bilayer high-$T_C$ cuprates. In multilayer systems, on the other hand, relatively strong three dimensionality may stabilize AF order.[@KK1; @Yamase2] This situation can be suitably treated by MF theories for the $t-J$ model by taking AF order into account. Actually MF calculations for the $t-J$ model predicted that AF survives up to $\delta \lesssim 0.1-0.15$ ($\delta$ being the doping rate) and it may coexist with SC,[@Inaba; @Yamase; @KK1; @Yamase2] and a similar result was obtained by the variational Monte Carlo method.[@Himeda] In this paper, we derive GL equations and the GL free energy microscopically from the two-dimensional $t-J$ model with extended transfer integrals (extended $t-J$ model) based on the SBMF approximation. In the MF approach the phase diagram will not be sensitive to the number of layers. It is the shape of the Fermi surface, in particular, the condition for the nesting that is crucial to determine the occurrence of the coexistent state, and an electronlike Fermi surface can lead to the experimentally observed phase diagram.[@KK1; @Yamase2] In multilayer cuprates we expect that such an electronlike Fermi surface may be stabilized as one of the Fermi surface due to strong hybridization between layers. This is the reason why we treat a single-layer (single-band) model, and we simulate the difference of the Fermi surface by including the extended transfer integrals. The derived GL theory can be used to investigate the spatial dependence of the AF and SC order parameters (OPs) in high-$T_C$ cuprates, and it may provide information on the electronic states in these systems. For example, near the surface or impurity the OPs are suppressed, and their recovery to the bulk values will provide the coherence length, which reflect the underlying electronic structures of each system. Although the GL theory is reliable only qualitatively except near $T_C$, it can give a simple and intuitive description of the coexistence and competition of multiple OPs. Thus, it is complementary to more sophisticated methods such as the Bogoliubov-de Gennes and quasiclassical Green’s function theory. Previously various models have been employed to derive GL equations microscopically; a continuum[@Ren] and tight-binding model[@Feder] with $s$- and $d$-wave SCOPs, Hubbard model with nearest-neighbor attractive interactions,[@KKYano] a model with a spin generalized BCS term and Heisenberg exchange term,[@Dahl] and the $t-J$ model (without taking AF order into account)[@KKSig]. The method of deriving GL equations in this work is based on that by Gor’kov[@Gorkov] with the extension to include AF order.[@KKYano] This paper is organized as follows. In $\S2$, we present the model and treat it by the SBMF approximation. GL equations and the GL free energy are derived in $\S3$. Section 4 is devoted to summary and discussion. Model and Mean-Field Approximation ================================== We consider the extended $t-J$ model on a square lattice whose Hamiltonian is given as $$\begin{aligned} \displaystyle H = -\sum_{j,\ell,\sigma} t_{j\ell} e^{i\phi_{j\ell}} {\tilde c}^\dagger_{j\sigma} {\tilde c}_{\ell\sigma} +J\sum_{\langle j,\ell\rangle} {\bf S}_j\cdot {\bf S}_\ell, %-\mu \sum_{j\sigma} c^\dagger_{j\sigma}c_{j\sigma}\end{aligned}$$ where the transfer integrals $t_{j\ell}$ are finite for the first- ($t$), second- ($t'$), and third-nearest-neighbor bonds ($t''$), and vanish otherwise. $J (>0)$ is the antiferromagnetic superexchange interaction and $\langle j,\ell \rangle$ denotes the nearest-neighbor bonds. The magnetic field is taken into account using the Peierls phase $\phi_{j,\ell} \equiv \frac{\pi}{\phi_0} \int_j^\ell {\bf A}\cdot d{\bf l}$, with ${\bf A}$ and $\phi_0 = \frac{hc}{2e}$ being the vector potential and flux quantum, respectively. ${\tilde c}_{j\sigma}$ is the electron operator in Fock space without double occupancy, and we treat this condition using the SB method[@Zou] by writing ${\tilde c}_{j\sigma}=b_j^\dagger f_{j\sigma}$ under the local constraint $\sum_{\sigma}f_{j\,\sigma}^{\dagger}f_{j\,\sigma} + b_j^{\dagger}b_j = 1$ at every $j$ site. Here $f_{j\sigma}$ ($b_j$) is a fermion (boson) operator that carries spin $\sigma$ (charge $e$); the fermions (bosons) are frequently referred to as spinons (holons). The spin operator is expressed as $%\displaystyle {\bf S}_j = \frac{1}{2}\sum_{\alpha,\beta} f^\dagger_{j\alpha} {\bf \sigma}_{\alpha\beta}f_{j\beta}$. We decouple Hamiltonian eq. (1) in the following manner. [@Inaba; @Yamase; @KK1; @Yamase2] The bond order parameters $\langle \sum_\sigma f^\dagger_{j\sigma}f_{\ell\sigma} \rangle$ and $\langle b^\dagger_j b_\ell\rangle$ are introduced and we denote $\chi_{j,\ell}/2 = \langle f^\dagger_{j\uparrow}f_{\ell\uparrow} \rangle = \langle f^\dagger_{j\downarrow}f_{\ell\downarrow} \rangle$ for the nearest-neighbor bond. Although the bosons are not condensed in purely two-dimensional systems at finite temperature ($T$), they are almost condensed at low $T$ and for finite carrier doping $\delta (\gtrsim 0.02)$.[@Inaba] Hence we approximate $\langle b_j \rangle \approx {\sqrt \delta}$ and $\langle b^\dagger_ib_j\rangle \approx \delta$. The magnetization is defined by $m_j = \frac{1}{2}\langle f^\dagger_{j\uparrow}f_{j\uparrow} - f^\dagger_{j\downarrow}f_{j\downarrow} \rangle $, and the superconducting OP on the bond $\langle j,\ell\rangle$ (under the assumption of the Bose condensation of holons) is given as $\Delta_{j,\ell} = \langle f_{j\uparrow}f_{\ell\downarrow}\rangle$. Fluctuations around the mean-field solutions in the slave-boson scheme can be treated as the gauge field. It is known that this gauge field may affect the physical properties of the solutions in a serious way.[@Nagaosa] However, in the SC and AF states the effect of the gauge field is strongly suppressed.[@LeeNag; @DKim] Since we are interested in these ordered states, we do not consider the effect of gauge-field fluctuations. In the following we are mainly interested in a region around the tetracritical point where the four states, AF, $d_{x^2-y^2}$-wave SC, their coexistence, and the normal states become identical. The onset temperature of the bond OPs is much higher than that for AF ($T_N$) and SC ($T_C$) in this doping region, so that they are almost independent of temperature near the tetracritical point. We consider only the spatial variations of $m_j$ and $\Delta_{j,\ell}$ assuming that $\chi_{j,\ell}$ is uniform in space. (Hereafter we denote it as $\chi$.) Then the mean-field Hamiltonian is given as $$\begin{array}{rl} H_{MFA} = & \displaystyle -\sum_{j,\sigma}\Big[ \sum_{\delta=\pm x,\pm y} \big(t\delta e^{i\phi_{j+\delta,j}} + \frac{3J}{8}\chi\big) f^\dagger_{j+\delta,\sigma}f_{j\sigma} + t'\delta\sum_{\delta=\pm x \pm y} e^{i\phi_{j+\delta,j}} f^\dagger_{j+\delta,\sigma}f_{j\sigma} \\ + & \displaystyle t''\delta\sum_{\delta=\pm 2x, \pm 2y} e^{i\phi_{j+\delta,j}} f^\dagger_{j+\delta,\sigma}f_{j\sigma} \Big] - \mu\sum_{j,\sigma} f^\dagger_{j\sigma}f_{j\sigma} +\frac{J}{2}\sum_j\sum_{\delta=\pm x,\pm y} m_{j+\delta} \big(f^\dagger_{j\uparrow}f_{l\uparrow} - f^\dagger_{j\downarrow}f_{l\downarrow}\big) \\ + & \displaystyle \frac{J}{2}\sum_j \sum_{\delta=\pm x,\pm y} \big[\Delta_{j,j+\delta}\big(f^\dagger_{j+\delta\uparrow}f^\dagger_{j\downarrow} -\frac{1}{2}f^\dagger_{j+\delta\downarrow}f^\dagger_{j\uparrow}\big) + h.c.\big] + E_0, \end{array}$$ with $$E_0= - J \sum_{\langle j,\ell \rangle} m_j m_\ell +J\sum_{\langle j,\ell\rangle}\Big(\frac{1}{2}\Delta_{j,\ell}\Delta^*_{\ell,j} +\frac{1}{4}|\Delta_{j,\ell}|^2\Big).$$ First we solve the self-consistency equations for $\chi$ and the chemical potential $\mu$ in the absence of $m$, $\Delta$, and ${\bf A}$. Self-consistency equations that determine $\chi$ and $\mu$ as functions of $T$ and $\delta$ are given as $$%\begin{array}{rl} \displaystyle \chi = \frac{1}{N}\sum_p (\cos p_x+\cos p_y)f(\xi_p) , \ \ \delta= 1 - \frac{2}{N}\sum_pf(\xi_p), %\end{array}$$ where $\xi_p = -(2t\delta + \frac{3J}{4}\chi)(\cos p_x+\cos p_y) -4t'\delta\cos p_x \cos p_y - 2t''\delta(\cos 2p_x+\cos 2p_y) -\mu$, with $f$ and $N$ being the Fermi function and the total number of lattice sites, respectively. (Lattice constant is taken to be unity.) In the next section we will carry out the GL expansion to obtain GL equations for $m$ and $\Delta$. For the values of $t'$ and $t''$ which reproduce the experimentally obtained phase diagram, incommensurate (IC) as well as commensurate (C) AF order may be possible around the tetracritical point depending on the choice of the parameters.[@Yamase2] (There are several distinct parameter sets which lead to similar phase diagrams.) Experimentally, since the NMR does not directly discriminate different ordering patterns of magnetism, at present it is not clear whether ICAF order exists. Then we will consider only the CAF state as a feasible candidate. \[sec:GL\]GL Equations and GL Free Energy ========================================= In this section we derive GL equations and the GL free energy. The procedure is essentially the same as that used in ref.20. Coupled equations for Green’s functions $G_\sigma(j,\ell,\tau) = -\langle T_\tau f_{j\sigma}(\tau)f_{\ell\sigma}^\dagger\rangle$ and $F^\dagger_{\sigma\sigma'}(j,\ell,\tau) = -\langle T_\tau f_{j\sigma}^\dagger(\tau)f_{\ell\sigma'}^\dagger\rangle$ can be derived from their equations of motion (Gor’kov equations) as $$\begin{array}{rl} G_\uparrow(j,\ell,i\varepsilon_n) = & \displaystyle {\tilde G}_0(j,\ell,i\varepsilon_n) + \frac{J}{2}\sum_{k,\delta_1} {\tilde G}_0(j,k,i\varepsilon_n) \\ & \displaystyle \times \Big[\Big(\Delta_{k+\delta_1,k}+\frac{1}{2}\Delta_{k,k+\delta_1}\Big) F^\dagger_{\downarrow\uparrow}(k+\delta_1,\ell,i\varepsilon_n) %\\ & \displaystyle +m_{k+\delta_1} G_\uparrow(k,\ell,i\varepsilon_n)\Big], \\ % G_\downarrow(j,\ell,i\varepsilon_n) = & \displaystyle {\tilde G}_0(j,\ell,i\varepsilon_n) - \frac{J}{2}\sum_{k,\delta_1}{\tilde G}_0(j,k,i\varepsilon_n) \\ & \displaystyle \times \Big[\Big(\Delta_{k,k+\delta_1}+\frac{1}{2}\Delta_{k+\delta_1,k}\Big) F^\dagger_{\uparrow\downarrow}(k+\delta_1,\ell,i\varepsilon_n) %\\ & \displaystyle + m_{k+\delta_1} G_\downarrow(k,\ell,i\varepsilon_n)\Big], \\ % F^\dagger_{\downarrow\uparrow}(j,\ell,i\varepsilon_n) = & \displaystyle - \frac{J}{2}\sum_{k,\delta_1} {\tilde G}_0(k,j,-i\varepsilon_n) \\ & \displaystyle \times \Big[\Big(\Delta^*_{k,k+\delta_1}+\frac{1}{2}\Delta^*_{k+\delta_1,k}\Big) G_\uparrow(k+\delta_1,\ell,i\varepsilon_n) %\\ & \displaystyle + m_{k+\delta_1} F^\dagger_{\downarrow\uparrow}(k,\ell,i\varepsilon_n)\Big], \\ % F^\dagger_{\uparrow\downarrow}(j,\ell,i\varepsilon_n) = & \displaystyle \frac{J}{2}\sum_{k,\delta_1} {\tilde G}_0(k,j,-i\varepsilon_n) \\ & \displaystyle \times \Big[\Big(\Delta^*_{k+\delta_1,k}+\frac{1}{2}\Delta^*_{k,k+\delta_1}\Big) G_\downarrow(k+\delta_1,\ell,i\varepsilon_n) %\\ & \displaystyle + m_{k+\delta_1} F^\dagger_{\uparrow\downarrow}(k,\ell,i\varepsilon_n)\Big], \\ \end{array}$$ where the summation on $\delta_1$ ($k$) is over $\pm {\hat x}$ and $\pm {\hat y}$ (all sites). Here, ${\tilde G}_0(j,\ell,i\omega_n)$ is Green’s function for the system without $\Delta$ and $m$ but with ${\bf A}$. ${\tilde G}_0(j,\ell,i\omega_n)$ is related to Green’s function for the system without ${\bf A}$, $G_0$, as $ {\tilde G}_0(j,\ell,i\varepsilon_n) \sim G_0(j,\ell,i\varepsilon_n) e^{i\phi_{j,\ell}}$, with $G_0(j,\ell,i\varepsilon_n)$ being the Fourier transform of $G_0({\bf p},i\varepsilon_n) = 1/(i\varepsilon_n-\xi_p)$. In the expression of $\xi_p$, the bond order parameter $\chi$ and the chemical potential $\mu$ determined by eq.(4) are substituted. Spin-singlet and spin-triplet SCOPs on the bond $(j,j+\eta)$ are expressed in terms of Green’s functions $F_{\uparrow\downarrow}^\dagger$ and $F_{\downarrow\uparrow}^\dagger$, $$\begin{array}{rl} \displaystyle (\Delta_\eta^{(S)}(j))^* \equiv & \displaystyle \frac{1}{2} \langle f_{j\uparrow}f_{j+\eta\downarrow} - f_{j\downarrow}f_{j+\eta\uparrow}\rangle^* = \frac{1}{2} \big(\Delta_{j,j+\eta}+\Delta_{j+\eta,j}\big)^* \\ = & \displaystyle \frac{T}{2} \sum_{\varepsilon_n} \Big[F^\dagger_{\uparrow\downarrow}(j+\eta,j,i\varepsilon_n) - F^\dagger_{\downarrow\uparrow}(j+\eta,j,i\varepsilon_n)\Big], \\ \displaystyle (\Delta_\eta^{(T)}(j))^* \equiv & \displaystyle\frac{1}{2} \langle f_{j\uparrow}f_{j+\eta\downarrow} + f_{j\downarrow}f_{j+\eta\uparrow}\rangle^* = \frac{1}{2} \big(\Delta_{j,j+\eta}-\Delta_{j+\eta,j} \big)^* \\ = & \displaystyle -\frac{T}{2} \sum_{\varepsilon_n} \Big[F^\dagger_{\uparrow\downarrow}(j+\eta,j,i\varepsilon_n) + F^\dagger_{\downarrow\uparrow}(j+\eta,j,i\varepsilon_n)\Big], \\ \end{array}$$ and the staggered magnetization $M_j \equiv m_j e^{i{\bf Q}\cdot{\bf r}_j}$ (${\bf Q} = (\pi,\pi)$) is similarly given using $G_{\uparrow}$ and $G_{\downarrow}$, $$\begin{array}{rl} M_j \equiv & \displaystyle \frac{1 }{2} \langle f^\dagger_{j\uparrow}f_{j\uparrow} - f^\dagger_{j\downarrow}f_{j\downarrow} \rangle e^{i{\vec Q}\cdot{\vec r}_j} \\ = & \displaystyle \frac{T}{2} \sum_{\varepsilon_n} \big[G_\uparrow(j,j,i\varepsilon_n)-G_\downarrow(j,j,i\varepsilon_n) \big] e^{i{\vec Q}\cdot{\vec r}_j}. \end{array}$$ We substitute eq. (5) into eqs. (6) and (7) iteratively and keep the terms up to the third order in OPs. In the coexistent state of AF and SC, spin-triplet SCOPs that oscillate in a similar manner as the staggered magnetization may occur[@Fenton; @Mura1; @Mura2; @Kyung; @Apens], and they are called the $\pi$-triplet SCOPs. The SCOPs of each symmetry, $\Delta_s$ ($s$-wave), $\Delta_d$ ($d$-wave), and $\Delta^{(\pi T)}_{px(y)}$ ($\pi$-triplet $px(y)$-wave), can be constructed by making a linear combination of eq.(6), $$\begin{array}{rl} \Delta_s(j) = & \displaystyle \frac{1}{4} \sum_{\eta=\pm{\hat x},\pm{\hat y}} \Delta_\eta^{(S)}(j) , \ \ \Delta_d(j) = \frac{1}{4} \Big[\sum_{\eta=\pm{\hat x}} \Delta_\eta^{(S)}(j) - \sum_{\eta=\pm{\hat y}} \Delta_\eta^{(S)}(j)\Big], \\ & \displaystyle \Delta^{(\pi T)}_{px(y)}(j) = \frac{1}{2} \big[\Delta^{(\pi T)}_{{\hat x}(y)}(j) + \Delta^{(\pi T)}_{-{\hat x}(y)}(j)\big]. \end{array}$$ Assuming that the SCOPs and $M$ are slowly varying, we take a continuum limit. The OPs in the linear terms are expanded in powers of derivatives up tp the second order, and the Peierls phase is also expanded in powers of ${\bf A}$ to the same order. Then after straightforward but lengthy calculations we get the following GL equations: $$\begin{array}{rl} & \displaystyle \alpha_s \Delta_s + 2\beta_s |\Delta_s|^2\Delta_s - K_s (D_x^2+D_y^2) \Delta_s - K_{ds}(D_x^2-D_y^2)\Delta_d \\ + & \displaystyle \gamma_1|\Delta_d|^2\Delta_s + 2\gamma_2\Delta_d^2\Delta_s^* + \gamma_3(|\Delta^{(\pi T)}_{px}|^2+|\Delta^{(\pi T)}_{py}|^2)\Delta_s + 2\gamma_5((\Delta^{(\pi T)}_{px})^2+(\Delta^{(\pi T)}_{py})^2)\Delta_s^* \\ + & \displaystyle \gamma_7(|\Delta^{(\pi T)}_{px}|^2 -|\Delta^{(\pi T)}_{py}|^2)\Delta_d + \gamma_8((\Delta^{(\pi T)}_{px})^2-(\Delta^{(\pi T)}_{py})^2)\Delta_d^* + \gamma_9(\Delta^{(\pi T)*}_{px}\Delta^{(\pi T)}_{py} + c.c.)\Delta_s \\ + & \displaystyle 2\gamma_{11}\Delta^{(\pi T)}_{px}\Delta^{(\pi T)}_{py}\Delta_s^* + \gamma_{ms}M^2\Delta_s + \gamma_{spm}M (\Delta^{(\pi T)}_{px}+\Delta^{(\pi T)}_{py}) = 0, \end{array}$$ $$\begin{array}{rl} & \displaystyle \alpha_d \Delta_d + 2\beta_d |\Delta_d|^2\Delta_d - K_d (D_x^2+D_y^2) \Delta_d - K_{ds}(D_x^2-D_y^2)\Delta_s \\ + & \displaystyle \gamma_1|\Delta_s|^2\Delta_d + 2\gamma_2\Delta_s^2\Delta_d^* + \gamma_4(|\Delta^{(\pi T)}_{px}|^2+|\Delta^{(\pi T)}_{py}|^2)\Delta_d + 2\gamma_6((\Delta^{(\pi T)}_{px})^2+(\Delta^{(\pi T)}_{py})^2)\Delta_d^* \\ + & \displaystyle \gamma_7(|\Delta^{(\pi T)}_{px}|^2 -|\Delta^{(\pi T)}_{py}|^2)\Delta_s + \gamma_8((\Delta^{(\pi T)}_{px})^2-(\Delta^{(\pi T)}_{py})^2)\Delta_s^* + \gamma_{10}(\Delta^{(\pi T)*}_{px}\Delta^{(\pi T)}_{py} + c.c.)\Delta_d \\ + & \displaystyle 2\gamma_{12}\Delta^{(\pi T)}_{px}\Delta^{(\pi T)}_{py}\Delta_d^* + \gamma_{md}M^2\Delta_d + \gamma_{dpm}M(\Delta^{(\pi T)}_{px}-\Delta^{(\pi T)}_{py}) = 0, \end{array}$$ $$\begin{array}{rl} & \displaystyle \alpha_{p1} \Delta^{(\pi T)}_{px(y)} + \alpha_{p2} \Delta^{(\pi T)}_{py(x)} + 2\beta_p |\Delta^{(\pi T)}_{px(y)}|^2\Delta^{(\pi T)}_{px(y)} \\ - & \displaystyle K_{p1}D_{x(y)}^2\Delta^{(\pi T)}_{px(y)} -K_{p2}D_{y(x)}^2\Delta^{(\pi T)}_{px(y)} - K_{p3}(D_x^2+D_y^2)\Delta^{(\pi T)}_{py(x)} \\ + & \displaystyle \gamma_{p1}|\Delta^{(\pi T)}_{py(x)}|^2\Delta^{(\pi T)}_{px(y)} + 2\gamma_{p2}(\Delta^{(\pi T)}_{py(x)})^2\Delta^{(\pi T)*}_{px(y)} \\ + & \displaystyle \gamma_{p3}(2|\Delta^{(\pi T)}_{px(y)}|^2\Delta^{(\pi T)}_{py(x)} + (\Delta^{(\pi T)}_{px(y)})^2\Delta^{(\pi T)*}_{py(x)} +|\Delta^{(\pi T)}_{py(x)}|^2\Delta^{(\pi T)}_{py(x)}) \\ + & \displaystyle (\gamma_3|\Delta_s|^2+\gamma_4|\Delta_d|^2) \Delta^{(\pi T)}_{px(y)} + 2(\gamma_5\Delta_s^2+\gamma_6\Delta_d^2) \Delta^{(\pi T)*}_{px(y)} \\ \pm & \displaystyle \gamma_7(\Delta_s\Delta_d^*+c.c.)\Delta^{(\pi T)}_{px(y)} \pm 2\gamma_8 \Delta_s\Delta_d\Delta^{(\pi T)*}_{px(y)} +(\gamma_9|\Delta_s|^2+\gamma_{10}|\Delta_d|^2)\Delta^{(\pi T)}_{py(x)} \\ + & \displaystyle (\gamma_{11}\Delta_s^2+\gamma_{12}\Delta_d^2) \Delta^{(\pi T)*}_{py(x)} +(\gamma_{mp1}\Delta^{(\pi T)}_{px(y)} +\gamma_{mp2}\Delta^{(\pi T)}_{py(x)})M^2 \\ + & \displaystyle (\gamma_{spm}\Delta_s \pm \gamma_{dpm}\Delta_d)M = 0, \end{array}$$ $$\begin{array}{rl} & \displaystyle \alpha_m M+ 2\beta_m M^3 -K_m(\nabla_x^2+\nabla_y^2)M \\ + & \displaystyle (\gamma_{ms}|\Delta_s|^2+\gamma_{md}|\Delta_d|^2)M + [\gamma_{mp1}(|\Delta^{(\pi T)}_{px}|^2+|\Delta^{(\pi T)}_{py}|^2) +\gamma_{mp2}(\Delta^{(\pi T)}_{px}\Delta^{(\pi T)*}_{py}+c.c.)]M \\ + & \displaystyle \frac{1}{2} \gamma_{spm}[\Delta_s^*(\Delta^{(\pi T)}_{px}+\Delta^{(\pi T)}_{py}) +c.c.] + \frac{1}{2} \gamma_{dpm}[\Delta_d^*(\Delta^{(\pi T)}_{px} -\Delta^{(\pi T)}_{py})+c.c.] = 0, \end{array}$$ where the coefficients appearing in eqs. (9)-(12) are given in the Appendix, and ${\bf D}$ is the gauge-invariant gradient defined as $ {\bf D} \equiv {\bf \nabla} +\frac{2\pi i}{\phi_0}{\bf A}$. Equations (9)-(12) are the coupled equations that determine SCOPs and the staggered magnetization self-consistently. The GL free energy $F$ up to the fourth order in OPs can be obtained from the above GL equations in such a way that the variations of $F$ with respect to OPs reproduce eqs. (9)-(12). The results are written as follows: $$\begin{array}{rl} % \displaystyle F = & \displaystyle F_S + F_T + F_{ST} +F_M + F_{SM} + F_{TM} + F_{STM}, \\ % F_S = &\displaystyle \int d^2{\bf r} \Big[ \alpha_s |\Delta_s|^2 + \beta_s |\Delta_s|^4 + K_s |{\vec D} \Delta_s|^2 + \alpha_d |\Delta_d|^2+ \beta_d |\Delta_d|^4 + K_d |{\vec D} \Delta_d|^2 \\ & \displaystyle + \gamma_1 |\Delta_s|^2|\Delta_d|^2 + \gamma_2 \big(\Delta_d^2(\Delta_s^*)^2 + c.c.\big) \\ & + K_{ds} \big((D_x\Delta_d)(D_x\Delta_s)^{*} - (D_y\Delta_d)(D_y\Delta_s)^{*} + c.c. \big)\Big], \\ % F_T = &\displaystyle \int d^2{\bf r} \Big[ \alpha_{p1}\big(|\Delta_{px}^{(\pi T)}|^2 + |\Delta_{py}^{(\pi T)}|^2\big) + \alpha_{p2}\big(\Delta_{px}^{(\pi T)}(\Delta_{py}^{(\pi T)})^* + c.c\big) + \beta_p\big(|\Delta_{px}|^4 + |\Delta_{py}|^4\big) \\ % & \displaystyle + \gamma_{p1} |\Delta_{px}^{(\pi T)}|^2|\Delta_{py}^{(\pi T)}|^2 + \gamma_{p2}\big((\Delta_{px}^{(\pi T)})^2(\Delta_{py}^{(\pi T)*})^2 + c.c.\big) \\ & \displaystyle + \gamma_{p3} \big(|\Delta_{px}^{(\pi T)}|^2 + |\Delta_{py}^{(\pi T)}|^2\big) \big(\Delta_{px}^{(\pi T)}(\Delta_{py}^{(\pi T)})^*+ c.c.\big) \\ % & \displaystyle + K_{p1}\big(|D_x\Delta_{px}^{(\pi T)}|^2 + |D_y\Delta_{py}^{(\pi T)}|^2\big) + K_{p2}\big(|D_y\Delta_{px}^{(\pi T)}|^2 + |D_x\Delta_{py}^{(\pi T)}|^2\big) \\ % & \displaystyle + K_{p3}\big((D_x\Delta_{px}^{(\pi T)})^{*} (D_x\Delta_{py}^{(\pi T)}) + (D_y\Delta_{px}^{(\pi T)})^{*}(D_y\Delta_{py}^{(\pi T)}) + c.c.\big)\Big], \\ % % F_{ST} = &\displaystyle \int d^2{\bf r} \Big[ \big(|\Delta_{px}^{(\pi T)}|^2 + |\Delta_{py}^{(\pi T)}|^2\big) (\gamma_3|\Delta_s|^2 + \gamma_4|\Delta_d|^2) \\ % & \displaystyle + \big\{\big((\Delta_{px}^{(\pi T)})^2 + (\Delta_{py}^{(\pi T)})^2\big) (\gamma_5(\Delta_s^{*})^2 + \gamma_6(\Delta_d^{*})^2) + c.c \big\} \\ % & \displaystyle + \gamma_7 \big(|\Delta_{px}^{(\pi T)}|^2 - |\Delta_{py}^{(\pi T)}|^2\big) \big(\Delta_s^{*}\Delta_d + c.c.\big) + \gamma_8 \big\{\big((\Delta_{px}^{(\pi T)})^2 - (\Delta_{py}^{(\pi T)})^2\big) \Delta_s^{*}\Delta_d^{*} + c.c.\big\}\Big] \\ & \displaystyle + \big((\Delta_{px}^{(\pi T)})^*\Delta_{py}^{(\pi T)}+c.c.\big) (\gamma_9 |\Delta_s|^2 + \gamma_{10} |\Delta_d|^2) \\ & \displaystyle + \big\{\Delta_{px}^{(\pi T)}\Delta_{py}^{(\pi T)} (\gamma_{11}(\Delta_s^*)^2 + \gamma_{12}(\Delta_d^*)^2) +c.c. \big\}\Big], \\ % F_M = &\displaystyle \int d^2{\bf r} \Big[ \alpha_m M^2 + \beta_m M^4 + K_m \big(\nabla M\big)^2\big], \\ % F_{SM} = &\displaystyle \int d^2{\bf r} \Big( \gamma_{ms} M^2 |\Delta_s|^2 + \gamma_{md} M^2 |\Delta_d|^2 \Big), \\ % F_{TM} = &\displaystyle \int d^2{\bf r} \Big[ \gamma_{mp1} M^2\Big(|\Delta_{px}^{(\pi T)}|^2 + |\Delta_{py}^{(\pi T)}|^2\Big) \\ & \displaystyle + \gamma_{mp2} M^2 \Big(\Delta_{px}^{(\pi T)} (\Delta_{py}^{(\pi T)})^* + c.c.\Big)\Big], \\ % \displaystyle F_{STM} = &\displaystyle \int d^2{\bf r} \Big[ \gamma_{spm} M \Delta_s\big(\Delta_{px}^{(\pi T)} + \Delta_{py}^{(\pi T)}\big)^* \\ & \displaystyle + \gamma_{dpm} M \Delta_d\big(\Delta_{px}^{(\pi T)} - \Delta_{py}^{(\pi T)}\big)^* + c.c.\Big]. \end{array}$$ Here, $F_S$, $F_T$, and $F_M$ are the free energy for the singlet and $\pi$-triplet SCOPs, and the staggered magnetization, respectively, while $F_{ST}$, $F_{SM}$, $F_{TM}$, and $F_{STM}$ describe their couplings. Note that $F$ is invariant under all the symmetry operations of the square lattice. $F_{SM}$ and $F_{TM}$ are the usual terms to represent the competition of SCOPs and $M$. $F_{STM}$ is a cubic term that couples spin-singlet SCOPs, staggered magnetization, and $\pi$-triplet SCOPS, and it induces $\pi$-triplet SCOPs in the coexistent state of AF and SC. Generally in the coexistent state of ferromagnetism and spin-singlet SC state, spin-triplet SCOPs may occur when OPs are not uniform in space.[@KK2; @Berg2; @Esch; @Buz; @Berg] In the GL theory this can be explained by a cubic term that has a gradient coupling of spin-singlet, triplet SCOPs, and the magnetization $m$.[@KKYano] In the AF state magnetization $m$ is oscillating (though the staggered magnetization $M$ is uniform) even in a uniform case, and thus $\pi-$triplet SCOP can arise irrespective of the spatial dependence of OPs. The important point of the present results is that the coefficients appearing in GL equations and the GL free energy are determined microscopically. These values depend on the parameters of the microscopic model and they reflect the evolution of the shape of the Fermi surface. This property can be used to study the material dependence of the coexistent states in various multilayer high-$T_C$ cuprates. \[sec:summary\]Summary and Discussion ===================================== We have derived GL equations and the GL free energy microscopically from the extended $t-J$ model using the slave-boson mean-field approximation. The derived GL theory can be used to investigate the spatial dependence of the AF and SC order parameters in high-$T_C$ cuprate superconductors. By analyzing the spatial variations of order parameters using the present results, information on the electronic states of high-$T_C$ cuprates may be extracted. A typical example to be studied is the state near the surface or impurity. The interface states of heterostructures composed of cuprate superconductors and magnetic materials are also worth studying. There the coexistence and competition of superconductivity and magnetism can occur in various ways depending on the materials used. Numerical study of the GL equations for the above situations assuming various band structure (by choosing the extended transfer integrals) may be interesting, and this problem will be examined separately. The author thanks H. Yamase for useful discussions. Coefficients in GL Equations and GL Free Energy =============================================== The coefficients appearing in GL equations \[eqs.(9)-(12)\] and the GL free energy \[eq.(13)\] are given as follows: $$\begin{array}{rl} % & \displaystyle\alpha_{s(d)} = 3J \Big(1-\frac{3J}{4N}\sum_p I_1(p) \omega_{s(d)}^2 \Big), \\ % & \displaystyle \beta_{s(d)} = \frac{81J^4}{32N}\sum_p I_2(p) \omega_{s(d)}^4, \\ % & \displaystyle \gamma_1= \frac{81J^4}{8N}\sum_p I_2(p) \omega_s^2 \omega_d^2, \ \ \ \gamma_2 = \frac{1}{4} \gamma_1, \\ % & \displaystyle K_{s(d)}= \frac{9J^2}{8N} \sum_p I_2(p) \Big(\frac{\partial \xi_p}{\partial p_x}\Big)^2 \omega_{s(d)}^2, \\ % & \displaystyle K_{ds}= \frac{9J^2}{8N} \sum_p I_2(p) \Big(\frac{\partial \xi_p}{\partial p_x}\Big)^2 \omega_s \omega_d, \\ % & \displaystyle \alpha_{p1} = -\frac{J}{2} \Big(1+\frac{J}{2N}\sum_p I_3(p) \cos^2p_x \Big) , \\ & \displaystyle \alpha_{p2} = -\frac{J^2}{4N}\sum_p I_3(p) \cos p_x \cos p_y, \\ % & \displaystyle \beta_p = \frac{J^4}{32N}\sum_p I_4(p) \cos^4p_x,\\ % & \displaystyle \gamma_{p1} = \frac{J^4}{8N}\sum_p I_4(p) \cos^2p_x\cos^2p_y, \ \ \ \gamma_{p2} = \frac{1}{4} \gamma_{p1}, \\ % & \displaystyle \gamma_{p3} = \frac{J^4}{16N}\sum_p I_4(p) \cos^3p_x\cos p_y, \\ % & \displaystyle K_{p1(2)} = - \frac{J^2}{8N}\sum_p I_4(p) \Big(\frac{\partial \xi_p}{\partial p_x}\Big)^2 \cos^2p_{x(y)}, \\ % & \displaystyle K_{p3} = - \frac{J^2}{8N}\sum_p I_4(p) \Big(\frac{\partial \xi_p}{\partial p_x}\Big)^2 \cos p_x\cos p_y, \\ % & \displaystyle \gamma_{3(4)} = \frac{9J^4}{8N} \sum_p I_5(p) \omega_{s(d)}^2 \cos^2p_x, \\ % & \displaystyle \gamma_{5(6)} = \frac{9J^4}{32N} \sum_p I_6(p) \omega_{s(d)}^2 \cos^2p_x, \\ % & \displaystyle \gamma_7 = \frac{9J^4}{8N} \sum_p I_5(p) \omega_s\omega_d \cos^2p_x, \\ % & \displaystyle \gamma_8 = \frac{9J^4}{16N} \sum_p I_6(p) \omega_s\omega_d \cos^2p_x, \\ % & \displaystyle \gamma_{9(10)} = \frac{9J^4}{8N} \sum_p I_5(p) \omega_{s(d)}^2 \cos p_x \cos p_y, \\ % & \displaystyle \gamma_{11(12)} = \frac{9J^4}{16N} \sum_p I_6(p) \omega_{s(d)}^2 \cos p_x \cos p_y, \\ % & \displaystyle \alpha_m = 2J\Big(1+\frac{2J}{N}\sum_p I_7(p)\Big), \\ % \end{array}$$ $$\begin{array}{rl} & \displaystyle \beta_m = \frac{8J^4}{N}\sum_p I_8(p), \\ % & \displaystyle K_m = \frac{4J^2}{N}\sum_p I_8(p) \Big(\frac{\partial \xi_p}{\partial p_x}\Big)^2, \\ % & \displaystyle \{ \gamma_{ms}, \ \gamma_{md}\} = - \frac{9J^4}{N} \sum_p [2I_9(p)+I_6(p)] \{ \omega_s^2, \omega_d^2\}, \\ % & \displaystyle \{\gamma_{mp1}, \ \gamma_{mp2}\} = - \frac{J^4}{N} \sum_p [2I_{10}(p)+I_6(p)] \{\cos^2p_x, \cos p_x\cos p_y\}, \\ % & \displaystyle \{ \gamma_{spm}, \ \gamma_{dpm}\} = -\frac{3J^3}{N} \sum_p I_{11}(p) \cos p_x \{\omega_s,\omega_d\}, \end{array}$$ where $\omega_s=\cos p_x+\cos p_y$ and $\omega_d=\cos p_x -\cos p_y$, and the summation on $p$ is taken over the first Brillouin zone. The functions appearing in the integrands are defined as $$\begin{array}{rl} I_1(p) = & \displaystyle T\sum_{\varepsilon_n} G_0(p,i\varepsilon_n)G_0(p,-i\varepsilon_n), \\ I_2(p) = & \displaystyle T\sum_{\varepsilon_n} G_0^2(p,i\varepsilon_n)G_0^2(p,-i\varepsilon_n), \\ I_3(p) = & \displaystyle T\sum_{\epsilon_n}G_0(p,-i\varepsilon_n)G_0(p+Q,i\varepsilon_n), \\ %= & \displaystyle %\frac{1}{2\mu}\big[f(\xi_{p+Q})-f(-\xi_p)\big], \\ % I_4(p) = & \displaystyle T\sum_{\epsilon_n}G_0^2(p,i\varepsilon_n)G_0^2(p+Q,-i\varepsilon_n), \\ %= & \displaystyle %\frac{1}{4\mu^2}\big[f'(\xi_p)+f'(\xi_{p+Q})\big] %+ \frac{1}{4\mu^3}\big[f(\xi_p)-f(-\xi_{p+Q})\big], \\ % I_5(p) = & \displaystyle T\sum_{\epsilon_n} G_0^2(p,i\varepsilon_n)G_0(p,-i\varepsilon_n)G_0(p+Q,-i\varepsilon_n), \\ %= & \displaystyle %-\frac{1}{4\mu\xi_p} %\Big[f'(\xi_p)+f(\xi_p) %\Big(\frac{1}{2\mu}-\frac{1}{2\xi_p}\Big)\Big] %+\frac{1}{4(\xi_p-\xi_{p+Q})}\Big[\frac{1}{\mu^2}f(-\xi_{p+Q}) %-\frac{1}{\xi_p^2}f(-\xi_p)\Big], \\ % I_6(p) = & \displaystyle T\sum_{\epsilon_n}G_0(p,i\varepsilon_n)G_0(p,-i\varepsilon_n) G_0(p+Q,i\varepsilon_n)G_0(p+Q,-i\varepsilon_n), \\ %= & \displaystyle %-\frac{1}{4\mu(\xi_p-\xi_{p+Q})} %\Big[\frac{\tanh\big(\frac{\xi_{p+Q}}{2T}\big)}{\xi_{p+Q}} %- \frac{\tanh\big(\frac{\xi_p}{2T}\big)}{\xi_p}\Big], \\ % I_7(p) = & \displaystyle T\sum_{\epsilon_n}G_0(p,i\varepsilon_n)G_0(p+Q,i\varepsilon_n), \\ %= & \displaystyle %\frac{f(\xi_p)-f(\xi_{p+Q})}{\xi_p-\xi_{p+Q}}, \\ % I_8(p) = & \displaystyle T\sum_{\epsilon_n}G_0^2(p,i\varepsilon_n)G_0^2(p+Q,i\varepsilon_n), \\ %= & \displaystyle %\frac{f'(\xi_p)+f'(\xi_{p+Q})}{(\xi_p-\xi_{p+Q})^2} %-2 \frac{f(\xi_p)-f(\xi_{p+Q})}{(\xi_p-\xi_{p+Q})^3}, \\ % I_9(p) = & \displaystyle T\sum_{\epsilon_n} G_0^2(p,i\varepsilon_n)G_0(p,-i\varepsilon_n)G_0(p+Q,i\varepsilon_n), \\ %= & \displaystyle %\frac{1}{2\xi_p(\xi_p-\xi_{p+Q})} %\Big[f(\xi_p)\Big(\frac{1}{\xi_p-\xi_{p+Q}}+\frac{1}{2\xi_p}\Big)-f'(\xi_p)\Big] %- \frac{1}{2\mu} %\Big[\frac{f(-\xi_p)}{4\xi_p^2} %-\frac{f(\xi_{p+Q})}{(\xi_p-\xi_{p+Q})^2}\Big], \\ % I_{10}(p) = & \displaystyle T\sum_{\epsilon_n} G_0^2(p,i\varepsilon_n)G_0(p+Q,i\varepsilon_n)G_0(p+Q,-i\varepsilon_n), \\ %= & \displaystyle %\frac{1}{2\mu(\xi_p-\xi_{p+Q})} %\Big[f'(\xi_p)-f(\xi_p)\Big(\frac{1}{\xi_p-\xi_{p+Q}} %-\frac{1}{2\mu}\Big)\Big] %+ \frac{1}{2\xi_{p+Q}}\Big[\frac{f(-\xi_{p+Q})}{4\mu^2} %-\frac{f(\xi_{p+Q})}{(\xi_p-\xi_{p+Q})^2}\Big], \\ % I_{11}(p) = & \displaystyle T\sum_{\epsilon_n} G_0(p,i\varepsilon_n)G_0(p,-i\varepsilon_n)G_0(p+Q,i\varepsilon_n). \\ %= & \displaystyle %-\frac{1}{2\xi_p} %\Big[\frac{f(\xi_p)}{\xi_p-\xi_{p+Q}}-\frac{f(-\xi_p)}{2\mu}\Big] %-\frac{f(\xi_{p+Q})}{2\mu(\xi_p-\xi_{p+Q})}. \end{array}$$ [9]{} Y. Kitaoka, H. Mukuda, S. Shimizu, S. Tabata, P. M. Shirage, and A. Iyo: J. Phys. Chem. Solids [**72**]{} (2011) 486. H. Mukuda, S. Shimizu, A. Iyo, and Y. Kitaoka: J. Phys. Soc. Jpn. [**81**]{} (2012) 011008. B. Keimer, N. Belk, B. J. Birgeneau, A. Cassanho, M. Greven, M. A. Kastner, A. Aharony, Y. Endoh, R. W. Erwin, and G. Shirane: Phys. Rev. B[**46**]{} (1992) 14034. S. Sanna, G. Allodi, G. Concas, A. D. Hillier, and R. De Renzi: Phys. Rev. Lett. [**93**]{} (2004) 207001. A, Iyo, Y. Tanaka, H. Kito, Y. Kodama, P. M. Shirage, D. D. Shivagan, H. Matsuhata, K. Tokiwa, and T. Watanabe: J. Phys. Soc. Jpn. [**76**]{} (2007) 094711. P. W. Anderson: Science [**235**]{} (1987) 1196. M. Ogata and H. Fukuyama: Rep. Prog. Phys. [**71**]{} (2008) 036501. P. A. Lee, N. Nagaosa, and X.-G. Wen: Rev. Mod. Phys. [**78**]{} (2006) 17. G. Kotliar and J. Liu: Phys. Rev. B[**38**]{} (1988) 5142. Y. Suzumura, Y. Hasegawa, and H. Fukuyama: J. Phys. Soc. Jpn. [**57**]{} (1988) 2768. Z. Zou and P. W. Anderson: Phys. Rev. B[**37**]{} (1988) 627. N. Nagaosa and P. A. Lee: Phys. Rev. Lett. [**64**]{} (1990) 2450. K. Kuboki, M. Yoneya, and H. Yamase: Physica C[**470**]{} (2010) S163. H. Yamase, M. Yoneya, and K. Kuboki: Phys. Rev. B[**84**]{} (2011) 014508. M. Inaba, H. Matsukawa, M. Saitoh, and H. Fukuyama: Physica C [**257**]{} (1996) 299. H. Yamase and H. Kohno: Phys. Rev. B[**69**]{} (2004) 104526. A. Himeda and M. Ogata: Phys. Rev. B[**60**]{} (1999) R9935. Y. Ren, J-H. Xu, and C. S. Ting: Phys. Rev. Lett. [**74**]{} (1995) 3680. D. L. Feder and C. Kallin: Phys. Rev. B[**55**]{} (1997) 559. K. Kuboki and K. Yano: J. Phys. Soc. Jpn. [**81**]{} (2012) 064711. E. K. Dahl and A. Sudbø: Phys. Rev. B[**75**]{} (2007) 144504. K. Kuboki adn M. Sigrist: J. Phys. Soc. Jpn. [**67**]{} (1998) 2873. L. P. Gor’kov: Sov. Phys. JETP [**9**]{} (1959) 1364. P. A. Lee and N. Nagaosa: Phys. Rev. B[**46**]{} (1992) 5621. D. H. Kim and P. A. Lee: Ann. Phys. [**272**]{} (1999) 130. G. C. Psaltakis and E. W. Fenton: J. Phys. C[**16**]{} (1983) 3913. M. Murakami and H. Fukuyama: J. Phys. Soc. Jpn. [**67**]{} (1998) 2784. M. Murakami: J. Phys. Soc. Jpn. [**69**]{} (2000) 1113. B. Kyung: Phys. Rev. B[**62**]{} (2000) 9083. A. Aperis, G. Varelogiannis, P. B. Littlewood, and B. D. Simons: J. Phys.: Condens. Matter [**20**]{} (2008) 434235. K. Kuboki: J. Phys. Soc. Jpn. [**68**]{} (1999) 3150. F. S. Bergeret, A. F. Volkov, and K. B. Efetov: Phys. Rev. Lett. [**86**]{} (2001) 4096. M. Eschrig, J. Kopu, J. C. Cuevas, and G. Schön: Phys. Rev. Lett. [**90**]{} (2003) 137003. A. I. Buzdin: Rev. Mod. Phys. [**77**]{} (2005) 935. F.S. Bergeret, A. F. Volkov, and K. B. Efetov: Rev. Mod. Phys. [**77**]{} (2005) 1321. [^1]: E-mail address: kuboki@kobe-u.ac.jp

--- author: - Uwe Lück title: | **Supplementary files belonging\ to the *lineno.sty* distribution\ Lazy `ASCII`$\to$`PDF` listings** --- \[2011/02/16 documenting supplementary files\] Preface {#preface .unnumbered} ======= `lineno.sty` is a macro package made by Stephan I. Böttcher for attaching line numbers to LaTeX documents. Some people have used it for revising submittings in collaboration with referees or co-authors. Documentations are nowadays preferred to be in Adobe’s `PDF`—so `lineno.sty`’s documentation is [ `lineno.pdf`[^1]]{}. `ednotes.sty` uses `lineno.sty` for critical editions, combining it with Alexander I. Rozhenko’s `manyfoot.sty`—this was Christian Tapp’s idea, who then hired me for adding the TeXnical details. In doing this, I had to change some internals of `lineno.sty`, so Stephan transferred maintenance to me; then some of my macro files that I originally had made for `ednotes.sty` wandered into the `lineno` directory of CTAN—because they turned out not to need `ednotes.sty`, just to work as extensions of `lineno.sty`. Now, I haven’t had the time for making `.dtx` versions of the `.sty` files for `ednotes`. Therefore, ordinary `.pdf` documentation for the remaining `.sty` files of `lineno` is missing. What you see here is nothing but a somewhat structured listing of the additional `.txt` and `.sty` files in `PDF`, deriving from the `verbatim` package and its [`"5Cverbatiminput`]{} command. I hope the high quality (scalable) output is worth it. By contrast, the new package `fnlineno.sty` added in 2011 for footnote line numbers is documented in [ `fnlineno.pdf`[^2]]{} in high quality, using the [ `nicetext`[^3]]{} bundle. *U.L.* The `.txt` files ================ Summary: `README.txt` --------------------- Licenses/Copyright: `COPYING.txt` --------------------------------- Update summaries: `CHANGEs.txt` ------------------------------- Source file infos: `SRCFILEs.txt` --------------------------------- Tabular and array environments ============================== `lineno.sty`’s package options `edtable`, `longtable`, and `nolongtablepatch` redefine LaTeX tabular and array environments such that `lineno` and `ednotes` commands can be used inside. The code for these options resides in separate files at present. We are listing them here. `edtable.sty` ------------- `ltabptch.sty` -------------- [`"5Clinelabel`]{} and notes from *math* mode: [ typeset@protect\ &gt;@ ]{} `ednmath0.sty` ================================================================= Extended line number references: `vplref.sty` ============================================= `vplref.sty` is input through the `lineno` package option `addpageno`. This adds page numbers to line number references to distant sides—using the `varioref` package from the LaTeX distribution. [^1]: <http://mirror.ctan.org/macros/latex/contrib/lineno/lineno.pdf> [^2]: <http://mirror.ctan.org/macros/latex/contrib/lineno/fnlineno.pdf> [^3]: <http://mirror.ctan.org/macros/latex/contrib/nicetext>

--- abstract: 'A simple new algorithm for the calculation of two-particle Bose-Einstein correlations from classical event generators is derived and discussed.' address: | Institut für Theoretische Physik, Universität Regensburg,\ D-93040 Regensburg, Germany author: - 'Q.H. Zhang[@home], U.A. Wiedemann, C. Slotta, and U. Heinz' title: 'Bose-Einstein Weights for Event Generators' --- -1cm For ultrarelativistic heavy-ion collisions, two-particle Bose-Einstein correlations between identical pions or kaons provide a unique possibility to reconstruct the geometry (size, temporal extension) ${\em and}$ dynamics (collective expansion flow) of the source at the point of hadron freeze-out. (For a recent theoretical review see [@H96].) This reconstruction is, however, not completely model independent; it requires the use of “reasonable” source parametrizations (e.g. [@H96; @CL96; @CN96]) whose parameters are then fixed by a simultaneous analysis of single-particle momentum spectra and two-particle momentum correlations [@CN96; @S96]. Invaluable help for the selection of “reasonable” source parametrization comes from microscopic event generators (e.g. VENUS [@W93], RQMD [@SSG89] or ARC [@PSK92]) which generate the phase-space distribution of hadrons at freeze-out from a dynamical Monte Carlo simulation of the (classical) kinetic phase-space evolution of the collision zone. Unfortunately, it was recently pointed out [@A97; @MKF96] that the direct computation of two-particle correlation functions from classical kinetic codes [@W93; @SSG89; @PSK92] is fraught with severe conceptual and practical problems. These can be simply understood by starting from the general relation [@S73] between the 2-particle correlation function $C(\bbox{q}, \bbox{K})$ and the (real) “emission function” (one-particle Wigner density at freeze-out) $S(x,K)$, \[1\] $$\begin{aligned} \label{1a} \FL C(\bbox{q},\bbox{K}) = 1 + \frac{{\left\vert\int d^{4}x\, S(x,K)\, e^{iq\cdot x} \right\vert}^{2}} {\int d^{4}x\, S(x,p_{a}) \,\int d^{4}y\, S(y,p_{b})} \, , \\ \label{1b} \FL \bbox{q}{=}\bbox{p}_a{-}\bbox{p}_b, \ q^{0}{=}E_{a}{-}E_{b},\ \bbox{K}{=}{\bbox{p}_a{+}\bbox{p}_b \over 2},\ K^{0}{=}{E_{a}{+}E_{b} \over 2}. \end{aligned}$$ $(\bbox{p}_a, E_{a})$ and $(\bbox{p}_b, E_{b})$ are the 4-momenta of the two observed particles. Eq. (\[1a\]) neglects final state interactions which we will leave out in this Letter in order to concentrate on the principal issues. The numerator in the second term of Eq. (\[1a\]) can be rewritten as $$\begin{aligned} \label{2} {\rm Num} (q,K) =&& \int d^4x\, d^4y \,S(x,K)\, S(y,K) \nonumber\\ && \ \times\, \cos(q{\cdot}(x-y)) \, . \end{aligned}$$ The problem is the construction of the Wigner density $S(x,K)$ from the output of the event generators. The latter consists of a set of phase-space points $(x_{i},p_{i})$ denoting the (on-shell) momenta $p_{i}$ and points of last interaction $x_{i}$ of the produced particles. According to (\[1b\]) $K$ is the average of two on-shell momenta, but not itself on-shell, $K^{0} \ne \sqrt{m^2+{\bbox{K}^2}}$. Therefore $S(x,K)$ cannot be directly related to the phase-space density generated by the distribution of points $(x_{i},p_{i})$. To overcome this difficulty one ususally imposes [@YK78; @P94] the “smoothness assumption” $S(x,\frac{p_a+p_b}{2}) \approx S(x,p_a) \approx S(x,p_b)$ and rewrites the expression (\[2\]) as $$\begin{aligned} \label{3} {\rm Num}(q,K) =&& \int d^{4}x\, d^{4}y \, S(x,p_a) \, S(y,p_b) \nonumber\\ && \times \cos\bigl((p_a-p_b){\cdot}(x-y)\bigr)\, . \end{aligned}$$ One now identifies $S(x,p)$ with the classical output distribution from the event generator, $$\label{4} S_{\rm class}(x,p) = \sum_{i=1}^{N} \delta^{(4)}(x-x_{i})\, \delta^{(4)}(p-p_{i})\, ,$$ where $N$ is the total number of pions of a given charge in the event. In the widely used Pratt algorithm [@P94] the expression which results after inserting (\[4\]) into (\[3\]) is simulated by the ad hoc prescription $$\label{5} {\rm Num} (q,K) \longmapsto \sum_{i,j \in {\rm bin}} \cos\bigl((p_i-p_j){\cdot}(x_i-x_j)\bigr) \, .$$ Here “bin” denotes a small bin in $(\bbox{q},\bbox{K})$ with $\bbox{p}_i-\bbox{p}_j \approx \bbox{q}$ and $(\bbox{p}_i+\bbox{p}_j)/2 \approx \bbox{K}$. (In practice the bin size depends on event statistics.) The prescription (\[5\]) has two severe problems: first, the positivity of Num$(q,K)$ got lost between Eqs. (\[2\]) and (\[3\]) when making the “smoothness assumption”. It was pointed out in Refs. [@CH94; @PRW94] and practically demonstrated in Ref. [@MKF96] that for sources with strong $x$-$p$-correlations (e.g. rapidly expanding sources) this can lead to unphysical oscillations of the simulated correlation function around unity. Second, the intuitive substitution law (\[5\]) is formally incorrect and results in a wrong selection of contributing pairs as well as an incorrect weighting factor for each pair. To prove this last point let us write in (\[2\]) $$\begin{aligned} \label{6} S(x,K)\!\!&&\!\!\! S(y,K) = \int d^4P_1\, d^4P_2 \, S(x,P_1)\, S(y,P_2) \nonumber\\ &\times&\, \delta^{(4)}(P_1-P_2) \, \delta^{(4)}\left(K-\frac{P_1+P_2}{2}\right)\, . \end{aligned}$$ Inserting the classical expression (\[4\]) one finds instead of (\[5\]) \[7\] $$\begin{aligned} \label{7a} {\rm Num}(q,K) =&& \sum_{i,j=1}^{N} \delta^{(4)}(p_i-p_j) \, \delta^{(4)}\left(K-\frac{p_i+p_j}{2}\right) \nonumber\\ && \qquad \times \cos\bigl(q{\cdot}(x_i-x_j)\bigr) \\ \label{7b} \longmapsto && \sum_{i,j \in {\rm bin}(K,\epsilon)} \cos\bigl(q{\cdot}(x_i-x_j)\bigr)\, . \end{aligned}$$ Here “${\rm bin}(K,\epsilon)$” denotes a small bin around $K$ with width $\epsilon$ in each of the four directions. The prescription (\[7b\]) can be derived rigorously and directly by first generating from Eq. (\[4\]) a piecewise constant function (“histogram”) through “binning”, $$\begin{aligned} \label{binning} \bar S (x,K) &=& \int_{{\rm bin}(K,\epsilon)} d^4p\, S_{\rm class}(x,p) \nonumber\\ &=& \int_ {K-\epsilon/2}^{K+\epsilon/2} d^4p\, S_{\rm class}(x,p)\, , \end{aligned}$$ and then inserting $\bar S(x,K)$ into Eq. (\[6\]). Eq. (\[binning\]) is a technical step required by finite event statistics; in practice, $\epsilon$ should be chosen as small as technically possible. Note that the selection of pairs in (\[7b\]) differs from the one in (\[5\]): for given $K$ the algorithm (\[7\]) selects pairs with $p_i \approx p_j \approx K$, independent of the value $\bbox{q}$ at which the correlation is to be evaluated. For different values of $\bbox{q}$ at fixed $\bbox{K}$, the correlator is obtained by weighting the [*same*]{} set of pairs with different weight factors $\cos\bigl(q{\cdot}(x_i-x_j)\bigr)$ which depend only on the spatial coordinates, but not on the momenta of the particles in the pair. This is consistent with the expectation from Eqs. (\[1a\],\[2\]) that the measured $\bbox{q}$-dependence of the correlator gives access to the distribution of relative distances $x_i-x_j$ in the source (at fixed $K$). Since the steps from Eq. (\[2\]) to Eq. (\[7\]) involve only identical transformations (the difference between (\[7a\]) and (\[7b\]) arising only from a different choice of emission functions (\[4\]) resp. (\[binning\])), they preserve the positivity of the second term in (\[1a\]). Please note that in contrast to (\[5\]), (\[7\]) is a continuous function of $\bbox{q}$, i.e. no binning in $\bbox{q}$ is required. We have checked for the simple analytically solvable model presented in [@MKF96] that the algorithm (\[7\]) indeed allows to reconstruct from a classical Monte Carlo simulation the correct analytical expression for $C(\bbox{q},\bbox{K})$; in particular it removes the unphysical oscillations found in [@MKF96]. In Fig. 1 we show various approximations for the correlation function for a 1-dimensional source in $z$-direction with emission function $$\label{model} S(z,t;K) = e^{-z^2/R^2}\, \delta(K-\alpha z)\, \delta(t)\, ,$$ with $R{=}10$ fm and $\alpha{=}0.02$ GeV/fm. This classical source features perfect $z$-$K$-correlations, $K{=}\alpha z$, which are, of course, quantum mechanically forbidden (see below). For the source (\[model\]), the exact correlator (\[1\]) can be calculated analytically, yielding $C(q) = 1 + \exp[q^2/(2\alpha^2 R^2)]$ (solid line in Fig. 1). The pathological rise of $C(q)$ [ *above*]{} the value 2 at $q{=}0$ is due to the violation of the uncertainty relation between $z$ and $p_z$ by the model (\[model\]); we selected this model because we believed that such a feature may be particularly difficult to reproduce in an event generator. Indeed, reconstructing the correlator from a Monte Carlo simulation of (\[model\]) via the Pratt prescription (\[5\]) yields the long-dashed line in Fig. 1 [@MKF96]; it can be calculated analytically from Eq. (\[3\]) as $C(q) = 1 + \cos(q^2/\alpha)$. This is always less than 2, but oscillates wildly around 1, becoming even 0 at regular $q$-intervals. This contradicts the positivity of the second term in (\[1\]) and is due to the smoothness approximation (\[3\]). – The two remaining lines in Fig. 1 show results from the same Monte Carlo simulation of (\[model\]) but reconstructing the correlator through the new algorithm (\[7b\]) instead of (\[5\]). For the dot-dashed line the bin width $\epsilon$ was chosen as $\epsilon= 10$ MeV, for the dotted line as $\epsilon = 5$ MeV. In both cases the simulated result deviates from the exact one (solid line); this is not a failure of the algorithm, but a result of the binning procedure (\[binning\]) applied to the source (\[model\]) – a purely technical step required by finite event statistics. As seen, the discrepancy decreases with decreasing bin width $\epsilon$. -4.0cm -2.5cm While Eq. (\[7\]) thus solves the technical problems of the prescription (\[5\]), it does not address the principal physical problem that the classical distribution (\[4\]) is not a valid Wigner density since it violates the uncertainty relation by simultaneously fixing the coordinates $x_{i}$ and momenta $p_{i}$ of the emitted particles. It has been repeatedly suggested [@PGG90; @Z96; @MP97] that this can be remedied by replacing the sharply localized $\delta$-functions in (\[4\]) by minimum-uncertainty wave packets. In the remainder of this Letter we will discuss how such a procedure will modify the algorithm for calculating single-particle spectra and two-particle correlations from event generators, thereby rendering it quantum mechanically consistent. Let us start from the folding relation for the emission function, derived in Ref. [@CH94] within the covariant current formalism [@GKW79]: $$\label{8} S(x,K) = \int d^4z \, d^4Q \, \rho(x-z,Q) \, S_{0}(z,K-Q) \, .$$ Here $$\label{9} S_{0}(x,p) = \int d^4v\, e^{-ip{\cdot}v}\, j_0^*\left(x+{\textstyle{v\over 2}}\right) \, j_0\left(x-{\textstyle{v\over 2}}\right)$$ is the Wigner density associated with an elementary source current amplitude $j_0(x)$, taken below as a Gaussian wavepacket, and $\rho(x,p)$ is a classical phase-space distribution for the centers $x_i$ and (average) momenta $p_i$ of these wave packets, here taken as $$\label{10} \rho(x,p) = \sum_{i=1}^N \delta^{(4)}(x-x_i) \, \delta^{(4)}(p-p_i)\, .$$ For the elementary source amplitude we make the ansatz $$\label{11} j_0(x) = {\cal N} \, \exp\left(-{\bbox{x}^2 \over 2\sigma^2}\right) \, \delta(x^0) \, ;$$ this source emits a Gaussian wave-packet with width parameter $\sigma$ at “freeze-out time” $x^0$. With this ansatz the elementary Wigner density $S_0$ becomes $$\label{12} S_0(x,p) = 8 (\pi\sigma^2)^{\frac{3}{2}} \,\vert{\cal N} \vert^2 \delta(x^{0}) \exp\left(-{\bbox{x}^2 \over \sigma^2} -\sigma^2\bbox{p}^2 \right) \, .$$ Inserting this and (\[10\]) into (\[8\]) one finds $$\begin{aligned} \label{13} S(x,K) = {\cal N'} \sum_{i=1}^N &&\delta(x^0-x_i^0) \, \exp\left(-{(\bbox{x}-\bbox{x}_i)^2 \over \sigma^2}\right) \nonumber\\ &\times& \exp\left(-\sigma^2(\bbox{K}-\bbox{p}_i)^2 \right)\, , \end{aligned}$$ with ${\cal N'} = 8(\pi\sigma^2)^{\frac{3}{2}} \vert {\cal N}\vert^2$. This generalizes the classical ansatz (\[4\]) into a quantum mechanically consistent source Wigner density; with the free parameter $\sigma$ one can choose the relative degree of localization in coordinate space $(\sigma \to 0)$ or momentum space $(\sigma \to \infty)$, always preserving $\Delta x \cdot \Delta p = \hbar$. The single particle spectrum, which occurs in the denominator of the second term of (\[1a\]), is now given by \[14\] $$\begin{aligned} \label{14a} E_a {dN\over d^3p_a} &=& \int d^4x \, S(x,p_a) = {\cal N''} \sum_{i=1}^{N} v_i(\bbox{p}_a) \, , \\ \label{14b} {\cal N''} &=& (2\pi\sigma^2)^2 \vert {\cal N} \vert^2 \, , \\ \label{14c} v_ i (\bbox{p}_a) &=& \exp\bigl(-\sigma^2 (\bbox{p}_a-\bbox{p}_i)^2\bigr) \, , \end{aligned}$$ and similarly for $\bbox{p}_b$. It is normalized to the total number $N$ of pions in the event; this fixes the normalization constant ${\cal N}$ above. The exchange term (\[2\]) in the two-particle spectrum (with $q$ and $K$ defined in Eq. (\[1b\])) is similarly derived as \[15\] $$\begin{aligned} \label{15a} {\rm Num}(\bbox{q},\bbox{K})\! &=& \!({\cal N''})^2 \exp(-{\textstyle{1\over 2}}\sigma^2\bbox{q}^2) \sum_{i,j=1}^N w_{ij}(\bbox{q},\bbox{K}) , \\ \label{15b} w_{ij}(\bbox{q},\bbox{K})\! &=& \! v_i(\bbox{K}) \, v_j(\bbox{K}) \, \cos\bigl(q{\cdot}(x_i-x_j)\bigr) . \end{aligned}$$ The normalization ${\cal N''}$ drops out in the correlator (\[1a\]). Eq. (\[15\]) should be compared with the classical expressions (\[5\]) and (\[7\]). Like (\[7\]) (but contrary to (\[5\])) it is positive definite and thus free of spurious oscillations around 0. The sum in (\[15a\]) is now over [*all*]{} pairs $(i,j)$; the sharp restriction to the bin “${\rm bin}(K,\epsilon)$” in (\[7\]) is replaced by the Gaussian weight factors $v_i(\bbox{K})$ and $\exp(-{1\over 2} \sigma^2 \bbox{q}^2)$. Please note, however, that the former also occur in the new definition (\[14a\]) for the single particle spectrum, and must be kept in both places for consistency. Eq. (\[15b\]) shares with Eq. (\[7\]) the property (which was already discussed) that the cosine weight factor for each pair $(i,j)$ depends only on $x_i-x_j$, but not on $p_i$ and $p_j$. By combining Eqs. (\[14\]) and (\[15\]) it can be shown that now the correlator $C(\bbox{q,K})$ is always between 1 and 2, i.e. that the pathological rise above 2 shown in Fig. 1 cannot happen for an emission function which respects the uncertainty relation. In Ref. [@Wetal] the results (\[14\]) and (\[15\]) (but not (\[7\])) were derived with different methods, including finite multiplicity corrections. Expressions (\[14\]) and (\[15\]) depend on one free parameter, the Gaussian width $\sigma$ of the wave-packets. It is instructive to discuss the two obvious limits, $\sigma \to 0$ and $\sigma \to \infty$. For $\sigma=0$ the elementary sources are sharply localized in space (cf. Eq. (\[11\])); as a result, the single-particle momentum spectrum (\[14\]) is completely flat. Furthermore, one sees from Eqs. (\[15\]) that the correlation function becomes $K$-independent, even for expanding sources of type (\[10\]) where the $x_i$ and $p_i$ are strongly correlated. Both features are clearly unrealistic. In the opposite limit, $\sigma \to \infty$, the Gaussian smearing factors in the single particle spectrum (\[14\]) disappear, and Eq. (\[14a\]) degenerates to a sum over $\delta$-functions; this is the usually employed algorithm for computing single-particle spectra from event generators with classical particle propagation [@W93; @SSG89; @PSK92]. The two-particle correlation term (\[15\]), on the other hand, is then sharply concentrated at $q=0$, i.e. the correlator $C(\bbox{q},\bbox{K})$ drops from 2 to 1 over a $q$-range of order $1/\sigma$. This translates into a source radius $\sim \sigma$ and reflects the diverging spatial extension of the elementary wavepackets in this limit, irrespective of the (localized) spatial distribution $\rho(x,p)$ of their centers. It is thus clear that in practice $\sigma$ must be kept finite, but non-zero. As pointed out in [@PGG90; @Z96; @MP97] this implies a broadening of the single-particle momentum distributions relative to the one derived from the classical phase-space distribution $\rho (x,p)$ of freeze-out points. Using the algorithms (\[14\],\[15\]) for a quantum mechanically consistent computation of spectra and Bose-Einstein correlations from classical event generators thus requires a retuning of the codes to elementary $e^+e^-$ and $pp$ collisions, using the same algorithms there. This last step imposes rather restrictive limits for the value of $\sigma$ [@PGG90]. Since the pions from high energy $pp$ and $e^+e^-$ collisions have an average transverse momentum $\langle p_\perp \rangle \simeq 0.35$ GeV, the Gaussian width in momentum space of the elementary wave packets must be below this value. This implies $\sigma \gtrsim 0.5$ fm. On the other hand, the effective source radii for such collisions extracted from 2-particle correlations are of the order of only $0.8 - 2$ fm [@L89]. This implies $\sigma \lesssim 1$ fm. It was already pointed out in the pioneering GGLP paper [@GGLP60] that this value roughly agrees with the pion’s Compton wavelength. This suggests the following interpretation of the measured pion spectra and Bose-Einstein correlations from elementary hadron-hadron and $e^+e^-$ collisions: each elementary collision produces a small number of elementary Gaussian wave packets with width $\sigma$, 0.5 fm $< \sigma <$ 1 fm. For $\sigma \sim 0.5$ fm the width of the Gaussian momentum distribution in (\[14c\]) nearly exhausts the measured $\langle p_\perp \rangle$; the measured single-pion spectra thus reflect mostly the intrinsic momentum distribution of the elementary pion wave packet. On the other hand, the HBT radius $R_{\perp}^{\rm HBT} = \sigma/\sqrt{2}$ corresponding to the $q$-Gaussian in (\[15a\]) nearly exhausts the values extracted from two-particle correlation measurements; the latter thus mirror the intrinsic spatial width of the wave packets. There is very little additional room in the data for random (thermal) motion of the wave packets relative to each other, nor for their spatial distribution over a larger volume. If the total source is much bigger than 1 fm, it must expand very rapidly, with homogeneity regions which are not much larger than the size of an elementary wave packet. In this sense, pion spectra from $e^{+}e^{-}$ and $pp$ collision measure the smallest sources compatible with the uncertainty relation. In summary, we have derived a new algorithm for the computation of single-particle spectra and two-particle correlations from classical event generators. The classical algorithm (\[7\]) removes the recently discovered deficiencies of the presently employed Pratt algorithm (\[5\]). The quantum mechanical algorithm (\[14\],\[15\]) additionally ensures that the uncertainty relation is not violated. We also showed how the free parameter $\sigma$ in the latter algorithm can be fixed from elementary $e^{+}e^{-}$ and hadron-hadron collisions. We acknowledge stimulating conversations with P. Foka, H. Kalechofsky, M. Martin, and S. Pratt, as well as many discussions during the HBT96 Workshop at the ECT\* in Trento, Sept. 16-27, 1996, which brought the problems with (\[5\]) into the open. This work was supported by BMBF, GSI and DFG. Q.H.Z. acknowledges support by the Alexander von Humboldt Foundation through a Research Fellowship. [99]{} Humboldt Research Fellow; on leave from China Center of Advanced Science and Technology (CCAST). U. Heinz, in: [*Correlations and Clustering Phenomena in Subatomic Physics*]{}, ed. by M.N. Harakeh, O. Scholtan and J.H. Koch, NATO ASI Series B, (Plenum, New York, 1997), in press (Los Alamos eprint archive nucl-th/9609029) T. Csörgő and B. Lörstad, Phys. Rev. C[**54**]{} (1996) 1396; and Nucl. Phys. A[**590**]{} (1995) 465c. S. Chapman and J.R. Nix, Phys. Rev. C[**54**]{} (1996) 866. S. Schönfelder, (NA49 Coll.), Ph.D. thesis, TU München, 1996. K. Werner, Phys. Rep. [**232**]{} (1993) 87. H. Sorge, H. Stöcker, and W. Greiner, Ann. Phys. (N.Y.) [**192**]{} (1989) 266. Y. Pang, T.J. Schlagel, and S.H. Kahana, Phys. Rev. Lett. [**68**]{} (1992) 2743; T.J. Schlagel, Y. Pang, and S.H. Kahana, Phys. Rev. Lett. [**69**]{} (1992) 3290. J. Aichelin, Nucl. Phys. A (1997), in press (Los Alamos eprint archive nucl-th/9609006). M. Martin, H. Kalechofsky, P. Foka, and U.A. Wiedemann, Los Alamos eprint archive nucl-th/9612023. E. Shuryak, Phys. Lett. B[**44**]{} (1973) 387; Sov. J. Nucl. Phys. [**18**]{} (1974) 667. F. Yano and S. Koonin, Phys. Lett. B[**78**]{} (1978) 556. S. Pratt et al., Nucl. Phys. A[**566**]{} (1994) 103c; and in [*Quark-Gluon Plasma 2*]{}, ed. by R.C. Hwa (World Scientific, Singapore, 1995), p. 700. S. Chapman and U. Heinz, Phys. Lett. B[**340**]{} (1994) 250. M. Plümer, L.V. Razumov, and R.M. Weiner, Phys. Rev. D[**49**]{} (1994) 4434; A. Timmermann, M. Plümer, L.V. Razumov, and R.M. Weiner, Phys. Rev. C[**50**]{} (1994) 3060. S. Padula, M. Gyulassy, and S. Gavin, Nucl. Phys. B[**329**]{} (1990) 357. J. Zimányi, talk given at the HBT96 Workshop, ECT\* Trento, Sep. 16-27, 1996. H. Merlitz and D. Pelte, Los Alamos eprint archive nucl-th/9702005. M. Gyulassy, S.K. Kauffmann, and L.W. Wilson, Phys. Rev. C[**20**]{} (1994) 2267. U.A. Wiedemann, P. Foka, H. Kalechofsky, M. Martin, C. Slotta, Q.H. Zhang, in preparation. B. Lörstad, Int. J. Mod. Phys. A[**12**]{} (1989) 2861. For a non-relativistic Boltzmann distribution, $\rho(x,p) \sim \exp[-\bbox{p}^2/(2mT)]$, the prescription (\[14\]) results in an effective Boltzmann distribution with an increased temperature $T_{\rm eff} = T + 1/(2m\sigma^2)$ [@Z96; @MP97]. G. Goldhaber, S. Goldhaber, W. Lee, and A. Pais, Phys. Rev. [**120**]{} (1960) 300.

--- abstract: 'Let $M$ be a smooth manifold, ${{\mathcal{S}}}$ the space of polynomial on fibers functions on $T^*M$ (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, ${\mathrm{Vect}}(M)$, of vector fields on $M$ with coefficients in the space of linear differential operators on ${{\mathcal{S}}}$. This cohomology space is closely related to the ${\mathrm{Vect}}(M)$-modules, ${{\mathcal{D}}}_\l(M)$, of linear differential operators on the space of tensor densities on $M$ of degree $\l$.' author: - 'P.B.A. Lecomte [^1]' - 'V.Yu. Ovsienko [^2]' title: Cohomology of the vector fields Lie algebra and modules of differential operators on a smooth manifold --- ł .1truein \[section\] \[thm\][Lemma]{}\[thm\][Corollary]{}\[thm\][Proposition]{}\[thm\][Example]{}\[thm\][Remark]{}\[thm\][Definition]{} Introduction and the Main Theorem ================================= Let $M$ be a smooth manifold and ${\mathrm{Vect}}(M)$ the Lie algebra of vector fields on $M$. The main purpose of this article is to study the cohomology of ${\mathrm{Vect}}(M)$ with coefficients in the space of linear differential operators acting on tensor fields. This cohomology is, actually, a natural generalization of the Gelfand-Fuchs cohomology (i.e., of ${\mathrm{Vect}}(M)$-cohomology with coefficients in the modules of tensor fields on $M$). The problem of computation of such cohomology spaces naturally arises if one considers *deformations* of the ${\mathrm{Vect}}(M)$-module structure on the space of tensor fields. The general theory of deformations of Lie algebra modules is due to Nijenhuis and Richardson [@NR; @Ric]. Let ${\mathfrak{g}}$ be a Lie algebra and $V$ a ${\mathfrak{g}}$-module, then the problem of deformation of the ${\mathfrak{g}}$-module structure on $V$ is related to the cohomology spaces: ${\mathrm{H}}^1({\mathfrak{g}};{\mathrm{End}}(V))$ and ${\mathrm{H}}^2({\mathfrak{g}};{\mathrm{End}}(V))$. More precisely, the first cohomology space classifies *infinitesimal* deformation, while the second one contains the obstructions to integrability of a given infinitesimal deformation. The origin of our investigation is related to the space of scalar linear differential operators on $M$ viewed as a module over ${\mathrm{Vect}}(M)$. It is quite clear a-priori that this module should be considered as a deformation of the corresponding module of *symbols* (i.e., of polynomial on fibers functions on $T^*M$). We are, therefore, led to study the first cohomology of ${\mathrm{Vect}}(M)$ with coefficients in the ${\mathrm{Vect}}(M)$-module of operators on the space of symbols. Differential operators on symmetric contravariant tensor fields {#SkSpace} --------------------------------------------------------------- Consider the space, ${{\mathcal{S}}}(M)$ (or ${{\mathcal{S}}}$ for short), of symmetric contravariant tensor fields on $M$ (i.e., ${{\mathcal{S}}}=\Gamma(STM)$). As a ${\mathrm{Vect}}(M)$-module it is isomorphic to the space of smooth functions on $T^*M$ polynomial on the fibers. Therefore, ${{\mathcal{S}}}$ is a Poisson algebra with a natural graduation given by the decomposition $$\label{decomp} {{\mathcal{S}}}=\bigoplus_{k=0}^\infty{{\mathcal{S}}}_k,$$ where ${{\mathcal{S}}}_k$ is the space of $k$-th order tensor fields. Obviously, ${{\mathcal{S}}}_0$ is isomorphic to $C^\infty(M)$ and ${{\mathcal{S}}}_1$ to ${\mathrm{Vect}}(M)$. The Poisson bracket on ${{\mathcal{S}}}$ is usually called the (symmetric) Schouten bracket (see e.g. [@Fuc]). The action of $X\in{\mathrm{Vect}}(M)$ on ${{\mathcal{S}}}$ is given by the Hamiltonian vector field $$\label{Hamilton} L_X= \frac{\partial{}X}{\partial\xi_i}\, \frac{\partial}{\partial{}x^i} - \frac{\partial{}X}{\partial{}x^i}\, \frac{\partial}{\partial\xi_i}\,,$$ where $(x,\xi)$ are local coordinates on $T^*M$ (we identified $X$ with the first-order polynomial $X=X^i\xi_i$; the summation over repeated indices is understood). Let us introduce the space, ${{\mathcal{D}}}({{\mathcal{S}}})$, of all linear differential operators on ${{\mathcal{S}}}$. This space is a ${\mathrm{Vect}}(M)$-module with a filtration $$\label{filtr1} {{\mathcal{D}}}^0({{\mathcal{S}}}) \subset {{\mathcal{D}}}^1({{\mathcal{S}}}) \subset\cdots\subset {{\mathcal{D}}}^r({{\mathcal{S}}}) \subset\cdots,$$ where ${{\mathcal{D}}}^r({{\mathcal{S}}})$ is the space of $r$-th order differential operators. In this article we compute the first cohomology space $$\label{DefCohom} {\mathrm{H}}^1({\mathrm{Vect}}(M);{{\mathcal{D}}}({{\mathcal{S}}})).$$ of ${\mathrm{Vect}}(M)$ acting on ${{\mathcal{D}}}({{\mathcal{S}}})$. Note that for $M=S^1$ this computation has been done in [@LO1; @BO] see also [@FF] for the case of the Lie algebra of formal vector fields on ${\mathbb{R}}$. Modules of differential operators on tensor densities ----------------------------------------------------- Let ${{\mathcal{F}}}_\l(M)$ (or ${{\mathcal{F}}}_\l$ in short) be the space of tensor densities of degree $\l$ on $M$ (i.e. the space of sections of the line bundle $\Delta_\l(M)=\mod{\Lambda^nT^*M}^{\otimes\l}$ over $M$). Clearly, ${{\mathcal{F}}}_0\cong{}C^\infty(M)$ as a ${\mathrm{Vect}}(M)$-module, any two ${\mathrm{Vect}}(M)$-modules of tensor densities are non-isomorphic (see also [@Fuc]). Denote ${{\mathcal{D}}}_\l$ the space ${{\mathcal{D}}}({{\mathcal{F}}}_\l)$ of linear differential operators on ${{\mathcal{F}}}_\l$. This space is an associative (and, therefore, a Lie) algebra with the filtration by the order of differentiation: $$\label{filtr1} {{\mathcal{D}}}^0_\l\subset{{\mathcal{D}}}^1_\l\subset\cdots\subset{{\mathcal{D}}}^k_\l\subset\cdots$$ The algebra ${{\mathcal{S}}}$ is naturally identified with the associated graded algebra ${\mathrm{gr}}({{\mathcal{D}}}_\l)$ that is, $$\label{principal} {{\mathcal{D}}}^k_\l/{{\mathcal{D}}}^{k-1}_\l\cong{{\mathcal{S}}}_k\,.$$ The corresponding projection $\s_k:{{\mathcal{D}}}^k_\l\to{{\mathcal{S}}}_k$ is called the (principal) *symbol*. The associative algebra ${{\mathcal{D}}}_\l$ can be naturally interpreted as a non-trivial deformation of ${{\mathcal{S}}}$ and constitutes one of the main objects considered in *deformation quantization*. We will be interested, however, only in the ${\mathrm{Vect}}(M)$-module structure on ${{\mathcal{D}}}_\l$ rather than in the whole associative (or Lie algebra) structure. The (tautological) Lie algebra embedding ${\mathrm{Vect}}(M)\hookrightarrow{{\mathcal{D}}}_\l$ $$\label{StandEmbed} X\mapsto{}L^\l_X,$$ where $L^\l_X$ is the Lie derivative on ${{\mathcal{F}}}_\l$, defines a ${\mathrm{Vect}}(M)$-module structure on ${{\mathcal{D}}}_\l$. [If $M$ is oriented by a volume form $\Omega$, then $$\label{LieDerDens} L^\l_X=L_X+\l\,{\mathrm{div}}_\Omega{X}.$$ Moreover, ${{\mathcal{D}}}_\l$ and ${{\mathcal{D}}}_\m$ are isomorphic associative algebras. However, as ${\mathrm{Vect}}(M)$-modules they are isomorphic if and only if $\l+\m=1$ [@DO; @LMT]. ]{} The Main Theorem ---------------- The space ${{\mathcal{D}}}({{\mathcal{S}}})$ is decomposed, as a ${\mathrm{Vect}}(M)$-module, into the direct sum: $$\label{Razlozhenie} {{\mathcal{D}}}({{\mathcal{S}}}) =\bigoplus_{k,\ell} {{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell),$$ where ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)\subset{\mathrm{Hom}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$. It would then suffice to compute the cohomology (\[DefCohom\]) with coefficients in each of these modules. Our main result is the following \[CohomThm\] If $\dim M\geq2$, then $$\label{CohomResult} {\mathrm{H}}^1({\mathrm{Vect}}(M);{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)) = \left\{ \begin{array}{lcl} {\mathbb{R}}& , & \hbox{if} \quad k-\ell=2\\[6pt] {\mathbb{R}}& , & \hbox{if} \quad k-\ell=1, \ell\neq0\\[6pt] {\mathbb{R}}\oplus{\mathrm{H}}^1_{\rm DR}(M) & , & \hbox{if} \quad k-\ell=0\\[6pt] 0 & , & \hbox{otherwise} \end{array} \right.$$ where ${\mathrm{H}}^1_{\rm DR}(M)$ is the first space of the de Rham cohomology of $M$. The proof will be given in Section \[TheProof\]. From now on we assume that $\dim{}M\geq2$. Differentiability {#peetre} ----------------- As a first step towards the proof of Theorem \[CohomThm\], we will prove now that any 1-cocycle on ${\mathrm{Vect}}(M)$ with values in the space of differential operators ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ is locally differentiable. Due to the well-known Peetre Theorem [@Pee], this means that for any $\g\in{\rm Z}^1({\mathrm{Vect}}(M);{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell))$, the bilinear map $(X,P)\mapsto\g(X)(P)$, where $X\in{\mathrm{Vect}}(M)$ and $P\in{{\mathcal{S}}}_k$, is local: $$\label{Sup} {\mathrm{Supp}}\,\g(X)(P) \subset{\mathrm{Supp}}\,{X}\cap\,{\mathrm{Supp}}\,{P}$$ \[PeetrePro\] Any 1-cocycle $\g$ on ${\mathrm{Vect}}(M)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ is local. Let $U\subset{}M$ be open and $X\in{\mathrm{Vect}}(M)$ vanish on $U$. We have to show that $ \g(X)_{|_U}=0. $ Let $x_0$ be any point in $U$. As well-known, there exists a neighborhood $V\subset{}U$ of $x_0$ and vector fields $X_i,X_i'$, $i=1,\ldots,r$ on $V$ such that $$X= \sum_{1\leq{}i\leq{}r} [X_i,X_i']$$ and $${X_i} _{|_V}={X_i'} _{|_V}=0,$$ where $r$ depends only on the dimension of $M$. One has, using the fact that $\g$ is a 1-cocycle $$\g(X)_{|_V}= \sum_{1\leq{}i\leq{}r} \left( L_{X_i}\g(X_i')_{|_V}-L_{X_i'}\g(X_i)_{|_V} \right)=0.$$ Non-trivial cohomology classes {#FormSection} ============================== Let us now describe a natural basis of the above cohomology spaces (\[CohomResult\]). Case $k=\ell$ ------------- Since ${\mathrm{Id}}\in{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_k)$ is ${\mathrm{Vect}}(M)$-invariant, $c\mapsto{}c\,{\mathrm{Id}}$ maps any cocycle $c$ to a cocycle and thus induces a homomorphism ${\mathrm{H}}({\mathrm{Vect}}(M);C^\infty(M))\to{\mathrm{H}}({\mathrm{Vect}}(M);{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_k))$. Theorem \[CohomThm\] states that it is an isomorphism in degree one. Recall that ${\mathrm{H}}({\mathrm{Vect}}(M);C^\infty(M))$ is well-known (see [@Fuc]). In particular, given a covariant derivation $\nabla$, the 1-cocycles are the maps $$\label{OneCocycle} c_{a,\omega}: X \mapsto a\,{\mathrm{div}}_\nabla(X)+i_X\omega,$$ where $a\in{\mathbb{R}}$ and $\omega$ is a closed 1-form, ${\mathrm{div}}_\nabla$ being the divergence associated to $\nabla$. The cocycle (\[OneCocycle\]) is a coboundary if and only if $a=0$ and $\omega$ is exact. Case $k=\ell+1$, $\ell\neq0$ {#Konstrukciya1} ---------------------------- Consider the exact sequence of ${\mathrm{Vect}}(M)$-modules $$\label{exact1} \begin{CD} 0 @> >>{{\mathcal{D}}}^{k-1}_\l @> >>{{\mathcal{D}}}_\l^k @> >>{{\mathcal{S}}}_k @> >>0. \end{CD}$$ Dividing out by ${{\mathcal{D}}}_\l^{k-2}$ leads to the exact sequence $$\label{exact1First} \begin{CD} 0 @> >>{{\mathcal{S}}}_{k-1} @> >>{{\mathcal{D}}}_\l^k/{{\mathcal{D}}}_\l^{k-2} @> >>{{\mathcal{S}}}_k @> >>0 \end{CD}$$ Assume $k\neq1$ and $\l\neq1/2$. The sequence (\[exact1First\]) does not split [@LO]. Its cohomology class is a non-zero element in ${\mathrm{H}}^1({\mathrm{Vect}}(M);{\mathrm{Hom}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-1}))$ (see Appendix). This class admits a representative with values in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-1})$, since the ${\mathrm{Vect}}(M)$-actions in (\[exact1First\]) are differential. It thus defines a non-trivial class in ${\mathrm{H}}^1({\mathrm{Vect}}(M);{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-1}))$ which, by Theorem \[CohomThm\], is a basis of this space. Case $k=\ell+2$ {#RelationSec} --------------- If $\l=1/2$, it is shown in [@LO] that the sequence (\[exact1First\]) is split and that the sequence $$\label{exact2} \begin{CD} 0@> >>{{\mathcal{S}}}_{k-2}@> >> {{\mathcal{D}}}_{1/2}^{k}/{{\mathcal{D}}}_{1/2}^{k-3}@> >> {{\mathcal{D}}}_{1/2}^{k}/{{\mathcal{D}}}_{1/2}^{k-2}@> >>0 \end{CD}$$ is not. Moreover, the splitting of (\[exact1First\]) is given by differential projectors. Since (\[exact1First\]) is split, the class $[{{\mathcal{D}}}^{k-1}_{1/2},{{\mathcal{D}}}_{1/2}^k]$ of (\[exact1\]) belongs to ${\mathrm{H}}^1({\mathrm{Vect}}(M);{\mathrm{Hom}}({{\mathcal{S}}}_{k},{{\mathcal{D}}}_{1/2}^{k-2}))$. Since (\[exact2\]) is not split, its projection ${\s_{k-2}}_\sharp\,[{{\mathcal{D}}}^{k-1}_{1/2},{{\mathcal{D}}}_{1/2}^k]$ is non-zero (see Lemma \[SequenceLemma\] from Appendix). As in the previous case, this projection is easily seen to admit a representative with values in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-2})$. Hence, it provides a basis of ${\mathrm{H}}^1({\mathrm{Vect}}(M);{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-2}))$. \[RemRes\] [In the above Subsections \[Konstrukciya1\] and \[RelationSec\] we have associated non-trivial cohomology classes to the exact sequences (\[exact1First\]) and (\[exact2\]). It is important to note that these classes are “natural” in the following sense. For any open subset $U\subset{}M$ their restrictions to $U$ are precisely the classes associated to the same sequences upon $U$. ]{} Projectively equivariant cohomology {#ProjectCohom} =================================== Throughout this section we put $M\cong{\mathbb{R}}^n$ and $n\geq2$. The Lie algebra of infinitesimal projective transformations {#Introducing} ----------------------------------------------------------- The main idea of our proof of Theorem \[CohomThm\] is to use the filtration with respect to the Lie subalgebra $$\label{Definitionsl2} {\mathrm{sl}}(n+1,{\mathbb{R}})\subset{\mathrm{Vect}}({\mathbb{R}}^n).$$ It is suggested by the fact that the exact sequence (\[exact1\]) that generate our cohomology is split as a sequence of ${\mathrm{sl}}(n+1,{\mathbb{R}})$-modules [@LO]. In some sense, this Lie subalgebra plays the same rôle in our approach as the linear subalgebra ${\mathrm{gl}}(n,{\mathbb{R}})$ in the traditional one (cf. [@Fuc]). Recall that the standard action of the Lie algebra ${\mathrm{sl}}(n+1,{\mathbb{R}})$ on ${\mathbb{R}}^n$ is generated by the vector fields $$\label{sl2} X_i= \frac{\partial}{\partial{}x^i}\,, \qquad X_{ij}= x^i\frac{\partial}{\partial{}x^j}\,, \qquad \bar{X}_i= x^i{\cal E}\,,$$ where $${\cal E} = x^i\frac{\partial}{\partial{}x^i}\,. \label{Euler}$$ Observe in particular that $X_i$ and $X_{ij}$ generate an action of the Lie algebra ${\mathrm{gl}}(n,{\mathbb{R}})\ltimes{\mathbb{R}}^n$. Computing the relative cohomology space --------------------------------------- In this section we will compute the first space of the so-called relative cohomology of ${\mathrm{Vect}}({\mathbb{R}}^n)$, i.e. the cohomology of the complex of ${\mathrm{Vect}}({\mathbb{R}}^n)$-cochains vanishing on the subalgebra ${\mathrm{sl}}(n+1,{\mathbb{R}})$. We will prove the following \[RelativeTheorem\] If $n\geq2$, then $$\label{RelatCohom} {\mathrm{H}}^1({\mathrm{Vect}}({\mathbb{R}}^n),{\mathrm{sl}}(n+1,{\mathbb{R}});{{\mathcal{D}}}(S_k,{{\mathcal{S}}}_\ell))= \left\{ \begin{array}{lcl} {\mathbb{R}}& , & \hbox{if} \quad k-\ell=2\\[6pt] {\mathbb{R}}& , & \hbox{if} \quad k-\ell=1, \ell\neq0\\[6pt] 0 & , & \hbox{otherwise} \end{array} \right.$$ Equivariance property {#EquivSection} --------------------- We begin the proof with a simple observation. Let ${\mathfrak{h}}\subset{\mathfrak{g}}$ be a Lie subalgebra and $V$ a ${\mathfrak{g}}$-module. If $c:{\mathfrak{g}}\to{}V$ is a 1-cocycle such that $c_{|_{\mathfrak{h}}}\equiv0$, then it is *equivariant* with respect to ${\mathfrak{h}}$ i.e. $$\label{equivCoc} L_X(c(Y))= c([X,Y]), \qquad X\in{\mathfrak{h}},$$ where $L$ stays for the ${\mathfrak{g}}$-action on the module $V$. Consequently, our strategy to compute the space of relative cohomology (\[RelatCohom\]) consists, first, in classifying the ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant linear maps $c:{\mathrm{Vect}}({\mathbb{R}})\to{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ vanishing on ${\mathrm{sl}}(n+1,{\mathbb{R}})$ and, second, to isolate among them the 1-cocycles. Commutant of the affine Lie algebra ----------------------------------- Consider the space of polynomials ${\mathbb{C}}[x,\xi]={\mathbb{C}}[x^1,\ldots,x^n,\xi_1,\ldots,\xi_n]$ as a submodule of ${{\mathcal{S}}}$ under the action of ${\mathrm{sl}}(n+1,{\mathbb{R}})$. We need to compute the *commutant* of the subalgebra ${\mathrm{gl}}(n,{\mathbb{R}})\ltimes{\mathbb{R}}^n$, i.e. the algebra of differential operators on ${\mathbb{C}}[x,\xi]$ commuting with the ${\mathrm{gl}}(n,{\mathbb{R}})\ltimes{\mathbb{R}}^n$-action. The differential operators on ${\mathbb{C}}[x,\xi]$ given by $$\label{Div} {\mathrm{E}}=\xi_i\, \frac{\partial}{\partial\xi_i},\qquad {\mathrm{D}}=\frac{\partial}{\partial x^i}\, \frac{\partial}{\partial\xi_i}$$ commute with the ${\mathrm{gl}}(n,{\mathbb{R}})\ltimes{\mathbb{R}}^n$-action. Let us recall the classical result of the Weyl invariant theory (see [@Wey]). \[WeylPro\] The algebra of differential operators on ${\mathbb{C}}[x,\xi]$ commuting with the action of the affine Lie algebra, is generated by ${\mathrm{E}}$ and ${\mathrm{D}}$. We will call the operators (\[Div\]) the Euler operator and the divergence operator respectively. The eigenspaces of ${\mathrm{E}}$ are obviously consist of homogeneous polynomials in $\xi$. \[NiceCor\] The operator ${\mathrm{D}}^{k-\ell}$ is the unique (up to a constant) ${\mathrm{gl}}(n,{\mathbb{R}})\ltimes{\mathbb{R}}^n$-equivariant differential operator from ${{\mathcal{S}}}_k$ to ${{\mathcal{S}}}_\ell$. Any differential operator on ${{\mathcal{S}}}_k$ is indeed determined by its values on the subspace ${\mathbb{C}}[x,\xi]$. The Euler operator ${\mathrm{E}}$ is clearly equivariant with respect to the whole ${\mathrm{Vect}}({\mathbb{R}}^n)$. We will need the commutation relations of the operator ${\mathrm{D}}$ with the quadratic generators of ${\mathrm{sl}}(n+1,{\mathbb{R}})$. For $\bar{X}_i$ as in (\[sl2\]), one has $$[L_{\bar{X}_i},{\mathrm{D}}] = \Big(2{\mathrm{E}}+(n+1)\Big)\circ\frac{\partial}{\partial{\xi_i}}\,, \label{relation}$$ : straightforward. Bilinear ${\mathrm{gl}}(n,{\mathbb{R}})\ltimes{\mathbb{R}}^n$-invariant operators --------------------------------------------------------------------------------- We also need to classify the bilinear ${\mathrm{gl}}(n,{\mathbb{R}})\ltimes{\mathbb{R}}^n$-invariant differential operators. For that purpose, let us use a natural identification $$\label{IdentBin} {\mathbb{C}}[x,\xi]\otimes{\mathbb{C}}[y,\eta] \cong {\mathbb{C}}[x,\xi,y,\eta].$$ There are, obviously, four invariant differential operators ${\mathrm{D}}_{(x,\xi)},{\mathrm{D}}_{(y,\eta)}$ (the divergence operators with respect to the first and the second arguments) and ${\mathrm{D}}_{(x,\eta)},{\mathrm{D}}_{(y,\xi)}$ (the operators of contraction in terms of tensors). Applying again [@Wey] one gets the following \[AffPro\] Every bilinear differential operator $$\label{Bilin1} {{\mathcal{S}}}_j\otimes{{\mathcal{S}}}_k\to{{\mathcal{S}}}_\ell$$ invariant with respect to the action of the affine Lie algebra, is a homogeneous polynomial in ${\mathrm{D}}_{(x,\xi)},{\mathrm{D}}_{(x,\eta)},{\mathrm{D}}_{(y,\xi)}$ and ${\mathrm{D}}_{(y,\eta)}$ of degree $j+k-\ell$. We are now ready to start the proof of Theorem \[RelativeTheorem\]. Bilinear ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant operators ---------------------------------------------------------------- In view of Section \[EquivSection\], we will now classify the ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant linear differential maps $$\label{LinearEquiv} c:{\mathrm{Vect}}({\mathbb{R}}^n)\to{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$$ vanishing on the subalgebra ${\mathrm{sl}}(n+1,{\mathbb{R}})\subset{\mathrm{Vect}}({\mathbb{R}}^n)$. We can, equivalently, consider the equivariant bilinear maps $$\label{Bilin} C:{{\mathcal{S}}}_1\otimes{{\mathcal{S}}}_k\to{{\mathcal{S}}}_{k-p},$$ where $p=k-\ell$. By Proposition \[AffPro\], any such operator is of the form $$\label{Ansatz} \begin{array}{rcl} C&=& \displaystyle\sum_{s=0}^{p+1} \left( \frac{\a_s}{s!(p-s+1)!}\, {\mathrm{D}}_{(x,\eta)}^{s}{\mathrm{D}}_{(y,\eta)}^{p-s+1}\right.\\[12pt] &&\qquad \left.\displaystyle+\frac{\b_s}{(s-1)!(p-s+1)!}\, {\mathrm{D}}_{(x,\xi)}{\mathrm{D}}_{(x,\eta)}^{s-1}{\mathrm{D}}_{(y,\eta)}^{p-s+1}\right)_{ \Big|\!\begin{array}{l} y=x \\ \eta= \xi \end{array} }\\[12pt] && {\displaystyle+\sum_{s=0}^p \frac{\g_s}{s!(p-s)!}\, {\mathrm{D}}_{(y,\xi)}{\mathrm{D}}_{(x,\eta)}^s{\mathrm{D}}_{(y,\eta)}^{p-s}}_{\Big|\! \begin{array}{l} y=x \\ \eta= \xi \end{array}} \end{array}$$ where $\a_s,\b_s,\g_s\in{\mathbb{R}}$. Moreover, $$\label{NulForSmall} \a_s=\b_s=\g_s=0 \qquad \hbox{for} \qquad s<2,$$ since $C$ vanishes on the affine subalgebra and $$\label{Zero} (k-p)\a_2+(n+1)\b_2+(p-1)\g_2=0,$$ since $C$ vanishes on the quadratic generators $\bar{X}_i$ of ${\mathrm{sl}}(n+1,{\mathbb{R}})$. For $k=p$, we have not to take into account the coefficients $\a_s$ in the expression (\[Ansatz\]) because the corresponding terms vanish when applied to ${{\mathcal{S}}}_1\otimes{{\mathcal{S}}}_k$. It is quite easy, using (\[relation\]) and analogous relations with the operators ${\mathrm{D}}_{(x,\eta)},{\mathrm{D}}_{(y,\xi)}$ and ${\mathrm{D}}_{(y,\eta)}$, to obtain the necessary and sufficient condition for the coefficients in (\[Ansatz\]) for $C$ to be equivariant. One gets the following recurrence relations: $$\begin{aligned} (s-1)\,\a_{s+1}-(2k+n-p+s-1)\,\a_s-\g_s &=&0 \label{sys1} \\ (s-1)\,\b_{s+1}-(2k+n-p+s-1)\,\b_s-\g_s &=&0 \label{sys2} \\ (s-2)\,\g_s-(2k+n-p+s-1)\,\g_{s-1} &=&0 \label{sys3} \\ (k-p)\,\a_{s+1}+(n+1)\,\b_{s+1}+(p-s)\,\g_{s+1}+(k-p+s)\,\g_s &=&0 \label{sys4}\end{aligned}$$ where $2\leq{}s\leq{}p$. (For $k=p$, equation (\[sys1\]) has not to be taken into account.) Now, to solve the system (\[Zero\]-\[sys4\]), we need the following technical \[Simplify\] If $\a_s,\b_s,\g_c$ verify the equations (\[sys1\]-\[sys3\]) and (\[Zero\]), then $\a_s,\b_s,\g_c$ verify the equation (\[sys4\]). (A similar result holds true when $k=p$.) Check that for $s=1$, the equation (\[sys4\]) coincides with (\[Zero\]), the result follows then by induction. It is now very easy to get the complete solution of the system (\[Zero\]-\[sys3\]). One has the following four cases. \(a) For $p=0$ and for $(p=1,k=1)$ there is no solution. \(b) For $(p=1,k\geq2)$ the system has a one-dimensional space of solutions spanned by $$\label{FirstCocycle} C_1={\frac{1}{2}}\,{\mathrm{D}}^2_{(x,\eta)}+ \frac{k-1}{n+1}\,{\mathrm{D}}_{(x,\xi)}{\mathrm{D}}_{(x,\eta)}\,,$$ which is, in fact, a just a solution of the equation (\[Zero\]). [One readily checks that the operator $c(X)\in{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-1})$ given by (\[FirstCocycle\]) coincides (up to a constant) with the operator of contraction with the tensor field $$\label{FirstCocycleTensor} c_1(X)= \left( \frac{\partial}{\partial{}x^i} \frac{\partial}{\partial{}x^j}(X^\ell) + \frac{2}{n+1}\, \d_j^\ell \frac{\partial}{\partial{}x^i} \frac{\partial}{\partial{}x^s}(X^s) \right) dx^i{}dx^j \otimes\xi_\ell$$ This expression is obviously a 1-cocycle. The expression (\[FirstCocycleTensor\]) is known in the literature as the Lie derivative of a flat *projective connection* (cf., e.g., [@kh]). ]{} \(c) For $p\geq2,k>p$, the system (\[sys1\]-\[sys3\]) under the condition (\[Zero\]), has a two-dimensional space of solutions parametrized by $(\a_2,\b_2)$. \(d) For $p=k\geq2$, the equation (\[sys1\]) should be discarded. The space of solutions is again one-dimensional. Projectively invariant cocycles ------------------------------- We will now determine which of the ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant maps (\[LinearEquiv\]) classified in the preceding section are 1-cocycles. Let us examine separately the cases (b)-(d). \(b) In the simplest case, $p=1$, one easily checks that the unique ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant map (\[FirstCocycle\]), indeed, defines a 1-cocycle on ${\mathrm{Vect}}({\mathbb{R}}^n)$ with values in ${{\mathcal{D}}}^0({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-1})$. \(c) The cocycle relation adds the equation $\b_3=2\b_2$ to the general system (\[sys1\]-\[sys3\]). In the case $p=2$, one checks by a straightforward computation, that the solutions are the constant multiples of the solution given by $$\label{CocSecondDef} \begin{array}{rcl} \a_2 &=& 2,\\[4pt] \a_3 &=& 2k+n+1,\\[4pt] \b_2 &=& 1, \\[4pt] \b_3 &=& 2, \\[4pt] \d_2 &=& \!\! -(2k+n-3). \end{array}$$ In the case $p>2$, the only solution of the system (\[sys1\]-\[sys3\]) together with the equation $\b_3=2\b_2$ is zero. \(d) If $k=p$, then the non-trivial solutions of the system are cocycles if and only if $k=p=2$. This cocycle is precisely of the form (\[CocSecondDef\]) disregarding $\a_2$ and $\a_3$. \[NontrivPro\] The 1-cocycles on ${\mathrm{Vect}}({\mathbb{R}}^n)$ defined by the formul[æ]{} (\[FirstCocycle\]) and (\[CocSecondDef\]) are non-trivial. This follows immediately from Sections \[Konstrukciya1\], \[RelationSec\] and the fact that the sequence (\[exact1\]) is split when restricted to ${\mathrm{sl}}(n+1,{\mathbb{R}})$, see [@LO]. Let us also give an elementary proof. Recall that a 1-cocycle on ${\mathrm{Vect}}({\mathbb{R}}^n)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ is a coboundary if it is of the form $X\mapsto{}[L_X,B]$ for some $B\in{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$. Moreover, the 1-cocycle vanishes on ${\mathrm{sl}}(n+1,{\mathbb{R}})$ if and only if $B$ is ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant. (cf. [@Lec1]). \[LemInv\] If $k\neq\ell$, there is no ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant operators $B\in{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ different from zero. In virtue of Corollary \[NiceCor\], the property of ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariance implies, in particular, that $B$ has to be proportional to ${\mathrm{D}}^{k-\ell}$. Now, the commutation relation (\[relation\]) shows that this operator can never be ${\mathrm{sl}}(n+1,{\mathbb{R}})$-equivariant. Proposition \[NontrivPro\] follows. Proof of Theorem \[RelativeTheorem\] ------------------------------------ We have shown that there exist unique (up to a constant) 1-cocycles $c_1$ and $c_2$ on ${\mathrm{Vect}}({\mathbb{R}}^n)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-1})$ and ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-2})$ respectively, vanishing on ${\mathrm{sl}}(n+1,{\mathbb{R}})$. These cocycles define non-trivial classes of relative cohomology. Theorem \[RelativeTheorem\] is proven. Proof of Theorem \[CohomThm\] {#TheProof} ============================= Using the filtration with respect to the subalgebra ${\mathrm{sl}}(n+1,{\mathbb{R}})$, we will first prove Theorem \[CohomThm\] in the case when $M$ is a vector space and then extend it to an arbitrary manifold. To that end, we need some more information about the cohomology of ${\mathrm{sl}}(n+1,{\mathbb{R}})$. Cohomology of ${\mathrm{sl}}(n+1,{\mathbb{R}})$ ----------------------------------------------- The cohomology of the Lie algebra ${\mathrm{sl}}(n+1,{\mathbb{R}})$ with coefficients in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ has been computed in [@Lec1]. \[SlThm\] The space of cohomology ${\mathrm{H}}({\mathrm{sl}}(n+1,{\mathbb{R}});{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell))$ is trivial for $k\neq\ell$, for $k=\ell$ it is isomorphic to the Grassman algebra of invariant functionals on ${\mathrm{gl}}(n,{\mathbb{R}})$: $$\label{CohSl} {\mathrm{H}}({\mathrm{sl}}(n+1,{\mathbb{R}});{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_k)) = \left( \bigwedge\textstyle{\mathrm{gl}}(n,{\mathbb{R}})^* \right)^{{\mathrm{gl}}(n,{\mathbb{R}})}$$ In particular, $$\label{CohSlParticul} {\mathrm{H}}^1({\mathrm{sl}}(n+1,{\mathbb{R}});{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell))= \left\{ \begin{array}{rl} {\mathbb{R}},& k=\ell \\ 0,& \hbox{otherwise} \end{array} \right.$$ and the class of the 1-cocycle $X\mapsto{\mathrm{div}}(X){\mathrm{Id}}$ spans that space in the case $k=\ell$. (In fact, it corresponds to the invariant function ${\mathrm{tr}}:{\mathrm{gl}}(n,{\mathbb{R}})\to{\mathbb{R}}$.) Note that this cocycle is just the restriction to ${\mathrm{sl}}(n+1,{\mathbb{R}})$ of the cocycle $c_{1,0}$, see (\[OneCocycle\]). The case of ${\mathbb{R}}^n$ {#VectorSection} ---------------------------- The restriction of a 1-cocycle $c:{\mathrm{Vect}}({\mathbb{R}}^n)\to{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ to ${\mathrm{sl}}(n+1,{\mathbb{R}})$ is a 1-cocycle on ${\mathrm{sl}}(n+1,{\mathbb{R}})$. If $k\neq\ell$, then this restriction is trivial and, therefore, $c$ is cohomological to a 1-cocycle on ${\mathrm{Vect}}({\mathbb{R}}^n)$ vanishing on ${\mathrm{sl}}(n+1,{\mathbb{R}})$; if $k=\ell$, then the restriction of $c$ to ${\mathrm{sl}}(n+1,{\mathbb{R}})$ is cohomological to $c_{1,0}$ and so $c-c_{1,0}$ is, again, cohomological to a 1-cocycle on ${\mathrm{Vect}}({\mathbb{R}}^n)$ vanishing on ${\mathrm{sl}}(n+1,{\mathbb{R}})$. The result then follows from Theorem \[RelativeTheorem\]. Theorem \[CohomThm\] is proven for the special case $M={\mathbb{R}}^n$. The general case ---------------- Let us now prove Theorem \[CohomThm\] for an arbitrary manifold $M$. Consider a 1-cocycle $c$ on ${\mathrm{Vect}}(M)$ with values in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$. \(a) If $k-\ell\neq0,1,2$, then in any domain of chart $U\cong{\mathbb{R}}^n$, the restriction $c_{|_U}$ is a coboundary, that is $c(X)_{|_U}=L_X(S_U),$ where $S_U\in{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ is some operator on $U$. But, on $U\cap{}V$, one has $c(X)_{|_{U\cap{}V}}=L_X(S_U)=L_X(S_V)$ and so the operator $S_U-S_V$ is invariant. Lemma \[LemInv\] implies $S_U-S_V=0$. Therefore, the $S_U$’s are the restrictions of some globally defined $S\in{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell)$ and $c$ is its coboundary. \(b) If $k-\ell=1$ or $2$, it follows from Theorem \[CohomThm\] for $M={\mathbb{R}}^n$ that the class of $c_{|_U}$ is determined up to a constant. In view of Remark \[RemRes\], one has thus $$\label{LocalEquation} c_{|_U} = \a_U\,\g_{|_U}+L_X(S_U),$$ for some $\a_U\in {\mathbb{R}}$ and $S_U$ as above, where $\g$ is a representative of one of the classes associated to the sequences (\[exact1First\]) and (\[exact2\]) respectively. On $U\cap{}V$ one obviously has $\a_U=\a_V$ and $S_U=S_V$ since $$\left( \a_U-\a_V \right) \g_{|_{U\cap{}V}} = \partial \left( S_U-S_V \right),$$ $\g_{|_{U\cap{}V}}$ is non-trivial and, as above, $S_U-S_V$ is invariant. \(c) If $k-\ell=0$, one has $$\label{LocalEquationBis} c_{|_U} = \a_U\,{c_{1,0}}_{|_U}+L_X(S_U).$$ Once again, $\a_U=\a_V\,(:=a)$ and $S_U-S_V$ is invariant, but any invariant operator in ${{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_k)$ is proportional to the identity so that $S_U-S_V=\b_{UV}\,{\mathrm{Id}}$, where $\b_{UV}$ is a constant. It is clear that the $\b_{UV}$’s define a $\check{\rm C}$ech 1-cocycle. If now $\omega$ is a closed 1-form representing the corresponding de Rham class, one easily sees that $c$ is cohomologous to $c_{a,\omega}$. Theorem \[CohomThm\] is proven. Cocycles associated to a connection {#ConnectCocycles} =================================== Using a torsion free covariant derivation $\nabla$, it is possible to construct globally defined cocycles spanning ${\mathrm{H}}^1({\mathrm{Vect}}(M);{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_\ell))$ for $k-\ell=1,2$. Lie derivative of a connection {#Konstrukciya1} ------------------------------ For each vector field $X$, the Lie derivative $$L_X(\nabla) : (Y,Z) \mapsto [X,\nabla_Y Z] - \nabla_{[X,Y]} Z-\nabla_Y[X,Z]$$ of $\nabla$ is well-known to be a symmetric $(1,2)$-tensor field. It yields a non-trivial 1-cocycle $$X \mapsto L_X(\nabla)$$ on ${\mathrm{Vect}}(M)$ with values in $\Gamma(\bigotimes_2^1TM)$. Therefore, for $k\geq2$, the contraction $$\label{TheFirstCoc1} \g^\nabla_1(X)(P) = \langle P,L_X(\nabla) \rangle, \qquad P\in{{\mathcal{S}}}_k,$$ defines a 1-cocycle on ${\mathrm{Vect}}(M)$ with values in ${{\mathcal{D}}}^0({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-1})$. Second-order cohomology class and the Vey cocycle {#Konstrukciya2} ------------------------------------------------- The last case, $\ell=k-2$, is directly related to deformation quantization. For any symplectic manifold $V$, there exists a non-zero class in ${\mathrm{H}}^2(C^\infty(V);C^\infty(V))$. It is given by so-called *Vey cocycle* usually denoted $S^3_\Gamma$ (see [@WL] and [@Rog] for explicit construction using a connection $\Gamma$ on $V$). In the particular, if $V=T^*M$ one can choose the connection so that $S^3_\Gamma$ is homogeneous of weight $-3$, namely, restricted to ${{\mathcal{S}}}\subset{}C^\infty(T^*M)$, $$\label{S3} S^3_\Gamma: {{\mathcal{S}}}_k\otimes{{\mathcal{S}}}_\ell\to{{\mathcal{S}}}_{k+\ell-3},$$ see [@WL1] (e.g. choosing $\Gamma$ as a lift of $\nabla$ to $T^*M$). It follows easily from (\[S3\]) that the map ${\mathrm{Vect}}(M)\to{{\mathcal{D}}}({{\mathcal{S}}}_k,{{\mathcal{S}}}_{k-2})$ defined by $$\label{VeyCoc1} \g_2^\nabla(X)(P)= S^3_\Gamma(X,P), \qquad P\in{{\mathcal{S}}}_k.$$ is a 1-cocycle. Appendix: approximations of the class of a short exact sequence of modules {#Appendix1} ========================================================================== Class of a short exact sequence of ${\mathfrak{g}}$-modules {#ShortExact} ----------------------------------------------------------- We will need some general information about short exact sequences of filtered modules. Let ${\mathfrak{g}}$ be a Lie algebra. Consider an exact sequence of ${\mathfrak{g}}$-modules $$\label{s1} \begin{CD} 0 @> >>A @>{i}>> B @>{j}>> C @> >>0 \end{CD}$$ It is characterized by an element of ${\mathrm{H}}^1({\mathfrak{g}};{\mathrm{Hom}}(C,A))$ (cf. [@Fuc], Sec. 1.4.5). It will be convenient to denote it $[A,B]$. Recall that if $\tau : C \to B$ is a section of $j$, then $[A,B]$ is the class of the 1-cocycle $\g^\tau: {\mathfrak{g}}\to{\mathrm{Hom}}(C,A)$ given by $$\label{AbstractCoc} \g^\tau(X)(T) = i^{-1}(X.\tau(T) - \tau(X.T)),$$ where $X\in{\mathfrak{g}}$ and $T\in{}C$ (this expression is well defined since $X.\tau(T) - \tau(X.T)\in\ker{j}$). Given a submodule $V$ of $A$, one has the following commutative diagram: $$\label{SubmodDiagramm} \begin{CD} @. 0 \\ @. @VV V \\ @. V \\ @.@VV{i_V}V\\ 0 @> >>A @> >> B @> >> C @> >>0 \\ @.@VV{\pi_A}V @VV{\pi_B}V @VV{{\rm Id}}V \\ 0 @> >>A/V @> >> B/V @> >> C @> >>0 \\ @. @VV V \\ @. 0 \end{CD}$$ where $i_V$ is the injection of $V$ into $A$ and $\pi_A$, $\pi_B$ are the projections. One has the relation $[A/V,B/V] = {\pi_A}_\sharp\,[A,B]$. Moreover, the left vertical of (\[SubmodDiagramm\]) leads to the exact triangle $$\label{TriangleDiagramm} \begin{array}{cl} \mathrm{H}({\mathfrak{g}};\mathrm{Hom}(C,V)) \\ & \searrow {i_V}_\sharp \\ \Delta\;\big\uparrow & \mathrm{H}({\mathfrak{g}};\mathrm{Hom}(C,A)) \\ & \swarrow {\pi_A}_\sharp \\ \mathrm{H}({\mathfrak{g}};\mathrm{Hom}(C,A/V)) \end{array}$$ where $\Delta$ is the connecting homomorphism. One easily obtains the following \[SequencePro\] (i) The class $[A/V,B/V]$ vanishes if and only if $[A,B]\in\mathrm{im}\,{i_V}_\sharp$. (ii) If $[A/V,B/V]=0$ then the class of the exact sequence $$\label{ExactB} \begin{CD} 0 @> >>V @>{i \circ i_V} >> B @> >> B/V @> >>0 \end{CD}$$ is $[V,B] = [V,A] + [A,B]$ and vanishes if and only if $[V,A]=[A,B]=0$. Case of a filtered module {#Filtr} ------------------------- Consider now a flag of filtered ${\mathfrak{g}}$-modules $A_0\subset{}A_1\subset\cdots\subset{}A_r\subset\cdots$ and put $S_r =A_r/A_{r-1}$. Let us study the classes $[A_r,A_{r+1}]$ of the sequences $$\label{BolshayaSeq} \begin{CD} 0 @> >>A_r @> >> A_{r+1} @> >> S_{r+1} @> >>0. \end{CD}$$ The quotient by $V= A_{r-1}$, leads to its “first approximation”: $$\label{s3} \begin{CD} 0 @> >>S_{r} @> >> A_{r+1}/A_{r-1} @> >> S_{r+1} @> >>0 \end{CD}$$ If the sequence (\[s3\]) is split, then $[A_r,A_{r+1}]\in{\mathrm{H}}^1({\mathfrak{g}};{\mathrm{Hom}}(S_{r+1},A_{r-1}))$ and so $$\label{SplitClass} [A_{r-1},A_{r+1}] = [A_{r-1},A_r]+[A_r,A_{r+1}]$$ by Proposition \[SequencePro\]. The next approximation is a result of the quotient by $A_{r-2}$. Let $\pi_r:A_r\to{}A_r/A_{r-1}$ be the projection to the quotient-module. \[SequenceLemma\] If the sequence (\[s3\]) is split for all $r>0$, but the sequences $$\label{SecondApprox} \begin{CD} 0 @> >>S_{r-1} @> >> A_{r+1}/A_{r-2} @> >>A_{r+1}/A_{r-1} @> >>0, \end{CD}$$ for $r>1$ are not split, then the class ${\pi_{r-1}}_\sharp[A_r,A_{r+1}]$ does not vanish. Since the sequence (\[s3\]) is split, one has ${\pi_{r-1}}_\sharp[A_{r-1},A_r]=0$. If in addition ${\pi_{r-1}}_\sharp[A_r,A_{r+1}] = 0$, then by Proposition \[SequencePro\] $$[A_{r-1}/A_{r-2},A_{r+1}/A_{r-2}]= {\pi_{r-1}}_\sharp [A_{r-1},A_{r+1}]= {\pi_{r-1}}_\sharp \left( [A_{r-1},A_r] +[A_r,A_{r+1}] \right) = 0$$ and the sequence (\[SecondApprox\]) is split. [*Acknowledgments*]{}. It is a pleasure to acknowledge numerous fruitful discussions with C. Duval. We are also thankful to M. De Wilde, V. Fock and C. Roger for helpful suggestions. [99]{} S. Bouarroudj & V. Ovsienko, [*Three cocycles on ${\mathrm{Diff}}(S^1)$ generalizing the Schwarzian derivative*]{}, IMRN [**1998**]{}, No.1, 25–39. M. De Wilde & P. Lecomte, [*Cohomologie $3$-différentiable de l’algèbre de Poisson d’une variété symplectique*]{}, Ann. Inst. Fourier, [**33:4**]{} (1983) 83–94. M. De Wilde & P. Lecomte, [*Star-products on cotangent bundles*]{}, Lett. in Math. Physics, [**7**]{} (1983) 235–241. C. Duval & V. Ovsienko, [*Space of second order linear differential operators as a module over the Lie algebra of vector fields*]{}, Adv. in Math. [**132**]{}: 2 (1997), 316–333. C. Duval, P. Lecomte & V. Ovsienko, [*Conformally equivariant quantization: Existence and uniqueness.*]{} Math.DG/9902032 B. L. Feigin & D. B. Fuchs, [*Homology of the Lie algebra of vector fields on the line*]{}, Func. Anal. Appl., 14 (1980), 201–212. D. B. Fuchs, [*Cohomology of infinite-dimensional Lie algebras*]{}, Consultants Bureau, New York, 1987. S. Kobayashi & C. Horst, [*Topics in complex differential geometry*]{},in: Complex Differential Geometry, Birkhäuser Verlag, 1983, 4–66. P. Lecomte, [*On the cohomology of ${\mathrm{sl}}(m+1,{\mathbb{R}})$ acting on differential operators and ${\mathrm{sl}}(m+1,{\mathbb{R}})$-equivariant symbol*]{}, Preprint Université de Liège, 1998. P. Lecomte, P. Mathonet & E. Tousset, [*Comparison of some modules of the Lie algebra of vector fields*]{}, Indag. Mathem., [**7:4**]{} (1996), 461–471. P. Lecomte & V. Ovsienko, [*Projectively invariant symbol map and cohomology of vector fields Lie algebras intervening in quantization*]{}, dg-ga/9611006. P. Lecomte & V. Ovsienko, [*Projectively invariant symbol calculus*]{}, math.DG/9809061. A. Nijenhuis & R.W. Richardson, [*Deformations of homomorphisms of Lie algebras*]{}, Bull. AMS [**73**]{} (1967) 175–179. J. Peetre, [*Une caractérisation abstraite des opérateurs différentiels,*]{} Math. Scand. [**7**]{} (1959), 211–218 and [**8**]{} (1960), 116–120. R.W. Richardson, [*Deformations of subalgebras of Lie algebras*]{}, J. Diff. Geom. [**3**]{} (1969) 289–308. C. Roger, [*Déformations algébriques et applications ˆ la physique*]{}, Gaz. Math. 1991, No. 49, 75–94. H. Weyl, The Classical Groups, Princeton University Press, 1946. [^1]: Institute de Mathématiques, Université de Liège, Sart Tilman, Grande Traverse, 12 (B 37), B-4000 Liège, BELGIUM, mailto:plecomte@ulg.ac.be [^2]: C.N.R.S., Centre de Physique Théorique, Luminy – Case 907, F–13288 Marseille, Cedex 9, FRANCE, mailto:ovsienko@cpt.univ-mrs.fr

--- author: - | Scott Aaronson[^1]\ UT Austin\ `aaronson@utexas.edu ` - | Daniel Grier[^2]\ MIT\ `grierd@mit.edu` - | Luke Schaeffer\ MIT\ `lrs@mit.edu` bibliography: - 'bibliography.bib' title: A Quantum Query Complexity Trichotomy for Regular Languages --- [^1]: Supported by a Vannevar Bush Fellowship from the US Department of Defense, a Simons Investigator Award, and the Simons “It from Qubit” collaboration. [^2]: Supported by an NSF Graduate Research Fellowship under Grant No. 1122374.

--- abstract: | We investigate routing on networks modeled as multiple access channels, when packets are injected continually. There is an energy cap understood as a bound on the number of stations that can be switched on simultaneously. Each packet is injected into some station and needs to be delivered to its destination station via the channel. A station has to be switched on in order to receive a packet when it is heard on the channel. Each station manages when it is switched on and off by way of a programmable wakeup mechanism, which is scheduled by a routing algorithm. Packet injection is governed by adversarial models that determine upper bounds on injection rates and burstiness. We develop deterministic distributed routing algorithms and assess their performance in the worst-case sense. An algorithm knows the number of stations but does not know the adversary. One of the algorithms maintains bounded queues for the maximum injection rate $1$ subject only to the energy cap $3$. This energy cap is provably optimal, in that obtaining the same throughput with the energy cap $2$ is impossible. We give algorithms subject to the minimum energy cap $2$ that have latency polynomial in the total number of stations $n$ for each fixed adversary of injection rate less than $1$. An algorithm is $k$-energy-oblivious if at most $k$ stations are switched on in a round and for each station the rounds when it will be switched on are determined in advance. We give a $k$-energy-oblivious algorithm that has packet delay ${\mathcal{O}}(n)$ for adversaries of injection rates less than $\frac{k-1}{n-1}$, and show that there is no $k$-energy-oblivious stable algorithm against adversaries with injection rates greater than $\frac{k}{n}$. An algorithm routes directly when each packet makes only one hop from the station into which it is injected straight to its destination. We give a $k$-energy-oblivious algorithm routing directly that has latency ${\mathcal{O}}\bigl(\frac{n^2}{k}\bigr)$ for adversaries of sufficiently small injection rates that are ${\mathcal{O}}\bigl(\frac{k^2}{n^2}\bigr)$. We develop a $k$-energy-oblivious algorithm routing directly that is stable for injection rate $\frac{k(k-1)}{n(n-1)}$, and show that no $k$-energy-oblivious algorithm routing directly can be stable against adversaries with injection rates greater than $\frac{k(k-1)}{n(n-1)}$. **Key words:** multiple access channel, energy cap, adversarial packet injection, routing, stability, latency author: - 'Bogdan S. Chlebus' - Elijah Hradovich - Tomasz Jurdziński - Marek Klonowski - 'Dariusz R. Kowalski' bibliography: - 'energy-mac-route.bib' title: Energy Efficient Adversarial Routing in Shared Channels --- 3ex 1ex   Introduction {#sec:introduction} ============ Energy efficiency has become a critical factor in the design of large scale distributed and networked systems [@BianzinoCRR12; @BollaBDC11; @FangYHLL15; @JonesSAJ01; @OrgerieAL13]. Ethernet is a popular communication technology used to implement local area networks. Its energy-efficient standard [@ChristensenRNBMM10] uses adaptive power modes that adjust to the amount of traffic. Multiple access channels are an abstraction of wireline Ethernet channels. This work considers communication on such channels that are subject to energy constraints. The underlying motivation is to design distributed communication algorithms that are efficient with respect to both packet latency and energy use. We study dynamic routing on multiple access channels when packets are injected continually. A channel is shared by a number of stations. Without energy considerations, the stations stay ready all the time to perform their communication tasks. We add an amount of energy available per round as additional component of the system. Formally, there is an upper bound on the number of stations attached to the channel that can be switched on simultaneously, which is interpreted as a cap on the available energy. Stations that are switched off cannot transmit nor receive messages from the channel, but they can have packets injected into them. A packet gets injected into some station before it is transmitted on the channel. A station storing injected packets will strive to transmit them eventually such that they are heard on the channel. A packet includes the name of a station it is addressed to. If a packet is heard while its destination station is switched off, then handling the packet might be taken over by some other station that would act as a relay for the packet. A packet can be repeatedly handed over among the stations such that it hops through a sequence of stations in a store-and-forward manner until eventually it is heard by its destination station, which consumes the packet. Packet injection is represented by adversarial models that determine upper bounds on injection rates and burstiness [@AndrewsAFLLK-JACM01; @BorodinKRSW-JACM01]. There is no stochastic component in the specification of how packets are injected. We develop routing algorithms that are distributed and deterministic. Their performance is measured by the worst-case latency and bounds on the number of queued packets. Performance bounds depend on the number of stations and the parameters of an adversary controlling packet injections. Routing algorithms know the size of the system but do not know the parameters of traffic generation. We consider the classes of routing algorithms defined by additional restrictions. These restricted algorithms may, for example, not use relay stations, or not use control bits in messages, or have the on-off status for each station scheduled in advance. #### A summary of the results. We develop deterministic distributed algorithms routing adversarial traffic on the multiple access channels and assess their efficiency in the worst-case sense, where performance bounds depend on the known number of stations $n$ and an unknown adversary. One of the algorithms maintains bounded queues for the maximum injection rate $1$ subject only to the energy cap $3$. This energy cap is provably optimal, in that obtaining the same throughput with the energy cap $2$ is impossible. Algorithms that have bounded latency for each fixed adversary of injection rate less than $1$ are said to be universal. We give universal algorithms subject to the minimum energy cap $2$ that have the latency polynomial in the number of stations $n$. One of these algorithms uses control bits in messages and has latency ${\mathcal{O}}(n^2)$ and another has stations transmit plain packets only and attains latency ${\mathcal{O}}(n^3\log^2 n)$. An algorithm is $k$-energy-oblivious if at most $k$ stations are switched on in a round and for each station the rounds it is switched on are determined in advance. We give a $k$-energy-oblivious algorithm that has latency ${\mathcal{O}}(n)$ for adversaries of injection rates less than $\frac{k-1}{n-1}$ and show that there is no $k$-energy-oblivious stable algorithm against adversaries with injection rate greater than $\frac{k}{n}$. An algorithm routes directly when it does not utilize relay stations, in that each packet makes only one hop straight to its destination from the station it is injected into. We give a $k$-energy-oblivious algorithm routing directly that has latency ${\mathcal{O}}\bigl(\frac{n^2}{k}\bigr)$ for adversaries of sufficiently small injection rates that are ${\mathcal{O}}\bigl(\frac{k^2}{n^2}\bigr)$. We develop a $k$-energy-oblivious algorithm routing directly that is stable for injection rate $\frac{k(k-1)}{n(n-1)}$ and show that no $k$-energy-oblivious algorithm routing directly can be stable against adversaries with injection rates greater than $\frac{k(k-1)}{n(n-1)}$. All the performance bounds of algorithms and impossibility results are tabulated in Table \[tbl:table\]. Algorithm/Impos. Sec. Injection Latency Queues Cap Properties ----------------------------------------------------------------------------- -------------------------------------- -------------------------------- -------------------------------------- ----------------------------- ----- -------------- *Max throughput:* [[<span style="font-variant:small-caps;">Orchestra</span>]{.nodecor}]{} \[subsect:algorithm-max-throughput\] $\rho=1$ $\infty$ $2n^3+\beta$ 3 NObl-Gen-Dir Impossibility \[subsect:impossibility-throughput\] $\rho=1$ Stable $2$ *Universality:* [[<span style="font-variant:small-caps;">Count-Hop</span>]{.nodecor}]{} \[subsec:general-universal\] $\rho<1$ $\frac{2(n^2+\beta)}{1-\rho}$ 2 NObl-Gen-Dir [[<span style="font-variant:small-caps;">Adjust-Window</span>]{.nodecor}]{} \[subsect:poly\] $\rho<1$ $\frac{18n^3\log^2n+2\beta}{1-\rho}$ 2 NObl-PP-Ind *Oblivious indirect:* [[<span style="font-variant:small-caps;">$k$-Cycle</span>]{.nodecor}]{} \[sect:oblivious-indirect\] $\rho< \frac{k-1}{n-1}$ $(32+\beta)\cdot n$ $k$ Obl-PP-Ind Impossibility \[sect:oblivious-indirect\] $\rho> \frac{k}{n}$ Stable $k$ Obl *Oblivious direct:* [[<span style="font-variant:small-caps;">$k$-Clique</span>]{.nodecor}]{} \[sect:oblivious-direct\] $\rho\le \frac{k^2}{2n(2n-k)}$ $8\frac{n^2}{k}(1+\frac{\beta}{2k})$ $k$ Obl-PP-Dir [[<span style="font-variant:small-caps;">$k$-Subsets</span>]{.nodecor}]{} \[sect:oblivious-direct\] $\rho=\frac{k(k-1)}{n(n-1)}$ $\infty$ $2\binom{n}{k} (n^2+\beta)$ $k$ Obl-Gen-Dir Impossibility \[sect:oblivious-direct\] $\rho>\frac{k(k-1)}{n(n-1)}$ Stable $k$ Obl-Dir #### Previous and related work. The surveys by Albers [@Albers10] and Irani and Pruhs [@IraniP05] discuss algorithms for managing energy usage. Routing and other communication primitives subject to energy constraints have been studied extensively, in particular by Chabarek et al. [@ChabarekSBETW08] and Andrews et al. [@AndrewsAZZ13; @AndrewsFZZ10]. Reducing network energy consumption via sleeping and rate-adaptation was addressed by Nedevschi et al. [@NedevschiPIRW08]. Bergamo et al. [@BergamoGTMMZ04] proposed distributed power control to improve energy efficiency of routing algorithms in ad hoc networks. Jurdziński et al. [@JurdzinskiKZ02] studied the problem of counting the number of active nodes in a single-hop radio network with the goal to simultaneously optimize the running time and the energy spent by each node, which is understood as the length of time interval when a node is awake. Kardas et al. [@KardasKP13] studied energy-efficient leader election in single-hop radio networks. Klonowski et al. [@KlonowskiKZ12] considered energy-efficient ways to alert a single hop network of weak devices. Efficiency of broadcasting in ad-hoc wireless networks subject to the number of transmissions a node can perform, interpreted as energy constraint, was studied in  [@GasieniecKKPS08; @KantorP16; @KarmakarKPS17]. Chang et al. [@ChangDHHLP18] as well as Klonowski and Pajk [@KlonowskiP18] studied tradeoffs between the time and energy for broadcasting in radio networks. Chang et al. [@ChangKPWZ17] studied the energy complexity of leader election and approximate counting in models of wireless networks. Herlich and Karl [@HerlichK] investigated saving power in mobile access networks when base stations cooperate to be active or passive in extending their range. Local area data networks implemented by the Ethernet are typically under-utilized; schemes for shutting down network interfaces for energy conservation when using the Ethernet were proposed by Gupta and Singh [@GuptaS07]. Gunaratne et al. [@GunaratneCNS08] investigated policies to adaptively vary the link data rate in response to the demand imposed on the link rate as a means of reducing the energy consumption in Ethernet installations. Adversarial communication in multiple-access channels was studied in [@BenderFHKL-SPAA05; @ChlebusKR-TALG12; @AnantharamuC15; @AnantharamuCR-TCS17], among others. Bender et al. [@BenderKPY16] considered the goal of minimizing the number of channel accesses for a constant throughput in the static case of broadcasting when a set of packets is given in advance. A broadcast algorithm in multiple access channels, that is stable for injection rate $1$ when the stations are switched on all the time, was given by Chlebus et al. [@ChlebusKR-DC09]. Technical Preliminaries {#sec:technical_preliminaries} ======================= We consider multiple access channels as a model of communication networks. There are $n$ stations attached to the channel; we refer to the number $n$ as the *size* of the system. Each station has a name assigned to it, which is a unique integer in the interval $[0,n-1]$. The network operates in a synchronous manner, in that an execution of a communication algorithm is partitioned into rounds. All the stations begin an execution of an algorithm in the same round. #### Messages. Stations may transmit messages on the channel. The duration of a round and the size of a message are mutually scaled such that it takes a round to transmit one message. We say that a station *hears* a transmitted message when the station is switched on and it receives the transmitted message successfully as feedback from the channel. If exactly one station transmits a message in a round then all the stations that are switched on in this round hear the message, including the transmitting station. When at least two stations transmit their messages in the same round then no station can hear any message in this round, including the transmitting stations. A round during which no message is transmitted is called *silent*. A station can be in one of two possible modes in a round. When a station is *switched on* then this means that it is fully operational, in being able to transmit a message and receive feedback from the channel. When a station is *switched off* then this means that it cannot transmit nor receive any communication from the channel. Each station is autonomous in when it is on and when it is off. In a round in which a station is switched on, the station can set its *timer* to any positive integer $c$, which results in the station spending the next $c$ rounds in the off-mode and returning to the on-mode immediately afterwards. It is a natural goal to design ways to save on the expenditure of energy while the communication system would maintain its functionality. We associate energy cost as an attribute of the communication infrastructure in the following manner. All the stations are connected to an external energy source, which has some output capacity and cannot provide more energy per round than this cap. Specifically, the following is assumed: (1) it costs one energy unit to keep a station switched on in a round, and (2) it costs a negligible amount of energy to keep a station switched off in a round. When representing the whole system’s expenditure of energy in a given round, we make it equal to the number of stations that spend this round switched on. The upper bound on the number of stations that can be switched on simultaneously in a round is the *energy cap* of the system. A multiple-access channel system is determined by the total number of available stations and the energy cap. The energy cap $2$ is minimum to make the tasks of point-to-point communication feasible in principle, since at least one transmitter and one receiver need to be switched on in a round. #### Dynamic packet generation. A packet $p=(d,c)$ consists of its *destination address $d$* and its *content $c$*. A destination is an integer in $[0,n-1]$ interpreted as a name of the station to which the packet needs to be delivered. A packet’s contents is the information that the packet carries, which does not effect how the packet is handled. An adversarial model imposes quantitative constraints on how packets get generated and injected. An adversary is determined by its *type $(\rho,\beta)$*, where $\rho$ and $\beta$ are numbers such that $0<\rho\le 1$ and $\beta\ge 1$. In each continuous time interval of length $t$, the adversary can generate and inject at most $\rho\cdot t + \beta$ packets. The parameter $\rho$ is interpreted as an *injection rate*. The maximum number of packets that can be generated in a single round is the adversary’s *burstiness*, which is $\lfloor \beta+\rho \rfloor$; we call $\beta$ the *burstiness coefficient*. When referring to an upper bound $\rho\cdot t + \beta$ on the number of injected packets, we say that $\rho t$ is the *injection-rate component* and $\beta$ is the *burstiness component* of the bound. This adversarial model of packet injection is called *leaky bucket*; it was used before to model traffic in shared channels, in particular in [@ChlebusKR-DC09; @AnantharamuCKR-JCSS19]. An adversary is restricted only by the number of packets it can generate in a time interval, as determined by its length. Once a packet is generated, the adversary injects it immediately into some station. Packets may be injected into any station, regardless of whether the station is switched on or off. Packets injected into a station can be stored in the station’s private memory, which is called this station’s *queue*. A station can transmit its queued packets in arbitrary order. A station can scan its queue and access any packet in negligible time. #### Routing packets. Each injected packet needs to be delivered to its destination station. We say that a packet $p=(d,c)$ gets *delivered* in a round $t$ when the following occurs: (1) a message containing packet $p$ is transmitted in round $t$ and is heard on the channel, (2) the destination station $d$ is switched on in round $t$. If a packet gets delivered then it is “consumed” by the destination station and disappears from the system. A packet may be transmitted and heard on the channel a number of times, which results in the packet hopping from station to station in a store-and-forward manner. If a station transmits a packet, which is then heard on the channel, then the packet may be removed from the queue of the transmitting station. If a message with a packet is heard on the channel but the packet is not delivered in this round then some station may *adopt* the packet by adding it to the queue; such a new station handling the packet becomes a *relay* for the packet and treats it as if it were injected directly by the adversary. The task of *routing* is defined as follows: while packets are continually generated and injected into the system, stations transmit them such that they are eventually delivered. The total number of packets that are queued in a round is referred to as the *queue size* in this round. The *delay of a packet $p$* is defined as the difference $t_2-t_1$ between the round $t_2$ in which packet $p$ gets delivered and the round $t_1$ in which packet $p$ was injected. #### Routing algorithms. Routing is performed by distributed algorithms that are executed by all the stations concurrently. Correctness of a routing algorithm means that each injected packet is eventually delivered to its destination and a delivery occurs exactly once for each packet. A station switched-on in a round either transmits a message or senses the channel (listens to it) in this round. A message consists of at most one packet and a string of control bits. The bits encoding packet’s destination address are not considered as control bits. Whether control bits are included in messages is a feature of algorithms. Algorithms that have a message consist of only a packet without any control bits are called *plain-packet* ones, they make a subclass of *general* routing algorithms. A station executing a plain-packet algorithm cannot send a message when it does not have a packet to route, since a message has to include a packet and nothing else. A station executing a general-routing algorithm may transmit a message without any packet but with control bits only. The destination address of a packet is just a station’s name, so it is represented by ${\mathcal{O}}(\log n)$ bits. We consider only routing algorithms that use the conservative amount of ${\mathcal{O}}(\log n)$ control bits per message. This restriction on the number of control bits transmitted per round makes coordinating actions among the stations reasonably costly, as reflected in the number of messages, and ultimately packet delays and size of queues. Routing algorithms that do not use relay stations are said to *route directly*, and otherwise they are said to *route indirectly*. Algorithms that route directly make each packet hop only once, from the station into which the packet got injected, straight to the destination. A routing algorithm is called *energy oblivious* when it determines in advance, prior to the start of an execution, for each station $i$ and each round $t$, whether station $i$ is on or off in round $t$. An energy-oblivious algorithm executed on a channel subject to an energy cap of at most $k$ is called *$k$-energy-oblivious*. The *queue size* measure, of an execution of a routing algorithm, is defined as a maximum number of queued packets in a round of this execution. The *latency* measure, of an execution of a routing algorithm, is defined as the maximum packet delay occurring in the execution. Both the queue size and latency are natural performance metrics of routing algorithms and are represented as functions of the size of the system and the type of an adversary. If the latency of a routing algorithm is bounded then queues are bounded as well, since a queue’s size at a station is always a lower bound on the delay of some packet handled by this station. We say that a routing algorithm is *stable*, against a class of adversaries, if the queue size is bounded, for a given number of stations and an adversary in this class. The *throughput* of an algorithm is the maximum injection rate for which it is stable, assuming such a rate exists. Throughput is always at most $1$ since at most one packet can be heard in a round. A routing algorithm has a *universally bounded latency* when latency is bounded against each adversary of injection rate less than $1$; we call such algorithms *universal*. Bounds on the latency of a routing algorithm against a specific adversary involve the size of the system $n$ and the type of an adversary $(\rho,\beta)$. #### Knowledge. We say that a property of a system is *known* when it can be used in codes of algorithms. It is assumed throughout that the size of the system $n$ and the energy cap $k < n$ are known, but the adversary’s type $(\rho,\beta)$ is not. Algorithms may have their correctness and performance bounds depend on the magnitudes of the unknown parameters of the communication environment. For example, an algorithm may be stable or have bounded latency for sufficiently small injection rates. Maximizing Throughput {#sec:maximizing-throughput} ===================== We present a direct-routing algorithm stable for injection rate $1$. This is the maximum throughput possible on multiple access channels. The algorithm requires energy cap to be at least $3$. We show that the number $3$ is best achievable in this sense by proving the impossibility of attaining throughput $1$ with energy cap $2$. An algorithm achieving maximum throughput {#subsect:algorithm-max-throughput} ----------------------------------------- An algorithm stable for injection rate $1$ which we give is called [[<span style="font-variant:small-caps;">Orchestra</span>]{.nodecor}]{}. It schedules at most three stations to be simultaneously switched on at any round, with at most one of them transmitting. The algorithm builds on the paradigms developed in [@ChlebusKR-DC09], which gave a broadcast algorithm with throughput $1$ for multiple-access channels without energy caps. We call a group of $n-1$ consecutive rounds of an execution a *season* if the last round $t$ of the group satisfies $t \equiv 0 \pmod{n-1}$. For each season, there is a unique station associated with it called a *conductor*. Stations that are different from a conductor are called *musicians*. A conductor for a season transmits a message in every round of this season, so there are no silent rounds. A round when a message heard on the channel does not include a packet but only control bits is called *light*. Every station keeps an ordered list of all the stations. These lists are the same in every station at the beginning of a season; at such a moment they represent one list, which we call the *baton list*. Initially, the baton list consists of all the stations ordered by their names. Stations assume the role of conductors in their order on the baton list. The first station on the list is assigned to serve as a conductor for the first season. The positions of stations on the baton list are understood as follows: the front entry of the list is considered as the first station on the list, then the consecutive stations have their positions increased by one, and the tail entry occupies the last $n$th position. In particular, if a station at position $i$ moves to become the head of the baton list, then this stations acquires position $1$ while each station at the original position $j<i$, which means closer to the front than $i$, gets its position incremented to $j+1$, so that its distance from the head of the list increases. The process of assigning conductors to seasons can be visualized as passing a virtual baton from station to station, such that a station holding the baton is a conductor. When a season ends then the baton is typically passed on to the next station on the baton list. The order determined by the list is understood in a cyclic sense, in that the first station becomes a conductor after the last one in the list has concluded its assignment. An exception for this process occurs when a conductor is moved to become the head of the baton list while keeping the baton. A conductor of a season is switched on in each round of the season. A musician switches on during a season either to learn or to receive, or possibly both. A message transmitted by a conductor contains control bits for a taught musician who is to learn, and a packet sent to a receiving musician, unless the message is light. We explain the actions of teaching/learning and sending/receiving next. The purpose of *learning* is to obtain information from the conductor, in particular one pertaining to a schedule to receive packets in the next season with the same conductor. A station learns by extracting control bits from a message transmitted by the conductor and interpreting them as round numbers to be switched on during the next season when the same stations acts as conductor. If a conductor conveys information to a learning station then we say that the conductor *teaches* the station this information. The purpose of *receiving* is to obtain packets injected into a conductor and destined for a receiving musician. A musician receives by extracting a packet included in a message from the conductor. If a conductor transmits a message with a packet then it *sends* this packet to the receiving station. For a message transmitted by a conductor to effectively serve its purpose, the following three involved stations need to be switched on simultaneously: the conductor, the learning musician, and the receiving musician. Musicians switch on during a season in the following manner. First, the musicians switch on to learn: they do it one by one, in the order of their names, for one round at a time. Second, the musicians switch on to receive: they do it according to the schedule taught during the latest previous season when the same station was a conductor. A packet injected into a station, when it acts as a conductor, stays *new* for the duration of this season, and after that becomes *old*. A packet injected into a musician becomes old immediately. In particular, when a new season begins, then all the packets queued in the stations are old. At the start of a season, when some station becomes a conductor, this station computes a schedule to send the old packets during the next season when it will become a conductor again. The schedule concerns only these old packets that have not been scheduled yet for the current season. A conductor schedules packets to send in the order of their injections. A station considers itself *big* if it has at least $n^2 -1$ old packets in its queue. A conductor that is big at the beginning of a season teaches each musician of this status, by suitably setting a toggle bit in messages. After a musician learns this information, it moves the conductor to the front of its private copy of the baton list. Such a season concludes with all the musicians having identical private lists representing the baton list, with the conductor at the front. A big conductor keeps the baton for the next season, after moving to the front of the baton list, and stays at the front as long as it is big. This mechanism allows for one station to act as a conductor for long stretches of seasons, possibly indefinitely, should the adversary inject packets into one station only. We group seasons into contiguous intervals of seasons, depending on the heaviness on traffic during these seasons. If the total number of packets in the queues at the beginning of a season is greater than $ n^3-2n+1$ then the season belongs to a *dense interval of seasons*, which means the traffic will be heavy. Otherwise, a season belongs to a *sparse interval of seasons*, which means that traffic might be light. We expect that there exist big stations when traffic is heavy. A station is *pre-big* in a round of an interval of seasons if it has not been big during this interval before the round. A station is *post-big* in a round of an interval of seasons if it is not big now but it has been big by this round during the interval. There are at most $2n^3+\beta$ packets queued in a round of an execution of algorithm [[<span style="font-variant:small-caps;">Orchestra</span>]{.nodecor}]{} against an adversary of injection rate $1$ and with a burstiness coefficient $\beta$. Let $D = n^3-2n+1=n(n^2-2)+1$ be the number of old queued packets used to differentiate between dense and sparse intervals of seasons. A big station has at least $n^2-1$ old packets in its queue. By the pigeonhole principle, there exists at least one big station during a season in a dense interval of seasons. We estimate the number of queued packets during a season depending on whether the season belongs to a dense or sparse interval. The adversary’s capability to inject packets due to the burstiness coefficient is accounted for only once at the end of a derivation of an upper bound on the number of queued packets. First, we consider sparse intervals. The system starts a season with empty queues, so the first season belongs to a sparse interval. If the system starts a season in a sparse interval then it has at most $D$ packets. The adversary can inject at most $(n-1)$ packets during a season. So there can be at most these many old packets in stations within a sparse interval: $$\label{eqn:sparse} D + (n-1) = n^3-2n+1 + n-1 = n^3 -n \ ,$$ not including burstiness. This is also an upper bound on the number fo queued packets when a sparse interval ends and a dense interval begins. Next, we consider dense intervals. In such an interval, the adversary can be assumed to inject at full power, namely, a packet per round. If a message with a packet is heard on the channel, then this does not affect the number of queued packets, since only one packet gets injected. Otherwise, if a light message is heard, this results in the number of queued packets incremented by $1$. This makes an upper bound on the number of light messages heard on the channel serve as an upper bound on the increase on the number of queued packets. We claim that neither big nor post-big stations can contribute light rounds when acting as conductors during dense intervals. It follows that only pre-big stations contribute light rounds. We prove this claim next. A big station has at least $n^2-1$ packets in its queue at the beginning of a season when it obtains the baton. It must have had at least $n-1$ old packets to schedule at the beginning of the previous season it was conducting, since the adversary could inject at most these many packets in the meantime during $n$ seasons, without accounting for burstiness: $$n(n-1)=n^2 -1-(n-1) \ .$$ A conductor that had at least $n-1$ old packets at the beginning of the previous season it conducted, has already scheduled a full season, so it sends a packet in each among the $n-1$ rounds of the current season. We conclude that a big station contributes no light rounds as long as it gains and maintains the status of a big conductor. Consider a station $i$ that is post-big but not big. Station $i$ has an opportunity to transmit only when there is no big station before it on the list, as such a station would be visited by the baton first and moved the baton back to front. When station $i$ receives the baton and $i$ is not big, then there is a big station after $i$, because such a station exists in every season of a dense interval. The first big station encountered by the baton is moved to the front of the baton list, thereby incrementing the $i$’s position to $i+1$. The position of a station cannot increase more than $n-1$ times in this way. The last time when $i$ was big, it had at least $n^2 -1$ packets in its queues and was placed at the front of the baton list. Station $i$ had at least these many packets when ending a season in which it was a big conductor: $$n^2-1-(n-1)= n(n-1) \ .$$ So the station can afford to increase its position up to $n$ times while consistently sending $n-1$ packets per season when serving as a conductor. We conclude that a post-big station contributes no light rounds during seasons in a dense interval. We are finally ready to count light rounds in a dense interval, all of which could be contributed by pre-big conductors only. There are at most $n-1$ pre-big stations in the system at the beginning of a dense interval, because at least one station is big. Light rounds occur in a season when the conductor has fewer than $n-1$ old packets scheduled to send; let us assume conservatively that such a conductor does not have any scheduled packets to send, to maximize the number of light rounds the season contributes. A pre-big station $i$ becomes a conductor only when the first big station on the baton list is behind station $i$. After the baton leaves and eventually reaches a big station, this big station advances to the front and the $i$’s position shifts by $1$. Such shifts can occur at most $n-1$ times. It follows that a pre-big station can contribute at most $(n-1)^2$ light rounds when acting as a conductor in a dense interval. Therefore all the pre-big stations together contribute at most $(n-1)^3$ light rounds as conductors. The bound  estimates the number of queued packets when a dense interval begins. This number can grow by at most the number of light rounds while the adversary injects at full power, plus burstiness. These three parts together contribute the following: $$n^3-1 +(n-1)^3 +\beta = (n-1)(2n^2 -2n +1)+\beta = 2n^3 -4n^2 +3n -1+\beta\le 2n^3+\beta \ .$$ This quantity serves as the ultimate upper bound on the number of queued packets. An impossibility for maximum throughput {#subsect:impossibility-throughput} --------------------------------------- Algorithm [[<span style="font-variant:small-caps;">Orchestra</span>]{.nodecor}]{} requires at least $3$ stations to be switched on in each round. We show that this is necessary for any algorithm to have throughput $1$. \[lem:executions\] Given an algorithm for a system of $n \geq 3$ stations, let us assume that we have defined an execution of the algorithm until some round $t_{i-1}$ such that the following holds: at least one station $s$ has no packets in its queues, no other station has packets to be delivered to $s$, and there are at least $i-1$ packets in the system. Then either the execution can be extended without bounds and the number of packets in the system grows unbounded or there exists a round $t_i>t_{i-1}$ such that the execution can be extended until $t_i$ in a way that the round $t_i$ satisfies the following conditions: 1. no packet is successfully transmitted at round $t_i$, and 2. by the end of round $t_i$ at least one station $s'$ has no packet in its queues and no other station has packets destined for station $s'$, and 3. after round $t_{i-1}$ and by the end of round $t_i$, one packet per round has been injected into the system on average and the burstiness was at most $1$. Let $t_0$ be the first round, and $i$ be an integer such that $i>0$. Let $s_1$ and $s_2$ be two stations different from $s$. We consider the following two possibilities to extend an execution, as determined by the adversary after round $t_{i-1}$. No packet is injected into station $s$, station $s_1$ gets one packet injected into it addressed to $s$ in each odd round and one packet addressed to $s_2$ in each even round: No packet is injected into station $s$, station $s_1$ gets one packet injected into it addressed to station $s_2$ in each round. For station $s$, these two cases to extend the execution are indistinguishable up to a round $t$ when station $s$ becomes switched on for the first time after round $t_{i-1}$. Now there are two possible continuations: such a round $t$ does not exist; in the execution determined by Case I the number of packets addressed to station $s$ grows unbounded. such a $t$ exists; then the execution determined by Case II extended to round $t_i = t$ satisfies the following: 1. no packet is heard at round $t$, as there are no packets involving station $s$ in the system; 2. station $s$ has no packets in its queues, since it had no packets in round $t_{i-1}$ and no packet in the system was addressed to it, and between the rounds $t_{i-1}$ and $t_i$ (inclusive) this has not changed; 3. the adversary injects exactly one packet per round. By the properties (a) through (c) above, and by the assumption that there are at least $i-1$ packets in the system by round $t_{i-1}$, the number of packets at the end of round $t_i$ is at least $i$. Lemma \[lem:executions\] gives a sequence of rounds $(t_i)_{i\ge 0}$ such that there are at least $i$ packets queued in the system at round $t_i$. No algorithm can be stable for energy cap $2$ and a system size greater than or equal to $3$ against leaky-bucket adversaries with injection rate $1$. Suppose that such an algorithm exists, to arrive at a contradiction. The argument is by induction on the round numbers. Consider the first round of an execution of the algorithm, to provide a base for induction. By the assumption about the energy cap, at least one station $s$ needs to be switched off. The assumptions of Lemma \[lem:executions\] are satisfied for $t_0=1$, so that $i=1$. Next we show the inductive step. Assume that the assumptions of Lemma \[lem:executions\] are satisfied for some $i\ge 1$, so that there is an execution determined up to some round $t_{i-1}$ such that the following holds: at least one station $s$ has no packets, no station has packets addressed for $s$, and there are at least $i-1$ packets in the system. By Lemma \[lem:executions\], this prefix of an execution up to round $t_{i-1}$ either could be extended to a full execution such that the number of queued packets grows unbounded, or there is a round $t_i$ and an extension that satisfy the assumption of Lemma \[lem:executions\] for $i+1$. To conclude, either there is $i$ such that from round $t_i$ there is an unstable extension of the execution, or we could continue extending the execution through rounds $t_j$, for all integers $j$. In the latter case, the number of packets in the system grows unbounded with $j$, and the resulting execution is unstable. Universal Routing {#sec:universal} ================= We give two routing algorithms with universally bounded latency. One routes directly while using control bits in messages to coordinate stations and the other routes indirectly with messages consisting of plain packets only. General universal routing {#subsec:general-universal} ------------------------- The direct-routing algorithm using control bits in messages is called [[<span style="font-variant:small-caps;">Count-Hop</span>]{.nodecor}]{}; it operates as follows. One station is dedicated to serve as a *coordinator* and the other stations are *workers*. An execution is structured into *phases*. Packets transmitted in a phase need to be *old*, in that they were injected in the previous phase. Packets injected in the current phase are *new* for the duration of the phase. At a round when a phase ends, all the packets available in the system become old for the next phase. These are the only old packets for the next phase, which means that each station knows which among its packets are old, for the duration of this phase, when a new phase begins. The first phase consists of $n$ rounds during which all the stations are switched off. Each of the following phases proceeds through *stages*, which are time intervals spent by the stations working to accomplish some task. A phase is partitioned into $n$ stages, one for each receiving station. Such a stage for each receiving station consists of three substages. During the first substage, each station, except for the receiving station $v$ and the coordinator, transmits once, sending a message with the number of old packets destined for $v$. This information allows the coordinator to assign to each station a time interval to transmit all its packets to $v$. The second substage consists of the coordinator transmitting the offset number for each station to be switched off waiting for its turn to transmit. Finally, the third substage has all the stations switch on one by one when the turn comes to transmit the old packets destined to $v$, while the station $v$ is switched on during the whole substage and the coordinator is switched off. \[thm:one-hop\] A direct-routing algorithm [[<span style="font-variant:small-caps;">Count-Hop</span>]{.nodecor}]{} requires the energy cap $2$, is stable for each injection rate $\rho<1$, and its latency for such an injection rate is at most the following: $$\frac{2(n^2+\beta)}{1-\rho} \ .$$ Each packet is delivered from the station of injection to the station of destination in one direct hop, by the algorithm’s design. There are $(n-1)^2$ rounds spent transmitting numbers during a phase but no packets. While such messages are transmitted, the adversary can inject new packets. These packets will extend the duration of the next phase by up to $\rho (n-1)^2$ rounds. This phenomenon can be iterated in a cascade-like manner, since when packets are transmitted, the adversary can use this time to inject even more packets. Taking into account all the possible extensions of phases, the duration of any phase is at most the following: $$(n^2+\beta)(1+\rho + \rho^2+\ldots ) = \frac{n^2+\beta}{1-\rho} \ .$$ A packet stays in a station during at most two consecutive phases. Plain-packet universal routing {#subsect:poly} ------------------------------ We describe an indirect-routing plain-packet algorithm that requires only a constant energy cap, but has universally bounded latency and attains packet delay ${\mathcal{O}}(n^3\log n)$. The algorithm is called [[<span style="font-variant:small-caps;">Adjust-Window</span>]{.nodecor}]{}. An execution of [[<span style="font-variant:small-caps;">Adjust-Window</span>]{.nodecor}]{} is structured into segments called *time windows*. The size of a time window may increase in the course of an execution. The current size of a window is denoted by $L$. All the stations use the same value of $L$ at each round. Packets injected before the current time window are called *old* and packets injected during the current time window are called *new* for the duration of the window. The goal to achieve during a window is to deliver all the outstanding old packets to their destinations. Whether or not this goal is accomplished in a particular time window may depend on the magnitude of $L$. Old packets that do not get delivered in a window remain old for the duration of the next window. If some old packets are not delivered in the current time window, then the window size $L$ gets doubled to become $2L$, which determines the duration of the next window. Otherwise, if all the old packets are successfully delivered in a window, then the window size $L$ stays the same, and so the duration of the next window stays the same as well. Algorithm [[<span style="font-variant:small-caps;">Adjust-Window</span>]{.nodecor}]{} works with energy cap $2$. An execution is organized such that in each round at most one station transmits. If a station $i$ transmits a message with packet in a round and another station $j$ is switched on, then we say that *station $i$ sends the transmitted packet to $j$*. If a station $i$ sends a packet to station $j$ and the packet is heard on the channel, then station $i$ removes the packet from its queue and station $j$ either consumes it, if it is addressed to $j$, or else adopts it and becomes its relay station. This means that packets may hop from station to station, and routing may be indirect. A time window is partitioned into three stages: Gossip, Main, and Auxiliary. The goal of a Gossip stage is to exchange information between stations regarding the numbers of old packets in their queues with particular destinations. In a Main stage, the stations transmit old packets directly to their destinations according to a schedule based on the information exchanged and learned during the preceding Gossip stage. A station knows the part of such a schedule relevant to its actions: it knows when to transmit messages to which destinations, and in which rounds to listen to messages addressed to it. It may happen that a station $i$ needs to convey some information to a station $j$ while station $i$ does not have packets with the destination $j$, then $i$ sends some packet(s) to $j$ whose destination is different from $j$. An Auxiliary stage deals with delivering such relayed packets to their destinations, as well as handling old packets at stations that could not participate in neither Gossip nor Main stages due to lacking sufficiently many packets. A message transmitted on the channel may only contain one plain packet without attached control bits. This means that numbers encoded as strings of bits cannot be piggybacked on messages. Instead, we design a protocol called *coded transfer (of bits)* to encode sequences of bits by way of sequences of transmissions of single packets, with rounds of transmissions possibly interspersed with silent rounds. One round of coded transfer can convey one bit. Coded transfer needs to overcome the following technical obstacle: a station that is supposed to transmit a packet to convey a bit needs to have at least one packet available in its queue, which after a successful transmission is removed from the queue. This implies that stations with empty queues cannot transmit messages and so their lack of transmission activity needs to be properly interpreted by the other stations. Coded transfer of bits works as follows. Suppose that a station $i$ is to transfer $r$ bits $B_1,B_2,\ldots,B_r$ to another station $j$, for $0\le i,j< n$, and the size of the queue of station $i$ is at least $r$. Then, in $r$ consecutive rounds, station $i$ sends a packet to $j$ in the $k$th consecutive round if and only if $B_k=1$, for $1\le k\le r$, while station $j$ listens to the channel. This approach makes the transmitting station $i$ use one packet for each transmitted bit $1$ and no packet for a $0$. Station $i$ may transmit packets not addressed to $j$, if packets addressed to $j$ are not available at $i$; if this occurs then station $j$ adopts them and becomes their relay. The stages of a window take a specific duration, depending on $n$ and $L$. Let $L_G$, $L_M$, $L_A$ denote the number of rounds of a Gossip stage, a Main stage and an Auxiliary stage, respectively. These three numbers sum up to $L$: $L_M+L_G+L_A=L$. The magnitudes of $L_G$ and $L_A$ are determined next, and the remaining part of $L$ rounds of a window is taken by a Main stage. We specify that stations without sufficiently many old packets do not participate either in Gossip stages or in Main ones. We categorize such stations as small. In what follows, the notation $\lg x$ stands for $\lceil \log_2 (x+1)\rceil$. Formally, a station is *small* in the considered window of size $L$ if the size of its queue at the beginning of that time window is less than $4n\lg L$; otherwise, the station is *large* in the window. A large station has sufficiently many packets to transmit should they be needed to convey bits by coded transfer. #### A Gossip stage. The goal of a Gossip stage is to share information among the stations about the contents of their queues at the beginning of the current time window. Such transmission of information is performed indirectly by coded transfer. A Gossip stage consists of $n^2$ *phases*, indexed by all pairs $(i,j)$ for $1\le i,j\le n$. Each phase takes $2+3\lg L$ consecutive rounds. Thus, a Gossip stage takes these many rounds: $$L_G=n^2(2+3\lg L) \ .$$ An $(i,j)$-phase for $i\neq j$ is structured as follows. The station $j$ listens to the channel in each round of a phase, as the only station that does so. If the station $i$ is small, it stays silent for the whole phase; otherwise, if $i$ is large, it conveys some information to $j$ as follows. The station $i$ sends a packet to $j$ in the first round of the phase to notify $j$ that $i$ is large. Then, in the second round of the phase, $i$ sends a packet to $j$ if and only if its queue size is greater than $L$. Finally, during the following $3 \lg L $ rounds of the phase, $i$ conveys the following three numbers to $j$ by coded transfer: 1. the minimum of its queue size and $L$, 2. the number of packets in its queue with destination $j$, or $L$ if the number of packets to $j$ is at least $L$, 3. the number of packets in its queue with destinations $k$ such that $k<j$, or $L$ if the number of such packets is at least $L$. At the end of a Gossip stage, each station $j$ knows one of the following about each station $i$, where $0\le i< n$ and the size of the queue of station $i$ is measured at the beginning of a Gossip stage: 1. the queue size of $i$ is less than $4n\lg L$, or otherwise 2. the queue of $i$ has more than $L$ packets to some destination, or otherwise 3. the exact size of the queue of $i$, the number of packets in $i$ with destinations $k$ such that $k<j$, and the number of packets in the station $i$ with the destination $j$, when none of the cases (a) nor (b) holds. This information determines the size of the next time window. Namely, if there is some station $i$ such that the queue size of $i$ is greater than $L$ then the window size is doubled to become $2L$. Similarly, if none among the queue sizes is greater than $L$ but the sum of the queue sizes of all the stations is greater than the length of the Main stage, the window size is also doubled to become $2L$. If none of the two conditions holds, the time-window size $L$ stays the same for the duration of the next time window. #### A Main stage. If it is known that some stations have their queue sizes greater than $L$, then the Main stage is dedicated to the station with the smallest name among them, which spends all the rounds transmitting its packets. Suppose otherwise that no station has the size of the queue greater than $L$. Let $m$ be the total number of packets queued in the stations, which is known by each station. The stations that are small in this window, meaning with fewer than $2+3n \lg L $ packets in queues at the beginning of the window, do not transmit in this stage, as if they had no packets. Based on the information collected in the Gossip stage, every station can compute on its own a comprehensive schedule for delivering the minimum of $L_M$ and $m$ packets from their queues that have been already stored in these queues at the beginning of the current time window. The schedule determines the sender of a packet and the destination of a packet for each round. Transmitting according to such a schedule completes the stage, where only a transmitter and receiver are switched on in each round. A Main stage, as given above, has stations operate based on the sizes of their queues at the beginning of the current time window. The actual numbers of old packets that the stations have in their queues, when a Main stage was planned, might have changed during the Gossip stage. This is because, as stations transmit old packets during a Gossip stage, these packets are not necessarily received by their destination stations, and so still need to be forwarded by the stations that received them and now should act as relays. This issue is taken care of by Auxiliary stages. #### An Auxiliary stage. The goal of this stage is to deliver all the old packets that are in the queues of small stations along with the packets received by the stations in a Gossip stage during the coded transfer that still need to be forwarded. This task is accomplished by the following round-robin style algorithm. A stage is structured into *phases* of $n^2$ rounds each, indexed by the pairs $(i,j)$ for $0\le i,j <n$. In a round $(i,j)$ of a phase, $j$ listens and $i$ sends a packet to $j$, provided that $i$ has such a packet in its queue. Since each small station has at most $4n \lg L$ packets at the beginning of a window, and a station can receive at most these many packets $$(2+3\lg L)\cdot(n-1)\le 4 n\lg L$$ during a Gossip stage, provided that $2\le \lg L$, it is sufficient to execute $8n\lg L$ phases to guarantee that all the considered packets are delivered to their destinations. We may specify that an Auxiliary stage takes these many rounds: $$L_A=n^2\cdot 8n\lg L \ .$$ To summarize, a Gossip stage consists of $n^2(2+3\lg L)\le 4n^2\log L$ rounds and an Auxiliary stage takes $8n^3\log L$ rounds. A Main stage takes the remaining rounds, their number being at least $$L-4n^2\lg L - 8n^3\lg L \ge L-9n^3\lg L \ ,$$ for sufficiently large $n$. We set the initial value of $L$ to the smallest natural number such that the following inequality holds: $$L-9n^3\lg L\ge\frac12L \ .$$ Thus the first Main stage takes at least half of the length of the first window, and so there is enough room for the first Gossip and Auxiliary stages to be completed. \[thm:adjust-window\] A plain-packet algorithm [[<span style="font-variant:small-caps;">Adjust-Window</span>]{.nodecor}]{} needs the energy cap $2$ and has the following latency, for each adversary of injection rate $\rho<1$ and burstiness $\beta$: $$\frac{18n^3\log^2n+2\beta}{1-\rho} \ ,$$ where $n$ is sufficiently large with respect to $\rho$ and $\beta$. It suffices to have such a window length $L$ that the duration of a Main stage is greater than the largest number of packets that might be injected in a window, which is $\rho L + \beta$. Let us assume temporarily that $\rho$ and $\beta$ are known to the stations and therefore the initial value of $L$ can be properly determined, based on these $\rho$ and $\beta$. This assumption may be dropped, as we show later. A Main stage has at least $L-9n^3\lg L$ rounds, so it suffices for a window size $L$ to satisfy the following inequality: $$L- 9 n^3 \lg L \ge \rho L + \beta \ .$$ The above inequality holds for $L=\frac{9n^3\lg^2 n+\beta}{1-\rho}$, for $n$ that is sufficiently large with respect to $\rho$ and $\beta$, as can be verified directly. The latency is at most $2L$ because a packet may spend two consecutive windows in a queue, first as a new packet and then as an old one. This completes the analysis in the case when $L$ is properly set at the beginning of an execution. Next we incorporate into the analysis the mechanism by which the length of the next window may get increased after a current window is over. If the window size is not increased at the end a time window $W$, then all packets injected before $W$ are delivered during $W$ and therefore only packets injected during $W$ are present in queues at the end of $W$. Then our estimates of window size from the beginning of the proof apply. Suppose the size of a time window $W$ is greater than the window size of the immediately preceding window. In general, let $W_1, W_2, W_3, \ldots, W_k$ be a sequence of consecutive windows, where $W_{i-1}$ occurs directly after $W_i$, for each $i$, and window $W$ occurs directly after $W_1$. Let moreover this sequence be such that the size of the window $W_{i-1}$ is greater than the window size of the window $W_i$, for each $1<i\le k$. This means that the window size was increased at the end of $W_i$, and also either $W_k$ is the first window of the considered execution or the size of the window preceding $W_k$ is equal to the size of the window $W_k$. Thus, all packets injected before $W_k$ are delivered during $W_k$. Therefore, as the window size of $W_i$ is twice as large as the window size of $W_{i+1}$ for $1\le i<k$, the number of packets in all queues at the beginning of the window $W$ is at most the following: $$\rho L\cdot \Bigl(\frac12+\frac1{2^2}+\cdots+\frac1{2^k}\Bigr)+\beta \le \rho L+\beta \ .$$ In each execution, the window size $L$ eventually becomes sufficiently large to provide that all packets injected before a window are transmitted within this window, and this final window size is at most $$L=\frac{9n^3\lg^2 n+\beta}{1-\rho} \ ,$$ by an argument applied as in the first part of this proof. The latency is again at most twice the length of such a longest window. Energy-Oblivious Indirect Routing {#sect:oblivious-indirect} ================================= Let an integer $k<n$ denote an energy cap. We present now a plain-packet $k$-energy-oblivious algorithm called [[<span style="font-variant:small-caps;">$k$-Cycle</span>]{.nodecor}]{}. The algorithm operates as follows. Up to $k$ stations are switched on in each round. The stations are partitioned into $\ell$ *groups* of size $k$ each. The $i$th group is denoted as $G_i$, for $1,\ldots,\ell$. Group $G_1$ consists of the $k$ stations $0,1,\ldots, k-1$, the next group $G_2$ comprises the station $k-1$ and the next $k-1$ stations $k, k+1,\ldots, 2k-2$, the next group $G_3$ includes station $2k-2$ and the next $k-1$ stations $2k-1,2k,\ldots, 3k-3$, and so on, with the last group padded with dummy stations if needed. The underlying idea is that a group consists of $k$ stations with consecutive numbers, and a group $G_{i+1}$ starts from the last station in group $G_i$ and includes the next $k-1$ stations. In general, the number of groups is at most $\ell\le \frac{n-1}{k-1}+1$. If $n\le 2k$ then $k$ gets decreased such that $2k=n+1$, which allows to keep fewer stations switched on. After this, we may assume that the inequality $2k\le n+1$ holds in general, which implies that there are at least two groups. Two consecutive groups share one station, called a *connector* of these groups, with group $G_\ell$ sharing station $0$ as a connector with group $G_1$. The stations in a group are ordered by their names into an ordered cycle. Groups themselves are also arranged into an ordered cycle as follows: group $G_{i+1}$ follows group $G_i$, for $i<\ell$, and group $G_1$ follows $G_\ell$. In each round $t$ of an execution, all the stations in some group $G_i$ are switched on, with the other stations switched off; we say that group $G_i$ is *active* in round $t$. The pattern of activity among the groups follows round robin according to the order cycle of the groups. A group is active for a time segment of these many rounds: $$\label{eqn:group-time-segment} \delta=\frac{4 (n-1) k}{n-k} \ .$$ When this time segment ends, the next group in the cyclic order takes over. Each group executes an algorithm related to the broadcast algorithm [[<span style="font-variant:small-caps;">Old-First-Round-Robin-Withholding (OF-RRW)</span>]{.nodecor}]{} during the consecutive rounds the group is active. Algorithm [[<span style="font-variant:small-caps;">OF-RRW</span>]{.nodecor}]{} was considered in [@AnantharamuCKR-JCSS19], we adapt it as a building block of routing algorithms; the details of the adaptation are given next. There is a conceptual token associated with each group. The actions of stations in a group are controlled by feedback from the channel. The feedback is the same for all the stations in a group, which allows to handle the token in such a manner that it is not duplicated nor lost. The token passes through all the stations in a group in a round-robin manner. When the token completes the whole cycle then this also ends a *phase*. Packets injected or adopted during a phase are *new* for this phase and otherwise they are *old* for this phase. When a station receives a token then it transmits all its old packets one by one. If there is no old packet to transmit by a station holding the token then this station does not transmit anything, which results in a silent round. A silent round triggers the token to advance to the next station in the group in their cyclic order. When a station holding the token of a group $G_i$ transmits then the message is heard on the channel by all the stations in the group $G_i$. If the destination station of this packet belongs to $G_i$ then the packet gets delivered and otherwise the station in $G_i$ that is a connector with $G_{i+1}$ adopts the packet and becomes its relay. This mechanism of handling packet implies that a packet may hop through all the $\ell$ groups until it reaches its destination station. \[thm:Cycle-of-Groups\] Algorithm [[<span style="font-variant:small-caps;">$k$-Cycle</span>]{.nodecor}]{} routes packets correctly, when the energy cap is at least $k$, and has latency at most $(32+\beta)\cdot n$ against an $(\rho,\beta)$-adversary such that $\rho< \frac{k-1}{n-1}$. A group operates as a virtual cycle of $k$ stations executing broadcast algorithm [[<span style="font-variant:small-caps;">OF-RRW</span>]{.nodecor}]{}. Such a cycle in isolation would have broadcast latency at most $$\label{eqn:cycle-1} \frac{2}{1-\rho}\cdot k+\beta(1+\rho)\le \frac{2k}{1-\rho}+2\beta \ ,$$ for an injection rate satisfying only the inequality $\rho<1$; see [@AnantharamuCKR-JCSS19]. A packet may perform at most $\frac{n-1}{k-1}$ hops through consecutive groups, which effectively amplifies injection rate by this factor. Therefore injection rates need to be less than $\frac{k-1}{n-1}$ to make routing stable. The bound  on packet delay has two parts, among which $\frac{2k}{1-\rho}$ applies to packets injected within the injection-rate component and $2\beta$ applies to the packets for which injection-rate component does not suffice and they need the adversary’s burstiness to justify their injection; see [@AnantharamuCKR-JCSS19]. We consider the delay of packets by categorizing the packets into two groups: those for which the injection-rate component $\frac{2k}{1-\rho}$ suffices and the remaining ones to which the burstiness component $2\beta$ needs to apply to justify their delay. This categorization of packets if for accounting purposes only. For packets accounted for as injected subject to the injection-rate constraint, the bound $\frac{2k}{1-\rho}$ on packet delay becomes at most $$\label{eqn:cycle-2} \frac{2 k(n-1)}{n-k}$$ after combining it with the upper bound $\rho<\frac{k-1}{n-1}$ on injection rates. Bound  on packet delay is less than the duration of a continuous segment of rounds of activity of a group of stations determined by . This implies that all the packets held by the stations in a group, and accounted for as injected subject to the injection-rate constraint, are heard on the channel when the group becomes active. This also means that these among such packets that are addressed to other groups will hop through the connector to the next group, while their current group is active. Such hops will continue without delay other than that incurred by the period of activity of the group where these packets reside. The bound  needs to be increased to the duration $\delta$ for a period of activity  of a group and then multiplied by the number of hops a packet can make, to obtain a bound on latency of routing. This yields the following estimate $$\frac{4 k(n-1)}{n-k}\cdot \frac{n-1}{k-1}\le \frac{8(n-1)^2}{n-k}\le 16 (n-1) \ ,$$ assuming $2k\le n+1$. This bound accounts for a full cycle of activity of all the groups, but a packet may spend another such cycle waiting for the group into which is got injected to become active. This means that $32\cdot n$ is a bound on latency, restricted to packets that can be accounted for as injected subject to the injection-rate restriction. Next, we estimate the delay of packets that need the adversary’s burstiness to account for their injection. A half of the duration $\delta=\frac{4 (n-1) k}{n-k}$ of group’s activity is needed for packets that can be accounted for as injected subject to the injection-rate restriction, as estimated by . What remains are $\frac{2 (n-1) k}{n-k}$ rounds that can be used to transmit a surplus of packets due to a burst of injections. Out of these many rounds, at most $k$ can be waisted because the token visits stations without packets. What remains are at least these many rounds: $$\label{eqn:burstiness} \frac{2 (n-1) k}{n-k} - k \ .$$ Observe that if $2k\le n+1$ then $\frac{2 (n-1) }{n-k}\ge 4$, and so the quantity  is at least $3k$. The packet delay of the considered packets is thus at most $\frac{2\beta}{3k}$ multiplied by the number of groups, which is at most the following: $$\frac{2\beta}{3k}\cdot \Bigl(\frac{n-1}{k-1}+1\Bigr) \le \frac{2\beta}{3k}\cdot\frac{n+k-2}{k-1} \le \frac{2\beta}{3k}\cdot\frac{\frac{3}{2}n-2}{k-1} \le \beta n \ .$$ The bound on latency is the maximum of the partial upper bounds $32n$ and $\beta n$. Next we give an impossibility which demonstrates that the bound on injection rate in Theorem \[thm:Cycle-of-Groups\] is very close to optimal. \[thm:indirect-oblivious\] For each $n$ and $k<n$, a $k$-energy-oblivious routing algorithm is unstable against adversaries with injection rates greater than $\frac{k}{n}$. If a station is switched on in a round then this contributes one *station-round*. A contiguous time interval $\tau$ of $|\tau|=t$ rounds can contribute at most $kt$ station-rounds. By the double-counting principle, there is some station $v$ which is switched on for at most $\frac{kt}{n}$ rounds during these $t$ rounds. The adversary with injection rate $\rho$ can inject at least $\rho t$ packets into station $v$ during these rounds. Even if $v$ transmits successfully in each round in $\tau$ then it can transmit at most $\frac{kt}{n}$ packets. If $\rho>\frac{k}{n}$ then there remain at least these many packets that need to be queued by $v$: $$\rho t-\frac{kt}{n} = t\Bigl(\rho-\frac{k}{n}\Bigr) \ .$$ This number can be made arbitrarily large for a suitably large $t$. There is no $k$-energy-oblivious universal algorithm when $k=c\cdot n$, for a constant $c<1$. By Theorem \[thm:indirect-oblivious\], a $k$-energy-oblivious algorithm is unstable for injection rates that are greater than the ratio $\frac{k}{n}=c<1$. Energy-Oblivious Direct Routing {#sect:oblivious-direct} =============================== Let an integer $k<n$ denote the energy cap. Now we present a plain-packet $k$-energy-oblivious algorithm that routes packets directly. It is called [[<span style="font-variant:small-caps;">$k$-Clique</span>]{.nodecor}]{}. There are up to $k$ stations switched on in each round. We assume that $k$ is even and divides $2n$, to simplify the notation. The stations are partitioned into $\frac{2n}{k}$ disjoint sets of size $\frac{k}{2}$ each. These sets are combined in $\frac{n}{k}(\frac{2n}{k}-1)$ *pairs* of size $k$ each. There are at least $3$ pairs, assuming $\frac{k}{2}\le \frac{n}{3}$. If $\frac{k}{2}> \frac{n}{3}$ then we can decrease $k$ by keeping fewer stations switched on, so that the inequality $k\le \frac{2n}{3}$ holds. In each round $t$ of an execution, all the stations in some pair are switched on, with the other stations switched off; we say that the pair is *active* in round $t$. The pairs are arranged into a virtual cycle to assign them the rounds of activity in a round-robin manner. A pair is active for one round at a time, and then the next pair takes over. When a pair is active then its stations execute an algorithm based on the principle of broadcasting algorithm [[<span style="font-variant:small-caps;">OF-RRW</span>]{.nodecor}]{}. A station that has the token transmits all the old packets whose destinations are among the stations that make up the pair. \[them:k-clique\] If algorithm [[<span style="font-variant:small-caps;">$k$-Clique</span>]{.nodecor}]{} is executed against a $(\rho,\beta)$-adversary then it has a bounded latency for injection rates $\rho<\frac{k^2}{n(2n-k)}$, and the latency is at most $8 \frac{n^2}{k}(1+\frac{\beta}{2k})$ if the injection rate $\rho$ is at most $\frac{k^2}{2n(2n-k)}$. Let $m=\frac{n(2n-k)}{k^2}$ be the number of pairs. We will use the estimate $m\le 2\frac{n^2}{k^2}$. A strategy for the adversary that maximizes queues and latency works by injecting packets into one pair with destinations in the same pair as well. Since a pair is allotted one round out of a segment of rounds equal to the number of pairs, an injection rate needs to be less than the inverse of the number of pairs, which is $\frac{1}{m}=\frac{k^2}{n(2n-k)}$. For each pair, when time is scaled only to the rounds which are assigned for the pair to execute [[<span style="font-variant:small-caps;">OF-RRW</span>]{.nodecor}]{}, the injection rate is less than $1$, so the algorithm has bounded latency. Suppose that the inequality $\rho<\frac{1}{m}$ holds. A pair of $k$ stations operating in isolation, and with time scaled only to the rounds assigned to the pair to be active, would have effective injection rate $m\cdot \rho$ and so its latency would be at most the following $$\label{eqn:clique} \frac{2}{1-m\rho}\cdot k+\beta(1+m\rho)\le \frac{2}{1-m\rho}\cdot k+2\beta \ ,$$ by applying the bound on broadcast latency of [[<span style="font-variant:small-caps;">OF-RRW</span>]{.nodecor}]{} derived in [@AnantharamuCKR-JCSS19]. The bound needs to be increased by a multiplicative factor of $m$, since a pair operates in one round only in a segment of $m$ rounds. This gives the following estimate on latency: $$\frac{2m}{1-m\rho}\cdot k+2\beta m \le \frac{2n^2}{k}\cdot \frac{2}{1-m\rho}+ \frac{4\beta n^2}{k^2} \ ,$$ which holds for any injection rate satisfying $\rho<m^{-1}$. Assuming additionally that the inequality $$\rho n(2n-k) \le \frac{k^2}{2}$$ holds, we can use the following estimate: $$\frac{2}{1-m\rho} = \frac{2}{1-\rho \frac{n(2n-k)}{k^2}} = \frac{2 k^2}{k^2-\rho n(2n-k)} \le 4 \ .$$ This yields $8 \frac{n^2}{k}(1+\frac{\beta}{2k})$ as a bound on latency. #### Maximum throughput of energy-oblivious direct routing. We describe a direct-routing $k$-energy-oblivious algorithm achieving the throughput $\frac{k(k-1)}{n(n-1)}$ for any burstiness $\beta$. The algorithm is called [[<span style="font-variant:small-caps;">$k$-Subsets</span>]{.nodecor}]{}. It uses algorithm [[<span style="font-variant:small-caps;">Move-Big-To-Front (MBTF)</span>]{.nodecor}]{} [@ChlebusKR-DC09] as a subroutine. [[<span style="font-variant:small-caps;">MBTF</span>]{.nodecor}]{} provides stability for injection rate $1$ with any burstiness, for a multiple access channel *without* any energy cap. Let us fix an enumeration of all the $k$-element subsets of the set $[n]$ in order: $A_0,\ldots,A_{\gamma-1}$, where $\gamma=\binom{n}{k}$. For an integer $i$ such that $0\le i \le \gamma-1$ and an integer $j\ge 0$, the rounds of the form $i+j\gamma$ make *thread $i$*. Algorithm [[<span style="font-variant:small-caps;">MBTF</span>]{.nodecor}]{} operates in $\gamma$ instantiations corresponding to the threads. Each such an instantiation has a dedicated queue in every station. The stations in $A_i$ are active during thread $i$ and process packets assigned to this thread. An execution is structured into *phases*, each of length $\gamma$, such that each thread has one round in a phase. A packet injected during a phase $j$ is treated by an instantiation of [[<span style="font-variant:small-caps;">MBTF</span>]{.nodecor}]{} that handles it as if it were injected “at round $j$.” These specifications mean that the algorithm is $k$-energy-oblivious. At the beginning of a phase, a station $v$ assigns all the packets it received in the previous phases to the threads in the following manner. For each station $w$, station $v$ keeps track of the numbers $x_0(w),\ldots,x_{\gamma-1}(w)$, which represent the respective numbers of packets addressed to $w$ and already allocated by $v$ to threads $0,\ldots,\gamma-1$. Station $v$ allocates the packets addressed to $w$ that it received in the previous phase such that the resulting allocation is as balanced as possible, subject to the constraint that a packet addressed to $w$ can be allocated to a thread $i$ only if both stations $v$ and $w$ are in the set $A_i$. The algorithm routes directly, since when a packet is heard transmitted in a round assigned to a thread $i$, the receiver is switched on by virtue of belonging to $A_i$. Obtaining balancing allocations means that we want the numbers $x_0(w),\ldots,x_{\gamma-1}(w)$ differ among themselves as little as possible; it follows that these numbers differ by at most $1$ at the beginning of each phase. \[thm:oblivious-direct-throughput\] For each $k<n$, algorithm [[<span style="font-variant:small-caps;">$k$-Subsets</span>]{.nodecor}]{} is stable against adversaries with injection rate $\frac{k(k-1)}{n(n-1)}$ and the number of queued packets is at most $2\binom{n}{k} (n^2+\beta)$ in every round. Let $\lambda=\frac{k(k-1}{n(n-1)}$ denote the injection rate we consider. Suppose that the algorithm is not stable for injection rate $\lambda$, to arrive at a contradiction. There exists a thread $i$ in which some queue corresponding to packets arriving at station $v$ with address $w$ and assigned to this thread grows unbounded. Since algorithm [[<span style="font-variant:small-caps;">MBTF</span>]{.nodecor}]{} is stable for injection $1$ and a fixed burstiness, there exists an infinite sequence of rounds $t_1,t_2,\ldots,t_j,\ldots$, for all $j\ge 1$, such that the number of packets from station $v$ to $w$ that get assigned to thread $i$ by round $j$ is at least $|t_j|/\gamma + j+2$. Indeed, each thread is executed once every $\gamma$ rounds, so the execution of algorithm [[<span style="font-variant:small-caps;">MBTF</span>]{.nodecor}]{} in thread $i$ would be stable if burstiness were bounded. The algorithm allocating packets to threads guarantees that the number of packets with the same pair $(v,w)$, of the source $v$ and destination $w$, assigned by $v$ to threads with stations $v$ and $w$ being active is almost balanced in each time period, in that the difference between any two of them is either $-1$ or $0$ or $1$. A thread with this property is determined by the stations different from $v$ and $w$, so their number equals the following: $$\binom{n-2}{k-2} = \binom{n}{k}\cdot \frac{k(k-1}{n(n-1)}=\gamma\lambda \ .$$ The number of packets injected to $v$ addressed to $w$ by round  $t_j$, for every $j\ge 1$, is at least $(t_j/\gamma + j+2)-1$ multiplied by the number of threads handling packets from $v$ to $w$. It follows that the following is a lower bound on the number of packets addressed to $w$ that are assigned by $v$ to some threads by round $t_j$: $$\lambda t_j + \gamma\lambda (j+1)=\lambda(t_j+\gamma) + \gamma\lambda j \ .$$ Therefore, the number of packets that arrive at $v$ by round $t_j+\gamma$ is at least $\lambda(t_j+\gamma) + \gamma\lambda j$. This contradicts the restrictions on the adversary for sufficiently large $j$, as the burstiness would be exceeded by round $t_j+\gamma$, and completes showing stability. A bound on the number of queued packets follows from the respective bound for algorithm [[<span style="font-variant:small-caps;">MBTF</span>]{.nodecor}]{} given in [@ChlebusKR-DC09], which is applied independently for each thread. It may occur in an execution of algorithm [[<span style="font-variant:small-caps;">$k$-Subsets</span>]{.nodecor}]{} that some packets never get delivered and so remain queued forever. The algorithm can be modified to prevent this as long as injection rates are less than $\frac{k(k-1}{n(n-1)}$. Namely, it is sufficient to replace algorithm [[<span style="font-variant:small-caps;">MBTF</span>]{.nodecor}]{}, used as a procedure in [[<span style="font-variant:small-caps;">$k$-Subsets</span>]{.nodecor}]{}, by [[<span style="font-variant:small-caps;">Round-Robin-Withholding (RRW)</span>]{.nodecor}]{}, see [@ChlebusKR-TALG12]. The resulting algorithm is stable for any injection rate less than $\frac{k(k-1)}{n(n-1)}$ and achieves bounded latency. By the performance bounds of [[<span style="font-variant:small-caps;">RRW</span>]{.nodecor}]{}, see [@AnantharamuCKR-JCSS19], the latency is $\Theta(\gamma \cdot (n+\beta))$, for a fixed adversary with injection rate less than $\frac{k(k-1)}{n(n-1)}$. The latency bound is at least $\gamma=\binom{n}{k}$, which is exponential in $n$ when $k$ is linear in $n$. #### A lower bound on throughput. We give a matching lower bound on throughput, which demonstrates that the throughput in Theorem \[thm:oblivious-direct-throughput\] is maximum achievable in the class of energy-oblivious algorithms. For each integer $n$ and $k<n$, and for any $k$-energy-oblivious algorithm routing directly, and for every adversary with an injection rate greater than $\frac{k(k-1)}{n(n-1)}$, some executions of the algorithm may be unstable against this adversary. We will count the following quantities: for each ordered pair $(x,y)$ of different stations $x$ and $y$, if they are switched on simultaneously in a round then this rounds contributes one *station-pair round*. A contiguous time interval $\tau$ of $|\tau|=t$ rounds can contribute at most $k(k-1)t$ station-pair rounds. By the double-counting principle, there is some ordered pair of stations $(w,z)$ such that $w$ and $z$ are switched on together for at most $\frac{k(k-1)}{n(n-1)}\cdot t$ rounds in the time interval $\tau$. The adversary with injection rate $\rho$ can inject at least $\rho t$ packets into station $w$ during these rounds. Let all these packets be destined for $z$. Even if $w$ transmits successfully in each round in $\tau$ such that $w$ is switched on along with $z$, then it can transmit at most $\frac{k(k-1)}{n(n-1)}\cdot t$ packets. If $\rho>\frac{k(k-1)}{n(n-1)}$ then there remain at least these many packets that need to be queued by $w$: $$\rho t-\frac{k(k-1)}{n(n-1)}t = t(\rho-\frac{k(k-1)}{n(n-1)}) \ .$$ This number can be made arbitrarily large for a suitably large $t$. Conclusion ========== There are several natural questions related to the presented topic that are open. One of them is to derive tradeoffs between latency and energy cap, for a class of algorithms. Such tradeoffs are natural to hold, since small energy cap is a constrain on scheduling transmissions in a distributed manner. Another group of questions pertains to minimizing latency so that it is ${\mathcal{O}}(n)$. Broadcast algorithms with such latency for all injection rates less than $1$ have been developed, in the case when there are no energy constraints on a channel. It is not known if there exists a universal routing algorithm for a non-trivial bound on energy cap that has ${\mathcal{O}}(n)$ latency for each injection rate less than $1$. It is not known if a plain-packet routing algorithm for a constant energy cap can have latency ${\mathcal{O}}(n)$ for a non-trivial range of injection rates. It is not known if an algorithm routing directly and subject to a non-trivial energy cap can have latency ${\mathcal{O}}(n)$.

--- abstract: 'We prove that a weakly ergodic, eventually strong Feller semigroup on the space of measures on a Polish space converges strongly to a projection onto its fixed space.' author: - Moritz Gerlach bibliography: - 'analysis.bib' title: | A Tauberian Theorem\ for Strong Feller Semigroups --- Introduction ============ In the study of Markov processes one is interested in semigroups of operators on the space of measures that describe the evolution of distributions. Of specific importance is the question under which conditions such a semigroup is stable in the sense that for every initial distribution the process converges to an invariant measure as time goes to infinity. A celebrated theorem by Doob asserts that a stochastically continuous Markov semigroup is stable if it admits an invariant measure, is irreducible and has the strong Feller property; see [@doob48], [@gerlach2014 Thm 4.4], [@gerlach2012], [@stettner1994] and [@seidler1997] for various versions and proofs of this result. We recall that a semigroup on the space of measures is said to have the strong Feller property if its adjoint maps bounded measurable functions to continuous ones. A necessary condition for stability of a semigroup is weak ergodicity, i.e. convergence of the Cesàro averages in the weak topology induced by the bounded continuous functions. In [@gerlach2013] M. Kunze and the author characterized ergodicity of semigroups on general norming dual pairs in the spirit of the classical mean ergodic theorem. In particular for eventually strong Feller Markov semigroups it was shown that they are weakly ergodic if the space of invariant measures separates the space of invariant continuous functions. In the present article we prove that for eventually strong Feller semigroups weak ergodicity is already sufficient for stability, i.e. pointwise convergence of the semigroup in the total variation norm. In comparison to Doob’s classical result, this Tauberian theorem even shows stability of not necessarily irreducible semigroups whose fixed space is of arbitrary high dimension. In the following section we show that the square of every strong Feller operator is a kernel operator in the sense that it belongs to the band generated by the finite rank operators. Section \[sec:main\] addresses Markov semigroups and their asymptotic behavior and contains the proof of our main result, Theorem \[thm:tauberian\]. Strong Feller and Kernel operators {#sec:kernelops} ================================== Throughout, $\Omega$ denotes a Polish space and $\mathscr{B}(\Omega)$ its Borel $\sigma$-algebra. We denote by $\mathscr{M}(\Omega), B_b(\Omega)$ and $C_b(\Omega)$ the spaces of signed measures on $\mathscr{B}(\Omega)$, the space of bounded, Borel-measurable functions on $\Omega$ and the space of bounded continuous functions on $\Omega$, respectively. We denote by ${\langle \; \cdot\;,\; \cdot\;\rangle}$ the duality between $B_b(\Omega)$ and $\mathscr{M}(\Omega)$. \[def:transkernel\] A *Markovian transition kernel* on $\Omega$ is a map $k: \Omega\times\mathscr{B}(\Omega)\to {\mathds{R}}_+$ such that (a) $A \mapsto k(x, A)$ is a probability measure for every $x\in \Omega$ and (b) $x \mapsto k(x,A)$ is a measurable function for every $A\in {\mathscr{B}}(\Omega)$. To each Markovian transition kernel $k$, one can associate a positive operator $T \in \mathscr{L}(\mathscr{M}(\Omega))$ by setting $$\begin{aligned} \label{eq.kernel} (T\mu)(A) {\mathrel{\mathop:}=}\int_\Omega k(x,A)\, {\mathrm{d}}\mu (x).\end{aligned}$$ for all $\mu \in {\mathscr{M}}(\Omega)$ and $A \in {\mathscr{B}}(\Omega)$. The following lemma characterizes operators of this form. \[lem:weaklycont\] For a positive operator $T \in \mathscr{L}(\mathscr{M}(\Omega))$ the following assertions are equivalent: (i) There exists a Markovian transition kernel $k$ such that $T$ is given by . (ii) The norm adjoint $T^*$ of $T$ leaves $B_b(\Omega)$ invariant and $T^* \mathds{1} = \mathds{1}$. (iii) The operator $T$ is continuous in the $\sigma(\mathscr{M}(\Omega), B_b(\Omega))$-topology. This follows from Propositions 3.1 and 3.5 of [@kunze2011]. If an operator $T\in {\mathscr{L}}({\mathscr{M}}(\Omega))$ satisfies the equivalent conditions from Lemma \[lem:weaklycont\], then $T$ is called *Markovian* and we write $T'$ for the restriction of $T^*$ to $B_b(\Omega)$. If a Markovian operator $T\in {\mathscr{L}}({\mathscr{M}}(\Omega))$ even satisfies $T' f \in C_b(\Omega)$ for all $f\in B_b(\Omega)$, then $T$ is called *strong Feller* and if, in addition, the family $$\{ T'f : f \in B_b(\Omega),\; \lvert f\rvert \leq c\mathds{1} \}$$ is equi-continuous for all $c> 0$, then $T$ is said to be *ultra Feller*. It is well know that the product of two strong Feller operators is ultra Feller, see [@revuz1975 §1.5]. We recall that for two Riesz spaces $E$ and $F$ a linear operator from $E$ to $F$ is called regular if it is the difference of two positive operators. If the Riesz space $F$ is order complete, the regular operators from $E$ to $F$ form itself a order complete Riesz space. By $E^*_{\text{oc}}$ we denote the order continuous linear functionals on $E$. \[def:kernelops\] Let $E$ and $F$ be Riesz spaces and $F$ be order complete. We denote by $E^*_{\text{oc}}\otimes F$ the space of order continuous finite rank operators from $E$ to $F$. The elements of $(E^*_{\text{oc}}\otimes F)^{\bot\bot}$, the band generated by $E^*_{\text{oc}}\otimes F$ in the regular operators from $E$ to $F$, are called *kernel operators*. Since $\mathscr{M}(\Omega)$ is a $L$-space, its norm is order continuous. Therefore, every bounded linear operator on $\mathscr{M}(\Omega)$ is regular and order continuous and ${\mathscr{M}}(\Omega)^*={\mathscr{M}}(\Omega)^*_{\text{oc}}$. Thus, $\mathscr{L}(\mathscr{M}(\Omega))$ is an order complete Banach lattice with respect to the natural ordering, see [@schaefer1974 Thm IV 1.5]. We use the following characterization of kernel operators on $L^\infty$-spaces due to Bukhvalov: \[thm:bukhvalov\] Let $\mu$ and $\nu$ be finite measures on $(\Omega,{\mathscr{B}}(\Omega))$ and $T$ a bounded linear operator from $L^\infty(\Omega,\nu)$ to $L^\infty(\Omega,\mu)$. Then $T$ is a kernel operator if and only if $\lim Tf_n =0$ $\mu$-almost everywhere for each bounded sequence $(f_n)\subset L^\infty(\Omega,\nu)$ satisfying $\lim \lVert f_n \rVert_{L^1(\Omega,\nu)} =0$. This follows from Bukhvalov’s theorem [@zaanen1983 Thm 96.5] and the identification of concrete and abstract kernel operators [@schaefer1974 Prop IV 9.8]. \[thm:ultrafellerkernel\] Let $T\in{\mathscr{L}}({\mathscr{M}}(\Omega))$ be an ultra Feller operator. Then $T$ is a kernel operator. Let $\mu \in {\mathscr{M}}(\Omega)_+$ and $\nu {\mathrel{\mathop:}=}T\mu$, then $T \{ \mu\}^{\bot\bot} \subset \{\nu\}^{\bot\bot}$. Let us denote by $T_\mu$ the restriction of $T$ to $\{\mu\}^{\bot\bot}$. By the Radon-Nikodym theorem, $\{\mu\}^{\bot\bot}$ and $\{\nu\}^{\bot\bot}$ are isometrically isomorphic to $L^1(\Omega,\mu)$ and $L^1(\Omega,\nu)$. Thus, we may consider $T_\mu$ as an operator from $L^1(\Omega,\mu)$ to $L^1(\Omega,\nu)$ and we prove that $T_\mu^* : L^\infty(\Omega,\nu)\to L^\infty(\Omega,\mu)$ is a kernel operator by applying Bukhvalov’s theorem in the version of Theorem \[thm:bukhvalov\]. It is easy to check that $$T_\mu^* [f] = [T' f]$$ for every $f\in B_b(\Omega)$, where $[f]$ denotes the equivalent class of $f$ in $L^\infty(\Omega,\mu)$. Let $(f_n) \subset L^\infty(\Omega,\nu)$ be a bounded sequence such that $\lim \lVert f_n \rVert_{L^1(\Omega,\nu)} =0$. By choosing representatives we may assume that every $f_n$ is a bounded measurable function. Moreover, we may assume that each $f_n$ vanishes on $\Omega\setminus \operatorname{supp}(\nu)$. Then $$0= \lim {\langle f_n,\nu\rangle}=\lim {\langle T_\mu^*f_n,\mu\rangle} = \lim {\langle T_\mu'f_n,\mu\rangle}.$$ Let $\omega \in \Omega$ such that $(T_\mu'f_n)(\omega)$ does not converge to $0$. Then there exists ${\varepsilon}>0$ and a subsequence $(f_{n_k})$ of $(f_n)$ such that $T_\mu'f_{n_k}(\omega) \geq {\varepsilon}$ for all $k\in{\mathds{N}}$. By the ultra Feller property of $T_\mu$, the family $$\{ T_\mu'f_{n_k} : k\in {\mathds{N}}\}$$ is equi-continuous. Therefore, we find an open neighborhood $U$ of $\omega$ such that $(T_\mu'f_{n_k})(s)\geq {\varepsilon}/2$ for all $s\in U$ and $k\in{\mathds{N}}$. Now we conclude from $$\mu(U)\frac{{\varepsilon}}{2} \leq \int_\Omega T_\mu'f_{n_k} {\mathrm{d}}\mu \to 0 \quad (k\to\infty)$$ that $\mu(U)=0$ and hence $U \subset \Omega\setminus \operatorname{supp}(\mu)$. This proves that $(T_\mu' f_n)(\omega)$ converges to $0$ for all $\omega\in \operatorname{supp}(\mu)$ and hence almost everywhere. Thus, it follows from Theorem \[thm:bukhvalov\] that $$T_\mu^*\in (L^\infty(\Omega,\nu)^*_{\text{oc}} \otimes L^\infty(\Omega,\mu))^{\bot\bot}.$$ By [@meyer1991 Prop 1.4.15], the order continuous functionals on $L^\infty(\Omega,\nu)$ are precisely $L^1(\Omega,\nu)$. Thus, $T_\mu^*\in (L^1(\Omega,\nu) \otimes L^\infty(\Omega,\mu))^{\bot\bot}$. Now we prove that $T_\mu \in (L^\infty(\Omega,\mu)\otimes L^1(\Omega,\nu))^{\bot\bot}$. Let $0\leq S_\alpha\leq T_\mu^*$, $\alpha\in\Lambda$, be an upwards directed net and $R_\alpha \in L^1(\Omega,\nu)\otimes L^\infty(\Omega,\mu)$ such that $\sup S_\alpha=T_\mu$ and $S_\alpha \leq R_\alpha$ for all $\alpha\in\Lambda$. Then $S_\alpha^* \leq R_\alpha^* \in L^\infty(\Omega,\mu)\otimes L^1(\Omega,\nu)$ for all $\alpha\in\Lambda$. Since $L^1(\Omega,\nu)$ is an ideal in the dual of $L^\infty(\Omega,\nu)$, we obtain that $${S_\alpha^*}\mid_{L^1(\Omega,\mu)} : L^1(\Omega,\mu) \to L^1(\Omega,\nu).$$ Now it follows from $$\sup {\langle (T_\mu-S^*_\alpha)f,g\rangle} = \sup {\langle f,(T_\mu^*-S_\alpha)g\rangle} = 0$$ for all $f\in L^1(\Omega,\mu)$ and $g\in L^\infty(\Omega,\nu)$ that $T_\mu = \sup S^*_\alpha$ and therefore $$T\mid_{ L^1(\Omega,\mu)} = T_\mu \in (L^\infty(\Omega,\mu)\otimes L^1(\Omega,\nu))^{\bot\bot}.$$ It follows that $TP_\mu \in ({\mathscr{M}}(\Omega)^*\otimes {\mathscr{M}}(\Omega))^{\bot\bot}$ for every $\mu\in {\mathscr{M}}(\Omega)_+$ where $P_\mu$ denotes the band projection onto $\{\mu\}^{\bot\bot}$. Thus, $$T = \sup \{ TP_\mu : \mu \in {\mathscr{M}}(\Omega)_+ \}$$ belongs to $({\mathscr{M}}(\Omega)^*\otimes {\mathscr{M}}(\Omega))^{\bot\bot}$ which completes the proof. Stability of ergodic strong Feller semigroups {#sec:main} ============================================= A *Markovian semigroup* on ${\mathscr{M}}(\Omega)$ is a family ${\mathscr{T}}=(T(t))_{t\geq 0} \subset {\mathscr{L}}({\mathscr{M}}(\Omega))$ of Markovian operators on ${\mathscr{M}}(\Omega)$ such that $T(t+s) = T(t)T(s)$ for all $t,s\geq 0$ and $T(0)=I$. A Markovian semigroup is called *stochastically continuous* if $t\mapsto {\langle T(t)\mu,f\rangle}$ is continuous for all $f\in C_b(\Omega)$ and $\mu \in {\mathscr{M}}(\Omega)$. Throughout, let ${\mathscr{T}}=(T(t))_{t\geq 0}$ be a stochastically continuous Markovian semigroup. It follows from [@kunze2011 Thm 6.2] that ${\mathscr{T}}$ is *integrable* in the sense of [@kunze2011 Def 5.1]. In particular, by [@kunze2011 Thm 5.8], for every $t>0$ there exists a Markovian operator $A_t \in {\mathscr{L}}({\mathscr{M}}(\Omega))$ satisfying $${\langle A_t \mu,f\rangle} = \frac{1}{t} \int_0^t {\langle T(s)\mu,f\rangle} {\mathrm{d}}s$$ for all $\mu\in {\mathscr{M}}(\Omega)$ and $f\in B_b(\Omega)$. We call the semigroup ${\mathscr{T}}$ $B_b$-*ergodic* if $\lim_{t\to\infty} A_t \mu$ exists in the $\sigma({\mathscr{M}}(\Omega),B_b(\Omega))$-topology for all $\mu\in{\mathscr{M}}(\Omega)$. The following proposition ensures that for every initial distribution the part on the disjoint complement of ${\mathrm{fix}}({\mathscr{T}})$ converges to zero if ${\mathscr{T}}$ is $B_b$-ergodic and eventually strong Feller. \[prop:stabilityonfixbot\] Let $P$ denote the band projection onto ${\mathrm{fix}}({\mathscr{T}})^{\bot}$. If ${\mathscr{T}}$ is $B_b$-ergodic and $T(t_0)$ is strong Feller for some $t_0>0$, then $$\lim_{t\to\infty} PT(t)\mu = 0$$ for all $\mu \in {\mathscr{M}}(\Omega)$. First note that, since ${\mathrm{fix}}({\mathscr{T}})^{\bot\bot}$ is ${\mathscr{T}}$-invariant, $R(t) {\mathrel{\mathop:}=}PT(t)$ defines a semigroup. Obviously, every operator $R(t)$ is positive and contractive. Fix $\mu \in {\mathscr{M}}(\Omega)_+$ and let $$\alpha {\mathrel{\mathop:}=}\lim_{t\to\infty} \lVert PT(t) \mu \rVert = \inf_{t\geq 0} \lVert R(t)\mu \rVert.$$ We pick $t_1>0$ such that $\nu {\mathrel{\mathop:}=}PT(t_1)\mu$ satisfies $\lVert \nu \rVert < \alpha+\frac{\alpha}{2}$. Since ${\mathscr{T}}$ is $B_b$-ergodic, $\lim A_t \nu {=\mathrel{\mathop:}}\tilde\nu$ exists with respect to the $\sigma({\mathscr{M}}(\Omega),B_b(\Omega))$-topology and $\tilde \nu \in {\mathrm{fix}}({\mathscr{T}})$ by [@gerlach2013 Lem 4.5]. In particular, $${\langle \tilde\nu,\mathds{1}\rangle} = \lim_{t\to\infty} \frac{1}{t} \int_0^t {\langle T(s)\nu,\mathds{1}\rangle}{\mathrm{d}}s = \lVert\nu\rVert \geq \alpha.$$ Let $t\geq t_0$. As $PT(2t)\nu$ and $\nu$ are disjoint, there exists a Borel set $B \subset \Omega$ such that $$(PT(2t)\nu)(B) = \tilde \nu (\Omega\setminus B)=0.$$ Since $\lVert T(2t)\nu \rVert \leq \lVert \nu\rVert < \alpha+\frac{\alpha}{2}$ and $$\lVert PT(2t)\nu\rVert = \lVert R(t_1+2t)\mu \rVert \geq \alpha$$ it follows from the additivity of the total variation norm that $\lVert (I-P)T(2t)\nu \rVert < \frac{\alpha}{2}$. Hence, $(T(2t)\nu)(B) < \frac{\alpha}{2}$. Let $f{\mathrel{\mathop:}=}T'(t)\mathds{1}_B$ and $g{\mathrel{\mathop:}=}T'(t)\mathds{1}_{\Omega\setminus B}$. Since $T(t)=T(t-t_0)T(t_0)$ is strong Feller and Markovian, $f,g \in C_b(\Omega)_+$ and $f+g = \mathds{1}$. It follows from ${\langle \tilde \nu,g\rangle}=0$ that $$A {\mathrel{\mathop:}=}\operatorname{supp}\tilde\nu \subset \{ g=0 \} = \{ f=1\},$$ i.e. $\mathds{1}_A \leq f$. Thus, $${\langle T(t)\nu,\mathds{1}_A\rangle} \leq {\langle T(t)\nu,f\rangle} = {\langle T(2t)\nu,\mathds{1}_B\rangle} < \frac{\alpha}{2}.$$ Since $t\geq t_0$ was arbitrary, we conclude that ${\langle T(t)\nu,\mathds{1}_A\rangle} < \frac{\alpha}{2}$ for all $t\geq t_0$ and hence $$\alpha = {\langle \tilde\nu,\mathds{1}_A\rangle} = \lim_{t\to\infty} \frac{1}{t} \int_0^t {\langle T(s)\nu,\mathds{1}_A\rangle} \leq \frac{\alpha}{2}.$$ Thus, $\alpha=0$. The assumption that ${\mathscr{T}}$ is eventually strong Feller cannot be dropped in Proposition \[prop:stabilityonfixbot\]. A counterexample is given by the rotation group on the Borel measures on the unit circle. Note that the situation is different for time-discrete semigroups. If $T$ is a positive and mean ergodic contraction on an $L$-space $E$ and $P$ is the band projection onto ${\mathrm{fix}}(T)^\bot$, then it is possible to prove that $$\lim_{n \to\infty} \lVert P T^n x \rVert = 0$$ for all $x\in E$. Our main tool for the proof of the desired Tauberian theorem is the following result from [@gerlach2012b Thm 4.2], a generalized version of [@greiner1982 Kor 3.11]. We recall that a family ${\mathscr{S}}= (S(t))_{t\geq 0} \subset {\mathscr{L}}(E)$ of positive operators on a Banach lattice $E$ is called a *positive strongly continuous semigroup* if $S(t)S(s) = S(t+s)$ for all $t,s\geq 0$, $S(0)=I$ and the mapping $t\mapsto S(t)x$ is continuous for all $x\in E$. A positive strongly continuous semigroup ${\mathscr{S}}= (S(t))_{t\geq 0}$ is called *irreducible* if $\{0\}$ and $E$ are the only closed ideals in $E$ that are invariant under the action of every operator $S(t)$. \[thm:greiner\] Let ${\mathscr{S}}= (S(t))_{t\geq 0} \subset {\mathscr{L}}(E)$ be a positive, bounded, irreducible and strongly continuous semigroup on a Banach lattice $E$ with order continuous norm such that ${\mathrm{fix}}({\mathscr{S}}) \not= \{0\}$. If $S(t_0)\in (E^*_{\text{oc}}\otimes E)^{\bot\bot}$ for some $t_0>0$, then there exists a positive $z^* \in {\mathrm{fix}}({\mathscr{S}}^*)$ and a positive $z \in {\mathrm{fix}}({\mathscr{S}})$ of $E_+$ such that $$\lim_{t\to\infty} S(t)x = {\langle z^*,x\rangle} z$$ for all $x\in E$. The following theorem shows that, if the semigroup ${\mathscr{T}}$ contains a kernel operator, every principal band $\{\mu\}^{\bot\bot}$ spanned by an invariant measure $\mu$ can be decomposed into countably many invariant bands such that the restriction of ${\mathscr{T}}$ to each of them is irreducible. \[thm:irreddecomp\] Let $\mu\in {\mathscr{M}}(\Omega)$ be a positive ${\mathscr{T}}$-invariant measure. If $T(t_0)$ is a kernel operator, then there exist at most countably many disjoint ${\mathscr{T}}$-invariant measures $\{ \mu_n \} \subset {\mathscr{M}}(\Omega)_+$ such that $\mu = \mu_1+\mu_2+\dots$ and the restriction of ${\mathscr{T}}$ to each $\{\mu_n\}^{\bot\bot}$ is irreducible. By the Radon-Nikodym theorem we may identify the band $\{\mu\}^{\bot\bot}$ with $L^1(\Omega,\mu)$ and thus consider ${\mathscr{T}}$ as a contractive semigroup on $L^1(\Omega,\mu)$. Since the measure $\mu$ is ${\mathscr{T}}$-invariant and corresponds to $\mathds{1} \in L^1(\Omega,\mu)$, $T(t)\mathds{1}_B \leq \mathds{1}$ for all $B\in {\mathscr{B}}(\Omega)$ and $t\geq 0$. In this proof, we call a Borel set $B\subset {\mathscr{B}}(\Omega)$ *invariant* if $T(t)\mathds{1}_B \leq \mathds{1}_B$ almost everywhere for all $t\geq 0$ and *irreducible* if for every invariant Borel set $A\subset B$ we have $\mu(A)=0$ or $\mu(A)=\mu(B)$. With this identification and notation, we have to find at most countable many disjoint invariant and irreducible Borel sets $B_1,B_2,\dots$ such that $B_1 \cup B_2 \cup \dots = \Omega$. First, we show that a Borel set $B$ is invariant if and only if $\Omega\setminus B$ is invariant if and only if $\mathds{1}_B, \mathds{1}_{\Omega\setminus B}\in {\mathrm{fix}}({\mathscr{T}})$. Let $B\in {\mathscr{B}}(\Omega)$ be invariant. Then, for every $t\geq 0$, $$T(t)\mathds{1}_{\Omega\setminus B} = T(t)\mathds{1} - T(t)\mathds{1}_B \geq \mathds{1} - \mathds{1}_B = \mathds{1}_{\Omega\setminus B} \quad \mu\text{-almost everywhere.}$$ Since $T(t)$ is contractive, $\mathds{1}_{\Omega\setminus B}$ is a fixed point of $T(t)$ and so is $\mathds{1}_B$. Next, we prove the existence of an irreducible Borel set of positive measure. Aiming for a contradiction, we assume that $\mu(B)=0$ for every irreducible $B\in {\mathscr{B}}(\Omega)$. For $n\in{\mathds{N}}$ define $${\mathscr{A}}_n {\mathrel{\mathop:}=}\left\{ \mathds{1}_A : A\subset {\mathscr{B}}(\Omega) \text{ is invariant and } \mu(A)\leq \frac{1}{n} \right\}$$ and let $B_n\in {\mathscr{B}}(\Omega)$ such that $$\mathds{1}_{B_n} = \bigvee_{A\in {\mathscr{A}}_n} \mathds{1}_A \quad \mu\text{-almost everywhere,}$$ where the supremum is taken in the order complete lattice $L^\infty(\Omega,\mu)$. Then $B_n$, hence by the above also $\Omega\setminus B_n$, is invariant. Since we assumed every irreducible set to be a null-set and $\mu(\tilde B)>\frac{1}{n}$ for every measurable invariant subset $\tilde B \subset \Omega\setminus B_n$, we conclude that $\mu(\Omega\setminus B_n)=0$. Therefore, $\mathds{1}_{B_n} = \mathds{1}$ almost everywhere. Let ${\mathscr{D}}_n$ be a maximal disjoint system in ${\mathscr{A}}_n$. By the countable sup property of $L^\infty(\Omega,\mu)$, see [@aliprantis2006 Thm 8.22], we obtain the existence of a countable subset $(\mathds{1}_{A_{k,n}})_{k\in{\mathds{N}}} \subset {\mathscr{D}}_n$ with $\sup_{k\in{\mathds{N}}} \mathds{1}_{A_{k,n}} = \mathds{1}$. Since the functions $\{\mathds{1}_{A_{k,n}} : k\in{\mathds{N}}\}$ are pairwise disjoint, it follows that $\lim_{k\to\infty} \lVert \mathds{1}_{A_{k,n}} \rVert_{L^1}=0$ for every $n\in{\mathds{N}}$. By ordering the sets $A_{k,n}$ decreasing in measure, we obtain a single sequence $(A_n)\subset {\mathscr{B}}(\Omega)$ that contains every set $A_{k,n}$ and satisfies $\lim \lVert \mathds{1}_{A_n} \rVert_{L^1} = 0$. Now it follows from Theorem \[thm:bukhvalov\] that $\lim T(t_0)\mathds{1}_{A_n} = 0$ almost everywhere in contradiction to $$\sup_{n\geq k} T(t_0) \mathds{1}_{A_n} = \sup_{n\geq k} \mathds{1}_{A_n} = \mathds{1}$$ for all $k\in{\mathds{N}}$. Thus, there exists an irreducible set $B\in{\mathscr{B}}(\Omega)$ with $\mu(B)>0$. Moreover, the same argument shows that every invariant set $\tilde \Omega \in{\mathscr{B}}(\Omega)$ of positive measure contains an irreducible Borel set of positive measure. Let ${\mathscr{D}}{\mathrel{\mathop:}=}\{ D \in {\mathscr{B}}(\Omega) : D \text{ is irreducible} \}$. Then $\sup\{ \mathds{1}_D : D \in {\mathscr{D}}\} = \mathds{1}$. By the countable sup property, we find a sequence $(D_n)\subset {\mathscr{D}}$ with $\sup \mathds{1}_{D_n} = \mathds{1}$. This proves the claim. Let us note that, by applying a general version of Bukhvalov’s theorem proven in [@grobler1980], Theorem \[thm:irreddecomp\] can be generalized to contractive semigroups containing a kernel operator on an arbitrary Banach lattice whose norm is strictly monotone and order continuous. Combining Greiner’s Theorem \[thm:greiner\] and the irreducible decomposition of Theorem \[thm:irreddecomp\], we obtain stability of ${\mathscr{T}}$ on the band spanned by its fixed space. \[prop:convfixed\] If $T(t_0)$ is a kernel operator for some $t_0>0$, then $\lim_{t\to\infty} T(t) \mu$ exists for all $\mu\in {\mathrm{fix}}({\mathscr{T}})^{\bot\bot}$. Since every $T(t)$ is a contraction and the total variation norm is strictly monotone on the positive cone ${\mathscr{M}}(\Omega)_+$, for all $\mu\in {\mathrm{fix}}({\mathscr{T}})$ it follows from $$\lvert \mu\rvert = \lvert T(t)\mu\rvert \leq T(t) \lvert \mu\rvert$$ that $T(t)\lvert \mu\rvert = \lvert \mu\rvert$. Hence, ${\mathrm{fix}}({\mathscr{T}})$ is a sublattice. Now let $\mu\in {\mathrm{fix}}({\mathscr{T}})^{\bot\bot}_+$ and denote by $P$ the band projection onto ${\mathrm{fix}}({\mathscr{T}})^{\bot}$. Let ${\mathscr{D}}$ be a maximal disjoint system in ${\mathrm{fix}}({\mathscr{T}})_+$. Since the total variation norm on ${\mathscr{M}}(\Omega)$ is a $L$-norm, i.e. it is additive on the positive cone ${\mathscr{M}}(\Omega)_+$, there exists an at most countable subset $\mathscr{C} \subset {\mathscr{D}}$ such that $\mu \in {\mathscr{C}}^{\bot\bot}$. In fact, for $\zeta\in {\mathscr{D}}$ let $P_\zeta$ denote the band projection onto $\{\zeta\}^{\bot\bot}$. Then for every $m\in{\mathds{N}}$ there exist only finitely many $\zeta\in {\mathscr{D}}$ such that $\lVert P_\zeta \mu \rVert \geq \frac{1}{m}$. This implies that there are at most countably many $\zeta\in {\mathscr{D}}$ such that $P_\zeta \mu >0$. Let $\mathscr{C} {\mathrel{\mathop:}=}(\zeta_k) {\mathrel{\mathop:}=}\{ \zeta \in {\mathscr{D}}$ and $(\mu_k) {\mathrel{\mathop:}=}(P_\zeta \mu)_{\zeta \in {\mathscr{C}}}$. Since $\mu$ is a fixed point of ${\mathscr{T}}$, the band $\{\mu \}^{\bot\bot}$ is ${\mathscr{T}}$-invariant. By Theorem \[thm:irreddecomp\], we may assume that the restriction of ${\mathscr{T}}$ to $\{\mu\}^{\bot\bot}$ is irreducible. Moreover, since ${\mathscr{T}}$ is stochastically continuous, this restriction is strongly continuous by [@hille2009 Thm 4.6]. Thus, for each $k\in {\mathds{N}}$, the limit $\nu_k {\mathrel{\mathop:}=}\lim_{t\to\infty} T(t)\mu_k \in \{ \zeta_k \}^{\bot\bot}$ exists by Theorem \[thm:greiner\] applied to the Banach lattice $\{\zeta_k\}^{\bot\bot}$. Next, we show that $\tau_n {\mathrel{\mathop:}=}\nu_1 + \dots + \nu_n$ is a Cauchy sequence. For a given ${\varepsilon}>0$ choose $n\in{\mathds{N}}$ such that $\sum_{k=n+1}^\infty \lVert \mu_k \rVert < {\varepsilon}$. Then $$\lVert \tau_n - \tau_m \rVert = \sum_{k=n+1}^m \lVert \nu_k \rVert \leq \sum_{k=n+1}^\infty \lVert \mu_k \rVert < {\varepsilon}$$ for all $m >n$. Therefore, $\tau {\mathrel{\mathop:}=}\lim \tau_m \in {\mathrm{fix}}({\mathscr{T}})^{\bot\bot}$ exists. We prove that $\lim T(t)\mu = \tau$. Let ${\varepsilon}>0$ and choose $n\in{\mathds{N}}$ such that $\sum_{k=n+1}^\infty \lVert \mu_k\rVert < {\varepsilon}$. Since $T(t)\mu_k$ converges to $\nu_k$ we find $s>0$ such that $\lVert T(t)\mu_k - \nu_k\rVert < {\varepsilon}/n$ for all $t\geq s$ and all $1\leq k\leq n$. Finally, we obtain that $$\begin{aligned} \lVert T(t)\mu - \tau\rVert &\leq \sum_{k=1}^\infty \lVert T(t)\mu_k - \nu_k \rVert \\ &\leq \sum_{k=1}^n \lVert T(t)\mu_k - \nu_k\rVert + \sum_{k=n+1}^\infty 2 \lVert \mu_k\rVert < n\cdot \frac{{\varepsilon}}{n} + 2{\varepsilon}\end{aligned}$$ for all $t\geq s$. This shows that $\lim_{t\to\infty} T(t)\mu = \tau$. Let us remark that, using a generalized version of Theorem \[thm:irreddecomp\], Proposition \[prop:convfixed\] remains true for every positive and contractive semigroup ${\mathscr{T}}=(T(t))_{t\geq 0}$ on a Banach lattice with strictly monotone and order continuous norm such that the restriction of ${\mathscr{T}}$ to $\{x\}^{\bot\bot}$ is strongly continuous for every $x\in {\mathrm{fix}}({\mathscr{T}})^{\bot\bot}$. Now we prove our main result. \[thm:tauberian\] If ${\mathscr{T}}$ is $B_b$-ergodic and $T(t_0)$ is strong Feller for some $t_0>0$, then $\lim_{t\to\infty} T(t)\mu$ exists for all $\mu\in{\mathscr{M}}(\Omega)$. Let $\mu \in {\mathscr{M}}(\Omega)_+$ and denote by $P$ the band projection onto ${\mathrm{fix}}({\mathscr{T}})^{\bot}$. By Proposition \[prop:stabilityonfixbot\], there exists an increasing sequence $t_n >0$ such that $\lVert PT(t_n)\mu \rVert < \frac{1}{n}$ for all $n\in{\mathds{N}}$. Define $$\mu_n {\mathrel{\mathop:}=}(I-P)T(t_n)\mu \in {\mathrm{fix}}({\mathscr{T}})^{\bot \bot}.$$ It follows from [@revuz1975 §1.5] that $T(2t_0)$ is ultra Feller and therefore a kernel operator by Theorem \[thm:ultrafellerkernel\]. Therefore, by Proposition \[prop:convfixed\], $\nu_n {\mathrel{\mathop:}=}\lim_{t\to\infty} T(t)\mu_n$ exists in ${\mathrm{fix}}(\mathscr{S})^{\bot\bot}$ for every $n\in{\mathds{N}}$. Hence, there exists an increasing sequence $s_n > 0$ such that $\lVert T(t)\mu_n -\nu_n\rVert < \frac{1}{n}$ for all $n\in{\mathds{N}}$ and $t\geq s_n$. This implies that for every $n\in{\mathds{N}}$ $$\begin{aligned} \lVert T(t+t_n)\mu - \nu_n \rVert &\leq \lVert T(t) (I-P) T(t_n)\mu - \nu_n \rVert + \lVert T(t) P T(t_n) \mu\rVert\\ &\leq \lVert T(t)\mu_n -\nu_n \rVert + \lVert PT(t_n)\mu \rVert < \frac{2}{n} \end{aligned}$$ for all $t \geq s_n$. Since $$\begin{aligned} \lVert \nu_n - \nu_m \rVert &\leq \lVert \nu_n - T(s_m+t_m)\mu\rVert + \lVert T(s_m +t_m)\mu -\nu_m\rVert <\frac{2}{n} + \frac{2}{m} \end{aligned}$$ for all $m\geq n$, $(\nu_n)$ is a Cauchy sequence. Let $\nu{\mathrel{\mathop:}=}\lim \nu_n \in {\mathrm{fix}}(\mathscr{S})^{\bot\bot}$. Then for every ${\varepsilon}>0$ there exists $n\in{\mathds{N}}$ such that $$\begin{aligned} \lVert T(t)\mu - \nu\rVert &\leq \lVert T(t)\mu - \nu_n \rVert + \lVert \nu_n -\nu \rVert < {\varepsilon}\end{aligned}$$ for all $t\geq t_n+s_n$ which proves the claim. Making use of the characterization of weak ergodicity in [@gerlach2013 Thm 5.7], we obtain the following Corollary. If $T(t_0)$ is strong Feller for some $t_0>0$, then the following are equivalent (i) ${\mathrm{fix}}({\mathscr{T}})$ separates ${\mathrm{fix}}({\mathscr{T}}') {\mathrel{\mathop:}=}\{ f\in C_b(\Omega) : T'(t)f=f \text{ for all } t\geq 0\}$. (ii) The semigroup ${\mathscr{T}}$ is weakly ergodic in the sense that $\lim_{t\to\infty} A_t \mu$ exists in the $\sigma({\mathscr{M}}(\Omega),C_b(\Omega))$-topology for all $\mu\in{\mathscr{M}}(\Omega)$. (iii) The semigroup ${\mathscr{T}}$ is $B_b$-ergodic. (iv) $\lim_{t\to\infty} T(t)\mu$ exists for each $\mu\in{\mathscr{M}}(\Omega)$. Let us assume (i) and pick $\mu \in {\mathscr{M}}(\Omega)$. It follows from [@gerlach2013 Thm 5.7] that there exists $\tilde \mu \in {\mathrm{fix}}({\mathscr{T}})$ such that $$\lim {\langle A_t \mu - \tilde \mu,f\rangle} = 0$$ for all $f\in C_b(\Omega)$, i.e. assertion (ii) holds. As explained in [@gerlach2013 Ex 3.6], one has that $$\lim_{t\to\infty} \lVert (T(t_0) - I)A_t \mu \rVert =0.$$ Since $T(t_0)$ is strong Feller, assertion (ii) implies that $$\lim_{t\to\infty} {\langle A_t \mu-\tilde \mu,f\rangle} = \lim_{t\to\infty} {\langle A_t \mu - T(t_0)A_t \mu,f\rangle} + {\langle A_t \mu - \tilde \mu,T'(t_0)f\rangle} =0$$ for all $f\in B_b(\Omega)$, i.e.  ${\mathscr{T}}$ is $B_b$-ergodic. Theorem \[thm:tauberian\] yields that (iii) implies (iv). In order to prove that (i) follows from (iv), we assume that $\lim T(t)\mu$ exists for each $\mu\in {\mathscr{M}}(\Omega)$. For $f\in {\mathrm{fix}}({\mathscr{T}}')$ choose $\mu\in {\mathscr{M}}(\Omega)$ such that ${\langle \mu,f\rangle} {=\mathrel{\mathop:}}\alpha \neq 0$. Let $\tilde\mu {\mathrel{\mathop:}=}\lim T(t)\mu \in {\mathrm{fix}}({\mathscr{T}})$. Then $${\langle \tilde \mu,f\rangle} = \lim_{t\to\infty} {\langle T(t)\mu,f\rangle} = \alpha \neq 0$$ which shows that ${\mathrm{fix}}({\mathscr{T}})$ separates ${\mathrm{fix}}({\mathscr{T}}')$.

--- abstract: 'We introduce a data-driven method and shows its skills for spatiotemporal prediction of high-dimensional chaotic dynamics and turbulence. The method is based on a finite-dimensional approximation of the Koopman operator where the observables are vector-valued and delay-embedded, and the nonlinearities are treated as external forcings. The predictive capabilities of the method are demonstrated for well-known prototypes of chaos such as the Kuramoto-Sivashinsky equation and Lorenz-96 system, for which the data-driven predictions are accurate for several Lyapunov timescales. Similar performance is seen for two-dimensional lid-driven cavity flows at high Reynolds numbers.' author: - 'M. A. Khodkar$^1$' - 'Pedram Hassanzadeh$^{1,2}$' - 'Athanasios Antoulas$^{3, 4, 5}$' bibliography: - 'Main.bib' title: 'A Koopman-based framework for forecasting the spatiotemporal evolution of chaotic dynamics with nonlinearities modeled as exogenous forcings' --- [^1] [^2] [^1]: mkhodkar@rice.edu [^2]: pedram@rice.edu

--- abstract: 'We consider a model in which a trader aims to maximize expected risk-adjusted profit while trading a single security. In our model, each price change is a linear combination of observed factors, impact resulting from the trader’s current and prior activity, and unpredictable random effects. The trader must learn coefficients of a price impact model while trading. We propose a new method for simultaneous execution and learning – the confidence-triggered regularized adaptive certainty equivalent (CTRACE) policy – and establish a poly-logarithmic finite-time expected regret bound. This bound implies that CTRACE is [*efficient*]{} in the sense that the $(\epsilon,\delta)$-convergence time is bounded by a polynomial function of $1/\epsilon$ and $\log(1/\delta)$ with high probability. In addition, we demonstrate via Monte Carlo simulation that CTRACE outperforms the certainty equivalent policy and a recently proposed reinforcement learning algorithm that is designed to explore efficiently in linear-quadratic control problems.' author: - | Beomsoo Park\ Electrical Engineering\ Stanford University\ [<beomsoo@stanford.edu>]{}\ - | Benjamin Van Roy\ Management Science & Engineering\ Electrical Engineering\ Stanford University\ [<bvr@stanford.edu>]{}\ bibliography: - 'adaptive\_execution.bib' title: 'Adaptive Execution: Exploration and Learning of Price Impact' --- [*Key words*]{}: adaptive execution, price impact, reinforcement learning, regret bound Introduction ============ A large block trade tends to “move the market” considerably during its execution by either disturbing the balance between supply and demand or adjusting other market participants’ valuations. Such a trade is typically executed through a sequence of orders, each of which pushes price in an adverse direction. This effect is called [*price impact*]{}. Because it is responsible for a large fraction of transaction costs, it is important to design execution strategies that effectively manage price impact. In light of this, academics and practitioners have devoted significant attention to the topic \[[@BertsimasLo98; @AlmgrenChriss00; @KissellGlantz03; @ObizhaevaWang05; @MoallemiParkVanRoy08; @AlfonsiSchiedSchulz07b]\]. The learning of a price impact model poses a challenging problem. Price impact represents an aggregation of numerous market participants’ interpretations of and reactions to executed trades. As such, learning requires “excitation” of the market, which can be induced by regular trading activity or trades deliberately designed to facilitate learning. The trader must balance the short term costs of accelerated learning against the long term benefits of an accurate model. Further, given the continual evolution of trading venues and population of market participants, price impact models require retuning over time. In this paper, we develop an algorithm that learns a price impact model while guiding trading decisions using the model being learned. Our problem can be viewed as a special case of reinforcement learning. This topic more broadly addresses sequential decision problems in which unknown properties of an environment must be learned in the course of operation (see, e.g., [@SuttonBarto98]). Research in this area has established how judicious investments in decisions that explore the environment at the expense of suboptimal short-term behavior can greatly improve longer-term performance. What we develop in this paper can be viewed as a reinforcement learning algorithm; the workings of price impact are unknown, and exploration facilitates learning. In reinforcement learning, one seeks to optimize the balance between exploration and exploitation – the use of what has already been learned to maximize rewards without regard to further learning. Certainty equivalent control (CE) represents one extreme where at any time, current point estimates are assumed to be correct and actions are made accordingly. This is an instance of pure exploitation; though learning does progress with observations made as the system evolves, decisions are not deliberately oriented to enhance learning. An important question is how aggressively a trader should explore to learn a price impact model. Unlike many other reinforcement learning problems, in ours a considerable degree of exploration is naturally induced by exploitative decisions. This is because a trader excites the market through regular trading activity regardless of whether or not she aims to learn a price impact model. This activity could, for example, be triggered by return-predictive factors, and given sufficiently large factor variability, the induced exploration might adequately resolve uncertainties about price impact. Results of this paper demonstrate that executing trades to explore beyond what would naturally occur through exploitation can yield significant benefit. Our work is constructive: we propose the [*confidence-triggered regularized adaptive certainty equivant*]{} policy (CTRACE), pronounced “see-trace,” a new method that explores and learns a price impact model alongside trading. CTRACE can be viewed as a generalization of CE, which at each point in time estimates coefficients of a price impact model via least-squares regression using available data and makes decisions that optimize trading under an assumption that the estimated model is correct and will be used to guide all future decisions. CTRACE deviates in two ways: (1) $\ell_2$ regularization is applied in least-squares regression and (2) coefficients are only updated when a certain measure of confidence exceeds a pre-specified threshold and a minimum inter-update time has elapsed. Note that CTRACE reduces to CE as the regularization penalty, the threshold, and the minimum inter-update time vanish. We demonstrate through Monte Carlo simulation that CTRACE outperforms CE. Further, we establish a finite-time regret bound for CTRACE; no such bound is available for CE. [*Regret*]{} is defined here to be the difference between realized risk-adjusted profit of a policy in question and one that is optimal with respect to the true price impact model. Our bound exhibits a poly-logarithmic dependence on time. Among other things, this regret bound implies that CTRACE is [*efficient*]{} in the sense that the $(\epsilon,\delta)$-convergence time is bounded by a polynomial function of $1/\epsilon$ and $\log(1/\delta)$ with high probability. We define the $(\epsilon,\delta)$-convergence time to be the first time when an estimate and all the future estimates following it are within an $\epsilon$-neighborhood of a true value with probability at least $1-\delta$. Let us provide here some intuition for why CTRACE outperforms CE. First, regularization enhances exploration in a critical manner. Without regularization, we are more likely to obtain overestimates of price impact. Such an outcome abates trading and thus exploration, making it difficult to escape from the predicament. Regularization reduces the chances of obtaining overestimates, and further, tends to yield underestimates that encourage active exploration. Second, requiring a high degree of confidence reduces the chances of occasionally producing erratic estimates, which regularly arise with application of CE. Such estimates can result in undesirable trades and/or reductions in the degree of exploration. It is also worth comparing CTRACE to a reinforcement learning algorithm recently proposed in [@Abbasi-YadkoriSzepesvari10] which appears well-suited for our problem. This algorithm was designed to explore efficiently in a broader class of linear-quadratic control problems, and is based on the [*principle of optimism in the face of uncertainty*]{}. [@Abbasi-YadkoriSzepesvari10] establish an $O ( \sqrt{T \log(1/\delta)} )$ regret bound that holds with probability at least $1-\delta$, where $T$ denotes time and some logarithmic terms are hidden. Our bound for CTRACE is on expected regret and exhibits a dependence on $T$ of $O(\log^2 T)$. We also demonstrate via Monte Carlo simulation that CTRACE dramatically outperforms this algorithm. To summarize, the primary contributions of this paper include: (a) We propose a new method for simultaneous execution and learning – the confidence-triggered regularized adaptive certainty equivalent (CTRACE) policy. (b) We establish a finite-time expected regret bound for CTRACE that exhibits a poly-logarithmic dependence on time. This bound implies that CTRACE is [*efficient*]{} in the sense that, with probability $1-\delta$, the $(\epsilon,\delta)$-convergence time is bounded by a polynomial function of $1/\epsilon$ and $\log(1/\delta)$. (c) We demonstrate via Monte Carlo simulation that CTRACE outperforms the certainty equivalent policy and a reinforcement learning algorithm recently proposed by [@Abbasi-YadkoriSzepesvari10] which is designed to explore efficiently in linear-quadratic control problems. The organization of the rest of this paper is as follows: Section \[sec:problemFormulation\] presents our problem formulation, establishes existence and uniqueness of an optimal solution to our problem, and defines performance measures that can be used to evaluate policies. In Section \[sec:CTRACE\], we propose CTRACE and derive a finite-time expected regret bound for CTRACE along with two properties: inter-temporal consistency and efficiency. Section \[sec:computationalAnalysis\] is devoted to Monte Carlo simulation in which the performance of CTRACE is compared to that of two benchmark policies. Finally, we conclude this paper in Section \[sec:conclusion\]. All proofs are provided in Appendix. Detailed proofs are available upon request. Problem Formulation {#sec:problemFormulation} =================== Model Description ----------------- **** We consider a trader who trades a single security over an infinite time horizon. She submits a market buy or sell order at the beginning of each period of equal length. $u_t \in \mathbb{R}$ represents the number of shares of the security to buy or sell at period $t$ and a positive (negative) value of $u_t$ denotes a buy (sell) order. Let $x_{t-1} \in \mathbb{R}$ denote the trader’s pre-trade security position before placing an order $u_t$ at period $t$. Therefore, $x_t = x_{t-1} + u_t, \,\, t \geq 1$. **** The absolute return of the security is given by $$\begin{aligned} \Delta p_{t} &= p_{t} - p_{t-1} = g^\top f_{t-1} + \lambda^* u_{t} + \sum_{m=1}^{M} \gamma_m^* (d_{m,t} - d_{m,t-1}) + \epsilon_{t} \nonumber \\ d_{m,t} &{\ensuremath{\triangleq}}\sum_{i=1}^{t} r_m^{t-i} u_i = r_m d_{m,t-1} + u_t, \quad d_t {\ensuremath{\triangleq}}[ d_{1,t} \,\,\, \cdots \,\,\, d_{M,t} ]^\top. \label{eqn:transientImpact}\end{aligned}$$ We will explain each term in detail as we progress. This can be viewed as a first-order Taylor expansion of a geometric model $$\log \left( \frac{ p_{t} }{ p_{t-1} } \right) = \tilde{g}^\top f_{t-1} + \tilde{\lambda}^* u_{t} + \sum_{m=1}^{M} \tilde{\gamma}_m^* (d_{m,t} - d_{m,t-1}) + \tilde{\epsilon}_{t}$$ over a certain period of time, say, a few weeks in calendar time, which makes this approximation reasonably accurate for practical purposes. Although it is unrealistic that the security price can be negative with positive probability, our model nevertheless serves its practical purpose for the following reasons: Our numerical experiments conducted in Section \[sec:computationalAnalysis\] show that price changes after a few weeks from now have ignorable impacts on a current optimal action. In other words, optimal actions for our infinite-horizon control problem appear to be quite close to those for a finite-horizon counterpart on a few week time scale. Furthermore, it turns out that in simulation we could learn a unknown price impact model fast enough to take actions that are close to optimal actions within a few weeks. Thus, learning based on our price dynamics model could also be justified. We will give concrete numerical examples later to support these notions. **** The term $\lambda^* u_t$ represents “permanent price impact” on the security price of a current trade. The permanent price impact is endogenously derived in [@Kyle85] from informational asymmetry between an informed trader and uninformed competitive market makers, and in [@Rosu09] from equilibrium of a limit order market where fully strategic liquidity traders dynamically choose limit and market orders. [@HubermanStanzl04] prove that the linearity of a time-independent permanent price impact function is a necessary and sufficient condition for the absence of “price manipulation” and “quasi-aribtrage” under some regularity conditions. The term $\sum_{m=1}^{M} \gamma_m^* d_{m,t}$ indicates “transient price impact” that models other traders’ responses to non-informative orders. For example, suppose that a large market buy order has arrived and other traders monitoring the market somehow realize that there is no definitive evidence for abrupt change in the fundamental value of the security. Then, they naturally infer that the large buy order came merely for some liquidity reason, and gradually “correct” the perturbed price into what they believe it is supposed to be by submitting counteracting selling orders. The dynamics of $d_{m,t}$ in (\[eqn:transientImpact\]) indicates that the impact of a current trade on the security price decays exponentially over time, which is considered in [@ObizhaevaWang05] that incorporate the dynamics of supply and demand in a limit order market to optimal execution strategies. In [@Gatheral10], it is shown that the exponentially decaying transient price impact is compatible only with a linear instantaneous price impact function in the absence of “dynamic arbitrage.” **** We assume that there are multiple observable return-predictive factors that affect the absolute return of the security as in [@GarleanuPedersen09]. Those factors could be macroeconomic factors such as gross domestic products (GDP), inflation rates and unemployment rates, security-specific factors such as P/B ratio, P/E ratio and lagged returns, or prices of other securities that are correlated with the security price. In our price dynamics model, $f_t \in \mathbb{R}^{K}$ denotes these factors and $g \in \mathbb{R}^{K}$ denotes factor loadings. The term $g^\top f_{t-1}$ represents predictable excess return or “alpha.” We assume that $f_t$ is a first-order vector autoregressive process $f_{t} = \Phi f_{t-1} + \omega_{t}$ where $\Phi \in {\ensuremath{{\ensuremath{\mathbb{R}}}}}^{K \times K}$ is a stable matrix that has all eigenvalues inside a unit disk and $\omega_t \in {\ensuremath{{\ensuremath{\mathbb{R}}}}}^{K}$ is a martingale difference sequence adapted to the filtration $\{ \mathcal{F}_t = \sigma(\{ x_0, d_0, f_0, \omega_1, \ldots, \omega_t, \epsilon_1, \ldots, \epsilon_t \}) \}$. We further assume that $\omega_t$ is bounded almost surely, $\text{i.e.}$ $\| \omega_t \| \leq C_\omega \,\, a.s. $ for all $t \geq 1$ for some deterministic constant $C_\omega$, and $\text{Cov}[\omega_t | \mathcal{F}_{t-1}] = \Omega \in {\ensuremath{{\ensuremath{\mathbb{R}}}}}^{K \times K}$ being positive definite and independent of $t$. **** The term $\epsilon_t$ represents random fluctuations that cannot be accounted for by price impact and observable return-predictive factors. We assume that $\epsilon_t$ is a martingale difference sequence adapted to the filtration $\{ \mathcal{F}_t \}$, and independent of $x_0$, $d_0$, $f_0$ and $\omega_\tau$ for any $\tau \geq 1$. Also, ${\ensuremath{\mathsf{E}}}[ \epsilon_t^2 | \mathcal{F}_{t-1}] = \Sigma_\epsilon \in {\ensuremath{{\ensuremath{\mathbb{R}}}}}$ being independent of $t$. Finally, each $\epsilon_t$ is assumed to be sub-Gaussian, $\text{i.e.}$, ${\ensuremath{\mathsf{E}}}[ \textrm{exp}(a \epsilon_{t}) | \mathcal{F}_{t-1}] \leq \textrm{exp}(C_\epsilon^2 a^2 /2), \,\, \forall t \geq 1, \,\, \forall a \in \mathbb{R}$ for some $C_\epsilon > 0$. **** A policy is defined as a sequence $\pi = \{ \pi_1, \pi_2, \ldots \}$ of functions where $\pi_t$ maps the trader’s information set at the beginning of period $t$ into an action $u_t$. The trader observes $f_{t-1}$ and $p_{t-1}$ at the end of period $t-1$ and thus her information set at the beginning of period $t$ is given by $\mathcal{I}_{t-1} = \{ x_0, d_0, f_0, \ldots, f_{t-1}, p_0, \ldots, p_{t-1} \}$. A policy $\pi$ is [*admissible*]{} if $z_t {\ensuremath{\triangleq}}[x_t \,\,\, d_{t}^\top \,\,\, f_t^\top]^\top$ generated by $u_t = \pi_t(\mathcal{I}_{t-1})$ satisfies $\lim_{T \rightarrow \infty} \| z_T \|^2/T = 0$. A set of admissible policies is denoted by $\Pi$. **** The trader’s objective is to maximize expected average “risk-adjusted” profit defined as $$\liminf_{T \rightarrow \infty} \,\, {\ensuremath{\mathsf{E}}}\left[ \frac{1}{T} \sum_{t=1}^{T} \left( \Delta p_{t} x_{t-1} - \rho \Sigma_\epsilon x_t^2 \right) \right]$$ where the first term $\Delta p_{t} x_{t-1}$ indicates change in book value and the second term $\rho \Sigma_\epsilon x_{t}^2$ a quadratic penalty for her non-zero security position in the next period that reflects her risk aversion. $\rho$ is a risk-aversion coefficient that quantifies the extent to which the trader is risk-averse. **** The following is a list of assumptions on which our analysis is based throughout this paper. Let $\theta^{*} {\ensuremath{\triangleq}}\left[ \lambda^{*} \,\, \gamma^{*}_{1} \,\, \ldots \,\, \gamma^{*}_{M} \right]^\top \in \mathbb{R}^{M+1}$. We will make two more assumptions as we progress. \[assum:one\] (a) The price impact coefficients $\theta^{*}$ are unknown to the trader. Note that they can be learned only through executed trades. (b) The factor loadings $g$ are known to the trader. This is a reasonable assumption since they can be learned by observing prices without any transaction. (c) The decaying rates $r {\ensuremath{\triangleq}}[r_1, \,\, \ldots, \,\, r_M]^\top \in [0, 1)^M$ of the transient price impact are known to the trader and all the elements are distinct. In practice, they are definitely not known a priori. However, it can be handled effectively for practical purposes by using a sufficiently dense $r$ with a large $M$ so that potential bias induced by modeling mismatch can be greatly reduced at the expense of increased variance, which can be reduced by regularization. (d) $\theta^{*} \in \Theta {\ensuremath{\triangleq}}\{ \theta \in {\ensuremath{{\ensuremath{\mathbb{R}}}}}^{M+1} : 0 \leq \theta \leq \theta_\text{max}, \,\, {\ensuremath{\mathbf{1}}}^\top \theta \geq \beta \}$ for some $\theta_\text{max} > 0$ component-wise and some $\beta > 0$. The constraint ${\ensuremath{\mathbf{1}}}^\top \theta \geq \beta$ is imposed to capture non-zero execution costs in practice. Note that $\Theta$ is compact and convex. **** $\| \cdot \|$ and $\| \cdot \|_F$ denote the $\ell_2$-norm and the Frobenius norm of a matrix, respectively. $a \vee b$ and $a \wedge b$ denote $\max\{ a, b\}$ and $\min\{ a, b\}$, respectively. For a symmetric matrix $A$, $A \succ 0$ means that $A$ is positive definite and $A \succeq 0$ means that $A$ is positive semidefinite. $\lambda_{\text{min}}(A)$ indicates the smallest eigenvalue of $A$. $(A)_{ij}$ of a matrix $A$ indicates the entry of $A$ in the $i$th row and in the $j$th column. $(v)_i$ of a vector $v$ indicates the $i$th entry of $v$. $\text{diag}(v)$ of a vector $v$ denotes a diagonal matrix whose $i$th diagonal entry is $(v)_i$. $A_{*,j}$ denotes the $j$th column of $A$ and $A_{i:j,k}$ indicates a segment of the $k$th column of $A$ from the $i$th entry to the $j$th entry. ${\ensuremath{\mathbf{1}}}\{ \mathcal{B} \}$ denotes an indicator function on the event $\mathcal{B}$. Existence of Optimal Solution {#sec:existenceOptSoln} ----------------------------- Now, we will show that there exists an optimal policy among admissible policies that maximizes expected average risk-adjusted profit. For convenience, we will consider the following minimization problem that is equivalent to maximize expected average risk-adjusted profit. $$\begin{aligned} \min_{ \pi \in \Pi} \,\, \limsup_{T \rightarrow \infty} \,\, {\ensuremath{\mathsf{E}}}\left[ \frac{1}{T} \sum_{t=1}^{T} \left(\rho \Sigma_\epsilon x_{t}^2 - \Delta p_{t} x_{t-1} \right) \right]\end{aligned}$$ We call the negative of average risk-adjusted profit “average cost.” This problem can be expressed as a discrete-time linear quadratic control problem $$\min_{ \pi \in \Pi } \,\, \limsup_{T \rightarrow \infty} \,\, {\ensuremath{\mathsf{E}}}\left[ \frac{1}{T} \sum_{t=1}^{T} \left[ \begin{array}{cc} z_{t-1}^\top & u_t \end{array} \right] \left[ \begin{array}{cc} Q & S \\ S^\top & R \end{array} \right] \left[ \begin{array}{c} z_{t-1} \\ u_t \end{array} \right] \right] \,\, \text{s.t.} \,\, z_t = A z_{t-1} + B u_t + W_t, \,\, u_t = \pi_t (\mathcal{I}_{t-1})$$ where $z_t = [ x_t \,\,\, d_t^\top \,\,\, f_t^\top ]^\top$, $v = [ 0 \,\,\, \gamma^{*\top} (\text{diag}(r) - I) \,\,\, g^\top ]^\top$, $\gamma^{*} = [ \gamma_1^{*} \,\,\, \cdots \,\,\, \gamma_M^{*} ]^\top$, $e_1 = [ 1 \,\,\, 0 \,\,\, \cdots \,\,\, 0 ]^\top$, $$Q = \rho \Sigma_\epsilon e_1 e_1^\top - \frac{1}{2} (v e_1^\top + e_1 v^\top), \quad S = \rho \Sigma_\epsilon e_1 - \frac{1}{2} (\lambda^* + \gamma^{*\top} {\ensuremath{\mathbf{1}}}) e_1, \quad R = \rho \Sigma_\epsilon,$$ $$A = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \text{diag}(r) & 0 \\ 0 & 0 & \Phi \end{array} \right], \quad B = \left[ \begin{array}{c} 1 \\ {\ensuremath{\mathbf{1}}}\\ 0 \end{array} \right], \quad W_t = \left[ \begin{array}{c} 0 \\ 0 \\ \omega_{t} \end{array} \right], \quad \tilde{\Omega} {\ensuremath{\triangleq}}\text{Cov}[W_t] = \left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \Omega \end{array} \right].$$ Note that $R$ is strictly positive but $Q$ is not necessarily positive semidefinite. Therefore, special care should be taken in order to prove the existence of an optimal policy. We start with a well-known Bellman equation for average-cost linear quadratic control problems $$\label{eqn:BellmanEqn} H(z_{t-1}) + h = \min_{u_t}\,\, {\ensuremath{\mathsf{E}}}\left[ \rho \Sigma_\epsilon (x_{t-1} + u_{t})^2 - \Delta p_{t} x_{t-1} + H(z_{t}) \right]$$ where $H(\cdot)$ denotes a differential value function and $h$ denotes minimum average cost. It is natural to conjecture $H(z_{t}) = z_{t}^\top P z_{t}$. Plugging it into (\[eqn:BellmanEqn\]), we can obtain a discrete-time Riccati algebraic equation $$\label{eqn:dare} P = A^\top P A + Q - (S^\top + B^\top P A)^\top (R + B^\top P B)^{-1} (S^\top + B^\top P A)$$ with a second-order optimality condition $R + B^\top P B > 0$. The following theorem characterizes an optimal policy among admissible policies that minimizes expected average cost, and proves existence and uniqueness of such an optimal policy. \[thm:optPolicyExistenceUniqueness\] For any $\theta^* \in \Theta$, there exists a unique symmetric solution $P$ to (\[eqn:dare\]) that satisfies $R + B^\top P B > 0$ and $\rho_{\text{sr}} (A + B L) < 1$ where $L = -(R + B^\top P B)^{-1} (S^\top + B^\top P A)$ and $\rho_{\text{sr}} (\cdot)$ denotes a spectral radius. Moreover, a policy $\pi = (\pi_1, \pi_2, \ldots )$ with $\pi_t(\mathcal{I}_{t-1}) = L z_{t-1}$ is an optimal policy among admissible policies that attains minimum expected average cost $\text{tr}(P \tilde{\Omega})$. For ease of exposition, we define some notations: $P(\theta)$ denotes a unique symmetric stabilizing solution to (\[eqn:dare\]) with $\theta^* = \theta$. $L(\theta) {\ensuremath{\triangleq}}-(R + B^\top P(\theta) B)^{-1} (S(\theta)^\top + B^\top P(\theta) A)$ denotes a gain matrix for an optimal policy with $\theta^* = \theta$, $G(\theta) {\ensuremath{\triangleq}}A + B L(\theta)$ denotes a closed-loop system matrix with $\theta^* = \theta$, and $U(\theta) {\ensuremath{\triangleq}}{\ensuremath{\mathbf{1}}}L(\theta) + \left[A-I \,\,\, O \right]$ denotes a linear mapping from $z_{t-1}$ to a regressor $\psi_t$ used in least-squares regression for learning price impact, $\text{i.e.}$ $\psi_t = U(\theta) z_{t-1}$. Having these notations, we make two assumptions about $L(\theta)$ as follows. Indeed, we can verify through closed-form solutions that these assumptions hold in a special case which will be discussed in Subsection \[subsec:closedForm\]. \[assum:optimalPolicy\] (a) There exists $C_L > 0$ such that $\| L(\theta_1) - L(\theta_2) \| \leq C_L \| \theta_1 - \theta_2 \| $ for any $\theta_1, \theta_2 \in \Theta$. (b) $(L(\theta))_1 \neq 0$ and $(L(\theta))_{M+2} \neq 0$ for any $\theta \in \Theta$ Using Assumption \[assum:optimalPolicy\], we can obatin an upper bound on $\| z_t \|$ uniformly over $\theta \in \Theta$ and $t \geq 0$. \[lem:uniformBound\] For any $0 < \xi < 1$, there exists $N \in \mathbb{N}$ being independent of $\theta$ such that $\| G^N(\theta) \| \leq \xi$ for all $\theta \in \Theta$. Thus, $\max_{0 \leq i \leq N-1} \sup_{\theta \in \Theta} \| G^i(\theta) \| {\ensuremath{\triangleq}}C_g$ is finite. For any fixed $\theta \in \Theta$, $\| z_t \| \leq C_g \| z_0 \| + C_g C_\omega / (\xi (1 - \xi^{1/N})) {\ensuremath{\triangleq}}C_z, \,\, \forall t \geq 0 \,\, a.s.$ where $z_t = G(\theta) z_{t-1} + W_t$. Moreover, $\sup_{\theta \in \Theta} \| U(\theta) \| \leq C_g + 1$. Note that Lemma \[lem:uniformBound\] can be applied only when $\theta$ is fixed over time. From now on, we assume $\| z_0 \| \leq 2 C_g C_\omega / (\xi (1-\xi^{1/N}))$ without loss of generality otherwise we can always set $C_g$ to be greater than $\| z_0 \| \xi (1-\xi^{1/N}) / (2 C_\omega)$. Finally, we present concrete numerical examples that support the validity of our price model as an approximation of the geometric model for practical purposes. As we discussed earlier, our numerical experiments conducted in Section \[sec:computationalAnalysis\] show that our infinite-horizon control problem could be approximated accurately by a finite-time control problem with a time horizon on a few week time scale. To be more precise, we define [*relative error for $P_0^{(T)}$*]{} as $\| P_{0}^{(T)} - P \| / \| P \|$ where $P_t^{(T)}$ denotes a coefficient matrix of a quadratic value function at period $t$ for a finite-horizon control problem with a terminal period $T$, and $P$ denotes a coefficient matrix of a quadratic value function for our infinite-horizon control problem. As shown in Figure \[fig:mixingLearningTime\], the relative error for $P_0^{(T)}$ appears to decrease exponentially in $T$ and the relative error for $P_0^{(300)}$ is almost $10^{-7}$ where $T = 300$ corresponds to 3.8 trading days. ![(Left) Relative error for $P_T$: $T = 300$ corresponds to 3.8 trading days. (Right) Relative error for $L(\theta_t)$ from CTRACE: Period 3000 corresponds to 38 trading days. The verical bars represent two standard errors. In both figures, the simulation setting in Section \[sec:computationalAnalysis\] is used. []{data-label="fig:mixingLearningTime"}](mixing_learning_time.eps) Furthermore, we could learn unknown $\theta^*$ fast enough to take actions that are close to optimal actions on a required time scale. An action from a current estimate could be quite close to an optimal action even if estimation error for the current estimate is large, especially in cases where a few “principal components” of $L(\theta)$ with large directional derivatives with respect to $\theta$ are learned accurately. To be more precise, we define [*relative error for $L(\theta_t)$*]{} as $$\frac{\text{E}[(L(\theta_t) z_{t-1}^* - L(\theta^*) z_{t-1}^* )^2]}{\text{E}[(L(\theta^*) z_{t-1}^*)^2]} = \frac{(L(\theta_t) - L(\theta^*)) \Pi_{zz}(\theta^*) (L(\theta_t) - L(\theta^*))^\top}{L(\theta^*) \Pi_{zz}(\theta^*) L(\theta^*)^\top}$$ where $z_t^*$ is a stationary process generated by $u_t^* = L(\theta^*) z_{t-1}^*$ and $\Pi_{zz}(\theta^*) = E[z_t^* z_t^{* \top}]$. The relative error for $L(\theta_t)$ indicates how different an action from an estimate $\theta_t$ is than an optimal action from the true value $\theta^*$. Figure \[fig:mixingLearningTime\] shows how the relative error for $L(\theta_t)$ evolves over time with two-standard-error bars when $\theta_t$’s are obtained from a new policy that we will propose in Section \[sec:CTRACE\]. As you can see, all the approximate 95%-confidence intervals lie within $\pm 3\%$ range after Period 2500 that corresponds to 32 trading days. It implies that actions from estimates learned over a few weeks could be sufficiently close to optimal actions. Closed-Form Solution: A Single Factor and Permanent Impact Only {#subsec:closedForm} --------------------------------------------------------------- When we consider only the permanent price impact and a single observable factor, we can derive an exact closed-form $P$ and $L$ as follows. $$P_{xx} = \frac{\lambda^* - \rho \Sigma_\epsilon + \sqrt{ 2 \lambda^* \rho \Sigma_\epsilon + (\rho \Sigma_\epsilon)^2}}{2}$$ $$P_{xf} = \frac{-g \lambda^*}{(1-\Phi) \lambda^* - \Phi \rho \Sigma_\epsilon + \Phi \sqrt{2 \lambda^* \rho \Sigma_\epsilon + (\rho \Sigma_\epsilon)^2} }$$ $$P_{ff} = \frac{- g^2 \Phi^2}{2 (1-\Phi^2) \left( (1-\Phi)^2 \lambda^* + (1+\Phi^2) \rho \Sigma_\epsilon + (1-\Phi^2) \sqrt{2 \lambda^* \rho \Sigma_\epsilon + (\rho \Sigma_\epsilon)^2} \right)}$$ $$L_x = \frac{-2 \rho \Sigma_\epsilon}{\rho \Sigma_\epsilon + \sqrt{2 \lambda^* \rho \Sigma_\epsilon + (\rho \Sigma_\epsilon)^2}}$$ $$L_f = \frac{g \Phi }{(1-\Phi) \lambda^* + \rho \Sigma_\epsilon + \sqrt{2 \lambda^* \rho \Sigma_\epsilon + (\rho \Sigma_\epsilon)^2}}$$ Although this is a special case of our general setting, we can get useful insights into the effect of permanent price impact coefficient $\lambda^*$ on various quantities. Here are some examples: - $| L_x |$ and $| L_f |$ are strictly decreasing in $\lambda^*$. - $\lim_{\lambda^* \rightarrow 0} L_x = -1$, $\lim_{\lambda^* \rightarrow \infty} L_x = 0$. - $\lim_{\lambda^* \rightarrow 0} L_f = g \Phi / (2 \rho \Sigma_\epsilon)$, $\lim_{\lambda^* \rightarrow \infty} L_f = 0$. - The expected average risk-adjusted profit $-P_{ff} \Omega$ is strictly decreasing in $\lambda^*$. - $\lim_{\lambda^* \rightarrow 0} (-P_{ff} \Omega) = g^2 \Phi^2 \Omega / ( 4(1-\Phi^2) \rho \Sigma_\epsilon)$, $\lim_{\lambda^* \rightarrow \infty} (-P_{ff} \Omega) = 0$. Performance Measure: Regret {#subsec:regret} --------------------------- In this subsection, we define a performance measure that can be used to evaluate policies. For notational simplicity, let $L^{*} = L(\theta^{*})$, $G^{*} = G(\theta^{*})$ and $P^* = P(\theta^{*})$. Using (\[eqn:dare\]), we can show that $$\begin{split} &J_T^\pi (z_0 | \mathcal{F}_{T} ) {\ensuremath{\triangleq}}\sum_{t=1}^{T} \left\{ \rho \Sigma_\epsilon (x_{t-1} + \pi_{t}(\mathcal{I}_{t-1}) )^2 - \Delta p_t x_{t-1} \right\} \\ & \quad = z_0^\top P^* z_0 - z_T^\top P^* z_T + 2 \sum_{t=1}^{T} (A z_{t-1} + B \pi_{t}(\mathcal{I}_{t-1}) )^\top P^* W_t + \sum_{t=1}^{T} W_t^\top P^* W_t - \sum_{t=1}^{T} x_{t-1} \epsilon_t \\ & \qquad + \sum_{t=1}^{T} (\pi_{t}(\mathcal{I}_t) - L^{*} z_{t-1})^\top (R + B^\top P^* B) (\pi_{t}(\mathcal{I}_{t-1}) - L^{*} z_{t-1}) \quad \text{for any policy } \pi. \end{split}$$ First, we define [*pathwise regret*]{} $R_T^\pi (z_0 | \mathcal{F}_{T} )$ of a policy $\pi$ at period $T$ as $J_T^\pi (z_0 | \mathcal{F}_{T} ) - J_T^{\pi^{*}} (z_0 | \mathcal{F}_{T} )$ where $\pi_t^{*}(\mathcal{I}_{t-1}) = L^{*} z_{t-1}^{*}$ and $z_t^{*} = G^* z_{t-1}^{*} + W_t$ with $z_0^{*} = z_0$. In other words, the pathwise regret of a policy $\pi$ at period $T$ amounts to excess costs accumulated over $T$ periods when applying $\pi$ relative to when applying the optimal policy $\pi^{*}$. By definition of $\pi^{*}$, the pathwise regret of a policy $\pi$ at period $T$ can be expressed as $$\begin{split} &R_T^\pi (z_0 | \mathcal{F}_{T} ) = z_T^{* \top} P^* z_T^{*} - z_T^\top P^* z_T + \sum_{t=1}^{T} (\pi_t(\mathcal{I}_{t-1}) - L^{*} z_{t-1})^\top (R + B^\top P^* B) (\pi_t(\mathcal{I}_{t-1}) - L^{*} z_{t-1}) \\ & \quad + 2 \sum_{t=1}^{T} ((A z_{t-1} + B \pi_t(\mathcal{I}_{t-1}) ) - (A + B L^{*}) z_{t-1}^{*} )^\top P^* W_t + \sum_{t=1}^{T} (x_{t-1}^{*} - x_{t-1}) \epsilon_t. \end{split}$$ Second, we define [*expected regret*]{} $\bar{R}_T^{\pi} (z_0)$ of a policy $\pi$ at period $T$ as ${\ensuremath{\mathsf{E}}}[ R_T^\pi(z_0 | \mathcal{F}_{T} ) ]$. Taking expectation of pathwise regret, we can obtain a more concise expression for expected regret because the last two terms vanish by the law of total expectation. Hence, we have $$\bar{R}_T^{\pi} (z_0) = {\ensuremath{\mathsf{E}}}[ z_T^{* \top} P^* z_T^{*} - z_T^\top P^* z_T ] + {\ensuremath{\mathsf{E}}}\left[ \sum_{t=1}^{T} (\pi_t(\mathcal{I}_{t-1}) - L^{*} z_{t-1})^\top (R + B^\top P^* B) (\pi_t(\mathcal{I}_{t-1}) - L^{*} z_{t-1}) \right].$$ Finally, we define [*relative regret*]{} $\tilde{R}_T^{\pi} (z_0)$ of a policy $\pi$ at period $T$ as $\bar{R}_T^{\pi} (z_0) / | \textrm{tr}(P^{*} \tilde{\Omega}) |$ where $\textrm{tr}(P^{*} \tilde{\Omega})$ is minimum expected average cost for $\theta^{*}$. Our choice of performance measure will be either expected regret or relative regret in the rest of this paper. Confidence-Triggered Regularized Adaptive Certainty Equivalent Policy {#sec:CTRACE} ===================================================================== Our problem can be viewed as a special case of reinforcement learning, which focuses on sequential decision-making problems in which unknown properties of an environment must be learned in the course of taking actions. It is often emphasized in reinforcement learning that longer-term performance can be greatly improved by making decisions that explore the environment efficiently at the expense of suboptimal short-term behavior. In our problem, a price impact model is unknown, and submission of large orders can be considered exploratory actions that facilitate learning. Certainty equivalent control (CE) represents one extreme where at any time, current point estimates are assumed to be correct and actions are made accordingly. Although learning is carried out with observations made as the system evolves, no decisions are designed to enhance learning. Thus, this is an instance of pure exploitation of current knowledge. In our problem, CE estimates the unknown price impact coefficients $\theta^*$ at each period via least-squares regression using available data, and makes decisions that maximize expected average risk-adjusted profit under an assumption that the estimated model is correct. That is, an action $u_t$ for CE is given by $u_t = L(\tilde{\theta}_{t-1}) z_{t-1}$ where $\tilde{\theta}_{t-1} = \operatorname*{argmin}_{\theta \in \Theta} \sum_{i=1}^{t-1} \left( (\Delta p_i - g^\top f_{i-1}) - \psi_i^\top \theta \right)^2$ with a regressor $\psi_i = [ u_i \,\,\, (d_i - d_{i-1})^\top ]^\top$. An important question is how aggressively the trader should explore to learn $\theta^*$. Unlike many other reinforcement learning problems, a fairly large amount of exploration is naturally induced by exploitative decisions in our problem. That is, regular trading activity triggered by the return-predictive factors $f_t$ excites the market regardless of whether or not she aims to learn price impact. Given sufficiently large factor variability, the induced exploration might adequately resolve uncertainties about price impact. However, we will demonstrate by proposing a new exploratory policy that executing trades to explore beyond what would naturally occur through the factor-driven exploitation can result in significant benefit. Now, let us formally state that exploitative actions triggered by the return-predictive factors induce a large degree of exploration that could yield strong consistency of least-squares estimates. It is worth noting that pure exploitation is not sufficient for strong consistency in other problems such as [@LaiWei86] and [@ChenGuo86]. \[lem:persistentExcitationFixed\] For any $\theta \in \Theta$, let $u_t = L(\theta) z_{t-1}$, $z_t = G(\theta) z_{t-1} + W_t$ and $\psi_t^\top = \left[ u_t \,\,\, (d_t - d_{t-1})^\top \right] = (U(\theta) z_{t-1})^\top$. Also, let $\Pi_{zz}(\theta)$ denote a unique solution to $\Pi_{zz}(\theta) = G(\theta) \Pi_{zz}(\theta) G(\theta)^\top + \tilde{\Omega}$. Then, $$\begin{aligned} \label{eqn:ergodic} \lim_{T \rightarrow \infty} \frac{1}{T} \sum_{t=1}^{T} \psi_t \psi_t^\top = U(\theta) \Pi_{zz}(\theta) U(\theta)^\top \succ 0 \,\,\, a.s. \end{aligned}$$ Moreover, we can show that $\Pi_{zz}(\theta)$ is continuous on $\Theta$ by proving uniform convergence of ${\ensuremath{\mathsf{E}}}\left[ \frac{1}{T} \sum_{t=1}^{T} z_{t-1} z_{t-1}^\top \right]$ to $\Pi_{zz}(\theta)$ on $\Theta$. Continuity leads to $\underline{\lambda}_{\psi \psi}^{*} {\ensuremath{\triangleq}}\inf_{\theta \in \Theta} \lambda_{\text{min}} \left( U(\theta) \Pi_{zz}(\theta) U(\theta)^\top\right) > 0$ which will be used later. \[cor:extremums\] $\Pi_{zz}(\theta)$ is continuous on $\Theta$ and $\underline{\lambda}_{\psi \psi}^{*} {\ensuremath{\triangleq}}\inf_{\theta \in \Theta} \lambda_{\text{min}} \left( U(\theta) \Pi_{zz}(\theta) U(\theta)^\top\right) > 0$. Lemma \[lem:persistentExcitationFixed\] implies that $\lambda_{\text{min}} \left( \sum_{t=1}^{T} \psi_{t} \psi_{t}^\top \right)$ increases linearly in time $T$ $\text{a.s.}$ asymptotically. In addition, we can obtain a similar result for a finite-sample case: There exists a finite, deterministic constant $T_1(\theta,\delta)$ such that $\lambda_{\text{min}} \left( \sum_{t=1}^{T} \psi_{t} \psi_{t}^\top \right)$ grows linearly in time $T$ for all $T \geq T_1(\theta,\delta)$ with probability at least $1-\delta$. This is a crucial result that will be used for bounding above “$(\epsilon,\delta)$-convergence time” later. It is formally stated in the following lemma. \[lem:linearGrowthFixed\] For any $\theta \in \Theta$, let $u_t = L(\theta) z_{t-1}$, $z_t = G(\theta) z_{t-1} + W_t$ and $\psi_t^\top = \left[ u_t \,\,\, (d_t - d_{t-1})^\top \right] = (U(\theta) z_{t-1})^\top$. Then, there exists an event $\mathcal{B}(\delta)$ such that on $\mathcal{B}(\delta)$ with $\text{Pr}(\mathcal{B}(\delta)) \geq 1-\delta$ $$\frac{7}{8} U(\theta) \Pi_{zz}(\theta) U(\theta)^\top \preceq \frac{1}{T} \sum_{t=1}^{T} \psi_{t} \psi_{t}^\top \preceq \frac{17}{16} U(\theta) \Pi_{zz}(\theta) U(\theta)^\top \quad \forall T \geq T_1(\theta,\delta) \quad \text{where}$$ [$$T_1(\theta,\delta) = 4 \left( \frac{32 (C_z C_g)^2 (M+K+1)}{\xi^2 (1-\xi^{\frac{2}{N}}) \lambda_{\text{min}}(\Pi_{zz}(\theta)) } \right)^2 \log \left( \frac{(M+K+2)^4}{432 \delta^2} \right) \vee 8 \left( \frac{32 (C_z C_g)^2 (M+K+1)}{\xi^2 (1-\xi^{\frac{2}{N}}) \lambda_{\text{min}}(\Pi_{zz}(\theta)) } \right)^3 \vee 216.$$ ]{} Furthermore, we can extend Lemma \[lem:persistentExcitationFixed\] in such a way that $\lambda_{\text{min}} \left( \sum_{t=1}^{T} \psi_{t} \psi_{t}^\top \right)$ still increases to infinity linearly in time $T$ for time-varying $\{ \theta_t \}$ adapted to $\{ \sigma(\mathcal{I}_{t}) \}$ as long as $\theta_t$ remains sufficiently close to a fixed $\theta \in \Theta$ for all $t \geq 0$. Here, $\sigma(\mathcal{I}_{t})$ denotes a $\sigma$-algebra generated by $\mathcal{I}_{t}$ and $\theta_t$ is $\sigma(\mathcal{I}_{t})$-measurable for each $t$. \[lem:persistentExcitationFloating\] Consider any $\theta \in \Theta$ and $\{ \theta_t \in \Theta \}$ adapted to $\{ \sigma(\mathcal{I}_t) \}$ such that $\| \theta_t - \theta \| \leq \frac{\eta}{\sqrt{M+1} C_L} \,\, a.s.$ $$\text{where} \quad \eta = \left( \frac{\nu^3 (1-\nu^{\frac{1}{N}})^3 \lambda_{\text{min}}(\Pi_{zz}(\theta))}{42 N C_g^{N+1} C_\omega^2} \wedge \frac{\nu^3 (1-\nu^{\frac{1}{N}})^3 \lambda_{\text{min}}(U(\theta) \Pi_{zz}(\theta) U(\theta)^\top)}{42 N C_g^{N+1} C_\omega^2 (1 + \| U(\theta) \|)^2} \wedge \frac{\nu - \xi}{N C_g^{N-1}} \right)$$ for all $t \geq 0$ and any $\nu \in (\xi, 1)$. Let $u_t = L(\theta_{t-1}) z_{t-1}$, $z_t = G(\theta_{t-1}) z_{t-1} + W_t$ and $\psi_t^\top = \left[ u_t \,\,\, (d_t - d_{t-1})^\top \right] = (U(\theta_{t-1}) z_{t-1})^\top$. Then, $$\liminf_{T \rightarrow \infty} \,\, \frac{1}{T} \sum_{t=1}^{T} \psi_t \psi_t^\top \succeq \frac{ \lambda_{\text{min}}( U(\theta) \Pi_{zz}(\theta) U(\theta)^\top)}{2} I \,\,\, a.s.$$ Similarly to Lemma \[lem:linearGrowthFixed\], we can obtain a finite-sample result for Lemma \[lem:persistentExcitationFloating\]. This result will provide with a useful insight into how our new exploratory policy operates in the long term. \[lem:linearGrowthFloating\] Consider $\{ \theta_t \in \Theta \}$ defined in Lemma \[lem:persistentExcitationFloating\]. Let $u_t = L(\theta_{t-1}) z_{t-1}$, $z_t = G(\theta_{t-1}) z_{t-1} + W_t$ and $\psi_t^\top = \left[ u_t \,\,\, (d_t - d_{t-1})^\top \right] = (U(\theta_{t-1}) z_{t-1})^\top$. Then, for any $0 < \delta < 1$ on the event $\mathcal{B}(\delta)$ in Lemma \[lem:linearGrowthFixed\] with $\text{Pr}(\mathcal{B}(\delta)) \geq 1 - \delta$ $$\lambda_{\text{min}} \left( \frac{1}{T} \sum_{t=1}^{T} \psi_t \psi_t^\top \right) \geq \frac{3}{8} \lambda_{\text{min}}(U(\theta) \Pi_{zz}(\theta) U(\theta)^\top), \quad \forall T \geq T_1(\theta, \delta) \vee \frac{3 \| z_0 \| (2 C_\omega + \| z_0 \|)}{C_\omega^2}.$$ $\theta_0$, $x_0$, $d_0$, $r$, $g$, $\kappa$, $C_v$, $\tau$, $L(\cdot)$, $\theta_{\text{max}}$, $\{ p_t \}_{t=0}^{\infty}$, $\{ f_t \}_{t=0}^{\infty}$ $\{ u_t \}_{t=1}^{\infty}$ $V_0 \gets \kappa I$, $t_0 \gets 0$, $i \gets 1$ $u_t \gets L(\theta_{t-1}) z_{t-1}$, $x_t \gets x_{t-1} + u_t$, $d_t \gets \text{diag}(r) d_{t-1} + {\ensuremath{\mathbf{1}}}u_t$ $\psi_t \gets [ u_t \,\,\, (d_t - d_{t-1})^\top]^\top$, $V_t \gets V_{t-1} + \psi_t \psi_t^\top$ $\theta_t \gets \operatorname*{argmin}_{\theta \in \Theta} \sum_{i=1}^{t} \left( (\Delta p_i - g^\top f_{i-1}) - \psi_i^\top \theta \right)^2 + \kappa \| \theta \|^2$, $t_i \gets t$, $i \gets i+1$ $\theta_t \gets \theta_{t-1}$ It is challenging to guarantee that all estimates generated by CE are sufficiently close to one another uniformly over time so that Lemma \[lem:persistentExcitationFloating\] and Lemma \[lem:linearGrowthFloating\] can be applied to CE. In particular, CE is subject to overestimation of price impact that could be considerably detrimental to trading performance. The reason is that overestimated price impact discourages submission of large orders and thus it might take a while for the trader to realize that price impact is overestimated due to reduced “signal-to-noise ratio.” To address this issue, we propose the [*confidence-triggered regularized adaptive certainty equivalent*]{} policy (CTRACE) as presented in Algorithm \[alg:CTRACE\]. CTRACE can be viewed as a generalization of CE and deviates from CE in two ways: (1) $\ell_2$ regularization is applied in least-squares regression, (2) coefficients are only updated when a certain measure of confidence exceeds a pre-specified threshold and a minimum inter-update time has elapsed. Note that CTRACE reduces to CE as the regularization penalty $\kappa$ and the threshold $C_v$ tend to zero, and the minimum inter-update time $\tau$ tends to one. Regularization induces active exploration in our problem by penalizing the $\ell_2$-norm of price impact coefficients as well as reduces the variance of an estimator. Without regularization, we are more likely to obtain overestimates of price impact. Such an outcome attenuates trading intensity and thereby makes it difficult to escape from the misjudged perspective on price impact. Regularization decreases the chances of obtaining overestimates by reducing the variance of an estimator and furthermore tends to yield underestimates that encourage active exploration. Another source of improvement of CTRACE relative to CE is that updates are made based on a certain measure of confidence for estimates whereas CE updates at every period regardless of confidence. To be more precise on this confidence measure, we first present a high-probability confidence region for least-squares estimates from [@Abbasi-YadkoriPalSzepesvari11]. \[prop:confidenceSet\] $$\text{Pr} \left( \theta^{*} \in \mathcal{S}_t (\delta), \,\, \forall t \geq 1 \right) \geq 1 - \delta \quad \text{where} \quad V_t = \kappa I + \sum_{i=1}^{t} \psi_i \psi_i^\top, \quad \hat{\theta}_t = V_t^{-1} \left( \sum_{i=1}^{t} \psi_i \psi_i^\top \theta^{*} + \sum_{i=1}^{t} \psi_i \epsilon_i \right),$$ $$\mathcal{S}_t (\delta) {\ensuremath{\triangleq}}\left\{ \theta \in {\ensuremath{{\ensuremath{\mathbb{R}}}}}^{M+1} : (\theta - \hat{\theta}_t )^\top V_t (\theta - \hat{\theta}_t) \leq \left( C_\epsilon \sqrt{2\, \textrm{log} \left( \frac{\textrm{det}(V_t)^{1/2} \textrm{det}(\kappa I)^{-1/2} }{\delta} \right)} + \kappa^{1/2} \| \theta_{\text{max}} \| \right)^2 \right\}.$$ This implies that for any $\theta \in \mathcal{S}_t(\delta)$ $$\| \theta - \hat{\theta}_t \|^{2} \leq \frac{1}{\lambda_{\textrm{min}}(V_t)} \left( C_\epsilon \sqrt{2\, \log \left( \frac{\textrm{det}(V_t)^{1/2} \textrm{det}(\kappa I)^{-1/2} }{\delta} \right)} + \kappa^{1/2} \| \theta_{\text{max}} \| \right)^2.$$ By definition, CTRACE updates only when $\lambda_{\textrm{min}}(V_t) \geq \kappa + C_v \, t$. $\lambda_{\textrm{min}}(V_t)$ typically dominates $\log \left( \textrm{det}(V_t) \right)$ for large $t$ because it increases linearly in $t$, and is inversely proportional to the squared estimation error $\| \hat{\theta}_t - \theta^* \|^{2}$. That is, CTRACE updates only when confidence represented by $\lambda_{\textrm{min}}(V_t)$ exceeds the specified level $\kappa + C_v \, t$. From now on, we refer to this updating scheme as confidence-triggered update. Confidence-triggered update makes a significant contribution to reducing the chances of obtaining overestimates of price impact by updating “carefully” only at the moments when an upper bound on the estimation error is guaranteed to decrease. The minimum inter-update time $\tau \in \mathbb{N}$ in Algorithm \[alg:CTRACE\] can guarantee that the closed-loop system $\{ z_t \}$ from CTRACE is stable as long as $\tau$ is sufficiently large. Meanwhile, there is no such stability guarantee for CE. The following lemma provides with a specific uniform bound on $\| z_t \|$. \[lem:uniformBoundCTRACE\] Under CTRACE with $\tau \geq N \log (2 C_g/\xi) / \log (1/\xi)$ $$\| z_t \| \leq \frac{(2 C_g + 1) C_g C_\omega}{\xi (1-\xi^{\frac{1}{N}})} {\ensuremath{\triangleq}}C_z^* \quad a.s. \quad \text{and} \quad \| \psi_t \| \leq \frac{(C_g +1)(2 C_g + 1) C_g C_\omega}{\xi (1-\xi^{\frac{1}{N}})} {\ensuremath{\triangleq}}C_{\psi} \quad a.s. \quad \forall t \geq 0.$$ Confidence-triggered update yields a good property of CTRACE that CE lacks: CTRACE is [*inter-temporally consistent*]{} in the sense that estimation errors $\| \theta_t - \theta^* \|$ are bounded with high probability by monotonically nonincreasing upper bounds that converge to zero almost surely as time tends to infinity. The following theorem formally states this property. \[thm:intertemporalConsistency\] Let $\{ \theta_t \}$ be estimates generated by CTRACE with $M \geq 2$, $\tau \geq N \log (2 C_g/\xi) / \log (1/\xi)$ and $C_v < \underline{\lambda}_{\psi \psi}^{*}$. Then, the $i$th update time $t_i$ in Algorithm \[alg:CTRACE\] is finite a.s. Moreover, $\| \theta_t - \theta^{*} \| \leq b_t, \,\, \forall t \geq 0$ on the event $\{ \theta^* \in \mathcal{S}_t(\delta), \,\, \forall t \geq 1 \}$ where $$b_t = \begin{cases} \frac{ 2 C_\epsilon \sqrt{(M+1) \log \left( C_\psi^2 \, t / \kappa + M+1 \right) + 2 \log \left( 1/\delta \right) } + 2\kappa^{1/2} \| \theta_{\text{max}} \| }{ \sqrt{C_v t }} & \text{if } t = t_i \text{ for some } i \\ b_{t-1} & \text{otherwise} \end{cases}, \quad b_0 = \| \theta_0 - \theta^{*} \|,$$ and $\{ b_t \}$ is monotonically nonincreasing for all $t \geq 1$ with $\lim_{t \rightarrow \infty} b_t = 0$ a.s. Moreover, we can show that CTRACE is [*efficient*]{} in the sense that its $(\epsilon,\delta)$-convergence time is bounded above by a polynomial of $1/\epsilon$, $\log(1/\delta)$ and $\log(1/\delta')$ with probability at least $1-\delta'$. We define [*$(\epsilon,\delta)$-convergence time*]{} to be the first time when an estimate and all the future estimates following it are within an $\epsilon$-neighborhood of $\theta^{*}$ with probability at least $1 - \delta$. If $\epsilon$ is sufficiently small, we can apply Lemma \[lem:persistentExcitationFloating\] and Lemma \[lem:linearGrowthFloating\] to guarantee that $\lambda_{\textrm{min}}(V_t)$ increases linearly in $t$ with high probability after $(\epsilon,\delta)$-convergence time and thereby confidence-triggered update occurs at every $\tau$ periods. This is a critical property that will be used for deriving a poly-logarithmic finite-time expected regret bound for CTRACE. By Theorem \[thm:intertemporalConsistency\], it is easy to see that the $(\epsilon,\delta)$-convergence time of CTRACE is bounded above by $t_{N(\epsilon,\delta,C_v)}$ where $N(\epsilon,\delta,C_v)$ is defined as $$N(\epsilon,\delta,C_v) = \inf \left\{ i \in {\ensuremath{{\ensuremath{\mathbb{N}}}}}: \frac{ 2 C_\epsilon \sqrt{(M+1) \log \left( C_\psi^2 \, t_i / \kappa + M+1 \right) + 2 \log \left( 1/\delta \right) } + 2 \kappa^{1/2} \| \theta_{\text{max}} \| }{ \sqrt{ C_v t_i }} \leq \epsilon \right\}.$$ The following theorem presents the polynomial bound on the $(\epsilon,\delta)$-convergence time of CTRACE. \[thm:efficiency\] For any $\epsilon > 0$, $0 < \delta, \delta' < 1$, $\tau \geq N \log (2 C_g/\xi) / \log (1/\xi)$ and $C_v < \frac{7}{8} \underline{\lambda}_{\psi \psi}^{*}$ on the event $\mathcal{B}(\delta')$ defined in Lemma \[lem:linearGrowthFixed\], $$t_{N(\epsilon,\delta,C_v)} \leq T_1^{*}(\delta') \vee \tau + T_2(\epsilon,\delta,C_v) \quad \text{where}$$ [$$T_1^{*}(\delta') = 4 \left( \frac{32 (C_z^* C_g)^2 (M+K+1)}{\xi^2 (1-\xi^{\frac{2}{N}}) \underline{\lambda}^{*}_{zz} } \right)^2 \log \left( \frac{(M+K+2)^4}{432 \delta^{'2}} \right) \vee 8 \left( \frac{32 (C_z^* C_g)^2 (M+K+1)}{\xi^2 (1-\xi^{\frac{2}{N}}) \underline{\lambda}^{*}_{zz} } \right)^3 \vee 216,$$ ]{} [$$T_2(\epsilon,\delta,C_v) = \left( \frac{ 8 C_\epsilon^2 C_\psi (M+1) + 4 \sqrt{ 4 C_\epsilon^4 C_\psi^2 (M+1)^2 + \kappa C_\epsilon^2 C_v \epsilon^2 \left( (M+1)^{3/2} + 2 \log (1/\delta) \right) } }{ \sqrt{\kappa} C_v \epsilon^2 } \right)^2 \vee \frac{( 4 \kappa \| \theta_{\text{max}} \|)^2}{ C_v \epsilon^2}.$$ ]{} Finally, we derive a finite-time expected regret bound for CTRACE that is quadratic in logarithm of elapsed time using the efficiency of CTRACE and Lemma \[lem:linearGrowthFloating\]. \[thm:finiteTimeBoundCTRACE\] If $\pi$ is CTRACE with $M \geq 2$, $\tau \geq N \log (2 C_g/\xi) / \log (1/\xi)$ and $C_v < \frac{7}{8} \underline{\lambda}_{\psi \psi}^{*}$, then for any $\nu \in (\xi, 1)$ and all $T \geq 2$, $$\begin{split} \bar{R}_T^{\pi}(z_0) &\leq 2 \| P^{*} \| C_z^{*2} + (R + B^\top P^* B) C_z^{*2} C_L^2 \Bigg( \left( \tau_1^{*}(T) + \tau_2^{*}(T) + 1 \right) \| \theta_{\text{max}} \|^2 + \tau_3^{*}(T) \epsilon^{2} \\ & \quad + \frac{ \tau \left( 2 C_\epsilon \sqrt{(M+1) \log \left( C_\psi^2 \, T / \kappa + M+1 \right) + 2 \log \left( 2 T \right) } + 2 \kappa^{1/2} \| \theta_{\text{max}} \| \right)^2 }{ \tilde{C} } \nonumber \\ & \quad \times \log \left( \frac{ \kappa + \tilde{C} (T-1) - (\tilde{C} - C_v)_{+} (\tau_1^{*}(T) + \tau_2^{*}(T))}{ \kappa + \tilde{C} (\tau^{*}(T) - 1) - (\tilde{C} - C_v)_{+} (\tau_1^{*}(T) + \tau_2^{*}(T)) } \right) {\ensuremath{\mathbf{1}}}\{ T > \tau^{*}(T) \} \Bigg) \end{split}$$ where $\tilde{C} {\ensuremath{\triangleq}}\frac{3}{8} \lambda_{\text{min}}(U(\theta^*) \Pi_{zz}(\theta^*) U(\theta^*)^\top )$, $\tau^{*}(T) = \tau_1^{*}(T) + \tau_2^{*}(T) + \tau_3^{*}(T)$, [$$\tau_1^{*}(T) = 8 \left( \frac{32 (C_z^* C_g)^2 (M+K+1)}{\xi^2 (1-\xi^{\frac{2}{N}}) \underline{\lambda}^{*}_{zz} } \right)^2 \log \left( \frac{(M+K+2)^2 T}{6 \sqrt{3}} \right) \vee 8 \left( \frac{32 (C_z^* C_g)^2 (M+K+1)}{\xi^2 (1-\xi^{\frac{2}{N}}) \underline{\lambda}^{*}_{zz} } \right)^3 \vee 216 \vee \tau,$$ ]{} [$$\begin{split} \tau_2^{*}(T) &= \left( \frac{ 8 C_\epsilon^2 C_\psi (M+1) + 4 \sqrt{ 4 C_\epsilon^4 C_\psi^2 (M+1)^2 + \kappa C_\epsilon^2 C_v \epsilon^2 \left( (M+1)^{3/2} + 2 \log (2 T) \right) } }{ \sqrt{\kappa} C_v \epsilon^2 } \right)^2 \vee \frac{( 4 \kappa \| \theta_{\text{max}} \|)^2}{ C_v \epsilon^2}, \end{split}$$ ]{} [$$\begin{split} \tau_3^{*}(T) &= 8 \left( \frac{32 (C_z^* C_g)^2 (M+K+1)}{\xi^2 (1-\xi^{\frac{2}{N}}) \lambda_{\text{min}}(\Pi_{zz}(\theta^*)) } \right)^2 \log \left( \frac{(M+K+2)^2 T}{6 \sqrt{3} } \right) \vee 8 \left( \frac{32 (C_z^* C_g)^2 (M+K+1)}{\xi^2 (1-\xi^{\frac{2}{N}}) \lambda_{\text{min}}(\Pi_{zz}(\theta^*)) } \right)^3 \vee 216 \\ &\quad \vee \frac{3 C_z^* (2 C_\omega + C_z^* )}{C_\omega^2}, \end{split}$$ ]{} [$$\epsilon = \frac{1}{\sqrt{M+1} C_L} \left( \frac{\nu^3 (1-\nu^{\frac{1}{N}})^3 \lambda_{\text{min}}(\Pi_{zz}(\theta^*))}{42 N C_g^{N+1} C_\omega^2} \wedge \frac{\nu^3 (1-\nu^{\frac{1}{N}})^3 \lambda_{\text{min}}(U(\theta^*) \Pi_{zz}(\theta^*) U(\theta^*)^\top)}{42 N C_g^{N+1} C_\omega^2 (1 + \| U(\theta^*) \|)^2} \wedge \frac{\nu - \xi}{N C_g^{N-1}} \right).$$ ]{} Note that $\tau_1^{*}(T)$, $\tau_2^{*}(T)$ and $\tau_3^{*}(T)$ are all $O(\log T)$. Therefore, it is not difficult to see that the expected regret bound for CTRACE is $O(\log^2 T)$. Computational Analysis {#sec:computationalAnalysis} ====================== In this section, we will compare via Monte Carlo simulation the performance of CTRACE to that of two benchmark policies: CE and a reinforcement learning algorithm recently proposed in [@Abbasi-YadkoriSzepesvari10], which is referred to as AS policy from now on. AS policy was designed to explore efficiently in a broader class of linear-quadratic control problems and appears well-suited for our problem. It updates an estimate only when the determinant of $V_t$ is at least twice as large as the determinant evaluated at the last update, and selects an element from a high-probability confidence region that yields maximum average reward. In our problem, AS policy can translate to update an estimate with $\theta_t = \operatorname*{argmin}_{\theta \in \mathcal{S}_t(\delta) \cap \Theta} \text{tr}(P(\theta) \tilde{\Omega})$ at each update time $t$. Intuitively, the smaller price impact, the larger average profit, equivalently, the smaller $\text{tr}(P(\theta) \tilde{\Omega})$ which is the negative of average profit. In light of this, we restrict our attention to solutions to $\min_{\theta \in \mathcal{S}_t(\delta) \cap \Theta} \text{tr}(P(\theta) \tilde{\Omega})$ of the form $\{ \alpha_t \hat{\theta}_{con,t} \in \mathcal{S}_t(\delta) \cap \Theta : 0 \leq \alpha_t \leq 1 \}$ where $\hat{\theta}_{con,t}$ denotes a constrained least-squares estimate to $\Theta$ with $\ell_2$ regularization. The motivation is to reduce the amount of computation needed for AS policy otherwise it would be prohibitive. Indeed, the minimum appears to be attained always with the smallest $\alpha_t$ such that $\alpha_t \hat{\theta}_{con,t} \in \mathcal{S}_t(\delta) \cap \Theta$, which is provable in the special case considered in Subsection \[subsec:closedForm\]. Note that $\alpha_t$ can be viewed as a measure of aggressiveness of exploration: $\alpha_t = 1$ means no extra exploration and smaller $\alpha_t$ implies more active exploration. $M$ 6 $K$ 2 --------------------- ------------------------------------------ ----------------------- ------------------------------------------------ Trading interval 5 mins Initial asset price \$50 Half-life of $r$ \[5, 7.5, 10, 15, 30, 45\] mins Half life of factor \[10, 40\] mins $r$ \[ 0.50, 0.63, 0.71, 0.79, 0.89, 0.93 \] $\Phi$ diag(\[0.707, 0.917\]) $\gamma$ (\$/share) \[0, 6, 0, 3, 7, 5\] $\times \, 10^{-8}$ $\lambda$ (\$/share) $2 \times 10^{-8}$ $\Sigma_\epsilon$ $0.0013$ (annualized vol. = 10%) $\Omega$ diag(\[1, 1\]) $\rho$ $1 \times 10^{-6}$ $\theta_{\text{max}}$ $(5 \times 10^{-7}) {\ensuremath{\mathbf{1}}}$ $\beta$ $5 \times 10^{-9}$ $g$ \[0.006, 0.002\] $T$ $3000$ ($\approx$ 38 trading days) Sample paths 600 : Monte Carlo Simulation Setting (1 trading day = 6.5 hours)[]{data-label="tab:simulationSetting"} Table \[tab:simulationSetting\] summarizes numerical values used in our simulation. The signal-to-noise ratio (SNR), which is defined as ${\ensuremath{\mathsf{E}}}[(\lambda u_t + \sum_{m=1}^{M} \gamma_m (d_{m,t} - d_{m,t-1}))^2 ]/{\ensuremath{\mathsf{E}}}[\epsilon_t^2]$ under $u_t = L(\theta^*) z_{t-1}$, is 0.058 and the optimal average profit is \$765.19 per period. $\epsilon_t$ and $\omega_t$ are sampled independently from Gaussian distribution even though $\omega_t$ is assumed to be bounded almost surely for the theoretical analysis. In fact, it turns out that the use of Gaussian distribution for $\omega_t$ does not make a noticeable difference from a bounded case. The regularization coefficient $\kappa$, the confidence-triggered update threshold $C_v$, the minimum inter-update time $\tau$ and the significance level $\delta$ are chosen via cross-validation with realized profit: For CTRACE, $\kappa = 1 \times 10^{11}$, $C_v = 600$ and $\tau = 1$. For AS policy, $\kappa = 1 \times 10^8$ and $\delta = 0.99$. The reason for smaller $\kappa$ and large $\delta$ for AS policy is to keep the radius of confidence regions small because the exploration done by AS policy tends to be more than necessary and thus costly. The left figure in Figure \[fig:regulConfTrig\] illustrates improvement of relative regret due to regularization. It shows the relative regret of CTRACE with varying $\kappa$ and fixed $C_v = 0$, $\text{i.e.}$ no confidence-triggered update. The vertical bars indicate two standard errors in both directions, that is, approximate 95% confidence intervals. It is clear that the relative regret is reduced as CTRACE regularizes more, and the improvement from no regularization to $\kappa = 1 \times 10^{11}$ is statistically significant with approximate 95% confidence level. The right figure in Figure \[fig:regulConfTrig\] shows improvement achieved by confidence-triggered update with varying $C_v$ but fixed $\kappa = 1 \times 10^{11}$. As you can see, update based on confidence makes a substantial contribution to reducing relative regret further. The improvement from $C_v = 0$ to $C_v = 600$ is statistically significant with approximate 95% confidence level. ![Relative regret with varying $\kappa$ and $C_v$: (Left) Varying $\kappa \in \{0, 2 \times 10^{10}, 1 \times 10^{11} \}$ with fixed $C_v = 0$. (Right) Varying $C_v \in \{ 0, 20, 600\}$ with fixed $\kappa = 1 \times 10^{11}$.[]{data-label="fig:regulConfTrig"}](reg_confTrig_5em9_SNR0p058_S600_T3000.eps) ![(Left) Relative regret of CTRACE and CE. (Right) Distribution of realized profit of CTRACE and CE. The red dotted line represents zero difference. []{data-label="fig:CTRACECE"}](CTRACE_CE_5em9_SNR0p058_S600_T3000.eps) As shown on the left of Figure \[fig:CTRACECE\], CTRACE clearly outperforms CE in terms of relative regret and the difference is statistically significant with approximate 95% confidence level. The dominance stems from both regularization and confidence-triggered update as shown in Figure \[fig:regulConfTrig\]. The figure on the right shows an empirical distribution of difference between realized profit of CTRACE and that of CE over 600 sample paths. Much more realizations are located to the right with respect to zero profit. It implies that CTRACE tends to make more profit than CE more frequently. Finally, we compare performance of CTRACE to that of AS policy in Figure \[fig:CTRACEAS\]. The left figure shows that CTRACE outperforms AS policy even more drastically than CE in terms of relative regret, and the superiority is statistically significant with approximate 95% confidence level. On the right, you can see an empirical distribution of difference between realized profit of CTRACE and that of AE over 600 sample paths. It is clear that CTRACE is more profitable than AS policy in most of the sample paths. This illustrates that aggressive exploration performed by AS policy is too costly. The reason is that AS policy is designed to explore actively in situations where pure exploitation done by CE is unable to identify a true model. In our problem, however, a great degree of exploration is naturally induced by observable return-predictive factors and thus aggressiveness of exploration suggested by AS policy turns out to be even more than necessary. Meanwhile, CTRACE strikes a desired balance between exploration and exploitation by taking into account factor-driven natural exploration. ![(Left) Relative regret of CTRACE and AS policy. (Right) Distribution of realized profit of CTRACE and AS policy. The red dotted line represents zero difference. []{data-label="fig:CTRACEAS"}](CTRACE_AS_5em9_SNR0p058_S600_T3000.eps) Conclusion {#sec:conclusion} ========== We have considered a dynamic trading problem where a trader maximizes expected average risk-adjusted profit while trading a single security in the presence of unknown price impact. Our problem can be viewed as a special case of reinforcement learning: the trader can improve longer-term performance significantly by making decisions that explore efficiently to learn price impact at the expense of suboptimal short-term behavior such as execution of larger orders than appearing optimal with respect to current information. Like other reinforcement learning problems, it is crucial to strike a balance between exploration and exploitation. To this end, we have proposed the confidence-triggered regularized adaptive certainty equivalent policy (CTRACE) that improves purely exploitative certainty equivalent control (CE) in our problem. The enhancement is attributed to two properties of CTRACE: regularization and confidence-triggered update. Regularization encourages active exploration that accelerates learning as well as reduces the variance of an estimator. It helps keep CTRACE from being a passive learner due to overestimation of price impact that abates trading. Confidence-triggered update allows CTRACE to have monotonically nonincreasing upper bounds on estimation errors so that it reduces the frequency of overestimation. Using these two properties, we derived a finite-time expected regret bound for CTRACE of the form $O(\log^2 T)$. Finally, we have demonstrated through Monte Carlo simulation that CTRACE outperforms CE and a reinforcement learning policy recently proposed in [@Abbasi-YadkoriSzepesvari10]. As extention to our current model, it would be interesting to develop an efficient reinforcement learning algorithm for a portfolio of securities. Another interesting direction is to incorporate a prior knowledge of particular structures of price impact coefficients, $\text{e.g.}$ sparsity, to an estimation problem. It is worth considering other regularization schemes such as LASSO. Proofs ====== **** Since the evolution of $f_t$ is not affected by $\{ x_t \}$, $\{ d_t \}$ and $\{ u_t \}$, it is not difficult to see that there exists a desired $P$ for our stochastic control problem if there exists $P$ with the same properties for a deterministic control problem having no $f_t$ and $g = 0$. Let $(\tilde{A},\tilde{B},\tilde{Q},\tilde{R},\tilde{S})$ denote reduced coefficient matrices for the deterministic problem of appropriate dimensions. Now, $(\tilde{A},\tilde{B})$ is controllable and this problem is a special case of the problem considered in [@Molinari75]. By Theorem 1 in [@Molinari75], there exists a desired $P$ if $\Psi(z) > 0$ for all $z$ on the unit circle where $$\Psi(z) {\ensuremath{\triangleq}}\left[ \begin{array}{cc} \tilde{B}^\top (I z^{-1} - \tilde{A}^\top)^{-1} & I \end{array} \right] \left[ \begin{array}{cc} \tilde{Q} & \tilde{S} \\ \tilde{S}^\top & \tilde{R} \end{array} \right] \left[ \begin{array}{c} (I z - \tilde{A})^{-1} \tilde{B} \\ I \end{array} \right].$$ In our problem, it is not difficult to check that for any $\phi \in (0,2 \pi)$, $\lambda \geq 0$ and $\gamma_i \geq 0$, $$\Psi(e^{i \phi}) = \frac{\rho \Sigma_\epsilon}{2(1 - \cos \phi)} + \frac{\lambda}{2} + \sum_{m=1}^{M} \frac{2 \gamma_m (1 - r_m \cos \phi )}{1 + r_m^2 - 2 r_m \cos \phi} > 0$$ and $\lim_{\phi \rightarrow 0} \Psi(e^{i \phi}) = \infty > 0$. Therefore, the desired result follows. Noting an upper block diagonal structure of the original closed-loop system matrix $A + B L$, we can easily see that the stability for the deterministic problem carries over to our original problem. The uniqueness of a stabilizing solution follows from the stability. For the optimality of $\pi$, we can use the same proof in Chapter 4 of [@Bertsekas05]. **** By Theorem \[thm:optPolicyExistenceUniqueness\], $\rho_{\text{sr}}(G(\theta)) < 1$ for all $\theta \in \Theta$. Since $\Theta$ is a compact set and Assumption \[assum:optimalPolicy\]-(a) implies the continuity of $L(\theta)$ and $G(\theta)$, it follows that $\sup_{\theta \in \Theta} \| G(\theta) \| < \infty$ and $\sup_{\theta \in \Theta} \rho_{\text{sr}}(G(\theta)) < 1$. Therefore, by Theorem in [@BuchananParlett66], $\{ G^n(\theta) \}$ uniformly converges to zero matrix. That is, for any $0 < \xi < 1$, there exists $N \in \mathbb{N}$ being independent of $\theta$ such that $\| G^N(\theta) \| \leq \xi$ for all $\theta \in \Theta$. Also, $\max_{0 \leq i \leq N-1} \sup_{\theta \in \Theta} \| G^i(\theta) \| < \infty$ by continuity of $G(\theta)$ and compactness of $\Theta$. For any $t \geq 0$, it is easy to see that $\| G^t(\theta) \| \leq C_g \xi^{\lfloor t/N \rfloor}$ by definition of $C_g$ and $N$. Since $z_t = G^t(\theta) z_0 + \sum_{i=1}^{t} G^{t-i}(\theta) W_t$, $$\begin{split} \| z_t \| &\leq \| G^t(\theta) \| \| z_0 \| + \sum_{i=1}^{t} \| G^{t-i}(\theta)) \| \| W_t \| \leq C_g \xi^{\lfloor t/N \rfloor} \| z_0 \| + \sum_{i=1}^{t} C_g \xi^{\lfloor (t-i)/N \rfloor} C_\omega \\ &\leq C_g \| z_0 \| + C_g C_\omega \sum_{i=1}^{t} \xi^{(t-i)/N-1} \leq C_g \| z_0 \| + C_g C_\omega / (\xi (1 - \xi^{1/N})) \,\, a.s. \end{split}$$ Since $U(\theta) = ( G(\theta) )_{1:M+1,*} - [I \,\,\, 0]$, it follows that $\| U(\theta) \| \leq \| (G(\theta) )_{1:M+1,*} \| + \| [ I \,\,\, 0] \| \leq C_g + 1$. **** For notational simplicity, let $G = G(\theta)$, $L = L(\theta)$ and $\Pi_{zz} = \Pi_{zz}(\theta)$. The almost-sure convergence in (\[eqn:ergodic\]) follows from Lemma 2 in [@AndersonTaylor79]. It is easy to see that $U(\theta)$ is full-rank since $(L)_1 \neq 0$. Therefore, it is sufficient to show that $\Pi_{zz}$ is positive definite. Since $G$ is a stable matrix and $\tilde{\Omega} \succeq 0$, $\Pi_{zz} = \sum_{i=0}^{\infty} G^i \tilde{\Omega} (G^\top)^i \succeq \sum_{i=0}^{M+K} G^i \tilde{\Omega} (G^\top)^i = H H^\top$ where $H = \left[ \tilde{\Omega}^{1/2} \,\, G \tilde{\Omega}^{1/2} \,\, \ldots \,\, G^{M+K} \tilde{\Omega}^{1/2} \right]$. Thus, it is sufficient to show that $H$ is full-rank. First, we will show that $\{ (G)_{1:M+1,M+2}, \ldots, (G^{M+1})_{1:M+1,M+2} \}$ is linearly independent. We can show by induction that $(G^i)_{*,M+2} = \left[ g_i(1) \,\,\, g_i(r_1) \,\,\, \cdots \,\,\, g_i(r_M) \,\,\, h_i \right]^\top$ where $g_i (r) = (L)_{M+2} \sum_{m=0}^{i-1} (\Phi^{m})_{1,1} r^{i-1-m}$ and $h_i = (\Phi^i)_{*,1}$. Since each $g_i(r)$ is a polynomial of degree $i-1$ and its leading coefficient is all $(L)_{M+2} \neq 0$, we can transform $[ (G)_{1:M+1,M+2}, \,\, \ldots \,\, (G^{M+1})_{1:M+1,M+2} ]$ into Vandermonde matrix through elementary row operations. Thus, $[ (G)_{1:M+1,M+2} \,\, \ldots \,\, (G^{M+1})_{1:M+1,M+2} ]$ is nonsingluar. Now, suppose $\alpha^\top H = 0$ for some $\alpha \in {\ensuremath{{\ensuremath{\mathbb{R}}}}}^{M+K+1}$. By definition of $H$ and $\tilde{\Omega}$, it implies $(\alpha)_{M+2:M+K+1} = 0$. Then, by nonsingularity of $[ (G)_{1:M+1,M+2} \,\, \ldots \,\, (G^{M+1})_{1:M+1,M+2} ]$, we may conclude $\alpha_{1:M+1}^\top = 0$. Therefore, $\alpha = 0$ and we may conclude that $H$ is full-rank. **** By Assumption \[assum:optimalPolicy\]-(a), $L(\theta)$ is continuous on $\Theta$ and so are $G(\theta)$ and $U(\theta)$. Uniform convergence of ${\ensuremath{\mathsf{E}}}\left[ \frac{1}{T} \sum_{t=1}^{T} z_{t-1} z_{t-1}^\top \right]$ to $\Pi_{zz}(\theta)$ on $\Theta$ follows from the fact that for any $\epsilon > 0$ $$\begin{aligned} &\left\| {\ensuremath{\mathsf{E}}}\left[ \frac{1}{T} \sum_{t=1}^{T} z_{t-1} z_{t-1}^\top \right] - \Pi_{zz}(\theta) \right\| = \left\| \frac{1}{T} z_{0} z_{0}^\top + \sum_{t=1}^{T-1} \frac{t-1}{T} G^{t-1} \tilde{\Omega} (G^\top)^{t-1} + \sum_{t=T}^{\infty} G^{t-1} \tilde{\Omega} (G^\top)^{t-1} \right\| \\ & \leq \frac{\| z_0 \|^2}{T} + \left\| \tilde{\Omega} \right\| \frac{C_g^2}{\xi^2} \left( \frac{1}{T} \frac{\xi^\frac{2}{N}}{(1 - \xi^\frac{2}{N})^2} + \frac{\xi^\frac{2(T-1)}{N}}{1-\xi^\frac{2}{N}} \right) \leq \epsilon \quad \text{for sufficiently large } T \text{ independent of } \theta.\end{aligned}$$ Since ${\ensuremath{\mathsf{E}}}\left[ \frac{1}{T} \sum_{t=1}^{T} z_{t-1} z_{t-1}^\top \right] = \frac{1}{T} z_0 z_0^\top + \frac{1}{T} \sum_{t=1}^{T-1} \sum_{i=0}^{t-1} G^i \tilde{\Omega} (G^\top)^i$ is continuous in $\theta \in \Theta$ for all $T \geq 1$, the limiting matrix $\Pi_{zz}(\theta)$ is continuous in $\theta \in \Theta$ component-wise. Thus, so is $U(\theta) \Pi_{zz}(\theta) U(\theta)^\top$. Finally, $\lambda_{\text{min}} \left( U(\theta) \Pi_{zz}(\theta) U(\theta)^\top\right)$ is continuous on $\Theta$. Since $\lambda_{\text{min}} \left( U(\theta) \Pi_{zz}(\theta) U(\theta)^\top\right) > 0, \,\, \forall \theta \in \Theta$ and $\Theta$ is a compact set, it follows from its continuity that $\inf_{\theta \in \Theta} \lambda_{\text{min}} \left( U(\theta) \Pi_{zz}(\theta) U(\theta)^\top\right) > 0$. **** Let $e_i \in {\ensuremath{{\ensuremath{\mathbb{R}}}}}^{M+K+1}$ denote an elementary vector whose entries are all zero except for $i$th entry being one and $\eta_{ij,k} {\ensuremath{\triangleq}}e_i^\top z_k z_k^\top e_j - e_i^\top {\ensuremath{\mathsf{E}}}[ z_k z_k^\top | \mathcal{F}_{k-1} ] e_j, \,\, 1 \leq i, j, \leq M+K+1$. Since $| \eta_{ij,k} | \leq 2 C_z^2 \,\, a.s.$, $\{ \eta_{ij,k} \}$ is an almost-surely bounded martingale difference process adapted to $\{ \mathcal{F}_k \}$ and thus it is conditionally sub-Gaussian with ${\ensuremath{\mathsf{E}}}[ \exp(\gamma \eta_{ij,k}) | \mathcal{F}_{k-1} ] \leq \exp \left( \gamma^2 (2 C_z^2)^{2} / 2 \right) \,\, a.s.$ Hence, if we use a special case of Corollary 1 in [@Abbasi-YadkoriPalSzepesvari11] with $m_k = 1$ for all $k$, then for all $1 \leq i, j \leq M+K+1$ and any $a > 0$ $$\text{Pr} \left( \left| \sum_{k=1}^{t} \eta_{ij,k} \right| \leq 2 C_z^2 \sqrt{ (a+t) \log \left( \frac{(M+K+2)^4 (a+t)}{4 a \delta^2} \right)} \quad \forall t \geq 1 \right) \geq 1 - \frac{2\delta}{(M+K+2)^2}.$$ Using $\eta_{ij,k} = \eta_{ji,k}$ and ${\ensuremath{\mathsf{E}}}[ z_k z_k^\top | \mathcal{F}_{k-1} ] = G(\theta) z_{k-1} z_{k-1}^\top G(\theta)^\top + \tilde{\Omega}$, it follows from the union bound that $\text{Pr} \Bigg( | \left( Y_t \right)_{ij} | \leq \epsilon, \,\, 1 \leq i, j \leq M+K+1, \,\, \forall t \geq t^*(\delta,\epsilon,a) \Bigg) \geq 1 - \delta$ where $Y_t {\ensuremath{\triangleq}}\frac{1}{t} \sum_{k=1}^{t} z_{k} z_{k}^\top - G(\theta) \left( \frac{1}{t} \sum_{k=1}^{t} z_{k-1} z_{k-1}^\top \right) G(\theta)^\top - \tilde{\Omega}$ and $$t^*(\delta,\epsilon,a) {\ensuremath{\triangleq}}4 \left( \frac{2 C_z^2}{\epsilon} \right)^2 \log \left( \frac{(M+K+2)^4 }{2 a \delta^2} \right) \linebreak \vee 8 \left( \frac{2 C_z^2}{\epsilon} \right)^3 \vee a \vee 216.$$ On the above event, $\| Y_t \| \leq \| Y_t \|_F \leq (M+K+1) \epsilon$ and $-(M+K+1) \epsilon I \preceq Y_t \preceq (M+K+1) \epsilon I, \,\, \forall t \geq t^*(\delta,\epsilon,a)$. We can rewrite $-(M+K+1) \epsilon I \preceq Y_t$ as $$\frac{1}{t} \sum_{k=1}^{t} z_{k-1} z_{k-1}^\top \succeq G(\theta) \left( \frac{1}{t} \sum_{k=1}^{t} z_{k-1} z_{k-1}^\top \right) G(\theta)^\top + \tilde{\Omega} - (M+K+1) \epsilon I + \left( \frac{1}{t} z_{0} z_{0}^\top - \frac{1}{t} z_{t} z_{t}^\top \right).$$ Repeating $n$ times a process of left-multiplying both sides with $G(\theta)$, right-multiplying with $G(\theta)^\top$ and adding the resulting inequality into the original one side-by-side, we obtain $$\begin{split} \frac{1}{t} \sum_{k=1}^{t} z_{k-1} z_{k-1}^\top &\succeq G^{n+1}(\theta) \left( \frac{1}{t} \sum_{k=1}^{t} z_{k-1} z_{k-1}^\top \right) G^{n+1}(\theta)^\top + \sum_{i=0}^{n} G^{i}(\theta) \tilde{\Omega} G^{i}(\theta)^\top \\ & \quad - (M+K+1) \epsilon \sum_{i=0}^{n} G^{i}(\theta) G^{i}(\theta)^\top + \sum_{i=0}^{n} G^{i}(\theta) \left( \frac{1}{t} z_{0} z_{0}^\top - \frac{1}{t} z_{t} z_{t}^\top \right) G^{i}(\theta)^\top. \end{split}$$ Note that $\left\| \sum_{i=0}^{n} G^{i}(\theta) \left( \frac{1}{t} z_{0} z_{0}^\top - \frac{1}{t} z_{t} z_{t}^\top \right) G^{i}(\theta)^\top \right\| \leq \sum_{i=0}^{n} \frac{2}{t} C_z^2 \| G^{i}(\theta) \|^{2} \leq 2 C_z^2 C_g^2 / (t \xi^2 (1 - \xi^{2/N}))$ and $\left\| \sum_{i=0}^{n} G^{i}(\theta) G^{i}(\theta)^\top \right\| \leq \sum_{i=0}^{n} \| G^{i}(\theta) \|^{2} \leq C_g^2 / (\xi^2 (1 - \xi^{2/N}))$. Taking limit over $n$ and using these two inequalities, we have with probability at least $1-\delta$ $$\frac{1}{t} \sum_{k=1}^{t} z_{k-1} z_{k-1}^\top \succeq \Pi_{zz}(\theta) - \left( \frac{C_g^2 (M+K+1)}{\xi^2 (1 - \xi^{\frac{2}{N}})}\epsilon + \frac{1}{t} \frac{2 C_z^2 C_g^2}{\xi^2 (1 - \xi^{\frac{2}{N}})} \right) I, \quad \forall t \geq t^*(\delta,\epsilon,a).$$ Setting $\epsilon = \xi^2 (1 - \xi^{2/N}) \lambda_{\text{min}}(\Pi_{zz}(\theta))/(16 C_g^2 (M+K+1))$ and $a = 216$, we have $\frac{1}{t} \sum_{k=1}^{t} z_{k-1} z_{k-1}^\top \succeq \Pi_{zz}(\theta) - \frac{\lambda_{\text{min}}(\Pi_{zz}(\theta))}{8} I$ for all $t \geq t^*(\delta,\epsilon,a) \vee 32 C_z^2 C_g^2 / (\xi^2 (1-\xi^{2/N}) \lambda_{\text{min}}(\Pi_{zz}(\theta)))$. It is easy to show that $ t^*(\delta,\epsilon,a) \geq 32 C_z^2 C_g^2 / (\xi^2 (1-\xi^{2/N}) \lambda_{\text{min}}(\Pi_{zz}(\theta))). $ Similarly, from $Y_t \preceq (M+K+1) \epsilon I$, we can obtain for all $t \geq t^*(\delta,\epsilon,a)$ $$\frac{1}{t} \sum_{k=1}^{t} z_{k-1} z_{k-1}^\top \preceq \Pi_{zz}(\theta) + \left( \frac{(M+K+1) C_g^2}{\xi^2 (1 - \xi^{\frac{2}{N}})}\epsilon - \frac{1}{t} \frac{2 C_z^2 C_g^2}{\xi^2 (1 - \xi^{\frac{2}{N}})} \right) I \preceq \Pi_{zz}(\theta) + \frac{ \lambda_{\text{min}}(\Pi_{zz}(\theta))}{16} I.$$ Since $\lambda_{\text{min}}(\Pi_{zz}(\theta)) I \preceq \Pi_{zz}(\theta)$, it follows that $\frac{7}{8} \Pi_{zz}(\theta) \preceq \frac{1}{t} \sum_{k=1}^{t} z_{k-1} z_{k-1}^\top \preceq \frac{17}{16} \Pi_{zz}(\theta)$ and thus $\frac{7}{8} U(\theta) \Pi_{zz}(\theta) U(\theta)^\top \preceq U(\theta) \frac{1}{t} \sum_{k=1}^{t} z_{k-1} z_{k-1}^\top U(\theta)^\top \preceq \frac{17}{16} U(\theta) \Pi_{zz}(\theta) U(\theta)^\top$. **** For notational convenience, let $G = G(\theta)$, $G_t = G(\theta_t)$, $U = U(\theta)$, $U_t = U(\theta_t)$, $\Pi_{zz} = \Pi_{zz}(\theta)$ and $\Pi(i,j) = G_i \cdots G_j$. By definition of $G$ and $\eta$, $\| G_t - G \| \leq \| B \| \| L(\theta_t) - L(\theta) \| \leq \sqrt{M+1} \, C_L \| \theta_t - \theta \| \leq \eta, \,\, \forall t \geq 0.$ Since $z_t$ can be expressed as $z_t = \Pi(0,t-1) z_0 + \sum_{i=1}^{t} \Pi(i,t-1) W_i$, we have $$\begin{split} \frac{1}{T} \sum_{t=1}^{T} z_{t-1} z_{t-1}^\top &= \frac{1}{T} \sum_{t=1}^{T} \left( G^{t-1} z_0 + \sum_{i=1}^{t-1} G^{t-i-1} W_i \right) \left(G^{t-1} z_0 + \sum_{j=1}^{t-1} G^{t-j-1} W_j \right)^\top \\ & \quad + \frac{1}{T} \sum_{t=1}^{T} \left( \Pi(0,t-2) z_0 z_0^\top \Pi(0,t-2)^\top - G^{t-1} z_0 z_0^\top G^{t-1 \top} \right) \quad \cdots \,\, (a) \nonumber \\ & \quad + \frac{1}{T} \sum_{t=2}^{T} \sum_{j=1}^{t-1} \left( \Pi(0,t-2) z_0 W_j^\top \Pi(j,t-2)^\top - G^{t-1} z_0 W_j^\top G^{t-j-1 \top} \right) \quad \cdots \,\, (b) \nonumber \\ & \quad + \frac{1}{T} \sum_{t=2}^{T} \sum_{j=1}^{t-1} \left( \Pi(j,t-2) W_j z_0^\top \Pi(0,t-2)^\top - G^{t-j-1} W_j z_0^\top G^{t-1 \top} \right) \quad \cdots \,\, (c) \nonumber \\ & \quad + \frac{1}{T} \sum_{t=2}^{T} \sum_{i=1}^{t-1} \sum_{j=1}^{t-1} \left( \Pi(i,t-2) W_i W_j^\top \Pi(j,t-2)^\top - G^{t-i-1} W_i W_j^\top G^{t-j-1 \top} \right) \quad \cdots \,\, (d) \nonumber \end{split}$$ Then, we can show that $$\| (a) \| \leq \frac{9 \eta N C_g^{N+1} \| z_0 \|^2 }{ T \nu^3 (1 - \nu^{2/N})^2}, \quad \| (b) \|, \| (c) \| \leq \frac{9 \eta N C_g^{N+1} C_\omega \| z_0 \|}{T \nu^3 (1 - \nu^{1/N})^3} \quad \| (d) \| \leq \frac{18 \eta N C_g^{N+1} C_\omega^2}{ \nu^3 (1 - \nu^{1/N})^3} \quad \text{and}$$ $$\| (a) + (b) + (c) + (d) \| \leq \frac{21 \eta N C_g^{N+1} C_\omega^2 }{ \nu^3 (1 - \nu^{1/N})^3} \leq \frac{\lambda_{\text{min}}(\Pi_{zz})}{2}, \quad \forall T \geq \frac{3 \| z_0 \| (2 C_\omega + \| z_0 \|)}{C_\omega^2}.$$ It follows that $$\frac{1}{T} \sum_{t=1}^{T} z_{t-1} z_{t-1}^\top \succeq \frac{1}{T} \sum_{t=1}^{T} \left( G^{t-1} z_0 + \sum_{i=1}^{t-1} G^{t-i-1} W_i \right) \left(G^{t-1} z_0 + \sum_{j=1}^{t-1} G^{t-j-1} W_j \right)^\top - \frac{\lambda_{\text{min}}(\Pi_{zz})}{2} I.$$ Taking $\liminf$ on both sides, $\liminf_{T \rightarrow \infty} \frac{1}{T} \sum_{t=1}^{T} z_{t-1} z_{t-1}^\top \succeq \Pi_{zz} - \frac{\lambda_{\text{min}}(\Pi_{zz})}{2} I \succeq \frac{\lambda_{\text{min}}(\Pi_{zz})}{2} I \,\, a.s.$ Likewise, we can show that $\liminf_{T \rightarrow \infty} \frac{1}{T} \sum_{t=1}^{T} \psi_t \psi_t^\top \succeq U \Pi_{zz} U^\top - \frac{\lambda_{\text{min}}(U \Pi_{zz} U^\top)}{2} I \succeq \frac{\lambda_{\text{min}}(U \Pi_{zz} U^\top)}{2} I \,\, a.s.$ **** Using the same techniques in the proof of Lemma \[lem:linearGrowthFixed\] and Lemma \[lem:persistentExcitationFloating\], we can obtain that on the event $\mathcal{B}(\delta)$ with $\text{Pr}(\mathcal{B}(\delta)) \geq 1 - \delta$, $$\begin{aligned} \frac{1}{T} \sum_{t=1}^{T} \psi_t \psi_t^\top & \succeq U(\theta) \frac{1}{T} \sum_{t=1}^{T} \left( G(\theta)^{t-1} z_0 + \sum_{i=1}^{t-1} G(\theta)^{t-i-1} W_i \right) \left(G(\theta)^{t-1} z_0 + \sum_{j=1}^{t-1} G(\theta)^{t-j-1} W_j \right)^\top U(\theta)^\top \\ & \quad - \frac{\lambda_{\text{min}}(U(\theta) \Pi_{zz}(\theta) U(\theta)^\top)}{2} I \\ & \succeq U(\theta) \left( \Pi_{zz}(\theta) - \frac{\lambda_{\text{min}}(\Pi_{zz}(\theta))}{8} I \right) U(\theta)^\top - \frac{\lambda_{\text{min}}(U(\theta) \Pi_{zz}(\theta) U(\theta)^\top)}{2} I \\ & \succeq \frac{3}{8} \lambda_{\text{min}}(U(\theta) \Pi_{zz}(\theta) U(\theta)^\top ), \quad \forall T \geq T_1(\| z_0 \|, \theta, \delta) \vee \frac{3 \| z_0 \| (2 C_\omega + \| z_0 \|)}{C_\omega^2}. \qed \end{aligned}$$ **** Using $\| z_{t_i + j} \| \leq C_g \xi^{j/N-1} \| z_{t_i} \| + C_g C_\omega / (\xi (1 - \xi^{1/N} ) ) \,\, a.s.$ for $j \leq t_{i+1} - t_i$ and $C_g \xi^{\tau/N - 1} \leq \frac{1}{2}$, we can show by induction that $\| z_{t_i} \| \leq 2 C_g C_\omega / (\xi (1-\xi^{1/N})) \,\, a.s.$ for all $i \geq 1$. For any $t_{i} < t < t_{i+1}$, $\| z_t \| \leq C_g \xi^{(t-t_i)/N-1} \| z_{t_i} \| + C_g C_\omega / (\xi (1-\xi^{1/N})) \leq C_z^* \,\, a.s.$ Finally, $\| \psi_t \| \leq \| U(\theta_{t-1}) \| \| z_{t-1} \| \leq (C_g + 1) C_z^* = C_\psi$. **** Given $C_v < \underline{\lambda}_{\psi \psi}^{*} \leq \lambda_{\text{min}}\left( U(\theta) \Pi_{zz}(\theta) U(\theta)^\top \right)$, it is easy to show that $\text{Pr}(t_i < \infty, \,\, \forall i \geq 1) = 1$. Using $\theta_{t_i} = \operatorname*{argmin}_{\theta \in \Theta} \,\, \sum_{j=1}^{t_i} \left( (\Delta p_j - g^\top f_{j-1}) - \psi_j^\top \theta \right)^2 + \kappa \| \theta \|^2 = \operatorname*{argmin}_{\theta \in \Theta} \,\, (\theta - \hat{\theta}_{t_i})^\top V_{t_i} (\theta - \hat{\theta}_{t_i})$ and Proposition \[prop:confidenceSet\], we can show that on the event $\{ \theta^* \in \mathcal{S}_t(\delta) \,\, \forall t \geq 1\}$ for any $i \geq 1$ [$$\begin{aligned} \| \theta_{t_i} - \theta^{*} \| \leq \| \theta_{t_i} - \hat{\theta}_{t_i} \| + \| \hat{\theta}_{t_i} - \theta^{*} \| \leq \frac{2 C_\epsilon \sqrt{(M+1) \log \left( C_\psi^2 \, t_i / \kappa + M+1 \right) + 2 \log \left( 1/\delta \right) } + 2 \kappa^{1/2} \| \theta_{\text{max}} \| }{ \sqrt{ C_v t_i}} = b_{t_i}.\end{aligned}$$ ]{} For any $t_i < t < t_{i+1}$, $\| \theta_{t} - \theta^{*} \| = \| \theta_{t_i} - \theta^{*} \| \leq b_{t_i} = b_t$. It is easy to show through elementary calculus that $b_{t_i}$ is strictly decreasing in $t_i \geq 1$ if $M \geq 2$. **** Using $\log(t+M+1) \leq \sqrt{t} + \sqrt{M+1}$ for all $t \geq 0$, we can show that $$\frac{ 2 C_\epsilon \sqrt{(M+1) \log \left( C_\psi^2 \, t / \kappa + M+1 \right) + 2 \log \left( 1/\delta \right) } + 2 \kappa^{1/2} \| \theta_{\text{max}} \| }{ \sqrt{ C_v t }} \leq \epsilon, \quad \forall t \geq T_2(\epsilon,\delta,C_v).$$ Suppose for contradiction that $t_{N(\epsilon,\delta,C_v)} > T_1^{*}(\delta') \vee \tau + T_2(\epsilon,\delta,C_v) {\ensuremath{\triangleq}}\tilde{T}^{*}$. Let $t_i$ be the last update time less than $T_2(\epsilon,\delta,C_v)$. Then, there is no update time in the interval $[t_i+1, \tilde{T}^{*}]$ by definition of $t_{N(\epsilon,\delta,C_v)}$ and $T_2(\epsilon,\delta,C_v)$. By definition of $t_i$ and Lemma \[lem:linearGrowthFixed\], $$\lambda_{\text{min}} \left( V_{\tilde{T}^{*}} \right) \geq \lambda_{\text{min}}( V_{t_i} ) + \lambda_{\text{min}} \left( \sum_{t = t_i + 1}^{\tilde{T}^{*}} \psi_t \psi_t^\top \right) \geq \kappa + C_v t_i + \frac{7}{8} \underline{\lambda}_{\psi \psi}^{*} ( \tilde{T}^{*} - t_i) \geq \kappa + C_v \tilde{T}^{*}.$$ It is clear that $\tilde{T}^{*} - t_i \geq \tau$. Consequently, $\tilde{T}^{*}$ is eligible for a next update time after $t_i$. It implies that $t_{N(\epsilon,\delta,C_v)} = \tilde{T}^{*}$ but this is a contradiction. **** Note that $$(R + B^\top P^* B) \sum_{t=1}^{T} ( (L(\theta_{t-1}) - L(\theta^{*})) z_{t-1} )^2 \leq (R + B^\top P^* B) C_z^{*2} C_L^2 \sum_{t=1}^{T} \| \theta_{t-1} - \theta^{*} \|^2.$$ Set $\delta = 1/T$. Then, on the event $\mathcal{A}(T) {\ensuremath{\triangleq}}\{ \theta^{*} \in \mathcal{S}_t(1/(2 T)) \,\,\, \forall t \geq 1\} \cap \mathcal{B}(1/(2 T))$ with $\text{Pr} \left( \mathcal{A}(T) \right) \geq 1 - 1/T$, we have $$\sum_{t=1}^{\tau_1^{*}(T) + \tau_2^{*}(T)} \| \theta_{t-1} - \theta^* \|^2 \leq (\tau_1^{*}(T) + \tau_2^{*}(T)) \| \theta_{\text{max}} \|^2, \quad \sum_{t=\tau_1^{*}(T) + \tau_2^{*}(T)+1}^{\tau^{*}(T)} \| \theta_{t-1} - \theta^* \|^2 \leq \tau_3^{*}(T) \epsilon^{2}.$$ By Lemma \[lem:linearGrowthFloating\], $$\begin{split} \lambda_{\text{min}} \left( V_{t-1} \right) &\geq \lambda_{\text{min}} \left( V_{t_{N(\epsilon,1/(2 T),C_v)}} \right) + \lambda_{\text{min}} \left( \sum_{i=t_{N(\epsilon,1/(2 T),C_v)}+1}^{t-1} \psi_i \psi_i^\top \right) \\ & \geq \kappa + \tilde{C} (t-1) - (\tilde{C} - C_v)_{+} (\tau_1^{*}(T) + \tau_2^{*}(T)), \quad \forall t \geq \tau^{*}(T). \end{split}$$ Therefore, [ $$\begin{aligned} &\sum_{t=\tau^{*}(T)+1}^{T} \| \theta_{t-1} - \theta^* \|^2 \leq \sum_{t=\tau^{*}(T)+1}^{T} \frac{ \tau}{\lambda_{\textrm{min}}(V_{t-1})} \left( 2 C_\epsilon \sqrt{2\, \log \left( \frac{\textrm{det}(V_{t-1})^{1/2} \textrm{det}(\kappa I)^{-1/2} }{\delta/2} \right)} + 2 \kappa^{1/2} \| \theta_{\text{max}} \| \right)^2 \\ & \leq \frac{ \tau \left( 2 C_\epsilon \sqrt{(M+1) \log \left( C_\psi^2 \, T / \kappa + M+1 \right) + 2 \log \left( 2 T \right) } + 2 \kappa^{1/2} \| \theta_{\text{max}} \| \right)^2 }{ \tilde{C} } \\ & \quad \times \log \left( \frac{ \kappa + \tilde{C} (T-1) - (\tilde{C} - C_v)_{+} (\tau_1^{*}(T) + \tau_2^{*}(T)) }{ \kappa + \tilde{C} (\tau^{*}(T) - 1) - (\tilde{C} - C_v)_{+} (\tau_1^{*}(T) + \tau_2^{*}(T)) } \right).\end{aligned}$$ ]{} Let $q = \text{Pr} \left( \mathcal{A}(T) \right)$, $L_{t} = L(\theta_{t})$ and $L^* = L(\theta^{*})$. Then, $$\begin{aligned} \bar{R}_T^{\pi} (z_0) &= {\ensuremath{\mathsf{E}}}[ z_T^{* \top} P^* z_T^{*} - z_T^\top P^* z_T ] + {\ensuremath{\mathsf{E}}}\left[ \sum_{t=1}^{T} (L_{t-1} z_{t-1} - L^{*} z_{t-1})^\top (R + B^\top P^* B) (L_{t-1} z_{t-1} - L^{*} z_{t-1}) \right] \\ & \leq 2 \| P^{*} \| C_z^2 + q {\ensuremath{\mathsf{E}}}\left[ \left. \sum_{t=1}^{T} (L_{t-1} z_{t-1} - L^{*} z_{t-1})^\top (R + B^\top P^* B) (L_{t-1} z_{t-1} - L^{*} z_{t-1}) \right| \mathcal{A}(T) \right] \\ & \quad + (1-q) {\ensuremath{\mathsf{E}}}\left[ \left. \sum_{t=1}^{T} (L_{t-1} z_{t-1} - L^{*} z_{t-1})^\top (R + B^\top P^* B) (L_{t-1} z_{t-1} - L^{*} z_{t-1}) \right| \mathcal{A}(T)^c \right] \\ & \leq 2 \| P^{*} \| C_z^2 + {\ensuremath{\mathsf{E}}}\left[ \left. \sum_{t=1}^{T} (L_{t-1} z_{t-1} - L^{*} z_{t-1})^\top (R + B^\top P^* B) (L_{t-1} z_{t-1} - L^{*} z_{t-1}) \right| \mathcal{A}(T) \right] \\ & \quad + (R + B^\top P^* B) C_z^2 C_L^2 \| \theta_{\text{max}} \|^2 \quad \left( \because \,\, 1 - \frac{1}{T} \leq q \leq 1 \right) \\ & \leq 2 \| P^{*} \| C_z^2 + (R + B^\top P^* B) C_z^2 C_L^2 \Bigg( \left( \tau_1^{*}(T) + \tau_2^{*}(T) + 1 \right) \| \theta_{\text{max}} \|^2 + \tau_3^{*}(T) \epsilon^{2} \\ & \quad + \frac{ \tau \left( 2 C_\epsilon \sqrt{(M+1) \log \left( C_\psi^2 \, T / \kappa + M+1 \right) + 2 \log \left( 2 T \right) } + 2 \kappa^{1/2} \| \theta_{\text{max}} \| \right)^2 }{ \tilde{C} } \\ & \quad \times \log \left( \frac{ \kappa + \tilde{C} (T-1) - (\tilde{C} - C_v)_{+} (\tau_1^{*}(T) + \tau_2^{*}(T)) }{ \kappa + \tilde{C} ( \tau^{*}(T) - 1) - (\tilde{C} - C_v)_{+} (\tau_1^{*}(T) + \tau_2^{*}(T)) } \right) {\ensuremath{\mathbf{1}}}\left\{ T > \tau^{*}(T) \right\} \Bigg). \qed\end{aligned}$$

--- abstract: 'We report density functional calculations of the electronic structure, Fermi surface, phonon spectrum, magnetism and electron-phonon coupling for the superconducting phase FeSe, as well as the related compounds FeS and FeTe. We find that the Fermi surface structure of these compounds is very similar to that of the Fe-As based superconductors, with cylindrical electron sections at the zone corner, cylindrical hole surface sections, and depending on the compound, other small hole sections at the zone center. As in the Fe-As based materials, these surfaces are separated by a 2D nesting vector at ($\pi$,$\pi$). The density of states, nesting and Fermi surface size increase going from FeSe to FeTe. Both FeSe and FeTe show spin density wave ground states, while FeS is close to an instability. In a scenario where superconductivity is mediated by spin fluctuations at the SDW nesting vector, the strongest superconductor in this series would be doped FeTe.' author: - Alaska Subedi - Lijun Zhang - 'D.J. Singh' - 'M.H. Du' title: 'Density functional study of FeS, FeSe and FeTe: Electronic structure, magnetism, phonons and superconductivity' --- introduction ============ Superconductivity was recently reported in $\alpha$-FeSe$_{1-x}$, with critical temperature $T_c \sim$ 8K. [@hsu] This is of interest both because of the fact that Fe containing superconductors are unusual and because this material shares in common square planar sheets of tetrahedrally coordinated Fe with the Fe-As based high temperature superconductors. [@kamihara; @ren; @ren2; @cwang; @sefat; @kito] $\alpha$-FeSe occurs in the PbO structure. This consists, as mentioned, of Fe square planar sheets, with Se atoms forming distorted tetrahedra around the Fe very similar to the structure of the FeAs planes in LaFeAsO, BaFe$_2$As$_2$ and LiFeAs, which are prototypes of the known families of Fe-As based high-T$_c$ superconductors. [@rotter1; @rotter2; @xcwang] Since the reported $T_c$ = 8K of doped $\alpha$-FeSe is modest, it is important first of all to establish the relationship between this material and the Fe-As based superconductors. We note that LaNiPO is also a superconductor and shares the crystal structure of LaFeAsO, [@watanabe; @tegel] but that it is apparently quite different electronically and can be understood in terms of standard electron-phonon theory,[@subedi] unlike the Fe-As based phases. [@boeri; @mazin] Here we report density functional calculations that show $\alpha$-FeSe and the other known Fe based chalcogenides in this structure to be very similar to that of the Fe-As based superconductors. [@singh-du] In particular the Fermi surface consists of small heavy hole cylinders near the zone center and lighter compensating electron cylinders around the zone corner. We show that the stoichiometric compounds are either very close to a spin density wave instability (FeS) or have an itinerant spin density wave instability without doping (FeSe and FeTe) similar to the Fe-As superconductors. [@cruz] We predict that this itinerant nesting driven magnetic state is strongest in FeTe and in addition that FeTe has the largest Fermi surface of the three compounds. Calculations of the electron-phonon coupling show that doped FeSe is not an electron-phonon superconductor, similar to what was found for the Fe-As phases. Within a spin-fluctuation driven picture of superconductivity the results indicate that FeTe with doping is a likely higher temperature superconductor. first principles methods and structure ====================================== Our calculations of the electronic structure and magnetic properties were performed within the local density approximation with the general potential linearized augmented planewave (LAPW) method, [@singh-book] including local orbitals, [@singh-lo] similar to our previous calculations for the Fe-As based superconductors. [@singh-du; @mazin; @singh-ba] We used LAPW spheres of radius 2.1 $a_0$ for Fe, Se and Te and 1.9 $a_0$ for S. These compounds occur in a simple tetragonal structure with one internal parameter, $z_X$ corresponding to the chalcogen height above the Fe square plane. The experimental lattice parameters [@hsu; @lennie; @finlayson] were employed and we relaxed the chalcogen height via energy minimization. The structural parameters used and some results are presented in Table \[tab-struct\]. The electron-phonon coupling and phonon dispersions were on the other hand obtained using linear response, again with the experimental lattice parameters, with the Quantum Espresso code [@qe; @qenote] within the generalized gradient approximation of Perdew, Burke and Ernzerhof [@pbe] as described for LaFeAsO and LaNiPO. [@subedi; @mazin]   $a$(Å)     $c$(Å)     $z_X$   $N(E_F)$ $m_{SDW}(\mu_B)$ $E_{SDW}$ ------ ------------ ------------ ----------- ---------- ------------------ ----------- FeS 3.6735 5.0328 0.2243 1.35 0.00 0 FeSe 3.765 5.518 0.2343 0.95 0.65 5 FeTe 3.8215 6.2695 0.2496 1.83 1.28 47 : Structural parameters and magnetic properties of PbO-structure Fe$X$. The lattice parameters are from experimental data, while the internal chalcogen structural parameter, $z_X$ is from LDA structure minimization. $m_{SDW}$ is the spin moment within the Fe LAPW sphere (radius 2.1 $a_0$) for the SDW state and $E_{SDW}$ is the energy per Fe of this state relative to the non-spin-polarized state in meV/Fe. $N(E_F)$ is the density of states at the Fermi energy in the non-spin-polarized band structure in eV$^{-1}$ on a per Fe both spins basis.[]{data-label="tab-struct"} electronic structure, phonons and magnetism =========================================== ![Band structures of FeS (top), FeSe (middle) and FeTe (bottom) from non-spin-polarized calculations with the LDA relaxed $X$ heights.[]{data-label="fig-bands"}](band-S.ps "fig:"){height="3.4in"} ![Band structures of FeS (top), FeSe (middle) and FeTe (bottom) from non-spin-polarized calculations with the LDA relaxed $X$ heights.[]{data-label="fig-bands"}](band-Se.ps "fig:"){height="3.4in"} ![Band structures of FeS (top), FeSe (middle) and FeTe (bottom) from non-spin-polarized calculations with the LDA relaxed $X$ heights.[]{data-label="fig-bands"}](band-Te.ps "fig:"){height="3.4in"} ![(color online) Electronic DOS and projection onto the LAPW Fe and chalcogen spheres indicating the Fe $d$ and chalcogen $p$ contributions for FeS (top), FeSe (middle) and FeTe (bottom) as in Fig. \[fig-bands\].[]{data-label="fig-dos"}](dos-S.ps "fig:"){height="3.4in"} ![(color online) Electronic DOS and projection onto the LAPW Fe and chalcogen spheres indicating the Fe $d$ and chalcogen $p$ contributions for FeS (top), FeSe (middle) and FeTe (bottom) as in Fig. \[fig-bands\].[]{data-label="fig-dos"}](dos-Se.ps "fig:"){height="3.4in"} ![(color online) Electronic DOS and projection onto the LAPW Fe and chalcogen spheres indicating the Fe $d$ and chalcogen $p$ contributions for FeS (top), FeSe (middle) and FeTe (bottom) as in Fig. \[fig-bands\].[]{data-label="fig-dos"}](dos-Te.ps "fig:"){height="3.4in"} Our main results for the electronic structure are given in Figs. \[fig-bands\], \[fig-dos\] and \[fig-fermi\], which show the non-spin-polarized band structures, electronic densities of states and Fermi surfaces of FeS, FeSe and FeTe. The calculated values of $N(E_F)$ are given in Table \[tab-struct\]. The phonon dispersions of FeSe are shown in Fig. \[fig-phonon\]. One interesting feature of the phonon dispersion is that they have little dispersion in the $k_z$ and e.g. are very flat along the tetragonal $\Gamma$-$Z$ direction. This presumably reflects anion-anion repulsion, which leads to long bonds between the FeSe layers. the result is that there may be an easy cleavage plane between the Se ions, which may facilitate preparation of clean surfaces for experiments such as photoelectron spectroscopy. The phonon density of states, $G(\omega)$ and electron-phonon spectral function, $\alpha^2 F(\omega)$ are given in Fig. \[fig-a2f\]. The electron-phonon coupling constant for FeSe as obtained in linear response is $\lambda$ = 0.17 with $\omega_{log}$=113 cm$^{-1}$. No superconductivity at any temperature even approaching 1K results with these values within standard electron-phonon theory even if very low values of the Coulomb parameter, e.g. $\mu^*=0.10$ are used in the Allen-Dynes equation. This is similar to what was found previously for LaFeAsO. [@boeri; @mazin] Therefore we conclude that FeSe is not a conventional electron-phonon superconductor. ![(color online) LDA Fermi surface of FeS, FeSe and FeTe from non-spin-polarized calculations with the LDA relaxed $X$ heights. The corners are $\Gamma$ points.[]{data-label="fig-fermi"}](fermi.ps){width="3.4in"} Turning to the electronic structure we find a strong qualitative similarity between these materials and the FeAs-based superconductors. In particular, we find these to be low carrier density metals, with high density of states. This arises from band structures that are closely related to those of the FeAs materials. The chalcogen $p$ states lie well below the Fermi level and are only modestly hybridized with the Fe $d$ states as may be seen from the projected DOS (Fig. \[fig-dos\]). Thus the electronic structure near the Fermi energy derives from metallic Fe$^{2+}$ layers, with direct Fe-Fe interactions. These are embedded inside a largely ionic background which imposes a competing tetrahedral crystal field. As in the Fe-As based materials, there is a pseudogap at an electron count of 6 $d$ electrons per Fe, and $E_F$ lies near the bottom of this pseudogap. We emphasize that this is not the position of a tetrahedral crystal field gap, which would be at 4 electrons, and emphasizes the fact that Fe chalcogen hybridization is not strong compared with the Fe-Fe interactions. This explains the similarity of the electronic structure to that of the FeAs-based materials, which were also found to be substantially ionic in similar calculations. [@singh-du] ![Calculated GGA phonon dispersions of non-spin-polarized FeSe. []{data-label="fig-phonon"}](FeSe_ep0.eps){width="3.4in"} ![Calculated GGA phonon density of state $G(\omega)$ and electron-phonon spectral function $\alpha^2 F(\omega)$ for FeSe. []{data-label="fig-a2f"}](phondos.eps){width="3.4in"} These band structures yield two intersecting elliptical cylindrical electron Fermi surfaces at the zone corner in all three materials. These are compensated by lower velocity hole sections at the zone center – two concentric hole cylinders, and in the case of FeS and FeTe a small closed hole section inside the inner cylinder. This is qualitatively very similar to the FeAs-based materials. It is important to note that cylinders at the zone center and zone corner, if they are the same size, would yield strong nesting peaked at the 2D ($\pi$,$\pi$) point. This will lead in general to enhanced spin fluctuations at the nesting vector, and if sufficiently strong will cause a spin density wave. In fact, an ordered SDW was found both in first principles calculations and in experimental studies for LaFeAsO and many other undoped FeAs-based compounds. [@mazin; @cruz; @dong; @chen; @zhao; @huang; @goldman; @nakai] There is a clear competition between the SDW and the superconducting state in that superconducting samples generally do not show the SDW, while samples with the SDW transition, generally do not show superconductivity. The ground state is an antiferromagnetic cell doubled along the \[11\] in plane direction to yield linear chains of nearest neighbor like spin Fe atoms arranged antiferromagnetically, although the values of the moments are dependent on details, especially the As height above the Fe plane. [@yildirim; @ishibashi; @yin; @mazin-2; @vildosola; @yildirim-2] discussion ========== Turning to the trends, the size of the pseudogap is approximately the same in FeSe and FeS, but is significantly smaller in FeTe. Specifically, there is a greater overlap between the hole and electron bands in the latter compound. This leads to larger Fermi surfaces. The value of $N(E_F)$=1.83 eV$^{-1}$ is also highest in this compound, although it is still lower than the 2.62 eV$^{-1}$ that is obtained for LaFeAsO by the same approach. We used a supercell approach to investigate the SDW with the chalcogen heights fixed to the values calculated by non-spin-polarized energy minimization. We find instabilities for FeSe and FeTe, but not for FeS. The spin moments and the energy of the SDW relative to the non-spin-polarized state are given in Table \[tab-struct\]. As may be seen the SDW is considerably stronger in FeTe than in FeSe, with an energy gain of 47 meV/Fe and a spin-moment of 1.3 $\mu_B$. The corresponding values for LaFeAsO, calculated in the same way are $E_{SDW}$=11 meV/Fe and $m_{SDW}$ = 1.0 $\mu_B$ in a 2.1 $a_0$ Fe sphere. Besides the SDW we find a borderline ferromagnetic tendency in FeTe, without doping when the SDW is not allowed, similar to LaFeAsO. [@singh-du] We do not find ferromagnetic instabilities in either FeSe or FeS, consistent with the lower values of $N(E_F)$ in those materials. The sensitivity of the moment size to the ordering pattern underscores that fact that these are itinerant magnetic systems in the LDA, as opposed to local moment magnets. This means that magnetic ordering is driven by electrons at and near the Fermi surface. On the other hand this is not to say that spin fluctuations are weak in the paramagnetic state above the SDW ordering temperature or the paramagnetic state as realized by doping. In fact, as noted, several authors have found that the As height in the Fe-As compounds is strongly coupled to magnetism and so strong spin fluctuations in the paramagnetic state would help rationalize the underestimated As height in non-spin-polarized LDA calculations. [@yildirim; @ishibashi; @yin; @mazin-2; @yildirim-2] In fact, there is evidence for strong spin fluctuations in the normal state of the FeAs compounds, e.g. from temperature dependent resistivity data indicating strong scattering above the SDW ordering temperature (note a drop in resistivity below the SDW ordering temperature even though the carrier density is strongly reduced), [@mcguire] as well as from spectroscopy. [@mannella] The size of the effects observed implies that these fluctuations should have large amplitudes and therefore unlike the SDW should be rather diffuse in ${\bf q}$-space, which might make them hard to directly observe. [@sf-note] Furthermore, the strong transport signatures especially the enhanced resistivity above the ordering temperature imply substantial coupling between spin flucuations and electrons at the Fermi energy, which is indicative of the itinerant nature of the magnetism. We emphasize also that our results in Table \[tab-struct\] are at the LDA relaxed atomic positions for the non-spin-polarized states. These systems become more magnetic as the chalcogen height is raised. In the case of FeS a recent refinement is available, and gives $z_{\rm S}$=0.2602. [@lennie] This puts the S ions 0.18 Å  higher than in the LDA structure. With this value we find a stable SDW for FeS, with a moment of 1.2 $\mu_B$/Fe. This is the same trend as in the Fe-As based superconducting materials. The result may be taken as an indication that in fact FeS may have a spin density wave as well and at least that there will be strong spin fluctuations in FeS as well as the other PbO-structure Fe chalcogenides. As mentioned, cylindrical Fermi surface sections of equal volume will be nested with nesting vector equal to the separation of the centers of these cylinders. This nesting can be reduced by imperfect matches in shape, three dimensionality, and size mismatch. Size mismatch can arise both from additional Fermi surface sections, such as the extra small hole sections obtained in FeS and FeTe, but not FeSe, or from doping. In particular, electron doping will reduce the size of the hole sections and increase the size of the electron sections consistent with Luttinger’s theorem. The mechanism for superconductivity in the Fe-As based phases has not yet been established. Nonetheless, there are indications that magnetism is associated with superconductivity. These include the modest electron phonon couplings in the materials, the proximity to magnetism and the phase diagrams which show an association between the SDW and superconductivity. We discuss our results within a general spin fluctuation mediated framework. [@moriya; @monthoux; @mazin; @kuroki] In general an itinerant SDW instability arises from a divergence of the real part of the susceptibility $\chi({\bf q})$ at a specific wavevector ${\bf q}$. Superconducting pairing is also associated with the real part of $\chi({\bf q})$ through an integral over the Fermi surface. $\chi({\bf q})$ for ${\bf q}$ connecting different parts of the Fermi surface can contribute to pairing. As a result, when the Fermi surfaces are small and disconnected as in these Fe based materials, the fluctuations associated with nesting will provide substantial interband pairing between the electron and hole sections, but will not provide substantial intraband pairing. [@mazin; @kuroki] As the nesting is reduced, e.g. by doping, $\chi({\bf q})$ will spread out while the peak value will be reduced, consistent with the destruction of the SDW. For circular cylinders of radii differing by $\delta q$, $\chi({\bf q})$ will show a plateau of high $\chi({\bf q})$ around the nesting vector with width $2 \delta q$ (this will persist until the radii differ by a factor of two at which point the center of the plataeu will dip). This means that the two cylinders will still be connected by spin fluctuations associated with their now reduced nesting, and therefore even though the SDW will be suppressed by the reduction in the maximum value of $\chi$ the associated spin fluctuations can still provide superconducting pairing. We emphasize that in this general framework the same parts of the Fermi surface are affected by the SDW and by the superconducting pairing. Thus these two Fermi surface instabilities compete for the same Fermi surface and therefore that there should be at best little coexistence of these two orders. summary and conclusions ======================= We report electronic structures, magnetic properties and electron-phonon calculations for Fe$X$, $X$=S,Se,Te. We find strong similarities to the Fe-As based superconductors, reflecting the ionic nature of the As and chalcogen atoms in these compounds. As in the arsenides, we find that the electron-phonon coupling cannot explain the superconductivity, and furthermore that these compounds display itinerant magnetism. These results imply a similar superconducting nature for the Fe-As phases and FeSe. The trend that we find in going from FeSe to FeTe is interesting in this context. In particular we find quite cylindrical Fermi surfaces and an SDW instability in both compounds. However, the strength of the SDW is substantially higher in FeTe as is the size of the Fermi surface. Within the general framework discussed above, FeTe would be expected to have stronger pairing, and therefore higher $T_c$ than FeSe assuming that the same mechanism applies in both materials, that both materials can be chemically doped to the optimum carrier density and that competing instabilities do not prevent superconductivity in that case. It will be of interest to experimentally probe the similarities of FeSe with those of the Fe-As phases and to search for superconductivity in doped FeTe and in the alloy Fe(Se,Te). We are grateful for helpful discussions with I.I. Mazin, D.G. Mandrus and B.C. Sales. This work was supported by the Department of Energy, Division of Materials Sciences and Engineering. note added ========== Mizuguchi and co-workers [@mizuguchi] recently reported observation of superconductivity at 27K in FeSe under pressure. This is consistent with the conclusion here regarding the relationship between FeSe and the Fe-As based materials. F.C. Hsu, J.Y. Luo, K.W. Yeh, T.K. Chen, T.W. Huang, P.M. Wu, Y.C. Lee, Y.L. Huang, Y.Y. Chu, D.C. Yan, and M.K. Wu, arXiv:0807.2369 (2008). Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am. Chem. Soc. [**130**]{}, 3296 (2008). Z.A. Ren, W. Lu, J. Yang, W. Yi, X.L. Shen, Z.C. Li, G.C. Che, X.L. Dong, L.L. Sun, F. Zhou, and Z.X. Zhao, Chin. Phys. Lett. [**25**]{}, 2215 (2008). Z.A. Ren, G.C. Che, X.L. Dong, J. Yang, W. Lu, W. Yi, X.L. Shen, Z.C. Li, L.L. Sun, F. Zhou, and Z.X. Zhao, Europhys. Lett. [**83**]{}, 17002 (2008). C. Wang, L. Li, S. Chi, Z. Zhu, Z. Ren, Y. Li, Y. Wang, X. Lin, Y. Luo, S. Jiang, X. Xu, G. Cao, and Z. Xu, arXiv:0804.4290 (2008). A.S. Sefat, M.A. McGuire, B.C. Sales, R. Jin, J.Y. Howe, and D. Mandrus, Phys. Rev. B [**77**]{}, 174503 (2008). H. Kito, H. Eisaki, and A. Iyo, J. Phys. Soc. Japan [**77**]{}, 063707 (2008). M. Rotter, M. Tegel, D. Johrendt, I. Schellenberg, W. Hermes, and R. Pottgen, arXiv:0805.4021 (2008). M. Rotter, M. Tegel, and D. Johrendt, arXiv:0805.4630 (2008). X.C. Wang, Q.Q. Liu, Y.X. Lu, W.B. Gao, L.X. Yang, R.C. Yu, F.Y. Li, and C.Q. Jin, arXiv:0806.1687 (2008). T. Watanabe, H. Yanagi, T. Kamiya, Y. Kamihara, H. Hiramatsu, M. Hirano, and H. Hosono, Inorg. Chem. [**46**]{}, 7719 (2007). M. Tegel, D. Bichler, and D. Jorendt, Solid State Sciences [**10**]{}, 193 (2008). A. Subedi, D.J. Singh, and M.H. Du, arXiv:0806.3785 (2008). L. Boeri, O.V. Dolgov, and A.A. Golubov, Phys. Rev. Lett. [**101**]{}, 026403 (2008). I.I. Mazin, D.J. Singh, M.D. Johannes, and M.H. Du, Phys. Rev. Lett. [**101**]{}, 057003 (2008). D.J. Singh and M.H. Du, Phys. Rev. Lett. [**100**]{}, 237003 (2008). C. de la Cruz, Q. Huang, J.W. Lynn, J. Li, W. Ratcliff II, J.L. Zaretsky, H.A. Mook, G.F. Chen, J.L. Luo, N.L. Wang, and P. Dai, Nature [**443**]{}, 899 (2008). D.J. Singh and L. Nordstrom, [*Planewaves, Pseudopotentials and the LAPW Method, 2nd. Ed.*]{} (Springer, Berlin, 2006). D. Singh, Phys. Rev. B [**43**]{}, 6388 (1991). D.J. Singh, arXiv:0807.2643 (2008). A.R. Lennie, S.A.T. Redfern, P.F. Schofield, and D.J. Vaughan, Mineralogical Magazine [**59**]{}, 677 (1995). D.M. Finlayson, D. Greig, J.P. Llewellyn, and T. Smith, Proc. Phys. Soc. B [**69**]{}, 860 (1956). S. Baroni, A. dal Corso, S. de Gironcoli, P. Gianozzi, C. Cavazzoni, G. Ballabio, S. Scandolo, G. Chiarotti, P. Focher, A. Pasquarello, et al., http://www.quantum-espresso.org. Calculations of the phonon spectra and electron-phonon coupling were done using ultrasoft pseudpootentials for Fe, and norm conserving pseudopotentials for Se. We used a 50 Ry basis set cutoff and a high 500 Ry cutoff for the charge density expansion. This was needed in order to obtain convergence in the linear response calculations. J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. [**77**]{}, 3865 (1996). J. Dong, H.J. Zhang, G. Xu, Z. Li, G. Li, W.Z. Hu, D. Wu, G.F. Chen, X. Dai, J.L. Luo, Z. Fang, and N.L. Wang, Europhys. Lett. [**83**]{}, 27006 (2008). G.F. Chen, Z. Li, D. Wu, G. Li, W.Z. Hu, J. Dong, P. Zheng, J.L. Luo, and N.L. Wang, Phys. Rev. Lett. [**100**]{}, 247002 (2008). J. Zhao, Q. Huang, C. de la Cruz, S. Li, J.W. Lynn, Y. Chen, M.A. Green, G.F. Chen, G. Li, Z. Li, J.L. Luo, N.L. Wang, and P. Dai, arXiv:0806.2528 (2008). Q. Huang, Y. Qiu, W. Bao, J.W. Lynn, M.A. Green, Y.C. Gasparovic, T. Wu, G. Wu, and X.H. Chen, arXiv:0806.2776 (2008). A.I. Goldman, D.N. Argyriou, B. Ouladdiaf, T. Chatterji, A. Kreyssig, S. Nandi, N. Ni, S.L. Budko, P.C. Canfield, and R.J. McQueeney, arXiv:0807.1525 (2008). Y. Nakai, K. Ishida, Y. Kamihara, M. Hirano, and H. Hosono, J. Phys. Soc. Japan [**77**]{}, 073701 (2008). T. Yildirim, arXiv:0804.2252 (2008). S. Ishibashi, K. Terakura, and H. Hosono, J. Phys. Soc. Japan [**77**]{}, 053709 (2008). Z.P. Yin, S. Lebegue, M.J. Han, B. Neal, S.Y. Savrasov, and W.E. Pickett, arXiv:0804.3355 (2008). I.I. Mazin, M.D. Johannes, L. Boeri, K. Koepernik, and D.J. Singh, Phys. Rev. B (in press). V. Vildosola, L. Pourovskii, R. Arita, S. Biermann, and A. Georges, arXiv:0806.3285 (2008). T. Yildirim, arXiv:0807.3936 (2008). M.A. McGuire, A.D. Christianson, A.S. Sefat, B.C. Sales, M.D. Lumsden, R. Jin, E.A. Payzant, D. Mandrus, Y. Luan, V. Keppens, V. Varadarajan, J.W. Brill, R.P. Hermann, M.T. Sougrati, F. Grandjean, and G.J. Long, arXiv:0806.3878 (2008). F. Bondino, E. Magnano, M. Malvestuto, F. Parmigiani, M.A. McGuire, A.S. Sefat, B.C. Sales, R. Jin, D. Mandrus, E.W. Plummer, D.J. Singh and N. Mannella, arXiv:0807.3781 (2008). In a system with strong magnetic tendencies ordering will normally occur at high temperature unless there is a large ${\bf q}$ space for fluctuations, e.g. due to frustration, itinerant electron physics or competing exchange interactions. It is difficult to avoid ordering with large amplitude spin fluctuations if they are they are strongly peaked, e.g. at the SDW nesting vector, or are subject to a simple exchange model that favors a specific ordering pattern, such as a nearest neighbor antiferromagnetic interaction on the square lattice (with 3D interactions or magnetoelastic coupling as a present here). Also, in the case where there is large phase space for fluctuations, local magnetic correlations will be very short range. This does not mean however that the fluctuations will necessarily be fast or only at high energy and in fact near a quantum critical point there will be slow fluctuations that extend to low energy. One experimental signature of such fluctuations besides the magneto-structural effects noted, will be via transport data and in particular data that show nearness to a quantum critical point through non-Fermi liquid scalings. T. Moriya and K. Ueda, Rep. Prog. Phys. [**66**]{}, 1299 (2003). P. Monthoux and G.G. Lonzarich, Phys. Rev. B [**63**]{}, 054529 (2001). K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H. Kotani, and H. Aoki, arXiv:0803.3325 (2008). Y. Mizuguchi, F. Tomioka, S. Tsuda, T. Yamaguchi, and Y. Takano, arXiv:0807.4315 (2008).

--- author: - Alexandre Gabard title: | Ahlfors circle maps and total reality:\ from Riemann to Rohlin --- -15pt [*Key words.*]{} [Eigenvalues of the Laplacian, bordered Riemannian surface, conformal transplantation. ]{} Introduction {#sec1} ============ [*Preliminary Warning.*]{} \[13.11.12\]—Despite its exorbitant size, the actual mathematical content of the present text is very limited. It focuses primarily on the Ahlfors map. Neither does our work have the pretence of being the logical sum of all knowledge accumulated in the past, nor will it give an accurate picture of real developments taking shape contemporaneously. Our intention was rather more to delineate a reasonably clear-cut perception of the early branches of the theory as to understand objectively the basic truths making Ahlfors theorem possible. Failing systematically, our pretence converted to that of throwing enough obscurantism on the whole theory as to motivate others to shed fresh lights over the edifice. Even the primary contribution to the field (that of Ahlfors 1950 [@Ahlfors_1950]) has not yet been fully assimilated by the writer (compare optionally Section \[Ahlfors-proof:sec\] for our fragmentary comprehension). We strongly encourage mathematicians having a complete mental picture of Ahlfors proof to publish yet another account helping to clarify the original one. We hope during the next months (or years) to be gradually able to improve the overall organization of this text, in case our understanding of classical results sharpens. All of our ramblings starts essentially in the big-bang of Riemann’s Thesis. It looks almost a triviality alike to expect that subsequent developments will involve a deeper interpenetration between the conformal and algebro-geometric viewpoints. One oft encounters in the field problems requiring serious combinatorial skills or geometric intuition. For instance how does the moduli space of bordered surfaces stratifies along gonalities; Sec.\[sec:profile-histogram\] guesses some scenarios via primitive methods. Riemann surfaces or the allied projective realizations offer an ornithological paradise requiring patience and observational skills from the investigator. This is especially stringent when the complexes are traded against the real number field, and inside this universe of $3g-3$ real dimensions one encounters with probability $1/3$ the so-called real orthosymmetric curves of Felix Klein (1876–1882) subsumed to the paradigm of [*total reality*]{}. This little third is actually all what our topic of the Ahlfors map is about. Last but not least, experimental studies point to a large armada of potential counter-examples menacing the improved bound $r+p$ announced in Gabard 2006 [@Gabard_2006]. It seems safe to declare as an open problem to either corroborate this bound (via other more analytic or algebraic treatments) or to reject it. \[19.03.13\] The main addition to the present edition (v.2) of our text, is an essay to connect Ahlfors’ theory with Hilbert’s 16th problem (at least the part thereof pertaining to the topology of real plane algebraic curves). Again our dancing queen is the paradigm of total reality (Riemann, Schottky, Klein, Bieberbach, Teichmüller, Ahlfors, Alling-Greenleaf, Geyer-Martens, etc.), but now reoriented as a missile against Hilbert’s 16th (quite akin to an asteroid menacing peaceful life on planet Earth). This trend is not new having been much foreseen in Rohlin’s seminal work 1978 [@Rohlin_1978] effecting a Verschmelzung[^1] between the conceptions of Klein and Hilbert (when it comes to real geometry). Rohlin never refers back to Ahlfors’ work (which he probably ignored?), yet the connection is very vivid through Rohlin’s (conjectural) philosophy that [*schemes*]{} of type I (not in Grothendieck’s highbrow sense, but merely Rohlin’s synonym for a [*distribution of ovals*]{} à la Zeuthen-Harnack-Hilbert) are necessarily [*maximal*]{} in the hierarchy of all schemes of some fixed degree. Through the lines of Rohlin’s text transpires the intuition that what detects (pure) orthosymmetry of schemes is a vertiginous phenomenon of [*total reality*]{} positing existence of adjoint pencils cutting only real points on the given curve. The byproduct is that total reality should act prohibitively upon all schemes enlarging those totally flashed by a pencil, which are so-to-speak already Bézout-saturated. Hence total reality should contribute to Hilbert’s problem (isotopic classification of curves), though this method can hardly be said to have been systematically exploited as yet (apart of course in the prophetical allusions in Rohlin 1978). This grand vision of Rohlin could benefit from the Klein-Ahlfors theory (which in our opinion has been much neglected in the tradition of the German-US-Italian-Russian school of real geometry involving such [*pointures*]{} as Hilbert 1891, Rohn 1888–1913, Ragsdale 1906, Brusotti 1910–50, Petrovskii 1933/38, Gudkov 1954–69, Arnold 1971, Rohlin 1972–78, Kharlamov, Viro, Marin, Fiedler, Shustin, Itenberg, etc.). Alternatively it could be reassessed through purely synthetical procedures that Rohlin himself envisioned (probably as consequences of deep 4D-topology, notably the type-I-forcing Kharlamov-Marin congruence modulo 8). Alas Rohlin’s synthetical proof even on the simplest prototype of sextics has never been published (and seems now to be lost forever), but was recently (partially) resuscitated in a tour de force of Séverine Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]. So the epoch seems ripe to dream about a big Ahlfors-Rohlin Verschmelzung with direct repercussions upon Hilbert’s 16th byway of prohibitions. It is already breathtaking (for a beginner) to contemplate how the Rohlin-Le Touzé total reality phenomenon for sextics explains nearly all prohibitions observed in Gudkov’s census (1969) of sextics curves (cf. Fig.\[Gudkov-TableTop:fig\]), which after all supplies (nothing less than) the [*complete*]{} solution to Hilbert’s problem in its original formulation (degree $m=6$). Those aspects are addressed in Sec.\[Klein-Rohlin-conj:sec\]&ff. For convenience we wrote a general overview in Sec.\[Hilbert’s16th-PartII:sec\] pointing to some open questions which looks semi-urgent to settle in order to build a more solid theory. All this second part, devoted to Hilbert’s 16th, could not have been written without the constant support and information generously shared by the leading experts (Viro, Marin, Kharlamov, Shustin, Orevkov, Le Touzé, Fiedler), whose instructive e-mails are reproduced in Sec.\[e-mail-Viro:sec\]. Needless to say we have not yet assimilated all their wisdoms and advices, but have reproduced faithfully their messages in the hope that other amateurs of the field can also beneficially profit from their invaluable expertise. \[11.04.13\] It seems (now) a firm conviction that a big piece of Hilbert’s 16th puzzle still remains to be fixed. This should be a fairly simple matter of assembly between the conceptions of Riemann-Schottky-Klein-Bieberbach-Teichmüller-Ahlfors about total reality and the theory of Hilbert-Rohn-Petrovskii-Gudkov-Arnold-Rohlin aiming to predict the distribution of ovals traced by algebraic curves (a God-given video game). Precisely, the isotopic classification of real plane curves should be regulated by a sole paradigm (total reality) itself piloted by the geometry of the canonical series (adjoint curves of order $m-3$) assigned to visit $(M-3)$ basepoints randomly selected among the most profound ovals of $(M-2)$-curves (alias the [*extended Rohlin-Le Touzé phenomenon*]{}, which in degree 4 boils down to the total reality of the Gürtelkurve quartic with 2 nested ovals). This looks special but wait a moment. It is conjectured (\[primitive-manifestation-of-tot-real:conj\]) that [*any*]{} [*primitive*]{} manifestation of the phenomenon of total reality on a plane curve is of this sort (i.e. a Rohlin-Le Touzé “adjunction” for $(M-2)$-curves subsumed to the eightfold periodicity $\chi\equiv_8 k^2+4$ of Rohlin-Kharlamov-Marin), except when it comes to the trivial case of $M$-curves, where total reality is nearly completely settled by an extension of (another) Le Touzé’s scholium (\[Le-Touzé-extended-in-odd-degree:scholium\]). The maximality of $M$-schemes being so trivial (Harnack-Klein inequality of 1876), this latter case looks a sterile syllogism not worth paying attention at, but this is probably not so via satellites. If [*not primitive*]{}, this is to say that the scheme is just derived as a [*satellite*]{} replicating the curve up to a certain multiplicity within its tube neighborhood. For instance satellites of a single oval of degree 2 reproduce the infinite series of deep-nests total under a pencil of lines \[by the way highly reminiscent to the “rondelles” of a certain artist known as Markus Schneider-Zeitler, Jura Suisse\]. Such deep-nests are Bézout-saturated hence extremal shapes in the Hilbert-Gudkov hierarchy. More generally, the magic formula reads $A+B=R m c^2$, i.e. Ahlfors plus Bézout implies Rohlin’s maximality conjecture (any scheme of [*type I*]{} kills all its enlargements). At this stage the architecture of higher Gudkov’s pyramids ($m\ge 7$) is completely predestined by (the felicity of) pure orthosymmetry à la Felix Klein, and (optionally) Rohlin’s theorem on the signature of spin $4$-manifolds governing the (Gudkov) $8$-fold periodicity via differential-topology. Paraphrasing, Hilbert’s 16th is virtually solved in [*all*]{} degrees, at least in its qualitative shape (prohibitions). It remains then of course to programme a (patchworking) machine doing all the constructions. This should be merely a matter of passive contemplation, requiring immortality and much patience from the investigator. It is evident that all this programme is not a novel idea, but much—not to say completely—anticipated by Rohlin 1978, safe that he does not seem to have been consciously aware of the Riemann-Ahlfors theory (nor perhaps the conjectural stability under satellites), but seemed rather adventurous enough to rediscover it [*ab ovo*]{} through purely synthetical processes, without any intrusion of analysis or transcendental gadgets, like Abelian integrals. Our messy text is just an invitation to inspect more exactly how the whole process will sediment itself within the next decades, probably via massive usage of Brill-Noether (i.e., Riemann for dummies). Then, the night of ignorance [*(les ténèbres de l’ignorance)*]{} allied to Hilbert’s combinatorial mess about the distribution of 16 ovals (recall that $M_7=11+5=16$) should be completely dissipated. Whether conversely, all this (ancient) geometry can acknowledge in feedback some impact upon 4D-differential-topology, e.g. the question of smooth structures on $S^4$ or $\CC P^2$ (so-called [*smooth Poincaré conjecture*]{}) is merely speculation of longstanding (reminding such names as Arnold, Maxwell, Milnor, Kuiper, Massey, Marin, Akbulut, Donaldson, Taubes, Finashin, Wang, Seiberg, Witten, etc.). But this is another story. \[22.03.13\] Lastly, we adopted (not deliberately but because we were not clever enough to proceed differently) Arnold’s philosophy of the mushroom (compare Arnold 2004 [@Arnold_2004]). That is, not just presenting overwhelming theorems (arid as they are) but the slow organical eclosion of truths through mistakes, conjectures, historical meanders, etc. The drawback is an intolerable inflation in size, also partly caused by the abundance of pictures, which in our opinion form the true core of any mathematical truth[^2]. We expect in the future to reorganize the material à la Bourbaki as to offer a cleaner view of what happens (after distillation, the factual content should be compressible to ca. 20 pages). Especially crucial is a rectification due to Fiedler of an erroneous theorem of mine that would have proved one-half of the (still open) Ragsdale conjecture for $M$-curves via the Thom conjecture (Sec.\[Thom:sec\]). $\star$$\star$$\star$ Works by Yamada 1978–2001, Gouma 1998 and Coppens 2011 suggest that fewer sheets than required in Ahlfors’ era is expectable for a clever placement of the basepoint(s) required to pose the extremal problem. E.g., is Coppens’ separating gonality (least degree of a circle map) always sustained by an Ahlfors function? How does the degree of the Ahlfors map fluctuates when the basepoint is dragged through the surface? A list of known applications is tabulated in the hope of guessing future ones. Some (e.g. Fraser-Schoen’s to Steklov eigenvalues) do not require the full swing of Ahlfors’ extremals, raising the hope that some improved control $\le r+p$ (Gabard 2004/06) on the degree of circle maps concretizing surfaces of genus $p$ with $r$ contours prompts some upgrades (e.g. in the corona problem with bounds of Hara-Nakai 1985). As to the foundation of the Ahlfors mapping theory itself, it is our partisan belief that much remains to be clarified both historically and logically. Albeit sembling a retrograde attitude, it is probably not since Ahlfors bound $r+2p$ certainly fails sharpness at least for low values of the invariants $(r,p)$. Apart from an abrupt claim by Teichmüller (1941) that everything (safe bounds) is to be found in Klein (what the writer was unable to certify from printed evidence), it is fair to admit that the bulk of the theory crystallized right after World War II. Several workers like Ahlfors 48/50, Matildi 45/48, Andreotti 50, Heins 50 (perhaps even Courant 39/40, not to mention Grunsky, one of the most brilliant protagonist albeit his work looks confined to the genus $0$ case) offered quite overlapping conclusions, but it seems fair to credit Ahlfors for having first expressed the story in the most clear-cut fashion. Quite shamefully, I confess that Ahlfors argument still escapes me slightly. A non-negligible amount of literature is devoted to reproving Ahlfors’ theorem: Heins 1950/75/85, Garabedian 1950, Kuramochi 1952, Read 1958 (student of Ahlfors), Mizumoto 1960 (topological methods), Royden 1962 (Hahn-Banach like Read), Forelli 1979 (extreme points and Poisson integral), Jenkins-Suita 1979 (Pick-Nevanlinna viewpoint), just to name those addressing the positive genus $p>0$ case. The actual tension between topological and analytic methods (made acute since the $r+p$ bound, granting its correctitude!), is possibly a temporary state of affairs destined to disappear after some examination of Ahlfors’ text (sharpening perhaps its ultimate convex geometry consideration). Another promising route is Meis’ work (1960) validating Riemann’s (semi)intuition of the $[\frac{g+3}{2}]$ gonality of closed genus $g$ surfaces via some Teichmüller-theoretic approach. It is likely that Meis’ approach to be transmutability to the bordered setting, reassessing thereby Ahlfors’ result (probably even with the sharp bound $r+p$). To put it briefly, it seems that the Grötzsch-Teichmüller mode of thinking has not yet fully penetrated the paradigm of the Ahlfors circle maps, more generally that of branched coverings. Dually, it also seems desirable to reprove the Riemann-Meis bound via topological methods (e.g. that used in Gabard 2006, which perhaps is nothing else than Riemann’s parallelogram method). Poincaré (1895) invented “homology” (modulo the Riemann-Betti heritage) with precisely function theory (Abelian functions) as one of the key motivation (beside celestial mechanics and the like). ${\it Warning.}$—This draft is a preliminary version, so avoid printing it for environmental reasons. It contains a list of hopefully clear-cut questions intended to encourage investigators. Several synoptic diagrams (authors names, keywords, etc.) should permit a quick optical scan of the whole content. The article contains no original insights, instead a series of failing attempts to contemplate the theory from different angles. A commented bibliography (of ca. 900 entries) tries to brush a photograph of old and contemporary trends including some ramifications (potential theory, minimal surfaces, spectral theory, analytic capacity, operator theory, Gromov’s filling conjecture, etc.). We have not attempted to reach any overwhelming mathematical density, but rather tried to dilute through historico-philosophical anecdotes, as well as spending some time trying to understand the available sources and their affiliation. Finally, we enjoyed speculating about some mechanical interpretation of the Klein-Ahlfors theory of real orthosymmetric curves (and the allied totally real maps) in terms of gravitational systems, positing a wild extension of Kepler’s planetary motions around ellipses. This is a prejudiced survey on the Ahlfors (extremal) function and (improvising terminology) the weaker [*circle maps*]{}, effecting the conformal representation upon the disc of an arbitrary differential-geometric membrane, alias [*compact bordered Riemann surface*]{}. Our jargon, borrowed from Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950], translates essentially the term [*Kreisabbildung*]{} used e.g., by Koebe 1915 [@Koebe_1915] and Bieberbach 1914 [@Bieberbach_1914]. Exciting works by Yamada 1978–2001 [@Yamada_1978], [@Yamada_2001], Gouma 1998 [@Gouma_1998] and Coppens 2011 [@Coppens_2011] suggest that fewer sheets than required in Ahlfors’ era is expectable, for a clever placement of the basepoint(s) required to pose the extremal problem. E.g., is Coppens’ (absolute) gonality of a membrane always sustained by an Ahlfors function? We also started tabulating a list of known applications in the hope of guessing future ones. Some applications (e.g. Fraser-Schoen’s recent one to Steklov eigenvalues [@Fraser-Schoen_2011]) do not require the full punch of Ahlfors’ extremals, raising the hope that the improved control $r+p$ on the degree of circle maps (predicted in Gabard’s Thesis 2004/06 [@Gabard_2006] for surfaces of genus $p$ with $r$ contours) could imply some ‘automatic’ upgrades (e.g. in the corona problem with bounds, as studied by Hara-Nakai 1985 [@Hara-Nakai_1985]). As to the foundation of the Ahlfors mapping theory itself, the issue that the naive qualitative approach (used in Gabard 2004/06 [@Gabard_2006]) affords a bound, $r+p$, quantitatively stronger than Ahlfors’ original $r+2p$ is somewhat surprising. It results a certain psychological tension between topological and analytical methods, which hopefully is just a superficial and temporary state of affairs destined to disappear after renewed examination of Ahlfors’ argument. The latter seems indeed to leave some free man[œ]{}uvring room, in its ultimate convex geometry portion (cf. Sec.\[Ahlfors-proof:sec\] for some strategy). It is our partisan belief that much remains to be clarified both historically and logically in the theory of the Ahlfors map. Albeit sembling a retrograde attitude, it is probably not since Ahlfors bound $r+2p$ certainly fails sharpness, at least for low values of the invariants $(r,p)$. (Consider for instance the topological type of Klein’s Gürtelkurve; i.e. $(r,p)=(2,1)$ where a projective realization (of the Schottky double) as a plane quartic with 2 nested ovals prompts existence of a total map of degree $3$ via projection from the inner oval. This beats by one unit Ahlfors’ bound $r+2p=4$.) Apart from an abrupt claim by Teichmüller 1941 [@Teichmueller_1941], that everything (safe bounds) is to be found in Klein (what the writer was unable to certify from printed evidence), it is fair to admit that the bulk of the theory crystallized right after World War II. Several workers like Ahlfors 1948/50 [@Ahlfors_1950], Matildi 1945/48 [@Matildi_1945/48], Andreotti 1950 [@Andreotti_1950], Heins 1950 [@Heins_1950] (perhaps even Courant 1939/40 [@Courant_1939], not to mention Grunsky 1937–40–41–42 [@Grunsky_1937], one of the most brilliant protagonist albeit his work looks confined to the genus $0$ case) offered quite overlapping conclusions. It seems fair however to give full credit to Ahlfors for having first expressed the story in the most clear-cut fashion. Quite shamefully, I confess that Ahlfors argument still escapes me slightly. A non-negligible amount of literature is devoted to reproving Ahlfors’ theorem: Heins 1950/75/85 [@Heins_1950] [@Heins_1975] [@Heins_1985-Extreme-normalized-LIKE-AHLF], Garabedian 1950 [@Garabedian_1950], Kuramochi 1952 [@Kuramochi_1952], Read 1958 [@Read_1958_Acta] (student of Ahlfors), Mizumoto 1960 [@Mizumoto_1960] (topological methods), Royden 1962 [@Royden_1962] (Hahn-Banach like Read), Forelli 1979 [@Forelli_1979] (extreme points and Poisson integral), Jenkins-Suita 1979 [@Jenkins-Suita_1979] (Pick-Nevanlinna viewpoint), just to name those authors addressing the positive genus case ($p>0$). Another promising route is Meis’ work 1960 [@Meis_1960] validating Riemann’s (semi)intuition of the $[\frac{g+3}{2}]$ gonality of closed genus $g$ surfaces via some Teichmüller-theoretic background. It is likely that Meis’ approach is transmutable to the bordered setting, reassessing thereby Ahlfors’ result (probably even with the sharp bound $r+p$ in case the latter is reliable). To put it briefly, it seems that the Grötzsch-Teichmüller mode-of-thinking (of the [*möglichst konform*]{} mapping) has not yet fully penetrated the paradigm of the Ahlfors circle map, more generally that of branched coverings, except of course in Meis’ memoir (alas notoriously difficult to access). Dually, it also seems desirable to reprove the Riemann-Meis bound via topological methods (e.g. that used in Gabard 2006 [@Gabard_2006], which perhaps is nothing else than Riemann’s parallelogram method). Poincaré’s “Analysis Situs” (1895 [@Poincare_1895-Analysis-Situs]) invented “homology” (modulo the Riemann-Betti=Brioschi \[sic!\] heritage) with precisely function theory (Abelian functions) as one of the key motivation (beside celestial mechanics and the like). This, jointly with the subsequent work of Brouwer, gives the basic conceptual framework for implementing such topological methods. [*User guide.*]{}—This draft is a preliminary version, so avoid printing it for environmental reasons. A list of hopefully clear-cut questions is given in Sec.\[sec:question\]. This is intended to challenge investigators. Several synoptic diagrams scattered as figures through the text should permit a quick optical scan of the whole content. More specifically, those includes: $\bullet$ an [*exhaustive*]{} list (Fig.\[Map:fig\]) of [*all*]{} articles supplying (or claiming to supply) a proof of Ahlfors theorem (existence of circle maps), $\bullet$ a list of keywords (Fig.\[Keyword:fig\]) tabulating concepts traditionally related to the Ahlfors map, $\bullet$ a comprehensive map (Fig.\[Geneal:fig\]) of authors involved in the theory (at least those cited in the bibliography). This essay, as already said, contains no original insights, instead a series of attempts to contemplate the theory from different angles. A commented bibliography (of ca. 900 entries) tries to brush a panorama of trends related to the Ahlfors map. This includes topics like Riemann surfaces, algebraic curves, conformal mapping, potential theory, Green’s functions, Dirichlet’s principle, Riemann mapping theorem, Kreisnormierung, parallel slit-maps, Bieberbach’s least-area map interpretation of the Riemann map, Bergman and Szegö kernels, minimal surfaces, Plateau’s problem, spectral theory, analytic capacity, removable singularities, corona problem, operator theory, Gromov’s filling conjecture, etc.). We have not attempted to reach any overwhelming mathematical density, but rather tried to dilute through historico-philosophical anecdotes. There is some interplay between Ahlfors maps and total reality of Klein’s orthosymmetric curves which gives rise to the gallery of pictures mentioned in the abstract. For a tourist view, browse the string of figures starting from Fig.\[Pencil:fig\] up to Fig.\[Fcubic3:fig\]. For “do-it-yourself” purposes, it is probably more valuable to describe the general recipe used to manufacture such pictures. Take any configuration of simple objects like lines and conics, and smooth it in an orientation-preserving sense to get a dividing curve (one is free to keep certain nodes unsmoothed). (Rohlin’s eminent student Thomas Fiedler (1981 [@Fiedler_1981]) ensures for us that the smoothed curve is dividing, alias orthosymmetric in the sense of Klein.) According to Ahlfors theorem there must be a totally real pencil of auxiliary curves cutting only real points on the given curve (plus maybe some imaginary conjugate basepoints). Geometric intuition usually tells us where to locate such a total pencil, roughly by assigning basepoints among the [*deepest*]{} ovals (in the sense of D. Hilbert’s 16th Problem). Albeit this is just a Plato cavern style extrinsic manifestation of Ahlfors theorem, the possibility of finding [*always*]{} such a total pencil reveals strikingly (in our opinion) some of the depth of Ahlfors theorem. (Incidentally it is not to be excluded that a deep understanding of extrinsic algebraic geometry (say à la Brill-Noether) could reprove the full Ahlfors theorem from within the Plato cavern.) In philosophical terms, [*real orthosymmetric curves behave on the reals as if they were complex varieties*]{}: all intersections prompted by Bézout are visible over the reals. This phenomenon is what we (and others, e.g. Geyer-Martens 1977 [@Geyer-Martens_1977]) call the paradigm of [*total reality*]{}. It seems evident that a global study of such pencils bears some close connection with Poincaré index theory, foliations à la Poincaré-Kneser-Ehresmann-Reeb, etc., and that both experimentally and theoretically much remains to be explored along the way. In particular we failed to make such totally real pictures for an $M$-quintic (Sec.\[sec:Total-reality-Harnack-max-case\]). This could be a challenging problem of computer visualization. As to our speculation about a mechanical interpretation of the Klein-Ahlfors theory of real orthosymmetric curves (and the allied totally real maps) in terms of gravitational systems, see Sec.\[sec:gravitation\]. This posits a broad extension of Kepler’s planetary motions around ellipses, enabling virtually all algebraic curves (and not just conic sections) to arise as the trajectories of a perfectly stable and periodic motion. Of course if such a grandiose connection between Klein-Ahlfors and Kepler-Newton-Coulomb-Poincaré is not verifiable, this may just be interpreted as a metaphoric language describing the dynamics of totally real morphisms prompted by Ahlfors theorem. In fact rather than mere gravitation, it is really a “[*dynamique de l’éléctron*]{}” which seems to be involved; for a toy example on the Gürtelkurve compare Fig.\[FGuert:fig\]. The resulting metaphysics is quite akin to Lord Kelvin’s speculations about the ultimate constitution (and stability) of matter (via the vortex atom geometrized by a knot), except that in our story the dancing queen is rather a (naked) bordered Riemann surface. Our partisan belief even in its most basic aspects the theory of the Ahlfors function (even when confined to the finite bordered realm) is far from being a closed (and transparent) chapter of geometry despite its sexagenary oldness, and the absence of significative achievements since Ahlfors 1950. Historical continuity invited us to rely massively upon primary sources, and this contributed to the prohibitive size of our output. Looking for logical alternatives, we oft rambled into dubious ‘historical revisionism’, wondering if other sources (maybe Klein—as cryptically suggested by Teichmüller 1941 [@Teichmueller_1941]—or perhaps Courant 1939/40 [@Courant_1939], [@Courant_1940-Acta]) do not anticipated the Ahlfors map (1948/50), at least at the qualitative level of circle maps. The Italian school (Cecioni and his students, including Matildi 1945/48 [@Matildi_1945/48] and the related work Andreotti 1950 [@Andreotti_1950]) should not be neglected (Ahlfors himself meticulously indexed those Italian references in the Ahlfors-Sario book (1960 [@Ahlfors-Sario_1960])). We warn again the present text is far from streamlined, so please avoid printing it for environmental reasons. It is rather intended to be an interactive browser, hopefully more structured than generic electronic databases, albeit the latter were of invaluable assistance to the present compilation. Human feedback is most welcome if some sloppy distortions look worth rectifying or the bibliography lacunary. Alas, we were soon confined to brush the large scale structure of the story without keeping close control over the logical details making the edifice. The field anyway is reputed for its overwhelming organical mode of growth, for Koebe proclaimed (im September 1921, Jena, DMV Jahresversammlung): “Es gibt viele Gebiete in der Mathematik, wo man sich durch Entdecken neuer Ergebnisse verdient machen kann. Es sind meistens lange und steile Gebirgshänge für meckernde Ziegen. Die Funktionentheorie ist aber mit einem saftigen Marschland zu vergleichen, besonders geeignet für dickes Rindvieh!” As usual, the paper is probably best read from the ending references, if one is patient enough. Else several synoptic diagrams (scattered through the text giving authors names and keywords) should permit a 5 minutes glimpse of the whole text. As the writer has zero analytic capacitance (=competence) this text should best be seen as a removable singularity in the realm of the literature devoted to the Ahlfors function, except possibly for our attempt to compile an up-to-date bibliography. Trying to wet some appetite out of the blue ------------------------------------------- Our aim is to discuss the following Riemann surface, giving an extremally biased view of some conformal mapping theory centering the discussion around the Ahlfors function. A long time ago (ICM 1908 Rome), Poincaré argued that in mathematics we need a strong principle of economy of thoughts by conceptualizing such notions as ‘uniform convergence’ as if the sole naming process would spare us repeating long intricate arguments. On the other hand, Felix Klein, asserted boldly “die Franzosen unhistorisch wie Sie sind” (exercise recover the source) and liked the motto “Zurück zur Natur, sie bleibt die grö[ß]{}te Lehrmeisterin”. Beside all those psychological tips of the masters of geometrization, we can safely agree with both of them that science requires—as a matter of conciliating the principle of economy with that of historical continuity (of course not so structurally incompatible as neo-expressionism seems to assess) —a certain amount of respectfulness about wisdoms accumulated during the past. This explains our ca. 900 references (albeit the explosion was mainly caused by my lack of internet connection occasioning a manual references chasing). In these notes we propose a (poorly guided) tour of some geometric function theory (GFT). The field is an old fashioned one, lying quite dormant with its old mysteries and legends (e.g., Koebe’s Kreisnormierungsprinzip, the exact determination of the Bloch constant in quasi-stagnation since Ahlfors-Grunsky 1937 [@Ahlfors-Grunsky_1937], etc.). Function-theory seems a volcano alike awaiting anxiously the next explosive eruption, whose pyroclastic rejections turned out to act (in the past at least) as a powerful fertilizer over neighbouring areas (like Riemannian and algebraic geometry, spectral theory, etc.). Actually Koebe had a more picturesque description, when proclaiming (im September 1921, Jena, Jahresversammlung der DMV[^3]): “Es gibt viele Gebiete in der Mathematik, wo man sich durch Entdecken neuer Ergebnisse verdient machen kann. Es sind meistens lange und steile Gebirgshänge für meckernde Ziegen. Die Funktionentheorie ist aber mit einem saftigen Marschland zu vergleichen, besonders geeignet für dickes Rindvieh!” The field itself (GFT) seems to be a strange cocktail of qualitative-flexible versus quantitative tricks, or as Gauss puts it [*geometria situs*]{} versus [*geometria magnitudinis*]{}. If topological methods look a priori quite foreign to the discipline, it was probably Riemann who first revealed: $\bullet$ the reactivity of the underlying topological substratum (anticipated maybe by Abel 1826 [@Abel_1826], who first introduced the [*genus*]{} (under a different name and the transcendant disguise of differentials of the first kind). \[The word [*Geschlecht*]{} is first coined in Clebsch 1865 [@Clebsch_1865 p.43]; and the allied [*Geschlechtsverkehr*]{}[^4] must have originated about the same period\] $\bullet$ the amazing plasticity (inherited from potential-theoretic considerations) of 2D-conformal mappings, leaving out moduli spaces of finite dimensionality after conformal evaporation of all metrical incarnation of a given surface. \[Gromov wrote in 1999 [@Gromov_1999]: [*Shall we ever reach spaces beyond Riemann’s imagination?*]{}\] Our text will soon be biased toward a single obsession, the so-called [*Ahlfors function*]{}, which is one (among several other possible) generalisation of the [*Riemann mapping theorem*]{} (RMT) to configurations of higher topological structure than the disc. Such configurations (compact bordered surfaces) are topologically determined by the number $r$ of boundary contours and the genus $p$ (number of handles) (see Fig.1a), as is well-known since the days of Möbius 1860/63 [@Moebius_1863] and Jordan 1866 [@Jordan_1866] (and of course very implicit in all of Riemann’s work). -35pt 0 -5pt0 The possibility of mapping any bordered surface to the disc conformally was pioneered by such towering figures as: $\bullet$ Riemann 1857/76 [@Riemann_1857_Nachlass] (manuscript not published during his lifetime), in which circular domains (hence $p=0$) of finite connectivity are mapped upon the disc. This fragment was edited by H. Weber and appeared in print only in 1876 in the first edition of Riemann’s Werke. The date of 1857 follows some oral tradition (Schwarz–Schottky), compare Bieberbach 1925 (Quote \[quote:Bieberbach-1925\] below), but conflicts slightly with Summer 1858 as estimated by Klein (cf. Quote \[Klein-1923:quote:Riemann-1858\]). \[11.08.12\] To pinpoint more about the exact date, should we recall that Riemann himself reports in the introduction of “Theorie der Abel’schen Functionen” 1857 [@Riemann_1857 p.116] his involvement with the topic of conformal mapping of multi-connected “surfaces” (Flächen) right after his Thesis (Fall 1851–Begin 1852), but was then sidetracked to another subject ([*ward aber dann durch einen andern Gegenstand von dieser Untersuchung abgezogen*]{}). $\bullet$ Schottky 1875–77 [@Schottky_1877] (=Dissertation under Weierstrass, Berlin, 1875), where a similar mapping is obtained for general real analytic contours. At first sight, it is natural to speculate that Schottky knew about Riemann’s Nachlass, but Schottky himself describes his trajectory as independent (cf. Quote \[quote:Schottky-1882\]). Apparently, it was Weierstrass’ special pupil, namely H.A. Schwarz who made Schottky aware of this connection, as reported in Bieberbach 1925 [@Bieberbach_1925], compare Quote \[quote:Bieberbach-1925\]. Albeit independent of Riemann’s, Schottky’s work was likewise physically motivated as emphasized by Klein 1923 [@Klein-Werke-III_1923 p.579]=Quote \[Klein-1923:quote:Riemann-1858\] below, or via Schottky’s own recollections (1882)=Quote \[quote:Schottky-1882\]. $\bullet$ Bieberbach 1925 [@Bieberbach_1925], found some elementary arguments (or just modernization) of the same Riemann–Schottky result, while emphasizing the trivial fact that the degree bound is optimum (apparently Schottky gave no bound), $\bullet$ Grunsky 1937–41 [@Grunsky_1937; @Grunsky_1941_KA], 1940–42–49 [@Grunsky_1940; @Grunsky_1942; @Grunsky_1950], who in a first series of papers rederived Bieberbach’s result and then switched to an extremal interpretation of the mapping problem. This terrible quantitative/competitive weapon (with historical precedents to be soon discussed) culminated, finally, in: $\bullet$ Ahlfors 1947 [@Ahlfors_1947], but it remained until Ahlfors 1950 [@Ahlfors_1950], to prove a generalization capable of including positive genera ($p>0$), superseding thereby quite dramatically the planarity (Schlichtartigkeit) where all previous efforts were perpetuated. (We shall attempt to ponder this absolute originality of Ahlfors, by comparing with others writers (e.g., Courant), but only with limited success due to my moderate competence with minimal surfaces and Plateau.) $\bullet$ Subsequent ramifications in the West (corona, operator theory, etc.), in Russia with Golusin, and Havinson (domains of infinite connectivity), and in Japan. For an overall picture of the roots plus some ramifications of Ahlfors, the reader may glance at the following map (Fig.\[Map2:fig\]) showing some of the links we are going to explore in this survey. We have opted for a Riemann surface style depiction of this histogram so as to give a quick-view of the varied [*troncs vivaces*]{} (in A. Denjoy’s prose when alluding to history of mathematics). Such trunks or handles are attached whenever some philosophical dependence (citation) is detected. Alas, it resulted a prolix accumulation of links creating a somewhat chaotical picture. For sharper pictures of the “Riemann galaxy”, we recommend Neuenschwander 1981 [@Neuenschwander_1981], Gray 1994 [@Gray_1994] and Remmert 1998 [@Remmert_1998]. -0pt0 -5pt0 -5pt0 [—The own contribution of the writer (Gabard 2006 [@Gabard_2006]) predicting an improved control $r+p$ upon Ahlfors’ degree $r+2p$ is enormously exaggerated, especially if it turns out to be false. Other distortions only reflects the writer’s poor understanding of this tentacular topic. For a more extensive compilation of authors involved in the theory, cf. Fig.\[Geneal:fig\]. If you are not cited on it, please send me an e-mail. ]{} As already said, our central hero will be Ahlfors, especially his paper of 1950 [@Ahlfors_1950]. In retrospect, it is not quite impossible that Riemann himself (or disciples like Schwarz, Schottky, Klein, Hurwitz, Koebe, Hilbert, Grötzsch, Teichmüller, etc., or also Bieberbach, Grunsky, Wirtinger, Courant, while not forgetting in Italy, Cecioni, Matildi 1945/48 [@Matildi_1945/48], Andreotti 1950 [@Andreotti_1950]) could have succeeded in proving such a version. Such speculations look not purely science-fictional especially in view of Ahlfors’ elementary argument in [@Ahlfors_1950 pp.124–126], which involves primarily only classical tricks (no deep extremal problem), like annihilating all the periods to ensure single-valuedness of the conjugate potential, and basic potential functions arising from the Green-Gauss-Dirichlet era. All these tricks are standard since Riemann’s days (cf. e.g. Riemann 1857 [@Riemann_1857 p.122], “[*so bestimmen da[ß]{} die Periodicitätsmoduln sämmtlich $0$ werden.*]{}”). Remember also, despite sembling dubious historical revisionism, that Teichmüller 1941 [@Teichmueller_1941](=Quote \[quote:Teichmueller-1941\]) seems to have possessed a clear-cut conception of the result at least without precise bound, while ascribing the assertion even back to Klein. $\bullet$ Mizumoto’s topological argument in 1960 [@Mizumoto_1960]; $\bullet$ Gabard’s topological argument in 2006 [@Gabard_2006]. However it took ca. 91 years—say from Riemann’s 1857/58 Nachlass up to the 1948 Harvard lecture held on the topic by Ahlfors, cf. Nehari’s Quote \[Nehari-1950:quote\] of 1950—until somebody puts it on the paper and it turned out to be no less an authority than Lars Valerian Ahlfors[^5]. It is true that Ahlfors moved in considerably deeper waters by solving as well a certain [*extremal problem*]{}. This extremal viewpoint is more punchy, yet arguably the corresponding extremals (so-called Ahlfors functions) are only circle maps of a special character. We gain in punch but loose in flexibility. The extremal functions do not substitute to—nor are substituted by—circle maps. Deciding which viewpoint is more useful is another question, probably premature to answer except for guessing a complementary nature depending on the problem at hand. Incidentally in Ahlfors paper (1950 [@Ahlfors_1950]), existence of circle maps is required as a preliminary step toward posing (non-nihilistically) the extremal problem. Ahlfors’ extremal problem stemmed surely not out of the blue, but was patterned along a tradition, whose first steps should probably be located in the following works. (We acknowledge guidance by Remmert’s book 1991 [@Remmert_1991 p.160–2, p.170–2], to which we refer for sharper historical details.) $\bullet$ Koebe’s elementary proof 1907, 1909, 1912 [@Koebe_1912], 1915 [@Koebe_1915] of the (RMT); ([*Quadratwurzeloperationen*]{}, [*Schmiegungsverfahren*]{}, etc.) $\bullet$ Carathéodory 1912 [@Caratheodory_1912]: similar iterative methods and convergence of his sequence via Montel’s theorem. This revitalized Koebe’s interest (cf. again Remmert’s description [@Remmert_1991 p.160, p.172]); in Carathéodory 1914 [@Caratheodory_1914] full details of the method were given in the Schwarz-Festschrift; $\bullet$ Fejér and F. Riesz 1922 obtain the Riemann mapping via an extremal problem for the derivative (published in Radó 1922/23 [@Rado_1922-3]). Montel’s normal families are also used, plus a tedious derivative computation eradicated in: $\bullet$ Carathéodory 1928 [@Caratheodory_1928] and Ostrowski 1929 [@Ostrowski_1929], where (independently) ultimate simplifications are provided. Carathéodory wrote about these developments: \[quote:Caratheodory-1928\] Nachdem die Unzulänglichkeit des ursprünglichen [*Riemann*]{}schen Beweises erkannt worden war, bildeten für viele Jahrzehnte die wunderschönen, aber sehr umständlichen Beweismethoden, die [*H.A. Schwarz*]{} entwickelt hatte, den einzigen Zugang zu diesem Satze. Seit etwa zwanzig Jahren sind dann in schneller Folge eine gro[ß]{}e Reihe von neuen kürzeren und besseren Beweisen \[von ihm selbst und von Koebe (Remmert’s addition); in the original Lindelöf 1916 is also quoted\] vorgeschlagen worden; es war aber den ungarischen Mathematikern [*L. Fejér*]{} und [*F. Riesz*]{} vorbehalten, auf den Grundgedanken von [*Riemann*]{} zurückzukehren und die Lösung des Problems der konformen Abbildung wieder mit der Lösung eines Variationsproblems zu verbinden. Sie wählten aber nicht ein Variationsproblems, das, wie das [*Dirichlet*]{}sche Prinzip, au[ß]{}erordentlich schwer zu behandeln ist, sondern ein solches, von dem die Existenz einer Lösung feststeht. Auf diese Weise entstand ein Beweis, der nur wenige Zeilen lang ist, und der auch sofort in allen neueren Lehrbüchern aufgenommen worden ist. \[Footnote 2: Siehe [*L. Bieberbach*]{}, Lehrbuch der Funktionentheorie, Bd.2 S.5.\] Mein Zweck ist nun zu zeigen, da[ß]{} man durch eine geringe Modifikation in der Wahl des Variationsproblems den [*Fejér-Riesz*]{}chen Beweis noch wesentlich vereinfachen kann. Let us quote thrice Ahlfors in this connection (the second of which occurred while celebrating the centennial of Riemann’s Thesis, 1851): \[Ahlfors-1961\] In complex function theory, as in many other branches of analysis, one of the most powerful classical methods has been to formulate, solve, and analyze extremal problems. This remains the most valuable tool even today, and constitutes a direct link with the classical tradition. \[Ahlfors-1953\] Very important progress has also been made in the use of variational methods. I have frequently mentioned extremal problems in conformal mapping, and I believe their importance cannot be overestimated. It is evident that extremal mappings must be the cornerstone in any theory that tries to classify conformal mappings according to invariant properties. \[Ahlfors-1958\] Es ist mir zugefallen, eine Übersicht über die Extremalprobleme in der Funktionentheorie zu geben. Seit der Formulierung des Dirichletschen Prinzips ist es klar gewesen, dass die Cauchy-Riemannschen Gleichungen nichts anderes sind als die Eulerschen Gleichungen eines Variationsproblems, und in diesem Sinne ist alle Funktionentheorie mit Extremaleigenschaften verbunden. Aber es ist nicht immer von vornherein klar, wie diese Probleme gestellt werden sollen, damit sie in wesentlicher Weise die tiefen Eigenschaften der analytischen Funktionen abspiegeln. Es gibt natürlich unzählige Maximaleigenschaften, etwa in der konformen Abbildung, die ganz nahe an der Oberfläche liegen. Von da aus soll man zu schwierigeren Problemen aufsteigen. Das geschieht nicht etwa so, dass man ein beliebiges, wenn auch verlockendes, Extremalproblem ins Auge fasst und es zu lösen versucht. Im Gegenteil, die Entwicklung ist so vor sich gegangen, dass man die Aufgaben stellt, die man lösen kann. Dadurch ist ein reiches Erfahrungsmaterial entstanden, und die Aufgabe des heutigen Funktionentheoretikers besteht darin, dieses Material zu klassifizieren und dadurch weiter zu entwickeln. \[…, and on page 7, of the same philosophical paper\] Carathéodory sagte einmal, dass er immer wieder zur Funktionentheorie zurückkehrt, weil man gerade dort die verschiedensten und verblüffendsten Methoden verwenden kann. Das ist sicher wahr, und eben deshalb ist die Funktionentheorie kein eng spezialisierter Zweig der Mathematik. Im Gegenteil, die Funktionentheorie scheint fast wie ein Miniaturbild der gesamten Mathematik, denn es gibt kaum eine Methode in der Geometrie, der Algebra und der Topologie, die nicht früher oder später in der Funktionentheorie wichtige Anwendung findet. \[…\] Such wisdoms cultivating the extremal philosophy—in particular as a growing mode for conformal mappings—presumably capture the deepest telluric part of the mushroom, out of which everything derives effortlessly. Alas, our survey is far from this ideal conception. In fact, we would be quite challenged if we were demanded to list a single application of Ahlfors’ extremal property, except of course in the planar case where one can easily mention all the activities centering around Painlevé’s problem. Applications ------------ The writer’s interest in the topic was recently revived by the article of Fraser-Schoen 2011 [@Fraser-Schoen_2011], where the Ahlfors function received a clear-cut interaction with spectral theory (Steklov eigenvalue) with a view toward minimal surfaces. At a more remote period of time, in the early 1950’s, when classification theory of open Riemann surfaces was a hot topic (especially in the Finnish and Japanese schools), Kusunoki 1952 [@Kusunoki_1952] proposed an application to the type problem, in the analytic sense of Nevanlinna’s Nullrand (null boundary). A (somewhat misleading but frequently used) synonym is [*parabolic type*]{} (not to be confused with the geometric sense of uniformization theory). This (analytic) sense of parabolicity is the one related to the transience of the Brownian motion (Kakutani, etc.) In view of the extremal rôle played by the (round) hemisphere as a vibrating membranes (compare Hersch 1970 [@Hersch_1970], and less relevantly Gabard 2011 [@Gabard_2011]), the author speculatively expected—yet failed dramatically to establish (Summer 2011)—the following: [(Gabard, April 2011, ca. 300 pages of sterile hand-written notes, unpublished)]{} There is a mysterious connection between the Ahlfors function and the (still open) [*filling area conjecture*]{} (FAC) of [Gromov 1983 [@Gromov_1983]]{}, whose genus zero case follows from the Thesis of [Pu 1952 [@Pu_1952]]{}, under Loewner 1949. More precisely, the filling area conjecture is true for all genus $p\ge 0$, and the proof will employ an Ahlfors map, at least as one of the ingredients \[others being Schwarz’s inequality, and group theoretical tricks à la Hurwitz–Haar–Loewner like in the $p=0$ case\]. The basic link is of course that conformal maps supply isothermic coordinates, yielding a way to compute areas via the infinitesimal calculus (of Newton–Leibniz, etc.). The best available result on FAC is still the hyperelliptic case handled by Bangert-Croke-Ivanov-Katz 2004 [@Bangert_2004], implying the full conjecture for $p=1$ (as in this case the double is of genus $g=2$, hence automatically hyperelliptic). Remember the formulation of the FAC problem: among all compact bordered (orientable?) Riemannian surfaces bounding the circle without shortening its intrinsic distance, the round hemisphere has the least possible area. The above “Ahlfors$\Rightarrow$Gromov” conjecture flashed my attention, after completing the note (Gabard 2011 [@Gabard_2011]) in view of the striking analogy between the isoperimetric rôle of the hemisphere both acoustically (spectral theory, like in Hersch 1970 [@Hersch_1970]) and geometrically in the Löwner-Pu-Gromov isosystolic ($\approx$filling) problem. Of course this analogy is already explicit in Gromov 1983 [@Gromov_1983], where Hersch 1970 () is cited. Incidentally, Gromov’s account also let play to Jenkins, Ahlfors’ student and Grötzsch’s admirator, a predominant logical rôle via the notion of “extremal length”. After more immature thinking (August 2012), it seems safer to formulate a relaxed version of the conjecture where the impulse does not necessarily come from the Ahlfors map but from some more ancestral source like the Green’s function (or the allied Gauss-Riemann isothermic coordinates). Also the (Lorenz-)Weyl’s asymptotic law enabling to “hear” the area of a drum from high-vibratory modes could be involved as well in FAC. When Marcel Berger describes Gromov’s systolic exploits (1983 ), he insinuates (surely with right) of them as lying at a much higher level of sophistication than 2D-conformal geometry (à la Gauss-Riemann, etc.). This acts as an optimism killer against anything like the above conjecture. Of course our conjecture or its relaxed variant “Conformal$\approx$Isothermic$\Rightarrow$Gromov” is far from prophetical, but only the expectation that the traditional methods (conformal theory and uniformization) which settled low-genus cases (Loewner 1949, Pu 1952 [@Pu_1952]) will extend soon or later to $p\ge 2$. Yet, who knows? Remember that even Marcel Berger, once validated (or at least quoted) an erroneous proof (ca. 1998) of the 2D-case of the filling conjecture in question. Compare his brilliant “Panoramic view” (2002 [@Berger_2002-A-Panoramic-view-of-RG]), or rather his likewise excellent survey in JDMV (1998 [@Berger_1998-JDMV p.147]): “[*The simplest filling volume, namely that for the circle $S^1$, was only obtained in (\[N.\] Katz, 1998).*]{}”, where the reference is given as (cf. p.196) “[*Katz, N. (1998). Filling volume of the circle.*]{}” This work has apparently never been published and probably turned out to contain a gap. This reference is still quoted in the “Panoramic view” (2002 [@Berger_2002-A-Panoramic-view-of-RG p.790]) modulo a puzzling shift of authorship from Neil N. Katz to Mikhail G. Katz: “Entry \[794\]=[M.G. Katz]{}, Filling volume of the circle, to appear, 1998.” In the text of “A Panoramic view…” this reference is apparently not cited, and at any rate on p.367 we read “[*Today there is not a single manifold whose filling volume is known, not even the circle (for which Gromov conjectures the value \[is\] $2\pi$).*]{}” Of course, probably no better guide than Ahlfors himself for listing applications of his method would have been desired. Alas it seems that the latter was suddenly sidetracked in the stratosphere of Teichmüller theory in the early 1950’s, leaving the Ahlfors map topic in some standby “in absentia” status. An exception is the later paper Ahlfors 1958 [@Ahlfors_1958], where Ahlfors discusses again extremal problems, though in a more philosophical way. Also the work of his student Read 1958 [@Read_1958_Acta] is described, which supplies another existence-proof of circle maps via a more abstract viewpoint (Hahn-Banach) inspired by other works like Macintyre-Rogosinski 1950 [@Macintyre-Rogosinski_1950], Rogosinski-Shapiro 1953 [@Rogosinski-Shapiro_1953], Rudin, etc. This Teichmüller shift in Ahlfors activities seems to coincide with the 100 years celebration of Riemann’s Thesis (in 1951), where L. Bers cames up with his list of urgent questions about Riemann surfaces. As a partial consolation, Grunsky worked out a brilliant book (1978 [@Grunsky_1978]) where much of the historical continuity is supplied. [*Quoting some first-hand sources.*]{}— We shall have to reproduce several quotations from primary sources as an attempt to observe the mutual influences among the variety of viewpoints. It resulted some inflation in size, but hopefully excusable as the information of some relevance to our topic is otherwise dispatched through a vast amount of literature. Those are given in the self-explanatory format [**Quote (Author, year)**]{}. Beside the historical aspect, the ultimate desideratum would be a logically optimal reconstruction of the theory. In the case at hand, our first task was just to list, locate (and sometimes understand) the several arguments already available. —We shall essentially touch the following aspects (all in reference to the Ahlfors mapping): \(1) Origins, background: prehistory of Ahlfors (Sec.\[Sec:Prehistory-Ahlfors\]); potential precursors (Sec.\[Sec:Precusors\]); \(2) How the writer came across this topic? (via Klein); cf. Sections \[Sec:Klein\] and \[Sec:Biased-recollections-of-Gabard\]; \(3) Potential theory vs. extremal problems (both from the same variational soup); \(4) Applications (Sec.\[Sec:Applications-of-the-Ahlf-map\]): equilibrium of electricity Riemann 1857, Painlevé’s problem, type problem, Carathéodory metric, corona problem, quadrature domains, spectral theory (Steklov or Dirichlet-Neumann); \(5) Open problems fictionally related to the Ahlfors function (Sec.\[Sec:Virtual-applications-Ahlf-map\]); \(6) (Partial) assimilation of Ahlfors or other works (logical reconstruction); via Green in Sec.\[Green:sec\] and via Ahlfors in Sec.\[Ahlfors-proof:sec\]; \(7) Sharpening Ahlfors work (for circle maps not necessarily subjected to the extremal problem). Roughly speaking our text splits as follows. A first half is devoted to historical aspects, while a second half (initiated by Sec.\[Sec:Starting-from-zero\] titled “Starting from zero knowledge”) is more “logical”, or rather liberal and futurist. This second part tries to explore what sort of mathematics lies beyond Ahlfors theorem. Of course it is hard going beyond Ahlfors without having digested his own work, and consequently much energy is spent to the original account. His result affords considerable information, especially the realizability of all gonalities lying above Ahlfors bound $r+2p$. (The [*gonality*]{} $\gamma$ is the least degree of a circle map tolerated by the given bordered surface.) Classically, some (episodic) penetrations beyond Ahlfors occurred by Garabedian, Heins, Royden, etc., and more recently in the spectacular progresses made by Yamada, Gouma on the extremal function. In the dual direction (of circle maps), Coppens’ work on the gonality is likewise penetrating deep behind the line fixed by Ahlfors, and raises several questions of primary importance. This includes that of describing how the moduli space of bordered surfaces (with fixed topological type $(r,p)$) stratifies along gonalities. Calculating dimensions of the varied strata is a first step toward quantifying by how much and how frequently one can expect to improve Ahlfors bound. We obtain so the [*gonality profile*]{}, that is, the function assigning to each gonality $\gamma$ (in the Coppens range $r\le \gamma \le r+p$, or outside it in case Gabard is wrong) the dimension of the moduli strata with prescribed gonality $\le \gamma$ (Section \[sec:profile-histogram\]). Describing this gonality profile appears to me a challenging (but hopefully reasonably accessible) problem. Another “futurist” problem is the one of describing the list of all degrees of circle maps tolerated by a given surface. This we call the [*gonality sequence*]{}. It is full above Ahlfors bound $r+2p$, but what can be said below? These are perhaps two typical kind of problems hinting at what sort of games we may encounter “beyond Ahlfors”. It seems also evident that a good understanding of (real) algebraic geometry could shed some light upon Ahlfors viewpoint. This interaction is alas quite rare to observe in the twenty century literature which is mostly split into specialized viewpoints. Bibliographic and keywords chart -------------------------------- The following chart (Fig.\[Map:fig\]) focuses on the tabulation of several articles where an existence-proof of Ahlfors circle maps is given. Such items are marked by full black circular symbols with eventual decorations. Applications are marked by triangular symbols. All entries of the picture (e.g. “Ahlfors 1950”) can unambiguously be located in the bibliography at the end of the paper. One counts essentially ca. 13 papers addressing the existential question of circle maps. Those includes: Ahlfors 1950 [@Ahlfors_1950], Garabedian 1950 [@Garabedian_1950], Heins 1950 [@Heins_1950], 1975 [@Heins_1975], 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF] and in the same spirit Forelli 1979 [@Forelli_1979]. Another trend is Nehari 1950 [@Nehari_1950] and Tietz 1955 [@Tietz_1955] (alas those works are a bit confusing, Tietz criticizes Nehari and is in turn attacked subsequently in Köditz-Timmann 1975 [@Koeditz-Timmann_1975]). The latter work (KT1975) actually offers an alternative existence-proof without degree control. In Japan we have Kuramochi 1952 [@Kuramochi_1952] and Mizumoto 1960 [@Mizumoto_1960]. (One should probably add several works of Kusunoki from the early 1950’s, but those are often confusing with subsequent errata, etc.) Another mouvance is the usage of Hahn-Banach in the papers Read 1958 [@Read_1958_Acta] and Royden 1962 [@Royden_1962]. Finally there is a work by the writer, Gabard 2006 [@Gabard_2006], which even claim a better control $r+p$ upon the degree of circle maps. Of course this work should still be better understood and its result should be either disproved or consolidated by alternative techniques. To this obvious list one can add some more telluric flows or possible forerunners: $\bullet$ Teichmüller’s claim (1941 [@Teichmueller_1941]) that everything is already in Klein. $\bullet$ Courant’s works starting say with Courant 1939 [@Courant_1939] where a Plateau-style approach à la Douglas is asserted to reproduce the Bieberbach-Grunsky “schlichtartig” case of Ahlfors. $\bullet$ Italian workers: Matildi 1945/48 [@Matildi_1945/48] and Andreotti 1950 [@Andreotti_1950]. If one has a good memory or a glouton working mode it is rather impressive to see the high level of branching and complexity of several neighboring fields. It will be soon flagrant that the writer’s competence will quickly break down in front of our too ambitious survey project. Again the writer would be enormously acknowledgeable if some specialists wants to tell their own vision of this tentacular circle of ideas. -25pt0 -5pt0 -5pt0 —Let us now put Ahlfors 1950 [@Ahlfors_1950] at the center of the universe, while trying to describe the portion of the cosmos visible from this perspective. Picturing in the non-Euclidean crystal, we obtain something like the following chart of keywords (Fig.\[Keyword:fig\]): a nebulosity of sidereal dusts gravitating in the immediate conceptual vicinity of the Ahlfors map. -25pt0 -5pt0 -5pt0 Mathematical questions {#sec:question} ---------------------- In this section we collect questions raised by our text. Most of our questions are of the retrograde sort “Can we reprove Ahlfors via …”, yet striving toward a perfect crystallography, where each result of the theory is certified by all methods ever imagined (compare optionally the kaleidoscopic Fig.\[Kaleidoscope:fig\] much below). $\bullet$ [**Klein $\Rightarrow$ Ahlfors?**]{} \[reported 04.11.12\] Is it possible to reprove existence of Ahlfors circle maps via Klein’s Rückkehrschnitttheorem (RST) (cf. Klein 1882 [@Klein_1882_Ruckkehrschnitt] or Klein 1923 Ges. Math. Abh. III [@Klein-Werke-III_1923 p.622–626])? This paradigm RST may be conceived as a positive genus case of the Kreisnormierung (of Koebe, but implicit in the Latin version of Schottky’s Thesis, cf. Klein’s Quote \[Klein-1923:quote:Riemann-1858\]). Further recall that Riemann (1857 [@Riemann_1857_Nachlass]) was able to produce circle maps for domains bounded by circles, and by analogy it seems plausible that Klein’s RST implies (modulo some work à la Riemann) the Ahlfors circle map. Of course Klein himself may not have been able to prove rigorously his RST, but the result was completed via some Brouwer-Koebe techniques ca. 1911/12 [@Klein-Brouwer-Koebe_1912]. (For a few more details about this strategy, cf. Sec.\[sec:Ruckkehrschnittthm\].) \[18.11.12\] An allied historical question is whether Teichmüller’s accreditation to Klein (1941 [@Teichmueller_1941]) of circle maps is based on the same stratagem (RST) as we are just suggesting. $\bullet$ [**Witt or Geyer $\Rightarrow$ Ahlfors?**]{} Can we reprove the theorem of Ahlfors via a purely algebraic method (say Abel, or Riemann-Roch) as Witt 1934 [@Witt_1934], Geyer 1964/67 [@Geyer_1964-67] or Martens 1978 [@Martens_1978] succeeded to do for the Witt mapping (of 1934)? For more on this, cf. Sec.\[sec:Witt\]. $\bullet$ [**Plateau $\Rightarrow$ Ahlfors?**]{} Can we reprove the theorem of Ahlfors via the method based on the Plateau problem (as Courant 1939 [@Courant_1939] did for the Riemann-Schottky-Bieberbach-Grunsky theorem, i.e. the schlichtartig case $p=0$ of Ahlfors). (See Sections \[sec:Courant\] and \[sec:Douglas\] for historical precedents (i.e., Douglas 1931 [@Douglas_1931-Solution]), and precise references about contemporary workers attacking related questions (Jost, Hildebrandt, von der Mosel). A closely related historical question is whether the works of Courant do not already contain (more-or-less explicitly) an existence-proof of Ahlfors circle maps. $\bullet$ [**Bergman $\Rightarrow$ Ahlfors?**]{} Idem via the method of the Bergman kernel function. This seems implicit in the literature (say especially by Bell, e.g. Bell 2002 [@Bell_2002], the great specialist of the technique), but to the writer’s knowledge no pedestrian account is available to the mathematical public (in the positive genus case). Compare Sec.\[sec:Bergman\] for some links to the literature. Of course behind Bergman 1922 [@Bergman_1922] one finds Bieberbach’s characterization (1914 [@Bieberbach_1914]) of the Riemann map via an extremal problem involving least area. This problems should be in some duality with Ahlfors extremal problem, more about this soon. $\bullet$ [**Behnke-Stein $\Rightarrow$ Ahlfors?**]{} \[reported 05.11.12\] The article (of Köditz-Timmann 1975 [@Koeditz-Timmann_1975 Satz 3, p.159]) seems to contain a qualitative version of Ahlfors’ theorem based upon an “Approximationssatzes von Behnke u. Stein”, yet without any bound on the degree. Can one improve the argument to get a quantitative control? As to Behnke-Stein 1947/49 [@Behnke-Stein_1947/49] (the famous paper going back to 1943), it contains the result that any open Riemann surface (arbitrary connectivity and genus) admits a non-constant analytic function. Is it possible conversely to deduce this theorem from Ahlfors theorem by exhaustion while pasting together various circle maps defined over a system of expanding compact subregions? $\bullet$ [**Other techniques?**]{} Koebe’s iteration, circle packings (cf. Rodin-Sullivan 1987 [@Rodin-Sullivan_1987]), Ricci flow, etc. Virtually any technique involved in the proof of the RMT (=Riemann mapping theorem) or the allied uniformization is susceptible to reprove the Ahlfors circle map. $\bullet$ [**Does Ahlfors imply Ahlfors?**]{} \[02.09.12\] This repetition is intentional and intended to emphasize that the writer was not able to digest Ahlfors argument in full details (compare Sections \[Green:sec\] and \[Ahlfors-proof:sec\]). If one remembers the proof of Koebe’s Kreisnormierung (say as implemented in Grunsky 1978 [@Grunsky_1978] or Golusin 1952/57 [@Golusin_1952/57]), then upon making abstraction of Koebe’s proof by iterative methods, it may be noticed that ultimately the proof depends on a topological principle (namely Brouwer’s invariance of domain). In comparison, Ahlfors’ proof of a circle map (1950 [@Ahlfors_1950]) makes no use of any topological principle, reducing rather to considerations of convex geometry (cf. Ahlfors 1950 [@Ahlfors_1950]). Should one deduce that the Ahlfors function lies somewhat less deep than Koebe’s Kreisnormierung? If not then maybe Ahlfors’ argument lacks a global topological character, and perhaps its validity needs to be reevaluated. (Of course this is only a superficial objection arising from my own frustration in not being able to catch the substance of Ahlfors text.) $\bullet$ [**Does Brill-Noether ($+$ Harnack’s trick) imply Ahlfors?**]{} \[26.10.12\] Upon using projective models of Riemann surfaces, especially birational models in the plane, it is common practice to understand the geometry on a curve via auxiliary pencils living on the ambient plane. Of particular importance are the so-called adjoint series passing through the singularities of the model which have the distinctive feature of cutting economical series of points on the curve. Such pencils are thus involved in the description of low-degree pencils living on the (abstract) smooth curve, hence morphisms to the line. Adapting this methodology to orthosymmetric curves one can evidently hope to reprove Ahlfors theorem, provided one is able to ensure total reality of the corresponding morphism. Details look quite formidable to implement. If such a proof exists it will probably be a happy hour for its discoverer. For more vague ideas about this strategy, see Sec.\[sec:Brill-Noether-approach-to-Ahlfors\]. $\bullet$ [**Does Ahlfors imply Gabard?**]{} \[09.09.12\] Upon using Ahlfors’ original argument in [@Ahlfors_1950] for the existence of a circle map of degree $r+2p$, it seems evident that one could append to Ahlfors argument a sharper geometric lemma which could produce a better control than Ahlfors’. Ideally one would like to recover Gabard’s bound $r+p$. For some evidence of why this should be possible compare Sec.\[Red’s-function:sec\]. $\bullet$ [**Gabard true? If, yes analytifiable?**]{} \[June 2012\] Is the bound $r+p$ predicted by the writer on the degree of a circle map true? And if yes is it accessible to more conventional analytical methods? Remember that the derivation in Gabard use some topological methods combined with the classical Abel theorem. $\bullet$ [**Gonality profile.**]{} \[June 2012\] Can we compute the dimension of the moduli spaces of membranes having fixed gonality $\gamma\le r+p$. (The [*gonality*]{} is the least degree of a circle map from the given bordered surface.) The similar question in the case of complex curves is well-known and easily predicted by a simple Riemann-Hurwitz count (but established rigorously much later). Slightly more on this in Sec.\[sec:profile-histogram\]. $\bullet$ [**Ahlfors extremals as economic as Gabard?**]{} \[March 2012\] Can the degree of the Ahlfors extremal function be made as economical as $r+p$, the circle map degree predicted by the writer, for a suitable location of the two points required to pose the extremal problem (resp. of a single point when considering the derivative maximizing variant of the problem)? $\bullet$ [**Ahlfors extremals as super-economic as Coppens?**]{} \[March 2012\] Same question for the sharper [*(separating) gonality*]{} introduced by Coppens 2011 [@Coppens_2011], that is, the minimum sheet-number required to concretize the bordered Riemann surface as a (holomorphic) branched cover of the disc. $\bullet$ [**Topology$\Rightarrow$Riemann-Meis complex gonality?**]{} \[21.06.12\] Can the topological method (irrigation) used in Gabard 2006 [@Gabard_2006] be adapted to prove that any complex curve of genus $g$ is $\le [\frac{g+3}{2}]$-gonal, meaning that there is always a morphism to ${\Bbb P}^1$ of degree $\le$ than the specified bound. (Perhaps this is already answered in the lectures of Gunning 1972 [@Gunning_1972], who uses Mattuck’s topological description of the symmetric powers of the curve). Conversely, there is a dual problem: $\bullet$ [**Grötzsch-Teichmüller-Meis$\Rightarrow$Ahlfors-Gabard separating gonality?**]{} \[16 June 2012\] According to secondary sources (e.g. Kleiman-Laksov 1974 [@Kleiman-Laksov_1974]), Meis’ proof (1960 [@Meis_1960]) of the complex gonality $\le [\frac{g+3}{2}]$ of genus-$g$ curves, is eminently Teichmüller-theoretic. By analogy, it should therefore be possible to prove the $(r+p)$-gonality of membranes (cf. Gabard 2006 [@Gabard_2006]) by using the same (Teichmüller-style) method as Meis. This would incidentally give an “analytic” proof (or if you prefer, a “geometria magnitudinis” proof of Gabard 2006 [@Gabard_2006]). Notice the fighting interplay between topology and analysis (or geometry) since Teichmüller amounts essentially to the “möglichst konform” map of Grötzsch. $\bullet$ \[05 June 2012\] Ozawa 1950 [@Ozawa_1950] presents a genuine extension of the Schwarz lemma to multiply-connected domain. Can we do the same job for a membrane of positive genus? $\bullet$ [**Ahlfors$\Rightarrow$Gromov?**]{} \[Mai 2011\] Does Ahlfors (or perhaps the non-orientable variant of Witt 1934 [@Witt_1934]) implies Gromov’s filling area conjecture? Any solution to this puzzling problem is rewarded by 50 Euros by Mikhail Katz (cf. his home web-page). Perhaps, some other ingredients than Ahlfors are required. We (already) loosely suggested, Weyl’s asymptotic law (acoustic proof) or perhaps a sort of duality between “Ahlfors” extremal problem and that of Bieberbach 1914 [@Bieberbach_1914] (more widely known for its connection to Bergman). Added \[02.09.12\], maybe it is enough to consider the isothermic coordinate generated by a single Green’s function (or a dipole avatar) instead of an Ahlfors function. $\bullet$ [**Gromov non orientable**]{} (Easier?) \[June 2011\] Is the Gromov filling conjecture also true (and meaningful) for non-orientable membranes? Can it be generalized to several contours (desideratum J. Huisman 2011, oral e-mail communication). We may also drift to related problems like KNP (Kreisnormierungsprinzip). This asserts that any domain (or planar Riemann surface) is conformally diffeomorphic to a domain bounded by circles (we suppose finite connectivity for simplicity). $\bullet$ [**Extremal problem$\Rightarrow$KNP?**]{} Inspired by the paper Schiffer-Hawley 1962 [@Schiffer-Hawley_1962], where (Koebe’s) Kreisnormierung (in finite connectivity) is derived from a minimum problem of the Dirichlet type, one may wonder if a suitable variant of Ahlfors extremal function may not be used to reprove the Kreisnormierung. More about this is Sec.\[sec:KNP\] (related to works by Grötzsch, and others.). $\bullet$ [**Bieberbach’s (least area) minimum problem.**]{} Bieberbach 1914 [@Bieberbach_1914] considers in a simply-connected domain $B$ the problem of minimizing the integral $\int\!\! \int_B \vert f(z)\vert^2 d\omega$ amongst analytic functions $f\colon B \to {\Bbb C}$ normed by $f'(t)=1$ at some fixed point $t\in B$ of the domain. He shows that the minimum gives the Riemann map. (It is well-known that this problem constitutes the origin of the Bergman kernel theory, cf. besides Bergman’s original paper of 1922 [@Bergman_1922], e.g. Behnke’s BAMS review of Bergman’s 1950 book [@Bergman_1950].) The naive question is what sort of maps are obtained when this problem is formulated on a multiply connected domain? Do we obtain a circle map? And if yes, does this $\beta$-function coincides with the Ahlfors map? Can the problem be extended to Riemann surfaces? More on this is discussed in Sec.\[Sec:Bieberbach-Bergman\]. Of course this is closely allied to the Bergman kernel, and was treated by several authors, cf. e.g. Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950]. However as far as the writer browsed the literature, the qualitative feature of this $\beta$-map appear to have not been explicitly described. In fact it seems that ultimately the answer is a bit disappointing in the sense that the least-area map may lack single-valuedness. This is well-explained in papers by Maschler (1956–59, e.g. [@Maschler_1956]), and was probably known earlier by Bergman, Schiffer, etc. $\bullet$ [**Heins’ proof?**]{} \[28.06.12\] Heins 1950 [@Heins_1950] proposes another existence-proof of circle maps à la Ahlfors, by using some theory of Martin and concepts from convex geometry (minimal harmonic functions and extreme points of convex bodies). Unfortunately, he does not keep a quantitative control upon the degree of the map so obtained. However, on p.571 Heins introduces the number $m$ (of loops generating the fundamental group), which is easily estimated as $2p+(r-1)$ for a surface of genus $p$ with $r$ contours. \[E.g., imagining contours as punctures, the first perforation liberates a free group of rank $2p$ (twice the genus), and each additional perforation creates a new generator.\] Since this must be augmented by one (cf. Heins’ lemma on p.568, i.e. essentially the issue that each point of a convex body in Euclidean $m$-space is expressible as a barycentric sum of $m+1$ extreme points of the body spanning an $m$-simplex) it seems probable that Heins’ proof reproduces the bound $r+2p$ of Ahlfors. More about this in Sec.\[sec:Heins\]. (Actually, Heins’ convex geometry argument looks quite akin to the one used “subconsciously” by Ahlfors 1950 [@Ahlfors_1950].) $\bullet$ \[22.10.12\] [**The gonality sequence.**]{} An emerging question of some interest is that of calculating for a given bordered surface $F$ (of type say $(r,p)$) the list of all integers arising as degrees of a circle map defined on the given surface. We call this invariant the [*gonality sequence*]{} of $F$. As a noteworthy issue Ahlfors upper bound $r+2p$ is always effectively realized, in sharp contrast to Gabard’s one $r+p$ which can fail to be. For some messy and premature thoughts on this problem cf. Sec.\[sec:gonality-sequence\]. Of course the problem looks a bit insignificant combinatorics, yet studying it properly seems to require both experimental contemplation of concrete Riemann surfaces and sharp theoretical analysis of the existence-proofs available presently. Asking fine quantitative questions should aid clarifying the qualitative existence theorems. $\bullet$ \[03.11.12\] [**Generalized Keplerian motions via Klein-Ahlfors?**]{} It is well known that the motion of a single planet around a star describes an orbit which is a certain algebraic curve, namely an ellipse (other conics do occur for cold comets escaping at infinity without periodicity). To visualize Ahlfors circle maps on real plane algebraic curves of dividing type (Klein’s orthosymmetry), one can contemplate totally real pencils of curves sweeping out the given curve along totally real collections of points. The prototypical example is the Gürtelkurve (quartic with two nested ovals) swept out by a pencil of lines whose center of perspective is located inside the deepest oval. All such lines cut the quartic in 4 [*real*]{} points (cf. Fig.\[FGuert:fig\]). This paradigm of total reality is the exact algebro-geometric pendant of Ahlfors theorem, and suggests looking at real dividing curves as orbits of planetary systems with dynamics governed by a total pencil. For instance the Gürtelkurve could occur as the orbit of a system of 4 electrons gravitating around a proton with electric repulsive forces explaining the special shape of the Gürtelkurve (cf. again Fig.\[FGuert:fig\] below). In Sec.\[sec:electrodynamics\] we explore the (overambitious?) idea positing that the real locus of [*any*]{} real orthosymmetric curve (in the Euclid plane or space) arises as the orbital structure of an electrodynamical system obeying Newton-Coulomb law’s of attraction/repulsion via a dynamics controlled by an Ahlfors circle map (incarnated by a totally real pencil). This gives quite an exciting interpretation affording plenty of periodic motions to the $n$ body problem. This idea probably requires to be better analyzed. Even if physically irrelevant, one can (by Ahlfors) trace for any orthosymmetric real curve (in the plane) a totally real pencil generating usually quite intriguing figures, especially when members of the pencil are varied through the full color spectrum to create some rainbow effect. Depictions of such totally real rainbows are given in Sec.\[sec:electrodynamics\], but we failed drastically to make serious pictures for Harnack-maximal curves. This represents perhaps a certain challenge for computer graphics? $\bullet$ \[27.12.12\] [**Green-Riemann imply Schoenflies?**]{} This question is quite outside our main track of 2D-conformal geometry, belonging really to highbrow unsettled differential topology. Remember first that the Riemann mapping theorem (and the closely allied Green’s function measuring the proliferation of a bacteria in a nutritive medium) is essentially the best approach toward the (topological) Schoenflies theorem (ca. 1906) stating that any plane Jordan curve bounds a disc. Compare the contributions of Osgood and Carathéodory 1912 [@Caratheodory_1912], plus the recent discussion in Siebenmann 2005 [@Siebenmann_2005]. When it comes to high-dimensional versions of Schoenflies, we know in the smooth category by combination of the topological version of Mazur-Brown with Smale’s $h$-cobordism theorem giving uniqueness of the smooth structure on high-dimensional balls ($\dim\ge 5$). However remind that presently differential-topologic methods failed to prove the (so-called) smooth Schoenflies conjecture in dimension $4$, that any smoothly embedded $S^3$ in ${\Bbb R}^4$ bounds a $4$-ball with its usual smoothness structure. It is tempting to wonder if the classical tools of potential theory (especially the Green’s function) are able to reprove at least the high-dimensional cases of Smooth-Schoenflies, and if so, if it is able to crack the residual remaining exceptional case resisting all efforts of topologists so far. More details and references in Sec.\[Schoenflies:sec\]. The intuition behind all this is that the bacteria expand from any given interior point as concentric circles (resp. spheres) in the infinitesimally small but soon realize where there is more free vital room for expanding more quickly in those directions (cf. Fig.\[Green:fig\]). In particular all bacteria reach the boundary spheroid simultaneously. Mathematically this is formalized by considering the Green’s function $G(z,t)$, where $z\in {\Bbb R}^n$ and $t$ is an interior point of the bounded component of $\Sigma$ the $S^{n-1}$ embedded in ${\Bbb R}^n$, defined as $\log \vert z -t \vert - u$ where $u$ is the unique harmonic function with boundary values given by $\log \vert z -t \vert$ on the boundary $\Sigma$. Studying the Green’s lines,that is the trajectory orthogonal to the levels of $G(z,t)$ should (possibly) enable one to establish the required diffeomorphism between the (sealed) interior of $\Sigma$ with the ball $B^{n}$ endowed with its usual smooth structure. (It is unknown if $B^4$ supports an exotic differential structure but that this another question a priori much harder to decide.) Some vague answers ------------------ This section tried to report question which looks exciting, and to which I tried some premature answer. It requires to be polished drastically and reorganized seriously. Hence it is probably safer to skip, but maybe readers fluent with techniques like Ahlfors extremals, Teichmüller extremal quasi-conformal maps, Plateau’s problem, etc. may find useful to clarify our vague ideas. $\bullet$ [**Quantum fluctuations of Ahlfors’ degree**]{} \[20.09.12\] The following problem is somewhat ill-posed, yet it is just an attempt to excite the imagination. Suppose given a compact bordered Riemann surface $F$ with $r\ge 1$ contours and of genus $p\ge 0$. For each interior point $a\in \rm{int} (F)$ there is a uniquely defined analytic Ahlfors function $f_a$ solving the extremal problem of making the derivative $f'(a)$ as large as it can be, while keeping this magnitude positive real and the range inside the unit disc. This extremal function is uniquely defined and independent of the local uniformizer used to compute the derivative. It is known by Ahlfors 1950 that each $f_a$ is a circle map of degree somewhere in the range from $r$ to $r+2p$, that is a (surjective) branched cover of the disc. According to Coppens 2011 [@Coppens_2011] the generic bordered surface has gonality $r+p$ so that one can considerably squeeze the Ahlfors range to the interval $r+p$ to $r+2p$. One would like to understand in geometric term (if possible?) what phenomena is responsible of the fluctuation of the Ahlfors degree. Of course, if $p=0$ there is no fluctuation just because of the Ahlfors squeezing: i.e. $\deg f_a$ is constant when the center of expansion $a$ is dragged throughout the surface. However if $p>0$, it is likely that some jump must occur albeit I know no argument. Gabard 2006 only showed that there is a circle map of degree $\le r+p$, but a priori there is no reason forcing such low degree maps to be realized as Ahlfors maps. Following Coppens we may define the gonality $\gamma$ of $F$ as the least degree of a circle map on $F$. By Gabard (2006 [@Gabard_2006]) $\gamma\le r+p$ (and trivially $r\le \gamma$). Coppens tell us that all intermediate values of $\gamma$ are realized (modulo the trivial exception that when $r=1$ and $p>0$, $\gamma=1$ cannot be realized). This gonality invariant infers a sharpened variability for the Ahlfors degrees, namely $r\le \gamma\le \deg f_a \le r+2p$, where $\gamma \le r+p$. A priori all intermediate values could be visited (between $\gamma$ and $r+2p$). However this scenario is incompatible with the case of hyperelliptic membranes studied in Yamada and Gouma, where the effective Ahlfors degrees are either maximal $r+2p$ or minimal (i.e. $2$). Those examples still indicate that despite a sparse repartition the degree distribution is in some sense extremal, occupying the maximum space at disposition. Is this a general behavior? This is the maximum oscillation (Schwankung) conjecture (MOC). If true, then Coppens gonality would always be sustained by an Ahlfors map and also Ahlfors upper bound $r+2p$ would be sharp for any surface, whatsoever its differential-geometric granularity. MOC displays the most naive scenario for the fluctuation of Ahlfors degree, and it would be a little miracle if it is correct. If not, then what can be said? A very naive idea idea would be that there is a sort of conservation law like in the Gauss-Bonnet theorem: whatsoever you bend the surface the Curvatura integra keeps constant. (Of course this holds for a closed surface but not for a bordered one, unless the geodesic curvature of the boundaries is controlled, e.g. by making it null.) The vague idea would be that if we think of the Ahlfors degree $\deg f_a$ as a sort of discrete curvature $\delta(a)$ assigned to the point $a$ then maybe $\int_F \delta (a) d\omega$ keeps a constant value (independent of the conformal structure). If so then at least in the cases where there is a hyperelliptic model (i.e. $r=1$ or $2$) one could conclude that the Ahlfors degree are somehow balanced. Yet recalling Yamada-Gouma’s investigations it seems that the maximum degree $r+2p$ occurs very sporadically for the center $a$ located on the finitely many Weierstrass points of the membrane, hence high values have little weight. So in the hyperelliptic case (with few contours $r=1$ or $2$) the Ahlfors degree are constantly very low $2$ with exceptional jump taking place on a finite set of points. Maybe this suggests a low energy scenario valid in general: given any (finite) bordered surface $F$ the Ahlfors degree is always equal to the gonality safe for some jump occurring on a finite set of points. Of course this must be perhaps refined suitably by saying that there is a stratification (decomposition) in pieces, where the lowest degree (i.e. the gonality) is always nonempty and containing the contours, and then as we penetrate more deeply inside the surface the degree may increase (eventually always reaching the extremum value $r+2p$?). $\bullet$ [**Quasiconformal doodlings**]{} \[02.10.12\] As is well known, Teichmüller 1939 [@Teichmueller_1939] exploited the flexibility of quasiconformal maps to put Riemann’s intuition of the moduli of conformal classes of differential-geometric surfaces (Riemannian surfaces) on a sound footing. The idea is both soft and flexible, yet with the devil of capitalism (geometria magnitudinis) cached just behind for one counts the distortion effected upon infinitesimal circles into ellipses. Using Grötzsch idea of the möglischt konform map relating two configurations produces an extremal map relating both configurations, and this least distortion gives the Teichmüller metric (a first step to endow the moduli “set” of a genuine space structure). Maybe this methodology is also fruitful in the theory of the (Ahlfors) circle maps. The first desideratum is to show existence of circle maps, and then the game refines in finding best possible bounds (over the degree of such maps). The framework is as follows (aping again Grötzsch-Teichmüller): given a finite bordered surface (and maybe also a mapping degree $d\ge r$) we look at all quasiconformal map (not necessarily schlicht), i.e. (full) branched cover of the disc (with the same topological feature as circle maps of taking the boundary to the boundary and the interior to the interior). Following Grötzsch’s idea we may look at the “möglischt konform” map, i.e. the most conformal quasiconformal map in the family (hoping eventually to find a beloved conformal one). Measuring distortion (largest eccentricity of the ellipses images of infinitesimal circles) one gets a numerical invariant $\varepsilon(F, d) \ge 0$, namely the infimum of the dilation among the class of all (differentiable) maps from the bordered surface $F$ to the disc. This invariant $\varepsilon(F, d)$ vanishes precisely when $F$ admits a (conformal) circle-map of degree $d$. Hence it vanishes if $d\ge r+2p$ by Ahlfors 1950 [@Ahlfors_1950 p.124–126], and even as soon as $d\ge r+p$ if one believes in Gabard 2006 [@Gabard_2006], where as usual $p$ is the genus and $r$ the number of contours of $F$. However we are rather interested to use the Grötzsch-Teichmüller theory to rederive an independent existence-proof. Of course in contrast with the classical setting of Teichmüller’s approach to the moduli problem, where one considers exclusively schlicht(=injective) maps, we tolerate now multivalent mappings, but this should not be an insurmountable obstacle. Our intuition is that it is not just a matter of measuring that is required, but one must somehow explore the pretzel underlying the surface to get an existence proof. Yet the flexible-quantitative viewpoint of measuring eccentricity probably gives an interesting numerical invariant which is now not a metric (Teichmüller metric), but rather a (potential) function on the moduli space. In fact we assign to a given (bordered) surface $F$ a series of number $\varepsilon(F,d)$ for $r\le d\le r+p$ (larger values of $d$ give $0$ by Gabard 2006 [@Gabard_2006]), which is probably decreasing (after eventually modifying the original problem by permitting all maps of degree $\le d$ instead of those having degree exactly $d$). So we get attached to $F$ a series of dilations $\varepsilon(F,r)\ge\varepsilon(F,r+1)\ge \dots \ge \varepsilon(F,r+p)=0$. Of course the sequence can crash to zero before the $r+p$ bound and indeed do so as soon as Coppens’ gonality $\gamma$ is reached (that is, the least degree of a circle map for the fixed $F$). \[Of course in the exact degree $d$ variant of the problem one can imagine more romantic behaviors with oscillation down to zero and then becoming positive again (touch-and-go phenomenology).\] Those $p$ invariants would refine Coppens gonality in a continuous fashion, yet fails to be “moduli” since there are $3g-3$ of them (Riemann-Klein) where $g$ is the genus of the double (that is $2p+(r-1)$), hence giving a total of $3g-3=3(2p+(r-1))-3=6p+3r-3$ free parameters which exceeds of course our $p$ parameters. But coming back to the basic existence problem, one can get started by observing that any topological type of membrane admits a circle map. One trick is to use symmetric membranes (cf. Chambéry section \[sec:Chambery\] below). This amounts to imagine a membrane in $3$-space symmetric under rotation by 180 degree so that the quotient as genus zero (cf. Fig.\[Chambery:fig\] below). Once the handles are killed one is reduced to the simple (planar) case of Ahlfors due to Bieberbach-Grunsky (and largely anticipated by Riemann, Schottky (no bound by Schottky?), and Enriques-Chisini (via Riemann-Roch and a continuity argument, cf. e.g. Gabard 2006 [@Gabard_2006 Sec.4]). The degree of the resulting map is easily computed (and of degree essentially equal to $(r/2) \cdot 2=r$ the minimum possible value, for the rotation identifies pairs of contours and gyrate all handles over themselves, cf. again Fig.\[Chambery:fig\], below). Thinking in the moduli space $M$ we have shown that the set $C$ of all circle-mappable surfaces is nonempty, and using the connectedness (of $M$) it would suffice to show that $C$ is [*clopen*]{} (i.e., closed and open). Checking openness, certainly requires enlarging the mapping degree to larger values. Now given an arbitrary bordered surface $F$ we can quasiconformally map it to our symmetric model $S$ and then compose with the circle-map. The dilatation is then controlled in term of the Teichmüller distance from $F$ to $S$, giving an upper bound over the eccentricity invariant $\varepsilon$ (for the appropriate degree). Of course this is still miles away from reproving even Ahlfors but maybe the idea is worth pursuing. In fact what is truly interesting is that we get for each $d$ a numerical function $\varepsilon_d$ (defined as $\varepsilon_d(F):=\varepsilon(F,d)$) on the moduli space $M_{p,r}$ of membranes of genus $p$ with $r$ contours, that vanishes precisely when $F$ has gonality $\le d$. Of course this sequence of functions is monotone decreasing when the index increases, and $\epsilon_d\equiv 0$ is identically zero (for $d\ge r+p$). According to Coppens result each of these functions (let us call them the Teichmüller potentials) vanishes somewhere. It is then perhaps interesting to look at the gradient flow $\varphi_d$ (w.r.t. Teichmüller metric) of these functions $\varepsilon_d$ affording a dynamical system (=flow) in which each bordered surface evolves in time to a sort of best possible surfaces for the prescribed gonality. (Morally each surface tries to improve its gonality along the trajectory of steepest descent.) If the global dynamics is simplest (say each trajectory finishes its life on a surface of gonality $d$) it is therefore reasonable to expect that the whole Teichmüller space is retracted by deformation to a sort of spine consisting of surfaces having the prescribed gonality $d$. Maybe one can deduce that the global topology of this spine is that of a cell (like the full Teichmüller space). Further it seems probable that the flows preserve the stratification by the gonality of $M_{p,r}$ since if $F$ has gonality say $d$ then its future $F_t$ has lower gonality. \[The situation looks analog to some works of René Thom (isotopy lemma, vector fields preserving a stratification, and “fonction tapissante” as it arise in the Thom-Mather problem of the stability of polynomial mappings??\] \[03.10.12\] Of course the above can be adapted to the case of closed (non-bordered) surfaces of genus say $g$, by replacing the target disc by the (Riemann) sphere. Likewise we define Teichmüller potentials $\varepsilon_d$, measuring the dilatation of the “möglichst konform” map of a fixed degree $d$ from the surface $F$ to $S^2$, and ideally one can imagine that the theory is able to reprove the famous (Riemann-Brill-Noether) bound $[\frac{g+3}{2}]$ first proved by Meis 1960 [@Meis_1960]. Hence all what we are trying to do is surely already well-known (alas I was never able to find a copy of Meis’ work, which is Teichmüller-theoretic according to other sources). Hence if Meis theory is just a sort of Teichmüller theory for branched covers of the sphere, with the ultimate miracle that Teichmüller not only affords a solution to Riemann’s moduli problem but also to the gonality question. A priori Meis’ theory should adapt to the bordered setting and arguably lead to another proof of the Ahlfors map, and optimistically with the sharp bound predicted in Gabard 2006 [@Gabard_2006]. Sharpness of the bound is due to Coppens 2011 [@Coppens_2011]. Recall that, Teichmüller himself was close to this (bordered) topic in the article Teichmüller 1941 [@Teichmueller_1941], yet the details (as well as exact bounds) are probably missing. $\bullet$ [**Ahlfors inflation/injection and generalized Ahlfors maps taking values outside the disc (alias, circle)**]{} \[09.10.12\] The theory of the Ahlfors function is primarily based upon the paradigm of maximizing the derivative (its modulus) within the family of maps with range confined to a (compact) container namely the unit disc. So it is primarily an inflation/injection (or pressurization) procedure (by opposition to the dual deflation/suction approach of Bieberbach-Bergman amounting to minimize the area among maps normed by $f'(z_0)=1$). Ahlfors 1950 [@Ahlfors_1950] showed that if the source object is any compact bordered Riemann surface and the target the unit disc then the Ahlfors (inflating) map turns out to be a circle map, i.e. a full covering of the unit circle taking boundary to boundary. This behavior is not surprising since maximizing the distortion (scaling factor) at a given basepoint forces the whole surface to be maximally stretched over the target, like an elastic skin pushed to its ultimate limit (in the Hollywoodian context of aesthetical surgery). The existence of Ahlfors maps relies on a Montel normal family argument, in substance inherited from the compactness of the disc. This suggests replacing the target disc by any compact bordered Riemann surface. We formulate then the following extremal problem: Given two finite bordered Riemann surfaces $F$ and $G$ and a given point $a \in F$ and $b\in G$, we look inside the family of all analytic maps $f\colon F \to G$ taking $a$ to $b$ at the map maximizing the modulus of the derivative $f'(a)$ computed w.r.t. local parameters introduced at $a$ and $b$. By analogy with the Ahlfors et ali theory, we expect that the extremal function exist (compactness of the receptacle $G$), is unique (this is either less evident or false for in the classical case $G=\Delta$ the argument relied heavily on the Schwarz lemma for the disc, so that our only hope in favor of uniqueness is that what actually counts is the universal covering). Arguably, even if lacking uniqueness extremals could still be interesting. Finally it is reasonable to expect that extremals are not oversensitive to the choice of local uniformizers. So we can speak of the map $f_{a,b}$ of extremum dilatation at $a,b$. Finally we are interested about knowing if the extremals are total maps in the sense of taking boundary to the boundary, as do the classical Ahlfors map in the circle/disc-valued case. Before proceeding to examples let us perhaps observe that in the special case where $F$ is given as a subsurface of $G$ and both points $a=b$ coincide, then the (complex) tangent space are readily identified so that $f'(a)$ has an intrinsic meaning as scaling factor of this complex line. Another special case of interest is when $G$ is a plane subregion, in which case the tangent bundle is trivialized so that one can consider a relaxed form of the problem without the constraint $f(a)=b$, in which no point $b$ is given but the sole extremalization of $f'(a)$ will actually dictate where $a$ has to be mapped. Albeit all we are saying looks a bit messy and unnatural (?), it should be noted that the whole game can be drastically simplified by just looking at avatars of circle maps, that is given two finite Riemann surfaces $F$ and $G$ when does there exist a total map (taking boundary to boundary) from the first to the second. (Of course this question is quite standard yet probably hard to answer precisely, cf. Landau-Osserman 1960 [@Landau-Osserman_1960], and Bedford 1984 [@Bedford_1984].) As we shall soon explain a vague answer is readily supplied by “algebraic geometry”, namely when the target $G$ is not the disc, and if $F$ has general moduli then in general there in not a single total map from $F$ to $G$. The moral is that circle maps enjoy a certain privilege due to their unconditional existence (by Ahlfors 1950 precisely). A basic obstruction arises from the Riemann-Hurwitz formula. Indeed given $f\colon F \to G$ a total map, it has no ramification along the boundary and is a full covering surface (cf. e.g. Landau-Osserman 1960 [@Landau-Osserman_1960 p.266, Lemma 3.1]). Denoting by $d\ge 1$ the degree of the map, we have $\chi(F)=d \chi (G)-b$, where $b\ge 0$ counts the branch points. When $d=1$, there is no branching and the topological types must agree. Another constraint says roughly that a total map can only simplifies the topology, precisely $\chi(F)=d \chi (G)-b\le d \chi (G)\le \chi(G)$, when $\chi(G)\le 0$. If $G$ is not the disc then the existence of a bordered map $f\colon F \to G$ implies that the Euler characteristic satisfies $\chi(F) \le \chi(G)$. (Of course the conclusion persists when $G$ is the disc for it maximizes the Euler characteristic among bordered surfaces.) Another simple constraint comes from the fact that a total map $f\colon F \to G$ induces a covering of the boundary $\partial f\colon \partial F \to \partial G$. Hence if $G$ has $r'$ contours then $F$ has at most $d \cdot r'$ contours, i.e. $r\le d \cdot r'$ where $r$ is the number of contour of $F$. On the other hand as $\partial f$ is onto, the surjection induced by $\partial f$ on the $\pi_0$ (=the arc-wise connected component functor from TOP to SET) implies that $r\ge r'$. Then there is a little zoology of cases to study. (Z1) Let us first suppose that the [*source*]{} is just the disc, then who is the (“Ahlfors”) extremal map? So we assume $F=\Delta$ and $G$ any bordered surface marked at $a=0$, $b\in G$ respectively. By uniformization (Koebe-Poincaré 1907) we know that the universal cover of the interior of any finite bordered surface is the disc. Now the extremal map $f_{a,b}\colon \Delta \to G$ (maximizing the distortion) may be lifted to the universal cover as say $F \colon \Delta \to \Delta$. Now by the Schwarz-Pick principle of hyperbolic contraction for analytic maps, the latter map contracts the hyperbolic metric implying the universal projection to effect a greater dilatation than the presumed extremal $f_{a,b}$. It follows that $F$ must be the identity (up to rotation) and the extremal function get identified to the universal cover. (Actually, works by Carathéodory and Grunsky actually manage to prove uniformization via the (Ahlfors) extremal problem, whereas we assumed it.) (Z2) Now consider the situation were both source and target have complicated topology. For instance the source is any bordered surface and the target an annulus. One may expect to get analogues of circle maps, i.e. [*total maps*]{} taking boundary to boundary (sometimes known as proper maps). (Such maps are called [*boundary preserving*]{} in Jenkins-Suita 1988 [@Jenkins-Suita_1988], cf. also Landau-Osserman 1960 [@Landau-Osserman_1960 p.265] who speak of maps “which takes the boundary into the boundary”, while ascribing to Radó 1922 [@Rado_1922-Z-Theorie-mehr] the basic result that such maps are full coverings taking each value of the image surface a constant number of times). Unfortunately, there is severe obstructions to boundary preservation of such (generalized) Ahlfors maps. One way to argue is via algebraic geometry and the Jacobians. It is indeed classic that a generic closed Riemann surface tolerates only nonconstant maps to the sphere (ruling out the trivial identity map or automorphisms available incidentally only for surfaces with specialized moduli). Assuming the Ahlfors map of $F$ to an annulus to be total, its symmetric extension to the Schottky-Klein double is a map from a closed surface to the torus, which for general moduli cannot exist at all! Of course all this requires better proofs, but is fairly well-known and classical (cf. e.g. Griffiths-Harris 1980 [@Griffiths-Harris_1980], who argue as follows (p.236–237): “[*A general curve $C$ of genus $g\ge 2$ cannot be expressed as a multiple cover of any curve $C'$ of genus $g'\ge 1$.*]{} This is readily seen from a count of parameters: the curve $C'$ will depend on $3g'-3$ parameters, and the $m$-sheeted covering $C\to C'$ depends on $b$ parameters, where \[$\chi (C) = m \chi(C')-b$, that is\] $$b=2g-2-m(2g'-2)$$ is the number of branch points of the cover. Thus if $m\ge 2$, $C$ will depend on $$b+(3g'-3)=b+\frac{3}{2}(2g'-2) =2g-2-\underbrace{\bigl(m-\frac{3}{2}\bigr)}_{\ge 1/2}\underbrace{(2g'-2)}_{\ge 0}\le 2g-2<3g-3$$ parameters, and so cannot be general.” (Another argument is given in the exercises of Arbarello-Cornalba-Griffiths-Harris 1985 [@Arbarello-Cornalba-Griffiths-Harris_1985-BOOK p.367, Ex.C-6], which of course we were not able to solve!) (Z3) Finally one can imagine a bordered surface embedded in a slightly larger one (say of the same topological type). Then the inclusion map is permissible in the extremal problem, so the extremal map will have distortion $\ge 1$ at some basepoint, and naively should expand the small surface into the larger one. However by the argument of (Z2) in general it is unlikely that the extremal will be total, and also a priori it not even clear that a true expansion can occur (try to lift the map to the universal cover a get maybe a conflict with the Schwarz-Pick principle of contraction??) But of course this looks dubious for when the subsurface is a disc expansion is possible. $\bullet$ [**Cyclotomic Riemann surfaces**]{} \[09.10.12\] (but similar examples in Chambéry Talk ca. 20 December 2004) At this stage we can do perhaps the following sort of experiment. As is well-known (Riemann-Prym-Klein 1882 [@Klein_1882]) a Riemann surface structure can also be defined in the most simplest way to visualize, namely as differential-geometric surface in $3$-space with metric (hence conformal structure) inherited by the Pythagorean/Euclidean line element. Consider a hemisphere in Euclidean 3-space surmounted by $m$ handles cyclotomically distributed as on Fig.\[Cyclo:fig\], joining themselves above the north pole. -5pt0 -5pt0 Ignoring the south hemisphere, we obtain so a bordered surface $F$ with one contour ($r=1$) and of genus $p=m-1$. (Notice here the standard psychological aberration that the genus is one less than the “handles”, for the first handle is not yet coupled to another one to create a real handle!) On rotating by angle $\frac{2\pi}{m}$ the configuration $F$ upon itself we obtain a map from $F$ to the disc (hence a circle map), because the fundamental domain of the rotation is glued over itself to give a disc. The circle map so obtained has degree $m=p+1$. This matches with the general bound $r+p$ predicted in Gabard 2006 [@Gabard_2006]. Let us now assume that a bubbling, i.e. an Euclid-Riemannian deformation of the metric takes place at one of the handle (yet not on the remaining ones) then the rotational symmetry is killed and it becomes much nontrivial that a circle map of same degree is still persistent. This experiment seems to damage the truth of Gabard 2006 [@Gabard_2006] (but hopefully is not?) A naive parade would be to use the (Riemann-Schwarz) uniqueness of the conformal structure on the closed 2-cell to resorb the cancerigenic bubbling. Yet this looks cavalier (for we are not living in the soft smooth $C^{\infty}$ category) and this would not settle the case of less localized cancerigenic degenerations not supported over a disc, but along a subregion having itself moduli. Then one cannot repair easily the deformation by a simple surgical lifting. At this stage we see that the result of Gabard 2006 [@Gabard_2006], if true at all, looks quite formidable for it should resist all those plastic deformations within the flexibility of conformal maps. Perhaps this might be not so surprising in view of the Riemann-Schwarz super-flexibility theorem telling us in particular that any simply-connected bordered surface (hence topologically the disc) is conformally the same as the disc or the hemisphere. Yet in our context where no simple connectivity is assumed an incarnation of the Riemannian miracle (highly counterintuitive but hopefully true)! It could be interesting (by adapting Yamada-Gouma) to study the degree of the Ahlfors map of such cyclotomic Riemann surfaces, especially when the basepoint is situated on the 3 fixed points of the rotation. $\bullet$ [**Special triangulations**]{} \[10.10.12\] Given a circle map of a bordered surface $F$, one can post-compose it with the map taking conformally the disc to an equilateral triangle (in the Euclid plane ${\Bbb C}$). (Recall that this can be done for any three point prescribed along the boundary). Upon subdividing the triangle in a mesh of equilateral triangles, and lifting via the conformal map we generate certain triangulations of $F$ which are almost equilateral. In fact if the mesh size is chosen so that all ramification points lye in the interior of the tiny triangles then the inverse image of such ramified triangles will be small hexagons. Try to study the differential geometry and specialize to Gromov’s Filling conjecture, or try to find a link with Belyi-Grothendieck (a Riemann surface is defined of $\overline{{\Bbb Q}}$ iff it admits an equilateral triangulation). Another special triangulation of the disc is the hyperbolic tessellation depicted on the front cover of Grunsky’s Collected papers (by equilateral triangles with angles $\pi/6$). \[This tessellation is supposed via the Ahlfors-Grunsky conjecture (1937 [@Ahlfors-Grunsky_1937]) to play an extremal rôle in the Bloch schlicht radius of maps $\Delta \to {\Bbb C}$ for it dominates the densest circle packing of the Euclidean plane.\] Try to understand if it is useful (or aesthetical) to lift this tessellation to the bordered surface via a circle map. $\bullet$ [**Plateau heuristics $\Rightarrow$ Ahlfors maps?**]{} \[17.10.12\] Soap film experiments of the Belgian physicist have a certain existential convincing power, albeit the rigorous mathematical existence proof (Douglas/Radó ca. 1930/31) required circa 30 years more delay than the allied Dirichlet principle (Hilbert 1900) itself interpretable at the equilibrium temperature distribution in a heat-conducting plate with assigned boundary values. Now Douglas 1931 [@Douglas_1931-Solution] observed that the Riemann mapping theorem (RMT) may be derived by specializing Plateau’s problem to the case where the contour degenerate to the plane, and Courant pushed the remark further so as to include the Riemann-Bieberbach-Grunsky theorem (=planar case of the Ahlfors map). On the other hand Douglas 1936 [@Douglas_1936-Some-new-results] envisaged the so-called Plateau-Douglas problem (PDP, or just PP) for membranes of higher topological structure. It should thus follow (either logically or intuitively) a physico-chemical existence proof of the Ahlfors map. Let $F$ a finite bordered Riemann surface of genus $p$ with $r\ge 1$ contours, and suppose also given a fixed circle in the plane interpreted as the prescribed wire frame of PP. More generally one can imagine a collection of $r$ contours to be given, and we look at the special case where all these coincide with the unit circle. Now cultivating the right intuition about PP it should be possible to deduce the existence of Ahlfors maps perhaps even with the degree control $r+p$ of Gabard. In fact it should even be possible to study wide extensions where not all frames are coincident with the unit circle. Can one take any frame prescription (e.g. disjoint round circles)? For instance take two unit circles with centers lying distance ten apart ($\vert z \vert=1$ and $\vert z-10 \vert=1$). Suppose the membrane to have the topological type of an annulus ($r=2$ and $p=0$). Then the minimal surface is something like a flat catenoid, where the inside of each circle is covered once and a certain tube connecting both circles is covered twice by the map. Yet notice that the apparent contour (where the map is folded) of such a film violates the local behavior of holomorphic maps. As we just saw the folding obstruction makes unlikely to span contours consisting of disjoint maximal circles. (Circles being ordered by inclusion of their interior in the plane.) In contrast a nested configuration of circles should cause no trouble to holomorphy. Thus it should be possible to render Ahlfors intuitively obvious via soap film experiments. Of course this was essentially done in Courant’s book (1950 [@Courant_1950]), yet the exact juncture with Ahlfors result probably deserves some extra working. Of course the real challenge would be to investigate if Plateau-style approaches are susceptible to vindicate the degree bound $r+p$ advanced by Gabard 2006 [@Gabard_2006]. Another idea is to imagine a Plateau problem with “wind” blowing through 3-space in some prescribed way (along a given vector field). For instance a soap film spanning a planar disc at rest could deform under a perpendicular wind into say a hemispherical membrane. Try to connect this with Gromov’s filling conjecture, yet unlikely due to the embedded nature of Plateau. Another more plausible connection would be with Gottschalk’s conjecture on flows in 3-space (no vector fields in 3-space having only dense trajectories). This is probably one of the most alienating open problem in the qualitative theory of dynamical systems. $\bullet$ [**(Ahlfors) circle maps of minimal degree**]{} \[19.10.12\] Given a finite bordered Riemann surface $F$ of genus $p$ with $r$ contours, there is always (by Ahlfors) a circle map. The set of (positive) integers being well-ordered there is a circle map of minimal degree. Call perhaps such maps [*minimal circle maps.*]{} We may ask to which extent such a map is unique and if not can we describe the “moduli space” of such maps. Of course in the most trivial case where $p=0$ and $r=1$ (topologically a disc) the Riemann map is essentially unique ignoring automorphisms of the disc. Likewise uniqueness holds for surfaces with hyperelliptic double provided the latter is not Harnack-maximal. Such hyperelliptic membranes have $r=1$ or $r=2$ and the hyperelliptic involution induces a totally real morphism of degree $2$. Our uniqueness assertion follows of course from the uniqueness for complex curves of the hyperelliptic involution when $g\ge 2$ and thus holds in our context provided $p\ge 1$ (recall that $g=(r-1)+2p$). When $p=0$ and $r=2$ uniqueness fails, for then the double has genus one and may be concretized as a smooth plane cubic with two circuits: one being a genuine “oval” bounding a disc in ${\Bbb P}^2({\Bbb R})$, the other being termed a pseudo-line. Projecting from any point located on the oval gives a totally real morphism of degree $2$, and correspondingly a circle map when restricted to the semi-Riemann surface. Another example is the Gürtelkurve, i.e. any smooth quartic with two nested ovals. Then the minimal degree of a circle map (for the half of the curve) is 3 (argue with the complex gonality of smooth plane curves), and such maps arise by projecting the curve from a real point located on the innermost oval of the nest. Hence there $\infty^1$ circle maps of minimum degree, those being parameterized by a circle $S^1$. Of course the problem does not depend only on the topology: the half of the Gürtelkurve belongs to the topological type $r=2$ and $p=1$, which contains also hyperelliptic representatives, those being circle mappable in a unique fashion via a map of degree 2. When $F$ is planar ($p=0$) then the double is Harnack-maximal and either the argument of Enriques-Chisini or that of Bieberbach-Grunsky shows that any divisor with one point on each oval moves in a linear system which is totally real (cf. e.g. Gabard 2006 [@Gabard_2006]). So we have now essentially a torus of dimension $r$ ($r$=number of contours) of circle maps of minimum degree. A details description is not so evident for such a divisor $D$ moves in a linear system of dimension $\dim \vert D \vert \ge \deg D-g$ (Riemann’s inequality, a direct consequence of Abel), where $g=r-1$ is the genus of the double. Thus $\dim \vert D \vert \ge r-(r-1)=1$ so that $D$ does not necessarily determines unambiguously a totally real pencil. Despite this difficulty it seems reasonable to assert that the set of circle maps for a planar membrane is a torus perhaps of dimension only $r-1$ for one has to unite divisors lying in the same pencil. Extrapolating such examples, we may wonder about structural properties of the set of (minimal) circle maps. Is it always compact? Always a manifold? Perhaps even always a torus. Is it always connected? Of course there are various way to formulate the question and there probably basic experiments giving quick answers to the naive connectedness assumption. Another question is to understand how the global degree $d$ of the circle map splits (partitioned) into the bordered degrees of the restriction to each contours. For instance in the case of the Gürtelkurve, albeit both ovals are perfectly equivalent from the viewpoint of analysis situs, it seems that on the Riemann surface the one corresponding to the inner oval can be mapped with degree 1 whereas the other is less “economic” requiring a wrapping of degree 2. Of course it would be nice to understand this in some intrinsic fashion? But how? (Perhaps via the uniformizing hyperbolic metric and the length of the corresponding ovals???) Let us try a naive approach to the connectedness problem (by actually trying to corrupt it). Consider in the “abstract quadric surface” ${\Bbb P}^1\times {\Bbb P}^1$ a configuration of bidegree say $(3,3)$. We have chosen both degrees equal so that both projections have the same degree. Imagine 3 lines in each ruling and smooth out the corresponding line arrangement to create a smooth curve $C_{3,3}$ (cf. Fig.\[Cyclo:fig\]b). Actually we have performed sense-preserving smoothings (cf. again the figure) so that the resulting curve is dividing (Fiedler type argument [@Fiedler_1981]). Contemplating the figure we count $r=3$ “ovals”. Both projections on the factors are totally real morphisms of degree 3 (the minimum possible degree in view of the trivial lower bound $\deg f \ge r$). However it seems unlikely that one can continuously deform one map into the other (while keeping its degree minimum). Hence this may give some evidence that the space of minimal circle maps (for the corresponding bordered surface, namely the half of the orthosymmetric Riemann surface underlying our dividing curve $C_{3,3}$) is not connected. However our argument is quite sloppy, having equally well applied to bidegree $(2,2)$ in which case the corresponding curve is Harnack-maximal \[recall that $g=(a-1)(b-1)$ for bidegree $(a,b)$\], hence subsumed to the connectivity principle. Of course it is probable that some basic complex algebraic geometry (gonality of complex curves) suffices to complete the above argument. Is it true that a smooth curve of bidegree $(n,n)$ is $n$-gonal in only two fashions (provided $n\ge 3$) via the natural projections? Of course the assertion is false for $n=2$, for then $g=1$ and a smooth plane cubic model creates $\infty^{1}$ pencils of degree 2. For another plane example seeming to violate the connectivity principle of minimal maps see Fig.\[Coppens:fig\](code 313). Some historical puzzles ----------------------- $\bullet$ The most scorching question is whether Klein really anticipate Ahlfors as suggested in Teichmüller 1941 [@Teichmueller_1941]? (Compare Sec.\[sec:Teichmueller\], especially Quote \[quote:Teichmueller-1941\].) Of course the question bears not only historical interest, but has some didactic importance for if a Kleinian argument ever existed (and not just in Teichmüller’s imagination), it is quite likely to be more geometric than Ahlfors’ (decidedly analytic) account. As we already said, it is possible that the Klein-Teichmüller proof rest upon the Rückkehrschnitt intuition of Klein. Even in case Klein himself never anticipated the Ahlfors circle maps, one may wonder from where Teichmüller derived it? In turn one may wonder if Ahlfors took inspiration by Teichmüller 1941 [@Teichmueller_1941]? Of course Ahlfors himself never quoted this Teichmüller work, except in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960], where also all the Italian workers are carefully listed (especially Matildi 1945/48 [@Matildi_1945/48] and Andreotti 1950 [@Andreotti_1950]). $\bullet$ Does Courant’s paper of 1939 [@Courant_1939] (and the somewhat earlier announcement of 1938 [@Courant_1938], plus the later book treatment of 1950 [@Courant_1950]) presage (modulo a suitable interpretation) any anticipation over the circle map result of Ahlfors 1950 [@Ahlfors_1950]? (For more, compare Sec.\[sec:Courant\].) The province of Felix Klein {#Sec:Klein} =========================== Felice Ronga and Felix Klein’s influence ---------------------------------------- In fact the writer himself came across (a weak version of) the Ahlfors function topic from a somewhat different angle, namely via Klein’s theory of [*real algebraic curves*]{} (spanning over the period 1876–92). For Klein this was probably just a baby case of his paradigm of the Galois-Riemann Verschmelzung (Erlanger Program 1873, friendship with Sophus Lie, Ikosaheder and its relation to quintic in one variable, etc.). Yet, real curves surely deserved special (Kleinian) attention as it provided a panoramic view (through the algebro-geometric crystal) of the just emerging topological classification of surfaces (Möbius 1863 [@Moebius_1863], Jordan 1866 [@Jordan_1866], etc.). This offered also a bordered (even possibly non-orientable) avatars of Riemann surfaces, as shown in the somewhat grandiloquent title chosen by Klein “[*Über eine neue Art der Riemannschen Flächen*]{}” (=title of 1874 [@Klein_1874], 1876 [@Klein_1876]) . Those works of Klein were probably not extremely influential (and still today represent only a marginal subbranch of the giant tree planted by Riemann). Klein himself lamented at several places his work not having found the quick impact he expected from. In 1892 [@Klein-Werke-II_1922 p.171] (ten years after his systematic theory presented in 1882 [@Klein_1882]), he writes: “[*Inzwischen hat noch niemand, so viel ich wei[ß]{}, die hier gegebene Fragestellung seither aufgegriffen, [\[…\]]{}*]{}”. About the same period in his lectures of 1891/92 [@Klein_1892_Vorlesung-Goettingen p.132], he wrote: [*Was ich bislang von diesen Theoremen publicirt habe (so die Einteilung der symmetrischen Flächen in meiner Schrift von 1881), hat nur wenig Anklang gefunden. Ich meine aber, da[ß]{} das nicht am Gegenstande der Untersuchung liegt, der mir viel mehr das grö[ß]{}te Intere[ß]{}e zu verdienen scheint, sondern an der knappen Form, mit der ich meine Resultate darstellte.* ]{} Of course this impact was first limited to his direct circle of students, where we count Harnack 1876 [@Harnack_1876], Weichold 1883 [@Weichold_1883] and Hurwitz 1883 [@Hurwitz_1883] (also a student of Weierstrass). Klein was also very proud that his results on real moduli supplied a natural answer to questions addressed (but not solved) at the end of Riemann Thesis. Klein insists twice on this issue in 1882 [@Klein_1882]=[@Klein-Werke-III_1923 p.572, §24] and in his subsequent lectures 1891/92 [@Klein_1891--92_Vorlesung-Goettingen p.154], where he writes: “[*Mit dieser Abzählung ist implicite die entsprechende Frage für beantwortet, was darum ein gewi[ß]{}es Intere[ß]{}e hat, weil diese Frage von Riemann in seiner Di[ß]{}ertation aufgeworfen, aber nicht zu Ende discutirt wird. Riemann denkt natürlich nur an berandete Flächen (nicht an Doppelflächen; dem deren Existenz wurde erst zehn Jahre später von Moebius bemerkt und wohl erst in meiner Schrift für funktionentheoretische Zwecke herangezogen).*]{}” From the very beginning 1876 [@Klein_1876]=[@Klein-Werke-II_1922 §7,p.154], Klein noticed that real curves are subjected to the dichotomy of being dividing or not, where the former case amounts to a separation of the complex locus through its real part (consisting of [*ovals*]{}, a jargon immediately suggesting Hilbert’s 16th problem, yet used much earlier, e.g. by Zeuthen 1874 [@Zeuthen_1874]). Zeuthen’s work seems to have much inspired Klein’s investigation on real curves, starting circa 1876, just two years later (cf., e.g. Klein 1892 [@Klein-Werke-II_1922 p.171]: “[*Ich hatte 1876 den Ausgangspunkt unmittelbar von den Kurven genommen. Das war bei $p=3$ möglich, wo ich zahlreiche geometrische Vorarbeiten, insbesondere diejenigen des Herrn Zeuthen \[…\] (1874), benutzen konnte.*]{}”) Perhaps, the more tenacious followers of Klein’s viewpoint came somewhat later and the real demographic explosion of the subject took place much later, say perhaps in the 1970’s. Here is a little chronology: $\bullet$ del Pezzo 1892 [@del-Pezzo_1892], where Klein’s trick of assigning the unique real point of an imaginary tangents is taken as the starting point of a study of curves of low genus. $\bullet$ Berzolari 1906 [@Berzolari_1906], who in an encyclopedia article surveyed in few pages Klein’s achievements and virtually coined the term “Klein surfaces” (Kleinsche Flächen) as a way to designate possibly non-orientable and eventually bordered avatar of Riemann surfaces. To say the least, this terminology was dormant during several decades until Alling-Greenleaf managed in 1969 [@Alling-Greenleaf_1969] a resurrection of Berzolari’s coinage, and since then the nomenclature gained in popularity. $\bullet$ Koebe 1907 [@Koebe_1907_UrAK] who studied uniformization of real algebraic curves taking advantage of Klein’s distinction orthosymmetric vs. diasymmetric. $\bullet$ Severi 1921 [@Severi_1921-Vorlesungen-u-alg.-Geom-BUCH p.230–6], who devotes some few pages of his book to Klein’s theory of real curves, \[Note: there Severi writes down the same formula as one used by Courant in his approach to conformal circle maps, ascribing it to Cauchy\]. $\bullet$ Comessatti 1924-25 [@Comessatti_1924/26] in Italy (full of admiration for Klein), who pushed the philosophy up to include a study of real abelian varieties, rational varieties, etc. (For this ramification we refer to the remarkable survey by Ciliberto-Pedrini 1996 [@Ciliberto-Pedrini_1996].) $\bullet$ several works of Cecioni in the late 1920’s ([@Cecioni_1929], [@Cecioni_1933], [@Cecioni_1935]), and his students (Li Chiavi 1932 [@Stella-li-Chiavi_1932]) makes direct allusion to Klein’s works. $\bullet$ In France, the work of Klein found a little echo in some passages of the book by Appell-Goursat whose second tome (1930) was apparently mostly written by Fatou. There, Klein’s orthosymmetry occurs at several places [@Appell-Goursat-Fatou_1930 p.326–332 and p.513–521]. $\bullet$ Witt 1934 [@Witt_1934], where a general existence theorem for [*invisible*]{} real algebraic curves (those with empty real locus like, e.g. $x^2+y^2=-1$) was established. This will be discussed later (Sec.\[sec:Witt\]), and is somehow quite akin to the Ahlfors function. Witt’s work makes explicit mention of Klein, and was subsequently elaborated by Geyer 1964/67 [@Geyer_1964-67], who arranged a purely algebraic interpretation of Weichold’s work. His pupil G. Martens, managed (1978 [@Martens_1978]) to determine the lowest possible degree of the Witt mapping; $\bullet$ (Jesse) Douglas 1936–39 makes a systematic use of Klein’s symmetric surfaces in his study of Plateau’s problem for configuration of higher topological structure. (We shall have to come back to this in Sec.\[sec:Douglas\].) $\bullet$ A marked influence of Klein upon Teichmüller 1939 [@Teichmueller_1939], 1941 [@Teichmueller_1941]. We shall try to explore this connection in greater detail later (Section \[sec:Teichmueller\]). Then different events occurred at a rather rapid pace with several schools penetrating into Klein’s reality paradigm through different angles: $\bullet$ Ahlfors 1950 [@Ahlfors_1950], who never quotes Klein. Probably with Lindelöf and Nevanlinna as teachers one is more inclined toward hard analysis à la Schwarz, than innocent looking geometry à la Klein. Of course Ahlfors quotes instead Schottky, as typified by the terminology Schottky differential, etc. used in Ahlfors 1950 (). It may then appear as a little surprise that Ahlfors’ result affords a purely algebraic (in term of real function fields) characterization of Klein’s orthosymmetric curves. However to my knowledge, this connection—as trivial as it is—was never emphasized in print until much later, namely in Alling-Greenleaf 1969 [@Alling-Greenleaf_1969]. $\bullet$ Schiffer-Spencer’s book 1954 [@Schiffer-Spencer_1954] (outgrowing from Princeton lectures held during the academic year 1949–50) where the book is started by recalling how Klein assimilated the full Riemannian concept after a 1874 discussion with Prym revealing him the ultimate secret of Riemann’s function theory developed over arbitrarily curved surfaces not necessarily spread over the plane. The original source reads as follows: \[quote:Klein-Prym\] [Ich wei[ß]{} nicht, ob ich je zu einer in sich abgeschlossenen Gesamtaufassung gekommen wäre, hätte mir nich Herr Prym vor längeren Jahren (1874) bei gelegentlicher Unterredung eine Mitteilung gemacht, die immer wesentlicher für mich geworden ist, je länger ich über den Gegenstand nachgedacht habe. Er erzählte mir,]{} [*da[ß]{} die Riemannschen Flächen ursprünglich durchaus nicht notwendig mehrblättrige Flächen über der Ebene sind, da[ß]{} man viel mehr auf beliebig gegebenen krummen Flächen ganz ebenso komplexe Funktionen des Ortes studieren kann, wie auf den Flächen über der Ebene.* ]{} From circa 1970 upwards, the study of so-called Klein surfaces (jargon of Berzolari [@Berzolari_1906]) [*per se*]{} enjoyed a rather exponential rate of growth as if the simple naming of them was a stimulus for a big expansion of the topic. After two decades an impressive body of knowledge has been accumulated (cf. e.g. the rich bibliography compiled in Natanzon’s survey 1990 [@Natanzon_1990/90]). Those developments can be roughly ranged into 3 main axes: $\bullet$ [*Foundational aspects.*]{}—Alling-Greenleaf 1971 [@Alling-Greenleaf_1971], and also in Romania with the numerous contribution of Andreian Cazacu (1986–88 [@Andreian-Cazacu_1986], [@Andreian-Cazacu_1988-Interior]) about the structure of morphism between them (interior influence of Stoilow). $\bullet$ [*Symmetry, automorphisms and NEC(=non-Euclidean crystallography).*]{}—This is especially active in the Spanish school but started somewhat earlier with Singerman 1971–88 (5 items), May 1975–88 (9 items), Bujalance 1981–89 (29 items) Costa, etc. $\bullet$ [*Moduli spaces of Klein surfaces.*]{} This starts of course in Klein 1882 [@Klein_1882], to reach a certain climax in Teichmüller 1939 [@Teichmueller_1939] and the Ahlfors-Bers school, Earle 1971, Seppäla 1978–89 (6 items on Teichmüller and real moduli), Silhol 1982–89 (Abelian varieties and Comessatti), Costa, Huisman 1998+, etc. All those works contributed to feel virtually as comfortable with real curves as with their complex grand sisters. We just mention one result of Seppälä 1990 (revisited by Buser-Seppälä-Silhol 1995 [@Buser-Seppala-Silhol_1995] and Costa-Izquierdo 2002 [@Costa-Izquierdo_2002]) to the effect that the moduli space of real curves is connected. (This sounds almost like a provocation to anybody familiar with the bio-diversity of topological types of symmetric surfaces listed by Klein). Of course the trick, here, is that those authors regard this moduli space projected down in that of complex curves (by forgetting the real involution). In other words we may deform the structure until new anti-conformal symmetries appears and switch from one to the other. Hence the subject is sometimes hard to grasp (due to varying jargon) and more seriously is full of real mysteries allied to the real difficulty of the subject. $\bullet$ [*Geometry of real curves*]{}. Here much of the impulse—very much in Klein’s tradition—came through the paper of Gross-Harris 1981 [@Gross-Harris_1981]. In this or related direction, we may cite authors like Natanzon, Ballico, Coppens, G. Martens, Huisman, Monnier, etc. This area proved very active since the 2000’s up to quite recently and a remarkable variety of difficult question are addressed giving the field arguably some maturity soon comparable to the complex hegemony. Of course, another line of thought is the interest aroused by Hilbert’s 16th problem (on the mutual disposition of circuits of real algebraic varieties esp. curves) especially among the early German, Italian and then mostly the Russian annexion of the subject. This captured and probably contributed to mask Klein’s more intrinsic viewpoint for a while. This axis includes the following workers (precise references listed in Gudkov 1974 [@Gudkov_1974/74]): $\bullet$ Hilbert 1891–1900–09, Rohn 1886–1911–11–13; (it is interesting to note that Hilbert’s first 1891 paper on the subject is quite synchronized with Klein’s lectures of 1891/92, which conjecturally may have stimulated Hilbert’s interest, yet not a single allusion to Klein in this paper, and recall also that Hilbert was still in Königsberg at that time). $\bullet$ Brusotti 1910–13–14–14–15–16–16–16–16–17–21–28–38/39–40–44/45–46–50/51–52–55–55 (characterized by “[*la piccola variazione*]{}”, i.e. the method of small perturbation permitting to construct real algebraic curves with controlled topology. The writer is indebted to Felice Ronga for this method, which of course has some historical antecedents older that Brusotti. In Klein 1873, footnote 2 in [@Klein-Werke-II_1922 p.11] the principle is traced at least back to Plücker 1839 [@Plücker_1839]. However Brusotti 1921 [@Brusotti_1921] may have been the first—modulo its reliance over work of Severi—to notice that the Riemann-Roch theorem admits as extrinsic traduction the possibility of smoothing independently the nodes of a plane curve. The main issue (as transmitted by Felice) is that the nodes a plane curve with nodal singularities impose independent conditions on curves of the same degree. Hence when the curve is being imagined as a point in the (projective) space of all curves, it sits on the discriminant hypersurface (parameterizing all singular curves) and nearby our nodal curve we see several transverse smooth branches crossing transversally. (In French or Italian, there are better synonyms like “falde analytiche” or “nappe”.) The net effect of transversality is that one can leave at will certain strata, while staying on others. This implies the independency of smoothing crossings, and thereby a rigorous foundation to the small perturbation method. (The resulting graphical flexibility of algebraic curves is a pleasant way to create Riemann surfaces, and we shall exploit it later in this text as a way to explore degrees of Ahlfors circle maps.) $\bullet$ Comessatti (more in the spirit of Klein) 1924–25–27/28–31–32–33, etc. $\bullet$ Petrovskii 1938–49 ([@Petrowsky_1938]), etc. many joint with Oleinik (real algebraic (hyper)surfaces and Betti numbers). $\bullet$ Gudkov 1954–54–62–62–62–65–66–69–69–73 (those works include in particular the spectacular discovery of a sextic whose oval configuration was expected to be impossible by Hilbert). $\bullet$ Arnold 1971–73. $\bullet$ Rohlin 1972–72–73. $\bullet$ Finally the long awaited (?) reunification of forces (call it maybe the Klein-Hilbert Verschmelzung) came in the work of Rohlin (himself apparently inspired by Arnold). Surprisingly, Rohlin took notice of Klein’s work quite late, ca. 1978 (compare Rohlin 1978 [@Rohlin_1978]). $\bullet$ Then real algebraic geometry exploded through the work of Kharlamov, Viro, Fiedler, Nikulin 1979, Orevkov, Finashin, etc. and in the west Risler, Marin, and many others gave a new golden age to a discipline reaching a certain popularity. Sometimes the real theory seems only to adapt over ${\Bbb R}$ whatever has been achieved over ${\Bbb C}$, yielding usually a kaleidoscopic fragmentation of truths into a real zoology. Thus for instance the Castelnuovo-Enriques classification of (algebraic) surfaces can be pushed through reality: K3 (Nikulin-Kharlamov), Abelian surfaces (Comessatti-Silhol), elliptic surfaces, etc. The topic is then strongly allied to deep methods in differential topology, Galois cohomology, symplectic geometry, Gromov-Witten, enumerative problems, tropical geometry, etc. The present number of active workers is so impressive and the recent connections so amazing (Okounkov, etc.) that we prefer to stop here our impressionist touristic overview of real algebraic geometry. Digression about Hilbert’s 16th problem (Klein 1922, Rohlin 1974, Kharlamov-Viro ca. 1975, Marin 1979, Gross-Harris 1981) -------------------------------------------------------------------------------------------------------------------------- The connection between Klein’s theory (especially the ortho- and diasymmetric dichotomy) with Hilbert’s 16th problem (plane curves in the projective plane ${\Bbb P}^2$) were profoundly investigated by the Russian school in the early 1970’s especially Arnold, Rohlin, Viro, Kharlamov, etc. Klein himself always dreamed of such a relationship , without really being able to formulate its precise shape. Here is a quote which Klein added (ca. 1922) to his Werke (cf. [@Klein-Werke-II_1922 p.155, footnote]): \[Klein-1922-immer-vorsgeschwebt:quote\] Es hat mir immer vorgeschwebt, dass man durch Fortsetzung der Betrachtungen des Textes Genaueres über die Gestalten der reellen ebenen Kurven beliebigen Grades erfahren könne, nicht nur, was die Zahl ihrer Züge, sondern auch, was deren gegenseitige Lage angeht. Ich gebe diese Hoffnung auch noch nicht auf, aber ich muss leider sagen, dass die Realitätstheoreme über Kurven beliebigen Geschlechtes (welche ich aus der allgemeinen Theorie der Riemannschen Flächen, speziell der “symmetrischen” Riemannschen Flächen ableite) hierfür nicht ausreichen, sondern nur erst einen Rahmen für die zu untersuchenden Möglichkeiten abgeben. In der Tat sind ja die doppelpunktslosen ebenen Kurven $n$-ten Grades für $n>4$ keineswegs die allgemeinen Repräsentanten ihres Geschlechtes, sondern wie man leicht nachrechnet, durch $(n-2)(n-4)$ Bedingungen partikularisiert. Da man über die Natur dieser Bedingungen zunächst wenig weiss, kann man noch nicht von vornherein sagen, dass alle die Arten reeller Kurven, die man gemäss meinen späteren Untersuchungen für $p={ n-1 \cdot n-2 \over 2}$ findet, bereits im Gebiete besagter ebener Kurven $n$-ter Ordnung vertreten sein mü[ß]{}ten, auch nicht, da[ß]{} ihnen immer nur [*eine*]{} Art ebener Kurven entspräche. K. It took several decades until the experimentally obvious conjecture (possibly anticipated by Klein, though he left no trace in print) that dividing curves in the plane have at least as many ovals as the half value of its degree found place in a paper of Gross-Harris 1981 [@Gross-Harris_1981 p.177, [*Note*]{}]. In fact, in a paper by Alexis Marin 1979/81 [@Marin_1979] this is stated as a corollary of a Rohlin formula (1978 [@Rohlin_1978]), involving intersection of homology classes deduced from the halves of the dividing curve capped off by the interiors of ovals in ${\Bbb P}^2({\Bbb R})$). In the case of $M$-curves (=the Russian synonym of Harnack-maximal coined by Petrowskii 1938 [@Petrowsky_1938]), this technique occurred earlier in Rohlin 1974/75 [@Rohlin_1974/75]. Moral: the tool missing to Klein was intersection theory of homology classes developed by Poincaré, Lefschetz, etc. In the little note Gabard 2000 [@Gabard_2000] it is verified that this Rohlin-Marin obstruction ($r\ge \frac{m}{2}$) is the only one, settling thereby completely the Klein-Gross-Harris question. This (simple) fact was known to Rohlin’s students Kharlamov and Viro which were familiar with this result as early as the middle 1970’s (as they both kindly informed me by e-mail). Of course the crucial ideas are due to Rohlin. A long unnoticed tunnel between Klein and Ahlfors (Alling-Greenleaf 1969, Geyer-Martens 1977) --------------------------------------------------------------------------------------------- More importantly, for our present purpose is to keep the abstract viewpoint of Klein (by opposition to the embedded Hilbert’s 16th problem), and to make the following observation. [(Klein?, Teichmüller 1941?, Ahlfors 1948/50, Matildi 1945/48?, Andreotti 1950?, who else?)]{} Dividing curves are precisely those admitting a real morphism (i.e., defined over the ground field ${\Bbb R}$) to the projective line ${\Bbb P}^1$ such that all fibers over real points consist entirely of real points. The non-trivial implication of this fact follows precisely from Ahlfors 1950 [@Ahlfors_1950] (but is made very explicit only in Alling-Greenleaf 1969 [@Alling-Greenleaf_1969], see also Geyer-Martens 1977 [@Geyer-Martens_1977]). To my actual knowledge there is no record in print of this fact prior to Ahlfors’ intervention, modulo the cryptical allusion in Teichmüller 1941 [@Teichmueller_1941] that the result was implicit in Klein’s works. Another related works are those of Matildi 1948 [@Matildi_1945/48] and Andreotti 1950 [@Andreotti_1950]. As we shall recall later Ahlfors’ result was exposed at Harward as early as 1948 (cf. Nehari 1950 [@Nehari_1950] reproduced here as Quote \[Nehari-1950:quote\]). It is however picturesque to notice that an analog result stating that a real curve without real points maps through a real morphism upon the empty curve $x^2+y^2=-1$ (or projectively $x_0^2+x_1^2+x_2^2=0$) was established as long ago as Witt 1934 [@Witt_1934]. Perhaps both problems are of comparable difficulty, and the method employed by Witt—namely Abelian integrals—turns out to be likewise relevant to the Ahlfors context (i.e. dividing curves). Hence in our opinion, there were no technological obstruction to Ahlfors result being discovered much earlier, say by Witt in the 1930’s, or by Bieberbach in 1925 [@Bieberbach_1925], or by Klein in the 1876–80’s, or even by Riemann in the late 1850’s (especially in view of his [*Parallelogramm methode/Figuren*]{}, cf. e.g., Haupt 1920 [@Haupt_1920]), and ultimately why not by Abel himself? (Of course all these peoples were probably involved with more urgent tasks, like some [*flüchtigen Versuche*]{} about the Riemann hypothesis, or regarding Klein the [*Grenzkreistheorem*]{} (in his health taking contest with Poincaré), which later became known as the uniformization theorem. The list of competent workers coming also very close to the paradigm ultimately discovered by Ahlfors could easily be elongated: especially Schwarz, Hurwitz (esp. in 1891 [@Hurwitz_1891-Uber-Riemannsche-Flachen]), Koebe, Courant (esp. in 1939 [@Courant_1939], 1940 [@Courant_1940-Acta] or 1950 [@Courant_1950]). As to the interesting result of Witt 1934 (on invisible real curves), we will try to discuss it later in more details (Sec.\[sec:Witt\]). Albeit, the writer lacks strong (analytical) competence, we shall attempt to delineate several schools (and teams of workers) scattered through the planet, which are still exploring this or related topics. Motivation (better upper bounds exist) -------------------------------------- Even though Ahlfors’ result is approaching 65 years (a venerable age for retirement) the basic result looks still grandiose, and mysterious enough if one wonders about the exact distribution of Ahlfors’ degrees (as suggested in Yamada-Gouma’s penetrating study (1978–1998–2001), discussed in Sec.\[Yamada-Gouma:subsec\]). The writer published a paper in 2006 [@Gabard_2006] where a circle map with fewer sheets (viz. $\le r+p$) than that proposed by Ahlfors (namely $\le r+2p$) is exhibited. This quantitative improvement is the motivation for much of this survey, and will hopefully excuses the bewildering variety of topics addressed. An obvious game is to renegotiate known application of the Ahlfors’ mapping involving a controlled degree in the hope to upgrade the bound. As tactically simple as it may look, we were not very successful in this game as it often already requires analytical skills beyond the competence of the writer. Yet we shall mostly content to list some articles where some upgrade could be expected (e.g., Hara-Nakai’s quantitative version of the corona with bounds [@Hara-Nakai_1985] looks to be a challenging place to test). Of course for this [*bound upgrading procedure*]{} to work it requires that the application in question does not use the full strength of the Ahlfors function, but only its qualitative property of being a circle map. A concrete instance were this was accomplished is Fraser-Schoen’s paper 2011 [@Fraser-Schoen_2011]. Alternatively we can dream of certain high powered applications requiring the full extremal power of the Ahlfors mapping. In this case it is known a priori (Yamada-Gouma) that we cannot lower the degree of the Ahlfors function, except possibly for very particular choices of base-points. So the main philosophical issue is roughly the following point: Is the Ahlfors extremal property truly required in applications, or just the arcane residue of those attempt to salvage the Dirichlet principle via extremal methods. Put differently, is the extremal problem just an artefact of the proof or something really worth exploiting in practice? Full coverings versus Ahlfors’ extremals ---------------------------------------- To avoid any confusion, one must from the scratch relativize strongly the importance of the recent contribution on the $r+p$ bound (Gabard 2006 [@Gabard_2006]) for several reasons. First the result is quite recent and probably not sufficiently verified as yet. In later sections when looking at explicit curves from the experimental viewpoint it seems that there is a large armada of potential counterexamples flying at high altitudes (flying fortresses). Next, Ahlfors’ upper bound $r+2p$ is known to be sharp within the realm of the extremal problem it solves. Indeed, Yamada 1978 [@Yamada_1978] has a rather simple argument showing that the Ahlfors function centered at the Weierstrass points of a hyperelliptic membrane has degree precisely $r+2p$ (and not less). Maybe it is an open question whether a similar sharpness holds for all membranes. Hence, one must keep in mind a subtle distinction between Ahlfors’ deep extremal problem (involving hard analysis via the paradigm of extremality) and the writer’s soft version ([@Gabard_2006]) which leads to a sharper bound but is based only upon (soft) topological methods, i.e. the Brouwerian degree and the allied criterion of surjectivity. To put it briefly, we must distinguish Ahlfors’ extremal function from the mere [*circle map*]{}, defined as follows (nomenclature borrowed from Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950 p.182]): A circle map is an analytic function from a compact bordered Riemann surface to the disc, expressing the former as a (generally branched) cover of the disc, say $f\colon W \to D=\{\vert z\vert \le 1\}$. Each interior points maps to an interior points of the disc (otherwise there is a problem as infinitesimally the mapping is a power map $z\mapsto z^n$, $n\ge 1$). Thus, the restricted covering $\partial W \to \partial D=S^1$ is unramified, whereupon it follows that $r\le \deg (f)$ (i.e. the number of contours is a trivial lower bound for the degree of a circle map). Varied synonyms (or closely allied designations) are used throughout the literature (here is a little sampling with citation of the relevant sources): $\bullet$ $n$ fach ausgebreitete Fläche, $n$ fach bedeckende Fläche (Riemann 1857–Weber 1876 [@Riemann_1857_Nachlass p.473]); $\bullet$ Schottky 1877 no clear cut terminology, and re-reading it (25.06.12) in details I realize that the statement about existence of circle maps is in fact not really proved (thus much of the written is somewhat biased), note that Bieberbach somewhat wrongly ascribe the result as well to Schottky, but that remains to be elucidated... In contrast, Grunsky never (?) credits Schottky, but rather Bieberbach 1925 [@Bieberbach_1925]; $\bullet$ mehrfach bedeckte Kreisscheibe, $n$-blättrige Kreisscheibe (Bieberbach 1925 [@Bieberbach_1925 p.6]); $\bullet$ mehrblättrige Kreise, $n$-blättrige Kreisscheibe (Grunsky 1937 [@Grunsky_1937 p.40]); $\bullet$ ein endlichvielblättriges Flächenstück über der oberen $z$-Halbebene mit endlich vielen Windungspunkten, das durch Spiegelung an der reellen Achse eine symmetrische geschlossene Riemannsche Fläche ergibt (Teichmüller 1941 [@Teichmueller_1941]); $\bullet$ cerchio multiplo (Matildi 1945/48 [@Matildi_1945/48 p.82], a student of Cecioni); $\bullet$ full covering surface of the unit circle (Ahlfors 1950 [@Ahlfors_1950 p.124, p.132]); $\bullet$ $(2g+m)$-sheeted unbounded covering surface of the unit disc (Encyclopedic Dictionary of Mathematics 1968/87 [@EDM_1968/87 p.1367]); $\bullet$ unbounded finitely sheeted covering surfaces of the unit disk (Nakai 1983 [@Nakai_1983 p.164]); $\bullet$ Schottky functions (Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949 p.214], Kühnau 1967 [@Kuehnau_1967 p.96], and earlier (yet without this appellation) in several works of Picard, e.g. Picard 1913 [@Picard_1913] and Cecioni, e.g. Cecioni 1935 [@Cecioni_1935]); $\bullet$ $p$-times covered unit-circle (Bergman 1950 [@Bergman_1950 p.87, line 5]); $\bullet$ $n$-times covered circle, multiply-covered circle (Nehari 1950 [@Nehari_1950 p.256, resp. p.267], Stanton 1971 [@Stanton_1971 p.289 and 293] Aharonov-Shapiro 1976 [@Aharonov-Shapiro_1976 p.60]); $\bullet$ Ahlfors mapping (Nehari 1950 [@Nehari_1950 p.256, p.267], Stanton 1971 [@Stanton_1971 p.289 and 293]; $\bullet$ Ahlfors function (Aharonov-Shapiro 1976 [@Aharonov-Shapiro_1976 p.60]); $\bullet$ Ahlfors map (Alling 1966 [@Alling_1966 p.345–6], Stout 1967 [@Stout_1967-Interpolation p.274], and then in many papers by Bell); $\bullet$ Ahlfors type function (Yakubovich 2006 [@Yakubovich_2006 p.31]); $\bullet$ Einheitsfunktionen (Carathéodory 1950 [@Caratheodory_1950_Buch_Funktionentheorie vol.II, p.12]), translated as: $\bullet$ [*unitary*]{} function in Heins 1965 [@Heins_1965 p.130], a jargon also adhered to by Fay 1973 [@Fay_1973 p.108, 111, etc.]; $\bullet$ unimodular function (Douglas-Rudin 1969 [@Douglas-Rudin_1969], Fisher 1969 [@Fisher_1969-BAMS-convex-combination-unimodular-fct], Gamelin 1973 [@Gamelin_1973-BAMS], Lund 1974 [@Lund_1974]); $\bullet$ many-sheeted disc (A. Mori 1951 [@Mori_1951]); $\bullet$ multi-sheeted circle (Havinson 1953 [@Havinson_1953]); $\bullet$ finitely sheeted disks (Hara-Nakai 1985 [@Hara-Nakai_1985]); $\bullet$ Vollkreisabbildung (Meschkowski 1951 [@Meschkowski_1951 p.121]); $\bullet$ (volle) $n$-blättrige (Einheits)Kreisscheibe (Golusin 1957 [@Golusin_1952/57 p.240, 412], as translated by Grunsky or Pirl); $\bullet$ interior mappings (Stoïlow, Beurling); $\bullet$ inner functions (Beurling 1949 [@Beurling_1949], Hoffman 1962 [@Hoffman_1962] (esp. p.74, where Beurling is credited of the coinage), Rudin 1969 [@Rudin_1969], Stout 1972 [@Stout_1972 p.343], Černe-Forstnerič 2002 [@Cerne-Forstneric_2002 p.686]). This concept usually refers to analytic functions with modulus a.e. equal to one along the boundary, but some writers corrupted this sense to mean a circle map, cf. Stout 1966/67 [@Stout_1966/67] which is followed by Fedorov 1990/91 [@Fedorov_1991 p.271]. $\bullet$ boundary preserving maps (Jenkins-Suita 1984 [@Jenkins-Suita_1988]); maps taking the boundary into the boundary (Landau-Osserman 1960 [@Landau-Osserman_1960]). $\bullet$ complete covering surfaces (cf. Ahlfors-Sario 1960 [@Ahlfors-Sario_1960 p.41–42, §21A]), i.e. one such that any point in the range has a neighborhood whose inverse image consists only of compact components; complete Klein coverings (Andreian Cazacu 2002 [@Andreian-Cazacu_2002]) (a direct extension of the former concept shown to be equivalent in the case of finite coverings to the next conception of Stoïlow). $\bullet$ total Riemann coverings (Stoïlow 1938 [@Stoilow_1938-Lecons]), i.e. one such that any sequence tending to the boundary has an image tending to the boundary. $\bullet$ unlimited covering surfaces (Nakai 1988 [@Nakai_1988], EDM=Japanese encyclopedia 1968/87 [@EDM_1968/87], Minda 1979 [@Minda_1979-hyperbolic-metric-and-coverings]) $\bullet$ proper (holomorphic) maps (onto the unit disc) (e.g., Bedford 1984 [@Bedford_1984 p.159], Bell 1999 [@Bell_1999-Ahlfors-maps p.329], Černe-Flores 2007 [@Cerne-Flores_2007], Fraser-Schoen 2011 [@Fraser-Schoen_2011]). $\bullet$ distinguished map (Jurchescu 1961 [@Jurchescu_1961-A-maximal]) $\bullet$ Myrberg surface over the unit disc (Stanton 1975 [@Stanton_1975 p.559, §2] uses this terminology for a Riemann surface $W$ admitting an analytic function $z\colon W \to \Delta$ realizing $W$ as an $n$-sheeted, branched, full covering surface of the unit disc $\Delta$). As no ramification appears along the boundary, explains the naming: $\bullet$ Randschlicht mapping (Köditz-Timann 1975 [@Koeditz-Timmann_1975]). In fact the writer came across this concept through real algebraic geometry where I used (2006 [@Gabard_2006]) the term [*saturated*]{}, whereas Coppens 2011 [@Coppens_2011] proposes the term [*separating*]{} morphism. In the same context, Geyer-Martens 1977 [@Geyer-Martens_1977] coined: $\bullet$ “total reell Morphismus”=totally real morphism/map, to which we shall adhere as it seems to be the most convenient terminology, especially when abridged just as “total maps”, which is quite in agreement with Stoïlow’s jargon. We shall attempt to reserve the designation [*Ahlfors maps/functions*]{} for those solving the extremal problem formulated in Ahlfors 1950 [@Ahlfors_1950]. The latter are known (since Ahlfors 1950 [@Ahlfors_1950]) to be circle maps, but the converse is wrong. Indeed, circle maps may have arbitrarily large degrees (post-compose with a power map $z\mapsto z^n$ for some large integer $n$), whereas Ahlfors maps have degrees $\le r+2p$ (in view of the deep result in Ahlfors 1950 [@Ahlfors_1950]). Are circle maps of degree compatible with Ahlfors’ bound always realizable via an Ahlfors map? The answer seems to be in the negative, at least if attention is restricted to infinitesimal Ahlfors maps. This follows from Gouma’s restriction (1998 [@Gouma_1998]) in the hyperelliptic case. Indeed consider a 2-gonal membrane, then post-composing with $z\mapsto z^n$ we get circle maps of degrees ranging through all multiples $2n$, whereas only 2 and $r+2p$ are realized as degrees of Ahlfors maps, by a result of Gouma 1998 [@Gouma_1998]. Note that Gouma restricts to ponctual Ahlfors maps and our claim is only firmly established in this context. A somewhat deeper question is whether any (or at least one) circle map of smallest degree arises via an Ahlfors map. We were not able to settle this question, but in a tour de force Yamada 2001 [@Yamada_2001] proved this in the hyperelliptic case. It amounts to know if the Ahlfors map is flexible enough to capture a circle map of the lowest possible degree (alias the gonality). Let us optimistically pose the conjecture, amounting to say that we can essentially take out the best of the two worlds: \[gonality:conj\] Any (or at least one) conformal mapping realizing the gonality arises as an Ahlfors extremal function $f_{a,b}$ (perhaps for coalescing two points yielding then the Ahlfors map $f_a$ maximizing the modulus of the derivative at $a$). Recent work by Marc Coppens 2011 [@Coppens_2011] supplies a sharp understanding of the gonality $\gamma$ as spreading through all permissible values $r\le \gamma \le r+p$ when the membrane is varied through its moduli space. Paraphrased differently the conjecture wonders if a suitable Ahlfors map always realizes the gonality. As yet we lack evidence, but the vague feeling that Ahlfors’ method is the best possible (being distilled by the paradigm of extremality) inclines one to believe that its economy should be God given. In contradistinction, it may be argued that Ahlfors maps depend on so few parameters (essentially one or two points on the surface), that they are perhaps not flexible enough to explore the full room of all circle maps. Such simple minded question exemplifies that the old subject of the Ahlfors’ map still deserves better understanding. A fine understanding of the Ahlfors map would truly be worth studying if we had some clear-cut applications in mind (taking full advantage of the extremal property of the map). In practice, one is often content with the weaker notion of circle maps, but in the long run it is likely that more demanding applications requires the full punch of the Ahlfors map. Sorting out applications: finite vs. infinite/compact vs. open -------------------------------------------------------------- As to applications (of the Ahlfors map), there are several ramifications, which —at the risk of oversimplification—may be ranked in two headings ([*in finito*]{} vs. [*in infinito*]{}). By this we have in mind essentially the sharp opposition between compact and non-compact Riemann surfaces. The later were intensively approached by several schools (mostly Finnish, Japanese and US), but the theory is certainly less complete than for compact surfaces, which from our viewpoint already represent a serious challenge. Furthermore it is evident that there is essentially one and only one road leading from the finite to the infinite namely the exhaustion process affording a cytoplasmic expansion of a compact bordered Riemann surface in some ambient open surface. Now let us enumerate such applications. \(A) [**Lifting truth from the disc via conformal transplantation.**]{} A reliable philosophy is roughly that a result known to hold good in the disc is lifted via the Ahlfors map to configurations of higher topological type. This is the strategy used by Alling 1964 [@Alling_1964] to transplant the corona of Carleson 1962 [@Carleson_1962] to Riemann surfaces. (The corona theorem amounts to say that the Riemann surface is dense in the maximal ideal space of its algebra of bounded analytic functions.) In spectral theory this method (systematically utilized by Polyá-Szegö) is known as “[*conformal transplantation*]{}”. Subsequent elaborations arose through the work of Hersch 1970 [@Hersch_1970] and Yang-Yau 1980 [@Yang-Yau_1980] (where branched covering are admitted, thereby diversifying widely the topology). Recently Fraser-Schoen 2011 [@Fraser-Schoen_2011] applied the Ahlfors mapping to spectral theory (Steklov eigenvalues). (This inspired a note of the writer [@Gabard_2011] extending Hersch 1970’s study of Dirichlet and Neumann eigenvalues on spherical membranes to arbitrary (compact) bordered surfaces.) Another spectacular work is due to Girouard-Polterovich 2012 [@Girouard-Polterovich_2012] where Fraser-Schoen’s work is extended to higher eigenvalues. \(B) [**Exhaustion and infinite avatars.**]{} Another philosophy (Nevanlinna, Ahlfors, etc.) is to exploit the fact that (infinite, i.e. open Riemann surface) may be exhausted by compact subregions (reminding somehow the finitistic slogan of André Bloch, [*“Nihil est in infinito...”*]{}) offering thereby a wide range of application of compact bordered Riemann surfaces to the more mysterious realm of open Riemann surfaces. This ramifies quickly to the so-called classification theory of Riemann surface (Nevanlinna 1941 [@Nevanlinna_1941], Ahlfors 46, Sario 46–49, Parreau 1951 [@Parreau_1951], Royden 1952 [@Royden_1952], etc.) much completed by the Japanese school (Tôki 1951, A. Mori, Kuramochi, Kuroda, etc.). Several books attempt to give a coherent account of this big classification theory, e.g. Ahlfors-Sario 1960 [@Ahlfors-Sario_1960], Sario-Nakai 1970 [@Sario-Nakai_1970], where the guiding principle (due to Sario 1946) is to classify surfaces according to the force of their ideal boundary. In another infinite direction, S.Ya. Havinson 1961/64 [@Havinson_1961/64] was the first (with Carleson 1967 [@Carleson_1967-book]) to extend the theory of the Ahlfors function to domains of infinite connectivity , and was followed by S. Fisher 1969 [@Fisher_1969], which propose some simplifications. The Slovenian school of complex geometry (Černe, Forstnerič, Globevnik, etc.) are also employing the Ahlfors function, often in connection with the open problem (Narasimhan, Bell, Gromov, etc.) of deciding if any open Riemann surface embeds properly in ${\Bbb C}^2$. In Forstnerič-Wold 2009 [@Forstneric-Wold_2009] reduced the full problem to a finitary question as to whether each compact bordered Riemann surface embeds holomorphically in the plane ${\Bbb C}^2$. (Maybe this is achievable by a suitable of Ahlfors functions, or more sophisticated variant thereof like (?) in the broader Pick-Nevanlinna context). As suggested by those authors, it is maybe enough to embed one representant in each topological type (this is possible, compare Černe-Forstnerič 2002 [@Cerne-Forstneric_2002 Theorem 1.1]) and try to use a continuity argument through the Teichmüller (moduli) space. Biased recollections of the writer {#Sec:Biased-recollections-of-Gabard} ================================== Klein’s viewpoint: real curves as symmetric Riemann surfaces (as yet another instance of the Galois-Riemann Verschmelzung) -------------------------------------------------------------------------------------------------------------------------- If the writer is allowed to recollect his own memories about his involvement with this circles of ideas, it started as follows. Maybe a natural point of departure is the (basic) algebraic geometry of curves. While reading Shafarevich’s Basic algebraic geometry (ca. 1998) one encounters some nice drawings of the real locus of a plane cubic into its complex locus materialized by a torus (as we know since time immemorial: Euler?, Abel?, Jacobi, Riemann, etc.). A torus of revolution reflected across a plane cutting the torus along two circles yields a plausible visualization of the embedding of $C({\Bbb R})$ into $C({\Bbb C})$ (even with the symmetry induced by the complex conjugation). Of course there are also real cubic curves whose real loci possess only one component. How to visualize the corresponding embedding? Lee Rudolph quickly helped us by just realizing that the Galois action (complex conjugation) acts over the torus $S^1\times S^1$ just by exchanging the two factors $(x,y)\mapsto (y,x)$ fixing thereby the diagonal (circle) $\{(x,x)\}\approx S^1$. More generally how to picture out the topology of a real curve? The first observation is that the complex locus $C({\Bbb C})$ is acted upon by complex conjugation $\sigma$ relative to some ambient projective space ${\Bbb P}^n({\Bbb C})$ (where after all the concrete curve is embedded). Therefore to each real curve $C$ is assigned a [*symmetric surface*]{} $(C({\Bbb C}), \sigma)=(X,\sigma)$ consisting of a pretzel $X$ together with an orientation reversing involution $\sigma\colon X\to X$. (For aesthetical reasons all of our algebraic curves are projective and non-singular, prompting thereby compactness of the allied Riemann surfaces.) With the invaluable assistance of (overqualified scholars) Claude Weber and Michel Kervaire, I learned how to classify such objects, according to the invariants $(g,r,a)$ where $g$ is the genus of $X$, $r$ the number of “ovals” (fixed under $\sigma$), and $a$ is the invariant counting mod 2 the number of components of $X-{\rm Fix}(\sigma)$. In other words $a=0$ corresponds to the separating (or dividing) case where ${\rm Fix} (\sigma)$ disconnects $X$, whereas $a=1$ means that the fixed locus does not induce a morcellation of the surface. I soon realized thanks to the paper Gross-Harris 1981 [@Gross-Harris_1981], that all this material was a well-known game for Felix Klein, who was essentially the first to classify symmetric surfaces taking advantage of the just established classification of compact bordered surfaces (Möbius 1863 [@Moebius_1863], Jordan 1866 [@Jordan_1866], etc.). The key trick is of course the yoga assigning to $(X,\sigma)$ its quotient $X/\sigma=:Y$ by the involution, and moving upward again via the orientation covering supplied by local orientations. If the point lies on the boundary then there is no duplication of the point by local orientations (alias “indicatrix” in older literature). [(Klein 1876 [@Klein_1876]=[@Klein-Werke-II_1922 p.154], explicit in Klein 1882 [@Klein_1882], Weichold 1883 [@Weichold_1883])]{} There is one-to-one correspondence between symmetric surfaces and compact bordered surfaces. Moreover the correspondence extends to the realm of conformal geometry, i.e. Riemann surfaces or Klein surfaces, if you prefer. [Modernized treatments of this Klein correspondence—say compatible with Weyl–Radó’s (1913/1925) abstract conception of the Riemann surface—are plenty, compare, e.g. Teichmüller 1939 [@Teichmueller_1939 p.99–101, Die Verdoppelung, §92, 93]=[@Teichmueller_1982], Schiffer-Spencer 1954 [@Schiffer-Spencer_1954 p.29–30, §2.2], Alling-Greenleaf 1971 [@Alling-Greenleaf_1971]. ]{} Via this dictionary, it is plain that the dividing case corresponds precisely to the orientable case. \[As a matter of terminology, Klein used (since Wintersemester 1881/82) the jargon [*orthosymmetrisch*]{} versus [*diasymmetrisch*]{} corresponding to the dividing respectively nondividing case. For instance Weichold 1883 [@Weichold_1883 p.322] writes: \[Weichold-1883:quote\] Was ferner die symmetrischen Riemann’schen Flächen anbelangt, deren Betrachtung die Grundlage der folgenden Untersuchung bildet, so sind auch diese schon mehrfach behandelt worden, wenn auch zum Theil unter ganz anderen Gesichtspunkten. Es hat sich nämlich Herr Professor Klein in den Bänden VII und X der Mathem. Annalen in den Aufsätzen mit dem Titel: “Über eine neue Art von Riemann’schen Flächen” mit diesen Flächen eingehender beschäftigt und daselbst auch schon die Hauptunterscheidung derselben in orthosymmetrische und diasymmetrische Flächen aufgestellt. Diese Bezeichnung findet sich allerdings noch in keiner Publication angewendet; sie wurde zuerst in einem in Wintersemester 1881/82 von Herrn Professor Klein abgehaltenen Seminar eingeführt, in welchem derselbe auch die weiter unten erwähnte weitergehende Classification mittheite und bei welchem auch der Verfasser die unmittelbare Anregung für die vorliegende Arbeit empfing. Perhaps it is worth tracking down further Klein’s motivation for this “savant” terminology; for this we supply the following extract: \[Klein-1923:quote\] Die Benennungen “diasymmetrisch” und “orthosymmetrisch” für die beiden Klassen symmetrischer Flächen wurden später von mir gerade wegen der im Text berührten Verhältnisse eingeführt; siehe Bd. 2 dieser Ausgabe, S.172. Vgl. auch Fu[ß]{}note $^{ 58)}$ auf S.565/566 im vorliegenden Bande. K. So this brings us at other places, the first cross-reference leads us to the following quote (whereas Fu[ß]{}note $^{ 58)}$ is merely a text written by Vermeil, not really worth reproducing here): \[Klein-1892/22:quote\] Reelle algebraische Kurven ergeben [*symmetrische*]{} Riemannsche Flächen und können umgekehrt allgemein gültig von letzteren aus defieniert werden, das ist der hier fundamentale Satz, den ich in §21 meiner Schrift entwickelte. Ich bezeichne dabei eine Riemannsche Fläche als symmetrisch, wenn sie durch eine konforme Abbildung zweiter Art von der Periode 2 in sich übergeführt wird (i.e. durch eine konforme Abbildung, welche die Winkel umlegt). Die symmetrischen Riemannschen Flächen eines gegebenen $p$ zerfallen, wie ich ebendort angab und Herr Weichold a.a.O. eingehender ausgeführt hat, nach der Zahl und Art ihrer “Symmetrielinien” in $[\frac{3p+4}{2}]$ Arten. Wir haben erstlich $[\frac{p+2}{2}]$ Arten [*orthosymmetrischer*]{} Flächen bez. mit $p+1, p-1, p-3, \dots$ Symmetrielinien; das sind solche symmetrische Flächen, welche längs ihrer Symmetrielinien zerschnitten, in zwei (zueinander symmetrische) Hälften zerfallen; — das einfachste (zu $p=0$ gehörige) Beispiel ist eine Kugel, welche durch “orthogonale” Projektion auf sich selbst bezogen ist —. Wir haben ferner $(p+1)$ Arten [*diasymmetrischer*]{} Flächen bzw. mit $p, p-1, \dots, 1, 0$ Symmetrielinien; das sind Flächen, die längs ihrer Symmetrielinien zerschnitten gleichwohl noch ein zusammenhängendes Ganzes vorstellen; — man vergleiche bei $p=0$, die durch eine “diametrale” Projektion auf sich selbst bezogene Kugel. — Hence to summarize this explanation of Klein, the fundamental dichotomy seems to be motivated by the basic case of genus $0$ (the sphere), which may be acted upon in two fashions by a sense-reversing involution (orthogonal vs. diametral). This basic motivation is even more emphasized in Klein’s lectures, worth reproducing (despite its very elementary character): \[quote:Klein-1891/92-ortho/dia\] Wir beginnen damit, anzugeben, auf wieviel verschiedene Weisen eine Kugel mit sich selbst symmetrisch sein kann (d.h. durch eine $\Sigma$ von der Periode $2$ in sich selbst übergehen kann). Das ist offenbar auf $2$ wesentlich verschiedene Arten möglich: das eine Mal bezieht man die Kugel auf sich selbst durch eine Centralprojection, deren Centrum au[ß]{}erhalb liegt: ($1,1';2,2'; \dots$ sind entsprechende Puncte), das zweite mal durch eine Centralprojection, deren Centrum sich innererhalb der Kugel befindet. Im ersten Falle giebt es auf der Kugel eine sogenannte , deren Puncte bei der Umformung sämmtlich festbleiben, das ist der Schnitt der Kugel mit der Polarebene des Projectionscentrums; im $2^{\rm ten}$ Falle giebt es eine solche Symmetrielinie nicht. Wir haben damit dasjenige Unterscheidungsmerkmal, nach welchem wir sogleich die symmetrischen Flächen einteilen: nach Erwähnen wir da gleich die Terminologie, welche ich anlä[ß]{}lich der Figuren 1 und 2 in Vorschlag gebracht habe. Figur 1 kann insbesondere so gezeichnet werden, da[ß]{} das Projectionscentrum unendlich weit liegt. Die Polarebene wird dann eine Diametralebene und die zugehörige Centralprojection eine orthogonale Projection. Ich sage dementschprechend überhaupt von der Figur 1, die Kugel sei bei der selben auf sich selbst bezogen. Die bei Figur 2 vorliegende Beziehung aber nenne ich , insofern bei ihr das Projectionscentrum, insbesondere in den Mittelpunkt der Kugel rücken kann, worauf je zwei diametrale Puncte der Kugel zusammengeordnet erscheinen. Diese Benennungen “orthosymmetrisch” u. “diasymmetrisch” übertrage ich dann demnächst in noch zu erklärender Weise auf die Flächen eines beliebigen $p$. Ahlfors result precisely affords a deeper function-theoretical propagation of this Kleinian paradigm: orthosymmetric surfaces are precisely those mapping in totally real way to the orthosymmetric sphere! The Russian school (Gudkov, Rohlin, Kharlamov, Viro, etc.) uses the (less imaginative) nomenclature Type I versus Type II, whose labelling is pure convention vintage; yet still a heritage from Klein’s initial nomenclature of 1876 [@Klein_1876]=[@Klein-Werke-II_1922 p.154] reproduced in the follwing: \[Klein-1876:quote\] Andererseits ergibt sich für die Kurven, deren Zügezahl $C>0$, $C<p+1$ eine bemerkenswerte Einteilung in zwei Arten. [*Die Kurven der ersten Art haben die Eigenschaft, da[ß]{} ihre Riemannsche Fläche, längs der $C$ Züge zerschnitten, zerfällt: bei den Kurven der zweiten Art findet ein solches Zerfallen nicht statt.*]{} Rohlin 1978 [@Rohlin_1978 p.90] refers explicitly to Klein as follows: \[Rohlin:quote\] Following Klein (see \[4\], p.154), we say that $\alpha$ belongs to type I if $A$ splits ${\Bbb C}A$ and to type II if $A$ does not split ${\Bbb C}A$. For example, $M$-curves obviously belong to type I. It is worth recalling that Rohlin made a surprisingly late discovery of Klein’s work as shown by the following extract: \[Rohlin2:quote\] As I learned recently, more than a hundred years ago, the problem of this article occupied Klein, who succeeded in coping with curves of degree $m\le 4$ (see \[4\], p.155). I do not know whether there are publications that extend Klein’s investigations. It is concomitant to speculate that the infamous [*Klein bottle*]{} (=[*Kleinsche Fläche*]{} which traversed the Atlantic as a “Flasche”) probably originated during Klein’s study of real curves. It just amounts to have a real curve of genus one without real points, whose complex locus will be a torus (of revolution) acted upon by a diametral involution $(x,y,z)\mapsto (-x,-y,-z)$. Criterion for Klein’s orthosymmetry=Type I, in Russian (Klein 1876–82; Rohlin 1978, Fiedler 1978 vs. Alling-Greenleaf 1969, Geyer-Martens 1977) ----------------------------------------------------------------------------------------------------------------------------------------------- Klein’s dichotomy for symmetric surfaces prompts for criterion detecting the dividing character of a real curve. The writer knows of essentially two methods: the first being genetic and the other qualifiable of synthetic. Despite their simplicity those criterions where overlooked by Klein, who relied upon more complicated arguments (cf. the following optional remark). [Besides, there are several other original methods due to Klein. One involves the dual curve, and more specifically a representation assigning to each imaginary point of the curve the real line passing through it and its conjugate. When the points becomes real the limiting position of this secant becomes the tangent. In this way Klein manages to visualize the complex locus of a plane curve living in the 4D-space ${\Bbb P}^2({\Bbb C})$ onto a the 2D real projective plane as a multiple cover, and to guess the type of the curve. Beautiful pictures are to be found in vol. II of his Ges. math. Abhandl. [@Klein-Werke-II_1922]). Another brilliant argument of Klein involves a degeneration to the hyperelliptic case.]{} [**Genetic method.**]{} This is essentially a [*surgery*]{} (if we may borrow the jargon of Thom, Milnor, etc.), and applies primarily to curves gained by small perturbation of two curves whose type is known. Maybe it is best explained on a specific example. Consider the [*Gürtelkurve*]{} as a small deformation of two conics having two nested ovals. ([*Gürtel*]{} means “belt”, a nomenclature coined by Klein in 1876 [@Klein_1876_Verlauf]=[@Klein-Werke-II_1922 p.111], presumably as a translation of the term “[*quartique annulaire*]{}” used by Zeuthen in 1874 [@Zeuthen_1874 p.417+Tafel I.,Fig.1].) Each conic corresponds to an equatorial sphere, and each smoothing amounts attaching a handle. During the process one can keep track of the two real braids to make a global drawing of the surface (compare right part of Figure \[Guertel-genetic:fig\]). Some contemplation of the drawing shows that when all smoothings are dictated by orientations then the resulting curve is dividing. Thus the Gürtelkurve is dividing. Indeed in this case all handles contains twisted braids and thus when travelling in the imaginary locus, say starting from position $A$ in the north (top) hemisphere of the left sphere and moving to the right sphere via an handle we reach position $B$ in the south hemisphere of the right sphere. Coming back to the left sphere, the twisting forces a return to the north hemisphere. We are thus never able to visit the south hemisphere of the left sphere. -5pt0 -5pt0 [**Synthetic method.**]{} Another way to see the dividing character of the Gürtelkurve involves looking at the pencil of lines through a point lying deepest inside the two nested ovals (Figure \[Guertel-saturated:fig\]). Since each real line of this pencil cuts the quartic $C_4$ along a totally real collection of points, this induces a map between the imaginary loci $C_4({\Bbb C})-C_4({\Bbb R})\to {\Bbb P}^1({\Bbb C})-{\Bbb P}^1({\Bbb R})$. It follows that $C_4$ is dividing since ${\Bbb P}^1$ is. (Just use the fact that the continuous image of a connected set is connected.) -5pt0 -5pt0 More generally, this argument gives the following criterion (which quite curiously seems to have escaped Felix Klein’s attention, cf. e.g. his lectures notes 1891–92 [@Klein_1892_Vorlesung-Goettingen p.168–69], where in our opinion Klein draws the orthosymmetric character of the Gürtelkurve from more complicated arguments than those just given): \[saturated:lemma\] If a real curve permits a morphism to the line whose fibers over real points are exclusively real, then the curve is dividing. Conversely, one may wonder if any dividing curve is expressible as such a totally real cover of the line. I clearly remember having asked this question at several experts (ca. 1999), yet without receiving clear-cut answers, and so decided to embark on a self-study of this question. Being a slow and superficial worker, I needed circa 2 years of work until getting an answer, which turned to be positive: \[saturated:Gabard\] [(Gabard 2001, first published in 2004)]{} Any dividing real curve admits a totally real morphism to the line. Moreover the degree of such a morphism can always be chosen $\le g+1$, where $g$ is the genus of the curve. Having completed this work, I started some detective work, and via papers of Geyer-Martens 1977 [@Geyer-Martens_1977] and Alling-Greenleaf 1969 [@Alling-Greenleaf_1969] (probably located via the bibliography of a survey by Natanzon 1990 [@Natanzon_1990/90]) realized that L.V. Ahlfors already proved this result in 1950 (and even exposed his results at Harvard in 1948 as reported in Nehari 1950 [@Nehari_1950]). This was a great deception, or rather more my first contact with the (glamorous) L.V. Ahlfors. [**Very anecdotic details:**]{} However as Ahlfors’ result was not fairly well-known (among the real algebraic geometry community) I received a nice invitation to expose this re-discovery in a RAAG-conference at Rennes in 2001. It was a great pleasure to meet for the first time great specialists like Johannes Huisman, Natanzon, Finashin, Viro, etc. My original proof involved an argument with incompressible fluids and Abel’s theorem to prove (\[saturated:Gabard\]). Some one week after the talk (or maybe even during the week of that conference yet preceding my talk), I confusedly realized that my argument was probably vicious, and reworked it completely to find a topological parade, amounting to the paragraphs 5,6 of Gabard 2006 [@Gabard_2006]. This argument looked more tangible and I was again invited to Rennes in 2001–2002 (by J. Huisman) to present it at a specialized seminar. At this stage I started to believe that one could improve the bound $g+1$ into $\frac{r+g+1}{2}$, which is the mean value of the number of ovals $r$ and the so-called [*Harnack bound*]{} $r\le g+1$. (In the abstract setting is truly a remark of Klein directly reducible to Riemann’s definition of the genus as the maximal number of retro-sections practicable on the pretzel without disconnecting, compare Klein 1876 [@Klein_1876 §7].) I needed some weeks (or months?) to establish this sharper version which gave a relative progress over Ahlfors. \[saturated\_new:Gabard\] [(Gabard 2002, published 2004, 2006 [[@Gabard_2006]]{})]{} Any dividing real curve admits a totally real morphism to the line ${\Bbb P}^1$ of degree $\le \frac{r+g+1}{2}$, where $g$ is the genus of the curve and $r$ the number of “ovals” (=reellen Züge). Using the Schottky(-Klein) double of a compact bordered Riemann surface (whose genus is visually seen to be $g=(r-1)+2p$) this can be translated as \[saturated\_new\_bordered:Gabard\] Any compact bordered Riemann surface with $r$ contours of genus $p$ is conformally representable as full covering of the disc of degree $\le r+p$. Dirichlet’s principle (Überzeugungskraft vs. mathematical comedy) {#Sec:Dirichlet} ================================================================= This section (with parenthetical title derived from jokes by Hilbert 1905 [@Hilbert_1905] and Monna 1975 [@Monna_1975] resp.) recalls the early vicissitudes of a principle supported by strong physical evidence (as early as Green 1828 [@Green_1828] in print), which Riemann placed as the grounding for the edification of the theory of conformal mappings (and the allied Abelian integrals). This section can be skipped without any further ado, but it fixes the context out of which emerged (simpler?) variational problems more suited to pure function-theoretical purposes. However, Dirichlet’s principle (after Hilbert’s resurrection) pursued his life (especially in the fingers of Courant) and merged again to our main topic of the Ahlfors mapping (at least in the schlichtartig situation handled by Riemann-Schottky-Bieberbach-Grunsky). Of course, this “Dirichlet” line of thought is very active today, e.g., by Hildebrandt and his collaborators. In short, Dirichlet’s principle flourished above any expectation by Riemann, was “killed” by Weierstrass, but resurrected by Hilbert, yet re-marginalized by extremal methods (Fejér-Riesz, Carathéodory, Ostrowski, Grunsky, up to Ahlfors) and re-flourished by Douglas and Courant as a (reliable) instrument for the existence of conformal mappings. Chronology (Green 1828, Gauss 1839, Dirichlet ca. 1840, Thomson 1847, Kirchhoff 1850, Riemann 1851–57, Weierstrass 1859/70, etc.) --------------------------------------------------------------------------------------------------------------------------------- Apart from a early contribution of Gauss 1825 [@Gauss_1825] about local isothermic parameters (conformal mappings in the small), the “global” theory of such mappings emerged from Riemann’s Thesis 1851 [@Riemann_1851] and his subsequent work 1857 [@Riemann_1857] on abelian functions. A landmark is the [*Riemann mapping theorem*]{} (RMT) (cf. Riemann 1851 [@Riemann_1851], and Riemann 1857 [@Riemann_1857]), derived from the so-called [*Dirichlet principle*]{}. This was apparently formulated by Dirichlet as long ago as the early 1840’s (lectures in Berlin, attended by Riemann in 1847/49). (The Göttingen 1856/57 version of those were published by Grube in 1876 as [@Dirichlet_1840-1876].) Independent formulations (or utilizations) of this principle are due to Gauss 1839 [@Gauss_1839], Thomson 1847 [@Thomson_1847] (popularizing the long neglected work of Green 1828 [@Green_1828]) and Kirchhoff 1850 [@Kirchhoff_1850]. It is known that Riemann knew all those works (when exactly in another question) from a manuscript estimated 1855/60 reproduced below (source=Neuenschwander 1981 [@Neuenschwander_1981 p.225]). Riemann does not cite Thomson and Kirchhoff in 1857 [@Riemann_1857]. Mit dem Namen des Dirichlet’schen Princip’s habe ich eine Mehode bezeichnet, um nachzuweisen, da[ß]{} eine Function durch eine partielle Differentialgleichung und geeignete lineare Grenzbedingungen völlig bestimmt ist, d.h.da[ß]{} die Aufgabe, eine Function diesen Bedingungen gemä[ß]{} zu bestimmen, eine Lösung und zwar nur eine einzige Lösung zulä[ß]{}t. Es ist diese Methode von William Thomson in seiner Note Sur une équation aux différences (Liouville.T.12.p.493.) und von Kirchhoff in seiner Abhandlung über die Schwingungen einer elastischen Scheibe angewandt worden, nachdem Gau[ß]{} schon vorher eine Aufgabe, welche als ein specieller Fall dieser Aufgabe betrachtet werden kann, ähnlich behandelt hatte (Allgemeine Lehrsätze.Art.29–34.) Ich habe diese Methode nach Dirichlet benannt, da ich von Hrn Professor Dirichlet erfahren hatte, da[ß]{} er sich dieser Methode schon $\langle$seit dem Anfang der vierziger Jahre (wenn ich nicht irre) \[Bl.66r\]$\rangle$ in seinen Vorlesungen bedient habe. There is also a letter of Riemann dated 30.Sept.1852 (cf.Neuenschwander 1981 [@Neuenschwander_1981-lettres]), where it is reported that Dirichlet supplied some references to Riemann. Here is the relevant extract, out of which we may speculate that Riemann learned the ref. to Thomson and Kirchhoff at this occasion (through Dirichlet). [Am Freitag Morgen, um in meinem Berichte fortzufahren, suchte Dirichlet mich in meinem Zimmer auf. Ich hatte ihn bei meiner Arbeit um Rath gefragt und er gab mir nun die dazu nöthigen Notizen so vollständig, da[ß]{} mir dadurch die Sache sehr erleichtert ist. Ich hätte nach manchen Dingen auf der Bibliothek sonst lange suchen können. D.\[irichlet\] war überhaupt äu[ß]{}erst nett theilte mir mit, womit er sich in den letzten Jahren beschäftigt hatte, ging meine Dissertation mit mir durch; und so hoffe ich, da[ß]{} er mich auch später nicht vergessen und mir seine Theilnahme schenken wird. ]{} As we know the principle was disrupted by the (non-fatal) Weierstrass’ critique 1870 [@Weierstrass_1870], but resuscitated by Hilbert in 1900-1 [@Hilbert_1900] [@Hilbert_1901/04] [@Hilbert_1905], after partial results by Neumann 1870, 1878 [@Neumann_1878], and 1884 [@Neumann_1884] Schwarz 1869/70 [@Schwarz_1869-70_Zur-Theorie-der-Abbildung], 1870 [@Schwarz_1870], 1872 [@Schwarz_1872] ([*alternierendes Verfahren*]{}) and Poincaré for fairly general boundary contours. Dirichlet’s principle (as Riemann christened it in 1857 [@Riemann_1857-DP]) amounts to solve the first boundary value problem for the Laplacian $\Delta u=0$ by minimizing the Dirichlet integral $$\int\!\!\int \Big\{ \bigl(\frac{\partial u}{\partial x}\bigr)^2+\bigl(\frac{\partial u}{\partial y}\bigr)^2\Big\} dx dy.$$ As a such the paradigm of [*extremality*]{} entered the arena of geometric function theory since its earliest day, and governed much of the subsequent developments. Other noteworthy hot spots in this realm are: $\bullet$ [*The Bieberbach conjecture*]{} (1916 [@Bieberbach_1916-BC]) $\vert a_n \vert \le n$ on the coefficients of schlicht (=univalent=injective) functions from the disc $\Delta=\{ \vert z \vert <1 \}$ to the (finite) plane ${\Bbb C}$ with Koebe’s function $k(z)=\frac{z}{(1-z)^2}=z+2z^2+3z^3+\dots$ as unique extremals among those satisfying the normalization $f(0)=0,f'(0)=1$. Completely solved by de Branges 1984. $\bullet$ Grötzsch-Teichmüller extremal quasi-conformal mappings (1928–1939 [@Teichmueller_1939]), i.e. the search of the “möglichst konform” mapping relating two configurations. This gave a sound footing to Riemann’s liberal study of the moduli spaces (1857 [@Riemann_1857]), and paved the way to the modern theory of deformation of complex structures (Kodaira-Spencer). Early suspicions about the Dirichlet principle (Weierstra[ß]{} 1859/70, Schwarz 1869, Prym 1871, Hadamard 1906) --------------------------------------------------------------------------------------------------------------- Weierstrass seems to have been the first to express doubts about the Dirichlet principle, pivotal to Riemann’s theory. Weierstrass lectured on his critique in 1870, and this appeared in print as late as 1894 in his Werke. However it is known that a meeting between Riemann and Weierstrass took place in Berlin, 1859, where this issue was discussed. Klein reports upon Riemann’s reaction at several places: Er \[Riemann\] erkannte die Berechtigung und Richtigkeit der Weierstra[ß]{}chen Kritik zwar voll an; sagte aber, wie mir Weierstra[ß]{} bei Gelegenheit erzählte: “er habe das Dirichletsche Prinzip nur als ein bequemes Hilfsmittel herangeholt, das gerade zur Hand war—seine Existenztheoreme seien trotzdem richtig.” Weierstra[ß]{} hat sich dieser Meinung wohl angeschlossen. Er veranla[ß]{}te nämlich seinen Schüler H.A. Schwarz, sich eingehend mit den Riemannschen Existenzsätzen zu befassen und andere Beweise dafür zu suchen, was durchaus gelang. Ich erinnere mich, da[ß]{} Weierstrass mir bei Gelegenheit erzählte, Riemann habe auf die Gewinnung seiner Existenzsätze durch das “Dirichletsche Prinzip” keinerlei entscheidenden Wert gelegt. Daher habe ihm auch seine (Weierstrass’) Kritik des “Dirichletschen Prinzips” keinen besonderen Eindruck gemacht. Jedenfalls ergab sich die Aufgabe, die Existenzsätze auf andere Art zu beweisen. Diese dürfte dann Weierstrass seinem Spezialschüler Schwarz übertragen haben, bei dem er die erforderliche Verbindung geometrisch-anschaulichen Denkens mit der Fähigkeit, analytische Konvergenzbeweise zu führen, bemerkt hatte. A more detailed chronology is roughly as follows (cf. Elstrodt-Ullrich 1999 [@Elstrodt-Ullrich_1999 p.285–6]): $\bullet$ In the late 1850s Weierstrass notices some gap in the Dirichlet principle (DP), and presents his objection to Riemann in 1859, who is not tremendously affected claiming that his existence theorems keep however their truths. $\bullet$ Thieme 1862, who met Riemann and requested from him some elucidations about his theory of Abelian functions, and the conversation turned to the foundation of the Dirichlet’s principle. This is materialized by a letter of Thieme to Dedekind of 1878 (reproduced in Elstrodt-Ullrich 1999 [@Elstrodt-Ullrich_1999 p.270–1], or as Quote \[quote:Thieme\] below) $\bullet$ Kronecker 1864, in a discussion with Casorati, also exposes some criticism of the (DP). This is materialized by notes taken by Casorati, and published by Neuenschwander 1978 $\bullet$ Schwarz 1869 [@Schwarz_1869-Ueber-einige-Abbildungsaufgaben p.120] expresses for the first time in print doubts about (DP) (compare Quote \[quote:Schwarz-1869\] below). $\bullet$ Heine February 1870 [@Heine_1870 p.360] also puts in print the reserves expressed by Weierstrass and Kronecker, specifically their objections to the assumption that a minimum must exist. $\bullet$ Weierstrass July 1870 [@Weierstrass_1870] presents a variational problem where the minimum is not attained. This note, however, appeared in print only in 1895 in the second volume of Weierstrass’s Werke [@Weierstrass_1870]. $\bullet$ Prym 1871 [@Prym_1871 p.361–4] gives the first (published) counterexample to the (DP) (as formulated, e.g. in Grube’s text 1876 [@Dirichlet_1840-1876] based upon Dirichlet’s lectures). Prym gives a continuous function on the boundary of the unit disc such that the Dirichlet integral for the associated harmonic solution to the Dirichlet problem is infinite. However Prym expressly emphasizes that Riemann never stated such a naive version of DP corrupted by Prym’s example. In fact Prym’s example seems rather to attack a vacillating attempt by Weber 1871 [@Weber_1870] to rescue the Dirichlet principle. \[quote:Schwarz-1869\] Dass es stets möglich ist, die einfach zusammenhängende Fläche, welche von einer aus Stücken analytischer Curven bestehenden einfachen Linie begrenzt ist, auf die Fläche eines Kreises zusammenhangend und in den kleinsten Theilen ähnlich abzubilden, hat [*Riemann*]{} mit Zuhülfenahme des sogenannten [*Dirichlet*]{}schen Principes zu beweisen gesucht. Da gegen die Zulässigkeit dieses Principes bei einem Existenzbeweise hinsichtlich der Strenge gegründete Einwendungen geltend gemacht worden sind, war es wünschenswerth, ein Beweisverfahren zu besitzen, gegen welches die bezüglich des [*Dirichlet*]{}schen Principes geltend gemachten Bedenken nicht erhoben werden konnten. \[quote:Thieme\] Vielleicht werden Sie sich meiner noch erinnern, als ich mich im Sommer 1862 in Göttingen aufhielt um bei Riemann Aufklärung über seine Theorie der Abel’schen Funct. zu erbitten. Ich traf Sie damals in der Krone, wo wir beide abgestiegen waren, und das Gespräch kam auf die, meiner damaligen Meinung nach (was seitdem vielseitig anerkannt), nicht ganz stichhaltige Begründung des Dirichlet’schen Princips, welches in der Riemann’sche Theorie fundamental ist. Philosophical remarks {#Sec:Philosophical} ===================== Flexibility of 2D-conformal maps -------------------------------- Maybe one way to enlarge slightly the discussion at the philosophical level is to observe some unifying plasticity in conformal maps. The underlying principle is roughly as follows: If there is no topological obstruction to a mapping problem, then a conformal mapping exist. This idea is very close to Koebe’s allgemeines Uniformisierungsprinzip in Koebe 1908 [@Koebe_1908_UbaK3], which is stated as follow [*Jedes Problem der im Sinne der Analysis situs eine Lösung hat kann auch funktiontheoretisch verwirklicht werden.*]{} Of course this is not quite true in view of say Riemann’s moduli for closed (or non closed) Riemann surfaces. However seminal instances where it works are the (RMT), the uniformization theorem (UNI) \[any simply-connected Riemann surface is biholomorphic to the sphere, the plane or the disc\], and the more general Koebe schlicht theorem to the effect that a schlichtartig Riemann surface is schlicht. Here the topological condition of “Schlichtartigkeit” (i.e. any Jordan curve divides) implies the stronger conformal embeddablility in the Riemann sphere. [(Koebe 1908 [@Koebe_1908_UbaK3], 1910 [@Koebe_1910_UAK2])]{} Any dichotomic[^6] (=schlichtartig) Riemann surface (i.e. one divided by any Jordan curve) embeds conformally in the Riemann sphere. Since simply-connected implies dichotomic this implies the (UNI) via (RMT). An even stronger assertion is Bochner 1928 that any Riemann surface of finite connectivity embeds in a closed Riemann surface. Ahlfors’ result about circle maps likewise illustrates the above principle (CP), especially if we interpret it in Klein’s theory of symmetric surfaces (compare Lemma \[saturated:lemma\]). Free-hand pictures of some Riemann-Ahlfors maps ----------------------------------------------- It would be nice if some general methodology for picturing such mappings could be developed. Let us try a naive look for domains (Riemann surface are harder but not hopeless). Maybe first a comment by Poritsky 1949–52 [@Poritsky_1949-52:Book p.21]: From the above it is clear that analytical methods, at least as developed thus far, have only limited power in solving the complicated field problems arising in electric machines. Electrical engineers have resorted extensively to the use of “flux plotting” or [*free-hand drawing*]{} of the flux lines and equipotentials. As is well known, these curves, when drawn for constant equal increments $\Delta \varphi=\Delta \psi$, form a curvilinear set of [*small squares*]{}. A certain aptitude, somewhat between mechanical drawing ability and artistic drawing, is required for successful flux plotting, and with practice people possessing such aptitude can learn to draw flux plots for a great variety of cases with relative ease. The picture below (Fig.\[Riemann:fig\]) is supposed to depict the pullback of the radial-concentric bi-foliation of the disc via a conformal representation of this 4-ply connected circle domain by a Riemann map to the disc. (Usually the term “Riemann map” is reserved for the simply-connected case, but recall that Riemann was the first to prove the existence of such maps, cf. Riemann 1857/76 Nachlass [@Riemann_1857_Nachlass]). Physically one may try to interpret it at the galvanic current generated by 4 batteries (electric charge) situated on a conducting plate. Whenever the potential generated by two charge enter in conflicts some saddle type singularity is generated (those can be counted via an Euler characteristic argument à la Riemann-Hurwitz). In the present case there is 6 saddles. In general $\chi D=d \chi (\Delta) -\deg (R)$, and $\chi D=2-r$ (holed sphere) and the degree $d$ is $d=r$, hence there is $\deg(R)=2r-2$ ramification points. (This was of course well-known to Riemann, compare his Nachlass, or Quote \[quote:Riemann\]). Dashed lines are equipotentials. -5pt0 -5pt0 This sort of picture as mystical as it is (the reader confesses to have had some trouble to generate it without grasping completely the possible physico-chemical interpretation) gives the impression of grasping slightly Riemann’s title to his Nachlass (Gleichgewicht der Electricität), i.e. equilibrium potential of electricity. Our figure is pure free-hand drawing without much scientific understanding. Thus it would be nice if the computer can do better pictures, maybe via the Bergman kernel (an eminently computable object, compare e.g. Bell papers). In particular albeit it looks physically obvious, it is not clear if the charge may be placed arbitrarily. (For instance it is not clear why the corresponding divisor should be linearly equivalent to its conjugate, compare Lemme 5.2 in Gabard 2006 [@Gabard_2006].) \[Some related references: Henrici Computational conformal map, Gaier, Konstruktive methoden in Konformen Abbildungen, etc... Or maybe Crowdy via the Klein’s prime\] Extracting some global understanding in the non-schlicht case of such isothermic coordinates may be of some relevance to Gromov’s filling conjecture. Further for less contours we may do similar pictures, and we then obtain the following figures (Fig.\[Annulus:fig\]). The fact that the boundary contours are circles is not crucial (but convenient for simple depiction). First we draw the electrical forces in the case of an annulus. Then we made two pictures for triply-connected with symmetrically disposed battery (electrical charge). Geometrically those are supposed to be the pull-back of the origin under the Riemann-Ahlfors map. Finally we would like to make a similar picture in the case where the charge distribution is not symmetric. Then the picturing becomes very difficult. Already in the symmetric cases it is hard to be convinced that what we are doing is really serious. There is a sort of subconscious algorithm to make such pictures: (1) first draw the thick black lines where the particles enter in collision, (2) draw at angle $\pi/4$ the dual saddle at those collision point, and then the filling by thin lines is essentially a matter of artistic feeling. Of course it is not always easy to arrange such that all lines meet perpendicularly, but experience gives some sort of algorithm to do this. Of course it is quite convenient to do such pictures on a computer rather than on the paper, as one can adjust trajectories by successive approximations. As we used a software Adobe Illustrator; with Bézier curves, thus the mathematical faithfulness of all this picture is highly questionable, but we hope that the picture are still of some qualitative value to help visualize such mappings, and to feel some sort of physical interpretation. One guess is that it amounts to have some positive electric charge at the marked point plus perhaps a distribution of such charges on the border. Then each positive particle is rejected by the charge and the border. Thus the particle move faster when there is much free-room in the plate. Alternatively one may have a biological interpretation where the source are bacteroides and the black line show the progression of the growing population which is faster in those direction where there is much vital room (imagine herbivores). Then the saddle amounts to junctions between various ethnical population, and at time one the full universe is explored. In this interpretation the proliferation of species can be slowed down either by proximity to the border (limitation of resources), or by vicinity of a competing population. -5pt0 -5pt0 Hard problems and the hyperelliptic claustrophobia -------------------------------------------------- Another unifying theme when it comes to hard problems regarding Riemann surfaces is the following constat: Several problems are fully settled in the hyperelliptic case, but horribly complicated otherwise. This is a paradigm well known since time immemorial. Probably one of the first problem were it came acute was Jacobi’s inversion problem occupying Jacobi, then Göpel and Rosenhain (hyperelliptic case) and only Weierstrass and above all Riemann 1857 [@Riemann_1857] could handle the general case. (Weierstrass never managed to put in print his own approach probably due to the extreme difficulty to follow an arithmetized path.) Another place is Klein’s trick of degeneresence to the hyperelliptic configurations (cf. Klein 1892 [@Klein_1892_Realitaet]). In some more contemporary problems we already addressed briefly this hyperelliptic barrier also delineate the current frontier of knowledge regarding: \(1) The Gromov filling area conjecture (compare the work by Bangert et al. [@Bangert_2004] where the conjecture is established in the hyperelliptic case, hence in particular for membranes of genus $p=1$). \(2) The Forstnerič-Wold conjecture [@Forstneric-Wold_2009] that compact bordered Riemann surface embeds in ${\Bbb C}^2$ (this is also known in the hyperelliptic case). \(3) The exact determination of Ahlfors degrees à la Yamada-Gouma (this is also settled in the hyperelliptic case, but not much seems to be known beyond those configurations). Topological methods ------------------- We started our Introduction by claiming that topological methods have some relevance to the field of function theory, Riemann surfaces, and the allied fields. The experience of the writer in this realm is rather modest and essentially reduces to his lucky stroke in Gabard 2006 [@Gabard_2006], about lowering the degree of a circle map upon the prediction made by Ahlfors for his extremal function. Such topological methods are quite common in function theory (Riemann, Klein, Poincaré, Brouwer, Koebe, etc.) albeit occupying a marginal place in comparison to potential-theoretic consideration or the allied quantitative extremum problems. Let us list some contributions using qualitative topological methods in the realm of classical function theory: \(1) The most famous (and probably important) example is the continuity method of Klein-Poincaré related to the uniformization problem. (Prior to this we may detect earlier trace of the continuity method, as one learns by reading Koebe 1912 [@Koebe_1912_BdKm], in the work of Schwarz-Christoffel and Schläfli.) \(2) The intuitions of Klein-Poincaré were put on a firm footing by Brouwer 1912 [@Brouwer_1912_Modulmannig], [@Brouwer_1912_top-Schwierig], using invariance of domain which he was the first able to prove via combinatorial topology. \(3) Closer to our main topic, we cite Garabedian 1949 [@Garabedian_1949] who also relies heavily on combinatorial topology to select appropriately certain auxiliary parameters. \(4) Mizumoto 1960 [@Mizumoto_1960], who reproves the existence of (Ahlfors-type) circle map of degree $r+2p$ (i.e. like Ahlfors bound). \(5) Gabard 2006 [@Gabard_2006], where the degree is lowered to $r+p$ (also via topological methods). Roughly, the philosophy is that Riemann surfaces are volatile objects when fluctuating through their moduli spaces, so that practically nothing is observable outside the inherent topological substratum which turns out to behave rather stably say w.r.t. the Abel-Jacobi mapping. At least this is philosophical substance of the proof in Gabard 2006 [@Gabard_2006]. Bordered Riemann surfaces and real algebraic curves --------------------------------------------------- It may seem at first that bordered surfaces are a bit borderline deserving less respectableness than the temple of closed Riemann surfaces. Likewise real algebraic geometry always appears like a provincial subdiscipline of pure complex algebraic geometry of the best stock. Perhaps, less is true. The rehabilitation of reality within algebraic geometry is in good portion, especially regarding the connection with bordered (possibly) non-orientable surfaces, the credit of Felix Klein (especially in 1882 [@Klein_1882]). Moreover, independently of the algebro-geometric analytic correspondence (à la Riemann, etc.) there is another simple reason for which bordered (Riemann) surfaces took gradually more-and-more importance during the 20th century, especially under the fingers of the Finnish and Japanese schools. This pivotal rôle of compact bordered objects results indeed merely from their intervention as building elements of general open surfaces. The latter being always exhaustible through such compact elements as follows easily from Radó’s triangulability theorem of 1925 [@Rado_1925]. Once again this illustrates basically the philosophy “Nihil est in infinito…”. Remind also that the device of exhaustion by finite (=compact) surface is somewhat older than those schools. It may have first occurred in Poincaré 1883 [@Poincare_1883] (where analytic curves or open Riemann surfaces are first taken seriously and subsumed to the uniformization paradigm) and then Koebe 1907 [@Koebe_1907_UbaK1] in same context. To caricature a bit Koebe’s proof, it amounts to use the RMT for compact discs (in a version cooked by Schwarz) and expand in the large. The exhaustion device is again used in Nevanlinna 1941 [@Nevanlinna_1941], where via exhaustions one constructs the corresponding so-called harmonic measure solving the Dirichlet problem for boundary values $0$ and $1$ on the initial resp. expanding contours of the exhaustion $F_n$, yielding the “Nullrand” dichotomy according to whether the $\omega_n$ flatten to $0$ or converge to a positive function. Ahlfors 1950 [@Ahlfors_1950] also uses (or planned to use) a similar technique for other problem. This was enough to launch the big classification programme of open Riemann surfaces. This philosophy is surely so well-known that it was probably not worth insisting on it so vaguely. Of course, in practice this is not a simple game but one which has experimented certain successes in the past. Lebesgue versus Riemann ----------------------- \[11.10.12\] This paragraph is free-style philosophical lucubration coming to me right after reading the fantastic paper Forelli 1978 [@Forelli_1979]. From a narrow minded viewpoint (the writer having zero measure theoretic knowledge) it seems that modernism, especially along the “capitalistic” line of thought involving measure theory, albeit initially quite concomitant with the (older complex) function theory, ultimately may have drifted a vast body of the vital fluid in a somewhat arid valley. (For a somewhat related diagnostic cf. Morse-Heins 1947 [@Morse-Heins_1947].) Let us be more specific. Circa 1898 the way was paved toward measure theory starting from function theoretic preoccupations (not to mention the earlier “Cantorism” starting from Fourier series). We have of course in mind E. Borel 1898 [@Borel_1898], and then the stream along Lebesgue 1902 [@Lebesgue_1902], Fatou 1906 [@Fatou_1906], the old brother F. Riesz 1907 (Fischer-Riesz effecting an Hilbert-Lebesgue unification, etc.). All those grandiose efforts/achievements may have polluted the pureness of (Riemann’s) geometric conceptions by charging the theory with complicated pathological paradigms not truly inherent to its geometric substance (at least in its finitistic aspects, which are not completely elucidated yet, e.g. Gromov’s filling conjecture). Of course the antagonism we are speaking about goes back to older generations, e.g. already acute in the Hermite vs. Jordan opposition, who were resp. anti- and pro-Lebesgue[^7]). This tension is also felt when it comes to prove existence of circle maps, where say proofs like Ahlfors’ 1950 [@Mizumoto_1960], Mitzumoto’s 1960 [@Mizumoto_1960], and many others (maybe even Gabard’s 2006 [@Gabard_2006]) proceeds along essentially classical lines, often emphasizing the soft topological category (very implicit by Riemann-Klein-Poincaré-Brouwer) instead of measure theory (again Borel-Lebesgue-Fatou-Riesz). Of course initially topology also arose from capitalism over the real line, namely the notion of metric (distance function). Yet ultimately the theory (be it axiomatically Bolzano-Cantor-Hilbert-Fréchet-Riesz-Hausdorff-Weyl or through educated intuition Riemann-Klein-Poincaré-Brouwer-Thurston) reached some higher romantic stratosphere producing some lovely science essentially the most remote from capitalistic preoccupation we were able to produce. Alas or fortunately, Grisha Perelman (and precursors Thurston/Yau-Hamilton) showed us that the likewise pleasant Riemannian geometry (albeit slightly more quantitative) turned to have some important topological impact (typically over Poincaré’s conjecture). In a survey article by Lebesgue (ca. 1927, easy to locate), a rather primitive mercantile metaphor is appealed upon to argue that his theory of integration supersedes Riemann’s. Lebesgue argues that when a huge amount of money (delivered as a chaotic mixture of pieces and bills) requires enumeration, his theory amounts to count things properly by first enumerating what has highest value and then paying attention to the more negligible money pieces. This procedure is tantamount to subdividing rather the range of the function as do Lebesgue instead of its domain as did Cauchy or Riemann. The bulk of the US production (Rudin, Gamelin, Forelli and many others) in the 1950-1970’s is much influenced by measure flavored analysis, and the art-form continues to prosper with deep paradigms allied to Painlevé’s problem (fully solved in Tolsa 2003 [@Tolsa_2003]). In contrast, some older workers, e.g. Koebe (cf. Gray’s 1994 paper [@Gray_1994]) as well as Lindelöf (cf. Ahlfors’ 1984 [@Ahlfors_1984-The-Joy]) (and probably more recent ones) were never full partisans of Lebesgue’s integral. Of course the latter theory added a mass of grandiose contributions, yet in some finitary problems like the one at hand (Ahlfors circle maps) its significance can probably be marginalized, or completely eliminated. So measure theory exists, but does it really capture the quintessence of the problematic we are interested in, which is more likely to be first of a [*qualitative*]{} nature (coarse existence theory). Arguably, the next evolution step is the [*quantitative*]{} phase (e.g. Ahlfors extremal problem, which is essentially solved modulo fluctuating incertitudes about degree variations of such maps). Finally any theory should culminate in the [*algorithmic*]{} era, that is claustrophobic (computer ripe) era. Remind that Riemann precisely disliked Jacobi’s approach, finding it too algorithmic and not conceptual enough (according to some forgotten source, try maybe Klein’s history [@Klein_1926-Vorlesungen-über-die-Entwicklung]: “Jacobi war ihm zu algorithmisch.” \[quoted by memory\]). At such a stage it is safer to let computers do the work, but of course it remains to find the algorithms. Will the machine not quickly be more fluent in this game as well? (Compare the little green men survey by David Ruelle in Bull. Amer. Math. Soc. ca . 1986, who tabulated on the imminence of machines cracking theorems with more ease than we are able to do. Hopefully so, since the goal of any science (indeed any living being) is to reach immortality. So if measure theory and general open (=non-compact) Riemann surfaces inclines much to Lebesgue (and the like), it seems evident that still much work must be clarified at the more basic (combinatorial) geometric level of simpler objects, e.g. super classical algebraic geometry should be cultivated again to penetrate more deeply in a variety of problems still unsolved. Prehistory of Ahlfors {#Sec:Prehistory-Ahlfors} ===================== This section attempts a fairly exhaustive tabulation of works antedating Ahlfors 1950 [@Ahlfors_1950], bearing more-or-less direct connection to it. In some critical cases, some of those may also be considered as (vague?) anticipations of the Ahlfors mapping by other “pretenders”. In chronological order, we shall discuss contributions of Riemann 1857–58–76, Schottky 1875–77, Klein ca. 1876–82–92, Koebe ca. 1907, Bieberbach 1925, Grunsky 1937–41–49, Courant 1937–39–50, Teichmüller 1941. Our history is not intended to be a smoothly readable account inclining to passive somnolence, but rather one inviting to further active searches to clarify several puzzling aspects, where in our opinion historical continuity is violently lacking. Historical turbulences arise mostly from several links hard to track down due to poor cross-referencing (especially in the case of Teichmüller 1941 [@Teichmueller_1941], who seems to credit Klein for a sort of qualitative version of the Ahlfors circle map, yet without bound upon the degree). In contrast, the first steps, i.e. the affiliation Riemann-Schottky-Bieberbach-Grunsky is well documented (but confined to planar surfaces, hence inferior to Ahlfors’ work). Courant’s contribution is more in the trend Dirichlet-Riemann-Plateau-Hilbert, but ultimately a bit sketchy when it comes to compare with Ahlfors. Regarding Koebe, he was quite influenced by Klein’s orthosymmetry (which bears a direct connection to Ahlfors’ conformal circle map via the algebro-geometric viewpoint), but was more involved with uniformization (in particular of real curves) and the [*Kreisnormierungsprinzip*]{} (rooted back in Schottky, if not Riemann). Koebe’s work concentrates more upon conformal diffeomorphisms than branched covers. Perhaps an exception concerns his later works ca. 1910 influenced by Hilbert, where he comes to investigate more closely non-schlichtartig surfaces. However in the overall we could not find (as yet) in the torrential series of Koebe’s papers a clear-cut anticipation of Ahlfors’ result. (Relevant works of Koebe will in fact rather be surveyed in the next section.) To summarize we have located essentially 3 potential forerunners of the Ahlfors circle map: \(1) Klein, through a citation (or rather allusion) of Teichmüller in 1941 (supplied without precise reference!) and to which we were not able to supply sound footing (despite long searches through Klein’s collected papers, plus his harder-to-find Göttingen lectures in 1891–92 [@Klein_1891--92_Vorlesung-Goettingen], [@Klein_1892_Vorlesung-Goettingen]). In case no trace is to be found in Klein’s work, it is conceivable that Teichmüller distorted somehow his memory about Klein, in which case Teichmüller should be regarded as the genuine forerunner. It may be imagined that a micro-tunnel (=logical wormhole) links Klein to Ahlfors, and this may have existed in Teichmüller’s brain (but as far as I know no proofs are to be found in print). \(2) Courant who makes a vague claim that the result of Riemann-Schottky-Bieberbach-Grunsky extend to configurations of higher genus. If Courant’s claim is correct, it would be of extreme interest to present the details, especially if it is possible to write down the bound arising from Courant’s argument (inspired from Plateau’s problem). \(3) Matildi 1945/48 [@Matildi_1945/48] and Andreotti 1950 [@Andreotti_1950]. Tracing back the early history (Riemann 1857, Schwarz, Schottky 1875–77, H. Weber 1876, Bieberbach 1925, Grunsky 1937–50, Wirtinger 1942) ----------------------------------------------------------------------------------------------------------------------------------------- From Grunsky’s papers (1937 [@Grunsky_1937], 1941 [@Grunsky_1941_KA], both cited in Ahlfors 1950’s paper) one can trace down the early history of Ahlfors theorem back to the very origin (i.e. Riemann) as follows. Grunsky was Bieberbach’s student. The latter proved a version of this theorem (yet without the extremal interpretation) for planar (schlichtartig membrane, i.e. $p=0$) in Bieberbach 1925 [@Bieberbach_1925]. In this paper, one detects an early influence of Schottky’s Dissertation (Berlin 1875, under Weierstrass) published 1877 [@Schottky_1877], as well as a Nachlass of Riemann estimated of 1857 (which was published in his Werke ca. 1876). Riemann apparently only handles the case of a [*Kreisbereich*]{} (circular domain), yet it seems that Heinrich Weber—who edited this Riemann’s Nachlass—may have considerably amputated the original manuscript. (Of course it would be a first class Leistung if some specialist of Riemann’s work would undertake the difficult project of producing a more all-inclusive account.) Let us reproduce the introduction of Bieberbach 1925 [@Bieberbach_1925]: \[quote:Bieberbach-1925\] [Es handelt sich in dieser Arbeit um die Abbildung eines mehrfach zusammenhängenden schlichten Bereiches auf eine mehrfach bedeckte Kreisscheibe. Insbesondere stelle ich mir die Aufgabe, zu beweisen, da[ß]{} ein $n$-fach zusammenhängender Bereich stets auf eine $n$-blättrige Kreisscheibe abgebildet werden kann. Die erste im Druck erschienene Arbeit, die sich mit diesen Fragen beschäftigt, ist die Dissertation von Schottky (Berlin 1875), die im 83. Bande des Crelleschen Journal abgedruckt ist. Die Frage nach der kleinstmöglichen Blätterzahl ist dort nicht behandelt, aber die Analogie und die Beziehung zur Theorie der algebraischen Funktionen und ihrer Integrale liegt den Betrachtungen zugrunde, und auch die Beziehung zur Theorie der linearen Differentialgleichungen 2. Ordnung kommt zum Vorschein. Wie mir Herr Schottky erzählte, machte bald darauf H.A. Schwarz darauf aufmerksam, da[ß]{} sich Riemann im Sommer 1857 bereits mit der eingangs erwähnten Frage beschäftigte. In der von H. Weber bearbeiteten Darstellung dieses Teils des Riemannschen Nachlasses findet sich freilich keine volle Erledigung der Frage. Ich finde, da[ß]{} auch nicht alle Gedanken des Riemannschen Manuskriptes zur Verwendung kamen. (Vrgl. Riemanns Werke 2.Auflage S.440–444) Riemann knüpft bei seinen Überlegungen an die Theorie der linearen Differentialgleichungen an. Die Theorie der algebraischen Funktionen wird nach der Weberschen Darstellung zur Lösung des Abbildungsproblems nicht herangezogen. Dagegen scheinen mir die Riemannschen Notizen zu lehren, da[ß]{} Riemann auch einen über die Theorie der Abelschen Integralen führenden Weg unabhängig von dem bei Weber dargestellten erwogen hat. Welcher von beiden Wegen der frühere ist, vermag ich nicht zu entscheiden. ]{} Hence the tension between Abelian integrals and potential theory seems to have always been surrounded by a little ring of mysteriousness, even in the passage of Bieberbach 1925’s article just quoted. Furthermore, after Grunsky completed in 1941 his series of papers on the question, it looked desirable to Wirtinger to publish 1942 [@Wirtinger_1942] his own interpretation of Riemann’s Nachlass which he probably knew since ca. 1899 during his duties as publisher of the second edition of Riemann’s Werke. \[Wirtinger:quote\] Die Abhandlung des Hrn. Helmut Grunsky, welche in diesen Berichten, Jahrgang 1941, Nr. 11, unter dem Titel “Über die konforme Abbildung mehrfach zusammenhängender Bereiche auf mehrblättrige Kreise II” erschienen ist, bringt mir Überlegungen wieder gegenwärtig, welche unmittelbar an die klassische Dissertation von F. Schottky (Berlin 1875) anschlie[ß]{}en, welche noch vor dem Bekanntwerden des Riemannschen Fragmentes über das Gleichgewicht der Elektrizität auf Zylindern von kreisförmigem Querschnitt (1876) erschienen ist. Zusammen mit dem dort entwickelten Symmetrieprinzip reicht die Theorie der algebraischen Funktionen vollkommen aus, um zu beweisen, da[ß]{} ein von $p+1$ Randkurven, welche völlig getrennt verlaufen und von denen keine sich auf einen Punkt reduziert, begrenzter Bereich sich konform auf die $p+1$fach überdeckte Halbebene der Variabeln $z=x+iy, y\ge 0$ abbilden lä[ß]{}t, wobei noch auf jeder Linie der dem Punkte $z=\infty$ entsprechende beliebig vorgegeben werden kann. In the above quote, Bieberbach also mentions that H.A. Schwarz was well acquainted with this Riemann’s Nachlass. In this connection, it can be reminded that the whole trend connected to the so-called [*Schwarz lemma*]{} involving Schwarz 1869–70 [@Schwarz_1869-70_Zur-Theorie-der-Abbildung p.109], Carathéodory 1907 [@Caratheodory_1907], 1912 [@Caratheodory_1912] (where the coinage “Schwarz lemma” is first used), Pick 1916, Ahlfors 1938, with some intermediate steps due to E. Schmidt ca. 1906 (as acknowledged in Carathéodory 1907 [@Caratheodory_1907]) is well known to have been another inspiring source for Ahlfors’ extremal problem. To be even more mystical, Carathéodory mentions—in his 1936 laudation to Ahlfors’ reception of the (chocolate) Fields medal (ICM 1936)—a certain “[*Ölfleckmethode*]{} of Schwarz, which seems to be related to all this. This intriguing terminology, probably refers to the common “Ölfleck” experiment consists of taking any “oil” region in a water recipient while exciting it slightly or even strongly with a thin instrument, yet preferably without causing a rupture of its connectedness. Observationally, one can then contemplate with which determination and structural stability the possibly highly distorted “Ölfleck” restores to the round circle-shape even if there are thin necks in the initial position. This seems to be one of the most beautiful way to visualize the Riemann mapping theorem in nature. Mathematically this Öfleck experiment bears perhaps more analogy to the normal curvature flow (Huisken, etc.), than the levels of the Riemann mapping function. One can wonder if there is an identity between the curvature flow and RMT. Schottky 1875–77 ---------------- All sources indicate that Schottky discovered the circle mapping for multiply-connected domains independently of Riemann’s Nachlass. Compare the next 3 quotes of Schottky (\[quote:Schottky-1882\]) and Klein (\[Klein-1923:quote:Schottky\]), (\[Klein-1923:quote:Riemann-1858\]). In 1882, during the hot Klein-Poincaré “competition” on automorphic functions vs. Fuchsian functions, Schottky’s Thesis came again to the forefront, with Klein asking its writer for some precision about its genesis. Besides, Schottky rectified some (historically) inaccurate statement made by Klein. It resulted a letter published 1882 in Math. Annalen [@Schottky_1882_Brief], which we reproduce in part: \[quote:Schottky-1882\] Dass übrigens Riemann bereits die mit dieser Figur in Zusammenhang stehenden Functionen und ihre Differentialgleichungen entdeckt hat, wird durch die Stelle pag. 413–416 seiner gesammelten Werke bewiesen. \[$\bigstar$Footnote: Gleichgewicht der Electricität auf Cylindern mit kreisförmigem Querschnitt und parallelen Axen.—Herr Weber fügt als Herausgeber diesem Aufsatze die Bemerkung zu: “Von dieser und den folgenden Abhandlungen \[des Riemann’schen Nachlasses\] liegen ausgeführte Manuscripte von Riemann nicht vor. Sie sind aus Blättern zusammengestellt, welche ausser wenigen Andeutungen nur Formeln enthalten.”$\bigstar$\] Indess möchte ich betonen, dass meine Dissertation ein Jahr vor der Publication von Riemann’s Nachlass erschienen ist. Auch erfuhr ich von Letzterem erst[^8], als meine Arbeit bereits in ihrer zweiten Fassung zum Druck übergehen war. Aber ich bin glücklich, mit Ihnen die Priorität der Entdeckung Riemann’s constatiren zu können. … Sie haben in freundlicher Weise den Wunsch geäussert, Genaueres über die Prämissen meiner damaligen Arbeit zu erfahren. Die Anregung zum selbständigen Eindringen in die Potentialtheorie verdanke ich Herrn Helmholtz. Das in der Arbeit behandelte Problem, der ursprünglichen Auffassung nach der Potentialtheorie gehörig, und wesentliche Anschauungen meiner Arbeit sind aus mathematisch-physikalischen Autoren geschöpft. Ich nenne neben den Vorlesungen und Schriften von Herrn Helmholtz insbesondere ein mir gütig von Herrn O.E. Meyer geliehenes Heft noch nicht publicirter Vorlesungen von Herrn F. Neumann, dann ferner ein Buch über Elektrostatik von Herrn Kötteritzsch, etc. Die Durchführung der so gewonnenen Ideen wurde mir sodann wesentlich erleichtert durch Herrn Weierstrass’ Vorlesungen über Abel’sche Functionen, sowie besonders durch die von Herrn Schwarz publicirten Untersuchungen über das Abbildungsproblem einfach zusammenhängender Flächen. Mit Rücksicht auf die letzteren wurde auf den Rath meines hochverehrten Lehrers, Herrn Weierstrass, der ursprünglich überreichte Entwurf der Arbeit so abgehändert, dass sich dieselbe in beiden veröffentlichten Fassungen an die Untersuchungen von Herrn Schwarz anschliesst. … This work of Schottky enjoyed early and great recognition among colleagues, and still today is frequently cited. The reasons of this success are multiple, but I cannot resist to quote first Le Vavasseur 1902 [@Le-Vavasseur_1902] \[since in Geneva there is a prominent artist bearing a similar name\], himself quoting Picard: Dans le Tome II de son [*Traité d’Analyse*]{}, page 285, M. Émile Picard écrit: “Deux aires planes $A$ et $A_1$, limitées chacune par un même nombre de contours, ne peuvent pas, en général, être représentées d’une manière conforme l’une sur l’autre. L’étude approfondie de ce problème a été faite par M. Schottky dans un beau et important Mémoire.” Plus loin même Tome, page 497, en note, M. Émile Picard écrit encore: “Nous avons déjà eu l’occasion de citer le beau travail de M. Schottky; c’est un Mémoire fondamental à plus d’un titre.” The enthusiasm for Schottky’s work diffused from France to Italy, cf. especially Cecioni 1908 [@Cecioni_1908], who may be credited for the first rigorous proof of the parallel slit map. The reason of Schottky’s popularity is the quite amazing novelty of his work in prolongation of Riemann’ ideas—but so in retrospect only for Schottky was not directly influenced by Riemann. The methods range from potential-theoretic to algebraic functions, flourishing into an breathtaking variety of results. Beside the circle map for multiply-connected domains, it contains both what later will be known as the Kreisnormierungsprinzip (KNP), plus the parallel-slit mappings (PSM). —[*Warning.*]{} In fact I am not sure that it contains KNP, but could easily have on the basis of a naive parameter count. Also it is never clear if material was amputated from the first 1875 edition of Schottky’s Thesis. According to Klein’s quote (\[Klein-1923:quote:Riemann-1858\]), it seems however that the first Latin edition (1875) of Schottky’s Thesis contains the statement of the Kreisnormierung, yet “[*nur auf Grund einer Konstantenzählung*]{}”.— At any rate, it contains (explicitly or in embryo) virtually all of the varied canonical conformal maps which will be re-studied by Koebe during the period 1904–1930, trying even to extend the results to infinite connectivity. As is notorious, this ramifies to deep waters still not completely elucidated today, cf. He-Schramm 1993 [@He-Schramm_1993], which is still the best result reached so far on the Kreisnormierung problem. Schottky’s Thesis also contains the idea of symmetric reproduction of such a domain, where Klein identifies one of the first instance of automorphic functions. The name “Schottky uniformization” is still of widespread usage today (e.g. Bers, Maskit, etc.). The influence of Schottky’s work is also apparent in the jargon “Schottky differentials” widely used in several of Ahlfors’ papers, especially Ahlfors 1950 [@Ahlfors_1950]. (From the algebro-geometric viewpoint this probably just amounts to a real differential.) Last but not least, the Schwarz principle of symmetry (1869 [@Schwarz_1869-Ueber-einige-Abbildungsaufgaben]) \[which afterwards Klein liked to identify in Riemann’s Nachla[ß]{} [@Riemann_1857_Nachlass] already, as testimonies the many brackets added in his collected papers, e.g. Klein 1923 [@Klein-Werke-III_1923 p.631, line 3]\] enables one to form the so-called [*Schottky double*]{}. All this appears first in this single work of Schottky. The admiration for Schottky’s Thesis propagated long through the ages, e.g.: An understanding of all identities between domain functions may be obtained by sustained application of Schottky’s theory of multiply-connected domains \[15\](=1877). Schottky proved that there is a close relation between the mapping theory of these domains and the theory of closed Riemann surfaces; the identities among domain functions have their complete analogue in the theory of Abelian integrals and might be proved by means of the latter. \[…, and on p.214\] [**Schottky functions and related classes.**]{} Schottky \[15\](=1877) was the first to consider the family $\frak R$ of all functions which are single-valued and meromorphic in $D$ \[a multiply-connected domain\] and have real boundary values on $C$ \[the full contour of $D$\]. He developed an interesting theory of conformal mapping of multiply-connected domains from the properties of this family and established by means of it the relation of this theory with the theory of closed Riemann surfaces. It is evident that functions $f(z)\in \frak R$ are very useful in the method of contour integration. Schottky’s Thesis originated in the ambiguous context of physical intuition vs. Weierstra[ß]{}ian rigor. It is notorious that the ultimate redaction was a hard gestation process subjected to incessant revisions demanded by Weierstra[ß]{}. As we know (from Schottky himself (\[quote:Schottky-1882\]), plus the next two quotes by Klein) the first impulse was physically motivated (Helmholtz, F. Neumann, the father of C. Neumann, etc.), and then only lectures of Weierstra[ß]{} and papers of Schwarz came to influence the mathematical treatment. It is notorious that Schottky’s Dissertation writing must have been a very difficult gestation process through Weierstra[ß]{} supervision, who drifted the methodology towards that of Schwarz. For Klein this excessive Weierstrassization is regarded from a sceptical angle (cf. again the next two quotes). It is a delicate question to wonder about the rigor reached in Schottky, despite its ultimate foundation over technology of Schwarz as a substitute to the Dirichlet principle. To ponder its ultimate rigor, it suffices to say that all of Schottky’s results where subsequently revisited, by the following workers: $\bullet$ Koebe for KNP in a (torrential) series of paper spread from 1906 to 1922. $\bullet$ Cecioni 1908 [@Cecioni_1908] for the PSM (=parallel-slit mapping); the latter even mark (discretely) the superiority of his proof by emphasizing that Schottky’s argument relies on a parameter count, whereas he proposes to prove PSM “[*direttamente”*]{} (cf. p.1). This technical “gap” was of course known to Klein, cf. the next Quote \[Klein-1923:quote:Riemann-1858\]. Also in Salvemini 1930 [@Salvemini_1930 p.3] (a student of Cecioni) the critique is made more explicit: “[*Questo risultato \[=PSM\] era stato enunciato dallo Schottky in base ad un computo di parametri, computo che non è poi esauriente.*]{}” $\bullet$ Bieberbach 1925 [@Bieberbach_1925] for the circle mapping problem. None of those writers attacks frontally the standards of rigor in Schottky (as based upon the complicated but solid foundations laid by Schwarz). Still, the technical complications was seen as a need to find simpler derivations of the geometrical results. After sufficiently time elapsed, the subsequent generation tends to ascribe the (rigorous) proof of Schottky’s result to this second wave of workers. E.g., Grunsky 1978 [@Grunsky_1978] ascribes Schottky’s circle maps to Bieberbach 1925 [@Bieberbach_1925] (cf. Quote \[quote:Grunsky-1978\]), and Bieberbach 1968 [@Bieberbach_1968-Das-Werk-Paul-Koebes] credits Koebe for the proof of KNP (in finite connectivity). All these redistributions are done without specific objections upon the original arguments Schottky’s. This is a usual loose process relegating methodologies just due to their cumbersomeness, as a sufficient reason for lack of rigor. In contrast, even more contemporary workers still credits Schottky for the first proof of the Kreisnormierung result (cf. e.g., Schiffer-Hawley 1962 [@Schiffer-Hawley_1962 p.183]). So it is a subtle socio-cultural game to pinpoint precisely about which writer furnished the first acceptable proof. Klein’s comments about Riemann-Schottky --------------------------------------- In the third volume of his collected papers Klein makes several comments about Riemann and Schottky Thesis. He insists first on the physical motivations of Schottky, which were progressively “censured” under Weierstrass’ influence. \[Klein-1923:quote:Schottky\] Ich greife gern noch einmal auf die wiederholt genannte Arbeit Schottkys in Crelle Journal, Bd. 83 (1877) zurück, zumal ich weiter unten (S. 578/579) ohnehin ausfürlicher auf sie zurückkommen mu[ß]{}. Die gro[ß]{}e Ähnlichkeit der auf einen besonderen Fall bezüglichen Schottkyschen Untersuchungen mit den allgemeinen meiner Schrift war mir von vornherein aufgefallen. Ich schrieb also damals an Herrn Schottky und fragte ihn nach der Enstehung seiner Ideen. Hierauf antwortete her mir in einem Briefe von Mai 1882 (der in Bd. 20 der Math. Annalen abgedruckt wurde), da[ß]{} er in der Tat ursprünglich auch von der Bertrachtung der Strömungen einer inkompressiblen Flüssigkeit ausgegangen sei und diesen physikalischen Ausgangspunkt nur auf Rat von Weierstrass bei der Drucklegung durch die Bezugnahme auf Schwarz’ Untersuchungen über konforme Abbildung ersetz habe. Then Klein recollects some more details in the following passage. This contains an anecdotic conflict (with Bieberbach 1925 [@Bieberbach_1925]=Quote \[quote:Bieberbach-1925\]) about the estimated date of Riemann’s Nachlass. More interestingly, Klein expresses the view that Schottky’s theorem (to the effect that a multiply-connected domain is conformal to a circle domain) may be seen as the planar case of Klein’s [*Rückkehrschnitttheorem*]{}, which in turn seems to be one of the weapon that Klein used in his early strategy toward uniformization (an approach not successfully completed until Brouwer-Koebe ca. 1911 [@Klein-Brouwer-Koebe_1912]). \[Klein-1923:quote:Riemann-1858\] Übrigens hat Riemann ja auch die andere Art automorpher Funktionen, die enstehen, indem man an einen von Vollkreisen begrenzten Bereich der Ebene an diesen Kreisen fortgesetzt symmetrisch reproduziert (Siehe das von H. Weber bearbeitete Fragment XXV in der ersten (1876 erschienenen) bzw. XXVI in der zweiten (1892 erschienenen) Auflage der Ges. math. Werke von Riemann.) Die Prüfung der Originalblätter hat ergeben, da[ß]{} Webers Mitteilungen den Vorbereitungen zu einer im Sommer 1858 gehaltenen Vorlesung entnommen sind. Und zwar geht Riemann dabei zunächst von der Aufgabe aus, für ein von mehreren Kugeln gebildetes Konduktorsystem das Gleichgewicht elektrostatischer Ladungen zu bestimmen. Hierfür war die Benutzung des Symmetrieprinzipes in den Arbeiten von W. Thompson vorgebildet, die als Briefe an Liouville in dessen Journal von 1845 an erschienen. Also auch hier sind die mathematischen Entwicklungen aus physikalischen Anregungen erwachsen. Auf dieselben Funktionen ist dann unabhängig in seiner Berliner Dissertation 1875 Herr Schottky gekommen. Von seinem physikalischen Ausgangspunkte ist schon oben auf S.573, die Rede gewesen. Im übrigen sind die Schicksale der Schottkyschen Arbeit, wie sie sich nach persönlicher Mitteilung des Verfassers ergeben, so merkwürdig, da[ß]{} ich gern die Gelegenheit ergreife, sie hier mitzuteilen. Es erfolgten nach einander drei verschiedene Redaktionen: a\) Eine lateinische Fassung, die nicht publiziert ist, sondern nur der Philosophischen Fakultät in Berlin vorgelegen hat, b\) Eine deutsche Bearbeitung, welche 1875 in Berlin als Dissertation gedruckt wurde, c\) Die umgearbeitete Darstellung in Crelles Journal, Bd. 83 (1877). Bei Niederschrift von a) hat der Verfasser noch keine Fühlung mit Weierstrass gehabt, dafür aber ganz seiner freien Ideenbildung folgen können. Aus dem Gutachten, da[ß]{} Weierstrass über a) seinerzeit für die Fakultät abgegeben hat und von dem ich durch die Freundlichkeit von Herrn Schottky eine Abschrift vor Augen habe, scheint mit Gewi[ß]{}heit hervorzugehen, da[ß]{} Schottky hier, freilich nur auf Grund einer Konstantenzählung, das “Rückkehrschnitttheorem” für den besonderen, von ihm betrachteten Fall ausgeschprochen hat, d. h. die Möglichkeit, einen von $p+1$ regulären Randkurven begrenzten eben Bereich auf einen von $p+1$ Vollkreisen begrenzten Bereich konform abzubilden (also das Rückkehrschnitttheorem für den obersten orthosymmetrischen Fall, wie ich mich ausdrücke). Die Redaktion b) ist dann durch eine erste Fühlungnahme mit Weierstrass bedingt. Bei der umfassenden Beherrschung ausgedehnter Teile der Mathematik und seiner stark ausgeprägten Persönlichkeit, die sich zu bestimmten Beweisgängen durchgearbeitet hatte, übte Weierstrass auf jüngere Forscher je nachdem einen au[ß]{}erordentlich fördernden, oder auch, wo ihm die Gedankengänge fremdartig waren, einen hemmenden Einflu[ß]{}. \[…\]. Schottky scheint ähnliche Erfahrungen gemacht zu haben, so da[ß]{} er in b) sich blo[ß]{} auf die Konstantenzählung beschränkt, ohne ihre Tragweite für das Fundamentaltheorem anzudeuten \[…\]. Die physikalische Ideenbildung aber, von der doch der Autor ausgegangen war, wird gänzlich ausgeschaltet und durch Zitate auf die das Existenzproblem der konformen Abbildungen betreffenden Arbeiten von Schwarz ersetzt. In c) endlich ist auch noch besagte Konstantenzählung weggeblieben. \[\[\[Fu[ß]{}note: Dagegen hat Schottky in c) (S.330 daselbst), wiederum auf Grund blo[ß]{}er Konstantenzählung, den Satz ausgeschprochen, da[ß]{} sich jedes ebene, von $p+1$ Randkurven begrenzte, Gebiet umkehrbar eindeutig konform auf die Vollebene mit Ausnahme von $p+1$ geradlinigen, zur $x$-Achse parallelen Strecken abbilden lä[ß]{}t. Bereiche der letzteren Art spielen in der modernen Literatur unter dem Namen [*Schlitzbereiche*]{} bekanntlich eine wichtige Rolle.\]\]\] Statt dessen finden sich wertvolle, vorher nicht publizierte, Angaben über die verschiedenen Normalformen, die Weierstrass bei den Gebilden $p>2$ unterschied; \[…\] Incidentally this [*Rückkehrschnitttheorem*]{}, may have some connection with the Ahlfors function albeit probably no direct link is evident, there is still some striking analogy developed in the next section. A historical puzzle: why Klein missed the Ahlfors circle mapping? ----------------------------------------------------------------- \[27.04.12\] After reading quite closely the above comments of Klein, plus having a vague idea of the content of Schottky’s Dissertation one is puzzled by how close Klein might have been (ca. 1882) to anticipate by circa 70 years the circle map of Ahlfors (1948–1950). Here is our reasoning. First, Schottky’s Thesis (and in cryptical form already Riemann’s Nachlass) contains two striking results: $\bullet$ the [*circle map*]{} (CM) of a (compact) multiply-connected domain to the disc, and beside $\bullet$ what later came to be known (in Koebe’s era, cf. e.g., Koebe 1922 [@Koebe_1922]) as the [*Kreisnormierungsprinzip*]{} (KNP) to the effect that any such domain is conformally equivalent to a circular domain. (Recall from Klein’s quote (\[Klein-1923:quote:Riemann-1858\]) that this occurs only in the original Latin version of Schottky’s Thesis.) Both results are natural extensions of RMT (=Riemann mapping theorem) either by allowing branched coverings or just by using faithful conformal diffeomorphisms (but then of course the target depends upon moduli). Now loosely speaking one may consider both results (CM and KNP) as lying at the same order of difficulty (at least both are to be found in Schottky’s Thesis). Next, Klein points out (cf. right above Quote \[Klein-1923:quote:Riemann-1858\]) that he was able in 1881–82 to prove an extension of (KNP) to positive genus $p>0$, which he calls (apparently with Fricke’s assistance—cf. Klein 1923 [@Klein-Werke-III_1923 p.623, footnote 4]) the [*Rückkehrschnitttheorem*]{} (RST). Klein was very proud of this result (cf. especially Klein 1923 [@Klein-Werke-III_1923 p.584], where this discovery is dated from September 1881 (Borkum)), comparing it (as a psychological experience) to Poincaré’s discovery of his general [*fonctions fuchsiennes*]{}. Thus, there is an obvious commutative diagram (Fig.\[KNP-RST:fig\]), and whatsoever the actual meaning of Klein’s (RST) should be, there is only a single natural candidate to fill in the diagram at the (triple) question-marks, namely the Ahlfors circle map. This accentuates once more why Klein may have been a serious candidate to anticipate the Ahlfors circle map, at least without extremal interpretation. -5pt0 Furthermore, in view of say Gabard 2006 [@Gabard_2006], the Ahlfors mapping amounts essentially to Jacobi’s inversion problem in the real case, and here again this was one of Klein’s major preoccupation (cf. e.g., Weichold’s Thesis 1883 [@Weichold_1883], Hurwitz’s work 1883 [@Hurwitz_1883], plus other sources, e.g. Klein 1892 [@Klein_1892_Realitaet]). Of course, it would be an excellent project to try getting acquainted with Klein’s techniques so as to inspect if they lead to another elementary existence-proof of Ahlfors maps. Again it should be recalled that even if Klein himself was never able to complete his programme some helping hand from Brouwer-Koebe ultimately vindicated all of Klein’s intuitions. Rückkehrschnittteorem (Klein 1881–82) {#sec:Ruckkehrschnittthm} ------------------------------------- Klein found this theorem in 1881, and published it 1882 in [@Klein_1882_Ruckkehrschnitt]. From the start the paper confesses to use some irregular methods. What does Klein in this paper? First he takes a closed Riemann surface of genus $p>1$ (w.l.o.g) and traces on it $p$ disjoint Rückkehrschnitten (retrosections) and asserts that the cutted Riemann surface may be mapped to a $2p$-ply connected domain on the sphere, whose corresponding boundaries $A_i'$, $A_i''$ are related by a linear substitution. He then uses these $p$ substitutions to reproduce the conformal mapping ad infinitum (a trick already present in Riemann’s Nachlass 1857 [@Riemann_1857_Nachlass]). He notes that the construction depends on the right number of free constants $3p$ compatible with Riemann’s moduli $3p-3$, thus yielding a sound evidence for some sort of uniformization. Of course it is not yet the standard uniformization as the reproduced domain filling more and more the sphere still avoids an infinite set (a Cantor set). In fact this construction gives an unramified infinite cover of the given closed Riemann surface by a subregion of the sphere (which is however not simply-connected). Then he applies a similar method to the case of symmetric Riemann surfaces by using a symmetric system of retrosections while showing that the above construction may be done equivariantly. For instance in the simpler to visualize dividing case the above dissection process leads to a similar symmetric domain, symmetric with respect to the orthogonal symmetry of the sphere (whence the name [*orthosymmetric*]{}). In the nondividing case the structural symmetry is rather the diametral one (antipodal map), whence the name [*diasymmetric*]{}. Logically it seems that Klein’s method depends on Schottky’s inasmuch as first doing the retrosections one is reduced to the schlichtartig case which turns out to be schlicht. (This result was extended by Koebe in 1908 [@Koebe_1908_UbaK3] to schlichtartig surfaces of infinite connectivity: schlichtartig implies schlicht!) Clearly something remains to be understood on this RST, and our guess that it is sufficiently strong to imply Ahlfors theorem is quite disputable. At any rate Klein seems to have had a clear-cut conception of how his dichotomy ortho- vs. diasymmetric is reflected into the Riemann sphere with its two real structures (equatorial symmetry vs. antipody). However the issue that dividing curves are precisely those mapping to the equatorial sphere in a totally real fashion may have escaped his attention and does not seem to be logically reducible to his RST. Yet since RST is supposed to be the positive genus case of KNP (cf. Klein’s quote (\[Klein-1923:quote:Riemann-1858\])) it may be expected that one first establishes KNP and from here one deduces a circle map, much like Riemann was able to do for the zero genus case in his Nachlass [@Riemann_1857_Nachlass]. This suggests yet another strategy to approach Ahlfors theorem. \[04.11.12\] A more naive idea could be to start from a bordered surface of type $(r,p)$, and make $p$ retrosections to get it planar (but with $r+2p$ contours). Then there is on the dissected surface a circle map of degree $r+2p$. Of course the map is a priori not assuming the same values on both ridges of the retrosections and even if we can arrange this, we would like those points to get mapped in the interior and not the boundary of the circle. Grunsky’s bibliographical notes (Grunsky 1978) ---------------------------------------------- Let us now reproduce Grunsky’s historical comments (in his monumental book 1978 [@Grunsky_1978 p.198]) about circle maps. (Brackets are ours additions. We added author’s names in front of the bracket-references to improve readability, plus page numbers, and finally inserted the symbol $\bigstar$ when disagreeing with Grunsky’s cross-references.) \[quote:Grunsky-1978\] Theorem 4.1.1. goes back to Riemann 1857/58/76 [@Riemann_1857_Nachlass], who gave some hints for the proof if \[the domain\] $D$ is bounded by circles. The first proof is due to Bieberbach 1925 [@Bieberbach_1925], who used the Schottky-double and deep results in the theory of algebraic functions. Elementary proofs were given by Grunsky 1937 [@Grunsky_1937], 1941 [@Grunsky_1941_KA]; for 4.1.3. \[a sort of auxiliary lemma in linear algebra\] see Furtwängler 1936, Bourgin 1939. Related proofs in Akira Mori 1951 [@Mori_1951], Komatu 1953 [@Komatu_1953] (containing generalizations), Tsuji 1956 [@Tsuji_1956]; cf. Golusin 1952/57 [@Golusin_1952/57], Tsuji 1959 [@Tsuji_1959-BOOK/Chelsea1975]. A proof based on the method for Plateau’s problem: Courant 1937 [@Courant_1937] \[$\bigstar$ in Gabard’s opinion this paper does not reprove the circle mapping, but rather the mapping to a Kreisbereich, due to Schottky–Koebe, cf. p.709 and p.717, of [*loc.cit.*]{}\], generalized in Courant 1939 [@Courant_1939]; cf. Courant 1950 [@Courant_1950] \[especially p.183–187\]. Another proof, using, like Bieberbach 1925 [@Bieberbach_1925], the Schottky double in Wirtinger 1942 [@Wirtinger_1942]; cf. also Rodin-Sario 1968 [@Rodin-Sario_1968] \[where ???\]. Triply connected domains: Limaye 1973 [@Limaye_1973]. Representation of the mapping function (Ahlfors function, see 4.3.) by an orthonormal system in Meschkowski 1952 [@Meschkowski_1952], by the Bergman kernel in Nehari 1950 [@Nehari_1950]. Proofs using extremal properties in papers quoted in 4.3. and 6. \[More about this below (Quote \[quote:Grunsky-1978-B\]).\] An extension to certain domains of infinite connectivity in Röding 1975 [@Roeding_1975]. A more general type of image domain for doubly connected domains in Bieberbach 1957 [@Bieberbach_1957]. Some generalizations, based on ideas used in the aforementioned papers, mainly concerning Riemann surfaces in Nehari 1950 [@Nehari_1950], Tietz 1955 [@Tietz_1955] (cf. Köditz-Timmann [@Koeditz-Timmann_1975]), Mizumoto 1960 [@Mizumoto_1960], Timmann 1969 (Diss., Hannover) [@Timmann_1969], Röding 1972 (Diss., Würzburg) [@Roeding_1972], Röding 1977 [@Roeding_1977_mero]. Cf. Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] \[$\bigstar$ where?\], Carathéodory 1950 [@Caratheodory_1950_Buch_Funktionentheorie] \[$\bigstar$ where?\], Sario-Oikawa 1969 [@Sario-Oikawa_1969] \[$\bigstar$ where?\]. [**Comments (Gabard, Mai 2012):**]{} Alas, regarding the three last books no pagination is supplied by Grunsky, and as far as I browsed through them, I failed to locate any place where Ahlfors’ circle mapping is established anew. Now we reproduce Grunsky 1978 [@Grunsky_1978 p.199]: \[quote:Grunsky-1978-B\] Theorem 4.3.1., a generalization of Schwarz’ lemma to multiply connected domains, is a special case of a more general theorem (individual bounds on each boundary component, prescribed zeros) proved by Grunsky in 1942 [@Grunsky_1942] (save for uniqueness, see Grunsky 1950 [@Grunsky_1950]). Cf. Hervé 1951 [@Herve_1951]. Another proof of 4.3.1. was given by Ahlfors in 1947 [@Ahlfors_1947], completed in Ahlfors 1950 [@Ahlfors_1950] (cf. Golusin 1952/57 [@Golusin_1952/57]) and the extremal function is called the “Ahlfors function”, a term frequently used in the broader sense of any function mapping \[the domain\] $D$ \[in a\] $(1,n)$ onto $U$ \[the unit disc\]; the result was carried on to characterization of the additional zeros of the extremal function. The method used by Ahlfors, Euler-Lagrange multipliers (also pointed out in Grunsky 1946 [@Grunsky_1946] and applied in Grunsky 1940 [@Grunsky_1940]) is likewise a basis for our §6. – For further proofs of our theorem see Nehari 1951 [@Nehari_1951] and Nehari 1952 [@Nehari_1952-BOOK pp.378 ff.], and some of the papers quoted for theorem 4.6.4. – Ahlfors function in a ring domain Kubo 1952 [@Kubo_1952]. – Applications of the Ahlfors function in Alenicyn 1956 [@Alenicyn_1959], 1961 [@Alenicyn_1961], (cf. Mitjuk 1965 [@Mitjuk_1965]). Italian school: Cecioni 1908, Stella Li Chiavi 1932, Matildi 1945/48, Andreotti 1950 {#sec:Italian-school} ------------------------------------------------------------------------------------ Of course in the overall Grunsky’s comments and references are essentially sharp (especially a deep knowledge of Russian/Ukrainian works). Maybe only some contribution of the Italian school are ignored. (Those are however meticulously listed in the book Ahlfors-Sario 1960 [@Ahlfors-Sario_1960].) For instance the simple continuity argument in the Harnack-maximal case based upon Riemann-Roch (without Roch) gives a simple proof in this case (compare e.g., Huisman 2001 [@Huisman_2001] or Gabard 2006 [@Gabard_2006]). This simple argument goes back to Enriques-Chisini seminal book 1915/18 [@Enriques-Chisini_1915-1918], and may have been implicit in Riemann’s original manuscript (not published), compare Bieberbach’s quote (\[quote:Bieberbach-1925\]). Further, closely allied work is to be found in works of Cecioni 1908 [@Cecioni_1908], and his students: Salvemini 1930 [@Salvemini_1930], Stella Li Chiavi 1932 [@Stella-li-Chiavi_1932], etc. Those works certainly deserve closer studying, but they do not seem to anticipate Ahlfors circle map. One notable exception is the article Matildi 1945/48 [@Matildi_1945/48] (discovered by the writer as late as \[13.07.12\]), where existence of circle maps for surfaces bounded by a single contour seems to be established. (This Italian work was known to Ahlfors (or Sario?) at least as late as 1960, again being quoted in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960].) Of course it would be interesting to see if Matildi’s method adapts to more contours, while trying to make (more) explicit the degree bound obtained by him. Andreotti 1950 [@Andreotti_1950] seems to go precisely in this sense by including several contours (alas the writer’s Italian declined fast enough to have failed understanding properly Andreotti’s achievements). Is there any precursor to Ahlfors 1950? {#Sec:Precusors} ======================================= What about Teichmüller 1941? {#sec:Teichmueller} ---------------------------- One can wonder about the content of Teichmüller’s Werke. Does it overlap with the Ahlfors function? While reading the long memoir of Teichmüller 1939 [@Teichmueller_1939] it transpires to anybody familiar with Klein’s work how strong the latter’s influence is; in particular Teichmüller gives a thorough account of the (now) so-called [*Klein surfaces*]{} (and their moduli). Of course such results were anticipated by Klein (at least at the heuristic level). Hence, it seems quite natural to wonder if Teichmüller anticipated the existence of Ahlfors function (for orientable membranes). Here is a report of those portions of Teichmüller’s works which looks closest to this goal, but it should still be debated how much of the Ahlfors circle maps was anticipated by Teichmüller. The most relevant passage in Teichmüller’s writings seems to be the following extract of Teichmüller 1941 [@Teichmueller_1941] (reedited in [@Teichmueller_1982 p.554–5]): \[quote:Teichmueller-1941\] Wir beschäftigen uns nur mit [**orientierten endlichen Riemannschen Mannigfaltigkeiten.**]{} Diese können als Gebiete auf geschlossenen orientierten Riemannschen Flächen erklärt werden, die von endlich vielen geschlossenen, stückweise analytischen Kurven begrenzt werden. Sie sind entweder geschlossen, also selbst geschlossene orientierte Riemannsche Flächen, die man sich endlichvielblättrig über eine $z$-Kugel ausgebreitet vorstellen darf, oder berandet. Im letzteren Falle, kann man sie nach Klein durch konforme Abbildung auf folgende Normalform bringen: ein endlichvielblättriges Flächenstück über der oberen $z$-Halbebene mit endlich vielen Windungspunkten, das durch Spiegelung an der reellen Achse eine symmetrische geschlossene Riemannsche Fläche ergibt; \[…\] (So lä[ß]{}t sich z.B. jedes Ringgebiet, d.h. jede schlichtartige endliche Riemannsche Mannigfaltigkeit mit zwei Randkurven, konform auf eine zweiblättrige Überlagerung der oberen Halbebene mit zwei Verzweigungspunkte abbilden.) Unfortunately, no precise cross-reference to Klein is given and one needs to browse Klein’s works (with the option of some Göttingen Lectures Note 1891/92 [@Klein_1891--92_Vorlesung-Goettingen], [@Klein_1892_Vorlesung-Goettingen] not reproduced in Klein’s collected papers). This absence of precise location is quite annoying. A charitable excuse is the World War II context in which the paper was written: “[*Weil mir nur eine beschränkte Urlaubzeit zur Verfügung steht, kann ich vieles nicht begründen, sondern nur behaupten.*]{}” (compare [*loc.cit.*]{} [@Teichmueller_1982 p.554] 2nd parag.) Detective work: Browsing Klein through the claim of Teichmüller --------------------------------------------------------------- Regarding Teichmüller’s cryptical allusion to Klein (as discussed in the previous section) we have the following candidates in Klein’s works (none of which at the present stage of our historical search truly corroborates Teichmüller’s crediting): \(1) Klein 1882 [@Klein_1882 p.75]=[@Klein-Werke-III_1923 p.567] where one reads: ** Man hat also eine komplexe Funktion des Ortes, welche in symmetrisch gelegenen Punkten geiche reelle, aber entgegengesestzt gleich imaginäre Werte aufweist. This looks quite close to the desired assignment, yet in reality only corresponds to the existence of a real morphism on any real curve; equivalently the existence for any (closed) symmetric Riemann surface of an equivariant holomorphic map to the sphere acted upon by the (usual) complex conjugation fixing an equator. Hence, in our opinion, this passage of Klein cannot be regarded as a genuine forerunner of the Ahlfors circle mapping. \(2) Another place where Klein comes quite close to Teichmüller’s assertion occurs in the same 1882 booklet “[*Über Riemanns Theorie …*]{}”, where Klein computes the moduli of real algebraic curves—equivalently symmetric Riemann surfaces (cf. [@Klein-Werke-III_1923 p.568–9]): \[quote:Klein-1882-moduli\] Indem wir uns jetzt zu den [*symmetrischen*]{} Flächen wenden, haben wir noch eine kleine Zwischenbetrachtung zu machen. Zunächst ist ersichtlich, da[ß]{} zwei solche Flächen nur dann “symmetrisch” aufeinander bezogen werden können, wenn sie neben dem gleichen $p$ dieselbe Zahl $\pi$ der Übergangskurven \[=real “ovals”\] darbieten und überdies beide entweder der ersten Art oder der zweiten Art angehören. \[This is the dichotomy ortho- vs diasymmetric.\] Im übrigen wiederhole man speziell für die symmetrischen Flächen die Abzählungen des §13 betreffs der Zahl der in eindeutigen Funktionen enthaltenen Konstanten unter der Bedigung, da[ß]{} nur solche Funktionen in Betracht gezogen werden, welche an symmetrischen Stellen konjugiert imaginäre Werte aufweisen. Hiermit kombiniere man sodann nach dem Muster des §19 die Zahl solcher über der $Z$-Ebene konstuierbarer mehrblättrigen Flächen, welche in bezug auf die Achse der reellen Zahlen symmetrisch sind. \[…\] Die Sache ist dann so einfach, da[ß]{} ich sie nicht speziell durchzuführen brauche. Der Unterschied ist nur, da[ß]{} die in Betracht kommenden, früher unbeschränkten Konstanten nunmehr gezwungen sind, entweder [*einzeln reell*]{} oder [*paarweise konjugiert komplex*]{} zu sein. Infolgedessen reduzieren sich alle Willkürlichkeiten auf die Hälfte. Wir mögen folgenderma[ß]{}en sagen: [*Zur Abbildbarkeit zweier symmetrischer Flächen $p>1$ aufeinander ist neben der Übereinstimmung in den Attributen das Bestehen von $(3p-3)$ Gleichungen zwischen den reellen Konstanten der Fläche erforderlich.*]{} If this passage sounds a bit sketchy to the reader, we may refer to Klein’s subsequent lecture notes of 1892 [@Klein_1892_Vorlesung-Goettingen p.151–4], where full details are given. The basic idea of this (Riemann-style) moduli count is to represent a given curve of genus $g$ as an $m$-sheeted cover of the line. If $m$ is large enough (so as to avoid exceptional cases of Riemann-Roch’s theorem), a group $g_m$ of $m$ points will move in a linear system of dimension $m-g$. To specify a map to ${\Bbb P}^1$ we may send the divisor $g_m=:D$ to $0$, say, and another $D'$ (linearly equivalent to the former) to $\infty$, leaving the possibility of a scaling factor. Thus the function depends on $2m-g+1$ constants. On the other hand by Riemann-Hurwitz such maps have $2m+2g-2$ branch points. Hence considering the totality of such covers modulo those yielding the same curve leaves $2m+2g-2-(2m-g+1)=3g-3$ essential constants. (cf. also Griffiths-Harris 1978 [@Griffiths-Harris_1978/94 p.256]). \[15.12.12\] It is legitimate to wonder if this method (à la Riemann-Klein) is powerful enough to compute the gonality profile (cf. Definition \[gonality-profile:def\]). Klein adapts this counting argument to the real case (again for full details we recommend Klein 1892 [@Klein_1892_Vorlesung-Goettingen p.151–4]). Doing so we may hope that he anticipates the Ahlfors mapping when the construction is particularized to the orthosymmetric case. Since a totally real morphism lacks real ramification, we must prescribe imaginary conjugate branch points. However this necessary condition is not sufficient as shown by a quartic smoothing a visible conic plus an invisible one like $x^2+y^2=-1$ (alternatively consider the Fermat curve $x^4+y^4=1$ projected from the inside of the unique oval). In this case the projection from the interior of the oval yields a real map without real ramification, but not totally real. We see no obvious link from Klein’s equivariant branched covers to the stronger assertion that fibres over real points consists only of real points, and consequently one of the orthosymmetric halves maps conformally to the upper half-plane (as Teichmüller credits to Klein). Of course it is not impossible that a suitable complement to Klein’s method yields something like an Ahlfors mapping. By a continuity argument in Gabard 2006 [@Gabard_2006 Lemme 5.2], it would be enough to chose $g_m=:D$ as an [*unilateral*]{} divisor, i.e. one supported entirely by one half of the curve. Then we would be finished if the symmetric divisor $D^{\sigma}$ is linearly equivalent to $D$. But this condition is far from automatic and involves probably some lucky choice in the position of the initial divisor $D$. Alternatively, one may try to specify the ramification and work out the Lüroth-Clebsch sort of argument to construct explicitly the finitely many conformal type of Riemann surfaces lying above the prescribed ramification. But the writer failed to draw any serious conclusion. More is less: Teichmüller again (1939) -------------------------------------- For those not overwhelmed by German prose, the following passage also bears some resemblances to the Ahlfors function: Falls $\frak M$ eine orientierte und berandete Mannigfaltigkeit ist, braucht man $f$ nur auf $\frak M$ zu kennen, um $f$ auf $\frak F$ berechnen zu können. \[The latter is of course the doubled surface.\] $f$ mu[ß]{} dann auf den Randkurven von $\frak M$, die ja zu sich selbst punktweise konjugiert sind, reelle Werte haben. Umgekehrt ist eine Funktion der Fläche, die in unendlich vielen Randpunkten von $\frak M$ reell ist, eine Funktion von $\frak M$, denn sie stimmt mit der konjugierten in unendlich vielen Punkten überein und ist darum gleich ihrer konjugierten Funktion. Ja, wir können die Funktionen $f$ von $\frak M$ sogar ganz auf $\frak M$ charakterisieren: Die Funktionen der orientierten berandeten endlichen Riemannschen Mannigfaltigkeit $\frak M$ sind genau die Funktionen $f$, die in $\frak M$ bis auf Pole regulär analytisch sind und die am Rande von $\frak M$ reell werden. D.h. die Punkte, wo die Funktion Werte eines abgeschlossenen Kreises der oberen oder der unteren Halbebene annimmt, sollen eine kompakte Menge im Innern von $\frak M$ bilden. In der Tat lassen sich diese Funktionen durch Spiegelung zu Funktionen von $\frak F$ machen, insbesondere sind sie auf den Randkurven von $\frak M$ stetig. In this passage we note that just adding the single word “nur” in the third line of the 2nd parag. to read “die nur am Rande von $\frak M$ reell werden” would essentially lead to an anticipation of Ahlfors 1950. However taken literally this assertion of Teichmüller is weaker than Ahlfors’ and indeed the previous Quote \[quote:Teichmueller-1941\] is perhaps just a logical distortion (through hasty writing!) of the above more precise (but logically weaker) formulation. Under this hypothesis then we agree perfectly with Teichmüller 1941 (cf. again Quote \[quote:Teichmueller-1941\]) that this reality behaviour of functions was known to Klein. The crucial distinction is between functions real on the boundary and those which are real only on the boundary. Now a priori a real function may be real on an interior point of the membrane, in which case the range (of the function) will not be contained in one of the half-plane, but overlap with both of them. In contrast a stronger reality behaviour arises when fibres of real points excludes imaginary conjugate points, in which case the range is contained in one of the half-plane, which is the context of Ahlfors’ circle mapping. Courant 1937, 1939, 1950 {#sec:Courant} ------------------------ In the paper Courant 1939 [@Courant_1939], one detects another approach to the existence of circle maps via the methods of Plateau’s problem (at least so is claimed by Grunsky 1978, cf. Quote \[quote:Grunsky-1978\]). We cite some portion of Courant’s introduction: \[quote:Courant-1939\] The theory of Plateau’s and Douglas’ problem furnishes powerful tools for obtaining theorems on conformal mapping. Douglas emphasized (1931) that Riemann’s mapping theorem is a consequence of his solution of Plateau’s problem; then he treated doubly connected domains and in a recent paper (1939) multiply connected domains. With a different method I gave in a paper on Plateau’s problem (1937) a proof of the theorem that every $k$-fold connected domain can be mapped conformally on a plane domain bounded by $k$ circles. The same method can be applied to the proof of the parallel-slit theorem and, as will be shown in the thesis of Bella Manel, to mapping theorems for various other types of plane normal domains. It is the purpose of the present paper first to give a simplification of the method by utilizing an integral introduced by Riemann in his doctoral thesis, and secondly, to prove a mapping theorem of a different character referring to normal domains which are Riemann surfaces with several sheets. \[…\] For the case $p=0$, the theorem was stated by Riemann, according to oral tradition. \[See Bieberbach 1925, where a proof is indicated; and Grunsky 1937, where another proof is given.\] It should still be elucidated if this work by Courant (officially overlapping with Bieberbach-Grunsky) may also be connected to the Ahlfors circle mapping. This is still not completely clear to the writer. The topic is addressed again in Courant’s book of 1950, e.g., as follows: [*Theorem 5.3:*]{} Every plane [\[footnote 12: As said before, in view of the general result of Chapter II the assumption that $G$ is a plane domain is not an essential restriction.\]]{} $k$-fold connected domain $G$ having no isolated boundary points can be mapped conformally onto a Riemann surface $B$ consisting of $k$ identical disks, e.g. interiors of unit circles, connected by branch points [\[footnote 13: The conformality of the mapping is of course interrupted at the branch points.\]]{} of total multiplicity $2k-2$. \[…\] This somewhat loose footnote 12 of Courant may advance him as a forerunner of the Ahlfors circle map. Courant does not specify the degree derived by his method, but reading him literally one recovers (quite strikingly!) Ahlfors’ bound $r+2p$ (compare the following numerology): The connectivity $k$ of a membrane of genus $p$ with $r$ contours is equal to $r+2p$ (each handles contributes 2 units to the connectivity). \[Alternatively, we may interpret the connectivity $k$ as $b_1+1$, where $b_1$ is the first Betti number. The Euler characteristic is $\chi=2-2p-r$, but also expressible as $\chi=1-b_1$ (since $b_2=0$). Back to the connectivity, we find $k=b_1+1=(1-\chi)+1=2-\chi=2-(2-2p-r)=2p+r$, as desired.\] Adopting Courant’s branching multiplicity $b:=2k-2$, we compute the corresponding degree $d$. By Riemann-Hurwitz $\chi= d \cdot \chi (D^2)-b$, hence $d=\chi+b=(2-2p-r)+(2k-2)=2k-2p-r=2(r+2p)-2p-r=r+2p$. [*q.e.d.*]{} This is pure numerology, without much control of the underlying geometry. More insight is suggested by Courant’s subsequent statement in [*loc.cit.*]{} [@Courant_1950 p.183–4, Thm5.3], which we reproduce: Moreover, an arbitrarily fixed point $F_{\nu}$ on each boundary circle $\beta_{\nu}$ can be made to correspond to a fixed boundary point $P_\nu$ on the boundary continuum $\gamma_\nu$ of $G$, and the position of one simple branch point in $B$ may be prescribed. The class $\frak N$ of these domains depends on $3k-6$ real parameters: the $2k-3$ freely variable branch points represent $4k-6$ coordinates, while fixing the points $F_\nu$ reduces the number of parameters by $k$. Extending this reasoning to (non-planar) membranes, we derive again Ahlfors’ bound, as follows: We assume the membrane $F_{r,p}$ (of genus $p$ with $r$ contours) conformally mapped as a $d$-sheeted cover of the disc $D^2$ with $b$ branch points. As usual the Riemann-Hurwitz relation reads $\chi=d\cdot \chi(D^2)-b$. From the $b$ branch-points, one of them can be normalized to a definite position (through a conformal automorphism of the disc). Now the fibre over a boundary point of the disc gives $d$ points on $\partial F$. Those $d$ boundary points can be thought of as having a prescribed image. Thus the mapping itself is fully determined by $2(b-1)-d$ real constants. On the other hand, we know since Klein 1882 (cf. our Quote \[quote:Klein-1882-moduli\]) that $F_{r,p}$ has $3g-3$ real moduli where $g$ is the genus of the double $2F$, i.e. $g=2p+(r-1)$. Positing the Ansatz that the family of $d$-sheeted covering surfaces has enough free-parameters to fill the full moduli space leads to the inequation $2(b-1)-d \ge 3g-3$. But $b=d-\chi$ and $2\chi=\chi(2F)=2-2g$. Hence $2(d-\chi-1)-d=d+(2g-2)-2\ge 3g-3$, i.e. $d\ge g+1$, which is Ahlfors’ bound $r+2p$. Of course, this happy numerology (noticed by the writer the \[20.05.12\]) is no substitute to a serious proof of the Ahlfors circle map. However Courant formulates a variational problem à la Plateau-Douglas (or Dirichlet-Riemann-Hilbert) affording existence of a circle map (presumably with the same bound as predicted by Ahlfors as prompted by our heuristic count). Unfortunately, in Courant’s book the presentation is not directly adapted to the case of general membranes of positive genus ($p>0$), making the reading somewhat hard to digest. Hopefully someone will manage in the future to present a self-contained account based upon Courant’s method. (This project involves some hard analysis and will be deferred to a subsequent technical section. ABORTED: I had not the time/force to adapt Courant’s text to higher genera as suggested by his sloppy footnote 12.) Of course in view of Carathéodory’s philosophy (cf. Quote \[quote:Caratheodory-1928\]) one may wonder which of Courant’s vs. Ahlfors approach enjoys methodological superiority? Further remind that Ahlfors (1950 [@Ahlfors_1950 p.125–6]) has also an elementary argument for circle mapping involving no extremal problem. Another puzzling feature of the above numerology is that it gives the impression that any $r+2p$ points prescribed on the contours may be mapped to a fixed point of the circle. Whether this is really true deserves to be investigated. Trying to read Courant’s book 1950 [@Courant_1950] with the focus of the Ahlfors circle map is not an easy task (in our opinion). We may then hope that reading the original 1939 article [@Courant_1939] is easier due to its more restricted content. Let us write down its main statement: \[quote:Courant\_1939:statement\] We consider a Riemann surface on a $u,v$-plane consisting of the interior of $k$ unit circles which are connected in branch points of total multiplicity $2k-2$; to this surface we affix $p\ge 0$ full planes with two branch points each. Thus we define a class of domains $B$ with the boundary $b$ on the plane of $w=u+iv$. Now our theorem is: Each $k$-fold connected domain $G$ in the $x,y$-plane with the boundary curves $g_1,g_2,\dots,g_k$ \[…\] can be mapped conformally on a domain $B$ of our class for any fixed $p$. In this mapping the branch points on the full planes and one more branch point may be arbitrarily prescribed and, moreover, on each boundary circle $b_\nu$ of $B$ a fixed point may be made to correspond to a fixed point of $g_\nu$. Personally, I find this statement hard-to-read for several reasons, I shall list subsequently. Moreover it is not clear if suitably interpreted, it really implies the Ahlfors circle mapping. How to interpret this statement of Courant? Here are some critics probably due to the writer’s incompetence (rigid brain)! On the one hand, we have $B$, which moves in a class of domains. Perhaps those are Riemann surfaces? For instance the operation of affixing $p$ full planes may give a surface of genus $p$, at least this is what is suggested by a latter publication of Courant 1940 [@Courant_1940-Acta], whose relevant portion we quote again for definiteness: On the basis of the previous results, the proof of the characteristic relation $\varphi(w)=0$ for the solution of the variational problem becomes very simple, if the underlying class of domains $B$ is chosen not as a domain in the plane but as a Riemann surface all of whose boundary lines are unit circles. This class is defined as follows: We consider for the case of genus zero a $k$-fold connected domain $B$ formed by the discs of $k$ unit circles which are connected in branch points of the total multiplicity $2k-2$. For higher genus $p$, we obtain domains $B$ by affixing to the $k$-fold circular disc $p$ full planes each in $4$ branch points \[footnote 2: Each such full plane represents a “handle” and increases the genus by $1$.\]. Well, but then the domain $B$ of Quote \[quote:Courant\_1939:statement\] would have genus $p$. Then how is it possible for him to get mapped conformally (in a one-to-one fashion?) to the domain $G$, which seems to be planar since its connectivity is equal to the number of boundaries! Perhaps $G$ should be assumed to be $(k+2p)$-fold connected (or put more briefly $G$ should have genus $p$ and $k$ contours)? If so then Courant gives a (conformal) one-to-one(?) map (=diffeomorphism) $G\to B$ onto a “normal” domain $B$. To make a link with Ahlfors, it would be desirable to know if $B$ maps to the disc even after the affixing of the $p$ full planes. (Incidentally, this operation is somewhat poorly defined, but perhaps better exposed in other publications, cf. e.g., Courant’s book 1950 [@Courant_1950 p.80 and ff.] or Courant 1949/52 [@Courant_1949-52:Book].) Hence the crucial point would be to know if $B$ is a many-sheeted cover of the disc, and if yes: how many sheets are required? Very naively $k+p$ could suffice, in which case Courant would not only compete with Ahlfors 1950 [@Ahlfors_1950], but also with Gabard 2006 …(NB: This $(k+p)$-sheeted-ness occurs again in Courant 1940 [@Courant_1940-Acta p.78], and it would be of interest to decide if this constitutes an anticipation of Gabard 2006.) If we push our misunderstanding of Courant to its ultimate limit, we may have the impression that what he do, is an attempt to mix the parallel-slit mapping he learned from Hilbert 1909 [@Hilbert_1909], with the Riemann-Schottky-Bieberbach-Grunsky theorem, but that the resulting surgery/transplantation does not lead to any really viable creature. Of course, probably much of our misunderstanding is caused not merely from the difficult mathematics but also from a shift in language (plus perhaps some inaccuracies due to the torrential number of publications?), yet we may still hope that either an appropriate reading (or reorganization) of Courant’s thoughts may lead to an anticipation of the Ahlfors circle map. Hence, we encourage strongly any reader able to take the defense of Courant to publish an account in this direction. Finally, we cite another papers of Courant about conformal maps, which could be of some relevance: $\bullet$ Courant 1937 [@Courant_1937], especially p.682, footnote 7, where we read: “[*If we assume the possibility of a conformal mapping on the unit circle for all surfaces admitted to competition \[…\]*]{}”. This could have some connection with Ahlfors circle maps, but probably does not. Later on, this article contains some conformal mapping theorems, which are only announced without proof. Perhaps, those could be of some relevance. Especially Fig.11, p.722, seems to be close to Klein’s Rückkehrschnitt-Theorem, and could eventually leads to a proof of Ahlfors? This paper also relates the ideas of J. Douglas about minimal surfaces (especially his extended version of the Plateau problem for surfaces of higher topological structures, where Douglas uses systematically Klein’s symmetric surfaces). One may therefore wonder if Ahlfors’ circle maps may somehow find application in this grandiose theory of minimal surfaces à la Plateau-Douglas-Radó-Courant, etc. As far as the writer knows no direct connection is presently available in print, despite the probable vicinity of both topics. $\bullet$ Courant 1938 [@Courant_1938], especially p.522 “[*Every plane $k$-fold connected domain can be mapped conformally to a $k$-fold unit circle*]{}”. Hence the result—we are mostly interested in—occurs here already in 1938. In contrast to the 1939 version [@Courant_1939], here neither Riemann, nor Bieberbach 1925 [@Bieberbach_1925], not even Grunsky 1937 [@Grunsky_1937] are cited. Did Courant rediscovered the result independently? $\bullet$ Finally we quote, Courant 1919 [@Courant_1919], where (under some influence of Hilbert 1909, and Koebe 1909) conformal mappings to “normal domains” are discussed for non-schlichtartig surfaces (of finite genus). This is also re-discussed in Courant’s book of 1950 [@Courant_1950]. Last but not least, it is perhaps relevant to remind that some doubts where expressed by Tromba 1983 [@Tromba_1983-PREPRINT] about the validity of Courant’s argumentation regarding higher genus cases of the Plateau-Douglas problem (compare also Jost 1985 [@Jost_1985]). It is not clear to the writer if Tromba’s objections compromise seriously the validity of Courant’s assertions (regarding higher genus conformal maps re-derived via the method of Plateau). This could be a another obstacle toward completing a Courant-style approach to the Ahlfors map. Douglas 1931–36–39 {#sec:Douglas} ------------------ Having discussed (very coarsely) Courant, it would be unfair to neglect J. Douglas. His resolution of Plateau’s problem interacts strongly with conformal mapping, with the distinctive attitude (partially successful) of not getting subordinated to the latter. As already pointed out (in Courant’s Quote \[quote:Courant-1939\]), Douglas re-derived the (RMT) as the 2D-case of Plateau (cf. Douglas 1931 [@Douglas_1931-Solution p.268]). Subsequently, Douglas extended his Plateau solution to configurations of higher topological structure (cf. Douglas 1936 [@Douglas_1936-Some-new-results], 1939 [@Douglas_1939-min-surf], 1939 [@Douglas_1939-The-most-general]). Thus, it is nearly natural to ask if Douglas (himself, or at least his methods) may anticipate/recover the Ahlfors circle map? Ironically, Douglas’ work relied on Koebe’s, and interestingly took a systematic advantage of (Klein’s) symmetric Riemann surfaces (e.g., orthosymmetry). Without entering the details of all those exciting connections, we just refer to the cited original works, plus the account of Gray-Micallef 2008 [@Gray-Micallef_2008], of which we quote some extracts: An unexpected bonus of Douglas’s method is a proof of the Riemann-Carathéodory-Osgood Theorem, which follows simply by taking $n=2$. \[…\] Douglas was rightly proud that his solution not only did not require any theorems from conformal mapping but that some such theorems could, in fact, be proved using his method. However, Douglas did have to use Koebe’s theorem in order to establish that his solution had least area among discs spanning $\Gamma$. He had hoped to fix this blemish, but he never succeeded. That had to wait for contributions from Morrey \[1948\] and, more recently, from Hildebrandt and von der Mosel \[1999\]. \[…\] Even before working out all the details for the disc case, Douglas was considering the Plateau problem for surfaces of higher connectivity and higher genus. \[…\] As early as 26 October 1929, Douglas announced that his methods could be extended to surfaces of arbitrary genus, orientable or not, with arbitrarily many boundary curves in a space of any dimension. He may well have had a programme at this early stage, but it is doubtful that he had complete proofs. Even when he did publish details in \[3\](=1939 [@Douglas_1939-min-surf]), the arguments are so cumbersome as to be unconvincing. One should remember that Teichmüller theory was still being worked out at that time and that the description of a Riemann surface as a branched cover of the sphere is not ideally suited for the calculation of the dependence of the $A$-functional on the conformal moduli of the surface. Courant’s treatment in \[7\](=1940 [@Courant_1940-Acta]) was more transparent but still awkward. The proper context in which to study minimal surfaces of higher connectivity and higher genus had to wait until the works of Sacks-Uhlenbeck \[19\](=1981), Schoen-Yau \[20\](=1979), Jost \[11\](=1985) and Tombi-Tromba \[21\](=1988). \[…\] Finally, we mention the recent work of Hildebrandt-von der Mosel 2009 [@Hildebrandt-von-der-Mosel_2009], plus the survey Hildebrandt 2011 [@Hildebrandt_2011]. Here we learn, that Morrey 1966 [@Morrey_1966] was the first to re-prove Koebe’s KNP (=Kreisnormierungsprinzip) via Plateau, modulo a gap fixed by Jost 1985 [@Jost_1985]. The ultimate exposition of 2009 (of loc.cit. [@Hildebrandt-von-der-Mosel_2009]) is intended to be “[*possibly simpler and more direct*]{}” (loc.cit., 2009, p.137) and “[*are complete analogs of the approach of Douglas and Courant*]{}” (loc.cit., 2011, p.77). As an agenda curiosity, the “Plateau-ization” of conformal mapping theorems does occur along diabolic chronological regularity. From Riemann 1851 [@Riemann_1851] to Douglas 1931 [@Douglas_1931-Solution], gives an elapsing period of 80 years. For circle maps, we have from Riemann 1858 to Courant 1939($-1$) also 8 decades, and from Koebe 1904 (announcement of KNP, in his Thesis talk, yet without convergence proof until 1907/08) to Jost 1984 [@Jost_1985] gives the same interval of time. Thus Ahlfors 1950 [@Ahlfors_1950] can safely wait up to 2030, before getting reproved via the method of Plateau? Again, from our focused viewpoint, the critical question is whether within the problem of Plateau (à la Douglas–Radó–Courant, etc.) germinates an alternative proof of the Ahlfors mapping. As far as we know, the paper closest to this goal his Courant 1939 [@Courant_1939]. Yet, we cannot readily claim that it includes the result of Ahlfors 1950. Cecioni and his students, esp. Matildi 1945/48, and Andreotti 1950 ------------------------------------------------------------------ Among several interesting works of Cecioni and his students (cf. Sec.\[sec:Italian-school\]) we point out especially the article by Matildi 1948 [@Matildi_1945/48] (discovered by the writer as late as \[13.07.12\]). In it the existence of an (Ahlfors-type) circle map in the special case of surfaces with a single contour seems to be established via classical potential-theoretic tricks, plus at the end some algebraic geometry. This work was known to Ahlfors (or Sario?) at least as late as 1960, being quoted in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960]. It would be interesting to see if Matildi’s method adapts to an arbitrary number of contours, and also try to make (more!) explicit the degree bound obtained by him. In that case Matildi should be considered as a serious forerunner of Ahlfors 1950 [@Ahlfors_1950], at least at the qualitative level (no extremal problem). Perhaps, il professore Cecioni himself has—and may have—several works (some of which we could not consult as yet) coming quite close to the circle mapping thematic à la Ahlfors. The idea that Matildi’s argument should extend easily to the case of several contours looks an accessible exercise. Andreotti 1950 [@Andreotti_1950] seems to go precisely in this sense. A global picture (the kaleidoscope) ----------------------------------- The place occupied by RMT (Riemann mapping theorem) is quite pivotal in conformal mapping with an organical explosion of results around it, like: $\bullet$ KNP=Kreisnormierungsprinzip (implicit in Riemann 1857/8, Schottky 1875 (Latin version of his Thesis, cf. Klein’s Quote \[Klein-1923:quote:Riemann-1858\]), in full by Koebe 1905-10-20). $\bullet$ RS=Riemann-Schottky mapping of a multiply-connected domain to the disc. (This is also known as the Bieberbach-Grunsky theorem, so RS$\approx$BG, if you want.) $\bullet$ AM=the Ahlfors mapping (of a compact bordered surface to the disc). $\bullet$ GKN=generalized Kreisnormierung (of a compact bordered surface to a circular domain inside a closed Riemann surface of constant curvature having the same genus $p$): apart from some anticipation for $p=1$ Strebel 1987 [@Strebel_1987] and Jost (unpublished), the full result is due to Haas 1984 [@Haas_1984] (existence), and Maskit 1989 [@Maskit_1989] (uniqueness). For an approach via circle packings, compare also He 1990 [@He_1990] and He-Schramm 1993 [@He-Schramm_1993]. $\bullet$ RST=Rückkehrschnitttheorem of Klein 1882 [@Klein_1882_Ruckkehrschnitt] is yet another form of generalized Kreisnormierung to positive genera, and for simplicity we identify it loosely to GKN. The first (rigorous) proof of RST is to be found in Koebe 1910 UAK2 [@Koebe_1910_UAK2], see also Bers 1975 [@Bers_1975] for a modern account via quasiconformal deformations. Of course (at least modulo some sloppiness) we have universal implications (just by specializing the topological structure) like $$\textrm{AM}\Rightarrow \textrm{RS} \Rightarrow \textrm{RMT} \Leftarrow \textrm{KNP} \Leftarrow \textrm{GKN}.$$ Besides it is desirable that GKN or RST$\Rightarrow$AM, at least this would resolve our big historical puzzle about Klein-Teichmüller as anticipating Ahlfors. This desideratum is a bit cavalier, yet akin to the implication KNP$\Rightarrow$RS, which is cryptical since Riemann’s Nachlass (1857 [@Riemann_1857_Nachlass]). On the other hand there is a large panoply of methods including: $\bullet$ algebraic functions (Abel 1826, Jacobi 1832, Riemann 1857, etc.), $\bullet$ potentials (Dirichlet ca. 1840, Green 1828, Gauss 1839, Thomson 1848, etc.), $\bullet$ iterative methods (Koebe, Carathéodory 1905–12), $\bullet$ extremal problems (Fejér-Riesz 1922, Carathéodory 1928, Ostrowski 1929, etc.), $\bullet$ orthogonal systems (Bergman kernel 1922, Szegö 1921) $\bullet$ Plateau-Douglas functionals (Plateau 1849, Douglas 1930, Courant 1939 via Dirichlet resurrected), $\bullet$ circle packings (originally in Koebe 1936, rediscovered by Andreev and Thurston 1985 with convergence proof by Rodin-Sullivan 1986), $\bullet$ Ricci flow (Hamilton 1988 [@Hamilton_1988], which specialized to 2D enables one to recover the uniformization theorem); idem via Liouville’s equation (desideratum Schwarz, followed by Picard 1890–93, Poincaré 1899, Bieberbach 1916 [@Bieberbach_1916-Delta-u-und-die-automorphen-Funkt], etc., cf. Mazzeo-Taylor 2002 [@Mazzeo-Taylor_2002] for a modern account), and also e.g., Zhang [*et al.*]{} 2012 [@Zhang-et-al_2012], where a mixed Ricci flow/Koebe’s iteration is advocated. Blending all these results with all those methods accessing them, we get the kaleidoscope depicted below (Fig.\[Kaleidoscope:fig\]) attempting to classify a body of results in a (more-or-less) systematic fashion. Black arrows stress out methods effective in solving a certain mapping problem, whose extremity points to the source (listed in our bibliography). Starting around RMT, arrows are propagated by translation to other locations (e.g., RS, or KNP). Arrows turn to white colored, if the corresponding method has not yet been applied to solve the relevant mapping problem. Of course several methods (like the balayage of Poincaré 1907, or some of Koebe’s method may be slightly outdated having few living practitioners). In contrast, Koebe’s iteration method is still quite popular due to its computational efficiency (see e.g., Zhang et al. 2012 [@Zhang-et-al_2012]), and presumably theoretically fruitful as well (where it is used in conjunction with the Ricci flow). -55pt 0 Of course the picture is hard to make completely reliable, yet it may aid feeling the power (or popularity) of some methods (e.g, the extremal problem method seems to apply quite universally, except presently to GKN). On the other hand some recent methods like circle packings look very powerful, and may not have as yet explored their full range of applicability (e.g. regarding AM). As discussed in the previous section, we do not know if the Plateau method could crack the AM. Another powerful method is that of the Bergman kernel, which probably also leads to a derivation of the AM. When reading papers of the golden period (1948–1950, Bergman, Schiffer, Garabedian, etc.) this seems to be almost folklore, as well as in some papers of Bell (e.g. 2002 [@Bell_2002]). While spending some time reading precisely what is put on the paper, the writer rather developed the feeling that the positive genus case is never handled in full details. (As a general lamentation, it is an easy challenge to cite about 20 papers where results proved in the planar case are followed by the apocryphal allusion that the proof works through [*mutatis mutandis*]{} without planarity proviso.) Digression on Bieberbach and Bergman {#Sec:Bieberbach-Bergman} ==================================== The Bergman kernel {#sec:Bergman} ------------------ Among the variety of methods mentioned in the previous section, one especially popular is the Bergman kernel function. This emerges in Bergman’s Thesis 1921/22 [@Bergman_1922]. The point of departure is an area minimal problem going back to Bieberbach 1914 [@Bieberbach_1914] capturing some salient geometric feature of the Riemann mapping. Interestingly, Bergman 1922 ( [@Bergman_1922 p.245]) confesses to be not able to reprove the RMT with this method: In dem betrachteten Spezialfall (Minimalabbildung durch analytische Funktion) ist die erhaltene Minimalfunktion die Kreisabbildungsfunktion. Wie oben gezeigt, kann man die Existenz der ersteren unabhängig von dem Hauptsatze der Funktionentheorie beweisen; es besteht somit die Möglichkeit, den Hauptsatz auf diesem Wege von neuem zu beweisen, was mir aber bis jetzt nicht gelungen ist. A similar lamentation is expressed by Bochner 1922 [@Bochner_1922 p.184]: Aus der Möglichkeit der Kreisuniformisierung eines einfach zusammenhängenden Bereiches folgt aber, wie Bieberbach bemerkt hat (l.c.), da[ß]{} die Minimalabbildung mit eben der Kreisabbildung identisch ist, indes ist es mir nicht gelungen, aus der Minimalabbildung der Kreisuniformisierung aufs neue herzuleiten. In a similar vein, some 3 decades later one among the prominent aficionados of the method wrote (source=Math.-Reviews for Lehto’s Thesis 1949 [@Lehto_1949]): Despite its great intrinsic elegance and its adaptability for numerical computations, the theory of complex orthonormal functions (centering about the concept of the Bergman kernel function) had the drawback of being a mere representation theory; the fundamental existence theorems had to be borrowed from other fields. In $\S 4$ the author fills this gap in one important instance by giving an existence proof for the parallel-slit mappings (in the case of simply-connected domains this is identical with the Riemann mapping theorem \[provided the slit is extended to $\infty$\]) within the framework of the orthonormal function theory. So somewhere in between 1922–1949 some technological turning point must have occurred amplifying dramatically the power of the Bergman kernel method. When and how did this occurred exactly? Probably through the Bergman–Schiffer collaboration in the 40’s, plus some fresh blood like Garabedian or Lehto. In several subsequent publications of Garabedian and Schiffer, it is emphasized that parallel-slit mappings are easier than circle maps (cf. Quotes \[quote:Garabedian-Schiffer\_1950\] and \[quote:Garabedian\_1949-52\]). However the Ahlfors circle mapping seems accessible to the Bergman-Szegö orthogonal system method as suggested in Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950], where only the planar case is handled in detail. Often in literature, yet not in the just cited paper, it is sloppily insinuated that a method implemented in the planar case extends to Riemann surfaces. A typical specimen is the earlier paper by Garabedian 1950 [@Garabedian_1950] claiming another proof of the full Ahlfors theorem by deploying a broad spectrum of techniques (yet not readily reducible to the Bergman kernel) ranging from Teichmüller 1939 [@Teichmueller_1939-Dreikreisesatzes], Grunsky 1941–42 [@Grunsky_1940], [@Grunsky_1942], Ahlfors 1947 [@Ahlfors_1947] and Schiffer’s inner variations. Inspecting back the Bergman method itself, it is not hard to understand why it is most readily implementable in the planar case. It seems indeed to require a sort global ambient coordinate system. Let us look at the beautiful original paper Bergman 1922 [@Bergman_1922 p.240]. Here the key idea is a characterization of the Riemann function $w\colon B \to \Delta$ (of a \[simply-connected\] domain $B$ to the disc) as the one whose range $w(B)$ has smallest possible area amongst all functions $f\colon B \to {\Bbb C}$ constrained by $f'(0)=1$ (and $f(0)=0$ after harmless translation so that $0\in B$). The area swept out by $f$ is calculated by the integral $$\int \int_B f'(z) \overline{f'(z)} d\omega,$$ where $d \omega$ is the surface element in the $B$-plane, while the integrand $\vert f'(z)\vert^2$ measures the distortion effected at $z$. Following Bieberbach 1914 [@Bieberbach_1914] (who in turn seems inspired by Ritz 1908 [@Ritz_1908]), Bergman plugs in place of $f(z)$ a polynomial (recall the finitistic motto of Bloch “[*Nihil est infinito …*]{}”): $$w_n(z)=a_0+a_1z+\dots+a_n z^n,$$ with coefficients determined as to minimize the above integral under the constraint $w_n'(0)=1$. There is always a unique such polynomial, which is computed by usual methods (finite extremum-problem). The limiting function $\lim_{n\to \infty} w_n$ gives—again Bieberbach is cited—the required mapping. The method is so simple and elegant that it is hardly conceived why it fails to reprove the RMT (which Bergman and others call the [*Hauptsatz der Theorie der konformen Abbildung*]{} (p.240)). The reason is however a quite simple vicious circle, namely that the above (Bieberbach) “areal” characterization of the Riemann function logically rests on RMT. Hence the minimum function (of Bergman) is eminently computable, but the resulting power series may not have a priori the required geometrical property of univalence and the right disc-range. I guessed the latter property follows from Bieberbach 1914 [@Bieberbach_1914], hence the real problem is univalence. However on \[13.06.12\], after reading Bergman 1947 [@Bergman_1947 p.32], the opposite looks true: namely univalence is easy but the disc-range issue is not. There are mentioned two contributions, one by Bergman 1932 [@Bergman_1932] and also Schiffer 1938 [@Schiffer_1938-CRAS-domaines-minima] where the desideratum (of reproving RMT) is established for starlike domains. So almost as importantly, this source (of 1947) points out that Bergman’s dream of 1922 (new proof of RMT via the area extremum problem) was not borne out until 1947, and therefore seems really to be credited to the newer generation like Garabedian and Lehto. Generally speaking, extremum problems are often solvable (even uniquely soluble), but it is another piece of careful analysis to control precisely the geometric behavior of solutions, e.g. in the hope to re-crack RMT. Of course, the problem was ultimately solved, cf. e.g., Garabedian 1950 [@Garabedian_1950] or the already mentioned Thesis of Lehto 1949 [@Lehto_1949], which are the first completed Bergman-style approaches to RMT. The point for re-exposing the hearth of the method is to emphasize the rôle of polynomials generated by $z^n$ as a preferred system of global functions on the domain $B$ out of which an ideal object is processed through an extremum procedure handled [*in finito*]{}. How can one adapt this on a Riemann surface where no global parameters are supplied a priori? This is a little puzzle to the writer \[06.06.12\], but the masters (Bergman, Garabedian, Bell, etc.) often claim the method to suit the broader context with minor changes. Compare, e.g., the following sources: $\bullet$ Bergman 1950 [@Bergman_1950 p.24, Remark] justifies in this book extensibility to Riemann surfaces by referring to results of Sario 1949–50. $\bullet$ Garabedian 1950 [@Garabedian_1950 p.361], where one reads “[*For the sake of a simple presentation of results we have merely stated the theorem for the case of schlicht domains of finite connectivity. However the theorem is true with only one change if $D$ is a Riemann surface [\[…\]]{}. The reader will easily verify that the proof which we shall give of the theorem carries over with minor changes to the more general situation.*]{}” If not pure bluff, it is sad that Garabedian did not write down the details at that time. If we believe in the unity of mathematics especially the algebro-geometric curves and analytic Riemann surfaces at the compact level, then the existential aspect of circle maps is frankly more trivial in the “schlicht” and even “schlichtartig” case, compare e.g., the argument in Gabard 2006 [@Gabard_2006] (reproduced below as Lemma \[Enriques-Chisini:lemma\]), which in substance is the one of Bieberbach 1925 [@Bieberbach_1925], Wirtinger 1942 [@Wirtinger_1942], but perhaps slightly streamlined by the mere usage of algebro-geometric language. Minimizing the integral vs. maximizing the derivative (suction vs. injection), i.e. Bieberbach 1914-Bergman 1921/22 vs. Fejér-Riesz 1922, etc. ---------------------------------------------------------------------------------------------------------------------------------------------- Trying to avoid the vicissitudes of life concomitant with the Dirichlet principle, the early 1920’s imagined two methods of attack to the RMT via extremum problems. Given $B \ni a$ a simply-connected domain in the complex plane ${\Bbb C}$, which is not the plane and therefore can easily be assumed to be bounded via a suitable transformation, RMT amounts to find a conformal map to the disc. The following (animalistic) acronyms are derived by contracting the contributors’ names: $\bullet$ (BIBER)=(Bieberbach 1914 [@Bieberbach_1914] and Bergman\[n\] 1922 [@Bergman_1922]). \[Biber=German for “beaver” (=“castor” in French).\] $\bigstar$ [*Amongst analytic functions $f\colon B \to {\Bbb C}$ normed by $f(a)=0$ and $f'(a)=1$ minimize the integral $\int \int_B \vert f'(z)\vert^2 d\omega$, where $d\omega$ is the surface element of the Euclidean metric.* ]{} $\bullet$ (FROG)=(Fejér-Riesz 1922, Carathéodory 1928 [@Caratheodory_1928]$\leftrightarrow$Ostrowski 1929 [@Ostrowski_1929], and Grunsky 1940 [@Grunsky_1940], Ahlfors 1947 [@Ahlfors_1947] in the multiply-connected context) $\bigstar$ [ *Amongst analytic functions $f\colon B \to \Delta$(= unit disc) normed by $f(a)=0$ maximize the modulus $ \vert f'(a)\vert$.*]{} As remembered in the previous section, the problem BIBER was not prompt in supplying an autonomous proof of RMT, while succeeding only in the late 1940’s (Garabedian’s or Lehto’s Thesis). Further this succeeded perhaps only under the proviso of smooth boundary (Jordan curve), cf. e.g. Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950 p.164]: “[*The most serious drawback in our method is, perhaps, that we must make assumptions upon the smoothness of the boundary of the domain we consider, so that the general case is reached only after a topological approximation argument is given.*]{}”. In contrast FROG met earlier success (cf. e.g., Ostrowski 1928/29 [@Ostrowski_1929] and Carathéodory 1928 [@Caratheodory_1928]) streamlining previous work of Fejér-Riesz 1922 (published in Radó 1923 [@Rado_1923-Uber-konf-Abb]). For extensions to multiple-connectivity, or even Riemann surfaces, we have the following contributions: $\bullet$ FROG leads to the works of Grunsky 1940–42 [@Grunsky_1940; @Grunsky_1942] (schlicht domains of finite connectivity) and Ahlfors 1950 [@Ahlfors_1950] (non-planar compact bordered Riemann surfaces), where the derivative $f'(a)$ is computed w.r.t. any local chart. In fact Ahlfors rather considers the variant where given two points $a,b$ the modulus of $f(b)$ has to be maximized amongst functions with $f(a)=0$. $\bullet$ BIBER is somewhat harder to formulate on a Riemann surface $F$ (taking the rôle of the domain $B$) as the magnitudes involved in the problem require something more than the Riemann surface structure. A Riemannian metric would make the problem meaningful, but which metric to choose? Of course there is the canonical conformal metric given by uniformization of the doubled membrane $F$. Of course we deviate slightly from a self-contained proof of RMT or Ahlfors (=AMT), but this is maybe not a dramatic concession. Thus, even in its basic formulation, some ideas are required to set a perfect analogue of the problem BIBER for a (bordered) surface. If this can be done, it is likely (or desirable) that the extremal function (whose existence and uniqueness is derived by Hilbert’s spaces arguments) is a circle map, i.e. effects a conformal representation over the disc. (This is a priori not the unit disc, but renormalize so.) In the simply-connected case, both extremals of BIBER and FROG (denoted $\beta$ and $\alpha$ respectively) yield the one and the same object, namely the Riemann mapping $B\to\Delta$ (again after a harmless scaling of $\beta$, cf. Bergman 1950 [@Bergman_1950 p.24] for its exact value in terms of the Bergman kernel). Hence, it is plausible that the least area map for the surface $F$ coincides with the Ahlfors function. So this would be a sort of conformal identity, perhaps of some practical significance. Of course, the primary interest would be to reobtain (via BIBER) a novel proof of Ahlfors 1950 [@Ahlfors_1950]. (This game may be already implicit in several works, as those of Bergman and Garabedian itemized in the previous section, but no pedestrian redaction is available in our opinion.) Yet, the real novelty would be the resulting “binocular view” of the one and same object (i.e., the Ahlfors extremal) through two different angles, yielding a sharper perception of it. Perhaps, this gives sharper differential-geometric insights about the Ahlfors map of a membrane, and may have some implications toward Gromov’s FAC(=filling area conjecture). Remember our naive conviction that this problem FAC should succumb just under the powerful methods of 2D-conformal geometry. Bergman kernel on Riemann surfaces ---------------------------------- \[13.06.12\] Consulting other sources (e.g. Weill 1962 [@Weill_1962]), it seems that the theory of the Bergman kernel can be developed over any Riemann surface. The idea is to use the Hilbert space structure on the space of analytic differentials. A complete exposition is e.g., Ahlfors-Sario 1960 [@Ahlfors-Sario_1960 p.302]. Whether or not this leads to another proof of Ahlfors circle maps is another question. \[15.06.12\] Other references for the Bergman kernel on Riemann surfaces include Nagura 1951 [@Nagura_1951], and Nehari 1950 [@Nehari_1950] where the Ahlfors function is expressed in terms of the Bergman kernel. \[25.06.12\] In fact the key observation is probably the [*conformal invariance*]{} of the integral involved in the minimum problem BIBER (of the previous section). Thus it may be hoped that this problem leads to an independent treatment of the Ahlfors mapping, treated from a Hilbert space \[of “areally” ([*aérolaire*]{}) square-integrable holomorphic functions\] viewpoint. This would give some culmination to the device of Bieberbach 1914 [@Bieberbach_1914]. So having in mind the possibility of extending the BIBER minimum area problem of the previous section to compact bordered Riemann surfaces (which looks reasonable in view of the conformal invariance of this area functional) we would like to reprove the existence of a circle map (à la Ahlfors 1950 [@Ahlfors_1950]). Relevant literature on this problem (but from our naive viewpoint not completely satisfactory) includes in chronological order: $\bullet$ Bieberbach 1914 [@Bieberbach_1914] (simply-connected schlicht case) $\bullet$ Bergman 1950 [@Bergman_1950 p.24], where the fact that the range of the minimizing function is a circle is considered as well-known (with reference to Bieberbach’s Lehrbuch (1945 edition) [@Bieberbach_1945-Lehrbuch]). Later in Bergman’s book 1950 [@Bergman_1950 p.87] the circle map $B\to \Delta$ is recovered through the function $F(z, \zeta)=\frac{\hat K(z, \bar \zeta)}{\hat L(z, \zeta)}$ defined on p.86, but it is not clear if this function solves the least area problem. (Perhaps the connection is easy to do.) $\bullet$ Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950 p.166–7] where the BIBER problem is again formulated, but somehow only in the purpose of showing existence of the reproducing kernel function, in the optic of re-deriving the PSM (parallel-slit maps). In particular one may wonder if it possible to show by a direct analysis if the minimum function is a circle map. Circle maps are reobtained later in the paper (p.182) however through a different procedure. $\bullet$ Nehari’s book 1952 [@Nehari_1952-BOOK] where the BIBER minimum problem appears on p.362 (for multiply-connected domain only) and its relation to the Bergman kernel is made explicit in the subsequent pages (esp. p.368-9). However I do not think that the issue about the circle mapping property of the minimum function of BIBER is handled. Later in the book (p.378) the Ahlfors extremal function is treated, yet a priori there is no clear-cut identity between the Bieberbach and Ahlfors extremal function. Nehari’s book borrows a lot of ideas from other writers without referring to them, thus it is an easy task to observe strong overlap with the previous literature (e.g. Bergman 1950 and Garabedian-Schiffer 1950). $\beta$ and $\alpha$ problems {#sec:beta-and-alpha-problems} ----------------------------- \[27.06.12\] As already discussed in Section 7.8, there are essentially two problems BIBER and FROG amounting respectively to minimize an integral and to maximize a derivative. We may rebaptize them respectively the $\beta$-problem (for Bieberbach-Bergman) and the $\alpha$-problem for Ahlfors (albeit this should truly be Fejér-Riesz 1922, for historical sharpness). For simplicity we restrict to domains, though the ultimate dream is to concoct didactic expositions pertaining to Riemann surfaces. Regarding the $\beta$-problem (of minimizing the areal integral) it has a direct Hilbert-space interpretation (recall the affiliation Dirichlet-Riemann-Hilbert-(Schmidt)-Ritz-Bieberbach-Bergman), as finding the vector of minimal length on the hyperplane defined by the prescription $f'(t)=1$, where $t$ is some fixed point (previously denoted $a$). Such minimization traduces into orthogonality to this hyperplane, yielding the so-called [*reproducing property*]{} while permitting to identify the $\beta$-extremal with the Bergman kernel (function). For a detailed execution, cf. e.g. Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950 p.166–7] (henceforth abridged GS50). Likewise the $\alpha$-problem received ultimately a similar treatment through Garabedian’s Thesis 1949 [@Garabedian_1949] (recast in the just cited Garabedian-Schiffer article), but the treatment is somewhat more involved appealing to the Szegö kernel instead, characterized via an orthogonalization taking place along the boundary of the domain (hence in substance the idea of length rather than area). It follows in particular an explicit formula for the derivative of the Ahlfors function $\vert f'(t) \vert=2\pi k(t,t)$ in term of the Szegö kernel. (Garabedian’s work is such a tour de force that it was represented in virtually all major texts of that period, e.g. Bergman 1950 [@Bergman_1950] and Nehari 1952 [@Nehari_1950], plus also the paper GS50.) Can we understand better the connection between both extremal problems? Our naive question is whether the $\beta$-map is a circle map. Remember that Bieberbach 1914 [@Bieberbach_1914] has an argument in the case where the domain is simply-connected (via his first Flächensatz saying that a map from the disc with normalized derivative expands the area of the disc unless it is the identity). Combining this with the Riemann mapping, Bieberbach argues that the $\beta$-map must be disc-ranged, for otherwise we could deflate the area by post-composing with the Riemann map, hence violating the minimum property. Alas, it seems that this argument is hard (impossible?) to extend to the multiply-connected case. Thus it is puzzling to wonder if the $\beta$-map is a circle map. If it is the case, then we could inject the $\beta$-solution into the Ahlfors problem and compare them. In view of the explicit formula of Garabedian we can even try a direct comparison of the respective derivatives at $t$ and hope to find an equality in which case by uniqueness we would have $\beta=\alpha$ (modulo scaling), i.e. a perfect coincidence between the Bieberbach and Ahlfors functions. Of course, ideally everything should be done geometrically from the extremal problem, without duelling with hard analysis. Recall that each problem has its allied reproducing kernel, which serves to express its solution. In particular we may hope to derive the circle mapping property of the $\beta$-function from the property of its allied (Bergman) kernel (cf. GS50, p.167). And if not, we may hope to connect the $\beta$ to the $\alpha$-map through a somewhat accidental identity between their kernels functions. As far as the writer knows this is not explicitly made, and perhaps wrong. Let us emphasize a naive duality between the $\alpha$- and $\beta$-problem. The first amounts to a maximal pressurization (inflation) within a limited container (the unit disc), whereas $\beta$ is a free vacuum deflation leading ineluctably toward a big-crunch to a point (constant map) if there were not the initial explosion sustained by the derivative normalization $f'(t)=1$. Hence it is not so surprising that the Ahlfors map is a circle-map but the same issue for the Bieberbach least-area map seems more like an isoperimetric miracle. We learned from Gaier’s 1978 survey [@Gaier_1978-JDMV p.34–35, §C] the following piece of information. Gaier’s article contains a proof of a striking fact due to Grötzsch 1931 (see also Gaier 1977 [@Gaier_1977-Roumaine], where the precise ref. is identified as Grötzsch 1931 [@Groetzsch_1931]) that a map (non-unique!) minimizing the area integral $\int\int \vert f'(z) \vert^2 d \omega$ (à la Bieberbach 1914 [@Bieberbach_1914]–Bergman\[n\] 1922 [@Bergman_1922], but extended to the multiply-connected setting) under the schlichtness proviso (and the normalization $f(a)=0, f'(a)=1$) maps the domain upon a [*circular slit disc*]{} (with concentric circular slits centered at the origin). According to Gaier, Grötzsch’s paper contains no details outside the indication of using his [*Flächenstreifenmethode*]{} (striptease method). Gaier’s proof combines Carleman’s isoperimetric property of rings (relating the modulus to the area enclosed by the inner contour) with Bieberbach 1914 [@Bieberbach_1914] (first area theorem) to the effect that a schlicht normalized map from the disc inflates area, unless it is the identity. A natural question \[13.07.12\] is what happens if we relax schlichtness of the map? Do we recover an Ahlfors circle map? As a historical curiosity, Gaier 1977 [@Gaier_1977-Roumaine] remarks that the above least-area problem for schlicht functions was reposed as a research problem as late as 1976 in the Durham meeting by Aharonov (compare for the exact ref. the Math. Review by Burbea of Gaier 1977 [@Gaier_1977-Roumaine]). It is apparently Kühnau (Grötzsch’s eminent student) who pointed Grötzsch’s priority in the reference just cited (Grötzsch 1931 [@Groetzsch_1931]). It should be remembered that several treatments existed in print (prior to Aharonov’s question), e.g. the one in Sario-Oikawa’s book of 1969 [@Sario-Oikawa_1969] (see pages as in MR of Gaier 1977 [@Gaier_1977-Roumaine]), which is inspired from Reich-Warschawski 1960 [@Reich-Warschawski_1960]. All these treatments are quite involved, and Gaier 1977 [@Gaier_1977-Roumaine] claims to simplify them. A paper related to Gaier’s and to this circle of ideas—i.e. Bieberbach’s area minimization, yet, alas not exactly furnishing our naive desideratum—is Alenicyn 1981/82 [@Alenicyn_1981/82]: this gives the exact reference to the relevant work of Carleman 1918 [@Carleman_1918] as well as to that of Vo Dang Thao 1976 [@Vo-Dang-Thao_1976] (the latter being however slightly criticized for mistakenly assuming the schlichtness of some function). Philosophically such Bieberbach-type area minimization problem amounts to a deflation as opposed to the inflation of Ahlfors-type problem maximizing the derivative. According to popular wisdom, both viewpoints could coincide since a semi-empty bottle is the same as a half-filled one. (This reminds the story of Ahlfors’ whiskey bottle used as a defense-weapon against an aggressor.) \[17.07.12\] We can also switch completely of extremal problem by looking at an Ahlfors (for short $\alpha$-type) extremal (inflationist) problem of maximizing the derivative among schlicht functions. Given $D$ a multiply-connected domain marked interiorly at the point $a\in D$, find among all schlicht functions $f\colon D \to {\Bbb C}$ bounded-by-one $\vert f\vert \le 1$ the one maximizing the modulus of the derivative $f'(a)$. It is reasonable to guess that “the” (unique?) extremal map will take $D$ upon the full circle with circular slits (schlichtness being only fulfilled on the interior). It seems that this behavior is the one described in Meschkowski 1953 [@Meschkowski_1953] (basing his analysis upon a distortion result of Rengel 1932 [@Rengel_1932-33]), and see also the treatment by Reich-Warschawski 1960 [@Reich-Warschawski_1960]. Added 27.07.12: Compare also Nehari 1953 [@Nehari_1953-Inequalities p.264–5], where another treatment of this problem is given, and credits is given to Grötzsch 1928 [@Groetzsch_1928] and Grunsky 1932 [@Grunsky_1932]. [**Optional digression:**]{} Asking schlichtness up to the boundary, we get maybe the Kreisnormierung of Koebe? This would be interesting since as pointed out in one of Meschkowski’s paper cited in the bibliography (locate where exactly!?, but anecdotic because cf. also Schiffer-Hawley 1962 [@Schiffer-Hawley_1962], Hejhal 1974 [@Hejhal_1974], etc.) there was in the 1950’s no clear-cut extremal problem leading to the Kreisnormierung (even in finite connectivity). Maybe the situation changed slightly after several works of Schiffer (and his collaborator Hawley) where some Fredholm eigenvalues came into the dance (compare several refs. cited below in the period 1959–1963). At this stage combining the analysis of Gaier 1978 [@Gaier_1978-JDMV] for the $\beta$-problem and that of Meschkowski/Reich-Warschawski for the $\alpha$-problem (refs. as in the penultimate paragraph) we see a perfect duality between the behavior of the extremal [*schlicht*]{} functions (at least qualitatively since both mappings carry the domain upon the same canonical region of a circular slit disc). Maybe one can even identify both functions (after harmless scaling). Those works raise some hope that the schlicht-relaxed $\beta$-problem (area minimization à la Bieberbach) produces again the Ahlfors map (or at least enjoys the same property of being a circle map). As far as we know \[20.07.12\], there is no such published account corroborating this intuition. This would be highly desirable to complete the symmetry of the picture below (Fig.\[alpha:fig\]) summarizing our discussion. \[22.07.12\] On reading Alenicyn 1981/81 [@Alenicyn_1981/81 p.202], 1981/82 [@Alenicyn_1981/82], where one is referred for the least-area problem back to Nehari’s book of 1952 [@Nehari_1952-BOOK], especially pp.340 (one can safely add p.341) and p.362. Nehari’s pages340–341 are perhaps not so relevant as it is merely a set of exercises. What is truly relevant is page 362, where the least area problem is posed and partially analyzed. In fact, this least area problem is handled earlier (with somewhat sharper information) in Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949 p.201] where the solution is represented as $M(z,a) \, M'(a,a)^{-1}=:M^{\ast}(z,a)$, where $M(z,a)=[A(z,a)-B(z,a)]/2$ is a combination of $A,B$ the two canonical parallel slit maps of the domain $B$ upon horizontal (resp. vertical) slit domains taking $a$ to $\infty$ as a simple pole with residue $+1$ (compare p.200). \[26.07.12\] In fact this solution is already announced in Grunsky’s Thesis 1932 [@Grunsky_1932 p.140]! As to the geometry of this map $M^{\ast}$, Garabedian-Schiffer (p.201) add the fact that it is at most $n$-valent ($n$ being the number of contours of the domain, equivalently, its connectivity). (This information is not to be found in Nehari 1952 [@Nehari_1952-BOOK].) Alas, Garabedian-Schiffer (1949 ) never seem to assert that the least-area map $M^{\ast}(z,a)$ is a circle map. On p.217, they show that any unitary function $E$ (=unit-circle map) may be expressed as a linear combination of the least-area maps $M(z, n_{\nu})$ centered at the $N$ zeros $n_{\nu}$ (‘Nullstellen’) of $E$ (assumed to be simple), compare Eq. (131) and (131’). Finally, on p.219 it is observed that the area of any such $E$, mapping the domain $D$ upon the unit-circle covered $N$ times, is exactly $N\cdot \pi$ (since area as to be counted with multiplicities). Of course, if our conjecture about the circle-mapping nature of least-area maps (there is one for each center $a$) is correct, then we could sharpen Garabedian-Schiffer’s assertion about the “at most $n$-valency” into an exact $n$-valency of those maps. \[27.07.12\] It could be the case, that our conjecture about the circle mapping nature of the least area map is settled in Lehto’s Thesis 1949 [@Lehto_1949] (see especially p.41). \[29.07.12\] However on consulting M. Maschler 1959 [@Maschler_1959] (esp. p.173) it seems to be asserted that the range of the least area maps are unknown for domains of connectivity higher than $2$. \[26.07.12\] To our grand surprise, we notice that the least-area problem is handled in full generality (i.e., for compact bordered Riemann surfaces) in Schiffer-Spencer 1954 [@Schiffer-Spencer_1954 p.135]. However again (as in Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949]) it is not shown that the resulting extremal function is a circle map. At this stage we see that there is a wide variety of extremal problems, and as a rough rule we may split the most common of them into the $\alpha$- and $\beta$-type (for Ahlfors and Bieberbach resp.) Each problem is hard to analyze precisely but there is a large body of wisdoms accumulated about them by the masters (Koebe, Carathéodory, Bieberbach, Grötzsch, Grunsky, Ahlfors, Schiffer, Garabedian, Golusin, etc.) Optionally by a nebulous bottle principle there may be a certain duality (even possibly an identity) between $\alpha$- and $\beta$-solutions. At least so is the case in the simplest simply-connected setting according to Bieberbach 1914 [@Bieberbach_1914], and apparently in the multi-connected setting we have at least coincidence of the range when considering the restricted schlicht problems. We may also speculate that a careful analysis of a suitable extremal problem may lead to a solution of the Gromov filling area conjecture. Finally we mention a related extremal problem treated in Schiffer 1938 [@Schiffer_1938], namely that of minimizing the maximum modulus in the family of schlicht functions $f\colon B \to {\Bbb C}$ normalized by $f(a)=0$ and $f'(a)=1$. The (or rather any) extremal is shown to map (conformally) the Bereich $B$ upon a circular slit disc. Least area problem vs. least momentum ------------------------------------- \[03.08.12\] The menagerie of extremal problems leading to the Riemann mapping can still be further enlarged. Each extremal problem exploits the ordered structure of the real line via some real-valued functional. One may incidentally get some feeling of regression about this massive usage of real numbers in complex geometry problems, but this is common and respectable practice since Dirichlet’s principle. Regarding the problem of circle maps [*per se*]{} it is not perfectly clear what is the [*ideally suited*]{} extremal problem (if any beside that of maximizing the derivative)? What is somehow missing is an extremal principle selecting the best extremal problem! The competitive nature of such extremal problems fascinated generations but requires strong classification aptitudes in view of the difficulty of each problem and the diversity of them. First the [*least-area problem*]{} consists in minimizing the area of the range of an analytic function counted by multiplicity. This is measured by the functional $A[f]=\int \int \vert f' \vert^2 d\omega$ (which seems much allied to the Dirichlet integral). (To extend the problem to Riemann surfaces one just needs to take notice of the conformal invariance of this integral upon conformal change of metrics.) To avoid the minimizers collapsing to the (uninteresting) constant functions, one imposes the side condition $f'(t)=1$ at some inner point $t$ of the domain $B$. The least-area map (which exists uniquely by Hilbert space theory) effects when $B$ is simply-connected nothing but than the Riemann mapping (Bieberbach 1914 [@Bieberbach_1914]). This viewpoint was widely pursued especially by Bergman, yielding in particular the concept of Minimalbereich. See for instance Bergman 1922 [@Bergman_1922], Bergman 1929 [@Bergman_1929-Hermite] where the concept seems to emerge, yet no precise definition. As noted in Maschler’s papers e.g. 1959 [@Maschler_1959] it seems that the nature of those minimal-domains was not completely elucidated in the late 1950’s. However, Maschler—extending a result of Schiffer 1938 [@Schiffer_1938-CRAS-domaines-minima]—observes that such minimal domains satisfy the mean property. Therefore on applying the result of Davis (as quoted in Aharonov-Shapiro 1976 [@Aharonov-Shapiro_1976]) characterizing the circle as the unique domain with a one-point quadrature identity (i.e. such that the mean value property holds for all harmonic functions) one may hope to infer our desideratum that the least area map has a range which is a disc. As we remarked it is not very clear what the range of this map is, and we may speculate about it being a (multiply-covered) circle. Another problem is that of the “least momentum” where one minimizes instead the integral $\int\int_B \vert f(z) \vert^2 d\omega$ (notice the suppression of the derivative) and again to avoid the trivial solution $f=0$ we impose $f'(t)=1$ at some point $t\in B$ of the domain. Another possible normalization is to ask $f(t)=1$, like in Fuchs 1945 [@Fuchs_1945/46]. Here again it seems reasonable to expect circularity of the range of the minimum mapping. The intuition being that the inertia-momentum of a rotating body gets minimized for a circular body (granting some atomical resistance avoiding a complete gravitational collapse of matter). \[07.08.12\] Of course all those problems are super-classical, yet we still find it hard to delineate the relevant clear-cut results among the super-massive literature. Our naive intuition would be that such least-area (or momentum) map are closely allied to circle maps. However it is not sure that this is the pure truth for non-simply-connected domains (and a fortiori for bordered surfaces). As we already said the relevant sources includes for the area problem: $\bullet$ Grunsky 1932 [@Grunsky_1932 p.140], alas no details, some more details in Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949] (but no assertion of circularity) only the Grunsky formula expressing the least-area map as combination of the two slit-maps. $\bullet$ for the least momentum see many works of Bergman starting from his Thesis 1922 [@Bergman_1922]. Perhaps it should be observed that the least-momentum problem is perhaps somewhat less easily extensible to Riemann surfaces in view of the lack of conformal invariance of its functional. Finally, we can mention Walsh’s 1935 survey (Mémorial [@Walsh_1935]) where all such problems are united under a generalized form where more points $z_1, \dots z_n$ are prescribed in the domain joint with some prescribed values $\gamma_1, \dots \gamma_n$ and one is required to find the map minimizing the functional under the interpolating condition $f(z_i)=\gamma_i$. Alas, in Walsh’s survey attention is confined to the simply-connected case and the multi-connected variants where at that time not systematically understood. A digression about Nehari’s paper of 1955 {#Nehari-digression:sec} ----------------------------------------- In Nehari 1955 [@Nehari_1955], the author presents a nice application of Bieberbach’s 1925 [@Bieberbach_1925] existence theorem of a circle map for an $n$-ply connected domain upon the disc of degree $n$. Precisely Nehari deduces a bound on the number of linearly independent solutions to a certain extremal problem (akin to those treated by Szegö 1921 [@Szego_1921]). It seems plausible that this Nehari argument is sufficiently universal to extend directly to the more general setting of compact bordered Riemann surfaces (membranes for short) upon invoking Ahlfors 1950 [@Ahlfors_1950] instead of Bieberbach 1925 [@Bieberbach_1925]. As the argument uses only the circle mapping nature of the Ahlfors map, we may even appeal to Gabard 2006 [@Gabard_2006] to obtain a sharper bound. In reality what is truly relevant is the absolute invariant of the (separating) gonality à la Coppens 2011 [@Coppens_2011]. Let us try to explore this connection, albeit some details require to be better worked out in order to really understand this technique of Nehari. We try first to go quickly to the hearth of Nehari’s ideas. The starting point is the following extremal problem formulated for $D$ a compact domain bounded by $n$ analytic curves (for simplicity) forming its complete boundary contour $C$. Further in the interior of $D$ a set $C_1$ consisting of a finite number of rectifiable Jordan arc and/or curves is given. \[Warning: in his paper [@Nehari_1955 p.29] Nehari writes “$C_1$ will stand for a subset of $C$”, which in our opinion is just a misprint! $C$ should be $D$!? Of course, our domain $D$ differs from Nehari’s as ours includes the contours.\] Let also $L^2=L^2(D)$ be the (Hilbert) space of analytic functions on $D$ with finite integral $\int_C \vert f(z) \vert^2 ds<\infty$ where $ds$ is the (Euclidean) length element. . Find the functions $f\in L^2$ minimizing the norm $\int_C \vert f(z) \vert^2 ds $ under the constraint $\int_{C_1} \vert f(z) \vert^2 ds=1$. This problem suggests looking at the functional $$J(f)=\frac{\int_C \vert f(z) \vert^2 ds}{\int_{C_1} \vert f(z) \vert^2 ds}$$ whose minimizers are (up to scaling) the solution of problem (P). Next Nehari sets up a certain integral equation whose eigenspace attached to the lowest eigenvalue parametrize the extremals of (P). We skip the details, but the key issue is just the linearity of the set of solutions to Problem (P). With this at hand, we can plunge directly to the core of Nehari’s argument, namely the: [(Nehari 1955 [@Nehari_1955 p.36])]{} Assuming (as above) the domain $D$ of connectivity $n$ (=number of contours), problem [(P)]{} admits at most $n$ linearly independent solutions. Nehari’s argument splits in 4 short steps: [**Step 1 (Bieberbach 1925)**]{} According to the latter ([@Bieberbach_1925]) there is a circle map $f\colon D \to \overline{\Delta}=\{\vert z\vert \le 1\} $ of degree $n$. This means that $\vert f(z) \vert=1$ exactly on the contours (i.e. $f^{-1}(\partial\overline{\Delta}=S^1)=C$) and upon changing the origin to an unramified place we may assume that $f$ has exactly $n$ zeroes, say $z_1,\dots,z_n$. [**Step 2 (Nehari’s trick in linear algebra)**]{} Assume by contradiction that (P) has $n+1$ linearly independent solutions $f_i$ ($i=1,\dots,n+1$). We consider the linear map $${\Bbb C}^{n+1} \to L^2 \to {\Bbb C}^n\,,$$ where the first arrow maps $(A_1,\dots, A_{n+1})\mapsto \sum_{i=1}^{n+1} A_i f_i$ and the second is the evaluation $ \varphi\mapsto (\varphi(z_1),\dots, \varphi(z_n))$ at the zeroes of the (Bieberbach) function $f$. For dimensionality reasons, there is a non-zero vector $(A_i)$ in the kernel which creates the function $f_0:=\sum_{i=1}^{n+1}A_i f_i$ vanishing at all $z_i$, yet without being identically $0$ (the $f_i$ being linearly independent). [**Step 3 (Nehari factorizes)**]{} The function $g$ defined by $g \cdot f=f_0$ is regular in $D$ (since writing $g=f_0/f$ we see that the zeroes of $f$ are cancelled out by those of $f_0$ which by construction englobe those of $f$). Now using the property of the circle map $f$ we find the following strict inequality $$J(f_0)=\frac{\int_C \vert f_0(z) \vert^2 ds}{\int_{C_1} \vert f_0(z) \vert^2 ds}=\frac{\int_C \vert g(z)\vert^2 \overbrace{ \vert f(z) \vert^2}^{=1} ds}{\int_{C_1} \vert g(z) \vert^2 \underbrace{\vert f(z) \vert^2}_{< 1} ds}>\frac{\int_C \vert g(z) \vert^2 ds}{\int_{C_1} \vert g(z) \vert^2 ds}=J(g)\,.$$ (Moreover reading backwards the numerators we see that the norm of $g$ equals that of $f_0$ so that $g\in L^2$.) The just obtained inequation $J(g)<J(f_0)$ shows that $f_0$ fails to solve (P). [**Step 4 (Using the linear structure)**]{} However the $f_i$ ($i=1,\dots,n+1$) solve (P), hence by virtue of the linear structure of the extremals to (P) \[which Nehari derives from an interpretation as the eigenspace attached to the lowest eigenvalue, but which perhaps may be derived more directly\] it follows that $f_0$ solves also (P) \[after scaling appropriately\], violating the conclusion of Step 3. Albeit our presentation is not completely polished (and Nehari’s maybe not perfectly organized for the beginner), we see that the basic trick looks sufficiently universal, as to extend to the following context. Instead of the finitely-connected domain $D$, we consider $F$ a compact bordered orientable Riemannian surface of genus $p$ and with $r$ contours. Now $ds$ denotes the induced length element attached to the (Riemannian) metric. As above, we specify a subset $C_1$ of the interior of $F$ consisting of a finite “drawing” of Jordan arcs and curves (perhaps they do not even need to be pairwise disjoint). Then we set up the extremal problem (P) in this context, and they above proof seems to work mutatis mutandis, except for trading Bieberbach 1925 [@Bieberbach_1925] by Ahlfors 1950 [@Ahlfors_1950] or Gabard 2006 [@Gabard_2006]. Precisely, we may consider a circle map $f\colon F \to \overline{\Delta}$ of least possible degree, say $\gamma$. By Gabard 2006 [@Gabard_2006] we know that $\gamma\le r+p$. So we arrive at the following statement: Let $F$ be a membrane of genus $p$ with $r$ contours. Assume that $F$ has the gonality $\gamma$, i.e. the least degree of a circle map to the disc. (We know $\gamma\le r+p$) Then the extremal problem [(P)]{} admits at most $\gamma$ linearly independent solutions. Ahlfors’ extremal problem ========================= Ahlfors extremal problem (Grunsky 1940–42, Ahlfors 1947–50) ----------------------------------------------------------- Ahlfors’ method involves solving the following extremal problem: \[Ahlfors-extremal:thm\] [(Ahlfors 1950 [@Ahlfors_1950])]{} Given any compact bordered Riemann surface (membrane for short) and two interior points $a,b$, find among all (analytic) functions bounded-by-one taking $a$ to $0$ the one maximizing the modulus $\vert f(b) \vert$. Such a function exists (normal families argument à la Vitali-Montel) and is unique up to a rotation (=multilication by an unimodular complex number $\omega=e^{i \theta}$). Hence it is unambiguously defined by the points $a,b$ if $f(b)$ is required to be positive real, and we denote $f_{a,b}$ the corresponding function. Furthermore Ahlfors’ extremal function $f_{a,b}$ concretizes the given surface as a full-covering of the disc $\Delta$, of degree $$r\le \deg f_{a,b} \le r+2p, \label{Ahlfors:pinch}$$ where $r$ is the number of contours and $p$ the genus (of the given membrane). It is nowadays quite customary—following (another) Russian school (Golusin, S.Ya. Havinson, etc.)—to call the extremal an [*Ahlfors function*]{}, albeit even Ahlfors seems to have been rather embarrassed by this probably unearned distinction (cf. his comments in Collected Papers [@Ahlfors_1982_Coll_papers p.438]). The same idea occurred somewhat earlier in works of Grunsky 1940–42 [@Grunsky_1940], [@Grunsky_1942], yet the latter confined attention to plane domains (as did Ahlfors 1947 [@Ahlfors_1947]). Being close colleagues—as materialized by their joint note (Ahlfors-Grunsky 1937 [@Ahlfors-Grunsky_1937]) about the best conjectural value for the [*Bloch constant*]{} (still open up to present days)—it is puzzling that both were not very aware of overlapping studies (admittedly imputable to the difficult World War II context). Semi-fictional reconstruction of Ahlfors’ background (Fejér-Riesz 1922, Carathéodory 1928, Ostrowski 1929) ---------------------------------------------------------------------------------------------------------- Where does Ahlfors’ extremal problem come from? This is surely a non-trivial question yet let us attempt to give some elements of answers. The narrative is made more plausible by looking a bit around while trying to keep track of the historical continuity. We shall thus use several indirect sources, especially Remmert. As notorious, the Dirichlet principle suffered ill-foundations during a long period of about 40 years (1860-1900). This was beneficial to Schwarz-Christoffel who developed some constructive methods for the RMT for polygons. Another trend involves directly rescuing the Dirichlet principle via the “alternierendes Verfahren” of Schwarz and the parallel work of C. Neumann. This influenced Picard’s [*méthodes des approximations successives*]{}, as well as Poincaré’s balayage. Then came Hilbert’s breakthrough. Yet, alternative methods circumventing the intricacies of potential theory seemed worth attention. As reported in Remmert 1991 [@Remmert_1991], one can ascribe to Fejér-Riesz ca. 1921 (published by Radó 1923 [@Rado_1923-Uber-konf-Abb]) the first purely complex variables (potential-theoretic free) proof of the RMT by using the extremal problem of making the modulus of the derivative as large as it can be. Several technical simplifications were then obtained by Carathéodory 1928 [@Caratheodory_1928] and Ostrowski 1929 [@Ostrowski_1929] (independently). This leads in principle to the most elementary proof of the RMT. Extending this idea to multiply-connected domains (say first of finite connectivity) leads directly to the extremal problem considered by Grunsky 1940–42 [@Grunsky_1940], [@Grunsky_1942], and Ahlfors 1947 [@Ahlfors_1947], and Ahlfors 1950 [@Ahlfors_1950] when extended to Riemann surfaces. In fact prior to Fejér-Riesz, it is fair to refer to Koebe’s (and Carathéodory’s) elementary proofs of the RMT, also via an extremal problem or at least iterative methods (compare e.g. Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950]). As a matter of digression, it can be recalled that this extremal viewpoint leads as well to a proof of the uniformization theorem (without potential theory). Compare Carathéodory 1950 [@Caratheodory_1950], plus several papers by Grunsky (easily located in his collected papers). Extremal problems and pure function-theoretic proofs of the RMT (Koebe, Carathéodory, Bieberbach) ------------------------------------------------------------------------------------------------- The previous section is a bit caricatural and the real history is marvellously detailed in Gray 1994 [@Gray_1994]. Let us summarize the chronology of this period, in the center of which there is probably one of the main inspiring force toward the Ahlfors extremal function (namely the [*Schwarz lemma*]{} as Carathéodory christened it in 1912). $\bullet$ Painlevé 1891 [@Painleve_1891]: boundary behavior of the Riemann mapping for a contour having an everywhere continuously varying tangent. $\bullet$ Harnack 1887 [@Harnack_1887] provides a satisfactory proof for solving a suitable version of Dirichlet’s principle, and states what has become known as [*Harnack’s theorem*]{} on monotone limits of harmonic functions. $\bullet$ Osgood 1900 [@Osgood_1900] applies Harnack’s theorem to draw the existence of a Green’s function for any simply-connected plane domain thereby resolving the Riemann mapping theorem (RMT). This dependance is eliminated in Koebe 1908 and Carathéodory 1912 (cf.items below), where Schwarz’s lemma is substituted. $\bullet$ Poincaré 1907 [@Poincare_1907] (and independently Koebe 1907, cf. below) proves uniformization (rigourously). For this Poincaré combines his [*méthode de balayage*]{} (of 1890 [@Poincare_1890]) and simplifies it using Harnack’s theorem. From the Green’s function he deduces the conformal map of a Riemann surface (à la Weierstrass) to the disc, and uses earlier works of Osgood. $\bullet$ Koebe 1907 also proves uniformization (UNI). In Koebe 1907c [@Koebe_1907_UbaK2] he compares his method to Poincaré’s. Like Poincaré he had relied on Schwarz’s method, but unlike him made a much more modest use of Harnack’s theorem. Koebe also insists upon his avoiding of the use of modular functions. $\bullet$ Koebe 1908 [@Koebe_1908_UbaK3] supplies another proof (of UNI) avoiding completely Harnack’s theorem. \[Subsequently Koebe interacted widely with Fricke’s attempt to modernize the original [*continuity method*]{} of Klein-Poincaré, and showed how this could be rigorized overlapping thereby with simultaneous work by Brouwer. This interaction with Brouwer seems to have ended quite contentiously.\] $\bullet$ Koebe 1909 [@Koebe_1909_UAK1], 1910 [@Koebe_1910_UAK2] proof of his [*Verzerrungssatz*]{} (distortion theorem). From it he derives, the first elementary proof of the (RMT) appealing to a long list series of name going back via Arzelà and Montel 1907 [@Montel_1907] to Ascoli 1883. $\bullet$ Carathéodory 1912 [@Caratheodory_1912 p.109] notes that [*Schwarz’s lemma*]{} (which he was the first to call by this name, and which he locates in Schwarz’s Ges. Abh., vol. 2, p.109) can act as a substitute to Harnack’s theorem (upon which Osgood 1900 relied heavily). \[Interrupting the present narrative this will have to play a major rôle in Ahlfors’ extremal problem.\] Using the Schwarz’s lemma and Montel’s theorem, Carathéodory obtains the Riemann mapping using an exhaustion of the domain $G$ by subdomains $(G_n)$ each mapped via $f_n$ to the disc and studied under which condition on $G_n$ the $f_n$ converges to a function $f$ giving the Riemann mapping (again without potential theory). $\bullet$ Carathéodory 1913a [@Caratheodory_1913a] proves Osgood’s conjecture, that the Riemann map extends to a homeomorphism of the boundary iff the boundary is a Jordan curve. In Carathéodory’s opinion this achievement is mostly a byproduct of Lebesgue’s far-reaching theory of integration (1902 [@Lebesgue_1902]), and the consequences drawn from it by Fatou 1906 [@Fatou_1906]. This reliance upon Lebesgue-Fatou was soon disputed by Koebe 1913 (cf. item below). $\bullet$ Carathéodory 1913b [@Caratheodory_1913b] discusses the boundary behavior when the boundary curve is not a Jordan curve. This paper is oft regarded as inaugurating the concept of [*prime ends*]{} (although earlier origins are in the work of Osgood, and related ideas in Study-Blaschke 1912 [@Study-Blaschke_1912]). $\bullet$ Koebe 1913 [@Koebe_1913] disputes the need for Lebesgue’s theory in Carathéodory’s treatment, showing how to generalize a theorem of Schwarz to the same effect. A similar result is claimed independently by Osgood-Taylor 1913 [@Osgood-Taylor_1913]. $\bullet$ Bieberbach 1913 [@Bieberbach_1913] wrote a short paper disputing the (in his opinion) excessive Carathéodory’s reliance on Schwarz’s lemma, proposing to use only Montel’s theorem. The next year Bieberbach 1914 [@Bieberbach_1914] invokes another extremum principle (area minimization of the range of the mapping suitably normalized) to simplify Carathéodory’s work. This freed the theory from any reliance upon Montel’s theorem (but uses instead ideas of Ritz). $\bullet$ Back to Koebe, in 1912 [@Koebe_1912] could not resist after the stimulus aroused by Carathéodory’s work to go back to some old idea of his own ([*Quadratwurzeloperationen*]{}) to create his [*Schmiegungsverfahren*]{} (squeezing methods) for solving the Riemann mapping by the iterated taking of square roots. This presentation was entirely elementary. $\bullet$ Carathéodory 1914 [@Caratheodory_1914] incorporated all these criticisms in his paper for the Schwarz Festschrift, which was to remain his final account until the newer methods of Perron were introduced. \[Here we may have also mentioned the argument of Fejér-Riesz 1921.\] $\bullet$ Bieberbach 1915 in his pocket book Göschen [@Bieberbach_1915 p.95] also proposes to deal entirely within pure function theory, while rejecting the potential-theoretic approach (despite Hilbert’s work). This actually presents a version of Koebe’s Schmiegungsverfahren and concludes to the Riemann mapping theorem via Koebe’s Verzerrungssatz (seen as a preferred alternative over Schwarz’s lemma). Interlude: Das Werk Paul Koebes ------------------------------- In this section we digress slightly from our main path to look closer at the monumental works of Koebe. A useful guide is Bieberbach’s overview of Koebe’s work in 1968 [@Bieberbach_1968-Das-Werk-Paul-Koebes]. The main point of overlap of Koebe with our main theme (Ahlfors) lies in the Riemann-Schottky mapping (albeit for Koebe the mapping to a Kreisbereich is given full attention neglecting thereby the circle mapping). Of course, the other main aspect of Koebe’s life is the uniformization theorem of (Klein-Poincaré-Schwarz). Again some chronology: $\bullet$ Riemann 1857–58 [@Riemann_1857_Nachlass] and Schottky 1877 [@Schottky_1877] (maybe only in the 1875 Latin version?) proved that any $n$-ply connected domain maps conformally to a Kreisbereich (circular domain). \[Bieberbach and indeed Koebe 1910 [@Koebe_1910_JDMV] ascribe this to Riemann, albeit we are not sure to be in total agreement with this assertion.\] $\bullet$ In Bieberbach’s opinion the above Riemann-Schottky Kreisbereich-mapping is first rigourously proved by Koebe in a series of four papers written in 1906, 1907, 1910, 1920 (which we attempt to summarize in more details): \(1) Koebe 1906 [@Koebe_1906_JDMV]: this starts with a rigidity result for two Kreisbereiche as being conformal to each other only through linear transformations. The proof uses potential theory (and the Cauchy integral). In the case of a Kreisbereich with two contours there is a pencil of orthogonal circles whose cross-ratio of any member affords the conformal invariant of the ring. It follows that $(\varrho+1)$-ply-connected Kreisbereiche depend upon $3\varrho -3$ essential constants when $\varrho \ge 2$, the same quantity as predicted by Schottky for general multiply-connected domains of the same connectivity. This yields some evidence for the possibility of mapping those to a Kreisbereich. Actually Koebe (p.150) reminds that the Kreisbereich mapping is (essentially) solved by Schottky and by Poincaré (referring loosely to the first volumes of Acta). \[In the next paper Koebe adopts a more critical position, and does not take this as granted.\] Next, he claims the result extends to schlichtartig surfaces. His argument amounts to fill the Riemann surface by discs, to get a closed surface of genus 0, and appeal to Schwarz 1870 [@Schwarz_1870] to map this to a sphere. Next, Koebe proposes to relax the schlichtartig character to formulate a similar result for positive genus. Again one fills the surface by discs to gain a closed surface of genus $p$. This can be mapped as a ramified cover of the sphere of degree $p+1$ (as well-known since Riemann, but for Koebe being Schwarz’s pupil Riemann is taboo and an ad hoc \[somewhat sketchy\] argument is supplied). At any rate the result is that any compact bordered Riemann surface of genus $p$ is conformally embeddable in a closed Riemann surface of genus $p$, hence representable as a $(p+1)$-sheeted cover of the sphere. Although this result concerns like Ahlfors 1950 [@Ahlfors_1950] compact bordered surfaces, it seems that this Koebe mapping lies not so deep as the image of the contours of the map are poorly controlled, in particular they need not coincide. \(2) Koebe 1907 [@Koebe_1907_UrAK]: this starts by quoting again his rigidity result of the previous paper. Then more critically Koebe notices that the mapping of a planar $(\varrho+1)$-ply connected domain upon a Kreisbereich of the same connectivity is not so easily established, making abstraction of the Klein-Poincaré [*Kontinuitätsmethode*]{} ([*méthode de continuité*]{}) not yet effective in 1907. (This had to wait until the work of Brouwer and Koebe ca. 1911 [@Klein-Brouwer-Koebe_1912], and Koebe 1912 [@Koebe_1912_BdKm].) The rigidity result affords an essentially unique solution of the mapping problem. Then Koebe proceeds to show that a Kreisbereich mapping exists for triply-connected domains ($\varrho=2$), and generally if the domain is symmetric under complex conjugation provided the real axis cuts all contours. For triply connected domains, he takes the Schottky double, which conformally maps to a closed Riemann surface of genus $2$ (via massive quotations to Schwarz, Ges. Abh. II, S.133–143, S.144–171, S.175–210). As any curve of genus 2 this is hyperelliptic (canonical mapping via holomorphic 1-forms). As to the more general case, the problem involves cutting the domain along the real axis, yielding a simply-connected region. This is mapped conformally to the upper half-plane, and symmetrically reproduced. Then Green’s function is constructed via Harnack’s theorem (quotation to Harnack 1887 [@Harnack_1887], Poincaré 1883 [@Poincare_1883], Osgood 1900 [@Osgood_1900] and Johansson 1905). \(3) Koebe 1910 [@Koebe_1910_JDMV]: the paper starts again with the objective to solve the [*Problem der konformen Abbildung eines $(p+1)$-fach zusammenhängenden Bereiches auf einen von $p+1$ Vollkreisen begrenzten Bereich*]{} (which he proposes to call [*Kreisbereich*]{} for short). Koebe recalls that the problem was first addressed by Schottky 1877 in his [*Doktordissertation*]{}, and earlier in Riemann’s Nachlass. He reminds from his first work \[item (1)\] that [*je zwei Kreisbereiche aufeinander nur durch lineare Funktionen konform abgebildet werden können*]{}. Then he repeats the two special cases he was able to solve previously, and now proposes to tackle the general case via two different methods (of his own): [*Überlagerungsfläche*]{} and [*iterierendes Verfahren*]{} \[cf. items (A) and (B) below\]. He proudly emphasizes that both methods have a larger applicability than to the present Kreisbereich problem, since their combination, allowed him to settle the whole series of classical mapping problems of Klein and Poincaré (1881–84) in their pioneering works on automorphic functions, and the allied uniformization. Hilbert’s 22th Problem (1900) is mentioned for reposing the uniformization question especially in connection to Poincaré 1883’s paper [@Poincare_1883]. Schwarz is again (justly) regarded as the father of the method [*der Überlagerungsfläche*]{}, which plays a key rôle in the newer developments in the automorphic theory, as exemplified through the work of Poincaré 1907 [@Poincare_1907] himself and Hilbert 1909 [@Hilbert_1909]. After these general remarks Koebe proceeds to prove the general Kreisbereich mapping. \[As warned in Bieberbach’s report, the present paper of Koebe does not contain full details, yet some lovely geometric ideas worth sketching. Complete details appear in the last contribution item (4), but then it is easy to get lost in technicalities.\] \(A) Koebe assumes the contours of the domain $B$ to be analytic curves. Via some abstract [*Spiegelungsprozesses*]{} (ascribed to Schwarz) he constructs via symmetric reproduction of $B$ a schlichtartig Riemann surface $B^{(\infty)}$. (One must imagine $B$ glued with replicas thought of as the back-side of the domain.) Then he can apply his [*allgemeines Abbildungsprinzip*]{} to the effect that schlichartig implies schlicht (first established in Koebe 1908 [@Koebe_1908_UbaK3], with subsequent approaches by Hilbert 1909 [@Hilbert_1909] and in Courant’s Thesis 1910/12 [@Courant_1912]). The new schlicht domain $B^{(\infty)'}$ is tesselated by replicas of the conformal copy $B'$ of $B$. Hence $B'$ admits a complete infinite system of symmetric reproduction. This is enough (for Koebe) to characterize a Kreisbereich. (Here we may agree with Bieberbach’s diagnostic that Koebe’s exposition is sketchy, but details were supplied later in Koebe 1920 [@Koebe_1920].) \(B) Then is exposed the promised [*iterierendes Verfahren*]{}. This is a beautiful device based upon successive applications of the RMT to circularize a specific contour and then reflecting by a [*Spiegelgung*]{} (inversion by reciprocal radii) the domain across this circularized contour. Koebe draws nice pictures (like below Fig.\[KoebeiV:fig\]) suggesting that this iteration scheme produces domains with [*sukzessive Steigerung der Spiegelungsfähigkeit des Bereichs*]{} whereupon it is made plausible that when repeated ad infinitum the resulting domain has an infinite aptitude of symmetric reproduction, hence must be a Kreisbereich. The convergence proof uses his [*Verzerrungssatz*]{} (distortion theorem). \(4) Koebe 1920 [@Koebe_1920], where full details are supplied. $\bullet$ In parallel, Koebe concentrates his efforts on the uniformization problem starting with Koebe 1907 [@Koebe_1907_UrAK] devoted to the uniformization of real algebraic curves, yet the real technological breakthrough occurs in the next paper. -35pt 0 $\bullet$ Koebe 1907 [@Koebe_1907_UbaK1] discovers a first version of his [*Verzerrungssatz*]{} (VZS), which turns out to be relevant both to the Riemann-Schottky Kreisbereich-mapping, as to uniformization. As forerunners of the (VZS) Bieberbach mentions the works of Landau, Schottky related to Picard’s theorem (1879 [@Picard_1879]). This Koebe’s paper also contains (what later came to be known) as the [*Viertelsatz*]{} to the effect that the range of any schlicht function on the unit disc normalized by $f(0)=0$ and $\vert f'(0) \vert=1$ contains a disc of some universal positive radius $\varrho$. The sharp value $\varrho=1/4$ is conjectured, but only established by Bieberbach 1915 [@Bieberbach_1915]. Armed with this Verzerrungssatz (yet without the precise bound) Koebe manages to prove uniformization. This represents a generalization of the RMT to simply-connected Riemann surfaces. Bieberbach recalls that according to oral tradition the trick of the universal covering surface is due to H.A. Schwarz (ca. 11. April 1882, as carefully reported in Klein’s Werke [@Klein-Werke-III_1923 p.584]). $\bullet$ Simultaneously and independently Poincaré 1907 [@Poincare_1907] also proves the uniformization theorem via his [*méthode de balayage*]{}. $\bullet$ Koebe 1907 [@Koebe_1907_UbaK2] inspects Poincaré’s proof and proposes a variant using Harnack’s theorem (in potential theory) circumventing thereby the Viertelsatz, as well as Poincaré’s balayage. $\bullet$ The new ingredient (Verzerrungssatz of Koebe) turned out to act usefully in other uniformization problems envisioned by Klein (e.g., the [*Rückkehrschnitttheorem*]{}, etc.) In Koebe’s formulation this resulted to the conformal mapping of a schlichtartig Riemann surface to a schlicht domain of the Riemann sphere. This result appears in Koebe 1908 [@Koebe_1908_UbaK3]. Its proof uses beside the Verzerrungssatz a general convergence theorem (à la Montel-Vitali), which Koebe discovered independently \[according to Bieberbach\]. $\bullet$ Koebe 1909 [@Koebe_1909_UAK1] gives a sharper version of the [*Verzerrungssatz*]{} and applications to Klein’s general uniformization problem (via groups of linear transformations). $\bullet$ Hilbert 1909 [@Hilbert_1909], using a variant of the Dirichlet principle, gives another method for the schlicht mapping of a schlichtartig surface (to the sphere), via a so-called parallel-slit mapping \[extending the Schottky-Cecioni result to infinite connectivity\]. $\bullet$ In response Koebe 1909 [@Koebe_1909_UbaK4], 1910 [@Koebe_1910_Hilbert] and independently Courant 1910/12 [@Courant_1912] proves anew the above Hilbert’s Ansatz about parallel-slit mappings. $\bullet$ Already Schottky 1877 [@Schottky_1877] tried \[in Bieberbach’s opinion\] to prove the \[Riemannian\] theorem that every $n$-ply connected planar domain conformal-maps bijectively to a parallel Schlitzbereich. Hilbert’s new method proves this for arbitrary schlichartig Riemann surfaces. Koebe in the aforementioned two works, sharpens Hilbert’s theorem by noticing that the range of the mapping fill the full plane save a set of measure zero. At this occasion Koebe also formulates his [*Kreisnormierungsprinzip*]{}, still open today, despite the spectacular progress by He-Schramm 1993 [@He-Schramm_1993]. $\bullet$ Bieberbach emphasizes that the [*iterierendes Verfahren*]{} may really have first emerged through the Kreisbereich mapping problem. \[This conflicts slightly with Koebe’s claim that he employed it earlier for uniformization.\] At any rate Bieberbach writes “[*Solche iterierenden Verfahren entwickelt Koebe über Jahrzehnte hin immer weiter, bis alle Uniformisierungsprobleme algebraischer Gebilde dem iterierenden Verfahren zugänglich werden.*]{}” $\bullet$ The proof of the (RMT) via repeated [*Quadratwurzelabbildungen*]{} itself constitutes an iterative method, which Koebe calls the [*Schmiegungsverfahren*]{}. Credit for this discovery is to be shared with Carathéodory. $\bullet$ A rigorous foundation to the [*Kontinuitätsmethode*]{} of Klein-Poincaré is paid by Koebe much attention in a torrential series of paper starting with 1912 [@Koebe_1912_BdKm], 1912 [@Koebe_1912_BdKm2], 1914 [@Koebe_1914_UAK4], etc. Those works overlaps (and then may supplement) the works of Brouwer on the invariance of domain (and dimension), and its application to Riemann surfaces. The resulting priority question is very intricate. Even Klein in 1923 [@Klein-Werke-III_1923 p.734] writes: [*Die entscheidende Wendung trat aber erst 1911/12 durch das Einsetzen der Untersuchungen von Brouwer und Koebe ein. (Ich halte um so mehr an der alphabetischen Reihenfolge fest, als die gegenseitige Beziehung der beiden Forscher nicht ganz geklärt ist.)*]{} Soon afterwards Klein also cites footnote 2) in Brouwer 1919 [@Brouwer_1919], where Brouwer seems to revendicate some priority over Koebe, while reporting some falsification of his own (Gött. Nachr.) article via a citation to Koebe added after proof-reading. Koebe and his relation to Klein or Ahlfors ------------------------------------------ In the overall Koebe’s monumental work is quite intricate with deep influences by methods of Schwarz (ca. 1870), results of Schottky (1875/77), visions of Klein and Poincaré (early 80’s), supplemented by methods of his own. The following chart (Fig.\[KoebeMap:fig\]) gives an Überblick maybe helping navigation through Koebe’s works and the logical links between his results. -85pt 0 From our Ahlfors’ biased viewpoint several points are worth noticing: \(1) Koebe frequently refers to Klein’s orthosymmetry for real algebraic curves. In view of the close connection between orthosymmetry and the Ahlfors circle mapping, it is tempting to wonder if Koebe was ever close to discover the Ahlfors circle mapping. Of course Koebe’s focus seems to have been more attracted by the uniformization problem (in particular for real algebraic curves), cf. Koebe 1907 [@Koebe_1907_UrAK]. However Klein’s orthosymmetry appears in many subsequent papers (e.g., 1919 [@Koebe_1919:47 p.29, p.35]), and we would not bet that one day someone discovers in Koebe some anticipation of the Ahlfors map (as it occurred say with the circles packing of Andreev–Thurston). If not directly, it could via the Rückkehrschnitttheorem of Klein (cf. Sec.\[sec:Ruckkehrschnittthm\]), which Koebe was the first to prove seriously (cf. Koebe 1910 UAK2 [@Koebe_1910_UAK2]). Hence schematically, there might exist a (harsh style) path like: $$\textrm{Koebe}\Rightarrow\textrm{Klein} \Rightarrow \textrm{Teichm\"uller} \Rightarrow \textrm{Ahlfors}.$$ \(2) Koebe also notices at several places (e.g., 1907 UbaK1 [@Koebe_1907_UbaK1 p.199]) that the orthosymmetry concept for real algebraic curves extend to analytic real curves. One can then wonder if there is likewise a function theoretical characterization of orthosymmetry in terms of (totally real) mapping to the sphere. This would amount to say that any bordered surface is expressible as a total cover of the disc (taking boundary to boundary). Of course this might be a bit fantasist, but perhaps deserves to be analyzed more carefully. (Maybe this fails already for planar domains, cf. a work of Heins ca. 1954.) Ahlfors’ background (Bergman 1941, Schiffer 1946, Schottky differentials) ------------------------------------------------------------------------- Let us quote the introduction of Ahlfors 1950 [@Ahlfors_1950]: \[Ahlfors-1950:quote\] In the handling of the extremal problems we are in close contact with the methods of Bergman \[1941\]=[@Bergman_1941] and Schiffer \[1946\]=[@Schiffer_1946], which they have developed for plane regions. A convenient tool for applying these methods to regions on Riemann surfaces is found in the class of Schottky differentials, and it was the recognition that Bergman’s kernel-functions are in fact Schottky differentials that led us to undertake this study. The second part of the paper (§§4–5) deals with an extremal problem that we have previously solved for plane regions. There are great simplification over my original proof for which I am partly indebted to my student P. Garabedian. An interesting point is that the extremal functions are again defined by means of Schottky differentials. As a complement, we may reproduce a passage of Ahlfors’ comments in his collected papers [@Ahlfors_1982_Coll_papers p.438]: \[Ahlfors-1982:quote\] The purpose of \[36\](=Ahlfors 1950 [@Ahlfors_1950]) was to study open Riemann surfaces by solving extremal problems on compact subregions and passing to the limit as the subregions expand. The paper emphasizes the use of harmonic and analytic differentials in the language of differential forms. It is closely related to \[35\](=Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950]), but differs in two respects: (1) It deals with Riemann surfaces rather than plane regions and (2) the differentials play a greater rôle than the functions. I regard \[36\] as one of my major papers. It was partly inspired by R. Nevanlinna, who together with P.J. Myrberg had initiated the classification theory of open Riemann surfaces, and partly by M. Schiffer (1943) and S. Bergman (1950), with whose work I had become acquainted shortly after the war[^9]. The paper also paved the way for my book on Riemann surfaces with L. Sario \[1960\], but it is probably more readable because of its more restricted contents. I would also like to acknowledge that when writing this paper I made important use of an observation of P. Garabedian to the effect that the relevant extremal problems occur in pairs connected by a sort of duality. This is of course a classical phenomenon, but in the present connections it was sometimes not obvious how to formulate the dual problem. The allied infinitesimal form of the extremal problem ----------------------------------------------------- The input required to pose Ahlfors’ extremal problem (Theorem \[Ahlfors-extremal:thm\]), is a membrane with two interior marked points, denoted $a,b$. When the point $b$ converges to the point $a$ (becoming infinitely close to it), we may think of a unique point of multiplicity two. This limiting process mutates the extremum problem into: Let $a$ be a single point in the membrane $W$. Among all functions $f$ analytic on $W$ with $\vert f \vert \le 1$ on $W$ it is required to find the one which makes the modulus of the derivative $f'(a)$ to a maximum. Here the derivative is computed w.r.t. any holomorphic chart. Its maximum value has no intrinsic meaning, yet the extremal function exists and is uniquely defined (up to a rotation) and denoted by $f_{a,a}=f_a$. It seems to be folklore that such functions are also circle maps subjected to the same Ahlfors bound $\deg f_{a}\le r+2p$. Presumably a continuity argument reduces to the case of (bipolar) functions $f_{a,b}$, or maybe adapt the whole argument in Ahlfors 1950 [@Ahlfors_1950]. At any rate, the result is taken for granted in Yamada 1978 [@Yamada_1978], Gouma 1998 [@Gouma_1998]. This can maybe deduced as a special case of Jenkins-Suita 1979 [@Jenkins-Suita_1979]. Higher extremal problems=HEP$\approx$High energy physics, alias Pick-Nevanlinna interpolation --------------------------------------------------------------------------------------------- What happens if we take more than two points? For instance three points $a,b,c$? Should we then maximize the area of the simplex spanned by the image points? If yes for which metric on the disc (Euclid vs. hyperbolic)? How does the problem reformulate when the 3 points coalesce at the subatomic level into a point affected by a multiplicity 3. Does the problem amount then to maximize the modulus of the first two derivatives? Maybe this brings us in the realm of Pick-Nevanlinna interpolation, a theory initially developed in the disc. Compare e.g. Garabedian 1949 [@Garabedian_1949], Heins 1950 [@Heins_1950], Jenkins-Suita 1979 [@Jenkins-Suita_1979]. Perhaps for any (effective) divisor $D=d_1 p_1+\dots +d_n p_n$ interior to the membrane there is an extremal problem denoted $EP(D)$. Then how much of Ahlfors’ theory extends: existence, uniqueness and qualitative circle mapping nature of the function, and estimates over the degree of the extremals. In the classic theory where $\deg(D)=2$ we have $\deg f_{a,b}=r+2p$. Maybe in general denoting by $f_D$ the extremal function allied to the divisor $D$ we find $\deg f_{D}\le r+ \deg(D) p$. Compare Jenkins-Suita 1979 [@Jenkins-Suita_1979] for more serious answers. If we could find a divisor of degree one then this would recover Gabard’s bound $r+p$. Maybe not a divisor is required but an ordered collections of points, as in Ahlfors’ original problem where $a$ seems to have a preferred rôle over $b$, getting mapped to zero. Such higher extremal problems depending upon a higher number of free parameters are probably more flexible in the sense that if the original $\deg(D)=2$ case of Ahlfors fails to realize the gonality, then maybe higher versions succeed. Perhaps there is even a universal quantum limit of such problem $EP\infty$ for a divisor of infinite degree, leading thereby to a branched (yet Randschlicht) version of the Bieberbach coefficient problem. This is to mean a version of the Ahlfors map where all derivatives are simultaneously maximized as a large convey? One can speculate about the existence of such an universal extremal problem whose solution would be a branched avatar (non schlicht) of the Koebe extremal function (involved in the Bieberbach-de Branges theorem). This would be for the given bordered surface the best circle mapping and arguably it ought to realize the gonality. \[05.11.12\] In the classic Bieberbach problem involving the disc the coefficients of schlicht power series are estimated by $\vert a_n\vert\le n$ when $f'(0)=1$. If we replace the disc by a finite bordered surface $F$ we could expect that all maps $F\to \Delta\to {\Bbb C}$ factorizing as a circle map (of minimal degree) followed by a schlicht map also admit universal estimates upon the coefficients w.r.t. to a chart. Perhaps the upper bounds sequence involved in Bieberbach-de Branges (regularly spaced integers $n$) has to be replaced by certain spectral eigenvalues of $F$ conceived as a vibrating membranes. So the problem is the following. Given a bordered surface $F$ marked interiorly at some point $a$. We look at all analytic maps $F\to {\Bbb C}$ with $f(a)=0$ and $f'(a)=1$ w.r.t. some chart. We develop $f$ in power series and expect some universal estimates on the coefficients at least when $f$ factorizes as a circle map of minimal degree followed by a schlicht map. The dream would be that there is a unique extremal function maximizing simultaneously all coefficients and this would be essentially the best possible Ahlfors map post-composed with the Koebe function. Of course it may happen that all this generality is not necessary in case the basic Ahlfors map $f_{a,b}$ is already the most ergonomic object, in the sense of realizing the gonality for suitable centers $a,b$. A more orthodox way to formulate higher versions of Ahlfors’ extremal problem involves the theory of Pick-Nevanlinna interpolation. Cf. for instance Jenkins-Suita 1979 [@Jenkins-Suita_1979]. The original theory being formulated in the disc $\Delta$, one may hope to lift things via an Ahlfors map but this probably leads nowhere. Genuine avatars of Ahlfors extremal problem are formulated by prescribing Taylor section (jets) at a given collection of points. Compare again Jenkins-Suita 1979 [@Jenkins-Suita_1979], building upon a paper of Heins 1975 [@Heins_1975]. In this extended context all features of the Ahlfors map persist: existence of an extremal (via normal families), uniqueness of the solution (Heins 1975), finite sheeted covering of the disc, and upper bound over the mapping degree. Again a crucial question is whether such problems always achieve the gonality. Ahlfors’ proof {#Ahlfors:sec} ============== \[January 2012\] This section is a superficial glimpse into Ahlfors’ original resolution of his extremal problem emphasizing that Ahlfors requires first the qualitative existence of a circle map. A more detailed analysis will be attempted later (Sec.\[Ahlfors-proof:sec\]). Soft part of Ahlfors 1950: circle maps with $\le r+2p=g+1$ sheets {#sec:Ahlfors-soft} ------------------------------------------------ When writing the paper Gabard 2006 [@Gabard_2006] (and a fortiori in my Thesis 2004 [@Gabard_2004]), I was very ignorant about the depth of Ahlfors’ paper (and the massive literature around it). To be honest I am still today quite ignorant having only a very fragmentary understanding of Ahlfors arguments. I take this opportunity, to rectify the arrogant claim (in [@Gabard_2006]) to the effect that a simplified proof of Ahlfors’ theorem is proposed. Of course, my paper only recovers the weaker assertion about existence of circle maps (in contradistinction to the deeper extremal problem analyzed by Ahlfors). Furthermore even in the weaker circle maps context, I only realized recently \[January 2012\] that a much shorter portion of Ahlfors’ paper achieves this goal (cf. Ahlfors 1950 [@Ahlfors_1950 p.124–126]), even with the $r+2p=g+1$ bound on the degree. We reproduce the relevant extracts (p.124 and then p.126): \[Ahlfors-1950-circle-map:quote\] \[p.124\] It must first be proved that the class of functions with $F(a)=0$ and $\vert F \vert=1$ on $C$ \[=the boundary contours\] is not empty. In other words, we must show that $\overline{W}$ can be mapped onto a full covering surface of the unit circle. \[p.126\] The function \[…\] maps $\overline{W}$ onto a covering surface of the unit circle\[=disk\], and a standard argument\[=just number conservation\] shows that every point is covered exactly $P+1$ times. \[$P$ is the genus of the double in Ahlfors’ notation\] Thus, we have the following historical: As early as Spring 1948, Ahlfors had an existence-proof of circle maps of degree $\le r+2p$. This conjecture is supported by the remarks made in Nehari 1950 [@Nehari_1950] (cf. our Quote \[Nehari-1950:quote\]). In contrast, the issue that the same upper bound $r+2p$ holds true for Ahlfors extremals may have required Garabedian formulation of the dual extremal problem for differentials. This is somehow in line with Jenkins-Suita 1979 [@Jenkins-Suita_1979], who speak of the [*Garabedian bound*]{} following a coinage of Heins 1975 [@Heins_1975 p.4]. At any rate, it seems first crucial to understand the easy part of Ahlfors’ argument (existence of a circle map of degree $\le r+2p$). Even here we failed as yet. [**Anecdote (skip!)**]{} Ahlfors’ argument bears some vague resemblance with the argument exposed by myself in the RAAG-conference of 2001. Here the game was that (in view of Riemann without Roch) any group of $g+1$ points on the curve moves. The orthosymmetric curve in question is of course the Schottky double of the given bordered surface. If such points are chosen on the real locus we are forced in the non-Harnack-maximal case ($r<g+1$, $r\equiv g+1 \pmod 2$) to select two points on the same oval (pigeon hole principle). All the subtlety is to ensure that those points will circulate along the complex orientation (as the border of one half) without doing collision repulsing them in the imaginary locus, and thereby violating total reality. Using Abel’s theorem plus some incompressible fluid argument I tried to argue that this is always possible for a clever choice of (totally real) divisor. However the argument was slightly vicious, and it would require me too hard work to repair it. If I have enough energy I should try to write down this argument, while trying to analyze it properly. In Ahlfors’ paper (1950 [@Ahlfors_1950]), one starts with a circle map of degree $\le r+2p$, and by a miraculous intervention of Garabedian the same bound turns out to be valid for all Ahlfors extremals. Let us refer to this vague principle as the Ahlfors-Garabedian divination (AGD). (Vagueness only alludes to my own poor understanding of their methods.) Now in view of Gabard 2006 [@Gabard_2006], as well as the deeper investigation in Coppens 2011 [@Coppens_2011], we know that circle maps of lower degrees $\le r+p$ exist. Thus, granting the AGD-divination, we may expect to find Ahlfors extremals of correspondingly low degrees. Of course this amounts to take the best from two different worlds, and is extremely far from a serious argument. Hence a thorough study of Ahlfors 1950 [@Ahlfors_1950] perhaps suitably adapted (and augmented by other tricks) could lead to a confirmation of the naive Conjecture \[gonality:conj\]. Of course this is pure speculation, and arguably the emphasis could be a study circle maps [*per se*]{} without getting obnubilated by Ahlfors extremal problem. Ahlfors hard extremal problem ----------------------------- We have nothing to add for the moment, suffices to say that Lagrange multipliers play a crucial rôle (as in earlier work of Grunsky). Yet it would be nice to summarize the idea (and the logical structure): \(1) [**Existence of extremals.**]{} Ahlfors first needs the existence of a circle map so as to arrange a nonempty set of competing functions (giving some ground under the foots to get started). Of course a function bounded-by-one would have been sufficient to get started, but Ahlfors achieves much more. Of course the normal families argument alone cannot supplant this preliminary study. In papers subsequent to Ahlfors’, namely Read 1958 [@Read_1958_Acta] and Royden 1962 [@Royden_1962] existence is derived via more abstract functional-analysis (Hahn-Banach). More on this in Sec.\[Read-Royden:sec\]. Other treatments Heins 1950 [@Heins_1950] appeals to Martin’s theory and elementary convexity consideration, which expressed in more highbrow setting essentially amounts to Krein-Milman existence of extreme points in convex bodies (cf. esp. Forelli 1979 [@Forelli_1979] and the discussion in Heins 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF]). \(2) [**Uniqueness of the extremal.**]{} Looks easy (essentially like when defining something by a universal property in category theory). Universal properties of category theory are essentially akin to extremal problems in geometry. This is not completely true for some natural extremum problems admits several solutions). In the case at hand uniqueness is essentially a version of Schwarz’s lemma. Other accounts of Ahlfors’ extremal problem =========================================== Ahlfors’ paper of 1950 [@Ahlfors_1950] aroused quick interest among the conformal mappers community (Nehari 1950, Heins 1950, Garabedian 1949–50, Schiffer, etc.). Numerous papers seems to reprove Ahlfors’ theorem along (better?) routes (e.g., Read 1958, slightly optimized in Royden 1962). The latter article seems to be among the most popular revision of Ahlfors 1950 [@Ahlfors_1950], with identic results but possible simplifications in the proof. The present section tries to review those (second generation) contributions while providing link to subsequent critiques (e.g., Nehari 1950 is criticized by Tietz 1955, who in turn is attacked by Köditz-Timmann 1975). Garabedian 1949, 1950 --------------------- Garabedian qualifies himself as a hard-worker, who could absorb simultaneously the influence of three giants: Ahlfors, Bergman and Schiffer. As a result, he seems to have exerted a notable influence over the final shape of Ahlfors 1950 [@Ahlfors_1950], and is even apparently able to reprove the full result of Ahlfors 1950 [@Ahlfors_1950] in the paper Garabedian 1950 [@Garabedian_1950 p.361]. (A little Riemann-Hurwitz computation is required to convince that Garabedian reobtains exactly the same degree $r+2p$ as Ahlfors.) The proof deploys a rich mixture of techniques (Teichmüller, Grunsky, Ahlfors, plus the variational method of Schiffer). Another point worth noticing is the following issue oft emphasized by Garabedian [@Garabedian-Schiffer_1950 p.182]: \[quote:Garabedian-Schiffer\_1950\] Thus our procedure leads to the existence of the circle mapping $F(z)$ which is associated with Schwarz’s lemma. It is to be noted that the existence of this function lies somewhat deeper than the existence of the slit mappings $\varphi(w)$ and $\psi(w)$ in multiply-connected domains, and therefore it is not too surprising that the present section is more difficult that the preceding ones. Of course, for $n=1$, $F(z)$ is just the function found in the elementary Koebe proof of the Riemann mapping theorem. Garabedian alone repeats a similar comment in Garabedian 1949 [@Garabedian_1949-52:Book p.207]: \[quote:Garabedian\_1949-52\] The conformal mappings which we obtain here are closely related to the generalization of Schwarz’s lemma to multiply connected domains in sharp form \[1, 7\] \[=resp. Ahlfors 1947 [@Ahlfors_1947], and Garabedian 1949, Duke Math. J.\], and their existence lies somewhat deeper than that of the more standard canonical maps in a multiply connected region. Nehari 1950, Tietz 1955, Köditz-Timann 1975 ------------------------------------------- Regarding the first two mentioned papers (Nehari 1950 [@Nehari_1950], Tietz 1955 [@Tietz_1955]), I suggested in Gabard 2006 [@Gabard_2006 p.946], that those papers may have conjectured the improved control $r+p$ on the degree of circle maps. (When discovering the $r+p$ bound ca. 2001/02, I was not influenced by those papers which I located only later in 2005 while polishing the ultimate shape of Gabard 2006 [@Gabard_2006].) Nehari 1950 [@Nehari_1950] does not seem to give a new proof of circle maps (Ahlfors’ theorem), but inspired by it proposes to describe canonical slit maps (incidentally those for which Garabedian seems to have a lesser esteem, cf. Quotes \[quote:Garabedian-Schiffer\_1950\] and \[quote:Garabedian\_1949-52\]). Nehari also shows how to express the Ahlfors function in terms of the Bergman kernel function. (If I understand well the situation, this is just a representation theory yet not an alternative existence-proof.) Nehari’s paper shows that Ahlfors was in possession of the degree $r+2p$ as early as Spring 1948, at least for a circle map. It is a delicate question if the same bound for extremal maps requires Garabedian’s remark about the dual extremal problem. Heins’ paper 1975 [@Heins_1975] using the term “Garabedian’s bound” may suggest a positive answer. The reader is not well placed to guess the answer, but remember that the (published) proof in Ahlfors 1950 [@Ahlfors_1950] requires (and acknowledges) Garabedian’s dual problem. Let us quote the crucial extract of Nehari: \[Nehari-1950:quote\] It was recently shown by Ahlfors \[1\](=L. Ahlfors, Material presented in a colloquium lecture at Harvard University in Spring 1948.) that the well known canonical conformal mapping of a schlicht domain of connectivity $n$ onto an $n$-times covered circle \[5,7\] (=Bieberbach 1925 [@Bieberbach_1925], Grunsky 1937–41 [@Grunsky_1937], [@Grunsky_1941_KA]) can be generalized, in the case of an open Riemann surface, in the following manner: an open Riemann surface of genus $g$ which is bounded by $n$ closed curves can be mapped conformally onto a multiply-covered circle, the number of coverings not exceeding $n+2g$. Soon afterwards, Tietz 1955 [@Tietz_1955 p.49] criticizes (slightly) some of Nehari’s asserted results: \[Tietz-1955:quote\] Bei der Herleitung seiner Schlitztheoreme kommt Herr Nehari ebenfalls auf diese Frage; sein Beweis für die genannte Vermutung ist jedoch unhaltbar. Nimmt man jedoch diese Neharische Behauptung als richtig an, so hie[ß]{}e das, da[ß]{} $R$ aus $p+r$, und damit $R^2$ aus $2p+r=G+1$ Blättern bestünde; … This seems to be a forerunner of the bound $r+p$ (by commutativity of addition!), at the conjectural level at least. \[Parenthetically, I do not understand Tietz’s claim about the sheet number of the double $R^2$. I believe that the degree keeps the same value $p+r$, as one has to double the map not just the space.\] Finally, Tietz concludes his paper [@Tietz_1955 p.49] as follows: So Tietz does not seem to be able reprove a result as strong as the one of Ahlfors 1950. In fact, the situation looks even worse, since even Tietz’s weak version is questioned in the paper by Köditz-Tillmann 1975 [@Koeditz-Timmann_1975 p.157], as shown in the following extract (parenthetical reference are ours addition): The extract is followed by a specific objection (not reproduced here). The article (of Köditz-Timmann 1975 [@Koeditz-Timmann_1975 Satz 3, p.159]) seems however to contain a proof of Ahlfors’ theorem based upon an “Approximationssatzes von Behnke u. Stein”, yet without any bound on the degree. A propos Behnke-Stein 1947/49 [@Behnke-Stein_1947/49] (the famous paper going back to 1943), it contains the result that any open Riemann surface (arbitrary connectivity and genus) admits a non-constant analytic function. Can one deduce this Behnke-Stein theorem by agglomerating Ahlfors extremals (or weaker circle maps) relative to compact subregions of an adequate exhaustion? In this connection let us remember the paper by Nishino 1982 [@Nishino_1982], where Ahlfors is applied to prove existence of (non-constant) analytic functions on certain complex surfaces (four real dimensions). Since this Nishino paper employs Ahlfors bound $r+2p$, it would be nice to understand it thoroughly to see if some better constant leads to some sharpened result. (Alas it seems that a subsequent paper of Nihino ca. 1983 proves a stronger result (pertaining to arbitrary complex dimension) while eradicating apparently any logical dependence upon Ahlfors 1950. The MR-reviewer, M. Hervé, seems to have been a bit overwhelmed by the work.) Heins 1950 {#sec:Heins} ---------- Heins being one of the most prolific and pleasant-to-read writers of the U.S. school (student of Walsh), it is not surprising to find several first classes contributions regarding our special Ahlfors map topic. Specifically, the paper Heins 1950 [@Heins_1950] reproves Ahlfors’ result in presumably its full strength (this even without quoting Ahlfors 1950 [@Ahlfors_1950] but the closely allied work Garabedian 1949 [@Garabedian_1949]). Remember that Ahlfors’ result was exposed at the Harvard seminar in Spring 1948 (cf. Nehari’s Quote \[Nehari-1950:quote\]), and must have widely circulated since then. Taking a closer look to Heins’ paper, it is at first sight not completely evident that a bound on the degree derives from his method but is quite likely to do since his quantity $m$ (number of generators of the fundamental group, cf. p.571) is easily recognized to be $2p+(r-1)$, where $p$ is the genus and $r$ the number of contours. Thus one certainly recovers exactly Ahlfors’ result with its bound. In some sense, Heins’ paper goes even deeper than Ahlfors by treating Pick-Nevanlinna interpolation. Several subsequent works in Heins’ spirit (overlapping with Ahlfors theorem) are worth mentioning: Heins 1975 [@Heins_1975], Forelli 1979 [@Forelli_1979] and Heins 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF]. Kuramochi 1952 -------------- The paper Kuramochi 1952 [@Kuramochi_1952] also seems to recover Ahlfors bound for circle maps using the extremal problem. This is maybe the sort of technical paper with too much [*notatio*]{} and not enough [*notio*]{}? (This is a joke of Hellmuth Kneser, compare his paper in JDMV.) Kuramochi’s work seems to be inspired by Ahlfors 1950 [@Ahlfors_1950] and by a 1951 paper by Nehari (confined to the planar case). Nehari offers a positive review (in MathReviews): \[Nehari-Kuramochi:quote\] [Generalizing a method developed by the reviewer for the case of plane domains \[Amer. J. Math. 73 (1951), 78–106\], the author discusses extremal problems for bounded analytic functions on open Riemann surfaces of positive genus. The procedure is illustrated by a detailed treatment of the case corresponding to the classical Schwarz lemma which had previously been discussed, by different methods, by L.V. Ahlfors \[1950\]. A complete characterization of the extremal function is obtained and Ahlfors’ positive differential is constructed.]{} Read 1958, Royden 1962 (via Hahn-Banach) {#Read-Royden:sec} ---------------------------------------- We start with: $\bullet$ Royden 1962 [@Royden_1962], where existence of a solution to Ahlfors’ extremal problem is achieved via conjunction of Hahn-Banach with Riesz’s representation theorem (circumventing thereby both Euler-Lagrange and normal families). Exploiting the duality pointed out by Garabedian (pair of extremal problems with a dualizing Schottky differential, i.e. one extending to the double), the control on the degree is achieved by the usual index formula $\deg(\vartheta)=2g-2$ (Poincaré 1881–85, but already in Riemann in the holomorphic case at hand). Ahlfors’ upper bound $\deg f_{a,b}\le r+2p$ follows. Royden’s paper is therefore quite remarkable for supplying alternatives arguments. It seems to have been inspired mostly by: $\bullet$ Read 1958 (two papers [@Read_1958_Fenn], [@Read_1958_Acta]). Read is also a student of Ahlfors (as one may learn in Ahlfors 1958 [@Ahlfors_1958]) and already relies on Hahn-Banach to prove existence of an Ahlfors function (but, as Royden observes, does not take care of making the argument with the Schottky differential so as to bound the degree). The technique employed (by Read) to prove extremals is to relate the dual extremal problems (à la Garabedian-Ahlfors, 1949–1950) to conjugate extremum problems of the Lebesgue classes $L_p$ and $L_q$, where $p^{-1}+q^{-1}=1$, where one maximizes an $L_p$-norm versus vs. minimizing an $L_q$-norm. Such problems classically reduce to Hahn-Banach. For this reduction of Garabedian-Ahlfors to Hahn-Banach, Read employs a converse to Cauchy’s theorem (itself an application of Stokes) due to Rudin 1955 [@Rudin_1955-class-Hp] in the planar case. Methods of Rogosinski–Shapiro 1953 [@Rogosinski-Shapiro_1953] are another ingredient to the proof. To summarize, the Read-Royden approach via Hahn-Banach (functional analysis, coinage of Hadamard) effects a little drift from the traditional Euler-Lagrange variational approach (used in Grunsky 1940–46 ([@Grunsky_1940], [@Grunsky_1946]), Ahlfors 1947 [@Ahlfors_1947], 1950 [@Ahlfors_1950]). As conceptually brilliant as it is, this new method does not lead to an improved degree control. The reason is quite simple, namely Ahlfors’ bound $r+2p$ is sharp within the extremal problem it solves (contribution of Yamada 1978 [@Yamada_1978] in the hyperelliptic case). The game naturally splits in existence of extremals (either via Montel’s normal families or via Hahn-Banach) and then to analyze its geometric properties. Ahlfors’ 1950 treatment (apparently influenced by Garabedian’s dual extremal problem) supplies the trick to bound the degree via a Schottky differential, and Royden’s argument looks, in this second geometric step, virtually osculant to Ahlfors’ original. Remember yet that Ahlfors’ original proof—presented in Spring 1948 at Harvard as reported in Nehari 1950 [@Nehari_1950 p.258, footnote]), and perhaps nearly similar with pages 124–126 of the published paper 1950 [@Ahlfors_1950]—manages without Garabedian’s influence to supply existence of circle maps of degree bounded by $r+2p$. Existence of (inextremal) circle maps ===================================== This section focuses on existence of circle maps on membranes (=finite bordered Riemann surfaces) without appeal to the extremal problem. In fact those are logically required (at least in Ahlfors’ account but not in Royden’s 1962 [@Royden_1962]) as a qualitative preparation to the analysis of the quantitative problem. Ahlfors 1948/50, Garabedian 1949 -------------------------------- \[09.06.12\] We mean the papers Ahlfors 1950 [@Ahlfors_1950] and Garabedian 1949 [@Garabedian_1949]. The additional 1948 date is intended to reflect that Ahlfors lectured on this material somewhat earlier, as shown by Nehari’s Quote \[Nehari-1950:quote\]. Those writers address the deeper extremal problem $\max \vert f'(a)\vert$ amongst functions bounded-by-one $\vert f\vert\le 1$, however it seems that they are well-acquainted with topological methods (e.g., Garabedian 1949 [@Garabedian_1949] cites Alexandroff-Hopf’s classical 1935 treatise “[*Topologie*]{}”). Such a qualitative topological inspection seems a logical prerequisite to their treatments of the quantitative extremal problem. Prior to posing any extremal problem, it is vital to ensure non-emptiness of the set of permissible competing functions. Perhaps the following trivial remarks are worth doing. For domains bounded by $r$ Jordan curves, we have clearly some function bounded-by-one (take the identity map suitably scaled to shrink the domain inside the unit-disc). For a general compact bordered surface, it is less obvious that such functions exist at all. Of course one can take the Schottky double to apply Riemann’s existence theorem (of a morphism to ${\Bbb P}^1$) and look at the image of the (compact) half. However the latter can still cover the full Riemann sphere, which is annoying for our purpose. \[05.11.12\] Using Klein’s work one can certainly find an equivariant map from the double to the sphere acted upon by orthosymmetry (standard complex conjugation), yet it may still be the case that the full sphere is covered by the half of the double. \[As a simple example we may take a conic $C_2$ with real points and project it from a real point $p$ outside of the unique oval. The corresponding map $C_2\to {\Bbb P}^1$ is equivariant and surjective when restricted to one half of the complex locus of the conic $C_2$. Indeed given a point of ${\Bbb P}^1$ is tantamount to give a line $L$ through the center of perspective $p$. This line $L$ cuts $C_2$ in two points (except for the two real tangents). If $L$ is a real line cutting the real locus $C_2({\Bbb R})$ we can take as antecedent a point on the border of the half Riemann surface. If $L$ does not cut $C_2({\Bbb R})$ its intersection with $C_2$ is a conjugate pairs of points one of them lying in the fixed half of $C_2$. Finally if $L$ is an imaginary line then its intersection with $C_2$ consists of two points distributed in both halves of $C_2$. Indeed $L$ can by continuity be degenerated to a real line $L_0$ missing the real locus of $C_2$ (recall that the pencil of line is just an equatorial sphere with equator corresponding to real lines) and since during the process no points of $L\cap C_2$ became real it follows that both $L$ and $L_0$ have the same distributional pattern when intersected with the conic.\] Hence in general some preparatory qualitative “topological” investigation is required to see that the extremal problem is non-vacuously posed. Remember that Ahlfors directly attacks the existence of a circle map, where it may have been sufficient to prove existence of a function bounded-by-one. His argument is in part topological inasmuch as it involves annihilating the periods of the conjugate differential of a suitable harmonic function, but also contains a great deal of non-trivial analysis, plus basic principles of convex geometry. We shall try later to penetrate in more details in Ahlfors proof. Regarding Garabedian 1949 [@Garabedian_1949], topological considerations also plays a vital rôle in conjunction with Abelian integrals, etc. We refer the reader to the original paper. In retrospect, it may just be too sad that this brilliant work was not directly written in the broader context of Riemann surfaces. Mizumoto 1960 ------------- This is the paper Mizumoto 1960 [@Mizumoto_1960] (which I discovered only in March 2012), yet it looks quite original making use of a topological argument involving (Brouwer’s) topological degree of a continuous mapping. So it is spiritually close to Gabard 2006 [@Gabard_2006]. However Mizumoto [@Mizumoto_1960 p.63, Thm 1, with $N$ defined on p.58] only recovers the old bound of Ahlfors $r+2p$. Gabard 2004–2006 ---------------- The proof published in the writer’s Thesis 2004 [@Gabard_2004] is essentially the same as the one in 2006 [@Gabard_2006] (modulo slight modifications suggested by the referee, presumably J. Huisman). In fact J. Huisman already on the 2004 version supplied some corrections about naive little mistakes that I made (esp. a wrong statement of Abel’s theorem forgetting to ask both divisors to be of the same degree). Of course it is to be hoped that the new bound $\le r+p$ will stay correct in the long run. In case the result is true, it would be desirable if alternative more conventional analytic methods are able to reprove this bound $r+p$. Recent results of Coppens 2011 [@Coppens_2011] show the bound $r+p$ to be best-possible, at least for generic curves in the moduli space. Coppens’ work actually supplies a much sharper understanding of all intermediate gonalities (compare Sec.\[Coppens:subsec\] for more). There is a little historical inaccuracy in Gabard 2006 [@Gabard_2006]. When writing the paper, I did not realized properly that Ahlfors has also a quite elementary proof of the existence of circle maps of degree $r+2p$. Alas, I still do not completely understand Ahlfors’ proof yet it is clear-cut that its elementary part does not use the extremal problem! Accordingly, the sentence in Gabard 2006, p.946 reading as follows is quite inaccurate: “[*[\[…\]]{} un résultat équivalent fut démontré par L.V. Ahlfors en 1950, qui déduit d’un problème d’extrémalisation la possibilité de représenter toute surface de Riemann à bord compacte comme revêtement holomorphe (ramifié) du disque.*]{} Related results ================ Some closely allied problems involves [*Parallelschlitzabbildung*]{} (parallel-slit mapping), the relationship with the Bergman kernel, etc. Although a bit outside our main theme of the Ahlfors map, the methods employed are quite similar and therefore a thorough knowledge of those proximate mapping problems can only reinforce the general understanding. In fact it is not to be excluded that the Kreisnormierung, or its positive genus case avatar, known as Klein’s Rückkehrschnitttheorem, is logically stronger than the Ahlfors mapping (but this is for the moment just a naked speculation). Parallel slit mappings (Schottky 1877, Cecioni 1908, Hilbert 1909) ------------------------------------------------------------------ Those mappings (abridged PSM) involve several tentacles using varied technologies tabulated as follows: $\bullet$ (Classical) Schottky 1877 [@Schottky_1877], Cecioni 1908 [@Cecioni_1908] (via methods of Schwarz, and Picard). Classically Schottky’s argument is criticized (by e.g. Klein, Cecioni, Salvemini, etc.) for depending only upon a constant count not fully sufficient to establish the mapping existence (this critique appears e.g., in Cecioni ) It is likely that subsequent rigorous continuity methods as developed by Brouwer upon topological ground can easily supplement Schottky’s heuristic argument (browse through Koebe’s works, etc.)\] $\bullet$ (Dirichlet resurrected) Hilbert 1909 [@Hilbert_1909], Koebe 1910 [@Koebe_1910_Hilbert], Courant 1910/12 [@Courant_1912] (those writers extend the PSM to domains of infinite connectivity) $\bullet$ (Extremal problem à la FROG Fejér-Riesz-Radó-(Carathéodory)-Ostrowski-\[Grunsky\]) de Possel 1931 [@de-Possel_1931], 1932 [@de-Possel_1932], Grötzsch, Rengel 1932/33 [@Rengel_1932-33], 1934 [@Rengel_1934], $\bullet$ (Bergman kernel) Nehari 1949 [@Nehari_1949], Lehto 1949 [@Lehto_1949], Meschkowski 1951 [@Meschkowski_1951], etc. A philosophical curiosity is that PSM is somewhat easier (according to specialists, cf. e.g. Garabedian’s Quote \[quote:Garabedian-Schiffer\_1950\] and Hejhal 1974 [@Hejhal_1974]) than the Kreisnormierung (KNP) (cf. next section), and this already in finite connectivity (cf. e.g. the very subtle approach to KNP imagined in Schiffer-Hawley 1962 [@Schiffer-Hawley_1962]). One may wonder about this sharp discrepancy of difficulty, since it is easily conceivable that for such canonical regions (bounded by elementary curves of the most elementary stock) one could easily pass from one normal-form to the other through explicit maps (at least in finite connectivity). \[Of course I do not claim that this is an easy game for me, but I suspect so for people like Schwarz-Christoffel or Schläfli it could be accessible. Of course there is maybe a difficulty in choosing the “accessory parameters” but this should be pulverizable through modern topological arguments à la Brouwer?\] Another striking asymmetry of the theory is that PSM hold true in infinite connectivity (since Hilbert 1909 [@Hilbert_1909] and the subsequent work of Koebe 1910 [@Koebe_1910_Hilbert]), whereas KNP is still wide open in infinite connectivity. (A very naive guess would be to deduce KNP$\infty$ from PSM$\infty$ through a continuity method for infinite dimensional manifolds. Maybe this suggests using Leray-Schauder theory as an infinite-dimensional avatar of the Brouwer degree? Regarding PSM, lucid remarks are made in Burckel 1979 [@Burckel_1979 p.357–8], namely: $\bullet$ the result in infinite connectivity is due to Hilbert, Koebe, Grötzsch, Rengel, de Possel (as we just said also), $\bullet$ excellent book expositions are credited to Bieberbach 193?/67 [@Bieberbach_1967-BUCH-Einfuehrung-in-die-konf-Abb], Golusin 1952/57 [@Golusin_1952/57] and Nehari 1952 [@Nehari_1952-BOOK], $\bullet$ de Possel’s proof in 1931 [@de-Possel_1931] (and the allied work by Rengel and Grötzsch) via an extremum problem is recognized as reminiscent of Fejér-Riesz’s proof of RMT. However at one point of the proof RMT is invoked. Later de Possel 1939 [@de-Possel_1939] found a (short) constructive way around this (see also Garabedian 1976 [@Garabedian_1976]). $\bullet$ for an approach to PSM, and the other canonical regions (radial or circular slits), via the Dirichlet principle see Ahlfors 1966 [@Ahlfors_1966-BOOK-Cplx-Anal]. Kreisnormierungsprinzip (Riemann 1857, Schottky 1875/77, Koebe 1906-08-10-20-22, Denneberg 1932, Grötzsch 1935, Meschkowski 1951–52, Strebel 1951–53, Bers 1961, Sibner 1965–68, Morrey 1966, Haas 1984, He-Schramm 1993) {#sec:KNP} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- This (cavalier?) principle (abridged KNP) starts with the fact that a multiply-connected domain of finite connectivity maps conformally to a circular domain. This was already implicit in Riemann’s Nachlass 1857/58/76 [@Riemann_1857_Nachlass] (according to Bieberbach 1968 [@Bieberbach_1968-Das-Werk-Paul-Koebes p.148–9] who apparently saw a copy of Riemann’s original manuscript, cf. our Quote \[quote:Bieberbach-1925\] reproducing Bieberbach 1925 [@Bieberbach_1925]; cf. also Koebe 1910 [@Koebe_1910_JDMV p.339]: “[*Den Hauptgegenstand dieser und des gegenwärtigen Vortrages bildet das Problem der konformen Abbildung eines $(p+1)$-fach zusammenhängenden Bereiches auf einen von $p+1$ Vollkreisen begrenzten Bereich, ein Problem, welches in der Literatur zuerst bei Schottky (Dissertation, Berlin 1875, umgearbeitet erschienen in Crelle 1877) in seiner bekannten Doktordissertation auftritt, jedoch früher bereits von Riemann in Betracht gezogen worden ist, wie aus seiner nachgelassenen Schriften hervorgeht.*]{}”). The statement resurfaced more explicitly in Schottky’s Thesis 1875/77 [@Schottky_1877] (at least in the Latin 1875 version). The latter’s argument rests again only on a naive parameter count of moduli. Indeed, a circular domain with $r$ contours depends upon $3r$ free parameters to describe centers and radii of those $r$ circles, while removing the 6 (real) parameters involved in the automorphism group of the (Riemann) sphere, we get $3r-6$ essential constants. Invoking the (Schottky) double of the domain, whose genus is $g=r-1$, this number agrees with Riemann’s count of $3g-3$ moduli (where of course attention is restricted to “real” moduli). This adumbrates why circular domains are flexible enough to conformally represent any domain. Such naive counting arguments usually turn into rigorous proofs by appealing to some topological principles (like Brouwer’s invariance of the domain) vindicating the so-called continuity method. This sort of game occupy several of Koebe’s papers, who probably arranged this already; see also Grunsky 1978 [@Grunsky_1978] for an implementation of KNP in 12 pages (p.114–126). Koebe devoted several papers to the KNP question in 1906 [@Koebe_1906_JDMV], 1907 [@Koebe_1907_JDMV], 1910 [@Koebe_1910_JDMV] (Überlagerunsfläche and iteration method), 1920 [@Koebe_1920]. As early as 1908 [@Koebe_1908_UbaK3], Koebe advanced conjecturally the validity of this principle for domains of infinite connectivity: an issue still undecided today (2012), but corroborated in He-Schramm 1993 [@He-Schramm_1993] in the case of countably many boundary components (via the method of circle packings). Most of the contributions (listed in our subtitle) are carefully referenced in He-Schramm’s paper just cited. Other proofs of the basic (finitary) KNP result are obtained by: $\bullet$ Courant 1950 [@Courant_1950] (via a Plateau-style approach) \[Micro-Warning: Hildebrandt-von der Mosel 2009 [@Hildebrandt-von-der-Mosel_2009] and also Hildebrandt 2011 [@Hildebrandt_2011] credit rather Morrey 1966 [@Morrey_1966] for the first rigorous proof, modulo yet another gap filled by Jost 1985 [@Jost_1985]\]. $\bullet$ Schiffer and Hawley in several papers: Schiffer 1959 [@Schiffer_1959] (via the Fredholm determinant) and Schiffer-Hawley 1962 [@Schiffer-Hawley_1962] (via an extremal problem of the Dirichlet type). It is common folklore that the Kreisnormierung, like the uniformization and even the Ahlfors circle map belong to a somewhat deeper class of problems than the parallel-slit mapping succumbing quickly to elementary techniques of potential theory. (Compare Garabedian-Schiffer’s Quotes \[quote:Garabedian-Schiffer\_1950\] and \[quote:Garabedian\_1949-52\], and also Hejhal 1974 [@Hejhal_1974 p.19] who makes similar remarks, for instance “[*We remark that the Koebe \[circular\] mapping is similar to the universal covering map, in that neither an explicit formula nor an explicit differential equation is known for it.*]{}”) Such higher stock problems make it challenging to ask whether KNP(finite) could not be handled via an extremal problem à la Ahlfors, or to be historically sharper in the spirit of FROG=Fejér-Riesz-(Carathéodory)-Ostrowski-Grunsky. \[$\bigstar$ Warning the sequel looks attractive yet erroneous, cf. the next paragraph for a rectification $\heartsuit$\] Maybe the relevant extremal problem (under educated guess) is to maximize the modulus of the derivative at a fixed point $a$ of the domain amongst functions bounded-by-one (in modulus) while imposing schlichtness to the mappings (otherwise we recover Ahlfors’ many-sheeted discs). Intuitively, this maximum pressurization exerted at the point $a$ ensures surjectivity of the mapping while filling most of the container in which the function in constrained by the condition $\vert f \vert \le 1$, yet roundness of the residual set of the image looks less intuitive. Speculating further, this “Ahlfors-schlicht” extremum problem could crack the fully general KNP in infinite connectivity (KNP($\infty$)). However, it suffices to remind that several complications are reported for the usual Ahlfors function in infinite connectivity (existence easy and uniqueness due to Havinson 1961/64 [@Havinson_1961/64], Carleson 1960/67 [@Carleson_1967-book], see also Fisher 1969 [@Fisher_1969]) by subsequent investigators like Röding 1977 [@Roeding_1977_Ahlfors], Minda 1981 [@Minda_1981-image-Ahlfors-fct], Yamada 1983–92 [@Yamada_1983-rmk-image-Ahlfors-fct] [@Yamada_1992-Ahlfors-fct-on-Denjoy], where the Ahlfors extremal function ceases to be a circle map and start to omit values). It is therefore quite overoptimistic to hope an Ahlfors-type (=FROG) strategy toward the prestigious KNP($\infty$). \[05.11.12\] $\heartsuit$ [*Correction.*]{}—The beginning of the previous paragraph is quite erroneous since the analogue of the Ahlfors map under the schlichtness proviso (=injectivity) is known to take a multi-connected domain not on a Kreisbereich but on a circular-slit disc. This result is due to Grötzsch 1928 [@Groetzsch_1928], Grunsky 1932 [@Grunsky_1932], Nehari 1953 [@Nehari_1953-Inequalities p.264–5] (another proof while crediting the just two cited works by Grötzsch and Grunsky), Meschkowski 1953 [@Meschkowski_1953] and finally Reich-Warschawski 1960 [@Reich-Warschawski_1960]. (Those references were already listed in Sec.\[sec:beta-and-alpha-problems\].) It is yet to be observed that such circular-slit-disc ranged maps fail schlichtness up to the boundary, and one can legitimately speculate about a suitable extremal problem akin to Ahlfors’ establishing KNP (in finite connectivity at least). Röding 1977 (still not read) ---------------------------- The paper Röding 1977 [@Roeding_1977_mero] is perhaps quite dangerous (for Gabard 2006 [@Gabard_2006]), yet I could not procure a copy as yet. Behavior of the Ahlfors function in domains of infinite connectivity -------------------------------------------------------------------- There is a series of works studying the behavior of the Ahlfors function for domains of infinite connectivity. Traditionally those works look more confined to the domain case. The basic existence and uniqueness result are addressed by Havinson 1961/64 [@Havinson_1961/64], Carleson [@Carleson_1967-book] with simplifications in Fisher 1969 [@Fisher_1969]. In contrast to the finite case, the image of the Ahlfors function does not necessarily fill the full unit circle (=disc). We just list some main contributions: Röding 1977 [@Roeding_1977_Ahlfors] (2 points are omitted), Minda 1981 [@Minda_1981-image-Ahlfors-fct] (fairly general discrete subset of omitted values), Yamada 1983 [@Yamada_1983-rmk-image-Ahlfors-fct] (omission of a fairly general set of logarithmic capacity zero), Yamada 1992 [@Yamada_1992-Ahlfors-fct-on-Denjoy] (characterization of omitted point-sets of the Ahlfors function in case of Denjoy domains). The quest of best-possible bounds ================================= The writers’s own contribution $r+p$ seems, at first glance, a dramatic improvement upon Ahlfors’ upper bound $r+2p$ (at least so sounded the diagnostic of the generous Zentralblatt reviewer of my article, namely Bujalance). In the long run it may be that Ahlfors’ extremals are always as good for suitable choices of points $a,b$, but only meagre evidence is presently available. Distribution of Ahlfors’ degrees (Yamada 1978–2001, Gouma 1998) {#Yamada-Gouma:subsec} --------------------------------------------------------------- The papers by Yamada 1978 [@Yamada_1978], 2001 [@Yamada_2001] and Gouma 1998 [@Gouma_1998] address the delicate question about the exact values realized as degrees of Ahlfors functions. Ahlfors’ pinching $r\le \deg(f_{a,b})\le r+2p$ collapses for planar surfaces ($p=0$) to an equality, and the question is trivially settled in this case. Yamada and Gouma rather consider the infinitesimal form of the problem where just a single interior point $a$ is prescribed while maximizing $\vert f'(a) \vert$. They obtain spectacular complete results for membranes having a hyperelliptic double ([*hyperelliptic membranes*]{}), yet without being planar ($p=0$) in which case we are in the trivial range already discussed. For a hyperelliptic membrane, the followings hold true: [(1) (Yamada 1978)]{} The ponctual Ahlfors function $f_a$ has degree $g+1$ at the fixed points of the hyperelliptic involution (so-called Weierstrass points). [(2) (Gouma 1998)]{} The degree of $f_a$ can only assume values $2$ or $g+1$. [(3) (Yamada 2001)]{} The case of degree $2$ is always realized at suitable points. \[05.11.12\] Gouma’s result shows large discrepancy between degrees taken by Ahlfors extremals and those of general circle maps. Of course the latter are more flexible with a specimen of degree 2 (just quotient by the hyperelliptic involution), whence circle maps exist in all even degrees (post-compose with a power map $z\mapsto z^k$). Those works promise a grandiose link between Ahlfors and the classic tradition of Weierstrass points, which probably also regulate the degree of Ahlfors maps for general (non-hyperelliptic) surfaces. Separating gonality (Coppens 2011) {#Coppens:subsec} ---------------------------------- In another direction of dramatic depth, Marc Coppens 2011 [@Coppens_2011] is able to show sharpness of the bound $r+p$ claimed in Gabard 2006 [@Gabard_2006]. Actually, Coppens establishes the more spectacular realizability of all intermediate values for the gonality. Even if Coppens’ result looks at first sight subsumed to that of Gabard, it is in reality logically independent, so that a possible misfortune of Gabard’s result should not necessarily affect the truth of Coppens’ one. To be more specific, we introduce the following definition: \[gonality:def\] The gonality (denoted $\gamma$) of a membrane (i.e. a compact bordered Riemann surface) is the least degree of a full (or total) covering map to the disc. \[05.11.12\] A full (or total) covering map can be defined just as non-constant analytic map taking boundary to boundary. Then it makes good sense to Schottky-double the map and classic theory ensures the local power-map $z\mapsto z^k$ character of analytic functions, whence the branched cover nature of the map, in particular its surjectivity (via a clopen argument). The jargon “total” is borrowed from Stoilow 1938 [@Stoilow_1938-Lecons] and quite compatible with the “total reality” jargon (of Geyer-Martens 1977 [@Geyer-Martens_1977]) incarnating the algebro-geometric pendant of Ahlfors circle maps. It is easy to show that a total map lacks ramification along the boundary. (Possible argument: Else it behaves locally like $z\mapsto z^2$ near a boundary uniformizer, but then the half-space is wrapped to a full domain expanding outside the permissible range of the map.) In particular such a total map induces a usual (unramified) cover of the circle $\partial W \to \partial D=S^1$, whereupon the trivial lower bound $r \le \gamma$ follows, where $r$ is the number of boundary contours of the membrane $W$. On the other hand Gabard’s main result in 2006 [@Gabard_2006] asserts the upper bound $\gamma \le r+p$, where $p$ is the genus of $W$. Coppens’s striking result states: [(Coppens 2011)]{} Practically, all intermediate values of the gonality compatible with the pinching $r\le\gamma\le r+p$ are realized as the gonality of a suitable membrane of topological type $(r,p)$. More accurately, there is a single trivial exception when $r=1$ and $p>0$, in which case the value $\gamma=1$ must be excluded. Taking $\gamma=r+p$ supplies sharpness of Gabard’s upper bound. On the other hand, Coppens’ theorem tightens considerably Ahlfors’ squeezing $$r\le \deg f_{a,b} \le r+2p$$ into $$r\le\gamma\le \deg f_{a,b}\le r+2p,$$ yielding a notable contribution to Yamada-Gouma’s general question on the distribution of Ahlfors degrees (cf. previous section). Of course the contraction becomes most stringent when the gonality $\gamma$ attains its maximum value (i.e., $\gamma =r+p$ if thrusting Gabard), as it does for generic membranes in the moduli space ${\cal M}_{r,p}$ (parameterizing isomorphism classes of bordered Riemann surfaces). Of course the moduli space stratifies through the gonalities. Imitating Riemann’s original count in our context it should be possible to predict dimensions of the varied strata. Such a deeper investigation looks desirable to complement the theory of Ahlfors circle maps. More on this in Sec.\[sec:gonality-sequence\]. \[05.11.12\] We are presently not aware of any total-bordered avatar of the simple Riemann-type counting argument, so efficient for closed surfaces in predicting correctly the gonality $[\frac{g+3}{2}]$ as well as the dimensions of moduli strata of lower gonalities. It is suspected that this asymmetry is inherent to the boundary behavior of total maps which causes certain difficulties. Of course the difficulty is somewhat akin to the intricacies arising when doing real instead of complex algebraic geometry. Yet the problem is certainly not insurmountable. Naive question: Ahlfors degree vs. the gonality ----------------------------------------------- All information mentioned so far is summarized in the string of estimates: $$r\le \gamma \le \begin{Bmatrix} \le\deg f_{a,b} \\ \le r+p \end{Bmatrix} \le r+2p.$$ An obvious question is whether inequality $\gamma \le \deg f_{a,b}$ is best-possible: Is Ahlfors extremal problem flexible enough that each membrane has an Ahlfors map $f_{a,b}$ of degree as low as the gonality $\gamma$ for suitable centers? Yamada’s deep result (2001 [@Yamada_2001]) positively answers the case of hyperelliptic membranes (those which are 2-gonal $\gamma=2$). Other sources (Fay 1973, Černe-Forstnerič 2002) ----------------------------------------------- In Fay 1973 [@Fay_1973 p.116], one reads the following assertion: \[Cerne-Forstneric-2002:quote\] It has been proved in \[3, p.126\](=Ahlfors 1950 [@Ahlfors_1950]) that there are always unitary functions with exactly $g+1$ zeroes [*all*]{} in $R$; and when $R$ is a planar domain, it is shown in Prop.6.16 that $S_{0,\dots,0}\cap \Sigma_a$ is empty for $a\in R$ and that the unitary functions holomorphic on $R$ with $g+1$ zeroes are parametrized by the torus $S_0$.” A similar comment is to be found in Černe-Forstnerič 2002 [@Cerne-Forstneric_2002 p.686] \[Cerne-Forstneric-2002:quote\] It is proved in Ahlfors 1950 [@Ahlfors_1950 pp.124–126] that on every bordered Riemann surface of genus $p$ with $r$ boundary components there is an inner function with multiplicity $2p+r$ (although the so-called Ahlfors functions may have smaller multiplicity). Actually, it seems that Ahlfors’ proof shows even the slightly stronger fact that each integer $d\ge r+2p$ do arise as the degree of a circle map. It is not perfectly clear to the writer how this claim must be interpreted: either as an exact degree $2p+r$ or as $\le 2p+r$. On page 684, Rudin 1969 [@Rudin_1969] is quoted. Also on page 693 we find an interesting stability of inner functions of degrees $\ge r+2p-1$. Applications of the Ahlfors mapping {#Sec:Applications-of-the-Ahlf-map} =================================== This section lists some of the known applications of Ahlfors maps. Those applications either require the extremal property or merely conformality and the essentially topological feature of circle maps. Gleichgewicht der Electricität (Riemann 1857) --------------------------------------------- This source (Riemann 1857/58/76 [@Riemann_1857_Nachlass]) is the very origin of all our story. Alas the physical applications Riemann had in mind were apparently only partially reproduced in H. Weber’s reconstruction of the original manuscript. Can someone imagine what Riemann had exactly in mind (eventually on the basis of the original manuscript, which must still be dormant somewhere in Göttingen)? Here are some well-known remarks concerning this posthumous fragment; compare the “original” (as edited by H. Weber and reproduced in part below) as well as the remarks in Bieberbach 1925 [@Bieberbach_1925 p.9, §7]. Interestingly, Riemann starts with the first boundary value problem for plane domains, and actually uses the conformal circle map to solve it, whereas the reverse engineering may look more natural in view of his Dirichlet principle philosophy. Strikingly, Riemann anticipates both the Schwarz symmetry/reflection principle (Schwarz 1869 [@Schwarz_1869-Ueber-einige-Abbildungsaufgaben p.106]) as well as the Schottky double (Schottky 1875–77 [@Schottky_1877]). Typical to Riemann, an equality sign is virtually put between potential theory and algebraic functions: the Green theorem is used and Abelian integrals (of the third species) and their periods (Periodicitätsmoduln) enter the scene. \[05.11.12\] Recall also that Bieberbach 1968 [@Bieberbach_1968-Das-Werk-Paul-Koebes] asserts that Riemann’s work also contains a trace of the Kreisnormierung, and so does earlier Koebe 1910 [@Koebe_1910_JDMV]. Besides, Bieberbach 1925 [@Bieberbach_1925] (cf. Quote \[quote:Bieberbach-1925\]) gives full credit to Riemann for the proof of circle maps in the planar case (both via potential theory and algebraic functions) emphasizing that Weber’s account is not completely faithful of the original manuscript. In contrast when based only on Weber’s account, reviewers of Riemann’s work tend to be more minimalist. E.g., Grunsky 1978 [@Grunsky_1978 p.198] writes: “[*Theorem 4.1.1. \[i.e. full covers of the disc for multi-connected domains\] goes back to Riemann, [\[423\]]{}, who gave some hints for the proof when $D$ is bounded by circles. The first proof is due to Bieberbach [\[88\](=1925 [@Bieberbach_1925])]{}, who used the Schottky-double and deep results in the theory of algebraic functions. Elementary proofs were given by Grunsky [\[195\](=1937–41); \[…\]]{}* ]{}” \[quote:Riemann\] Das Problem, die Vertheilung der statischen Electricität oder der Temperatur im stationären Zustand in unendlichen cylindrischen Leitern mit parallelen Erzeugenden zu bestimmen, vorausgesetzt, dass im ersteren Fall die vertheilenden Kräfte, im letzteren die Temperaturen der Oberfächen constant sind längs geraden Linien, die zu den Erzeugenden parallel sind, ist gelöst, so bald eine Lösung der folgenden mathematischen Aufgabe gefunden ist: In einer ebenen, zusammenhängenden, einfach ausgebreiteten, aber von beliebigen Curven begrenzten Fläche $S$ eine Funktion $u$ der rechtwinkligen Coordinaten $x,y$ so zu bestimmen, dass sie im Innern der Fläche $S$ der Differentialgleichung genügt: $$\frac{\partial^2 u}{\partial x^2} +\frac{\partial^2 u}{\partial y^2}=0$$ und an den Grenzen beliebige vorgeschriebene Werthe annimmt. \[So this is ‘just’ the first boundary value problem, alias Dirichlet problem.\] Diese Aufgabe lässt sich zunächst auf eine einfachere zurückführen: Man bestimme eine Function $\zeta=\xi + \eta i$ des complexen Arguments $z=x+iy$, welche an sämmtlichen Grenzcurven von $S$ nur reell ist, in je einem Punkt einer jeden dieser Grenzcurven unendlich von der ersten Ordnung wird, übrigens aber in der ganzen Fläche $S$ endlich und stetig bleibt. Es lässt sich von dieser Function leicht zeigen, dass sie jeden beliebigen reellen Werth auf jeder der Grenzcurven ein und nur einmal annimmt, und dass sie im Innern der Fläche $S$ jeden complexen Werth mit positiv imaginärem Theil $n$mal annimmt, wenn $n$ die Anzahl der Grenzcurven von $S$ ist, vorausgesetzt, dass bei einem positiven Umgang um eine der Grenzcurven $\zeta$ von $-\infty$ bis $+\infty$ geht. Durch diese Function erhält man auf der obern Hälfte der Ebene, welche die complexe Variable $\zeta$ repräsentirt, eine $n$fach ausgebreitete Fläche $T$, welche ein conformes Abbild der Fläche $S$ liefert, und welche durch die Linien begrenzt ist, die in den $n$ Blättern mit der reellen Axe zusammenfallen. Da die Fläche $S$ und $T$ gleich[^10] vielfach zusammenhängend sein müssen, nämlich $n$-fach, so hat $T$ in seinem Innern $2n-2$ einfache Verzweigungspunkte (vgl. Theorie der Abelschen Functionen, Art.7, S.113) und unsere Aufgabe ist zurückgeführt auf die folgende: Eine wie $T$ verzweigte Function des complexen Arguments $\zeta$ zu finden, deren reeller Theil $u$ im Innern von $T$ stetig ist und an den $n$ Begrenzungslinien beliebige vorgeschriebene Werthe hat. Kennt man nun eine wie $T$ verzweigte Function $\widetilde{\omega}=h+ig$ von $\zeta$, welche in einem beliebigen Punkt $\varepsilon$ im Innern von $T$ logarithmisch unendlich ist, deren imaginärer Theil $ig$ ausser in $\varepsilon$ in $T$ stetig ist und an der Grenze von $T$ verschwindet, so hat man nach dem Greenschen Satze (Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse Art.10. S.18f.): $$u_{\varepsilon}=-\frac{1}{2\pi} \int u \frac{\partial g}{\partial \eta} d \xi\,,$$ wo die Integration über die $n$ Begrenzungslinien von $T$ erstreckt ist. Die Function $g$ aber lässt sich auf folgende Art bestimmen. Man setze die Fläche $T$ über die ganze Ebene $\zeta$ fort, indem man auf der unteren Hälfte (wo $\zeta$ einen negativ imaginären Theil besitzt) das Spiegelbild der oberen Hälfte hinzufügt. Dadurch erhält man eine die ganze Ebene $\zeta$ $n$fach bedeckende Fläche, welche $4n-4$ einfache Verzweigungspunkte besitzt und welche sonach zu einer Klasse algebraischer Functionen gehört, für welche die Zahl $p=n-1$ ist. (Theorie der Abel’schen Functionen Art.7 und 12, S.113, 119.) Die Function $ig$ ist nun der imaginäre Theil eines Integrals dritter Gattung, dessen Unstetigkeitspunkte in dem Punkt $\varepsilon$ und in dem dazu conjugirten $\varepsilon'$ liegen, und dessen Periodicitätsmoduln sämmtlich reell sind. Eine solche Function ist bis auf eine additive Constante völlig bestimmt und unsere Aufgabe ist somit gelöst, sobald es gelungen ist, die Function $\zeta$ von $z$ zu finden. Wir werden diese letztere Aufgabe unter der Voraussetzung weiter behandeln, dass die Begrenzung von $S$ aus $n$ Kreisen gebildet ist. Es können dabei entweder sämmtliche Kreise ausser einander liegen, so dass sich die Fläche $S$ ins Unendliche erstreckt, oder es kann ein Kreis alle übrigen einschliessen, wobei $S$ endlich bleibt. Der eine Fall kann durch Abbildung mittelst reciproker Radien leicht auf den andern zurückgeführt werden. Ist die Function $\zeta$ von $z$ in $S$ bestimmt, so lässt sich dieselbe über die Begrenzung von $S$ stetig fortsetzen, dadurch dass man zu jedem Punkt von $S$ in Bezug auf jeden der Grenzkreise den harmonischen Pol nimmt und in diesem der Function $\zeta$ den conjugirt imaginären Werth ertheilt. Dadurch wird das Gebiet $S$ für die Function $\zeta$ erweitert, seine Begrenzung besteht aber wieder aus Kreisen, mit denen man ebenso verfahren kann, und diese Operation lässt sich ins Unendliche fortsetzen, wodurch das Gebiet der Function $\zeta $ mehr und mehr über die ganze $z$-Ebene ausgedehnt wird. \[…\] This last paragraph is the one where Klein identifies (by Riemann) early examples of “automorphic functions” (compare Quote \[Klein-1923:quote:Riemann-1858\]). Painlevé’s problem (Painlevé 1888, Denjoy 1909, Besicovitch, Ahlfors 1947, Ahlfors-Beurling 1950, Vitushkin, Melnikov, Garnett, Marshall, Jones, Tolsa 2003) ------------------------------------------------------------------------------------------------------------------------------------------------------------ This connection is first explored in Ahlfors 1947 [@Ahlfors_1947]. The point of departure is usually identified (modulo notorious sloppiness on finding the modern formulation) in Painlevé’s Thesis 1888 [@Painleve_1888] concerned with generalizations of Riemann’s removable singularity theorem: [*when do all bounded analytic functions defined in the vicinity of a compactum extend across the compactum?*]{} Riemann’s theorem settles removability of singletons. A necessary and sufficient condition for removability is the vanishing of a certain numerical invariant directly attached to the Ahlfors function, the so-called [*analytic capacity*]{}. This is nothing but the maximum possible distortion $\vert f'(\infty)\vert$ measured at infinity among all analytic functions defined on the complement of the compactum and bounded-by-one there. This characterization (due to Ahlfors 1947 [@Ahlfors_1947]) is not regarded as a satisfactory answer to Painlevé problem requiring a purely geometric (quasi-optical) recognition procedure of removable sets. If the compact set lies on a rectifiable curve of the plane, removability is tantamount to zero length (Denjoy’s conjecture 1909 [@Denjoy_1909-Painleve/Sur-les-fct-anal-unif-a-sing-discontinues], initially a theorem which turned out to be “gapped”, but confirmed via Calderón 1977 [@Calderon_1977] in Marshall [@Marshall_1978?]). In the general case, Vitushkin 1967 [@Vitushkin_1967] proposed a characterization via “invisible sets” (due to Besicovitch in the 1930’s), i.e. those sets having orthogonal projections of zero Lebesgue measure along almost all directions. Verdera 2004 [@Verdera_2004] explains brilliantly a metaphor with ghost objects virtually impossible to photography. Alas, Vitushkin’s expectation turned out to be not entirely correct, cf. Jones-Murai 1988 [@Jones-Murai_1988] for a counterexample, yet it gives already an approximate idea of the whole problem. For instance, the prototypical example is the [*one-quarter Cantor set*]{}: a unit-square subdivided in $4\times 4=16$ congruent subsquares whose only 4 extreme “corner-squares” are kept, with this operation iterated ad infinitum (Fig.\[Cantor:fig\]). The resulting Cantor set turns out to be removable (Garnett 1970 [@Garnett_1970]), but has positive (Hausdorff) length since its projection on the line of slope $1/2$ fills a whole interval (again Fig.\[Cantor:fig\]). Here the 1/2-slope photography of the set is Lebesgue massive (hence “visible”), yet most other projections give sets of zero measures, in accordance with the removability of the set. Of course Garnett argues differently using in particular the classic analytic theory of Ahlfors-Garabedian. The Painlevé problem engaged many investigators (Painlevé 1888 [@Painleve_1888], Denjoy 1909 [@Denjoy_1909-Painleve/Sur-les-fct-anal-unif-a-sing-discontinues], Urysohn, Besicovitch 1930’s, Ahlfors 1947 [@Ahlfors_1947], Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950], Vitushkin 1958 [@Vitushkin_1958], 1967 [@Vitushkin_1967], Garnett 1970 [@Garnett_1970], Melnikov 1967 [@Melnikov_1967], 1995 [@Melnikov_1995], Calderón 1977 [@Calderon_1977], 1978 [@Calderon_1978-ICM], G. David 1998 [@David_1998], and many others up to its ultimate solution in Tolsa 2003 [@Tolsa_2003]. This tour de force blends a vast array of technologies (Melnikov’s Menger curvature, stopping processes à la Carleson already involved in the corona, etc.) As a naive question how much of this theory extends to Riemann surfaces, using say Ahlfors 1950 [@Ahlfors_1950] instead of Ahlfors 1947 [@Ahlfors_1947]. \[06.11.12\] Of course it may be argued that most compactums of interest are phagocytable in a chart or a schlichtartig region hence planar via Koebe’s theorem. However it may seem that non-planar compactums exist as well on Riemann surfaces? What is the simplest example if any? Of course I certainly miss(ed) something trivial. A naive example is to take the $1/4$-Cantor set and project it down to the torus ${\Bbb R}^2/{\Bbb Z}^2$, but of course the latter set may be planarized again (via suitable retrosections). Classical literature discussing Painlevé’s problem includes: $\bullet$ Ahlfors 1947 [@Ahlfors_1947] $\bullet$ Garabedian 1949 [@Garabedian_1949] Type problem (Kusunoki 1952) ---------------------------- In a 1952 paper [@Kusunoki_1952], Kusunoki found a clever application of the Ahlfors function to the type of open Riemann surfaces. Beware that the type of open Riemann surfaces is here understood in the analytic sense due the Finnish school (Myrberg 1933 [@Myrberg_1933], Nevanlinna 1941 [@Nevanlinna_1941]) of having a Nullrand. More precisely, Nevanlinna 1941 () introduced a notion of surfaces with [*null-boundary*]{} (Nullrand). This amounts to exhaust the open Riemann surface by compact subregions $F_n$, while solving via Dirichlet (rescued by Schwarz, Hilbert, etc.) the boundary problem $\omega_n$ equal $0$ on $\Gamma_0=\partial F_0$ and equal to $1$ on $\Gamma_n=\partial F_n$. As the subregions $F_n$ expand to infinity, two scenarios are possible: $\bullet$ either the $\omega_n$ converges to $0$, or $\bullet$ the sequence $\omega_n$ converges to a positive harmonic function, $\omega$. In the first case, the open Riemann surface $F$ is said to have null-boundary, and in the second case to have positive boundary. Null-boundary is equivalent to having no Green’s function, or a recurrent Brownian motion. More relevant to Kusunoki’s work is Nevanlinna’s equivalent formulation in terms of the convergence to $0$ of the Dirichlet integral $d_n=D[\omega_n]$. Now, Kusunoki proves the following estimate (yielding a null-boundary criterion in case the right hand-side explodes to infinity): [(Kusunoki, 1952)]{} $$\frac{1}{2\pi \lambda_n} \log{\frac{1}{\bar{r}_n}} \le \frac{1}{d_n},$$ where $\lambda_n\le r_n+2p_n$ is the degree of an Ahlfors function $f_n\colon F_n\to D$ and $\bar{r}_n$ the maximum value of $f_n$ achieved on the Anfangsbereich $F_0$ of the exhaustion. Kusunoki’s argument does not seem to use in any fundamental way the extremal property of the Ahlfors function. Thus perhaps any circle map (of possibly lower degree, e.g. $\le r_n+p_n$ via Gabard 2006 [@Gabard_2006]) accomplishes the job as well. This option is also corroborated by the fact that Kusunoki also appeals to Bieberbach 1925 [@Bieberbach_1925] where no extremal property is put in the forefront. Accordingly there is some hope to derive a sharper Kusunoki’s estimate. Alas the magnitudes $r_n$ change during the process so the net bonus is hard to quantify. Carathéodory metric (Carathéodory 1926, Grunsky 1940, etc.) ----------------------------------------------------------- Cf. for instance Grunsky 1940 [@Grunsky_1940 p.232, §3], Burbea 1977 [@Burbea_1977-Caratheodory]. Corona (Carleson 1962, Alling 1964, Stout 1964, Hara-Nakai 1985) ----------------------------------------------------------------- In Alling 1964 [@Alling_1964], the explicit degree bound $r+2p$ of the Ahlfors map is [*not*]{} employed. In fact any “innocent” circle map (of finite degree and not necessarily solving Ahlfors’ extremal problem) suffices to transplant the truth of Carleson’ corona theorem (1962 [@Carleson_1962]) from the disc to any finite bordered Riemann surface. Assuming that Ahlfors circle mapping theorem is really involved to prove, or speculating on a very apocalyptic earthquake destroying simultaneously all the ca. 13 proofs presently available, it is still true that the Alling/Stout extension of the corona persists all such crashes. Recall that Köditz-Timmann 1975 [@Koeditz-Timmann_1975] prove existence of a circle map (via a Behnke-Stein approximation theorem) without any control on the mapping degree. This weak form of Ahlfors is enough to complete Alling’s proof. In contrast, Hara-Nakai 1985 [@Hara-Nakai_1985] exploit fully Ahlfors bound $r+2p$ for the finer [*corona problem with bound*]{}. The obvious problem is whether one can produce better corona bounds using circle maps of lowered degrees (e.g. those in Gabard 2006 [@Gabard_2006]). What probably plagues the game is that even in the disc case sharp estimation of the best corona constant is still an open difficult matter. Cf. e.g. Treil 2002 [@Treil_2002], where the best upper estimate of Uchiyama 1980 (Preprint) is supplemented by a lower bound improving one of Tolokonnikov 1981. Literature includes: $\bullet$ For the disc: Carleson 1962 [@Carleson_1962], Hörmander, Gamelin 1980 [@Gamelin_1980-Wolff's-proof] (Wolff’s proof), Garnett’s book 1981 [@Garnett_1981-BOOK], etc. $\bullet$ For bordered surfaces: Alling 1964 [@Alling_1964], Hara-Nakai 1985 [@Hara-Nakai_1985], Oh 2008 [@Oh_2008]. Quadrature domains (Aharonov-Shapiro 1976, Sakai 1982, Gustaffson 1983, Bell 2004, Yakubovich 2006) ---------------------------------------------------------------------------------------------------- This is another discipline bearing deep connections with the Ahlfors function. For instance Aharonov-Shapiro 1976 [@Aharonov-Shapiro_1976] prove that Ahlfors maps associated to quadrature domains are algebraic. Combining this with works by Gustafsson 1983 [@Gustafsson_1983-Quadrature], Bell 2005 [@Bell_2005-Quadrature-domains] arrives at the striking conclusion: “[It is proved that quadrature domains are ubiquitous in a very strong sense in the realm of smoothly bounded multiply connected domains in the plane. In fact they are so dense that one might as well assume that any given smooth domain one is dealing with is a quadrature domain, and this allows access to a host of strong conditions on the classical kernel functions associated to the domain.]{}” Compare also Yakubovich 2006 [@Yakubovich_2006], and the references therein. Wilson’s optical recognition of dividing curves (Gabard 2004) ------------------------------------------------------------- \[30.12.12\] Another highbrow (yet poorly explored) application of Ahlfors theorem was sketched in Gabard’s Thesis (2004 [@Gabard_2004 p.7]). This was an answer to Wilson’s question (1978 [@Wilson_1978 p.67]) on deciding the dividing character of a plane curve by sole inspection of its real locus. Here again Ahlfors theorem affords an answer: a real curve is dividing iff it admits a total pencil (with possibly imaginary conjugate basepoints). Yet it must be admitted that the answer, albeit perfectly geometric, has probably little algorithmic value unless complemented by further insights. Steklov eigenvalues (Fraser-Schoen 2010, Girouard-Polterovich 2012) ------------------------------------------------------------------- Compare the paper by Fraser-Schoen 2010/11 [@Fraser-Schoen_2011] where, for the first time, the Ahlfors map is applied to spectral theory (the first Steklov eingenvalue). Of course the basic trick of conformal transplantation is akin to the closed case (Yang-Yau 1980 [@Yang-Yau_1980]), yet in the bordered case it seems that the Ahlfors map respects precisely what should be, when it comes to take care of the Neumann boundary condition. In this respect the Fraser-Schoen contribution looks extremely original. Building upon a paper of Payne-Polya-Schiffer, Girouard-Polterovich 2012 [@Girouard-Polterovich_2012] are able to extend the (Fraser-Schoen) estimate to higher eigenvalues. Other (Dirichlet-Neumann) eigenvalues (Gabard 2011) --------------------------------------------------- Inspired by Fraser-Schoen exciting paper, I also tried the game with the modest arXiv note Gabard 2011 [@Gabard_2011], where the second inequality of Hersch 1970 [@Hersch_1970] is adapted to configurations of higher topological structure. Note that the other two remaining inequalities of Hersch are probably likewise extensible (involving the quadrant and octant of a sphere). Klein’s intuition (Klein 1876, Marin 1988, Viro 2013, Gabard 2013) ------------------------------------------------------------------ Another little application of the Ahlfors map can be given to Klein’s intuition that a orthosymmetric (i.e. dividing or of type I) curve in the plane cannot acquire a solitary double point by progressive variation of its coefficients. This goes back to Klein 1876, and was probably justified by several workers though in a somewhat different shape from this original statement (e.g. Marin 1988 [@Marin_1988], and based upon him Viro 1986/86 [@Viro_1986/86-Progress]). For a clear-cut arguement using the deep Ahlfors theorem, cf. our Lemma \[Klein-via-Ahlfors(Viro-Gabard):lem\] below, which was essentially suggested to me by Viro (though in the modern Marin-Viro formulation differing somewhat from Klein’s original assertion). Our lemma below (\[Klein-via-Ahlfors(Viro-Gabard):lem\]) is however exactly Klein’s assertion, though proved by the device of Ahlfors maps. Even if Teichmüller 1941 [@Teichmueller_1941] should be right by ascribing to Klein the existence of Ahlfors circle maps, it is quite unlikely that Klein disposed of this as early as 1876 (the critical range being rather ca. 1882 right before the psychological collapse of Klein due to overwork). Accordingly our (Viro inspired) proof of Klein’s assertion via Ahlfors might be a bit too eclectic, yet it is quite hygienical while requiring little topological concentration. Eclectic applications of the Ahlfors map {#Sec:Virtual-applications-Ahlf-map} ======================================== Those are only oneiric applications of Ahlfors maps, i.e. topics bearing only vague analogy to our main focus. Filling area conjecture (Loewner 1949, Pu 1952, Gromov 1983) ------------------------------------------------------------ This was already discussed in the Introduction. One may wonder whether the FAC is also meaningful (and true) for non-orientable membranes. It seems so, imagine, e.g, a hemisphere surmounted by a microscopic cross-cap over a “glass-of-wine shaped” protuberance at the north pole (Fig.\[Wineglass:fig\]). This membranes satisfy FAC, for it effects no shortening of the intrinsic distance of the circle while having an area slightly larger than that of the hemisphere. One (possibly more accessible) question is whether the FAC holds true for membrane having the topological structure of a Möbius band (equivalently a disc with a single cross-cap). This case in view of simplicity of the topological structure is perhaps already known, or at least accessible via the traditional methods of Loewner-Pu, etc. (Alas, I am not aware of a specific reference.) Another option is to generalize Gromov’s problem to membranes filling several contours (as suggested by J. Huisman ca. Sept. 2011). Arguably, a disjoint union of hemispheres is the best filling, at least when the contours are completely insulated (at infinite distance). Perhaps specifying some finite distance-functions $\rho_{i,j}$ between each pair of circles one can expect a least-area connected filling (without shortenings), but I have presently no clear view on how to pose properly such generalized problems. Open Riemann surfaces embed in ${\Bbb C}^2$ (Narasimhan, Gromov, Slovenian school, etc.) {#Open-RS-embed-in-C2:sec} ---------------------------------------------------------------------------------------- The Slovenian school of complex geometry (Černe, Forstnerič, Globevnik, etc.) are also frequently employing the Ahlfors map. One among the most notorious elusive open problem (Narasimhan, Gromov, Forstnerič, Wold, etc.) is: \[Narasimhan:conj\] Any open Riemann surface embeds properly in ${\Bbb C}^2$ (equivalently such that the image is a closed set). In Forstnerič-Wold 2009 [@Forstneric-Wold_2009], the full problem (or at least the case of interiors of finite Riemann surfaces) is reduced to the following finitary version, seemingly much more accessible, yet apparently still out of reach: [(FW2009)]{} \[FW2009:conj\] Each compact bordered Riemann surface $F$ embeds holomorphically in the plane ${\Bbb C}^2$. \[06.11.12 (based on ideas of ca. Sept. 2011)\] Such an embedding is possible whenever the corresponding real curve $C$ (namely the Schottky double of $F$) admits a [*totally real pencil of lines*]{}. This is for instance the case for Klein’s Gürtelkurve (any real plane quartic with 2 nested ovals). Fig.\[Coppens:fig\] below provides plenty of other baby examples (alas most of them being only immersed). Indeed in this situation (total pencil of lines) the corresponding projection is totally real and the allied morphism $C\to {\Bbb P}^1$ induces a continuous map between the imaginary loci, i.e. $C({\Bbb C})-C({\Bbb R})\to {\Bbb P}^1({\Bbb C})-{\Bbb P}^1({\Bbb R})$. It follows that an imaginary line of the pencil cuts the curve $C$ [*unilaterally*]{} (i.e. only along one half of the orthosymmetric Riemann surface). Removing such an imaginary line from ${\Bbb P}^2({\Bbb C})$ leaves a replica of ${\Bbb C}^2$ containing entirely the original bordered surface $F$. This simple method fails miles-away from the full Forstnerič-Wold desideratum. Indeed Ahlfors theorem (1950 [@Ahlfors_1950]) only implies existence of a totally real pencil but a priori involving auxiliary curves of order higher than one. On the other hand when starting from the abstract bordered surface (and its double) we may have first a projective model in ${\Bbb P}^3$, which projected down to the plane ${\Bbb P}^2$ may develop singularities. Hence the model in question is only immersed in general. Our naive approach only helps grasping the notorious difficulty of the question, yet still permits to settle a limited collection of special cases. Actually the method, requiring a totally real pencil of lines, applies only to real dividing smooth curves of order $m=2k$ having a deep nest of profundity $k$ (that is, higher order avatars of the Gürtelkurve). Another classical idea was to exhibit the required embedding $F\hookrightarrow {\Bbb C}^2$ via a suitable pair of Ahlfors circle maps (not necessarily extremals). This works in special cases, e.g. hyperelliptic configurations; see Černe-Forstnerič 2002 [@Cerne-Forstneric_2002], and also the related paper Rudin 1969 [@Rudin_1969]. Maybe more sophisticated variants of Ahlfors maps arising in the broader Pick-Nevanlinna context could do the job, but this looks extremely delicate. Another natural strategy is to embed one representant in each topological type (this is actually possible by Černe-Forstnerič 2002 [@Cerne-Forstneric_2002 Theorem 1.1]), while trying to use a continuity argument inside Teichmüller (moduli) space as suggested in Forstnerič-Wold 2009 [@Forstneric-Wold_2009]. Naive approaches to the Forstnerič-Wold question ------------------------------------------------ This section tries (unsuccessfully) to connect some highbrow geometry on the isometric resp. conformal embedding problem with the FW-desideratum of the previous section. Available are some rather formidable weapons cooked resp. by Nash-Kuiper-Gromov and Teichmüller-Garsia-Rüedy-Ko, which alas lack some rigid analytic character upon the image model as to assess anything like the FW-conjecture. Of course the conformal embedding technique is most likely to pierce the hearth of the FW-problem, yet the merely smooth character of the conformal model hinders realizability as a holomorphic curve. However it is still conceivable that a suitable tour de force, somewhat akin to Garsia’s (1962/63 [@Garsia_1962/63-algebraic-surfaces]) conformal realizability as a real algebraic surface in $E^3$, is able to unlock the secret of the FW-problem. \[06.11.12\] Another little puzzle is whether there is a connection with (Gromov’s and probably others) question as to whether [*any Riemannian surface embeds isometrically in Euclidean $4$-space $E^4$*]{}. (Compare Gromov 1999 [@Gromov_1999] delightful preprint “Spaces and questions”, note yet the article Gromov-Rohlin 1969 [@Gromov-Rohlin_1969] where the (real) projective plane with its round “elliptic” geometry is shown to lack such an embedding.) Thus orientability is required. Via a bordered version, we can probably embed our Riemann surface $F$ (equipped with a conformal Riemannian metric) in $E^4$ isometrically hence conformally. Via hasty thinking, the FW-desideratum (\[FW2009:conj\]) follows, but alas it does not due to the lacking rigid analytic nature of the image-model. \[16.11.12\] A more rigid constant curvature version of Gromov’s isometric embedding conjecture would be actually sufficient: \[Space-form-embedding:conj\] [(Space-forms embedding)]{} Any orientable bordered Riemannian surface of constant Gaussian curvature $K\equiv -1$ (and totally geodesic boundary) isometrically embeds in $E^4$. When combined with the uniformization theorem, one should be able to deduce the FW2009 conjecture (but again this is illusory unless one is able to ensure complex analyticity of the image). The space-form embedding [(\[Space-form-embedding:conj\])]{} implies the [FW2009]{} conjecture (and perhaps the full proper embedding problem [(\[Narasimhan:conj\])]{} via an exhaustion trick). Given the bordered Riemann surface, we take its double $2F$, which is acted upon by a canonic involution $\sigma$. On this closed Riemann surface, there is by the uniformization theorem (Poincaré–Koebe 1907 [@Poincare_1907], [@Koebe_1907_UbaK1]) a conformal hyperbolic metric (whenever $\chi (F)=\frac{1}{2} \chi(2F)<0$) and the involution $\sigma$ becomes isometric. (This equivariant uniformization is due to Koebe 1907 [@Koebe_1907_UrAK], cf. also Jost 1985 [@Jost_1985] for another approach via Plateau.) It follows that the boundary of $\partial F$ are geodesics. On applying the space-form embedding (\[Space-form-embedding:conj\]) to $F$ we get the FW2009 desideratum. Naively it seems that bordered hyperbolic space-forms already embed isometrically in $E^3$, cf. Fig.\[SpaceForms:fig\] for some qualitative pictures. Of course finding a hyperbolic model for a membrane of type $(r,p)=(1,1)$ is more tricky to visualize. However on tessellating the hyperbolic pants one would (under suitable junctures) get probably trouble with the Cebyshev-Hilbert obstruction to realizing the hyperbolic geometry in 3-space. So maybe one must still accept variable (negative) curvature. More flexible and suited to the problem at hand is the theory of Teichmüller-Loewner-Garsia-Rüedy realizing in the vicinity of any smoothly embedded closed surface in $E^3$ any conformal type of Riemann surface having the same topology via normal deformations. In particular: \[Garsia’s-thm\] [(Garsia 1961 [@Garsia_1961])]{} Any closed Riemann surface embeds conformally in Euclidean $3$-space $E^3$. (The image model can also be made real-algebraic by techniques à la Nash, etc., cf. [Garsia 1962/63 [@Garsia_1962/63-algebraic-surfaces]]{}.) (Rüedy 1971 [@Ruedy_1971] extended the result to open Riemann surfaces, and may also have contributed to the embedded version of Garsia, if the latter only showed an immersion, at least so is claimed in Ko 1993 [@Ko_1993-finite-type], yet not so in the papers by Rüedy.) A direct bordered version of Garsia’s result is the following: \[Conformal-embeddings:conj\] [(Conformal embeddings)]{} Any compact bordered Riemann surface embeds conformally in $E^3$. This conjecture would still be sufficient to answer the FW2009 conjecture. In fact this last conjecture (\[Conformal-embeddings:conj\]) may appear as a direct consequence of Garsia’s theorem upon taking the Schottky double: Any compact bordered Riemann surface embeds conformally in $E^3$. Let $F$ be the given surface. Take its (Schottky) double, to get the closed Riemann surface $2F$. By Garsia’s theorem (\[Garsia’s-thm\]) $2F$ is conformally diffeomorphic to a classical surface in $E^3$. We conclude by restricting this embedding to the original half of the orthosymmetric Riemann surface $2F$. The Garsia-Rüedy theorem climaxes the Riemann-Prym-Klein conception of the Riemann surface seen as a classic (Euclid-Gauss) differential-geometric curved surface in $3$-space (compare the introduction of Klein 1882 [@Klein_1882], equivalently Quote \[quote:Klein-Prym\]). Next there is a series of papers by Ko starting with his Thesis in 1989 where the Garsia-Rüedy conformal embedding is extended by trading ambient 3-space by an arbitrary preassigned Riemannian manifold of dim $\ge 3$. Specifically, he obtains the following results: [(Ko 1989, 1991, 1999, 2001)]{} Given any ambient orientable Riemannian manifold $\frak M$ of dimension $\ge 3$, then any Riemann surface $F$ embeds conformally in $\frak M$ provided: [(1)]{} $F$ is compact(=closed) [(Ko’s Thesis 1989 [@Ko_1989-compact] reissued as Ko 2001 [@Ko_2001])]{}. [(2)]{} $F$ has finite topological type, i.e. $\pi_1$ is of finite generation or equivalently homeomorphic to a finitely many punctured closed surface [(Ko 1993 [@Ko_1993-finite-type])]{}; [(3)]{} [nothing!]{}, i.e. $F$ is a completely arbitrary open surface [(Ko 1999 [@Ko_1999-open])]{}. Specializing (1) to $\frak M=E^4$ seems to approach the desideratum of FW2009. However there is a serious plague, for when applied to a closed surface we get a conformal embedding in ${\Bbb R}^4={\Bbb C}^2$, while complex-analyticity of the image is inhibited by the lack of non-constant bounded analytic functions on closed Riemann surfaces. Again the whole point is that the conformal model (of the Garsia-Rüedy-Ko=GRK theory) are only smooth $C^{\infty}$-surfaces and not holomorphic curves in ${\Bbb C}^2$. While it is impossible to ensure complex-analyticity in the closed case, there is no evident obstruction in the (compact) bordered realm, except that the maximum modulus of any linear projection on a complex line must take its maximum modulus on the boundary. \[27.12.12\] Assuming $F$ holomorphically embedded in ${\Bbb C}^2$, we get a family of holomorphic maps $\pi_t\colon F\to {\Bbb C}_t$ parameterized by the Riemann sphere of all (complex) lines through the origin. Since $F$ is compact bordered each such holomorphic mapping has compact range with maximum modulus reached on a boundary point. Of course the image is a priori not a disc (in which case we would have a circle map), but some more complicated shadow of the Riemann surface $F$. Concentrating much on this geometry it may be hoped that for some $F$ (alas not all remind the half of the Gürtelkurve) some obstruction is detected and FW is false. Alternatively effecting a linear projection on the Riemann sphere of all lines through a point $p=(x,y)\in{\Bbb C}^2$ gives the family of projections $\lambda_p\colon F \to {\Bbb P}^1_p$, where the latter symbol is the pencil of lines through $p$. Using translation in ${\Bbb C}^2$ all such pencils identify to ${\Bbb P}^1_0$, where $0=(0,0)$ is the origin, and we get $\Lambda_p\colon F \to {\Bbb P}^1_0={\Bbb P}^1({\Bbb C})$ a holomorphic family parameterized by $p\in{\Bbb C}^2$. Again some obstruction could occur here, but it is hard to capture. Last one could look at the tangent line assignment yielding a sort of Gauss mapping $F\to {\Bbb P}^1({\Bbb C})$ and explore its geometry in the hope of detecting some fine geometric obstruction. All this is merely canary singing without tangible grounding. In the other optimistic direction it can be hoped that a much boosted version of the Garsia-Rüedy-Ko theory proves the FW-conjecture. \[19.12.12\] First we know (from Černe-Forstnerič 2002 [@Cerne-Forstneric_2002]) that any topological type of finite bordered surface contains a representative holomorphically embedded in ${\Bbb C}^2$. Applying the high-dimensional version of Garsia (due to Ko 1989 [@Ko_1989-compact], plus subsequent articles) we can realize all Riemann surfaces within a normal tubular neighborhood via an (infimal) normal variation. This is akin to a cellulite bubbling, alas destroying a priori the holomorphic character of the initial model. However it is not to be excluded that better controlled vibrations of the pudding permit to explore the full moduli space. This would assess the full Forstnerič-Wold conjecture. Of course what we are saying here is nothing new that was not already said in FW2009, and one requires serious new idea to make progresses. A naive idea would be to take a holomorphic tube around the bordered surface $F\subset {\Bbb C^2}$, i.e. a neighborhood $N$ of $F$ together with a framing, i.e. a biholomorphic trivialization $N \to F\times \Delta$ where $\Delta$ is the unit disc. (I hope that a such exists but I am not sure!) One may also suppose with a compact tube involving the closed disc $\overline{\Delta}$, so $t\colon \overline N \to F\times \overline \Delta$. Via this trivialization any holomorphic normal variation amounts to a circle map, provided the amplitude of the variation is maximum along the boundary. Conversely given a circle map $f\colon F \to \overline \Delta$ we construct a holomorphic deformation by considering the image under the map $$F \buildrel{id\times f}\over{\longrightarrow} F \times \overline \Delta \buildrel{t^{-1}}\over{\to} \overline N.$$ However the image curve is biholomorphic to the original $F$ and our variation is trivial. Perhaps one should use quasi-conformal avatars of circle maps (QCM for short) to perform a genuine variation of the complex structure. Such maps clearly exist, and we get so perhaps a tangible strategy toward Forstnerič-Wold 2009. Note however that the normal variation effected by a QCM destroys the analytic character of the image. On the other hand it is not essential to work with circle maps to get normal deformations so perhaps there is some freedom to be gained here. As another vague idea on how to construct the required trivialization of the normal bundle (or thickening $N$) one could imagine that it is fixed in the smooth category and then sliced by the normal 2-planes orthogonal to $F$. Each slice would be essentially a simply-connected domain and one would construct the trivialization by a version of the Riemann mapping theorem with parameters. This looks dubious but maybe leads somewhere? It is likely now that the trivialization $t$ is not holomorphic globally but only so in restriction to each slice (=fibre of the normal bundle). This is good for varying moduli, but of course disrupting the holomorphic character of the deformation. Let us try to summarize a naive strategy toward FW2009: [*Step 1*]{}.—Fix a semi-holomorphic trivialization $t$ of the normal neighborhood of $F$ in ${\Bbb C}^2$. (Seems accessible via a Riemann mapping theorem with parameters). [*Step 2*]{}.—Among all normal deformations (cellulite bubbling) described by $C^\infty$ maps $f$ to the disc $\overline \Delta$ those inducing “rigid” analytic curves under the above displayed map. [*Step 3*]{}.—By an avatar of the Teichmüller-Garsia-Rüedy technique try to calculate the moduli of the resulting deformations. Ko’s theorem ensures that all moduli are realized via soft $C^{\infty}$ deformations, but what happens when we restrict to the class of rigid perturbations? Pick-Nevanlinna interpolation ----------------------------- Compare the paper Jenkins-Suita 1979 [@Jenkins-Suita_1979]. Klein-Rohlin maximality conjecture(s) (Gabard 2013) --------------------------------------------------- \[11.01.13\] The first paper were the notion of dividing curves appeared (namely Klein 1876 [@Klein_1876]) is concluded by some cryptical allusions which Klein might have derived from experimental data or by a theoretical argument involving his deep geometric intuition of Riemann surfaces. Those intuitions were nearly forgotten for ca. 102 years until Rohlin picked them up again in his seminal paper 1978 [@Rohlin_1978] enriching the complete solution ca. 1969 by Gudkov’s of Hilbert’s 16th problem for sextics by the data of complex characteristics, i.e. Klein’s types I/II (erster und zweiter Art). This Rohlin achievement was in good part made possible by Arnold 1971 [@Arnold_1971/72] breakthrough of filling the half of an $M$-curves (or more generally) one of type I by discs bounding the ovals. Once this is in place Klein’s assertion or intuition found quite spectacular evidence and were somehow distorted by Rohlin in a related but somewhat more Hilbertian and grandiose conjecture: “a real scheme is of type I iff it is maximal”. One of the application was apparently destroyed in Shustin 1985/85 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin], yet the direct half “type I$\Rightarrow$maximal” could still be true. A somewhat elusive desideratum would be that this reputed difficult conjecture of Rohlin follows from Ahlfors theorem. More about this in Sec.\[Klein-Rohlin-conj:sec\] below. Starting from zero knowledge {#Sec:Starting-from-zero} ============================ As yet the text was mostly historiographical, but from now on our intention is to elevate to the higher sphere of complete mathematical arguments. (Of course the title of this section is borrowed from a joke of academician V.I. Arnold.) The Harnack-maximal case (Enriques-Chisini 1915, Bieberbach 1925, Wirtinger 1942, Huisman 2001) ----------------------------------------------------------------------------------------------- The theorem of Ahlfors (existence of circle maps) is easier in the planar case (and due in this case to Riemann-Schottky-Bieberbach-Grunsky, etc.). Using the corresponding Schottky double which is a real curve (of Harnack-maximal type), the assertion follows quite immediately from Riemann-Roch (Riemann’s inequality) via a simple continuity argument. This argument is implicit in Enriques-Chisini 1915 [@Enriques-Chisini_1915-1918] (perhaps even in Riemann 1857/58 manuscript [@Riemann_1857]), and was then rediscovered by many peoples including Bieberbach 1925 [@Bieberbach_1925], Wirtinger 1942 [@Wirtinger_1942], Johannes Huisman 1999/01 [@Huisman_2001], and myself Gabard 2006 [@Gabard_2006]. The nomenclature [*Bieberbach-Grunsky theorem*]{} used say by much of the Japanese school (e.g. A. Mori 1951 [@Mori_1951], etc.) is thus slightly in jeopardy. \[Enriques-Chisini:lemma\] [(Riemann 1857/8, Schottky 1875/77 ($\pm$), Enriques-Chisini 1915 ($\pm$), Bieberbach 1925, Wirtinger 1942, etc.)]{} Any planar bordered Riemann surface with $r$ contours has a circle map of degree $r$. Moreover the fibre over a boundary point may be prescribed as any collection of points having one point on each contour. Double the surface to get a closed one of genus $g=r-1$. On the corresponding Harnack-maximal curve (i.e. $r=g+1$), pick one point $p_i$ on each oval to get a divisor $D_0$ of degree $g+1$. Riemann’s inequality states $\dim \vert D \vert \ge d-g$, where $\vert D \vert$ is the complete linear system spanned by the divisor $D$ and $d$ is its degree. (This is Riemann-Roch without Roch, and follows easily from Abel’s theorem.) It follows that the divisor $D_0=\sum_{i=1}^{g+1} p_i $, moves in its linear equivalence class. We may thus choose in the linear system $\vert D_0 \vert$ a line (classically denoted) $g^1_d$, consisting of groups of $d=g+1$ points. Subtracting eventual basepoints, this $g^1_\delta$ ($\delta\le d$) induces a totally real morphism to ${\Bbb P}^1$, since by continuity the points $p_i$ cannot escape their respective ovals. Indeed looking at Fig.\[Enriques:fig\] while imagining one point evading the real locus $C({\Bbb R})$ another one must instantaneously jump to locate himself symmetrically w.r.t. the (Galois-Klein) symmetry $\sigma$ induced by complex conjugation. Since a totally real morphism has degree $\ge r$, the final degree $\delta$ must be $g+1=r$. -35pt 0 Gabard’s argument: circle maps of $\deg\le r+p$ {#sec:Sketch-of-Gabard} ----------------------------------------------- The basic principle used in Gabard 2006 [@Gabard_2006] to prove existence of circle maps is some topological stability of the embedding of a closed Riemann surface into its Jacobian via the Abel map, which is quite insensitive to variations of the complex structure. This is how we derived universal existence theorem valid for all Riemann surfaces with upper control on the degree of such maps. We suspect that the same method (suitably adapted to closed surfaces) enables one to recover the Riemann-Meis bound $[\frac{g+3}{2}]$ for the minimal sheet number concretizing a genus $g$ curve as a branched cover of the line ${\Bbb P}^1({\Bbb C})$ (cf. Riemann 1857 [@Riemann_1857] and Meis 1960 [@Meis_1960]). Yet we failed presently to write down the details. \[22.10.12\] Let us sketch rapidly the argument in Gabard 2006 [@Gabard_2006], to which we refer for more details. \[Gabard:thm\] Any bordered surface $\overline{F}=\overline{F}_{r,p}$ of type $(r,p)$ supports a circle map of degree $\le r+p$. Using the Schottky double $C=2 \overline{F}$, it is enough locating an unilateral divisor $D$ (i.e. one supported in the interior denoted $F$) linearly equivalent to its conjugate $D^{\sigma}$. By a simple continuity argument the pencil spanned by the pair $D, D^{\sigma}$ is totally real, hence induces a circle map; compare Lemme 5.2 in Gabard 2006 [@Gabard_2006] which we reproduce now: The morphism induced by a pencil spanned by an unilateral pair of linearly equivalent divisors $D, D^{\sigma}$ is totally real. Consider $g^1_d$ the linear series spanned by $D, D^{\sigma}$. It is readily verified to be real. To check total reality imagine $D$ degenerating toward a point $D_0$ on the equator $g^1_d({\Bbb R})$ of the pencil $g^1_d$ (cf. Fig.\[Continuity-Gab:fig\]). -5pt0 As long as $D$ stays imaginary it cannot acquire a real point (else as the morphism induced by $g^1_d$ is real it would have a real image). Therefore $D$ is so-to-speak magnetically confined to the original half, hence itself unilateral. Yet when $D$ becomes real it corresponds to a symmetric divisor (invariant under the involution $\sigma$), which must be the limit of unilateral divisors. The only possibility is for $D_0$ to be totally real. Since in a sphere, any point of the equator is accessible from the north pole, it follows that $D_0$ is always totally real. This completes the proof. The task is thus reduced to exhibit an unilateral divisor such that $D\sim D^{\sigma}$ (linear equivalence on the curve $C$). Using Abel’s map $\alpha\colon C \to J$ to the Jacobian (variety) this amounts to say that $\alpha(D)$ is a real point of the Jacobian. Looking in the quotient $J/J({\Bbb R})$ this amounts to express zero as a sum of unilateral points. Taking any point $x_d$ in $F$, we search points $x_i\in F$ so that $$x_1+\dots +x_{d-1}=-x_d.$$ To solve this equation we use a principle of topological irrigation (subsumable to Brouwer’s theory of the mapping degree), but whose essence lies in the periodic behavior of the Abel map. Specifically, we know that $\alpha$ induces an isomorphism on the first homology. In a similar way (cf. Fig.\[Orthosym-basis:fig\]), the $r-1$ semi-cycles $\beta_1^+, \dots, \beta_{r-1}^+$ (linking one contour to the others) and the $2p$ cycles $\widetilde{\alpha}_1, \dots, \widetilde{\alpha}_p,\widetilde{\beta}_1, \dots, \widetilde{\beta}_p$ winding around the $p$ handles form a basis of the first homology of the quotient $T^g:=J/J({\Bbb R})$, a $g$-dimensional (real) torus. Note that the extremity of the semi-cycles $\beta_i^+$ are pasted together when passing to the quotient. -5pt0 The irrigation principle says that if we have $g$ cycles representing a basis of the 1st-homology of a $g$-dimensional torus $T^g$ then any point of the torus is expressible as the sum of at most $g$ points situated on the given cycles. Applying this, we can solve the above equation for $d-1\le (r-1)+2p$, i.e. $d\le r+2p$ recovering Ahlfors bound. Since we are presently unable to reprove Ahlfors theorem via his original argument, let us state this as an independent theorem. (Note at the didactic level that our proof merely use Abel 1826 [@Abel_1826], and perhaps some Riemann in as much as we use that a curve of genus $g$ supports $g$ many holomorphic $1$-forms involved in the definition of the Abel map, yet nothing like say Green 1828 [@Green_1828] which is pivotal in Ahlfors’ implementation, although this is only stressed subconsciously.) [(Ahlfors 1950, via Gabard’s method)]{} \[Ahlfors-via-Gabard:thm\] Any bordered surface $\overline{F}=\overline{F}_{r,p}$ of type $(r,p)$ supports a circle map of degree $\le r+2p$. Now our points $x_i$ are situated on curves traced in advance around the handles. This constraint is not inherent to our problem, where only unilaterality is required. Thus the points enjoy more freedom and this is how we discovered (ca. 2002) the possibility of improving Ahlfors. More formally, we can imagine instead of the two cycles $\widetilde{\alpha}_i,\widetilde{\beta}_i$ winding around a handle a 2-cycle $\widetilde{\alpha}_i \star\widetilde{\beta}_i$ having the shape of a 2-torus. The latter torus is not traced on our surface $F$, but a vanishing cycle operation makes the torus visible. This torus is interpreted as a cycle with stronger irrigating power. Summarizing, we have in the quotient $T^g$ the $(r-1)$ semi-cycles and $p$ many 2-tori of stronger irrigating power. An (evident) variant of the irrigation principle gives solubility of the above equation for $d-1\le (r-1)+p$, i.e. $d\le r+p$ (Gabard’s bound). [*Warning.*]{}—\[06.11.12\] Presenting the full details in some less intuitive manner occupies the last 7 pages of Gabard 2006 (). It is hoped that the $r+p$ result is correct, but it should not be excluded that something wrong happened (or at least that the proof is not convincing enough). Thus more investigations require to be made to assess or disprove the above theorem. Of course the first part, where only Ahlfors’ bound $r+2p$ is recovered, seems less subjected to “corrosion”, because the irrigating cycles are readily traced on the bordered surface (without appeal to vanishing cycles, homologies, etc.). Assigning zeroes and the gonality sequence {#sec:gonality-sequence} ------------------------------------------ \[22.10.12\] Here we explore some little new ideas inspired by the irrigating method discussed in the previous section. Alas, details are a bit messy (mostly due to severe degradations of the little I knew about algebraic curves). Most propositions of this section suffer the plague of hypothetical character. We hope that, despite vagueness of conclusions, the thematic addressed is worth clarifying. A general question of some interest is that of calculating for a given bordered surface the list of all integers arising as degrees of circle maps tolerated by the given surface. We call this invariant the [*gonality sequence*]{}. Another noteworthy issue is that apparently Ahlfors’ upper bound $r+2p$ is always effectively realized, in sharp contrast to Gabard’s one $r+p$ which can fail to be. In the above argument (proof of (\[Gabard:thm\])) we may replace the point $x_d\in F$ by a collection of $k$ points say $z_1, \dots, z_k\in F$. By the irrigation principle it is still possible to solve the following equation in the group $T^g=J/J({\Bbb R})$ $$x_1+\dots +x_{d-1}=-(z_1+ \dots+ z_k)$$ for $d-1\le (r-1)+p$. Alas, if the divisor $z_1+ \dots+ z_k$ is linearly equivalent to its conjugate the right hand side vanishes in $T^g$, and all $x_i$ could lye on the boundary of the semi-cycle (violating the unilaterality condition). However, in this case there is a circle map of degree $k$ exactly given by the divisor $D=\sum_{i=1}^k z_i$. Thus, we can still conclude the following: (Circle maps with assigned zeroes) Given any collection $z_i$ of $k$ points in a bordered surface $\overline{F}$ of type $(r,p)$ there is a circle map of degree $\le (r-1)+p+k$ vanishing on the assigned points $z_i$. It must just be observed that the pencil through $D,D^{\sigma}$, where $D=x_1+\dots +x_{d-1}+(z_1+ \dots+ z_k)$ is basepoint free due to the unilaterality of this divisor. (This holds true even if some of the $x_i$ or $z_i$ come to coincide.) It seems even that there exists circles maps of any degree $d\ge r+p$, but I am not sure about this point. Checking the truth of this requires the assertion that any point in the torus is expressible as the sum of the exact number of cycles available in the irrigating system. At first glance, this looks untrue in the trivial irrigating system for the flat 2-torus ${\Bbb R}^2/{\Bbb Z}^2$ consisting of the 2 factors. Yet the origin may be redundantly expressed as sum of two points. Idem for a point on the vertical axis, there is an expression as that point plus the origin. So maybe it works. The general (hypothetical) statement would be: (Hypothetical lemma=Sharp irrigation principle) Given cycles $\gamma_1, \dots, \gamma_k$ of dimensions (say one and two, yet this is certainly not essential) in a $g$-torus $T^g$ such that their Pontrjagin product $\gamma_1 \star \dots \star \gamma_k$ represents the fundamental class of $T^g$. Then any point of $T^g$ is expressible as the sum of $k$ points $x_i$, one situated on each $\gamma_i$. (Some $x_i$ may coincide.) Granting this we seem to get a sharpener version of the previous proposition. (Very hypothetical!!!) \[hypothetical:prop\] Given any collection $z_i$ of $k$ points in a bordered surface $F$ of type $(r,p)$ there is a circle map of degree exactly $(r-1)+p+k$ vanishing on the assigned points $z_i$. In particular there exists circles maps of any degree $d\ge r+p$. In fact the real problem is that our irrigating system involves the $r$ semi-cycles on $F$ (which close up into $J/J({\Bbb R})$). If the sum involves points located on the boundary of those semi-cycles, then those points must be discarded to ensure unilaterality of the divisor. Thus our method gives only an upper bound on the degree of the final map, but never an exact control. Basic examples show that special Riemann surfaces may well admit circle maps of degree $d<r+p$ (cf. e.g. Fig.\[Chambery:fig\]). The [*gonality*]{} $\gamma=\gamma(F)$ of a compact bordered Riemann surface $F$ is the least possible degree of a circle map tolerated by $F$. Evidently $r\le \gamma\le r+p$ (the second estimate being Gabard’s claim). One can ask if each value $d\ge \gamma$ above the gonality occurs as the degree of a circle map. Alas, the above irrigation technique fails close to imply this. Our guess is that the response is in the negative, that is, there may be “gaps” in the sequence of all circle mapping degrees. Thus to detect a gap it is natural to look among “special” surfaces of small gonality in comparison to its generic value $r+p$. A rapid glance at the combinatorics of Fig.\[Coppens:fig\] (below) helps us identify the simplest such example as a hyperelliptic surface with $(r,p)=(2,1)$. Then $\gamma=2<r+p=3$. Borrowing an idea of Klein, we can think of the corresponding real curve as a doubled conic. This occurs actually via the so-called canonical mapping (of algebraic geometry) which fails injectiveness for hyperelliptic curves. (Note: we switch constantly from bordered surfaces to real dividing curves, committing oft slight abuses of language.) Klein regards this doubled conic as a degeneration of the general Gürtelkurve (with two nested ovals) when both of them come to coalesce. This projective model of the hyperelliptic surface suggests that when projected from the doubled curve it has degree 2, but if the center of projection moves in the inside of the conic then the projection acquires degree 4 suddenly, without visiting the value 3. However substituting to the bordered surface this double conic is a bit fraudulent, e.g. because the latter is reducible and correspond rather to a disconnected Riemann surface. Also the doubled conic looks 2-gonal in $\infty^1$ ways whereas the original surface is uniquely 2-gonal. Thus another more reliable argument requires to be given. (This must probably be akin to the lemma proving uniqueness of the hyperelliptic involution.) ($\bigstar$) If I remember well there is a lemma saying that any basepoint free pencil $g^1_d$ on a hyperelliptic curve is composed with the hyperelliptic involution $g^1_2$. In more concrete words, any morphism to the line factors through the hyperelliptic projection, and so has even degree. If this is correct, Prop.\[hypothetical:prop\] is corrupted since the gonality sequence is exactly the set of even integers $2{\Bbb N}$. This remark would equally apply to any hyperelliptic membrane with $r=1$ or $2$, $p$ arbitrary. However this conclusion conflicts with the Černe-Forstnerič claim (cf. 2002 [@Cerne-Forstneric_2002]) that Ahlfors proved any surface to exhibit a circle map of degree $r+2p$ exactly (take $r$ odd equal to 1). Of course the mistake is mine and to be found in the parag. ($\bigstar$) right above, as shown by the following example. (Convention) Below and in the sequel, we shall often say just total morphism instead of totally real morphism. [**Example 1.**]{} Consider a quartic with one node (so of genus $g=2$). This is hyperelliptic (alias 2-gonal) when projected from the node. However the curve also admits maps to the line of degree 3 (projection from a smooth point). Manufacturing a real picture gives the picture nicknamed 112 on Fig.\[F112:fig\] deduced via sense-preserving smoothings of both ellipses (ensuring the dividing character of the resulting curve by Fiedler 1981 [@Fiedler_1981]). The dashed circle indicates the node left unsmoothed. To avoid any mysticism, our nicknaming coding consists in writing the 3 invariants $r,p, \gamma$ as the string $rp\gamma$. -5pt0 Picture 112 shows total morphisms (i.e. with totally real fibers over real points) of degree 2 (projection from the node), of degree 3 (projection from the inner loop) and of degree 4 (projection from inside the inner loop). One would like to know if $5$ is also the degree of a circle map, etc. We believe the answer to be positive: Ahlfors exhibits circle maps of degree exactly $r+2p$, and of all higher values too (as follows from his convexity argument). Hence the gonality sequence seems to be $\gamma=2=r+p, 3=r+2p, \dots$, where the dots mean all subsequent gonalities do occur after Ahlfors bound. This being a bit notationally messy, we introduce a graphical punching-card system on the figure, where the gonality sequence 234…is decorated by a triangle for $r+p$, an underlining of Ahlfors’ bound $r+2p$ (after which the gonality sequence is full), and the arrow indicating the least position from whereon the sequence is full. The given example does [*not*]{} confirm our initial guess about gaps in the gonality sequence, so let us examine another example. [**Example 2.**]{} Consider a hyperelliptic model of type $(r,p)=(2,1)$. Then the genus $g$ of the double is $g=(r-1)+2p=1+2=3$. This prompts looking at plane smooth quartics having the right genus $3$, but alas the wrong gonality 3 (not 2). Thus we move to quintics (“virtual” smooth genus 6) and to lower down to $g=3$ we introduce one triple point (counting like 3 double points since perturbing slightly 3 coincident/concurrent lines creates 3 ordinary nodes). This gives the correct gonality $5-3=2$. Doing a real picture one may draw picture 212 on Fig.\[F112:fig\]. (Keep in mind the orientation-consistent smoothing ensuring the dividing=orthosymmetric character of the curve). It has $r=2$, and $p=\frac{g-(r-1)}{2}=1$. Notice total maps of degrees 2 (projection from the “tri-node”=triple point), degree 4 (projection from the inner circuit) and degree 5 (projections from the inside of this inner circuit). Yet we missed degree 3. Over the complexes such a curve is not 3-gonal (because it is 2-gonal from the tri-node and 4-gonal when projected from a smooth point). Consequently, the allied bordered surface has circle maps of degrees $2,4,5$ but not $3$, which is missing. Hence this example probably corrupts our naive Prop.\[hypothetical:prop\]. Also, Gabard’s bound $r+p$ needs [*not*]{} to be exactly the degree of a circle map. Further this example shows the gonality sequence to be gapped in general. Now one general question is to wonder what can be said about the following invariant. \[def:gonality-sequence\] The [*gonality sequence*]{} $\Lambda=\Lambda(F)$ consists of the ordered list $\gamma<\gamma_1<\gamma_2<\dots$ of all integers occurring as degrees of circle maps tolerated by the given bordered Riemann surface $F$. Fragmentary information includes the following facts, gathered as a theorem. (To nuance reliability of the varied constituents we assign them some percentages of truth likelihood, with frankly Schopenhauerian scepticism!) For any bordered Riemann surface with topological invariant $(r,p)$ (viz. number of contours $r$ and genus $p$) and gonality $\gamma$ (i.e. the least degree of a circle map), the following estimates hold good ([*en principe[^11]*]{}): [\[100%\] $\bullet$ (T) (Trivial)]{} $r\le\gamma$. [\[99%\] $\bullet$ (KTA)]{} $\Lambda$ is nonempty or equivalently $\gamma<\infty$ is finite [(Ahlfors 1950 [@Ahlfors_1950], or Teichmüller 1941 [@Teichmueller_1941] crediting Klein for the result; cf. also Köditz-Timmann 1975 [@Koeditz-Timmann_1975] for a proof via Behnke-Stein)]{}. [\[100%\] $\bullet$ (Semigroup property)]{} the set $\Lambda$ is “multiplicative”, i.e. whenever it contains an element $\lambda \in \Lambda$ it contains all integral multiples $k \lambda$. (This follows by composing the corresponding circle map by a power map $z\mapsto z^k$ from the disc to itself.) In particular [(KTA)]{} implies that $\Lambda$ is always infinite. [\[98%\] $\bullet$ (A50) (Ahlfors 1950)]{} $\gamma\le r+2p$. [\[75%\] $\bullet$ (G06) (Gabard 2006)]{} $\gamma\le r+p$. [\[79%\] $\bullet$ (C11) (Coppens 2011)]{} $\gamma $ takes all intermediate values $r \le\gamma\le r+p$ (if $r=1$ the lower bound $r$ must be modified as $2$, excepted when $p=0$). [\[97%\] $\bullet$ (AFCF) (Ahlfors 1950 [@Ahlfors_1950 p.126], adhered to in Fay 1973 [@Fay_1973 p.116] and Černe-Forstnerič 2002 [@Cerne-Forstneric_2002])]{} Ahlfors bound $r+2p\in \Lambda$ always belongs the gonality sequence; and so do all higher values. The last assertion follows from Ahlfors proof (1950 [@Ahlfors_1950 pp.124–126]) where the origin is expressed as convex sum of points lying on a collection of circuits in ${\Bbb R}^g$. This is always feasible for $g+1=r+2p$ points, and a fortiori for more points. We shall try to digest Ahlfors argument in subsequent sections. In contrast to (AFCF), Example 2 (=212 on Fig.\[F112:fig\]) above shows (or at least indicates strongly) that Gabard’s bound $r+p$ is not necessarily in the gonality sequence. Further evaluations of the gonality sequence are tabulated on Fig.\[Coppens:fig\] as bold fonts. As before, the underlined number is Ahlfors (universal) bound $r+2p$, after which all gonalities are realized. The position pointed onto, by a triangle, is Gabard’s bound $r+p$. The little arrow is a pointer indicating the lowest integer after which the gonality sequence is full. At an early stage of the tabulation, it seemed realist to advance the following. \[conj:full-above-Gabard\] (Naive, destroyed by Example $4$) Strictly above $r+p$ each gonality occurs. This is pure guessing, but if true it would considerably lower Ahlfors’ universal lower bound $r+2p$ for “fullness”. The next example still supports the guess, but the next Example 4 ought to violate it. Consider, within the topological type $(r,p)=(1,2)$ where $g=(r-1)+2p=4$, a hyperelliptic model ($\gamma=2$). Looking at quintics (virtual genus 6) requires 2 nodes to correct the genus, but then the (complex) gonality is still 3 (and not 2 as we would like). The trick is (like in Example 2) to increase further the degree to permit a high order singularity lowering drastically the gonality. So we move to sextics (virtual genus 10) with a 4-node (counting for 6 ordinary nodes) decreasing correctly the genus to 4. As initial configuration we consider 3 coincident lines plus a conic through the coincidence and another line (in general position). An appropriate smoothing generates picture 122 on Fig.\[F122:fig\] with $r=1$ (all real circuits being connected through $\infty$). The gonality sequence seems to be $2, 5, 6, \dots$. However 4 must be added to the list (being a multiple of 2). Hence the true sequence is $2,4,5,6, \dots$. Gabard’s bound is $r+p=3$, and strictly above it all values are realized (Ahlfors bound is $r+2p=5$). -10pt0 \[23.10.12\]—[*Vague philosophy.*]{} An interesting feature of this example (122 on Fig.\[F122:fig\]) is that when gonality is very low in comparison to topological complexity, the Riemann surface, after having dispensed much energy to reach such a low gonality, seems falling into some dormant state without creating many new gonalities (missing the value 3). Perhaps this is a general phenomenon prompted by a principle of energy conservation. \[28.12.12\]—[*Warning.*]{} On looking carefully at picture 122 above (Fig.\[F122:fig\]) it is seen that as the center of perspective is dragged from the $4$-node toward the two red loops or even their insides we may loose total reality for some lines of the pencil become tangent to the circuit somewhere (cf. dashed lines on picture 122) so that a suitable perturbation let disappear two intersections in the imaginary locus. Accordingly, it is not even evident form the picture that degrees $5, 6$ occur as degrees of total maps. We now consider, in the topological type $(r,p)=(2,2)$ for which $g=(r-1)+2p=5$, again a hyperelliptic model. Looking at sextics with smooth genus $10$, we must use a correction by 5 (alas not a triangular number as those involved in multiple points). Thus we move to septics (order $m=7$) of smooth genus $\tilde{g}=\frac{(m-1)(m-2)}{2}=15$, and a 5-node (counting for $1+2+3+4=10$ ordinary nodes) effects the desired correction upon the genus. Smoothing a suitable configuration gives picture 222 on Fig.\[F122:fig\] with $r=2$ (two real circuits red and green colored). The gonality sequence includes the values $2,4,6=r+2p, \dots$. Six being Ahlfors bound the sequence is full from there on. When projected from the 5-node the degree is 2. Dragging the center of perspective along one of the two red loops gives total maps of degrees $7-1=6$ ([*warning:*]{} this is not even true, cf. again the dashed line on the picture!). The value 4 is not visible on the projective model, yet arises by the semigroup property. Studying the gonality over the complexes, it seems evident that 3 and 5 are not even complex gonalities, and we should be able to conclude that $2,4,6,\dots$ is the exact gonality sequence. (Here the “dots” refer again to the issue that all higher values belong to the gonality list, according to Ahlfors.) But then our conjecture \[conj:full-above-Gabard\] is violated (as $5$ does not belong the list). Incidentally, this example shows sharpness of Ahlfors bound $r+2p$ as the place from where the sequence is full. Those examples can be iterated for higher values of the invariant $(r,p)$ while staying in the hyperelliptic realm. The arithmetical issue is the possibility to compensate the genus by a high-order singularity. We obtain for $(r,p)=(1,3)$, hence $g=6$, the figure 132 on Fig.\[F122:fig\], an octic (smooth genus 21) with a 6-node (counting for $\frac{6\cdot 5}{2}=15$ ordinary nodes) hence lowering down the genus to $6$. The gonality sequence is $2,4,6,7=r+2p, \dots$. Similarly, for $(r,p)=(2,3)$, $g=7$. Browsing through increasing degrees the genus are $10, 15, 21, 28, \dots$, whereas the nodes give the list $1,3,6,10,15,21, \dots$. The right pair is thus $28-21=7$. So we take a 9-tic (smooth genus $\tilde g=\frac{(9-1)(8-1)}{2}=\frac{56}{2}=28$) with a 7-node. We construct easily picture 232 on Fig.\[F122:fig\], a curve whose gonality sequence is $2,4,6,8=r+2p, \dots$. (Note that in this case Ahlfors bound is sharp for the fullness of the sequence, but it was not in the previous example. It may again be observed that in the first example the $r+p$ bound occurs as a gonality, but it does not in the second example.) The real outcome of these constructions is that for (certain, all?) hyperelliptic curves we can be totally explicit about the gonality sequence. Iterating ad infinitum we have: For any topological type $(r,p)$ there is a surface of hyperelliptic type $(r,p)$ (with $r=1$ or $2$) whose gonality sequence $\Lambda$ is known explicitly. Namely, $\bullet$ if $r=1$, then $\Lambda=\{2,4,6, \dots, 2p, r+2p, \dots \}$, where the first “dots” runs through even values and the second means fullness after Ahlfors bound $r+2p$. $\bullet$ if $r=2$, then $\Lambda=\{2,4,6, \dots, r+2p, \dots \}$, where the first “dots” runs through even values and the second means fullness after Ahlfors bound $r+2p$. The natural question is of course to know if this spectrum distribution is specific to our models or generally valid for all hyperelliptic surfaces. (This looks likely, we think, maybe just by counting moduli.) Of course it is evident by the semigroup property that the gonality sequence contains the value listed, and is full after Ahlfors bound, yet the assertion that it reduces to this requires some argument. A conjecture about fullness --------------------------- \[23.10.12\] At this stage the situation is admittedly a bit messy. We try to clarify it by bringing into the picture the [*fullness invariant*]{} $\varphi$, that is the least integer from whereon the gonality sequence is full. (On the pictures discussed this is nothing but the little arrow used previously.) We have the string of inequalities: $$r\le \gamma \le \begin{Bmatrix} \buildrel{{\rm Ga}}\over{\le} r+p \le \\ \hskip4pt \le \hskip8pt \varphi \hskip8pt \buildrel{{\rm Ah}}\over{\le} \end{Bmatrix} \le r+2p.$$ Is any comparison possible between $r+p$ and $\varphi$? On example 212 of Fig.\[Coppens:fig\] $r+p=3$ beats the fullness $\varphi=4$. Many examples on Fig.\[Coppens:fig\] do satisfy $r+p\le \varphi$, but there is also several counter indicators, e.g. pictures 313, 414 or 223. The following is a trivial consequence of inequation $\gamma\le \varphi$: Fullness below Gabard’s bound (i.e. $\varphi < r+p$) implies low-gonality (i.e. $\gamma < r+p$). The converse fails, see pictures 212 or 222. On the pictures of Fig.\[Coppens:fig\] the fullness $\varphi$ is indicated by a little upward arrow. Examining examples on this figure it seems that when the surface has generic gonality (i.e. $\gamma=r+p$) then its fullness coincides with the gonality (i.e. $\varphi=\gamma$). It would be interesting to know if a general theorem hides behind this experimental observation. (Pressing up and down: fullness conjecture) \[conj:fullness\] If $\gamma=r+p$, then $\varphi=\gamma$. In other words if $\gamma$ achieves maximum value (granting the truth of Gabard’s bound!) then $\varphi$ collapses to its minimum value (namely $\gamma$). In particular the gonality sequence of a generic surface would be perfectly explicit, as being full from $r+p$. This would also show that generically Ahlfors bound $r+2p$ for fullness can be drastically lowered. It seems plausible that an adaptation of Gabard 2006 could prove this conjecture. (Ahlfors’ original proof can also be useful.) The idea would be that in the irrigation method the equation $x_1+\dots +x_{d-1}=-x_{d}$ which is soluble for $d\le (r-1)+p$ points is, by the assumption made on $\gamma$, not soluble for fewer points. One would then like to extend this “exact solubility” to the equation $x_1+\dots+ x_{d-1}=-(z_1+\dots+z_k)$, where the $z_i$ is a collection of points assigned in $F$. A vague idea is then that if some $x_i$ (or their lifts to $\overline{F}$ belong to the border) then upon dragging the $z_i$ we may hope to displace them to avoid this circumstance (incompatible with unilaterality). This would construct an unilateral divisor of any assigned degree $\ge r+p$, producing in turn circle maps of all such degrees. The conjecture would follow. Another basic phenomenon is that even when two surfaces have the same invariants $(r,p)$ and the same gonality $\gamma$ their gonality sequences may differ. (See for such an example both pictures 324 on Fig.\[F324:fig\]). Interestingly the left figure 324 is 4-gonal in $\infty^1$ ways (projection from the inner oval), whereas its companion 324bis, is 4-gonal in only 4 ways (projection from the nodes). Again some conservation law seems involved for all the energy absorbed by the many pencils of degree 4 living on the first model seems to provoke the missing of pencils of degree 6. It could be imagined that the right curve is totally real when swept out by the pencil of conics through the 4 nodes of degree $2\cdot 6 -4 \cdot 2=12-8=4$, but it fails to be total for the circular conics through the 4 nodes certainly misses the outer oval. -5pt0 To investigate the fullness conjecture (\[conj:fullness\]) further, we test curves of higher topological structure. $\bullet$ For $(r,p)=(3,2)$, we seek a surface with maximum gonality $\gamma=r+p=5$. If we imagine this gonality arising via linear projection it is natural to look at a sextic having a deep nest. The virtual genus is then $10$, but we want genus $g=(r-1)+2p=2+4=6$. Thus we introduce 4 nodal singularities. To keep the gonality maximum those nodes must not be accessible from the inner oval, and consequently we distribute the dashed circles (indicating unsmoothed nodes) in the “periphery”. We thus obtain curve nicknamed 325 (on Fig.\[Generic:fig\]). It has $\gamma=5$ and the gonality sequence is $5,6,7=r+2p,\dots$. In fact $\gamma$ could be $<5$ via some nonlinear pencil harder to visualize. A pencil of conics with 4 basepoints matching with the $4$ nodes creates a series of degree $2\cdot 6-4\cdot 2=12-8=4$, more economical than our $5$. However looking at picture 325, the special conic consisting of two horizontal lines fails to intersect the inner oval. Thus this pencil is not total, and we safely conclude that $\gamma=5$, exactly. In particular, the fullness conjecture (\[conj:fullness\]) is verified on this example. -5pt0 $\bullet$ For $(r,p)=(4,2)$, we seek a surface with maximum gonality $\gamma=r+p=6$. Imagine again this gonality arising via linear projection, we consider a septic with a deep nest. The virtual genus is then $15$, but we want $g=(r-1)+2p=3+4=7$, hence we conserve 8 nodal singularities. We obtain so the curve labelled 426 on Fig.\[Generic:fig\]. It has $\gamma=6$ and the gonality sequence is $6,7,8=r+2p,\dots$. The fullness conjecture (\[conj:fullness\]) seems verified on this example. Warning \[25.10.12\]: now the claim $\gamma=6$ is possibly an optical illusion, for a pencil of cubics with basepoints assigned on the nodes has degree $3\cdot 7-8\cdot 2=21-16=5<r+p$. If the latter is total then $\gamma\le 5$, violating our claim $\gamma=6$. Of course tracing pencil of cubics is not an easy game. Experience tell us that total pencils arise when basepoints are deeply rooted inside the deepest ovals. In the case at hand (curve 426), this feature is not fulfilled. The 8 basepoints of the cubics pencil lye outside the inner oval, yet, it could be that the 9th basepoint falls (by a lucky stroke) inside this oval. In fact, it is enough to observe that the cubic, consisting of the ellipse through the 6 points lying highest on figure 426, plus the line through the remaining 2 points (lying lowest on the same figure), fails to cut the inner oval. This gives evidence that our pencil of cubics is not total. We conclude $\gamma=6$, exactly. In fact one must check that the 8 assigned basepoints impose independent conditions on cubics, and so our pencil is forced to contain the special reducible cubic just described. Independence is checked by the usual stratification method, where one imposes more and more conditions while verifying that each extra condition drops dimension by checking that the corresponding inclusion is strict. The method seems to apply to our situation, and we conclude $\gamma=6$ (with reasonable self-confidence). Of course another detail that must be taken care is our somewhat tacit supposition that the gonality (or the gonality sequence) do not depend tremendously on the choice of smoothing. The classical method of small perturbation (Brusotti, etc.) asserts existence of a curve effecting the assigned smoothings, but there is an infinitude of choices for the coefficients. A priori the fine gonality invariants are sensitive to the choice effected. Remember that Brusotti’s method relies on the fact that the initial curve (thought of as a point in the discriminant hypersurface) has a neighborhood consisting of several “falde analytice”, i.e. a divisor with normal crossings each branch of which corresponding to preserving a certain node. This explains the liberal way to smooth away nodes of our initial configurations. Yet more hazardous is the claim of a smoothing conserving the exact location of all nodes. This remark hinders slightly the previous argument made on figure 426. $\bullet$ We next test the invariant $(r,p)=(5,2)$, and within it seek again a representative of maximum gonality $\gamma=r+p=7$. Using the same device as above, we are inclined to look at an octic (order $m=8$) with an interior oval kept protected from intrusion of singularities. The smooth genus is then $\tilde g=\frac{7\cdot 6}{2}=21$, but need be lowered down to $g=(r-1)+2p=8$. We thus consider a distribution of 13 nodes distant from the inner oval to produce the curve nicknamed 337 (on Fig.\[Coppens:fig\], see also Fig.\[Gabard:fig\] for a larger depiction). This curve has $r=3$ (not 5 as desired!). This means that I am a bad experimentalist, but the curve 337 is worth looking at closer. Since $g=8$ by construction, and $r=3$ we have $p=3$ (recall $p=\frac{g-(r-1)}{2}$). When projected from a point on the inner oval the curve is 7-gonal. This degree is [*larger*]{} than Gabard’s bound $r+p=6$! The example seems to violate Gabard’s bound $r+p$. [*Summary.*]{}—While testing the fullness conjecture, we rather arrived to a counterexample to Gabard’s estimate $\gamma\le r+p$. We thus switch slightly of game, but try to keep in mind the fullness problem for later. Potential counterexamples to Gabard 2006 ($\gamma\le r+p$) ---------------------------------------------------------- \[24.10.12\] The curve just discussed (337) seems a potential violation of the theorem $\gamma\le r+p$ asserted in Gabard 2006 [@Gabard_2006]. Can we solve this paradoxical situation? Either Gabard’s bound $r+p$ is false or something wrong happened. A possible explanation is that we were too cavalier when claiming $\gamma=7$; in fact the total lines pencil on curve 337 just shows $\gamma\le 7$. A priori there might be optical illusion about evaluating gonality. For instance sweeping our octic with 13 nodes by a pencil of cubics with 9 basepoints located on the nodes gives a linear series of degree $3\cdot 8-9\cdot 2=24-18=6$, rescuing the $r+p=6$ bound. Of course, it is another story to convince that such a map can be chosen total. Thus 337 still represents a severe aggression against $\gamma\le r+p$. Similar counterexamples (be they illusory or real) can be manufactured in lower topological complexity. Starting with a configuration of 3 conics, we conserve the deep nest, but keep the maximum number of singularities in the periphery so as to lower the genus as much as possible. Keeping 7 nodes unsmoothed, but smooth away all others crossings in a sense-preserving fashion (to ensure the dividing character of the curve), we obtain the curve 215 with $r=2$ (on Fig.\[Gabard:fig\]). Its genus is $g=10-7=3$. Thus the genus of the half (complex locus split by the real one) is $p=\frac{g-(r-1)}{2}=\frac{3-1}{2}=1$. Projecting from the interior oval gives $\infty^{1}$ total maps of degree $5$, and the hasty guess is that $\gamma=5$. Since $r+p=3$, Gabard’s bound $\gamma\le r+p$ looks again corrupted. -25pt 0 However the curve at hand (215) having $g=3$ (and being dividing), elementary knowledge of Klein’s theory prompts that the canonical map $C\to {\Bbb P}^{g-1}$, here ${\Bbb P}^2$, will exhibit the curve as a “Gürtelkurve”, i.e. a quartic with two nested ovals. Then the gonality is reevaluated as $\gamma=3$, and Gabard’s bound is vindicated again (by the rating agency!). Another way to argue, would be to take a pencil of cubics with 7 basepoints assigned on the 7 nodes and another basepoint on the curve. The degree is then $3\cdot 6- 7\cdot 2- 1\cdot 1=18-14-1=3$. The bound $r+p$ looks rescued again. Yet some hard work is required to check total reality of a suitable pencil. Perhaps there is some conceptual argument, else one really requires tracing carefully the pencil after an educated guess of where to place the extra assigned basepoint. It is even possible to construct a quintic with “visual” gonality exceeding $r+p$. The cooking recipe is the same as above. Start from a configuration of 2 conics and one line, keep the inner oval while maximizing the number of peripheral singularities. It results picture 214 on Fig.\[Gabard:fig\]. We see $r=2$ real circuits. The genus is $g=6-3=3$ (3 nodes must be subtracted), and thus $p=1$. The naive gonality seems to be $4$, exceeding (hence violating) Gabard’s bound $r+p=3$. Again to resolve the paradox one can either argue via the canonical map carrying the curve to a Gürtelkurve, or find a total linear series of lower degree. Here this would involve a pencil of conics through the 3 nodes plus one assigned basepoint inside the deep oval. The resulting series has degree $2\cdot 5 - 3\cdot 2- 1\cdot 1=10-6-1=3$, in agreement with the $r+p$ bound. A drawback of figure 214 is that the 3 remaining nodes are nearly collinear, rendering nearly impossible the depiction of the conics pencil. (In reality the 3 nodes are not aligned, else the line through them cuts the quintic in 6 points.) It is convenient to consider rather a related quintic $214bis$ on Fig.\[Pencil:fig\], where the line has penetrated the inner oval (yet without destroying it). All invariants $r,p$ (as well as the naive gonality) keep the same value as on the previous example 214. The new curve makes it easier to trace a total series cut out by a pencil of conics, where the extra basepoint has been chosen most symmetrically. Each member of it has beside the 4 assigned basepoints (counting for $3\cdot 2+ 1\cdot 1=7$ intersections) 3 moving points which are permanently all real, as follows (only?) through patient inspection of the picture. -45pt 0 -35pt0 Fig.\[Pencil:fig\] attempts to show various members of the conics pencil, as well as the 3 mobile points of the series. Those sections are depicted by the same letters e.g. 1,1,1 corresponds to the section by the ellipse invariant under symmetry about the vertical axis. Ultimately, the whole figure has to be extended by this symmetry, but this is better done mentally for not surcharging the figure. The dynamics (circulation) is quite tricky to understand, but the motion looks much accelerated (hence hard-to-follow) when the (red) curve crosses the basepoints of the pencil. This is a bit if the particle motion would be much accelerated by a gravitational black hole. Once the picture is carefully analyzed, it is evident that all 3 mobile points stay permanently real. Thus our quintic curve has gonality $\gamma\le 3$. (Gabard’s bound is rescued on this simple example.) Perhaps similar miracles (via high-order pencils hard-to-visualize) produce for all other pseudo-counterexamples to $\gamma \le r+p$. Yet this probably requires considerable work even just for the previous curves (of Fig.\[Gabard:fig\]). In full generality, some divine act of faith is required to imbue with chimeric respect the last vestiges of truth imputable to Gabard’s result. Note that pencil of cubics are required for the examples (Fig.\[Gabard:fig\]): even our sextic with 7 nodes achieves, via conics, only gonality $2\cdot 6- 4\cdot 2=4$, not as economic as the $r+p$ bound. [*Summary.*]{}—Two scenarios are possible: either Gabard’s bound $\gamma\le r+p$ is false (which is not quite improbable as its proof is intricate and there is an infinite menagerie of potential counterexamples), or it is true in which case it might be proved extrinsically by a highbrow extension of the last example described (Fig.\[Pencil:fig\]). This brings us to the next section, which albeit not very tangible in our fingers is perhaps technically implementable (at least at the level of Ahlfors bound $\gamma \le r+2p$). Brill-Noether-type (extrinsic) approach to Ahlfors via total reality {#sec:Brill-Noether-approach-to-Ahlfors} -------------------------------------------------------------------- \[26.10.12\] Neutralizing all virtual counterexamples (of the previous section) to $\gamma\le r+p$ amounts a sort of high-powered Brill-Noether theory for totally real pencils able to reprove Ahlfors theorem $\gamma \le r+2p$ (and then optionally to corroborate Gabard’s $\gamma\le r+p$) in a purely synthetic way. This section touches superficially this grandiose programme, we are quite unable to complete. Let us be more explicit. Any smooth projective curve (or, what is the same, closed Riemann surface) embeds in ${\Bbb P}^3$. A generic projection will realize the curve (like a knot projection) as a nodal model in the plane ${\Bbb P}^2$ having at worst ordinary double points. Specializing to real (orthosymmetric) curves we get a model in the plane, on which one can hope to first prove existence of a total pencil while evaluating the least degree of such a pencil. This should amount considering adjoint curves passing through the nodes so as to lower most the degree. The procedure would be as follows. Let $F$ be a bordered Riemann surface of invariant $(r,p)$. We consider its Schottky double $C=2F$, interpreted as a real orthosymmetric curve of genus $g=(r-1)+2p$ with $r$ real circuits. Using a generic immersion in the plane gives a model $\Gamma_m$ of the curve $C_g$ of order $m$ having $r$ real circuits, and a certain number of nodes $\delta$. For simplicity assume the nodes to be simple, though the more general situation must perhaps not be excluded. We have of course $g=\frac{(m-1)(m-2)}{2}-\delta$. Let $\Delta$ be the divisor of double points of $\Gamma_m$. (Those can occur in conjugate pairs under complex conjugation.) Consider in the complete linear system $\vert kH\vert:=\vert {\cal O}_{\Bbb P^2}(k)\vert$ of all curves of degree $k$, a linear pencil $L$ of curves passing through the nodes $\Delta$ of $\Gamma$ (adjunction condition). The resulting series has degree $\le k\cdot m- 2\cdot \delta$. In fact a better control must be possible. First $k$ has to be chosen large enough so that the adjunction condition is possible at all. Since $\dim \vert k H\vert=\binom{k+2}{2}-1$, the integer $k$ is chosen as the least integer such that this dimension exceeds $\delta$. Then we may have some excess permitting to assign other (simple) basepoints. Let us be even more explicit (we work first over the complex, for simplicity). So assume given $C$ a curve of genus $g$. We look first at the canonical embedding $\varphi\colon C \to {\Bbb P}^{g-1}$. The image curve has degree $2g-2$. We manufacture a plane model via successive projections from points chosen on the curve. This lowers the degree by one unit after each projection. We arrive ultimately at a nodal model $\Gamma_m\in {\Bbb P}^2$ of degree $m=(2g-2)-[(g-1)-2]=g+1$. Experimental study or an inspired guess suggests considering adjoint curves of degree $k=m-3$. This value is calibrated so that our $k$-tics have enough free parameters to visit all $\delta$ nodes of $\Gamma$. Indeed $$\begin{aligned} \dim \vert k H\vert =\binom{k+2}{2}-1=\frac{(k+2)(k+1)}{2}-1&=\frac{(m-1)(m-2)}{2}-1\cr &\ge\frac{(m-1)(m-2)}{2}-g=\delta.\end{aligned}$$ We look at all curves of degree $k$ going through the nodes $\Delta$ of $\Gamma$. Denote $\frak d=\vert kH(-\Delta)\vert$ the corresponding linear system, and let $\varepsilon$ be its dimension. Obviously $$\varepsilon \ge \dim\vert kH\vert-\delta.$$ (In fact since nodes of an $m$-tic impose independent conditions upon adjoint curves of degree $m-3$ this is an equality. But we do not this deep fact essentially equivalent to Riemann-Roch.) Both displayed formulas show that $\varepsilon\ge 0$, and we may thus impose to our $k$-tics to pass through $\varepsilon-1$ extra points, while still moving inside a linear system of dimension $\ge 1$ (a so-called pencil). This gives a pencil $L\subset \frak d$ of degree $$\begin{aligned} &\le k\cdot m- 2\delta -1 \cdot (\varepsilon-1)\cr &\le k\cdot m- 2\delta-\Bigl[\binom{k+2}{2}-1-\delta\Bigr]+1\cr &= k\cdot m- \delta-\binom{k+2}{2}+2\cr &= k\cdot m+ g-\binom{m-1}{2}-\binom{k+2}{2}+2 \quad \textrm{[now recall $m=g+1$]}\cr &= (m-3) m+ (m-1)-2\binom{m-1}{2}+2\cr &= (m-3) m+ (m-1)-(m-1)(m-2)+2\cr &= (m-3) m+ (m-1)\underbrace{[1-(m-2)]}_{-(m-3)}+2\cr &= (m-3) [m-(m-1)]+2\cr &= m-1=g. $$ This proves that any curve of genus $g$ admits a pencil of degree $\le g$, which made basepoint-free induces a map of, eventually, lower degree. (Of course our assertion fails when $g=1$, but true otherwise granting some knowledge.) This “degree $g$” bound is a bit sharper than the usual degree $g+1$ prompted by Riemann(-Roch)’s inequality, but much weaker than the Riemann-Meis bound $[\frac{g+3}{2}]$ for the complex gonality. A natural wish is obtaining the Riemann-Meis bound via the above strategy, hoping that special configurations of $\varepsilon-1$ points on the curve impose less conditions than expected, leaving some free room for additional constraints lowering further the degree. This is essentially what Riemann was able to do (at least heuristically) via transcendental methods, and (exactly) what Brill-Noether’s theory is about at the pure algebro-geometric level. Recall, yet, that both works apparently fail satisfying modern standards, cf. e.g. Kleiman-Laksov 1972 [@Kleiman-Laksov_1972] and H.H. Martens 1967 [@Martens_Henrik_1967], where the problem was not yet solved apart via Meis’ analytic (Teichmüller-style) approach. At this stage, starts the difficulties. The big programme would be to adapt the above trick to real orthosymmetric curves, in order to tackle Ahlfors theorem. The latter prompts the bound $g+1$ rather, but this little discrepancy should not discourage us. So in some vague sense a “real” Brill-Noether theory is required, combining probably also principles occurring in Harnack’s proof (1876 [@Harnack_1876]) of the after him named inequality. From the real locus $\Gamma({\Bbb R})$ one shall identify deep nests, and it is favorable to choose them as the extra basepoints to ensure total reality of the pencil we are trying to construct. Then there is also a foliation on the projective plane induced by the members of the pencil. Inside each oval, the foliation must exhibit singularities (otherwise total reality is violated). In fact total reality imposes the foliation to be transverse to the real circuits. Hence if there is no singularity we would have a foliation of the disc which is impossible. Perhaps Poincaré’s index formula is also required. To be brief there is some little hope that a very careful analysis of the geometry establishes existence of a total pencil of degree $g+1=r+2p$, recovering so Ahlfors result. This would be pure geometry (or the allied devil of algebra) without intrusion of either potential theory, neither transcendental Abelian integrals, nor even topological principles. Perhaps only elementary topological tricks are required to ensure total reality by gaining extra intersections via a continuity argument akin to Harnack’s. This offers maybe another approach to Ahlfors, yet it requires some deep patience. It looks perhaps somewhat cavernous as (extrinsic) plane curves with singularities are just a “Plato cavern”-style shadow of the full Riemannian universe. If this dream of a synthetic proof of Ahlfors theorem is possible, then it would be nice (if possible) to boost the method at the deeper level of special groups of points to gain the sharper Riemann-Brill-Noether-Meis sharp control upon the gonality, whose real orthosymmetric pendant is expected to be the $r+p$ bound (of Gabard). Last, I know (only through cross-citations) the work of Chaudary 1995 [@Chaudary_1995-Thesis] where a real Brill-Noether theory is developed. This probably helps clarifying the above ideas. [*Philosophical remark.*]{}—Everybody experimented difficulties when playing with extrinsic models of Riemann surfaces. A typical instance occurs with Harnack’s inequality $r\le g+1$, whose extrinsic proof (Harnack 1876 [@Harnack_1876]) is pretty more intricate than Klein’s intrinsic version (same year 1876 [@Klein_1876]) based on Riemann’s conception of the genus. By analogy, one can predict that any synthetic programme toward Ahlfors will ineluctably share some unpleasant features of Harnack’s proof. The substance of the latter is a spontaneous creation of additional intersection points forced by topological reasons, leading to an excess violating Bézout. Arguments similar to Harnack’s might be required to ensure total reality of a well chosen pencil. Instead of being obnubilated by real loci (of both the curve and the plane), it is sometimes fruitful to move in the “complex domain” to understand better reality. A typical example is Lemme 5.2. (in Gabard 2006 [@Gabard_2006]) about unilateral divisors linearly equivalent to their conjugates. This was one of the key in my approach to Ahlfors maps. Perhaps this lemma is also relevant to the problem at hand ensuring total reality quite automatically. In the series of adjoint curves $\frak d$, one then imposes passing not through deeply nested ovals, but rather through imaginary points all located on the same half. The difficulty is of course showing existence of such a curve intersecting the fixed one only along one half (unilaterality condition), except eventually for some assigned basepoints (either real or imaginary conjugate). Extrinsic significance of Ahlfors theorem ----------------------------------------- \[07.11.12\] Another (less retrograde) desideratum is to explicit the extrinsic significance of Ahlfors theorem for real algebraic (immersed) plane curves. We touched this already in the Slovenian section \[Open-RS-embed-in-C2:sec\] but now a sharper idea is explored. The point is delicate to make precise and already quite implicit in my Thesis (2004 [@Gabard_2004], especially p.7 second “bullet”) plus of course in Rohlin 1978 [@Rohlin_1978] (albeit the latter may never have been aware of Ahlfors theorem). Today I discovered a certain complement which is perhaps worth presenting. First Ahlfors theorem traduces in the following. Any real orthosymmetric (=dividing) algebraic curve admits a totally real morphism to the line. The half of the dividing curve is a bordered surface. By Ahlfors 1950, the latter tolerates a circle map, which Schottky-doubled gives the required total map. For another proof cf. e.g. the first half of Gabard 2006 [@Gabard_2006]. This pertains to abstract curves (equivalently Riemann surfaces) but it acquires some extra flavor when the curve becomes concrete. Of course the ontological problem of concreteness is that there are plenty of ways for an abstract object to become concrete. Thus concreteness is oft the opposite extreme of canonicalness. Arguably, there is perhaps still a preferred “Plato cavern” namely the projective plane which can be used as an ambient space where to trace all Riemann surfaces provided we accept nodal singularities. Concretely this is done via generic projections from a higher projective space (${\Bbb P}^3$ actually suffices to embed any abstract curve), and then projecting down to the plane ${\Bbb P}^2$ gives a nodal model. All this being pure synthetic geometry it transpierces matters regarding fields of definition (A. Weil’s jargon) and so adapts to the reality setting. As yet just trivialities, but now we aim interpreting synthetically the (non-trivial) Ahlfors theorem. Starting from a real dividing curve in some projective space, suitable projections exhibit a birational model, $C$, in the plane as a nodal curve. Existence of a total morphism traduces into that of a total pencil, i.e. one all of whose member cut only real points on the curve $C$, at least as soon as they are mobile. A priori basepoints may include conjugate pairs of points. (A simple example arises when we look at the pencil of circles through 2 points. Recall that circles always pass to the so-called cyclic points at $\infty$, but this is just an affine conception). In extrinsic terms, Ahlfors theorem takes essentially the following form. [(IAS=Immersed Ahlfors via Kurvenscharen)]{} Given a dividing (real algebraic) curve $C$ immersed nodally in the plane ${\Bbb P}^2$. There is a totally real pencil of (auxiliary) curves of some order $k$, all of whose members cut on $C$ solely real points plus eventually imaginary conjugate pairs of basepoints. This reduces to the basic theorem that any abstract morphism of algebraic geometry admits a concrete description in terms of ambient linear systems when the abstract object is projectively concretized. In substance this is just the spirit of Riemann (algebraic curves=Riemann surfaces) but extended to the realm of morphisms. So the required theorem is just basic algebraic geometry but I forgot all the foundations. Historically add to Riemann, certainly Cayley-Bachach, Brill-Noether, (Klein?), all the Italians, and finally Weil, Grothendieck, plus of course many others. Now the new observation \[07.11.12\] is that we may always assume $k=1$ (in the theorem IAS) up to changing of birational nodal model. The idea is that we may first reembed the curve $C$ via the complete linear system of all curves of degree $k$ (alias Veronese embedding) in some higher space ${\Bbb P}^N$, where $N=\binom{k+2}{2}-1$. Then the image curve $C'$ is (totally) swept out by a pencil of hyperplanes corresponding to the original total pencil $L$ of $k$-tics in the plane ($k$-tics=curves of degree $k$). If we project from the base locus of the hyperplane pencil which is a linear variety of codimension 2 we arrive down again in ${\Bbb P}^2$, but now with a new model total under a pencil of lines. It seems to me that this trick works and we get the: [(IAP=Immersed Ahlfors via lines pencils)]{} Given an abstract dividing (real algebraic) curve, there is always a nodal(ly immersed) model in the plane ${\Bbb P}^2$ which is total under a pencil of lines. This permits to remove one of the obstruction in our discussion of the Forstnerič-Wold problem (already touched in Sec.\[Open-RS-embed-in-C2:sec\]). We now deduce the stronger assertion: Any finite bordered Riemann surface immerses in ${\Bbb C}^2$. Let $F$ be the bordered surface, and $C:=2F$ be its Schottky double which is real orthosymmetric. By the theorem (IAP) we find a nodal model in the plane ${\Bbb P}^2$ total under a pencil of lines. The pencil being real its unique basepoint $p$ is forced to be real. Since the allied morphism (projection) is total the fibre of an imaginary point is an unilateral divisor, i.e. confined to one half of the curve. This means that all imaginary lines through the basepoint cuts unilaterally the curve. It suffices thus to remove (from ${\Bbb P}^2({\Bbb C})$) an imaginary line through $p$ to obtain an immersed replica of $F$ in ${\Bbb C^2}$. Note that if $p$ lies on the (nodal) curve then only the open half (interior of $F$) is so embedded, but we can probably arrange this by displacing slightly the center of perspective $p$ outside the curve while conserving total reality. The net bonus is that the whole bordered surface (boundary included) is in ${\Bbb C}^2$. Of course this is still millions of lightyears away from Forstnerič-Wold postulated embedding (for all finite bordered surfaces), yet represents already a nice application of Ahlfors. Of course the corollary is also the special (finitary) case of the famous Gunning-Narasimhan theorem (1967 [@Gunning-Narasimhan_1967]), immersing any open Riemann surface in ${\Bbb C}^2$. Maybe their immersions are proper also, whereas ours are not. Maybe the Fatou-Bieberbach trick arranges this issue always, cf. e.g. Forstnerič-Wold 2009 [@Forstneric-Wold_2009]. Anyway using the quantitative form of Ahlfors (not used as yet) one can go perhaps further, maybe saying things on the degree of the model. Note also that the viewpoint of nodal model of orthosymmetric curves affords another numerical invariant, namely: [(quite implicit in Matildi 1945/48 [@Matildi_1945/48])]{} Given an abstract dividing real curve $C$. The least order $\delta$ of a nodal birational model of $C$ is termed (by us) the nodality of the curve $C$. Via Schottky-doubling this invariant also makes sense for finite bordered Riemann surfaces. Projecting down to ${\Bbb P}^2$ the canonical model in ${\Bbb P}^{g-1}$ of a curve of genus $g$, we get a nodal model of degree $g+1=(2g-2)-[(g-1)-2]$ (each projection from a point on the curve decreases the degree by one unit). Hence $\delta \le g+1$. If the theorem (IAP) is correct, one could also try to define the linear gonality of a bordered surface (or the allied orthosymmetric double) as the least degree of a nodal plane model totally real under a pencil of lines. This gives perhaps yet another invariant $\lambda$, which seems to satisfy $\gamma\le \lambda+1$. Another dream of longstanding (Gabard’s Thesis 2004 [@Gabard_2004]) is whether Ahlfors’ theorem implies Rohlin’s inequality $r\ge m/2$ for a smooth dividing curve of order $m$. If such a curve $C=C_m$ is total under a pencil of lines, then sweeping out the curve by the pencil gives collections of $m$ real points. When rotating the line around the basepoint, those $m$ points never enter in collision (else smoothness is violated), nor do they disappear in the imaginary locus (else total reality is violated). After a 180 degree rotation already, the line returns to its initial position while the group of $m$ points recover its initial position giving raise to a monodromy permutation. Total reality forces each circuit of the curve $C$ to be transverse to the foliation underlying the pencil of lines. It follows that the monodromy transformation is an involution (order 2) and we deduce: [(Rohlin essentially)]{} Let $C_m$ be a smooth real curve of order $m$ totally real under a pencil of line. Then the real locus $C_m({\Bbb R})$ consists of a deep nest of $m/2$ ovals when $m$ is even, and if $m$ is odd there is as usual one pseudoline and ovals distribute in a nest of depth $(m-1)/2$. In particular Rohlin’s inequality $r\ge m/2$ follows in this special case where total reality is given by a pencil of lines. The general case of Rohlin still appeals to some formidable work, but perhaps may be derived via a linear pencil on a nodal model. Alas we are unable to complete this project. Let us however try to be more explicit. Given a smooth dividing $C_m$. Let $L$ be a total pencil of $k$-tics given by Ahlfors (theorem IAS). Then one can either try to study directly the corresponding foliation appealing to Poincaré’s index formula, and hope to mimic the above argument. Alternatively one can try to use the reembedding trick, where we use another model total under a pencil of lines. Now on the new nodal model of degree say $\lambda$, we apply the same sweeping procedure. We see on one initial line $L_0$ (assumed generic, i.e. avoiding the nodes) $\lambda$ points all real. When rotating by a half-twist the line we see groups of $\lambda$ points which now may cross themselves, but one can still assign a monodromy permutation. Naively any point finishes its trajectory on the other side of the basepoint (alas this makes no sense since a projective real line is a circle not disconnected by a puncture). The number of real circuits $r$ of the curve $C$ is equal to the number of cycles of the monodromy permutation, but a priori the latter number can be very low since crossings are permitted. (Imagine e.g. a spiral which after several growing revolution times closes up to form a single circuit.) Note that we do not yet exploited the smoothness hypothesis of the original model $C_m$. A naive way to exploit this is via the complex gonality $\gamma_{\Bbb C}$. We have indeed $m-1=\gamma_{\Bbb C}\le \gamma$ ($C$ being smooth). On the other hand $\gamma\le \lambda-1$. Hence $m\le \lambda$. This is interesting yet certainly not enough to conclude Rohlin’s inequality. So we give up the question for the moment. The gonality spectrum --------------------- An idea perhaps worth exploring is to enrich the gonality sequence (Definition \[def:gonality-sequence\]) into what could be called the [*gonality spectrum*]{}. This would just be the former weighted by the dimension of the space of all circle maps having the prescribed degree. As we already observed earlier (hyperelliptic examples) it seems that when a surface has a very low gonality then it “somnolates” without creating new gonalities. Thus more generally, the intuition behind this spectrum invariant would be a conservation law somewhat akin to Gauss-Bonnet: whatever the Riemannian incarnation of a topological surface the curvatura integra keeps constant value equal to the Euler characteristic ($ \int_F K d\omega= 2\pi \chi(F)$). Of course experiments requires to be made (using e.g. the specimens on Fig.\[Coppens:fig\]). Alas I had not presently the time to do serious investigations about this spectrum. It seems also expectable that from a certain range on, the spectrum is independent from the conformal structure. (At least so is the case for the gonality sequence which is always full after $r+2p$.) Of course some convention is required, probably consider only maps up to automorphisms of the disc. Example the only example where the spectrum is very easy to describe is the disc: in this case the $\gamma$-sequence is full starting from 1, and there is essentially only one map of degree one (the Riemann map). Given any unilateral group $D$ of $d$ points in the disc, thought of as the north hemisphere of the Riemann sphere the pencil through $D$ and its complex conjugate $D^{\sigma}$ induces a totally real map. (cf. Lemme 5.2 in Gabard 2006 [@Gabard_2006]). Conversely, given the map its fibre over 0 gives an unilateral divisor, which up to a range automorphism may be assumed to contain 0. Normalizing by a rotation there are thus the map depends upon $2d-3$ real constants. (Make this more precise…). Such maps are (in the complex function literature) often called finite Blaschke products. Once the setting is well understood, this gonality spectrum encodes valuable information upon all circle maps. Of course one perhaps still want to know more; e.g. to understand the incidence relation among the varied maps, especially how high-degree maps may degenerate to lower degree ones. Fig.\[Coppens:fig\] shows some interesting examples. Considering e.g. picture 313 we see that both maps of degree 3 are limit of maps of degree 4 (actually can be connected by such), and both of them are also limits of maps of degree 5. Looking at picture 112 (again on Fig.\[Coppens:fig\]) we see that the unique (total) map of degree 2 is also the limit of maps of degrees 3 and 4. The gonality sequence $2,3,4,\dots$ can be enriched by weighting by dimensions to get $2_0,3_1, 4_2, \dots$. Beware that probably there are other maps of degree 4 than those visible on the picture as linear projection, namely the unique 2-gonal map post-composed by circle maps of degree 2 from the disc to itself. Our guess is that such Blaschke maps may degenerate to their originator (the hyperelliptic projection) but not to maps of degree 3. More lowbrow counterexamples to $\gamma\le r+p$ ----------------------------------------------- \[27.10.12\] We now pursue the project of multiplying and diminishing further the order of virtual counterexamples to Gabard’s estimate $\gamma\le r+p$ (cf. Fig.\[Gabard:fig\] and Fig.\[Pencil:fig\]). There we found curves (via an uniform recipe) seemingly violating the gonality upper bound $r+p$. The simplest example had order 5, but it is easy to get examples of order 4. The game is again to depict total pencils vindicating Gabard’s bound. Albeit very modest corroboration of the bound, we found instructive to visualize the corresponding total pencils. First remind the general recipe: to manufacture an (at least virtual) counterexample to $\gamma\le r+p$, we leave tranquil the inner oval but maximize the number of singularities, so as to lower the genus $g=(r-1)+2p$, and hence $(r,p)$. Having left quiet the inner oval the virtual gonality via linear projection is one less than the degree, but $r+p$ may go lower down this value. We first consider a configuration of order 5 consisting of 2 conics plus one line, see picture 304 below (Fig.\[F304:fig\]). Smoothing it as dictated by orientations while keeping unsmoothed the dashed circles gives a curve with $r=3$ real circuits of genus $g=6-4=2$. Hence $p=\frac{g-(r-1)}{2}=0$. The virtual gonality is $\gamma^\ast=4$ (projection from the inner oval). This seems to violate $\gamma\le r+p=3$. Looking at the pencil of conics through the 4 nodes gives a series of degree $2\cdot 5-4\cdot 2=10-8=2$. This violates the trivial bound $r\le \gamma$, but of course this pencil is not total: e.g. the conic consisting of the 2 horizontal (or better oblique) lines misses the inner oval. Assigning instead one of the 4 basepoints on the inner oval gives a pencil of degree $3$, which is claimed to be total. Totality of the morphism requires examining (patiently) that each conic of the pencil cuts only real points on the quintic $C_5$. This is depicted on the large part of Fig.\[F304:fig\], where each triad of moving points of the series are labelled by triples $1,1,1$, then $2,2,2$, etc. Let us start from the conic consisting of the oblique line through $1,1$, plus the horizontal line. The latter cut the red pseudoline at infinity. This pair of lines deforms to a hyperbola cutting the triad $2,2,2$. This hyperbola is in turn pinched toward a pair of lines cutting the group $3,3,3$, etc, up to $7,7,7$. From here on, things becomes harder to visualize. (Alas our picture is not optimally designed.) The conic of the pencil now becomes very close to the primitive conic involved in the generation of the quintic $C_5$ via small perturbation. The net effect is that points on the green branch nearly “osculated” by the primitive ellipse are (violently) accelerated (like in CERN’s particles accelerator). At this stage it is quite delicate to make a consistent picture, but total reality seems to work: all particles stay real during the motion without disappearing as ghost in the imaginary locus (as conjugate pairs of points under Galois). We promised a similar example of degree 4; this will be pictured later (Sec.\[sec:degree-four\]), being now sidetracked to another topic which looks more exciting. Some crazy ideas about gravitation and unification of forces {#sec:gravitation} ============================================================ From gravitation to electrodynamics {#sec:electrodynamics} ----------------------------------- Now we arrive at the following crazy interpretation (discovered the 27.10.12 at ca. 13h58). It would be nice if there is some relation of the Ahlfors maps with periodic solutions to the $n$ body problem in gravitation (celestial mechanics). The 4 basepoints of Fig.\[F304:fig\] may be thought of as supermassive black-holes, so massive that there is no interaction between them (imagine purely static objects lying in different sheets of the multiverse). Dually, the moving points of the linear system are imagined as massless microparticles (electrons, or better photons). There is also no gravitational interaction between them. Thus the sole interactions reigning are those between black holes and photons. It is also imagined that a photon can traverse a black-hole (without captivation). As a wild speculation, the trajectories described by the 3 photons on Fig.\[F304:fig\] may satisfy exactly Newton’s law of gravitation. In particular the full trajectory would be the real locus of an algebraic curve! This would of course be a wide extension of Kepler’s law (on the rôle of conic sections in the simplest case of one sun and one planet). If this is true we see a deep connection between Klein’s orthosymmetric curves, Ahlfors maps of conformal geometry and the totally real circulations positing periodic stable motions along circuits of an orthosymmetric curve. Exaggerating a bit this should explain the ultimate constitution of matter (and its relative stability) not via knots (as Lord Kelvin desired via Helmholtz vortices) but via bordered Riemann surfaces (probably quite ubiquitous already in the so-called string theory). Note that our basic experiment (with Fig.\[F304:fig\]) is—as far as speed of motion is concerned—quite in line with this interpretation. Let us look at one of the simplest example of orthosymmetric curve, namely the (Zeuthen-Klein) Gürtelkurve (aka [*courbe annulaire*]{}). This is a quartic with two nested ovals arising by smoothing two transverse ellipses having 4 intersections. The picture is given below (Fig.\[FGuert:fig\]). One can convincingly argue that the shapes of trajectories (especially the outer oval) are unlikely to be gravitational orbits. It seems that some hidden force repulses the particles (labelled 1 on the figure). Invoking some other (electric) force effecting repulsion between particles, then the trajectories of the Gürtelkurve look again physically tolerable. Thus the “physical” model should include two types of interactions: gravitational and electromagnetic. -10pt0 Of course one can drag the position of the sun while still having a totally real pencil. This gives the next figure (Fig.\[FGuert2:fig\]). Note that we did not changed the curve, yet it is still plausible that for suitable initial conditions (velocity vectors) the orbits of our 4 bodies follows exactly the same quartic curve. -5pt0 -10pt0 We arrive at the following metatheorem \[14h57\]: [(Kepler generalized?)]{} \[metatheorem:thm\] Given any orthosymmetric real (algebraic) curve embedded (or immersed) in the Euclid plane ${\Bbb R^2}$ and a totally real pencil (existence ensured by Ahlfors theorem). There exists initial conditions (velocity vectors) such that the trajectories of particles obeying the inverse square law of Newtonian attraction resp. Coulombian repulsion match exactly the real circuits (“ovals”) of the given real algebraic curve. Further the dynamics (speed of motion) is dictated by the pencil. In particular there is plenty of periodic solutions to the $n$-body problem, essentially one for each such curve. How to prove this? Philosophically, algebraicity might be not so surprising: recall Laplace’s potential-theoretic interpretation of Newton, and from Laplace there is just one step to Riemann, hence to Klein. The miracle should be essentially akin to Riemann’s existence theorem prompting any closed Riemann surface (an a priori completely fluid object) to rigidify canonically as an algebraic curve. Even if true the metatheorem is quite modest because in practice (meteorites, apocalyptic black holes scenarios, etc.) one is given the initial conditions and the goal is to predict the future evolution of the system. Here in contrast, we know in advance the trajectories (hence the destiny) while claiming existence of initial conditions compatible with the orbital structure. Generally, integrating the differential equations governing some motion, we meet a highly complex dynamical system subjected to the paradigm of chaotic determinism à la Poincaré. Note that a Euclidean model of the projective curve is required to give sense to Newton’s inverse square law. Several questions naturally occurs assuming the truth of the metatheorem. The theorem affords plenty of periodic motions. Essentially we obtain as many periodic motions as there are real orthosymmetric curves. Even more than that, one requires an Ahlfors circle map (equivalently a totally real morphism à la Klein-Teichmüller). A first naive question is: do this recipe exhausts all periodic motions? Certainly not, try Euler and Lagrange’s periodic motions. Roughly all algebraic motions are periodic, but the converse has no chance to be true. Observationally, Fig.\[FGuert2:fig\] looks anomalous because the series 1,2,3,4 closest to the sun looks much slowed down, whereas we are accustomed (Kepler) to rapid motions near a massive star. One requires perhaps a third type of interaction, say the [*strong interaction*]{}, to explain this. Namely both particles the proton and the electron are of a dualistic nature, hence they tend to “love” themselves like partners staying close together over a long period of time. This third force would have the net effect of diminishing the real speed by a factor proportional to the (squared?) distance separating the bodies. What is then the fourth force, alias [*weak interaction*]{} in contemporary physics? Maybe none is required in our model? Perhaps dually, particles of the same nature (namely electrons) dislike themselves like competitors and the [*weak force*]{} just produces some acceleration of the motion when they are in close vicinity. Visually this behavior is perhaps observed near the groups labelled 2,3 on the top part of Fig.\[FGuert2:fig\]. We have now a model with 4 fundamental forces. One must of course still define time. This would, on our example, just be the angular parameter of the pencil. Presumably the metatheorem should take into account these two extra forces, becoming somewhat sophisticated, yet probably still completely deterministic and hopefully reasonably easy to integrate. The miracle would be that it admits dividing (=orthosymmetric) real curves as periodic orbits. Of course to relativize, one can do similar games with real diasymmetric curves, but then there is no total reality prompted by Ahlfors theorem and particles sometimes disappear in the imaginary locus. We leave to the reader’s imagination appropriate physical interpretations (ghost particles, anti-matter, etc.) Perhaps there is a more elementary way to explain slowness of the motion near the star (without appealing to the exotic forces at the subatomic level). Recall Kepler’s law in the elliptic case, that identic sectorial areas are swept out during the same amount of time. This suggests that the time parameter is not the angular parameter but the areal one. Of course one gets other troubles since the distant electron is supposed to move synchronously with the one closer to the proton (cf. Fig.\[FGuert:fig\]). Some little objections ---------------------- \[28.10.12\] Another objection to our metatheorem (\[metatheorem:thm\]) is the following one. Assume the given orthosymmetric curve to be of the simplest stock, namely a line swept out by a total pencil of lines. Then one must assume that there is no forces between the two bodies to explain the rectilinear motion. A more serious objection arises when $C$ is an ellipse swept out by a total pencil of lines through the middle of both foci. If all (four) fundamental forces involved satisfy the inverse square law, then so does the resulting force. Hence all interactions reduce to a single one which is attractive (to get an elliptic trajectory). However according to Kepler the orbit must be an ellipse with the sun located at one of the foci. Hence our geometric model where the basepoint of the total pencil lies at the center of the ellipse is not physically relevant. This example suggests that the metatheorem requires corrections. Maybe one is only given in advance the orthosymmetric curve but not the total pencil, while the metatheorem states existence of a pencil physically observable. For an ellipse we would only be allowed to take pencil of lines through one of both foci; if the ellipse degenerates to a circle only the center would be permissible. \[30.12.12\] At this stage it might be relevant to remind that there is a vast theory of foci for high-order algebraic curves, due it seems to Plücker first and then Siebeck 1864 [@Siebeck_1864], etc. cf. e.g. Casas-Alvero 2013 [@Casas-Alvero_2013]. Of course this Kepler obstruction should not preclude physical systems obeying more complicated interactions laws with say several fundamental forces, maybe not all subsumed to the inverse square law. Such could validate exotic orbital structures, e.g. an ellipse with a sun at its center, as physically reasonable. \[28.12.12\] Another possible objection comes from the following curve Fig.\[F324bis:fig\]. This possesses a total pencil of lines, yet some particles do not repulse, rather crossing themselves unsensitive of each other. On relabelling the particles one can posit a repulsion acting rather as a bounce of billiard balls (elastic shock). Another explanation could involve some quantumchromodynamics like assigning spins to the electrons neutralizing interaction between some of them. -10pt0 Back to insignificant geometry ------------------------------ Let us now leave such complex modelling question, to contemplate more complicated systems arising from other curves than the Gürtelkurve, especially some of higher order. First staying of order 4 there is, dual to the Gürtelkurve, the curve arising by reversing orientation of one of the ellipses (cf. arrows on Fig.\[F4oval:fig\]). This gives a quartic with 4 ovals when smoothing compatibly with the prescribed orientations. A total pencil arises from all conics through 4 basepoints distributed inside the ovals (Fig.\[F4oval:fig\]). -10pt0 Initially the point $a_1$ animated by a suitable horizontal velocity vector is mostly subjected to the attraction of the nearby star (=upper basepoints of the conics pencil). If $a_1$ and this star were to be alone in the universe, $a_1$’s orbit would be close to the dashed ellipse of “vertical eccentricity”, provided the upper star coincides with the focus of this ellipse. Yet in reality, as the body $a_1$ arrives near position $d_1$ and meanwhile body $a_8$ reached position $d_8$, electric repulsion is becoming predominant causing a (finally violent) deviation from the elliptic trajectory. Instead of appealing to gravitation one can just imagine the basepoints (alias “stars” previously) as positively charged protons, the whole system reducing to an electrodynamical one obeying only Coulomb’s law of attraction resp. repulsion. The fixed protons would however not repulse, maintaining their fixed positions due to some nuclear cohesion (strong/weak forces). It is easy to produce examples of higher topological complexity via curves of higher orders. Instead of starting with two ellipses, take three of them and smooth the configuration in a sense-preserving way to get Fig.\[Fsextic:fig\]. -10pt0 Reversing orientation of one of the ellipses (say that with horizontal major axis) gives the more interesting Fig.\[Fsext2:fig\] requiring a pencil of conics to exhibit total reality. -10pt0 Again we use the same labelling as before, namely the first (cyan=pale blue colored) conic consisting of the vertical and horizontal lines cuts on the sextic $C_6$ the group of points labelled $a_1,\dots, a_{12}$, all of them being real. Moving clockwise from the top, a subsequent conic (blue colored) cuts the series denoted $b_i$, etc. One checks easily all conics of the pencil to cut only real points on the $C_6$. Looking at the corresponding dynamical process, we note that $a_1$ is first repulsed against $a_2$ being rejected as far as $c_1$, then attraction of the 4 protons (mostly the North and East one) track back the orbit to position $d_1$ where a repulsion against $d_{12}$ takes place, deflecting again the orbit along the way of the North proton, but then vicinity of $d_2$ causes another repulsion towards $e_1$ and $f_1$, which is finally gently repulsed by $a_{11}$. etc. The sequel of the story reproduces symmetrically. It is now fairly evident how to construct similar dynamical systems of ever increasing complexity. It may be observed that the totally real map induced by the pencil gives a circle map of degree 12. Now the topological invariants are $r=5$ and $g=10$. Hence the half-genus is $p=\frac{g-(r-1)}{2}=3$. Hence this map has degree exceeding Ahlfors bound $r+2p=11(=g+1)$. However a parietal degeneration of the 4 basepoints against the ovals immediately enclosing them (cf. “squigarrows” on Fig.\[Fsext2:fig\]) exhibits a total map of degree $2\cdot 6-4\cdot 1=12-4=8$. This is actually in accordance with the $r+p$ bound predicted in Gabard 2006 [@Gabard_2006]. It is tempting to consider the (mildly singular) foliation induced by the pencil (of conics). It seems clear from the picture that there is a relation between the sum of Poincaré indices extended to the interior of an oval and the number of points circulating on the oval. Observe also that the foliation is transverse to the boundary of the disc bounding the oval. This property is general and follows at once from the fact that totally real maps lack real ramification points. Using Ahlfors total reality paradigm combined maybe with Poincaré’s index formula we suspect that some old (and perhaps new?) information on the topology of real plane (dividing) curves can be re-derived. In particular we suspect that it must be possible to recover Rohlin’s inequality. This states $r\ge m/2$, i.e. any smooth dividing plane curve of order $m$ has at least so many circuits as the half value of its order. This is a fantastic project, but we leave it aside for now. \[vague details p.32 of hand-notes\]. \[08.11.12\] Another highbrow (yet poorly explored) application of Ahlfors theorem was sketched in Gabard’s Thesis (2004 [@Gabard_2004 p.7]). This was an answer to Wilson’s question (1978 [@Wilson_1978 p.67]) on deciding the dividing character of a plane curve by sole inspection of its real locus. Here again Ahlfors theorem affords an answer: a real curve is dividing iff it admits a total pencil (with possibly imaginary conjugate basepoints). Yet it must be admitted that the answer, albeit perfectly geometric, has probably little algorithmic value unless complemented by further insights. Of course another question is to decide the dividing character from the sole data of a ternary form (homogeneous polynomial in 3 variables with real coefficients). The simplest case of Wilson’s question is that of a deep nest, i.e. a smooth curve $C_m$ of say even degree $m=2k$ with a completely nested collection of $k$ ovals. Then linear projection from a point on the deepest oval is total of degree $m-1$. Since the complex gonality is also $m-1$, we deduce that the gonality $\gamma$ is also $m-1$. On the other hand the topological invariants are $r=k$ and $g=\frac{(m-1)(m-2)}{2}$. Hence in this case Ahlfors bound $r+2p=g+1$ is strongly beaten by the gonality $\gamma=m-1<\!\!<g+1=[1+2+3+\dots+(m-2)]+1$. Gabard’s bound $r+p$ is also much greater than the exact $\gamma=m-1$; indeed $r+p$ is nothing but the mean value of $r$ and $g+1$ and in the case at hand the former is $m/2$ but the latter is quadratic in $m$. \[29.10.12\] We consider next an octic (Fig.\[Foctic:fig\]) arising from a sense-preserving perturbation of 4 ellipses rotated by 45 degrees. Of course if all ellipses are oriented clockwise we get a nest of depth 4 and accordingly a total pencil of lines through the innermost oval. Here instead, we reverse some orientations to create 16 ovals and no nesting (cf. black curve on Fig.\[Foctic:fig\]). The theorem of Ahlfors predicts existence of a total pencil. The general principle is to impose basepoints inside the deepest ovals, hence the desired pencil must have degree 4. At this stage depiction can be a fairly difficult artform (reminiscent of gothical “rosaces”= rosewindows). Our trick was to use a ground ellipse of pretty large eccentricity so that oblique line of (angular) slope different from $\pi/4$ (the green and lilac colored ones) also passes through the deep nests. Of course this trick is not supposed to affect the generality of the method (i.e. Ahlfors theorem) but just intended to simplify the artwork! -10pt0 As to the arithmetics, recall that (plane) quartics depends upon $\binom{4+2}{2}-1$ parameters (coefficients counting), hence one is free to assign 13 basepoints. On the other hand, our dividing octic $C_8$ has genus $g=\frac{(m-1)(m-2)}{2}=\frac{7\cdot 6}{2}=21$ and $r=16$ ovals, thus the genus of the half (semi Riemann surface) is $p=\frac{g-(r-1)}{2}=3$. Imagine now that among all 16 basepoints of the pencil 13 moves against the ovals, then a series of (reduced) degree $4\cdot 8- 13\cdot 1=32-13=19$ is obtained. This matches with the $r+p$ bound on the degree of circle maps predicted in Gabard 2006 [@Gabard_2006]. Geometrically it is pleasant to observe that certain members of the pencil are Gürtelkurven (see the lilac-colored curve). Those are not connected. Hence total reality of a pencil is not necessarily allied to connectedness of the auxiliary curves. For the fun of depiction, one can increase the number of curves of the pencil while sweeping out more and more of the full color spectrum, creating a sort of rainbow effect (cf. Fig.\[Foctic2:fig\]). -10pt0 At this stage one gets the impression that the theory (or rather the pictures) works only for highly symmetric patterns. However the strength of Ahlfors result lies in its universal validness for all curves regardless of symmetry. This imbues some suitable respect plus a certain feeling of vertigo about the whole Ahlfors result. Of course there is another possible orthosymmetric smoothing of our configuration of 4 ellipses. This is given by reversing one of the orientations of the ellipses, and we obtain the black-traced curve on Fig.\[Foctic3:fig\]. This times there is only 4 deep nests and a pencil of conics suffices to exhibit total reality. -10pt0 As to arithmetic matters, this octic has still $g=21$ but now only $r=6$ ovals. Hence the semi-genus $p=\frac{g-(r-1)}{2}=8$. Dragging the 4 basepoints against the deep ovals gives a total map of degree $2\cdot 8-4\cdot 1=16-4=12$. This is more economical that the $r+p$ bound, here equal to $14$. Finally there is yet another smoothing of our 4 ellipses producing Fig.\[Foctic4:fig\] with 4 nests of depth 2. A pencil of conics suffices to show total reality. -10pt0 Regarding the topological invariants we have $r=8$, hence $p=7$. As before there is a total map of degree 12 (via parietal degeneration), which is better that Gabard’s bound $r+p=15$. Naively this relative improvement over the previous example (in comparison to the $r+p$ bound) could be explainable by the higher symmetry of the new curve probably reflecting a further particularization of the “moduli”. (Recall that if one believes Gabard 2006 [@Gabard_2006] and especially Coppens 2011 [@Coppens_2011] a bordered surface of type $(r,p)$ has generically gonality $\gamma =r+p$.) Note yet that our total pencil of conics persists for any octic with 4 nests of depth 2, hence the symmetry of the pattern can be greatly damaged by large deformation of the coefficients without affecting the (estimated) gonality. So we certainly have the: Any octic curve with $4$ nests of depth $2$ has gonality $\gamma\le 12$ (and presumably not lower, yet this remains to be elucidated). Having clearly exhausted the smoothing options of our 4 ellipses, one is somehow disappointed that pencils of cubics were not yet required. Looking on p.7 of my Thesis [@Gabard_2004] I rediscover a simple such example involving only a sextic. Let me reproduce this with the rainbow technology. We start now from a configuration of 3 ellipses one of which is a circle and get Fig.\[Fcubic:fig\]. -10pt0 The sextic has $g=10$ and $r=9$ (hence pre-maximal amond dividing curves), and thus $p=1$. Cubics depends on $\binom{3+2}{2}-1=10-1=9$ parameters, hence 8 basepoints may be freely assigned. Pushing them along ovals gives a total map of degree $3\cdot 6-8\cdot 1=10$. This matches with Gabard’s bound $r+p$, hence the curve should be considered has having general moduli. Of course if the smoothing is done very symmetrically and if moreover we play with the radius of the initial circle, we can perhaps arrange that all 9 basepoints lands on the sextic curve in which case the gonality would lover to 9 the minimum value (recall $r\le \gamma$). Starting from the above sextic, one can perform a large deformation of the coefficients staying inside the space of all smooth sextic curves. The real locus picture may then undergo drastic change of shape yet its topological type keeps unaltered and so in particular the orthosymmetric character. It is not clear anymore that our simple minded pencil of cubics (spanned by 2 pairs of 3 lines) suffices to exhibit total reality. This amounts essentially to the claim that for any 8 basepoints distributed among the ovals then the ninth basepoint luckily falls into the remaining one. This luckiness phenomenon becomes even more hazardous when it comes to vindicate Gabard’s bound by a synthetical procedure. The latter seems equivalent to the claim that given any such curve (orthosymmetric with 9 non-nested ovals it is always possible to choose 8 points one on each oval) so that the pencil through them creates an extra basepoint inside the remaining oval. This lucky-stroke phenomenon should perhaps be further explored either as an application of the $r+p$ bound or as a way to disprove it. \[08.11.12\] Let us fail to be more specific as follows. Remember first that a real sextic curve with 9 unnested ovals needs not to be dividing, cf. e.g. Gabard’s Thesis 2004 [@Gabard_2004] p.8, but this is of course well-known since at least the Rohlin-Fiedler era, e.g. Rohlin 1978 [@Rohlin_1978]. Second it is not even clear a priori that the conditions “dividing plus 9 unnested ovals” specifies a unique rigid-isotopy type of curves, i.e. a unique chamber in the space of all smooth sextics. This is a projective space of dimension $\binom{6+2}{2}-1=28-1=27$ parcelled into chambers by the discriminant hypersurface of degree $3(m-1)^2=3 \cdot 5^2=75$. (Inserted \[24.01.13\]: However this is true by a deep result of Nikulin 1979 [@Nikulin_1979/80].) (very hypothetical!!) Any dividing sextic with $9$ unnested ovals admits a total pencil of cubics with $8$ basepoints on the sextic and the $9$th basepoint inside the remaining oval. (pseudo-proof!) Since the curve is dividing we know by Ahlfors that there is a total pencil. We have very poor control on the degree of the curves of the pencil. We only know Ahlfors bound $r+2p=g+1=11$, Gabard’s one $r+p=10$ and the complex gonality $\gamma_{\Bbb C}=5$ which is completely useless. Stronger information comes from the trivial bound $r\le \gamma$. So the gonality $\gamma$ is fairly well squeezed as $9=r\le \gamma \le r+p=10$. A priori a least degree total map could be given by a pencil of quartics. Then the degree could be as low as $4\cdot 6-16=24-16=8$; for quintics as low as $5 \cdot 6-25=5$; for sextics as low as $6\cdot 6-36=0$; septics $7\cdot 6-49=-7$; $k$-tics $k\cdot 6- k^2$ highly negative! Hence we have virtually no control on the degree of (members of) a total pencil, despite the bounds on the degree of the abstract total map. Let us thus shamefully postulate that the pencil in question can be chosen among cubics. For foliated reasons it is clear that the nine basepoints (elliptic points or “foyers” of Poincaré index $+1$) must be surjectively distributed among the 9 ovals. Indeed the total pencil is transverse to the real circuits and the disc bounding an oval cannot be foliated transversely (Euler-Poincaré obstruction). Hence we have the: All basepoints of a total cubics pencil on a smooth sextic with $9$ unnested ovals are real, distinct, and surjectively(=equitably) distributed between the $9$ ovals (either in their insides or their periphery). Applying the parietal degeneration trick we can take any 8 of the basepoints and drag them to the ovals. During the process we get new pencils (of possibly jumping dimension?) while the 9th basepoint could a priori escape its enclosing oval. The difficulty looks so insurmountable that we have to abort the project. In fact the following principle is worth noticing. It gives a basic lower bound on the degree of total pencils, yet as we saw the real difficulty is rather upper bounds! As a matter of annoying nomenclature crash, note that the degree of the pencil is not that of the allied map but that of its constituting curves, so we should perhaps rather speak of the order of a (total) pencil. [(Poincaré-style lower bound on the order of total pencils)]{}\[Poincare-lower-bound\] Given a (smooth) (dividing) plane curve with a total pencil of $k$-tics with $D$ many deepest ovals (i.e. the minimal elements of the nesting ordered structure). Then $D\le k^2$ or $k\ge \sqrt{D}$. Each deep oval must enclose at least one singularity of the foliation. Remember that the latter is transverse to the curve by total reality. Poincaré’s index formula (1882/85) says that the sum of all indices equates the Euler characteristic. Applied to the disc bounding a deepest oval this forces the latter to encloses at least one singularity of index +1. Warning: one must explain why the disc could not be foliated by say two singularity of index $1/2$, so-called thorn singularities. The pencil has at most $k^2$ singularities of the foyer type (index=+1) materialized by the basepoints. Thus $D\le k^2$. Indeed for each deepest oval chose one foyer inside it. We get a map from the set of deepest oval to that of basepoints, which is injective since the deepest ovals are disjoint at least for a smooth curve. Try to clarify if smoothness is really required as a hypothesis! \[30.10.12\] Let us look at another intriguing example. Start again with 2 ellipses invariant under rotation by 90 degrees, and add a concentric circle as the dashed one on Fig.\[Fcubic:fig\], but shrink its radius slightly beyond the critical radius where the circle passes through the 4 intersections of the 2 ellipses. Smoothing this configuration along our choice of arrows gives Fig.\[FcubicA:fig\]: a sextic with $r=9$ ovals one of them enclosing all others. -10pt0 The picture has the annoying property that ovals are pretty small, challenging a bit the visual perception of homo habilis. Since the curve is dividing, Ahlfors theorem predicts the existence of a total map. It is evident that no pencil of lines, nor of conics, is total. (This is either optically clear or deduced from Poincaré’s bound $k\ge \sqrt{D}=\sqrt{8}=2.828\dots$, i.e. Lemma \[Poincare-lower-bound\]). The 8 deep ovals prompts seeking among pencil of cubics. Of course we may just assign 8 basepoints inside those deep ovals and hope for total reality. Yet to manufacture a concrete picture it is natural to assign basepoints in the most symmetric way. Once this is done one try to identify special singular curves passing through the 8 points. We find 4 degenerate cubics consisting of a line plus a conic (cf. colored curves on Fig.\[FcubicA:fig\]). Once those are detected it is an easy matter to interpolate between them (by continuity) to trace a qualitative picture of the pencil (Fig.\[Fcubic3:fig\]). -10pt0 This archipelago sextic $C_6$ has $g=10$ as usual, and $r=9$, thus $p=1$. The total pencil can be lowered to degree $3\cdot 6- 8\cdot 1=10$, as predicted by the $r+p$ bound. \[08.11.12\] Again several questions poses themselves naturally. (The sequel uses some jargon of Rohlin 1978 [@Rohlin_1978], for instance the [*real scheme*]{} of a smooth plane real curve is the isotopy class of the embedding of its real locus in the real projective plane): \(1) Is any sextic $C_6$ belonging to the real scheme of the archipelago (i.e. 8 unnested ovals altogether surrounded by an outer oval) of dividing type? The answer is probably known to Rohlin and his students, especially if there is a nondividing counterexample? \[Update 24.01.13: yes there is one and this was well-known at least since Rohlin 1978 [@Rohlin_1978], yet his article was far from explicit when it comes to constructions. Personally I understood this point only after reading Marin 1979 [@Marin_1979], compare our Fig.\[GudHilbMarin:fig\] much below (virtually copied from Marin), which is a (clever) variant of Hilbert’s method of vibration.\] Rohlin distinguishes real schemes as definite or indefinite depending on whether all its representatives belong to the same type or not, w.r.t. Klein’s dichotomy (ortho- vs. diasymmetric). (cf. Rohlin 1978 [@Rohlin_1978]) \(2) What is the exact gonalities occurring in this archipelago scheme (of course restricting attention to dividing models in case the scheme is indefinite)? If we believe in Gabard’s bound $\gamma\le r+p$, we have $9=r\le \gamma \le r+p=10$. Perhaps answers are to be searched along the following direction. Maybe it is true that for any 8 basepoints (injectively) distributed in the 8 deepest ovals the corresponding cubics pencil is total. On counting intersections, we get roughly $8\cdot 2=16$ many coming from the 8 deep ovals and the outer oval should also contributes for 2 intersections. This is at least evident if the real part of the cubics are connected since the real circuit of each such cubic has to go at “infinity” (in the sense of moving outside the outer oval, for otherwise it would be contractible inside the bounding disc of the latter, whereas we know the cubic circuit to realize an “odd” nontrivial class in the fundamental group $\pi_1({\Bbb R}P^2)$ or just the allied homology). On the other hand, the cubic circuit must also visit the 8 assigned basepoint inside the outer oval, and so is forced to intercepts the latter. We arrive at a total of 18 real intersections, the maximum permissible by Bézout ($3\cdot 6=18$). Total reality would follow. I remind vaguely of a standard result claiming that for a generic collection of 8 points there is a pencil of rational (hence connected) cubic interpolating them. (Cf. e.g. Kharlamov-Degtyarev survey ca. 2002). Now if all this is true, the archipelago scheme is dividing, and any such curve admits plenty of total cubics pencil of degree $3\cdot 6- 8\cdot 1=10$ (essentially one for each selection of 8 points on the deep ovals). It seems however hard to lower the gonality $\gamma$ up to the absolute minimum $r=9$, but I know no argument. Total reality in the Harnack-maximal case {#sec:Total-reality-Harnack-max-case} ----------------------------------------- \[08.03.13\] Much of this section is by now much illuminated by Le Touzé’s observation in Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics], where it is remarked that a very simple prescription of basepoints ensure total reality of a pencil of cubics on an $M$-quintic. \[ca. 31.10.12\] Quite paradoxically it is much harder to depict total pencils on Harnack-maximal curves, alias $M$-curves (in Russia since Petrovskii 1938 [@Petrowsky_1938], cf. Gudkov 1974 [@Gudkov_1974/74 p.18]), especially when the order is $m\ge 5$. (For lower orders $m\le 4$ everything is essentially trivial: since $m=4$ just requires a pencil of conics passing through the 4 ovals of the quartic (with $g=3$).) Recall indeed that Ahlfors theorem is much easier in the planar case $p=0$, where it goes back to Bieberbach-Grunsky, if not earlier. Logically the argument simplifies much via Riemann-Roch and the absence of collision, cf. e.g. Gabard 2006 [@Gabard_2006 Prop.4.1] or Lemma \[Enriques-Chisini:lemma\] above in this text. Shamefully, the following section climaxes the poor level of organization of the present text. Of course the game is quite outside the main stream of our subject (Ahlfors theorem), yet we think that some phenomena require to be clarified. In particular we were not able to make any reliable picture of a total pencil on a Harnack-maximal (smooth) plane curve of order $m\ge 5$. After some three days of pictorial tergiversation we found a sort of weak obstruction to manufacturing such pictures involving a basic type of pencil spanned by two special cubics. This obstruction is described at the end of the section, which otherwise reduces to a messy gallery of failing attempts of the desired easy depiction! Yet the abstract theorem of Bieberbach-Grunsky implies the existence of total pencil but they probably involve delicate-to-visualize pencil of cubics (in the quintic case). We would like to challenge gifted amateurs to picture them appropriately. Let us first recall the construction of such $M$-curves due to Harnack (in the variant of Hilbert). We start with degree 5. Consider as primitive configuration an ellipse $E_2$ plus a line $L_1$. Take further 3 parallel lines $l_1,l_2,l_3$. There is some psychological difficulties to know if we should first smooth $E_2 \cup L_1$ and then perturb along $l_1\cup l_2\cup l_3$ or if we can directly perturb $E_2 \cup L_1$ without taking care of smoothing. Let us adopt the shorter route (actually so do Hilbert) by putting directly $C_3=(E_2\cup L_1)+\varepsilon \vartheta_3$. This cubic (in black thick stroke) oscillates across the ellipse $E_2$ meeting it in the maximum number of 6 real points. Next smoothing their union (=product) $C_3 \cup E_2$ we get the (red-colored) quintic $C_5$ realizing the maximum number $r=7(=g+1)$ of ovals (one of them being in fact a pseudoline i.e. a Jordan curve in ${\Bbb R}P^2$ not bounding a disc). -10pt0 \[31.10.12\] Now the (perpetual) game is to find a total pencil on this dividing curve $C_5$ (recall that Harnack-maximal curves are always dividing). As usual the recipe is to distribute imposed basepoints $p_1, \dots, p_6$ in the deepest ovals. Those are fixed once for all and marked by black points on Fig.\[Harna1:fig\]. Since there are 6 ovals, pencil of lines or conics are not flexible enough to reveal the total reality of our $C_5$. We thus have to look among pencils of cubics. In view of the (vertical) symmetry of the curve $C_5$ it is natural to seek a symmetric pencil. We shall define them by specifying two of its members. A first vertically symmetric cubic through the 6 basepoints is the union of the 3 cyan-colored lines. This special (cyan) cubic $C_3$ cuts our quintic $C_5$ twice along each oval and once on the pseudoline, hence in $12+1=13$ points. Those are at finite distance but looking at infinity both horizontal cyan lines cuts the pseudoline branch of $C_5$ in two extra points, yielding a total of 15 point, the maximum possible (all of them being real). Beside, we consider another vertically symmetric cubic, namely the red-colored cubic $R_3$ consisting of the red ellipse through 5 points $p_i$ plus the red horizontal line (denoted $C$) through the remaining $p_i$. We can now consider the corresponding pencil spanned by the cyan and red cubics (equation $\lambda C_3+ \mu R_3=0$). Unfortunately, the red cubic cuts $C_5$ along $2\cdot 6=12$ points on the ovals and only once at infinity. Indeed the pseudoline branch of $C_5$ is asymptotic to the line $D$ which in transverse to the red line $C$. Hence the intersection $R_3\cap C_5$ is not totally real. Of course this defect does not prevent us from tracing the corresponding (non-total) pencil. -10pt0 [*Note.*]{}—A pencil of cubic may be defined by assigning 8 basepoints. By letting degenerate those against the 6 ovals (or the pseudoline) we get a series of degree $3\cdot 5 - 8\cdot 1=15-8=7$ as predicted by Bieberbach-Grunsky (cf. e.g. Lemma \[Enriques-Chisini:lemma\]). But it is far from evident to ensure total reality. Of course a coarse calculation would stipulate that the 6 ovals contributes for $2\cdot 6=12$ many intersections and imposing 2 extra basepoints on the pseudoline gives 2 additional intersection, totalizing 14 many hence the last man surviving is forced to be real as well. This argument certainly holds good if we know that all cubics of the pencil are connected but a priori a cubic may well have an oval which could be nested in one of the tiny ovals of our sextic. If so is the case then this one cubic’s oval only visits one of the 8 basepoints, without spontaneous creation of intersection on one oval of the quintic $C_5$. Maybe this scenario is quite improbable but I missed some argument. A modest improvement over our previous attempt is to take a red-colored cubic satisfying total reality. This is given by changing the red-colored ellipse by taking the one passing through the 5 “highest” (relatively to our figure Fig.\[Harna2:fig\]) black-colored basepoints. Symmetry forces us then to take an additional red-colored line passing through the “lowest” basepoint. We obtain the following Fig.\[Harna2:fig\]. Alas it is not evident that total reality is satisfied. -10pt0 A third option is to change the cyan configuration of 3 lines and we get the following Fig.\[Harna3:fig\], which alas again seems to fail total reality. -15pt0 \[01.11.12\] Of course we would like ultimately to extend the game to sextic. Let us first reproduce a picture in Hilbert 1909 [@Hilbert_1909-Ueber-die-Gestalt-sextic]. The idea is again that a union of two ellipses is vibrated into a quartic $C_4$ oscillating across one of the ellipse $E_2$ (which is a circle on Fig.\[Hilb1:fig\], left), and next $E_2\cup C_4$ is smoothed to a sextic with 11 ovals (compare Fig.\[Hilb1:fig\], right). -10pt0 Again the challenge would be to trace a total pencil of curves on this $C_6$. We have 10 deep ovals, thus pencils of cubics look overwhelmed already with their only 8 assignable basepoints (and maximally 9 of them). Quartics have $\binom{4+2}{2}-1=15-1=14$ free parameters hence we can impose 13 basepoints. Choosing them in the deep ovals and doing a parietal degeneration gives a series of degree $4\cdot 6- 13 \cdot 1=24-13=11$. This matches with the Bieberbach-Grunsky bound, however it is far from evident that total reality is ensured. In general if $C_m$ is a Harnack-maximal curve of order $m$, the previous examples (with $m=5,6$) suggest to consider auxiliary curves of degree $m-2$ forming a space of dimension $\binom{(m-2)+2}{2}-1=\binom{m}{2}-1$ and thus assigning $\binom{m}{2}-2$ basepoints will define a pencil. By parietal degeneration the resulting series has degree $(m-2)m-[\binom{m}{2}-2]$, and this is easily calculated as being equal to $$\begin{aligned} (m-2)m-[\binom{m}{2}-2]&=(m-2)m-\frac{m(m-1)}{2}+2\cr &=\frac{1}{2}[2(m-2)m-m(m-1)+2]+1=\frac{1}{2}[m^2-3m+2]+1\cr &=\frac{(m-1)(m-2)}{2}+1=g+1, \end{aligned}$$ where $g$ is the genus. This again agrees with the Bieberbach-Grunsky theorem, but of course does not reprove it, be it just for the simple reason that smooth plane curves have specialized moduli among all curve sof the same genus. Still it would be exciting to manufacture tangible pictures of such total pencils in the planar case. Now let us try again to do better pictures of the $M$-quintic. Any such $M$-quintic has 6 ovals and one pseudoline. By Bézout no three ovals can be aligned (otherwise 6 intersection with a line). Thus the six ovals are somehow distributed along a configuration resembling a hexagon. This raises some hope to draw reasonable pencil of cubics spanned by two configurations of 3 lines according to one of the following patterns (left of Fig.\[Hilb2:fig\]). This suggested to draw another model whose 6 ovals are nearly situated like a regular hexagon. Using cyclotomy, we get quickly the right part of Fig.\[Hilb2:fig\]. -40pt0 A little piece of comment on the last Fig.\[Hilb2:fig\]: of course we started with a circle divided primarily in 6 equal parts, and have chosen the 3 horizontal lines as passing through the cyclotomic points. Those three lines are those used for the Harnack-Hilbert vibration trick, and the rest of the picture should be self-explanatory. Alas the bottom portion is quite difficult to observe. Yet a clear-cut portrait of Lars Valerian clearly emerges: the bottom oval is the mouth, then just above two big eyes “with an air of determination”, as well as some hairs emanating from the beret. In fact the portrait looks more like an alien, but the resemblance with Lars is much more flagrant when the circle is depicted as a “vertically oblong” ellipse. \[I apologize for adding some extra prose as otherwise the figures desynchronize from the text.\] Now we consider the following pencil spanned by the cyan and red collections of lines (Fig.\[Hilb3:fig\]). Alas it fails to be totally real, for it contains the green cubic cutting only 13 real points on the quintic $C_5$. Of course the advantage of our pencil is that it is simple to draw, yet its disadvantage is that it has only 6 among the 8 assignable points located on the quintic. Somehow one should try to conciliate both properties. -25pt0 Testing the other configuration (of 2 pairs of 3 lines through the hexagon) one gets Fig.\[Hilb4:fig\]. The situation is not much improved. Now the 3 additional basepoints (intersection of pairs of parallel lines) are ejected at infinity but are not lying on the (black-colored) quintic curve $C_5$ whose pseudoline is asymptotic to the horizontal line. The corresponding pencil of cubics (spanned by the cyan and red colored lines) is probably not total, for it should contain a nearly circular ellipse through the hexagon plus the line at infinity, and the aggregated corresponding cubic seems to cut the $C_5$ only along 12, plus one at infinity, so a total of only 13 real points!? -0pt -10pt0 One can also make the following picture Fig.\[Hilb5:fig\], where the 3 additional basepoints are marked by circles, one of them lying, alas, quite outside the range of the picture. A possible, yet delicate, desideratum would be to distort the configuration (pair of 3 lines arrangements) so that 2 of those circled basepoints lands on the quintic $C_5$. Then we would get a good candidate for an easy to depict total pencil of cubics on our quintic. Evidently this desideratum is probably impossible to arrange (a so-called “Irrweg”). -0pt -10pt0 Maybe another arrangement worth looking at is the following Fig.\[Hilb6:fig\]. Now among the 3 extra basepoints at least one (that one corresponding to the intersection of both horizontal lines) is located on the quintic $C_5$ (at infinity). Hence 13 points are ensured to be real for all members of the pencil. It is easily checked that both fundamental curves of the pencil (cyan and red cubics) cut the $C_5$ in a totally real fashion (15 real points). For symmetry reasons (along the axe at 120 degrees) the nearly circular ellipse through the 6 points at finite distance plus the line at angle 120 degrees belongs to the pencil, but alas its intersection with the $C_5$ it hard to understand. Note by the way that the hexagonal configuration of 6 points is slightly perturbed thus there is no perfectly well defined such ellipse. At this stage the whole exercise is akin to a dolorous acupuncture session. Note that our symmetry deduced member of the pencil has the wrong behavior through the basepoints at infinity, hence the right curve belonging to the pencil includes rather the line at infinity (or at least a slight perturbation thereof). Thus we count 12 intersections with the oval coming from the nearly circular circuit, and just one intersection at infinity. This underscored total of 13 seems to indicate that this pencil again fails total reality. -10pt0 Albeit our exposition is not from the best stock, we hope at least to have demonstrated that the synthetic construction of total pencils on $M$-curves is not an easy matter. Of course it is not improbable that I missed something fairly easy! [**Isoperimetric digression.**]{} During the session I wondered if the following problem makes sense. One of the notorious difficulty when trying to do real pictures of algebraic curves is that some ovals tend to be microscopic (especially for Harnack-maximal curves). Is there some optimal curve best suited for depiction? Admittedly the problem makes sense only for Euclidean affine models as opposed to projective curves (which could be pictured on the sphere up to a double cover). One could for instance ask the curves to enclose maximum area for a given length of the circuits. (Of course this makes sense only for curves of even degrees, except if we neglect the pseudoline.) This would be a sort of isoperimetric problem for curves competing among algebraic ones (of some fixed degree). Of course for degree two the isoperimetric solution is the circle. What about degree 4? A candidate is perhaps the Fermat curve $x^4+y^4=1$ whose real picture is somewhere between a circle $x^2+y^2=1$ and a square $x^{\infty}+y^{\infty}=1$. Of course one could argue that the optimal quartic is just a circle counted by multiplicity 2, but then the length of the circuit has to be counted twice. We have no certitude that our problem is well posed, nor that it is truly interesting. The naive scenario would be that the optimum is always the Fermat curves of higher even orders, yet what about $M$-curves? Maybe we need to restrict the problem to them, and ask for the best Euclidean realization of an $M$-curve? So for instance what is the best $M$-quartic? The best $M$-quintic? Does it looks like Ahlfors’ portrait (on Fig.\[Hilb2:fig\])? Let us a last time return to our main problem of tracing a totally real pencil for an $M$-quintic. Remember once more that theoretical existence is ensured by the baby case (Bieberbach-Grunsky) of Ahlfors theorem on circle maps. Our dream would be that for such a quintic there is a simple-to-draw pencil generated by 2 configurations of 3 lines. Psychologically it is helpful to reverse the viewpoint. Instead of starting from the quintic $M$-curve $C_5$ and trying hard to depict the pencil, we shall start from the pencil and try to construct a curve tailored to it. So we consider the pencil generated by 2 systems of parallel lines (colored cyan and red) with 9 basepoints (multicolored intersections) and try to build around this perfectly explicit pencil (cf. the previous Fig.\[Fcubic:fig\] Fig.\[Hilb6b:fig\]b below) a quintic having the following schematic picture (Fig.\[Hilb6b:fig\]a). This is to mean that each of the 6 ovals encloses one of the 9 basepoints, with the Bézout restriction that no aligned triad are enclosed (else 6 intersections in $C_5$ with a line) and further the pseudoline passes through 2 other basepoints. If such a “real scheme” (Rohlin’s jargon) exists then each curve of the pencil will cut on the $C_5$ a total of 15 real points. Indeed the 6 ovals contribute each for twice (now Fig.\[Hilb6b:fig\]b ensures connectedness of all cubics forming the pencil!) and the pseudoline for 2, hence a total of 14 and the last one is forced to be real as well (for algebraic “Galois theoretic” reasons). -10pt0 So exhibiting this scheme would complete our goal. Note the absence of Bézout-type obstruction to the posited real scheme (Fig.\[Hilb6b:fig\]a). Yet maybe there is deeper topological obstructions involving say the foliation underlying the pencil. In fact the argument is more modest. The two basepoints connected by the pseudoline are separated by the green ellipse. So the arc joining them (choose one!) is forced to have an extra intersection with the green ellipse (on Fig.\[Hilb6b:fig\]b). Topology forces the creation of a second intersection (intuitively the pseudoline once trapped in the green ellipse has to escape it). Thus we arrive at a total of $12$ (6 ovals), plus the $2$ assigned basepoints on the pseudo-line and plus the 2 extra-points just created. This gives 16 intersections between $C_5$ and the green cubic (enough to overwhelm Bézout). This prohibits the desired scheme. Another (a priori) tangible real scheme is the one depicted on Fig.\[Hilb6b:fig\]c. Then it seems that arguing with the lilac conic we may repeat something like the previous argument. More precisely, if the pseudoline never penetrates inside the lilac ellipse $L_2$ then it has to be tangent to it at the 2 assigned basepoints but this gives already 4 extra-points which added to the 12 God-given produce an excess $16>15$! Thus we may assume the pseudoline $P$ to penetrate in the lilac ellipse (total of 13 intersection). Then several cases may occur. If $P$ tries to evade from the lilac ellipse $L_2$ then we have $14$ intersections, yet it must still pass to the second basepoint and (being now outside the $L_2$) this creates at least 2 intersections (counted by multiplicity). So eventually the pseudoline $P$ is forced to reach the other basepoint while staying inside the lilac $L_2$, and hence to cut the lilac axis of this ellipse. The latter axis being contained in the inside of the green ellipse, we get again 4 extra intersections with the green cubic (beside the 12 arising de facto from the ovals); too much for Bézout. All this (if correct?, and suitably simplified!) should prove the following: It is impossible to sweep out in a totally real fashion an $M$-quintic via a basic pencil of cubics spanned by two arrangements of parallel lines. If true and suitably generalized to other configurations (see $\bigstar$ right below) this explains perhaps why we had so much trouble to make an appropriate depiction of the desired pencil. Again totally real pencils exist in abstracto hence in concreto, yet are probably of a somewhat more elaborated vintage. \[02.11.12\] $\bigstar$ For instance it should be noticed that there is another possible scheme (distribution of 6 ovals) satisfying the “no-three-in-line” condition prompted by Bézout. This is depicted on Fig.\[Hilb6b:fig\]d which is admissible provided the horizontal diagonal is not aligned. Hence the real picture looks rather like Fig.\[Hilb6b:fig\]e. Of course it would be too cavalier to claim that the previous obstruction to the case at hand as the ellipses were destroyed during the process. We leave the problem in this very unsatisfactory state of affairs, but let us perhaps try to motivate why the explicit depiction project could be fruitful! From the viewpoint of gravitational systems (cf. the previous Sec.\[sec:electrodynamics\]) the interest of $M$-curves is that they express in some sense the most complex orbital structure permissible for a given genus (at least the maximum number of real circuits). Hence if Metatheorem \[metatheorem:thm\] is reliable such $M$-curves should display some remarkable motions. The intricacy of the trajectories is already suggested by Hilbert’s $M$-sextic on Fig.\[Hilb1:fig\]. However until the total pencil (of Bieberbach-Grunsky-Ahlfors) is not made explicit the dynamics of the electrons is imbued by mystery and darkness. Remind from Bieberbach-Grunsky (=Lemma \[Enriques-Chisini:lemma\]) that Hilbert’s $M$-sextic is not only static object but one animated by a circulation (total pencil) having one electron on each oval. We can from the static picture vaguely try to guess where repulsions take places and arrive at something like Fig.\[Hilb7:fig\]. 0 -10pt0 On Fig.\[Hilb7:fig\], italics numbers enumerate ovals while roman numbers indicates positions at various times $1,2,3,4$. Note that our Harnack-maximal curve being dividing, it has a complex orientation (as the border of one half). This orientation agrees with that inherited from the smoothing. Further it has to be respected by the circulation due to the holomorphic character of the (Bieberbach-Grunsky) circle map. Having this is mind it is straightforward to make the picture above (Fig.\[Hilb7:fig\]) using the rule that whenever a repulsion is observed then electrons must be in close vicinity and thus any pair of points minimizing the distance between two neighboring ovals must be synchronized, hence labelled by the same time unit. In contrast when two close ovals do not repulse them (like ovals [*1*]{} and [*10*]{}) then they must be anti-synchronized in the sense that both particles do not visit the contiguity zone at the same moment. For instance there is also a repulsion between electrons on ovals [*1*]{} and [*11*]{} at time 1. So far so good. However on completing the picture one sees between ovals [*6*]{} and [*11*]{} some anomalous (asynchronic) repulsion. Maybe one can explain this via distant repulsion involving other particles of the system (especially the electron on oval [*10*]{}). All this is very informal and saliently illustrates the sort of obscurantism caused by a lack of explicit knowledge of the total pencil. This perhaps motivates once more to complete the programme of the present section (construction of total pencils in Harnack-maximal cases). Ultimately one could dream of a computer program showing in real time the circulation of electrons prompted by the Bieberbach-Grunsky Kreisabbildung(en) along an Hilbert $M$-sextic. Let us finally observe that there are other $M$-sextics (Harnack’s, Hilbert’s and even Gudkov’s). Basically the one we depicted (Hilbert’s) is gained by smoothing the configuration $E_2\cdot C_4=0$ consisting of an ellipse $E_2$ (circle on the picture) and an $M$-quartic $C_4$ one of whose oval oscillates across the ellipse $E_2$. It may be noticed that the oscillating oval lies mostly inside the ellipse (cf. the left-top part of Fig.\[Hilb8:fig\]). \[This schematic—yet Bézout compatible—style of depiction is borrowed from Gudkov 1974 [@Gudkov_1974/74 p.20].\] One can reverse this situation, by putting the vibrating oval outside the ground ellipse to get another $M$-sextic (cf. the right-top part of Fig.\[Hilb8:fig\]). A concrete construction this is achieved on the bottom part of Fig.\[Hilb8:fig\]). -10pt0 This curve has one “big” oval enclosing nine “small” ovals and the other lies outside. Of course if our metatheorem (\[metatheorem:thm\]) is plausible then it is challenging to interpret the dynamics especially the orbit along this long oval enclosing all others but one. Of course this would essentially boils down to visualize a total pencil for this $C_6$. Finally let us make a little remark. We see that there must a deep reaching connection between Ahlfors theory of circle maps and the extrinsic geometry of real dividing curves, the link being given by the notion of total pencil. Another basic application of total pencils could arise in curve plotting problems. Assume given an algebraic equation $f(x,y)=0$ and a machine supposed to make a plot of the real locus. Suppose e.g. that the polynomial has degree 5, defines a smooth curve and that we have already traced within reasonable accuracy 2 ovals and a pseudoline and finally that both ovals are nested. Then the theory of total maps (but in fact Bézout suffices) ensures that the real locus has already been exhausted and we may stop the “root finding” algorithm. Of course the story becomes even more grandiose on appealing to Newton-Cayley iteration method and the allied fractals appearing as attracting basins. Likewise if an octic has 4 nests of depth 2 its real locus has already been exhausted (compare Fig.\[Foctic4:fig\]). Indeed in that case the pencil of conics through the 4 deeply nested ovals imposed to pass through another hypothetical point would create an excess of $8\cdot 2+1=17>16$ intersection points. A baby pseudo-counterexample in degree 4 {#sec:degree-four} ---------------------------------------- We now give an example in degree 4. The recipe is always is the same and we get the example 102 below (Fig.\[F102:fig\]). It has $g=0$, $r=1$, thus $p=0$. At first glance the visual gonality as measured via a pencil of lines is $\gamma^{\ast}=2$ (projection from one of the nodes). This seems of course to violate Gabard’s bound $\gamma\le r+p$. However using a pencil of conics passing through the 3 nodes plus the point (labelled $8$ on the figure) gives a total pencil of the right degree. Of course the example is a paroxysm of triviality, yet it is still a nice case to visualize the fairly complex dynamics of total pencils. The forward semi-orbit of the series is depicted by points $1,2,\dots, 8$ after which the motion reproduces symmetrically. Another example arises when we keep less singularities unsmoothed. We obtain so a linkage of “heartsuits” (cf. the middle picture 202 on Fig.\[F102:fig\]). Now $r=2$, $g=3-2=1$, and so $p=0$. A linear projection from one of the 2 nodes suffices to exhibit total reality, and so the gonality is $\gamma=2$. One can still trace pencils of conics through the nodes plus 2 extra points on the curve to get series of degree $2\cdot 4- 2\cdot 2-2\cdot 1=8-4-2=2$. Those gives more maps realizing the gonality. Of course one can also materialize such a curve as a smooth plane cubic, in which case we also see $\infty^1$ total pencils induced by linear projection from the unique oval. (Projecting from the pseudo-line, the oval of the cubic has some “apparent contour” and total reality fails.) One can also get the bottom picture 202 on Fig.\[F102:fig\], which has the same invariants. Low-degree circle maps in all topological types by Harnack-maximal reduction {#sec:Chambery} ---------------------------------------------------------------------------- \[Source=Gabard 2005, Chambéry talk (unpublished as yet)\] Once Ahlfors theorem is known in the simple Harnack-maximal case (cf. Lemma \[Enriques-Chisini:lemma\]) one can easily exhibit in any topological type some very special surfaces (in Euclid’s 3-space) admitting a circle map to the disc having very low degree. Of course this is far remote from reassessing the full Ahlfors theorem, yet it is an interesting construction, which perhaps could lead to a general proof when combined with some Teichmüller theory. But this is only a vague project we shall not be able to pursue further. Let us start with a membrane in Euclidean 3-space (endowed with the conformal structure induced by the Euclidean metric). Suppose the surface invariant under a symmetry of order two (cf. Fig.\[Chambery:fig\]). The key feature of this figure is that the axis of rotation “perforates” each “hole” of the pretzel. Hence, when taking the quotient all handles are killed, and we get a proper(=total) morphism to a schlichtartig configuration (i.e. of genus $p=0$). This in turn admits a circle map of degree equal to the number of contours (by the Bieberbach-Grunsky theorem=Lemma \[Enriques-Chisini:lemma\]). The composed mapping gives a circle map of degree $2\cdot \frac{r}{2}=r$ when $r$ is even, and of degree $2\cdot \frac{r+1}{2}=r+1$ if $r$ is odd. (Compare again Fig.\[Chambery:fig\].) Of course this has little weight in comparison to the general theorem of Ahlfors (1950 [@Ahlfors_1950]), yet it is a simple example showing that the degree of circle maps can be fairly lower than the degrees $r+2p$ or even $r+p$. Experimental evidence for Coppens’ gonality ------------------------------------------- \[24.03.12\]/\[19.10.12\] In this section we discuss Coppens result (2011 [@Coppens_2011]) on the realizability of all gonalities compatible with the $r+p$ bound (Gabard 2006 [@Gabard_2006]) on the degree of an Ahlfors circle map. Our superficial approach will not recover Coppens full result, yet is worth presenting for it enhances the depth of Coppens’ result. Looking at explicit projective models of Riemann surfaces always makes Riemann-type existence theorems (like Ahlfors maps) look quite formidable jewels (not to say miracles) when looked at experimentally through the Plato cavern of extrinsic algebraic geometry. The game is also pleasant because sometimes one gets the impression that Gabard’s bound $r+p$ looks blatantly violated. Also interesting is the issue that such basic experimental studies (akin to the CERN particles collider at a modest scale) are quite useful for understanding the failure of connectivity of the space of minimal circle maps (those of lowest possible degree). Further experiments should contribute to add some valuable insights over Ahlfors’ theory. (A. Einstein puts it as follows: “Any knowledge of the world starts and ends with experiments.”) Coppens’ result is the following. To stay closer to Ahlfors’ viewpoint, we paraphrase it in the language of [*compact bordered Riemann surfaces*]{} (abridged membranes) instead of that of real dividing curves. Albeit most of our examples are derived via algebraic geometry, we will never have to write down any (boring) equation due to the graphical flexibility of plane curves à la Brusotti/Klein/Plücker (reverse historical order). So we are drifted to a sort of synthetic geometry. [(Coppens 2011 [@Coppens_2011])]{} Given any two integers $r\ge 1$ and $p\ge 0$, and any integer $\gamma$ satisfying $\max\{2, r \}\le \gamma \le r+p$, there is membrane $F_{r,p}$ with $r$ contours of genus $p$ whose gonality is the assigned value $\gamma$. Recall that the gonality of the membrane is understood as the least degree of a circle map from the given membrane (to the disc). $\bullet$ For $(r,p)=(1,0)$, the statement becomes vacuous, but of course we can alter the range of permissible values as $r\le \gamma\le r+p$. $\bullet$ When $p=0$, $\gamma$ can take only the value $r$ and the latter is realized via the Bieberbach-Grunsky theorem (Lemma \[Enriques-Chisini:lemma\]). $\bullet$ For $(r,p)=(1,1)$, the double has genus $g=(r-1)+2p=2$ hence is hyperelliptic. This actually proves the existence of a circle map of degree 2 ($=r+p$) in accordance with Gabard’s bound $r+p$. Coppens’s realizability theorem is trivially verified in this case for $\gamma$ can only assume value 2. $\bullet$ For $(r,p)=(2,1)$, the range of $\gamma$ is $2\le \gamma \le 3$. The value $\gamma=2$ is realized by a hyperelliptic model. The value $\gamma=3$ is obtained by considering a smooth quartic $C_4$ with two nested ovals while projecting it from a point on the innermost oval. This gives a [*totally real*]{} morphism of degree $3$. Total reality means that fibers above real points consists entirely of real points. We use also the abridged jargon [*total map*]{} which is quite in line with terminology used by Stoïlow 1938 [@Stoilow_1938-Lecons] or Ahlfors-Sario 1960 [@Ahlfors-Sario_1960], who use “complete coverings”. -75pt 0 $\bullet$ For $(r,p)=(3,1)$, the genus of the double is $g=(r-1)+2p=2+2=4$. This is not the genus $g=\frac{(m-1)(m-2)}{2}$ of a smooth plane curve of order $m$ which belongs to the list $0,1,3,6,10, \dots$ of triangular numbers, yet suggests looking at a quintic $C_5$ with two nodes. We thus consider a configuration of two conics plus a line and smooth it out in a orientation preserving way (so as to ensure the dividing character of the curve by a result of Fiedler 1981 [@Fiedler_1981]). We obtain so the curve depicted on Fig.\[Coppens:fig\] bearing the nickname 313. This actually encodes the value of the invariant $(r,p, \gamma)$ written as the string $rp\gamma$, yet a priori the gonality $\gamma$ is not known and its value must be justified. On that figure 313 the dashed circles indicate those crossings that were [*not*]{} smoothed. The half of this curve is a bordered surface of type $(r,p)=(3,1)$, since $p=\frac{g-(r-1)}{2}$. It remains to evaluate its gonality. The idea is always to look at the curve from the innermost oval. In the case at hand, we project the curve from one of the two nodes to get a total morphism of degree $5-2=3$. Since $r=3$ is a lower bound on the gonality $\gamma$, it follows that $\gamma=3$, exactly. Note that this example seems to [*answer in the negative our question about the connectivity of the space parameterizing minimal circle maps*]{}. Further one can drag one point to the other while travelling only through [*total*]{} maps of degrees 4 (namely projections from points located in the intersection of the interiors of the blue resp. red ovals). \[09.11.12\]—[*Warning.*]{} Remember that a similar picture (Fig.\[F102:fig\], right-middle part) gave an example where the curve looked 2-gonal in only 2 ways, but another model of the curve (as a plane cubic) prompted the same gonality in $\infty^1$ fashions. So some deeper argument is required either to assess (or disprove) the italicized assertion. Next, still for the same topological invariants $(r,p)=(3,1)$, we would like to find a membrane of gonality $\gamma=4$. This may be obtained from the same initial arrangement while moving the location of the dashed circles (of inert crossings) to get picture labelled 314 on Fig.\[Coppens:fig\]. The corresponding quintic projected from a point situated on the inner (blue-colored) oval has $\gamma\le 4$. Over the complexes, this quintic has gonality $\gamma_{\Bbb C}=3$ (projection from one of the nodes) and this is the only way for the curve to be trigonal. Yet over the real picture (our 314) none of these (trigonal) projections is total (since the inner oval has an apparent contour, i.e. some tangent to it passes through the node). It follows that $\gamma=4$, exactly. $\bullet$ Let us next examine $(r,p)=(4,1)$. Then $g=(r-1)+2p=5$, so we look at quintics with one node. To create as many ovals, it proves convenient to reverse the orientation of one of the conics. We obtain so the figure coded 415. After noting that $r=4$, we project the curve via a pencil of conics assigned to pass through 4 points chosen in the innermost ovals (asterisks on the figure). Letting those 4 points degenerate against the ovals while exploiting the possibility of pushing one of them toward the node (so as to lower by 2 units the degree) we find $\gamma\le 2\cdot 5- 3\cdot 1 -1\cdot 2=10-3-2=5$. Over the complexes, the curve at hand (uninodal quintic) is trigonal only when seen from its unique node and 4-gonal only when projected from a smooth point. Inspection of the figure shows that none of these maps is total. It follows that $\gamma=5$ exactly. It remains to find an example with $\gamma=4$. For this we just drag below the dashed circle (cf. label 414 on Fig.\[Coppens:fig\]), do the prescribed smoothing (always in the orientation consistent way). The resulting curve has $r=4$ (as it should). The novel feature is that the node is now accessible from 2 basepoints of the pencil of conics assigned in the deep ovals. This permits a lowering of the degree to $\gamma\le 2\cdot 5- 2\cdot 1 -2\cdot 2=10-2-4=4$. Remarking that the unique morphism of lower degree 3 (linear projection from the node) is not total we deduce that $\gamma=4$ exactly. The other morphisms of degrees 4 (namely projections from real points on the curve) obviously fails to be total, thus we infer that the curve (or the allied membrane) is uniquely minimal (i.e. there is a unique circle map of minimum degree). Before embarking on larger values of the invariants $(r,p)$, we make a general remark, related to the previous Sec.\[sec:Chambery\]. There a suitable membrane in 3-space invariant under rotation by $\pi=180^{0}$ with a totally vertical array of handles (cf. Fig.\[Chambery:fig\]) showed the following: [(Barbecue/Bratwurst principle)]{}\[Barbecue:lem\] $\bullet$ If $r$ is even, there is for any value of $p$ a membrane of type $(r,p)$ admitting a circle map of degree $r$ (the minimum possible value), whose gonality is therefore $\gamma=r$ exactly. $\bullet$ If $r$ is odd ($p$ arbitrary), there is a membrane of type $(r,p)$ admitting a circle map of degree $r+1$, whose gonality $\gamma$ is therefore $r\le \gamma\le r+1$. (Alas, the exact value remains a bit undetermined!) This lemma fills quickly several positions of our Fig.\[Coppens:fig\], namely those marked by a square. In the special case $r=1$ (belonging to the indefinite odd case), we can get rid off the annoying indetermination, because as soon as $p\ge 1$ the minimal value $r$ of the range $r\le \gamma\le r+1$ cannot be attained. Corresponding invariants are reported by rhombuses (squares rotated by $\pi/4$) on Fig.\[Coppens:fig\]. $\bullet$ Next we study $(r,p)=(5,1)$. Then $g=(r-1)+2p=6$, prompting to look at smooth quintics (without nodes). Consider the curve denoted 516 on Fig.\[Coppens:fig\], which has $r=5$. When projected via a pencil of conics through the assigned 4 basepoints (depicted by asterisks on the figure) and letting them degenerate toward the ovals gives a total map of degree $2\cdot 5- 4\cdot 1 =10-4=6$. Hence $\gamma \le 6$. Morphisms of lower degrees exist in degree 4 (linear projection from a point situated on the curve), and degree 5 (projection from points outside the curve). Clearly none of these maps is total, so that $\gamma=6$ exactly. Of course the minimal degree maps considered are plenty (no uniqueness), yet their parameter space is connected. Next we require a specimen with $\gamma=5$. It seems evident that we have exhausted the patience of quintics (at least for the given arrangement), hence let us move to sextics of genus 10 (when non-singular). To get the right genus $g=6$, we have to conserve 4 nodes. Starting from a configuration of 3 conics suitably oriented and smoothed we obtain the figure denoted 515 with $r=5$ (still on Fig.\[Coppens:fig\]). Using a pencil of conics with 4 assigned basepoints (asterisks on the figure) gives a (probably total) map of degree $2\cdot 6-1\cdot 2-3\cdot 1=12-2-3=7$. This agrees with Ahlfors bound $r+2p$, but seems to challenge Gabard’s bound $r+p=6$. Maybe a pencil of cubics is required instead. Such a cubics pencil has 9 basepoints but only 8 of them may be assigned. Hence creating some 4 new basepoints (denoted by bold letters [**1,2,3,4**]{} on the figure) and letting them degenerate to the ovals or better the nodes (when some are accessible) gives a map of degree $3\cdot 6-3\cdot 1- 5\cdot 2=18-3-10=5$, rescuing Gabard’s $r+p=6$ and also giving the desired gonality $\gamma=5$. Admittedly this example is quite complex and perhaps not the best suited to illustrate Coppens’ gonality result. Its interest is still that it seems to corrupt Gabard’s bound $r+p$, and the latter can only be rescued by appealing to fairly sophisticated pencils. Of course it could be the case a priori that our curve (515) admits a pencil of conics of lower degree than 7, but under the totality condition basepoints must be distributed in the deep ovals by a Poincaré index argument (cf. Lemma \[Poincare-lower-bound\]). This impedes a lowering of the degree via a more massive degeneration of the base-locus to the nodes of picture 515 on Fig.\[Coppens:fig\]. Admittedly the predicted total pencil of cubics ought to be described more carefully. [*Summary of the situation.*]{}—Of course one should still work out the higher values of $r$ while keeping $p=1$. As you notice our method is far from systematic. (All the difficulties encountered so far already enhance the power of Coppens’ result.) $\bullet$ Then one must also handle higher values of $p$, starting with $(r,p)=(1,2)$. The case $\gamma=2$ is easy (via the barbecue construction, Lemma \[Barbecue:lem\]). For $\gamma=3$ we can imagine a surface with 3-fold rotational symmetry (cf. picture 123X on Fig.\[Coppens:fig\]). For it $\gamma\le 3$, but how to show equality? Alternatively, one may consider an algebraic model. Since $g=(r-1)+2p=4$, we look among quintics with 2 nodes. A suitable smoothing gives figure named 123, with $r=1$ (one circuit). Linear projection for the “inner” node gives a total map of degree $1\cdot 5-1\cdot 2=5-2=3$, so $\gamma\le 3$. But the complex gonality of such a quintic is $\gamma_{\Bbb C}=3$. Since $\gamma_{\Bbb C}\le \gamma$ it follows that $\gamma=3$ exactly. $\bullet$ Let us next explore $(r,p)=(2,2)$. Then $g=5$. A surface with $\gamma=2$ is easily found (barbecue rotational symmetry). To realize the other gonalities we look among quintics with one node. We first meet figure 223, which has a total morphism of degree 3 (projection from the node). Hence $\gamma\le 3$ which is in fact an equality, since 3 is also the complex gonality of an uninodal quintic. To get a curve with $\gamma=4$ we just drag the unsmoothed singularity to get figure 224. Projection from the node is not total anymore, but a total map arises when projecting from the (green) oval giving rise to degree $5-1=4$. Since such a quintic is uniquely trigonal (via projection from the unique node, which which failed to be total), we infer that $\gamma=4$, exactly. Coppens’s theorem is verified for this topological type. [**Premature conclusion/State of the art.**]{} It is clear that one can continue the game to tackle higher and higher values of the invariants. Instead of looking solely in ${\Bbb P}^2$ it is also pleasant to trace curves in ${\Bbb P}^1\times{\Bbb P}^1$, albeit ${\Bbb P}^2$ is a universal receptacle (any Riemann surfaces nodally immerses in the projective plane). However it is clear that our naive approach is quite time consuming and as yet we did not deciphered a combinatorial pattern permitting to boost the speed of the procedure to the level of an inductive process. (Curves or Riemann surfaces of higher topological structures are like [*homo sapiens*]{}, the result of a long, intricate morphogenesis.) Coppens proved the full result in one stroke by somehow penetrating the genetic code governing the evolution of all species. Minimal sheet number of a genus $g$ curve as a cover of the line {#Minimal-sheet:sec} ---------------------------------------------------------------- It is classical (since Riemann 1857 [@Riemann_1857 §5, p.122–123]) that a general curve of genus $g$ is expressible as a branched cover of the sphere ${\Bbb P}^1$ of degree the least integer $\ge \frac{g}{2}+1$ (equivalently of degree $[\frac{g+3}{2}]$). \[Indeed if $g$ is even $g=2k$ the first value is $k+1$ and $[\frac{g+3}{2}]=[(2k+3)/2]=(2k+2)/2=k+1$; if $g=2k+1$ is odd then the first value is $ g/2+1$ which rounded from above gives $(2k+2)/2+1=k+2$, and $[\frac{g+3}{2}]=[(2k+4)/2]=k+2$.\] Riemann’s truly remarkable argument (involving Abelian integrals) is beautifully cryptical (I should still study it properly). It is not clear (to me) if it includes the stronger assertion that [*any*]{} curve of genus $g$ admits a sphere-map of degree $\le [\frac{g+3}{2}]$. At any rate, all modern specialists agree that the first acceptable proof of this pièce de résistance is Meis’ account (1960 [@Meis_1960]). (Meanwhile the algebro-geometric community devised several alternative approaches.) Another allied (but different?) argument is the one to be found in Klein’s lectures 1892 [@Klein_1891--92_Vorlesung-Goettingen p.98–99], cf. also Griffiths-Harris 1978 [@Griffiths-Harris_1978/94 p.261]. The latter’s argument works as follows. Assume there is a $d$-sheeted map $C_g \to {\Bbb P}^1\approx S^2$ of a genus $g$ surface to the Riemann sphere. Then Euler characteristics are related by $\chi(C_g)= d \chi(S^2)-b$, where $b$ is the number of branch points. This gives $b$ ramified positions, whose locations determine the overlying Riemann surface up to finitely many ambiguities. So the $d$-sheeted surface depends upon $b-3$ essential parameters (after substraction of the 3 arising from the linear transformations on ${\Bbb P}^1$). This quantity has to be $\ge 3g-3$ the number of moduli of genus $g$ curves. This implies $b\ge 3g$, i.e. $2d-\chi(C_g)\ge 3g$, or $2d\ge 3g+(2-2g)=g+2$. [*q.e.d.*]{} So far as we know, a similar computation as never been written down for the case of a bordered Riemann surface expressed as a $d$-sheeted cover of the disc (i.e., the context of Ahlfors circle maps). The reason is probably quite mysterious, yet also quite simply that the naive parameter count seems to lead nowhere. Let us attempt the naive computation. Suppose $F_{r,p}\to D^2$ to be a membrane of genus $p$ with $r$ contours expressed as a $d$-sheeted cover of the disc. Euler characteristics are related by $\chi(F)=d\chi(D^2)-b$, where $b$ is the number of branch points. The group of conformal automorphisms of the disc as (real) dimension $3$. Hence our $d$-sheeted surface depends upon $2b-3$ real constants, whereas the membrane $F$ itself depends on $3g-3$ real constants, where $g$ is the genus of the double (cf. Klein 1882 [@Klein_1882]). The Ansatz $2b-3\ge 3g-3$ gives $2b\ge 3g$, and since $b=d-\chi$ and $g=(r-1)+2p$, this gives $2d \ge 3g+2\chi=3[(r-1)+2p]+2(2-2p-r)=r+2p+1$, equivalently $d\ge (r+1)/2+p$. This beats the value $r+p$ predicted in Gabard 2006 [@Gabard_2006], but looks blatantly overoptimistic. For instance taking $p=0$, gives the degree $\frac{r+1}{2}$ violating the absolute lower bound $r$ on the degree of a circle map. The provisory conclusion is that the naive parameters count leads nowhere in the bordered case. Does somebody know an explanation? \[21.10.12\] A crude attempt of explanation is that the above count merely uses the Euler characteristic which is a complete topological invariant only for closed surfaces, but not for bordered ones. (Since $\chi(F_{r,p})=2-2p-r$, trading one handle against two contours leaves $\chi$ unchanged.) Of course the above counting uses also $g$ (the genus of the double $2F$) but the latter is also uniquely defined by $\chi(F)$, via the relation $2-2g=\chi(2F)=2\chi(F)$. Thus it is maybe not so surprising that Riemann(-Hurwitz)’s count predicts correctly the gonality of closed Riemann surfaces but fails seriously to do so in the bordered case. It could be challenging to find a moduli count existence-proof of Ahlfors circle maps supplemented probably by an adequate continuity method. For an (unsuccessful) attempt cf. Sec.\[Hurwitz-type\]. A very naive (numerological) parade is to introduce a new bound $\nu:=\max\{p+ \frac{r+1}{2}, r \}$ between the one found above and $r$ the absolute minimum of a total morphism. However a simple example probably shows this to be overoptimistic as well. Consider the plane quintic $C_5$ derived via a sense preserving smoothing of the depicted configurations of $2$ conics and a line (cf. Fig.\[quintic:fig\]). Its genus is $g=\frac{(m-1)(m-2)}{2}=\frac{4\cdot3}{2}=6$, and we see $r=5$ real circuits. The relation $g=(r-1)+2p$ gives $p=1$ (genus of the half). Hence the new bound is $p+ \frac{r+1}{2}=1+3=4$, but $r=5$ so $\nu=\max=5$. However the membrane (corresponding to one half of the dividing curve $C_5$) cannot be represented with 5 sheets over the disc. Indeed a morphism of degree 5 from $C_5$ to the line ${\Bbb P}^1$ can arise through linear projection of the quintic $C_5$ from a point not on the curve (else degree $4$), but no such projection is totally real (compare central part of Fig.\[quintic:fig\], or argue via the Poincaré index, cf. Lemma \[Poincare-lower-bound\]). \[09.11.12\] [*WARNING about the underlined “only”.*]{}—This argument looks at first sight quite convincing, yet it appears to be insufficient, and possibly the assertion itself on the gonality $\gamma(C_5)=6$ is erroneous. First, a total morphism of degree $r+p=5+1=6$ (as predicted in Gabard 2006 [@Gabard_2006]) should exist. This is corroborated by taking a pencil of conics through 4 points inside the 4 ovals of the above depicted $C_5$ (cf. Fig.\[FGuerN:fig\], left part) and letting them degenerate against the ovals, giving a total map of degree $2 \cdot 5-4=6$. This tell us only $\gamma \le 6$. A priori, it could be the case that higher order pencils access the low degree $5$, and with some good-fortune do it in a totally real way. In that case the gonality lowers down to $\gamma=5$ (the minimum permissible as $r=5$). Let us quickly discuss how this could happen, at least over the complexes. A priori pencils of cubics may have degrees as low as $3\cdot 5-3^2=6$ (hence not violating the previous token); quartics as low as $4\cdot 5- 4^2=4$, but quartics have dimension $\binom{4+2}{2}-1=14$ so that in reality only 13 basepoints may be assigned freely, hence the right value is $4\cdot 5- 13=7$; for quintics this is as low as $5\cdot 5- 5^2=0$ (yet all values $<4$ violates already the complex gonality of a smooth quintic, cf. e.g. Arbarello et al. 1985 [@Arbarello-Cornalba-Griffiths-Harris_1985-BOOK p.56, Exercise 18]). In fact the dimension of quintics is $\binom{5+2}{2}-1=20$ and thus the minimum degree is $5\cdot 5- 19=6$. For sextics the degree is as low as $6\cdot 5- 6^2=-6$, but since the sextics dimension is $\binom{6+2}{2}-1=27$, the real minimum degree is $30-26=4$ (and this beats linear projections from outside the curve). Recall incidentally that this is the value of the universal Riemann-Meis bound $[\frac{g+3}{2}]=[9/2]=4$, which was already attained by linear projections from the curve but nobody will exclude a priori a second return. Actually all 26 assigned basepoints fails to impose independent conditions on sextics, because our quintic $C_5$ aggregated to any line is a sextic meeting the requirement and varying among $\infty^2$ parameters (and not just the expected $\infty^1$ pencil). Thus we seem to fail getting a genuine pencil, but contrast this with the just remembered Riemann-Meis gonality. The situation is quite more tricky than initially expected. Another torpedo against the naive belief that a smooth $C_5$ has only $\infty^1$ series of type $g^1_5$ is the existence theorem of Brill-Noether-Kempf-Kleiman-Laksov theory (cf. e.g. Arbarello et al. [@Arbarello-Cornalba-Griffiths-Harris_1985-BOOK p.206]). The latter states the following. Let $C$ be a (complex) curve of genus $g$. Every component of the variety $G^r_d$ parameterizing all linear series $g^r_d$ of dimension $r$ and degree $d$ has dimension at least equal to the so-called [*Brill-Noether number*]{} $\rho$, symbolically: $$\dim_{\ast} G^r_d\ge \rho:=g-(r+1)(g-d+r).$$ In particular when the latter number $\rho$ is $\ge 0$ the variety $G^r_d$ is nonempty. In the case at hand it follows that $\dim_{\ast} G^1_5 \ge 6-(1+1)(6-5+2)=6-2\cdot2=6-4=2$. Hence there are other pencils of degree 5 than those readily visualized on the projective realization! This shows how vicious the Plato cavern is! Of course our appeal to the above general theorem, is a violation against the principle of do-it-yourself-ness, since low genus cases are in best treated by hand (cf. Arbarello et cie [@Arbarello-Cornalba-Griffiths-Harris_1985-BOOK p.209–211] for a possible treatment, alas not perfectly self-contained). The following summarizes the swampy situation (while trying to extend the generality): [(To be clarified with percentages of truth)]{} $\bullet$ \[100 %\] Any smooth real quintics $C_5$ with $r=5$ (hence $4$ ovals and one pseudoline) is unnested (otherwise the line through the nest plus another oval gives 6 intersections, corrupting Bézout). $\bullet$ \[80 %\] Furthermore taking a pencil of conics through the $4$ nests gives a total pencil (why exactly? clear on the Fig.ref[FGuerN:fig]{}(left part) but why in general?). $\bullet$ \[79 %\] Assuming the previous point, the gonality is $\gamma\le 2\cdot5-4\cdot 1= 6$ (in accordance with Gabard’s bound $r+p$, but it is preferable to mistrust this!). $\bullet$ \[100 %=0 %\] Alas it is not clear a priori that pencils of orders $\ge 6$ do not induce total pencils of possibly lower degree $=5$. (Recall that $r=5$ is an absolute lower bound for total maps!) The only delicate point is the assertion about total reality of the pencil of conics. In fact this is fairly evident on the figure above, but in the general case I see no reason. \[10.11.12\] In the light of the Kempf-Kleiman-Laksov existence theorem of special divisors (ESD) in the case of complex curves one may wonder about its relativization in the Ahlfors context of total maps. The point is of course that for $g^1_d$’s the existence theorem (ESD) boils down to the Riemann-Meis bound $\gamma_{\Bbb C}\le [\frac{g+3}{2}]$ for the gonality of complex curves. (Plug $d\ge g/2+1$ in the Brill-Noether number $\rho$ and notice its non-negativity.) Since Ahlfors 1950 $\gamma\le r+2p$ or maybe Gabard 2006 $\gamma \le r+p$ is to be considered as the genuine bordered (or orthosymmetric) avatar of the Riemann-Meis theorem one can dream of an orthosymmetric(=dividing) version of the whole special of divisor theory. It is not clear how to extend total reality for higher series $g^r_d$ which are not pencils $g^1_d$. Of course one can ask that all real members are totally real but this seems too restrictive. Is there any example at all? Perhaps not for simple dimension reason. For $g^2_d$’s this would amount to a plane model of the curve cut by all real lines in real points only. This looks overambitious by just perturbing a tangent at a non-inflection point outside the sense of curvature. At any rate the theory surely works for pencils and the bonus is that we have a certain variety akin to $G^1_d$ parameterizing all total pencils of degree $d$ on a given dividing curve. How to denote it? I never understood for what the “$g$ or $G$” of resp. $g^1_d$ or $G^1_d$ is standing? (Candidates: groups of points, Gerade, Gebilde, Grassmann, ?) Improvising notation, we define $T^1_d$ the variety of total linear series of degree $d$ on a given dividing curve. We dream about repeating all the phenomenology of the classic theory, cf. e.g. p.203 of ACGH 1985 [@Arbarello-Cornalba-Griffiths-Harris_1985-BOOK]: “[*A genus $g$ curve depends on $3g-3$ parameters, describing the so-called moduli. Our goal is to describe how the projective realizations of a curve vary with its moduli, and what it means to say that a curve is general or special. Accordingly, we would like to know, what linear series can we expect to find on a general curve and what the subvarieties of the moduli space corresponding to curves possessing a series of specified type look like. \[…\] A natural question is, how can we tell one curve from another by looking at these configurations \[$G^r_d$\], or more precisely, what do these look like in general, and how—and where—can they degenerate?*]{}” For our “totality” varieties $T^1_d$ of total pencils we would gather them into a “telescope” $T^1:=\cup_{d=1}^{\infty} T^1_d$ naturally embedded in $C^{(\infty)}$, the infinite symmetric power of the (dividing) curve $C$. We have the degree function $\deg\colon C^{(\infty)}\to {\Bbb N}$, and the image of $T^1$ is nothing but than the gonality sequence $\Lambda$ (Definition \[def:gonality-sequence\]), whose least member is the (separating) gonality $\gamma$ (of Coppens). One would like to understand how total pencils may degenerate to lower degrees w.r.t the natural topology induced by $C^{(\infty)}$. We probably get a sort of telescope with high strata attached to lower dimensional ones (like in a CW-complex) and the game would be to understand the geometry or combinatoric of this tower. Understanding all this is arguably the most refined form of Ahlfors theorem one could desire. One would then like to know not only the gonality spectrum telling one the dimension of each strata $T^1_d$, but also know how they can degenerate to lower strata. Degeneration could still be encoded combinatorially in a simplicial-complex $\Lambda^{\ast}$ with vertices $\Lambda$ (gonality sequence). Two vertices $d_1<d_2\in \Lambda$ are linked by an edge if a total $g^1_{d_2}$ can degenerate to a $g^1_{d_1}$. More generally $d_1<d_2<\dots<d_{k+1}\in \Lambda$ form a $k$-simplex whenever each integer of the sequence admits a representative $g^1_d$ degenerating to its immediate predecessor, hence to all predecessors. Working out this explicitly looks tedious already for simple example. For the Gürtelkurve (any smooth quartic $C_4$ with 2 nested ovals) the variety $T^1_3$ is a circle and $T^1_4$ is a 2-cell attached to the former in a natural way. Of course when a total $g^1_4$ degenerates to a total $g^1_3$ it acquires a basepoint, which as to be deleted (particle destruction). Total $g^1_d$ will ultimately be denoted as $t^1_d$’s. In view of the Brill-Noether theorem (ESD) the variety $G^1_4$ has dimension $\ge \rho=3-2(3-4+1)=3$ and so we have a priori more than the $\infty^2$ evident total pencils $t^1_4$ arising via projection from the inner oval. For instance pencils of conics may have degree as low as $2\cdot 4- 4 \cdot 1=8-4=4$. Can they be total? I would have guessed not, but it seems that they can. Compare Fig.\[FGuerN:fig\] below. It would be desirable if some continuity principle can ensure total reality, e.g. if the 4 basepoints are distributed both inside and outside the nested resp. unnested oval. Then like a salesman traveller, the conic has to visit all 4 basepoints and thus creates at least $8$ real intersections. Our picture would just be the limiting position of such a bipartite pencil, and the variety $T^1_4$ would be $\infty^4$, a much larger dimension than initially expected. Further if 3, among the 4 basepoints, become collinear then it may be argued that the conics pencil specializes to one of lines (after removing the static line). All this remains to be better analyzed. Heuristic moduli count to justify Ahlfors or Gabard (Huisman 2001) ------------------------------------------------------------------ It is still plausible that one may gain some evidence in favor of the Ahlfors circle map (either with Ahlfors $r+2p$ or preferably the improved Gabard’s bound $r+p$) by arguing via a moduli count. (The reader reminds to have discussed orally this option with Natanzon and Huisman in Rennes in Summer 2001, resp. December 2001.) I do not know if it is possible to supply a better count than the unrealistic one of the previous section. \[14.10.12\] In fact at a time when I only conjectured the bound $r+p$, Huisman (December 2001 or 2002?) reacted instantaneously with a parameter count giving some evidence to the conjecture. Let me reproduce this faithfully from hand written notes. We adopt the viewpoint of dividing real (algebraic) curves. So let $C$ be a such with $r\ge 1$ ovals and of genus $g$. I mentioned to Johannes Huisman the intuition that there is a totally real morphism $C\to {\Bbb P}^1$ (i.e. inverse image of real locus contained in the real locus) whose degree is the barycenter of $r$ and $g+1$, that is $\frac{r+(g+1)}{2}$. (The heuristic reason behind this 2001 intuition are given in Gabard 2006 [@Gabard_2006], and in its most primitive form in the previous Section\[sec:Sketch-of-Gabard\].) “Let us count parameters!”. Thus spoke Huisman, like Zarathustra. First the Riemann-Hurwitz relation written for the Euler characteristic is $\chi(C)= d \cdot \chi({\Bbb P}^1)-b $, where $d$ is the degree and $b$ the number of branch points (with multiplicity). Now we count real moduli. The ramification divisor of any totally real morphism actually lies in the imaginary locus of the sphere (not on the equator), but is of course symmetric w.r.t. the involution. Hence we may imagine the $b/2$ branch points prescribed only in the north hemisphere, thus depending on $2\cdot (b/2)=b$ real constants. The curve itself depends on $b-3$ moduli (subtract the dimension of the automorphism group of ${\Bbb P}^1$ defined over ${\Bbb R}$), that is $$\begin{aligned} b-3= d \cdot \chi({\Bbb P}^1)-\chi(C)&=\frac{r+g+1}{2} \cdot 2-\chi(C)-3\cr &=(r+g+1)-(2-2g)-3=3g-4+r\ge 3g-3.\end{aligned}$$ This prompts enough free parameters to sweep out the full moduli space. Of course this does not reprove the existence of circle maps of the prescribed degree, yet give some evidence to the assertion. \[15.10.12\] A notable defect of this Huisman count is that it is a posteriori, giving no hint why the degree value should be given by our Ansatz. It is thus preferable to make the same computation in a more organical way. As above the curve $C$ depends on $b-3$ real moduli, and we demand $b-3\ge 3g-3$. This gives $d\cdot \chi(S^2)-\chi(C)\ge 3g$, i.e. $2d\ge 3g+(2-2g)=g+2$, or $d\ge g/2+1$. Two remarks are in order. The above is exactly the same heuristic calculation as the that (going back to Riemann) to be found in Griffiths-Harris for the complex gonality of a curves, and which we remembered before. (The least integer $d\ge (g+2)/2$ is $[\frac{g+3}{2}]$, obvious for $g$ even and also obvious when $g$ is odd.) Hence in substance this modification of Huisman’s count truly just assert that Gabard’s bound is compatible with the gonality of the underlying complex curve, yet does not predict the bound $(r+g+1)/2$. Perhaps there is a better way to count, compare the section devoted to Courant (Sec.\[sec:Courant\]). Since Riemann 1857 we know that a complex curve of genus $g$ depends on $3g-3$ moduli. For real curves the same assertion holds true by virtue of Klein 1882 [@Klein_1882]. (The modernized treatment is of course due to Teichmüller 1939 [@Teichmueller_1939].) In view of this we may formulate the Ansatz of an (Ahlfors) circle map to the disc and try to compute the minimal number of sheet and the allied ramification points required to supply the branched Riemann surface with enough free parameters so as to paint out the full moduli space. I tried recently to redo this computation, but was not very successful. So we leave this as an easy project to be clarified at the occasion. Other application of the irrigation method (Riemann 1857, Brill-Noether 1874, Klein, etc.) ------------------------------------------------------------------------------------------ The method used in Gabard 2006 [@Gabard_2006] is primarily based upon an irrigation principle in a torus, which in turn is logically reducible to the surjectivity criterion via the (Brouwer) topological degree of a mapping to a manifold. Via this method we obtained (in ) the existence of an (Ahlfors) circle map of degree $\le r+p$. As pointed out there, the method also supplies a purely topological proof of Jacobi inversion theorem, to the effect that the Abel-Jacobi mapping from the symmetric powers $C^{(d)}$ of a complex curve to its Jacobian is surjective as soon as dimension permits (that is for $d\ge g$). Of course the complex (or closed) avatar of the Ahlfors mapping is just the mapping of a closed genus $g$ surface as a branched cover of the sphere. In this situation it is classically known since Riemann 1857 [@Riemann_1857] and Brill-Noether 1874 [@Brill-Noether_1874] (but disputed by the modern writers) that the most economical sheet number required is $[\frac{g+3}{2}]$. Contributions on this problem is vast (and according to the modern consensus first [*rigorously*]{} proved in Meis 1960 [@Meis_1960] for linear series of dimension one, whereas some classic references includes the more case of arbitrary dimensional series, esp. Brill-Noether and Severi) $\bullet$ Riemann 1857 (Theorie der Abel’schen Functionen) [@Riemann_1857 §4], $\bullet$ Brill-Noether 1874 [@Brill-Noether_1874] (working with plane curves with singularities, so a pure algebraization of Riemann’s theory if one does not fell claustrophobic in the Plato cavern.) $\bullet$ Klein’s lectures of 1891 [@Klein_1891--92_Vorlesung-Goettingen p.99] (based on Abelian integrals and Riemann-Roch, essentially akin to Riemann’s original derivation) $\bullet$ Hensel-Landsberg 1902 [@Hensel-Landsberg_1902 Lecture 31] (probably quite similar to Brill-Noether or inspired by Dedekind-Weber) $\bullet$ Severi 1921 [@Severi_1921-Vorlesungen-u-alg.-Geom-BUCH Anhang G] Then the modern era begins with: $\bullet$ Meis 1960 [@Meis_1960] (Teichmüller theoretic) \[alas, this monograph is notoriously difficult to obtain\] $\bullet$ H.H. Martens 1967 [@Martens_Henrik_1967] (no proof, but a remarkable study of the geometry assuming non-emptiness) $\bullet$ Kempf 1971 [@Kempf_1971] the first existence proof (simultaneous with the next contributors) of special divisors in general case (linear series of arbitrary dimension, extending thereby the pencil case first established by Meis 1960) $\bullet$ Kleiman-Laksov 1972–74 [@Kleiman-Laksov_1972] [@Kleiman-Laksov_1974] (using resp. Schubert calculus, plus Poincaré’s formula and resp. singularity theory à la Thom, Porteous) $\bullet$ Gunning 1972 [@Gunning_1972] using MacDonald computation of the homology of the symmetric power of the curve $\bullet$ Griffiths-Harris 1978 [@Griffiths-Harris_1978/94 p.261], where the heuristic count à la Riemann-Klein is reproduced; and latter a rigorous argument (p.358) is supplied (along the line of Kempf’s Thesis ca. 1970). In view of the interest aroused by this Riemann-Meis bound, and the apparent difficulty to prove it (appealing to a variety of ingenious devices), it seems reasonable to wonder if there is not a much simpler argument based upon the same “irrigation method” as the one used by the writer in relation with the Ahlfors map. This would merely use simple homology theory and the allied surjectivity criterion in term of the Brouwer degree. Heuristically, this amounts to see the genus $g$ pretzel inside its Jacobian and let it homologically degenerate over a bouquet (wedge) of $g$ $2$-tori irrigating the Jacobian. Thus it seems evident that with roughly $g/2$ points we may find a pair of (effective) divisor of that degree collapsing to the same point of the Jacobian. This pair of disjoint divisors serves to define the desired morphism to ${\Bbb P}^1$. The writer as yet did not found the energy to write down the details, but is quite confident that the strategy is worth paying attention. Of course it could be the case that this merely boils down (up to phraseological details) to the already implemented attack of Gunning 1972 [@Gunning_1972]. (Shamefully, I did not yet had the time to consult this properly.) Thus, it seems rather obvious that this result of Riemann-Brill-Noether-Meis as well (at least the existence of a mapping with such a degree) permits a proof via the irrigation method employed in Gabard 2006 [@Gabard_2006]. Of course “irrigation” would not establish the sharpness of Meis’ bound (which is another question), but could predict its value as universal upper-bound upon the gonality. Another application: Complex manifolds homeomorphic to tori ----------------------------------------------------------- This section deviates from the mainbody of the text, but serves to illustrate another spinoff of the irrigation method. The writer wondered about the following naive question (ca. 2001/2?). Assume given a complex (analytic) manifold (arbitrary dimension), and suppose also the underlying manifold to be homeomorphic to a torus. [*Must such a manifold be biholomorphic to a complex torus, i.e. ${\Bbb C}^n$ modulo a lattice?*]{} The answer is easy in dimension one (Abel essentially). In general the answer is negative, by virtue of a construction of Blanchard (Thesis ca. 1955) closely allied to the Penrose twistor. Basically there is over $S^2$ a certain bundle parametrizing quaternionic structures, and taking a fiber product with an elliptic curve yields on the torus $T^6$ (of 6 real dimensions) a complex structure which turns out to be not Kähler. This answers negatively the question when the complex dimension is 3. (For more details cf. also work by Sommese (ca. 1978), etc.) All this is rather exotic complex geometry, but one may wonder if the assertion becomes true under the Kähler assumption. Then Hodge theory applies, and we dispose of a bona fide analog of the Abel mapping (sometimes called the Albanese mapping). The latter is also a map to a complex torus (called Albanese) and using the irrigation principle it is easy to show that $\alpha$ induces an isomorphism on the top-dimensional homology. First, it induces an isomorphism on the $H_1$, but the latter elevates up to the top-dimension since tori have a total homology $H_{\ast}$ modelled upon the exterior algebra over the $H_1$. By the Brouwer degree argument (irrigation intuitively), it follows that $\alpha$ is surjective. Then one can show that it is injective as well (I have forgotten the exact argument, but essentially if Albanese collapse a submanifold then like by Abel it collapses linear varieties which are simply-connected projective spaces, hence liftable to the universal cover of the Albanese torus). Any torus shaped Kähler manifold is biholomorphic to its Albanese torus. Of course this is surely well-known, but we just wanted to remember this as another high dimensional—but baby—application of the irrigation principle. Further Kodaira’s classification of (complex analytic) surfaces plus a deformation argument of Andreotti-Grauert (which I learned from R. Narasimhan) implied also a positive answer to the basic question in (complex) dimension 2. But I take refuge in my failing memory, and to not remember the exact details. Thus in principle, Blanchard’s 3-dimensional counterexample is sharp. Invisible real curves (Witt 1934, Geyer 1964, Martens 1978) {#sec:Witt} ----------------------------------------------------------- Ahlfors’ theorem bears some analogy with Witt’s theorem (1934 [@Witt_1934]) stating that a (smooth) real curve without real points admits a morphism (defined over the reals ${\Bbb R}$) to the invisible real line (materialized by the conic $x_0^2+x_1^2+x_2^2=0$). The analogy is again that when there is no topological obstruction, then a geometric mapping exists. Subsequent works along Witt’s direction are due to: $\bullet$ Geyer 1964/67 [@Geyer_1964-67] (alternative proof of Weichold, and Witt via Galois cohomology and Hilbert’s Satz 90); $\bullet$ Martens 1978 [@Martens_1978], where the precise bound on the degree of the Witt mapping has been determined. Philosophically, it seems challenging to examine if such strongly algebraic techniques (Riemann-Roch algebraized à la Hensel–Landsberg 1902, Artin, etc.) are susceptible to crack as well the Ahlfors mapping? Geyer, Martens or others are perhaps able to address this challenge? (So far as we know, no such account exist in print.) Martens’ statement (quantitative version of Witt) is the following. Given a closed non-orientable Klein surface with algebraic genus $g$ (i.e. the genus of the orientable double cover[^12]) there is a morphism to the projective plane of degree $\le g+1$. Moreover this is the best we can hope for, i.e. for each $g$ there is a Klein surface not expressible with fewer sheets. Perhaps the first portion of the statement is already in Witt 1934 [@Witt_1934]. Of course this can—via the Schottky-Klein Verdoppelung—also be stated in term of symmetric Riemann surfaces (equivalently real algebraic curves) as follows: Given a symmetric Riemann surface of genus $g$ without fixed point, there is an equivariant conformal mapping to the diasymmetric sphere of degree $g+1$. Moreover the bound is sharp. This formulation of Martens’s result also appears in Ross 1997 [@Ross_1997 p.3097], who supplies additional comments which are quite in accordance with our own sentiments, especially the issue that the short argument by Li-Yau 1982 [@Li-Yau_1982 p.272] does not appear as very convincing. Moreover Ross supplies some attractive differential geometric applications of this Witt-Martens mapping theorem, e.g. to the effect that the totally geodesic ${\Bbb R}P^2$ is the only stable minimal surface in ${\Bbb R}P^3$. The three mapping theorems (Riemann 1857, Ahlfors 1950, Witt 1934) ------------------------------------------------------------------ From the conformal viewpoint we have thus three basic mapping theorems enabling a gravitational collapse of all compact surfaces to their simplest representatives (the sphere, the disc or the projective plane) depending on whether the original surface is: $\bullet$ closed orientable (Riemann 1857 [@Riemann_1857]); $\bullet$ compact bordered orientable (Ahlfors 1950 [@Ahlfors_1950]); $\bullet$ closed non-orientable (Witt 1934 [@Witt_1934]). None of those results tells what to do with a compact bordered non-orientable surface (whose simplest specimen is the Möbius band/strip). The latter does not carry positive curvature, which implies finiteness of the fundamental group for complete metrics (else punctured sphere). Alternatively the orientable double cover of Möbius is the torus, which has already moduli. Hence it is quite clear that the above three theorems form an exhaustive list of truths positing a fundamental trichotomy (of definitive crystallized shape). The motto “Alle guten Dinge sind drei”, is quite ubiquitous in life and mathematics! It is reasonable to expect that each of those mappings will pursue to find valuable applications in the future, yet much work remain to be done as to the stratification of the moduli space induced by the degree of such representations, etc. For each of these 3 concretization problems one is interested in the exact determination of the lowest possible sheet number. In principle the answer is already known as follows: For all $3$ types of conformal mapping to elementary surfaces of positive Euler characteristics $\chi >0$ (including $\chi (S^2)=2, \chi (\Bbb R P^2)=1, \chi (\Delta=D^2)=1$) the sharp universal bound on the degree of such representation is known. More precisely, $\bullet$ $[\frac{g+3}{2}]$ always concretizes closed genus $g$ surfaces expressed as cover of the sphere [(Riemann, Meis 1960 [@Meis_1960])]{}, and the bound is sharp [(again Meis 1960 [@Meis_1960])]{}. $\bullet$ $g+1$ always concretizes non-orientable closed surface of algebraic genus $g$ (i.e. genus of the orientation double cover) expresses as cover of the projective plane [(Witt 1934 [@Witt_1934])]{}, and the bound is sharp [(Martens 1978 [@Martens_1978])]{}. $\bullet$ $r+p$ always concretizes bordered orientable surfaces with $r$ contours and $p$ handles as (full or total) cover of the disc [(Gabard 2006 [@Gabard_2006])]{}, and the bound is sharp [(Coppens 2011 [@Coppens_2011])]{}. Adhering to Klein’s viewpoint of symmetric surfaces, one can always interpret such objects as real curves of some genus $g$ (the first class is an exception except if one tolerates disconnected surfaces). In the third bordered case $g=(r-1)+2p$. The $r+p$ bound can be rewritten as $\frac{r+(g+1)}{2}$. If $r$ is lowest, i.e. $r=1$, this is statistically equal to $g/2$, as so is the first Riemann-Meis bound. In contrast the Witt-Martens bound looks much higher. Of course if $r=g+1$ is highest (Harnack-maximality) then $r+p=r+0=g+1$, agreing with Witt-Martens’s bound. In the overall it may be argued that both Martens’ and Gabard’s bound are fairly less economical that Riemann-Meis’, and that this is due to the equivariance or even total reality of the corresponding maps. On the other hand Ahlfors bound $r+2p=g+1$ looks much more compatible with Martens’ and if one is sceptical about Gabard’s version one could imagine that Ahlfors is asymptotically sharp for large values of the invariants. This scenario remains hypothetically possible in case we are unable to reassess through other mean Gabard’s $r+p$ or able to disprove its validity. The following tabulation summarizes the key contributions: \(1) Riemann 1857: any (or at least the general) closed Riemann(ian) surface maps conformally to the sphere with $\le [\frac{g+3}{2}]$ sheets, where $g$ is the genus. It is not clear-cut if Riemann showed sharpness of the bound. Related works includes (in chronological order): $\bullet$ Brill-Noether 1874 [@Brill-Noether_1874]; $\bullet$ Klein 1891 [@Klein_1891--92_Vorlesung-Goettingen p.99]; $\bullet$ Severi 1921 [@Severi_1921-Vorlesungen-u-alg.-Geom-BUCH]; $\bullet$ B. Segre 1928 [@Segre_1928]; $\bullet$ Meis 1960 [@Meis_1960]; $\bullet$ Kempf 1971 [@Kempf_1971] and Kleiman-Laksov 1972–74 [@Kleiman-Laksov_1972] [@Kleiman-Laksov_1974]; $\bullet$ Gunning 1972 [@Gunning_1972]; $\bullet$ Griffiths-Harris 1978 [@Griffiths-Harris_1978/94 p.261]; $\bullet$ Arbarello-Cornalba 1981 [@Arbarello-Cornalba_1981]. This sharp bound $[\frac{g+3}{2}]$ as applied to spectral theory is observed in El Soufi-Ilias 1983/84 [@El-Soufi-Ilias_1983/84] (Yang-Yau 1980 [@Yang-Yau_1980] contented themselves with the weaker value $g+1$.) An interesting aspect of the Italian works is that they not only focus on the gonality upper bound, but also compute the dimensions of the lower dimensional strata for a prescribed gonality. Of course, the answer is the expected one (as easily predicted by Riemann-Hurwitz). \[The above Italian works, especially Segre has however a little objection to the simplicity of the exercise.\] We point out this is issue as it could be interesting to make a similar count for the Ahlfors circle map (bordered case). This topic will be briefly addressed in the next Sec.\[sec:profile-histogram\]. \(2) Ahlfors 1950 [@Ahlfors_1950]: any compact bordered Riemann surface maps conformally to the disc with $\le r+2p$ sheets (where as usual $r$ is the number of boundary contours and $p$ the genus). This bound is not sharp (at least for low values of the invariants $(r,p)$, e.g. for the Gürtelkurve type $(r,p)=(2,1)$). Modulo a mistake by the writer (in Gabard 2006 [@Gabard_2006]), Ahlfors bound can be improved as $\le r+p$. The latter is in turn sharp according to Coppens 2011 [@Coppens_2011]. \(3) Witt 1934 [@Witt_1934]: any closed non-orientable surface maps conformally to the projective plane ${\Bbb R}P^2$. Witt does not specify a bound (?), or maybe he does but sharpness was obtained by Martens 1978 [@Martens_1978]. Witt’s result received, arguably, only sporadic spectral applications, except in the article Li-Yau 1982 [@Li-Yau_1982], which however does not quote Witt, but whose authors were apparently able to reprove the result by their own \[compare their argument on p.272\]. (As already mentioned, Ross 1997 [@Ross_1997] does not seem to be convinced by the Li-Yau argument.) Of course all this “diaporama” is the direct heritage of Riemann (plus maybe indirectly some Abel!) the first result being often called Riemann’s existence theorem. The 2 avatars of Ahlfors and Witt are akin to the absolute case of Riemann, via the trick of the Schottky-Klein double (or Verdoppelung as Teichmüller calls it) but then some equivariance or total reality is required, acting as a sort of boundary condition explaining probably why those versions took longer to emerge. Of course such equivariance/or boundary behaviors just hide a reality condition encoded in the field of definition of the allied Riemann surfaces. All this is best summarized diagrammatically: $$\begin{aligned} i(p^*,{\cal F}^*)=1-\frac{c_*-c_*'}{4}&=1-\frac{2c-2c'}{4}\cr &=2\bigl( 1-\frac{c-c'}{4}\bigr)-1 =2j(p, {\cal F})-1.\end{aligned}$$ The gonality profile, moduli strata and the Ahlfors space {#sec:profile-histogram} --------------------------------------------------------- \[10.11.12\] Heuristically (cf. e.g. Segre 1928 [@Segre_1928]) one can count the dimension of closed Riemann surfaces expressible as coverings of degree $d$ of the sphere as follows. By Riemann-Hurwitz $\chi(C_g)=d \chi(S^2)-b$. Hence there are $b-3$ free parameters, that is $$b-3=(2d-\chi)-3=2d-(2-2g)-3=2d+2g-5.$$ In particular the strata of given gonality $\gamma=d$ increases its dimension along a very simple arithmetic progression (as a function of $d$) until the full moduli space is exhausted for $d$ the least integer $\ge g/2+1$ (Riemann-Meis bound). The smallest strata is the [*hyperelliptic locus*]{} ($d=2$) of dimension $4+2g-5=2g-1=(2g+2)-3$, in accordance with the $2g+2$ ramification points visible as invariant points of an half twist acting upon a purely vertical pretzel in 3-space. I do not know if such a regularity occurs for bordered surfaces. Coppens’s theorem states another form of regularity, namely full realizability of all intermediated gonalites, but it does not pertain to the dimensions of the corresponding moduli strata. On behalf of Coppens’s theorem the situation could be as follows. For a given topological type $(r,p)$, Coppens tells us that all intermediate $r\le \gamma \le r+p$ are realized. So we have $p+1$ possible gonalities, the largest of which $\gamma=r+p$ fills the full moduli space of real dimension $3g-3$ (Klein’s count conjointly with Gabard’s bound). As usual $g=(r-1)+2p$, so $3g-3=6p+3r-6$. If we knew the number of moduli of the minimal strata $\gamma=r$ we could try a linear interpolation as a possible scenario for the dimensions increments of the gonality strata. Naively our rotationally invariant picture (Fig.\[Chambery:fig\]) could act as a bordered substitute to the hyperelliptic closed case (at least for $r$ even). If so is the case can we count its moduli? Everything would be determined by the quotient planar surface with $r/2=r'$ contours. This planar surface (whose double has genus $g'=r'-1$) depends on $3g'-3$ moduli. This expressed in terms of $r$, gives he following $3g'-3=3r'-6=3/2 \cdot r-6$. This a candidate for the dimension of the lowest strata. Looking for a progression in $p$ steps toward the maximum value, we consider the difference $[6p+3r-6]-[3/2 \cdot r-6]=6p+3/2\cdot r=1/2[12p+3r]$, which is not easily divided by $p$. $\bullet$ In fact we have looked at the quotient but barely omitted the branched locus. Taking this into account we get rather a dependance on $3g'-3+2(2p+2)$ (real) moduli for the lowest strata. Expressing this in terms of $(r,p)$, gives $3/2 \cdot r+4p-2$. Hence the difference of the top and lowest strata would be $2p+3/2\cdot r-1$, which is alas still not nicely divisible by $p$. $\bullet$ Another idea is just to use maps from $F_{r,p}$ to the disc of minimum degree $r$. Then we have $\chi(F)=r \chi(\Delta)-b$. Hence there are $2b-3$ free real parameters. Expressed in terms of $(r,p)$, this is $2b-3=2(r-\chi)-3=2(r-(2-2p-r))-3=4r+4p-7$. Hence the difference between the top dimensional and the lowest dimensional strata is $\delta=(6p+3r-6)-(4r+4p-7)=2p-r+1$, which is not even positive in general. It looks again dubious to divide this in $p$ equal parts as suggests Coppens result. Again this just confirms what we already noticed (earlier in the text) that the Riemann-Hurwitz count looks seriously jeopardized in the bordered case, at least as long as we apply it so naively as we do. One can reverse the game: instead of speculating on the size of the lowest strata we can speculate on the increment as being by 2 real units (like in the complex case) and draw the dimension $\lambda$ of the lowest strata. This would give $\lambda=\dim ({\rm top \;strata})-p\cdot 2=(6p+3r-6)-2p=4p+3r-6$. Testing this on the type of the Gürtelkurve $(r,p)=(2,1)$ gives $\lambda=4+6-6=4$, whereas the hyperelliptic model depends on $2g+2-3=2\cdot3+2-3=5$ real parameters. Hence the later has codimension 1 in the full moduli of the Gürtekurve type, which as dimension $3g-3=3\cdot 3-3=9-3=6$. This motivates modifying the increment to one of only 1 unit. This leads to the following Ansatz: $\lambda=5p+3r-6$. This gives for $(r,p)=(2,1)$, $\lambda=5+6-6=5$ the correct number. But if we look at the type $(r,p)=(2,2)$ we get $\lambda =10+6-6=10$; but on the other hand the hyperelliptic models have $2g+2-3=2\cdot 5+2-3=9$ moduli conflicting the new Ansatz for $\lambda$. Of course the real scenario about the increments might be pretty more complicated than the linear progression observed in the complex case (corresponding to closed Riemann surface). Another more neutral way to look at the question is as follows. Given is $(r,p)$ a pair of integers. Allied to this there is a moduli space ${\cal M}_{r,p}$ of all bordered (Riemann) surfaces of type $(r,p)$. Its dimension is $3g-3$ (Klein 1882 [@Klein_1882]), where $g$ is the genus of the double. We imagine the range of all possible gonalities $r\le\gamma\le r+p$ as a horizontal array of entries above each of which is reported the dimension of the moduli space of curve having gonality $\le \gamma$. This is depicted as a vertical bar. At first, only the top dimension attached to $\gamma=r+p$ is known as $3g-3$. By Coppens we know that there will be $p$ descents of this highest bar to the lower gonalities between $r$ and $r+p$. Pause at this stage to notice that assigned to the sole data $(r,p)$ there is assigned unambiguously such a histogram of gonalities (cf. Fig.\[Histogr:fig\]). One special case in which we can hope to be more explicit regarding the lowest strata is when $r$ equals 1 or 2. In this case we know that the moduli space contains hyperelliptic membranes. Assuming $p$ large enough ($p\ge 1$) the lowest gonality is $\gamma=2$. It is tautological that the hyperelliptic locus has this gonality, and conversely. So we control explicitly the dimension of the lowest strata. We find $(2g+2)-3$ real constants. Thus the dimension difference $\delta$ of the top and lowest strata is $\delta=3g-3-[(2g+2)-3]=g-2$. This rewritten in terms of $(r,p)$ is also $g-2=(r-1)+2p-2=r+2p-3$. $\bullet$ If $r=1$, this gives $\delta=2p-2=2(p-1)$. Positing linearity of the increment, this ought to be divided in $p-1$ equals parts (since $r=1$ itself is not a gonality when $p\ge 1$), and we get exactly a progression by 2 units. Hence under the Ansatz of linearity the histogram would be completely known. $\bullet$ If $r=2$, this gives $\delta=2p-1$. Assuming linearity of the increment, this ought to be divided in $p$ equals parts, and we get something like a progression by 2 units. However the non-divisibility implies that in this case it is impossible to have a linear progression of the histogram. Hence some jumps must occurs. So in these cases there is some hope to be completely explicit about the histogram attached to $(r,p)$. It would essentially suffices to decide where occur some irregular jumps. Let us formalize a bit. Given a pair of integers $(r,p)$, we have a moduli space ${\cal M}:={\cal M}_{r,p}$ of all bordered Riemann surfaces of type $(r,p)$. (To allege notation with omit the indices $(r,p)$, as the topology is fixed once for all.) Its dimension is invariably $3g-3$, where $g=(r-1)+2p$ is the genus of the double. \[gonality-profile:def\] [Inside the full moduli space ${\cal M}:={\cal M}_{r,p}$, consider the sublocus $M_d$ of all surfaces with gonality $\gamma\le d$, and let $\mu_d=\dim M_d$ be its dimension. The histogram we were speaking about is essentially the function $d\mapsto \mu_d$, which we call the [*gonality profile*]{}.]{} It is evidently monotone but a priori not strictly. Misinterpreting Coppens’s result one would guess strict monotony, but Coppens states only that each gonality is exactly realized, hence in symbols that $M_d-M_{d-1}$ is non-void (at least for $d$ in the range $[r,r+p]$). Thus a priori it could be the case that when incrementing the parameter $d$ we get new surfaces but their variety is not of larger dimension. Of course this scenario may look a bit unlikely due to the algebro-geometric character of the whole topic, but I do not know an argument. The domain of our function $d\mapsto \mu_d$ is the set of all integers but the interesting range is $[r, r+p]$ at least taking Gabard for granted. The latter amounts to say that $\mu_{r+p}=3g-3$. Now if $r=1$ or $2$, then the moduli space ${\cal M}={\cal M}_{r,p}$ contains hyperelliptic representatives, and the latter exhaust the locus $M_2$. We calculate easily $\mu_2=(2g+2)-3=2g-1$ and deduced the difference $\delta=\mu_{r+p}-\mu_2=(3g-3)-(2g-1)=g-2$. From here we inferred that: $\bullet$ when $r=1$ (and $p\ge 1$) then $r$ itself is not a gonality and so there is really only $p-1$ descents. Since $\delta=g-2=r+2p-3=2p-2=2(p-1)$, we can divide (without rest) this by the number of $p-1$ descents, to get a statistical increment of 2 units. If one believes in the linearity regularity then the histogram would be completely known in that case. $\bullet$ when $r=2$ then $r$ is a gonality, and we have exactly $p$ admissible descents along the range $[r,r+p]$. Now $\delta=g-2=r+2p-3=2p-1$, which is not divisible by $p$. We infer an obstruction to the scenario of linearly evolving histogram. (In other words the function is not linear on the segment $[r, r+p]$.) Perhaps it is just doing a gentle seesaw at some early place? At this stage we may have exhausted all what can be said on trivial arithmetical grounds. Going further probably requires some geometric impetus, like looking at explicit models (extending the hyperelliptic case). So one needs probably to describe large families of $d$-gonal surfaces for $d\ge 2$. If a general result describing the gonality profile $d\mapsto \mu_d$ looks out of reach, one can start examining low values of $(r,p)$ to explore the situation. \[11.11.12\] [*Examples.*]{}—$\bullet$ E.g. for $(r,p)=(2,1)$ (thus $g=3$) (the Gürtelkurve type) then the profile is completely known, namely $\mu_2=5$ (hyperelliptic locus of dimension $(2g+2)-3$) and $\mu_3=6$ (equal to $3g-3$). Of course Gabard’s $\gamma\le r+p$ follows in this case via the canonical embedding realizing the curve as a Gürtelkurve in ${\Bbb P}^2$. $\bullet$ For $(r,p)=(2,2)$ (thus $g=5$), we have again the hyperelliptic locus giving $\mu_2=(2g+2)-3=9$. The top locus $M_{r+p}=M_4$ has dimension $\mu_4=\mu_{r+p}=3g-3=15-3=12$ (Gabard is used but maybe there is an argument by hand). What about $\mu_3$? To seek an answer we refer back to the table of Fig.\[Coppens:fig\], where we traced a picture (label 223) of an uninodal quintic with gonality $\gamma=3$. Quintics depends on $\binom{5+2}{2}-1=\frac{7\cdot 6}{2}-1=20$ parameters, but modulo the collinearity group $PGL(3)={\rm Aut}({\Bbb P}^2)$ of $3^2-1=8$ dimensions, this boils down to 12 effective parameters. Of course the uninodal quintic we consider is really compelled to live on the smaller discriminant hypersurface of dimension $19$ and so our curve 223 truly depends on only 11 essential parameters. Assuming that a full neighborhood of curve $223$ consists of curves keeping the same gonality $\gamma=3$ suggests therefore the value $\mu_3=11$ (at least as a lower bound). Observe that the picture 223 is total under a pencil of lines, and it seems reasonable to expect that when the curve is slightly perturbed along the discriminant hypersurface, total reality of the pencil persists on the ground of some topological [*stability*]{}. Remember e.g., that total reality amounts to the transversality of the foliation (induced by the pencil) along the curve, and transversality is the mother of any topological stability (Thom-style philosophy). Note of course that our curve (being uninodal) represents actually a smooth point of the discriminant and so we safely dispose of the required parameters of deformation. This is perhaps worth saying if one remembers certain plane cubics (or even conics) as examples of real algebraic varieties having an isolated real point. Maybe the above stability argument adapts to situations where there are several nodes via Brusotti’s theorem describing the infinitesimal structure of the discriminant near a multi-nodal curve (with say $\delta$ nodes) as an union of smooth branches crossing transversally (normal crossing). $\bullet$ For $(r,p)=(3,2)$ (thus $g=6$), we have no hyperelliptic locus. The top locus $M_{r+p}=M_5$ has dimension $\mu_5=\mu_{r+p}=3g-3=18-3=15$ (Gabard is used but maybe there is an argument by hand). What about $\mu_3$ and $\mu_4$? We look again back to Fig.\[Coppens:fig\], where we find curve 324. This is merely a smooth quintic with 2 nested ovals hence with gonality $\gamma=4$. Remember that smooth plane $m$-tics have in general complex gonality $(m-1)$. As quintics depends on 12 essential parameters, the above stability argument shows that the strata $M_4$ contains the locus of all such quintics, and we infer $\mu_4\ge 12$. Is this an equality? How to estimate $\mu_3$? Due to time limitation, we have to leave all this (in our opinion) exciting topic at a fragmentary stage. Perhaps a last word, if we use picture 324bis (still on Fig.\[Coppens:fig\]), which is a sextic with 4 nodes also having $\gamma\le 4$ (projection from the node), then we get a model depending on $\binom{6+2}{2}-1=\frac{8\cdot 7}{2}-1=27$ constants, minus the $8$ coming from $PGL(3)$ gives $19$, of which must be subtracted 4 units (using Brusotti’s normal crossings description). The final result is 15. Repeating the above stability argument implies that $\mu_4\ge 15$. This is a much stronger lower estimate, which in fact must be an equality since we have already attained the dimension of the full moduli space. Hence we conclude $\mu_4=15$; strikingly as big as $\mu_5$! This answer is quite intriguing in case it is correct at all? It would show that the gonality profile does not need to be strictly increasing! \[11.12.12\] Alas all of our counts are completely erroneous in view of some basic examples shown in the next section. Of course the mistake is that not all birational (conformal) equivalence giving rise to Riemann’s moduli space need to be induced by a collineation as an automorphism of the ambient plane ${\Bbb P}^2$. \[11.11.12\] Finally, it is perhaps fruitful to keep a view on the space of all (total) circle maps. This is the [*Ahlfors space*]{} (improvised jargon) quite akin to so-called Hurwitz spaces. All what we were concerned with in this subsection is arguably just a shadow of this larger space dominating the moduli space ${\cal M}_{r,p}$. Precisely, the [*circle maps (or Ahlfors) space*]{} $C_{r,p}$ consists for a fixed pair $(r,p)$ (number of contours and handles resp.) of all circle maps $f\colon F \to \Delta$ on a “variable” bordered Riemann surface of specified topological type $(r,p)$. Forgetting the circle map $f$ induces a natural map $C_{r,p}\to {\cal M}_{r,p}$ to the moduli space. Of course one must consider the space $C$ modulo the equivalence relation of a conformal diffeomorphism commuting with the maps to the disc. The strata $M_d$ of all surfaces of gonality $\gamma\le d$ appear then as the projections of the fibres of the degree function on $C_{r,p}$. The fibre of the map $C\to M$ (indices omitted) is the space of all total maps on a fixed bordered Riemann surface $F$. Correcting the previous section ------------------------------- \[11.12.12\] There are many counterexamples to our naive moduli count via plane nodal models. For instance considering curves of $g=2$, and using the projective realization as a quartic with one node, we get the dependence upon $\dim \vert 4H \vert=\binom{4+2}{2}-1=\frac{6\cdot 5 }{2}-1=14$ parameters, of which must be subtracted one unit to be on the discriminant (due to the presence of the node) and finally one has to remove the $8$ dimensions of $PGL(3)$. The end result is $14-1-8=5$, which exceeds by $2$ units Riemann’s $3g-3=3$. Of course this excess is due to the fact that we moded out only by (linear) automorphisms of the plane whereas there might be more mysterious birational equivalence relating to configurations of our family of uninodal quartics. This demonstrates that the estimate we got in the previous section are completely erroneous and unreliable, and one must find some completely new ideas (i.e. old stuff à la Riemann) to really penetrate the intrinsic nature of the problem. For the moment I have no idea on how to attack the problem of describing the size (=dimensions) $\mu_d$ of the varied gonality strata $$M_d=\{ F\in {\cal M}_{r,p} : \gamma (F)\le d \}.$$ Let us try anew to attack this problem of describing the gonality profile $d\mapsto \mu_d$ for each pair $(r,p)$. \[12.12.12\] First complete information is obtained in the easy case of planar membranes ($p=0$) as a consequence of the Bieberbach-Grunsky theorem (Lemma \[Enriques-Chisini:lemma\]). When $p=0$ the gonality profile is a skyscraper concentrated at the single place $d=r=r+p=r+2p$, i.e. $\mu_r=3g-3$, where $g=(r-1)+2p=r-1$ is the genus of the double. This follow at once from the trivial lower bound $r\le \gamma$ on the degree of circle maps (or the allied gonality $\gamma$), and the Bieberbach-Grunsky theorem (Lemma \[Enriques-Chisini:lemma\]). (Notice that neither Ahlfors 1950 [@Ahlfors_1950] ($\gamma \le r+2p$), nor Gabard 2006 [@Gabard_2006] ($\gamma \le r+p$) is required.) After that let us examine the cases with $p=1$. We start with: $\bullet$ $(r,p)=(1,1)$: then we have $1=r\le \gamma\le r+p=2$ (using Gabard). However since the genus of the double is $g=(r-1)+2p=2$, the curve is hyperelliptic and we may avoid Gabard. The lower bound $r=1$ cannot be realized, since $p\neq 0$. We deduce: For $(r,p)=(1,1)$, the gonality profile is a skyscraper concentrated at $d=2$, i.e. $\mu_d=3g-3=3$ for $d=2$ and $\mu_d=0$ elsewhere $(d\neq 2)$. $\bullet$ $(r,p)=(2,1)$ (with $g=(r-1)+2p=3$): then we have $2=r\le \gamma\le r+p=3$ (using Gabard). However one can dispense Gabard by using the canonical embedding taking the double of the bordered Riemann surface to a Gürtelkurve $C_4 \subset {\Bbb P}^2$, i.e. a quartic with 2 nested ovals. This proves $\gamma \le 3$ (via projection from the inner oval). So $\mu_3=3g-3=6$. Of course we have also a hyperelliptic locus, whose dimension is $(2g+2)-3=5$, so $\mu_2=5$. This proves the: For $(r,p)=(2,1)$, the gonality profile is a “twin tower” concentrated at two places $d=2,3$, and $\mu_3=3g-3=6$ whereas $\mu_2=5$ (all other $\mu_d$ are zero). $\bullet$ $(r,p)=(3,1)$ (with $g=(r-1)+2p=4$): then we have $3=r\le \gamma\le r+p=4$ (using Gabard). Without using Gabard, one can look at the canonical model in ${\Bbb P}^{g-1}={\Bbb P}^{3}$ of degree $2g-2=6$. This is probably a complete intersection of a cubic surface with a quadric, weighting bidegree $(a,b)=(3,3)$ on the latter, hence of genus $g=(a-1)(b-1)=2\cdot 2 = 4 $ (the expected value). One can then draw a picture by smoothing two pairs of 3 lines distributed in each ruling. When the lines are oriented in the most trivial way (each inducing the same integral homology class on the torus ${\Bbb P}^1({\Bbb R})\times {\Bbb P}^1({\Bbb R})$) we get a total map of degree $3$ by projection on the factors of ${\Bbb P}^1\times {\Bbb P}^1$ (do a picture). Taking (somewhat cavalier) Gabard for granted we get $\mu_4=3g-3=9$. How to estimate $\mu_3$? Let us try several strategies: \(1) [*Extrinsic plane projective realizations*]{}.—The naive idea is to look at Fig.\[Coppens:fig\] (picture 313). This is a quintic with 2 nodes and $\gamma\le 3$ (hence equal to $3$ by the trivial lower bound $r\le \gamma$). If we count the (naive) moduli of such a curve we obtain: $\dim \vert 5H \vert=\binom{5+2}{2}-1=\frac{7\cdot 6}{2}-1=20$, of which must be subtracted $2$ for the two nodes, and $8=\dim PGL(3)$ to get $10$. This exceeds by one unit the full moduli space $3g-3$, and so we get an alienating count. Of course as already said the reason is that we only took into account linear collineations (ambient automorphisms) whereas one should mod out by all inherent isomorphisms of the family of curves. One way to remedy the situation would be to look at Cremona transformations (birational transformations of the plane), but it is not even evident that this would give the right answer on abstract moduli. Another idea is to look at higher order plane models with $\delta$ many nodes as to adjust the genus to $g=4$. For instance sextics with 6 nodes $C^6_6$, septics with 11 nodes $C_7^{11}$, octics with 17 nodes $C_8^{17}$, etc. However the same calculation shows that such family of curves depends on $\dim C^6_6=\frac{8\cdot 7}{2}-1-\delta-8=13$, $\dim C^{11}_7=\frac{9\cdot 8}{2}-1-\delta-8=16$, $\dim C^{17}_8=\frac{10\cdot 9}{2}-1-\delta-8=19$, etc. It seems that there is perpetually an increment by 3 units, and never get something realistic via naive counting. \(2) [*Extrinsic projective realization as a branched cover of the line (or the disc), i.e. circle maps via Riemann-Hurwitz.*]{}—We fix as an Ansatz $\gamma=3$ (inside our fixed topological type $(r,p)=(3,1)$), and by the Riemann-Hurwitz relation applied to a circle map $F \to \overline{\Delta}$ of degree $3$, we find $\chi (F)= 3 \chi (\overline \Delta)-b$, so $b=3-\chi (F_{3,1})=3-(-3)=6$ many branch points. Moving those 6 points arbitrarily in the disc, and quotienting by automorphisms of the disc we arrive at $2\cdot 6 - 3=9$ (real) moduli. This looks again anomalous for we receive the same answer as for the full $3g-3$ moduli. (We already experimented this failure of Riemann-Hurwitz in the bordered setting, and we are in depressive mode.) The mystery is perhaps that we do not enjoy complete freedom in moving branch points in the bordered setting, but I lack any understanding of which sort of geometric restrictions have to be taken into account. \(3) [*Intrinsic count à la Nielsen-Fenchel.*]{}—Another possible strategy, is to adapt the Nielsen-Fenchel count of moduli via a decomposition in pants. Remember that this works at the perfection to predict the dimension of the full moduli space (cf. e.g. our Sec.\[Nielsen-Fenchel:sec\] below). The idea would be that if we prescribe a lower gonality then an appropriate decomposition in pants (somehow calibrated on the circle map) should predict the moduli dimension of the restricted class too. Alas for $\gamma=3$, I do not really see how to proceed, but let us first experiment the method on the simpler hyperelliptic case. Consider e.g. a membrane with $(r,p)=(2,p)$, $p$ arbitrary, of gonality $\gamma=2$ (hyperelliptic case). On drawing the configuration, and decomposing it into pants invariant under the hyperelliptic involution (visualized as a half-twist rotation) we obtain Fig.\[Pants-hyper:fig\] (left part). 0 -5pt0 Introducing on the surface its uniformizing metric of constant curvature $-1$ (alias hyperbolic metric), we count moduli as the lengths of loops bounding pants affected by certain twist parameters. We get (reading contributions from the top to the bottom of the figure): $$2+2 \cdot 2 p-1=4p+1$$ free parameters. Indeed the first term ($2$) arises from the top loop (its length plus its twisting aptitude). Next we see $p$ shaded pants whose contours exhaust all junctures of the pants decomposition. However all bottom parts of the shaded pants are permuted via the hyperelliptic involution (half-turn rotation), hence of the same length. So each shaded pants really contributes for $2$ lengths each susceptible of a twist, whence the second term ($2 \cdot 2 p$). As to the last term ($-1$), notice that the very bottom contours of the surface have no gluing companion (to be twisted with), so one unit must be subtracted. The announced count follows. On the other hand, such hyperelliptic curves depend (via a count à la Riemann-Hurwitz) on $(2g+2)-3$ parameters where $g=(r-1)+2p=2p+1$. Hence on $(2g+2)-3=[2(2p+1)+2]-3=4p+1$, in accordance with the result as calculated via the pants method. A similar count works for hyperelliptic membranes with $(r,p)=(1,p)$, cf. right part of Fig.\[Pants-hyper:fig\]. In that case we obtain $$2+2\cdot 2(p-1)+1=4p-1$$ moduli, and on the other hand $(2g+2)-3=[2(2p)+2]-3=4p-1$ parameters. Both counts are again in accordance. Of course we could even dream that the pants dissection method (cf. Fig.\[Pants:fig\], right part) affords yet another full proof of Ahlfors circle maps, but this looks a bit tricky to implement. Perhaps even more ambitious one could hope that pants dissection affords a proof of the Forstnerič-Wold 2009 [@Forstneric-Wold_2009] desideratum that each finite bordered surface embeds holomorphically in ${\Bbb C}^2$. (Notice that this is much stronger (viz. complex analytic) than the conformal embedding in $E^4$ prompted by the Garsia-Rüedy-Ko theory as implemented in Ko 1999 and 2001 [@Ko_2001]. At least the latter shows that there is no conformal obstruction to the Gromov conjecture/question (1999 [@Gromov_1999]) that any Riemannian surface should isometrically embed in $E^4$.) Coming back to our problem of calculating $\mu_3$ for membranes of type $(r,p)=(3,1)$ we severely lack any reliable technique of calculation. Existence of Ahlfors maps via the Green’s function (and the allied Dirichlet principle) {#Green:sec} ======================================================================================= All what follows is extremely classical, yet the writer confesses to have assimilated (the first steps of the argument) as late as the \[04.08.12\]! First it is well-known that the solubility of the Dirichlet problem (say on a bordered Riemann surface) is tantamount to the existence of the Green’s function $G(z,t)$ with pole at $t$, for each $t$. (Actually, we primarily need that the former implies the latter.) This “Dirichlet-to-Green” mechanism will be recalled below along with the definition and some geometric (biochemical) intuition about the Green’s function. The latter has also strong electrostatic or hydrodynamic connotations. The definition of the Green’s function is somewhat easier in the case of plane domains, and its extension to bordered surface—while still laying in the range of Dirichlet—implicates some conceptual difficulties. The [*Green’s function*]{} $G(z,t)$ with pole at $t$ (a fixed interior point) is a completely canonical function characterized by the properties: it is harmonic off $t$, vanishes along the boundary and its germ near has the singular behavior prescribed by the function $\log\vert z-t\vert$ in any local uniformizer $z$. It will be verified that $G(z,t)$ is negative on the interior of the bordered surface (consequence of Gauss’ mean value property of harmonic function and the resulting maximum principle). Then we shall try to approach the existence of the Ahlfors function by duplicating the Green-type proof of the Riemann mapping theorem (simply-connected case), which just amount to write down the magic formula $f(z)=e^{G(z,t)+i G^{\ast}(z,t)}$, where $G^{\ast}$ is the conjugate potential. Note that $G(z,t)\le 0$ ensures $\vert f(z) \vert=e^{G(z,t)}\le 1$ with equality precisely along the boundary. The main difficulty about extending this “Green-to-Riemann” trick to the multiply-connected setting is to arrange single-valuedness of the conjugate potential $G^{\ast}$. This amounts to kill all periods of the $1$-form $dG^{\ast}$ from which $G^{\ast}$ arises through line-integration. To achieve this one is invited to introduce enough free parameters in the problem by considering a superposition of various Green’s functions $\sum_i \lambda_i G(z, t_i)$ for several poles $t_i$ sufficiently abundant so as to enable the killing of all periods (via linear algebra). Since a planar domain with $r$ contours has $r-1$ essential cycles (up to homology) and attaching $p$ handles creates 2 new essential cycles, we need annihilating $(r-1)+2p$ periods. Taking one more pole (raising the total number to $r+2p$) supplies enough parameters for linear algebra to ensure existence of a non-trivial solution in the kernel of the period mapping. This prompts (almost) the existence of an Ahlfors circle map of degree $r+2p$ (as predicted in Ahlfors 1950 [@Ahlfors_1950]). Alas, a serious technical difficulty occurs, namely ensuring the positivity of all $\lambda_i$. Ignoring this issue, any $r+2p$ points (in the interior) could be the zeroes of a circle map. Presently, we lack a complete existence of an Ahlfors map through this procedure. Of course it would be even more challenging to arrive at Gabard’s bound (mapping degree $\le r+p$) through this classical strategy (à la Green, Riemann, Grunsky, Ahlfors, Kuramochi, etc.). In Riemann the trick of annihilating periods appears of course very explicitly in the following jargon: “[*so bestimmen da[ß]{} die Periodicitätsmoduln sämmtlich $0$ werden.*]{}” (cf. e.g. Riemann 1857 [@Riemann_1857 p.122]). The core of Heins’ argument 1950 [@Heins_1950] is also exactly in this spirit and Heins seems able to complete the program via consideration of convex geometry. Our intention is first to recall the basic procedure, and we hope to be able later to settle the positivity problem. A priori it is not evident that the latter condition is always achieved for an arbitrary selection of poles $t_i$ of Green’s functions (which will mutate into zeroes of the “Riemann-Ahlfors map” $f$ after exponentiation). \[25.08.12\] [*Corrigendum.*]{}—The above linear superposition $\sum_i \lambda_i G(z,t_i)$ on Green’s functions is maybe somewhat too continuous in nature. This may be seen by exponentiating and looking at the local behavior of $f$. Near some $t_i$, $G(z,t)\sim \lambda_i G(z, t_i) \sim \lambda_i \log \vert z \vert$ so that $\vert f(z)\vert \sim \exp(\lambda_i \log\vert z\vert)=\vert z \vert^{\lambda_i}$ so that $f$ has not the character of a holomorphic function when $\lambda_i$ is not integral. Another way to argue in the same sense is suggested by Ahlfors 1950 [@Ahlfors_1950 p.126–7, §4.3]. Assume that $f(z)$ is a circle-map $f\colon F \to \overline \Delta$ with zeros at $t_1,\dots, t_d$ (counted with multiplicities), then upon post-composing with the function $\log\vert z \vert$ (harmonic off the origin) we get the function $\log \vert f (z) \vert$ harmonic on $F$ save at the $t_i$ where it has logarithmic poles. Therefore this function must coincide with superposition $G:=\sum_{i=1}^d G(z,t_i)$ of Green’s potentials. Indeed, the difference $\log \vert f (z) \vert-G$ is throughout $F$ harmonic (cancellation of singularities) and vanishes along the border $\partial F$, hence is identically zero. \[NB: the above remark is to be found in Ahlfors (), who (in our opinion) fails to insist on the assumption that $f$ is a circle-map (i.e. $\vert f \vert =1$ along the border), which is crucial to ensure that $\log \vert f (z) \vert$ vanishes along the border $\partial F$.\] So given a circle-map $f$ with $d$ zeros $t_i$ we have the formula $$\log \vert f (z) \vert=\textstyle\sum_{i=1}^d G(z,t_i).$$ Conversely, given points $t_i$, we may consider the right-hand side of the previous equation $$\label{Green-super:eq} G:=\textstyle\sum_{i=1}^d G(z,t_i)$$ and the following formula will define a circle-map $$f(z)=e^{G+iG^{\ast}}$$ provided $dG^{\ast}$ (the conjugate differential of $G$) has all its periods integral-multiples of $2\pi$. (It follows incidentally, that a circle-map is uniquely determined up to a rotation by the geographic location of its zeros. This can also be seen algebro-geometrically, by considering the Schottky double, where the divisor of zeros $D$ becomes linearly equivalent to its symmetric conjugate $D^{\sigma}$, spanning together a pencil $g^1_{d}$ defining a total morphism to ${\Bbb P}^1$ of degree $d$, cf. Lemme 5.2 in Gabard 2006 [@Gabard_2006]. The desired integrality of periods resembles a [*Diophantine condition*]{} (at least is qualified as a such by Ahlfors 1947 [@Ahlfors_1947 p.1]), emphasizing from the outset the relative difficulty of the problem. All of our freedom relies on dragging the points $t_i$ through the surface $F$ hoping that for a lucky constellation the $1$-form $dG^{\ast}$ acquires simultaneous integrality of all its periods along $\gamma_1, \dots, \gamma_g$ the $g:=(r-1)+2p$ many essential $1$-cycles traced on $F$ (cf. Fig.\[Green:fig\]e). As a personal trouble, $dG^{\ast}$ seems to have singularities where $G$ does, but maybe they disappear. Bypassing this point, Ahlfors’ Diophantine problem (1947) looks well-posed and one may hope a direct attack upon arranging integrality of all periods. (Ahlfors 1950 [@Ahlfors_1950] (p.127) first reformulates the condition in term of Schottky differentials and then switches quickly to the extremal problem, so does not seem to attack directly the Diophantine question. In fact, its elementary proof on p.124–126 follows a somewhat different route by constructing a half-space map involving avatars of Green’s function with poles situated along the boundary. We shall come back to this subsequently.) [**Trying a direct attack.**]{} Assuming the problem well-posed, we can consider a period mapping $$\wp\colon R^d\longrightarrow {\Bbb R}^g \longrightarrow ({\Bbb R} / 2\pi {\Bbb Z})^g=:T^g,$$ where $R={\rm int} (F)$ is the interior of the bordered surface $F$, and the first map takes the periods along the fixed basis of the first homology $\gamma_i$ of the $1$-form $dG^{\ast}$ corresponding to the points $(t_1, \dots, t_d)\in R^d$ via formula . The second map is just the natural quotient map. Now one may hope to apply the usual surjectivity criterion for a continuous map to a closed manifold (here $R^d\to T^g$) saying that if the representation induced on the top-dimensional homology of the target-manifold is non-zero then the mapping is surjective. For definiteness we recall its statement and short proof. Let $f\colon X \to T$ be a continuous map from a (topological) space $X$ to a (target) manifold $T$ of dimension $n$, say. It is assumed that $T$ is closed (i.e. compact borderless). It is also essential to assume that $T$ is a Hausdorff manifold. If the induced homomorphism $H_n(f)$ is non-zero, then $f$ is onto. One considers the map induced on the homology $H_n$ of dimension $n$ equal to that of the manifold $T$. If $f$ fails to be surjective, it factors through the punctured manifold $X \to T-\{t\}$ for some point $t$. Now it is a simple fact that the top-dimensional homology of a (Hausdorff) manifold vanishes, so in particular $H_n(T-\{t\})$ is trivial. By functoriality it follows that $H_n(f)=0$, violating our assumption. In particular $0=(0,\dots, 0)\in T^g$ would be the image of some $(t_1, \dots , t_d)\in R^d$ and the corresponding potential $G$ given by would have a conjugate differential $dG^{\ast}$ meeting the Diophantine requirement. This strategy requires a good understanding of the mapping $\wp$ perhaps in the sense that when one pole $t_i$ is dragged along the cycle $\gamma_j$ then the image winds once around the corresponding factor of the torus $T^g$. Choosing $d=g$ and in the Künneth factor of $H_g(R^d)$ the element $\gamma_1 \otimes \dots \otimes \gamma_g$ which has the correct weight $g$ so as to be an element of $H_g(R^g)$ whose image would be the fundamental class of the torus $T^g$. This would establish the surjectivity of $\wp$ for $d=g$. Alas, this is a bit too optimistic in the planar case ($p=0$). So our argument must be foiled at some place. The reasonable result to be expected is $d=g+1$ (like Ahlfors 1950 [@Ahlfors_1950]) and boosting the method upon choosing $\gamma_1 \otimes\dots \otimes \gamma_{r-1} \otimes (\alpha_1 \star \beta_1)\otimes \dots \otimes(\alpha_p \star \beta_p)$ where the $\alpha_i, \beta_i$ are the cycles winding around the handles (cf. Fig.\[Green:fig\]e) one may expect to achieve $d=r+p$ as predicted in Gabard 2006 [@Gabard_2006]. Digression on Dirichlet (optional) ---------------------------------- The Dirichlet solution may be interpreted as the permanent equilibrium state of temperature in a heat-flow conducting medium. Arguably (physico-chemical intuition?), this phenomenology is completely insensitive to the topology. Hence Dirichlet’s problem is always soluble whatever the topological complexity of the bordered manifold is. One only requires a Riemannian metric to give a good sense to the (Beltrami) Laplacian (or the allied mean value property). Hence any metric bordered smooth manifold, say compact to stay in the reasonable realm of finiteness is suitable to pose and solve the first boundary value problem. \[Remember maybe that there is vast jungle of non-metric manifolds, those of Cantor 1883 and Prüfer 1922 being the most prominent examples, but the latter do not enter the scene of function theory at least in complex dimension 1.\] What is the most general context where the Dirichlet problem is soluble? Our guess is for any compact bordered Riemannian manifold, eventually non-orientable. Hence Dirichlet makes sense also on non-orientable manifolds, but the case of immediate interest is that of compact bordered Riemann surfaces ([*ipso facto*]{} orientable). Solid existence proofs were primarily devised by H.A. Schwarz, alternating method (ca. 1870), etc. with many subsequent extensions, e.g. Nevanlinna 1939 [@Nevanlinna_1939], several works of Ahlfors, H. Weyl 1940 [@Weyl_1940] (method of orthogonal projection), not to mention Neumann, Poincaré, Korn-Lichtenstein, etc., cf. e.g. Neumann 1900 [@Neumann_1900]). Another source is Hilbert-Courant’s book cited e.g. for this purposes in Royden’s Thesis 1950/52 [@Royden_1952]. \[For those inclined toward modern expressionism, there is surely a concept of “Dirichlet space” (Brelot, Beurling, Deny, etc.) which should englobe any bordered Riemannian manifold and much more.\] In the appropriate Hilbert space, minimizing the Dirichlet integral amounts to minimize the length of a vector lying on a certain hypersurface $M$ corresponding to the boundary data $f\colon\partial F\to {\Bbb R}$. A priori this hypersurface could spiral around the origin impeding existence of a minimum or be bumpy enough as to violate uniqueness. But one rather imagine it to be a linear manifold implying a unique minimum of the distance function (norm). Of course the hypersurface in question (corresponding to a certain boundary prescription) is readily shown to have linear character, as subtracting any member of it, its translate through the origin identifies with the set of functions vanishing along the boundary. The latter is vectorial, being the kernel of a linear mapping (restriction to the boundary). Dirichlet principle looks thus immediately imputable to an Euclid-Hilbertian conception of space, yet with difficulty concentrating on the existence question of a member (=point) in this hypersurface $M$ (i.e., of a function matching the boundary prescription having with finite Dirichlet integral). As we know Hilbert’s solution primarily involved the compactness paradigm, formalized as a such some few years later by Fréchet. The naive minimization procedure is not fairly evident, and indeed plagued by the counterexample of Hadamard 1906 [@Hadamard_1906], and the earlier one of Prym 1871 [@Prym_1871]. Prym (1871 ) describes a continuous function on the boundary of the unit disc such that the Dirichlet integral for the associated harmonic extension of the boundary function is infinite. \[The latter harmonic extension is known to exist independently of the Dirichlet principle, e.g. on the ground of Poisson’s formula which solves Dirichlet in the disc-case.\] Later Hadamard (1906 ) gave a similar example where any (continuous) function matching the boundary data has infinite Dirichlet integral. (Perhaps, any Prym data is also explosive in the sense of Hadamard?) The moral is quite subtle to grasp: roughly the Dirichlet principle fails but not the Dirichlet problem which is always uniquely soluble! Hilbert’s solution (ca. 1900 [@Hilbert_1900], [@Hilbert_1901/04]) under special hypotheses (involving only the space and not the boundary data?!) is certainly sufficient for the purpose at hand. Hilbert’s hypothesis where weakened in subsequent works by B. Levi 1906 [@Beppo-Levi_1906], Fubini 1907 [@Fubini_1907], Lebesgue 1907 [@Lebesgue_1907], compare also the historiography in Zaremba 1910 [@Zaremba_1910]). For practical purposes (e.g. for the construction of the Green’s function) one can probably restrict attention to reasonable boundary data, as those arising via geometric construction (e.g., the logarithmic charge allied to the construction of the Green’s function of a plane smoothly bounded domain). Possibly, for tame boundary data the original Dirichlet principle remains an efficient tool for a direct variational treatment of the boundary value problem. Alternatively, of Dirichlet-Riemann-Hilbert one may use the classical but cumbersome alternating method of Schwarz (or Neumann’s variant) to solve the Dirichlet problem. To summarize we need the result: [(Dirichlet, Riemann, Schwarz 1870, Hilbert 1900, etc.)]{} Given a compact bordered Riemann surface $F$, and a continuous boundary function $f\colon \partial F \to {\Bbb R}$. There is a unique harmonic function $u\colon F \to {\Bbb R}$ extending $f$. First (rigourously) obtained in Schwarz 1870 [@Schwarz_1870-alternirendes-Verfahren] via the alternating method. Variation of this technique Picard’s method of successive approximation (cf. Picard, Zaremba, Korn, Lichtenstein). Another variant of proof is Hilbert’s resurrection of the Dirichlet principle (direct variational method). Reference in book form cf. Hilbert-Courant. Another more modern trend is to use Perron’s method which affords great simplification. Compare for instance Ahlfors-Sario 1960 [@Ahlfors-Sario_1960 p.138–141, esp. 11G] for an execution of Perron’s method (joint with Harnack’s principle) in the context of abstract Riemann surfaces. From Green to Riemann {#sec:From-Green-to-Riemann} --------------------- In term of the Green function for a simply-connected domain one may write down the Riemann map as $$f(z)=e^{G+iG^{\ast}}\,,$$ where $G^{\ast}$ is the conjugate potential (satisfying the Cauchy-Riemann equations). \[This is basically the second proof given by Riemann in 1857 [@Riemann_1857-DP], and see also e.g. Picard 1915 [@Picard_1915].\] That $f$ is a circle map follows from $G\le 0$ with vanishing precisely on the boundary, and the fact that $G^{\ast}$ is single-valued since the domain is simply-connected. Details are supplied during the next Steps, where we examine the more delicate multiply-connected domains or even general compact bordered Riemann surfaces. The conjugate $G^{\ast}$ potential is defined by the desideratum that $G+iG^{\ast}$ is holomorphic, i.e. ${\Bbb C}$-linearizable in the small. This gives the Cauchy-Riemann equations $$\frac{\partial G}{ \partial x}=\frac{\partial G^{\ast}}{ \partial y}, \qquad \frac{\partial G^{\ast}}{ \partial x}=-\frac{\partial G}{ \partial y}.$$ Writing formally $G^\ast$ as the integral of its differential, gives $$\begin{aligned} G^{\ast}= \int dG^{\ast}=\int (\frac{\partial G^{\ast}}{ \partial x} dx+\frac{\partial G^{\ast}}{ \partial y}dy) =\int (-\frac{\partial G}{ \partial y} dx+\frac{\partial G}{ \partial x}dy),\end{aligned}$$ whose integrand (a $1$-form) coincides actually with the $dG$ twisted by multiplication by $i$ on the tangent bundle. Therefore $dG^{\ast}$ is a genuine $1$-form canonically attached to the function $G$. ([*Warning*]{}.—The symbol $G^\ast$ (taken alone) as no intrinsic meaning at least as a single-valued function unless $dG^{\ast}$ is period free.) The Green’s function -------------------- But what is the Green’s function at all about? It is a sort of logarithmic potential attached to an electric charge placed at $t$. It is easier to define in the case of a plane domain bounded by smooth curves. The case of ultimate interest (compact bordered Riemann surfaces) will be discussed later. Given a domain $B\subset {\Bbb C}$ (smoothly bounded) marked at an (interior) point $t$ one considers the function $\log \vert z-t\vert$ which induces (by restriction) a charge (temperature) on the boundary $\partial B=C$ and one solves the Dirichlet(=first boundary-value) problem for this data. It results an (everywhere regular) harmonic function $u=u(z,t)$, which subtracted from the original logarithmic potential gives the (so-called) [*Green’s function with pole at $t$*]{} $$( \log\vert z-t \vert )- u(z)=:G(z,t). \label{Green:eq}$$ By construction, it vanishes along the contour $\partial B$ and possesses a logarithmic singularity near the point $t$. This is a canonical function attached to the sole data of $B$ and a certain interior point $t$. Note that $G(z,t)$ tends to $-\infty$ as $z$ approaches $t$, so one may think of the Green’s function as a black hole centered at $t$ with a vertiginous sink plunging into deep darkness. One can interpret this Green’s function as some electric potential (Galvanic current) on a conducting plate. If one prefers a biological metaphor one can visualize $G(z,t)$ as the proliferation of bacteroides originating from $t$ while expanding through the medium $B$ driven by an apparent global knowledge of the shape of the universe. To be more concrete, the expansion is more rapid where more free resources are available. In particular all bacteria reach synchronously the boundary having consumed all resources of the nutritive substratum in what looks to be the most equitable way. Compare the pictures in the simply-connected case (Fig.\[Green:fig\]a) and then for a multi-connected region (Fig.\[Green:fig\]b). Trying to imagine the same proliferation occurring on a bordered surface realized say in Euclidean $3$-space we get something like Fig.\[Green:fig\]d. 0 -5pt0 Now it is clear that the above formula $f(z)=f(z,t)=e^{G(z,t)+iG^{\ast}(z,t)}$ supplies the Riemann map with $f(t)=0$ and $f(z)\in S^1$ (unit circle) whenever $z$ lies on the boundary, where $G$ vanishes. Of course the map is only defined up to rotation, coming from an arbitrary additive constant in $G^{\ast}$. \[Compare for instance Riemann 1857 [@Riemann_1857-DP], Picard 1915 [@Picard_1915], etc.\] If one tries to adapt this proof to multi-connected domains one meets the notorious difficulty that the conjugate potential $G^{\ast}$ is not single-valued, a priori. So the efforts focus on eliminating the periods of its differential $dG^{\ast}$ by choosing appropriately some accessory parameters. \[This universally known device goes back at least to Riemann 1857 [@Riemann_1857 p.122] Schottky 1877 [@Schottky_1877], see also Picard 1913 [@Picard_1913], Koebe 1922, Julia 1932 [@Julia_1932], Grunsky 1937 [@Grunsky_1937], etc.\] Using this idea we may concoct a circle map $B\to \overline{\Delta}$. \[cf. Grunsky 1937 [@Grunsky_1937] or Grunsky 1978 [@Grunsky_1978] and also Ahlfors\]. The natural trick is probably to take several poles $t_i$ (say $d$ many). Those will ultimately become the zeroes of the circle map we are looking for as $e^{-\infty}=0$. One now form the combination of the corresponding Green’s functions $$G(z):=\textstyle\sum_{i=1}^d \lambda_i G(z,t_i) \quad (\lambda_i\in {\Bbb R}).$$ This gives a (finite) constellation of black holes scattered through the domain $B$ and we shall try to choose the constants $\lambda_i$ so that $dG^\ast$ has no period. Since the combination $G$ vanishes on the contour $\partial B$ (being a superposition of Green’s functions) the allied function $f(z)=f(z; t_i, \lambda_i):=e^{G(z)+iG^{\ast}(z)}$ will map $\partial B$ onto $S^1$. To arrange it as a circle map $f\colon B \to \overline{\Delta} $ requires the basic remarks of the next section, plus the more delicate issue of being able to choose positive $\lambda_i>0$. Quasi-negativity of Green ------------------------- The following property of the Green’s function is basic, yet important. Each Green function $G_t(z):=G(z,t)$ is quasi-negative (i.e. $\le 0$ throughout the domain and strictly $<0$ in its interior). From its definition it is clear that $G_t(z)\to -\infty$ as $t$ approaches the pole $t$. Thus choosing a very large negative (real) constant $C<0$ the corresponding level line $L_C$ of Green $G_t^{-1}(C)$ will be a nearly circular (Jordan) curve enclosing the pole $t$ in its interior. Further it looks evident that for $C<0$ large enough (in absolute value) this Jordan curve bounds a (topological) disc in the domain. (One could uses the general Schoenflies theorem requiring just to check that $L_C$ is null-homotopic in the domain $D$.) Next it is intuitive (but need to be arithmetized) that within this sufficiently small disc-shaped domain (i.e. the inside of $L_C$ for $C<0$ sufficiently large) the Green function $G_t$ is negative (indeed $\le C$). Cutting away from the domain $D$ the interior of $L_C$ we obtain an excised domain $D^{\ast}$ with one more contour. On this new domain, the Green’s function $G_t$ solves Dirichlet (first boundary-value) problem for the data $0$ on all contours but $C<0$ on the newly created contour $L_C$. We now conclude via the next lemma. \[Depressive:lem\] [(Depressiveness of Dirichlet, or rather the allied harmonic functions)]{} Let $F$ be a compact bordered Riemann surface. If the (continuous) boundary data function $f\colon \partial F \to {\Bbb R}_{\le 0}$ is non-positive, then so is its Dirichlet solution $u:=u(f)$, i.e. $u\le 0$ throughout $F$. If not then $u(z_0)>0$ (positive) at some interior point $z_0$ of the surface $F$. By compactness $u$ achieves its maximum, which is positive. Since $f\le 0$ the latter would not be achieved on the boundary violating the maximum principle (compare the next lemma). [(Maximum principle)]{} Any harmonic function $u$ on a compact bordered surface $F$ achieves its maximum on the boundary $\partial F$. In fact, if the maximum is achieved at some interior point then the function $u$ is constant. Assume $z_0$ to be an interior point realizing the maximum $M$ of the harmonic function $u$ defined on $F$. We trace a little (metric) circle about $z_0$ of sufficiently small radius as to lye entirely inside $F$ (together with its interior disc $D$). Harmonicity may be characterized via the [*mean-value property*]{} (Gauss, it seems): $$\int \int_D u(z) d\omega = area(D) \cdot u(z_0). \label{mean-value-prop:eq}$$ As $u(z)\le M$, we get $M\cdot area (D)\ge \int \int_D u(z) d\omega = area(D) \cdot u(z_0)$. Since $M= u(z_0)$, both extreme members coincide and so does the last inequality. This forces constancy on the little disc $D$ ($u$ being continuous). It follows by ‘propagation’ that $u$ is globally constant. (Alternatively use general topology: the set of points where $u$ achieves its maximum is both nonempty (compactness), closed and open.) Indeed choosing a path from $z_0$ to any point $z\in F$ covered by a chain of little discs $D_1, \dots, D_k$, each $D_i$ centered on the border of the previous one $D_{i-1}$, one argues that two successive discs have enough overlap to ensure constancy over the next disc. [ \[11.08.12\] [There is a Garabedian paper 1951 (A PDE..., p.486) were it is asserted that the Green’s function of a convex clamped plate need not be of one sign; but of course this is not relevant to our matter were we use the usual the Laplacian $\Delta$ and not the bi-Laplacian $\Delta^2$ corresponding to clamped plated, instead of vibrating membranes. This is the seminal work of Garabedian (but others were also involved) were the famous Hadamard conjecture on the bi-Laplacian was disproved.]{} ]{} Killing the periods ------------------- The previous section ensures that any superposition of Green’s functions $G:=\sum_i \lambda_i G(z,t_i)$ will be likewise quasi-negative provided all $\lambda_i$ are positive. In this circumstance the function $f=e^{G+iG^{\ast}}=e^G \cdot e^{i G^{\ast}}$ (whose modulus is $e^G$) is a unit-circle map ($\vert f \vert \le 1$), because the real exponential takes nonpositive values $(-\infty, 0]$ to $(0,1]$. It is consistent by continuity to send the $t_i$ on $0$. If $r$ is the connectivity of the domain $B$ (number of its contours) then there are homologically $r-1$ non-trivial loops $\gamma_1, \dots \gamma_{r-1}$ running around the $r-1$ holes in our domain (cf. Fig.\[Green:fig\]c illustrating the case $r=3$). We consider the linear period mapping $$\begin{aligned} \label{period-mapping:eq} {\Bbb R}^{d} &\longrightarrow {\Bbb R}^{r-1} \cr (\lambda_1, \dots \lambda_d) &\mapsto (\textstyle\int_{\gamma_1} dG^{\ast}, \dots, \int_{\gamma_{r-1}} dG^{\ast})\end{aligned}$$ By linear algebra if $d$ is large enough (precisely already for $d=r$) we have enough free constants so as to find non-trivial $\lambda_i$ extincting all periods. \[Heuristically the electric poles of the multi-battery in the electrolytic tank (nomenclature as in e.g. Courant 1950/52 (Conformal book)) are affected by suitable charges so as to generate an “ideal” potential with single-valued conjugate.\] Exponentiating gives $f=e^{G+iG^{\ast}}$ a circle map with $d=r$ zeroes, provided one is able to ensure all $\lambda_i >0$. Without taking care of this last proviso, one may reach too hastily the impression that we have complete freedom in prescribing the location of the $d=r$ poles (of the Green’s functions, which convert ultimately to zeroes of the related circle map). The linear-algebra argument gives only a real-line inside the kernel of the (linear) period-mapping , but a priori this line could miss the “octant” ${\Bbb R}_{>0}^{d}$ consisting of totally positive coordinates. In fact upon letting vanish some of the $\lambda_i$ what is only required is a non-trivial penetration of this line $\ell$ into the closed octant $\overline{O}={\Bbb R}_{\ge 0}^{d}$, i.e. the intersection $\ell \cap \overline{O}$ should not reduce to the origin. A true penetration of this line in the interior of $O$, or a degenerate one where the line meet along one of its face would be enough to complete the existence-proof. The latter case amounts to extinct some Green’s “batteries” by assigning a vanishing coefficient $\lambda_i=0$. The net effect would be degree lowering of the circle map $f$. Beware, that for planar domains (which correspond to Harnack-maximal Schottky doubles) no such lowering of the degree is possible for simple topological reasons ($r\le \gamma$). However the described theoretical eventuality may well happen in the non-planar case to be soon discussed. Understanding how and why to arrange degenerate penetrations could well offer a strategy toward improving Ahlfors $r+2p$ bound. Extra difficulties in the surface case -------------------------------------- It is obvious that the above method via Green’s functions adapts to bordered Riemann surface $F=F_{r,p}$ of (positive) genus $p$ with $r$ contours (Rand). Remember however that at this stage we did not offered a complete treatment of the planar case ($p=0$). First note a conceptual difficulty regarding Green’s function, which, in the plane case of a domain $B\subset {\Bbb C}$, is constructed via $\log\vert z-t \vert$ appealing to a global coordinate system. In the abstract bordered setting, there is no such ambient medium. One could try to work with a (conformal) Riemann metric and the allied logarithmic distribution $ \log \varrho (z,t), $ where $\varrho$ is the intrinsic distance (defined as usual as the infimum of lengths of rectifiable pathes joining two given points). Note however that this construction specialized to the domain case does not duplicate the former, since the intrinsic distance $\varrho(z,t)$ does not coincide with the extrinsic one $\vert z-t\vert$, unless the domain $B\subset {\Bbb C}$ is starlike about $t$. Bypassing this difficulty \[which will be resolved later\], we first note that each handle creates two $1$-cycles yielding a total of $(r-1)+2p$ many essential loops (compare Fig.\[Green:fig\]e). Thus introducing $d:=r+2p$ poles $t_i$ we dispose of enough free parameters to arrange (via linear algebra) the vanishing of all periods of the conjugate differential $dG^{\ast}$ of the potential $G=\sum_{i=1}^{d}\lambda_i G_{t_i}$. This explains quite clearly why Ahlfors discovered (about 1948) the upper-bound $r+2p$ for the degree of a circle map. Of course there is still the subtlety of explaining why it is possible to choose all $\lambda_i>0$ at least for a clever choice of the poles $t_i$. All this is probably when suitably interpreted the quintessence of the Ahlfors mapping (of degree $r+2p$). Again the writer does not mask his happiness after having understood this point (as late as the 04.08.12). Now it is evident to reconstruct (even if somewhat fictionally) what must have happened in Ahlfors’ brain (at least as early as 1948, and presumably much earlier, yet no record in print). With this piece of information and, on the other hand, being well-aware of the modern purely function-theoretic proofs of RMT (à la Koebe-Carathéodory, Fejér-Riesz 1922 (published by Radó 1923), Carathéodory 1928 and Ostrowski 1929) it must have seemed highly desirable (or trendy) to reinterpret the above (somewhat heuristic but fruitful potential theory) in terms of a function-theoretic extremal problem. This leads e.g. to the problem we discussed at length of maximizing either the modulus of the derivative at some inner point $t=a$, or to maximize the distance of two points $a,b$ where the first maps to $0$ and the second is repulsed at maximum distance from the origin. In both case the competing functions are analytic and bounded-by-one in modulus $\vert f \vert \le 1$. So we get the Ahlfors function $f_a$ or $f_{a,b}$. It seems obvious that all those Ahlfors functions are included in the above trick à la Green-Riemann (GR), and thus subsumed to an electrolytic interpretation. Yet the exact dependance and location of the corresponding logarithmic poles of Green’s $G$ (becoming the zero of Riemann’s $f$, after exponentiation) must be a transcendentally sublime business. Also the corresponding degree of the Ahlfors function is another mystery. It is conceivable that less than the $r+2p$ generically required poles suffices in case the linear period mapping ${\Bbb R}^d \to {\Bbb R}^{(r-1)+2p}$ along fundamental loops has a degenerate image permitting to economize some poles $t_i$. The task is reduced to find the lowest $d$ such that the kernel of the period map is non-trivial and contains a non-zero element all of whose coordinates are $\ge 0$. Remember, that Gabard 2006 [@Gabard_2006] showed—using another method, based on a topological argument of irrigation (Riemann-Betti-Jordan-Poincaré’s homologies, and Brouwer’s degree plus some basic Pontrjagin theory in the Jacobian torus as a very special commutative Lie group—that there is a circle map of degree $\le r+p$ (i.e. with one unit economized for each handle). Assuming that any circle map is allied to a Green-Riemann map there would be a fewer number namely $d\le r+p$ of batteries required to generate this mapping. Of course, the first part of the assertion looks evident: given a degree $d$ circle map $f$ with zeroes at $t_i$, then $\log \vert f(z) \vert$ coincides with $\sum_{i=1}^d G(z,t_i)$. This is Ahlfors formula following from the fact that both functions vanishes on the border and have the same singularities. [*Philosophy.*]{} \[08.08.12\] Modulo elusive details, it is fair to resume the situation by saying that the Ahlfors circle maps (if not all existence theorems of function theory) derives form the Dirichlet principle (or the allied Green’s functions). \[This was of course best incarnated by Riemann, 1851 and 1857, where in bonus the whole algebraic geometry of curves was subsumed to this principle!\] Conversely one could hope that the Ahlfors function could be used to lift the Dirichlet solubility on the disc (via Poisson integral formula) to an arbitrary bordered surface. However it seems obvious that there is no way to descend the boundary function to the disc since the Ahlfors branched covering is multi-valent. We arrive at the conclusion that the true mushroom is the Dirichlet principle, while Ahlfors function being just one tentacle of the mushroom. Of course, the only paradigm susceptible of competing with Dirichlet are the function-theoretic extremal problems à la Koebe-Carathéodory-Fejér-Riesz-Bieberbach-Ostrowski, etc. For plane domains the Kreisnormierung (instead of the Ahlfors map) may be used as normal domains where the Dirichlet problem is easier to solve. This is akin to Poisson’s formula for the round disc case of Dirichlet, and quite implicit in Riemann’s Nachlass 1857 [@Riemann_1857_Nachlass] (cf. also Bieberbach 1925 [@Bieberbach_1925]). A similar reduction of Dirichlet for bordered surfaces occurs is also likely on the ground of Klein’s Rückkehrschnitttheorem (cf. Section \[sec:Ruckkehrschnittthm\]), supposed to be an extension of the Kreisnormierung. Regarding the detailed execution of the removal of the period as to construct an Ahlfors-type mapping one should compare also the paper of Heins 1950 [@Heins_1950], Kuramochi 1952 [@Kuramochi_1952] and (albeit confined to planar domains) the paper by D. Khavinson 1984 [@Khavinson-Dimitri_1984], whose argument is considered by its author akin to the arguments of Grunsky. The Green’s function of a compact bordered Riemann surface=CBRS --------------------------------------------------------------- \[14.08.12\] This section examines the issue that the Green’s function $G(z,t)$ with pole at $t$ is a canonically defined function in the generality of a CBRS. This is super-classical, cf. e.g., the treatises Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] or Schiffer-Spencer 1954 [@Schiffer-Spencer_1954]. It is to be expected to find older treatments by Riemann, Schwarz, Klein, Koebe, etc. Several accounts by Nevanlinna proceed via Schwarz’s alternating method, a viewpoint which looked most convenient to adhere with. As already noticed, the case of a plane domain $B\subset {\Bbb C}$ (bounded by smooth curves) it is easy to define Green’s function $G(z,t)$ via the (logarithmic) potential $\log \vert z-t \vert$ from which we subtract the Dirichlet solution matching the logarithmic potential restricted to the boundary $\partial B$. Alas, for a CBRS $F$ one lacks an ambient space like ${\Bbb C}$ permitting an analogous construction. Of course, $\log \vert z-t \vert$ bears some significance only locally within a uniformizer chart about $t$. Taking another local chart, one may argue that in the small the expression will mutate into $\log \vert \alpha (z-t) \vert$ for some $\alpha\in{\Bbb C}^{\ast}$ incarnating the derivative of the transition between the two charts. Thus the log-potential w.r.t. the new chart is $\log\vert \alpha \vert + \log \vert z-t \vert$, hence equal to the old one modulo an additive constant. Presumably some philosophical argument can corroborate the vague feeling that the asymptotic of the logarithmic pole is unaffected by such additive constant. \[Added in proof: compare Pfluger 1957 [@Pfluger_1957 p.110, 28.3] for an accurate formulation, or Farkas-Kra 1980/1992 [@Farkas-Kra_1980/1992 p.182, Remark].\] It seems then meaningful to set: The Green’s function of a CBRS $F$ with pole at $t$ (an interior point of $F$) is the unique harmonic function on $F$ save $t$ with singularity $\log\vert z-t \vert$ near $t$ which vanishes continuously on the boundary $\partial F$. Compare (modulo a different sign convention) Ahlfors-Sario 1960 [@Ahlfors-Sario_1960 p.158, 4B]. Uniqueness is considered as evident there. Indeed, a chart change affect the logarithmic potential by an additive constant and harmonic functions are quite rigid (being determined by their values on any open disc). Hence knowledge of the function on any punctured chart about $t$ via $\log \vert z-t \vert$ determines it uniquely. The delicate point is existence. Choose around $t$ a nice analytic Jordan curve $J$ and via RMT construct a holomorphic chart taking $D$ (the “sealed” interior of $J$, i.e. $J$ included) to the unit disc $\overline{\Delta}$. Consider $\log\vert z\vert$ in the unit circle and transplant to $D\subset F$ and then after adding an additive constant we try to solve a Dirichlet-Neumann problem on $F-D$ piecing together smoothly the logarithmic piece with the Dirichlet-Neumann solution. In this procedure the Green’s function looks highly non-unique depending on the “ovaloidness” of the Jordan curve $J$ chosen. In fact $J$ cannot be chosen at will but must somehow be a level-line of Green (still undefined). Infinitesimally $J$ should be a perfect circle, and this is perhaps the key to put the naive pasting argument on a sound basis via a convergence procedure. (Infinitesimal circles are well-defined on Riemann surfaces via the conformal structure.) Existence and uniqueness look then plausible, but involve a considerable sophistication over the plane-case where the Green’s function reduced straightforwardly to the Dirichlet problem. Let us paraphrase the above more formally. Take any chart $\varphi\colon U \to \Delta$ about the “pole” point $t$ (sending $t$ to the origin $0\in {\Bbb C}$), write down $\log\vert z\vert$ in that chart and shrink gradually attention to the (round) disc $\Delta_{\varepsilon}$ of radius $\varepsilon$. Let $D_{\varepsilon}$ be $\varphi^{-1}(\Delta_{\varepsilon})$. For each (positive) value of $\varepsilon$ one can solve the Dirichlet problem in $F-{\rm int} D_{\varepsilon}$ with boundary value $0$ on $\partial F$ and $\log \varepsilon$ on $\partial D_{\varepsilon}$. Denote by $u_\varepsilon $ the corresponding solution. By construction $u_\varepsilon$ pasts continuously with the $\varphi$-pullback of the log-potential (i.e. $(\log \vert z\vert) \circ \varphi$). Of course this glued function is a Frankenstein creature lacking a smooth juncture. For instance, if $\varepsilon=1$ then $u_{\varepsilon}$ is identically zero, whereas in $D_1$ we have the logarithmic “trumpet” with derivative $1$ along the normal direction. However as $\varepsilon$ decreases from $1$ to $0$, $u_\varepsilon$ becomes $\le 0$ (having prescribed the negative value $\log \varepsilon$ on $\partial D_{\varepsilon}$) and the dependence of $u_{\varepsilon}$ is perhaps monotonic. So it seems arguable (Harnack?) that while $\varepsilon \to 0$ (say via dyadic numbers $\varepsilon_n=1/2^n$) the $u_n$ converges to a harmonic function on $F-t$ which is the desired Green’s function $G(z,t)$. It seems evident (since $\partial D_n$ becomes more and more circular in $F$ as $n$ grows to infinity) that the limit is harmonic and independent of the gadgets used along the way (chart $\varphi$, dyadic sequence $\varepsilon$). This vaguely explains existence and uniqueness can maybe be derived by a similar trick (combined with a “leapfrog” argument). Try to locate a reference along this naive line: maybe Schwarz?, Klein? Koebe? Weyl? Pfluger? and otherwise try Ahlfors-Sario [@Ahlfors-Sario_1960], Sario-Oikawa [@Sario-Oikawa_1969]. (Sometimes Sario’s formalism of the normal/principal operator is a bit awkward to digest.) For treatments of the Green’s function on a CBRS cf. Schiffer-Spencer 1954 [@Schiffer-Spencer_1954 p.33, and 93–94]. See also Sario-Oikawa 1969 [@Sario-Oikawa_1969 p.49–50]. We summarize the discussion by the Given a CBRS $F$ and an interior point $t$, there is a uniquely defined Green’s function $G(z,t)$ with pole $t$ which is characterized by the following conditions: it is harmonic on $F-t$, vanishes (continuously) on the boundary $\partial F$ and it has the prescribed singularity $\log \vert z-t \vert$ near $t$. For complete details, compare several sources: $\bullet$ first Ahlfors-Sario 1960 [@Ahlfors-Sario_1960 p.158] and Sario-Oikawa 1969 [@Sario-Oikawa_1969 p.50] (both via Perron’s method, and Sario’s formalism of the normal operator). $\bullet$ Then also Pfluger 1957 [@Pfluger_1957 p.110, end of §28.2, as well as p.110, §28.3 and last 3 lines of p.111] $\bullet$ Schiffer-Spencer 1954 [@Schiffer-Spencer_1954 §4.2, p.93–94] $\bullet$ Nevanlinna 1953 [@Nevanlinna_1953-Uniformisierung] via Schwarz’s alternating method (SAM). We detail this argument in the next section. Schwarz’s alternating method to construct the Green’s function of a compact bordered surface (Nevanlinna’s account) ------------------------------------------------------------------------------------------------------------------- \[15.08.12\] As promised, in this section we attempt to understand Nevanlinna’s exposition of the existence of Green’s functions on compact bordered surfaces. All pagination given refers to Nevanlinna 1953 [@Nevanlinna_1953-Uniformisierung], the book “Uniformisierung”. Nevanlinna follows Schwarz’s alternating method (SAM) quite closely. The argument is a bit tedious but quite elementary. It uses merely the following facts on a bordered surface $F$: \(1) if $f\le g$ on $\partial F$ then the associated Dirichlet solution $u(f)\le u(g)$ (compare Lemma \[Depressive:lem\]). \(2) maybe Harnack’s theorem is required? In Nevanlinna’s book the relevant information is scattered at two places (at least) so we attempted to compactify the presentation for our own understanding. First Nevanlinna introduces a concept of “Kreisbereich”. Alas the jargon is not very fortunate being already consecrated by Koebe in a different context, so let us speak rather of a “celluloid” (or a “Kreisgebilde”). This is \[cf.p.142\] a connected finite union of (closed) discs in a Riemann surface (whose images by a (parametric) chart are round discs in ${\Bbb C}$). On each such disc the first boundary value problem (abridged DP=Dirichlet problem) is soluble via Poisson’s formula. Assuming the contours of each pair of discs to have finite intersection, SAM enables one to solve DP on the union, hence on any celluloid. So for instance it is clear that any CBRS is a celluloid. (A formal proof certainly requires Radó’s triangulation theorem 1925 [@Rado_1925].) To absorb the boundary in one stroke one could add annular regions where the DP is also soluble by an explicit recipe, sometimes ascribed to Villat 1912 [@Villat_1912]. The Green’s function will be obtained by specializing the following technical lemma \[cf.p.148\] \[due to Schwarz and probably related to what Koebe’s calls the “gürtelförmige Verschmelzung”($\approx$belt-shaped fusion)\]. Intuitively, the lemma amounts to construct a harmonic function $u$ with prescribed boundary values and with prescribed singularity $u_0$ near a point $t$, or rather on a ring enclosing the pole $t$. (At first, it is not perfectly transparent how to deduce the Green’s function from the lemma, but we shall try elucidate this issue later.) Let $F$ be a compact bordered Riemann surface and $t\in {\rm int} F$ an interior point. Let $U$ be a neighborhood of $t$ mapped to the unit-disc $\Delta=\{ \vert z \vert <1\}$ via a chart. In $U$, let $K$ be the ring corresponding to $r_1\le \vert z \vert \le r<1$. [\[It seems that $r_1=0$ is permissible and needed for the application to the Green’s function.\]]{} Let further $X$ be a celluloid containing the external contour of the ring $K$ $(\vert z \vert=r)$ in its interior, as well as the boundary $\partial F$ but missing the small disc in $U$ corresponding to $\vert z\vert \le r_1$. Set finally $A:=X \cup K$ [(compare Fig.\[Nevanl:fig\]]{}). Then given $u_0 \in H (K)$ (harmonic on the ring $K$) and $f\colon \partial F \to {\Bbb R}$ continuous, there is a unique $u \in H(A)$ such that $u_{\vert \partial F}= f$ and with $u-u_0$ extending harmonically to $B$, the disc corresponding to $\vert z\vert\le r$. -5pt0 -5pt0 Before detailing the proof, let us see how this helps defining $G(z,t)$ the Green’s function. \[The difficulty is just so trivial to be not completely explicit in Nevanlinna \[p.198–199, §2, Art.6.4\].\] First we impose $f\equiv 0$. Then we choose the singularity function $u_0=\log\vert z\vert$ which has to be defined on $K$, hence we shrink $r_1$ to $0$ via a sequence of dyadic radii $r_n=1/2^n$. On applying the lemma we get a sequence of solution $u=u_n$ defined on $A_n$ a sequence of expanding subsurfaces (the outsides of the shrinking discs $\vert z\vert< 1/2^n$). Now observe that $u_n$ for a large $n$ solves the problem of the lemma for all smaller values of $n$: just take the restriction (and use uniqueness). Consequently all the $u_n$ form a telescopic system of functions (each restricting to all its predecessors) defined on larger and larger compact subregions $A_n$ ultimately expanding to the punctured surface $F-t$. The very constant (indeed completely monotone) limit of those $u_n$ gives the desired Green’s function $G(z,t)$. It is harmonic on $F$ save $t$, vanishes on the boundary and $G(z,t)-\log\vert z \vert$ extends harmonically through $t$ (on a little neighborhood). It remains to check that those 3 properties defines $G(z,t)$ unambiguously. This is again the same sort of argument. Assume there were two Green’s solutions $G_1,G_2$, then $G_i-u_0=:h_i$ harmonic on some neighborhoods $V_i$ of $t$. So $G_1-G_2=(h_1+u_0)-(h_2+u_0)=h_1-h_2$ which is harmonic on the intersection $V_1 \cap V_2$. Hence the difference $G_1-G_2$ is harmonic throughout $F$, but with vanishing boundary value on $\partial F$. Consequently it must be identically zero (by the uniqueness part of Dirichlet) which follows from the maximum principle. This is a matter of implementing Schwarz’s alternating method \[see p.148–150\] and we follow exactly Nevanlinna’s text (annotating our copy by the symbol $\bigstar$ to indicate the sole cosmetic difference). $\bullet$ [*Uniqueness*]{} Assuming the existence of two functions $u_1, u_2$ solving the problem, their difference $u_1-u_2$ will be harmonic on $A$, and 0 on $\gamma:=\partial F$. But each difference $u_i-u_0=:h_i\in H(B)$ extends harmonically across $B$ ($i=1,2$). Hence on $B$, $u_1-u_2=(h_1+u_0)-(h_2+u_0)=h_1-h_2 \in H(B)$, and therefore $u_1-u_2 \in H(A\cup B)$, and $A\cup B$ is all of $F$. It follows (Dirichlet’s uniqueness) that $u_1-u_2$ vanishes identically. $\bullet$ [*Optional remark.*]{} It is clear that the case $f\equiv 0$ is typical, since the general case just requires adding the Dirichlet solution for the data $f$. \[This explains why I had the impression to find many misprints!\] $\bullet$ [*Existence (after Schwarz)*]{} First it is observed that DP is solvable on both $A$ and $B$ ($B$ is just a ball and $A$ is a celluloid, yet of the general type involving a ring). Of course $A$ is also a compact bordered surface and therefore one is ensured of Dirichlet solvability, thereby bypassing the concept of a celluloid, and accordingly one can shorten slightly the statement of Nevanlinna’s lemma, with the direct bonus that one can make abstraction of all the little discs drawn on the picture. \[CAUTION: here it is perhaps NOT permissible to take $r_1=0$\] We denote by $\alpha$ and $\beta$ the internal resp. external contour of $K$, and let $\gamma:=\partial F$. Set first $v_0\equiv 0$. We define inductively sequences $u_n\in H(A)$ and $v_n\in H(B)$ by their boundary values ($n\ge 1$) $$\label{SAM:eq} u_n=\begin{cases} v_{n-1}+u_0 \quad &\textrm{on } \alpha, \cr 0 [\bigstar or f] \quad &\textrm{on } \gamma, \end{cases}$$ and $$v_n=u_n-u_0 \quad \textrm{on } \beta.$$ \[This Ansatz comes a bit out of the blue, but notice that passing to the limit both definitions leads to the identity $u-u_0=v$ holding on $\alpha \cup \beta$ which is the full contour of the ring $K$, so that anticipating harmonicity this will hold throughout $K$, and $v$ will afford the required extension of $u-u_0$ (only defined on $A\cap K=K$) to the disc $B$ containing the ring $K$. Of course, it is also crucial to notice that both sequences $u_n,v_n$ are “interlocked” or “leapfrogged” requiring an alternating progression of one term to go one step further with the other.\] The successive differences are given by $$\label{success-diff:eq} u_{n+1}-u_n=\begin{cases} v_{n}-v_{n-1} \quad &\textrm{on } \alpha \cr 0 \quad &\textrm{on } \gamma \end{cases}$$ and $$v_{n+1}-v_n=u_{n+1}-u_{n} \quad \textrm{on } \beta.$$ Let us write $$M_n:=\max_{\beta} \vert u_{n}-u_{n-1}\vert=\max_{\beta} \vert v_{n}-v_{n-1}\vert,$$ then by the maximum- and minimum-principle $ \vert v_{n}-v_{n-1}\vert\le M_n$ in $B$, and so in particular on $\alpha$. Hence by , $ \vert u_{n+1}-u_{n}\vert\le M_n$ on $\alpha$. Further, the difference $u_{n+1}-u_{n}$ vanishes on $\gamma$ (cf. ), and so it is bounded on the boundary of $A$ (and therefore throughout $A$) by the potential $M_n \cdot \omega$, where $\omega$ is the harmonic function vanishing along $\gamma$ and equal to $1$ on $\alpha$. Hence $$\label{diff:eq} \vert u_{n+1}-u_n \vert\le M_n \cdot \omega\quad \textrm{in } A.$$ In the interior of $A$, one has $0<\omega < 1$. If $q$ is the maximum of $\omega $ on $\beta$, then $0<q<1$. Further on $\beta$ we have $$\vert u_{n+1}-u_n \vert\le q \cdot M_n,$$ and also (by definition of $M_n$) $$M_{n+1}\le q \cdot M_n.$$ By induction, it follows that $$M_{n+1}\le q^{n} \cdot M_{1},$$ and recalling again the definition of $M_n$ we get (first on $\beta$ and thus on $B$) $$\vert v_{n+1}-v_n \vert \le M_{n+1}\le q^{n} \cdot M_{1}.$$ When particularized to $\alpha$, this implies in view of $$\vert u_{n+1}-u_n \vert \le q^{n-1} \cdot M_{1} \quad \textrm{in } \alpha,$$ and by the maximum principle this extends to $A$ (recall that $\partial A= \alpha \cup \gamma$ and the function $u_{n+1}-u_n$ vanishes on $\gamma$). Consequently, both series $\sum_{n}(u_{n+1}-u_{n})$ and $\sum_{n}(v_{n+1}-v_{n})$ converges uniformly on $A$ resp. $B$. The limiting functions $u$ and $v$ of $u_n$ resp. $v_n$ are therefore harmonic on $A$ resp. $B$, and taking the limit in the definition of $u_n$ (see ) we see that $u$ vanishes on $\gamma$ \[$\bigstar$ equals $f$ on $\gamma$\]. We show finally that $u-u_0=v$ on $B$ \[$\bigstar$ $K$ probably?\]. Indeed, taking the limit in the first line of gives $u=v+u_0$ on $\alpha$, and the definition of $v_n$ pushed to its limit gives $v=u-u_0$ on $\beta$. Therefore the same identity $u-u_0=v$ holds on both contours of the ring $K$, and consequently its validity propagates throughout $K$. Finally, as $v$ is harmonic on $B$ we are happy to conclude that $u$ fulfills all of our requirements: namely $u\in H(A)$, $u=f$ on $\gamma=\partial F$ and $u-u_0$ defined on $A \cap K=K$ coincide there with $v$ defined on the larger set $B\supset K$, yielding the asserted harmonic extension. \[NAIVE AND WRONG—see rather the argument given above\] Finally, \[compare p.198–199\] one obtains the Green’s function $G(z,t)$ by taking $u_0=\log \vert z \vert$, $f\equiv 0$ and $r_1=0$ \[Caution: this point is not made explicit in Nevanlinna\]. For this choice of $r_1$, note that $A=F-t$. The lemma supplies a unique $u\in H (F-t)$ such that $u_{\vert \partial F}=0$ and so that $u-\log\vert z\vert=:h$ is harmonic on $F$. The function $u$ is the desired Green’s function $G(z,t)$. Little green’s men dreams (extraterrestrial applications of Green’s) ==================================================================== The following three subsections are optional reading containing more questions than answers. The reader interested primarily in the Ahlfors map should preferably skip them. From Green to Gromov? (directly bypassing Riemann and Löwner) {#sec:Green-to-Gromov} ------------------------------------------------------------- To mention once more a deep frustration (the Gromov filling conjecture) it looks not completely crazy to hope that a careful examination of the Green’s function and the allied isothermic coordinates could prompt a solution of this problem. We tried quickly the \[14.08.12\] but failed dramatically as usual (along with circa 10 attempts of essentially the same vein). Roughly the idea would be to look at the streamlines of Green and its equipotentials, and remove every trajectory ending to the (finitely many) critical points of Green while attempting to estimate area via this (isothermic) parametrization. Of course, Schwarz’s inequality enters into the game but I only arrived at weak estimates like $\pi$ or $\pi/2$ (in place of $2\pi$!) upon doing highly fallacious calculus. Schoenflies via Green? {#Schoenflies:sec} ---------------------- A notorious topological paradigm is the so-called Schoenflies theorem to the effect that a reasonably embedded sphere $S^{n-1}$ in ${\Bbb R}^n$ bounds a topological ball $B^n$. (There is a large debate (cf.e.g.Siebenmann 2005 [@Siebenmann_2005]) about who (and more broadly speaking which community) proved first the case $n=2$. In the topological-combinatorial realm there is a contribution of Schoenflies reaching full maturity ca. 1906, and somewhat earlier there is the contribution of Osgood which may have reached full stability with Carathéodory 1912 [@Caratheodory_1912]. Of course the statement (for $n=2$ and maybe even $n=3$) was largely anticipated heuristically by other workers, e.g. Moebius 1863 [@Moebius_1863]. Schoenflies’s theorem was extended to higher dimensions by J.W. Alexander ($n=3$ ca. 1922), B. Mazur and M. Brown (all $n$ ca. 1960) for any locally flat (e.g. smooth) hypersphere in ${\Bbb R}^n$. From Thom or Smale’s $h$-cobordism theorem (early 1960’s) it is inferred that the closed ball $B^n$ carries a unique smooth structure when $n\neq 4$ (the case $n=4$ being still largely unsettled). It follows that the interior of the smoothly embedded sphere is a ball differentiably. Another unsolved problem of longstanding is the truth of the same conclusion for $n=4$ (the so-called [*smooth Schoenflies*]{} in dimension $4$, SS4, see e.g. papers by Scharlemann). Naive physical (or bacteriological, cf. Fig.\[Green:fig\]) intuition about the Green’s function makes hard to visualize why there should be any anomaly for $n=4$, yet nobody ever succeeded to prove or disprove SS4. This belongs to the charming mysteries of low-dimensional differential topology at the critical dimension $n=4$. One may speculate about a naive approach to SS4 through the ca. 200 years older potential theory (of Laplace, Poisson, Green, Gauss, Dirichlet and Riemann’s era). Alas, there is few records in print of analysts feeling confident enough about the explorative aptitudes of the Green’s function (compare Fig.\[Green:fig\]) to claim the required diffeomorphism with $B^4$. Of course in the very small vicinity of the pole $t$ the levels of $G(z,t)$ (now $z\in {\Bbb R}^n$) look alike round spheres, and by the synchronization principle stating that each bacteria reaches the boundary at the same moment it may look immediate how to write down the diffeomorphism. Can somebody explain why this Green’s strategy fails to establish SS4. Less ambitiously can somebody reprove SSn (for $n \neq 4$) via the Green’s function. If yes with some little chance his/her proof will possibly include the case $n=4$. Green, Schoenflies, Bergman and Lu Qi-Keng ------------------------------------------ \[06.08.12\] As discussed in the previous section, a dream would be to show SS4 (smooth Schoenflies conjecture) via the Green’s function in 4D-space ${\Bbb R}^4$. On reading an article by Boas 1996 (PAMS), where Suita-Yamada 1976 [@Suita-Yamada_1976] is cited we see a potential connection between both problems. The problem of Lu Qi-Keng asks for domains where the Bergman kernel is zero-free (so-called Lu Qi-Keng=LQK-domains). Since Schiffer 1946 [@Schiffer_1946], there is an identity connecting the Bergman kernel to the Green’s function. It seems that the zeros of Bergman corresponds to the critical points of Green. Of course the latter is forced to have critical points as soon as the topology is complicated (not a disc). Suita-Yamada’s result that the Bergman kernel necessarily exhibits zeroes for membranes which are not discs looks nearly obvious. Hence LQK-bordered surfaces are precisely those topologically equivalent to the disc. Now Boas in 1986 found a counterexample showing that no topological characterization of LQK-domains holds in higher dimensions: there exists in ${\Bbb C}^2$ a bounded, strongly pseudoconvex, contractible domain with $C^{\infty}$ regular boundary whose Bergman kernel does have zeroes. \[Addendum \[18.09.12\]: in fact upon reading Boas original paper (1986), Boas’ domain is diffeomorphic to the ball $B^4$.\] [**Optimistic scenario (Green implies Schoenflies)**]{} It would be interesting to know what the topology of Boas’ hypersurface $S=\partial \Omega$ is. In view of Poincaré-Alexander-Lefschetz duality $S$ must be a homology sphere, if I don’t mistake. Now upon speculating that SS4 is true (by naive geometric intuition), and even more that it is provable via the streamlines of Green’s function, and granting a persistence of Schiffer’s Green-Bergman identity (in the realm of two complex variables), it may seem that Boas’s counterexample must have an “exotic” boundary (not diffeomorphic to $S^3$). \[Of course, not so in view of the just given Addendum.\] [**Pessimistic scenario (Green does not implies Schoenflies).**]{} The other way around, assuming that Boas’ boundary is the 3-sphere, there would be critical points of the Green’s function $G(z,t)$ and Boas’s example may foil any naive attempt to reduces SS4 to the streamlines of the Green’s function. But even so maybe the Green-Bergman identity of Schiffer is specific to one complex variable, leaving some light hope that there is a potential-theoretic proof of the differential-topology puzzle of SS4. So a bold conjecture (somewhat against Boas’ philosophy that there is no topological characterisation of LQK-domains) would be that any domain in ${\Bbb C}^2$ bounded by a smoothly embedded 3-sphere is a LQK-domain (i.e. its Bergman function is zero-free). \[This is wrong in view of Boas 1984 (addendum just mentioned)\]. However it could be true that the Green’s functions $G(z,t)$ for any center $t$ located in the inside of $\Sigma$ is critical point free, whereupon an elementary integration of its gradient flow should establish a diffeomorphism of the inside the spheroid with the ball $B^4$ with its usual differential structure. (Recall that it is yet another puzzle of low-dimensional topology, whether the $4$-ball has a unique smooth structure! All others balls (maybe except the five-dimensional one) do enjoy uniqueness by virtue of Smale’s $h$-cobordism theorem.) Note that the Bergman kernel is defined without reference to a basepoint whereas Green’s function requires a basepoint (its pole). Arithmetics vs. Geometry (Belyi-Grothendieck vs. Ahlfors) {#sec:Belyi-Grothendieck} --------------------------------------------------------- \[10.08.12\] Closed Riemann surfaces are subsumed to the (alienating) theorem of Belyi-Grothendieck, that [*a surface is defined over $\Qbar$ iff it admits a morphism to the line ${\Bbb P}^1$ ramified at only $3$ points*]{} (so-called [*Belyi map*]{}). Another characterization (due to Shabat-Voevodsky 1989 [@Shabat-Voevodsky_1989/89]) is the possibility to triangulate the surface by equilateral triangles (with or without respect to the hyperbolic uniformizing metric). Basically this follows as one may sent homographically the 3 points to the vertices of the regular tetrahedron inscribed in the sphere. (Compare Belyi 1979/80 [@Beyli_1979/80], Grothendieck 1984 [@Grothendieck_1984/1997-esquisse-d'un-programme] “Esquisse d’un progamme”, Shabat-Voevodsky 1989 [@Shabat-Voevodsky_1989/89], Bost 1989/92/95 [@Bost_1989/92/95] (p.99–102), Colin de Verdière-Marin, etc.) Is there an analog of this result for bordered surfaces in the context of Ahlfors (circle) mapping to the disc, and if so what is its precise shape? In the Riemann sphere any 3 points are transmutable through a Moebius rigid motion. The analog statement in the disc involves either one boundary point plus one interior point or 3 boundary points. Those are of course just the (heminegligent) hemispherical trace of real triads on the equatorial sphere corresponding to ${\Bbb P}^1$ with its standard real structure. (Remember that there is an exotic twisted real structure projectively realized by the invisible conic $x_0^2+x_1^2+x_2^2=0$.) This lack of canonical choice of a real triad on ${\Bbb P}^1$ could plague slightly an appropriate bordered version of Belyi-Grothendieck. \[12.11.12\] More seriously the ubiquity of real points in both those triads of the disc looks incompatible with Ahlfors maps lacking real ramification (when Schottky doubled to the realm of Klein’s orthosymmetric curves). Of course since bordered surfaces are in bijective correspondence with real orthosymmetric curves, one may expect first an answer along the line: [*a real orthosymmetric curve is defined over $\Qbar\cap {\Bbb R}\supset {\Bbb Q}$ iff it admits a totally real map ramified solely at $3$ real points or at one real point and $2$ imaginary conjugate points*]{}. Remember yet that total reality means that the inverse image of the real line is the real locus of the (orthosymmetric) curve, and since such maps lack real ramification our naive real version of Belyi-Grothendieck looks foiled. There seems to be a structural incompatibility between Belyi-Grothendieck and Klein-Ahlfors. Of course our desideratum of a simultaneous realization of Belyi-Grothendieck and arithmetization of Ahlfors may well just be a nihilist folly. By an “arithmetization of the Ahlfors map” we just mean something in much the same way as Belyi-Grothendieck arithmetizes Riemann’s existence theorem (any closed Riemann surface admits a morphism to the sphere ${\Bbb P}^1({\Bbb C})$). Possibly, one should be content with a reality version of Belyi-Grothendieck without bringing Ahlfors’ total reality into the picture. Then we have something like [*a real curve is defined over $\Qbar$ iff it admits a real morphism to the line ramified above only one of the two real triads, i.e. $0,1, \infty$ or $0,\pm i$.*]{} A priori this statement tolerates both types of real curves (ortho- and diasymmetric) and thus be more liberal than Ahlfors theorem (which tolerate only orthosymmetric curves). Adhering instead to the geometric interpretation of Belyi-Grothendieck (due to Shabat-Voevodsky 1989/89 [@Shabat-Voevodsky_1989/89]) in terms of equilateral triangulations might be more appealing. For instance one can imagine an orthosymmetric real curve with an equilateral triangulation invariant under (complex) conjugation. A such would according to BG be defined over $\Qbar$. It is clear that such a triangulation would contain the real circuits as subcomplex of the triangulation. In particular what is the significance of the corresponding vertices, e.g. as rational points of the curve. Also the tetrahedron plays some rôle in Belyi-Grothendieck-Shabat-Voevodsky and what are the rôle of the other Platonic solids? In particular the octahedron looks particularly well suited for getting pull-backed by the Ahlfors map? etc. \[14.11.12\] Of course invariant equilateral triangulability is not reserved to orthosymmetric patterns, as shown e.g. by the sphere acted upon by the antipodal map endowed with a Platonic triangulation invariant under the involution (octahedron and icosahedron). One can also consider in genus $1$ a rhombic lattice in ${\Bbb C}$ leading to a diasymmetric (non dividing) curve with $r=1$ real circuit. When the lattice is equilateral say spanned by 1 and $\omega$ a cubic root of $-1$, we have an obvious invariant equilateral triangulation by 8 triangles (with vertices at $0, 1/2,1,\omega/2,\omega/2+1/2,\omega$ and their conjugates). \[10.08.12\] Back to the closed case, we know (Mordell-Faltings ca. 1981) that when the genus is $g\ge 2$ then the curve has finitely many rational points in any number field (finite extension of $\Bbb Q$). Of course this fails if we raise up to the full $\Qbar$ (as slicing a plane model by rational lines gives infinitely many $\Qbar$-points on the curve). One can dream on a connection between the “canonical” equilateral triangulation (ET) and the finitely many rational points evaluated in the various number fields. Of course given an ET of an arithmetic (Riemann) surface we can imagine a subdivision into another ET. Given a Euclidean equilateral triangle it is obvious how to subdivide it in 4 smaller equilateral triangles (bisecting the edges). Is there an equivalent subdivision for hyperbolic equilateral triangles? (I cannot see one...) Thus maybe there is some rigidity. At any rate among all ET of an arithmetic $\Qbar$-surface there is one involving the least number of triangles. This gives an integer invariant for any Riemann surface defined over $\Qbar$. Can this value be related to the finitely many rational points when $g\ge 2$? By Gauss(-Bonnet) \[$\alpha+\beta+\gamma=\pi+\int_T K dA$\] which reduces to $3\alpha=\pi-area$ for an equi-triangle in constant negative curvature equal to $-1$ we see a direct relation between the area and its angle of an equi-triangle. \[11.12.12\] For a more lucid Real Belyi theory than our vague ideas, compare the account in Köck-Singerman 2006 [@Koeck-Singerman_2006], where however the Ahlfors maps does not seem not enter the arena. Ahlfors’ proof {#Ahlfors-proof:sec} ============== \[27.08.12\] This section is our modest attempt to examine and understand Ahlfors’ existence proof of a circle map (of degree $\le r+2p$). Alas we failed this basic goal, but it is perhaps of some interest to discuss the original text while trying to capture some mental pictures (made real) which may have circulated in Ahlfors’ vision. More objectively we also try to identify if Ahlfors argument can be boosted to reassess the prediction of maps with smaller controlled degree $\le r+p$ (Gabard 2006 [@Gabard_2006]). We emphasize once more that Gabard’s result is potentially false, but even if so, it is evident that for low values of the invariants $(r,p)$ Ahlfors bound $r+2p$ fails sharpness. Near its completion, Ahlfors proof takes a geometric “tournure” (convex geometry) where there seems to be some free room suitable for improvements. We tried to imagine some (topological) strategy which could possibly sharpen Ahlfors result along his method (at least for low invariants). This is, apart from didactic interest, the only original idea of the present section. In the original paper Ahlfors 1950 [@Ahlfors_1950 p.124–126], the existence proof, we are interested in, occupies only a short 2 pages argument which looks essentially self-contained albeit not quite easy to digest. I would (personally) be extremely grateful if someone understanding Ahlfors proof could publish a more pedestrian account than Ahlfors’, explaining it in full details. Some of the background required is dispatched earlier in the text (esp. p.103–105 in ), hence trying some rearrangement could improve readability. We were personally not able to follow all the (boring) computations or formulas required by Ahlfors. Alas, big masters tend to give only cryptical output of boring computations. Ahlfors is further typical for his annoying (arrogant?) style “it is clear that”, etc. and one often suffers a lot just to fill some details. Of course, nothing is clear in mathematics especially when it comes to follow mechanical computations. Maybe the presence of those just reveals a lack of conceptual grasp over the underlying geometry. Trying to be more optimistic and less severe due to frustration, it would be nice—I repeat myself intentionally—if somebody could take the defense of Ahlfors by presenting an argument as close as possible to the original (meaning perhaps just eradication of misprints, if any?) which further would be completely mechanical, i.e. where each identity is decorated by the appropriate tag referring to the formula under application. Of course, Ahlfors’ proof seems to involve nothing more than the formalism of differential forms (à la Cartan, de Rham, etc., which he learned from A. Weil’s visit in Scandinavia during World War II), plus Stokes’ formula (already a nightmare to prove, at least for Bourbaki) and the allied integration-by-part formula (consequence of Leibniz’s rule). We were personally unable to produce a perfectly pedestrian (accessible to anybody, in particular myself!) exposition of Ahlfors’ account, lacking both intelligence and patience to make his text perfectly intelligible. The writer probably read this Ahlfors’ argument several times in diagonal (since ca. 2001/02), but never completely understood the details. My motivation for looking at it more closely became more acute, after realizing (August 2012) that it is not completely trivial to complete the Green’s function strategy to the problem (cf. previous Section \[Green:sec\]). It should be noted that Ahlfors’ argument does not employ exactly the Green’s function, but a close relative cousin with pole located on the boundary instead of the interior. As a matter of joking we refer to it as the [*Red’s function*]{}, and as far as we know there is no (standardized) terminology to refer to this object! Accordingly, Ahlfors rather constructs an half-plane map instead of a circle map. Of course both moneys are ultimately convertible, yet both geometrically and analytically this implies a little alteration of the viewpoint. One may then may get a bit confused about wondering on the optimal strategy. Finally, remember that several workers in Japan or the US seem to have found necessary to rework Ahlfors’ proof in a more do-it-yourself fashion. Several other authors, having to cite Ahlfors work, often cross-cited those alternative proofs, like those produced by Heins 1950 [@Heins_1950] or Royden 1962 [@Royden_1962] (cf. e.g. Stout 1972 [@Stout_1972] or Gamelin 1973 [@Gamelin_1973-Extremal-I p.3], who both cite Royden for the piece of work originally due to Ahlfors). For a more complete list of “dissident” authors drifting from Ahlfors’ account as the optimal source compare Sec.\[dissident:sec\]. The latter tabulation is supposed to illustrate that I may not be isolated in having missed the full joy of complete satisfaction with Ahlfors’ output. Yet, personally we still would like to believe that Ahlfors account is superior in geometric quintessence to all of what followed, but only regret to have missed some crucial details. As far as we know, nobody ever raised a fatal objection against Ahlfors’ proof. (Personally, I only criticize a lack of details in the execution, plus a matter of organization[^13] and finally a lack of geometric visualization.) It may also be speculated that the argument published by Ahlfors 1950 [@Ahlfors_1950] (and reproduced below) is not the way Ahlfors originally discovered the statement (as early as 1948, cf. Nehari 1950 [@Nehari_1950]), which looks more intuitive when approached from the Green’s function viewpoint, or just bare Riemann-Roch theorem (yet with dangerous probability of collision, cf. the remark in Gabard 2006 [@Gabard_2006 p.949]). In the sequel we shall attempt to conciliate Ahlfors’ analytic treatment with the geometric intuition behind it. The goal is (as usual) to prove: [(Ahlfors 1950 [@Ahlfors_1950 p.124–126])]{} Let $\overline{W}$ be a compact bordered Riemann surface of genus $p$ with $r\ge 1$ contours. Then there exists a circle map $f\colon \overline{W}\to \overline{\Delta}$ of degree $\le r+2p=g+1$, where $g:=(r-1)+2p$ can be either interpreted as the genus of the (Schottky) double or as the number of essential $1$-cycles on $F$ considered up to homologies (the so-called Betti number). The core of Ahlfors’ argument ----------------------------- For the proof Ahlfors uses the concept of a Schottky differentials. Those are differentials on the bordered surface which extends to the Schottky double. The following subclass plays a special rôle: $$S_r=\textrm{ the space of analytic Schottky differentials which are real along $C=\partial \overline{W}$}.$$ \[bipole:lemma\] Given $g+1$ distinct points $z_j$ on the contour $C=\partial \overline{W}$ and corresponding reals $A_j \in {\Bbb R}$, it is possible to construct an analytic differential $\theta_0$ which is real on[^14] $C$ and whose only singularities are double poles at the $z_j$ with singular parts: $$A_j \frac{dz}{(z-z_j)^2},$$ where the local variable $z$ at $z_j$ is chosen so as to map $C$ onto the real-axis ${\Bbb R}$ and inner points of $W$ into the upper half-plane. Further such a differential $\theta_0$ is uniquely determined up to a differential $\theta\in S_r$, and for a proper choice of the latter we can make vanish the periods and half-periods of the imaginary-part $\Im \theta_0$. Ahlfors prefers to construct instead of a circle map a upper half-plane mapping $F\colon \overline W \to \overline H=\{ \Im z \ge 0\}$ which will ultimately arise through the equation $\theta_0=dF$, after arranging exactness of $\theta_0$ for a suitable location of the $z_j$ and some $A_j\ge 0$. Once this is achieved we may write $\theta_0=dF$ for some analytic function $F$ on $\overline W$. The latter is uniquely defined modulo an additive constant and can be chosen real on $C=\partial \overline W$, except at the $z_j$ where $\Im F$ becomes positively infinite. The maximum principle ensures $\Im F >0$ on the whole interior $W$, and therefore $F$ is the desired half-plane mapping of degree $\le r+2p$. This is the bare strategy of the argument, but it is time to adventure into the details. A first ingredient is the fact (compare the second Corollary on p.109): \[g-dimensional:lemma\] The real vector space $S_r$ (of Schottky differentials real along the border) has real dimension $g$. This looks rather plausible upon thinking with the Schottky double and explains the second (uniqueness) clause of the above lemma. Notice indeed that there is $(r-1)$ half-periods corresponding to pathes on the bordered surface $\overline W$ joining a fixed contour $C_1$ to the remaining ones $C_2, \dots, C_r$ and $2p$ full periods arising by winding around the $p$ handles. To arrange exactness of $\theta_0$, Ahlfors employs the inner product $(\theta_0, \theta)$ and a corresponding criterion for exactness in terms of orthogonality to the space $S_r$ (cf. Lemma \[orthogonality:lemma\] below). (The reader can skip the proof of the next two lemmas to move directly to the core of the argument which in our opinion is Lemma \[clever-choice:lemma\].) Before attacking the proof we first recall the pertinent definitions. The [*inner product*]{} of two differentials on a Riemann surface is defined by: $$(\omega_1, \omega_2)=\int_W \omega_1 \overline{\omega_2}^{\ast},$$ where the star denotes the [*conjugate differential*]{} and the bar is the [*complex conjugate*]{} (compare Ahlfors, p.103). (Locally if $\omega=a\, dx+b\,dy$ then $\omega^{\ast}=-b\, dx+a\,dy$ and $\overline\omega=\bar a\, dx+\bar b\, dy$) Further we need probably Stokes $$\int_W d\omega=\int_C \omega,$$ which combined with Leibniz $$d(f \omega)=df \cdot \omega + f d \omega.$$ gives the so-called integration by parts formula $$\int_W (df \cdot \omega + f d \omega)\buildrel{\rm \tiny{Leibniz}}\over=\int_W d(f\omega) \buildrel{\rm \tiny{Stokes}}\over{=}\int_C f\omega,$$ which can be rewritten as $$\int_W df \cdot \omega =\int_C f\omega- \int_W f d \omega,$$ which is hopefully the exact form used (subconsciously) in the sequel. Further he requires an expression of this inner product in term of local variables. Namely the following: \[loc-formula:lemma\] If $\theta=\alpha dz$ near $z_j$, then we have the following formula for the inner product $$\label{loc-formula:eq} (\theta_0, \theta)=-\pi\textstyle\sum_{j=1}^{g+1}A_j \alpha(z_j),$$ where $\theta_0$ is the differential of Lemma \[bipole:lemma\]. As in the first lemma, once we have arranged vanishing of the period and the half-period of the imaginary part $\Im \theta_0$ we may write something like $$\theta_0-\overline{\theta_0}=i \, dG,$$ where $G$ vanishes on $C$ except at the $z_j$. Then brute-force computation gives $$\label{inner-product:eq} (\theta_0, \theta)\buildrel{?}\over{=}(\theta_0-\overline{\theta_0}, \theta)=(i \, dG, \theta)=\dots=-\int_C G \bar \theta,$$ where the “dots” indicates steps left un-detailed by Ahlfors. Of course one should first apply the definition of the inner product and then use integration-by-part, as we just recalled, while noticing that the second term vanish involving the differential of an analytic function. \[Alas, the writer had not the energy to complete the detailed computation.\] Now writing $\theta=\alpha dz$ near $z_j$, Ahlfors claims the following local expression for $G$ $$G \sim i\, A_j (\frac{1}{z-z_j}-\frac{1}{\bar z-z_j}),$$ whereupon he claims that the singularity at $z_j$ contributes the amount $-\pi A_j \alpha(z_j)$ to the last integral of . The announced formula should follow easily. \[orthogonality:lemma\] $\theta_0$ is exact iff $(\theta_0, \theta)=0$ for all $\theta \in S_r$. A priori we could expect to save forces by proving only sufficiency (i.e. the implication $[\Leftarrow]$), but alas Ahlfors’ proof requires the direct sense as well, plus the previous lemma involving the rather (unappealing) computation in local coordinate. Enough philosophy and lamentation, and let us follow along Ahlfors’ exposition. $[\Rightarrow]$ Write $\theta_0=dF$. Then Ahlfors write cryptically $$(\theta_0,\theta)\buildrel{?}\over{=}(\theta_0,\theta+\bar \theta)=\int_W dF \overline{\cdots}=i\int_C F(\bar \theta- \theta)=\pi \textstyle\sum_{j=1}^{g+1 } A_j \alpha(z_j),$$ and comparison with Equation  shows that $(\theta_0, \theta)=0$, as required. $[\Leftarrow]$ Conversely, suppose $(\theta_0, \theta)=0$ for all $\theta\in S_r$, and let $\varphi$ be the analytic Schottky differential making $\theta_0-\varphi$ exact. Then by the former implication[^15] $( \theta_0-\varphi, \theta)=0$ and so $(\varphi, \theta)=0$ for all $\theta \in S_r$. This implies $\varphi=0$, and we conclude that $\theta_0$ is exact. Combining both those lemmas, the exactness of $\theta_0$ is reduced to the following (tricky) lemma, involving a mixture of convex geometry and Stokes formula (which Ahlfors calls the [*fundamental formula*]{} probably due its anticipation by Green or Gauss and others). \[clever-choice:lemma\] It is possible to choose the $z_j$ and the $A_j\ge 0$ so that $$\label{Ahlfors_sum-Aj:eq} \textstyle\sum_{j=1}^{g+1}A_j \alpha(z_j)=0$$ for all $\theta \in S_r$ locally expressed as $\theta=\alpha dz$. Let $\theta_i\in S_r$ ($i=1, \dots ,g$) be a basis of the $g$-dimensional space $S_r$ (cf. Lemma \[g-dimensional:lemma\]). Locally we can write $\theta_i= \alpha_i dz$ near $z_j$. Equation  can be satisfied with $A_j\ge 0$ iff the simplex with vertices $$(\alpha_1(z_j), \dots, \alpha_g(z_j))\in {\Bbb R}^g \qquad\textrm{ for } j=1, \dots, {g+1}$$ contains the origin $0\in {\Bbb R}^g$. If this condition is not full-filled for any choice of the $z_j$, the convex-hull of the set of points $$K:=\{(\alpha_1(t), \dots, \alpha_g(t)) : \textrm{ for } t\in C \}$$ would fail to contain $0$. (One can think of this set as a sort of link (in the sense of knot theory) traced in ${\Bbb R}^g$ with $r$ components. However the latter is not perfectly canonical since the $\alpha_i(t)$ depends on the local chart. [*Expressing some naive doubts.*]{} So here Ahlfors argument looks a bit fragile (or at least sketchy) as one probably requires to fix a finite system of holomorphic charts covering the full contour of the bordered surface). \[We do not have a specific objection, yet it should be noted that the whole Ahlfors theory even that of the refined extremal problem depends on the non-emptiness of the class of bounded functions, hence upon the present argument! In principle even if there should be a global crash of Ahlfors’ proof here, then the theorem should conserves its validity in view of several subsequent treatments hopefully logically more reliable, we cite: $\bullet$ Kuramochi 1952 [@Kuramochi_1952] (alas quite unreadable?), $\bullet$ Mizumoto 1960 [@Mizumoto_1960], $\bullet$ Royden 1962 [@Royden_1962] (alas a bit functional-analytic, whereas the statement sentimentally belongs to pure geometric function theory), and maybe $\bullet$ Gabard 2006 [@Gabard_2006] (hopefully reliable, at least it first part not improving Ahlfors’ $r+2p$). However it is likely that the set $K$ can be defined according to the totality of possible $\alpha_i(t)$ arising through a fixed system of permissible charts covering the contour $C$. Now a (Euclidean) set of ${\Bbb R^g}$ whose convex-hull misses the origin is contained in a closed half-space \[maybe even an open half-space?\]. Thus there exists scalars $a_1, \dots, a_g \in {\Bbb R}$ (not all zero) so that $$\textstyle\sum_{i=1}^g a_i \alpha_i(t)\ge 0 \quad \textrm {for all } t\in C.$$ (Geometrically, this is to be interpreted as the scalar product with the vector $(a_1, \dots, a_g) \in {\Bbb R}^g$ orthogonal to the hyperplane whose half contains the set $K$.) Hence the corresponding differential $\theta= \sum_{i=1}^g a_i \theta_i$ is $\ge 0$ along $C$. \[Maybe strict???\] However this violates the fact that $\int_C \theta=0$, as prompted by Stokes’ formula $$\int_{C=\partial \overline W} \theta=\int_{\overline W} d \theta,$$ and the fact that $\theta$ belongs to $S_r$, hence analytic, and thus closed, i.e. $d\theta =0$. Geometric interpretation as dipoles ----------------------------------- \[28.08.12\] Let $F$ be a membrane (=compact bordered Riemann surface), then Ahlfors constructed (cf. previous subsection) a half-plane map $F\to \overline H:=\{\Im z\ge 0 \}$ to the closed upper-half plane. We get a circle map after post-composing with the natural conformal map to the unit-disc $ \overline H \to \overline{\Delta}$. Under such a map, the horizontal lines transforms to a pencil of circles tangent to the boundary and vertical lines mutate to arc of circles orthogonal to the boundary. (cf. Fig.\[Dipole:fig\]a). One recognizes essentially the so-called Hawaiian earrings (cf. Fig.\[Dipole:fig\]b). Given a circle map, one can pull-back the isothermic (=right-angled) Hawaiian bi-foliation to obtain a graphical representation of the circle map. -5pt0 -5pt0 Starting with with a (doubly-connected) ring, one obtains Fig.\[Dipole:fig\]c or Fig.\[Dipole:fig\]d. Going to higher connectivity on gets for instance Fig.\[Dipole:fig\]e. The Bieberbach-Grunsky theorem (or just Riemann-Roch, cf. e.g. Lemma \[Enriques-Chisini:lemma\]) tell us that we can prescribe a point on each contour and there is a circle map taking all those points to the same image in the unit-circle $S^1=\{ \vert z \vert=1\}$. Hence, we enjoy complete freedom in picturing the isothermic bi-foliation of circle maps, at least in the planar case. This situation is to be contrasted with the situation for the zeros, where some hidden symmetry requires to be fulfilled (compare e.g. Gabard 2006, where we have the condition $D\sim D^{\sigma}$ of linear equivalence of the divisor with its conjugate, an also Fedorov 1991 [@Fedorov_1991] who speaks of an opaque condition that must be satisfied to prescribe the zeros). Of course, the contemplation (and manufacture) of such pictures raises more questions than clarifying the perception of Ahlfors’ theorem. One can hope some guidance via physical intuition (if one feels comfortable with the mineral world) or appeal again to the metaphor about proliferation of bacteria in some nutritive medium. We do not repeat the long discourse we made already for Green (cf. Sec.\[Green:sec\], esp. Fig.\[Green:fig\]) where one had radial expansions emanating from an inner point. Presently, the bacteria are rather located on the boundary, whereupon their local expansion is more of the Hawaiian type, or if you prefer look alike the Doppler effect at the critical speed of sound. The dipole of our title would occur upon considering the symmetric Schottky double of the membrane. This new Hawaiian/Doppler mode of expansion can again be explained via lacking nutritive resources caused by the boundary where the world stops. On Fig.\[Dipole:fig\]f, we have attempted to picture the pull-back of the Hawaiian foliation under a circle-map of degree $r+p=1+1=2$ (for the value $r+p$ predicted by Gabard). This picture looks anomalous for the following reason. Letting grow the population, there is a first junction of the 2 populations right “under” the handle, then there is 2 self-junction at 2 points aside the handle. From now on the bacteria starts invading the handle from both “sides” and will actually merge on the core circle of it. This is problematic since ultimately the expansion should finish along the boundary contours (by definition of a circle-map). It easy to manufacture a picture where no such anomaly occurs (cf. e.g. Fig.\[Dipole:fig\]g which admittedly requires some little effort of concentration to contemplate its morphogenesis). Of course, similar pictures can be made by prescribing less boundary points than the degree of circle-maps predicted by Ahlfors $r+2p$ or $r+p$, e.g. by a choosing a single dipole, cf. Fig.\[Dipole:fig\]h and Fig.\[Dipole:fig\]i. However those patterns cannot correspond to circle-map due to obvious topological obstructions: first the degree of a circle-map must be $\ge r$ impeding Fig.\[Dipole:fig\]h to be allied to a circle-map. As to Fig.\[Dipole:fig\]i the degree would be one, implying the circle-map to be unramified and covering theory (of the simply-connected disc) implies the membrane $F$ to be the disc, violating its genus $1$ nature. We stop this graphical discussion at this primitive stage, yet it is to be hoped that a deeper study of such figures could lead to some theoretical results complementing the understanding of the Ahlfors maps. Perhaps such (dipole) isothermic drawings are of some relevance to Gromov’s filling conjecture, as we already suggested in the case of Green’s function (Sec.\[sec:Green-to-Gromov\]). \[29.08.12\] In fact there is a another more convincing obstruction impeding Fig.\[Dipole:fig\]f to represent a circle-map. This consists in identifying the counter-images of the growing Hawaiian circles past the critical levels while checking if they contribute to the correct numerical multiplicity permissible with the degree of the branched covering. To be concrete we enumerate a series of typical smooth levels on Fig.\[Dipole:fig\]f. The first one denoted $1$ consists of $2$ little circles. Past the first critical level, we see the curve $2$ with $1$ component. After the next critical level, we pick a curve $3$, which has $3$ components. This is too much for our mapping to be of degree 2. This proves that Fig.\[Dipole:fig\]f do not correspond to a circle-map. In contrast repeating the same counting exercise for Fig.\[Dipole:fig\]g, no such excess occurs. The level $1$ has 2 components, level 2 (chosen after the first critical level) has one component, level 3 has 2 components and finally level 4 has 1 component. Thus the picture looks topologically coherent, but it is evident that it is far from metrically realist. Naively speaking we were forced to distort the propagation so has to have a virtually planar mode of depiction for the levels. Trying to recover Ahlfors from the Red’s function {#Red's-function:sec} ------------------------------------------------- \[29.08.12\] Let $F$ denote a finite (=compact) bordered Riemann surface of genus $p$ and with $r$ contours. From the previous section, it seems evident that there is some canonical function akin to the Green’s function yet with pole pushed to the boundary (dipole singularity when doubled). Call them perhaps the [*Red’s function*]{} as an [*ad hoc*]{} acronym honoring writers like Riemann, Schwarz, Klein, Koebe, Ahlfors, etc. Such a Red’s function denoted $R(z,t)=R_t(z)$ with (di)pole at $t\in \partial F$ (a boundary-point) is defined by the property of being harmonic, null along $\partial F$ save at $t$ where it becomes positively infinite according to a specific local singularity (maybe like ${\rm Re}(1/z^2)$). \[18.10.12\] As a more intrinsic definition one can define $R_t$ as the unique positive harmonic function vanishing continuously along $\partial F-\{ t\}$. The function then looks unique up to scalar multiple. Note however that Heins (in e.g. Heins 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF p.241, right after Thm3.1]) defines the function $u_{\zeta}$ our $R_t$ by adding the requirement of minimality (in the sense of Martin 1941 [@Martin_1941]). A positive function $u$ is minimal if whenever there is a smaller function $0<v<u$, $v$ is a constant multiple of $u$. The sudden explosion of $R_t$ at just one boundary point looks at first almost paradoxical, but see again our previous Fig.\[Dipole:fig\]b-h-i) for a depiction of their levels and one can of course imagine such a function just as a “borderline” degeneration of the usual Green’s function. Now one can attempt to construct a half-plane-map (HP-map, for short), by considering a superposition $R(z):=\sum_{i=1}^{d} R(z, t_i)$ of such Red’s functions $R(z,t_i)$ for several points $t_i$ on the border. The formula $$\varphi:=R+i R^{\ast},$$ where $R^{\ast}$ is the conjugate function would then define the HP-map provided the conjugate potential is single-valued in other word that the conjugate differential of $R$, $(d{R})^{\ast}$ is period free. Since $F$ has $(r-1)+2p=:g$ essential cycles (homologically independent), a parameter count suggests that if $d=g+1$ there is enough freedom to annihilate all the $g$ periods of $d{R}^{\ast}$. Maybe this approach (which presumably differs not very much from Ahlfors’) has some technical advantage over the Green’s technique (presented in Sec.\[Green:sec\]). First it seems that the dipole singularity has some linear character contrasting with the arithmetical rigidity of the logarithmic singularity. Thus it is permissible to form a more general linear combination $$R(z):=\textstyle\sum_{i=1}^{d} \lambda_i R(z, t_i),$$ with some reals $\lambda_i$ which must however be $\ge 0$. Hence killing the periods essentially reduces to linear algebra. Another advantage over the Green’s approach stems from the fact that in the interior we meet no singularity thus the period mapping looks less dubious. As usual we write down the period mapping by integrating the $1$-form $dR^{\ast}$ along the $g$ many 1-cycles $\gamma_1, \dots, \gamma_g$ and obtain for each fixed $t_1, \dots t_{g+1}\in \partial F$ a linear map ${\Bbb R}^{g+1} \to {\Bbb R}^g$. Thus there is some non-zero vector in the kernel, and the corresponding $(\lambda_i)$ would solve the problem, provided one is able to check that they can be chosen $\ge 0$. This is non-trivial and a priori it is not evident (and nobody ever asserted) that this can be done for any choice of the $(g+1)$-tuple $t_i$. So it is just here that the difficulty starts, and that some idea is required to complete the proof. \[04.09.12\] Due to a lack of creativity/energy, I was blocked here for a couple of days. So let me make a list of writers who seem to have grasped the geometric quintessence of Ahlfors’ argument: $\bullet$ Gamelin-Voichick 1968 [@Gamelin-Voichick_1968 p.926]: “According to \[1, §4.2\](=Ahlfors 1950 [@Ahlfors_1950]), there exist $r+1$ ($r=g$ in our notation) points $w_1, \dots, w_{r+1}$ on $bR$ such that if $B_j$ is the period vector of the singular function $T_j$ corresponding to a unit point mass at $w_j$, then $B_1, \dots, B_{r+1}$ are the vertices of a simplex in ${\Bbb R}^r$ which contains $0$ as an interior point.” \[10.09.12\] $\bullet$ Fisher 1973 [@Fisher_1973 p.1187/88]: “By a theorem of Ahlfors \[A1; §4.2\] there is a set of $r+1$ points $p_j$ in $\Gamma$ such that if $v_j$ is the period vector of a unit mass at $p_j$, then $v_0,\dots, v_r$ form the vertices of a simplex in ${\Bbb R}^{r}$ which contains the origin as an interior point.” \[this looks alike verbatim copy of the previous source, yet reinforce confidence in the viewpoint\] \[07.09.12\] In fact some little hope to complete the argument is raised by borrowing ideas of convex geometry used by Ahlfors, yet in our context which is perhaps not so reliable (albeit it seems to match with the Gamelin-Voichick twist of Ahlfors). Alas, we failed to recover Ahlfors statement, but we see obvious room for improving upon Ahlfors by using essentially his method of proof augmented by some further geometric tricks. Ideally one would like to recover the bound predicted in Gabard 2006 [@Gabard_2006] by using an argument very close to Ahlfors’. Let us now be more concrete. Again we fix some $d$ points $t_1, \dots t_d$ on the boundary $\partial F$, with at least one point one each contour $C_i$ (forming the boundary $\partial F$). For any point $t\in \partial F$ the function $R_t(z):=R(z,t)$ is uniquely defined once a chart around $t$ is specified (otherwise it is unique only up to a positive scaling factor). Let us assume $R_t$ fixed once for all with a continuous dependence over the parameter $t$. (Alas the writer has no clear-cut justification of this possibility. \[09.09.12\] Maybe use a boundary uniformizer for an annular tubular neighborhood of each contour, cf. e.g. Hasumi 1966 [@Hasumi_1966 p.241], also Gamelin-Voichick 1968 [@Gamelin-Voichick_1968 p.926]. \[18.10.12\] Of course since $R_t$ is unique up to scalar multiple, we are somehow choosing a section of a ray-bundle and even if after winding once around an oval of $\partial F$ the $R_t$ should not return to its initial position $R_{t_0}$, it seems easy to apply a sort of “closing lemma” so that $R_t$ comes back to the original choice.) We now introduce $\Pi(t)$ the period of $(dR_t)^{\ast}$ along the fixed representatives $\gamma_1, \dots, \gamma_g$ of the first homology, that is, $$\Pi(t)=(\textstyle\int_{\gamma_1} (dR_t)^{\ast}, \dots, \int_{\gamma_g}(dR_t)^{\ast}) \in {\Bbb R}^g.$$ We seek $R$ of the form $R=\sum_{i=1}^{d} \lambda_i R_{t_i}$ with $\lambda_i>0$ such that the conjugate differential $(dR)^{\ast}$ is period-free. Period-freeness amounts to say that [*the simplex of ${\Bbb R}^g$ spanned by the $\Pi(t_1), \dots, \Pi(t_d)$ contains the origin in its interior*]{}[^16]. Then positive masses $\lambda_i$ can be assigned to the $\Pi(t_i)$ so that the origin occurs as barycenter of this masses distribution. The italicized condition is equivalent to saying that the convex-hull of the set $X:=\Pi(\partial F)$ contains the origin (say then that the set $X$ is balanced). Balancedness paraphrases also into the condition that the set is not contained in a half-space delimited by a hyperplane through the origin. Ahlfors derives his result from the following simple lemma applied to $X=\Pi(\partial F)$. Let $X$ be a subset of some number space ${\Bbb R}^g$. Any point in the convex-hull of $X$ is the barycenter (=convex combination involving positive coefficients) of at most $g+1$ points of $X$. Of course the lemma is sharp in general: consider $X\subset {\Bbb R}^2$ a set of 3 points in general position (not collinear) then any point chosen in the interior of the convex-hull of $X$ (a simplex) requires all 3 points in a barycentric combination. However if $X$ is a more continuous shape like a topological circle in ${\Bbb R}^2$ it is clear that 2 points situated on $X$ will suffice (cf. Fig.\[Convex:fig\]a). Indeed, imagine first that $X$ is a Jordan curve and that the point lies in its interior. Any line through the point intercepts the Jordan curve in at least 2 points which can be used for a convex combination of the given point. If the point is not in the interior, one can meet an “U-shaped” Jordan curve where the point is situated near the top of the “U” (Fig.\[Convex:fig\]b), yet still expressible as the barycenter of 2 points on the top of the “U”. This already raises some hope upon improving Ahlfors, and optimistically a careful inspection could recover the $r+p$ bound of Gabard 2006 [@Gabard_2006]. -5pt0 -5pt0 Let us summarize the situation. The lemma shows is that if the convex-hull of $\Pi(\partial F)$ contains the origin $0$, then one can certainly find $g+1$ points $t_i$ (eventually fewer) and corresponding $\lambda_i>0$ such that $R=\sum \lambda_i R_{t_i}$ has a period-free conjugate differential. This implies the existence of a half-plane map (via $f=R+iR^{\ast}$) of degree $\le g+1=r+2p$, recovering therefore Ahlfors’ result of 1950. Thus the problem splits in two parts: $\bullet$ Step (1): explain why the convex-hull of $\Pi(\partial F)$ contains the origin $0$ (implying Ahlfors’ $r+2p$ bound); (Ahlfors is able to do this, yet hopefully the ambient context of his argument can be slightly simplified to our present setting which is closer say to Heins’ accounts in 1950 [@Heins_1950] or 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF]) $\bullet$ Step (2): try to lower Ahlfors degree $r+2p$ by taking advantage of the fact that $X=\Pi(\partial F)$ is not an arbitrary set but the continuous image of $r$ circles; (ideally try to recover the $r+p$ upper bound predicted in Gabard 2006 [@Gabard_2006], or at least partial improvements of Ahlfors bound $r+2p$ for low values of the invariants $(r,p)$). As to the first point (1), we notice that if it is violated then the set $\Pi(\partial F)$ is contained in a half-space of ${\Bbb R}^g$. Thus there is a non-zero vector $a=(a_1,\dots, a_g)\in {\Bbb R}^{g}$ such that the scalar product $(a, \Pi(t))>0$ for all $t\in \partial F$. This means $$\sum_{i=1}^g a_i \int_{\gamma_i}(dR_t)^{\ast}>0 \textrm{ for all } t\in \partial F.$$ Alas, the writer failed to find a reason why this should be a contradiction. (In Ahlfors’ presentation Stokes’ theorem plays a crucial rôle.) Even if the present geometric strategy (cooked by the writer via slow assimilation of the very classical strategy of annihilating periods) should be impossible to complete, nothing forbids to switch again to the original treatment of Ahlfors, and apply our Step (2), whose tangibleness relies on Fig.\[Convex:fig\]. The essential point is that ultimately the geometric setting is invariably the one and same problem of convex geometry, whether we start from Ahlfors “analytic” approach or from our more geometric reformulation via the Red’s functions. Let us be more explicit. We have a map $\Pi\colon \partial F=:C \to {\Bbb R}^g$. (“$C$” for contours, like in Ahlfors notation.) In Ahlfors’ paper this occurs as the map $C\ni t \mapsto (\alpha_1(t), \dots , \alpha_g(t))$ cf. p.125 of his article. (From the algebro-geometric viewpoint this must probably be the vectorial lift of the so-called [*canonical map*]{} $\varphi\colon C \to {\Bbb P}^{g-1}$ (usually ascribed to Noether or Klein) allied to the canonical series $\vert K \vert$ living over the curve $C$, obtained by doubling the bordered Riemann surface.) We try to address the second issue (2). The setting is a map $\Pi\colon C \to {\Bbb R}^g$ whose image is balanced (i.e. the convex-hull of the image contains the origin, or equivalently the set $\Pi(\partial F)$ is not contained in any open half-space of ${\Bbb R}^g$ delimited by a hyperplane through the origin). The whole problem is then reduced to the following geometric question. \[problem:Ahlfors-circuit\] Given two integers $r\ge 1$ and $p\ge 0$. Let $g:=(r-1)+2p$, and suppose given in the corresponding Euclidean space ${\Bbb R}^g$ a collection of $r$ (possibly singular) circles $C_1, \dots, C_r$. It is assumed that the union of all these circles is balanced. Find the minimum cardinality of a group of $d$ points with at least one point on each $C_i$ spanning a simplex containing the origin. The previous lemma solves the problem for degree $d=g+1=r+2p$ (recovering Ahlfors’ result). To do better we start from such a group and try to move the vertices, while taking care that the simplex still contains $0$. From the $r+2p$ points, we imagine $r$ many as essentially fixed and the other coupled in $p$ many pairs. The initial simplex is top-dimensional matching the dimension $g$ of the ambient number-space ${\Bbb R}^g$. Moving vertices, it looks reasonable that we may coalesce two points of the $g$-simplex to get a $(g-1)$-simplex still containing $0$. This presupposes both coalescing points being located on the same circuit $C_i$ (try to argue with the pigeon hole principle). After $p$ such collisions (one for each pair) we reach the degree $r+p$ predicted by Gabard 2006 [@Gabard_2006]. Alas this “piano mover” argument is not easy to believe, nor to prove. Perhaps a less naive variant involving an adequate trick (most probably of a topological nature akin say to the Borsuk-Ulam proof of the ham-sandwich theorem) could recover the $r+p$ bound. Less optimistically, it may happen that the above problem is not always soluble with $d\le r+p$, but only for circuits $C_i$ arising from bordered Riemann surfaces via the period map recipe. At any rate, we see the prominent rôle of convex geometry in the question of the least possible degree of the Ahlfors function. In principle there is a canonically defined set $\Pi(C)\subset {\Bbb R}^g$ (we shall call the [*Ahlfors figure*]{}) whose spanning simplices going through the origin affords a complete understanding (in theory at least) of the minimal degree of a circle map concretizing the given bordered surface $F$. \[11.09.12\] Perhaps one can solve the above problem (\[problem:Ahlfors-circuit\]) for $d=r+p$ by an inductive procedure. Let us sketch an attempt that fails (reasonably close to the goal). Recall that given two integers $(r,p)$ and a balanced configuration of $r$ circles $C_i$ in ${\Bbb R}^g$, where $g:=(r-1)+2p$. We would like to show that the origin is the barycenter of at most $r+p$ points with at least one on each $C_i$. Of course the assertion holds true when $p=0$, because we know (by the lemma) that $d\le g+1=r+2p=r$ and on the other hand we have the trivial lower-bound $r\le d$ imposed by the fact that each circle supports at least one point. It follows that $d=r=r+p$, and the claim is vindicated. Thus one can try an induction reducing to the “planar case” $p=0$. This can be done in several ways via the moves $(r,p)\mapsto (r,p-1)$, or $(r,p)\mapsto (r+1,p-1)$ or finally $(r,p)\mapsto (r+2,p-1)$. The latter of which has the advantage that the new value of $g$, denoted $g'$ stays invariant. Now given a geometric configuration of type $(r,p)$ in the number-space ${\Bbb R}^g$ we construct one of type $(r+2, p-1)$ in the same ${\Bbb R}^g$, maybe naively just by duplicating two of the circles (i.e., assigning them a multiplicity). This new configuration is still balanced, so by induction hypothesis the origin is expressible as the barycenter of $r'+p'=(r+2)+(p-1)=r+p+1$ points located on the $C_i$. Alas, this exceeds by one unit the desired $r+p$. \[18.10.12\] Low-dimensional examples may help to give some weak evidence toward solving Problem \[problem:Ahlfors-circuit\] with Gabard’s bound $d=r+p$. Let us discuss this aspect. If we take $(r,p)=(1,1)$, then $g=2$. So geometrically we have one circuit in the plane ${\Bbb R}^2$. In this situation our Fig. \[Convex:fig\] prompts solubility of the problem with $d=2$. Note the agreement with Gabard’s bound $r+p$. This proves the (modest) theorem that [*a bordered surface with one contour and of genus one always admits a circle map of degree $2$*]{}, whereas Ahlfors only predicts degree $r+2p=1+2=3$. Another evidence comes from the well-known hyperellipticity of genus 2 curves. Indeed the double of such a membrane having genus 2, it is hyperelliptic and can therefore be visualized in 3-space as something like Fig.\[Convex2:fig\]a. Doing a rotation of angle $\pi$ we find the required circle map of degree 2 (look at the Figs.\[Convex2:fig\]b and \[Convex2:fig\]c). -5pt0 -5pt0 Let us next examine the case $(r,p)=(2,1)$, then $g=(r-1)+2p=1+2=3$. So we have 2 circuits in space ${\Bbb R}^3$ (like in knot or link theory). Since the set of circuits is balanced, we have something like Fig.\[Convex2:fig\]d (assuming no knotting for simplicity). Balancing amounts picturesquely to say that if you dispose of a 180 degrees angular vision (like any respectable homo sapiens) you will never be able from the origin to contemplate the full link. Paraphrased differently, whatever the direction you choose to focus your vision the link will always move in your back. It seems plausible that, instead of the 4 points prompted by Ahlfors’ top-dimensional $3$-simplex, 3 points actually suffices to span a $2$-simplex passing through the origin (see again Fig.\[Convex2:fig\]d). Justifying this intuition could again corroborate the $r+p$ bound (at least for low invariants). Of course the genus (of the double) being now $g=3$ there is no hyperelliptic reduction, yet appealing to the canonical map $C_g\to {\Bbb P}^{g-1}$ (an embedding precisely when the curve is not hyperelliptic) our curve is concretized as a plane quartic (the canonical divisor $K$ having degree $2g-2$). Some basic knowledge of Klein’s theory then prompts that our orthosymmetric real quartic with $r=2$ must consist of two nested ovals. Projecting from a real point on the inner oval gives a totally real morphism of degree $4-1=3$, in accordance again with the $r+p$ bound. All these little experiments raise the hope that Ahlfors original approach suitably sharpened by a geometric lemma about balanced collections of circuits in ${\Bbb R}^g$ should enable some improvements, and eventually confirm the prediction of the $r+p$ bound. However we confess that the required positive solution to Problem \[problem:Ahlfors-circuit\] with $d=r+p$ looks difficult to obtain and perhaps only true for special circuits arising through period maps. It is quite hard to connect Ahlfors method with the one in Gabard 2006 [@Gabard_2006] in which Abel’s map was exploited more systematically. Since both maps, $\Pi$ an Abel, involve periods, a natural guess is that [*Ahlfors’ figure*]{}, that is the set $\Pi(\partial F)\subset {\Bbb R}^g$, is closely related to the Abel map or at least the so-called (Noether-Klein) canonical map $C\to {\Bbb P}^{g-1}$ which is just the Gauss map of the Abel map: each tangent to the curve seen in its Jacobian is reported to the origin via translation in the Jacobi torus. If so interpretable, it is perhaps no surprise that Ahlfors approach is cumbersome because one is working in the Plato cavern where the essence (embedded-ness) of things is lost. Still, the Ahlfors figure is perhaps useful for other questions. For instance if we take a top-dimensional spanning simplex with $g+1=r+2p$ vertices containing $0$ in its interior, it is clear that we may perturb slightly the vertices keeping the origin inside the simplex. This shows a sort of topological stability of Ahlfors maps having degrees $r+2p$. (This phenomenon is not new, compare Černe-Forstnerič 2002 [@Cerne-Forstneric_2002].) The same stability cannot be expected with the more economical $r+p$ bound, for a slight perturbation of our hypothetical simplex will generally miss the origin. Ahlfors’ figure also shows existence of circle maps for each degree $\ge r+2p$. For those of degrees $>r+2p$ there is a menagerie of convex combinations expressing $0$ and accordingly plenty of circle maps having the same fibre above a boundary point. Such results look not easily accessed via Gabard’s method (in Gabard 2006 [@Gabard_2006]). Trying to make the last “menagerie” point more accurate could lead to interesting result. For simplicity imagine ${\Bbb R}^g$ as the plane ${\Bbb R}^2$ and in it a $2$-simplex spanning the origin. If we have more than $(g+1)$ points, say $g+2=4$ then we may interpret the convex-hull of those 4 points as the shadow (projection) of a $3$-simplex living in ${\Bbb R}^3$. Hence above the origin there is a segment in this higher $3$-simplex each elements of which is a convex sum of the 4 vertices. Hence we get $\infty^1$ circle maps having the same 4 points as prescribed value. This requires of course to be better presented but should be straightforward application of Ahlfors method. Strip mappings (Nehari, Kuramochi) ---------------------------------- \[31.08.12\] As we saw instead of a circle map, Ahlfors 1950 [@Ahlfors_1950] prefers to construct a half-plane map. Ultimately this amounts to the same except that the disc instead of being decorated by the polar coordinates it is by the Hawaiian dipole (Fig.\[Dipole:fig\]a). A third option is to envisage (as Nehari and Kuramochi 1952 [@Kuramochi_1952]) a strip mapping to the strip $S:=\{z: -1\le\Re(z)\le 1\}$. When rectangular coordinate on the strip are transplanted to the disc we obtain a dipole looking like a mitosis. This yields yet another isothermic system on the disc. To synthesize, the disc can be decorated by 3 types of isothermic coordinates (systems): \(1) the monopole attached to an inner point of the disc, which when the pole is the center is just the foliation by concentric circles plus the orthogonal rays. We may from here drag the pole away from the center to get other isothermic systems best interpreted as the geodesic expansion w.r.t. to the hyperbolic metric on the disc. Upon letting degenerate the pole to the boundary circle we get: \(2) the dipole depicted on Fig.\[Dipole:fig\]a and finally upon disintegrating this source of multiplicity 2 into two separate elements of multiplicity one we get: \(3) a genuine dipole which ultimately can be the mitosis about antipodal points of the circle. In principle to each of these geometric decoration of the disc corresponds an existence-proof of the Ahlfors function differing so-to-speak just in the “cosmetic details”. Finally, each isothermic system suggests an angle of attack to Gromov’s filling conjecture. Eventually, it seems plausible that the totality of those isothermic systems could be exploited collectively upon using an averaging process (somehow reminiscent to Löwner-Pu’s trick). Hurwitz type proof of Ahlfors maps? {#Hurwitz-type} =================================== \[21.10.12\] This section wonders about an elementary existence-proof of circle maps via a continuity method reinforcing some naive moduli count. As we noted (in Sec.\[Minimal-sheet:sec\]) the disaster with bordered surfaces is that their gonality is not prompted by a naive moduli count, and thus the project looks from the scratch a bit hazardous. However it is not impossible that we missed something crucial. The general philosophy would be not to fix a surface and try hard to find a map, but rather to look at all possible maps and lift the complex structure of the disc while hoping that if the degree is large enough there are sufficiently many free parameters to paint the full moduli space. Hence any Riemann surface would be expressible as a branched cover of the disc of some controlled degree. (Natanzon suggested to me this strategy during an oral conversation at the Rennes conference 2001, and I came again to this idea by reading Natanzon et al. 2001 [@Natanzon-Shapiro-Vainshtein_2001/XX].) The basic idea may be formalized as follows. We fix a topological type $(r,p)$ encoding the number of contours and the genus. We introduce the (Hurwitz) space $$H_{r,p}^d:=\textrm{ set of all circle maps from surfaces of type } (r,p) \textrm{ having degree } \le d.$$ An element of this natural set (hence a space!) is a branched cover of which we may keep in mind only the “total space”. This gives a map $$\tau\colon H_{r,p}^d\to {\cal M}_{r,p},$$ to the moduli space of bordered surfaces of type $(r,p)$. We want to show that this mapping is surjective for $d$ sufficiently large (but controlled à la Ahlfors). First, we know (since Klein essentially) that ${\cal M}_{r,p}$ is connected. Thus it would be enough to find a suitable $d$ so that the $\tau$-image is closed, open and nonempty. As $(r,p)$ is fixed we may omit it from the notation. Of course $H^d:=H_{r,p}^d$ is empty when $d<r$. The example of rotational surfaces (cf. Fig.\[Chambery:fig\]) shows that $H^d$ is non-void for $d=r$ or $d=r+1$ when $r$ is even resp. odd. It seems also trivial (since we have defined $H^d$ by the condition $\deg(f)\le d$) that the image $\tau(H^d)$ is closed for any $d$. Intuitively a map can degenerate to a map of lower degree, but will never degenerate to one of higher topological complexity. Observationally, this is well seen on the example of the Gürtelkurve (plane quartic with two nested ovals): when projected from a point in the interior of the oval we get a total map of degree 4, which can degenerate to one of degree 3 if the center of projection is specialized toward the inner oval. However, if we take a sequence of maps of degree 3 given by such projections the limit will be a similar projection (the oval being closed) and we never reach a map of degree 4. Of course an abstract explanation requires be given (perhaps just by compactness of $H^d$). The hard part is to show that $\tau$ is open for some large $d$. Naively one could hope to do this via Brouwer’s invariance of the domain requiring something like $\tau=\tau_d$ being étale for a suitable $d$. Another idea is perhaps to factorize $\tau$ by taking the fibre of the circle map $f\colon F \to \Delta$ ($\Delta$=closed disc, here!) over the origin $0$ of the disc to get a surface marked by a group of $d$ points. The nice feature is that $(F,f^{-1}(0))$ permits one to recover uniquely (up to rotation) the map $f$ (cf. Lemma 5.2 about unilateral divisors in Gabard 2006 [@Gabard_2006]). Taking instead the fibre over the real unit $1\in \Delta$ gives a surface marked by a group of $d$ distinct along the boundary. Taking simultaneously the fibre over $0$ and $1$ gives a surface marked by $d$ points on both the interior and the border. So we have 3 natural spaces of marked surfaces living above the moduli space ${\cal M}={\cal M}_{r,p}$, namely $I^d$ (interior marking); $B^d$ (bordered marking); and $M^d$ (mixed marking). Forgetting the markings gives varied arrows descending to $M$. The map $\tau$ factorizes through all these marked moduli space. An idea could be to show that the lift of $\tau$ (which is an embedding especially when we factor through the mixed marking) is sufficiently horizontal w.r.t. to the fiber bundle projection afforded by the forgetful map. Alas, this is not very evident and should of course hold for some special value of $d$. Another route to explore is to make a Lüroth-Clebsch/Hurwitz type analysis of trying to understand from ramification and monodromy how one reconstruct the Riemann surface. Miscellaneous ============= Moduli counts via dissection in pants (Klein, Fricke, Nielsen, Fenchel, etc.) {#Nielsen-Fenchel:sec} ----------------------------------------------------------------------------- \[25.11.12\] This section presents a well-known argument to count moduli of Riemann surfaces, which applies both to the closed and bordered cases. The argument uses a decomposition in pants and the hyperbolic metric, so differs somewhat from the original arguments of Riemann 1857 [@Riemann_1857] and Klein 1882 [@Klein_1882], respectively. \[Nielsen-Fenchel:thm\] [(Riemann 1857 [@Riemann_1857], Klein 1882 [@Klein_1882], Teichmüller 1939 [@Teichmueller_1939])]{} The closed genus $g$ surface $F_g$ depends on $3g-3$ complex moduli or $6g-6$ real moduli, while compact bordered Riemann surfaces $F_{r,p}$ with $r$ contours and $p$ handles depend upon $3g-3$ real moduli, where $g=(r-1)+2p$ is the genus of the double. First consider the closed case. Introduce on $F_g$ a uniformizing metric of curvature $K\equiv -1$ and choose a decomposition in [*pants*]{} (alias [*trinion*]{} by Möbius 1860/63 [@Moebius_1863]). Each pant is a bordered surface with $3$ contours and of genus $p=0$. The conformal structure is unambiguously defined by the lengths of the contours, plus some twisting parameters (rotation like) permissible at the junctures of pants. Looking at the left-hand side of Fig.\[Pants:fig\], we count $(g-2)$ shaded pants each contributing for 3 lengths, and one must add one loop on the top and two on the bottom part of the figure. We arrive at a total of $$1+3(g-2)+2=3g-3$$ many loops. Since each such loop is a juncture we add as many twisting parameters to get finally the dependence upon $$2(3g-3)=6g-6$$ real moduli. -0.0cm0 -0.0cm0 -5pt0 In the bordered setting, we proceed similarly by looking at a pants decomposition of the bordered surface $F_{r,p}$ as depicted on the right-hand side of Fig.\[Pants:fig\]. Counting from the top to the bottom we get $$1+3(p-1)+(r-1)+r=3p+2r-3$$ many loops. Each of these loops is twistable by a parameter, except the $r$ boundary loops which have no companion loops. So we get $3p+r-3$ additional parameters, hence a total of $$6p+3r-6$$ real moduli. On the other hand the genus of the double of $F_{r,p}$ is $g=(r-1)+2p$, so that the above quantity is nothing but $3g-3$. This completes the proof. Of course a more algebro-geometric count do as well the job while using the reality paradigm of the Galois-Riemann Verschmelzung. More concretely inside the complex moduli space one defines an antiholomorphic involution, and the moduli of “real surfaces” appears as the real (=fixed-point) locus of that involution so has half dimensionality. Such an argument has the advantage of encompassing directly the diasymmetric case, which leads to non-orientable Klein surfaces. For more details, cf. Klein 1882 [@Klein_1882], Teichmüller 1939 [@Teichmueller_1939], Earle 1971 [@Earle_1971-On-the-moduli], Seppäla 1978 [@Seppala_1978-Teich-spaces-of-Klein-surfaces], Huisman, etc. Part II: Hilbert’s 16th {#Hilbert's16th-PartII:sec} ======================= General overview ---------------- \[26.03.13\] As announced in the introduction, we enter now in the second part to our text dealing with Hilbert’s 16th problem. The switch from Ahlfors to Hilbert’s 16th flashed us when reading in more details Rohlin’s work of 1978 (cf. the next Sec.\[Klein-Rohlin-conj:sec\]). Since our assimilation of the material evolved in slow organical mode (with several mistakes of ours), it seems worth summarizing which waters were investigated and what seems to be urgent open problems in the field. This section should thus replace the reading of all the sequel which needs severe reorganization at several places. Further what we understood is still miles away of the fine jewellery reached by Russian scholars in this field, but to defend our messy text we also feel that the philosophy à la Ahlfors or Rohlin has not yet been fully exploited, nor elucidated. First, Hilbert’s 16th includes the topological classification of real algebraic (smooth=non-singular) curves in $\RR P^2$. In its original formulation the critical degree was $m=6$ (sextics), where Hilbert’s intuition produced both the best (the Ansatz that an $M$-curve[^17] cannot have all its $11$ ovals unnested) as well as a misconception that persisted through several decades, until being refuted through Gudkov’s seminal 1969 construction of the curve $\frac{5}{1}5$ (5 ovals enveloped in a larger oval, plus 5 ovals outside). This was a big surprise as Hilbert expected that $M$-curves appear only along the scheme $\frac{1}{1}9$ discovered by Harnack 1876, and the one constructed by himself $\frac{9}{1}1$ in Hilbert 1891 (compare the top-row of Fig.\[Gudkov-Table3:fig\]). The Gudkov symbol $\frac{x}{1}y$ encodes a distribution of ovals where $x$ ovals are directly nested in one oval, while $y$ unnested ovals are lying outside (compare again Fig.\[Gudkov-Table3:fig\] if necessary). Petrovskii’s own scepticism about the unexpected twist of Gudkov’s solution, launched the work of Arnold 1971, and Rohlin 1972, where Gudkov hypothesis $\chi \equiv_8 k^2$ went verified through revolutionary insights on the “complexification”. Here $\chi$ always denotes the Euler characteristic of Ragsdale’s orientable membrane bounding the curve from “inside”, while $k=m/2$ is the semi-degree of a curve of even degree $m=2k$. The modern era of real algebraic geometry was launched, characterized by deep interconnections with 4D-differential topology (Rohlin’s early work on spin 4-manifolds, etc.) What has this topic to do at all with Ahlfors maps? To say the least very few factual links have been tied up presently, but we can dream of a big connection. The sequel is our attempt to enhance the rôle which Ahlfors theory could play in Hilbert’s 16th. We should warn the reader that our viewpoint is much partisan (biased by what produced such masters as Ahlfors and Rohlin) and it may well be the case that the real mathematical terrain is not as plastic and smooth as the expressed in the next lines. First, we should stress that there is no anachronism in expecting such a connection with Hilbert since (modulo technical details) the quintessence of Ahlfors theory truly goes back to the Riemann-Schottky-Klein era (resp. 1851/57–1875/77–1876/82), which is much prior to Hilbert (1862–1943), and a fortiori to Hilbert’s geometrical period ca. 1891—when he left Algebra, Invariant theory, Number theory—to move in the softer realms of geometry, calculus of variations, or “functional” analysis, especially Dirichlet, Fredholm, etc. To be honest, our connection was already envisioned by Rohlin 1978 [@Rohlin_1978] (apparently completely unaware of Ahlfors work, as we were ourselves ca. 2001 when rediscovering the result independently), but who also used implicitly what we call [*total reality*]{} as a tool detecting the dividing character of curves. More strikingly, in a genius stroke without any antecedents, Rohlin asserts a phenomenon of total reality for certain $(M-2)$-sextics explaining a posteriori (nearly) all prohibitions of Gudkov’s table of periodic elements (=Fig.\[Gudkov-Table3:fig\]). The latter table affords nothing less than the complete solution to Hilbert’s 16th by way of a curious pyramidal structure encoding all possible distributions of ovals realized by algebraic curves of degree 6 with real coefficients. Rohlin’s (unproved) synthetical assertion stayed dormant for more than 3 decades until Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics] recently managed to establish a slightly weaker form thereof. We can thus now feel confident in expecting that Ahlfors theory will have to play some major rôle in the future destiny of Hilbert’s 16th, i.e. for curves of degree $m\ge 8$. (The case $m=8$ looks nearly settled if one is expert enough in the field and willing to sacrifice a long period of his time to assemble many bits of knowledge scattered through the literature.) Hilbert’s problem (like any existential puzzle) splits naturally into constructions versus prohibitions. Now the rôle of Ahlfors could be as follows. If one has a distribution of ovals (à la Hilbert) such that all curves representing it are dividing(=type I=orthosymmetric) in the sense of Klein (what Rohlin calls [*a scheme of type I*]{}), then it seems a reasonable folly to expect the phenomenon of [*total reality*]{}, namely existence of a pencil of “adjoint” curves cutting only real points on the given curve. At least Ahlfors theorem implies no conformal obstruction to the scenario. , while general topology (i.e. the image of a connected set is connected) implies an absence of topological obstruction to existence of such a total pencil. Incidentally, it should be no surprise that both Ahlfors 1950 and Rohlin’s maximality claim (1978) refers back to a common denominator, namely works of Felix Klein. In Ahlfors’ case this is indirect since reference is more readily confessed to Schottky’s results somewhat prior to Klein’s (but also more schlichtartig than Klein’s). Via Teichmüller 1941 [@Teichmueller_1941] some return to Klein is implicit though poorly cross-referenced. In Rohlin’s case the analogy with Klein is inherent though disputed in Viro’s survey 1986 [@Viro_1986/86-Progress] via Marin’s assessment of Klein’s assertion that curves of type I cannot gain an oval by crossing the discriminant. Apart from those details it is evident that Klein (and before him Riemann) gave the impulse for all what followed, and the fusion awaited upon is probably merely a matter of reunifying the original conception of Riemann-Klein before it diverged into pure conformal geometry (Schwarz, Schottky, Klein, Koebe, Bieberbach, Grötzsch, Ahlfors, Grunsky, Teichmüller, Ahlfors again) versus plane curves in Hilbert’s 16th (Harnack, Hilbert, Ragsdale, Rohn, Brusotti, Petrovskii, Gudkov, Arnold, Rohlin). One can wonder how much knowledge went lost just through older generations passing away and how much time consuming it will be for us to revive old wisdoms that are probably the key to most of our naive questions. Our 1st fundamental problem is to decide if Ahlfors theorem particularized to the setting of plane curves implies existence of such a total pencil. I personally always thought this being a triviality (see optionally Gabard 2004 [@Gabard_2004 p.7]), but recently Marin warned me that life might not be so easy (cf. letter in Sec.\[e-mail-Viro:sec\]). Alas, meanwhile I forgot nearly all the little I ever knew about the foundations of algebraic geometry, so that what I thought to be trivial is now floating in some suspense (“ombre propice” as would say Thom). Even if not true (or rather implementable), synthetical procedures à la Rohlin-Le Touzé (=RLT) could redeliver the phenomenon of total reality [*ab ovo*]{} (independently of Ahlfors conformal geometry). This seems to require a vertiginous and lengthy verification process climbing ad infinitum. In more gently slope, one can expect a gradual propagation of RLT from degree 6 to 8, and so on, that could be relevant to detect new prohibitions in Hilbert’s 16th. Why so optimistic? As exemplified by the case of sextics $m=6$ (much influential upon Gudkov, Arnold, Rohlin, etc. and as demonstrated by those smart guys fairly typical of the general case $m=2k$) it is likely that a scheme of type I is totally real under a suitable pencil (or viceversa) and this should in turn imply the scheme being maximal in the hierarchy of all schemes. This produces prohibitions in Hilbert’s 16th, which a priori could be new, and governed by an uniform paradigm. So a 2nd fundamental problem is to decide if Rohlin’s maximality conjecture (RMC) positing that “type I implies maximal” is true. At first sight, it seems that a positive solution to the 1st problem implies this as a byproduct, but there seems to be severe obstacles in completing the programme. For explicitness, it is worth sketching the (naive, uncomplete) argument. Given a scheme of type I, there is by Ahlfors a total pencil, which cuts only real points on the curves. Hence the curve is already saturated, and cannot be enlarged by adding an additional oval without violating Bézout. The difficulty however is that the enlargement is not a priori involving the same (or even a nearby curve) augmented by some other ovals, but can be a priori very distant of the original curve. ([*Added in proof*]{} \[13.03.13\], for a loose strategy using isotopies, cf. Sec.\[RMC-via-Mangler:sec\]\].) However a 3rd route is that whenever we encounter a synthetic phenomenon of total reality à la Rohlin-Le Touzé looks (akin to a concretization of Ahlfors abstract theorem within the Plato cavern of Hilbert’s 16th involving only plane curves), then it seems evident (via Bézout-saturation) that Rohlin’s maximality conjecture will hold true for this specific scheme. Again the proof is not easy to formalize, but it is perhaps realist to expect a positive solution in the case of curves of degree 6, and hopefully somewhat higher as to produce new truths. This brings us to the 4th problem. How to extend Rohlin’s total reality claim to high-degree curves $m> 6$. Is there any algorithm telling one where to assign basepoints in order to assure total reality of the corresponding pencil? One could dream that this can be done from the sole knowledge of Rohlin’s complex orientations, cf. optionally Sec.\[Galton-brett:sec\]. For plane $M$-curves of degree $m$, we prove below a basic Theorem \[total-reality-of-plane-M-curves:thm\] stipulating a total pencil swept by curves of order $(m-2)$. This merely traduces the so-called Bieberbach-Grunsky theorem, which (apart from phraseological details like Dirichlet’s illness) truly belongs to Riemann 1857, Schottky 1875–77, Enriques-Chisini 1915, and only then Bieberbach 1925, Grunsky 1937, Wirtinger 1942 [@Wirtinger_1942], etc. A structural asymmetry appears: while $M$-curves are crudely-put reputed hardest-to-construct within Hilbert’s 16th, their conformal geometry is most trivial, due to the planar=schlichtartig character of the half of the curve. Total reality is simplest to ensure in the $M$-setting, just because it is like having one train on each track, hence no risk of collision. Precisely, the trick is just to choose one point on each oval getting so a group of $g+1$ points which moves (by Riemann-Roch or Abel), and total reality is automatically granted (cf. Lemma \[Enriques-Chisini:lemma\] for more details). Making this abstract argument concrete proves the theorem. Can we extend this to non-maximal curves? The risk is then an overpopulation of $g+1$ points scattered on $r<g+1$ ovals, hence 2 of them are forced to cohabit on the same oval (pigeonhole principle due to Dirichlet apparently) exposing us to a possible collision jeopardizing total reality! So what is demanded is controlling a dextrogyration of points when moving along linear equivalence. Ahlfors 1950 [@Ahlfors_1950] or Gabard 2006 [@Gabard_2006] affords basic tricks to achieve dextrogyration in the abstract setting. Can we transplant them directly inside the Plato cavern of plane curves, as we just managed to do for $M$-curves? As yet we never succeeded, but this should not discourage more serious attempts. If we think more concretely à la Rohlin-Le Touzé (or perhaps to go back earlier in history à la Brill-Noether), numerological reasons make evident that $(M-2)$-curves of type I (or even schemes of type I) and degree $m$ will have their total reality exhibited by a pencil of curves of degree $(m-3)$. Evidence is given later in this text, but readily follows by analogy with Rohlin-Le Touzé and a simple constants count (cf. Remark \[M-2-curve-degree-like-Gabard:rem\], which is just the end product of numerological coincidences observed for $m=6,8,10, \dots$). Remind perhaps at this stage old Italian works (recognized as possible competitors to Ahlfors 1950), like Matildi 1945/48 [@Matildi_1945/48], Andreotti 1950 [@Andreotti_1950]. Those could already anticipate our present desideratum. Since already Rohlin’s proof (which is lost) and that of Le Touzé (2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]) is quite delicate (or the sequel of this text which contains ca. 30 pages of unsuccessful attempts to prove Rohlin’s original claim), we are not claiming that total reality will be easy to prove in full generality but perhaps for degree $8, 10$ this remains manageable (at least within the next 4 decades). By experience we are accustomed in the field to slow progresses (remind Hilbert, Gudkov, etc.), and it is quite unlikely (but not impossible) that a new Abel or Riemann will crack the full puzzle in a single stroke. Some part of our text tries to take the census of all such schemes of type I in degree 8. Alas our optical faculties tend to be much more limited than those of aliens, like insects with 8 eyes looking at their preys via pencil of cubics, chameleons with mobile ocular systems, or any sort of creature with 19 eyes (when it comes to look at the world through a pencil of quintics, …), and generally, $M-3$ basepoints (i.e. 19 when $m=8$, $34$ for $m=10$). Accordingly, we are presently (and probably for the rest of our life) confined to deduce total reality not from optical skills but via boring arithmetics, namely the subliminal (Rohlin?)-Kharlamov-Marin congruence $\chi\equiv k^2+4 \pmod 8$, which forces type I under this “shifted” Gudkov-style congruence mod 8. This is a crucial weapon (whose proof we have not yet studied in full details). This harpoon detects for us a menagerie of schemes of type I, all possibly subsumed to total reality, and conjecturally (via RMC=Rohlin’s maximality conjecture) acting prohibitively upon all schemes(=distribution of ovals) pretending to enlarge the given one. Such schemes crystallize therefore Bézout-extremal (or saturated) shapes of Hilbert’s 16th, which as would say Klein cannot develop further without exploding the latent degree. Call an RKM-scheme any scheme satisfying the RKM-congruence $\chi\equiv_8 k^2+4$. It is not clear to the writer, and the experts were a bit silent on this aspect as yet, if conversely any $(M-2)$-scheme of type I is forced to respect the RKM-congruence. This deserves perhaps to be clarified at the occasion. \[[*Added in proof*]{} \[13.04.13\] An answer is probably implicit in Rohlin 1978, Art.3.5, on p.93 (extremal property of Zvonilov-Wilson).\] With some sloppiness, we arrive at some big picture along the following philosophy (in our opinion fairly implicit in Rohlin 1978). Any scheme of type I is detected: $\bullet$ either trivially because it is an $M$-scheme whose total reality is exhibited à la Bieberbach-Grunsky (yet no direct impact upon Hilbert’s 16th by virtue of Harnack’s bound (1876), or more simply its intrinsic variant $r\le g+1$ due to Klein 1876 proved via retrosections à la Riemann), \[but some indirect impact by using satellites!! (13.04.13)\] $\bullet$ or it is an $(M-2)$-scheme verifying the RKM-congruence, in which case total reality is flashed by a pencil of $(m-3)$-tics. $\bullet$ or finally it arises as “satellite” of a scheme of lower degree dividing the given degree. The idea of satellites arises simply by noting (or expecting) that total reality propagates when the curve is doubled, tripled and so on, by replicating several copies of the curve within a tube-neighborhood of it (\[satellite-total-reality:sec\]). For the conic with a single oval (unifolium) this just leads to the series of deep nests total under a pencil of lines, while for a quartic with 4 ovals (quadrifolium in the jargon of Zeuthen 1874 [@Zeuthen_1874] who inspired much Klein 1876) this leads to the series of curves in degrees multiple of $4$, totally real under a pencil of conics. The case of degree 8 is explicitly mentioned in Rohlin 1978, being just the double of the quadrifolium. We expect that satellites do extend to curves of odd degrees (\[Satellite-odd-degree:sec\]), yielding some interesting prohibitions on schemes of degree 10 when applied to an $M$-quintic doubled. Likewise doubling the Rohlin-Le Touzé sextic $\frac{6}{1}2$ (or its mirror $\frac{2}{1}6$) gives a scheme of degree 12 which should be maximal (hence killing all extensions of it). The general philosophy is now clear. Total reality (basically due to Ahlfors 1950, though Teichmüller 1941 ascribes it to Klein directly) acts as an upper-bound on the complexity of Hilbert’s 16th problem, by killing all distributions of ovals adventuring above one totally flashed by a pencil. In substance everything boils down to a phenomenon of Bézout-saturation, with in the background of the scene an extension of the Riemann mapping theorem to surfaces of higher topological structures (so-called Ahlfors maps). This looks a fundamental truth (or philosophy?) since it seems robust, and implementable when the flashing is as explicit as Rohlin-Le Touzé’s as opposed to the abstract nonsense of Ahlfors. If skeptical, the just predicted maximality of satellites in degree 10 and 12 should be tested against highbrow methods of constructions of the modern era (Viro-Itenberg). If the Ahlfors-Rohlin philosophy resists the shock against this structural test, then some experimental evidence is gained that the Ahlfors-Rohlin Verschmelzung is a deep reality governing a substantial part of Hilbert’s 16th at the universal scale (all degrees). If not, then the whole story of the 16th problem could be even more chaotical and unruly than it presently is, i.e. just a combinatorial mess only worth deserving the attention of computing machines. Of course the latter are quite likely to show us hidden patterns of symmetries, maximality, etc. that were not yet appreciated due to a lack of experimental data. More pragmatically, it must feasible to inspect if in degree 8, the Ahlfors-Rohlin scenario of total reality and the allied extremal principle of saturated schemes is compatible with factual data, and optimistically even able to preclude schemes that were not yet prohibited. Alas, we are not expert enough in the field to tell an answer, but peoples like Viro, Fiedler, Korchagin, Orevkov, Le Touzé, must already have a clear-cut vision along this idea. In degree 6 it is clear that the saturation principle of Rohlin is entirely covered (or re-explain) by the congruences mod 8 due to Gudkov and Gudkov-Krakhnov-Kharlamov, but it is not clear to me if the same subordination holds true in degree 8 (maybe in general). On the other hand if the RKM-congruence fails to detect a type I scheme, then there could be some sporadic phenomenon of total reality explaining it, and this would be a new source of saturation (perhaps not covered by the congruences mod 8). This is the main-body of our quest, but during the trip we went sidetracked to other connected topics. Here are some aspects perhaps worth putting in evidence centering around the theme of rigid-isotopy, and the allied contraction principles where the end-point of the path is permitted to touch the discriminant (parameterizing all singular curves). Total reality takes its simplest incarnation for the deep nests swept out by a total pencil of lines. A theorem by Nuij 1968 [@Nuij_1968] (later revisited by Dubrovin 1983 [@Dubrovin_1983/85]) states that such deep nests are rigid in the sense than one can pass from any 2 curves representing it by a continuous deformation of the coefficients without encountering any singular curve during the deformation. Such large deformation pertains to what is called [*rigid-isotopies*]{}, which actually refines Hilbert’s 16th problem. This topic always attracted geometers even prior to Hilbert’s era, e.g. Schläfli (apparently known for having the most massive human brain ever weighted with ca. 1.936 kg for only 157 cm of body height), or Zeuthen and Klein adding several contributions regarding curves and surfaces of low orders (quartic and cubics resp.). For quartic curves Klein established (1876 [@Klein_1876_Verlauf]) that the rigid-isotopy class is fully determined by the real scheme already. It seems natural to ask if Nuij’s rigidity result (for deep nests) has equivalents whenever total reality holds true. Alas this fails by the Marin-Fiedler locking technique which refutes this Ansatz for $M$-curves of degree 7 (cf. Fig.\[Marin:fig\], for a hopefully lucid exposition of Marin’s trick). Despite this disruption of the naive scenario, it seems to us likely that rigidity holds true for satellites of the quadrifolium. We confess however to have not yet studied Nuij’s proof, nor do we know (a fortiori) if his proof extends mutatis mutandis. Though rigid-isotopy merely involves the $\pi_0$ (=nullest homotopy group measuring the arcwise-connected components) of the space of curves excised along the discriminant, very little is known on such problems. A naive conjecture of us—based essentially on the failure of the Marin-Fiedler locking device, plus the fact (subsequent to Rohlin’s formula) that curves of type I have $r\ge m/2$ ovals (also valid if $m$ is odd)—postulates that curves with few ovals are necessarily [*rigid*]{}, i.e., are unambiguously determined up to a large deformation by their sole real schemes. Precisely this could hold true as soon as the curve has strictly less ovals than $DEEP+2$, where $DEEP:=\Delta:=[(m+1)/2]$ is the number of circuits of the deep nest of degree $m$. The intuition is simply that by Rohlin’s formula (\[Rohlin-formula:thm\]) this is the first dividing scheme encountered (as $r$ the number of ovals increases), and two units above this ($r=\Delta+2$) it is a simple matter to exhibit a scheme of indefinite type (Rohlin’s jargon to say that there is curves of both types I.vsII realizing a prescribed configuration of ovals). This conjecture (called LARS, for low-altitude-rigidity-conjecture) is merely a cavalier extension of: \(1) Nuij’s theorem of 1968, which is not specific to curves (but valid for algebraic hypersurfaces, where there is an evident notion of deep nest via concentric spheres, plus an eventual pseudo-plane). \(2) Nikulin’s rigid classification of sextics in 1979/80 [@Nikulin_1979/80] implying the case $m=6$ of our LARS, and telling much more namely the fact that the real scheme (as tabulated on Gudkov’s table) enhanced by the data of Klein-Rohlin’s types affords a complete system of invariants under rigid-isotopy. Hence for $m=6$, Nikulin is stronger than LARS as it prompts rigidities at all altitudes. However as soon as $m\ge 7$ this is foiled (cf. again Marin’s example=Fig.\[Marin:fig\]). \(3) A unofficial conjecture of Rohlin (reported in a Viro letter in Sec.\[e-mail-Viro:sec\]) that curves of odd degrees with a single (pseudoline) component are rigid-isotopic, cf. also Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical]. By analogy, curves of even degrees with a single oval could be rigid-isotopic. Those questions are settled in degrees $m\le 6$ ($m=4$ Klein 1876 [@Klein_1876_Verlauf], $m=5$ Kharlamov 1981 [@Kharlamov_1981/81], $m=6$ Nikulin 1979/80 [@Nikulin_1979/80]), but still resist in degrees $\ge 7$. Hence our conjecture LARS appears very presumptuous, and it may be a more reasonable challenge trying to disprove it. Alas the locking method of Fiedler-Marin looks (as far as we experimented in the sequel) quite impuissant to destroy LARS. What techniques could be used to prove LARS or more modest rigidity conjectures? Our naive idea is that geometric flows (amounting to look at orthogonal trajectories of suitable functionals like calculating the length or area of ovals) could prove this and related results of rigid-isotopies. This would give some intrusion of differential-geometric methods in problems of rigid-isotopies, a priori of a purely algebraic nature. Presently we were never able to complete any serious proof along this way, but our text contains ca. 20 pages of (dubious) trials along such lines. Viro’s survey 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical] also contains a brief desideratum to know more about geometric properties of curves, and this could evidently pertain to rigid-isotopies, in a way perhaps reminiscent of the Ölfleck of H.A. Schwarz (where the Riemann mapping theorem is visualized by an oil-flake restoring to the circular shape), or the eclectic Ricci flow of Yau-Hamilton-Perelman, where a similar phenomenology appears in the abstract Riemannian setting (convergence to the round metric, with the well-known bonus about Poincaré’s conjecture). Affiliated to those rigid-isotopy questions there is a conjecture of Itenberg-Viro (cf. Viro’s preface of the volume containing Itenberg 1994 [@Itenberg_1994]) to the effect that some empty oval of any curve can always be shrunk toward a solitary node. \[[*Added in proof*]{} \[13.04.13\].—Similar (but more vague) ideas are actually ubiquitous in Klein, e.g. 1892 [@Klein_1892_Realitaet].\] This is still wide open, but Itenberg’s article just cited establishes the case $m=6$. Again one could hope that the flow minimizing the length of an oval could achieve such a contraction. Inspired by this conjecture we advanced a strengthened version CCC(=collective contraction conjecture) saying that all empty ovals can simultaneously contract toward solitary nodes. (This is like a perfect landing in Flight-Simulator v.18.5 with aircraft Antonov 72 having its 94 wheels touching the ground simultaneously!) If this (unlikely) miracle is true, one gets e.g. a 2-seconds proof of Hilbert’s Ansatz of the non-existence of an $M$-sextic without nesting by reduction to Bézout. Of course this is also more hygienically derived from Rohlin’s formula (or Arnold’s congruence mod 4), which involves softer homological intersection theory à la Poincaré-Lefschetz, etc. We were not as yet able to disprove our strong CCC-version of Itenberg-Viro. Its real impact being still obscure we did not pursued this issue in any serious fashion. A philosophical consequence, of large deformations is that they should (like total reality) act prohibitory, whereas small perturbations are classically exerting their swings at the constructive level (Harnack 1876, Hilbert 1891, Brusotti 1914/21, Gudkov 1969/72, Viro 1980, Itenberg 1993, etc.) Of course a clear-cut realm of where to corrupt CCC could be a dividing curve without nesting, for those could after [*strangulation*]{} be split into two complex-conjugate halves intersecting in as many points as there were ovals initially. Alas either Thom (\[Thom-Ragsdale:thm\]) or better Rohlin’s formula (\[Rohlin-formula:thm\]) forces such a curve to have $\chi=r\le k^2$ resp. exactly $r=k^2$ ovals, hence we fail to corrupt Monsieur Étienne Bézout. It looks so quite challenging to kill CCC, albeit its truth looks very fragile, as it incarnates an extreme flexibility of algebraic objects reputed “rigid” in the large. Yet it should be remembered (though at some more local viz. regional scale) that Brusotti’s theorem gives via Riemann-Roch-Brill-Noether-Severi a remarkable flexibility of algebraic curves (independence of the smoothing of nodes). So one should not be surprised at last, if sometimes algebraic curves appear more plastic than expected a priori. However, as we shall soon discuss, Shustin disproved (in degree 8) a flexibility conjecture of Klein (1876) that nondividing curves can always acquire a solitary node through continuous variation of the coefficients ([*champagne bubble phenomenon*]{}). In slight contrast, building over the previously cited works of Nikulin 1979, Itenberg 1994, and the whole diagrammatic of the Gudkov-Rohlin table (1969–78), we think that Klein’s intuition of champagne bubbling is correct in degree 6 (cf. Prop.\[Klein-vache-deg-6:prop\]). The philosophical impact of Shustin’s disproof of Klein (though his aim was refuting a related assertion of Rohlin) is that we cannot expect to have solely topological obstructions regulating large algebraic deformations. A last theme involves the impact of Thom’s conjecture (meanwhile Kronheimer-Mrowka’s theorem 1994) upon Hilbert’s 16th (Sec.\[Thom:sec\]). A classical trick (called the Arnold surface) is to fill Klein’s half of a dividing curve $C_m$ (of degree $m=2k$) by the real Ragsdale orientable membrane bounding the curve from inside. This gives a homology class of half-degree $k=m/2$, smoothly represented (after rounding corners, if necessary). This object looks ideally suited to an application of Thom’s genus estimate. Taking for granted orientability of the Arnold surface, we found an erroneous estimate $\chi\le k^2$ (for all dividing plane curves of degree $2k$ (cf. (\[Thom-Ragsdale:thm\])). Albeit wrong in general (as Fiedler kindly pointed out to us) it holds in special cases when all (primitive) pairs of ovals are positive in the sense of Rohlin, i.e. when complex and real orientations match together. Real vs. complex orientations may even disagree yet along pieces not connected by the Ragsdale membrane (cf. Lemma \[Arnold-surface-orientable-iff-oddly-charged:lem\]). If optimistic Thom or even Rohlin’s formula gives a way to attack the (still open) Ragsdale’s conjecture for $M$-curves, which amounts to $\vert \chi \vert \le k^2$ (\[Thom-implies-one-half-of-Ragsdale:lem\]). Is the Arnold surface (=Klein’s half glued with the Ragsdale membrane) of an $M$-curve always orientable? If yes, then the proof of our (erroneous) Theorem \[Thom-Ragsdale:thm\] implies Ragsdale’s conjecture via Thom’s estimate on the genus. Unfortunately, Arnold’s surface is nonorientable already for Hilbert’s $M$-sextic, cf. Lemma \[disproof-orientability-Arnold-M-curve:lem\]. Maybe the theorem à la Bieberbach-Grunsky specialized to plane $M$-curves (i.e. our Theorem \[total-reality-of-plane-M-curves:thm\]) could give (via dextrogyration[^18]) enough control on complex orientations of $M$-curves as to imply Ragsdale, either via Thom or directly via Rohlin’s formula $2(\pi-\eta)=r-k^2$ (where $\pi:=\Pi^+$, $\eta:=\Pi^-$ to abridge notation). (This amounts then to check that $\pi-\eta\le n$, the number of negative ovals.) This admittedly looks naive, but we cannot exclude such a coarse strategy for the moment. More modestly, it may be noted that filling Klein’s half with Ragsdale’s membrane of an $M$-sextic without nesting reduces Hilbert’s nesting Ansatz to the “baby” case of Thom for homology classes of degree 3, acquitted by Kervaire-Milnor 1961 [@Kervaire-Milnor_1961] building upon Rohlin’s early work 1951 on spin $4$-manifolds. As said, our erroneous estimate $\chi\le k^2$ was corrected by Fiedler in a series of letters where he learned us the Petrovskii estimates on $\chi=p-n$, and Arnold’s strong avatars thereof involving hyperbolic ovals. This is again closely connected to the Ragsdale conjecture, which is still a [*pièce de resistance*]{} in the case of $M$-curves. Moreover though our estimate $\chi\le k^2$ was erroneously founded it turned out to be quite difficult to find an explicit counterexample. At least we failed via classical methods (cf. Figs.\[HilbGab1:fig\]–\[HilbGab4:fig\]), which rather inclined to think that $\chi\le k^2$ was sharp if true at all. Namely using Hilbert’s construction we find an infinite series of $M$-curves or $(M-2)$-curves such that $\chi=k^2$ exactly, but failed to beat $k^2$. We presume this is exactly the sort of experiments that led Ragsdale to her conjecture. However the story does not finish here, and the big surprise arrives now. It is notorious that a marvellous construction of Viro-Itenberg (patchwork and $T$-construction, cf. Fig.\[Itenberg:fig\]) killed the Ragsdale conjecture in degree 10 (even in its relaxed shape of Petrovskii), yet leaving intact the $M$-curves case. The Itenberg-Viro construction supplied us with the apparently simplest counterexample to our erroneous estimate $\chi \le k^2$ (for type I curves of degree $2k$). It produces namely an $(M-2)$-curves of degree 10 with $\chi=29 \nleqslant k^2=25$, hence necessarily dividing by the RKM-congruence $\chi\equiv_8 k^2+4$. This was the fatal stroke (coup de grâce) against our estimate $\chi \le k^2$, which is quite robust as it seems incorruptible via Harnack-Hilbert and challenging to refute in the $M$-case. Last but not least, there is a disproof due to Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin] of one-side of Rohlin’s maximality conjecture namely “type I$\Leftarrow$ maximal”. This disproof is not so dramatic for Rohlin’s prohibitive programme which uses rather the converse (still hypothetical) implication “type I$\buildrel{?}\over\Rightarrow$ maximal”. Shustin’s note looks historically pivotal as it kills the second part of Klein’s intuition, pertaining to large deformations of nondividing curves as always admitting the apparition of a champagne bubble created by crossing the discriminant through a solitary node. Due to its extreme concision we had first not understood Shustin’s argument (and [*unduly*]{} mistrusted his result for a while). Finally, we understood its logic, but confess to have not yet assimilated all the results required to complete its proof. It suffices to say that Shustin’s work exploits Viro’s construction on the one side, and also advanced Bézout-style obstructions due to Fiedler and extended by Viro. Some details perhaps assisting beginners to grasp the structure of Shustin’s proof are to be found in Sec.\[Shustin-understood:sec\]. This a brief summary of the territories we managed to explore in ca. 3 months of investigation. Besides our text may have some didactic value on the following aspects. (1).—We give a self-contained account of Gudkov solution to Hilbert’s 16th problem in degree $m=6$, by exposing the original constructions of Harnack, Hilbert and Gudkov. Those issues are well-known and described in Gudkov’s seminal survey (and at several other places like A’Campo’s Bourbaki survey 1979 [@A'Campo_1979], etc.). Yet not all species are always accompanied by decent pictures requiring sometimes clever twists of Harnack’s construction (oft messy to implement if one wants to realize a type given in advance). So we had long hours of trials with computer-assisted depictions. This can hopefully be of some use to some nonspecialist readers. Our intention was to reproduce all (including the infructuous) trials, but that generated “microfilm” pictures often too heavy for the purpose of arXivation. By the way our microfilm though still readable in pdf-format at 600 dpi resolution will still be hard to contemplate on the screen. \[[*Added in proof*]{} \[13.04.13\] This technical problem was settled by shrinking the size of pictures in the Adobe software, permitting so to economize much memory space, yet without altering the optical size of pictures.\] (2).—We give also full details (and a graphical view=Fig.\[Gudkov-Table3:fig\]) of Rohlin’s enhancement of Gudkov’s census of sextics by adding the complex topological characteristics of Klein (that were much neglected during the era of Hilbert, Ragsdale, Rohn, Petrovskii, Gudkov) up to the Arnold-Rohlin revival of the complexification (which turned to be the conceptual key to explain Gudkov’s experimental phenomenology). This is merely a simple exercise yet that can be quite time-consuming if one starts from zero-knowledge. Of course an excellent account of this, differing form ours only in the minor details, is already given in the masterpiece Marin 1979 [@Marin_1979]. (Our account differs just in using more primitive configurations of 3 ellipses.) (3).—We give in Sec.\[Prohibitions:sec\] a reasonably exhaustive list of classical obstructions, especially a (nearly complete) proof of Rohlin’s formula (\[Rohlin-formula:thm\]). In the original source (Rohlin 1974 [@Rohlin_1974/75]) this is not presented in its full generality (only $M$-curves), though the adaptation to general dividing curves is very minor. This Rohlin’s formula looks extremely fundamental as it appears as the most universal obstruction that can be derived by nearly abstract nonsense (i.e. using very little from the assumption of algebraicity), yet still affording strikingly precise information while staying completely elementary. For instance it covers Hilbert’s Ansatz of nesting, and extends it to all degrees $m\ge 6$. It also formally implies the Arnold congruence mod 4, which is a weak form of Gudkov hypothesis for $M$-curves, yet an extension thereof to arbitrary dividing curves. Then there is a series of avatars of the Gudkov congruence mod 8, that truly requires more advanced topological tools, essentially in the spirit of Rohlin 1952 [@Rohlin_1952-4-manifolds]. Those more advanced results are not proved in our text, and we hope to be able to offer a lucid view on them in the future. Hence, to assimilate the marvellous congruences due to Rohlin, Gudkov-Krakhnov/Kharlamov, Kharlamov-Marin, etc., our reader is invited to consult the original sources (Rohlin 1972 [@Rohlin_1972/72-Proof-of-a-conj-of-Gudkov] (with a gap but essentially correct and repaired by Marin-Guillou, e.g. 1986 [@Guillou-Marin_1986] or Marin 1979 [@Marin_1979]), and also Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91], giving a synthesized view). Challenging vs. less challenging open problems {#Challenging-open-prob:sec} ---------------------------------------------- \[28.03.13\] This section summarizes what looks to us major open questions in the field investigated. The reader is warned that our list is a mixture of hard Soviet conjectures of longstanding with newcomers (due to myself), therefore probably much easier to settle down when not ill-posed. To distinguish among them the symbol $\bigstar$ marks venerable Russian conjectures, while our more modest variants are marked by “$\bullet$”. $\bigstar$ (R6) Can somebody reconstruct Rohlin’s lost proof that the $(M-2)$-sextics with schemes $\frac{6}{1}2$ or $\frac{2}{1}6$ are totally real under a pencil of cubics assigned to pass through 8 points distributed on (or inside) the empty ovals. This is nearly solved in Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics], but she uses the RKM-congruence (\[Kharlamov-Marin-cong:thm\]) to infer a priori the curve being dividing. It could be more natural to draw dividingness from total reality via a purely synthetical procedure [*a priori*]{}. At any rate the conjunction of the RKM-congruence ($\chi\equiv_8 k^2+4$) with Le Touzé’s result implies that Rohlin’s assertion is true. Hence, it should be already “safe terrain” to explore. If much more pessimistic Rohlin’s claim is wrong and then either RKM, or Le Touzé is false, which is very unlikely. $\bigstar$ (RLT6$\to$RMC6) Can someone complete the proof that the Rohlin-Le Touzé phenomenon of total reality (RLT6) prevents all sextic schemes extending those described in the previous problem (R6), so as to infer nearly all obstructions of Hilbert’s 16th via the paradigm of total reality (TR) and the allied phenomenon of Bézout-saturation. Cf. the diagrammatic of the Gudkov table (Fig.\[Gudkov-Table3:fig\]) to appreciate this issue in degree $m=6$. Of course this problem can be considered as very implicit in Rohlin 1978 [@Rohlin_1978], but in our opinion not solved there. $\bullet$ (RLT$m>6$) How does the Rohlin-Le Touzé phenomenon described in (R6) above extend to higher degrees $m>6$? Cf. Sec.\[total-(M-2)-schemes:sec\] for a germ of answer. $\bullet$ (A50$\to$R78) How valuable is the abstract theory of Ahlfors to assess Rohlin’s vision of total reality? Cf. e.g. Sec.\[Esquisse-dun-prog-deja-esquiss:sec\] for some scenarios. In particular is it true that any dividing plane curve admits a total pencil (i.e. whose real members cut only real points)? If yes, is it always of degree $\le (m-2)$ when the given curve has degree $m$? For the case of $M$-curves, cf. (\[total-reality-of-plane-M-curves:thm\]) which gives a total pencil of order $(m-2)$. $\bullet$ (R78$\to$G13) Is it true as conjectured in our text (\[M-2-curve-degree-like-Gabard:rem\]) that any curve belonging to an $(M-2)$-scheme of type I and degree $m$ has its total reality exhibited by a pencil of curves of degree $(m-3)$. Further what is the exact rôle of Riemann, and Brill-Noether adjoint curves, in this game? Notice still in (\[M-2-curve-degree-like-Gabard:rem\]) a strange concomitance between Rohlin-Le Touzé’s role of cubics and Gabard’s $r+p$ bound on the gonality. The latter improves Ahlfors by replacing $g+1$ by the mean-value of Harnack’s bound $g+1$ and the number $r$ of real circuits. All this numerology looks to match too nicely for this being merely a fortuitous coincidence. In particular for quartics, quintics, sextics, etc. the total reality of $(M-2)$-curves of type I seems always exhibited by such a pencil of degree $(m-3)$. $\bullet$ (RKM$\leftarrow$type I) The RKM-congruence $\chi\equiv_8 k^2+4 $ detects many $(M-2)$-schemes of type I, but does it detect all of them? The answer is yes for $m=6$ (cf. the Gudkov-Rohlin table Fig.\[Gudkov-Table3:fig\]). Hence $m=8$ is the first place to look for a counterexample. Assuming there is one, then it could be that total reality detects type I schemes at places where RKM fails. \[[*Added in proof*]{} \[13.04.13\].—The answer to this question must be implicit in Rohlin 1978 [@Rohlin_1978 p.93, Art.3.5] (and due to Zvonilov-Wilson).\] $\bigstar$ (RMC) Is Rohlin’s maximality conjecture true, i.e. all schemes of type I are maximal in the hierarchy of all schemes of fixed degree? Can this be disproved by the Viro-Itenberg patchwork, as it was possible to refute the converse sense of Rohlin’s conjecture (cf. Shustin’s note 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]). $\bullet$ (SAT) Are satellites of schemes of type I still of type I? For instance what about the 2nd satellite of the Rohlin-Le Touzé sextics of point (R6). Can this be disproved via patchwork? Assuming a positive answer to the first question (even in a special case) points to potential place where to corrupt RMC. Personally, we would be much more happy if RMC holds true, as then, and only then, there is some chance to make a big Riemann-Hilbert or Ahlfors-Rohlin synthesis. $\bigstar$ (H8) Complete the solution of Hilbert’s 16th in degree 8, and analyze objectively if Rohlin’s maximality principle (RMC) has some things to say in this realm, as it did in degree $m=6$. In particular does Rohlin’s maximality conjecture (RMC) still persists in degree $8$. (Some hints are given in Orevkov’s letter in Sec.\[Orevkov:sec\].) $\bullet$ (LARS) Can someone disprove our low-altitude rigidity speculation (LARS) positing that a curve with less ovals than 2 units above the deep nest is entirely determined up to large deformations by its real scheme. Cf. (\[LARS:conj\]). $\bullet$ (URS) \[02.04.13\] The unnested rigidity speculation (URS) is akin to LARS, and posits that any unnested curve is rigid provided the number of ovals is not the square of the semi-degree ($r\neq k^2$, and assume $m$ even). Motivation comes from Rohlin’s formula (which forces such curves being of type II), and the case $m=6$ which follows from Nikulin. Another (weak) evidence comes from the fact that the locking technique of Fiedler-Marin seems to have little grip on such schemes as there is no way to choose a canonical triangle (moving frame). $\bigstar$ (OOPS)=(One oval postulation).—In particular what about the much more modest (than LARS or URS, yet still wide open) rigidity conjecture for curves having only one component. Are such unifolium curves rigid as conjectured by Rohlin, Viro, etc. How useful are geometric flows to do this? Cf. (\[OOPS:one-oval-rigid-isotopic:conj\]). Actually Viro ascribes to Rohlin (cf. e-mail in Sec.\[e-mail-Viro:sec\]) the rigidity of curves of odd degree with a unique component, but the case of even degrees looks likewise open. Further it seems evident (at least for Viro, cf. the same letter) that OOPS is implied by CC, i.e. the Itenberg-Viro contraction conjecture for empty ovals. By analogy it seems evident that our CCC (cf. right below) implies URS. Sketch of proof: contract all ovals simultaneously (which are all supposed empty) as to reach the connected empty chamber, and do this twice. Of course when $r=k^2$ the real scheme can be of both types, and this case has to be ruled out (or optimistically the type is the sole obstruction to rigid isotopy). $\bullet$ (CCC)=(Collective contraction conjecture).—Can someone disprove our strong version CCC (of the Itenberg-Viro contraction principle for empty ovals) positing a simultaneous and collective contraction of all empty ovals toward solitary nodes. Cf. (\[CCC:conj\]). $\bigstar$ (CC) \[30.03.13\] There is a (still open) conjecture of Itenberg-Viro (cf. Itenberg 1994 [@Itenberg_1994] and Viro’s preface of the same volume) positing that [*any*]{} empty oval of a real plane curve can be contracted to a point (solitary node). This is true in degree 6 as proved by Itenberg (), and weaker than our (CCC) above (cf. (\[CCviaCCC-Brusotti:lem\])). $\bigstar$ (CC vs. TR) A noteworthy consequence of CC is that all obstructions in degree 6 derived (clumsily) via total reality (TR) are likewise derived by this contraction principle of Itenberg (CC6) (modulo knowledge of the RKM-congruence and Klein’s Thesis which is fairly easy to prove since Marin 1988 [@Marin_1988]). The problem is first to decide which method “total reality” versus “contraction” is more easily implemented in degrees $\ge 8$, while trying to make a comparative study of the prohibitions resulting from both procedures. In particular one may wonder if the Itenberg-Viro conjecture implies (formally or not) Rohlin’s maximality conjecture. Sketch of proof: Take any scheme of type I, and a curve enlarging it. Contract an empty oval so as to recover the initial scheme (note here an obvious difficulty, namely the additional oval of the extended scheme is not necessarily an empty one!), and conclude via Klein’s Thesis (a curve of type I cannot champagne-bubble). $\bullet$ (Refuting CC via Shustin?) \[31.03.13\] By the proof of Prop. \[Klein-vache-deg-6:prop\], we see that the Itenberg contraction principle combined with the diagrammatic of the Gudkov-Rohlin table (of all typed-schemes) implies [*Klein-vache*]{} (KV), i.e. the possibility for diasymmetric curve to acquire a solitary node and then a new oval ([*comme surgit du néant*]{}). Now as Klein-vache is disproved in degree 8, it seems that it is just a matter of waiting completion of Hilbert’s 16th problem in degree 8, until the Itenberg-Viro contraction conjecture get refuted. This is merely a crude scenario but of course one needs to keep track of a massive diagrammatic to get an extension in degree 8 of Rohlin’s theorem (\[Rohlin-type:thm\]) classifying all sextics according to their types. $\bullet$ (GR8)=(Gudkov-Rohlin census in degree 8).—Assume someone has completed Hilbert’s 16th in degree 8 (i.e. isotopy classification of real schemes), how difficult will it be to complete the corresponding Rohlin table enhancing schemes by their types I or II. Assume this information available, does it follow (by analogy with our proof of Prop.\[Klein-vache-deg-6:prop\]) that under the contraction principle (CC), Klein-vache holds true in degree 8? If yes, then Shustin 1985 would refute the Itenberg-Viro contraction conjecture in degree 8 (CC8). $\bigstar$ (RAG)=(Ragsdale).—While our erroneous Thom-style estimate $\chi\le k^2$ (cf. \[Thom-Ragsdale:thm\]) is disproved by the Itenberg-Viro $(M-2)$-curve (Fig.\[Itenberg:fig\]), is this estimate still true for $M$-curves? This amounts to one-half of Ragsdale conjecture $\vert \chi\vert \le k^2$ (still open in the $M$-context). A priori a “random” computer-assisted search along the Itenberg-Viro method could detect an $M$-curve refuting Ragsdale. How difficult is it to program a machine adventuring blindly and by brute force in such a random quest? In contradistinction, how difficult is it to write down a proof of Ragsdale’s conjecture in case it should be true. Could it be that a clever use of Thom, or Rohlin’s formula and even some knowledge of complex orientations derived maybe from our synthetic version (\[total-reality-of-plane-M-curves:thm\]) of the Bieberbach-Grunsky theorem (planar case of Ahlfors) assesses the full puzzle. If feasible this would be a spectacular application of conformal geometry to the Hilbert-Ragsdale-Petrovskii 16th problem, boiling down in quintessence to Riemann’s Nachlass 1857 [@Riemann_1857_Nachlass]. Of course we do not claim this to be an easy project. $\bullet$ In degree 6, it may be observed that among the trinity of congruences mod 8 (due to GR, GKK, RKM, where G=Gudkov, R=Rohlin, K=Krakhnov or Kharlamov (twice), M=Marin, cf. (\[Gudkov-hypothesis:thm\]), (\[Gudkov-Krakhnov-Kharlamov-cong:thm\]), (\[Kharlamov-Marin-cong:thm\])), the latter, i.e. RKM, implies the 2 formers, when combined with Rohlin’s maximality principle (RMC). Is this subsuming a general feature due to trivial geographical/arithmetical reasons? If yes can we condense, i.e. proceed to an unification of forces by reducing nearly all prohibitions of Hilbert’s 16th to the phenomenon of total reality. $\bullet$ Can we write down an explicit ternary form with integral coefficients $F\in \ZZ[x_0,x_1,x_2]$ whose real locus is Gudkov’s curve $\frac{5}{1}5$, and estimate the smallest size of the coefficients involved? As discussed in Sec.\[Diophantine-and-proba:sec\], can we compute the natural masses (w.r.t. Lebesgue measure on the space of coefficients) of each of the 64 chambers (past the discriminant) of smooth sextic curves given by the census of Gudkov-Rohlin-Nikulin (i.e. Fig.\[Gudkov-Table3:fig\]). $\bullet$ \[31.03.13\] A more modest but fundamental problem is to publish (in the West side of Ural) an avatar in degree 8 of the Gudkov-Rohlin table (Fig.\[Gudkov-Table3:fig\]). For a partial depiction of just the simplest planar face of this 4D-pyramid, cf. Fig.\[Degree8:fig\]. I presume that one can by mean of an Atlas consisting of ca. 20 pages dress a list of all combinatorially possible schemes after taking into account the obvious Bézout-style obstructions (Bézout, Zeuthen, Hilbert’s bounds on the depth of nest, Gudkov, plus the total reality obstructions allied to the deep nest and doubled quadrifolium, etc.). Once this atlas of all octics is made available it should be a trivial matter to appreciate: —how far/close we are to solve Hilbert’s 16th in degree 8 (soft-isotopy); —how the paradigm of total reality (resp. the contraction principle) explain the prohibitions, and finally, —whether the contraction principle (CC) implies Klein-vache, in which case CC would be disproved by Shustin’s refutation of Klein-vache in degree 8. $\bullet$ (CG6)=(Contiguity graph for $m=6$) \[01.04.13\] Can we describe all the contiguity relation realizable via algebraic Morse surgeries on the Gudkov-Rohlin table of periodic elements (in degree 6). To be more specific, is some result along our Conjecture \[eversion-and-other surgeries:conj\] true. The proof of this could be merely a matter of adapting the work by Nikulin, and Itenberg, yet it seems quite challenging to decide precisely which eversions are realized algebro-geometrically. $\bullet$ (KV7) Klein-vache (KV) was disproved in degree 8 by Shustin 1985 via a conjunction of Viro’s method and advanced Bézout obstructions due to Viro (and Fiedler). On the other hand, we prove below (\[Klein-vache-deg-6:prop\]) that KV is true in degree 6. So one may wonder about the case $m=7$, where to my knowledge KV is undecided. $\bullet$ (II/II) Is the “toutou” conjecture true? This posits that any scheme of type II and even degree $2k$ augmented by a pseudoline to a scheme of degree $2k+1$ is of type II too. Cf. (\[toutou:conj\]) for some surgical motivation (à la Fiedler) and inspiration coming from reading Gross-Harris 1981, who were unable to settle the case of quintics with 2 unnested ovals (and a pseudoline of course). Perhaps a general solution of this problem merely follows from a conjunction of Rohlin’s and Mishachev’s formulae. If not, then one could use a large deformation principle. $\bullet$ (Klein’s bipolarity conjecture). Is it possible for two real plane curves to have distinct distributions of ovals, yet conformally equivalent underlying symmetric Riemann surfaces (under an equivariant diffeomorphism). This can be paraphrased in the algebro-geometric language as the quest of two real planes curves with distinct distribution of ovals, but bi-rationally equivalent over $\RR$ as abstract curves. For more see (\[Klein-bipolarity:conj\]), where it is explained that the first place where to look for this (hypothetical but likely) phenomenon is degree $m=6$. It would be interesting to see if this question due to Klein 1922 (safe misunderstanding on my side) can be settled via Cremona transformations not inducing diffeomorphisms of $\RR P^2$. $\bullet$ (RIG/SAT) Is rigidity stable under satellites? This is a wild speculation based on Nuij’s rigidity of the deep nest caricatured as reducible via satellite to the rigidity of the conic (known since time immemorial). Likewise the more highbrow rigidity result of Klein 1876 for quartics could induce rigidity of all satellites of the 6 possible quartic schemes, in particular of the quadrifolium (whose satellites are totally real under a pencil of conics). Further Nikulin’s rigidity result for sextics could imply also a vast array of rigidity results in degrees $6k$, by satellitosis of all schemes of degree 6 which are not of indefinite type (and which are explicitly known $64-2\cdot 8=48$ types by the Gudkov-Rohlin table=Fig.\[Gudkov-Table3:fig\]). Perhaps all this stability of rigidity under satellites has to be combined with total reality, in which case the analogy with Nuij’s rigidity is still deeper. In that case we would only take satellites of Rohlin’s $(M-2)$-schemes of degree $6$, cf. (\[satellite-Rohlin-(6)-schemes-rigid:conj\]), and for quartics only the quadrifolium (and the deep nest) would be permissible. $\bullet$ (ANTI-GAB) It seems that the case of $(M-4)$-sextics of type I offers a possible corruption of Gabard’s bound $r+p$. Compare Scholium \[(M-4)-sextics-corrupt-Gabard:scholium\]. If not, this is at least a [*pièce de résistance*]{} against the principle that any abstract pencil is concrete, and therefore Ahlfors abstract theorem is unlikely to apply without friction in Hilbert’s 16th problem. In other words Riemann’s canary feels claustrophobic in the Plato cavern of Brill-Noether-Hilbert. $\bullet$ (LETOUZE-SCH) Inspired by a Scholium of Le Touzé 2013 (\[LeTouze-quintic:scholie\]), we extended her result to all $M$-curves of odd degrees, cf. Theorem \[Le-Touzé-extended-in-odd-degree:scholium\]. It seems of interest to extend her method to even degrees as well. We had just the time to treat the case of degree 6, cf. Lemma \[Le-Touzé-scholium-deg-6:lem\] which uses imaginary basepoints yet without affecting total reality. It could be challenging to see if this method of total pencil (becoming more and more explicit) could be used to reprove the deep prohibitions for $M$-curves due to Hilbert-Rohn-Gudkov-Rohlin. $\bullet$ (RMC via Mangler 1939 and Ahlfors 1950, maybe implicit in Rohlin 1978).—-Rohlin’s maximality conjecture looks nearly implied by Ahlfors, safe for the difficulty that the enlargement of the type I scheme is a priori very distant from the enlarged curve realizing the orthosymmetric scheme. Using triviality of the mapping class group of $\RR P^2$ (Mangler 1939, probably a student of H. Kneser?), one can try to isotope the distant enlargement to make it identic with the original curve. The latter being swept out by a total pencil (Ahlfors 1950, plus epsilon!), one could get a corruption of the homological version of Bézout (i.e. intersection theory à la Poincaré, Lefschetz, etc.) Of course one requires a procedure to extend the (Mangler) isotopy to $\CC P^2$, and one may object that our sketch of proof equally well applies to curves of type I whose scheme is however of indefinite type (but non-maximal). So there is perhaps some obstruction to extend Mangler’s isotopy as to preserve positivity of intersection-indices. Understanding this obstruction, and supposing one able to show its vanishing in case of a scheme of type I, could procure a proof of the elusive RMC. It seems very likely that Rohlin thought about this strategy, but never wrote something down. Perhaps experts like Marin can complete this game? Cf. Sec.\[RMC-via-Mangler:sec\] for slightly more details. To keep some slight control on all these conjectures, see Fig.\[CCvsCCC:fig\] showing how they interact and their validity range. -5pt0 -5pt0 \[30.03.13\] Let us conclude with a historical remark. It should always be remembered, and amazing to rediscover everyday, that “everything” in this topic goes back to Klein. Himself expected that the type of the symmetric Riemann surface (underlying a real curve acted upon by complex conjugation) has some interplay with Hilbert’s problem on the distribution of ovals. Compare a footnote added ca. 1922 in his Ges.Math.Abhdl., reproduced as Quote \[Klein-1922-immer-vorsgeschwebt:quote\], but of which we now reproduce the most prophetical side: This is worth translating (in the poor English of the writer): It always puzzled me, to infer more about shapes of real plane curves of arbitrary degrees by pursuing considerations of the text, not only regarding the number of circuits, but also their mutual dispositions. I do not abort this hope, but must alas confess, that the reality theorems on curves of arbitrary genus (which I deduce from the general theory of Riemann surfaces, specially that of symmetric Riemann surfaces), are not sufficient for this purpose, affording instead merely a framework for the menagerie of possibilities to be investigated. It is striking to notice how this Kleinian prose remains very much actual, reflecting best our own frustration to make the Ahlfors-Rohlin Verschmelzung, we are dreaming about, a true reality. It shows also how much Klein would have appreciated the developments made possible in the 1970’s by Gudkov, Arnold, and especially Rohlin, etc., and perhaps even more, something like anticipating the vision of total reality by Rohlin. \[05.04.13\] The last sentence of this same footnote, reads: Da man über die Natur dieser Bedingungen zunächst wenig weiss, kann man noch nicht von vornherein sagen, dass alle die Arten reeller Kurven, die man gemäss meinen späteren Untersuchungen für $p={ n-1 \cdot n-2 \over 2}$ findet, bereits im Gebiete besagter ebener Kurven $n$-ter Ordnung vertreten sein mü[ß]{}ten, auch nicht, da[ß]{} ihnen immer nur [*eine*]{} Art ebener Kurven entspräche. K. Here, one realizes that Klein anticipated the simple phenomenon of what Rohlin calls schemes of indefinite type, i.e. that the real scheme alone (i.e., distribution of ovals) does not need to determine the type (i.e. dividingness or not). Klein also emphasizes the issue that not all topologically permissible symmetric Riemann surfaces have to appear in the plane. In both cases the first examples appear in degree 5, and then massively in degree 6. For instance a quintic with only one pseudoline cannot be of type I , albeit since its genus is even (namely 6) the corresponding Riemann surface exists. (Compare (\[Klein-Marin-quintic:lem\]) which is based on Klein-Marin, or Gross-Harris argument via theta-characteristics discussed at the same place that was probably known to Klein in 1892. Finally, it is also puzzling to see that Klein 1892 anticipated somewhat the contraction conjecture of Itenberg-Viro, cf. historical note right after (\[Itenberg-Viro-contraction:conj\]). Albeit Klein missed some basic modern tricks (like Rohlin’s formula, or Fiedler surgical smoothing law), he also mastered perfectly the Riemannian theory (conformal maps, the allied circle maps and total reality as credited by Teichmüller 1941, theta-characteristics, allied deep enumerative problems of bitangents to quartics à la Plücker-Zeuthen). Further he appealed to contraction principles, as well as his own singular geometric method to represent complex loci as multiple cover of the projective plane as to infer the “complexified” topology of real curves (in the 1874–76 articles “Über eine neue Art der Riemannschen Flächen”). Hence Klein’s legacy on the topic is massive (ca. 300 pages, if one counts the Göttingen lectures [@Klein_1892_Vorlesung-Goettingen]), and much remains to be learned from it. The Klein-Rohlin conjecture on real schemes of type I {#Klein-Rohlin-conj:sec} ===================================================== \[01.01.13\] A fascinating question is raised by (the master) V.A. Rohlin in his 1978 [@Rohlin_1978 p.95] survey looping back directly to a (prophetic) allusion of Klein. Remember that Rohlin was fluent with German language, being involved during World War II as translator on the front-line, cf. Guillou-Marin’s book 1986 [@Guillou-Marin_1986 p.ix]: “[*En 1941, quand l’Allemagne attaqua l’U.R.S.S., Rohlin rejoignit le corps des volontaires du Peuple (unités militaires non entraînées). Son unité fut encerclé et Rohlin fait prisonnier par les allemands. Ensuite il réussit à s’échapper, à rejoindre l’armée soviétique et finit la guerre comme traducteur militaire (Rohlin parlait couramment l’allemand). Immédiatement après la guerre Rohlin fut emprisonné par la sécurité de l’armée (comme ce fut le cas pour de nombreux anciens prisonniers de guerre) mais fut libéré à la fin de l’année 1945.*]{}” . [A study of the available factual material suggests that possibly a real scheme belongs to type I iff it is [*maximal*]{}, that is, it is not part of a larger real scheme of the same degree. This conjecture is true for $m\le 6$, and there is much to be said in its favour[^19] for $m>6$. There is an allusion to it in Klein: see \[4\], p.155 (=Klein 1922=Ges. Math. Abh. II [@Klein-Werke-II_1922]).]{} The passage Rohlin had in mind is unambiguously identified as the following (going back actually to Klein 1876 [@Klein_1876]), which is worth reproducing albeit it is first quite hard to interpret (cf. also Viro 1986/86 [@Viro_1986/86-Progress p.67–68] or Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.785], and especially Marin 1988 [@Marin_1988], clarifying earlier work partially incorrect of Cheponkus 1976 [@Cheponkus_1976]): \[Klein\_1876-niemals-isolierte:quote\] \[17.01.13\] It is essential to note that Klein’s quote contains two very distinct parts. The first part on which Klein is affirmative may be translated as the assertion that a dividing (=type I or orthosymmetric) plane curve cannot acquire a new oval by transgressing the discriminant at a solitary node (with imaginary conjugate tangents like the germ $x^2+y^2=0$). With the strong word “kann …niemals” (=never never!!), Klein emphasizes his complete self-confidence about the truth of his assertion. Alas no proof (as far as I know) were ever given by him, even in his Göttingen lectures 1892 [@Klein_1892_Vorlesung-Goettingen]. The first proof had to wait 112 years until Marin 1988 [@Marin_1988] write down a two-lines argument (of a somewhat stronger assertion). A recent e-mail exchange with Viro suggested that Klein’s Ansatz may easily be deduced from the Ahlfors map (cf. Lemma \[Klein-via-Ahlfors(Viro-Gabard):lem\]). The second part of Klein’s text, starting with “während die Kurven …”, is pretty subtle to interpret and definitively less categoric. It is suggested that curves of type II are in contrast susceptible of acquiring new ovals springing [*ex nihilo*]{} from a solitary double point like a champagne bubble. The vague wording “sozusagen noch entwicklungsfähig” (=“so-to-speak still developable”) emphasizes that Klein did not saw any nondividing curve champagne-bubbling, but merely that he found no (topological) obstruction to such an eventuality. As we shall see, this second clause which we shall call “Klein-vache” is refutable (in degree $8$) via the disproof of one half of Rohlin’s conjecture by Shustin 1985 [@Shustin_1985]. Alas, we have not yet completely digested Shustin’s work, which relies on deep Bézout-style obstructions due to Fiedler-Viro. In the positive sense, we proved via a cocktail of Russian results that “Klein-vache” holds true in degree 6 (cf. Prop.\[Klein-vache-deg-6:prop\]). \[01.01.13\] Rohlin’s conjectural criterium looks a pearl of observational skills. What does it mean, or rather how practical is it if true at all? One should make a list of all real schemes (i.e. isotopy classes) of curves of a given order. Then assuming one competent and patient enough to have tabulated the exhaustive list one could detect the dividing types by inspecting maximum elements in the lattice ordered by inclusion. ([*Insertion*]{} \[28.03.13\].—This is not really what happens in practice, especially since Shustin’s disproof, and it seems more likely that the residual half of Rohlin’s conjecture acts by means of prohibitions, that are anyway required to dress a table of all schemes.) Let us work out low-order examples to gain some experimental evidence Rohlin is referring to. First in degree $1$ there is just the line, which is of type I (=dividing). Then in degree 2, there is two isotopy classes represented either by the circle $x^2+y^2=+1$ and the invisible conic $x^2+y^2=-1$ (empty real locus). (This follows e.g. from Sylvester’s law of inertia, alias diagonalization of quadratic forms, also to be found earlier by Jacobi, and presumably many others? and on the case of 2 variables this can safely goes back to ancient Greeks, Euclid, etc.) In order 3 we have cubics (extensively studied by Newton and Plücker), but up to isotopy the story becomes much simpler and we have two isotopy classes differentiated merely by the number of real circuits $r=1,2$; the latter being dividing while the other is not. This follows readily from the abstract Klein-Weichold classification of symmetric surfaces (the latter being merely a mirror image of the Möbius-Jordan classification of abstract topological surfaces). The little zoo of all quartics (Plücker 1839, Zeuthen 1874, Klein 1876, Rohlin 1978) ------------------------------------------------------------------------------------ \[01.01.13\] Next it comes to quartics (order 4). Here the number of real circuits $r$ fails to classify isotopy classes for there exist quartics with 2 ovals being either nested or not. The first basic thing-to-do (going back apparently to Plücker 1839 [@Plücker_1839]) is exploring varied examples by smoothing a pair of conics with 4 intersections (cf. Fig.\[KleinRohlin-quartic:fig\]). -5pt0 -5pt0 Among all those curves only those marked by the attribute “dividing” are dividing as they result from sense-preserving smoothings. One can also remember Klein’s congruence $r\equiv g+1 \pmod 2$ in the dividing case to detect nondividing curves, e.g. those with $r=3,1$. Also under Harnack-maximality, i.e. $r=g+1$ (here $4$) then dividingness is automatic (by Riemann-Klein 1876 [@Klein_1876] or the more tedious synthetic argument of Harnack 1876 [@Harnack_1876]). So $r=2$ is the only ambiguous value. Here, as shown on the figure, the 2 ovals can either be nested or not. In the first case the curve is dividing (due to total reality under a pencil of lines through the innermost oval), while the unnested case is always nondividing (as Klein knew as early as 1876). \[Klein-unnested-quartic-nondividing:lem\] [(Klein 1876)]{} All quartics with $r=2$ unnested ovals (and not just the two traced above) are nondividing. This is already a nontrivial result. We sketch several proofs: \(1) How Klein derived the result? Maybe as follows. Klein knew as early as 1876 [@Klein_1876_Verlauf] (basing himself on deep works of Schläfli and Zeuthen on cubic surfaces and their apparent contours which are quartics) that quartics are rigidly classified by the real scheme. This is to mean that any two quartic curves having the same distribution of ovals can be continuously deformed through a large deformation of the coefficients without ever meeting a singular curve. Hence Klein had only to check the nondividing character of a specific quartic to get that of all curves with 2 unnested ovals. Klein used a special device of representation of the curve as a branched cover of the projective plane by assigning to each point (of the complexification) the unique real point of the tangent and so could see the curve. Nowadays the surgical recipe of Fiedler looks also best suited to do this. For an elementary graphical proof compare our Fig.\[Guertel-genetic:fig\] earlier in this text. \(2) Perhaps one way to argue could involve Ahlfors theorem, yet some nontrivial details deserve being worked out. \[03.01.13\] Assuming Gabard there is a total map of degree $r+p=3$ if dividing (and not less via the complex gonality), yet since the ovals are unnested it cannot be induced by a pencil of lines. So the auxiliary curves are of order at least two. Assume first the order to be two, so we have a pencil of conics. Since the degree of the morphism is 3, we have $2\cdot 4-3=5$ basepoints on the $C_4$, but a pencil of conics has only 4 basepoints by Bézout. (Note that Ahlfors bound $r+2p=g+1=4$ would not be strong enough for this purpose!) If the auxiliary curve are of order $3$, then we must have $3\cdot 4-3=9$ basepoints on the $C_4$. No basic corruption is detected? \(3) Another argument via theta-characteristics is implicit in Klein 1892 [@Klein_1892_Realitaet] (see also his Göttingen lectures 1891/92 [@Klein_1892_Vorlesung-Goettingen]) and appears in modernized form in Gross-Harris 1981 [@Gross-Harris_1981]. \(4) Another more elementary (and purely topological) proof follows from Rohlin’s formula 1974–78 (valid for dividing curves), cf. Sec.\[Rohlin-formula:sec\]. This formula reads $2(\pi-\eta)=r-k^2$, where $\pi$, $\eta$ are the number of positive, resp. negative pairs of ovals. This distinction appears by comparing the orientation induced as boundary of the half of the Riemann surface underlying the curve, with that of the annuli bounding a pair of nested ovals. In our case there is no nesting hence $\pi=\eta=0$, and so $r=k^2=4$ violating the assumption $r=2$. \(5) A related proof involves Arnold congruence 1971 [@Arnold_1971/72] for $M$-curves of degree $2k$ (with an obvious extension to dividing curves in Wilson 1978 [@Wilson_1978]). This reads $\chi:=p-n \equiv k^2 \pmod 4$ and suffices. Here $p, n$ are notation coined in Petrovskii 1938 [@Petrowsky_1938 p.190] for positive and negative ovals, also interpretable as the number of even and odd ovals. An oval is said to be [*even*]{} if it is lying within an even number of consecutive ovals. For the case at hand (2 unnested ovals), both are even (being subsumed to zero ovals), hence $p-n=2-0\equiv k^2=4 \pmod 4$ is violated, and the nondividing character of the curve follows. The difference $p-n$ is readily interpreted as the Euler characteristic $\chi$ of the Ragsdale membrane bounding (orientably) the curve from inside. In the case at hand, the Ragsdale membrane is the disjoint union of 2 discs, whence obviously $\chi=2$. Now using the theorem of Bézout, it is clear (cf. Zeuthen 1874) that our picture above (Fig.\[KleinRohlin-quartic:fig\]) exhaust all possible shapes traced by quartics. For instance a such cannot have 2 ovals nested in a third one, etc. So it is a simple matter to convince that we have listed all real schemes of quartics (with all of them safe the empty curve $x^4+y^4=-1$ arising through small perturbation of 2 ellipses). Of course a priori a quartic could have 5 ovals but this was precluded by Harnack 1876 [@Harnack_1876], and of course already by Zeuthen 1874 [@Zeuthen_1874 p.411] using a prototype of Harnack’s device. Indeed if a quartic had 5 ovals (or more) the conic through them would cut it in $5\cdot 2=10>8=2\cdot 4$ overwhelming Bézout. At any rate Klein’s argument of 1876 via the underlying Riemann surface gives the general Harnack bound $r\le g+1$ in some more intrinsic fashion. Even stronger is the following result (due to Klein 1876 [@Klein_1876_Verlauf], though his proof makes a détour through surfaces and it could be interesting to find a more direct argument). Perhaps the transition through cubic surfaces is necessary as it rationalize the irrationality of quartics curves, though Klein in 1876 seems to have add a direct argument staying in the realm of curves, but he did not exposed details. The real scheme is a complete invariant for rigid-isotopy classes of quartics. (Rigid isotopy refers to the morcellation of the space of all curves of some fixed order effected by the discriminant hypersurface parametrizing singular curves.) Modulo such knowledge one can draw the lattice of all real schemes (right part of Fig.\[KleinRohlin-quartic:fig\]) on which the (Klein-)Rohlin intuition is verified: a real scheme is dividing (or of type I) iff it is maximal. \[28.03.13\] In fact it is tempting to make a baby Gudkov table in degree 4 (inspired by the case of degree 6, cf. Fig.\[Gudkov-Table3:fig\]) as to visualize the situation. Here the Gudkov symbol $\frac{x}{1}y$ is merely a symbolical way to mean that $x$ ovals are nested in a big oval (the denominator $1$), while $y$ ovals are lying outside. It is noteworthy that Rohlin’s maximality principle is fully validated here and prohibits all the schemes lying above the configuration $\frac{1}{1}$ of the nest of depth 2, which is already Bézout-saturated. It is also pleasant to notice the presence already of the highbrow Gudkov-Rohlin sawtooth (dashed on the picture) so typical of the solution of Hilbert’s 16th in degree 6 (cf. again Fig.\[Gudkov-Table3:fig\]). This extends to all degrees by the congruences of Gudkov-Rohlin $\chi\equiv_8 k^2=4$ for $M$-curves (\[Gudkov-hypothesis:thm\]), Gudkov-Krakhnov-Kharlamov for $(M-1)$-curve and Kharlamov-Marin for $(M-2)$-curves of type I). This sawtooth, which looks like a piecewise linear sine-curve, forces the scheme below its depressions, to be of type I. Of course it tends to pass unnoticed here ($m=4$) as it is such a trivial consequence of Bézout with lines. -5pt0 -5pt0 As already announced we conjecture in general that the whole sawtooth can be explained by invoking the phenomenon of total reality for $(M-2)$-curves via adjoint curves of order $(m-3)$. At this stage a comparison with degree 6 (Fig.\[Gudkov-Table3:fig\]) makes it puzzling to wonder if all schemes lying below the sawtooth are always realized, yielding a sort of denseness below the sawtooth (alias Gudkov line). The answer is no in degree 8 (cf. Fig.\[Degree8:fig\]), where Petrovskii’s estimate of 1933/38 (\[Petrovskii’s-inequalities:thm\]) starts to act prohibitively. Yet taking this and other conjectural hypothesis of Ragsdale into account, one can still wonder about the question of denseness of schemes, namely the issue as to whether prohibition are essentially confined to the high level of the pyramid (i.e. above $(M-2)$-curves), or if in contrast there is some sort of porism (or lacunae) killing schemes at low altitudes. As we shall see latter if the conjectural maximality principle of Rohlin as well as our stability of type I under satellites holds true, it is likely that for high degree $m$ (especially when the integer $m=2k$ as a rich factorization into primes) then there will be a myriad of cone-like [*no man’s land*]{} zone where schemes are killed because they extend a Bézout-Ahlfors-Rohlin saturated scheme subsumed to the paradigm of total reality. Of course the situation of low degrees $m=4,6$ may give the wrong impression that the whole paradigm of obstruction by the saturation allied to total reality are already explained by the trinity of Russian congruences mod 8 (of all the workers already cited starting with Gudkov-Arnold-Rohlin). Yet in reality this is not even true for low degrees because under the disguise of Bézout it is already total reality which assures the planar character of the lowest Gudkov tables $m\le 6$. Otherwise we had to consider a menagerie of other schemes with more nested structures. So this gives some intuition a priori that Rohlin’s maximality principle (in our opinion much allied to Ahlfors) will not be subsumed to the trinity of congruence mod 8, albeit the lowest of it pertaining to $(M-2)$-curves may act as vivid generator of total reality phenomena. Quintics (Klein 1892?, Rohlin-Mishachev 1976, Fiedler 78, Marin 79, Gross-Harris 1981) {#quintic-table-Klein-Gudkov:sec} -------------------------------------------------------------------------------------- \[04.04.13\] This survey has some repugnancy against curves of odd order for reasons hard-to-explain, perhaps allied to the cumbersomeness of the avatar of Rohlin’s formula (due to Mishachev). However the theory especially our main focus of total pencils works as well in this case. Let us take a small look at the “Gudkov-Rohlin” table in degree $m=5$. We recommend however to skip this section on first reading as our understanding is lacunary (in part because we do not discuss Mishachev’s formula, or because we do not entered into the Klein-Gross-Harris theory of real theta characteristics). Yet, the case of quintics and more generally curves of odd degrees (especially those of the shape $m=5+4n$, else Klein’s congruence suffices) offer a pleasant application of the Klein-Marin principle, when it comes to check that the scheme with only one circuit is of type II (see Lemmas \[Klein-Marin-quintic:lem\] and \[Klein-Marin-odd-degree:lem\]). First, in degree $m=5$, Harnack’s bound is $M=g+1=7$, since $g=\frac{(m-1)(m-2)}{2}=(4\cdot 3)/2=6$. In odd degrees there is always a unique pseudoline (Möbius 18XX [@Moebius_18XX], von Staudt 18XX [@von-Staudt_18XX], Zeuthen 1874 [@Zeuthen_1874], Harnack 1876 [@Harnack_1876], Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege], etc.), and we may omit it from the Gudkov symbols. Hence on the table below (Fig.\[Gudkov-Table-quintic:fig\]) we suppress the pseudoline from the real scheme depiction. As usual one of the most noteworthy configuration is the deep nest, here $\frac{1}{1}=(1,1)$ which is total under a pencil of lines, hence of type I. [*Bibliographical puzzle*]{}.—This argument looks to us much more elementary than the one of Gross-Harris 1981 [@Gross-Harris_1981 p.175] via theta-characteristics which goes back probably to Klein 1892. Despite its extreme elementariness, it seems also to have escaped Klein’s attention. In fact Klein uses it in 1892 (p.177 of Ges. Math. Abh, II) but only after contracting the empty oval of the Gürtelkurve. We cite the relevant passage: Sollen wir diese geometrischen Verhältnisse durch Beispiele belegen, so nehmen wir vielleicht zunächst den Fall der Gürtelkurve $p=3$. Hier hat es ersichtlich keine Schwierigkeit, das innere Oval auf einen Punkt zusammenzuziehen. Von diesem aus projizieren wir jetzt die Kurve auf eine gerade Linie. Die Gerade wird dann nach ihrer ganzen Erstreckung von den Bildpunkten doppelt überdeckt, so zwar, da[ß]{} dabei kein reeller “Scheitel” auftritt[^20]. Das entspricht in der Tat dem orthosymmetrischen Falle $\lambda=1$ des Geschlechtes $p=2$. Later, the argument of total reality seems to have escaped the attention of another great master, namely Alexis Marin 1979 [@Marin_1979], compare especially on p.56 his complicated argument for “N’existe pas” in the bottom-right angle of the tabulation, as well as the question p.59: “Est-ce qu’une courbe ayant cette disposition sépare sa complexifiée.” This total reality (or Bézout-saturation) kills all schemes enlarging it (cf. unframed white-colore schemes on Fig.\[Gudkov-Table-quintic:fig\]). Apart from this obstruction there is essentially no other. First we can construct an $M$-curve necessarily unnested (by Bézout-saturation) with symbol $6$ (again the pseudoline $J$ is omitted). This is constructed via a Hilbert-method on Fig.\[Gudkov-Table-quintic:fig\]a (for a less schematic picture cf. Fig.\[Harna0:fig\]). We presume that nobody knew existence of such a curve prior to Harnack 1876 (but this is just a historical challenge, perhaps Plücker, Zeuthen, Klein before but not sure). -10pt0 Next Fig.b. shows a quintic with 5 ovals (and one pseudoline), which arises by slight perturbation of 2 ellipses plus a line. This was probably known to Plücker 1839 [@Plücker_1839], or earlier. The curve constructed is of type II by Fiedler’s signs law, or just by Klein’s congruence $r\equiv_2 g+1$. The latter forces actually all the schemes $5,3,1$ being of type II. Below we have the scheme $4$. This admits realizations of both types (as knew Rohlin 1978, and perhaps Klein?), as shown by Fig.c and Fig.d (using Fiedler’s smoothing law). The curve of Fig.c has actually a total pencil of conics assigned to pass through the 4 ovals (which was depicted earlier in this text). Again it is interesting to note that Gross-Harris (p.176–177) used a somewhat more synthetic and complicated argument than just Fiedler’s law, to show existence of curves in both types I/II. [*Historical note*]{}.—The above phenomenon is the first instance of where the type of a curve is not determined by the distribution of ovals. It admits as a simple consequence the fact that there exists obstructions to rigid-isotopy lying beyond the real scheme (remark due to Rohlin 1978). It is clear however that Klein knew (or at least suspected) this basic phenomenon, compare his footnote of 1922: \[Inserted 05.04.13\].—In fact this can be interpreted either à la Rohlin, by saying that a real scheme can have realizations in both types (I/II=ortho- or diasymmetric). Somewhat more crazy would be the following interpretation. \[Klein-bipolarity:conj\] [(Kleinian bipolarity—Klein 1922, Gabard 2013)]{}.—An abstract symmetric Riemann surface (SRS) can admit plane realizations with distinct distributions of ovals. Actually I do not know if this phenomenon of “bipolarity” can occur. Of course it does trivially occur, with a line and a conic both representing the Riemann sphere with its standard real structure (equatorial involution). In degree 3, it cannot occur since the real scheme determines the type, and likewise in degree 4 (cf. previous section). In degree 5, predestination of the type by the real scheme is not true any more, yet the combinatorics of Fig.\[Gudkov-Table-quintic:fig\] is sufficiently simple as to preclude bipolarity. Indeed if the SRS is fixed, hence in particular the number $r$ of real circuits, the only height at which there are several distributions of ovals is $r=3$, where we have the nested ($\frac{1}{1}$) and the unnested ($2$) schemes. Yet both of them are differentiated by the type. Hence the first place to look for bipolarity is degree 6. Here we have (see the Gudkov-Rohlin table Fig.\[Gudkov-Table3:fig\]) a myriad of sextics having the same underlying (topological) symmetric surface. It is unclear if they can be conformally equivalent while exhibiting different distributions of ovals. It could be imagined that a Hilbert sextic is sometimes conformally diffeomorphic to one of Harnack, or even Gudkov. This question looks a bit artificial or puzzling, yet has perhaps of some importance if one likes to link with the abstract theory of Ahlfors taking into account only the abstract Riemann surface. To settle the bipolarity question we can look at the natural map from the hyperspace of smooth plane curves to the real moduli space of SRS, i.e. $\mH-\disc \to M_g$. Perhaps then two chambers may have overlapping images. As noted by the old Felix Klein (aged 73 at the moment of his 1922 footnote) plane curves have specialized moduli. Hence the images in question are fairly small subloci of the moduli space, but this does not prevent overlap. Another approach is to use Cremona transformations of the plane defined over $\RR$ which do not induce diffeomorphisms of the plane $\RR P^2$ (this remembers works by Ronga-Vust ca. 2002, or their student J. Blanc). By this procedure we can perhaps alter the distribution of ovals, yet without distorting the conformal structure, as the curve and its image are in birational equivalence. Can this vague idea be implemented? Otherwise the approach can be the Teichmüller-theory of the map described above from concrete plane curves to the moduli space of abstract real(=symmetric) Riemann surfaces, while trying to study exactly the coincidences of this mapping. One could try to determine exactly which among the 64 chambers of sextics (Nikulin’s theorem (\[Nikulin:thm\])) are in bipolarity, i.e. contains conformal replicas of the same symmetric Riemann surface. This defines an additional graph structure on the Gudkov-Rohlin table where edges are traced whenever two vertices(=chambers) contains curves abstractly isomorphic over $\RR$. Of course the edges have to preserve the height $r$ on the Gudkov pyramid, as well as the types. Those (topological) obstructions to bipolarity could be the sole ones, in which case the graph in question would show plenty of edges. In particular it restricts to the complete graph on each levels at height $r$ not congruent to $g+1 \pmod 2$ where diasymmetry reigns ubiquitously (Klein’s congruence). Few other levels have pure types too, e.g. $M$-curves (type I only), and the level $r=3$ (types II only) (via Rohlin’s formula), see again Fig.\[Gudkov-Table3:fig\]. At those levels the bipolar graph could be complete too. After this digression, we return to the basic classification of quintics. The scheme $3$ is of type II by Klein’s congruence (as we already noted), and exists as shown by Fig.e. Next the scheme 2 exists in type II as shown by Fig.f. It is however more tricky to prove that the scheme $2$ is of type II. This follows either from the avatar in odd degree of Rohlin’s formula (i.e. Mishachev’s formula). Perhaps there is a more elementary argument, say by using a pencil of lines while trying to permute 2 imaginary points during a sweeping. Also Gross-Harris have probably an argument via theta-characteristics, but alas those authors confess being not able to prove this compare p.175 of Gross-Harris 1981 [@Gross-Harris_1981] where we read: “In the non-nested case, we suspect that $a(X)$ is always 1 \[i.e. type II, or nondividing\] but have no proof”. Was Felix Klein (1892 paper [@Klein_1892_Realitaet] or lectures [@Klein_1891--92_Vorlesung-Goettingen]) able to tackle this case? A crude principle that do this work is the postulation that whenever we add a pseudoline to a scheme of type II, it remains of type II. Recall that we know since Klein that the quartic scheme $2$ is of type II, cf. Lemma \[Klein-unnested-quartic-nondividing:lem\]. Some evidence comes from surgeries on the Riemann surface while noticing that diasymmetry is a dominating character in the genetical sense. This is implicit in Fiedler’s law of smoothing and really a simple matter of visualizing the corresponding Riemann surfaces. So let us posit the: \[toutou:conj\] [(Gabard 2013, but probably standard by Rohlin-Fiedler, if not erroneous)]{}.—When a scheme of even degree $2k$ is of type II, then the same scheme of degree $2k+1$ augmented by a pseudoline is of type II too. (Il y a trop de toutous dans la langue anglaise, mon ostie!) Proving this could again involve a large deformation principle (as discussed in the sequel) like minimizing the length of the pseudoline as to make it a line. There will then be a strangulation of the Riemann surface and we are reduced to Fiedler’s genetic law. Perhaps there is an elementary proof of the conjecture based on a conjunction of Rohlin’s and Mishachev’s formulae. Note also that the conjecture holds true for the empty scheme by Lemma \[Klein-Marin-odd-degree:lem\] below. Another idea to show that the quintic (unnested) scheme $2$ is of type II could be to use Klein’s 1876 remark that a curve of type I cannot acquire a solitary node, cf. below for a proof (\[Klein-Marin:lem\]) essentially along the lines of Marin 1988. Then we are reduced to showing that any quintic with scheme 2 (again we omit the pseudoline $J$) can indeed acquire a new oval. This looks a priori hard, but in view of the diagrammatic of the table Fig.\[Gudkov-Table-quintic:fig\] (mostly prompted by Bézout) we could just make a deformation along a pencil spanned by the curve plus a curve with more ovals (e.g. Harnack’s or just $5$ of Fig.b). The difficulty however is that the deformation is not forced to raise immediately the number of ovals, as it may first lower down the number of ovals. Incidentally if this argument via Klein-Marin would have worked it would also have prohibited the type I realization of the scheme 4. Next we have the scheme $1$ forced to be of type I, by Klein’s congruence, and easily constructed (e.g. by a slight alteration of the picture Fig.\[Gudkov-Table-quintic:fig\]f above). Finally, the scheme $0$ poses again a little problem, but can also be shown to be of type II. This follows either from Rohlin-Mishachev, or via theta-characteristics. In this case Gross-Harris 1981 [@Gross-Harris_1981 p.175] were able to conclude type II via theta-characteristics (see their proof of Prop.7.1, p.173, which contains some minor misprints, namely “Prop.4.1” should be “Prop.5.1”, and “$h^0({\bf a})=(d^2-1)/2$” should be “$h^0({\bf a})=(d^2-1)/8$”). A somewhat more conceptual argument can be based on Klein’s Thesis (as Viro calls it) of 1876 to the effect that [*a curve of type I cannot gain an oval (at least when crossing a solitary node)*]{}. This was perhaps historically the first known proof, albeit Klein did not mentioned this consequence explicitly in print (1876 paper, nor later). On writing down the proof below, we realized that one needs the stronger version due to Marin 1988 of Klein’s Thesis relaxing the parenthetical proviso above. Hence our claim of historical priority is somewhat sloppy, but in substance Klein could have anticipated it. \[Klein-Marin-quintic:lem\] [($\approx$Klein 1876, 1892, Rohlin-Mishashev ca. 1974–76, Gross-Harris 1981, Marin 1988, Gabard 2013 trying to assembly all this today)]{}.—Any quintic with only one pseudo-line is necessarily of type II (i.e. nondividing or diasymmetric). Take such a curve $C_5$ (with only a pseudoline) and any auxiliary (smooth) curve with at least one oval (and so $r\ge 2$). Pass a line through both curves (in the hyperspace of curves) and perturb it slightly to ensure transversality w.r.t. the discriminant. Since the initial curve $C_5$ has the least possible number of real circuit (namely one), the first contact (along one of the 2 possible pathes inside the pencil) with the discriminant will be a “Morse” surgery (jargon Thom-Milnor) [*forced to increase the number of ovals*]{} ($\bigstar$). This last (italicized) assertion ($\bigstar$) requires perhaps more substantiation. Let us admit it to conclude quickly. If the new oval raises from a solitary-node then Klein’s Thesis of 1876 (alas left unproven by the great geometer) suffices to conclude. If not, e.g. if the pseudoline self-collides with itself as to split off a new oval (Fig.\[Eversionpseudo:fig\]a), then Marin’s version of Klein completes the proof, cf. (\[Klein-Marin:lem\]) or Marin 1988 [@Marin_1988]. To justify better ($\bigstar$) we should check that all Morse surgeries of a pseudoline forces an augmentation of the number of circuits. In the case of an oval this is not true due to “eversions” (cf. Sec.\[Eversion:sec\] especially Fig.\[Eversion:fig\]), whence our extreme prudence. However doing naive experimental pictures deforming a pseudoline, it seems impossible to evert a pseudoline (Fig.\[Eversionpseudo:fig\]b). It remains of course to find a theoretical explanation. -5pt0 -5pt0 It is clear that the above lemma extends to all other odd degrees: \[Klein-Marin-odd-degree:lem\] Any curve with a unique real circuit is of type II, safe if it is a line or a conic (degree $m=1,2$). [*Remark*]{}.—The argument of (Klein-)Gross-Harris only works under the (extraneous and stringent) assumption $m \equiv 5 \pmod 8$ (cf. their Prop.7.1, p.173). The case of odd degrees follows by the same method using the Klein-Marin theorem. The case $m=3$ is of course more elementary and can be reduced to the uniformization of elliptic curves e.g. à la Weierstrass via the doubly-periodic $\wp$-function defined on a rhombic lattice. Sorry, it suffices actually to use Klein’s congruence, or to remember—if you do not want to sell your soul to the devil of arithmetics—that a symmetric torus with one fixed circuit is forced to be $S^1\times S^1$ acted upon by exchange of both factors (while fixing the diagonal circle). Actually Klein’s congruence $r\equiv_2 g+1$ settles the lemma whenever $m=3+4n$. Indeed then $g=\frac{(m-1)(m-2)}{2}=\frac{(2+4n)(1+4n)}{2}=(1+2n)(1+4n)=1+6n+8n^2$, which is odd, and so Klein’s congruence (forced by type I) is corrupted, whence type II. For the other cases $m=1+4n$, Klein’s congruence tells nothing and one make appeal to the Klein-Marin argument instead. For even degrees, one can again treat half of the cases via Klein’s congruence, namely when $m=4, 8, 12, \dots$, i.e. $m=4n$ as then $g=\frac{(4n-1)(4n-2)}{2}=(4n-1)(2n-1)$ which is odd, and so Klein congruence is violated for $r=1$. For the other cases $m=2+4n$, the congruence tells nothing. However we can still conclude type II (of course provided $m\ge 4$), either via Rohlin’s formula (\[Rohlin-formula:thm\]) or maybe a variant of the Klein-Marin argument. However now the configuration with one circuit has not the minimal number of circuits and so we may first descend to the empty chamber and the Klein-Marin method looks impuissant. Of course perhaps some extra trick can ensure that we can increase the number of component immediately yet I do not see any obvious argument. The impressive landscape of all sextics (Harnack 1876, Hilbert 1891/00/09, Rohn 1911/13, Petrowskii 1933/38, Gudkov 1948/54/69, Arnold 1971, Rohlin 1972/74/78) --------------------------------------------------------------------------------------------------------------------------------------------------------------- \[31.12.12\] Perhaps the Klein-Rohlin conjecture follows from Ahlfors theorem interpreted in terms of total reality. Intuitively having a total pencil, no real circuit can be added without corrupting Bézout (more on this in Sec.\[Rohlin-via-Ahlfors\]). Yet perhaps this is too naive as shown by an example of order 6 to be found in Gabard’s Thesis 2004 [@Gabard_2004 p.8] (I should acknowledge Kalla-Klein 2012 [@Kalla-Klein_2012-Computation-cite-Gabard] for reminding me that my Thesis contained this example). -5pt0 -5pt0 This shows that the real scheme alone fails to determine the dividing character (alias type I=erster Art in Klein 1876 [@Klein_1876]). At first I thought this corrupts Rohlin’s assertion that his conjecture is true in degree 6. Of course Rohlin 1978 [@Rohlin_1978] knew very much this phenomenon, which he calls “real schemes of indefinite type” (on p.94 of ), i.e. real schemes admitting representatives of both types (dividing or not). Hence it was first puzzling to wonder why he made such a basic mistake, or more likely why we first failed to interpret correctly his simple message? Understanding the Klein-Rohlin conjecture requires some more mature thinking. One should list all schemes dominating the “nine unnested ovals” scheme. On p.95 Rohlin 1978 [@Rohlin_1978] refers to the census (tabulation) set up by Gudkov 1974 [@Gudkov_1974/74 p.40] listing all the logically possible (real schemes of) sextic curves (taking into account Bézout for lines). This is worth reproducing as Fig.\[Gudkov-Table3:fig\]. Recall that in Gudkov’s symbolism, $\frac{x}{1} y$ denotes the scheme consisting of $x$ ovals enclosed by one “big” oval, while there is $y$ ovals living outside. This gives a total of $1+2+3+\dots+11=\frac{12 \cdot 11}{2}=6\cdot 11=66$ logically possible curves (counting inside the “triangle”), to which must be added the empty real scheme (denoted $0$) and the deep nest of depth $3$ (denoted $(1,1,1)$ or $\frac{1}{{\frac{1}{1}}}$). We get so the 68 schemes ([*année érotique*]{}) mentioned by Gudkov (p.40). -5pt0 -5pt0 Not all of those 68 schemes are actually realized. If they would this would roughly mean that all obstructions are Bézout-like prompted by tracing a single line. However the plane is swept out by a myriad of other curves. Quite eclectically, the architecture of Hilbert-Gudkov’s table of elements is a bit like a pharaohs pyramid (turn Fig.\[Gudkov-Table3:fig\] upsidedown) and those are known to have sanctuary galleries forming tunnels. The first to have spotted this porousness of the pyramid is no less an authority than Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege], albeit it took several decades until his work got consolidated (especially by Rohn 1911, Petrovskii 1933/38) and pushed forward to its ultimate perfection (thanks to the efforts of Gudkov). Soon afterwards, Arnold and Rohlin offered quite dramatic simplifications based on pure topology, and extensions of the prohibitions to all degrees. We suspect this Hilbert-Gudkov pyramid to have some overlap—not yet much elucidated except perhaps for allusions in Rohlin 1978 [@Rohlin_1978 p.94]—with some more ancient force, namely Abel-Riemann-Klein-Teichmüller-Ahlfors and their circle maps. The latter are of course specific to dividing curves concomitant with the paradigm of total reality. In the case at hand (real plane smooth sextics), total reality is exhibited according to a Rohlin’s claim 1978 (not yet fully understood by the writer) via total pencil of cubics for certain schemes, e.g. $\frac{6}{1}2$ and its mirror $\frac{2}{1}6$. More trivial is the nest of depth $3$, totally real under a pencil of lines. This can be interpreted as a prohibition of diasymmetry for those schemes. Likewise the porous portion of Gudkov’s pyramid (=white cases above the broken line on Fig.\[Gudkov-Table3:fig\]) can be prohibited (and this is how I understand vaguely the Hilbert-Rohn method) by pure synthetic geometry. Paraphrasing, not merely linear Bézout obstructions do exist, but also those via the menagerie of all other curves grooving nonlinearly the plane. More than that, not just static curves but dynamical collections of such (e.g., pencils) have to be considered. It is charming to note a strong parallel between Hilbert’s and Rohlin’s claims that pure geometry is able to prohibit schemes, especially as both look insufficiently justified, but intuitively plausible. How much Rohlin’s synthetic proof of the type I of the schemes $\frac{6}{1}2$ and its mirror has in common with Hilbert-Rohn’s method? A last word of caution for pyramids builders: one of the first ever constructed in Ancient Egypt had a somewhat pathetic destiny. Once arriving near the $2/3$ of the planned final size, fissures started to appear menacing the whole foundations to crack under pressure. The only reasonable option left to the engineers was to diminish the slope for the last third as to lower pressure. It is not known if this sufficed to ensure immortality of the Pharaoh. Thus, it should be no surprise that the most telluric part of the pyramid (near the funerary chamber where the pressure is highest) is the most secrete part of the edifice. This needed to wait the contribution of Gudkov 1969 [@Gudkov-Utkin_1969/78] who exhibited the most elusive schemes $\frac{5}{1}5$ of Fig.\[Gudkov-Table3:fig\] (cf. also Fig.\[GudkovCampo-5-15:fig\] for the explicit construction). At this stage Hilbert’s 16th problem was completely solved (at least for sextics which is arguably the official context of Hilbert’s question). Now let us be more formal. As explained by Gudkov (1974 ), Kahn 1909 [@Kahn_1909] and Löbenstein 1910 [@Löbenstein_1910] published dissertations under Hilbert’s direction—(if I understood well the story, vgl. Hilbert 1909 [@Hilbert_1909-Ueber-die-Gestalt-sextic], both were feminine candidates)—attempting to prohibit sextics with 11 unnested ovals. (Challenge: try to prove this via Ahlfors 1950 or rather via Bieberbach-Grunsky (1925/1937). Philosophically, this would just, as it should, put Little Hilbert in the baskets of Big Riemann!) This follows also from Rohlin’s formula of 1974–78, cf. Sec.\[Rohlin-formula:sec\], or from Arnold’s congruence of 1971. Soon later Rohn 1911–1913 [@Rohn_1913] devoted two articles attempting by the same method to exclude sextics of type $\frac{10}{1}$ or $11$, making a big contribution to the development of Hilbert’s idea. The resulting prohibition method was christened by Gudkov (1974 ) the [*Hilbert-Rohn method*]{}. In Gudkov’s view, even Rohn’s proof is not perfectly sound due to some messy combinatorics impeding Rohn to take care of all logically possible cases. Gudkov then mentions several more Western attempts, by Wright 1907 (same idea as Hilbert, but not rigorous prohibition of type $11$). In Donald 1927, the same non-rigorous attempt is repeated (apparently without knowledge of Kahn, Löbenstein or Rohn’s work). Hilton 1936 devoted a paper criticizing Donald’s work. The next step is essentially Gudkov’s work (yet do not miss what did Petrovskii 1933/38 though its impact upon the case of sextics is nearly covered by Hilbert-Rohn). Ultimately Gudkov was able to prohibit in 1969 and probably much earlier (Gudkov-Utkin 1969 [@Gudkov-Utkin_1969/78]) all schemes above the broken line of Fig.\[Gudkov-Table3:fig\]. This breakthrough originated in 1948 when Andronov suggested (to Gudkov) applying the concept of [*roughness*]{} (also known later as [*structural stability*]{} in the West since Lefschetz, and adhered to by Thom, etc.) to the topology of real algebraic surfaces. Petrovskii’s advice (1950) suggested focusing rather on the case of sextic curves. Combining those novel Russian ideas with the Hilbert-Rohn method, enabled Gudkov in 1954 [@Gudkov_1954] to get solid prohibitive proofs above the critical line, and even beyond (sic! cf. lilac schemes $\frac{5}{1}y$, $3\le y \le 5$ on Fig.\[Gudkov-Table3:fig\]) but that turned out to be too massive amputation for the pyramid to support its own structural mass. Nowadays there are simpler proofs from the Arnold-Rohlin era (early 1970’s) or via Rohlin’s complex orientation formula 1974–78 which prohibit only a portion of those (namely those [*not*]{} lying on the continuation of the lattice by blue rhombs and red circles on Fig.\[Gudkov-Table3:fig\]). However Rohlin’s proof (1972/72 [@Rohlin_1972/72-Proof-of-a-conj-of-Gudkov]) of the Gudkov hypothesis inhibits all $M$-schemes above the broken line, but the price to pay is highbrow differential topology à la Rohlin from the early 1950’s (cf. Sec.\[Gudkov-hypothesis:sec\]). Related work by Gudkov-Krakhnov/Kharlamov prohibits all the four $(M-1)$-schemes above the broken line. What happens under the critical line? Short-cutting a century of efforts, the answer is rapid: all of them are realized. In fact all specimens (except the 3 lilac-colored ones) are easily construct by (slight variants) of Harnack’s and Hilbert’s method. At least this is what we read in Gudkov’s survey 1974 [@Gudkov_1974/74], yet the cases of $\frac{4}{1}5$, $\frac{3}{1}5$ are a bit tricky (but see our Fig.\[HarnaGudkov4-15:fig\] and Fig.\[HarnaGudkov3-15XXL:fig\] resp.). The three remaining ones $\frac{5}{1}5$, $\frac{5}{1}4$, $\frac{5}{1}3$ needed to wait until Gudkov’s trick (1969–1973) of using some Cremona transformations. Beware yet that historically, the very first argument of Gudkov’s Thesis 1969 was a pure existence proof along the line of Hilbert-Rohn’s method (ca. 20 pages long and extremely hard-to-follow according to Russian experts, cf. Polotovskii 1996 [@Polotovskii_1996-D-A-Gudkov], Viro, etc.), and was not constructive at all. Remind also that Gudkov himself at some early stage, in 1954, asserted incorrectly inexistence of those 3 difficult birds, quite in line with Hilbert’s intuition at the Paris Congress of 1900. As like to emphasize Arnold, it seems that Petrovskii himself was at first very skeptical about the twist taken by Gudkov’s solution. Harnack’s and Hilbert’s constructions ------------------------------------- First Fig.\[Harnack-original:fig\](left) recalls Harnack’s method of construction (trying to keep reasonably close to the original 1876 [@Harnack_1876 p.195], but making more explicit pictures (assisted by Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical p.188]). The top-right of Fig.\[Harnack-original:fig\] reproduces Hilbert’s more expeditious way to realize this scheme. The bottom-right is the new scheme discovered by Hilbert in 1891 [@Hilbert_1891_U-die-rellen-Zuege] (yet no pictures until Hilbert 1909 [@Hilbert_1909-Ueber-die-Gestalt-sextic], who traces only the top-right picture, whose scheme is Harnack’s). If The bottom nice picture is borrowed from A’Campo 1979 [@A'Campo_1979 p.08–09]. -2.3cm0 -5pt0 -5pt0 Once all the knowledge synthesized in Gudkov’s table (Fig.\[Gudkov-Table3:fig\]) is understood (or admitted for short) one starts appreciating Rohlin’s maximality claim. Indeed having listed all real schemes (there remains $68-12=56$ many below the broken line) it is an easy matter to spot maximal elements in the lattice. We find 6 types red-circled on Fig.\[Gudkov-Table3:fig\] corresponding to $\bullet$ the three $M$-schemes $\frac{9}{1}1$, $\frac{5}{1}5$, $\frac{1}{1}9$ of Hilbert, Gudkov, Harnack respectively, $\bullet$ plus two $(M-2)$-schemes namely $\frac{6}{1}2$ and $\frac{2}{1}6$, $\bullet$ and finally one $3$-schemes $(1,1,1)$ corresponding to the deep nest. Rohlin’s assertion is that those (distinguished) 6 schemes are precisely those which are definite of type I (i.e. universally orthosymmetric). Of course the assertion is trivial for the 3 possible $M$-schemes (since Harnack 1876 or via Klein’s 1876 intrinsic proof of Harnack’s inequality via the topology of Riemann surfaces). The deep nest of weight 3 is likewise trivially of type I, for it is enough to sweep it out by a total pencil of lines. It remains thus to analyze the two $(M-2)$-schemes with $r=9$, i.e. $\frac{6}{1}2$ and $\frac{2}{1}6$ by showing that they are definite of type I. At first one can imagine to prove this via pencil of conics (or maybe cubics pencils?). A complete argument must be given in Rohlin. [*Insertion.*]{}—\[28.03.13\] In fact Rohlin claimed a proof which is now lost via pencils of cubics, so that there is strictly speaking presently still only one known proof which involves a congruence modulo 8 due to Rohlin-Kharlamov-Marin (\[RKM-congruence-reformulated:thm\]). Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics] was able to validate the total reality assertion of Rohlin, yet only after supposing the curve dividing. It should however not be impossible that methods of Le Touzé suitably modified establish the full Rohlin claim. This seems to be an urgent problem to deal with. We personally tried a lot but failed dramatically. This sort of problem seems to require extreme cleverness. Rohlin 1978 [@Rohlin_1978] states the following theorem summarizing all those efforts (up to Gudkov plus his own input reconciliating with Klein’s viewpoints): \[Rohlin-type:thm\] [(Rohlin 1978)]{} The $56$ possible real schemes for sextics (Harnack, Hilbert, Rohn, Gudkov) split as follows according to Klein’s types: $\bullet$ There are $6$ schemes of type I (red-circles on Gudkov’s table=Fig.\[Gudkov-Table3:fig\]); $\bullet$ There are $42$ schemes of type II (green-squares on Gudkov’s table=Fig.\[Gudkov-Table3:fig\]); $\bullet$ There are $8$ schemes of indefinite(=mixed) type (blue-rhombs on Gudkov’s table=Fig.\[Gudkov-Table3:fig\]). Note that the resulting distribution of types to be nearly symmetric (on Fig.\[Gudkov-Table3:fig\]), modulo some anomaly at the place $5$ (five unnested ovals). A similar remark is made in Fiedler 1981 [@Fiedler_1981 p.13]: “[*Bemerkung. Eventuell ist die Tabelle nicht vollständig. Aber es ist schon ersichtlich, da[ß]{} die Tabelle der zerteilenden Kurven im Unterschied zur Tabelle aller existierender Typen von singularitätfreien Kurven sechster Ordnung (vgl. [\[1\](=Gudkov 1974 [@Gudkov_1974/74])]{}) nicht symmetrisch ist.*]{}”. Indeed the asymmetry we (and Fiedler) notice (but of course implicit in Rohlin’s survey) is the (unique) symmetry breaking occurring between the schemes $\frac{4}{1}$ and the scheme ${5}$. The latter turns out to be of type II, as it cannot satisfy Rohlin’s formula (\[Rohlin-formula:thm\]), whereas the former scheme is easily seen to be indefinite (cf. Fig.\[R4-1:fig\]). Note that Arnold’s congruence $5-0=p-n=k^2 \pmod 4=3^2=9=5 \pmod 4$ is not fine enough to detect this break of symmetry. Let us try to understand this spectacular statement of Rohlin (\[Rohlin-type:thm\]). \(1) From the easy Klein congruence $r\equiv g+1 \pmod 2$ if type I, we draw that all schemes with an even number $r$ of ovals belong to type II (this explains all the green-squares at heights $r=0,2,4,6,8,10$, cf. Fig.\[Gudkov-Table3:fig\]). \(2) As already explained all $M$-schemes (here $r=11$) are trivially of type I (Harnack’s inequality or Klein’s argument of 1876 [@Klein_1876]). For another reason the scheme $(1,1,1)$ (deep nest of profundity 3) is easily shown to be of type I (total pencil of lines). \(3) For similar reasons (but deeper) is the assertion that the 2 circled schemes with $r=9$ belongs to type I. This is truly the work of Rohlin, albeit philosophically akin to Klein-Teichmüller-Ahlfors’ total reality. [*Insertion*]{}—\[28.03.13\] As just said this is still unproven synthetically, and the only proof available involves deep differential topology (Rohlin-Kharlamov-Marin). \(4) Appurtenance to the indefinite type is usually easy requiring merely exhibiting two curves, one in each type. So for instance the scheme $9$ (consisting of $9$ outer ovals without nesting) is indefinite (cf. our Fig.\[KleinRo-sextic:fig\]). Of course here the basic theoretic tool is Fiedler’s observation that the type is governed by the smoothing effected in the Plücker-Klein-Brusotti method of small perturbation. Full details are worked out in Sec.\[indefinite-types:sec\]. \(5) Another piece of information (now purely Rohlinian) is Rohlin’s inequality $r\ge m/2$ for a smooth plane curve of degree $m$. (Remind this to follow for Rohlin’s formula, in turn derived by a intersection theory argument of halves of the dividing curve capped off by real ovals and brought into general position by perturbing via a vector field normal to the real locus). Conceptually this involves Poincaré homology theory, and the allied intersection theory (e.g. by Lefschetz, etc.). From Rohlin’s inequality, one deduces that the scheme with $r=1$ is of type II. \(6) Using the stronger Rohlin formula, one must be able to treat all schemes with $r=3$ to belong to type II (except the deep nest) and likewise assess type II for all other schemes. [*Insertion*]{} \[28.03.13\].—Yes this is essentially true. More precisely Rohlin’s formula admits the Arnold congruence as corollary (cf. \[Rohlin-implies-Arnold:lem\]), and the latter $\chi\equiv_4 k^2=9\equiv 1$, forces a curve of type I to live on the grid formed by blue rhombs (and red-circles) of Fig.\[Gudkov-Table3:fig\]. So Rohlin’s assertion is evident safe for the scheme $5$, $\frac{1}{1}1$ and $1$. But all those cases are prohibited by Rohlin’s formula $2(\pi-\eta)=r-k^2$. Indeed in case of no-nesting Rohlin’s formula reduces to $r=k^2=9$, hence rules out the schemes $1$ and $5$. For $\frac{1}{1}1$, we have only one pair so $\pi+\eta=1$, while Rohlin’s formula says $\pi-\eta=-3$, whence $2\pi=-2$, which is impossible as $\pi$ is a cardinal (namely the number of positive pairs). At this stage the proof of Rohlin’s theorem (\[Rohlin-type:thm\]) is complete. What can be proved via Ahlfors? ------------------------------- \[17.01.13\] From the viewpoint of our survey, it is of some interest to decide which results of the theory aroused from Hilbert’s 16th problem (Hilbert-Rohn, etc. and the Russian school Gudkov-Arnold-Rohlin, just to quote the 3 supermassive black holes) can be (re)proved via the Ahlfors map. As we said already all this section was actually motivated by the guess that Ahlfors could be used to prove the still unsettled Rohlin maximality conjecture (at least what remains thereof post Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]). However as yet we failed to complete this grandiose project. Another more didactic aspect (yet perhaps not to be neglected as a first step toward subsequent progresses) would be to see if Ahlfors implies the (Gudkov-)Arnold congruence mod 4: $\chi=p-n=k^2 \pmod 4$ for dividing curves of degree $2k$. [*Insertion*]{} \[02.04.13\].—This game looks quite artificial since Arnold’s congruence is, e.g., a fairly trivial consequence of Rohlin’s formula, cf. (\[Rohlin-implies-Arnold:lem\]). The method would be to examine the Ahlfors foliation, i.e. that induced by the total pencil of curves while trying to apply Poincaré(-Bendixson-Kneser-Hamburger) index formula for foliations. Recall that $\chi$ is the Euler characteristic of the “Ragsdale-Petrovskii” (orientable) membrane of ${\Bbb R}P^2$ bounding the ovals. Of course it looks hard for Ahlfors to beat the elegance of Arnold’s argument based on intersection theory and the divisibility by 8 of the signature of spin manifold (as prompted by the algebra of integral quadratic symmetric form). However, as just mentioned, there is an even simpler proof of Arnold based on Rohlin’s formula. Of course there is a myriad of sub-Arnoldian truths that could be treated via the Ahlfors foliation, e.g. Hilbert-Rohn prohibition of an $M$-sextic without nesting, or the type II of a quartic with two unnested ovals (all these assertions being implied by Arnold’s congruence). Another game out of reach to Arnold, but proved via Rohlin’s formula is the prohibition of the sextic scheme $5_I$ of five unnested ovals in the type I case. This could perhaps also be proved via the Ahlfors map. The only point which we managed (presently) to prove via Ahlfors is the (easy sense) of Klein’s Ansatz that a dividing curve cannot gain an oval while crossing the discriminant through a solitary node (with a complex conjugate pair of tangents). Compare for this Lemma \[Klein-via-Ahlfors(Viro-Gabard):lem\] suggested by a letter of Viro. However, we always use as a premiss the issue that for a plane curve the abstract total map of Ahlfors extends to the ambient projective plane. We should acknowledge a letter of Marin (cf. Sec.\[e-mail-Viro:sec\]) for having made us aware of this subconscious short cut. We still hope this to be true via basic algebraic geometry, of which we forgot all the foundations. Klein’s Ansatz (1876) can also be proved without Ahlfors by using some Picard-Lefschetz and Dehn stuff, or rather just some “Anschauung” that might have been folklore as early as 1876. Here is an argument (cf. also the next Sec.\[Klein-Marin:sec\]). At the level of the complexification, one can only explain the apparition of a solitary node as the strangulation of some vanishing cycle $\beta$ on the Riemann surface. Then we analyze all possibilities. By the reality of our deformation, the cycle $\beta$ must be invariant under conjugation $\sigma$, hence either be a real circuit (or “oval”[^21]) (pointwise invariant under $\sigma$), or an ortho-cycle (2 fixed points under $\sigma$) or a dia-cycle (no fixed point under $\sigma$). This is exhaustive via the classification of involutions on the circle, which via the quotient map and covering theory, reduces to the classification of one-dimensional manifolds (Hausdorff and metric). As we already noted a dia-cycle cannot exist in the orthosymmetric case (Lemma \[antioval:lem\]). For an “oval” it can indeed shrink to a point (hence a solitary node) but then disappear of course (cf. Fig.\[Klein-Marin:fig\], right). If we have an ortho-cycle then two cases are to be distinguished. It can either cross two distinct “ovals”, in which case both ovals merges together after the Dehn twist (cf. Fig.\[Klein-Marin:fig\], left). The last possibility is an ortho-cycle cutting only one “oval”. In this case Fig.\[Orthoovals2:fig\] shows that $r$ stays constant, and of course we do not cross a solitary node in that case, but rather a non-isolated one with 2 real tangents. This “proves” Klein’s Ansatz (modulo some Picard-Lefschetz theory), and even Marin’s stronger assertion that a dividing curve cannot increase its number component when crossing the discriminant. Indeed in all 3 cases analyzed, either $r$ drops by one unity (first two cases), or stays constant. Remark however that Marin 1988 [@Marin_1988] has a more conceptual proof. What is Picard-Lefschetz theory in our context? Since any crossing of the discriminant can be interpreted as a smooth arc traversing the discriminant transversally, we may (in the small) always replace this little arc by a linear pencil, and are reduced to classical Picard-Lefschetz theory, where in our case we have holomorphic fibration of the plane by a pencil of curves. The theory in question tell us the geometric monodromy when winding around a singular member of the pencil, but also gives the Dehn twist description of what happens when we (more cavalier) cross frontally the singularity. Recall that Picard’s thesis (the first work of Picard on another subject) is dated 1879 [@Picard_1879], while Klein’s Ansatz (no proof but probably Klein had one) is dated 1876 (3 years younger). So clearly our approach is somewhat historically contorted. Still, it is not impossible that Klein (and many others) were aware of the geometry behind our argument (via Dehn twists, ca. 1910). It is also possible that Klein’s argument was closer to Marin’s, albeit the latter result is perhaps slightly different (and of course stronger). Rohlin’s conjecture almost implied by Klein-Marin (Klein 1876, Marin 1988, Viro 1986) {#Klein-Marin:sec} ------------------------------------------------------------------------------------- \[04.01.13\] This section presents another tactic (pseudo-proof) of Rohlin’s conjecture that probably everybody had in mind (especially Rohlin and Marin), yet nobody write it down as it fails blatantly. Although being a “pot-pourri” it is worth presenting as it helps clarifying the relation (or absence thereof) between Klein’s original assertion 1876 [@Klein_1876] as interpreted by modern workers, notably Marin 1988 [@Marin_1988] and Viro 1986/86 [@Viro_1986/86-Progress] (the latter being based on a “private communication” of the former). First there is a remarkable observation of Klein 1876 that Marin 1988 [@Marin_1988] was probably the first to supply with a proof. To be perfectly accurate we believe that Marin’s result is slightly stronger than Klein’s original asserting only that a dividing curve cannot acquire a new oval like a champagne bubble emanating from a solitary node (compare Klein’s Quote \[Klein\_1876-niemals-isolierte:quote\] especially the phraseology “isolierte reelle Doppeltangente”). In Marin 1988 article, full credit is ascribed to Klein, either by over-modesty or because Marin overlooked to notice the little nuance between his and Klein’s weaker assertion. (Compare the recent e-mail exchanges in Sec.\[e-mail-Viro:sec\].) \[Klein-Marin:lem\] [($\approx$ Klein 1876, but in the formulation of Marin 1988)]{}.—A (plane) dividing curve cannot increase its number of ovals when crossing a node (non-degenerate double point). For Viro (1986 ) the curve does not actually need to be plane. Perhaps Klein gained evidence from the case of quartics. Imagine a Gürtelkurve (quartic with 2 nested ovals), then there cannot be created a new oval without violating Bézout. Hence either both ovals amalgamate or the inner oval evanishes. In both cases the number of ovals decreases (by one unit). \[02.04.13\] This is not an exhaustive discussion, for there can be also an eversion (Sec.\[Eversion:sec\]), keeping $r=2$ constant. Bringing into the picture the Riemann surface (of orthosymmetric type) underlying the dividing curve, then, as the latter traverses the discriminant (at some smooth point of it) our curve becomes uninodal via some vanishing cycle pinching the Riemann surface. -0.2cm0 -1.2cm0 -5pt0 -5pt0 This vanishing cycle can actually be an oval of the Riemann surface: once shrunk to a point it disappears and one oval gets lost (cf. Fig.\[Klein-Marin:fig\], right). This is not the only possibility as shown by the Gürtelkurve whose 2 ovals may coalesce. It is just a little harder to visualize the corresponding surgery on the Riemann surface. The key is to imagine an anti-invariant vanishing cycle $\beta$ whose contraction is depicted on Fig.\[Klein-Marin:fig\], left. The two ovals traversed by the cycle $\beta$ have merged together. [*Insertion*]{} \[02.04.13\].—Further there is a 3rd possibility, of when the ortho-cycle $\beta$ intersects only one oval (cf. Fig.\[Orthoovals2:fig\]). Then the corresponding Morse surgery is an eversion keeping $r$ constant but destroying the dividing character of the curve. (Marin’s proof in the dirty fingers of Gabaredian[^22]) Marin’s proof is somewhat different and in substance as follows (please refer to the French original for faithfulness). The initial curve is orthosymmetric. Such curves satisfy the Klein congruence $r\equiv g+1 \pmod 2$. On the other hand when traversing the discriminant the curve is uninodal and the real part undergoes a “Morse surgery” which alter the number of ovals by one unit. [*Insertion*]{} \[02.04.13\].—Warning. Possibly $r$ can stay constant in case of an eversion, cf. Fig.\[Orthoovals2:fig\], but then the post-critical Riemann surface becomes diasymmetric. At any rate, the new curve (past the discriminant) is necessarily diasymmetric, either by Klein’s congruence when $r$ moves by one, or by the Dehn-twist argument of Fig.\[Orthoovals2:fig\] when $r$ is kept constant. Marin concludes by arguing that a path between two conjugate points avoiding the real locus subsists in all nearby curve. (Alas I confess to have not properly understood this argument which is presumably much superior to the above via vanishing cycles.) Compare also Marin’s e-mail, where he explained us more details. (NB: Marin’s argument is also repeated in Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.785, 4.6.8], and was apparently always easily digested by Russian scholars, Viro included.) Armed with the lemma, let us try to attack Rohlin’s conjecture. (Pseudo-proof of Rohlin via Klein-Marin) Suppose $S_1$ to be a scheme of type I which is not maximal, say embeddable in $S_2$. Take algebraic models $C_i$ of each $S_i$ ($i=1,2$). By general position we may assume that the line $L$ through $C_1$ and $C_2$, in the hyperspace of all curves of degree $m$ (à la Cayley, etc.), crosses transversally the discriminant hypersurfaces (in smooth points of it) at uninodal curves. This unique node is necessarily real (when we look at real members of the pencil $L$). Hence when we join $C_1$ to $C_2$ we get real curves (finitely many of them being singular). Whenever we cross the discriminant the real locus undergoes a “Morse surgery” which is (up to reversing time) is either $\bullet$ the death of an oval (shrinking to a point) $\bullet$ the fusion of two unnested ovals $\bullet$ the fusion of two nested ovals. Each operation effects a fluctuation of $\pm 1$ on the number $r$ of ovals. (Warning \[02.04.13\].—This is not even true due to eversions!) So we can imagine a staircase starting from $C_1$ to $C_2$ recording the history of the varied fluctuations of $r$ during the transition from $C_1$ to $C_2$ along the pencil $\lambda C_1 + \mu C_2=0$ (cf. Fig.\[Klein-Marin:fig\], center-bottom). By Klein-Marin the first staircase is moving downwards, and as we ultimately reach $C_2$ having more ovals, we are naively inclined to claim that we shall revisit the same scheme $S_1$ at some step after which $r$ only increases. This would be true if a scheme would be completely encoded by its number $r$ of circuits. In this naive world, we get an intermediate curve $C_1'$ also representing the scheme $S_1$ (hence dividing since $S_1$ is of type I) and after which $r$ only increases. This would violate the Klein-Marin theorem. The moral is that Klein-Marin seems to imply, but fails implying, the Rohlin maximality conjecture (for an explicit objection see the little diagrammatic of ovals on Fig.\[Klein-Marin:fig\], bottom). Nonetheless, the Klein-Marin lemma certainly implies the: The chambers past the discriminant corresponding to orthosymmetric curves are local maxima of the function $r$ counting the number of real circuits. Further all chambers adjacent to an orthosymmetric chamber are diasymmetric. The last assertion follows directly from Klein’s congruence $r\equiv g+1 \pmod 2$. [*Insertion*]{} \[02.04.13\] This is true when $r$ varies (by one unit), but if it stays constant one has to invoke the Dehn-twist argument of Fig.\[Orthoovals2:fig\]. Hence orthosymmetric chambers are never contiguous (along a wall of codimension 1), but a priori they could still have closures with non-void intersections. Back to degree 6: Rigid isotopy (Nikulin 1979 via K3’s, Torelli of Pyatetsky-Shapiro-Shafarevich 1971) ------------------------------------------------------------------------------------------------------ \[05.01.13\] Even if Rohlin’s conjecture (type I $\Rightarrow$ maximal) looks out of reach, it might be easier in degree 6 (we mean by a theoretical argument independent of Rohlin’s census). In that case granting orthosymmetry of the schemes $\frac{6}{1}2$ and $\frac{2}{1}6$ one recovers all of Gudkov’s obstructions (prohibition of the semi-hexagons above those schemes, cf. Fig.\[Gudkov-Table3:fig\], safe those that were established by Hilbert and Rohn, namely the schemes $11$ and $\frac{10}{1}$). Of course this is nothing new, yet methodologically distinct from the topological arguments à la Arnold-Rohlin explaining the Gudkov hypothesis. So we are asking for a fighting interplay between pure geometry and topology. Also in view of the Morse surgery inherent in the Klein-Marin theorem, one can ask several questions about the contiguity of chambers in the space of all sextics and correlate this with the diagrammatic of Gudkov’s table (Fig.\[Gudkov-Table3:fig\]). The first basic point is that when we cross a wall $r$ fluctuates by $\pm 1$. Hence we do not have complete freedom to random-walk on the triangular lattice underlying Gudkov’s table (all horizontal edge cannot be used). [*Insertion*]{} \[02.04.13\].—This is a naive misconception, since in fact there is also eversion (cf. Sec.\[Eversion:sec\]) keeping $r$ constant! Define the [*distance*]{} between two chambers as the minimum number of walls needed to be crossed to join them (by a path transverse to the discriminant). Another “distance” is defined by restricting to pathes along (linear) pencils of curves. A great miracle (specific to order 6) is the following result due to joint efforts of Kharlamov and Nikulin 1979/80 [@Nikulin_1979/80]: [(Nikulin 1979)]{}\[Nikulin:thm\] The real scheme enhanced by the (Klein-Rohlin) type affords a complete invariant of the rigid-isotopy class of sextics. Thus via Rohlin’s classification (Theorem \[Rohlin-type:thm\]) there is $6+42+2\cdot 8=56+8=64=2^{8}$ “typed” schemes and so many rigid-isotopy classes. (This number being a power of 2 is perhaps just good fortune? probably because for quartics the number of chambers is $6$.) This is yet another “tour de force”. It uses (but strangely does note cite!) the topological classification of Rohlin 1978 [@Rohlin_1978] (making already a fusion between Klein 1876 and Hilbert 1891/1900’s 16th problem as solved by Gudkov 1969 [@Gudkov-Utkin_1969/78]). But that is not all! It also combines this with the complex geometry of K3 surfaces (Kummer-Kähler-Kodaira as coined by Weil), especially the contribution of Pyatetsky-Shapiro–Shafarevich 1971/71 [@Pyatetsky-Shapiro-Shafarevich_1971/71] on the global Torelli theorem, as well as the surjectivity of the period mapping (Kulikov 1977 [@Kulikov_1977]). Rohlin, and especially Kharlamov’s rôle in this proof seems to have been quite pivotal (and acknowledged as a such). The Gudkov-Rohlin-Nikulin pyramid and the contiguity graph ---------------------------------------------------------- [*Warning*]{} \[02.04.13\].—This section is a miscellany of mistakes about the combinatorial structure of the hyperspace of all sextics. We kept our text intact in its original shape (modulo Insertions and corrections) since we think that it is more important to avoid the basic mistake than to reach the ultimate verity of what is quite likely to become a combinatorial mess if pushed to its ultimate perfection. We still encourage the indulgent reader to follow our output as it may contain interesting problems. In particular is it possible for a pencil of sextics to visit only a single chamber? This could be the case if there is a chamber contiguous to itself. You move in your chamber and tries to get out of it by traversing a wall, but alas fall again trapped in the same room as you started where. \[05.01.13\] Looking (once again) at Gudkov’s table Fig.\[Gudkov-Table3:fig\] (while duplicating all the blue-rhombs) we get a complete picture of all chambers past the discriminant (i.e. rigid-isotopy classes of smooth curves). One would like to understand their contiguity relation to enhance this set into the [*contiguity graph*]{}. Basically we have 6 moves prompted by the equilateral lattice underlying Gudkov’s table. But as all Morse surgeries amounts to the creation or destruction of an oval we can rule out the two horizontal moves (keeping $r$ unchanged and corresponding resp. to the evasion or encapsulation of an oval). Next certain Morse surgeries are of course incompatible with Bézout (e.g. that depicted on the top-right of Fig.\[Gudkov-contig:fig\]). A little moment thought shows that all admissible Morse surgeries correspond to one of the $4$ legal moves. A further obstruction comes from the Klein-Marin theorem (\[Klein-Marin:lem\]) impeding the $2$ creationist moves (going up $r\mapsto r+1$) as soon as the chamber is of type I. -1.2cm0 -5pt0 -5pt0 Naively one is tempted to say that this is a complete list of legal moves. If so is the case then we would have a complete description of the contiguity graph (whose edges are depicted by red strokes on Fig.\[Gudkov-contig:fig\]). As we shall see later (eversions) this answers is quite unlikely to be the definitive answer. (Exercise: count the number of edges of this graph: counting edges going to the North-West: we have $2\cdot 9+6+3\cdot 5+2+3\dot 1=18+6+15+2+3=44$. This must be doubled by symmetry to $88$. Then adding those edges going to the $8$ indefinite types adds $2\cdot 8-3=13$ edges. Next the empty scheme $0$ gives one edge and the deep nest for 2. In definitive, $88+13+3=104=2\cdot 52=2^2\cdot 26=2^3 \cdot 13$ edges. This is the number of contiguity zones of the discriminant hypersurface.) It seems also that the most remote pair of vertices are the schemes $\frac{9}{1}$ and $10$ lying at distance 18 apart. On the other hand the discriminant has degree $\delta=3(m-1)^2$. (This can be proved via a Euler characteristic count in the fibration induced by a pencil of $m$-tics after blowing up the basepoints, and is also implied by the so-called Zeuthen-Segre formula in the algebro-geometric community, which is merely the avatar of Riemann-Hurwitz in one more dimension.) For sextics $m=6$ this gives $\delta=3 \cdot 5^2=75$. It follows that the linear distance between two chambers (as measured inside a linear pencil) is at most $[75/2]=[37.5]=37$ (the temperature of the human body). Probably this bound is far from sharp (except of course if there is a line hitting the discriminant 75 times on the reals), and one could try to find a least upper bound. Given two chambers (=rigid-isotopy classes) define their distance $\delta$ as the minimum number of wall-crossings separating them. This is also the combinatorial distance in the contiguity graph. $\bullet$ Define also their écart $\varepsilon$ as the minimum number of wall-crossings in a generic pencil (transverse to the discriminant) through two curves belonging to the given chambers. We have always $\delta \le \varepsilon$; and when $m=6$, $\delta \le 18$ and $\varepsilon \le 37$. Since the degree of the discriminant is 75 ($\delta=3(m-1)^2$ is odd whenever $m$ is even) any pencil of sextics intersects the discriminant (in a real point) and so the curves undergo at least one Morse surgery, and assuming genericity there is an odd number of such surgeries. However the structure of the graph only permits loops of even length (the girth=systole of the graph is 4). This is almost a contradiction in mathematics. How to resolve it? [*Insertion*]{} \[21.01.13\].—Just look at eversions, cf. Sec.\[Eversion:sec\]. Further using eversions it is likely that $\delta$ is much smaller than above, and we predict rather something like $\delta\le 11$ (cf. \[eversion-and-other surgeries:conj\] and the semi-conjectural Prop.\[Erdos-number-of-sextics=11:prop\]). Then there is a host of combinatorial-geometric questions arising. E.g. is there a pencil cutting 75 times the discriminant (a sort of total reality of Bézout). If not what is the maximal number $\mu$ of real intersections a line can have with the discriminant? (Since the distance between the extreme $(M-1)$-schemes is 18, taking the line joining them we get a pencil with $18+18=36$ (aller-retour, no one way ticket!) real intersections, to which one can safely add one unit due to oddness of the degree, so $\mu \ge 37$ the temperature of the human body.) [*Insertion*]{} \[01.04.13\].—Alas this argument is foiled as it does not take into account eversions. With eversions the maximal distance seems to be 11 (between $M$-curves and the empty one), and so arguing as above gives only $\mu\ge 22+1=23$. What is the least (resp. maximum) number of chambers visited by a pencil of sextics? Denote them $\alpha$ resp. $\omega$. Naively a pencil could stay entirely inside a chamber, but this is precluded by Bézout as $\deg \disc=75$ is odd (so $\alpha\ge 2$). A priori among the 64 chambers all could be visited since the discriminant has degree 75. Same question for the length of the loop induced in the contiguity graph by a (generic) pencil. (A priori this length can be as long as $75$, but not longer.) Is this loop always non-contractible (in the contiguity graph)? Can it be embedded (i.e. visits only once each chambers it visits)? Is there a pencil visiting only diasymmetric chambers? (It is evident from Klein’s congruence and surgeries affecting $r$ by $\pm 1$ that a pencil cannot visit only orthosymmetric chambers.) What is the maximum number of orthosymmetric chambers visitable by a pencil? (Of course at least two, take the line spanned by 2 points in two ortho-chambers, but probably some lucky Stonehenge alignment exists.) What is the minimum height of a pencil? The height being just the invariant $r$, number of real circuits. An interesting result is the theorem of Cheponkus-Marin (cf. Marin 1988 [@Marin_1988 p.192]): [(Cheponkus-Marin 1988)]{} In any generic (linear) pencil of curves of even degree $m> 2$ there is a curve having at most $M-3$ components ($r\le M-3$). Looking at Gudkov’s table (Fig.\[Gudkov-Table3:fig\]) this looks almost evident, but is not. Since $M-3$ is the highest line full of squares, this Marin result implies that a pencil cannot confine its visits in one of the 3 regions lying above this line. One can define the depth $d$ of a pencil as the lowest value of $r$. So Marin’s result implies $d\le M-3$. Is this sharp at least for $m=6$? Define three chambers as aligned if there is a line hitting them simultaneously. In view of Gudkov’s table enhanced by the Klein-Marin theorem one sees that there are $3+1=4$ special chambers which are contiguous to a single chamber (vertices of valency 1 in the contiguity graph), namely those of type I with schemes $\frac{8}{1}$, $9$, $\frac{4}{1}$ as well as the empty scheme $0$ ($\bigstar$). It follows that the triad consisting of any of those four, plus its unique neighbor and any chamber are aligned. It would be interesting to find a triad of chamber which are [*not*]{} aligned. [*Insertion*]{} \[02.04.14\] Again the assertion right before the ($\bigstar$) above, looks foiled due to eversions. It looks more realist to expect that the empty scheme is the unique chamber contiguous to a single chamber. It could be interesting to describe the chambers adjacent to only 2 chambers. By the theory of eversion (developed latter), those includes the three $M$-schemes, and 2 Rohlin maximal $(M-2)$-schemes, plus apparently the 3 orthosymmetric chambers corresponding to symbols on the “boundary” of the pyramid, namely $\frac{8}{1}$, $9$, $\frac{4}{1}$. However it should not include a scheme like $10$, which by eversion is potentially related to $\frac{9}{1}$ (cf. Fig.\[Gudkov-eversion:fig\]). The above list could be exhaustive, but beware that the median schemes in type I, like $\frac{4}{1}4$, $\frac{3}{1}3$, $\frac{2}{1}2$ have also only two connections except for being potentially related to themselves under eversion. Yet later we shall see that eversion necessarily destroy the orthosymmetry, so that those schemes are eversively related to their type II twins lying below the sheet of paper. So those schemes have really valency 3. This raises however the question if a chamber can be contiguous to itself. By the diagrammatic of all Morse surgeries (eversion included) a necessary condition is that the chamber lies on the median line of the Gudkov table. By what as been said (orthosymmetry destroyed by eversions), the sole candidate for self-contiguity are $\frac{1}{1}1$ and $1$. Under this phenomenon of self-contiguity it could be the case that a pencil of sextics stays entirely within such a chamber safe for a quick perforation of the discriminant (forced by by Bézout), yet bringing us directly back to the same chamber. In that case the invariant $\alpha$ discussed above could be as low as $1$. \[13.01.13\] Another aspect of Nikulin’s isotopic classification of sextics via the Rohlin pyramid is that it affords a broad generalization of the Nuij 1968 [@Nuij_1968] and Dubrovin 1983/85 [@Dubrovin_1983/85] theorem stating that the deep nest schemes represents a unique rigid-isotopy class of curves. (Actually Nuij’s theorem holds in arbitrary dimension.) By Nikulin’s theorem this uniqueness determination by the real scheme holds true more generally for all sextic schemes which are not hermaphrodite (i.e. of indefinite type). All this just amount to the connectivity of the chambers, yet one may wish to know more on their individual topology (in the large). One obvious tool is the monodromy representation $$\pi_1(\textrm{ some chamber} ) \to {\frak S}(\textrm{ovals})$$ acting upon the ovals by permutation while following a loop inside some fixed chamber. Now for a scheme having both inner and outer ovals there is an obvious constraint preventing the permutation to shuffle inner ovals with outer ovals. In other words for a scheme of type $\frac{k}{1}\ell$ the range of monodromy would lye inside ${\frak S}_k\times {\frak S}_{\ell}$. A (naive?) conjecture would be that this are the sole restrictions on the monodromy (i.e. the restricted morphism is epimorphic). If so is the case then all chambers are not simply-connected, except perhaps the 5 ones corresponding to the non-permutable schemes, i.e. $0$, $1$, $\frac{1}{1}$, $\frac{1}{1}1$ and $(1,1,1)$. For those schemes the monodromy of ovals is a trivial representation, and so there is no obstruction for those chambers to be simply-connected. Can one of those chambers even be contractible? A natural tactic is to ask if it can be starlike, in the sense of having a special viewpoint (curve) inhabiting the chamber so that each curve of the same chamber is accessible by the half-circle of the line joining the base curve to the “variable” one. Some obvious candidate are the Fermat equations $x^6+y^6=-1$ for chamber $0$ and $x^6+y^6=+1$ for chamber $1$, yet it is not clear at all if those are “visibility curve”. [*Insertion*]{} \[02.04.13\].—Much sharper and complete results of the monodromy of sextics are due to Itenberg 1994 [@Itenberg_199X-monodromy-deg-6] extending results of Kharlamov. We shall come back this this latter. Weak reformulation à la Marin-Viro of the Klein-Rohlin maximality conjecture ---------------------------------------------------------------------------- \[13.01.13\] After some discussions with Marin (12–13 Jan. 2013, cf. Sec.\[e-mail-Viro:sec\]), the following issue came quite clear. First let us contemplate once more the Gudkov-Rohlin pyramid as depicted as the contiguity figure \[Gudkov-contig:fig\]. On it we imagine the blue rhombs schemes doubled with a “$\Lambda$” shaped pair of edges raising to the orthosymmetric chambers (provided not on the periphery of the pyramid), whereas the diasymmetric chamber have generically a $X$-shaped quadruplets of edges in the contiguity graph. We can consider the POSET of all real schemes enhanced by the type I/II of Klein. This is essentially what did Rohlin 1978, safe that instead of declaring indefinite those “hermaphrodite” schemes tolerating both type of representatives (type I and II) we duplicate those schemes to see them as independent elements. This amounts considering all Gudkov’s symbols decorated by signs $\pm$ telling the ortho/dia-symmetry, and of course only those realized algebro-geometrically. This is a well-defined finite set of $64=2^8$ elements. How to define an ordered structure to make it into a POSET? Answer just as the picture Fig.\[Gudkov-contig:fig\] suggests, namely a type I scheme (alias ortho-scheme) has two legs going down (some leg may be amputated if the scheme is peripheral), whereas dia-schemes have two legs (going down) and two arm (going up), except if it lies in the periphery. For instance the scheme $0$ is maximally amputated having one arm but no legs. Of course this order structure looks somewhat ad hoc yet quite in line with the remarks of Klein and the theorem of Marin 1988, which is a stronger variation thereof (apparently Marin did not noticed that his statement looks stronger than Klein’s original statement, compare our discussion in Sec.\[e-mail-Viro:sec\]). Let me call the purified pyramid this poset. Paraphrasing Rohlin’s census (diagrammatically encoded in Fig.\[Gudkov-contig:fig\]) we plainly have: The maximal elements of the purified pyramid of sextic ortho- and dia-schemes are exactly the orthosymmetric ones. It seems evident that this statement extends trivially to all degrees as a mere paraphrase of Marin’s theorem (1988 [@Marin_1988]). Yet is non trivial to make a picture even for degree $8$. Yet this is worth trying to depict at the occasion (cf. Fig.\[Degree8:fig\]). An eclectic proof of Rohlin’s conjecture via Ahlfors {#Rohlin-via-Ahlfors} ---------------------------------------------------- \[04.01.13\] Let us summarize the situation. Rohlin in 1978 [@Rohlin_1978] advanced (taking some indirect inspiration by Klein 1876) the bold conjecture that $$\textrm{a scheme is of type~I iff it is maximal,}$$ in the hierarchy of all real schemes of some fixed degree. One sense of the conjecture turned wrong, in degree 8 by a conjunction of Polotovskii 1981 [@Polotovskii_1981] and Shustin 1985/85 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin] works (see also the remarks in Viro’s survey 1986/86 [@Viro_1986/86-Progress p.67–68]). Namely Shustin showed existence of a maximal scheme in degree 8 of type II. So the “$\Leftarrow$” implication of Rohlin’s conjecture is disrupted. It remains the hope that the “$\Rightarrow$” implication is correct (still open in 2013): [(Rohlin’s maximality conjecture—post Shustin)]{} Fix any integer $m\ge 1$, and consider only schemes of that degree $m$. If a real scheme is of type I, then it is maximal in the lattice (POSET) of all real schemes. This is perhaps a trivial consequence of Ahlfors theorem: [(Gabard 31.12.12 and 04.01.13)]{} If a real scheme is of type I, then it is maximal (among all schemes of the same degree). Suppose the given scheme, say $S_1$, to be of type I. By contradiction assume it non-maximal so that it embeds in some larger scheme $S_2$ as a strict subset. But our schemes are algebraically realized by real algebraic curves say $C_1$ and $C_2$ (defined over ${\Bbb R}$) so that the inclusion $C_i({\Bbb R})\subset {\Bbb P}^2({\Bbb R})$ belongs to the respective isotopy classes of $S_i$ ($i=1,2$). Since $S_1$ is of type I, $C_1$ is dividing, and thus there is by Ahlfors 1950 [@Ahlfors_1950] a total pencil $\pi$ of auxiliary curves all of whose real members cut only real points on $C_1$ (at least as soon as they are mobile). Now $C_2$ has at least one extra real circuit over $C_1$ (which in fact must be an oval, as curves of odd order necessarily have a pseudoline). Naively, one would like to choose any point $p_0$ on $C_2({\Bbb R})$ and let pass through it a curve of the pencil $\pi$, say $\Gamma_0\ni p_0$ while arguing that this curve has supernumerary intersection with $C_2$, violating thereby Bézout. This works (effortlessly) if we could assume $C_1({\Bbb R})\subset C_2({\Bbb R})$, but this corrupts rigidity of algebraic curves. We see that Ahlfors nearly implies Rohlin, but some gigantic gap requires to be filled. Obviously the problem has to be embedded in some more flexible medium so as to bridge the gap between algebraic rigidity and softness of isotopy classes à la Hilbert-Rohlin. \[10.01.13\] The little flash on how to complete the argument came to me ca. \[05h20\] in the morning after some too early waking up. It is as follows. Suppose our curve $C_1$ to be of type I. By Ahlfors there is a total pencil of curves. If the scheme of $C_1$ is not maximal it can be enlarged, so there is a curve $C_2$ with larger scheme. But $C_1$ is transverse to the foliation induced by the total pencil, and transversality is a robust feature (structural stability à la Thom, etc.) Accordingly a small perturbation of $C_1$ towards $C_2$ is still maximally cut by the curves of the pencil. Propagating this so forth we see (assuming genericity of the pencil) that the first Morse surgery decreases the number of ovals. (This was anticipated by Viro yesterday \[09.01.13\] (cf. Sec.\[e-mail-Viro:sec\]), and goes back to Klein 1876.) However one would more, namely that $C_2$ cannot have more ovals than $C_1$. To be more precise one should compare the pencil $L$ spanned by $C_1, C_2$ to the total pencil for $C_1$, while understanding perhaps the filmography of the deformation in reference to this foliation. Thinking of the latter as locally vertical, the first Morse surgery is like the hyperbola $(x-y)(x+y)=x^2-y^2=\varepsilon<0$ transverse to the vertical foliation while degenerating to the pair of lines of slope $\pm 1$ and then becoming another hyperbola $x^2-y^2=\varepsilon>0$ no longer transverse to the vertical foliation. (In fact this is only the scenario of when the node is not a solitary one.) The effect of crossing the first critical level is that some member of the pencil loose their total reality. Yet not all of the total reality is lost. In fact merely an interval of “imaginariness” is inserted. Pushing the analysis in the large (several Morse surgeries), while also treating the other case one may hope to ensure that when $C_2$ is reached still some totally real curve persists in the pencil. In fact Ahlfors theorem (only) implies Klein’s thesis (cf. Klein’s Quote \[Klein\_1876-niemals-isolierte:quote\]): \[Klein-via-Ahlfors(Viro-Gabard):lem\][(Klein 1876, in a presentation of Gabard inspired by Viro, while using Ahlfors)]{} When a dividing curve crosses the discriminant it cannot acquires a solitary double point. In particular all (discriminantal) walls bounding an orthosymmetric chamber correspond to nodes with real tangents. (This was anticipated the \[10.01.13\] by Oleg Viro (e-mail communication in Sec.\[e-mail-Viro:sec\]), but I only understand it now after ca. 10 hours of delay and some sleep in between.) Imagine the curve moving, with suddenly, a solitary double point appearing in the real locus, like an ufo(=unidentified flying object) raising into the blue sky. If not believing in extraterrestrial flying saucers, think of this as the ex-nihilo creation of a champagne bubble from a point inflating slightly to a little oval. The initial curve $C_{-1}$ is dividing, hence admits by Ahlfors 1950 [@Ahlfors_1950] a total pencil of curves[^23]. The later induces a (mildly singular) foliation of the real projective plane, which we call the [*Ahlfors (or total) foliation*]{}. It can be assumed transverse to the given curve $C_{-1}$. A priori the (Ahlfors) foliation may be singular along the curve, yet upon dragging away the center(s) of perspectives it should always be possible to avoid this (compare upper row of Fig.\[Tube:fig\] for an implementation on the Gürtelkurve). W.l.o.g. suppose the curve (already) close to the discriminant and the deformation $C_t$ ($t\in[-1,1]$) to be a small one traversing that hypersurface. By continuity, transversality to the Ahlfors foliation persists after the critical level, and so a nascent champagne bubble would violate Bézout. Indeed, passing a curve of the total pencil through a point inner to the newly created oval (or on that oval) gives an excessive intersection with the post-critical curve $C_{+1}$. Formalizing requires to fix a tubular neighborhood of the initial curve while noticing that its product structure is the trace of the total foliation. If the degree $m$ is odd then there is one pseudoline whose tubular neighborhood is a twisted bundle (Möbius band). -5pt0 -5pt0 As long as the discriminant is avoided, a slight continuous variation of the coefficients engenders a small perturbation of the real locus (continuity lemma for rigid-isotopies). When the discriminant is crossed at a solitary node (ordinary double point with 2 imaginary conjugate tangents), the real locus acquires a (single) champagne bubble, while the rest of the curve is isotoped within the prescribed tube (compare Fig.\[Tube:fig\] bottom). It can be assumed (for simplicity, but not vital) that the deformed curves $(C_t)_{t\in[-1,+1]}$ stay transverse to the foliation. Inside of the tube, the intersection of the curve $\Gamma$ of the total pencil through the solitary node, with the post-critical curve $C_{+1}$ is in natural bijection with that of the pre-critical curve $C_{-1}$ . Yet the former intersection $C_{-1} \cap \Gamma$ is totally real, whereas the second contains two additional points when $\Gamma$ cuts the new created oval of $C_{+1}$ (better ask $\Gamma$ to pass through a point of this new oval). Bézout is violated, and our (Viro inspired) proof of Klein’s assertion is complete. Klein’s thesis (as discussed in Marin 1988 [@Marin_1988] or Viro 1986) is the stronger assertion that a dividing curve cannot see its number of circuits increase when crossing the discriminant. This probably also follows from Ahlfors after some suitable thinking, yet perhaps is less close to Klein’s original statement. It is only now that Klein’s allusion (Quote \[Klein\_1876-niemals-isolierte:quote\]) appears to me quite transparent (yet via the powerful Ahlfors theorem). [*Insertion*]{} \[30.03.13\].—It is unlikely that this was the original proof of Klein (despite the fact that Teichmüller 1941 ascribes to Klein the theorem usually ascribed to Ahlfors). Klein’s original reasoning (alas unpublished in details, but only claimed in Klein 1876 [@Klein_1876]) might have been rather purely topological, essentially like Marin’s (though the latter’s statement is somewhat stronger). More on this will be discussed below, especially in (\[Klein-Marin:lem\]). \[11.01.13\] One may wonder if, conversely, a nondividing curve can always acquire a solitary node and so a new oval. This is also implicit in Klein 1876 intuition, and probably true up to degree $6$ (cf. Gudkov’s table=Fig.\[Gudkov-Table3:fig\] for some evidence, while a rigorous proof probably rests on Nikulin’s rigid-isotopy classification via Rohlin’s enhanced Gudkov table by complex characteristics). (For a verification of Klein’s intuition in degree $6$, see Prop.\[Klein-vache-deg-6:prop\], and for a disproof in degree 8, cf. Shustin 1985 and our accompanying comments in Sec.\[Shustin-understood:sec\].) Gudkov’s table shows however that the location for the apparition of a bubble cannot be chosen in advance. Indeed starting say from the scheme $10$ we could by bubbling create the scheme $11$ violating the Hilbert-Rohn-Petrovskii-Gudkov theorem, that such a scheme is not realized algebraically. Another more obvious argument is just to take any scheme (not on the “visible faces” of Gudkov’s pyramid, equivalently, such that $\frac{1}{1}1$ is a subscheme) and create a bubble inside the outer oval so that the new real scheme contains the subscheme $\frac{1}{1}\frac{1}{1}$ consisting of 2 nests of depth 2, which violates Bézout. Prohibitions {#Prohibitions:sec} ============ \[28.03.13\] From now on, we do not follow historical order, but rather logical necessity. Admittedly there is no universal measure of simplicity as it depends much on the background of the investigator. From a radical viewpoint, the unique measure of simpleness could be the natural historical time-arrow. Yet sometimes big surprises arise. Arguments extremely powerful and strikingly simple (nearly stemming from nowhere) tend to trivialize much of the past efforts. Such an example is Rohlin’s formula discussed below, which bears some antecedents only by Arnold, plus the topological heritage of Riemann, Betti, Poincaré, Lefschetz, Weyl, Pontryagin, etc (homological intersection theory). The source of prohibitions in Hilbert’s 16th problem, are multiple. Albeit we are not expert in the field let us brush a brief historical sketch. First there are evident restrictions coming from Bézout. Those were used by Zeuthen 1874 [@Zeuthen_1874], and exploited in full in Hilbert 1891 (boring bounds on the depth of ovals). A major prohibition (directly affiliated to Zeuthen) is the Harnack inequality $r\le g+1=\frac{(m-1)(m-2)}{2}$ of 1876, in no way specific to plane curves. Klein 1876, then aged 27, (but already Harnack’s teacher) gave a more intrinsic justification boiling down to a basic fact on the topology of surfaces directly imputable to Riemann’s definition of the genus (or rather its allied connectivity). Recall that the jargon of the genus is due to Clebsch. Harnack’s inequality is something very robust, as it extends to all dimensions via Smith theory, as was noticed by Thom and Milnor, yielding something like $b_\ast(\RR X)\le b_\ast (\CC X)$, for $b_\ast$ the total Betti number. We shall not need this as we confine attention to curves (where enough work remains to be done). As discussed above (\[Klein-unnested-quartic-nondividing:lem\]), Klein 1876 also used large deformations (rigid-isotopies), to prove e.g. that a quartic with 2 unnested ovals is nondividing. Later he also exploited theta-characteristics. The first method is unlikely to extend to curves of higher orders (despite Nikulin’s rigid classification), while the second has been poorly explored further since Klein 1892, and Gross-Harris 1981 [@Gross-Harris_1981], and does not seem able to compete seriously with information distilled by Rohlin’s formula. Perhaps those old Jacobi-Riemann-Klein methods deserve to be revived. As to Nikulin, it seems at first that it will tell nothing being rather built upon the Gudkov-Rohlin classification by types (cf. Fig.\[Gudkov-Table3:fig\]). However in the fingers of Itenberg 1994 [@Itenberg_1994] (contraction theorem of empty ovals), we can expect (at least if this strengthens to our CCC=(\[CCC:conj\])) to rederive via strangulation the diasymmetry(=type II) of the schemes $1$ and $5$ of degree 6 (gaining so some analogy with Klein’s rigid-isotopy argument for the bifolium $2$ in degree 4). Yet, the difficulties are so great that this looks quite artificial as compared to the topological straightforwardness of Rohlin’s formula. By the way this would miss the scheme $\frac{1}{1}1$. Further this seems much limited to degree 6 as we lack precise information on rigid-isotopy in high-degrees. Then the connection with K3-surfaces is lost, and so the tool making Nikulin’s theorem possible (deep transcendental algebraic geometry, global Torelli theorem, etc.). After Zeuthen-Harnack-Klein, came Hilbert’s 1891 intuition[^24] that an $M$-sextic is forced to nest. This has no antecedents (as far as I know), yet it could be challenging to reprove it via conformal geometry (i.e. the Riemann-Schottky-Bieberbach-Grunsky theorem). This has never been implemented and is probably a hard game, if feasible at all. After Hilbert came several things like Ragsdale 1906, and Rohn 1911 who consolidated Hilbert’s method. This involves a deep analysis of the stratification of the space of curves and the usage of pencils. In Gudkov’s fingers, this produced an exhaustive list of prohibitions in degree 6. Perhaps an extension of this method also implies (or rather converges) with Rohlin-Le Touzé’s phenomenon of total reality. At least the diagrammatic of the Gudkov table (Fig.\[Gudkov-Table3:fig\]) strongly suggests this. In degree 6, all the information gained via Hilbert-Rohn is recovered for topological reasons à la Gudkov-Arnold-Rohlin (and extended to all degrees). It is not clear to me if this subsuming of HR to GAR is specific to degree 6 or a general feature. Probably not if I remember well a seminal talk by Orevkov (Geneva, ca. 2011) where Hilbert-Rohn was still much on the appetizer. After all it is unlikely that deep geometrical methods get completely phagocytozed by topological ones. Enriques-Chisini 1915 [@Enriques-Chisini_1915-1918] gave a proof of Harnack’s inequality based on Riemann-Roch and a continuity argument (compare our Lemma \[Enriques-Chisini:lemma\]). This is much akin to the phenomenon of total reality, and need to be extended to less trivial cases. Recall that from the viewpoint of total reality, $M$-curves constitute the trivial case. This desideratum is the main motivation of the present text yet we still have very few factual things to present. The next great step is Petrovskii 1933/38, who seems to be the first to find universal obstructions (valid in all degrees). This is based on Euler-Jacobi-Kronecker’s interpolation formula plus some Morse theory. Then there is Gudkov breakthrough (apparition of congruences mod 8 as opposed to mere estimates), and the theorists Arnold, Rohlin, etc. validating them via 4D-topology or Atiyah-Singer. In this move we have the trinity of congruences modulo 8 for $M$, $(M-1)$ and $(M-2)$-curves due to GR, GKK, RKM, respectively. Here G=Gudkov, R=Rohlin, 1st K=Krakhnov, 2nd and 3rd K=Kharlamov, while M=Marin. The importance of those can hardly be underestimated. First, the conjunction of GR and GKK explains all prohibitions in degree 6 on real schemes (i.e. Hilbert’s 16th), while the 3rd GKK (forcing orthosymmetry(=type I) of schemes with $\chi\equiv_8 k^2+4$) seems even to imply (via the hypothetical Rohlin maximality conjecture=RMC) the conjunction of GR+GKK. Even without the elusive RMC, it can be that explicit instances of total reality (e.g., Rohlin-Le Touzé’s) imply in low-degrees (say $m=6,8$) the truth of RMC in special situations. This looks after all plausible, since totality involves a geometrization of the type I topological condition by a stronger geometric property (total pencil). Here and in the sequel, we shall often abridge “total reality” by “totality”. Then appears Rohlin’s formula 1974–78. This is very strong and completely elementary. In degree 6, it rules out all schemes above the broken-line of Gudkov’s table (Fig.\[Gudkov-Table3:fig\]) safe 6, namely the 2 triangles involving the symbols $\frac{7}{1}3$, $\frac{7}{1}2$, $\frac{6}{1}3$ and its mirror $\frac{3}{1}7$, $\frac{2}{1}7$, $\frac{3}{1}6$. Rohlin’s formula is very powerful, yet somewhat too elementary to grasp the full mystery. It need therefore to be complemented by more advanced weapons like the Gudkov congruence (GR), and GKK, or by Rohlin’s maximality principle allied to total reality. In 1978, we have Rohlin’s maximality principle (RMC), still conjectural and not yet fully exploited in our opinion. This could loop-back to conformal geometry à la Riemann, Schwarz, Schottky, Klein, Koebe, Bieberbach, Grunsky, Teichmüller, Ahlfors. As said above, if RMC looks impossible to implement in universal generality it could be verifiable in special cases by using totality as a geometric strengthening of the (topological) type I-condition. For instance Rohlin-Le Touzé’s totality should suffice (either with or without RKM) to kill all expansions of the 2 orthosymmetric $(M-2)$-schemes of degree 6. This would unify all prohibitions in degree 6 safe the schemes $11$ and $\frac{10}{1}$ (easily ruled out via Rohlin’s formula). Ca. 1978–80, we have advanced Bézout-style obstructions à la Fiedler-Viro (\[Viro-Fiedler-prohibition:thm\]) that really pertains to curves of degree $8$. Those plays a pivotal rôle in Shustin’s disproof of Klein’s champagne bubbling principle for nondividing curves, as well as the disproof of the reverse implication of Rohlin’s maximality principle. More generally those look indispensable in the higher cases $m=7,8$ of Hilbert’s 16th. We have also the locking trick of Marin-Fiedler (also founded on Bézout for lines) that provides obstruction to rigid-isotopy on $M$-curves of degree $\ge 7$. Here the idea is that if we have a triangle (3 lines) which is Bézout-saturated and canonically attached to a scheme (typically a disc with 3 holes), then during a rigid-isotopy ovals cannot traverse this moving frame. Hence the distribution of ovals past such a fundamental triangle is an invariant of the rigid-isotopy class. Of course this method is not a method of prohibition of schemes, but prohibits the existence of pathes in the hyperspace of smooth curves. Finally, we have probably a rôle of Thom’s conjecture on genus-bound (verified since Kronheimer-Mrowka 1994 [@Kronheimer-Mrowka_1994]), yet whose role is not so clear-cut as initially expected. The simple case of Thom, due to Kervaire-Milnor 1961 [@Kervaire-Milnor_1961], may be used to settle Hilbert’s nesting “theorem” for $M$-sextics. In general the role of Thom, is perhaps marginalized by Rohlin’s formula and other strong results, yet seems to give new information in the work of Mikhalkin 1994 [@Mikhalkin_1994-adjunction-Thom] when it comes to split curves (communication of Fiedler, not yet digested by the writer=Gabard). This is a brief overview of nearly all what exists. In contrast one can ask for more conciseness when it comes to explain all the prohibitions of Hilbert’s problem (in degree 6) to a classroom. As often repeated, nearly everything could reduce to the (Klein-Ahlfors-)Rohlin-Le Touzé’s phenomenon of total reality. Remind that 2 technical points are still obscure, but philosophically trivial. The first is a complete proof of Rohlin’s claim (preferably without employing the RKM congruence mod 8). The second is to verify that Rohlin-Le Touzé’s total reality is strong enough to imply maximality of the two Rohlin’s $(M-2)$-schemes. Assuming this settled, we still miss the prohibition of Hilbert’s unnested scheme $11$, and Rohn’s maximally nested scheme $\frac{10}{1}$. This is paradoxical inasmuch as those 2 guys were historically the first ruled out by the Hilbert-Rohn method. The 1st scheme $10$ can be killed by the Kervaire-Milnor 1961 [@Kervaire-Milnor_1961] elementary case of Thom’s conjecture in degree $k=3$, but the second $\frac{10}{1}$ fails to succumb under Thom. However both of them are killed by Rohlin’s formula. Hence a good cocktail (for the classroom or the economical reader) is to mix total reality with Rohlin’s formula. This reduces all prohibitions in degree 6 to only 2 paradigms. As far as we know, apart form the Hilbert-Rohn method (as developed by D.A. Gudkov) there is no universal force unifying all prohibitions in a single one (even in degree 6). A substitute to Thom-Kervaire-Milnor is to use Petrovskii 1933/38 (\[Petrovskii’s-inequalities:thm\]). This rules out $11$ but not Rohn’s scheme $\frac{10}{1}$. The latter is not even killed by the strong Petrovskii inequality of Arnold (\[Strong-Petrovskii-Arnold-ineq:thm\]), i.e., $n-p^-\le \frac{3}{2}k(k-1)=\frac{3}{2}3\cdot 2=9$, where $p^{-}=1$ is the number of hyperbolic positive ovals, so $n\le 10$ while Rohn’s scheme has $n=10$. Obstructions via Rohlin’s formula (Rohlin 1974, 1978) {#Rohlin-formula:sec} ----------------------------------------------------- \[03.01.13\] We repeat the proof of the following pivotal result (whose proof puzzled me a lot as I was young, and still imbues some suitable respect[^25] when getting older). Crudely put, Rohlin’s formula is nothing less than the most universal obstruction that one may derive by abstract non-sense (i.e. using virtually nothing from the algebraicity assumption). [(Rohlin 1974–78)]{} \[Rohlin-formula:thm\] For any (real, smooth, algebraic, plane) dividing curve of even order $m=2k$ (odd orders were treated by Rohlin’s student Mishachev), the following equation holds: $$2(\Pi^+ -\Pi^-)=r-k^2, \label{Rohlin-formula:eq}$$ where $r$ is the number of ovals, while $\Pi^{\pm}$ are the number of positive (resp. negative) pairs of nested ovals. Each pair of nested ovals bounds a ring=annulus in ${\Bbb R}P^2$, and upon comparing with the complex orientation (as the border of the semi-Riemann surface) one defines a positive pair when both orientations (complex vs. real) agree, and a negative pair when they disagree (cf. Fig.\[Rohlinsformula:fig\]). -0.3cm0 -5pt0 -5pt0 The idea involves computing the self-intersection of the half of the dividing curve after capping off by discs bounding the ovals in the real projective plane, or rather the intersection with the conjugate capped off membrane. This argument seems inspired by a similar device used by Arnold in 1971 [@Arnold_1971/72], but now slightly more punch is acquired. The proof is very elementary using merely intersection of homology classes (available since the days of Poincaré, Lefschetz, Hopf, Pontryagin, etc.) and Poincaré’s index formula (available since Gauss?, Kronecker 1868, Poincaré 1885), plus some basic trick about the “Lagrangian property” of real parts of algebraic varieties (jargon used in Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000]). One can then nearly wonder why such a formula escaped Felix Klein’s attention, but this is of course just historical slowness of the revelation of brutal combinatorial truths. \[06.03.13\] The detailed proof is given in Rohlin 1974/75 [@Rohlin_1974/75 p.332], but stated there only for $M$-curves. The adaptation to the general case requires only minor notational changes, even simplifying a bit Rohlin’s original. For convenience let us thus copy Rohlin’s prose (while adapting it to the broader context, brackets are our additions): Denote by $B_C\subset \RR P^2$ the bounding disc for the oval $C$. Complete the half $C^+$ of the dividing curve to a closed surface $\Sigma$ by adding nonintersecting copies of the disc $B_C$. Let $T$ be the closed surface obtained from the other half $C^{-}$ by the same procedure, and let $\varphi \colon \Sigma \to \CC P^2$ and $\psi \colon T \to \CC P^2$ be mappings fixed[^26] on $C^+$ and $C^-$ and superimposing copies of $B_C$ onto these discs. Further, let $\xi$, $\eta$ be elements of the (integral) homology group $H_2(\CC P^2)$ determined by the mappings $\varphi$ and $\psi$ and the natural orientations on the pretzels $\Sigma$ and $T$ (i.e. the orientation obtained from $C^+$ and $C^-$). We shall establish Eq. by computing the intersection index $\xi \eta$ by two procedures \[geometrically and algebraically\]. 1.—The first procedure is based on the fact that $\xi \eta$ can be interpreted as the algebraic number of points in the intersection of the oriented singular pretzels $\varphi\colon \Sigma \to \CC P^2 $ and $\psi \colon T \to \CC P^2 $. This number cannot be determined directly, since the intersection consists of wholes disks, and we begin by applying a deformation to $\varphi$, making the intersection more regular\[=nearly transverse\]. Let $u$ be some tangent vector field on $\RR P^2$ with a finite number of zeros, not having zeros on $A:=C_m(\RR)$ and normal to $A$ on $A$. Since the field $iu$ is normal to $\RR P^2$ in $\CC P^2$ and normal to $\CC A:=C_m(\CC)$ on $A$, it can be normally extended to some field $v$ on $\RR P^2 \cup C^+$ (the latter, of course, will have zeros inside of $C^+$); let $\gamma\colon \RR P^2 \cup C^+ \to \CC P^2$ be a geodesic translation defined by the field $\delta v$, where $\delta$ is a sufficiently small positive number, and $\varphi'\colon \Sigma\to \CC P^2$ be the mapping defined by the formula $\varphi'(x)=\gamma(\varphi(x))$. For $\varphi'$ the algebraic number of points of intersection with $\psi$ is determined directly and can be found in the following way. Since the sum index of the singularities of $u$ in each of the disks $B_C$ is equal to 1 and multiplication by $i$ anti-isomorphically maps the tangent bundle of $\RR P^2$ onto its normal bundle in $\CC P^2$, the sum index of $v$ on each of the disks $B_C$ is equal to $-1$. Consequently, the contribution added by the pair of disks $B_C$ and $B_C'$ to the algebraic number \[$\xi \eta$\] of intersection points that we are interested in is equal to: $\bullet$ $+1$ if $C=C'$; $\bullet$ $+2$ if the pair $C,C'$ is negative, and equal to $\bullet$ $-2$ if the pair $C,C'$ is positive. This number itself is thus equal to $r-2(\Pi^+-\Pi^-)$. Since $\varphi'$ is homotopic to $\varphi $, the index $\xi \eta$ is also like that and thus $$\xi \eta=r-2(\Pi^+-\Pi^-).$$ 2.—The second procedure reduces to two remarks. First the class of $\xi + \eta$ is realized by the surface $\CC A$ and therefore coincides with $2k \alpha$, where $\alpha$ is the natural generator of the group $H_2(\CC P^2)$. Second, since the homomorphism $conj_\ast\colon H_2(\CC P^2)\to H_2(\CC P^2)$ represents multiplication by $-1$ \[as it flips the orientation of the generator interpreted as the fundamental class of a line defined over $\RR$\] and takes $\xi$ to $-\eta$, we have $\xi= \eta$. From these remarks it follows that $\xi=k\alpha$, $\eta=k\alpha$ and $\xi \eta=k^2$. Comparing the last equations obtained along each procedure, we obtain the announced formula . We list some consequences. First a (promised) remark about quartics: [(Klein 1876, Rohlin 1978)]{} Any quartic with $2$ unnested ovals is nondividing. Since there is no nesting there in no pairs of ovals and the left-side of Rohlin’s formula vanishes, while the right-side is equal to $r-k^2=2-2^2=2-4=-2$. The sextic scheme $5$ (five unnested ovals) is of type II. More generally the sextic scheme $r$ ($0\le r\le 11$ excepted $r=9$) is of type II (actually $11$ is not realized by Hilbert, Kahn 1909, Löbenstein 1910, Rohn 1911–13, Petrovskii 1938, Gudkov, but a more limpid proof follows from Rohlin’s formula). (due to Rohlin 1978 [@Rohlin_1978], also in Fiedler 1981 [@Fiedler_1981 p.13]). Since there is no nesting $\Pi^{\pm}$ are both zero, while the left-side $r-k^2=r-3^2$ of Rohlin’s formula vanishes only for $r=9$. \[Rohlin’s-inequality:cor\] [(Rohlin’s inequality)]{} A dividing plane curve of (even) order $m$ has at least $r\ge m/2$ ovals. Further if equality $r=m/2$ holds (and the curve is dividing) then its real scheme must be a deep nest (i.e. $m/2$ ovals each pair of them being nested). (explicit in Marin 1979 [@Marin_1979], or Gabard 2000 [@Gabard_2000 p.148], but due to Rohlin). Let $\Pi=\Pi^+ +\Pi^-$ be the total number of nested pairs of ovals. We have $$\Pi\le \textstyle\binom{r}{2},$$ (binomial coefficient counting the number of pair of a finite set of size $r$). Equality occurs only for a deep nest! Rohlin’s formula gives: $$r=k^2+2(\Pi^+-\Pi^-)\ge k^2-2\Pi^-\ge k^2-2\Pi\ge k^2-2 \textstyle\binom{r}{2}=k^2-r(r-1),$$ whence (looking at the extremities) $r^2\ge k^2$, i.e. $r\ge k$. If an equality each intermediate estimates crunch to equality, in particular the estimate $\Pi\le \binom{r}{2}$, which is fulfilled only for a deep nest. The sequel studies Rohlin’s consequence in degree 6. This is a bit pedestrian, and can be omitted as we gave a somewhat more conceptual explanation before, by noticing that Rohlin implies Arnold, etc. Assume again no-nesting ($\Pi=0$). Then Rohlin’s formula gives $0=2(\Pi^+-\Pi^-)=r-k^2=r-9$, it follows $r=9+ 0= 9$ (in accordance with Fig.\[Gudkov-Table3:fig\]). It is quite remarkable to notice that this gives an instant proof of Hilbert’s conjecture (and semi-theorem of his students Kahn-Löbenstein and Rohn, etc.) to the effect that there is no $M$-curve with 11 unnested ovals. Next we assume that there is one pair of nested ovals ($\Pi=1$). Then Rohlin’s formula gives $\pm 2=2(\Pi^+-\Pi^-)=r-k^2=r-9$, it follows $r=9\pm 2= 11, 7$ (in accordance with Fig.\[Gudkov-Table3:fig\]). Next suppose $2$ nested pairs ($\Pi=2$). Hence $\{ 4, 0,-4 \} \ni 2(\Pi^+ -\Pi^-)=r-k^2=r-9$, it follows $r=9+ \{ 4,0,-4 \}= 13, 9,5$ (in accordance with Fig.\[Gudkov-Table3:fig\]). For 3 nested pairs, $\{ 6, 2,-2, -6 \} \ni 2(\Pi^+ -\Pi^-)=r-k^2=r-9$, it follows $r=9+ \{ 6, 2,-2, -6 \}= 15, 11, 7,3$ (in accordance with Fig.\[Gudkov-Table3:fig\]). For 4 nested pairs, $\{ 8, 4, 0, -4,-8 \} \ni 2(\Pi^+ -\Pi^-)=r-k^2=r-9$, it follows $r=9+ \{ 8, 4, 0, -4,-8 \}= 17, 13, 9,5,1$ (in accordance with Fig.\[Gudkov-Table3:fig\]). For 5 nested pairs, $\{ 10, 6, 2, -2,-6, -10 \} \ni 2(\Pi^+ -\Pi^-)=r-k^2=r-9$, it follows $r=9+ \{ 10, 6, 2, -2,-6, -10 \}= 19, 15, 11, 7, 3, -1$ (in accordance with Fig.\[Gudkov-Table3:fig\]). Etc., at this stage it is clear how to link the arithmetics of Rohlin’s formula to the geometry of Gudkov’s table enhanced by Rohlin’s data, and we have proven Rohlin’s claim: All green-squared schemes on Gudkov’s table=Fig.\[Gudkov-Table3:fig\] are of type II. One noteworthy feature of the diagrammatic is that Rohlin’s formula gives a tiling by squares rooted on our blue-rhombs plus the red circles of Fig.\[Gudkov-Table3:fig\]. Hence all the schemes not situated on this grid are necessarily of type II. In particular since $M$-schemes are forced to type I it follows from Rohlin that all $M$-schemes not on the square grid are prohibited as real schemes (yielding a significant contribution to Hilbert’s 16th problem). Explicitly we have: All the $M$-schemes outside the grid are not realized (algebraically), that is $\frac{10}{1}$, $\frac{8}{1}2$, $\frac{6}{1}4$, $\frac{4}{1}6$, $\frac{2}{1}8$, $11$. However Rohlin’s formula alone fails to prohibit the schemes $\frac{7}{1}3$ and $\frac{3}{1}7$ (which are situated on the grid). Those are however prohibited either by the Hilbert-Rohn-Gudkov method, or by the Gudkov hypothesis proved by Rohlin 1972/72 (as detailed in the next Sec.\[Gudkov-hypothesis:sec\]). This last corollary helps the beginner to catch the substance of the following remark by Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.736]: [*“Another fundamental result difficult to overestimate is Rokhlin’s formula for complex orientations. The notion of complex orientation of a dividing real curve (see below), as well as Rokhlin’s formula and its proof, seem incredibly transparent at first sight. The formula settles, for example, two of Hilbert’s conjectures on 11 ovals of plane sextics, which Hilbert himself tried to prove in a very sophisticated way and then included in his famous problem list (as the sixteenth problem).”*]{} To remind it seems that Hilbert conjectured (wrongly) that only the schemes $\frac{9}{1}1$ and $\frac{1}{1}9$ do exist among $M$-schemes. This turned out to be wrong when Gudkov exhibited the scheme $\frac{5}{1}5$. So Rohlin’s formula settles actually six (!) of Hilbert’s conjectures (if taken as individual prohibition). Presumably what Degtyarev-Kharlamov had in mind were the extreme schemes $\frac{10}{1}$ and $11$ (eleven unnested ovals). The philosophical outcome of this spectacular Rohlin formula is how much information can be derived by basic topological methods, basically emanating from the Riemann-Betti-Poincaré tradition, yet to which workers like Klein or Hilbert were not enough familiar with. Of course a first class topologist like Rohlin was needed to reveal this truth. Gudkov hypothesis (Gudkov 1969, Arnold 1971, Rohlin 1972, etc.) {#Gudkov-hypothesis:sec} --------------------------------------------------------------- \[07.01.13\] For $M$-curves, the congruence $\chi\equiv_8 k^2$ was conjectured by Gudkov on the basis of experimental data gathered along his Hilbert-Rohn approach for sextics, and of course by looking as well to higher degrees via the Harnack-Hilbert construction. Figs.\[HilbGab1:fig\], \[HilbGab2:fig\], \[HilbGab4:fig\] below illustrate with which metronomic precision the Hilbert construction always produce $M$-curves respecting the congruence $\chi\equiv_8 k^2$. As pointed somewhere in Viro’s writings, nothing could thus have impeded Miss Ragsdale to detect this congruence in 1906 already. Yet it is the full-credit of Gudkov to have spotted this regularity. Once Arnold knew this, it was just a matter of hard-work toward elaborating the right strategy of proof, and some extra-skills of Rohlin turned to be indispensable. So the full proof belongs to Rohlin 1972/72 [@Rohlin_1972/72-Proof-of-a-conj-of-Gudkov] (alas contain a little bug), boosting ideas initiated by Arnold 1971 [@Arnold_1971/72] (who got the weaker congruence mod 4). Rohlin’s proof extract his punch not just from algebra (divisibility by 8 of an even integral unimodular quadratic form) but from the deeper divisibility by 16 coming from his own old “grand cru” of 1952 (Rohlin 1952 [@Rohlin_1952-4-manifolds]) on the signature of spin smooth $4$-manifolds. It is notorious that Rohlin’s proof (1972/72 ) contains a mistake that was repaired by Guillou-Marin 1977 [@Guillou-Marin_1977] (compare e.g. Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.736] and also Wilson 1978 [@Wilson_1978], who seems to have noticed (the same?) gap). \[Ragsdale-Petrovskii:notatio\] [(Ragsdale 1906 [@Ragsdale_1906], Petrovskii 1938 [@Petrowsky_1938]) Given a plane curve of even order (or more generally a real scheme of ovals), it is customary to denote by $p$ the number of [*even ovals*]{} (those included in an even number of ovals) and by $n$ the number of [*odd ovals*]{} (defined analogously). The difference $p-n$ can always be interpreted as the Euler characteristic $\chi$ of the orientable membrane of $\RR P^2$ bounding the curve. The notation $p,n$ are Petrovskii’s, intended to stand for positive and negative ovals.]{} \[Gudkov-hypothesis:thm\] [(Gudkov hypothesis/conjecture =Rohlin’s theorem of 1972, modulo a correction by Guillou-Marin 1977), and another proof in Rohlin 1974]{} A plane $M$-curve of degree $m=2k$ satisfies the Gudkov-Rohlin congruence: $$\chi=p-n\equiv k^2 \pmod 8.$$ The technique is akin to the subsequent Rohlin’s complex orientation formula of 1974–78, namely fill the halves of the orthosymmetric curve to a closed membrane and calculate the resulting intersection. However here the proof use (an extension of) the seminal Rohlin theorem 1952 [@Rohlin_1952-4-manifolds] on the divisibility by 16 of the signature of spin 4-manifolds. (At this stage there is a huge constellation of coincidence around Hilbert’s heritage: the 16th problem and as well his student, H. Weyl, whose “Analisis situs combinatorio” of 1922 is the first place where the signature of $4k$-manifolds is defined). So a breathtaking connection between differential topology and the more rigid algebraic geometry is accomplished in the Arnold-Rohlin era. Logically this proof is a bit tricky to implement for one is warned of some mistakes in Rohlin’s initial paper. Hence pivotal is the Guillou-Marin extension of Rohlin’s signature formula, for a full exposition cf. Guillou-Marin 1986 [@Guillou-Marin_1986]. Once this is understood its application to Gudkov’s hypothesis is exposed in A’Campo 1979 [@A'Campo_1979] (following a presentation due to Marin). Among all sextic $M$-schemes only those of Harnack, Hilbert and Gudkov exist. Originally the proof was achieved by Gudkov 1954 [@Gudkov_1954] via the Hilbert-Rohn(-Gudkov) method, i.e. supplemented by the concept of roughness coming from the Andronov-Pontrjagin theory of dynamical (structural stability). However here we derive it rather from the above theorem (Rohlin 1972). For schemes of degree 6, written in Gudkov’s notation $\frac{k}1 \ell$, we obviously have $p=\ell+1$ and $n=k$. Some boring computation is required to check that this prohibit all $M$-schemes above the broken line in Gudkov’s table. Indeed: $\bullet$ for $\frac{10}{1}$, $p-n=1-10=-9=-1\neq +1=k^2 \pmod 8$. $\bullet$ for $\frac{9}{1} 1$, $p-n=2-9=-7= +1=k^2 \pmod 8$ (no obstruction), $\bullet$ for $\frac{8}{1} 2$, $p-n=3-8=-5\neq +1=k^2 \pmod 8$, $\bullet$ for $\frac{7}{1} 3$, $p-n=4-7=-3\neq +1=k^2 \pmod 8$, etc. (progression by 2 units), so the rest is better done mentally on looking at Gudkov’s table (Fig.\[Gudkov-Table3:fig\]) of which we reproduce the top portion below (Fig.\[Gudkov-TableTop:fig\]). -5pt0 -5pt0 Interestingly, prohibiting sextic $M$-schemes is much easier (no deep differential topology à la Rohlin) for the schemes not situated at the two centers of the semi-hexagon of Gudkov’s table, i.e. $\frac{k}{1} \ell$ with $k$ even, whereas the “hexagonal” schemes $\frac{7}{1}3$ and its mirror $\frac{3}{1}7$ are much harder to disprove (at least in the modern Arnold-Rohlin theory). For sextics, one may wonder what is more elementary: Hilbert-Rohn-Gudkov or Rohlin 1952–1972. Remember that Arnold 1971 [@Arnold_1971/72] proved the weaker congruence modulo 4 of Gudkov’s hypothesis: [(Arnold 1971, Wilson 1978)]{} For $M$-curves of degree $2k$ (or more generally dividing curves, cf. [Wilson 1978 [@Wilson_1978 p.67–69]]{}), we have $$\chi=p-n\equiv k^2 \pmod 4.$$ This theorem of Arnold is more elementary than Gudkov-Rohlin, while prohibiting exactly the same $M$-schemes as those excluded by Rohlin’s formula (i.e. fails to exclude the $\frac{7}{1}3$ and its mirror $\frac{3}{1}7$). (Of course this is not so surprising as Rohlin owed some inspiration from Arnold.) Note also that Arnold’s congruence forces all schemes on the square-grid (extending the red-hexagons where $p-n\equiv -1 \pmod 4$) to be of type II as do Rohlin’s formula. The latter is however a bit stronger for ascribing type II to the schemes $5$, $\frac{1}{1}1$, and $1$. \[06.03.13\] In fact, as suggested in Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.737]: \[Rohlin-implies-Arnold:lem\] Rohlin’s formula [(\[Rohlin-formula:thm\])]{} implies straightforwardly the (extended) Arnold congruence $\chi\equiv k^2 \pmod 4$ (for dividing curves). This involves some abstract nonsense (yet pleasant) combinatorics. Using the usual notation of Petrovskii (cf. \[Ragsdale-Petrovskii:notatio\]), we have $\chi=p-n$ (Euler characteristic of the Ragsdale membrane), $r=p+n$ (total number of ovals split into $p$ even ones and $n$ odd ones), and Rohlin’s formula $2(\Pi^+-\Pi^-)=r-k^2$. Assembling this gives $$\begin{aligned} \label{Rohlin-to-Arnold:eq} \chi=p-n=(p+n)-2n&=r-2n\cr &=[2(\Pi^+-\Pi^-)+k^2]-2n\cr &=k^2+2(\Pi^+-\Pi^- -n).\end{aligned}$$ It remains to check that the “corrector term” $(\Pi^+-\Pi^- -n)$ is even. Modulo 2 we have, $ \Pi^+-\Pi^- \equiv_2 \Pi^+ +\Pi^- = \Pi$. Hence we can ultimately ignore Rohlin’s complex orientations. The following lemma concludes the proof, via the usual construction (like on Fig.\[Stalin3:fig\]) assigning to a plane curve its nested hierarchy of ovals ordered by inclusion of their insides (i.e. the unique bounding disc of the oval afforded by “Schoenflies theorem” applied in $\RR P^2$). Recall that a Jordan curve on any surface is null-homotopic iff it bounds a disc. (Cf. e.g. Reinhold Baer’s proof ca. 1927, Thesis under H. Kneser, reproduced in Gabard-Gauld 2010 [@Gabard-Gauld_2010-Jordan-and-Schoenflies].) -5pt0 -5pt0 \[Stalin:lemma\] Given a finite tree with a directed structure upward so that the tree really looks like the roots of a tree (or better a mushroom in Arnold’s metaphor). Formally we have a finite POSET where each element admits at most one superior, i.e. an element larger than it and minimal with this property (like in capitalistic or feodal hierarchies). Then the number $\Pi$ of pairs $x<y$ and the number $n$ of vertices lying at odd depths are congruent modulo $2$: $$\Pi\equiv n \pmod 2.$$ By additivity we may assume the tree connected. Then there is a unique maximal element in the hierarchy (Stalin), and we can draw from him all his subordinated elements as a “tree” growing downwards (cf. Fig.\[Stalin:fig\]) with several elements lying at different depths(=combinatorial distance to Stalin). -5pt0 -5pt0 Let $p_0, p_2, p_4, \dots$ be the number of elements at even depths $0, 2, 4, \dots$, and $n_1, n_3, n_5, \dots$ be those at odd depths $1,3,5, \dots$ respectively. To count $\Pi$ the number of subordinations of the hierarchy, we range them by order of importance (proximity to Stalin). Since an element has as many superiors as its depth, this gives $$\Pi=n_1+2p_2+3n_3+4p_4+5n_5+\dots\equiv_2 n_1+n_3+n_5+\dots=n.$$ This enumeration clearly exhausts all possible hierarchical pairs, and the proof is complete. Gudkov-Rohlin congruence via Rohlin’s formula? {#Gudkov-hyp-via-Rohlin's-formula:sec} ---------------------------------------------- \[11.03.13\] In fact the programme of this section looks extremely dubious, just by virtue of the diagrammatic of the Gudkov table in degree 6 (=Fig.\[Gudkov-Table3:fig\]). Indeed for a scheme like $\frac{7}{1}3$ the Rohlin equation is trivially soluble (cf. e.g. Theorem \[no-chance-to-reduce-Gudkov-to-Rohlin:thm\]). Therefore there is little chance to reduce Gudkov hypothesis to Rohlin’s formula and the sole signs-law on the Rohlin tree, unless one is able to infer sharper information on complex orientations from geometrical considerations, maybe via total pencils that are fairly easy to construct (cf. Theorem \[total-reality-of-plane-M-curves:thm\]) but it is probably another matter to visualize their dynamics. Hence we recommend to skip reading this section. \[08.03.13\] In view of the previous reduction of Arnold’s mod 4 congruence to Rohlin’s formula, an evident idea is to get better control on the residue modulo 4 of the term $(\Pi^+- \Pi^- -n)$ occurring in Equation  to draw sharper congruences (than Arnold’s). In the above proof we ignored completely (and could do so) the sign of Rohlin’s pairs, yet there is an evident composition law when the Hilbert tree of the scheme is decorated by signs (dictated by Rohlin’s pairs with complex orientations, see Fig.\[Rohlinsformula:fig\]). It seems however unlikely that one can boost the method up to include a proof of Gudkov hypothesis based on Rohlin’s formula. (If feasible, this would have certainly been mentioned in the Degtyarev-Kharlamov survey [@Degtyarev-Kharlamov_2000]). If optimistic one could use a total pencil (like in Theorem \[total-reality-of-plane-M-curves:thm\]) as to control complex orientations. This could give an elementary proof of Gudkov hypothesis via basic algebraic geometry instead of highbrow topology (like Rohlin 1952, or Marin 1977–79). Let us look if Rohlin’s formula implies the Gudkov-Rohlin congruence. As above, we start from the Rohlin-to-Arnold equation $$\chi=k^2+2(\Pi^+-\Pi^- -n),$$ and try now to control the residue modulo $4$ of the corrector term $(\Pi^+-\Pi^- -n)$ under the assumption that the curve $C_{m=2k}$ is an $M$-curve. (En passant, it seems that this corrector term is always $\le 0$ by Thom’s conjecture, cf. Theorem \[Thom-Ragsdale:thm\]. \[29.03.13\] Warning: this is false!) If one is able to show that this corrector term is $0$ modulo 4 then Gudkov hypothesis follows. As in the previous reduction of Arnold-to-Rohlin, we consider the hierarchy of the scheme (alias tree or mushroom), but now one takes into account complex orientations. The latter induce a distribution of signs on all injective pairs of the tree. (An injective pair is any hierarchical pair $x>y$ like in Hegel’s dialectic “du maître et de l’esclave” where $x$ is not necessarily the direct superior of $y$.) \[Signs-law:lem\] [(Signs-law)]{}.—In the Rohlin tree with (injective) pairs decorated with signs $\pm 1=\sigma_{x,y}$ we have given two (composable) pairs $x<y$, and $y<z$ the following “twisted” signs-law: $$\sigma_{x,z}=(-1)\sigma_{x,y} \cdot \sigma_{y,z}.$$ This looks a priori exotic, as it amounts to say that $+\times +=-$, $-\times-=-$, i.e. consanguinity is bad, while mixing the genes is good, i.e. $+\times -=+$, $-\times+=+$. This exotic signs law is justified by looking at Fig.\[Signs-law-dyad:fig\]: -5pt0 -5pt0 Of course the boring sign $(-1)$ in the lemma could be avoided if we flipped the convention in Rohlin’s definition, but we are too conservative to risk such a modification. Actually, Rohlin’s convention is perfectly sound, cf. again Fig.\[Rohlinsformula:fig\]. Next, it may be observed that the difference $\Delta \Pi=\Pi^+-\Pi^-$ computed locally in reference to a pair of consecutive edges is $\sigma_{x,y}+\sigma_{y,z}+\sigma_{x,z}= \sigma_{x,y}+\sigma_{y,z}+(-1)\sigma_{x,y} \cdot \sigma_{y,z}$ which is always either $+1$ or $-3$ (compare again Fig.\[Signs-law-dyad:fig\]), hence always $+1$ modulo 4. Globally on the whole Rohlin tree, we have the formula $$\Delta \Pi:=\Pi^+-\Pi^-= \sum_{all\; edges (x<y)} \sigma_{x,y}.$$ One could hope via the signs-law to evaluate this modulo 4, and all should boil down to $n$ modulo $4$ under the assumption of an $M$-curve. Along each triad $x<y<z$ the contribution is $+1$ modulo 4. However it becomes soon a combinatorial mess, and one cannot hope this in full generality, as otherwise the Gudkov-Rohlin congruence mod 8 would hold for all dividing curves and not merely for $M$-curves. This violates experimental knowledge, e.g. the Gudkov-Rohlin table in degree 6 (Fig.\[Gudkov-Table3:fig\]). A naive idea is to write down a cumbersome formula evaluating $\Delta \Pi$, but alas this still does not use the $M$-curve assumption. Maybe this is a first necessary step unless one has some better idea. Denote as before $p_0,n_1,p_2,n_3,p_4,\dots$ the number of ovals at depths $0,1,2,3,4,\dots$ respectively. Using Rohlin’s signs we define an oval at depth $\ge 1$ as positive if the edge immediately above it is a positive pair and as negative otherwise. Accordingly we get splittings: $$\begin{aligned} \label{splitting-rel:eq} n_1&=n_1^+ + n_1^-\cr p_2&=p_2^+ + p_2^- =p_2^{++}+p_2^{+-}+p_2^{-+}+p_2^{--} \cr n_3&=n_3^+ + n_3^- =n_3^{++}+n_3^{+-}+n_3^{-+}+n_3^{--}=n_3^{+++}+n_3^{++-}+etc., $$ where $n_1^+$ is the number of oval at depth 1 which are positive, $p_2^{++}$ is the number of ovals at depth 2 such that the 2 edges right above it are positive, while $p_2^{+-}$ is the number of ovals at depth 2 surmounted by 2 edges of signs $+$ and $-$ (in this order when moving up), and so on. Once all this notation is introduced we can write down a cumbersome formula for $\Delta \Pi$ enumerating all edges (=injective pairs) weighted by their signs according to the depth of their starting-point: $$\begin{aligned} \Delta \Pi =& n_1^+ - n_1^-\cr &+p_2^+ -p_2^- + (-p_2^{++}+p_2^{+-}+p_2^{-+}-p_2^{--})\cr &+n_3^+ -n_3^- + (-n_3^{++}+n_3^{+-}+n_3^{-+}-n_3^{--})\cr &+(+n_3^{+++}-n_3^{++-}-n_3^{+-+}+n_3^{+--} -n_3^{-++}+n_3^{-+-}+n_3^{--+}-n_3^{---})\cr &+etc.\end{aligned}$$ Using the splitting relations above this can be somewhat condensed as $$\begin{aligned} \Delta \Pi =& n_1^+ - n_1^-\cr &+ 2p_2^{+-}-2p_2^{--}\cr &+(+n_3^{+++}-n_3^{++-}+n_3^{+-+}-n_3^{+--} -n_3^{-++}+n_3^{-+-}-n_3^{--+}-3n_3^{---})\cr &+etc.\end{aligned}$$ alas some intelligence is required to decipher the hidden structure. Even if properly done we still require to put into action the $M$-curve assumption. Since each non maximal vertices of the Rohlin tree defines a unique edge above it we have the relation $r=p_0+ number\; of\; edges$. This is only a weak grip. All this mess is just given as to motivate someone to arrange a combinatorial proof of the Gudkov hypothesis $\chi\equiv k^2 \pmod 8$ on the basis of Rohlin’s formula alone. This seems a serious combinatorial challenge. Since all classical proofs—(i.e., the first erroneous one of Rohlin 1972, the latter one by Rohlin 1974 via Atiyah-Singer, plus the Marin-Guillou rescue of Rohlin’s original misproof, yet still via an extension of Rohlin’s deep result on signatures of spin 4-manifolds)—use some deep results it is quite unlikely that our naive programme can be completed. Still someone gifted in combinatorics with a clever idea on how to exploit the $M$-curve assumption (Harnack maximality) can perhaps crack the problem in a very elementary fashion. A vague suggestion is to exploit the total reality result for $M$-curves given in Theorem \[total-reality-of-plane-M-curves:thm\] prompting perhaps some information on complex orientations via the usual dextrogyration argument. Another project along this reductionism to Rohlin’s formula would be to attack our naive converse conjecture to the RKM-congruence, cf. Conjecture \[RKM-converse:conj\]. But this is merely a naive conjecture quite unlikely to hold true. A trinity of congruences: Gudkov-Krakhnov-Kharlamov and (Rohlin)-Kharlamov-Marin -------------------------------------------------------------------------------- \[07.01.13\] To prohibit $(M-1)$-schemes (above Gudkov’s broken line) one can use (beside the Hilbert-Rohn-Gudkov method) the following analogue congruence (paralleling that of Gudkov-Rohlin) due to Kharlamov 1973/73 [@Kharlamov_1973/73] and independently Gudkov-Krakhnov 1973/73 [@Gudkov-Krakhnov_1973/73]: [(Kharlamov 1973, Gudkov-Krakhnov 1973)]{} \[Gudkov-Krakhnov-Kharlamov-cong:thm\] A plane $(M-1)$-curve of degree $m=2k$ satisfies the congruence $$\chi=p-n\equiv k^2\pm 1 \pmod 8.$$ Several proof are available: $\bullet$ The original sources just referred to. $\bullet$ Since $(M-1)$-curves are not dividing the technique is different from the capping-off trick à la Arnold-Rohlin. However Marin is able to get an unified proof (à la Rohlin) by using the Guillou-Marin extension of Rohlin’s signature formula. For an exposition cf. A’Campo 1979 [@A'Campo_1979]. \[11.01.13\] We have also the following remarkable congruence (due independently to Kharlamov and Marin (first reported in print in Rohlin 1978 [@Rohlin_1978 3.4] and the first detailed proof is given in Marin 1979/80 [@Marin_1979]): [(Kharlamov 197?, Marin 1979/80, first reported in print in Rohlin 1978)]{} \[Kharlamov-Marin-cong:thm\] A plane $(M-2)$-curve of degree $m=2k$ and type II satisfies the congruence $$\chi=p-n\equiv k^2 \textrm{ or } k^2\pm 2 \pmod 8.$$ This can be paraphrased by saying that an $(M-2)$-curve with $\chi\equiv k^2+4 \pmod 8$ is necessarily of type I. Compare Rohlin 1978 [@Rohlin_1978 3.4] or Marin 1979/80 [@Marin_1979], or also Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91] for an unified account (and various approaches). For the paraphrase either look at the Gudkov table in degree 6, or more seriously do some boring arithmetics, cf. (\[RKM-congruence-reformulated:thm\]), which we reproduce quickly. The paraphrase follows from the fact that [*an $(M-2)$-curve of order $m=2k$ verifies universally $\chi \equiv k^2 \pmod 2$.*]{} This is easy to prove using the relation $\chi=p-n$, $r=p+n=M-2$ $$\chi=p-n=(p+n)-2n\equiv_2 p+n = r=M-2,$$ while by Harnack’s bound and the genus formula $g=\frac{(m-1)(m-2)}{2}$ we have $$M=g+1=\textstyle\frac{(2k-1)(2k-2)}{2}+1=(2k-1)(k-1)+1=2k^2-3k+2,$$ whence $$\chi\equiv_2 M-2=2k^2-3k \equiv_2 k \equiv_2 k^2.$$ Specializing to sextics ($m=6$, so $k=3$) implies the following (compare Fig.\[Gudkov-TableTop:fig\]): [(Rohlin 1978)]{} The two real sextic schemes $\frac{6}{1}2$ and $\frac{2}{1}6$ are of type I. It seems that this result had no classical counterpart à la Hilbert-Rohn-Gudkov prior to the Arnold-Rohlin revolution. Nonetheless Rohlin 1978 [@Rohlin_1978] mentions the possibility of a synthetic argument involving pencil of cubics. More on this in Sec.\[total-(M-2)-schemes:sec\]. \[10.03.13\] In fact this synthetic argument of Rohlin is now lost but was partially reconstructed by Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]. This issue of Rohlin-Le Touzé should have a strong interaction with Ahlfors theorem, while affording the first nontrivial phenomenon of total reality. We will have the occasion to dwell more on this later in this text. \[10.03.13\] Another little remark (valid in degree 6 but perhaps more universally) is the: Once we know Arnold’s congruence and the Gudkov-Krakhnov-Kharlamov(=GKK) congruence then Gudkov hypothesis follows formally. Indeed contemplating Gudkov’s table (Fig.\[Gudkov-Table3:fig\]), the Arnold congruence prohibits all while-colored $M$-schemes safe those at the center of the semi-hexagons (i.e. $\frac{7}{1}3$ and it mirror $\frac{3}{1}7$). Further the GKK-congruence prohibits all white-colored $(M-1)$-schemes on that same Gudkov table. Hence if one of the two schemes $\frac{7}{1}3$ (or its mirror) existed, it would appear as an isolated island in the ocean. Yet, by transversality (à la Bertini-Morse-Whitney-Sard-de Rham-Thom[^27]) a generic pencil of curves through an (hypothetical) algebraic representant and any other curve with less oval, e.g. the anti-Fermat (invisible) curve with zero oval (equation $x_0^6+x_1^6+x_2^6=0$) would produce a combinatorial path on the Gudkov table. This violates isolation. (NB: even an eversion (Sec.\[Eversion:sec\]) can only take the scheme to its mirror, and so elementary Morse surgeries necessarily create an elementary path on the Gudkov table). \[10.03.13\] A more radical intuition of Rohlin 1978 (now partially justified by Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]) is that owing to their total reality the $(M-2)$-schemes $\frac{6}{1}2$ (and its mirror) are maximal. This explains all the prohibitions materialized by the (white) semi-hexagons on the Gudkov Table (Fig.\[Gudkov-Table3:fig\]), safe the 2 schemes $11$ and $\frac{10}{1}$ that were prohibited since the Hilbert-Rohn era (at least modulo some German sloppiness, made perfectly rigorous by Academician D.A. Gudkov). Nowadays prohibiting them is a trivial consequence of either Arnold’s congruence or Rohlin’s formula (\[Rohlin-formula:thm\]). So at least in degree 6, we see that the phenomenon of total reality acts as a strong unifying principle for classical prohibitions. Rohlin probably had the intuition that this phenomenon perpetuates in higher degrees. More along this vertiginous idea (potentially allied to Ahlfors theorem) will be discussed in Sec.\[Esquisse-dun-prog-deja-esquiss:sec\]. Total reality of the two maximal sextic $(M-2)$-schemes (Rohlin 1978, Le Touzé 2013) {#total-(M-2)-schemes:sec} ------------------------------------------------------------------------------------ \[03.01.13\] This is akin to Ahlfors’ theorem, yet somewhat different and actually the hard part of the game. Rohlin 1978 [@Rohlin_1978 p.94] writes the following cryptical note: \[Rohlin1978-total-reality:quote\] [(Rohlin 1978)]{} After a suitable modification, these arguments can be applied to some other schemes. For example, when we replace real lines by real curves of degree 2 we can establish that a real scheme of degree 8 consisting of 4 nests of depth 2 lying outside one another belongs to type I[^28], and when we apply it to curves of degree 3, we can establish (in a rather complicated way) that the schemes $\frac{6}{1}2$ and $\frac{2}{1} 6$ of degree 6 \[considered below in §3.8\][^29] belong to type I. However, all the schemes that we have so far succeeded in coping with by means of these devices are covered by Theorem 3.4 and 3.5. What is crucial here is the parenthetical comment “(in a rather complicated way)”. This is highly reminiscent of some Hilbertian prose “[*freilich auf einem au[ß]{}erordentlich umständlichen Wege*]{}”, cf. Hilbert 1891 (p.418, in Ges. Abh., Bd.II)): “[*Diesen Fall $n=6$ habe ich einer weiteren eingehenden Untersuchung unterworfen, wobei ich—freilich auf einem au[ß]{}erordentlich umständlichen Wege— fand, da[ß]{} die elf Züge einer Kurve 6-ter Ordnung keinesfalls sämtlich au[ß]{}erhalb un voneinander getrennt verlaufen können. Dieses Resultat erscheint mir deshalb von Interesse, weil er zeigt, da[ß]{} für Kurven mit der Maximalzahl von Zügen der topologisch einfachste Fall nicht immer möglich ist.*]{}” Of course both problems are slightly different but perhaps there is some common difficulty in both games, while it is not impossible that Rohlin’s made a direct winking at Hilbert’s prose. So Rohlin claims being able to prove the following (on the basis of pure geometry): [(Rohlin 1978, no published proof)]{} The two real sextic $(M-2)$-schemes $\frac{6}{1}2$ ($6$ ovals encapsulated in one oval and $2$ outsides) and $\frac{2}{1} 6$ ($2$ ovals encapsulated in one oval and $6$ outsides) are of type I, i.e. any smooth real curve realizing one of those schemes is necessarily orthosymmetric (=dividing) in the sense of Klein. On the basis of Rohlin’s Quote (right above) one guesses that the proof involves looking at a pencil of cubics through 8 points inside the deep ovals while checking total reality of the resulting morphism to the line. (As usual it results a circle map à la Ahlfors, which is of degree $3\cdot 6-8=10$ after degenerating the basepoints on the ovals. This quantity coincides with Gabard’s bound $r+p=\frac{r+(g+1)}{2}=\frac{(g-1)+(g+1)}{2}=g=10$.) Of course, it is quite sad that Rohlin did not found the place to write down the details. Naively the proof could be as follows. Take a cubic in the pencil based at some 8 points inside the $2+6=8$ deep ovals (equivalently those containing no ovals). If the cubic is connected then it visits all 8 points. Counting intersection we have $2\cdot 8=16$ intersections coming from the deep ovals, plus two intersections coming from traversing the enclosing oval. This gives 18 the maximum permitted by Bézout, whence the desired total reality. This looks simple, but this by no mean a complete argument. What to do if the cubic is not connected? One could of course try to arrange a pencil of connected cubics. Recall that the discriminant parametrizing singular cubics has degree $3(k-1)^2=3 \cdot 2^2=12$ of even degree. Thus there is no “Galois” obstruction to finding a line in the space of cubics $\vert 3 H \vert \approx {\Bbb P}^9$ missing the discriminant. Another objection is that our simpleminded proof equally well applies to all other $(M-2)$-schemes excepted $9$. Indeed this is clear for all of them since the ovals are split in two packets by the enclosing oval. In the case of $\frac{8}{1}$ the enclosing oval is also necessarily cut twice, since $C_3({\Bbb R})$ is not null-homotopic. This would imply that all $(M-2)$-schemes safe the unnested one ($9$) are of type I. This is however too radical and incompatible with experience (or with theory, e.g. Arnold’s congruence). For instance it is easy to alter Hilbert’s method to get the scheme $\frac{7}{1}1$ in a nondividing way as switched some signs of smoothing (cf. Fig.\[TypeII:fig\]). -5pt0 -5pt0 \[08.01.13\] Maybe a fruitful idea is to look at special pencils of cubics spanned by 2 reducible (split) cubics well-understood. For instance one could try with reducible cubics aggregating a conic and a line. (This looks too rigid as a conic can pass only through 5 points while the 3 remaining one are not necessarily aligned.) Another idea is to look for rational cubics (uninodal). Inside any cubics pencil (e.g. one based on the 8 deep ovals), we have by the theorem of Cheponkus-Marin 1988 (cf. Marin 1988 [@Marin_1988 p.192]) a curve with at most $M-3=(g+1)-3=(1+1)-3=-1$ components. This looks exotic and wrong for cubics for it would lack real circuits violating thereby Galois-Bézout! Shame on me, I forgot that Cheponkus-Marin assume the pencil to be even degree $>2$! However it is a well-known fact that a generic pencil of cubics has a curve with $r=1$, which sounds quite likely (though not forced by the even degree 12 of the discriminant). Even accepting this we get only one curve of the pencil which is connected, then of course a family of such, but we are by no mean ensured that all will be so. The case of the Gürtelkurve, or that of the sextic having a nest of depth 3, inclines to believe that the total reality of Rohlin’s sextic schemes enjoy some structural stability in the sense that it is enough to assign the 8 basepoints inside the deep nests to ensured total reality. So we are looking for something quite rare yet reasonably robust. Suppose we have some connected cubics in our pencils. When moving in the pencil it may splits in two components. This can occur (assuming genericity, i.e. transversality to the discriminant) either through the birth of an oval after crossing a solitary node or by a self-coalescence of the pseudoline of the cubic crossing in this case a non-solitary node (with two real branches). $\bullet$ In the first case the newly created oval cannot contribute to additional intersections. Here is our argument requiring some hypothesis. First our generic pencil will exhibit at most 12 uninodal curves (either solitary or non-solitary nodes). By general position it may be assumed that those 12 curves (as well as the allied 12 nodes) are not located on the given sextic $C_6$. When crossing the discriminant, the solitary node will inflate into a little surrounding oval (or at least nearby ovals). Thus by continuity this little oval do not interact with the $C_6$. In fact what is important is that our 12 nodes are not on the 8 basepoints. Hence our just born oval do not contribute visiting the base locus, which is therefore entirely visited by the residual pseudoline. This is connected and so total reality is preserved (even after crossing the wall), at least at some instantaneous future right after the crossing. Then the little bubble (oval) can inflate, yet as the number of real intersections is already maximum via the pseudoline, the oval cannot cross the $C_6$. Its motion is in some sense confined to its complement (of the $C_6$). Then two scenarios are possible, either the oval deflates again to some solitary points or it merges with the pseudoline. In both cases we come back to a curve with one circuit and total reality is ensured for topological reasons. This story has to be repeated perhaps 12 times but we seem finished, modulo analyzing the other case. $\bullet$ In the second case (real normal crossing) our pseudoline of the moving cubic self-collides with itself and then splits in two branches. Of course it may then result a loss of total reality. Imagine the crossing (non solitary node) to be located on the $C_6$, then at the critical time there are locally two intersections and soon afterwards these may disappear loosing two intersections. Yet assuming by general position that our nodes are never located on the $C_6$ (after eventually perturbing the 8 basepoints) we still have right after the critical time two real intersections, and total reality is conserved in the short run. Now our curve is decomposed in two branches, and accordingly so are the basepoints. If the pseudoline (at the post-critical level) contains a mixture of points both inside and outside the nonempty oval of $C_6$ we are happy for two extra intersections are created while entering in and evading out this separating oval. If not, then the oval of $C_3$ visits the 6 inner ovals of $C_6$ and the pseudoline the 2 outer oval of $C_6$. (The other situation is also possible but then total reality is obvious for the pseudoline must evade the surrounding oval.) Now our oval in the long run may loose two intersections. However as the intersection $C_3\cap C_6$ was totally real, this can only occur by a retraction of a tongue slipping inside the separating oval, and then the 6 inner ovals of $C_6$ are trapped inside an oval of a cubic. The latter is reasonably rigid and convex. This oval still has $2\cdot 6=12$ intersection with the 6 inner ovals. In particular it cannot shrink to a point. It cannot also evolve to an ellipse, as otherwise the 6 inner basepoints would be co-elliptic, which can be avoided by general position. OF COURSE the proof is still not finished and some idea need to be discovered. Perhaps let pass a conic through 5 of the inner basepoints, while noting that a 6th intersection is created by Galois-Bézout, etc... Of course all this needs much more substantiation! \[08.01.13\] The end of Rohlin’s Quote (above) shows that there must be alternative non-synthetic but crudely speaking topological proofs of the theorem. This is indeed what was discussed in the previous section. E-mail correspondence {#e-mail-Viro:sec} ===================== \[09.01.13\] This section gathers some responses given by experts (Viro, Marin, Orevkov, Kharlamov, Shustin, Le Touzé, etc.) to some naive questions of mine about the work of Rohlin. Here are the original messages in chronological order (inserted with the tacit approval of their authors). I acknowledge most sincerely their authors for the stimulating atmosphere it created and their generous sharing of knowledge. \[09.01.13\] Two naive questions on Rohlin 1978 Dear Viatcheslav, Alexis, Oleg, Stefan and Grisha, Sorry for disturbing so many experts among yours with some little aspect of the work of academician Vladimir Abramovich. (I should have written this message in French, yet cannot remember exactly about Oleg’s progresses over the last 6 years in that language.) I was those last days quite fascinated by reading Rohlin’s 1978 survey on complex topological characteristics of real curves in some more detail. As you all know, he gave a quite spectacular enhancement of Gudkov’s pyramid for all schemes of sextics by enriching it with the data of Klein’s type I/II (1876). (Compare optionally the attached pdf file giving a graphical snapshot view of Rohlin’s classification.) My two questions are as follows. \(1) First Rohlin (1978) claims to have a certain synthetic argument (via pencils of cubics) able to show the type I of the schemes 6/1 2 and 2/1 6. He confesses however his argument to be a complicated one. Let me cite Rohlin exactly: “...when we apply it to curves of degree 3, we can establish (in a rather complicated way) that the schemes $\frac{6}{1}2$ and $\frac{2}{1} 6$ of degree 6 belong to type I. However, all the schemes that we have so far succeeded in coping with by means of these devices are covered by Theorem 3.4 and 3.5.” My first question is whether Rohlin’s synthetic argument has ever been published (assuming its truth of course)? I suspect the proof to be quite beautiful, but I am myself not able to prove it for the moment. Did one of you ever worked out the argument in detail, or remember about some exposition during Rohlin’s lectures? Is it of the same order of difficulty as the Hilbert-Rohn method, requiring “roughness” á la Andronov-Pontrjagin to turn round? Would it be didactically useful to publish (on the arXiv) an account of Rohlin’s argument if one is able to reconstruct it? \(2) The second question is of course the general Rohlin’s maximality conjecture (a scheme is of type I iff it is maximal in the hierarchy of all real schemes of some fixed degree). As reported in Viro’s survey (1986 Progresses over the last 6 years) it seems that one implication was disproved by Polotovskii and Shustin (combined efforts ca. 1982, 1985). Yet one implication looks still possible, namely type I implies maximal (if I am not wrong). It seems to me that this (last vestige of the) Rohlin conjecture could be proved somewhat eclectically in two lines via Ahlfors theorem (1950) on the total reality of orthosymmetric curves (alias type I). Namely if the curve is of type I, then there is a pencil of curves cutting only real points on the curve, so its real scheme cannot be enlarged without violating Bezout. q.e.d. Some more thinking shows of course this argument to be insufficient but maybe there is a (clever) way to complete it. Qu’en pensez-vous? Many thanks for your attention, and also for all your fantastic papers (I am presently trying to digest, so do not take the pain to answer me properly if my questions look too naive.) I apologize again for this collective message, but as the material is quite old, most of you probably forgot some details. So I hoped to maximize some chance of getting an answer from a collective chat room. Best regards, Alex PS: In attachment I send you a copy of an informal text of mine on the Ahlfors map. Section 24 (pp. 205–229) is more specifically devoted to Rohlin’s conjecture, yet contains nothing original (while being quite poorly organized). $\bullet$ On Wed, 9 Jan 2013 13:33:23 +0100 (alexandregabard@hotmail.com) wrote to Kharlamov, Marin, Viro, Fiedler, Orevkov, and Mikhalkin a collective e-mail titled “Two naive questions on Rohlin 1978”: Dear Viatcheslav, Alexis, Oleg, Thomas, Stepan and Grisha, Sorry for disturbing so many experts among yours with some little aspect of the work of academician Vladimir Abramovich. (I should have written this message in French, yet cannot remember exactly about Oleg’s progresses over the last 6 years in that language.) I was those last days quite fascinated by reading Rohlin’s 1978 survey on complex topological characteristics of real curves in some more detail. As you all know, he gave a quite spectacular enhancement of Gudkov’s pyramid for all schemes of sextics by enriching it with the data of Klein’s type I/II (1876). (Compare optionally Fig.71 on page 208 of the attached pdf file giving a graphical snapshot view of Rohlin’s achievement.) My two questions are as follows. \(1) First Rohlin (1978) claims to have a certain synthetic argument (via pencils of cubics) able to show the type I of the schemes 6/1 2 and 2/1 6. He confesses however his argument to be a complicated one. Let me cite Rohlin exactly: “...when we apply it to curves of degree 3, we can establish (in a rather complicated way) that the schemes $\frac{6}{1}2$ and $\frac{2}{1} 6$ of degree 6 belong to type I. However, all the schemes that we have so far succeeded in coping with by means of these devices are covered by Theorem 3.4 and 3.5.” My first question is whether Rohlin’s synthetic argument has ever been published (assuming its truth of course)? I suspect the proof to be quite beautiful, but I am myself not quite able to write it down for the moment. Did one of you ever worked out the argument in detail, or remember about some exposition during Rohlin’s seminar? Is it of the same order of difficulty as the Hilbert-Rohn method, requiring “roughness” à la Andronov-Pontrjagin to turn round? Would it be didactically useful to publish (on the arXiv) an account of Rohlin’s argument if one is able to reconstruct it? Many thanks if you have some ideas (or recent references) on those or related questions... \(2) The second question is of course the general Rohlin’s maximality conjecture (a scheme is of type I iff it is maximal in the hierarchy of all real schemes of some fixed degree). As reported in Viro’s survey (1986 Progresses over the last 6 years) it seems that one implication was disproved by Polotovskii and Shustin (combined efforts ca. 1982, 1985). Yet one implication looks still possible, namely “type I implies maximal” (if I am not wrong). It seems to me that this (last vestige of the) Rohlin conjecture could be proved (somewhat eclectically) in two lines via Ahlfors theorem (1950) on the total reality of orthosymmetric curves (alias type I). Namely if the curve is of type I, then there is a pencil of curves cutting only real points on the curve, so its real scheme cannot be enlarged without violating Bézout. q.e.d. Alas, some more thinking shows of course this argument to be insufficient but maybe there is a (clever) way to complete it. Qu’en pensez-vous? Many thanks for your attention, and also for all your fantastic papers (I am presently trying to digest). So do not take the pain to answer me properly if my questions sound too naive. I apologize again for this collective message, but as the material is quite old, most of you probably forgot some details. So I hoped to maximize some chance of getting an answer from a collective chat room. Best regards, Alex (Gabard) PS: The attachment[^30] is a copy of an informal text of mine on the Ahlfors map. Section 24 (pp. 205–229) is more specifically devoted to Rohlin’s conjecture, yet contains nothing original (except being poorly organized). $\bullet$ \[Viro’s answer the same day (09.01.13) ca. 20h00, additional footnotes are mine (Gabard)\] Dear Alexandre, Thank you for your message and manuscript. I was not aware about the Ahlfors theorem[^31]. It seems to be very interesting. I doubt though if it can be used for proving the half of Rokhlin conjecture. It gives a proof for impossibility of raising the number of components of a type I curve by a single algebraic Morse modification (what I called Klein’s thesis). I do not remember if I even ever heard about Rokhlin’s proof that you ask about, but the fact follows from the congruence. Slava[^32] did not mention it when he proved the corresponding congruence (at the moment the type was not yet considered). I learned this theorem from Slava in September 1977 and wrote down Slava’s proof to my notebook then. I guess the first proofs was[^33] published by Slava Nikulin (among many other statements) and Alexis Marin. Marin’s proof looks simpler, but requires Pin- structures. Best regards, Oleg $\bullet$ Gabard’s reply \[Same day (09.01.13) ca. 21h00\] Dear Oleg, Many thanks for your rapid and illuminating responses, plus all the historical details. If you see no objection, I would be very happy to cut-and-paste them in my survey. I still need to assimilate some congruences of the early phase (Rohlin, Gudkov-Krakhnov-Kharlamov, etc.) Hence you cannot imagine how your hints are illuminating my modest understanding of that golden period. Regarding Ahlfors, as you say, there is little hope to crack the big fish, yet of course I shall keep you informed if I get not too depressed by the immense difficulty. All the best, and so many thanks again, Alex $\bullet$ 10 Jan 2013 (Marin’s answer) Cher Gabard En plein déménagement, je met un peu plus de temps à vous répondre que Viro. Comme Viro, je ne connais pas la preuve de Rohlin pour votre première question (c’est pourquoi j’avais imaginé la preuve dont parle Viro qui est dans “Quelques remarques sur les courbes algébriques planes réelle”, votre référence 742) Cependant ce séminaire de Paris VII est dans un carton et y restera tant que je n’aurai pû trouver un nouvel appartement assez grand pour contenir ma bibliothèque et, n’ayant le temps d’aller à la bibliothèque, ma mémoire ne me permet pas de vous en dire plus que Viro. Pour la seconde question par contre je peux vous répondre, c’est à dire lever votre aveux d’incompréhension en fin (p. 226) de preuve du Lemme 24.20[^34]. Soit une courbe séparante gagnant un ovale de plus après franchiment d’un point quadratique ordinaire. Un argument de congruence (utilisant $d > 2$ dans le cas plan ou une hypothèse dans le cas général donnant que la désingularisée de cette courbe de franchiment est irréductible : l’ensemble de ses points complexe est connexe) donne que cette désingularisée de la courbe de franchiment est non séparante. Ainsi deux points non réels conjugués de la courbe de franchiment sont lié par un arc évitant la partie réelle, en particulier le point singulier, et par extension des isotipie[^35] un tel arc subsite dans toute déformation vers l’un des des deux côtés du discriminant, en particulier avant le franchiment la courbe est non séparante ce qui contredit l’hypothèse. Par contre si le franchiment du discriminant se fait en un point singulier plus compliqué il me semble que l’on peut augmenter le nombre de composantes connexes d’une courbe séparante. Je crois me souvenir que selon les constructions de Viro (ou peut être seulement après avec la présentation Itenbergienne de cette méthode de Viro) il y a une courbe singulière de degré 6 dont tout voisinage contient tous les types. N’étant plus familier du sujet depuis plus de 20 ans je ne peux vous en dire plus, par contre pour les surfaces de degré 3 vous trouverez dans le second tome des oeuvres de Klein un magnifique article illustré de non moins magnifiques figures où il établi que tous les types de surface cubique s’obtiennent par déformation de la (unique à changement projectif de coordonnées) surface cubique qui a 4 points quadratiques ordinaire. Merci de votre long article que j’essayerai de lire quand déménagement, vente,.... seront terminés. Bien cordialement et bonne année. Alexis Marin PS 1 Je trouve Viro un peu “oublieux” d’écrire “ (at the moment the type was not yet considered)”: en parcourant le second tome des oeuvres de Klein vous vous appercevrez qu’un sciècle avant Viro “tout” était chez Klein! 2 Vous trouverez un article historique, beaucoup plus court\* et sur un autre sujet en mettant dans la boite de recherche d’Arxiv le mot clef “troupeau” \*il fait 6 pages table des matières comprise et tout est dit (de façon “autocontenue” ) dans le résumé en français de la première page, mais si vous remontez à toutes les références\*\* des commentaires bibliographiques celà peut vous prendre un peu de temps. \*\*accesibles à travers la "bibliothèque des sophomores http://alexis.marin.free.fr/BIB/ $\bullet$ Gabard’s answer \[12.01.13 ca. 23h00\] Cher Alexis, C’est avec une immense joie que j’ai reçu votre message. N’ayant pas d’internet à la maison, je l’ai seulement découvert ce soir en visitant mon père, qui lui est connecté. Je vais donc tenter d’assimiler toutes vos remarques savantes, et si vous le permettez, de les intégrer dans mon survey, en spécifiant bien sûr qu’il s’agit de vos contributions. De mon côté, je me demande si une courbe non-séparante peut toujours acquérir un point double ordinaire solitaire. (C’est semi-implicite dans Klein 1876 qui écrivait “noch entwicklungsfähig”, mais il me semble que ça contredit le résultat de Shustin 1985 (contre-exemple à la conjecture de Rohlin), dont la logique m’échappe quelque peu, mais j’ai sûrement raté une subtilité). Grâce à vos commentaires je devrais pouvoir produire prochainement une version plus solide et limpide de la section correspondante du survey, que je vous enverrai dès que possible. L’interaction avec Ahlfors me semble aussi prometteuse... Amitiés, et merci infiniment pour votre message, Alex PS J’espère que le déménagement se passe bien. Restez-vous à Grenoble, ou bien s’agit-il d’une opération plus conséquente? PPS: J’ai bien à la maison votre article de Paris VII, qui a toujours été mon meilleur compagnon (en 1999-2000), et je suis content de le retrouver pour ce point encore plus profond. PPPS: je me suis procuré une copie de l’article sur “il capo”, qui me semble fabuleux. Merci beaucoup. C’est exactement l’analyse que l’on rencontre à proximité de Dirichlet, etc jusqu’à Ahlfors, et Rogosinski, et que je dois essayer à l’occasion d’apprivoiser... PS 1 Je trouve Viro un peu “oublieux” d’écrire “ (at the moment the type was not yet considered)”: en parcourant le second tome des oeuvres de Klein vous vous appercevrez qu’un sciècle avant Viro “tout” était chez Klein! Vous avez parfaitement raison, et je suis moi même très “spécialisé ” dans l’oeuvre de Klein. Cependant le gros quiz, c’est l’assertion de Teichmüller 1941, qui prétend que Klein 18XX? anticipe Ahlfors 1950, de 70 ans environ. Toute courbe séparante (ou surface de Riemann orthosymmétrique, pour reprendre le jargon kleinéen) admet un morphisme réel vers la droite dont les fibres au dessus des points réels sont toutes exclusivement formées de points réels. C’est cet énoncé fondamental qui me semble être sous-exploité! Evidemment comme la noté Viro, il implique la partie facile de l’assertion de Klein (1876): une courbe séparante ne peut gagner un ovale spontanément comme une bulle de champagne surgit du néant. $\bullet$ Réponse de Marin (le lendemain 13 Jan 2013 ca. 09h00) de les intégrer dans mon survey, en spécifiant bien sûr qu’il s’agit de vos contributions. A part l’explication de votre doute (où relativement à l’article que vous citez il n’y a que les mots “extension des isotopies” en plus) ce ne sont que de très vagues souvenirs que je vous conseille de vérifier (éventuellement auprès de plus compétent : Viro, Itenberg,... avant de les intégrer) De mon côté, je me demande si une courbe non-séparante peut toujours acquérir un point double ordinaire solitaire. voulez-vous dire dont les deux directions tangentes sont complexes conjuguée? celà me parait très très optimiste. (C’est semi-implicite dans Klein 1876 qui écrivait “noch entwicklungsfähig”, Êtes vous sûr que c’est ce que pensait Klein, ou incluait-il dans ce terme les modification par franchiment d’une courbe ayant un unique point double qui est quadratique ordinaire à tangentes réelles “apparu en rapprochant deux points d’un même ovale”? mais il me semble que ça contredit le résultat de Shustin 1985 (contre-exemple à la conjecture de Rohlin), dont la logique m’échappe quelque peu, mais j’ai sûrement raté une subtilité). PS J’espère que le déménagement se passe bien. oui mais c’est long, à ce propos, vous trouverez sur http://alexis.marin.free.fr/BIB/papier/ la liste des livres que j’ai en plusieurs exemplaires et (sauf ceux dont la colonne “héritier” est remplie (par Vinel et/ou Guillou)) qui sont à la disposition de qui (en particulier vous) les demande. Restez-vous à Grenoble, ou bien s’agit-il d’une opération plus conséquente? Je reste près de Grenoble (mon adresse est dans la signature électronique ci-dessous PPPS: je me suis procuré une copie de l’article sur “il capo”, Voulez vous dire “Le capo”? Cependant le gros quiz, c’est l’assertion de Teichmüller 1941, qui prétend que Klein 18XX? anticipe Ahlfors 1950, de 70 ans environ. Toute courbe séparante (ou surface de Riemann orthosymmétrique, pour reprendre le jargon kleinéen) admet un morphisme réel vers la droite dont les fibres au dessus des points réels sont toutes exclusivement formées de points réels. Voulez-vous dire revêtement d’espace total l’ensemble des ovales? Il y a-t-il quelque chose de plus précis sur le degré et sa répartition parmis les ovales? Les références sont-elles dans votre article? C’est cet énoncé fondamental qui me semble être sous-exploité! Evidemment comme la noté Viro, il implique la partie facile de l’assertion de Klein (1876): une courbe séparante ne peut gagner un ovale spontanément comme une bulle de champagne surgit du néant. Soyez plus précis pourquoi un tel morphisme admettrait-il une déformation le long de la modification d’adjonction d’un ovale? Amitiés. Alexis – http://le-tonneau-de-thales.tumblr.com/ Alexis Marin, chez Danielle Bozonat 6 Allée de la roseraie, 38240 Meylan fixe : 04 76 00 96 54 port. : 06 38 29 33 99, 00351925 271 040 $\bullet$ Gabard 13 Jan 2013 ca. 13h30 Cher Alexis, Merci pour votre message. Je vais en effet essayer d’intégrer vos commentaires de manière ciblée et prudente. De toute manière avant d’arXiver une nouvelle version d’ici six mois environ, j’aurai l’occasion de vous montrer précisement la prose que je vous aurez emprunté. J’essaye maintenant de répondre à vos questions: De mon côté, je me demande si une courbe non-séparante peut toujours acquérir un point double ordinaire solitaire. voulez-vous dire dont les deux directions tangentes sont complexes conjuguée? celà me parait très très optimiste. REPONSE: Oui, exactement à tangentes imaginaires conjuguées. Cela me parait aussi très optimiste. Klein semble le prétendre semi-implicitement (du moins qu’il n’ y a a priori pas d’obstruction topologique à la formation de telles bulles de champagne). Cependant si ce truc fou (“Klein-vache”) est vrai alors un des sens de la conjecture de Rohlin 1978 (type I iff maximal real scheme) est vérifié. Malheureusement, ce que donne “Klein-vache” est le sens de Rohlin détruit par Shustin 1985 (dont je n’ai cependant pas compris l’argument). Mais vous avez surement raison “Klein-vache” est probablement beaucoup trop optimiste... Êtes vous sûr que c’est ce que pensait Klein, ou incluait-il dans ce terme les modification par franchiment d’une courbe ayant un unique point double qui est quadratique ordinaire à tangentes réelles “apparu en rapprochant deux points d’un même ovale”? REPONSE: je pense que oui, car Klein précise “isolierte reelle Doppeltangente”, comparez ma Quote 24.2[^36] page 205 de mon survey (si vous n’avez pas le volume 2 de Klein sous la main). Ainsi il me semble que votre interprétation moderne (Marin 1988) diffère un peu de l’original Kleinéen, en étant toutefois plus puissant que l’assertion d’origine. PS J’espère que le déménagement se passe bien. oui mais c’est long, à ce propos, vous trouverez sur http://alexis.marin.free.fr/BIB/papier/ la liste des livres que j’ai en plusieurs exemplaires et (sauf ceux dont la colonne “héritier” est remplie (par Vinel et/ou Guillou)) qui sont à la disposition de qui (en particulier vous) les demande. C’est une magnifique liste de trésor. Je voudrais bien les acquérir, mais je me demande si mon hygiène de vie (overwork) rend une telle acquisition raisonable...(Il faudrait que je passe à Grenoble avec la camionnette de mon oncle pour récupérer les “invendus”. Il est préférable en effet de trouver des preneurs plus compétents que moi. Si en dernier recours, vous ne trouvez pas de preneurs je pourrais récupérer les volumes restants en vrac...Merci infiniment pour cette généreuse proposition. Moi même je suis très marginal financièrement et spatialement, petit appartement à Genève partagé avec ma mère (avec environ 8 tonnes de littérature mathématique), mais dans le futur je pourrai peut être m’installer dans une ferme fribourgoise, où il reste de l’espace pour expandre la bibliothèque...) PPPS: je me suis procuré une copie de l’article sur “il capo”, Voulez vous dire “Le capo”? Oui, j’essayais d’improviser en italien, mais c’est une langue plus subtil que vous utilisez... Cependant le gros quiz, c’est l’assertion de Teichmüller 1941, qui prétend que Klein 18XX? anticipe Ahlfors 1950, de 70 ans environ. Toute courbe séparante (ou surface de Riemann orthosymmétrique, pour reprendre le jargon kleinéen) admet un morphisme réel vers la droite dont les fibres au dessus des points réels sont toutes exclusivement formées de points réels. Voulez- vous dire revêtement d’espace total l’ensemble des ovales? Il y a-t-il quelque chose de plus précis sur le degré et sa répartition parmis les ovales? Les références sont-elles dans votre article? OUI, toute surface de Riemann à bord (=membrane compacte) s’exprime comme revêtement holomorphe ramifié du disque. C’est juste une version relative (à bord) du théorème d’existence de Riemann qui concrètise toute surface de Riemann close comme revetement conforme de la sphère (ronde). Il a fallut toutefois attendre la contribution d’Ahlfors 1950 qui donne en plus un contrôle sur le degré d’un tel revêtement conforme, à savoir r+2p, où r est le nombre d’“ovales” (mieux le nombre de contours de la membrane), et p son genre. La Thèse de moi-même (Gabard 2004, et l’article de 2006 au Commentarii Math. Helv.) donne un meilleur contrôle, à savoir $r+p$, en économisant donc une cartouche pour chaque anse. Les références précises sont dans le survey. L’énoncé d’Ahlfors était vachement anticipé dans le cas $p=0$ (membrane planaire ou schlichtartig pour reprendre la terminologie de Paul Koebe) par la grande lignée Riemann 1857 (Nachlass), Schottky 1875-77, Bieberbach 1925 et Grunsky 1937. Lorsqu’on passe au double de Schottky-Klein de la surface à bord on obtient (via Ahlfors) une courbe séparante avec un morphisme totalement réel vers la droite projective. Inversement toute courbe séparante est totalement réelle, puisqu’il suffit d’appliquer Ahlfors à une des moitiés orthosymétrique de Klein. \[Gabard\] C’est cet énoncé fondamental qui me semble être sous-exploité! Evidemment comme la noté Viro, il implique la partie facile de l’assertion de Klein (1876): une courbe séparante ne peut gagner un ovale spontanément comme une bulle de champagne surgit du néant. \[Marin\] Soyez plus précis pourquoi un tel morphisme admettrait-il une déformation le long de la modification d’adjonction d’un ovale? \[Gabard\] Je pense que ça marche car lorsque la courbe est plongée dans le plan, le morphisme total d’Ahlfors admet une réalisation projective comme un pinceau de courbes planes dont tous les membres découpent seulement des points réels sur la courbe orthosymmétrique (=séparante). Par conséquent, en traçant la courbe du pinceau total qui passe par un point de l’oval spontanément créé, on obtient une contradiction avec Bézout. Donc Ahlfors 1950 implique Klein 1876, mais votre démonstration de 1988$-\varepsilon$ (votre preuve est déjà mentionnée dans Viro 1986) est surement plus intrinsèque et voisine de l’argument d’origine de Klein (s’il en avait un au delà de la pure contemplation empirique des quartiques notamment...) Merci infiniment pour vos messages, et d’ici tout bientôt (3-4 jours) je vous enverrai une version mise-à-jour du survey qui clarifiera peut-être les assertions précédentes. Toutefois les grands problèmes et plein de détails m’échappent encore dans la pyramide Gudkovo-Rohlinienne. Quelle splendide pyramide qui joint à la perfection Klein et Hilbert! Un détail qui m’échappe, c’est le fait que le discriminant est de degré $3(m-1)^2=75$ pour $m=6$, tandis que que du point de vue des chirurgies “de Morse” il y a des cycles de longueur 4 dans la pyramide de Gudkov. Donc il y a un problème de parité si on déforme le long d’un pinceau générique (transverse au discriminant)...Désolé, de vous embêter avec ces détails que j’ai honte de ne pas réussir à clarifier depuis quelques jours. Amitiés, et bon courage pour la suite du déménagement, Alex $\bullet$ \[16h40 15.01.13\] Cher Alexis, Merci encore pour vos messages et vos remarques fascinantes que je dois encore bien digérer. De mon côté, j’ai fait de minimes progrès, et vous envoie malgré votre déménagement une version ajournée de mon survey. Il me semble que le truc fou dont nous parlions il y a quelques jours, que j’appele depuis “Klein-vache”, i.e. la possiblilité de faire naitre un noeud solitaire (à tangentes conjuguées) depuis n’importe quelle courbe diasymétrique est vrai pour les sextiques. Pour cela j’utilise un argument qui combine Rohlin 1978, Klein-Marin 1988, et Nikulin 1979 (classification isotopique) et un résultat relié de Itenberg 1994 (possibilité de contracter n’importe quel ovale vide, i.e. sans autre ovales dans son intérieur, sur un tel noeud isolé). Les détails de la preuve sont exposés dans la Prop.24.24\[meanwhile this is \[Klein-vache-deg-6:prop\]\], page 235 du fichier ci-joint. ? Evidemment, en principe “Klein-vache” n’a aucune chance d’être vrai en degré supérieur. Cependant la seule obstruction que je connaisse est ce résultat de Shustin 1985, dont je ne comprends toujours pas la logique de base (sans même parler du fait que c’est fondé sur la méthode de Viro, dissipation de singularités tacnodales..., une technologie que je n’ai jamais maitrisée). Mes objections naives à l’argument de Shustin se trouvent en page 248 (dans le paragraphe qui précède la Figure 94\[=meanwile Fig.\[Shustin:fig\]\]). Dans cette figure, je ne sais pas comment prohiber le $(M-1)$-schémas encadré par le carré vert (à mi-hauteur de la figure), et dans son article de 1985 Shustin n’est pas trés explicite. Mais bon, il s’agit la d’une question assez ennuyeuse et en fait je vais peut-être prendre l’initiative d’écrire un nouveau message collectif pour clarifier ce point d’ici quelques heures. Merci infiniment encore pour vos messages, et meilleurs voeux de courage pour la suite du déménagement, Amitiés, Alex PS: Pour l’instant j’ai inégré en vrac tous nos échanges e-mail dans le survey (p.219 et suivante), mais bien entendu dès que possible je censurerai les remarques plus confidentielles..., et masquerai les répététitions, voire l’intégralité de la discussion si je parviens bien à résumer votre apport malgré mon anglais catastrophique. Cependant en relisant vos remarques, elles apportent une prose substantielle que je ne saurais jamais reproduire en anglais, donc je trouverais très dommage de censurer vos souvenirs en vracs!!! Evidemment rien ne presse et je suis désolé de vous avoir dérangé durant cette délicate opération du déménagement inter-grenoblois. Amitiés, encore, et je vous tiens au courant d’éventuelles progrès...Je suis surtout curieux des réponses de Shustin (et Viro) s’ils parviennent à éclairer ma lanterne. PPS: Je joins une copie de la note de Shustin, si jamais, mais je ne veux pas vous distraire de votre tâche prioritaire... $\bullet$ \[15.01.13–18h30\] Dear Evgenii, Ilia, Oleg and Alexis (and Felix Klein), I was much fascinated those last days by Evgenii’s counterexample to (one part of) Rohlin’s maximality conjecture to the effect that a real scheme is of type I iff it is maximal in the hierarchy of all schemes. Quite interestingly this work of you (Shustin) also destroys an old (semi-)conjecture of Klein (1876) positing that any nondividing plane curve can acquire a solitary node by crossing only once the discriminant (the resulting Morse surgery then sembling like the formation a champagne bubble arising like a blue sky catastrophe of little green men’s coming with flying saucers). Alas from Shustin’s note of 1985 (in its English translation), I was not quite able to understand your proof (compare optionally the attached file, on page 248, in the paragraph right before Figure 94\[=Fig.\[Shustin:fig\]\]). In fact I do not know how to prohibit the $(M-1)$-scheme $4/1 2/1 1/1 11$ enlarging Shustin’s (M-2)-scheme. Alas I am not an expert in the field and I feel quite shameful disturbing you with such a detail. Despite having myself full Leningradian origins (through my father), I do not master the Russian language so that it may well be the case that the original Russian text is more detailed than its translation. Of course it is much more likely that I missed something well-known, that you perhaps may not have made completely explicit in the note? (Incidentally I send you a copy of Shustin’s note for convenience!) I apologize for this question of detail, yet it seems quite important to me for your result of 1985 is the only obstruction (I am aware of) to the naive desideratum of truth about Klein’s conjecture. Klein himself is extremely clever and quite ambiguous about stating this as a conjecture or as a result (compare optionally Klein’s original quote reproduced on page 206 of the attachment). Today I managed as a simple exercise to check the truth of Klein’s hypothesis in degree 6, via an armada of Russian results (especially Itenberg 1994 contraction principle for empty ovals), plus the Klein-Marin theorem (for the details of this exercise cf. optionally Prop.24.24\[=\[Klein-vache-deg-6:prop\]\] on page 235 of the attached text). You, Oleg Viro, in the preface of that volume presenting Itenberg’s article (1994) advanced the (crazy?) conjecture that one might always be able to contract empty ovals!!! Do you know if there is meanwhile some counterexample (in high degrees)? Of course there is some vague parallelism between Itenberg’s contraction and the one required to implement Klein’s hypothesis (which must amount shrinking an anti-oval, i.e. an invariant circle acted upon antipodically by conj). Sorry again for disturbing you with all these naive questions, and do not take the pain answering me properly if you are overwhelmed by other more important duties. Many thanks for all your attention. Sincerely yours, Alex (Gabard) $\bullet$$\bullet$$\bullet$ \[16.01.13–02h57: Oleg Viro\] Dear Alexandre, I do not mind to pose crazy conjectures. I do not mind if my crazy conjecture would be disproved. However, I suspect that my conjecture is not as crazy as possibility of shrinking of an anti-oval. The difference between the oval and an anti-oval is that the oval is assumed to exist and be empty, i.e., not linked with the complex curve in whatever sense, while the anti-oval apparently has none of these properties. I am not aware about any counter-examples that you ask about. I do not bet that they do not exist, but find the question stimulating, and better motivated than the conjecture that was proven to be wrong. Best, Oleg $\bullet$$\bullet$$\bullet$ \[16.01.13–14h56: Stepan Orevkov\] A small remark: It is wrong that $11 U 1<1> U 1<2> U 1<4>$ is not a part of an $(M-1)$-scheme. It is[^37]. Moreover, there is no known example of $(M-2)$-curve of type II which cannot be obtained from an $(M-1)$-curve by removing an empty oval. In contrary, there are $(M-1)$-curves of degree $8$ (which are necessarily of type II) which do not come from any $M$-curve. These are: $3<6>$ $4 U 1<2> U 2<6>$ $8 U 2<2> U 1<6>$ $12 U 3<2>$ Constru\[r\]ction (inspired by Shustin’s construction of $4 U 3<5>$): Consire\[der\] a tricuspidal quartic $Q_{sing}$ symmetric by a rotation $R$ by $120$ degree and perturb\[e\] is\[=it\] so that each cusp gives an oval (we assume that this perturbation is very small). Let $Q$ be the perturbed curve. Two flex points appear on $Q$ near each cusp of $Q_{sing}$. We chose flex points $p_0, p_1, p_2$ (one flex point near each cusp) so that $R(p_0)=p_1, R(p_1)=p_2, R(p2)=p_0$. We choose homogeneous coordinates $(x_0 : x_1 : x_2)$ so that the line $x_i = 0$ is tangent to $Q$ at $p_i$ $(i = 0,1,2)$. Let $C$ be the image of $Q$ under the Cremona transformation $(x_0 : x_1 : x_2) \mapsto (x_1x_2 : x_2x_0 : x_0x_1)$. Then $C$ has 3 singular points, each singular point has two irreducible local branches: a branch with $E6$ and a smooth branch which cuts it “transversally”. By a perturbation of $C$ we obtain all the four curves mentioned above. The fact that these curves cannot be obtained from $M$-curves immediately follows from the fact that, for any $M$-curve of degree 8 of the form $b U 1<a_1> U 1<a_2> U 1<a_3>$, all the numbers $a_1$, $a_2$, $a_3$ are odd[^38]. Best regards Stepa O $\bullet$ \[17.01.13 ca. 23h00\] Dear Oleg and Stepa, Many thanks for all your fascinating remarks and detailed answers. I look forward digesting them carefully tomorrow. Sorry for my late reply as I have no internet at home and was quite busy trying to understand some basic facts, notably that one may have some “eversion” of a real scheme when the oval explodes at infinity undergoing a Morse surgery not affecting its connectedness. This implies that there is some hidden passages in the Gudkov-Rohlin pyramid of all sextics changing a Gudkov symbol $k/l \ell$ to its mirror $\ell/1 k$. The resulting combinatorics of this graph looks quite formidable and I wonder if it is known whether each of those secret edges corresponding to eversions (except those linking $M$-curves) can be explored algebraically. Perhaps the problem is related to Ilia’s shrinking process for empty ovals, but seems to involve yet another species of “anti-ovals”, namely those with two fixed points under conj, yet located on the same oval. All what I am saying is for sure well-known to you since time immemorial, yet I was very happy to understand this point which solved several paradoxes of mine, notably those related to the degree of the discriminant and the contiguity graph between chambers residual to the discriminant under elementary algebraic Morse surgeries, as Oleg says. Of course, I shall send you an updated version of my file, when I manage to reorganize slightly the exposition. Many many thanks for all your excellent answers! All the best, Alex $\bullet$ \[18.01.13 ca. 10h00, Viatcheslav Kharlamov\] Dear Alex, I followed rather attentively the discussions, but kept silence since had no much to add to the reaction of the others. This “eversion”, as you call it, played some important role in the prehistory of the Gudkov conjecture. As you probably know, the first classification declared by Gudkov was wrong, and it is one of his “thesis referees”, Prof. Morosov, who had objected the first classification exactly because of a small irregularity with respect to “eversion” of the answer. Repairing this asymmetry Gudkov came to his final result, and, if my memory is correct, in particular, at this stage discovered the missing $M$-curve. If honestly, I don’t remember did somebody ever before discussed seriously any conceptual explanation to this “eversion”. However, it was implicitly present in all results obtained through $K3$ and their lattices. Recently, studying the shadows of cubic surfaces with Sergey Finashin and having proven, to our own surprise, for them a very similar “symmetry”, which we have called “partners relation”, we have formalized it as follows. First level of explanation is coming from lattices of double coverings: the partner relation consists indeed in transferring an $U$-summand (unimodular even lattice of rang $2$ and signature $0$) from one eighenlattice to another. Second level of explanation is coming from moduli in terms of periods: each partner in the partner pair can be deformed to a triple conic, near the triple conic the family looks as $Q^3+tbQ^2+t^2cQ+d=0$, and switching of the sign of $t$ (passing through the triple conic) replace curves of one deformation class by curves from the partners class: moreover, such degenerations are deformationally unique. Literally the same explanation (and with much easier proofs at the both levels) works for nonsingular sextics (the shadows are sextic curves with $6$ cusps on a conic; remarkably, in many respects they behave in a way more similar to that of nonsingular sextics, than other sextics with singularities). Yours, Viatcheslav Kharlamov $\bullet$ \[18.01.13, Kharlamov, title of message=Correction\] Writing the message a bit in a hurry I did not describe fully and appropriately the partner relation at the lattice language. The summand $U$ does play a crucial role, and it should be moved from one eighenlattice to another, but then additionally one should exchange the eighenlattices. In fact this $U$ contains indeed the $2$-polarization vector, $h¨2=2$, and thus the eighenlattice containing this distinguished U is aways $(-1)$-eighenlattice. The existing exception to the partner relation (as I remember, in the nonsingular case, there is only one) is the case when the $(-1)$-eighenlattice does not contain such a pair $(U,h)$. Sorry, for being in a hurry, but I should stop at this point. Hope that now it is more clear. $\bullet$ \[18.01.13, Gabard, ca. 21h00\] Dear Viatcheslav, Oleg and Stepa (and all the others), So many thanks for all the excellent comments, especially on Morosov. There was some allusion to this issue in Viro’s survey from 2006, in Japanese Journal of Math, as to the lack of symmetry in Gudkov’s initial answer. Yet Morosov was not mentioned if I remember well... On my side I was quite stimulated by the last letter from Oleg, about the contraction conjecture, as looking indeed much more realist than Klein’s Ansatz on the champagne bubbling in any nondividing curve. I attempted today to imagine what sort of proof one could expect to find for this fascinating Itenberg-Viro contraction conjecture of empty ovals. After some trials with orthogonal trajectories to the functional computing the area of the empty oval, I arrived at some sort of strategy (probably completely fantasist) consisting in using the Riemann mapping theorem as applied to the interior of the empty oval. Naively as the contour is algebraic so is the Riemann map and hence the concentric sublevels of it ought to be algebraic curves of the same degree!!??? This would give the shrinking. I am sure that tomorrow while checking more carefully the details all this argument will crash down. Hence sorry for this premature message. Some more details about this and my naive understanding of “eversions” are in the attached file, especially Section 24.15 (p.251) and p.242 (Section 24.12 for eversions). Regarding eversions I wonder which edges in the Gudkov pyramid are actually realized algebro-geometrically? All, except those connecting the $M$-schemes is my naive guess, yet it is probably too optimistic... Many thanks again for sharing all your knowledge on that fascinating topic, and all your exciting letters. All the best, Alex \[21.01.13, ca. 20h00\] Dear real geometers, Thank you again, Oleg, Alexis, Stepa, and Viatcheslav, for all your messages which I have carefully integrated in my TeX-notes, and to which I frequently refer for citation in my text. Your messages suggested me several ideas I would never have explored without your precious hints. On my side, I noticed of course that the cavalier Riemann mapping strategy toward the (Itenberg-Viro 1994) contraction conjecture (CC) of empty ovals fails blatantly (cf. Section 25.7(=\[CC-via-Riemann:sec\]), pages 255-258, roughly even if the Riemann map of an algebraic oval would be algebraic then its degree seems to be twice as big as it should, or better the polynomials arising as norms of algebraic Riemann maps are not the most general representatives of their degree!!!). Perhaps the Riemann method works for special ovals, but of course they are unlikely to be interspersed in all chambers of the discriminant! This failure drifted me toward another formulation of the contraction conjecture which I call CCC, for collective contraction conjecture. This posits that all empty ovals of a real algebraic curve can be contracted simultaneously toward solitary nodes (by a path having solely its end-point in the discriminant). This looks even more “crazy” than CC, but I found no counterexamples (in my pockets). I would much appreciate if you already thought about this natural variant, especially if you detected some counterexample (perhaps arising from the Viro-Itenberg patchworking method or the dissipation of higher singularities, with which I am alas still unfamiliar with, like in Shustin’s counterexample to Rohlin’s maximality conjecture). Here are the trivialities I managed to prove. Via Brusotti 1921, it is plain that CCC implies the usual contraction conjecture (CC) (cf. details in Lemma 25.22(=\[CCviaCCC-Brusotti:lem\]) on page 263 of the attached file). On the other hand CCC implies (as a large deformation principle) several well-known prohibitions. E.g. a two-seconds proof of the Hilbert-Rohn-Petrovskii prohibition of the sextic $M$-scheme $11$ (eleven ovals without nesting), as well as Rohlin’s prohibition of the sextic scheme $5$ of type I (by the way causing the unique asymmetry in the Gudkov-Rohlin table of sextics). Under CCC, all these facts appear as trivial consequences of Bézout (compare Section 25.8(=\[CCC:sec\]) on pages 258-259). I found this simplicity quite exciting (even though it leads to nothing new as compared to Arnold-Rohlin). One can wonder if Hilbert already used this, at least as a heuristic tool??? Philosophically, I found also interesting that such large deformation conjectures produce prohibitions, in contradistinction to small perturbations as being primarily a method of construction (Harnack-Hilbert-Brusotti, etc.). There is accordingly some nice duality between Luigi Brusotti and Ilia-Oleg’s contraction conjecture. Of course you surely noted this issue a long time ago, yet for me it was a happy discovery (yesterday). Perhaps CCC and CC are actually equivalent, yet this looks more hazardous but maybe not completely improbable... (One would just have to synchronize the death of all ovals posited by CC.) This is all the modest news I have collected during the week-end. Of course I still have some naive hope that CCC (hence CC) could be attacked via some gradient flow, but it looks quite difficult to locate the right functional (or Morse function). Looking at the area (or length) of all empty ovals is probably too naive...Perhaps some “degree of roughness” à la Gudkov could be projectively more intrinsic and useful... Thank you so much for your attention and all your brilliant letters and answers, while apologizing me for sending you only easy doodlings. Best regards, Alex \[26.01.13, ca 20h00\] Dear Oleg, Stepa, Viatcheslav, Evgenii, Alexis, Thomas, etc. I continued my naive investigations of real plane curves. What a beautiful story! I finally “understood” and studied in detail the marvellous construction of Gudkov $\frac{5}{1} 5$ (ca. 1971-73), as to understand the more tricky (but related) construction proposed by Stepa, which I attempted to depict on Fig. 111(=\[Orevkov2:fig\]) of the attached file. (I did not as yet assimilated the full details but feel on the good way. In fact I tried to use the dissipation of $Z_15$ in Viro’s survey from 1989/90 in Leningrad Math. J., which I hope is the same as the $E_6$ advocated by Stepa. Sorry for being very ignorant about singularities...) Yesterday, I also finally understood the correctedness of Evgenii’s argument. (As helped by Stepa’s e-mail, the point which I missed is this obstruction of Viro extending that of Fiedler) for $M$-schemes of degree 8 as having necessarily “odd content”. On the other hand, I was scared (since three days) by the fact that something which I subconsciously thought as evident (or rather which I was sure to have read somewhere) is perhaps not true. My (naive) question is whether two empty curves are necessarily rigid-isotopic? This looks at first between metaphysical nonsense and “triviality”? Maybe it is unknown, when $m$ is large enough. (m=6 follows from Nikulin 1979, and as far as I know there is not a simpler proof, say valid for all (even) degrees). So I am quite shameful asking you about this point: Is the empty chamber always connected? I tried a dynamical approach (to this problem) in Section 25.12(\[rigidity-empty-scheme-via-dyna:sec\]), but it is not very convincing. On the other hand, if the empty room is connected, then maybe the space of all curves with one component is also connected? (Naively one would apply the Itenberg-Viro contraction conjecture, to reduce to the empty case, move there for a while to resurface at the other curve (the contraction thereof). Perturbing this path in the “visible world” would conclude the proof modulo some difficulties...) Again, you Oleg, in your wonderful survey of 2008 (in Japanese J. Math) lists as an open problem the question of deciding the rigid-isotopy of curves of odd degree having a unique real circuit. As you emphasize the word “odd degree”, I wondered if the case of even degree (again with only one oval) is already settled? In Section 25.10(=\[CCCviaDynamics:sec\]), I have attempted a naive dynamical approach to the collective contraction conjecture(CCC). This states that we can shrink simultaneously all the empty ovals toward solitary nodes. This is a bit like a perfect landing in flight simulator where all wheels touch the ground simultaneously. My naive strategy is just to study the gradient lines of the functional measuring the total area of all empty ovals, but it is surely not serious. It would be exciting, in my opinion, to describe a counterexample to CCC if there is one. Many thanks for the attention, all your patience about my naive reasonings, and above all for the brilliant answers you already gave me. Best regards, Alex $\bullet$$\bullet$$\bullet$ samedi 26 janvier 2013 20:15:54, the prompt response of Eugenii Shustin: Dear Alex, The chamber of empty curves of a given (even) degree is indeed connected: two such curves can be defined by homogeneous polynomials, positive for any real not all zero variables, and their linear homotopy $(1-t)P+tQ$, $0\le t\le 1$, gives a path in the chamber of empty curves. By the way, another (well) known connected chamber consists of hyperbolic curves (i.e. those which have totally real intersection with lines of certain pencil) - this is a consequence of Nuij W. A note on hyperbolic polynomials. Math. Scandinavica 23 (1968), no. 1, 69–72. With best wishes, Eugenii $\bullet$$\bullet$$\bullet$ samedi 26 janvier 2013 21:08:27, Oleg Viro: Dear Alex, The counterpart of the Rokhlin conjecture[^39] about rigid-isotopy of any two curves of odd degree with one component is the obvious observation described by Evgenii, about empty curves of even degree. The question about curves of even degree with a single oval is equivalent to the question about removing this single oval by an algebraic Morse modification. I don’t think it was ever discussed, but I could miss it. $Z_15$ is not $E_6$. The easiest way to construct the Gudkov $M$-curve is by perturbing two $J_10$ singularities of the union of 3 non-singular conics tangent to each other at 2 points.[^40] Best, Oleg $\bullet$ \[28.01.13, lundi 28 janvier 2013 20:03:58\] Gabard wrote euphorically[^41] an e-mail titled “Some more metaphysical non-sense about the rigid-isotopy of empty curves?”: Dear Eugenii, Oleg, Viatcheslav, Alexis, Stepa, etc. So many thanks, Eugenii, for putting me again on the right track, and recalling me the argument which I shamefully forgot about. Yesterday, I was quite excited by trying to digest your argument (albeit it seems so simple). In fact the little detail that worried me is that I do not know why during the linear homotopy $(1-t)P+tQ$ the variable curve could not acquire (while staying of course empty if $P,Q$ have both the same sign) a pair of conjugate nodal singularities. This puzzled me for a while, and then using systematically your argument, I arrived at the somewhat opposite conclusion that the empty (smooth) chamber must be disconnected (for all even degrees $m\ge 4$)!!! This violates all what we know since Rohlin 1978 (and surely Gudkov as well??), while the former refers directly back to the argument of Klein 1876 based on Schläfli cubics surfaces $F_3$’s and Zeuthen correspondence between cubic surfaces and quartics (via the apparent contour). Klein’s proof is a bit tricky and uses as well his rigidification (Klein 1873) of Schläfli’s isotopic classification. Needless to say I could not follow Klein’s reasoning completely, as I just studied it today for ca. 2 hours. Marin informed me recently that he, in contrast, was able to digest all of those Kleinian works! So using your method of linear homotopy, one sees quickly that the (cone) space $C^+$ of positive anisotropic (=not representing zero) forms is contractile (convex actually) hence simply-connected. Its projection in the space of curves is the invisible locus $I$ consisting of all empty curves. Since the latter is merely a quotient of $C^+$ (by positive homotheties) it follows that it is also simply-connected (via the exact homotopy sequence of a fibering). But the discriminant is visible inside this invisible locus $I$, since it is a simple matter via Brusotti (1921) to construct empty curves with a pair of conjugate nodes. Thus we see inside the simply-connected manifold $I$ a certain hypersurface (namely a portion of the discriminant), which by Jordan-Brouwer (or a slight extension thereof) should separate this manifold $I$ in pieces (at least so is my naive intuition). It follows that our empty chamber (consisting of smooth curves) is disconnected!!!! This is my proof in its broad lines (for more details, compare Section 25.13, page 282-283 of the attachement, Theorem 25.29 and its proof on page 283). This is just one page long... Since this conclusion contradicts violently what is asserted by Klein 1876 (and approved by Rohlin 1978), it is of course very likely that my proof contains a serious flaw, or at least that I am confusing somehow the basic conceptions. However presently I do not see where is my mistake! Of course, my pseudo-theorem also violates the part of Nikulin 1979 concerned with the rigid-isotopy of the empty chamber of sextics. Many thanks again for your attention, and sorry for overflowing your mail boxes with my naive questions (and dubious reasonings). Thank you again so much for all your excellent and detailed responses (especially on $E_6$ and $Z_15$). Best wishes, Alex PS: I send you a copy of my TeX-file in case someone would like to work out a specific passage. At the occasion I would also be happy to send you my figures in zipped format so that one of you can continue the project in case I make a fatal bicycle accident (like Academician V.I. Arnold?) $\bullet$ \[30.01.13, 18h10\] Dear Oleg, Eugenii, and the other experts, I think that I found the mistake in my “proof” of the disconnection of the empty locus (that you certainly noticed meanwhile in case my explanation is the correct one). The reason seems to be simply that the discriminant inside the invisible locus has only real codimension 2, hence cannot separate anything. I have attempted to explain this in Section 25.14 on page 288. If this is not wrong it seems that the next natural question is to decide which chambers residual to the principal stratum of the discriminant contains such smaller pieces of the discriminant shrunk to codimension 2. I think to have found a topological obstacle for $M$ and $(M-1)$-curves, and conjecture (very naively) this to be the sole obstruction. In more geometric terms, this amounts essentially to decide which smooth curves can acquire a pair of imaginary conjugate nodes. Many thanks, Eugenii and Oleg, for your detailed answers. As you said, it seems that the (Itenberg-Viro) contraction conjecture of empty ovals implies the rigidity conjecture for even order curves with a unique oval. However I should probably still try to understand this implication in some more details. Perhaps it is somehow related to the previous codimension 2 phenomenon inside the “invisible” chamber. Sorry for all my confusing messages, and many thanks again for all your kind efforts in trying to educate myself. All the best, Alex $\bullet$ \[01.02.13, ca. 20h00\] Obstruction to rigid-isotopy (strictly) below height DEEP+2? Dear Oleg, Eugenii, Alexis, Thomas, Stepa, etc. Many thanks for all your brilliant messages and articles I am still slowly trying to assimilate properly. I hope not taking too much of your precious time. Albeit I met all of you only rarely, I remind very accurately your brilliant talks (in Geneva or Rennes), and so it is a special pleasure to remind each of yours while trying to explore this fantastic topic. On my side I was those last two days fascinated by the conjecture that the one-oval scheme ought to be rigid, as Oleg or Rokhlin conjectures. (Let me say that a scheme is rigid, if all the curves representing it are rigid-isotopic.) Given a degree $m$, one may wonder what is the smallest height $r(m)$ at which there is a non-rigid scheme. (For me the height of a scheme just means its number of components.) For any degree $m$, there is of course the deep nest with $r=[(m+1)/2]=:DEEP$ real branches. Two units above the latter’s height, it is easy to construct (for each $m$) curves having the same real scheme yet different types (I vs. II) hence not rigid-isotopic. (This is a simple iteration of Rohlin’s construction in degree $5$, cf. Figs. 102, 103 in my file). Using the Marin-Fiedler method of the lock it is even possible to exhibit at this height $DEEP+2$ curves of degree 7 or 9 having the same real scheme and the same type II, yet not rigid-isotopic (Figs. 105, 104). (Probably the method extends to all other odd degrees.) However, it seems much more tricky (and the lock-method seems ineffective) to detect obstruction below this height $DEEP+1$ (i.e. one unit above the height of the deep nest). Could it be the case that all schemes at or below this height are rigid? Of course this looks super-optimistic as we do not even know rigidity at height one, but I was unable to find a counterexample. I would be very happy if you know one? If there is a simple candidate, I hope to detect it alone during the next few days…. So do not take care answering me if my question is trivial. (As I just work on this since two days, I probably missed something accessible.) Paraphrasing slightly, I found quite puzzling, that the very explicit function $r(m)$ measuring the smallest height of a non-rigid scheme is only subsumed to the large pinching $1 \le r(m)\le [(m+1)/2]+2=DEEP+2$. Of course a better lower bound seems out of reach, but perhaps you know better upper bounds. I also wondered if there is an extension of the Nuij-Dubrovin rigidity of the deep nest to, say, the totally real scheme of degree 8 consisting of 4 nests of depth 2. I should think more seriously on this at the occasion. Many thanks for your attention, and sorry again for all my enthusiastic and naive e-mails. All the best, Alex $\bullet$ \[written 08.02.13 and sent 09.01.13\] Dear Oleg, Eugenii, Stepa, Viatcheslav, etc. I still continued my trip through real plane curves and cannot say that my curiosity is starting to fade out. I tried for several days to find a counter-example to the conjecture (of mine so probably quite wrong) that all schemes below height $DEEP+2$ are rigid, where $DEEP=[(m+1)/2]$ is the number of branches of the deep nest of degree $m$. At least the method of the lock (Fiedler-Marin) seems quite inoperant to detect an obstruction to rigid-isotopy at such low altitudes. If true, the proof probably involves a geometric flow collapsing either the pseudoline to a line (by shortening its length like a systole) or improving the rotundity of some oval to a circle (via an isoperimetric functional?). If all this works, it would reduce the low-altitude rigidity conjecture to Nikulin’s theorem (or maybe even Klein’s on $C_4$) as the starting step of a big recursive process. Of course this seems still quite out reach (canary music) unless one feels very motivated! Next I tried to corrupt the truth of Slava’s remarkable rigid-isotopic classification (Nikulin 1979) of sextics via the Marin-Fiedler locking argument using Bézout saturation. Of course I have nothing against Slava, but this was rather intended to test experimentally the power of Nikulin’s result. Specifically I looked at sextic schemes of the form $3/1 \ell$, and wondered if for some specific curves the distribution of the $\ell$ outer ovals away the fundamental triangle traced through the 3 inner ovals (those enveloped by the unique nonempty oval) could be different for different curves. On all examples I tested it seems that the outer ovals are never separated by the “deep” triangle. So we find no violation of Nikulin’s theorem, and the latter rather implies that as soon as we are able to visualize the distribution for a single curve it will be the same for all curves belonging to this scheme. The case most tricky to understand is the maximal permissible, namely $3/1 5$. I managed to construct it à la Harnack (as preconized in Gudkov 1974 or 1954). But being quite unable to decide from this model the distributional question of the outer ovals past the fundamental triangle, I decided to switch to Oleg’s method of construction via dissipation of the singularities of a triplet of coaxial ellipses. I played this game yesterday but could not decide the distributional question for this Viro curve (cf. especially Fig.126 on page 320 and the hypothetical Theorem 26.29 on page 322). In fact today I tried again to inspect directly Harnack construction and found Lemma 26.27 on page 319 whose proof seemed to me very transparent until I found the little warning, which I think is not fatal. In conclusion I believe now that there is no separation by the fundamental triangle!!!?? Of course I imagine that, if I am not completely wrong, what I am investigating must be quite familiar to you. I would much appreciate if you know if this hypothetical theorem (26.29 page 322) is true. It amounts essentially to check whether in Viro’s construction of $3/1 5$ the triangle through the 3 deep inner ovals does not separate the 5 outer ovals. I find this question quite attractive as it seems to require some understanding of the geometric location of the microscopic ovals arising in Viro’s method (optionally compare Fig. 127 (page 322) which shows a scenario with the two bottom micro-ovals aligned vertically in which case the fundamental triangle would separate the outer ovals). This scenario seems to me quite unlikely but it does not seem to be impeded by naive Bézout obstructions. Many thanks for your attention, and sorry again for all my naive and confuse questions. Thank you very much again for your precious guidance and answers. All the best, Alex $\bullet$$\bullet$$\bullet$ (10.02.13) Bonjour Alexandre, Thomas m’a transmis ta question. La réponse est toute simple: soient $A$, $B$, $C$ trois ovales intérieurs et $D$, $E$ deux ovales exterieurs de ta sextique. Le triangle fondamental $ABC$ est entiérement contenu dans l’ovale non-vide. Si $D$ et $E$ sont dans deux triangles $ABC$ (non-fondamentaux) différents, alors la conique passant par $A$, $B$, $C$, $D$, $E$ coupe la sextique en $14$ points, contradiction. Avec des coniques, on montre plus généralement que: Les ovales vides de la sextique sont distribués dans deux chaines (int, ext), l’ordre cyclique est donné par les pinceaux de droites basés dans les ovales interieurs. Les ovales interieurs sont disposés en position convexe dans l’ovale non-vide. Bon dimanche,   Séverine $\bullet$ \[12.02.13\] Is the Gudkov chamber simply-connected? Dear Séverine, Viatcheslav, Ilia, Oleg, and all the other experts, First many thanks, Séverine, for your excellent answer on my distribution question of ovals of sextic, and sorry for my late reply on it as I lack an Internet connection at home. I tried today to understand when a dividing (plane) curve admits a transmutation, i.e. a rigid-isotopy permuting both halves of the curve. I also studied the weaker notion of mutation of when there is a linear automorphism of the plane permuting both halves. Using the Kharlamov-Itenberg calculation of the monodromy of sextics I think that I managed to get some obstruction to mutability, especially for the 3 dividing curves which have trivial monodromies (compare Lemma 26.6, page 288, which is hopefully correct). However I don’t know if the Gudkov chamber (or the 2 other related “antidromic” chambers, i.e. having trivial monodromies) is simply-connected. I hoped to detect some non simple-connectivity by looking at the monodromy induced on the halves instead of the ovals. At least this works of course for the deep-nest chamber which is not simply-connected since there is a symmetric model which can be mutated. So my (hopefully not too naive) question is the following: is it known whether or not the Gudkov chamber is simply-connected? (equivalently is the Gudkov curve transmutable?) The same question looks attractive for the other 2 antidromic curves, i.e. the left wing “Rohlin curve” $6/1 2$ and $4/1 4$ in type I. Thank you so much for all your attention and patience, and in advance for your answer if it is known. Best regards, Alex $\bullet$$\bullet$ $\bullet$ mercredi 13 février 2013 04:34:12 Dear Alexandre, If I understand correctly the question then the answer is not, if I state the question appropriately then the answer is yes. I mean the following precise statements. Let consider the part of the projective space of real sextics that is represented by maximal sextics of Gudkov’s type. Then the fundamental group of this part is $Z/2$. It becomes simply connected after taking quotient by the natural action of $SL(3,R)$. In fact, before factorization it is a fibration over contractible base with the fiber $SL(3,R)$. These results (and there analogs for other maximal sextics and certain curves of lower degree) are contained in my talk On monodromies of real plane algebraic curves at one of Petrovsky seminars in 80th, I guess (short summary should be found in Russian Surveys). The proof (in the case of sextics) is rather straightforward as soon as based on the $K3$ surfaces periods uniformization. As it happens rather often with this approach, to treat the maximal curves is extremely easy, since the corresponding eighenlattices become unimodular. In general the period domain, which is the product of two polyhedra in the real case, represents the studied sextics (or associated K3 surfaces) only up to codimension 2. Which makes laborious to treat the fundamental group. But, surprise, in the case of maximal curves there are no codimension 2 phenomena, since such holes appear only as traces of $(-2)$-cycles having nontrivial components in the both eighenspaces, which is impossible since in the maximal case the components are integral and the eighenlattices are even. I don’t remember by heart the final result for other maximal sextics. It should be pointed in the same summary and by the way easy to get following the same approach I have pointed. The key is that even if it is no more a pure fibration - it has special fibers which are quotients of $SL(3,R)$ by the corresponding monodromy group (which indeed coincides with the maximal possible group of symmetries for the given type of sextics) - its fundamental group is exactly the fundamental group of this special quotient. Yours VK $\bullet$ mercredi 13 février 2013 11:46:20 Dear Colleagues, Am I alone who did not receive a copy of Severine’s letter? I would be happy to know its content :) Yours VK $\bullet$ mercredi 13 février 2013 13:43:10 Dear colleagues, I had written only to Alexandre, sorry! My answer was this: let $A, B, C$ be three inner ovals, and $D, E$ be two outer ovals of the sextic. The fundamental triangle $ABC$ is entirely contained in the non-empty oval. If $D$ and $E$ are in two different (non-fundamental) triangles $ABC$, then the conic through $A, B, C, D, E$ cuts the sextic at 14 points, contradiction. Using conics, one proves more generally that there is a natural cyclic ordering of the empty ovals, given by the pencils of lines based at the inner ovals. The empty ovals are distributed in two consecutive chains (inner, outer). The inner ovals lie in convex position in the non-empty oval. Best regards, Séverine $\bullet$ \[14.02.13\] Dear Viatcheslav, Séverine and all the other colleagues, Thank you very much for this beautiful answer on the Gudkov chamber. I look forward to digest properly all that incredible technology that you and Nikulin developed. Again many thanks also to Séverine for the clever argument which I digested yesterday with great pleasure, and integrated in my notes in Section 26.10(=\[LeTouze:sec\]) pages 332–334. This gave me yesterday some motivation again to attack the very first question of all our chat room, namely Rohlin’s claim that the pencil of cubics through the 8 deep basepoints located inside the 8 empty ovals of any sextic curve of type $6/1 2$ or its mirror $2/1 6$ is totally real, hence of type I (also called orthosymmetry by Klein ca. 1881-82 and his student Weichold 1883). In fact I (naively) hoped to prove this Rohlin claim via Poincaré’s index theorem, yet the qualitative picture (Fig. 133 on page 337) rather inclined me to believe that the proof cannot reduce to mere combinatorial topology of foliations (i.e. Poincaré’s index formula of 1885). So I am still puzzled, but perhaps an argument like Séverine’s one do the job. At any rate I would be very excited if someone manages to reconstruct this proof asserted by Rohlin (1978) if it is not too tantalizing for the brain. Otherwise I am also much frustrated by failing to visualize totally real pencil on the three $M$-sextics, whose existence is I think predicted by Ahlfors theorem of 1950 (or better the special zero-genus case thereof known to Riemann 1857, and reworked by Schottky 1875-77, or even Bieberbach 1925 and his more respectable student Grunsky 1937). Marin warned me recently that the transition from the abstract Riemann surface viewpoint to the planar context “of Hilbert’s 16th problem” may be not so easy as I always assumed subconsciously. (If necessary, all the correspondence I received from all the colleagues is gathered in Section 24.6, p.221). Overpassing this difficulty (which I hope is not fatal) there should be on all $M$-curves (more generally dividing curves) auxiliary pencils which are totally real. Alas for $M$-sextics (even $M$-quintics), I am completely unable to trace them and know nothing about the degree of the curves involved (in the pencil). I hope to be able to tackle such questions in the future, but perhaps you have better ideas (or motivations) than I do have. Thank you very much again for all your brilliant answers, and kind messages. All the best, Alex $\bullet\bullet\bullet$ samedi 16 février 2013 17:54:55 Dear Alexandre, dear other colleagues, I have managed to prove that a pencil of cubics with eight base points distributed in the eight empty ovals of a sextic $2 \cup 1(6)$ is necessarily totally real. Details will follow soon in a paper. Yours, Séverine $\bullet$ \[16.02.13,19h41\] Dear Séverine and colleagues, Congratulations for this fantastic achievement. I am sure the proof must be very beautiful. On my side I tried to work out for all sextics of type I an optical recognition procedure of the type by some synthetical procedure akin to Rohlin’s claim, yet this is still much in embryo. In particular the case of $(M-4)$-sextics is quite puzzling as it seems to contradict the version of Ahlfors theorem due to myself (existence of a totally real map of degree the mean value the number of ovals and Harnack’s bound). I hope to send you more palatable material soon, but confess that the questions look quite hard and I seem much less efficient than Séverine. So I suppose that Rohlin’s claim is one among several other (less pure) total reality result. So I look forward with great interest to see Séverine’s article. All the best, Alex $\bullet$ \[19.02.13\] Dear colleagues, Many congratulations again to Séverine for your fantastic achievement. Sorry to have been brief in my last letter, as I wrote (lacking an internet connection at home) from a friend of mine who had a romantic party with his girlfriend, and I do not wanted to disturb too long his romantic evening. Meanwhile I also tried hard to concentrate on a proof of the Rohlin-le Touzé’s theorem, which still overwhelms my intelligence. The last things that I have written are on pages 336–352 (Sections 27.1, 27.2, and 27.3), but this is poorly organized and supplies no serious proof of the Rohlin-Le Touzé’s theorem. Some few days ago, I got Theorem 27.5\[=\[total-reality-of-plane-M-curves:thm\]\] (on page 346), which (if it is correct) answers one of the question I asked in my penultimate e-mail (as well as desideratum of Alexis), namely the question of estimating the order of curves involved in a total pencil on an $M$-curve. It seems that there is always such a pencil of order $(m-2)$, i.e. two units less than the given degree $m$ of the $M$-curve. In fact, the proof is a nearly trivial adaptation of the abstract argument going back to several peoples (in chronological order Riemann 1857, Schottky 1875, Enriques-Chisini 1915, Bieberbach 1925, Grunsky 1937, Courant 1939, Wirtinger 1942, Ahlfors 1947, 1950, a myriad of Japaneses, a myriad of Russians including Golusin 1953/57, etc....., up to Huisman 2000, and Gabard 2001/2006, who else?). The point is that total reality is trivial in the case of $M$-curves since we have one point circulating on each oval (such a group moves by Riemann-Roch!!!) and so we have like a train-track with only one train on each track, hence no collision can occur and total reality is automatic. If we work with plane curves we only need to take curves of order $(m-2)$ which have enough free parameters to pass through any given distribution of $M$ points (one on each oval), and this works by looking at the residual group of points (details in the proof on page 346). So this is quite interesting but probably only a first step toward deeper things. (One could dream to recover all the Gudkov-Rohlin/Arnold congruence via this method but that looks hard work...) After this little discovery I focused again on the Rohlin-Séverine theorem, yet without any success. So I have not more to report for the moment. Thanks a lot for the attention, and all my congratulations again to Séverine for your deep advance. Best wishes, Alex $\bullet$ 19.02.13 Dear Alex, Let me ask you a question from your previous field of interest. Do you know any example of a non-Hausdorff 1-manifold which does not admit a differential structure? I heard about existence and could easily construct examples of exotic, i.e., homeomorphic but not diffeomorphic non-Hausdorff 1-manifolds. See http://www.map.mpim-bonn.mpg.de/1-manifolds Sincerely, Oleg 20.02.13 Dear Oleg, David and Mathieu, Many thanks, Oleg, for your lovely question, and best greetings to the other friends. Alas my memory is failing quite dramatically, so my answer will be of poor quality. If I remember well I asked myself the same question some 3-4 years ago, but I cannot record to have ever found an answer. Thus I forward your question to David and Mathieu, the leading experts of non-metric surfaces who perhaps will supply a better answer. On my side I hope to think more seriously to your question when I see clearer with Rohlin-Le Touzé’s sextics. Maybe a first idea is that there ought to be a (non-canonical) “twistor construction” assigning to each non-Hausdorff curve a Hausdorff surface fibered by (real) lines. This construction should go back to Haefliger’s very first note in the colloque de topologie de Strasbourg ca. 1955-1956 (yet it is not very detailed). In substance it is like a train-track construction à la Penner-Thurston…(some intuition about this is given in my article ‘Ebullition and gravitational clumping, arXiv, 2011). Do not worry if you don’t understand me, as I myself remember only vague souvenirs and are not so convinced by what I am saying!!! In fact Haefliger (ca. 1956) claims this construction only for second countable curve (even with a proviso on the fundamental group), but when I was in touch with the subject I was fairly convinced that it must work universally. OPTIONAL REMARK: Haefliger, and Haefliger-Reeb 1957 use this construction to prove that any simply-connected curve (second countable) arises as the leaf space of a foliation of the plane. (Sketch of proof: take the twistor of the given curve which is by the exact sequence of a fibering 1-connected and (by Poincaré-Volterra) second countable, hence it is the plane, q.e.d) So the idea would be to descend a smooth structure on the surface to get one on the curve. Alas, it is a well-known open problem whether any (non-metric but Hausdorff) surface admits a smooth structure (Spivak 1971, Nyikos, etc.) However quite puzzlingly Siebenmann 2005 (Russian Math Surveys) claims (and even prove in some details) that a PL structure exists universally on all such surfaces, merely as a consequence of Schoenflies theorem. So perhaps Siebenmann argument work as well for DIFF structures, and the metaphysical problem of Spivak-Nyikos is cracked. If this works (ask maybe Siebenmann, or an Indian in the States(=Ramachandran) who albeit not an expert was fairly convinced that there should be no asymmetry between PL and DIFF in dimension 2), then there is perhaps some chance to get a smooth structure on all non-Hausdorff curves. Of course there is perhaps a more direct strategy without transiting through surfaces. Otherwise, regarding exotic smooth structures on curves the original reference is Haefliger-Reeb 1957 article in L’Enseignement Math. Perhaps you could quote this in your brilliant web-page. Sorry for this vague answer, but at the moment my brain is much concentrated on this Rohlin-Le Fiedler total reality claim which still puzzles me a lot!!! Best greetings to all, as well as to Rachel and Chiara. All the best, Alex $\bullet$ \[22.02.13\] Dear colleagues (especially Séverine), I worked hard (but without success) on the Le Touzé’s theorem, at least for 8 basepoints assigned on the nonempty ovals of a sextic of type $6/1 2$. If I understood well Séverine’s announcement, you rather handle the case of $2/1 6$ and assign more generally the points in the insides of the empty ovals (but of course I suppose that your argument adapts to $6/1 2$). Even in my weaker form I am not really able to conclude but send you my last thinking on the question (Section 27.4, p.352–356, esp. Fig.141). Ultimately I found a method which I call “barrages”. A special rôle is played by nodal cubics of the pencil, and I try to get a corruption with Bézout by looking at nodal curves with a barrage, i.e. such that 4 arcs of some other cubic joins pairwise the 8 basepoints distributed on the loop of the original cubic. (By the loop of a nodal cubic, I mean the unique path from the node to itself which is null-homotopic in the plane $RP^2$.) Of course I am not sure that details can be decently completed, but for the moment it is the only reasonable strategy I could imagine. I am sure that Séverine’s argument is much more elegant and convincing. My reasoning is completely conditioned by Fig.141, and I am probably too naive in believing that it reflects the general situation. Sorry for sending you this very coarse material, and of course do not take the pain to react to this message. Many thanks again a lot to all for sharing so generously your knowledge and for all your answers. Best regards, Alex \[25.02.13\] Dear real geometers, I was still much fascinated by the Rohlin-Le Touzé theorem (RLT) albeit still not able to prove it. Being frustrated by my failing attempts (probably due to a lack of stubbornness and competence in algebraic geometry) I decided to speculate a bit of why it is so important or at least to explore how the statement could generalize. In its most elementary incarnation involving pencil of lines and conics, the phenomenon of total reality occurs along infinite series stable under the operation of satellite of a real scheme (of even order). Satellite just amounts to trace each oval with a certain multiplicity $k$ (jargon obviously borrowed from knot theory). So the unifolium scheme of degree 2 (allied to a conic) gives rise to the deep nests, and the quadrifolium scheme of degree 4 gives rise by taking its satellites to an infinite series of schemes of order multiples of 4 which are totally real under a pencil of conics (assigned to pass through the deepest ovals). It seems therefore natural to ask if the satellites (e.g. the second satellite) of the Rohlin’s scheme $6/1 2$ (or its partner $2/1 6$) are also totally real (and hence of type I) under the “same” pencil of cubics as posited by the Rohlin-Le Touzé theorem. Alas I was not even able to settle this question. (Of course this seems evident (granting RLT) for a small perturbation of the algebraic double (essentially $F \cup F+\epsilon$), since total reality forces transversality of the foliation induced by the pencil with the curve.) Next, I tried to understand what are the higher order avatars of the RLT-theorem (in the hope that it is not an isolated phenomenon as vaguely suggested by Ahlfors theorem). I found using the Rohlin-Kharlamov-Marin congruence ensuring the type I-ness (=orthosymmetry) of some $(M-2)$-schemes an (obvious) infinite series of avatars of the Rohlin’s $(M-2)$-schemes of degree 6 . Those are also $(M-2)$-schemes and total reality seems to be possible for a pencil of curves of order $(m-3)$, exactly like for the Gürtelkurve of Zeuthen-Klein (bifolium quartic with 2 nested ovals totally flashed by a pencil of line through the deep nest) or for the Rohlin’s sextic (flashed by a pencil of cubics). So it seems that the theory of adjoint curves of order $(m-3)$ plays some special rôle in this question of Rohlin-Séverine. I would be very happy if one of you knows if it is reasonable to expect an extension the RLT total reality theorem to all this schemes whose type I ness is ensured by Rohlin-Kharlamov-Marin congruence (sorry if I am not hundred percent right in crediting as I could not extract the exact history of this subliminal result). Specifically I have Conjectures 27.17 and 27.18 (page 365 and 367 resp.) which list some candidate-schemes for total reality in degree 8 and 10. If the conjectures are right, it would be of great interest to know if Séverine’s proof adapts to them. Sorry if I am too naive about the real difficulty of such problems, but I found exciting to wonder if there is something more general behind the cryptical allusion of Rohlin. Of course I presume that he derived the synthetic result a posteriori from highbrow topology (or Kähler geometry in Kharlamov’s case?), but perhaps there is a simple explanation with (“basic”) algebraic geometry and total reality as Séverine was able to do? As Oleg knows my problem is that I wasted too much time with non-metric manifolds and so forgot all the little I ever knew about algebraic geometry. During the way, I think to have found a counterexample to the conjecture of mine (inspired by the Itenberg-Viro contraction conjecture of empty ovals), and according to which all empty ovals could be contracted simultaneously to solitary nodes. This counter-example is Thm 27.16 on page 364 (which I hope is correct and sharp as far as the degree is concerned). Thanks a lot for the attention, and sorry for all the modest news (you surely thought about in sharper form already). All the best, Alex PS: The material summarized in this message occupies page 357-367 (Sections 27.5, 27.6, 27.7), as usual I had not much time to polish, but I hope it is still readable. \[27.02.13\] A census of 100 octic $(M-2)$-schemes of type I satisfying the RKM-congruence, plus a little addendum for Oleg’s non-Hausdorff curves Dear colleagues, I have pursued some preliminary study toward the total reality phenomenon, yet merely in its combinatorial aspect prompted by the modulo 8 RKM-congruence (for Rohlin-Kharlamov-Marin) ensuring the type I of $(M-2)$-schemes of degree $2k$ with $\chi = k^2+4 \pmod 8$. Accordingly, I call an RKM-scheme any $(M-2)$-scheme satisfying this congruence. While any RKM-scheme is of type I, I do not know alas whether the converse statement is true. If it is known I would be extremely grateful if someone can tell me (and our collective chat room) the answer. Further I noticed that the list given in my previous e-mail of RKM-schemes of degree 8 can be much enlarged. If I am not too bad in combinatorics, there are precisely 100 such schemes in degree 8, all of them being potentially subsumed to the phenomenon of total reality under a pencil of quintics akin to the Rohlin-Le Touzé theorem (for sextics flashed by cubics). This modest material is to be found in Section 27.8, p.368-373 (especially Fig. 146 page 370 and Lemma 27.24, p.372, plus all the 36 Gudkov symbols on page 372). I hope of course that I missed nobody in this catalogue. Extrapolating a bit using the (hypothetical) converse statement to RKM, I would say that there are precisely 100 schemes of type I which are $(M-2)$-schemes. Is this well-known and correct? Actually, I do not really know if all these 100 schemes are realized algebraically, but presume that most of them (all?) are. Possibly I am much too naive. Of course it is quite amazing to see that the only two RKM-schemes of degree 6 (namely $6/1 2$ and $2/1 6$) demographically explodes to a menagerie of 100 such schemes in degree $8$, but that should be no surprise for you much acquainted with the higher cases of Hilbert’s 16th problem. It would be even more crazy if all those 100 schemes (or at least a good portion thereof) are subsumed to the phenomenon of total reality. If you have some ideas on those circle of ideas, I would be extremely thankful. Many thanks again for the patience and attention, and I hope that what I am telling is nearly correct (not too surrealist). Very best regards, Alex PS: For Oleg, regarding my loose answer on smooth structures on non-Hausdorff 1-manifolds, I would like to add another philosophical remark related to the method of Haefliger’s “twistor”. This is of course like a thickening along a normal bundle except that there is no ambient manifold (safe the ether) and so the construction must be intrinsic. To my knowledge it was never exposed in details (albeit Haefliger’s 1st article ca. 1955-56 in Colloque de Topologie de Strasbourg uses implicitly this construction). Now my point is that albeit the twistor method looks somewhat indirect, I think that it is fairly useful. For instance, I was since 2006-07 puzzled by the naive question if the fundamental group of a one-manifold is always a free group. (Of course such non-Hausdorff curves resemble somehow graphs, whence some intuition). For instance the line with 2 origins has $\pi_1=Z$ as follows quickly from Seifert-van Kampen (and if 3 origins or 2 doubled origins then $\pi_1=F_2$ is free of rank 2). Ultimately in 2011 I found a general answer to this “freeness” puzzle by using the Haefliger twistor construction, while showing first that all open (non-metric) surfaces have free fundamental groups. (This is actually a very modest extension of the metric case, which to my knowledge is first treated in Ahlfors-Sario book of 1960, albeit it may have belonged to the folklore much earlier, say Kerekjarto, H. Kneser, Rado, in the 1920’s, Papakyriakopoulos in the 1940’s???). This material is exposed in some details in my arXiv note of ca. 2011 (Ebullition in Foliated surfaces versus gravitational clumping). I hope that those results are nearly correct but they certainly require more professional treatments and exposition than I was able to do. I hope this little remark makes perhaps more plausible that the approach via the (Hausdorffizing) Haefliger twistor is also reasonable for your problem of DIFF structures. $\bullet\bullet\bullet$ vendredi 1 mars 2013 19:07:12 Dear Alexandre, dear other colleagues, here is the note I had promised to send you. There are still many open questions, as Alexandre wrote. It would be also interesting to know whether one could find a totally real pencil with respect to the dividing $M-2$-sextics with real scheme of indefinite type. I will think about it when I have more time. Best regards, Séverine (01.03.13, 22h15) Dear Séverine and the other colleagues, So many thanks Séverine for sending us your splendid article. I am much excited to read the details tomorrow, as myself started today to doubt about the whole result (at least in the strong form that any points 8 points distributed on the empty ovals ensures total reality). (If I am not wrong the whole phenomenon depends upon the location of the 9th base point, namely the pencil is totally real iff the 9th base point lands in the inside of the nonempty oval.) So I was much depressed and lost in my poorly organized thoughts. So your sending arrives as a true deliverance for my brain. Many congratulations again to Séverine for this fantastic work. Very best regards, Alex \[02.03.13\] Can total reality fail for a distribution of 8 points on the empty ovals? Dear Séverine and the other geometers (especially Professor Nikulin), I enjoyed much a detailed look at your splendid article full of illuminating remarks. I will probably need much more time to digest the impressive technology you use, and need to print the material to make a deeper reading (especially of the former works upon which your argument seems to depend). So many thanks again for sending us your work in so rapid delay. I wrote some naive reactions in Section 27.11, where I mostly copied your sayings, and tried to add hopefully pertinent footnotes. Regarding your question “Can conversely any dividing curve be endowed with some totally real pencil?”, I still wonder if a positive answer is not a trivial consequence of Ahlfors theorem (compare very optionally Gabard’s Thesis 2004, page 7). However since Marin warned me in January 2013 (cf. Section of e-mails) it may be the case that the transition from the abstract conception of Riemann-Schottky-Klein to the embedded viewpoints of Hilbert-Gudkov-Rohlin is not so easy. Yet I am still confident (or naive enough) to believe that it holds true. The point seems to be primarily a matter of projective algebraic geometry, namely the question if any abstract morphism on a concrete plane curve to the line $\PP^1$ is induced by a (linear) pencil of ambient curves. This is either trivially true or trivially wrong, but alas I do not know the answer due to my failing memory about the foundations of algebraic geometry. Your article already helped much as I suffered under the misconception that your result states that any distribution of 8 points on the empty ovals induces a totally real pencil. Your statement is much more subtle, yet personally I do not know if this stronger (universal) form of total reality is wrong! If you know a counterexample foiling universal total reality I would be very happy. It could then still be the case that there is some special sextics for which universal total reality holds true, i.e. for all octuplets distributed on the empty ovals. (Perhaps reading more carefully your article, especially the aspect related to Nikulin-Kharlamov’s rigid-isotopic classification already answers those questions?) (The newest material of mine (as usual confusing and poorly organized) occupies Section 27.9–27.11 on pages 373–378. Here I attempted a topological approach to the existence of octuplets inducing a totally real pencil, but alas was not able to conclude, presumably because I know too little on the predestination process creating the 9th basepoint as a function of the 8 assigned ones.) Many congratulations again to Séverine for this breakthrough. Best regards, Alex dimanche 3 mars 2013 18:07:57 a new version with small corrections? Dear Alexandre, dear other colleagues, I owe you some apologies: the Theorem was slightly incorrect, as Alexandre pointed out. I let you discover this new version, where I have reformulated the Theorem, and added a few words in the end of the proof. Best regards, Séverine (04.03.13) Dear Séverine and the other colleagues, Many thanks for the new version. In fact, it seems that the main change is that you now assign the 8 basepoints ON the empty ovals instead of IN their insides. Rereading my previous message, I realize that I misstated your original statement and so it is pure chance that assignation on the ovals turned out to be “more correct”. Your fascinating article gave me new forces to think about the problem, but alas still without success. For instance, I still do not know if there exist octuplets (on the empty ovals) failing to induce a totally real pencil. Of course assigning them in the insides gives more freedom, but presently it looks to me harder to ensure total reality. So despite your correction, it could still be the case (in my modest understanding) that the pencil is total for all octuplets chosen in the insides of the empty ovals. Perhaps you know a counterexample to this strongest form of the statement? Many thanks again for the article, which guided much my thinkings. I hope to send you more exciting news soon, but the whole problem which you call “the lost proof of Rohlin” seems to me still much out of reach. All the best, Alex $\bullet$$\bullet$$\bullet$ answer to Alexandre’s questions (mardi 5 mars 2013 13:30:42) Dear Alexandre, dear other colleagues, let me try to answer the question with a new formulation. Assume first that the base points are distributed [*inside*]{} of the empty ovals. Applying your nice “dextrogyration argument” to all nine ovals gives the following lemma: [*The pencil is totally real iff 9 lies inside of the non-empty oval $O$ and outside of the empty ovals.*]{} If 9 is outside of $O$, the bad cubics are as shown in Figure 2 of the paper. If 9 is inside of an empty oval $X$, the bad cubics have an oval passing through the two base points 9 and X only, and this oval is entirely contained in the empty oval denoted also $X$. To get rid of this latter possibility, it suffices to take the base points [*on*]{} the empty ovals. In ii), I give an explicit description of the pencil, valuable for any generic choice of the eight base points [*inside*]{} of the eight empty ovals. (It turns out that the only possible non-generic situation is that of a pencil with a double base point $9=2$, this means that the points 1, ..8 lie on a nodal cubic with node at 2.) Recall that 2 is the base point chosen in the extreme inner oval forming a positive pair with $O$. For this pencil, the only possibly bad cubics are those with an oval passing through 9 and 2 only. To grant total reality, it suffices to choose the base point 2 [*on*]{} the corresponding empty oval, the other base points lie arbitrarily in the inside discs of the other empty ovals. Thus, your conjecture 27.29\[=\[SRLT:conj\]\] is true, and an even stronger result holds for the sextic with six inner ovals. Best regards, Séverine \[07.03.13\] Little news from Alex, and so many thanks to Séverine for the answer Dear Colleagues, First many thanks to Séverine for your very detailed answer (which I will study in detail tomorrow). Sorry for being always a bit differed in time due to my lack of internet at home. I added some material in my loose notes. In Section 28.1–28.2 (pp.384–392), I tried once more to explore the grand programme that Rohlin might have had in mind, namely total reality and its connection with his maximality conjecture. As I often said it seems to me that the missing link could be played by Ahlfors theorem, or perhaps Rohlin had a grand vision that he could arrange total reality by purely synthetical processes extending in all degree the already tricky theorem of Rohlin-Le Touzé in degree $m=6$. This idea when explored in full looks to me extremely vertiginous, but its net impact would be a sort of upper bound upon the complexity of Hilbert’s 16th problem, and in some sense subsume all prohibitions (à la Gudkov et cie.) to the paradigm of total reality. All this necessitates to be made much more precise, but I \[have\] attempted to make a psychoanalysis of what Rohlin may have had in the brain, without that he himself ventured to put it on the paper due to his own modesty and pragmatism. Next I discovered the little Theorem 28.7\[=\[Thom-Ragsdale:thm\]\] (p.393), which is just a matter of making explicit the consequence of Thom’s conjecture (=Kronheimer-Mrowka theorem) as it pertains to Hilbert’s 16th problem. The result is the lovely estimate[^42] $\chi \le k^2$ for a curve of type I and degree $2k$. With this I realized that my former counterexample (with the scheme $20$ in degree 8) to CCC(=collective contraction conjecture) is actually killed by Thom, and realized (later only!!) that it is also killed by Rohlin’s formula. So CCC is again resuscitated but probably not for long!? Then I tried to make a comparative study of Rohlin’s formula versus the Thom obstruction. It seems that the latter is often implied by Rohlin’s formula, but not always. More in Section 28.4 (p.393). It seems however that at least for degree $m \ge 10$ there is some cases where Thom really affords new information not covered by Russian congruences or Rohlin’s formula (cf. Thm 28.11, p.396). Finally using the Gudkov table in degree 10 (=Fig.148 on page 395), I got some naive hope to disprove the Rohlin maximality conjecture, but this quickly turned into disillusion (cf. Point 3 on p.396–397). Sorry for all these messy remarks, yet I found the rôle of Thom quite pleasant. I am sure that this is not new, and that I read it somewhere, but again could not recover where precisely. (I thought it was in Degtyarev-Kharlamov 2000’s survey but apparently not, though Kronheimer-Mrowka is alluded to.) If you remember some anecdotes about the rôle of Thom’s conjecture in Hilbert’s 16th problem, and who puts it first into action as a such, I would be extremely happy to insert your remarks in my (messy) survey. Thanks a lot for the attention, Best regards, Alex Constructions ============= \[29.03.13\] Construction of algebraic curves seems a syllogism since they are nearly God-given. Perhaps the word contemplation looks more appropriate, but clumsy. Despite existence of Gods, the art of tracing of algebraic (plane) curves goes back to time immemorial Newton, Plücker 1839 [@Plücker_1839], Zeuthen 1874 [@Zeuthen_1874], with the modern era usually identified by Harnack 1876 [@Harnack_1876], Klein 1876, Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege]). The game is especially interesting over $\RR$, else nearly everything follows from Riemann. Much of the elementary aspects can be treated by the primitive method of small perturbation, which nearly gives a good picture of what happens in degree 6. This is how worked Plücker, Klein, Harnack, Hilbert, Ragsdale, Brusotti, etc. However already in degree 6 the classical method starts showing some limitation. Albeit the Gudkov curve can be distilled by small perturbation, it requires an extra twist by means of Cremona transformations (at first difficult to visualize). It took the community ca. 8 decades (including such masters as Hilbert, Ragsdale, Brusotti, Petrovskii, Gudkov first not an exception) until to discover the fairly trivial picture traced by Gudkov ca. 1972 (Fig.\[GudkovCampo-5-15:fig\]) exhibiting a curve with topology $\frac{5}{1}5$. Ca. 1980 Viro described how to dissipates more complicated singularities, allowing experts to create more funny curves refuting most of the conjectures erected along the primitive method. For instance also Gudkov’s sextic appears fairly trivially when one knows how to smooth a triplets of ellipses tangent at 2 points, compare Fig.\[Viro3-15:fig\]c. A variant of Viro’s patchwork due to Itenberg (called the $T$-construction) is purely combinatorial and permitted to disprove severely the Ragsdale conjecture (cf. Fig.\[Itenberg:fig\]), as well as our naive Thom estimate $\chi\le k^2$ for dividing curves. Constructing the two maximal $(M-2)$-schemes {#const-total-(M-2)-schemes:sec} -------------------------------------------- \[05.01.13\] As to the existence of the two maximal $(M-2)$-schemes (namely $\frac{6}{1}2$ and $\frac{2}{1}6$), they can be constructed (as observed in Gudkov 1974/74 [@Gudkov_1974/74 p.42]) by a slight modification of Hilbert’s method. Let us reproduce his figure (Fig.4, p.16). This gives (after smoothing) the left-side of Fig.\[Gudkov-Hilbert-modified:fig\]. Alas, this is not the desired scheme. Is there a mistake in Gudkov at this place? Apparently not as it seems approved in A’Campo 1979 [@A'Campo_1979] (alas no detail). -5pt0 -5pt0 The right-part of Fig.\[Gudkov-Hilbert-modified:fig\] is just a variant inspired from Hilbert’s configuration. This has again the wrong real scheme. Since Gudkov does not seem to give exactly what he claims, we must rely on some do-it-yourself endeavor. A naive idea gives Fig.\[GudHilb2:fig\], failing again to have the correct scheme. -5pt0 -5pt0 Let us now work more systematically. The key is first to make Walt-Disney pictures of Hilbert’s method à la Gudkov. This involves a simplified art-form, far from geometrically realist, but topologically faithful and more malleable. This produces the following pictures (Fig.\[GudHilb3:fig\]). The trick of Hilbert’s method is to let oscillate an oval across an ellipse while smoothing their union (cf. left part of Fig.\[GudHilb3:fig\]). Such oscillations are Bézout compatible: each oscillating quartic intersects 8 times the ellipse. A posteriori it is a simple matter (Hilbert’s method) to realize such oscillation by rigid algebraic curves suitably perturbed by lines arrangements. Yet it is valuable first exploring the softer topological figures as to understand which oscillation is able producing a prescribed topology (e.g. $\frac{6}{1}2$ and its companion $\frac{2}{1} 6$). -1.0cm0 -5pt0 -5pt0 The variant of Hilbert’s construction involves letting oscillate various ovals across the ground ellipse. On the middle row of Fig.\[GudHilb3:fig\] we let oscillate thrice one oval and once the opposite oval with respect to some ground ellipse. Smoothing gives some $(M-2)$-curves not realizing the desired schemes. Choosing instead a triple oscillation of the upper oval of the quartic combined with a simple oscillation of the nearby oval gives the desired schemes (right row of Fig.\[GudHilb3:fig\]) with either 6 outer ovals (top) or 6 nested ovals (bottom). Gudkov was right albeit his discourse was not in perfect adequation with his picturing. It remains to geometrize such oscillations à la Hilbert. This is an easy matter, except that realist pictures require judicious scalings to make things visible. The first mode of vibration leading to $\frac{2}{1}6$ is geometrized on Fig.\[GudHilb5:fig\] below. -5pt0 -5pt0 Some few comments on this figure: first one has the two blue ellipses forming a quartic $C_4$. Next one has 4 dashed lines (another quartic). Perturbing slightly the former along the other (within the pencil spanned by both) gives another $C_4$ traced in black. This has the virtue of oscillating across the circular ellipse (in blue). Finally, smoothing their union gives the sextic in red realizing the desired real scheme $\frac{2}{1} 6$ (i.e. 2 ovals captured in one and 6 outside). The other scheme $\frac{6}{1}2$ is obtained similarly via the following system of oscillations (Fig.\[GudHilb6-12:fig\]) geometrizing the bottom-right part of Fig.\[GudHilb3:fig\]: -5pt0 -5pt0 -5pt0 -5pt0 Using either the schematic pictures or the more geometric one it is an easy matter to see that both curves just traced are dividing. This follows as usual (Fiedler’s law) by checking that all smoothings are compatible with complex orientations (cf. Fig.\[GudHilbdividing:fig\] below). -5pt0 -5pt0 Hence according to Ahlfors theorem there must be a total pencil of curves. Actually Rohlin claims much more that any sextic realizing those schemes is totally real under a pencil of cubics, but his argument has never been published[^43]. A bit like Hilbert in 1900, Rohlin 1978 says that his proof is too cumbersome to be written down. As we know Hilbert’s 2nd assertion that there is only two $M$-sextics was refuted by Gudkov some 7 decades latter, so it is not impossible that Rohlin’s claim is fallacious as well. Of course it can also be the case that Rohlin’s claim on the type I of the maximal $(M-2)$-schemes is correct, but that total reality via a pencil of cubics is erroneous. However as we noted cubics leads to a mapping-degree of $3 \cdot 6 -8=10$, in adequation with Gabard’s bound $r+p$ on the degree of circle maps. At this stage the naivest thing-to-do is to convince that there is no trivial counterexample to Rohlin’s claim. So we trace more oscillations to get the following pictures (Fig.\[GudHilb8:fig\]). Some noteworthy species appear especially the remarkable scheme $\frac{4}{1}4$, occupying the central position of Gudkov’s table (Fig.\[Gudkov-Table3:fig\]). Our specimen is dividing and it looks hard to get the same scheme in the nondividing way (though Rohlin 1978 asserts its existence). Speculating that this scheme is of type I, while admitting the truth of Rohlin’s maximality conjecture, then all 3 sextics schemes dominating $\frac{4}{1}4$ would agonize along a blue sky catastrophe! Gudkov would be wrong and Hilbert right! Of course this seems a too apocalyptic scenario, yet up to now our text does not entail this option! Also difficult to find are the schemes $\frac{5}{1}3$ and its mirror $\frac{3}{1}5$. Apart those exceptions, Hilbert’s method offers all possible $(M-2)$-schemes. [*Insertion*]{} \[08.02.13\] For a Harnack method realization of $\frac{3}{1}5$, see Fig.\[HarnaGudkov3-15XXL:fig\] much below, while $\frac{5}{1}3$ truly requires the method of Gudkov (cf. Fig.\[GudkovCampo-5-15:fig\]). If, via a small perturbation, one merges together $2$ small ovals on Harnack’s or Hilbert’s curve (cf. détail on Fig.\[GudHilb8:fig\]), then one gets the $(M-1)$-schemes $\frac{1}{1}8$ resp. $\frac{8}{1}1$. The other $(M-1)$-schemes of Gudkov’s table are somewhat harder to exhibit, except of course if one is aware of a large deformation able to extinct the inner oval (case of Harnack) or the outer oval (in Hilbert’s case). This contraction of empty ovals is actually possible via Itenberg 1994 [@Itenberg_1994]—using the apparatus of Nikulin’s (1979/80 [@Nikulin_1979/80]) (rigid-isotopy classification via K3 surfaces)—but of course this is surely not the most economical argument for our purpose (known to Gudkov 1969 or earlier). -1.2cm0 -5pt0 -5pt0 It seems clear that we have exhausted the faculty of Hilbert’s method (and its variation where the vibration is dissipated on several ovals). Some naive questions: what can be obtained by perturbing an arrangement of lines (in general position)? To go further one is helped once more by Gudkov 1974/74 [@Gudkov_1974/74 p.42] asserting that the scheme $\frac{4}{1} 5$ can be gained by a modification of Harnack’s method. Alas, Gudkov makes no picture but was aware of this at least since 1954. [*Insertion*]{} \[08.02.13\].—For a picture of this cf. Fig.\[HarnaGudkov4-15:fig\] much below. \[07.01.13\] Of course it is also possible to apply Hilbert’s oscillations to a Gürtelkurve (or other quartics), cf. Fig.\[GudHilb9:fig\]. Alas the list of schemes so obtained is not very exciting (no new species over the previous vibrations). -5pt0 -5pt0 Constructing the indefinite types (Brusotti, Rohlin 1978, Fiedler 1978, Marin 1979, plus do-it-yourself) {#indefinite-types:sec} -------------------------------------------------------------------------------------------------------- \[02.01.13\] Recall the definition (Rohlin 1978), a scheme is of [*indefinite type*]{} if it admits representatives of both types I and II, in the sense of Klein 1876, i.e. curves which are both dividing and not. This section aims to construct all schemes of indefinite type in degree 6 as to understand in full details Rohlin’s theorem (\[Rohlin-type:thm\]) enhancing Gudkov’s table by the data of Klein’s types. All the strategic information is tabulated in Fig.\[Gudkov-Table3:fig\], but each bit of coloring involves a little fight with the geometrical substratum. Again the ideas are purely those of Rohlin and his school, especially Fiedler. In Rohlin’s 1978 survey [@Rohlin_1978] detailed constructions are not given. After completion of this section, we noted that full details are given in Marin 1979 [@Marin_1979 p.57–58], whose constructions differ slightly from ours, but settling one case we failed to detect alone, namely the type $\frac{8}{1}_{II}$. $\bullet$ First, consider the scheme $\frac{2}{1}2$. Smoothing positively a triad of conics gives the dividing curve on the left of Fig.\[R2-12:fig\]. -5pt0 -5pt0 On the other hand, starting from a triangular configuration of ellipse (center part of Fig.\[R2-12:fig\]) one may by free-hand drawing (without taking care of orientation) arrange the real scheme to be the prescribed one. After reporting signs of our chosen smoothing we find them to be all negatives. At this stage I thought the curve to be dividing. However, right below one of the orientation is reversed but the smoothing effected left unchanged. Now it is of mixed signs, so the curve is in fact nondividing. On the right-top part of Fig.\[R2-12:fig\] is depicted another free-hand drawing realizing the given given real scheme. Mixture of signs implies the nondividing type of this curve. $\bullet$ On smoothing positively the configuration on the center-bottom part of Fig.\[R2-12:fig\] we get the (dividing) curve on the left-bottom of Fig.\[R2-12:fig\] which belongs to the real scheme $\frac{1}{1}5$. Next starting from the 3 ellipses, we got the miniature figure on the bottom of Fig.\[R2-12:fig\] who alas had not the right number of ovals. We thus started anew for the “radioactive” (triangular) triad of ellipses to find the right-bottom curve on Fig.\[R2-12:fig\] which has the correct real scheme and is nondividing (as it involves mixed signs). $\bullet$ On smoothing positively the “radioactive triad” of ellipses for the prescribed orientation gives the dividing curve on the left of Fig.\[R4-1:fig\]. This belongs to the scheme $\frac{4}{1}$ (i.e. 4 ovals nested in one big oval and nothing outside). It is easy to trace the same scheme using as template the “atomic triad” of ellipses, cf. middle-part of Fig.\[R4-1:fig\], and checking orientation one finds a mixture of signs imposing the nondividing character of this curve. Another option also yielding a nondividing curve is given on the right-part of Fig.\[R4-1:fig\] -5pt0 -5pt0 $\bullet$ Consider now a triad of ellipses with two ellipses invariant under rotation by $90$ degrees, plus one circle pinched in between. A positive smoothing creates the archipelago sextic on Fig.\[R8-1:fig\], which is dividing and of real scheme $\frac{8}{1}$ (i.e. $8$ ovals captured in a bigger one and nothing outside). It remains to find a nondividing realization of this scheme, cf. for this Marin’s picture=Fig.\[GudHilbMarin:fig\] below. -5pt0 -5pt0 $\bullet$ Dragging down the archipelago circle gives a configuration of ellipses smoothable positively to the scheme $\frac{3}{1}3$, cf. Fig.\[R3-13:fig\](left). After several infructuous attempts (depicted as miniatures) one finds the strange triad of conics on the right-part of Fig.\[R3-13:fig\] which admits a smoothing belonging to the same real scheme, but which is nondividing due to mixed signs. -5pt0 -5pt0 $\bullet$ After some patience and many trials (especially if one is tired) one finds another configuration of ellipses smoothable positively to the scheme $\frac{5}{1}1$, cf. left of Fig.\[R5-11:fig\]. Besides, one finds quickly the right-part of Fig.\[R5-11:fig\] belonging to the same real scheme, yet nondividing due to mixed signs. -5pt0 -5pt0 [**What remains to be constructed?**]{} At this stage we are nearly finished (compare the list of schemes we explored with those marked by rhombs on Fig.\[Gudkov-Table3:fig\]). It remains us to find the scheme $\frac{4}{1}4$. As we did not found it presently as a perturbation of 3 ellipses, and since this lies quite near (on Gudkov’s table=Fig.\[Gudkov-Table3:fig\]) to Gudkov’s $M$-sextic (notoriously difficult to construct) one is imbued of some suitable respect. Possibly it is impossible to exhibit as a deformation of (transverse) 3 ellipses. Notice yet that the curve $\frac{4}{1}4$ exists as shown by a variant of Hilbert’s method (cf. Fig.\[GudHilb8:fig\]). However presently this only realizes the scheme in the dividing way, whereas Rohlin claims this type to be indefinite. \[12.01.13\] A somewhat mystical way to solve this question involves taking a curve lying just above the scheme $\frac{4}{1}4$, while contracting an empty oval via passage through a solitary node. (Remember this to be possible by Itenberg 1994 [@Itenberg_1994].) Reading the deformation backward it follows from Klein’s remark (1876)(=Marin’s theorem 1988 [@Marin_1988]) that the resulting curve has type II. However there is probably a more elementary proof by looking at the scheme $\frac{4}{1}5$, which according to Gudkov (1974 [@Gudkov_1974/74]) can be exhibited by a variant of Harnack’s method, while its mirror $\frac{5}{1}4$ is harder to construct (Gudkov 1954 [@Gudkov_1954] even claiming erroneously its non-existence). \[07.02.13\] One elementary way to realize $\frac{4}{1}4$ in type II involves a modification of Harnack’s method depicted below (Fig.\[indef414:fig\]). (This is inspired by Gudkov’s text, but alas no picture there). This is the sort of bird hard to tackle down if one is tired. Beware also that in practicing Harnack’s method one never finds directly what one is seeking (I found this while searching $\frac{4}{1}5$.) The trick here is that we leave much room between the vertical lines effecting Harnack’s oscillations. So we start with the 3 high vertical lines, and a slight perturbation of the circle union the horizontal line produces a cubic $C_3$ oscillating thrice about the horizontal line $L$. The reducible quartic $C_3\cup L$ is then perturbed by a quadruplet of lines, which again is much stretched so as to effect another intermediate vibration. Then we have an $M$-quartic $C_4$ oscillating 4 times across $L$. Via the same trick $C_4\cup L$ is perturbed by a quintuplet of vertical lines to produce a $C_5$ oscillating 5 times across $L$. Then using Brusotti’s theorem (that German workers used subconsciously it seems or used ad hoc tricks to complete their perturbations) we have two ways to smooth $C_5\cup L$ to get a smooth $C_6$. Taking caring of orientations, the first depicted choice leads a curve of type I, whereas the second involves a negative sign and therefore produces type II. It is easily checked that both curves belong to the scheme $\frac{4}{1}4$. -2.2cm0 -15pt0 -5pt0 In a similar way, we do not have yet constructed the type II incarnation of the scheme $\frac{8}{1}$. Again in somewhat sloppy fashion, one could argue by contracting successively two empty ovals in Hilbert’s $M$-curve (scheme $\frac{9}{1}1$), namely the one outside and one inside the nonempty oval. Granting such a deformation through two (successive) solitary nodes, Klein’s remark implies the resulting curve being of type II, and we are done. Yet I presume there must be a more elementary construction. \[17.01.13\] Indeed one such is sketched in Marin 1979 [@Marin_1979 p.57, very bottom left of the table]. Let us reproduce Marin’s picture as Fig.\[GudHilbMarin:fig\]b: -5pt0 -5pt0 Marin’s trick here is to start from 2 ellipses tangent at one point but transverse elsewhere (Fig.a). Perturbing this by a suitable quadruplet of lines as in Hilbert’s method gives a quartic $C_4$ oscillating as depicted on Fig.a and with 3 ovals only. Hence the $C_4$ is nondividing (Klein’s congruence), and so is a fortiori the resulting sextic $C_6$ (as the nondividing character is dominant in the genetic sense), which realizes the $(M-1)$-scheme $\frac{9}{1}$ (Fig.a). A simple conjunction of two inner ovals yields the $(M-2)$-scheme $\frac{8}{1}$ (Fig.b), we were really interested in (and which is again of type II for the same genetical reason). With this trick we can construct several other curves, depicted on the second row of Fig.\[GudHilbMarin:fig\], in particular we get the scheme $10$ as well as $\frac{8}{1}_{II}$ via a variant of Marin avoiding tacnodality. The little price to pay is that we concede two imaginary intersections between the ground ellipses so that the nondividing character of the $C_4$ (unnested) has to be derived by some ad hoc argument (e.g. Klein’s in (\[Klein-unnested-quartic-nondividing:lem\]), or Arnold’s congruence $2=\chi=p-n\equiv k^2 \pmod 4$, or Rohlin’s formula $0=2(\Pi^{+}-\Pi^{-})=r-k^2$ or even Bézout modulo the highbrow contraction conjecture CCC, cf. Sec.\[CCC:sec\]). Of course there must also be an elementary argument by noticing that the two imaginary intersections of both ellipses are “connecting” different halves, so that when smoothed as shown the resulting curve is nondividing. Once this $C_4$ is known to be nondividing the depicted $C_6$ is likewise by virtue of the genetical dominance of nondividingness. All this argument looks tricky but is in reality trivial (think-yourself, and compare optionally Rohlin 1978, Fiedler 1981 [@Fiedler_1981], Marin 1979 [@Marin_1979], and maybe Gabard 2000 [@Gabard_2000]). As knowledge advances it will perhaps become as difficult to find new truths as to discover old mistakes. E.g., is Falting’s proof of Mordell correct? Is Freedman’s proof à la Bing reliable? Is Perelman’s proof of Poincaré really eclectic? If not should we retire him the million. No because because it was never accepted. Finding mistakes in those venerable implementations will perhaps be as challenging as claiming new truths? At any rate the game is always pleasant. Gudkov’s sextic $\frac{5}{1}5$ (Gudkov 1969, 1973, etc.) {#Gudkov:sec} -------------------------------------------------------- \[24.01.13\] Several constructions are available, but first some historical remarks. $\bullet$ The very first treatment appears in D.A. Gudkov’s Doctor Thesis (1969 [@Gudkov_1969-Doctor's-Thesis]) under Petrovskii and the liberal supervision of Arnold (apparently none of the supervisors were able to digest the full swing of the candidate Dmitrii Andreevich). Upon this Polotovskii 1996 [@Polotovskii_1996-D-A-Gudkov] comments as follows: “It is interesting to remark that the first proof of this fact in \[18\](=1969 [@Gudkov_1969-Doctor's-Thesis]) was extraordinarily complicated. It takes up $28$ pages of text, is a “pure existence proof”, and was obtained by means of a combination of the Hilbert-Rohn method with quadratic transformations. Shortly after D.A. Gudkov suggested significantly simpler [*constructions*]{} of curves having this scheme, see \[19\](=1971 [@Gudkov_1971-const-new-ser-M-curv]), \[21\](=1973 [@Gudkov_1973-const-curve-deg-6-type-515]), \[23\](=1974/74 [@Gudkov_1974/74]).” $\bullet$ This complicated proof was published in Gudkov-Utkin 1969/78 [@Gudkov-Utkin_1969/78] (English transl. issued in 1978). $\bullet$ New simpler constructions, are due to Gudkov and to be found in Gudkov 1971 [@Gudkov_1971-const-new-ser-M-curv], or in [@Gudkov_1973-const-curve-deg-6-type-515]), reproduced in his survey Gudkov 1974 [@Gudkov_1974/74]. $\bullet$ This is also reexposed in A’Campo 1979 [@A'Campo_1979]. $\bullet$ Viro 1989/90 [@Viro_1989/90-Construction p.1076] also emphasizes Gudkov’s initial construction “was rather complicated” (an euphemism as compared to Polotovskii’s prose above). His second proof reduces “to the first stage of Brusotti’s construction, i.e., the classical small perturbation of the union of the curve and the line.” Yet the whole difficulty is to find a quintic oscillating 5 times across the line while enveloping 5 ovals in one “wave oscillation” while leaving one oval outside (cf. Viro’s figure 12 in , p.1077). According to Viro (, p.1076): “It was only in 1971 that Gudkov \[11\](=1971 [@Gudkov_1971-const-new-ser-M-curv]) found an auxiliary curve of degree 5 that did this.” Hence Gudkov had two constructions of $\frac{5}{1}5$: $\bullet$ A first very complicated one, published in Gudkov-Utkin 1969/78 [@Gudkov-Utkin_1969/78] (English transl. published in 1978) or Gudkov 1973 [@Gudkov_1973-const-curve-deg-6-type-515] (using Cremona), and $\bullet$ a much more elementary one (à la Brusotti) given in Gudkov 1971 [@Gudkov_1971-const-new-ser-M-curv], by finding an $M$-quintic oscillating appropriately about a line, while smoothing their union. (I don’t know upon which version is based A’Campo’s account, presumably the second simpler variant, yet A’Campo uses Cremona.) Of course since Viro in the early 1980’s, Gudkov’s sextic may also be exhibited by Viro’s patchwork; or as a perturbation of three ellipses tangent at 2 points like Hawaiian earrings. This involves yet a deep understanding of how to dissipate such higher singularities. The interested reader can look at Fig.\[Viro3-15:fig\]c. Now let us describe once more Gudkov’s trick (source used Gudkov 1974 [@Gudkov_1974/74 p.42–43] and some more détail in A’Campo 1979 [@A'Campo_1979 p.12–13]). This is artwork of the best stock (cf. Fig.\[GudkovCampo-5-15:fig\]). -5pt0 -5pt0 $\bullet$ The first step is easy and consists to perturb a line at one of the 8 flexes of the quartic $C_4$ with $r=4$ slightly so that it creates 3 nearby intersections. Look at the fourth intersection, and from here trace two secant intercepting some other oval as shown, while cutting 2 nearby point on the “large” oval such that the line through them cut a little teats on the large oval. That all this can be achieved is already clever and explained in detail, first in Gudkov 1971 [@Gudkov_1971-const-new-ser-M-curv] or in A’Campo 1979 [@A'Campo_1979]. $\bullet$ The 2nd picture right below is merely a qualitative redrawing of the first. $\bullet$ The 3rd picture shows the transformation of the $C_4$ experimented under the Cremona transformation centered at the 3 points $4,5,6$, mutating it into a quintic $C_5$. One way to argue is via the birational invariance of the genus, keeping the value $g=3$ constant. Hence as the image curve has 3 nodes (arising as the intersection of the fundamental triangle through $4,5,6$ with $C_4$), it must be a quintic. Another way to argue is to remember the definition of Cremona as the projective (rational) map induced by the linear system of conics through the 3 basepoints (located on the large oval). Hence the pullback of a line is a member of the system, cutting the $C_4$ along 8 points, but 3 of them being assigned, we find $5$ for the degree of the image of $C_4$. Likewise the image of the diagonal line $L$ intersecting only the large oval $\alpha$ is of degree 1, hence a line. To understand the Cremona-map picture of Gudkov, one must keep in mind that Cremona contracts any edge of the triangle $4,5,6$ to the opposite edge of this triangle (and viceversa its explodes each basepoint to the opposite side of the triangle). The map being actually an involution (order 2). So the 4 chambers residual to the triangle are preserved. It is then fairly easy to check that Gudkov’s picture is realist, where tildes are images under Cremona. Life becomes easier if we number some few points on the $C_4$, while denoting by the same letters their images under Cremona (omitting the tilde for simplicity), compare Fig.\[GudkovCampo-5-15:fig\]. It remains to convince that the location of $\tilde{\gamma}$, $\tilde{\delta}$ is as depicted by Gudkov. The line $L$ is imagined as invariant under Cremona. In fact if we remove the 3 fundamental lines it remains 4 open triangles (homeomorphic to a cell ${\Bbb R}^2$) which are preserved. An involution of the plane has necessarily a fixed point (Brouwer, Kerekjarto, Smith, etc.) in fact a line or a singleton of fixed-points depending on whether it reverse or preserve orientation. The usual formula for Cremona $$(x_0, x_1, x_2)\mapsto(x_0 x_1, x_1 x_2, x_0 x_2)$$ shows that $(1,1,1)$ is fixed, and solving the fixed point equation (outside the fundamental triangle whence all $x_i\neq 0$) gives $(1,1,1)=\lambda(x_1, x_2, x_0)$ as unique solution. So the fundamental triangle splits ${\Bbb R}P^2$ in 4 chambers preserved under Cremona. How are they permuted? If we normalize the sign of the first coordinate as positive then we have the following signs distribution corresponding to the chambers $$I=(+,+,+),\; II=(+,+,-),\; III=(+,-,+),\; VI=(+,-,-).$$ The first chamber is preserved by Cremona. The second mutates to the fourth. The third maps to the second, and the fourth maps to the second. This looks a bit anomalous for by functoriality we would have expected that an involution induces an involution on the set of components (functor $\pi_0$). Changing the formula to the one written down in Gudkov 1974 [@Gudkov_1974/74 p.43] gives $$(x_0,x_1,x_2)\mapsto (x_1 x_2, x_0 x_2, x_0 x_1),$$ and we get $$I\mapsto I, II\mapsto(-,-,+)=II, III\mapsto (-,+,-)=III, \textrm{ and } VI\mapsto (+,-,-)=VI,$$ which is more pleasant. Thus after mutation the ovals $\gamma, \delta$ stays in the same chamber. Yet this is not enough for if they would lye like $ \gamma, \delta$ (without tilde) then we would get the scheme $\frac{3}{1}7$, which is prohibited either by Hilbert-Rohn-Gudkov or by Rohlin 1972 [@Rohlin_1972/72-Proof-of-a-conj-of-Gudkov] proof of Gudkov’s conjecture. (Remind Rohlin’s original proof to contain a little flow, repaired either by Rohlin via Atiyah-Singer or by Guillou-Marin!). So here we are quite close to adding another dramatic twist in the Hilbert-Gudkov saga. However it is more realist that a more thorough examination of the Cremona map shows the location of $ \tilde\gamma, \tilde\delta$ to be the one depicted by Gudkov. Indeed the chamber (say $III$) containing $ \gamma, \delta$ is invariant (like any other). However the line $L$ is also invariant and divides the chamber $III$ in two pieces which have to be exchanged by the Cremona involution. It suffices indeed to use the topological classification of involutions in the plane ${\Bbb R}^2$ à la Brouwer, to notice that in all cases (orientation reversing or not) the involution is either a reflection about a line or about a point (rotation). In both cases the residual components of an invariant line are exchanged. Hence chamber $III$ splits in two halves ${III}_{+}$ and ${III}_{-}$, where the former contains $\gamma, \delta$. Their images have to lye in the other chamber ${III}_{-}$, and Gudkov’s depiction is verified. In fact looking at the image of the point $x$ as mapped to $6=7=8$ shows that Cremona restricted to $III$ acts as a rotation (having one fixed points). More algebraically, solving the fixed-point equation $(x_0,x_1,x_2)=\lambda (x_1x_2, x_0 x_2, x_0x_1)$ shows that $$x_0=\lambda x_1 x_2=\lambda^2 x_0 x_2^2$$ so that $x_2^2=1/\lambda^2$, and likewise—by repeating the calculation or anticipating it by symmetry—we find $x_0^2=1/\lambda^2$, and $x_1^2=1/\lambda^2$. Thus up to homothety we have $(x_0,x_1,x_2)=(1,1,1)$ modulo the 4 possible variations of signs $(+,+,+)$, $(+,+,-)$, $(+,-,+)$ and $(+,-,-)$. We conclude that Cremona has exactly 4 fixed points (one in the barycenter of each chamber). So in particular Cremona is orientation preserving (within each chamber). $\bullet$ The fourth picture (of Fig.\[GudkovCampo-5-15:fig\]) contains also a little trick, namely the possibility to smooth the node (at $7=8$) of the trinodal quintic $C_5$ is such a way that its pseudoline penetrates slightly inside the line $\widetilde L$. Once this is done it suffices to smooth à la Brusotti $C_5 \cup \widetilde{ L}$ to obtain the desired Gudkov sextic. And the miracle is full. Why did it took so long (ca. one century from Harnack up to Gudkov) to discover this curve? Why Hilbert missed it? Admittedly the construction is quite tricky, but completely elementary. Up to our knowledge there is not any further simplification in this second Gudkov proof, apart perhaps via Viro’s patchwork or dissipation method of higher singularities, which probably require more highbrow technologies making them didactically hard to compete with Gudkov’s construction[^44]. As a last (sentimental) outcome, look how the quintic $C_5$ of Fig.\[GudkovCampo-5-15:fig\] resemble a portrait of its happy discoverer, especially $\tilde \gamma, \tilde \delta$ are like the eyes, and $\tilde \beta$ the smiling mouth of Gudkov near to crack the centennial problem. Finally it is plain from Gudkov’s curve to derive curves with less ovals, especially the $(M-1)$-curve $\frac{5}{1}4$ and the $(M-2)$-curve $\frac{5}{1}3$ (e.g. by changing the smoothing at the nodes $2$ and $1$). Those curves were notoriously hard to construct, and no construction independent of Gudkov’s is known. Using Fiedler’s signs-law it is plain that the curve $\frac{5}{1}3$ so constructed is of type II, as it should by virtue of say Arnold’s congruence. If instead we change the smoothing in the inside of the oval along the smiling mouth $\tilde \beta$ of Gudkov, then we get the $(M-1)$-scheme $\frac{4}{1}5$, and the $(M-2)$-scheme $\frac{3}{1}5$. Those were however much easier to construct by a variant of Harnack’s method (as reported in Gudkov 1974); compare indeed our Fig.\[HarnaGudkov4-15:fig\] and \[HarnaGudkov3-15XXL:fig\]. Finally we note that we may also obtain the $(M-2)$-scheme $\frac{4}{1}4$ in type II by smoothing the Gudkov configuration $C_5\cup \widetilde{L}$. However there is surely a more elementary approach via $\frac{4}{1}5$ constructed by a variant of Harnack; yes indeed compare Fig.\[indef414:fig\]. Diophantine and probabilistic aspects {#Diophantine-and-proba:sec} ------------------------------------- \[26.01.13\] Why did it took so long to discover Gudkov’s sextic? Is it only because it is the most secret part of the pyramid (Fig.\[Gudkov-Table3:fig\]), or because we have difficulty to visualize Cremona transformations? Is there some more intrinsic reason. One boring algebro-arithmetic game is to think of curves as ternary forms $F(x_0,x_1,x_2)=\sum_{i,j,k:i+j+k=m} a_{i,j} x_0^i x_1^j x_2^{k}$ with real coefficients. Up to rounding a bit the real coefficients randomly we may assume them rational numbers in ${\Bbb Q}$, and this can be done without affecting the topology nor the rigid-isotopy class. So we find nearby the given curve a smooth one defined over ${\Bbb Q}$, and we may put all coefficients in ${\Bbb Z}$ after scaling. As usual we may chase the common divisor of the equation to get a Diophantine equation with coefficients primes together $(\gcd (a_{i, j})=1)$. This we call the reduced equation of the rational curve (in the sense of Diophante as opposed to having genus $0$). It is unique up to sign. In particular there is a [*height*]{} defined as the largest coefficient of the equation. Then there is a myriad of question. For instance, given an isotopy type of real curve (or even a rigid-isotopy class) what is the smallest height of a Diophantine equation realizing this type? To make this concrete imagine the case of sextics. The Fermat equation $x_0^m+x_1^m-x_2^m=0$ shows that the corresponding chamber (unifolium) has always height 1. Similar remark for the invisible curve $x_0^m+x_1^m+x_2^m=0$ (anti-folium) when $m$ is even (empty real locus). However it is unknown if the curve with $r=1$ real branches always corresponds to a unique chamber of the discriminant (cf. Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical]). What is the height of Gudkov’s curve? Can we write down (the best) explicit equation? Another question is to look for some fixed integer $N$ (altitude) the set of all Diophantine equation $F(x_0,x_1,x_2)\in {\Bbb Z} [x_0,x_1,x_2]$ of height $\le N$ and consider how they distribute between the chambers of the discriminant. If $m=6$ there is 64 many chambers by Klein-Rohlin-Kharlamov-Nikulin 1979 [@Nikulin_1979/80]) encoded by the chromatic Gudkov table of Rohlin (Fig.\[Gudkov-Table3:fig\]). Of course some sporadic equations may land on the discriminant. Now count the corresponding $65$ (or rather $64$, maybe I added one for the discriminant but this will tend to zero) frequencies and consider the corresponding probabilities $p_{i,N}$ (indexed by the Gudkov symbols $i=\frac{k}{1} \ell$ (plus $(1,1,1)$ deep nest) enhanced sometimes by the type as on Fig.\[Gudkov-Table3:fig\]). Is the probability assigned to Gudkov’s chamber $\frac{5}1 5$ particularly low, say as compared to Hilbert’s or Harnack chamber? Paraphrasing slightly, how long would it take to a stupid computer to discover Gudkov’s sextic by merely tracing with clever algorithms the real locus of an explicit Diophantine equation, while randomly trying one equation after the other. In contrast one may expect that when $N \to \infty$ there is some equidistribution, with all probabilities tending to be equal. Perhaps some special rôle is played by the empty chamber which is connected by Nikulin 1979 [@Nikulin_1979/80], or better by the more elementary argument valid in all degrees, cf. (\[empty-chamber-connected-Shustin:lem\]). Of course a priori it is not even clear that the limiting probabilities converge as $N\to \infty$. What about the height of Gudkov’s chamber, i.e. the least size of the coefficient of a defining equation. Idem for Harnack and Hilbert’s chambers. Are they lower? Can we estimate the heights from above using the classical constructions made effective over ${\Bbb Q}$? Of course all these questions look perhaps a bit unnatural or somewhat out of reach. Also they depend on the height function (maximum coefficient), while there is perhaps other more natural ways to measure the complexity of an equation, e.g. by the Pythagorean distance (sum of all spares of the coefficient $\sum_{i,i} a_{i,j}^2$). This grows like a ball instead of like a cube, but perhaps the corresponding probabilities are independent of the exhaustion process? In that case there would be canonical probabilities and their estimation could be interesting. All this seems out of reach even when $m=6$, e.g. because we lack serious algorithms to detect the type from the equation (compare e.g. Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical]). Of course the asymptotic probability as $N\to \infty$ of landing in the discriminant will tend to zero (being a hypersurface of Lebesgue measure zero). So we should really have a distribution between $64$ numbers $p_i\in [0,1]$ (some possibly zero? yet unlikely) weighting the Gudkov-Rohlin pyramid (Fig.\[Gudkov-Table3:fig\]) by real masses. Are those probabilities all equal (equidistribution), rational numbers, etc.? Is the empty chamber much more heavy than the other? A crude intuition is that when coefficients get larger and larger, we get some thermodynamic excitation with all topological schemes (as complicated as they may be) fairly represented. Another less arithmetical way to pose the question of the frequency (e.g. of curves as Gudkov’s) is just to put the natural(?) round elliptic volume element à la Riemann-Lebesgue on the space $\vert mH \vert\approx {\Bbb P}^N$ of all curves-coefficients dominated by the round (unit) sphere $S^N$. The latter is calibrated to volume $2$ as to arrange unity volume for its quotient ${\Bbb R}P^N$. Each of the 64 chambers (when $m=6$) has then a (natural) mass, which demands only to be explicitly determined. It would be again exciting to compare the mass of Gudkov’s $M$-chamber with those of Hilbert’s or Harnack’s. Now it is clear that the discriminant has measure zero being a hypersurface, whereas all other chambers are affected by positive masses. How does a random equation (curve) look alike? Letting $p_i$ ($i=1,\dots, 64$) be the probabilities assigned to each of the (Rohlin-Kharlamov-Nikulin) chambers. Those are either all equal (equidistribution) which is quite unlikely, or some “curve” occurs more frequently? From zero-knowledge all what can be said is that some $p_i\ge 1/64$. What is the largest $p_i$? Maybe the empty chamber is the most massive? Of course then there is also refined questions about the Riemannian geometry of those chambers. Assume for simplicity equidistribution of masses. Then the whole hotel $\vert mH \vert-\frak D$ is shared by 64 families having chambers of the same volume, yet perhaps some are much more comfortable to live in. Annoying might be chambers highly contorted where there is little room to plug mobiliary inside. For instance we could look at the largest Riemannian ball expansible inside a given chamber, etc. Perturbation of lines (Plücker 1839, Klein 1873, Finashin 1996) {#Line-perturbation:sec} --------------------------------------------------------------- \[08.04.13\] This short section can be skipped. It was written at an early stage when we had not yet found all schemes asserted by Gudkov-Rohlin, primarily because we did not mastered sufficiently the Harnack method. So we attempted to realize schemes by perturbing lines. In this primitive context it could still be of interest to understand precisely what schemes are realized. If I remember well Felice Ronga (ca. 1999) once mentioned this problem as one challenging his imagination. Perhaps it is worth at the occasion trying to understand what can be said. \[12.01.13\] As yet we missed several schemes whose existence is asserted in Rohlin 1978 [@Rohlin_1978]. This is one motivation for trying to look at what is obtainable by perturbing an arrangements of lines. Of course some more ancestral motivation like the work of Plücker 1839 [@Plücker_1839] as credited for by Klein 1873 [@Klein_1873-Uber-Flächen-dritter-Ordn] gives also such a motivation. In fact this section was motivated by a figure in Finashin 1996 [@Finashin_1996 Fig.10] which we shall now reproduce while trying to explore other choices. The general question could be which (typed) schemes of the Gudkov-Rohlin table (Fig.\[Gudkov-Table3:fig\]) can be realized by perturbing a line arrangement. -5pt0 -5pt0 Contraction conjectures (Klein 1876, Rohlin 1978, Shustin 1985, Itenberg 1994, Viro 1994) ========================================================================================= “Klein-vache”: Nondividing implies champagne bubbling? (Klein 1876, disproof Shustin 1985) ------------------------------------------------------------------------------------------ \[14.01.13\] As early as 1876 [@Klein_1876], Klein asserted the firm conviction that curves of type I cannot gain an oval by crossing a solitary node. It required ca. 110 years until Marin 1988 took the pain to write down a proof of a somewhat stronger assertion (cf. Sec.\[Klein-Marin:sec\]). In the same paper, Klein (1876) speculated about a much more metaphysical converse allowing any nondividing curve to gain an oval after crossing a solitary node. This was never rigidly asserted by the cautious Felix Klein, but disproved 99 years later by Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]. Personally, we have not yet assimilated the full details of Shustin argument, as it uses much technology, but all experts (Shustin, Viro, Fiedler, Orevkov, etc.) have validated this disproof. [(Klein’s hypothesis of 1876, abridged “Klein-vache” in the sequel, disproved in Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin])]{} \[Klein-1876:conj-noch-entwicklungsfahig\] Given any nondividing plane curve of arbitrary degree $m$, it is possible to let it cross the discriminant through a solitary node via a path of curves $(C_t)_{t\in [-1,+1]}$ traversing only once the discriminant. In other words any diasymmetric chamber bounds a solitary wall. \[15.01.13\] The conjecture is nearly evident when $m=6$ in view of Rohlin’s enrichment of Gudkov’s table by types and the subsequent rigid-isotopic classification of Nikulin 1979/80 [@Nikulin_1979/80] (Theorem \[Nikulin:thm\]). With this data available one gets a bijection between chambers past the discriminant and Rohlin’s enriched schemes (cf. Fig.\[Gudkov-contigBIS:fig\] below). -5pt0 -5pt0 A moment contemplation of this table shows that all diasymmetric chambers admit at least one edge moving upwards in the hierarchy incrementing the number of ovals $r$ by one unit. Of course a priori such an increment does not necessarily correspond to the formation of a solitary node (isolated double point) but can also traduce the subdivision of an oval shrinking to a lemniscate. Hence more work is required, yet we believe strongly that “Klein-vache” holds true for sextics. In fact here is a complete proof: \[Klein-vache-deg-6:prop\] [(Gabard 15.01.13, but a trivial corollary of Rohlin 1978, Nikulin 1979/80, Itenberg 1994 [@Itenberg_1994] and Klein-Marin 1876–1988)]{}.—The conjecture “Klein-vache” [(\[Klein-1876:conj-noch-entwicklungsfahig\])]{} holds true for $m=6$, i.e. any nondividing sextic can acquire a solitary double point by a rigid-isotopy crossing only once the discriminant transversally. It is first a matter of paying attention to the combinatorics of Rohlin’s classification into types (Fig.\[Gudkov-contigBIS:fig\] above). The rest of the proof is then nearly self-explanatory. In slight contrast to Rohlin 1978 [@Rohlin_1978] we forbid the “hermaphrodite” [*indefinite schemes*]{} (allowing projective realizations of both types I/II) but rather imagine them as two superposed (but distinct) elements, with the dividing schemes (especially the blue rhombs on Fig.\[Gudkov-contigBIS:fig\]) levitating slightly above the sheet of paper of that figure. By Nikulin’s theorem 1979/80 [@Nikulin_1979/80] those combinatorial symbols (with levitating twins above the blue-rhombs) are in one-to-one correspondence with the chambers past the discriminant. Now imagine on Fig.\[Gudkov-contigBIS:fig\] a sort of random flow moving downwards along the red-edges of that figure. Let us be more precise. By a result of Itenberg 1994 [@Itenberg_1994 Prop.2.1, p.196] (based upon techniques used by Nikulin ()) [*each empty oval of a sextic can be contracted to a solitary node before disappearing in the blue sky*]{}. (An oval is said to be [*empty*]{} if it contains no oval in its interior.) Pick a curve in each chamber and pick two contractions (given by Itenberg) shrinking either an outer oval or an inner oval, provided both are available on the real scheme. If only inner or outer ovals are available, pick only one contraction. This can be visualized as a “random” vector field moving downward along the diagrammatic of Fig.\[Gudkov-contigBIS:fig\]. Each Itenberg contraction necessarily lands in type II (diasymmetric) chambers. Else if landing in an orthosymmetric (=dividing) chamber, then reading the Morse surgery backwards corrupts the Klein-Marin theorem (even in its weak original formulation of Klein 1876 [@Klein_1876], though the latter gave no proof but see (\[Klein-via-Ahlfors(Viro-Gabard):lem\]), or (\[Klein-Marin:lem\]), or the 1st hand source Marin 1988 [@Marin_1988]). Hence our random vector field has its “trajectories” ending on the bottom sheet of paper (as we imagine orthosymmetric chambers levitating somewhat above the sheet of paper, see again the blue-rhombs on Fig.\[Gudkov-contigBIS:fig\]). It is plain now that all diasymmetric chambers (green squares on Fig.\[Gudkov-contigBIS:fig\], plus those lying behind the blue rhombs) do occur as extremities of our vector field encoding the varied Itenberg contractions chosen. Interpreting this process backward-in-time proves “Klein-vache” in degree $6$. The proof is complete. [*Insertion*]{} \[30.03.13\] It should be noted that Itenberg’s contraction theorem affords in degree 6 another proof (independent of total reality) of Rohlin’s maximality principle (in degree 6), at least if we take for granted the RKM-congruence (\[Kharlamov-Marin-cong:thm\]). This prompts another strategy toward Rohlin’s maximality conjecture (independent of total reality) and perhaps worth exploring further. Of course the hearth of the problem seems to be the Itenberg-Viro contraction conjecture for any empty oval (\[Itenberg-Viro-contraction:conj\]), but this does not seem to imply Rohlin’s maximality conjecture. In contrast to “Klein-vache” the Itenberg-Viro contraction conjecture is still open and certainly worth investigating further. It is also worth noting that at the earth of the above proof (\[Klein-vache-deg-6:prop\]) we have Itenberg’s contraction theorem. Thus roughly Itenberg implies Klein-vache, yet this is not the sole ingredient for otherwise in degree 8 Shustin’s disproof of Klein-vache would refute the contraction principle (which is still open in degree 8). So the above proof really uses more than just the contraction principle. In some sense it uses results by Nikulin but only as a mean to get Itenberg contractions. What looks more pivotal is the role of the Gudkov-Rohlin table. One may thus wonder if in degree 8, we can get sufficient grasp on the Gudkov-Rohlin table as to infer the logical move from the contraction principle to Klein-vache. If feasible, then Shustin’s disproof (1985) would refute the Itenberg-Viro contraction conjecture (1994) in degree 8. This scenario looks a priori quite likely and requires perhaps just completing the full diagrammatic of Hilbert’s 16th in degree 8, plus the extra-data of types. (This is perhaps available within the next decade, if we appreciated correctly the optimism of experts). Factually, the above proof can be summarized by saying “Itenberg contraction+Gudkov-Rohlin diagrammatic$\Rightarrow$Klein-vache”, yet without that it is crucial to have a bijection between typed-schemes and rigid-isotopy classes à la Nikulin. This correspondence being disrupted in degree 7 (and so probably 8) by Marin 1979 (cf. Fig.\[Marin:fig\]). Hence it seems likely that a completion of the Gudkov-Rohlin table in degree 8, will imply a refutation of the Itenberg-Viro contraction conjecture. The above proof of Klein-vache (in degree 6) is quite attractive, but to be really sublime it should extend to higher orders. Several obstacles arise. First Itenberg’s contraction principle becomes conjectural for $m>6$ (compare Viro’s preface in the same volume). Next our argument rests on the deep combinatorial classification of Rohlin 1978 [@Rohlin_1978], plus Nikulin’s rigid-isotopy classification via real schemes enriched by the type data (I/II). This ceases to be true for orders $m\ge 7$ (Marin 1979/80 [@Marin_1979], Fiedler 1982/83 [@Fiedler_1982/83-Pencil]). Thus the above proof looks jeopardized for higher orders. Of course, if one believes in Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]), then “Klein-vache” is actually false when $m=8$. Historiographically, it is of course quite improbable that Klein’s (weak) intuition about “Klein-vache” was based upon the above procedure (Torelli for K3’s being needed by Nikulin), yet it is also not completely impossible that a more elementary proof than the one above exists (cf. optionally Sec.\[Klein-vache-proof:sec\]). At any rate Klein’s power of prediction via geometric intuition is once more quite amazing. More modestly, it should be stressed that Klein, interpreted in the lowbrow fashion, merely asserts that there is no topological obstacle toward implementing “Klein-vache”, yet he is prudent enough in not claiming this as a theorem (compare again Klein’s original Quote \[Klein\_1876-niemals-isolierte:quote\] which is beautifully ambiguous). \[11.01.13\] A first natural question is whether Klein-vache implies the direct sense of Rohlin’s 1978 conjecture (i.e. “type I implies maximal”). In fact Klein-vache shows rather that if a scheme is not of type I (so contains a nondividing representative) then it is non-maximal. Paraphrasing, “type I is implied by maximal”. This is however the part of Rohlin’s conjecture that was refuted by Shustin 1985/85 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]. So indirectly Shustin’s counterexample also destroys (the hard half of) Klein’s intuition (i.e. Klein-vache). Shustin’s result is somewhat stronger: [(Shustin 1985)]{} \[Shustin:thm\] There exists a maximal scheme of degree $8$, which is of type II. [(copied from the source)]{} Shustin proves first the following assertion. There exists $(M-2)$-curves of degree $8$ with the schemes $10 \sqcup 1 \langle 1 \rangle \sqcup 1 \langle 2 \rangle \sqcup 1 \langle 4 \rangle$, and $6 \sqcup 1 \langle 2 \rangle \sqcup 1 \langle 4 \rangle \sqcup 1 \langle 5 \rangle$, in the notation of Viro (i.e. the notation $1 \langle k \rangle$ means one ovals enveloping directly $k$ empty ovals). One starts with a certain quintic $C_5$ having controlled topology with respect to the $3$ axes (constructed in Polotovskii 1977 [@Polotovskii_1977/77]). Then applying a quadratic transformation gives a singular octic with “complicated” singularities. On dissipating such complicated singularities (Viro’s method 1980) one may create the 2 required schemes. [*End of the proof of Theorem \[Shustin:thm\] (compare also Sec.\[Shustin-understood:sec\] for our slow assimilation of Shustin’s proof)*]{}. Applying a result of Viro 1983 [@Viro_1983/84-new-prohibitions], the $(M-2)$-schemes constructed above are of type II. It remains now to check that they are maximal. [*Insertion*]{} \[31.03.13\].—The Euler-Ragsdale $\chi$ of the first scheme is $\chi=10+(1-1)+(1-2)+(1-4)=6$, while $k^2=16$. Hence Arnold’s congruence mod 4 (or the allied Rohlin’s formula) suffices to establish type II of the curve. For the second, $\chi =6+(1-2)+(1-4)+(1-5)=-2$, and again Arnold/Rohlin suffices to show type II. First Shustin says that the $(M-1)$-schemes obtained from them by addition of an oval (if they exist) are (always) of type II, referring to Rohlin 1978 [@Rohlin_1978 point 3.2]. Needless to say, this is actually a trivial consequence of Klein’s congruence (1876) $r\equiv g+1 \pmod 2$. Yet more seriously it seems to me (Gabard) that we do not need only to know these schemes being of type II, but rather that they do not exist at all!? So in my opinion there may be a trivial misconception here? In fact we can apply the Gudkov-Krakhnov-Kharlamov congruence (Theorem \[Gudkov-Krakhnov-Kharlamov-cong:thm\]) for $(M-1)$-curves to all possible enlargements (cf. Sec.\[Degree8:sec\] for details) yet this fails prohibiting a specimen. Shustin’s argument looks uncomplete at this stage, or presumably rests on stronger obstructions used subconsciously by the author!?) (\[24.01.13\] Compare again Sec.\[Shustin-understood:sec\] for our assimilation of Shustin’s proof; what is required is a prohibition of Viro.) Next Shustin argues that the $M$-schemes obtained from the given ones by the addition of two ovals are forbidden by the extremal comparison in Rohlin 1978 [@Rohlin_1978 point 1.3], and Viro 1980 [@Viro_1980-degree-7-8-and-Ragsdale Theorem 4]. [*Conclusion.*]{}—Beside Polotovskii 1977, Shustin’s result relies massively on Viro’s revolutionary technique of construction via dissipation of complicated singularities (which came to be known as “patchworking”). Yet the basic logics of Shustin’s reasoning looks a bit elusive and perhaps flawed. (\[24.01.13\] Not all, cf. again Sec.\[Shustin-understood:sec\].) Hence it is not clear to me if it really destroys the hard-half of Klein’s intuition (i.e. Conjecture \[Klein-1876:conj-noch-entwicklungsfahig\]). Let us repeat once more the crucial quote of Klein 1876: [*Z. B. kann bei den Kurven der ersten Art durch allmähliches Ändern der Konstanten niemals eine isolierte reelle Doppeltangente neu enstehen, um dann einen $(C+1)$-ten Kurvenzug zu liefern; während die Kurven der zweiten Art in dieser Richtung nicht beschränkt sind. Die Kurven der zweiten Art sind sozusagen noch entwicklungsfähig, während es die Kurven der ersten Art nicht sind. Doch soll hier auf diese Verhähltnisse noch nicht näher eingegangen werden.*]{} Translated in English (while adhering to Russian notation and jargon) gives something like: [*For instance, for curves of type I an isolated solitary node can never rise as to produce a new real circuit through progressive variations of the coefficients; whereas curves of type II are not restricted in this way. Curves of type II are so-to-speak still developable, while those of the first type are not.*]{} This demonstrates that Klein only cautiously asserted that curves of type II are not obstructed to acquire a solitary node, yet not claiming something so radical as our Conjecture \[Klein-1876:conj-noch-entwicklungsfahig\], albeit his second sentence goes closer to suggesting this interpretation. \[24.01.13\] At any rate this Ansatz of Klein turns out to be corrupted by Shustin’s article, relying heavily on the new prohibition detected by Viro (cf. again Sec.\[Shustin-understood:sec\] for our ultimate assimilation of this). Degree 8: the Grand pyramid of Gizeh {#Degree8:sec} ------------------------------------ \[12.01.13\] Can we picture out the Gudkov-Rohlin pyramid in order 8? Since $m=8$ we have $g=\frac{(m-1)(m-2)}{2}=\frac{7\cdot 6}{2}=7\cdot 3=21$. So $M=g+1=22$. It is first quite easy to extend upwards the Gudkov symbols as to build a larger pyramid (Fig.\[Degree8:fig\]). Yet this contains only schemes with 1 (or less) nonempty oval. One can easily report the modulo 8 prohibitions coming from Gudkov-Rohlin, etc., as discussed in Sec.\[Gudkov-hypothesis:sec\]. -3.2cm0 -5pt0 -5pt0 [*Insertion*]{} \[02.04.13\].—A naive trick is to think of the whole pyramid as fibred over the depicted one (Fig.\[Degree8:fig\]) which shows the range of $(\chi, r)$. So a basic procedure is to start from the given elementary configuration with symbol $\frac{x}{1}y$ and to make a menagerie of transfer of ovals conserving $\chi$. This involves a Bonsai-cutting art-form of the Hilbert tree. Yet this does not really solve our puzzle of making a good chart of all possible schemes in degree 8. The others schemes having $\ge 2$ nonempty ovals are a bit messy to report. In particular it seems unrealistic willing to report all schemes on a single table! Should we try several charts, but then how to track their interrelations and overlaps? Can we split in several classes? Let us try to use the number $N$ of nonempty ovals as a splitting recipe. $\bullet$ if $N=0$ we have the schemes $\ell$ ($0\le\ell\le 22$), so 23 schemes. $\bullet$ if $N=1$ we have the schemes $\frac{k}{1}\ell$. $\bullet$ if $N=2$ we have $\frac{k}{1}\frac{\ell}{1}m$ (if $m=0$ this is still pictured as the left semi-triangle, yet for larger $m$’s one must imagine several layers lying above the sheet of paper). Of course it may be assumed $k\ge \ell$. $\bullet$ if $N=3$ we have $\frac{k}{1}\frac{\ell}{1}\frac{m}{1}n$ $\bullet$ if $N=4$ we have $\frac{k}{1}\frac{\ell}{1}\frac{m}{1}\frac{n}{1}o$, but using a pencil of conics we see that $o=0$, and that $k,\ell, m, n \le 1$ and so we have unique such scheme, namely 4 nests of depth 2. $\bullet$ Schemes with $N\ge 5$ are prohibited by Bézout with conics. Okay but all this is a bit overwhelming to depict (except if one is able to visualize a pyramid in 4D!). Yet we could ask if there is a reasonable classification of all schemes according to their 3 types (as did Rohlin 1978 for $m=6$). Apart from the obvious schemes of type I, and the natural consequences of Arnold-Rohlin, etc. giving a complete answer looks again a herculean effort. Incidentally, it is an open problem as still some few cases are resisting to the experts of Hilbert’s 16th. Yet we can ask more specific questions like (as did Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]) to corrupt one half of Rohlin’s maximality conjecture. The trick is that under an enlargement of the scheme the number of nonempty ovals can only increase. So to see what lies above Shustin $(M-2)$-schemes with $N=3$, it is enough to contemplate the face of the pyramid with $N=3$, since $N=4$ is nearly empty. Now writing one of Shustin’s scheme in Gudkov’s notation gives $\frac{4}{1}\frac{2}{1}\frac{1}{1}10$. Note that $\chi=p-n=(1-4)+(1-2)+(1-1)+(10)=+6=-2 \pmod 8$ so the Kharlamov-Marin congruence (\[Kharlamov-Marin-cong:thm\]) says nothing, but as observed above the more elementary Arnold congruence forces type II. (Elementary B.A.-BA, hence skip).—To compute the value of $p-n$ (positive minus negative ovals also called even\[=pair in French\] and odds) one may use the trick of filling the ovals by an orientable membrane in ${\Bbb R}P^2$ bounding them in the obvious way, i.e. we take the interior of all the outer ovals, then remove the interior of the subsequent generation of ovals immediately nested inside, and aggregate again the inside of the next generation, etc. One has then the psychologically useful formula $p-n=\chi$, where $\chi$ is the Euler characteristic of this orientable planar membrane (which Möbius would call a reunion of binions, trinions, etc.) Let us now examine the enlargements of Shustin’s scheme. First, we find four $(M-1)$-schemes ruling out those which are not Bézout permissible (cf. Fig.\[Shustin:fig\]). One of them $\frac{4}{1}\frac{2}{1}\frac{1}{1}11$ (framed on the figure) is not prohibited by the Gudkov-Krakhnov-Kharlamov-congruence $\chi=p-n\equiv k^2\pm 1 \pmod 8$ (Theorem \[Gudkov-Krakhnov-Kharlamov-cong:thm\]). Whether this scheme is actually realized is another question. If it is then Shustin’s result (1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]) would be erroneous. \[\[24.01.13\] No sorry this is a misconception of Gabard, cf. Sec.\[Shustin-understood:sec\] for a clarification, but detailed right now for the impatient reader. The point is that if this $(M-1)$-scheme is realized, then it will be the counterexample to Rohlin (hence to Klein) for there is nothing above it by a Viro prohibition (stating that $M$-schemes of degree 8 have an odd content trough out, cf. (\[Viro-Fiedler-prohibition:thm\])). Further if it does not exist then the $(M-2)$-scheme $\frac{4}{1}\frac{2}{1}\frac{1}{1}10$ will be a counter-example to Rohlin’s conjecture, since it would be maximal but of type II. So we get the promised disproof of one-half of Rohlin and of Klein-vache, without having to know precisely what happens above Shustin’s $(M-2)$-scheme. As a matter of fact it seems, the first alternative correspond to reality, i.e. the $(M-1)$-scheme $\frac{4}{1}\frac{2}{1}\frac{1}{1}11$ is realized.\] On reading again Shustin argument () he does not explain convincingly why this $(M-1)$-scheme enlarging his one does not exist algebraically, but merely notice that all $(M-1)$-schemes must be of type II (a triviality of course since Klein’s congruence $r\equiv g+1 \pmod 2$). So again I cannot follow Shustin’s reasoning, and perhaps it is not a serious torpedo against Rohlin’s maximality conjecture! Is there a flaw in this Shustin note??? -5pt0 -5pt0 Of course one can play the same game for the other scheme proposed by Shustin, namely $\frac{5}{1}\frac{4}{1}\frac{2}{1}6$, and we get the following Fig.\[Shustin2:fig\]. Alas again one larger $(M-1)$-scheme is not prohibited by Gudkov-Krakhnov-Kharlamov. -5pt0 -5pt0 Of course it can be the case that the $(M-1)$-scheme lying above Shustin’s scheme (we write in the singular as we fix attention to one of his scheme) can be prohibited by a stronger prohibition that Gudkov-Krakhnov-Kharlamov, which Shustin might have used subconsciously. Yet with what is written down in the article I could not verify his argument. So perhaps “Klein-vache” is still true (cf. Conjecture \[Klein-1876:conj-noch-entwicklungsfahig\]). In fact Shustin’s paper is the only (published) obstruction I am aware of against Klein’s intuition. \[15.01.13\] Idea to explore (but skip as it leads nowhere \[02.04.13\]).—Maybe Shustin used subconsciously some inequalities stronger than the congruences, perhaps those à la Ragsdale-Petrovskii. (Those are discussed in Rohlin 1978 [@Rohlin_1978 p.87].) Let us try the Ragsdale conjecture. It states $$p\le \frac{3}{2} k(k-1)+1.$$ Rohlin asserts () that the first chance of refuting this inequality is the case $m=10$ (cf. indeed the breakthrough of Itenberg-Viro of Fig.\[Itenberg:fig\]). So Ragsdale should be true for $m=8$ (despite having been subsequently disproved by Itenberg via Viro’s patchwork of a variant thereof (again Fig.\[Itenberg:fig\]). Calculating on the first Shustin’s enlarged $(M-1)$-scheme we find $p=3+11=14$ versus $\frac{3}{2} k(k-1)+1=\frac{3}{2} 4(4-1)+1=2\cdot 9+1=19$. So Miss Ragsdale is far from violated. Another idea would be to use a pencil of cubics through some deep ovals of the enlarged $(M-1)$-schemes. Yet some easy counting shows that we may force 22 real intersection by taking a connected cubics through 8 basepoints specified inside the deep ovals, but this not enough to overwhelm Bézout accepting $3\cdot 8=24$ intersections. Finally understanding Shustin’s argument (with the help of Orevkov’s letter) {#Shustin-understood:sec} ---------------------------------------------------------------------------- \[24.01.13\] Thanks to a letter of Stepa Orevkov, and a survey of Viro 1989/90 [@Viro_1989/90-Construction p.1126], we learned the presence of new Bézout-like prohibitions on $M$-schemes derived in Viro 1983 [@Viro_1983/84-new-prohibitions]. Those have the special feature of not being of topological origins, but rather algebro-geometric. (This also enabled Viro in 1979 to complete the (soft) isotopic classification of septics solving thereby the next case of Hilbert’s 16th, compare e.g. Viro 1989 [@Viro_1989/90-Construction p.1124].) Here is the relevant Viro’s result of which we actually just need the first clause (transcribed in conservative Gudkov’s notation, having in our opinion a slight advantage of compactness when it comes to put the symbols on a pyramid): [(Viro 1983 [@Viro_1983/84-new-prohibitions])]{} \[Viro-Fiedler-prohibition:thm\] $\bullet$ ($M$)—If $ \frac{\alpha}{1}\frac{\beta}{1}\frac{\gamma}{1} \delta$ is the real scheme of an $M$-curve of degree $8$ with $\alpha, \beta$ and $\gamma$ nonzero, then $\alpha, \beta$ and $\gamma$ are odd. $\bullet$ ($M-2$)—If $ \frac{\alpha}{1}\frac{\beta}{1}\frac{\gamma}{1} \delta$ is the real scheme of an $(M-2)$-curve of degree $8$ with $\alpha, \beta$ and $\gamma$ nonzero and with $\alpha+\beta+\gamma\equiv 0 \pmod 4$, then two of the numbers $\alpha, \beta, \gamma$ are odd and one is even. —\[25.01.13\] We postpone the proof to a latter occasion, and merely reproduce now the remark to be found in Viro 1983 [@Viro_1983/84-new-prohibitions p.416]: “The special case of Theorem 2.2.E when $\delta=0$ and $\beta=1$ is due to Fiedler \[11\](=Fiedler 1982/83 [@Fiedler_1982/83-Pencil]). Theorem 2.2E was stated as a conjecture by A.B. Korchagin in connection with my results on realization of the real schemes of $M$-curves of degree $8$. The theorem rules out $40$ real schemes which are not ruled out by Theorems 2.2.A–2.2.D (of these forty, four are ruled out by the special case of Theorem 2.2.E which was proved by Fiedler).” The proof is completed on p.422 of Viro’s text. It starts as follows: Let $C=C_8$ denote a smooth octic with real scheme $\la \alpha \vc 1\la \beta\ra \vc 1 \la \gamma\ra \vc 1\la \delta \ra \ra$, where $\beta, \gamma$ and $\delta$ are nonzero. The crucial result is Theorem 4.2 in Viro, which itself is based upon Fiedler. So the proof looks too technical to be reproduced here. A self-contained account encompassing Fiedler and Viro’s article would require several pages, and we postpone this to a future occasion. One may wonder if the special case implemented by Fiedler does not suffice actually to corrupt “Klein-vache”, i.e. Klein’s Ansatz that nondividing curve can bubble out a new solitary node out of the blue sky. However Fiedler’s result prohibit only the scheme $ \la 1 \la 1\ra \vc 1 \la \alpha \ra \vc 1\la \beta\ra \ra$ with even nonzero $\alpha$ and $\beta$ (cf. e.g. Viro 1983 [@Viro_1983/84-new-prohibitions p.420]), and a priori this is not enough to prohibit the enlargeability of some suitably chosen $(M-1)$-scheme (compare e.g. the constructions proposed by Orevkov in the next Sec.\[Orevkov:sec\]). The first assertion prohibits the remaining $M$-scheme of Fig.\[Shustin:fig\]. As to the second clause of (\[Viro-Fiedler-prohibition:thm\]) pertaining to $(M-2)$-schemes, I do not know what to do with it. At this stage I read again Orevkov’s letter (cf. Sec. \[e-mail-Viro:sec\]), which I have some pain to interpret properly. Let us reproduce it right below for convenience, while adding some brackets of mine. Before completing this reading, I finally understood Shustin’s argument. The point is that whether or not the $(M-1)$-scheme (framed on Fig.\[Shustin:fig\]) exists do not matter. Indeed if it does exist (algebraically) then it is of type II (by Klein’s trivial congruence) and maximal (by Viro’s prohibition in the above theorem), whereas if does not exist then Shustin’s $(M-2)$-scheme is maximal but of type II, by construction (or by Arnold). So in both cases Rohlin’s reverse implication “type I$\Leftarrow $ maximal” is foiled. [*Insertion*]{} \[02.04.13\].—Of course it would be interesting to know if Shustin’s $(M-1)$-scheme enlargement do exist (algebraically), i.e. the scheme $\frac{4}{1}\frac{2}{1}\frac{1}{1}11$. If I interpret correctly the letter below of Orevkov (while removing a little misprint from it, namely trading the “11” for a “10”), it seems that the $(M-1)$-scheme written above is realized algebraically. Stepa Orevkov’s letter {#Orevkov:sec} ---------------------- We now reproduce Orevkov’s letter (brackets=\[ \], are our additions): $\bullet$$\bullet$$\bullet$ \[16.01.13–14h56: Stepa Orevkov\] A small remark: It is wrong that $11 \cup 1\langle 1\rangle \cup 1\langle 2\rangle \cup 1\langle 4\rangle $ is not a part of an $(M-1)$-scheme. It is. \[Not clear how to interpret this? Does it mean that Shustin’s claim is wrong, or simply that this scheme is an $(M-1)$-scheme. My question was whether this $(M-1)$-scheme is realized algebraically, of course. Yet, I admit that my question was a bit ill posed. In fact I wonder if Orevkov not intended to write a “10” instead of the above eleven.\] Moreover, there is no known example of $(M-2)$-curve of type II which cannot be obtained from an $(M-1)$-curve by removing an empty oval. \[So Klein looks still plausible for $(M-2)$-schemes, while Shustin looks wrong. No sorry, in fact I misunderstood Shustin for a long time, as he does not claim that the framed $(M-1)$-scheme does not exist.\] In contrary, there are $(M-1)$-curves of degree $8$ (which are necessarily of type II) which do not come from any $M$-curve. These are[^45]: $3\langle 6\rangle $[^46] $4 \cup 1\langle 2\rangle \cup 2\langle 6\rangle $ $8 \cup 2\langle 2\rangle \cup 1\langle 6\rangle $ $12 \cup 3\langle 2\rangle $ Construction (inspired by Shustin’s construction of $4 \cup 3\langle 5\rangle $ \[should locate the reference\]): Consider a tricuspidal quartic $Q_{sing}$ symmetric by a rotation $R$ by $120$ degree and perturb is\[=it\] so that each cusp gives an oval (we assume that this perturbation is very small). Let $Q$ be the perturbed curve. Two flex points appear on $Q$ near each cusp of $Q_{sing}$. We chose flex points $p_0, p_1, p_2$ (one flex point near each cusp) so that $R(p_0)=p_1, R(p_1)=p_2, R(p_2)=p_0$. We choose homogeneous coordinates $(x_0 : x_1 : x_2)$ so that the line $x_i = 0$ is tangent to $Q$ at $p_i$ $(i = 0,1,2)$. Let $C$ be the image of $Q$ under the Cremona transformation $(x_0 : x_1 : x_2) \mapsto (x_1x_2 : x_2x_0 : x_0x_1)$. Then $C$ has 3 singular points, each singular point has two irreducible local branches: a branch with $E6$ and a smooth branch which cuts it “transversally”. By a perturbation of $C$ we obtain all the four curves mentioned above. The fact that these curves cannot be obtained from $M$-curves immediately follows from the fact that, for any $M$-curve of degree 8 of the form $b \cup 1\langle a_1\rangle \cup 1\langle a_2\rangle \cup 1\langle a_3\rangle $, all the numbers $a_1$, $a_2$, $a_3$ are odd[^47]. Best regards Stepa O This letter helped me much to understand finally Shustin’s proof, and is of course worth studying for its own (especially to make a picture of it). It gives another counterexample to Rohlin’s maximality conjecture, hence to Klein’s Ansatz of champagne bubbling nondividing curves. \[25.01.13\] Now here is an attempt to vizualize Orevkov’s example. As he said we start with a tricuspidal quartic. This is known since time immemorial (maybe Euler 1745, Steiner 1857, cf. e.g. Briekorn-Knörrer 1981/86 [@Brieskorn-Knörrer_1981/1986 p.32] where it is described as a hypocycloid, cf. also Lawrence p.135, where it is called the Deltoid). This being given we smooth out the cusps to create some little ovals. I presume this can be done by hand, otherwise there is a theorem of Gudkov 1962 [@Gudkov_1962] extending to cusps that of Brusotti 1921. The more difficult task is to understand what happens under the Cremona transformation. Here I was much aided by the prototype of Gudkov’s example (cf. Sec.\[Gudkov:sec\]), which is the first place where Cremona maps were applied to topology of real varieties. Remind that Orevkov’s example, is inspired from Shustin, himself being a direct student of Gudkov. So the first steps are fairly easy (say classical for Gudkov’s era), yet it took me some times to trace appropriately the Cremona transform of the $C_4$. I hope my picture is correct (ask Orevkov if needed)? It is imagined (I presume) that (like in Gudkov’s construction, cf. again Fig.\[GudkovCampo-5-15:fig\]) the flecnodal tangent is slightly perturbed to become transverse to the $C_4$. This implies then the funny behaviour “forth-back-and-forth” of the image $C_8$ at the place $1$, say. So there is an octic as depicted. To trace the picture it is useful to keep in mind that the Cremona map takes edges of the fundamental triangle to the opposite vertices of the triangle, while preserving the 4 residual component of the triangle. -3.7cm0 -5pt0 -5pt0 Now we arrive at the hard step, namely the dissipation of such singularities. Here comes the contribution of Viro (if we do not misunderstood history). As explained say in Viro 1989/90 [@Viro_1989/90-Construction p.1111–12], there is a myriad of high order singularity, and one would like to understand their dissipation. The singularity at hand in our case has 4 branches, while 3 of them have a second order contact (tangency) at the singular point. For this specific singularity, he quotes Korchagin (1988). So we have a table of possible template of dissipation, which may be locally glued in place of the singularity (exactly as in Brusotti’s method of small perturbation which amounts to the simplest singularity “$A_1$”). Substituting one of this template, one may hope to find the schemes announced in Orevkov’s letter. Yet, it must be hoped that I use the right singularity (???, their naming being non-canonical apparently?), and as yet I failed Worse if we take $\alpha=5$ and $\beta=1$ and the very first dissipation of Viro’s picture (right of Fig.\[Orevkov2:fig\]) (which is permissible in view of the congruence mod 4), then we get a curve with $18+3+3=24$ ovals violating frankly Harnack’s bound $M=g+1=(7\cdot 6/2)+1=21+1=22$. I got something wrong!!! \[26.01.13\] Another explanation could be that for higher singularities there is no analog of Brusotti’s theorem on the independence of simplification. The latter is brilliantly explained in Gudkov 1974 [@Gudkov_1974/74], as reducing ultimately to Riemann-Roch, but also a theorem of Max Noether, and even special series. Note that in this passage of Gudkov, he seems to be not completely up-to-date with the problem of special series on curves, as was solved by Meis 1960 [@Meis_1960] in the special case of pencils, and by Kempf 1971 [@Kempf_1971], and Kleiman-Laksov 1972 [@Kleiman-Laksov_1972] independently in the early 1970’s. Mistrusting Shustin 1985, while trying to prove “Klein-vache” 1876 (via Garsia-Rüedy or vanishing cycles à la Poincaré-Severi-Lefschetz-Deligne-Mumford, etc.) {#Klein-vache-proof:sec} -------------------------------------------------------------------------------------------------------------------------------------------------------------- \[06.03.13\] All of our (initial) mistrusting of Shustin’s proof is not really justified anymore, being in part clarified above (Sec.\[Shustin-understood:sec\]) modulo assimilation of the Viro-Fiedler advanced Bézout-style prohibition (\[Viro-Fiedler-prohibition:thm\]). Hence the sequel has to be read with suitable discernment, but was not completely censured as it may contains geometric ideas worth exploring further, and other issues of independent interest. \[13.01.13\] Could it be that Shustin 1985 was wrong, while Klein 1876 is correct!? If so how to prove Klein-vache (\[Klein-1876:conj-noch-entwicklungsfahig\])? Of course this amounts to an amazing topological flexibility of Riemann surfaces as flying-saucers moving in the Plato cavern of plane projective geometry, where smooth curves are known to have “particular” moduli. More concretely one could imagine that this is always possible via pure Anschauung, namely the process dual to the subsequent Fig.\[Klein-Marin:fig\](left) read in reversed time. One would take a (globally) invariant cycle (=circle) traced on the diasymmetric surface which is however acted upon without fixed point by the symmetry (antipodal map on the circle). Such circles deserve a name: \[antioval:def\] [An [*antioval*]{} of a symmetric (Riemann) surface is a topological circle traced on the surface invariant under the involution and acted upon antipodically by the symmetry.]{} First, note as a trivial topological issue, the following. \[antioval:lem\] Antiovals only exist on diasymmetric surfaces, all of them admitting one. Assume the surface orthosymmetric (i.e. dividing) and containing an antioval. By definition an antioval lacks real points, being acted upon antipodically (by Galois). Take one point of the antioval and its conjugate (which is distinct) and look at an arc of the antioval linking $p$ to $p^\sigma$. This arc is in the imaginary locus, yet connects two conjugate points, violating the orthosymmetry assumption. Conversely suppose given a diasymmetric surface $(S, \sigma)$, hence the quotient $S/\sigma$ is non-orientable. Choose a loop reversing the indicatrix (local orientation) and avoiding the boundary of $S/\sigma$. The counter-image of this circle in $S$ gives a circle $C$, since the orientation reversing loop lifts to an arc via the quotient map which is a genuine double cover outside the boundary (alias contour by analysts). Since the symmetric surface is recovered from the quotient via the orientation cover (Klein-Weichold yoga), the circle $C$ is the desired antioval. By Klein 1876 (and Riemann), the number $r$ of ovals (better real circuits) is bounded by $r\le g+1$ (so-called Harnack bound, under the supervision of Klein who found a more intrinsic reason). It is natural asking about a similar bound for antiovals. The antipodal sphere $S^2$ shows that each great circle is an antioval, whence an infinity of such. Consider next an antipodal torus of revolution in 3-space invariant under rotation about the $z$-axis and acted upon by central symmetry $(x,y,z)\mapsto (-x,-y,-z)$. We see 2 evident antiovals by sectioning with the horizontal plane $z=0$ (Fig.\[Antioval-torus:fig\]a). -5pt0 -5pt0 Varying the slope of this plane gives an infinitude of antiovals until we reach the critical tangent plane (Fig.\[Antioval-torus:fig\]b) after which the 2 ovals are not nested anymore. The argument certainly generalizes to any genus upon placing some symmetric pretzels in 3-space. So any diasymmetric surface without fixed points has infinitely many antiovals. In fact the above proof (part 2) shows that infinitude is a general feature without resorting to a Euclidean realization of Klein’s symmetric surfaces. After this topological triviality let us try to attack Conjecture \[Klein-1876:conj-noch-entwicklungsfahig\], i.e. “Klein-vache” (allusion to Lefschetz’ vache coined by Grothendieck? Weil? and used by Deligne, etc.) The idea we try to exploit (but we are unable to complete the argument) involves another crazy intuition of Klein validated by Garsia-Rüedy building over works by Teichmüller, namely the fact that any Riemann surface admits a Euclidean realization in 3-space. Suppose given a diasymmetric (real) plane curve $C_m$ of (arbitrary) degree $m$. We may as usual look at the underlying Riemann surface. According to Klein’s intuition (validated by Garsia and Rüedy, building over a contribution of Teichmüller) we know that all closed Riemann surfaces admit a conformal model in Euclidean $3$-space. Let us dream that this adapts as well in some equivariant form for symmetric surfaces (with an anticonformal involution). Note en passant: of course closed non-orientable surfaces do not embedded in $E^3$ (Euclidean $3$-space), but if bordered they do. So the only boring diasymmetric surfaces are the invisible ones (no fixed points) but those luckily enough admit a centrally symmetric model in $E^3$. Choose a conformal model of $C_m({\Bbb C})$ in $E^3$ supplied by Garsia-Rüedy. By the lemma we know that there is an antioval on the diasymmetric surface. Shrink the latter to a point via a (plastical) deformation in $E^3$ akin to Fig.\[Klein-Marin:fig\](left) read in reverse sense. Note at this stage that not all antiovals pinch to a “connected” surface (e.g. a pretzel of genus 2 with a belt dividing into two pieces). So even the topological aspect deserves to be precised, by looking at “good” (i.e. nondividing antiovals). Let us assume that those always exists. Next look at our isotopic deformation in $E^3$ to a pinched pretzel, generating a one-parameter family of Riemann surfaces. The difficulty is to ensure that they stay planar (embeddable in ${\Bbb P}^2$) during the deformation. This looks a priori quite implausible, but we have not yet exploited the full punch of Garsia-Rüedy. Their result states that all Riemann surfaces arise in the tubular vicinity of any classical surface in $E^3$ via a normal deformation of arbitrarily small amplitude. Picturesquely, if you have any old woman(=Riemann) surface, but feel erotically bored by her due to an acute case of cellulitis just let vibrate her skin to get any girl you ever dreamed about. In the oldest lady hides any beautified young girl with taught epiderm, at least so conformally! So there is some chance that even if our initial plastical shrink deviate outside the realm of plane curves (seen as a stratum $\Pi$ in the moduli space ${\cal M}_g$, where $g=\frac{(m-1)(m-2)}{2}$) we can still rectify the trajectory so as to stay scotched along the planarity manifold $\Pi$ (for each time). We get so an “abstract isotopy”, i.e. a path in $\Pi$ the planarity manifold. Next we have a canonical map $\vert mH \vert-\disc \to \Pi \subset {\cal M}_g$ from the space of smooth curves of order $m$ to the moduli space ($\disc$ being the discriminant hypersurface). It should be easy to lift our abstract isotopy to $\vert m H \vert$ while having only the extremity ending in $\disc$ (necessarily at a solitary node by construction). Then one continues by letting emerge an oval. If all this is feasible (taking further better care of the involution) then Garsia-Rüedy implies “Klein-vache”. Perhaps the above strategy requires to be adapted in $E^4$ to gain more flexibility and more care about the symmetry. Also if the Garsia-Rüedy trick in $E^3$ is not best suited to the problem at hand, a more direct approach could be to stay in ${\Bbb C}P^2$. Recall indeed that Ko 2001 [@Ko_2001] has a fairly general extension of the theorem to any ambient Riemannian manifold. Alternatively more classic algebro-geometric methods (Severi’s Anhang F, etc., e.g. as modernized by Harris, etc.) are perhaps quite likely to imply “Klein-vache” if such methods of degeneration adapt to the reality context (equivariance w.r.t. Galois which is quite a rule when pure synthetic geometry is involved). But I must seriously refresh my memory on those works. Another tactic toward “Klein-vache” via Itenberg-Viro suggesting a general evanescence principle ------------------------------------------------------------------------------------------------ \[14.01.13\] Is there a relation between “Klein-vache” and the natural Itenberg-Viro conjecture (cf. Itenberg 1994 [@Itenberg_1994], and the preface of that volume by Viro) positing that: \[Itenberg-Viro-contraction:conj\] [(Itenberg-Viro 1994, abridged CC=contraction conjecture).]{}—Any empty oval of a (real, smooth) plane curve can be contracted to a point (solitary node) via a rigid-isotopy. [*Historical note*]{} \[04.04.13\].—In Klein 1892 [@Klein_1892_Realitaet] (p.176 in the pagination of Ges. Math.Abh. 1922 [@Klein-Werke-II_1922]) there is discussed what he calls the “Doppelpunktsmethode” amounting essentially to contract any symmetry-line of the Riemann surface. This seems to anticipate the Itenberg-Viro contraction principle. It is not clear however that Klein ever formulated something as precise as the above conjecture (specific to plane curves). On p.176–177, Klein’s prose extracted from its context sounds a bit overoptimistic, namely: “Bei allen anderen Fällen hat die Durchführung des genannten Prozesses und damit die Zusammenziehung eines beliebiegen Ovals der Kurve zu einem isolierten Doppelpunkte keine Schwierigkeit.” This seems to trivialize the Itenberg-Viro conjecture but probably does not because Klein thinks really with abstract Riemann surfaces where there is much more flexibility than with plane curves. However it is not impossible that refining Klein’s argument/ideas could prove CC, but it is also quite likely that CC is false. At first sight one may expect a direct logical subsuming of “Klein-vache” to “Itenberg-Viro’s contraction conjecture”. However some moment thought shows that there is no such direct “rapport de force”, i.e. “Klein-vache” is not implied, nor does it imply, the Itenberg-Viro contraction of empty ovals. However Prop. \[Klein-vache-deg-6:prop\] gives a logical subordination of Klein-vache to the contraction principle in presence of additional combinatorial knowledge available in degree $6$. Via Nikulin’s theorem (\[Nikulin:thm\]) on the rigid-isotopy classification of sextics it is nearly evident that Conjecture \[Itenberg-Viro-contraction:conj\] holds true for sextics. This is actually the object of Itenberg’s article just cited. [*Insertion*]{} \[31.03.13\] In view of Viro’s isotopy classification in degree 7, and the philosophy that contraction plus combinatorial knowledge implies Klein-vache, one can also wonder if Klein-vache holds true in degree 7. Alas we lack a tool like K3’s in degree 7, and so the situation is somewhat obscure in degree 7. Possibly, Shustin’s disproof of Klein-vache descends from degree 8 to 7, and then maybe that the contraction principle is already disrupted in degree 7. Recall, that presently the contraction principle is wide open in degree 8, yet perhaps disprovable via Shustin (and a completed classification). The true relationship between “Klein-vache” and Itenberg-Viro contraction hypothesis (\[Itenberg-Viro-contraction:conj\]) could be rather an analogy in the principle of proof that one might naively develop, namely the possibility of shrinking a cycle invariant under Galois(=complex conjugation). Indeed “Klein-vache” amounts essentially to shrink an antioval (cf. Def. \[antioval:def\]), whereas Itenberg-Viro amounts shrinking an empty oval. Hence it may be suspected that there is a general strangulation principle specializing to both “Klein-vache” and “Itenberg-Viro” stipulating the following: [(Shrinking principle)]{} \[shrinking-principle-vague:conj\] Any (Galois) invariant cycle(=circle) on a smooth plane curve of degree $m$ can be strangulated through a path in the hyperspace of curves crossing only once the discriminant at a smooth point of the latter (whenever there is no topological obstruction to do so). The parenthetical proviso is required, for one cannot shrink a nonempty oval without shrinking all its inner ovals, creating thereby a singularity of higher complexity than nodal. The proof of (\[shrinking-principle-vague:conj\]) could be similar to the eclectic one sketched for “Klein-vache” in the previous section, i.e. either via Garsia-Rüedy (hence Teichmüller theoretic) or algebro-geometric via vanishing cycles à la Poincaré-Picard-Lefschetz-Severi, etc. Let us examine the combinatorial possibilities for such a Galois-cycle. Being (by definition) invariant under complex conjugation $\sigma$, it can either be: \(1) an oval (pointwise fixed by the Galois-Klein symmetry $\sigma$); \(2) an antioval or dia-oval (acted upon antipodically by $\sigma$); \(3) a pseudo-oval or ortho-oval (acted upon by $\sigma$ with two fixed points, hence like $(x,y)\mapsto (x,-y)$ on $S^1=\{x^2+y^2=+1\}$). Our terminology ortho- and dia-oval is directly inspired by the figure of Klein 1892 (reproduced in our Quote \[quote:Klein-1891/92-ortho/dia\]), where given a circle and a point outside it one considers the involution of $S^1$ exchanging the 2 intersections of each line of the pencil. When the point lies inside the circle we get a diasymmetry (antipode like), while if it is outside an orthosymmetry (mirror with 2 fixed points). Given a real curve (equivalently a symmetric Riemann surface in the sense of Klein), an oval exists except in the lowest diasymmetric case $r=0$ (of Klein’s classification). A dia-oval exists only in the diasymmetric case (Lemma \[antioval:lem\]). An ortho-oval can exist in both the dia- and orthosymmetric cases. An example of an ortho-oval is traced as the cycle $\beta$ on Fig.\[Klein-Marin:fig\]. Specializing the shrinking principle (\[shrinking-principle-vague:conj\]) to an oval implies the Itenberg-Viro contraction hypothesis (\[Itenberg-Viro-contraction:conj\]), to a dia-oval implies “Klein-vache” (\[Klein\_1876-niemals-isolierte:quote\]). Finally shrinking an ortho-oval leads to another natural: Any two contiguous ovals can coalesce after crossing an ordinary node with real tangents. [*Contiguous*]{} means here that both ovals can be joined in ${\Bbb R}P^2$ by an arc having only its extremities on the ovals. Two contiguous ovals can either be directly nested or unnested yet unseparated by a larger oval. One should not forget the possibility of a single oval subdividing himself. The latter operation is subsumed to no topological obstruction, except that one might enter in conflict with Bézout. So we may dream of such an unifying principle explaining the perfect topological flexibility of “rigid-isotopies” permitted to traverse only once the discriminant transversally. In some sense (to be made precise) our shrinking conjecture asserts that any Galois-cycle shrinks provided there is no topological obstruction either in ${\Bbb R}P^2$ nor in the complex locus. Alas our crude principle does not seem compatible with the: \[Finashin-obstruction-to-coalesce-Harnack:lem\] [(Admitted, but not understood!, to whom is it due? Stated in Finashin 1996)]{} Harnack’s (sextic) scheme $\frac{1}{1}9$ can only degenerate toward the scheme $10$ by contraction of the inner oval, yet not by coalescence of the two nested ovals. Cf. e.g. Finashin 1996 [@Finashin_1996 p.68, proof of Thm 6.2], who alas does not give a precise reference for this assertion. So here we have a clear-cut example of a Galois-cycle (namely an ortho-oval) linking the inner oval with the nonempty one of Harnack’s curve (Fig.\[Finashin:fig\]), yet which cannot be shrunk. Why is it so? Remember in contrast that the Gürtelkurve $C_4$ (quartic with 2 nested ovals) can see both its ovals coalesce (Fig.\[Finashin:fig\]). What is the difference between Zeuthen-Klein $C_4$ and Harnack’s $C_6$? If we take the pain of tracing the complex orientation (by Fiedler’s algorithm) we get the following pictures. It is seen that for the Gürtelkurve $C_4$ the complex orientation (in red-arrows) disagree from the orientation as the boundary of the annulus (grey-shaded), while for Harnack’s $C_6$ the ${\Bbb C}$-orientation matches that as boundary of the ring. Could positive pairs of ovals be an obstruction to coalescence? -5pt0 -5pt0 \[16.01.13\] As we said we may also take an ortho-oval cutting twice the same oval. Shrinking this would effect a (cellular) subdivision of the oval. A first example is a hyperbola pinching to a pair of lines to become another hyperbola. Projectively we have permanently a conic with a single oval, so there is no naive minded subdivision like that of a cell in the naive organical sense. Incidentally 2 ovals for a conic corrupt either Bézout or Harnack, especially in the formulation of Klein. Likewise the unique oval of a cubic (if available) cannot be subdivided (without corrupting either Bézout or Harnack-Klein $r\le g+1$). However an oval of a quartic can sometimes subdivides (cf. Fig.\[Subdivide:fig\]). (If this figure is realist it is tempting to create an octic by small perturbation with $16$ unnested oval, yet let us not be sidetracked by this.) -5pt0 -5pt0 Visualizing the corresponding surgeries on the Riemann surface must be a pleasant exercise. If the curve is dividing (hence its Riemann surface orthosymmetric) then a subdivision is impossible without corrupting the strong form of Klein’s Ansatz proved by Marin 1988 [@Marin_1988]. It is still tempting to imagine an ortho-oval on a orthosymmetric curve, especially if it cuts only one oval. A myriad of choices are possible. Let us depict few of them. Fig.\[Orthoovals:fig\]a shows an ortho-oval dividing the surface. Hence when contracted we would get a reducible curve. As long as our naive picture (be it embedded or abstract) is respected this is incompatible with Bézout (or if you prefer the intersection theory of ${\Bbb P}^2({\Bbb C})$ whose generator $H$ of the second homology $H_2$ satisfy $H^2=+1$) unless both sides of the cycle $\beta$ have genus $0$. This proves the following lemma, whose significance is of course not confined to the case of real curves. \[strangulation-impossible:lem\] A dividing cycle on a smooth plane curve $C_m$ of degree $m\ge 3$ cannot be strangulated by a rigid-isotopy crossing only once the discriminant. By contradiction, assume strangulability possible along the given dividing cycle via a path of curves $(C_t)_{t\in [-1, +1]}$ starting from the given curve, i.e. $C_{-1}=C_m$ and so that only $C_0$ is singular and uninodal (smooth point of the discriminant). Denote by $S_t$ the corresponding Riemann surfaces, $S_t=C_t({\Bbb C})$, where of course $S_0$ is mildly singular. Then the strangulated surface $S_0$ splits in two (smooth) orientable surfaces $S_1, S_2$ each porting a fundamental class $\sigma_i$ in $H_2({\Bbb C}P^2)\approx {\Bbb Z}$ ($i=0,1,2$). Hence we get in homology $\sigma_0=\sigma_1+\sigma_2$, and so taking respective degrees $m=m_0=m_1+m_2$, where $m_i=\deg \sigma_i$ (degree in the homological sense). The intersection $\sigma_1 \cdot \sigma_2$ computes as $m_1 \cdot m_2$, which have to be equal to $1$ (as the critical curve $C_0$ as just one normal crossing). It follows that $m_1=m_2=1$, violating the assumption $m\ge 3$. The case $m=2$ is entirely different as a conic may degenerate to a pair of lines. The interesting option is to take a nondividing ortho-cycle $\beta$, as depicted on Fig.\[Orthoovals:fig\]b. -5pt0 -5pt0 Let us now shrink such a nondividing cycle $\beta$ to a point getting something like Fig.\[Orthoovals2:fig\]b, which is a nodal curve (still irreducible, because its Riemann surface is connected). After the critical level we could expect to find Fig.\[Orthoovals2:fig\]d, but this is impossible for the genus drops by one (remind that all smooth curves of some fixed degree have the same genus, since on the complexes the discriminant does not disconnect having real codimension $2$). In fact as soon as the handle is strangulated by the vanishing cycle it reappears instantaneously as depicted on Fig.\[Orthoovals2:fig\]c. On meditating slightly this occurs like a twisting, compare the miniature figures below depicting the pre- and post critical levels near the singularity. During the twisting one see that the north hemisphere of the orthosymmetric surface is suddenly connected with the south hemisphere forcing the diasymmetric nature of the post critical curve. (All this phenomenology is of course allied to the name of Picard-Lefschetz and Dehn.) -5pt0 -5pt0 Discovering eversions (Gabard 16.01.13, but surely in Möbius, von Staudt, Hilbert, Morosov, Gudkov, Kharlamov, Finashin, etc.) {#Eversion:sec} ------------------------------------------------------------------------------------------------------------------------------ \[16.01.13\] How can this process (Fig.\[Orthoovals2:fig\]) occur at all if it is supposed to occur in the plane? In the naive Euclidean plane ${\Bbb R}^2$, any self-coalescence of a Jordan curve leads to a subdivision (compare the center of Fig.\[Klein-Marin:fig\] read backwardly) increasing the number of real circuits. However during our Riemann surface surgery (again Fig.\[Orthoovals2:fig\]) the number of real circuits is kept constant. Hence there seems to be a basic topological obstruction to our shrinking process, yet some more mature thinking shows this not to be the case. In reality we live in the projective plane ${\Bbb R} P^2$, so one oval may well expand “to infinity” to self-coalesce while keeping one component after having been “Morse surgered”. For varied depictions of this phenomenon, see Fig.\[Eversion:fig\] where as usual ${\Bbb R} P^2$ is depicted as a disc with contour antipodically identified. -5pt0 -5pt0 Thus an oval can be Morse surgered without splitting off a new oval. Let us call this process [*eversion*]{}[^48] of an oval. Note that after the eversion all the inside of the oval appears suddenly outside of it! This basic phenomenon resolves several misconceptions or paradoxes that foiled for ca. 6 days my understanding of that theory, especially when it comes to a parity anomaly between the degree of the discriminant for sextics $3(m-1)^2=75$ and the legal moves in the Gudkov pyramid (Fig.\[Gudkov-Table3:fig\]) encoding all real schemes combinatorially. If eversions are overlooked, the contiguity graph between chambers permits only closed circuits of even length, whereas by Bézout or Galois a generic pencil of sextics (defined of ${\Bbb R}$) has to cut an odd number of times the discriminant of degree 75. Nearly a contradiction in mathematics if eversion would not exist! Consequently, \[eversion-deg-6-or-more-forced-by-loops:lem\] Any generic pencil of real curves of even order $m$ contains at least one eversion. At least this is clear for $m=4,6$, and hopefully correct in general. (inserted \[01.04.13\], but contains a gap!) The discriminant has odd degree $3(m-1)^2$ when $m$ is even. Our pencil is generic in the sense of being transverse to the discriminant, hence induces a sequence of Morse surgeries. Those surgeries (if not “eversive”) can be visualized on the Gudkov table (Fig.\[Gudkov-Table3:fig\]) in degree 6 as moves along the lattice of red-rhombs which permits only closed pathes of even length, whence the assertion for $m=6$. In general we can introduce the invariants $(\chi, r)$ and notice that any Morse surgery which is not an eversion acts as one of the 4 transformations $(\chi, r)\mapsto ( \chi \pm 1, r \pm 1)$ where signs can be chosen independently. (Not all of them being possible as shown for $m=6$.) Alas this does not seem to be enough to conclude, because those sole invariants $(\chi, r)$ amounts to a planar projection of the whole pyramid (which in general is not a “planar” object say for $m=8$). So one really needs to understand the crystallography of higher pyramids which hopefully still contain merely loops of even length when eversions are omitted. Hopefully our lemma is still true (cf. maybe a related argument in Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000]). Of course an oval belonging to a certain real scheme can be (topologically) everted iff it is [*maximal*]{} (i.e. not included in any larger oval). Let us consider some examples. Suppose the given scheme to be $\frac{1}{1} 9$, i.e. Harnack’s $M$-sextic. Then there are 10 maximal ovals available. Everting the unique nonempty oval of Harnack’s scheme gives Hilbert’s scheme $\frac{9}{1} 1$ (cf. Fig.\[Eversion2:fig\]), while everting of of the 9 outer ovals leads to a configuration which is not Bézout-tolerable. The net consequence is that in the hyperspace of all curves two schemes (better chambers) may be in reality much closer than they look far apart on Gudkov’s pyramid (Fig.\[Gudkov-Table3:fig\]). There seems to be some secret passages permitting quick travelling in the pyramid. Of course whether Harnack’s chamber is really contiguous to that of Hilbert is another story! It would be fine so for them to sleep in good company, yet more mature thinking bring us back to the Riemann surface picture. The ortho-cycle effecting the strangulation is (since both are $M$-curves) necessarily dividing hence not strangulable by Lemma \[strangulation-impossible:lem\]. For instance one can imagine the top orthosymmetric surface with $r=g+1$ as the double of a planar domain $D$ (with $r$ contours). Suppose given on this an ortho-cycle $\beta$ meeting twice the same oval. The image of $\beta$ in one half (our plane domain) is an arc $\beta^{+}$ joining twice the same contour, and some moment thought shows that $\beta$ divides the surface. Indeed the arc $\beta^+$ can be completed to a Jordan curve in $D$ by aggregating an arc of the boundary and we apply Jordan. Better, argue that $\beta^+$ divides $D$ because we may shrink to a point the contour containing the extremities of $\beta^{+}$, and then apply Jordan separation. The separation effected by $\beta^+$ readily implies that by $\beta$. We have proven: \[eversion-impossible-for-M-curves:lem\] An $M$-curve (of degree $m\ge 3$) cannot undergo an eversion (while crossing normally the discriminant). In particular Harnack’s chamber in the hyperspace of sextics is not contiguous to Hilbert’s. If it could be everted, the corresponding path of curves would be materialized by the evanescence (strangulation) of an ortho-cycle $\beta$ cutting twice the same oval. But $M$-curves correspond to the top-orthosymmetric case with planar half. Thus by Jordan separation our cycle $\beta$ divides and therefore cannot be strangulated (by Lemma \[strangulation-impossible:lem\]). -5pt0 -5pt0 One can imagine more complicated eversions of Harnack’s scheme (cf. bottom of Fig\[Eversion2:fig\]), yet the result is still the same. Everything depends merely on the oval being everted, for what was inside becomes outside and conversely. For instance Gudkov’s scheme $\frac{5}{1}5$ turns into itself under eversion of the nonempty oval, whereas everting a nonempty oval leads to a scheme enlarging $(1,1,1)$, the deep nest of depth 3, hence Bézout incompatible. Is the Gudkov chamber self-contiguous to itself via an eversion? Again this is merely a topological possibility, but it requires a deeper investigation to see if it is really so. This would imply Gudkov’s chamber to be highly contorted like a banana-shaped, and it is quite likely that its closure is not simply-connected. However the lemma above (\[eversion-impossible-for-M-curves:lem\]) precludes a self-contiguity of the Gudkov chamber to itself. The real option however is that there are two (non-maximal) chambers past the discriminant related by eversion, and actually we know this phenomenon to exist a priori in view of the degree argument of Lemma \[eversion-deg-6-or-more-forced-by-loops:lem\]. It seems of interest to understand the secrete passages between Gudkov symbols of Gudkov’s pyramid, at least those topologically permissible under eversion. For sextics we get the following enhancement of the Gudkov-Rohlin pyramid with curvilinear-edges amounting to the varied eversion (Fig.\[Gudkov-eversion:fig\]). Note that we may only evert the nonempty oval without corrupting Bézout (maximality of the deep nest $(1,1,1)$). The sole exception arise with the unnested schemes, plus the scheme $\frac{1}{1} {1}$ whose empty-oval eversion is precisely the deep nest, whereas the nonempty-oval eversion flips back the scheme to itself. We get something like the following messy picture (Fig.\[Gudkov-eversion:fig\]) attempting to keep track of all logically possible eversions. -5pt0 -5pt0 The question is to decide which among those are effectively realized algebraically. We already know that those interconnecting $M$-schemes cannot be realized so due to a topological obstruction (Lemma \[eversion-impossible-for-M-curves:lem\]). As usual blue-rhombs have to be duplicated according to their types and really correspond to 2 distinct chambers of the discriminant. If a dividing curve (of degree $m\ge 3$) undergoes an eversion then the post-critical curve is nondividing. Fig.\[Orthoovals2:fig\] nearly proves this via the occurrence of a Dehn-twist in the Picard-Lefschetz transformation. Again the proviso $m\ge 3$ is evident since Dehn twisting an equatorial sphere leads to the same equatorial sphere which is still orthosymmetric. Incidentally this gives another proof of the impossibility of everting $M$-curves (Lemma \[eversion-impossible-for-M-curves:lem\]) since the latter are necessarily of type I (by Klein 1876 [@Klein_1876]). [*Insertion*]{} \[01.04.13\] Moreover the lemma implies that both Rohlin’s chambers $\frac{6}{1}2$ and $\frac{2}{1}6$ are not connected by an eversion, since those schemes are of type I (either by the RKM-congruence (\[Kharlamov-Marin-cong:thm\]) or via Rohlin’s (lost) proof of total reality via a pencil of cubics). Note here that the newly discovered version of Le Touzé is not strong enough for this purpose (as it uses RKM). Thus if we imagine the type I chambers levitating somewhat higher than the sheet of paper, the eversion starting from a dividing chamber always moves down to the ground floor of the diagram. Can we conclude that conversely the diasymmetric type always rises up to orthosymmetric via eversion? As already noticed, eversions are impossible for $M$-curves (except of course if $m=2$, i.e. conics). Thus the 2 top $M$-curves eversions are actually impracticable. Looking one stage lower at $(M-1)$-curves we see 3 eversions. Examining the corresponding Riemann surface, we can imagine something like Fig.\[Orthoovals2:fig\], i.e. an orthocycle cutting only one real circuit while Dehn twisting the handle it is strangulating. So the real picture is exactly the same as Fig.\[Orthoovals2:fig\] safe that one real circuit has to be imagined missing. It seems plain that the eversion will conserve the diasymmetric character. At this stage the problem becomes quite fascinating: for instance we could via eversion travel from $5_{II}$ to $\frac{4}{1}_{II}$ but not to $\frac{4}{1}_{I}$. Hence $\frac{4}{1}_{I}$ could not be everted to $5_{II}$. This anti-commutativity looks a bit puzzling, since any path can be travelled backwardly or forwardly. All this properly understood could help unravel the mystery allied to the break of symmetry prompted by Rohlin’s complex orientation formula, forbidding the scheme $5_{I}$. Despite this and other intricacies, it seems reasonable to put forward the: \[eversion-and-other surgeries:conj\] All the red-edges of Fig.\[Gudkov-eversion:fig\] (except the top curvilinear ones linking $M$-curves, and the one linking the Rohlin’s $(M-2)$-schemes $\frac{6}{1}2$ and $\frac{2}{1}6$) can be realized by an eversion (crossing only once transversally the discriminant). This would give a complete picture of the contiguity graph between chambers residual to the discriminant via elementary algebraic Morse surgeries. Nothing forbids that some edges actually correspond to various Morse surgeries, hence different wall crossings (e.g. coalescing two inner ovals amounts to coalesce one inner oval with the nonempty oval). Hence it is clear that our conjecture is only a crude approximation, for one might really want to catalogue all walls between chambers (including self-contiguous one) while describing the corresponding Morse surgeries. This number of walls is very likely to be much bigger than the number of edges depicted on Fig.\[Gudkov-eversion:fig\]. Modulo little adjustments about the combinatorics, such a spectacular result is perhaps not completely out of reach, as its proof could be akin to that implemented in Itenberg 1994 [@Itenberg_1994] for contracting empty ovals of sextics. (The latter implies our conjecture (\[eversion-and-other surgeries:conj\]) for all the [*straight*]{} edges.) The rough philosophy (at least for sextics) is that when there is no topological obstruction to shrink, one can strangulate in a rigid-isotopic way. However it is quite evident that there must be some extra obstruction, at least if Finashin’s claim (\[Finashin-obstruction-to-coalesce-Harnack:lem\]) that the schematic move $\frac{1}{1}9 \mapsto 10$ (from Harnack’s scheme to the configuration with 10 unnested ovals) can only be accomplished via contraction of the nested oval but not by its coalescence with the nonempty oval. Perhaps Finashin’s claim is merely subsumed to a topological obstruction, which we did not yet understood properly. Strangulation principle (infarctus, etc.) ----------------------------------------- \[27.03.13\] Infarctus=hearth-attack seems to be the generic mortality cause by geometers too much in love with their topis (Dirichlet, Gudkov, Rohlin, etc.) \[18.01.13\] A message from Viro (16.01.13, cf. Sec.\[e-mail-Viro:sec\]) suggested us the following naive remark. Assume an oval (or a priori just a real circuit) to be contracted to a solitary node via a rigid-isotopy $C_t \in \vert m H \vert$ (having only one extremity in the smooth locus of the discriminant parametrized by uninodal curves). Call such a path a [*pseudo-isotopy*]{} for short. Of course our contraction supplies a homotopy shrinking the oval to a point, hence our circuit is null-homotopic in ${\Bbb R}P^2$ and so forced to be an oval. (By an extension of the theorem of Schoenflies ca. 1906 a Jordan curve in a surface is null-homotopic iff it bounds a disc; compare e.g. R. Baer 1928, Epstein 1966, or Gabard-Gauld 2011 [@Gabard-Gauld_2010-Jordan-and-Schoenflies], etc.) The presence of this canonical bounding disc gives some evidence to the Itenberg-Viro contraction conjecture (IVO) of empty ovals, which supplies some membrane for the strangulation to occur. [*Added in proof*]{} \[01.04.13\] If pessimistic, it may be also the case that the contraction conjecture is violently false (and perhaps deducible via Shustin 1985), along the line our Prop.\[Klein-vache-deg-6:prop\] which shows crudely that contraction plus classification implies Klein-vache. Note at this stage that if there were some dividing plane curve with only one oval and of degree $m\ge 3$ then strangulating this oval would be impossible by Lemma \[strangulation-impossible:lem\]. Quite fortunately such curves do not exist by Rohlin’s inequality $r\ge m/2$ (\[Rohlin’s-inequality:cor\]). So the contraction conjecture for empty ovals (CCEO) suggests perhaps having some bounding disc for a cycle to be strangulable, as to refine slightly the statement of the Shrinking principle (\[shrinking-principle-vague:conj\]): [(Strangulation principle)]{} \[shrinking-principle:conj\] Any (Galois) invariant cycle (=circle) $\beta$ on a smooth plane curve $C$ of degree $m$ can be strangulated (through a path in the hyperspace of curves crossing only once the discriminant at a smooth point of the latter), whenever there exist in ${\Bbb C}P^2$ a smooth disc $D$ bounding $\beta$ which is invariant under complex conjugation, and intersects the complexification only along $\beta$ (i.e., $D\cap C({\Bbb C})=\beta$). Say in this case that $\beta$ is fillable. This is of course a true extension of (CCEO) perhaps susceptible to imply “Klein-vache”, i.e. any nondividing curve can acquire a solitary node by a pseudo-isotopy. Of course any nondividing curve admits an anti-oval (just lift an orientation reversing loop from the non-orientable quotient $C(\Bbb C)/\sigma$), but is another story to find one which is fillable. If so and if the above conjecture extending Itenberg-Viro’s conjecture is right we could deduce “Klein-vache”, which is however disproved in Shustin 1985 [@Shustin_1985]. If there is a filling disc $D$ then as it is invariant under conj=$\sigma$, we have an involution on the disc (so with a fixed point by Brouwer). In the case of an anti-oval acted upon by antipody, it seems that the involution has to act as an antipody on the whole disc. In general involutions on the disc are of 3 types (identity of order 1, orthosymmetry fixing a diameter, and antipody fixing the center). (This must be ex/implicit in work by Brouwer, Kerékjártó ca. 1914-1922.) As argued by Viro’s e-mail, CCEO looks more natural that “Klein-vache” since the filling is virtually God-given, just taking the (sealed) inside of the oval. Yet perhaps this has some analog for an anti-oval in term of differential-geometric fillings, e.g. minimal surfaces in ${\Bbb C}P^2$ endowed with its “round” Fubini-Study metric coming from $S^5\to {\Bbb C}P^2$ (Hopf fibering). One problem with anti-ovals is that there are plenty of them (not just finitely many like for ovals), and so one’s idea could be to select some preferred one, maybe as “the” systole. In the case of the diasymmetric sphere this is not enough to ensure finiteness, but perhaps suffices to single out some natural class of anti-ovals. Recall that systoles are geodesics, and so are usual ovals. A recipe could be as follows: given a nondividing curve $C_m$ of some degree. Endow it with the natural Fubini-Study metric of ${\Bbb C}P^2$ to get a Riemannian metric on the Riemann surface $C_m(\Bbb C)$. Since the curve is diasymmetric it contains an antioval (invariant circle acted upon by antipody by $\sigma$). Hence by compactness there is also a such of minimal length, the so-called systole, not perfectly unique of course, but choose one such systolic antioval. Consider the latter as a circuit in the ambient ${\Bbb C}P^2$ and solve the Plateau problem for that contour, in its classical setting of soap films diffeomorphic to the disc. Plateau makes also sense for membranes of higher topological structure, but ignore them to stay closest to the Itenberg-Viro conjecture. Plateau is always soluble but the notorious difficulty is to ensure embeddedness of the solution. Perhaps this is true in $E^3$ and also in ${\Bbb C}P^2$ due to some simple-connectivity, or perhaps the special systolic properties of the boundary data. (\[21.01.13\] Beware that a minimal surface has vanishing mean curvature, while the natural Itenberg-Viro “reality” membrane is positively curved. But the former assertion is specific to $E^3$…) 2[${\Bbb C} P^2$]{} As the given contour is invariant under $\sigma$ (an isometry of ${\Bbb C} P^2$) it is likely that Plateau’s solution enjoys a similar invariance, and we would be essentially finished (modulo the difficulties enumerated). At this stage we would have a perfect analog of the bounding disc of Itenberg-Viro’s empty oval, via our Plateau filling of “the” systole realizing the anti-oval of shortest length. For the analogy to be perfect one should ensure that the Plateau film intersects the Riemann surface $C_m({\Bbb C})$ only along the contour (systolic anti-oval). This looks either hard or trivial. For instance recall (from Wirtinger, cf. also Mumford’s book [@Mumford_197X-BOOK-complex-alg-var]) that algebraic subvarieties of ${\Bbb C}P^n$ endowed with Fubini-Study are (precisely?) minimal surfaces. So there is perhaps some chance to prove disjointness. (If they intersect interiorly then try to build a canal surface by surgering a piece of $C_m({\Bbb C})$ to the Plateau film, trying so to violate its area minimization ….) Maybe some interesting twist of Plateau’s problem is that one may be able to reconstruct the whole complex locus via Plateau if we are given only the real locus of the curve. Of course as there is now handles (except for $M$-curves) and several contours this will necessarily involve the so-called Plateau-Douglas problem permitting membranes of higher topological structure (than the disc). As hazardous as it is, this claim would perhaps only work for dividing curves. Assume all this to work then we have a perfect analogy with Itenberg-Viro, but it is still not explained why the empty oval or our anti-ovals are strangulable via a pseudo-isotopy of algebraic character. Naively from the given data consisting of minimal film bounding the cycle $\beta$ we can hope to shrink concentrically the disc (put via the Riemann mapping in conformal equivalence with the round disc) to its center. Solving the higher Plateau-Douglas problem for this shrinking contour gives a minimal surface (which by the converse of Wirtinger) would be an algebraic curve realizing the deformation we are looking for. Since this concentric shrinking respect the symmetry (of the round disc whatever its type, i.e. identity, antipodal diasymmetry or orthosymmetric mirror like $z\mapsto \bar z$), the given smaller contours are invariant under $\sigma$ in 2 and so we get real curves by solving Plateau-Douglas, and therefore the desired pseudo-isotopy from $C_m$ to a nodal curve (with a solitary point). All this is a bit reminiscent of Riemann’s spirit (except for being of lesser vintage) yet dreaming like a canary is quite pleasant. The above strategy (with all its gaps) suggests even that in the Itenberg-Viro shrinking of an empty oval it could be arranged that the subsequent curves all have their ovals progressively shrinking inside the initial one. Whether this stronger conjecture has some chance to hold true is unknown to me. The above argument suggests that a real algebraic curve should be reconstructible from a single oval. This is certainly true via something like the Nullstellensatz, yet the assertion that all this algebra can be supplanted by differential geometry à la Plateau looks a bit doubtful for we are not controlling the full contours of the membrane. Actually the dividing case looks psychologically more comfortable to a direct appeal of Plateau. As “Klein-vache” is probably false (cf. Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin] and our partial discussion in Sec.\[Shustin-understood:sec\]), we shall from now on concentrate on the Itenberg-Viro contraction conjecture for empty ovals, which is still non refuted (hence more likely to be true) and easier technically due to the canonicalness of the film bounding the vanishing cycle. So suppose given any empty oval of a smooth plane real algebraic curve. Goal: strangulate it algebraically via a pseudo-isotopy, i.e. a rigid-isotopy except for its extremity which is a solitary nodal curve. To the disc bounding the (marked) empty oval, apply the Riemann mapping theorem to take it conformally to the unit disc $\Delta:=\{z\in {\Bbb C}: \vert z\vert \le 1\}$ via a map $f\colon D\to\Delta$. This mapping is canonical up to conformal automorphism of the disc, hence unique once a center and a boundary point are chosen (variant choose $3$ boundary points). Consider the pullback $C_\rho:=f^{-1}(\Gamma_\rho)$ of the circles $\Gamma_\rho: \vert z\vert=\rho$ of radii $\rho$ ($0\le \rho\le 1$). Question: are those still algebraic ovals when $\rho<1$? In other words are the Riemann levels $C_\rho$ of an algebraic oval (which is empty) still algebraic curves (at least part thereof)? The truth of this assertion is of course a necessary condition for our above strategy of constructing the pseudo-isotopy via Plateau. If the initial oval is a circle (or even an ellipse), algebraicity of the Riemann levels looks evident (at least classical I think). For the ellipse it could involve Schwarz’s explicit solution to the Riemann mapping. Another idea: it would we nice if there is some flow effecting the contraction conjectured by Itenberg-Viro. One idea could be to take any empty oval, and look at its normal curvature flow à la Huisken. Usually this is presented in $E^2$ but there is surely a variant in $S^2$ the double cover of ${\Bbb R}P^2$ on which Fubini-Study induces the round metric (I think). Is it true that the normal curvature flow preserves algebraicity of ovals? If so, the flow would shrink one of the empty oval to a point (yet not necessarily one oval chosen in advance like by Itenberg-Viro), and perhaps it will shrink all ovals in some succession it is alone able to decide until all get shrunk. This could prove a weak form of the contraction conjecture. Toward a naive dynamical treatment of the Itenberg-Viro conjecture {#CC-via-dynamics:sec} ------------------------------------------------------------------ \[18.01.13\] How large can an oval be? If we imagine a real projective curve as traced on the sphere $S^2$ we can wonder what the area or length of an oval can be. In degree 2 a quadratic cone can be as large as we please and so the inside area of the oval can be as close to $2\pi$ as we please, but of course not larger as its “twin” occupy the antipodal area of the sphere. If we restrict to even degrees then we have the Ragsdale orientable membrane bounding the ovals (that one with $\chi=p-n$). How large can its area be when lifted to $S^2$? For quartics and when $r=1$ we can enlarge the Fermat equation $x^4+y^4=+1$ as much as we want by taking $x^4+y^4=R^4$ which conserve the same shape (homothety), while covering more and more space of the sphere. So here the upper bound is again $4\pi$ (the full sphere). (We count now the full lift to $S^2$). But what about other quartics, e.g. the Gürtelkurve. Again we may imagine the latter just as a perturbation of two concentric circles. While the outer circle enlarges the inner contracts and so we get the Ragsdale membrane of nearly full area. It seems that there is little chance to find nontrivial upper bound. But what about an $M$-quartic with 4 ovals. How large can the area of its interior be? Again we can imagine a configuration of 2 transverses ellipses one very large but of small eccentricity and one fairly small but orthogonal and smooth it in the usual way to get 4 ovals; one very large and all 3 remaining fairly small. Expanding this at infinity shows that the Ragsdale membrane can cover nearly all the sphere and the upper bound is again $4\pi$. Such consideration could extend to all curves constructed say via Hilbert’s method. But how to treat the general case? At any rate to each rigid-isotopy class of curves we can consider the supremum of all areas of the corresponding Ragsdale membrane. Is this always equal to $4\pi$? Looking at the area of the Ragsdale membrane assigns to each curve a numerical value in $]0, 4\pi[$. Perhaps it is interesting to look at the orthogonal trajectories of this Ragsdale function? The allied gradient flow could provide a dynamical flow shrinking some empty ovals. There is plenty of other functionals perhaps better suited to a dynamical treatment of the Itenberg-Viro conjecture. For instance instead of Ragsdale area we could look at the [*empty area*]{} defined as the cumulated area of all empty ovals. Looking at the descending orthogonal trajectories of this function is likely to shrink ovals. Another choice is the function looking at the area of the smallest oval. Perhaps this has the drawback of lacking smoothness in case two ovals enter in competition for the infimum? Another strategy more suited to the Itenberg-Viro contraction problem is that we are given an empty oval, and during a rigid-isotopy we can follow him continuously. Hence given a curve with a marked empty oval we can define in the whole chamber residual to the discriminant (alias the rigid-isotopy class of the curve) the functional ascribing the area of the inside of this marked but moving oval. Of course when dragging the curve around a loop in its chamber the oval can be to another oval, so the function looks multivalued. Yet we get it single-valued on the space of curves with a marked oval. So the space of curves with a marked oval is actually an $r$ sheeted cover of the usual space of smooth curves (with $r$ variable on the different chambers of course). As long as we keep the marked oval into view there is a way of steepest descent diminishing maximally the area of the oval. For this to make good sense we require orthogonal trajectories and so a metric on the space of all curves. The canonical choice seems to be the elliptic geometry on $\vert m H \vert \approx {\Bbb P}^N({\Bbb R})$ the space of coefficients double covered by the round sphere $S^N$. Now follow the corresponding trajectory of steepest descent. What can happen? By construction our marked oval will decrease in area, but will it docilely shrink to a point? Here are some evident difficulties (D.$n$, $n=1,2,\dots$) (D.1) [*Wrong attractor (stable equilibrium).*]{} First one can imagine that our function as a sink trapping us into some “depression” like the basin of a lake yet not at zero altitude (e.g. lake Baikal). Then our motion stops and the goal fails blatantly, having only reached an algebraic curve realizing a local minimum of the area yet still positive. Perhaps some clever argument precludes such depression (e.g. if our functional turned out to be harmonic by some miracle?) (D.2) [*Saddles points (unstable equilibrium).*]{} We may of course also reach something like a saddle point, where we need then to choose quite randomly one of the two (or more if not Morse) way of steepest descent. Generically up to perturbing the initial curve, we can avoid such accidents. (D.3) [*Controlling the limit.*]{} Hence let us assume that the area shrinks to zero (assuming (D.1) to have been overcome). Naively one can imagine the oval shrinking to a complicated dendrite (though Bézout unlikely) or to a segment (again algebro-geometrically improbable). The sole possible limit seems to be a point. (This seems an easy task via implicit function theorem, Bézout, etc.) (D.4) [*Choosing the right functional (i.e. arranging a “convex” or “harmonic” landscape).*]{} We have as yet only considered the area functional yet the length functional looks nearly as appropriate or better? Or even one could use a mixture of both like the isoperimetric ratio. Note that nearby a solitary node the behavior become nearly circular or at least elliptical like in the local model $x^2+y^2=\varepsilon^2$ or $ax^2+by^2 = \varepsilon^2$ with $\varepsilon \to 0$. Among all difficulties the most stringent seems to be (D.1) which of course as to be settled by playing with (D.4), i.e. choice of the functional. To settle (D.1) it is enough showing that nearby all curves there is one of smaller “energy”. For the area functional one could imagine an oscillation by perturbing slightly like in the Harnack-Hilbert method our marked oval by a collection of $m$ lines. Alas it result a vibration of the oval slaloming across its initial position so it is not obvious how to decrease area. So to impede getting blocked by (D.1) we are reduced to the “local” problem of finding an appropriate function which can always be decreased by small (algebraic) perturbations. What about the total curvature of the oval (or the inverse thereof as to go to zero for a shrinking circle), etc. If we work with area functional and if the oval is nearly circular, we can plug in it a smaller circle and taking this equation $k$ times (assume $m=2k$ even for simplicity). Perturbing the curve along this multiple circle may decrease area. This is of course just a very special case. It seems to apply to the case when the curve is a deep nest in which case the innermost (=empty) oval as to be “convex” (else Bézout is violated, compare Zeuthen 1874 [@Zeuthen_1874]). The whole problem seems reduced to that of finding a good functional $\varphi$ which has no local minimum. Above it is not fundamental that the oval is convex, to plug a circle inside it. This just uses the fact that the interior of the oval is open, and tracing a little circle inside the oval while taking its $k$th multiple gives another curve $k\cdot E_2$, along which to deform inside the spanned linear pencil $\lambda C_m +\mu (k\cdot E_2)$. Alas nothing ensure the oval to stretch within its interior. Another idea is to apply the Riemann mapping theorem and shrink the radius of the representing circle, while hoping that this new smaller Riemann level is still an algebraic curve of the [*same*]{} degree. If this work we are able to decrease the area functional. This reminds me some work of Bell and Aharonov-Shapiro [@Aharonov-Shapiro_1976] to the effect that the Riemann (and more generally Ahlfors) map of a quadrature domain is algebraic, and that quadrature domains are dense in the space of all domains so that virtually any Riemann map is algebraic. But in our context we have an algebraic contour(=oval) and the following would simplify life: \[Riemann’s-level-algebraic:conj\] Given a nonempty oval of a real plane algebraic curve of degree $m$ and suppose the corresponding spherical calotte conformally mapped (via Riemann) to the unit disc $\{z: \vert z\vert \le 1 \}$. Then the pullbacks of the smaller circumferences $\vert z\vert =r $ are still algebraic curves of the same degree $m$!(???) If so then we can decrease area thus solving difficulty (D.1), and perhaps the whole conjecture of Itenberg-Viro. Of course this looks a bit optimistic (due to the a priori highly transcendental nature of the Riemann map), but at least the dynamical strategy looks quite stimulating. Needless to say we have not proved the Itenberg-Viro conjecture, but in case it is true, perhaps the above vague ideas are quite close (at least in broad lines) to its ultimate technical solution. We Summarize the discussion as follows: Let $\vert m H\vert$ be the space of all real algebraic curves of fixed degree $m\ge 1$ and $\frak D$ be the corresponding discriminant parametrizing singular curves. The complement $S_m=\vert mH\vert - \frak D$ is the space of smooth curves. Suppose given some real positive-valued smooth functional $\alpha$ on the space MEO of all curves with a marked empty oval which has \(H) no stable equilibrium (local minimum) and such that if $\alpha(C)\to 0$ then the curve $C\in S_m$ tends to a solitary nodal curve. Then the trajectory of steepest descent (gradient flow) always converges toward a curve with a solitary node, after possibly slight perturbation of the initial data (permissible as we work up to rigid-isotopy). In particular the Itenberg-Viro contraction conjecture holds true. Of course this is just the formal aspect of the story (i.e. imputable to the theory of ordinary differential equations). Yet the real problem is to find a functional $\alpha $ suiting hypothesis (H). A candidate is to take $\alpha$ the area of the marked oval, and then hypothesis (H) could follow from the optimistic Conjecture \[Riemann’s-level-algebraic:conj\] on the algebraicity of Riemann’s level. Call for an attack via the Riemann mapping (yet another Irrweg=aberration?) {#CC-via-Riemann:sec} --------------------------------------------------------------------------- \[18.01.13\] Let us do some experiments. Suppose given some real algebraic curve and take an empty oval on it. Mark an interior point and consider the Riemann map $f$ taking the domain $D$ interior of the oval to the unit disc $\Delta$. Pull-backing polar coordinates on the disc gives an isothermic system of coordinates on $D$. Fig.\[ItenbergViroRiem:fig\] gives some qualitative pictures for an ellipse or with ovals of a cubic or even of some quartics. -5pt0 -5pt0 Naively the levels $\vert f \vert = \rho$ of the Riemann mapping look again like curves of the same degree. This is especially striking for the contorted quartic with one oval of Fig.\[ItenbergViroRiem:fig\]c. It is like a volcano spreading its lava on the whole territory available in the island interior to the oval of this $C_4$. If algebraicity is true and degree conserved, then the levels of the Riemann map gives directly the contraction of the Itenberg-Viro conjecture (\[Itenberg-Viro-contraction:conj\]). Is there such a miracle? Maybe giving to Riemann an algebraic contour, the Riemann map itself is algebraic of the same degree and so are all its sublevels $\vert f\vert = \rho= const$. Let us first consider the more basic converse assertion that if the Riemann map is algebraic then so is its contour. So let $f\colon D \to \Delta$(=unit disc) be a Riemann map which is algebraic, i.e. the power series of this analytic function is finite, i.e. a polynomial of finite degree $f(z)=c_0+c_1 z+c_2 z^2+ \dots +c_n z^n$. Then as $f^{-1}(\partial \Delta=S^1)=\partial D$ we see that $$f(z)\overline{f(z)}=1$$ identically on the contour $\partial D=:\Gamma$. Doing the usual splitting in real and imaginary parts by letting $z=x+iy$ and $c_n=a_n+ib_n$ one sees that the product $f(z)\overline{f(z)}$ becomes a polynomial $P(x,y)+iQ(x,y)$, where $P(x,y)$ has degree $n$ \[[*Warning*]{}.—this is disproved in the sequel!\] while $Q(x,y)$ is identically zero. It follows that $P(x,y)-1=0$ identically on the contour $\Gamma$ which is therefore algebraic. Of course the same holds true for any sublevels of the Riemann map, i.e. sets $f(z)\overline{f(z)}=\rho$ are also algebraic. This proves the \[Riemann-level-algebraic\] If the Riemann map $f$ is algebraic then the contour is a real algebraic curve and so are all sublevels. Further the degree of all this Riemann levels $f \bar f =\rho$ ($0\le \rho \le 1$) have the same degree as $f$. This is the trivial sense. What about the converse? Suppose given a contour which is an oval of some algebraic curve can we conclude that the Riemann map is algebraic? This is what Arnold would call pure Riemannian predestination. We think this to be true as follows: \[Riemann-map-algebraic:thm\] Suppose given an (empty) oval $\Gamma$ of a real algebraic curve $C_m$ of degree $m$. Mark any point $p$ inside this oval and let $D$ be the sealed interior of the oval. Let $f\colon D \to \Delta$ be the Riemann mapping taking $p$ to the origin $0\in \Delta$ (unique up to rotation). Then the Riemann map is a polynomial of degree $m$. (vague sketch) The idea is simply to look at the function $f \bar f$ which is analytic and vanishes on an algebraic locus, namely $\Gamma$. So by an appropriate Nullstellensatz (Arnold’s predestination? or Bloch’s slogan “Nihil est in finito quod non prius fuerit in finito”) it must follow that the function $f \bar f$ is itself algebraic. The proof is complete. (???) At this stage we would have deduced Itenberg-Viro’s contraction conjecture (as a simple corollary of the Riemann mapping theorem!!!): \[Itenberg-Viro-via-Riemann:cor\] Any nonempty oval of a plane real algebraic smooth curve (of degree $m$) can be contracted to a solitary node. Apply Theorem \[Riemann-map-algebraic:thm\] to the nonempty oval under consideration, to obtain a Riemann map $f$ which is algebraic. By Lemma \[Riemann-level-algebraic\] all the levels $\vert f\vert=\rho$ are algebraic curves of degree $m$, while shrinking the radius to $\rho=0$ the curve acquires a solitary node. Some objections to the method (or details to be filled): (DET.1) How to ensure that the nodal curve so obtained has only this solitary node as sole singularity? (DET.2) We have worked as if the curve were affine and not projective. This is usually a harmless nuance via the usual yoga, (des)homogenization of the equations. More severe is our supposition that our oval can be put in some affine chart! As I learned from (the late) Felice Ronga (ca. 1999–02), there is a sextic with one oval only, such that any line cuts it in at least two (real) points. In other words no line avoids this sextic. This is simple to construct by perturbing à la Plücker-Brusotti a configuration consisting of two concentric circles, plus the two axes of coordinates. Smoothing this sextic arrangement quite randomly (slalom as much as you can), one gets easily the required curve (Fig.\[ItenbergViroRiem:fig\]d). \[Ronga’s original picture is to be found as the front cover of his book “Analyse réelle post-élémentaire, 1999 après J.Christ.” [@Ronga_1999-BOOK].\] Such a “Ronga curve” causes some trouble to our procedure, which requires putting the oval in an affine chart which we identify to the complex plane. Perhaps this is not fatal as the Riemann mapping theorem has some more intrinsic character, namely any Riemannian membrane topologically equivalent to the disc is conformal to the unit disc. So maybe one should use this more general version, and adapt the above affine argument in this more global setting. Algebraically this would amount to work always with homogeneous coordinates and think with cones in ${\Bbb R}^3$. We may then apply the Riemann mapping theorem to the spherical calotte bounding the oval (there is two of them, but choose one), and use as cut function homogeneous polynomials. A variant is perhaps just to pass from the sphere covering ${\Bbb R}P^2$ to the complex plane via stereographic projection (from a point outside the oval). This projection is conformal, but does it preserve the degree of polynomials? \[19.01.13\] Here is a more fatal destruction of the above pseudo-proof of the contraction conjecture via the Riemann mapping theorem. The key is a little computation of $f(z)\overline{f(z)}$, where $$f(z)=c_0+c_1z+c_2 z^2+\dots +c_n z^n,$$ $$\overline{f(z)}=\overline{c_0}+\overline{c_1}\bar z+\overline{c_2} {\bar z}^2+\dots +\overline{c_n} {\bar z}^n.$$ When expanding this product it is useful to write it as a “diamond”: $$\begin{aligned} c_0 &\overline{c_0} \cr c_0 \overline{c_1} \bar z + & c_1 \overline{c_0} z \cr c_0 \overline{c_2} {\bar z}^2 +c_1 \overline{c_1}& z \bar z+ c_2 \overline{c_0} {z}^2 \cr c_1 \overline{c_2} z {\bar z}^2 + & c_2 \overline{c_1} z^2 \bar z \cr c_2 \overline{c_2}& z^2 {\bar z}^2.\end{aligned}$$ This is the end result when $n=2$, but otherwise it will expand to larger rhombs. At any rate, the weightiest term is $c_n \overline{c_n} z^n\bar{z}^n=\vert c_n\vert^2 (x^2+y^2)^n$, and so we get indeed a polynomial, but one of degree twice as big as that of $f$. Now suppose the given contour to be an ellipse (which is not a circle), then even if the qualitative part of Theorem \[Riemann-map-algebraic:thm\] ought to be true, i.e., if the Riemann map $f$ of an algebraic contour is algebraic, then the degree of $f$ cannot be $1$ (for then $f$ is a similitude which preserves circles). Hence the degree of the Riemann map $f$ of an ellipse is at least $2$, hence by the above computation the degree of $f\bar f$ is at least $4$, and so the sublevels of the Riemann map are at least quartics, and not ellipses as we initially imagined (compare e.g. the misleading Fig.\[ItenbergViroRiem:fig\]a). Hence even granting algebraicity of the Riemann mapping of an algebraic contour, the corresponding levels would not be of the same degree. Actually the above argument shows that our strategy fails for a conic, but is it really a disproof in general? Assume the initial curve $C_m$ to be a quartic, then the degree of the Riemann map could be $n=2$, and thus the degree of $f\bar f$ is four and so the sublevels would stay a deformation within quartics. More generally this numerology makes sense for any $C_{2k}$ curve of even degree $2k\ge 4$, by taking $n=k$. A more severe objection is surely, as already apparent by looking at the highest power $c_n \overline{c_n} z^n\bar{z}^n=\vert c_n\vert^2 (x^2+y^2)^n$, the fact that $f\bar f-1=0$ is not the most general curve of degree $m=2n$ (e.g. the monomial $x^{2n-1}y$ is missing). Hence there is no chance the Riemann map of any curve $C_{m=2k}$ being algebraic of degree $k$, and our strategy is definitively foiled. As a modest consolation one can apply the above method to some few ad hoc curves with equations of the shape $f\bar f-1=0$ where $f$ is some algebraic Riemann map. The corresponding curves (say “Riemann curves”) could be contracted by the above recipe. This is of course far from settling the initial desideratum of Itenberg-Viro. What can be retained from this attempt? Let us start with a polynomial $f(z)\in {\Bbb C}[z]$ of degree $n$. This induces a holomorphic map ${\Bbb C}\to {\Bbb C}$ of degree $n$ (Gauss–D’Alembert fundamental theorem of algebra) which is a branched covering. Sometimes it turns out that the unit disc is a trivializing open set for this covering. In the language of complex analysts (Bloch, Landau, Ahlfors, etc.) this is also what they would call a schlicht unit disc. Taking one among the $n$ many sheet lying over $\Delta$ gives a simply connected domain which by $f$ is conformally mapped to the disc, and so we recover a Riemann map. For all those domains (which are algebraic of degree $2n$ via the equation $f\bar f-1=0$ but with $n$ unnested ovals), we may a priori implement our contraction algorithm. (Overlooking the unnested condition, we could hope that such Riemann curves are spread in all chambers of the discriminant and hope a general attack.) Now back to our setting it must be noticed that during the shrinking $f \bar f=\rho$, with $\rho \to 0$ our algebraic curve sees all its $n$ ovals being simultaneously shrunk. Hence in this very favorable setting, the solitary node condition fails blatantly. Conclusion: our strategy via Riemann leads nowhere, if it pertains to implement the contraction conjecture of empty ovals. CCC: collective contraction conjecture, as an avatar of Itenberg-Viro (Gabard 2013) =================================================================================== [*Insertion*]{} \[04.04.13\].—As we noticed only today, in substance it seems that the conjecture posited below bears some analogy with Klein 1892 [@Klein_1892_Realitaet] (p.177 of Ges.Math.Abhd.) who wrote: “Wir könnten z.B. mehrere Züge unserer Kurve gleichzeitig in isolierte Doppelpunkte überführen”. More generally Klein 1892 discusses at this place (p.176) what he calls the “Doppelpunktsmethode” amounting essentially to contract any symmetry-line of the Riemann surface, and this of course seems to anticipate what we called before the Itenberg-Viro contraction principle. It is not clear however that Klein ever formulated something as precise as the Itenberg-Viro contraction conjecture (specific to plane curves). On p.176–177, Klein writes something which taken out from its context looks a bit overoptimistic namely: “Bei allen anderen Fällen hat die Durchführung des genannten Prozesses und damit die Zusammenziehung eines beliebiegen Ovals der Kurve zu einem isolierten Doppelpunkte keine Schwierigkeit.” Failing with Riemann suggests a variant of Itenberg-Viro, viz. CCC=collective contraction conjecture: deformation in the large as a method of prohibition {#CCC:sec} --------------------------------------------------------------------------------------------------------------------------------------------------------- \[19.01.13\] The above discussion suggests the following variant of the contraction conjecture, which has maybe some spontaneous appeal and independent interest. Suppose given a projective smooth plane real curve $C_m$. Look at all empty ovals simultaneously. Is it possible to shrink all of them in one single stroke toward solitary nodes (via a deformation of smooth curves sole for its end-point being in the discriminant)? It seems likely that the contraction conjecture (\[Itenberg-Viro-contraction:conj\]) implies this, roughly by shrinking one oval, and then the second, etc. Of course one then needs to arrange a bit things so that prior to extinct one oval completely, one waits until the second empty oval becomes “small” enough, etc. Finally one synchronizes the ultimate “coup de grâce” to kill all the empty ovals at the same time (“time” being just the parameter of the path $[0,1] \to \vert mH \vert$ in the space of all curves of order $m$). [*Insertion*]{} \[02.04.13\].—It may help reading the sequel to remarked first that the reverse process, of deducing an individual (solitary) contraction (à la Itenberg-Viro) from our collective one, is much easier and a trivial consequence of Brusotti, if we did no mistake, cf. Lemma \[CCviaCCC-Brusotti:lem\] below. So the conjecture posited right below is stronger than the one of Itenberg-Viro (yet perhaps equivalent, or at least easier to disprove). So let us (somewhat cavalier) formulate the: [(Collective contraction conjecture=CCC, \[19.01.13, 22h40\])]{}\[CCC:conj\] Given any smooth real curve $C_m$ of degree $m$, it is possible to shrink all the empty ovals simultaneously toward solitary nodes. (Solitary but synchronized death of all ovals.) This is obviously true for $m=2,3$ (being actually equivalent to the individual contraction principle) since there is at most one empty oval available. The case $m=4$ is already more tricky, yet still compatible with Bézout. If $r=4$ ($M$-quartic), we would have a quartic with four isolated (solitary) nodes. This exists just take an imaginary conic $C$, and aggregate it with its conjugate $C \cdot C^\sigma$ (this is real but a priori the four intersections need not all be real). More simply take two transverse conics, look at signs and arrange a level so that there are 4 isolated points by making a naive picture of the graph of $E_2 \cdot F_2$. Since the real scheme encodes completely (in degree $m=4$) the rigid-isotopy class (Klein 1876, etc.) it follows that CCC holds true in degree $m=4$. The case $r=2,3$ are treated similarly by looking at the graph of a special equation and passing a plane tangent to the 3 (or lesser) hills. Now what about degree 6? Deciding the truth of the above conjecture in degree 6 (CCC6), is already more tricky. Perhaps this follows from Itenberg’s CC6, if not formally by the method used therein, i.e. Nikulin’s theory with $K3$-surfaces. As said at the start it could be that CC implies CCC in all generality. In the sequel we assume CCC as granted and look what can be derived from it. Let us first suppose that there is an $M$-sextic $C_6$ with 11 unnested ovals (what Hilbert, Rohn, Petrovskii, Gudkov, Arnold, etc. were fighting hard against). Shrink all of them to a point according to CCC (\[CCC:conj\]). Then the Riemann surface is strangulated along all its oval in two (algebraic) pieces which are topological spheres. Since the nodes are supposed to be solitary these two pieces are smooth curves of genus 0 intersecting transversally. Therefore (via the genus formula $p=\frac{(d-1)(d-2)}{2}$) they are of degree $1$ or $2$, but have to intersect in 11 points. Bézout is overwhelmed! (Alternatively the genus formula is corrupted, since we have a degeneration of $C_6$ toward two cubics $C_3$ and its conjugate $C_3^\sigma$!) This gives a new “proof” of Hilbert-Rohn-Petrovskii via CCC. Of course it is quite tempting to wonder if Hilbert (or subsequent workers) did not knew about this argument at least as a heuristic tool. More generally: [(like Hilbert 1891)]{} Under axiom CCC, a smooth $M$-curve (of even degree) cannot have all its ovals unnested unless its degree $m$ is less than four ($m\le 4$). In particular an $M$-sextic cannot have all its $11$ oval unnested (which is Hilbert’s original claim as early as Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege].) Shrinking collectively all the empty ovals of $C_m$ (via CCC) gives a splitting $C_m \to C_d \cup C_d^\sigma$ in two algebraic curves of degree $d=m/2$ intersecting transversally in $r$ points. So by Bézout $d^2=r$. Since both strangulated halves have genus $p=0$ (for we started from an $M$-curve), their common degree $d$ can only be $1$ or $2$. Hence $d\le 2$, and so $r=d^2\le 4$. Since $r=g+1$ ($M$-curve assumption) it follows $g\le 3$ and so $m\le 4$. (Variant: conclude more directly via $d=m/2$.) As a matter of philosophical dilettantism (?), it may be wondered, whether Hilbert himself used the above argument, at least as a heuristic tool. To my knowledge there is no record in print along this sense. Yet, Hilbert, say unlike Poincaré was a formalist, in particular never writing down crazy ideas. Thus, it may be not be impossible (our subjective speculation) that Hilbert may have argued along this route. In fact it is probably more realist that Hilbert argued along another idea, cf. e.g. the passage of Gudkov 1974 [@Gudkov_1974/74], where Hilbert’s method is described as implemented by his students Kahn and Löbenstein. Even more generally: Under CCC, a smooth dividing curve (of even degree $m=2k$) cannot have all its ovals empty (equivalently unnested) unless: [(1)]{} its number of ovals $r$ is a square ($1,4,9,\dots$), and actually the square of its semi-degree $k$[^49]; [(2)]{} some stringent arithmetical conditions (say predestination or coincidence) are verified, namely all the displayed formulas in the proof below have to be satisfied. By CCC shrink all empty ovals of the curve $C_m$. The Riemann surface $C_m({\Bbb C})$ (“complexification”) has genus $$g=(m-1)(m-2)/2.$$ Once strangulated, it splits in two Riemann surfaces of genus $$p=[g-(r-1)]/2$$ (since $g=(r-1)+2p$ by visualizing the orthosymmetric surface). Both halves are algebraic smooth curves intersecting in $r$ points. Being actually interchanged by conjugation, they have some common degree, say $d$, verifying $$p=(d-1)(d-2)/2.$$ So by Bézout (or homological intersection theory) we infer $$d^2=r.$$ Hence $r$ must be a square. In fact a sharper argument based on the degeneration $C_{2k}\to C_k\cup C_k^\sigma$ shows that $r=k^2$ directly by applying Bézout to both halves of the limiting curve of the collective contraction. A this stage “Eureka” \[23h41\] we have already proved that the sextic scheme $5$ is necessarily of type II (as followed first from Rohlin’s complex orientation formula). Likewise the sextic schemes $\ell$ ($\ell = 9$ excepted) cannot admit a type I incarnation (though this was already implied by Klein’s congruence modulo 2, and Arnold’s congruence mod 4), safe for $\ell =1$ where either Rohlin or our suggestive geometric argument do instead the job. Indeed if $r=1$ and $m=6$, then both strangulated parts have genus $p=5$ (imagine the Riemann surface of genus $g=10$ split by the one oval), but this is not even the genus of a smooth curve. Hence strangulation is impossible violating axiom CCC. Of course the philosophy behind CCC is quite akin to the filling trick of Arnold-Rohlin safe that the closing is God given by some postulated (but hypothetical) shrinking procedure in the rigid algebraic category. In general it remains the boring task of extracting the exact arithmetical consequences of CCC, while checking if it is really compatible with factual data. In degree $6$, CCC seems to live in perfect harmony with Rohlin’s enhancement of Gudkov’s table (Fig.\[Gudkov-Table3:fig\]). Since the usual (individual) contraction conjecture CC holds true in degree $6$ (by Itenberg 1994 [@Itenberg_1994]), it is likely that the collective variant CCC holds good as well. Of course all the arithmetical relations are in reality less stringent that they look at first glance, since they are all coming from the genus formula which itself may be interpreted as a surgical process regulated by Bézout. (Recall the simple proof of the genus formula based on the morphogenesis of lines getting smoothed under surgeries.) Applying CCC to Harnack’s sextic configuration leads nowhere since the Riemann surface keeps connected. Another exercise: assume there is a dividing quartic with two unnested ovals. Apply CCC to both ovals. Then the Riemann surface of genus $g=3$ is strangulated in two surfaces of genus $p=1$, hence of degree $d=3$. But the latter cut themselves in 9 points and not two. This contradiction reproves (modulo CCC) the well-known fact (\[Klein-unnested-quartic-nondividing:lem\]) due to Klein 1876, Arnold 1971, Rohlin 1972–1978, Wilson 1978, Marin 1979, Gross-Harris 1981, etc., that a quartic with two unnested ovals is necessarily nondividing. (Variant of the argument: $r=2$ is not a square.) The principle emerging is that large deformations prompted by contraction conjectures affords a puissant method of prohibition, as opposed to the method of small perturbations which is merely a toolkit for construction. \[20.01.13\] At this stage, the method CCC looks quite powerful, at least as a heuristic tool, reducing to Bézout several deep assertions and results of Klein, Hilbert, Rohlin, etc. However as yet the method is quite limited to the case where all the ovals are empty so that the strangulation really implies an algebraic splitting of the dividing Riemann surface. Perhaps the method can be extended beyond this proviso. For instance after shrinking the first generation of all empty ovals and making them effectively disappear from the real locus, a second generation of empty ovals appears, which would be contracted in turn to solitary nodes, etc. Iterating so collective contractions would contract all ovals and so achieve a splitting of the dividing Riemann surface. For the method to be effective it seems that the solitary nodes of the first generation ought to resurface as such right after the contraction of the second generation of empty ovals. All this looks very dubious, yet some more clever intelligence can perhaps extract something from this procedure. Consider a specific example to make the difficulty more concrete. Consider the sextic scheme $\frac{10}{1}$, and let us call it (improvising terminology) the Rohn scheme (for Rohn 1911–13 [@Rohn_1913]) was the first attempting to disprove its existence via a substantiation of Hilbert’s method. Let us contract all empty ovals. The resulting Riemann surface is still connected, and we lack a splitting suited to an application of Bézout. The obvious idea is to make first an eversion of the nonempty oval so as to reduce to the unnested scheme $11$. (For the definition of “eversion” cf. Sec.\[Eversion:sec\].) So we need another highbrow large deformation principle, dual to the contraction principle stating that any maximal oval can be everted provided the resulting scheme is not prohibited by Bézout. Alas, eversions are not permitted for $M$-curves by virtue of Lemma \[eversion-impossible-for-M-curves:lem\], and this stratagem looks jeopardized. [*Insertion*]{} \[01.04.13\].—After the collective strangulation of all empty ovals of a Rohn curve of scheme $\frac{10}{1}$, we would get a Riemann surface of genus 0 and degree 6, but with 10 nodes. Alas this is still permissible! Of course our argument being merely abstract (i.e. using the abstract topology) would equally well apply to the veritable $M$-curves, and this explains that. Consider next the sextic $M$-scheme $\frac{2}{1}8$. Shrink all empty ovals to get a curve of genus $p=0$ with one oval (once it is desingularized). So it is really just a rational curve of the dividing type (like our orthosymmetric equatorial planet Earth). By a trivial case of Ahlfors (actually the Riemann mapping theorem) this has a unique conformal structure. So there is a total map of degree $1$. This in turn gives a total pencil of curves, of order say $k$. It seems clear that all solitary nodes of our (contracted) $C_6$ must be in the base locus of the pencil. Indeed else the pencil is sweeping out some node, and so the curve through it has one intersection (counting double) but zero nearby whence a disappearance in the imaginary locus (violating total reality). So our pencil must exhibit at least $10$ basepoints. Naively the disc inside the nonempty oval looks foliated by two foyers (index$=+1$) violating Poincaré’s index formula, but this looks too naive because it would kill as well Hilbert’s or Gudkov’s sextics. So we are again confronted to some complicated foliation argument which we know to be quite difficult to implement. We could dispense using CCC, by applying instead directly Bieberbach-Grunsky to the smooth curve $C_6$ (i.e. the genus zero case of Ahlfors 1950 [@Ahlfors_1950]). All this looks difficult and let us abort here shamefully. \[23.01.13\] Let us insert here an optional side remark. It is tempting to wonder what follows from Thom’s conjecture (Kronheimer-Mrowka theorem meanwhile of 1994) to the effect that algebraic curves minimize the genus among smooth orientable surfaces embedded in ${\Bbb C}P^2$ realizing the same fundamental homology class. If one fills by the half of a hypothetical real $M$-sextic of real scheme $11$ (eleven unnested ovals) by the interior of all ovals, one obtains a smooth surface of genus $0$ (round the corners) of degree $6/2=3$. This violates Thom-Kronheimer-Mrowka, which therefore implies again the Hilbert-Rohn-Petrovskii prohibition. [*Insertion*]{} \[01.04.13\].—The degree 3 case of Thom is really due to Kervaire-Milnor 1961 [@Kervaire-Milnor_1961], based on deep works by Rohlin, ca. 1952. Some thinking shows however that Thom’s conjecture does not imply much more, for the filled membrane has then genus $\ge 1$. ++[C++]{} Do iterated contractions (++) imply Rohlin’s formula? ----------------------------------------------------- \[02.04.13\] We recommend to skip this section which is neither exciting nor seriously written. \[22.01.13\] One may wonder if there is not a stronger mode of degeneration (alias contraction) of a real smooth plane curve $C_m=C_{2k}$ than CCC yielding the Rohlin complex orientation formula $2(\Pi^{+}-\Pi^{-})=r-k^2$ as a corollary of Bézout. In the case of no nesting this is precisely what did the previous section. Indeed the solitary node degeneration of CCC implies symbolically $C_{2k}\to C_k \cup C_k^\sigma$ (a topologically dividing curve divides algebraically!), whence by Bézout $r=k^2$. This coincides with Rohlin’s formula since there is no nesting. Let us call such a hypothetical mode of degeneration C++ (like Turbo Pascal?). If this exists this would be a geometrization of Rohlin’s formula, in the sense that topology (homological intersection theory à la Poincaré-Lefschetz-Weyl(1923)-Pontrjagin-de Rham (1930), who else?) would be subsumed to Monsieur Étienne Bézout (ca. 1768) alone. This would also posit a wide extension of the Itenberg-Viro conjecture. It seems evident that such a contraction C++ should exist. It remains only to be not overwhelmed by the combinatorics. So suppose given a dividing curve $C_{2k}$ in the plane. The idea is to contract all its ovals so as to split the curve in two algebraic pieces exchanged by Galois(=conj), all this being just caused by a strangulation of the underlying Riemann surface. Of course we apply first CCC to contract all empty ovals toward solitary nodes. Then it appears a second generation of nearly empty ovals (those which formerly were at height 1 in the tree of the nesting structure). We may hope to shrink those in turn while necessarily coalescing together all the solitary nodes inside this oval. Perhaps we can do this while keeping the tangents distinct at those solitary nodes gravitationally clumped together. One continues this big crunch process and once all ovals have been contracted one get a splitting $C_{2k}=C_k\cup C_k^{\sigma}$. Counting properly intersections with Bézout should give Rohlin’s formula $$k^2=r-2(\Pi^+ - \Pi^{-}).$$ Of course we need to be much more explicit (as if Rohlin would not have influenced us). Recall that $\Pi^{+}$ is the number of positive (injective) pairs of ovals that is with complex orientation matching that of the bounding annulus of ${\Bbb R}P^2$, and likewise $\Pi^{-}$ being the number of pairs with disagreeing orientation when induced from the complexes versus the real bounding annulus. Recall that all pairs are taken into consideration not just oval succeeding themselves immediately. On applying first CCC we can shrink all empty ovals to solitary nodes. Naively one would then like to shrink all the nearly empty ovals containing only solitary nodes, and so on. So we would have a degeneration $C_{2k}\to C_k\cup C_k^\sigma$. Computing the intersection $C_k\cap C_k^\sigma$ with Bézout gives $k^2$ algebraically. Geometrically, as each oval is shrunk to a pair of conjugate lines, and this explains the presence of the term $r$ on the RHS of Rohlin’s formula (as displayed above). Further the Riemann surface of the reduced curve can be naively imagined in 3-space as a pair of paraboloid of revolution together with their orthosymmetric replicas. Each branches intersect its conjugate in one point, but those contribution where already taken into account. So it remains to count the intersection of the top small paraboloid with the bottom large paraboloid, and vice-versa the large top with the bottom small. So we get 2 additional intersections, and this explains the term $+2\Pi^{-}$ of Rohlin’s formula. (Alas the term $-2\Pi^+$ looks much harder to explain.) More clarification is required. A negative pair of ovals can be shrunk simultaneously. An example is provided by the Gürtelkurve, quartic $C_4$ with two nested ovals. Either via Fiedler or by Ahlfors it is plain that this $C_4$ has a negative pair of ovals. Looking at an equation like two concentric circles and perturbing slightly to get away from the reducible locus (and the discriminant) we have $$(x^2+y^2-\rho^2) (x^2+y^2-R^2)=0,$$ and if $0\le \rho<R$ then we can shrink $R\to 0$ and obtain the required multi-contraction. This example obviously extends to deep nest in any (even) degree, as to shrink negative towers of ovals. Our guess is in contrast that positive pair of ovals resist simultaneous shrinking. Probably there is an evident topological obstruction which I missed to notice as yet. If so then there is no possibility to reduce Rohlin’s formula to Bézout via a super strong contraction principle C++ reducing the whole curve to a microcosm of solitary nodes. If so, the whole curve under the action of some gravitational clumping would truly reduce to a constellation of isolated points with real scheme condensed at the atomic scale. (Imagine points and then infinitesimal circle surrounding the first generation of point, etc.) It seems more likely that there is an obstruction to shrink everything (algebraically), and so Rohlin’s proof is surely the best one can implement. Yet there could still be some geometry behind it suggested by the contraction principle. [*Insertion*]{} \[01.04.13\] The latter is true in degree 6, and actually stronger than Rohlin’s formula, when combined with RKM (\[Kharlamov-Marin-cong:thm\]) since it rules out all schemes lying above the $(M-2)$-schemes of type I (e.g. $\frac{7}{1}3$), what Rohlin’s formula is unable to do alone. Failing to reduce Rohlin to Bézout suggests again a dynamical approach {#CCCviaDynamics:sec} ---------------------------------------------------------------------- \[22.01.13\] What is this geometry and is it worth paying attention at? Before trying answering this, note that even if a positive pair of ovals resists to shrinking it could undergo another type of Morse surgery, namely coalescence to a figure eight (lemniscate looking like a “sweetheart”, i.e. with one branch lying inside the other). Yet this operation corresponds to the contraction of an ortho-cycle without disconnecting the Riemann surface so as to produce an algebraic splitting $C_{2k}\to C_k \cup C_k^\sigma$. Let us turn to the geometric aspect. Our goal is essentially to shrink the ovals at least those which are empty (CCC), and then eventually push further the deformation as to shrink the negative pairs (memno-technic trick imagine negative=depressive=shrinkable). The other positive pairs may offer some resistance (due to a topological obstruction, which we should still understand better). To achieve such a shrinking it looks natural to look at the length-functional of [*all*]{} ovals (not just the empty ones). Consider the round metric on the unit sphere $S^2$ lying above ${\Bbb R}P^2$, and measure lengths in this metric. Given $C_m$ a curve (i.e. a homogeneous ternary form $F_m(x_0,x_1,x_2)$ with real coefficients up to homothety), look at the set $(F_m=0)\cap S^2$ which is obviously rectifiable (Lebesgue, Jordan, Riemann, Gauss, Archimedes, etc.). Denote its length by $\lambda (C_m)$. This is zero iff $C_m({\Bbb R})$ is empty or contains merely isolated points. Further there is an obvious way to take into account the multiplicity of branches; e.g. a conic (degree 2) consisting of a double line has length not just $2\pi $ but twice that quantity. This is crucial to ensure continuity of $\lambda$ on the parameter space of all $m$-tics. Since the latter space is compact (actually an ${\Bbb R}P^N$, $N=\binom{m+2}{2}-1$) the length functional $\lambda$ reaches a maximum. How long can an $m$-tics be? By the above compactness argument there is some universal constant $L_m$ bounding the length of all curves $C_m$ of some fixed degree $m$: $\lambda(C_m) \le L_m$, and the maximum is actually realized (a priori not by smooth curve). A configuration of $m$ lines produces the lower estimate $2\pi m \le L_m$, and by Brusotti 1921 [@Brusotti_1921] there are smooth curves $C_m$ of length as close as we please to $2\pi m$, but slightly longer. Imagine a crossing getting smoothed then the geodesic of $S^2$ are entailed by curvilinear arcs which are longer (triangle inequality or Pythagoras in the small). So there certainly exist longer curves! But how long can an $m$-tic be? This is probably very difficult to answer. When $m=2$, the above argument via Brusotti still makes sense. If we imagine quadratic cones in 3-space $E^3$ (say with elliptical affine cross-section at $x_2=1$), then they may cut strange ovals on $S^2$ possibly longer that $2\pi \cdot 2$??? When $m$ is odd then there is a pseudoline (or at least a circuit possibly singular) not null-homotopic in ${\Bbb R}P^2$. Obviously its length is at least $2\pi$, which is the lower bound of the functional $\lambda$ when $m=2k+1$ is odd. The estimation of $L_m$ is surely an attractive problem, but let us try to be not sidetracked by this. Our goal would be rather to study the gradient flow of $\lambda$ as a dynamical process susceptible to implement the collective contraction conjecture CCC, or more elaborated versions thereof if feasible (e.g. some iterated contractions like C++). Assume first $m$ even. Then $\lambda$ vanishes identically on the chamber corresponding to empty curves, as well as on its adherence which consists of curves with isolated real points (either solitary double points or of higher multiplicity necessarily even). Thus $\lambda $ cannot be analytic, but seems rather being $C^\infty$. (As usual with distance functions (e.g. $\vert x \vert$) they sometimes lack even smoothness until tacking their squares. So perhaps take $\lambda^2$ squared.) Consider the gradient flow of this functional $\lambda$, while hoping that the corresponding trajectories materialize the collective contraction conjecture (CCC). Usually ovals fails severely to be geodesics on $S^2$, but are perhaps so when we look them in the Riemann surface $C_m({\Bbb C})$ endowed with the Fubini-Study(=FS) metric on ${\Bbb C}P^2$. Is the corresponding length of the ovals the same, in other words does FS induce the round elliptic metric on $S^2$? It is (always) tempting to regard ${\Bbb C}P^2$ as the variety of groups of two points on the Riemann (round) sphere. In this model how to describe the FS-metric? Another idea, at least if we restrict to smooth curves, is to take the uniformizing hyperbolic metric (of Schwarz-Klein-Poincaré-Koebe) on $C_m({\Bbb C})$ with curvature $K\equiv -1$, when $m\ge 4$ (so $g\ge 3$). Then we get another measure of length of the ovals, which we shall denote $h$. The problem here is that this length functional is not a priori defined on the full hyperspace of curves $\vert m H\vert$. As soon as we look also in the complex domain, there is a myriad of other functionals like the systole of the Riemann surface, the area of one half in the dividing case, etc. We just remark that from the systolic viewpoint there might by an ortho-cycle of much shorter length than the real ovals, and which dynamically might be advantageous being first contracted. Of course the technical difficulties look immense, but the problem involves a mixture of Poincaré-Morse versus Hilbert-Petrovskii, i.e. a synthesis thereof. So the game is certainly worth paying attention at. What seems called upon is a dynamical study of algebraic equations governed by motions regulated by (natural) geometric functionals on the corresponding varieties (zero loci). In particular find appropriate functionals whose trajectories converge (generically) to curves with solitary nodes as to implement the Itenberg-Viro contraction conjecture or its collective variant CCC. To shrink all empty ovals simultaneously it seems not so fruitful to shrink the shortest oval. More collective optimization is asked as if one had to bring fastest to the harbor a convey of ships each carrying rough materials involved in the manufacture of some complex end-product (Polyà’s metaphor for Rayleigh eigenvalues). Here we are in a similar situation. If all empty ovals have to dye simultaneously (scenario posited by CCC), it is important to shrink faster the longer ovals. Perhaps this suggests looking and $\lambda^2$ the squared length functional penalizing longer ovals. It is also tempting to look at the area $\alpha$ (of the interior of all empty ovals measured on $S^2$), and to play perhaps with the isoperimetric inequality. For instance the functional $\lambda^2/\alpha$ looks natural, and is bounded from below in the small by $(2\pi \rho)^2/(\pi \rho^2)=4\pi$, so it admits a finite limit when it shrinks. The isoperimetric functional $\iota=\lambda^2/ \alpha$ intuitively forces ovals to dye in a round manner, penalizing agonies along eccentric ellipses. If optimistic, integrating the gradient flow of either $\lambda$ or $\alpha$, length resp. area of the empty ovals directly leads to a solution of CCC. The serious obstacle is that there may be a sink, i.e. a local minimum of the functional preventing convergence to a curve with solitary nodes arising as contractions of the $r_0$ empty ovals. One naive idea is to let vibrate the ovals via a configuration of lines (as in the Harnack-Hilbert method), hoping to decrease area through this perturbation. The oval then oscillates inside and outside itself but on a larger portion it would move inside himself, hence area decreases (cf. Fig.\[Vibrate:fig\]a). The similar assertion for the length functional looks even more fantasist. Hence the area functional looks better suited to the problem. [*Insertion*]{} \[01.04.13\] One problem is that if we have several ovals it is not clear that decreasing the area of one will not enlarge area of the other empty ovals. This problem dissipates somewhat if we look only at the usual Itenberg-Viro conjecture, but of course also the latter is subject to doubts, e.g. those allied with Shustin’s disproof of Klein-vache. -5pt0 -5pt0 A more naive idea is to look at some sublevel of the equation. For simplicity assume the given curve completely inside some affine chart (cf. however Ronga’s counter-example on Fig.\[ItenbergViroRiem:fig\]) with equation $f(x,y)=0$, and w.l.o.g. positive on the outside unbounded (in ${\Bbb R}P^2$ nonorientable) component residual to the curve. Then the sublevel $f(x,y)=-\varepsilon<0$ (small negative constant) ought to have a smaller area $\alpha$ (compare Fig.\[Vibrate:fig\]b). Let us be more precise. Look at all empty ovals of the curve $C_m$. Then following Ragsdale-Petrovskii, some are positive and some negative (or, even and odd depending on the parity of the number of ovals surrounding it). Maximal ovals are even (being surrounded by zero ovals), those immediately inside them are odd, etc. Under our sign convention for the equation $f$ (positive outside) we see that even ovals decrease in area when considering the sublevel $f=-\varepsilon$, while odd ovals increase in area (compare Fig.\[Vibrate:fig\]b). The net bilan is hard to quantify, but on the situation of the picture where the even empty oval is much larger than the other empty ovals there is some chance to decrease the functional $\alpha$. Is there some chance to deduce a general argument from our naive picture? Split all empty ovals in even and odd ones (denoted resp. 0 and 1 depending on their class modulo 2). Look which of both collections has more massive total area, i.e. compare $\alpha_0$ vs. $\alpha_1$. If $\alpha_0>\alpha_1$ then take $\varepsilon$ positive (and vice-versa if $\alpha_0<\alpha_1$ then take $\varepsilon$ negative). Of course if unlucky both magnitudes $\alpha_i$ are equal, in which case we are a bit lost (perhaps avoidable by genericity, as we work up to isotopy, hence can always perturb slightly the data). Let us assume to be in first case $\alpha_0>\alpha_1$ (as on the picture). By a classical continuity lemma (cf. e.g. Gudkov 1974 [@Gudkov_1974/74]) we can preassign a tubular neighborhood of the curve in which the perturbation will stay confined. Thus we can probably control from above the area expansion of the odd ovals. It seems indeed that the tubular expansion is maximum for a circle as follows from the isoperimetric inequality. However it is not clear that the deflation of area of the even ovals ought to supersede this as it may be a very thin penetration (imagine the big south island as very mountainous hence poorly affected by a raise of the ocean level). This leads to the idea of looking at the normal derivative of the defining (polynomial) function $f(x,y)$ across the sea level ($f=0$). The inflation of area resp. deflation of area of the empty ovals ought to be proportional to this normal slope and the length of those ovals via some explicit formula given by differential calculus. So we compare both quantities for even an odd ovals, and choose the right sign for $\varepsilon$ in order to create a deflation of area while keeping the degree constant. Of course if both quantities coincide we are a bit disturbed, but perhaps avoidable by genericity (two random real numbers are generically distinct.) If this trick works there is some hope to show that the area functional $\alpha$ lacks local minima, and the corresponding orthogonal trajectories of steepest descent ought to converge toward curves with solitary nodes. Note another phenomenon: imagine one empty oval shrinking prematurely before the others. Soon after this death another oval (formerly nonempty) may become suddenly empty implying a large jump of the area functional $\alpha$ which looks therefore discontinuous. The only reasonable parade against this catastrophe is that the trajectory of steepest descent will not kill abruptly the small oval as it is much more profitable to shrink first the voluminous ovals. Intuitively, the $\alpha$-flow would promote a collective contraction. Nonetheless the $\alpha$-functional can be discontinuous with big jumps across walls of the discriminant, as caused by the fact that we measure only empty ovals, which in contrast to all ovals, is subsumed to violent fluctuation. Have we proved something? Maybe yes if quite sloppy. Let us resume some of the difficulties: (D.1) First Ronga’s example of a $C_6$ not confined to an affine chart is presumably not a severe obstacle. For even degrees the sign of the projective equation $F(x_0,x_1,x_2)$ is always well-defined so that there is a variation $C_m^{\varepsilon}$ with disjoint real locus ($C_m\cap C_m^{\varepsilon}=\varnothing$). (For odd degrees nothing similar can be done so easily, but we confine attention to even degrees. That is already hard enough.) (D.2) We look at the penetration index under a small perturbation as measured by the normal derivative of the landscape pondered against the length element of the oval. More precise, calculate along each point of $C_m$ the normal derivative $ \frac{\partial f}{\partial n}, $ and then integrate this against the length element $ds$ of some fixed oval $O$ to get $$\int_{O=an oval} \frac{\partial f}{\partial n} ds=: \pi (O).$$ This real number measures the rate of area change under a flood (variation of $\varepsilon$). Of course the normal derivative above can be interpreted as the gradient of $f$ on the coast line $f=0$. (If it is big in norm then the slope of the coast is low hence the territory much affected by floods, while if small then the coast slope is steep and the island has little to fear from inundations). Call the above quantity $\pi(O)$ the “piaf” (protection index against floods). It is well-defined for any oval (up to sign and anodyne choices effecting a collective change). It seems to make also good sense in the projective context. Next look at all the empty ovals $O_1, \dots, O_{r_0}$ of some smooth curve $C_m$, splits them in even and odd, and look which of both collections have the highest piaf. Depending on this knowledge, an appropriate choice of $\varepsilon$ create a variation of the curve with smaller inner area $\alpha$ . (D.3) What to do exactly when both (even and odd) piafs are equal? Can we avoid this just by slight perturbation, i.e. is there always a small perturbation making them different? (Perhaps Petrovskii thought about such questions…) Assume now that (D.3) can be overcome. Then the functional $\alpha$ has no stable equilibrium (local minimum) and we interpret it as a Morse function on the space $\vert mH \vert$ of all curves. In this generality it may rather look like a Grand canyon with big ravine when one cross a solitary node due to the brusque change of empty oval. Two attitudes are possible: (A.1)—either we localize $\alpha$ to one chamber of the discriminant[^50] or (A.2)—we look at the whole space of curves hoping that the trajectories dictated by the functional never cross such ridges (=ravine of $\alpha$). After having introduced a natural metric on $\vert m H \vert$, e.g. the elliptic geometry available on each projective space, we look at the trajectories of steepest descents w.r.t. the function $\alpha$ (area of the nonempty ovals). This is as usual obtained by integrating the vector field ${\rm grad} \alpha $. By the above (D.2), for almost every initial condition $C_m$ it decreases endlessly up to reach level $\alpha=0$, which must necessarily be a curve with solitary nodes. Could the trajectory starts oscillating like a $\sin (1/x)$ curve without reasonable convergence? Looks unlikely due to the algebraic nature of our problem, but requires perhaps an argument. If the trajectory of $C_m$ converges to a saddle point (unstable critical point of $\alpha$) it suffices to perturb slightly the initial condition $C_m$ (which is allowable up to small rigid-isotopic perturbation). (In fact it is likely that such exceptional saddles correspond precisely to curves having the same even and odd piafs, especially if we have a rigorous proof that there is no local minimum for $\alpha$, as we tried to argue in Step D.2.) At this stage we believe the proof would be completed (no additional difficulties) and we would conclude: (Hypothetical!!!) Given any (non-void) curve $C_m$ (of even degree $m$ for simplicity), the trajectory of the gradient flow of the empty-ovals area $\alpha$ generically converges to a curve with solitary nodes in finite time, while the empty ovals themselves converges to the solitary nodes. If not then it finishes its trajectory to an unstable equilibrium and it suffices to perturb slightly to ensure convergence toward a solitary nodal curve (soliton for short, as compression of solitary and singleton). In particular, CCC holds true, i.e. there is a path in the space of curves such that all empty ovals contract to solitary nodes. That the extinction of all the empty ovals occurs in finite time merely follows from the fact that the time parameter of any gradient flow is just the “height” function, here the functional $\alpha\colon \vert mH \vert \to {\Bbb R}$ but taking value in $[0, 4\pi]$ (where $4\pi$ is the area of the full sphere or $2\pi$ if you count this area divided by two). Here we have looked at the empty-oval area functional $\alpha$. What happens if we look the same functional for all ovals. A priori the functional looks more continuous but be careful with eversions. \[23.01.13\] Metaphor.—Problems of rigid-isotopy (or large deformations of curves) are like a video game in the sense that there is a joystick upon which one may act by freewill by varying the coefficients while there is in reaction a canonical picture emerging on the screen (the corresponding real locus of the algebraic curve conceived as an optical object). In some sense it is like a flight simulator (you move the “manche à balais” and the aircraft responds accordingly). The contraction conjecture CCC says that using the full freedom of the joystick one can always shrink the empty ovals simultaneously. The above theorem states roughly that there is some predestination, i.e. that a very sleepy autopilot or video game player suffices to land safely the aircraft, while performing actually a perfect landing (all wheels touch the ground simultaneously!). Of course in reality the autopilot in question is very well programmed for its action is governed by a principle of least action. The only little impulse required is when the aircraft arrives at critical points (global maximum of $\alpha$ or its saddle critical points), where some jiggling is required to perturb the initial condition. Some few other applications of CCC {#application-of-CCC:sec} ---------------------------------- \[23.01.13\] What can be deduced from CCC? Quite a lot and alas no so much, compare the case of $M$-sextics. Applying it to Gudkov’s $C_6$ gives a rational (genus 0) real sextic with 10 solitary nodes equidistributed as $5$ inside and $5$ outside the unique oval. Of course there is no obstruction given by the genus formula to such a eventuality. In general, given any curve $C_m$ with say $r$ ovals, then can be split as $r_0\le r$ empty ovals, which can be contracted to solitary nodes. Then applying Brusotti 1921 [@Brusotti_1921] we can let them disappear all, and so appears a new generation of empty ovals, to which the collective contraction process can be applied again, and so on. So we can reach the empty curves (or a pseudoline if $m$ is odd) after some few iterated contraction (as many as the height of the oval-graph encoding the nested structure of the original $C_m$). This we call the height of the curve $C_m$, denoted $h(C_m)$ This implies (as a crude estimate) that any two curves $C_m, D_m$ can be related by a rigid-isotopy crossing only $h(C_m)+h(D_m)$ times the discriminant. Of course it is not a transverse crossing in general for our critical curves have several solitary nodes. However by perturbing slightly we may cross the discriminant transversally, and each initial crossing through a multi-solitary curves implies as many intersection with $\frak D$ as there are nodes. So counting properly we deduce: [(modulo CCC and the connectedness of invisible curves=CIC)]{} Any two smooth curves $C_m, D_m$ of even degree $m$ can be joined by a path in the hyperspace of curves transverse to the discriminant while crossing it exactly $r(C_m)+r(D_m)$ times, where $r$ is the number of ovals (composing the real locus). Applies iteratively CCC (conjointly with Brusotti) to both curves, to derive two curves with empty real locus. The latter are known to form a unique chamber of the discriminant, in other words to be rigid-isotopic by Lemma \[empty-chamber-connected-Shustin:lem\]. A similar assertion holds perhaps true in case of odd degrees, however it is still unknown whether two curves of the same odd degree are rigid-isotopic provided their real loci reduce to a pseudoline (compare Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical p.199]). This goes in the sense of showing that the contiguity graph of chambers residual to the discriminant is a “small world”, in the sense that is has high connectivity and much “consanguinity”. How good is the above estimate? More precisely the distance $\delta$ (or Erdös number) in the contiguity graph (of chambers) is majored by $\delta(C,D) \le r(C)+r(D)$. This is fairly good as compared to the estimate coming from the degree of the discriminant $3(m-1)^2$, which implies $\delta(C,D)\le 3/2(m-1)^2$, or the integral part thereof, as we may always choose the one side of the circle hitting less many times the discriminant. For $m=6$, the discriminant estimate gives $\delta\le [75/2]=[37.5]=37$, while the CCC estimate gives $\delta\le 11+11=22$. We can be more economical by not going down to the empty chamber but that having only one oval, which form already a unique rigid-isotopy class by Nikulin 1979 [@Nikulin_1979/80]. This raises the following question: we know either by Rohlin’s formula (or less rigourously by CCC) that any curve with one oval (hence of even degree) is nondividing provided $m=2k\ge 4$. Indeed if dividing apply CCC to get a splitting $C_{2k}\to C_k\cup C_k^\sigma$, whence by Bézout $C_k \cap C_k^\sigma=k^2=r$, where $r=1$ whereas $k^2\ge 4$. Thus there is no obstruction to rigid-isotopy given by the Klein’s type between any two curves having only one oval, and extrapolating (violently) we arrive at the: \[OOPS:one-oval-rigid-isotopic:conj\] [(OOPS=One oval postulation)]{} Any two smooth curves having only one oval are rigid-isotopic. (“Oval” is interpreted here in the strong sense of a Jordan curve which is null-homotopic, hence our curves are of even degree.) (Perhaps there is an obstruction à la Fiedler-Marin, but unlikely as it seems to require a splitting of ovals, cf. Marin’s argument exposed below.) Remind also from Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical] that replacing above “oval” by pseudoline is still an open problem. Further Viro in his e-mail (dated 26.01.13 in Sec.\[e-mail-Viro:sec\]) confirmed me that this is still an open problem and goes back to Rohlin. [*Insertion*]{} \[02.02.13\].—A naive approach to this would be to assert that any $C_{2k+1}$ may degenerate by a large deformation to $L_1\cup C_{2k}$ a line $L_1$ union an invisible curve $C_{2k}$ of even degree. Perhaps one can demand that both curves in the limit are transverse. Call this process a rectification of the pseudoline. Now given two smooth curves whose real scheme consist of a unique pseudoline, one may apply twice rectification, and then isotope both corresponding empty curves. This could imply the Rohlin-Viro conjecture. In degree 6 the Erdös number of the graph (supremum of $\delta$) is probably much smaller in view of Fig.\[Gudkov-eversion:fig\], i.e. Gudkov’s table with eversions. Of course this figure posits that all logically possible eversions are realized geometrically (safe those linking $M$-curves and both $(M-2)$-schemes of type I). Under this circumstance the Erdös number looks hardly greater than 8, i.e. $\delta \le 8$ universally. It seems indeed that the maximal distance is realized by Hilbert vs. Gudkov or Gudkov vs. Harnack. Naively on the table (without eversions) Hilbert’s scheme and Harnack’s looks far apart, but using the eversion $\frac{8}{1}1 \to \frac{1}{1}8$ shows that their real distance is only 3, i.e. $\delta (Hilbert, Harnack)=3$. This is the answer if we confine attention to the top of the table, but of course the most distant chambers are the $M$-curves as separated from the empty scheme $0$. Those are at distance $11$ apart. So the correct Erdös number is $\delta=11$, with this maximal distance being realized thrice (empty vs. Hilbert, Gudkov, Harnack respectively). Morally eversions do not shorten the vertical distance, and we have proven the following (modulo Conjecture \[eversion-and-other surgeries:conj\]). [(Semi-conjectural)]{} \[Erdos-number-of-sextics=11:prop\] The Erdös number of the contiguity graph of sextics is actually equal to the Harnack bound $M=11$. It is tempting to wonder if it so in general. Perhaps there is some little chance to answer this without having to work out the exact rigid-isotopy classification in each degree (an insurmountable task!?). The trick would be that eversions collapse sufficiently horizontal distances, so as to make only the vertical chain the only plausible candidate for maximizing $\delta$. As to the above OOPS conjecture (\[OOPS:one-oval-rigid-isotopic:conj\]), one could of course also imagine a dynamical proof. The whole task reduces to finding the right functional. Heuristically the obvious attractor ought to be a circle (possibly multiple). This inclines to look at the isoperimetric functional looking the length squared divided by the area of the unique oval. More pragmatically, [(but apparently validated by Viro, cf. comments right-after the proof)]{}.—It seems clear that CCC (and of course the weaker formulation CC) implies the one oval postulate (OOPS). (pseudo, of course!) Indeed take any two curves having only one oval. By CCC (CC suffices) each of them can be shrunk to a solitary node. By Brusotti there is slight perturbations to empty curves. The latter can be linked by a path, by virtue of the connectivity of the empty chamber (Lemma \[empty-chamber-connected-Shustin:lem\]). Since the empty chamber of the discriminant is a manifold with corners (like probably any other chamber by the way) which topologically is a manifold with boundary there is such a path staying close to the boundary. (Actually the boundary of the empty chamber has faces consisting of uninodal solitary curves with one isolated real point.) Pushing this path slightly outside the empty chamber would give the required isotopy. However there is a serious difficulty, if our path meets another wall of ${\frak D}$ outside the empty chamber. However by genericity of this path (as avoiding sets of codimension 2) we may assume this crossing to be a transverse one of a wall which must keep $r=1$ constant since we stay in the vicinity of the empty chamber with $r=0$. Note of course by CCC as applied to quartics for instance that chambers with higher $r$’s are also contiguous to the empty chamber, yet the are like cubes hitting the empty chamber imagined as a cube at some vertices of codimension 2. So we have some wall crossing keeping $r=1$ constant, which as a Morse surgery must correspond to an eversion. But this means that there is an eversive wall falling down to the boundary of the empty chamber like a tripod. This looks incompatible with the local structure of algebraic sets? Maybe there is a more direct argument in ${\Bbb R}P^2$. Another obstruction comes also from Brusotti’s description of the discriminant as branches with normal crossing. All this is very confuse, we confess. \[30.01.13\] As kindly informed by Viro (cf. his e-mail, dated 26.01.13 in Sec.\[e-mail-Viro:sec\]), it seems that the implication CC$\Rightarrow$OOPS causes no problem. It would be nice to write down complete details. Perhaps this simply follows from the fact that inside the empty locus the discriminant has real codimension 2 (cf. Lemma \[invisible-discriminant-codim-2:lem\]). Hence there is no wall inside it, hence no wall outside it by the “implicit function theorem”. [*Insertion.*]{} \[01.04.13\] If CC$\Rightarrow$OOPS causes no troubles it would be interesting to extend the method to higher schemes (having more ovals than one). \[URS:conj\] (URS)=(Unnested rigidity speculation).—The unnested configurations (which are of type II provided $r<k^2$ or even $r\neq k^2$, as shown by Rohlin’s formula) are always rigid (i.e. any $2$ curves representing the unnested scheme are rigid-isotopic provided $r\neq k^2$). When $r\neq k^2$ it could be that the type is the sole obstacle to rigid isotopy. This holds true in degree 6 by Nikulin’s theorem (\[Nikulin:thm\]). Further it looks hard to disprove this by the Fiedler-Marin method as we lack a canonical choice for the fundamental triangle. Finally, it could be that the same argument as above shows that CCC implies URS. [*Sketch of proof that CCC$\Rightarrow$URS*]{}.—Contract all ovals simultaneously to land in the connected empty chamber of curves without real points. So given 2 curves connect them by respective contractions to the empty locus, and therein by a path of invisible curves. The hard part is then to push this path by a small deformation again in the visible locus, and this in such a way that we never meet the discriminant. This looks feasible as the (closured) empty locus seems to be a bordered manifold, with connected boundary (being essentially fibred over $\RR P^2$ via assignment of the unique solitary node). So the joining-path in the empty locus can be pushed on the boundary and then further inside the chamber. Of course it remains the difficulty of ensuring that we do not meet other nappes of the discriminant. As above the loose argument is that since the discriminant as codimension 2 inside the empty locus, it will appear outside along the same dimension, and not effect any separation. This would supply the required rigid-isotopy between our pair of unnested curves. Of course it is essential to assume $r\neq k^2$ since otherwise there is curves of both types, yet this condition did not as yet appeared frankly in our argument, which is far from a serious proof. Of course one could expect that for $r=k^2$ the type is sole additional obstruction. Looking around (in vain?) for counterexamples to CCC (=collective contraction conjecture) ----------------------------------------------------------------------------------------- \[20.01.13\] An a priori easier game is to test if CCC (\[CCC:conj\]) has really some chance to be true. As usual experimentation is required. We may first consider a curve $C_8$ arising through perturbation of 4 ellipses rotated by $180/4=45$ degrees (cf. Fig.\[CCCRoses:fig\]b). -1.2cm0 -5pt0 -5pt0 The smoothing being compatible with orientations, this curve $C_8$ is dividing (Fiedler). It has $r=16$ unnested ovals (cf. Fig.\[CCCRoses:fig\]b). Assume it shrinkable via CCC, it will degenerate and decomposes as $C_8 \to C_d \cup C_d^{\sigma}$, where $C_d$ has genus $p=\frac{g-(r-1)}{2}=\frac{21-(16-1)}{2}=3$, hence of degree $d=4$ (this can be inferred more directly via the degree of the degeneration). The relation $d^2=r$ is verified and there is no numerical obstruction to CCC. Note in contrast that in the same construction for 3 or 5 ellipses, we cannot arrange all ovals unnested while smoothing in a sense preserving way. Look and see! (Figs.\[CCCRoses:fig\]a and c). The construction of our $C_8$ generalizes whenever the degree $m=4\ell$ is a multiple of four. Indeed rotate an ellipse by $\pi/ \ell$. Orient the ellipses “alternatively” and smooth in a sense preserving way. The resulting curve has $r=4 \ell^2$ unnested ovals (as easily counted by extrapolating the figures $C_8$ and $C_{12}$ of Fig.\[CCCRoses:fig\], while noting that the ovals in $\ell=m/4$ couches containing each $m=4\ell$ ovals). Shrinking $C_m$ via CCC gives two curves of genus $p=\frac{g-(r-1)}{2}$, where $g=\frac{(m-1)(m-2)}{2}=(4\ell-1)(2 \ell-1)$. Hence $$\begin{aligned} p =[(4\ell-1)(2 \ell-1)-(4 \ell^2-1)]/2 &=[(2\ell-1) [(4\ell-1)-(2\ell+1) ]]/2 \cr &=[(2 \ell-1) (2\ell-2)]/2, $$ so that the half has degree $2\ell$, and intersects its conjugate in $4\ell^2$ points, which is precisely $r$. This little numerical miracle implies an absence of numerical obstruction to CCC via our rosewindows constructions. Of course, all the above computation can be shortcuted by noticing a degeneration $C_{m=4\ell} \to C_d \cup C_d^{\sigma}$, where $d=2\ell$ necessarily (since the conjugation $\sigma$ preserves the degree of equations as it just acts upon the coefficients). As yet the center of rotation was chosen inside the ground ellipse. Another series of picture arise when choosing it outside instead (but sorry this is probably not the right thing to do). Let us instead manufacture the dividing $C_6$ with $r=9$ unnested ovals as on Fig.\[CCCRoses2:fig\]a. Next we tried to construct a $C_{10}$ with 25 unnested ovals but failed somewhat. Note that this is not obstructed by Arnold’s congruence $\chi=p-n=k^2 \pmod 4$, as both sides are 25. -1.2cm0 -5pt0 -5pt0 Assume there is such a $C_{10}$ with real scheme $25$ (Gudkov’s notation) which is dividing. On applying CCC we find a splitting $C_{10} \to C_5 \cup C_5^{\sigma}$. The genus $g=9\cdot 8/2=9\cdot 4 =36$, hence $p=[g-(r-1)]/2=(36-24)/2=6$ which is indeed the genus of a quintic. So no obstruction on this side. Note also that Rohlin’s formula $2(\Pi^+ -\Pi^-)=r-k^2$ implies no obstruction, since $\Pi^{\pm}=0$ (no nesting) and so $r=k^2$ with $k=5$. After our stupid trials we find ultimately the right ground configuration of ellipse as Fig.\[CCCRoses2:fig\]y, which is dividing with $25$ ovals. It is clear now how to extend this in an infinite series of curves $C_{4 \ell +2=2( \ell +1)}$ with $(\ell+1)^2$ ovals lying outside each other, i.e. the scheme $(\ell+1)^2$ in type I. Fig.\[CCCRoses2:fig\]z gives the case of a $C_{14}$. All this pictures are pleasant, yet they do not help at all to corrupt the conjecture CCC. Of course the latter has some chance to be true especially if the Itenberg-Viro conjecture (\[Itenberg-Viro-contraction:conj\]) is true. A dividing curve without nesting has to satisfy $r=k^2$ by Rohlin’s formula, and therefore applying CCC gives a degeneration $C_{2k}\to C_{k} \cup C_{k}^{\sigma}$ yielding no chance to corrupt Bézout. Also there is no chance to corrupt Klein $p=[g-(r-1)]/2$. Indeed $g=(2k-1)(2k-2)/2$, hence $$\begin{aligned} p =[(2k-1)( k-1)-(k^2-1)]/2 &=[(k-1) [(2k-1)-(k+1) ]]/2 \cr &=[(k-1) (k-2)]/2, $$ so that the half has degree $k$, as it should. CCC versus Brusotti: large deformations vs. small perturbations --------------------------------------------------------------- \[20.01.12\] Philosophically, it seems that such contraction conjectures (Itenberg-Viro (\[Itenberg-Viro-contraction:conj\]) or our collective version thereof (\[CCC:conj\]))—if they turn out to be true by some lucky stroke—incarnate a sort of large deformation principle illustrating once more the perfect graphical flexibility of algebraic curves despite their intrinsic rigidity. In some sense this is an avatar in the large of the small perturbation method à la Plücker-Klein-Harnack-Hilbert-Brusotti-Viro. Hence it seems indeed (as Viro advocated) being of some primary interest to establish the contraction conjectures. Flexibility in the small gives rise to the perturbation method which is primarily a method of construction, whereas flexibility in the large (contraction principles) implies as a byproduct prohibitions (as we superficially experimented at the beginning of Sec.\[CCC:sec\] and more convincingly because Itenberg’s contraction theorem for sextics re-explain all Gudkov-style prohibitions by reduction to the RKM-congruence). At this stage we feel some big duality: local versus global and constructions versus prohibitions (to be or not to be). What would be the net impact of the contraction principle for Hilbert’s 16th problem (in the extended sense of high degrees)? Somewhat optimistically it would reduce the whole task (or rather adventure) to a combinatorial video game best suited for machines. So exaggerating slightly, the contraction conjecture seems quite close to reveal the ultimate secret of the whole problem. Of course some supplementary large deformation principles ought also to complete the picture, e.g. certain permissible eversions compatible with Bézout, and more generally the full morphogenesis of all algebraic Morse surgeries. If all this is available, the video game solving Hilbert’s 16th problem would show in real time all the possible perestroikas which the real loci of projective curves of some fixed degree can undergo, while dragging at free will the joystick in the parameter space. As sketched in the previous Sections \[CC-via-dynamics:sec\] and \[CC-via-Riemann:sec\]), in order to prove the contraction conjectures (CC or CCC), we could either imagine a dynamical proof via orthogonal trajectories, hence akin to Morse theory, or a direct intervention of conformal geometry à la Riemann (albeit our implementation failed seriously). Whatsoever the exact details it is quite likely that the proof of CC or CCC will employ the calculus of variation in the large over which practically every deep geometrical theorem is based upon (from the brachystochrone, to the Dirichlet principle, via the Riemann mapping theorem up the recent solution of the Poincaré conjecture via the Ricci flow.) Note finally another very modest piece of evidence in favor of CCC. Remember that for small perturbations à la Brusotti, there is complete freedom to smooth the nodes of a plane curve with normal crossings (compare e.g. Gudkov 1980 [@Gudkov_1980/80-Brusotti]) in the sense that all crossings may be smoothed away or some may be conserved. By analogy CCC is just the case where all empty ovals are contracted simultaneously, while Itenberg-Viro’s CC is just the contraction of a single empty oval. Of course there ought to be the full panoply of intermediate contractions. CCC implies CC (i.e. Gabard stronger than Itenberg-Viro) -------------------------------------------------------- \[21.01.13\] Let us now observe that CCC implies CC, just via Brusotti’s theorem (1921 [@Brusotti_1921]): \[CCviaCCC-Brusotti:lem\] Suppose given a collective contraction of all the empty ovals of a smooth real curve $C_m$, then it is possible to construct all partial contractions via Brusotti. In particular if CCC holds true then so does the Itenberg-Viro contraction conjecture [(\[Itenberg-Viro-contraction:conj\])]{}. Let $C_m$ be a smooth real curve. W.l.o.g. let us assume it having some empty ovals, say $r_0\ge 1$ many (otherwise the curve just reduces to a pseudoline or to empty real locus). By CCC there is a path $c\colon [0,1] \to \vert mH \vert$ such that $c(0)=C_m$ and $c(t)\in \frak D$(=the discriminant) only for $t=1$ where $c(1)$ is a nodal curve with solitary nodes only ($r_0$ many). By Brusotti’s theorem the neighborhood of the nodal curve $c(1)$ consists of $r_0$ “falde analytiche” (=analytic branches or better [*nappes*]{}) meeting transversally at $c(1)$. Further each of those nappes corresponds to the conservation of some node in the vicinity. In other words the chamber of $C_m$ looks like manifold-with-corner near the nodal curve $c(1)$, locally diffeomorphic to ${\Bbb R}^N$ with $r_0$ many distinguished hyperplanes of coordinates. It is now plain how to construct all other contractions, in particular all the ones of Itenberg-Viro CC contracting just a single empty oval (compare Fig.\[CCCBrusotti:fig\]). -5pt0 -5pt0 So if optimistic about the truth of CC, it may even seem that CCC is a fairly reasonable angle of attack. Of course one then needs a good functional (e.g. the area of all empty ovals). Alas, it is less evident that CC implies CCC, as we suggested at the beginning of the investigation. This could be slightly easier if there is a contraction principle extended to solitary nodal curves while keeping the “solitons” in place. Naively, one may dream of a contraction principle effecting a retraction of a whole chamber (past the discriminant) to its boundary. Yet compact bordered manifolds never retract to their boundaries (as shown by homology mod 2, as we learned from J.-C. Hausmann). This is evidently no obstacle against the contraction conjectures, for the retraction may be undefined on small loci, as since we work up to isotopy such equilibrium points can be avoided. More lucidly nobody ever asserted that the contractions should depend continuously on their initial point(=curve). Imagine as a very naive picture, the chamber as being a disc with a radial projection upon the boundary. Then it is undefined on the center of the disc but this is not a problem for perturbing it slightly it will get mapped somewhere. The whole analogy with retraction of bordered manifolds is not extremely pertinent as in general the chamber will have a boundary consisting not merely of faces touching the empty chamber. Hence under a retraction a curve close to the discriminant could first coalescence 2 ovals instead of shrinking one empty oval. To show CC or even CCC we could employ a dynamical system (continuous flow) spreading nearly all curves toward the boundary of this chamber at curves having solitary nodes. This could occur as the orthogonal trajectories of some functional. (Another idea would be to look at the Green function of the chamber yet the Green’s lines, streamlines of the flow, would often finish at curves with non-solitary nodes.) Our flow should be strongly attracted by the multi-solitary nodal curves manifold, whereas all other walls corresponding to non-isolated singularities have to repulse the flow. A qualitative picture is given on Fig.\[CCCflow:fig\]. -5pt0 -5pt0 This picture proves nothing, safe maybe the absence of topological obstruction (a priori) to find such a flow (especially if it is allowed some equilibriums when the chamber has complicated topology). Again to get a proof it is likely that one should consider the gradient flow of some real-valued functional on the chamber. This could be the area or length of all empty ovals (as measured on the round metric of $S^2$ double covering ${\Bbb R}P^2$). Note that CCC implies a little technical simplification over our previous pseudo-proof of CC, where we were troubled by marking one oval. As already discussed, the main obstacle occurs if our function has some global attracting basin inside the chamber preventing us to reach the desired multi-solitary nodal curve shrinking all empty ovals. This is basically the sole difficulty yet it looks quite insurmountable. At least for the area or length functional, we saw no obvious way to produce small variations diminishing the “energy”. Perhaps there is some more clever (projective) invariants like degree of roughness à la Gudkov (cf. e.g. Gudkov 1974 [@Gudkov_1974/74]). For instance one could look at the largest (quadratic) cones in ${\Bbb R}^3$ which can be nested inside the ovals, and the corresponding area intercepted on the unit sphere. This is another functional measuring the conical area of the empty ovals. Can we show that this functional is “good”, i.e. no sink inside the chamber? Naive idea: trace inside each oval some maximal ellipse and try to deform the $C_m$ along suitable multiples of those ellipses. Now the problem looks reduced to a fantastic game probably only soluble by such authorities as Andronov, Leontovich, Gudkov, etc. mixing the qualitative theory of differential equations with that of algebraic curves. So it is truly a Poincaré-Hilbert Verschmelzung(=fusion in Klein’s prose) which seems demanded to settle CCC (or its avatar CC). \[03.04.13\] Of course, it also safe to say pessimistic and expect maybe that Shustin’s disproof of Klein-vache in degree 8 also implies a disproof of CC. We do not repeat our vague strategy for this, but refer to Sec.\[Challenging-open-prob:sec\]. Problems of rigid-isotopy ========================= \[09.02.13\] Two (real, plane) curves are said to be [*rigid-isotopic*]{} if one can pass from one to the other by continuous deformation of the coefficients of the defining equations which avoids the discriminant. This involves again the paradigm of [*large deformations*]{} like the contraction conjectures discussed in the previous section. In fact there should be some direct connections between both topics. As a rule very little is known about the phenomenon of rigidity. All what is trivial is that any topological characteristic persists during a rigid-isotopy, so for instance the [*real scheme*]{} (i.e. the isotopy class of $C_m(\RR)\subset \RR P^2$) as well as Klein’s type I, II measuring the situation of the curve in its complexification. Such invariance were intuitively clear since the era of Schläfli, Zeuthen, Klein 1876 [@Klein_1876_Verlauf], but requires perhaps Ehresmann’s lemma that a locally trivial fibering over a contractible (paracompact$\approx$metric) base is globally trivial. Paracompactness is essential as shown by the simply-connected (indeed contractible) [*Prüfer surface*]{} (cf. works by Prüfer 1922, Radó 1925 [@Rado_1925], Calabi-Rosenlicht 1953, Spivak’s book on Diff. Geom., Vol. I, Appendix, Baillif). Up to degree $\le 4$ the real scheme (or Klein’s types I/II) suffices to encode the rigid-isotopy class as knew Klein 1876 [@Klein_1876_Verlauf], building over Schläfli and Zeuthen’s works. In degree 5, and 6, the same real scheme plus Klein’s type suffices to ensure a rigid-isotopy. This spectacular result is joint work of Nikulin, with the collaboration of Kharlamov building over two pillars, namely: \(1) the Gudkov-Rohlin census solving Hilbert’s 16th problem (for sextics) while revitalizing the earlier Riemannian conceptions of Klein about the complexification and, \(2) the theory of K3 surfaces (Torelli, etc.). The situation changes drastically from degree 7 upwards as shown by the Fiedler-Marin trick using a locking triangle which consists of $3$ Bézout-saturated lines, hence which cannot be crossed by ovals during a rigid-isotopy. Here the basic idea is that if one can associate to a curve in some canonical manner an auxiliary curve called the lock then the distribution of ovals past the lock is rigid-isotopically invariant. A typical example in degree 7 is the lock consisting of 3 lines through the inner ovals of a curve having the scheme $\frac{3}{1}\ell J$ for some $\ell$. When $\ell 2$ the fundamental triangle can separate in different ways the $\ell$ outer ovals, and curves having different splittings past the lock will not be rigid-isotopic. This obstruction to rigid-isotopy requires several ovals, and does not seem suited to curves with few ovals where the rigidity problem looks fully open. It is presently very unclear which sort of scenario is to be expected. For instance it is still undecided for $m\ge 7$ whether curves having only one real circuit are always rigid-isotopic. As informed by Viro, this is at least for odd degrees a (confidential) conjecture due to Rohlin. More generally we posit the following speculation (not that we strongly believe in it, but just as a way to confess our ignorance): \[LARS:conj\] (LARS).—Curves of degree $m$ with less than $DEEP+2$ real branches, where $\Delta(m)=DEEP(m)=[(m+1)/2]$ is the number of components of the deep nest of degree $m$, are always rigid-isotopic provided they have the same real scheme. The basic motivation for this conjecture is that below altitude $r\le \Delta+1$ all schemes are of type II except the deep nest which is of type I (by total reality under a pencil of lines). This follows from Rohlin’s formula (\[Rohlin-formula:thm\]), especially its corollary known as Rohlin’s inequality (\[Rohlin’s-inequality:cor\]), as well as Klein’s congruence $r\equiv_2 g+1$ forcing dividing curves to have their numbers of real circuits $r$ jumping by quanta of 2 units, hence we gain one type II level right above the deep nest. Further the deep nest is known to be rigid by Nuij 1968 [@Nuij_1968]. Apart form those elementary facts, we have little evidence for this “low-altitude rigidity speculation” (LARS), except suspecting that if the assertion is true it will use a geometric flow permitting a degeneration to curves of lower orders after splitting off a line or a conic. For instance given a curve of odd degree it is tempting to look at the flow shortening the length of the unique pseudoline. Orthogonal trajectories of this functional should abut to a curve splitting off a line, which after all is the shortest pseudoline. This could be the basis of a grand inductive process reducing the rigidity of curves with few branches ($r\le \Delta(m)+1$) to that of curves of lower orders. It seems that much can be explored along this line of geometric flows, somewhat reminiscent of say Möbius 1863, Poincaré, Morse, etc., up to Perelman’s proof of Poincaré’s conjecture, except that in our case the dynamics lives merely on a finite-dimensional manifold (the hyperspace of all algebraic curves of some fixed degree). Another source of rigidity comes from the empty scheme, which is rigid. In fact I started to doubt about this issue, until Shustin kindly remembered me the following simple argument. If we have two curves with empty real locus ([*invisible curves*]{} for short), then after choosing equations of the same sign, the linear deformation $(1-t)P+tQ$ will connect both curves while conserving the same sign (provided $0\le t\le 1$), hence producing a path of invisible curves. However it is not a priori (nor a posteriori!?) evident that our path avoids curves with singularities, which could occur in imaginary conjugate pair. Hence the complete proof seems to use the fact that empty curves with singularities form a locus of codimension 2, since there are two conjugate nodes generically. Once rigidity of the empty scheme is known, rigidity of curves with one oval should follow simply by contracting the one oval and letting it then disappear. This gives the basic connection with the former section (which is primarily a remark of Viro). Also as we remarked earlier, the stronger version CCC of the contraction conjecture could accomplish stronger rigidity results like URS, cf. (\[URS:conj\]). [*Insertion*]{} \[03.04.13\].—Conversely if the one-oval scheme (unifolium) is rigid then it suffices to contract the Fermat curve (of even degree) to establish CC, but alas only for this unifolium chamber. Incidentally the validity of CC even for the unifolium scheme is not completely obvious, for taking the linear pencil between such a curve and an empty one leads (after assuming general position w.r.t. the discriminant, i.e. transversality) to a sequence of Morse surgeries a priori much more complex than just the death of the oval. \[10.02.13\] Another question of didactic interest is to study the interplay between Fiedler-Marin locking method and Nikulin’s rigid classification. For sextic schemes of the form $\frac{3}{1}\ell$ we have an obvious lock given by the 3 lines through pairs among the 3 deep inner ovals. Each such line cuts twice the ovals it visits and twice the (nonempty) surrounding oval, hence is Bézout-saturated. Further this fundamental triangle is canonically assigned to the configuration in the sense that the position of 3 points in the insides of the ovals is parametrized by the 3rd symmetric power of a cell which is a contractible space. Hence the distribution of the outer ovals past the deep (Bermudian) triangle is invariant under rigid-isotopy. This adumbrates a strategy toward corrupting Nikulin’s theorem, but in reality the latter rather implies an invariance of this Bermudian distribution of outer ovals for all curves having the same real scheme (and the same type in Klein’s sense). It is therefore of interest to determine this outer distribution past the fundamental triangle for some specific curves as it will imply the same for all isotopic curves. This question is elaborated in Sec.\[Nikulin-corruption:sec\]. [*Insertion*]{} \[03.04.13\].—All this problematic went in decrepitude after an illuminating message of Le Touzé (cf. Sec.\[LeTouze:sec\]) yielding a conceptual explanation of why it is impossible to corrupt Nikulin via Fiedler-Marin. The reason is a simple chromatic law for conics passing through $5=3+2$ points with 3 of them black-colored (situated or defining a triangle), while the location of the 2 remaining points (white colored) past the triangle will determine how the sequence of 5 points distributes on the conic interpolating them. When the 2 white-points belongs to different component of the (black) triangle the distribution will be dichromatic in the sense that the 2 white points are not standing nearby, but separated by black points ($1$ or $2$ depending on the path chosen on the topological circle underlying the conic). Applying this lemma on conics to the above setting, shows that all ovals of a sextic enlarging the scheme $\frac{3}{1}$ are necessarily not separated by the deep triangle, for otherwise we can trace a conic with 4 transitions black-and-white (i.e. inside-vs.-outside of the nonempty oval) with therefore $5\cdot 2+4=14>12=2\cdot 6$ real intersections violating Bézout. A last phenomenon is the rigidity of the deep nest established in Nuij 1968 [@Nuij_1968] or Dubrovin 1983 [@Dubrovin_1983/85]. This seems connected with Ahlfors total reality, since the deep nest is totally real under a pencil of lines. Extrapolating the Nuij-Dubrovin rigidity one can speculate that curves (or schemes) totally real under other pencils are likewise rigid. (A real scheme is [*rigid*]{} if all curves belonging to it are rigid-isotopic.) For instance the scheme of degree 8 consisting of 4 nests of depth 2 is totally really under a pencil of conics and thus could be rigid. Note however that total reality in the abstract sense which is actually (by Ahlfors theorem) synonymous to “type I” is not sufficient to ensure rigidity as exemplified by Marin’s construction of two isotopic $M$-septics, yet not rigid-isotopic (cf. Fig.\[Marin:fig\] below). Therefore if there is any connection between total reality and rigidity it must be a more subtle one. This theme is explored in Sec.\[Nuij-Dubrovin-extended:sec\], but we lack any serious result presently. Problems of rigid-isotopy amount studying the residual components past the discriminant $\disc$ which is a hypersurface of degree $3(m-1)^2$ in the hyperspace of all curves of degree $m$. Call such components, [*chambers*]{} “of” the discriminant. When $m\le 6$ virtually everything is known, e.g. there are precisely 64 chambers of sextics by Nikulin’s theorem built upon the Gudkov-Rohlin census (cf. Fig.\[Gudkov-Table3:fig\]). A myriad of questions occur which are hard to handle systematically. For instance how many chambers in function of $m$? Is there an universal upper bound on the number of chambers residual to a hypersurface in function of its degree and dimension. This seems perhaps accessible via a conjunction of Harnack-Klein-Smith-Thom-Milnor and Jordan-Brouwer separation (plus Phragmen?). Asymptotic results in this sense were studied by Kharlamov-Orevkov. One would like to describe the contiguity graph between chambers where edges label Morse surgeries while crossing the discriminant transversally along a principal stratum of codimension 1 (so-called [*walls*]{}). One can also investigate the topology of the varied chambers. Here one tool is the monodromy representation encoding how ovals permute when the curve is travelled along a loop in the given chamber. This and other issues is the object of next section, which is probably not extremely relevant to our main topic of the Ahlfors map, yet pleasant for its own. It can be left with loss of continuity. The topology of chambers, symmetry, monodromy and transmutation (Kharlamov 1980, Itenberg 1994) ----------------------------------------------------------------------------------------------- \[10.02.13\] To each real scheme is attached a (Zeuthen)-Hilbert (multi-)tree (“forest”) with vertices the ovals and with edges whenever there is a nesting. Since any oval is immediately enveloped in at most one other oval this forest looks like a forest of pines (or a mushroom if you prefer). Hence, it is a directed set branching only downwards. The monodromy acts on this tree respecting its combinatorial structure. So for instance the deep nest is a “naked” tree having only a trunk but no branches. The automorphism group of this trivial tree is trivial, and so must be the monodromy representation. There is no obstruction to the deep-nest chamber having a simply-connected topology, and we can conjecture it to be simply-connected. What about the empty chamber $E$? Define the [*invisible locus*]{} $I$ as the set of all curves with empty real locus. The empty chamber is $E=I-\disc$. A simple argument (detailed in the sequel) shows that $I$ is simply-connected and even contractible. Another simple argument based on Brusotti shows that $\disc \cap I$ is nonempty (when $m\ge 4$). Further $I\cap \disc$ has real codimension 2 in the hyperspace of curves $\mH$ (or in $I$) and so is like a knot (possibly with singularity). In any case, it seems to follow ($I$ being noncompact) that the fundamental group $\pi_1(E)$ is non-trivial despite triviality of the monodromy. Perhaps $E$ is an Eilenberg-MacLane space $K(\pi, 1)$, i.e. aspheric. Can we compute $\pi_1(E)$ as a function of $m$? What about $m=6$ or even $m=4$? Likewise for the deep chamber $D$ it seems hasty to expect simple-connectivity from trivialness of the monodromy. Consider for instance the Gürtelkurve $C_4$ quartic (with 2 nested ovals) and assume it a very symmetric perturbation of 2 transverse ellipses rotated by $90$ degrees. Assume further the existence of say a symmetry $\tau$ about the vertical axis. Since the group $G=PGL(3, \RR)$ is connected we can connected the identity to $\tau$ by a continuous path $c$. This $c$ induces a loop $\gamma$ in the space of quartics from $C_4$ to itself. As $C_4$ belongs to the deep chamber $D$ upon which the group $G$ acts, it makes sense to ask whether $\gamma $ is trivial or not in $\pi_1(D)$. For $c$ we may choose the path in $SO(3)$ given by $180$ degrees gyration about the vectorial line of $\RR^3\ni (x,y,z)$ parallel to the axis of symmetry of $C_4$ (viewed in the affine chart $z=1$). (Warning actually it seems that the line orthogonal to that is required!) Actually choosing any path $c$ from $id$ to $\tau$ in $G=PGL$, its image in $D\subset \mH$ is a loop $\gamma$ likely to be not null-homotopic. Alas the map from $G$ to the orbit of $C_4$ is not really a covering, the argument looks a bit sloppy. A more convenient way to argue is to consider the double cover of the deep chamber $D$ by polarized curves, i.e. with a preferred half of the underlying orthosymmetric Riemann surface. Polarizing amounts specifying a complex orientation à la Rohlin (by taking the oriented boundary of the preferred half w.r.t. the canonical orientation induced by the complex structure). The loop $\gamma$ based at $C_4$ lifts—w.r.t. the polarized cover—to a [*non-closed*]{} path, since $\tau$ exchanges both halves of the complexification. Imagine indeed the symmetric surface underlying the Gürtelkurve as a pretzel of genus 3 with 2 ovals acted upon by an involution ($\tau$) with 4 fixed points then it must necessarily be a rotation by a half-twist about a line in 3-space perforating the ovals in 4 points. It is clear that this argument extends to all deep nests and we obtain the: For any integer $m$ (odd or even do not matter) the chamber of the deep nest (alias deep chamber) is not simply-connected. Any curve in the deep chamber $D$ is of type I (Klein’s orthosymmetry) since there is a totally real pencil of lines. (This is the trivial sense of Ahlfors theorem so-to-speak.) We consider for each plane orthosymmetric curve the two possible ways to paint one half of the curve in black, and call the corresponding painted object a polarized curve (or Riemann surface). If $O$ is the union of all orthosymmetric chambers, we have a natural way to topologize the space $O_2$ of all polarized curves to turn it in a double cover of $O$, the orthosymmetric locus. In particular we have a double cover of the deep chamber $D_2\to D$. Any member of the deep chamber admits a representative $C_m$ with a mirror involution $\tau$ given as $(x,y)\mapsto (-x,y)$ in affine coordinates. It suffices indeed to define $C_m$ as a small perturbation of an union of concentric circles (plus a horizontal line outside them when $m$ is odd). Either by inspecting the Riemann surface or just by noticing that $\tau$ reverses orientation of the ovals (and the pseudoline if $m$ odd) we infer that $\tau$ takes the polarized curve to its opposite (where the other half is preferred). Using connectedness of $G=PGL(3, \RR)$ (more generally $PGL(n, \RR)$ is connected whenever $n$ is odd because then both components of $GL(n, \RR)$ given by the sign of the determinant coalesce together since the identity matrix $I_n$ and its opposite $-I_n$ are homothetic yet of opposite determinants), we infer existence of a path $c$ in the Lie group $G$ connecting $id$ to the symmetry $\tau$. Applying this path to $C_m$ gives a loop $\gamma$ in the deep chamber $D$ based at $C_m$. Lifting this loop to the $O_2$ cover continuously amounts tautologically to apply the path $c$ to the polarized curve, whose end-point $c(1)$ is the opposite polarization as the one we started with. Hence the lift of the loop $\gamma$ is not a loop, and covering theory tell us that $\gamma$ is not null-homotopic. If now $O$ denotes a specific orthosymmetric chamber we also have the double cover $O_2\to O$ (by polarized Riemann surfaces) and it is likely that the above argument extends to all or at least some orthosymmetric chambers having a representative with a mirror symmetry $\tau$. Each such chamber would not be simply-connected. Abstractly an orthosymmetric surface can always be rotated by an half-twist permuting both halves. However it is not evident that this can be done in the plane, at least we know about no general argument. Thus we retract to examples in degree $m=6$, where due to the combined efforts of Harnack-Hilbert-Gudkov-Rohlin we know exactly what happens (cf. the Gudkov table=Fig.\[Gudkov-Table3:fig\]). In each orthosymmetric chamber we look for a symmetric representant under an involution fixing a line. Consulting this table and gathering earlier constructions on a single plate (Fig.\[Symmetry:fig\] below) gives the following symmetric realizations of dividing sextics: $\bullet$ the $M$-schemes of Hilbert and Harnack can both be given a symmetric realization as evidenced by the picture below. In both cases the invariant line intercepts 3 ovals. $\bullet$ for Gudkov’s scheme $\frac{5}{1}5$ the existence of a symmetry is less obvious in Gudkov’s original construction (cf. Fig.\[GudkovCampo-5-15:fig\]). The situation appears more pleasant on Viro’s construction of the latter (compare Fig.\[Viro3-15:fig\]c), but alas since we cannot choose $\alpha=\beta=2$ both in V1 and V2 (please refer to the notation of that figure) we cannot conclude the existence of a global symmetry. Should we conjecture that Gudkov is somehow asymmetric? $\bullet$ $\frac{8}{1}$ admits a symmetric realization as shown by a variant of Hilbert’s method (cf. figure below). Notice also the model with double (dihedral) symmetry. Again 3 ovals are intercepted by the (vertical) axis of symmetry. $\bullet$ $\frac{6}{1}2$ in Hilbert’s realization below severely lacks symmetry. Appealing to Viro’s method (Fig.\[Viro3-15:fig\]c or below) does not aid (V5 twice looks promising yet do not confuse the values of $\alpha,\beta$!). $\bullet$ $\frac{4}{1}4$ along Hilbert’s realization again lacks symmetry, and Viro’s method does not seem to help. $\bullet$ $\frac{2}{1}6$ in Hilbert’s realization is asymmetric. Via Viro’s method this is realized by taking in V1 bottom $(\alpha, \beta)=(1,1)$ and in V2 top $(\alpha, \beta)=(0,4)$. Alas this is highly asymmetric. $\bullet$ $9$ is symmetric under Hilbert’s construction, or a more elementary (Plücker-style) deformation of 3 ellipses, which is even more symmetrical. $\bullet$ $\frac{5}{1}1$ is symmetric as shown below via a primitive perturbation of ellipses à la Plücker-Klein (pre Harnack-Hilbert oscillation trick). Again 3 ovals are intercepted by the symmetry-axis. $\bullet$ $\frac{3}{1}3$ is symmetric by a perturbation of ellipses depicted below. $\bullet$ $\frac{1}{1}5$ is likewise symmetric as shown by the depiction below. $\bullet$ $\frac{4}{1}$ is highly symmetric as shown by a perturbation of ellipses below. $\bullet$ $\frac{2}{1}2$ is symmetric as shown by the perturbation of ellipses below (2 models). This is the exhaustive list of sextics of type I, modulo the omission of the deep nest (which is certainly symmetric). When taking 3 concentric circles one gets the impression of a continuous Lie group of symmetries, yet any perturbed curve will be more rigid (recall finiteness of automorphisms due to Schwarz-Klein-Poincaré-Hurwitz and the bound $84(g-1)$). -1.2cm0 -5pt0 -5pt0 From this investigation it follows the: All orthosymmetric chambers of sextics are not simply-connected except perhaps the $4$ “antechambers” nearby Gudkov’s scheme that is $\frac{5}{1}5$, $\frac{6}{1}2$, $\frac{4}{1}4$, and $\frac{2}{1}6$. In fact all sextic orthosymmetric chambers $O$ have nontrivial polarization covering $O_2\to O$, safe perhaps the $4$ above asymmetrical schemes. \[11.02.13\] Perhaps there is an obstruction for those 4 schemes to admit a symmetry about a line. In case of a Gudkov curve (of type $\frac{5}{1}5$), the symmetry has to leave invariant 3 ovals for the unique nonempty oval has to be preserved while the number of inner and outer ovals are odd. At this stage (or earlier) it is pleasant to visualize the Riemann surface in 3-space. Besides the horizontal orthosymmetry imagine a rotational symmetry under half-twist (180 degrees) leaving 3 ovals invariant while the 8 remaining one are pairwise exchanged. Of course per se this is no obstruction since Hilbert or Harnack have such a symmetry. So the obstruction is necessarily a subtle one if it exists perhaps say à la Arnold-Rohlin. Another idea is to smooth the Gudkov curve along the axis of symmetry and hope to get a septic violating Bézout (Fig.\[Symmetry:fig\]d), but looks improbable. Let us look at Rohlin’s formula $2(\Pi^+ -\Pi^-)=r-k^2$ (see (\[Rohlin-formula:thm\])). Applying it to a Gudkov curve we have $r=11$ and $k^2=9$, hence $(\Pi^+ -\Pi^-)=1$. But on Gudkov’s curve we have $5$ injective pairs of ovals, i.e. $5=\Pi=\Pi^{+}+\Pi^{-}$, and it follows $2\Pi^+=5+1=6$, whence $\Pi^{+}=3$ and $\Pi^{-}=2$. Can it be inferred that there is no symmetry? A priori not, since the 5 inner ovals could have their complex orientations being reversed by the symmetry while one oval is kept invariant. Doing the same calculation for Hilbert’s curve we find $(\Pi^+ -\Pi^-)=1$ and $9=\Pi=\Pi^{+}+\Pi^{-}$, so $2\Pi^+=9+1=10$ and $\Pi^{+}=5$ while $\Pi^{-}=4$. The symmetry could be not a reflection about a line but a rotation about a point. Yet from the projective viewpoint this seems to be equivalent. At any rate Gudkov’s curve in Viro’s realization is anyway not symmetric under a rotation. Let us apply Rohlin’s formula to the $(M-2)$-schemes which are potentially asymmetric, e.g., $\frac{6}{1}2$. Then $2(\Pi^+ -\Pi^-)=9-k^2=0$, hence $\Pi^+ -\Pi^-=0$, but $\Pi^+ +\Pi^-=6$ so that $2\Pi^+=6$, and $\Pi^+=3=\Pi^{-}$. Hence the symmetry cannot exchange the $6$ inner ovals in pairs without fixing any of them. This requires some explanation. Recall we are looking for holomorphic involutions of some plane dividing curve $C_m$ induced by an element of $PGL(3, \RR)$ exchanging both halves. Call such an involution a [*mutation*]{}. If a curve has a mutation then its chamber $O$ has nontrivial polarized covering $O_2\to O$. For sextics (except the deep nest and the unnested curves $1,2,\dots,10$) there is a unique nonempty oval. Distinguished as a such, this must be preserved by the mutation $\tau$, which must reverse its orientation. If not, orientation is preserved and $\tau$ acts as a rotation on this circle. Taking an invariant tube-neighborhood one deduces that both halves are preserved as $\tau$ respects orientation of the surface (hence of this tube), violating the mutating assumption. Supposed fixed a complex orientation of the dividing curve. The mutation reverses orientation of the nonempty oval, and also the complex orientations of all other ovals because $\tau$ preserves orientation but exchanges both halves. Symbolically, we may see this by writing $\tau(\partial C^+)=\partial (\tau C^+)=\partial (C^-)=-\partial (C^+)$. At this stage we are ripe for picturing. Imagine the mutation given by a symmetry about a line (this is probably no loss of generality in projective geometry, as the other candidate namely a rotation about a point fixes the line at infinity). Consider the following schematic pictures (Fig.\[Sym2:fig\]). The first (Fig.a) is not mutating the orientation, hence precluded. Fig.b is mutating the complex orientation, but violates Rohlin’s formula. In fact the mutation condition in case where no inner ovals are invariant imposes an even number of positive injective pairs of ovals and Rohlin’s formula cannot be fulfilled. Hence Rohlin’s formula forbids a mutation without invariant inner oval. Fig.c shows a configuration where both mutation and Rohlin’s formula are satisfied. This is good schematically but bad theoretically, as it fails obstructing mutability of a curve of type $\frac{6}{1}2$. We are not much advanced in our problem. The other figures of Fig.\[Sym2:fig\] show that for each of the other asymmetric types there is always a schematic symmetry compatible with both mutation and Rohlin’s formula. So Rohlin fails to detect any structural asymmetry. Paraphrasing, asymmetry may just be a defect of our models (Hilbert and Viro) yet not an intrinsic property of the chamber. -5pt0 -5pt0 In fact, the above Rohlin’s formula argument only shows for the two $(M-2)$-schemes of type I ($\frac{6}{1}2$ and its mirror), that if they have a mutation the latter must preserves 2 inner ovals. \[12.02.13\] $\bullet$ A smooth dividing plane curve $C_m$ defined over $\RR$ is transmutable if there is a rigid-isotopy switching its half (called a transmutation). $\bullet$ A mutation is a linear automorphism $\tau \in G=PGL(3,\RR)$ of the curve $C_m$ permuting both halves of the curve, say in this case that the curve is mutable. Since the group $PGL(3,\RR)$ is connected any mutation induces (non canonically) a transmutation. Indeed choose a path $c$ in $G$ joining $id$ to $\tau$ and its operation upon $C_m$ defines a loop in the corresponding chamber of the space of curves (past the discriminant) which is a transmutation. It is not essential that $\tau$ has order 2, but then speak of a $2$-mutation. So any $2$-mutable (dividing) curve is mutable, and in turn transmutable. The converses looks a priori quite improbable. Are all dividing plane curves transmutable? or even mutable, or $2$-mutable after some rigid-isotopy? The question looks of interest because a non transmutable curve would have a preferred half (privileged so-to-speak) which looks a bit against the flavor of Galois-theory and French revolution “égalité, fraternité, etc.”. A mutation (like any self map of $\RR P^2$) has a (real) fixed point (e.g. via Lefschetz fixed point theorem using homology over ${\Bbb Q}$), and any $2$-mutation is a mirror about a line fixing also a real isolated point, as inferred from linear algebra (existence of real eigenvalues for an endomorphism of a real vector space of odd dimension). Perhaps the above questions can be handled via C. Segre’s classification of real structures on projective spaces, especially the fact that the plane ${\Bbb P}^2$ has a unique real structure, but looks unlikely as the curves are not taken into account. One way to approach the problem in general would be to look at the action of $G=PGL(3, \RR)$ on the chamber past the discriminant containing a dividing curve. Existence of a mutation in each such chamber amounts this action being never free, i.e. with nontrivial isotropy subgroup $G_{C}$ at some suitable curve $C=C_m$. This is not enough for the automorphism in question needs not permute both halves. Omitting this difficulty, there would be a free Lie group operating, hence an induced foliation of the chamber by leaves of dimension 8 (=$\dim G$). Alas such a foliation also exists when the action is only locally-free, i.e. discrete isotropy are allowed. It would be nice to know if all chambers of the discriminant (not only the orthosymmetric chambers) contains a curve with (linear) automorphism in themselves. For orthosymmetric chambers we would further like to know if there is such an automorphism permuting the halves of the curve (i.e. a mutation). In a remarkable article extending earlier work by Kharlamov, Itenberg 199X [@Itenberg_199X-monodromy-deg-6] is able to compute the monodromy groups of each chamber of sextics. Extracting from his tabulation, only the type I cases gives the: [(Kharlamov, Itenberg)]{} The monodromy groups of smooth sextics of dividing type are given by the following list (where $\triv$ is the trivial group, $S_n$ the symmetric group on $n$ letters, and $D_n$ the dihedral group): $\bullet$ $\frac{9}{1}1 \rightsquigarrow \ZZ_2 $, $\frac{5}{1}5 \rightsquigarrow \triv $, $\frac{1}{1}9 \rightsquigarrow S_3 $, $\bullet$ $\frac{8}{1} \rightsquigarrow D_4 $, $\frac{6}{1}2 \rightsquigarrow \triv $, $\frac{4}{1}4 \rightsquigarrow \triv $, $\frac{2}{1}6 \rightsquigarrow \ZZ_2 $, $9 \rightsquigarrow S_9 $, $\bullet$ $\frac{5}{1}1 \rightsquigarrow \ZZ_2 $, $\frac{3}{1}3 \rightsquigarrow \ZZ_2 $ $\frac{1}{1}5 \rightsquigarrow D_5 $, $\bullet$ $\frac{4}{1} \rightsquigarrow S_3 $, $\frac{2}{1}2 \rightsquigarrow \ZZ_2 \times \ZZ_2 $, $\bullet$ $(1,1,1) \rightsquigarrow \triv$. It is interesting to compare this result with our picture Fig.\[Symmetry:fig\], as sometimes the whole monodromy group can be realized by rigid projective motions. Besides, it seems interesting to compare this monodromy of ovals to the monodromy upon the halves. Albeit the latter viewpoint is less rich in general (being only a representation on the group with 2 elements) it is sometimes of complementary nature in detecting non-triviality of the fundamental group of the fixed chamber. For the moment, our halves-monodromy is only more sensitive in the deep-nest case. (Question: does the $\pi_1$ of the deep chamber reduces to $\ZZ_2$?) Finally, note that the oval-monodromy is also fairly small for the 4 exceptional schemes, asymmetric in Hilbert’s (or Viro’s) realization (again Fig.\[Symmetry:fig\]). Hence both methods oval-monodromy and half-monodromy (at least via rigid symmetries) fails to detect nontrivial elements in $\pi_1$ of the corresponding chamber for the schemes of Gudkov $\frac{5}{1}5$, of left-Rohlin $\frac{6}{1}2$ and $\frac{4}{1}4$. Can we extrapolate that those chambers are simply-connected? If yes then those 3 curves are not transmutable, hence not mutable and therefore structurally asymmetric (i.e. there is no model invariant under a mirror). More factually, Itenberg’s calculation prevents a curve of type $\frac{6}{1}2$ to accept a mirror like Fig.\[Sym2:fig\]c. Indeed otherwise if $\tau$ is such a mirror, it suffices to take a path in $PGL(3, \RR)$ joining the identity to this $\tau$ to get a loop in the space of curves with non trivial monodromy. Likewise no curve of type $\frac{4}{1}4$ can accept a mirror like Figs.\[Sym2:fig\]d,e, and no curves of the Gudkov type $\frac{5}{1}5$ can accept a mirror like Figs.\[Sym2:fig\]i. Hence Itenberg’s calculation implies the following answer to one of our basic question: None of the $3$ monodromically-trivial dividing curves—as listed by Itenberg, i.e. Gudkov’s, the left Rohlin curve $\frac{6}{1}2$, and that of type $\frac{4}{1}4$—can support a mirror. A mirror is a linear involution of $PGL(3,\RR)$ which fixes a line (plus a point at $\infty$). By Bézout, the fixed line can intercept at most 3 ovals which are then invariant. Examining the following Fig.\[Sym3:fig\] shows that there is always some pair of ovals permuted by the mirror. -5pt0 -5pt0 Alas, an oval can be invariant under the mirror without having to intersect the fixed line, say by running to infinity. Then one can try to argue with the line at infinity and Bézout. A better way to argue is that the nonempty oval of the sextic has to be invariant under the mirror, hence at most 2 inner ovals can be left invariant under the mirror (else Bézout corrupted). So we infer existence of inner ovals permuted under the mirror $\tau$. Taking a path from $id $ to $\tau $ yields a loop in the chamber whose monodromy is nontrivial. This contradicts Itenberg’s calculation of the monodromy. In particular this shows existence of curves without a $2$-mutation, since a $2$-mutation is certainly a mirror. However it is not clear if our 3 curves lack a mutation. Given a mutation (i.e. a linear automorphism) permuting the halves it is not a priori of order 2. However it has finite order (by Klein-Poincaré-Hurwitz finiteness of the automorphism group of closed Riemann surfaces of genus $\ge 2$). Since $\tau$ permutes the halves its order must be even say $n=2 e$. Hence $\tau^e$ is an involution, and a mutation if $e$ is odd. Alas in the other case, i.e. when $n$ is divisible by $4$, we cannot say much. Assume given a mutation $\tau$ on a Gudkov curve, then it has to preserve the unique nonempty oval, and the inner and outer ovals have to be respected. Further $\tau$ (being a mutation) it has to reverse the complex orientation. As we may assume the order $n$ of $\tau$ divisible by $4$, it should follow that $\tau$ permutes the ovals along a 4-cycle, etc. Via some group theory there is presumably an obstruction to mutate the Gudkov curve? Alternatively assume there is a mutation on some Gudkov curve $C_6$ then it will permute the ovals while preserving the nonempty one and the inner/outer ovals subdivision. Further it must reverse the complex orientation. If any permutation of the ovals is detected (which is precluded by the Kharlamov-Itenberg’s calculation of the monodromy as being trivial), we are finished. Assume so that the mutation preserves all ovals. Since it has to reverse their orientations, the mutation has 2 fixed points on each of them (an orientation reversing transformation of the circle has 2 fixed points e.g. via Lefschetz or via covering theory). But globally our mutation is of finite order (since it induces an automorphism of the Riemann surface of genus $10\ge 2$), and any element of finite order in $PGL(3, \RR)$ preserves a line. So we get a line intersecting the $C_6$ in 22 points, overwhelming Bézout. As another variant, once our mutation is known to have 2 fixed points on each ovals we have $2(g+1)$ of them, which is the maximum permitted by Lefschetz trace formula. We would like to conclude that $\tau $ is the hyperelliptic involution, which cannot exist on a smooth $C_6$ (which is only $5$-gonal). Let us clarify this argument with the: [A curve is [*antidromic*]{} if it monodromy group is trivial. A [*symmetry*]{} of a plane real curve is a linear automorphism of the plane (i.e. an element of $PGL(3,\RR)$) preserving globally the curve.]{} [(1)]{} Any symmetry of an antidromic curve must leave each oval invariant. Hence all the $3$ antidromic sextics listed by Itenberg can only admit a mutation preserving all the ovals. [(2)]{} In particular all the $3$ antidromic dividing sextics of Itenberg (the deep nest being excluded) lack a mutation. So Gudkov’s curve, and the left-wing Rohlin curve $\frac{6}{1}2$ and the dividing curve $\frac{4}{1}4$ are asymmetric at least under a mutation (i.e. they cannot mutate). \(1) If there is a symmetry $\tau$ permuting somehow the ovals, then the path in $PGL(3,\RR)$ connecting $id$ to $\tau$ induces a loop in the space of curves with nontrivial monodromy (namely the ovals-permutation induced by $\tau$). (2) As to the second assertion, our oval-preserving mutation must necessarily invert the orientation of all ovals since it exchanges both halves. (Recall that a mutation reverses the complex orientation in the sense of Rohlin, since it preserves the orientation induced by the complex structure while exchanging both halves of the dividing curve.) Since a sense-reversing transformation of the circle has 2 fixed points as follows from Lefschetz’s fixed-point formula. (Indeed the Lefschetz trace number is $(+1)-(-1)=+2$ so there is a fixed point and removing it one obtains an orientation reversing homeomorphism of the line which has another fixed point, e.g. by applying Bolzano to the graph of this continuous decreasing function.) (Is it true that any sense-reversing transformation of the circle is an involution? We do not need this anyway!) Since Itenberg’s antidromic curves have $r=11$ or $r=9$ ovals, we get 22 or 18 fixed points created. Next we have: A linear automorphism in $PGL(3, \RR)$ that fixes $4$ (or more) points of $\RR P^2$ fixes either a line plus an isolated point or is the identity. This follows by looking at the 4 corresponding eigen-lines in $\RR^3$, two of which have to correspond to the same eigenvalue (pigeonhole principle), and so there is the required fixed projective line. The third eigenvalue left (necessarily real) gives a third eigen-line and the announced alternative follows depending on whether this 3rd eigenvalue differs or coincides with the former double eigenvalue. It follows that among our 22 or 18 many fixed-points (at least so many less one) are aligned, but this corrupts Bézout. A priori it is much harder to detect an obstruction to transmute the Gudkov curve (or its 2 antidromic cousins), and likewise hard to show that it may be transmuted. Assume there is a transmutation. Then since by Kharlamov-Itenberg’s lemma the curve is antidromic the induced permutation of ovals is trivial, but the complex orientation is reversed. Perhaps some obstruction can be deduced from this... \[15.02.13\] Two days ago, Kharlamov informed me that the Gudkov chamber has fundamental group $\ZZ_2$ compare his letter (dated \[13.02.13\]) in Sec.\[e-mail-Viro:sec\]. So I presume that the Gudkov curve can be transmuted, and that the isomorphism $\pi_1\approx \ZZ_2$ may be realized as the monodromy acting upon halves. Kharlamov’s messages also emphasize the issue that the $M$-curves case is somewhat easier than the other cases. Hence while the (oval)-monodromies are completely calculated by Itenberg, it may be the case that the determination of the fundamental group of each chamber is somewhat harder to obtain. Some naive questions are as follows. We presume that the deep-nest chamber has $\pi_1=\ZZ_2$. Are the (fundamental) of chambers always finite? This would follow (theorem of Myers, Synge, Hopf, etc.) if there is a complete metric of positive curvature on the hyperspace of curves (which is the case) yet the natural elliptic metric is not complete when restricted to the chambers. Further since Gudkov’s chamber is not simply-connected, any Gudkov curve is presumably transmutable (this amounts to say that the Kharlamov isomorphism $\pi_1=\ZZ_2$ is realized by the monodromy of halves). If so is the case, perhaps that even all dividing plane curves are transmutable. Isotopic rigidity of the empty scheme: connectedness of invisible curves {#rigidity-empty-scheme-via-dyna:sec} ------------------------------------------------------------------------ \[08.04.13\] We now come to a tortuous revelation of a basic truth, namely the fact that the empty chamber is connected, i.e. any 2 real smooth plane curves with empty real locus can be connected by a path of similar curves (avoiding the discriminant). This problematic covers no less than $6$ sections up to Sec.\[invisible-discriminant-codim-2:sec\], where it is elucidated that the portion of the discriminant inside the locus of empty(=invisible) real curves has codimension 2 hence cannot effect a disconnection. Pivotal in this search was a kind letter by Shustin explaining the linear homotopy argument between empty curves, which shows that the empty locus (including possibly singular curves) is connected, and actually much more like being contractible. So we warn the reader that those six sections are far from a geodesic toward the goal, but we had not the courage to censure any bit of our poorly organized material as it often ramifies toward considerations of independent interest. \[23.01.13\] In fact to be honest I realize that even the foundation of our reasoning (in the previous section[^51]) is not completely sound, namely the following fact: [(Folklore??? is it really true? If yes where is it proved?)]{} Any two empty curves are rigid-isotopic. [*Insertion*]{} \[08.04.13\].—Folklore probably! True, certainly, compare Shustin’s letter, but also the codimension 2 lemma (poorly) established in our Lemma \[invisible-discriminant-codim-2:lem\]. Where it is proved? We still lack a detailed reference, but apparently the fact is so trivial that nobody took care writing down a complete proof. It would be of interest to make a deeper historical search of who knew first this simple result. Possible candidates: Schläfli 1863 [@Schlaefli_1863], Cayley, Klein 1873–1925, C. Segre, Hilbert ca. 1891, Berzolari 1906 [@Berzolari_1906], Severi e.g. 1921 [@Severi_1921-Vorlesungen-u-alg.-Geom-BUCH], Brusotti 1921 or earlier, Petrovskii 1933, etc. (Sometimes instead of saying empty curves we shall say invisible curve, when the real locus $C_m({\Bbb R})$ is empty.) I was sure this to be known (but completely forgot where I read this in case I am remembering well!!!) At first sight this looks trivial but is not. One could imagine the empty chamber to be starlike or even convex, but this is not even evident. Somehow invisible curves could be like the immersed half of an iceberg, hence connected, while the visible part of the iceberg may consist of several islands (peaks) corresponding to the menagerie of chambers past the discriminant, well-known in Hilbert’s 16th. Alas even if this metaphor ought to contain some truth, it is easy to construct an iceberg with disconnected immersed locus. Take a letter “E”, rotate it by $-\pi/2$ to get the symbol ($\Pi\!\!\Pi$) considered as a tripod with 3 legs immersed in the water. To go in 3D, just take the body of revolution of this symbol to get an iceberg with 2 immersed components[^52]. After having being puzzled for while one might suspect this to be a result à la Hilbert, that a form not representing zero is something like a sum of squares...(not clear). \[24.01.13\] More geometrically, one can look at the distance between the complex locus of a real curve $C_m({\Bbb C})$ and the real plane ${\Bbb R}P^2$. It is natural to work with the Fubini-Study metric on ${\Bbb C}P^2$. Then look at “the” point of the real plane closest to $C_m({\Bbb C})$, which must be generically unique. If we let $E$ be the empty chamber (a priori not connected), we obtain so a random-map $\pi\colon E \to {\Bbb R}P^2$ taking an invisible curve to its closest “projection” in ${\Bbb R}P^2$. Up to removing some subset of $E$, one could arrange $\pi$ to be single-valued, and one would check that $\pi$ is akin to a fibration with connected fibres. The connectedness of $E$ could follow. It is also tempting to imagine a flow driving invisible curves to solitary nodes. This would be just the gradient flow of the functional distance to the real locus ${\Bbb R}P^2$. The corresponding trajectories of steepest descent could converge to a curve with a unique solitary node (generic case). At the level of the Riemann surface this isotopy (given by the path of trajectory) really amounts to the contraction of an anti-oval toward a solitary node. So this is just a special case of Klein’s Ansatz (\[Klein-1876:conj-noch-entwicklungsfahig\]), that a nondividing curve can acquire a novel solitary node (by a large deformation). (Of course recall this to be erroneous by Shustin 1985 [@Shustin_1985], yet it is perhaps true for empty curves). In fact: Klein’s Ansatz is trivially true for empty curves (just by general position and surgery of the real locus). Indeed, given any invisible curve $C_m$, take any pencil through it passing through a visible curve $D_m$ (with nonempty real locus), then making this line transverse to the discriminant we get Morse surgeries the first of which must necessarily be a solitary node formation. [*Insertion*]{} \[08.04.13\] Maybe one can object again this proof, by arguing that the first contact with the discriminant could be through a pair of imaginary nodes. Paraphrasing a bit we could imagine that travelling along the pencil spanned by $C_m,D_m$ we hit the discriminant but then fall again in the empty chamber. Presumably both scenarios can be avoided if we know that the invisible discriminant has real codimension 2 (as we shall see in the sequel, cf. Lemma \[invisible-discriminant-codim-2:lem\].) Our flow would precisely do this contraction yet in some more organic(ized) fashion (i.e. no choices). Yet notice that we could make the above pencil argument by choosing once for all some visible curve $D_m$, while driving all the invisible curves $C_m$ along the line spanned by $C_m$ and $D_m$. If this does not work look at the flow (discussed above). Optimistically, this method may suffice to establish connectedness of the empty locus $E \subset \vert m H\vert$. One may wonder if a variant of the argument could not also establish Viro’s open problem on the connectedness of the pseudoline locus $P\subset \vert m H \vert$ when $m$ is odd. (Added \[08.04.13\].—It seems that this conjecture really goes back to Rohlin, if we interpreted correctly a letter of Viro in Sec.\[e-mail-Viro:sec\], dated \[26.01.13\].) Return yet to the case of the empty locus $E$ (non void only for curves of odd degree). By what could it be disconnected? A curve in the discriminant $\frak D$ may well have two imaginary conjugate singularities. But this locus has codimension 2, so it cannot disconnect $E$. Does this suffices to prove connectedness of $E$? Probably not as a priori it may have several components lying “far apart”. Another idea is to fix an invisible conic, e.g. the “canonical” one $E_0\colon x_0^2+x_1^2+x_2^2=0$ (on which conj acts like an antipodal map). To each point of the plane one can attach the apparent contour of this ellipse (polar lines) as seen from the given point to get a group of two points on this ellipse which is a Riemann sphere. (This is the most synthetic way to establish the well-known isomorphism between ${\Bbb C}P^2$ and the second symmetric power of ${\Bbb C}P^1$.) Points of $E_0$ correspond to groups of superposed two points, while real points in ${\Bbb R}P^2$ maps to antipodal pair (invariant under conj when seen as a pair). Define a function $\rho$ on ${\Bbb C}P^2$ which given a pair measures the distance on the round sphere $S^2$ between the corresponding 2 points. This is equal to $\pi=3.14\dots$ on ${\Bbb R}P^2$ and vanishes on $E_0\colon x_0^2+x_1^2+x_2^2=0$. Now given any compact sublocus of ${\Bbb C}P^2$ (in particular an invisible curve $C_m$) one can look at the maximum of $\rho$ on $C_m({\Bbb C})$, which is $<\pi$. This gives the functional $$\theta:=\max \rho\colon E \to [0,\pi[,$$ whose ascending gradient lines should converge to solitary nodal curves (perhaps with several such nodes). Note that the ground (invisible) ellipse $k \cdot E_0$ (counted $k=m/2$ times) is the unique absolute minimum of this functional. This $\theta$ is quite likely to be a Morse function (or a slight generalization thereof with Monkey saddles, etc.), yet the critical points (causing annoying stagnation of the dynamics) ought to be isolated (codimension 2 suffices), hence not affecting the connectivity of $E$. By construction our flow tends to make an invisible curve more “visible” by pushing it progressively closer to the real plane. In the limit we expect something visible having solitary nodes and generically just one should emerge. So upon excising from $E$ a small set (of codimension 2, since at least two nodes is bad) we find a (dense) subregion $E^{\ast}$ which maps ${\Bbb R}P^2$ by assigning the unique solitary node of the limit of $C_m \in E^{\ast}$ under the flow at time $\theta=\pi$ (using $\theta$ as time parameter as usual for gradient flow). Now the fibre of this map $E^{\ast}\to{\Bbb R}P^2$ is the same as the [*bassin d’attraction*]{} of the flow which is cone-like formed by several trajectories abutting to the same solitary node. So this cone is connected by the end point, and connectivity of $E^{\ast}$ (hence $E$) should follow. Still, the main difficulty is (as usual) to show that the $\theta$-functional lacks a local maximum preventing convergence to a visible curve. So given any invisible curve one should produce a small perturbation with larger $\theta$. This is probably not too hopeless. Naively one could perturb $C_m$ inside the pencil spanned by $k E_0$ and $C_m$. Since $k E_0$ is the most invisible curve a deformation along it should decrease $\theta$, while one \[deformation\] away \[of\] it should increment $\theta$. Of course there is some objection to this, since in a projective (real) line (a circle) it is never clear what means “along” and “away”. All this is somewhat confuse and unconvincing. Perhaps also there is a much more elementary argument without gradient lines. As we said this could involve deforming all empty curves along some fixed visible curve. But which direction of retraction should we choose in the corresponding pencil? Since $m=2k$ is even, $\vert m H \vert$ is of dimension $\binom{m+2}{2}-1=(k+1)(2k+1)-1=2k^2+3k=k(2k+3)=:N$. So when $k$ is odd there must be another singular point in the foliation induced by the [*faisceau*]{} (sheaf, bundle) of all lines through $D_m$ (by Poincaré-Hopf). Can we orient this foliation? No because when $N=2$, we have a Möbius strip after puncturing the basepoint of the pencil. The situation would be somewhat simpler if we could find in the hyperspace of curves $\vert m H \vert \approx {\Bbb R}P^N$ a hyperplane $H\ni C_m$ avoiding the empty locus $E$. I do not know whether this is possible? Then there would be a nice way to retract the whole complement of $H$ toward the point $C_m$ in some canonical way. In particular all points of $E$ (invisible curve) would mark a first impact on the real locus ${\Bbb R}P^2$. Alas it is not even obvious that connexity of $E$ follows. \[23.01.13\] Recall that the related question for odd degrees is still an open problem. [(Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical p.199])]{}.—[Are all non-singular real projective curves of a given [*odd degree*]{} with connected set of real points [*rigid-isotopic*]{} to each other?]{} The emphasis is Viro, and may suggest that without odd degree the assertion is known to be false!? If so then our (OOPS) conjecture (\[OOPS:one-oval-rigid-isotopic:conj\]) would be oops! [*Insertion*]{} \[08.04.13\].—The answer were given in Viro’s letter dated \[26.01.13\] in Sec.\[e-mail-Viro:sec\], and may be summarized as follows. First in the even degree case, the problem of rigidity of the curve with a unique oval is still open (but probably reducible to the contraction conjecture (\[Itenberg-Viro-contraction:conj\])). Second, as we already said, the question ascribed above to Viro (2008) truly goes back to Rohlin (unpublished as far as we know). Rigidity of the empty scheme (Shustin’s letter) ----------------------------------------------- \[27.01.13\] This section treats the following desideratum: given two empty (plane) curves (hence of even degree), it is always possible to find a path of curves linking them while avoiding the discriminant. In fact, I read about this fact a long time ago (ca. 2000) but could not remember from which source. Recently (24.01.13) failing to recover the source, we started to doubt about the truth of this assertion. Very kindly Shustin communicated us the simple (forgotten) argument giving a positive answer. Alas we have little idea of who proved this first. Shustin’s proof looks at first sight extremely trivial, but on more mature thinking the story looks a bit more tricky than expected. \[empty-chamber-connected-Shustin:lem\] Any two empty curves are rigid-isotopic. (Courtesy of Eugenii Shustin \[26.01.13\]) The chamber of empty curves of a given (even) degree is connected. Two such curves are defined by homogeneous polynomials $P,Q$, supposed (w.l.o.g.) positive for all real variables not all simultaneously zero. The linear homotopy $(1-t)P+tQ$, $0\le t\le 1$ gives then a path in the chamber of empty curves. Indeed the linear path between two positive numbers consists of positive numbers, and so all intermediate curves of this homotopy are empty curves. A priori one can imagine that some intermediate curve of this homotopy (while staying empty) crosses the discriminant by acquiring a conjugate pair of nodes. (This eventuality was not mentioned in Shustin’s letter, but we think that is is a slight obstacle to the argument. However it seems to be not fatal as we shall discuss at length.) Maybe the above argument should be supplemented by an examination of the partial derivatives $\frac{\partial P}{\partial x_i}$. Smoothness of a curve amounts the 3 partial derivatives (of the defining equation) lacking a common zero. Recall Euler’s relation $m F=\sum_i \frac{\partial F}{\partial x_i} x_i$ valid for any homogenous polynomial (form) of degree $m$. (This requires only to be checked on monomials $x_0^i x_1^j x_2^k$ such that $i+j+k=m$.) Combining this we infer that if there is some time $t\in [0,1]$ such that $(1-t)P+tQ$ is singular, then all its 3 partials vanishes simultaneously at some point $(x_0,x_1,x_2)$, which by Euler’s relation would be also a zero of $(1-t)P+tQ$. This violates however the emptiness of this curve. However for this argument to hold good it is essential for the point $(x_0,x_1,x_2)$ to be real, which is however not the case a priori. Another more qualitative argument would be to first perturb slightly $P$ and $Q$ so that the linear pencil spanned by them is transverse to the discriminant. In that case if some member of the pencil acquires a singularity it will be a simple node, which consequently must be real. This violates emptiness of the intermediate curves $(1-t)P+tQ$, $0\le t\le 1$. In summary the lemma looks true, yet not in the strong sense that any two empty smooth curves are linked by a linear homotopy consisting only of smooth curves. This strong form amounts to convexity of the empty chamber, and not just connectedness. Convexity is perhaps wrong, as the curve may traverse a pair of conjugate nodes during the linear deformation. Can we corrupt convexity of the empty chamber? We think yes as follows. For the empty chamber to be nonempty, assume the degree $m$ even. Suppose given an empty curve $C_m$ with a pair of conjugate nodes. We shall later explain how to construct this, but one can already imagine in 3-space (or in the ether) a diasymmetric Riemann surface acted upon antipodically without fixed point, on which two symmetric vanishing cycles are contracted. Now through the given curve $C_m$ (considered as a point in the hyperspace of $m$-tics), trace a little rectilinear segment transverse to the discriminant. Both extremities of the segment will be smooth empty curves, yet the linear homotopy connecting them hits the discriminant (hence fails to be entirely in the empty chamber). This argument works fine provided the linear homotopy coincides with our little segment, instead of being actually the “long” residual pieces of it in the pencil (which is a circle). When $m=2$, a conic can only have one node, when degenerating to a pair of lines. As we require $2$ conjugate nodes, let us look at quartics. Start with a pair of empty conics with transverse complexifications intersecting in 4 points $p,p^\sigma, q, q^\sigma$ (cf. Fig.\[ShustinEmpty:fig\]a for a schematic view). By Brusotti 1921 [@Brusotti_1921] (or earlier workers like Plücker, Klein, etc.) we can smooth $q,q^\sigma$ away to create an empty quartic $\Gamma_4$ with 2 nodes nearby $p,p^\sigma$. The corresponding (singular) Riemann surface (complex locus $\Gamma_4({\Bbb C})$) is visualized as a genus 3 surface (acted upon by antipody) with 2 handles shrunk to points $p, p^\sigma$ exchanged by conjugation (Fig.\[ShustinEmpty:fig\]b). We can also imagine the structure of the discriminant near $\Gamma_4$ as being a single nappe (Fig.\[ShustinEmpty:fig\]a). Take a little rectilinear segment transverse to this nappe. Both extremities of this segment are smooth curves $C_4$, $D_4$ which are empty. This gives our counterexample provided the linear homotopy $(1-t) C_4 + t D_4$, $t\in [0,1]$ visits $\Gamma_4$. -5pt0 -5pt0 It is worth trying to clarify the above proof. Given two forms $P,Q$ of degree $m$, define the [*linear homotopy*]{} as the path of forms $(1-t) P+t Q$ where $t\in [0,1]$. Denote it symbolically $P\to Q$. If we look at curves (i.e. homothety classes of forms) then between any two curves $C, D$ of degree $m$ there is a pencil of curves $\lambda C +\mu D$ which is the line $\overline{CD}$ through both points seen in the hyperspace of curves. A little drawing (Fig.\[ShustinEmpty:fig\]c) shows that if $P, Q$ represents $C,D$ resp., then the linear homotopy $P\to Q$ projects to one piece of the line $\overline{PQ}$, while the linear homotopy $P\to -Q$ describes the other road of access in the circle $\overline{CD}$. Shustin’s argument shows that if two forms $P, Q$ not representing zero have the same sign then the linear homotopy $P\to Q$ consists of forms not representing zero. (Recall that the sign of even degree forms is well-defined, because $F(\lambda x_0, \dots, \lambda x_n)=\lambda^m F( x_0, \dots, x_n)$.) However it does not say that if $P,Q$ represents nonsingular curves, then so are all members of the linear homotopy $P\to Q$. In our example with $C_4$ and $D_4$ we could argue that the corresponding polynomials $P,Q$ are very near (by Brusotti’s construction) so of the same sign, and further that the linear homotopy $P\to Q$ really passes through $R$ the defining equation of $\Gamma_4$. In that case Shustin’s argument would be in slight jeopardy. Does our counter-argument work? We think yes we start from $R$ a form defining $\Gamma_4$, and perturb slightly the coefficients of $R$ by Brusotti to get the polynomials $P$ and $Q$ so that the linear homotopy $P\to Q$ passes through $R$. Note that $R$ has some well-defined sign on ${\Bbb R}P^2$, and by smallness of the perturbation $P$ and $Q$ have the same sign as $R$. Thus even when $P$ and $Q$ do have the same sign we are not ensured that the linear homotopy $P\to Q$ transits only through nonsingular curves. In conclusion given two smooth[^53] empty curves $C,D$ and choose representing forms $P,Q$ (resp.) of the same sign, then the projection of the linear homotopy $P\to Q$ in the space of curves $\vert m H\vert$ correspond to empty curves yet not necessarily smooth. [*Related literature for Shustin’s argument.*]{} Maybe Wilson, Shustin ICM, etc... Of course even if the above Brusotti-type construction is correct, it does not prove that the locus of empty curves is disconnected, but merely that the proof via linear homotopies is insufficient. One possible critique to our argument is that because $\Gamma_4$ has $2$ nodes it is not on a principal stratum[^54] (wall) of the discriminant of codimension 1. Yet in reality this is a pair of conjugate points so really one point in the sense of Grothendieck’s schemes (to which we are from adhering). Perhaps our segment not transverse but rather tangent to the nappe of $\frak D$. However this looks not so realist by construction. The key issue is to decide whether our binodal curve $\Gamma_4$ is a smooth point of the discriminant, which looks likely if we regard only real curves. Of course in the complexified discriminant $\frak D({\Bbb C})$ there is two nappes of passing through the binodal curves $\Gamma$. If our reasoning is correct, we see that the problem of the connectedness of the empty chamber is not settled by the linear homotopy argument. Perhaps the empty locus is even disconnected? How to approach the problem? Let $m$ be some fixed even degree. Consider $\vert m H \vert$ the space of all real curves, and $\frak D$ be the discriminant parametrizing real singular curves. Denote by $I$ the [*invisible locus*]{}, consisting of all empty curves, and let $E$ be the [*empty locus*]{} consisting of all smooth empty curves. Obviously $E\subset I$. In fact $E=I - \frak D$. The linear homotopy argument shows that $I$ is connected (even convex in some sense), but a priori the hypersurface $\frak D$ could split $I$ in several pieces. In general a hypersurface does not need splitting a manifold (consider e.g. a (pseudo)line in ${\Bbb R}P^2$ or a meridian/parallel in a torus). In our case $\partial I$ the boundary (or frontier) of $I$ consists primarily of solitary nodal curves (principal strata) and further subsequent lower strata. Hence clearly, $\partial I \subset \frak D$. But what about $\frak D \cap \overline{I}$? A priori this does not reduce to $\partial I$. One can imagine additional nappes of $\frak D$ moving inside $I$, or even that $\frak D$ contains spheroids (or other closed manifolds) not directly connected to the boundary $\partial I$ (cf. Fig.\[ShustinEmpty:fig\]d). Extending our Brusotti argument to $k=m/2$ pairs of empty conics having transverse complexifications shows that for any even integer $m\ge 4$, there is an empty curve $\Gamma_m$ of degree $m$ with 2 conjugate nodes $p,p^\sigma$. Such a curve belongs to $\frak D \cap {I}$, i.e. is both singular and invisible. Can such a curve be connected to $\partial I$ by a path in $\frak D$, and hence to the “visible world” $V:=\vert mH \vert- I$. The answer would be yes if $\frak D$ is connected. The latter is a real algebraic hypersurface, a priori with several components. Looking at the (singular) Riemann surface (Fig.\[ShustinEmpty:fig\]b), we can try to contract algebraically the anti-oval winding around the middle hole toward a solitary node. This would give a path as required. This sort of problem was already discussed at length in another strangulation section, yet we lack a serious procedure. Assume now the opposite, i.e., $\frak D$ disconnected in the sense of having a component inside $I$. The linear homotopy argument shows that $I$ is convex in the sense that between any two of its points the projective line joining them has one half contained in $I$. It follows that $I$ is a contractible manifold! (Warning since Whitehead 1936 do not draw hastily that $I$ is homeomorphic to ${\Bbb R}^N$). Such manifolds (more generally those which are simply-connected, or even under weaker homological condition) are subsumed to Jordan-Brouwer separation. Under our supposition that $\frak D$ has some component inside $I$, it would result a separation of $I$ by $\frak D$. In fact quite independently of this supposition even if $\frak D$ is connected there is still a separation. Of course we need some lemma extending Jordan separation caused by a manifold, to a separation caused by a stratified variety (not necessarily smooth). This looks true either by Anschauung (cf. Fig.\[Shustin2:fig\]d) or by a reduction to the manifold case by selecting adequately strata as to manufacture first a topological (but piecewise smooth) manifold out of the strata (Fig.\[Shustin2:fig\]e). Of course this should rest upon Brusotti’s description of the discriminant. Whatever the method used we get a morcellation of $I$ by the discriminant. Quite ironically the linear homotopy argument (reminded by Shustin) seems to do exactly the opposite job than its primary intention. More precisely it implies that $I$ is contractible, hence subsumed to Jordan-Brouwer separation. On the other hand our Brusotti-type construction shows that $\frak D$ appears inside $I$ (provided $m\ge 4$), hence must divide the invisible locus. Modulo details, we believe to have proved the following: [(Revolutionary if true, but false!)]{}.—For any even integer $m\ge 4$ the empty smooth locus (past the discriminant) is disconnected. If true this would wash up several misconceptions in the literature, e.g. that the rigid-isotopy type of quartics is unambiguously determined by the real scheme (this would be false for the empty scheme). This rigidity is due to Klein 1876 [@Klein_1876_Verlauf] and well-known to Russian geometers (e.g. Rohlin 1978, Viro 1984, 1989, 2008, etc.) Likewise it would corrupt the same assertion in degree 6, which is included in Nikulin’s theorem (1979 [@Nikulin_1979/80]). Let us again examine our argument to find our probable mistake. It decomposes in 3 distinct steps. Remember that $I$ denotes the invisible locus consisting of all curves having empty real locus. \(1) Linear homotopy implies that the invisible locus $I$ is a contractible manifold. \(2) The discriminant $\frak D$ is visible inside the invisible locus $I$ for $m\ge 4$. (This follows via simple application of Brusotti.) \(3) Jordan separation holds true in a contractible manifold (or more generally a simply connected one). This can be proved in several ways, either by homology or directly by building a certain double cover out of the hypersurface by a polarization trick going back to Riemann (cf. e.g. Gabard 2011 [@Gabard_2011-Ebullition], arXiv, “Ebullition in foliated surfaces vs. gravitational clumping”). [*Insertion*]{} \[06.04.13\].—At the risk of killing some dialectic suspense, there is a 4th issue namely the codimension of the discriminant inside the invisible locus, as not being 1 but 2 instead! The step which looks most fallacious is Step (1). The reason could be the following. While there is between any two $m$-forms $P,Q$ a path $(1-t)P+tQ$ of $m$-forms (called the linear homotopy $P\to Q$), and which sweeps out forms not representing zero if $P$ and $Q$ have the same never changing sign, it is not clear that given (invisible) curves $C,D\in I$ there is always a consistent choice of sign for representing forms ensuring a global retraction of $I$ to a point. Claiming this amounts finding a section of the evident (tautological) bundle. Let ${\cal F}_m$ be the set of all forms (=homogeneous polynomials) of degree $m$. If we include the zero polynomial this becomes a vector space, with the space of $m$-tics $\vert m H \vert$ being its projectivization. Denote by $$\pi\colon {{\cal F}}_m \to \vert mH \vert$$ the corresponding projection. Choose a basepoint $D$ in $I$ (e.g. the class of the form $Q= x_0^m+x_1^m+x_2^m$). (This is akin to Fermat’s equation $x^n+y^n=z^n$ except for lacking real points.) $Q$ has positive sign on ${\Bbb R}P^2$. Given any point $C\in I$ choose a representing form $P$ which has positive sign. Then the linear homotopy $h_P\colon P\to Q$ defined by $h_P(t)=(1-t)P+t Q$ stays in $I$, which projected down to $\vert m H \vert$ gives a path joining $C$ to $D$. This path is actually independent of the chosen representative $P$ of $C$ (by a variant of Thalès, alias linearity). Define now $$H\colon I \times [0,1] \to I, \quad H(C,t)=\pi (h_P(t) ).$$ This would be the required contraction (retraction to a point) showing that $I$ is contractible. However the subtlety is whether we can choose $P=s(C)$ continuously as a function of $C$. This amounts asking if $\pi $ (the tautological projection) admits a continuous section above $I$. Of course $\pi$ lacks a (global) section by looking at the fundamental group $\pi_1$ while using functoriality. Indeed the base of the fibration has $\pi_1={\Bbb Z}_2$, while the total space has trivial $\pi_1$. Over the smaller subregion $I$ the situation is less obvious. Can one compute $\pi_1(I)$? If it is non trivial then we cannot find a section, and we are annoyed. Can we construct a section geometrically? We can look at the counter-image $\pi^{-1}(I)$ interpreted as the cone of forms not representing zero (so-called [*anisotropic*]{} forms, if we remember well some highbrow arithmetical jargon[^55]). While on ${\cal F}_m$ the sign of a form is well-defined at a point, on $\pi^{-1}(I)$ it is well-defined globally. So our cone $C:=\pi^{-1}(I)$ splits in two components $C^+, C^-$, each being connected by the linear homotopy argument. Now choose the hyperplane $\Pi$ through $Q$ which is orthogonal to $Q$ seen as a vector. This could give a section. In fact both $C^+$ and $C^-$ are contractile, being actually starlike and even convex by the linear homotopy argument. Each of them is fibred by rays (semi-lines=orbits under scaling by the positive reals ${\Bbb R}_{>0}$) and the quotient of each of these cones by the multiplicative group of positive reals ${\Bbb R}_{>0}$ is naturally identified with $I$. Abusing geometric intuition we could nearly conclude that $I$ is contractible. Yet, this is not so evident as we lack a global cross-section, e.g. by cutting by a hyperplane selecting globally a point in each fibres. This looks hazardous, so let us concede some little algebraic détour or rather homotopy theory (presumably the quintessence of topology since Jordan 1865, Klein 1882 [@Klein_1882], Poincaré 1895, Dehn-Heegard 1907 who coined the term, and then Brouwer, H. Hopf ca. 1926–30, Hurewicz, Borsuk ca. 1935, J.H.C. Whitehead, who else?[^56]). The exact homotopy sequence of a fibration ${\Bbb R_{>0}} \approx fibre \to C^+\to I$ gives $$0=\pi_1(fibre)\to \pi_1(C^+) \to \pi_1(I) \to \pi_0(fibre)=0,$$ and implies that $\pi_1(I)=0$ is trivial. So there is no algebraic obstruction to find a section over $I$ (but algebra is never enough to ensure geometric existence!). Pursuing in that way with the exact homotopy sequence of a fibration of the early 1940’s (Hurewicz, Hopf[^57], Stiefel, Eckmann, Steenrod, G.W. Whitehead, Pontrjagin, etc.) we get $$0=\pi_i(fibre)\to \pi_i(C^+) \to \pi_i(I) \to \pi_{i-1}(fibre)=0,$$ and so $\pi_i(I)=0$ for all $i=0,1,2,\dots, \infty$ (modulo “nihil est infinito”!). Note that $I$ is connected being the image of the connected set $C^+$. Now our space $I$ is not a bad one (remember Viro’s talk “Compliments to bad spaces”). More precisely, $I$ is a manifold (being an open set in the manifold $\vert m H \vert\approx {\Bbb R}P^N$). This manifold $I$ is metric moreover, hence it has the homotopy type of a CW-complex in the sense of J.H.C. Whitehead[^58] (compare Hanner, Borsuk, Milnor 1959 [@Milnor_1959], Palais 1962, Gabard 2006/08 [@Gabard-2006/08]). By a theorem of J.H.C Whitehead 1949, it follows that $I$ is contractile. —From the 1940’s (Ehresmann-Feldbau=Laboureur[^59]-Hopf-Stiefel-Pontrjagin), etc., any locally trivial fibration over a contractile base (which is paracompact[^60]) is globally trivial, hence admits a continuous section. Applying this to $C^+ \to I$ gives the required section permitting to contract $I$ via $H$. However all this optional remark is not really logically required. This proves the following: The space $I$ of invisible curves is contractible, and so it is separated by the discriminant $\frak D$ in several components. In particular the “chamber” of empty curves $E_m$ is never connected as soon as $m\ge 4$. So call it rather the empty locus. (Insertion \[06.04.13\].—This last clause is probably erroneous.) The determination of the number of components $\iota={\rm card} (E_m)$ is probably another pleasant game. (Let us guess that $\iota_4=2$, and $\iota_6=3$?) This theorem (especially its second clause) contradicts nearly everything what has been said about the empty locus. It shows (despite being a pure existence proof using primarily the exact homotopy sequence of a fibration and Whitehead) that there are obstructions to rigid-isotopy lying beyond the pure optical level. It is of course a marginal contribution to Hilbert’s 16th problem, who primarily asked the right opposite extreme (isotopy classification especially of $M$-curves). Here we live in the opposite invisible part of the mushroom (Arnold’s prose) of what could be called (by analogy with Petrovskii 1933/38 [@Petrowsky_1933], [@Petrowsky_1938]) $m$-curves, where $m$ stands for Harnack “minimal” or minimalist artwork (empty locus like Mark Rothko’s monochromes[^61]). In some sense our result of disconnectedness is reminiscent (albeit different in method) to Marin’s disproof (1979 [@Marin_1979]) of the rigidity of $M$-schemes in degree 7. In both cases the real scheme fails determining unambiguously a chamber of the discriminant, and this in situations where there is no duplication by Klein’s types I/II (what Rohlin 1978 [@Rohlin_1978] calls schemes of indefinite type). Simplifying the previous section: disconnection of the empty locus via Jordan separation and the exact homotopy sequence {#Disconnection-of-the-empty-locus:sec} ------------------------------------------------------------------------------------------------------------------------ \[27.01.13\] As the former section reflects our discovery process (as “meandering” as it may be) we prefer to keep its shape unchanged. Since our conclusion contradicts all what was asserted about the empty locus $E$ (especially Klein 1876, and Nikulin 1979), we shall here try to be more formal and direct, leaving aside historical considerations, and actually simplifying much the proof (in particular Whitehead’s criterion of contractibility via the vanishing of homotopy groups $\pi_i$ is not needed). \[28.01.13\] Our intention is to prove the following disconnection of the empty locus of plane curves of even order $m\ge 4$. \[Gabard-anti-Klein-Rohlin:thm\] [(Gabard, 27.01.13, but certainly false)]{} The “empty locus” $E$ of all real smooth plane curves having empty real parts of some fixed degree $m\ge 4$ is disconnected. In other words there exists, for any even integer $m\ge 4$, two empty smooth curves of degree $m$ which are not rigid-isotopic. We shall emphasize that this conclusion is quite unexpected. It seems to contradict much that has been said about rigid-isotopy of empty (smooth) curves. In particular, it is incompatible with the assertion going back to Klein 1876 [@Klein_1876_Verlauf] (see also Rohlin 1978 [@Rohlin_1978 p.96], or Viro’s surveys 1986, 1989, 2008) that the real scheme of a quartic curve determines uniquely its rigid-isotopy class. (More on this at the end of this section.) It conflicts also with Nikulin’s result (1979 [@Nikulin_1979/80]) that for sextic real curves the real scheme enhanced by the type data (I/II) of Klein (1876 [@Klein_1876]) suffices to determine the rigid-isotopy class. Hence it is very likely that our theorem contains a serious misconception, either at the conceptual level of definitions, and if not so, there must be a bug in the proof below. In fact it is well known that the empty chamber is connected. Eugenii Shustin was kind enough to recall us the simple argument of linear homotopy between two forms of the same sign. This is supposed to show connectedness, yet exploiting it systematically we arrived ironically at the opposite conclusion. In part, this discrepancy is merely a matter of deciding what we like to call the empty locus. The linear homotopy argument shows connectedness of what we call the [*invisible locus*]{} $I$ (consisting of all empty curves), whereas by the [*empty locus*]{} $E$ we really mean the sublocus of $I$ consisting of smooth curves. The latter space is the more relevant one when it comes to problems of rigid-isotopy, where the game is to travel as much as we can while avoiding the discriminant (i.e., never strangulate the underlying Riemann surface). To avoid any misunderstanding, let us fix our jargon more precisely. A [*ternary form*]{} is a homogeneous polynomial in three variables of some degree $m$. We shall only consider those with real coefficients, and call them [*real forms*]{}. A [*real plane curve*]{} is a homothety class of real forms under scaling of the coefficients. This is nothing else that what A. Weil would call a plane curve defined over ${\Bbb R}$. Denote by ${\cal F}_m$ the set of all real forms of degree $m$, and by $\vert mH \vert$ the space of all real curves of degree $m$ (the latter being merely the projectivization of the former). Denote by $\pi\colon {\cal F}_m \to \vert mH \vert$ the tautological projection which is an ${\Bbb R}^{\ast}$-bundle over $\vert mH \vert\approx {\Bbb R}P^N$, where $N=\binom{m+2}{2}-1$. A plane curve is [*smooth*]{} (or [*nonsingular*]{}) if the three partial derivatives of any defining form do not vanish simultaneously on ${\Bbb C}^3-\{0 \}$; else it is said to be [*singular*]{}. The set of all real singular curves forms the [*discriminant*]{} (hypersurface) denoted $\frak D$. Elimination theory (or better some counting argument) shows the latter set to be an algebraic hypersurface in the hyperspace $\vert mH \vert$ of all $m$-tics. Note that a singular real curve may well have a smooth real locus (in the sense of differential topology), yet it will then have conjugate pairs of singularities exchanged by ${\rm conj}\colon {\Bbb C} P^2 \to {\Bbb C} P^2$, $(x_0,x_1,x_2)\mapsto (\overline{x_0},\overline{x_1},\overline{x_2})$. A real form is [*anisotropic*]{} if it does not represents zero (non-trivially), i.e. the sole real solution of the equation $P(x_0,x_1,x_2)=0$ is $(x_0,x_1,x_2)=(0,0,0)$. This is tantamount to emptiness of the real locus $C({\Bbb R})$ of the corresponding curve. Say in this case that the real curve is [*empty*]{} or [*invisible*]{}. Intersecting with any line defined over ${\Bbb R}$, one sees that any odd degree curve has non-void real locus. Let $I$ be the set of empty (invisible) curves. This is nonempty iff $m$ is even, and $\pi^{-1}(I)=C$ is the cone of anisotropic forms. Such a form has a well-defined sign $\pm$, and accordingly the cone $C$ splits in two halves $C^+, C^-$ invariant under ${\Bbb R}_{>0}$-scalings. (We overuse the letter $C$, for being the cone, or the curve but no confusion should arise.) The proof of our (dubious) theorem (\[Gabard-anti-Klein-Rohlin:thm\]) decomposes in 3 short steps. $\bullet$ [*Step 1*]{} (Simple-connectivity of the invisible locus $I$).—We consider the fibration $\pi\colon C^+ \to I$, whose base is the set of invisible curves, while the total space is the space of positive-definite anisotropic form. The fibre is the space ${\Bbb R}_{>0}$ of positive reals. The space $C^+$ is convex. Whenever we choose 2 points in it, say $P, Q \in C^+$, the barycentric combination $(1-t)P+tQ$ for $t\in [0,1]$ belongs to $C^+$. Accordingly $C^+$ is certainly contractile, and in particular [*simply-connected*]{}. (Perhaps $C^{+}$, being starlike, is even diffeomorphic to a genuine cell, but we do not need that presently. This follows perhaps from J.W. Alexander’s lemma on isotopy, ask L. Siebenmann or A. Marin?) The first stage of the exact homotopy sequence of the fibering ${\Bbb R}_{>0}\approx F\to C^+\to I$ reads $$0=\pi_1(fibre)\to \pi_1(C^+) \to \pi_1(I) \to \pi_0(fibre)=0,$$ and it follows that the space $I$ is also simply-connected. $\bullet$ [*Step 2*]{} (Construction of invisible curves with singularities).—By a simple application of Brusotti 1921 [@Brusotti_1921], it is easy to construct invisible real curves $C_m$ of degree $m\ge 4$ having a pair of conjugate nodes, i.e. ordinary double points (cf. lemma below for details). (Abstractly, from the Riemann complexification viewpoint, imagine a pretzel acted upon by antipody with two handles strangulated to a pair of points $p, p^{\sigma}$ exchanged by conj.) $\bullet$ [*Step 3*]{} (Jordan-Brouwer separation of the invisible locus $I$ by the discriminant ${\frak D}$).—Paraphrasing Step 2 in our notation, this means that $\frak D \cap I$ is nonempty. The space, we are really interested in, is the empty locus $E$ consisting of all smooth empty curves. By definition we have $E=I-{\frak D}$. So the empty locus $E$ arises from the invisible locus $I$ by removing a certain real algebraic hypersurface (or at least the portion ${\frak D} \cap I $ visible in $I$). Since the manifold $I$ is simply-connected (Step 1), it follows from the Jordan-Brouwer separation theorem that $\frak D \cap I$ disconnects $I$. We conclude that the residual set $E=I-{\frak D}$ has at least 2 components whenever $m=2k\ge 4$. We now make more explicit the lemma required in Step 2 (for a schematic picture in the case $m=4$, cf. Fig.\[ShustinEmpty:fig\]a.): \[Brusotti-binodal-invisible-curves:lem\] Given any even integer $m\ge 4$, there exists a degree $m$ real plane curve which is invisible (empty real locus) with $2$ ordinary nodes exchanged by complex conjugation. Take a collection of $k=m/2$ real conics $E_1, \dots, E_k$ (degree 2) each having empty real locus such that the union of their complexifications has only normal crossings (ordinary nodes). By Brusotti’s theorem (1921 [@Brusotti_1921]), we may smooth away from $E_1\cup \dots\cup E_k$ all pairs of conjugate points safe one $p, p^{\sigma}$, where $\sigma={\rm conj}$ is complex conjugation. (Since $k\ge 2$, there is at least 4 nodes on our configuration of ellipses by Bézout.) The resulting binodal curves is real, invisible (being manufactured by small perturbation of an invisible curve). The proof of the lemma is complete. In Step 3 of the proof of Theorem \[Gabard-anti-Klein-Rohlin:thm\], we use a slightly extended form of Jordan separation imposed by a variety (possibly singular) and not just by a manifold. In fact we may imagine that the structure of the discriminant permits one to deduce a sublocus of ${\frak D} \cap I$ which is a genuine topological manifold, yet piecewise smooth (nothing so crazy as Bing-Casson-Freedman). This would involve aggregating suitably some principal strata of the discriminant exploiting perhaps Brusotti’s description of the latter, or just general properties of algebraic sets. It also conceivable that there is a direct proof by applying directly the homological apparatus involved in Jordan separation to the case of an algebraic hypersurface. Probably this is already implemented somewhere (maybe by H. Kneser, Bieberbach, Whitney, Thom, Milnor, Tognolli, Marin, Bochnak-Coste-Roy, etc.). Alas we do not know a more precise reference. Evidently the proof of separation within the simply-connected locus $I$ should just use some very basic properties of real algebraic hypersurfaces. Crudely speaking a real algebraic hypersurface cannot “stop” like a manifold with boundary (via the implicit function theorem), and so really effects a separation in the large (at least within the simply-connected subregion $I$). This elementary property of real algebraic variety was known for long (e.g. when Zeuthen 1874 [@Zeuthen_1874] speaks of a “branche complète”, etc.) Rigid isotopy of quartics: classical sources (Schläfli, Zeuthen 1874, Klein 1873–76, Rohlin 1978) {#Klein-rigidity-of-quartics:sec} ------------------------------------------------------------------------------------------------- \[28.01.13\] This section discusses in some more details some masterpieces of classical literature conflicting strongly with our conclusion (Theorem \[Gabard-anti-Klein-Rohlin:thm\]). A first place is Rohlin 1978 [@Rohlin_1978 p.96], who ascribes the rigid-isotopy classification for $m=4$ to Klein, while writing the following: “§4. Isotopy.—4.1. The classical problem. By virtue of the definition of a real plane projective algebraic curve of degree $m$, such curves form a real projective space of dimension $m(m+3)/2$. Singular curves, that is, curves with real or imaginary singularities, fill out in this space a hypersurface of degree $3(m-1)^2$, and non-singular curves fill out the complementary open set, which splits into a finite number of components[^62]. It is clear that curves that belong to one component have the same real scheme, that is, the class of all non-singular curves with a given real scheme consists of whole components. The investigation of these components is a very old problem, like the investigation of the classes themselves. It was known more than hundred years ago that for $m\le 4$ the components coincide with the classes[^63] (the least trivial case $m=4$ was considered by Klein; see \[4\](Klein 1922=Klein 1876 [@Klein_1876_Verlauf]), p.112). From the results of the previous section it follows that for $m\ge 5$ this is not so; for the complex scheme is constant on each component, but is capable of changing within one class for $m\ge 5$. \[…\]” As usual Rohlin quotes Klein’s GMA=Ges.Math.Abhandl. (1922), yet the original source is the 1876 paper “Über den Verlauf der Abelschen Integrale bei den Kurven vierten Grades” [@Klein_1876_Verlauf]. Klein’s prose is as usual quite magical (like that of Rohlin), and reads as follows: “ Eine wesentliche Eigenschaft dieser Einteilung der Kurven vierter Ordnung in sechs Arten ist in dem folgenden Satze ausgeschprochen, der weiterhin eine fundamentale Bedeutung für die Tragweite unserer Untersuchungen gewinnt: [*Von jeder allgemeinen[^64] Kurve vierter Ordnung kann man zu jeder anderen, die derselben Art angehört, durch allmähliche reelle Änderung der Konstanten übergehen, ohne da[ß]{} bei dem Übergangsprozesse Kurven mit Doppelpunkt oder gar allgemeine Kurven, die einer anderen Art angehören, überschritten zu werden brauchten.*]{} Ein direkter Beweis dieses Satzes hat keine Schwierigkeit[^65], aber er ist weitläufig. Es soll hier um so mehr Abstand genommen werden, als die bei ihm nötig werdenden Betrachtungen mit diejenigen, die im gegenwärtigen Aufsatze zu entwicklen sind, wenig Beziehungspunkte haben. \[So roughly Klein says that there is little connections between rigid-isotopy and Abelian integrals!\] Dagegen sei angedeutet, da[ß]{} man ihn vermöge kurzer Zwischenbetrachtungen führen kann, wenn man auf frühere Untersuchungen von Zeuthen und mir zurückkgreift. Ich habe \[in Abh. XXXV, S.24, 25\] gezeigt, da[ß]{} ein ähnlicher Satz gilt für die fünf Arten, welche man nach Schläfli bei den allgemeinen Flächen dritter Ordnung zu unterscheiden hat. Es hat dann Zeuthen bewiesen (Math. Ann., Bd.7 (1874), S.428), da[ß]{} die Arten der Kurven vierter Ordnung den fünf Flächen Arten in sehr einfacher Weise entschprechen. Projiziert man die $F_3$ von einem ihrer Punkte aus stereographisch auf eine Ebene, so tritt als scheinbare Umhüllung bei den Arten I, II, III, IV von Schläfli eine vierteilige, drei-, zwei-, einteilige Kurve vierter Ordnung auf. Die Art V ergibt, bei analoger Konstruktion, je nachdem man den Projektionspunkt auf ihrem unpaaren oder paaren Teile annimmt, die Gürtelkurve oder die imaginäre Kurve. Umgekehrt kann auch jede Kurve vierter Ordnung aus der entschpechenden Flächenart in der angegebenen Weise gewonnen werden. Hierin liegt der vor uns gewünschte Beweis. Um ihn völlig zu führen, hat man nur noch die Modifikationen zu untersuchen, welche die scheinbare Umhüllungskurve erfährt, wenn der Projektionspunkt auf der fest gedachten Fläche beliebig verschoben wird. Aber auch dieses hat Zeuthen ausgeführt \[Études des propriétés de situation des surfaces cubiques; Math Annalen, Bd.8, (1874/75).\]” Let us summarize Klein’s proof of the rigid-isotopy of $C_4$’s: first Schläfli in 1863 [@Schlaefli_1863] found five isotopy class of real cubic surfaces $F_3$, and Klein showed them to be rigid-isotopic (in Klein 1873 [@Klein_1873-Uber-Flächen-dritter-Ordn]). Then Klein exploits the yoga of Zeuthen (which goes back to Geiser, compare Zeuthen 1874 [@Zeuthen_1874 p.428]) of looking at the apparent contour of the $F_3$ (projected from a point on the surface) to get a $C_4$ (all of them arising so). In fact the $F_3$ with 2 components produces both the Gürtelkurve or the empty $C_4$ depending on whether the center of vision is located on the pseudo-plane or on the spherical component. This is easily visualized if one imagine $F_3$, a small perturbation of a sphere union an equatorial plane (Fig.\[Zeuthen-Klein:fig\]). Note that the empty apparent contour arises from a phenomenon of total reality of the bundle of lines through the spherical component (this being again reminiscent of “Ahlfors”). -5pt0 -5pt0 So a smooth $F_3$ with a marked point gives rise to a $C_4$, and all quartics arise so. One must still study the rôle of the marked point, and there is not just the five classes of Schläfli-Klein but one more due to the marking (being either on the pseudo-plane or the spheroid). Hence Klein’s argument looks quite convincing but we should understand the details more precisely. Maybe there is little gap in this proof when setting up explicitly the $F_3\leftrightarrows C_4$ correspondence. Is the contour apparent of a smooth $F_3$ always a smooth $C_4$ (and viceversa a singular cubic to a singular quartic), so that the Zeuthen-Klein correspondence really sets up a dictionary between the corresponding discriminants (hence rigid-isotopy classes). In particular to what sort of $F_3$ corresponds the empty quartic with a pair of conjugate nodes. Those are the essential guys which in our Theorem \[Gabard-anti-Klein-Rohlin:thm\] causes the disconnection of the invisible locus $I$. Further as the center of projection is moving in 3-space the ZK-correspondence is somewhat non-canonical, i.e. there is not a fixed ${\Bbb P}^2$ on which to project. So the reduction proposed by Klein is perhaps foiled somewhere, at least requires to be modernized (and detailed) seriously. Naively our Theorem \[Gabard-anti-Klein-Rohlin:thm\] (if correct) could be an obstruction to completing Klein’s proof. If the ZK-correspondence is sound, it could be that Klein’s 1873 rigid-isotopy classification of $F_3$’s (cubic surfaces) is foiled. Fixing the paradox ------------------ \[29.01.13\] Can it be that the discriminant $\frak D$ while penetrating inside the invisible locus $I$ appears there with (real) codimension 2 hence without separating $I$? Recall that real loci of algebraic hypersurfaces (=primals in old British jargon, e.g. Semple-Roth) may look anomalously small. For instance a solitary node on a plane cubic curve is merely an isolated point of real dimension $0$ (hence real codimension $2$). This phenomenon could foil Step 3 of the proof of Theorem \[Gabard-anti-Klein-Rohlin:thm\]. However via Brusotti description of the discriminant one could still hope that ${\frak D} \cap I$ has real-codimension $1$. Naively the principal stratum corresponding to a curve with a conjugate pair of nodes looks at first of codimension 2 because there is two nodes. However by the reality condition one of them is forced and so it is really one closed point in the sense of Grothendieck’s schemes. Without Grothendieck such arguments of reality counting also abound in Klein, e.g. when it comes to coverings of the Riemann sphere with complex conjugate ramification. Here a complex ramification point count for 2 real dimensions, while a real branch point affords one freedom parameter. Yet under the symmetry condition both cases actually contribute to the same. It is with this sort of argument that Klein managed to compute the dimension of the moduli space of real curves by aping what did Riemann over the complexes. Alas one can argue that the stratum of the complexified discriminant $\disc(\CC)$ with two nodes $x,y$ nearby $p,p^\sigma$ has geometric codimension 2 over the complexes, and it would follow that the real locus $\frak D$ has at most real codimension 2 when $x,y$ lye symmetric under $\sigma=$conj. Another way is to start the dimension count of the discriminant from the scratch. So we look at plane curves with a node or a higher singularity marked on it. This gives an incidence variety $(C,p)$ consisting of curves $C$ singular at $p$. Saying that $p$ is singular of $C$ amounts (via the Euler relation) to say that the 3 partials of a defining equation vanish at $p$, yielding 3 linear conditions on the coefficients. So looking at the projections $p\leftarrowtail (C,p) \mapsto C$ we see fibres of codimension $3$, while moving the point $p$ gives codimension $1$ (hypersurface) in the space of curves. Adapting this dimension count near our binodal empty curve $C$ shows that we have a pair of projections $p\leftarrowtail (C,p, p^{\sigma}) \mapsto C$. Imposing a singularity at $p$ gives 3 linear conditions, but the point $p$ being any imaginary point of the plane its location depend upon 2 complex parameters (4 real parameters). When $C$ is defined over ${\Bbb R}$ the 3 equations for the partials are again linear (in the coefficients), but involve complex constants. Thus splitting into real and imaginary parts yields twice so many linear equations, hence 6 of them. Those being satisfied then $p^\sigma$ is also a node by symmetry, and so the variety ${\frak B}$ of (binodal) curves having a conjugate pair of nodes has real (co)dimension $4-6=-2$ in the space of curves. In that case it seems that the principal stratum ${\frak D}\cap I$ consisting of empty curves with a pair of conjugate node has also real codimension 2 in $\vert mH \vert$, and this would foil our theorem (\[Gabard-anti-Klein-Rohlin:thm\]). For short let us call ${\frak D}\cap I$ the [*invisible discriminant*]{}, abridged $I$-discriminant. The above argument has to be polished by checking that the 6 linear equations are really independent conditions. The main issue is therefore to calculate the dimension of the invisible discriminant. If it has codimension 2 then $E=I-\frak D$ is connected, and so Klein-Rohlin were right. (Otherwise, if of codimension $1$, then it separates and Klein-Rohlin were wrong.) If the $I$-discriminant has codimension 2, it is like a knot, and it seems interesting to compute its fundamental group (of its complement), and to look at the Picard-Lefschetz monodromic transformation arising when winding once around a meridian of this $I$-discriminant. Before adventuring we should first solve (more rigorously) the dimension problem of the $I$-discriminant. This must surely be done in Brusotti 1921 [@Brusotti_1921] (and known to Gudkov, maybe in the 1974 survey [@Gudkov_1974/74]). Perhaps this was already known in the era of Zeuthen-Klein-Harnack-Hilbert. The invisible discriminant has codimension $2$ {#invisible-discriminant-codim-2:sec} ---------------------------------------------- \[29.01.13\] The goal of this section is to resolve our paradox, that the locus of empty smooth curves is disconnected (violating thereby well assessed knowledge of Klein, Rohlin-Nikulin, etc.). It seems that our sole mistake was based on the linguistical misconception of thinking that the discriminant-hypersurface is a hypersurface (throughout)! Names and terminologies are often misleading in mathematics. In fact we shall try to convince that inside the invisible locus (of curves with empty real loci) the discriminant has only codimension 2, hence too small to effect any Jordan-Brouwer separation. Once this is observed this raises some little questions about knowing which chambers residual to the principal strata of the discriminant (where it has really of codimension 1) contains such smaller strata of codimension 2. In more geometric terms, this amounts essentially deciding which smooth curves can acquire a pair of conjugate nodes. Fix some even integer $m=2k$, and consider only real curves of fixed degree $m$. Let $\vert mH \vert\approx {\Bbb R}P^N$ be the corresponding parameter space of real $m$-tics. In this space we pay special attention to the space $I$ of [*invisible curves*]{} (those with empty real locus). This is clearly an open set in $\vert mH \vert$. Let $\frak B$ be the variety of invisible real plane curves with at least one singular point (hence necessarily at least a pair thereof). We have $\frak B = {\frak D}\cap I$, the so-called [*invisible discriminant*]{} (abridged $I$-discriminant). \[invisible-discriminant-codim-2:lem\] The $I$-discriminant has (real) codimension $2$ in the hyperspace of all curves. Consider the incidence relation $B=\{ (C,p) \colon C \in I, p \in Sing C \}$. We have natural projections $${\frak B}\buildrel{\pi_1}\over\longleftarrow B \buildrel{\pi_2}\over{\longrightarrow} {\Bbb P}^2({\Bbb C}).$$ First study the fibre of the second projection $\pi_2\colon (C,p)\mapsto p$ . This amounts to look at all curves having a prescribed singularity at $p$ an imaginary point. This imposes 3 linear equations (vanishing of the 3 partials, which suffices by Euler equation for the point to be on the curve). Splitting in real and imaginary parts gives 6 linear conditions, which looks linearly independent. So the fibre $\pi_2^{-1}(p)\approx (I \cap {\Bbb R}P^{N-6})\times\{p\}$. Since $\pi_2$ is surjective onto ${\Bbb P}^2({\Bbb C})-{\Bbb P}^2({\Bbb R})$ it follows that the real dimension of $B$ is $\dim_\RR B= 4+(N-6)=N-2$. As the first projection $\pi_1$ is generically 2-to-1 (or finite-to-one except over special curves with multiple irreducible components), it follows that $\dim_\RR {\frak B}=N-2$, hence of codimension 2 in $I$ (or in $\vert m H \vert$). More generally, imagine a visible curve (i.e. $C_m(\RR)\neq \varnothing$) acquiring a conjugate pair of nodes. This will not affect the real scheme (=soft isotopy class of the embedding $C_m(\RR)\subset \RR P^2$). A priori inside each “class” can penetrate a portion of the discriminant of codimension 2. Let us be more formal. Split the discriminant $\disc=\disc^+\sqcup \disc^{-}$ in two parts depending on whether $Sing C$ contains a real point or not. Precisely define $\disc^{-}$ as the set of singular curves lacking real singularities. The argument of the above lemma shows that $\disc^{-}$ has real codimension two. We call it hence the [*hypo-discriminant*]{}. The set $\disc^+$ is defined as its complement, i.e. $\disc^+=\disc- \disc^{-}$. The latter has codimension 1 by a variant of the above argument. (Indeed a real singularity imposes 3 linear conditions, but moving the point create 2 dimensions, whence the defect of $-1$.) We call $\disc^+$ therefore the [*hyper-discriminant*]{}. The assignment $\vert mH \vert -\disc \to \frak S$ of the real scheme to a non-singular equation is more generally defined on the larger space $\vert m H\vert-\disc^+$ residual to the hyper-discriminant, while being locally constant there. Crudely put, in problems of rigid-isotopies the hypo-discriminant can be neglected (being only of codimension 2, hence effecting no additional separations). However when we would like not only to study the connectivity of the chambers but also their topology then the hypo-discriminant ought to be considered again. A first question is whether any chamber residual to the hyper-discriminant (abridged [*hyper-chamber*]{}) intersects the hypo-discriminant. (This is true for the empty chamber residual to $\disc^+$, by our Brusotti-style lemma \[Brusotti-binodal-invisible-curves:lem\].) The general problem looks again to involve a contraction principle of Riemann surfaces, now under a symmetric pair of vanishing cycles. For $M$-curves, there seems to be a topological obstruction, since strangulating two imaginary cycles $\beta, \beta^{\sigma}$ causes a disconnection of the Riemann surface in 2 algebraic pieces $C_m\to C_k\cup C_l$ of degree $k, l$ (hence cutting themselves in $k\cdot l$ points by Bézout). But $C_k, C_l$ cuts transversally in $2$ points only, hence $k=2$, $l=1$ (up to renumbering). Hence, this eventuality can only occur for $m=3$ (cubics), where it does occur when an $M$-cubic degenerates to $E_2\cup L$ a conic union a disjoint line. This curve $E_2\cup L$ belongs to the hypo-discriminant since it lacks real singularities. (\[06.04.13\] Further the dimension of such split cubics is $5+2=7$, which is indeed of codimension 2 in the hyperspace of cubics of dimension $\binom{3+2}{2}-1=\frac{5\cdot 4}{2}-1=9$.) Given a curve in the hypo-disc $\disc^-$, it seems likely that by genericity we may assume the latter to be a binodal curve with a conjugate pair of nodes. By Brusotti 1921 [@Brusotti_1921], smooth them away. Interpreting the process backward in time we see a pair of imaginary conjugate vanishing cycle $\beta, \beta^{\sigma}$ strangulating toward the nodes $p,p^{\sigma}$. If this argument holds true we see that each hypo-discriminantal component gives rises to a bistrangulation along imaginary cycles. In particular: Safe for $m= 3$, the hyper-chamber of an $M$-curve of degree $m$ never contains the hypo-discriminant. Perhaps this is the sole obstruction, in the sense that any other hyper-chamber (than those of $M$-curves) intersects the hypo-discriminant. In fact there is perhaps still such an obstruction for $(M-1)$-curves. Naively the latter look like an $M$-curve safe that one “oval” is masked. Still if we imagine the corresponding symmetric Riemann surface it seems that a pair of imaginary cycles must divide. \[30.01.13\] So we are led to the following general topological question: \[bicycle-existence:ques\] (Existence of bicycles).—[Given a symmetric surface in the sense of Klein 1876 (i.e. an oriented closed surface $X$ with an orientation reversing involution $\sigma$), when is it possible to find a pair $\beta,\beta^{\sigma}$ of imaginary cycles(=Jordan curves) such that $\beta \cup \beta^{\sigma}$ does not divide $X$?]{} Here imaginariness means that $\beta$ has no point fixed under $\sigma$. It is also required that $\beta$ is disjoint from its conjugate $\beta^{\sigma}$. We call such a pair an (imaginary) [*bicycle*]{}. As well-known since Klein 1876 [@Klein_1876], symmetric surfaces are either ortho- or diasymmetric (equivalently dividing or not, or of type I resp. II). The type together with the number $r$ of “ovals” (pointwise fixed circuit) and the genus $g$ fixes the equivariant topology of a symmetric surface. As to our question it is plain that in the type I (dividing) case then there is a bicycle provided the surface is not maximal $r=g+1$ (“$M$-surface”). In the maximal case such a bicycle is not available, because the quotient $X/\sigma$ is planar hence schlichtartig (i.e. divided by any Jordan curve). Literally “schlichtartig” means planar like. Incidentally recall the implications: \[Gabard-five-lemma:lem\] “simply-connected$\Rightarrow$schlichtartig$\Rightarrow$ orientable” for a topological surface. Exaggerating a bit the only rigorous proof, we are aware of, uses the five lemma and homology, cf. e.g. Gabard-Gauld 2011 [@Gabard-Gauld_2011-Dynamics Lemma 4.17], Dynamics of non-metric manifolds. It works universally without having even to assume the surface metrizable. Hausdorffness is however crucial (consider a branched plane and in it a Jordan curve on the upper sheet, then you can evade via the lower sheet so as to reach the outside of the upper sheet.) The five lemma implies that when we have a Jordan curve $J\subset U \subset M$ included in two nested spaces, then if it divides the large space $M$, it must divide the small one $U$, and viceversa provided $H_1(U)\to H_1(M)$ is onto. This latter fact prompts the first implication “$\Rightarrow$” by taking $U$ the tubular neighborhood of $J$ (which must be trivial since otherwise there would be an indicatrix-reversing loop violating the assumption $\pi_1(M)=0$). The second implication “$\Rightarrow$” follows from the first fact, namely a division in the large $M$ implies a division in the small $U$, hence is particular of the tubular neighborhood which must therefore be trivial. When applied to the quotient of a symmetric surface, Lemma \[Gabard-five-lemma:lem\] gives for the latter: “ortho-sphere$\Rightarrow M$-surface$\Rightarrow$ dividing”. Note also that if there is a bicycle on $(X,\sigma)$, then its projection in the quotient $\bar X=X/\sigma$ is a cycle $\bar \beta$ interiorly traced which does not divide and preserves the indicatrix (=local orientation). (The non-division just follows from the fact that the image of the connected set $X-(\beta \cup \beta^{\sigma})$ has to be connected.) It remains to answer our question (\[bicycle-existence:ques\]) in the diasymmetric case. The lowest diasymmetric case $r=0$ is easy since then $(X,\sigma)$ may be visualized in 3-space as a pretzel invariant under central symmetry (antipody). Hence a bicycle exists provided the genus $g\ge 2$. (For $g=1$ there is a pair of cycle exchanged, but collectively they do divide. It cannot be otherwise by Riemann’s definition of the genus.) For the other cases there are several models. One way due to Klein-Weichold-Kervaire (private communication of the latter in 1999) amounts to look at a Möbius band embedded in 3-space (make holes in it) and look at a thickening of the normal bundle of thickness vanishing along the boundary (Fig.\[Kervaire:fig\]a). -5pt0 -5pt0 Alternatively, we may start from the Harnack-maximal case visualized as a planar membrane with $r=g+1$ contours (Fig.\[Kervaire:fig\]b), and kill successively contours by cross-capping them (à la von Dyck 1888 [@von-Dyck_1888 p.479], another of Klein’s student). This operation does not alter the Euler characteristic $\chi$, and so keep the genus of the double unchanged as $\chi (X)=2 \chi (X/\sigma)$. The symmetric surface is constructed abstractly via the usual process of the double orientation cover (without duplication of the boundary points by local orientations). So imagine a disc with $g$ holes while cross-capping them successively. If no cross-cap we have an $M$-curve[^66], if one cross-cap an $(M-1)$-curve, if two cross-caps an $(M-2)$-curve of type II, etc. As soon as there is $2$ cross-caps, connect them by a path and closing it back (Fig.\[Kervaire:fig\]d) gives a cycle $\bar \beta$ which preserves the indicatrix (as it traverses twice the cross-caps) and which does not divide $\bar X$ (because its apparent inside is connected with the outside via the cross identifications). Lifting $\bar \beta$ to $X$ gives the desired bicycle. Another, but slightly weaker argument, is that as soon as there is 3 cross-caps available, they can be traded against one cross-cap and one handle (as both contribute identically to the Euler characteristic). Then it is enough to take the meridian (or parallel) of that handle (Fig.\[Kervaire:fig\]e). The case of where there is only one cross-cap is a bit more tricky, and it seems that we cannot find a nondividing cycle $\bar\beta$ which is indicatrix-preserving. (Trace a picture which can be either like a figure “8” crossing the cross-cap (Fig.\[Kervaire:fig\]f) or like a figure “$\omega$” with extremity linked together (Fig.\[Kervaire:fig\]g). This is abstractly just the figure 8, except that the one loop envelopes the other one. In both cases it is seen that a division is produced.) Here is the obstruction: An $(M-1)$-surface (with $r=g$) cannot have a bicycle. Since the bicycle $\beta\cup \beta^{\sigma}$ does not divide the surface $X$, its image $\bar \beta$ in the quotient $\bar X$ does not divide it. The covering $X-{\rm Fix}(\sigma)\to \bar X-\partial \bar X$ restricted to the complement of $\bar \beta$ shows that $X-{\rm Fix}(\sigma)-(\beta \cup \beta^{\sigma})$ has at most $2$ components (exchanged by $\sigma$). Yet it suffices to add one oval (of ${\rm Fix}(\sigma)$) to make it connected. Hence we have $(g-1)+2=g+1$ retrosections not disconnecting the surface, violating Riemann’s definition of the genus. If such a $\bar beta$ existed then lifting it to the symmetric surface would give 2 cycles $\beta,\beta^{\sigma}$ on $X$ of genus $g$. Since $\bar\beta$ does not divide $\bar X$, the covering $X-{\rm Fix}(\sigma)\to \bar X-\partial \bar X$ shows that $X-{\rm Fix}(\sigma)-(\beta \cup \beta^{\sigma})$ is still connected since it is the orienting cover of the connected surface $(\bar X-\partial \bar X)-\bar \beta$ which is still non-orientable. If orientable then the double orientation cover would be trivial on both pieces hence on their union. In conclusion we exhibit $(g-1)+2=g+1$ cycles on $X$ the surface of genus $g$ not disconnecting it. This overwhelms Riemann’s definition of the genus. In summary we have proven: A symmetric surface admits a bicycle iff $g\ge 2$ and $r\le g-1$. In other words iff it is not an $M$-surface nor an $(M-1)$-surface. Via Brusotti this seems to afford obstructions to the presence of the hypo-discriminant in certain hyper-chambers. More precisely: Inside a pre-maximal hyper-chamber (i.e. $r\ge g$) the hypo-discriminant is vacuous. In particular a premaximal curve (i.e. an $M$- or an $(M-1)$-curve) cannot acquire a conjugate pair of nodes by continuous variations of its coefficients (among smooth curves safe for the extremity). \[Warning: the case $m=3$ is the sole exception.\] More risky is the (converse) assertion that via a suitable (but very hypothetical) contraction principle the topological presence of a bicycle suffices to create an algebraic deformation toward a curve $C_m$ with an imaginary pair of nodes. If optimistic about the freedom of the joy-stick this supports the: \[hypo-discriminant:conj\] Any (smooth real plane) curve $C_m$ which is not premaximal can acquire a conjugate pair of nodes (bi-node) via continuous deformation among smooth curves safe for its extremity. In particular the hypo-discriminant appears in all the corresponding (“ante-maximal”) hyper-chambers. This is another large-deformation principle a bit akin to the Itenberg-Viro contraction conjecture (\[Itenberg-Viro-contraction:conj\]) for empty ovals. To get serious prohibitions one would perhaps even require a strengthened collective form of it. For instance if $r=2$ and if we are dividing, we could contract several bicycles which collectively split the Riemann surface. Browsing through increasing degrees $m=3,4,5,6,7, \dots$ gives the genus $g=1,3,6,10,15,\dots$. By virtue of Klein’s congruence $r\equiv_2 g+1$ we look especially at $m=4$ or $m=7$ (or $m=3k+1$). When $m=4$, we find no obstruction to the splitting (since $g=3$ and we have 4 vanishing cycles contracting to $4=2\cdot 2$ points in agreement with Bézout, see Fig.\[Pretzel:fig\]a). If $m=7$, then $g=15$ and so there is $8$ bicycles (Fig.b) which strangulated toward nodes gives 16 (simple) intersections between both pieces of the degeneration $C_7\to C_k\cup C_l$, where $k+l=7$. Testing all values $(k,l)=(1,6),(2,5),(3,4)$ gives always the wrong number of intersections $k\cdot l=6,10,12$ never equal to $16$. This contradiction with Bézout reproves that a dividing septic cannot have $r=2$. (Of course this is best proved as a consequence of Rohlin-Mishachev’s formula). -5pt0 -5pt0 Extending this to all degrees requires another form of contraction. In the case of $C_5$’s: then assuming $r=1$, there is 3 bicycles and 1 ortho-cycle (Fig.c). We have a splitting $C_5\to C_k\cup C_l$, and as $g=6$ we have 7 intersections in $C_k\cap C_l$ after strangulation of the Riemann surface. But this is never equal to $k\cdot l$ for $(k,l)=(1,4), (2,3)$. So this would prove again that a quintic with one circuit ($r=1$) cannot be dividing. Compare (\[Klein-Marin-quintic:lem\]) for another proof. [*Insertion*]{} \[06.04.13\].—Applying the same method to a quintic with $r=3$, while imagining the underlying Riemann surface orthosymmetric and spliced by 2 bicycles and 1 orthocycle (Fig.d), the strangulation process leads to 2 algebraic pieces $C_k\cup C_l$ intersecting in 5 points. This is again not of the form $k\cdot \ell$ (equal as above to $4$ or $6$). This reasoning proves that a quintic with $r=3$ cannot be dividing, which is [*nonsense*]{} (remind the deep nest and its total reality). So the methodology (of such imaginary contractions) appears jeopardized. It seems at first that by “rotating” the pretzel of genus $6$ as to make the five cycles into reals circuit prompts a corruption of Itenberg-Viro (\[Itenberg-Viro-contraction:conj\]) by the same device. This is not so because one of the circuit is a pseudoline (since we are in degree $m=7$ odd). But of course we could imagine an example in even degree. In fact what protects a direct corruption of the Itenberg-Viro conjecture (\[Itenberg-Viro-contraction:conj\]) is Rohlin’s formula which in case of no-nesting forces $r=k^2$, and intersecting the half of the strangulated Riemann surface (now of degree $k$ since exchanged by Galois) is concomitant with Bézout. More generally, one can perhaps by this method get another derivation of the Rohlin-Marin inequality $r\ge m/2$ for a plane dividing curve of degree $m$. One could also imagine more radical degeneration by a bicycle $\beta, \beta^{\sigma}$ such that already $\beta$ divides $C_m({\Bbb C})$ (fig.f). A such is easy to visualize in the dividing case and would separate all imaginary handles from the real contours. However in the case of a $C_4$ of type I, such a pair $\beta,\beta^\sigma$ would strangulate the surface of genus $3$ in three pieces of genus 1, so the degree must be at least $3+3+3=9$, which is much greater than $4$. Hence such contractions are unlikely to exist algebraically. \[31.03.13\] When $m=4$, the above conjecture (\[hypo-discriminant:conj\]) looks trivial, e.g. because all ante-maximal schemes $r\le M-2=2$ admits realization as pair of conics (either nested or disjoint). Rigidity index -------------- \[31.03.13\] Another naive remark concerns the rigidity of the “one-oval scheme” $1$. Once the empty scheme is known to be rigid, then via the contraction conjecture CC (\[Itenberg-Viro-contraction:conj\]), the one-oval scheme ought to be rigid as well. Naively via CC one could pursue inductively and all schemes would be rigid (which is not true as best and first shown by Rohlin via Klein’s type for $m\ge 5$). So there are subtle obstructions coming from separation between chambers at the next level. Despite such difficulties we call this method the [*rigidification procedure by reduction to the empty chamber*]{}. So we start from the empty chamber $0$, and then there is the chamber $1$ (which should be still connected). Then “attached” to this there is the “chamber” $\frac{1}{1}$ and $2$, etc. Of course here “chamber” should rather be “isotopy class” and it should be proved that such schemes are rigid, i.e. that their respective isotopy classes correspond to a unique chamber of the discriminant. For a fixed integer $m$, the smallest integer $r=r(m)$ such that all real schemes of degree with less than $r$ ($\le r$) real branches are rigid, is called the rigidity index in degree $m$. As usual in mathematics, when we are unable to prove something we just introduce a: [The [*bifurcation index*]{} $r(m)$ in degree $m$ is the smallest integer $r=\rig(m)$ such that there is a real scheme of degree $m$ with $r$ real branches which is non-rigid, i.e. represented by 2 real curves of degree $m$ which are not rigid-isotopic. It is set equal to $+\infty$ if all schemes are rigid. If finite, and diminished by one unit it could be called the [*rigidity index*]{}, since below it all schemes would be rigid. (Our terminology is a bit awkward, because a high rigidity index truly means that the video game is flexible.)]{} Basically $\rig(m)$ measure the critical level at which the above rigidification algorithm fails surely. Very little is known on it as exemplified by the (still open) conjecture on the rigidity of the one-oval scheme (\[OOPS:one-oval-rigid-isotopic:conj\]), which traduces into the assertion $\rig(m)\ge 2$ for all even $m$. Even this modest estimate is pure speculation to present knowledge. It is trivial that $\rig(1)=+\infty$, $\rig(2)=+\infty$ (because $PGL(3,\RR)$ acts transitively on lines or conics provided the latter are defined by quadratic forms with the same signature). $\rig(3)=+\infty$, i.e. all schemes of degree 3 are rigid is already somewhat more sophisticated since there are moduli. Yet this must follow either form Newton-Plücker or from the theory of elliptic functions (Euler-Legendre-Abel-Weierstrass, etc.) It is probably not completely trivial to write down the details, but looks evident if we keep in mind a reduction to the Weierstrass normal form $y^2=(x-a_1)(x-a_2)(x-a_3)$, where the distinct $a_i$ can either be all real or two of them imaginary conjugate. The result of Schläfli-Zeuthen-Klein (cf. Klein 1876 [@Klein_1876_Verlauf]) implies that $\rig(4)=+\infty$. This is already less evident, and was first proved by Klein via cubic surfaces as discussed in Sec.\[Klein-rigidity-of-quartics:sec\]. For $\rig(5)$ we have a scheme of indefinite type (namely $4\sqcup J$) which is elementary to find (see Fig.\[Gudkov-Table-quintic:fig\]) and first described in Rohlin’s era (cf. e.g. 1974 [@Rohlin_1974/75] and 1978 [@Rohlin_1978]). It suffices to smooth a pair of conics plus a line in two different ways as indicated on the left of Fig.\[Indefiniteodd:fig\]. It follows that $\rig(5)\le 5$. By Rohlin’s inequality ($r\ge m/2$ if type I) and Klein’s congruence holding right above, the above example is clearly minimal to detect an obstruction to rigid-isotopy via Klein’s types (see also the argument in Sec.\[quintic-table-Klein-Gudkov:sec\]). Thus it is fairly clear that $\rig(5)=5$, but we know no proof without appealing to Kharlamov’s version (1981/81 [@Kharlamov_1981/81]) of Nikulin’s classification in degree 6 (cf. next item). Looking at the Gudkov-Rohlin table (Fig.\[Gudkov-Table3:fig\]) of sextics, it is clear that $\rig(6)=5$. The proof of this rests the deep result of Nikulin 1979 [@Nikulin_1979/80] that the type enhanced real scheme suffices to encode the rigid-isotopy class. It would be interesting to know if the CC conjecture is able to reprove the rigidity of all sextic schemes lying below $r< 5$ the bifurcation index. Of course Nikulin tells much more. In view of the Rohlin-Marin inequality $r\ge m/2$ for a dividing curve, Klein’s types afford no obstruction to rigid-isotopy for curves with few branches. A naive optimist can expect no bifurcation below this value. By a bifurcation of a scheme we simply mean it being stretched apart in two chambers of the discriminant. So $m/2$ is a sort of ebullition temperature, below which everything is frozen, i.e. only type II schemes are represented apart from the deep nest scheme with $r=m/2$. The latter scheme is (pure) of type I by the simplest form of total reality, hence does not cause a type bifurcation, while being actually rigid by Nuij’s theorem (1968 [@Nuij_1968]). The critical temperature $r=m/2$ can be augmented by unit, because the next $r$ is forced belonging type II by Klein’s congruence. (All this extends to the case where $m$ is odd by taking as critical temperature $r=(m+1)/2$ the number of branches of the deep nest). Hence all schemes with $r\le [(m+1)/2]+1=[(m+3)/2]$ are necessarily of type II, safe for the deep nest. In particular there is no indefinite schemes below this level, and the first such indefinite scheme is expected to be found at height $[(m+1)/2]+2=[(m+5)/2]$. (The height of a scheme is merely its number of components, a jargon suggested by the diagrammatic of the Gudkov table Fig.\[Gudkov-Table3:fig\].) [The [*type bifurcation index*]{} (or just [*indefiniteness*]{}) $\indef=\indef(m)$ is the minimal height of an indefinite scheme. It is set equal to $+\infty$ if all schemes of degree $m$ are definite (i.e. either of type I or II in the sense of Rohlin 1978).]{} For $m\ge 5$ it seems evident that indefinite schemes always exist, but this requires some proof. (This and more will follow from Figs. \[Indefinite:fig\] and \[Indefiniteodd:fig\] below.) The following is all what can be said at first sight: We have $1 \le \rig(m)\le \indef(m)$, and $\indef(m)\ge [(m+5)/2]$. (In fact it is a simple matter to show that the latter is sharp for $m\ge 5$, cf. Figs.\[Indefinite:fig\] and \[Indefiniteodd:fig\].) \(1) The first estimate is trivial when $m$ odd, and when $m$ is even it follows from the rigidity of the empty scheme, which is a consequence of the fact that the invisible discriminant has real codimension 2 (cf. Lemma \[invisible-discriminant-codim-2:lem\]). \(2) The second estimate is a trivial consequence of the fact that a rigid-isotopy induces an equivariant isotopy between the allied symmetric Riemann surfaces. Formally the proof may require the Ehresmann-Feldbau-Pontrjagin, etc. trivialization of a fiber bundle over a contractible base (which is paracompact, else false tangent bundle to the (simply-connected) Prüfer surface). Actually it requires an equivariant version thereof, but by passing to the quotient and reconstructing the symmetric surface as the double orientation cover we can reduce to the classical setting. \(3) The third estimate follows from Rohlin’s inequality, conjointly with Klein’s congruence, interpreted as obstructing type I right above the height of the deep nest (of type I being totally real under a pencil of lines). The second estimate could be sharp. This dream is still much out of reach, and so is the nightmare of refuting this via the Fiedler-Marin method. At first the problem may look tractable, yet it certainly does not follow from Gabard 2000 [@Gabard_2000] where a sole classification of symmetric surfaces realizable as plane curves is given. So below the height $[(m+3)/2]$ things are nearly pure and frozen (no indefinite types) and naively we could expect that all schemes are rigid below this altitude, i.e. when $r\le [(m+3)/2]$. Hence: All schemes of degree $m$ with $r\le [(m+3)/2]$ are rigid. (Check if this was not disproved by Fiedler, but we do not think so.) We know (e.g. by the conceptual argument of Morse surgeries as exposed in Viro 1989 [@Viro_1989/90-Construction]) that all values of $r$ (number of real circuits) below Harnack’s bound are realized, by taking a generic pencil between an $M$-curve (e.g. Harnack’s) and an empty or Fermat curve with $r=1$. Likewise either by the pedestrian argument in Gabard 2000 [@Gabard_2000] or perhaps a variant of Viro’s conceptual argument we know that for all intermediate values there is a representative of type II (safe for $M$-curves). Of course the conceptual argument involves the Klein-Marin theorem (\[Klein-Marin:lem\]), since when lowering its number of component through a Morse surgery a curve of type II cannot become of type I. (Note yet that this is not enough to reprove the little theorem of Kharlamov-Viro-Gabard exposed in Gabard 2000 [@Gabard_2000] in a conceptual Morse-theoretic fashion.) At any rate it shows that the type II is ubiquitous at all levels of the “pyramid” as measured by the basic invariant $r$ (number of real circuit), safe at the maximal $M$-level ($M=g+1$). The behavior of the function $\rig(m)$ is highly mysterious. It gives a measure of the flexibility of the video game allied to Hilbert’s sixteenth problem. You see on the screen $\RR P^2$ (essentially our retina) two curves of some fixed degree $m$ presenting the same isotopic topology (i.e. distribution of ovals), can you pass continuously from one to the other by moving the joystick in the hyperspace of all curves while avoiding the discriminant $\disc$? If you can [*always*]{} achieve this goal $\rig(m)=+\infty$ (you win always the game). If not $\rig(m)$ measure the smallest number of “ovals” (better real branches) where you can loose the game. A priori $\rig(m)$ could be as low as $\rig(m)=1$ when $m$ is large say $m>10^3=1000$, but some topologist expect the Hilbert-video-game to be a more flexible one, e.g. Rohlin-Viro-Itenberg positing rather that $\rig(m)\le 2$ for all $m$. What is (inside each chamber of the discriminant) the curve with largest systolic ratio, i.e. the most healthy against infarctus (=hearth attack). Of course all this would be computed w.r.t. the Fubini-Study metric, or maybe the uniformizing metric. It seems likely that flows allied to such functionals ought to give some information on the above problem, essentially because when the systole shortens we approach the discriminant. We learned the following from an e-mail of O.Viro (dated \[26.01.13\] in Sec.\[e-mail-Viro:sec\]): If the one-oval scheme (unifolium) is rigid then the contraction conjecture holds true for curves with one oval. Let $C_{2k}$ be an even order smooth curve with one oval of degree $2k$. It is enough to construct a contraction for a specific curve $F_{2k}$ of degree $2k$. One can consider the Fermat contraction (in affine equation) $x^{2k}+y^{2k}=\rho^{2k}$ with $\rho\to 0$. The latter shrink to a point but alas the tangent cone is not really that of an ordinary solitary node but rather possess $2k$ branches which form $k$ conjugate pairs corresponding to the $2k$-th roots of $-1$. So we need a slightly different contraction toward a solitary node. This can probably be done explicitly or more loosely by taking a curve of the form a circle $E_2$ union a $D_{2k-2}$ which is invisible, while shrinking the radius of $E_2^{\rho}\colon x^2+y^2=\rho^2$ and smoothing by Brusotti the union $E_2^{\rho}\cup D_{2k-2}$. Now let us estimate the indefiniteness $\indef(m)$, i.e. the lowest altitude at which there is a scheme of indefinite type. This amounts to construct an indefinite scheme of minimum height. This is fairly easy as shown by the following series of type I curves of even degree (upper row of Fig.\[Indefinite:fig\]). The bottom row of this figure exhibits type II curves with the same schemes. This is obtained by starting with a type II sextic while adding conics. Since type II is a genetically dominant character, all successors are also of type II. Hence all those schemes are indefinite and have the minimal height permissible by Rohlin’s inequality (\[Rohlin’s-inequality:cor\]), namely two units above the corresponding deep nest. -5pt0 -5pt0 A similar series is easy to find in odd degrees (Fig.\[Indefiniteodd:fig\]) and hardly requires any further comments. This proves the: -5pt0 -5pt0 For $m\ge 5$, $\indef(m)=[(m+1)/2]+2=[(m+5)/2]+2$. For $m\ge 5$, we have the estimate $\rig(m)\le [(m+5)/2]$. This implies a certain rigidity in the video game, and is merely a consequence of Rohlin’s work. Of course it would be miraculous if this estimate is sharp, prompting a maximal flexibility of the video game. For $m=5,6$ it is certainly sharp by the work of Kharlamov 1981 [@Kharlamov_1981/81] and Nikulin 1979 [@Nikulin_1979/80], respectively. So we must concentrate on degree 7, or 8, and by Rohlin’s inequality the required obstruction must necessarily be of a somewhat deeper nature than via Klein’s types. Searching obstructions to rigid-isotopy below height $DEEP+2$ ------------------------------------------------------------- \[31.01.13\] Here the technology is due to Marin-Fiedler and involves the lock allied to the subscheme $S$ of degree 7 of symbol $\frac{3}{1}\sqcup J$ (3 ovals enveloped in one oval and a pseudoline $J$ outside). If we trace the triangle of 3 lines through the 3 empty ovals (Fig.\[Locks:fig\]a), each line has 7 real intersections (saturating Bézout). It follows that 2 schemes of degree 7 enlarging $S$ cannot be rigid-isotopic as soon as the distribution of the remaining ovals past the 3 lines is different. Alas $\indef(7)=4+2=6$, hence to beat this we must find a pair of isotopic but non rigid-isotopic curves with $r\le 5$ circuits, which is already the height of the Marin-Fiedler lock $S$. Hence this method seems not suited to our goal. -5pt0 -5pt0 We could change the lock into $\frac{2}{1} 1\sqcup J$ (Fig.\[Locks:fig\]b), but this has still height $5$. Another choice is Fig.\[Locks:fig\]d but this has also height 5. Another lock could involve a conic through 5 ovals (Fig.\[Locks:fig\]e) but this is not locked as $10< 2\cdot 7=14$. Another idea is to use [*pseudo-locks*]{} like on the second row. Alas a line is not dividing $\RR P^2$. Perhaps one can construct a lock by aggregating the pseudoline $J$ to the lock (cf. Fig.\[Locks:fig\]f), whence our name [*pseudo-lock*]{}. Then the red line union the pseudoline $J$ divides $\RR P^2$, and so the location of a fifth oval could be an obstruction to rigid-isotopy. Of course Fig.\[Locks:fig\]g is not interesting being saturated (maximal scheme). Fig.\[Locks:fig\]h could be employed as the former Fig.f. For this to work one should have an isotopic-invariant way to distinguish both residues to the [*augmented-lock*]{} consisting of the red line plus the pseudoline. Alas in view of the symmetry of the lock it seems that there is little chance to distinguish invariantly both halves (of the augmented lock). One could imagine to move from the empty oval to the deep oval (on Fig.\[Locks:fig\]f) along the line while choosing the route not intersecting the pseudoline $J$. W.r.t. this oriented segment there would be a left and right hand side residual to the lock. This concept is perhaps invariant under isotopy, and there is some little chance to detect 2 septics with $r=5$ which are isotopic but not rigid-isotopic. Such a pair of septics is constructed on Figs.\[Locks:fig\]i,j, where the remaining oval lies either of the left (Fig.i) or on the right (Fig.j) of the oriented red segment from [*the*]{} empty unnested oval to the empty nested oval. Does this prove both curves being not rigid-isotopic? Maybe not since both are mirror images under a symmetry in $G=PGL(3,\RR)$, which is a connected group ($\RR P^2$ being non-orientable there is no way to reverse orientation) and so there is a path in $G$ from the identity to the mirror transformation. Applying this path to the first curve yields a rigid-isotopy to the second curve. So where is our former argument faulty? Culpability seems to be the italicized “the” some few line above. Indeed there is on Fig.i no canonical choice for the origin of the arrow, and if instead we had chosen it in the other (outer) oval then the free (unlocked) oval would of course sit on the right (instead of left) of the red arrow. We can try to remedy this defect by allowing only one outer oval, but then there is another inner oval and there is no canonical way to choose it (cf. Fig.\[Locks:fig\]k). One could hope that one of both inner ovals is distinguished, say by complex orientations (but no chance as we are in the “post deep-nest case” $r=5=4+1$ hence nondividing). \[01.02.13\] The situation becomes more favorable if we look at locks in degree 9, especially the one depicted on Fig.\[Locksdeg9:fig\]l. Then there is a canonical way to trace an arrow between the deep ovals (say from the less profound to the more profound one as on Fig.\[Locksdeg9:fig\]l). This is invariantly defined in case the remaining oval (dashed) lies outside the largest nonempty oval (as on Fig.\[Locksdeg9:fig\]l). Then the choice of this arrow is canonical and it is hoped that the position of the dashed oval on the left versus right of the arrow (augmented by the pseudoline) affords an obstruction to rigid-isotopy. It is easy to manufacture an algebraic curve realizing this schematic lock, cf. Fig.\[Locksdeg9:fig\]m where the free oval is righthanded. It causes no trouble to find a similar picture with the free oval lefthanded. This would give a nontrivial obstruction to rigid-isotopy below Rohlin’s temperature $\indef(m=9)=5+2=7$, namely at $r=6$. In particular Fig.m would not be rigid-isotopic to its mirror image. This violates however the above argument using connectedness of the group $PGL(3,\RR)$. Of course our mistake is that in the nonorientable $\RR P^2$ there is no consistent way to distinguish the left from the right. More precisely while it is possible to orient the red line from the less massive to the deepest oval, when the latter intercept the pseudoline there is no way to choose a left or right sense to bifurcate as the pseudoline itself lacks a preferred orientation. -5pt0 -5pt0 The method becomes more effective if we permit one more ovals. Then the two “free” (dashed) ovals can either be separated by the augmented lock $L\cup J$ or not (Fig.\[Locksdeg9:fig\]n). Both cases do occur as shown by Figs.\[Locksdeg9:fig\]o,p. Both depicted curves are of type II (inspect the little 3 arrows and the negative smoothing right above it on Fig.\[Locksdeg9:fig\]o). The same local pattern appears on Fig.\[Locksdeg9:fig\]p, which is thus also of type II. However both curves are not rigid-isotopic, because during the rigid-isotopy the two free “dashed” ovals of Fig.\[Locksdeg9:fig\]n cannot traverse the red line which is Bézout-saturated nor can they traverse the pseudoline. This is a little success of the Fiedler-Marin method, alas occurring at the same height as the indefiniteness $\indef(m=9)=7$. A similar example can be found already in degree 7, since we do not actually require to orient the line, compare Fig.\[Locks2:fig\]a,b which should be self-explanatory. Note again that both septics on Fig.\[Locks2:fig\]c,d are of type II, yet not rigid isotopic. This would be worth stating as a lemma since it is a little variant of the Fiedler-Marin method (with now separation caused by the added pseudoline). However this does not answer our puzzle of detecting obstruction to rigid isotopy below the critical temperature $DEEP+2$. It is then tempting to lower to degree 5, while considering the lock Fig.\[Locks2:fig\]e,f, but then alas we lack a canonical choice for the red line. One can try other locks in degree 7, like Fig.\[Locks2:fig\]g, but then we lack again canonicalness. Still one could make some choice and propagate it consistently during the isotopy. So we get Figs.\[Locks2:fig\]h,i and arrive at the fallacious conclusion that the curve is not rigid-isotopic to itself. This nonsense helps emphasizing the importance of the lock being somehow God-given by the curve, and we (human beings) making minimalist intervention upon the creation. -5pt0 -5pt0 It seems that detecting obstructions to rigid-isotopy beyond Klein’s type and below the critical temperature (=indefiniteness $\indef(m)$) is a hard business requiring completely new ideas, or at least some better acquaintance with the Marin-Fiedler obstruction. \[01.02.13\] Paraphrasing, the method of the lock does not seem to obstruct rigid-isotopies below the indefiniteness $\indef(m)$, i.e. the lowest height of an indefinite scheme. So perhaps the first obstruction to rigid-isotopy is given by Klein’s type and occurs at height $\indef(m)=[(m+1)/2]+2=[(m+5)/2].$ In that case the rigidity index $\rig(m)$ would be highest possible equal to the indefiniteness $\indef(m)$. \[02.02.13\] Let us summarize the discussion. For any degree $m$, there is a deep nest with $r=[(m+1)/2]=:DEEP$ real branches. Two units above the latter’s height it is easy to construct curves having the same real scheme yet different types (I vs. II) hence not rigid-isotopic. Using the method of the lock it is even possible to exhibit at this height curves of degree 7 or 9 having the same real scheme and the same type II, yet not rigid-isotopic. Probably the method described extend to all other odd degrees. However, it seems much more tricky and actually the locking method seems incapable detecting obstruction below this height, starting thus at height one unit above the height of the deep nest. Could it be that all schemes at or below this height are rigid, i.e. any two curves representing it are rigid-isotopic. [*Minor question (skip)*]{}.—As a minor problem we suspect that for all odd degrees $m\ge 7$ there is a non-rigid scheme at height $DEEP+2$ containing a pair of type II curves which are not rigid-isotopic. This is probably easy and merely involves extending into series the examples of Figs.\[Locks2:fig\]c,d and \[Locksdeg9:fig\]o,p. \[03.02.13\] [*Main problem*]{}.—So we first focus on the case $m=7$. Let us denote by $\Delta=DEEP=[(m+1)/2]$ the height of the deep nest. Our goal is to find obstruction to rigid-isotopy (strictly) below height $\Delta(m)+2$. For $m=7$, we have $\Delta=4$, and so we look at schemes with height $r=5$. Several cases occur and are primarily the schemes $$\frac{3}{1}J,\quad \frac{2}{1}1J,\quad \frac{1}{1}2J,\quad 4J,$$ where we use Gudkov’s notation and $J$ denotes the pseudoline (unique up to isotopy). The corresponding schemes are depicted on Fig.\[Locksdeg7:fig\], where the locks are depicted as red thick-lines which are Bézout saturated, while dashed-lines are not. The philosophy of the locking method is that a free oval cannot traverse during a rigid-isotopy the lock (without violating Bézout) and therefore the distribution of additional ovals [*among the residual components of the lock*]{} ([*past the lock*]{} for short) has to be respected. If is [*not*]{}, then we have an obstruction to rigid-isotopy. The dramaturgy in our case, where the height is as low as $r=\Delta+1$, is that we do not have any such additional ovals available (all having been consumed by the lock so-to-speak). On Fig.a we could kill the nonempty oval, but then we loose Bézout-saturation of the red-lines. -5pt0 -5pt0 So we need some much deeper idea. One idea is to look how the locked ovals themselves are separated by the lock. This seems however to lead nowhere. Indeed examine the case of the scheme $\frac{1}{1}2J$ (i.e. Fig.\[Locksdeg7:fig\]c), where there is a menagerie of possible disposition of the pseudoline $J$ (Fig.\[Locksdeg7:fig\]e). To effect a nice separation we include the pseudoline into the lock. The pseudoline plus the 2 thick Bézout-saturated lines effects a separation in 4 zones, yet whatever the situation of $J$ the disposition of ovals in those zones is still the same. At least so are the number of residual components in each of these zones, weighted on Fig.\[Locksdeg7:fig\]e by the corresponding number of components $2,3,3b,4$. One can even play more sophisticated games by choosing one of this zone in some invariant manner. For instance given the 2 points of intersections of the thick lines with $J$, we may link them to the deep nest along the thick lines while choosing the way avoiding the dashed line, and close this by the piece of $J$ cutting the dashed line an even number of times (counted by multiplicity). Since this canonical curve $J_0$ cuts the dashed lines an even number of times, it is null-homotopic and bounds a unique disc, which is our canonical region. Alas one checks (experimentally) on Fig.\[Locksdeg7:fig\]e that it always contains 2 components of the scheme. There is a dual curve constructed by taking the segments linking the points of $J \cap L_i$ to the deep nest via the path cutting once the dashed line, and aggregating the same portion of $J$ as above. This Jordan curve still cuts $L_3$ (dashed line) an even number of times, and so bounds a unique disc. The latter (alas) always contains 4 components of the curve. Another little idea we had, is to mark for each point in $L_i\cap J$ the vertices of the locking triangle which looks closest to the intersection point while travelling on the given $L_i$. However as shown by Figs.\[Locksdeg7:fig\]f,g this is completely insensitive to a variation of the position of the pseudoline. Repeating ourselves, it seems that the [*method of the lock*]{} fails to detect any obstruction to rigid-isotopy at height $\Delta+1$ (or below). Accordingly one may suspect that there is no such obstruction. Here is an idea. Given a smooth $C_7$, there is a unique pseudoline $J$. Let us speculate about a large deformation $C_7 \to C_6 \cup L_1$ toward a sextic plus a line. This is supposed to be a path in the space of curves avoiding the discriminant sole for its extremity. In particular the split curve is isotopic to the original $C_7$. We call this the rectification conjecture: [(Rectification conjecture=RC)]{} \[rectif-conj:conj\] Given any (smooth, real, plane) curve of odd order $C_{2k+1}$ there is a deformation in the large toward a curve $C_{2k}\cup L_1$ where $L_1$ is a line. [*Objection*]{} \[07.04.13\] Already for quintics, this formulation is sloppy: take an $M$-quintic (hence with symbol $6J$), while quartics can have at most 4 ovals. If this large deformation is implementable (more about this soon), then we deduce that $r(C_7)=r(C_6)+1$. If the given septic scheme has height $r\le \Delta(7)+1=4+1=5$, then the sextic has $r\le 4=\Delta(6)+1$. But in this low range sextic schemes are rigid by Nikulin’s rigid classification enhancing the Gudkov-Rohlin table (cf. Fig.\[Gudkov-Table3:fig\]). Hence we are inclined to think that septic schemes are rigid below height $\Delta+1$. Indeed given 2 septics which are (soft) isotopic, i.e. belong to the same real scheme, we apply the rectification conjecture (\[rectif-conj:conj\]) twice to deduce sextics with the same real scheme and of low height $r\le 4$, hence rigid-isotopic. Now using a path between the split curves of degree $6+1$ and using a version of Brusotti’s theorem with parameters (yet to be formulated) one could argue that the 2 given septics are rigid-isotopic. The proof would be completed. A brief word in favor of the conjecture (\[rectif-conj:conj\]). Given an odd order curve there is a unique pseudoline, and one may measure its length (w.r.t. the round elliptical geometry on the real projective plane $\RR P^2$). Obviously the (genuine) line is the pseudoline of minimum length, namely $\pi=3.14\dots$ if we work on the unit sphere as preferred double cover of $\RR P^2$. Hence for this functional (length of the pseudoline) the gradient lines ought to converge toward curves splitting off a line. (Maybe one can also look at the total geodesic curvature of the pseudoline as another competing functional doing the same job.) Having this we may dream of a grand inductive process reducing the whole problem of rigid-isotopy (at least below the range $\Delta+1$) to Nikulin’s seminal theorem on sextics (itself relying on deformation theory of K3 surfaces). This would lead to a sharp estimation of the rigidity index $\rig(m)$ of the previous section as being equal to $\Delta(m)+2$. However even for degree 8, this looks hazardous. One could imagine two modes of deformation of a $C_8$ to either a septic plus a line $C_7\cup L_1$ or a $C_6\cup E_2$. The latter looks dubious for the (8)-scheme consisting of 3 nests of depth 2 (of height $r=6=\Delta+2$), since removing one oval one has still the line through the two remaining nests creating 8 intersections (too much for a $C_6$). Yet the latter is precluded as we restrict to schemes of height $\le \Delta+1$. Listing all of them we find in Gudkov’s notation the following list of schemes (cf. Fig.\[Locksdeg8:fig\]): $$\frac{4}{1},\rau\frac{3}{1}1,\rau\frac{2}{1}2, \rau\frac{1}{1}3,\rau 5,\rau (1,\frac{1}{1}2), \rau (1,\frac{3}{1}), \rau (1,\frac{2}{1}1), \rau (1,\frac{2}{1})1, \rau (1,\frac{1}{1}1)1, \rau (1,\frac{1}{1})2,\rau \frac{1}{1}\frac{1}{1}1, \rau \frac{2}{1}\frac{1}{1}.$$ -5pt0 -5pt0 Albeit messy, our picture (Fig.\[Locksdeg8:fig\]) is supposed to take the census of all possible degenerations $C_8\to C_6\cup E_2$ which are Bézout permissible. Of course we do not claim that all these moves are algebraically realized, but at least Bézout gives no obstruction. Alas it is far from obvious (unlike in the odd degree case) which functional is capable effecting the large structural deformation (LSD) of “conification” $C_{2k}\to C_{2k-2}\cup E_2$ splitting off a conic $E_2$. Naively we may expect that it is always some of the empty oval which shrinks to a solitary node, but that soon before getting extinct he splits off an infinitesimal circle (or ellipse). This could involve a sort of isoperimetric functional measuring rotundity of ovals, and the allied lines of steepest descent (or ascension). Note however that for the $8$-scheme $(1,\frac{3}{1}), (1,\frac{2}{1}1),(1,\frac{2}{1})1$ (the 3 firsts of the third row on Fig.\[Locksdeg8:fig\]) we cannot “conify” the empty ovals without violating Bézout. Indeed removing one of the 3 possible empty ovals leads to scheme containing the deep nest of depth 3 as a (strict) subscheme. Of course those (8)-schemes really exist, as depicted on Fig.\[Locksdeg8:fig\]b. A priori nothing precludes a degeneration like Fig.\[Locksdeg8:fig\]c, where a nonempty oval would be “conified”. [(Conical/ellipticity conjecture=EC) \[inserted 05.02.13\]]{} Given a (smooth, real, algebraic, plane) curve $C_{2k}$ of even degree $m=2k$ with few ovals (i.e. $r\le \Delta(m)+1$ where $\Delta(r)=k$ is the number of ovals of the deep nest of degree $m$) there is a deformation (=rigid-isotopy safe its extremity) toward a curve $C_{2k-2}\cup E_2$ where $E_2$ is an ellipse, or equivalently a circle up to projectivity. Alas this cannot always occur by extinction of an empty oval, but sometimes by inflation of a large oval (perhaps via an isoperimetric gradient-flow). The difficulty with this conjecture is that unlike for its odd degree avatar (\[rectif-conj:conj\]) we lack a canonical functional to be minimized like the length of the pseudoline. (The line is the shortest pseudoline, and being non-null-homotopic it is like a systole.) In the even degree case all ovals are null-homotopic and there is no systole in $\RR P^2$. Of course there could be a systole on the Riemann surface of the complexification. Alternatively one may replace the systolic problem by an isoperimetric one taking also area into account. Let us introduce the isoperimetric ratio ($\isop$) of an oval as its length squared divided by the area of its bounding disc, all in reference to the round elliptical geometry on $\RR P^2$. In Euclidean geometry this is minimum for a circle $(2\pi \rho)^2/(\pi \rho^2)=4 \pi=12.566\dots$. For a large circle near the equator this can be as close as we please to $(2\pi)^2/ (2 \pi)=2\pi=6.28\dots$, which is smaller. This is probably the absolute infimum if we demand the oval on the unit sphere to be disjoint from its antipode. Now we could hope that the minimum isoperimetric ratio of all ovals leads to a functional whose gradient lines tend to inflate the most rotund oval toward an ellipse (rotundity being measured by the isoperimetric ratio). This could give the required degeneration. Perhaps in the limit the most rotund oval degenerate to a pair of lines (double line) and suppressing one of those leads to a odd degree curve of degree one less. This would give the other mode of degeneration: Given any $C_{2k}$ of even degree of height $\le \Delta+1=k+1$, there is a rigid-isotopy safe extremity toward a curve $C_{2k-2}\cup (L_1)^2$ splitting off a double line $L_1$. More precisely the orthogonal trajectories of the rotundity functional (measured by the minimum isoperimetric ratio) drives any such curve toward such a curve in a canonical fashion. If this conjecture holds true then we would have a sharp estimate of the rigidity index $\rig(m)$ for all degrees \[end insert 05.02.13\]. All what we are saying sounds very optimistical, and we are still very far from having a decent understanding of this problem of rigid-isotopy (strictly) below the height $\Delta+2$. We can hope that the method of the lock is more efficient in degree 8 than it was in degree 7 (still confining our attention to heights $\le \Delta+1$). Fig.\[Locksdegree8:fig\] depicts some of them. The method of the lock is a jewel discovered in the late 1970’s by Marin and Fiedler independently (all being inspired by V.A. Rohlin’s work). It involves basically the idea of attaching in the most canonical way to a given curve a certain red configuration acting as a separator. More precisely special attention is paid to red thick lines which are Bézout-saturated, so that the remaining ovals of the curve cannot traverse this line during the isotopy. So basically we choose a triad of points inside some “deep” ovals and link them by a triangle of lines. Of course the choice of the points is not perfectly canonical, but we choose them inside the disc bounding an empty oval. The Marin-Fiedler trick is quite reminiscent of what Grothendieck calls “le principe des choix anodins” (in Esquisse d’un programme 1984 [@Grothendieck_1984/1997-esquisse-d'un-programme]) that whenever we make some choices within a contractible space the construction is nearly canonical, hence robust and fruitful. It is also reminiscent of the moving-frame method of Darboux-Cartan (repère mobile), since during the rigid isotopy will really move the whole triangle. -5pt0 -5pt0 On the 3 first pictures of the 2nd row of Fig.\[Locksdegree8:fig\] we have a perfect lock by a triangle consisting of 3 lines which are Bézout-saturated. Alas we have no more ovals left to separate and the method looks inoperative. So let us look at the next degree $9$, and list all the schemes at height $\Delta+1=5+1=6$. It seems plain that this merely amounts to add a pseudoline to the former configurations listed in degree 8 (cf. Fig.\[Locksdegree9:fig\]). -5pt0 -5pt0 Again the big deception is that no elementary obstruction given by locking appears in view. Of course one could interpret the figure as a rectification (\[rectif-conj:conj\]) toward octics which are (hypothetically) rigid at height $\Delta+1$ (say via a reduction to sextics), and so would be our curves of degree $9$. The next real jump in complexity involves degree 10. Let us tabulate all schemes at the critical height $\Delta+1=5+1=6$ while avoiding any Gudkov symbolism (cf. Fig.\[Locksdegree10:fig\]). This is elaborated as follows. Start from any configuration, especially the maximum elements sembling highly concentric and protected medieval settlements like $(1,1,1,1,1,1)$, or $(1,1,1,1)(1,1)$, $(1,1,1)(1,1,1)$, $(1,1)(1,1)(1,1)$, and then apply basically two moves freeing an oval. Vertical moves correspond to liberating a deep oval, while horizontal moves freed a superficial oval (of small depth). Sometimes there are ovals at 3 different depths so that we have also a 3rd oblique move. The red framed schemes are prohibited by Bézout, yet are useful as generator (under the described moves) of other schemes that otherwise are easily overlooked. -5pt0 -5pt0 \[04.02.13\] It seems evident at this stage that there is a combinatorial law (which overwhelms my intelligence) impeding the the locking method to act as an obstruction to rigid-isotopy. Looking at all possible locks on Fig.\[Locksdegree10:fig\], no obvious obstruction to rigid-isotopy strikes the vision. In contrast on the basis of the same picture, one may argue that erasing a suitable oval all our (10)-schemes reduce to one of degree 8, and if this cancellation is geometrized via the conification (elliptization) conjecture we could deduce rigidity of all the (10)-schemes at height $\Delta(10)+1=6$, from that of the corresponding (8)-schemes, which in turn was reduced to (6)-schemes where low-height rigidity holds true by virtue of Nikulin’s theorem (1979 [@Nikulin_1979/80]). As a little experiment imagine the curve of degree $2k$ to have 3 nests of depth $d_1,d_2,d_3$. Since we are at height $\Delta(2k)+1=k+1$, we have $k+1=r=d_1+d_2+d_3$. Let $L_1,L_2,L_3$ be 3 lines passing through the deep nests and suppose them Bézout-saturated, then $d_1+d_2,d_2+d_3,d_1+d_3$ are all equal to $2k$, and thus summing and dividing by two we infer that $d_1+d_2+d_3=3k$, which is much greater than $r=k+1$. If instead of 3 deep nests we have one deep nest containing 3 little ovals, then the 3 lines through them supposed Bézout-saturated cut the curves in $4+2d=2k$ real points where $d$ is the depth of the nest. Hence $r=d+3=(d+2)+1=k+1$, so that all ovals are exhausted by the lock (and nothing remains left to be separated). Such arguments seem to extend to all other schemes of Fig.\[Locksdegree10:fig\]. There sometimes we lock with only two totally real lines like e.g. on the scheme $(1,1,1,1)2$ lying near the center of Fig.\[Locksdegree10:fig\] (right above the 2 anti-Bézoutian schemes). Two lines suffice to separate the plane $\RR P^2$, but here again the construction of the lock consumes all the ovals at disposal. In summary it seems hopeless to find an obstruction to rigid-isotopy at or below the height $\Delta(m)+1$ (at least via the lock-method of Marin-Fiedler). As a last chance, consider the (10)-scheme of Fig.\[Locksdegree10:fig\] right before the “mild” arrow, that is $(1,(1,\frac{1}{1}1)1)$. This is distinguished by having 3 empty ovals at different depths. So we can link them by a $2$-simplex with boundary oriented as going from the deepest to the middle deep and then to the less profound oval closed back to the deepest one. This would induce a certain orientation on the inside of the largest oval (as usual ovals being ordered by inclusion of their insides). The problem however is that while the 2 lines through the deepest oval are saturated (hence there is preferred pathes joining them in the inside of the maximal oval), the third is not and so there is no preferred way to join the middle empty oval to the less deep one (compare Fig.\[Lock10:fig\]). However we could argue that whatsoever the way chosen we get the same orientation (compare Figs.\[Lock10:fig\]b and c). On the latter figures we follow the line until reaching the maximal oval $O_m$ and then follow the latter. The problem is which direction to choose when we meet $O_m$. A priori there is no preferred sense to bifurcate, but we may choose the path such that the circuit $1\to 2\to 3 \to 1$ does [*not*]{} enclose the deep oval of depth 3 (i.e. the one containing the point $1$). This has no intrinsic meaning unless we take the precaution of first rounding the corner at the vertices 1 as shown on Figs.\[Lock10:fig\]e,f. Note that there is a unique way to put near $1$ an arrow circulating on the deepest oval in such a way that we do not intercept the lines $1,2$ and $1,3$ too frequently (i.e. only twice instead of 4 times). This as an intrinsic meaning since those lines are saturated. -5pt0 -5pt0 As a result any ten-ics $C_{10}$ belonging to the discussed scheme would have a canonical orientation of the inside disc of its maximum oval (hence of the latter as well). Of course this (semi-)local orientation of the maximal oval propagates continuously through a rigid-isotopy, but it seems that as $\RR P^2$ is nonorientable no obstruction to rigid-isotopy can be derived from this complicated trick. So even if two such curves $C_{10}$ would have opposed canonical orientation over some region of overlap of there maximal discs (bounding the maximal oval) this would not impede them being rigid-isotopic. Notwithstanding since the maximal disc (the inside of the maximal oval) is oriented canonically, we may look at the deep line $1,2$ through the deepest empty ovals. This line does not separate $\RR P^2$, but certainly separates the maximal disc. Further the deep line is oriented by going from $1$ to $2$ while staying inside the maximum oval. Using the canonical orientation there is a left and right hand side of this deep line inside the maximal disc. The location of the superficial empty oval as being right- or left-sided could give an obstruction (since the 3rd superficial oval $O_3$ is not permitted to traverse the deep line during the rigid-isotopy). So if like on Fig.\[Lock10:fig\]a the superficial (empty) oval $O_3$ is left-sided with respect to the oriented line $1,2$ and the canonical orientation it will stay so during for all curves explored by the isotopy, in particular for the end curve. So naively it would suffices to apply a horizontal axis $1,2$ symmetry to Fig.\[Lock10:fig\]a (and realize the scheme geometrically which causes no difficulty via Brusotti) as to find a curve with $O_3$ sitting on the other (right) side. However we must really work with the canonical orientation of the maximal disc, looking at Fig.\[Lock10:fig\]abis shows that the oval $O_3$ really sits on the left albeit sembling on the right (where of course left has to be interpreted as the half pointed by the canonical orientation). With all these confusing remarks, it should be clear that there is no hope to detect an obstruction to rigid-isotopy. \[05.02.13\] We can also study the embryology of the scheme as shown on Fig.\[Lock10bis:fig\] depicting a nearly exhaustive list of collision which an oval can acquire with the non-saturated line $2,3$ through the 2 most superficial ovals. This represents the possible cytoplasmic expansions of the ovals, but does not [*per se*]{} afford obstructions to rigid-isotopy since all configurations are linked to the initial one in some starlike fashion. -5pt0 -5pt0 At the opposite extreme of such Bézout permissible moves, we have the following 3 motions forbidden by Bézout where one of the empty oval cannot traverse the saturated thick red line (Fig.\[Lock10tris:fig\]). So if we transgress the Bézout obstruction by letting the oval traverse the dead-line then we get the configuration of the second row of Fig.\[Lock10tris:fig\] which are priori could be non-rigid-isotopic to the initial one. Alas there is still this argument of symmetry using the connectedness of the group $PGL(3, \RR)$ which prevents one to conclude that the configuration pre- and post-transgression are not rigid-isotopic. -5pt0 -5pt0 Trying in vain to corrupt Nikulin (via Marin-Fiedler) {#Nikulin-corruption:sec} ----------------------------------------------------- \[05.02.13\] It is quite tempting (for dummies) to see if the method of the lock (Marin-Fiedler 1979–1980) can parasite Nikulin’s rigid-isotopy classification of sextic (1979 [@Nikulin_1979/80]). Of course this is not to palish the glory of Nikulin’s theorem which is perhaps the deepest jewel ever obtained along the lines of Hilbert’s 16th problem, but rather an experimental game emphasizing the profundity of Nikulin’s result. Usually, the more a theorem looks unbelievable, the deeper it stands. For instance we may start with the basic scheme $\frac{3}{1}$ of degree 6 (locked by the triad of lines through the 3 pairs of deep ovals), and enhance it by adding 2 outer ovals to get the scheme $\frac{3}{1}2$. We look at the distribution of outer ovals past the locking triangle, which a priori can be as on Fig.\[Lock6:fig\] either monopartite or bipartite. If one is capable to exhibit two curves $C_6$ with distinct distributions then both curves are not rigid-isotopic, for during a rigid-isotopy the unlocked ovals cannot traverse the (moving) triangle which is already Bézout-saturated. (Of course the locking triangle works as well for $\frac{2}{1}1$, but then there is nothing to separate, and if we add ovals then canonicalness of the triangle is spoiled.) On tracing explicit sextic curves $C_6$ via the small perturbation method applied to configurations of 3 conics we always find the same mono-partite arrangement where both ovals lies in the same component residual to the triangle (Fig.\[Lock6:fig\]). -5pt0 -5pt0 It seems impossible to corrupt Nikulin’s result. As the scheme $\frac{3}{1}2$ has height $r=6=\Delta+3$, three units above the deep nest it is necessarily of type II (by Klein’s congruence) and therefore Nikulin’s theorem actually implies the: Any sextic $C_6$ belonging to the scheme $\frac{3}{1}2$ is such that the triangle through the $3$ deep ovals does not separate the outer ovals. (We do not know whether this can be proved in an elementary fashion without using the technological arsenal behind Nikulin’s theorem.) \[07.04.13\] Update: yes we can, cf. Le Touzé in Sec.\[LeTouze:sec\]. \[06.02.13\] Of course we may also add to $\frac{3}{1}$ more ovals and examine the resulting distributions past the deep triangle. That is we consider the schemes $\frac{3}{1}\ell$, where $2\le \ell \le 5$ according to Gudkov’s table (Fig.\[Gudkov-Table3:fig\]). Consider first the scheme $\frac{3}{1}3$. Smoothing 3 ellipses we can realize this scheme in two fashions either of type I or II (Fig.\[Lock6bis:fig\]). However in both cases the locking triangle through the deep (odd) ovals does not separate the 3 outer ovals. This is quite surprising as both curves are not rigid-isotopic, one could have expected that the lock-method to detect the obstruction. We may also realize this scheme via a variant of Hilbert’s oscillation method, but again the distribution of the 3 outer ovals is the same mono-partite one (at least on the Walt-Disney depiction of Hilbert “à la Gudkov”). -5pt0 -5pt0 Since both types I, II have representatives with the same distribution, Nikulin’s theorem implies the: Any sextic $C_6$ with scheme $\frac{3}{1}3$ (be it dividing or not) is such that the triangle through the $3$ deep ovals does not separate the $3$ outer ovals. Next examine the scheme $\frac{3}{1}4$. Here we start with a schematic picture à la Hilbert-Gudkov producing the curve $\frac{4}{1}4$ (cf. Fig.\[Lock3-14:fig\]a) which has too much inner ovals (4 instead of the 3 desired). Such a Hilbert-vibration is realized by Fig.b. A suitable smoothing gives Fig.c. The latter has actually a companion generated by smoothing differently the 3 inner nodes. In both cases however the deep triangle does not separate the 4 outer ovals. Fig.d depicts a Hilbert vibration perturbing the union of both ellipses to the Zeuthen-Klein Gürtelkurve, but the quartic $C_4$ would then intersect too frequently (at least 10 times) the conic. Such a vibration is therefore precluded. The dual vibration however (Fig.e) is Bézout compatible (as the $C_4$ intersect $8$ times the two conics). It is questionable if such a vibration exists as the dual does not. Anyway let us (somewhat liberally) smooth Fig.e to get Fig.f, a somewhat funny curve belonging to the scheme $\frac{3}{1}3$. Tracing the triangle through the deepest ovals is somewhat challenging, but does not seem to effect a division of the 3 outer ovals. A priori the depiction could be like on the surrealist détail (i.e., the median oval lying on the “left” of the line through the other 2 inner ovals), but this does not even seem to affect our issue about distribution of outer ovals past the lock. Fig.g depicts another mode of vibration which still overwhelms Bézout. Fig.h depicts yet another mode of vibration essentially dual of Fig.b, but which also overwhelms Bézout. It is a bit puzzling that not any admissible vibration seems to admit a dual vibration. -5pt0 -5pt0 At any rate if we believe in Nikulin’s theorem (as we should since it is Soviet mathematics of the best stock) our sole Fig.\[Lock3-14:fig\]c suffices to imply (since by Klein’s congruence our scheme $\frac{3}{1}4$ is of type II) the following: Any sextic $C_6$ belonging to the scheme $\frac{3}{1}4$ is such that the triangle through the $3$ deep ovals does not separate the $4$ outer ovals. Next we consider the scheme $\frac{3}{1}5$. For this we can either look at Gudkov’s construction (Fig.\[GudkovCampo-5-15:fig\]) or at the easier construction via a variant of Harnack’s method. In Gudkov’s setting, we must presumably consider the pull-back of the triangle under the Cremona transformation and this a bit tricky to depict. This should be manageable if one is in good form but perhaps there is a more elementary direct construction via the Harnack method. -5pt0 -5pt0 So it seems fundamental to construct the over-scheme $\frac{4}{1}5$ via the variant of Harnack’s method mentioned in Gudkov 1974 [@Gudkov_1974/74 p.42], where alas no details are supplied. The smaller scheme $\frac{3}{1}5$ we are interested in should then easily be deduced by taming the smoothing. Since this has some independent interest we devote the next section to the topic. Gudkov’s variant of Harnack: construction of the $(M-1)$-scheme $\frac{4}{1}5$ ------------------------------------------------------------------------------ \[06.02.13\] We now try to fix Gudkov’s claim (in 1974 [@Gudkov_1974/74 p.42]) that a suitable variant of Harnack’s method produces the $(M-1)$-scheme $\frac{4}{1}5$. [*Per se*]{} this is not extremely original for we already managed (on the shoulder of Gudkov’s “original” construction of $\frac{5}{1}5$, cf. Fig.\[GudkovCampo-5-15:fig\]) to exhibit this scheme, yet now a more elementary method is demanded. Despite elementariness, if one is not so clever (like the writer) this game can be pretty time consuming as demonstrated by the following section. This consisted in a sequence of failing trials, and alas TeX forced us to censure most of these instructive trials as otherwise our text was not anymore synchronized with the images. For convenience the first picture (Fig.\[Harnack-Gudkovvariant:fig\]a) reminds the classical implementation of Harnack’s method of 1876 [@Harnack_1876] (little warning: in the original paper the depiction is much left to the imagination of the writer, and our picture though standard is really inspired by nice drawings available in Viro’s papers). A first idea is to put the oscillation inside the ground circle, but this looks too naive and we recover exactly Harnack’s scheme $\frac{1}{1}9$ (cf. Fig.\[Harnack-Gudkovvariant:fig\]b). -10pt0 -5pt0 Next we may consider the following variant (Fig.\[Harnack-Gudkovvariant3:fig\]a) but after much effort it only leads to the scheme $9$. Now we try another variant, and after much efforts manufacture the following picture (Fig.\[Harnack-Gudkovvariant3:fig\]b), upon which one arrives at a meta-mathematical contradiction. Namely we have constructed a sextic (of even degree) with a pseudoline! What we do is really real time mathematics with mistakes, irritation, dirty fingers, etc. At this stage the moral is double: first Harnack’s original method is a quite boring inductive process, and further it is not so easy to implement the variant advocated by Gudkov. Probably it is not so surprising that himself found not the place in his survey to make a decent picture. Geometry without pictures is not geometry it becomes alchemical arithmetico-symbolism impossible to assimilate. So we need to make more pictures and more careful ones. In fact the mistake we did is that we omitted to neutralize the curve $C_4$ of the previous step. What a grotesque mistake! -5pt0 -5pt0 So let us make a better picture, and correcting the previous one gives Fig.\[Harnack-Gudkovvariant5:fig\]a. Alas the scheme so obtained is $\frac{1}{1}8$. Our next idea was to invert the order of some oscillation as on Fig.\[Harnack-Gudkovvariant5:fig\]b, but alas this does not help much, and we obtain the same $(M-1)$-scheme $\frac{1}{1}8$ close to Harnack’s. -5pt0 -5pt0 So it seems we are in need of a more drastic form of intertwining the oscillation. This lead us to the following Fig.\[Harnack-Gudkovvariant7:fig\]a which alas still represent the same scheme. (At this stage we really feel unclever or at least unlucky.) The next part b. of the figure still remains to be done by slight permutation of the oscillation. -5pt0 -5pt0 \[07.02.13\] In fact one may argue that if by our choice of starting from an $(M-1)$-cubic we would arrive at the scheme $\frac{4}{1}{5}$ which is pre-Gudkovian (i.e. $\frac{5}{1}{5}$) then we would also be likely to realize the latter by a Harnack construction. This seems unlikely, at least experimentally and there is perhaps a known theoretical obstruction. Accordingly it is perhaps better to start from the maximal vibration and loose one oval latter through the disposition of vibrations. -5pt0 -5pt0 If one is somewhat tired (or hung over) one is even able to produce via Fig.\[Harnack-Gudkovvariant7:fig\]b the scheme $\frac{2}{1}8$ corrupting Gudkov’s knowledge or theoretical avatars like Arnold 1971’s congruence mod 4. -5pt0 -5pt0 And in reality we only get the scheme $8$: -5pt0 -5pt0 After much efforts and trials we ultimately found (the next day \[07.02.13\]) the solution as Fig.\[HarnaGudkov4-15:fig\]. The trick is to leave much room in between the vertical lines effecting the oscillation of Harnack’s method, so as to place the subsequent vibration in between. One of the difficulty we encountered before finding the solution is that since the desired configuration $\frac{4}{1}5$ is an $(M-1)$-curve one is tempted to start with an $(M-1)$-cubic. Then one can apparently loose much energy in the desert. Instead we start form a Harnack-maximal cubic obtained by slight perturbation of an ellipse $E_2\cup L$ union the horizontal line $L$ and perturb this by a triplet of vertical lines. It results the black depicted $C_3$ on the first row of Fig.\[HarnaGudkov4-15:fig\] intersecting thrice the horizontal line. The quartic curve $C_3\cup L$ is then perturbed by a quadruplets of lines. Those could be a priori be located everywhere, but we choose them in between as depicted on the figure. Here it seems quite crucial that as the number of lines is even we may concentrate the vibration on a single oval. After this vibration the large central oval looks like a pair of Ray-Ban eyeglasses (viewed in perspective). We have now a $C_4$ oscillating 4 times across the (horizontal) line $L$ and we perturb again the union. How to do this? Always by the same method but we are free to choose the location of the vibrator. A priori since an oval can vibrate an even number of times across a line we may want to choose only 4 vibrations in the “nearby glass” of the Ray-Ban and one outside. This leads to something interesting namely the scheme $\frac{4}{1}4$ (compare Fig.\[indef414:fig\], which we transported earlier in this text). Here instead we keep the 5 vibratory lines close together (this usually maximizes their vibratory impact), and all inside the big glass of the Ray-Ban, cf. second row of the figure. It remains to depict the resulting smoothing of $C_4\cup L$. The result is the red curve $C_5$ depicted but it is essential to choose this oscillation (and not the opposite one) in which case you destroy many ovals (this will be depicted concretely on the next Fig.\[HarnaGudkov3-14:fig\]). So there is something like a snake visiting the nearby glass of the Ray-Ban. This gives a $C_5$ traversing 5 times the line $L$. Smoothing the union $C_5\cup L$ produces the sextic of the 3rd row (of Fig.\[HarnaGudkov4-15:fig\]). Note that the two (red-colored) branches nearby the horizontal line are linked together at $\infty$ to form a single circuit, which we call the median circuit. More generally all branches going to infinity are connected with the diametrically opposite branch. The median circuit of the $C_6$ is clearly the unique nonempty oval. What appears naively in its interior is in reality a Möbius band (due to the diametral identification), hence its interior really contains 4 ovals. This shows that the constructed curve realizes the desired scheme $\frac{4}{1}5$. -10pt0 -5pt0 Somewhat against our expectation this curve cannot be simplified toward the scheme $\frac{3}{1}5$ we were interested in (in the previous section), as the 4 inner ovals are not coming from a vibration. Nonetheless during our exploration up to finding this premaximal scheme $\frac{4}{1}5$ nearly Gudkovian, we found a variant of the exposed construction yielding the scheme $\frac{3}{1}5$ (cf. Fig.\[HarnaGudkov3-15XXL:fig\] below). First let us choose the opposite mode of vibration as the lucky one we first depicted. This gives Fig.\[HarnaGudkov3-14:fig\]. Now the snake oscillates around the nearby glass, wind around the nose of the investigator, to loop around the second (distant) glass of the Ray-Ban, etc. (As we must optically smooth the union of both black curves $C_4\cup L$, we could a priori hope to close up an oval with the bottom half of the first close glass, but this forces a 6th intersection in $C_5\cap L$ violating Bézout.) On smoothing $C_5\cup L$ we find a curve realizing the scheme $\frac{3}{1}4$. Although not so exciting as $\frac{3}{1}5$, this is already interesting for the purpose of the previous section (namely trying to corrupt Nikulin). So the game is to trace the triangle through the 3 inner ovals and look upon the separation it effects upon the 4 outer ovals. Of course our picture has poor metrical qualities as we blew it up topologically as to see what happens within the viscera of Harnack’s method. Notwithstanding the naive green triangle depicted seems to leave unseparated the 4 outer ovals (which to me remembered appears “inside”). However upon dragging the upper vertex below while staying in the outer oval residual to the upper semi-circle, we can easily (at least on our topological picture) effect a separation. Remind (from the reasoning of the previous section) that if such a division occurs, then the rigid-isotopy classification of Slava Nikulin 1979 [@Nikulin_1979/80] is violated. Can we infer anything serious from such a topological picture of Harnack method? Maybe we can via a mental contraction of some ovals restore some metrical faithfulness in the depiction as to be sufficiently accurate to answer the (non)separation question by the fundamental triangle through the 3 (deep) inner ovals. Let us start with the observation that the initial cubic $C_3$ looks on our distorted picture (Fig.\[HarnaGudkov3-14:fig\]) more like a quintic (consider a line “parallel” to the horizontal one passing through the unique oval of the $C_3$). So in the real picture the central oscillating bump of the cubic is much less pronounced. Imagine the oval of the cubic as a sun radiating light, then there cannot be shadow lying behind the hill formed by this bump (otherwise 4 intersections with a line too much for Bézout). So the real picture is heuristically like the 4th row of Fig.\[HarnaGudkov3-14:fig\]. In particular the Ray-Ban glasses (=vibrating oval of the $C_4$) is much stretched vertically. One may argue that the Ray-Ban glass traps the oscillation, and also the resulting 4 outer ovals created in the last step of Harnack’s iteration. Accordingly it seems sufficient to use the $C_4$ as a sort of envelope. Since the 3 ovals of the quartic $C_4$ distinct from the Ray-Ban are actually (modulo infinitesimal perturbations) the 3 inner ovals of the final sextic $C_6$, and noting also that the line through two of them cannot intersect the Ray-Ban oval (Bézout), we may conclude that any triad of lines through the inner ovals of our $C_6$ does not separate the 4 outer ovals. This no-separation scenario is in accordance with our previous depiction of such a curve via the more user friendly Hilbert’s method (cf. Fig.\[Lock3-14:fig\]). Alas or fortunately our reasoning does not foil Nikulin’s theorem. We summarize this trapping argument by the: For the sextic curve $C_6$ of scheme $\frac{3}{1}4$ realized via Harnack’s method the fundamental triangle through the deep ovals does not separate the $4$ outer ovals. -10pt0 -5pt0 At this stage, it is with a mixture of happiness and disappointment that Nikulin still seems to resist our naive aggression via the Fiedler-Marin method. It remains however to look at the scheme $\frac{3}{1}5$. \[08.02.13\] How to realize it? Again several tests are required and usually we (at least the writer) lack an understanding of the predestination governing Harnack’s method. Using the technique of the microcosmic vibration “in between” we realized the schemes $\frac{4}{1}5$, $\frac{4}{1}4$ and $\frac{3}{1}4$ all hitting quite central positions of Gudkov’s pyramid (Fig.\[Gudkov-Table3:fig\]). But how to get $\frac{3}{1}5$ lying more “eccentric” on this table? Incidentally one could dream that this Harnack method we are using leads to the eclectic Gudkov scheme $\frac{5}{1}5$. Of course this would corrupt experimental evidence assembled along centennial working tradition (Harnack 1876, Hilbert 1891, Rohn 1888–1913, Brusotti 1910–1945, Gudkov 1954–1973, etc.). However we do not know (personally) a theoretical obstruction impeding Harnack’s method to produce Gudkov’s scheme. Arguably if well assimilated Harnack’s method reduces to a finite collection (for a fixed degree say $m=6$) of combinatorially distinct locations for the vibratory lines. So it suffices to explore all choices and notice that Gudkov’s scheme never appears. We do not claim to be clever enough to complete this boring exercise, but our microfilm picture perhaps contributes to this (cf. Fig.\[HarnaGudkov3-15MICRO:fig\]). \[08.04.13\] It is not clear at this stage if this picture will be publishable in the arXiv due to size limitations. But let us return to our main duty of exhibiting $\frac{3}{1}5$. Here a series of tests given in micro-film format (Fig.\[HarnaGudkov3-15MICRO:fig\] only consultable on a PC where one can zoom and alas unreadable on the paper). Alas we cannot give the pictures in decent format for otherwise the flow of pictures overrun dramatically what we have to say on the topic. We are in the realm of pure geometry were only pictures have some weight, but alas this does not seem to please my TeX-compilator, who accept at most two pictures per page. Here the second column picture of this microfilm shows an interesting variant of the scheme $\frac{3}{1}4$ where the 4 outer ovals are not directly enveloped by the “Ray-Ban” oval, and so our former argument does not readily apply here. It seems however dubious to expect a corruption of Nikulin. Without getting sidetracked by this issue, keep in mind our goal of realizing $\frac{3}{1}5$. -10pt0 -5pt0 After several trials (cf. again the microfilm Fig.\[HarnaGudkov3-15MICRO:fig\]) we arrived at the idea of using the same vibratory configuration of lines as for $\frac{4}{1}5$ safe that instead of starting from an $M$-cubic we start from an $(M-1)$-cubic. This seems to require locating one of the vibratory line inside the circle. Our final picture is Fig.\[HarnaGudkov3-15XXL:fig\]. It hardly deserves to be commented upon once it is found, except for saying that the initial cubic is to be thought of as a small perturbation of the circle $E_2$ union the line, despite sembling a large deformation thereof. The trick in tracing Harnack’s curves is always to exaggerate small perturbations as to create some free room to depict the next stage of the inductive process (vibratory pudding). This is of course possible due to the malleability of the continuum $\RR$ of real numbers. On this figure (Fig.\[HarnaGudkov3-15XXL:fig\]) we recognize again our Ray-Ban oval except that it has now acquired a “branch” (compare 2nd row of Fig.\[HarnaGudkov3-15XXL:fig\]). Again our interest is to apply the lock method of Fiedler-Marin. So we trace the triplet of lines through 3 points in the deep (inner) ovals $1,2,3$, and examine whether and how this triangle splits apart the outer ovals [*1,2,3,4,5*]{}. (Notice the importance of italicization in our notation: italics are outer ovals while roman-arabic numbers are the inner ovals.) In contrast to the Harnack curve realizing $\frac{3}{1}4$ where all the 4 outer ovals were encapsulated in the Ray-Ban oval, we notice now that the oval [*5*]{} lies outside this (Ray-Ban) oval. So our former argument does not readily apply. Notice also that the inner oval $2$ lies inside the Ray-Ban. All this is a bit puzzling but should not discourage us attempting to study the division of the outer ovals by the locking triangle for our Harnack-modified curve $C_6$. Note incidentally that the latter is not perfectly well defined as a curve unless we specify exactly the deformation constants involved in Harnack’s small perturbation method. Yet it seems natural to expect that the combinatorial data involved in our Harnack-Gudkov style description is enough to determine unambiguously a rigid-isotopy class. Hence by the Fiedler-Marin locking argument (involving merely Bézout saturation) we infer that the distribution of outer ovals within the 4 components past the lock is well-defined. It remains only to determine it. -10pt0 -5pt0 In the sequel we shall often speak of “the line passing through two disjoint ovals”. This is a slight abuse of language for such a line is not uniquely defined, and is really intended to mean choose 2 points in the insides of the 2 disjoint ovals and trace the line joining them. Of course the phrasing “the line” becomes somewhat sloppy, but when the two ovals are inner ovals then any such line is Bézout saturated, and so from the viewpoint of analysis situs there is some canonicalness. Since “the” (or a) line through the ovals $1,3$ regarded on the quartic $C_4$ cannot cut more times the $C_4$, we infer that it does not cut the Ray-Ban oval of the $C_4$ (the one oscillating 4 times across the horizontal line $L$). Next the line through the ovals $2,3$ interpreted on the quintic $C_5$ cannot cut more this curve safe for a point on its pseudoline. A similar remark holds for the line through $1,2$. All this looks a bit sterile and we really need the geometry of the picture to understand the distributional question. For this purpose, look at the 3 green-colored lines on Fig.\[HarnaGudkov3-15XXL:fig\], while enlarging slightly oval 3 as to adjust the picture. It seems then that the triangle separates oval [*5*]{} from the ovals [*1,2,3,4*]{}. Of course upon stretching further oval $3$, we could arrange that the line $2,3$ passes above oval [*5*]{} in which case the locking triangle effects no subdivision of the outer ovals. Which of both scenarios corresponds the reality? A priori the first scenario looks more likely (at least in line with our picture). Remind however the slogan (anonymous, Poincaré, etc.) “La géométrie c’est l’art de bien raisonner sur des figures mal dessinées”. Let us attempt a more realist depiction on the following figure (Fig.\[HarnaGudkov3-15XXLTEST:fig\]). Even the first right-side picture (fig.d) is not Bézout permissible (the green-line cut the cubic $C_3$ five times). Further it may be observed that the line through $2,3$ may pass “below” the series of ovals [*1,2,3,4*]{}. This is a third possible scenario in which there is no subdivision. -10pt0 -5pt0 Admittedly this question looks quite tricky to decide and requires some good idea or high optical acuity. Harnack’s method seems not ideally suited to clinch the matter. Hilbert’s method would be more convenient, yet does not seem capable producing the scheme $\frac{3}{1}5$ which is slightly more on the Harnack right-hand side of Gudkov’s pyramid (Fig.\[Gudkov-Table3:fig\]). This is surely no intrinsic reason since Harnack’s scheme itself is accessible to Hilbert’s method. Inspecting carefully our former Fig.\[GudHilb8:fig\] cataloging several variants of Hilbert’s method it is pretty clear why Hilbert’s method fails producing the scheme $\frac{3}{1}5$. Indeed what comes closest to $\frac{3}{1}5$ is the scheme $\frac{4}{1}4$ depicted near the center of Fig.\[GudHilb8:fig\], and one may argue that the vibrating oval has always an even number of (cytoplasmic) expansions coming across the fundamental ellipse $E_2$ of Hilbert’s construction thereby creating an odd number of ovals. So to have 3 inner ovals requires 2 inner expansions like on the scheme $\frac{3}{1}3$ on the bottom of Fig.\[GudHilb8:fig\] but this dissipates too much of the oscillating energy and not enough outer ovals are created. As an attempt to corrupt Nikulin it would be interesting to detect a second realization of the scheme $\frac{3}{1}5$ besides Harnack’s presented above, and study on it the distribution of outer ovals past the fundamental triangle through the deep ovals. Even without any scepticism about Nikulin the net effect is that if the latter is correct the determination of this distribution for a single curve $C_6$ belonging to the scheme $\frac{3}{1}5$ which is of type II would by the Fiedler-Marin argument determines this distribution for all curves belonging to the scheme. (That such a curve is of type II necessarily, follows from Arnold’s congruence (\[Rohlin-implies-Arnold:lem\]) $(3=1-3+5=)\chi=p-n\equiv k^2 \pmod 4(=3^2=9\equiv 1)$ valid for all type I curves.) We get so some nice geometric theorem as a consequence of the truth of Nikulin. (Insertion \[08.04.13\]: a much more elementary argument is given by Le Touzé in Sec.\[LeTouze:sec\].) \[09.02.13\] We now present some tricky argument in favor of non-separation of the outer ovals on the Harnack model constructed above. Referring to Fig.\[HarnaGudkov3-15XXLTEST:fig\]a, let us look at the quartic $C_4$ with $r=3$ ovals occurring as an intermediate step of Harnack’s construction. The unique oval of the $C_4$ which oscillates 4 times across the horizontal line $L$ is referred to as the [*Ray-Ban oval*]{}. Label the intersection points in $C_4\cap L$ as $p_1,p_2,p_3,p_4$. Consider the 3 green-lines through the 3 inner ovals of the sextic $C_6$ (Fig.\[HarnaGudkov3-15XXLTEST:fig\]c) but imagined traced on this $C_4$ (i.e. on plate Fig.a). Since the oval $2$ of the $C_6$ is enveloped by the Ray-Ban oval (of the $C_4$) it may be inferred that the line $2,3$ does not intercept the Ray-Ban oval outside of the arc $p_3,p_4$ of the $C_4$ on Fig.a). Hence the line $2,3$ is actually much more horizontal than on our Fig.c. More precisely we infer the following. Since the ovals [*1,2,3,4*]{} are encapsulated in the Ray-Ban oval (of the $C_4$), the line $2,3$ passes below them. (Of course “below” as no absolute sense in projective geometry, but here it has since we have another line $L$ as reference.) Even better than that consider also the lines $1,2$ and $1,3$ as depicted on Fig.c. Here we see clearly (and if in doubt argue as above with Bézout saturation w.r.t. the Ray-Ban oval) that all 5 outer ovals [*1,2,3,4,5*]{} are not separated by the union $1,2 \cup 1,3$ of those lines. Once those both lines are removed passing below really means not separating them. Another way to formalize this is to look at the intersection $2,3 \cap L$ which should be either very negative or positive when $L$ is identified to the real line. This again not so appealing. In fact instead of trying to define formally what means passing below, we should merely say avoid the Ray-Ban oval of $C_4$ and hence cannot separate $1,2,3,4$. In fact the inside of the Ray-Ban smashed along $p_3,p_4$ linearly is a topological disc (say $D$) away the 3 green-lines, and containing the 4 ovals [*1,2,3,4*]{}. Further it also passes below oval [*5*]{}, for otherwise it passes above but having to avoid the Ray-Ban it would then have to lounge the nasal portion of the Ray-Ban while passing between oval [*5*]{} and the curvilinear arc $p_2,p_3$ of $C_4$. (Recall that our line $2,3$ is de facto Bézout-saturated (w.r.t. to $C_6$ or even w.r.t. the $C_5$), hence cannot intercept any outer oval, in particular [*5*]{}.) But then the line $2,3$ would intersect twice the horizontal line $L$, violating the simplest case of Bézout. In conclusion the line $2,3$ passes below all outer ovals [*1,2,3,4,5*]{}, and the real picture could be more like Fig.\[HarnaGudkov3-15XXLTEST:fig\]e. Alas this depiction does not seem possible because the line $2,3$ already crosses 6 times the sextic so the oval $1$ cannot cross this line. Since it moves above it on the right side of the picture it must resurface on the right side below the line $2,3$, which is not the case on our picture (Fig.e). Accordingly Fig.\[HarnaGudkov3-15XXLTEST:fig\]f might be more realistic, and the conclusion would be that the fundamental triangle does not separate the outer ovals. Have we proved anything? Let us say “yes” and state the following lemma of which we shall supply a more formal proof right below. The fundamental triangle consisting of the $3$ lines passing through the $3$ inner ovals of Harnack’s curve (depicted above as Fig.\[HarnaGudkov3-15XXL:fig\] or Fig.\[HarnaGudkov3-15XXLTEST:fig\]) realizing the scheme $\frac{3}{1}5$ does not separate the $5$ outer ovals. The trick toward a more formal proof is to consider the topological disc $D$ obtained from the inside of the Ray-Ban oval by expanding it at $p_2,p_3$ linearly while smashing it inside at $p_3,p_4$ (compare the shaded region on Fig.\[HarnaGudkov3-15XXLTEST:fig\]a). Since this region contains the 5 outer ovals [*1,2,3,4,5*]{} it suffices to check that this disk avoids the 3 green lines through the 3 inner ovals. This is clear for the line $1,3$ which is Bézout-saturated on the $C_4$, hence can only attack our modified disc $D$ through the arc $p_2,p_3$, but as the inside of the Ray-Ban oval is avoided it results a second intersection with $L$ violating Bézout. The same argument works for the remaining two lines $1,2$ and $2,3$ after noticing that since those lines intercept twice the oval $2$ of the sextic, it may be assumed that they intercept twice the curvilinear arc $p_3,p_4$ of $C_4$. This follows merely from the nature of the method of small perturbation. [*Warning*]{}.—In fact a priori we could imagine that the line $2,3$ penetrates in the oval $2$ much more vertically than on Fig.c meaning really that it intercepts the segment $p_3,p_4$ of $L$, but in that case too, it is clear that the fundamental triangle does not separate the outer ovals. In fact in this case our line $2,3$ cuts the Ray-Ban oval only twice, and we may excise from $D$ the half of the trace of our line on $D$ containing $p_4$, plus a little tubular neighborhood thereof. During this excision it is clear that we do not loose the covering of the outer ovals, since the line $2,3$ is Bézout-saturated on $C_6$. Now each of our lines through oval $2$ is Bézout-saturated with the $C_4$ hence must avoid the inside of the Ray-Ban. However our line cannot penetrate the arc $p_2,p_3$, for otherwise a 2nd intersection with $L$ is created, hence has void intersection with $D$. Viro’s construction specialized to the scheme $\frac{3}{1}5$ ------------------------------------------------------------ \[08.04.13\] This section explores other realizations than Harnack’s (especially of the scheme $\frac{3}{1}5$) which is somewhat cumbersome. It is primarily a matter of exploring Viro’s method, but the latter turns out to be not much more suited than Harnack’s model to fix the distribution question. As we already said the royal road is Le Touzé’s argument in Sec.\[LeTouze:sec\]. Of course Viro’s method has supernatural appeal too, but our exposition is far from explaining the true core of the dissipation method which is a secret to us. Hence this section can be omitted with loss of continuity. \[08.02.13\] A first idea is to use Marin’s variant of Hilbert’s method but this seems only able to produce the scheme $\frac{3}{1}4$ (cf. Fig.\[Viro3-15:fig\]a). Another idea is to use Viro’s dissipation of 3 ellipses tangent at 2 points. This being again a small perturbation method like Harnack’s it is not a priori clear that we will be in a better position to tackle the distribution question past the deep triangle. The charming feature of Viro’s method is its ability to create nearly all sextics as perturbation of this configuration of 3 coaxial ellipses. To implement this, look at Fig.29 in Viro 1989/90 [@Viro_1989/90-Construction p.1103] showing all the possible dissipations of a germ of curve singularity of type $J_{10}^-$ consisting of 3 real branches having a second order tangency like on our global model of the 3 coaxial ellipses. This Viro figure is reproduced as Fig.\[Viro3-15:fig\]b below, which includes 5 modes of dissipation denoted by us V1,V2, …, V5 (V standing for Viro of course). Each of them admits an array of permissible values for spontaneous “champagne bubbling” of ovals created out of the blue. Then we can patchwork such smoothing [*independently*]{} at both singular points of the configuration of 3 ellipses to create a global curve with controlled topology. (This is of course highly reminiscent of Brusotti-Gudkov’s independence of smoothing, based on Severi and in turn upon Riemann(-Roch) via possible détours through the Plato cavern of Brill-Noether. Compare Brusotti 1921 [@Brusotti_1921] and Gudkov 1974 [@Gudkov_1974/74].) -10pt0 -5pt0 For instance the dissipation V1-V2 with $(\alpha,\beta)=(4,0)$ and $(\alpha,\beta)=(4,0)$ resp. yields Hilbert’s scheme $\frac{9}{1}1$. Choosing instead for V1, $(\alpha,\beta)=(0,4)$ and for V2 $(\alpha,\beta)=(0,4)$ yields Harnack’s scheme $\frac{1}{1}9$. If we choose for V1, $(\alpha,\beta)=(4,0)$ and for V2 $(\alpha,\beta)=(0,4)$ yields Gudkov’s scheme $\frac{5}{1}5$. Pause a little moment at this stage, to be puzzled by the fact that Gudkov’s rare bird—which escaped the attention of all experts during 8 decades (from Hilbert to Gudkov)—appears in the fingers of Viro as a species not much more tropical than common birds like Harnack and Hilbert. Perhaps this banalization of Gudkov by Viro is against the philosophy expressed in Sec.\[Diophantine-and-proba:sec\] that Gudkov’s curve(s) ought to have some statistical and Diophantine rarity. How to realize $\frac{3}{1}5$? It suffices to choose at V1 $(\alpha,\beta)=(2,0)$ and at V2 $(\alpha,\beta)=(0,4)$ (cf. Fig.c, right). Does Viro’s method help to solve our distributional problem. A priori not as shown by our naive depiction Fig.\[Viro3-15:fig\]d exhibiting both distribution (separating or not) as logically possible a priori. Recall yet that on behalf of Nikulin’s theorem both options cannot occur simultaneously. Of course naive geometric intuition tells us that the upper part of Fig.d is more likely with small ovals spread horizontally as a vestige of the horizontal tangent line at the singular point of the initial configuration of 3 coaxial ellipses. But of course even in this situation where the 2 microcosmic ovals (generated by the dissipation of the bottom singularity “V1”) are nearly horizontal (and so is the line through them) this does not prevent the “vertical” green-lines to separate the upper series of 4 ovals. But again on ground of some microscopical geometric intuition it seems realist to argue that even if the top vertex of the green-triangle is chosen very near to the top of the banana oval (i.e. the large inner oval of the curve $C_6$ resulting from the fusion of the two branches $2,3$ of ellipses when labelled from left to right), the little 4 top ovals will condense themselves as to be non-separated by the deep triangle. Actually if a separation would occur, then by sweeping the nearly vertical green-line inside the pencil of lines rooted at a basepoint in the bottom oval gives by continuity an intermediate line with 8 intersections with the sextic $C_6$ overwhelming Bézout (or the smoothness of the $C_6$). Indeed if the line through one of the bottom micro-oval has slope of ca. 135 degree it cuts twice the micro-oval, twice the banana and twice the (largest) nonempty oval, hence 6 times the curve. If we let this angle diminishes to 90 degree (plus $\varepsilon$) by dragging the upper (banana) vertex up to the top of the banana while supposing that the pair of lines effects a division of the 4 top micro-ovals for a suitably small value of $\varepsilon $ then both lines have again 6 intersections, but in between $135$ degrees and $90+\varepsilon$ degrees there must be a line cutting 8 times the curve $C_6$ namely the one line sweeping the separated oval. Sorry for this messy argument. Of course the key is just to observe that the green-lines are Bézout-saturated, hence the distribution of outer ovals cannot change. By the same sort of argument, precisely by tracing the line through two points near the bottom of the inner banana and the outer banana we get a line which is already Bézout-saturated. Pushing this line to its ultimate confinement we get the bitangent through the most “meridional” points of both bananas. Following this motion by continuity implies that the 2 bottom ovals are pressed down below this bitangent, and so looks nearly horizontal. Alas this does not prevent the situation of the bottom half of Fig.\[Viro3-15:fig\]d where both microscopic ovals are sitting nearly one above the other but of course at much lesser height than depicted, that is below the bitangent to the most meridional portion of both bananas. All this is quite exciting for the imagination, but does not seem to answer our puzzle on the distribution of outer ovals past the fundamental triangle. To analyze better the situation we should introspect in more detail the quantitative geometric aspects of Viro’s construction which certainly includes answers to our basic question. Ignoring that issue for the moment, we can introduce the concept of the bundle spanned by two disjoint ovals. This is the collection of all lines traced through a pair of points chosen inside the respective ovals (“boundaries” included). This bundle is often supported by (or spanning) a bordered surface homeomorphic to a Möbius band. If we think of both ovals as celestial bodies (like Earth and Moon) then this bundle (or rather its support, i.e. what is swept out by this collection of lines) is essentially the region where one oval masks the other (at least partially) like during an eclipse. We call thus this region the [*eclipsus*]{} of both given ovals, or just their mutual [*shadow*]{}. The shadow of the 2 bottom micro-ovals in case their mutual disposition is nearly vertical (like on the bottom half of Fig.\[Viro3-15:fig\]d) cannot intercept any further oval than the 3 obvious one (each of the 2 protagonists plus the nonempty oval enclosing them). In particular in that case of nearly vertical alinement of both “meridional” micro-ovals their shadow must find its way out through the little room left vacant between the top part of the outer banana and the top 4 micro-ovals. This is actually possible as suggested by Fig.\[Viro3-15shadows:fig\]c, provided both ovals really live at the microscopic scale. Fig.\[Viro3-15shadows:fig\]a depicts the shadow of both bananas. Each line in this shaded region is Bézout-saturated, hence no ovals can survive in this region, hence the situation is forced to be like on Fig.\[Viro3-15shadows:fig\]b. Once we have Fig.c then we must still analyze further shadows, but some thinking at the nanoscale should convince the reader that the verticality scenario posited by Fig.c is not further obstructed by Monsieur Étienne Bézout. Philosophically algebraic plane curves are like celestial configurations not liking to have their horizon too much saturated by galactic nebulosity. They express a principle of economy and purity. -10pt0 -5pt0 Our conclusion is that the topological aspect of Viro’s method alone does not seem sufficient to settle the distributional question of a $C_6$ of type $\frac{3}{1}5$. However it is more likely that for some Viro curve $C_6$ the distribution of the bottom ovals is nearly horizontal, and therefore that the fundamental triangle through the 3 inner ovals does not separate the 5 outer ovals of this $C_6$. If this is true (and Nikulin’s theorem also) then we deduce first the following lemma and next the following theorem by uniting the forces of all lemmas of the previous Sec.\[Nikulin-corruption:sec\]: Any sextic $C_6$ with scheme $\frac{3}{1}5$ is such that the fundamental triangle through the $3$ inner ovals does not separate the five outer ovals. \[Nikulin-Fiedler-Marin-Gabard-no-separa:thm\] Any sextic $C_6$ belonging to a scheme of the form $\frac{3}{1}\ell$ (with $0\le \ell \le 5$ according to Gudkov’s table) is such that the fundamental triangle through the $3$ inner ovals does not separate the outer ovals. If this theorem is true, one may of course wonder if there is an elementary proof circumventing the highbrow intervention of K3 surfaces, Torelli, etc. i.e. all the technology involved in Nikulin’s proof. \[13.02.13\] Such an elementary proof is given in the next section (\[LeTouze:sec\]) and was communicated by Le Touzé. Fiedler-Le Touzé’s answer (conical chromatic law) {#LeTouze:sec} ------------------------------------------------- \[13.02.13\] Three days ago, I received the following answer from Séverine Fiedler-Le Touzé (née Le Touzé, and often abridged as a such too avoid confusion with her husband Fiedler, who also worked in the field “a long time ago”). $\bullet$$\bullet$$\bullet$ \[10.02.13\] Bonjour Alexandre, Thomas m’a transmis ta question. La réponse est toute simple: soient $A$, $B$, $C$ trois ovales intérieurs et $D$, $E$ deux ovales exterieurs de ta sextique. Le triangle fondamental $ABC$ est entièrement contenu dans l’ovale non-vide. Si $D$ et $E$ sont dans deux triangles $ABC$ (non-fondamentaux) différents, alors la conique passant par $A$, $B$, $C$, $D$, $E$ coupe la sextique en $14$ points, contradiction. Avec des coniques, on montre plus généralement que: Les ovales vides de la sextique sont distribués dans deux chaines (int, ext), l’ordre cyclique est donné par les pinceaux de droites basés dans les ovales interieurs. Les ovales interieurs sont disposés en position convexe dans l’ovale non-vide. Bon dimanche, Séverine Translated in my poor English this gives the: \[LeTouze:lem\] Let $C_6$ be a sextic with $3$ inner ovals, and at least $2$ outer ovals, then the latter are in the same component past the fundamental triangle consisting of the $3$ lines through the deep ovals. In particular any sextic of type $\frac{3}{1}\ell$ has all its ovals distributed in the same component past the fundamental triangle. [*Insertion*]{} \[08.04.13\].—As a loose idea it could be interesting to see if the method can be boosted as to prohibit the scheme $\frac{3}{1}6$. Of course this follows also via total reality of the scheme $\frac{2}{1}6$ also due to Le Touzé. Assume on the contrary that the 2 outer ovals are in different subregions past the fundamental triangle. The idea is to look at the conic passing through the $3+2=5$ ovals (3 being deep and 2 being outer ovals). This conic certainly cuts the $C_6$ in at least $(5+1)\cdot 2=12$ points, like say on Fig.\[LeTouze:fig\]a albeit of course no depiction is required since those intersections are so-to-speak topologically forced. However from our supposition that the 2 outer ovals are separated by the fundamental triangle the real picture is rather like Fig.\[LeTouze:fig\]b or Fig.\[LeTouze:fig\]c yielding $2\cdot 5+4=14$ intersections. Bézout is overwhelmed. -5pt0 -5pt0 It remains to find the intrinsic reason of why this holds true. The key is to look at the order of the 5 assigned points on the conic. On Fig.a the 3 inner points (in black) are not separated by the outer ones (white-colored), while on Fig.b the 3 inner points are separated by the 2 outer points. This explains the formation of extra intersections whenever we have to salesman-travel from the inside to the outside of the nonempty oval. More precisely when the 3 black inner points are separated by the 2 white outer points then we see 4 elliptical arcs with dichromatic boundary, each of which contributing for an intersection (with the nonempty oval separating the outer from the inner ovals). So the whole story is reduced to the following Hilfssatz somewhat hard to state elegantly: [(Chromatic law for conics)]{}.—\[LeTouze-Gabard-Hilfssatz:lem\] Let us be given $3$ black-colored points in the plane $\RR P^2$ which are not aligned, and let $T$ be the triangle through them. Take $2$ additional (white colored) points outside the triangle. Then the unique conic $E$ through the $5$ points considered is smooth. Further if both white points are in the same component past the triangle $T$, then the $5$ points are monochromatically distributed on the conic as white-white-black-black-black (with only $2$ chromatic transitions when reading cyclically). If instead both white points are separated by the triangle, then the distribution is dichromatic as white-black-white-black-black, which read cyclically gives $4$ chromatic transitions (from black to white or viceversa). Applying this Hilfssatz concludes the proof of the lemma (\[LeTouze:lem\]). The philosophical outcome of this brilliant argument (communicated by Séverine Le Touzé) is that we cannot hope any corruption to Nikulin by the Fiedler-Marin locking technique. In fact the distribution of outer ovals past the fundamental triangle is always monopartite. In particular we get a more conceptual and lucid proof of several lemmas that we tried hard to establish on the cumbersome models of Harnack, Viro. In particular, we get an elementary proof of Theorem \[Nikulin-Fiedler-Marin-Gabard-no-separa:thm\] without the whole transcendental apparatus behind Nikulin’s theorem (K3 surfaces, Torelli, etc.). Trying to extend Nuij-Dubrovin rigid-isotopy of the deep nest via total reality {#Nuij-Dubrovin-extended:sec} ------------------------------------------------------------------------------- \[23.01.13\] Apart from the beautiful result of Nikulin on sextics, stating that the real scheme enhanced by Klein’s types suffices to determine unambiguously the rigid-isotopy class (\[Nikulin:thm\]) and the rigidity of the empty scheme, the only positive general result available is Nuij-Dubrovin’s theorem stating that the deep nest constitutes a unique rigid-isotopy class. \[rigid-scheme:defn\] [Let us say that an ($m$)-scheme (i.e. a scheme of order $m$) is [*rigid*]{} if any two $m$-tics curves representing the scheme are rigid-isotopic.]{} Since the deep nest is totally real (à la Ahlfors) under a pencil of lines one might wonder if other totally real schemes also enjoy rigidness. Examples of such total schemes include all the $M$-schemes by Bieberbach-Grunsky, the $(M-2)$-schemes $\frac{6}{1}2$ and its mirror $\frac{2}{1}6$ by an unpublished argument of Rohlin (nobody is able to reconstruct). More easily it includes the scheme of degree 8 $\frac{1}{1}\frac{1}{1}\frac{1}{1}\frac{1}{1}$ (4 nests of depth 2) which is total under a pencil of conics. Conjecturally for a scheme, type I implies maximal (Rohlin 1978). $\bullet$ A scheme of degree $m$ is [*total of order $k$*]{} if any curve $C_m$ representing the scheme admits a total pencil of $k$-tics. (For instance the $2k$-scheme $(1,1,\dots,1)$ consisting of $k$ nested ovals is total of order 1.) $\bullet$ Say that a scheme of degree $m$ is total if any curve representing the scheme admits a total pencil. It seems natural to expect the: If a scheme of degree $m$ is total then it is total of order $k$ for some universal $k$ depending only on $m$. At first glance this could follow from Ahlfors 1950 [@Ahlfors_1950]. At any rate we have the implications: Total of some order $k$ $\Rightarrow$ total $\Rightarrow$ type I (perhaps implying maximal). Marin convinced me that the transition from the abstract to embedded viewpoints might be not so easy, hence the converse of the second arrow might not be a trivial corollary of Ahlfors 1950, but we naively still hope so. One may speculate on an extension of Nuij-Dubrovin’s theorem as follows: For a scheme, totality of order $k$ (for some fixed $k$) implies rigidness. [*Insertion*]{} \[08.04.13\].—This conjecture (like the subsequent one (\[M-schemes-rigid\])) is probably also disrupted by Marin’s obstruction (discussed in the next Sec.\[Marin:sec\]). Indeed by our extrinsic variant of Riemann-Bieberbach-Grunsky (\[total-reality-of-plane-M-curves:thm\]) the total reality of plane $M$-curves of degree $m$ is exhibited by a pencil of curves of degree $(m-2)$, hence we have totality for some universal degree, namely $k=m-2$ (depending only on the degree $m$ of the scheme and not on the geometry of the curve), yet no rigidity can be observed by Marin’s obstruction. Perhaps it suffices to assume type I, or maybe even maximal implies rigidity (in increasing order of hazardousness). Here maximality is interpreted in the sense of Rohlin (as opposed to Harnack’s more specialized sense). Of course a priori there is very little evidence for a direct correlation between those concepts. (Again try to look if Fiedler, Marin give some counterexample, more on Marin soon.) In particular we may have something like: \[M-schemes-rigid\] [(Too Naive!!!, completely false as shown by Marin, Fiedler)]{}.—Any $M$-scheme is rigid. A priori if the devil of algebra does well his job this ought to be completely false in high degrees $m\ge 8$, or say perhaps $m\ge 10^3=1000$. Marin’s lock 1979: obstruction to rigid-isotopies {#Marin:sec} ------------------------------------------------- \[23.01.13\] Alas it seems that the above conjecture (\[M-schemes-rigid\]) is completely wrong for $m=7$ already, compare Marin 1979 [@Marin_1979 p.60–61]. Alas I was on a bad day and could not completely understand his argument, which looks however fairly simple involving the prose: “La distinction des deux courbes se fait en étudiant la position des ovales extérieurs par rapport aux droites joignant les ovales impairs.” (cf. p.60, of ). At first this looks sloppy justification as we are not just playing with projectivities but with rigid-isotopies which (in marked contrast to their names) are completely soft pathes in the residue of the discriminant. Yet I am sure that Marin is right (as usual) but his argumentation is for highbrow readers? Ah yes the argument must be that when dragging the curve in the parameter space (with the joystick) while choosing a triangle through the deep ovals as on Marin’s picture (reproduced as Fig.\[Marin:fig\]), then the forced intersection with the pseudo-line gives already total reality (or Bézout-saturation) of these lines with the septic $C_7$ preventing the remaining (outer) ovals to traverse the triangle during the motion (=rigid-isotopy). Hence the distribution of ovals past the deep triangle is an invariant of the rigid-isotopy class, i.e. any 2 curves exhibiting distinct distributions of ovals past the fundamental triangle cannot be deformed into the other. -1.2cm0 -5pt0 -5pt0 So this is a bit like the moving frame of E. Cartan, and seems indeed to corroborate Marin’s clever observation!!! (Compare also Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000] who call this trick a “lock”, while ascribing it as well to Fiedler.) Note that even the complex orientations agree on Marin’s example. This method of the lock (or moving frame/traingle) affords therefore an obstruction (à la Bézout) to rigid-isotopy. It uses the fact that a triangle in the projective plane subdivides it in 4 pieces. One can wonder if other (more complicated) locks are also useful. This method surely deserves to be better explored and assimilated (as remarked in Degtyarev-Kharlamov ). For instance one can wonder if it is enough to a lock with a pair of lines which suffices to separate the outer ovals of the top figure. During the isotopy we can keep track of them (at least the ovals where they are passing through). Of course Marin’s choice has the advantage of canonicalness. What is crucial is that the lock do not degenerate during the isotopy, which is ensured by the fact that the 3 inner ovals cannot become aligned without violating Bézout. We have proved Marin’s result: [(Marin 1979, or Fiedler)]{} There is two isotopic $M$-septics, i.e. having the same real scheme (and in fact the same complex orientations, but that requires adding the arrows on Fig.\[Marin:fig\]), yet which are not rigid-isotopic (i.e., belong to distinct chambers past the discriminant). [*Insertion*]{} \[08.04.13\].—This raises of course the question of counting the number of septics chambers corresponding to this scheme (of degree 7). Perhaps variants of Marin’s figure (Fig.\[Marin:fig\]) produce more than 2 chambers, but we are not sure. Still some link between totality and rigidity? Highbrow Nuij’s principle ------------------------------------------------------------------------ \[23.01.13\] Is there still some link between total reality and rigidity? A priori imagine the simplest situation of total reality under a pencil of line (ensured whenever we have a deep nest). Then very naively one could imagine to contract progressively the curve to some normal form like concentric circles and then drag it as a such toward the other center of perspective and blow it up again along the radial foliation toward the other curve. Of course doing so we meet reducible curves hence the discriminant yet perturbing the path there is some hope to avoid it completely, proving thereby the Nuij theorem [@Nuij_1968]. Though surrealist this argument is the best we can give in favor of a connection between totality and rigidity. How does Nuij or Dubrovin prove their fantastic results? Can we “do-it-yourself” by making precise the above idea? One trick would be to take a total pencil (vision from the innermost oval) and perturb the nest toward concentric circles around the center of perspective. Since both curves are (softly) isotopic there is some chance that one path along the pencil (this being a circle abstractly there is 2 such pathes) does not cross the discriminant while affording a rectilinear rigid-isotopy. Then one finishes as above. $\bigstar$ [*Long Insertion*]{} \[08.04.13\].—In quintessence, the deep nest for which rigidity holds true by Nuij 1968, is the satellite of the conic (cf. Sec.\[satellite-total-reality:sec\]) whose rigidity can be nearly ascribed to ancient Greeks (or Descartes, Newton, Sylvester’s law of inertia for quadratics forms, etc.). By analogy the quadrifolium schemes of degree $m=4k$ consisting of 4 nests of depth $k$ are total under a pencil of conics (as is fairly trivial, and explicitly remarked in Rohlin 1978 at least for $m=8$). This in turn is the satellite of the quartic quadrifolium (degree $m=4$ with $r=4$ ovals) whose rigidity is known since Klein 1876 (Sec.\[Klein-rigidity-of-quartics:sec\]). Hence this gives some evidence that totality in degree 2 implies rigidity. Extrapolating further along a stability of rigidity under satellites it could follow from the Rohlin-Le Touzé $(M-2)$-schemes of degree 6 (whose rigidity is ensured by Nikulin’s theorem deeper than Klein but sharing with it the rôle of surfaces, viz. K3 quartics vs. cubics for Klein) that: \[satellite-Rohlin-(6)-schemes-rigid:conj\] All satellites of Rohlin’s sextic schemes $\frac{6}{1}2$ (and its mirror $\frac{2}{1}6$) are rigid. Of course if rigidity is stable under satellites this would not only applies to the Rohlin (totally real) schemes but to all schemes of degree 6 which are of definite type as tabulated on the Gudkov-Rohlin table (Fig.\[Gudkov-Table3:fig\]). There are precisely $64-2\cdot 8=48$ many such schemes. $\bigstar$ [*End insertion*]{}. Another method of proof (of Nuij’s theorem) could be dynamical like the one (we attempted) for CCC (\[CCC:conj\]), yet involving another functional a priori. Very loosely the functional ought to have a unique attractor consisting of a series of concentric circles (or perhaps ellipses). If the flow can be shown to have this unique attractor (itself a certain manifold) then every nested curve converges there and going forth and then back we link rigidly our both curves. Can we adapt the above synthetic method (or find an even more synthetic method) to construct the (hypothetical) rigid-isotopy between two (8)-schemes consisting of 4 nests of depth 2? The naive canonical form would be the same with circle or ellipses yet its degree is twice too big (namely 16). ([*Added*]{} \[08.04.13\].—Perhaps as suggested above, the canonical form is the satellite of a quartic which has the correct degree.) We can first observe that such an octic curve has, like the deep nest, all the nested ovals forming negative pairs (this can be seen either à la Fiedler on a model or à la Ahlfors via the total pencil which forces the orientation to gyrate in the same sense as swept by the pencil). As a such there is no topological obstruction to shrink them at the microscopic scale (or apply alternatively a variant of CCC). Once contracted at the microscopic scale our configuration moves without resistance in the free vacuum and then may be re-expanded at the next curve. This is very sloppy heuristic of course, sembling much like inter-sidereal travelling, but there may be some truth in this. At least one sees a connection with CCC. If this works we get a proof of the: \[rigidity-sat-quadrifolium:conj\] The $8$-scheme $4\times\frac{1}{1}$ of $4$ nests of depth $2$ is rigid (i.e. any two of its representatives are rigid-isotopic). More generally the $4k$-scheme consisting of 4 nests of depth $k$ is rigid. This could be the “degree 2” avatar of Nuij’s theorem, and ought to be proved by an iterated variant of CCC (like C++). As yet the total reality of the scheme was only involved to ensure that all the ovals belonging to some nest gyrate in the same sense according to complex orientation (what Rohlin 1978 calls negative pairs). Those negativity may be interpreted as some depressiveness permitting precisely the collapse to the microscopic scale where then we can travel without friction within the “ether”. Given any collection of deep nests with negative pairs of ovals (gyrating in the same sense), we may hope to contract them at the nanoscale (via CCC or C++) and then travel and re-expand to reach any other curve with the same complex orientation. This looks topologically plausible yet the drawback of ignoring the total pencil (‘à la Ahlfors-Rohlin) is that our assumption does not only pertain on the real scheme but also upon its complex characteristics. So the assertion gain in generality but loose some elegance. As yet we have merely considered schemes which are towers (i.e. without branching in their nesting graph). However Rohlin (1978) claims that the $(M-2)$-schemes of degree 6 $\frac{6}{1}2$ and its mirror $\frac{2}{1}6$ are total of order 3. Can we deduce that those schemes are rigid by a method independent of Nikulin 1979 [@Nikulin_1979/80], and analog to the one sketched above? One should be in position to visualize the cubic pencil so as to draw the complex orientation. Bypassing this difficult task we may appeal to Rohlin’s formula to deduce the complex orientation. Assume the scheme to be $\frac{2}{1}6$, hence $2(\Pi^+ -\Pi^-)=r-k^2=9-9=0$ tell us that one pair is positive and the other negative. So both inner ovals gyrate “differently” (in accordance with Fig.\[GudHilb8:fig\]). At this stage one is quite puzzled, i.e. one does not see how to bring the curve at the nanoscale via contractions. Of course granting CCC we can shrink the empty ovals, but a priori cannot shrink the nonempty one. So our heuristic method looks here quite impuissant! Another quasi-paradox of our heuristic method arises for a curve of type I belonging to the scheme $9$, which under a total pencil of cubics (easy to visualize, cf. Fig.\[Fcubic:fig\]) could be isotoped to the other curve(s) of this scheme of type II. Yet of course the second curve lacks a total pencil to re-expand. All this is just supposed to illustrate that we see no direct relation between total reality and rigidity, at least via the naive contraction approach. However this does not preclude a deeper relationship. That would maybe involve exploiting more the total pencil as a tool to construct a first reduction to some normal shape, which ought to be then easily tele-transported and then re-expanded via the second total pencil. So the total pencil should act as some sort of (contracting) wormhole or as a railway guiding the curve to some canonical shape easier to tele-transport (at the speed of light). Alas this is much too vague to convince us about any implication like “total $\Rightarrow$ rigid”. \[26.01.13\] After Shustin’s e-mail, who remembered me the reason why the empty scheme is rigid, one can suspect two basic scenarios ensuring rigidity. Taking the deep nest as prototype, where rigidity holds true by Nuij 1968 [@Nuij_1968] one could suspect that rigidity is causal either of total reality à la Ahlfors or by the proximity to the empty chamber (which is connected, and actually baricentrically “convex”). Both phenomena could explain Nuij’s rigidity of the deep nest while affording basic intuition about guessing further rigidity results. For instance total reality could explain the rigidity of schemes swept out by pencil of conics (e.g. $4\times \frac{1}{1}$ in degree 8), while the proximity to the empty locus could via CCC prompt rigidity of the scheme with one oval (in even degree at least). Another naive idea I had (but which is now quite outdated) is that while total reality could imply rigidity via Ahlfors, the avatar of the latter for empty curves (namely Witt 1934 [@Witt_1934]) could be involved in the rigidity of the empty scheme. More total reality ================== Another attempt to prove Rohlin’s total reality claim ----------------------------------------------------- [*Editorial note*]{} \[08.04.14\].—The prose of this section starts a bit abruptly, due to a permutation of section. Prior this material came right after Le Touzé’s section (Sec.\[LeTouze:sec\]), explaining why the chromatic law for conics impedes a direct corruption of Nikulin by Fiedler-Marin’s trick. This miracle of extra intersection created by dichromatism gave me some hope to attack Rohlin’s highbrow claim, but the difficult turned out to be immense to fill. Further the impact of this method of extra intersections gained by dichromatism must probably also be the key behind Rohlin’s proof of the universal orthosymmetry of the sextic schemes $\frac{6}{1}2$ and $\frac{2}{1}6$. We call any curve having one of these schemes a [*Rohlin curve*]{} as the latter Academician in his 1978 article [@Rohlin_1978] was the first (and actually the unique creature in the universe except for possible extraterrestrial intelligences) to state the universal orthosymmetry of such curves. Through the 8 deep ovals (equivalently the empty ones) of such a Rohlin curve we let pass a pencil of cubics. As above (\[LeTouze-Gabard-Hilfssatz:lem\]) we imagine the inner basepoints black colored while the outer basepoints are white colored. Let $C_3$ be any cubic of the pencil, which we assume smooth for simplicity. If $C_3(\RR)$ is connected, then $C_3$ intersects $8\cdot 2=16$ plus twice the nonempty oval of $C_6$, and so the intersection is totally real. If $C_3$ is not connected then its splits an oval and a pseudoline. A priori the oval could visit the 6 inner points while the pseudoline the 2 outer ovals. In this case there is no forced extra intersections. -5pt0 -5pt0 By Le Touzé’s lemma \[LeTouze:lem\] we may infer that the triangle through any 3 of the 6 inner ovals (assuming $\frac{6}{1}2$) does not separate the 2 outer ovals (Fig.\[LeTouzeRohlin:fig\]c). There are $\binom{6}{3}=20$ such triangles. But this does not seem very useful. Let us start again. Consider a $C_6$ with scheme $\frac{6}{1}2$, a so-called [*left-wing Rohlin curve*]{} (in view of its position on the Gudkov table, Fig.\[Gudkov-Table3:fig\]). Consider the pencil of cubics through 8 deep basepoints selected inside the $6+2=8$ empty ovals of the $C_6$. We claim (with Rohlin 1978 [@Rohlin_1978]) that this pencil is totally real (and consequently the curve is of type I). Total reality of $C_3\cap C_6$ is clear when the cubic is connected for then there are $2\cdot 6+2=18$ intersections, the last two being created while crossing the nonempty oval. So let us assume the cubic disconnected. Two cases are possible either it is smooth or not. If singular then the cubic may either have a solitary point or even be a conic union a disjoint line. However since we are free to perturb the 8 basepoints the pencil can probably be assumed to be transverse to the discriminant so that its singular members are uninodal curves. This rules out the second case. In the first case of a solitary cubic then the solitary node cannot be one of the 8 basepoints, and so the connected pseudoline of this singular cubic visits all 8 points, creating thereby 2 additional intersections with the nonempty oval. So may assume the cubic smooth and as soon as its pseudoline visits both inner and outer points we are finished (2 bonus intersections are created). On the other hand if the pseudoline visits only inner points then it must evade out of the nonempty oval (otherwise it would be null-homotopic) and so we score again 2 extra points, and gain total reality. Hence we may assume: We have to show that this is contradictory, but are still far from the goal. In fact at the time of writing these lines the writer does not know if he will ever be able to complete this argument. A first remark is that our pencil of cubic has another (non-assigned basepoint). Where is it? We think that it must be on the pseudoline of $C_3$ for simple vibratory reasons. Indeed look at Fig.\[LeTouzeRohlin:fig\]d and imagine a nearby cubics $C_\epsilon$ in the real locus of the pencil. The corresponding oval will have to oscillate about that of $C_3$ and since 6 basepoints are on the oval the oscillation closes up perfectly. (In savant terms the oval has a trivial tubular bundle.) Another thing natural to do is to cut out the inside of the oval $O$ of the cubic $C_3$ of Fig.d in order to apply the Poincaré index formula to the foliation ${\cal F}$ induced by the pencil. Then the situation is a bit messy but as follows. The pencil hits the discriminant of cubics 12 times over the complexes as $\deg \disc_3=3(m-1)^2=12$ for $m=3$. A priori not all intersections are real, but can occur in conjugate pairs. Those singular curves which are real are either solitary cubics or have two real branches crossing transversally (“real bitangent” and we call them [*nodal cubics*]{}). Denote their respective number $\sigma$ and $\beta$ (where $\sigma$ stands for “solitary” and $\beta$ for “bitangent”). We have $\sigma+\beta=12-2k$. Using the Poincaré local index formula $j=1+\frac{I-E}{2}$ where $I$, $E$ are the number of internal resp. external tangencies of the foliation with a small circle surrounding the singularity, it is a simple matter to compute indices. Of course a basepoint gives a foyer of index $+1$, a solitary cubic gives a “centre” of index +1 (since $I=E=2$), while a nodal cubic gives a hyperbolic saddle of index $-1$ (as $I=0, E=4$). Applying Poincaré’s index formula (cf. Poincaré 1885 or Gabard 2011 [@Gabard_2011-Euler-Poincare-obst-pretzel-long-tentacles], arXiv, “long tentacles”) it follows $$9+\sigma - \beta=\chi (\RR P^2)=1.$$ It may be deduced that $\beta \ge 8$, that is the following: \[nodal-cubics-8-many-in-a-pencil:lem\] Any generic pencil of cubics contains at least $8$ nodal cubics. Further we easily tabulate the possible value as $(\beta, \sigma)=(8,0),(9,1), (10,2)$. It would be however probably more interesting to apply the Poincaré formula in the inside of the oval $O$ (of the cubic $C_3$) doubled to get a sphere. (This doubling is merely a trick yet useful to eliminate the boundary.) The difficulty in doing so is that we do not really know a priori how the singularities of the foliation $\cal F$ induced by the pencil are distributed inside the oval $O$. So let us denote with subscript “naught=0” the corresponding quantity of singularities inside the oval $O$. Then we have $$6+2 \sigma_0- 2 \beta_0=\chi(S^2)=2,$$ where we used implicitly the fact that the 9th basepoint is not inside our oval (nor on its periphery since it is rather located on the pseudoline). Alas at this stage the situation looks confuse. One idea is to imagine a solitary node inside the oval $O$. Then there is a unique time direction so that this oval inflates while moving inside the pencil. Since the oval $O$ has a tube neighborhood like Fig.\[LeTouzeRohlin:fig\]e this oval cannot hit the oval $O$, and must rather collide with the pseudoline component to form a nodal singularity of type $\beta$. So to each solitary node is canonically assigned a non-solitary node (all this occurring inside $O$). It seems evident that the corresponding map is injective, and so $\sigma_0\le \beta_0$. Alas this gives no contradiction when injected in Poincaré’s relation displayed above. \[scholie:Rohlin-does-not-boils-to-Poincare\] [\[14.02.13\]]{}—In fact it seems unlikely that there is a proof of Rohlin’s total reality assertion (for $\frac{6}{1}2$ and its mirror) using Poincaré’s formula only. We arrived at this conclusion after tracing a rather complicated foliation of the plane $\RR P^2$ containing the bad cubic $C_3$ as a leaf. (More about the discussion of the relevant pictures soon.) The bad cubic is one which is not totally real, hence whose oval is necessarily enclosed in the nonempty oval of the $C_6$. In reality such a foliation is fairly easy to construct just by starting with the bad cubic and then merging its components together to a nodal cubic and pursuing the depiction in a more or less canonical fashion. At each step Bézout for $C_3\cap C_6$ is respected and Poincaré index formula is of course verified. What should we deduce? Could it be that Rohlin’s total reality assertion is false, while its theorem on the type I of his schemes is right as follows from some highbrow topological congruence (due to himself, Kharlamov and Marin, cf. (\[Kharlamov-Marin-cong:thm\]))? If the latter super-classical congruence is correct then via Ahlfors theorem the total reality assertion is likely to hold true yet perhaps not for [*all*]{} pencil of cubics through the 8 empty ovals (or a priori for pencils involving curves of higher order). Perhaps the proof should involve Abel’s theorem applied on the cubics, yet they vary so seems unlikely. Perhaps Abel has to be used on the $C_6$? All this is puzzling and we frankly confess our poor understanding which calls for a synthesis between the abstract viewpoint of Riemann-Klein-Ahlfors and the embedded viewpoint of Harnack-Hilbert-Gudkov-Rohlin. This subdivision of our science is still vivid today, compare e.g. in Russia the tradition along Natanzon vs. Kharlamov-Viro. As we noticed earlier in this text (very optionally see Sec.\[sec:Total-reality-Harnack-max-case\]), it is also tantalizing to trace a total pencil on the $M$-sextics, already on those of Harnack and Hilbert. It would be of interest to know what is the degree of curves forming a total pencil in the $M$-case, whose existence follows in principle (modulo Marin’s private communication objection) from Ahlfors even in the simple schlichtartig variant of Bieberbach-Grunsky. ([*Added*]{} \[08.04.13\].—This question should now be settled via (\[total-reality-of-plane-M-curves:thm\]).) Since an $M$-sextic has 10 empty ovals it looks quite improbable that total reality is exhibited by a pencil of cubics which has only 9 foyer-type singularities. This must, we believe, easily follow from Poincaré’s index formula. Indeed each empty oval must contain a singularity of positive index while thorn singularities of index $+1/2$ are precluded for an algebraic pencil. Now let us discuss our picture leading to the announced Scholium \[scholie:Rohlin-does-not-boils-to-Poincare\]. The game is to foliate the plane by “flexible” cubics in the sense that we depict only the singular nodal curves abstractly like a figure “8”. A cubic cannot have a node plus an oval. Using this we sometimes have the impression that there is an obstruction to complete the foliated structure like on Fig.\[LeTouzeRohlin:fig\]. (Could it be the case that Rohlin made such a mistake, in the sense of a too hasty inference?) -5pt0 -5pt0 However deleting curves and starting again one finds Fig.\[LeTouzeRohlin2:fig\] which is topologically admissible. On it each cubic looks like a cubic, Bézout is respected as well as Poincaré’s index formula (as it should). In fact our solution shows no solitary cubics. Of course we do not claim that this free-hand drawing foliation is algebraic (in which case Rohlin’s claim would be erroneous), but we are also not able to exclude this eventuality. What this picture really shows is that we cannot expect to prove Rohlin’s assertion via the sole apparatus of the combinatorial topology of foliations (à la Poincaré). So if true Rohlin’s statement has some deeper geometric significance, and it is quite tantalizing to imagine its complexity. Further by Ahlfors theorem it is likely that Rohlin’s claim is just the top of the iceberg of a plethora of another phenomena of total reality in higher degrees which must all be very delightful to visualize if not quickly overburdening any human intelligence. Again our prophecy (cf. Introd. of this text) is that this is linked to the stability of matter at the nano-scale, or at least that such totally real pencils describe the dynamics of electrons about an atomic nucleus, as ellipses described the trajectory of Mars about the Sun in Kepler’s days (ca. 1605). Total reality from the elementary viewpoint ------------------------------------------- \[15.03.13\] Broadly speaking our main Leitmotiv is the question of examining if there is any relation between total reality à la Ahlfors (or rather Riemann-Schottky-Klein-Bieberbach-Grunsky-Ahlfors, etc.) and Hilbert’s 16th problem. Rohlin’s claim about his sextic curves posits such a deep relation. In this section we try to explore more systematically this relation. Before entering into the details let us pose some of the guiding questions. Given an (abstract) dividing curve $C$ there is according to Ahlfors 1950 [@Ahlfors_1950], a totally real map $f\colon C\to \PP^1$ of the curve to the projective line $\PP^1$ (i.e. $f^{-1}(\PP^1(\RR))=C(\RR)$). When this curve is plane does this map extends to the projective plane $\PP^2$ as to be induced by a pencil of curves? The question is actually pure geometry primarily meaningful over the complexes. Given a plane curve $C_m$ defined over $\CC$, and a holomorphic map $C_m(\CC) \to \PP^1(\CC)$ is it true that there is a pencil of curves so that the map induced by the pencil is the given holomorphic map. If this is true then it is certainly true equivariantly and Ahlfors theorem implies that [*any plane dividing curve admits a total pencil of curves.*]{}[^67] The next question is then to trace such pencils, and try to control its order (i.e. the degree of the curve constituting it). Tracing them in case of $M$-curves looks an especially hard exercise. (\[08.04.13\] Okay but see (\[total-reality-of-plane-M-curves:thm\]).) Without appealing to Ahlfors theorem one can also study totally real pencils [*per se*]{}, as a tool to detect the dividing character of curves. Actually it is this trivial criterion (as applied to the Gürtelkurve, or also hyperelliptic curves) which lead the writer to discover Ahlfors’ theorem in ca. 2000–01 independently and prior of knowing about Ahlfors’ work. So whenever a curve $C_m$ is swept out by a total pencil of curves it is dividing. So one can examine which sort of curves are exposed to such a total pencil to derive in principle an infinite series of orthosymmetry criterions. The prototype is the case of the deep nest totally swept out by a pencil of lines through the deepest oval. Then one would like to study pencil of conics, cubics, quartics, etc. Some extra difficulty arises from the distinction between universal total reality where it is forced by the sole knowledge of the real scheme as in the case of deep nests or Rohlin’s sextics $\frac{6}{1}2$ and it mirror $\frac{2}{1}6$, and “versal” total reality where the detailed geometry of the curve is required to exhibit total reality. This is of course much allied to what Rohlin calls schemes of indefinite type. For instance the octic scheme $4\times \frac{1}{1}$ consisting of 4 nest of depth 2 is universally totally real, under the pencil of conics through 4 basepoints inside the empty ovals. This example easily extends to schemes of degree $4k$ having 4 nests of depth $k$. Existence of such curves for each $k$ is demonstrated by Fig.\[Total:fig\]. (\[14.03.13\] A somewhat more conceptual reason is given by taking the algebraic satellites, i.e. nearby levels of the quadrifolium quartics, so $P_4\cup P_4+\epsilon_2\cup\dots \cup P_4+\epsilon_k$ and smoothing this union of reducible curve.) -5pt0 -5pt0 [*Insertion*]{} \[08.04.13\].—It may be observed (Fiedler’s smoothing law) that those curves are of type I for surgical reasons, providing another proof independent of total reality. Further the total pencil induces the complex orientations due to the holomorphic character of the underlying total map. It suffices then to imagine the pencil of conics to see that the intersection series will move along Fiedler’s arrows (this we shall vaguely refer to as dextrogyration). Understanding this properly in general could be the source of some progresses in the field, maybe? Any curve $C_{4k}$ of degree $4k$ whose real scheme consists of $4$ nests of depth $k$ (for short) is total under the pencil of conics through the $4$ empty ovals. Let $C_2$ be any conic of the pencil. First, $C_2(\RR)$ is connected, e.g. because it is a rational curve, aka as [*unicursal*]{} in Cayley’s jargon, cf. optionally the Introd. of Harnack 1876 [@Harnack_1876], which is of course not really required on the case at hand since it suffices like in Antiquity to sweep out the conic by lines from one of its point. It follows that $C_2(\RR)$ has to cut our curve $C_{4k}$ in $4\cdot 2k$ real points for topological reasons. But this is the maximum permissible according to Bézout, hence the pencil is totally real. It seems likely that the converse statement holds true, namely any curve $C_{4k}$ totally real under a pencil of conics with 4 real basepoints has this scheme of 4 nests of depth $k$. Note at least that the depth of the ovals cannot be distributed otherwise without violating Bézout for lines. For instance if a $C_{12}$ instead of having 4 nests of depth 3, had nests of depths say $2,4,3,3$ then the line through the nests of depth $4$ and $3$ would have too much intersection (namely $8+6=14>12$). It is worth noticing that the degree of such total maps are in accordance with the bound $r+p$ announced in Gabard 2006 [@Gabard_2006]. Indeed if $C_{4k}$ is a quadrifolium, then $r=4k$. Hence by the obvious Klein relation $g=(r-1)+2p$ and the genus formula $g=\frac{(m-1)(m-2)}{2}$, where $m$ is the degree, we find $$g=\textstyle\frac{(4k-1)(4k-2)}{2}=(4k-1)(2k-1), \quad \textrm{ and }$$ $$p=\textstyle\frac{g-(r-1)}{2}=\textstyle\frac{(4k-1)(2k-1)-(4k-1)}{2}= \textstyle\frac{(4k-1)(2k-2)}{2}=(4k-1)(k-1).$$ On the other hand by letting degenerate the 4 basepoints against the deep oval we find a total morphism of degree $2\cdot 4k-4=8k-4=4(2k-1)$. This has to be compared with the $r+p$ bound $$r+p=4k+(4k-1)(k-1),$$ which is indeed much greater as shown e.g. by evaluating for $k=1,2,\dots$. We find for $k=1$, $4(2k-1)=4\le r+p=4$. For $k=2$, $4(2k-1)=12\le r+p=8+7=15$, and so on due to quadratic growth of $r+p$. In fact the gonality of such curves is probably $\gamma=4(2k-1)$ (at least majored by this quantity) and so significantly lower than the universal upper bound $r+p$ stated in Gabard 2006 [@Gabard_2006]. In fact it is worth testing the truth of this $r+p$ bound on sextics already. Then we shall basically pass into review all the dividing curves of the Gudkov-Rohlin table Fig.\[Gudkov-Table3:fig\]. It would be natural to start from the top of this table but as $M$-curves are paradoxically tricky to understand from the viewpoint of total reality, we start from the bottom. The paradox is that the total reality phenomenon for abstract $M$-curves is basically the schlichtartig ($p=0$) case of Ahlfors theorem which is pretty much easier than the positive genus case. ([*Added*]{} \[08.04.13\].—This paradox is settled via (\[total-reality-of-plane-M-curves:thm\]).) So starting from the bottom we have first the deep nest $(1,1,1)$. Then $r=3$ and $p=[g-(r-1)]/2=[10-2]/2 = 4$. The gonality $\gamma=5 \le r+p=7$ is exhibited by the pencil of lines through a point on the deepest oval. For the scheme $\frac{4}{1}$ (when of type I) total reality comes from the conics pencil through the deep nest. It leads to a total series of degree $\gamma \le 2\cdot 6-4=8 \le r+p=5+3=8$ and Gabard’s bound is sharply realized. (Trick: while it is sometimes boring to compute $r+p$ it may be remembered that this is also $\frac{r+(g+1)}{2}$, i.e. the mean between $r$ and Harnack’s bound $g+1$.) In fact checking total reality involves here the exact geometry and not merely knowledge of the real scheme. More specifically the 2 prototypes of curves of type $\frac{4}{1}$ are depicted on Fig.\[R4-1:fig\]. One would like to have a geometric criterion for deciding a priori the type. In both cases we have 4 deep (=empty) ovals. Now we may choose 4 points in them. Given such a tetrad there are 2 cases to be distinguished (cf. Fig.\[Total2:fig\]a). Crudely put, either one of the 4 points can be inside the triangle spanned by the 3 other or not. However projectively this is a misconception as shown by Fig.b. Yet as we are given a curve of type $\frac{4}{1}$ it may look either like one of the 2 versions of Fig.c. Here the deep triangles (those traced through 3 deep ovals) have always a distinguished 2-simplex traced inside the nonempty oval. We call any such a (fundamental) simplex and there are 4 of them. Now two cases are possible: either one fundamental simplex is the union of the 3 others or not. Alternatively one oval is contained inside a fundamental simplex or not. (Check that this is well-defined requires keeping Bézout in the background memory.) Hence Fig.a recovers some intrinsic significance (and amounts essentially to the 2 possible visions we may have of a 3D-tetrahedron when projected on our 2D-retina). We call the second option a (as a short cut for stable tetrad like a prism stably posed on the sheet of paper, in contrast to the unstable tetrad posed on its edge hence in unstable equilibrium). -5pt0 -5pt0 This being said we have the following recognition lemma of Klein-Rohlin’s type by pure geometry: If the $4$ empty ovals of a sextic curve of type $\frac{4}{1}$ form a then the pencil of conics through the deep (empty) ovals is total, and in particular the curve is dividing (type I). Let us choose any conic $C_2$ of the pencil (through the 4 deep points inside the empty ovals). Like on Fig.e this curve will appear inside the largest fundamental simplex. Then the idea is to surger the conic into 2 pseudolines $C_2=J_1+J_2$ as shown on Fig.f or g. There is several way to do this but choose one. This surgery amounts to aggregate a certain edge of the tetrahedron which is exempt of intersection with the nonempty oval $N$ (because it is already Bézout-saturated). Therefore $C_2\cap N=(J_1\cup J_2)\cap N=(J_1\cap N)\cup ( J_2\cap N)$, but as each $J_i$ is a pseudoline each must intersect twice the nonempty oval $N$, and we gain 4 extra intersections. On the other hand $C_2$ intersect twice each empty oval and so we totalize $2\cdot 4+ 4=12$ intersections the maximum permitted by Bézout. Total reality follows, and the proof is complete. Albeit not perfectly hygienical our proof shows how to gain extra intersections by this splitting method. (Ideally we could hope that this method is also the key to Rohlin’s total reality claim for the curve $\frac{6}{1}2$, but this is not clear a priori. Imagine the bad cubic whose oval is contained inside the nonempty oval of the $C_6$, and what to do next!!!?) Next we would like a similar optical recognition criterion of the type for the sextic scheme $\frac{2}{1}2$. Here looking at Fig.\[R2-12:fig\] reproduced below as Fig.\[Total2-12:fig\]a,b below (which I borrowed from Degtyarev-Kharlamov’s survey 2000 [@Degtyarev-Kharlamov_2000]) suggests that what distinguishes both types is whether the line through the 2 outer ovals separates or not the two inner ovals within the inside of the nonempty oval. -5pt0 -5pt0 Somewhat more formally, let us extract from both prototypical curves (Fig.a, b resp.) some combinatorial datum. We mark in black inner ovals by choosing a point inside, and choose also 2 white points in the outer ovals. Unlike in the previously studied case, there is no preferred fundamental simplex inside the nonempty oval $N$, but we have a $1$-simplex entirely traced inside the nonempty oval $N$, which we mark by a double stroke. So we extract Fig.c resp.d and what distinguishes both is the issue that the line through the white vertices intercepts the line through the black vertices along its double marking or not. Let us call the first case (like Fig.c) a [*crucifix*]{} and then we have the: If a sextic curve $C_6$ of real scheme $\frac{2}{1}2$ has a crucifix, then the pencil of conics through the empty ovals is totally real (and the curve is of type I). We have defacto $4\times 2=8$ real intersections coming from the empty ovals. Take any conic of the pencil. Two cases may appear. Either the conic is [*dichromatic*]{}, that is when we follow it along some orientation we visit the 4 basepoints in the sequence black-white-black-white (BWBW) in this alternating way, or it can be monochromatic if this sequence reads BBWW (compare Fig.e). In the dichromatic case 4 intersections are created, and total reality is ensured. In the monochromatic case, we apply the splitting method which decomposes the conic $C_2$ as an union of two pseudolines $J_1,J_2$ traced on Fig.f obtained by cutting the conic at the two black points and adding the fundamental $1$-simplex linking both black vertices in the inside of $N$ (the nonempty oval). Since this $1$-simplex does not cut $N$, the intersection with $N$ remains the same after this surgery, but each pseudoline forces 2 intersections with $N$, and we gain the 4 required extra intersections. Total reality of the whole pencil is proved. [*Insertion*]{} \[09.04.13\].—On a second reading of this proof, it is not clear what prevents to apply the same argument to the other configuration. Try to clarify this at the occasion. To summarize the proof is the same as the previous one safe for an intervention of chromatism (black and white reflecting the inner and outer ovals). Note also that as $r$ is the same as in the previous case, Gabard’s bound $r+p$ is likewise verified (at least not quashed=invalidated)! $\bullet$ Can we continue this game? According to Gudkov-Rohlin’s table (Fig.\[Gudkov-Table3:fig\]) the next specimen to study is $\frac{5}{1}1$. As we have now $6$ empty ovals it is evident that a pencil of conics will not exhibit total reality. We probably have to move to cubics with 8 basepoints assignable. Let us use the same naive device of combinatorial extraction from two prototypes (cf. Fig.\[Total5-11:fig\]). We get some beautiful bi-pyramid (or octahedron) and one should imagine each face (or interface) shaded whenever the corresponding triangle is fundamental (i.e. included in the outer oval of the sextic $C_6$). Alas both configurations so obtained look combinatorially equivalent, and we feel puzzled. The next idea that comes to mind is to look at the conic through the deep black points rooted in the inner ovals. It seems that what distinguishes both types (I vs. II) is the location of the outer points as being resp. inside or outside this conic. Another feature distinguishing both models is the absence resp. presence of a line through the outer oval missing the nonempty oval $N$. -5pt0 -5pt0 In fact even on the model it is quite difficult to guess which pencil of cubics will exhibit total reality of the type I configuration (as predicted by Ahlfors’ theorem). One could take the horizontal line through the white point which cut the sextic 6 times, and take two extra basepoints on this line (perhaps in the 2 lower wings of the butterfly). Then at least the split cubic consisting of the conic through the 5 black and 3 white points would be totally real. Another puzzling point is that such a pencil will have mapping-degree $3\cdot 6- 8=10$ when the 8 basepoints degenerate on the curve $C_6$, whereas Gabard predicts one of degree $r+p=(r+g+1)/2$ the mean value of $r$ and Harnack’s bound that is $(7+11)/2=9$. So perhaps this constitutes a (potential) counterexample to Gabard 2006 [@Gabard_2006], at least if all abstract pencils are concrete and realized by cubics pencils. If we look at quartics pencil with $\binom{4+2}{2}-1-1=13$ free basepoints then the degree would be $4\cdot 6- 13=11$, still higher than Gabard’s bound. For quintics there are $\binom{5+2}{2}-1-1=19$ free basepoints and so the degree is $5\cdot 6-19=11$, for sextics $\binom{6+2}{2}-1-1=26$, so the degree is $6\cdot 6-26=10$, for septics $\binom{7+2}{2}-1-1=34$, so the degree is $7\cdot 6-34=8$. Gabard seems rescued, yet it looks quite tantalizing to understand the geometry of such a total pencil if it exists. (If Gabard’s bound is true and the Riemann-Hilbert transition from the abstract to the concrete viewpoints equally holds true then such a total pencil should exist of order at least seven!) [*Insertion*]{} \[09.04.13\] Let us summarize this as follows: \[(M-4)-sextics-corrupt-Gabard:scholium\] $(M-4)$-sextics of type I are perhaps a good place where to corrupt Gabard’s bound $r+p$. And if not it is at least a pièce de résistance against the principle that any abstract pencil is concrete, and therefore Ahlfors abstract theorem is unlikely to apply without friction in Hilbert’s 16th problem. In other words Riemann’s canary feels claustrophobic in the Plato cavern of Brill-Noether-Hilbert. Perhaps the above example merely corrupts the conception that the mapping-degree of a total pencil is minimized when the order of its constituting curves is. However it could still be true that any dividing plane curve of degree $m$ has its total reality exhibited by a pencil of order $(m-2)$ (or less), compare e.g. [(\[total-reality-of-plane-M-curves:thm\])]{} for the case of $M$-curve. \[16.02.13\] Let us leave aside this problematic concerning the truth of Gabard’s bound $r+p$ to concentrate on the existence on a cubics pencil which is total on our sextic of symbol $\frac{5}{1}1$. Of course the existence of the latter is more an act of faith than a truth a priori, as it is not obviously implied by Ahlfors’ theorem. The latter probably gives the existence of a total pencil and one may wonder what is the least possible order of the curves in the pencil. $\bullet$ Then we can look at the next curve $\frac{3}{1}3$ of the Gudkov tabulation (Fig.\[Gudkov-Table3:fig\]). Two models are depicted on Fig.\[Total3-13:fig\]c, and one may hope to distinguish them by some combinatorial recipe (perhaps by looking at the inner fundamental simplex and some outer simplex). -5pt0 -5pt0 Another idea is that since our curve has 6 empty ovals one should look at the corresponding hexagon and at pencils of cubics spanned by 2 triangles. Specifically, we may choose a hexagon which visits the 3 black inner points and the 3 white outer points in dichromatic alternation (BWBWBW), cf. e.g. Fig.d. Then we may expect that the pencil of cubics spanned by the red and green cubics is total. Alas Fig.e refutes this expectation. Of course there are other dichromatic hexagons but this is unlikely to be the right method. For instance Fig.f is another dichromatic hexagon, yet the corresponding pencil is still not total as shown by Fig.g. In conclusion those $(M-4)$-schemes are a bit puzzling from the viewpoint of total reality as there is no (obvious) canonically defined pencils since we have 6 empty ovals, which is not the number of basepoints of a pencil of plane curves, namely $4$ for conics and 8 for cubics. The two extra virtual basepoints for cubics could be chosen as high-order contacts imposed to the pencil and this done properly could exhibit total reality. It remains however to understand the natural geometric condition that are so-to-speak imposed by the geometrical vision of the curve. Remind indeed that total reality always amounts to place the ocular system “inside” of the glass so that the latter has no apparent contour (compare the baby case of the Gürtelkurve, Fig.\[Guertel-saturated:fig\].) [*$(M-2)$-curves*]{}.—We may hope that the situation is improved when moving to $(M-2)$-curves. The first case to study is the scheme $\frac{8}{1}$. Fig.\[Total8-1:fig\] shows models of both types I vs. II, but it is again quite puzzling to decide which intrinsic criterion distinguishes both configurations. Of course a loose answer could be that the type I configuration is characterized by the fact that the pencil of cubics through the 8 empty ovals is total, however one could desire a more optical recognition algorithm. Perhaps what distinguishes the type I is the possibility of tracing a convex octagon through the empty ovals. Note that convexity has some meaning since given two points in the nonempty oval $N$ we shall always select the half-projective line (segment) which is inside this oval $N$. ([*Added*]{} \[09.04.13\].—But the oval $N$ can be non-convex, and so this is meaningful only when the 2 points are inside the deep ovals of course.) -5pt0 -5pt0 For this scheme $\frac{8}{1}$, one could nearly argue that the pencil of cubics through the deep 8 points is always total, for we have $8\cdot 2$ automatic intersections, plus the 2 coming from the fact that the cubic is not null-homotopic hence must cut twice the nonempty oval. However this would contradict Rohlin’s remark that this scheme is indefinite as shown by the above constructions (pictures). However the sole obstruction to total reality of a cubic in this deep pencil is that the cubic has a small oval entirely inside some of the empty ovals. Indeed if the cubic is connected total reality is clear as $2\cdot 8+2=18=3\cdot 6$, and if not yet the oval of the $C_3$ visits at least two ovals of the sextic then each oval visited contribute for 2 intersections and total reality is evident. [*Insertion*]{} \[09.04.13\].—This “soleness” looks inexact: another obstruction occurs when the cubic splits off an oval visiting all 8 basepoints and the pseudoline stays confined outside the nonempty oval of the $C_6$. One would like to show that under a suitable geometric hypothesis (capturing the essence of the type I scheme) this sole obstacle cannot occur. \[17.02.13\] All these questions are fairly delicate and have to be extended to all type I schemes listed by Rohlin (see again the Gudkov-Rohlin Table=Fig.\[Gudkov-Table3:fig\]). Precisely what is demanded is an optical recognition procedure of the type in the sense of Klein (orthosymmetry vs. diasymmetry) via a synthetical device ensuring total reality of a certain class of pencils naturally attached to the curve (or its schemes). This would extend somehow Rohlin’s claim of the absolute orthosymmetry of the sextic schemes $\frac{6}{1}2$ and $\frac{2}{1}6$, which is the purest manifestation of the phenomenon. Meanwhile (yesterday), Séverine Fiedler-Le Touzé informed us (and several other colleagues, cf. letter in Sec.\[e-mail-Viro:sec\] dated \[16.02.13\]) that she was able to prove Rohlin’s claim for the scheme $\frac{2}{1}6$. Probably her argument contains crucial ideas that solve as well our slightly generalized problematic. [*Insertion*]{} \[09.04.13\].—This is nearly true, safe that it turned out that Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics] proves a slightly weaker assertion than the full Rohlin claim, namely she relies on the RKM-congruence ensuring type I a priori. Concretely we have then the scheme $\frac{4}{1}4$ for which one need to formulate an optical recognition, and the scheme $9$. For the latter it seems that the type I configuration is characterized by the fact that the pencil of cubics through 8 deep points inside some of $8$ ovals is such that the 9th basepoint lands in the 9th oval. A lucky stroke! [*Insertion*]{} \[09.04.13\].—It is nearly implicit from the above that we posit: Any dividing $(M-2)$-sextic has a total pencil of cubics. In a stronger shape: any cubics-pencil assigned to pass in the $8$ deep ovals is total, and it is permissible to let degenerate the basepoints on the ovals themselves. Then it is also plain to see that this is much in line with Gabard’s bound $r+p=10$, since $3\cdot 6-8=10$ is the mapping degree when the 8 basepoints are degenerated upon the ovals. Total reality of plane $M$-curves --------------------------------- \[17.02.13\] Next we have the case of $M$-sextics. This puzzled me for a while, yet it seems clear that now cubics pencils will not exhibit total reality. The reason is that we have 10 empty ovals but only 9 basepoints available. On the other hand it seems evident that each empty oval must contain a singularity of foyer-type corresponding to a basepoint of the pencil. So this follows from Poincaré’s index formula applied to the foliation induced by the pencil (cf. Lemma \[Poincare-lower-bound\] much earlier in this text, but restituted below in perhaps clearer fashion). This being said we shall move to pencil of quartics. The crucial idea is to remind the synthetic proof in the abstract context of the schlichtartig avatar of Ahlfors’s theorem, i.e. the theorem due to Riemann 1857-Schottky 1875–77-Bieberbach 1925-Grunsky 1937. More precisely we have in mind the simple argument via Riemann-Roch rediscovered by Huisman and Gabard, yet first clearly enunciated in Enriques-Chisini 1915. Bypassing all these historical details, the logical argument is simply given in our Lemma \[Enriques-Chisini:lemma\] (prior in this text). The idea is merely that if one has an abstract $M$-curve (not necessarily plane), then choosing one point on each oval (=real circuit which is linguistically better in this abstract context) one has a group of $g+1$ points which therefore move in its linear equivalence class by Riemann(-Roch), or just by Abel’s theorem since there are $g$ Abelian differentials (holomorphic one-forms) imposing magneto-hydrodynamical constraints upon the motion of a divisor in its linear equivalence class. So our effective divisor of degree $g+1$ moves on the curve of genus $g$. Since there is only one point on each “oval”, it is like a miniature railroad, in which there is only one train one each track, and so there cannot be collisions and total reality is automatic. Now when the $M$-curve is plane, I was frustrated to know nothing on the degree of such total maps. However the answer is “[*toute simple*]{}”(=very simple). Indeed inspired by the abstract proof, choose one point on each oval (and also one on the pseudoline if there is one). Then there is a standard recipe to construct the linear series spanned by a given group of points on a plane curve $C_m$ (due to Brill-Noether 1873/74?, Enriques-Chisini’s book 1915, Severi’s book 1921 [@Severi_1921-Vorlesungen-u-alg.-Geom-BUCH], van der Waerden’s book 1939/73 [@van-der-Waerden_1939/73], Walker’s book 1950 [@Walker_1950/62], who else?): just choose an integer $k$ large enough so as to have enough free parameters to pass a $k$-tics through the given group of points. Choose such a curve $C_k$ and look at the residual intersection with the curve $C_m$. Consider then all $C_k$’s passing through this residual intersection and the latter cut on the curve groups whose mobile part are divisors equivalent to the given one. This method clearly belongs to the genre of a sweeping method (balayage). Applying this to an $M$-curve leads to the following very modest theorem (stated as a such just because it escaped my attention for several months, if not years): \[total-reality-of-plane-M-curves:thm\] Given any plane $M$-curve of degree $m$ there is a total pencil of $k$-tics of degree $k=m-2$ (two units less than the given degree $m$). In fact exactly like in the abstract Bieberbach-Grunsky theorem, any equidistribution of points (i.e., one point on each real circuit) moves in a linear system of dimension $\ge 1$ and induces a totally real pencil by the sweeping method. In particular each $M$-sextic is total under a pencil of quartics. As the proof involves some arithmetical nonsense it is didactic to first handle the case of sextics. Then Harnack’s bound (in Petrovskii’s notation) is $M=g+1=\frac{(m-1)(m-2)}{2}+1=11$. The space of $k$-tics has dimension $\dim \vert k H \vert=\binom{k+2}{2}-1$, that is $5$ for conics, $9$ for cubics, 14 for quartics, etc. Choose an equidistribution of $11$ points one on each oval of the $C_6$. Then quartics have enough freedom to visit them. Choose a $C_4$ passing through the 11 points, and the residual group has $4\cdot 6-11=24-11=13$ points. But this is exactly one less than the dimension of all quartics, and so the residual series—consisting of all curves passing through the residual group—gives the required pencil. The total reality of the latter follows by the non-collision principle involving the continuity argument implicit in Enriques-Chisini’s anticipation of the Bieberbach-Grunsky theorem. Of course the impossible-to-beat anticipation is Riemann’s 1857 Nachlass [@Riemann_1857_Nachlass]! The general case is merely the same numerological coincidence worked out in general. Given any $M$-curve $C_m$ of degree $m$, Harnack’s bound is $M=g+1=\frac{(m-1)(m-2)}{2}+1$. Choose $M$ points on the real locus $C_m({\RR})$, one on each oval. Locate the least integer $k$ such that $\dim \vert k H \vert \ge M$. Since both the genus and the dimension of this complete linear system are given by binomial coefficients, this traduces into $\binom{k+2}{2}-1 \ge \binom{m-1}{2}+1$ which is first satisfied for $k=m-2$ (but not at $k=m-3$). Indeed this amounts to $\binom{m}{2}-1 \ge \binom{m-1}{2}+1$ which is plain as $\binom{m}{2}=1+2+3+\dots+(m-1)$. Now the residual intersection of a $C_k$ through the $M$ points with $C_m$ gives so many points as the following expression, which turns out to be the dimension of the system $\vert k H \vert $ less one unit, as shown by the following boring calculation: $$\begin{aligned} k\cdot m-M&=(m-2)m-\textstyle\frac{(m-1)(m-2)}{2}-1 \cr &\textstyle =(m-\frac{m-1}{2})(m-2)-1 =(\frac{m+1}{2})(m-2)-1=\dots=\dim \vert k H \vert -1.\end{aligned}$$ Somewhat more elegantly, $$\begin{aligned} k\cdot m-M &=(m-2) m-[1+2+\dots+(m-2)]-1 \cr &=[(m-1)+(m-2)+\dots+2]-1=\textstyle\binom{m}{2}-2=\dim\vert kH \vert-1,\end{aligned}$$ for $k=m-2$. Several questions arise as usual after discovering a trivial truth. (Derrière les montagnes encore des montagnes: Proverbe des iles créoles, if I remember well). In the case of sextics one can probably say therefore (modulo a more careful analysis of the case of $(M-4)$-sextics that all dividing sextics have their total reality exhibited by a pencil of degree $\le 4$). Perhaps it is true in general that: Any dividing $m$-tic has its total reality exhibited by a pencil of curves of order less than $(m-2)$. A more serious game would be to see if the above theorem essentially due to Riemann-Enriques-Chisini-Bieberbach-Grunsky-Wirtinger-Huisman-Gabard does not imply when suitably complemented by foliation theory à la Poincaré the well-known obstruction of Hilbert-Rohn-Petrovskii-Gudkov for $M$-sextics (e.g. the highbrow Gudkov-Rohlin congruence mod 8 or at least the weak version thereof mod 4 due to Arnold $\chi=p-n\equiv k^2 \pmod 4$). For higher degrees one may even dream of new results along this method, but all this requires more serious work. One could even dream that the method extends outside the realm of $M$-curves, as to recover e.g. Rohlin’s claim (meanwhile Le Touzé’s theorem) but this is unlikely because the continuity principle of no collision meets then serious difficulties, which are precisely those making Ahlfors theorem harder than Bieberbach-Grunsky’s theorem. Gabard 2006 gives an abstract topological algorithm overcoming this difficulty of collisions, but it looks hard to transplant this to the context of Hilbert’s 16th problem. [*Added*]{} \[09.04.13\].—The key is of course that if one has an overpopulation of trains circulating on a track one must ensure dextrogyration so as to avoid collisions. Then total reality is granted. \[02.03.13\] As a lovely special instance of the above theorem (\[total-reality-of-plane-M-curves:thm\]), Le Touzé (1 March 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]) observes the: [(Le Touzé 2013)]{} \[LeTouze-quintic:scholie\] “For an $M$-quintic $\langle J \sqcup 6 \rangle$, one finds a suitable pencil of cubics with six basepoints distributed on the six ovals, and two further chosen on the odd component $J$. As this component must cut any cubic an odd number of times, the required $15$ real intersections are granted.” —[Gabard’s addendum \[09.04.13\]:]{} As it will be observed in the sequel, but can already be noted here, one can also avoid this topological argument with the pseudoline, by noting that since $2\cdot 6+2=14$ intersections are granted, the remaining one is forced to reality by algebra (Galois-Tartaglia[^68] involution). [*Long Insertion (ca. $2\frac{1}{2}$ pages)*]{} \[09.04.13\].—How to generalize this Le Touzé’s Scholium to an $M$-septic? Pencil of quintics have 19 basepoints assignable. Assign 17 of them on ovals and 2 on the pseudoline, then there is $34+3=37$ real intersections granted, overwhelming the $5\cdot 7=35$ of Bézout. But by Harnack $M_7=g_7+1=\frac{6\cdot 5}{2}+1=16$, so we do not have as many ovals as 17. Actually our sloppy argument reproves Harnack’s bound with one unit less. This is, by the way, not so surprising as Enriques-Chisini 1915’a purpose was precisely to re-derive a proof of Harnack via Riemann-Roch. Okay, but this sounds too modest and we can surely expect more clever generalizations of Le Touzé’s Scholium. Indeed, by Harnack we have $15$ ovals on the $C_7$ (recall from Möbius-von Staudt that odd order curves have exactly one pseudoline). Distribute the 19 basepoints on the $15$ many ovals, plus 4 on the pseudoline. Then $30+4=34$ real intersections are granted by topology, and the last one is forced by algebra (Galois-Tartaglia symmetry of complex conjugation). Hence total reality is granted. We have proven the: Given any $M$-septic $C_7$, the pencil of quintic assigned to pass through the inside of all $15$ ovals (warning: it is more prudent to assign them directly on the ovals themselves) of $C_7$ and $4$ points marked on the pseudoline is totally real. Two questions arise. A deep one is whether this can be used to infer something about Hilbert’s 16th (distribution of ovals solved for $m=7$ essentially by a single hero, Viro ca. 1979). Another question is whether Le Touzé’s Scholium extends to all (odd) degrees. More philosophically, it seems that our abstract argument of Theorem \[total-reality-of-plane-M-curves:thm\] works in all degrees and gives the required total pencil of degree $(m-2)$, yet Le Touzé’s scholium (\[LeTouze-quintic:scholie\]) looks sharper as it tells precisely where to assign basepoints, without using the sweeping method and the Restsatz à la Brill-Noether. In degree 6, Le Touzé’s method suggests looking at a Harnack-maximal $C_6$ swept by $C_4$’s. Those (quartics) have 13 basepoints assignable. Distribute $11$ of them on the ovals, but where to place the remaining 2? A priori only 22 real intersections are granted. Of course we still have 2 more basepoints, but they do not force new intersections. How to ensure total reality in this case? Let us look at degree $m=9$. Then septics have $\binom{7+2}{2}-2=34$ basepoints assignable (for a pencil). For $m=9$, Harnack’s bound is $M_9=\frac{8\cdot7}{2}+1=29$. So distribute the 34 bases on the 28 ovals plus 6 on the pseudoline, granting so $2\cdot 28+6 = 56+6=62$ intersections (just one less than Bézout’s $7\cdot 9=63$), but algebra forces reality of the last man. At this stage it is evident that Le Touzé’s Scholium extends to all odd degrees as follows: \[Le-Touzé-extended-in-odd-degree:scholium\] (Extended Le Touzé’s Scholium).—Given any $M$-curve of odd degree $m$, the pencil of $(m-2)$-tics assigned to visit once all ovals and with residual collection of basepoints assigned on the pseudoline is totally real. Further, the mobile part of the pencil has exactly one point moving on each real circuit, and the allied circle map has lowest possible mapping-degree namely the number $r=M$ of real circuits (exactly like in the Riemann-Bieberbach-Grunsky theorem). What is first chocking is that when $m=5$ (Le Touzé’s Scholium) there is really an extra intersection gained by topology (of the pseudoline), while for $m=7,9$ our argument merely uses algebra. So a priori the argument could split in two cases depending on some (sordid) periodicity modulo 4. The geometer dislikes intrusion of capitalism and arithmetics in his garden. However it should be observed that even in Le Touzé’s argument one can use algebra, which is alas more capitalistic than her geometric argument. So it is still reasonable to expect an unified proof (without mod 4 stories) along pure arithmetical nonsense. This is as follows: let $C_m$ be our $M$-curve of odd degree $m=2k+1$. We look at curves $C_{m-2}$. Those can be assigned $\binom{m}{2}-2=:B$ many basepoints. But by “Möbius-von Staudt” $C_m$ has $M-1$ ovals (just omit the pseudoline of course), and by Harnack $M-1=g=\binom{m-1}{2}$. We distribute the $B$ basepoints on the $g$ ovals and the $B-g$ remaining ones on the pseudoline. Let us calculate $B-g=[(1+2+\dots+(m-1))-2]-[1+2+\dots+(m-2)]=(m-1)-2=m-3$. So we have $2g+(B-g)$ real intersections granted by topology (of the ovals), and this is equal to $(m-1)(m-2)+(m-3)=m(m-2)-1$. But this is one unity less than Bézout’s number $m(m-2)$ of complex intersections in $C_m\cap C_{m-2}$, so that the last intersection is forced to reality too! Did we used the assumption that $m$ is odd in any dramatic fashion? I would say no, but we did! Probably the argument adapts to even degrees as well if one is a bit more clever than we were for $m=6$. As to the last clause, it is evident by construction. Indeed since one basepoint is assigned on each oval, some extra (mobile) intersection is created on the oval. Further the last intersection granted by the Galois-Tartaglia symmetry conj, is forced to live on the pseudoline, since each real circuit contains at least one mobile point (by an evident sweeping principle or just the fact that any point has a well defined image). Hence a problem of interest is to understand the even degree case, and we hope that someone will easily tackle this question. It is quite beautiful at this stage to feel some big harmony between Riemann, Harnack, Brill-Noether as well as Le Touzé, or Rohlin, at least a sort of unity between conformal and algebraic geometry. Poincaré wrote something like the following: [*“La pensée n’est qu’un éclair dans la nuit, mais c’est ce qui éclaire tout”*]{}. In fact, when $m=6$ we have 13 basepoints (for quartics) and 11 ovals on the $C_6$. Could it be useful to impose the 2 additional basepoints as imaginary conjugate on the curve $C_6$. Those points will not be real, yet since they are statical they do not spoil total reality which merely involves dynamical points. (Note: this is not a new idea, cf. e.g. p.7 of Gabard’s Thesis 2004 [@Gabard_2004].) Note also that in the above proof (\[total-reality-of-plane-M-curves:thm\]) adapting Bieberbach-Grunsky to plane $M$-curves nothing grants that basepoints are real. In fact we start from any group of $g+1$ points equidistributed on the $M=g+1$ circuits, pass a curve of sufficiently large degree (i.e. $(m-2)$) through them and look at the residual intersection, which is a priori not totally real (but just real, stable under conj). This gives perhaps some evidence that we should for $m=6$ permit a pair of conjugate basepoints. Doing so we really have 11 real points moving on each oval in accordance with the train-track principle (i.e. Bieberbach-Grunsky, or Enriques-Chisini, etc.) So we state: \[Le-Touzé-scholium-deg-6:lem\] Given any $M$-sextic, the pencil of quartics assigned to visit any $11$ points marked on the $11$ ovals and a pair of conjugate points of $C_6$ is totally real and induces a (circle) map of degree $11$. Quartics depend upon $\binom{4+2}{2}-1=14$ parameters and so $13$ basepoints may be assigned. Distribute them on the 11 ovals available and fix the 2 remaining ones as a conjugate pair of points of the $C_6$. For topological reason each real curve of the pencil cuts once more each oval (usual closing lemma for ovals), and so $2\cdot 11=22$ real intersections are granted. By Bézout there is a total of $4\cdot 6=24$ intersections. Hence, [*all*]{} intersections are under control, i.e. either the 22 real ones or 2 imaginary ones which are statical. The latter do [*not*]{} perturb total reality, since they are not moving “electrons”. Probably the statement extends to all other even degrees (by working properly the arithmetics eventually by using what we already calculated in the odd degree case). More geometrically (and returning to $m=6$), one may wonder if we could not by continuity push the 2 imaginary basepoints on the real locus so as to impose a tangential contact in the limit (zusammenrücken). Then the modest advantage is that all basepoints would again be visible on the reals, and total reality should be conserved by continuity. So we arrive at the: For any $M$-sextic, the pencil of quartics assigned to visit any $11$ points marked on the $11$ ovals and tangent at a $12$th point of the $C_6$ is totally real and induces a (circle) map of degree $11$. Hence some contact can be imposed at any point, and the resulting foliation will look like a dipole at this point of tangency. It would be interesting to see if we can infer any of the deep classical obstructions on the distributions of ovals due to Hilbert, Rohn, Gudkov, from this method. Another idea (to be explored better than what follows) is to assign such imaginary basepoints on the puzzling case of $(M-4)$-sextics (of type I), cf. Scholium \[(M-4)-sextics-corrupt-Gabard:scholium\]. Then we had a pencil of cubics, of which we distribute the 8 basepoints on the 6 empty ovals and 2 points remain left. A priori we could imagine that an imaginary pair counts just for one linear condition (since after all the passage through one of them forces passing through the conjugate). So we could impose 2 imaginary pairs of additional basepoints, and the mapping-degree of the pencil would be $3\cdot 6- 6 -4=8$, again in accordance with Gabard’s bound $r+p=9$ (best interpreted as the mean of $r$ and Harnack’s bound). This looks however dubious since the pencil of cubics would then have $6+4=10$ basepoints overwhelming Bézout. This can be repaired if we impose only 5 real points and 2 imaginary pairs, and then Gabard’s bound is (exactly) verified, since $3\cdot 6-5-4=9$. However it another piece of work to check that total reality can be ensured. So let us be happy with a conjectural (and admittedly vague) statement: Any $(M-4)$-sextic of type I admits a total cubics-pencil of degree $9$ (like Gabard) with $5$ basepoints assigned on the oval and $2$ pairs of imaginary basepoints assigned on the curve. Here only $10$ real intersections are granted (among the $18-4=14$ which are moving) and so much work remains to be done to ensure totality under suitable assumptions. If we impose 6 real basepoints then 12 are granted, etc. [*End Long insertion.*]{} \[18.02.13\] Let us try the following strategy. \[We now come back to Rohlin’s total reality problem.\] Suppose the sextic to have the scheme $\frac{6}{1}2$. Let us choose 8 points $p_i$ on the $8$ empty ovals, one on each oval, and consider the corresponding pencil of cubics. We would like to show total reality of this pencil. It is clear that any cubic cuts at least $2\cdot 8=16$ times the $C_6$. This is because the $C_3$ can be either tangent at $p_i$ to $C_6$ or transverse. In the first case, intersection multiplicity is two, while in the second, one side of an infinitesimal analytic arc of the curve is inside the oval while the outer is out, hence an extra intersection is gained by closing the real circuit. Take any cubic in the pencil which is [*connected*]{} (and smooth). The latter is clearly totally real as 2 bonus intersections are created on the nonempty oval of the $C_6$ (which acts as a separator between the inner and outer basepoints). Note that we use the lemma that any pencil of cubics contains a connected cubics (which we nearly proved via Lemma \[nodal-cubics-8-many-in-a-pencil:lem\] showing that a pencil contains in general 8 nodal cubics). Now a simple idea ensuring total reality would be to look at a nearby cubic $C_3'$ and look if the 2 extra intersections [*gyrate in the same sense*]{} (dextrogyrate) on the nonempty oval $N$. In case of dextrogyration, total reality follows because there would be no collision between both points on $N$. Indeed when we move the cubic curve in the pencil then the intersection points move continuously and none of them can suddenly change its sense of motion, for otherwise there would be 2 curves of the pencil (nearby hence distinct) passing through the same point. So it suffices checking that the 2 intersections $p',q'$ of $C_3'\cap C_6$ located on $N$ move in same sense (dextrogyrate) on $N$, equivalently that $p',q'$ are separated by the corresponding 2 intersections $p,q$ for $C_3\cap N$. Since the curve $C_3'$ is a small perturbation of $C_3$ it oscillates about it (in a slaloming fashion). Now a simple picture shows that the gyration is good (occurs in the same sense, or dextrogyre) iff the number of basepoints inside $N$ is odd. So the whole question reduces to knowing if the 9th (non-assigned) basepoint of the pencil is located inside $N$ (or not). If it is inside then we are finished and total reality follows. Alas I know about of no argument ensuring the inside-ness of the 9th basepoint. The above argument (or rather strategy) relies on the existence of a connected cubic in the pencil which must be a simple matter. This can be bypassed if we argue differently. It is clear that the sole obstruction to total reality is a disconnected cubic whose oval lies inside $N$. Such a cubic is smooth except if it has a solitary node, yet in that case total reality is evident for the pseudoline of the solitary cubics has to connect an inner and outer point so contribute for an extra 17th intersections. Then either by algebra or topology the 18th intersection is real too. Given such a bad cubic $C_3$ which is smooth and whose ovals lies inside $N$, we can again look at a small perturbation $C_3'$ which will oscillate about $C_3$, and so do the corresponding ovals. Now it is clear by a slaloming argument that the oscillation is possible iff the number of inner basepoints (inside $N$) is even. Hence again we would have a contradiction, if we knew that the 9th basepoint of the cubics-pencil lies inside $N$. Whatever the strategy adopted, Rohlin’s claim seems to require innerness of the 9th basepoint. So this gives a 2nd reduction of Rohlin’s claim. As a metaphor it seems that such total reality proofs à la Rohlin-Le Touzé are akin to an Eiger-Nordwand ascension. There are several base-camps where to rest, but as the climbing goes on they become rarer and rarer and one is forced to follow a nearly canonical route, \[more and more vertiginous and perilous, by the way.\] It should be noted yet that our approach is slightly weaker than the Rohlin-Le Touzé claim for we do not check total reality of all pencils with 8 deep basepoints [*inside*]{} the empty ovals, but merely the case when the latter 8 points are located [*on*]{} the ovals. Our weaker variant suffices yet to detect total reality and so the type I of such Rohlin’s schemes (e.g. $\frac{6}{1}2$). Our tactic looks simpler, since when basepoints are assigned in the interior of ovals, they do not create defacto real intersections, because the cubic’s oval could be microscopically nested inside one oval of the sextic. [*A dubious strategy with the hexagon (skip the next 2 paragraphs)*]{}.—Another idea was suggested by a naive look at the Hilbert-style construction of $\frac{6}{1}2$. Here it seems that the hexagon through the inner points is not convex. $\bigstar$[*Inserted Objection*]{} \[09.04.13\].—This is so on the naive Walt-Disney picture Fig.\[GudHilb8:fig\], but less clear on the more realist picture Fig.\[GudHilb6-12:fig\].$\bigstar$ On the other hand if there is a bad cubic (one whose oval is inside the nonempty oval of the $C_6$) then we know (since at least Zeuthen 1874 [@Zeuthen_1874]) that the oval of the cubic is convex. This is to mean that whenever we join two points inside the oval by the rectilinear segment inside the oval it stays entirely inside the oval. This is not well phrased and should be formulated by saying that whenever we take 2 points inside the cubic-oval the line through it dissected in 2 pieces by the 2 intersection points with the oval is such that the half not meeting the pseudoline is entirely within the inside of the oval. So Rohlin’s claim would follow if it can be shown that the fundamental hexagon of our $C_6$ of type $\frac{6}{1}2$ is non-convex. Here the fundamental hexagon is defined as the union of all fundamental 2-simplices (triangles). Recall that given 3 points on the inner ovals (inside $N$) the lines joining them are Bézout-saturated and there is a unique full-triangle traced inside $N$, which we call fundamental. The fundamental hexagon of our $C_6$ (with $8$ marked points $p_i$ on the empty ovals) is the union of all these fundamental triangles rooted at the 6 inner points. Another strategy is as follows. Choose any 8 points on the sextic $C_6$, one on each empty oval. To show: the pencil of cubics through them is totally real. [*Step 0*]{}.—The sole obstruction to total reality is the presence of a bad cubic, i.e. one with an oval entirely traced inside the nonempty oval of $C_6$. [*Step 1*]{}.—Assume that there is a bad cubic then the 6 inner points are hexagonally distributed on the oval. [*Step 2*]{}.—Imagine the 6 inner points black and the 2 outer points white-colored. Then try to infer existence of a conic $C_2$ through 3 inner points and the 2 outer points which is dichromatic, i.e. such that the 2 white points split the 3 black points in 2 groups. [*Step 3*]{}.—Such a conic has 4 transitions from black to white, hence cuts the $C_6$ in $10+4=14>12=2\cdot 6$ violating Bézout. The difficult step is Step 2. To exhibit a dichromatic conic it suffices by Le Touzé’s lemma (\[LeTouze:lem\]) that some fundamental triangle through 3 black points separates the 2 white points. So by contradiction assume that all black triangles does not separate the 2 white points. But alas I do not know why this circumstance (which is actually forced by Le Touzé’s lemma) implies a contradiction with the bad cubic assumption. Yet another strategy via long run evolution of the bad cubic ------------------------------------------------------------ \[19.02.13\] In this section we explore another strategy toward a proof of the Rohlin-Le Touzé claim of total reality for Rohlin’s curve $\frac{6}{1}2$. The argument perhaps adapts to its mirror $\frac{2}{1}6$, yet we concentrate on $\frac{6}{1}2$ for simplicity. As above, we rather attack the somewhat weaker total reality assertion for a pencil with basepoints assigned [*on*]{} the ovals of the curve. By letting basepoints degenerate on the ovals, this is probably logically implied by the Rohlin-Le Touzé’s theorem (of which at the time of writing we have not seen a proof). It seems also that Le Touzé proves rather the case of the mirror $\frac{2}{1}6$ but probably her argument adapts to $\frac{6}{1}2$. Our argument is just a strategy far from a complete proof, trying to study the dynamical evolution of a pencil lacking total reality while hoping to detect a contradiction. So it is a dynamical approach, but perhaps the real proof (of Rohlin and Le Touzé) is more clear-cut or based perhaps on the same idea. Start by recalling certain trivialities, which we repeat for convenience. In all this section, $C_6$ denotes a “Rohlin curve” of type $\frac{6}{1}2$, i.e. 6 ovals enveloped in a larger one with 2 ovals outside. The 6 ovals are said to be inner ovals and the 2 ovals outside called outer ovals. A pencil of cubics passing through $8$ basepoints injectively distributed on the $8$ empty ovals is said to be deep. The following statement (for me still hypothetical) is a variant of the Rohlin 1978–Le Touzé 2013 theorem, and probably weaker than it, yet which seems to us easier to prove as there is not the possibility of microscopic ovals passing through the assigned basepoints yet without creating real intersections. Albeit weaker it is sufficient for detecting the type I of the given scheme, and in some sense stronger as it yields circle maps (or totally real maps) of smaller mapping-degree. Any deep pencil on a sextic $C_6$ of real scheme $\frac{6}{1}2$ is totally real, i.e. any (real) curve of the pencil cuts only real points on the $C_6$. The sequel is an (unsuccessful) attempt of proof of this (hypothetical) statement. It is divided in several steps, each justified in the subsequent paragraph. $\bullet$ [*Step 1*]{}.—Each cubic $C_3$ of such a deep pencil has at least 16 (real) intersections with the $C_6$. Indeed $C_3$ is assigned to pass through 8 points $p_i$ distributed on the 8 empty ovals of $C_6$. Two cases can occur. Either the cubic is tangent to the sextic at $p_i$ in which case we have intersection multiplicity $2$, or the $C_3$ is transverse to $C_6$ in which case there is through $p_i$ a small analytic arc of the $C_3$ with extremities both inside and outside the corresponding oval of $C_6$. By basic properties of algebraic projective curves, this arc of curve has to close up itself and so a 2nd (real) intersection with $C_6$ is created. A priori $C_3$ could visit $p_i$ via just a solitary node (isolated real ordinary double point). In that case the intersection multiplicity is still $2$, and by the way I suspect that this case cannot occur by elementary properties of pencils which have a foyer-type singularity at the basepoints preventing an isolated singularity to appear there. (All this is clumsy due to a lack of profound algebro-geometric knowledge of the writer.) Step 1 shows that we are quite close to total reality, where [*each*]{} cubic curve is required to have 18 real intersections (counted by multiplicity) with the sextic $C_6$. $\bullet$ [*Step 2*]{}.—The sole obstruction to total reality is the presence of a [*bad cubic*]{}, i.e. a smooth cubic with 2 components whose oval is contained in the nonempty oval $N$ of the $C_6$. If the cubic $C_3$ is connected (i.e. $C_3(\RR)$ is connected), then as it must visit both inner and outer points a 17th intersection is created and the 18th follows either by algebra or topology. If $C_3$ is not connected then it is either smooth with 2 components, or a solitary cubic with a solitary node. In the latter case the solitary node passes at most through one of the eight $p_i$ (though this is improbable), yet even in that case the pseudoline of the solitary cubics visits both inner and outer points so has to be total. Hence the sole curve possibly failing total reality is a smooth cubic with 2 real branches. It has further to be monochromatic in the sense that the outer and inner points $p_i$ have to be “purely” distributed on both real circuits of the $C_3$. Else if both an inner and an outer point among the $p_i$ land on a same circuit of $C_3$ then a 17th intersection is created by topology, and so an 18th one by algebra. Further if the inner points are on the pseudoline of $C_3$, then topology forces a 17th intersection (else the pseudoline would be contractible inside the bounding disc of the nonempty oval $N$). So the inner points are on the oval of $C_3$, and Step 2 is completed. So from now on we shall assume that our deep pencil contains a bad cubic $C_3$, and try to infer a contradiction. Several basic remarks are perhaps useful. 1\. The unique oval of a cubic with 2 components is convex in some obvious sense. (Perhaps this already implies a contradiction, but need to be detailed.) 2\. The oval of our bad cubic $C_3$ will vibrate during an infinitesimal motion along the (deep) pencil $\Pi$. As $6$ basepoints are assigned on the oval $O$ of $C_3$, a vibratory (slaloming) principle implies that the oval oscillates an even number of times across itself. (Of course this may also be reduced to homological intersection mod 2.) It follows that the 9th basepoint of the pencil $\Pi$ is located on the pseudoline of the bad cubic $C_3$. Now our strategy is the naive one of studying the long-run evolution of the bad cubic as time evolves, i.e. as the cubic is dragged along the pencil. Probably the real argument of Rohlin-Le Touzé is more clear-cut Bézout-style obstruction without dynamical process. So what may happen to our bad cubic as time evolves? The discriminant of plane cubics has alas even degree $3(m-1)^2=12$ for $m=3$ (or more generally when $m$ is odd) so that we cannot infer presence of a singular curve in the pencil for basic degree reasons. Yet there is surely a deeper argument either like Klein-Marin (1876–1988 [@Marin_1988]) or via Poincaré’s index formula (1885) prompting the existence of a connected curve in any pencil of cubics. Cf. e.g. (\[nodal-cubics-8-many-in-a-pencil:lem\]). Accordingly two scenarios may occur when the bad cubic is dragged along one of the two possible sense along the real locus of the pencil: SC1.—The bad cubic has its oval coalescing with its pseudoline. SC2.—The bad cubic sees its oval shrinking to a solitary node. Of course SC2 seems unlikely since the oval of the bad $C_3$ passes through the 6 inner points so a shrinking looks impossible at least in the near future of $C_3$. So SC1 is the first thing to occur when the bad cubic is propagated along the deep pencil. A qualitative picture (without high precision tracing instrument) may give something like Fig.\[Total-qualitative:fig\]a showing the coalescence of the oval of the bad (black) cubic with its pseudoline via transition through a nodal cubic (in red). On tracing naively the next lilac curve one seems to get a corruption with Bézout as the lilac curve seems intersecting 4 times the horizontal line. This is fairly naive and there must be ways to avoid such a trivial accident. Another optical illusion is the following. On looking Fig.\[Total-qualitative:fig\] one may get the impression that in the transition from the red curve to the lilac one along the segment $A,B$ the cubic must necessarily split off the line $A,B$ (and accordingly a so-called residual conic $C_2$). If so, then the 6 remaining (assigned) basepoints have to lie on the residual conic $C_2$ which intersects $12+2=14$ times the $C_6$, since 2 bonus intersections are forced with $N$ (by dichromatism). (Note also to complete the argument that none of the 3 (assigned) basepoints can be aligned as then we get $6+2=8>6$ intersection of $C_6$ with a line.) However on zooming (violently) the segment $A,B$ one arrives at Fig.\[Total-qualitative:fig\]b showing a transition from red to lilac by an undulating family of (qualitative) cubics respecting Bézout (at least as far as the intersection with line $A,B$ is concerned). During this undulation no splitting off of a line is forced. -5pt0 -5pt0 A similar depiction could settle the pseudo contradiction with Bézout of Fig.a (involving the line $C,D$ and the lilac curve). This is suggested on our loose picture Fig.c. In reality nobody tell us that the picture is like this, being possibly rather like Fig.d or even different. It is clear at this stage that the argument becomes much involved if possible to complete at all. Philosophically the drawback of our strategy is that it is indirect (by contradiction). One could dream of a direct argument, but this surely requires different ideas. Our indirect argument requires solid consolidations perhaps by enumerating carefully the several Morse surgeries implied by the evolution. By genericity those could be assumed of elementary type (uninodal curves only). Also during the time the oval of the bad cubic stays an oval, its expansion seems, by convexity, confined within the fundamental triangle of Fig.a. Finally, in the limit when we encounter the first nodal curve of the pencil, the inside of this loop (which is also the geometric limit of the insides of the ovals past the bad cubic) has also to be convex and therefore contained in the fundamental triangle of Fig.a, plus its companion (forming a “David star”). Another strategy to Rohlin-Le Touzé’s phenomenon ------------------------------------------------ \[21.02.13\] (based on hand-notes of the past 3 days). We consider again a sextic $C_6$ of type $\frac{6}{1}2$. We distribute $8$ points on the empty ovals of the $C_6$. The phenomenon in question claims that the pencil of cubics through those 8 points is totally real, i.e., each real curve of the pencil cuts only real points on $C_6$. First one notices that each curve of the pencil (denoted $\Pi$) cuts at least 16 points on the empty ovals. (Here and in the sequel, intersections are always counted by multiplicity.) Denote by $N$ the nonempty oval of $C_6$. If the pencil $\Pi$ is not totally real, then it contains a bad cubic $C_3$, i.e. such that $C_3(\RR)\cap N=\varnothing$. If all cubics of $\Pi$ cut $N$ then all have 2 extra intersections located on $N$, and so the pencil is totally real. Such a bad cubic is necessarily smooth, because singular cubics are either connected or have a solitary node, but in the latter case the real pseudoline connects inner and outer points so an interception of $N$ is forced by continuity. Assume (by contradiction) that there is a bad cubic in $\Pi$. One idea is to look at the future of this bad conic along the pencil $\Pi$. One can introduce the projection induced by the pencil as the map $$\pi\colon C_6 \to \Pi$$ taking a point of the curve to the unique curve of the pencil passing through it. $\pi(N)=:G$ is the set of good conics, whose complement is $B$ the set of bad conics. Under our assumption that $B\neq \varnothing$, it is clear that $G$ is a (compact) interval in the circle $\Pi$. In fact as the pencil is defacto nearly total with 16 real intersections (over the 18 maximum permissible), the map $\pi\colon N\to G$ is two-to-one. Hence given $C_3$ our (initial) bad cubic we may let it degenerate toward one of the 2 extremities of the interval $G$ along 2 pathes consisting only of bad cubics safe for their extremities. As only 2 extra intersections are possible, this may occur in 3 fashions only (by a simple continuity argument): \(I) Inner touch: the oval of $C_3$ inflates inside $N$ and ultimately touch it from the interior. \(D) Double touch: the oval of $C_3$ inflates from inside and collides with the pseudoline of $C_3$ on a point of $N$. \(O) Outer touch: the pseudoline of $C_3$ touches $N$ (necessarily from outside) while the oval stays inside $N$ disjoint from it. Further when dragging the curve along $\Pi$, at some stage (first touch or contact) two real points eventually appear on $N$, and subsequently move apart along $N$ (without possible return by the property of linear systems or holomorphic maps) to merge again on the opposite first contact of $C_3$ with good cubics. This looks attractive but is probably only a first step toward a contradiction. In fact a simple picture (Fig.\[Tot1:fig\]a) shows that such a scenario is perfectly permissible, topologically at least. -5pt0 -5pt0 This figure suggested another idea as follows. While the above picture (Fig.\[Tot1:fig\]a) is topologically legal, the thick traced blue curve seems to violate Bézout upon tracing a line through its node intercepting it 4 times. A tactic would be to argue by Poincaré’s index formula (applied to the inside of the egg $E$, i.e. the unique oval of our bad cubic $C_3$) that there is necessarily such a nodal curve in the pencil (with node located inside $E$), and by some messy combinatorial argument such a curve would necessarily corrupt Bézout, heuristically because it has to visit too many points forcing high-contortion like the thick blue curve above. As to the Poincaré argument, look at the inside $E^{\ast}$ of the egg $E$ with the (mildly singular) foliation induced by $\Pi$ and double it to a sphere, $2E^{\ast}\approx S^2$. We see on the boundary (doubled!) 6 foyers of index $+1$ (locally like the pencil of lines through a point). A priori there could be centers (locally like concentric circles) with index $+1$ and arising from a solitary cubic. Finally nodal cubics (with non isolated ordinary singular point) contributes for (hyperbolic) saddles (locally like the levels of $x^2-y^2$) which are of index $-1$. Poincaré’s index formula tells the sum of indices being equal to the Euler characteristic of the manifold. Hence $6-2 S\ge \chi(S^2)=2$, where $S$ is the number of saddles, and we deduce that there is at least two of them inside $E$. (As $E$ is smooth they cannot be located on the boundary of $E^{\ast}$.) The above programme sounds good (albeit requiring alienating combinatorics!) until the moment, one realizes that a nodal cubic is able to salesman-travel through the 6 basepoints on $E$ without being contorted. Remember at this stage that the 8 (assigned) basepoints of $\Pi$ determine (by Bézout) a 9th one, which for vibratory reasons has to be outside $N$ (otherwise a slight perturbation of $E$ would intercept an odd number of times $E$, violating the depiction or if you prefer homological intersection modulo 2.) Of course if the 9th unassigned basepoint lands on $N$ then total reality is evident. Let us now depict such a nodal cubic able to visit the 6 inner points without being contorted (Fig.\[Tot1:fig\]b). \[22.02.13\] On the latter all the (rational) nodal cubics occurring in the pencil have relatively decent looks. To formalize the lack of contortion of such a cubic one can uses the pencil of lines through the node which cuts a group of 3 points with 2 of them statically monopolized by the node while the 3rd moving along the curve. So when one looks from a nodal cubic from its node one always see at most (an in fact exactly one) point forced to be real. Our idea was that at least one of the nodal cubics (ensured via Poincaré’s index formula) would be contorted, i.e. violating this tightness of nodal cubics, yet our Fig.\[Tot1:fig\]b gives little hope to complete this. Another strategy also jeopardized by the above pictures (Fig.\[Tot1:fig\]) is that there ought to be always a cubic of the pencil which is dichromatic in the sense that the 6 inner points (black colored) and 2 outer points (white colored) are lying mixed on some suitable cubic of the pencil $\Pi$ with the 2 white points separating the collection of all 6 black points. If so is the case, 4 extra intersections are gained on the nonempty oval $N$, and Bézout is violated. Perhaps this strategy is the right one but requires more geometrical argument à la Le Touzé. Yet another idea is that by using the nodal cubics of the system we may infer that the outer basepoints are strongly stretched apart, while by contrast Le Touzé’s lemma (chromatic law for conics, cf. \[LeTouze-Gabard-Hilfssatz:lem\]) forces them to be much condensed, in the sense of not being separated by any triangle through any triplet among the 6 inner points. Remember (from Le Touzé’s Sec.\[LeTouze:sec\]) that if a separation occurs then the conics through the 3 corresponding inner points and the 2 outer points is dichromatic (with the 2 white points separating the 3 black points) so that the corresponding conic has $10+4$ intersections with $C_6$ (violating Bézout). Another idea is that since the 9th basepoint is outside $N$ (for the vibratory reasons already explained), all cubics of the pencil have to oscillate about those 3 points. This is perhaps incompatible with the tightness of (rational) nodal cubics. Yet another idea was that the 9th basepoint of our pencil $\Pi$ (almost canonically assigned to the $C_6$) has to land inside $N$ and this would contradict the vibratory properties of a bad cubic. However this miraculous property looks logically much stronger (i.e. not logically equivalent) to the Rohlin-Le Touzé total reality claim, so that this is perhaps not a realistic strategy, at least we were not able to implement it. Maybe what is required is an avatar of the chromatic law for cubics instead of the version for conics (Lemma \[LeTouze-Gabard-Hilfssatz:lem\]). The logics would be as follows. Trace the “diamond” of all $\binom{6}{2}=15$ lines through the 6 inner points. By the chromatic law for conics, this diamond does not separate the 2 outer points. So by a hypothetical chromatic law for cubics it could follow that the pencil of cubics through the 8 points is dichromatic, i.e. contains a dichromatic cubic. The latter would overwhelm Bézout. Of course all this if it works should use the assumption of a bad cubic which implies an hexagonal (convex) distribution of the 6 inner points on the egg-shaped oval $E$ of the bad cubic. The bottom foliation (Fig.\[Tot1:fig\]c) extends the right-part (Fig.b) of that figure (while changing slightly the colorimetry), and shows again that there is no topological obstruction in the large. So it seems that the contradiction (if it exists, i.e. if Rohlin-Le Touzé are right) must really involve some deeper geometry (presumably at the level of Bézout, or maybe Cayley-Bacharach, Jacobi, etc.). Of course our global picture shows some new nodal cubics which are highly contorted, for instance the thick-blue curves. Reminding tightness of nodal cubics, the inside of the loop of that cubic must be convex. By the [*loop*]{} of a nodal cubic we mean the unique arc joining the node to itself via the half which is null-homotopic in $\RR P^2$. This being said, we may from the node of the blue-thick curve $B_3$ trace a rectilinear segment joining the top-point of the loop of $B_3$, and lying entirely in the inside of the loop of $B_3$ (cf. dashed line on Fig.\[Tot1:fig\]c). This segment which is linear (despite the appearances!) cuts for topological reasons at least 4 times the lilac-colored cubic, hence Bézout is corrupted, and perhaps the Rohlin-Le Touzé theorem is nearly proved. Let us formalize the argument. Consider the pencil $\Pi$ of cubics through 8 basepoints (injectively) distributed on the 8 empty ovals of the $C_6$. If $\Pi$ is not totally real, there is a bad cubic $C_3$ whose real part is disjoint from $N$, the nonempty oval of $C_6$. Denote by $E$ the unique oval of this bad cubic which is necessarily smooth. By applying Poincaré’s index formula to $E$ (or the double of its inside) we infer that there is at least 2 saddle points inside $E$. On applying it to $\RR P^2$ we infer that there is at least 8 saddle points on the whole projective plane. Such saddle points correspond to nodal cubics (and perhaps it is convenient to assume some genericity of the pencil after dragging slightly the 8 assigned basepoints). By Bézout recall that $N$ is at most intercepted twice by each $C_3$ of the pencil, and actually exactly twice for each cubic which is not bad (i.e. which intersects $N$). Accordingly we get an involution with 2 fixed points on $N$ (namely the first contact of the bad cubic with good conics). This permits to fold the boundary of $N$ to get a topological sphere. Now depending on whether the first contact with good cubics are inner touch, or outer touch, or double touch (as discussed earlier) we get by the folding different type of singularities. Specifically an inner touch induce no singularity, and so do a double touch, while a outer touch induces a center (do some simple local pictures to get convinced). So we may apply Poincaré inside $N$ (folded) and deduce that there is at least 4 saddles inside $N$ (in accordance with the picture) and perhaps at most 6 saddles (compare picture or think hard). All this to ensure that there is at least one saddle outside $N$ and the corresponding nodal cubic ought to have always a loop enveloping 4 transverse arcs of another nodal cubic with inner node (as on the picture). Remind that the existence of a lilac-colored cubic seems to be forced by Bézout. If all this works then we are finished and the general case is so-to-speak always reducible to the one depicted. Of course we need to be slightly formal (and clever) for instance by defining the concept of a barred-pair of nodal cubics (or barrage for short). This is a pair of nodal cubics such that one of them appears 4 times inside the loop of the other. (For an example cf. again the thickest curves of Fig.\[Tot1:fig\]c.) It remains then to show that such a barred pair always exists, which requires some abstract self-confidence in combinatorics or a long discourse. Note on the picture (at least) that if we consider instead of the thick lilac curve the red one then there is also a barrage consisting of 4 disjoint arcs inside the loop of the blue curve. Hence the proof could decompose in the following 2 steps: Step 1.—Show that there is always a nodal cubic whose loop visits all the 8 assigned basepoints. Step 2.—Show that there is always another nodal cubic forming a barrage w.r.t. the nodal cubic of step 1, i.e. which appears in the inside of its loop as 4 pairs of transverse arcs joining the 8 basepoints in pairs. This is perhaps a universal property of pencil of cubics (or maybe valid only in our special situation, where the 8 basepoints are on a $C_6$, with six of them hexagonally distributed of the convex egg of the bad cubic $C_3$). Universality would be better as then the proof could be simpler, but this looks too optimistic for in that case pencil of cubics would just not exist. Further if Step 1 looks too hard, one could imagine other types of barrages like the one depicted on the 3rd row of Fig.\[Tot1:fig\]d. We hope that \[all\] this \[mess\] can be made clearer and perhaps there is a simpler argument (maybe Le Touzé’s proof). Albeit difficult to make formal the above proof (if it is one!) shows the special rôle played by nodal cubics in the pencil which have lowest complexity from the viewpoint of algebraic geometry. Those are perhaps the unique “brèche par laquelle on puisse entrer dans une place réputée jusqu’ici imprenable”. Doubling, Satellites and total reality {#satellite-total-reality:sec} -------------------------------------- \[23.02.13\] As discussed at length, Rohlin-Le Touzé’s theorem is somewhat elusive to prove but let us assume it to be correct. Why is it so important? Why is it fairly difficult to prove? How does it generalize? As a last remark we note that any proof using the bad cubic tends to be indirect, and this makes any proof a bit frustrating. One could dream of a direct proof using maybe the fact that any cubic of the pencil is dichromatic in the sense of having both inner and outer points one the same component of the cubic. This would give a direct proof but of course still much remains to be justified. Though quite unable to complete the proof, we may try to speculate of what comes next, and what is the true phenomenology governing such phenomena of total reality. According to Ahlfors theorem (1950 [@Ahlfors_1950]) what is behind total reality is basically the orthosymmetric character of the curve. More concretely (or in the spirit of Rohlin 1978 [@Rohlin_1978]), total reality seems to be sometimes forced by the sole knowledge of the real scheme. For instance, we have the prototype of the deep nest of depth $k$ and degree $2k$ which is totally real under a pencil of lines. The point here is that topology forces so many intersections as algebra permits whence total reality. Idem for a quadrifolium nest consisting of 4 nests of depth $k$ and degree $4k$ which is total under a pencil of conics assigned to pass through any 4 points distributed in the deepest ovals. Modulo technicalities, some higher intelligence should be able to perceive total reality of Rohlin’s sextics with the same ease as in the above two examples. Of course certain aspects changes radically, like the 9th unassigned basepoint, as well as the issue that cubics are possibly disconnected, concomitantly with their irrationality, or positive genus of the underlying Riemann surfaces. Is this a sufficient reason to mistrust the ubiquitousness of the phenomenon of total reality, say as (partially) evidenced by Ahlfors theorem at the abstract level? Typical to the basic cases of total reality—sweeping of deep nests via pencil of lines through a deep center of perspective, or the vision of quadrifolia through conics—is some concentric paradigm, namely an infinite series of species totally real under the same pencil. So one can start from a conic and imagine its unique oval (unifolium) doubled, then tripled, etc., and so we get the series of deep nests. The same “satellitosis” occurs by starting from the quadrifolium quartic and doubling each of its ovals in a tube neighborhood, to get an octic totally real, a twelve-tic, etc. [Given a real scheme $S$ (of degree $m$) with only ovals (=nullhomotopic curves) we may abstractly define its [*$k$th satellite*]{} by replicating each oval up to a certain multiplicity $k\ge 1$ and get so the scheme $k\times S$ of degree $km$. (In Rohlin’s sense, a [*real scheme*]{} is primarily an isotopy class of embedding of a disjoint union of circles plus some integer $m$ given in the background memory, the so-called [*degree*]{} of the scheme.)]{} Note that this abstract operation can be aped algebraically just by taking an equation of even degree realizing the scheme $S$ (we assume this to be possible) and then taking $k$ nearby levels close to zero while perturbing the union to get a smooth algebraic curve realizing the $k$th satellite schemes. This makes sense because the sign of an even degree form is well-defined. In particular we can take Rohlin’s sextic scheme $\frac{6}{1}2$ and double it (second satellite) to get the scheme $2 \times \frac{6}{1}2$ of degree 12, or triple it, and so on. It seems clear that this satellite is totally real under the same pencil of cubics as in Rohlin’s (unproven) phenomenon. This is evident when the satellite is realized as a small algebraic perturbation of parallel levels, because in that case we have on the original sextic curve a foliation transverse to the real locus \[\[10.04.13\] this is a bit sloppy but nearly true\], and transversality is topologically stable. So we get an infinite series of curves of type I (as forced by total reality), and it is likely that not merely the algebraic satellites are of type I but all the curves belonging to the schemes. This would be the case if the schemes were known to be rigid, i.e. each forming a unique rigid-isotopy class (\[rigid-scheme:defn\]). More pragmatically, the fact that total reality is exhibited by a synthetic procedure (namely by assigning 8 basepoints on the 8 empty ovals of the $C_6$ of Rohlin’s type or over any satellite of this scheme) makes that we have some robust recipe ensuring total reality. So it is likely that Rohlin-Le Touzé’s theorem implies the following: [(Hypothetical)]{}.—Any $k$th satellite of Rohlin’s schemes of degree $6$ (they are 2 of them namely $\frac{6}{1}2$ and $\frac{2}{1}6$) is again total under a pencil of cubics and so of type I. In particular the $2$nd satellites of Rohlin’s schemes are schemes of degree $12$ which are of type I. With some good faith (or pessimism) one could fear that this implies a corruption of Rohlin’s maximality conjecture (type I implies maximal). The idea would be to take a fairly complicated configuration of 6 ellipses and smooth it à la Brusotti to get a curve whose scheme enlarges $2\times( \frac{6}{1}2)$. Fig.\[Tot2:fig\] includes inconclusive attempts along this naive tactic. -5pt0 -5pt0 Adhering to the opposite attitude, the 2nd satellite of any one of both Rohlin’s $6$-schemes are $12$-schemes (denoted $2\times R$ or just $2R$, cf. Fig.\[Tot2:fig\]a) which are totally real in some geometric way (pencil of cubics through the 8 empty ovals). Hence it is likely that those schemes cannot be enlarged without corrupting Bézout. More precisely assume a real $12$-scheme $S$ enlarging $2R$, then select in $S$ a replica of $2R$ and construct the allied total pencil. Let pass a curve through one of the deleted oval of $S$, and get a corruption with Bézout. More generally if a scheme is of type I, one may expect its representing curves to be totally real under a pencil of curves in some geometrically controlled way. This posits both a concretization of Ahlfors theorem as well as an extension of Rohlin-Le Touzé’s theorem. The byproduct would be a general proof of Rohlin’s maximality conjecture. At this stage, we confess to have first understood the full swing of Rohlin’s prophetical allusion when formulating his maximality conjecture: “there is much to say in its favor” (cf. Rohlin 1978 [@Rohlin_1978]). This idea will be developed in the next section. We can also look at the 3 possible $M$-sextics permitted by Gudkov’s classification and take their satellites to get schemes of type I, actually total under a pencil of quartics via (\[total-reality-of-plane-M-curves:thm\]). The emerging philosophy is that the phenomenon of total reality should be stable under satellitoses and possesses a series of minimal (or primitive) models in each degree. Pencils of lines correspond to deep nests. Pencils of conics (with 4 real basepoints) correspond to the quadrifolium quartics and its satellites. Pencil of cubics have two minimal models with Rohlin’s sextics. Pencil of quartics have (at least) 3 minimal models given by the $M$-sextics (cf. Theorem \[total-reality-of-plane-M-curves:thm\]), etc. All this is quite vague and need perhaps strong correction, but our intention is to suggest the idea of a big tower of total pencils, of which the Rohlin-Le Touzé phenomenon should just be one of the very first cornerstones supporting a big cathedral. Admittedly the latter may reach such altitudes, that its higher structure is still completely dissimulated behind the clouds. The motive behind total reality seems to be a topological predestination forcing reality of all intersections. So the phenomenon ought to be fairly robust. Now if we are given a scheme of type I, then any curve representing it is totally real by Ahlfors theorem. \[$\star$ Not even obvious!\] Yet to attack Rohlin’s maximality conjecture (RMC) we need more namely total reality forced by topological reasons. This amounts essentially to a synthetic knowledge a priori of the location of the basepoints. In this case let us say that the scheme is photovoltaic, more precisely: A real scheme is photovoltaic (PV) if there is a canonical recipe($\approx$algorithm$\approx$Turing machine) exhibiting a total pencil of curves on it. When the recipe is as simple as saying “by assigning basepoints on the empty ovals” of any representing curve of the scheme, the scheme is said to be photographic. We have “photovoltaic” implies “type I”, and even “photovoltaic” implies “maximal”. Of course the problem is that our “canonical recipe” is poorly defined, but one may just understand some algorithm. For instance Rohlin’s $6$-schemes are photographic by the Rohlin-Le Touzé’s theorem, while the $M$-schemes of degree 6 are photovoltaic since there is an algorithm to construct a total pencil via some residual series (cf. Theorem \[total-reality-of-plane-M-curves:thm\]). \[$\star$ But compare also (\[Le-Touzé-extended-in-odd-degree:scholium\]) showing that, in odd degree at least, there is a more concrete recipe for the total reality of $M$-schemes.\] One chance to go around the conceptual difficulty of the ill-posedness of our definition would be the following miracle: All schemes of type I which are not $M$-schemes are actually photographic. \[$\star$ Again in view of (\[Le-Touzé-extended-in-odd-degree:scholium\]) it is likely that $M$-schemes have not to be excluded.\] This is true for sextics (granting the Rohlin-Le Touzé theorem), and deserves to be investigated in general. If the conjecture is true in general, type I implies photovoltaic (by virtue of Theorem \[total-reality-of-plane-M-curves:thm\]), and hence maximal, and Rohlin’s conjecture would be settled. Of course we are using the implication “PV” implies maximal. It looks hard to prove it because “PV” is ill-defined, but we really may avoid this concept since $M$-schemes are automatically maximal (Harnack 1876), while the other are photographic (by the conjecture) so that Rohlin’s maximality conjecture follow form the: (Hypothetical!!!) If a scheme is photographic then it is maximal.  \[$\star$ too vague!\] Suppose by contradiction that $S\subset E$ is an enlargement of the photographic scheme $S$. Choose $E_m$ a real curve representing the scheme $E$, and select a sublocus $\Sigma_m$ of $E_m$ realizing the scheme $S$. Alas we loose algebraicity doing so. However this sublocus $\Sigma_m$ is total under a pencil of curves with basepoints assigned (say) on the empty ovals of $\Sigma_m$ for “robust” topological reasons. This is to mean that any curve of the pencil of $k$-tics cuts $k\cdot m$ points on $\Sigma_m$ for topological reasons (e.g., like for the deep nests). Then we could conclude to a contradiction with Bézout by letting pass a curve of the pencil through the extra oval of $E_m$. The above clumsy proof imposes a refinement of the definition making the above lemma true with “photogenic” instead of the “photographic” assumption. An $m$-scheme is photogenic if any (differentiable, or real analytic) curve $\Sigma$ representing it admits a “total” pencil of $k$-tics such that each curve of the pencil cuts at least $k\cdot m$ points on $\Sigma$. It may even be assumed that the basepoints of such a pencil are assigned in the insides of the empty ovals. This “photogeny” is a violent evasion outside the algebro-geometric realm, yet the deep nest as well as the quadrifolium of depth $k$ are photogenic schemes in this sense. It would be interesting to know if the Rohlin’s $6$-schemes are photogenic amounting to say that Rohlin-Le Touzé’s theorem accepts a purely topological proof. This seems already quite unlikely, and the right part of Fig.\[Tot3:fig\] supplies a simple counterexample. Here we consider a smooth cubic curve $C_3$ and triad of lines $D_3$ (both black colored). We trace (in the smooth category) the blue curve $C_6$ realizing the $6$-scheme $\frac{6}{1}2$ with 8 small ovals about 8 of the 9 intersections of the cubics, plus one large oval enveloping the oval of the smooth cubic. Now the pencil of cubics spanned by $C_3$ and $D_3$ may be interpreted as the pencil of cubics assigned to pass through the insides of the 8 empty ovals of the flexible curve $C_6$, yet it fails to be total as $C_3\cap C_6$ has only 16 points (and not $18$ the product of their degrees). -5pt0 -5pt0 So we cannot expect to be so naive as to be photogenic. This relates to the fact that pencil of cubics (or higher order curves) generally contains disconnected curves. One crude way to ensure total reality could be to use degenerate pencils lying entirely in the discriminant and more than that consisting only of rational curves (forcing via Harnack-Klein or less, like Lüroth-Clebsch, or Cayley) the curve to be connected. However this looks overspecialized and probably not even suited to detect the universal orthosymmetry of Rohlin’s $6$-schemes. It remains to clarify several aspects. Is total reality stable under satellites? In particular is there an infinite series of examples above Rohlin-Le Touzé’s phenomenon of total reality. What are the higher order avatars of Rohlin-Le Touzé’s theorem, and how frequent is the phenomenon? More precisely which schemes are photographic? This looks of course extremely hard requiring a highbrow extension of the Rohlin-Le Touzé’s theorem. Are photographic schemes stable under satellites? If yes this is the trivial part of an iterative propagation of each total reality phenomenon. A priori one can speculate that photographic schemes are quite rare and essentially exhausted by pencil of lines, conics and cubics. In contrast one may expect the phenomenon to be ubiquitous and so frequent that all schemes of type I (safe perhaps some $M$-schemes) \[$\star$ this proviso looks not justified anymore, cf. (\[Le-Touzé-extended-in-odd-degree:scholium\])\] are photographic. In that case there is some little chance to tackle Rohlin’s maximality conjecture (the part thereof post-Shustin’s disproof). Alas even that looks difficult. One may also wonder how frequent are schemes of type I, again rarity versus abundance is quite puzzling. Stability of type I under satellites ------------------------------------ \[24.02.13\] Are schemes of type I stable under satellites? The first case to test is $2\times R$ the 2nd satellite of Rohlin’s $6$-scheme $R:=\frac{6}{1}2$. Of course taking a perturbation of the double of Hilbert’s realization of $\frac{6}{1}2$ (Figs.\[GudHilb8:fig\] or \[GudHilb6-12:fig\]) it is likely that we find a dividing curve, and perhaps Rohlin-Le Touzé’s theorem is sufficiently robust as to imply universally the type I of this $12$-scheme. If not it may be that the $12$-scheme $2\times R$ is indefinite. A priori curves of degree 12 could be sufficiently messy as to allow a type II realization of the 12-scheme $2R$, or in contrast Rohlin-Le Touzé’s phenomenon could be sufficiently robust as to propagate to satellites. The data of a curve plus a totally real pencil of “adjoint” curves is called a flash, and we say that the curve is flashed by the pencil. If a curve of even degree is flashed by a pencil then the doubled curve (and more generally its $k$th satellite) obtained by small perturbation of $k$ concentric levels is flashed by the same pencil. Note that for an algebraic satellite to be defined it is convenient to take an affine chart in which the whole curve is visible. This works certainly for Hilbert’s realization of Rohlin’s $6$-schemes. Hence it is clear that the 12-scheme $2R$ contains a representatives of type I (hence is not a scheme of type II). The question is to decide if this scheme is of type I or indefinite. One idea could be to realize the 8 nests of depth 2 by an octic and then add two ellipses to get $2R$. However, passing a (connected) cubic through the 8 deep nests creates $4\cdot 8=32>3\cdot 8=24$ many intersections, and Bézout is much overwhelmed. Replacing the octic by a curve of degree 10 is still insufficient $(32>3\cdot 10=30)$. A priori one could hope to find a type II realization of the $12$-scheme $2R$ by perturbing an arrangement of 12 lines. This is a bit messy to depict. The most approaching object we could trace is shown on Fig.\[Tot4:fig\]. This is rather akin to the double of the (other) Rohlin scheme $\frac{2}{1}6$, but alas there is not enough free room left to build the prescribed configuration. -5pt0 -5pt0 Another idea is to use Hilbert’s method, but the latter does not seem ideally suited for the generation of nest of depth 2 (Fig.\[Tot5:fig\] of very poor quality). Let us shamefully leave this delicate question, as we sincerely hope that total reality is ubiquitous (in particular stable under satellites). -5pt0 -5pt0 Satellites of curves of odd degrees {#Satellite-odd-degree:sec} ----------------------------------- [*Inserted*]{} \[16.03.13\].—It seems evident that the construction of satellites extends to curve of odd degrees. Of course there is a slight complication coming from the fact that the pseudo-line lacks a trivial tube-neighborhood, and so we cannot replicate so canonically as in the even degree case. As a simple example consider a cubic with 2 circuits (one oval and a pseudoline). Doubling its oval and “doubling” its pseudoline will lead to a curve of degree 6 which (for a suitable smoothing) will be a nest of depth 3, hence again totally real under a pencil of lines. By analogy if we look at the next odd degree, namely 5, we have examples of total reality given by the $M$-quintics (cf. Le Touzé’s Scholie \[LeTouze-quintic:scholie\]). So when taking its satellite we are supposed to find a nice example of total reality in degree 10 for a scheme of the form $(1,6\times 1)$, i.e. 6 nests of depth 2 enveloped in a larger oval. So it is natural to conjecture that this scheme of degree 10 is of type I. If this is possible to prove this is quite interesting because the scheme in question has $r=13$ ovals which is fairly low in comparison to Harnack’s bound $M=37$, when $m=10$ as $g=\frac{(m-1)(m-2)}{2}=\frac{9\cdot 8}{2}=9\cdot 4=36$. Of course, the type I of this scheme is not covered by the RKM-congruence for $(M-2)$-curves. So this gives a certain addendum to Rohlin’s (somehow denigrating) remark that the method of total reality is somehow subsumed to the RKM-congruence (cf. his remark in 1978 which reads “However, all the schemes that we have so far succeeded in coping with by means of these devices are covered by Theorem 3.4 and 3.5. \[i.e., the congruences\]”, compare (\[Rohlin1978-total-reality:quote\]) for the integral citation. Of course total reality (hence type I) is also observed for satellites of the unifolium or quadrifolium having the same property of being at lesser altitude than $(M-2)$-schemes. Thus the phenomenon under examination is formally not new, but those examples being so trivial they were probably not taken seriously enough. So: \[satellite-of-M-quintic-total:conj\] The scheme of degree $10$ of symbol $(1, 6\times \frac{1}{1})$ (cf. Fig.\[satellite-of-Harnack’s-quintic:fig\]a) arising as the 2nd satellite of Harnack’s $M$-quintic with symbol $6 \sqcup J$ (a unique rigid-isotopy class by Kharlamov-Nikulin) is totally real under a pencil of cubics, hence in particular of type I. To prove this we use the method of Le Touzé’s scholie, namely to assign the 8 basepoints of a cubics-pencil on the 6 ovals of the quintic plus 2 on the pseudoline. Then we have 14 intersections and the last one is forced to reality either by algebra (Galois-Tartaglia) or topology (Möbius-von Staudt). [*Optional side remark*]{}.—It may be observed that the scheme in question is not prohibited by Rohlin’s formula. Hint: decomposes the Hilbert tree of the scheme in $x$ and $y$ many branches of length 2 (so $x+y=6$) which are resp. positively or negatively charged. By Rohlin’s formula we have $2(\pi-\eta)=r-k^2=13-25=-12$. By the signs-law (cf. Fig.\[Signs-law-dyad:fig\]) we find $\pi-\eta=x-3y$, and thus $x-3y=-6$, $x+y=6$. Eliminating $x$ gives $-4y=-12$, so $y=3$ and $x=3$. Rohlin’s equation with signs is thus soluble. A priori Le Touzé’s total reality should adapt to the double. Formally we assign 6 basepoints on the deep ovals of the 6 nests and 2 on the maximal oval. Crudely speaking we await $6\cdot 4+3\cdot 2=24+6=30=3\cdot 10$ and total reality would be granted. However in reality we get less than that on basic topological grounds. However it is clear that for a small deformation of the doubled quintic we can expect total reality and the hope is that this propagates to the full scheme (chamber which a priori is not even known to be connected!) Maybe there is some deep reason ensuring total reality like in Le Touzé’s argument. [*Insertion*]{} \[10.04.13\].—It also interesting to compute the mapping-degree of the allied circle map. The number of mobile points of Le Touzé’s series will be $6+12+4=22$, with 1 point circulating on each of the $6$ deep ovals, 2 on the 6 ovals immediately surrounding them and 4 moving on the doubled pseudoline (maximal oval). This degree of $22$ can be compared with Gabard’s bound $(r+M)/2=(13+37)/2=25$ and turns out to be compatible with it. A validation of the conjecture (\[satellite-of-M-quintic-total:conj\]) could be of interest for the following reason related to Rohlin’s maximality conjecture(=RMC). If we think globally at the satellite operation and the arithmetics of small integers factorized into primes ($1,2,3,4=2\cdot 2,5,6=2\cdot 3,7,8=2^3,9=3^2,10=2\cdot 5,11,12=2^2 \cdot 3=2\cdot 6$, etc.), we remark that the first nontrivial satellite [*not*]{} totally real under a pencil of lines or conics truly arises in degree 10. Degree 9 involves the prime $3$, yet the 3rd satellite of the cubic with 2 components is merely the deep nest $4\sqcup J$ (totally real under lines). In degree 6 we have indeed the 2 total realities of Rohlin-Le Touzé yet they are primitive manifestations (not satellites), and of course in adequation with Gudkov’s classification and Rohlin’s maximality conjecture. Hence degree 10 is the first case where (granting our conjecture) we get a type I scheme which possibly is not maximal (in case we are skeptical about the truth of RMC). Of course this would be against our own philosophy that Ahlfors has much to say about Hilbert’s 16th. Yet we must keep in mind this eventuality. Thus we can ask for a curve of degree 10 enlarging the doubled $M$-quintic (with 6 bifolia, plus one large oval, Fig.\[satellite-of-Harnack’s-quintic:fig\]a). The naive construction of Fig.\[satellite-of-Harnack’s-quintic:fig\]b does not even reach more than 4 bifolia nested in a larger oval. So other techniques of construction are demanded, perhaps Harnack, Hilbert or Viro. -5pt0 -5pt0 If the posited phenomenon of total reality holds true, and is sufficiently explicit to imply maximality (as nearly evident by Bézout) then our scheme of degree 10 (Fig.\[satellite-of-Harnack’s-quintic:fig\]a called say the closed sextibifolia) would be maximal in the hierarchy of all schemes of degree 10. Hence this scheme could not be enlarged and it results, in one stroke, a myriad of prohibitions upon Hilbert’s 16th problem in degree 10 (still wide open), basically by virtue of the sole idea of total reality which goes back virtually to Riemann’s thesis 1851/57, then Schottky, Klein, Bieberbach, Teichmüller, Ahlfors just to name the heros. So the question looks nephralgic. As a philosophical detail, while in Hilbert’s 16th there is some traditional focus upon curves maximizing the number of ovals (so-called $M$-curves since Petrovskii 1933/38), we see here in contrast that the lower the number of ovals is (for a curve subsumed to total reality) the stronger will be its prohibitive impact upon the higher stages of the pyramid. Of course the prototype is the deep nest, but this is merely the trivial case. In degree 5 there are also $(M-2)$-curves which are totally real, typically under a pencil of conics, yet the corresponding scheme is not of type I. Its double will be of degree 10 and have a bi-quadrifolium (double couche) nested in a larger oval. Toward a census of all type I (totally real) schemes or at least extension of the Rohlin-Le Touzé’s phenomenon prompted by the Rohlin-Kharlamov-Marin congruence for $(M-2)$-schemes {#census-and-extension-of-Rohlin-Le-Touzé:sec} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ \[24.02.13\] Another problem is to list all schemes of type I, or at least those totally real under a pencil of curves. We restrict attention to even degrees schemes ($m=2k$) and call [*order*]{} the degree $d$ of curves involved in the total pencil. In degree $m=2$ we have a single oval (unifolium, denoted $1$ in Gudkov’s notation) which is total under a pencil of lines $d=1$ with center of perspective chosen inside the oval. In degree $m=4$, we have the nest of depth 2 (denoted $\frac{1}{1}=(1,1)$ in Gudkov’s symbolism) total under a pencil of lines, and the quadrifolium $4$ total under a pencil of conics. For $m=6$, we have the nest of depth 3 (Gudkov symbol $(1,1,1)$) total for $d=1$, and for $d=3$ the 2 Rohlin’s schemes $\frac{6}{1}2$, $\frac{2}{1}6$ (Rohlin-Le Touzé’s theorem), as well as the three $M$-schemes $\frac{9}{1}1$, $\frac{5}{1}5$, $\frac{1}{1}9$ of Hilbert, Gudkov, Harnack respectively. The latter are total for $d=4$ (cf. Theorem \[total-reality-of-plane-M-curves:thm\]). Before attacking the case $m=8$, some remarks are in order. What is expected is that sometimes total reality is ensured by Bézout. This is the case when $d=1$, or $d=2$ where we have the extra knowledge of the connectivity of all members of the pencil due to the rationality (unicursality) of all genus $0$ curves. This property is lost when passing to high-orders $d\ge 3$ pencils. Still we can hope a priori that some other reasons prompt total reality in some favorable cases, which would be more elementary than the Rohlin-Le Touzé’s theorem for $(m,d)=(6,3)$. It is with this naive hope that we feel encouraged to adventure in the jungle of $m=8$. Another vague motivation is that schemes of type I are conjecturally maximal, so we are really attacking a sort of simplified Hilbert’s 16th problem (or rather surfing on its upper envelope). A last remark is that we (sentimentally) expect via Ahlfors theorem that the dividing character of curves is always exhibited by a [*linear*]{} pencil inducing a map to the projective line (whose complexification is the Riemann sphere with its standard “equatorial” real structure like our planet Earth). [*Optional (non-linear pencils).*]{}—However any totally real map to more complicated dividing curve suffices to exhibit the dividing character of the covering curve. So perhaps we should keep in mind to explore also such nonlinear pencil. How do they arise concretely is another question. A naive guess of mine was via a plane cubic $E_3$ with 2 circuits and its dual curve (variety of tangents). We could hope that any point of some curve $C_m$ determines a unique tangent to $E_3$ but alas there are (generally) 6 of them passing through a given point (intersect with the polar curve, a conic here). Perhaps a suitable adaptation of this idea leads somewhere. However it is fairly standard, and we briefly discussed this (in Part I devoted to the abstract theory of Riemann-Klein-Ahlfors) that generally speaking curves mapping to irrational curves of positive genus have specialized moduli. Hence it is quite unlikely that we shall gain a general methodology, though plane curves themselves are modularly confined. $\bigstar$ For $m=8$, we have the deep nest of depth 4, denoted $(1,1,1,1)$ total for $d=1$, and the doubled quadrifolium $\frac{1}{1}\frac{1}{1}\frac{1}{1}\frac{1}{1}=2\times 4$ which is total for $d=2$. $\bullet$ We examine next $d=3$ (cubics-pencils). We have then 8 basepoints available assumed all real, and so we look at curves with this number of empty ovals. Imposing the 8 basepoints on the empty ovals, we are ensured for twice so many intersections (i.e. $16$) but this is still much less than $3\cdot 8=24$ (Bézout’s upper bound). Total reality looks hard to ensure. Of course we may envelop our empty ovals by some nonempty ovals. Remember that there is at most 4 nonempty ovals (as the doubled quadrifolium $2\times 4$ is total under a pencil of conics hence maximal). So we may range our 8 empty ovals in 4 groups of ovals and consider the scheme $\frac{k}{1} \frac{\ell}{1} \frac{m}{1} n$, where $k+\ell+m+n=8$. If optimistic each of the 3 nonempty ovals contributes for 2 intersections, and we arrive at $16+6=22$ which is still less than $24$. Hence it seems nearly impossible to find a (naive) phenomenon of total reality for $(m,d)=(8,3)$, but this does not of course exclude the possibility of such a phenomenon prompted by deep geometric reasons à la Rohlin-Le Touzé. Note also that the case $m=6$ showed that there is no (absolute) total reality for $d=2$, and so we cannot expect a priori to observe the phenomenon for each preassigned order and therefore let us skip the present value $d=3$. $\bullet$ Assume next $d=4$. We have then $13$ basepoints assignable. Remember indeed that the space $\vert 4 H \vert$ of all quartics has dimension $\binom{4+2}{2}-1=\frac{6\cdot 5}{2}-1=14$, so that 13 conditions leave the mobility of a pencil. So we are again directed toward curves with 13 empty ovals (over which we shall as usual distribute our 13 basepoints), ensuring so 26 intersections, which is less than the $d m=4\cdot 8=32$ required. Enveloping our ovals in the at most 3 possible nonempty ovals (if 4 of them we reduce to the doubled quadrifolium) we get the schemes $\frac{k}{1} \frac{\ell}{1} \frac{m}{1} n$, where $k+\ell+m+n=13$. So the number of ovals is $r=13+3=16$ and we have an $(M-6)$-scheme (as $M=g+1=22$ for $m=8$). Perhaps like in the case $(m,d)=(6,3)$ some of them are total for deep geometrical reasons. If lucky, the 3 nonempty ovals creates 6 additional intersections, so reaching $26+6=32$ Bézout’s bound, and total reality would be granted. Remember yet that for $(m,d)=(6,2)$ there is no phenomenon of total reality, at least of the purest form where only knowledge of the real scheme is required. Repeating ourselves, we cannot expect a priori that total reality prevails for each value of $d$ given a fixed $m$. $\bullet$ Examine next $d=5$. Then $\dim \vert 5 H \vert = \frac{7\cdot 6}{2}-1=20$ so that 19 basepoints are assignable. By the same token, we look at schemes $\frac{k}{1} \frac{\ell}{1} \frac{m}{1} n$, where $k+\ell+m+n=19$, ensuring $2\cdot 19=38<40=d m=5 \cdot 8$ intersections. So in fact outside from the 19 empty ovals it is enough to have one nonempty oval being intercepted to gain total reality. We list the following candidates: $\bullet$ $\frac{k}{1}\ell$, with $k+\ell=19$ and $r=20=M-2$. $\bullet$ $\frac{k}{1}\frac{\ell}{1}m$, with $k+\ell+m=19$ and $r=21=M-1$ so cannot be total by Klein’s congruence $r\equiv g+1 \pmod 2$. $\bullet$ $\frac{k}{1} \frac{\ell}{1} \frac{m}{1} n$, with $k+\ell+m+n=19$ and $r=19+3=22=M$, which are $M$-curves. In the first class of $(M-2)$-schemes we may appeal to the Rohlin-Kharlamov-Marin congruence (\[Kharlamov-Marin-cong:thm\]) to select the serious candidates. (It seems fairly plausible that this mode of reasoning was the true motivation behind Rohlin’s Ansatz of total reality for the $6$-scheme $\frac{6}{1}2$ and its mirror, and that he found his (unpublished) proof a posteriori of this deeper knowledge.) This RKM-congruence states that an $(M-2)$-curve of degree $m=2k$ and type II satisfies the congruence $\chi=p-n\equiv k^2$ or $k^2\pm 2 \pmod 8$. This looks a priori undigest, but as merely to be interpreted as a deviation from Gudkov’s hypothesis(=congruence proved by Rohlin), compare, e.g., the diagrammatic in degree 6 (Fig.\[Gudkov-Table3:fig\]). So when the congruence is violated a scheme of type I is granted. Since $\chi=1-k+\ell$ and $k^2=4^2=16\equiv_8 0$ (beware the overuse of the letter $k$ but no risk of confusion). Recall the Swiss cheese algorithm for the Euler characteristic $\chi$ (that whenever we make a hole in the sense of removing a disc, $\chi$ drops by one unit, cf. Listing-Klein-von Dyck 1888, etc.). Starting with the scheme $\frac{19}{1}$, we find $\chi=1-19=-18\equiv_8 -2$. Then we have $\frac{18}{1}1$ for which $\chi=1-18+1$ equal to the former plus 2 units, and there is always an increment of 2. Running through the full list of such schemes we find that the condition $\chi\equiv_8 4$ ensuring type I occurs with periodicity 4 for the following schemes: $$\label{octics-five-examples-RKM:eq} \frac{16}{1}3,\quad \frac{12}{1}7,\quad \frac{8}{1}11,\quad \frac{4}{1}15,\quad 20.$$ The latter case (of the scheme $20$) seems to disprove our collective contraction conjecture (\[CCC:conj\]). \[CCC-conj-disproof:thm\] [(ERRONEOUS—cf. $\bigstar$ below)]{}.—The collective contraction conjecture (of Gabard positing a wild extension of the contraction conjecture of Itenberg-Viro) is false in degree $8$ already. $\bigstar$ \[05.03.13\] [*Corrigendum*]{}.—It follows easily from the so-called Thom conjecture that the scheme $20$ is not realized by an algebraic curve of degree 8 (necessarily of type I if it existed by the RKM-congruence), compare Theorem \[Thom-Ragsdale:thm\]. So the given argument is not a disproof of CCC. \[07.03.13\] In fact a simpler obstruction of this scheme comes from Rohlin’s formula (\[Rohlin-formula:thm\]), as $2(\pi-\eta)=r-k^2$ but the left-side is zero, so $r=k^2=16$, which is no the case. \[10.04.13\] Further this scheme is also prohibited (and this was historically the first proof available) by Petrovskii’s inequality (\[Petrovskii’s-inequalities:thm\]), which reads $\chi\le \frac{3}{2}k(k-1)+1=18+1=19$. This in contrast to the proofs via Thom or Rohlin does not use the dividing character of the curve prompted by RKM. Further it should be noted that our Thom-style theorem cited above is erroneous in the generality stated, yet sufficient to imply the present application as in the case at hand the filled surface is orientable (since we only glue disc to the half, and so there is no risk to create a twisted handle like in Klein’s bottle). For definiteness, let us briefly work out the argument. Since $20$ is an $(M-2)$-curve (of type I by RKM), we may split the Riemann surface and fill one half by the $20$ discs bounding the ovals. It will result a surface of genus $1$, whose fundamental class has degree $8/2=4$. However Thom conjectured (and Kronheimer-Mrowka, and others, proved) that the genus is minimized by algebraic (smooth) curves, hence at least $3$ in degree 4. Since our surface beats this bound, the real curve $20$ is prohibited.  \[Outdated, but keep in mind the 2nd part of the proof (strangulation argument), which under CCC would provide another obstruction of the scheme $20$, of a fairly intuitive character, though hard to implement with present technology.\] —It seems clear that this 8-scheme $20$ is realized by (a variant of) Hilbert’s method.[^69] (I should still work out this in some more detail.) The resulting curve is of type I by the just cited congruence of Rohlin-Kharlamov-Marin, which is essentially based either on the deep Rohlin’s signature theorem for spin 4-manifolds, or perhaps on Kähler geometry in the presentation of Kharlamov (unpublished?). Alternatively on the model at hand (via Hilbert’s method) the type I of this curve realizing $20$ may be checked more elementarily via Fiedler’s sense-preserving smoothing law (elementary surgeries). However the resulting curve cannot be contracted collectively by shrinking simultaneously all its ovals to points, for if it could, then $C_8\to C_4 \cup C_4^\sigma$ would degenerate to a pair of conjugate quartics obtained by strangulating the Riemann surface $C_8(\CC)$ along all the separating ovals, and so $C_4\cap C_4^\sigma$ would consist of 20 solitary nodes. Bézout is overwhelmed. Moreover the above 5 schemes are avatars of the total reality claim of Rohlin-Le Touzé for $(m,d)=(6,3)$, i.e. sextics flashed by cubics, while now octics are flashed by quintics. In both cases we note the rôle of curves of order 3 units less than the given degree $m$, and one seems being sidetracked to the theory of adjoint curves à la Brill-Noether, etc. Recall indeed that adjoints of order $(m-3)$ cut out the so-called [*canonical series*]{} on the given plane curve, and thus there is perhaps some conceptual reason ensuring total reality of all these linear systems. This is perhaps the royal road to attack the Rohlin-Le Touzé’s assertion/theorem (and extension thereof prompted by the RKM-congruence). In both cases $m=6$ or $8$ we have $(M-2)$-curves which are of type I, and swept out by a pencil of order $d=m-3$. The latter cuts the canonical series of the curve $C_m$ of degree $2g-2$ and dimension $(2g-2)-g=g-2$ or rather $g-1$?? In view of Gabard 2006 [@Gabard_2006] we may expect to find a total morphism of degree the mean value of $r=M-2$ and Harnack’s bound $M$, hence of degree $M-1$. So we could choose so many points on the ovals of $C_m$ while putting two of them on the nonempty oval. It is then hoped that the 2 points situated on the same oval will [*dextrogyrate*]{} (i.e. move along one orientation of the oval without entering in collision) and then total reality is ensured. While any collection of $M=g+1$ points on a curve of genus $g$ moves in its linear equivalence class, only special collections of $M-1=g$ points will move but perhaps this is enough to ensure total reality hence recover the type I of the above list of schemes predicted by the RKM-congruence (\[Kharlamov-Marin-cong:thm\]). It is not entirely clear if the Rohlin-Le Touzé’s phenomenon is true in full generality or only for special groups of points (at least this is the naive intuition coming from the abstract Ahlfors and Gabard viewpoint). Note in this respect that Rohlin’s claim is a priori less strongly formulated than Le Touzé’s assertion, in claiming only that a pencil of cubic exhibit total reality and not that all of them with deeply assigned 8 basepoints are total (compare Rohlin 1978 [@Rohlin_1978 p.94] with Le Touzé’s 2013 announcement in Sec.\[e-mail-Viro:sec\]). ($\star$ \[10.04.13\]—Meanwhile see also the article [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics].) It is likely (say by analogy with the trivial case $m=4$) that Rohlin had in mind the strong assertion of Le Touzé, and that it is only the extreme compression of Rohlin’s exposition that forced him to his somewhat looser version of the statement. For $m=8$ we may assign 19 basepoints on the empty ovals of any of the $(M-2)$-scheme listed above (the scheme $20$ is exceptional in a sense that remains to be clarified[^70]). Then we can pass a quintic $C_5$ through these points and one extra point. Choose the latter on the nonempty oval $N$ of the $C_8$. Another intersection is created by topology. So we see 21 real points. The residual intersection of $C_5\cap C_8$ consist of $40-21=19$ points, etc... Of course the difficulty looks immense but let us postulate the following avatar of Le Touzé’s theorem (i.e. strong form of Rohlin’s total reality claim): \[tot-real-for-octics:conj\] For any octic representing one of the $8$-schemes listed above (Eq. \[octics-five-examples-RKM:eq\]) the pencil of quintics assigned to pass through the $19$ empty ovals (or even their insides) is totally real. (Of course the scheme $20$ deserves a modified statement of which we do not know yet the exact shape.)[^71] If so is the case we get a total morphism $C_8\to \PP^1$ of degree $40-19=21$, which is the mean value of $r=20$ and $M=g+1=22$, in accordance with Gabard 2006 [@Gabard_2006]. Of course the latter affords only weak evidence as its result is subsumed to high suspicion. It could be expected (granting Gabard’s result as correct) that a suitable interpretation thereof (at the level of extrinsic algebraic geometry à la Brill-Noether) could supply a proof of the conjecture at least in the weak form of a special configuration of $19+2=21$ points two of them being distributed on the same oval (while dextrogyrating). However ideally we would like a purely synthetical proof say as elementary as Bézout without incursion of such transcendental philosophers like Abel-Riemann. This is perhaps possible as some highbrow variant of the Rohlin-Le Touzé’s theorem but remains to be explored and is quite likely to be extremely elusive. On the other hand it could be of vital interest that the Abel-Riemann abstract viewpoint may help to see clearer what happens in the Plato cavern of Hilbert’s 16th problem as twisted by Rohlin’s synthesis with Klein’s Riemannian viewpoint. In this optimistic scenario we may hope to get when enlarging further $m$ above $m=8$ an infinite series of schemes of type I totally flashed by pencil of $(m-3)$-curves. So the phenomenon of total reality à la Rohlin-Le Touzé would be fairly frequent (and in some sense an extrinsic reflection of Ahlfors abstract theorem). Perhaps Le Touzé’s proof (2013) adapts to give the above the conjecture, but alas as I do not know yet the details. It can also be the case that additional difficulties occur while the combinatorics viz. geometry becomes more involved and the argument more tedious. The argument is likely to start as follows. As we have $2\cdot 19=38$ intersections granted, only two are missing to reach total reality at $40=5\cdot 8$. The sole obstruction is therefore a bad quintic $C_5$ in the pencil disjoint from the nonempty oval $N$ of the $C_8$. But this mean that there is an Abelian differential without zero on this oval, and try to derive a contradiction. Remember that the differential has $2g-2$ zeros (Riemann-Poincaré index formula) and we may hope to infer something. When we look at the trajectories of a (generic) holomorphic $1$-form we see only hyperbolic saddles of index $-1$ explaining the degree of the canonical class as boiling down to Poincaré’s index formula (1881/85). In the case $m=6$ for the scheme $\frac{6}{1}2$ we have 8 zeros assigned each creating a companion on the same oval (so 16) and a total of $3\cdot 6=18$ zeros, in accordance with $2g-2$ for $g=10$ the genus of sextics. The above looks a numerical miracle alike, but is not for $g=\frac{(m-1)(m-2)}{2}$ so that $\deg K=2g-2=(m-1)(m-2)-2=m(m-3)$, showing that adjoint curves of degree $m-3$ are indeed involved in the canonical class. There is of course a more intrinsic reason allied to the adjunction formula. The dream would be that there is some metaphysical/toplogical principle ensuring total reality on the basis of holomorphic 1-forms which can often be interpreted in terms of incompressible fluids. Note yet that the argument cannot be too abstract (else could imply that all $(M-2)$-curves are of type I regardless of the isotopy class, a nonsense already for $m=4$), yet ideally it would be as simple as in the case $m=4$ so $d=m-3=1$ where we really see total reality on the Gürtelkurve $C_4$ with 2 nested ovals. Here we see some potential place of action for old stuff à la Abel-Riemann-Klein (\[10.04.13\] or maybe also Thurston’s argument in Gross-Harris 1981), but alas this escaped much from my memory or my curriculum. And still there is always this objection that the argument should really use the assumption on the real scheme, and so ought to be more in the Arnold-Rohlin spirit. Have we listed all schemes of degree 8 of type I? Probably not as our search was far from exhaustive. It remains of course to list $M$-schemes (and this is a classical still open problem for some few exceptional cases). Compare works by Viro, Korchagin, etc. It is however likely that our list is exhaustive for $(M-2)$-schemes. (\[08.03.13\] Not even true, as we shall soon see!) $\bigstar$ What is next? Degree $m=10$ of course. Here we have the deep nest of depth 5, totally flashed by a pencil of lines. The quadrifolium $4$ does not give nothing by taking its satellite (as our $m=10$ is not a multiple of 4). Next we move directly to adjoint curves of order $d=m-3=7$ (septics). One has $\dim \vert 7H \vert= \binom{7+2}{2}-1=\frac{9\cdot 8}{2}-1=35$. So $34$ basepoints may be assigned freely, and we can force $34 \cdot 2=68<70=7\cdot 10$ nearly all points to be real by assigning the basepoints to be located on distinct ovals. Again it is a reasonably folly (by analogy with Rohlin-Le Touzé’s assertion) to expect that under adding an extra nonempty oval enveloping some of the ovals and if furthermore the RKM-congruence is satisfied that the resulting scheme (being of type I) is totally real under the described pencil. Precisely we look at the schemes $\frac{k}{1} \ell $, with $k+\ell =34 $. So we have $r=35$ ovals and $M=g+1=\frac{(m-1)(m-2)}{2}+1=\frac{9\cdot8}{2}+1=37$ is 3 units above the number of basepoints (no surprise as the dimension of the space of curves and the genus both involve the same binomial coefficient). So we are again in presence of $(M-2)$-curves. The RKM-congruence says that type II forces $\chi$ to be either $k^2=25\equiv_8 1, k^2\pm 2\equiv 3,-1\equiv 7 \pmod 8$ so that $\chi\equiv_8 5$ forces type I. Applying the Swiss cheese recipe to $\frac{34}{1}$, we find $\chi= 1-34=-33\equiv_8 -1 $ and then running through all subsequent schemes $\frac{33}{1}1$, etc., $\chi$ always increases by two units. So we first met $\chi=5$ for $\frac{31}{1}3$, and find using fourfold periodicity the following list of schemes (potentially totally real): $$\label{RKM-schemes-deg-10:planar:eq} \frac{31}{1}3, \quad\frac{27}{1}7, \quad\frac{23}{1}11, \quad\frac{19}{1}15, \quad\frac{15}{1}19, \quad\frac{11}{1}23,\quad\frac{7}{1}27,\quad\frac{3}{1}31.$$ (Like for $m=6$ (but unlike the case $m=8$) the list is symmetrical under the evident mirror of partnership in the jargon of Kharlamov-Finashin.) Again we expect the following total reality: All curves $C_{10}$ of degree $10$ representing any one of the schemes of the previous display are totally real under the pencil of septics assigned to visit $34$ basepoints injectively distributed among the $34$ empty ovals of the ten-ics $C_{10}$. Actually the last item $\frac{3}{1}31$ of the list is prohibited by either Thom [(\[Thom-Ragsdale:thm\])]{} or by Rohlin’s formula $2(\Pi^+-\Pi^-)=r-k^2=35-25=10$, since $\Pi^+-\Pi^-\le \Pi:=\Pi^+ +\Pi^-=3$. [*Insertion*]{} \[10.04.13\].—The last scheme of the series is (alas) [*not*]{} prohibited by Petrovskii’s inequality (\[Petrovskii’s-inequalities:thm\]), but it is by the strong Petrovskii estimate of Arnold (1971), cf. (\[Strong-Petrovskii-Arnold-ineq:thm\]). This states $p-n^{-}\le \frac{3}{2}k(k-1)+1$, where $n^{-}=0$ here (negative hyperbolic ovals), so $p\le 31$ while our scheme as $p=32$. (It may be useful—if you are better in geography than in arithmetics—to visualize all this on Fig.\[Degree10:fig\]. Crudely put, Arnold is as strong as the Ragsdale conjecture.) Further our assertion regarding the prohibition by Thom is certainly foiled as there is no reason ensuring orientability of the Arnold surface in the present case, as we are really attaching a 3-holed disc to the half of the complexification. (More explanations in Sec.\[Thom:sec\].) It is evident (due to the little arithmetical coincidence between the genus and the dimension of the curve-hyperspace, plus the universal validity of the RKM-congruence) that this series of $(M-2)$-schemes propagates in each degree $m\ge 10$, and so we get an infinite (nearly tautological) repetition of the above conjectures for each even integer $m$. [*Can all these conjectures be proven in a single stroke?*]{} This would be a highbrow extension of the Rohlin-Le Touzé’s theorem. This would give an infinity and certain abundance to the phenomenon of total reality as one could have suspected from Ahlfors’ theorem. Of course we do not claim that this will supply an exhaustive list of the phenomenon, but perhaps it is modulo the operation of satellites. More precisely: \[primitive-manifestation-of-tot-real:conj\] Any primitive manifestation of the phenomenon of total reality on a curve $C_m$ of (even) degree $m$ arises either as an $(M-2)$-scheme totally real under a pencil of adjoint curves of order $m-3$ assigned to pass through the empty ovals of $C_m$, or as a pencil of curves of order $m-2$ if $C_m$ is an $M$-curve (cf. [Theorem \[total-reality-of-plane-M-curves:thm\]]{}). If not primitive then the scheme is a satellite of either: $\bullet$ the unifolium scheme $1$ of degree $2$ total under a pencil of lines (this gives the series of deep nests which exist in all degrees $m$), or $\bullet$ the quadrifolium $4$ of degree $4$ total under a pencil of conics, with satellites in all degrees multiples of $4$, or $\bullet$ the other $(M-2)$-schemes of Rohlin $\frac{6}{1}2$ or $\frac{2}{1}6$ with satellites in all degrees multiple of $6$, and so on inductively as the satellites of $(M-2)$-schemes predicted by the Rohlin-Kharlamov-Marin congruence, or finally $\bullet$ as satellites of $M$-schemes of lower degrees always dividing the given one $m$. If this conjecture is true we would have a complete classification of the phenomenon of total reality for plane curves. This is surely somewhat premature and probably requires some slight adjustments to reach more respectableness. [*Insertion*]{} \[10.04.13\].—In particular, one must probably also takes into account satellites of curves of odd orders, cf. Sec.\[Satellite-odd-degree:sec\]. For instance in degree $m=10$, there is probably the 2nd satellite of Harnack’s $M$-quintic playing a rôle. At any rate note that our initial expectation that some phenomenon of total reality is purely prompted by Bézout in a very primitive way is apparently never borne out. It seems rather that apart from the satellites of the elementary schemes (unifolium and quadrifolium) flashed resp. by the trivial pencil of lines and conics the phenomenon of total reality is at least as hard as Rohlin-Le Touzé’s theorem, but perhaps not much harder. At least both ought to be connected by the geometry of the canonical series. More $(M-2)$-schemes in degree 8 of type I {#RKM-schemes-deg-8-MORE:sec} ------------------------------------------ \[26.02.13\] In fact it is clear that even for $m=8$ we have not listed all $(M-2)$-schemes of type I for we have only considered those with one nonempty oval, but we must also consider those with 2, or 3 nonempty ovals. Tabulating a complete list is merely an exercise of combinatorics. Geometrically, it may not be essential to assign basepoints on empty ovals but some can be located on nonempty ovals, and we may expect total reality provided the RKM-congruence is fulfilled. The sole problem is that we then lack some recipe to assign basepoints, and so the game becomes somewhat obscure \[but quite challenging\]. First the RKM-congruence (\[Kharlamov-Marin-cong:thm\]) can be more conveniently paraphrased as: \[RKM-congruence-reformulated:thm\] [(Rohlin 1978-Kharlamov 197?-Marin 1979)]{} Any $(M-2)$-scheme of degree $m=2k$ such that $\chi\equiv k^2+4 \pmod 8$ is of type I. \[08.03.13\] [*Little Warning*]{}.—There is a minor metaphysical trouble with this statement. Indeed when $m=8$ (or for larger $m$) we have the scheme $20$ which satisfies the RKM-congruence, but which is not realized algebraically as follows either from Thom (\[Thom-Ragsdale:thm\]) or from Rohlin’s formula $2(\Pi^+-\Pi^-)=r-k^2=20-16=4$, since $\Pi^+-\Pi^-\le \Pi:=\Pi^+ +\Pi^-=0$ due to the absence of nesting in $20$. There is two ways to go around this trouble, either add the assumption that the scheme is algebraic, or interpret Rohlin’s definition of the types of schemes by declaring (usual logical nonsense allied to the empty set) that a non-realized scheme is simultaneously of type I and type II (but not of indefinite type which needs algebraic representatives in both types). Of course when tracing pyramids, e.g. the Gudkov-Rohlin table (Fig.\[Gudkov-Table3:fig\]) we ascribe the type I label (red-circle) only to those schemes which are of type I in the concrete sense that the scheme is [*algebraic*]{} (and all its realizations are of type I). The RKM-congruence (Theorem \[Kharlamov-Marin-cong:thm\]) says that an $(M-2)$-curve of degree $m=2k$ and type II satisfies the congruence $\chi\equiv k^2 $ or $k^2 \pm 2 \pmod 8$. So the theorem follows after checking the following basic fact: [*an $(M-2)$-curve of order $m=2k$ verifies universally $\chi \equiv k^2 \pmod 2$.*]{} (From the diagrammatic of pyramids, e.g. Fig.\[Gudkov-Table3:fig\], this is the fairly trivial matter that the rhombic equilateral lattice underlying the pyramid is adjusted at $k^2$, and one may infer that the claimed congruence holds more generally on all $(M-2i)$-levels.) This is easy to prove using the relations $\chi=p-n$, $r=p+n=M-2$ $$\chi=p-n=(p+n)-2n\equiv_2 (p+n) = r=M-2,$$ and by Harnack’s bound and the genus formula $g=\frac{(m-1)(m-2)}{2}$ we have $$M=g+1=\textstyle\frac{(2k-1)(2k-2)}{2}+1=(2k-1)(k-1)+1=2k^2-3k+2,$$ whence $$\chi\equiv_2 M-2=2k^2-3k \equiv_2 k \equiv_2 k^2.$$ Is the converse statement true? Remember that if a scheme of degree $m=2k$ is of type I, then it satisfies Arnold’s congruence $\chi\equiv k^2 \pmod 4$. Hence $\chi\equiv k^2, k^2+4 \pmod 8$, and the second option leads to type I, but I do not know if the first option necessarily implies type II or indefinite type. We know only that this converse holds true for $m\le 6$ by the Gudkov-Rohlin table (=our Fig.\[Gudkov-Table3:fig\]) which involve explicit constructions. So the RKM-congruence detects many $(M-2)$-schemes of type I, but it is not clear (to me) if it detects all of them. \[RKM-converse:conj\] An $(M-2)$-scheme of degree $m=2k$ which is of type I necessarily satisfies the RKM-congruence $\chi\equiv k^2+4 \pmod 8$. [*Insertion*]{} \[11.04.13\].—Perhaps an answer can be found in Rohlin 1978 [@Rohlin_1978 p.93–94], esp. Art. 3.5 and the end of 3.6, where it seems that an extremal property of the strong Arnold’s inequalities observed by Zvonilov-Wilson prompts type I in situation apparently not covered by the congruence. Alas, I had not yet the time to assimilate this properly, but look forward with great excitement to do so in the future (after some long editorial duty). \[08.03.13\] Again there is little worry about definitions. For instance when $m=6$ we can consider the (non-algebraic) scheme $(1,1,1)6$ which is thus of type I (in the logical sense but of course also of type II), yet with $\chi=(1-1+1)+6=7\neq 3^2+4\equiv_8 5$. So we tacitely assume the scheme of type I in the strong sense that it is algebraically realized. Of course the conjecture \[RKM-converse:conj\] is true for $m=6$ (look at the Gudkov-Rohlin Table=Fig.\[Gudkov-Table3:fig\]), which depends upon explicit construction of curves of type II for all schemes which are not RKM. Already for $m=8$, the conjecture seems to demand a menagerie of construction. One could hope that there is some theoretical argument. Let us leave this question aside, as we merely want to list schemes of type I potentially subsumed to the phenomenon of total reality. Let us now tackle the combinatorial aspect of dressing the list of all $(M-2)$-schemes of degree $m=8$ satisfying the RKM-congruence (hence of type I). We may start with schemes with zero or one empty ovals and list all of them using the fourfold periodicity as we already did. Yet to be more systematic we start with $\frac{16}{1}3$ expand its Gudkov’s symbol as $\frac{16}{1}\frac{0}{1}\frac{0}{1}1$ to be of the shape $\frac{x}{1}\frac{k}{1}\frac{\ell}{1}m$ and then we trace a cubical lattice (Fig.\[RKM-schemes-deg-8:fig\]) in 3-space encoding all variations of this symbol for varied values of $(k,\ell,m)$. To aid visualization it turned useful to ascribe colors to the different levels: the ground floor is orange, the 1st floor is lilac, the 2nd floor blue, the 3rd floor is cyan, the 4th floor is yellow-green. As we are interested in $(M-2)$-schemes we have the relation $x+k+\ell+m+3=M-2=20$. The crucial point is that when $k$ or $\ell$ increases by one unity, then one hole is traded against another hole, so that $\chi$ is left unchanged. In contrast an increment of $m$ reduces $x$ by one, and so a hole in “$x$” is traded against a disc outside, so that $\chi$ increases by two. Hence as the RKM-congruence is modulo 8, we have a 4-fold periodicity until reaching the same value for $\chi$. (This explains the vertical motion along the cubical lattice.) Those symbols surrounded by dashed lines are doubloons (non-normalized Gudkov’s symbol), yet useful to stop the combinatorial proliferation. Underbraced symbols are those whose Gudkov’s symbol admits a shorter expression given below the brace (when enough room is left available). -5pt0 -5pt0 All those schemes are avatars of the 2 Rohlin’s $(M-2)$-schemes of degree 6 (subsumed to total reality). It is a simple matter to count them. First collect on the nearby face of Fig.\[RKM-schemes-deg-8:fig\] all schemes lying in perspective beyond the red/thick numbers. Adding them vertically gives the blue/big numbers on the bottom row, yielding a total of $5 \cdot 11+9= 55+9=64$ schemes. (That this is a power of 2, incidentally the same as Rohlin’s count of all schemes of degree 6 decorated by types, is probably a mere coincidence.) This is somewhat amazing combinatorics, and the geometrical conjecture would be that all these schemes are total under a pencil of quintics with suitably assigned 19 basepoints. The case which looks most appealing is when there are exactly 19 empty ovals. Those corresponds to the 4 schemes forming the vertical left 1-simplex of Fig.\[RKM-schemes-deg-8:fig\], which we call the monolith. The monolith has some obvious structure of a 3-simplex, stratified as follows into sub-simplices: $\bullet$ 0-simplex corresponding to the scheme $20$ with zero nonempty oval, $\bullet$ $1$-simplex corresponding to the 4 schemes with 1 nonempty oval and so admitting a Gudkov’s symbol $\frac{k}{1}\ell$, $\bullet$ $2$-simplex corresponding to the 20 schemes with 2 nonempty ovals admitting a Gudkov writing $\frac{k}{1}\frac{\ell}{1}m$, $\bullet$ $3$-simplex corresponding to the 39 schemes with 3 nonempty ovals admitting a Gudkov writing $\frac{k}{1}\frac{\ell}{1}\frac{m}{1}n$. Only the schemes forming the one simplex have exactly 19 empty ovals. For the other categories (with resp. 20, 19, 18, 17 empty ovals) one may assign the 19 basepoints among the nonempty ovals (by choosing say the most massive nonempty oval, i.e. containing the largest number of empty ovals). Of course this is pure speculation and maybe the exact opposite has to be done. There is probably here work for several generations of computing machines, unless one is able to crack all total reality phenomenon in a single stroke. Somewhat brutally in comparison to our low understanding of where to assign basepoints, we posit that whenever the RKM-congruence is verified then there is a phenomenon of total reality: Suppose given an $(M-2)$-curve of degree $m=2k$ verifying the RKM-congruence $\chi\equiv k^2+4 \pmod 8$. Then the pencil of adjoint curves of order $(m-3)$ ascribed to pass through the empty ovals plus some other points distributed on the nonempty ovals is totally real. As to the crude arithmetics, remember that the number $B$ of basepoints assignable to adjoints of order $(m-3)$ is given by the binomial coefficient $$B=\textstyle\binom{(m-3)+2}{2}-1-1,$$ while the pre-Harnack bound $$M-2=(g+1)-2=\textstyle\binom{m-1}{2}+1-2,$$ so that $$B=M-3.$$ This means that we have one basepoint less than the number of ovals, and so we may canonically distribute them when there is one nonempty oval. \[M-2-curve-degree-like-Gabard:rem\] [\[03.03.13\] Assuming we are capable to ensure total reality of the pencil, it may be observed that the degree of the induced total map to $\PP^1$ would be in accordance with Gabard’s bound $r+p$, which is also the mean value of $r$ and $M=g+1$. This follows from a simple calculation. First $$2B=2(M-3)=2(\textstyle\frac{(m-1)(m-2)}{2}+1-3)=(m-1)(m-2)-4=m(m-3)-2,$$ and so the degree of the map is $$m(m-3)-B=2B+2-B=B+2=M-1,$$ which is Gabard’s bound i.e. the mean of $r=M-2$ and $M$. ]{} Of course if one as some self-confidence in Gabard 2006 [@Gabard_2006], then there is a total map of that degree on each dividing $(M-2)$-curve (in particular those satisfying the RKM-congruence which are universally of type I). By some concretization yoga this map would be induced by a total pencil, and by a dubious reverse engineering of the above arithmetics this would be a pencil of $(m-3)$-tics. This gives some very weak evidence for the: Any dividing $(M-2)$-curve of degree $m$ is totally real under a pencil of curves of order $(m-3)$. \[27.02.13\] In fact we are not even sure that the above cubical lattice (Fig.\[RKM-schemes-deg-8:fig\]) gives an exhaustive list of RKM-schemes in degree 8, where we use the jargon: A scheme of degree $2k$ is an RKM-scheme if it is an $(M-2)$-scheme satisfying the Rohlin-Kharlamov-Marin congruence $\chi\equiv k^2+4 \pmod 8$ which forces type I (alias orthosymmetry) of the scheme, i.e. that the real locus disconnects the complex one. While any RKM-scheme is of type I, we do not know whether the converse holds true (for $(M-2)$-schemes). It is true for $m=6$ as follows from the Gudkov-Rohlin classification (Fig.\[Gudkov-Table3:fig\]). It seems that degree 8 is a perfidious iceberg killing any naive conjecture arising from contemplation of low order curves (say $\deg \le 6$). Specific illustration of this vague principle are Shustin’s disproof of the one-half of Rohlin’s maximality conjecture, and concomitantly the disproof of Klein’s Ansatz that nondividing curves may always acquire a solitary node. Another remark along the same line is the disproof (using the RKM-congruence) of the naive CCC-conjecture, cf. Theorem \[CCC-conj-disproof:thm\]. (\[06.03.13\] Alas this disproof of CCC is disproved by Thom’s conjecture, as remarked there!) Now back to our classification of RKM-schemes of degree 8 we may wonder if there is one containing $(1,1,1)$ the nest of depth 3. As we focus on $(M-2)$-schemes and since $M=22$ when $m=8$, we may start with this configuration plus 17 outer ovals. In Gudkov’s notation this is the scheme $(1,1,1)17$. Any scheme $S$ of even degree is bounded by the Ragsdale orientable membrane $S^{\ast}$ with $\chi(S^{\ast})=p-n$. In our case $\chi((1,1,1)17^{\ast})=1-1+1+17=18\equiv 2 \pmod 8$ and not 4 as posited by the RKM-congruence. If we trade outer ovals against inner ovals lying deepest then $\chi$ is left unchanged. However if the trading is made for ovals at intermediate depth then the outer discs of the Ragsdale membrane becomes holes and $\chi$ diminishes by two. So the RKM-congruence is first arranged for the scheme $(1,\frac{1}{1}3)14$ (with $\chi=12$), and then using 4-fold periodicity the list is augmented as: $$\label{RKM-scheme-deg-8-four-primitive-type:eq} (1,\frac{1}{1}3)14, \quad (1,\frac{1}{1}7)10, \quad (1,\frac{1}{1}11)6, \quad (1,\frac{1}{1}15)2,$$ which are all RKM-schemes (containing the nest of depth 3). Once the Euler characteristic is adjusted to satisfy the RKM-congruence, we may trade outer ovals with innermost oval at depth 3 without altering $\chi$. So each of these schemes produces a list of derived schemes also RKM. Namely the 15 schemes $$\begin{aligned} (1,\frac{1}{1}3)14,& (1,\frac{2}{1}3)13, (1,\frac{3}{1}3)12, (1,\frac{4}{1}3)11, (1,\frac{5}{1}3)10, (1,\frac{6}{1}3)9, (1,\frac{7}{1}3)8, (1,\frac{8}{1}3)7, \cr (1,\frac{9}{1}3)6,& (1,\frac{10}{1}3)5, (1,\frac{11}{1}3)4, (1,\frac{12}{1}3)3, (1,\frac{13}{1}3)2, (1,\frac{14}{1}3)1, (1,\frac{15}{1}3),\end{aligned}$$ and then the 11 schemes $$\begin{aligned} (1,\frac{1}{1}7)10,& (1,\frac{2}{1}7)9, (1,\frac{3}{1}7)8, (1,\frac{4}{1}7)7, (1,\frac{5}{1}7)6,\cr (1,\frac{6}{1}7)5,& (1,\frac{7}{1}7)4, (1,\frac{8}{1}7)3, (1,\frac{9}{1}7)2, (1,\frac{10}{1}7)1, (1,\frac{11}{1}7),\end{aligned}$$ and likewise the 7 schemes $$\begin{aligned} (1,\frac{1}{1}11)6,\quad (1,\frac{2}{1}11)5,\quad (1,\frac{3}{1}11)4,\quad (1,\frac{4}{1}11)3,\quad (1,\frac{5}{1}11)2,\quad (1,\frac{6}{1}11)1,\quad (1,\frac{7}{1}11), $$ and finally, the 3 schemes $$\begin{aligned} (1,\frac{1}{1}15)2,\quad (1,\frac{2}{1}15)1,\quad (1,\frac{3}{1}15).\end{aligned}$$ (All together this gives $15+11+7+3=36$ additional schemes to be added to the 64 tabulated on Fig.\[RKM-schemes-deg-8:fig\], hence a total of $64+36=100$.) If we neglect the largest nonempty oval we have 19 ovals and we may expect total reality of the quintic pencil ascribed to pass trough any (injective) distribution of the 19 basepoints on those 19 ovals (which albeit not all empty are the deepest items of the combinatorial scheme). At this stage we hope to have exhausted the RKM-schemes of degree 8: \[RKM-schemes-deg-8-census:lem\] Any RKM-scheme of degree $8$ has either depth $\le 2$ in which case it is catalogued as one of the $64$ schemes of Fig.\[RKM-schemes-deg-8:fig\], or it has depth $3$ in which case it is one of the $4$ displayed schemes or one of the $36$ derived products where an outer oval is traded against an innermost oval (cf. the last 4 display formulae tabulating the corresponding $36$ Gudkov’s symbols). In particular there are exactly $64+36=100$ schemes of degree $8$ which are RKM, and hence of type I (and therefore potentially subsumed to the phenomenon of total reality). \[06.03.13\] Addendum: One of them (at least), namely $20$ is not realized as it violates the Thom conjecture (cf. [\[Thom-Ragsdale:thm\]]{}), or better the Rohlin formula. \[13.03.13\] [*Warning*]{}.—The list of 100 schemes is far from exhaustive, cf. remarks right after that Lemma \[RKM-scheme-ruled-out:lem\]. (pseudo-proof) Alas we are not even sure that this list is now exhaustive albeit it might be likely by using the concept of depth of a scheme (the longest chain of ovals totally ordered by inclusions of their insides, i.e. the unique bounding disc given by the Schoenflies theorem in its smooth variant implicit in Möbius 1863 [@Moebius_1863], Hilbert (tacit), Dehn ca. 1899 (unpublished), Osgood 1902, Schoenflies 1906, etc., cf. e.g. Siebenmann 2005 [@Siebenmann_2005] for some historical background and the literature cited therein). Given any scheme of degree $8$, its depth is at most 4. If equal to 4 it contains the deep nest and so the scheme is saturated (i.e. it cannot be enlarged without corrupting Bézout). If the depth is 3 then its contains $(1,1,1)$ the nest of depth 3, and if we were not too bad in combinatorics our recipe of 36 schemes above was exhaustive. For the same vague reason, when the depth is $\le 2$ then the catalogue of 64 schemes is exhaustive. So we have one RKM-scheme of depth 1, $63$ such schemes of depth $2$, and 36 RKM-schemes of depth 3, while the unique scheme of depth 4 (deep nest (1,1,1,1)) is not an $(M-2)$-scheme hence not an RKM-scheme. Further, it could be that sophisticated Bézout-style obstructions à la Fiedler-Viro (\[Viro-Fiedler-prohibition:thm\]) prohibit the realizability of some of those schemes in the algebraic realm. So perhaps several items albeit schemes in the abstract sense of Rohlin are not algebraically realized. (Improvising terminology and to conflict even more with the Grothendieck-Rohlin collapse of jargon we could speak of a Hilbert-scheme (H-scheme) when the scheme is realized algebraically.) So I do not know if the 100 RKM-schemes listed above are H-schemes. ([*Update*]{} \[06.03.13\] At least one of them $20$ is not realized as follows from Thom’s conjecture, cf. Theorem \[Thom-Ragsdale:thm\], or better just apply Rohlin’s formula.) Taking another naive look at the Gudkov-Rohlin table for degree $m=6$ (Fig.\[Gudkov-Table3:fig\]) shows that $(M-2)$-schemes are subjected to no restriction and so we can speculate the same for $m=8$, in which case all our 100 schemes would be $H$-schemes. At any rate we note that enlarging $m=6$ by just 2 units, involve a de-multiplication by the factor $50$ of all RKM-schemes. (Skip this paragraph.) Have we really listed every RKM-schemes? We could start from another elementary configuration like 2 nests of depth 2 (Gudkov symbol $\frac{1}{1}\frac{1}{1}=(1,1)(1,1)$) and then add 16 outer ovals to get $(1,1)(1,1) 16$. Then $\chi= 1-1+1-1+16\equiv 0 \pmod 8$, while the good RKM-value is 4. So we trade outer ovals for inner ovals at depth 1, and so $\chi$ diminishes by 2 units. Hence we find first $(1,3)(1,1)14=\frac{3}{1}\frac{1}{1}14$ or $(1,2)(1,2)14=\frac{2}{1}\frac{2}{1}14$. Those are already catalogued on Fig.\[RKM-schemes-deg-8:fig\]. Then as the Euler characteristic is adjusted we may apply the same trick of trading outer ovals for innermost ovals without changing $\chi$, yet doing so we create schemes with 2 nests, one of depth 3 and one of depth 2, so that Bézout is violated by tracing the line through their “centers”. So it seems that no new candidates for total reality occurs along this way. How to assign basepoints? {#How-to-assign-basept:sec} ------------------------- \[01.03.13\] As a matter of extending the Rohlin-Le Touzé theorem (still unpublished and abridged RLT) to degree 8, we would like to know where to assign basepoints on each item of our list of 100 RKM-schemes. A priori not all of them are totally real in some uniform way despite the presence of Ahlfors theorem. Recall that our census of 100 RKM-schemes may be interpreted as five families: (1).—the scheme $20$, (not realized by Thom \[Thom-Ragsdale:thm\], or Rohlin’s formula). (2).—4 schemes of the form $\frac{k}{1}\ell$, (3).—20 schemes of the form $\frac{k}{1}\frac{\ell}1 m$, (4).—39 schemes of the form $\frac{k}{1}\frac{\ell}1 \frac{m}{1} n$, (5).—36 schemes enlarging the nest of depth 3. The class (2) consist precisely of those elements having 19 empty ovals. And those are the most direct candidates for an avatar of the RLT-theorem. However in the class (5) there is also 19 preferred deep ovals, namely all those which are either empty or if nonempty which are not maximal (for the usual order on ovals given by inclusion of their bounding discs). So there is a family of 40 schemes where a direct extension of the RLT-theorem is straightforward (at least to state, but maybe not to prove). On the other hand it could be the case that there is an extended formulation including all those 100 schemes. At least one idea would be to consider the notion of dextrogyre oval (abridged dextro-oval). For a dividing curve, we say that an oval $O$ is a dextro-oval if its porous-inside $O^{\star}$, that is the inside minus the insides of all ovals directly inside it, has complex orientation matching $\partial O^{\star}$ that arising as boundary of the porous-inside. Of course any empty oval is dextro. As an example consider the Gürtelkurve $C_4$ of degree 4 with 2 nested ovals. Then either by using the total pencil of lines through a center in the innermost of the nest or by Fiedler’s law of positive smoothings the complex orientation consist of 2 concentric circles with the “same” orientation. So the nonempty oval is not dextro, while of course the inner oval is (being empty). Here we see that the pencil of lines is total precisely when its basepoint is assigned on the dextro-oval. We may therefore expect that the pencil of quintics on our curves of degree 8 is total whenever the $19$ basepoints are distributed on $19$ dextro-ovals supposed available. Consider e.g. the scheme $\frac{3}{1}\frac{1}{1} 14$. Then by Rohlin’s formula $2(\Pi^{+}-\Pi^{-})=r-k^2=20-16=4$, the difference $\Pi^{+}-\Pi^{-}$ is 2, while $\Pi^{+}+\Pi^{-}=4$, so that $\Pi^{+}=3$ and $\Pi^{-}=1$. From this one infers (picture) that there is at least one dextro-oval which is not empty. And so we have here precisely 19 dextro-ovals. Obviously one can extend this to some other schemes, and running through the catalogue one could dress an exhaustive list of all schemes with 19 dextro-ovals (of course 20 will not belong to it), and expect the phenomenon of total reality for the latter. Of course this method is somewhat ad hoc as it uses the dividing character of the curve while in its purest form (say as a way of taking independence of the RKM-congruence) one would like to avoid this knowledge. Back to degree 6: weak form of RLT {#RLT:sec} ---------------------------------- \[01.03.13\] Obviously we were moving too fast by looking at degree 8 and need to return to degree 6, to get rid off of those combinatorial difficulties while concentrating on the geometry of total reality. Our point now is that we may be pessimistic about Rohlin-Le Touzé’s theorem as being false in the generality announced by Le Touzé 2013[^72] (or at least difficult to prove). Recall moreover that Rohlin’s cryptical statement is not as strong as Le Touzé (at least leaves some free room for interpretation). Even if Le Touzé’s claim of total reality is correct, it could be that total reality is easier to prove for special 8 assigned basepoints. This weaker statement would still be sufficient to detect the dividing character of curves having an RKM-scheme, i.e. $\frac{6}{1}2$ and $\frac{2}{1}6$. \[9th-basepointI:lem\] Suppose given a sextic $C_6$ of type $\frac{6}{1}2$. Assign $8$ basepoints on the empty ovals, and look at the corresponding pencil $\Pi$ of cubics. Then there is a 9th basepoint $p_9$ of $\Pi$. To ensure total reality of $\Pi$ it is enough that $p_9$ is either on $N$ the nonempty oval of $C_6$ or more generally in its inside $N^{\ast}$. As 16 intersections are forced by topology, it remains only to gain 2 extra intersections for totality. If $p_9\in N$ then this is clear, and if $p_9\in N^{\ast}$ then by taking a cubic $C_3$ of $\Pi$ through a point $p\in N$ which is connected then it is easy to see for vibratory reasons that the other point $q$ of $C_3\cap N$ (whose existence is ensured either by topology or by algebra) will dextrogyrate on $N$. This is to mean that when $C_3$ is slightly perturbed both points $p,q$ will move on $N$ along the same orientation. This suffices to ensure total reality, as both points cannot then enter in collision to disappear in the imaginary locus. This argument uses existence of a connected cubic in any pencil of cubics. Another slight variant is to argue by contradiction by appealing to a bad cubic $C_3$, i.e. disjoint from $N$. If $p_9$ is inside $N$ then the oval of $C_3$ cannot vibrate properly, and we reach a contradiction. So the whole problem of Rohlin-Le Touzé in weakened form can be reduced to the: Given any $C_6$ of type $\frac{6}{1}2$, there exists an (injective) distribution of $8$ points $p_1,\dots, p_8$ on the $8$ empty ovals such that the pencil of cubics $\Pi$ interpolating them has its $9$th basepoint located in the (sealed) inside $N^{\ast}$ (i.e. $N$ included) of the nonempty oval. (In particular such a calibrated pencil is totally real.) One could hope to prove this weaker assertion by pure topology without having to enter in fine Bézout-style considerations (upon which Le Touzé’s proof is likely to rest). How to prove this conjecture? One very naive way would be just to use that $N$ divides the plane and so it would suffice to find two octuplets $p_i$ such that $p_9$ is resp. inside and outside $N$. By continuity of the 9th basepoint as a function $\beta$ of the 8 assigned one, we would be ensured of another intermediate octuplet such that $p_9\in N$. A less naive way would be to assume by contradiction that $p_9$ always misses $N^{\ast}$ (the sealed interior) and then retract on the core of the residual Möbius band. This seems to give an essential (non null-homotopic) map. On the other hand as $\beta$ extends to a (16-dimensional) cell (given by allowing the $p_i$ to explore the insides of the 8 empty ovals) it must be null-homotopic. Alas the first step of argument is not easy to complete. At any rate, designating by $E_i$ the 8 empty ovals of the $C_6$, we can define the Rohlin body of the $C_6$ as the image of the Rohlin map $$\beta \colon E_1\times\dots\times E_8 \to \RR P^2$$ taking the 8 assigned basepoints of $\Pi$ to the 9th unassigned basepoint $p_9$. This map is well-defined by virtue of the following easy lemma. \[independent-cond:lem\] Our $8$ basepoints impose independent conditions on the space of cubics, and this holds true more generally when the basepoints are allowed to vary in the insides $E_i^{\ast}$ of the empty ovals $E_i$. Let $\Pi_i$ be the linear system of curves passing through the first $i$ points $p_1,\dots,p_i$, $i=1,\dots, 8$. We have a filtration $$\vert 3H \vert \supset \Pi_1 \supset \Pi_2 \supset \Pi_3 \supset \Pi_4\supset \Pi_5\supset \Pi_6\supset \Pi_7\supset \Pi_8=\Pi,$$ and one checks that all inclusions are strict by exhibiting an appropriate curve. Strictness of the first inclusions is trivial by taking appropriate configuration of 3 lines (Fig.\[TotFiltra:fig\]). For strictness of $\Pi_6\supset \Pi_7$, consider a conic $C_2$ passing through 5 inner points plus a line through the 6th inner point, yet missing the 2 outer points $7,8$. We have to check that this $C_3$ does not pass through $7$. If it does then $C_2\cap C_6$ would consist of $5\cdot 2+2+2=14>2\cdot 6$ points as two extra intersections are created with $N$, so that Bézout is violated. For the last strictness $\Pi_7\supset \Pi_8$ one takes the same $C_2$ and aggregate the line $L$ through the points $7,8$. -5pt0 -5pt0 Denoting by $E:=E_1\times \dots \times E_8$ the Cartesian product of the empty ovals and by $E^\ast:=E_1^{\ast}\times \dots \times E_8^{\ast}$ that of their insides $E_i^{\ast}$, we have a factorization of $\beta$ as $$\beta\colon E \to E^{\ast}\to \RR P^2,$$ where the first map is the inclusion and the second $\beta^{\ast}$ is given by the same recipe as $\beta$. It follows that $\beta$ is null-homotopic (since $E^{\ast}$ is a 16-cell). On the other hand we may hope to show that if $\beta$ avoids $N^{\ast}$ the inside of the empty oval that the induced co-restriction map $\bar\beta\colon E\to \RR P^2-N^\ast$ whose target is homotopically a circle is [*essential*]{} (i.e. not null-homotopic). For this it would be enough to show that the induced morphism $\pi_1(\bar\beta)$ hits a odd multiple of the generator of the $\pi_1$ of $\RR P^2-N^\ast$ which is a Möbius band. This contradiction would prove the conjecture and so total reality of a suitable pencil. However this strategy demands some geometric understanding that presently elude us. Another more naive strategy (using less topology) would be that the map $\beta\colon E \to \RR P^2$ is open (say as a vestige of the holomorphic character of the underlying complexification). Then it would be plain that $\beta(E)$ is compact (hence closed) and open, hence equal to all $\RR P^2$ by connectedness of the latter space. (As to be soon discussed this surjectivity of $\beta$ would however conflict with the Le Touzé theorem) At any rate we see that the problem of total reality à la Rohlin-Le Touzé in its weakened form due to us (or perhaps Rohlin depending on the interpretation of his cryptical statement) is fairly basic at first sight. We are given a curve $C_6$ of type $\frac{6}{1}2$. Given any 8 points on the empty ovals $E_i$ we have some predestination mapping $\beta$ assigning to them the 9th basepoint (by a somewhat elusive recipe) and total reality of the pencil $\Pi$ of curves passing through the 8 points is ensured if $p_9$ lands in the inside $N^{\ast}$ of the nonempty oval $N$ of $C_6$. So crudely speaking we have one chance over two that total reality holds true, for a given configuration of point. We would like to show that it is always possible to have total reality for a clever configuration of 8 points, while the stronger Le Touzé’s announcement claims it for all choices of 8 points. Our hope is that independently of whether this stronger statement is right or false there ought to be a simpler proof of the weaker assertion by say essentially topological methods. By using the dextrogyration argument it is clear that we have the: \[9th-basepoint-totalII:lem\] The pencil $\Pi$ is total iff $p_9$ its non-assigned basepoint belongs to $N^{\ast}$ (the sealed inside of the nonempty oval $N$). If $p_9\in N$ total reality is clear. If $p_9$ is in the open inside (interior) of $N$, then we have an odd number of basepoints insides. Looking at the oscillation of a connected member $C_3$ of the pencil about its basepoints it results that both points of $C_3\cap N$ (there cannot be more than two by Bézout) will dextrogyre, i.e. move along the same orientation of $N$ (compare Fig.\[Dextrogyre:fig\]a). Hence no collision can occur in the long run, since by the holomorphic character of the map a point cannot reverse spontaneously its sense of motion as the curve $C_3$ is dragged along the real locus of the pencil. [*Insertion*]{} \[11.04.13\].—We can dispense connectedness of the $C_3$ (though quite easy to arrange) as follows. Choose any point on $N$ and consider the cubic $C_3$ through this point. If $C_3\cap C_6$ is not totally real, the oval of $C_3$ is necessarily inside $N$. For vibratory reasons this oval must visit an even number of basepoints, and actually must visit all 6 assigned inner basepoints (otherwise total reality of $C_3\cap C_6$ is granted). The 9th basepoint cannot be located on the oval of the $C_3$ (else it cannot vibrate properly), hence it is situated on its pseudoline and we may again conclude dextrogyration by the slaloming argument across an odd number of basepoints (compare Fig.\[Dextrogyre:fig\]b). -5pt0 -5pt0 Conversely if the 9th basepoint $p_9$ lies outside of $N$ then there is an even number of basepoints inside $N$ and the 2 points of a connected $C_3$ located on $N$ will anti-dextrogyre. In that case there will be a collision in the long run and total reality is foiled. Hence the option $\beta$ surjective would contradict the Le Touzé’s theorem. Of course this is not a serious objection against her theorem because usually holomorphic maps restricted to real loci fails blatantly to be surjective (a key prototype of this phenomenon opposite to total reality, is when one projects on a line an ellipse from an outer point). [*Optional paragraph.*]{}—Another slight variant: we may assign 7 basepoints on all safe one empty ovals and the 8th one $p_8$ on $N$. Then a little advantage is that whenever $p_8$ is collinear with 2 points of the first seven $p_i$, we get a special split cubic in the pencil $\Pi$, namely the line plus a residual conic. It is not very clear if this little advantage is really useful, and leave open this discussion. New meditation after reception of Le Touzé’s article ---------------------------------------------------- \[02.03.13\] Yesterday (01.03.13), we conjointly received (with several other Russian colleagues Kharlamov, Viro, Nikulin, etc.) a copy of Le Touzé’s article vindicating Rohlin’s cryptical assertion of total reality of the RKM-sextics of type $\frac{6}{1}2$ and $\frac{2}{1}6$ under a pencil of cubics. This fascinating paper helped me to rectify several misconceptions of mine about the content of her earlier announcement. In particular, she writes the following illuminating remark (Le Touzé[^73] 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics p.3]): “By a congruence due to Kharlamov \[8\](=Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91]), the real schemes $\la 2 \vc 1 \la 6\ra\ra$ or $\la 6 \vc 1 \la 2\ra\ra$ are both of type I. We use this fact in the proof, so we confirm that the sextics with this two schemes do not contradict Rokhlin’s conjecture. Rokhlin claimed that he could prove the very same statement [*without*]{} using the fact that the sextics are dividing. It’s a stronger result. Unfortunately, his proof was never published and is now lost[^74].” The Rokhlin conjecture alluded to by Le Touzé is made explicit in (p.2–3 of ), which is again worth quoting: “The $M$-curves are clearly dividing. The so-called hyperbolic curves are also dividing. A hyperbolic curve of degree $m=2k$ or $2k+1$, consists in $k$ nested ovals, plus one pseudo-line if $m$ is odd. A pencil of lines whose base point is chosen in the innermost oval sweeps out the curve in such a way that the $m$ intersections are always real. One says that this pencil of lines is totally real with respect to the hyperbolic curve. Starting from this observation, Rokhlin \[10\](=1978 [@Rohlin_1978]) presents a beautiful argument[^75] proving that if an algebraic curve is swept out by a totally real pencil of lines, then this curve is dividing. The argument generalizes to pencils of curves of higher degrees. Can conversely any dividing curve be endowed with some totally real pencil?[^76] A weaker conjecture suggested implicitly in \[10\](=Rohlin 1978 [@Rohlin_1978]) is that any curve whose real scheme is of type I may be endowed with a totally real pencil[^77]. It turns out that the $M$-curves may indeed be endowed with suitable pencils of degree $(m-2)$, see \[6\], page 348(=Theorem \[total-reality-of-plane-M-curves:thm\] in this work=Gabard 2012/13 [@Gabard_2012/13], pagination may have fluctuated meanwhile).” Le Touzé’s article clarified several misinterpretation of mine about her announced result (dated 16 février 2013, cf. Sec.\[e-mail-Viro:sec\]) but which we reproduce now as we misunderstood it: $\bullet\bullet\bullet$ samedi 16 février 2013 17:54:55 Dear Alexandre, dear other colleagues, I have managed to prove that a pencil of cubics with eight base points distributed in the eight empty ovals of a sextic $2 \cup 1(6)$ is necessarily totally real. Details will follow soon in a paper. Yours, Séverine In fact my misconception was to think that Le Touzé claims total reality for [*any*]{} such pencil. So shame on me for not having read her message more carefully as she expressly writes [*a pencil*]{}. For me it is still unclear if the stronger claim of total reality for all such pencils holds true. Such a strong form of total reality holds in the basic cases (pencil of rational curves of degree $\le 2$) but perhaps the case of cubics is completely different for such curves need not being connected and also there is a 9th predestined basepoint which cannot be freely assigned (magneto repulsion well-known at least since Euler 1748 as reported e.g. in Griffiths-Harris 1978 [@Griffiths-Harris_1978/94 p.673]). Taking (deliberately) the risk of being too cavalier let us put forward the strongest form of total reality (à la Rohlin-Le Touzé but perhaps too coarsely interpreted than what those authors ever wrote): [*Added in proof*]{} \[08.03.13\].—Meanwhile Le Touzé validated the following conjecture, for explanations, cf. her message (dated 5 March 2013 in Sec.\[e-mail-Viro:sec\]). \[SRLT:conj\] [(SRLT=Strong Rohlin-Le Touzé total reality, as misinterpreted by Gabard)]{} Given any (smooth) sextic $C_6$ of type $\frac{6}{1}2$ or $\frac{2}{1}6$ and any (injective) distribution of $8$ points on the $8$ empty oval of $C_6$, the pencil of cubics through the $8$ assigned basepoints is totally real. First, the basic Lemma \[independent-cond:lem\] (on independency of conditions) seems to imply that the given linear system is really a pencil. The next step would be to prove (or disprove) this strong conjecture of total reality. The (more cautious) statement of Le Touzé does not imply the conjecture, yet it would be interesting to know if Le Touzé is aware of an obstruction refuting the conjecture. Let us reproduce Le Touzé’s theorem (cf. Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]): \[LeTouze-big-thm-01-March-13:thm\] [(Le Touzé 1st March 2013, announced 17 February 2013)]{} Any $(M-2)$-sextic with real scheme $\frac{2}{1}6$ or $\frac{6}{1}2$ may be endowed with a totally real pencil of cubics with $8$ basepoints distributed in the $8$ empty ovals. Also very interesting (though somewhat disappointing) is the issue that Le Touzé’s proof uses the RKM-congruence (as brilliantly nuanced by herself), and so does not recover the [*synthetic a priori*]{} character of Rohlin’s claim (which so becomes even more cryptical than it ever was). This dependence of Le Touzé’s proof on the RKM-congruence (which she ascribes to Kharlamov alone, like in the original text Rohlin 1978) has some overlap with thoughts I had yesterday (especially Sec.\[How-to-assign-basept:sec\]). Le Touzé’s work makes very acute the desire to find the so-called [*lost proof of Rohlin*]{}. We thought on the question from various angles yet our present understanding is very confused due to overwork and nervous collapse (i.e., plethora of strategies and panoply of statements of varying strength). The naive strategy (toward conjecture SRLT=(\[SRLT:conj\]) or just a weakened form thereof) at which we arrived yesterday (Sec.\[RLT:sec\]), was to use the map $\beta$ assigning to the 8 assigned basepoints the 9th (unassigned) basepoint. With this the lost proof of Rohlin could be harpooned by an essentially topological argument, yet we failed to overcome the last difficulty. For convenience let us repeat some of our ideas along this topological tactic. Given a $C_6$ say of type $\frac{6}{1}2 $ for simplicity. Denote by $E_i$ the 8 empty ovals. For any injective distribution of points on those 8 ovals we have a pencil $\Pi$ of cubics through them (Lemma \[independent-cond:lem\]), and the latter is totally real iff the 9th basepoint of $\Pi$ lands in the (sealed) inside $N^{\ast}$ of $N$ the nonempty oval of $C_6$ (cf. Lemmas \[9th-basepointI:lem\] and \[9th-basepoint-totalII:lem\]). Formally we can so introduce the [*$9$th basepoint map*]{} $$\beta\colon E_1\times \dots \times E_8 \to \RR P^2$$ taking the octuplet $(p_1, \dots , p_8) $ to the 9th basepoint of the pencil $\Pi$ of cubics passing through $p_1, \dots, p_8$. A priori we could hope this mapping to be onto for reasons of Brouwer’s topological degree of a map (in homology modulo 2). Precisely if the induced morphism $H_2(\beta, \ZZ_2)$ is non-trivial, the mapping $\beta$ would be surjective (for otherwise a point is missed and the map factorizes through a punctured (hence open) Hausdorff manifold whose top-dimensional homology vanishes). Of course all this general theory can be dispensed as we just have a punctured projective plane homotopically equivalent to a circle. In this surjectivity scenario for $\beta$, it would suffice to take an inner point $p_9\in N^{\ast}$ and any lift of it via $\beta$ yields an octuplet inducing a total pencil on the $C_6$. In contrast, taking a point outside $N$ would imply that the strong form of total reality (SRLT)=(\[SRLT:conj\]) for all octuplets fails. However it is unlikely that this Brouwer-style surjectivity criterion works because the map $\beta$ extends (still by virtue of Lemma \[independent-cond:lem\]) to the (sealed) insides $E_i^{\ast}$ (bounding discs) of the $E_i$ as to give the map $$\beta^{\ast}\colon E_1^{\ast}\times \dots \times E_8^{\ast} \to \RR P^2,$$ also defined by taking the 8 assigned basepoints to the 9th one. Since the source of the map $\beta^{\ast}$ is a contractible $16$-cell (hypercube) the induced map $H_2(\beta, \ZZ_2)=0$ is trivial. Still there is some hope to do something good. If the range(=image) of the map $\beta\colon E:=\times_{i=1}^8 E_i \to \RR P^2$ meets $N^{\ast}$ we have total reality at least in the weak form (RLT). (Asserting the strong form SRLT amounts knowing that $\beta(E)\subset N^{\ast}$.) Hence to prove RLT it suffices to show that the option $\beta(E)$ disjoint form $N^{\ast}$ leads to a contradiction. Our naive idea is then that the map $\beta$ would have its range confined to the Möbius band $M:=\RR P^2-N^{\ast}$ residual to $N^\ast$. If one is able to show that $\beta\colon E \to M$ is not null-homotopic (e.g. by showing that it hits an odd multiple of the generator of $\pi_1(M)=\ZZ$), then it would follow that $\pi_1 (\beta)$ is nontrivial, violating the above factorization $\beta^{\ast}$ through a contractible 16-cell. This contradiction would prove the weak form of RLT, i.e. existence of an octuplet inducing a total pencil. Of course the above tactic to be completed requires a proof of the hypothetical fact that there is a loop in the 8-torus $E$ taken by $\beta$ to the nontrivial element of $\pi_1(\RR P^2)$. This seems hard to prove and requires at any rate some geometric understanding of pencil of cubics, especially of the predestination process creating the 9th basepoint as a function of the 8 assigned ones. More pragmatically we could define the [*Rohlin body*]{} of a given $C_6$ (of RKM-type) as the image $B:=\beta (E)$. This compactum and especially its location w.r.t. the nonempty oval $N$ will govern much of the total reality question of the $C_6$. Essentially we have a trichotomy of alternatives: $\bullet$ Either $B \subset N^{\ast}$ in which case the $C_6$ is strongly totally real, in the sense that any octuplet (in $E$) induces a totally real pencil, or $\bullet$ $B$ overlaps $N^{\ast}$ without being contained in it, in which case some octuplets induce a totally real pencil and some other do not, or finally $\bullet$ $B$ is disjoint of $N^{\ast}$ in which case all octuplets fail inducing a total pencil. (This is nearly incompatible with the RLT total reality phenomenon). In the first scenario we could say that the sextic $C_6$ is [*strongly totally real*]{}, in the second that is [*(weakly) totally real*]{}, and in the 3rd case that is “anti-real”. We believe that a continuity/degeneration argument applied to Le Touzé’s theorem (2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]) prevents the 3rd option, by letting degenerate the 8 basepoints to the empty ovals. Alas it does not seem that Le Touzé’s theorem (\[LeTouze-big-thm-01-March-13:thm\]) gives sufficiently many inner permissible basepoints so as to ensure a degeneration to the ovals. As to the first two scenarios, we do not know if both of them do occur, and if not which one is ubiquitous. Paraphrasing, we do not know if there is a single sextic of type $\frac{6}{1}2$ (or it mirror) such that any octuplet chosen on the empty ovals induces a totally real pencil of cubics, nor can we preclude the option that all sextics verify this property. [*Insertion*]{} \[12.04.13\].—If we interpreted correctly the last news of Le Touzé, it seems that total reality holds in the strongest possible sense, i.e. for all octuplets and even when the latter are inside of the ovals and not on themselves directly. Compare her article (2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics], plus eventually her letter in Sec.\[e-mail-Viro:sec\]. Esquisse d’un programme (déjà esquissé): the Ahlfors-Rohlin Verschmelzung {#Esquisse-dun-prog-deja-esquiss:sec} ========================================================================= \[07.03.13\] The programme in question was probably nearly implicit in Rohlin 1978 [@Rohlin_1978], safe that he seems to have missed the possible connection with Ahlfors theorem. It is only in this respect that our programme bears some originality, yet presently we are unable to substantiate it in any serious fashion. Large scale structure of total reality as it pertains to Hilbert’s 16th problem ------------------------------------------------------------------------------- \[04.03.13\] Let us brush a sloppy summary of the situation. Rohlin 1978 [@Rohlin_1978] in somewhat cryptical fashion asserted existence of a totally real pencil of cubics on all sextics curves of type $\frac{6}{1}2$ or its mirror, yielding therefore a geometrization of the RKM-congruence asserting that $\chi\equiv k^2+4 \pmod 8$ forces the curve of degree $2k$ being of type I. After consulting several specialists (Viro, Marin, Kharlamov, Fiedler, Le Touzé), it seems that this proof is now lost forever (or dormant in some celestial Eden). It seems extremely challenging to rediscover it if it ever existed, i.e. if Rohlin’s argument was sound and complete, as opposed to a Fermat-style cryptical allusion destined to challenge geometers. On reading the survey Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000], it seems that this is not the sole prophetical allusion of Rohlin in his fantastic 1978 survey. Also the question of estimating the number of points through which one can pass a rational connected curve is also sloppily stated by Rohlin without proof, and Degtyarev-Kharlamov consider the problem as still open. Perhaps the status of this problem evolved meanwhile. Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics] supplied the first (and unique) written proof of a weak form of Rohlin’s total reality phenomenon, yet assuming the RKM-congruence, and so the curve to be of type I. As suggested in Le Touzé’s article (2013 ), Rohlin seems to have cultivated a large expansion of the phenomenon of total reality, as follows: [TOR=Total reality (Le Touzé 2013, Gabard 2004 [@Gabard_2004], who ascribed this as implicit in Ahlfors 1950, and Teichmüller 1941 [@Teichmueller_1941], who loosely claims this to be found in Klein’s works).]{}—Any dividing (plane) curve admits a totally real pencil of curves. [ROTOR=Rohlin’s total reality (implicit in Rohlin 1978 according to Le Touzé 2013)]{}.—Any dividing curve representing a scheme of type I admits a totally real pencil. [RMC=Radio Monte Carlo=Rohlin’s maximality conjecture]{}.—Any real scheme of type I is maximal in the hierarchy of all schemes (of some fixed degree). As noted by Le Touzé (always ), TOR implies ROTOR, and it is always tempting to believe that the latter implies RMC. At the very source of that string of implications, we should have the Ahlfors theorem (ATR=abstract total reality) (which Teichmüller 1941 [@Teichmueller_1941] ascribes to Klein) [ATR=Abstract total reality (Ahlfors 1950, Gabard 2004–06).]{}—Any (abstract) dividing curve $C$ (or what Klein calls an orthosymmetric Riemann surface) admits a totally real map $C\to \PP^1$ to the projective line, i.e. such that $f^{-1}(\PP^1 (\RR))=C(\RR)$. Further the degree of such a map can be arranged $\le g+1$, where $g$ is the genus of $C$. According to Gabard 2006, but still deserves to be better analyzed, there should even be such a total map of degree $\le \frac{r+(g+1)}{2}$ where $r$ is the number of real circuits of the curve $C$. ATR should imply TOR modulo some little difficulties. A first difficulty is merely due my incompetence, of knowing if an abstract morphism from a plane curve to the line is necessarily induced by a pencil of curves. Another little difficulty is that a priori some basepoints (of a total pencil given by Ahlfors) may be imaginary conjugate, so that total reality has perhaps to include a slightly broader definition (admittedly more cumbersome) than the one used in Le Touzé. So probably the correct definition of a totally real pencil has to involve total reality of the moving points of the linear series, while the (statical) basepoints themselves being permitted to be non real(=imaginary, as we say since Tartaglia-Cardano (1535/39), essentially). At the very end of the string ATR$\Rightarrow$TOR$\Rightarrow$ROTOR$\Rightarrow$RMC, we seem to have (accepting the elusive RMC) a sort of subordination of all highbrow congruences modulo 8, that is the Gudkov-Rohlin congruence for $M$-curves, the Gudkov-Krakhnov-Kharlamov congruence for $(M-1)$-curves (\[Gudkov-Krakhnov-Kharlamov-cong:thm\]), to the (Rohlin)-Kharlamov-Marin congruence (\[Kharlamov-Marin-cong:thm\]). To appreciate this fact in degree 6, contemplate once more the Gudkov-Rohlin Table(=Fig.\[Gudkov-Table3:fig\]). Then we see that (virtually) all congruences are subordinated to that of Kharlamov-Marin modulo the truth of RMC, safe the prohibition of the schemes $11$ (Hilbert) and $\frac{10}1$ (Rohn). The latter may be prohibited either via Arnold’s congruence or Rohlin’s formula. So in crude approximation (and modulo RMC) “all” the prohibitions of Hilbert’s 16th problem (say perhaps apart from refined Bézout-style obstructions à la Fiedler-Viro virulent in degree $8$ or higher) are subsumed to the Kharlamov-Marin congruence ensuring type I whenever $\chi\equiv k^2+4 \pmod 8$ (abridged RKM, where R stands for Rohlin, albeit I am not sure about his exact contribution, while most writers credit Kharlamov and Marin only, cf. e.g. Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91]). ([*Insertion*]{} \[12.04.13\].—In Rohlin’s 1978 [@Rohlin_1978 p.93] the result is credited to Kharlamov alone, but it is remarked that the proof of this theorem was still unpublished.) Accepting the above string of implications, we arrive essentially at the conclusion that all of Hilbert’s 16th problem could be governed by the phenomenon of total reality. This would be especially true if there is a geometrization of the RKM-congruence via total reality. This requires a highbrow extension of the Rohlin-Le Touzé theorem to all other $(M-2)$-schemes verifying the RKM-congruence. This looks hard but according to the previous sections it is likely that the total pencil is always of order $(m-3)$ involving the so-called adjoint curves of Brill-Noether incarnating the canonical series. The resulting degree of the map would then also be in accordance with Gabard’s version of Ahlfors theorem (compare Remark \[M-2-curve-degree-like-Gabard:rem\]). To caricature a bit, as to emphasize our philosophy (especially as it pertains to the title of the present text devoted to Ahlfors), we formulate the: All (or most) of Hilbert’s 16th problem can be reduced to the phenomenon of total reality, due to various authors. In in its most primitive schlichtartig form, this involves primarily Riemann 1857 [@Riemann_1857_Nachlass], Schottky 1875–77 [@Schottky_1877], Bieberbach 1925 [@Bieberbach_1925], Grunsky, etc. and in general to Klein according to Teichmüller 1941 [@Teichmueller_1941], Ahlfors 1950 [@Ahlfors_1950]). Basically the phenomenon of total reality involves a linearization in the large of the curve via the concept of branched coverings, where any dividing curve is reduced to its most primitive incarnation, namely the line $\PP^1$. Any extraterrestrial planet, possibly with handles and several equators ([*exoplanet*]{}), yet not so exotic as to share with our planet Earth the character of orthosymmetry(=type I=dividing) involving 2 distinguishable hemispheres, can be conformally shrunk to our equatorial sphere so that fibers above the equator $\PP^1(\RR)$ are totally real, i.e. on the exo-equators of the exoplanet[^78]. This is an abstract theorem yet it should imply a concrete result of total reality like the above TOR. This process could be termed [*descent*]{} from the Riemannian universe (Gromov’s prose) to the terrestrial Plato cavern (of Hilbert’s 16th problem). Especially important would be a quantitative control on the order of those total pencils obtained by descent of the abstract Riemann-Schottky-Klein-Teichmüller-Ahlfors circle maps (equivalently total maps to $\PP^1$). For a simple implementation in the case of $M$-curves, cf. our Theorem \[total-reality-of-plane-M-curves:thm\], which is a trivial adaptation of the no collision principle of Riemann-Enriques-Chisini-Bieberbach-Wirtinger, etc. Next when we move down to $(M-2)$-curves the first concrete (and nontrivial) phenomenon of total reality is the Rohlin-Le Touzé theorem (abridged RLT after their author’s names or for (total) ReaLiTy). Historiographically, it is noteworthy that Rohlin does not seem to have ever been aware of Ahlfors theorem, and it seems that the gap between both traditions (Riemann vs. Hilbert) as not yet been fully bridged. It is also notorious (either from the viewpoint of geometric function theory à la Riemann, or the algebro-geometric perspective) that total reality is much easier for $M$-curves as it involves a schlichtartig semi-Riemann surface (planar orthosymmetric half). This is also evidenced by the fact that there is no collision between a group of $g+1$ distributed on the $M=g+1$ ovals. In the non-Harnack maximal case, any group of $g+1$ points moves (Riemann-Roch) but then several of them being distributed on the same oval a risky collision can occur in the long run, foiling total reality. A subtle condition of dextrogyration must be ensured to gain total reality. So what remains to be done? Project 1.—Try to clarify the above logical implications between ATR, TOR, ROTOR, RMT. This is basically a Riemann-ification of Hilbert’s 16th problem. Project 2.—Try to understand better the lost proof of Rohlin, and how the statement extends to curves of higher order. Two routes toward (Project 2) seems a priori possible making the exploration somehow elusive or at least time consuming. Either work from the scratch in the Plato cavern of plane curves grooving the plane $\PP^2$ where Hilbert’s problem is formulated, or attempt a descent of the abstract result à la Riemann-Schottky-Klein-Enriques-Chisini-Bieberbach-Grunsky-Wirtinger-Teichmüller-Ahlfors-Gabard. The method of descent looks a priori delicate but worked for (plane) $M$-curves, cf. again Theorem \[total-reality-of-plane-M-curves:thm\]. The drawback of this method of descent is that it would (like Le Touzé’s proof) depend upon a knowledge a priori of the dividing character of the curve. If optimistic it could be that Gabard’s theorem (2004-2006 [@Gabard_2006]) could imply total reality for dividing $(M-2)$-curves of degree $m$ via a pencil of (adjoint) curves of degree $(m-3)$ (cf. again Remark \[M-2-curve-degree-like-Gabard:rem\] for some weak evidence). As a special case it could be the case that Gabard’s theorem implies a weak-form of the RLT-theorem, yet much work along ATR$\Rightarrow$TOR is required to bridge the gap. Rohlin’s intuition vindicable via Ahlfors? Algorithmic rôle of RMC for plotting machines ---------------------------------------------------------------------------------------- \[05.03.13\] All what we have to say is now essentially well-known, but alas still in embryonal state. Let us try to make the philosophy behind the grand dessin imagined by Rohlin more palatable, than in the previous section. What is Hilbert’s 16th problem at all about? Answer: Topology of real plane algebraic curves. But in reality we are geometers and topology is merely a weakening of what wanted to study earlier geometers (say Diophante, Euclid, Archimedes, the algebro-geometrization of Fermat, Descartes, Newton, etc.) What is called upon is an understanding of the big video game where given an equation one traces the corresponding curve in real time. One should imagine a powerful enough machine showing us in real time the curve evolving when dragging with a joystick the coefficients of the equation. Any machine able to do this presumably request at an algorithm telling when to stop the tracing procedure as the real locus has been represented within sufficient accuracy as to infer the exact topology of the curve. It is at this stage that Rohlin’s maximality conjecture may enter into the scene. Indeed for a given degree there exists certain distinguished schemes (in the sense of Rohlin) representing so-to-speak fully crystallized extremal shapes not susceptible of any further apparition of ovals (as small as they may be). As popularized by Rohlin 1978, this intuition of saturated schemes truly goes back to Klein 1876, yet in some primitive sense distinct from Rohlin’s interpretation. Consider for instance a curve of degree 6 whose real locus contains a deep nest of depth 3, then it is already saturated and there cannot be any further oval. More generally we have certain schemes of type I, which according to Rohlin’s intuition ought to be maximal, hence incarnating the maximum topological complexity permissible for the given degree. The problem of understanding this ontological truth splits in two parts: Why is it true, and why is it useful? First we try answering the second question. As already pointed out, Rohlin’s maximality conjecture (RMC) incarnates a sort of stopping process for the plotting machine (realizing the dynamical video game or just its statical variant). From the viewpoint of Hilbert’s 16th problem, the fact that the deep nest of depth 3 is a maximal scheme of degree 6 forbids a menagerie of other schemes enlarging it which a priori could exist, but do not essentially by virtue of Bézout’s theorem. Hence Rohlin’s type I schemes (granting their maximality) are like advanced sentinels prohibiting schemes of higher topological complexity. Without such prohibitions Hilbert’s 16th problem would be even more intractable than it already is. Hence Rohlin’s maximality conjecture is a sort of upper bound for the complexity of Hilbert’s 16th problem. This should be sufficient reasons for answering the utilitarian aspect. As to the first question, it is somehow ironical that Klein seems to have been much in touch with both aspects of our question. First, as we said he is regarded (by Rohlin himself) as a precursor of Rohlin’s maximality conjecture. Second (but this is more elusive to testify with high accuracy), Klein is credited by Teichmüller 1941 [@Teichmueller_1941] as the true forerunner of Ahlfors theorem. At some broader scale, all goes back to Riemann (especially if his life would not have been so short). Rohlin’s maximality conjecture should according to our intuition (Gabard ca. 1st January 2013) reduces to Ahlfors theorem, via what Le Touzé calls the Rohlin total reality conjecture, cf. (ROTOR) of the previous section. So it is quite interesting to see that the extremal shapes (maximal schemes) of Rohlin are induced by schemes of type I, and what makes this possible is the phenomenon of total reality. Behind the latter there is of course Ahlfors circle maps, and so basically an extension of the Riemann mapping theorem. This in turn is governed by potential theory, itself concomitant of the calculus of variation of Euler-Lagrange as applied to Laplace’s equation. All this détour to make apparent that the algebro-geometric extremal principle posited by Rohlin’s maximality conjecture seems governed by another extremal principle, namely those ensuring solvability of Dirichlet’s principle. So be it via Abel (and what some like to call Hodge theory) or directly via Euler-Lagrange-Laplace-Dirichlet-Riemann-etc-Ahlfors we have the phenomenon of total reality for curves of type I, and when the scheme itself is of type I this phenomenon acquires an extra punch of universality, making the scheme maximal in the Hilbert-Gudkov-Rohlin hierarchy. Coarsely, our thesis could be that what missed to Rohlin to complete his programme (amounting essentially to bound the complexity of Hilbert’s 16th problem) is merely a rather simple theorem of function theory (due basically to Riemann-Schottky-Klein-Teichmüller-Ahlfors). The latter in turn being not much more than a bordered avatar of Riemann existence theorem exhibiting any closed Riemann surface as a branched cover of the projective line. In other words any abstract Riemann surface (à la Riemann-Prym-Klein-Weyl-Radó) becomes a concrete one (à la Abel-Galois-Cauchy-Puiseux-Riemann-Weierstrass, etc.) Next it comes to make this Ahlfors-Rohlin fusion a reality. Here starts alas some little difficulties which we hope to able to overcome in the future. The problem breaks in two steps: $\bullet$ Step 1. Make the Ahlfors theorem concrete by specializing it to dividing plane curves, and conclude the existence of a total pencil of “adjoint” curves exhibiting total reality. $\bullet$ Step 2. Prove that a totally real pencil (often abridged total pencil) implies maximality of the scheme. Start with a scheme of type I of degree $m$. This means by definition that any curve $C_m$ representing it is of type I. By ROTOR (i.e. a concretized version of Ahlfors theorem), there is a total pencil of curves $\Pi$ of order say $k$. This means that each curve $C_k$ of the pencil cuts $C_m$ only along real points \[as soon as they are mobile\]. \[[*Brackets Added*]{} \[12.04.13\].—Probably imaginary basepoint have to be allowed, yet the mobile part of the pencil should be totally real.\] Such a balayage seems actually to supply a fast algorithm to trace the real locus by reduction to a problem in one variable (consider e.g. the Gürtelkurve swept out by a pencil of lines), which in turn could involve the Newton root-finding algorithm via linearization. Of course in general, curves of the pencil are not rational and so this asks for a tricky extension of Newton. Once existence of a total pencil is granted we seems nearly finished, because if there are more ovals then passing a curve of the pencil through the additional point would corrupt Bézout. The notorious bug of that argument is that we assume implicitly the over-scheme to be realized by an augmentation of the given algebraic curve, which is priori not the case. \[12.04.13\] Further a total pencil exist as well on some type I curves belonging to indefinite (yet non-maximal) schemes, compare the case of sextics. This is of course another obstruction to completing crudely the just sketched argument. However what makes that the argument works for the deep nest or the satellites of the quadrifolium? Is it the fact that the pencil curves are rational? or is it a sort of geometric intuition of the pencil, or perhaps some canonicalness of it? Maybe to get the RMC we need not only existence of a total pencil but some sort of uniqueness (up to anodyne choices like center of perspectives chosen in the “deepest” ovals). It seems that a scheme of type I incarnates a family of curves which are so-to-speak totally real in some canonical way, and the total pencil is virtually God-given. (Beware yet that the family in question is not necessarily connected in the hyperspace of curves, cf. Marin 1979 [@Marin_1979] or Fig.\[Marin:fig\].) At this stage it seems important to remember a metaphor allied to total reality. Total reality means that all intersections are visible on the reals. Using a pencil means essentially that we choose a mode of vision of the curve. Basepoints are eyes of some insect having several eyes and curves of the pencil are optical rays enhancing how the animal perceives the curve (Gebilde). What is strange is that total reality amounts saying that the vision is purely transverse and so the object is in reality invisible (no apparent contour). To make this concrete consider the example of the Gürtelkurve $C_4$ (2 nested ovals) projected from an innermost point inside the deepest oval. Paraphrasing in a real life metaphor, looking at a glass of wine from outside you see its apparent contour, but when placed inside of it, it suddenly becomes invisible. This is total reality. It is tantalizing that total reality (via Ahlfors theorem) seems so close to prove RMC(=Rohlin’s maximality conjecture) but apparently fails. As we (or better Rohlin) suspected the assumption that the scheme is of type I must impose the corresponding curves being strongly harpooned by total reality. But how to make this idea precise? We can imagine the space of all curves representing the scheme, and think about this a universal curve of type I. There should then be a version of Ahlfors theorem for family of curves or (as Teichmüller, Ahlfors, Bers, liked to say) for a [*variable*]{} Riemann surface. The net effect would be that total reality is genetically imbued in the curve(s) itself in such a strong fashion that the scheme is maximal. So any scheme of type I has a canonical vision making it totally real, amounting essentially to look at the world form inside the glass (or bottle) of wine. This is akin to the photoelectric effect. (Compare with the known examples of the unifolium and its satellites, alias deep nests in the jargon of Hilbert and the Russian school, or the quadrifolium, and its satellite total under a pencil of conics, plus the (elusive) Rohlin-Le Touzé phenomenon for sextics). Once this photoelectric vision of the curve is given then nothing more can appear in the blue sky and so the scheme is maximal. This is the intuition of why RMC holds true, but how to convert this in a mathematical proof. What seems to be in demand is a mechanism which from the shape alone of the scheme identifies the total vision of the curve. This we call the photorealism or photogenism. If a scheme is of type I then it is photogenic, and then it must be maximal. This seems to request for a general mechanism of where to assign basepoints which would extend the total reality of unifolium, quadrifolium, and 9-folium of Rohlin-Le Touzé flashed resp. by by pencil of lines, conics and cubics. As we discussed in a earlier section the case of degree 8 schemes looks a bit puzzling, where by the RKM-congruence we have plenty of $(M-2)$-schemes of type I (ca. 100 if we were not too bad in counting). The center of vision (basepoints) are then quite hard to predict. In general there are $B=M-3$ of them (where $M$ is Harnack’s bound), so $B=19$ for $m=8$, and alas it is presently not very clear where to assign them in full generality. Making all this explicit could solve the question of giving a precise sense to our notion of photogeny, and as a by product crack the RMC. Is this a realist strategy? Is there a more abstract argument? If not, we really need some highbrow extensions of the Rohlin-Le Touzé theorem dictating us for all schemes of type I where to assign basepoints. This seems to call first for a classification of schemes of type I. [*Long (paragraph) Insertion*]{} \[12.04.13\].—To tell the truth it should be remarked that even in the case of Rohlin-Le Touzé (degree $m=6$) we lack presently a proof of the desideratum that the vision of total reality (via the pencil of cubics) is strong enough as to ensure maximality of the scheme. (This conclusion is of course true via the Gudkov census but we lack a direct proof along the philosophy of total reality.) Perhaps Rohlin knew a proof, but as far as we know it was not published too.) Note two things. First, the more naive principle of maximality of Klein-Marin 1876/1988, when combined with Itenberg’s contraction affords another approach to the problem of RMC, which is perhaps easier to implement (though in general only based on the conjectural principle of contraction). Second, it seems that in Rohlin’s approach we lack some flexible medium to carry the enlarged scheme to the original one. This could involve trying to approximated a diffeomorphism of $\RR P^2$ by something more algebraic (maybe a Cremona transformation), but then it looks hard to finish the job. So maybe $A+B+C=Rmc^2$, i.e. Ahlfors, plus Bézout, plus Cremona implies Rohlin’s maximality conjecture. One may also wonder if there is not a much more flexible proof of RMC say akin to Rohlin’s formula where merely soft topology is used (while avoiding any contraction principle). We have then basically 2 curves, one enlarging the other, and one of which universally of type I. So one could fill the half à la Rohlin, i.e. by all discs (like in the proof of Rohlin’s formula) and inspect the intersection of the homology class of degree $k$ with the homology class of the enlarging curve. The sequel is certainly hard to complete. (It is at this stage that we had the idea of using Mangler to isotope the enlarged curve back to the original, cf. Sec.\[RMC-via-Mangler:sec\] for more detail on this strategy to attack RMC via Ahlfors, plus Mangler.) Here we know quite little, but as said earlier in this text, it could be the case that the RKM-congruence is a universal detector of $(M-2)$-schemes of type I, while all other type I schemes arise as satellites of schemes of type I of lower orders (dividing the given degree $m$). So when $m=2p$ is twice a prime number there should be no such satellite (except that of the unifolium) and all new type I schemes would be concentrated at the $(M-2)$-level. Of course we can always make abstraction of the $M$-schemes where RMC holds trivially true. So we see some sort of higher arithmetic structure emerging in Hilbert’s 16th problem as boosted by Rohlin’s conceptions, namely a sort of inductive process that could progressively step-by-step enumerate all schemes of type I, merely as $(M-2)$-schemes of type I satisfying the RKM-congruence mod 8, or as satellite of earlier such schemes, and for all of them expect a synthetical revelation of the type by a canonical pencil à la Rohlin-Le Touzé incarnating primitive forms of the phenomenon of total reality. This deserves nearly the name of Rohlin’s divination. As the list of photogenic schemes increases at each step $m$, we may conclude RMC by having exhibited in some ad hoc fashion the total reality of all type I schemes, and the RMC would follow step-by-steps. Needless to say this requires an immense effort, and the induction required to validate RMC in all degrees looks a priori extremely tricky. Furthermore one could imagine that all this ascension effected in autarchy from Ahlfors theorem by using rather ad hoc optical recognition procedure via total reality. This would be parallel to the evident total reality of the satellites of the unifolium (alias deep nests) and idem for the quadrifolium, or Rohlin’s schemes of degree 6 (modulo the lost proof of Rohlin). In that case the theory would be purely Rohlinian and this is probably essentially what Rohlin envisioned. In contradistinction, when attacking RMC, we know a priori the scheme being of type I so there could be some inference of Ahlfors theorem permitting to shortcut the (pure) total reality vision of Rohlin. This inference could increase the (ascensional) speed conceding some abstractness in the verification of RMC. Yet, as observed, even this looks hard unless we get a better grip upon the abstract total reality of Ahlfors. A first modest (but nontrivial) exercise is to write down a clear version of Rohlin-Le Touzé’s total reality claim, and using it deduce the maximality of those 2 schemes. Here again notice that exploiting the type I assumption as do Le Touzé is not a concession since we are interested in RMC. So here total reality seems sufficiently strong (canonical) to ensure maximality of the schemes and we rederive so from Le Touzé’s result the prohibition of Gudkov, etc. (compare Table \[Gudkov-Table3:fig\]). In particular RMC holds true in degree 6 for some intrinsic reason allied to total reality, as opposed to being a byproduct of the full classification of Gudkov. A more highbrow project would be to inject the function theory à la Ahlfors in the problem. Assume given a scheme of type I, we can for each representing curve $C_m$ choose a total pencil $\Pi$ which is a line in the space $\vert kH\vert$ of $k$-tics curves. It seems plausible that the dependence can be made continuous. Then we have a universal family of photoelectric effects on $C_m$ and its deformations (possibly in different chambers of the discriminant) in which case the line $\Pi$ may jump, a priori even in different hyperspaces indexed by different $k$. On applying the $k$-tuple Veronese embedding $v_k$—i.e. the holomorphic map $\PP^2\to \PP^N$ induced by the linear system of all k-tics—the total reality of $v_k(C_m)$ would appear under a pencil of hyperplane, hence the curve would be total real under a pencil and therefore located as several spires gyrating around the base locus (plane of codimension 2 in $\PP^N$). Now it may be expected that the phenomenon of total reality is as evident as it was for the deep nest (i.e. reduction to the case of a linear pencil) and that we may conclude maximality from Bézout (applied of course now in the Veronese hyperspace). After this little psychoanalysis of Rohlin’s secret garden, we see that “la réalité totale nous colle à la peau.” In some sense the phenomenon ought to be so inherent to a curve belonging to a scheme of type I that maximality of the scheme should follow via the photoelectric effect. By the latter we really mean that the total pencil being saturated nothing more is allowed to appear in the blue sky without corrupting Bézout. (Prototype: a deep nest with a pencil of lines through the deepest oval, or satellites of the quadrifolium in degree $4k$.) As a foundational detail, I always thought that possibly imaginary basepoints have to be permitted in the definition of a totally real pencil, so that merely moving points of the series are real (cf. Gabard 2004, p.7). Now I am not sure that this is really required. In all basic examples of total reality (i.e. the deep nests interpretable as satellites of the unifolium or the quadrifolium and its satellites) the permissible basepoints ensuring total reality are always varying through a contractible union of cells as they are located inside the deepest ovals. This is probably also true for the sextics of Rohlin-Le Touzé. If this is a general phenomenon then this is a bit in line with our desideratum that the total pencil ought to be almost canonically associated to the dividing scheme of type I. If instead we are interested in total maps of lowest possible degree then we are inclined to let degenerate the basepoints on the ovals themselves and so the total pencils of this sort are parametrized rather by tori. Another idea \[developed in Sec.\[Thom:sec\]\] is to fill the plane curve by the (orientable) membrane of $\RR P^2$, to get a certain smooth surface in $\CC P^2$ whose fundamental class is $kH\in H_2(\CC P^2, \ZZ)$. Smoothing its corner and applying Thom’s conjecture (=Kronheimer-Mrowka’s theorem) could lead to some interesting consequence. (More about this soon, cf. Theorem \[Thom-Ragsdale:thm\].) Of course this is basically related to the ideas of Arnold and Rohlin. Following our main theme, the idea would be that there is always for a scheme of type I some preferred (up to the ambiguity of a contractible space of parameters) total pencil, which we call a photon. This would naively speaking be obtained by assigning basepoints among the deepest ovals. All this works good for degrees $\le 6$. In degree $8$, the RKM-scheme 20 already affords a little problem as quintics have 19 basepoints assignable and it is not clear which ovals have to be used as “anchor” basepoints. \[But this scheme is prohibited by Thom, cf. again Theorem \[Thom-Ragsdale:thm\], or argue via Rohlin’s formula.\] Once we have a photon (i.e. a canonical total pencil) then we would like to argue that its satisfies the photoelectric effect, and RMC would follow. In the case of $(M-2)$-curves of type I we have $B=M-3$ basepoints for a pencil of $(m-3)$-tics that are freely assignable. This is one unit less than the number of ovals and it is not clear which one can be dispensed of being marked by a basepoint. We could imagine that we could always dispense the oval whose porous inside has the most negative Euler characteristic. To make this serious compare Fig.\[RKM-schemes-deg-8:fig\], where we find however schemes where such a dispensed oval is not uniquely defined, e.g. $\frac{6}{1}\frac{6}{1}\frac{4}{1}1$. So our recipe is certainly dubious. We hope to have made the nature of the question clear enough. It seems first that there is no direct reduction of RMC to Ahlfors theorem, except perhaps if one as some deeper grasp upon the geometry of a total pencil (photon) so as to ensure via the photoelectric effect the RMC. As we said at the beginning of the section, the net impact would be a sort of upper bound upon the complexity of Hilbert’s 16th problem. In fact it would really be the clef de voûte yielding some insights upon the architecture of the pyramid of all schemes of some fixed degree $m$, namely type I schemes ought to be maximal element (yet not the sole ones cf. Shustin 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]). This conjectural maximality explains nearly all prohibitions (at least in the case $m=6$). Despite our obsession to harpoon RMC via Ahlfors’ total reality (what Viro was sceptical about) and we have only the vague suggestions of the present section to propose \[see also maybe the next Sec.\[RMC-via-Mangler:sec\]\], it may be argued that even if it worked by a clever trick, it would perhaps not be as satisfactory as the full and slow progression along the menagerie of all schemes that must be tabulated along the higher order cases of Hilbert’s 16th problem. In other words we would like to know the whole pyramid and not just its maximal element. From the viewpoint of the French revolution we would like to know the whole folk and not just the aristocrats. More seriously we want to “see” the exact geometry of the phenomenon of total reality, and not just its capitalistical/hierchical impact via the photoelectric effect upon a validation of the RMC. \[12.04.13\] Also interesting is the question of the density of schemes below the aristocrats. Of course it seems that the pyramids as dense below the maximal elements: philosophically because an aristocrat seems unable to provide for its wants without the force of all its servitors. An isotopic attack on RMC via Mangler 1939 {#RMC-via-Mangler:sec} ------------------------------------------ \[12.04.13\] As we often experimented RMC would follow from Ahlfors if we knew that the enlarged scheme lies in a tube neighborhood of the given curve of type I. The difficulty is that a priori the enlarging curve is very distant and hard to compare to the original one. Crudely put one could expect to reduce always to the easy case by using an isotopy. So one could try to isotope the diffeomorphism taking the small curve to its enlargement to the identity. Recall this to be possible since the mapping class group of $\RR P^2$ is trivial) \[Mangler 1939 [@Mangler_1939], often used by Teichmüller 1939\]. One could then perhaps try to extend this isotopy to $\CC P^2$ so as to get reduced to the case where the enlargement is a small perturbation (actually the identity). In this case Ahlfors suffice to imply RMC, since Ahlfors’s pencil affords something like a transverse structure, and one gets an easy corruption with Bézout by letting pass a curve of the pencil through the new oval (not within the tube neighborhood). The difficulty is of course to check that Bézout (which is some something algebraic rigid) is conserved during the very plastical deformation of isotopy. Yet perhaps we may reinterpret intersections homologically as to gain more flexibility. Further the isotopy could be compatible with reality (equivariant and respecting $\RR P^2$ and its complement of imaginary points). Finally due to the geometric interpretation of intersection numbers (in homology) their values will be clearly conserved by the isotopy. So we arrive at the: There is perhaps a trivial proof of RMC via isotopy of $\RR P^2$ equivariantly extended to $\CC P^2$, so that RMC reduces trully to Ahlfors. Suppose $C_m$ to be a curve of degree $m$ belonging to a scheme of type I. Let $D$ be a curve (of the same degree) whose scheme enlarges that of $C_m$. Fix a diffeomorphism $f$ of pair $(\RR P^2, C_m)\to (\RR P^2, D_{\ast})$ where $D_{\ast}$ is $D$ less one oval (w.l.o.g. or more ovals in general). By Mangler 1939, we can isotope $f$ to the identity of $\RR P^2$. Now it seems reasonable to expect that there is a natural way to extend an isotopy of $\RR P^2$ to one of $\CC P^2$. This does not need to be strongly unique but merely to exist in some sense that it preserves real parts and maybe can be chosen equivariant w.r.t. conj. I.e. carrying a point along the isotopy up to time $t\in[0,1]$ commutes with the symmetry conj. Knowing that the quotient $\CC P^2/ conj$ is $S^4$ could be of valuable assistance to construct the extended-isotopy. So we have $f_t$ an isotopy of $\RR P^2$ say with $f_1=f$ and $f_0=id$, and $F_t$ and extension thereof to $\CC P^2$. The map $f$ pushes injectively the ovals of the first good curve $C_m$ into those of the hypothetical enlargement $D$. So operating backward in time along the isotopy $F_t$ we may retract the complexified curve $D(\CC)$ so that its real part becomes close to that of $C_m$ (and even identic to it). Denote $D_0$ this “temporal retraction”, which is a “flexible” Riemannian surface, with fundamental class still of degree $m$, by homotopy-invariance of homology. Now by total reality the first curve being of type I it admits a total pencil, all of whose members have Bézout saturated intersections with the curve. Taking a curve $P_k$ of the pencil passing through the additional oval of $D$ isotoped backward in time ($t=0$), create one extra intersection (that will count positively because the extended isotopy is orientation preserving). All other other intersections also counts positively if we are capable arranging the large isotopy $F_t$ to respect somehow the normal bundle of $\RR P^2$. Hence the pull-back $D_0$ will have excessive number of intersections with $P_k$. The homological Bézout (i.e. Poincaré, Lefschetz, etc.) is therefore corrupted. Rohlin’s maximality conjecture would be proved by “soft topology” plus some Ahlfors. The critique to this argument however is that a priori it applies to any dividing curve supporting a total pencil and those can be of indefinite type (yet not maximal), cf. the case of degree $m=6$ where there is plenty of such examples (Fig.\[Gudkov-Table3:fig\]). So a serious gap requests to be filled. Maybe this will be an easy game for Alexis Marin? To be optimistic, our argument looks so close to prove the big desideratum (=RMC) that it is certainly worth exploring further. In particular since the argument is spoiled by the objection of indefinite schemes there must be (for instance in degree $6$ where the Ahlfors total pencil are very easy to describe explicitly, e.g. for the scheme $9$ where we have the simple Fig.\[Fcubic:fig\]) there must be some obstruction to extend the Mangler isotopy to $\CC P^2$ (at least in a fashion that positivity of intersections are conserved). Understanding this obstruction, and assuming that one capable to show that it vanishes if the scheme is of type I could afford a proof of Rohlin’s maximality conjecture. The proof is likely to involve some 4D-topology (say à la Marin-Siebenmann-Alexander). Additional remarks on Rohlin-Le Touzé total reality for $(M-2)$-sextics of RKM-type ----------------------------------------------------------------------------------- \[04.03.13\] We concentrate again on the case of a sextic $C_6$ of type $\frac{6}{1}2$. Before entering into the elusive technical details we recall the basic problematic yet quite elusive for the moment. We would like to show a phenomenon of total reality, yet its exact shape is still obscure to us. Either we can use the RKM-congruence to infer a priori that the curve is of type I. Then we could apply the (abstract) result of either Ahlfors 1950, or Gabard 2006 and hope to effect a descent in the plane, to get a total pencil (hopefully of cubics). This descent probably requires some theory à la Brill-Noether as a Plato-cavern-style reflection of Riemann’s work. Alternatively, we can try to follow the (direct concrete) route proposed by Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics] depending upon a detailed analysis of pencils of cubics. Finally we could dream to recover the lost proof of Rohlin, i.e. without assuming the dividing character while trying directly to ensure total reality of such a curve under a pencil of cubics with suitably assigned $8$ basepoints. Here the problem is that we know presently very little of how general the phenomenon of total reality is, i.e. which octuplets induce total reality. More about this soon. A very first (extremely basic remark) that we all use subconsciously is a sort of closing lemma for algebraic curves: \[Closing-lemma:lem\] [(Closing lemma)]{}.—Given any real plane projective curve, its real locus (if non empty) consists of closed circuits (Jordan curves in $\RR P^2$) or possible poly-cycles like figure 8, etc, or eventually an isolated singularity. Topologically it is always locally a multi-node consisting of a certain number of branches crossing transversally, or an isolated point. When the point of the real curve is non-isolated then there is at least one Jordan curve based on the given point. This is a mixture of algebraic-geometry, implicit function theorem, and topological compactness of $\RR P^2$, and abstract classification of compact (Hausdorff) $1$-manifolds (just the circle), and some singular “graph” avatars. This pertains to the very first step of our total reality story as follows: Assume given 8 basepoints distributed on the empty ovals $E_i$ of our $C_6$. Let $\Pi$ be the pencil of cubics passing through the 8 points. Then all curves of $\Pi$ cut the curve $C_6$ in at least 16 real points (i.e. just 2 units less than the maximum permissible by Bézout). Say in that case that the pencil is quasi-total. Our 8 basepoints forces 8 real intersections, but by the closing lemma each intersection has at least one companion (possibly the same yet then with a tangency and so counted with multiplicity 2). More precisely we look at one of the basepoint and choose any curve of the pencil. First note that the basepoint cannot be an isolated real point of the curve $C_3$, and so there is by the closing lemma a topological Jordan curve in the real locus of $C_3$ which has thus at least 2 intersections with the given oval. So to reach Rohlin’s (lost) theorem just 2 real intersections are missing (quasi-total) but the gap toward total reality is still immense. Let us first observe the following extension where basepoints are located inside the ovals, as opposed to the former case where they were directly imposed on the ovals themselves: (Interior distribution quasi-total).— If the $8$ basepoints are assigned in the insides of the empty ovals $E_i$, then the pencil is also quasi-total, i.e. $C_3\cap C_6$ has $16$ real intersection for all $C_3\in \Pi$. Suppose given such a $C_3$, hence visiting the basepoints $p_i$ labelled as to be in the insides of the $E_i$. Each $p_i$ forces 2 intersections in $C_3\cap C_6$, except if the circuit of $C_3$ through $p_i$ is a small oval inside $E_i$. But then the residual pseudoline of $C_3$ has to visit all 7 remaining points, and so is forced to intercept $N$ the nonempty oval of $C_6$. We count then $14+2=16$ intersections. [*Addendum*]{}.—Further in the above situation of a small oval of $C_3$ inside $E_i$, then for vibratory reasons the 9th basepoint of $\Pi$ has to be on it. Hence if $p_i$ is an inner point of $N$, then (even) total reality is fulfilled (Lemmas \[9th-basepointI:lem\] and \[9th-basepoint-totalII:lem\]). If $p_i$ is an outer point then the residual pseudoline of $C_3$ will intercept (twice) $N$, and we have again $14+2=16$ (real) intersections. The above [*Addendum*]{} is not formally required for the proof of the lemma but we include it as it nearly give a hope to attack Rohlin’s total reality claim (abridged RTR in the sequel). Beware that Rohlin’s statement is very loose in the original paper (Rohlin 1978 [@Rohlin_1978]) and so RTR should not be given a too strong connotation from the scratch. Part of the problem is to decide with which level of generality Rohlin’s claim is correct. With our zero knowledge, we can distinguish several layers of interpretation for RTR. In its strongest form this would be the assertion: [(Inside total reality)=ITR]{} Denote by $E_i^{\ast}$ the (sealed) insides of the empty ovals $E_i$ of the $C_6$, then the pencil $\Pi$ of cubics through any points $(p_1,\dots, p_8)\in E_1^{\ast}\times \dots\times E_8^{\ast}$ is totally real. Then there are several weaker variants, namely the same conclusion under the assumption that the $p_i$ belong to the ovals $E_i$ themselves. The corresponding statement is called OTR, for oval total reality. Another weakening is to relax the conclusion by claiming only total reality of $\Pi$ for a suitable octuplet, as opposed to claiming it for all of them. This relaxed form induces statements called WITR resp. WOTR, where the “W” stands for weak total reality. Though being weak this would be enough to geometrize the degree $m=6$ case of the RKM-congruence. Le Touzé’s theorem is essentially WITR, i.e. weak inside total reality modulo the fact that Le Touzé assumes (or infers from the RKM-congruence) the dividing character of the curve. Of course we have formal implications like the following commutative square $$\begin{array}{rcl} \textrm{ITR} & \Rightarrow & \textrm{OTR}\\ \Downarrow \quad & &\quad \Downarrow\\ \textrm{WITR} & \Leftarrow & \textrm{WOTR}.\\ \end{array}$$ Alas we know very little about those statements. We do not know if the strong versions (upper row) are true, and if foiled it could a priori still be the case that they hold true for special sextics $C_6$. To be factual at the time of writing (and modulo an understanding of Le Touzé’s proof) the only available knowledge is that the weakest form WITR holds true, and even as we said under the assumption that the curve is of type I as may be inferred from the RKM-congruence. Hence of course the whole square can be extended to a cube with another square face of statements assuming the dividing character of the curves. Modulo RKM-both squares are actually formally equivalent, but a very purist could prefer eliminating this dependency. As asserted (but never proved) by Rohlin 1978, one could hope to do more and prove one of the above statement [*ex nihilo*]{} (without reliance upon RKM). As usual in mathematics (or in the world of bird of preys) one should always start attacking the weakest prey, namely WITR. This is a bit strange because quasi-total reality is slightly easier to establish when the basepoints are located on the ovals. Further keep in the subconscious part of the brain, that Rohlin’s hints are so vague that it is not even clear that our 4-fold strategy covers all what is permissible (for instance it could be useful to assign the basepoint not on the empty ovals but one also on $N$. This looks exotic, but perhaps useful in extreme case of desperation). So what is a reasonable strategy toward WITR? We may start from the observation that the pencil $\Pi$ is totally real iff the 9th basepoint $p_9$ of $\Pi$ is in the sealed inside $N^{\ast}$ of the $C_6$ (cf. Lemma \[9th-basepoint-totalII:lem\]). In reality this lemma holds true for basepoints assigned on the ovals, but probably extends to the broader setting. This leads to the following: (Hypothetical lemma) Assume $p_9$ to be in $N^{\ast}$ and the $p_i\in E_i^\ast$ in the (sealed) insides. Then $\Pi$ is totally real (abridged total). Let us show where the naive proof breaks down. Assume given any curve $C_3$ of the pencil. A priori $C_3$ may pass through an inner point $p_i$ (i.e. inside $N$) via a microscopic oval $E$ of $C_3$ entirely inside $E_i$, thereby creating no real intersections. Of course then the residual pseudoline of $C_3$ (i.e. $J=C_3(\RR)-E$) intercepts (twice) the nonempty oval $N$, but this affords altogether only $14+2=16$ real intersections. Note that $p_9$ has for vibratory reasons necessarily to be located on $E$, yet this is no contradiction. Perhaps I missed something and there is a more clever argument establishing this modest technical conjecture. (Le Touzé probably has some idea.) Let us skip this conjecture, while attacking rather the stronger looking WOTR proposition, as in the latter case total reality is easier to ensure. Of course doing so we loose some freedom for the parameters as the large 16-dimensional cell $E^\ast=\times_{i=1}^8 E_i^\ast$ is traded against the 8-dimensional torus $E=\times_{i=1}^8 E_i$ but perhaps this suffices to conclude. Further the advantage would be to get a total map of lower degree, namely one corroborating Gabard’s bound. So it is a delicate matter to decide which strategy “insides of the oval versus the ovals themselves” is more likely to give a proof of RTR (=Rohlin’s total reality claim). For the moment we have no better idea than the topological approach sketched in one of the previous section, i.e. to ensure that the 9th basepoint lands in $N^{\ast}$, and so abort this delicate question. Thom’s conjecture vs. Hilbert’s 16th {#Thom:sec} ==================================== \[21.03.13\] [*Warning.*]{}—All this Sec.\[Thom:sec\] is poorly organized for reasons to be soon explained. In particular it contains several mistakes, but also such fundamental results as Petrovskii inequalities, the strong-Petrovskii-Arnold inequalities. Some higher Gudkov tables of periodic elements (e.g. Fig.\[Degree10:fig\]) show the geographical impact of Petrovskii-Arnold as compared to Ragsdale’s conjecture (briefly discussed in Sec.\[Ragsdale-conj:sec\]). The importance of the Petrovskii-Arnold results was pointed out to me by Thomas Fiedler, who corrected several benign mistakes and one much more fatal bug of mine. This section should thus be read with extreme discernment, as it mixes both the best (Petrovskii-Arnold and even the marvellous construction of Itenberg-Viro) and the worst (Gabard). Several footnotes and WARNINGS should aid the reader to avoid going into the same pitfall as I did. All those WARNINGS are due to kind letters of Fiedler who fixed all my misconceptions and posed me challenging problems. We hope in the future to be able to reorganize the text in a more decent fashion after exploring in more depth a possible fascinating interplay between Hilbert, Ragsdale, Thom=Kronheimer-Mrowka 1994 [@Kronheimer-Mrowka_1994] (independently Morgan-Szabó-Taubes 1995/96 [@Morgan-Szabo-Taubes_1996]), and the work of Petrovskii-Arnold (1938–1971). Of course the interested reader is invited to consult more professional sources, notably Mikhalkin 1994 [@Mikhalkin_1994-adjunction-Thom]. \[21.03.13\] This section was built around the fundamental result $\chi\le k^2$ for any dividing curve of even degree $2k$ (Theorem \[Thom-Ragsdale:thm\]) directly inferred from the so-called [*Thom conjecture*]{} (which he humbly considered himself as rather belonging to the folklore, compare footnote in Lee Rudolph 1984 [@Rudolph_1984]). This should have implied a clear-cut impact of Thom upon Hilbert’s 16th problem. The summit of our fictional “Gabard-Thom” theory went so far as to establish one-half of Ragsdale’s conjecture, (still open for $M$-curves) (cf. Lemma \[Thom-implies-one-half-of-Ragsdale:lem\]) and to show that René Thom was on his 31, i.e. can be stronger than the conjunction of all Russian estimates, congruences and formulas (due primarily to Petrovskii 1938, Gudkov 1969, Arnold 1971 and Rohlin 1972–74–78), cf. Theorem \[Alsatian-scheme-Thom-strong-Petrov-Arnold:thm\]. This would have refuted a belief of Th. Fiedler (cf. his letter ca. 13 March in Sec.\[e-mail-Viro:sec\]). $\bigstar\bigstar\bigstar$ Fortunately, Fiedler brought us back to reality by showing that our reasoning is wrong as it overlooks the issue that despite being constructed by pasting two orientable pieces—namely Klein’s orthosymmetric half married with Miss Ragsdale’s membrane bounding the curve from inside—the so-called [*Arnold surface*]{} (1971) does [*not*]{} need to be orientable. My mistake is thus nearly as basic as having overlooked that one can create (like in Klein’s bottle) non-orientable objects merely by pasting a handle to itself in a twisted fashion (this reminds me some lovely pictures in the Fuks-Rohlin “Beginner’s course on topology”). This being confessed, most of this section is foiled and we are much indebted to Th. Fiedler for having catched our mistake at the right moment and stimulated our investigations. Albeit much of the sequel is foiled we have decided to keep it for didactic reasons. In our case it was so pathetic to write ca. 40 pages based upon a misconception without noticing anything (prior to Fiedler’s correction) that we would by no mean that somebody else do the same mistake. More positively many questions arises through Fiedler’s correction. \(1) Where (in particular in which degree) lives the first counterexample to the Gabard-Thom bound $\chi\le k^2$ (no false modesty in calling it so since it is false) for all dividing curves? \[The sole counterexample I know is the Itenberg-Viro curve corrupting Ragsdale. This is a beautiful picture, see Fig.\[Itenberg:fig\].\] \(2) If “Gabard-Thom” is obviously false for theoretical reasons, why does it look nearly true as implying one-half of the vestiges of Ragsdale’s conjecture? \(3) Under which condition is the estimate $\chi\le k^2$ still true? As noted by Fiedler, the answer seems rather clear namely iff the Arnold surface is orientable. This is in turn the case iff all primitive pairs are positive. The last condition is perhaps not an “iff”. In crude approximation one could say that the Arnold surface is orientable iff the Rohlin tree is positively charged throughout. At least if this is the case (a rather stringent condition) then Arnold’s surface is orientable, and the estimate $\chi\le k^2$ holds true (cf. the limpid proof of the erroneous Theorem \[Thom-Ragsdale:thm\]). Further it seems evident that when the tree is positively charged throughout then the Rohlin mass $\pi-\eta$ is maximized (We always set $\pi:=\Pi^+$, $\eta:=\Pi^-$ to abridge Rohlin’s notation). Alas even that is false (more details soon). So Thom’s conjecture has still something to say on Hilbert’s 16th yet its impact is much more subtle than expected when doing the fundamental mistake. Finally, all this section on “Thom” contains above all comparative study with the strength of Rohlin’s formula $2(\pi-\eta)=r-k^2$ (see (\[Rohlin-formula:thm\])), hence a cuneiform formalism that really pertains to Rohlin’s complex orientations as it electrifies the Hilbert tree by putting charges (distributions of signs on the edge). Albeit our exploration of this topic was completely random (and biased by our erroneous Thom estimate), it should have some independent interest. Reorganizing all this material, without loosing any bit of information will take us several weeks, and cannot be done on the present edition. We hope in the future being able to give a more structured exposition of this cuneiform formalism (Hilbert’s tree with signs, alias Rohlin’s trees) and our messy account can motivate others to clarify this. What can be salvaged after Fiedler’s earthquake? ------------------------------------------------ \[22.03.13\] As spotted by Fiedler, we overlooked that Arnold’s surface (arising by pasting Klein’s half with Ragsdale’s membrane) is not necessarily orientable though both its constituents are. Hence one cannot apply Thom so straightforwardly. Incidentally we hope that Thom applies without trouble coming from the necessity of rounding corners. This is folklore but it would be nice to find adequate reference (Thom?, Cerf?, Hirzebruch? Milnor?, Wall?, etc.) Though Thom was a heuristic way to discover the inequality $\chi\le k^2$ it could be that it holds true for more elementary reasons directly rooted in Rohlin’s formula. Let us briefly explain how. \[$\bigstar$ Non-sense (!) by the Itenberg-Viro counterexample in degree $10$, Fig.\[Itenberg:fig\].\] In the sequel we use the jargon of the [*Rohlin tree*]{} which is simply Hilbert’s tree (encoding the distribution of ovals via a POSET whose order relation comes from the inclusion of the insides of the ovals), plus a decoration of its edges by signs coming from Rohlin’s complex orientations of a curve of type I (in the sense of Klein, also called latter by him orthosymmetric curves). So if any plane real curve has a Hilbert tree, dividing curves have an extra distribution of signs on the primitive edges (those of length one) which by the signs-law (\[Signs-law:lem\]) propagates consistently along the whole injective pairs of the tree. First if Rohlin’s tree is positively charged (i.e. if all primitive pairs of ovals are positive in the sense of Rohlin) then Arnold’s surface $A=C^+\cup R$ (=Klein’s half $C^+$ pasted with Ragsdale’s (orientable) membrane $R$) is orientable too! In that case via Thom we have $\chi\le k^2$ (cf. Theorem \[Thom-Ragsdale:thm\] and its simple proof). \[$\bigstar$ Okay, but as we shall soon see this may also be inferred from Rohlin’s formula!\] \(1) Moreover for a positively charged tree we have $\pi-\eta=n$, i.e. the Rohlin mass $\pi-\eta$ is equal to the number $n$ of negative=odd ovals (compare Lemma \[Rohlin-mass-of-a-positively-charged-tree:lem\] below). \(2) Rohlin’s formula reads $2(\pi-\eta)=r-k^2$, hence fixes the Rohlin mass $\mu:=\pi-\eta$ which is something coming from the “complexification” (i.e. the Riemannian) in terms of real characteristics ($r$ being the number of ovals and $k=m/2$ the semi-degree). This formula can be used to express the Euler-Ragsdale characteristic $\chi=\chi(R)$ as follows $$\begin{aligned} \label{Rohlin-to-Arnold-tris:eq} \chi=p-n=(p+n)-2n&=r-2n\cr &=[2(\pi-\eta)+k^2]-2n\cr &=k^2+2[(\pi -\eta) -n].\end{aligned}$$ Combining (1) and (2) gives the following lemma (incidentally remarked in Fiedler’s letter): If all primitive pairs are positive (equivalently if the Rohlin tree is positively charged) then $\chi=k^2$. Now it seemed to us realist to posit a sort of “positive mass conjecture” (POSMASS) stipulating that the Rohlin mass $\mu:=\pi-\eta$ (of a signed tree) is maximized when all primitive edges/pairs (of the tree) are positively charged. (We formulated this some 7 days ago, cf. optionally \[positive-mass-conjecture:conj\] dated \[15.03.13\].) This looks quite appealing, albeit we have little evidence for the truth of this principle which if optimistic could be pure combinatorics (i.e. valid for any signed tree with charges respecting the signs-law) or be merely valid for such Rohlin signed trees arising via dividing curves. Of course it could be also false in this restricted case. Our evidence for POSMASS is presently only derived from the case of chains of length $\le 4$ (or so), compare the signs-law for dyads (Fig.\[Signs-law-dyad:fig\]), that for triads (Fig.\[Signs-law-triad:fig\]), and that for tetrads (Fig.\[Signs-law-tetrad:fig\]). If POSMASS is true, then $\pi-\eta\le n$ and so the “Euler-Rohlin formula” implies $\chi\le k^2$, i.e. the so-called Gabard-Thom (dubious) estimate. The striking issue is that the Gabard-Thom theorem would be still true but merely as a logical consequence of Rohlin’s formula (plus some combinatorics required to validate POSMASS). In particular the intervention of Thom could be completely dispensed. This scenario looks risky, since Fiedler claims our Gabard-Thom theorem to be wrong. However, it is\[=was\] not clear to me if there is really a counter-example to the theorem, or if Fiedler just stated wrongness of our reasoning. More ironically I forgot to remember that even 5 days before formulating POSMASS, I had a simple counterexample to it in the combinatorial setting (cf. Theorem \[Garidi-mass-conj-is-FALSE:thm\]). Hence there is no chance to prove the Gabard-Thom estimate via pure combinatorics. Of course, it is also likely that the POSMASS conjecture is false also for Rohlin’s trees arising as dividing curves but that deserves an explicit example. Of course the method should be to ape algebraically the combinatorial structure of a batônnet that foils the mass conjecture (cf. Fig.\[Garidi-mass-false:fig\]a). A bâtonnet is merely a usual tree with a trunk that ramifies strongly into several branches at depth 2 (look at that picture Fig.\[Garidi-mass-false:fig\]a.) Accordingly, my first idea was to look back in Gabard 2000 [@Gabard_2000 Fig.13, p.154] where is traced a classical (variant of) Hilbert’s construction of an $M$-curve of degree 8, which has nearly the required bâtonnet structure. We shall soon reproduce this and related pictures. Moreover why degree 8? Simply because in degree 6 the Thom-Gabard estimate $\chi\le k^2$ (for dividing curves) is trivially true as follows by glancing at Gudkov’s table (=Fig.\[Gudkov-Table3:fig\]) of which we merely use Hilbert’s intuition/theorem that the unnested $M$-scheme (symbol $11$) is not algebraic, plus the fact that the unnested $(M-1)$-scheme (symbol $10$) has no dividing realization (as follows from Klein’s congruence $r\equiv_2 g+1$). This being said let us do an iterated Hilbert construction (Fig.\[HilbGab1:fig\]). This gives first the well-known $M$-sextic $C_6$ of Hilbert (symbol $\frac{9}{1}1$), and then an $M$-octic $C_8$ with $\chi=16=k^2$, and then an $M$-curve of degree 10, $C_{10}$ with only $\chi=9$, yet still $\le k^2=25$ (and congruent to it mod 8 as it should by virtue of Gudkov hypothesis). Of course the Gabard-Thom estimate $\chi\le k^2$ has little chance to be corrupted so, since it formally implies one-half of Ragsdale’s conjecture (which is still open for $M$-curves, cf. Lemma \[Thom-implies-one-half-of-Ragsdale:lem\]). A more intrinsic reason is of course that Ragsdale conjectures were calibrated along a deep contemplation of the Harnack-Hilbert method. Hence historical continuity is fighting against our attempt to corrupt Gabard-Thom via Hilbert’s construct. As the (maximal) $M$-Ragsdale conjecture is still open (not succumbing even to the Viro-Itenberg method, cf. Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale]), it is much more likely to corrupt Gabard-Thom by using non-maximal curves. Keep this idea in mind for later. -2.8cm0 -5pt0 -5pt0 Before adventuring outside the realm of $M$-curves, let us do also a plate for Harnack $M$-curves constructed à la Hilbert (Fig.\[HilbGab2:fig\]). Those Harnack-style curves have a priori a larger $\chi$, so better suited to corrupt Gabard-Thom. Precisely Harnack’s $M$-sextic has $\chi=9=k^2$, then we get a $C_8$ with $\chi=16=k^2$, and then a $C_{10}$ with $\chi=25=k^2$. Our depicted $C_{12}$ (right) is not the most natural choice as we switched to an “internal vibration”, while in the first steps $C_6\rightsquigarrow C_{8}\rightsquigarrow C_{10}$ we consistently opted for an external oscillation (typical of Harnack’s curves reckoned à la Hilbert). It should be noted than even the natural choice of an external vibration does not lead to a $C_{12}$ with $\chi=36$, but we were only able to get one with $\chi=28$ (in accordance with Gudkov hypothesis). \[[*Added in proof*]{}.—THIS IS A MISTAKE, due to the fact that I reported ovals at the wrong place\] However Miss Ragsdale in 1906 was surely much more clever than we are \[THIS IS NOT PERTINENT ANY MORE\], and I presume she was able to reach always the “Gabard-Thom upper-bound” $\chi\le k^2$, since this really amounts to (one-half of) her conjecture (cf. again Lemma \[Thom-implies-one-half-of-Ragsdale:lem\]). -2.8cm0 -5pt0 -5pt0 \[23.03.13\] Correcting my mistake, an extension of this figure (Fig.\[HilbGab2:fig\]) shows the following result (compare Hilbert 1891, and Ragsdale 1906): There is an infinite series of $M$-curves of degree $2k$ with $\chi=k^2$. Hence if Ragsdale’s conjecture $\chi\le k^2$ (for $M$-curves) is true, then it is sharp. Look at the first steps of Fig.\[HilbGab2:fig\] and do not commit the mistake of making an inner vibration, but choose always external vibrations as we did for $C_6, C_8$. On looking at the Hilbert trees of $C_6, C_8, C_{10}$ one easily derive the general evolution of the Hilbert tree of $C_{2k}$. Namely the number of outer ovals $9,17,27$ augments along the increment $+8, +10, +12, etc.$, while the tree itself is always pushed down one step deeper while acquiring new branches on its top, compare the windows on the figure for $C_6, C_8, C_{10}$. In view of the extreme regularity of the construction it is easy to extrapolate the nested structure of $C_{2k}$. Writing down a general formula looks not even necessary, and the Gudkov symbol will be something like $$(1,(2k-6) (1, 2k-8 (1, 2k-10) \dots )) (9+8+10+\dots+2k).$$ In more geometric terms this means that the Hilbert tree of $C_{2k}$ has $9+8+10+\dots+(2k-2)+(2k)$ outer ovals, and a trunk of length $2k-7$ with branches hanging on. It should be an easy matter to compute directly the Euler characteristic of $C_{2k}$ by the evolution rule for the tree. \[23.03.13\] Perhaps instead of using Hilbert’s method one must really uses the more time-consuming Harnack original method which amounts to oscillate around a ground-line instead of the ellipse used in Hilbert’s method. (Incidentally, I wonder if one wants a fast-Hilbert method, if it is possible to vibrate across a split quartic, union of 2 ellipses.) All this is good but will perhaps only confirm the intuition (of Ragsdale) that her estimate $\chi\le k^2$ is sharp for $M$-curves (if true at all). Our object is somewhat different namely to refute the Gabard-Thom estimate $\chi\le k^2$ for all dividing curves. Of course it could be possible to disprove the Ragsdale $M$-estimate $\chi\le k^2$ yet this deserves highbrow methods as even the powerful Viro-Itenberg method apparently failed as yet in that game (compare Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale]). So how to construct non-Harnack-maximal dividing curves? As we know from degree 6 (Fig.\[GudHilb8:fig\]), the trick is just to disperse the vibratory energy on several ovals, as opposed to the monopole of Hilbert’s vibration where a single oval is oscillating. By Klein’s congruence we look at $(M-2)$-curves, and in degree 6 the specimen with largest $\chi$ is the scheme with symbol $9$, which can be obtained by (a variant of) Hilbert’s method where the vibration is dispatched on 2 ovals (cf. the earlier Fig.\[GudHilb8:fig\] or right below Fig.\[HilbGab3:fig\]). It is now hoped that when iterating the construction to higher degrees we get a refutation of Gabard-Thom. Applying this idea we get the following series of $(M-2)$-curves (Fig.\[HilbGab3:fig\]), all of type I by Fiedler’s law of smoothing dictated by (and dictating) complex orientations. Of course at the higher steps we choose again a monopolized vibration as otherwise we descend further the energetic level and reach curves with less ovals that $(M-2)$. We use now a quicker depiction mode where vibration and smoothing are depicted on the same plate at each step of Hilbert’s inductive process. Further on the diagram standing right below each curve, we depict Hilbert’s nested tree encoding the distribution of ovals, and some easy calculation of topological characteristics of the curves so constructed. The conclusion is that we get an infinite series of $(M-2)$-curves with $\chi=k^2$ as shown by the figure up to $k=7$, and the regularity of the procedure is so evident that this property easily follows for all $k$. In particular it may be observed that the number of outer ovals evolves along the progression $9, 17, 27, 39, 53,\dots$, incrementing along the progression $+8,+10,+12, +14, \dots$, while the nested portion of the tree is simply pushed down at each step, with $(2k-6)$ new branches arising on the top. -2.8cm0 -5pt0 -5pt0 Hence, modulo some arithmetical nonsense, we have proved the: There exists an infinite series of dividing $(M-2)$-curves $C_{2k}$ of degree $2k$ with $\chi=k^2$. Alas, this does not refute the Gabard-Thom estimate, but rather show its sharpness in case it would be correct. (Another proof of the sharpness can be derived by perturbing ellipses, cf. Remark right after Thm \[Thom-Ragsdale:thm\]). (The contrast between the present regularity and the lack thereof for $M$-curves was so striking that it permitted us to correct the earlier mistake that we did above.) But where to find a counter-example to the Gabard-Thom-bound (as promised by Fiedler’s claim of erroneousness)? Lacking some imagination let us redo the Hilbert vibration for $M$-curves more systematically by always vibrating from “outside”. This gives Fig.\[HilbGab4:fig\]. The same regularity is observed while the general pattern becomes evident after some few iterations. Hence one can reduce to the depiction of the trees. The latter get always more profound by one unit as $k$ increments, while the number of deepest ovals belongs to the series $9, 17, 27, 39, \dots$ which regularly increments by $+8,+10, +12, \dots$ so that one can predict the future evolution of the tree, compare the very bottom row. This shows that one times over two we will attain the Gabard-Thom bound $\chi=k^2$, while of course the sign of $\chi$ oscillates between negative and positive values. It is evident that we will not get a counterexample to Gabard-Thom in this fashion. -2.8cm0 -5pt0 -5pt0 Then we can still vary more the constructions, e.g. by starting with a Harnack-like outer vibration of the $C_6$ like on Fig.\[HilbGab2:fig\] and perform the dissipation (leading to $(M-2)$-curves) at the next step on the octic $C_8$. Alas we still found $\chi=k^2$ for the $C_8$ and even the $C_{10}$ (details of the picture on p.AR-114 of my hand-notes). At this stage one gets a bit depressed. It seems hard to corrupt Gabard-Thom by construction à la Hilbert-Harnack. Maybe I missed something, or perhaps one should appeal to more sophisticated constructions like Viro-Itenberg. \[This turned out to be the good idea, more soon.\] At this stage I cannot therefore preclude the option that the Gabard-Thom estimate is true (of course for another reason than the gapped proof given in Theorem \[Thom-Ragsdale:thm\]). So a priori 4 scenarios may happen: (1).—Either there is an elementary counterexample to Gabard-Thom(=GT) via elementary constructions à la Harnack-Hilbert (and we missed it due to lack of cleverness). (2).—There is a counterexample to Gabard-Thom via highbrow constructions à la Viro-Itenberg. In particular it is not impossible that the counterexample described by them to Ragsdale conjecture also supplies a counterexample to Gabard-Thom. This requires controlling the type in their construction (which is an issue known to them). More on this in the next Sec.\[Itenberg-Viro-disprove-Gab-Thom:sec\]. (3).—There is a counterexample to GT via another construction not covered by Viro-Itenberg. (To my knowledge there is no theorem stating that any algebraic curve is constructible via their method.) (4).—GT is true for another reason than the one exposed in Theorem \[Thom-Ragsdale:thm\]. This would however conflict with Fiedler’s assertion that our theorem is wrong. (Again, it is not clear if Fiedler merely stated wrongness in the proof or of the statement.) The next section gives a clear-cut answer to this puzzle. A formal disproof of Gabard-Thom via Itenberg-Viro’s patchwork and Kharlamov-Marin {#Itenberg-Viro-disprove-Gab-Thom:sec} ---------------------------------------------------------------------------------- \[23.03.13\] Here we say “formal” just because we are not yet familiar with the patchwork method due to Viro, and elaborated by Itenberg later into the so-called $T$-curves context. Of course the method has nothing formal: it is crystallography of the best stock as we shall soon see. As we failed along strategy (1) (cf. previous section), let us look at (2) which invites to take a better look on Itenberg-Viro seminal paper (1996 [@Itenberg-Viro_1996-disproves-Ragsdale]). There on Fig.2 (p.20) we find a remarkable picture reproduced below as Fig.\[Itenberg:fig\]. (Our sole change is to have traced the curve with less thickness to see better what happens than on the downloaded black-and-white pdf.) Note that the underlying triangulation (by triangles all of area one-half of the unit square) is symmetrical under the dihedral group (Vierergruppe $D_4\approx \ZZ_2\times \ZZ_2$, cf. Fig.a). I did not noticed this for a while when reproducing this figure! Then there is a signs-distribution that gives the red curve via a bisection procedure of all triangles. Interpret this merely as a piecewise-linear random walk if you have zero-knowledge like the writer. Tracing this curve is a miracle, very enjoying if one is computer assisted. Representing the curve alone gives Fig.b (a simple cut-and-past operation for the computer). As usual in projective geometry the boundary of the rhombs must be identified antipodically. Hence the 5 semi-ovals on the bottom left-side of the rhombs are really just capping off (closing) the long contorted oval occupying the oriental (Siberian) part of the rhombs. We count on Fig.c precisely 29 outer ovals. Hence the curve in question (which admits an algebraic realization by a deep theorem of Viro-Itenberg) has the scheme depicted on Fig.d, hence Gudkov symbol $(1, (1,2)2)29$. The total number of oval is $r=29+6=35$, i.e. 2 unit less than $M=37$ (temperature of the human body), so its an $(M-2)$-curve. Hence there no obstruction (via Klein’s congruence) for the curve being dividing. Looking optionally at the corresponding Hilbert tree (Fig.e) gives quickly $\chi=29+1-3+2=29$ (variant look at the scheme Fig.d and apply a Swiss cheese recipe à la Euler-Listing, etc.). Arnold’s congruence mod 4 ($\chi\equiv_4 k^2$ if type I) is verified and so there still no obstruction for the curve being dividing. In the fact, the wonderful RKM-congruence $\chi\equiv k^2+4 \pmod 8$ (Rohlin-Kharlamov-Marin, see (\[RKM-congruence-reformulated:thm\])) implies this scheme being of type I (and so is in particular any curve representing it). Hence the Itenberg-Viro curve is of type I, and it gives the long searched counter-example to our Gabard-Thom pseudo-theorem. Probably, there is a more elementary (organical) way to deduce the dividing character of the curve by an avatar of Fiedler’s signs-law in the realm of the Viro method. (This goes back to the early 1980’s, and perhaps in the present $T$-curve context is due to Itenberg. Parenti’s thesis 199X [@Parenti_199X] looks also involved in this topic.) -0.4cm0 -5pt0 -5pt0 So we are at this stage nearly in the paradise! To add some suspense note that this curve being of type I (even in the very strong sense that its scheme is) a vague conjecture of us (founded on Ahlfors) posits that there should be a total pencil of $(m-3)$-adjoint curves exhibiting the dividing character of the curve (as in the Rohlin-Le Touzé 1978–2013 theorem for sextics, cf. Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]). According to our (hypothetical) extension of the Le Touzé theorem, there should be a pencil of septics cutting only real points on the Itenberg-Viro curve. In general, the degree of such a pencil should be in accordance with Gabard’s bound $r+p$ on the degree of circle maps (compare Sec. \[census-and-extension-of-Rohlin-Le-Touzé:sec\], the numerology therein and especially Remark \[M-2-curve-degree-like-Gabard:rem\]). Pencil of septics have 34 basepoints freely assignable (in general this is $M-3$), whereas the Itenberg-Viro curve has 33 empty ovals. So where to assign the remaining basepoint? This is again a bit puzzling. However it is likely by the philosophy of dextrogyration that Rohlin’s complex orientation plays some rôle there. Rohlin’s formula applied to the Itenberg-Viro curve gives $2(\pi-\eta)=r-k^2=35-25=10$, hence $\pi-\eta=5$, while $\pi+\eta=7$ as is apparent from the Hilbert tree (Fig.e, where one counts 2 additional pairs of length 2). Hence $\pi=6$ and $\eta=1$. The presence of a negative pair is no surprise, as Fiedler noticed this being the gap in our (erroneous) proof of the Gabard-Thom estimate (Theorem \[Thom-Ragsdale:thm\]). By the signs-law (\[Signs-law:lem\]) it is clear that the (unique) negative pair must have length one (primitive pair), otherwise all primitive edges are positively charged but then there are 2 negative pairs (those of length 2). Since there is always a bijective correspondence between primitive edges of Hilbert’s tree and (non-maximal) ovals by taking the bottom of the edge which is always uniquely defined. So there is one negative oval (i.e. whose edge) and thinking more with the signs-law it seems that the negative pair is forced to be the trunk (i.e. the edge that ramifies at depth 2, cf. Fig.e and lemma below). So this ovals (negatively charged right above it) is perhaps the good candidate of where to assign the remaining basepoint. All this deserves of course to be better understood, and is digressed upon in the next section (\[Galton-brett:sec\]). Let us verify the simple: The Rohlin tree of the Itenberg-Viro curve has a unique negative charge that is forced to be on the trunk. As noted above Rohlin’s formula implies that there is a unique negative pair ($\eta=1$). Since in Rohlin’s arithmetics $+\times +=-$ (i.e. consanguinity is bad) the deepest subtree of length 2 (“Y”-shaped modulo horizontal mirror) cannot be positively charged else there would be 2 negative pairs of length 2. So a primitive minus-charge must be located on the “Y”-subtree. If it is on one of the 2 branches (as opposed to being on the trunk at depth $0-1$) then by uniqueness, the trunk and the (other) branch are positively charged, so that it results a minus-charge on their concatenation (of length 2), violating $\eta=1$. So the negative charge must be located on the trunk as asserted. Where to assign basepoints to ensure total reality: toward a Galton-Brett algorithm? {#Galton-brett:sec} ------------------------------------------------------------------------------------ \[24.03.13\] The above fantastic example of Itenberg-Viro (Fig.\[Itenberg:fig\]) raises again the general problem of deciding where to assign basepoints as to ensure total reality of an adjoint pencil. This means as usual a pencil of curves cutting only real points on a dividing curve. By general topology (the image of a connected set is connected) a dividing curve presents no obstruction to the existence of such a pencil. More than that, Ahlfors theorem says that there is no conformal obstruction to do this (being after all just an extended Riemann mapping theorem for bordered surfaces of higher topological structure that the disc). If one has a total map to the line $\PP^1$ then we have a branched cover taking boundary to boundary and interior to interior, and the map restricted to the real locus is an (unbranched) covering. Accordingly there is a phenomenon of dextrogyration, i.e. when the image-point circulates once around the fundamental circle $\PP^1 (\RR)$ the counter-images (fibre) circulate along the complex orientation of the abstract curve, i.e. as the boundary of the half. (This follows of course from the holomorphic, hence sense preserving, character of Ahlfors maps.) If the curve is plane ($C\subset \PP^2$) we would like a procedure predicting where to assign basepoints. By the above dextrogyration principle, there should be some relation with Rohlin’s complex orientations, which measure merely the (abstract) complex orientations as compared with those of rings (annuli) bounding (injective) pair of ovals in the plane $\RR P^2$. By the signs law it suffices to determine Rohlin’s signs for primitive pairs of ovals. A vague idea is as follows. Given a plane curve we have the Hilbert tree and Rohlin’s complex orientations decorate its edges with signs (pluses, minuses). We can imagine the resulting Rohlin tree as a “Galton Brett”, i.e. Galton’s table where billiard-balls fall downwards along an inclined table interspersed with a distribution of nails. Whenever meeting one of those nails the ball is deflected left or right with probability one-half. For the usual equilateral distribution we recover so the Chinese-Pascal binomial distribution. Our naive idea is to interpret the Rohlin tree as a Galton-Brett, while putting balls at the top of Hilbert’s tree and looking where they stabilize to an equilibrium. It is imagined that negative pairs are inclined so that balls fall gravitionally along them. Consider the example of the deep nest. We know then either from Rohlin’s formula or via the dextrogyration argument applied to the obvious pencil of lines through the deep nest that all primitive pairs are negative. Here the Galton-Brett reduces to a simple track (without branching) always negatively charged, and the ball descends right up to its bottom. This is in agreement with the fact that total reality of a lines-pencil is ensured when assigning the basepoint in the deepest oval. Similar considerations hold for the quadrifolium and its satellites, i.e. curves of degree divisible by four and totally really under a pencil of conics. On the next example of Rohlin-Le Touzé’s sextics, e.g. that of type $\frac{6}{1}2$, the Rohlin tree has 6 branches emanating from an oval and Rohlin’s formula $2(\pi-\eta)=r-k^2=9-9=0$ shows that $\pi=\eta=3$ since we have a total $\pi+\eta=6$ of six pairs. So our tree has 3 negative and 3 positive edges. Our metaphor of the Galton-Brett already looks dubious on that example, since it would prescribe imposing basepoints only on the $3$ ovals surmounted by a negative charge (and of course the 2 outer ovals). So it remains to understand if an improved Galton-Brett principle permits to understand where to assign basepoints in function of a knowledge of Rohlin’s complex orientations. Maybe an improved rule is to let balls fall-down regardless of signs along the Rohlin tree, and some few of them could stay in levitation (unstable equilibrium being blocked by a needle) with special signs-property, like being surmounted by a negative charge while branching down below (so-called hyperbolic ovals). This complicated condition comes to mind when looking at the deep nest plus the above Itenberg-Viro curve, where the trunk (of the tree) is negatively charged, which is the only reasonable signs-distribution compatible with Rohlin’s formula (cf. lemma above). So on the tree of Fig.\[Itenberg:fig\]e balls would fall along the Hilbert tree and stabilize of course in each “deep” ovals (aka empty ovals), but some nontrivial equilibrium arises at the vertex at depth 2 which branches further (alias hyperbolic oval). Of course hyperbolicity alone is not enough as shown by Rohlin-Le Touzé’s theorem. However hyperbolicity plus a negative charge above it could give an equilibrium, i.e. a place where to assign a basepoint. Though a bit complicated this looks even reasonable from the viewpoint of Galton’s Brett. Namely hyperbolic ovals are those where there is an indetermination (bifurcation) when falling down, while negativity of the edge above is a sort of kinetic impulse giving the particle some momentum forcing it to move against the bifurcation, whence an “unstable” equilibrium (crystallizing thereby in the formation of a basepoint). Of course all this need to be further explored, and to be related to more intrinsic properties of dividing plane curves. For the Viro-Itenberg curve this algorithm would assign basepoints of the septics-pencil on the 33 empty ovals (stable equilibrium) plus one unstable equilibrium materialized by the unique hyperbolic oval at depth 1. This would give the 34 basepoints required in a pencil of septics, and total reality could follow (assuming that our Galton algorithm is somehow compatible with dextrogyration or perhaps indexes formulae à la Gauss-Kronecker-Poincaré-von Dyck). Let us look at more examples. For octics we have 4 basic schemes listed in Eq. \[octics-five-examples-RKM:eq\] which satisfy the RKM-congruence (cf. also the Gudkov table in degree 8, Fig.\[Degree8:fig\]). Those were $$\frac{16}{1}3,\quad \frac{12}{1}7,\quad \frac{8}{1}11,\quad \frac{4}{1}15,\quad 20.$$ The last of which is precluded as it violates either Petrovskii’s inequality (\[Petrovskii’s-inequalities:thm\]) or the Thom estimate $\chi\le k^2$ which is valid when there is no nesting (or even the more elementary Rohlin’s formula). However for all other schemes it may be reasonable to expect total reality for a pencil of quintics which has 19 basepoints (recall that $B=M-3$ for the number of basepoints in terms of Harnack’s bound $M$) and all of them are ascribed on the empty ovals (in accordance with our Galton principle). All those 4 RKM-schemes are just the top of the iceberg depicted on Fig.\[RKM-schemes-deg-8:fig\]. On that tabulation we find e.g. the scheme $\frac{3}{1}\frac{1}{1}14$. This has 18 empty ovals, and we need a 19th basepoint. Our algorithm of the negative hyperbolic oval fails to give it since looking at the Hilbert tree of the scheme we see a unique hyperbolic oval, and this has no edge above it! So our method fails and deserves to be further improved. Less likely, our method could be right and then it could preclude existence of those schemes in type I. (Note that our Galton method could have killed the scheme $20$, since we expect 19 basepoints but there are 20 stable equilibriums.) Of course all this must be further explored. Summarizing, a fundamental question seems to be: Is there a general algorithm telling one where to ascribe basepoints in terms of the combinatorics of Rohlin’s tree encoding his complex orientations? If so then we get a mechanical device extending the Rohlin-Le Touzé phenomenon of total reality for $(M-2)$-sextics satisfying the RKM-congruence. Viceversa, suppose zero-knowledge on the complex orientations one could argue that the principle of total reality via dextrogyration is a good recipe to infer a knowledge of them (e.g. as it is flagrant in the trivial deep nest case). Presently very little seems to be known in general, and this is of course much reminiscent of the lost proof of Rohlin’s (last) theorem. When is Arnold’s surface orientable and Ragsdale via Bieberbach-Grunsky? {#Ragsdale-via-Riemann-Bieberbach-Grunsky:sec} ------------------------------------------------------------------------ \[27.03.13\] Here we propose a (naive) attack upon one half of the Ragsdale conjecture for $M$-curves. This may be translated as the condition $\chi\le k^2$ (cf. Lemma \[Thom-implies-one-half-of-Ragsdale:lem\]). As shown by the proof of the erroneous Theorem \[Thom-Ragsdale:thm\], under the additional assumption that the Arnold surface is orientable, Thom applies and gives promptly the (upper) Ragsdale estimate $\chi\le k^2$. (Note: The full Ragsdale amounts to the pinching $-k^2\le \chi \le k^2$, equivalently $\vert \chi \vert \le k^2$.) So the core of the question is to know when Arnold’s surface is orientable. We shall discuss this soon. The net impact could be as follows: \[Arnold-surface-M-curves-orient:conj\] The Arnold surface of a (plane) $M$-curve is always orientable. If this is true then the upper-Ragsdale estimate $\chi\le k^2$ follows from Thom (cf. proof of (\[Thom-Ragsdale:thm\])). $\bigstar$ Alas, the sequel shows that this naive conjecture fails already for Hilbert’s $M$-sextic, cf. Fig.\[Ragsdale:fig\]c. Recall that the [*Arnold surface*]{} of a dividing plane curve of even degree $2k$, is simply Klein’s half of the curve filled by Ragsdale’s membrane in $\RR P^2$ bounding the curve from inside. It will be orientable iff the orientation coming from the complexification and the real Ragsdale membrane match together in some sense made precise below. First it is plain that if Rohlin’s tree is positively charged (on all its primitive edges) then Arnold’s surface is orientable (cf. Fig.\[Ragsdale:fig\]a). This positive-charge assumption is very stringent and implies actually much more, namely that the Rohlin mass $\pi-\eta$ equals $n$ (cf. Lemma \[Rohlin-mass-of-a-positively-charged-tree:lem\]). Via Rohlin’s formula rewritten as $\chi=k^2+2[(\pi-\eta)-n]$, this implies in turn that $\chi=k^2$ exactly. However to derive the (elusive) upper-Ragsdale-estimate $\chi\le k^2$ from Thom, it suffices that the Arnold surface is orientable. A small picture (Fig.\[Ragsdale:fig\]b) convinces one that this holds more generally whenever Rohlin’s tree is positively charged on odd edges. Here we always define the depth of an edge in reference to that of its bottom vertex (the latter being uniquely defined by—and defining uniquely—the given edge). -5pt0 -5pt0 So we get the: \[Arnold-surface-orientable-iff-oddly-charged:lem\] The Arnold surface is orientable iff the Rohlin tree is positively charged on odd edges (say then that it is oddly charged). This condition is much weaker, but still implies Ragsdale via Thom. So the upper Ragsdale $M$-conjecture (URMC) reduces to the: \[oddly-charged-M-curves:conj\] Any $M$-curve is oddly charged. A guess could be to use our translation (Theorem \[total-reality-of-plane-M-curves:thm\]) of the Bieberbach-Grunsky theorem (truly due to Riemann 1857). The idea is that for $M$-curves we have a fairly explicit way to construct a total pencil via curves of degree $(m-2)$ assigned to pass through any distribution of $g+1$ points (one on each oval) and then by looking at the residual group of points, while assigning them as basepoints. For more details cf. proof of Theorem \[total-reality-of-plane-M-curves:thm\], but we repeat the general recipe of the construction of a total series on a plane $M$-curve $C_m$ of degree $m$: 1.—Choose any distribution $D$ of $g+1$ points one on each oval. 2.—Let pass a curve $\Gamma_{m-2}=:\Gamma$ of degree $(m-2)$ through $D$. 3.—Consider $R$ the residual intersection $\Gamma\cap C$ less the points of $D$. 4.—Assign $R$ as the basepoints to the system of curves of degree $(m-2)$, and get (or choose) a pencil $\Pi$ putting the initial group $D$ into motion. 5.—By continuity the pencil $\Pi$ is total since there is only one point one each circuit hence no risk of collision. Total reality follows. The dream would be that this procedure is sufficiently explicit as to control complex orientations, especially the issue that the Rohlin tree is oddly charged. If this is possible we get a proof of the upper-half $\chi\le k^2$ of Ragsdale’s conjecture. Concretely, once the distribution $D$ is fixed we are assured that the curve $\Gamma$ will cut $C$ once more along each ovals (by the closing lemma for algebraic circuits \[Closing-lemma:lem\]). So we have $2(g+1)$ real intersections in $\Gamma\cap C$, i.e. $2(g+1)=2(\frac{(m-1)(m-2)}{2}+1)=(m-1)(m-2)+2=m^2-3m+4$. This is less than the $m(m-2)$ expected. \[disproof-orientability-Arnold-M-curve:lem\] Alas, the oddly-charged conjecture (\[oddly-charged-M-curves:conj\]) or equivalently the orientability of the Arnold surface of an $M$-curve (\[Arnold-surface-M-curves-orient:conj\]) fails already in degree 6, e.g. for Hilbert’s $M$-sextic as shown on Fig.\[Ragsdale:fig\]c. Indeed reporting complex orientation via Fiedler’s transmission-law we get Fig.\[Ragsdale:fig\]c. Here we report first the orientation induced on the quadrifolium quartic $C_4$ from the dashed ellipse, and then smooth $C_4\cup E_2$ (where $E_2$ is the thick-ellipse) along positive orientation and receive so the complex orientations of Hilbert’s sextic. We see that among the 9 nested ovals, 5 are dominated by a positive pair, while 4 are by a negative pair. This is in accordance with Rohlin’s formula, $2(\pi-\eta)=r-k^2=11-9=2$, i.e. $\pi-\eta=1$ while $\pi+\eta=9$. Hence $2\pi=10$, i.e. $\pi=5$ and $\eta=4$. However this refutes our very naive conjecture (\[oddly-charged-M-curves:conj\]), which diagrammatically amounts saying that the Rohlin tree is positively charged on edges at odd depths. Of course the equivalent formulation in terms of the orientability of the Arnold surface (\[Arnold-surface-M-curves-orient:conj\]) is killed in the same stroke. So our naive strategy fails severely but of course Hilbert’s sextic has $\chi$ very negative ($\chi=2-9=-8$). Hence there is perhaps a refined argument, that can establish Ragsdale. Alas it seems that what we just did kill definitively an approach via Thom which requires orientability of the Arnold surface. Of course one could expect a tricky case distinction along the sign of $\chi$, and a strengthened conjecture stating orientability of the Arnold surface (of an $M$-curve) provided $\chi>0$. Even this is easily disproved, e.g. by looking at the $M$-curve $C_{10}$ with $\chi=9$ of Fig.\[HilbGab1:fig\], and reporting the complex orientations via Fiedler’s law. This is a bit tedious but straightforward and gives Fig.\[HilbGab1bis:fig\]. We find that the 11 edges at depth 3 splits into 7 positive pairs and 4 negative ones. Rohlin’s formula can be verified via the signs-law. However Rohlin’s tree is not positively charged at the odd depth 3. -3.5cm0 -5pt0 -5pt0 Of course one could still expect that Rohlin’s tree is oddly-charged when $\chi>k^2$, and this would suffices via Thom to prove the upper Ragsdale conjecture, but we are obviously playing a sterile arithmetical game, without much geometrical penetration. Alternatively, if not via Thom one could hope to use directly Rohlin’s formula, but again some external information on complex orientations must be gained via some deep geometric procedure. As we said one can dream that a version of the Bieberbach-Grunsky theorem do the job, but that deserves investigating with much more care and patience than we are presently able to do. Good luck to anybody who still feel optimistic. Of course it may also be that Ragsdale’s upper bound is just false by a highbrow variant of the Itenberg-Viro construction (though since Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale] nobody apparently ever succeeded), and it is evident that we (personally) lack experimental data to feel really secure in claiming the Ragsdale bound. Hence we abort this problem for the moment. A sporadic (?) obstruction via Thom (Kronheimer-Mrowka 1994) ------------------------------------------------------------ \[22.03.13\] It should be noted that the first examples of Thom’s conjecture as applied to Hilbert’s 16th where we fill an unnested curve by discs are not affected by Fiedler’s correction. So in particular the “elementary” degree 3 case of Thom due to Kervaire-Milnor 1961 (yet relying massively upon Rohlin’s early work ca. 1951–52) really implies e.g. a purely topological proof of Hilbert’s intuition of nesting for $M$-sextics. This is detailed below. More generally, Thom’s conjecture forbids in all (even) degrees $m=2k\ge 6$ the possibility of an unnested $M$-curve (symbol $M$). This was first proved by Petrovskii 1938, and can also be deduced from Rohlin’s formula $0=2(\pi-\eta)=r-k^2$, since $M$ is strictly larger than $k^2$ when $k\ge 3$. \[05.03.13\] Now just a little remark along the Thom conjecture (=the Kronheimer-Mrowka theorem 1994 [@Kronheimer-Mrowka_1994], abridged as “Thom” in the sequel). If we look at the $(M-2)$-scheme $20$ of degree 8, and fill one half by the canonical orientable membrane we get a surface of genus $p=1$ whose homology class is $4H$ (where $H$ is the natural generator of $H_2(\CC P^2,\ZZ)=\ZZ$, the so-called hyperplane-section, here a line). By Thom the genus should be at least as big as that of a (smooth) quartic, hence 3. So we get the: The scheme $20$ is not realized algebraically by a curve of degree $8$ (necessarily of type I by the RKM-congruence \[Kharlamov-Marin-cong:thm\]). If we take the scheme $\frac{4}{1}15$ (cf. Fig.\[RKM-schemes-deg-8:fig\]), then the genus of the filled membrane will be $1+4=5$ and so Thom’s principle is not violated. No other scheme of that table are prohibited by Thom. If we look at $m=6$, and the Gudkov table Fig.\[Gudkov-Table3:fig\], especially the seminal intuition of Hilbert ca. 1891–1900, that the unnested scheme $11$ does not exist algebraically, then again we see that this may be inferred from Thom’s conjecture (meanwhile a theorem). Indeed making the canonical filling of the half Riemann surface by the canonical (Ragsdale) membrane (often denoted $\RR P^2_+$) we get a surface of genus of 0 realizing the homology class $3H$, hence violating Thom’s conjecture. [*Insertion*]{} \[21.03.13\].—As a matter of fact, this special degree 3 case of Thom’s conjecture was first established by Kervaire-Milnor 1961 [@Kervaire-Milnor_1961 Cor.2, p.1652], basing themselves much upon Rohlin’s work of 1951–52. Thom is alas not mentioned there (KM61). So historiographically, it is worth emphasizing that the Kervaire-Milnor paper afforded so-to-speak the first purely topological proof of Hilbert’s 1891 semi-intuition/semi-theorem that a Harnack-maximal sextic cannot have all its 11 ovals unnested! Prior to this we had only available: (1) the algebro-geometric (stratificational) proofs of Hilbert (1891 unpublished) and Rohn (1911–13), plus technical refinements of the same method by Gudkov ca. 1954 and (2) the proof of Petrovskii 1933/38 half analytical (Euler-Jacobi analytical interpolation) and half topological (Morse theory). Nowadays we have of course the proof via Rohlin’s formula (1974 [@Rohlin_1974/75]), which is more elementary and purely topological (or the related one via Arnold’s congruence mod 4). Several ideas arise. (1).—Of course this Thom argument is not the most elementary prohibition of the scheme $10$ of degree 6, but maybe it is a good way to prohibit the scheme $20$ in degree $8$ (at least I know no other method for the moment). [*Update*]{} \[07.03.13\]: the prohibition of this scheme $20_{8}$ follows more elementarily from Rohlin’s formula $2(\Pi^+-\Pi^-)=r-k^2$ (\[Rohlin-formula:thm\]), since the absence of nesting implies vanishing of the left-hand side, hence $r=k^2$ has to be a square (even $16$ as $k=4$). Private anecdote, I missed this consequence of Rohlin, and noticed it while completing the Gudkov Table in degree 8 (cf. Fig. \[Degree8:fig\]). \[21.03.13\] Another way to prohibit this scheme $20$, I presume the first historically found, involves the Petrovskii inequalities, cf. (\[Petrovskii’s-inequalities:thm\]). More generally what schemes can be prohibited by Thom, and did it affords new obstructions (not known before Kronheimer-Mrowka)? The first question is answered by Theorem \[Thom-Ragsdale:thm\] below \[alas erroneous!\], while the second is perhaps answered via Theorem \[Alsatian-scheme-Thom-strong-Petrov-Arnold:thm\]. (2).—Looking at this canonical membrane filling might be a good device toward understanding the RKM-congruence. (But this is merely a matter of reading once carefully the Kharlamov or Marin arguments.) At least it is tempting to calculate the self-intersection of this filled membrane with itself (or its companion) to get some numerical relation. Doing so we probably obtain nothing new but what exactly? (guess the Arnold congruence). (3).—Despite having corners this filled Riemann surface is perhaps a good object to do conformal geometry with (compare especially works by H.A. Schwarz ca. 1870 and his student Koebe ca. 1906–07, also that of Hilb ca. 1907, NB: Hilb is not Hilbert misprinted but a less well-known conformal geometer of that period). (4).—As this Thom argument prohibits the scheme $20$ \[true despite Fiedler’s correction\], which was an obstacle toward assigning the 19 basepoint, try to pursue the game of understanding the order 8 avatars of the Rohlin-Le Touzé theorem. About (2), let $C_m$ be a dividing curve of even degree $m=2k$. Denote by $F$ the closed surface $C_m^{+}\cup R$, where $R:=\RR P^2_+$ is the canonical “Ragsdale” membrane (my own jargon but historically justified I think after reading Viro’s admiration for Miss Ragsdale, as US-Studentin of Klein-Hilbert). On the one hand $F$ is homologous to $kH$. To compute the self-intersection $F^2$ we use a vector field on the canonical membrane $R$, which is either transverse or tangent along the boundary and with finitely many zeros inside $R$. We have by Kronecker-Poincaré’s index formula $\sum indices= \chi$, where $\chi$ is the Euler characteristic of the membrane $R$. Multiplying this vector field by $i=\sqrt{-1}$ permits to push one replica of $F$ in general position. Naively it seems to follow that $k^2=F^2=\chi$, but one needs to count better indices.... Another remark is to write down the Thom conjecture inequality for the filled surface, and this gives the following (which is certainly not new \[$\bigstar$ but alas false!!\], cf. maybe Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000], but after a rapid check it does not seem to be explicitly stated there): \[Thom-Ragsdale:thm\]—[ERRONEOUS AT LEAST IN THE WEAK SENSE THAT THERE IS A BASIC BUG, YET NO CONCRETE EXAMPLE KNOWN TO ME[^79]]{} [(Thom applied to Klein-Hilbert-Ragsdale)]{}.—Let $C_m$ be a dividing plane curve of degree $m=2k$. Then $\chi \le k^2$, where $\chi$ denotes (as usual) the Euler characteristic of the Ragsdale membrane. [*Insertion.*]{}\[14.03.13\].—For each $k$, it is a simple matter to convince that the estimate is sharp, compare Figs.\[CCCRoses:fig\] and \[CCCRoses2:fig\]. [*Inserted (optional reading).*]{} \[16.03.13\].—At first sight this result looks so limpid \[$\bigstar$ outdated now!\] that one may wonder if it extends to higher dimensions, e.g. to algebraic surfaces in $\PP^3$. First one requires an extension of the Thom conjecture for surfaces in $\CC P^3$. This is perhaps quite straightforward, by replacing the genus by the Euler characteristic and arguing experimentally that surgeries (aka spherical modifications increase the topological complexity, yet without changing the homology class). However the second step of the proof fails blatantly as we lack a natural extension of the concept of dividing curves to surfaces though several peoples (notably Viro) proposed extensions requiring e.g. that the homology class of the real part mod 2 vanishes in the complexification. Alas, the real locus of an algebraic surface, having (real) codimension 2 in its complexification, never divides. Even if it would the Ragsdale membrane has only (real) dimension 3, hence not ideally suited to cap off the 4D-half (if it existed). This could be remedied by looking at surfaces in $\PP^4$ instead, yet we still lack a way to split the complexification by the real locus. Of course all this failure is somewhat akin to the lack of a good extension of Rohlin’s formula to surfaces as deplored upon, e.g. in Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000]. Basically, the ideas behind Rohlin and the Thom estimate above are very similar, namely to fill the “imaginary” half by a “real” membrane coming from the real locus (either the Ragsdale membrane or bounding discs for ovals). (of (\[Thom-Ragsdale:thm\])).—We paste to the (bordered) half $C_m^+$ of the dividing curve $C_m$ the Ragsdale membrane $R$ which is the orientable surface bounding $C_m(\RR)$. The resulting closed surface $F$ is orientable \[HERE IS THE MISTAKE (22.03.13)!!!\] and realizes the homology class $kH$ (of halved degree) in the group $H_2(\CC P^2, \ZZ)\approx \ZZ$. It is plain[^80] that we can smooth the “corners” arising along the “cut-and-paste-locus” to get a nearby smooth surface still denoted $F$. By Thom’s conjecture (=meanwile the Kronheimer-Mrowka theorem 1994 [@Kronheimer-Mrowka_1994]) we infer that the genus of $F$, say $f:=g(F)$, is at least as big as that of a smooth curve of the same degree, i.e., $$f\ge g(k)=\textstyle\frac{(k-1)(k-2)}{2}.$$ On the other hand we have by additivity of the characteristic $$\chi(F)=\chi (C_m^+)+\chi (R).$$ For the same reason $2\chi(C_m^+)=\chi(C_m)=2-2g(m)$, and so $$\begin{aligned} \chi&:=\chi(R)=\chi(F)-\chi (C_m^+)=(2-2f)-1+g(m)=1-2f+g(m) \cr &\le 1-(k-1)(k-2)+\textstyle\frac{(2k-1)(2k-2)}{2}\cr &=1-(k-1)(k-2)+(2k-1)(k-1) =1+(k-1)(k+1)=k^2.\end{aligned}$$ As already discussed, this has some interesting applications, e.g. to the prohibition of Hilbert’s (unnested) scheme $11$ of degree 6, and to the scheme $20$ in degree 8. (However all this can be more elementarily deduced from Rohlin’s formula.) \[21.03.13\] Yet compare Theorem \[Alsatian-scheme-Thom-strong-Petrov-Arnold:thm\] below for an example showing that Thom’s estimate is sometimes stronger than the conjunction of several powerful prohibitions of the Russian school (Petrovskii 1938, Gudkov 1969, Arnold 1971, Rohlin 1972/74). Also pleasant is the direct link of this estimate with those conjectured decades prior to Thom by Virginia Ragsdale in 1906 (cf. Sec.\[Ragsdale-conj:sec\]). As I was informed by Th. Fiedler, it seems that it is Mikhalkin who first investigated systematically the repercussion of Thom-Kronheimer-Mrowka upon Hilbert’s 16th. \[07.03.13\] When we look back at Gudkov’s Table (Fig.\[Gudkov-Table3:fig\]) we see that we get a nearly complete system of prohibition by using total reality and the Rohlin maximality conjecture (RMC), while combining it with the Thom obstruction. Remember that RMC ought to be a reliable principle whenever total reality is exhibited in some concrete fashion as in the Rohlin-Le Touzé theorem. Hence what misses is a prohibition of the scheme $\frac{10}{1}$ in degree 6. One may thus wonder if there is an avatar of Thom’s conjecture for non-orientable surfaces in $\CC P^2$, able to prohibit the sextic scheme $\frac{10}{1}$ (of Rohn). Cavalier, one could put forward something like the: Every prohibition of Hilbert’s 16th problem, is either interpretable via total reality and the allied Rohlin maximality principle to the effect that a scheme flashed by a total pencil is maximal, or is a consequence of Thom’s conjecture plus an avatar thereof including non-orientable membranes. Of course for the sextic scheme $\frac{10}{1}$, the idea would be to fill by the non-orientable membrane (residual to the Ragsdale membrane). Further our conjecture is certainly much premature unless it takes into account advanced Bézout-style prohibitions à la Fiedler-Viro (cf. Theorem \[Viro-Fiedler-prohibition:thm\]), and Petrovskii-Arnold style prohibitions (cf. Theorems \[Petrovskii’s-inequalities:thm\] and \[Strong-Petrovskii-Arnold-ineq:thm\]). Ragsdale’s conjecture (Ragsdale 1906, Petrovskii 1938, Viro 1979/80, Itenberg 1993, Thom 19XX-Kronheimer-Mrowka 1995, and still open, Fiedler) {#Ragsdale-conj:sec} ---------------------------------------------------------------------------------------------------------------------------------------------- \[18.03.13\] As I was made (personally) aware by Fiedler (cf. his 9 March 2013 letter reproduced in Sec.\[e-mail-Viro:sec\]) there ought to be some connection between Thom’s and Ragsdale’s conjecture, which is still open for $M$-curves, despite the disproofs due to Viro 1979 (=Viro 1980/80 [@Viro_1980-degree-7-8-and-Ragsdale]) and Itenberg 1993 [@Itenberg_1993-ctrex-a-Ragsdale] (cf. also Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale]). I can also remember an oral discussion with Thomas Fiedler (Geneva ca. 2011), where Thomas alluded to all the effort he invested on the Ragsdale problem (for $M$-curves). At that time (and arguably still today), I could not appreciate the full swing of this investment. This section makes no pretence of any breakthrough in the field. It is rather a humble attempt to get familiarized with the topic. Despite our incompetence, let us make some remarks. From our viewpoint of total reality much allied to Ahlfors theorem, which quite paradoxically seems more familiar to complex/conformal geometers than purely real ones (having in mind the antagonism between Riemann-Schottky-Klein-Bieberbach-Teichmüller-Ahlfors versus Harnack-Hilbert-Ragsdale-Rohn-Petrovskii-Gudkov-Arnold-Rohlin, etc.) we could expect a connection of Ragsdale’s conjecture to our paradigm of total reality, e.g. via Theorem \[total-reality-of-plane-M-curves:thm\] as a first basic step. This vague suggestion should probably not be taken too seriously. Another vague idea is to wonder if there is some connection of Ragsdale with the contraction conjectures of Itenberg-Viro, or perhaps our version thereof called CCC, cf. (\[CCC:conj\]). After those abrupt remarks, let us be more pedestrian. First what is Ragsdale’s conjecture at all about? What is known on it and what is not? Especially does it connect to 4D-dimensional topology as the whole Hilbert problem was realized to be since Arnold’s breakthrough 1971 [@Arnold_1971/72] and the deeper investigations of Rohlin (e.g., the validation of Gudkov’s hypothesis $\chi\equiv_8 k^2$). In particular how does Ragsdale connect with Thom’s conjecture which is basically a problem of embedded differential topology of smooth surfaces in the complex projective plane $\CC P^2$ (arguably the 4-manifold simplest to visualize as the configuration space of all unordered pairs grooving on the 2-sphere). As we shall see, the link Thom-Ragsdale is very clear-cut, at least for one half of the Ragsdale conjecture (cf. Lemma \[Thom-implies-one-half-of-Ragsdale:lem\]). \[$\bigstar$ Okay, but alas this is based on our erroneous estimate $\chi\le k^2$!\] First, Virginia Ragsdale, coming from the U.S. was a student of both Klein and Hilbert in Göttingen ca. 1906. Building upon a careful inspection of the features of Harnack’s and Hilbert’s constructions (of small vibratory perturbations), she posited a conjecture on the numbers $p,n$ of even resp. odd ovals of real plane algebraic curves. Although the chance of deriving any transcendental truth from such a specific mode of generation looks a priori very meagre, the conjecture in question turned out to be extremely robust requiring at least ca. 7 decades up to being disproved \[Viro, Itenberg\]. Yet some respectable vestiges remains open, and deserves further efforts. Via some naive acquaintance with Gudkov’s Table in degree 6 (Fig.\[Gudkov-Table3:fig\]) (the little that we personally have at disposal) and the allied geometry of pyramids, it seems that the Harnack and Hilbert constructions explore only the superficies of the pyramid, while the profound part of the puzzle is cracked in Gudkov’s revolution (ca. 1969–72) constructing the scheme $\frac{5}{1}5$ (which is so-to-speak the Pharaoh chamber). The Ragsdale conjecture (in modernized shape) may be stated as the estimate $\vert \chi \vert \le k^2$, hence it is perhaps not too surprising that the particular methods of Harnack and Hilbert lead to sharp estimates at least for $M$-curves. What appears historically first is likely to be the most superficial objects, hence extremalizing the functional $\vert \chi\vert$ which roughly measures the level of superficiality in the pyramid. We state now precisely Ragsdale’s original statement (compare Ragsdale 1906 [@Ragsdale_1906]). \[Ragsdale-conj:conj\] [(Ragsdale 1906, disproved for the number $n$ of odd ovals by Viro 1979, and in general by Itenberg 1993)]{}—For any curve of even degree $m=2k$, we have $$p\le \textstyle\frac{3}{2}k(k-1)+1, \qquad n\le \textstyle\frac{3}{2}k(k-1)$$ As explained in Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale] (especially p.24) this was refuted by Viro in 1979 for the number $n$ of odd ovals, and by Itenberg in general. Moreover it is explained (in ) that Petrovskii made similar conjectures, being apparently unaware of Ragsdale’s paper. In particular the so-called [*Petrovskii inequality*]{} (cf. (\[Petrovskii’s-inequalities:thm\])) is considered there as having been formerly conjectured by Ragsdale as a weak form of her conjecture. Finally it is remarked that Petrovskii himself (1938) formulated a version of Ragsdale’s conjecture (\[Ragsdale-conj:conj\]), yet more cautious by one unit than Ragsdale’s on the number $n$, so that both bounds are identic equal to $\textstyle\frac{3}{2}k(k-1)+1$. Despite the disproof (by Viro-Itenberg) of both the Ragsdale and the weaker Petrovskii conjectures, the interesting quick is that the case of $M$-curves is still open (at least in the weaker formulation of Petrovskii). Precisely [(Ragsdale’s conjecture on $M$-curves 1906, still open)]{}—For any $M$-curve of even degree $m=2k$, the Euler characteristic $\chi=p-n$ of the Ragsdale membrane is bounded by the square of the semi-degree $k$, i.e. $$\vert \chi \vert \le k^2.$$ We borrowed this from Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale p.24] (cf. also Kharlamov-Viro (undated) [@Kharlamov-Viro_XXXX-UNDATED p.15]). It should be remarked that one half of this conjecture (namely the estimate $\chi\le k^2$) follows directly (cf. Theorem \[Thom-Ragsdale:thm\]) from Thom’s conjecture proved by Kronheimer-Mrowka in 1994 [@Kronheimer-Mrowka_1994]. Curiously, this is not pointed out in the Itenberg-Viro 1996 article (presumably due to backlog reasons). Actually as noted in Theorem \[Thom-Ragsdale:thm\], the estimate $\chi\le k^2$ holds more generally for dividing curves. $\bigstar\bigstar$ [*Insertion.*]{} \[23.03.13\] This historical puzzle is now completely fixed by Fiedler’s correction of my mistake of overlooking that the Arnold surface is not necessarily orientable. Some few words are required to understand why the above conjecture $\vert \chi \vert \le k^2$ is termed Ragsdale’s conjecture. (It seems to me that Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale p.24] contains a serious misprint at this place, specifically on p.24 in the statement of the Ragsdale conjecture on $M$-curves the equivalent conditions $p\ge \frac{(k-1)(k-2)}{2}$ and $n\ge\frac{(k-1)(k-2)}{2}$ looks to me erroneous; and the same misprint appears in Kharlamov-Viro [@Kharlamov-Viro_XXXX-UNDATED p.15]) Let us clarify this as follows: \[Thom-implies-one-half-of-Ragsdale:lem\] For $M$-curves of degree $2k$, the condition $\vert \chi\vert\le k^2$ is equivalent to Petrovskii’s cautious version of the Ragsdale conjecture, i.e. $$p\le \textstyle\frac{3}{2}k(k-1)+1, \qquad n\le \textstyle\frac{3}{2}k(k-1)+1.$$ More precisely the upper estimate on $\chi$ (i.e. $\chi\le k^2$) is equivalent to the bound on $p$, while the lower estimate $-k^2\le \chi$ is equivalent to the bound on $n$. Further the first upper bound $\chi\le k^2$ follows from Thom’s bound (cf. [Theorem \[Thom-Ragsdale:thm\]]{} valid more generally for any dividing curve), while the other is perhaps still open, though one could dream reducing it to Thom too, after taking maybe an orientable cover (but looks dubious), or maybe by reducing it via differential geometry to Gauss-Bonnet and Wirtinger (as discussed below). Start from the condition $-k^2\le\chi\le k^2$. By definition $\chi=p-n$, and $M=r=p+n$. As usual Harnack’s bound is $M=g+1=\frac{(2k-1)(2k-2)}{2}+1=(2k-1)(k-1)+1=2k^2-3k+2$. Adding $\chi=p-n\le k^2$ to $p+n=M=2k^2-3k+2$ gives $$2p\le k^2+2k^2-3k+2=3k^2-3k+2=3k(k-1)+2,$$ whence Ragsdale’s bound on $p$. On the other hand, the lower estimate on $\chi$, i.e. $-k^2\le \chi=p-n$, rewritten as $k^2\ge n-p$, gives when added to $p+n=M$ the 2nd Ragsdale estimate on $n$. Again the text of Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale p.24] which reads as follows, seems not perfectly up-to-date \[SORRY MY MISTAKE!\] in view of Kronheimer-Mrowka’s validation of Thom’s conjecture: [“Which of Ragsdale’s questions are still open now? The inequalities[^81] $$p\le \textstyle\frac{3}{2}k(k-1)+1, \qquad n\le \textstyle\frac{3}{2}k(k-1)+1.$$ have been neither proved[^82] nor disproved for $M$-curves.”]{} As shown by Lemma \[Thom-implies-one-half-of-Ragsdale:lem\] (implicit in Itenberg-Viro’s article modulo the misprint), Thom’s conjecture implies one half of Ragsdale conjecture \[ALAS NOT TRUE\], namely the “positive” half concerning $p$ where it matches exactly with Petrovskii’s subsequent rediscovery of the conjecture. However the second half looks much out of reach, as it requires the estimate $-k^2\le\chi=p-n$ which seems to take care of the non-orientable (“anti-Ragsdale”) membrane (not ideally suited to Thom). First without any idea the lower bound on $\chi=\chi(B^+)$ the characteristic of the (orientable) Ragsdale membrane $B^+$, i.e. $-k^2\le \chi$, can be transmuted using $B^+\cup B^-=\RR P^2$ into $k^2\ge\chi(B^-)-1$. So one seems forced to study this non-orientable membrane $B^-$. One idea to explore is to arrange an orientable membrane via the usual trick of the double orienting cover (essentially due to Gauss, Möbius, Klein, Teichmüller 1939, etc.) but which have now to be implemented in some embedded fashion. This looks dubious as we do not know what to do along the boundary of $B^-$. At least the surface we get (granting that there is some natural way to construct an oriented Verdoppelung=double) would have tripodal singularities along the boundary. This is common in soap film experiment, yet a priori outside the tolerance permitted in Thom’s conjecture. As a completely different strategy there could be a result (dual to Thom’s) stating that for smooth surfaces the genus cannot be too big when attention is confined to smooth surfaces arising by rounding corners of the half of a dividing curve capped off by the Ragsdale membrane. Of course in general the genus of a smooth surface of prescribed (homological) degree can be made as large as we please (just attach small handles), yet perhaps the surfaces that arise by the “Ragsdale filling procedure” of Klein’s orthosymmetric half (a method truly inaugurated by Arnold 1971, and Rohlin 1974, etc.) are of a special type subsumed to an upper-bound upon the genus in terms of the degree. Recall the Wirtinger’s inequalities stating that complex projective varieties (or even Kähler manifolds) minimize the volume among differential-geometric submanifolds in a given homology class. Perhaps this combined with Gauss-Bonnet can supply the required upper-estimate upon the genus dual to Thom’s estimate, hence validating the remaining half of Ragsdale’s conjecture (as modified by Petrovskii 1938). \[19.03.13\] In fact I do not know if the lower estimate of the pinching $-k^2\le \chi\le k^2$ holds true more generally for dividing curves as do the upper estimate $\chi \le k^2$ by virtue of Thom’s (genus) bound (Theorem \[Thom-Ragsdale:thm\]) \[FALSE, cf. Itenberg-Viro’s curve on Fig.\[Itenberg:fig\]\]. It is worth first noting that the Petrovskii jargon (1938 [@Petrowsky_1938]) of $p$ and $n$ as positive and negative ovals is quite good as they contribute positively resp. negatively to $\chi$ of the Ragsdale membrane. \[Of course the sign of $\chi$ is a matter of convention that varied through the ages, but at least now we seem to all agree about its sign ($\chi(pt)=+1$).\] Then it is also useful to keep in mind the geography of the generic Gudkov pyramid (cf. Fig.\[Pyramidragsdale:fig\]) though this is a coarse simplification of the real one which is a multidimensional (non-planar) object as soon as $m\ge 8$. Now the point is that as shown by Lemma \[Thom-implies-one-half-of-Ragsdale:lem\] the Thom estimate $\chi\le k^2$ and its dual $-k^2\le\chi$ formally implies for $M$-curves the Ragsdale-Petrovskii estimates $p\le P$, and $n\le P$ respectively, where we set $P:=\frac{3}{2}k(k-1)+1$ (for Petrovskii’s bound). Diagrammatically, this amounts saying that [*Ragsdale’s zone*]{} ($p,n\le P$) arises from Thom’s vertical strip $\vert\chi\vert\le k^2$ by reflecting vertical rays at angle of 60 degrees (cf. Fig.\[Pyramidragsdale:fig\]). We get so a pentagonal diamond (the Ragsdale diamond) that was supposed to contain all algebraic schemes by virtue of the Ragsdale-Petrovskii conjecture, which alas turned out wrong by Itenberg-Viro \[Fig.\[Itenberg:fig\]\]. However on the top face of the diamond ($M$-curves) the conjecture is still robust. Further, Thom’s strip $-k^2\le \chi \le k^2$ is perhaps a container for all dividing curves \[FALSE, cf. again the Itenberg-Viro Fig.\[Itenberg:fig\]\]. This holds true on the right positive side (still Theorem \[Thom-Ragsdale:thm\]) \[FALSE!\] but the lower estimate is presently more dubious. A look on Gudkov’s table(=Fig.\[Gudkov-Table3:fig\]) shows that $-k^2\le\chi$ holds true in degree $6$ (actually for all curves regardless of being dividing), yet this low-degree case is probably atypical. -5pt0 -5pt0 By analogy with Thom’s conjecture, let us put forward the: \[anti-Thom-conj:conj\] [(Anti-Thom conjecture, due to Gabard hence probably not serious at all)]{} Any plane dividing curve of even degree $m=2k$ respects a Thom-style lower bound $$-k^2\le \chi.$$ If true this would imply the remaining half of the Ragsdale conjecture for $M$-curves (via Lemma \[Thom-implies-one-half-of-Ragsdale:lem\]). It would be interesting to know if the Viro-Itenberg method (viz. counter-examples) already disproves this naive conjecture. \[QUITE PROBABLE!\] If not, one would like to imagine a proof along the above sketched line (Gauss-Bonnet-Wirtinger), or via an oriented double cover of the non-orientable Ragsdale membrane $B^-$. Another third strategy would be to use an eversion (cf. Sec. \[Eversion:sec\]) to reduce to the case of Thom, yet this looks hazardous as it requires a large deformation (for which very little is known apart vague speculation of us that differential-geometric flows could do such jobs). At this stage it is wise to contemplate the higher Gudkov’s pyramids in degree 8 and 10 (cf. resp. Figs.\[Degree8:fig\] and \[Degree10:fig\]). In degree 8 (Fig.\[Degree8:fig\]) we see that the anti-Thom line $-k^2\le \chi$ is well adjusted to the blue-rhombs materializing Arnold’s congruence mod 4 for dividing curves. Hence there is little chance to corrupt our anti-Thom conjecture (\[anti-Thom-conj:conj\]). In contrast in the degree 10 table (Fig.\[Degree10:fig\]) there is a myriad of 7 schemes adventuring outside the anti-Thom line. Those are given by the symbols $\frac{34}{1}2$, $\frac{33}{1}1$, $\frac{32}{1}$ and $\frac{31}{1}3$, $\frac{30}{1}2$, $\frac{29}{1}1$, $\frac{28}{1}$. If any one of those schemes admits a type I(=orthosymmetric) realization our conjecture is faulty. This problem can either be approached by Harnack or Hilbert’s method of vibrations or by the Viro-Itenberg patchworking. Note that it is unlikely that those schemes (in type I) are prohibited by Rohlin’s formula. \[21.03.13\] However the first 3 listed (with $\chi=-31$) are prohibited by Petrovskii’s inequality (\[Petrovskii’s-inequalities:thm\]). For the 4 remaining schemes (with $\chi=-27$) one can expect to do naive Hilbert constructions like on Fig.13 of Gabard 2000 [@Gabard_2000], and look what happens. That requires some concentration and is differed to latter. \[21.03.13\] It may be noted that the strong Petrovskii inequality (\[Strong-Petrovskii-Arnold-ineq:thm\]) $n-p^- \le \frac{3}{2}k(k-1)=30$ specialized to the range of our diagram (Fig.\[Degree10:fig\]) involving simple symbols of the form $\frac{x}{1}y$ where $p^-=1$ (one hyperbolic oval, i.e. which ramifies in Hilbert’s tree) implies that $n\le 31$ and so Ragsdale line is corroborated. This does not kill any of the 4 schemes in candidature above. Note further that it must be a general issue that the strong Petrovskii-Arnold estimate gives Ragsdale in the “planar” range of the pyramid involving symbols $\frac{x}{1}y$. On the right-side of Fig.\[Degree10:fig\], the (other dual) strong Petrovskii-Arnold estimate $p-n^-\le \frac{3}{2}k(k-1)=30+1$ (with now $n^-=0$) also kills all schemes lying “above” Ragsdale’s line. This implies a severe crumbling in the corners of the pyramid (cf. Fig.\[Degree10:fig\]) on both the right and left side of it, yet note that a priori the scheme $\frac{31}{1}3$ (if it exists) seems to imply some mysterious asymmetry in the architecture. Further this scheme, being of type I, could be an interesting place to look for a counterexample to Rohlin’s maximality conjecture. Thom versus Rohlin ------------------ \[07.03.13\] As noticed above the Thom obstruction (\[Thom-Ragsdale:thm\]) is at least for degree 6 (and to some extend in degree 8) subsumed to Rohlin’s formula (\[Rohlin-formula:thm\]). The latter also prohibits the scheme $\frac{10}{1}$ (cf. Fig.\[Gudkov-Table3:fig\]) of degree 6 without that we have to worry about a dubious non-orientable extension of Rohlin’s formula. One can wonder if in general the information derived from Thom (\[Thom-Ragsdale:thm\]) is always subsumed to Rohlin’s formula. Let us notice the following: The scheme $20$ cannot be realized in degree $8$. Let $C_8$ be a (hypothetical) octic of type $20$. By the RKM-congruence (\[Kharlamov-Marin-cong:thm\]) or better its reformulation as (\[RKM-congruence-reformulated:thm\]) the curve has to be of type I, but then its existence is ruled out by Rohlin’s formula (\[Rohlin-formula:thm\]). \[21.03.13\] Another proof of the lemma (historically sharper) is to appeal to Petrovskii’s inequality (1933/38), cf. (\[Petrovskii’s-inequalities:thm\]). It is worth then comparing the Gudkov table in degree 8 (Fig.\[Degree8:fig\]), which shows that several obstructions are not readily interpreted via total reality and Rohlin’s allied principle of maximality (look especially at the upper-right corner of that figure). Now we turn to the question of deciding if Thom is subsumed to Rohlin (at least in the realm of Hilbert’s 16th problem). Glancing at Fig.\[Degree8:fig\] the answers seems to be yes for degree 8 (at least for schemes of the form $\frac{x}{1}y$ as those represented on that figure). If true in general this should follow from a simple combinatorial argument. If we look at the degree $m=10$ table, we find the following structure (Fig.\[Degree10:fig\]). This picture is built as usual. First one compute Harnack’s bound $M=g+1=\frac{(m-1)(m-2)}{2}+1=\frac{9\cdot 8}{2}+1=36+1=37$. So one extends the previous pyramid in degree 8, up to that level $37$. Then there is the sawtooth broken line à la Gudkov. Its upper undulations have to be adjusted at the Gudkov-Rohlin congruence $\chi\equiv k^2 \pmod 8$, here $k^2=25$. Remind that on such pictures Euler-Ragsdale’s $\chi$ may (always) be interpreted as the abscissa (“$x$-axis”, i.e. horizontal axis). So we have the $M$-schemes lying at the top of the sawtooth broken line, while in their depressions (“creux”) we have the RKM-schemes with $\chi\equiv k^2+4 \pmod 8$, that are forced being of type I. All this can be extended into the lattice of blue-rhombs where type I curves are forced to live (Arnold’s congruence mod 4). \[[*Warning.*]{}—On our Fig.\[Degree10:fig\] this lattice of blue rhombs is correct on the upper half, but need to be adjusted (mentally) on the lower part, where we just copied the pyramid in degree 8. However we thought it would be more instructive to see this lower object intact as to appreciate better the growing mode of pyramids.\] By Thom (\[Thom-Ragsdale:thm\]) we have $\chi\le k^2=25$ for curves of type I. It seems (at first sight) that several schemes permissible for Rohlin are prohibited by Thom, yet the ultimate answer will be nearly the opposite one. -3.2cm0 -5pt0 -5pt0 First the $M$-scheme $\frac{2}{1}34$ is prohibited by Thom \[DUBIOUS INFERENCE OF THOM!\] though it is not by the Gudkov-Rohlin congruence. (Incidentally this scheme is prohibited by Petrovskii (\[Petrovskii’s-inequalities:thm\]).) Is this scheme prohibited by Rohlin’s formula $2(\Pi^+-\Pi^-)=r-k^2$? Here we have two nested pairs hence $\Pi=\Pi^+ +\Pi^-=2$, and $r-k^2=37-25=12$. So $\Pi^+-\Pi^-\le \Pi=2$ and Rohlin’s formula cannot be fulfilled. So Thom brings nothing new. Then there is the scheme $\frac{1}{1}33$, here $\Pi=1$, so $\Pi^+-\Pi^-\le \Pi=1$, but $r-k^2=35- 25=10$ and Rohlin’s formula cannot hold. For the scheme $33$, Rohlin’s formula cannot hold as well (here $\Pi=0$ so $\Pi^+-\Pi^-=0$) and $r$ has to be a square. Next we have the RKM-scheme $\frac{3}{1}31$. Here $\Pi=3$ and so $\Pi^+-\Pi^-\le 3$, while $r-k^2=35-25=10$ and again Rohlin’s formula cannot be verified. Below the former we have $\frac{2}{1}30$. Here $\Pi=2$ and so $\Pi^+-\Pi^-\le 2$, while $r-k^2=33-25=8$ and Rohlin cannot be fulfilled. Below, we have $\frac{1}{1}29$, where $\Pi=1$ and so $\Pi^+-\Pi^-\le 1$, while $r-k^2=31-25=6$ and Rohlin cannot be fulfilled. Below, we have $29$ which is not realized in type I, as $r$ is not a square. Conclusion: [*all schemes prohibited by Thom are actually also prohibited by Rohlin.*]{} (at least within the range of Fig.\[Degree10:fig\]). Perhaps Rohlin even prohibits more than Thom. The next boy is the scheme $\frac{6}{1}30$. Here $\Pi=6$, and so $\Pi^+-\Pi^-\le 6$, while $r-k^2=37-25=12$ and Rohlin can by now be fulfilled. So no obstruction. All this little experiments points out to a subsumation of Thom to Rohlin (at least for schemes of the form $\frac{x}{1}y$). For schemes of the type $\frac{x}{1}y$, Thom’s inequality is subsumed to Rohlin’s formula. For a scheme of this form the total number of pairs (denoted $\Pi$) is $\Pi= x$, hence $\Pi^+-\Pi^-\le x$. By Rohlin’s formula (\[Rohlin-formula:thm\]) we infer $$2x\ge 2(\Pi^+ -\Pi^-)=r-k^2=(1+x+y)-k^2.$$ Calculating $\chi$ gives $$\chi=1-x+y\le k^2,$$ by the above estimate. Perhaps this is even true in general, but this deserves another argument. (Update the sequel, will show that the contrary is true, cf e.g. Theorem \[Alsatian-scheme-Thom-strong-Petrov-Arnold:thm\].) First the argument extends to schemes of the form $\frac{x}{1}\frac{y}{1}z$. Then $\Pi=x+y$, and so $\Pi^+-\Pi^-\le \Pi=x+y$. Then writing down Rohlin’s formula $$2(x+y)\ge 2(\Pi^+-\Pi^-)=r-k^2=(2+x+y+z)-k^2,$$ from which it is inferred that $$\chi=1-x+1-y+z=2-(x+y)+z\le k^2,$$ i.e. Thom’s estimate. And so on, it seems that the passage from Rohlin to Thom will always succeed as long as the depth is at most one. So for a counterexample we shall investigate deeper schemes. For instance in degree 8, we can look at an extension of the $3$-nest (of depth 3). For instance the $M$-scheme $(1,1,1)19=(3\times 1) 19 $ (in our satellite notation). Here $\Pi=3$, and Rohlin’s formula $2(\Pi^+ -\Pi^-)=r-k^2=22-16=6$ is verified for $\Pi^+=3$, and $\Pi^-=0$. But Thom’s estimate $\chi\le k^2=16$ is not (as $\chi=1-1+1+19=20$). So here we get an example where Thom is not subsumed to Rohlin. However our example is artificial being prohibited by the Gudkov-Rohlin congruence $20=\chi\equiv k^2=16 \pmod 8$. Yet our example makes unlikely a general subordination of Thom to Rohlin’s formula alone. A basic idea is to adjust $\chi=20$ at $\chi=16$ to make it Gudkov-Rohlin compatible. Starting from the above scheme, we may trade an outer oval for one at depth 1 in the $3$-nest. Each such trading diminishes $\chi$ by 2, and so two trades are required to adjust $\chi=16$. Doing this we get the scheme $(1,\frac{1}{1}2)17$ but alas now Thom’s inequality is verified. (It is also easy to check that Rohlin’s formula is satisfied.) So our game becomes: find a [ *French scheme*]{}, i.e. one prohibited by Thom yet not succumbing under the armada of Russian prohibitions (Gudkov, Arnold, and above all Rohlin, and its companions especially Kharlamov-Marin). Let us look at degree 10, and to a deep scheme, say extending the $4$-nest. As the Harnack bound is $M=g+1=9\cdot 4+1=37$, we look at the scheme $(1,1,1,1)33=(4\times 1)33$. Now $\chi=(1-1+1-1)+33=33\equiv k^2=25 \pmod 8$, i.e. the Gudkov-Rohlin congruence is fulfilled. However the scheme is prohibited by Thom’s estimate $\chi \le k^2$. Further Rohlin’s formula $2(\Pi^+ -\Pi^-)=r-k^2=37-25=12$, and $\Pi=\binom{4}{2}=6$ (count all pair in the $4$-nest) so that $\Pi^+=6$ and $\Pi^-=0$. So Rohlin’s formula affords no prohibition. We have proven the: \[French-scheme:thm\] [(ERRONEOUS, cf. Corrigendum right below, and for a corrected version cf. Theorem \[French-scheme-corrected:thm\] below)]{} There exists a French scheme, i.e. where Thom is not subsumed to Rohlin’s formula nor to the Gudkov-Rohlin congruence mod $8$. The scheme in question is even an $M$-scheme of degree $10$, namely $(1,1,1,1)33$. However for schemes of depth $\le 2$, Rohlin’s formula is as strong (and of course stronger) than the Thom obstruction. \[10.03.13\] [*Corrigendum*]{}.—Th. Fiedler objected as follows to the above theorem (compare his letter dated \[09.03.13\] in Sec.\[e-mail-Viro:sec\]): “The $M$-curve of degree 10 mentioned in your Thm 28.11\[=\[French-scheme:thm\]\] is in fact ruled out by Rokhlin’s formula. I think that you have mixed $\Pi^+$ with $\Pi^-$. In a positive couple the orientations are just opposite. So, four nested ovals can contribute at most $+2$ to Rokhlin’s formula.” After some hesitation, Gabard realized of course that Fiedler is perfectly right. Let me paraphrase his explanation differently. This is of course allied to the signs-law of Fig.\[Signs-law-dyad:fig\], but let us be more specific. If we consider a 4-nest and choose on it complex orientations forming positive pairs at each immediately successive nested ovals then we get Fig.\[Fiedler-correction:fig\]a. Pairs of length 2 becomes negative, while the unique pair of length 3 is positive again. (This is seen either by looking at the picture or if one like extracting an arithmetical law one finds the twisted signs-law of (\[Signs-law:lem\]) akin to usual arithmetics modulo a twisted sign, so $+\times +=-$, $+\times -=+$, $-\times +=+$, $-\times -=-$. This is best memorized by saying that “mixing the genes is good, while consanguinity is bad”!) So the contribution to $\Delta \Pi:=\Pi^{+}-\Pi^{-}$ is at most 2, and certainly never equal to $6$ (though it can be $-6$ as on Fig.\[Fiedler-correction:fig\]b). -5pt0 -5pt0 It remains now to see how Fiedler’s remark generalizes as to see if Thom’s estimate $\chi\le k^2$ is a formal consequence of Rohlin’s formula. If not then it remains to find another (more serious!) French scheme. The naive scenario would be that Rohlin always implies Thom (at least within the realm of Hilbert’s 16th). The only chance to prove this seems to involve an estimation of the corrector term in the “Rohlin-to-Arnold formula” , which we reproduce for convenience $$\begin{aligned} \label{Rohlin-to-Arnold-bis:eq} \chi=p-n=(p+n)-2n&=r-2n\cr &=[2(\Pi^+-\Pi^-)+k^2]-2n\cr &=k^2+2(\Pi^+ -\Pi^- -n).\end{aligned}$$ So setting $\Delta \Pi:=\Pi^+ -\Pi^-$ we would like to show the: [(Garidi[^83] mass conjecture.)]{}—It holds universally $\Delta \Pi \le n$. Of course the conjecture is true (at least if one believes in the Kronheimer-Mrowka validation of the Thom conjecture) \[THIS REASONING IS DUBIOUS BEING BASED ON OUR FALSE THOM ESTIMATE\], so that the true meaning of our conjecture is an independent derivation of the estimate via pure combinatorics. Let us be more precise. [A signed or [*Rohlin tree*]{} is a combinatorial object consisting of a (finite) directed set plus a distribution of signs $\pm$ on its edges such that the signs-law (of Lemma \[Signs-law:lem\]) is verified. By a directed set we mean a finite POSET such that each element as at most one superior, i.e. an element larger and minimal with this property. Recall also that the signs-law can be easily remembered by saying that consanguinity is bad, i.e. $+\times +=-$, $-\times -=-$, while mixing the genes is good $+\times -= +$ and $-\times += +$ (this exotic signs-law is the exact opposite of the usual convention).]{} Of course this concept arises naturally when taking a smooth dividing plane (algebraic) curve $C_m$ of even degree $m=2k$ and assigning to it its Hilbert tree (encoding the distribution of ovals), while decorating the edges with signs coming from the complex orientations as in Rohlin’s formula (\[Rohlin-formula:thm\]). More precisely, given an oriented real scheme (i.e. an isotopy class of embedding of a disjoint union of circles in $\RR P^2$ supplied with an orientation), we can assign to it a Rohlin tree. Conversely it is clear that any Rohlin tree arises in this fashion. To a real plane dividing curve is assigned a complex orientation (uniquely defined up to reversal of all orientations), yet this leaves invariant the concept of positive or negative pairs as defined by Rohlin. Hence to be very formal, we have first the map taking a dividing real curve to its real scheme with complex orientation, which is a “projectively” oriented real scheme (weel-defined up to reversing all orientations), which in turn defines unambiguously a Rohlin tree. Diagrammatically, $$\textrm{dividing plane} \to \textrm{oriented real schemes } \to \textrm{Rohlin trees}.$$ -23pt $$\hskip-80pt\textrm{curves ($m=2k$) \hskip0.5cm (mod reversion) }$$ So the precise meaning of the above conjecture is that any Rohlin tree (not necessarily induced by a real algebraic curve) satisfies the above estimate $\Delta \Pi \le n$. After several hours of attempting to prove this “Garidi mass conjecture”, one finds a simple counterexample as follows. To keep $n$ small (say $n=1$), we consider a tree with only one vertex at depth 1, but ramifying (violently) at depth 2 (cf Fig.\[Garidi-mass-false:fig\]a). -5pt0 -5pt0 Let us introduce the sign distribution of Fig.a, and denote by $p_2$ the number of (even) vertices at depth 2. We find by using the twisted signs-law: $$\Delta \Pi= (-1)+p_2+p_2=2p_2-1.$$ (Here the first $-1$ comes from the top edge (visible on Fig.a), the second term $p_2$ is the contribution of the $+$-signs visible on Fig.a), while the 3rd term $p_2$ comes from the $p_2$ pairs of length 2 obtained by concatenation of elementary edges. The signs-law in question (based on Fig.\[Signs-law-dyad:fig\]) is the same as the usual one modulo a twist by $-1$. So here $+\times -$ gives $+$ (the opposite of the usual sign rule!). The displayed formula is justified.) Now as soon as $p_2\ge 2$, the above $\Delta \Pi $ will be $\ge 3$, foiling thereby the mass conjecture. [*Insertion*]{} \[22.03.13\] For later reference, let us state this as a: \[Garidi-mass-conj-is-FALSE:thm\] The Garidi mass conjecture is false, and therefore the positive mass conjecture (\[positive-mass-conjecture:conj\]) is erroneous too. For a simple counterexample cf. Fig.\[Garidi-mass-false:fig\]a right above. This basic corruption aids us to detect a more serious French scheme (where Thom is not subsumed to Rohlin). As above we consider curves of degree $m=10$. Harnack’s bound is $M=g+1=\frac{9\cdot 8}{2}+1=9\cdot 4 +1=37$ (temperature of the human body). Our tree converts then to the scheme of Fig.\[Garidi-mass-false:fig\]b, where we see 2 nested ovals containing 35 unnested ones. The characteristic of the “Ragsdale membrane” is $\chi=(1-1)+35=35 \nleqslant 25=k^2$, so that the scheme is prohibited by Thom. Is this scheme (with Gudkov symbol $(1,1,35)$) prohibited by Rohlin’s formula? Remember that Rohlin’s formula implies Arnold’s congruence (cf. (\[Rohlin-implies-Arnold:lem\])), so the answer is an (indirect) yes since $\chi\equiv k^2=25\neq 35 \pmod 4$. As we have an $M$-scheme, let us even adjust to the (stronger) Gudkov congruence mod 8: $\chi\equiv k^2=25=33 \pmod 8$. So in order to diminish $\chi$ by 2 (from 35 to 33), let us trade a deep oval (at depth 2) against one at depth 1 (cf. transition from Fig.b to Fig.d). We have now $\chi=33$, and so Thom is still violated. Is this new scheme (symbol $(1,1(1,34))$) prohibited by Rohlin’s formula? We count (e.g. via Fig.e) that the total number of pair is $\Pi=2+34+34=70$ (this can be viewed as an application of the formula $\Pi=n_1+2p_2+3n_3+4p_4+ etc$, cf. proof of (\[Stalin:lemma\])). To abridge Rohlin’s (heavy “Cyrillic”) notation let us set $\pi:=\Pi^+$, and $\eta:=\Pi^-$, and Rohlin’s formula then reads $2(\pi-\eta)=r-k^2=37-25=12$. So we get the pair of equations $\pi-\eta=6$ and $\pi+\eta=\Pi=70$. Adding them gives $2\pi=76$, whence $\pi=38$, and $\eta=32$. So Rohlin’s is (formally) soluble but is there a distribution of signs compatible with the signs law? To answer this let us consider a “variable” distribution of signs like on Fig.f with $x$ many $+$ and $y$ many $-$ for the edges rooted “at depth 1”, while both edges rooted at depth 0 have signs $-$. Of course we assume $x+y=34$. Counting the number of positive pairs $\pi$ we find $\pi=x+x=2x$ (were the second $x$ term comes from the sign-law $+\times -= +$ the opposite of the usual convention!). For the number $\eta$ of negative pairs we find $\eta=2+y+y=2+2y$, where the 3rd $y$ term comes again from the exotic sign-law (“of Rohlin”). Combining with the previous paragraph, gives $x=19$ and $y=15$. All equations are then verified! Conclusion there is a distribution of signs on the tree which satisfies Rohlin’s formula, which therefore does nor prohibit the scheme under examination, i.e. $(1,1(1,34))$. The latter is therefore a French scheme. So we hope to have this time proven the: \[French-scheme-corrected:thm\] There exists a “French” $M$-scheme of degree $10$, namely that with Gudkov symbol $(1,1(1,34))$ (cf. [Fig.\[Garidi-mass-false:fig\]d]{} above), i.e. which is prohibited by Thom but not by the armada of Russian congruences (especially Gudkov’s) nor by Rohlin’s formula. [*Insertion*]{}—\[17.03.13\] Thomas Fiedler kindly reacted as follows to this statement, cf. his \[12.03.13\]-letter in Sec.\[e-mail-Viro:sec\] reproduced below for convenience (our brackets are just automatized updates of labels): “sorry, but all your $M$-schemes of degree 10 in Thm 30.14\[=\[French-scheme-corrected:thm\]\] and 30.15\[=\[French-scheme-corrected-bis:thm\]\] have $n=2$ and are ruled out simply by Petrovskis inequality. I don’t think that genus bounds give anything new for real schemes alone[^84] but they definitely do so for configurations of several real curves. Just take a look on Mikhalkin’s paper.” Though a pertinent remark (since Petrovskii is not a French guy), Fiedler’s remark does not affect the modest truth of our statement but points to the Petrovskii inequality as another sharp Russian weapon. (Shamefully, I confess to have not properly appreciated this fundamental statement prior to Fiedler’s comment.) The latter states: [(Petrovskii 1933/38 [@Petrowsky_1938])]{} \[Petrovskii’s-inequalities:thm\] For any real plane smooth curve of even degree $m=2k$ we have the (so-called) Petrovskii inequalities (which are pure jewels nearly coming out of the blue safe for having been apparently anticipated conjecturally by Miss Ragsdale in 1906) $$- \textstyle\frac{3}{2} k(k-1)\le \chi \le \textstyle\frac{3}{2} k(k-1)+1.$$ (An analogous but more complicated statement holds for curves of odd degrees.) We make just some few remarks. [*Historical substance.*]{}—Petrovskii 1938 [@Petrowsky_1938 p.191] comments that his method of proof is based on two ingredients: \(1) a formula of Jacobi-Euler 1768–70 (and also cite en passant Kronecker, which as we know is also one of the forerunner of Poincaré’s index theory via Hermite’s transmissive rôle), and, \(2) on the consideration of the deformations of lines $F(x,y)=C$ when $C$ crosses the critical values of $F(x,y)$. These last investigations being identified as analogous to those of Morse (1925 [@Morse_1925?]) on the critical points of a function. [*Neo-expressionist proofs*]{}.—Another proof of Petrovskii’s inequalities namely the “Preuve d’Arnol’d dans une présentation de A. Marin” is given in A’Campo 1979 [@A'Campo_1979 p.537–17], where the result is stated as: $$\vert 2\chi-1 \vert \le \textstyle\frac{3m^2}{4}-\frac{3m}{2}+1,$$ which (after setting $m=2k$) is readily seen to be equivalent to the above formulation. Indeed $\vert 2\chi-1 \vert \le \textstyle\frac{3(2k)^2}{4}-\frac{3(2k)}{2}+1=3k^2-3k+1=3k(k-1)+1$. Hence $3k(k-1) \le 2\chi\le 3k(k-1)+2$, and the equivalence is now obvious. [*The original statement differs slightly.*]{}—On adapting to our (fairly standard modern) notations, Petrovskii’s original result is stated as follows (cf. p.190 of Petrovskii 1938 [@Petrowsky_1938]) $$\vert p-n \vert \le \textstyle\frac{3m^2-6m}{8}+1.$$ As $m=2k$ and $\chi=p-n$, this gives indeed $$\vert \chi \vert\le 3 \textstyle\frac{(2k)^2-2(2k)}{8}+1=\textstyle\frac{3}{2}(k^2-k)+1 =\textstyle\frac{3}{2}k(k-1)+1,$$ which is essentially the announced bound modulo a discrepancy on the lower-bound by one unit. In fact we copied the stated lower bound from Rohlin 1978 and hope that there is no misprint there. Let us look at the example $k=3$ of sextics. Then $\frac{3}{2} k(k-1)=\frac{3}{2} 3\cdot 2=9$, and so $-9\le \chi $, hence even the stronger version written down by Rohlin 1978 (also in Wilson 1978, p.55) does not prohibit “Rohn’s scheme” $\frac{10}{1}$ (cf. Gudkov’s Table=Fig.\[Gudkov-Table3:fig\]). Applied to our situation $k=5$, Petrovskii’s theorem shows that $\chi \le \frac{3}{2} 5\cdot 4+1=31$ and so our scheme with $\chi=33$ is prohibited by Petrovskii. In general, for an even degree $m=2k$ we have Harnack’s bound $M=g+1=(2k-1)(k-1)+1=2k^2-3k+2\approx 2k^2$, the universal Petrovskii’s bounds $-P\le \chi\le P+1$, where $P=\frac{3}{2} k(k-1)\approx \frac{3}{2} k^2$, and finally Thom’s bound $\chi \le k^2 $ for dividing curves (only). So when $k$ is large the “Hilbert-Petrovskii-Gudkov” pyramid looks as follows (Fig.\[PyramidPetrov:fig\]), and of course Thom will asymptotically be stronger than Petrovskii (at least for dividing curves and on the right-wing of the pyramid where $\chi$ is positive). -5pt0 -5pt0 So I am not sure not adhere completely with Fiedler’s illuminating comment, because if we take a larger $m=2k$ than $10$ then it will be possible to arrange Petrovskii’s bound yet not Thom’s one, while further taking care of respecting Gudkov’s congruence and Rohlin’s formula. Let us first take $m=12$ (so $k=6$) then Petrovskii’s upper-bound is $P+1=\frac{3}{2} 6\cdot 5+1=46$, while Thom’s is the sharper $k^2=36$. Arranging Gudkov’s hypothesis $\chi\equiv k^2 \pmod 8$ permits to take $36+8=44$ which is still lower than Petrovskii’s upper-bound. Now $M=g+1=\frac{11\cdot 10}{2}+1=56$. Hence to arrange $\chi=44$, we transplant 6 ovals of the unnested configuration at depth 1 (each such move drops $\chi$ by 2 units) to get the (12)-scheme[^85] $(1,6)49$ (with $\chi=44$). By Rohlin’s formula $2(\pi-\eta)=r-k^2=56-36=20$, hence $\pi-\eta=10$. As there are no deep nesting the signs-law is negligible and we merely have $\pi+\eta=6$, so that $2\pi=16$, whence $\pi=8$ and we get an obstruction. \[17.03.13\][*Optional reading (skip if you do not want to loose the main-flow and move to $\clubsuit\clubsuit$)*]{}.—This argument extends to the following formal consequence of Rohlin (probably subsumed to Thom, yet much more elementary). \[Rohlin-consequence-for-M-curves:lem\] An $M$-curve of degree $2k$ and of type $\frac{x}{1}y$ with few nested ovals in the sense that $x<\frac{(k-1)(k-2)}{2}$ is prohibited by Rohlin’s formula. In particular there is no unnested $M$-curve provided $k\ge 3$. First recall that Harnack’s bound is $r=M=g+1= (2k-1)(k-1)+1=2k^2-3k+2$. By Rohlin’s formula $2(\pi-\eta)=r-k^2=k^2-3k+2=(k-1)(k-2)$. So $\pi-\eta=\frac{(k-1)(k-2)}{2}=:\binom{k-1}{2}$. But $\pi+\eta=x$, so that $2\pi=x+\binom{k-1}{2}$, and hence $\pi=(x+\binom{k-1}{2})/2$. Yet the equation $\pi+\eta=x$ is impossible whenever $\pi>x$, that is when $\pi=(x+\binom{k-1}{2})/2>x$, i.e. as $(x+\binom{k-1}{2})>2x$, so when $x<\binom{k-1}{2}$, which is the asserted condition. $\clubsuit\clubsuit$ \[18.03.13\] Back to our main object of the (12)-scheme $(1,6)49$, our idea is to remove this Rohlin obstruction by injecting more freedom gained by transferring some ovals at depth 2 (leaving thus $\chi$ unchanged). So starting from the (12)-scheme $(1,6)49$, whose tree is depicted as Fig.\[Fied3:fig\]a, we transplant ovals at depth 2 to get Fig.b with a certain quantity $x+y$ of ovals at depth 2. By Rohlin’s formula $\pi-\eta=10$, and from Fig.b we have $\pi+\eta=6+2(x+y)$. Adding the last equations gives $2\pi=16+2(x+y)$, whence $\pi=8+(x+y)$. The condition $\pi\le \pi+\eta$ becomes so $8+(x+y)\le 6+2(x+y)$, i.e. $x+y\ge 2$, which is necessary to solve Rohlin’s equation. -5pt0 -5pt0 Now introduce signs as on Fig.b by putting $a$ many pluses on the 5 edges at depth 1 and $b$ many minuses (so $a+b=5$), and a $+$-sign on the trunk ([*Warning.*]{}—This choice is fatally bad as we shall see, and a minus sign is more fruitful as we shall experiment soon). Applying the signs-law (cf. Fig.\[Signs-law-dyad:fig\]) to Fig.b we find by sorting out contribution according to their edge-length (which appears underbraced) $$\pi-\eta=\underbrace{a-b+1+x-y}_1+\underbrace{(-x+y)}_2=a-b+1.$$ But as $\pi-\eta=10$, we deduce the system $a-b=9$, $a+b=5$, whence $2a=14$ and $a=7$ which is incompatible with $a+b=5$. (Then we repeated such calculation in higher degrees $14, 16$ finding always the same obstruction, though for $m=16$ there are even two possible values of $\chi$ permissible under Gudkov and Petrovskii). However if we take a $-$-sign on the trunk (Fig.c), then the signs-law gives $$\pi-\eta=\underbrace{a-b-1+x-y}_1+\underbrace{(x-y)}_2= a-b-1+2(x-y).$$ As by Rohlin we still have $\pi-\eta=10$, this gives the system $a-b=11-2(x-y)$, $a+b=5$, whence $2a=16-2(x-y)$, i.e. $a=8-(x-y)$. Let us fix $x-y=3$, so that $a=5$, $b=0$. Since $x+y\ge 2$ (cf. necessary condition discussed few lines above), we may choose $x=3$ and $y=0$. It is worth at his stage checking that the resulting signs distribution indeed solves Rohlin’s equation, and we have proven: \[Alsatian-schemes:thm\] There exists an Alsatian scheme, i.e. a French scheme which furthermore respects Petrovskii’s inequalities. More precisely, there is an $M$-scheme of degree $12$ for instance $(1, 5(1,3))46$ (cf. [Figs.\[Fied3:fig\]d,e]{}) which respects both the Petrovskii bound, the Gudkov congruence and Rohlin’s formula, yet which is prohibited by Thom’s bound. [*Insertion*]{}.—\[20.03.13\] Thomas Fiedler was kind enough to object (once more) to this result as follows (cf. his message dated 19 March 2013 in Sec.\[e-mail-Viro:sec\], yet reproduced here for convenience): “sorry again, but your curve has $p=50$ and is ruled out by Arnold’s inequality : $p\le 3/2 k(k-1) + 1 + n_-$[^86], which is $47$ in this case. In fact Arnold’s inequalities are by fare the strongest result in the whole field.” We will try to react to this objection right below the proof, cf. $\bigstar\bigstar$ below. The assertion is clear by our search, but as it is easy to make mistakes, let us do an ad hoc self-contained verification. The scheme in question is depicted on Fig.\[Fied3:fig\]e. It has $\chi=(1-6+3)+46=44$. Petrovskii’s inequalities says $-P\le\chi\le P+1$, where $P:=\frac{3}{2}k(k-1)=\frac{3}{2}6\cdot 5=45$, and this is satisfied by our scheme. Gudkov’s congruence $\chi\equiv_8 k^2=6^2=36$ is also verified. Finally Rohlin’s formula is fulfilled when like on Fig.\[Fied3:fig\]d all signs are positive safe that on the trunk at depth 1 which has further ramifications at depth 2. (The depth of an edge is that of the unique vertex below it.) In that case the signs-law gives $$\pi-\eta=\underbrace{5-1+3}_1+\underbrace{3}_2,$$ where contributions are underbraced along the length of the pairs. Hence $\pi-\eta=10$ in accordance with Rohlin’s formula $2(\pi-\eta)=r-k^2=56-6^2=20$. However the scheme in question is prohibited by Thom’s estimate $\chi\le k^2=36$. It should be easy to extend the result to other schemes but it looks artificial to strive toward maximum generality as our purpose was merely to find an example where Thom affords valuable information. $\bigstar\bigstar$ [*Trying to fix Fiedler’s new objection based on Arnold’s strong Petrovskii inequalities*]{}.—The result mentioned by Fiedler is the following sometimes called the [*strong Petrovskii inequalities*]{}. Those are really due to Arnold 1971 [@Arnold_1971/72], and sharper than Petrovskii’s original inequalities of 1933/38. Apparently (cf. Rohlin 1978 [@Rohlin_1978], p.87, footnote), Arnold’s original statement contained further unnecessary restrictions that were relaxed in Rohlin 1974 [@Rohlin_1974/75]. The final shape of the result is as follows: \[Strong-Petrovskii-Arnold-ineq:thm\] [(Strong Petrovskii inequalities, aka Arnold inequalities).—(Arnold 1971, Rohlin 1974)]{}.—For any curve of even degree $m=2k$, $$n-p^- \le \textstyle\frac{3}{2} k(k-1), \qquad p-n^- \le \textstyle\frac{3}{2} k(k-1)+1,$$ where $p^-, n^-$ are the number of positive=even resp. negative=odd ovals which are hyperbolic (cf. [Definition \[hyperbol-ovals:def\]]{} right below). \[hyperbol-ovals:def\] $\bullet$ An oval of a plane real algebraic curve (or a scheme=distribution of ovals) is [*elliptic*]{}, [*parabolic*]{} or [*hyperbolic*]{}[^87] depending on whether its poros[^88] (cf. below) has positive, zero or negative Euler characteristic[^89]. $\bullet$ The [*poros*]{} of an oval of a plane curve $C_m(\RR)$ is the inside of the oval minus the insides of all ovals immediately nested in the given one (equivalently remove the insides of all subordinated ovals). An oval is hyperbolic iff the Hilbert tree of the scheme ramifies at the corresponding vertex. This basic remark is the key to the little problem suggested by Fiedler’s objection. Now our Alsatian question is again: is there a scheme where Thom is stronger than the conjunction of strong-Petrovskki=Arnold, Gudkov’s hypothesis and Rohlin’s formula. Our basic algorithm to do this is always same: \(1) Start from the unnested configuration of $M$-ovals. \(2) Then adjust $\chi$ to $k^2$ as to verify Gudkov’s hypothesis. (This can be done by transplanting outer ovals at depth 1, dragging them say inside a fixed oval.) \(3) Then Thom’s estimate $\chi\le k^2$ is verified, but corrupt it by incrementing $\chi$ by 8. Do this as many times as Petrovskii’s bound $\chi\le \frac{3}{2}k(k-1)+1$ permits. \(4) Next without changing $\chi$ transplants outer ovals at depth 2 (as many as you want), while introducing a branched structure making Rohlin’s equation soluble. (This being essentially inspired by our counter-example to the Garidi mass conjecture discussed above.) Let us be more explicit. Suppose $k=6$ (so $m=2k=12$). Harnack’s bound is $M=g+1=\frac{11\cdot 10}{2}+1=55+1=56$. Adjust to $\chi=k^2=36$ (Gudkov) and increment by 8 to get $44,52$. Petrovskii 1938 says $\chi \le \frac{3}{2}k(k-1)+1=3\cdot 3\cdot 5+1=46$. So we consider $\chi=44$. Hence we transplant from the unnested scheme $M=56$, precisely $(M-\chi)/2=(56-44)/2=6$ ovals at depth 1 to get adjusted at $\chi=44$. This gives the scheme on Fig.\[Fied4:fig\]a with symbol $(1,6)49$. This scheme is prohibited by Rohlin’s formula $2(\pi-\eta)=r-k^2=56-36=20$, hence $\pi-\eta=10$, while $\pi+\eta=6$. This is of course soluble as $\pi=8$ and $\eta=-2$ (ruled out as $\eta\ge 0$ is a cardinal!). Variant: imagine signs on the edges of the tree (of Fig.a) we can have at most 6 pluses and so $\pi-\eta\le 6$ hence cannot be $10$. To arrange Rohlin the idea is to transplant ovals at depth 2 creating thereby more freedom to solve Rohlin’s equation. -5pt0 -5pt0 The extra challenge is to take care of the (strong) Petrovskii-Arnold estimate. Note first that our configuration $(1,6)49$ has $p=50$ and this will not change under transfers at depth 2. Hence to respect Arnold’s estimate $$p\le \textstyle\frac{3}{2}k(k-1)+1+n^-=46+n^-,$$ it suffices to arrange $n^-=4$. Remember that $n^-$ counts the number of hyperbolic negative(=odd) ovals, so we just have to transplant ovals at 4 different places of Fig.a to get a tree like Fig.b with at least two extra branches growing at each 4 places. To ensure hyperbolicity it is sufficient to have branchings of “order 2”, so we look to Fig.c where the 3 first branches ramifies by 2, while the fourth by a magnitude $x+y$ (yet undetermined) safe for $x+y\ge 2$. On the left of the tree (still Fig.c), we put $+$-signs at $a$ many places (w.l.o.g. on the “left” though this has no intrinsic meaning here), and $-$-signs at $b$ many places. Hence $a+b=2$. Likewise on the right of the tree we introduce $x$ many $+$ resp. $y$ many $-$ as depicted on Fig.c. On the “center” of the tree where we have 3 branches like “Y”-letters inverted, we plug everywhere $-$-signs as those depressive guys are simplest to calculate by the signs-law ($-\times -=-$). This rigidification looks a reasonable Ansatz for there should be already enough free parameters available with $x,y$ and $a,b$ (subjected to $a+b=2$). Once the combinatorics of the tree is fixed and the signs-distribution too (modulo the free parameters) we can compute the Rohlin mass $\pi-\eta$ according to the signs-law. First note that each 3 central subtrees (“Y-shaped”) contributes for 5 pairs (3 visible, plus 2 concatenation) each being negatively charged. Hence the contribution of each such subtree is $-5$. Globally on the whole tree, we find therefore (upon remembering the signs-law (\[Signs-law:lem\]) saying that mixing the genes is good so $+\times -=+$, while consanguinity is bad, e.g. $-\times -=-$) $$\pi-\eta=a-b-15-1+2x-2y=a-b-16+2(x-y).$$ By Rohlin’s formula $\pi-\eta=10$, and we get the system $a-b=26-2(x-y)$, $a+b=2$. Hence $2a=28-2(x-y)$, i.e. $a=14-(x-y)$. Choose (freewill vs. predestination!) $x-y=12$, so that $a=2$, $b=0$, and choose again $x=12$, and $y=0$ as a special solution. This proves the: \[Alsatian-scheme-Thom-strong-Petrov-Arnold:thm\] There exists an Alsatian $M$-scheme where Thom is stronger than the conjunction of (strong) Petrovskii-Arnold 1971, Gudkov hypothesis 1969–72 (proved by Rohlin-Marin), and Rohlin’s formula. Specifically, there is such a scheme in degree $12$, namely the one allied to [Fig.\[Fied4:fig\]c]{} for $x+y=12$, whose scheme is depicted on [Fig.d]{} called “René Thom sur son $31$” since the nested portion of the tree involves (counting along increasing depths) $p_0+n_1+p_2=1+6+18=25$ ovals so that it remains left $56-25=31$ outer ovals. The $M$-scheme in question has Gudkov symbol $(1,2(1,2)(1,2)(1,2)(1,12))31$. Let us do again an ad hoc verification. From the scheme (Fig.c with $x+y=12$ or Fig.d) we have $\chi=1-6+18+31=44$. So Gudkov $\chi\equiv_8 k^2$ is happy. Petrovskii 1938, i.e. $\chi \le \frac{3}{2}k(k-1)+1=45+1=46$ is also satisfied. Now the strong version of Petrovskii due to Arnold, reads $$p-n^-\le \textstyle\frac{3}{2}k(k-1)+1=46.$$ But our scheme has (cf. Fig.c or d) altogether $r^-=5$ hyperbolic ovals (i.e. containing immediately at least 2 other ovals in their insides), yet only 4 of them are at odd depth 1, so $n^-=4$. Though equivalent, working with the tree (as opposed to the scheme) looks more convenient to see this (at least when one is tired). On the other hand our scheme has either by its construction (cf. Fig.a) or by counting $p=50$, since $p=p_0+p_2=(1+31)+(6+12)=32+18=50$. Hence the Petrovskii-Arnold estimate is verified. Note also that the other strong Petrovskii inequality $n-p^-\le \frac{3}{2}k(k-1)=45$ is verified, as $n=6$ (either from Fig.c or from $p+n=r=M=56$, where $r$ is as usual in our notation the number of “reellen Züge”, denoted $l$ in Russia). Although not needed it may be noted that $p^-=1$ either via Fig.c or via the relation $p^- + n^-=r^-$ (splitting hyperbolic ovals according to their parities). Finally Rohlin’s formula is verified for the signs-distribution of Fig.c where $x=12$, $y=0$ and $a=2$. Indeed the signs-law gives $\pi-\eta=2-3\cdot 5+(-1)+12+12=2-16+24=10$, in accordance with Rohlin’s formula $2(\pi-\eta)=r-k^2=56-36=20$. The verification is complete. Old material (to skip or reorganize) ------------------------------------ [*Sequel of my text (prior to Fiedler’s objection(s) via Petrovskii, and then via Arnold)*]{}.—One can paraphrase the statement (\[French-scheme-corrected:thm\]) by saying that there is a complex scheme (i.e. with orientation) which satisfies Rohlin’s formula, but which is not realized algebraically (being ruled out by Thom’s $\chi\le k^2$). Our example is of degree 10, and is simply the complex scheme associated to the Rohlin tree of Fig.\[Garidi-mass-false:fig\]f for $(x,y)=(19,15)$, hence is representable as the complex scheme of Fig.\[Garidi-mass-false:fig\]g. In fact such schemes already exist in degree 6 (cf. optionally Theorem \[no-chance-to-reduce-Gudkov-to-Rohlin:thm\]), where it is just a matter of solving Rohlin’s equation for the two $M$-schemes of degree 6 which are prohibited by Gudkov’s hypothesis. Of course the example proposed probably belongs to a larger list of such French schemes. More about this soon. On the other hand it could be nice to know if there is a French scheme in degree 8 already. [*Degree $m=8$*]{}.—Then the Harnack bound is $M=g+1=\frac{7\cdot 6}{2}+1=22$. Applying the same method, we start with the $M$-scheme $(1,1,20)$ with $\chi=20$. This has to be adjusted to the Gudkov hypothesis $\chi\equiv_8 k^2=16$, but then Thom’s inequality $\chi\le k^2$ is satisfied, except if we could move up to $\chi=24$ but this violates the basic estimate $\chi\le M$. \[Proof: $\chi=p-n\le p+n=r\le M$ by Harnack.\] So it seems that there is no French in degree 8, but a more systematic study is required. As a loose evidence for the absence of French scheme in degree 8, we note that the RKM-congruence for $(M-2)$-schemes $\chi\equiv k^2+4 \pmod 8$ (ensuring type I) forces under Thom’s inequality ($16\le \chi \le M=22$ and Harnack’s bound) to have $\chi=20$. But then $p-n=20$ and $p+n=r=20$, so that $2p=40$, hence $p=20$ and $n=0$. So our scheme is forced to be unnested and is $20$, which is prohibited by Rohlin’s formula. [*More French schemes in degree $10$*]{}.—By the above method we now proceed to find more French schemes. The idea is merely that starting from the scheme on Fig.d we may move innermost ovals at depth 2 outside at depth $0$ without changing $\chi=33$ (as forced by the Gudkov hypothesis). So we consider a scheme whose tree is like Fig.\[Fiedler3:fig\]a where there are $z$ outer ovals which are empty. We introduce $x$ and $y$ many free signs plus and minus resp. as on Fig.\[Fiedler3:fig\]a, with both top signs negative. To get an $M$-scheme we impose $x+y+z=34$. As before we seek to solve the Rohlin’s equation. Recall that we abridge $\pi:=\Pi^+$, $\eta:=\Pi^-$. By Rohlin’s formula $2(\pi-\eta)=r-k^2=37-25=12$ so $\pi-\eta=6$. But on the other hand by the signs-law, we have (by looking at Fig.a) $$\pi=2x \qquad \textrm{and} \qquad \eta=2+2y.$$ If $z$ is given (a priori in the range $0\le z \le 34$, but we shall soon see that some more constraint are required), we solve in $x,y$ as to satisfy Rohlin’s formula. This gives $\pi-\eta=6$ and $\pi+\eta=2(1+x+y)$, so adding $2\pi=8+2(x+y)$, hence $\pi=4+x+y=38-z$. So $\eta=32-z$. Finally we find $x=\frac{\pi}{2}=19-\frac{z}{2}$. This requires so the assumption $z$ even ($\bigstar$!!!), which looks anomalous but more about this soon. And finally $y=\frac{\eta-2}{2}=\frac{30-z}{2}$, so that we must assume $z\le 30$. ($\bigstar$ HYPOTHESIS to add!) Now when $z$ is odd we proceed similarly, but the trick is to change one of the top sign as on Fig.b into a plus. We still have $x+y+z=34$, but now by the signs-law applied to the new diagram (Fig.b): $$\pi=1+2x \qquad \textrm{and} \qquad \eta=1+2y.$$ By Rohlin’s formula we still have $\pi-\eta=6$, and now $\pi+\eta=2(1+x+y)$ (actually like above!) and so repeating the above $\pi=38-z$, $\eta=32-z$. Solving gives $x=\frac{\pi-1}{2}=\frac{37-z}{2}$ and $y=\frac{\eta-1}{2}=\frac{31-z}{2}$. So we assume $z\le 31$ ($\bigstar$ Hypothesis!). -5pt0 -5pt0 All this should prove the admittedly insignificant following result: \[French-scheme-corrected-bis:thm\] For any integer $0\le z \le 31$, the $M$-scheme $(1,1(1,34-z))z$ (compare Fig.\[Fiedler3:fig\]c) is a French scheme, i.e. it is prohibited by Thom but not by Gudkov’s hypothesis nor by Rohlin’s formula. From the proof $z=31$ is sharp with this property, but it is perhaps tranquilizing to check this more experimentally. Then the tree reduces to Fig.d, and the total number of pair is $\Pi=4+2=6$, but as $\pi-\eta=6$ (by Rohlin’s formula) we are forced to have only positive pairs, but this is impossible by the signs-law (since in Rohlin’s arithmetics $+$ times $+$ is minus!) What to do next? A naive game would be to classify all French scheme in degree 10. Another more serious problem would be to detect some universal rule as to understand better the prohibitions given by Rohlin’s formula, or the lack thereof when the Hilbert tree can be given a sign distribution so that Rohlin’s formula is satisfied. All this looks a bit unappealing combinatorics, yet the lack of conceptualization in our above account surely ask for a better understanding of the Rohlin tree. One would like to understand all Rohlin trees of dividing curves. Any such must satisfy the Rohlin formula, but as we saw this is not the sole obstruction (as sometimes Thom imposes additional constraints). As a more specific goal we could try to find a French $(M-2)$-scheme in degree 10. As noted earlier this is impossible to do on the “planar” face of the Gudkov pyramid (cf. our Fig.\[Degree10:fig\]). There Rohlin’s formula is always as strong as Thom. (This follows also from the truth of the Garidi mass principle for such simple schemes. We leave as a loose-end exercise to exhibit classes of schemes for which the mass conjecture holds true, albeit disproved in general for “batônnet” like schemes, cf. Fig.\[Garidi-mass-false:fig\].) We now proceed to find $(M-2)$-schemes of degree 10 where Thom is stronger than Rohlin’s formula. The method is similar as above, but we repeat the detail by unifying somewhat the proof. The batônnet structure of schemes violating the mass conjecture suggests looking at the scheme $(1,1,33)$ (cf. Fig.\[Fiedler4:fig\]a). Here $\chi=33$, but in order to apply Thom we have to ensure type I, and the best known recipe to do this is to adjust to the RKM-congruence $\chi\equiv k^2+4=25+4=29 \pmod 8$. So as $\chi=p-n\le p+n=r=35$, we cannot move up to 37, but instead lower down $\chi=33$ to $29$. This is achieved by delocalizing two ovals at depth 2 toward ovals at depth 1, and we get the scheme $(1,2(1,31))$ (cf. Fig.\[Fiedler4:fig\]b). The scheme $(1,2(1,31))$ and more generally its companions $(1,2(1,31-z))z$ (cf. Fig.\[Fiedler4:fig\]c) are by RKM of type I, but prohibited by Thom $\chi\le k^2=25$. However, provided $z\le 29$, all these schemes are not prohibited by Rohlin’s formula. As above we prove that there is a distribution of signs on the Hilbert tree of the scheme compatible with the signs-law and with Rohlin’s formula. For this we consider the diagram of Fig.\[Fiedler4:fig\]d, where we have free parameters $x,y$ counting the number of positive resp. negative edges at depth 2. We introduce also $\epsilon, \delta$ counting positive resp. negative signs on the only 2 available edges at depth 1 (lacking prolongation). So $\epsilon+\delta=2$. The edge at depth 1 prolonging to depth 2 is given the sign $-1$. Further we have $z$ isolated vertices at depth 0 corresponding to the outer empty ovals of the scheme of Fig.c. We have thus $x+y+z=31$. . -5pt0 -5pt0 By Rohlin’s formula $2(\pi-\eta)=r-k^2=35-25=10$, we find $\pi-\eta=5$. By applying the signs-law to Fig.d, we find $$\pi=2x+\epsilon \qquad \textrm{and} \qquad \eta=2y+\delta+1.$$ So $\pi+\eta=2(x+y)+3$, and thus $2\pi=8+2(x+y)$, whence $\pi=4+(x+y)=35-z$ and $\eta=\pi-5=30-z$. So we solve $$x=\frac{\pi-\epsilon}{2}=\frac{35-z-\epsilon}{2} \quad \textrm{and} \quad y=\frac{\eta-\delta-1}{2}=\frac{29-z-\delta}{2}.$$ It remains just to ensure integrality by choosing appropriately the free parameters. Specifically, $\bullet$ if $z$ is odd, then for $x$ to be integral choose $\epsilon$ even (and then $\delta$ is even and so $y$ as well, but sorry this parenthetical stuff is automatic!); $\bullet$ if $z$ is even, choose $\epsilon$ odd. Note that $z\le 29$ (to ensure $y\ge 0$), while for $z=29$ the system of equations is soluble for $\delta=0$, $y=0$ and $\epsilon=2$, whence $x=\frac{35-29-2}{2}=2$. (Optionally, one can check diagrammatically that Rohlin’s formula is verified on this extremal example, cf. Fig.e. where we have $\pi=4+2=6$ and $\eta=1$, so $\pi-\eta=5$ as it should by Rohlin.) What is the moral of all these messy calculations with this exotic signs-law of Rohlin? Is there some respectable way to extract a general result. A vague moral is that it is always quite boring to apply Rohlin’s formula and it is quite easy to make mistakes, as was pointed out by Fiedler. Perhaps some higher intelligence than me is able to discern some order in this chaos of Rohlin trees and proves valuable corollaries of Rohlin’s formula. However it is also clear that the latter has some limitation in failing to prohibit schemes ruled out by Thom. In the same spirit we wondered (earlier in this text, Sec.\[Gudkov-hyp-via-Rohlin’s-formula:sec\]) if Rohlin’s formula could imply the Gudkov hypothesis. Perhaps a variant of the method used right above can detect a counterexample, i.e. a scheme prohibited by Gudkov but not by Rohlin. Actually we have candidates in degree 6 already, namely the schemes $\frac{7}{1}3$ and its mirror $\frac{3}{1}7$. Taking the first its diagram-tree is like on Fig.\[Fiedler4:fig\]f, where we have $x$ many positive edges and $y$ many negative ones. Of course $x+y=7$. Since here we have no adjacent edges to compose, and we simply have $\pi=x$ and $\pi=y$ and Rohlin’s equation $2(\pi-\eta)=r-k^2=11-9=2$ is trivial to solve. Indeed, $x-y=1$ and $x+y=7$ gives $2x=8$, whence $x=4$ and $y=3$. This proves the: \[no-chance-to-reduce-Gudkov-to-Rohlin:thm\] There is no chance to reduce Gudkov’s hypothesis to Rohlin’s formula, unless one is able to put more stringent restriction on the complex orientations (via geometric procedures). $\bigstar$—[*Old stuff, pre-Fiedler’s correction, hence to be read with discernment.*]{}—Several questions arise: 1.—On working out more carefully the combinatorics, try to understand if a French scheme already exists in degree 8. 2.—It seems that as the degree increases French schemes will be more and more frequent and so the rôle of Thom increases more and more and becomes a valuable complement to the Russian congruences, and Rohlin’s formula. 3.—Glancing at the Gudkov table in degree 10 (Fig.\[Degree10:fig\]), we see that the famous Gudkov broken line is subjected to a severe deformation. More precisely, the $M$-schemes $\frac{2}{1}34$ is prohibited by either Rohlin’s formula or by Thom \[or even by Petrovskii’s inequality (\[Petrovskii’s-inequalities:thm\])\], and so is the $(M-2)$-scheme $\frac{3}{1}31$ for the same reasons plus (the full punch of) the RKM-congruence (\[RKM-congruence-reformulated:thm\]). The diagrammatic consequence is a distortion of the Gudkov front-line which is penetrated from above to become the lilac line on Fig.\[Degree10:fig\]. Hence directly below $\frac{3}{1}31$ we have 2 schemes (whose symbols are not even worth writing down), and which according to the diagrammatic of Fig.\[Degree10:fig\] are not maximal schemes. In contrast below the scheme $\frac{2}{1}34$ we have the two schemes $\frac{2}{1}33=:S_1$ and $\frac{1}{1}34=:S_2$. Those guys are interesting birds because they are maximal (at least in the planar model of the Hilbert-Gudkov pyramid, as follows from Rohlin’s formula), yet they are certainly not of type I (by Klein’s congruence $r\equiv_2 g+1$ of 1876). So there is some little opportunity here to corrupt Rohlin’s maximality conjecture (RMC). Of course as all what we are using here was well-known to Rohlin, our expectation will certainly quickly turn to disillusion. [*Insertion.*]{} \[19.03.13\]—Apart from the fact that (as discussed in the sequel) it is hard to prohibit all extensions of any one of those schemes $S_1,S_2$, a sharper look at the Russian architecture of Fig.\[Degree10:fig\] shows that both those schemes are simply ruled out by Petrovskii’s inequality (\[Petrovskii’s-inequalities:thm\]) telling us that $\chi\le \frac{3}{2}k(k-1)+1=\frac{3}{2}5(5-1)+1=3\cdot 5\cdot 2+1=31$. A first question is whether those two $(M-1)$-schemes $S_1,S_2$ are realized[^90]. If so, then it remains to check that they are maximal. So we have to list all their possible enlargements (extensions). Let us do this for $S_2=\frac{1}{1}34$ (which looks more appealing as it is “less nested”), we have (up to isotopy) 4 possible extensions depending upon the additional oval is added: \(1) outside the nest, which option leads to $\frac{1}{1}35$, which is prohibited by the Gudkov-Rohlin congruence; \(2) inside the nest at depth 1, which option leads to $\frac{2}{1}34$, which is prohibited by Rohlin’s formula (or Thom); \(3) inside the nest at depth 2, which option leads to $(1,1,1)34$. Then $\chi=(1-1+1)+34=35\neq k^2=25 \pmod 8 $ so that the scheme is prohibited by the Gudkov-Rohlin congruence, while one can even note that its forerunner namely the Arnold congruence mod 4 suffices; \(4) inside an outer oval, which option leads to $\frac{1}{1}\frac{1}{1}33$. Then $\chi=(1-1)+(1-1)+33=k^2=25\pmod 8$ so that Gudkov-Rohlin is not violated. However Thom’s inequation is violated, and so this scheme is not realized. As to Rohlin’s formula we have $\Pi=2$, and so $2(\Pi^+-\Pi^-)=r-k^2=37-25=12$ cannot be verified as $(\Pi^+-\Pi^-)\le(\Pi^++\Pi^-)=\Pi$. Intuitively we see that when $\chi$ is large Rohlin’s formula forces some nesting. At this stage we have clearly explored all possible extensions of $S_2$ and prohibited all of them via the classical congruence (Gudkov-Arnold-Rohlin) or the Rohlin formula. Yet even that is wrong as the additional oval needs not to be a small one injected as above, but it can also surround other ovals. Hence we have at least the following two families of schemes enlarging $S_2$: \(5) $(1,\frac{1}{1}x)y$ where $x+y=34$, and \(6) $\frac{1}{1} \frac{x}{1} y $, where $x+y=34$. This can of course be much diminished by the Gudkov-Rohlin congruence, as follows. For (5) and $x=0$, we have the primitive scheme $(1,1,1)34$ with $\chi=(1-1+1)+34=35$ which 2 unit above $33=25 \pmod 8$. Hence by trading outer “$y$” ovals against “$x$” ovals at depth 1 we decrease $\chi$ by 2, so that the first scheme with correct $\chi$ is $(1,\frac{1}{1}1)33$, and then the list extends by using 4-fold periodicity as $$(1,\frac{1}{1}1)33, \quad, (1,\frac{1}{1}5)29, etc.$$ with resp. $\chi=33, 25, etc.$ (descent by 8). Hence the first is still prohibited by Thom, yet the subsequent schemes are unlikely to be prohibited by Thom nor by Rohlin’s formula. At this stage our naive project to corrupt Rohlin’s maximality conjecture breaks down. For instance the second listed Thom compatible scheme $(1,\frac{1}{1}5)29$ has $\Pi=\binom{3}{2}+5=8$ and so Rohlin’s formula $2(\Pi^+-\Pi^-)=r-k^2=37-25=12$ is soluble for the pair $\Pi^{\pm}=(7,1)$. At this stage the only vestiges of expectance to foil RMC via our naive strategy would be that all those extended schemes are prohibited by a degree-10-extension of the Fiedler-Viro theorem (cf. Theorem \[Viro-Fiedler-prohibition:thm\] in degree 8), but that looks a dubious expectation. Of course it is much more likely that either Harnack, Hilbert or Viro’s method of construction realize one of those schemes, and so that our scheme $S_2$ (and likewise $S_1$) are not maximal. Some intuition behind Hilbert, Thom, Rohlin? -------------------------------------------- \[12.03.13\] The theory of abstract real algebraic curves is essentially a closed chapter of mathematics. More precisely since Riemann 1857 [@Riemann_1857] paper on Abelian integrals, we have a clear-cut vision of all complex curves and their moduli. Avatars thereof in the bordered setting or even nonorientable realm were worked out by Klein 1882 [@Klein_1882] (upon the heritage of Gauss, Möbius, Listing the discoverers of “il nastro di Möbius”). Teichmüller 1939 [@Teichmueller_1939] put the nearly final touch by approaching the moduli problem via quasiconformal maps along the philosophy developed by Grötzsch 1928, Lavrentieff 1929, Ahlfors ca. 1930–35. Notwithstanding our thesis is that from the viewpoint of total reality (or what amounts to the same branched covers of the disc), some quantitative aspects have not yet attained their ultimate perfection and sharpness, though it may be only a matter of assimilating the heritage of previous generations. This is discussed at length in the Introduction of this text, but briefly we may recall that in the closed case the ultimate perfection regarding Riemann surfaces expressed as branched cover of the line(=Riemann sphere) is achieved in Riemann 1857, Brill-Noether 1874 [@Brill-Noether_1874], and especially Meis 1960 [@Meis_1960] (which I had never the occasion to consult). When it comes to bordered surfaces (equivalently orthosymmetric real curves), the first steps belongs to Riemann 1857 (Nachlass) [@Riemann_1857], Schottky 1875–1877, Bieberbach 1925, Grunsky 1937, Ahlfors 1950, etc., up to perhaps Gabard 2004 [@Gabard_2004], 2006 [@Gabard_2006], whose result still deserves better introspection, but whose sharpness is adhered upon (perhaps too hastily) in several works (e.g. Fraser-Schoen 2011 [@Fraser-Schoen_2011], Coppens 2011 [@Coppens_2011]. Actually Coppens’ result although adhering to the truth of Gabard’s is so logically independent that its truth may not be jeopardized by a disproof of Gabard’s bound $\gamma\le r+p$. Next we may move in the Plato cavern of plane curves and contemplate the so-called [*Hilbert’s problem*]{} on the topology of real plane algebraic curves. Here most of the difficulty is allied to a certain combinatorial mess arising from the nested structures of ovals and their distribution. The whole point is to look at a certain hierarchical structure (POSET) arising from the inclusion between the insides of ovals. We call it the [*Hilbert tree*]{}. As early as 1891, Hilbert was the first to formulate the intuition that this Hilbert tree has a certain verticality, in the sense that an algebraic curve cannot reduce to an unnested collection of ovals. He formulates this for $M$-curves of degree 6, and of course the assertion holds only for maximal curves (cf. the Gudkov table=Fig.\[Gudkov-Table3:fig\] or for a specific example Fig.\[KleinRo-sextic:fig\]). The Hilbert tree of such an unnested scheme (Gudkov’s symbol $M$, where $M=g+1$ is Harnack’s bound) just reduce to a collection of vertices at depth 0, without nesting (hence without edges). At some stage Thom had the following intuition. [(Thom conjecture, Kronheimer-Mrowka theorem 1994, and independently Morgan-Szabó-Taubes 1995/96 [@Morgan-Szabo-Taubes_1996])]{}.—The genus of a smooth embedded oriented surface in $\CC P^2$ is at least as large than that of “the” algebraic smooth curve realizing the same homology class, alias the degree. This prompts a wide extension of Hilbert’s verticality principle \[or nesting Ansatz\] that the Hilbert tree cannot be to flat. This is alas a bit like in human feodal systems where a certain concentration of power, slavery and subjection is observed. Precisely this is given by Theorem \[Thom-Ragsdale:thm\] stating that any dividing curve of degree $m=2k$ satisfies $\chi \le k^2$. As the degree increases to 8, or 10, etc. this prohibits more and more schemes on the right wing of the Gudkov pyramid (cf. e.g. Fig.\[Degree10:fig\]), yet to some extend those schemes can also be more elementarily ruled out by Rohlin’s formula. Since the Harnack bound is $M=g+1=\frac{(2k-1)(2k-2)}{2}+1=(2k-1)(k-1)+1=2k^2-3k+2$, the asymptotic location as $m=2k\to \infty$ of Thom’s bound $k^2$ \[ALAS INCORRECT, MY MISTAKE CORRECTED BY FIEDLER\] is the half value of Harnack’s bound $M\approx 2k^2$, and so nearly one quarter of the $M$-curves are prohibited by Thom (cf. e.g. Fig.\[Degree10:fig\]). However this figure only represents schemes with one nonempty oval so that the real pyramid is much less amputated than its planar sheet leads one to suspect. On the other hand the Hilbert hierarchies cannot be too deep. Indeed one cannot observe in degree $m$ a nested chain of ovals of depth larger than $k=m/2$ as follows directly from Bézout for lines. More generally using conics through 5 points there cannot be schemes with 5 chains of total length larger that $2m$. This is the well-known topics of Hilbert’s bound on the depth of nests. Further when those deepest configurations are attained then the curve is necessarily of type I via the phenomenon of total reality. Summarizing we see that the Hilbert tree cannot be too deep (as follows from reality consideration and Bézout) nor can it be too superficial (as follows from the complexification, e.g. Thom’s principle or the Petrovskii inequalities \[Petrovskii’s-inequalities:thm\]). Further the total reality phenomenon ought to play some big rôle as envisioned by Rohlin. Of course all this needs to be made more precise (further explored). On an arithmetical problem valorizing Thom in the detriment of Rohlin (and sometimes Gudkov): yet another numerical coincidence regarding Hilbert’s 16th -------------------------------------------------------------------------------------------------------------------------------------------------------- \[13.04.13\] This section (and its title!) is somewhat naive and misleading (as I failed to exploit the full punch of Rohlin’s formula), but diverges to a lovely basic arithmetic problem (on which I have little grasp, but must be very classical, surely Gauss? or probably much older Diophante?). Hence it was not censured but can safely be omitted. \[16.03.13\] As we said, but repeat it once more, Hilbert arrived ca. 1891 (vgl. Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege]) at the intuition (at least for sextics) that ovals of algebraic curves necessarily exhibit some feodal structure of nesting impeding all the ovals lying outside each others. Is this really true in general? \[OF COURSE, E.G., VIA ROHLIN’S FORMULA, AND FIRST KNOWN TO PETROVSKII 1933/38.\] Of course Hilbert posited this for $M$-curves (of order $m\ge 6$) and nearly proved it only for $m= 6$, yet presumably we are allowed to extrapolate his thoughts. In fact, the original text proceeds more carefully and reads as follows (cf. Hilbert 1891, , Fussnote, S.418, in Ges. Abhandl., Bd.II, Algebra, Invariantentheorie, Geometrie): “Diesen Fall $n=6$ habe ich einer weiteren eingehenden Untersuchung unterworfen, wobei ich—freilich auf einem au[ß]{}erordentlich umständlichen Wege— fand, da[ß]{} die elf Züge einer Kurve 6-ter Ordnung keinesfalls sämtlich au[ß]{}erhalb und voneinander getrennt verlaufen können. Dieses Resultat erscheint mir deshalb von Interesse, weil er zeigt, da[ß]{} für Kurven mit der Maximalzahl von Zügen der topologisch einfachste Fall nicht immer[^91] möglich ist.” Hilbert addressed this again, and more generally the question of elucidating the isotopic classification of sextic curves (or even quartics surfaces), into his well-known 16th problem at the Paris Congress of 1900. Circa 7 decades came the deep semi-experimental work of Academician Dimitrii Andreevich Gudkov solving Hilbert’s isotopic problem for sextics curves, though according to Arnold (e.g. 1997/00 [@Arnold_1997/200X-Symplectization-Complexification]) this left the supervising teacher Petrovskii quite dubitative, not to say skeptical). Later V.A. Rohlin made a series of jokes by noticing that his 1952 theorem on the divisibility by 16 of the signature (coined by Weyl 1923, in Analisis Situs Combinatorio written in Spanish with the assistance of his linguist wife) of a spinorial smooth $4$-manifold turns out to imply the Gudkov hypothesis $\chi\equiv k^2 \pmod 8$ for $M$-curves (extrapolating widely the phenomenology observed in degree $6$), where $\chi$ denotes as usual the Euler characteristic of the “Ragsdale” orientable membrane bounding the ovals from “inside”. In the same elan, Rohlin 1974 (and 1978) wrote down the [*Rohlin formula*]{} $2(\Pi^+-\Pi^-)=r-k^2$ (cf. \[Rohlin-formula:thm\]) which is so fundamental that it seems worth (to save ink) to abridge notation by letting $\pi:=\Pi^+$ and $\eta:=\Pi^-$. When the curve has no nesting this formula implies that $r=k^2$ is a square (whenever the curve is dividing, as it is automatically the case for $M$-curves). [*Warning.*]{}—\[17.03.13\] The sequel is much ill-posed as I (very shamefully) missed to notice that $r$ is not any square, but that of the semi-degree $k$. Yet since it seems to involve pleasant arithmetics, hence we did not censured it! Restricting attention to $M$-curves of even degree $m=2k$, we are therefore invited to study the following arithmetical problem as a way to corrupt Hilbert’s feudalistic intuition (forced presence of nesting for $M$-curves of high-degrees $m\ge 6$). [(Quadrature of the Harnack bound).]{}—Given any integer $k\ge$, set $m=2k$ (interpreted as the order of the curve) and let $M=g+1=\frac{(2k-1)(2k-2)}{2}+1=(2k-1)(k-1)+1=2k^2-3k+2$ be the corresponding Harnack bound. For which values of $k$ is $M$ predestined to be a square, so that Rohlin does not prohibit the unnested $M$-scheme (THIS IS FALSE), and additionally try to arrange the Gudkov congruence $M=\chi\equiv k^2 \pmod 8$. Assume that there is such an integer $k$ then the unnested $M$-scheme (Gudkov symbol $M$ also!) is not prohibited by Gudkov nor by Rohlin and so constitutes a potential violation of Hilbert’s principle. To kill the suspense, as far as I know this scenario could not have been precluded until the Kronheimer-Mrowka validation of Thom’s conjecture. Recall (from \[Thom-Ragsdale:thm\]) that Thom implies $\chi\le k^2$ and so $\chi$ cannot be as large as $M$ which is asymptotically twice so big (except of course for low degrees $m\le 4$). So curiously I would say (personal feeling probably foiled due to a lack of Russian wisdoms) that before Thom-Kronheimer-Mrowka 1994 [@Kronheimer-Mrowka_1994] Hilbert’s intuition could have been completely wrong depending upon a resolution of the above arithmetical problem. Of course I arrived at the problem by modest acquaintance with the geometry of Gudkov’s pyramid which look basically as follows (Fig.\[Pyramid:fig\]): -5pt0 -5pt0 Now being as bad in arithmetics than in combinatorics, we propose to tackle the arithmetical problem naively. We want to know first of all when Harnack’s bound is a square. We know that Harnack’s bound evolves like the genus of plane curves as a quadratic function (Gauss freshman calculus at 5 years old) namely $g=g(m)=1+2+3+\dots+(m-2)=\frac{(m-1)(m-2)}{2}$. Here we concentrate upon even degrees $m=2k$ and we may tabulate the values of $M$ by adding a linear progression of increment 4 (cf. Fig.\[Pyramid:fig\] right-side table). Then we compare this with the list of all squares $1,4,9, 16\dots$. Apart from the trivial values $k=1,2$, the Harnack bound is never a square until we reach the value $k=17$ where $M=529$ is the square of $23$. Since $17=16+1$ this is yet another numerological coincidence in Hilbert’s 16th (yet surely not as deep of those of Gudkov-Rohlin). In view of this I naively expected that the next quadrature of Harnack’s bound occurs at $k=2\cdot 16+1=33$. But a simple calculation shows this to be a wrong intuition. What about $k=4 \cdot 16+1=65$, also not good! At this stage we stopped guessing and continued the tabulation by hand, and found the next square Harnack bound at $k=46$, namely $M=\frac{91\cdot 90}{2}+1=91\cdot 45+1=4096$ which is $64^2$. It is now time looking at the Gudkov congruence. For $k=17$, the Harnack bound $M=529$ (which is also $\chi$ the characteristic of the Ragsdale membrane as we suppose no nesting) reduces modulo 8 to $529\equiv_8 49 $ (after removing $480$) and then to $1$ (after removing 48). On the other hand $k^2\equiv_8 17^2\equiv_8 1^2 \equiv_8 1$. Hence Gudkov’s congruence is satisfied. So the $M$-scheme (with Gudkov symbol $529$) could be a potential counter-example to Hilbert’s intuition, if Thom would not salvage it! For the next value $k=46$, $\chi=M=4096\equiv_8 96 \equiv_8 16 \equiv_8 0$, whereas $k^2=(46)^2\equiv_8 (-2)^2=4$. So Gudkov hypothesis is not verified and here Hilbert’s intuition is already vindicated by Gudkov (and Rohlin who proved it). Notice that Arnold’s congruence would be not enough not rescue Hilbert. This is essentially all what we have to say on this problem. Though the arithmetical problem looks attractive, I do not know how to solve it in general. Of course it looks an easy Diophantine equation $$x^2=2k^2-3k+2,$$ of degree 2 but this does not boils down to studying the rational points on a conic (via the usual method of sweeping like for Pythagorean triplets), as we are here really concerned with integral solutions of an equation laking homogeneity. Of course I presume it is a well-known and easy problem of arithmetics, yet actually its solution has little impact upon Hilbert’s sixteenth problem, since the corresponding (unnested) schemes are defacto prohibited by Thom. Still the particular solution exhibited above $(k,x)=(17, 23)$ offers another instance of French scheme where Thom is stronger that the conjunction of Gudkov and Rohlin. I do not if prior to Kronheimer-Mrowka one was able to prohibit the existence of the corresponding $M$-scheme of degree $34$ with $M=529$ unnested ovals. \[17.03.13\] In fact the answer is a trivial consequence of Rohlin’s formula, since $0=2(\pi-\eta)=r-k^2$, but $r=529$ is not the square of $17$ (as $17^2=289$). More generally, all the problematic of this section is spoiled since I omitted at the beginning of the discussion to notice that Rohlin’s formula does not merely implies in the unnested case that $r$ is the square, but is indeed the square of the semi-degree $k$. In fact as shown, e.g. by Lemma \[Rohlin-consequence-for-M-curves:lem\] it is a trivial consequence of Rohlin’s formula that Hilbert’s nesting intuition occurs for $m\ge 6$. The simplest way to argue is as follows. \[Hilbert’s-nesting-intuition:lem\] [(Hilbert’s nesting intuition, validated via Rohlin 1974 or via Petrovskii 1938)]{} Any $M$-curve of even degree $m=2k$ with $m\ge 6$ cannot be unnested, i.e. have all its ovals outside another. Actually, a dividing unnested curve is forced to have $r=k^2$ ovals. A simple way to argue is via Rohlin’s formula 1974 (\[Rohlin-formula:thm\]). We have $0=2(\pi-\eta)=r-k^2$, hence $r=k^2$. But by the $M$-curve assumption $r=M=g+1=(2k-1)(k-1)+1=2k^2-3k+2$, which is strictly larger than $r=k^2$ as soon as $k\ge 3$. The first clause also follows by specializing Petrovskii’s upper-bound $\chi\le \frac{3}{2}k(k-1)+1=:P$ valid for all curves. It is plain indeed to check that $P<M$ (“Petrovskii is sharper than Harnack”) for $k\ge 3$. Indeed the difference $M-P=(2k^2-3k+2)-(\frac{3}{2}k(k-1)+1)=\frac{1}{2}k^2 -\frac{3}{2}k+\frac{1}{2}=\frac{1}{2}\underbrace{(k^2-3k+1)}$. The roots of the underbraced quadratics are $k_{1,2}=\frac{3\pm \sqrt{9-4}}{2}=\frac{3\pm \sqrt{5}}{2}\le \frac{3+ \sqrt{9}}{2}=3$. Variant: $\sqrt{5}\approx 2.24$, so $k_1\approx 2.62$. \[17.03.13\] Misha Gabard (my father) and his skills in Excel calculated the next values of $k$ making $M=2k^2-3k+2$ into a square. The complete list for $M\le 1'000'000=10^6$ is $$k=1,2,17,46,553,1538,18761,52222,637297,$$ where $M$ is resp. the square of $$x=1,2,23,64,781,2174,26531,73852,901273.$$ On looking at the successive ratio of $k$, one finds the list $$R:=\frac{k_{n+1}}{k_n}=2, 8.5, 2.71, 12.02, 2.78, 12.20, 2.7835, 12.2036.$$ So naively the progression oscillates between a factor of about 2.78 and one about 12.20. So there seems to some extreme regularity, and one can predict in advance the size of the solutions. The long quest of a Caucasian scheme, i.e. where Rohlin is stronger than Thom and Gudkov (united) ------------------------------------------------------------------------------------------------- \[12.03.13\] The following section has been written in realm time out of paper notes, and we have reproduced our long march toward the trivial truth. The pressed reader can directly jump to Theorem \[Caucasian-scheme:thm\]. (WHICH IS ACTUALLY FALSE) So one must really read the first few sections which collects loosely organized thoughts on the problem of finding a scheme prohibited by Rohlin but not by Thom. Of course to avoid trivialities one must add more assumptions like the curve being an $M$-curves satisfying the Arnold or even Gudkov congruence. \[12.03.13\] As a consequence of Thom’s estimate $\chi\le k^2$ (\[Thom-Ragsdale:thm\]), it seems that the Gudkov pyramids are much amputated on their right wings causing thereby a certain asymmetry of them. It is at this stage of some interest to wonder what happens on the opposite left-wings where Thom becomes useless but perhaps Rohlin’s formula has still some prohibitive impact. In view of Fig.\[Degree10:fig\] let us look at the scheme $\frac{34}{1}2$ (not prohibited by Gudkov’s hypothesis). Here $\chi=1-34+2=-31$ and we have no prohibition by Thom. Is this scheme prohibited by Rohlin’s formula? Certainly not because as the Rohlin tree is simple there is no signs-law and the Rohlin equation $2(\pi-\eta)=r-k^2=37-25=12$ is soluble under the obvious additional relation $\pi+\eta=34$. Indeed $2\pi=40$, so $\pi=20$ and $\eta=14$. (Optional question: Is this scheme realized algebraically?) (\[21.03.13\] Answer: negative as follows from Petrovskii 1933/38 (\[Petrovskii’s-inequalities:thm\])!) Without changing $\chi$ we may delocalize the 2 outer islands to make them islands in a lake. We find so the scheme $(1,(1,2)33)$ (cf. Fig.\[Signs-law-triad:fig\]b). Is this schemes prohibited by Rohlin? We think the answer is no and the proof proceeds along the usual algorithm of solving the Rohlin equation under the signs-law. Recall that the latter can be easily remembered by saying that consanguinity is bad, i.e. $+\times +=-$, $-\times -=+$, while mixing the genes is good, i.e. $+\times -= +$ and $-\times += -$. This exotic signs law is the exact opposite of the usual convention. Fig.c depicts the Hilbert tree of the scheme of Fig.b, and we decorate it with a sign distribution as depicted, i.e. with $x$-many plus, and $y$-many minus, and likewise $\epsilon$ plus and $\delta$ minus at the indicated place. We have thus $x+y=33$, and $\epsilon+\delta=2$. Using the signs-law we find for the number of positive $\pi:=\Pi^+$ resp. negative pair $\eta:=\Pi^-$ the following expressions: $$\pi=x+2\epsilon, \quad \textrm{and} \quad \eta =y+2\delta+1.$$ Adding gives $\pi+\eta=(x+y)+4+1=38$. By Rohlin’s formula $2(\pi-\eta)=r-k^2=37-25=12$, so $\pi-\eta=6$. Hence $2\pi=44$, so $\pi=22$ and $\eta=\pi-6=16$. Solving finally in $x$ gives $x=\pi-2\epsilon=22-2\epsilon$. We are free to choose say $\epsilon=2$, then $x=22-4=18$ and $y=15$. So Rohlin’s equation is soluble. -5pt0 -5pt0 At this stage the question becomes under which condition is Rohlin’s equation not soluble? A priori it may be remarked that Rohlin’s formula $2(\pi-\eta)=r-k^2$ is coupled with the formula $\pi+\eta=\Pi$ the total number of pair. Both right-hand sides of this pair of equations are entirely determined by the real scheme without having to worry about orientations. So under reasonable hypothesis, like $r-k^2$ even which is a disguised (Russian) version of Klein’s congruence $r\equiv g+1 \pmod 2$—as follows from the boring calculation $$r\equiv_2 g+1=(2k-1)(k-1)+1=2k^2-3k+2\equiv_2 3k \equiv_2 k \equiv_2 k^2,$$ —we could expect that Rohlin’s equation is always soluble, especially if the (concomitant) Arnold congruence is satisfied. However this is not the case as shown by Fiedler’s corrigendum. There we considered the $M$-scheme $(1,1,1,1)33$ (of degree 10) which is prohibited by Rohlin’s formula since $\pi-\eta\le 2$ by the signs-law for triad (compare Fig.\[Signs-law-triad:fig\]d). But on the other hand Rohlin’s formula forces $2(\pi-\eta)=r-k^2=37-25=12$, hence $\pi-\eta=6$. Hence the philosophy behind Fiedler’s corrigendum seems to be that Rohlin’s formula imposes restriction, when there is much predestination forced by the signs-law, and this is naively speaking the case when there is much nesting like on the example just given. It seems of course to be of some interest to generalize the estimate $\Delta \Pi=\pi-\eta\le 2$ for triads to tetrads, etc. So the scheme $(1,1,1,1)33$ is prohibited by Rohlin (in its strong form of the signs-law), but it is also by Thom as $\chi=33 \nleqslant k^2=25$. Now however we would rather be interested in schemes prohibited by Rohlin but not by Thom. A first idea we had was to start from the scheme on Fig.b, and move $z$-many ovals inside to get Fig.e. As to maximize the “predestination” it seems wise to move the empty oval outside, and to assume $z=33$. But still under these circumstances it turned out that the Rohlin equation is soluble for a suitable distribution of sign, cf. Fig.f, from which we infer by the signs-law the following (where underbraced is the length of the corresponding edges=pairs) $$\pi-\eta=\underbrace{x-y+2}_{1}+\underbrace{(-x+y-1)}_{2} +\underbrace{(+x-y)}_{3}=x-y+1.$$ (In this calculation it is useful to remind the consanguinity law, and the Fig.d for a triad.) Hence as $\pi-\eta=6$ (by Rohlin), we $x-y=5$, and $x+y=33$. So $2x=38$, whence $x=19$, and $y=14$. So Rohlin’s equations is soluble. Philosophically it seems to be that whenever we have the free parameters $x,y$ then we can solve despite the predestination forced by the signs law. Keep in mind that our question is whether Rohlin implies prohibitions on the left wing of the pyramid (i.e. for $\chi \ll 0$ much negative) where Thom tells nothing. In fact $\chi \le k^2$ is enough for Thom to be non-prohibitive. Hence our next idea was to start from Fiedler’s example $(1,1,1,1)33$ with $\chi=33$, and lower down to $\chi=25$. This lowering may be achieved by trading the (oceanic) outer islands against lakes, i.e. ovals at odd depths, and this requires to be done four times. (This can be done in several ways, cf. Fig.g.) However on applying the usual algorithm of sign distributions, we were always able to solve the Rohlin equation in a way compatible with the signs-law. Details on p.AR91 of my hand-notes, but the philosophy seems to be basically that Rohlin’s formula gives one equation and the signs-law another but as soon as there free-parameter $x,y$ counting the number of signs on branches there is enough freedom to solve all equations consistently. In fact let us write down the argument. Starting from Fiedler’s example $(1,1,1,1)33$ with $\chi=33$, we lower down to $\chi=25$ by dragging 4 outer ovals at odd depth. This can be done in several fashions as we said, for instance like on Fig.\[Fied9:fig\]a by putting the 4 ovals at depth 1. However this new $M$-scheme is not prohibited by Rohlin’s formula. The latter says that $2(\pi-\eta)=r-k^2=37-25=12$, $\pi-\eta=6$. On the other hand by using the signs-law, and splitting the 4 empty ovals at depth 1 into $4=x+y$, where $x$ are positive, and $y$ are negative (w.r.t. the sign of the edge right above it), and assuming further for simplicity that the trunk is everywhere positive, we can still solve the equation. Indeed by the signs-law we have $\pi-\eta=2+x-y$, and so $x-y=4$, $x+y=4$ whence $x=4$, $y=0$. If instead we drag the 4 ovals at depth 3 we get Fig.\[Fied9:fig\]b. Now the signs-law becomes more involved, but we find splitting according to the length of the pairs (underbraced index) the following expression (assuming for simplicity $+$-signs already fixed on the trunk and as above there are $x$ many $+$ and $y$ many $-$): $$\pi-\eta=\underbrace{3+x-y}_{1}+ \underbrace{(-2-x+y)}_{2}+ \underbrace{(+1+x-y)}_{3}=2+x-y,$$ where we used the signs-laws for triad (Fig.\[Signs-law-triad:fig\]d) As $\pi-\eta=6$, this gives $x-y=4$, $x+y=4$, which is soluble. -5pt0 -5pt0 Then we can also disperse the 4 ovals at different heights like on Fig.\[Fied9:fig\]c where 2 are at depth 1 and the 2 others at depth 3. Instead of applying the method of indeterminate signs, we content to give a solving sign distribution as depicted with only $+$-signs. One checks that $\pi-\eta=\underbrace{7}_{1}+\underbrace{(-4)}_{2}+ \underbrace{(+3)}_{3}=6$, in agreement with Rohlin’s formula. Finally it remains to analyze the case of Fig.d, where we rigidify already some signs. Here $x+y=3$. Calculating via the signs-law gives $$\pi-\eta=\underbrace{4+x-y}_{1}+\underbrace{(-3)}_{2}+ \underbrace{(+2)}_{3}=3+x-y,$$ whence, as $\pi-\eta=6$, the system $x-y=3$, $x+y=3$, which is soluble (integrally) as $x=3$, $y=0$. (One checks mentally that this everywhere positive distribution works, as one do quickly mistakes in such calculation!) Last the case of Fig.\[Fied9:fig\]e, is also handled by the same method. Here $$\pi-\eta=\underbrace{4+x-y}_{1}+\underbrace{(-2-x+y)}_{2}+ \underbrace{(+1+x-y)}_{3}=3+x-y,$$ like above (!) hence soluble. Some further idea would be to increase the “predestination” by adding one or more triads as on Fig.f,g, yet such schemes are already prohibited by Bézout. Actually to lower $\chi$ down to Thom’s range $\chi\le k^2$ we look at Fig.g, but the latter is not even prohibited by Rohlin as $\pi-\eta=T_1+T_2+T_3$ is contributed by 3 trunks each $T_i\in\{2,0,-6\}$ (by Fig.\[Signs-law-triad:fig\]d) hence soluble as by Rohlin $\pi-\eta=6$. On Fig.f instead we have only 2 trunks so the scheme is prohibited by Rohlin (of course more elementarily by Bézout), yet it is also by Thom. So it does not solve our problem of finding a scheme where Rohlin is stronger than Thom. Hence Fiedler’s example looks a typical case of predestination of the Rohlin mass $\Delta \Pi=\pi-\eta$. Of course it may be generalized in higher degrees than $10$, by looking at $M$-schemes of the form $(1,1,1,1)M-4$. Then by Rohlin $2(\pi-\eta)=r-k^2=M-k^2=(g+1)-k^2=(2k-1)(k-1)+1-k^2 =k^2-3k+2=(k-1)(k-2)$. But on the other hand by the signs-law for triad (Fig.\[Signs-law-triad:fig\]d) we have $\Delta\Pi:=\pi-\eta\le 2$. It follows that the scheme considered is prohibited as soon as $\pi-\eta=(k-1)(k-2)/2$ is $\ge 3$ that is for $k\ge 4$. (The case $k=4$ being quite stupid for Bézout would have sufficed.) Okay, but such schemes are also prohibited by Thom. One could also try to deepen the nest as the degree increase. Yet our goal is really to find a “Caucasian” scheme, i.e. obstructed by Rohlin, but not by Thom (nor by Gudkov or Arnold). Here is an example that I discovered later and of degree 8 already. It is the $M$-scheme of degree 8 with symbol $\frac{1}{3}\frac{1}{17}$. Now Rohlin’s formula says $2(\pi-\eta)=r-k^2=22-16=6$, so $\pi-\eta=3$. On the other hand $\pi+\eta=20$, so $2\pi=23$ which is insoluble. Besides we have $\chi=(1-3)+(1-17)=-2-16=-18$ SO GUDKOV NOT VERIFIED \[14.03.13\] So let us approach this problem more systematically. First when $m=6$ it is clear that there is no Caucasian $M$-scheme as follows from Gudkov’s table (=Fig.\[Gudkov-Table3:fig\]). Indeed all the Thom permissible $M$-schemes are prohibited by Gudkov. So we move to $m=8$, so $M=22$. Here we start with the $M$-scheme $(1,1,1)19$ with $\chi=20$ (cf. Fig.\[Fied1:fig\]a). As the 3-nest corresponds to a dyad (2 pairs of length 1) they contribute by the signs-law to at most 1 to $\pi-\eta$. Hence Rohlin’s formula $2(\pi-\eta)=r-k^2=22-16=6$ (i.e. $\pi-\eta=3$) cannot be solved. Of course the scheme in question is also prohibited for deeper reasons (at via deeper results) like Gudkov hypothesis $\chi\equiv k^2=16\pmod 8$, or Thom’s inequality $\chi \le k^2$. Our goal is to find an $M$-scheme where the “elementary” Rohlin formula becomes stronger than the conjunction of 2 deep results (Gudkov hypothesis proved by Rohlin-Rohlin/Atiyah-Singer-Marin and Thom proved by Kronheimer-Mrowka). -5pt0 -5pt0 Can we adjust the invariants as to neutralize Gudkov and Thom? A first idea is to drag one outer oval at depth 1. Then the contribution to the Rohlin (signed) mass $\pi-\eta$ is still $\le 2$, and so his formula cannot be solved. Yet doing so we have $\chi=18$ and the scheme is also prohibited by Gudkov or Thom. If we delocalize one more outer oval at depth 1 we get Fig.c, where now Rohlin’s formula can be satisfied, and as $\chi=16$ both Gudkov and Thom are happy. So we see some annoying (as far as our Caucasian policy is concerned) concomitance between the 3 forces involved Rohlin, Gudkov, Thom. This phenomenon is not specific to degree 8 and repeats itself in degree $m=10$, where $M=37$ (temperature of the body of a primate). Let us experiment this concretely. As Bézout now permits, we start now with a deep nest of profundity 4 (4 nested ovals) and add 33 outer ovals to reach Harnack’s bound $M=37$ (cf. Fig.d, which involve a “triad” chain with 3 consecutive edges). By Rohlin’s formula $2(\pi-\eta)=r-k^2=37-25=12$, so $\pi-\eta=6$. However by the signs-law for triad the contribution of the triad is at most $2$ (cf. Fig.\[Signs-law-triad:fig\]), and thus the scheme $(1,1,1,1)33$ is prohibited by Rohlin. As $\chi=33\equiv_8 k^2=25$, the scheme is not prohibited by Gudkov, but it is by Thom. So our Caucasian goal is not achieved. To improve the situation we have to lower $\chi$ down to $k^2=25$. As far as the signs-law is involved we can transfer at most 3 oval at depth 1 (like on Fig.e), so as to have a contribution to $\pi-\eta$ still $\le 2+3=5<6$. Doing so $\chi=27$, and the scheme is still prohibited by Thom (and anew by Gudkov). Another idea is to use 3 dyads as on Fig.f as the latter also contribute to at most $1$ to Rohlin’s mass $\pi-\eta$. Alas doing so does not diminish $\chi$ in the Thom range, and actually violates Bézout (cf. Fig.g). This can be remedied if we abort the triad, and look at a configuration with 5 dyads (the maximum possible while still taking care to making Rohlin’s equation $\pi-\eta=6$ impossible). This gives Fig.h with alas still $\chi=27$. So this schemes is prohibited by both Rohlin, Gudkov, and Thom (but as far as I see not by Bézout even for conics). If we nest one more outer oval we may get Fig.i with $\chi=25$, but suddenly Rohlin’s equation is now soluble (choose e.g. $+$-signs throughout). Likewise we may consider Fig.j, but Rohlin is likewise soluble. Considering Fig.k still calibrated as to make Rohlin’s equation impossible (as each triad contribute for $\le 2$), we only reach $\chi=27$ (of course this configuration is anti-Bézout). We can push it further to Fig.l, which despite being prohibited by Bézout it is not by Gudkov nor by Thom, yet alas not by Rohlin since $2+2+2=6$ and so Rohlin’s equation is soluble taking $+$-signs on all edges. Our naive cuneiform construction can still be more varied, yet it seems unlikely that we will ever find a Caucasian scheme by this method. We still consider Fig.m (not interesting). Next look at Fig.n with $\chi=25$. Denoting by $x$, $y$ the number of $+$’s resp. $-$’s on the edge right above the corresponding letter, the signs-law gives (after fixing $+$ on the trunks) $$\pi-\eta=\underbrace{4+x-y}_{1}+\underbrace{(-3)}_{2} +\underbrace{(+2)}_{3}=3+x-y,$$ whence the system $x-y=3$, $x+y=3$ soluble as $(x,y)=(3,0)$. So the scheme is not obstructed by Rohlin. We can also transmute Fig.d into Fig.o, where again 3 branches are added so as to keep Rohlin’s formula impossible, but again $\chi$ drops only to $27$ (and not $25$). Of course such a scheme is defacto prohibited by the Bézout-Hilbert bound on the depth of nests (cf. Fig.p). If it is not possible to find a Caucasian scheme in degree $10$, what about degree 12. First we need to extend the signs-law to tetrads. While the latter involves for dyads a square (4 possible products of two signs), and for triads a cube (with 8 possible signs combinations), we have now a 4D-hypercube with $2^4=16$ combinations. The signs-law for tetrads is depicted below (Fig.\[Signs-law-tetrad:fig\]). It corroborates the a-priori reasonable expectation that the maximum contribution arises when all 4 signs are $+$, in which case the contribution to Rohlin’s mass $\pi-\eta$ is 2. Still a priori we may expect that our (Caucasian) game will not become easier since Harnack’s bound $M=g+1=(2k-1)(k-1)+1=2k^2-3k+2$ increases much faster than Thom’s bound $k^2$ and so we will have more pain to lower down $\chi$ in Thom’s range. Despite these objections [*a priori*]{} let us track down our prey more slowly. -5pt0 -5pt0 \[15.03.13\] Hence: The contribution to Rohlin’s mass $\pi-\eta$ of a deep nest is: $\bullet$ for a dyad either $+1$ or $-3$, $\bullet$ for a triad either $+2,0$ or $-6$, $\bullet$ for a tetrad either $+2,0,-2$ or $-10$. With some combinatorial ingeniousness it should be easy to extend to the general case. However let us first tackle our Caucasian problem in degree 12. Again we resort to the cuneiform formalism used above. When $m=12$, $M=g+1=\frac{11\cdot 10}{2}+1=55+1=56$. We consider first Fig.\[Fied2:fig\]a. Here $\chi=52$, while Gudkov says $\chi\equiv_8 k^2=36=44=52$, which is verified. By Rohlin’s formula $2(\pi-\eta)=r-k^2=56-36=20$, so $\pi-\eta=10$, but the contribution of the tetrad is at most 2 by the signs-law, and thus the scheme is prohibited by Rohlin. Of course it is also by Thom $\chi \le k^2$. As above the game is to lower $\chi$, by transferring outer ovals at depth 1. As to keep Rohlin in defeat, we may add at most 7 branches as on Fig.b, and obtain a scheme with $\chi=52-2\cdot 7=52-14=38$, again 2 unit above Thom’s bound. The next idea is to let branches of dyads (dyadic branches) hang on. Each contributes at most $1$ to Rohlin’s mass $\pi-\eta$, and so keeping Rohlin’s formula in check we may add 7 of them, but of course $\chi$ remains unchanged to $\chi=38$ (as we merely traded outer ovals at depth 0 for ones at depth 2). Next Fig.d involves only dyads contributing for at most 1, so keeping Rohlin check-mate we can plug 9 of them, and the resulting $\chi$ is still $38$. Using instead triadic branches as on Fig.e which contributes for 2 (at most) we may plug 4 of them and the remaining unit is consumed by inserting a dyadic branch, and we find of course again $\chi=38$. Considering only monadic branches as on Fig.f (und zwar 9 of them to Rohlin in check) yields again $\chi=38$. If like on Fig.g the monadic branches are not subsumed to a single dominator we find again introducing 9 of them, $\chi=38$. Using instead 9 dyadic branches of the same sort, we get Fig.h, where still $\chi=38$. So it looks again hard to find a Caucasian scheme where Rohlin is stronger than Thom (and Gudkov united). Either we are looking at the wrong place or there is some subsumation of Rohlin to Thom, for some trivial arithmetical reasons. That is assume you have an $M$-scheme with $\chi\equiv_8 k^2$ (Gudkov) and $\chi\le k^2$ then Rohlin’s equation is always soluble. Indeed write formally $2(\pi-\eta)=r-k^2$. By the $M$-curve assumption $r-k^2$ is $r-k^2=(2k-1)(k-1)+1-k^2=k^2-3k+2$, etc. -5pt0 -5pt0 One could also ask if for $M$-curves the Thom and Gudkov obstructions are the sole one, but this is probably corrupted by Bézout-style obstruction à la Fiedler-Viro (in degree 8 already). Some weak evidence against Caucasian schemes -------------------------------------------- \[15.03.13\] In the previous section we tried (hard) to find a “Caucasian” scheme, i.e. prohibited by Rohlin’s formula $2(\pi-\eta)=r-k^2$ but not by Thom $\chi\le k^2$, but failed. Of course as Rohlin’s formula formally implies Arnold congruence it is actually a simple matter to find such a scheme, e.g. any $M$-scheme violating Arnold’s congruence but not Thom do the job. For instance in degree 6, the $M$-scheme $\frac{2}{1}8$ is prohibited by Rohlin, but not by Thom. Likewise for the $(M-2)$-scheme $\frac{1}{1}7$ in type I. However if we add the Gudkov hypothesis as a side condition (or perhaps just the Arnold congruence) then it seemed difficult to find a Caucasian scheme where Rohlin is stronger than Thom. The sequel tries to give some evidence that it is impossible to find a Caucasian scheme, but our argument will be somewhat loose. The main difficulty is the mess arising with the signs-law and so the difficulty looks merely combinatorial. Of course it is not impossible that we missed a trivial counter-example that impedes the completion of the present programme. First we described in the previous section the mess arising form the signs-law applied on dyad, triads, up to tetrads, which are totally ordered chains. In general the situation is more tricky as the Hilbert tree of the scheme may be highly branched. Recall that to a dividing curve is assigned complex orientations (up to global reversion of all of them), which in turn decorates the Hilbert tree (of the curve) with a signs-distribution (abridged charge) making it into what we call the Rohlin tree. (Of course the tree can be a “forest”, i.e. have several components, and it really “branches downwards” as to look more like Arnold’s paradigm of the mushrooms.) Each such (Rohlin) signed tree is completely determined by the signs ascribed to the edges of length 1 as it then extends to longer edges by the signs-law. Further each such tree has a Rohlin mass $\mu=\pi-\eta$ which is the difference between $\pi:=\Pi^+$ the number of positive pairs and $\eta:=\Pi^-$ the number of negative pairs. When the Rohlin tree is the one induced by a curve of degree $m=2k$, (the marvellous) Rohlin’s formula $2(\pi-\eta)=r-k^2$ fixes the mass in function of $r$ the number of ovals and $k$ the semi-degree. Further $\pi+\eta=\Pi$ is the total number of pair, and so one gets the wrong impression that Rohlin’s equation is always soluble but there is some hidden rôle played by the mass $\mu:=\pi-\eta$ which for certain configuration can only be very low (especially on chains), cf. Fiedler’s example with $(1,1,1,1)33$ in degree 10. So the core of the problem is to understand the behavior of the Rohlin mass $\mu$. In the previous section we nearly understood this for vertical chains. Especially easy, is the case where all signs are positive, in which case it is a simple matter to show the: Given an $n$-chain of $n$ consecutive edges all positively charged then the Rohlin mass of the chain is the integer part $[\frac{n+1}{2}]$ Look and see (i.e. make pictures). Indeed looking at the Signs-laws for triad ($n=3$) or tetrad ($n=4$), we get resp. $\mu=3-2+1=1+1=2$ and $\mu=(4-3)+(2-1)=1+1=2$. For a 5-chain this extends as $\mu=(5-4)+(3-2)+1=3$, for a 6-chain as $\mu=(6-5)+(4-3)+(2-1)=3$, and so on. In general for a signed tree there ought to be a sort of skein relation permitting an iterated evaluation of the Rohlin mass $\mu$, based on the formula that $\mu$ of an inverted “Y” looking like a $\Lambda$ surmounted by a chain is equal to $\mu$ of the left maximal chain in the inverted “Y” plus the right chain, minus $\mu$ of the common trunk. If this is not clear, please compare Fig.\[Fied2:fig\]i. This formula is of course a formal consequence of the inclusion-exclusion principle in combinatorics or measure theory. Now in sloppy fashion first, the idea is that under certain assumptions (like the conjunction of Gudkov and Thom’s $\chi\le k^2$) one could show the existence of a charge (=distribution of signs) solving Rohlin’s equation. Perhaps this can be done via a sort of linear algebra modulo 2. First the charge in question is merely a $\pm 1$-valued function on the set $E$ of all edges of the tree. The set of all such distributions denoted $\cal E$ can be turned into a vector space over the field ${\Bbb F}_2$ with 2 elements. Define indeed the sum of two charges $\epsilon, \delta$ as $(\epsilon+\delta)(e)=\epsilon(e)\times \delta(e)$, where $\times$ is the Rohlin product given by the signs-law (i.e. the opposite of the usual sign convention for products). The neutral element is $\epsilon_0$ the minus distribution, as $(\epsilon+\epsilon_0)(e)=\epsilon(e)\times (-1)=\epsilon(e)$. Also each element has order 2, as it should. (More boring details in p.AR99=hand-notes, especially the inversed charge where all signs are switched is not the inverse charge!) The multiplication by a scalar is naturally defined. The Rohlin mass is the function $\mu\colon \cal E \to {\Bbb Z}$. Now that we have a good vector space, we could hope our problem reducible to linear algebra! Intuitively if $\chi\le k^2$ then there must be enough edges as to solve Rohlin’s equation. More precisely Rohlin’s formula fixes the mass via $2\mu =r-k^2$, and via the skein relation the mass of the tree reduces to that of chains which in turn can be reduced to that of edges via the signs-law, i.e. the knowledge of the charge $\epsilon$ itself. If one is good in combinatorics there is a little hope to show that each skein relation induces a linear equation and count that there is enough free parameter as to solve the equation. [*Warning.*]{}—As remarked in more details latter, already in degree 6 we have the scheme $5$ whose type I realization is prohibited by Rohlin but not by Thom, so there is no chance to complete this programme, unless extra assumptions are added, e.g. that of being an $M$-curves (which further must satisfy the Arnold congruence, else prohibited by Rohlin but not by Thom). As a trivial example we may extend the observations of the previous section. Consider an $M$-scheme without nesting. Then $\chi=M$. Now to diminish $\chi$ we introduce edges (i.e. nested pairs), cf. Fig.\[Fied2:fig\]g for an example. We have $r=M=g+1=(2k-1)(k-1)+1=2k^2-3k+2$, so $r-k^2=k^2-3k+2=(k-1)(k-2)$. By Rohlin’s formula $\mu:=\pi-\eta=(k-1)(k-2)/2$. Hence to keep Rohlin’s formula in default, we introduce only $\mu-1$ edges. Then we compute $\chi$, and find $\chi=r-2(\mu-1)$ and a boring calculation shows this to be $k^2+2$. So if obstructed by Rohlin then also by Thom. (More details in p.AR.99 of the hand-notes.) Of course this is not the general case as the scheme has a very specific structure akin to Fig.\[Fied2:fig\]g. Can we generalize, perhaps but requires to work out some messy combinatorics. A somewhat more appealing idea is that if the scheme is obstructed by Rohlin then it is because its mass $\mu$ is strictly less than $\pi-\eta=(r-k^2)/2$ even when the tree is positively charged, and conjecturally this should maximize the mass. All this is vague but points to the right direction. Namely it gives the idea of computing the Rohlin mass of a tree with positive charges only. The answer turns out to be simple and elegant: \[Rohlin-mass-of-a-positively-charged-tree:lem\] The Rohlin mass $\mu$ of a positively charged (signed) tree $T$ is equal to $$\mu(T)=n_1+n_3+n_5+\dots=n,$$ the number of ovals at odd depths, where $n_1$ counts those at depth $1$, $n_3$ at depth $3$, and so on. Make a picture of a tree with possibly several components. Put plus signs everywhere as stipulated. By additivity we may focus attention on a single component of the tree. Each vertex at depth $\ge 1$ has exactly one edge above it. All pairs are enumerated by starting from vertices at depth 1 and looking at edges above them gives the contribution $n_1$. Next we look at the $p_2$ many vertices at depth 2, each inducing two pairs above it (of length 1 and 2 resp.). The first contribute for $+1$, while the other has sign $-1$ by the signs-law $+\times +=-$ (recall that consanguinity is bad). So ovals at depth 2 contributes for $p_2-p_2=0$. Continuing in this fashion we find: $$\mu=n_1+(p_2-p_2)+(n_3-n_3+n_3)+(p_4-p_4+p_4-p_4)+\dots,$$ which implies the announced formula. As implicit above we posit the: \[positive-mass-conjecture:conj\] The Rohlin mass of a signed (Rohlin) tree is maximized when the tree is positively charged throughout. Some evidence comes from the case of chains (as tabulated on the signs-law tables, e.g. Fig.\[Signs-law-tetrad:fig\]). There is perhaps a simple argument. But do we really need this? Let us make another observation based on the lemma. Assume that Thom holds, i.e. $\chi\le k^2$. In general, we have: $$\chi=p_0-n_1+p_2-n_3+p_4-\dots,$$ where each symbol $p_i, n_i$ counts the number of ovals at depth $i$, where $p,n$ are just “residue” of Petrovskii notations for positive and negative but best interpreted in terms of even or odd depth resp. (The notation are nearly consistent in French-Swiss-German, where “even=pair” and “odd=uNgerade”.) Besides, the total number of ovals, denoted $r$, is expressible as $$r=p_0+n_1+p_2+n_3+p_4+\dots,$$ so that subtracting the double of the Rohlin mass $\mu$ of the positively charged Rohlin tree (as calculated in the lemma) gives the relation: $$r-2\mu=\chi,$$ which holds universally when the tree is positively charged. So if Thom is verified, i.e. $\chi\le k^2$, we find $2\mu=r-\chi\ge r-k^2$. This means that there is no obstruction [*a priori*]{} to solve Rohlin’s equation, since Rohlin’s mass is as large as it should by virtue of Rohlin’s formula $2\mu=r-k^2$. Paraphrasing, Rohlin’s equation is virtually soluble. If Thom’s estimate $\chi\le k^2$ is fulfilled, then there is no “quantitative” obstruction to solve Rohlin’s equation. Yet beware that there may of course be finer arithmetical reasons impeding solubility as with the scheme $5$ of degree $6$ which has no type I realization. This prompts some evidence toward the: There is no Caucasian scheme, where Rohlin is stronger than Thom (at least modulo adding some suitable hypotheses, e.g. that of an $M$-scheme). Can we find a formal proof? A crudely idea is to notice that if we charge the tree negatively throughout then by the signs-law all pairs are negative as $-\times -=-$. So the Rohlin mass of the negatively charged tree is $-\Pi$, where $\Pi$ is the total number of pairs. Hence by a dubious mean-value theorem (in the discontinuous realm of the arithmetics of quanta) we would like to infer existence of a charge fulfilling Rohlin’s formula. Another idea is to introduce indeterminate signs and try to solve a system of linear equations. We did this frequently formerly, but we had some grasp on the geometry of the tree. Whether this can be done in abstracto is not clear to me, and may of course converge to the first strategy using linear algebra on the spaces of all charges (plus the skein-relation). Of course recall that we have the Rohlin-Marin inequality that a dividing curve $C_{m=2k}$ has $r\ge m/2=k$, i.e. at least as many ovals as the semi-degree. This is a formal consequence of Rohlin’s formula and precludes in degree $6$ a type I incarnation of the scheme $1$ (unifolium), which is not prohibited by Arnold’s congruence mod 4 (cf. the Gudkov-Rohlin table=Fig.\[Gudkov-Table3:fig\]). This example (or also the scheme $5$ in degree 6) are of course trivial counterexamples to the above conjecture (freed from the parenthetical proviso). The latter gains however some more credibility when the curve is assumed to be an $M$-curve (or perhaps even an $(M-2)$-curve). It is evident that our whole problem is somewhat ill-posed, yet we hope to have demonstrated that some complicity between Rohlin and Thom requires to be elucidated. The simple example of the scheme $5$ in degree 6, where Thom’s estimate $\chi\le k^2$ as well as Arnold’s congruence are fulfilled, but whose realizability in type I is precluded by Rohlin’s formula (as $r$ is not a square and there is no nesting hence $\Pi=0$, and so a fortiori $\pi=\eta=0$) shows that our above desideratum of solving Rohlin’s equation under the sole assumption of being in Thom’s range is not realistic. So one must really add some extra assumptions, typically that of being an $M$-curve, which can perhaps be somewhat relaxed. Let us close the discussion via precise conjectures: $\bullet$ An $M$-scheme verifying the Gudkov congruence $\chi\equiv k^2 \pmod 8$ and the Thom estimate $\chi\le k^2$ is never prohibited by Rohlin’s formula. $\bullet$ An $(M-2)$-scheme verifying the RKM=Rohlin-Kharlamov-Marin congruence $\chi\equiv k^2 +4 \pmod 8$ (ensuring the scheme to be of type I) and the Thom estimate $\chi\le k^2$ is never prohibited by Rohlin’s formula. A basic mistake in the search of a Caucasian scheme --------------------------------------------------- DO NOT READ THE SEQUEL IT IS FALSE! Our next idea was to look at an $M$-scheme extending 5 nests of depth 2 (cf. Fig.\[Fiedler6:fig\]a,b). By Rohlin’s formula we still have $\pi-\eta=6$, but $\pi-\eta\le \pi+\eta=\Pi=5$ so Rohlin is violated, but alas $\chi=27$ hence the scheme is also prohibited by Thom. Can we diminish $\chi$? Yes as usual by trading an outer oval at depth 0 against one at depth 0, cf. Fig.c. But then the corresponding tree (Fig.d) has $\Pi=6$ pairs and so Rohlin’s formula is soluble with $\pi=6$ (i.e. all edges positive). Finally, we started from the scheme $(1,1)35$ (cf. Fig.e) with $\chi=35$ and to lower to Thom’s range $\chi=25$, we make 5 moves to get Fig.f and its allied tree on Fig.g. But again we have $\Pi=6$ and so can solve Rohlin’s equation by putting only positive signs. From this last configuration, we decided to drag one of the ovals at depth 2 to get Fig.h, which has still $\chi=25$. By Rohlin $\pi-\eta=6$. Now introducing free variables $x,y$ counting positive resp. negative signs on the 5 edges and choosing any distribution of signs on the trunk of length 2, we know that the latter will contribute for at most $\le 2$ to $\Delta\Pi:=\pi-\eta$ (by Fig.\[Signs-law-triad:fig\]d) and its contribution $T$ is either $2,0,-6$, which is at any rate even. WARNING: HERE I MADE A BASIC CONFUSION IN THE LENGTH OF THIS CHAIN!!! On calculating $\pi-\eta$ by the signs-law we get $$\pi-\eta=T+x-y,$$ so $x-y=6-T$ and $x+y=5$ so that $2x=11-T$ which is impossible modulo 2! So we found our first scheme prohibited by Rohlin but not by Thom, an therefore: \[Caucasian-scheme:thm\] (WARNING=ERRONEOUS) There exists a “Caucasian” scheme[^92], where Rohlin is stronger than Gudkov and Thom. More precisely the $M$-scheme of degree 10 of Fig.h that is $(1,(1,1)5)29$—in Gudkov’s notation—is prohibited by Rohlin’s formula, but not by Gudkov’s hypothesis $\chi=k^2 \pmod 8$ nor by Thom’s inequality $\chi\le k^2$. The example has $\chi=25$ (by construction). -5pt0 -5pt0 Of course the above argument extends to the case where we drop 3 ovals at depth 2 (Fig.i) so that we have $x+y=3$ still odd. (This could even be $x+y=1$.) Indeed denoting by $T_1,T_2,T_3$ the contributions of the 3 trunks of length 2 on Fig.i and by $T$ their sum (which is even by signs-law for triad), we find by the signs-law $$\pi-\eta=T+x-y,$$ and therefore as $\pi-\eta=6$ (by Rohlin’s formula) we have $x-y=6-T$ and $x+y=3$, so that summing $2x=9-T$, which is impossible modulo 2. The phenomenon just discovered is probably not new and perhaps related to Slepyan’s law (also a formal consequence of Rohlin’s formula), cf. perhaps Rohlin 1978 or Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000]. \[13.03.13\] It is quite evident that we may generalize somewhat the result. What seems essential to the argument is that the Hilbert tree of the scheme as the structure of Figs.h.i with an odd number $x+y$ of empty ovals at depth 1, so to be like Fig.\[Fiedler7:fig\]a with $x+y$ odd. In fact let us be more general and leave degree 10. So suppose to have an $M$-scheme of (arbitrary) degree $2k$ (or more generally a scheme of type I, but we reserve this for latter) so that the dividing character is granted and therefore Rohlin’s formula applies. We suppose additionally that the scheme is like Fig.a with $x+y$ denoting the number of empty ovals at depth 1, partitioned into $x$ many positive pairs and $y$ many negative pairs (when looking at the unique edge right above those ovals). The argument is then to compute $\pi-\eta$ in two fashions. One way involves Rohlin’s formula $2(\pi-\eta)=r-k^2$, while the other route involves the signs-law for triad (cf. Fig.\[Signs-law-triad:fig\]d) and gives $$\pi-\eta=T+x-y,$$ where $T$ is the contribution of the trunks of length 2 which is necessarily even (again by Fig.\[Signs-law-triad:fig\]d). Now if $r-k^2\equiv 0\pmod 4$, by Rohlin $\pi-\eta$ is even and the signs-law equation is corrupted if $x-y\equiv_2 x+y$ is odd. Viceversa if $r-k^2$ is not divisible by 4, then Rohlin says that $\pi-\eta$ is odd, but the signs-law that it is even provided $x+y$ is even. Finally for $M$-curves it is a simple matter to check that $r-k^2\equiv 0\pmod 4$ iff $k\equiv 1,2 \pmod 4$. Indeed $$r-k^2=(g+1)-k^2=(2k-1)(k-1)+1-k^2=k^2-3k+2,$$ which is mod 4 for $k=1$, $1-3+2=0$ and for $k=2$, $4-6+2=0$, while for $k=3$, it is $9-9+2=2$ and for $k=0=4$, it is $2$. Hence we have proved the: Define a dendritic scheme as one like depicted on Fig.\[Fiedler7:fig\]a, i.e. with Gudkov symbol of the form $(1,(1,1)\dots(1,1) x+y ) z$. If $k\equiv 1,2 \pmod 4$, there is no $M$-curve of degree $2k$ with dentritic scheme having an odd number of empty ovals at depth 1. If instead $k\equiv 1,2 \pmod 4$, there is no such curve having an even number of empty ovals at depth 1. Some additional remarks are in order, which we detail right after. 1.—First it is a simple matter to see that Caucasian schemes exists already in degree 8. 2.—Second it seems clear that we may formulate of the theorem for $(M-2)$-schemes satisfying the RKM-congruence ensuring type I. 3.—Third, we could expect to extend the result to the case where there are several dendrites, or deeper nests. 1.—Indeed for definiteness we may assume that there is a single trunk of length 2 (like on Fig.b). For $m=8$, $M=22$ and so for an $M$-curve we have $2(\pi-\eta)=r-k^2=22-16=6$, so that $\pi-\eta$ is odd. Yet calculating via the signs-law prompts that $\pi-\eta=T+x-y$ which is even provided the number $x+y$ of branches of length 1 is even. (THIS IS AGAIN A MISTAKE CAUSED BY CONFUSION IN THE LENGTH OF THE CHAIN: BASICALLY IT INVOLVES 3 OVALS BUT THE LENGTH IS 2!!!) So we get schemes prohibited by Rohlin along the series depicted on Fig.c, which traduced in Gudkov’s symbols gives the list $$(1,1,1)19, (1,(1,1)2)17, (1,(1,1)4)15, (1,(1,1)6)13, (1,(1,1)8)11, \dots, (1,(1,1)18)1,$$ where $\chi$ is first $\chi=(1-1+1)+19=20$ (hence the scheme is also prohibited by Gudkov or Thom) and then successively drops by 4 units, so that the second listed scheme $(1,(1,1)2)17$ has $\chi=16$ (hence not prohibited by Gudkov nor by Thom, but prohibited by Rohlin), and so on. At $\chi=8$ we find another Caucasian scheme $(1,(1,1)6)13$. -5pt0 -5pt0 2.—For $(M-2)$-schemes satisfying the RKM-congruence $\chi\equiv k^2+4 \pmod 8$ we are ensured to be of type I. We consider again a dentritic scheme like on Fig.\[Fiedler7:fig\]a and repeat the above argument. By Rohlin $2(\pi-\eta)=r-k^2$, while on the other hand $\pi-\eta=T+x-y$ where $T$ is the contribution from the trunks which is even. As $x+y\equiv_2 x-y$, we get a contradiction as soon as $x+y$ and $(r-k^2)/2$ have opposite parities. When $r=M-2$, $r-k^2$ is congruent to 0 mod 4 precisely when $k\equiv 0,3 \pmod 4$ (just shift by 2 the previous calculation). To get an example, consider $m=8$ and take just one trunk like on Fig.c, but now consider the $(M-2)$-scheme $(1,1,1)17$. It has $\chi=18$. But we impose the RKM-congruence, hence adjust $\chi$ to $12$. Hence we consider $(1,(1,1)3)14$, and this is prohibited by Rohlin. Indeed his formula becomes $2(\pi-\eta)=r-k^2=20-16=4$, which is divisible by 4 (as predicted above), while by the signs-law $\pi-\eta=T+x-y$ is odd since there are $x+y=3$ little branches hanging on. \[21.03.13\] THIS IS ERRONEOUS BUT MAYBE CAN BE CORRECTED BY CHANGING OF PARITY, as the trunks contributes for an odd number. So the real outcome of this method of prohibition based on Rohlin’s formula and the signs-law seems to be a powerful tool for prohibition. It remains of course to examine its exact significance, and how it generalize to nest of deeper structure. It may be noted that earlier in this text we attempted a complete classification of RKM-schemes of degree 8. This was rather a census, i.e. a weak form of combinatorially possible schemes yet without any claim of realizability. Now with the present method we see that some of them are prohibited. It remains to understand which of them are prohibited by Rohlin enhanced by the signs-law. Clearly the argument given above extends and implies the following lemma. \[RKM-scheme-ruled-out:lem\] (ERRONEOUS) The four primitive types of RKM-schemes, i.e. $$(1,(1,1)3)14, (1,(1,1)7)10, (1,(1,1)11)6, (1,(1,1)15)2,$$ already listed in Equation \[RKM-scheme-deg-8-four-primitive-type:eq\], are prohibited by Rohlin’s formula. However their derived products looks harder to prohibit as they ramify and do not anymore belong to the dendrite type. For instance for the scheme $(1,(1,2)3)13$ one can easily solve Rohlin’s equation with a distribution of sign, and so probably for all other derived products. That remains to be checked. On the other hand, it is clear that our census of 100 schemes was far from exhaustive. Indeed we may consider a dendrite with 2 trunks like on Fig.d. This has $\chi=16$, and to adjust the RKM-congruence $\chi\equiv_8 k^2+4=20\equiv_8 12$ we move down to $\chi=12$ by transferring 2 outer ovals at depth 1 to get Fig.e. This scheme is not prohibited by Rohlin. Indeed $2(\pi-\eta)=r-k^2=20-16=4$ (so $\pi-\eta=2$), and by the signs-law $\pi-\eta=T_1+T_2+x-y$ where each trunks contributes to $T_i\in \{2,0,-6\}$ by Fig.\[Signs-law-triad:fig\]d. Even if we impose $T_1=T_2=2$, the system is still soluble, being $x-y=2$ and $x+y=2$, hence $(x,y)=(2,0)$. As usual, the RKM-scheme of Fig.e whose (Gudkov) symbol is $(1,\frac{1}{1}\frac{1}{1}2)13$ has a myriad of companions. One can either: $\bullet$ without changing $\chi$ transfer outer ovals at depth 2 (in various ways); $\bullet$ drop $\chi$ by 8 units by transfer quanta of 4 ovals at depth 1. All this is a bit messy to write down, and this will still not be exhaustive as we can start from the configuration with 3 trunks (Fig.f) which has $\chi=14$, and to adjust to RKM we make one transfer at depth 1 (Fig.g) and get so another RKM-scheme. This times the scheme is prohibited by Rohlin as $2(\pi-\eta)=r-k^2=20-16=4$ (so $\pi-\eta=2$), but by the signs-law $\pi-\eta=T+x-y$ where each 3 trunks contributes evenly while $x-y\equiv_2 x+y=1$ is odd. Likewise if from Fig.g we transfer quanta of 4 outer ovals at depth 1 the number of branches $x+y$ is still odd, and so those schemes are prohibited too. However those schemes derived from Fig.g by transferring outer ovals at depth 2 are probably not prohibited. Then there is still the cases of 4, 5, etc. trunks and the classification looks quite messy to obtain. Fortunately the story the story as soon as the tree contains 4 disjoint edges, since this correspond to 4 nest of depth 2 (a configuration which is saturated by Bézout or better Rohlin’s maximality principle). Note incidentally that this principle also prohibits the schemes like Fig.g with 3 trunks since there are extra branches, so that the above prohibition via Rohlin’s formula can be subsumed to total reality. \[Warning this last sentence looks dubious!\] Albeit messy, it could be of primary importance to get a good view of what happens along the way to extend the Rohlin-Le Touzé phenomenon of total reality from degree 6 to degree 8. So a pivotal question is whether there is any reasonable way to list all RKM-schemes of degree 8? If so then make some cleaning by ruling out those prohibited by Rohlin’s formula (enhanced by the signs-law), and finally try to understand which are realized algebraically (simplified form of Hilbert’s 16th nearly solved by the experts, but by far by myself, as one requires certainly the Viro method). Once this s achieved try to understand if all those schemes are subjected to the phenomenon of total reality (probably under pencil of quintics, as we discussed earlier). If so then there is some chance that Rohlin’s maximality conjecture holds true in degree $8$. All this requires either Herculean forces or some good idea. Toward a complete census of RKM-schemes of degree 8 --------------------------------------------------- \[13.03.13\] Our goal is to list all RKM-schemes of degree 8. Those are $(M-2)$-scheme satisfying the RKM-congruence $\chi\equiv k^2+4=20 \pmod 8$. Of course there is a menagerie of them, but we have also upper bound given by the saturation principle of Rohlin allied to total reality, once the depth is 4 the configuration is saturated and cannot develop further. Actually the nest of depth 4, is not an $(M-2)$-scheme and so the depth is at most 3 (or 2 depending on the way you count). Likewise the pencil of conics shows that that there can be at most 4 nests of depth 2, or when translated in the cuneiform language of Hilbert’s tree there is at most 4 edges which are disjoint. With this upper bound in mind, there is some little hope to make a complete classification. Further as the number of ovals $r$ is fixed to $M-2=20$, we may kill all empty ovals lacking a superior (so-called outer ovals) and thus condense a bit notation. So to each scheme is assigned a “skeleton” (kill the outer ovals) from which we may recover the scheme unambiguously. We abort this project as it is quite overwhelming. Trying to corrupt Rohlin’s maximality conjecture (RMC) ------------------------------------------------------ WARNING DO NOT READ: FULL OF MISTAKES. But try to correct at the occasion. Compare p.AR95–96 for the original, and keep in MIND the example of p.AR96. This is the $M$-scheme of degree 8 with symbol $\frac{1}{3}\frac{1}{17}$ which is prohibited by Rohlin, since $\pi-\eta=3$ (via Rohlin), but $\pi+\eta=20$, so $2\pi=23$ which is not soluble integrally. BUT yet another MISTAKE, this scheme is already prohibited by Gudkov (or even Arnold!) as $\chi=(1-3)+(1-17)=-18$ \[14.03.13\] Another basic question is whether from all schemes (in particular $M$-schemes) that we prohibited via Rohlin (and the signs-law), if it is not possible to corrupt RMC by finding an $(M-1)$-scheme right below which is maximal, but of course not of type I by Klein’s congruence $r\equiv g+1 \pmod 2$ for dividing curves. One such scheme in degree 8 was given by $(1,\frac{1}{1}2)17$ (cf. Fig.\[Fiedler7:fig\]c) with $\chi=16$. WARNING THIS IS A MISTAKE, as Rohlin’s equation can be solved with all signs positive!!! Of course there is a myriad of other such $M$-schemes verifying the Gudkov congruence $\chi\equiv k^2 \pmod 8$ but prohibited for different reasons (Rohlin with signs-law, or by Thom (\[Thom-Ragsdale:thm\])). So the vague idea is that if the scheme is French or Caucasian (i.e. prohibited by Thom resp. Rohlin but not by Gudkov) then the scheme is nearly realized in the sense that killing one of its oval then the GKK-congruence $\chi\equiv k^2\pm 1 \pmod 8$ (cf. \[Gudkov-Krakhnov-Kharlamov-cong:thm\]) is satisfied and so there is some hope to construct some $(M-1)$-curve. Further, and this is the most dubious part of the game, we would like that the resulting $(M-1)$-curve cannot be enlarged, which requires to inspect a menagerie of schemes. Of course we play this game [*à contre coeur*]{} as it is against our philosophy that the phenomenon of total reality is ubiquitous, and as posited by Rohlin 1978, that it governs the saturation principle (alias Rohlin’s maximality conjecture) saying that a scheme of type I is maximal in the hierarchy of all schemes. Note actually our logical MISTAKE, namely our project only disproof the half of RMC already disproved by Shustin, as it will exhibit a maximal $(M-1)$-scheme which is not of type I. However the harder game is to find a scheme of type I which is not maximal. Let us however work out an example of this disproof strategy for RMC as a potential application of the Rohlin formula and the signs-law obstruction. We start with an $M$-scheme which is prohibited by Rohlin though not by Gudkov. Our (fairly random) candidate is, as said above, the $M$-scheme in degree 8 given by $(1,\frac{1}{1})$ (cf. Fig.\[Fiedler7:fig\]c or Fig.\[Fiedler8:fig\]) with $\chi=16$. The latter can be diminished to an $(M-1)$-schemes, 3 typical ways being depicted on Fig.b, where either the trunk is killed, or a branch or an outer oval. Of course one could also kill the maximal nonempty oval like on Fig.c, but then the GKK-congruence is not verified as $\chi=19$. The 3rd specimen of Fig.b... WARNING A THIS STAGE I HAD TO STOP AS I NOTICED AN EARLIER MISTAKE!!!! -5pt0 -5pt0 E-mail correspondence {#e-mail-Viro:sec} ===================== \[09.01.13\] This section gathers responses given by experts (Viro, Marin, Orevkov, Kharlamov, Shustin, Le Touzé, Fiedler, etc.) to some naive questions of mine about the work of Rohlin. Here are the original messages in chronological order (inserted with the tacit approval of their authors). I acknowledge most sincerely their authors for the stimulating atmosphere it created and their generous answers. Messages are left in their original shapes safe for adding brackets \[ \] supplying electronically-updated label-links to the present text. \[09.01.13\] Two naive questions on Rohlin 1978 Dear Viatcheslav, Alexis, Oleg, Stefan and Grisha, Sorry for disturbing so many experts among yours with some little aspect of the work of academician Vladimir Abramovich. (I should have written this message in French, yet cannot remember exactly about Oleg’s progresses over the last 6 years in that language.) I was those last days quite fascinated by reading Rohlin’s 1978 survey on complex topological characteristics of real curves in some more detail. As you all know, he gave a quite spectacular enhancement of Gudkov’s pyramid for all schemes of sextics by enriching it with the data of Klein’s type I/II (1876). (Compare optionally the attached pdf file giving a graphical snapshot view of Rohlin’s classification.) My two questions are as follows. \(1) First Rohlin (1978) claims to have a certain synthetic argument (via pencils of cubics) able to show the type I of the schemes 6/1 2 and 2/1 6. He confesses however his argument to be a complicated one. Let me cite Rohlin exactly: “...when we apply it to curves of degree 3, we can establish (in a rather complicated way) that the schemes $\frac{6}{1}2$ and $\frac{2}{1} 6$ of degree 6 belong to type I. However, all the schemes that we have so far succeeded in coping with by means of these devices are covered by Theorem 3.4 and 3.5.” My first question is whether Rohlin’s synthetic argument has ever been published (assuming its truth of course)? I suspect the proof to be quite beautiful, but I am myself not able to prove it for the moment. Did one of you ever worked out the argument in detail, or remember about some exposition during Rohlin’s lectures? Is it of the same order of difficulty as the Hilbert-Rohn method, requiring “roughness” á la Andronov-Pontrjagin to turn round? Would it be didactically useful to publish (on the arXiv) an account of Rohlin’s argument if one is able to reconstruct it? \(2) The second question is of course the general Rohlin’s maximality conjecture (a scheme is of type I iff it is maximal in the hierarchy of all real schemes of some fixed degree). As reported in Viro’s survey (1986 Progresses over the last 6 years) it seems that one implication was disproved by Polotovskii and Shustin (combined efforts ca. 1982, 1985). Yet one implication looks still possible, namely type I implies maximal (if I am not wrong). It seems to me that this (last vestige of the) Rohlin conjecture could be proved somewhat eclectically in two lines via Ahlfors theorem (1950) on the total reality of orthosymmetric curves (alias type I). Namely if the curve is of type I, then there is a pencil of curves cutting only real points on the curve, so its real scheme cannot be enlarged without violating Bezout. q.e.d. Some more thinking shows of course this argument to be insufficient but maybe there is a (clever) way to complete it. Qu’en pensez-vous[^93]? Many thanks for your attention, and also for all your fantastic papers (I am presently trying to digest, so do not take the pain to answer me properly if my questions look too naive.) I apologize again for this collective message, but as the material is quite old, most of you probably forgot some details. So I hoped to maximize some chance of getting an answer from a collective chat room. Best regards, Alex PS: In attachment I send you a copy of an informal text of mine on the Ahlfors map. Section 24 (pp. 205–229) is more specifically devoted to Rohlin’s conjecture, yet contains nothing original (while being quite poorly organized). $\bullet$ On Wed, 9 Jan 2013 13:33:23 +0100 (alexandregabard@hotmail.com) wrote to Kharlamov, Marin, Viro, Fiedler, Orevkov, and Mikhalkin a collective e-mail titled “Two naive questions on Rohlin 1978”: Dear Viatcheslav, Alexis, Oleg, Thomas, Stepan and Grisha, Sorry for disturbing so many experts among yours with some little aspect of the work of academician Vladimir Abramovich. (I should have written this message in French, yet cannot remember exactly about Oleg’s progresses over the last 6 years in that language.) I was those last days quite fascinated by reading Rohlin’s 1978 survey on complex topological characteristics of real curves in some more detail. As you all know, he gave a quite spectacular enhancement of Gudkov’s pyramid for all schemes of sextics by enriching it with the data of Klein’s type I/II (1876). (Compare optionally Fig.71\[=\[Gudkov-Table3:fig\]\] on page 208 of the attached pdf file giving a graphical snapshot view of Rohlin’s achievement.) My two questions are as follows. \(1) First Rohlin (1978) claims to have a certain synthetic argument (via pencils of cubics) able to show the type I of the schemes 6/1 2 and 2/1 6. He confesses however his argument to be a complicated one. Let me cite Rohlin exactly: “...when we apply it to curves of degree 3, we can establish (in a rather complicated way) that the schemes $\frac{6}{1}2$ and $\frac{2}{1} 6$ of degree 6 belong to type I. However, all the schemes that we have so far succeeded in coping with by means of these devices are covered by Theorem 3.4 and 3.5.” My first question is whether Rohlin’s synthetic argument has ever been published (assuming its truth of course)? I suspect the proof to be quite beautiful, but I am myself not quite able to write it down for the moment. Did one of you ever worked out the argument in detail, or remember about some exposition during Rohlin’s seminar? Is it of the same order of difficulty as the Hilbert-Rohn method, requiring “roughness” à la Andronov-Pontrjagin to turn round? Would it be didactically useful to publish (on the arXiv) an account of Rohlin’s argument if one is able to reconstruct it? Many thanks if you have some ideas (or recent references) on those or related questions... \(2) The second question is of course the general Rohlin’s maximality conjecture (a scheme is of type I iff it is maximal in the hierarchy of all real schemes of some fixed degree). As reported in Viro’s survey (1986 Progresses over the last 6 years) it seems that one implication was disproved by Polotovskii and Shustin (combined efforts ca. 1982, 1985). Yet one implication looks still possible, namely “type I implies maximal” (if I am not wrong). It seems to me that this (last vestige of the) Rohlin conjecture could be proved (somewhat eclectically) in two lines via Ahlfors theorem (1950) on the total reality of orthosymmetric curves (alias type I). Namely if the curve is of type I, then there is a pencil of curves cutting only real points on the curve, so its real scheme cannot be enlarged without violating Bézout. q.e.d. Alas, some more thinking shows of course this argument to be insufficient but maybe there is a (clever) way to complete it. Qu’en pensez-vous? Many thanks for your attention, and also for all your fantastic papers (I am presently trying to digest). So do not take the pain to answer me properly if my questions sound too naive. I apologize again for this collective message, but as the material is quite old, most of you probably forgot some details. So I hoped to maximize some chance of getting an answer from a collective chat room. Best regards, Alex (Gabard) PS: The attachment[^94] is a copy of an informal text of mine on the Ahlfors map. Section 24 (pp. 205–229) is more specifically devoted to Rohlin’s conjecture, yet contains nothing original (except being poorly organized). $\bullet$ \[Viro’s answer the same day (09.01.13) ca. 20h00, additional footnotes are mine (Gabard)\] Dear Alexandre, Thank you for your message and manuscript. I was not aware about the Ahlfors theorem[^95]. It seems to be very interesting. I doubt though if it can be used for proving the half of Rokhlin conjecture. It gives a proof for impossibility of raising the number of components of a type I curve by a single algebraic Morse modification (what I called Klein’s thesis). I do not remember if I even ever heard about Rokhlin’s proof that you ask about, but the fact follows from the congruence. Slava[^96] did not mention it when he proved the corresponding congruence (at the moment the type was not yet considered). I learned this theorem from Slava in September 1977 and wrote down Slava’s proof to my notebook then. I guess the first proofs was[^97] published by Slava Nikulin (among many other statements) and Alexis Marin. Marin’s proof looks simpler, but requires Pin- structures. Best regards, Oleg $\bullet$ Gabard’s reply \[Same day (09.01.13) ca. 21h00\] Dear Oleg, Many thanks for your rapid and illuminating responses, plus all the historical details. If you see no objection, I would be very happy to cut-and-paste them in my survey. I still need to assimilate some congruences of the early phase (Rohlin, Gudkov-Krakhnov-Kharlamov, etc.) Hence you cannot imagine how your hints are illuminating my modest understanding of that golden period. Regarding Ahlfors, as you say, there is little hope to crack the big fish, yet of course I shall keep you informed if I get not too depressed by the immense difficulty. All the best, and so many thanks again, Alex $\bullet$ 10 Jan 2013 (Marin’s answer) Cher Gabard En plein déménagement, je met un peu plus de temps à vous répondre que Viro. Comme Viro, je ne connais pas la preuve de Rohlin pour votre première question (c’est pourquoi j’avais imaginé la preuve dont parle Viro qui est dans “Quelques remarques sur les courbes algébriques planes réelle”, votre référence 742\[=[@Marin_1979]\]) Cependant ce séminaire de Paris VII est dans un carton et y restera tant que je n’aurai pû trouver un nouvel appartement assez grand pour contenir ma bibliothèque et, n’ayant le temps d’aller à la bibliothèque, ma mémoire ne me permet pas de vous en dire plus que Viro. Pour la seconde question par contre je peux vous répondre, c’est à dire lever votre aveux d’incompréhension en fin (p. 226) de preuve du Lemme 24.20[^98]. Soit une courbe séparante gagnant un ovale de plus après franchiment d’un point quadratique ordinaire. Un argument de congruence (utilisant $d > 2$ dans le cas plan ou une hypothèse dans le cas général donnant que la désingularisée de cette courbe de franchiment est irréductible : l’ensemble de ses points complexe est connexe) donne que cette désingularisée de la courbe de franchiment est non séparante. Ainsi deux points non réels conjugués de la courbe de franchiment sont lié par un arc évitant la partie réelle, en particulier le point singulier, et par extension des isotipie[^99] un tel arc subsite dans toute déformation vers l’un des des deux côtés du discriminant, en particulier avant le franchiment la courbe est non séparante ce qui contredit l’hypothèse. Par contre si le franchiment du discriminant se fait en un point singulier plus compliqué il me semble que l’on peut augmenter le nombre de composantes connexes d’une courbe séparante. Je crois me souvenir que selon les constructions de Viro (ou peut être seulement après avec la présentation Itenbergienne de cette méthode de Viro) il y a une courbe singulière de degré 6 dont tout voisinage contient tous les types. N’étant plus familier du sujet depuis plus de 20 ans je ne peux vous en dire plus, par contre pour les surfaces de degré 3 vous trouverez dans le second tome des oeuvres de Klein un magnifique article illustré de non moins magnifiques figures où il établi que tous les types de surface cubique s’obtiennent par déformation de la (unique à changement projectif de coordonnées) surface cubique qui a 4 points quadratiques ordinaire. Merci de votre long article que j’essayerai de lire quand déménagement, vente,.... seront terminés. Bien cordialement et bonne année. Alexis Marin PS 1 Je trouve Viro un peu “oublieux” d’écrire “ (at the moment the type was not yet considered)”: en parcourant le second tome des oeuvres de Klein vous vous appercevrez qu’un sciècle avant Viro “tout” était chez Klein! 2 Vous trouverez un article historique, beaucoup plus court\* et sur un autre sujet en mettant dans la boite de recherche d’Arxiv le mot clef “troupeau” \*il fait 6 pages table des matières comprise et tout est dit (de façon “autocontenue”) dans le résumé en français de la première page, mais si vous remontez à toutes les références\*\* des commentaires bibliographiques celà peut vous prendre un peu de temps. \*\*accesibles à travers la "bibliothèque des sophomores http://alexis.marin.free.fr/BIB/ $\bullet$ Gabard’s answer \[12.01.13 ca. 23h00\] Cher Alexis, C’est avec une immense joie que j’ai reçu votre message. N’ayant pas d’internet à la maison, je l’ai seulement découvert ce soir en visitant mon père, qui lui est connecté. Je vais donc tenter d’assimiler toutes vos remarques savantes, et si vous le permettez, de les intégrer dans mon survey, en spécifiant bien sûr qu’il s’agit de vos contributions. De mon côté, je me demande si une courbe non-séparante peut toujours acquérir un point double ordinaire solitaire. (C’est semi-implicite dans Klein 1876 qui écrivait “noch entwicklungsfähig”, mais il me semble que ça contredit le résultat de Shustin 1985 (contre-exemple à la conjecture de Rohlin), dont la logique m’échappe quelque peu, mais j’ai sûrement raté une subtilité). Grâce à vos commentaires je devrais pouvoir produire prochainement une version plus solide et limpide de la section correspondante du survey, que je vous enverrai dès que possible. L’interaction avec Ahlfors me semble aussi prometteuse... Amitiés, et merci infiniment pour votre message, Alex PS J’espère que le déménagement se passe bien. Restez-vous à Grenoble, ou bien s’agit-il d’une opération plus conséquente? PPS: J’ai bien à la maison votre article de Paris VII, qui a toujours été mon meilleur compagnon (en 1999-2000), et je suis content de le retrouver pour ce point encore plus profond. PPPS: je me suis procuré une copie de l’article sur “il capo”, qui me semble fabuleux. Merci beaucoup. C’est exactement l’analyse que l’on rencontre à proximité de Dirichlet, etc jusqu’à Ahlfors, et Rogosinski, et que je dois essayer à l’occasion d’apprivoiser... PS 1 Je trouve Viro un peu “oublieux” d’écrire “ (at the moment the type was not yet considered)”: en parcourant le second tome des oeuvres de Klein vous vous appercevrez qu’un sciècle avant Viro “tout” était chez Klein! Vous avez parfaitement raison, et je suis moi même très “spécialisé ” dans l’oeuvre de Klein. Cependant le gros quiz, c’est l’assertion de Teichmüller 1941, qui prétend que Klein 18XX? anticipe Ahlfors 1950, de 70 ans environ. Toute courbe séparante (ou surface de Riemann orthosymmétrique, pour reprendre le jargon kleinéen) admet un morphisme réel vers la droite dont les fibres au dessus des points réels sont toutes exclusivement formées de points réels. C’est cet énoncé fondamental qui me semble être sous-exploité! Evidemment[^100] comme la noté Viro, il implique la partie facile de l’assertion de Klein (1876): une courbe séparante ne peut gagner un ovale spontanément comme une bulle de champagne surgit du néant. $\bullet$ Réponse de Marin (le lendemain 13 Jan 2013 ca. 09h00) de les intégrer dans mon survey, en spécifiant bien sûr qu’il s’agit de vos contributions. A part l’explication de votre doute (où relativement à l’article que vous citez il n’y a que les mots “extension des isotopies” en plus) ce ne sont que de très vagues souvenirs que je vous conseille de vérifier (éventuellement auprès de plus compétent : Viro, Itenberg,... avant de les intégrer) De mon côté, je me demande si une courbe non-séparante peut toujours acquérir un point double ordinaire solitaire. voulez-vous dire dont les deux directions tangentes sont complexes conjuguée? celà me parait très très optimiste. (C’est semi-implicite dans Klein 1876 qui écrivait “noch entwicklungsfähig”, Êtes vous sûr que c’est ce que pensait Klein, ou incluait-il dans ce terme les modification par franchiment d’une courbe ayant un unique point double qui est quadratique ordinaire à tangentes réelles “apparu en rapprochant deux points d’un même ovale”? mais il me semble que ça contredit le résultat de Shustin 1985 (contre-exemple à la conjecture de Rohlin), dont la logique m’échappe quelque peu, mais j’ai sûrement raté une subtilité). PS J’espère que le déménagement se passe bien. oui mais c’est long, à ce propos, vous trouverez sur http://alexis.marin.free.fr/BIB/papier/ la liste des livres que j’ai en plusieurs exemplaires et (sauf ceux dont la colonne “héritier” est remplie (par Vinel et/ou Guillou)) qui sont à la disposition de qui (en particulier vous) les demande. Restez-vous à Grenoble, ou bien s’agit-il d’une opération plus conséquente? Je reste près de Grenoble (mon adresse est dans la signature électronique ci-dessous PPPS: je me suis procuré une copie de l’article sur “il capo”, Voulez vous dire “Le capo”? Cependant le gros quiz, c’est l’assertion de Teichmüller 1941, qui prétend que Klein 18XX? anticipe Ahlfors 1950, de 70 ans environ. Toute courbe séparante (ou surface de Riemann orthosymmétrique, pour reprendre le jargon kleinéen) admet un morphisme réel vers la droite dont les fibres au dessus des points réels sont toutes exclusivement formées de points réels. Voulez-vous dire revêtement d’espace total l’ensemble des ovales? Il y a-t-il quelque chose de plus précis sur le degré et sa répartition parmis les ovales? Les références sont-elles dans votre article? C’est cet énoncé fondamental qui me semble être sous-exploité! Evidemment comme la noté Viro, il implique la partie facile de l’assertion de Klein (1876): une courbe séparante ne peut gagner un ovale spontanément comme une bulle de champagne surgit du néant. Soyez plus précis pourquoi un tel morphisme admettrait-il une déformation le long de la modification d’adjonction d’un ovale? Amitiés. Alexis – http://le-tonneau-de-thales.tumblr.com/ Alexis Marin, chez Danielle Bozonat 6 Allée de la roseraie, 38240 Meylan fixe : 04 76 00 96 54 port. : 06 38 29 33 99, 00351925 271 040 $\bullet$ Gabard 13 Jan 2013 ca. 13h30 Cher Alexis, Merci pour votre message. Je vais en effet essayer d’intégrer vos commentaires de manière ciblée et prudente. De toute manière avant d’arXiver une nouvelle version d’ici six mois environ, j’aurai l’occasion de vous montrer précisement la prose que je vous aurez emprunté. J’essaye maintenant de répondre à vos questions: De mon côté, je me demande si une courbe non-séparante peut toujours acquérir un point double ordinaire solitaire. voulez-vous dire dont les deux directions tangentes sont complexes conjuguée? celà me parait très très optimiste. REPONSE: Oui, exactement à tangentes imaginaires conjuguées. Cela me parait aussi très optimiste. Klein semble le prétendre semi-implicitement (du moins qu’il n’ y a a priori pas d’obstruction topologique à la formation de telles bulles de champagne). Cependant si ce truc fou (“Klein-vache”) est vrai alors un des sens de la conjecture de Rohlin 1978 (type I iff maximal real scheme) est vérifié. Malheureusement, ce que donne “Klein-vache” est le sens de Rohlin détruit par Shustin 1985 (dont je n’ai cependant pas compris l’argument). Mais vous avez surement raison “Klein-vache” est probablement beaucoup trop optimiste... Êtes vous sûr que c’est ce que pensait Klein, ou incluait-il dans ce terme les modification par franchiment d’une courbe ayant un unique point double qui est quadratique ordinaire à tangentes réelles “apparu en rapprochant deux points d’un même ovale”? REPONSE: je pense que oui, car Klein précise “isolierte reelle Doppeltangente”, comparez ma Quote 24.2[^101] page 205 de mon survey (si vous n’avez pas le volume 2 de Klein sous la main). Ainsi il me semble que votre interprétation moderne (Marin 1988) diffère un peu de l’original Kleinéen, en étant toutefois plus puissant que l’assertion d’origine. PS J’espère que le déménagement se passe bien. oui mais c’est long, à ce propos, vous trouverez sur http://alexis.marin.free.fr/BIB/papier/ la liste des livres que j’ai en plusieurs exemplaires et (sauf ceux dont la colonne “héritier” est remplie (par Vinel et/ou Guillou)) qui sont à la disposition de qui (en particulier vous) les demande. C’est une magnifique liste de trésor. Je voudrais bien les acquérir, mais je me demande si mon hygiène de vie (overwork) rend une telle acquisition raisonable...(Il faudrait que je passe à Grenoble avec la camionnette de mon oncle pour récupérer les “invendus”. Il est préférable en effet de trouver des preneurs plus compétents que moi. Si en dernier recours, vous ne trouvez pas de preneurs je pourrais récupérer les volumes restants en vrac...Merci infiniment pour cette généreuse proposition. Moi même je suis très marginal financièrement et spatialement, petit appartement à Genève partagé avec ma mère (avec environ 8 tonnes de littérature mathématique), mais dans le futur je pourrai peut être m’installer dans une ferme fribourgoise, où il reste de l’espace pour expandre la bibliothèque...) PPPS: je me suis procuré une copie de l’article sur “il capo”, Voulez vous dire “Le capo”? Oui, j’essayais d’improviser en italien, mais c’est une langue plus subtil que vous utilisez... Cependant le gros quiz, c’est l’assertion de Teichmüller 1941, qui prétend que Klein 18XX? anticipe Ahlfors 1950, de 70 ans environ. Toute courbe séparante (ou surface de Riemann orthosymmétrique, pour reprendre le jargon kleinéen) admet un morphisme réel vers la droite dont les fibres au dessus des points réels sont toutes exclusivement formées de points réels. Voulez- vous dire revêtement d’espace total l’ensemble des ovales? Il y a-t-il quelque chose de plus précis sur le degré et sa répartition parmis les ovales? Les références sont-elles dans votre article? OUI, toute surface de Riemann à bord (=membrane compacte) s’exprime comme revêtement holomorphe ramifié du disque. C’est juste une version relative (à bord) du théorème d’existence de Riemann qui concrètise toute surface de Riemann close comme revetement conforme de la sphère (ronde). Il a fallut toutefois attendre la contribution d’Ahlfors 1950 qui donne en plus un contrôle sur le degré d’un tel revêtement conforme, à savoir r+2p, où r est le nombre d’“ovales” (mieux le nombre de contours de la membrane), et p son genre. La Thèse de moi-même (Gabard 2004, et l’article de 2006 au Commentarii Math. Helv.) donne un meilleur contrôle, à savoir $r+p$, en économisant donc une cartouche pour chaque anse. Les références précises sont dans le survey. L’énoncé d’Ahlfors était vachement anticipé dans le cas $p=0$ (membrane planaire ou schlichtartig pour reprendre la terminologie de Paul Koebe) par la grande lignée Riemann 1857 (Nachlass), Schottky 1875-77, Bieberbach 1925 et Grunsky 1937. Lorsqu’on passe au double de Schottky-Klein de la surface à bord on obtient (via Ahlfors) une courbe séparante avec un morphisme totalement réel vers la droite projective. Inversement toute courbe séparante est totalement réelle, puisqu’il suffit d’appliquer Ahlfors à une des moitiés orthosymétrique de Klein. \[Gabard\] C’est cet énoncé fondamental qui me semble être sous-exploité! Evidemment comme la noté Viro, il implique la partie facile de l’assertion de Klein (1876): une courbe séparante ne peut gagner un ovale spontanément comme une bulle de champagne surgit du néant. \[Marin\] Soyez plus précis pourquoi un tel morphisme admettrait-il une déformation le long de la modification d’adjonction d’un ovale? \[Gabard\] Je pense que ça marche car lorsque la courbe est plongée dans le plan, le morphisme total d’Ahlfors admet une réalisation projective comme un pinceau de courbes planes dont tous les membres découpent seulement des points réels sur la courbe orthosymmétrique (=séparante). Par conséquent, en traçant la courbe du pinceau total qui passe par un point de l’oval spontanément créé, on obtient une contradiction avec Bézout. Donc Ahlfors 1950 implique Klein 1876, mais votre démonstration de 1988$-\varepsilon$ (votre preuve est déjà mentionnée dans Viro 1986) est surement plus intrinsèque et voisine de l’argument d’origine de Klein (s’il en avait un au delà de la pure contemplation empirique des quartiques notamment...) Merci infiniment pour vos messages, et d’ici tout bientôt (3-4 jours) je vous enverrai une version mise-à-jour du survey qui clarifiera peut-être les assertions précédentes. Toutefois les grands problèmes et plein de détails m’échappent encore dans la pyramide Gudkovo-Rohlinienne. Quelle splendide pyramide qui joint à la perfection Klein et Hilbert! Un détail qui m’échappe, c’est le fait que le discriminant est de degré $3(m-1)^2=75$ pour $m=6$, tandis que que du point de vue des chirurgies “de Morse” il y a des cycles de longueur 4 dans la pyramide de Gudkov. Donc il y a un problème de parité si on déforme le long d’un pinceau générique (transverse au discriminant)...Désolé, de vous embêter avec ces détails que j’ai honte de ne pas réussir à clarifier depuis quelques jours. Amitiés, et bon courage pour la suite du déménagement, Alex $\bullet$ \[16h40 15.01.13\] Cher Alexis, Merci encore pour vos messages et vos remarques fascinantes que je dois encore bien digérer. De mon côté, j’ai fait de minimes progrès, et vous envoie malgré votre déménagement une version ajournée de mon survey. Il me semble que le truc fou dont nous parlions il y a quelques jours, que j’appele depuis “Klein-vache”, i.e. la possiblilité de faire naitre un noeud solitaire (à tangentes conjuguées) depuis n’importe quelle courbe diasymétrique est vrai pour les sextiques. Pour cela j’utilise un argument qui combine Rohlin 1978, Klein-Marin 1988, et Nikulin 1979 (classification isotopique) et un résultat relié de Itenberg 1994 (possibilité de contracter n’importe quel ovale vide, i.e. sans autre ovales dans son intérieur, sur un tel noeud isolé). Les détails de la preuve sont exposés dans la Prop.24.24\[meanwhile this is \[Klein-vache-deg-6:prop\]\], page 235 du fichier ci-joint. ? Evidemment, en principe “Klein-vache” n’a aucune chance d’être vrai en degré supérieur. Cependant la seule obstruction que je connaisse est ce résultat de Shustin 1985, dont je ne comprends toujours pas la logique de base (sans même parler du fait que c’est fondé sur la méthode de Viro, dissipation de singularités tacnodales..., une technologie que je n’ai jamais maitrisée). Mes objections naives à l’argument de Shustin se trouvent en page 248 (dans le paragraphe qui précède la Figure 94\[=meanwile Fig.\[Shustin:fig\]\]). Dans cette figure, je ne sais pas comment prohiber le $(M-1)$-schémas encadré par le carré vert (à mi-hauteur de la figure), et dans son article de 1985 Shustin n’est pas trés explicite. Mais bon, il s’agit la d’une question assez ennuyeuse et en fait je vais peut-être prendre l’initiative d’écrire un nouveau message collectif pour clarifier ce point d’ici quelques heures. Merci infiniment encore pour vos messages, et meilleurs voeux de courage pour la suite du déménagement, Amitiés, Alex PS: Pour l’instant j’ai inégré en vrac tous nos échanges e-mail dans le survey (p.219 et suivante), mais bien entendu dès que possible je censurerai les remarques plus confidentielles..., et masquerai les répététitions, voire l’intégralité de la discussion si je parviens bien à résumer votre apport malgré mon anglais catastrophique. Cependant en relisant vos remarques, elles apportent une prose substantielle que je ne saurais jamais reproduire en anglais, donc je trouverais très dommage de censurer vos souvenirs en vracs!!! Evidemment rien ne presse et je suis désolé de vous avoir dérangé durant cette délicate opération du déménagement inter-grenoblois. Amitiés, encore, et je vous tiens au courant d’éventuelles progrès...Je suis surtout curieux des réponses de Shustin (et Viro) s’ils parviennent à éclairer ma lanterne. PPS: Je joins une copie de la note de Shustin, si jamais, mais je ne veux pas vous distraire de votre tâche prioritaire... $\bullet$ \[15.01.13–18h30\] Dear Evgenii, Ilia, Oleg and Alexis (and Felix Klein), I was much fascinated those last days by Evgenii’s counterexample to (one part of) Rohlin’s maximality conjecture to the effect that a real scheme is of type I iff it is maximal in the hierarchy of all schemes. Quite interestingly this work of you (Shustin) also destroys an old (semi-)conjecture of Klein (1876) positing that any nondividing plane curve can acquire a solitary node by crossing only once the discriminant (the resulting Morse surgery then sembling like the formation a champagne bubble arising like a blue sky catastrophe of little green men’s coming with flying saucers). Alas from Shustin’s note of 1985 (in its English translation), I was not quite able to understand your proof (compare optionally the attached file, on page 248, in the paragraph right before Figure 94\[=Fig.\[Shustin:fig\]\]). In fact I do not know how to prohibit the $(M-1)$-scheme $4/1 2/1 1/1 11$ enlarging Shustin’s (M-2)-scheme. Alas I am not an expert in the field and I feel quite shameful disturbing you with such a detail. Despite having myself full Leningradian origins (through my father), I do not master the Russian language so that it may well be the case that the original Russian text is more detailed than its translation. Of course it is much more likely that I missed something well-known, that you perhaps may not have made completely explicit in the note? (Incidentally I send you a copy of Shustin’s note for convenience!) I apologize for this question of detail, yet it seems quite important to me for your result of 1985 is the only obstruction (I am aware of) to the naive desideratum of truth about Klein’s conjecture. Klein himself is extremely clever and quite ambiguous about stating this as a conjecture or as a result (compare optionally Klein’s original quote reproduced on page 206 of the attachment). Today I managed as a simple exercise to check the truth of Klein’s hypothesis in degree 6, via an armada of Russian results (especially Itenberg 1994 contraction principle for empty ovals), plus the Klein-Marin theorem (for the details of this exercise cf. optionally Prop.24.24\[=\[Klein-vache-deg-6:prop\]\] on page 235 of the attached text). You, Oleg Viro, in the preface of that volume presenting Itenberg’s article (1994) advanced the (crazy?) conjecture that one might always be able to contract empty ovals!!! Do you know if there is meanwhile some counterexample (in high degrees)? Of course there is some vague parallelism between Itenberg’s contraction and the one required to implement Klein’s hypothesis (which must amount shrinking an anti-oval, i.e. an invariant circle acted upon antipodically by conj). Sorry again for disturbing you with all these naive questions, and do not take the pain answering me properly if you are overwhelmed by other more important duties. Many thanks for all your attention. Sincerely yours, Alex (Gabard) $\bullet$$\bullet$$\bullet$ \[16.01.13–02h57: Oleg Viro\] Dear Alexandre, I do not mind to pose crazy conjectures. I do not mind if my crazy conjecture would be disproved. However, I suspect that my conjecture is not as crazy as possibility of shrinking of an anti-oval. The difference between the oval and an anti-oval is that the oval is assumed to exist and be empty, i.e., not linked with the complex curve in whatever sense, while the anti-oval apparently has none of these properties. I am not aware about any counter-examples that you ask about. I do not bet that they do not exist, but find the question stimulating, and better motivated than the conjecture that was proven to be wrong. Best, Oleg $\bullet$$\bullet$$\bullet$ \[16.01.13–14h56: Stepan Orevkov\] A small remark: It is wrong that $11 U 1<1> U 1<2> U 1<4>$ is not a part of an $(M-1)$-scheme. It is[^102]. Moreover, there is no known example of $(M-2)$-curve of type II which cannot be obtained from an $(M-1)$-curve by removing an empty oval. In contrary, there are $(M-1)$-curves of degree $8$ (which are necessarily of type II) which do not come from any $M$-curve. These are: $3<6>$ $4 U 1<2> U 2<6>$ $8 U 2<2> U 1<6>$ $12 U 3<2>$ Constru\[r\]ction (inspired by Shustin’s construction of $4 U 3<5>$): Consire\[der\] a tricuspidal quartic $Q_{sing}$ symmetric by a rotation $R$ by $120$ degree and perturb\[e\] is\[=it\] so that each cusp gives an oval (we assume that this perturbation is very small). Let $Q$ be the perturbed curve. Two flex points appear on $Q$ near each cusp of $Q_{sing}$. We chose flex points $p_0, p_1, p_2$ (one flex point near each cusp) so that $R(p_0)=p_1, R(p_1)=p_2, R(p2)=p_0$. We choose homogeneous coordinates $(x_0 : x_1 : x_2)$ so that the line $x_i = 0$ is tangent to $Q$ at $p_i$ $(i = 0,1,2)$. Let $C$ be the image of $Q$ under the Cremona transformation $(x_0 : x_1 : x_2) \mapsto (x_1x_2 : x_2x_0 : x_0x_1)$. Then $C$ has 3 singular points, each singular point has two irreducible local branches: a branch with $E6$ and a smooth branch which cuts it “transversally”. By a perturbation of $C$ we obtain all the four curves mentioned above. The fact that these curves cannot be obtained from $M$-curves immediately follows from the fact that, for any $M$-curve of degree 8 of the form $b U 1<a_1> U 1<a_2> U 1<a_3>$, all the numbers $a_1$, $a_2$, $a_3$ are odd[^103]. Best regards Stepa O $\bullet$ \[17.01.13 ca. 23h00\] Dear Oleg and Stepa, Many thanks for all your fascinating remarks and detailed answers. I look forward digesting them carefully tomorrow. Sorry for my late reply as I have no internet at home and was quite busy trying to understand some basic facts, notably that one may have some “eversion” of a real scheme when the oval explodes at infinity undergoing a Morse surgery not affecting its connectedness. This implies that there is some hidden passages in the Gudkov-Rohlin pyramid of all sextics changing a Gudkov symbol $k/l \ell$ to its mirror $\ell/1 k$. The resulting combinatorics of this graph looks quite formidable and I wonder if it is known whether each of those secret edges corresponding to eversions (except those linking $M$-curves) can be explored algebraically. Perhaps the problem is related to Ilia’s shrinking process for empty ovals, but seems to involve yet another species of “anti-ovals”, namely those with two fixed points under conj, yet located on the same oval. All what I am saying is for sure well-known to you since time immemorial, yet I was very happy to understand this point which solved several paradoxes of mine, notably those related to the degree of the discriminant and the contiguity graph between chambers residual to the discriminant under elementary algebraic Morse surgeries, as Oleg says. Of course, I shall send you an updated version of my file, when I manage to reorganize slightly the exposition. Many many thanks for all your excellent answers! All the best, Alex $\bullet$ \[18.01.13 ca. 10h00, Viatcheslav Kharlamov\] Dear Alex, I followed rather attentively the discussions, but kept silence since had no much to add to the reaction of the others. This “eversion”, as you call it, played some important rôle in the prehistory of the Gudkov conjecture. As you probably know, the first classification declared by Gudkov was wrong, and it is one of his “thesis referees”, Prof. Morosov, who had objected the first classification exactly because of a small irregularity with respect to “eversion” of the answer. Repairing this asymmetry Gudkov came to his final result, and, if my memory is correct, in particular, at this stage discovered the missing $M$-curve. If honestly, I don’t remember did somebody ever before discussed seriously any conceptual explanation to this “eversion”. However, it was implicitly present in all results obtained through $K3$ and their lattices. Recently, studying the shadows of cubic surfaces with Sergey Finashin and having proven, to our own surprise, for them a very similar “symmetry”, which we have called “partners relation”, we have formalized it as follows. First level of explanation is coming from lattices of double coverings: the partner relation consists indeed in transferring an $U$-summand (unimodular even lattice of rang $2$ and signature $0$) from one eighenlattice to another. Second level of explanation is coming from moduli in terms of periods: each partner in the partner pair can be deformed to a triple conic, near the triple conic the family looks as $Q^3+tbQ^2+t^2cQ+d=0$, and switching of the sign of $t$ (passing through the triple conic) replace curves of one deformation class by curves from the partners class: moreover, such degenerations are deformationally unique. Literally the same explanation (and with much easier proofs at the both levels) works for nonsingular sextics (the shadows are sextic curves with $6$ cusps on a conic; remarkably, in many respects they behave in a way more similar to that of nonsingular sextics, than other sextics with singularities). Yours, Viatcheslav Kharlamov $\bullet$ \[18.01.13, Kharlamov, title of message=Correction\] Writing the message a bit in a hurry I did not describe fully and appropriately the partner relation at the lattice language. The summand $U$ does play a crucial rôle, and it should be moved from one eighenlattice to another, but then additionally one should exchange the eighenlattices. In fact this $U$ contains indeed the $2$-polarization vector, $h¨2=2$, and thus the eighenlattice containing this distinguished U is aways $(-1)$-eighenlattice. The existing exception to the partner relation (as I remember, in the nonsingular case, there is only one) is the case when the $(-1)$-eighenlattice does not contain such a pair $(U,h)$. Sorry, for being in a hurry, but I should stop at this point. Hope that now it is more clear. $\bullet$ \[18.01.13, Gabard, ca. 21h00\] Dear Viatcheslav, Oleg and Stepa (and all the others), So many thanks for all the excellent comments, especially on Morosov. There was some allusion to this issue in Viro’s survey from 2006, in Japanese Journal of Math, as to the lack of symmetry in Gudkov’s initial answer. Yet Morosov was not mentioned if I remember well... On my side I was quite stimulated by the last letter from Oleg, about the contraction conjecture, as looking indeed much more realist than Klein’s Ansatz on the champagne bubbling in any nondividing curve. I attempted today to imagine what sort of proof one could expect to find for this fascinating Itenberg-Viro contraction conjecture of empty ovals. After some trials with orthogonal trajectories to the functional computing the area of the empty oval, I arrived at some sort of strategy (probably completely fantasist) consisting in using the Riemann mapping theorem as applied to the interior of the empty oval. Naively as the contour is algebraic so is the Riemann map and hence the concentric sublevels of it ought to be algebraic curves of the same degree!!??? This would give the shrinking. I am sure that tomorrow while checking more carefully the details all this argument will crash down. Hence sorry for this premature message. Some more details about this and my naive understanding of “eversions” are in the attached file, especially Section 24.15 (p.251) and p.242 (Section 24.12 for eversions). Regarding eversions I wonder which edges in the Gudkov pyramid are actually realized algebro-geometrically? All, except those connecting the $M$-schemes is my naive guess, yet it is probably too optimistic... Many thanks again for sharing all your knowledge on that fascinating topic, and all your exciting letters. All the best, Alex \[21.01.13, ca. 20h00\] Dear real geometers, Thank you again, Oleg, Alexis, Stepa, and Viatcheslav, for all your messages which I have carefully integrated in my TeX-notes, and to which I frequently refer for citation in my text. Your messages suggested me several ideas I would never have explored without your precious hints. On my side, I noticed of course that the cavalier Riemann mapping strategy toward the (Itenberg-Viro 1994) contraction conjecture (CC) of empty ovals fails blatantly (cf. Section 25.7(=\[CC-via-Riemann:sec\]), pages 255-258, roughly even if the Riemann map of an algebraic oval would be algebraic then its degree seems to be twice as big as it should, or better the polynomials arising as norms of algebraic Riemann maps are not the most general representatives of their degree!!!). Perhaps the Riemann method works for special ovals, but of course they are unlikely to be interspersed in all chambers of the discriminant! This failure drifted me toward another formulation of the contraction conjecture which I call CCC, for collective contraction conjecture. This posits that all empty ovals of a real algebraic curve can be contracted simultaneously toward solitary nodes (by a path having solely its end-point in the discriminant). This looks even more “crazy” than CC, but I found no counterexamples (in my pockets). I would much appreciate if you already thought about this natural variant, especially if you detected some counterexample (perhaps arising from the Viro-Itenberg patchworking method or the dissipation of higher singularities, with which I am alas still unfamiliar with, like in Shustin’s counterexample to Rohlin’s maximality conjecture). Here are the trivialities I managed to prove. Via Brusotti 1921, it is plain that CCC implies the usual contraction conjecture (CC) (cf. details in Lemma 25.22(=\[CCviaCCC-Brusotti:lem\]) on page 263 of the attached file). On the other hand CCC implies (as a large deformation principle) several well-known prohibitions. E.g. a two-seconds proof of the Hilbert-Rohn-Petrovskii prohibition of the sextic $M$-scheme $11$ (eleven ovals without nesting), as well as Rohlin’s prohibition of the sextic scheme $5$ of type I (by the way causing the unique asymmetry in the Gudkov-Rohlin table of sextics). Under CCC, all these facts appear as trivial consequences of Bézout (compare Section 25.8(=\[CCC:sec\]) on pages 258-259). I found this simplicity quite exciting (even though it leads to nothing new as compared to Arnold-Rohlin). One can wonder if Hilbert already used this, at least as a heuristic tool??? Philosophically, I found also interesting that such large deformation conjectures produce prohibitions, in contradistinction to small perturbations as being primarily a method of construction (Harnack-Hilbert-Brusotti, etc.). There is accordingly some nice duality between Luigi Brusotti and Ilia-Oleg’s contraction conjecture. Of course you surely noted this issue a long time ago, yet for me it was a happy discovery (yesterday). Perhaps CCC and CC are actually equivalent, yet this looks more hazardous but maybe not completely improbable... (One would just have to synchronize the death of all ovals posited by CC.) This is all the modest news I have collected during the week-end. Of course I still have some naive hope that CCC (hence CC) could be attacked via some gradient flow, but it looks quite difficult to locate the right functional (or Morse function). Looking at the area (or length) of all empty ovals is probably too naive...Perhaps some “degree of roughness” à la Gudkov could be projectively more intrinsic and useful... Thank you so much for your attention and all your brilliant letters and answers, while apologizing me for sending you only easy doodlings. Best regards, Alex \[26.01.13, ca 20h00\] Dear Oleg, Stepa, Viatcheslav, Evgenii, Alexis, Thomas, etc. I continued my naive investigations of real plane curves. What a beautiful story! I finally “understood” and studied in detail the marvellous construction of Gudkov $\frac{5}{1} 5$ (ca. 1971-73), as to understand the more tricky (but related) construction proposed by Stepa, which I attempted to depict on Fig. 111(=\[Orevkov2:fig\]) of the attached file. (I did not as yet assimilated the full details but feel on the good way. In fact I tried to use the dissipation of $Z_15$ in Viro’s survey from 1989/90 in Leningrad Math. J., which I hope is the same as the $E_6$ advocated by Stepa. Sorry for being very ignorant about singularities...) Yesterday, I also finally understood the correctedness of Evgenii’s argument. (As helped by Stepa’s e-mail, the point which I missed is this obstruction of Viro extending that of Fiedler) for $M$-schemes of degree 8 as having necessarily “odd content”. On the other hand, I was scared (since three days) by the fact that something which I subconsciously thought as evident (or rather which I was sure to have read somewhere) is perhaps not true. My (naive) question is whether two empty curves are necessarily rigid-isotopic? This looks at first between metaphysical nonsense and “triviality”? Maybe it is unknown, when $m$ is large enough. (m=6 follows from Nikulin 1979, and as far as I know there is not a simpler proof, say valid for all (even) degrees). So I am quite shameful asking you about this point: Is the empty chamber always connected? I tried a dynamical approach (to this problem) in Section 25.12(\[rigidity-empty-scheme-via-dyna:sec\]), but it is not very convincing. On the other hand, if the empty room is connected, then maybe the space of all curves with one component is also connected? (Naively one would apply the Itenberg-Viro contraction conjecture, to reduce to the empty case, move there for a while to resurface at the other curve (the contraction thereof). Perturbing this path in the “visible world” would conclude the proof modulo some difficulties...) Again, you Oleg, in your wonderful survey of 2008 (in Japanese J. Math) lists as an open problem the question of deciding the rigid-isotopy of curves of odd degree having a unique real circuit. As you emphasize the word “odd degree”, I wondered if the case of even degree (again with only one oval) is already settled? In Section 25.10(=\[CCCviaDynamics:sec\]), I have attempted a naive dynamical approach to the collective contraction conjecture(CCC). This states that we can shrink simultaneously all the empty ovals toward solitary nodes. This is a bit like a perfect landing in flight simulator where all wheels touch the ground simultaneously. My naive strategy is just to study the gradient lines of the functional measuring the total area of all empty ovals, but it is surely not serious. It would be exciting, in my opinion, to describe a counterexample to CCC if there is one. Many thanks for the attention, all your patience about my naive reasonings, and above all for the brilliant answers you already gave me. Best regards, Alex $\bullet$$\bullet$$\bullet$ samedi 26 janvier 2013 20:15:54, the prompt response of Eugenii Shustin: Dear Alex, The chamber of empty curves of a given (even) degree is indeed connected: two such curves can be defined by homogeneous polynomials, positive for any real not all zero variables, and their linear homotopy $(1-t)P+tQ$, $0\le t\le 1$, gives a path in the chamber of empty curves. By the way, another (well) known connected chamber consists of hyperbolic curves (i.e. those which have totally real intersection with lines of certain pencil) - this is a consequence of Nuij W. A note on hyperbolic polynomials. Math. Scandinavica 23 (1968), no. 1, 69–72. With best wishes, Eugenii $\bullet$$\bullet$$\bullet$ samedi 26 janvier 2013 21:08:27, Oleg Viro: Dear Alex, The counterpart of the Rokhlin conjecture[^104] about rigid-isotopy of any two curves of odd degree with one component is the obvious observation described by Evgenii, about empty curves of even degree. The question about curves of even degree with a single oval is equivalent to the question about removing this single oval by an algebraic Morse modification. I don’t think it was ever discussed, but I could miss it. $Z_{15}$ is not $E_6$. The easiest way to construct the Gudkov $M$-curve is by perturbing two $J_{10}$ singularities of the union of 3 non-singular conics tangent to each other at 2 points.[^105] Best, Oleg $\bullet$ \[28.01.13, lundi 28 janvier 2013 20:03:58\] Gabard wrote euphorically[^106] an e-mail titled “Some more metaphysical non-sense about the rigid-isotopy of empty curves?”: Dear Eugenii, Oleg, Viatcheslav, Alexis, Stepa, etc. So many thanks, Eugenii, for putting me again on the right track, and recalling me the argument which I shamefully forgot about. Yesterday, I was quite excited by trying to digest your argument (albeit it seems so simple). In fact the little detail that worried me is that I do not know why during the linear homotopy $(1-t)P+tQ$ the variable curve could not acquire (while staying of course empty if $P,Q$ have both the same sign) a pair of conjugate nodal singularities. This puzzled me for a while, and then using systematically your argument, I arrived at the somewhat opposite conclusion that the empty (smooth) chamber must be disconnected (for all even degrees $m\ge 4$)!!! This violates all what we know since Rohlin 1978 (and surely Gudkov as well??), while the former refers directly back to the argument of Klein 1876 based on Schläfli cubics surfaces $F_3$’s and Zeuthen correspondence between cubic surfaces and quartics (via the apparent contour). Klein’s proof is a bit tricky and uses as well his rigidification (Klein 1873) of Schläfli’s isotopic classification. Needless to say I could not follow Klein’s reasoning completely, as I just studied it today for ca. 2 hours. Marin informed me recently that he, in contrast, was able to digest all of those Kleinian works! So using your method of linear homotopy, one sees quickly that the (cone) space $C^+$ of positive anisotropic (=not representing zero) forms is contractile (convex actually) hence simply-connected. Its projection in the space of curves is the invisible locus $I$ consisting of all empty curves. Since the latter is merely a quotient of $C^+$ (by positive homotheties) it follows that it is also simply-connected (via the exact homotopy sequence of a fibering). But the discriminant is visible inside this invisible locus $I$, since it is a simple matter via Brusotti (1921) to construct empty curves with a pair of conjugate nodes. Thus we see inside the simply-connected manifold $I$ a certain hypersurface (namely a portion of the discriminant), which by Jordan-Brouwer (or a slight extension thereof) should separate this manifold $I$ in pieces (at least so is my naive intuition). It follows that our empty chamber (consisting of smooth curves) is disconnected!!!! This is my proof in its broad lines (for more details, compare Section 25.13, page 282-283 of the attachement, Theorem 25.29 and its proof on page 283). This is just one page long... Since this conclusion contradicts violently what is asserted by Klein 1876 (and approved by Rohlin 1978), it is of course very likely that my proof contains a serious flaw, or at least that I am confusing somehow the basic conceptions. However presently I do not see where is my mistake! Of course, my pseudo-theorem also violates the part of Nikulin 1979 concerned with the rigid-isotopy of the empty chamber of sextics. Many thanks again for your attention, and sorry for overflowing your mail boxes with my naive questions (and dubious reasonings). Thank you again so much for all your excellent and detailed responses (especially on $E_6$ and $Z_15$). Best wishes, Alex PS: I send you a copy of my TeX-file in case someone would like to work out a specific passage. At the occasion I would also be happy to send you my figures in zipped format so that one of you can continue the project in case I make a fatal bicycle accident (like Academician V.I. Arnold?) $\bullet$ \[30.01.13, 18h10\] Dear Oleg, Eugenii, and the other experts, I think that I found the mistake in my “proof” of the disconnection of the empty locus (that you certainly noticed meanwhile in case my explanation is the correct one). The reason seems to be simply that the discriminant inside the invisible locus has only real codimension 2, hence cannot separate anything. I have attempted to explain this in Section 25.14 on page 288. If this is not wrong it seems that the next natural question is to decide which chambers residual to the principal stratum of the discriminant contains such smaller pieces of the discriminant shrunk to codimension 2. I think to have found a topological obstacle for $M$ and $(M-1)$-curves, and conjecture (very naively) this to be the sole obstruction. In more geometric terms, this amounts essentially to decide which smooth curves can acquire a pair of imaginary conjugate nodes. Many thanks, Eugenii and Oleg, for your detailed answers. As you said, it seems that the (Itenberg-Viro) contraction conjecture of empty ovals implies the rigidity conjecture for even order curves with a unique oval. However I should probably still try to understand this implication in some more details. Perhaps it is somehow related to the previous codimension 2 phenomenon inside the “invisible” chamber. Sorry for all my confusing messages, and many thanks again for all your kind efforts in trying to educate myself. All the best, Alex $\bullet$ \[01.02.13, ca. 20h00\] Obstruction to rigid-isotopy (strictly) below height DEEP+2? Dear Oleg, Eugenii, Alexis, Thomas, Stepa, etc. Many thanks for all your brilliant messages and articles I am still slowly trying to assimilate properly. I hope not taking too much of your precious time. Albeit I met all of you only rarely, I remind very accurately your brilliant talks (in Geneva or Rennes), and so it is a special pleasure to remind each of yours while trying to explore this fantastic topic. On my side I was those last two days fascinated by the conjecture that the one-oval scheme ought to be rigid, as Oleg or Rokhlin conjectures. (Let me say that a scheme is rigid, if all the curves representing it are rigid-isotopic.) Given a degree $m$, one may wonder what is the smallest height $r(m)$ at which there is a non-rigid scheme. (For me the height of a scheme just means its number of components.) For any degree $m$, there is of course the deep nest with $r=[(m+1)/2]=:DEEP$ real branches. Two units above the latter’s height, it is easy to construct (for each $m$) curves having the same real scheme yet different types (I vs. II) hence not rigid-isotopic. (This is a simple iteration of Rohlin’s construction in degree $5$, cf. Figs. 102, 103 in my file). Using the Marin-Fiedler method of the lock it is even possible to exhibit at this height $DEEP+2$ curves of degree 7 or 9 having the same real scheme and the same type II, yet not rigid-isotopic (Figs. 105, 104). (Probably the method extends to all other odd degrees.) However, it seems much more tricky (and the lock-method seems ineffective) to detect obstruction below this height $DEEP+1$ (i.e. one unit above the height of the deep nest). Could it be the case that all schemes at or below this height are rigid? Of course this looks super-optimistic as we do not even know rigidity at height one, but I was unable to find a counterexample. I would be very happy if you know one? If there is a simple candidate, I hope to detect it alone during the next few days…. So do not take care answering me if my question is trivial. (As I just work on this since two days, I probably missed something accessible.) Paraphrasing slightly, I found quite puzzling, that the very explicit function $r(m)$ measuring the smallest height of a non-rigid scheme is only subsumed to the large pinching $1 \le r(m)\le [(m+1)/2]+2=DEEP+2$. Of course a better lower bound seems out of reach, but perhaps you know better upper bounds. I also wondered if there is an extension of the Nuij-Dubrovin rigidity of the deep nest to, say, the totally real scheme of degree 8 consisting of 4 nests of depth 2. I should think more seriously on this at the occasion. Many thanks for your attention, and sorry again for all my enthusiastic and naive e-mails. All the best, Alex $\bullet$ \[written 08.02.13 and sent 09.01.13\] Dear Oleg, Eugenii, Stepa, Viatcheslav, etc. I still continued my trip through real plane curves and cannot say that my curiosity is starting to fade out. I tried for several days to find a counter-example to the conjecture (of mine so probably quite wrong) that all schemes below height $DEEP+2$ are rigid, where $DEEP=[(m+1)/2]$ is the number of branches of the deep nest of degree $m$. At least the method of the lock (Fiedler-Marin) seems quite inoperant to detect an obstruction to rigid-isotopy at such low altitudes. If true, the proof probably involves a geometric flow collapsing either the pseudoline to a line (by shortening its length like a systole) or improving the rotundity of some oval to a circle (via an isoperimetric functional?). If all this works, it would reduce the low-altitude rigidity conjecture to Nikulin’s theorem (or maybe even Klein’s on $C_4$) as the starting step of a big recursive process. Of course this seems still quite out reach (canary music) unless one feels very motivated! Next I tried to corrupt the truth of Slava’s remarkable rigid-isotopic classification (Nikulin 1979) of sextics via the Marin-Fiedler locking argument using Bézout saturation. Of course I have nothing against Slava, but this was rather intended to test experimentally the power of Nikulin’s result. Specifically I looked at sextic schemes of the form $3/1 \ell$, and wondered if for some specific curves the distribution of the $\ell$ outer ovals away the fundamental triangle traced through the 3 inner ovals (those enveloped by the unique nonempty oval) could be different for different curves. On all examples I tested it seems that the outer ovals are never separated by the “deep” triangle. So we find no violation of Nikulin’s theorem, and the latter rather implies that as soon as we are able to visualize the distribution for a single curve it will be the same for all curves belonging to this scheme. The case most tricky to understand is the maximal permissible, namely $3/1 5$. I managed to construct it à la Harnack (as preconized in Gudkov 1974 or 1954). But being quite unable to decide from this model the distributional question of the outer ovals past the fundamental triangle, I decided to switch to Oleg’s method of construction via dissipation of the singularities of a triplet of coaxial ellipses. I played this game yesterday but could not decide the distributional question for this Viro curve (cf. especially Fig.126 on page 320 and the hypothetical Theorem 26.29 on page 322). In fact today I tried again to inspect directly Harnack construction and found Lemma 26.27 on page 319 whose proof seemed to me very transparent until I found the little warning, which I think is not fatal. In conclusion I believe now that there is no separation by the fundamental triangle!!!?? Of course I imagine that, if I am not completely wrong, what I am investigating must be quite familiar to you. I would much appreciate if you know if this hypothetical theorem (26.29 page 322) is true. It amounts essentially to check whether in Viro’s construction of $3/1 5$ the triangle through the 3 deep inner ovals does not separate the 5 outer ovals. I find this question quite attractive as it seems to require some understanding of the geometric location of the microscopic ovals arising in Viro’s method (optionally compare Fig. 127 (page 322) which shows a scenario with the two bottom micro-ovals aligned vertically in which case the fundamental triangle would separate the outer ovals). This scenario seems to me quite unlikely but it does not seem to be impeded by naive Bézout obstructions. Many thanks for your attention, and sorry again for all my naive and confuse questions. Thank you very much again for your precious guidance and answers. All the best, Alex $\bullet$$\bullet$$\bullet$ (10.02.13) Bonjour Alexandre, Thomas m’a transmis ta question. La réponse est toute simple: soient $A$, $B$, $C$ trois ovales intérieurs et $D$, $E$ deux ovales exterieurs de ta sextique. Le triangle fondamental $ABC$ est entiérement contenu dans l’ovale non-vide. Si $D$ et $E$ sont dans deux triangles $ABC$ (non-fondamentaux) différents, alors la conique passant par $A$, $B$, $C$, $D$, $E$ coupe la sextique en $14$ points, contradiction. Avec des coniques, on montre plus généralement que: Les ovales vides de la sextique sont distribués dans deux chaines (int, ext), l’ordre cyclique est donné par les pinceaux de droites basés dans les ovales interieurs. Les ovales interieurs sont disposés en position convexe dans l’ovale non-vide. Bon dimanche,   Séverine $\bullet$ \[12.02.13\] Is the Gudkov chamber simply-connected? Dear Séverine, Viatcheslav, Ilia, Oleg, and all the other experts, First many thanks, Séverine, for your excellent answer on my distribution question of ovals of sextic, and sorry for my late reply on it as I lack an Internet connection at home. I tried today to understand when a dividing (plane) curve admits a transmutation, i.e. a rigid-isotopy permuting both halves of the curve. I also studied the weaker notion of mutation of when there is a linear automorphism of the plane permuting both halves. Using the Kharlamov-Itenberg calculation of the monodromy of sextics I think that I managed to get some obstruction to mutability, especially for the 3 dividing curves which have trivial monodromies (compare Lemma 26.6, page 288, which is hopefully correct). However I don’t know if the Gudkov chamber (or the 2 other related “antidromic” chambers, i.e. having trivial monodromies) is simply-connected. I hoped to detect some non simple-connectivity by looking at the monodromy induced on the halves instead of the ovals. At least this works of course for the deep-nest chamber which is not simply-connected since there is a symmetric model which can be mutated. So my (hopefully not too naive) question is the following: is it known whether or not the Gudkov chamber is simply-connected? (equivalently is the Gudkov curve transmutable?) The same question looks attractive for the other 2 antidromic curves, i.e. the left wing “Rohlin curve” $6/1 2$ and $4/1 4$ in type I. Thank you so much for all your attention and patience, and in advance for your answer if it is known. Best regards, Alex $\bullet$$\bullet$ $\bullet$ mercredi 13 février 2013 04:34:12 Dear Alexandre, If I understand correctly the question then the answer is not, if I state the question appropriately then the answer is yes. I mean the following precise statements. Let consider the part of the projective space of real sextics that is represented by maximal sextics of Gudkov’s type. Then the fundamental group of this part is $Z/2$. It becomes simply connected after taking quotient by the natural action of $SL(3,R)$. In fact, before factorization it is a fibration over contractible base with the fiber $SL(3,R)$. These results (and there analogs for other maximal sextics and certain curves of lower degree) are contained in my talk On monodromies of real plane algebraic curves at one of Petrovsky seminars in 80th, I guess (short summary should be found in Russian Surveys). The proof (in the case of sextics) is rather straightforward as soon as based on the $K3$ surfaces periods uniformization. As it happens rather often with this approach, to treat the maximal curves is extremely easy, since the corresponding eighenlattices become unimodular. In general the period domain, which is the product of two polyhedra in the real case, represents the studied sextics (or associated K3 surfaces) only up to codimension 2. Which makes laborious to treat the fundamental group. But, surprise, in the case of maximal curves there are no codimension 2 phenomena, since such holes appear only as traces of $(-2)$-cycles having nontrivial components in the both eighenspaces, which is impossible since in the maximal case the components are integral and the eighenlattices are even. I don’t remember by heart the final result for other maximal sextics. It should be pointed in the same summary and by the way easy to get following the same approach I have pointed. The key is that even if it is no more a pure fibration - it has special fibers which are quotients of $SL(3,R)$ by the corresponding monodromy group (which indeed coincides with the maximal possible group of symmetries for the given type of sextics) - its fundamental group is exactly the fundamental group of this special quotient. Yours VK $\bullet$ mercredi 13 février 2013 11:46:20 Dear Colleagues, Am I alone who did not receive a copy of Severine’s letter? I would be happy to know its content :) Yours VK $\bullet$ mercredi 13 février 2013 13:43:10 Dear colleagues, I had written only to Alexandre, sorry! My answer was this: let $A, B, C$ be three inner ovals, and $D, E$ be two outer ovals of the sextic. The fundamental triangle $ABC$ is entirely contained in the nonempty oval. If $D$ and $E$ are in two different (non-fundamental) triangles $ABC$, then the conic through $A, B, C, D, E$ cuts the sextic at 14 points, contradiction. Using conics, one proves more generally that there is a natural cyclic ordering of the empty ovals, given by the pencils of lines based at the inner ovals. The empty ovals are distributed in two consecutive chains (inner, outer). The inner ovals lie in convex position in the nonempty oval. Best regards, Séverine $\bullet$ \[14.02.13\] Dear Viatcheslav, Séverine and all the other colleagues, Thank you very much for this beautiful answer on the Gudkov chamber. I look forward to digest properly all that incredible technology that you and Nikulin developed. Again many thanks also to Séverine for the clever argument which I digested yesterday with great pleasure, and integrated in my notes in Section 26.10(=\[LeTouze:sec\]) pages 332–334. This gave me yesterday some motivation again to attack the very first question of all our chat room, namely Rohlin’s claim that the pencil of cubics through the 8 deep basepoints located inside the 8 empty ovals of any sextic curve of type $6/1 2$ or its mirror $2/1 6$ is totally real, hence of type I (also called orthosymmetry by Klein ca. 1881-82 and his student Weichold 1883). In fact I (naively) hoped to prove this Rohlin claim via Poincaré’s index theorem, yet the qualitative picture (Fig. 133 on page 337) rather inclined me to believe that the proof cannot reduce to mere combinatorial topology of foliations (i.e. Poincaré’s index formula of 1885). So I am still puzzled, but perhaps an argument like Séverine’s one do the job. At any rate I would be very excited if someone manages to reconstruct this proof asserted by Rohlin (1978) if it is not too tantalizing for the brain. Otherwise I am also much frustrated by failing to visualize totally real pencil on the three $M$-sextics, whose existence is I think predicted by Ahlfors theorem of 1950 (or better the special zero-genus case thereof known to Riemann 1857, and reworked by Schottky 1875-77, or even Bieberbach 1925 and his more respectable student Grunsky 1937). Marin warned me recently that the transition from the abstract Riemann surface viewpoint to the planar context “of Hilbert’s 16th problem” may be not so easy as I always assumed subconsciously. (If necessary, all the correspondence I received from all the colleagues is gathered in Section 24.6, p.221). Overpassing this difficulty (which I hope is not fatal) there should be on all $M$-curves (more generally dividing curves) auxiliary pencils which are totally real. Alas for $M$-sextics (even $M$-quintics), I am completely unable to trace them and know nothing about the degree of the curves involved (in the pencil). I hope to be able to tackle such questions in the future, but perhaps you have better ideas (or motivations) than I do have. Thank you very much again for all your brilliant answers, and kind messages. All the best, Alex $\bullet\bullet\bullet$ samedi 16 février 2013 17:54:55 Dear Alexandre, dear other colleagues, I have managed to prove that a pencil of cubics with eight base points distributed in the eight empty ovals of a sextic $2 \cup 1(6)$ is necessarily totally real. Details will follow soon in a paper. Yours, Séverine $\bullet$ \[16.02.13,19h41\] Dear Séverine and colleagues, Congratulations for this fantastic achievement. I am sure the proof must be very beautiful. On my side I tried to work out for all sextics of type I an optical recognition procedure of the type by some synthetical procedure akin to Rohlin’s claim, yet this is still much in embryo. In particular the case of $(M-4)$-sextics is quite puzzling as it seems to contradict the version of Ahlfors theorem due to myself (existence of a totally real map of degree the mean value the number of ovals and Harnack’s bound). I hope to send you more palatable material soon, but confess that the questions look quite hard and I seem much less efficient than Séverine. So I suppose that Rohlin’s claim is one among several other (less pure) total reality result. So I look forward with great interest to see Séverine’s article. All the best, Alex \[19.02.13\] Dear colleagues, Many congratulations again to Séverine for your fantastic achievement. Sorry to have been brief in my last letter, as I wrote (lacking an internet connection at home) from a friend of mine who had a romantic party with his girlfriend, and I do not wanted to disturb too long his romantic evening. Meanwhile I also tried hard to concentrate on a proof of the Rohlin-le Touzé’s theorem, which still overwhelms my intelligence. The last things that I have written are on pages 336–352 (Sections 27.1, 27.2, and 27.3), but this is poorly organized and supplies no serious proof of the Rohlin-Le Touzé’s theorem. Some few days ago, I got Theorem 27.5 (on page 346), which (if it is correct) answers one of the question I asked in my penultimate e-mail (as well as desideratum of Alexis), namely the question of estimating the order of curves involved in a total pencil on an $M$-curve. It seems that there is always such a pencil of order $(m-2)$, i.e. two units less than the given degree $m$ of the $M$-curve. In fact, the proof is a nearly trivial adaptation of the abstract argument going back to several peoples (in chronological order Riemann 1857, Schottky 1875, Enriques-Chisini 1915, Bieberbach 1925, Grunsky 1937, Courant 1939, Wirtinger 1942, Ahlfors 1947, 1950, a myriad of Japaneses, a myriad of Russians including Golusin 1953/57, etc....., up to Huisman 2000, and Gabard 2001/2006, who else?). The point is that total reality is trivial in the case of $M$-curves since we have one point circulating on each oval (such a group moves by Riemann-Roch!!!) and so we have like a train-track with only one train on each track, hence no collision can occur and total reality is automatic. If we work with plane curves we only need to take curves of order $(m-2)$ which have enough free parameters to pass through any given distribution of $M$ points (one on each oval), and this works by looking at the residual group of points (details in the proof on page 346). So this is quite interesting but probably only a first step toward deeper things. (One could dream to recover all the Gudkov-Rohlin/Arnold congruence via this method but that looks hard work...) After this little discovery I focused again on the Rohlin-Séverine theorem, yet without any success. So I have not more to report for the moment. Thanks a lot for the attention, and all my congratulations again to Séverine for your deep advance. Best wishes, Alex $\bullet$ 19.02.13 Dear Alex, Let me ask you a question from your previous field of interest. Do you know any example of a non-Hausdorff 1-manifold which does not admit a differential structure? I heard about existence and could easily construct examples of exotic, i.e., homeomorphic but not diffeomorphic non-Hausdorff 1-manifolds. See http://www.map.mpim-bonn.mpg.de/1-manifolds Sincerely, Oleg 20.02.13 Dear Oleg, David and Mathieu, Many thanks, Oleg, for your lovely question, and best greetings to the other friends. Alas my memory is failing quite dramatically, so my answer will be of poor quality. If I remember well I asked myself the same question some 3-4 years ago, but I cannot record to have ever found an answer. Thus I forward your question to David and Mathieu, the leading experts of non-metric surfaces who perhaps will supply a better answer. On my side I hope to think more seriously to your question when I see clearer with Rohlin-Le Touzé’s sextics. Maybe a first idea is that there ought to be a (non-canonical) “twistor construction” assigning to each non-Hausdorff curve a Hausdorff surface fibered by (real) lines. This construction should go back to Haefliger’s very first note in the colloque de topologie de Strasbourg ca. 1955-1956 (yet it is not very detailed). In substance it is like a train-track construction à la Penner-Thurston…(some intuition about this is given in my article ‘Ebullition and gravitational clumping, arXiv, 2011). Do not worry if you don’t understand me, as I myself remember only vague souvenirs and are not so convinced by what I am saying!!! In fact Haefliger (ca. 1956) claims this construction only for second countable curve (even with a proviso on the fundamental group), but when I was in touch with the subject I was fairly convinced that it must work universally. OPTIONAL REMARK: Haefliger, and Haefliger-Reeb 1957 use this construction to prove that any simply-connected curve (second countable) arises as the leaf space of a foliation of the plane. (Sketch of proof: take the twistor of the given curve which is by the exact sequence of a fibering 1-connected and (by Poincaré-Volterra) second countable, hence it is the plane, q.e.d) So the idea would be to descend a smooth structure on the surface to get one on the curve. Alas, it is a well-known open problem whether any (non-metric but Hausdorff) surface admits a smooth structure (Spivak 1971, Nyikos, etc.) However quite puzzlingly Siebenmann 2005 (Russian Math Surveys) claims (and even prove in some details) that a PL structure exists universally on all such surfaces, merely as a consequence of Schoenflies theorem. So perhaps Siebenmann argument work as well for DIFF structures, and the metaphysical problem of Spivak-Nyikos is cracked. If this works (ask maybe Siebenmann, or an Indian in the States(=Ramachandran) who albeit not an expert was fairly convinced that there should be no asymmetry between PL and DIFF in dimension 2), then there is perhaps some chance to get a smooth structure on all non-Hausdorff curves. Of course there is perhaps a more direct strategy without transiting through surfaces. Otherwise, regarding exotic smooth structures on curves the original reference is Haefliger-Reeb 1957 article in L’Enseignement Math. Perhaps you could quote this in your brilliant web-page. Sorry for this vague answer, but at the moment my brain is much concentrated on this Rohlin-Le Fiedler total reality claim which still puzzles me a lot!!! Best greetings to all, as well as to Rachel and Chiara. All the best, Alex $\bullet$ \[22.02.13\] Dear colleagues (especially Séverine), I worked hard (but without success) on the Le Touzé’s theorem, at least for 8 basepoints assigned on the nonempty ovals of a sextic of type $6/1 2$. If I understood well Séverine’s announcement, you rather handle the case of $2/1 6$ and assign more generally the points in the insides of the empty ovals (but of course I suppose that your argument adapts to $6/1 2$). Even in my weaker form I am not really able to conclude but send you my last thinking on the question (Section 27.4, p.352–356, esp. Fig.141). Ultimately I found a method which I call “barrages”. A special rôle is played by nodal cubics of the pencil, and I try to get a corruption with Bézout by looking at nodal curves with a barrage, i.e. such that 4 arcs of some other cubic joins pairwise the 8 basepoints distributed on the loop of the original cubic. (By the loop of a nodal cubic, I mean the unique path from the node to itself which is null-homotopic in the plane $RP^2$.) Of course I am not sure that details can be decently completed, but for the moment it is the only reasonable strategy I could imagine. I am sure that Séverine’s argument is much more elegant and convincing. My reasoning is completely conditioned by Fig.141, and I am probably too naive in believing that it reflects the general situation. Sorry for sending you this very coarse material, and of course do not take the pain to react to this message. Many thanks again a lot to all for sharing so generously your knowledge and for all your answers. Best regards, Alex \[25.02.13\] Dear real geometers, I was still much fascinated by the Rohlin-Le Touzé theorem (RLT) albeit still not able to prove it. Being frustrated by my failing attempts (probably due to a lack of stubbornness and competence in algebraic geometry) I decided to speculate a bit of why it is so important or at least to explore how the statement could generalize. In its most elementary incarnation involving pencil of lines and conics, the phenomenon of total reality occurs along infinite series stable under the operation of satellite of a real scheme (of even order). Satellite just amounts to trace each oval with a certain multiplicity $k$ (jargon obviously borrowed from knot theory). So the unifolium scheme of degree 2 (allied to a conic) gives rise to the deep nests, and the quadrifolium scheme of degree 4 gives rise by taking its satellites to an infinite series of schemes of order multiples of 4 which are totally real under a pencil of conics (assigned to pass through the deepest ovals). It seems therefore natural to ask if the satellites (e.g. the second satellite) of the Rohlin’s scheme $6/1 2$ (or its partner $2/1 6$) are also totally real (and hence of type I) under the “same” pencil of cubics as posited by the Rohlin-Le Touzé theorem. Alas I was not even able to settle this question. (Of course this seems evident (granting RLT) for a small perturbation of the algebraic double (essentially $F \cup F+\epsilon$), since total reality forces transversality of the foliation induced by the pencil with the curve.) Next, I tried to understand what are the higher order avatars of the RLT-theorem (in the hope that it is not an isolated phenomenon as vaguely suggested by Ahlfors theorem). I found using the Rohlin-Kharlamov-Marin congruence ensuring the type I-ness (=orthosymmetry) of some $(M-2)$-schemes an (obvious) infinite series of avatars of the Rohlin’s $(M-2)$-schemes of degree 6 . Those are also $(M-2)$-schemes and total reality seems to be possible for a pencil of curves of order $(m-3)$, exactly like for the Gürtelkurve of Zeuthen-Klein (bifolium quartic with 2 nested ovals totally flashed by a pencil of line through the deep nest) or for the Rohlin’s sextic (flashed by a pencil of cubics). So it seems that the theory of adjoint curves of order $(m-3)$ plays some special rôle in this question of Rohlin-Séverine. I would be very happy if one of you knows if it is reasonable to expect an extension the RLT total reality theorem to all this schemes whose type I ness is ensured by Rohlin-Kharlamov-Marin congruence (sorry if I am not hundred percent right in crediting as I could not extract the exact history of this subliminal result). Specifically I have Conjectures 27.17 and 27.18 (page 365 and 367 resp.) which list some candidate-schemes for total reality in degree 8 and 10. If the conjectures are right, it would be of great interest to know if Séverine’s proof adapts to them. Sorry if I am too naive about the real difficulty of such problems, but I found exciting to wonder if there is something more general behind the cryptical allusion of Rohlin. Of course I presume that he derived the synthetic result a posteriori from highbrow topology (or Kähler geometry in Kharlamov’s case?), but perhaps there is a simple explanation with (“basic”) algebraic geometry and total reality as Séverine was able to do? As Oleg knows my problem is that I wasted too much time with non-metric manifolds and so forgot all the little I ever knew about algebraic geometry. During the way, I think to have found a counterexample to the conjecture of mine (inspired by the Itenberg-Viro contraction conjecture of empty ovals), and according to which all empty ovals could be contracted simultaneously to solitary nodes. This counter-example is Thm 27.16 on page 364 (which I hope is correct and sharp as far as the degree is concerned). Thanks a lot for the attention, and sorry for all the modest news (you surely thought about in sharper form already). All the best, Alex PS: The material summarized in this message occupies page 357-367 (Sections 27.5, 27.6, 27.7), as usual I had not much time to polish, but I hope it is still readable. \[27.02.13\] A census of 100 octic $(M-2)$-schemes of type I satisfying the RKM-congruence, plus a little addendum for Oleg’s non-Hausdorff curves Dear colleagues, I have pursued some preliminary study toward the total reality phenomenon, yet merely in its combinatorial aspect prompted by the modulo 8 RKM-congruence (for Rohlin-Kharlamov-Marin) ensuring the type I of $(M-2)$-schemes of degree $2k$ with $\chi = k^2+4 \pmod 8$. Accordingly, I call an RKM-scheme any $(M-2)$-scheme satisfying this congruence. While any RKM-scheme is of type I, I do not know alas whether the converse statement is true. If it is known I would be extremely grateful if someone can tell me (and our collective chat room) the answer. Further I noticed that the list given in my previous e-mail of RKM-schemes of degree 8 can be much enlarged. If I am not too bad in combinatorics, there are precisely 100 such schemes in degree 8, all of them being potentially subsumed to the phenomenon of total reality under a pencil of quintics akin to the Rohlin-Le Touzé theorem (for sextics flashed by cubics). This modest material is to be found in Section 27.8, p.368-373 (especially Fig. 146 page 370 and Lemma 27.24, p.372, plus all the 36 Gudkov symbols on page 372). I hope of course that I missed nobody in this catalogue. Extrapolating a bit using the (hypothetical) converse statement to RKM, I would say that there are precisely 100 schemes of type I which are $(M-2)$-schemes. Is this well-known and correct? Actually, I do not really know if all these 100 schemes are realized algebraically, but presume that most of them (all?) are. Possibly I am much too naive. Of course it is quite amazing to see that the only two RKM-schemes of degree 6 (namely $6/1 2$ and $2/1 6$) demographically explodes to a menagerie of 100 such schemes in degree $8$, but that should be no surprise for you much acquainted with the higher cases of Hilbert’s 16th problem. It would be even more crazy if all those 100 schemes (or at least a good portion thereof) are subsumed to the phenomenon of total reality. If you have some ideas on those circle of ideas, I would be extremely thankful. Many thanks again for the patience and attention, and I hope that what I am telling is nearly correct (not too surrealist). Very best regards, Alex PS: For Oleg, regarding my loose answer on smooth structures on non-Hausdorff 1-manifolds, I would like to add another philosophical remark related to the method of Haefliger’s “twistor”. This is of course like a thickening along a normal bundle except that there is no ambient manifold (safe the ether) and so the construction must be intrinsic. To my knowledge it was never exposed in details (albeit Haefliger’s 1st article ca. 1955-56 in Colloque de Topologie de Strasbourg uses implicitly this construction). Now my point is that albeit the twistor method looks somewhat indirect, I think that it is fairly useful. For instance, I was since 2006-07 puzzled by the naive question if the fundamental group of a one-manifold is always a free group. (Of course such non-Hausdorff curves resemble somehow graphs, whence some intuition). For instance the line with 2 origins has $\pi_1=Z$ as follows quickly from Seifert-van Kampen (and if 3 origins or 2 doubled origins then $\pi_1=F_2$ is free of rank 2). Ultimately in 2011 I found a general answer to this “freeness” puzzle by using the Haefliger twistor construction, while showing first that all open (non-metric) surfaces have free fundamental groups. (This is actually a very modest extension of the metric case, which to my knowledge is first treated in Ahlfors-Sario book of 1960, albeit it may have belonged to the folklore much earlier, say Kerekjarto, H. Kneser, Rado, in the 1920’s, Papakyriakopoulos in the 1940’s???). This material is exposed in some details in my arXiv note of ca. 2011 (Ebullition in Foliated surfaces versus gravitational clumping). I hope that those results are nearly correct but they certainly require more professional treatments and exposition than I was able to do. I hope this little remark makes perhaps more plausible that the approach via the (Hausdorffizing) Haefliger twistor is also reasonable for your problem of DIFF structures. $\bullet\bullet\bullet$ vendredi 1 mars 2013 19:07:12 Dear Alexandre, dear other colleagues, here is the note I had promised to send you. There are still many open questions, as Alexandre wrote. It would be also interesting to know whether one could find a totally real pencil with respect to the dividing $M-2$-sextics with real scheme of indefinite type. I will think about it when I have more time. Best regards, Séverine (01.03.13, 22h15) Dear Séverine and the other colleagues, So many thanks Séverine for sending us your splendid article. I am much excited to read the details tomorrow, as myself started today to doubt about the whole result (at least in the strong form that any points 8 points distributed on the empty ovals ensures total reality). (If I am not wrong the whole phenomenon depends upon the location of the 9th base point, namely the pencil is totally real iff the 9th base point lands in the inside of the nonempty oval.) So I was much depressed and lost in my poorly organized thoughts. So your sending arrives as a true deliverance for my brain. Many congratulations again to Séverine for this fantastic work. Very best regards, Alex \[02.03.13\] Can total reality fail for a distribution of 8 points on the empty ovals? Dear Séverine and the other geometers (especially Professor Nikulin), I enjoyed much a detailed look at your splendid article full of illuminating remarks. I will probably need much more time to digest the impressive technology you use, and need to print the material to make a deeper reading (especially of the former works upon which your argument seems to depend). So many thanks again for sending us your work in so rapid delay. I wrote some naive reactions in Section 27.11, where I mostly copied your sayings, and tried to add hopefully pertinent footnotes. Regarding your question “Can conversely any dividing curve be endowed with some totally real pencil?”, I still wonder if a positive answer is not a trivial consequence of Ahlfors theorem (compare very optionally Gabard’s Thesis 2004, page 7). However since Marin warned me in January 2013 (cf. Section of e-mails) it may be the case that the transition from the abstract conception of Riemann-Schottky-Klein to the embedded viewpoints of Hilbert-Gudkov-Rohlin is not so easy. Yet I am still confident (or naive enough) to believe that it holds true. The point seems to be primarily a matter of projective algebraic geometry, namely the question if any abstract morphism on a concrete plane curve to the line $\PP^1$ is induced by a (linear) pencil of ambient curves. This is either trivially true or trivially wrong, but alas I do not know the answer due to my failing memory about the foundations of algebraic geometry. Your article already helped much as I suffered under the misconception that your result states that any distribution of 8 points on the empty ovals induces a totally real pencil. Your statement is much more subtle, yet personally I do not know if this stronger (universal) form of total reality is wrong! If you know a counterexample foiling universal total reality I would be very happy. It could then still be the case that there is some special sextics for which universal total reality holds true, i.e. for all octuplets distributed on the empty ovals. (Perhaps reading more carefully your article, especially the aspect related to Nikulin-Kharlamov’s rigid-isotopic classification already answers those questions?) (The newest material of mine (as usual confusing and poorly organized) occupies Section 27.9–27.11 on pages 373–378. Here I attempted a topological approach to the existence of octuplets inducing a totally real pencil, but alas was not able to conclude, presumably because I know too little on the predestination process creating the 9th basepoint as a function of the 8 assigned ones.) Many congratulations again to Séverine for this breakthrough. Best regards, Alex dimanche 3 mars 2013 18:07:57 a new version with small corrections? Dear Alexandre, dear other colleagues, I owe you some apologies: the Theorem was slightly incorrect, as Alexandre pointed out. I let you discover this new version, where I have reformulated the Theorem, and added a few words in the end of the proof. Best regards, Séverine (04.03.13) Dear Séverine and the other colleagues, Many thanks for the new version. In fact, it seems that the main change is that you now assign the 8 basepoints ON the empty ovals instead of IN their insides. Rereading my previous message, I realize that I misstated your original statement and so it is pure chance that assignation on the ovals turned out to be “more correct”. Your fascinating article gave me new forces to think about the problem, but alas still without success. For instance, I still do not know if there exist octuplets (on the empty ovals) failing to induce a totally real pencil. Of course assigning them in the insides gives more freedom, but presently it looks to me harder to ensure total reality. So despite your correction, it could still be the case (in my modest understanding) that the pencil is total for all octuplets chosen in the insides of the empty ovals. Perhaps you know a counterexample to this strongest form of the statement? Many thanks again for the article, which guided much my thinkings. I hope to send you more exciting news soon, but the whole problem which you call “the lost proof of Rohlin” seems to me still much out of reach. All the best, Alex $\bullet$$\bullet$$\bullet$ answer to Alexandre’s questions (mardi 5 mars 2013 13:30:42) Dear Alexandre, dear other colleagues, let me try to answer the question with a new formulation. Assume first that the base points are distributed [*inside*]{} of the empty ovals. Applying your nice “dextrogyration argument” to all nine ovals gives the following lemma: [*The pencil is totally real iff 9 lies inside of the non-empty oval $O$ and outside of the empty ovals.*]{} If 9 is outside of $O$, the bad cubics are as shown in Figure 2 of the paper. If 9 is inside of an empty oval $X$, the bad cubics have an oval passing through the two base points 9 and X only, and this oval is entirely contained in the empty oval denoted also $X$. To get rid of this latter possibility, it suffices to take the base points [*on*]{} the empty ovals. In ii), I give an explicit description of the pencil, valuable for any generic choice of the eight base points [*inside*]{} of the eight empty ovals. (It turns out that the only possible non-generic situation is that of a pencil with a double base point $9=2$, this means that the points 1, ..8 lie on a nodal cubic with node at 2.) Recall that 2 is the base point chosen in the extreme inner oval forming a positive pair with $O$. For this pencil, the only possibly bad cubics are those with an oval passing through 9 and 2 only. To grant total reality, it suffices to choose the base point 2 [*on*]{} the corresponding empty oval, the other base points lie arbitrarily in the inside discs of the other empty ovals. Thus, your conjecture 27.29\[=\[SRLT:conj\]\] is true, and an even stronger result holds for the sextic with six inner ovals. Best regards, Séverine \[07.03.13\] Little news from Alex, and so many thanks to Séverine for the answer Dear Colleagues, First many thanks to Séverine for your very detailed answer (which I will study in detail tomorrow). Sorry for being always a bit differed in time due to my lack of internet at home. I added some material in my loose notes. In Section 28.1–28.2 (pp.384–392), I tried once more to explore the grand programme that Rohlin might have had in mind, namely total reality and its connection with his maximality conjecture. As I often said it seems to me that the missing link could be played by Ahlfors theorem, or perhaps Rohlin had a grand vision that he could arrange total reality by purely synthetical processes extending in all degree the already tricky theorem of Rohlin-Le Touzé in degree $m=6$. This idea when explored in full looks to me extremely vertiginous, but its net impact would be a sort of upper bound upon the complexity of Hilbert’s 16th problem, and in some sense subsume all prohibitions (à la Gudkov et cie.) to the paradigm of total reality. All this necessitates to be made much more precise, but I \[have\] attempted to make a psychoanalysis of what Rohlin may have had in the brain, without that he himself ventured to put it on the paper due to his own modesty and pragmatism. Next I discovered the little Theorem 28.7\[=\[Thom-Ragsdale:thm\]\] (p.393), which is just a matter of making explicit the consequence of Thom’s conjecture (=Kronheimer-Mrowka theorem) as it pertains to Hilbert’s 16th problem. The result is the lovely estimate[^107] $\chi \le k^2$ for a curve of type I and degree $2k$. With this I realized that my former counterexample (with the scheme $20$ in degree 8) to CCC(=collective contraction conjecture) is actually killed by Thom, and realized (later only!!) that it is also killed by Rohlin’s formula. So CCC is again resuscitated but probably not for long!? Then I tried to make a comparative study of Rohlin’s formula versus the Thom obstruction. It seems that the latter is often implied by Rohlin’s formula, but not always. More in Section 28.4 (p.393). It seems however that at least for degree $m \ge 10$ there is some cases where Thom really affords new information not covered by Russian congruences or Rohlin’s formula (cf. Thm 28.11, p.396). Finally using the Gudkov table in degree 10 (=Fig.148 on page 395), I got some naive hope to disprove the Rohlin maximality conjecture, but this quickly turned into disillusion (cf. Point 3 on p.396–397). Sorry for all these messy remarks, yet I found the rôle of Thom quite pleasant. I am sure that this is not new, and that I read it somewhere, but again could not recover where precisely. (I thought it was in Degtyarev-Kharlamov 2000’s survey but apparently not, though Kronheimer-Mrowka is alluded to.) If you remember some anecdotes about the rôle of Thom’s conjecture in Hilbert’s 16th problem, and who puts it first into action as a such, I would be extremely happy to insert your remarks in my (messy) survey. Thanks a lot for the attention, Best regards, Alex $\bullet\bullet\bullet$ Thomas Fiedler wrote (samedi 9 mars 2013 17:32:59) Dear Alexandre, I am no longer in business in this field, but let me just make some remarks which could be perhaps helpful. The $M$-curve of degree 10 mentioned in your Thm 28.11\[=\[French-scheme:thm\]\] is in fact ruled out by Rokhlin’s formula. I think that you have mixed $Pi^+$ with $Pi^-$. In a positive couple the orientations are just opposite. So, four nested ovals can contribute at most $+2$ to Rokhlin’s formula. It is an interesting idea to apply the Thom conjecture to real algebraic curves. To my knowledge the only new result obtained this way is contained in G. Mikhalkin “Adjunction inequality for real algebraic curves”. Let me formulate the problem (which exists certainly already somewhere). GENERALIZED THOM PROBLEM. Let $X$ be a simply connected smooth closed $4$-manifold and let $h$ be a non trivial integer $2$-dimensional homology class. Let $F$ be a smoothly embedded oriented surface which represents $h$ and such that the components of $F$ represent classes which are linearly independent over $Z/2Z$. What is the maximal Euler characteristic of $F$? In the complex projective plane this boils down to the Thom problem, because evidently the surface F has to be connected in this case. However, it becomes interesting in a more general complex surface. It is an easy matter to make a non connected surface $F$ connected but the opposite is quite hard. It is equivalent to finding an embedded “membrane” with trivial normal Euler number. I don’t know wether Seiberg-Witten theory nowadays can give a sufficient criterium to ensure the existence of such a membrane (as stretching the neck of the surface $F$). But it seems to me that this is the place to look at. If one considers as $X$ the double cover of the projective plane ramified in the complexification of an $M$-curve of even degree then one can consider as $F$ the fix point set of one of the two induced anti-holomorphic involutions on $X$. We know $2b_0(F)+b_1(F)$ from Harnacks equality. Hence the Euler characteristic $2b_0(F)-b_1(F)$ is maximal when $b_1(F)$ is minimal, i.e. the numbers $p$ or $n$ of the real curve are maximal. So this problem is closely related to the still open Ragsdale conjecture for $M$-curves. Best regards, Thomas $\bullet$ \[Gabard, 09.03.13, ca. 21h00\] Does the Gudkov hypothesis (mod 8) reduces to Rohlin’s (complex orientation) formula Dear Colleagues, On reading recently the Degtyarev-Kharlamov 2000 survey, I learned the (simple) fact that the Arnold congruence mod 4 (weak Gudkov hypothesis) can be reduced to Rohlin’s formula. I wrote down a proof of this simple issue in Lemma 24.16\[=\[Rohlin-implies-Arnold:lem\]\], on page 219 of the messy survey. Of course then I wondered if a sharpened combinatorial argument taking into account the signs distribution on the edges of the “Hilbert tree” (encoding the distribution of ovals) prompted by Rohlin’s complex orientations could likewise subsume the Gudkov hypothesis to Rohlin’s formula. The combinatorics becomes much more messy and I lack a good idea on how to exploit the $M$-curve assumption (apart from the dubious idea of using the dextrogyration allied to a totally real pencil, but this looks somewhat ad hoc!!!?). If feasible (so or otherwise), the net impact would be that the highbrow topology used in the Rohlin-Rohlin/Atiyah-Singer-Marin proofs could be replaced perhaps by basic algebraic geometry. This issue is merely didactic of course, yet it looks perhaps technically challenging to implement this modest “dream”. My (unsuccessful) attempt to tackle this reduction is given in Section 24.5, pages 220–223. Of course, I am sure that you already tried hard along this way and that this is a pot-pourri naive problem. So I write you the letter, only in the hope that someone already worked this reduction successfully, though I doubt (as otherwise it would certainly have been mentioned in the Degtyarev-Kharlamov 2000 survey). Many thanks again for all your attention and all your precious hints, Alex PS: I apologize much to Séverine for not having yet found the time to study carefully enough the last brilliant explanations, but look forward doing so in the best delay. PPS: Many thanks also to Thomas for the brilliant answer on Thom and Mikhalkin, Seiberg-Witten, etc. I will include your letter in (the next version of) my notes so that anybody can contemplate it. $\bullet$ \[11.03.13\] Dear Thomas and the other Colleagues, So many thanks to Thomas for having catched my mistake regarding Thom versus Rohlin (existence of schemes prohibited by Thom but not by Rohlin’s formula). I have written down a corrected version of the Theorem (whose first clause is I think still true despite Thomas’s corrigendum) as Thm 30.14 on page 382. I hope this time it is correct! The subsequent Thm 30.15 and Lemma 30.16 describe larger family of such schemes. Alas, what I have written is not extremely appealing, but I hope still readable for such experts as you are (though I confess that it is not extremely exciting!). The philosophy is just of course that there is no subsumation of Thom’s estimate $\chi\le k^2$ to Rohlin’s formula. Many thanks again to all for all your kind answers, and especially to Thomas for taking care to bring me on the right track! Best regards, Alex PS: I have drifted the Section of your e-mails at the end of the text, in particular the last letter of Thomas is to be found on page 435. Alas, I had not yet the time to digest its full swing, but the idea looks very promising! $\bullet\bullet\bullet$ mardi 12 mars 2013 13:56:45 Dear Alexander, sorry, but all your $M$-schemes of degree $10$ in Thm 30.14 and 30.15 have $n=2$ and are ruled out simply by Petrovskis inequality. I don’t think that genus bounds give anything new for real schemes alone but they definitely do so for configurations of several real curves. Just take a look on Mikhalkin’s paper. Best regards Thomas \[16.03.13\] Dear Geometers, Many thanks again to Thomas for his former correction. I got so drifted in a sort of cuneiform formalism of trees with signs but mostly lost myself into dubious combinatorics of (what I call) the Rohlin tree (=Hilbert’s nested tree with signs materializing of course Rohlin’s complex orientations). Ultimately after several basic combinatorial mistakes, I did NOT even succeeded in finding a Caucasian $M$-scheme where Rohlin’s formula is stronger than Thom $\chi \le k^2$. This problem is discussed in Sections 30.7 and 30.8 (pages 393–402). It seems to me that there is (for $M$-curves at least) a certain concomitance between Rohlin and Thom, i.e. you cannot corrupt Rohlin’s formula without corrupting simultaneously Thom’s bound. So maybe it is reasonable to conjecture that if Thom’s estimate is fulfilled then Rohlin’s equation is always soluble for a suitable distribution of signs on the edges of the tree. Sorry that my summary is vague as I do not myself understand properly what happens. Today, I switched on a somewhat more pleasant arithmetical problem exposed in Section 30.6, p.391–392. This is hopefully more readable, and you surely studied this a long time ago. Here the question is to find an $M$-scheme without nesting of high-degree (say $m\ge 6$ assuming zero-knowledge). As we know, Hilbert posited the intuition that $M$-curves are forced to exhibit nesting. (In fact on reading this afternoon more carefully Hilbert’s text, he is not so categoric but let us assume so, to add some suspense to our story!) In view of Rohlin’s formula (which forces the number of ovals of an unnested dividing curve to be a square) and the Gudkov congruence mod 8, one is invited to ask when the Harnack bound $M=g+1$ of a degree $2k$ curve is a (perfect) SQUARE? This leads to a little arithmetical problem which admits a nontrivial solution at $k=17$ for $M=529$ which is by a lucky stroke equal to $23$ squared. (Of course I found this just by an “exhaustive” tabulation.) (Note at this stage that $17=16+1$ yet another subconscious coincidence in Hilbert’s numbering!!??) It turns out moreover that the Gudkov congruence is then fulfilled! So this leads me to ask if, you in Russia, knew a way (prior to Thom-Kronheimer-Mrowka 1994) to prohibit this $M$-scheme of degree $2k=34$ with 529 ovals (all lying outside another, i.e. no nesting). Personally, I would be very happy if someone can tell me an answer (in case I did not foiled the arithmetics!!!) Incidentally, I am so ignorant in that field that I do not know how to solve in general the Diophantine problem of the quadrature of Harnack’s bound, i.e. for which $k$ is Harnack’s bound $M=g(2k)+1$ a square (=Problem 30.19, on page 391). Finally, this morning, building upon Séverine’s (elementary) remark on $M$-quintics totally real under a pencil of cubics (nothing so hard as the Rohlin-Le Touzé theorem for sextics), I had the idea to extend the construction of satellites to curves of odd degrees, and so found a scheme of degree 10 (the 2nd satellite of Harnack $M$-quintic). This schemes, according to the philosophy of total reality, should be of type I (stability under satellites). This material is exposed in Section 29.8 page 352–353. The scheme in question has 13 ovals one of them enclosing 6 nests of depth 2 (cf. Fig 149 p.353). Assuming that this scheme is of type I, it could be a counterexample to Rohlin’s maximality conjecture, in case someone ever saw a curve of degree 10 enlarging it. Otherwise, more in line with Rohlin’s philosophy, this scheme being totally real in some explicit way it should be maximal and it results a myriad of prohibitions on all schemes enlarging it! If you know some experimental construction à la Viro-Itenberg that may help to see clearer, I would be extremely thankful. Of course, it would also be very interesting to know if Séverine thinks that there is some good chance that total reality holds true for this satellite. Many thanks for your attention, and I hope my questions are not too ill-posed for such experts as you are! All the best, Alex PPS: Many thanks for the message meanwhile received from Thomas which I hope make my message not to obsolete. Best regards. PPPS: Dear Viatcheslav, I try twice to send you my message as I received delivery failure notification! I hope I am not overloading your mail-box. I try now with a zipped file hopefully toujours lisible pour vous. Amitiés. Alexandre $\bullet\bullet\bullet$ mardi 19 mars 2013 17:54:39 Dear colleagues, my note “totally real pencils. . .” is now available on the archiv, here is the reference: http://arxiv.org/abs/1303.4341 Best regards, Séverine $\bullet$ \[19.03.13, ca. 19h30\] An Alsatian scheme and planning to put a version on arXiv in ca. 10 days (please confirm me if you accept the insertion of your kind letters in my messy survey) Dear Colleagues, Many thanks to Thomas again for pointing out the marvellous Petrovskii’s inequalities (which I confess I had not assimilated properly before, despite all the brilliant surveys available). Shame on me! So of course most of the questions of my previous message were rather stupid. In particular that relating to the arithmetical problem was completely ill-posed as I missed to use the full punch of Rohlin’s formula. I apologize much for all these inconsistencies. Meanwhile I have attempted to find what I call an Alsatian scheme where Thom is stronger than the conjunction of Petrovskii 1933/38, Gudkov hypothesis (1969–72=Rohlin’s semi-proof) and Rohlin’s formula (1974–78). I think that such an Alsatian scheme do exist (cf. Thm 30.22 on page 392). This answers (I hope correctly) a question raised by Thomas in the message reproduced right below, where if I understand it well Thomas expected that Thom says nothing new already known earlier in Russia. Then in view of Thomas’ stimulating messages and also the marvellous Itenberg-Viro 1996 Math. Intelligencer article, I adventured slightly in the province of Miss Ragsdale. Here first I learned that the Ragsdale conjecture for $M$-curves really boils down to Thom (at least one-half thereof)[^108]. I found strange that Kronheimer-Mrowka 1994, was not cited in Itenberg-Viro 1996 probably due to backlog reasons (by Intelligencer)[^109]? Further it seems to me that there is a misprint in the statement of the RAGSDALE CONJECTURE ON $M$-CURVES (on p.24 of Itenberg-Viro 1996). At any rate, I tried to write down my own naive account on Ragsdale in Sect. 30.4\[=\[Ragsdale-conj:sec\]\] (p.380–384) which details more slowly what I understood (hopefully correctly). In fact, I wonder at my (premature) stage if the full Ragsdale conjecture could not follow à la Thom via the lower estimate $-k^2 \le \chi$ which I very naively conjecture to be true for all dividing curves? (I confess that I made no experiments even with the methods of constructions I am aware of, i.e. Hilbert-Harnack). So as you see, life really starts becoming exciting. Alas I fear that I will be strongly interrupted due to editorial reasons of our Journal L’Enseign. Math. in Geneva of which I am the TeX-editor. So perhaps I should stop thinking and try to polish a bit the big mess I produced (during ca. 10 days), before putting it on the arXiv (prior to my long editorial job). Alas, I had already great difficulties to submit the previous version of my file (due to size limitation policy of arXiv), but received an exceptional derogation to do so. Now my text on Ahlfors + Rohlin is twice as large as it was before (600 vs. 300 pages) so there is little chance that I get accepted. I will try to see if an arXiv administrator looks optimistic. Meanwhile, I would be very happy if you all confirm me that you accept the integration of your marvellous letters in my modest text. Of course answer me only in case of objection. I plan to submit the new version ca. the 1 March 2013. Of course my text is so messy that your letters alas are not properly pushed into evidence, but I expect to produce a better text in the next months or years. Many thanks again to all for your letters, indulgence, patience and kind answers! Best regards, Alex $\bullet$$\bullet$$\bullet$ [*Reproduction of Fiedler’s former letter (already above but the other colleagues did not saw it)*]{} mardi 12 mars 2013 13:56:45 Dear Alexander, sorry, but all your $M$-schemes of degree $10$ in Thm 30.14 and 30.15 have $n=2$ and are ruled out simply by Petrovskis inequality. I don’t think that genus bounds give anything new for real schemes alone but they definitely do so for configurations of several real curves. Just take a look on Mikhalkin’s paper. Best regards Thomas $\bullet\bullet\bullet$ mardi 19 mars 2013 22:17:27 Dear Alexandre, sorry again, but your curve has $p=50$ and is ruled out by Arnold’s inequality: $p\le 3/2 k(k-1) + 1 + n_-$, which is $47$ in this case. In fact Arnold’s inequalities are by fare the strongest result in the whole field. Best regards Thomas $\bullet$ \[20.03.13\] “René Thom sur son 31?” Lieber Thomas (and the other colleagues), Es wird jadoch immer spannender, oder wat? Many thanks for the new challenge raised by Thomas (which I received as a special gift on my birthday date, today 20st of March). As Thomas demonstrated yesterday evening (cf. message reproduced below), my example of Alsatian scheme (where Thom is stronger than the Soviet Red army) (cf. Thm 30.22\[=\[Alsatian-schemes:thm\]\], p.392) collapses under the strong Petrovskii inequality of Academician Vladimir Igorevich Arnold. It took me just some few hours (as I was quite tired) to find a stronger candidate of Alsatian scheme where Thom looks stronger that the conjunction of (strong) Petrovskii-Arnold 1971, Gudkov 1969 (proved by Rohlin 1972-Marin ca. 1977), and Rohlin’s formula (1974–78). For the exact statement cf. Thm 30.25\[=\[Alsatian-scheme-Thom-strong-Petrov-Arnold:thm\]\] on page 395. I hope I made no mistake (I checked the details twice). By (3-fold) experience, I am quite confident that Thomas will find a new obstruction in his pockets killing this new example. Hence many thanks again to Thomas for all this precious guidance that oriented much my modest working. As I said I am now under strong temporal constraints, and will not be able to pursue any thinking for a long period of circa 1 month. So please feel free to elaborate more upon the direction indicated by Thomas if it looks hard stopping the inertia. Myself find the Alsatian topic quite pleasant, yet it would (I presume) be interesting to penetrate deeper in the geographical question (by really understanding the diagrammatic impact of Petrovskii-Arnold, Ragsdale, etc. upon the higher Gudkov’s tables of periodic elements) I made several pictures of big pyramids (e.g Fig. 153\[=\[Degree10:fig\]\] p. 386) and that could be a first step toward understanding better what happens. Of course all this must be familiar to you, yet personally I still lack a good algorithm to make good (color-plates) maps evidencing “all” obstructions. My main worry now, is how to publish (and polish) the 600 pages long article that I have produced, especially in view of stringent size restrictions imposed by the arXiv administrators. Many thanks that you seem to all approve the insertion of your letters in my survey, which looks to me essential as you influenced much my (chaotical) trajectory. Of course in the future we will have the occasion to clean better in case some bugs are detected, which is quite unlikely (apart of course in my own letters full of inconsistencies). All the best, Alex $\bullet\bullet\bullet$ jeudi 21 mars 2013 07:51:45 Dear Alexandre, unfortunately your Theorem 30.7\[=\[Thom-Ragsdale:thm\]\] is wrong. Half of the curve together with R is usually called Arnold’s surface. It is an orientable surface iff R has an orientation which induces the complex orientation on its boundary, i.e. the real curve. Hence if there is a negative pair of ovals in in the boundary of one component of R then Arnold’s surface is not orientable. By the way it is well known and easy to prove that if Arnold’s surface is orientable then $p-n=k^2$. Best regards, Thomas $\bullet$ \[21.03.2013, ca. 22h00\] Dear Thomas and the other Colleagues, Many thanks to Thomas for having spotted out my fundamental mistake. I apologize much hence for all the dubious letters that I sent you the former days. I hope I will still be able to repair a bit the situation in the next weeks, though this will require dramatic changes in my messy text! So many thanks again to Thomas for having detected this great Harnaque!!! All the best, Alex $\bullet\bullet\bullet$ vendredi 22 mars 2013 07:54:47 Dear Alexandre, don’t worry. Just read Mikhalkins paper and find other interesting applications of his method. Having a reducible dividing curve you can switch the canonical orientation of exactly one component. This allows to construct immersed surfaces which are far from being complex curves. Then indeed genus bounds start to work. Best regards, Thomas $\bullet$ \[24.03.13\] Is there a simpler counter-example to “Gabard-Thom” than via Itenberg-Viro mirabilis $(M-2)$-curve of degree 10 disproving Ragsdale Dear Colleagues, As pointed out by Thomas (cf. my last message) my proof of the “Gabard-Thom” estimate $\chi \le k^2$ for all dividing curves of degree $2k$ was highly fraudulent as it was based on the (erroneous) assumption that the Arnold surface (=Klein’s half married with Miss Ragsdale) is always orientable. Of course Thom has still something to say on Hilbert’s 16th e.g. in the very special case of no nesting. As a historical curiosity one can notice that the elementary case due to Kervaire-Milnor (1961) of Thom in degree 3 gives in my opinion the first purely topological proof of Hilbert’s Ansatz of nesting for $M$-sextics. Prior to that we had only Hilbert-Rohn (stratificational as explained by Eugenii), and Petrovskii 1933/38 which involves the Euler-Jacobi-Kronecker stuff (interpolation formula). Of course Kervaire-Milnor is merely Rohlin’s early work 1951 disguised! Next I was a bit puzzled by Thomas’s claim that my “Gabard-Thom” theorem is wrong, since I was not able to find an explicit counter-example. I tried a while with elementary constructions à la Harnack-Hilbert but could not find a single counterexample. So I found the pleasant plates Fig. 152-153-154-155\[=\[HilbGab1:fig\]–\[HilbGab4:fig\]\] on pages 382 and ff. Here we get nice infinite families of curves with $\chi=k^2$ that were surely known to Hilbert and Ragsdale. Alas this inclined more toward thinking that Gabard-Thom is sharp, rather than disproving it. Then of course I had the idea to take a closer look to Itenberg-Viro 1996’s article (disproof of Ragsdale), which came like a deliverance and killed in the same stroke the Gabard-Thom dubious estimate. I used the Kharlamov-Marin congruence to check type I of the Itenberg-Viro curve (which I reproduced on Fig. 156\[=\[Itenberg:fig\]\], p.387), though there is surely a more elementary argument à la Fiedler-Viro-Itenberg-Parenti. Many congratulations by the way to Ilia and Oleg for this geometric paradise, which I contemplated with much pleasure and extreme admiration yesterday evening!!! A naive question of mine, is whether we really need to resort to the patchwork method to disprove Gabard-Thom, in the sense that perhaps I missed (a non-maximal) counterexample via Harnack-Hilbert. Finally this marvellous curve (Itenberg-Viro’s) poses again the question of total reality, this time under a pencil of septics (in general I conjecture total reality for adjoint curves of order $m-3$ when it comes to $(M-2)$-curves). The question is where to assign exactly the anchor basepoints! I tried some guesses in Sec. 31.3\[\[Galton-brett:sec\]\] (p. 388). This deserve perhaps much deeper investigations than what I am presently able to do. Perhaps Séverine and Thomas already have some good ideas. My own dream is that there should be a combinatorial recipe telling one from the sole knowledge of the Rohlin tree (=Hilbert’s one with signs given by complex orientations) where to assign basepoints. One should perhaps imagine the Rohlin tree as a Galton-Brett=table with billiards balls falling downwards to the empty ovals and perhaps stabilizing at some other (unstable) equilibriums when they meet a “nail”, which in first approximation could be a hyperbolic oval negatively charged on the edge right above it. All my thanks again to Thomas for having catched my fundamental mistake at the right moment. I hope to polish a bit the text during the next days before submitting it to the arXives. Then I must move toward doing more boring editorial duties for the Swiss journal L’ Ens. Math. (close to collapse by the way). Good Sunday to all, and best regards, Alex PS: Many thanks also for Thomas’s last letter which I just discover now. It is an invitation to read Mikhalkin’s article, which alas I had not yet the time to do properly... Synoptic tabulations ==================== This is an attempt to gather information scattered through the literature. The first synoptic project compiles a list of nomenclatures. A second tabulation reflects how Ahlfors work (existence of circle maps) has been appreciated by subsequent workers of a slightly dissident nature in the sense that they cite conjointly other sources. Nomenclature project -------------------- This section tries to get sharp lower bounds on the basic nomenclature of our topic. As Poincaré tried to convince Felix Klein “[*Name ist Schall und Rauch*]{}” (cf. e.g. Klein 1923 [@Klein-Werke-III_1923 p.611]), but it is somehow pleasant to investigate the historical background of some jargons to use them hopefully appropriately. $\bullet$ (Gauss 1825/1844, F.T. Schubert earlier?) [**Conformal mapping=konforme Abbildung**]{}, maybe the first non-trivial result is to be found in Gauss 1825 [@Gauss_1825], yet the word “conformal” itself appears in Gauss 1844 in the first paper on higher geodesy: “[*ich werde daher dieselben conforme Abbildungen oder Übertragungen nennen, indem ich diesem sonst vagen Beiworte eine mathematisch scharf bestimmte Bedeutung beilege*]{}” \[Werke IV, p.262\]. As noted in Struik 1933 [@Struik_1933 p.164] (via Cantor), the word “conformal” is already used prior to Gauss by F.T. Schubert in “De projectione sphaeroidis ellipticae geographica”, [*Nova Acta Petr.*]{}, p.130–146. $\bullet$ (1865) [**Riemann surface**]{}, maybe first coined by C. Neumann 1865 [@Neumann_1865], followed by Lüroth 1871 [@Lueroth_1871], Clebsch 1872 [@Clebsch_1872], Klein 1874–76 [@Klein_1876], [@Klein_1876], Clifford 1877 [@Clifford_1877] and then too many to record. $\bullet$ [**Berandete (Riemannsche) Flächen, Compact bordered Riemann surfaces, finite Riemann surface, membranes**]{}. The first appellation appears often in Klein 1882 [@Klein_1882] (reprint in Klein 1923 [@Klein-Werke-III_1923 p.569,§23]) and others. The second appellation is coined and popularized in Ahlfors-Sario’s 1960 book [@Ahlfors-Sario_1960], whereas the third competing name is used in Schiffer-Spencer’s book of 1954 [@Schiffer-Spencer_1954]. The term membrane also occurs (in this context) by Klein in his lecture notes. $\bullet$ (1907?) [**Uniformization**]{} probably a coinage of Poincaré. In 1883, just the word “fonction uniforme” appears and the word “uniformization” as a such, came in vogue ca. two decades latter in Poincaré 1907 [@Poincare_1907] and Koebe 1907 [@Koebe_1907_UbaK1]. $\bullet$ (1908) [**Kreisnormierungsprinzip**]{} coined and proved (in fairly general special cases: finite connectivity and symmetric under complex conjugation) by Koebe in 1908 [@Koebe_1908_UbaK3]. $\bullet$ (1912) [**Schwarz’s lemma**]{}. The coinage as a such appears first in Carathéodory 1912 [@Caratheodory_1912], but already published in the modern fashion in 1907 by the same writer [@Caratheodory_1907], acknowledging the argument of E. Schmidt. $\bullet$ (1916) [**Extremal problems=Extremalprobleme**]{} used in function theory by Bieberbach 1916 [@Bieberbach_1916]. $\bullet$ (ca. 1914) [**Circle mapping=Kreisabbildung.**]{} This is used (at least) since Bieberbach 1914 [@Bieberbach_1914 p.100], Koebe 1915 [@Koebe_1915], Bergman\[n\] 1922 [@Bergman_1922 p.238], Bochner 1922 [@Bochner_1922 p.184], with the English translation appearing first in Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950]. $\bullet$ (ca. 1975–1977) [**Total reality, “total reell”, etc.**]{} The adjective “total reell” (totally real) is first used (in the generality) in Geyer-Martens 1977 [@Geyer-Martens_1977 p.101, p.103], where the connection with Ahlfors theorem is made explicit along the line already suggested in Alling-Greenleaf 1969 [@Alling-Greenleaf_1969]. Geyer-Martens ascribe (cf. p.101) the concept of total reality (when paraphrased in the language of field extensions) to J.T. Knight 1969 [@Knight_1969]. Somewhat earlier in lesser generality of Galois coverings “total reell” appears already in Martens 1975 [@Martens_1975]. Meanwhile I think that “totally real” is quite widespread, especially in the growing field of real enumerative algebraic geometry (e.g. works by Sottile). $\bullet$ [**Erster Art=Type I, orthosymmetric and dividing curves**]{} All this jargon is due to Klein. More precisely, Erster Art appears in Klein 1876 [@Klein_1876] and is much used in Russian literature (meanwhile diffusing in the west), cf. e.g. Rohlin 1978 [@Rohlin_1978] (and Gudkov 1974 [@Gudkov_1974/74]???). I remember some irony of Grisha Mikhalkin during a talk by Orevkov, where he found Klein’s subsequent jargon “orthosymmetric” (first in print in Weichold 1883 [@Weichold_1883]) quite awkward. Yet, Klein himself turned to be quite proud of this more intrinsic coinage. This turned to be quite influential, adhered by eminent workers like Koebe, J. Douglas, etc. albeit quite in desuetude today. The reason is mostly due to synonyms like dividing (or separating) curves. Summarizing, the following words are used resp by: $\bullet$ “Erster Art=Type I” first coined in Klein 1876 [@Klein_1876], and adhered to by Rohlin 1978 [@Rohlin_1978 p.90], and then much of the subsequent Russian literature, $\bullet$ orthosymetrisch Klein ca. 1882 (lectures), adhered to Weichold 1883 [@Weichold_1883] (first occurrence in print), then Klein 1891/92 [@Klein_1891--92_Vorlesung-Goettingen] (Vorles. Göttingen), and followed by Koebe 1907 [@Koebe_1907_UrAK] (etc.), Fatou 1930 (in Appel-Goursat 1930 [@Appell-Goursat-Fatou_1930]), Julia 1932 [@Julia_1932], Douglas 1936 [@Douglas_1936-Some-new-results]–1939 [@Douglas_1939-The-most-general], etc. $\bullet$ “zerteilend vs. nichtzerteilend” in Fiedler 1981 [@Fiedler_1981 p.7] $\bullet$ “divide” alone is briefly mentioned in Arnold 1971 [@Arnold_1971/72] (yet only as a property of $M$-curves) $\bullet$ “dividing curves” is used by Wilson 1978 [@Wilson_1978 p.66], Viro 1986/86 [@Viro_1986/86-Progress p.58] Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91 p.359], Gilmer 1991 [@Gilmer_1991], Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.736,737]. $\bullet$ “separating curves”, occurs in Fiedler 1982/83 [@Fiedler_1982/83-Pencil p.162], Dubrovin 1983/85 [@Dubrovin_1983/85], Nikulin 1983/84 [@Nikulin_1983/84], Benedetti-Risler 1990 [@Benedetti-Risler_1990], Natanzon 1990 [@Natanzon_1990/90] or 1999 [@Natanzon_1999-Moduli-real-alg-surf.superanal-differ-spinors], Coppens 2011 [@Coppens_2011], $\bullet$ “courbes séparantes” in Marin 1979 [@Marin_1979], “courbe qui sépare sa complexifiée in Marin 1988 [@Marin_1988], Gabard 2006 [@Gabard_2006] (alas in French it sounds strange to say “courbe divisante”). The following concept is a priori foreign to our survey, albeit it would be interesting to see if the methods of Grötzsch-Teichmüller are of some relevance to the Ahlfors mapping of 1950. This is another mathematical question, but here we content ourselves with a point of terminology: $\bullet$ (1928/1935) [**Quasiconformal mappings=quasikonforme Abbildungen**]{}. This nomenclature is usually ascribed to Ahlfors 1935, who however could not remember precisely from where he borrowed the jargon, according to Kühnau 1997 [@Kuehnau_1997 p.133]), which is worth quoting: Der Name Grötzsch ist wohl bei vielen vor allem mit der Theorie der quasikonformen Abbildungen verbunden, die er ab 1928 begründete. Die Bezeichnung “Quasikonforme Abbildungen” wurde allerdings erst später von L.V. Ahlfors eingeführt. (Freilich sagte mir Ahlfors Februar 1992 in Oberwolfach, da[ß]{} er diese Bezeichnung bei jemandem “gestohlen” habe, er wisse nur nicht mehr bei wem.) Maybe it contributes to the question to remember that the jargon “[*quasikonform*]{}” appears already in 1914, und zwar bei Carathéodory 1914 [@Caratheodory_1914 §16](=page 294 in the pagination of the Ges.Math.Schriften,Bd.3). Dissidence from Ahlfors {#dissident:sec} ----------------------- \[31.08.12\] Sec.\[Ahlfors-proof:sec\] attempted to present Ahlfors’ proof in full details, but failed to digest the details. This deplorable issue motivated us to tabulate a list of “dissident” authors, who instead of quoting the original source Ahlfors 1950 [@Ahlfors_1950] adhered to subsequent treatments. Two accounts emerge with high rating, namely: $\bullet$ Heins 1950 [@Heins_1950] $\bullet$ Royden 1962 [@Royden_1962] Of course, our “dissident” writers (quoting beside Ahlfors some derived product) never (as far as I know) criticizes directly the 1950 work of Ahlfors. At least there dissidence may suggest that themselves were not completely happy with (resp. convinced by) the original text finding more convenient another implementation. Albeit nobody ever expressed frontal objections against Ahlfors 1950 [@Ahlfors_1950], it is not to be excluded (yet of very low probability ca. $10^{-14}$) that somebody once detected some little bug, explaining perhaps the numerous initiatives to reprove Ahlfors’ result from different viewpoints. (We mention again the articles by Mizumoto 1960 [@Mizumoto_1960] and Kuramochi 1952 [@Kuramochi_1952] (undigest?), and refers for a extensive tabulation of such initiatives to the circled item of Fig.\[Map:fig\]). Here is a sample of dissident authors (grouped according to their preferred source) with relevant extracts in “…”: $\bullet$ Stout 1972 [@Stout_1972 p.345]: “…a theorem of Ahlfors \[2\](=Ahlfors 1950 [@Ahlfors_1950]) shows that $\cal H(R)$ contains many inner functions. (See also the elegrant \[sic!\] construction of Heins \[15\](=Heins 1950 [@Heins_1950]) as well as the earlier paper of Bieberbach \[3\](=Bieberbach 1925 [@Bieberbach_1925]) which deals with the case of planar domains.)” $\bullet$ Khavinson 1984 [@Khavinson-Dimitri_1984 p.377]: “The following theorem is a classical result of Bieberbach and Grunsky (see \[6\](=Golusin 1952/57 [@Golusin_1952/57]), \[8\](=Grunsky 1978 [@Grunsky_1978])). For a different approach due to L. Ahlfors, see \[1\](=Ahlfors 1950 [@Ahlfors_1950]). Our proof, although discovered independently, is almost the same as that due to M. Heins in \[11\](=preprint=now published as Heins 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF]) or H. Grunsky in \[8\](=Grunsky 1978 [@Grunsky_1978]). [Theorem 3.]{} [*Let $\zeta_1, \dots, \zeta_n$ be arbitrary fixed points on $\gamma_1, \dots,\gamma_n$ respectively. Then, for each $j$, $\phi(z)$ is the unique function giving a conformal mapping of $G$ onto an $n$-sheeted right half-plane such that $\phi(\zeta_j)=\infty$, for all $j$, $\phi(z_0)=1$.*]{} $\bigstar$ admittedly, this Khavinson’ extract in not hundred percent pertinent to our present purpose inasmuch as the Bieberbach-Grunsky theorem is confined to the planar case. $\bullet$ Stout 1965 [@Stout_1965]: “In order to establish our result, we shall need to make use of a result of Ahlfors \[1\](=Ahlfors 1950 [@Ahlfors_1950]). (For an alternative proof, one may consult Royden \[15\](=Royden 1962 [@Royden_1962].) Theorem 3.1 [*There exists a function $P$ holomorphic on a neighborhood of $\bar R$ which maps $R$ onto the open unit disc in an one-to-one manner for some $n$ and which satisfies $\vert P \vert =1$ on $\partial R$.*]{}” $\bigstar$ Of course the above “one-to-one” is a typo to be read as “$n$-to-one”. $\bullet$ Alling 1966 [@Alling_1966 p.346]: “Finally, I am indebted to Professor Royden for his excellent paper, [*The boundary values of analytic and harmonic functions*]{}, \[24\](=Royden 1962 [@Royden_1962]), which not only gave a new proof of the existence of the Ahlfors’ map, but also gave generalizations of the classical boundary value theorems over the disc. …” $\bullet$ Stout 1966/67 [@Stout_1966/67 p.366]: “Let $R$ be a finite open Riemann surface whose boundary $\Gamma$ consists of $N$ analytic, pairwise disjoint, simple closed curves. Let $\eta$ be an analytic mapping from $R$ onto $U$, the open unit disc which is holomorphic on a neighborhood of $\overline R$ and which is of modulus one on $\Gamma$. That such functions exists was first established by Ahlfors \[1\](=Ahlfors 1950 [@Ahlfors_1950]); another proof of their existence is in the paper \[12\](=Royden 1962 [@Royden_1962]).” $\bullet$ Stout 1967 [@Stout_1967-Interpolation]: “It is convenient to make use of an [*Ahlfors map*]{} for $R$, i.e., a function continuous on $\overline R$ and holomorphic in $R$ which is constantly of modulus one on $\Gamma$. The existence of such function was established by Ahlfors in \[1\](=Ahlfors 1950 [@Ahlfors_1950]); an alternative proof of their existence is in \[4\](=Royden 1962 [@Royden_1962]).” $\bullet$ O’Neill-Wermer 1968 [@O'Neill-Wermer_1968]: “Let $W$ be a region on some Riemann surface whose boundary is the union of a finite number of analytic simple closed curves and with $W$ having compact closure. In “Open Riemann surfaces and extremal problems on compact subregions”, (1950), L. Ahlfors considers the following extremal problem: [*Problem*]{} I. [*Let $a,b$ be points of $W$. among the functions $F$ analytic on $W$ with $\vert F(z)\vert\le 1$ on $W$ and $F(a)=0$, it is required to find the one which makes $\vert F(b)\vert$ a maximum.*]{} He shows that this problem has a unique solution[^110] $f$ which maps $W$ in an $n$-to-$1$ fashion[^111] onto the unit disk, for some $n$. His method of proof depends on a certain associated extremal problem introduced by P.R. Garabedian in his Thesis. (See Garabedian 1949 [@Garabedian_1949]). Another proof is given by H. Royden, “The boundary values of analytic and harmonic functions,” Math. Z. 78 (1962), 1–24.” $\bullet$ Stanton 1971 [@Stanton_1971 p.293]: “Our argument rests on the following theorem of Ahlfors \[1\](=1950). [Theorem.]{} [*There is a function $f$ which is analytic on $W\cup \Gamma$ and which maps \[the interior\] $W$ onto $U$ and $\Gamma$ onto $T$.*]{} This theorem is also proved in Royden \[7\](=1962). A function $f$ of the kind described in this theorem is called an [*Ahlfors mapping*]{}.” $\bigstar$ Upon recalling, that Stanton is a Royden student this may eventually be counted as a self-voting. $\bullet$ Hejhal 1972 [@Hejhal_1972 p.119]: “Suppose first of all that $W$ is the interior of a compact bordered surface $\overline W$. L. Ahlfors \[2\](=1950) and H. Royden \[24\](=1962) have studied the present linear extremal problem on such $W$ at least for the case $\chi \equiv {\rm constant}$ and $\frak L [f]=f(b)$ with $b\in W$. …” $\bullet$ Gamelin 1973 [@Gamelin_1973-Extremal-I p.3]: “…the paper of H.L. Royden deals with finite bordered Riemann surfaces.” $\bullet$ Gamelin 1973 [@Gamelin_1973-BAMS p.1105]: “For dual extremal problems on Riemann surfaces, see \[2\](=Ahlfors 1950 [@Ahlfors_1950]) and \[36\](=Royden 1962 [@Royden_1962]).” $\bullet$ Fisher 1973 [@Fisher_1973 p.1183]: “A similar problem \[…\] has been investigated by L. Ahlfors \[A1\], H. Royden \[R\], and others. In that case, the class of competing function is convex, the solution is unique, is analytic across the boundary $\Gamma$, and has modulus one on $\Gamma$.” And further on page 1187: “Let $F$ be the solution to the Ahlfors-Royden extremal problem described in the introduction. …” $\bullet$ Lund 1974 [@Lund_1974 p.495]: “Let $U$ be the open unit disk in ${\Bbb C}$. We call $F$ an unimodular function if $F$ is analytic in a neighborhood of $\overline{R}$ and maps $\overline R$ onto $\overline U$ so that $F$ is $n$-to-one if we count the multiplicity of $F$ where $dF$ vanishes. If $T$ is the unit circle, then $F$ maps $\Gamma$ onto $T$. The existence of such a function was first proved by Ahlfors \[1\](=1950). Later, Royden \[4\](=1962) gave another proof of this result.” $\bullet$ Kirsch 2005 [@Kirsch_2005]: “Ahlfors generalized Garabedian’s result to regions on Riemann surfaces \[2\](=Ahlfors 1950 [@Ahlfors_1950]); see Royden’s paper \[159\](=Royden 1962 [@Royden_1962]) for another treatment as well as further references to the literature.” $\bullet$ Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000 p.240]: “It has been shown by Ahlfors \[4\](=Ahlfors 1950 [@Ahlfors_1950]) that such a function \[=ramified $n$-sheeted covering of the unit disk\] always exists, and it may be chosen to have the minimal possible degree $g+1$; see also \[5\](=Alling-Greenleaf 1971 [@Alling-Greenleaf_1971]), \[19\](=Fay 1973 [@Fay_1973]), and \[21\](=Fedorov 1991 [@Fedorov_1991]).” Apart from the fact that the writer (Gabard) does not adhere with Alpay-Vinnikov’s claim about $g+1$ being the minimal possible degree for such a mapping ($g$ is the genus of the double, cf. op.cit. p.230), the three proposed references are in our opinion not perfectly adequate as substitute to Ahlfors 1950 [@Ahlfors_1950]. Alling-Greenleaf [@Alling-Greenleaf_1971 p.16, Thm1.3.6] only states Ahlfors’ result yet without reproving it, whereas both Fay and Fedorov recover the result in the planar case only. Acknowledgements ---------------- The author wishes to thank the following long list of geometers (in chronological order of their interaction with the writer in connection to the present text) $\bullet$ Felice Ronga (ca. 1997/98 for his explanation of Brusotti’s theorem), $\bullet$ Claude Weber, Michel Kervaire (for their explanations on how to classify Klein’s symmetric surfaces by looking at the quotient bordered surface) $\bullet$ Frédéric Bihan for pleasant discussions about real algebraic geometry, $\bullet$ Lee Rudolph (ca. 1999 for explaining to us what is the natural topological model for a real elliptic curve with only one “oval”, namely just a torus $S^1\times S^1$ acted upon by factor permutation $(x,y)\mapsto (y,x)$ fixing thereby the diagonal circle), $\bullet$ Alexis Marin, Viatcheslav Kharlamov, Oleg Viro, Jean-Jacques Risler, Thierry Vust, Michel Kervaire, Pierre de la Harpe, John Steinig (for their comments and corrections improving the shape of the article Gabard 2000 [@Gabard_2000]) $\bullet$ Ragahavan Narasimhan, Jacek Bochnack (ca. 1999 for [*not*]{} having been in touch with Ahlfors’ result of 1950 [@Ahlfors_1950] enabling me some free gestation about thinking on the problem) $\bullet$ Manfred Knebusch for his kind interest in the modest work Gabard 2000 [@Gabard_2000], $\bullet$ Johannes Huisman for his constant interest (2001–04–06), and his care about correcting bugs in both my Thesis and the article Gabard 2006 [@Gabard_2006], $\bullet$ Sergei Finashin for an exciting discussion in Rennes 2001, $\bullet$ Jean-Claude Hausmann (ca. 2000/01) for telling me about the standard surjectivity criterion via the Brouwer degree, which was decisive to complete Gabard 2006 [@Gabard_2006], $\bullet$ Antonio Costa, for his fascinating talks in Geneva, $\bullet$ Bujalance for his surely over-enthusiastic Zentralblatt review of my article (Gabard 2006 [@Gabard_2006]), $\bullet$ Fraser-Schoen, whose brilliant work revived my interest in the theory of the Ahlfors’ mapping (ca. the 13 March 2011) at a stage were I was mostly sidetracked by “non-metric manifolds” thanks to efforts of Mathieu Baillif and David Gauld. $\bullet$ Stepan Orevkov, Oleg Viro (2011) for their talks and pleasant discussions, $\bullet$ Marc Coppens (2011–12) for e-mails, and his work on the separating gonality (2011 [@Coppens_2011]) adumbrating sharper insights on the degree of the Ahlfors function (or rather the more general allied circle maps). His turning-point result appeals to a better conciliation of the analytic theory of Ahlfors with the algebro-geometric viewpoint. $\bullet$ (2011/12) Hugo Parlier, Peter Buser, Alexandre Girouard, Gerhard Wanner und Martin Gander are acknowledged for their recent e-mail exchanges. $\bullet$ (Oct. 2012) Daniel Coray for enlarging the capacity of my TeX-compilator, and for his lovely (Cambridge-style) teaching about the geometric Galois action in ca. 1999. $\bullet$ (16 Nov.2012) Franc Forstnerič for pointing out his text with Wold (2012 [@Forstneric-Wold_2012]) showing the state of the art on the proper holomorphic embedding problem. $\bullet$ (09 Jan. 2013) Oleg Viro for his excellent answers on some naive questions on Rohlin’s paper (cf. Sec.\[e-mail-Viro:sec\]). $\bullet$ (10 Jan. 2013) Alexis Marin for his invaluable insights on Klein’s intuitions and more (cf. Sec.\[e-mail-Viro:sec\]). $\bullet$ (January-February-March 2013) Viatcheslav Kharlamov, Stepa Orevkov, Eugenii Shustin, Séverine Le Touzé, Thomas Fiedler for all their letters reproduced in Sec.\[e-mail-Viro:sec\]. Bibliographic comments ====================== The writer does not pretend that the following bibliography is complete (nor that he absorbed all those fantastic contributions in full details). More extensive bibliographies (overlapping ours), but covering more material include those of: $\bullet$ Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] (ca. 40 pages times 25 items per pages=1000 entries covering such topics as the Dirichlet problem, extremal problems, the type problem, the allied classification theory, etc.); $\bullet$ Grunsky 1978 [@Grunsky_1978] (=562 refs, including 48 Books). Most entries of our bibliography are followed by some comment explaining briefly the connection to our primary topic of the Ahlfors map. The following symbolism is used: $\clubsuit$ serves to point out a special connection to Ahlfors 1950 (especially alternative proofs). $\spadesuit$ gives other comments (attempting to summarize the paper contents or to explicit the connection in which we cite it). $\P$ signals papers not quoted in the main-body of the text but connected to our topic. $\bigstar$ marks sources, I could not as yet procure a copy. 47 50 60 78 $\bullet$ the stickers/sigles 60, 78 are assigned when the source has already been cited in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] resp. Grunsky 1978 [@Grunsky_1978]. $\bullet$ 50 designates those references citing the paper Ahlfors 1950 [@Ahlfors_1950] (there represents circa 106 articles on “Google”), and occasionally 47 those quoting Ahlfors 1947 [@Ahlfors_1947]. [**$\heartsuit$n**]{} is something like the indicator of the US rating agency (to be read “liked by $n$”). It indicates the cardinal number [**n**]{} of citations of the paper as measured by “Google Scholar”. The latter machine often misses cross-citations, especially those in old books, or old articles with references given in footnotes format. Many sources cited in Grunsky’s book (1978 [@Grunsky_1978]) are never cited electronically. Accordingly, those rating numbers only supply a statistical idea of the literature ramifications lying beyond a given entry. Also low-citation articles are sometimes the most polished product ripe for museum entrance. Forelli 1979 [@Forelli_1979] is typical: self-contained, elegant and polished proof of Ahlfors result, yet only rated by 3. Our bibliography is somewhat conservative with comparatively few modern references. Our excuse is two-fold: modern expressionism is sometimes harder to grasp, and recent references are usually well detected through computer search. (Papers are listed in alphabetical, and then chronological order, regardless of shared co-authorship.) The primary focus is on the Ahlfors map and the weaker (but more general) circle maps. As a such the topic overflow slightly over the territory of real algebraic geometry. Ahlfors-Sario’s book 60 address Riemann surfaces, whereas Grunsky’s book 78 focuses to the case of planar domains. Hence both bibliographies 60, 78 are quite complementary, and ours is essentially a fusion of both, but we gradually included more and more recent contribution. Still additional references are welcome. For conformal maps, it is helpful when browsing the vast literature to keep in mind the basic question: [*what result through which method?*]{} [**Results.**]{} Objects traditionally range along increasing order of generality through: simply-connected regions, multiply-connected ones and finally Riemann surfaces. We often add a humble compactness proviso, as the passage to open objects is traditionally achieved through the exhaustion trick (going back at least to Poincaré 1883 [@Poincare_1883], and see also Koebe 1907 [@Koebe_1907_UbaK1]), and active in recent time (e.g. Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950].) As to the mappings, they may all be interpreted in some way or another as ramification of RMT (Riemann’s mapping to a circle=disc). We distinguish primarily: $\bullet$ CM=circle maps (usually not univalent, but multi-sheeted disc with branch, or winding points=Windungspunkte) $\bullet$ KNP=Kreisnormierung(sprinzip) (univalent map to a circular domain) $\bullet$ SM=slit mappings for various types of them (parallel, circular, radial, logarithmic spiral, etc.). Those are all allied to certain natural foliation of the sphere, and some extreme generality in this respect is achieved in Schramm’s Thesis where any foliation is permitted as support for the slits. [**Methods.**]{} They may be classified in two broad classes quantitative vs. qualitative (each having some branchings): $\bigstar$ (Quantitative) variational methods, including: $\bullet$ DP=Dirichlet principle (or more broadly speaking, potential theory=PT, centering around such concepts as the Green’s function, harmonic measures (i.e. harmonic function with special null/one boundary prescription of the various contour), etc. Of course, there is a standard yoga between Dirichlet and Green, so all this is essentially one and the same method. $\bullet$ IM=Iterative methods (originators: Koebe and Carathéodory), and by extension this may proliferate up to including the circle packings. $\bullet$ EP=extremal problems (e.g. the one of maximizing the derivative amongst the class of function bounded-by-one) and leads to the Ahlfors map. $\bullet$ BK=Bergman kernel (or Szegö kernel), here the fundamental ideas rest upon Hilbert’s space methods, and the idea of orthogonal system. Initially, the method is also inspired by Ritz, and Bieberbach extremal problem (1914 [@Bieberbach_1914]) for the area swept out by the function. Since the middle 1940’s, there were found several conformal identities among so-called domain functions (Green’s, Neumann’s, etc.) and the kernel functions so that virtually this is now highly connected to DP$\approx$PT. Also the Ahlfors map is expressible in term of the Bergman kernel (cf. e.g., Nehari 1950 [@Nehari_1950]) so that this heading is strongly connected to EP. $\bullet$ PP=Plateau problem style methods (for RMT, this starts with the observation of Douglas 1931 [@Douglas_1931-Solution]). This strongly allied to DP, albeit some distinction is useful to keep in mind just for cataloguing purposes. $\bigstar$ (Qualitative) topological methods: $\bullet$ the [*continuity method*]{}, as old as Schläfli, (as Koebe notices somewhere) is involved in the accessory parameters of Schwarz-Christoffel, in Klein-Poincaré’s uniformization through automorphic functions, Brouwer (invariance of the domain), Koebe, etc., e.g. Golusin 1952/57 [@Golusin_1952/57]) $\bullet$ Brouwer topological degree and the allied surjectivity criterion (cf. e.g., Mizumoto 1960 [@Mizumoto_1960], Gabard 2006 [@Gabard_2006]). Here the idea is that there is some topological stability of the embedding of a curve into its Jacobian via the Abel mapping in the sense that its homological feature are unsensitive to variation of the complex (analytic) structure (moduli), and this enables one to draw universal statement by purely topological considerations. Finally we have attempted to manufacture a genealogy map showing the affiliation between the authors. The picture turned out to be so large that TeX prefers reject it at the very end of the file. -1.2cm0 -4.2cm0 -5pt0 -5pt0 \[15.10.12\] When I reached 884 references, I unfortunatel met the so-called “TeX capacity exceeded, sorry.” obstruction (cf. Knuth’s “The TeX Book”, p.300 for more details). Thus I had to deactivate some references which are not used for cross-citation, albeit they clearly belong to our topic. \[16.10.12\] This problem was ultimately solved by my advisor Daniel Coray, to whom I express my deepest gratitude for enlarging the TeX capacity of my compilator. [9]{} H. Abe, [*On some analytic functions in an annulus*]{}, Kodai Math. Sem. Rep. 10 (1958), 38–45. \[$\spadesuit$ quoted in Minda 1979 [@Minda_1979] in connection with the theta function expression of the Ahlfors function of an annulus\]  N.H. Abel, [*Mémoire sur une propriété générale d’une classe très étendue de fonctions transcendantes*]{}, présenté à l’Académie des Sciences à Paris le 30 octobre 1826; published (only) in: [*Mémoires présentés par divers savants*]{}, t. VII, Paris, 1841. Also in [Œ]{}uvres, t. I, 145–211. \[$\spadesuit$ first occurrence of Abel’s theorem, which in Gabard 2004/06 [@Gabard_2006] is used as the main weapon toward proving existence of Ahlfors circle maps\]  N.H. Abel, [*Remarques sur quelques propriétés générales d’une certaine sorte de fonctions transcendantes*]{}, Crelle J. Reine Angew. Math. 3 (1828). \[$\spadesuit$\]  N.H. Abel, [*Démonstration d’une propriété générale d’une certaine classe de fonctions transcendantes*]{}, Crelle J. Reine Angew. Math. 4 (1829), 201–202. \[$\spadesuit$\]  W. Abikoff, [*The Real Analytic Theory of Teichmüller spaces*]{}, Lecture Notes in Math. 820, Springer, 1980. \[$\spadesuit$\]  M.B. Abrahamse, [*Toeplitz operators in multiply connected domains*]{}, Amer. J. Math. 96 (1974), 261–297. \[$\spadesuit$ extend to finite Riemann surfaces? try Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000]\]  M.B. Abrahamse, R.G. Douglas, [*A class of subnormal operators related to multiply connected domains*]{}, Adv. Math. 19 (1976), 106–148. \[$\spadesuit$ for extension to finite Riemann surfaces? try Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000], Yakubovich 2006 [@Yakubovich_2006]\]  M.B. Abrahamse, J.J. Bastian, [*Bundle shifts and Ahlfors functions*]{}, Proc. Amer. Math. Soc. 72 (1978), 95–96. 47 \[$\spadesuit$ the Ahlfors function of a domain $R$ with $n$ contours is applied to the calculation of a bundle shift (that is, a pure subnormal operator with spectrum contained in the closure of $R$ and normal spectrum contained in the boundary of $R$)\]  M.B. Abrahamse, [*The Pick interpolation theorem for finitely connected domains*]{}, Michigan J. Math. 26 (1979), 195–203. \[$\spadesuit$ extend to finite Riemann surfaces? try Heins 1975 [@Heins_1975], Jenkins-Suita 1979 [@Jenkins-Suita_1979], both works subsuming in principle Ahlfors 1950 [@Ahlfors_1950]\]  N. A’Campo, [*Sur la première partie du seizième problème de Hilbert*]{}, Sém. Bourbaki 537 (1979). \[$\spadesuit$ contains a nice picture of Hilbert’s method in degree 6 $\spadesuit$ an exposition of Gudkov’s construction of the $M$-scheme $\frac{5}{1}5$ omitted by Hilbert\]  R. Accola, [*The bilinear relation on open Riemann surfaces*]{}, Trans. Amer. Math. Soc. ?? (1960), ??–??. 50 \[$\spadesuit$\]  J. Agler, J. Harland, B.J. Raphael, [*Classical function theory, operator dilation theory, and machine computation on multiply-connected domains*]{}, Mem. Amer. Math. Soc. 191 (2008), 159pp. 78 \[$\spadesuit$ cite Grunsky 1978 [@Grunsky_1978] and gives via the Grunsky-(Ahlfors) extremal function an interpretation of the Herglotz integral representation via the Kreĭn-Milman theorem $\spadesuit$ \[06.10.12\] contains also a nice desription of circle maps (in the form of half-plane maps, which seems to be directly inspired from Heins’ treatment (1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF]) who probably offers an alternative derivation of Ahlfors’ bound $r+2p$) $\spadesuit$ “In three chapters the authors first cover generalizations of the Herglotz representation theorem, von Neumann’s inequality and the Sz.-Nagy dilation theorem to multiply connected domains. They describe the fist through third Herglotz representation and provide an …”\]  D. Aharonov, H.S. Shapiro, [*Domains on which analytic functions satisfy quadrature identities*]{}, J. Anal. Math. 30 (1976), 39–73. 78 \[$\spadesuit$ includes the result that the Ahlfors map of a quadrature domain is algebraic, see also papers by Gustafsson, Bell, etc.\]  P.R. Ahern, D. Sarason, [*On some hypo-Dirichlet algebras of analytic functions*]{}, Amer. J. Math. 89 (1967), 932–941. \[$\spadesuit$\]  P.R. Ahern, D. Sarason, [*The $H^p$ spaces of a class of function algebras*]{}, Acta Math. 117 (1967), 123–163. \[$\spadesuit$ “This paper is a study of a class of uniform algebras and of the associated Hardy spaces of generalized analytic functions. It is a natural continuation of a number of similar studies which have appeared in recent years; see Bochner \[7\], Helson and Lowdenslager \[15\], ...”\]  P.R. Ahern, [*On the geometry of the unit ball in the space of real annihilating measures*]{}, Pacific J. Math. 28 (1969), 1–7. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited on p.4, yet not exactly for the result we have in mind, but see also the related paper Nash 1974 [@Nash_1974] where the fascinating study of the geometry of the convex body of representing measures is continued $\spadesuit$ \[13.10.12\] it could be fascinating to penetrate the geometry of this body in relation to the “link” (collection of circles in the “same” Euclidean space ${\Bbb R}^g$, where $g$ is the genus of the double), as it occurs in Ahlfors original proof 1950 [@Ahlfors_1950]). Understanding this is probably the key to a sharper understanding of circle maps, in particular of their lowest possible degrees\]  L.V. Ahlfors, H. Grunsky, [*Über die Blochsche Konstante*]{}, Math. Z. 42 (1937), 671–673. \[$\spadesuit$ not directly relevant to this text, except for the hardness of some extremal problems. Compare the front cover of Grunsky’s Coll. Papers for a depiction of the hyperbolic tessellation allied to the conjectured extremal function. Basically one consider in the Euclidean plane ${\Bbb C}$ the triangulation by equilateral triangles and above it in the hyperbolic disc one manufactures a equilateral triangle of angle the half value namely $\pi/6$. Mapping conformally (via Schwarz-Christoffel) this hyperbolic triangle to the fatter Euclidean triangle, while reproducing this map by the (Riemann)-Schwarz principle of reflection yields a function conjectured to have extremal Bloch constant. The latter amounts to have the largest schlicht disc avoiding the ramification. In our case the densest packing (à la Kepler) is clearly the equilateral triangulation of ${\Bbb C}$, so the above Ahlfors-Grunsky function is the natural candidate for having the maximum Bloch radius, alas nobody ever succeeded to show this. Similar problems make sense for other geometries, e.g. planar to spherical, in which case the Bloch constant was estimated in the breakthrough of Bonk-Eremenko, ca. 2002\]  L.V. Ahlfors, [*Bounded analytic functions*]{}, Duke Math. J. 14 (1947), 1–11. 60, 78 \[$\clubsuit$ the planar case of Ahlfors 1950 [@Ahlfors_1950], cite Grunsky 1940–42 [@Grunsky_1940; @Grunsky_1942] as an independent forerunner $\spadesuit$ albeit less general than the next item (Ahlfors 1950 [@Ahlfors_1950], which includes the case of positive genus) this article is more quoted that its successor essentially due to the intense activity centering around analytic capacity and the Painlevé null-sets implying a super vertical series of workers like Vitushkin, Melnikov, Garnett, Calderón 1977 [@Calderon_1977], David, etc. culminating to Tolsa’s 2002/03 [@Tolsa_2003] resolution of the Painlevé problem $\spadesuit$ as a detail matter, it may be recalled that the present article contains a minor logical gap fixed in Ahlfors 1950 [@Ahlfors_1950] (cf. the later source, especially footnote p.123) and also the “Commentary” in the Collected papers Ahlfors 1982 [@Ahlfors_1982_Coll_papers p.438]: “When writing the paper I overlooked a minor difficulty in the proof. This was corrected in \[36\](=Ahlfors 1950 [@Ahlfors_1950]).” $\spadesuit$ compare also the comment in the German edition of Golusin 1952/57 [@Golusin_1952/57 p.415, footnote 2]: “Der biesherige Beweisgang erlaubt es nicht, zu schlie[ß]{}en, da[ß]{} keine der $n-1$ Nullstellen eine mehrfache ist. Diese Lücke des [Ahlfors]{}schen Beweises wurde von P.R. Garabedian in seiner Dissertation (=1949 [@Garabedian_1949]), beseitigt., Anm.d.Red.d.deutschenAusgabe.”\]  L.V. Ahlfors, [*Open Riemann surfaces and extremal problems on compact subregions*]{}, Comment. Math. Helv. 24 (1950), 100–134. 60, 78 \[$\clubsuit$ the central reference of the present article $\clubsuit$ contains the first existence-proof of a circle map on a general compact bordered Riemann surface $\clubsuit$ in fact both a qualitative existence result as well as a quantitative extremal problem are presented\] 60, 78  L.V. Ahlfors, A. Beurling, [*Conformal invariants and function-theoretic null sets*]{}, Acta Math. 83 (1950), 101–129. 60, 78  L.V. Ahlfors, [*Development of the theory of conformal mapping and Riemann surfaces through a century*]{}. In: [*Contributions to the Theory of Riemann Surfaces. Centennial Celebration of Riemann’s Dissertation*]{}, Annals of Math. Studies 30, 3–13, Princeton 1953; or [*Collected Papers*]{}, Vol.1, 1929–1955, Birkhäuser, 1982. \[$\spadesuit$ a colorful historical survey of Riemann, Schwarz, Poincaré, Koebe, Nevanlinna, Grötzsch, Grunsky and Teichmüller\]  L.V. Ahlfors, [*Variational methods in function theory*]{}. Lectures at Harvard University, 1953 transcribed by E.C. Schlesinger. \[$\spadesuit$ cited in Read 1958 [@Read_1958_Acta]. Does this contains another (more pedestrian) treatment of Ahlfors 1950 [@Ahlfors_1950]?\]$\bigstar$$\bigstar$  L.V. Ahlfors, [*On quasiconformal mappings*]{}, J. Anal. Math. (Jerusalem) 3 (1953–54), 1–58. ($+$Erratum) \[$\spadesuit$\]  L.V. Ahlfors, [*Extremalprobleme in der Funktionentheorie*]{}, Ann. Acad. Sci. Fenn., A.I., 249/1 (1958), 9 pp. \[$\spadesuit$ survey like, but pleasant philosophy\]  L.V. Ahlfors, [*The complex analytic structure of the space of closed Riemann surfaces*]{}, In: Analytic functions, Princeton Univ. Press, 1960, 45–60. \[$\spadesuit$\]  L.V. Ahlfors, L. Sario, [*Riemann Surfaces*]{}, Princeton Univ. Press, 1960.  L.V. Ahlfors, [*Classical and contemporary analysis*]{}, SIAM Review 3 (1961), 1–9.  L.V. Ahlfors, [*Complex Analysis*]{}, McGraw-Hill Book Co. (2nd ed.), 1966. \[$\spadesuit$ p.243–253 proof of the PSM (and other radial/circular avatars) via the Dirichlet principle\]  L.V. Ahlfors, [*Lectures on Quasiconformal mappings*]{}, Van Nostrand, Princeton, NJ, 1966, 146pp. \[$\spadesuit$\]  L.V. Ahlfors, [*Collected Papers, Vol. 1, 1929–1955*]{}, Birkhäuser, 1982. \[$\spadesuit$ p.438 is worth quoting in extenso: “The point of departure in \[30\](=Ahlfors 1947 [@Ahlfors_1947]) is Painlevé’s problem: Given a compact set $E\subset {\Bbb C}$, when does there exist a nonconstant bounded analytic function $f(z)$ on ${\Bbb C}\setminus E$? I was really interested in the function with the smallest upper bound of $\vert f(z)\vert$ when normalized so that $f(z)\sim 1/z$ at $\infty$. This smallest maximum is now called the [*analytic capacity*]{}[^112] of $E$, and the Russians[^113] used to refer to the extremal function, if it exists[^114], as the “Ahlfors function”, an unexpected and probably unearned distinction. In this form Painlevé’s problem is closely related to the precise form of Schwarz’s lemma[^115] for an arbitrary region, and that is what the paper is actually about. To be specific: if $\Omega\subset {\Bbb C}$ is a region and if $\vert f(z)\vert \le 1$ in $\Omega$ while $f(z_0)=0$ for a given $z_0\in \Omega$, exactly how large can $\vert f'(z_0)\vert$ be?—The difficulty lies in the fact that while $u=\log\vert f(z)\vert$ is a harmonic function with a logarithmic pole at $z_0$, the single-valuedness of $f$ translates into diophantine conditions on the conjugate harmonic function $\nu$. Quite obviously this makes the problem much harder than if only the absolute value $\vert f(z)\vert$ were required to be single-valued.—In my paper I restrict myself to a region $\Omega$ of finite connectivity $n$, and my aim is to describe the extremal function $f(z)$. I show that $\vert f(z)\vert=1$ on the boundary and that $f$ has exactly $n-1$ zeros[^116]. In other words, $f$ maps $\Omega$ on an $n-1$ times covered disk[^117]. In addition there are conditions on the location of the zeros[^118]. When writing the paper I overlooked a minor difficulty in the proof. This was corrected in \[36\](=Ahlfors 1950 [@Ahlfors_1950]).—The purpose of \[36\](=Ahlfors 1950 [@Ahlfors_1950]) was to study open Riemann surfaces by solving extremal problems on compact subregions and passing to the limit as the subregions expand. The paper emphasizes the use of harmonic and analytic differentials in the language of differential forms. It is closely related to \[35\](=Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950]), but differs in two respects: (1) It deals with Riemann surfaces rather than plane regions and (2) the differentials play a greater role than the functions.—I regard \[36\] as one of my major papers. It was partly inspired by R. Nevanlinna, who together with P.J. Myrberg (1954[^119]) had initiated the classification theory of open Riemann surfaces, and partly by M. Schiffer (1943 [@Schiffer_1943]) and S. Bergman (1950 [@Bergman_1950]), with whose work I had become acquainted shortly after the war. The paper also paved the way for my book on Riemann surfaces with L. Sario (1960 [@Ahlfors-Sario_1960]), but it is probably more readable because of its more restricted contents.—I would also like to acknowledge that when writing this paper I made important use of an observation of P. Garabedian to the effect that the relevant extremal problems occur in pairs connected by a sort of duality. This is of course a classical phenomenon[^120], but in the present connections it was sometimes not obvious how to formulate the dual problem.”\]  L.V. Ahlfors, [*The Joy of function theory*]{}, ca. 1984. \[$\spadesuit$ p.443: “It has been customary to write about the joy of everything, from the joy of cooking to the joy of sex, so why not the joy of function theory?” $\spadesuit$ p.444: “I remember vividly how he \[=Lindelöf\] encouraged me to read the collected papers of Schwarz and also of Cantor, but he warned me not to become a logician, for which I am still grateful. Riemann was considered too difficult, and Lindelöf never quite approved of the Lebesgue integral.” $\spadesuit$ p.445: “It is impossible to change an analytic function at or near a single point without changing it everywhere. This crystallized structure is a thing of great beauty, and it plays a great role in much of nineteenth-century mathematics, such as elliptic functions, modular functions, etc. On the other hand, it was also an obstacle, perhaps most strongly felt in what somewhat contemptuously was known as “Abschätzungsmathematik”. Consciously or subconsciously there was a need to embed function theory in a more flexible medium. For instance, Perron used the larger class of subharmonic functions to study harmonic functions, and it had also been recognized, especially by Nevanlinna and Carleman, that harmonic functions are more malleable that analytic functions, and therefore a more useful tool.”\]  V.B. Alekseev, [*Abel’s Theorem in Problems and Solutions*]{}, Nauka, Moscow, 1976. (Russian).$\bigstar$  Ju.E. Alenicyn, [*On some estimates for functions regular in a region of finite connectivity*]{}, (Russ.) Mat. Sb. N. S. 49 (1959), 181–190. 78 $\bigstar$  Ju.E. Alenicyn, [*An extension of the principle of subordination to multiply connected regions*]{}, (Russ.) Trudy Mat. Inst. Steklov 60 (1961), 5–21; Amer. Math. Soc. Transl. (2) 43, 281–297. 78 $\bigstar$  Ju.E. Alenicyn, [*Conformal mappings of a multiply connected domain onto many-sheeted canonical surfaces*]{}, (Russ.) Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964), 607–644. 78  Ju.E. Alenicyn, [*An estimate of the derivative in certain classes of function, analytic in a multiply connected domain*]{}, Zap. Nauch. Sem. Leningrad Otdel. Mat. Inst. Steklov 24 (1972), 6–15; English transl. (1974), 565–571. 78 \[$\spadesuit$ Ahlfors extremal problem is recalled and extended to a more general setting, where instead of considering functions bounded-by-one in modulus, there is some continuous positive function $\lambda(z)$ defined on the contour which acts as the upper-bound over the permissible modulus via $\lim\sup_{z\to z_0\in \partial D} \vert f(z) \vert \le \lambda(z_0)$\]  Ju.E. Alenicyn, [*Inequalities for generalized areas for multivalent conformal mappings of domains with circular cuts*]{}, Translated from Matematicheskie Zametki 29 (1981) 387–395; \[$\spadesuit$ extension of the result of Vo Dang Thao 1976 [@Vo-Dang-Thao_1976] and Gaier 1977 [@Gaier_1977-Roumaine] (which the latter ascribes to Grötzsch 1931 [@Groetzsch_1931]) $\spadesuit$ this is close to (but not exactly) the desideratum that Bieberbach’s minimum problem (1914 [@Bieberbach_1914]) yields another interpretation of the Ahlfors circle map when extended to multiply-connected domains $\spadesuit$ p.202 a cross-reference to Nehari 1952 [@Nehari_1952-BOOK] is given, but this does not really answer our question whether the minimal map (of least area) is a circle map (it is just observed that in higher-connectivity it is [*not*]{} schlicht)\]  Ju.E. Alenicyn, [*Least area of the image of a multiconnected domain of $p$-sheeted conformal mappings*]{}, Translated from Matematicheskie Zametki 30 (1981) 807–812; \[$\spadesuit$ extension of the result of Vo Dang Thao 1976 [@Vo-Dang-Thao_1976] and Gaier 1977 [@Gaier_1977-Roumaine] (which Gaier 1978 [@Gaier_1978-JDMV] ascribes to Grötzsch 1931) $\spadesuit$ this is close to (but not exactly) the desideratum that Bieberbach’s minimum problem (1914 [@Bieberbach_1914]) yields another interpretation of the Ahlfors circle map when extended to multiply-connected domains\]  J.W. Alexander, [*Functions which map the interior of the unit circle upon simple regions*]{}, Ann. of Math. (2) 17 (1915), 12–22. \[$\clubsuit$\]  N.L. Alling, [*A proof of the corona conjecture for finite open Riemann surfaces*]{}, Bull. Amer. Math. Soc. 70 (1964), 110–112. \[$\clubsuit$ applies Ahlfors 1950 [@Ahlfors_1950] to the corona as a lifting procedure of the disc-case established in Carleson 1962 [@Carleson_1962] $\spadesuit$ for an alternative proof of the same result avoiding the Ahlfors map but using uniformization instead, cf. Forelli 1966 [@Forelli_1966]\]  N.L. Alling, [*Extensions of meromorphic function rings over non-compact Riemann surfaces. I*]{}, Math. Z. 89 (1965), 273–299. \[$\clubsuit$ idem as Alling 1964 [@Alling_1964] with full details\]  N.L. Alling, [*Extensions of meromorphic function rings over non-compact Riemann surfaces. II*]{}, Math. Z. 93 (1966), 345–394. 50 \[$\clubsuit$ p.346: “Finally, I am indebted to Professor Royden for his excellent paper, [*The boundary values of analytic and harmonic functions*]{}, \[24\](=Royden 1962 [@Royden_1962]), which not only gave a new proof of the existence of the Ahlfors’ map, but also gave generalizations of the classical boundary value theorems over the disc. …” $\spadesuit$ p.345: “As in Alling 1965, theorems are frequently proved for $\overline{X}$ \[=a finite open Riemann surface\] by lifting the corresponding classical result for the disc, using the Ahlfors map in conjunction with various algebraic facts. For example, Fatou’s theorem and Nevanlinna’s theorem (about functions of bounded characteristics) are easily proved in this way.”\]  N.L. Alling, in MathReviews, Review of Stout 1965, Bounded holomorphic functions on finite Riemann surfaces. \[$\clubsuit$ quoting an extract of the text: “It is now clear that a great many of the results for the disc $U$, which can be found, for example in K. Hoffman’s book (=1962 [@Hoffman_1962]), also hold for $R$ \[=the interior of a compact bordered surface\]. The choice of technique to extend such results depends then on the ease of proof, the intuition generated by the setting, and the predisposition of the investigator. Uniformization and the algebraic approach \[based upon Royden’s idea (1958=[@Royden_1958]) of a lifting procedure along an Ahlfors map\] seem to have an advantage over annular analysis in that they treat the whole space and the whole ring simultaneously. Still, special advantages in using uniformization and in using the algebraic approach persist. For example, the theory of the closed ideals in the standard algebra on $R$, ${\cal A} (R)$, and the theory of invariant subspaces have been worked out by M. Voichick (=1964 [@Voichick_1964]), using uniformization, but has not been achieved yet using the algebraic approach. (See also Voichick 1966 [@Voichick_1966], and a paper by Hasumi now in preprint \[=Hasumi 1966 [@Hasumi_1966]\], all of which deal with the invariant subspace problem.)” $\spadesuit$ \[13.10.12\] for an upgrade giving full answer to Alling’s desideratum of an Ahlfors-map proof of the closed ideals, see Stanton 1971 [@Stanton_1971] $\spadesuit$ the review is concluded with the following: “Finally, concerning the corona problem,as far as the reviewer knows, no one has given a new proof of Carleson’s theorem or re-proved it on $R$; everyone, to generalize it to $R$, has merely lifted the result to $R$ \[Or “descended” in the case of the uniformizing approach.\]. A substantially simpler and more lucid proof of Carleson’s theorem still remains the most challenging question in this subject.” $\spadesuit$ possible upgrades the new proofs à la Hörmander/Wolff (cf. e.g. Gamelin 1980 [@Gamelin_1980-Wolff's-proof]), and also the localization of the corona done by Gamelin 1970 [@Gamelin_1970-Localization-of] should be satisfactory answers. Yet our impression is that eventually any sharper understanding of the geometry of Ahlfors map (e.g. Gabard’s improved bound (2006 [@Gabard_2006]) on the degree of the Ahlfors circle maps) could implies modest quantitative refinements in the corona with bounds (cf. Hara-Nakai 1985 [@Hara-Nakai_1985] and Oh 2008 [@Oh_2008])\]  N.L. Alling, N. Greenleaf, [*Klein surfaces and real algebraic function fields*]{}, Bull. Amer. Math. Soc. 75 (1969), 869–872. \[$\clubsuit$ the first paper (to the best of my knowledge) which makes explicit the link between Ahlfors 1950 [@Ahlfors_1950] and the much older Kleinian theory (1876–82) of orthosymmetric (=dividing) real algebraic curves, see Klein 1876 [@Klein_1876] and Klein 1882 [@Klein_1882]\]  N.L. Alling, N. Greenleaf, [*Foundations of the Theory of Klein Surfaces*]{}, Lecture Notes in Math. 219, Springer-Verlag, Berlin, 1971. \[$\clubsuit$ repeat the same Klein-Ahlfors connection (cf. comments to the previous entry Alling-Greenleaf 1969 [@Alling-Greenleaf_1969]), and develop a systematic theory of Klein surfaces, a new jargon derived from Berzolari 1906 [@Berzolari_1906] $\clubsuit$ Ahlfors’ theorem (compare p.16, Theorem 1.3.6) is stated as follows: “Theorem 1.3.6 (Ahlfors \[${\rm A}_1$\]). Let $\frak X$ be \[a\] compact, connected, orientable Klein surface with non-void boundary. There exists $\underline{f}\in E(\frak X)$ such that $\partial X=\Gamma_{\underline f}$. $\spadesuit$ if I do not mistake Ahlfors’ result is only stated but not reproved in the text (perhaps quite contrary to the hope borne out by the cross-citation in Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000]) $\spadesuit$ personal reminiscence \[03.09.12\]: I can remind clearly that I knew this famous Alling-Greenleaf text quite early (ca. Spring 1999), but did not appreciated directly the significance of Ahlfors result, and had to rediscover it later (ca. 2001) after some intense own work (ca. 2 years of efforts) $\spadesuit$ this is a bit ironical for showing how one can severely miss a crucial information through quick reading, but permitted me to develop an independent approach which ultimately turned out to give a sharper result than Ahlfors’ $\spadesuit$ so this is probably a perfect illustration of how a poor knowledge of the literature is sometimes beneficial for the creativity in “young men games” (if we can borrow Hardy’s bitter joke)\]  N.L. Alling, B.V. Limaye, [*Ideal theory on non-orientable Klein Surfaces*]{}, Ark. Mat. ?? (1972), 277–292. \[$\clubsuit$ extension of the Beurling-Rudin result for the disc to non-orientable bordered surfaces, hence cannot employ the Ahlfors map (whose existence is confined to the orientable case), for which case see Stanton 1972 [@Stanton_1971] who uses the Ahlfors map for the same purpose (extension of Beurling-Rudin)\]  N.L. Alling, [*Analytic geometry on real algebraic curves*]{}, Math. Ann. 207 (1974), 23–46. \[$\spadesuit$\]  D. Alpay, V. Vinnikov, [*Indefinite Hardy spaces on finite bordered Riemann surfaces*]{}, J. Funct. Anal. 172 (2000), 221–248. 50 \[$\clubsuit$ p.240 Ahlfors 1950 [@Ahlfors_1950] is cited and other references are given namely Alling-Greenleaf 1971 [@Alling-Greenleaf_1971] (where however no existence-proof is given), Fay 1973 [@Fay_1973] (where perhaps only the schlicht case is treated?), and finally Fedorov 1991 [@Fedorov_1991] (where probably only the planar case is treated) $\spadesuit$ still on p.240 it is asserted that $g+1$ is the minimal possible degree for expressing a compact bordered Riemann surface as ramified covering of the unit-disc ($g$ being as usual the genus of the double, cf. p.230) $\spadesuit$ if correct this assertion would (blatantly) corrupt the main result of Gabard 2006 [@Gabard_2006] (which by virtue of the incertitude principle could be false) $\spadesuit$ however the sharpness of $g+1$ in general is easily corrupted on the basis of simple concrete example of Klein’s Gürtelkurve (real quartic with two nested ovals) projected from a real point situated in the inner oval (cf. Figure 6 in Gabard 2006 [@Gabard_2006 p.955]), and therefore Alpay-Vinnikov’s assertion looks slightly erroneous. NB: this little misconception about the sharpness of Ahlfors bound seems to originate in Fay’s book, cf. Fay 1973 [@Fay_1973] $\spadesuit$ of course this sloppy detail does not entail at all the intrinsic beauty of this paper namely the study of Hardy spaces on finite bordered Riemann surface: “Furthermore, each holomorphic mapping of the finite bordered Riemann surface onto the unit disk (which maps boundary to boundary)determines an explicit isometric isomorphism between this space \[a certain Kreĭn space\] and a usual vector-valued Hardy space on the unit disk with an indefinite inner product defined by an appropriate Hermitian matrix.”\]  E. van Andel, [*Extending Riemann mapping capabilities for the sage mathematics package*]{}, Calvin College, 2011. \[$\spadesuit$ p.1: “computation and visualization tools for the Riemann mapping”, “Ahlfors spiderweb”; p.3: “the Ahlfors map conformally maps multiply-connected regions to the unit circle. \[Of course in this case it is more traditional (at least correct) to speak of the Bieberbach-Grunsky map.\] This map is such that for a region with $n$ holes, $n+1$ points in the original region will map to $1$ point in the unit circle.”\]  C. Andreian Cazacu, [*On the morphisms of Klein surfaces*]{}, Rev. Roumaine Math. Pures Appl. 31 (1986), 461–470. \[$\clubsuit$ inspired by Alling-Greenleaf 1971 [@Alling-Greenleaf_1971] and Stïlow 1938 [@Stoilow_1938-Lecons] $\spadesuit$ \[17.10.12\] for another (more elementary) proof of this result, cf. a paper by Cirre 1997\]  C. Andreian Cazacu, [*Interior transformations between Klein surfaces*]{}, Rev. Roumaine Math. Pures Appl. 33 (1988), 21–26. \[$\clubsuit$ from the Introd.: “The interior transformations were introduced by Simion Stoilow in order to solve Brouwer’s problem: the topological characterization of analytic functions. By means of these transformations he founded a vast topological theory of analytic functions with essential implications in the study of Riemann surfaces \[8\](=Soilow 1938 [@Stoilow_1938-Lecons]). In this paper we show that interior transformations are a powerful tool in Klein surfaces theory \[2\](=Alling-Greenleaf 1971 [@Alling-Greenleaf_1971]) too. \[$\dots$\]”\]  C. Andreian Cazacu, [*Complete Klein coverings*]{}, Bull. Soc. Sci. Lett. [Ł]{}ódź Sér. Rech. Déform. 37 (2002), 7–14. \[$\clubsuit$ the notion of the title is introduced as a generalization of the Ahlfors-Sario notion of complete Riemann coverings (1960 [@Ahlfors-Sario_1960 p.42, §21A]), i.e. any point in the range has a neighborhood whose inverse image consists only of compact components. For the case of coverings with a finite number of sheets, it is shown that a Klein covering is complete iff it is total, in the sense of Stoïlow (1938 [@Stoilow_1938-Lecons]), that is any sequence tending to the boundary has an image tending to the boundary. $\spadesuit$ \[13.10.12\] such purely topological conceptions are mentioned for they subsume the topological behaviour of Ahlfors circle maps (i.e. full covering of the circle, alias disc)\]  A. Andreotti, [*Un’applicazione di un teorema di Cecioni ad un problema di rappresentazione conforme*]{}, Ann. Sc. Norm. Super. Pisa (3) (1950), 99–103. 60, but not in 78 \[$\clubsuit$ seems to extend the result of Matildi 1948 [@Matildi_1945/48] to the case of several contours, hence could be an (independent) proof of the existence of a circle map (than that of Ahlfors 1950 [@Ahlfors_1950]) $\clubsuit$ in fact the writer (Gabard) was not able to follow all the details of Andreotti’s proof but I have no specific objection to make (it would be a good challenge if somebody is convinced by the argument to translate it in English to make the argument more generally accessible, ask maybe Coppens or Huisman) $\clubsuit$ it would be interesting to see which degree is obtained by this method (presumably the genus of the double plus one, i.e. $p+1$ cf. p.101, where $k>p$ \[by Riemann-Roch\]) $\clubsuit$ maybe a last comment is that in Andreotti’s result it is not perfectly clear if the circumference can be arranged to coincide, so has to get an Ahlfors circle map\]  P. Appell, E. Goursat, [*Théorie des fonctions algébriques*]{}, Paris, 1895. \[$\spadesuit$\]  P. Appell, E. Goursat, [*Théorie des fonctions algébriques d’une variable et des transcendantes qui s’y rattachent*]{}, Deuxième édition revue et augmentée, Tome II, Fonction automorphes, par Pierre Fatou, Paris, Gauthier-Villars, 1930. 60 \[$\spadesuit$ discusses Klein’s orthosymmetry\]  E. Arbarello, M. Cornalba, [*Footnotes to a paper of Beniamino Segre. The number of $g^1_d$’s on a general $d$-gonal curve, and the unirationality of the Hurwitz spaces of $4$-gonal and $5$-gonal curves*]{}, Math. Ann. 256 (1981), 341–362.  E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris, [*Geometry of algebraic curves, Volume I*]{}, Grundlehren der math. Wiss. 267, Springer-Verlag, 1985. \[$\spadesuit$ p.217: “The existence theorem for $g_1^d$’s was first proved by Meis \[1\](=Meis 1960 [@Meis_1960]). Later, at a time when the techniques of enumerative geometry were better understood, the first fundamental theorem of the theory was established with a completely modern approach. In fact (partly under the influence of unpublished work of Mumford) simultaneously Kempf and Kleiman–Laksov gave the first rigorous proof of the Existence Theorem, and of Theorem (1.3). (See Kempf \[1\](=1971/72 [@Kempf_1971]), Kleiman–Laksov \[1,2\](=1972 [@Kleiman-Laksov_1972], 1974 [@Kleiman-Laksov_1974]))” $\spadesuit$ \[09.10.12\] again one may wonder if this enumerative geometry technology is susceptible to adapt to the context of the Ahlfors map, which amounts to real curves of the orthosymmetric(=dividing) type (ideally the goal would be to adapt the Kempf/Kleiman–Laksov proof to recover the bound of Gabard 2006 [@Gabard_2006] interpreted as a bordered avatar of Meis 1960 [@Meis_1960]) $\spadesuit$ another reason for quoting this book in connection with the Ahlfors map is the issue about generalized Ahlfors maps taking values not in the disc but in another finite bordered Riemann surface. Then there is a certain evidence that such Ahlfors maps generally fail to be full covering surface, for the doubled map relates two closed Riemann surfaces. But the latter are severely restricted and generally not existing. This can be either argued via a moduli count as in Griffiths-Harris 1980 [@Griffiths-Harris_1980] or via Exercise C-6. given on p.367 (of the book under review): “From the preceding exercise and the theorem on global monodromy proved in Chapter X conclude that a general curve of genus $g\ge 2$ does not admit a nonconstant map to a curve of positive genus.” $\spadesuit$ of course the statement is a bit sloppy for there is always the identity map as a trivial counterexample, but probably maps of non-unity degree are excluded tacitly. The proof given seems to use the fact that given a branched covering of curves the fundamental class of the inverse image of the Jacobian variety of the image curve is not a rational multiple of $\theta^{g-h}$, where $\theta$ is the theta divisor and $g,h$ are the resp. genuses of the curves\]  R. Arens, [*The closed maximal ideals of algebras of functions holomorphic on a Riemann surface*]{}, Rend. Circ. Mat. Palermo 7 (1958), 245–260.\[$\spadesuit$\]  V.I. Arnold, [*Distribution of ovals of the real plane algebraic curves, the involutions of four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms*]{}, Funkt. Anal. Prilozen 5 (1971), 1–9; English transl., Funct. Anal Appl. 5 (1972), 169–175. \[$\spadesuit$ some revolutionary ideas preparing the terrain for Rohlin’s breakthrough\]  V.I. Arnold, [*Index of a singular point of a vector field, the Petrovskii-Oleinik inequalities, and mixed Hodge structures*]{}, Funkt. Anal. Prilozen 12 (1978), 1–14; English transl., Funct. Anal. Appl. 12 (1978), 1–12. \[$\spadesuit$\]  V.I. Arnold, O.A. Oleinik, [*The topology of real algebraic varieties*]{}, Vestnik Moscov.Gos. Univ. Ser. 1 (1979), 7–17; English transl., Moscow Univ. Math. Bull. 34 (1979), 5–17. \[$\spadesuit$ a survey (?) oft cited, e.g. in Viro 1986/86 [@Viro_1986/86-Progress]\]  V.I. Arnold, [*The branched covering ${\Bbb C}P^2\to S^4$, hyperbolicity and projective topology*]{}, Sibirsk. Mat. Zh. 29 (1988), 36–47; English transl., Sib. Math. J. 29 (1989), 717–726. \[$\spadesuit$ compare also Anosov’s “obituary of Pontrjagin” where this famous homeomorphism ${\Bbb C}P^2/ {\rm conj}\approx S^4$ (Kuiper-Massey-Marin) is ascribed back to Pontrjagin $\spadesuit$ Rohlin expressed (orally) the same opinion, cf. e.g. Finashin 1995/98 [@Finashin_1995/98] $\spadesuit$ yet according to Arnold this result can be traced back to Maxwell\]  V.I. Arnold, [*Topological content of the Maxwell theorem on multipole representation of spherical functions*]{}, Topol. Methods Nonlinear Anal. 7 (1996), 205–217. \[$\spadesuit$ cited in Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.757]: “as explained in \[5\](=Arnold 1996=this entry [@Arnold_1996]), this beautiful explicit proof was essentially known to Maxwell; \[…\]”\]  V.I. Arnold, [*Symplectization, complexification and mathematical trinities*]{}, Fields Inst. Communications ?? (20??), ?–?. \[$\spadesuit$ p.7: “Near 1970 Petrovsky asked me to help in evaluating a thesis of a mathematician Gudkov from Nizhni Novgorod (it was Gorky at that time). He was studying the Hilbert problem 16, the question on the plane algebraic curves of degree 6: what are the possible shapes of the set $f(x,y)=0$, if $\deg f=6$?—The classical answers for degree $2$ were extended to degrees $3$ and $4$ by Newton and Descartes. But then the difficulties starts. Hilbert was unable to solve the case of degree $6$, and this problem was explicitly formulated in his list. One may also consider the affine version but it is more complicated and instead we may consider the projective one, dealing with \[the\] curves in ${\Bbb R}P^2$. Even to this, easier question no answer was known at Hilbert’s time.—The only known thing was the celebrated theorem of Harnack \[…\] Gudkov claimed to obtain the complete possible configurations list of ovals of degree $6$ curves but Petrovsky was doubtful of his result. Let us describe it. The list contains three $M$-curves. \[…\] And this relation $\chi\equiv k^2 \pmod 8$ was observable in all examples of $M$-curves of degree $2k$ which Gudkov was able to construct for higher degrees. But there were no explanations for this behavior.—I was aware that congruences modulo $8$ were standard in $4$-dimensional topology. So my idea was that there existed somewhere a $4$-dimensional manifold which governed the topology of the real plane curve. But how to construct it? This was the place where the complexification came into the game and became very helpful. \[…\] p.14 [**Question.**]{} [Did Gudkov get the recommendation for his thesis?]{}—[**Answer**]{} The thesis was of course defended even though I was never able to read it. But as a result I invented all the matter I have explained to you: I was working hard for a month and after this I proved his conjecture modulo 4. The most difficult thing was some lemma which I was able to guess but not to prove. I always had very good undergraduate students and at that time I asked Varchenko to help me. \[…\] Unfortunately Varchenko had declined to sign the final paper as a coauthor.—D.A. Gudkov became the leader of a strong team in real algebraic geometry at Nizhni Novgorod (Utkin, Polotovskii, Shustin, …). Some of the results of Gudkov and his student were recently rediscovered by C.T.C. Wall. \[…\]”\]  V.I. Arnold, [*I.G. Petrovskii, Hilbert’s topological problems and modern mathematics*]{}, Uspekhi Math. Nauk 57 (2002), 197–207; English transl., Russain Math. Surveys 57 (2002), 833–845. \[$\spadesuit$ p.197: “The content of Hilbert’s problem is [*to give a topological classification of real algebraic curves and manifolds (of fixed degree)*]{}. It is one of the principal and eternal problem of mathematics which is also important for many of its application (where these curves and manifolds describe laws of nature). [*What algebraic curves look like*]{}; even today this is unnown, even for plane curves of degree $8$ consisting of 22 connected ovals \[…\].” $\spadesuit$ p.834: “There are 1812 [*topologically possible*]{} arrangements of $11$ ovals in the plane. Hilbert’s result stated that of all these arrangement only two are realised by [*algebraic*]{} curves of degree $6$.—This result of Hilbert is wrong, as was shown 70 years later by D.A. Gudkov, who was a student of both Petrovskii and the physicist A.A. Andronov. Gudkov showed that there are three, not two, realizable arrangements.”\]  V.I. Arnold, [*From Hilbert’s superposition problem to dynamical systems*]{}, Amer. Math. Monthly 111 (2004), 608–624. \[$\spadesuit$ p.608: “Some people, even though they study, do so without enough zeal, and therefore live long.”—Archbishop Genady of Novgorod, ca. 1500. $\spadesuit$ p.608: philosophy of the mushroom $\spadesuit$ p.622: “[**Question**]{} (J. Milnor) You often told us about important mathematical work in Russia that we did not know about and you gave another example today. I wonder if you can explain to us how to locate something interesting in the literature starting with zero information.”—[**Answer**]{}. \[…\] I usually start with the German [*Encyclop[æ]{}dia …*]{}. In Klein’s [*Vorlesungen über die Entwicklung der Math. im 19. Jahrhundert*]{} there is a lot of information on whatever happened in the nineteenth century and before. \[…\]”\]  (On) V.I. Arnold, by A.A. Davydov, S.M Gusein-Zade, Yu.S. Ilyashenko, M.E. Kazaryan, A.G. Khovanskii, A.G. Kushnirenko, S.K. Lando, A.N. Varchenko, V.A. Vassiliev, and V.M. Zakayukin, [*Vladimir Igorevich Arnold in the eyes of his students*]{}, Proc. Steklov Inst. Math. 259 (2007), 1–5. \[$\spadesuit$ p.3: “Arnold’s seminar covered everything, for example real algebraic geometry. Hilbert spent a lot of effort constructing real plane algebraic curves of a given degree that have the maximum possible number of ovals. Unsuccessful attempts to construct such curves with an a priori possible topology of arrangement on the projective plane convinced him that not all possibilities are feasible. Hilbert collected open problems of real algebraic geometry in his 16th problem. D.A. Gudkov solved one of these problems for curves of degree $6$; however, the general picture remained unclear. Arnold general surprisingly fine topological obstacles showing that many a priori possible arrangements of curves with the maximal number of ovals cannot be realized. Arnold’s studies were picked up by V.A. Rokhlin, D.A. Gudkov, and their students. As a result real algebraic geometry has reached a completely new modern level.”\]  V.I. Arnold, [*Topological properties of eigenoscillations in mathematical physics*]{}, Proc. Steklov Inst. Math. 273 (2011), 25–34. \[$\spadesuit$ discussion of Courant’s theorem on the number of residual component of the nodal hypersurface of an oscillating manifold (vibrating membrane) and its relationship with Hilbert’s 16th problem $\spadesuit$ precisely, Abstract: “Courant proved that the zeros of the $n$th eigenfunction of the Laplace operator on a compact manifold $M$ divide this manifold into at most $n$ parts. He conjectured that a similar statement is also valid for any linear combination of the first $n$ eigenfunctions. However, later it was found out that some corollaries to this generalized statement contradict the results of quantum field theory. Later explicit counterexamples were constructed by O. Viro. \[…\] ”\]  D.S. Arnon, S. McCallum, [*A polynomial time algorithm for the topological type of a real algebraic curve*]{}, J. Symb. Comput. 5 (1988), 213–236. \[$\spadesuit$ cited in Kalla-Klein 2012 [@Kalla-Klein_2012-Computation-cite-Gabard]\]  N. Aronszajn, [*Theory of reproducing kernels*]{}, Trans. Amer. Math. Soc. 68 (1950), 337–404. \[$\spadesuit$ abstract unified view on the theory of the reproducing kernel containing the special cases of Bergman and Szegö, etc.\]  C. Arzelà, [*Sul Principo di Dirichlet*]{}, Nota letta alla R. Accademia delle Scienze dell’Instituto di Bologna nell’Adunza del 24 Gennaio 1897. \[$\spadesuit$ cited in Zaremba 1910 [@Zaremba_1910] as a precursor of Hilbert’s resurrection of the Dirichlet principle\]$\bigstar$$\bigstar$$\bigstar$  M.F. Atiyah, [*Riemann surfaces and spin structures*]{}, Ann. Sci. École Norm. Sup. (4) 4 (1971), 47–62.  G. Aumann, C. Carathéodory, [*Ein Satz über die konforme Abbildung mehrfach zusammenhängender ebener Gebiete*]{}, Math. Ann. 109 (1934), 756–763. 78, but not in 60  H.F. Baker, [*Abel’s theorem and the allied theory including the theory of the theta function*]{}, Cambridge Univ. Press, Cambridge, 1897. \[$\spadesuit$\]  J.A. Ball, [*Operators of class $C_{00}$ over multiply-connected domains*]{}, Michigan Math. J. 25 (1978), 183–196. 47 \[$\spadesuit$ p.187, Ahlfors 1947 [@Ahlfors_1947] is cited for the following result (in fact due to Bieberbach 1925 [@Bieberbach_1925] in this formulation): “If $R$ is a domain in the complex-plane bounded by $n+1$ nonintersecting analytic Jordan curves, there exists a complex-valued inner function on $R$, which is analytic on a neighborhood of $\bar R$, has precisely $n+1$ zeros in $R$, and wraps each component of the boundary of $R$ once around the unit disk \[sic, but “disk” should rather be “circle”\] $\spadesuit$ this theorem (of Bieberbach-Grunsky-Ahlfors) is then applied to a problem in operator theory\]  J.A. Ball, [*A lifting theorem for operator models of finite rank on multiply-connected domains*]{}, J. Operator Theory 1 (1979), 3–25. 47 \[$\spadesuit$ Ahlfors 1947 [@Ahlfors_1947] is cited on p.11 (in a context where perhaps Bieberbach 1925 [@Bieberbach_1925] would have been logically sufficient) $\spadesuit$ again the philosophy of the paper seems to transplant via the Ahlfors function a certain lifting theorem for operator models on the disc (due to Sz.-Nagy-Foiaş) to the more general case of a multi-connected domain $\spadesuit$ one can of course wonder about extension on bordered Riemann surfaces, probably established meanwhile (?)\]  J.A. Ball, K. Clancey, [*Reproducing kernels for Hardy classes on multiply-connected domains*]{}, Integral Equations Operator Theory 25 (1996), 35–57. \[$\spadesuit$ extension to finite bordered Riemann surface, try Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000]\]  E. Ballico, [*Real algebraic curves and real spanned bundles*]{}, Ricerche di Matematica 50 (2001), 223–241. \[$\spadesuit$\]  E. Ballico, G. Martens, [*Real line bundles on $k$-gonal real curves*]{}, Abh. Math. Sem. Univ. Hamburg 71 (2001), 251–255. \[$\spadesuit$ real curves, “symmetric” Riemann surfaces, diasymmetric/orthosmmetric, p.251: “One cannot expect irreducible moduli for real curves of genus $g$. But as already indicated by Klein (in permitting shrinkage of components of $X({\Bbb R})$ to isolated double points, \[9\]=(Klein 1892 [@Klein_1892_Realitaet])), the set of (real or complex, whatsoever) isomorphism classes of real stable curves of arithmetic genus $g\ge 1$ can be equipped with a topology making it a connected space (\[14\]=Seppälä 1991 [@Seppala_1991-Moduli]). $\spadesuit$ p.252: “In particular the classical bound $k_{{\Bbb C}}\le (g+3)/2$ may be false for the real gonality $k$ (\[5\](=Chaudary 1995 [@Chaudary_1995-Thesis]), \[10\](=Martens 1978 [@Martens_1978])). If $n(X)>0$ one knows (\[3\]=Ballico 2003 [@Ballico_2003]) that $k\le (g+3)/2+3$; it seems not known if this is sharp for some $g$.”\]  E. Ballico, [*Gonality and Clifford index for real algebraic curves*]{}, Collectanea Math. 53 (2002). \[$\spadesuit$\]  E. Ballico, [*Codimension 1 subvarieties of ${\cal M}_g$ and real gonality of real curves*]{}, Czechoslovak Math. J. 53 (2003), 917–924. \[$\spadesuit$ some results seem to be reanalyzed in Coppens-Huisman 2010...\]  E. Ballico, [*Real curves with fixed gonality and empty real locus*]{}, Le Matematiche 60 (2005), 129–131. \[$\spadesuit$\]  E. Ballico, [*Real ramifications points and real Weierstrass points of real projective curves*]{}, Glasnik Mat. 41 (2006), 233–238. \[$\spadesuit$\]  S. Banach, [*Théorie des opérations linéaires*]{}, Warsaw, 1932. \[$\spadesuit$ cited in Read 1958 [@Read_1958_Acta], where the Hahn-Banach theorem is put in connection to the Ahlfors map\]  C. Bandle, M. Flucher, [*Harmonic radius and concentration of energy; hyperbolic radius and Liouville’s equations $\Delta U= e^U$ and $\Delta U=U^{\frac{n+2}{n-2}}$*]{}, SIAM Review 38 (1996), 191–238. 47 \[$\spadesuit$ Ahlfors 1947 [@Ahlfors_1947] is cited on p.200 as follows: “Corollary 4 extends Liouville’s formula to multiply connected planar domains and so does the following formula from Mityuk’s monograph \[79\](=1985). Denote by $f\colon \Omega \to B$ an [*Ahlfors map*]{} of $\Omega$ (cf. Ahlfors \[1\](=Ahlfors 1947 [@Ahlfors_1947]), obtained as a solution of the same extremal problem that we used for the definition of the Riemann map (§1). Then the inner radius of $\Omega$ is given by $r(x)=\frac{1-\vert f(x)\vert^2}{\vert f'(x)\vert} \exp\bigl(-2\pi \sum_{\{y\neq x: f(y)=x\}} G_x(y)\bigr)$, provided $f'(x)\neq 0$. Note that on a $k$-connected domain the Ahlfors map is a $k$-sheeted branched covering. A modified formula involving higher derivative of $f$ holds at the branch points. The proof is similar to that of Corollary 4.” $\spadesuit$ perhaps instead of the mentioned monograph, the original articles of Mitjuk already contain this (multi-connected) extended formula, cf. Mitjuk 1965 [@Mitjuk_1965-inner-radius] (and also Mitjuk 1968 [@Mitjuk_1968] for the statement (in English), yet without the proof)\]  V. Bangert, C. Croke, S. Ivanov, M. Katz [*Filling area conjecture and ovalless real hyperelliptic surfaces*]{}, Geom. Funct. Anal. ?? (2005), ?–?. \[$\spadesuit$ solve the hyperelliptic case of the filling area conjecture due to Gromov, hence in particular the genus-one case $p=1$ $\spadesuit$ the hearth of the argument seems to be an old result of Hersch\]  W.H. Barker II, [*Kernel functions on domains with hyperelliptic double*]{}, Trans. Amer. Math. Soc. 231 (1977), 339–347. (Diss. under M.M. Schiffer) \[$\spadesuit$ p.345, the Ahlfors (extremal) function of a domain is discussed by referring to Bergman 1950 [@Bergman_1950], Heins 1950 [@Heins_1950], and also the original treatment due to Ahlfors 1947 [@Ahlfors_1947] and that of Garabedian 1949 [@Garabedian_1949]\]  E. Bedford, [*Proper holomorphic mappings*]{}, Bull. Amer. Math. Soc. (N.S.) 10 (1984), 157–19?. 50 \[$\spadesuit$ p.159, Ahlfors 1950 is quoted as follows: “The existence of many proper mappings is given by a result of Grunsky \[55\](=[@Grunsky_1937]) and Ahlfors \[1\](=1950 [@Ahlfors_1950]). [Theorem]{}. [*If $M$ is a finite Riemann surface with nondegenerate boundary components, then there exists a proper mapping $f\colon M \to \Delta$.*]{} In general, however, given two Riemann surfaces $M$ and $N$, it does not seem easy to say whether there exists a proper mapping $f\colon M \to N$.”\]  H. Behnke, H. Zumbusch, [*Konforme Abbildung von Bereichen auf in ihnen liegenden Bereiche*]{}, Semester-Ber. math. Sem. Münster 8 (1936), 100–121. \[$\spadesuit$ quoted in Grunsky 1940 [@Grunsky_1940 p.233], in connection with the definition of the Carathéodory metric (first appearance in Carathéodory 1926 [@Caratheodory_1926]) for multi-connected domains\]  H. Behnke, K. Stein, [*Entwicklungen analytischer Funktionen auf Riemannschen Flächen*]{}, Math. Ann. 120 (1947/49), 430–461. \[$\spadesuit$ proves that any open Riemann surface carries a nonconstant analytic function $\spadesuit$ in the case where the Riemann surface is the interior of a compact Riemann surface this also follows form Ahlfors 1950 [@Ahlfors_1950] in the much sharper form of a branched covering of the disc $\spadesuit$ naive question \[01.10.12\]: by using an exhaustion of the open Riemann surface by finite bordered ones what sort of functions can be constructed on the whole surface? Is it in particular possible to subsume the Behnke-Stein theorem to that of Ahlfors? (looks a bit naive I confess)\]  H. Behnke, F. Sommer, [*Theorie der analytischen Funktionen einer komplexen Veränderlichen*]{}, Die Grundlehren der math. Wiss. in Einzeldarstellungen, Bd.77, Springer-Verlag, Berlin, 1955; Third Edition, Springer-Verlag, New York, 1965. \[$\spadesuit$ pp.581–2 is quoted in Černe-Forstnerič 2002 [@Cerne-Forstneric_2002] for the “(Schottky) double” $\spadesuit$ other sources for this purposes are Klein 1882 [@Klein_1882] (in romantic pre-axiomatic style), else Koebe 1928 [@Koebe_1928-Acta], or Teichmüller 1939 [@Teichmueller_1939] and of course also Springer’s book 1957 [@Springer_1957-BOOK-Introd-to-RS] or Schiffer-Spencer 1954 [@Schiffer-Spencer_1954]\]  S.R. Bell, [*Numerical computation of the Ahlfors map of a multiply connected planar domain*]{}, J. Math. Anal. Appl. 120 (1986), 211–217. \[$\spadesuit$ from the Introd.: “N. Kerzman and E.M. Stein discovered in \[6\](=1978 [@Kerzman-Stein_1978]) a method for computing the Szegö kernel of a bounded domain $D$ in the complex plane with $C^{\infty}$ smooth boundary. In case $D$ is also simply-connected, the Kerzman-Stein method yields a powerful technique for computing the Riemann mapping function associated to a point $a\in D$ (see \[6\](=Kerzman-Stein 1978 [@Kerzman-Stein_1978]), \[7\](=Kerzman-Trummer 1984)). In this note, we show how the Kerzman-Stein method can be generalized to yield a method for computing the Ahlfors map associated to a point in a finitely connected, bounded domain in the plane with $C^2$ smooth boundary. The Ahlfors map is a proper holomorphic mapping of $D$ onto the unit disc which maps each boundary component of $D$ one-to-one onto the boundary of the unit disc.—The Ahlfors map might prove to be useful in certain problems arising in fluid mechanics. For example, in the problem of computing the transonic flow past an obstacle in the plane, a conformal map of the outside of the obstacle onto the unit disc is used to set up a grid which is most convenient for making numerical computations (see \[5\](=Jameson 1974, “Iterative solution of transonic flows over airfoils and wings, including flows at Mach $1$”)). The Ahlfors map could be used in a similar way in problems of this sort in which more than one obstacle is involved. \[…\]”\]  S.R. Bell, [*The Szegö projection and the classical objects of potential theory in the plane*]{}, Duke Math. J. 64 (1991), ?–?. \[$\spadesuit$ quoted in McCullough 1996 [@McCullough_1996] for the result that the Ahlfors function acquires distinct (simple) zeros when the center $a$ (the place where the derivative is maximized) is chosen near enough the boundary of the domain $\spadesuit$ a probably related result is to be found in Ovchintsev 1996/96 [@Ovchintsev_1996/96] $\spadesuit$ question \[20.09.12\]: does this result extend to bordered surfaces\]  S.R. Bell, [*The Cauchy transform, potential theory, and conformal mapping*]{}, CRC Press, Boca Raton, Florida, 1992. 50 \[$\spadesuit$\]$\bigstar$  S.R. Bell, [*Complexity of the classical kernel functions of potential theory*]{}, Indiana Univ. Math. J. 44 (1995), 1337–1369. \[$\spadesuit$\]  S.R. Bell, [*Ahlfors maps, the double of a domain, and complexity in potential theory and conformal mapping*]{}, J. d’Anal. Math. 78 (1999), 329–344. \[$\spadesuit$ proof that generically two Ahlfors maps suffice to generate the field of meromorphic function of the double of the domain (so-called primitive pairs)\]  S.R. Bell, [*Complexity in complex analysis*]{}, Adv. Math. 172 (2002), 15–52. 50  S.R. Bell, [*Möbius transformations, the Carathéodory metric, and the objects of complex analysis and potential theory in multiply connected domains*]{}, Michigan Math. J. 51 (2003), 352–361. \[$\spadesuit$ p.361: “It is also a safe bet that many of the results in this paper extend to the case of Riemann surfaces. I leave these investigations for the future.”\]  S.R. Bell, [*Quadrature domains and kernel function zipping*]{}, Ark. Math. 43 (2005), 271–287. \[$\spadesuit$ p.271 (Abstract): “It is proved that quadrature domains are ubiquitous in a very strong sense in the realm of smoothly bounded multiply connected domains in the plane. In fact they are so dense that one might as well assume that any given smooth domain one is dealing with is a quadrature domain, and this allows access to a host of strong conditions on the classical kernel functions associated to the domain.”\]  S.R. Bell, [*The Green’s function and the Ahlfors map*]{}, Indiana Univ. Math. J. 57 (2008), 3049–3063. \[$\spadesuit$ yet another fascinating paper among the myriad produced by the author, where now a striking expression is given for the Green’s function of a finitely connected domain in the plane in terms of a single Ahlfors mapping answering thereby (see third page of the introd.) a subconscious desideratum of Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949]\]  S.R. Bell, [*The structure of the semigroup of proper holomorphic mappings of a planar domain to the unit disc*]{}, Complex Methods Function Theory 8 (2008), 225–242. \[$\spadesuit$ a description of all circle maps is given by returning to the original papers of Bieberbach and Grunsky\]  S.R. Bell, [*The Szegö kernel and proper holomorphic mappings to a half plane*]{}, Comput. Methods Funct. Theory 11 (2011), 179–191. 50 \[$\spadesuit$ construction (for domains) of proper holomorphic maps of arbitrary mapping degree, reminiscent of Heins’ argument 1950 [@Heins_1950] about positive harmonic functions\] $\bigstar$$\bigstar$  E. Beltrami, [*Saggio di interpretazione della geometria non-euclidea*]{}, Giornale di Matematiche 6 (1868), 262–280. \[$\spadesuit$\]  E. Beltrami, [*Teoria fundamentale degli spazii di curvatura costante*]{}, Annali di Matematica (2) 2 (1868), 232–255. \[$\spadesuit$\]  R. Benedetti, J.-J. Risler, [*Real Algebraic and Semi-algebraic Sets*]{}, Hermann, Paris, 1990. \[$\spadesuit$ contains much material (and where from I initially learned the Brusotti theorem in “un lavoro di Diploma” under the guidance of Felice Ronga) $\spadesuit$ p.260: elementary properties of separating curves $\spadesuit$ exposition (not always with complete proofs) of some results of the Germano-Russian school: Harnack, Hilbert, Gudkov, Arnold, Rohlin, etc. $\spadesuit$ p.288: “…it can easily by\[=be\] (sic!) proved that any configuration under the broken line of figure 5.24, can be realized by a smooth curve of degree 6.” This is a bit sloppy, for Gudkov’s skill is required!\]  M. Berger, [*Riemannian geometry during the second half of the twentieth century*]{}, Jber. d. Dt. Math.-Verein. 100 (1998), 45–208. \[$\spadesuit$ p.147: “The simplest filling volume, namely that for the circle $S^1$, was only obtained in (\[N.\] Katz, 1998).”), where the reference is (cf. p.196) “Katz, N. (1998). Filling volume of the circle.” $\spadesuit$ This work, presaging a complete solution to Gromov’s filling conjecture, has apparently never been published and probably turned out to contain a gap.\]  M. Berger, [*A Panoramic View of Riemannian geometry*]{}, Springer, 2002. \[$\spadesuit$\]  S. Bergman\[n\], [*Über die Entwicklung der harmonischen Funktionen der Ebene und des Raumes nach Orthogonalfunktionen*]{}, Math. Ann. 86 (1922), 238–271. (Thesis, Berlin, 1921.) \[$\spadesuit$ formulates—like Bieberbach 1914 [@Bieberbach_1914]—the desideratum that the function minimizing the area integral $\int \int \vert f'(z)\vert^2 d\omega$ is the Kreisabbildung (alias Riemann mapping) $\clubsuit$ this desideratum will be only accomplished in the late 1940’s, i.e. Garabedian/Lehto’s era\]  S. Bergman\[n\], [*Über eine Darstellung der Abbildungsfunktion eines Sternbereiches*]{}, Math. Z. 29 (1929), 481–486. \[$\spadesuit$ Minimalbereich in a special case\]  S. Bergman\[n\], [*Über unendliche Hermitesche Formen, die zu einem Bereiche gehören, nebst Anwendunden auf Fragen der Abbildung durch Funktionen von zwei komplexen veränderlichen*]{}, Math. Z. 29 (1929), 641–677. \[$\spadesuit$ p.641 formulates—inspired by Bieberbach 1914 [@Bieberbach_1914]—the concept of a Minimalbereich, by referring to 3 of his previous work (alas no precise cross-references)\]  S. Bergman\[n\], [*Eine Bemerkung über schlichte Minimalabbildungen*]{}, Sitzgsber. Berliner Math. Ges. 30 (1932) \[$\spadesuit$ quoted in Lehto 1949 [@Lehto_1949] for yet another formulation—like Bieberbach 1914 [@Bieberbach_1914]—of the desideratum that the function minimizing the area integral $\int \int \vert f'(z)\vert^2 d\omega$ is the Kreisabbildung (alias Riemann mapping) $\clubsuit$ this desideratum will be only accomplished in the late 1940’s, i.e. Garabedian/Lehto’s era, cf. Lehto 1949 [@Lehto_1949 p.46]\]  S. Bergman\[n\], [*Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande*]{}, J. Reine Angew. Math. 169 (1933), 1–42; and 172 (1934), 89–128. \[$\spadesuit$ p.3 footnote 2 contains some brief indication on the case of multi-connected domains (in one complex variable) and a cross-ref. to Zarankiewicz 1934 [@Zarankiewicz_1934-numerisches]\]  S. Bergman, [*Partial differential equations, Advanced topics*]{} (Conformal mapping of multiply connected domains), Publ. of Brown Univ., Providence, R.I., 1941. \[$\spadesuit$ probably completely incorporated in Bergman 1950 [@Bergman_1950]\] $\bigstar$$\bigstar$$\bigstar$  S. Bergman, [*A remark on the mapping of multiply-connected domains*]{}, Amer. J. Math. 68 (1946), 20–28. \[$\spadesuit$ uniformize via the Bergman kernel domains of finite connectivity, and via Koebe (1914/15) can be used for the Kreisnormierung.\] 78  S. Bergman\[n\], [*Sur les fonctions orthogonales de plusieurs variables complexes avec les applications à la théorie des fonctions analytiques*]{}, Mémorial des Sci. Math. 106 (1947), 1–63. \[$\spadesuit$ p.32 points out that the old desideratum of Bieberbach-Bergman 1922 [@Bergman_1922] of reproving RMT via the problem of least area was still not achieved until this date of 1947, except for the special case of starlike domains (Bergman 1932 [@Bergman_1932] and Schiffer 1938 [@Schiffer_1938-CRAS-domaines-minima]). The breakthrough may have occurred only by Garabedian and Lehto’s Thesis, cf. e.g. [@Lehto_1949]\]  S. Bergman\[n\], [*Sur la fonction-noyau d’un domaine et ses applications dans la théorie des transformations pseudo-conforme*]{}, Mémorial des Sci. Math. 108 (1948). \[$\spadesuit$ quoted in Maschler 1956 [@Maschler_1956] for the theory of minimal domains $\spadesuit$ p.41, Kufarev [@Kufareff_1935/37] is credited for the issue that for a doubly-connected domain the least area map is not univalent(=schlicht) $\spadesuit$ of course it looks evident that univalence fails as well in higher connectivity, cf. e.g. Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949] $\spadesuit$ yet nobody seems to claim that the range is a circle\]  S. Bergman, [*The kernel function and conformal mapping*]{}, Mathematical Surveys 5, Amer. Math. Soc., New York, 1950. 50, 60, 78 \[$\spadesuit$ p.87 existence of a circle map for domains via an explicit formula (p.86) as a ratio of two kernel functions $\spadesuit$ a second revised edition was published in 1970\]  S. Bergman, M. Schiffer [*Kernel functions and conformal mapping*]{}, Compositio Math. 8 (1951), 205–249. 60, 78  A. Bernard, J.B. Garnett, D.E. Marshall, [*Algebras generated by inner functions*]{}, J. Funct. Anal. 25 (1977), 275–285. \[$\spadesuit$ p.282, the Ahlfors function is briefly mentioned as follows: “To show the inner functions separates the points of $X$ we modify the well-known construction of the Ahlfors function for a Denjoy domain.” $\spadesuit$ the bulk of the paper is devoted to the question of knowing when the unit ball of an uniform algebra (typically $H^{\infty}(\Omega)$ for $\Omega$ a finitely connected domain) is the closed convex hull of the inner functions $\spadesuit$ Corollary 5.2 (p.285) gives this conclusion provided the inner functions separate the points of the Shilov boundary, but the authors seem to confess that they do not know whether this proviso is automatically fulfilled (note: of course the simple argument of Stout 1966 [@Stout_1966/67 p.375] saying that inner functions separates points on the Riemann surface (just because the Ahlfors function based at the two given points do separate them) does not apply here, as we are truly looking at mystical points of the Shilov boundary) $\spadesuit$ p.276, one reads: “Minor modifications of the proof for this case \[i.e. finitely connected plane domain\] will give the result when $\Omega$ is a finite open Riemann surface, but we leave those details to the interested reader.” $\spadesuit$ conclusion: since the whole paper actually seeks for an extension of a disc result of Marshall (asserting precisely that the unit ball of the disc algebra $H^{\infty}(\Delta)$ is the closed convex hull of the inner functions), one could wonder if there is not a more naive strategy exploiting more systematically the Ahlfors function\]  L. Bers, [*Quasiconformal mappings and Techmüller’s theorem*]{}, in: Analytic functions, Princeton Univ. Press, 1960, 89–119. \[$\spadesuit$ modernized account of Teichmüller theory\]  L. Bers, [*Uniformization by Beltrami equation*]{}, Comm. Pure Appl. Math. 14 (1961), 215–228. \[$\spadesuit$ contain striking results on the Kreisnormierung\]  L. Bers, [*Automorphic forms for Schottky groups*]{}, Adv. Math. 16 (1975), 332–371. \[$\spadesuit$ modernized proof via quasiconformal deformation of the retrosection theorem (alias Rückkehrschnitttheorem=RST) going back to Klein 1882 [@Klein_1882_Ruckkehrschnitt] and first seriously proved in Koebe 1910 UAK2 [@Koebe_1910_UAK2] $\spadesuit$ a question of some interest is whether the bordered avatar of this RST implies the Ahlfors circle mapping\]$\bigstar$  L. Berzolari, [*Allgemeine Theorie der höheren ebenen algebraischen Kurven*]{}, in: Enzyklopädie der math. Wissenschaften, Bd. III, 2, 1, 313–455, Leipzig, 1906. \[$\spadesuit$ a short survey of Klein’s theory of symmetric surfaces while coining first the designation “[*Klein’s surfaces*]{}” made popular much later by Alling-Greenleaf 1971 [@Alling-Greenleaf_1971].\]  L. Berzolari, [*Algebraische Transformationen und Korrespondenzen*]{}, in: Enzyklopädie der math. Wissenschaften, Bd. III, C, 1, 1, 1781–2218, Leipzig, 1933. \[$\spadesuit$\]  E. Betti, [*Sopra gli spazi di un numero qualunque di dimensioni*]{}, Annali di Matematica (2) 4 (1871), 140–158. \[$\spadesuit$ inspired from Riemann 1852/53 Fragment aus der Analysis situs [@Riemann_1852/53-Fragment-aus-der-Analysis-Situs], and will in turn inspire (modulo being misspelled as Brioschi) Poincaré 1895 [@Poincare_1895-Analysis-Situs] $\spadesuit$ this well-known line of thoughts leads to “homology theory” and is inasmuch relevant to the present article for the issue that several peoples (starting with Riemann, Klein, Poincaré, Brouwer, etc.) used topological methods in function theory, and in the specialized context of Ahlfors circle maps (similar inferences were used by Garabedian 1949 [@Garabedian_1949], Mizumoto 1960 [@Mizumoto_1960], and Gabard 2004/06 [@Gabard_2006])\]  A. Beurling, [*Sur un problème de majoration*]{}, Thèse, Upsala, 1935, 109 pp.  A. Beurling, [*On two problems concerning linear transformation in Hilbert space*]{}, Acta Math. 81 (1949), 239–255. \[$\spadesuit$ the so-called Beurling’s invariant subspaces theorem $\spadesuit$ for an extension to finite bordered Riemann surface see M. Hasumi 1966 [@Hasumi_1966] (and also related work by Voichick 1964 [@Voichick_1964]), yet without using the Ahlfors map, but cite Royden 1962 [@Royden_1962] which is closely allied $\spadesuit$ \[03.10.12\] one can wonder if like for the corona problem/theorem there is a direct inference of the Ahlfors map upon Beurling’s invariant subspaces (as Alling 1964 [@Alling_1964] managed to do for the corona)\]  G.V. Beylĭ, [*On Galois extensions of the maximal cyclotomic field*]{}, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 269–276; English transl., Math. USSR Izv. 14 (1980), 247–256. \[$\spadesuit$ proof that a closed surface is defined over $\Qbar$ iff it ramifies only above 3 points of the sphere $\clubsuit$ can this be extended to bordered surfaces in the context of Ahlfors maps?\]  G.V. Beyli, [*A new proof of the three points theorem*]{}, Sb. Math. 193 (2002), 329–332. \[$\spadesuit$\]  L. Bieberbach, [*Über ein Satz des Herrn Carathéodorys*]{}, Gött. Nachr. (1913), 552–560.  L. Bieberbach, [*Zur Theorie und Praxis der konformen Abbildung*]{}, Rend. del Circolo mat. di Palermo 38 (1914), 98–112. \[$\spadesuit$ this had some influence over Bergman’s Thesis 1921/22 [@Bergman_1922], and is in turn inspired by W. Ritz ca. 1908–09 $\spadesuit$ p.100, first formulation of the principle that the function minimizing the area integral $\int \int \vert f'(z)\vert^2 d\omega$ is the Kreisabbildung (alias Riemann mapping), and the hope is expressed of getting an independent proof of its existence through this least area problem $\clubsuit$ this desideratum (vividly sustained in Bergman’s Thesis 1921/22 [@Bergman_1922] and Bochner’s 1922 [@Bochner_1922]) will be only achieved in the late 1940’s, i.e. Garabedian/Lehto’s era (see Lehto 1949 [@Lehto_1949]) $\clubsuit$ another desideratum (Gabard 16-$\varepsilon$ June 2012, but perhaps already known) would be that such an extremal problem (closely allied to the theory of the Bergman kernel) yields an alternative proof of the Ahlfors mapping $\clubsuit$ even more since it is eminently geometric can we crack—via this Bieberbach-Bergman philosophy—the Gromov filling area conjecture? (Recompense 50 Euros)\]  L. Bieberbach, [*Einführung in die konforme Abbildung*]{}, Sammlung Göschen, Berlin, 1915. \[$\spadesuit$ pp.94–108 deal specifically with Bieberbach’s minimizing principle (cf. Bieberbach 1914 [@Bieberbach_1914])\] L. Bieberbach, [*$\Delta u= e^u$ und die automorphen Funktionen*]{}, Math. Ann. 77 (1916), 173–212. \[$\clubsuit$ p.175 speaks of (Klein’s) orthosymmetry, and write a sentence (which when read ouside of its context) bears strange resemblance with the Ahlfors circle map: “[*Wir nehmen die Fläche orthosymmetrisch an, d.h. sie möge durch diese Symmetrielinien in zwei symmetrische Hälften zerlegt werden, so da[ß]{} es sich also im Hauptkreisfalle um die konforme Abbildung eines berandeten Flächenstückes—der einen Flächenhälfte—auf das Innere des Einheitskreises handelt.*]{}” $\spadesuit$ the real issue in this paper is to implement Schwarz’s desideratum (Göttinger Preisaufgabe 1889) (primarily followed by Picard and Poincaré) of uniformizing (compact) Riemann surfaces via the Liouville equation whose geometric interpretation amounts searching a conformal metric of constant Gaussian curvature $\spadesuit$ \[03.10.12\] probably the above should not be interpreted as an Ahlfors map but rather as the fact that the interior of any compact bordered Riemann surface is uniformized by the unit disc (i.e. the universal covering of the interior is the unit disc)\]  L. Bieberbach, [*Über einige Extremalprobleme im Gebiete der konformen Abbildung*]{}, Math. Ann. 77 (1916), 153–172. \[$\spadesuit$ includes (among other nice geometrical things) the first proof of Koebe’s Viertelsatz with the sharp constant $1/4$ upon the radius of a disc contained in the range of a schlicht map of the unit disc normed by $\vert f'(0) \vert =1$\]  L. Bieberbach, [*Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln*]{}, S.-B. Preuss. Akad. Wiss. Berlin (1916), 940–955. \[$\spadesuit$ where the Bieberbach/coefficient conjecture is first formulated. Solution: de Branges 1984/85.\]  L. Bieberbach, [*Über einen Riemannschen Satz aus der Lehre von der konformen Abbildung*]{}, Sitz.-Ber. Berliner Math. Ges. 24 (1925), 6–9. 60, 78 (also cited in Courant 1939 [@Courant_1939]) \[$\clubsuit$ the schlicht(artig) case of Ahlfors 1950 [@Ahlfors_1950] is proved, and earlier work by Riemann 1857/58 [@Riemann_1857_Nachlass] and Schottky 1877 [@Schottky_1877] is put in perspective\]  L. Bieberbach, [*Lehrbuch der Funktionentheorie*]{}, vols. 1 and 2, Berlin, New York, 1945. (Photographic reprint of the 4th edition of Band I (1934) and the 2nd edition of Band II (1931)) \[$\spadesuit$ cited by Bergman 1950 [@Bergman_1950 p.24] for the proof that the minimum function for the problem $\int \int_{B} \vert f'(z)\vert^2 d\omega$ has circle range; of course the original source is Bieberbach 1914 [@Bieberbach_1914]\] $\bigstar$$\bigstar$$\bigstar$  L. Bieberbach, [*Conformal mapping*]{}, Chelsea, New York, 1953. \[$\spadesuit$\]$\bigstar$  L. Bieberbach, [*Eine Bemerkung zur konformen Abbildung zweifach zusammenhängender Gebiete*]{}, Math. Z. 67 (1957), 99–102. 78  L. Bieberbach, [*Einführung in die konforme Abbildung*]{}, Sammlung Göschen Bd. 768/768a, Walter de Gruyter and Co. (6th ed., 1967). 78 \[$\spadesuit$ includes a proof of the Hilbert-Koebe PSM (in infinite connectivity) $\spadesuit$ presumably an earlier edition (as the one cited in Burckel 1979 [@Burckel_1979]) do the job as well\]  L. Bieberbach, [*Das Werk Paul Koebes*]{}, Jahresber. Deutsche Math.-Verein. 70 (1968), 148–158. 78 \[$\spadesuit$ contains a complete tabulation of Koebe’s work\]  E. Bishop, [*Subalgebras of functions on a Riemann surface*]{}, Pacific J. Math. 9 (1959), 629–642. \[$\spadesuit$\]  E. Bishop, [*Abstract dual extremal problems*]{}, Notices Amer. Math. Soc. 12 No.1 (1965), 123. \[$\spadesuit$  cited in O’Neill-Wermer 1968 [@O'Neill-Wermer_1968] for an abstract version of Ahlfors’ extremal problem pertaining to a function algebra over a compact space $X$\]  E. Biswas, [*On line bundles over real algebraic curves*]{}, Bull. Sci. Math. 134 (2010), 447–449. \[$\spadesuit$\]  A. Bloch, [*La conception actuelle de la théorie des fonctions entières et méromorphes*]{}, L’Enseign. Math. 25 (1926), 83–103. \[$\spadesuit$ great French prose and finitistic philosophy à la Kronecker, culminating to the slogan “Nihil est infinito…”\]  A. Bloch, [*Les fonctions holomorphes et méromorphes dans le cercle-unité*]{}, Mémorial des Sci. Math. 20 (1926), 1–61.  , Dokl. Akad. Nauk SSSR 295 (1987), 268–272; English transl., Soviet Math. Dokl. 36 (1988), 38–42. \[$\spadesuit$\]  Some propositions concerning the geometric representation of imaginaries. [*Ann. of Math. 7*]{} (1892/93), 70–76. \[$\spadesuit$\]  J. Bochnack, W. Kucharz, [*A characterization of dividing real algebraic curves*]{}, Topology (1996), 451–455. \[$\spadesuit$ definition of dividing curves à la Klein $\spadesuit$ albeit the title could be perfectly adequate to reflect the Ahlfors circle map existence theorem, the paper treat another characterization of dividing curves in terms of “regular mapping” (in the sense of real algebraic geometry) and their Brouwer’s topological degree, plus the Hopf’s theorem (classification of sphere valued mappings up to homotopy by the Brouwer degree)\]  J. Bochnack, W. Kucharz, R. Silhol, [*Morphisms, line bundles and moduli spaces in real algebraic geometry*]{}, Publ. Math. IHES (1997 ca.), 5–65. \[$\spadesuit$ p.12: “dividing curves” appear, and occur in some problems about approximation of the smooth category by the algebro-geometric one\]  S. Bochner, [*Über orthogonale Systeme analytischer Funktionen*]{}, Math. Z. 14 (1922), 180–207. (Thesis, Berlin, 1921.) \[$\spadesuit$ p.184: like Bieberbach 1914 [@Bieberbach_1914] and Bergman 1922 [@Bergman_1922] the author confesses to be not able to reprove the RMT via Bieberbach’s minimum problem (least area map) $\spadesuit$ this problem will be cracked (independently) in Garabedian and Lehto’s Thesis (cf. Garabedian 1949 [@Garabedian_1949] and Lehto 1949 [@Lehto_1949])\]  S. Bochner, [*Fortsetzung Riemannscher Flächen*]{}, Math. Ann. 98 (1927), 406–421. \[$\spadesuit$ any Riemann surface of finite genus embeds conformally into a closed Riemann surface of the same genus $\spadesuit$ any Riemann surface embeds into a non-prolongeable Riemann surface\]  C.F. Bödigheimer, [*Configuration models for moduli spaces of Riemann surfaces with boundary*]{}, Abh. Math. Seminar Hamburg (2006). \[$\spadesuit$\]  M.D. Bolt, S. Snoeyink, E. van Andel, [*Visual representation of the Riemann and Ahlfors maps via the Kerzman-Stein equation*]{}, Involve 3 (2010), 405–420. \[$\spadesuit$ from MR: “The paper provides an elementary description of the Riemann and Ahlfors maps using the Szegö kernel. It further describes a numerical implementation of the maps.”\]$\bigstar$$\bigstar$  E. Borel, [*Leçons sur la théorie des fonctions*]{}, Gauthier-Villars, Paris, 1898. \[$\spadesuit$ complex function theory, but also the starting point of modern measure theory (influence over Lebesgue)\]  J.B. Bost, [*Introduction to compact Riemann surfaces, Jacobians and Abelian varieties*]{}, in: From Number Theory to Physics, Springer-Verlag, 1992, Second Corrected Printing 1995. \[$\spadesuit$ p.99–104 contains an account of the Belyi-Grothendieck theorem as well as its geometric traduction in terms of equilateral triangulations\]  M. Brandt, [*Ein Abbildungssatz für endlich-vielfach zusammenhängende Gebiete*]{}, Bull. de la Soc. des Sciences et des Lettres de Łódź XXX, 3 (1980). \[$\spadesuit$ extension of Koebe’s KNP to shapes with arbitrary contours; similar result in Harrington 1982 [@Harrington_1982] $\spadesuit$ variant of proof in Schramm 1996 [@Schramm_1996]\] $\bigstar$$\bigstar$$\bigstar$  M. Brelot, G. Choquet, [*Espaces et lignes de Green*]{}, Ann. Inst. Fourier 3 (1951), 199–263. 60 \[$\spadesuit$ the paper is started with a result of Evans (1927) that the streamlines of the Green’s function $G(z,t)$ \[with pole at $t$\] in a simply-connected plane domain have almost all (in the angular sense about $t$) finite length and therefore converge to a frontier-point $\spadesuit$ this is adapted to domains of arbitrary connectivity (as well as to “superior spaces”) $\spadesuit$ presumably as well as to bordered surfaces: \[11.08.12\] incidentally one could dream of a proof of Gromov’s FAC just via the Green’s function, while using the corresponding isothermic coordinates to calculate the area\]  M. Brelot, [*La théorie moderne du potentiel*]{}, Ann. Inst. Fourier 4 (1952), 113–140. \[$\spadesuit$ p.114 “Mais si l’on peut dire que tout est dans l’[œ]{}uvre de Gauss, il apparut bientôt que la [*rigueur*]{} était insuffisante” $\spadesuit$ a historical survey starting from Poisson, then Gauss 1840 (who considers as evident that the minimum energy is attained, electrostatic influence, problème du balayage), and the culmination of Frostman’s Thesis (1935); meanwhile Dirichlet, Riemann and Hilbert; and also Neumann, Schwarz, Harnack and Poincaré’s balayage; next Fredholm’s theory (1900) and its application to Dirichlet and Neumann; Zaremba’s works; Lebesgue’s integral found an application in Fatou’s study of the Poisson integral and Evans introduced the Radon integral in potential theory; Perron and Wiener renewed the Dirichlet problem; F. Riesz introduced subharmonic functions (precursors like Poincaré and Hartogs are signaled on p.134); ca. 1930 de la Vallée Poussin took up again the méthode du balayage to study “les masses balayées”, etc.\]  E. Brettler, [*Absolute Galois groups of real function fields in one variable*]{}, Diss. McGill, Univ. Montréal 1972. \[$\spadesuit$ quoted in Geyer-Martens 1977 [@Geyer-Martens_1977]\] E. Brieskorn, H. Knörrer, [*Ebene algebraische Kurven*]{}, Birkäuser Boston, 1981; English translation: [*Plane Algebraic Curves*]{}, Trans. from the German by John Stillwell, Birkäuser Basel, 1986. \[$\spadesuit$ p.ii: “Es ist die Freude an der Gestalt in einem höheren Sinne, die den Geometer ausmacht. (Clebsch, in memory of Julius Plücker, Gött. Abh. Bd. 15).”\] A. Brill, M. Nöther, [*Ueber die algebraischen Functionen und ihre Anwendung in der Geometrie*]{}, Math. Ann. 7 (1874), 269–310. \[$\spadesuit$\] A. Brill, M. Noether, [*Bericht über die Entwicklung der Theorie der algebraischen Functionen in älterer und neuerer Zeit*]{}, Jahresber. Deutsche Math.-Verein. 3 (1894), 107–566. \[$\spadesuit$ a monumental historiography spreading over more than 400 pages, from Descartes to Riemann and much more\]  A. Brill, [*Über dden Weierstra[ß]{}chen Vorbereitungssatz*]{}, Math. Ann. 64 (1910), 538–549. \[$\spadesuit$\]  L.E.J. Brouwer, [*Über die topologischen Schwierigkeiten des Kontinuitätsbeweises der Existenztheoreme eindeutig umkehrbarer polymorpher Funktionen auf Riemannschen Flächen*]{}, Gött. Nachr. (1912), 603–606. 60 \[$\spadesuit$ topological methods as applied to uniformization\]  L.E.J. Brouwer, [*Über die Singularitätenfreiheit der Modulmannigfaltigkeit*]{}, Gött. Nachr. (1912), 803–806. 60 \[$\spadesuit$ idem\]  L.E.J. Brouwer, [*Ueber eineindeutige, stetige Transformationen von Flächen in sich (6. Mitt.)*]{}, KNAW Proceedings 21 (1919), 707–710. \[$\spadesuit$ Brouwer seems to vindicate his priority over Koebe for a topological proof of uniformization via the continuity method\]  L. Brusotti, [*Sulla generazzione di curve piane algebriche reali mediante “piccola variazione” di una curva spezzata*]{}, Annali di Mat. (3) 22 (1913), 117–169. \[$\spadesuit$ systematic small perturbation method for the independent smoothings of nodal plane curves (based upon an extrinsic version of Riemann-Roch (??), worked out over ${\Bbb C}$ by Severi $\spadesuit$ forerunners Plücker 1839 [@Plücker_1839], Klein 1873 [@Klein_1873-Uber-Flächen-dritter-Ordn], Harnack 1876 [@Harnack_1876], etc\]  L. Brusotti, [*Curve generatrici e curve aggregate nella costruzione di curve piane d’ordine assegnato dotate del massimo numero di circuiti*]{}, Rend. Circ. Mat. Palermo 42 (1917), 138–144. \[$\spadesuit$ cited in Mikhalkin 2000 [@Mikhalkin_2000]\]  L. Brusotti, [*Sulla “piccola variazione” di una curva piana algebrica reale*]{}, Rend. Rom. Acc. Lincei (5) 30 (1921), 375–379. \[$\spadesuit$ systematic small perturbation method for the independent smoothings of nodal plane curves (based upon an extrinsic version of Riemann-Roch, worked out over ${\Bbb C}$ by Severi $\spadesuit$ forerunners Plücker 1839 [@Plücker_1839], Klein 1873 [@Klein_1873-Uber-Flächen-dritter-Ordn], Harnack 1876 [@Harnack_1876], etc. $\spadesuit$ immediate follower Gudkov 1962 [@Gudkov_1962] (extension to cusps), Gudkov 1980/80 [@Gudkov_1980/80-Brusotti] (extension to other surfaces), etc.\]  L. Brusotti, [*Su talune questioni di realita nei loro metodi, resultati e problemi*]{}, in: Colloque sur les questions de réalité en géométrie, (Liège 1955), Georges Johne, Liège, et Masson, Paris, 1956, 105–129. \[$\spadesuit$ briefly discussed in Viro’s survey 1989/90 [@Viro_1989/90-Construction] and also cited in Viro 1986/86 [@Viro_1986/86-Progress]\]  E. Bujalance, J.J. Etayo, J.M. Gamboa, G. Gromadzki, [*Automorphisms groups of compact bordered Klein surfaces*]{}, Lecture Notes in Math. 1439, Springer-Verlag, Berlin, 1990. \[$\spadesuit$\]  E. Bujalance, A. Costa, S. Natanzon, D. Singerman, [*Involutions of compact Klein surfaces*]{}, Math. Z. 211 (1992), 461–478. \[$\spadesuit$\]  E. Bujalance, G. Gromadzki, D. Singerman, [*On the number of real curves associated to a complex curve*]{}, Proc. Amer. Math. Soc. 120 (1994), 507–513. \[$\spadesuit$\]  J. Burbea, [*The Carathéodory metric in plane domains*]{}, Kodai. Math. Sem. Rep. 29 (1977), 159–166. \[$\spadesuit$ application of the Ahlfors function to a curvature estimate of the Carathéodory metric (defined via the analytic capacity) along the line of Suita’s works $\spadesuit$ Abstract: “Let $D\notin O_{AB}$ be a plane domain \[i.e., supporting non-constant bounded analytic functions\] and let $C_D(z)$ be its analytic capacity at $z\in D$ \[that is the maximum distortion of a circle-map centered at $z$\]. Let ${\cal K}_{D}(z)$ be the curvature of the Carathéodory metric $C_D(z) \vert dz \vert$. We show that ${\cal K}_D(z)<-4$ if the Ahlfors function of $D$ w.r.t. $z$ has a zero other than $z$. For finite \[domains\] $D$, ${\cal K}_D(z) \le -4$ and equality holds iff $D$ is simply-connected. As a corollary we obtain a result proved first by Suita, namely, that ${\cal K}_D(z)\le -4$ if $D\notin O_{AB}$. Several other properties related to the Carathéodory metric are proven.” $\spadesuit$ a little anachronism is noteworthy, here logically the Ahlfors function and the allied analytic capacity (1947 [@Ahlfors_1947]) precedes the Carathéodory metric (1926 [@Caratheodory_1926] and 1927 [@Caratheodory_1927]), but of course in view of the real history, especially Carathéodory 1928 [@Caratheodory_1928] the definitional aspect is essentially compatible with the historical flow\]  J. Burbea, [*The curvatures of the analytic capacity*]{}, J. Math. Soc. Japan 29 (1977), 755–761. \[$\spadesuit$ p.755: Ahlfors function à la Havinson 1961/64 [@Havinson_1961/64], i.e. for domains $D\notin O_{AB}$, analytic capacity, method of the minimum integral w.r.t. the Szegö kernel\]  J. Burbea, [*Capacities and spans on Riemann surfaces*]{}, Proc. Amer. Math. Soc. 72 (1978), 327–332. \[$\spadesuit$ p.329: “Ahlfors function” is mentioned (in connection with the analytic capacity, yet it is not clear to me \[03.10.12\] if this definition is meaningful not for a domain but also on a finite Riemann surface)\]  J. Burbea, [*The Schwarzian derivative and the Poincaré metric*]{}, Pacific J. Math. 85 (1979), 345–354. \[$\spadesuit$\]  J. Burbea, [*The Cauchy and the Szegö kernels on multiply connected regions*]{}, Rend. Circ. Mat. Palermo (2) 31 (1982), 105–118. \[$\spadesuit$ Ahlfors function mentioned on p.106 an p.116\]  R.B. Burckel, [*An Introduction to Classical Complex Analysis*]{}, Vol.1, Mathematische Reihe 64, Birkhäuser, 1979. \[$\spadesuit$ p.357 some nice comments upon the literature about PSM\]  W. Burnside, [*On functions determined from their discontinuities, and a certain form of boundary condition*]{}, Proc. London Math. Soc. 22 (1891), 346–358. \[$\spadesuit$ detected \[30.07.12\] via W. Seidel’s bibliogr. (1950/52), who summarize the paper as: a method is given for mapping a region bounded by $m$ simple closed curves $C_i$ on an $n$-sheeted Riemann surface over the $w$-plane, where the curves $C_i$ correspond to rectilinear slits $\spadesuit$ surprisingly this paper is not quoted in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] nor in Grunsky 1978 [@Grunsky_1978] $\spadesuit$ the topic addressed bears some vague resemblance with the Bieberbach-Grunsky-Ahlfors paradigm of the circle map\]  W. Burnside, [*On a class of automorphic functions*]{}, Proc. London Math. Soc. (3) 23 (1892), 49–88. \[$\spadesuit$\]  P. Buser, M. Seppälä, R. Silhol, [*Triangulations and moduli spaces of Riemann surfaces with group actions*]{}, Manuscr. Math. 88 (1995), 209–224. \[$\spadesuit$ connectedness of the moduli space of real curves when projected down in the complex moduli\]  A.P. Calderón, [*Cauchy integrals on Lipschitz curves and related operators*]{}, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1324–1327. \[$\spadesuit$ implies a resolution of the so-called Denjoy conjecture, according to which a subset of a rectifiable curve is removable for the class of bounded analytic functions (alias Painlevé null-sets) iff it has zero length $\spadesuit$ the explicit link from Calderón-to-Denjoy is made explicit in Marshall [@Marshall_1978?], upon combining a long string of previous works (Garabedian, Havinson, Davie 1972 [@Davie_1972])\]  A.P. Calderón, [*Commutators, singular integrals on Lipschitz curves and applications*]{}, ICM Helsinki 1978, 85–96. \[$\spadesuit$ the Denjoy’s conjecture is mentioned as an application of Calderón 1977 [@Calderon_1977], as follows (p.90): “Now let us turn to applications. Let $\Gamma$ be a simple rectifiable arc in the complex plane. Then the function $G(z)=\frac{1}{2\pi i}\int_\Gamma \frac{f(w)}{w-z} dw$, where $f(w)$ is a function on $\Gamma$ which is integrable w.r.t. arc length, has a limit almost everywhere in $\Gamma$ as $z$ approaches nontangentially a point of $\Gamma$. \[…\] Another application is the following result due to D.E. Marshall (personal communication) which confirms an old conjecture of A. Denjoy (1909 [@Denjoy_1909-Painleve/Sur-les-fct-anal-unif-a-sing-discontinues]): the analytic capacity $\gamma(E)$ of a compact subset $E$ of a rectifiable arc in the complex plane is zero if and only if its one-dimensional Hausdorff measure vanishes.” $\spadesuit$ for the detailed proof see Marshall [@Marshall_1978?] (and maybe also Melnikov 1995 [@Melnikov_1995])\]  A.P. Calderón, [*Acceptance speech for the Bôcher Price*]{}, Notices A.M.S. 26 (1979), 97–99. \[$\spadesuit$ the solution to the Denjoy’s conjecture is mentioned as one of the most significant application of the article Calderón 1977 [@Calderon_1977]\]$\bigstar$$\bigstar$  A. Candel, [*Uniformization of surface laminations*]{}, Ann. Sci. Ecole Norm. Sup. (1993). \[$\spadesuit$ ... To have the same relation between Riemann surface laminations and oriented surface laminations with riemannian metric we then need a regularity theorem for the Beltrami equation depending on parameters. This is precisely what Ahlfors and Bers proved in their classical ... \]  C. Carathéodory, [*Sur quelques applications du théorème de Landau-Picard*]{}, C.R. Acad. Sci. Paris 144 (1907), 1203–1204; also in: Ges. Math. Schriften, Band 3, 6–9. \[$\spadesuit$ first modern proof of the Schwarz lemma, acknowledging E. Schmidt, compare footnote 2: “Je dois cette démonstration si élégante d’un théorème connu de M. Schwarz (Ges.Abh.,t.2,p.108) à une communication orale de M. E. Schmidt.”\]  C. Carathéodory, [*Über die Variabilitätsbereich der Koeffizienten von Potenzreihen die gebebene Werte nicht annehmen*]{}, Math. Ann. 64 (1907), 95–115. \[$\spadesuit$ this and the next entry where the first work bringing together Minkowski’s theory of convex sets with complex function theory $\spadesuit$ for an extension of this Carathéodory theory to finite Riemann surface, cf. Heins 1976 [@Heins_1976]\]  C. Carathéodory, [*Über die Variabilitätsbereich der der Fourierschenkonstanten von positiven harmonischen Funktionen*]{}, Rend. Circ. Mat. Palermo 32 (1911), 193–217. \[$\spadesuit$\]  C. Carathéodory, L. Fejér, [*Über den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und über den Picard-Landauschen Satz*]{}, Rend. Circ. Mat. Palermo 32 (1911), 218–239. \[$\spadesuit$ for a (vague?) interconnection of this article with the Ahlfors map, cf. Jenkins-Suita 1979 [@Jenkins-Suita_1979]\]  C. Carathéodory, [*Untersuchungen über die konformen Abbildungen von festen und veränderlichen Gebieten*]{}, Math. Ann. 72 (1912), 107–144. 78  C. Carathéodory, [*Über die gegenseitige Beziehung der Ränder bei der konformen Abbildung des Inneren einer Jordanschen Kurve auf einen Kreis*]{}, Math. Ann. 73 (1913), 305–320.  C. Carathéodory, [*Über die Begrenzung einfach zusammenhängender Bereiche*]{}, Math. Ann. 73 (1913), 323–370.  C. Carathéodory, [*Elementarer Beweis für den Fundamentalsatz der konformen Abbildung*]{}. In: Mathematische Abhandlungen, Hermann Amandus Schwarz zu seinem fünfzigjähren Doktorjubiläum am 6. August 1914 gewidmet von seinen Freunden und Schülern, 19–41; also in: Ges. Math. Schriften, Band 3, 273–299. \[$\spadesuit$ p.294 perhaps the first usage of the jargon “[*quasikonform*]{}”, compare Ahlfors’ memory failure reported in Kühnau 1997 [@Kuehnau_1997] $\spadesuit$ more importantly the classical square-root procedure is developed in detail\]  C. Carathéodory, [*Über das Schwarzsche Lemma bei analytischen Funktionen von zwei komplexen Veränderlichen*]{}, Math. Ann. 97 (1926), 76–98. \[$\spadesuit$ quoted in Grunsky 1940 [@Grunsky_1940 p.233], who discusses the connection between the Carathéodory metric and the “Ahlfors” function (which in the present connection should be definitively better called the “Grunsky-Ahlfors function”)\]  C. Carathéodory, [*Über eine spezialle Metrik, die in der Theorie der analytischen Funktionen auftritt*]{}, Atti Pontifica Acad. Sc., Nuovi Lincei 80 (1927), 135–141. \[$\spadesuit$ where the so-called Carathéodry metric is first defined (but see also the previous entry Carathéodory 1926 [@Caratheodory_1926]), which in turn turned out to be closely related to the Ahlfors function, cf. e.g. Grunsky 1940 [@Grunsky_1940], Simha 1975 [@Simha_1975], Burbea 1977 [@Burbea_1977-Caratheodory], Krantz 2008 [@Krantz_2008]\]  C. Carathéodory, [*Bermerkungen zu den Existenztheoremen der konformen Abbildung*]{}, Bull. Calcutta Math. Soc. 20 (1928), 125–134; also in: Ges. Math. Schriften, Band 3, 300–310. 60 \[$\clubsuit$ along lines initiated by Fejér-Riesz (published by Radó 1922/23 [@Rado_1922-3]) a new proof of RMT is given via an extremal problem, which is a simply-connected prelude to Ahlfors 1950 [@Ahlfors_1950] $\clubsuit$ as pointed out by Remmert 1991 [@Remmert_1991] Carathéodory’s elegant proof appears rarely in book form (exception Narasimhan’s book), and is somewhat less popular than the variant of Fejér-Riesz-Bieberbach-Ostrowski $\spadesuit$ the article involves (cf. p.303) the extremal problem $\max \vert f(z_1) \vert$ of maximizing the modulus of the function at an auxiliary point $z_1$, whereas the other method (Fejér-Riesz, etc.) maximizes the derivative at the basepoint $z_0$ $\spadesuit$ it is precisely Carathéodory’s version which is extended in Ahlfors 1950 [@Ahlfors_1950], but of course the other formulation lead likewise to a circle map\]  C. Carathéodory, [*Conformal representation*]{}, Cambridge Tracts in Math. and Math. Physics 28, London 1932. (2nd edition 1958) 60, 78 \[$\spadesuit$ an introduction to problem of conformal mapping\]  C. Carathéodory, [*On Dirichlet’s problem*]{}, Amer. J. Math. 59 (1937), 709–731. \[$\spadesuit$ surprisingly this item is not quoted in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] $\spadesuit$ p.710: “In the foregoing chapter, I have tried to give a very elementary treatment of the principal properties of harmonic functions culminating in the existence proof for Dirichlet’s problem devised by O. Perron \[=1923 [@Perron_1923]\] and very much simplified by T. Radó and F. Riesz \[=1925 [@Rado-Riesz_1925]\]. I have done this in order to show how the whole theory can be condensed if one puts systematically from the outset Poisson’s Integral in the limelight.”\]  C. Carathéodory, [*A proof of the first principal theorem on conformal representation*]{}, Studies and Essays presented to R. Courant on his 60th birthday, Jan. 8, 1948, Interscience Publ., 1948, 75–83; also in: Ges. Math. Schriften, Band 3, 354–361. 60, 78 \[$\spadesuit$ another proof of RMT through an iterative method, using square-roots operations, Schwarz’s lemma and Montel $\spadesuit$ naive question, although this might be more in line with the earlier approach ca. 1910 of Koebe-Carathéodory, this approach looks more involved than the extremum problem in the previous item [@Caratheodory_1928], and perhaps less susceptible of extension to Riemann surfaces\]  C. Carathéodory, [*Funktionentheorie, I, II*]{}, Birkhäuser, Basel, 1950. 60, 78  C. Carathéodory, [*Bemerkung über die Definition der Riemannschen Flächen*]{}, Math. Z. 52 (1950), 703–708. 60 \[$\spadesuit$ purist approach to uniformization via extremal problems, similar ideas in several papers by Grunsky not listed here, cf. his Coll. Papers $\spadesuit$ the (Grenzkreis) uniformization appears also in Carathéodory 1928 [@Caratheodory_1928]\]  A.L. Carey, K.C. Hannabuss, [*Infinite dimensional groups and Riemann surfaces field theories*]{}, Comm. Math. Phys. 176 (1996), 321–351. \[$\spadesuit$\]  T. Carleman, [*Über ein minimal Problem der mathematischen Physik*]{}, Math. Z. 1 (1918), 208–212. 78 \[$\spadesuit$ used in Gaier 1978 [@Gaier_1978-JDMV] and Alenycin 1981/82 [@Alenicyn_1981/82] in connection with an extension of the (Bieberbach 1914) minimum area problem to multiply-connected regions\]  T. Carleman, [*Sur la représentation conforme des domaines multiplement connexes*]{}, C.R. Acad. Sci. Paris 168 (1919), 843–845. 78 \[$\spadesuit$ another proof of KNP=Kreisnormierungsprinzip, originally due to Koebe 1907/1920, if not (implicit in) Schottky 1877 [@Schottky_1877]\]  T. Carleman, [*Über die Approximation analytischer Funktionen durch linear Aggregate von vorgegebenen Potenzen*]{}, Arkiv för mat., astron. o. fys. 17 (1922). \[$\spadesuit$ credited in Lehto 1949 [@Lehto_1949 p.8] for some work (independent of Bergman 1922 [@Bergman_1922] and Bochner’s 1922 [@Bochner_1922]) inaugurating the usage of orthogonal systems in the theory of conformal mappings\]  L. Carleson, [*On bounded analytic functions and closure problems*]{}, Ark. Mat. 2 (1952), 283–291. \[$\spadesuit$\]$\bigstar$$\bigstar$  L. Carleson, [*Interpolations by bounded analytic functions and the Corona problem*]{}, Ann. of Math. (2) 76 (1962), 547–559. \[$\spadesuit$ one of the super-famous problem solved by Carleson, and which received (thanks Alling 1964 [@Alling_1964] and others) an extension from the disc to any compact bordered Riemann surface via the Ahlfors circle map\]  L. Carleson, [*Selected problems on exceptional sets*]{}, Van Nostrand, Princeton, 1967. \[$\spadesuit$ p.73–82 uniqueness of the Ahlfors extremal function \[the one maximizing the derivative at a fixed point amongst functions bounded-by-one\] for domains of infinite connectivity; similar work in Havinson 1961/64 [@Havinson_1961/64] and simplifications in Fisher 1969 [@Fisher_1969]\]  L. Carleson, [*Lars Ahlfors and the Painlevé problem*]{}. In: [*In the tradition of Ahlfors and Bers*]{} (Stony Brook, NY, 1998), 5–10. Contemp. Math. 256, Amer. Math. Soc., Providence, RI, 2000. Nostrand, Princeton, 1967. \[$\spadesuit$ survey of the theory of removable sets for bounded analytic functions (a.k.a. Painlevé null-sets) from Painlevé, Denjoy to G. David, via Ahlfors (analytic capacity), Garabedian and Melnikov (Menger curvature). Future research is suggested along the 3 axes: (i) develop a theory of periods for the conjugate of positive harmonic functions (ii) sharper study of the extremal function (and induced measure) that appear in Garabedian 1949 [@Garabedian_1949] (iii) to continue the study of the Cauchy integrals in relation with Menger curvature and rectifiability\]  F. Carlson, [*Sur le module maximum d’une fonction analytique uniforme. I*]{}, Ark. Mat. Astron. Fys. 26 (1938), 13pp. 78 \[$\spadesuit$ quoted (joint with Teichmüller 1939 [@Teichmueller_1939] and Heins 1940 [@Heins_1940-Extremal]) in Grunsky 1940 [@Grunsky_1940] as one of the precursors of the extremal problem for bounded analytic functions\] $\bigstar$$\bigstar$$\bigstar$  H. Cartan, [*Théorie él’ementaire des fonctions analytiques d’une ou plusieurs variables complexes*]{}, Hermann, Paris, 1961. \[$\spadesuit$\]  E. Casas-Alvero, Roots of complex polynomials and foci of real algebraic curves, L’Enseign. Math. 2013. \[$\spadesuit$\]  A.L. Cauchy, [*Mémoire sur la théorie des intégrales définies*]{}, communicated to the Paris Academy in 1814, and first published in 1827. \[$\spadesuit$ first occurrence of the Cauchy-Riemann equations, as criterion for the analyticity (holomorphy) of functions of a complex variable, modulo earlier occurrences in the work of Euler on hydrodynamics, and even earlier in the work of Jean le Rond d’Alembert (1717–1783) (on behalf of p.12 of Monastyrsky 1987/99 [@Monastyrsky_1987/99/08-even-1979], who do not give the exact sources)\]  A.L. Cauchy, [*Mémoire sur les intégrales définies prises entre des limites imaginaires*]{}, De Bure Frères, Paris, 1825, posthumous papers not published until 1874, in: [Œ]{}uvres de Cauchy, 1876, Série II, tome XV, 41–89. \[$\spadesuit$ definition of the integral of a function in the complex domain, including the case of singularity in which case the integral may depend on the path $\spadesuit$ first appearance of the notion of residue\]  A.L. Cauchy, [*???*]{}, work completed in 1831, published in 1836. \[$\spadesuit$ power series expansion of an analytic function and the integral representation of $f(z)$ inside a circle (Cauchy formula)\]  A.L. Cauchy, [*Considérations nouvelles sur les intégrales définies qui s’étendent à tous les points d’une courbe fermée, et sur celles qui sont prises entre des limites imaginaires*]{}, C.R. Acad. Sci. Paris 23 (1846), 689. \[$\spadesuit$ Cauchy’s residue theorem\]  F. Cecioni, [*Sulla rappresentazione conforme delle aree piane pluriconnesse su un piano in cui siano eseguiti dei tagli paralleli*]{}, Rend. Circ. Mat. Palermo 25 (1908), 1–19. 78 \[$\spadesuit$ another derivation of the parallel-slit map of Schottky 1877 [@Schottky_1877], via several citation to Picard’s book for the foundational aspects $\spadesuit$ as Schottky’s proof depends on a heuristic moduli count, this paper of Cecioni may well be regarded as the first rigorous existence proof of PSM (cf., e.g., Grunsky 1978 [@Grunsky_1978 p.185])\]  F. Cecioni, [*Sulla rappresentazione conforme delle aree pluriconnesse appartenenti a superficie di Riemann*]{}, Annali delle Università Toscane 12, nuova serie (1928), 27–88. \[$\spadesuit$ cited via Matildi 1948 [@Matildi_1945/48]; WARNING: this entry looks much like the next item, yet differs in the pagination\] $\bigstar$$\bigstar$  F. Cecioni, [*Sulla rappresentazione conforme delle aree pluri-connesse appartenenti a superficie di Riemann*]{}, Rend. Accad. d. L. Roma (6) 9 (1929), 149–153. 60 \[$\spadesuit$ cited via Ahlfors-Sario 1960 [@Ahlfors-Sario_1960]\] $\bigstar$$\bigstar$  F. Cecioni, [*Osservazioni sopra alcuni tipi aree e sulle loro curve caratteristiche nella teoria della rappresentazione conforme*]{}, Rend. Palermo 57 (1933), 101–122. \[$\spadesuit$ la parole “curve catteristiche” means the Schottky(-Klein) double $\spadesuit$ contains several nice remarks about the Klein correspondence when particularized to orthosymmetric curve tolerating a direct-conformal involution which is sense reversing on the ovals\]  F. Cecioni, [*Un teorema su alcune funzioni analitiche, relative ai campi piani pluriconnessi, usate nella teoria della rappresentazione conforme*]{}, Ann. Pisa (2) 4 (1935), 1–14. 78  M. Černe, J. Globevnik, [*On holomorphic embedding of planar domains into ${\Bbb C}^2$*]{}, J. Anal. Math. 81 (2000), 269–282. 50 \[$\spadesuit$ Koebe’s Kreisnormierungsprinzip is combined with the Ahlfors function to show that every bounded, finitely connected domain of ${\Bbb C}$ without isolated boundary points embeds properly holomorphically into ${\Bbb C}^2$ $\spadesuit$ of course, those are not the sole ingredients for otherwise the method would probably extend to positive genus surfaces in view of Ahlfors 1950 [@Ahlfors_1950], and positive genus extensions of KNP due to Haas 1984 [@Haas_1984]/Maskit 1989 [@Maskit_1989]\]  M. Černe, F. Forstnerič, [*Embedding some bordered Riemann surfaces in the affine plane*]{}, Math. Research Lett. 9 (2002), 683–696. 50 \[$\spadesuit$ Ahlfors 1950 is cited at several places $\spadesuit$ p.684: “On each smoothly bounded domain $\Omega\Subset {\Bbb C}$ with $m$ boundary components there exists an inner function $f$ with $\deg(f)=m$ \[Ahl\](=Ahlfors 1950 [@Ahlfors_1950])[^121]. The map $F(x)=(f(x),x)\in{\Bbb C}^2$ for $x\in \overline\Omega$ satisfies the hypothesis of Theorem 1.2 and hence $\Omega$ embeds in ${\Bbb C}^2$. This is the theorem of Globevnik and Stens[ø]{}nes \[GS\](=1995).” $\spadesuit$ p.684: “We shall call a bordered Riemann surface $\cal R$ hyperelliptic if its double is hyperelliptic. Such \[an\] $\cal R$ has either one or two boundary components[^122] and it admits a pair of inner functions $(f,g)$ which embed ${\rm int}{\cal R}$ in the polydisc $U^2$ such that $b{\cal R}$ is mapped to the torus $(bU)^2$; moreover, one of the two functions has degree $2g_{\cal R}+m$ and the other one has degree $2$ (see \[Ru1\](=Rudin 1969 [@Rudin_1969]) and sect.2 in \[Gou\](=Gouma 1998 [@Gouma_1998])). Thus $\cal R$ is of class ${\cal F}$ and we get:—[**Corollary 1.3**]{} [*If ${\cal R}$ is a hyperelliptic bordered Riemann surface then ${\rm int} {\cal R}$ admits a proper holomorphic embedding in ${\Bbb C}^2$. In particular, each torus with one hole embeds properly holomorphically into ${\Bbb C}^2$.*]{} $\spadesuit$ p.686: “[**Comments regarding class ${\cal F}$.**]{} It is proved in \[Ahl,pp.124–126\](=Ahlfors 1950 [@Ahlfors_1950]) that on every bordered Riemann surface $\cal R$ of genus $g_{\cal R}$ with $m$ boundary components there is an inner function $f$ with multiplicity $2 g_{\cal R}+m$ (although the so-called Ahlfors functions may have smaller multiplicity). A generic choice of $g\in A^1({\cal R})$ gives an immersion $F=(f,g)\colon {\cal R}\to \overline U \times {\Bbb C}$ with at most finitely many double points (normal crossings). The main difficulty is to find $g$ such that $F=(f,g)$ is injective on ${\cal R}$. We do not know whether such $g$ always exists as Oka’s principle does not apply in this situation (Proposition 2.2).” $\spadesuit$ Ahlfors 1950 is cited once more on p.687 during the proof of Theorem 1.1 stating that there is no topological obstruction to holomorphic embeddability in ${\Bbb C}^2$, in the following sense (p.683) “[**Theorem 1.1**]{} [*On each bordered surface $\cal R$ there exists a complex structure such that the interior ${\rm int} {\cal R}={\cal R}\setminus \partial {\cal R}$ admits a proper holomorphic embedding in ${\Bbb C}^2$.*]{} $\spadesuit$ p.693: “[**Remark.**]{} As already mentioned, Ahlfors \[Ahl\](=1950) constructed inner functions of multiplicity $2g_{\cal R}+m$ on any bordered Riemann surface. Proposition 4.1 shows that such functions are stable under small perturbations of the complex structure. On the other hand this need not be true for the Ahlfors function $f_p$ which maximizes the derivative at a given point $p\in {\cal R}$ since the degree of $f_p$ may depend on $p$.” $\diamondsuit$ \[28.09.12\] maybe there is a somewhat more elementary approach to the main result (no topological obstruction) by looking at some real algebraic models in ${\Bbb P}^2$ or ${\Bbb P}^1\times {\Bbb P}^1$, for instance taking a saturated pencil on the Gürtelkurve (cf. Gabard 2006 [@Gabard_2006]) and removing an imaginary line of this pencil one gets an embedding of the bordered surface (half of the real quartic $C_4$) into ${\Bbb C}^2$\]  M. Černe, [*Nonlinear Riemann-Hilbert problem for bordered Riemann surfaces*]{}, Amer. J. Math. 126 (2004), 65–87. 50 \[$\spadesuit$\]  M. Černe, M. Flores, [*Generalized Ahlfors functions*]{}, Trans. Amer. Math. Soc. 359 (2007), 671–686. \[$\spadesuit$ a promising generalization of the Ahlfors function is given where the (static) unit-circle is replaced by a dynamical family $\{\gamma_z \}_{z\in \partial F}$ of Jordan curves enclosing the origin parametrized by the boundary of the bordered surface $F$\]  M. Černe, F. Forstnerič, [*Embedding some bordered Riemann surfaces in the affine plane*]{}, Math. Res. Lett. 9 (2002), 683–696. \[$\spadesuit$ it is shown that there is no topological obstruction to embed membranes in ${\Bbb C}^2$ $\spadesuit$ this is a good starting toward the difficult conjecture about proper embedding of open Riemann surfaces in the same recipient\]  S. Chaudary, [*The Brill-Noether theorem for real algebraic curve*]{}, Ph.D. Thesis. Duke University, 1995. \[$\spadesuit$\]$\bigstar$  ?. Chebotarev, [*?*]{}, ? , 1948. \[$\spadesuit$ a textbook on analytic functions and Riemann surfaces cited in Gudkov 1974/74 [@Gudkov_1974/74] for the intrinsic proof of Harnack’s inequality $\spadesuit$ this entry is possibly the source of some historical confusion crediting Hurwitz instead of Klein for this result (compare comments after Gudkov 1974/74 ) $\spadesuit$ Hurwitz extended to the case of singular curves, but modulo the normalization (desingularization) everything reduces to the smooth case\]  A.L. Cheponkus, [*On nests of real plane algebraic curves*]{}, Litovsk. Mat. Sb. 16 (1976), 239–243, 257; English transl., Lithuanian Math. J. 16 (1976), 634–637. \[$\spadesuit$ cited in Rohlin 1978 [@Rohlin_1978], cf. also a paper by Marin 1988 [@Marin_1988] for another proof $\spadesuit$ in fact Cheponkus’ proof turned out to be incorrect (cf. e.g. Viro 1986/86 [@Viro_1986/86-Progress p.68] and especially Marin 1988 [@Marin_1988] for a specific objection)\]  I.V. Cherednik, [*Reality condition in “finite-zone integration”*]{}, Dokl. Akad. Nauk SSSR 252 (1980), 1104–1108; English transl., Soviet Phys. Dokl. 25 (1980), 450–452. \[$\spadesuit$ cited in Dubrovin 1983/85 [@Dubrovin_1983/85]\]  S.S. Chern, P. Hartman, A. Wintner, [*On isothermic coordinates*]{}, Comment. Math. Helv. 28 (1954), 301–309. \[$\spadesuit$\]  S.S. Chern, [*Complex Manifolds without Potential Theory*]{}, Second Edition. Springer-Verlag, Berlin, 1979. \[$\spadesuit$\]  C. Chevalley, [*Introduction to the Theory of Algebraic Functions of One Variable*]{}, Math. Surveys 6, Amer. Math. Soc., New York, 1951, 188 pp. \[$\spadesuit$\]  Yu.S. Chislenko, [*Pencils of real algebraic curves*]{}, Leningrad Topology Conference, 1982, 28. \[$\spadesuit$ correct proof of a special case of Cheponkus’ theorem, namely through 13 points in general position in the real projective plane passes a connected quartic $\spadesuit$ for a generalization based upon Klein’s remark (1876), cf. Marin 1988 [@Marin_1988]\]  Yu.S. Chislenko, [*$M$-curves of degree $10$*]{}, Zap. Nauch. Sem. Leningrad 122 (1982), 142–161; English transl., 1984. \[$\spadesuit$\]  M. Christ, [*A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral*]{}, Colloq. Math 60/61 (1990), 1367–1381. \[$\spadesuit$\]  E. Christoffel, [*Ueber die Abbildung einer $n$-blättrigen einfach zusammenhängender ebenen Fläche auf einen Kreise*]{}, Gött. Nachr. (1870), 359–369. \[$\spadesuit$ the so-called Schwarz-Christoffel formula effecting the (one-to-one) conformal representation of a polygon upon the disc $\spadesuit$ more precisely Schwarz 1869 [@Schwarz_1869-Ueber-einige-Abbildungsaufgaben] stated that the formula is easily generalized to the case of a multi-sheeted domain bounded by straight lines and containing branch points, and Christoffel considers here this generalization in some detail $\spadesuit$ \[07.10.12\] can we connect the Schwarz-Christoffel theory with that of the Ahlfors map? try perhaps Kühnau 1967 [@Kuehnau_1967]\]  Y.-B. Chung, [*The Ahlfors mapping function and an extremal problem in the plane*]{}, Houston J. Math. 263 (1993), 263–273. \[$\spadesuit$\]$\bigstar$  Y.-B. Chung, [*The Bergman kernel function and the Ahlfors mapping in the plane*]{}, Indiana Univ. Math. J. 42 (1993), 1339–1348. \[$\spadesuit$ Ahlfors mapping, Bergman kernel, etc.\]  Y.-B. Chung, [*Higher order extremal problem and proper holomorphic mapping*]{}, Houston Math. J. 27 (2001), 707–718. 50 \[$\spadesuit$ Ahlfors extremal problem (in the domain-case) with multiplicity (i.e. some first derivatives are imposed to be $0$ at some base-point $a$)\]$\bigstar$$\bigstar$$\bigstar$\[MR-OK\]  Y.-B. Chung, [*The Bergman kernel function and the Szegö kernel function*]{}, J. Korean Math. Soc. 43 (2006), 199–213. \[$\spadesuit$ the “Ahlfors map” of a smoothly bounded domain in the plane occurs several times through the paper\]  C. Ciliberto, C. Pedrini, [*Annibale Comessatti and real algebraic geometry*]{}, Rend. Cont. Circ. Mat. Palermo (2) Suppl. 36 (1994), 71–102. \[$\spadesuit$\]  C. Ciliberto, C. Pedrini, [*Real abelian varieties and real algebraic curves*]{}. In: Lectures in Real Geometry, F. Broglia (ed.), de Gruyter Exp. in Math. 23 (1996), 167–256. \[$\spadesuit$ a modernized (neoclassical) account of the theories of Klein, Weichold and Comessatti\]  K. Clancey, [*Representing measures on multiply connected planar domains*]{}, Illinois J. Math. 35 (1991), 286–311. \[$\spadesuit$ what about Riemann surface? try Alpay-Vinnikov 200 [@Alpay-Vinnikov_2000], and also Nash 1974 [@Nash_1974]\]  A. Clebsch, [*Ueber die Anwendung der Abelschen Functionen in der Geometrie*]{}, J. Reine Angew. Math. 63 (1863), 189–243. \[$\spadesuit$\]  A. Clebsch, [*Ueber diejenigen ebenen Curven, deren Coordinaten rationale Functionen eines Parameters sind*]{}, J. Reine Angew. Math. 64 (1865), 43–65. \[$\spadesuit$ coins the nomenclature genus, conceptually put in the limelight by Riemann (plus maybe Abel in some algebraic disguise)\]  A. Clebsch, P. Gordan, [*Theorie der Abelschen Functionen*]{}, Teubner, Leipzig, 1866. \[$\spadesuit$\]  A. Clebsch, [*Zur Theorie der Riemann’schen Flächen*]{}, Math. Ann. 6 (1872), 216–230. 60 \[CHECK date for Ahlfors-Sario 1960, it is 1873?\]  C.H. Clemens, [*A Scrapbook of Complex Curve Theory*]{}, Plenum Press, New York, 1980, 186 pp. \[$\spadesuit$\]  W.K. Clifford, [*On the space-theory of matter*]{}, Cambridge Philos. Society’s Proc. 2 (1876), 157–158. \[$\spadesuit$\]  W.K. Clifford, [*On the canonical form and dissection of a Riemann’s surface*]{}, Proc. London Math. Soc. 8 (1877), 292–304. \[$\spadesuit$ not cited in Ahlfors-Sario 1960!\]  R. Coifman, G. Weiss, [*A kernel associated with certain multiply connected domains and its application to factorization theorems*]{}, Studia Math. 28 (1966), 31–68. 78 \[$\spadesuit$ p.31: “Our main result is a generalization of the classical factorization theorem for function in the Nevanlinna class of the unit disc.”\]  R.R. Coifman, A. McIntosh, Y. Meyer, [*L’opérateur de Cauch définit un opératuer borné sur $L^2$ pour les courbes lipschitziennes*]{}, Ann. of Math. (2) 116 (1982), 361–387. \[$\spadesuit$\]  Y. Colin de Verdière, A. Marin, [*Triangulations presques équilatérales des surfaces*]{}, J. Differential Geom. 32 (1990), 199–207. \[$\spadesuit$\]  H. Comessatti, [*Fondamenti per la geometria sopra le superficie razionali dal punto di vista reale*]{}, Math. Ann. 43 (1912), 1–72. \[$\spadesuit$\]  H. Comessatti, [*Sulla connessione delle superficie razionali reale*]{}, Math. Ann. ?? (1914), 215–283. \[$\spadesuit$\]  H. Comessatti, [*Sulle varietà abeliane reali, I, II*]{}, Ann. Mat. Pura Appl. (4) 2 (1924), 67–106; (4) 4 (1926), 27–71.  H. Comessatti, [*Sulla connessione delle superficie algebriche reali*]{}, Verhandl. Internat. Math. Kongress Zürich, vol.2, p.129. \[$\spadesuit$\]  H. Comessatti, [*Reelle Fragen in der algebraischen Geometrie*]{}, Jahresb. d. Deutsch. Math. Verein. 41 (1932), 107–134. \[$\spadesuit$\]  H. Comessatti, [*Problemi di realtà per le superficie e varietà algebriche*]{}, Reale Accad. Ital. (Fondaz. A. Volta), Atti dei Convegni, vol.9 (1939), Rome 1943, 15–41. \[$\spadesuit$ cited in Nikulin 1983/84\]  . Oxford University Press, London, 1931. M. Coppens, [*One-dimensional linear systems of type II on smooth curves*]{}, Ph.D. Thesis, Utrecht, 1983. \[$\spadesuit$\]  M. Coppens, G. Martens, [*Linear series on general $k$-gonal curves*]{}, Abh. Math. Sem. Univ. Hamburg 69 (1999), 347–371. \[$\spadesuit$\]  M. Coppens, [*Totally non-real divisors in linear ssystems on smooth real curves*]{}, Adv. Geometry 8 (2008), 551–555. \[$\spadesuit$\]  M. Coppens, G. Martens, [*Linear pencils on real algebraic curves*]{}, J. Pure Appl. Algebra 214 (2010), 841–849. 50 \[$\clubsuit$ Ahlfors 1950 [@Ahlfors_1950] is cited in the following fashion (p.843): “Let $X$ be a real curve of genus $g$ with $s\ge 1$ real components and $g^1_d$ be a basepoint free pencil on $X$. Since $X({\Bbb R})\neq \varnothing$ the image curve $X'$ of the morphism $\varphi$ induced by the pencil is the rational real curve ${\Bbb P}^1_{\Bbb R}$. Assume that the fibre of $\varphi$ at every real point of $X'$ consists entirely of real points of $X$ (or, what is the same, that $\varphi$ separates conjugate points of $X_{\Bbb C}$: $\varphi(\sigma P)\neq \varphi(P)$ for any non-real point $P\in X_{\Bbb C}$); we call such a $g^1_d$ [*totally real*]{}. Then $\varphi$ is a ramified covering of bordered real surfaces (in the topological sense, cf. \[7, part3\](=Geyer-Martens 1977 [@Geyer-Martens_1977])), and the induced covering $X({\Bbb R})\to X'({\Bbb R})\cong S^1$ is unramified. In particular, $s\le d$. Since $X'=({\Bbb P}^1_{\Bbb C} {\rm\; mod\; conjugation})$, a half-sphere with boundary, is an orientable real surface it follows that also the Klein surface \[of\][^123] $X$ must be orientable which implies $s\not\equiv g \mod 2$ (cf. \[7, part.2\](=Geyer-Martens 1977 [@Geyer-Martens_1977]))[^124]. Hence the assumed property that every divisor of $X$ in the $g^1_d$ is entirely made up by real points puts severe restrictions on $X$. So we cannot expect to find such a pencil on every real curve. More precisely, by a result of Ahlfors \[10\](=1950 [@Ahlfors_1950]) there is a totally real pencil of degree $g+1$ on $X$ iff the Klein surface $X$ is orientable thus giving an interesting algebraic characterization of a topological property.”\]  M. Coppens, J. Huisman, [*Pencils on real curves*]{}, arXiv (2011). \[$\clubsuit$\]  M. Coppens, [*The separating gonality of a separating real curve*]{}, arXiv (2011); or Monatsh. Math. 2012. \[$\clubsuit$ the spectacular result is proven that all intermediate gonalities compatible with Gabard’s bound ($\le r+p$) are realized by some compact bordered Riemann surface $\spadesuit$ the work is written in the language of real algebraic geometry, especially dividing (or separating) curve and is a tour de force involving several techniques: Kodaira-Spencer deformation theory, Meis’ bound and its phagocytose into modernized Brill-Noether theory, stable curves à la Deligne-Mumford (1969), geometric Riemann-Roch, Hilbert scheme, etc.\]  M. Coppens, [*Pencils on separating $(M-2)$-curves*]{}, arXiv (2012). \[$\clubsuit$\]  A.F. Costa, [*On anticonformal automorphisms of Riemann surfaces with nonembeddable square*]{}, Proc. Amer. Math. Soc. 124 (1996), 601–605. \[$\spadesuit$\]  A.F. Costa, [*Embeddable anticonformal automorphisms of Riemann surfaces*]{}, Comment. Math. Helv. 72 (1997), 203–215. \[$\spadesuit$ outgrowth of the work by Garsia/Rüedy on conformal embeddings, in particular Prop.1.1. [*Let $f$ be an anticonformal involution of a Riemann surface $S$ then $f$ is embeddable iff either $S/f$ is orientable or $S/f$ is non-orientable without boundary.*]{} $\spadesuit$ Since the problem of conformal embeddings of Reimann surface as classical surface was first posed by Klein, it is easy to imagine how this result is a double Kleinian synthesis, which would have much pleased the “magister ludis”\]  A.F. Costa, M. Izquierdo, [*On the connectedness of the locus of real Riemann surfaces*]{}, Ann. Acad. Sci. Fenn. Math. 27 (2002), 341–356. \[$\spadesuit$ a new proof is offered of a result due Buser-Seppälä-Silhol 1995 [@Buser-Seppala-Silhol_1995], stating the connectedness of the projection of the real moduli down to the complex one (upon forgetting the anti-holomorphic involution) $\spadesuit$ intuitively this means that any symmetric Riemann surface can be deformed so as to create a new symmetry and one can explore the full real moduli space (doing some jump when one switch the symmetry)\]  A.F. Costa, M. Izquierdo, [*On real trigonal Riemann surfaces* ]{}, Math. Scand. (2006). \[$\spadesuit$ A closed Riemann surface X which can be realized as a $3$-sheeted covering of the Riemann sphere is called trigonal, and such a covering will be called a trigonal morphism. A trigonal Riemann surface $X$ is called real trigonal if there is an anticonformal involution (symmetry) ...\]  R. Courant, [*Über die Anwendung des Dirichletschen Prinzipes auf die Probleme der konformen Abbildung*]{}, Math. Ann. 71 (1912), 141–183. 78 \[$\spadesuit$ Diese Arbeit ist bis auf einige redaktionelle Änderungen ein Abdruck meiner Inauguraldissertation, Göttingen 1910.\]  R. Courant, [*Über eine Eigenschaft der Abbildungsfunktion\[en\] \[sic!?\] bei konformer Abbildung*]{}, Gött. Nachr. (1914), 101–109. 78 \[$\spadesuit$ this work is regarded by Gaier 1978 [@Gaier_1978-JDMV p.43] (and probably many others) as the first apparition of the length-area principle, which will be largely exploited by Grötzsch (Flächenstreifenmethode) and Ahlfors-Beurling (extremal length), etc., and which in the long run should obviously constitutes one of the key to the resolution of the Gromov filling conjecture $\spadesuit$ uses also the area integral $\int \int \vert f'(z)\vert^2 dx dy$ like Bieberbach 1914 [@Bieberbach_1914]\]  R. Courant, [*Über konforme Abbildung von Bereichen, welche nicht durch alle Rückkehrschnitte zerstückelt werden, auf schlichte Normalbereiche*]{}, Math. Z. 3 (1919), 114–122. 60  R. Courant, D. Hilbert, [*Methoden der mathematischen Physik. I*]{},Springer-Verlag, Berlin, 1931. (Reedited 1968) \[$\spadesuit$ cited e.g. in Simha 1975 [@Simha_1975] for an explicit formula for Jacobi theta function, the latter being involved in an explicit description of the Ahlfors map and the Carathéodory metric\]  R. Courant, [*Plateau’s problem and Dirichlet’s principle*]{}, Ann. of Math. 38 (1937), 679–725.  R. Courant, [*Remarks on Plateau’s and Douglas’ problem*]{}, Proc. Nat. Acad. Sci. U.S.A. 24 (1938), 519–522. \[$\clubsuit$ this is first place where the theorem of Bieberbach-Grunsky is reproved via Plateau, yet without citing them $\spadesuit$ a more detailed proof is given in the next entry (Courant 1939 [@Courant_1939])\]  R. Courant, [*Conformal mapping of multiply-connected domains*]{}, Duke Math. J. 5 (1939), 814–823. 60, 78 \[$\clubsuit$ the Bieberbach-Grunsky theorem is re-proved à la Plateau; now Bieberbach 1925 [@Bieberbach_1925] and Grunsky 1937 [@Grunsky_1937] are cited as well as Riemann (as an oral tradition)\]  R. Courant, [*The existence of minimal surfaces of given topological structure under prescribed boundary condition*]{}, Acta Math. 72 (1940), 51–98. \[$\spadesuit$ specializing to the case of ambient dimension 2 might perhaps reprove a theorem like the Ahlfors circle map $\spadesuit$ recall however that Tromba 1983 [@Tromba_1983-PREPRINT] seems to express doubts about the validity of Courant’s proof, compare also Jost 1985 [@Jost_1985]\]  R. Courant, M. Manel, M. Shiffman, [*A general theorem on conformal mapping of multiply connected domains*]{}, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 503–507. 78 \[¶Result generalized in Schramm’s Thesis ca. 1990, cf. arXiv\]  R. Courant, [*The conformal mapping of Riemann surfaces not of genus zero*]{}, Univ. Nac. Tucumán Revista A. 2 (1941), 141–149. 60 \[$\spadesuit$ detected only the 13.06.2012, via Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] $\spadesuit$ alas Gabard could not a find a copy of this article, and it seems unlikely that the article contains material not overlapping with previous and subsequent work by Courant, especially it is unlikely that the paper contains an existence of circle maps à la Ahlfors\] $\bigstar$  R. Courant, [*Dirichlet’s principle, Conformal Mapping, and Minimal Surfaces*]{}, with an appendix by M. Schiffer. Pure and appl. math. 3, New York, Interscience Publishers, 1950. 60, 78 \[$\clubsuit$ overlap much with the previous ref., but somehow hard to read due to its large content and mutatis mutandis type proof, in particular it is not clear if p.183 contains another proof of the circle map of Ahlfors $\spadesuit$ p.169 contains a proof of the Kreisnormierung in finite connectivity\]  R. Courant, [*Flow patterns and conformal mapping of domains of higher topological structure*]{}. In: [*Construction and Applications of Conformal Maps*]{}, Proc. of a Sympos. held on June 22–25 1949, Applied Math. Series [*18*]{}, 1952, 7–14.  D. Crowdy, J. Marshall, [*Green’s functions for Laplace’s equation in multiply connected domains*]{}, IMA J. Appl. Math. (2007), 1–24. \[$\spadesuit$ p.13–14, contains beautiful pictures of the levels of the Green’s function on some circular domains\]  D. Crowdy, [*Conformal mappings from annuli to canonical doubly connected Bell representations*]{}, J. Math. Anal. Appl. 340 (2008), 669–674. \[$\spadesuit$ p.670, the Ahlfors map is briefly mentioned in connection with the work of Jeong-Oh-Taniguchi 2007 [@Jeong-Oh-Taniguchi_2007] on deciding when Bell’s doubly-connected domain $A(r)=\{ z\in{\Bbb C} : \vert z+z^{-1} \vert<r \}$ is conformally equivalent to the Kreisring $\Omega(\rho^2)=\{ \zeta \in {\Bbb C}: \rho^2<\vert \zeta \vert <1 \}$\]  G. David, [*Unrectifiable $1$-sets have vanishing analytic capacity*]{}, Rev. Mat. Iberoam. 14 (1998), 369–479. 47 \[$\spadesuit$ p.369: “[**Abstract.**]{} We complete the proof of a conjecture of Vitushkin that says that if $E$ is a compact set in the plane with finite $1$-dimensional Hausdorff measure, then $E$ has vanishing analytic capacity iff $E$ is purely unrectifiable (i.e., the intersection of $E$ with any curve of finite length has zero $1$-dimensional Hausdorff measure). \[…\]” $\spadesuit$ \[29.09.12\] this is quite close to a solution of Painlevé’s problem, but just not so due to the proviso $H^1(E)<\infty$, which cannot be relaxed for p.370: “Actually Vitushkin’s conjecture also said something about the case when $H^1(K)=+\infty$[^125], but this part turned out to be false (\[Ma1\]=(Mattila 1986 [@Mattila_1986]))”\]  G. David, [*Analytic capacity, Calderón-Zygmund operators, and rectifiability*]{}, Publ. Mat. 43 (1999), 3–25. 47 \[$\spadesuit$\]  A.M. Davie, [*Analytic capacity and approximation problems*]{}, Trans. Amer. Math. Soc. 171 (1972), 409–414. \[$\spadesuit$ a reduction is effected of the Denjoy conjecture (on removable sets lying on rectifiable curves) to the case where the supporting curve is $C^1$, giving one of the ingredient toward the ultimate solution of Denjoy’s conjecture (compare Marshall [@Marshall_1978?]), where the last piece of the puzzle is the contribution of Calderón 1977 [@Calderon_1977]\]  A.M. Davie, B. [Ø]{}ksendal, [*Analytic capacity and differentiability properties of finely harmonic functions*]{}, Acta Math. 140 (1982), 127–152. \[$\spadesuit$\]  P. Davis, H. Pollak, [*A theorem for kernel functions*]{}, Proc. Amer. Math. Soc. 2 (1951), 686–690. \[$\spadesuit$ parallel-slit mapping via Bergman kernel\] 78 B. Deconinck, M. van Hoeij, [*Computing Riemann matrices of algebraic curves*]{}, Physica D 152/153 (2001), 28–46. \[$\spadesuit$\]$\bigstar$  R. Dedekind, [*??*]{}, Abhandlungen der Königl. Gesellchaft der Wiss. zu Göttingen 13 (1868). \[$\spadesuit$ first published report of Riemann’s Habilitationsvortrag (1854 [@Riemann_1854-Habilitation/Ueber-die-Hypothesen]) analyzing primarily the mathematical side of Riemannian geometry $\spadesuit$ in particular the so-called Laplace-Beltrami operator ought to be discussed here; there is somewhere a commented version in French\]  R. Dedekind, H. Weber, [*Theorie der algebraischen Funktionen einer Veränderlichen*]{}, Crelles J. 92 (1879). \[$\spadesuit$ “arithmetized” account of algebraic function and the allied Riemann surfaces, cf. also in the same spirit Hensel-Landsberg 1902 [@Hensel-Landsberg_1902] and H. Weber 1908 [@Weber_1908]\]  A. Degtyarev, V. Kharlamov, [*Topological properties of real algebraic varieties: Rokhlin’s way*]{}, Uspekhi Mat. Nauk 55 (2000) 129–212; English transl., Russian Math. Surveys 55 (2000), 735–814. \[$\spadesuit$ p.736: “Another fundamental result difficult to overestimate is Rokhlin’s formula for complex orientations. The notion of complex orientation of a dividing real curve (see below), as well as Rokhlin’s formula and its proof, seem incredibly transparent at first sight. The formula settles, for example, two of Hilbert’s conjectures on 11 ovals of plane sextics, which Hilbert himself tried to prove in a very sophisticated way and then included in his famous problem list (as the sixteenth problem).” $\spadesuit$ p.739: “Note that the topological properties of abstract, not embedded, real curves are simple and have been understood completely since Klein’s time; see for example, \[87\](=Natanzon 1990 [@Natanzon_1990/90]) and \[99\](=Rohlin 1978 [@Rohlin_1978]).” $\spadesuit$ p.757: “According to Arnol’d, the following result is due to Maxwell.—[**3.2.1 Theorem**]{}. The orbit space ${\Bbb P}^2/{\rm conj}$ is diffeomorphic to $S^4$.” $\spadesuit$ p.785: “[**4.6.8. Klein’s statement.**]{} If a real curve of type I undergoes a Morse surgery through a non-degenerate double point, then the number of connected components of this curve cannot increase. \[…\]” (For related literature see Klein 1876 [@Klein_1876]), Rohlin 1978 [@Rohlin_1978], and Marin?, etc. $\spadesuit$ p.788: “[**Digression: real rational curves**]{}. As far as we know, the following problem is still open: is it possible to draw an irreducible real rational curve (or more precisely a connected component of it) of degree $q$ through any set of $3q-1$ real points in general position? In \[99\](=Rohlin 1978 [@Rohlin_1978]) the question is answered in the affirmative; however, the proof has never been published; possibly it contained a gap. \[…\] The first non-trivial case (and the only case in which the complete answer is known) is $q=3$, namely through 8 generic points one can draw 12 rational cubics; depending on the arrangement of the points, the number of real cubics among them can be $8,10$ or $12$. (All three va,ues occur; the 12 rational cubics in a pencil are real iff the pencil contains 2 cubics with a solitary real double point.)”\]  K. de Leeuw, W. Rudin, [*Extreme points and extremum problems in $H_1$*]{}, Pacific J. Math. 8 (1958), 467–485. \[$\spadesuit$ gives a characterization of the extreme points of the unit ball of the disc-algebra $H^1(\Delta)$, an analogue of which for the same algebra attached to a finite bordered Riemann surface will be given in Gamelin-Voichick 1968 [@Gamelin-Voichick_1968] upon making use of the Ahlfors map or at least techniques closely allied to its existence-proof (as given by Ahlfors 1950 [@Ahlfors_1950])\]  P. Deligne, D. Mumford, [*Irreducibility of the space of curves of given genus*]{}, Publ. Math. Inst. Hautes. Études Sci. 36 (1969), 75–109. \[$\spadesuit$\]  A. Denjoy, [*???*]{}, C.R. Acad. Sci. Paris 14? (1907), 258–260. 60 \[$\spadesuit$ yet another Denjoy conjecture (not to be confounded with that of the next entry) on the number of asymptotic values of entire functions of finite order $\spadesuit$ formulated by Denjoy at age 21, it was solved by Ahlfors in 1928 (at age 21), 21 years after its formulation (arithmetical curiosity noticed by Denjoy)\]$\bigstar$ A. Denjoy, [*Sur les fonctions analytiques uniformes à singularités discontinues*]{}, C.R. Acad. Sci. Paris 149 (1909), 258–260. 60 \[$\spadesuit$ the following theorem is proved (or rather asserted since a gap was later located in proof) but Denjoy’s assertion turned out to be ultimately correct via Calderón 1977 [@Calderon_1977] and Marshall [@Marshall_1978?]: “a closed set of positive length lying on a rectifiable arc is unremovable in the class of bounded analytic functions” $\spadesuit$ this became the famous “Denjoy conjecture” $\spadesuit$ partial positive results on it where obtained by Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950] in the case where the supporting arc is a segment (for this case they credit Denjoy himself) and then they extend the result to an analytic curve via conformal mapping $\spadesuit$ Ivanov treated the case of curves slightly smoother than $C^1$ $\spadesuit$ Davie 1972 [@Davie_1972] proved that it sufficed to assume the curve $C^1$ (i.e. the rectifiable case of Denjoy can be reduced to the $C^1$ case) $\spadesuit$ then, Caderón 1977 [@Calderon_1977] proved that the Cauchy integral operator, for $C^1$ curves, is bounded on $L^p$, $1<p<\infty$ $\spadesuit$ at this stage, Marshall [@Marshall_1978?] put the “touche finale” by writing a note explaining how Calderón implies Denjoy via classical results of Garabedian, Havinson and finishing the proof with Davie’s reduction to the $C^1$-case, validating thereby Denjoy’s assertion announced ca. 7 decades earlier $\spadesuit$ Calderón himself was first not aware of the relevance of his work to Denjoy’s (as one learns e.g. from Verdera 2004 [@Verdera_2004 p.29]), but in the acceptance speech for the Bôcher price (see Calderón 1979 [@Calderon_1979]), Calderón mentions the solution to the Denjoy conjecture as one of the most significative application of his article (see also Calderón 1978 [@Calderon_1978-ICM], ICM lecture)\]$\bigstar$ H. Denneberg, [*Konforme Abbildung einer Klasse unendlich-vielfach zusammenhängender schlichter Bereiche auf Kreisbereiche*]{}, Ber. Verhd. Sächs. Akad. Wiss. Leipzig 84 (1932), 331–352. 60, 78 \[$\spadesuit$ a contribution to KNP\]$\bigstar$ J. Dieudonné, [*Cours de géométrie algébrique*]{}, Presses universitaires de France, Paris, 1974. \[$\spadesuit$ as mixture of Bourbakist pesanteur mixed with the usual charming touch of the gifted “God-given” writer\] J. Diller, [*Green’s functions, electric networks, and the geometry of hyperbolic Riemann surfaces*]{}, Illinois J. Math. 45 (2001), 453–485. \[$\spadesuit$ p.456, Swiss cheese description of Hardt-Sullivan’s work (1989 [@Hardt-Sullivan_1989]) on the Green’s function for a bordered Riemann surface given as a branched cover of the unit-disc $\spadesuit$ so this Hardt-Sullivan work may possibly interact with the Ahlfors function\] P.G. Lejeune Dirichlet, [*??*]{}, Crelle’s Journal 1829. \[$\spadesuit$ first convergence proof of the Fourier series toward the given (continuous) function\]  P.G. Le Jeune Dirichlet, [*Vorlesungen über die im umgekehrten Verhältniss des Quadrats der Entfernung wirkenden Kräfte*]{}, herausgegeben von Dr. F. Grube, Leipzig, 1876. \[$\spadesuit$ first version of DP available in print under (essentially) Dirichlet’s own pen, as Grube reproduced a Dirichlet Göttingen lecture (ca. 1856) $\spadesuit$ alas Dirichlet’s formulation was a bit ill-posed, as came very apparent through the example of Prym 1871 [@Prym_1871] (compare also Elstrodt-Ullrich 1999 [@Elstrodt-Ullrich_1999])\]$\bigstar$  S. Donaldson, [*Yang-Mills invariants of smooth four-manifolds*]{}, in: Geometry of low-dimensional manifolds, vol. 1, Cambridge Univ. Press, 1990, 5–40. \[$\spadesuit$ contains some trick to transmute to holomorphic the antiholomorphic involution induce by Galois on a real quartic surface (or more general K3 surfaces=Kummer-Kähler-Kodaira in Weil’s designation)\]  S. Donaldson, [*Complex curves and surgery*]{}, Publ. Math. Inst. Hautes Ét. Sci. 1989, 91–97. \[$\spadesuit$ a discussion of Thom’s conjecture, p.91: “An entrancing problem in Geometric Topology, usually ascribed to R. Thom, asks whether $C$ minimises the genus among all $C^\infty$ representatives for the homology class.” p.92 Lee Rudolph (1984 [@Rudolph_1984]) counterexamples for “topologically locally flat” surfaces are mentioned $\spadesuit$ Kirby’s 1970 list [@Kirby_1970--95] of problems already contains (conjecturally) an extension of Thom to any complex projective surface\]  J. Douglas, [*Solution of the problem of Plateau*]{}, Trans. Amer. Math. Soc. 33 (1931), 263–321. \[$\spadesuit$ a new proof of RMT is given via Plateau, including the Osgood-Carathéodory refinement about the boundary behaviour of the Riemann map $\spadesuit$ reduces the mapping problem to that of minimizing a functional (named after Douglas by now)\]  J. Douglas, [*Some new results in the problem of Plateau*]{}, Amer. J. Math. 61 (1939), 590–608.  J. Douglas, [*Minimal surfaces of higher topological structure*]{}, Ann. of Math. (2) 40 (1939), 205–298. 78  J. Douglas, [*The most general form of the problem of Plateau*]{}, Amer. J. Math. 61 (1939), 590–608. 60  R.G. Douglas, W. Rudin, [*Approximation by inner functions*]{}, Pacific J. Math. 31 (1969), 313–320. \[p.314 the Ahlfors function (in the very trivial case of an annulus $D=\{z \colon r_1<\vert z\vert < r_2\}$) is involved in the proof of the following theorem: the set of all quotients of inner functions is norm-dense in the set of unimodular functions\]  B. Drinovec Drnovšek, [*Proper discs in Stein manifolds avoiding complete pluripolar sets*]{}, Math. Res. Lett. 11 (2004), 575–581. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited\]  B. Drinovec Drnovšek, F. Forstnerič, [*Holomorphic curves in complex spaces*]{}, Duke Math. J. 139 (2007), 203–252. \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited in the bibliography, but apparently not within the text $\spadesuit$ yet the connection with Ahlfors is evident in view of the following extract of the review of the paper: “Since the early 1990’s a series of papers, motivated mainly by works J. Globevnik and of Forstnerič, has been devoted to constructing holomorphic discs $f\colon \Delta \to M$ in complex manifolds that are proper. The article under review offers a culmination of the subject, lowering as much as possible the convexity assumptions, working on complex spaces with singularities, and “properizing” not only discs, but general open Riemann surfaces whose boundary consists of a finite number of closed Jordan curve. \[…\]”\]  V.N. Dubinin, S.I. Kalmykov, [*A majoration principle for meromorphic functions*]{}, Sbornik Math. 198 (2007), 1737–1745. \[p.1740 a majoration principle is specialized to the Ahlfors function upon using the formula expressing the logarithm of the modulus of the Ahlfors function as a superposition of Green’s functions with poles at the zeros of the Ahlfors function\]  B.A. Dubrovin, I.M. Krichever, S.P. Novikov, [*The Schrödinger equation in a periodic field and Riemann surfaces*]{}, Dokl. Akad. Nauk SSSR 229 (1976), 15–18; English transl., ?? ? (197?), ?–?. \[$\spadesuit$ cited in Dubrovin 1983/85 where some connection with Klein’s orthosymmetry/separating type I is given\]  B.A. Dubrovin, S.M. Natanzon, [*Real two-zone solutions of the sine-Gordon equation*]{}, Funkt. Anal. Prilozhen. 16 (1982),27–43; English transl., Funct. Anal. Appl. 16 (1982), 21–33. \[cited in Vinnikov 1993 [@Vinnikov_1993] who claims a simplified proof\]  B.A. Dubrovin, [*Matrix finite-zone operators*]{}, (Itogi Nauki i Tekhniki) 23 (1983), 33–78; English transl., Contemporary problems in math. ? (1985), 20–50. \[$\spadesuit$ cited in Vinnikov 1993 [@Vinnikov_1993 p.478] for a proof of the rigid-isotopy of any two smooth plane real curves having a deep nest (a result first established by Nuij 1968 [@Nuij_1968]) $\spadesuit$ p.48: “We shall now list themost important properties of real Riemann surfaces. A Riemann surface is called real if on it there is given an antiholomorphic involution \[…\] There are two possible cases: I) the union of real ovals decomposes $\Gamma$ into two components \[…\]; or II) the union of ovals does not decompose $\Gamma$. Surfaces of type I we call surfaces of separating type, while those of type II we call surfaces of nonseparating type.” $\spadesuit$ p.43: “The Riemann surface $\Gamma$ with antiinvolution $\tau$ belongs to \[the\] separating type. (the proof given on p.43–44 seems to use a sort of total reality?) $\spadesuit$ p.41: “separating type” $\spadesuit$ p.42: Fay 1973 [@Fay_1973] is cited, yet not clear to Gabard \[12.01.13\] if Dubrovin’s paper has any dep connection with Ahlfors 1950 [@Ahlfors_1950] $\spadesuit$ p.43: Rohlin 1978 [@Rohlin_1978] is cited for the simple fact that a plane curve with a deep nest is separating\]  B.A. Dubrovin, [*Theory of operators and real algebraic geometry*]{}, in: Global Analysis and Math. Physics, III, Voronezh State Univ., 1987; English transl., Lecture Notes in Math. 1334, 1988, 42–59. \[$\spadesuit$\]$\bigstar$  B.A. Dubrovin, S.M. Natanzon, [*Real theta-function solutions of the Kadomtsev-Petviashvili equation*]{}, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 267–286; English transl., Math. USSR Izv. 32 (1989), 269–288. \[$\spadesuit$\]$\bigstar$  B.A. Dubrovin, S.P. Novikov, A.T. Fomenko, [*Modern Geometry: Methods and Applications*]{}, 3rd ed., Nauka, Moscow, 1986; English transl., Part I, II, III, Springer, 1984, 1985, 1990. \[$\spadesuit$\]$\bigstar$  C.J. Earle, A. Marden, [*On Poincaré series with application to $H^{p}$ spaces on bordered Riemann surfaces*]{}, Illinois J. Math. 13 (1969), 202–219. \[$\spadesuit$ cited in Forelli 1979 [@Forelli_1979], where the automorphic uniformization is employed to construct the Poisson kernel of a finite bordered Riemann surface, which in turn is involved in a new derivation of Ahlfors circle maps of controlled degree $\le r+2p$\]  C.J. Earle, A. Schatz, [*Teichmüller theory for surfaces with boundary*]{}, J. Differential Geom. 4 (1970), 169–185. \[$\spadesuit$\]  C.J. Earle, [*On the moduli of closed Riemann surfaces with symmetries*]{}, In: Advances in the Theory of Riemann Surfaces, Annals of Math. Studies 66, Princeton Univ. Press and Univ. of Tokyo Press, Princeton, N.J., 1971, 119–130. \[$\spadesuit$ modernized account of Klein 1882 [@Klein_1882] and Teichmüller 1939 [@Teichmueller_1939], cf. also related works by Natanzon and Seppälla 1978 [@Seppala_1978-Teich-spaces-of-Klein-surfaces] $\spadesuit$ [*Warning*]{}. According to Natanzon 1999 [@Natanzon_1999-Moduli-real-alg-surf.superanal-differ-spinors p.1101], Earle’s description of the topological structure of the components of the moduli space of real algebraic curves (as being each diffeomorphic to ${\Bbb R}^{3g-3}/ {\rm Mod}_{g,r,\varepsilon}$ for a suitable discrete modular group) while being correct, its proof (using the theory of quasiconformal maps) is not, since it relies on a Kravetz (1959 [@Kravetz_1959]) theorem “which turned out latter to be wrong”. Still according to Natanzon () “A correct proof based on the theory of quasiconformal maps was obtained in Seppälä 1978 [@Seppala_1978-Teich-spaces-of-Klein-surfaces].”\]  T. Ekedahl, S. Lando, M. Shapiro, A. Vainshtein, [*Hurwitz numbers and intersections on moduli spaces*]{}, Invent. Math. 146 (2001), 297–327. \[$\spadesuit$ a new derivation of Hurwitz’s count of the number of branched coverings of the sphere having prescribed ramification\]  A. El Soufi, S. Ilias, [*Le volume conforme et ses applications d’après Li et Yau*]{}, Sém. Théo. Spectrale Géom. (1983/84), 15pp. \[$\spadesuit$ exploits the optimum $[\frac{g+3}{2}]$ gonality (Riemann-Brill-Noether-Meis) in the realm of spectral theory\]  J. Elstrodt, P. Ullrich, [*A real sheet of complex Riemannian function theory: a recently discovered sketch in Riemann’s own hand*]{}, Historia Math. 26 (1999), 268–288.  , edited by Kiyosi Itô, Vol. II, Second Edition, English transl. (1987) of the third (Japanese) edition (1968) \[sic!\]. 50 \[$\spadesuit$ on p.1367 the result of Ahlfors 1950 [@Ahlfors_1950] is quoted as follows (with exact source omitted but given on the next page p.1368): “L. Ahlfors proved that a Riemann surface of genus $g$ bounded by $m$ contours can be mapped conformally to an at most $(2g+m)$-sheeted unbounded covering surface of the unit disk.”\]  F. Enriques, [*Sul gruppo di monodromia delle funzioni algebriche, appartenti ad una data superficie di Riemann*]{}, Rom. Acc. L. Rend. 13 (1904), 382–384. 60 \[$\spadesuit$ just quoted to ponder a bit the severe diagnostic to be found in the next entry (i.e. Ahlfors was of course by no mean ignorant about the Italian algebro-geometric community)\]  F. Enriques, O. Chisini, [*Lezioni sulla Teoria Geometrica delle Equazioni e delle Funzioni Algebriche*]{}, Zanichelli, Bologna, 1915–1918–1924. \[$\spadesuit$ appears to the writer as a clear-cut forerunner of both Bieberbach 1925 [@Bieberbach_1915] and Wirtinger 1942 [@Wirtinger_1942], as argued in Gabard 2006 [@Gabard_2006 p.949] (cf. also Huisman 2001 [@Huisman_2001] for a similar proof) $\spadesuit$ actually Enriques-Chisini give another derivation of Harnack’s bound (on the number of components of a real curve) via Riemann-Roch, but their argument supplies an immediate proof of the so-called [*Bieberbach-Grunsky theorem*]{} (cf. Bieberbach 1925 [@Bieberbach_1925], Grunsky 1937 [@Grunsky_1937] and for instance A. Mori 1951 [@Mori_1951]), that is, the planar version of the Ahlfors map $\spadesuit$ as far as I know this little anticipation of Enriques-Chisini over Bieberbach-Grunsky has never been noticed (or admitted?) by the function-theory community (say Bieberbach, Grunsky, Wirtinger, Ahlfors, A. Mori, Tsuji, …) showing an obvious instance of lack of communication between the analytic and geometric communities\]  F. Enriques, [*Sulle curve canoniche di genere $p$ dello spazio a $p-1$ dimensioni*]{}, Rend. Accad. Sci. Ist. Bologna 23 (1919), 80–82. \[$\spadesuit$ so-called canonical curve termed “Normalkurve der $\varphi$” in Klein 1892 [@Klein_1892_Realitaet] and also studied by M. Noether\]  B. Epstein, [*Some inequalities relating to conformal mapping upon canonical slit-domains*]{}, Bull. Amer. Math. Soc. ?? (1947), ??–??. \[$\spadesuit$\]  B. Epstein, [*The kernel function and conformal invariants*]{}, J. Math. Mech. 7 (1958). \[$\spadesuit$ quoted in Gustafsson 2008\]  $\bigstar$$\bigstar$$\bigstar$ L. Euler, [*Instit. Calc. Integr. Petrop.*]{}, 1768–70, 2, 1169. \[$\spadesuit$ cited in Petrowsky 1938 [@Petrowsky_1938] as one of the tool used in the proof of the Petrovskii’s inequalities via the so-called Euler-Jacobi interpolation formula (Kronecker also involved) concerning solutions of systems of algebraic equations and yielding a highbrow extensions to curves of higher orders of the results of Hilbert-Rohn for sextics\]  G. Faber, [*Neuer Beweis eines Koebe-Bieberbachschen Satzes über konforme Abbildung*]{}, Sitz.-Ber. math.-phys. Kl. Bayer. Akad. Wiss. (1916), 39–42. \[$\spadesuit$ related to the so-called area principle of Gronwall 1914/15 [@Gronwall_1914/15], Bieberbach 1916 [@Bieberbach_1916]\]  $\bigstar$$\bigstar$$\bigstar$ G. Faber, [*Über den Hauptsatz aus der Theorie der konformen Abbildung*]{}, Sitz.-Ber. math.-phys. Kl. Bayer. Akad. Wiss. (1922), 91–100. 78 \[$\spadesuit$ must be another proof of RMT $\spadesuit$ regarded in Schiffer 1950 [@Schiffer_1950-Appendix-Courant p.313] as one of the originator of the method of [*extremal length*]{} (jointly with Grötzsch (1928) and Rengel 1932/33 [@Rengel_1932-33]), cf. also the introductory remarks of Bieberbach 1957 [@Bieberbach_1957] $\spadesuit$ maybe another origin is Courant 1914 [@Courant_1914] (at least for the length-area principle), cf. e.g. Gaier 1978 [@Gaier_1978-JDMV]\] $\bigstar$$\bigstar$$\bigstar$  G. Faltings, [*Endlichkeitssätze für abel’sche Varietäten über Zahlkörpern*]{}, Invent. Math. 73 (1983), 349–366. \[$\spadesuit$ proof of the so-called Mordell conjecture that a curve defined over ${\Bbb Q}$ (or a more general number field, i.e. a finite extension of ${\Bbb Q}$) has only finitely many rational points provided the genus $g$ of the underlying complex curve has genus $g\ge 2$ $\spadesuit$ it would we interesting to detect if the finer Kleinian invariants allied to real curves also have some similar arithmetical repercussion (to my knowledge nothing is known along this way, even at the conjectural level)\]  G. Faltings, [*Real projective structures on Riemann surfaces*]{}, Compos. Math. 48 (1983), 223–269. \[$\spadesuit$ p.231: “Any Riemann surface may be considered as an algebraic curve defined over ${\Bbb C}$. Sometimes this algebraic variety is already definable over the real numbers. This happens precisely if there exist an antiholomorphic involution on the surface, and these involutions correspond bijectively to the different real models of the curve.—The basic example here is the double of a Riemann surface with boundary, which has a canonical real structure. The real points of this real curve are the fixed-points of the involution, hence the points in the boundary of our original Riemann surface.—Not every real curve is of this form, since for example there exist curves $X$ over ${\Bbb R}$ for which $X({\Bbb C})-X({\Bbb R})$ is connected. ($X({\Bbb C})$, $X({\Bbb R})$ denote the ${\Bbb C}$-respectively ${\Bbb R}$-valued points of a real algebraic curve $X$.) We shall see that all counterexample are of this form.” (Okay but all this is of course trivial since Felix Klein.) $\spadesuit$ \[20.12.12\] an evident “Jugendtraum” of mine (and probably of many others, Gross, Faltings, etc.?) since ca. 1999/2000 is whether the finer topological invariants of Klein of a real curve (as opposed to the sole Riemannian genus fixing the topology of the underlying complex curve) have any arithmetical repercussion, à la Mordell-Faltings, namely finiteness of the rational points $C({\Bbb Q})$ whenever the genus $g\ge 2$. To my knowledge not a single result of the sort is known and it is quite hard to speculate about any such topologico-arithmetical connection. Crudely speaking the implication could be of the format if the curve is dividing then the cardinality of $C({\Bbb Q})$ is even, but this is surely wrong\]  H.M. Farkas, I. Kra, [*Riemann surfaces*]{}, Second Edition, Grad. Texts in Math. 71, Springer, 1992. (1st edition published in 1980)  P. Fatou, [*Séries trigonométriques et séries de Taylor*]{}, Acta Math. 30 (1906), 335–400. \[$\spadesuit$ influenced by Lebesgue, and will in turn influence F. Riesz (so called Fischer-Riesz theorem)\]  J. Fay, [*Theta functions on Riemann surfaces*]{}, Lecture Notes in Math. 352, Springer, 1973. 47, 50 \[$\spadesuit$ cite Ahlfors 1950 [@Ahlfors_1950] and write down explicit formulas for the Ahlfors function (at least in the planar case) in terms of theta-functions $\spadesuit$ gives perhaps another proof of Ahlfors 1950 (cf. Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000]) but this hope is probably not borne out (Fay probably only recovers the Ahlfors circle map in the planar case) $\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited thrice in this booklet $\spadesuit$ on p.108 (just for the double) $\spadesuit$ on p.116: “It has been proved in \[3, p.126\](=Ahlfors 1950 [@Ahlfors_1950]) that there are always unitary functions with exactly $g+1$ zeroes [*all*]{} in $R$; and when $R$ is a planar domain, it is shown in Prop.6.16 that $S_{0,\dots,0}\cap \Sigma_a$ is empty for $a\in R$ and that the unitary functions holomorphic on $R$ with $g+1$ zeroes are parametrized by the torus $S_0$.” \[Added by Gabard \[10.09.12\]: of course one can wonder how much of this is anticipated in Bieberbach 1925 [@Bieberbach_1925]\] $\spadesuit$ p.129: “Using this result, a solution can be given to an extremal problem for bounded analytic functions as formulated in \[3, p.123\](=Ahlfors 1950):” where the Ahlfors function is expressed in terms of the theta function and the prime form, yet it should be noted that unfortunately at some stage Fay’s exposition is confined to the case of planar domains $\spadesuit$ somewhat earlier in the text (in a portion not yet confined to the planar case) we read on p.114: “The spaces $S_{\mu}$ parametrize the generic unitary functions on $C$ with the minimal $(g+1)$ number of zeroes:”, maybe this claim of minimality is erroneous as it could be incompatible with Gabard 2006 [@Gabard_2006], and even if the latter is incorrect there is basic experimental evidence violating this minimality claim on the bound $g+1$, compare our remarks after Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000]\]  S.I. Fedorov, [*Harmonic analysis in a multiply connected domain, I*]{}, Math. USSR Sb. 70 (1991), 263–296. \[$\spadesuit$ credited by Alpay-Vinnikov 2000 [@Alpay-Vinnikov_2000 p.240] (and also Yakubovich 2006 [@Yakubovich_2006]) for another existence-proof of the Ahlfors map (at least for planar domains), cf. p.271–275 $\spadesuit$ on p.272 it is remarked that one cannot prescribe arbitrarily the $n$ zeroes of a circle-map on an $n$-connected domain of minimum degree $n$ as follows: “Unfortunately we cannot prescribe $n$ points on $\Omega_{+}$ arbitrarily in such a way that their union will be the set of zeros of an $n$-sheeted inner function $\theta$ of the form (3), since the zeros of an $n$-sheeted function $\theta$ must satisfy the rather opaque condition $\sum_{k=1}^n \omega_s(z_k), \; s=1, \dots, n-1,$ where $\omega_s$ is the harmonic measure of the boundary component $\Gamma_s$.” $\spadesuit$ \[26.09.12\] it seems to the writer (Gabard) that this condition already occurs (at least) in A. Mori 1951 [@Mori_1951] $\clubsuit$ it would be interesting to analyze carefully Fedorov’s argument (or Mori’s) to see if it can be extended to the positive genus case (this is perhaps already done in Mitzumoto 1960 [@Mizumoto_1960]) $\spadesuit$ p.272 desideratum of a constructive procedure for building all $n$-sheeted inner functions on an $n$-connected domain, which is answered on p.274 via “Theorem 1. Let $z_1, \dots, z_n$ be arbitrary points with $z_k\in \Gamma_k$, $k=1,\dots, n$. Then there exist positive numbers $\lambda_1,\dots, \lambda_n$ such that the function $w=\int_{z_\Gamma}^z \sum_{j=1}^n \lambda_j \nu_{z_j}$, $z_\Gamma\in \Gamma$, $z_\Gamma\neq z_j$, $j=1,\dots,n$, is a single-valued $n$-sheeted function on $\hat \Omega$, real-valued on $\Gamma$, with positive imaginary part on $\Omega_+$. The function $\theta=\frac{w-i}{w+i}$ is an $n$-sheeted inner function.” $\spadesuit$ of course in substance (or essence) this is nothing but what Japaneses calls the Bieberbach-Grunsky theorem (cf. Mori 1951 [@Mori_1951] or Tsuji 1956 [@Tsuji_1956])\]  J.L. Fernandez, [*On the existence of Green’s function in Riemannian manifolds*]{}, Proc. Amer. Math. Soc. 96 (1986), 284–286. \[$\spadesuit$\]  $\bigstar$ Fiedler, student of Rohlin, ca. 1978. T. Fiedler, [*Eine Beschränkung für die Lage von reellen ebenen algebraischen Kurven*]{}, Beiträge Algebra Geom. 11 (1981), 7–19. \[$\spadesuit$ the eminent DDR student of Rohlin, who seems to have been the first to notice the simple fact that orientation-preserving smoothings conserve the dividing character of curves, compare also Rohlin 1978 [@Rohlin_1978] where the contribution of Fiedler is already mentioned\]  T. Fiedler, [*Geraden Büschel und die Topologie der reellen algebraischen Kurven*]{}, Dissertation, 1981. \[$\spadesuit$ (in part) reproduced in the next entry Fiedler 1982/83 [@Fiedler_1982/83-Pencil]\]  T. Fiedler, [*Pencils of lines and the topology of real algebraic curves*]{}, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 853–863; English transl., Math. USSR Izv. 21 (1983), 161–170. \[$\spadesuit$ p.161 (Abstract): “\[…\] a new invariant of the strict isotopy type of the curve is given, which in particular distinguishes some seventh degree $M$-curves with the same complex scheme.”\]  T. Fiedler, [*New congruences in the topology of real plane algebraic curves*]{}, Dokl. Akad. Nauk SSSR 270 (1983), 56–58; English transl., Sov. Math. Dokl. 27 (1983), 566–568. \[$\spadesuit$\]  T. Fiedler, [*New congruences in the topology of singular real plane algebraic curves*]{}, Dokl. Akad. Nauk SSSR 286 (1986), 1075–1079; English transl., Sov. Math. Dokl. 33 (1986), 262–266. \[$\spadesuit$\]  T. Fiedler, [*Additional inequalities in the topology of real plane algebraic curves*]{}, Izv. Akad. Nauk SSSR Ser. Mat. ?? (1986), ?–?; English transl., Math. USSR Izv. 27 (1986), 183–191. \[$\spadesuit$\]  T. Fiedler, [*Real points on complex plane curves*]{}, Math. Ann. 284 (1989), 267–284. \[$\spadesuit$\]  S. Fiedler-Le Touzé, [*Orientations complexes des courbes algébriques réelles*]{}, Thèse doctorale 2000. \[$\spadesuit$ cited in the entry Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]\]  S. Fiedler-Le Touzé, [*Cubics as tools to study the topology of $M$-curves of degree $9$ in ${\Bbb R}P^2$*]{}, J. London Math. Soc. (2) 66 (2002), 86–100. \[$\spadesuit$ p. dividing curves\]  S. Fiedler-Le Touzé, [*Pencils of cubics with eight base points lying in convex position in $\RR P^2$*]{}, arXiv, v2, 53 pages, Sept. 2012. \[$\spadesuit$ contains foundations required in the next entry Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]\]  S. Fiedler-Le Touzé, [*Totally real pencils of cubics with respect to sextics*]{}, a marvellous preprint received the 1 March 2013 (v.1), and a second version (v.2) the 3 March 2013 (where the basepoints are assigned on the ovals instead of in their insides like in v.1). Final version on arXiv 18–19 March 2013. \[$\spadesuit$ a seminal work containing proofs of Rohlin’s 1978 (unproven) total reality assertion for certain $(M-2)$-sextics totally swept out by suitable pencil of cubics $\spadesuit$ this is the first non-trivial (i.e. not involving pencil of lines or conics) extrinsic manifestation of Ahlfors theorem $\spadesuit$ another but much more modest phenomenon of total reality occurs for $M$-curves (as slowly discovered by Gabard, cf. Theorem \[total-reality-of-plane-M-curves:thm\]) but this is merely at the level of the Bieberbach-Grunsky theorem, i.e. the genus zero case of Ahlfors theorem $\spadesuit$ \[20.03.13\] as brilliantly explained in the paper in question (p.3), Le Touzé proves actually a slightly weaker statement that Rohlin’s original claim, namely the dividing character is not deduced a priori from total reality (as Rohlin claimed being able to do), but rather the dividing character is taken as granted via the Rohlin-Kharlamov-Marin congruence while total reality of the pencil of cubics is built upon this preliminary knowledge. Hence it could still be of some interest to reconstruct a proof [*purely a priori*]{} assuming of course that there is a such. This looks quite likely, yet apparently quite elusive to implement.\]  S.M. Finashin, [*The topology of the complement of a real algebraic curve in ${\Bbb C}P^2$*]{}, Zap. Nauch. Sem. LOMI 122 (1982), 137–145; English transl., J. Soviet Math. 26 (1984), 1684–1689. \[$\spadesuit$ briefly discussed in Viro 1986/86 [@Viro_1986/86-Progress]\]  S.M. Finashin, [*Differential topology of quotients of complex surfaces by complex conjugation*]{}, Zap. Nauch. Sem 231 (1995), 215–221; English transl., J. Math. Sciences 91 (1998), 3472–3475. \[$\spadesuit$ p.3472: “A well-known example is $X={\Bbb C} P^2$, for which \[the quotient by conj is\] $Y \cong S^4$. According to V.A. Rohlin, the last equality was quite widely known in the mathematical folklore, in any case to those who reflected on the four-dimensional Poincaré conjecture, for example, to Pontryagin. However, the author knows no mention of this account before Arnold’s paper \[2\](=1971 [@Arnold_1971/72]) and no published proofs before the papers of Kuiper \[9\](=1974) and Massey \[10\](=1973).” $\spadesuit$ according to some subsequent publication by Arnold, the result goes back to Maxwell!\]  S.M. Finashin, [*Rokhlin conjecture and quotients of complex surfaces by complex conjugation*]{}, J. reine angew. Math. 481 (1996), 55–71. \[$\spadesuit$ p.68: some remarks on sextics, e.g. Fig.10 gives the $(M-1)$-scheme $10$ via a perturbation of a line arrangement $\spadesuit$ p.68: “It is well known and not difficult to see directly from the Hilbert and Gudkov constructions of nonsingular real sextics (cf. \[V\](=Viro 1986/86 [@Viro_1986/86-Progress])), that the ones with schemes $\langle \alpha \sqcup 1 \langle \beta \rangle$ can be deformed to the both schemes \[having resp. one less outer oval or inner oval\] by passing through a cross-like real node which connects the ambient oval with an exterior oval resp. with an interior one. The only exceptio is the scheme $\langle 9 \sqcup 1 \langle 1 \rangle$ \[ of Harnack\] which can be reduced to $\langle 10 \rangle$ only by contracting the inner oval.”\]  S.M. Finashin, [*On the topology of real plane algebraic curves with nondegenerate quadratic singularities*]{}, Algebra i Analiz 8 (1996), 186–204; English transl., St. Petersburg Math. J. 8 (1997), 1039–1051. \[$\spadesuit$\]  A.E. Fischer, A.J. Tromba, [*On a purely “Riemannian” proof of the structure and dimension of the unramified moduli space of a compact Riemann surface*]{}, Math. Ann. (1983). \[$\spadesuit$ ... Ahlfors \[2\]. The space of extremal quasi-conformal maps between two Riemann surfaces (the so-called Teichmüller space) is in fact a ramified covering of the space of conformal classes of Riemann surfaces of prescribed genus (the real moduli space). In \[4\] Ahlfors shows that ...\]  S.D. Fisher, [*Exposed points in spaces of bounded analytic functions*]{}, Duke Math. J. 36 (1969), 479–484. 50 \[$\spadesuit$ cite Ahlfors 1950 [@Ahlfors_1950] and the following result is obtained: the exposed points of the algebras $A(\overline R)$ (resp. $H^{\infty}(R)$) are uniformly dense in the unit sphere of the respective space\]$\bigstar$$\bigstar$$\bigstar$  S.D. Fisher, [*Another theorem on convex combination of unimodular functions*]{}, Bull. Amer. Math. Soc. ?? (1969), 1037–1039. \[$\spadesuit$ finite Riemann surfaces, inner functions and it is proved that the closed convex-hull of the inner functions is the unit ball (for the sup norm) of the algebra $A(R)$ of analytic functions continuous up to the border $\spadesuit$ this is proved via an interpolation lemma due to Heins 1950 [@Heins_1950], which is closely allied to the Ahlfors function (plus maybe some Garabedian) $\spadesuit$ this is stated as: “Lemma 1: Let $z_1,\dots,z_N$ be distinct points of $R$ (=a finite Riemann surface) and let $h$ be an analytic function on $R$ bounded by $1$. Then there is an inner function $f$ (i.e. of modulus one on the boundary $\partial R$) in $A(R)$ with $f(z_j)=h(z_j)$, $j=1,\dots, N$.” $\spadesuit$ one can take $h\equiv 1$ then $f$ looks strange for it maps inner points to the boundary point 1, yet still $f=1$ works $\spadesuit$ the question is of course whether this reproves Ahlfors 1950, but this looks unlikely especially as no control is supplied on the degree, but see Heins 1950 [@Heins_1950], which suitably modified should recover Ahlfors result by controlling appropriately the bound involved\]  S.D. Fisher, [*On Schwarz’s lemma and inner functions*]{}, Trans. Amer. Math. Soc. 138 (1969), 229–240. 47, 78 \[$\spadesuit$ after Havinson 1961/64 [@Havinson_1961/64] and Carleson 1967 [@Carleson_1967-book], study the Ahlfors map for domains of infinite connectivity $\spadesuit$ subsequent ramifications in Röding 1977 [@Roeding_1977_Ahlfors], Minda 1981 [@Minda_1981-image-Ahlfors-fct], Yamada 1983–92 [@Yamada_1983-rmk-image-Ahlfors-fct; @Yamada_1992-Ahlfors-fct-on-Denjoy]\]  S.D. Fisher, [*The moduli of extremal functions*]{}, Michigan Math. J. 19 (1972), 179–183. 47 \[$\spadesuit$ the Ahlfors function of a domain (supporting nonconstant bounded analytic functions) is shown to be of unit modulus on the Šilov boundary of $H^{\infty}$\]  S.D. Fisher, [*Non-linear extremal problems in $H^{\infty}$*]{}, Indiana Univ. Math. J. 22 (1973), 1183–1190. 50 \[$\spadesuit$ p.1183/7 speaks of the “Ahlfors-Royden extremal problem” $\spadesuit$ the author explains that in Ahlfors extremal problem the class of competing functions is convex, explaining uniqueness of the soution and studies a variant of the problem with a side-condition amounting to require “no other zeros” which leads to a non-convex problem lacking uniqueness $\spadesuit$ p.1187/88, grasp of the geometric quintessence of Ahlfors’ argument: “By a theorem of Ahlfors \[A1; §4.2\] there is a set of $r+1$ points $p_j$ in $\Gamma$ such that if $v_i$ is the period vector of a unit mass at $p_j$, then $v_0,\dots, v_r$ form the vertices of a simplex in ${\Bbb R}^{r}$ which contains the origin as an interior point.”\]  S.D. Fisher, [*Function theory on planar domains*]{}. A second course in complex analysis. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1983.  S.D. Fisher, D. Khavinson, [*Extreme Pick-Nevanlinna interpolants*]{}, Canad. J. Math. 51 (1999), 977–995. \[$\spadesuit$ Ahlfors function (in the domain case only), its connection with Blaschke products and the Green’s functions, Pick bodies (jargon of Cole, Lewis, Wermer) and interpolation\]  H. Florack, [*Reguläre und meromorphe Funktionen auf nicht geschlossenen Riemannschen Flächen*]{}, Schr. Math. Inst. Univ. Münster no. 1 (1948), 34pp. 60 \[$\spadesuit$ cited also in Royden 1962 [@Royden_1962] (yet not within the text?) and briefly summarized in a ICM talk ca. 1954 of Behnke\]$\bigstar$$\bigstar$$\bigstar$  F. Forelli, [*Bounded holomorphic functions and projections*]{}, Illinois J. Math. 10 (1966), 367–380. \[$\spadesuit$ the universal covering method is employed to derive another proof of the corona theorem for interiors of compact bordered Riemann surfaces, relativizing thereby the ubiquitousness of the Ahlfors function given in Alling 1964 [@Alling_1964] $\spadesuit$ Forelli’s proof uses the following tools: $\bullet$ (p.368) “measure and Hilbert space theory, and the harmonic analysis that goes with the Hilbert space $H^2$” $\bullet$ (p.373,374) existence of analytic differentials with prescribed periods on the Schottky double (via Pfluger 1957 [@Pfluger_1957]) $\bullet$ Beurling’s invariant subspace theorem (p.366), but this can be dispensed in the compact bordered case by appealing to a holomorphic function continuous up to the border “whose zeros are the critical point of the Green’s function with pole at $t(0)$” (p.377)\]  F. Forelli, [*Extreme points in $H^1(R)$*]{}, Canad. J. Math. 19 (1967), 312–320. \[$\spadesuit$\]$\bigstar$$\bigstar$  F. Forelli, [*The extreme points of some classes of holomorphic functions*]{}, Duke Math. J. 46 (1979), 763–772. \[$\spadesuit$ study of the extreme points of the family of analytic functions with positive real part on a given finite Riemann surface normalized to take the value $1$ at a given point $\spadesuit$ the paper Heins 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF] supplements the results of Forelli by precise characterizing results for the case where the genus of $S$ is positive $\clubsuit$ \[11.10.12\] in fact this Forelli paper is a jewel (that I was only able to read today=\[11.10.12\], shame on me!) $\clubsuit$ despite presenting itself too humbly as a modest appendix to Heins 1950 [@Heins_1950], its main result (Theorem 3.2, p.766) gives the chain of inclusions $N_q(W, \zeta)\subset \partial N(W, \zeta) \subset \bigcup_{q}^{2p+q} N_k(W, \zeta)$, which readily implies a new proof of circle maps of degree $\le 2p+q$ (like Ahlfors 1950 [@Ahlfors_1950]). To understand this point, first recall Forelli’s notation: $\overline{W}$ is a compact bordered Riemann surface of genus $p$ with $q$ contours, $W$ is of course its interior; $N(W, \zeta)$ is the class of holomorphic functions $f$ on $W$ with positive[^126] real part (${\rm Re} f>0$) normalized by $f(\zeta)=1$ at some fixed $\zeta \in W$ (it is easily verified that $N(W, \zeta)$ is convex and compact in the compact-open topology) \[notion due to Arens/Fox, if I remember well???\]; the symbol $\partial$ used above refers [*not*]{} to the boundary but to the set of all extreme points of a convex body, i.e. those points of the body not expressible as convex (=barycentric) combination $t x + (1-t)y$ ($t\in [0,1]$) of two (distinct) points $x,y$ of the body. This is also the smallest subset of the body permitting its complete reconstruction via the convex-hull operation; finally $N_k(W, \zeta)$, for $k>0$ a positive integer, is the subclass of $N(W,\zeta)$ consisting of functions that cover the right half-plane $k$ times. $\clubsuit$ having explained notation, it is plain to deduce Ahlfors’ result. Indeed from the cited properties of convexity and compactness for $N(W,\zeta)$ one deduces (via Krein-Milman) existence of extreme points, i.e. $\partial N(W,\zeta)\neq \varnothing$ (this issue is not explicit in Forelli’s paper, but so evident that it is tacit, cf. e.g., Heins’ commentary in 1985 [@Heins_1985-Extreme-Pick-Nevanlinna p.758]: “My paper \[7\](=Heins 1950 [@Heins_1950]) showed the existence of minimal positive harmonic functions on Riemann surfaces using elementary standard normal family results without the intervention of the Krein-Milman theorem[^127] and gave applications to qualitative aspects of Pick-Nevanlinna interpolation on Riemann surfaces with finite topological characteristics and nonpointlike boundary components.” $\clubsuit$ Now Forelli’s second inclusion implies immediately the desideratum (existence of circle maps of degree $d$ such that $q\le d\le 2p+q$) $\spadesuit$ note of course that the first set of the string, that is $N_q(W, \zeta)$, can frequently be empty. Consider e.g. $\overline W$ be one-half of [*Klein’s Gürtelkurve*]{}[^128], that is any real plane smooth quartic, $C_4 \subset {\Bbb P}^2$, with two nested ovals, then $q=2$ but quartics and more generally smooth plane curves of order $m$ are known to be $(m-1)$-gonal). For an even simpler example, consider any bordered surface $W$ with only one contour ($q=1$) and of positive genus $p>0$, then there cannot be a circle-map of degree $d=q=1$ for a such would be an isomorphism (by the evident branched covering features of analytic maps), violating the topological complexity prompted by $p>0$ $\spadesuit$ several questions arise naturally form Forelli’s work. A first one is the perpetual question about knowing if the method can recover the sharper bound $p+q$ ($\approx r+p$) of Gabard 2006 [@Gabard_2006]. (Here and below $\approx $ refers to notational conversion from Forelli’s notation to the one used in the present paper). Again it is our belief that the ultimate convex geometry reduction of the problem (already explicit in Ahlfors) could be slightly improved so as to do this (compare below for more details). Another problem is to understand the distribution of degrees corresponding to extreme points of Forelli’s convex body $\partial N(W, \zeta)$ (maybe call it the Carathéodory-Heins-Forelli body to reflect better the historical roots of the technique, brilliantly discussed in Heins 1985 [@Heins_1985-Extreme-Pick-Nevanlinna]). For instance is the least degree half-plane map (equivalently circle map) always an extreme point, as the nebulous principle of economy ($\approx$ least effort) could suggest? (Nature always tries to relax itself along an equilibrium position necessitating the minimum existential stress-tensor!??) Finally one would like to see the connection between Ahlfors extremals and the extreme points of Heins-Forelli. Of course there is a little tormenting routine to switch from the one to the others via a Möbius-Cayley transformation from the disc to the half-plane. Yet loosely it seems that Ahlfors functions are a subclass of the extreme points, for they former depend on less parameters. For instance as noted by Forelli in the special planar case $p=0$, the above chain of inclusions collapses to give the clear-cut equation $\partial N(W, \zeta)=N_q(W,\zeta)$ characterizing the set of extreme points in, essentially, purely topological terms. Yet the Bieberbach-Grunsky theorem (1925 [@Bieberbach_1925], or A. Mori [@Mori_1951]) tell us that circle maps are in this case ($p=0$) fairly flexible insofar that we can preassign one point on each contour and find a circle map (of degree $q$) taking those points over the same boundary point[^129]. Hence for large values of $q$ such minimal degree circle maps depends on essentially $q$ real parameters, whereas for Ahlfors maps we can only specify the basepoint undergoing maximum distortion (hence just 2 real free parameters). $\clubsuit$ Finally some words about Forelli’s method of proof: It uses some “functional analysis” in the form of measure theory. Specifically Radon measures are mentioned, and a proposition permitting to express extreme points of a body $B$ specified by $n$ linear integral conditions as combination of $(n+1)$ extreme probability measures (cf. Prop.2.1 for the exact statement identified as dating back to Rosenbloom 1952 [@Rosenbloom_1952], \[but in geometric substance a similar lemma is already employed in Heins 1950 [@Heins_1950], as well as in Ahlfors 1950 [@Ahlfors_1950])\]. This is then specialized to the case where the space $X$ is the boundary of the bordered surface $\partial W$[^130], and the $n$ conditions amounts essentially to ask the vanishing of the periods along representatives of a homology basis of $\overline W$, consisting of $n:=2p+(r-1)$ cycles. The crucial potential theory is done via the Poisson integral inducing a bijective map $\#\colon P(\partial W) \to h_+(W, \zeta)$ between probability measures on the boundary and positive harmonic functions normalized by taking $\zeta$ to $1$. It is defined by $\mu^{\#}(w)=\int_{\partial W} Q(w,y) d\mu (y)$, where $Q(w,y)$ is the Poisson kernel of $W$ ($w\in W, y\in \partial W$). Now to find and describe (extreme) half-plane maps in $\partial N(W, \zeta)$, we are reduced via the above correspondence to a special set $B$ of measure verifying $n$ integral equations. On applying (Rosenbloom’s) proposition, the measure $\mu$ defined by $\mu^{\#}={\rm Re} f$ where $f\in N(W, \zeta)$ is decomposed as a convex sum (i.e. with positive coefficient $t_k$) of Dirac measures $\mu=\sum_1^m t_k \delta_k$ concentrated at some boundary points $y_k\in \partial W$, where $m\le n+1$. It follows by calculation (Poisson+Dirac’s trick) that ${\rm Re} f(z)=\sum_1^m t_k Q(w, y_k)$ (because integrating a function against the Dirac measure concentrated at some point just amounts evaluating the function at that point). Of course notice at this stage that the Poisson function $Q(w,y)$ is nothing else than the Green function with pole pushed to the boundary (so the object that we manipulated during our attempt to decipher Ahlfors’ proof). At this stage the proof is essentially finished. $\spadesuit$ as a matter of details Forelli further discuss the construction of the Poisson kernel taking inspiration from techniques of Earle-Marden 1969 [@Earle-Marden_1969-On-Poincare], using primarily the uniformization of Poincaré-Koebe. To sum up Forelli’s is able to reprove existence of circle maps but needs uniformization, admittedly in a simple finitistic context. Of course Ahlfors proof seems to avoid this dependance, which is anyway perhaps not so dramatic. $\spadesuit$ The latter issue should of course not detract us from the geometrical main aspect of the proof. First Forelli’s proof uses heavily a little yoga between measures and harmonic functions converting the one to the others via the Poisson integral. This technique involves so Poisson, then Stieltjes and finally the so-called Herglotz-Riesz (1911 [@Herglotz_1911-U-Potenzreihen]) (representation) theorem, a special incarnation of Fischer-Riesz (1907). Of course the yoga in question boils down to the Dirichlet principle when the measure has continuous density so that Herglotz-Riesz is just the Dirichlet problem enhanced by Lebesgue integration. Of course all this is beautiful, yet probably not fully intrinsic to the problematic of half-plane (or the allied circle) maps, which can probably be arrived upon via more classical integration theories (and in particular the classical Dirichlet problem, plus the allied potential functions, Green’s, Poisson’s or whatever you like to call them). I personally used the term Red’s function (somewhere in this text) as colorful contrast to evergreens tree, honoring George Green, but of course Poisson’s function might be historically more accurate. (After all, human beings descend from fishes rather than vegetables, and Green himself quotes of course Poisson, and Dirichlet was a Poisson student). $\clubsuit$ but now the key issue would be to penetrate even deeper in the geometry of Forelli’s proof. Again the hearth of the problem is the possibility of expressing a certain point as convex combination of [*at most*]{} $(n+1)$ points; in Forelli’s treatment cf. Prop.2.1, where however the “at most” proviso is not explicit but implicitly used later in the proof of Theorem 3.2. Like in our attempt to push Ahlfors proof down to recover Gabard’s bound, we believe that a better inspection of this convex geometry could corroborate the possibility of locating half-plane maps of lower degree. The situation we have in mind is the following (to which we were reduced by reading carefully Ahlfors 1950 [@Ahlfors_1950]): suppose we are given in ${\Bbb R}^(n\approx g)$ a collection of $q\approx r$ curves forming a balanced configuration (all $\approx$ signs just amounts to conversion from Forelli’s notation to the one used in the present text), in the sense that the convex hull encloses the origin, then it is of course possible to express the origin as convex sum of $\le n+1\approx g+1$ point (recovering thereby Ahlfors’ result). However it must be also possible to be more economical by using a more special, lower-dimensional simplex, able to cover the origin with a smaller quantity of points. We hope that this is a problem of pure (Euclid/convex/Minkowski) geometry (perhaps involving some topological tricks like in the Borsuk-Ulam (ham-sandwich) theorem, which can concomitantly be proved via more simple center of masses considerations, cf. e.g. Fulton’s book on “topology”). Alas I can only try to convince the reader by looking at the (very special) case where $n\approx g=2$ coming (via $g=2p+(r-1)$) from the values $p=1, r=1$. Then we have one balanced circle in the plane ${\Bbb R}^2$. If we follow Ahlfors, we just have the plain remark that there is $g+1=r+2p=1+2\cdot 1=3$ points spanning a simplex covering the origin (which is trivial for dimensional reason), however it is evident that a more special and lucky constellation (Stonehenge alinement) of two points situated on the topological circle (Jordan curve) corresponding to the contour of the bordered surface, suffice to cover the origin with a $1$-simplex, giving existence of a circle map of degree $2$, like the $r+p$ bound predicted in Gabard 2006 [@Gabard_2006] $\spadesuit$ of course all we are saying does not detract the possibility that the extreme points studied by Forelli always contain an element landing in the highest possible degree $2p+q \approx 2p+r=g+1$\]  J.E. Fornaess, N. Sibony, [*Some open problems in higher dimensional complex analysis and complex dynamics*]{}, Publ. Mat. 45 (2001), 529–547. \[$\spadesuit$ p.539: “Question 3.16. [*Can one embed all Stein Riemann surfaces as closed complex submanifolds of ${\Bbb C}^2$? (See [\[GS\]=Globevnik-Stensones 1995 [@Globevnik-Stensones_1995]]{})*]{}”\]  O. Forster, [*Riemannsche Flächen*]{}, Springer, Berlin, 1977, 223 pp; English trans., available. \[$\spadesuit$ sheaf-theoretic approach\]  F. Forstnerič, E.F. Wold, [*Bordered Riemann surfaces in ${\Bbb C}^2$*]{}, J. Math. Pures Appl. 91 (2009), 100–114. \[$\spadesuit$ reduction of the big problem of embedding open Riemann surfaces in the affine plane to that of embedding compact bordered surfaces, which looks more tractable due to its finitary nature, yet apparently completely out of reach\]  F. Forstnerič, E.F. Wold, [*Embeddings of infinitely connected planar domains in ${\Bbb C}^2$*]{}, arXiv (2012). \[$\spadesuit$ “Abstract. We prove that every circled(=circular) domain (=Koebe’s Kreisbereich) in the Riemann sphere admits a proper holomorphic embedding (=PHE) in ${\Bbb C}^2$.” This is yet another spectacular advance on the proper embedding problem, giving insights on how to crack the general problem. (This may be restated as, p.1: “[*Is every open Riemann surface biholomorphic to a smoothly embedded, topologically closed complex curve in ${\Bbb C}^2$.*]{}”) Of course, when combined the He-Schramm 1993 [@He-Schramm_1993] uniformization result this gives the “Theorem 1.1.—[*Every domain in the Riemann sphere with at most countably many boundary components, none of which are points, admits a PHE in ${\Bbb C}^2$*]{}.” p.2: This result gives a wide extension of the similar statement in finite connectivity due to Globevnik-Stens[ø]{}nes 1995 [@Globevnik-Stensones_1995]. p.17: “There exists a Cantor set in ${\Bbb P}^1$ whose complement embeds PH into ${\Bbb C}^2$ (Orevkov 2008 [@Orevkov_2008]), but it is an open problem whether this holds for each Cantor set.”\]  J. Fourier, [*Théorie analytique de la chaleur*]{}, 1822. \[$\spadesuit$ trigonometric series expansion of an arbitrary function (so-called Fourier series), despite some earlier appearance of them in works by by Clairaut and Euler $\spadesuit$ Fourier’s first work on the topic was presented to Paris Academy in 1807, yet rejected by Lagrange, Laplace and Legendre\]  W.F. Fox, [*Harmonic functions with arbitrary singularity*]{}, Pacific J. Math. (1961), 153–164. \[$\spadesuit$ discusses and rederives old results of Schwarz 1870, Koebe while pointing out to the developments made by Sario $\spadesuit$ p.153 probably corroborates the intuition that the solvability of the Dirichlet principle on a compact bordered Riemann surface was first treated by Schwarz 1870\]  A. Fraser, R. Schoen, [*The first Steklov eigenvalue, conformal geometry, and minimal surfaces*]{}, Adv. in Math. 226 (2011), 4011–4030. 50 \[$\clubsuit$ applies Ahlfors 1950 [@Ahlfors_1950] (and even Gabard 2006 [@Gabard_2006]) to spectral theory, especially first Steklov eigenvalue. For higher eigenvalues, cf. Girouard-Polterovich 2012 [@Girouard-Polterovich_2012], and for Dirichlet-Neumann eingenvalues, cf. Gabard 2011 [@Gabard_2011].\]  I. Fredholm, [*Sur une classe d’équations fonctionnelles*]{}, Acta Math. 27 (1903), 365–390. \[$\spadesuit$ early influence of Abel (1823), then Neumann’s approach to the Dirichlet problem and Volterra (1896) where Neumann’s method was successfully applied to an integral equation\]  R. Fricke, F. Klein, [*Vorlesungen über die Theorie der automorphen Functionen*]{}, Two volumes, Teubner, Leipzig, 1897, 1912, 634 pp., 668 pp.; Reprinted by Johnson Reprint Corp., New York and Teubner, Stuttgart, 1965. \[$\spadesuit$ contains versions of RST (=Rückkehrschnitttheorem), while the completion of the second volume seem to have received some helping hand from Paul Koebe $\spadesuit$ p.180ff. contains anothe account of the classification of Klein’s syymmetric Riemann surfaces\]  G. Fubini, [*Il Principio di minmo i teoremi di esistenza per i problemi al contorno relativi alle equazioni alle derivate parziale di ordini pari*]{}, Rend. Circ. Mat. Palermo (1907). \[$\spadesuit$ cited in Zaremba 1910 [@Zaremba_1910] as another extension (beside Beppo Levi 1906 [@Beppo-Levi_1906] and Lebesgue 1907 [@Lebesgue_1907]) of Hilbert’s resurrection of the Dirichlet principle\]  B. Fuchs, [*Sur la fonction minimale d’un domaine, I, II*]{}, Mat. Sbornik N.S. 16 (58) (1945); 18 (60) (1946). \[$\spadesuit$ quoted in Lehto 1949 [@Lehto_1949] and consider the problem of least momentum, i.e. minimizing $\int\!\!\int_B \vert f(z)\vert^2 d\omega$ under the side-condition $f(t)=1$ at some interior point\] $\bigstar$$\bigstar$$\bigstar$\[part I OK, part II still not found\]  A. Gabard, [*Topologie des courbes algébriques réelles: une question de Felix Klein*]{}, L’Enseign. Math. 46 (2000), 139–161. \[$\spadesuit$ furnish a complete answer to a question raised by Klein as a footnote to his Coll. Papers, using an inequality due to Rohlin 1978 [@Rohlin_1978]. Previous (unpublished) work on the same question due to Kharlamov-Viro in the Leningrad seminar of topology supervised by V.A. Rohlin. Confirms incidentally a desideratum of Gross-Harris 1981 [@Gross-Harris_1981].\]  A. Gabard, [*Sur la topologie et la géométrie des courbes algébriques réelles*]{}, Thèse, Genève, 2004. 50 \[$\clubsuit$ includes the improved bound $r+p$ upon the degree of a circle map of a membrane of genus $p$ with $r$ contours. Up to minor redactional change this is the same as the next entry Gabard 2006 [@Gabard_2006]\]  A. Gabard, [*Sur la représentation conforme des surfaces de Riemann à bord et une caractérisation des courbes séparantes*]{}, Comment. Math. Helv. 81 (2006), 945–964. 50 (This result also appeared previously in the Ph.D. Thesis of the author published in 2004, cf. the previous item.) \[$\clubsuit$ proposes an improved bound upon Ahlfors 1950 [@Ahlfors_1950], as discussed in the previous item $\spadesuit$ for an update regarding the question about the sharpness of the bound so obtained see Coppens 2011 [@Coppens_2011] $\spadesuit$ \[03.10.12\] all this is fairly good yet a certain discrepancy with Ahlfors viewpoint is annoying and much remains to be clarified $\spadesuit$ \[03.10.12\] further one can wonder if there is not a Teichmüller-theoretic proof of the existence of such circle maps, parallelling that of Meis 1960 [@Meis_1960] in the case of closed surfaces, and conversely one can of course wonder if Meis cannot be proved via the topological method used in the present entry (Gabard 2006 [@Gabard_2006])\]  A. Gabard, [*A separable manifold failing to have the homotopy type of a $CW$-complex*]{}, arXiv 2006, and another (simpler?) proof suggested by the referee in, Archiv der Math. (2008). \[$\spadesuit$ this little note was primarily intended to give a counterexample to an assertion made by Milnor in 1959, to the effect that all separable manifolds have the homotopy type of a CW-complex. Alas, this is completely wrong (as soon as one familiar with the Prüfer surface 1922–25). Notwithstanding, more mature knowledge of mine (ca. 2009) I realized that Milnor was not wrong at all, except that for him separable meant at that time second countable or metrizable (compare for instance sone of his preprint ca. 1958–59 available on the net). So the explanation is simply that the term “separable” had a different meaning in the first half of the 20th century (up to some residues moving as high as Milnor’s 1959 article [@Milnor_1959])\]  A. Gabard, D. Gauld, [*Jordan and Schoenflies in non-metrical analysis situs*]{}, arXiv 2010. \[$\spadesuit$\]  A. Gabard, D. Gauld, [*Dynamics of non-metric manifolds*]{}, arXiv 2011. \[$\spadesuit$ this is just cited for the proof of the implication: simply-connected $\Rightarrow$ schlichtartig $\Rightarrow$ orientable, which may be reduced to the five lemma\]  A. Gabard, [*Compact bordered Riemannian surfaces as vibrating membranes: an estimate à la Hersch-Yang-Yau-Fraser-Schoen*]{}, arXiv 2011. 50 \[$\clubsuit$ inspired by Fraser-Schoen 2011 [@Fraser-Schoen_2011], this adapts Hersch 1970 [@Hersch_1970] (isoperimetric property of spherical vibrating membranes) to configurations of higher topological structure using the Ahlfors circle map with the bound of Gabard 2006 [@Gabard_2006] $\spadesuit$ notice an obvious (but superficial) connection with Gromov’s filling area conjecture (FAC) (1983 [@Gromov_1983]) positing the minimality of the hemisphere among non-shortening membranes, hence it would be fine that conformal geometry/transplantation—enhanced perhaps by Weyl’s asymptotic law for the high vibratory modes (out of which we can ‘hear’ the area of the drum)—affords a proof, either geometric or acoustic, of FAC. This would maybe be a spectacular application of the Ahlfors map, or maybe some allied conformal maps, e.g. that of Witt-Martens [@Witt_1934], [@Martens_1978], for non-orientable membranes. Recall indeed Gromov’s trick of cross-capping (à la von Dyck) the boundary contour of the membrane reduces the filling area problem (in genus zero) to Pu’s systolic inequality for the projective plane\]  A. Gabard, [*Ebullition in foliated surfaces vs. gravitational clumping*]{}, arXiv 2011. \[$\clubsuit$ not relevant to the present topic, but just cited for a Jordan separation argument via covering theory that can be ascribed to Riemann with some imagination\]  A. Gabard, [*Euler-Poincaré obstruction for pretzels with long tentacles à la Cantor-Nyikos*]{}, arXiv, Dec. 2011. \[$\clubsuit$ not relevant to the present topic, but just cited for some rudiment about Poincaré’s index formula for foliations\]  A. Gabard, [*Ahlfors circle maps: historical ramblings*]{}, arXiv 2012. \[$\clubsuit$ this is the present article available on the arXiv, but which soon afterward (2013) was expanded so has to reinforce the connection with Rohlin’s work on Hilbert’s 16th problem. The pivotal motivation for this junction between Ahlfors-Rohlin is Rohlin’s cryptical claim of total reality for certain $(M-2)$-sextics having a real scheme forcing the type I. Also instrumental for this expansion (of material) is the Rohlin conjecture (or at least the vestige thereof post Shustin) that a scheme of type I is maximal. This problem (still open) bears some connection with earlier speculations of Klein (1876) which however could not resist Shustin’s work 1985 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin] (much based upon the Viro revolution as well as deep Bézout-style obstructions coming from Fiedler-Viro)\]  D. Gaier, [*Konforme Abbildung mehrfach zusammenhängender Gebiete durch direkte Lösung von Extremalproblemen*]{}, Math. Z. 82 (1963), 413–419. 78 \[$\spadesuit$ what sort of maps via which (extremal) method? Essentially the PSM via the Ritz-Ansatz (ca. 1908), à la Bieberbach-Bergman (1914/22), plus Nehari’s 1949 integral representation of such slit mappings\]  D. Gaier, [*Konstruktive Methoden der konformen Abbildung*]{}, Ergebnisse d. Angew. Math. 3, Springer, Berlin, 1964. 78 \[$\spadesuit$ Chap.III discusses in details the extremal properties of the Riemann mapping for a plane simply-connected region (distinct of ${\Bbb C}$), namely that the range of the map normalized by $f'(z_0)=1$ has minimal area (first in Bieberbach 1914 [@Bieberbach_1914]) or that the boundary of the range has minimal length (probably first in Szegö 1921 [@Szego_1921]) $\spadesuit$ this material was also presented (in book format) by Julia 1931 [@Julia_1931]\]$\bigstar$  D. Gaier, [*Über ein Flächeninhaltsproblem und konforme Selbstabbildungen*]{}, Rev. Roumaine Math. Pures Appl. 22 (1977), 1101–1105. \[$\spadesuit$ cited for the same reasons as the next item and complement some details of it (especially in the sharpness of cross-references)\]  D. Gaier, [*Konforme Abbildung mehrfach zusammenhängender Gebiete*]{}, Jber. d. Dt. Math.-Verein. 81 (1978), 25–44. \[$\spadesuit$ p.34–35, §C, brilliant proof (of a fact discovered and briefly handled by Grötzsch 1931 \[alas no precise cross-ref.\]) via his [*Flächenstreifenmethode*]{} that “the” (non-unique!) map minimizing the area integral $\int\int \vert f'(z) \vert^2 d \omega$ (à la Bieberbach 1914 [@Bieberbach_1914]–Bergman\[n\] 1922 [@Bergman_1922], but extended to the multiply-connected setting) under the schlichtness proviso (and the normalizations $f(z_0)=0, f'(z_0)=1$) maps the domain upon a circular slitted disc (with concentric circular slits centered about the origin) $\spadesuit$ Gaier’s proof is based upon a Carleman isoperimetric property of rings relating the modulus to the area enclosed by the inner contour, plus Bieberbach 1914 [@Bieberbach_1914] (first area theorem) to the effect that a schlicht normed map ($f'(a)=1$) from the disc inflates area, unless it is the identity $\clubsuit$ a natural (naive?) question of the writer (\[13.07.12\]) is what happens if we relax schlichtness of the map? Do we recover an Ahlfors circle map? Try maybe to get the answer from the entry Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949]\]  T.W. Gamelin, M. Voichick, [*Extreme points in spaces of analytic functions*]{}, Canad. J. Math. 20 (1968), 919–928. 50 \[$\spadesuit$ Ahflors 1950 [@Ahlfors_1950] is quoted several times through the paper, the most relevant being $\clubsuit$ p.926: “According to \[1, §4.2\], there exist $r+1$ ($r=g$ in our notation) points $w_1, \dots, w_{r+1}$ on $bR$ such that if $B_j$ is the period vector of the singular function $T_j$ corresponding to a unit point mass at $w_j$, then $B_1, \dots, B_{r+1}$ are the vertices of a simplex in ${\Bbb R}^r$ which contains $0$ as an interior point.” $\clubsuit$ This is indeed the geometric heart of Ahlfors’ existence proof of a circle map of degree $\le g+1=r+2p$ $\spadesuit$ \[28.09.12\] the obvious game is whether one can lower the number of $w_j$ to recover the degree predicted in Gabard 2006 [@Gabard_2006] $\spadesuit$ as to the content of this entry, it is involved with an extension of the de Leeuw-Rudin (1958 [@deLeeuw-Rudin_1958]) characterization of the extreme points of the unit ball of the disc-algebra $H^1(\Delta)$ as the outer functions of norm $1$, and as usual this is obtained upon appealing to the Ahlfors map, or techniques closely allied to its existence-proof\]  T.W. Gamelin, [*Embedding Riemann surfaces in maximal ideal spaces*]{}, J. Funct. Anal. 2 (1968), 123–146. \[$\spadesuit$ p.130: “Let $R$ be a finite bordered Riemann surface with boundary $\Gamma$. Let $A$ be the algebra of functions continuous on $R\cup \Gamma$ and analytic on $R$. Let $\varphi$ be the evaluation at some point $z_0$ of $R$. Then the harmonic measure for $z_0$ on $\Gamma$ is a unique Arens-Singer measure for $\phi$ on $\Gamma$. The spaces $N_c$ consists of the boundary values along $\Gamma$ of the analytic differentials on the doubled surface of $R$, the so-called Schottky differentials of $R$. The space $N_c$ is finite-dimensional.” $\spadesuit$ p.133: “Since $P$ admits a finite-sheeted covering map over $\{\vert \lambda \vert<1\}$, $P$ must be one-dimensional.” $\spadesuit$ it is not clear (to Gabard) if this Gamelin argument makes tacit use of the Ahlfors map\]  T.W. Gamelin, [*Uniform algebras*]{}, Prentice Hall, 1969. \[$\spadesuit$ p.195–200, analytic capacity as the first coefficient in the Laurent expansion of the Ahlfors function $\spadesuit$ p.197, existence and uniqueness of the Ahlfors function for a general open set in the plane $\spadesuit$ p.198, proof of the following convergence property of the Ahlfors function $f_E$ of a compact plane set $E$ (meaning the one, centered at $\infty$, of the outer component of $E$, i.e. the component of the complement of $E$ containing $\infty$): if $E_n$ is decreasing sequence of compacta with intersection $E$, and $f_n$ be the Ahlfors functions of $E_n$, then $f_n$ converge to $f$ uniformly on compact subsets of the outer component of $E$, and the corresponding analytic capacities converge $\gamma(E_n) \to \gamma(E)$ $\spadesuit$ \[21.09.12\] this reminds perhaps one the famous conjecture (e.g. of Bing) about knowing if a descending sequence of plane (topological) discs must necessarily converge to a compactum satisfying the fixed-point property, even when the latter has the ugliest possible ‘dendrite’ shape $\spadesuit$ one may wonder if function theory, especially boosted version of RMT, could crack the problem (this is of course just a naive challenge)\]$\bigstar$  T.W. Gamelin, [*Localization of the corona problem*]{}, Pacific J. Math. 34 (1970), 73–81. 78  T.W. Gamelin, J. Garnett, [*Distinguished homomorphisms and fiber algebras*]{}, Amer. J. Math. ?? (1970), 455–474. \[$\spadesuit$ p.474 Ahlfors function mentioned as follows: “It is more difficult to relate the Shilov boundary of $H^{\infty}(D)$ to the Shilov boundaries of the fiber algebras. The problem is to decide whether the distinguished homomorphisms $\phi_\lambda$ lie in the Shilov boundary of $H^{\infty}(D)$. This question was resolved negatively by Zalcman \[11\](=1969 [@Zalcman_1969-TAMS]) for the domains he considered, because in this case the Ahlfors function of $D$ could be seen to have unit modulus on the Shilov boundary of $H^{\infty}(D)$.”\]  T.W. Gamelin, [*The algebra of bounded analytic functions*]{}, Bull. Amer. Math. Soc. 79 (1973), 1095–1108. 47, 50, 78 \[$\spadesuit$ p.1104: “The Ahlfors function tries hard to be unimodular on the boundary of an arbitrary domain.” The following result of Fisher is quoted (and reproved) “The Ahlfors function for a bounded domain $D$ in ${\Bbb C}$ has unit modulus on the Šilov boundary of $H^{\infty}(D)$.” $\spadesuit$ circa 12 occurrences of “Ahlfors function” throughout the paper $\spadesuit$ p.1104: “Incidentally, the preceding proof \[via the Šilov boundary\] also establishes the uniqueness of the Ahlfors function.” $\spadesuit$ p.1104: “Combined with cluster value theory, Fisher’s theorem yields information on the Ahlfors function which is already sharper than that which had been obtained by classical means.” p.1106–07: “if the harmonic measure for $D$ is carried by the union of an at most countable number of boundary components of $D$, then the Ahlfors function $G$ for $D$ is inner; that is, the composition $G\circ \pi$ with the universal covering map $\pi\colon \Delta \to D$ has radial boundary values of unit modulus a.e. ($d\theta$). Without the hypothesis on the harmonic measure, the Ahlfors function needs not be inner, and an example is given in \[17\](=Gamelin, to appear) of a domain $D$ with Ahlfors function $G$ satisfying $\vert G \circ \pi \vert<1$ a.e. ($d\theta$) on $\partial \Delta$.” $\clubsuit$ the paper Ahlfors 1950 [@Ahlfors_1950] is quoted in the following brief connection: “For dual extremal problems on Riemann surfaces, see \[2\](=Ahlfors 1950) and \[36\](=Royden 1962).”\]   T.W. Gamelin, [*Extremal problems in arbitrary domains*]{}, Michigan Math. J. 20 (1973), 3–11. 50, 78 \[$\spadesuit$ quoted in Hayashi 1987 [@Hayashi_1987] for the issue that the following property: “the natural map of a Riemann surface $R$ into its maximal ideal space $\frak M (R)$ (this is an embedding if we assume that the algebra $H^{\infty}(R)$ of bounded analytic functions separates points) is a homeomorphism onto an open subset of $\frak M (R)$” has some application to the uniqueness of the Ahlfors function, as well as to its existence via Hayashi 1987 [@Hayashi_1987] $\spadesuit$ Royden 1962 [@Royden_1962] is cited instead of the original work Ahlfors 1950 [@Ahlfors_1950] for the treatment of extremal problems on finite bordered Riemann surfaces\]  T.W. Gamelin, [*Extremal problems in arbitrary domains, II*]{}, Michigan Math. J. 21 (1974), 297–307. 78 \[$\spadesuit$ p.297, Ahlfors function is quoted as follows: “Hejhal proof’s depends on the methods developed by Havinson 1961/64 [@Havinson_1961/64], who proved the uniqueness of the Ahlfors function of arbitrary domains. Now there is in \[4\](=Gamelin 1973 [@Gamelin_1973-Extremal-I]) an economical proof of Havinson’s theorem that depends on function-algebraic techniques (see also \[3\](=Gamelin 1972, La Plata Notas) and \[5\](=Gamelin 1973 [@Gamelin_1973-BAMS])\]  T.W. Gamelin, [*The Shilov boundary of $H^{\infty}(U)$*]{}, Amer. J. Math. 96 (1974), 79–103. \[$\spadesuit$ p.79, the Ahlfors function is cited and the author finds a bounded domain in the plane whose Ahlfors function fails to be inner (violating thereby a guess formulated, e.g. in Rubel 1971 [@Rubel_1971]) $\spadesuit$ let us quote the text (p.79): “Let $U$ be a bounded domain in the plane, and let $H^{\infty}(U)$ be the algebra of bounded analytic functions on $U$, and $\frak M (U)$ be its maximal ideal space. Our object here is to study the Shilov boundary $S(U)$ of $H^{\infty}(U)$. It will be shown that $S(U)$ is extremely disconnected, and that every positive continuous function on $S(U)$ is the modulus of a function in $H^{\infty}(U)$. Fisher \[7\](=1972 [@Fisher_1972-The-moduli-of-extremal-fctions]) has shown that there exist nonconstant functions in $H^{\infty}(U)$ with unit modulus on $S(U)$. In fact, he proves that the Ahlfors function for $U$ is unimodular. We will show that there is an abundant supply of unimodular functions in $H^{\infty}(U)$, sufficiently many to separate $S(U)$ from the points of ${\frak M} (U )\setminus S(U)$ which are adherent to $U$. In the negative direction, we show that the property of having unit modulus on the Shilov boundary of $H^{\infty}(U)$ does not yield a great deal of information concerning the classical boundary values of functions in $H^{\infty}(U)$. In fact, an example is given of a reasonably well-behaved domain $U$ with the following property: If $f$ is any nonconstant function in $H^{\infty}(U)$ such that $\|f\|\le 1$, then the lift of $f$ to the open unit disc via the universal covering map has radial boundary values of modulus $<1$ a.e. ($d\theta$).” $\spadesuit$ the latter assertion specialized to an Ahlfors function (at some center) shows that the latter can fail to be inner (indeed not even hypo-inner in the sense of Rubel)\]  T.W. Gamelin, J.B. Garnett, L.A. Rubel, A.L. Shields, [*On badly approximable functions*]{}, J. Approx. Theory 17 (1976), 280–296. \[$\spadesuit$ if $F$ is a finite bordered Riemann surface, let $A(F)$ be the algebra of functions, analytic in the interior with continuous extension to the boundary $\Gamma:=\partial F$. The boundary value map $A(F)\to C(\Gamma)$ is injective (upon splitting into real/imaginary parts and applying the uniqueness of the Dirichlet problem). The algebra $C(\Gamma)$ (complex-valued functions on the boundary $\Gamma$) is endowed with the sup-norm $\|\varphi\|=\sup_{z\in \Gamma} \vert \varphi(z) \vert$. Now given any $\varphi \in C(\Gamma)$ there must be a best analytic approximant $f\in A(F)$, that is minimizing $\| \varphi - f\|$. The authors (following Poreda 1972) call $\varphi\in C(\Gamma)$ [*badly approximable*]{} if its distance $d(\varphi, A(D))$ to the space $A(D)$ is equal to the norm $\|\varphi\|$. This amounts saying that the best analytic approximant of $\varphi$ is $0$ (zero function). $\spadesuit$ \[01.10.12\] of course such badly approximable function are the opposite extreme of the boundary-values of an Ahlfors function (or of a circle map), since the latter coincide with their best analytic approximant. Despite this contrast, badly approximable functions are shown to have constant modulus along the boundary (Theorem 1.2, p.281) sharing a distinctive feature of circle maps, but deviates from them by having a small index (=winding number), namely ${\rm ind} (\varphi)<2p+(r-1)$, where $p$ is the genus and $r$ the contour number of $F$. Precisely Theorem 8.1 (p.294) states: “[*If $\varphi\in C(\Gamma)$ is badly approximable, then $\varphi$ has nonzero constant modulus, and ${\rm ind}(\varphi)<2p+(r-1)$.*]{}” The proof involves the theory of Toeplitz operators and reduces ultimately to the theory of Schottky differentials (forming a real vector space of dimension equal to the genus $g$ of the double which is precisely the upper bound involved above). Hence the connection with Ahlfors 1950 [@Ahlfors_1950] is evident (at least at some subconscious level), and accentuated by the numerous citations to the allied paper Royden 1962 [@Royden_1962]. $\spadesuit$ finally, let us maybe observe that the converse of the above statement (Theorem 8.1) can be foiled as follows: via Gabard 2006 [@Gabard_2006] there is always a circle map $f$ of degree $d\le r+p$. Its boundary restriction $\partial f=:\varphi$ has index equal to this degree ${\rm ind } (\varphi) =d\le r+p\buildrel{!}\over{<}2p+(r-1)$, provided $p>1$. Yet the map $\varphi$ is not badly approximable, for by construction it admits a perfect analytic approximant.\]  T.W. Gamelin, [*Cluster values of bounded analytic functions*]{}, Trans. Amer. Math. Soc. 225 (1977), 295–306. \[$\spadesuit$ several aspects of the Ahlfors function are discussed, and some new property (extending a result of Havinson) is given. To be more precise, we quote some extracts $\spadesuit$ p.296: Recall that the Ahlfors function $G$ of $D$, depending on the point $z_0 \in D$, is the extremal function for the problem of maximizing $\vert f'(z_0)\vert$ among all $f\in H^{\infty}(D)$ satisfying $\vert f \vert \le 1$; $G$ is normalized so that $G'(z_0)>0$, and then $G$ is unique. If $\zeta$ is an essential boundary point of $D$, then $\vert G \vert =1$ on $\amalg\!\!\!\,\amalg_{\zeta}$ (Šilov boundary). Furthermore, either $\lim_{D\ni z \to \zeta} \vert G(z)\vert=1$ or ${\rm Cl}(G, \zeta)= \overline{\Delta}$(=closed unit disc). S.Ya. Havinson \[7, Theorem 28\] has proved that $G$ assumes all values in $\Delta$, with the possible exception of a subset of $\Delta$ of zero analytic capacity. $\spadesuit$ p.297: we conclude the following sharper version of Havinson’s Theorem. [**1.2 Corollary.**]{} Let $G$ be the Ahlfors function of $D$, and let $\zeta$ be an essential boundary point of $D$ such that ${\rm Cl}(G, \zeta)=\overline{\Delta}$. Then values in $\Delta$ are assumed infinitely often by $G$ in every neighborhood of $\zeta$, with the exception of those lying in a set of zero analytic capacity.\]  T.W. Gamelin, [*Wolff’s proof of the corona theorem*]{}, Israel J. Math. ?? (1980), ??–??. \[$\spadesuit$ “Abstract. An expository account is given of T. Wolff’s recent elementary proof of Carleson’s Corona Theorem (1962). The Corona Theorem answers affirmatively a question raised by S. Kakutani (1957) as to whether the open unit disc in the complex plane is dense in the …”\]  T.W. Gamelin, M. Hayashi, [*The algebra of the bounded analytic functions on a Riemann surface*]{}, J. Reine Angew. Math. 382 (1987), 49–73. \[$\spadesuit$ p.72 some sophisticated (but lucid) questions about the Grunsky-Ahlfors (abridged Grahl=Graal=Sangreal) extremal problem of maximizing the derivative $f'(p)$ among functions bounded-by-one $\vert f \vert \le 1$ (where $p$ is a given point and the derivative is taken w.r.t. a fixed local coordinates). The following questions are posed under the proviso that $H^{\infty}(R)$ separates points. [**Problem 1.**]{} For a fixed $p\in R$, is there an $f\in H^{\infty}(R)$ such that $f'(p)\neq 0$? If such an $f$ exists, then any extremal function for the Grahl-problem normalized so that $f'(p)>0$ is termed an [*Ahlfors function*]{} corresponding to $p$. [**Problem 2.**]{} For fixed $p\in R$, assume the Grahl-extremal problem is non-trivial. Is the Ahlfors function unique? Does it have unit modulus on the Shilov boundary of $H^\infty(R)$? $\spadesuit$ the writer (Gabard) is not aware of any update on those questions, yet it may be emphasized that partial answers are sketched in Hayashi 1987 [@Hayashi_1987], namely that under the assumption that the natural map of $R$ to its maximal ideal space $\frak M (R)$ takes $R$ homeomorphically onto an open set of $\frak M (R)$, then existence and uniqueness of the Ahlfors function is ensured\]  M. Gander, G. Wanner, [*From Euler, Ritz and Galerkin to modern computing*]{}, (2012), 49–73. \[$\spadesuit$ a historical survey about Galileo, Bernoulli, Euler, Lagrange, Chladni,…, Ritz, Galerkin and their influence upon modern computing\]  P.R. Garabedian, M.M. Schiffer, [*Identities in the theory of conformal mapping*]{}, Trans. Amer. Math. Soc. 65 (1949), 187–238. 60, 78 \[$\spadesuit$ p.201, the problem of least area is considered (i.e. minimization of $\int\!\!\int \vert f'(z) \vert^2 d\omega$) among [*all*]{} (not necessarily schlicht) mappings $f$ normed by $f(a)=0, f'(a)=1$ defined on an $n$-connected domain $\spadesuit$ it should be emphasized that the solution of this problem was stated (without proof) by Grunsky 1932 [@Grunsky_1932 p.140]; Grunsky’s influence is recognized in the introduction (p.188), yet not made explicit at the relevant passage (p.201, Problem I.) for the specific result of the least area map $\spadesuit$ assert (without detailed proof) that the solution is at most $n$-valent $\spadesuit$ alas it is not asserted that those least-area maps are circle maps (which looks a natural conjecture)\]  P.R. Garabedian, [*Schwarz’s lemma and the Szegö kernel function*]{}, Trans. Amer. Math. Soc. 67 (1949), 1–35. 60, 78 \[$\spadesuit$ includes the formula $f'(t)=2\pi k(t,t)$ for the derivative of the Ahlfors function in terms of Szegö’s kernel function, other expositions of the same result in Bergman 1950 [@Bergman_1950], Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950] and Nehari 1952 [@Nehari_1952-BOOK] $\spadesuit$ at several crucial stage this paper makes use of topological arguments (hence a possible connection with Gabaredian 2006 [@Gabard_2006] remains to be elucidated)\]  P.R. Garabedian, [*The sharp form of the principle of hyperbolic measure*]{}, Ann. of Math. 51 (1950), 360–379. 60, 78 \[$\clubsuit$ claims to recover the full Ahlfors (1950 [@Ahlfors_1950]) theorem on existence of circle maps (by deploying a large array of techniques blending from Teichmüller 1939 [@Teichmueller_1939-Dreikreisesatzes], Grunsky 1940–42 [@Grunsky_1940], [@Grunsky_1942], Ahlfors 1947 [@Ahlfors_1947] and the variational method of Schiffer/Hadamard), but the detailed execution is limited to the planar case, and only the same bound as Ahlfors 1950 [@Ahlfors_1950] is obtained\]  P.R. Garabedian, [*The class $L_p$ and conformal mapping*]{}, Trans. Amer Math. Soc. 69 (1950), 392–415. \[$\clubsuit$\]  P.R. Garabedian, M.M. Schiffer, [*On existence theorems of potential theory and conformal mapping*]{}, Ann. of Math. (2) 52 (1950), 164–187. 78 \[$\spadesuit$ reprove RMT via the Bergman kernel (for smooth boundary, p.164), but the general case follows by topological approximation (exhaustion) $\spadesuit$ p.182 points out that circle maps lye somewhat deeper than slit mappings $\spadesuit$ p.181 recover the circle map for domains (of finite-connectivity) $\spadesuit$ recover also the parallel-slit mappings and cite Lehto 1949 [@Lehto_1949] for equivalent work $\spadesuit$ p.182 coins the designation “circle mapping”, to which we adhere in this survey.\]  P.R. Garabedian, [*A new proof of the Riemann mapping theorem*]{}. In: [*Construction and Applications of Conformal Maps*]{}, Proc. of a Sympos. held on June 22–25 1949, Applied Math. Series [*18*]{}, 1952, 207–213. \[$\spadesuit$ consider a (strange) least area problem yet without making very explicit the range of the geometry of the extremal function\]  P.R. Garabedian, [*Univalent functions and the Riemann mapping theorem*]{}, Proc. Amer. Math. Soc. 61 (1976), 242–244. \[$\spadesuit$ yet another new proof of RMT via an extremal problem and normal families $\spadesuit$ also cited for the reasons annotated after de Possel 1939 [@de-Possel_1939], namely the issue of avoiding the use of RMT in the extremal proof of PSM\]  L. Garding, [*The Dirichlet problem*]{}, Math. Intelligencer 2 (1979/80), 43–53. \[$\spadesuit$ historical survey of the Dirichlet problem with Poisson, Gauss 1839 [@Gauss_1839], its influence upon Thomson 1847 [@Thomson_1847], Stokes (credited for the maximum principle!?), Dirichlet, Riemann, Weierstrass, Schwarz, Neumann, Poincaré (balayage) and its modern ramification by Perron [@Perron_1923] and Radó-Riesz 1925 [@Rado-Riesz_1925], up to Frostman, Beurling-Deny\]  J. Garnett, [*Positive length but zero analytic capacity*]{}, Proc. Amer. Math. Soc. 24 (1970), 696–699. \[$\spadesuit$ simplifies the example of Vitushkin 1957 [@Vitushkin_1959] by taking advantage of the homogeneity of the compactum which is a simple planar Cantor set obtained by keeping only the 4 corner squares of a subdivision of the unit-square in $4\times 4$ congruent subsquares, and iterating ad infinitum $\spadesuit$ compare Murai 1987 [@Murai_1987] for another direct strategy (via Garabedian instead of Ahlfors) which is supposed to give more insight about the general problem\]  J. Garnett, [*Analytic capacity and measures*]{}, Lecture Notes in Math. 297, Springer, Berlin, 1972, 138 pp. \[$\spadesuit$ p.18, Ahlfors function $\spadesuit$ p.36, Denjoy conjecture (cf. for its resolution Marshall [@Marshall_1978?] via Calderón mostly)\] 78  J.B. Garnett, [*Bounded analytic functions*]{}, Pure and Appl. Math. 96, Academic Press, New York, 1981. \[$\spadesuit$ includes proofs of the corona theorem\]  J. Garnett, J. Verdera, [*Analytic capacity, bilipschitz mappings and Cantor sets*]{}, Math. Res. Lett. 10 (2003), 515–522. \[$\spadesuit$\]  A.M. Garsia, [*Calculation of conformal parameters for some imbedded Riemann surfaces*]{}, Pacific J. Math. 10 (1960), 121–165. \[$\spadesuit$\]  A.M. Garsia, [*Imbeddeding of Riemann surfaces by canal surfaces*]{}, Rend. Circ. Math. Palermo (2) 9 (1960), 313–333. \[$\spadesuit$\]  A.M. Garsia, E. Rodemich, [*An imbeddeding of Riemann surfaces of genus one*]{}, Pacific J. Math. 11 (1961), 193–204. \[$\spadesuit$ p.193: “Theorem. [*Any compact Riemann surface of genus one can be $C^\infty$ embedded in $3$-space.*]{}” $\spadesuit$ inspiration=Teichmüller 1944 [@Teichmueller_1944-Beweis-der-analytischen-Abhaengigkeit] and Ahlfors 1953/54 [@Ahlfors_1953/54-On-quasiconformal-mappings] $\spadesuit$ extension of the result in the next entry Garsia 1961 [@Garsia_1961]\]  A.M. Garsia, [*An imbedding of closed Riemann surfaces in Euclidean space*]{}, Comment. Math. Helv. 35 (1961), 93–110. \[$\spadesuit$ yet another brilliant student of Loewner; it is shown that any closed Riemann surface admits a conformal model in Euclid’s $3$-space $E^3$. \[10.12.12\] Upon taking taking the Schottky double, the same assertion holds true for bordered Riemann surfaces, and this is perhaps enough when $E^3$ is replaced by the more generous $E^4$ to settle the Forstnerič-Wold 2009 [@Forstneric-Wold_2009] desideratum that any compact bordered Riemann surface embeds holomorphically in ${\Bbb C}^2$. \[11.12.12\] Warning: not at all enough for the image is merely a smooth surface, and not a complex analytic curve $\spadesuit$ p.94: “The main result of the present paper is a proof that there exists in Euclidean space a conformally equivalent $C^{\infty}$ model for every compact Riemann surface of genus $g\ge 2$.” Compare also Rüedy 1968 [@Ruedy_1968] $\spadesuit$ “The methods that we have followed are essentially an extension of those in \[9\]. However, here certain devices introduced by J. Nash in \[13\], together with some results of L. Ahlfors \[2\] and L. Bers \[3\] on spaces of Riemann surfaces are quite crucial…” $\spadesuit$ to be fair the main technique permitting the breakthrough on this almost centennial problem conjectured by Klein (realizability of all Riemann surfaces as classical surface in $E^3$) is primarily Teichmüller theory, especially the 1944 paper [@Teichmueller_1944-Beweis-der-analytischen-Abhaengigkeit]. $\spadesuit$ \[19.12.12\] Garsia’s result can be given the following metaphoric interpretation (for single people having the Riemann($\approx$ woman) surface) as sole sentimental partner during their whole life, e.g. Koebe who never married): in the vicinity of any surface embedded in Euclid’s $3$-space $E^3$ one can realize any conformal structure via small variations confined to the normal bundle of the initial surface. This holds true for arbitrarily small thicknesses $\varepsilon$ of the tubular neighborhood. Metaphorically, this amounts to say that if the Riemann surface becomes a woman surface (materialized by the skin of some naked woman) then a minim variation of the skin permits to explore all other (women) surfaces by epidermic bubbling, alas generically akin to a cellulite formation. $\spadesuit$ This metaphor seems again to say something on the Forstnerič-Wold 2009 [@Forstneric-Wold_2009] desideratum. First we know (from Černe-Forstnerič 2002 [@Cerne-Forstneric_2002]) that any topological type of bordered surface contains a representative holomorphically embedded in ${\Bbb C}^2$. Applying the high-dimensional version of Garsia (due to Ko 1989 [@Ko_1989-compact], plus subsequent articles) we can realize all Riemann surfaces within a normal tubular neighborhood via an (infimal) normal variation. This is akin to a cellulite bubbling, alas destroying a priori the holomorphic character of the initial model. However it is not to be excluded that better controlled vibrations of the pudding[^131] permit to explore the full moduli space.\]  A.M. Garsia, [*On the conformal type of algebraic surfaces in euclidean space*]{}, Comment. Math. Helv. 37 (1962–63), 49–60. \[$\spadesuit$ “It has been an open question for some time whether or not the classical (i.e. $C^2$ surfaces) of Euclidean space (3-dimensional) exhaust all possible conformal types. In the non compact case the question is still open. \[UPDATE: Rüedy 1971 [@Ruedy_1971]\] In the compact case it can be shown (see \[1\]=Garsia-Rodemich 1961 [@Garsia-Rodemich_1961] and \[2\]=Garsia 1961 [@Garsia_1961]) that among the $C^\infty$ surfaces of Euclidean space there are surfaces conformally equivalent to any given compact Riemann surface.—In this paper we are to improve the results in \[1\] and \[2\]. It will be shown that for any given compact Riemann surface conformally equivalent models can be found among the algebraic surfaces of ordinary space. Here by “algebraic surface” we mean a surface satisfying an equation of the type $F(x,y,z)=0$, where $F(x,y,z)$ is a real polynomial in its argument.—\[...\] it is not known whether or not the affine images of the tori of revolution contain all conformal types of surfaces of genus one.—Perhaps it should be noted that the ease with which the results in \[2\] and specially those of this paper are obtained illustrate once more the power of the Teichmüller results on quasiconformal mappings and the usefulness of the concept of Teichmüller space for the study of families of compact Riemann surfaces. $\spadesuit$ of course in view of Garsia’s result one would like to bound the degree of the representing algebraic surface..., for more on this cf. Pinkall 1985 [@Pinkall_1985]\]  C.F. Gauss, 1811 (unpublished) correspondence with F.W. Bessel. \[$\spadesuit$ complex integration and the formula $\frac{1}{2\pi i} \int_l \frac{dz}{z}=1$, where $l$ is a circle enclosing the origin\]  C.F. Gauss, Allgemeine Auflösung der Aufgabe: die Theile einer gegebnen Fläche auf einer andern gegebnen Fläche so abzubilden, dass die Abbildung dem Abgebildeten in den kleinsten Theilen ähnlich wird. Als Beantwortung der von der königlichen Societät der Wissenschaften in Copenhagen für 1822 aufgegebnen Preisfrage, in: [*Schumacher’s Astronomische Abhandlungen*]{}, Drittes Heft, pp. 1–30, Altona 1825. (Also in: Werke, Bd.4, 189–216.) \[$\spadesuit$ This is probably the only record in print which may be regarded as a weak (very local) forerunner of the RMT. This text was of course known to Riemann, while adumbrating the conformal plasticity of 2D-mappings $\spadesuit$ in fact this Gauss text 1822/25 is the pre-big-bang, for it is the only reference cited in Riemann’s Thesis 1851 [@Riemann_1851], who however had several other inspirators like Dirichlet, Cauchy, etc. $\spadesuit$ some antecedents of this Gauss work is that by Lagrange 1779 [@Lagrange_1779] involved with a cartography problem, yet failing to prove, as Gauss do (in op.cit.), that locally any surface is conformally flat\]  C.F. Gauss, Disquisitiones generales circa superficies curvas, 1827. \[$\spadesuit$ concept of Gaussian (total) curvature, theorema egregium (the curvature $K$ is isometry-invariant, e.g. under bending), etc.\]  C.F. Gauss, [*Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnisse des Quadrates der Entfernung wirkenden Anziehungs- und Abstossungs-Kräfte*]{}, Magnetischer Verein (1839). Werke vol. 5, 195–242. \[$\spadesuit$ a forerunner of the Dirichlet principle. This text was known to Riemann $\spadesuit$ this Gauss work is supposed to lack in rigor, yet encompass the substance of the all potential theory (compare Brelot 1952 [@Brelot_1952] for a modern appreciation)\]  P.M. Gauthier, M. Goldstein, [*From local to global properties of subharmonic functions on Green spaces*]{}, J. London Math. Soc. (2) 16 (1977), 458–466. \[$\spadesuit$ p.465, includes the following application of the Ahlfors function. Let $\overline{\Omega}$ be a compact bordered Riemann surface with interior $\Omega$ and contour $C=\partial \overline{\Omega}$. Given $f\colon C \to \overline{\Bbb R}=[-\infty,+\infty]$ an extended real-valued continuous function, one says that $f$ is Dirichlet soluble if it continuously extends to $\overline{\Omega}$ so that its restriction to the interior $\Omega$ is harmonic. In this case, $f^{-1}(+\infty)$ is a closed set of HMZ(=harmonic measure zero). Now the authors shows the converse statement. Indeed, given $E\subset C$ closed and of HMZ, its image under an Ahlfors function (cf. Ahlfors 1950 [@Ahlfors_1950]) $F\colon \overline{\Omega} \to \overline{\Delta}$ is a subset of the circumference $S^1=\partial\overline{\Delta}$ of measure zero (Sard required?). According to Fatou 1906 [@Fatou_1906] any null-set of the circle occurs as $u_1^{-1}(\infty)$ for a continuous function $u_1\colon \overline{\Delta}\to \overline{{\Bbb R}}$ harmonic in the interior. The composed map $u_1\circ F$ has the desired properties $\spadesuit$ note however that the trick of the Ahlfors function seems not well suited for reducing the Dirichlet problem (even with non-extended boundary values) on a compact bordered Riemann surface to the case of the disc where it is soluble via the Poisson integral (albeit this may have been a partial intention in Bieberbach 1925 [@Bieberbach_1915])\]  W.-D. Geyer, [*Ein algebraischer Beweis des Satzes von Weichold über reelle algebraische Funktionenkörper*]{}, In: [*Algebraische Zahlentheorie*]{} (Ber. Tagung Math. Forschungsinst. Oberwolfach, 1964), 83–98. \[$\spadesuit$ includes a new proof of the theorem of Witt 1934 [@Witt_1934]\]  W.-D. Geyer, G. Martens, [*Überlagerungen berandeter Kleinscher Flächen*]{}, Math. Ann. 228 (1977), 101–111. 50 \[$\clubsuit$ after Alling-Greenleaf 1969 [@Alling-Greenleaf_1969], Ahlfors 1950 [@Ahlfors_1950] is also interpreted in terms of Klein’s orthosymmetric real curves, specifically p.106: “Gewisserma[ß]{}en als Umkehrung von a) ist das resultat von Ahlfors (\[1\], §4) anzusehen, wonach jede Kleinsche Fäche vom Typ $+(g,r)$ mit $r>0$ eine $(g+1)$-blättrige verzweigte Überlagerung der zur reellen projectiven Geraden ${\Bbb P}_1$ gehörenden Kleinschen Fläche $\overline{\Bbb C}/\sigma$ (=Riemannsche Zahlenhalbkugel) ist.” $\spadesuit$ p.101: “Seit Klein \[6, 12\] zieht man zum Studium reeller algebraischer Funktionenkörper $F$ einer Variablen mit Erfolg die zur Komplexifizierung von $F$ gehörige Riemannsche Fläche, versehen mit einer antiholomorphen Involution $a$, heran, oder auch die Kleinsche Fläche \[…\].—Unter den algebraischen Körpererweiterungen $E\vert F$ gibt es gewisse, durch ihr Realitätsverhalten ausgezeichnete Typen, die zuerst von Knight in \[7\](=1969 [@Knight_1969]) betrachtet wurden. Wir nennen $E\vert F$ total reell, wenn über reellen Stellen von $F$ nur reelle Stellen von $E$ liegen. Da die reellen Stellen von $F$ auf $\frak R$ eine disjunkte Vereinigung endlich vieler Kreise $Z1, \dots, Z_r$ bilden, induziert eine total reelle Erweiterung $E\vert F$ (unverzweigte) Überlagerungen der $Z_i$.” $\clubsuit$ p.102: “Die Kleinsche Fläche $\frak K=\frak R / \sigma$ is ebenfalls kompakt und zusammenhängend; sie ist genau dann orientierbar wenn $\frak R-\cal C ({\Bbb R})$ unzusammenhängend ist \[3,6\](=Alling-Greenleaf 1971 [@Alling-Greenleaf_1971], Klein 1882 [@Klein_1882]). Eine algebraische Kennzeichnung der Orientierbarkeit gab Ahlfors in \[1\](=1950 [@Ahlfors_1950]): $\frak K$ ist genau dann orientierbar, wenn es eine Funktion $f\in F$ gibt, die nur auf $\cal C ({\Bbb R})$ reelle Werte annimt.” $\spadesuit$ p.103: “Ein Morphismus $\varphi\colon \cal B\to \cal C$ reeller Kurven hei[ß]{}t [*total reell*]{}, wenn $\varphi^{-1} {\cal C} ({\Bbb R})={\cal B} ({\Bbb R})$ ist. Dann ist also eine Erweiterung $E\vert F$ reeller Funktionenkörper total reell, wenn jede reelle Stelle von $F$ nur reelle Fortsetzungen hat, und ein Morphismus Kleinscher Flächen is total reell, wenn der Rand respektiert wird, d.h. nur Randpunkte auf Randpunkte abgebildet werden.”\]  W.-D. Geyer, [*Reelle algebraische Funktionen mit vorgegeben Null- und Polstellen*]{}, Manuscripta Math. 22 (1977), 87–103. \[$\spadesuit$ p.91: “Ein Morphismus $\varphi\colon Y\to X$ reeller Kurven hei[ß]{}e total reell, wenn $\varphi^{-1} X(K)=Y(K)$ ist, d.h. wenn alle reellen Stellen von $F=K(X)$ nur relle Forsetzungen in $E=K(Y)$ haben.”\]  P. Gilmer, [*Algebraic curves in $RP^1 \times RP^1$*]{}, Proc. Amer. Math. Soc. 113 (1991), 47–52. \[$\spadesuit$ switching the Hilbert problem to ${\Bbb P}^1\times {\Bbb P}^1$\]  P. Gilmer, [*Real algebraic curves and link cobordism*]{}, Pacific J. Math. 153 (1992), 31–69. \[$\spadesuit$ promise (in the next entry Gilmer 1996 [@Gilmer_1996-Link-cobordismII]) a new derivation of the Gudkov-Rohlin congruence for $M$-curves, as well as the related congruence for $(M-1)$-curves\]  P. Gilmer, [*Real algebraic curves and link cobordism, II*]{}, in: Topology of Real Algebraic Varieties and Related Topics, ed. by V. Kharlamov et al., Amer. Math. Soc. Transl. Ser. 2 173, Amer. Math. Soc.., Providence, Ri, 1996, 73–84. \[$\spadesuit$ new derivation of the Gudkov-Rohlin congruence for $M$-curves, as well as the related congruence for $(M-1)$-curves\]  A. Girouard, I. Polterovich, [*Steklov eigenvalues*]{}, arXiv (2012). 50 \[$\clubsuit$ extension of Fraser-Schoen 2011 [@Fraser-Schoen_2011] to higher eigenvalues\]  A.M. Gleason, [*Function algebras*]{}, Seminar on analytic functions, Institute for Advanced Study, Princeton, N.J., 1957. \[$\spadesuit$ where the Gleason parts are defined as the equivalence classes of the following relation $\spadesuit$ for an arbitrary function algebra $A$ on a compact metrizable space $X$, let $M$ be its maximal ideal space and $S$ its Shilov boundary. Realizing $A$ as a function algebra on $M$, two points $m_1, m_2 \in M $ are (Gleason) equivalent if $\sup \{ \vert f(m_1)\vert: f(m_2)=0, \| f\| \le 1 \}<1$. $\spadesuit$ for a connection with the Ahlfors map cf. e.g. O’Neill-Wermer 1968 [@O'Neill-Wermer_1968]\]  J. Globevnik, B. Stens[ø]{}nes, [*Holomorphic embeddings of planar domains in ${\Bbb C}^2$*]{}, Math. Ann. 303 (1995), 579–597. \[$\spadesuit$ it is show that any plane domain of finite connectivity without point-like (punktförmig) boundaries has a proper holomorphic embedding in the affine complex plane $\spadesuit$ for a wide extension to infinite connectivity, cf. Forstnerič-Wold 2012 [@Forstneric-Wold_2012]\]  G.M. Golusin, [*Auflösung einiger ebener Grundaufgaben der mathematischen Physik im Fall der Laplaceschen Gleichung und mehrfach zusammenhängender Gebiete, die durch Kreise begrenzt sind*]{}, (Russian, German Summary) Mat. Sb. 41 (1934), 246–276. 78 \[$\clubsuit$ Seidel’s summary: a harmonic function $U$ of two real variables is sought exterior to the circles $C_1, \dots, C_n$, with $U(\infty)$ finite, which on $C_k$ assumes preassigned continuous values $f_k$. The problem is reduced to the solution of a finite system of functional equations which are solved by successive approximations. The method is applied to solve Neumann’s problem and other similar problems for Laplace’s equation and for regions of the above type. The Green’s functions of such regions and the functions which map them on slit planes are determined\]$\bigstar$  G.M. Golusin, [*Sur la représentation conforme*]{}, (French, Russian Summary) Mat. Sb.=Rec. Math. 1 (43) (1936), 273–282. 78 \[$\clubsuit$ p.273, Lemme 1 gives another proof of a basic lemma about areas of rings under conformal maps $\spadesuit$ Pólya-Szegö 1925 are cited, but it should go back to Carleman 1918 [@Carleman_1918] $\spadesuit$ for the relevance of this lemma to the least area problem of multi-connected under schlicht maps see Gaier 1977 [@Gaier_1977-Roumaine] where a dissection process shows that a solution (non-unique!) to this problem effects a representation upon a circular slit disc $\spadesuit$ incidentally the proof of Thm 1, p.274 looks very akin to Gaier’s argument of 1977 [@Gaier_1977-Roumaine]\]  G.M. Golusin, [*Iterationsprozesse für konforme Abbildungen mehrfach zusammenhängender Bereiche*]{}, (Russian, German Summary) Mat. Sb. N.S. 6 (48) (1939), 377–382. 78 \[$\clubsuit$ Iterative methods are established by means of which a schlicht conformal map of regions of finite connectivity on some canonical domains is reduced to a sequence of conformal maps of simply-connected regions\]$\bigstar$  G.M. Golusin, [*Geometrische Funktionentheorie*]{}, Übersetzung aus dem Russischen. Hochschulbücher f. Math. Bd. 31, Berlin, VEB Deutscher Verlag d. Wiss., 1957. English transl.: Geometric theory of functions of a complex variable, 1969. (Russian original published in 1952.) 60, 78 \[$\spadesuit$ p.240–4, proof of a circle map in the schlicht(artig) case following Grunsky 1937–41 (potential-theoretic) $\spadesuit$ p.412–8, the extremal approach is presented (Ahlfors 1947 [@Ahlfors_1947] is cited, and ref. to Grunsky 1940–42 [@Grunsky_1940; @Grunsky_1942] where added by the German editors (probably Grunsky himself) $\spadesuit$ p.200–217 present a proof of Koebe’s KNP via the continuity method (approached via Brouwer’s invariance of the domain)\]  T. Gouma, [*Ahlfors functions on non-planar Riemann surfaces whose double are hyperelliptic*]{}, J. Math. Soc. Japan 50 (1998), 685–695. 50 \[$\clubsuit$ detailed study of the degrees of the Ahlfors map in the hyperelliptic case $\clubsuit$ a complement (tour de force) is to be found in Yamada 2001 [@Yamada_2001] $\spadesuit$ for an application to proper holomorphic embeddings in ${\Bbb C}^2$, cf. Černe-Forstnerič 2002 [@Cerne-Forstneric_2002] $\spadesuit$ Köditz’s summary (MathReviews): “Let $R$ be a finite bounded \[=bordered\] Riemann surface with genus $p$ and $q$ contours and let $P$ be a point in $R$. The author studies the set of Ahlfors functions on $R$. These functions are the extremal functions obtained by maximizing the derivative $\vert f'(P) \vert$ (in some local parameter at $P$) in the class of holomorphic functions on $R$ bounded by one. Each Ahlfors function has modulus $1$ on the boundary of $R$ and gives a complete[^132] covering of the unit disk. It is known that the degree $N$ of any Ahlfors function satisfies $q\le N \le 2p+q$ (Ahlfors, 1950=[@Ahlfors_1950]). The set of degrees $N(R)$ of Ahlfors functions on a given Riemann surface $R$ is not well known. In this paper, the author deals with Ahlfors functions on non-planar Riemann surfaces whose doubles are hyperelliptic. Among others, examples for such Riemann surfaces with $N(R)=\{2, 2p+q\}$ are constructed.”\]  W.H. Gottschalk, [*Conformal mapping of abstract Riemann surfaces*]{}, Published by the author, Univ. of Pennsylvania, Philadelphia, 1949, 77p. $\bigstar$ 60  L.B. Graĭfer, S.Ja. Gusman, V.V. Dumkin, [*An extremal problem for forms with singularities on Riemannian manifolds*]{}, Perm. Gos. Univ. Učen. Zap. 218 (1969), 47–52. \[$\spadesuit$ from MathReview (by Kiremidjian): “In 1950, Ahlfors showed that a number of extremal problems on compact subregions of open Riemann surfaces could be solved by studying the class of Schottky differentials \[Ahlfors 1950 [@Ahlfors_1950]; errata, MR 13, p.1138\]. In recent years, certain aspects of Ahlfors’ work were investigated in the case of $n$-dimensional orientable differentiable manifolds \[the second author, 1966\]. I the present paper, the authors study the class of Schottky-Ahlfors forms with singularities.” $\spadesuit$ so those cited works constitute a rare but foolhardy attempt to extend Ahlfors’ theory to higher dimensions $\spadesuit$ perhaps one is prompted by the (naive!) question if one could formulate a theory able to (re)prove the famous 3D-conjecture of Poincaré-Perelman in its bordered incarnation: any compact bordered $3$-manifold is topologically equivalent to the $3$-ball, provided it is contractible or simply-connected and bounded by the $2$-sphere (of course the modest antecedent being the fact that one can prove the Schoenflies theorem via RMT thanks to Osgood/Carathéodory) $\spadesuit$ the Ahlfors function $W^3\to \Delta^3$ has then perhaps to be a harmonic map with maximal distortion at some basepoint, and if the contours are surfaces distinct from the sphere then there is no chance to have a covering along the boundary, but otherwise e.g. for $W^3$ the interspace of two concentric spheres it is not difficult to visualize a 3D-avatar of the Ahlfors map (just by taking the revolution of a map from a annulus to the disc, cf. our Fig.\[Annulus:fig\])\]  J. Gray, [*On the history of the Riemann mapping theorem*]{}, Rend. Circ. Mat. Palermo (2) 34 (1994), 47–94. \[$\spadesuit$ from Riemann to Koebe’s area, through Osgood, etc.\]  J. Gray, M. Micallef, [*The work of Jesse Douglas on minimal surfaces*]{}, Bull. Amer. Math. Soc. (N.S.) 45 (2008), 293–302. \[$\spadesuit$ contains several critiques (mostly raised by Tromba 1983 [@Tromba_1983-PREPRINT]) about the rigor of the work of Douglas/Courant on the Plateau problem, especially when it comes to higher topological structure\]  H. Grassmann, [*Die lineare Ausdehnungslehre*]{}, 1844. \[$\spadesuit$ summarized in Briekorn-Knörrer 1981/86 [@Brieskorn-Knörrer_1981/1986 p.122]: “A very important contribution to the development of the manifold concept was made by H. Grassmann in \[t\]his work, in which he spoke of $n$-tuply extended manifolds for the first time and developped, among other things, modern $n$-dimensional analytic geometry anfd linear algebra, in which the mathematical theory structure is worked out in a coordinated-free way, allowing the simplest treatment of problems in $n$-dimensional geometry, and in other fields as well.”\]  G. Green, [*An essay on the application of mathematical analysis to the theories of electricity and magnetism*]{}. Printed for the Author by Whellhouse T. Nottingham, 1828, 72 pp. Also in: Mathematical Papers of George Green, Chelsea Publishing Co., 1970, 1–115; and reprinted in three parts in J. Reine Angew. Math. 39 (1850), 73–89; 44 (1852), 356–374; 47 (1854), 161–221. \[$\spadesuit$ this Crelle reprint was organized by W. Thomson $\spadesuit$ contains a form of the Dirichlet principle, presumably the first ever put in print $\spadesuit$ as to the connection with our problem of the Ahlfors map, the connection is evident and implicit in Ahlfors 1950 paper [@Ahlfors_1950], albeit the latter employs a variant of the Green’s function with “dipole” singularity placed at a boundary point\]  P. Griffiths, J. Harris, [*Principles of Algebraic Geometry*]{}. Wiley, New York, 1978, 813 pp.; Wiley Classic Library edition, 1994; Russian transl., Vol.1, Mir, Moscow, 1982. \[$\spadesuit$ contains both an heuristic and formal proof of the $[\frac{g+3}{2}]$ gonality of closed Riemann surfaces of genus $g$, a result predicted since Riemann 1857 [@Riemann_1857] but only firmly validated in the modern era through the work of Meis 1960 [@Meis_1960] $\spadesuit$ see especially p.358, (special linear systems) and the proof presented is presumably quite close (??) to that of Kempf 1971 [@Kempf_1971]\]  P. Griffiths, J. Harris, [*On the variety of special linear systems on a general algebraic curve*]{}, Duke Math. J. 47 (1980), 233–272. \[$\spadesuit$ p.236/7 gives a parameter count argument (via Riemann-Hurwitz and Riemann’s $3g-3$ moduli) to show that “a general curve $C$ of genus $g\ge 2$ cannot be expressed as a multiple cover of any curve $C'$ of genus $g'\ge 1$.” $\spadesuit$ this can be employed to show that the avatar of the Ahlfors map with range not a disc but a membrane of higher topological complexity fails generally to share the property of the usual circle-valued Ahlfors map of taking the boundary to the boundary\]  M. Gromov, V.A. Rohlin, [*Embeddings and immersions in Riemannian geometry*]{}, Russian Math. Surv. 25 (1969), 1–57. \[$\spadesuit$ p.14: “In Appendix 4 we show that the real projective plane with a metric of positive curvature, in particular, the elliptic plane, cannot be isometrically $C^2$-embedded in ${\Bbb R}^4$.”\]  M. Gromov, [*Filling Riemannian manifolds*]{}, J. Differential Geom. 18 (1983), 1–147. \[$\spadesuit$ present a modernized proof of the Loewner-Pu isosystolic inequality, by quoting Jenkins, hence indirectly Grötzsch, so back to Koebe–Poincaré, genealogically. Of course the uniformization required for Loewner (torus) and Pu (projective plane) are of a simpler nature, (Abel and Riemann, Schwarz resp.).\]  M. Gromov, [*Spaces and questions*]{}, Preprint (1999). T.H. Gronwall, [*Some remarks on conformal representation*]{}, Ann. of Math. (2) (1914/15), 72–76. \[$\spadesuit$ probably one of the first usage of the area-principle, cf. also Bieberbach 1914 [@Bieberbach_1914], Bieberbach 1916 [@Bieberbach_1916] and Faber 1916 [@Faber_1916]\]  B. Gross, [*Real algebraic curves and their Jacobians*]{}, preprint, 1979. \[$\spadesuit$ this is cited in Jaffee 1980[@Jaffee_1980], yet probably never appeared as a such but might have been phagocytosed in the next entry Gross-Harris 1981 [@Gross-Harris_1981]\]  B.H. Gross, J. Harris, [*Real algebraic curves*]{}, Ann. Sci. École Norm. Sup. (4) 14 (1981), 157–182. \[$\spadesuit$ modern account of Klein’s theory of real curves with many innovative ideas and viewpoints $\spadesuit$ the question posed on p.177 about the number of ovals for dividing plane smooth curves easily follows from the ideas of Rohlin[^133] 1974/75 [@Rohlin_1974/75], 1978 [@Rohlin_1978], compare Gabard 2000 [@Gabard_2000] for a detailed discussion\]  A. Grothendieck, [*Techniques de construction en géométrie analytique*]{}, Sém. H. Cartan 1960/61, Exp.7, 9–17, Paris, 1962. \[$\spadesuit$ Teichmüller theory à la Grothendieck\]  A. Grothendieck, [*Techniques de construction et théorèmes d’existence en géométrie algébrique IV. Les schéms de Hilbert.*]{} Sém. Bourbaki 221, 1960/61. \[$\spadesuit$\]  A. Grothendieck, [*Techniques de construction et théorèmes d’existence en géométrie algébrique V. Les schéms de Picard.*]{} Sém. Bourbaki 232, 1960/61. \[$\spadesuit$\]  A. Grothendieck, [*Esquisse d’un programme*]{}, 1984; reproduced in: L. Schneps and P. Lochak (eds), Geometric Galois Actions I. Around Grothendieck’s Esquisse d’un programme, London Math. Soc. Lecture Note Ser. 242, Cambridge Univ. Press, 1997, 5–48. \[$\spadesuit$ Teichmüller, Thurston, legos, etc. plus the Belyi-Grothendieck theorem that a closed Riemann surface is defined over $\Qbar$ iff it has only 3 ramifications over the sphere\]  H. Grötzsch, [*Über einige Extremalprobleme der konformen Abbildung* ]{}, Ber. Verh. Sächs. Akad. Wiss. Leipzig 80 (1928), 497–502. 60, 78 \[$\spadesuit$ credited by Nehari 1953 [@Nehari_1953-Inequalities] for the solution of maximizing the derivative (distortion) at a given point of a multi-connected domain among schlicht functions bounded-by-one (extremals mapping upon a circular slit disc)\]  H. Grötzsch, [*Über konforme Abbildung unendlich vielfach zusammenhängender schlichter Bereiche mit endlich vielen Häufungsrandkomponenten*]{}, Ber. Verh. Sächs. Akad. Wiss. Leipzig (1929), 51–86. 60, 78 \[$\spadesuit$ first proof of the circular slit disc mapping in infinite connectivity, see also Reich-Warschawski 1960 [@Reich-Warschawski_1960] for more subsequent references\]  H. Grötzsch, [*Das Kreisbogenschlitztheorem der konformen Abbildung schlichter Bereiche*]{}, Ber. Verh. Sächs. Akad. Wiss. Leipzig (1931), 238–253. 60, 78 \[$\spadesuit$ another proof of the circular slit disc mapping in infinite connectivity, compare Grötzsch 1929 [@Groetzsch_1929]\]  H. Grunsky, [*Neue Abschätzungen zur konformen Abbildung ein- und mehrfach zusammenhängender Bereiche*]{}, (Diss.) Schriften Math. Semin., Inst. angew. Math. Univ. Berlin 1 (1932), 95–140. \[$\spadesuit$ \[26.07.12\] p.140, Grunsky announces (without proofs) the result that a suitable combination $c(\frak x (\zeta;z) - \frak y (\zeta;z))$ of the horizontal $\frak x$ (resp. vertical $\frak y$) \[those being the fraktur letters for $x$ resp. $y$!\] slit-maps affords the solution to the problem of least area among all analytic functions normed by $f'(z)=1$ $\spadesuit$ on reading the rest of the paper it seems that the image might fail to be a disc, compare esp. p.135 where a similar least area problem is handled $\spadesuit$ this topic is addressed again in Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949] and Nehari 1952 [@Nehari_1952-BOOK]\] 60, 78  H. Grunsky, [*Über die konforme Abbildung mehrfach zusammenhängender Bereiche auf mehrblättrige Kreise*]{}, Sitzungsber. Preu[ß]{}. Akad. (1937), 40–46. 60, 78 \[$\clubsuit$ new potential-theoretic proof of the circle map for domains\]  H. Grunsky, [*Über die konforme Abbildung mehrfach zusammenhängender Bereiche aud mehrblättrige Kreise, II*]{}, Abh. Preu[ß]{}. Akad. Wiss. Math.-nat. Kl. 11 (1941), 1–8. 60, 78 \[$\clubsuit$ idem as the previous item\]  H. Grunsky, [*Eindeutige beschränkte Funktionen in mehrfach zusammenhängenden Gebieten I*]{}, Jahresb. d. Deutsch. Math.-ver. 50 (1940), 230–255. 78 \[$\clubsuit$ extremal-problem description of circle maps for domains\]  H. Grunsky, [*Eindeutige beschränkte Funktionen in mehrfach zusammenhängenden Gebieten II*]{}, Jahresb. d. Deutsch. Math.-ver. 52 (1942), 118–132. 78 \[$\clubsuit$ sequel of the previous item\]  H. Grunsky, [*Zur Funktionentheorie in mehrfach zusammenhängenden Gebieten*]{}, Ber. Mathematikertagung Tübingen (1946), 68–69; in Coll. Papers, 245–6. 78  H. Grunsky, [*Nachtrag zu meinen Arbeiten über “Eindeutige beschränkte Funktionen in mehrfach zusammenhängenden Gebieten”*]{}, Math. Z. 52 (1950), 852. 78  H. Grunsky, [*Über die Fortsetzung eines auf einer berandeten Riemannschen Fläche erklärten meromorphen Differentials*]{}, Math. Nachr. 39 (1969), 87–96. \[$\clubsuit$ one of the rare work by Grunsky concerned with bordered surfaces, yet it does not seem to reprove the existence of a circle map à la Ahlfors\]  H. Grunsky, [*Lectures on Theory of Functions in Multiply Connected Domains*]{}, Studia Mathematika, Skript 4, Vandenhoeck and Ruprecht in Göttingen, 1978. \[$\clubsuit$ all inclusive account but focusing to the case of domains (no Riemann surfaces)\]  $\bigstar$ Gudkov (1918–1992), student of Andronov and Petrovskii. Famous for his sextic solution to Hilbert’s 16th problem (1 st part thereof on the mutual arrangements of ovals), and in particular for its refutation of Hilbert’s conjecture that there is only two possible configurations in degree $6$ (his own and the earlier one of Harnack). Influential over Arnold 1971 [@Arnold_1971/72], hence also over Rohlin, etc. Gudkov’s students includes Utkin, Polotovskii, Shustin, Korchagin. D.A. Gudkov, [*Establishing all existing types of non-singular plane algebraic curves of the sixth order with real coefficients*]{}, Ph.D. Dissertation, Gorki, 1952. \[$\spadesuit$ the title sounds slightly overambitious, compare comment in the next entry Gudkov 1954 [@Gudkov_1954]\]  D.A. Gudkov, [*The complete topological classification of non-singular real algebraic curves of the sixth order in the projective plane*]{}, Dokl. Akad. Nauk SSSR 98 (1954), 521–524. \[$\spadesuit$ albeit this paper contains some mistakes (too prohibitive), it must contain the first serious prohibitions via the Hilbert-Rohn method consolidated by Andronov-Pontrjagin theory of roughness (structural stability) $\spadesuit$ thus the method is still of interest, and its charming power is perhaps only eclipsed by more topological variants of the Arnold-Rohlin era, e.g. Arnold 1971 [@Arnold_1971/72] or Rohlin 1972/72 [@Rohlin_1972/72-Proof-of-a-conj-of-Gudkov]\]  D.A. Gudkov, [*On certain questions in the topology of plane algebraic curves*]{}, Mat. Sb. (N.S.) 58 (1962), 95–127. \[$\spadesuit$ some overlap with Brusotti 1921 [@Brusotti_1921], yet an extension thereof to curves having “turning points” (=cusps)\]  D.A. Gudkov, [*On the topology of plane algebraic curves*]{}, Doctor’s Thesis, Gor’kii, 1969, 1–351. \[$\spadesuit$ first disproof of Hilbert’s conjecture about the nonexistence of the scheme $\frac{5}{1}5$. According to Polotovskii 1996 [@Polotovskii_1996-D-A-Gudkov]’s overview: “It is interesting to remark that the first proof of this fact in \[18\](=this entry) was extraordinarily complicated. It takes up $28$ pages of text, is a “pure existence proof”, and was obtained by means of a combination of the Hilbert-Rohn method with quadratic transformations. Shortly after D.A. Gudkov suggested significantly simpler [*constructions*]{} of curves having this scheme, see \[19\](=1971 [@Gudkov_1971-const-new-ser-M-curv]), \[21\], \[23\]. ”\]  D.A. Gudkov, [*Complete topological classification of the disposition of ovals of a sixth order curve in the projective plane*]{}, Gor’kov. Gos. Univ. Ucen. Zap. Vyp. 87 (1969), 118–153. \[$\spadesuit$ where Hilbert got corrected?!\]  D.A. Gudkov, G.A. Utkin, [*The topology of sixth-order curves and fourth-order surfaces*]{}, Gor’kov. Gos. Univ. Učen. Zap. Vyp. 87 (1969), 154–211; English transl., in: Nine papers on Hilbert’s 16th problem, Amer. Math Soc. Transl. (2) 112 (1978). \[$\spadesuit$ where Hilbert got corrected by Gudkov: disproof of one of one of Hilbert’s conjectures on the arrangement of ovals of plane $M$-sextic (that is Harnack-maximal). Gudkov corrected Hilbert’s conjecture by including the newly discovered so-called Gudkov curve and showed that only the trinity Harnack 1876 [@Harnack_1876], Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege] and Gudkov’s 1969 can exist, yielding a complete classification up to isotopy of $M$-sextic (when combined with the prohibition à la Hilbert-Rohn which Gudkov was the first to implement correctly in 1954 [@Gudkov_1954] upon combining the classical method with “roughness” à la Andronov-Pontrjagin) $\spadesuit$ more generally this work contains a complete classification of all 56 isotopy classes realized by plane sextic (hence a complete solution to Hilbert’s 16th problem), but the story does not end here (cf. e.g. Rohlin’s 1978 enhancement by complex characteristics à la Klein, hence concomitant with Ahlfors, cf. our discussion in Sec.\[Klein-Rohlin-conj:sec\]) $\spadesuit$ another subsequent step is Nikulin 1979/80 [@Nikulin_1979/80] stronger classification of $C_6$ up to rigid-isotopy showing the completeness of the Klein-Rohlin invariants (the proof uses the whole apparatus of the complex geometry of K3 surfaces)\]  D.A. Gudkov, [*Construction of a new series of $M$-curves*]{}, Dokl. Akad. Nauk SSSR 200 (1971), 1269–1272; English transl., Soviet Math. Dokl. 12 (1971), 1559–1563. \[$\spadesuit$ Gudkov construction is reproduced (at least in abridged form) in Gudkov 1974/74 [@Gudkov_1974/74], in A’Campo 1979 [@A'Campo_1979]\]$\bigstar$$\bigstar$$\bigstar$  D.A. Gudkov, [*Construction of a curve of degree $6$ of type $\frac{5}{1}5$*]{}, Izv. Vyssh. Uchebn. Zaved. Mat. 3 (1973), 28–36; English transl., Soviet Math. (Iz. VUZ) (1973). \[$\spadesuit$ simplification in the disproof of Hilbert’s conjecture on the arrangement of ovals of plane $M$-sextics (that is Harnack-maximal). Gudkov corrected Hilbert’s conjecture by including the newly discovered so-called Gudkov curve (this contribution appears first in Gudkov’s Doctor Thesis (1969 [@Gudkov_1969-Doctor's-Thesis]), yet in a much more sophisticated way\]  D.A. Gudkov, A.D. Krakhnov, [*On the periodicity of the Euler characteristic of real algebraic $(M-1)$-manifolds*]{}, Funkt. Anal. Prilozhen. 7 (1973), 15–19; English transl., Funct. Anal. Appl. 7 (1973), 98–102 \[$\spadesuit$ same result as in Kharlamov 1973 [@Kharlamov_1973/73], i.e. an obstruction on $(M-1)$-schemes, e.g. for plane curves of degree $2k$, which is independent of the Hilbert-Rohn-Gudkov geometric method\]  D.A. Gudkov, [*The topology of real projective algebraic varieties*]{}, Uspekhi Mat. Nauk 29 (1974), 3–79; English transl., Russian Math. Surveys 29 (1974), 1–79. \[$\spadesuit$ a masterpiece survey full of historical details and mathematical tricks $\spadesuit$ contains an extensive bibliography (157 entries) of early real algebraic geometry (in Germany, Italy and Russia), mostly in the spirit of Hilbert (by contrast to Klein’s more Riemannian approach) $\spadesuit$ p.2 and p.17 contain in my opinion a historical inaccuracy which imbued alas some of the subsequent literature (e.g. A’Campo 1979 [@A'Campo_1979 p.01], Jaffee 1980 [@Jaffee_1980 p.82]), namely Hurwitz 1891–92 is jointly credited for the intrinsic proof of Harnack’s inequality ($r\le g+1$), while it goes back of course to Klein 1876 [@Klein_1876] (and not only Klein 1892 lectures as cited by Gudkov)\]  D.A. Gudkov, [*On the topology of algebraic curves on a hyperboloid*]{}, Uspekhi Mat. Nauk 34 (1979), 26–32; English transl., Russian Math. Surveys 34 (1979), 27–35. \[$\spadesuit$ p.27: “Algebraic curves on a hyperboloid of one sheet have been studied for a long time. A start was made by Plücker and Chasles in the mid-nineteenth century (see \[1\], Ch.IV)=(Klein 1926 [@Klein_1926-Vorlesungen-über-die-Entwicklung]). In a fundamental article \[2\](=Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege]) Hilbert proved \[…\]”\]  D.A. Gudkov, [*Generalization of a theorem of Brusotti for curves on a surface of second order*]{}, Funkt. Anal. Prilozhen. 14 (1980), 20–24, 96; English transl., Funct. Anal. Appl. 14 (1980), 15–18. \[$\spadesuit$ p.15: “Brusotti \[1\](=1921 [@Brusotti_1921]) proved the following assertion in 1921.—[*Brusotti’s theorem.*]{} If all the singular points of a curve $F$ in the complex projective plane ${\Bbb C}P^2 (x_0:x_1:x_2)$ are simple double points, then the simplifications of these singular points are independent.—This means that by adding an arbitrarily small term of degree $m$ (with real coefficients) to the polynomial $F$, it is possible to get a curve $\Phi$ such that each real singular point of the curve $F$ either remains or else simplifies in one of the two possible ways (depending on our choice, cf. \[5\]), and each pair of imaginary conjugate singular points either remains or vanishes (depending on our choice).”\]  D.A. Gudkov, G.M. Polotovskii, [*Stratification of a space of fourth-order curves. Contiguity of strata*]{}, Uspekhi Mat. Nauk 42 (1987), 152. \[$\spadesuit$\]  D.A. Gudkov, [*Plane real projective quartic curves*]{}, in: Lecture Notes in Math. 1346, Springer, Berlin, 1988, 341–347. \[$\spadesuit$ cited in Shustin 1990/91 [@Shustin_1990/91-Geom-of-discr-alg-curve] for a complete description of the discriminant of quartics $\spadesuit$ presumably some overlap with the previous entry $\spadesuit$ is this description of Gudkov compatible with our crazy disconnection result in Sec.\[Disconnection-of-the-empty-locus:sec\]\]  D.A. Gudkov, [*N.I. Lobachevskii. Biographical enigmas*]{}, Nizhnii Novgorod, 1992 (monograph in print). \[$\spadesuit$\]  L. Guillou, A. Marin, [*Une extension d’un théorème de Rohlin sur la signature*]{}, C.R. Acad. Sci Paris Ser. A 285 (1977), 95–98. \[$\spadesuit$ useful in correcting Rohlin’s proof of the Gudkov hypothesis $p-n\equiv k^2 \pmod 8$ (compare Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000]) $\spadesuit$ this paper is an announcement and more details are to be found in the book Guillou-Marin 1986 [@Guillou-Marin_1986 p.97–118]\]  L. Guillou, A. Marin, [*A la recherche de la topologie perdue, I Du côté de chez Rohlin, II Le côté de Casson*]{}, Progress in Math. 62, Birkäuser, Boston, Basel, Stuttgart, 1986. \[$\spadesuit$ contains French translation of Rohlin’s ground-breaking works in low-dimensional differential topology, plus some of its applications to real algebraic geometry\]  R.C. Gunning, [*Lectures on Riemann surfaces*]{}. Princeton Acad. Press, Princeton, 1966. \[$\spadesuit$\] $\bigstar$  R.C. Gunning, R. Narasimhan, [*Immersion of open Riemann surfaces*]{}, Math. Ann. 174 (1967), 103–108. \[$\spadesuit$ no directly visible connection with Ahlfors 1950, but there must be some link in the long run\]  R.C. Gunning, [*Lectures on Riemann surfaces: Jacobi varieties*]{}, Princeton Univ. Press, Princeton, N.J., 1972, 189 pp. \[$\spadesuit$ new (essentially topological?) proof of Meis’ result upon the gonality of complex curves (=closed Riemann surfaces) $\spadesuit$ \[21.06.12\] the following extract of H.H. Martens’s review in MathReviews is capital for it brings the hope to gain a Teichmüller theoretic approach to the existence of circle maps with the best possible bounds (hence hinting how to recover Ahlfors and even Gabard 2006 [@Gabard_2006] by an analytic (or rather geometric!) approach competing seriously with the naive topological proof of the writer): “A pièce de résistance is served in the appendix in the form of a proof of the existence of functions of order $\le [\frac{1}{2}(g+3)]$ on any closed Riemann surface. This result was previously obtained by T. Meis 1960 [@Meis_1960] using Teichmüller space techniques, and it is a special case of the more general results of Kleiman-Laksov 1972 [@Kleiman-Laksov_1972] and Kempf 1971 [@Kempf_1971].”\] $\bigstar$$\bigstar$$\bigstar$  B. Gustafsson, [*Quadrature identities and the Schottky double*]{}, Acta Appl. Math. 1 (1983), 209–240. \[$\spadesuit$ \[13.10.12\] can the theory be extended to non-planar domains?\]  B. Gustafsson, [*Applications of half-order differentials on Riemann surfaces to quadrature identities for arc-length*]{}, J. Anal. Math. 49 (1987), 54–89. \[$\spadesuit$\]  A. Haas, [*Linearization and mappings onto pseudocircle domains*]{}, Trans. Amer. Math. Soc. 282 (1984), 415–429. \[$\spadesuit$ Koebe’s Kreisnormierungsprinzip for positive genus, uniqueness complement in Maskit 1989 [@Maskit_1989]\]  J. Hadamard, [*Sur le principe de Dirichlet*]{}, Bull. Soc Math. France (1906). \[$\spadesuit$ p.135 an example is given of a continuous function on the boundary of a domain such that none functions satisfying the boundary prescription has finite Dirichlet integral $\spadesuit$ a similar example was given in Prym 1871 [@Prym_1871], where a continuous function is given on the circle such that the harmonic function matching this boundary data (whose existence is derived by another procedure, e.g. the Poisson integral) has infinite Dirichlet integral $\spadesuit$ of course, heuristically any Prym’s boundary data must be of the Hadamard type (precisely by virtue of the just corrupted Dirichlet principle!): if the harmonic solution explodes any vulgar solution (hence less economical) must explode as well\]  J. Hadamard, [*Mémoire sur le problème d’analyse relatif à l’équilibre de plaques élastiques encastrées*]{}, Mémoires présentés par divers savants à l’Académie des Sciences 33 (1908), 128pp. \[$\spadesuit$ Discussion of the famous method, named after Hadamard, of variation of domains $\spadesuit$ further developed by Schiffer especially\]  G. Halphen, [*Mémoire sur la classification des courbes gauche algébriques*]{}, J. École Polytech. 52 (1882), 1–200. \[$\spadesuit$ sharing the price with M. Noether\]  R.S. Hamilton, [*The Ricci flow on surfaces*]{}, In: Mathematics and General Relativity (Santa Cruz, CA, 1986). Contemporary Mathematics 71, Amer. Math. Soc., Providence, 1988, 237–262. \[$\spadesuit$ uniformization of surfaces via the 2D-Ricci flow (at least in the compact case)\] $\bigstar$$\bigstar$$\bigstar$ M. Hara, M. Nakai, [*Corona theorem with bounds for finitely sheeted disks*]{}, Tôhoku Math. J. 37 (1985), 225–240. 50 \[$\clubsuit$ applies Ahlfors mapping in a quantitative fashion (making use of its degree in contrast to Alling 1964 [@Alling_1964]) $\clubsuit$ naive question (ca. Sept. 2011) can we improve the bounds by appealing instead to Gabard 2006 [@Gabard_2006]\]  A. Harnack, [*Ueber die Vieltheiligkeit der ebenen algebraischen Curven*]{}, Math. Ann. 10 (1876), 189–198. \[$\spadesuit$ a proof is given (via Bézout’s theorem) that a smooth plane real curve of order $m$ possesses at most $g+1=\frac{(m-1)(m-1)}{2}+1$ components (reellen Züge) and such Harnack-maximal curves are constructed for each degree via a method of small perturbation $\spadesuit$ as everybody knows a more intrinsic proof was given by Klein 1876 [@Klein_1876] by simply appealing to Riemann’s definition of the genus as the maximum number of retrosections not morcellating the surface $\spadesuit$ a more exotic derivation of the Harnack bound (using Riemann-Roch) is to be found in Enriques-Chisini 1915 [@Enriques-Chisini_1915-1918], whose argument actually supplies a proof of the so-called Bieberbach-Grunsky theorem (cf. Bieberbach 1925 [@Bieberbach_1925], Grunsky 1937 [@Grunsky_1937] and for instance A. Mori 1951 [@Mori_1951]) which is the planar version of the Ahlfors map\]  A. Harnack, [*Die Grundlagen der Theorie des logarithmischen Potentiales, und der eindeutigen Potentialfunktionen in der Ebene*]{}, Teubner, Leipzig, 1887.  R. Hardt, D. Sullivan, [*Variation of the Green function on Riemann surfaces and Whitney’s holomorphic stratification conjecture*]{}, Publ. Math. I.H.E.S. (1989), 115–138. \[$\spadesuit$ \[10.08.12\] the starting point of the paper (p.115) is a representation of a Riemann surface as a $k$-sheeted branched covering of the unit disc (denoted $B$) with branch point $a_1, \dots, a_l$ in $B_{1/2}$ (ball of radius one-half) $\spadesuit$ this situation resembles sufficiently to Ahlfors 1950 [@Ahlfors_1950] to ask if a precise connection can be made $\spadesuit$ of course one may notice that a map of the type required (by Hardt-Sullivan) exists for any interior of a compact bordered Riemann surface: indeed take a Ahlfors map or just a circle map (existence ensured by Ahlfors 1950 [@Ahlfors_1950], or other sources, e.g. Gabard 2006 [@Gabard_2006]) and then upon post-composing by a power-map $z\mapsto z^n$ we may contract the modulus of the branch points to make them as small as we please upon choosing $k$ large enough $\spadesuit$ perhaps the dual game of looking at largest possible winding points should relate to the problem of finding the circle maps of lowest possible degrees $\spadesuit$; at least one should be able to define a conformal invariant of a bordered surface $F$ by looking at the largest possible modulus of a branch point of a circle map (of course composing with a disc-automorphism, the branch point can be made very close to $1$, so one requires a normalization, e.g. mapping a base-point of $F$ to $0$) $\spadesuit$ this defines a $[0,1)$-valued numerical invariant of a marked compact bordered Riemann surface $(F,b)$; how does it depends on $b$ when the latter is dragged through the (fixed) surface and does this invariant takes the value $0$ only for when $F$ is the disc $\spadesuit$ as another variant without marking, we may always assume that $0$ is nor ramified, and we may look for the largest radius free of ramification, this defines another numerical invariant taking values in $]0,1]$; obviously it takes the value one only when $F$ is topologically a disc (Riemann mapping theorem maybe in the variant firmly established by Schwarz) $\spadesuit$ maybe in the spirit of Bloch there is an absolute (strictly) positive lower bound on this “schlicht radius” at least for prescribed topological characteristic (i.e. the invariant $p$ and $r$ counting the genus and the contours) $\spadesuit$ call this constant $B_{p,r}$: how does it depend on $p,r$ asymptotically (maybe convergence to $0$ if $p,r \to \infty$); further is the infimum achieved by some surfaces, if so can we describe the extremal surfaces (naive guess the ramification is then cyclotomic); compare maybe work of Minda ca. 1983 for related questions\]  G.H. Hardy, [*On the mean modulus of an analytic function*]{}, Proc. London Math. Soc. 14 (1915), 269–277. \[$\spadesuit$\]  A.N. Harrington, [*Conformal mappings onto domains with arbitrarily specified boundary shapes*]{}, J. d’Anal. Math. 41 (1982), 39–53. \[$\spadesuit$ extension of Koebe’s KNP; similar result in Brandt 1980 [@Brandt_1980] $\spadesuit$ method: potential theory and (algorithmic) Brouwer’s fixed point $\spadesuit$ variant of proof in Schramm 1996 [@Schramm_1996]\]  J. Harris, [*On the Severi problem*]{}, Invent. Math. 84 (1986), 445–461. \[$\spadesuit$ based on virtually the same idea as Severi 1921, and Brusotti 1921, cf. e.g. Shustin 1990/91 [@Shustin_1990/91-Geom-of-discr-alg-curve]\]  R. Hartshorne, [*Algebraic Geometry*]{}, Grad. Texts in Math. 49, Springer-Verlag, 1977. \[$\spadesuit$ some elementary aspects of curves and surfaces via the sheaf theoretic approach (Leray, etc.)\]  M. Hasumi, [*Invariant subspaces for finite Riemann surfaces*]{}, Canad. J. Math. 18 (1966), 240–255. \[$\spadesuit$ extension of Beurling’s theorem (1949 [@Beurling_1949]) for the disc to the case of finite bordered Riemann surface, yet without using the Ahlfors map, but cite Royden 1962 [@Royden_1962] which is closely allied\]  O. Haupt, [*Ein Satz über die Abelschen Integrale 1. Gattung*]{}, Math. Z. 6 (1920), 219–237. \[$\spadesuit$ only cited for the Riemann parallelogram method, which bears (perhaps?) some resemblances with Gabard 2006 [@Gabard_2006] $\spadesuit$ work is influenced by Prym, tries to answer a question of Klein, further influence of Wirtinger, etc. $\spadesuit$ for modern ramification cf. Gerstenhaber 1953 [@Gerstenhaber_1953]\]  S.Ya. Havinson, [*On an extremal problem in the theory of analytic functions*]{}, (Russian) Uspekhi Mat. Nauk. 4 (1949), 158–159. \[$\spadesuit$\]$\bigstar$  S.Ya. Havinson, [*On extremal properties of functions mapping a region on a multi-sheeted circle*]{}, Doklady Akad. Nauk. SSSR (N.S.) 88 (1953), 957–959. (Russian.) 60 $\bigstar$  S.Ya. Havinson, [*Extremal problems for certain classes of analytic functions in finitely connected regions*]{}, Mat. Sb. (N.S.) 36 (78) (1955), 445–478; Amer. Math. Soc. Transl. 5 (1957), 1–33. 78 \[$\spadesuit$ generalized linear extremal problems (finite connectivity), i.e. maximization of the modulus of the derivative replaced by an arbitrary linear functional\]$\bigstar$  S.Ya. Havinson, G.C. Tumarkin, [*Existence of a single-valued function in a given class with a given modulus of its boundary values in multiply connected domains*]{}, (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 543–562. \[$\spadesuit$ quoted in Khavinson 1984 [@Khavinson-Dimitri_1984 p.378], and the same problem of prescribing the boundary modulus had been already treated by Grunsky 1942 [@Grunsky_1942]\]$\bigstar$  S.Ya. Havinson, [*Analytic capacity of sets, joint nontriviality of various classes of analytic functions and the Schwarz lemma in arbitrary domains*]{}, Mat. Sb. 54 (96) (1961), 3–50; English transl., Amer. Math. Soc. Transl. (2) 43 (1964), 215–266. 47, 50, 78 \[$\spadesuit$ uniqueness of the (Ahlfors) extremal function for domains of infinite connectivity (similar work in Carleson 1967 [@Carleson_1967-book]); but Khavinson’s work goes deeper (according to Hejhal 1972 [@Hejhal_1972]) into the study of the behavior of the extremal function\]  S.Ya. Havinson, [*Factorization theory for single-valued analytic functions on compact Riemann surfaces with boundary*]{}, Uspekhi Math. Nauk. 44 (1989), 155–189; English transl., Russian Math. Surveys 44 (1989), 113–156. \[$\spadesuit$ p.117 explains the usual trick of annihilating the $2p+(h-1)$ periods along essential cycles on a finite Riemann surface, for which we may take any $r-1$ of the boundary contours, as well as meridians and parallels taken along each handle $\spadesuit$ so this is quite close to our naive attempt to reprove Ahlfors’ theorem, compare Sec.\[Ahlfors-proof:sec\]\]  S.Ya. Havinson, [*Duality relations in the theory of analytic capacity*]{}, St. Petersburg Math. J. 15 (2004), 1–40. (Russian version published in 2003.) \[$\spadesuit$ Ahlfors function appears on pp.2,11,13,20 $\spadesuit$ the terminology “analytic capacity” (or “Ahlfors capacity”) is credited to V.D. Erokhin’s: “In accordance with V.D. Erokhin’s proposal (1958), the quantity $\gamma(F)$ has been called the [*analytic capacity*]{} or the [*Ahlfors capacity*]{} since that time.”\]  N.S. Hawley, M. Schiffer, [*Half-order differentials on Riemann surfaces*]{}, Acta Math. 115 (1966), 199–236. \[$\spadesuit$\]  N.S. Hawley, M. Schiffer, [*Riemann surfaces which are double of plane domains*]{}, Pacific J. Math. 20 (1967), 217–222. 78 \[$\spadesuit$\]  N.S. Hawley, [*Weierstrass points of plane domains*]{}, Pacific J. Math. 22 (1967), 251–256. 78 \[$\spadesuit$ addresses the question of the distribution of Weierstrass points upon the Schottky double of a plane domain. Precisely, for a planar membrane with hyperelliptic double, all W-points are located on the boundary. The author gives an example, derived from a real quartic with 4 ovals, whose W-points are not confined to the boundary. Such questions make good sense over positive genus membranes and are perhaps worth investigating further. Probably updates are already known, and one would like to explicit any possible relation between W-points and the degree of the Ahlfors function. Compare for this issue, Yamada 1978 [@Yamada_1978]\]  M. Hayashi, [*The maximal ideal space of the bounded analytic functions on a Riemann surface*]{}, J. Math. Soc. Japan 39 (1987), 337–344. \[$\spadesuit$ the following property: “the natural map of a Riemann surface $R$ into its maximal ideal space $\frak M (R)$ (this is an embedding if we assume that the algebra $H^{\infty}(R)$ of bounded analytic functions separates points) is a homeomorphism onto an open subset of $\frak M (R)$” has some application to the uniqueness of the Ahlfors function (cf. Gamelin 1973 [@Gamelin_1973-Extremal-I]), as well as to its existence $\spadesuit$ the bulk of this paper consists in giving examples where this property fails answering thereby a question of Gamelin 1973\]  Z.-X. He, [*Solving Beltrami equation by circle packing*]{}, Trans. Amer. Math. Soc. 322 (1990), 657–670. \[$\spadesuit$ includes another proof of GKN (generalized Kreisnormierung) where a compact bordered Riemann surface is conformally mapped upon a circular domain in a space-form (=constant curvature) \[of the same genus?\] $\spadesuit$ similar statement obtained by Haas 1984 [@Haas_1984] and Maskit 1989 [@Maskit_1989] (curiously non-cited here)—maybe also Jost 1985 [@Jost_1985] $\spadesuit$ perhaps the “syntax” of the main result (Thm 5.1, p.669) must be slightly corrected, probably by assuming the contours of $\partial{\overline{\Omega}}$ to bounds discs in the surface $M$ (equivalently to be null-homotopic)\]  Z.-X. He, O. Schramm, [*Fixed points, Koebe uniformization and circle packings*]{}, Ann. of Math. (2) 137 (1993), 369–406. \[$\spadesuit$ the deepest advance upon the KNP=Kreisnormierungsprinzip (raised by Koebe 1908 UbaK3 [@Koebe_1908_UbaK3]), which is established for countably many boundary components $\spadesuit$ The general case is still unsettled today (2012), and maybe undecidable within ZFC? (just a joke, of course)\]  Z.-X. He, O. Schramm, [*On the convergence of circle packings to the Riemann map*]{}, Invent. Math. 125 (1996), 285–305. \[$\spadesuit$ improvement and generalization of the Rodin-Sullivan proof (1987 [@Rodin-Sullivan_1987]), making it logically independent of RMT (thus reproving it via the technology of circle packings) $\spadesuit$ \[08.10.12\] what about the same game for the Ahlfors map?\]  E. Heine, [*Ueber trigonometrische Reihen*]{}, J. Reine Angew. Math. 71 (1870), 353–365. \[$\spadesuit$ the rôle of uniform convergence is emphasized (i.e. Weierstrass’ notion, yet first only familiar to his direct circle of students)\]  M.H. Heins, [*Extremal problems for functions analytic and single-valued in a doubly connected region*]{}, Amer. J. Math. 62 (1940), 91–106. 78 \[$\spadesuit$ quoted (joint with Carlson 1938 [@Carlson_1938] and Teichmüller 1939 [@Teichmueller_1939-Dreikreisesatzes]) in Grunsky 1940 [@Grunsky_1940] as one of the forerunners of the extremal problem for bounded analytic functions (alias Ahlfors map, subsequently)\]  M. Heins, [*On the iteration of functions which are analytic and single valued in a given multiply connected region*]{}, Amer. J. Math. 63 (1941), 461–480. 78 \[$\spadesuit$ regarded by Minda 1979 [@Minda_1979 p.421] as the proper originator of the [*annulus theorem*]{} (i.e., an analytic self-map of an annulus can take the generator of the fundamental group only upon a $0$ or $\pm 1$ multiple of itself, and the $\pm 1$ case forces the map to be a conformal automorphism)\]  M. Heins, [*A lemma on positive harmonic functions*]{}, Ann. of Math. (2) 52 (1950), 568–573. 60, 78 \[$\spadesuit$ may contain another proof of the existence of the Ahlfors function (at least a circle map), yet not very clear which degree Heins’ argument supplies $\clubsuit$ in fact since the quantity $m$ appearing on p.571 for a generating system of the fundamental group is easily found to be $m=2p+(r-1)$ (where $p$ is the genus and $r$ the number of contours) it is quite likely (albeit the writer has no certitude!) that Heins’ method may reproduce (by specialisation) exactly Ahlfors upper bound upon the degree of a circle map $\spadesuit$ \[06.10.12\] for a possible corroboration of this intuition, check also the subsequent paper Heins 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF] which truly seems to get again the $r+2p$ bound of Ahlfors (1950) $\spadesuit$ treats Pick-Nevanlinna interpolation for a bordered surface (extending the work of Garabedian 1949 [@Garabedian_1949])\]  M. Heins, [*Symmetric Riemann surfaces and boundary problems*]{}, Proc. London Math. Soc. (3) 14A (1965), 129–143. \[$\spadesuit$ looks closely allied to Ahlfors 1950 [@Ahlfors_1950], which is not cited, but so are some direct descendants, Read 1958 [@Read_1958_Acta] and Royden 1962 [@Royden_1962] $\spadesuit$ enters into the category of “transplanting papers” where some result for the disc is lifted to a compact bordered surface (=membrane) $\spadesuit$ in the present case M. Riesz’s theorem on the conjugate Fourier series, and the unique decomposition of $f\in L^p$ into interior/exterior Fatou boundary functions of functions in $H_p$\]  M. Heins, [*Hardy classes on Riemann surfaces*]{}, Lecture Notes in Mathematics 98, Springer, 1969. \[$\spadesuit$ p.59–65 contains a re-exposition of Heins 1950 [@Heins_1950], yielding probably an alternate proof of the Ahlfors circle maps\]$\bigstar$$\bigstar$$\bigstar$ M. Heins, [*Nonpersistence of the Grenzkreis phenomenon for Pick-Nevanlinna interpolation on annuli*]{}, Ann. Acad. Sci. Fenn. Ser. A. 596 (1975), 1–21. 50, 78 \[$\spadesuit$ cited in Jenkins-Suita 1979 [@Jenkins-Suita_1979] $\spadesuit$ from MathReviews: “Let $A$ be a subset of the open unit disc $\Delta$. Consider the family of functions $f$ regular in $\Delta$ such that $\vert f\vert \le 1$ on $\Delta$ and at each point of $A$, a specified initial Taylor section is assigned. For $b\in \Delta$, let $W(b)$ denote the set of values assumed by the functions of the family at $b$. \[As Heins explains in the original article “W(b) is termed the [*Wertevorrat*]{} of the family at $b$.”\] The Pick-Nevanlinna-Grenzkreis phenomenon asserts that if there is more than one function in the family and $b\in \Delta-A$, then the set $W(b)$ is a closed circular disc of positive radius. The author constructs a counter example to show that this is no longer true for multiply connected domains. Let $\Omega$ be the annulus $r<\vert z\vert< r^{-1}$ and let $B(c)$ denote the set of functions $f$, analytic in $\Omega$, such that $\vert f\vert \le 1$, $f(-1)=0$ and $f'(-1)=c$, where $c$ is small and positive. The author shows that in this case $W(b)$, $b\neq -1$, is a set with nonempty interior but is not a circle.—A result of Garabedian 1949 [@Garabedian_1949] asserts that if $\Omega$ is a domain of finite connectivity such that no boundary component reduces to a point and if the values of the function are assigned at a finite number of points, then the unique extremal function which takes at $b$ a given value on ${\rm Fr} W(b)$ maps $\Omega$ onto $\Delta$ with constant valency. The author shows that this remains true for his example although the initial Taylor section assigned is of order one at $z=-1$. There is also a general discussion of the problem in the general setting of Riemann surfaces with finite topological characteristics.” $\spadesuit$ \[07.10.12\] as a modest task one may wonder if Heins’ paper reproves Ahlfors’ existence of circle maps of degree $\le r+2p$. As a pessimistic remark it seems that there is a wide variety of extremal problems, somehow reflecting our mankind capitalistic/competitive aberration, making it unclear what the God given problem is, especially the one capturing circle maps of lowest possible degree $\spadesuit$ more optimistically it is clear that there is a fascinating body of knowledge among such problems (interpolation by prescribed Taylor section). Given a finite Riemann surface $\overline F$ (bordered), choose a finite set $A$ each point being decorated by a Taylor section (w.r.t. a local uniformizer), look at all functions bounded-by-one matching the Taylor data. For any $b\in F-A$, define $W(b)\subset \Delta$ as the set of values assumed at $b$ by functions of the family. $\spadesuit$ as above we look at the function $f_{b,w}$ taking at $b$ a given value $w$ of the frontier of $W(b)$. Q1. Is then Garabedian’s result on the constant valency of $f_{b,w}\colon F \to \Delta$ true in this non-planar setting? If yes what is the degree of the corresponding circle map (Q2). Of course the case where $A=\{a\}$ is a singleton with Taylor section $f(a)=0$ ($b\neq a$) and $w$ chosen so as to maximize the modulus in the set $W(b)$ gives exactly the Ahlfors map $f_{a,b}$ studied in Ahlfors 1950 [@Ahlfors_1950]. This induces (via the assignment $\overline{F}\mapsto \vert f_{a,b}(b)\vert$) a real-valued function ${\cal M}_{r,p}\to ]0,1[$ on the moduli space of surfaces with two marked points. One can dream about understanding the Morse theory of this function. $\spadesuit$ The answer to our two naive questions (Q1, Q2) is apparently already in Heins’ paper, for Jenkins-Suita 1979 [@Jenkins-Suita_1979 p.83] write: “Quite recently Heins \[10\](=1975 [@Heins_1975]) proved uniqueness of the extremal function $f_0$ which maximizes ${\rm Re}(e^{i\theta} f(z_0))$ among the class of analytic functions $f$ bounded by unity and with given Taylor sections \[…\] on a compact bordered Riemann surface $\Omega$. He also proved the extremal $f_0$ maps $\Omega$ onto a finite sheeted covering of the unit disc and gave a bound on the number of sheets called the [*Garabedian bound*]{}.” $\spadesuit$ \[07.10.12\] as a micro-objection the terming “Garabedian bound” is probably slightly unfair for Ahlfors as the latter probably knew it (in the case of a single interpolating point) without Garabedian’s helping hand (at least for circle maps, yet arguably not for the Ahlfors’ extremals) (cf. of course the acknowledgments to be found in Ahlfors 1950 [@Ahlfors_1950], but see also Nehari 1950 [@Nehari_1950] where the Ahlfors upper bound $r+2p$ is credited back to Ahlfors’ Harvard lectures in Spring 1948) $\clubsuit$ \[12.10.12\] Heins’ statement is as follows (p.18): “(3) [*The Garabedian bound*]{}. We consider a determinate Pick-Nevanlinna problem relative to $\Omega$ with a finite set of data and denote the solution by $f$. \[…\] For an interpolation point $b$ we let $\nu(b)$ denote the order of interpolation at $b$ augmented by one. We let $\nu$ denote the sum of the $\nu(b)$ taken over the interpolation points $b$. The Euler characteristic of $\Omega$ will be denoted by $\chi$. We shall show—[**Theorem 8.2**]{} [*f has at most $\nu+\chi$ zeros counted by multiplicity.*]{} $\clubsuit$ this statement subsumes the upper estimate of Garabedian, but also that of Ahlfors: indeed Ahlfors extremal problem is the case where there is a single interpolating point of order zero. So $\nu=0+1=1$. Now given a bordered surface $\Omega$ of genus $p$ with $r$ contours, we have $\chi(\Omega)=2-2p-r$ \[beware that Heins seems to work with the old convention about the sign of the Euler characteristic, hence just change his formula to $\nu-\chi$\]. So we get $\deg f \le \nu-\chi= \nu-2+2p+r\approx r+2p$ (note a little arithmetical discrepancy from Ahlfors, surely easily explained) $\clubsuit$ Heins’ proof uses the following tools: $\bullet$ basic facts concerning Hardy classes on Riemann surfaces for which one is referred to Heins 1969 [@Heins_1969-LNM-Hardy] $\bullet$ a variational formula of F. Riesz 1920 [@Riesz_1920-Ueber-Potenzreihen] $\bullet$ the theorem of Cauchy-Read (cf. Read 1958 [@Read_1958_Acta]) $\bullet$ the Fatou boundary function, $\bullet$ the Green’s function $\bullet$ the qualitative Harnack inequality $\spadesuit$ a slightly different proof of a much related result (on “Garabedian bound”) is given as Theorem 3 of Jenkins-Suita 1979 [@Jenkins-Suita_1979], which uses maybe less machinery (?), an instead of Read the closely allied paper Royden 1962 [@Royden_1962]. Yet Jenkins-Suita’s proof depend on Heins’ proof when it comes to the “interpolation divisor”\]  M. Heins, [*Carathéodory bodies*]{}, Comm. in honorem Rolf Nevanlinna LXXX annos nato, Ann. Acad. Sci. Fenn. Ser. A.I, Math. 2 (1976), 203–232. \[$\spadesuit$ extension to the setting of finite Riemann surfaces of Carathéodory’s theory on the “Variabilitätsbereich” (1907 [@Caratheodory_1907-Variabilitaetsbereich], 1911 [@Caratheodory_1911-Variabilitaetsbereich]) of coefficient of analytic functions with positive real part (bringing together Minkowski’s theory of convex sets with complex function theory), while encompassing interpolation problems subsuming those of Pick-Nevanlinna type\] $\bigstar$  M. Heins, [*Extreme normalized analytic functions with positive real parts*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 10 (1985), 239–245. 50 \[$\spadesuit$ localized via Bell 2009/11 [@Bell_2009_PREPRINT/2011] $\spadesuit$ also quoted in Khavinson 1984 [@Khavinson-Dimitri_1984 p.377] for another proof of the Bieberbach-Grunsky theorem $\spadesuit$ Heins handles the more general non-planar case recovering probably the Ahlfors circle maps of 1950, and so seems indeed to be the case according to MathReviews (translated from Hervé’s review in Zentralblatt): “Let $P$ be the family of holomorphic functions $f$ on a given Riemann surface $S$ satisfying ${\rm Re f}>0$ on $S$ and $f(a)=1$ for a given point $a\in S$. If $S$ is the unit circle, the extremal elements of $P$ are the functions $z\to (\eta+z)/(\eta-z)$, $\vert \eta \vert=1$. If $S$ is a bounded open plane region whose boundary consists of $c$ analytic Jordan curve $\Gamma_1,\dots, \Gamma_c$, the author associates the extremal elements $f_\zeta$ of $P$ with the system $\zeta=(\zeta_1, \dots, \zeta_c)\in \Gamma_1\times \dots \Gamma_c$; ${\rm Re} f_{\zeta}$ is an appropriate linear combination of minimal harmonic functions $>0$ on $S$ with poles $\zeta_k$, $k=1,\dots, c$. This results extends to the case in which $S$ is an open region of a compact Riemann surface of genus $g$, but here the real parts of the extremal element of $P$ are linear combination of \[AT MOST\][^134] $2g+c$ minimal positive harmonic functions on $S$.” $\spadesuit$ \[06.10.12\] so it seems that this new work of Heins, albeit quite close to Heins 1950 [@Heins_1950], may be a bit more explicit and truly include the existence of (Ahlfors) circle map with the bound $r+2p$ like Ahlfors 1950 [@Ahlfors_1950] $\spadesuit$ \[06.10.12\] it would be of course of primary importance to study if Heins’ methods is susceptible of recovering the sharper $r+p$ bound asserted in Gabard 2006 [@Gabard_2006] $\spadesuit$ \[12.10.12\] after reading the original text, it must alas recognize that Heins’ proof is not perfectly satisfactory, for when it comes to the case of positive genus, he writes simply (p.243): “the corresponding developments of Section 3 \[=planar case\] may be paraphrased.” $\spadesuit$ hence the pedestrian reader will not find it easy to recover even Ahlfors basic (but deep) result from Heins’ account. So let me try once to degage the substance of the argument, while trying to locate “en passant” those critical steps which in our opinion is not made explicit in Heins’ exposition. (I shall use my notation hopefully for convenience of the reader.) We start as usual with $\overline{F}$ a compact bordered Riemann surface of genus $p$ and with $r$ contours. Let $a\in F$ be some fixed interior point. Heins considers $P$ the set of analytic functions $f$ on $F$ with ${\rm Re} f>0$ and $f(a)=1$. (The family $P$ is convex and compact, hence admits extreme points by Krein-Milman. Actually we shall probably not need this, albeit being an interesting viewpoint.) Let $g:=2p+(r-1)$ and $\gamma_1, \dots, \gamma_g$ be representatives of the homology group $H_1(F)$. For $u$ harmonic on $F$, let $\pi(u)$ be the period vector given by $\pi(u)=(\int_{\gamma_1} \delta u, \dots, \int_{\gamma_g} \delta u)$, where $\delta u$ is a certain abelian differential given by some local recipe. In fact it is perhaps more natural (and equivalent?) to define $\delta u$ as the conjugate differential $(du)^{\ast}$. For $\zeta \in \partial F$, Heins considers (p.241) $u_{\zeta}$ the minimal positive harmonic function on $F$ vanishing on $\partial F-\{\zeta\}$ and normalized by $u_\zeta(a)=1$. \[Maybe here Heins still relies subconsciously on Martin 1941 [@Martin_1941], yet arguably this is nothing else that the Green’s function with pole pushed to the boundary, what I called a Red’s function, but perhaps calls it a Poisson function, as may suggest the paper Forelli 1979 [@Forelli_1979].\] We seek to construct a half-plane map $f$ by taking a combination $u=\sum_{k=1}^d \mu_k u_{\zeta_k}$ of such elementary potentials, with $\mu_k>0$ while trying to arrange the free parameters (e.g. the $\zeta_k\in \partial F$) so as to kill all periods of $(du)^{\ast}$. If this can be achieved for some $d$, then $f=u+iu^{\ast}$ (where $u^{\ast}$ is defined by integrating the differential $(du)^{\ast}$) supplies a half-plane map of degree $d$. (Recall indeed that $u$ vanishes continuously on the boundary $\partial F$, except at the $\zeta_k$ which are catapulted to $\infty$. Hence the map is boundary preserving and has therefore constant valency, here $d$.) To kill all periods, we may look at the map $\varphi\colon \partial F \buildrel{u}\over{\to} h(F) \buildrel{\pi}\over{\to} {\Bbb R}^g$, where $u(\zeta)=u_\zeta$ and $h(F)$ denotes the space of harmonic functions. At this stage it must be explained that the image $\varphi(\partial F)$ is “balanced”, i.e. not situated in a half space of ${\Bbb R}^g$. \[I am not sure that Heins explains this in details.\] If so then it is plain that there is a collection of $d\le g+1$ points (assume $d=g+1$ if you want) on $\varphi(\partial F)$ spanning a simplex containing $0$. This is just the principle that in Euclidean space of some dimension, a collection of one more points than the given dimension span a top-dimensional simplex with optimum occupation property of the territory (=Euclid space). Thus expressing the origin as convex combination of those $g+1$ points we find scalars $\mu_k>0$, which injected in the formula defining $u$, gives us an $u$ meeting the requirement. This reproves Ahlfors 1950, but alas I still do not have a simple explanation for the balancing condition. Next the challenge, is of course to improve the geometry by remarking that clever placements of points may span a lower dimensional simplex yet still covering the origin. Hopefully one may reprove the $r+p$ upper bound of Gabard 2006 [@Gabard_2006], along this path (which is essentially Ahlfors’ original approach).\]  M. Heins, [*Extreme Pick-Nevanlinna interpolating function*]{}, J. Math. Kyoto Univ. 25–4 (1985), 757–766. \[$\spadesuit$ p.758: “It is appropriate to cite instances of convexity considerations related to the present paper. The pioneer work of Carathéodory \[2\](=1907 [@Caratheodory_1907-Variabilitaetsbereich]),\[3\](=1911 [@Caratheodory_1911-Variabilitaetsbereich]) on coefficient problems for analytic functions with positive real part is, as far I am aware, the first bringing together of the Minkowski theory of convex sets and complex function theory. Extreme points are present in the fundamental work of R.S. Martin \[12\](=1941 [@Martin_1941]) on the representation of positive harmonic functions as normalized minimal positive harmonic functions. My paper \[7\](=Heins 1950 [@Heins_1950]) showed the existence of minimal positive harmonic functions on Riemann surfaces using elementary standard normal families results without the intervention of the Krein-Milman theorem and gave application to qualitative aspects of Pick-Nevanlinna interpolation on Riemann surfaces with finite topological characteristics and nonpointlike boundary components. Such Riemann surfaces will be termed [*finite*]{} Riemann surfaces henceforth. In \[8\](=Heins 1976 [@Heins_1976]) the Carathéodory theory cited above was extended to the setting of finite Riemann surfaces for interpolation problems subsuming those of Pick-Nevanlinna type. Forelli \[5\](=1979 [@Forelli_1979]) has studied the extreme points of the family of analytic functions with positive real part on a given finite Riemann surface $S$ normalized to take the value $1$ at a given point of $S$. In my paper \[9\](=Heins 1985 [@Heins_1985-Extreme-normalized-LIKE-AHLF]) the results of Forelli were supplemented by precise characterizing results for the case where the genus of $S$ is positive.”\]  D.A. Hejhal, [*Linear extremal problems for analytic functions*]{}, Acta Math. 128 (1972), 91–122. 50, 78 \[$\spadesuit$ generalized extremal problem, existence as usual via normal families, and so uniqueness is given “a reasonably complete answer” (p.93) $\spadesuit$ p.119 Royden 1962 [@Royden_1962] is cited for another treatment of Ahlfors’ extremal problem\]  D.A. Hejhal, [*Theta functions, kernel functions and Abelian integrals*]{}, Memoirs Amer. Math. Soc. 129, 1972. \[$\spadesuit$ quoted in Burbea 1978 [@Burbea_1978-Capacities], where like in Suita 1972 [@Suita_1972] an application of the Ahlfors function is given to an estimation of the analytic capacity\] $\bigstar$  D.A. Hejhal, [*Some remarks on kernel functions and Abelian differentials*]{}, Arch. Rat. Mech. Anal. 52 (1973), 199–204. 78  D.A. Hejhal, [*Universal covering maps for variable regions*]{}, Math. Z. 137 (1974), 7–20. 78 \[$\spadesuit$ just quoted for the philosophical discussion on p.19, especially the issue that the (Koebe) Kreisnormierung (=circular mapping) \[not to be confused with our circle maps!\] is “[*somewhat more involved*]{} than the other canonical mappings, esp. the PSM\]  D.A. Hejhal, [*Linear extremal problems for analytic functions with interior side conditions*]{}, Ann. Acad. Sci. Fenn. Ser.A 586 (1974), 1–36. 78 \[$\spadesuit$ cited in Jenkins-Suita 1979 [@Jenkins-Suita_1979]\]$\bigstar$  D.A. Hejhal, [*On Schottky and Teichmüller spaces*]{}, Adv. Math. 15 (1975), 133–156. \[$\spadesuit$\]$\bigstar$  B. Heltai, [*Symmetric Riemann surfaces, torsion subgroups and Schottky coverings*]{}, Proc. Amer. Math. Soc. 100 (1987), 675–682. \[$\spadesuit$\]  ?. Henoch, [*De Abelianarum Functionum Periodis*]{}, Inaugural-Dissertation, Berlin. \[$\spadesuit$ cited in both Hurwitz 1883 [@Hurwitz_1883] and Weichold 1883 [@Weichold_1883], who both relax hyperellipticity from Henoch results on the periods of Abelian integrals on real algebraic curve, while extending also Klein’s general version for $g=3$ (cf. Klein 1876 [@Klein_1876_Verlauf])\]  K. Hensel, W. Landsberg, [*Theorie der algebraischen Funktionen einer Variabeln und ihre Anwendung auf algebraische Kurven und Abelsche Integrale*]{}, Leipzig, 1902. \[$\spadesuit$ contains the sharp estimate $[\frac{g+3}{2}]$ of Riemann-Brill-Noether upon the gonality of a closed surface, but the treatment is not considered as convincing (to contemporary scientists) until the work of Meis 1960 [@Meis_1960], compare e.g., H.H. Martens 1967 [@Martens_Henrik_1967] and Kleiman-Laksov 1972 [@Kleiman-Laksov_1972]\]  G. Herglotz, [*Über Potenzreihen mit positivem reellen Teil im Einheitskreis*]{}, Ber. Verhandl. Sächs. Ges. Wiss. Leipzig 93 (1911), 501–511. \[$\spadesuit$ yet another theorem in the disc susceptible (???) of a transplantation to finite bordered Riemann surface, try e.g Agler-Harland-Raphael 2008 [@Agler-Harland-Raphael_2008] (multi-connected planar domains), or Heins 1985 [@Heins_1985-Extreme-Pick-Nevanlinna] $\spadesuit$ Herglotz’s representation theorem is concerned with the so-called Poisson-Stieltjes representation for analytic functions on the unit disc $\Delta$ with $\ge 0$ real part (simultaneous work by F. Riesz), and cf. the above cited work of Heins for an application (of Herglotz-Riesz 1911) to a description of extreme points of a class $I$ of analytic function arising from a Pick-Nevanlinna interpolation problem involving functions $\Delta \to \{{\rm Re} z>0 \}$: “Theorem 1. The extreme points of $I$ \[in the sense of convex geometry\] are precisely the members of $I$ having constant valence on $\{Re z>0\}$, the value $\nu$ of the valence satisfying $1+n<\nu<1+2n$.” \[$n$ being the number of interpolation points\] $\spadesuit$ such results are probably extensible to the situation where the source (=disc $\Delta$) is replaced by a finite bordered Riemann surface, and the resulting theory probably interacts with the Ahlfors map $\spadesuit$ \[13.10.12\] in fact the Poisson-Stieltjes-Herglotz-Riesz representation formula is rather involved in another proof of Ahlfors circle maps, see for this Forelli’s brilliant account (1979 [@Forelli_1979])\]  J. Hersch, [*Quatres propriétés isopérimétriques de membranes sphériques homogènes*]{}, C. R. Acad. Sc. Paris 270 (1970), 1645–1648. \[$\spadesuit$ contains 4 spectral (eigenvalues) inequalities for disc-shaped membranes emphasizing the extremality of resp. the round sphere, the hemisphere, the quarter of sphere and of the octant of sphere $\spadesuit$ the first inequality has been extended via conformal transplantation to closed surfaces of higher topological type by Yang-Yau 1980 [@Yang-Yau_1980] (who failed to take advantage of the well-known sharp gonality bound of Riemann-Meis (1960 [@Meis_1960]), but see El Soufi-Ilias 1983/84 [@El-Soufi-Ilias_1983/84]) $\spadesuit$ the first inequality has been extended by Gabard 2011 [@Gabard_2011] upon using the Ahlfors map $\spadesuit$ \[08.10.12\] of course it would be also interesting trying to get extensions of the two remaining Hersch’s inequalities (involving the quarter of sphere and its octant resp.)\]  J. Hersch, L.E. Payne, M.M. Schiffer [*Some inequalities for Steklov eigenvalues*]{}, Arch. Rat. Mech Anal. 57 (1973/74), 99–114. \[$\spadesuit$ contain estimates of Steklov eigenvalues along the method of Szegö 1954 [@Szego_1954], Weinstock 1954 [@Weinstock_1954], Weinberger 1956 [@Weinberger_1956], etc. (i.e. conformal transplantation) $\spadesuit$ in the light of Fraser-Schoen’s article (2011 [@Fraser-Schoen_2011]) one can remark two things: (1) it is strange that the present paper does not cite the planar avatar of the Ahlfors map (that is the Bieberbach-Grunsky theorem); this is perhaps done subconsciously in §5.2, p.106 (2) of course (at least since Fraser-Schoen’s paper 2011 [@Fraser-Schoen_2011]) it is obvious that the result can (via the Ahlfors map) be extended to bordered Riemann surfaces; for an exact implementation cf. Girouard-Polterovich 2012 [@Girouard-Polterovich_2012]\]  M. Hervé, [*Quelques propriétés des transformations intérieures d’un domaine borné*]{}, Ann. sci. École norm. sup. (3) 68 (1951), 125–168. 78 \[$\spadesuit$ Grunsky’s works, as well as Ahlfors 1947 [@Ahlfors_1947] are cited, and it could be nice to look for extensions to bordered surfaces\]  R.A. Hidalgo, A.F. Costa, [*Anticonformal automorphisms and Schottky coverings*]{}, Ann. Acad. Sci. Fenn 26 (2001), 489–508. \[$\spadesuit$\]  R.A. Hidalgo, [*Schottky uniformization of stable symmetric Riemann surfaces*]{}, Notas de la Soc. Mat. de Chile (N.S.) 1 (2001), 82–91. \[$\spadesuit$\]  R.A. Hidalgo, [*Real surfaces, Riemann matrices and algebraic curves*]{}, In: [*Complex Manifolds and Hyperbolic Geometry*]{}, Guanajuato 2001, Contemp. Math. 311, Amer. MAth. Soc., Providence, 2002. \[$\spadesuit$ a neoclassical account on the Rückkehrschnitttheorem of Klein 1882 $\spadesuit$ question: is it sufficient to reprove existence of Ahlfors circle maps?\]  R.A. Hidalgo, B. Maskit, [*On Klein-Schottky groups*]{}, Pacific J. Math. 220 (2005), 313–328. \[$\spadesuit$\]  R.A. Hidalgo, B. Maskit, [*On neoclassical Schottky groups*]{}, Trans. Amer. Math. Soc. 358 (2006), 4765–4792. \[$\spadesuit$\]  R.A. Hidalgo, [*On the retrosection theorem*]{}, Proyecciones 27 (2008), 29–61. \[$\spadesuit$ a neoclassical account on the Rückkehrschnitttheorem of Klein 1882 $\spadesuit$ question: is it sufficient to reprove existence of Ahlfors circle maps?\]  R.A. Hidalgo, [*On the inverse uniformization problem: real Schottky uniformization*]{}, Rev. Mat. Complut. 24 (2011), 391–420. \[$\spadesuit$ p.394: “The reciprocal is valid by the retrosection theorem \[13\](=Koebe 1910 UAK2 [@Koebe_1910_UAK2]) (see \[4\]=(Bers 1975 [@Bers_1975]) for a modern proof using quasiconformal deformation theory).”\]  D. Hilbert, [*Über die reellen Züge algebraischen Kurven*]{}, Math. Ann. 38 (1891), 115–138; or Ges. Abhandl., Bd.II. \[$\spadesuit$ where Hilbert’s 16th problem (Paris 1900) starts taking shape, in the sense of asking for the isotopy classification of plane smooth real sextics in ${\Bbb R}P^2={\Bbb P}^2 ({\Bbb R})$ $\spadesuit$ a method of oscillation is given permitting to exhibit a new scheme of $M$-sextic not available via Harnack’s method of 1876 (this is nowadays called Hilbert’s method) which is quite powerful (but not omnipotent) to analyze the topology of plane (real) sextics $\spadesuit$ in particular Hilbert develops the intuition that a sextic cannot have 11 unnested ovals, so must be nested yielding some noteworthy form of complexity of algebraic varieties $\spadesuit$ a complete proof of this assertion will have to wait for a longue durée series of attempt by his own students Kahn 1909 [@Kahn_1909] Löbenstein 1910 [@Löbenstein_1910] and especially Rohn 1911–13 [@Rohn_1913]. All these attempts where judged unconvincing, and from the Russian rating agency not judged as sufficiently rigorous until the intervention of Petrovskii 1933–38 [@Petrowsky_1938] and Gudkov 1948–1969, cf. e.g. Gudkov 1974/74 [@Gudkov_1974/74] $\spadesuit$ p.418 (in Ges. Abh., Bd.II): “Diesen Fall $n=6$ habe ich einer weiteren eingehenden Untersuchung unterworfen, wobei ich—freilich auf einem au[ß]{}erordentlich umständlichen Wege— fand, da[ß]{} die elf Züge einer Kurve 6-ter Ordnung keinesfalls sämtlich au[ß]{}erhalb un voneinander getrennt verlaufen können. Dieses Resultat erscheint mir deshalb von Interesse, weil er zeigt, da[ß]{} für Kurven mit der Maximalzahl von Zügen der topologisch einfachste Fall nicht immer möglich ist.” $\spadesuit$ for the next episode in Hilbert’s pen, cf. Hilbert 1909 [@Hilbert_1909] where Hilbert ascribes to his students a complete proof of the result (inexistence of the unnested scheme of 11 ovals)\]  D. Hilbert, [*Mathematische Probleme*]{}, Arch. Math. Phys. (3) 1 (1901), 213–237; also in Ges. Abh., Bd.III. \[$\spadesuit$ includes Hilbert’s 16th problem on the mutual disposition of ovals of plane curves (especially sextics), completely solved by Gudkov ca. 1969, cf. Gudkov-Utkin 1969 [@Gudkov-Utkin_1969/78]\]  D. Hilbert, [*Über das Dirichletsche Prinzip*]{}, Jahresb. d. Deutsch. Math.-ver. 8 (1900), 184–188. \[$\spadesuit$ \[08.10.12\] the technological breakthrough based upon the “direct method” in the calculus of variation, where one directly minimizes the integral (without transiting to its first variation, alias Euler-Lagrange equation) via the idea of minimizing sequences implying a topologization of the space of test functions while checking its compactness (=Fréchet’s jargon) of the resulting family $\spadesuit$ the method also differs from its predecessors Schwarz-Neumann-Poincaré where the problem was first solved for the disc and combinatorial tricks permitted proliferation to higher topological complexity\] 60  D. Hilbert, [*Über das Dirichletsche Prinzip*]{}, Math. Ann. 59 (1904), 161–186. (Abdruck aus der Festschrift zur Feier des 150jährigen Bestehens der Königl. Gesellschaft der Wissenschaften zu Göttingen 1901.) 60  \[$\spadesuit$ p.161 (Introd.): “Unter dem Dirichletschen Prinzip verstehen wir diejenige Schlu[ß]{}weise auf die Existenz einer Minimalfunktion, welche Gauss (1839)\[=[@Gauss_1839]\], Thomson (1847)\[=[@Thomson_1847]\], Dirichlet (1856)\[=of course much earlier, at least as early as when Riemann studied in Berlin, ca. 1849–50!\] und andere Mathematiker zur Lösung sogennanter Randwertaufgaben angewandt haben und deren Unzulässigkeit zuerst von Weierstrass erkannt worden ist. \[…\] Durch das Dirichletsche Prinzip hat insbesondere Riemann die Exitenz der überall endlichen Integrale auf einer vorgelegten Riemannschen Fläche zu beweisen gesucht. Ich bediene mich im folgenden dieses klassischen Beispiels zur Darlegung meines strengen Beweisverfahrens.”\]  D. Hilbert, [*Über das Dirichletsche Prinzip*]{}, J. Reine Angew. Math. 129 (1905), 63–67. (Abdruck eines Vortrages aus dem Jahresb. d. Deutsch. Math.-ver. 8 (1900), 184–188.) 60  D. Hilbert, [*Zur Theorie der konformen Abbildung*]{}, Gött. Nachr. (1909), 314–323; Ges. Abh. 3, 73–80. 78 \[$\spadesuit$ parallel-slit mapping including positive genus (and infinite connectivity) $\spadesuit$ influenced much Courant, and also Koebe 1910 [@Koebe_1910_Hilbert]\]  D. Hilbert, [*Über die Gestalt einer Fläche vierter Ordnung*]{}, Gött. Nachr. (1909), 308–313; Ges. Abh. 2, 449–453. \[$\spadesuit$ contains a good picture for the construction of Harnack-maximal sextic $\spadesuit$ p.453, Hilbert ascribes to his students G. Kahn 1909 [@Kahn_1909] and Löbenstein 1910 [@Löbenstein_1910] a complete proof that a real sextic cannot have 11 unnested ovals (but that was not judged solid enough by subsequent workers, e.g. Rohn, Petrovskii, and Gudkov 1974 [@Gudkov_1974/74]): “\[…\] eine ebene Kurve 6-ter Ordnung hervorgehen, die aus elf au[ß]{}erhalb voneinander getrennt verlaufenden Zügen bestände. Da[ß]{} aber eine solche Kurve nicht existiert, ist einer der tiefstliegenden Sätze aus der Topologie der ebenen algebraischen Kurven; derselbe ist kürzlich von G. Kahn und K. Loebenstein (Vgl. die Göttinger Dissertationen derselben Verfasserinnen.) auf einem von mir angegebenen Wege bewiesen worden.” $\spadesuit$ nowadays there is five-minute proof of what Hilbert called one of the deepest problem in the topology of plane curves, via Rohlin’s formula ca. 1974–78 (cf. e.g. our Sec.\[Rohlin-formula:sec\]), yet we believe that there is perhaps also a proof via the Ahlfors map (in the special case due to Riemann-Schottky-Bieberbach-Grunsky). This would be a fantastic project\]  D. Hilbert, S. Cohn-Vossen, [*Anschauliche Geometrie*]{}, Springer, Berlin, 1932. (Translation: Geometry and the Imagination) \[$\spadesuit$\]  S. Hildebrandt, H. von der Mosel, [*Conformal mapping of multiply connected Riemann domains by a variational approach*]{}, Adv. Calc. Var. 2 (2009), 137–183. \[$\spadesuit$ new proof of the Kreisnormierung for (planar) domains via Plateau-style method $\spadesuit$ question: can we apply the same method for the (Ahlfors) circle map? (cf. Courant 1939 [@Courant_1939] for the planar case \[$p=0$\]) $\spadesuit$ “Abstract. We show with a new variational approach that any Riemannian metric on a multiply connected schlicht domain in ${\Bbb R}^2$ can be represented by globally conformal parameters. This yields a “Riemannian version” of Koebe’s mapping theorem.”\]  S. Hildebrandt, [*Plateau’s problem and Riemann’s mapping theorem*]{}, Milan J. Math. (2011), 67–79. \[$\spadesuit$ survey putting in perspective several recent developments, including the previous item\]  F. Hirzebruch, [*Topological methods in algebraic geometry*]{}, Springer, 1978; translated from the Original German text ca. 1955. \[$\spadesuit$ how to put Pontrjagin, Thom, cobordism, the signature theorem, etc. into action to get the generalized Riemann-Roch theorem\]  W. Hodge, [*The Theory and Applications of harmonic integrals*]{}, Cambridge, 1941.  K. Hoffman, [*Banach spaces of analytic functions*]{}, Prentice-Hall (Englewood Cliffs), 1962; Dover Reprint, 1988.  M. Homma, [*Separable gonality of a Gorenstein curve*]{}, Math. Contemp. ca. 2004. \[$\spadesuit$ cited in Ballico 2003 [@Ballico_2003]\]  M. Horikawa, [*On deformations of holomorphic maps I*]{}, J. Math. Soc. Japan 25 (1973), 372–396. \[$\spadesuit$\]  J. Huisman, [*Real quotient singularities and nonsingular real algebraic curves in the boundary of the moduli space*]{}, Compos. Math. 118 (1999), 42–60. \[$\spadesuit$\]  J. Huisman, [*Real Teichmüller spaces and moduli of real algebraic curves*]{}, Contemp. Math. 253 (2000), 145–179. \[$\spadesuit$\]  J. Huisman, [*On the geometry of algebraic curves having many real components*]{}, Rev. Mat. Complut. 14 (2001), 83–92. \[$\spadesuit$ p.87, Prop.3.2 contains an algebro-geometric proof of the so-called Bieberbach-Grunsky theorem (for antecedent along similar lines compare Enriques-Chisini 1915/18 [@Enriques-Chisini_1915-1918], Bieberbach 1925 [@Bieberbach_1925], and Wirtinger 1942 [@Wirtinger_1942]) $\spadesuit$ of course Huisman’s paper goes much deeper by exploring the properties of linear series on Harnack-maximal curves (alias $M$-curves)\]  J. Huisman, [*Non-special divisors on real algebraic curves and embeddings into real projective spaces*]{}, Ann. di Mat. 182 (2003), 21–35. \[$\spadesuit$\]  A. Hurwitz, [*Über die Perioden solcher eindeutiger, $2n$-fach periodischer Funktionen, welche im Endlichen überall den Charakter rationaler Funktionen besitzen und reell sind für reelle Werte ihrer $n$ Argumente*]{}, J. Reine Angew. Math. 94 (1883), 1–20. (Math. Werke, Bd. I) \[$\spadesuit$\]  A. Hurwitz, [*Über Riemannsche Flächen mit gegebenen Verzweigungspunkten*]{}, Math. Ann. 39 (1891), 1–61. \[$\spadesuit$ \[13.10.12\] if we fix a ramification divisor in the sphere of degree $b$ and a mapping degree $d$ there is finite number of Riemann surfaces $F$ of Euler characteristic $\chi (F)=d \chi(S^2)-b$ having the prescribed topological behaviour (Hurwitz is able to make a fine study, using of course the monodromy and so to get upper bounds on the number of admissible maps). It seems evident that the game should extend in the bordered setting in the context of Ahlfors circle maps, which are truly (upon doubling) real maps of a special kind (totally real, saturated or separating) from Klein’s orthosymmetric real curves to the real projective line. Then one can try to adventure into similar group theoretical (combinatorial) games as did Hurwitz in the complex case (in fact Hurwitz himself give close attention to reality questions) $\spadesuit$ a more modest question is whether a careful variation of branch points does not produce a quick “action-painting” or “sweeping out” proof of the existence of circle maps of lowest possible degree. $\spadesuit$ as yet I was never able to proceed along this way, which looks yet a reasonable strategy for in the complex case such argument yield at least the right prediction about the gonality of complex curves as divinized by Riemann 1857 (cf. e.g. the heuristic count in Griffiths-Harris 1978 [@Griffiths-Harris_1978/94]). I remind clearly that this idea was suggested by Natanzon (Rennes ca. 2001), and in Rennes 2001/02 (Winter) Johan Huisman also presented to me a simple moduli parameter count somehow comforting the bound $r+p$ (when I suggested him the possibility of the sharpened $r+p$ bound); for the details of Huisman’s count cf. \]  A. Hurwitz, [*Über algebraische Gebilde mit eindeutigen Transformationen in sich*]{}, Math. Ann. 41 (1893), 403–442; or Math. Werke, Bd.I, Funktionentheorie. \[$\spadesuit$ it is proved that if a conformal self-map of a closed Riemann surface of genus $>1$ induces the identity on the first homology group then the self-map is the identity. Historically, one may wonder how this formulation borrowed from Accola ca. 1966 is reliable for the language of homology was not yet “invented” at least in this precise context (recall Poincaré 1895, but of course a myriad of people used the term “homology” in different contexts, e.g. Jordan) $\spadesuit$ despite this detail the assertion is correct\]  A. Hurwitz, R. Courant, [*Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen*]{}, Grundl. der. Math. Wiss. 3, Springer-Verlag, Berlin, 1922. (Subsequent editions 1929, 1964, 706 pp.) \[$\spadesuit$ contains another proof of the KN(=Kreisnormierung) in finite connectivity, according to Schiffer-Hawley [@Schiffer-Hawley_1962], also quoted for this purpose in Stout 1965 [@Stout_1965] $\spadesuit$ Ahlfors once said (recover the source!!) that it this in this book that he learned the length-area principle so fruitful in the theory of quasi-conformal maps (roughly the pendant of Grötzsch’s Flächenstreifenmethode)\]  Y. Imayoshi, [*Holomorphic families of Riemann surfaces and Teichmüller spaces*]{}, Ann. of Math. Stud, 1981. \[$\spadesuit$\]  M.S. Ioffe, [*Extremal quasiconformal embeddings of Riemann surfaces*]{}, Sib. Math. J. (1975), 520–537; English transl. 1976. \[$\spadesuit$ Teichmüller theory for finite bordered surfaces (with optional punctures)\]  I.V. Itenberg, [*Curves of degree 6 with one nondegenerate double point and groups of monodromy of nonsingular curves*]{}, in: Real Algebraic Geometry, Proceedings, Rennes 1991, Lecture Notes in Math. 1524, Springer-Verlag, Berlin, 1992. \[$\spadesuit$\]  I.V. Itenberg, [*Contre-exemples à la conjecture de Ragsdale*]{}, C.R. Acad. Sci. Paris (Sér.I) 317 (1993), 277–282. \[$\spadesuit$ where Ragsdale conjecture is seriously destroyed, yet it remains open the case of $M$-curves, compare also Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale]\]  I.V. Itenberg, [*Groups of monodromy of non-singular curves of degree 6*]{}, in: ???, 199?. \[$\spadesuit$ extending a private communication of Kharlamov, the monodromy groups (ovals permutation) of each chamber of the discriminant is described $\spadesuit$ relies on Nikulin’s classification via K3 surfaces and uses Coxeter, Vinberg, etc.\]  I.V. Itenberg, [*Rigid isotopy classification of curves of degree 6 with one nondegenerate double point*]{}, in: Topology of Manifolds and Varieties, Advances in Soviet Math. 18, Amer. Math. Soc., 1994, 193–208. (English) \[$\spadesuit$ p.196: “Proposition 2.1. [*Each empty oval of a nonsingular curve of degree $6$ can be contracted and there is only one rigid-isotopy class of the result of such a degeneration.*]{}” $\spadesuit$ the method employed seems to depend upon “Nikulin’s approach for obtaining rigid-isotopy classification of nonsingular curves of degree $6$” (cf. p.193) $\spadesuit$ as noted by Viro (in the same volume, p.xiii): “I only want to formulate a conjecture, suggested by Itenberg’s Prop. 2.2: each empty oval of a nonsingular real algebraic plane projective curve can be contracted by a deformation of the curve in the class of curves of the same degree. According to Prop. 2.2, this is true for curves of degree $6$. It is easy to check for curves of degree $\le 5$. The first case for which it is unknown is the case of degree $7$.” $\spadesuit$ \[08.01.13\] In the same vein one can perhaps conjecture that any two ovals lying at the same depth can always coalesce to a single one.\]  I.V. Itenberg, O. Viro, [*Patchworking algebraic curves disproves the Ragsdale conjecture*]{}, The Math. Intelligencer 18 (1996), 19–28. \[$\spadesuit$ self-explanatory title $\spadesuit$ contains (besides some fascinating historical sketches) in particular a formulation of the last vestige of Ragsdale’s conjecture which is still open for $M$-curves (as I learned personally from Th. Fiedler)\]  C. Jacob, [*Sur le problème de Dirichlet dans un domaine plan multiplement connexe et ses applications à l’hydrodynamique*]{}, J. Math. Pures Appl. 18 (1939), 363–383. 78 \[$\spadesuit$ cf. also the next entry\]  C. Jacob, [*Introduction mathématique à la mécanique des fluides*]{}, 1959. (ca. 1286 pp.) \[$\spadesuit$\]  C.G.J. Jacobi, [*Considerationes generales de transcendentibus Abelianis*]{}, Crelle J. Reine Angew. Math. 9 (1832), 394–403. \[$\spadesuit$ Jacobi inversion problem, and first place where jargon like Abelian integrals are employed\] C.G.J. Jacobi, [*Gesammelte Werke, III*]{}. \[$\spadesuit$ cited in Petrovskii 1938 [@Petrowsky_1938] for the so-called Euler-Jacobi formula, as being one of the tool towards Petrovskii’s proof of the extended Hilbert-Rohn theorem forcing the presence of nesting in $M$-curves (though Petrovskii’s inequalities have a universal validity)\] S. Jacobson, [*Pointwise bounded approximation and analytic capacity of open sets*]{}, Trans. Amer. Math. Soc. 218 (1976), 261–283. \[$\spadesuit$ the Ahlfors function appears on p.261, 272, 274 in the context of analytic capacity, which is examined from the angle of the semi-additivity question (Vitushkin) $\spadesuit$ the latter aspect has meanwhile been settled in the seminal breakthrough of Tolsa 2003 [@Tolsa_2003] giving also a complete (geometric) solution to Painlevé’s problem\]  A. Jaffe, S. Klimek, L. Lesniewski, [*Representation of the Heisenberg algebra on a Riemann surface*]{}, Comm. Math. Phys. 126 (1990), 421–433. \[$\spadesuit$\]  H. Jaffee, [*Real algebraic curves*]{}, Topology 19 (1980), 81–87. \[$\spadesuit$ p.82: “We state Harnack’s Theorem in a slightly strengthened form which is probably due to Hurwitz:—[Theorem 2]{}. [*Let $(X, \rho, g,r)$ be as in §$1$. The number $c$ of components of the space $X-X^{\rho}$ is at most $2$. If $c=2$, then $r\le 1+g$ and $g-r$ is odd. If $c=1$, then $r\le g$.*]{}” $\spadesuit$ of course this is historical non-sense, read “Klein” in place of “Hurwitz” $\spadesuit$ \[05.01.13\] the explanation for this historical mistake (alas quite widespread in literature) seems to find its origin in Gudkov’s survey 1974/74 [@Gudkov_1974/74], where Klein’s priority is not sufficiently emphasized! $\spadesuit$ otherwise the paper is quite pleasant albeit quite elementary, especially it cites (p.86) a preprint of Gross 1979 [@Gross_1979-PREPRINT] which probably was phagocytosed in Gross-Harris 1981 [@Gross-Harris_1981]\]  P. Järvi, [*On some function-theoretic extremal problems*]{}, Complex Variables Theory Appl. 24 (1994), 267–270. \[$\spadesuit$ related to the Ahlfors function\]$\bigstar$  J.A. Jenkins, [*On the existence of certain general extremal metrics*]{}, Ann. Math. (2) (1957). \[$\spadesuit$\]  J.A. Jenkins, [*Some new canonical mappings for multiply-connected domains*]{}, Ann. Math. (2) 65 (1957), 179–196. 60, 78 \[$\spadesuit$ new derivation of the parallel-slit maps (and radial avatar) in the slightly extended context of rectangular multi-connected domains (resp. radioactive) domains bounded respectively by rectangles or by rectangles in polar coordinates $\spadesuit$ technique: the classical continuity method à la Brouwer-Koebe, but augmented by some quasi-conformal technology (à la Grötzsch, etc.)\]  J.A. Jenkins, N. Suita, [*On the Pick-Nevanlinna problem*]{}, Kōdai Math. J. 2 (1979), 82–102. \[$\clubsuit$ includes probably an extension and thus also a new derivation of the Ahlfors circle map, compare also Heins 1975 [@Heins_1975] who probably already achieves this goal\]  J.A. Jenkins, N. Suita, [*On analytic maps of plane domains*]{}, Kōdai Math. J. 11 (1988), 38–43. \[$\clubsuit$ for $D$ a plane bordered surface, an analytic map $f\colon D \to \Delta$ to another bordered surface is called [*boundary preserving*]{} it it takes boundary to boundary. “A boundary preserving map $f\colon D \to \Delta$ covers the image domain finitely many times. It can also be extended to the doubles as $\hat D\to \hat \Delta$. Now the Seveli-deFranchis’ theorem[^135] states finiteness for the number of nonconstant analytic maps between two closed Riemann surfaces of genuses both $>1$, hence we get as a dividend finiteness for the above boundary preserving maps, as soon as the genus of the doubles are $>1$. $\spadesuit$ p.40: “Since $f$ is boundary preserving, $f$ has no branch points on the boundary.”, this is completely akin to the Ahlfors map $\spadesuit$ of course the problematic addressed by Jenkins-Suita extends directly to bordered surface of positive genus, and it could be nice to work out corresponding bounds\]  M. Jeong, [*The Szegö kernel and a special self-correspondence*]{}, J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 5 (1998), 101–108. \[$\spadesuit$ the Ahlfors map is briefly mentioned in the following connection: “Since the zeroes of the Szegö kernel are parts of the zeroes of the Ahlfors map and give rise to a particular basis for the Hardy space $H^2(b\Omega)$ (see \[5\]=Bell 1995 [@Bell_1995-Complexity-Indiana]), they can be the powerful tools for getting the properties of the mapping for planar domains.”\]  M. Jeong, M. Taniguchi, [*Bell representations of finitely connected planar domains*]{}, Proc. Amer. Math. Soc. 131 (2002), 2325–2328. \[$\spadesuit$ a problem posed by Bell (1999/2000) is given a positive answer, even in the following sharper form: “Theorem 1.2. [*Every non-degenerate $n$-connected planar domain with $n>1$ is mapped biholomorphically onto a domain defined by $\{ \vert z+ \sum_{k=1}^{n-1}\frac{a_k}{z-b_k}\vert<1\}$ with suitable complex numbers $a_k$ and $b_k$.*]{}” $\spadesuit$ the philosophy of such Bell’s domain is a sort of reverse engineering: instead of constructing the Ahlfors function of a given domain one first gives the function $f$ and define the domain as $\vert f(z) \vert <1$ $\spadesuit$ p.2326, it is observed that Bell’s domains depend on $2n-2$ complex parameters (so $4n-4$ real parameters) exceeding the $3g-3=3(n-1)-3=3n-6$ real moduli predicted by Riemann-Schottky-Klein-Teichmüller $\spadesuit$ this discrepancy is explained by the fact “that every Bell domain is actually associated with an $n$-sheeted branched covering of the unit disk”, for knowing the $a_k,b_k$ we may construct the circle-map $f(z)=z+\sum_{k=1}^{n-1}\frac{a_k}{z-b_k}$ (call it the Bell representation) $\spadesuit$ of course the question arise of describing the “Ahlfors locus” (within the Hurwitz space) of those parameters $a_k,b_k$ such that the Bell representation is actually an Ahlfors extremal function $\spadesuit$ this problem is reposed again in Taniguchi 2004 [@Taniguchi_2004] $\spadesuit$ from the more traditional point view, starting form an $n$-connected domain (with contours $C_1, \dots, C_n$) one can construct a circle map (of minimal degree $n$) by prescribing boundary points $p_i\in C_i$ mapping to $1\in S^1$ (Bieberbach-Grunsky), thus roughly speaking circle maps depends over $n$ parameters, whereas the Ahlfors functions $f_a$ (or $f_{a,b}$) depend only on 2 real parameters (resp. $4$) $\spadesuit$ compare maybe Agler-Harland-Raphael [@Agler-Harland-Raphael_2008] (and its MathReview summary) for a description of the Grunsky functions as the extreme points of the compact convex set of holomorphic functions with positive real parts normalized by $f(z_0)=1$ for some fixed interior point\]  M. Jeong, [*The exact Bergman kernel and the extremal problem*]{}, Kangweon-Kyungki Math. J. 13 (2005), 183–191. \[$\spadesuit$ the Ahlfors map appears twice on p.185–6\]  M. Jeong, J.-W. Oh, M. Taniguchi, [*Equivalence problem for annuli and Bell representations in the plane*]{}, J. Math. Anal. Appl. 325 (2007), 1295–1305. \[$\spadesuit$ the Ahlfors function is employed in the problem of determining the parameter for which a certain doubly connected domain of Bell, namely $\vert z+z^{-1}\vert < r$, is conformal to a circular (concentric) ring\]  G. Jones, D. Singerman, [*Belyi functions, hypermaps and Galois groups*]{}, Bull. London Math. Soc. 28 (1996), 561–590. \[$\spadesuit$\]  P.W. Jones, D.E. Marshall, [*Critical points of Green’s function, harmonic measure, and the corona problem*]{}, Ark. Mat. 23 (1985), 281–314. 47, 50 \[$\spadesuit$ p.293–4 the Ahlfors function enters into the dance as follows: “We mention one more method for solving the corona problem. The previous methods have the drawback that Green’s function does not ignore subsets of $\partial {\cal R}$ which have zero analytic capacity and positive logarithmic capacity. To avoid this we can use Ahlfors’ function, $A$, instead. Ahlfors’ function for a point $\zeta_0 \in{\cal R}$ is defined by $A'_{\zeta_0}(\zeta_0)=\sup \{{\rm Re} f'(z_0) : f\in H^{\infty}({\cal R}), \|f\|_{\infty} \le 1\}$. \[…\] Ahlfors \[1\](=1947), \[2\](=1950) has shown that for our “nice” Riemann surfaces $\vert A_{\zeta_0}(\zeta)\vert =\exp \{ -\sum_{j=0}^{n-1} g(\zeta, \zeta_j)\}$ for some points $\zeta_1, \dots, \zeta_{n-1}\in{\cal R}$. \[…\] Then all of the results of this section hold for the critical points of Ahlfors’ function $\{w_{j,k}\}$ as well as for the critical points of $G$. One can easily construct Riemann surfaces where $\sum_k G(\zeta_k, \zeta')=\infty$, so that the methods using the critical points of $G$ will not work, yet this method using Ahlfors’ function gives solution to the corona problem. \[…\] We remark that we chose the Ahlfors function here because of its natural association with $H^{\infty}({\cal R})$, but we could have chosen any function $F\in H^{\infty}({\cal U})$ with $-\log \vert F(z)\vert=\sum_{j=1}^m G(\pi(z), \alpha_j)$, $\alpha_j\in {\cal R}$.” $\spadesuit$ p.286: “If $\cal R$ is a planar domain, then it is a simple consequence of the argument principle that $G(\zeta, \pi(0))$ has $N-1$ critical points (counting multiplicity), where $N$ is the number of closed boundary curves. See e.g. \[33\](=Nehari 1952 [@Nehari_1952-BOOK])[^136] More generally, the number of critical points of $G$ isthe first Betti number, or the number of generators of the first singular homology group, of $\cal R$ \[46\](=Widom 1971 [@Widom_1971]), and hence is finite. See Walsh \[44, Chap.VII\](=Walsh 1950 [@Walsh_1950-The-location:AMS-Colloq.-Publ.]) for more information concerning the location of the critical points.”\]  P.W. Jones, T. Murai, [*Positive analytic capacity but zero Buffon needle probability*]{}, Pacific J. Math. 133 (1988), 99–114. \[$\spadesuit$ self-explanatory, i.e. a counter-example to the Vitushkin conjecture (that a plane compactum is a Painlevé null-set iff it is invisible, i.e. a.e. projection of the set have zero length) $\spadesuit$ note: the Buffon needle problem was solved by Crofton in 1868: if $E$ is a compactum in the plane, let $\vert P_{\theta}(E)\vert$ be the Lebesgue measure of the orthogonal projection of $E_{\theta}$ on the line of angular slope $\theta$ and define the Crofton invariant as $CR(E)=\int_{0}^{2\pi}\vert P_{\theta}(E)\vert d\theta$. This quantity may be interpreted as the probability of the body $E$ falling over a system of parallel lines equidistantly separated by the diameter of $E$\]  P.W. Jones, [*Square functions, Cauchy integrals, analytic capacity, and harmonic measure*]{}, in: Proc. Conf. on Harmonic Analysis and Partial Differential Equations, El Escorial 1987, Lecture Notes in Math. 1384, Springer-Verlag, 1989, 24–68. \[$\spadesuit$\]  P.W. Jones, [*Rectifiable sets and the travelling salesman problem*]{}, Invent. Math. 102 (1990), 1–16. \[$\spadesuit$\]  C. Jordan, [*Sur la déformation des surfaces*]{}, J. Math Pures Appl. (2) 11 (1866), 105–109. \[$\spadesuit$ after Möbius 1863 [@Moebius_1863] in the closed case, discuss a classification of compact orientable bordered surfaces, by the genus and the number of contours $\spadesuit$ quoted in Klein’s lectures 1892/93 [@Klein_1892_Vorlesung-Goettingen p.150], and in Weichold 1883 [@Weichold_1883 p.330], who need the non-orientable case as well\]  J. Jost, [*Conformal mappings and the Plateau-Douglas problem in Riemannian manifolds*]{}, J. Reine Angew. Math. 359 (1985), 37–54. \[$\spadesuit$ reprove some results about conformal mapping (uniformization of real orthosymmetric curves) surely well-known since Koebe’s era (and conjectured by Klein) via the method of Plateau $\spadesuit$ then attack and solve a very general case of Plateau’s problem in a generality unifying the desire of Douglas (positive genus) and Morrey (curvy ambient Riemannian manifold instead of flat Euclid) $\spadesuit$ reports also some of Tromba’s critics over the solution of Courant to the Plateau-Douglas problem of higher genus $\clubsuit$ it is not clear to the writer if such critics (of Tromba) affects as well the whole content of Courant’s book 1950 [@Courant_1950] especially regarding the varied type of conformal maps $\spadesuit$ at any rate Jost propose a parade using techniques of Mumford and Schoen-Yau, but the “meandreousness” of the resulting proof is slightly criticized in Hildebrandt-von der Mosel 2009 [@Hildebrandt-von-der-Mosel_2009]\]  J. Jost, [*Two-dimensional geometric variational problem*]{}, Wiley, New York, 1991. \[$\spadesuit$ from Sauvigny’s review in BAMS: “Chapter 3 deal with conformal representation of surfaces homeomorphic to the sphere $S^2$, circular domains, and closed surfaces of higher genus. The proof is given by direct variational methods and not as usual by uniformization, completing a fragmentary proof of Morrey.\]  G. Julia, [*Sur la représentation conforme des aires simplement connexes*]{}, C.R. Acad. Sci. Paris 182 (1926), 1314–1316. \[$\spadesuit$ another characterization of the (Riemann) mapping function by a minimum principle\]  G. Julia, [*Développement en série de polynômes ou de fonctions rationelles de la fonction qui fournit la représentation conforme d’une aire simplement connexe sur un cercle*]{}, Ann. Éc. Norm. Sup. 44 (1927), 289–316. \[$\spadesuit$ Seidel’s summary: a determination of a sequence of polynomials is given which converges to the properly normed (Riemann) mapping function of a simply-connected region\]  G. Julia, [*Leçon sur la représentation conforme des aires simplement connexes*]{}, Gauthier-Villars, Paris, 1931. \[$\spadesuit$ one among the early book format exposition of the extremal properties of the Riemann mapping for a plane simply-connected region (distinct of ${\Bbb C}$), namely that the range of the map normalized by $f'(z_0)=1$ has minimal area (first in Bieberbach 1914 [@Bieberbach_1914]) or that the boundary of the range has minimal length (probably first in Szegö 1921 [@Szego_1921]) $\spadesuit$ for both those extremal principles see also the detailed treatment in the book Gaier 1964 [@Gaier_1964-BOOK-Konstruktive-methoden]\]  G. Julia, [*Sur la représentation conforme des aires multiplement connexes*]{}, Ann. Sc. Norm. Sup. Pisa (2) 1 (1932), 113–138. 78 \[$\spadesuit$ still in great admiration for Schottky 1877 [@Schottky_1877] and use Klein’s jargon of orthosymmetry, yet confined to the case of domains $\spadesuit$ however the main purpose is the study of a new sort of mapping introduced by de la Vallée Poussin (and which will in turn fascinate Walsh and Grunsky)\]  G. Julia, [*Reconstruction d’une surface de Riemann $\sigma$ correspondant à une aire multiplement connexe $\cal A$*]{}, C.R. Acad. Sci. Paris 194 (1932), 423–425. 60  G. Julia, [*Prolongement d’une surface de Riemann $\sigma$ correspondant à une aire multiplement connexe $\cal A$*]{}, C.R. Acad. Sci. Paris 194 (1932), 580–583. 60  G. Julia, [*Leçon sur la représentation conforme des aires multiplement connexes*]{}, Gauthier-Villars, Paris, 1934, 94 pp. 60, 78 \[$\spadesuit$\]  G. Julia, [*La représentation conforme des aires multiplement connexes*]{}, L’Enseign. Math. 33 (1935), 137–168. 78 \[$\spadesuit$ survey from Riemann, Schottky 1877 [@Schottky_1877] through Hilbert 1909 [@Hilbert_1909], Koebe, up to the extremal treatments by de Possel and Grötzsch (slit mappings in infinite connectivity)\]  G. Julia, [*Quelques applications fonctionnelles de la topologie*]{}, Reale Accademia d’Italia Fondazione A. Volta, Att dei Convegni 9 (1939), Rome, 1943, 201–306. 60 \[$\spadesuit$ cited in Ahlfors-Sario 1960\]$\bigstar$$\bigstar$$\bigstar$  M. Juurchescu, [*A maximal Riemann surface*]{}, ???? ?? (1961?), 91–93. \[$\spadesuit$ p.91, a map between bordered Riemann surfaces taking boundary to boundary is termed [*distinguished*]{}\]  G. Kahn, [*Eine allgemeine Methode zur Untersuchung der Gestalten algebraischer Kurven*]{}, Inaugural Dissertation, Göttingen, 1909. \[$\spadesuit$ Dissertation under Hilbert (cf. e.g. Hilbert 1909 [@Hilbert_1909-Ueber-die-Gestalt-sextic]), attempting to prohibit the real sextic scheme consisting of 11 unnested ovals $\spadesuit$ considered non-rigorous in Gudkov 1974 [@Gudkov_1974/74] $\spadesuit$ historical anecdote: Kahn’s work as well as the related Thesis by Löbenstein 1910 [@Löbenstein_1910] were instead considered as rigorous in Hilbert 1909 [@Hilbert_1909-Ueber-die-Gestalt-sextic])\]  S. Kakutani, [*Rings of analytic functions*]{}, Lectures on functions of a complex variable, 71–83, Univ. of Michigan Press, Ann Arbor, 1955. \[$\spadesuit$\]$\bigstar$  C. Kalla, Ch. Klein, [*On the numerical evaluation of algebro-geometric solutions to integrable equations*]{}, Nonlinearity 25 (2012), 569–596. \[$\spadesuit$\]  C. Kalla, Ch. Klein, [*Computation of the topological type of a real Riemann surface*]{}, arXiv (2012). \[$\spadesuit$\]  L.V. Kantorovič, [*FOUR ARTICLES IN FRENCH in the period 33–34 including multi-connected and potentially based upon Bieberbach’s method*]{} $\bigstar$$\bigstar$$\bigstar$  L.V. Kantorovič, V.I. Krylov, [*Methods for the approximate solution of partial differential equations*]{}, Leningrad–Moscow, 1936, Russian. \[$\spadesuit$ Chap.V is devoted to conformal mapping. §1 is introductory. §2 takes up the method of Bieberbach (1914 [@Bieberbach_1914]) which reduces the problem of conformal mapping to a minimum principle (for the area). This is then solved by Ritz’s method. In §3 a second extremal property for mapping functions is discussed and Ritz’s method is again applied §4 takes up orthogonal polynomials of Szegö and Bochner-Bergman types and applies them to the above minimizing problems.\]$\bigstar$$\bigstar$$\bigstar$  V. Karimipour, A. Mostafazadeh, [*Lattice topological field theory on nonorientable surfaces*]{}, J. Math. Phys. 38 (1997), 49–66. \[$\spadesuit$\]  M.G. Katz, S. Sabourau, [*Hyperellipticity and systoles on Klein surfaces*]{}, Geom. Dedicata 159 (2012), 277–293. \[$\spadesuit$\]  S. Katz, C.-C.M. Liu, [*Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc*]{}, Geom. Topol. Monogr. 8 (2006), 1–47; reproduced from: Adv. in Theoret. Math. Phys. 5 (2002), 1–49. \[$\spadesuit$\]  M.V. Keldysh, M.A. Lavrentief, [*Sur la représentation conforme des domaines limités par des courbes rectifiables*]{}, Ann. Sci. Éc. Norm. Sup. 54 (1937), 1–38. \[$\spadesuit$ only the case of simply-connected domains bounded by a rectifiable Jordan curve in the plane, but deep questions about the boundary behavior of the Riemann map $\varphi$ as well as the Smirnov problem of deciding when the harmonic function $\log \vert \varphi'(w)\vert$ is representable in the unit-disc by the Poisson integral of its (limiting) values on the circumference\]  M.V. Keldysh, [*Sur la résolubilité et la stabilité du problème de Dirichlet*]{}, C.R. Acad. Sci. URSS 18 (1938), ???–??? (French). \[$\spadesuit$ quoted in Walsh-Sinclair 1965\]$\bigstar$$\bigstar$$\bigstar$  M.V. Keldysh, [*Sur l’approximation en moyenne quadratique des fonctions analytiques*]{}, Mat. Sb. (N.S.) 5 (1939), 391–401. \[$\spadesuit$ quoted in Walsh-Sinclair 1965\]$\bigstar$$\bigstar$$\bigstar$  M.V. Keldysh, [*Conformal mappings of multiply connected domains on canonical domains*]{}, (Russian) Uspehi Mat. Nauk 6 (1939), 90–119. 78 \[$\spadesuit$ a survey of the developments in the field, up to 1939\]  O.D. Kellogg, [*Foundations of potential theory*]{}, Grundl. d. math. Wiss. 31, Springer, Berlin, 1929. \[$\spadesuit$ “Introduction to fundamentals of potential functions covers: the force of gravity, fields of force, potentials, harmonic functions, electric images and Green’s function, sequences of harmonic functions, fundamental existence theorems, the logarithmic potential, and much more.”\]  G. Kempf, [*Schubert methods with an application to algebraic curves*]{}, Stichting mathematisch centrum, Amsterdam, 1971. \[$\spadesuit$ the first (simultaneous with Kleiman-Laksov 1972 [@Kleiman-Laksov_1972]) existence proof of special divisors in the general case, extending thereby the result of Meis 1960 [@Meis_1960]\]  B. Kerékjártó, [*Vorlesungen über Topologie I, Flächen Topologie*]{}, Springer, Berlin, 1923. \[$\spadesuit$ a seminal work (Part II never occurred) with parcelled appreciation (disliked by Lefschetz but admired by Weyl) $\spadesuit$ cited in Natanzon 1993 [@Natanzon_1993 p.268] for the basic result that one may lift a complex structure under a branched covering $\spadesuit$ \[30.12.12\] boosting somewhat the method one could hope to reprove so the Ahlfors theorem\]  M. Kervaire, J. Milnor, [*On $2$-spheres in $4$-manifolds*]{}, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1651–1657. \[$\spadesuit$ as noted in Kronheimer-Mrowka 1994 [@Kronheimer-Mrowka_1994]\]  N. Kerzman, E.M. Stein, [*The Cauchy kernel, the Szegö kernel, and the Riemann mapping function*]{}, Math. Ann. 236 (1978), 85–93. \[$\spadesuit$ quite influential, especially over Bell\]  G. Khajalia, [*Sur la représentation conforme des domaines doublement connexes*]{}, (French) Mat. Sb. N.S. 8 (1940), 97–106. 78 \[$\spadesuit$ Seidel’s summary: the problem of mapping a doubly connected finite region on a circular ring is reduced to minimizing an area integral for a certain class of functions. If the region is accessible from without, then a sequence of minimal rational fractions converges uniformly to the desired mapping function $\spadesuit$ in fact the condition in question seems to ensure the least area map (minimizing $\int\int_{B}\vert f' (z)\vert^2 d\omega$) to be schlicht and maps it upon the concentric circular ring $1<\vert w-w_0\vert<R$, thus the problem is different from that à la Bieberbach-Bergman handled in Kufareff 1935/37 [@Kufareff_1935/37] where the least area map is not univalent $\spadesuit$ a naive question \[05.08.12\] is whether Khajalia’s method could perform the Kreisnormierung in higher connectivity\]  V.M. Kharlamov, [*The maximum number of components of a surface of degree $4$ in ${\Bbb R}P^3$*]{}, Funkt. Anal. Prilozhen. 6 (1972), 101; English transl., Funct. Anal. Appl. ? (197?), ?–?. \[$\spadesuit$ besides Gudkov 1969, yet another solution to one part of Hilbert’s 16th problem (or even Rohn 1886 [@Rohn_1886]) asking for the maximum number of “sheets” of the 4th order in three dimensional space (Kharlamov’s answer $10$, already in his Master Thesis 1972)\]  V.M. Kharlamov, [*New congruences for the Euler characteristic of real algebraic manifolds*]{}, Funkt. Anal. Prilozhen. 7 (1973), 74–78; English transl., Funct. Anal. Appl. 7 (1973), 147–150. \[$\spadesuit$ cited in Wilson 1978 [@Wilson_1978] for an $(M-1)$-avatar (i.e. $p-n=k^2\pm 1 \pmod 8$) of the Gudkov-Rohlin congruence modulo 8 (i.e. $p-n=k^2 \pmod 8$).\]  V.M. Kharlamov, [*The topological type of nonsingular surfaces in ${\Bbb R}P^3$ of degree four*]{}, Funkt. Anal. Prilozhen. 10 (1976), 55–68; English transl., Funct. Anal. Appl. 10 (1976), 295–305. \[$\spadesuit$ topological classification of nonsingular quartics surfaces in 3-space resting on the theory of K3 surfaces (via Tyurina, but [*not*]{} via Torelli’s theorem of Pyatetsky-Shapiro-Shafarevich 1971/71 [@Pyatetsky-Shapiro-Shafarevich_1971/71])\]  V.M. Kharlamov, [*Isotopic types of nonsingular surfaces of fourth degree in ${\Bbb R}P^3$*]{}, Funkt. Anal. Prilozhen. 12 (1978), 86–87; English transl., Funct. Anal. Appl. ? (197?), ?–?. \[$\spadesuit$\]  V.M. Kharlamov, [*Real algebraic surfaces*]{}, Proc. Internat. Congr. of Mathematicians, Helsinki, 1978, 421–428. \[$\spadesuit$\]  V.M. Kharlamov, O.Ya. Viro, [*Congruences for real algebraic curves with singularities*]{}, Uspekhi Mat. Nauk 35 (1980), 154–155; English transl., ?? (198?), ?–?. \[$\spadesuit$ cited in Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91]\]  V.M. Kharlamov, [*Rigid classification up to isotopy of real plane curves of degree $5$*]{}, Funkt. Anal. Prilozhen. 15 (1981), 88–89; English transl., Funct. Anal. Appl. 15 (1981), 73–74. \[$\spadesuit$ as a historical curiosity the same result in degree $6$ was effected earlier in Nikulin 1979/80 [@Nikulin_1979/80]\]  V.M. Kharlamov, [*On the classification of non-singular surfaces of degree $4$ in ${\Bbb R}P^3$ with respect to rigid*]{}, Funkt. Anal. Prilozhen. 18 (1984), 49–56; English transl., Funct. Anal. Appl. 18 (1984), 49–56. \[$\spadesuit$\]  V.M. Kharlamov, O. Viro, [*Extensions of the Gudkov-Rohlin congruence*]{}, in: Topology and Geometry, Rohlin Seminar, edited by O.Ya. Viro, 1984–86, Lecture Notes in Math. 1346, Springer (1988 or 1991? CHECK DATE), 357–406. \[$\spadesuit$ p.359: “type I or dividing”\]  V.M. Kharlamov, O. Viro, [*Easy reading on topology of real plane algebraic curve*]{}, UNDATED but (ca. 1978–2013), i.e. a shortened version of the book planned (but apparently never completed) by Rohlin-Kharlamov-Viro. \[$\spadesuit$ \[21.03.13\] p.15, contains valuable information on Ragsdale, yet overlapping with Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale]. It seems to me (Gabard) that one-half of the Ragsdale conjecture follows from Thom[^137], cf. Lemma \[Thom-implies-one-half-of-Ragsdale:lem\], and so answers one half of question 10.E posed on p.15 (where incidentally it seems to me that there is the same misprint as in Itenberg-Viro 1996 [@Itenberg-Viro_1996-disproves-Ragsdale])\]  D. Khavinson, [*On removal of periods of conjugate functions in multiply connected domains*]{}, Michigan Math. J. 31 (1984), 371–379. 50 \[$\spadesuit$ p.377 reproves the Bieberbach-Grunsky-Ahlfors theorem in the planar case while quoting Heins 1950 [@Heins_1950] and using the classical device of annihilating “the periods of the conjugate function”\]  R. Kirby, [*Problems in low-dimensional topology*]{}, 1970, updated in 1995 (available on the net). \[$\spadesuit$ Thom’s conjecture is mentioned as Problem 4.36, where the proof of Kronheimer-Mrowka 1994 and Morgan-Szabó-Taubes 1995 are cited\]  G. Kirchhoff, [*Über das Gleichgewicht und die Bewegung einer elastischen Scheibe*]{}, J. Reine Angew. Math. 40 (1850), 51–88. \[$\spadesuit$ Riemann was aware of this ref. in connection to the Dirichlet principle (cf. Neuenschwander 1981 [@Neuenschwander_1981]), yet never mentions it in print $\spadesuit$ the next big revolution is Ritz, see Gander-Wander 2012 [@Gander-Wanner_2012] for a thorough “mise en perspective”\]  S. Kirsch, [*Transfinite diameter, Chebyshev constant and capacity*]{}, in: Handbook of Complex Analysis, Elsevier, 2005. 50 \[$\spadesuit$ extract from the web (whence no page): “Ahlfors generalized Garabedian’s result to regions on Riemann surfaces \[2\](=Ahlfors 1950 [@Ahlfors_1950]); see Royden’s paper \[159\](=1962 [@Royden_1962]) for another treatment as well as further references to the literature.” $\spadesuit$ compare (if you like) our (depressive) Sec.\[dissident:sec\] for a complete list of “dissident” authors having apparently (like me) some pain to digest Ahlfors proof, and therefore cross-citing often Royden $\spadesuit$ “Abstract. The aim of the present chapter is to survey alternate descriptions of the classical transfinite diameter due to Fekete and to review several generalizations of it. Here we lay emphasis mainly on the case of one complex variable. We shall generalize this notion…”\]$\bigstar$$\bigstar$  S.L. Kleiman, D. Laksov, [*On the existence of special divisors*]{}, Amer. J. Math. 94 (1972), 431–436. \[$\spadesuit$ cite Riemann 1857 [@Riemann_1857], Hensel-Landsberg 1902 [@Hensel-Landsberg_1902] for linear series of dimension $1$, and Brill-Noether 1874 [@Brill-Noether_1874], Severi 1921 [@Severi_1921-Vorlesungen-u-alg.-Geom-BUCH] in the general case $\spadesuit$ supplies an existence proof of its title via Schubert calculus, Poincaré’s formula, some EGA (=Grothendieck), and a bundle constructed in Kempf’s Thesis $\spadesuit$ compare Kempf 1971 [@Kempf_1971] for a simultaneous solution of the same fundamental problem $\spadesuit$ \[08.10.12\] since this Kempf-Kleiman-Laksov result includes as a special case the result of Meis 1960 [@Meis_1960], it enables one eradicating Teichmüller theory from the gonality problem (this is not so surprising for Poincaré’s formula is essentially “homology theory” (intersection theory) specialized to the Jacobian variety, and the theta-divisor, image the $(g-1)$-symmetric power $C^{(g-1)}$ of the curve into the Jacobian via the Abel map $\spadesuit$ thus roughly speaking (and with some imagination) we are back to the method used in Gabard 2006 [@Gabard_2006] $\spadesuit$ for less arrogant looseness it would be nice to adapt the methods of Kempf/Kleiman-Laksov to the problem of the Ahlfors mapping with sharp bounds (i.e. like in Gabard 2006 [@Gabard_2006] granting of course the latter to be correct, else)\]  S.L. Kleiman, D. Laksov, [*Another proof of the existence of special divisors*]{}, Acta Math. 132 (1974), 163–176. \[$\spadesuit$ cite Gunning’s work of 1972 [@Gunning_1972] as an alternative to Meis’ (for linear series of dimension $1$) $\spadesuit$ novel proof via the theory of singularities of mappings (Thom polynomial, Porteous’ formula, plus influence of Mattuck) $\spadesuit$ \[08.10.12\] like in the previous entry, try again to specialize the Thom-Porteous technique to the context of real algebraic geometry (orthosymmetric curve à la Klein) so as to recover the circle maps of Ahlfors 1950 [@Ahlfors_1950], optionally with the bound of Gabard 2006 [@Gabard_2006] $\spadesuit$ of course the view point of special divisors (=essentially those moving in linear systems $g_d^r$ of dimensions higher than predicted by Riemann’s inequality $\dim \vert D \vert \deg D -g$ (due to the $g$ constraints imposed by Abelian differentials) seems to indicate that the theory of the Ahlfors map is just the top of a much larger iceberg, probably already partially explored by experts (Coppens, Huisman, Ballico, Martens, Monnier, etc.)\]  F. Klein, [*Über die sogenannte Nicht-Euklidische Geometrie*]{}, Math. Ann. 4 (1871), also in Ges. math. Abh. I, 244–253. \[$\spadesuit$\]  F. Klein, [*Über Flächen dritter Ordnung*]{}, Math. Ann. 6 (1873), also in Ges. math. Abh. II, 11–62. \[merely cited for Plücker 1839 [@Plücker_1839] as being the oldest user (recorded) of the method of “small perturbation”, compare also Gudkov 1974/74 [@Gudkov_1974/74] whose first entry in his Refs. list is Plücker 1839 $\spadesuit$ “Wenn eine Kurve mit Doppelpunkten gezeichnet vorliegt, so kann man aus ihr Kurven derselben Ordnung ohne Doppelpunkt oder mit weniger Doppelpunkten schematisch ableiten, indem man die in den Doppelpunkten oder einigen derselben zusammensto[ß]{}enden Kurvennäste durch ähnlich verlaufende, sich nicht treffende ersetzt. Nach diesem ebenso einfachen als fruchtbaren Prinzip \[footnote=Wer diese Prinzip zuerst verwertet hat, lä[ß]{}t sich bei dessen gro[ß]{}er Selbstverständlichkeit wohl kaum festellen. Dem Verf. is dasselbe, sowie namentlich das Beispiel der Erzeugung einer Kurve $n$-ter Ordnung aus $n$ geraden Linien, von Plücker her bekannt: vgl. z.B. dessen Theorie der algebraischen Kurven (1839), in welcher fortwärend ähnliche Überlegungen angewandt werden.\] erhählt man z.B. ohne weiteres die beiden Grundformen der ebenen Kurven dritter Ordnung, wenn \[…\]”\]  F. Klein, [*Bemerkungen über den Zusammenhang der Flächen*]{}, (zwei Aufsätze aus den Jahren 1874 und 1875/76), Math. Ann. 7, 9 (1874, 1875/76), also in Ges. math. Abh. II, 63–77. \[$\spadesuit$ some discussions with Ludwig Schläfli about the topology of surfaces (especially in the non-orientable case) $\spadesuit$ taken together with the earlier works of Riemann, Möbius 1860/63 [@Moebius_1863] and Jordan 1866 [@Jordan_1866] this constitutes a complete classification of finite(=compact) surfaces be they orientable or not, bordered or closed $\spadesuit$ this classification is of course instrumental to Klein’s classification of the topology of real algebraic curves (equivalently symmetric Riemann surfaces), as discussed in Klein 1876 [@Klein_1876], Klein 1882 [@Klein_1882] or Klein 1892 [@Klein_1892_Realitaet], as well as in Weichold 1883 [@Weichold_1883]\]  F. Klein, [*Über eine neue Art der Riemannschen Flächen*]{} (Erste Mitteilung), Math. Ann. 7 (1874), also in Ges. math. Abh. II, 89–98. \[$\spadesuit$ first apparition of some “new” types of Riemann surface, which later will evolve to the concept of “Klein surfaces”, but at this stage this is merely a synthetic visualization of the complex locus of a plane curve defined over the reals upon the real projective plane via the map assigning the unique real point of an imaginary line. Also this is not yet “[*was ich den “echten” Riemann zu nennen pflege*]{}” as Klein expresses himself in the Introd. to volume 2 of his Coll. Papers [@Klein-Werke-III_1923 p.5] $\spadesuit$ however it is obvious that this mode of representation is almost forgotten by now and perhaps it could be useful in the future (e.g., to reprove the Rohlin inequality saying that plane dividing curves have at least as many ovals than their half degree, cf., e.g., Gabard 2000 [@Gabard_2000] for more details and the original refs.)\]  F. Klein, [*Über den Verlauf der Abelschen Integrale bei den Kurven vierten Grades*]{} (Erster Aufsatz), Math. Ann. 10 (1876); also in Ges. math. Abh. II, 99–135.  F. Klein, [*Über eine neue Art von Riemannschen Flächen*]{} (Zweite Mitteilung), Math. Ann. 10 (1876), also in Ges. math. Abh. II, 136–155. \[$\spadesuit$ p.154 the first place where the dichotomy of “dividing” curves appears, under the designation “Kurven der ersten Art/zweiten Art” depending upon whether its Riemann surface is divided or not by the real locus (this is from where derived the Russian terminology type I/II) \[hopefully Klein came up later with the better terminology ortho- vs. diasymmetric!\] $\spadesuit$ p.154 contains also the first intrinsic proof of the Harnack inequality (1876)\]  F. Klein, [*Ueber die conforme Abbildung von Flächen*]{}, Math. Ann. 19 (1882), 159–160. \[$\spadesuit$ a lovely announcement of the next item [@Klein_1882], showing a little influence of Schwarz (Ostern 1881). NB: item not reproduced in the Ges. math. Abh.\]  F. Klein, [*Über Riemann’s Theorie der algebraischen Funktionen und ihrer Integrale*]{} B.G. Teubner, Leipzig, 1882. 60 \[$\spadesuit$ a masterpiece where Klein’s theory reaches full maturity $\spadesuit$ long-distance influence upon Teichmüller 1939 [@Teichmueller_1939] (moduli problems including the case of possibly non-orientable surfaces, alias Klein surfaces since Alling-Greenleaf), and Douglas 1936–39 [@Douglas_1936-Some-new-results; @Douglas_1939-The-most-general] and also Comessatti 1924/26 [@Comessatti_1924/26], Cecioni 1933 [@Cecioni_1933], etc. $\clubsuit$ evident (albeit subconscious) connection with Ahlfors 1950 [@Ahlfors_1950], yet first made explicit (in-print) only by Alling-Greenleaf 1969 [@Alling-Greenleaf_1969] (to the best of the writer’s knowledge)\]  F. Klein, [*Über eindeutige Funktionen mit linearen Transformationen in sich. Erste Mitteilung.*]{} Math. Ann. 19 (1882); also in [*Gesammelte mathematische Abhandlungen. Dritter Band*]{}. 1923, Reprint Springer-Verlag, 1973, 622–626.  F. Klein, [*Über eindeutige Funktionen mit linearen Transformationen in sich. Zweite Mitteilung.*]{} Math. Ann. 20 (1882); also in [*Gesammelte mathematische Abhandlungen. Dritter Band*]{}. 1923, Reprint Springer-Verlag, 1973, 627–629.  F. Klein, [*Neue Beiträge zur Riemannschen Funktionentheorie*]{}, Math. Ann. 21 (1882/83); also in [*Gesammelte mathematische Abhandlungen. Dritter Band*]{}. 1923, Reprint Springer-Verlag, 1973, 630–710.  F. Klein, [*Über Realitätsverhältnisse bei der einem beliebigen Geschlechte zugehörigen Normalkurve der $\varphi$,*]{} Math. Ann. 42 (1892), 1–29. \[$\spadesuit$ this means the canonical embedding by holomorphic differentials into ${\Bbb P}^{g-1}$, which is like the Gauss map of the Abel embedding normalized through translation within the Jacobi torus $\spadesuit$ an incredible interplay between the intrinsic geometry of the symmetric Riemann surface (including its topological characteristics) and the real enumerative issues allied to the canonical embedding, compare Gross-Harris 1981 [@Gross-Harris_1981] as the most cited best modern counterpart\]  F. Klein, [*Riemannsche Flächen, I.*]{} Vorlesung, gehalten während des Wintersemester 1891–92, Göttingen 1892, Neuer unveränderter Abdruck, Teubner, Leipzig 1906. 60  F. Klein, [*Riemannsche Flächen, II.*]{} Vorlesung, gehalten während des Sommersemester 1892, Göttingen 1893, Neuer unveränderter Abdruck, Teubner, Leipzig 1906. 60 \[$\spadesuit$ for those not overwhelmed by German prose and handwritings, these lecture notes gives a very exciting view over Klein’s lectures and a good supplement to his papers. NB: these 2 items are [*not*]{} reprinted in the Ges. math. Abh., and somewhat hard-to-find in Switzerland but well-known in Russia, cf. e.g. Gudkov [@Gudkov_1974/74] and Natanzon 1990 [@Natanzon_1990/90], plus also in some US references, of course\]  F. Klein, [et al.]{} [*Zu den Verhandlungen betreffend automorphe Funktionen, Karlsruhe am 27. September 1911. Vorträge und Referate von F. Klein, L.E.J. Brouwer, P. Koebe, L. Bieberbach und E. Hilb*]{}. Jahresb. d. Deutsch. Math.-verein. 21 (1912), 153–166. \[$\spadesuit$ an account of the dramatic events occurring in 1911, when Brouwer was able to re-crack the uniformization (of Poincaré-Koebe, at least in the reasonable near to compact context) via topological methods (viz. invariance of domain) implementing thereby the old dream of Klein-Poincaré (or vice versa if you prefer)\]  F. Klein, [*Gesammelte mathematische Abhandlungen. Zweiter Band*]{}. 1922, Reprint Springer-Verlag, 1973. 60  F. Klein, [*Gesammelte mathematische Abhandlungen. Dritter Band*]{}. 1923, Reprint Springer-Verlag, 1973. 60, 78  F. Klein, [*Vorlesungen über die Entwicklung der Mathematik im 19.Jahrhundert, Teil I*]{}. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Bd.24, Springer-Verlag, Berlin, 1926; Russian transl., [*Lektsii o razvitii matemiki v XIX stoletti*]{}, ONTI, Moscow-Leningrad, 1937. \[$\spadesuit$ where according to the legend Arnold learned all his background about mathematics $\spadesuit$ often cited, e.g. by Arnold, Gudkov, etc.\]  T. Klotz, [*Imbedding compact Riemann surfaces in $3$-space*]{}, Pacific J. Math. 11 (1961), 1035–1043. \[$\spadesuit$ cited in Garsia 1961 [@Garsia_1961] as follows: “Some interesting results on $C^{\infty}$ imbeddings in the higher genus case have been obtained by T. Klotz in \[10\](=this paper). This author is almost successful in proving that the set of Riemann surfaces of a given genus $g\ge 2$ which can be imbedded in Euclidean space is open \[footnote 2: using the results of Kuiper it could be shown that it is dense.\] in the Teichmüller topology. Perhaps we should point out that from some of the results of the present paper one obtains the arguments that are needed to complete her proof.”\]  M. Knebusch, [*On real algebraic curves over real closed fields. I*]{}, Math. Z. 150 (1976), 49–70. \[$\spadesuit$\]  M. Knebusch, [*On real algebraic curves over real closed fields. II*]{}, Math. Z. 151 (1976), 189–205. \[$\spadesuit$\]  J.T. Knight, [*Riemann surfaces of field extensions*]{}, Proc. Cmabridge Philos. Soc. 65 (1969), 635–650. \[$\spadesuit$ cited in Geyer-Martens?, Monnier 2007\]  S.-K. Ko, [*Embedding Riemann surfaces in Riemannian manifolds*]{}, University of Connecticut, Dissertation, Aug. 1989. \[$\spadesuit$ it is shown that every compact (=closed) Riemann surface admits a conformal embedding in any preassigned Riemannian manifold of dimension $\ge 3$. Compare also the treatment in Ko 2001 [@Ko_2001]\]  S.-K. Ko, [*Embedding bordered Riemann surfaces in Riemannian manifolds*]{}, J. Korean Math. Soc. 30 (1993), 465–484. \[$\spadesuit$ §0, Introd.: “Around 1960, A. Garsia (\[6\]=1961 [@Garsia_1961]) proved that every compact Riemann surface can be conformally immersed in Euclidean $3$-space ${\Bbb R}^3$. He stated that he had found a realization of every compact surface as a classical surface although Klein required that classical surfaces be embedded. \[Garsia’s proof uses Teichmüller’s idea, results, and constructions inspired by Nash’s embedding theorem and Brouwer’s fixed point theorem.\][^138]—In 1970, Rüedy extended Garsia’s result to open Riemann surfaces $S$ by applying Garsia’s techniques to compact exhaustions of $S$ (\[16\]=Rüedy 1971 [@Ruedy_1971]) and later[^139] he proved that every compact Riemann surface can be conformally embedded in ${\Bbb R}^3$ (\[17\]=Rüedy 1971 [@Ruedy_1971-BOOK], \[18\]=Rüedy 1968 [@Ruedy_1968]).” $\spadesuit$ \[10.12.12\] It is not clear (to Gabard) if this reflects the real history, for Rüedy himself seems always to ascribe the full embedded result to Garsia, yet perhaps by over-modesty in case Ko’s description is correct!?? $\spadesuit$ next: “In 1989, author apply[^140] Teichmüller theory to prove that we can find a conformally equivalent model surface in an orientable Riemannian manifold $\frak M$ of $\dim \frak M \ge 3$ for every compact Riemann surface (\[8\]=Ko 1989 [@Ko_1989-compact]).—Here we prove the [*extension of the Embedding theorem for compact Riemann surfaces (Ko \[8\]=Ko 1989 [@Ko_1989-compact]) to finite topological type Riemann surfaces in orientable Riemannian manifolds.*]{}\]  S.-K. Ko, [*Embedding open Riemann surfaces in Riemannian manifolds*]{}, J. Geom. Anal. 9 (1999), 119–141. \[$\spadesuit$ like in the previous entry the author persists in his assertion that Garsia only obtained immersed conformal maps to classical surfaces, while ascribing the embedded results again to Rüedy. $\spadesuit$p.119 (abstract): “Any open Riemann surface has a conformal model in any orientable Riemannian manifold. Precisely, we will prove that, given any open Riemann surface, there is a conformally equivalent model in a prespecified orientable Riemannian manifold \[of $\dim\ge 3$\].”\]  S.-K. Ko, [*Embedding compact Riemann surfaces in Riemannian manifolds*]{}, Houston J. Math. 27 (2001), 541–577. \[$\spadesuit$ seems to be a published account of the result arrived at in the Ph.D. Dissertation of the writer (Ko 1989 [@Ko_1989-compact]); i.e. p541 (abstract): “Any compact Riemann surface has a conformal model in any orientable Riemannian manifold. Precisely, we will prove that, given any open Riemann surface, there is a conformally equivalent model in a prespecified orientable Riemannian manifold \[of $\dim\ge 3$\]. The techniques we use include Garsia’s Continuity Lemma, Brouwer’s Fixed Point Theorem along with techniques from Teichmüller theory.”\]  S. Kobayashi, N. Suita, [*On analytic diameters and analytic centers of compact sets*]{}, Trans. Amer. Math. Soc. 267 (1981), 219–228. 47, 50 \[$\spadesuit$ Ahlfors function and the allied conceptions of Vitushkin (analytic diameter and center), plus negative answers to several of Minsker’s questions (cf. Minsker 1974 [@Minsker_1974])\]  S. Kobayashi, [*On analytic centers of compact sets*]{}, Kodai Math. J. 5 (1982), 318–328. 47, 50 \[$\spadesuit$ second derivative variant of the Ahlfors function developed along conceptions of Vitushkin (analytic diameter and center) and Minsker\]  B. Köck, D. Singerman, [*Real Belyi theory*]{}, Quarterly J. Math. 58 (2007), 463–478. \[$\spadesuit$ “Abstract. We develop a Belyi-type theory that applies to Klein surfaces, that is (possibly non-orientable) surfaces with boundary which carry a dianalytic structure. In particular, we extend Belyi’s famous theorem from Riemann surfaces to KLein surfaces.”\]  P. Koebe, [*Über konforme Abbildung mehrfach zusammenhängender ebener Bereiche, insbesondere solcher Bereiche, deren Begrenzung von Kreisen gebildet wird,*]{} Jahresb. d. Deutsch. Math.-Ver. 15 (1906), 142–153. \[$\spadesuit$ special cases of the KN=Kreisnormierung\]  P. Koebe, [*Über konforme Abbildung mehrfach zusammenhängender ebener Bereiche*]{}, Jahresb. d. Deutsch. Math.-Ver. 16 (1907), 116–130. \[$\spadesuit$ special cases of the KN=Kreisnormierung\]  P. Koebe, [*Über die Uniformisierung reeller algebraischer Kurven*]{}, Gött. Nachr. (1907), 177–190. \[$\spadesuit$ self-explanatory and relies heavily on Klein’s ortho- and diasymmetry\]  P. Koebe, [*Über die Uniformisierung beliebiger analytischer Kurven*]{}, Gött. Nachr. (1907), 191–210. \[$\spadesuit$ joint with Poincaré 1907 [@Poincare_1907], the first acceptable and accepted proof of uniformization of open Riemann surfaces (alias analytical curves, by opposition to algebraic reflecting compactness, in the jargon of Fréchet) $\spadesuit$ key ingredient the “Verzerrungssatz”, for which Koebe confess some little “coup de pouce” from the colleague Carathédory\]  P. Koebe, [*Über die Uniformisierung beliebiger analytischer Kurven, (2. Mitt.)*]{}, Gött. Nachr. (1907), 633–669. \[$\spadesuit$ another proof of the general uniformization inspired by the reading of Poincaré’s account, and using methods of Schwarz (esp. the Gürtelförmigeverschmelzung)\]  P. Koebe, [*Über die Uniformisierung beliebiger analytischer Kurven, (3. Mitt.)*]{}, Gött. Nachr. (1908), 337–358. \[$\spadesuit$ discusses other types of uniformizations, and put forward the KNP, which he is already able to prove (in finite connectivity, or even in infinite connectivity under special symmetry), but no detailed arguments\]  P. Koebe, [*Über die Uniformisierung der algebraischen Kurven durch automorphe Funktionen mit imaginärer Substitutionsgruppe*]{}, Gött. Nachr. (1909), 68–76. \[$\spadesuit$ announce other types of uniformization formulated by Klein\]  P. Koebe, [*Über die Uniformisierung beliebiger analytischer Kurven, (4. Mitt.)*]{}, Gött. Nachr. (1909), 324–361.  P. Koebe, [*Ueber die Uniformisierung der algebraischen Kurven, I*]{} Math. Ann. 67 (1909), 145–224. \[$\spadesuit$ detailed proof\]  P. Koebe, [*Über die konforme Abbildung mehrfach zusammenhängender Bereiche*]{} Jahresb. d. Deutsch. Math.-Ver. 19 (1910), 339–348. \[$\spadesuit$ contains the general case of the KN in finite connectivity, via 2 methods: Überlagerungsfläche and the so-called Koebe iteration method $\spadesuit$ again no complete proof but the convergence is ensured by the “Verzerrungssatz” $\spadesuit$ full details only much latter in 1920–22? [@Koebe_1922] (according, e.g., to Bieberbach 1968 [@Bieberbach_1968-Das-Werk-Paul-Koebes]) $\spadesuit$ p.339: “Den Hauptgegenstand dieser und des gegenwärtigen Vortrages bildet das Problem der konformen Abbildung eines $(p+1)$-fach zusammenhängenden Bereiches auf einen von $p+1$ Vollkreisen begrenzten Bereich, ein Problem, welches in der Literatur zuerst bei Schottky (Dissertation, Berlin 1875, umgearbeitet erschienen in Crelle 1877) in seiner bekannten Doktordissertation auftritt, jedoch früher bereits von Riemann in Betracht gezogen worden ist, wie aus seiner nachgelassenen Schriften hervorgeht.”\]  P. Koebe, [*Über die Hilbertsche Uniformisierungsmethode*]{}, Gött. Nachr. (1910), 59–74.  P. Koebe, [*Ueber die Uniformisierung der algebraischen Kurven, II*]{} Math. Ann. 69 (1910), 1–81.  P. Koebe, [*Begründung der Kontinuitätsmethode im Gebiete der konformen Abbildung und Uniformisierung. (Voranzeige)*]{}, Nachr. Königl. Ges. Wiss. Gött., Math.-phys. Kl. (1912), 879–886. \[$\spadesuit$ self-explanatory, but compare the practically simultaneous work of Brouwer 1912 [@Brouwer_1912_top-Schwierig], plus the announcements in 1911 [@Klein-Brouwer-Koebe_1912]\] P. Koebe, [*Ueber eine neue Methode der konformen Abbildung und Uniformisierung,*]{} Nachr. Königl. Ges. Wiss. Gött., Math.-phys. Kl. (1912), 844–848. \[$\spadesuit$ introduction of the Schmiegungsverfahren (osculation method?)\]  P. Koebe, [*Begründung der Kontinuitätsmethode*]{}, Ber. Math. Math.-phys. Kl. Sächs. Akad. Wiss. Leipzig 64 (1912), 59–62.  P. Koebe, [*Ränderzuordnung bei konformer Abbildung*]{}, Gött. Nachr. (1913), 286–288. \[$\spadesuit$ contests the heavy reliance upon Lebesgue’s measure theory in Carathéodory’s proof (1912) of the boundary behavior of the Riemann mapping for Jordan curves, by appealing to a device of Schwarz\]  P. Koebe, [*Ueber die Uniformisierung der algebraischen Kurven, IV (Zweiter Existenzbeweis der allgemeinen kanonischen uniformisierenden Variablen: Kontinuitätsmethode)*]{}, Math. Ann. 75 (1914), 42–129.  P. Koebe, [*Abhandlungen zur Theorie der konformen Abbildung, I, die Kreisabbildung des allgemeinsten einfach und zweifach zusammenhängenden schlichten Bereichs und die Ränderzuordnung bei konformer Abbildung*]{}, J. Reine Angew. Math. 145 (1915), 177–223. \[$\spadesuit$ uses the word “Kreisabbildung” which is perhaps first coined in Bieberbach 1914 [@Bieberbach_1914]\]  P. Koebe, [*Abhandlungen zur Theorie der konformen Abbildung, IV*]{}, Acta Math. 41 (1918), 305–344. \[$\spadesuit$ first existence proof of the circular/radial slit maps for domains of finite connectivity (general case in Grötzsch 1931 [@Groetzsch_1931]); subsequent proof in Reich-Warschawski 1960 [@Reich-Warschawski_1960]\]  P. Koebe, [*Über die Strömungspotentiale und die zugehörenden konformen Abbildungen Riemannscher Flächen*]{}, Gött. Nachr. (1919), 1–46. P. Koebe, [*Abhandlungen zur Theorie der konformen Abbildung. VI. (Abbildung mehrfach zusammenhängender schlichter Bereiche auf Kreisbereiche. Uniformisierung hyperelliptischer Kurven. Iterationsmethoden)*]{}, Math. Z. 7 (1920), 235–301.  P. Koebe, [*Abbildung beliebiger mehrfach zusammenhängender schlichter Bereiche auf Kreisbereichen*]{}, Math. Z. 7 (1922), 116–130.  P. Koebe, [*????*]{}, Acta Math. ?? (1928), ??–??.  P. Koebe, [*Das Wesen der Kontinuitätsmethode*]{}, Deutsche Math. 1 (1936), 859–879. 78 \[$\spadesuit$ survey-like with many refs.\] $\bigstar$$\bigstar$$\bigstar$  H. Köditz, St. Timmann, [*Ranschlichte meromorphe Funktionen auf endlichen Riemannschen Flächen*]{}, Math. Ann. 217 (1975), 157–159. 78 \[$\clubsuit$ supply a proof of a circle map (without bound) using techniques of Behnke-Stein $\clubsuit$ criticizes and demolishes an earlier argument of Tietz 1955 [@Tietz_1955] intended to give another treatment of the Ahlfors circle map\]  G. Kokarev, [*Variational aspects of Laplace eigenvalues on Riemannian surfaces* ]{}, arXiv:1103.2448, 2011. \[$\spadesuit$ Abstract: We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem (and its higher eigenvalue versions) via the direct ...\]  J. Kóllar, [*The topology of real and complex algebraic varieties*]{}, Adv. Stud. Pure Math. 31 (2001), Math. Soc. Japan, 127–145.  Y. Komatu, [*Identities concerning canonical conformal mappings*]{}, Kōdai math. Sem. Rep. 3 (1953), 77–83.  W. Koppelman, [*The Riemann-Hilbert problem for finite Riemannian surfaces*]{}, Comm. Pure Appl. Math. 12 (1959), 13–35. \[$\spadesuit$ work oft cited in the investigation of the Slovenian school, see e.g. Černe-Forstnerič 2002 [@Cerne-Forstneric_2002] $\spadesuit$ “The problem of finding a function, analytic in some domain $D$, for a given relation between the limiting values of its real and imaginary parts on the boundary of $D$ was originally mentioned by Riemann in his dissertation \[12\]. Here we shall treat the special case where …”\]$\bigstar$$\bigstar$$\bigstar$  A.B. Korchagin, [*$M$-curves of degree $9$: New prohibitions*]{}, Math. Notes 39 (1986), 277–293. \[$\spadesuit$\]$\bigstar$  A. Korn, [*Application de la méthode de la moyenne arithmétique aux surfaces de Riemann*]{}, C.R. Acad. Sci. Paris 135 (1902), 94–95. 60 \[$\spadesuit$\]$\bigstar$$\bigstar$$\bigstar$  A. Korn, [*Sur le problème de Dirichlet pour des domaines limités par plusieurs contours (ou surfaces)*]{}, C.R. Acad. Sci. Paris 135 (1902), 231–232.$\bigstar$$\bigstar$$\bigstar$60  A. Korn, [*Über die erste und zweite Randwertaufgabe der Potentialtheorie*]{}, Rend. Circ. Mat. Palermo 35 (1913), 317–323. \[$\spadesuit$ application of the authors’s theory of the asymmetrical kernel to the first and second boundary value problem of potential theory and its resolution by the method of the arithmetical mean (C. Neumann, Robin) leading anew to the solution predicted by Poincaré 1896 [@Poincare_1896], which the author first succeeded in 1901 after appealing to a result of Zaremba (1901)\]  I. Kra, [*Maximal ideals in the algebra of bounded analytic functions*]{}, ???? 31 (1967), 83–88. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is applied to a characterization of “point-like” maximal ideals in the function algebra $\spadesuit$ more precisely Ahlfors is cited on p.85 as follows (yet without precise control on the degree except for its finiteness): “Lemma 5. Let $X$ be a finite domain of the Riemann surface $W$. Then for each discrete sequence $\{x_n\}\subset X$, there exists an $f\in B(X)$ \[=the ring of bounded holomorphic functions, cf. p.83\] such that $\lim_{n\to \infty} f(x_n)$ does not exist.—[*Proof*]{}. Ahlfors \[1\](=1950 [@Ahlfors_1950]) has shown that there exists a mapping $p$, analytic in a neighborhood of ${\rm Cl} X$, that is an $N$-to-one covering of the closed unit disc, for some positive integer $N$. Moreover $p\vert X$ is an $N$-to-one covering of the interior of the closed unit disc, and $p\vert {\rm Cl} X-X$ is an $N$-to-one covering of the unit circle. Because ${\rm Cl} X$ is compact we may assume (by choosing a subsequence) that $x_n\to x \in {\rm Cl} X-X$. Then $p(x_n)\to 1$ \[modulus missing??\] and $\vert p(x_n)\vert<1$. Again, we may choose a subsequence such that $p(x_n)$ is distinct \[???\] and infinite and constitutes an interpolating sequence (see Hoffman \[6,pp.194–204\]). Choose a bounded analytic function $f$ on the unit disc such that $f(p(x_{2n+1}))=0$ and $f(p(x_{2n}))=1$ for $n=0,1,2,\dots$. Then $f\circ p \in B(X)$, and $\lim_{n\to \infty}(f\circ p)(x_n)$ does not exist. \[q.e.d.\]” $\spadesuit$ p.87: “Theorem 2 is a generalization of Theorem 1, because every boundary point of a finite domain is an essential singularity for some bounded holomorphic function. The unit disc certainly has this property. The general case is reduced to the unit disc [*via*]{} any Ahlfors maps. (See the proof of Lemma 5.)”\]  I. Kra, [*Automorphic Forms and Kleinian Groups*]{},Benjamin, Reading, Mass., 1972, 464 pp. \[$\spadesuit$\]  S.G. Krantz, [*The Carathéodory and Kobayashi metrics and applications in complex analysis*]{}, Amer. Math. Monthly 115 (2008), 304–329. \[$\spadesuit$ p.311 brief mention of the Ahlfors function and as it is connected to the Carathéodory metric; for more on the Ahlfors function the reader is referred to Fisher 1983 [@Fisher_1983] or the book Krantz (2006)\]  V.A. Krasnov, [*Albanese mapping for real algebraic varieties*]{}, Mat. Zametki 32 (1982), 365–374; English transl., Math. Notes 32 (1983), 661–666. \[$\spadesuit$\]  V.A. Krasnov, [*Albanese map for GM${\Bbb Z}$ varieties*]{}, Mat. Zametki 35 (1984), 739–747; English transl., Math. Notes 35 (1984), 391–396. \[$\spadesuit$\]  D. Kraus, O. Roth, [*Critical points, the Gauss curvature equation and Blaschke products*]{}, arXiv (2011). 47, 78 \[$\spadesuit$ p.15 the Ahlfors map is mentioned\]  S. Kravetz, [*On the geometry of Teichmüller spaces and the structure of their modular groups*]{}, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 278 (1959), 1–35. \[$\spadesuit$ according to Natanzon 1999 [@Natanzon_1999-Moduli-real-alg-surf.superanal-differ-spinors p.1101], this Kravetz’s paper was employed in Earle’s 1971 [@Earle_1971-On-the-moduli] description of the topological structure of the components of the moduli space of real algebraic curves as being each diffeomorphic to ${\Bbb R}^{3g-3}/ {\rm Mod}_{g,r,\varepsilon}$ for a suitable discrete (modular) group. Earle’s proof used the theory of quasiconformal maps, but relied on a Kravetz’s theorem “which turned out latter to be wrong”. Still according to Natanzon () “A correct proof based on the theory of quasiconformal maps was obtained in Seppälä 1978 [@Seppala_1978-Teich-spaces-of-Klein-surfaces].”\]  ?. Krazer, [*Lehrbuch der Thetafuntionen*]{}, Teubner, 1903. \[$\spadesuit$ student of Prym, in turn student of Riemann\]  I.M. Krichever, S.P. Novikov, [*Virasoro-Gelfand-Fuchs type algebras, Riemann surfaces, operator theory of closed strings*]{}, J. Geom. Phys. 5 (1988), 631–661. \[$\spadesuit$\]  L. Kronecker, [*Über die Diskriminante algebraischer Funktionen*]{}, Crelles J. 91 (1881). \[$\spadesuit$\]  L. Kronecker, [*Über einige Interpolationsformeln für ganze Funktionen mehrer Variabeln*]{}, Werke Leipzig 1895, Bd.I, 133–141. \[$\spadesuit$ quoted in Petrovskii 1938 [@Petrowsky_1938] as one of the tool involved in the proof of the Petrovskii’s inequalities, where Kronecker’s work connects to the so-called Euler-Jacobi (interpolation) formula\]  P.B. Kronheimer, T.S. Mrowka[^141], [*The genus of embedded surfaces in the projective plane*]{}, Math. Res. Letters 1 (1994), 797–808. \[$\spadesuit$ a proof of the Thom conjecture on the genus of smooth surfaces embedded in the complex projective plane, via Gauge theory (Donaldson theory, etc.) $\spadesuit$ this has some modest relevance to Hilbert’s 16th problem, cf. e.g. Theorem \[Thom-Ragsdale:thm\] in this text $\spadesuit$ the special degree $3$ case of Thom’s conjecture was known to Kervaire-Milnor 1961 [@Kervaire-Milnor_1961]\]  V. Krylov, [*Une application des équations intégrales à la démonstration de certains théorème de la théories des représentations conformes*]{}, (Russian, French Summary) Rec. Math. de Moscou \[Mat. Sb.\] 4 (1938), 9–30. 78 \[$\spadesuit$ Seidel’s summary: the problem of mapping conformally a region of connectivity $n$, bounded by $n$ analytic contours, on various canonical domains is reduced to the problem of solving a system of simultaneous integral equations\]  T. Kubo, [*Bounded analytic functions in a doubly connected domain*]{}, Mem. Coll. Sci. Univ. Kyoto, A. 26 (1951), 211–223.  T. Kubota, [*Über konforme Abbildungen. I.*]{}, Science Reports Tôhoku Imperial Univ. ser. I, 9 (1920), 473–490. \[$\spadesuit$ quoted in Grunsky 1932 [@Grunsky_1932 p.135] for the simply-connected case of an extension of Bieberbach’s first Flächensatz\]$\bigstar$$\bigstar$  T. Kubota, [*Über konforme Abbildungen. II.*]{}, Science Reports Tôhoku Imperial Univ. ser. I, 10 (1921). \[$\spadesuit$\]$\bigstar$  P. Kufareff, [*Über das zweifach zusammenhängende Minimalgebiet*]{}, Bull. Inst. Math. et Mec. Univ. de Tomsk 1 (1935–37), 228–236. \[$\spadesuit$ quoted in Lehto 1949 [@Lehto_1949] and Bergman 1950 [@Bergman_1950], and akin to the works of Zarankiewicz 1934 [@Zarankiewicz_1934; @Zarankiewicz_1934-numerisches] $\spadesuit$ Seidel’s summary: a minimal problem is set up for functions analytic and single-valued in a circular ring and the mapping effected by the minimizing function is discussed\] $\bigstar$ $\bigstar$$\bigstar$  R. Kühnau, [*Über die analytische Darstellung von Abbildungsfunktionen, insbesondere von Extremalfunktionen der Theorie der konformen Abbildung*]{}, J. Reine Angew. Math. 228 (1967), 93–132. 78 \[$\spadesuit$ p.95–96 proposes a contribution to a question raised by Garabedian-Schiffer 1949 [@Garabedian-Schiffer_1949] related to the representation of the so-called Schottky function (via [*Normalabbildungsfunktionen*]{}) $\spadesuit$ Kühnau alludes to several (subsequent) work of Schottky where the circle maps should appear again? (no precise refs. hence requires some detective work)\]  R. Kühnau, [*Geometrisch-funktionentheoretische Lösung eines Extremalsproblems der konformen Abbildung, I, II*]{}, J. Reine Angew. Math. 229 (1967), 131–136; 237 (1969), 175–180. 78 \[$\spadesuit$\]  R. Kühnau, [*Herbert Grötzsch zum Gedächtnis*]{}, Jber. d. Dt. Math.-Verein. 99 (1997), 122–145. \[$\spadesuit$ alas, cited merely for the matter of the “quasi-conformal” jargon, as occuring apparently first (the jargon, not the concept) in Carathéodory 1914 [@Caratheodory_1914]\]  V.S. Kulikov, [*Epimorphicity of the period map for $K3$ surfaces*]{}, Uspekhi Mat. Nauk 32 (1977), 257–258. (Russian) \[$\spadesuit$ employed in Nikulin’s (1979/80 [@Nikulin_1979/80]) rigid-isotopy classification of plane real sextics\]  Z. Kuramochi, [*A remark on the bounded analytic function*]{}, Osaka Math. J. 4 (1952), 185–190. 50, 60 \[$\clubsuit$ p.189 seems to reprove the result of Ahlfors 1950 [@Ahlfors_1950] about the existence of a circle map of degree $\le r+2p$ by using the Green’s function (while generalizing a method of Nehari 1951 [@Nehari_1951] for the case of plane domains) $\spadesuit$ unfortunately Kuramochi’s paper is written in some mysterious tongue (the Nipponglish), and despite its moderate size (of ca. 5 pages) it contains several dozens of misprints obstructing seriously its readability $\spadesuit$ despite our critical comments this work is quoted in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] so should probably be not completely science-fictional $\spadesuit$ it would perhaps be desirable (in case this paper emerged from some solid underlying structure) to undertake a polishing of this Kuramochi paper to improve its readability\]  A. Kuribayashi, [*On analytic families of compact Riemann surfaces with non-trivial automorphisms*]{}, ??? ?? (1966), 119–165. \[$\clubsuit$ Teichmüller theory à la Ahlfors-Bers, plus an influence of Shimura $\spadesuit$ p.133: “Thm 2.17. [*There exists one and only one Riemann surface up to conformal equivalence which has group of automorphism of order 168 among compact Riemann surfaces of genus $3$.”*]{} $\spadesuit$ is this uniqueness new? perhaps already in Hurwitz?\]  Y. Kusunoki, [*Über die hinreichenden Bedingungen dafür, dass eine Riemannsche Fläche nullberandet ist,*]{} Mem. Coll. Sci. Univ. Kyoto 28 (1952), 99–108. 60 \[also cited in Sario-Nakai [@Sario-Nakai_1970] CHECK $\spadesuit$ an application of Ahlfors 1950 [@Ahlfors_1950] (and the older predecessor Bieberbach 1925 [@Bieberbach_1925]) is given to the type problem\]  Y. Kusunoki, [*Contributions to Riemann-Roch’s theorem*]{}, Kyoto J. Math. ? (1958), ?–?. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited\]  Y. Kusunoki, [*Square integrable normal differentials on Riemann surfaces*]{}, J. Math. Kyoto Univ. 3 (1963), 59–69. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited on p.64, in the following connection (as usual many “notatio”): “If $R$ is a bordered surface with $p$ contours, $\{A_n,B_n, C_{\nu}\}_{n=1, \dots, g; \nu 1, \dots, p-1}$ is admissible for $\Gamma_0=\Gamma_{aS}=\Gamma_{AB} \oplus\Gamma_C$ and $P_\gamma$ gives a one-to-one mapping of $\Gamma_0=\Gamma_0'$ to $(2g+p-1)$-dimensional vector space by (II) (Ahlfors \[1\](=1950 [@Ahlfors_1950])).”\]  M.P. Kuvaev, P.P. Kufarev, [*An equation of Löwner’s type for multiply connected regions*]{}, Tomskiĭ gos. Univ. Uč. Zap. Mat. Meh. 25 (1925), 19–34. 78 $\bigstar$ $\bigstar$$\bigstar$  J.-L. Lagrange, [*??*]{}, 1779. \[$\spadesuit$ a source often quoted e.g. by Koebe \[ca. 1910, in Math. Ann.\], and Monastyrsky 1987/99 [@Monastyrsky_1987/99/08-even-1979] where one reads (p.15): “It is noteworthy that Joseph-Louis Lagrange (1736–1813) obtained the Cauchy-Riemann equations conditions also in 1779, also in connection with the solution to a cartographic problem.”\]  J.-L. Lagrange, [*Mecanique analitique*]{}, 2 volumes, Paris, 1788. \[$\spadesuit$\]  J.-L. Lagrange, [*Theorie des fonctions analytiques*]{}, Paris, 1797, 2ème édition 1813. \[$\spadesuit$\]  E. Landau, [*Einige Bermerkungen über schlichte Abbildung*]{}, Jahresb. Dt. Math.-Verein. 34 (1926), 239–243. \[$\spadesuit$\]  H.J. Landau, R. Osserman, [*On analytic mappings of Riemann surfaces*]{}, J. Anal. Math. (1960), 249–279. \[$\spadesuit$ p.266 contains the basic lemma that an analytic map taking the boundary to the boundary is a (full) branched covering (this follows directly from the local behavior of such maps and bears a certain relevance to the Ahlfors circle map) $\spadesuit$ however the paper does not seem to supply an existence proof of the Ahlfors map $\spadesuit$ in fact it is worth reproducing the text faithfully (p.265): “We now turn to the problem of mapping one Riemann surface into another. We shall need a lemma which, in a special case, was proved by Radó \[12\](=Radó 1922 [@Rado_1922-Z-Theorie-mehr]). Let us recall that a sequence of points in a Riemann surface is said to tend to the boundary if the sequence has no limit points in $R$[^142]. We shall say that a map $f$ of one Riemann surface $R_1$ into another, $R_2$, takes the boundary into the boundary if for every sequence of points in $R_1$ which tends to the boundary, the image sequence tends to the boundary of $R_2$. Let us note that if $R_1$ and $R_2$ are relatively compact regions on Riemann surfaces, the above definition coincides exactly with the usual notion of mapping the boundary into the boundary.—[**Lemma 3.1:**]{} [*Let $R_1$ and $R_2$ be any two Riemann surfaces and $f$ an analytic map of $R_1$ into $R_2$ which takes the boundary into the boundary. Then $f$ maps $R_1$ onto $R_2$, and every points of $R_2$ is covered the same number of times, counting multiplicities.*]{}” $\spadesuit$ for this basic lemma see also the treatments in Stoïlow 1938 [@Stoilow_1938-Lecons Chap.VI] and Ahlfors-Sario 1960 [@Ahlfors-Sario_1960 p.41, 21B.]\] G. Landsberg, [*Algebraische Untersuchungen über den Riemann-Roch Satz*]{}, Math. Ann. 50 (1898), 333–380. \[$\spadesuit$ drifting the transcendental theory toward arithmetization\] G. Landsberg, [*Über das Analogon des Riemann-Roch Satz in der Theorie der algebraischen Zahlen*]{}, Math. Ann. 50 (1898), 577–582. \[$\spadesuit$\] M. Lavrentieff, [*On the theory of conformal mapping*]{}, Trav. Inst. phys.-math. Stekloff 5 (1934), 159–245. \[$\spadesuit$ cited in Schiffer 1950 [@Schiffer_1950-Appendix-Courant]\] $\bigstar$$\bigstar$$\bigstar$  P.D. Lax, [*Reciprocal extremal problems in function theory*]{}, Comm. Pure Appl. Math. 8 (1955), 437–453. 78 \[$\spadesuit$ extract from Rogosinski’s review (MathReview): “This principle is dual to one used for similar problems by the reviewer and H.S. Shapiro (=Rogosinski-Shapiro 1953 [@Rogosinski-Shapiro_1953]); both principles are easy interpretations of the Hahn-Banach extension theorem in the complex case. \[…\] This important paper is somewhat marred by numerous misprints and a rather loose presentation.” \] $\bigstar$  R.F. Lax, [*On the dimension of the varieties of special divisors*]{}, Illinois J. Math. 19 (1975), 318–324. \[$\spadesuit$ extract from H.H. Martens’s review (MathReview): “The proof is inspired by the methods of T. Meis 1960 [@Meis_1960], and the paper contains, in addition to the author’s results, a very useful review of Meis’ monograph, which is rather difficult to obtain.” \] $\bigstar$  R. Le Vavasseur, [*Sur la représentation conforme de deux aires planes à connexion multiple, d’apès M. Schottky*]{}, Ann. Fac. Sci. Toulouse (2) 4 (1902), 45–100. 78 \[$\spadesuit$ re-expose the results of Schottky 1877 [@Schottky_1877]\]  H. Lebesgue, [*Intégrale, Longueur, Aire*]{}, Annali di Mat. 7 (1902), 231–358. \[$\spadesuit$ Lebesgue’s Thesis, where Lebesgue’s integration and the allied geometry is introduced (yet another descendant of Riemann) $\spadesuit$ Fatou 1906 [@Fatou_1906], F. Riesz 1907 and Carathéodory 1912 [@Caratheodory_1912] are the best illustration of the rôle of measure theory in (complex) function theory, a rôle disputed by Koebe at least in the early steps (compare Gray’s fine analysis [@Gray_1994])\]  H. Lebesgue, [*Sur le principe de Dirichlet*]{}, Rend. Circ. Mat. Palermo 24 (1907), 371–402. \[$\spadesuit$ extension of Hilbert’s solution to the Dirichlet problem by allowing general boundaries, cf. also Zaremba 1910 [@Zaremba_1910] for possible simplifications and (Beppo Levi 1906 [@Beppo-Levi_1906] and Fubini 1907 [@Fubini_1907] for related contributions of the same period $\spadesuit$ further (quasi-ultimate) simplifications in Perron 1923 [@Perron_1923], in turn simplified in Radó-Riesz 1925 [@Rado-Riesz_1925]\]  S. Lefschetz, [*On certain numerical invariants of algebraic varieties with applications to Abelian varieties*]{}, Trans. Amer. Math. Soc. 22 (1921), 327–482. \[$\spadesuit$\]  S. Lefschetz, [*L’analysis situs et la géométrie algébrique*]{}, Gauthier-Villars, 1924. \[$\spadesuit$\]  J. Lehner, M. Newman, [*On Riemann surfaces with maximal automorphism groups*]{}, Glasgow Math. J. 8 (1967), 102–112. \[$\spadesuit$\]  O. Lehto, [*Anwendung orthogonaler Systeme auf gewisse funktionentheoretische Extremal- und Abbildungsprobleme*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. 59 (1949), 51 pp. \[$\spadesuit$ new existence proof of parallel-slit mappings via the Bergman kernel (and so in particular of RMT, answering thereby the old desideratum of Bieberbach 1914 [@Bieberbach_1914]-Bergman 1922 [@Bergman_1922]-Bochner 1922 [@Bochner_1922]); equivalent work in Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950] $\spadesuit$ p.48 reproves the identity $B(z)=1/S(w(z)-w^{\ast}(z))$ (expressing the least area map as combination of the two Schlitzfunktionen $w$ and $w^{\ast}$) announced by Grunsky 1932 [@Grunsky_1932] $\clubsuit$ p.41 seems to show that the least area map is a circle map\]  O. Lehto, K.I. Virtanen, [*Quasiconformal mappings in the plane*]{}, 2nd edition, Springer-Verlag, Berlin, 1973. \[$\spadesuit$\]  O. Lehto, [*Univalent functions and Teichmüller spaces*]{}, Springer-Verlag, Berlin, 1986. \[$\spadesuit$\]  O. Lehto, [*On the life and work of Lars Ahlfors*]{}, Math. Intelligencer (1998), 4–8. \[$\spadesuit$ p.7: “In this same paper (1953/54), Ahlfors also defined the notion which he called Teichmüller space.”\]  G.W. Leibniz, [*Characteristica Geometrica*]{}, 1679. \[$\spadesuit$ beside Descartes (pseudo?)-anticipation of the Euler characteristic theorem for spherical polyhedrons ($V-E+F=2$), this is oft regarded as the first “topological” text, summarized as follows in Monastyrsky 1987/99 [@Monastyrsky_1987/99/08-even-1979], p.89: “In 1679 Leibniz published \[t\]his famous book \[…\], in which (in modern terms) he tried to study the topological rather than the metric characteristics of properties of figures. He wrote that, aside from the coordinate representation of figures, ‘we are in need of another analysis, purely geometric or linear, which also defines the position (situs), as algebra defines magnitude.’ It is interesting to note that Leibniz tried to interest Christiaan Huygens (1629–1695) in his work, but the latter showed little enthusiasm. This was the first (albeit unsuccessful) attempt to interest a physicist in topology.” $\spadesuit$ so this Leibniz’s text must be the first place where the term “analysis situs” appears in embryo, then rebaptized “Topologie” in Listing 1847 [@Listing_1847-Vorstudien-zur-Topologie] (yet receiving only slow acceptance, say in the 1920’s, e.g. Riemann, Poincaré, etc. used exclusively the term “analysis situs”).\]  F. Leja, [*Une méthode de construction de la fonction de Green appartenant à un domaine plan quelconque*]{}, C.R. Acad. Sci. Paris 198 (1934), 231–234. \[$\spadesuit$ Seidels’ summary: a method for constructing the Green’s function of an arbitrary region is given. The approximating functions are closely related to Lagrange polynomials)\]  F. Leja, [*Construction de la fonction analytique effectuant la représentation conforme d’un domaine plan quelconque sur le cercle*]{}, Math. Ann. 111 (1935), 501–504. \[$\spadesuit$ Seidel’s summary: for a given bounded simply-connected \[sic!?\] region in the plane (of the complex variable $z$), a sequence of elementary functions is constructed which tends to the \[Riemann\] mapping function of the region\]  F. Leja, [*Sur une suite de polynômes et la représentation conforme d’un domaine plan quelconque sur le cercle*]{}, Annales Soc. Polonaise de Math. 14 (1936), 116–134. \[$\spadesuit$ Seidel’s summary: a set of polynomials is obtained by means of which the mapping function of a region $D$, with $z=\infty$ as interior point, on $\vert w\vert>1$ can be expressed. If $D$ is simply-connected, the map is one-to-one (schlicht) $\spadesuit$ question (of Gabard) and if not, does it relates to the map of Riemann-Bieberbach-Grunsky-Ahlfors (cf. e.g. Bieberbach 1925 [@Bieberbach_1925] and Ahlfors 1947 [@Ahlfors_1947])\]$\bigstar$$\bigstar$  J. Lewittes, [*Automorphisms of compact Riemann surfaces*]{}, Amer. J. Math. (1963), 738–752. \[$\spadesuit$\]$\bigstar$  J. Lewittes, [*Riemann surfaces and the theta function*]{}, Acta Math. 111 (1964), 37–61. \[$\spadesuit$\]  B. Levi, [*Sul Principo di Dirichlet*]{}, Rend. Circ. Mat. Palermo (1906). \[$\spadesuit$ cited in Zaremba 1910 [@Zaremba_1910] an extension of Hilbert’s resurrection of the Dirichlet principle\]  P. Li, S.-T. Yau, [*A new conformal invariant and its application to the Willmore conjecture and the first eigenvalue of a compact surface*]{}, Invent. Math. 69 (1982), 269–291. \[$\spadesuit$ p.272 claims a result along the line of the Witt-Martens mapping theorem for symmetric surfaces without fixed points, but the Li-Yau argument appears as sketchy, or maybe even invalid according to Ross 1997 [@Ross_1997]\]  J.-L. Lions, [*Remarks on reproducing kernels and some function spaces*]{}, In: Function Spaces, Interpolation Theory and Related topics, Proceedings, Lund, Sweden, 2000, Walter de Gruyter, 2002, 49–59. \[$\spadesuit$ present the definition of the reproducing kernel in the general setting due to Aronszajn (p.50): “This definition is due to N. Aronszajn \[1\](=Aronszajn [@Aronszajn_1950]) who studied general properties of reproducing kernels.—In particular cases, such notions have been introduced by S. Bergman \[2\](=1922 [@Bergman_1922]), G. Szegö \[11\](=1921 [@Szego_1921]), M. Schiffer \[9\](=1946 [@Schiffer_1946]), S. Zaremba \[12\](=1908 [@Zaremba_1908-calcul-numerique]), where the corresponding reproducing kernels are computed and estimated; cf. N Aronszajn, loc.cit., and P. Garabedian \[3\](=1949 [@Garabedian_1949]).” $\spadesuit$ p.56: “All these kernels can be computed by the same strategy as above. But we have not been able to recover by this method the results of P. Garabedian \[3\](=1949 [@Garabedian_1949]), which give the connection between $S(x,b)$ and $B(x,b)$.”\]  J.B. Listing, [*Vorstudien zur Topologie*]{}, Göttinger Studien, 1847. \[$\spadesuit$ influenced by Descartes, Leibniz 1679 [@Leibniz_1679], Euler, Gauss, etc. and influential upon Riemann (who attended in 1850 a seminar on mathematical physics run by W.Weber, Listing, Stern, and Ulrich), cf. e.g. p.9 of Monastyrsky 1987/99 [@Monastyrsky_1987/99/08-even-1979], as well as upon Tait and Kelvin\]  M.S. Li Chiavi, [*Sulla rappresentazione conforme delle aree pluriconnesse appartenti a superficie di Riemann su un’opportuna superficie di Riemann su cui siano eseguiti dei tagli paralleli*]{}, Rend. Sem. Mat. Univ. Padova 3 (1932), 95–107. 60 \[$\spadesuit$ Maria Stella Li Chiavi is a student of Cecioni\]  L. Lichtenstein, [*Randwertaufgaben der Theorie der linearen partiellen Differentialgleichungen zweiter Ordnung vom elliptischen Typus. II*]{}, J. Reine Angew. Math. 143 (1913), 51–105. \[$\spadesuit$ quoted in Nevanlinna 1939 [@Nevanlinna_1939] for Schwarz’s alternating procedure recasted as the solution of an integral equation through successive approximations\]  L. Lichtenstein, [*Zur Theorie der konformen Abbildung nichtanalytischer, singularitätenfreier Flächenstücke auf ebene Gebieten*]{}, Bull. Internat. Acad. Sci. Cracovie, Cl. Sci. Math. Nat. Ser. A. (1916), 192–217. 60 \[$\spadesuit$ an extension of Gauss 1825 [@Gauss_1825] (local isothermic coordinates), simultaneous work by Korn\] $\bigstar$$\bigstar$$\bigstar$  L. Lichtenstein, [*Zur konformen Abbildung einfach zusammenhängender schlichter Gebiete*]{}, Archiv der Math. u. Physik 25 (1917), 179–180. \[$\spadesuit$ Seidels’ summary: the problem of mapping conformally on a circle a simply-connected region bounded by a simple closed curve with continuous curvature is reduced to the solution of a linear integral equation\] $\bigstar$$\bigstar$$\bigstar$  L. Lichtenstein, [*Neuere Entwicklung der Potentialtheorie. Konforme Abbildung*]{}, Encykl. d. math. Wiss. II, 3., 1. Hälfte, 177–377. Leipzig, B.G. Teubner, 1919. 78 \[$\spadesuit$ should contain another proof of the Kreisnormierung in finite connectivity, according to Hawley-Schiffer 1962 [@Schiffer-Hawley_1962]\] $\bigstar$  B.V. Limaye, [*Blaschke products for finite Riemann surfaces*]{}, Studia Math. 34 (1970), 169–176.\[$\spadesuit$ the paper starts with a little manipulation amounting to annihilate the periods of a conjugate differential, hence quite in line with say Ahlfors 1950 [@Ahlfors_1950], yet does not seem to reprove the existence result of a circle map\]  B.V. Limaye, [*Ahlfors function on triply connected domains*]{}, J. Indian Math. Soc. 37 (1973), 125–135. 78 $\bigstar$  I. Lind, [*An iterative method for conformal mappings of multiply-connected domains*]{}, Ark. Mat. 4 (1963), 557–560. 78 \[$\spadesuit$ another proof of PSM (due to Schottky 1877, Cecioni 1908 [@Cecioni_1908], Hilbert 1909 [@Hilbert_1909], Koebe, etc.) via iterative scheme à la Koebe (who uses rather this device for the harder Kreisnormierung)\]  E. Lindelöf, [*Memoire sur la théorie des fonctions entières de genre fini*]{}, 1902. \[$\spadesuit$\]  M. Lindner, [*Über Mannigfalfigkeiten ebener KKurven mit Singularitäten*]{}, Archiv. Math. 28 (1977), 603–606. \[$\spadesuit$ cited e.g. in SHustin 1990/91 [@Shustin_1990/91-Geom-of-discr-alg-curve] $\spadesuit$ seems to contain a variant of the Severi-Brusotti description of the discriminant\]  F. Lippich, [*Untersuchungen über den Zusammenhang der Flächen im Sinne Riemann’s*]{}, Math. Ann. 7 (1874), 212–230. 60 \[$\spadesuit$ topology of surfaces, overlaps slightly with Möbius and Jordan 1866 [@Jordan_1866], but no cross-citations\]  K. Löbenstein, [*Über den Satz, dass eine ebene algebraische Kurve 6. Ordnung mit $11$ sich einander ausschliessenden Ovalen nicht existiert*]{}, Inaugural Dissertation, Göttingen, 1910. \[$\spadesuit$ Dissertation under Hilbert, attempting to prohibit the real sextic scheme consisting of 11 unnested ovals $\spadesuit$ considered non-rigorous in Gudkov 1974 [@Gudkov_1974/74], albeit it was in Hilbert’s view (cf. Hilbert 1909 [@Hilbert_1909-Ueber-die-Gestalt-sextic])\]  O. Lokki, [*Über Existenzbeweise einiger mit Extremaleigenschaft versehenen analytischen Funktionen*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. 76 (1950), 15 pp. 60, 78 $\bigstar$  B. Lund, [*Subalgebras of finite codimension in the algebra of analytic functions on a Riemann surface*]{}, Pacific J. Math. 51 (1974), 495–497. 60 \[$\spadesuit$ quotes Ahlfors’ existence of a circle map (termed therein unimodular function) and cite also Royden’s proof of 1962 [@Royden_1962] $\spadesuit$ the paper itself is devoted to the following result: if a uniform subalgebra $A$ of $A(R)$ the algebra of all analytic functions on the interior of a compact bordered Riemann surface $\bar R$ and continuous up to its boundary included contains a circle map, then $A$ has finite codimension in $A(R)$ $\spadesuit$ question: what about the converse? If the codimension is zero this boils down to Ahlfors 1950 [@Ahlfors_1950]\]  J. Lüroth, [*Note über Verzweigungsschnitte und Querschnitte in einer Riemann’schen Fläche*]{}, Math. Ann. 3 (1871), 181–184. 60 \[$\spadesuit$ considered as sketchy by Clebsch 1872 [@Clebsch_1872], and consequently supplemented with more details\]  A.M. Macbeath, [*On a theorem of Hurwitz*]{}, Proc. Glasgow Math. Assoc. 5 (1961), 90–96. \[$\spadesuit$ construction of infinite families of surfaces for which Hurwitz’s bound $84(g-1)$ is attained\]  A.M. Macbeath, [*Generators of the linear fractional groups*]{}, Proc. Sympos. Pure Math. (Houston, 1967). \[$\spadesuit$ construction of infinite families of surfaces for which Hurwitz’s bound $84(g-1)$ is attained\]  A.M. Macbeath, [*The classification of non-euclidean plane crystallographic groups*]{}, Canad. J. Math. 19 (1967), 1192–1205. \[$\spadesuit$\]  A.J. Macintyre, W.W. Rogosinski, [*Extremum problems in the theory of analytic functions*]{}, Acta Math. 82 (1950), 275–325. \[$\spadesuit$ this enters into our specialized picture as follows: this paper, joint with Rogosinski-Shapiro 1953 [@Rogosinski-Shapiro_1953], and Rudin 1955 [@Rudin_1955-class-Hp] constitutes a stream influencing the production of the paper of Read 1958 [@Read_1958_Acta] and Royden 1962 [@Royden_1962], where a new existence-proof of the Ahlfors map is given via functional analytic tools (Hahn-Banach) $\spadesuit$ challenge \[30.09.12\] upon assuming that Gabard 2006 [@Gabard_2006] is true, prove it via Hahn-Banach (good luck!)\] $\bigstar$  F. Maitani, Y. Kusunoki, [*Canonical functions on open Riemann surfaces*]{}, Complex Variables and Elliptic Equations (1992). \[$\spadesuit$ The canonical functions are meromorphic functions with a finite number of poles and their real parts are, roughly speaking, constant on each ideal boundary component of an open Riemann surface. The existence and geometrical: properties of such functions have been ...\]  B. Manel, [*Conformal mapping of multiply connected domains on the basis of Plateau’s problem*]{}, Revista Univ. Nac. Tucuman 3 (1942), 141–149. 78 \[$\spadesuit$ title essentially self-explanatory modulo the question of knowing which types of mappings are handled: Kreisnormierung, circle map, or some slit mappings?\] $\bigstar$  W. Mangler, [*Die Klassen von topologischen Abbildungen einer geschlossenen Fläche auf sich*]{}, Math. Z. 44 (1939), 541–554. \[$\spadesuit$ oft quoted, e.g. by Teichmüller\]  Yu.I. Manin, [*Superalgebraic curves and quantum strings*]{}, Trudy Mat. Inst. Steklov. 183 (1990), 126–138; English transl., Proc. Steklov Inst. Math. 183 (1991), 149–162. \[$\spadesuit$\]  A. Marden, [*On homotopic mappings of Riemann surfaces.*]{} Ann of Math. (2) (1969), 1–8. \[$\spadesuit$ Lemma 2 (on unlimited branched covering surfaces) is probably akin to the well-known lemma ascribed to Radó 1922 [@Rado_1922-Z-Theorie-mehr], compare Landau-Osserman 1960 [@Landau-Osserman_1960]\]  A. Marin, [*Quelques remarques sur les courbes algébriques planes réelles.*]{} In: Séminaire sur la géométrie algébrique réelle. Publ. Math. Univ. Paris VII, 1979, 51–86. \[$\spadesuit$ where the writer (Gabard) learned about the Rohlin inequality $r\ge m/2$, which does not appear in print by Rohlin although quite easily derived from the Rohlin formula in Rohlin 1978 [@Rohlin_1978]. For more details, cf. Gabard 2000 [@Gabard_2000] and the refs. therein\]  A. Marin, [*Sur un théorème de Cheponkus*]{}, in: Real Analytic and Algebraic Geometry, Proceedings Trento, 1988 (Eds. M Galbiati, A. Tognoli). Springer, Lecture Notes in Math., 1420, 191–193. \[$\spadesuit$ contains a corrected version of a theorem of Cheponkus (unsuitably proved in the original), as well as a proof of Klein’s intuition that curves of type I cannot acquire a new oval by continuous deformation of the coefficients. $\spadesuit$ \[26.03.13\] in fact it seems to me that Marin’s statement is slightly stronger than Klein’s original (unproved) assertion inasmuch as Klein 1876 [@Klein_1876] supposed that the curve traverses the discriminant across an isolated double point (with imaginary conjugate tangents), alias solitary double points in the Russian jargon of Arnold, Viro, etc.\]  D.E. Marshall, [*An elementary proof of the Pick-Nevanlinna interpolation theorem*]{}, Mich. Math. J. \[$\spadesuit$\]  D.E. Marshall, [*Removable sets for bounded analytic functions*]{}, In: [*Linear and complex analysis problem book*]{}. Lecture Notes in Mathematics 1043. Springer, Berlin, 1984, 2233–2234. CHECK PAGINATION incompatible with Murai 1990/91 [@Murai_1990/91-ICM] \[$\spadesuit$ if I do not mistake this is the first place where it is explained why Calderón’s achievement (Calderón 1977 [@Calderon_1977] on the $L^p$-continuity of the singular integral operator with a Cauchy kernel on a smooth curve) implies the so-called Denjoy conjecture (Denjoy 1909 [@Denjoy_1909-Painleve/Sur-les-fct-anal-unif-a-sing-discontinues]) about the removability of a closed set lying on a rectifiable curve being equivalent to the vanishing of its length $\spadesuit$ historical detail: Murai 1990/91 [@Murai_1990/91-ICM p.904–905] seems to ascribe the Denjoy conjecture to Calderón-Havin-Marshall using the (cryptical) abbreviation CHM on p.905 (but quotes only the present text of Marshall) $\spadesuit$ the Calderón-to-Denjoy implication is obtained by combining classical results of Garabedian, Havinson with Davie’s reduction of the Denjoy conjecture to the $C^1$-case, completing thereby the proof of Denjoy’s conjecture $\spadesuit$ in fact, Denjoy announced this as a theorem, but his proof turned out to be erroneous (compare Marshall [@Marshall_1978?] or Melnikov 1975/76 [@Melnikov-Sinanyan_1975/76 p.691] or Verdera 2004 [@Verdera_2004 p.29]) $\spadesuit$ alas, people rarely say explicitly who located the gap in Denjoy’s claim (this is a non-trivial historical quiz), but maybe Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950 p.122] are good candidates, yet they do not criticize directly Denjoy but rather establish the special case of Denjoy’s conjecture for linear and then analytic curves\]  G. Martens, [*Komponentenzerfällende abelsche Erweiterungen reeller algebraischer Funktionenkörper einer Variablen*]{}, Diss. Univ. Heidelberg, 1974. \[$\spadesuit$ cited in Martens 1975 [@Martens_1975], and might be the first place where the phrasing “total reell” appears in print\]  G. Martens, [*Galoisgruppen über aufgeschlossenen reellen Funktionenkörpern*]{}, Math. Ann. 217 (1975), 191–199. \[$\spadesuit$ p.197: total reality (“total reelle Galois Erweiterungen” are defined in the context of Galois extensions of real function fields)\]  G. Martens, [*Minimale Blätterzahl bei Überlagerungen Kleinscher Flächen über der projectiven Ebene*]{}, Archiv der Math. 30 (1978), 481–486. \[$\spadesuit$ sharp bound upon the degree of the Witt mapping $\spadesuit$ differential geometric application in Ross 1999 [@Ross_1997]\]  G. Martens, [*Die Zerlegungscharaktere abelscher total reeller Erweiterungen reeller Funktionenkörper einer Variablen*]{}, J. Reine Angew. Math. ?? (ca. 1978), ??–??. \[$\spadesuit$ quoted in Geyer-Martens 1977 [@Geyer-Martens_1977]\]  G. Martens, [*Funktionen von vorgegebener Ordnung auf komplexen Kurven*]{}, J. Reine Angew. Math. 320 (1980), 68–85. H.. Martens, [*A new proof of Torelli’s theorem*]{}, Ann. of Math. (2) 78 (1963), 107–111. \[$\spadesuit$\]  H.H. Martens, [*On the varieties of special divisors on a curve*]{}, J. Reine Angew. Math. 227 (1967), 111–120. \[$\spadesuit$ self-explanatory title, but do not prove the existence of special divisors\]  R.S. Martin, [*Minimal positive harmonic functions*]{}, Trans. Amer. Math. Soc. 49 (1941), 137–172. 60 \[$\spadesuit$ plays a fundamental rôle in Heins 1950 [@Heins_1950], who seems to offer an alternative proof of the existence of a circle map as the one of Ahlfors 1950 [@Ahlfors_1950]\]  M. Maschler, [*Minimal domains and their Bergman kernel function*]{}, Pacific J. Math. 6 (1956), 501–516. \[$\spadesuit$\]  M. Maschler, [*Classes of minimal and representative domains and their kernel functions*]{}, Pacific J. Math. 9 (1959), 763–782. \[$\spadesuit$ contain (according to entry Maschler 1959 [@Maschler_1959 p.173]) a description of the geometric shapes of minimal domains (i.e. essentially the range of the least area maps) in the case of doubly-connected domains\]  M. Maschler, [*Analytic functions of the classes $L^2$ and $l^2$ and their kernel functions*]{}, Rend. Circ. Mat. Palermo (2) 8 (1959), 163–177. \[$\spadesuit$ p.173 seems to assert that the range of the least area maps are unknown for domains of connectivity higher than $2$ $\spadesuit$ still on p.173 Kufarev 1935/37 [@Kufareff_1935/37] is credited for establishing that the least area map in the case of doubly-connected domains is not univalent, but schlicht upon a (two sheeted) Riemann surface\]  B. Maskit, [*The conformal group of a plane domain*]{}, Amer. J. Math. 90 (1968), 718–722. 78 \[$\spadesuit$ proves two results to the effect that any plane domain (resp. Riemann surface of \[finite\] genus $g$) conformally embeds into either the sphere or a closed Riemann surface of the same genus so that, under this embedding, every conformal automorphism of the original surface is the restriction of one of the compactified closed surface $\spadesuit$ the proof proceeds (via exhaustion) by reduction to the case of finite Riemann surfaces, previously established by the author\]  B. Maskit, [*Moduli of marked Riemann surfaces*]{}, Bull. Amer. Math. Soc. (1974). \[$\spadesuit$\]  B. Maskit, [*Canonical domains on Riemann surfaces*]{}, Proc. Amer. Math. Soc. 106 (1989), 713–721. \[$\spadesuit$ Kreisnormierung for surfaces supplementing the uniqueness lacking in the existence proof in Haas 1984 [@Haas_1984]\]  J. Mateu, X. Tolsa, J. Verdera, [*The planar Cantor sets of zero analytic capacity and the local $T(b)$-theorem*]{}, J. Amer. Math. Soc. 16 (2002), 19–28. \[$\spadesuit$ a complete characterization of the sets in the title is given via a little incursion of the Ahlfors function (on p.25) $\spadesuit$ \[23.09.12\] since Vitushkin and especially Garnett 1970 [@Garnett_1970] it is known that the $1/4$-Cantor set has $\gamma=0$ (zero analytic capacity), but positive length. More generally one may consider a $\lambda$-Cantor set $E(\lambda)$ for $0<\lambda<1/2$ (obtained by keeping only the four subsquares of length $\lambda$ pushed to the 4 corners of the unit-square and iterating ad infinitum) and ask about the ‘critical temperature’, i.e. the smallest $\lambda$ such that $\gamma(E(\lambda))>0$ $\spadesuit$ the critical value $\lambda$ is precisely $\lambda=1/4$, as follows from Theorem 1 of the cited work, describing more generally the case of a Cantor set $E(\lambda)$ associated to a sequence $\lambda=(\lambda_n)_{n=1}^{\infty}$ with variable $\lambda_n$ (in the range \]0,1/2\[) $\spadesuit$ naive question: in the case where $\lambda$ is constant (self-similar Cantor set) can we describe the behavior of $\gamma(E(\lambda))$ as a function $]0, 1/2[ \to [0,+\infty)$: is it monotone, bounded, analytic or at least derivable (especially at the critical value)? $\spadesuit$ naive answers: monotone most probably, bounded also certainly namely by $\gamma$ of the unit square corresponding to $E(1/2)$\]  P. Matildi, [*Sulla rappresentazione conforme di domini appartenenti a superficie di Riemann su di un tipo canonico assegnato*]{}, Ann. Scuola Norm. Super. Pisa (2) 14 (1945), (1948), 81–90. 60 \[$\clubsuit$ this paper (read by writer only the 13.07.12) seems to establish the existence of a circle map ([*cerchio multiplo*]{}) for compact bordered Riemann surface having only [*one*]{} contour. Thus with some imagination this may be regarded as a precursor of the Ahlfors circle map. (Recall that Ahlfors was well aware of this paper at least subsequently for it is cited in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960], alas without detailed comment.) Matildi also proposes a bound on the degree of the mapping whose dependence upon the topology is, however, not made completely explicit. He proposes namely, the degree $\lambda\le n(2n-3)$, where $n$ is the minimum degree of a projective-plane model for the Schottky-double of the given membrane. Perhaps it would be useful to estimate his bound purely in term of the topology (via basic algebraic geometry) $\clubsuit$ an extension of Matildi’s work to the case of membranes having several contours is claimed in Andreotti 1950 [@Andreotti_1950], but it still hard to decide if ti really cover the Ahlfors theorem of 1950\]  S. Matsuoka, [*Nonsingular algebraic curves in $RP^1\times RP^1$*]{}, Trans. Amer. Math. Soc. ?? (1991–93 ca. CHECK), ?–?. \[$\spadesuit$ Hilbert’s problem in ${\Bbb P}^1\times {\Bbb P}^1$, while extending to this context the Gudkov-Rohlin congruence and inequalities of Petrovskii and Arnold\]  S. Matsuoka, [*The configuration of the $M$-curves of degree $(4,4)$ in $RP^1\times RP^1$ and periods of real K3 surfaces*]{}, Hokkaido Math. J. 19 (1990), 359–376. \[$\spadesuit$ Hilbert’s problem in ${\Bbb P}^1\times {\Bbb P}^1$, while extending to this context the Gudkov-Rohlin congruence and inequalities of Petrovskii and Arnold\]  P. Mattila, [*Smooth maps, null-sets for integralgeometric measure and analytic capacity*]{}, Ann. of Math. (2) 123 (1986), 303–309. \[$\spadesuit$ includes a counterexample to the original formulation of Vitushkin’s conjecture ($E$ removable iff purely unrectifiable, i.e. the intersection with any curve of finite length has zero $1$-dimensional Hausdorff measure $H^1$) $\spadesuit$ Mattila’s counterexample has $H^1(E)=\infty$ (infinite length) $\spadesuit$ for the validation of Vitushkin’s conjecture in the case $H^1(E)<\infty$, see G. David 1998 [@David_1998]\]  P. Mattila, M.S. Melnikov, J. Verdera, [*The Cauchy integral, analytic capacity and uniform rectifiability*]{}, Ann. of Math. (2) 144 (1996), 127–136. 47 \[$\spadesuit$ analytic capacity, Ahlfors function and a step forward in understanding Painlevé null-sets geometrically $\spadesuit$ for the complete solution see Tolsa 2003 [@Tolsa_2003]\]  R. Mazzeo, M. Taylor, [*Curvature and uniformization*]{}, Israel J. Math. 130 (2002), 323–346. \[$\spadesuit$ uniformization via Liouville’s equation (Schwarz’s strategy, cf. also Bieberbach 1916 [@Bieberbach_1916-Delta-u-und-die-automorphen-Funkt]), as we know since Koebe (Überlagerungsfläche) this gives then the Kreisnormierung, cf. e.g. Bergman 1946 [@Bergman_1946]\]  S. McCullough, [*The trisecant identity and operator theory*]{}, Integr. Equat. Oper. Theory 25 (1996), 104–127. \[$\spadesuit$ pp.113–5: discussion of the Ahlfors function along the lines of Fay 1973 [@Fay_1973] and p.125 mentions Bell’s result 1991 [@Bell_1991-Szego] that the zeros of Ahlfors function are distinct if the center $a$ is chosen near enough the boundary $\spadesuit$ \[20.09.12\] as we already observed once, it could be interesting to investigate if Bell’s result extends to bordered surfaces (compact) of positive genus $p>0$. Of course in this case the degree of the Ahlfors map may jump from points to points (within the Ahlfors range $r\le \deg\le r+2p$), and this phenomenology is probably connected with deep algebro-geometric or differential-geometric invariants of the surface (Weierstrass points, etc.), compare the work of Yamada 2001 [@Yamada_2001] and Gouma 1998 [@Gouma_1998] for the hyperelliptic context (phenomenology of the mutation of the Ahlfors maps and their fluctuating degree upon dragging away the basepoint) $\spadesuit$ maybe it could be also worth looking if Bell’s result is somehow connected to Solynin’s result (2007 [@Solynin_2007]) about the confinement of the zeros of the Green’s function inside a compactum when the pole is dragged through the surface $\spadesuit$ at first sight, this looks quite plausible in view of Ahlfors formula (cf. 1947 [@Ahlfors_1947] and 1950 [@Ahlfors_1950]) that if $f$ is a circle map with zeros at $t_1, \dots , t_d$ then $\log \vert f(z)\vert$ matches a superposition of Green’s functions with poles at the $t_i$, i.e. $\log \vert f(z)\vert=\sum_{i=1}^d G(z,t_i)$, since both functions vanish on the contours and present the same singularities at the $t_i$ $\spadesuit$ since it is not the critical points of the individual Green’s function, but those of the superposed Green’s functions which are responsible for the ramification of $f$, a direct application of Solynin looks hazardous $\spadesuit$ still, one may wonder if the ramification of the Ahlfors map stay likewise trapped within a compactum upon dragging the center $a$ of the Ahlfors map $f_a$ through the membrane\]  A.D. Mednykh, [*Nonequivalent coverings of Riemann surfaces with a prescribed ramification type*]{}, Sibirsk Mat. Zh. 25 (1984), 120–142. \[$\spadesuit$\]$\bigstar$  A.D. Mednykh, [*Determination of the number of nonequivalent coverings on a compact Riemann surface*]{}, Doklady SSSR 239 (19??), 269–271. \[$\spadesuit$ cited in Natanzon 1993 [@Natanzon_1993] as akin to Hurwitz 1891 [@Hurwitz_1891-Uber-Riemannsche-Flachen]\]$\bigstar$  A.D. Mednykh, [*Branched coverings of Riemann surfaces whose branch orders coincide with the multiplicity*]{}, Comm. in Algebra 18 (1990), 1517–1533. \[$\spadesuit$\]$\bigstar$  Th. Meis, [*Die minimale Blätterzahl der Konkretisierung\[en\] einer kompakten Riemannschen Fläche*]{}, Schriftenreihe des Math. Inst. der Univ. Münster, Heft 16 (1960). \[$\spadesuit$ a much quoted—but hard-to-find—source where the gonality of a general closed Riemann surface of genus $g$ is found to be the bound predicted by Riemann, Brill-Noether, etc., namely $[\frac{g+3}{2}]$ $\diamondsuit$ Meis belongs to the Münster school (Behnke–Stein, etc.) $\clubsuit$ \[04.10.12\] it seems probable that the technique employed by Meis (which involves Teichmüller theory according to secondary sources, e.g. H.H. Martens’ MathReview of Gunning 1972 [@Gunning_1972]) could be adapted to the context of bordered surfaces and thus lead to a new proof of the Ahlfors map, even perhaps with the sharp bound given in Gabard 2006 [@Gabard_2006] $\spadesuit$ this seems to us to be a task of primary importance, but lacking a copy of Meis article we were relegated to make some general speculations (cf. Sec.\[sec:question\] which we summarize briefly) $\spadesuit$ the basic idea is to develop a “relative” Teichmüller theory not for pairs of Riemann surfaces of the same topological type (hence relatable by a “möglichst konform” diffeomorphism effecting the minimum distortion upon infinitesimal circles), but for just one Riemann surface which we try to express as a branched cover of the sphere (or the disc) for a fixed mapping degree $d$, while exhibiting the (quasiconformal) map of least distortion. Measuring this least dilatation, we get instead of the usual Teichmüller metric (distance) on the moduli space, a Teichmüller temperature $\varepsilon_d$ (or potential) whose vanishing amounts to the possibility of expressing the given surface as a (conformal!) branched cover of the disc (or the sphere), thereby resolving the Ahlfors mapping problem (or the Riemann-Meis problem) depending on the bordered or closed context. \[As a matter of convention the distortion (eccentricity of infinitesimal ellipses is $\ge 1$ and this is converted in values $\ge 0$ upon taking the logarithm)\] $\spadesuit$ in fact upon looking at the gradient flow of the Teichmüller temperature (trajectories of steepest descent orthogonal to the isothermic hypersurfaces $\varepsilon_d={\rm const.}$) we get a flow on the moduli space ($M_g$, if closed or $M_{p,r}$, if bordered) with the net effect of improving the gonality of each individual surface during its evolution $\spadesuit$ as the Teichmüller space is a cell one can hope to derive the existence of stagnation point of the flow by the usual Poincaré-Brouwer-Hopf index formula giving so an existence-proof of a conformal map. However this is a bit artificial for the existence of low degree maps is usually evident (looking at hyperelliptic surfaces and their bordered avatars). Of course it must perhaps be ensured that the flow only stagnates when the temperature vanishes (i.e. no saddle points nor sinks of positive temperature) $\spadesuit$ in such favorable circumstances any closed surface of genus $g$ would flow toward a hyperelliptic model representing the smallest possible gonality (=two) $\spadesuit$ likewise, in the bordered context one expects that any membrane of type $(p,r)$ converges to a membrane of least possible gonality, that is $r$ (excepted when $r=1$ and $p>0$ where the least topological degree is $2$) $\spadesuit$ admittedly, all this does not readily reprove Meis’ gonality (nor that of Ahlfors-Gabard) but maybe it is a first step toward a solution along this path, which—we repeat—should be found in the work of Meis (which in substance is nothing else than a relative (or ramified) version of classical Teichmüller theory) $\spadesuit$ perhaps the flow we are speaking about is not logically needed in Meis’ proof but it can certainly enhance the game $\spadesuit$ basically for each $d$ the continuity of the temperature function shows that the set of $d$-gonal surfaces is closed in the moduli space $M_g$, and since the set is nonempty as soon as $d\ge 2$ (hyperelliptic models) it suffices to show that it is open when $d$ is appropriately large. The expected value for $d$ is $[\frac{g+3}{2}]$ (resp. $r+p$ in the bordered case of $M_{p,r}$), yet it is precisely here that some idea is required $\spadesuit$ naively if the degree is high enough one disposes of enough free parameters to make variations exploring locally the full moduli space $\spadesuit$ alternatively one can perhaps argue that the temperature function $\varepsilon_d$ is real-analytic on $M_g$ so that it would suffice to check its vanishing on a small parametric (open) ball consisting of Riemann surfaces with explicitly given equations (this resembles perhaps Meis’ approach through the little I know of it via indirect sources, e.g. R.F. Lax 1975 [@Lax_1975-special-divisors-II])\]$\bigstar$  M.S. Melnikov, [*Structure of the Gleason part of the algebra $R(E)$*]{}, Funkt. Anal. Prilozhen. 1 (1967), 97–100; English transl. (1968), 84–86. \[$\spadesuit$ p.86, Ahlfors function via Vitushkin 1958 [@Vitushkin_1958] $\spadesuit$ the paper itself is devoted to giving another proof (via the apparatus of analytic capacity) of Wilken’s theorem that the Gleason part of the algebra $R(E)$ (of uniform limits on a compactum $E\subset {\Bbb C}$ of the rational functions of the variable $z$) consists either of one point (and is then a peak point), or it has positive area\]  M.S. Melnikov, S.O. Sinanyan, [*Aspects of approximation theory for functions of one complex variable*]{}, Itogi Nauki i Tekhniki 4 (1975), 143–250; English transl. in J. Soviet Math. 5 (1976), 688–752. \[$\spadesuit$ Vitushkin’s theory (i.e., uniform approximation by rational functions) and its relation to the Ahlfors function and the allied analytic capacity\]  M.S. Melnikov, [*Analytic capacity: discrete approach and curvature of measure*]{}, Sb. Math. 186 (1995), 827–846. 47 \[$\spadesuit$ analytic capacity (p.827), Ahlfors function (p.830, 838) and introduction of the concept of the curvature of a (positive Borel) measure in the plane \[Menger curvature\], which enables a new proof of Denjoy’s conjecture (without using Calderón’s $L^2$-estimates for the singular Cauchy integral) $\spadesuit$ this technique of Melnikov is also instrumental in Tolsa’s solution (2003 [@Tolsa_2003]) of the (full) Painlevé problem\]  M.S. Melnikov, J. Verdera [*A geometric proof of the $L^2$ boundedness of the Cauchy integral on Lipschitz graphs*]{}, Internat. Math. Research Notices 7 (1995), 325–331. \[$\spadesuit$ another approach to Calderón 1977 [@Calderon_1977]\]  O. Mengoni, [*Die konforme Abbildung, gewisser Polyeder auf die Kugel*]{}, Monatsh. f. Math. u. Phys. 44 (1936), 159–185. \[$\spadesuit$ Seidel’s summary: the paper is a contribution to the problem of conformal mapping of simply-connected closed polyhedra upon the sphere. According to H.A. Schwarz, this problem can be reduced to the determination of a number of constants from a set of transcendental equations. It is shown that the explicit solution can be determined in a number of cases not considered by Schwarz. The paper concludes with a discussion of the results form the viewpoint of numerical computations\] $\bigstar$  S.N. Mergelyan, [*Uniform approximations to functions of a complex variable*]{}, Amer. Math. Soc. Transl. 1001 (1954).  H. Meschkowski, [*Beziehungen zwischen den Normalabbildungsfunktionen der Theorie der konformen Abbildung*]{}, Math. Z. 55 (1951), 114–124. 78  H. Meschkowski, [*Über die konforme Abbildung gewisser Bereiche von unendlich hohen Zusammenhang auf Vollkreisbereiche, I*]{}, Math. Ann. 123 (1951), 392–405. 60, 78 \[$\spadesuit$ some infinite connectivity cases of KNP, via iterative methods à la Koebe and area estimates due to Rengel 1932/33 [@Rengel_1932-33]\]  H. Meschkowski, [*Über die konforme Abbildung gewisser Bereiche von unendlich hohen Zusammenhang auf Vollkreisbereiche, II*]{}, Math. Ann. 124 (1952), 178–181. 60, 78 \[$\spadesuit$ sequel of the previous paper, building again over a Rengel (1932/33 [@Rengel_1932-33]) area estimate for 4-gons and using Grötzsch’s (1929 [@Groetzsch_1929]) mapping of a domain of infinite connectivity upon a Kreisschlitzbereich, reducing therefore the general study to this special case\]  H. Meschkowski, [*Einige Extremalprobleme aus der Theorie der konformen Abbildung*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. 117 (1952), 12pp. 60, 78 \[$\spadesuit$ which mappings?, essentially all types, but relies heavily on previous works of Garabedian-Schiffer and Nehari $\spadesuit$ mention the issue that there is no known extremal problem yielding the Koebe Kreisnormierung, for an update on this question see several works of Schiffer via Fredholm eigenvalues\]  H. Meschkowski, [*Verzerrungssätze für mehrfach zusammenhängende Bereiche*]{}, Compositio Math. 53 (1953), 44–59. 78 \[$\spadesuit$ Kreisbogenschlitz map, and somewhat relevant to the point discussed in Gaier 1978 [@Gaier_1978-JDMV] $\spadesuit$ more precisely shows that the Ahlfors-type problem of maximizing the derivative among [*schlicht*]{} function bounded-by-one gives a conformal map upon a Kreisschlitzbereich (=circular slit disc). See also Reich-Warschawski 1960 [@Reich-Warschawski_1960]\]  G. Mikhalkin, [*Adjunction inequality for real algebraic curves*]{}, arXiv (1994). \[$\spadesuit$ as pointed out by Th. Fiedler (letter of the 9 March in Sec.\[e-mail-Viro:sec\]), this paper of Mikhalkin seems to be one of the earliest application of the Thom conjecture to Hilbert’s 16th\]  G. Mikhalkin, [*The complex separation of real surfaces and extensions of Rokhlin congruence*]{}, Invent. Math. 118 (1994), 197–222. \[$\spadesuit$\]  G. Mikhalkin, [*Adjunction inequality for real algebraic curves*]{}, Math. Res. Lett. 4 (1997), 45–52. \[$\spadesuit$\]  G. Mikhalkin, [*Real algebraic curves, the moment map and amoebas*]{}, Ann. of Math. (2) 151 (2000), 309–326. \[$\spadesuit$\]  I.P. Millin, [*The method of areas for schlicht functions in finitely connected domains*]{}, (Russian) Trudy Mat. Inst. Steklov 94 (1968), 90–121. 78 \[$\spadesuit$ cited in Grunsky 1978 [@Grunsky_1978 p.185], hence possibly relevant to the issue discussed in Gaier 1978 [@Gaier_1978-JDMV]\] $\bigstar$$\bigstar$$\bigstar$  J. Milnor, [*On spaces having the homotopy type of $CW$complexes*]{}, Trans. Amer. Math. Soc. ? (1959), ?–?. \[$\spadesuit$ states the result that a metric manifolds has always the homotopy type of a $CW$-complex, building over result of the Polish and Swede school (Kuratowski, Borsuk, Hanner), cf. also the more detailed implementation in Palais 1962 $\spadesuit$ be careful about terminology, Milnor states this result under the assumption of separability, yet if this means (as it does presently) existence of a denumerable dense part, then there is some simple counter-example of Prüfer described in Gabard 2006/08 [@Gabard-2006/08] \]  J. Milnor, [*On the Betti numbers of real varieties*]{}, Proc. Amer. Math. Soc. 15 (1964), 275–280. \[$\spadesuit$ cf. also Thom 1965 [@Thom_1965]\]  C.D. Minda, [*The Aumann-Carathéodory rigidity constant for doubly connected regions*]{}, Kodai Math. J. 2 (1979), 420–426. 47 \[$\spadesuit$ p.422 an elementary existence-proof of the Ahlfors function is given in the case of an annulus $A:=\{z: 1/R<\vert z\vert < R\}$ conjointly with the fact that the map is uniquely prescribed by its two zeros $a,b$ (up to rotation) subjected to the relation $\vert a b\vert=1$ $\spadesuit$ one can wonder if in this case the circle maps of minimum degree (here $r=2$, i.e. two contours) coincide exactly with the Ahlfors map $f_a$ maximizing the distortion at $a$ (both depends upon $2$ real parameters) $\spadesuit$ p.424, still in the annulus case an explicit expression of the Ahlfors function is given in terms of the theta function, in a way analogous to work of Robinson 1943 [@Robinson_1943] and Abe 1958 [@Abe_1958] (\[03.10.12\] compare maybe also Golusin 1952/57 [@Golusin_1952/57]) $\spadesuit$ this is then applied to give an explicit formula for the Aumann-Carathéodory rigidity constant (1934 [@Aumann-Caratheodory_1934]) $\spadesuit$ p.420, another proof of the so-called annulus theorem is given (quoting the variety of proofs due to H. Huber 1951, Jenkins 1953, Kobayashi 1970, Landau-Osserman 59/60 [@Landau-Osserman_1960], Reich 1966, Schiffer 1946), but emphasizing that the present proof is patterned along Heins 1941 [@Heins_1941-iteration] showing “that the annulus theorem should properly be traced back to Heins’ work”\]  C.D. Minda, [*The hyperbolic metric and coverings of Riemann surfaces*]{}, Pacific J. Math. 84 (1979), 171–182. 50 \[$\spadesuit$ Ahlfors 1947 [@Ahlfors_1947] and 1950 [@Ahlfors_1950] are cited as follows (p.180): “A function $\tilde{f}$ in ${\cal B}(X)$ which maximizes $\vert\tilde{f}'(p)\vert$ is called an Ahlfors function (\[1\](=1947), \[2\](=1950)) and $c_B(p)=\max \{ \vert\tilde{f}'(p)\vert : \tilde{f} \in {\cal B}(X)\}$ is called the analytic capacity metric.” $\spadesuit$ from the abstract (freely and perhaps loosely reproduced): given two Riemann surfaces $X,Y$ endowed with their hyperbolic metrics, the principle of hyperbolic metric (aka Schwarz-Pick-Ahlfors lemma) says that any analytic map $f\colon X\to Y$ is a contraction. “Moreover, equality holds if and only if $f$ is an (unbranched, unlimited) covering of $X$ onto $Y$” $\spadesuit$ \[04.10.12\] the latter property is essentially topological so applies to any Ahlfors map (even in the extended sense of—what we call—circle maps). We could then lift the hyperbolic (Riemann-Poincaré) metric on the disc to the bordered surface. Alas the ramification creates singularities (in this metric attached to a circle map), so that we certainly do not recover the hyperbolic metric on the interior of the bordered surface. The other way around we may assume uniformization (recall that the interior of any bordered surface is hyperbolic) and try to investigate the metric properties of varied circle maps. In particular is there any special feature related to the circle maps of smallest degree (alias the separating gonality in Coppens 2011 [@Coppens_2011])? Also, given a point $p\in F$ (in the interior) there is a unique Ahlfors map $f_p$ from $F$ to the disc maximizing the derivative and since $f_p$ is “étale” at $p$ we get the above mentioned capacity metric which is more negatively curved that the hyperbolic metric (cf. Suita and Burbea’s papers). Unfortunately the degree of the Ahlfors function is quite mysterious (being subjected to spontaneous quantum fluctuations), but since everything is encoded in the hyperbolic metric there must be an algorithm which given the input of $F$ with the marked point $p$ computes the degree of $f_p$ in terms of the intrinsic geometry of $F$ $\spadesuit$  Some very vague guesses: given $p$ there is a homology basis consisting of loops all based at $p$, and by compactness a smallest “systolic-type” system of such curves of minimal total length probably individually consisting of geodesics; this gives a real number and \[pure guess\] its integer part is the degree of $f_p$. Variant: there is a compact bordered surface capturing all the homology (plus the given point $p$) whose finite volume(=area) $\spadesuit$ Further once the hyperbolic metric is introduced on $F$ any Ahlfors map at $p$ gives a stretching factor at $p$ which by the principle of contraction is $\le 1$, and we get a (probably continuous) function $\delta\colon F\to ]0,1]$ of $F$ measuring this distortion. Does the function extends to the boundary $\partial F$? (and could it be harmonic??). Intuitively when the degree of $f_p$ is low one may expect that the distortion is high. On the other hand there is largest schlicht disc centered at the origin where $f_a$ is unramified, but beware that ramification may come from another point than $p$ lying above the origin. So the right viewpoint is that there is a maximal disc centered at $p$ which is ramificationless. \]  C.D. Minda, [*The image of the Ahlfors function*]{}, Proc. Amer. Math. Soc. 83 (1981), 751–756. \[$\spadesuit$ Ahlfors function for domains of infinite connectivity $\spadesuit$ p.751: “Ahlfors \[1\](=1947 [@Ahlfors_1947]) showed that $h(\Omega)=B$ \[i.e. the Ahlfors function is surjective on the disc $B$\] for regions $\Omega$ of finite connectivity that have no trivial boundary components. More precisely, he proved that $h$ expresses $\Omega$ as an $n$-sheeted branched covering of $B$, where $n$ is the order of connectivity of $\Omega$. In the general situation Havinson 1961/64 [@Havinson_1961/64] and Fisher 1969 [@Fisher_1969] demonstrated that $B\setminus h(\Omega)$ has analytic capacity zero; \[…\]. It is not difficult to give an example of a region $\Omega$ such that $B\setminus h(\Omega)\neq \varnothing$. For example, let $K$ be a closed set of $B$ which has analytic capacity zero and $\Omega=B\setminus K$. If $0\in \Omega$, then the Ahlfors function $h$ for $\Omega$ and $0$ is the identity function, so $h(\Omega)=B\setminus K$. The question of the size of $B\setminus h(\Omega)$ becomes more interesting if it is required that $\Omega$ be a maximal region for bounded holomorphic functions in the sense of Rudin 1955 [@Rudin_1955]. \[$\to$ Recall Rudin’s definition (p.333): “A boundary point $x$ of $D$ \[=domain in the Riemann sphere\] is said to be [*removable*]{} if for every $f\in B(D)$\[=bounded analytic function\] there exists a neighborhood $V$ of $x$ such that $f$ can be extended to $V$. By an [*essential*]{} boundary point of $D$ we mean one that is not removable. If every boundary point of $D$ is essential, we say that $D$ is [*maximal*]{}.”\] For such a maximal region $\Omega$, Fisher 1972 [@Fisher_1972-The-moduli-of-extremal-fctions] raised the question of whether the Ahlfors function must map $\Omega$ onto $B$. Röding 1977 [@Roeding_1977_Ahlfors] answered this question in the negative by exhibiting a maximal region $\Omega$ and a point $p\in \Omega$ such that the Ahlfors function for $\Omega$ and $p$ omitted two values in $\Omega$. We shall extend Röding’s result by showing that an Ahlfors function for a maximal region can actually omit a fairly general discrete set of values in $B$.” $\spadesuit$ p.755: “Therefore, it is still an open question whether the Ahlfors function for a maximal region can actually omit an uncountable set of zero analytic capacity.” $\spadesuit$ \[05.10.12\] an update (positive answer) is implied by Yamada 1992 [@Yamada_1992-Ahlfors-fct-on-Denjoy] where an example is given where the omitted set of the Ahlfors function has positive logarithmic capacity (hence uncountable, because sets of logarithmic capacity zero are stable under countable unions, see e.g. Tsuji 1959 [@Tsuji_1959-BOOK/Chelsea1975])\]  C.D. Minda, [*Bloch constant for meromorphic functions*]{}, Math. Z. ?? (1982), ??–??. \[$\spadesuit$ “Our geometric approach to the construction of an upper bound is more elementary and clearly shows the analogy with the Ahlfors-Grunsky example. \[…\] Let $XXXX$ be a compact bordered Riemann surface with genus $g$ and $m$ boundary components.”\]  S. Minsker, [*Analytic centers and analytic diameters of planar continua*]{}, Trans. Amer. Math. Soc. 191 (1974), 83–93. \[$\spadesuit$ the Ahlfors function is mentioned twice on p.91, 92 $\spadesuit$ the paper itself contains results about analytic centers and analytic diameters (concepts arising in Vitushkin’s work on rational approximation)\]  N.M. Mishachev, [*Complex orientations of plane $M$-curves of odd degree*]{}, Funkt. Anal. Prilozhen. 9 (1975), 77–78; English transl., Funct. Anal. Appl. 9 (1975), 342–345. \[$\spadesuit$ adaptation of Rohlin’s complex orientation formula to odd degrees\]  I.P. Mitjuk, [*The principle of symmetrization for multiply connected regions and certain of its applications*]{}, (Russ.) Ukrain. Mat. Ž 17 (1965), 46–54; Amer. math. Soc. Transl. 73, 73–85. 78 $\bigstar$  I.P. Mitjuk, [*The inner radius of a region and various properties of it*]{}, (Russ.) Ukrain. Mat. Ž 17 (1965), 117–122; Amer. math. Soc. Transl. ??, ??–??. \[$\spadesuit$ a formula expressing the inner radius $r(G,0)$ of a domain $G$ containing the origin and bounded by $n$ analytic Jordan curves is given in terms of the Ahlfors function $F_G(z,0)$ (normalized as usual by $F_G(0,0)=0$ and $F'_G(0,0)>0$) and the Green’s function $g_G(z,0)$ $\spadesuit$ the formula reads $r(G,0)=\frac{1}{F'_G(0,0)}\exp(\sum_{k=1}^{n-1}g_G(0,z_k))$, where $z_k$ are the $n-1$ extra zeros of the Ahlfors function $\spadesuit$ related material in Bandle-Flucher 1996 [@Bandle-Flucher_1996]\] $\bigstar$$\bigstar$$\bigstar$  I.P. Mitjuk, [*Extremal properties of meromorphic functions in multiply connected domains*]{}, Ukrain. Mat. Ž 20 (1968), 122–127; Amer. math. Soc. Transl. 76, 116–120. 47 \[$\spadesuit$ Ahlfors’ function occurs thrice: twice on p.116 and once on p.117 and is applied to obtain a connection between the inner radius and the transfinite diameter\]  Y. Miyahara, [*On relations between conformal mappings and isomorphisms of spaces of analytic functions on Riemann surfaces*]{}, J. Math. Soc. Japan 31 (1979), 373–389. 50 \[$\clubsuit$ on p.375, Ahlfors 1950 [@Ahlfors_1950] is cited for a result on the existence of a basis of analytic Schottky differentials whose periods along a canonical homology basis are calibrated to Kronecker’s delta. Hence the discussion is not directly relevant to the circle map, yet the general construction is quite akin (Green’s function, period of the conjugate differential, etc.) $\spadesuit$ p.380, one reads: “Let $g$ be a nonconstant function in $A(S')$ satisfying $\vert g\vert=1$ on $\partial S'$. (This is a so-called inner function.)” $\spadesuit$ the existence of a such a map follows (perhaps) from Ahlfors 1950, and if so the author perhaps fails to emphasize this issue adequately\]  Y. Miyahara, [*On local deformations of a Banach space of analytic functions on a Riemann surface*]{}, J. Math. Soc. Japan 40 (1988), 425–443. 50 \[$\clubsuit$ on p.436, Ahlfors 1950 [@Ahlfors_1950] is cited in essentially the same context as for the previous entry, i.e. Miyahara 1979 [@Miyahara_1979]\]  H. Mizumoto, [*On conformal mapping of a Riemann surface onto a canonical covering surface*]{}, Kōdai Math. Sem. Rep. 12 (1960), 57–69. 50, 78 \[$\clubsuit$ an essentially topological proof of (Ahlfors) circle maps is given, recovering the same degree $r+2p$ as Ahlfors 1950 [@Ahlfors_1950] $\clubsuit$ for a (possible) improvement to $r+p$, cf. Gabard 2006 [@Gabard_2006] $\spadesuit$ in case Mizumoto’s argument is solid, this seems to be a much underestimated paper as it is quoted by $0$ according to the electronic counters, but it is in Grunsky 1978 [@Grunsky_1978]\]  A.F. Möbius, [*Über die Grundformen der Linien der dritten Ordnung*]{}, ?? (18??), ??–??. (Möbius Werke II, S.89). \[$\spadesuit$ cited in Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege] for the elementary property of what we call now “ovals” versus “pseudolines” (then called “paare und unpaare Curvenzüge”) and also in Harnack 1876 (p.190), but not in Zeuthen 1874 [@Zeuthen_1874] who only cites von Staudt 18XX [@von-Staudt_18XX]\]  A.F. Möbius, [*Theorie der elementaren Verwandschaft*]{}, Ber. Verhandl. Königl. Sächs. Gesell. d. Wiss., mat.-phys. Klasse 15 (1863), 18–57. (Möbius Werke II). \[$\spadesuit$ a revolutionnary paper fixing the bases of “Morse theory” and classifying en passant the closed orientable surfaces, $\spadesuit$ followed by Jordan 1866 [@Jordan_1866], and vital to Klein’s theory of symmetric surfaces. Of course according to Klein (cf. 1892 [@Klein_1892_Vorlesung-Goettingen]), this topological classification must have been known to Riemann\]  M. Monastyrsky, [*Riemann, Topology and Physics*]{}, 1st edition 1987; Second Edition 1999, Reprint in 2008, Modern Birkhäuser Classics (Part I of the 1st ed., Moscow, 1979). \[$\spadesuit$ p.20: “The concluding remarks in the dissertation show that the general nature of the problem of analytic functions on arbitrary multiconnected domains was already clear to Riemann.” $\spadesuit$ compare for similar remarks Klein 1882 [@Klein_1882], Klein 1892 [@Klein_1892_Vorlesung-Goettingen] $\clubsuit$ p.72: “Riemann’s note, “Equilibrium of electricity on circular cylinders”, evidently dates to this same period. The problem of the distribution of electrical charge in cylindrical conductors leads to the purely mathematical problem of solving Laplace’s equation in a simply connected[^143] domain with prescribed boundary condition. Here for the first time automorphic functions arise.” $\spadesuit$ \[26.12.12\] quoting Weierstrass: “To the question, Can one really obtain anything directly applicable from those abstract theories with which today’s contemporary mathematicians occupy themselves?, I can answer that Greek mathematicians studied the properties of conic sections in a purely theoretical way long before the time when anyone could foresee that these curves represent the paths along which the planets move. I believe that many more functions with such properties will be found; for example, the well-known $\theta$-functions of Jacobi make it possible, on the one hand, to find the number of squares into which any given number decomposes, thereby making it possible to rectify an arc of ellipse, and, on the other hand, they make it possible to find the true law of the oscillations of a pendulum.”\]  A.F. Monna, [*Dirichlet’s principle. A mathematical comedy of errors and its influence on the development of analysis*]{}, Oosthoek, Scheltema, and Holkema, Utrecht, 1975. $\bigstar$  J.-P. Monnier, [*Divisors on real curves*]{}, Adv. Geom. 3 (2003), 339–360. \[$\spadesuit$ compare for a partial rejection of Monnier’s conjecture the discussion in Coppens-Martens 2010 [@Coppens-Martens_2010]\]  P. Montel, [*Sur les suites infinies de fonctions*]{}, Ann. École Norm. Sup. (3) 4 (1907), 233–304. \[$\spadesuit$ Montel’s Thesis building over Arzelá, Vitali, Lebesgue, etc. leading to the concept of “normal families”, pivotal in the resolution of extremal problems involving bounded functions (e.g. the so-called Ahlfors function) $\spadesuit$ the nomenclature “normal families” was coined afterwards in Montel 1913 [@Montel_1913-CRAS] $\spadesuit$ simultaneous related work appeared independently by Koebe ca. 1907 in relation with his distortion theorem, compare e.g. the historical analysis of Bieberbach 1968 [@Bieberbach_1968-Das-Werk-Paul-Koebes p.150–151] who writes: “Beim Beweis wird nun neben dem Viertelsatz ein allgemeiner Konvergenzsatz benutzt. Das ist nichts anderes als das, was man in Montels Theorie der Normalen Funktionenfamilien, heute kurz den [*Vitalischen Reihensatz*]{} nennt. Koebe hat ihn selbständig entdeckt \[11\](=Koebe 1908 UbaK3 [@Koebe_1908_UbaK3]). Er leitet ihn aus der Wurzel ab, die auch den anderen Forschern die Anregung gab: Hilberts Arbeit über das Dirichletsche Prinzip (1901) und die vierte Mitteilung über Integralgleichung (1906) des gleichen Forschers. \[…\]”\]  P. Montel, [*???*]{}, C.R. Acad. Sci. Paris 153 (1911), 996–998. \[$\spadesuit$ where the nomenclature “normal families” appears first in the literature\]  P. Montel, [*Leçons sur les familles normales de fonctions analytiques et leurs applications*]{}, Gauthier-Villars, Paris, 1927. \[$\spadesuit$ Montel’s treatise on the subject which appeared 20 years after the subject began\]  G. Moore, N. Seiberg, [*Classical and quantum conformal field theory*]{}, Commun. Math. Phys. 123 (1989), 177–254. \[$\spadesuit$ p.178: “The Riemann surface can be formed by sewing a number of three holed spheres (a.k.a. trinions).” \[this jargon is due to Möbius 1860/63 [@Moebius_1863]\]\]  J.W. Morgan, Z. Szabó, C.H. Taubes, [*The generalized Thom conjecture*]{}, Preprint 1995; cf. also next entry \[$\spadesuit$ cited in Kirby’s list 1970–95 [@Kirby_1970--95], as another (independent of Kronheimer-Mrowka’s) proof of the Thom conjecture\]  J.W. Morgan, Z. Szabó, C.H. Taubes, [*A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture*]{}, J. Differ. Geometry 44 (1996), 706–788. \[$\spadesuit$ proof of a generalized Thom conjecture for smooth holomorphic curves with $C\cdot C\ge 0$ (nonnegative self-intersection) in an arbitrary Kähler surface (or even a symplectic $4$-manifolds) $\spadesuit$ p.707: “The Thom conjecture and very similar generalizations of it have been established independently by Kronheimer-Mrowka; see \[4\](=Kronheimer-Mrowka 1994 [@Kronheimer-Mrowka_1994]).”\]  A. Mori, [*Conformal representation of multiply connected domain on many-sheeted disc*]{}, J. Math. Soc. Japan 2 (1951), 198–209. 60, 78 \[$\clubsuit$ reprove the circle map (ascribed to Bieberbach/Grunsky) via potential theory (Green’s function), plus a mixture of linear algebra and topology (homology) $\spadesuit$ Lemma 1 gives also an “iff” condition for a group of points in the interior to be the fibre of a circle map (in terms of harmonic measure) (compare Fedorov 1991 [@Fedorov_1991] for a similar game) $\spadesuit$ \[26.09.12\] it would be nice(?) to extend such a characterization to the positive genus case, and try to recover the Gabard bound $r+p$ by this procedure\]  C.B. Morrey, [*Multiple Integrals in the Calculus of Variations*]{}, Grundlehren der math. Wissenschaften 130, Springer-Verlag, Berlin, 1966. \[$\spadesuit$ includes a proof of Koebe’s Kreisnormierung via a Plateau-style approach (extending thereby Douglas’ derivation (1931 [@Douglas_1931-Solution]) of the RMT) $\spadesuit$ however some little gaps in the execution are noticed (but filled) by Jost 1985 [@Jost_1985], cf. also Hildebrandt-von der Mosel 2009 [@Hildebrandt-von-der-Mosel_2009] $\spadesuit$ \[07.10.12\] it is tempting to conjecture that the Plateau-style approach should also have something to say about the Ahlfors circle maps (cf. Courant 1939 [@Courant_1939] for the planar case, i.e. the Bieberbach-Grunsky theorem), however to (my knowledge) it was never attempted to tackle the case of positive genus ($p>0$)\]  M. Morse, [*?*]{}, Trans. Amer. Math. Soc. 27 (1925?). \[$\spadesuit$ crtical point theory, cited in Petrowsky 1938 [@Petrowsky_1938] as one of the tool used in the proof of the Petrovskii’s inequalities\]  M. Morse, M. Heins, [*Topological methods in the theory of a function of a complex variable*]{}, Bull. Amer. Math. Soc. ?? (1947), 1–14. \[$\spadesuit$ p.1: “The modern theory of meromorphic functions has distinguished itself by the fruitful use of the instruments of modern analysis and in particular by its use of the theories of integration. It success along the latter line has perhaps diverted attention from some of the more finitary aspects of the theory which may be regarded as fundamental.”\]  M. Morse, [*La construction topologique d’un réseau isotherme sur une surface ouverte*]{}, J. Math. Pures Appl. (9) 35 (1956), 67–75. 60 $\bigstar$  D. Mumford, [*Theta characteristics of an algebraic curve*]{}, Ann. Sci. École Norm. Sup. (4) 4 (1971), 181–192. \[$\spadesuit$\]  D. Mumford, [*Curve and Their Jacobians*]{}, The University of Michigan Press, Ann Arbor, 1975, 104 pp; reprinted in the “Red Book of Varieties and Schemes”, 2nd Edition, Lecture Notes in Math. 1358, Springer, 1999. \[$\spadesuit$\]  D. Mumford, [*Complex algebraic varieties*]{}, ??, Springer, 197X. \[$\spadesuit$\]  T. Murai, [*Construction of $H^1$ functions concerning the estimate of analytic capacity*]{}, Bull. London Math. Soc. 19 (1987), 154–160. \[$\spadesuit$ p.154 mentions the Ahlfors function (via Garnett’s book 1972 [@Garnett_1972 p.18]) and its indirect rôle in Garnett’s 1970 [@Garnett_1970] exposition (of Vitushkin’s 1959 example [@Vitushkin_1959] of a set of positive length but vanishing analytic capacity), but then Murai prefers to switch to the so-called Garabedian function to derive a direct proof of the vanishing of the analytic capacity\]  T. Murai, [*Analytic capacity for arcs*]{}, In: Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990, 901–911, The Mathematical Soc. of Japan, 1991. 47 \[$\spadesuit$ 3 occurrences of the Ahlfors function, on p.902 (via Garnett 1970 [@Garnett_1970 p.24]), p.904, p.905 $\spadesuit$ seems to ascribe the Denjoy conjecture to Calderón-Havin-Marshall using the (cryptical) abbreviation CHM on p.905 (but quotes only Marshall [@Marshall_1978?])\]  T. Murai, [*The arc-length variation of analytic capacity and a conformal geometry*]{}, Nagoya Math. J. 125 (1992), 151–216. 47 \[$\spadesuit$ 4 occurrences of the Ahlfors function, on p.152, 159, 191, 199 $\spadesuit$ analytic capacity (of a compact plane set) and its variation under a small change of the compactum $E$ (theory of Hadamard-Schiffer), with apparently a connection to Löwner’s differential equation\]  P.J. Myrberg, [*Über die Existenz der Greenschen Funktionen auf einer gegebenen Riemannschen Fläche*]{}, Acta Math. 61 (1933), 39–79. 60 \[$\spadesuit$\]  S. Nagura, [*Kernel functions on Riemann surfaces*]{}, Kōdai Math. Sem. Rep. (9) 35 (1951), 73–76. 60 \[$\spadesuit$ theory of the Bergman kernel on a Riemann surface using an exhaustion by compact bordered subregions with analytic boundaries\]  M. Nakai, [*The corona problem on finitely sheeted covering surfaces*]{}, Nagoya Math. J. 92 (1983), 163–173. 50 \[$\spadesuit$ p.164: “As is well known these surfaces \[=finite open Riemann surfaces\] are represented as unbounded finitely sheeted covering surfaces of the unit disk $\Delta$ (cf. e.g. Ahlfors \[1\](=1950 [@Ahlfors_1950])).” $\spadesuit$ comment of Gabard \[12.09.12\]: it may appear as a bit unfair that Alling’s works are omitted in the bibliography of this work, and more specifically Gamelin’s accreditation of the bordered corona on p.164 looks historically erroneous in view of the earlier work of Alling 1964 [@Alling_1964], and Alling 1965 [@Alling_1965] (for full details)\]  M. Nakai, 1985 see Hara-Nakai 1985 [@Hara-Nakai_1985]. M. Nakai, [*Valuations on meromorphic functions of bounded type*]{}, Trans. Amer. Math. Soc. 309 (1988), 231–252. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited in the following context (of valuation-theoretic stability) on p.240: “The following is due to Frank Forelli \[4\](=private communication) to whom the author is very grateful for many valuable suggestions and information:—[Example 1.]{} [*Any finitely sheeted disc is stable*]{}.—The result follows immediately from Theorem 1 \[[*an unlimited finite covering surface is stable iff its base is*]{}\] and Theorem 2 \[[*the open unit disc is stable*]{}\]. Plane regions bounded by finitely many mutually disjoint nondegenerate continua are finitely sheeted disks by the Bieberbach-Grunsky theorem (cf. e.g. \[16\](=Tsuji 1959/75 [@Tsuji_1959-BOOK/Chelsea1975])) or more generally finite open Riemann surfaces are finitely sheeted disks by the Ahlfors theorem \[1\](=Ahlfors 1950 [@Ahlfors_1950]). Here a finite open Riemann surface is a surface obtained from a closed surface by removing a finite number of mutually disjoint nondegenerate continua. Hence as a special case of the above example we have—[Corollary.]{} [*Finite open Riemann surfaces are stable.*]{}” $\spadesuit$ the notion of stability involved is the following (p.231): “Any valuation on the field $M(W)$ of single-valued meromorphic functions on a Riemann surface $W$ is a point valuation (Iss’ssa 1966). What happens to valuations on subfields of $M(W)$? An especially interesting subfield in this context is the field $M^{\infty}(W)$ of meromorphic functions of bounded type on $W$ (cf. \[2\](=Alling 1968)) $\spadesuit$ the exact definition is given on p.232: “A single-valued meromorphic function $f$ on a Riemann surface $W$ is said to be [*of bounded type*]{} if $f=\frac{g}{h}$ on $W$ where $g$ and $h$ are bounded holomorphic functions on $W$ with $h \not\equiv 0$.” $\spadesuit$ p.232/4: “We say that a Riemann surface $W$ is [*stable*]{} if $M^{\infty}(W)$ is nontrivial and any valuation on $M^{\infty}(W)$ is a point valuation.” $\spadesuit$ \[29.09.12\] roughly it seems that this notion of stability leads to a theory quite parallel to that of the corona problem, for the above positive (finitistic) result of Nakai is quite parallel to that of Alling 1964 [@Alling_1964] in the “coronal realm” and further the open question are similar e.g. p.241: “[Open problem 2]{}. [*Is there any stable plane region of infinite connectivity?*]{}” $\spadesuit$ however in the Corona problem it is still an open problem whether any plane region satisfies the corona theorem, but here Nakai (p.241) gives a nonstable plane region “obtained from the punctured open unit disc $\Delta_0$ by removing a sequence of mutually disjoint closed disks with centers on the positive real axis that accumulates only at $z=0$ (a \[so-called\] Zalcman $L$-domains \[17\](=Zalcman 1969 [@Zalcman_1969-TAMS]))\]  M. Namba, [*Geometry of Projective Algebraic Curves*]{}, Marcel Dekker, New York and Basel, 1984. \[$\spadesuit$ a textbook on curves via a mixture of transcendant and algebro-geometric recipes (browsed through it ca. 1998–2000, so cannot remember exactly the content)\]  D. Nash, [*Representing measures and topological type of finite bordered Riemann surfaces*]{}, Trans. Amer. Math. Soc. 192 (1974), 129–138. (Dissertation Berkeley, Advisor: Sarason) 50 \[$\spadesuit$ cite Ahlfors 1950 [@Ahlfors_1950], yet apparently not within the main-body of the text $\spadesuit$ given $\overline R$ a finite bordered surface, let $A$ be the usual hypo-Dirichlet algebra consisting of functions continuous on the bordered surface and holomorphic on its interior $R$. For a point $a\in R$, let $e_a$ be the corresponding evaluation. A [*representing measure for $e_a$*]{} is a positive Borel measure $m$ of total mass one supported on $\partial R$ such that $f(a)=\int_{\partial R} f dm$ for all $f\in A$. The collection of all such measures form a compact convex set ${\frak M}_a$. The paper shows some connections between the topology and even the conformal type of the surface $R$ and the geometry of the convex body ${\frak M}_a$ of representing measures. It is shown that if ${\frak M}_a$ has an isolated extreme point, then $R$ must be a planar surface. $\spadesuit$ let $g$ be the genus of $R$ and $s$ the number of contours, Theorem 1.2 states: “If $g=0$ and $s=3$, then ${\frak M}_a$ has precisely four extreme points if $a$ lies on one of three distinguished analytic arcs, and ${\frak M}_a$ is strictly convex if $a$ lies off these arcs. If $g=s=1$, then ${\frak M}_a$ is strictly convex for all $a\in R$.” $\spadesuit$ \[28.09.12\] it seems evident that this article (using such concepts as harmonic measure, Green’s function, Schottky differentials, convex bodies, etc.) must bear some close connection with Ahlfors 1950 [@Ahlfors_1950], and it would be nice if the degree of the Ahlfors map $f_a$ (at $a$) could somehow be related to the geometry of the body ${\frak M}_a$\]  S.M. Natanzon, [*Invariant lines of Fuchsian groups and moduli of real algebraic curves*]{}, Candidate (Ph.D.) dissertation, Moscow, 1974. (Russian) \[$\spadesuit$\]  S.M. Natanzon, [*Moduli of real algebraic curves*]{}, Uspekhi Mat. Nauk 30 (1975), 251–252. (Russian) \[$\spadesuit$ it is shown (in line with Klein’s intuition or Teichmüller’s work 1939 [@Teichmueller_1939]) that all real algebraic curves of a given topological type $(g,k,\varepsilon)$ (viz. genus, invariant “ovals” and the “dividing” type) form a connected space of dimension $3g-3$ $\spadesuit$ for an English translation see also Natanzon 1978/80 [@Natanzon_1978/80]\]  S.M. Natanzon, [*Automorphisms of the Riemann surface of an $M$-curve*]{}, Funct. Anal. Appl. 12 (1978), 228–229. \[$\spadesuit$\]  S.M. Natanzon, [*Moduli spaces of real curves*]{}, Trudy Moskov. Mat. Obshch. 37 (1978), 219–253; English transl., Trans. Moscow Math. Soc. 37 (1980), 233–272. \[$\clubsuit$ modernized account of the theory of Klein 1882 [@Klein_1882] and Teichmüller 1939 [@Teichmueller_1939] $\spadesuit$ compare also nearly parallel work by Seppäla 1978 [@Seppala_1978-Teich-spaces-of-Klein-surfaces]\]$\bigstar$$\bigstar$  S.M. Natanzon, [*Spaces of real meromorphic functions on real algebraic curves*]{}, Dokl. Akad. Nauk SSSR 279 (1984), 803–805; English transl., Soviet. Math. Dokl. 30 (1984), 724–726. \[$\clubsuit$ contains a topological description of real meromorphic function, cf. also the subsequent note Natanzon 1987/88 [@Natanzon_1987/88] and full details in Natanzon 1993 [@Natanzon_1993]\]  S.M. Natanzon, [*Topological classification of pairs of commuting antiholomorphic involutions of Riemann surfaces*]{}, Russian Math. Surveys 41 (1986), 159–160. \[$\spadesuit$ p.159: “It is well known that the topological equivalence class of a pair $(P, \alpha)$ consisting of a compact \[orientable\] surface $P $ and an orientation-reversing involutory homeomorphism $\alpha\colon P \to P$ is determined by the genus $g=g(P)$ of $P$, the number of ovals $k= \Vert P^\alpha \Vert$, and whether the set $P-P^\alpha$ is connected $(\varepsilon=0)$ or not $(\varepsilon=1)$. The triple $(g,k,\varepsilon)$ is called the [*topological type*]{} of $(P, \alpha)$. For such triples the Weichold \[read Klein to be slightly more accurate\] relation hold (see \[4\](=Weichold 1883 [@Weichold_1883]), \[5\](=Natanzon 1978 [@Natanzon_1978/80])):—(1) $0\le k \le g$ when $\varepsilon=0$,—(2) $1\le k\le g+1$ and $k \equiv g+1 \pmod 2$ when $\varepsilon=1$.\]  S.M. Natanzon, [*Real meromorphic functions on real algebraic curves*]{}, Dokl. Akad. Nauk SSSR 297 (1987), ?–?; English transl., Soviet. Math. Dokl. 36 (1988), 425–427. \[$\clubsuit$ contains a fine topological study of real meromorphic functions (using the method of Clebsch 1873 [@Clebsch_1872]), yet (apparently) without reproving Ahlfors theorem $\spadesuit$ \[30.12.12\] the proofs seem only sketched,but it is of utmost interest to assimilate better this and subsequent works by Natanzon (e.g. Natanzon 1993 [@Natanzon_1993])\]  S.M. Natanzon, [*Finite groups of homeomorphisms of surfaces and real forms of complex algebraic curves*]{}, Trudy Moskov. Mat. Obsh. 51 (1988), 3–53. \[$\spadesuit$ inspiration Clebsch 1873 [@Clebsch_1872] and Hurwitz 1891 [@Hurwitz_1891-Uber-Riemannsche-Flachen]\]$\bigstar$$\bigstar$  S.M. Natanzon, [*Spinor bundles over real algebraic curves*]{}, Uspekhi Mat. Nauk 44 (1989), 165–166; English transl., Russian Math. Surveys 44 (1989), 208–209. \[$\clubsuit$\]  S.M. Natanzon, [*Prymians of real curves and their applications to the effectivization of Schrödinger operators*]{}, Funkt. Anal. Prilozhen. 23 (1989), 41–56; English. transl., Funct. Anal. Appl. 23 (1989), 33–45. \[$\clubsuit$\]$\bigstar$$\bigstar$  S.M. Natanzon, [*Klein surfaces*]{}, Uspekhi Mat. Nauk 45 (1990), 47–90; English transl., Russian Math. Surveys 45 (1990), 43–108. \[$\clubsuit$ contains an extensive bibliography, through which—if I remember accurately—I discovered circa 2001 the papers Alling-Greenleaf 1969 [@Alling-Greenleaf_1969] and Geyer-Martens 1977 [@Geyer-Martens_1977] which pointed out to me the connection between Klein’s dividing curves and the Ahlfors map of Ahlfors 1950 [@Ahlfors_1950] (i.e. circle maps) $\spadesuit$ “The structure of a Klein surface is an analogue of the complex-analytic structure for surfaces with boundary and non-orientable surfaces. Similar to the way in which the theory of compact Riemann surfaces gives an adequate language for the description of complex …”\]  S.M. Natanzon, [*Topology of $2$-dimensional coverings and meromorphic functions on real and complex algebraic curves*]{}, Selecta Math. (formerly Sovietica) 12 (1993), 251–291; Originally published in: Trudy Sem. Vektor. Tenzor. Anal. 23 (1988), 79–103; and ibidem 24 (1991), 104–132. \[$\spadesuit$\]  S.M. Natanzon, [*Moduli of Riemann surfaces, Hurwitz-type spaces, and their superanalogs*]{}, Uspekhi Mat. Nauk 54 (199?), 61–116; English transl., Russian Math. Surveys 54 (1999), 61–117. \[$\spadesuit$\]  S.M. Natanzon, [*Moduli of real algebraic surfaces, and their superanalogues. Differentials, spinors, and Jacobian of real curves*]{}, Uspekhi Mat. Nauk 54 (199?), 3–60; English transl., Russian Math. Surveys 54 (1999), 1091–1147. \[$\spadesuit$ real algebraic curves (à la Klein-Weichold), antiholomorphic involution and its action upon all structures allied to the Riemann surface (vector bundles, Jacobians, Prymians and so on), topological invariants and the corresponding moduli spaces, inspiration=mathematical physics (solitons, string theory, etc.) $\spadesuit$ p.1092: “According to standard definitions, a [*real algebraic curve*]{} is a pair $(P,\tau)$, where $P$ is a complex algebraic curve (that is, a compact Riemann surface) and $\tau\colon P\to P$ is an antiholomorphic involution. The category of real algebraic curves is isomorphic to the category of Klein surfaces \[1\](=Alling-Greenleaf 1971 [@Alling-Greenleaf_1971]), \[35\](=Natanzon 1990 [@Natanzon_1990/90]). Investigations of real algebraic curves were started by Klein \[25\] (=1892=Vorles. Gött.[^144] [@Klein_1891--92_Vorlesung-Goettingen], [@Klein_1892_Vorlesung-Goettingen]) and Weichold 1883 [@Weichold_1883]. For a long time thereafter researchers studied only plane algebraic curves[^145], that is, real curves embedded in ${\Bbb R P}^2$. The systematic study of “general” real algebraic curves was renewed only in the seventies \[1\](=Alling-Greenleaf 1971 [@Alling-Greenleaf_1971]), \[16\](=Earle 1971 [@Earle_1971-On-the-moduli]), \[20\](=Gross-Harris 1981 [@Gross-Harris_1981]), \[31\]–\[33\](=Natanzon 1974 [@Natanzon_1974-PhD], 1975 [@Natanzon_1975], 1978/80 [@Natanzon_1978/80]), \[48\](=Seppälä 1978 [@Seppala_1978-Teich-spaces-of-Klein-surfaces]). The method of algebraic-geometric integration of works by S.P. Novikov and his school, posed a number of new problems in the theory of real curves and significantly stimulated the development of this theory \[10\](=Cherednik 1980 [@Cherednik_1980/80]), \[12\]–\[14\](=Dubrovin 1987/88 [@Dubrovin_1987/88], Dubrovin-Natanzon 1982 [@Dubrovin-Natanzon_1982/82], Dubrovin-Natanzon 1988 [@Dubrovin-Natanzon_1988/89]), \[34\](=Natanzon 1989 [@Natanzon_1989/89]), \[37\](=Natanzon 1992), \[42\](=Natanzon 1995). Conformal field theory and, in particular, string theory \[9\](=Carey-Hannabuss 1996 [@Carey-Hannabuss_1996]), \[23\](=Jaffe-Klimek-Lesniewski 1990 [@Jaffe-Klimek-Lesniewski_1990]), \[24\](=Karimipour-Mostafazadeh 1997 [@Karimipour-Mostafazadeh_1997]), \[49\](=Vajsburd-Radul 1991 [@Vajsburd-Radul-1991]) has become another area of applications of real curves.” \]  S.M. Natanzon, B. Shapiro, A. Vainshtein, [*Topological classification of generic real rational functions*]{}, arXiv (2001) or J. Knot Theory Ramif. 11 (2002), 1063–1075. \[$\clubsuit$ §3.1, p.7 (arXiv pagination) titled “On the space of branched covering of a hemisphere by a Riemann surface with boundary” should evidently bears some strong connection with Ahlfors theory. In fact the authors describe the “set ${\cal H}_{g,m}^k$ of all generic degree $m$ branched coverings of the form $f\colon P \to \Lambda^{+}$” where $P$ is a topological surface of genus $g$ with $k$ contours and $\Lambda^{+}$ is the upper hemisphere $\{ z \in \overline {\Bbb C}: {\rm Im} (z)\ge 0 \}$. $\spadesuit$ \[21.10.12\] this space is of course thought of as a Hurwitz space and it may be partitioned according to the varied multi-degrees of the restricted maps along the $k$ contours, which are indexed by partitions $(m_1,\dots, m_k)$ of $m$. The corresponding subspace of the Hurwitz space having fixed bordered degree $(m_1, \cdot, m_k)$ is shown to be connected (via an extension of the Lüroth-Clebsch theorem). $\spadesuit$ alas, it is not clear to me (Gabard) if the article shows an Ahlfors-type existence result, amounting to the non-emptiness of ${\cal H}_{g,m}^k$ for $m$ sufficiently large (cf. Ahlfors 1950 [@Ahlfors_1950], or Gabard 2006 [@Gabard_2006]). But note that the surface is here only topological, so that the viewpoint is different! Yet perhaps compatible if one lifts the complex structure of the disk/hemisphere via all topological maps obtaining a “variable” Riemann surface with enough free moduli to realize them all, recovering so perhaps Ahlfors’ theorem via an Hurwitz-type strategy. (I clearly remember to have discussed this idea with Natanzon in a 2001 Rennes conference, but as yet never managed to deduce an existence proof corroborating either Ahlfors 1950 or Gabard 2006.) The argument could start as follows: set ${\cal H}_{g}^k$ the set of all branched covers of the disc (without specified degree). Lifting the complex structure, gives a map ${\cal H}_{g}^k \to M_{g,k}$ to the moduli space of bordered surfaces of type $(g,k)$ (=genus, number of contours). The latter is probably continuous and one would like to show (by a topological argument akin to the continuity method made rigorous by Brouwer-Koebe) that the map is onto when restricted to the Hurwitz space of degree $m$, for some suitable value of $m$. Of course the lack of compactness of the moduli space may suggest to invoke a Deligne-Mumford compactification? Alternatively one can maybe avoid compactification via a clopen argument based on Brouwer’s invariance of the domain\]  Z. Nehari \[né Willi Weisbach\], [*Analytic functions possessing a positive real part*]{}, Duke Math. J. 15 (1948), 165–178. 78 \[$\spadesuit$ cites the result of Bieberbach 1925 [@Bieberbach_1925], Grunsky 1937–41 [@Grunsky_1937; @Grunsky_1941_KA], Ahlfors 1947 [@Ahlfors_1947], i.e. only planar domains via extremal methods\]  Z. Nehari, [*The kernel function and canonical conformal maps*]{}, Duke Math. J. 16 (1949), 165–178. 60, 78 \[$\spadesuit$ integral representation of the varied slit-mappings (parallel/circular slits or circular holes) via the Bergman kernel\]$\bigstar$$\bigstar$  Z. Nehari, [*The radius of univalence of an analytic function*]{}, Amer. J. Math. 71 (1949), 845–852. 78 \[$\spadesuit$ application of the Ahlfors function 1947 [@Ahlfors_1947] and of Garabedian’s identity $2\pi F'(z)=K(z,z)$ (Szegö kernel) to the problem of determining the radius of univalence to some families of analytic functions on multi-connected domains, generalizing thereby sharp estimates of Landau for bounded functions in the unit-circle\]  Z. Nehari, [*On bounded analytic functions*]{}, Proc. Amer. Math. Soc. 1 (1950), 268–275. 60, 78 \[$\spadesuit$ alternative (simplified, but lucky-guess type) derivation of Ahlfors 1947 [@Ahlfors_1947] and Garabedian 1949 [@Garabedian_1949] results around the Schwarz’s lemma via potential theory (Green’s function) and the Szegö kernel\]  Z. Nehari, [*Conformal mapping of open Riemann surfaces*]{}, Trans. Amer. Math. Soc. 68 (1950), 258–277. 60, 78 \[$\clubsuit$ the paper starts with the historically interesting fact that the main result in Ahlfors 1950 [@Ahlfors_1950] was already presented in Spring 1948 at Harvard (multiply-covered circle with number of sheets not exceeding $(r+2p)$) $\clubsuit$ contains various type of slit mappings (parallel vs. circular or radial), where the first type is given an elementary proof whereas the second requires Jacobi inversion (cf. Ahlfors’ in MathReviews) \[Incidentally one may wonder whether the first (parallel-slit) result is not already implicit in Hilbert 1909 [@Hilbert_1909]?\] $\clubsuit$ $\spadesuit$ p.267: “Representation of the Ahlfors mapping in terms of the kernel function.” $\spadesuit$ NB: some part of this paper are criticized by Tietz 1955 [@Tietz_1955], but himself is critiqued later so it is not clear who (and what) is right and how reliable those papers are $\clubsuit$ the writer asserted in Gabard 2006 [@Gabard_2006 p.946], that Nehari and Tietz may have conjectured the improved bound $r+p$ upon the degree of a circle map, yet on more mature thought this assignment may be a bit cavalier. We leave the competent readers make their own opinion\]  Z. Nehari, [*Bounded analytic functions*]{}, Bull. Amer. Math. Soc. 57 (1951), 354–366. 50, 78 \[$\clubsuit$ an interesting survey of the Ahlfors’ extremal function (the name appears on p.357) emphasizing its relation to other domain functions such as the kernel functions and the Green’s function\]  Z. Nehari, [*Extremal problems in the theory of bounded analytic functions*]{}, Amer. J. Math. 73 (1951), 78–106. 78 \[$\spadesuit$ only multiply-connected domains, but the methodology is extended to the positive genus case by Kuramochi 1952 [@Kuramochi_1952], which seems to recover Ahlfors’ 1950 result [@Ahlfors_1950] with the same upper-bound\]  Z. Nehari, [*Conformal Mapping*]{}, Mac Graw-Hill, New York, 1952. (Dover reprint 1975.) 60, 78 \[$\spadesuit$ only the planar case (domains)\]  Z. Nehari, [*Some inequalities in the theory of functions*]{}, Trans. Amer. Math. Soc. 75 (1953), 256–286. 78 \[$\spadesuit$ p.264–65 another derivation of the fact (ascribed to Grötzsch 1928 [@Groetzsch_1928] and Grunsky 1932 [@Grunsky_1932]) that the mapping maximizing the derivative at some inner point of a multi-connected domain amongst schlicht functions bounded-by-one (i.e. $\vert f \vert \le 1$) is a circular slit mapping\]  Z. Nehari, [*An integral equation associated with a function-theoretic extremal problems*]{}, J. Anal. Math. 4 (1955), 29–48. \[not quoted in 60 nor in 78\] \[$\spadesuit$ p.36 cite Bieberbach 1925 [@Bieberbach_1925] (i.e. existence of a circle map of degree equal to the number of contours for a planar domain) and find a brilliant application of it to bound the the number of linearly independent solutions of a certain extremal problem. It seems realist to expect that this Nehari argument could be widely generalized by using Ahlfors 1950 [@Ahlfors_1950] (and optionally Gabard 2006 [@Gabard_2006]) in place of Bieberbach 1925 (). However the writer \[Gabard, 30.07.12\] does not understand why the inequality advanced by Nehari on p.36 ought to be strict (as the integration is taking place within the contours where the modulus of the Bieberbach(-Ahlfors) function is unity! Hence try to locate the bug... $\spadesuit$ in fact helped by an article of Leung 2007 (On an isoperimetric …) it seems that Nehari’s argument is hygienical modulo correcting the misprint on p.29 that $C_1$ should be a subset of the (open) domain $D$ (instead of the asserted contour $C$) \[this is in agreement with the reviews generated by MR and ZB\] $\spadesuit$ then everything looks more plausible, and there is some hope to extend Nehari’s arguments to the more general setting of bordered surfaces—compare our treatment in Sec.\[Nehari-digression:sec\]\]  E. Neuenschwander, [*Lettres de Bernhard Riemann à sa famille*]{}, Cahiers du séminaire d’histoire des mathématiques 2 (1981), 85–131.  E. Neuenschwander, [*Über die Wechselwirkungen zwischen der französischen Schule, Riemann und Weierstra[ß]{}. Eine Übersicht mit zwei Quellenstudien*]{}, Arch. History Exact Sci. 24 (1981), 221–255. \[$\spadesuit$ Cauchy, Puiseux 1850, Weierstrass and the geometrization by Riemann\]  C. Neumann, [*Das Dirichletsche Prinzip in seiner Anwendung auf die Riemannschen Flächen*]{}, Leipzig bei B.G. Teubner, 1865. $\bigstar$ \[$\spadesuit$ probably—together with the next item—one of the first place where the jargon “Riemann surface” is used in history\]  C. Neumann, [*Vorlesungen über Riemanns Theorie der Abelschen Integrale*]{}, Leipzig bei B.G. Teubner, 1865. \[$\spadesuit$ (For the Zweite Auflage, cf. 1884 [@Neumann_1884]. $\spadesuit$ seems to post- resp. anti-cipate what is called nowadays the Riemann-Hurwitz relation (cf. e.g. the discussion in Scholz 1999 [@Scholz_1999])\]  C. Neumann, [*Neumann’s Untersuchungen über das Logarithmische und Newton’sche Potential*]{}, (Referat des Verfasser). Math. Ann. 13 (1878), 255–300.  C. Neumann, [*Vorlesungen über Riemanns Theorie der Abelschen Integrale*]{}, Zweite Auflage, 1884, 472 pp. 60 $\bigstar$ \[$\spadesuit$ contains, e.g., the first purely topological proof of the (so-called) Riemann-Hurwitz relation, according to Laugel’s French translation of Riemann’s Werke, p.164.\]  C. Neumann, [*Über die Methode des arithmetischen Mittels insbesondere über die Vervollkommnungen, welche die betreffende Poincaré’schen Untersuchungen in letzter Zeit durch die Arbeiten von A. Korn und E.R. Neumann erhalten haben*]{}, Math. Ann. 54 (1900), 1–48. 60 $\bigstar$  R. Nevanlinna, [*Ueber beschränkte analytische Funktionen, die in gegebenen Punkten vorgeschriebene Werte annehmen*]{}, Ann. Acad. Sci. Fenn. BXV (1919), 71pp. \[$\spadesuit$ Nevanlinna’s first paper on the so-called Pick-Nevanlinna interpolation $\spadesuit$ for a connection with the Ahlfors map (or generalization thereof) cf. e.g. Jenkins-Suita 1979 [@Jenkins-Suita_1979] $\spadesuit$ as to Pick’s work cf. Pick 1916 [@Pick_1916-U-d-Beschraekungen]\]  R. Nevanlinna, [*Ueber beschränkte analytische Funktionen*]{}, Comm. in honorem Ernesti Leonardi Lindelöf, Ann. Acad. Sci. Fenn. A XXXII (1929), 75pp. \[$\spadesuit$ Nevanlinna’s second paper on the so-called Pick-Nevanlinna interpolation $\spadesuit$ same comment as for the previous entry [@Nevanlinna_1919-U-beschr-anal-Funkt]\]  R. Nevanlinna, [*Das harmonische Mass von Punktmengen und seine Anwendung in der Funktionentheorie*]{}, C.R. Huitième Congr. Math. Scand., Stockholm, 1934, 116–133. 60 $\bigstar$ \[$\spadesuit$ presumably the first place where the name “harmonic measure” appears, the concept going back at least to H.A. Schwarz (compare, e.g. Sario-Nakai 1970 [@Sario-Nakai_1970])\]  R. Nevanlinna, [*Eindeutige analytische Funktionen*]{}, 1936. 60 \[$\spadesuit$\]$\bigstar$$\bigstar$  R. Nevanlinna, [*Über die Lösbarkeit des Dirichletschen Problems für eine Riemannsche Fläche*]{}, Nachr. zu Gött. 1 (1939), 181–193. \[$\spadesuit$ cited in Brelot-Choquet 1951 [@Brelot-Choquet_1951], but the case of open Riemann surfaces\]$\bigstar$$\bigstar$\[ZB OK\]  R. Nevanlinna, [*Über das alternierende Verfahren von Schwarz*]{}, J. Reine Angew. Math. 180 (1939), 121–128. \[$\spadesuit$ Seidel’s summary: the convergence of the alternating procedure of Schwarz is proved under more general conditions on the boundary of the region than those considered by Schwarz and the problem is reformulated as a method of successive approximation applied to a certain integral equation\]  R. Nevanlinna, [*Quadratisch integrierbare Differentiale auf einer Riemannschen Mannigfaltigkeit*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. 1 (1941), 34 pp. 60 \[$\spadesuit$ an indispensible prerequisite to understand Kusunoki 1952 [@Kusunoki_1952]: application of the Ahlfors mapping to the type problem.\] R. Nevanlinna, [*Über die Neumannsche Methode zur Konstruktion von Abelschen Integralen*]{}, Comment. Math. Helv. 22 (1949), 302–316. 60 R. Nevanlinna, [*Uniformisierung*]{}, Zweite Auflage, Grundlehren der math. Wiss. 64, Springer, 1953, 391 pp. (The Second edition to which we refer, published in 1967) 60 \[$\spadesuit$ p.148–150, contains a very illuminating implementation of Schwarz’s alternating method applied to the problem of constructing harmonic functions with prescribed singular behavior, in particular the Green’s function of a compact bordered surface\]  D.J. Newman, [*???*]{}, Trans. Amer. Math. Soc. 92 (1959), 501–507. \[$\spadesuit$ like the very deep corona problem, Newman’s characterization of interpolating sequence (also studied by Carleson, cf. e.g. Hoffman 1962 [@Hoffman_1962] for more historical details) is yet another paradigm which can be lifted from the disc to more general finite bordered Riemann surface via appeal to the Ahlfors map, as shown by Stout, cf. e.g. his second implementation in Stout 1967 [@Stout_1967-Interpolation]\]  I. Newton, [*The method of fluxions and infinite series with applications to the geometry of curves*]{}, in: The Mathematical Papers of Isaac Newton, Cambridge Univ. Press, 1967. \[$\spadesuit$\]  V.V. Nikulin, [*Integral symmetric bilinear forms and some of their applications*]{}, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 111–177; English transl., Math. USSR Izv. 14 (1980), 103–167. \[$\spadesuit$ cited by many (e.g. Fiedler 1982/83 [@Fiedler_1982/83-Pencil p.168]) for “the strict/rigid-isotopy classification of curves of degree six” showing that the real scheme enhanced by the type in the sense of Klein 1876 (and Rohlin 1978) affords a complete invariant of the rigid-isotopy class of sextics $\spadesuit$ the proof employs the apparatus of complex K3 surfaces, especially the version of Torelli’s theorem due to Pyatetsky-Shapiro–Shafarevich 1971/71 [@Pyatetsky-Shapiro-Shafarevich_1971/71] as well via remarks of Kharlamov the profound description in Rohlin 1978 [@Rohlin_1978] of complex topological characteristics (i.e. Klein’s orthosymmetry) in the realm of real plane sextic $\spadesuit$ quite strangely Rohlin’s 1978 paper is not even cited in Nikulin’s albeit it is logically used (for the assertion made on p.107), namely: “As a supplement to Gudkov’s isotopic classification \[42\](=Gudkov-Utkin 1969 [@Gudkov-Utkin_1969/78]) of plane sextics, we shall show that this classification differs from the coarse projective classification(=rigid-isotopy) only for the following sequence of ovals(=real schemes): $\frac{8}{1}, \frac{4}{1}4, 9, \frac{5}{1}1, \frac{3}{1}3, \frac{1}{1}5, \frac{4}{1}, \frac{2}{1}2$, while each of these listed ovals corresponds to precisely two coarse projective equivalence classes (see \[…\]).” $\spadesuit$ this is, of course, precisely the list of indefinite schemes as listed in Rohlin 1978 [@Rohlin_1978] (upon which Nikulin rests without reproving it)\]  V.V. Nikulin, [*Involutions of integral quadratic forms and their applications to real algebraic geometry*]{}, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), 109–188; English transl., Math. USSR Izv. 22 (1984), 99–172. \[$\spadesuit$ just cited for the nomenclature “separating” (on p.158)\]  T. Nishino, [*L’existence d’une fonction analytique sur une variété analytique complexe à deux dimensions*]{}, Publ. RIMS, Kyoto Univ. 18 (1982), 387–419. 50 \[$\spadesuit$ applies Ahlfors 1950 [@Ahlfors_1950] to complex surfaces (4 real dimensions), and specifically the existence of an analytic function under a suitable assumption $\spadesuit$ Nishino’s result was quickly extended by himself to arbitrary dimensions, yet during the process it seems that the relevance of Ahlfors 1950 [@Ahlfors_1950] disappeared\]  M. Noether, [*Zur Grundlegung der Theorie der algebraischen Raumcurven*]{}, Verlag d. Königl. Akad. d. Wiss., Berlin, 1882, 120 pp. \[$\spadesuit$ price shared with Halphen 1882 [@Halphen_1882]\]  W. Nuij, [*A note on hyperbolic polynomials*]{}, Math. Scand. 23 (1968), 69–72. \[$\spadesuit$ proof that two smooth plane curves with a deep nest are rigidly isotopic in the space of all algebraic curves $\spadesuit$ cited in Vinnikov 1993 [@Vinnikov_1993], who point out also the proof of Dubrovin 1983 [@Dubrovin_1983/85] and also in Viro 1986/86 [@Viro_1986/86-Progress p.74]: “In conclusion I state an old result on rigid-isotopy, which for a long time was not known to experts in the topology of real algebraic manifolds. In 1968, Nuij \[24\](=this entry) proved that any two hypersurfaces of degree $m$ in ${\Bbb R}P^n$ containing $[m/2]$ spheres totally ordered by inclusion are rigidly isotopic. Recently Dubrovin \[5\](=1983 [@Dubrovin_1983/85]) obtained this result for the case of plane curves by a different method.” $\spadesuit$ (from an e-mail of Shustin \[26.01.13\]) By the way, another (well) known connected chamber consists of hyperbolic curves (i.e. those which have totally real intersection with lines of certain pencil) - this is a consequence of Nuij W. A note on hyperbolic polynomials. Math. Scandinavica 23 (1968), no. 1, 69–72. \]  B.G. Oh, [*A short proof of Hara and Nakai’s theorem*]{}, Proc. Amer. Math. Soc. 136 (2008), 4385–4388. 50 \[$\spadesuit$ Ahlfors 1950’s result on circle maps is used in a quantitative version of the corona $\spadesuit$ question of the writer (since Sept. 2011): is it possible to exploit the improved bound of Gabard 2004/06 [@Gabard_2006] in this sort of game $\spadesuit$ p.4387, Ahlfors 1950 [@Ahlfors_1950] is cited as follows: “[**Theorem 3**]{} (Ahlfors \[1\](=Ahlfors 1950 [@Ahlfors_1950])). [*Suppose $R$ is a finitely[^146] bordered Riemann surface with $g(R)=g$ and $b(R)=b$. Then there exists an $m$-sheeted branched covering map $f\colon R \to {\Bbb D}$, called the [*A*hlfors map]{}, such that $b\le m\le 2g+b$.*]{}”\]  M. Ohtsuka, [*Dirichlet problems on Riemann surfaces and conformal mappings*]{}, Nagoya Math. J. 3 (1951), 91–137. 60 \[$\spadesuit$\]  O.A. Oleinik, [*Estimates of the Betti numbers of real algebraic hypersurfaces*]{}, Mat. Sb. 28 (1951), 635–640 (Russian). \[$\spadesuit$\]  O.A. Oleinik, [*On the topology of real algebraic curves on an algebraic surface*]{}, Mat. Sb. 29 (1951), 133–156 (Russian). \[$\spadesuit$\]  B.V. O’Neill, Jr., J. Wermer [*Parts as finite-sheeted coverings of the disk*]{}, Amer. J. Math. 90 (1968), 98–107. 50 \[$\spadesuit$ p.98, the paper is started by citing Ahlfors 1950 [@Ahlfors_1950] and mentions the alternative proof of Royden 1962 [@Royden_1962] $\spadesuit$ the Ahlfors’ function is given an application to Gleason parts (certain analytic discs in the maximal ideal space) extending thereby a previous disc-result of Wermer 1964 $\spadesuit$ p.98, it is emphasized that E. Bishop 196 5 [@Bishop_1965] gave an abstract version of Ahlfors’ extremal problem in the context of function algebra on a compact space $X$ (i.e. an algebra of complex-valued continuous functions containing the constants, separating the points, and closed under uniform convergence)\]  S.Yu. Orevkov, [*Link theory and oval arrangements of real algebraic curves*]{}, Topology 38 (1999), 779–810. \[$\spadesuit$ can be realized holomorphically, pseudo-holomorphically, holomorphically, pseudo-holomorphically, etc.\]  S.Yu. Orevkov, [*Riemann existence theorem and construction of real algebraic curves*]{}, Ann. Fac. Sci. Toulouse Math. 12 (4) (2003), 517–531. \[$\spadesuit$\]  S.Yu. Orevkov, [*Proper analytic embedding of ${\Bbb CP}^1$ minus a Cantor set into ${\Bbb C}^2$*]{}, Uspehki Math. Nauk 63 (2008), 155–156; English transl.: Russian Math. Surveys 63 (2008), 168–169. \[$\spadesuit$ shows the result of the title, but as pointed out in Forstnerič-Wold 2012 [@Forstneric-Wold_2012 p.17] it is an open problem whether this holds for each Cantor set.\]  D. Orth, [*On holomorphic families of holomorphic maps*]{}, Nagoya Math. J. 39 (1970), 29–37. \[$\spadesuit$ p.33, Ahlfors 1950 is cited as follows: “Ahlfors \[1\](=1950 [@Ahlfors_1950]) has shown the existence of a holomorphic map $f$ from a bordered Riemann surface with finite genus and a finite number of boundary components onto a full covering surface $S\buildrel{\pi}\over{\longrightarrow} D$ of the unit disk. N. Alling \[2\] has shown that $\pi\circ f\vert U$ is a covering map of $D$ near $\partial D$ for some open neighborhood $U$ of $\partial X$. Theorem 2.–4. can be thought of as concerning holomorphic families of such maps.”\]  B. Osgood, [*Notes on the Ahlfors mapping of a multiply connected domain*]{}, Unpublished (?) manuscript (available from the web), undated (estimated date in the range 1993/2005). \[$\spadesuit$ pleasant re-exposition of the neo-expressionist sort of the Ahlfors-Garabedian theory (inspired by Bell, Kerzman-Stein, etc.), in particular the formula for the Ahlfors function as the ratio of the Szegö kernel divided by the Garabedian kernel\]  W.F. Osgood, [*On the existence of the Green’s function for the most general simply connected plane region*]{}, Trans. Amer. Math. Soc. 1 (1900), 310–314. 60  W.F. Osgood, [*Jordan curve of positive area*]{}, Trans. Amer. Math. Soc. 4 (1903), 107–112. \[$\spadesuit$ shows how pathological Jordan curve can be\]  W.F. Osgood, E.H. Taylor, [*Conformal transformations on the boundary of their regions of definition*]{}, Trans. Amer. Math. Soc. 14 (1913), 277–???.  W.F. Osgood, [*Existenzbeweis betreffend Funktionen, welche zu einer eigentlichen diskontinuierlichen automorphen Gruppe gehören*]{}, Palermo Rend. 35 (1913), 103–106. 60  R. Osserman, [*Riemann surfaces of class $A$*]{}, Trans. Amer. Math. Soc. 82 (1956), 217–245. \[$\spadesuit$\]  R. Osserman, [*A hyperbolic surface in $3$-space*]{}, Proc. Amer. Math. Soc. 7 (1956), 54–58. 60 \[$\spadesuit$ example of a function ${\Bbb R}^2\to {\Bbb R}$, whose graph (endowed with the Euclidean metric) defines a surface of hyperbolic type, i.e. conformally equivalent to the disc, answering thereby a question of Ch. Loewner, reported by L. Bers in 1951 on the occasion of the 100th Birthday of Riemann’s Thesis\]  A. Ostrowski, [*Mathematische Miszellen XV. Zur konformen Abbildung einfach zusammenhängender Gebiete*]{}, Jahresb. Deutsch. Math.-Ver. 38 (1929), 168–182. \[$\clubsuit$ omitted in both 60 and 78; however this (joint with Carathéodory 1928 [@Caratheodory_1928]) is the simply-connected version of the Ahlfors map\]  K. Ott, [*Über die Konstruktion monogener analytischer Funktionen mit vorgegebenen Unstetigkeitsstellen auf der Riemann’schen Fläche*]{}, Monatsh. Math. 4 (1893), 367–375. 60 $\bigstar$  M.P. Ovchintsev, [*Optimal recovery of functions of class $E_p$, $1\le p\le \infty$, in multiply connected domains*]{}, Siberian Math. J. 37 (1996), 288–307. \[$\clubsuit$ p.293, three occurrences of “Ahlfors function” for $m$-connected domains; in particular Prop.1 asserts the existence of neighborhoods of the boundary contours such that if $z_0$ lies in one of these neighborhood then the extra zeros of the Ahlfors function lie one-by-one in the other domains; in particular it seems likely that such neighborhoods can be chosen pairwise disjoint, in which case we recover a result of Bell 1991 [@Bell_1991-Szego]\]  M. Ozawa, [*On bounded analytic functions and conformal mapping, I*]{}, Kōdai Math. J. (1950), 33–36. 78  M. Ozawa, [*A supplement to “Szegö kernel function on some domains of infinite connectivity”*]{}, Kōdai Math. J. 13 (1961), 215–218. 78 \[$\spadesuit$ p.215: “Let $D$ be an $n$-ply connected analytic domain and ${\frak B}(D) $ be the class of regular functions in $D$ whose moduli are bounded by the value $1$. In ${\frak B}(D) $ there exists, up to rotation, a unique extremal function by which the maximum $\max_{{\frak B}(D) } \vert f'(z_0)\vert$ for a fixed point is attained. This extremal function $F(z, z_0)$ maps $D$ onto the $n$ times covered unit disc \[1\](=Ahlfors 1947 [@Ahlfors_1947]), \[3\](=Garabedian 1949 [@Garabedian_1949]), \[4\](=Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950]), \[9\](=Nehari 1950 [@Nehari_1950-PAMS-On-bounded-anal-fcts]), \[11\](=Schiffer 1950 [@Schiffer_1950-Duke]). In ${\frak B}(D) $ there exists an infinite number of essentially different functions which map $D$ onto the $n$ times covered unit disc \[2\](=Bieberbach 1925 [@Bieberbach_1925]), \[5\](=Grunsky 1937 [@Grunsky_1937]), \[8\](=Mori 1951 [@Mori_1951]).”\]  P. Painlevé, [*Sur les lignes singulières des fonctions analytiques*]{}, Ann. Fac. Sci. Toulouse 2 (1888), 130 pp. 78 \[$\spadesuit$ the classical Painlevé problem, interest revived through the work of Ahlfors 1947 [@Ahlfors_1947] and complete solution in Tolsa 2003 [@Tolsa_2003]\]  P. Painlevé, [*Sur la théorie de la représentation conforme*]{}, C. R. Acad. Sci. Paris 112 (1891), 653–657. \[$\spadesuit$ one of the first study of the boundary behavior of the Riemann mapping for a domain bounded by a smooth Jordan curve $\spadesuit$ same holds true for a general (topological) Jordan domain, cf. Osgood and Carathéodory\]  H. Pajot, [*Analytic capacity, rectifiability, Menger curvature and the Cauchy integral*]{}, Lecture Notes in Math. 1799, Springer-Verlag, Berlin, 2002. \[$\spadesuit$\]$\bigstar$  P. Parenti, [*Combinatorics of dividing $T$-curves*]{}, Tesi di dottorato, Pisa, (1996), 133 pp. Tutori: Galbiati, Itenberg \[$\spadesuit$ combinatorial construction of curves with a control of the type, building upon Viro’s method (early 1980’s) and the special case thereof called the $T$-construction $\spadesuit$ contains a combinatorial version of Rohlin’s formula for $T$-curves\]$\bigstar$  S. Paris, [*An extremal property of Rokhlin’s inequality for real algebraic curves*]{}, Math. Ann. 304 (1996), 613–620. \[$\spadesuit$\]$\bigstar$  J. Parkkonen, V. Ruuska, [*Finite degree holomorphic covers of compact Riemann surfaces*]{}, Acta Math. Sinica, English Ser. 23 (2007), 89–94. \[$\spadesuit$ “A conjecture of Ehrenpreis (1970) states that any two compact Riemann surfaces of genus $\ge 2$ have finite degree unbranched holomorphic covers that are arbitrarily close in moduli space. Here we prove a weaker result …”\]  M. Parreau, [*Sur les moyennes des fonctions harmoniques et analytiques et la classification des surfaces de Riemann*]{}, Ann. Inst. Fourier (Grenoble) 3 (1951), 103–197. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is briefly cited in two footnotes $\spadesuit$ the work also contains a study of Hardy classes on Riemann surfaces extending the classical Hardy-Riesz’s brothers theory for the disc, and some overlap is to be found with the (subsequent) work of Rudin 1955 [@Rudin_1955-class-Hp]\]$\bigstar$  O. Perron, [*Eine neue Behandlung der ersten Randwertaufgabe fur $\Delta u=0$*]{}, Math. Z. 18 (1923), 42–54. 60 \[$\spadesuit$ a new solution to the Dirichlet problem (using Poisson and Lebesgue’s integration) yielding the result in the same generality on the boundary (cf. p.53–54) as those obtained by Lebesgue 1907 [@Lebesgue_1907], Courant 1914 [@Courant_1914] and Lichtenstein (1916), but further very much simplified in Radó-Riesz 1925 [@Rado-Riesz_1925] (according to e.g., Carathéodory 1937 [@Caratheodory_1937-On-Dirichlet's-problem p.710]) $\spadesuit$ the paper is concluded by the simple remark (already made by Zaremba 1910 [@Zaremba_1910]) that the Dirichlet problem does not permit isolated boundary component (reducing to an isolated point), e.g. the punctured disc with boundary prescription $1$ on the circumference and $0$ at the center does not admit a harmonic extension, since otherwise the mean value property would be violated (intuitively a punktförmig radiator is too insignificant to induce a heat flow equilibrium) $\spadesuit$ on the other hand this paper tolerates non-schlicht surfaces covering multiply the plane and therefore may be regarded as a suitable treatment of the Dirichlet problem on a compact bordered Riemann surface (given abstractly à la (Riemann-Prym-Klein)-Weyl-Radó), compare for this well-known affiliation the following ref. given backwardly in time: Radó 1925 [@Rado_1925], Weyl 1913 [@Weyl_1913], and Klein 1882 [@Klein_1882]\]  K. Petri, [*Über die invariante Darstellung algebraischer Funktionen einer Veränderlichen*]{}, Math. Ann. 88 (1923), 242–289. \[$\spadesuit$\]  I.G. Petrovsky \[Petrovskii\], [*Sur la topologie des courbes réelles et algébriques*]{}, C.R. Acad. Sci. Paris 197 (1933), 1270–1273. \[$\spadesuit$ announcement of results with proofs detailed in the next entry (Petrowsky 1938 [@Petrowsky_1938])\]  I.G. Petrowsky \[Petrovskii\], [*On the topology of real plane algebraic curves*]{}, Ann. of Math. (2) 39 (1938), 189–209. (in English of course.) \[$\spadesuit$ where the jargon $M$-curve is coined, and where some obstruction is given (using the Euler-Jacobi interpolation formula), yielding perhaps the first proof, e.g. of the fact (first enunciated by Hilbert, Rohn, etc.) that a plane sextic cannot have 11 unnested ovals $\spadesuit$ note however that Petrovskii validates Rohn’s proof of 1911 by writing on p.189: “After a series of attempts the above mentioned theorem announced by Hilbert was at last proved in 1911 by K. Rohn(=Rohn 1911 [@Rohn_1911]).” This contrast with Gudkov’s latter diagnostic (e.g. in Gudkov 1974 [@Gudkov_1974/74]) that even Rohn’s proof was not logically complete, though the method fruitful when suitably consolidated with Russian conceptions of roughness.\]  I.G. Petrovskii, [*On the diffusion of waves and the lacunas for hyperbolic equations*]{}, Mat. Sb. 17 (1945), 289–370. (in English.) \[$\spadesuit$ cited in Viro 1986/86 [@Viro_1986/86-Progress p.58] as a forerunner of Rohlin’s complex orientations for dividing curves\]  I. Petrovskii, O.A. Oleinik, [*On the topology of real algebraic surfaces*]{}, Izv. Akad. Nauk SSSR, Ser. Mat. 13 (1949), 389–402. (Russian) \[$\spadesuit$\]  P. del Pezzo, [*Sulle superficie di Riemann relative alle curve algebrice*]{}, Palermo Rend. 6 (1892), 115–126. 60 \[$\spadesuit$ presumably one among the first reaction to the reality works of F. Klein outside his direct circle of student (Harnack, Hurwitz, Weichold)\]  A. Pfluger, [*Ein alternierendes Verfahren auf Riemannschen Flächen*]{}, Comment. Math. Helv. 30 (1956), 265–274. 60 \[$\spadesuit$\]  A. Pfluger, [*Theorie der Riemannschen Flächen*]{}, Grundlehren der math. Wiss. 89, Springer, Berlin, 1957, 248pp. 50, 60, 78 \[$\spadesuit$ quotes the article Ahlfors 1950 [@Ahlfors_1950] at several places (p.126, 181, 185, 202) yet never in close connection with the circle map paradigm $\spadesuit$ of course the book itself is a masterpiece of Swiss-German architecture and we do not attempt to summarize its broad content\]  E. Picard, [*Sur une propriété des fonctions entières*]{}, C.R. Acad. Sci. Paris 88 (1879), 1024–1027. \[$\spadesuit$ where the famous Picard theorem appears first (a nonconstant entire function (on ${\Bbb C}$) omits at most one value, for otherwise lifting to the universal covering $\Delta$ of $S^2-\{3 {rm pts}\}$ we get ${\Bbb C}\to \Delta$ a bounded analytic function violating Liouville’s theorem) $\spadesuit$ widespread influence over Borel 1896, Schottky, Landau 1904, Lindelöf 1902 [@Lindeloef_1902], Phragmén, Iversen, Montel, Bloch, Littlewood, Nevanlinna 1923, Ahlfors, Sario, etc. $\spadesuit$ \[07.10.12\] since ${\Bbb C}$ is the punctured sphere and Liouville’s theorem may be interpreted as Riemann’s removable singularity for bounded analytic function, one can also state that any analytic function defined on a punctured closed Riemann surface omits at most 3 values, but this is completely wrong for the monodromy principle does not apply anymore\]  E. Picard, [*De l’équation $\Delta u=k e^u$ sur une surface de Riemann fermée*]{}, J. Math. Pures Appl. (4) 9 (1893), 273–291. 60 \[$\spadesuit$ supply an attempt to uniformize via the so-called Liouville equation, such a strategy seems to follow a problem suggested by H.A. Schwarz; for a modern execution of this programme cf. Mazzeo-Taylor 2002 [@Mazzeo-Taylor_2002] (and also a related work of Bieberbach 1916 [@Bieberbach_1916-Delta-u-und-die-automorphen-Funkt])\]  E. Picard, [*Traité d’analyse, Vol.II, Fonctions harmoniques et fonctions analytiques. Introduction à la théorie des équations différentielles, intégrales abéliennes et surfaces de Riemann*]{}, Gauthier-Villars, Paris 1892. Reedited 1926, 624pp. 60 \[$\spadesuit$ contains a treatment of Schottky’s theory of 1877 (cited e.g. in Le Vavasseur 1902 [@Le-Vavasseur_1902], Cecioni 1908 [@Cecioni_1908] and Schiffer-Spencer 1954 [@Schiffer-Spencer_1954])\]  E. Picard, [*Sur la représentation conforme des aires multiplement connexes*]{}, Ann. École Norm. (3) 30 (1913), 483–488. 78 \[$\spadesuit$ a brilliant re-exposition of Schottky 1877 [@Schottky_1877], which was much appreciated by Julia 1932 [@Julia_1932]\]  E. Picard, [*?????*]{}, Ann. École Norm. (3) 30 (1915), 483–488. \[$\spadesuit$ yet another brilliant re-exposition of the Riemann mapping theorem via the Green’s function\]  G. Pick, [*Ueber die Beschränkungen analytischen Funktionen, welche durch vorgegebene Funktionswerte bewirkt werden*]{}, Math. Ann. 77 (1916), 7–23. \[$\spadesuit$ the beginning of so-called Pick-Nevanlinna interpolation, and see Heins 1975 [@Heins_1975] or Jenkins-Suita 1979 [@Jenkins-Suita_1979] for an extension to finite bordered Riemann surface offering an overlap (indeed an extension) of the Ahlfors map\]  U. Pinkall, [*Hopf tori in $S^3$*]{}, Invent. Math. 81 (1985), 379–386. \[$\spadesuit$ p.379: “Corollary. [*Every compact Riemann surface of genus one can be conformally embedded in ${\Bbb R}^3$ as an algebraic surface of degree $8$.*]{}—Garsia \[2\](=1962/63 [@Garsia_1962/63-algebraic-surfaces]) had shown that every compact Riemann surface (of any genus) can be conformally embedded in ${\Bbb R}^3$ as an algebraic surface, but his method of proof was not constructive and he therefore did not obtain bounds for the degree of this surface.” $\spadesuit$ this result does not seem to answer the Garsia question (1962/63 [@Garsia_1962/63-algebraic-surfaces]) if the image can always be chosen among torus of revolution twisted by an affine transformation of $3$-space. In this case the degree would be four. $\spadesuit$ for each genus $g$ we can define the Garsia degree $d(g)$ as the smallest integer $d$ such that each surface $F_g$ conformally embeds as an algebraic surface of degree $\le d$. In fact from Garsia’s theorem (1962/63 ) it is not perfectly clear that there is a uniform bound depending only on the topology. (So in general $d(g)$ is possibly ill-defined.) Of course $d(0)=2$ (every sphere is conformal to the round $2$-sphere, Riemann, Schwarz 1870); $d(1)\le 8$ (Pinkall 1985, [*op.cit.*]{}), but is this sharp?, in general do somebody know a bound on $d(g)\le ???$\]  U. Pirl, [*Über isotherme Kurvenscharen vorgegebenen topologischen Verlaufs und ein zugehöriges Extremalproblem der konformen Abbildung*]{}, Math. Ann. 133 (1957), 91–117. 78 \[$\spadesuit$\] (another well-known student of Herbert Grötzsch) J.A.F. Plateau, [*Statique expérimentale et théorétique des liquides soumis aux seules forces moléculaires*]{}, Gauthier-Villars, Paris, 1873. \[$\spadesuit$\]  J. Plemelj, [*Ein Ergänzungssatz zur Cauchy’schen Integraldarstellung analytischer Funktionen, Randwerte betreffend*]{}, Monats. f. Math. u. Phys. 19 (1908), 205–210. \[$\spadesuit$ quoted in Nehari 1955 [@Nehari_1955]\]  J. Plücker, [*System der analytischen Geometrie*]{}, Berlin, 1835. \[$\spadesuit$ quoted in Brieskorn-Knörrer 1981/86 [@Brieskorn-Knörrer_1981/1986]\] J. Pücker, [*Theorie der algebraischen Curven*]{}, Bonn 1839. \[$\spadesuit$ cited by all the masters, e.g. Zeuthen 1874 [@Zeuthen_1874 p.415], Klein 1873 [@Klein_1873-Uber-Flächen-dritter-Ordn], Gudkov 1974/74 [@Gudkov_1974/74] $\spadesuit$ according to Klein 1873 [@Klein_1873-Uber-Flächen-dritter-Ordn] might be one of the first place where the method of small perturbation is mentioned $\spadesuit$ p.253, contains a conjecture on the number of real bitangents to a quartic as taking only the values $28, 16, 8, 4,0$, the last case of which was prohibited in Zeuthen 1874 [@Zeuthen_1874]\]  H. Poincaré, [*Mémoire sur les fonctions fuchsiennes*]{}, Acta Math. 1 (1882), 193–294. 60  H. Poincaré, [*Sur un théorème général de la théorie des fonctions*]{}, Bull. Soc. Math. France 11 (1883), 112–125. 78 \[$\spadesuit$ proposes (and succeeds partially) to uniformize not only algebraic, but also analytic curves (=open, a priori highly transcendental, Riemann surfaces). Programm completed in Poincaré 1907 [@Poincare_1907], independently Koebe 1907 [@Koebe_1907_UbaK1].\]  H. Poincaré, [*Sur les groupes des équations linéaires*]{}, Acta Math. 4 (1884), 201–311. \[$\spadesuit$\]  H. Poincaré, [*Sur les équations aux dérivées partielles de la physique mathématique*]{}, Amer. J. Math. 12 (1890), 211–294. \[$\spadesuit$ where the [*méthode du balayage*]{} is first introduced\]  H. Poincaré, [*Analysis Situs*]{}, J. École Polytechnique 1 (1895), 1–121. \[$\spadesuit$ embryo of modern homology theory, quite relevant to problems of conformal mappings (especially circle maps), e.g. in Gabard 2006 [@Gabard_2006]\]  H. Poincaré, [*Sur la méthode de Neumann et le problème de Dirichlet*]{}, C.R. Acad. Sci. Paris 120 (1895), 347–352. 60 $\spadesuit$  H. Poincaré, [*La méthode de Neumann et le problème de Dirichlet*]{}, Acta Math. 20 (1896), 59–142. 60 \[$\spadesuit$ it seems that the method in question, may in turn goes back to Gauss 1839 [@Gauss_1839]\] $\spadesuit$  H. Poincaré, [*Sur l’uniformisation des fonctions analytiques*]{}, Acta Math. 31 (1907), 1–63. 60, 78 \[$\spadesuit$ simultaneously with Koebe 1907 [@Koebe_1907_UbaK1] uniformize arbitrary complex analytic curves (equivalently open Riemann surfaces), completing the 1883 desideratum of Poincaré in [@Poincare_1883], revived in Hilbert’s 22th problem\]  G.M. Polotovskii, [*Problem of topological classification of the disposition of ovals of nonsingular algebraic curves in the projective plane*]{}, in: Methods of the Qualitative Theory of Differential Equations \[in Russian\], Vol.1, Gorki (1975), 101–128. \[$\spadesuit$\]  G.M. Polotovskii, [*A catalogue of $M$-reducible curves of order $6$*]{}, Dokl. Akad. Nauk SSSR 236 (1977), 548–551; English transl., Soviet Math Dokl. 18 (1977), 1241–1245. \[$\spadesuit$\]  G.M. Polotovskii, [*$(M-2)$-curves of order $8$ and some conjectures*]{}, Uspekhi Mat. Nauk SSSR 36 (1981), 235–236. \[$\spadesuit$ contains some observation on Rohlin’s conjecture, that were ultimately employed in Shustin 1985/85 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin] to disprove one implication of Rohlin’s conjecture (in degree 8)\]$\bigstar$$\bigstar$$\bigstar$  G.M. Polotovskii, [*Dimitrii Andreevich Gudkov*]{}, in: Topology of Real Algebraic Varieties and Related Topics, Amer. MAth. Soc. Transl. 173, 1996, 1–9. \[$\spadesuit$ survey of Gudkov’s contributions with an exhaustive list of his scientific works\]  Ch. Pommerenke, [*Über die analytische Kapazität*]{}, Archiv der Math. 11 (1960), 270–277. \[$\spadesuit$ some estimates of the analytic capacity (defined as in Ahlfors 1947 [@Ahlfors_1947]) and its connection to Schiffer’s span 1943 [@Schiffer_1943] $\spadesuit$ uses heavily Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950] and Nehari 1952 [@Nehari_1952-BOOK]\]  H. Poritsky, [*Some industrial applications of conformal mapping*]{}. In: [*Construction and Applications of Conformal Maps*]{}, Proc. of a Sympos. held on June 22–25 1949, Applied Math. Series [*18*]{}, 1952, 207–213. \[$\spadesuit$ quoted for a joke about free-hand drawings\]  R. de Possel, [*Sur le prolongement des surfaces de Riemann*]{}, C.R. Acad. Sci. Paris 186 (1928), 1092–1095. 60 \[$\spadesuit$ problem of deciding when an (open) Riemann surface can be continued to a larger one $\spadesuit$ relates to work of Radó 1924 [@Rado_1924-Uber-nicht-fortsetzbare], and Bochner 1927 [@Bochner_1927]\]  R. de Possel, [*Sur le prolongement des surfaces de Riemann*]{}, C.R. Acad. Sci. Paris 187 (1929), 98–100. 60 \[continuation of the previous work in the spirit of Radó and Bochner\]  R. de Possel, [*Zum Parallelschlitztheorem unendlich-vielfach zusammenhängender Gebiete*]{}, Gött. Nachr. (1931), 199–202. 60, 78 \[$\spadesuit$ proof of the parallel-slit mapping à la Schottky 1877 [@Schottky_1877]-Cecioni 1908 [@Cecioni_1908]-Hilbert 1909 [@Hilbert_1909]-Koebe 1910 [@Koebe_1910_Hilbert]-Courant 1910/12 [@Courant_1912], via an extremal problem (method analogous to Carathéodory 1928 [@Caratheodory_1928], but uses also the Flächensatz of Bieberbach) $\spadesuit$ of course Schottky-Cecioni are not cited as they only treats the case of finite connectivity $\spadesuit$ it is noteworthy that the similar problem for the Kreisnormierung is still unsolved in full generality. This supports once more the philosophy advanced by Garabedian-Schiffer 1950 [@Garabedian-Schiffer_1950] that parallel-slit mappings are easier than circle maps\] R. de Possel, [*Quelques problèmes de représentation conforme*]{}, J. École Polytech. (2) 30 (1932), 1–98. 60, 78 \[$\spadesuit$ parallel (as well as radial) slit maps in the case of domains via an extremal problem $\spadesuit$ some little details of it seem to be criticized in Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950]\]  R. de Possel, [*Sur quelques propriétés de la représentation conforme des domaines multiplement connexes, en relation avec le théorème des fentes parallèles*]{}, Math. Ann. 107 (1932), 496–504. 60, 78 \[$\spadesuit$ again parallel-slits via an extremal problem, overlap with work by Grötzsch\]  R. de Possel, [*Sur les ensembles de type maximum, et le prolongement des surfaces de Riemann*]{}, C.R. Acad. Sci. Paris 194 (1932), 98–100. 60 \[$\spadesuit$ still relates to work of Radó, and Bochner and reports some mistakes in the previous notes\]  R. de Possel, [*Sur la représentation conforme d’un domaine à connexion infinie sur un domaine à fentes parallèles*]{}, J. Math. Pures Appl. (9) 18 (1939), 285–290. 60, 78 \[$\spadesuit$ as noted in Burckel 1979 [@Burckel_1979], this de Possel paper affords a trick to circumvent the reliance upon RMT in his 1931 proof of the PSM through an extremum problem, similar trick in Garabedian 1976 [@Garabedian_1976]\] $\bigstar$$\bigstar$$\bigstar$  W. Pranger, [*Extreme points in the Hardy class $H^1$ of a Riemann surface*]{}, Canad. J. Math. 23 (1971), 969–976. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is quoted twice: on p.975 for certain decompositions and on p.976: “On a compact bordered surface $R$ the periods of the conjugate of a function which is harmonic on $R$ and continuous on its closure may be specified arbitrarily (see \[1, p.110\]=Ahlfors 1950 [@Ahlfors_1950 p.110])\]  F.E. Prym, [*Zur Integration der Differentialgleichung $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$*]{}, J. Reine Angew. Math. 73 (1871), 340–364. \[$\spadesuit$ p.361–364, an example is given of a continuous function on the unit-circle whose harmonic extension to the disc has infinite Dirichlet integral! (The existence of such an extension is established directly in the first part of the paper, independently of Schwarz’s 1870 [@Schwarz_1870-Zurich-15ter-Jahrgang] solution based upon Poisson’s integral.) This Prym’s example is nothing less than a counterexample to the Dirichlet principle (as formulated, e.g., in Grube’s text [@Dirichlet_1840-1876]=redaction of Dirichlet’s lectures). Compare Elstrodt-Ulrich 1999 [@Elstrodt-Ullrich_1999 p.285]. Prym emphasizes at the end of his paper (p.364) that Riemann himself never committed such a “basic” mistake, but (still on p.364) Prym formulates an implicit critique to all contemporary attempts to rescue the Dirichlet principle based on the tacit assumption of finiteness of the Dirichlet integral, presumably including the one of Weber 1870 [@Weber_1870] (who is however not directly attacked for diplomatique reasons) $\spadesuit$ a related example (where any continuous function matching the boundary data has infinite Dirichlet integral) is due to Hadamard 1906 [@Hadamard_1906] $\spadesuit$ such counter-example affects directly the Dirichlet-Riemann argument of minimizing the Dirichlet integral, and seems to destroy as well H. Weber’s attempt (1870 [@Weber_1870]) to consolidate Riemann’s proof $\diamondsuit$ student of Riemann, who played a pivotal rôle as well in explaining to Klein, that Riemann himself did not confined his attention to surfaces spread over the plane but included a more organical mode leading to the “abstract” Riemann surfaces, compare Klein 1882 [@Klein_1882]\]  P.M. Pu, [*Some inequalities in certain nonorientable Riemannian manifolds*]{}, Pacific J. Math. 2 (1952), 55–71. \[$\spadesuit$ includes a proof of the isosystolic estimates for the projective plane ${\Bbb R}P^2$ (by adapting the method of Loewner 1949 for the torus) stating that the round elliptical metric has the best systolic ratio (i.e. is the more robust less susceptible to dye from a “Herzinfarkt”: that is $sys^2/area\le (\pi)^2/2 \pi=\pi/2=1.570796327\dots$) $\spadesuit$ this results implies directly the Gromov filling conjecture for genus $p=0$, upon cross-capping the boundary contour (cf. Gromov 1983 [@Gromov_1983])\]  V.A. Puiseux, [*Recherches sur les fonctions algébriques*]{}, Journal de Math. 1 15 (1850), 365–480. \[$\spadesuit$ study of the algebraic equation $f(z,u)=0$ ($f$ a polynomial), poles, branch points, concept of essential singularities (where the Laurent series expansion contains an infinity of negative terms, e.g. $e^{1/z}$ at $z=0$) $\spadesuit$ independent investigations of the same material by Weierstra[ß]{}\]  I.I. Pyatetsky-Shapiro, I.R. Shafarevich, [*A Torelli theorem for algebraic surfaces of type K3*]{}, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572; English transl., Math. USSR Izv. 5 (1971), ?–?. \[$\spadesuit$ used by Nikulin 1979/80 [@Nikulin_1979/80] in his rigid-isotopy classification of plane sextic via Klein-Rohlin’s type (I/II=ortho- vs. dia-symmetry)\]  T. Radó, [*Zur Theorie der mehrdeutigen konformen Abbildung*]{}, Acta Szeged 1 (1922), 55–64. 78 \[$\spadesuit$ quoted in Landau-Osserman 1960 [@Landau-Osserman_1960], who ascribe to Radó the basic fact that an analytic map taking boundary to boundary is a full covering surface taking each value a constant number of times $\spadesuit$ hence this Radó bears an obvious connection to the Ahlfors map, albeit it does not reprove its existence when the target surface is the disc\]  T. Radó, [*Über die Fundamentalabbildung schlichter Gebiete*]{}, Acta Sci. Math. Szeged 1 (1922/23), 240–251; cf. also Fejér’s Ges. Arb. 2, 841–842. 78 \[$\clubsuit$ supplies in print an argument of Fejér-Riesz proving RMT via an extremal problem (maximization of the derivative), perfected in Carathéodory 1928 [@Caratheodory_1928] and Ostroski 1929 [@Ostrowski_1929] $\clubsuit$ this constitutes the underlying background for the extremal methods used by Grunsky and Ahlfors, leading ultimately to Ahlfors 1950 [@Ahlfors_1950]\]  T. Radó, [*Bemerkung zu einem Unitätssatz der konformen Abbildung*]{}, Acta litt. ac. scient. Univ. Hung. 1 (1923), 101–103. 78  T. Radó, [*Über die konforme Abbildung schlichter Gebiete*]{}, Acta litt. ac. scient. Univ. Hung. 2 (1924), 47–60. 78  T. Radó, [*Über eine nicht fortsetzbare Riemannsche Mannigfaltigkeit*]{}, Math. Z. 20 (1924), 1–6. 60  T. Radó, [*Über den Begriff der Riemannschen Fläche*]{}, Acta Szeged 2 (1925), 101–121. 60 \[$\spadesuit$ aside from Weyl 1913 [@Weyl_1913] (“sheaf theoretic”) this supplies the first (modern) definition of an “abstract” Riemann surface, modulo Klein who anticipated the “atlas” idea quite explicitly in [@Klein_1891--92_Vorlesung-Goettingen] (“Dachziegelige Überdeckung”). \[CHECK, pages\] Klein knew it essentially since Prym indicated him how Riemann saw the story, as reported, e.g., in the introduction of Klein 1882 [@Klein_1882]. $\spadesuit$ compare also the discussion in Remmert 1998 [@Remmert_1998] $\spadesuit$ besides the article contains a bunch of results: triangulability of surfaces (via what is nowadays known as the Schoenflies theorem), existence of non-metric surfaces following the (unpublished) construction of Prüfer (ca. 1922)\]  T. Radó, F. Riesz, [*Über die erste Randwertaufgabe für $\Delta u=0$*]{}, Math. Z. 22 (1925), 41–44. \[$\spadesuit$ supplies drastic simplifications over Perron’s method (Perron 1923 [@Perron_1923]) according to Carathéodory 1937 [@Caratheodory_1937-On-Dirichlet's-problem p.710]\]  T. Radó, [*Subharmonic functions*]{}, Berlin, 1937.  V. Ragsdale, [*On the arrangement of the real branches of plane algebraic curves*]{}, Amer. J Math. 28 (1906), 377–404. \[$\spadesuit$ formulation of the Ragsdale conjecture saying that if $m=2k$, and $p,n$ are the number of even resp. odd ovals then $p\le \frac{3k(k-1)}{2}+1$ and $n\le \frac{3k(k-1)}{2}$. This conjecture was disproved by Itenberg in 2000 using Viro’s patchworking (in degree 10)\]$\bigstar$  Z. Ran, [*Families of plane curves and their limits: Enriques’ conjecture and beyond*]{}, Ann. of Math. (2) 130 (1989), 121–157. \[$\spadesuit$\]  H.E. Rauch, [*Weierstrass points, branch points, and the moduli of Riemann surfaces*]{}, Comm. Pure Appl. Math. 12 (1959), 543–560. \[$\spadesuit$\]$\bigstar$  H.E. Rauch, [*A transcendental view of the spaces of algebraic Riemann surfaces*]{}, Bull. Amer. Math. Soc. 71 (1965), 1–39. \[$\spadesuit$ the cream of the theory (Riemann, Teichmüller, Ahlfors, etc. revisited)\]  A.H. Read, [*Conjugate extremal problems of class $p=1$*]{}, Ann. Acad. Sci. Fenn., A.I., 250/28 (1958), 8 pp. 60, 78  A.H. Read, [*A converse to Cauchy’s theorem and applications to extremal problems*]{}, Acta Math. 100 (1958), 1–22. 50, 78 \[$\clubsuit$ an alternative proof of Ahlfors 1950 [@Ahlfors_1950] is given via Hahn-Banach $\clubsuit$ subsequent work via a similar approach in Royden 1962 [@Royden_1962] $\diamondsuit$ we probably do not need to recall that both Royden and Read were students of Ahlfors\]  E. Reich, S.E. Warschawski, [*On canonical conformal maps of regions of arbitrary connectivity*]{}, Pacific J. Math. 10 (1960), 965–985. 78 \[$\spadesuit$ like Meschkowski 1953 [@Meschkowski_1953] (which is not cited!) shows that the Ahlfors-type problem of maximizing the derivative among [*schlicht*]{} function bounded-by-one gives a conformal map upon a Kreisschlitzbereich (=circular slit disc). This analysis is also based upon Rengel’s inequality, or a variant thereof closer to Grunsky’s Thesis 1932\]  H.J. Reiffen, [*Die differentialgeometrischen Eigenschaften der invarianten Distanzfunktion von Carathéodory*]{}, Schrift Math. Inst. Univ. Münster 26 (1963). \[$\spadesuit$ quoted e.g. in Burbea 1977 [@Burbea_1977-Caratheodory]\]$\bigstar$  R. Remmert, [*Funktionentheorie 2*]{}, Grundwissen Mathematik [*6*]{}, Springer-Lehrbuch, 1991. (1. unveränderter Nachdruck 1992 der 1. Auflage.) R. Remmert, [*From Riemann surfaces to complex spaces*]{}, Séminaire et Congrès 3, Société Math. de France, 1998, 203–241.  E. Rengel, [*Über einige Schlitztheoreme der konformen Abbildung.*]{} (Diss.), Schriften math. Semin., Inst. angew. Math. d. Univ. Berlin 1 (1932/33), 140–162. 60, 78 $\bigstar$  E. Rengel, [*Existenzbeweise für schlichte Abbildungen mehrfach zusammenhängender Bereiche auf gewisse Normalbereiche*]{}, J.-Ber. Deutsche Math.-verein. 44 (1934), 51–55. 60, 78 \[$\spadesuit$ via the extremal problem method in vogue at the time obtain the exitence of the circular/radial slit maps for domain of finite connectivity (cf. also de Possel, and Grötzsch) $\spadesuit$ the terminology “Normalbereiche” goes back to Weierstrass, compare Schottky’s Thesis 1877 [@Schottky_1877] $\spadesuit$ this paper shows the existence of a schlicht mapping of a finitely-connected domain upon a circular slit disk $\spadesuit$ antecedent in Koebe 1918, see also Reich-Warschawski 1960 [@Reich-Warschawski_1960]\]  H. Renggli, [*Zur konformen Abbildung auf Normalgebiete*]{}, (Diss. ETH Zürich) Comment. Math. Helv. 31 (1956), 5–40 60, 78 \[$\spadesuit$ limited to plane domains, where the various slit mappings are reproved via an extremal problem involving the extremal length, Montel’s normal families are used\]  M. von Renteln, [*Friedrich Prym (1841–1915) and his investigations on the Dirichlet problem*]{}, Suppl. Rend. Circ. Mat. Palermo 44 (1996), 43–55 \[$\spadesuit$ detailed discussion of Prym’s counterexample to the (naive) Dirichlet principle (compare Prym 1871 [@Prym_1871])\]$\bigstar$$\bigstar$$\bigstar$  S. Richardson, [*Hele-Shaw flows with time-dependent free boundaries involving a multiply-connected fluid region*]{}, European J. Appl. Math. 12 (2001), 571–599 \[$\spadesuit$\]  B. Riemann, [*Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse*]{}. Inauguraldissertation Göttingen, 1851. In: Ges. math. Werke 1876/1892/1990 [@Riemann_1990]. \[$\clubsuit$ first proof of RMT, some bad tongues claim that the proof is dubious (even abstraction made of the difficulty allied to the Dirichlet principle), whereupon Riemann reacted with [@Riemann_1857-DP]\]  B. Riemann, [*Über die Hypothesen, welche der Geometrie zugrunde liegen*]{}. Habilitationsvortrag 1854, first published in: Abhandlungen der Königl. Ges. d. Wiss. Göttingen 13 (1867), reproduced in Ges. math. Werke [@Riemann_1990]. \[$\spadesuit$ a breathtaking generalization of geometry, ramifying to the eclectic topic of Riemannian geometry, Dedekind, Beltrami, Ricci, etc., up to Gromov, Perelman, etc.\]  B. Riemann, [*Fragment aus der Analysis Situs*]{}. circa 1852/53. Published In: Ges. math. Werke [@Riemann_1990]. \[$\spadesuit$ a first attempt to generalize the connectivity number to high-dimensional manifolds, leads to the work of Betti, and Poincaré 1895 [@Poincare_1895-Analysis-Situs]\]  B. Riemann, [*Theorie der Abel’schen Functionen*]{}, Crelle J. Reine Angew. Math. 54 (1857), ?–?. In: Ges. math. Werke [@Riemann_1990 88–142]. \[$\spadesuit$ contains in particular the statement that any (or at least one with general moduli?) closed Riemann surface of genus $g$ maps conformally to the sphere with $\le[\frac{g+3}{2}]$ sheets $\spadesuit$ this assertion not accepted by modern geometers until Meis 1960 [@Meis_1960] $\clubsuit$ p.116, some historical hints given by Riemann shows an involvement with conformal maps of multiply-connected regions (maybe even surfaces) as early as Fall 1851 (up to Begin 1852), but then he was sidetracked to another topic\]  B. Riemann, [*Bestimmung einer Function einer veränderlichen complexen Grösse durch Grenz- und Unstetigkeitsbedingungen*]{}, Crelle J. Reine Angew. Math. 54 (1857), 111–114. \[$\clubsuit$ after Riemann 1851 [@Riemann_1851] the second (more solid, but less romantic) proof of RMT, of course in retrospect not sound until Hilbert’s resurrection of the Dirichlet principle\]  B. Riemann, [*Gleichgewicht der Electricität auf Cylindern mit Kreisförmigem Querschnitt und parallelen Axen. Conforme Abbildung von durch Kreise begrenzten Figuren*]{} (Nachlass XXVI). In: Ges. math. Werke [@Riemann_1990 p.472–476]. 78 \[$\clubsuit$ the first version of the “Ahlfors map” in the planar case (perhaps confined to the case of circular domains) $\spadesuit$ for subsequent works see primarily Schottky 1875/77 [@Schottky_1877], Bieberbach 1925 [@Bieberbach_1925], Grunsky 1937–41/40–42, Ahlfors 1947–50 [@Ahlfors_1950]\]  B. Riemann, [*Ueber das Verschwinden der Thetafunctionen*]{}, ?? ?? (1865), ?–?. In: Ges. math. Werke [@Riemann_1990 ?–?]. \[$\spadesuit$ another complete solution to the problem of inverting Abelian integrals $\spadesuit$ Schottky’s problem (1903): what conditions must be imposed on the Riemann matrices to arise as period matrices $\spadesuit$ full effective solution in Shiota 1986\]  B. Riemann, [*Gesammelte mathematische Werke, wissenschaftlicher Nachlass und Nachträge,*]{} Nach der Ausgabe von H. Weber und R. Dedekind, Teubner, Leipzig, 1876; neu herausgegeben von R. Narasimhan, Springer-Verlag, Berlin, 1990. \[$\spadesuit$ the first edition 1876 (as well as the subsequent editions) contains the first publication of Riemann’s Nachlass ([@Riemann_1857_Nachlass] estimated 1857/58), where existence of circle maps is proven for planar surfaces, especially in the case of a domain bounded by circles\] 60  F. Riesz, [*Ueber Potenzreihen mit vorgeschriebenen Anfangsgliedern*]{}, Math. Z. 18 (1923), 87–95. \[$\spadesuit$ cited in Heins 1975 [@Heins_1975], who employs a Riesz variational formula to derive another proof of Ahlfors’ circle maps with upper control upon the degree $\spadesuit$ in fact the cited variational formula of F. Riesz, was given by him for the case $p=1$ (Hardy classes index) and for the disc $\Delta$. However it is available suitably modified for any (finite) bordered Riemann surface and all possible Hardy classes indexes $1\le p <\infty$. (source=p.20 of the just cited Heins work, where for details one must probably browse Heins 1969 [@Heins_1969-LNM-Hardy])\]  F. Riesz, [*Über die Randwerte einer analytische Funktion*]{}, Math. Z. 18 (1923), 87–95. \[$\spadesuit$\]  F. Riesz, [*Sur les fonctions subharmoniques et leur rapport à la théorie du potentiel*]{}, Acta Math. 54 (1930), 321–360. \[$\spadesuit$ somehow inspired by Perron 1923 [@Perron_1923]\]  J.-J. Risler, [*Les nombres de Betti des ensembles algébriques réels, une mise au point*]{}, Gazette des math. ?? (1992), ?–?. \[$\spadesuit$\]  E. Ritter, [*Die multiplicativen Formen auf algebraischem Gebilde beliebigen Geschlechts mit Anwendung auf die Theorie der automorphen Formen*]{}, Math. Ann. 44 (1894), 261–374. \[$\spadesuit$\]  W. Ritz, [*???*]{}, ??? ?? (1908), ??–??. \[$\spadesuit$ inspired Bieberbach 1914 [@Bieberbach_1914], who in turn inspired Bergman 1922 [@Bergman_1922], which had some influence on Ahlfors 1950 [@Ahlfors_1950] $\spadesuit$ for a brilliant discussion of Ritz work [*per se*]{} cf. Gander-Wanner 2012 [@Gander-Wanner_2012]\]  ??. Robertson, [*On the theory of univalent functions*]{}, Ann. of Math. 37 (1936), 374–408. \[$\spadesuit$ contains a simple derivation of the Bieberbach conjecture $\vert a_n\vert \le n \vert a_1 \vert$ for starlike regions via the Schwarz-Christoffel formula\]  G. Robin, [*Sur la distribution de l’éléctricité à la surface des conducteurs fermés et des conducteurs ouverts*]{}, Ann. Sci. École Norm. Sup. 3 (1886), 3–58. \[$\spadesuit$\]  R.M. Robinson, [*Analytic functions in circular rings*]{}, Duke Math. J. 10 (1943), 341–354. 78 \[$\spadesuit$ quoted in Minda 1979 [@Minda_1979] in connection with the theta function expression of the Ahlfors function of an annulus $\spadesuit$ for this see also Golusin 1952/57 [@Golusin_1952/57] $\spadesuit$ also quoted in Jenkins-Suita 1979 [@Jenkins-Suita_1979]\]  R.M. Robinson, [*Hadamard’s three circles theorem*]{}, Bull. Amer. Math. Soc. 50 (1944), 795–802. 78 \[$\spadesuit$\]$\bigstar$$\bigstar$  G. Roch, [*Ueber die Anzahl der willkürlichen Constanten in algebraischen Functionen*]{}, Crelle J. Reine Angew. Math. 64 (1865), 372–376.  R. Rochberg, [*Almost isometries of Banach spaces and moduli of Riemann surfaces*]{}, Duke Math. J. ?? (1973), ??–??. \[$\spadesuit$ compact bordered Riemann surfaces\]  R. Rochberg, [*Deformation of uniform algebras on Riemann surfaces*]{}, Pacific J. Math. 121 (1986), 135–181. 50 \[$\spadesuit$ on p.142 Ahlfors 1950 [@Ahlfors_1950] is cited as follows: Ahlfors has shown that given $S$ in $\cal S$ \[the set of all connected finite bordered Riemann surfaces, cf. p.135\] and $x,y$ in $S\setminus \partial S$ there is a function $F=F_{x,y}$ in $A(S)$ which has $\vert F \vert=1$identically on $\partial S$, $F(x)=0$, $F(y)\neq 0$, and $F$ maps $S$ onto the closed unit disk in an $m$ to one manner (counting multiplicity). Furthermore, if $g$ denotes the genus of $S$ and $c$ the number of components of $\partial S$, then $F$ can be selected so that $m$ satisfies $c\le m \le 2g+c$. $\spadesuit$ on the same page the Ahlfors’ bound ($r+2p$ in our notation) is applied to a problem a bit too technical to be summarized here, and naively one could ask if the improved bound $r+p$ of Gabard 2006 [@Gabard_2006] could be applied to Rochberg’s work. This is not evident because a lowest possible degree map does not a priori separates two points prescribed in advance (hence we have not pursued the issue further)\]  B. Rodin, L. Sario, [*Principal functions*]{}, Princeton, van Nostrand, 1968. 78 $\bigstar$$\bigstar$ B. Rodin, [*The method of extremal length*]{}, Bull. Amer. Math. Soc. 80 (1974), 587–606. 78 \[$\spadesuit$ p.590 Teichmüller listed (without reference!) amongst the contributor to the Löwner-Pu systolic inequality? $\spadesuit$ if this is true it would be nice to localize the precise source\]  B. Rodin, D. Sullivan [*The convergence of circle packings to the Riemann mapping*]{}, J. Differ. Geom. 26 (1987), 349–360. \[$\spadesuit$ building over work of Koebe 1936 (not cited), Andreev 1970 and Thurston 1985, develop a convergence proof of (finitistic) approximation by circle packings of the Riemann mapping $\clubsuit$ an obvious desideratum would be to implement a similar proof for the case of the Ahlfors function on compact bordered Riemann surface\]  E. Röding, [*Konforme Abbildung endlicher Riemannscher Flächen auf kanonische Überlagerungsflächen der Zahlenkugel*]{}, Diss. Würzburg, 1972, 71 S. 78 \[$\spadesuit$ this entry is cited on the “critical” page 198 of Grunsky 1978 [@Grunsky_1978], according to which it gives a generalization to Riemann surfaces of the Bieberbach-Grunsky theorem (i.e. circle map in the planar case) $\spadesuit$ in particular, it could be the case that Röding reproves the existence of an Ahlfors circle map, yet probably this is not the case $\spadesuit$ perhaps this aspect has been subsequently published in Röding 1977 [@Roeding_1977_mero]\] $\bigstar$  E. Röding, [*Nichtschlichte konforme Abbildung\[en\] unendlich vielfach zusammenhängender Teilgebiete der Ebene*]{}, Arch. d. Math. 26 (1975), 391–397. 78 \[$\spadesuit$ infinite connectivity analog of the “Riemann-Bieberbach” mapping theorem.\]  E. Röding, [*Über die Wertannahme der Ahlforsfunktion in beliebigen Gebieten*]{}, Manuscr. Math. 20 (1977), 133–140. 78  E. Röding, [*Über meromorphe Funktionen auf endlichen Riemannschen Flächen vom Betrag eins auf den Randlinien*]{}, Math. Nachr. 78 (1977), 309–318. 78  M. Roggero, [*Real divisors on real curves*]{}, Le Matematiche 54 (1999), 67–76. \[$\spadesuit$ “...every divisor \[on a smooth real algebraic curve having a nonempty real part\], which is linearly equivalent to its conjugate, is also equivalent to a divisor supported on a set of real points.” $\spadesuit$ this resembles slightly the reformulation of Ahlfors theorem given in Gabard 2006 [@Gabard_2006], but differs substantially for Roggero’s result applies also to diasymmetric curves (with real points) $\spadesuit$ p.75–76: an example is given of a smooth real plane quartic such that every line intersect the (supposed nonempty) real locus in at most 2 points; evidently such a curve has at most one oval and another such example is the Fermat quartic $x^4+y^4=1$\]  W.W. Rogosinski, H.S. Shapiro, [*On certain extremum problems for analytic functions*]{}, Acta Math. 90 (1953), 287–318. \[$\spadesuit$ this article pertains to our topic (of the Ahlfors map) inasmuch as it may have influenced some new generation existence-proof (of “abstract” functional analytic character) of the Ahlfors map (where Hahn-Banach takes over the rôle of Euler-Lagrange), like those of Read 1958 [@Read_1958_Acta], and the popular version of Royden 1962 [@Royden_1962]\]  V.A. Rohlin, [*New results in the theory of $4$-dimensional manifolds*]{}, Dokl. Akad. Nauk SSSR 84 (1952), 221–224; French transl. available in Guillou-Marin 1986 [@Guillou-Marin_1986]. \[$\spadesuit$ seminal result of 4D-differential topology on the divisibility by 16 of a simply-connected manifold with even intersection form $\spadesuit$ this (suitably generalized) turned out to be relevant to Hilbert’s 16th problem yielding a proof of the Gudkov hypothesis, cf. Rohlin 1972/72 [@Rohlin_1972/72-Proof-of-a-conj-of-Gudkov]\]  V.A. Rohlin, [*Proof of a conjecture of Gudkov*]{}, Funkt. Anal. Prilozhen. 6 (1972), 62–64; English transl., Funct. Anal. Appl. 6 (1972), 136–138. \[$\spadesuit$ the congruence in question (nowadays known as the Gudkov-Rohlin congruence) states that a plane $M$-curve of order $2k$ satisfies $p-n\cong k^2 \pmod 8$ $\spadesuit$ when particularized to degree 6 it affords a new “elementary” solution to Hilbert’s 16th problem (free from the vicissitudes allied to the Hilbert-Rohn-Gudkov method) $\spadesuit$ alas Rohlin’s first proof contains a little flaw (cf. next $\spadesuit$) though being essentially correct using the seminal Rohlin’s divisibility by 16 of the signatures of spin $4$-manifolds (even forms of intersection on the $2$-dimensional homology) $\spadesuit$ from Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91 p.361]: “Three proof of the Gudkov-Rohlin congruence have been published. They are due to V.A. Rohlin \[16\](=1972/72 [@Rohlin_1972/72-Proof-of-a-conj-of-Gudkov]=Proof of Gudkov’s hypothesis), \[17\](=[@Rohlin_1972/72-Cong-mod-16]=Congruence modulo $16$ in Hilbert’s 16th problem) and A. Marin \[12\](=1979/80 [@Marin_1979]). The third \[12\](=Marin ) appears to be an improvement of the first. The example considered by Marin \[12\](=) seems to show that there is no correct proof of (1.A)\[=Gudkov’s hypothesis\] which is closer to Rohlin’s argument than Marin’s proof.—Marin’s \[12\] and Rohlin’s second \[17\] approaches \[are\] based on quite different techniques. Rohlin’s proof work in any dimension while no generalization of Marin’s proof to higher dimensions is known. Nevertheless the approaches seem to be closely related. Rohlin asked his students to find a relation and said that an understanding of it might lead to essential progress.” $\spadesuit$ from Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000 p.736]: “In Rokhlin’s first paper \[97\](=this entry) there is a mistake in the proof of Gudkov’s conjecture. However the approach in the paper, namely, using characteristic surfaces in a $4$-manifold to evaluate the signature $\mod 16$, became a powerful method in the study of real algebraic curves. It was used by Marin, who together with Guillou (see \[46\](=Guillou-Marin 1977 [@Guillou-Marin_1977])) extended Rokhlin’s signature formula to non-orientable characteristic surfaces and thus corrected the mistake.”\]  V.A. Rohlin, [*Congruence modulo 16 in Hilbert’s sixteenth problem*]{}, Funkt. Anal. Prilozhen. 6 (1972), 58–64; English transl., Funct. Anal. Appl. 6 (1972), 301–306. \[$\spadesuit$ severe restriction upon the isotopy classification of $M$-curves reinforcing earlier work of Petrovskii 1938 [@Petrowsky_1938] and Arnold 1971 [@Arnold_1971/72]\]  V.A. Rohlin, [*Congruence modulo 16 in Hilbert’s sixteenth problem, II*]{}, Funkt. Anal. Prilozhen. 7 (1973), 91–92; English transl., Funct. Anal. Appl. ? (197?), ?–?. \[$\spadesuit$\]  V.A. Rohlin, [*Complex orientations of real algebraic curves*]{}, Funkt. Anal. Prilozhen. 8 (1974), 71–75; English transl., Funct. Anal. Appl. 8 (1974), 331–334. \[$\spadesuit$ present a general method of closing the one half of a dividing real plane curve by piecing together real discs to construct a closed membrane, whose (fundamental) homology class yields via intersection theory a certain numerical relation known as Rohlin’s complex orientation formula. The latter implies the striking fact that a dividing curve exhibits at least as many ovals as the half value of its degree(=order). This answers a question of Klein, made more explicit in Gross-Harris 1981 [@Gross-Harris_1981]. Compare Gabard 2000 [@Gabard_2000] for more details. NB: In this seminal paper, Rohlin treats only the case of $M$-curve(=Harnack-maximal) (the general formula being written down in the next entry Rohlin 1978 [@Rohlin_1978], but the proof is easy to adapt). $\spadesuit$ Rohlin’s formula also prohibits many (but not all) $M$-schemes of sextics (e.g. that consisting of eleven unnested ovals) supplying so a 5 minutes proof of the tricky theorem of Hilbert (1891–00–08), which he was never able to complete himself (or with his numerous students)\]  V.A. Rohlin, [*Complex topological characteristics of real algebraic curves*]{}, Uspekhi Mat. Nauk. 33 (1978), 77–89; translation: Russian Math. Surveys 33 (1978), 85–98. \[$\spadesuit$ shows strikingly that Rohlin discovered Klein’s work at a very late stage (despite the fact that Klein is generously quoted e.g. in Gudkov 1974 [@Gudkov_1974/74]), but with great happiness apparently (p.85): “As I learned recently, more than hundred years ago, the problems of this article occupied Klein, who succeeded in coping with curves of degree $m\le 4$ (see \[4\](=Klein 1922 [@Klein-Werke-II_1922]), p.155).” $\spadesuit$ p.93–94 prove the result that a real plane curve with a nest of maximal depth is dividing, via an argument which (in our opinion) can be slightly simplified as follows $\spadesuit$ given $C_m\subset {\Bbb P}^2$ a nonsingular curve of degree $m$ with a deep nest then projecting the curve from any point chosen in the innermost oval gives a morphism $C_m \to {\Bbb P}^1$ whose fibers over real points are totally real. Hence there is an induced map between the imaginary loci $C_m({\Bbb C})-C_m({\Bbb R}) \to {\Bbb P}^1({\Bbb C})-{\Bbb P}^1({\Bbb R})$ and it follows that $C_m$ is dividing (just by using the fact that the image of a connected set is connected). q.e.d. (this argument avoids the consideration of the canonical fibering ${\rm pr}\colon {\Bbb C} P^2- {\Bbb R} P^2 \to S^2$ envisaged by Rohlin) $\spadesuit$ p.94: “If $A_1$ and $A_2$ belong to type I, then the question is rather complicated, in general, but Fiedler first noted that everything is radically simplified when $s=m_1 m_2$ \[i.e. all intersections are real\]. Namely, in this situation, $A$ belongs to type I” $\spadesuit$ some interesting question is raised on p.95: “[**3.9 A conjecture about real schemes of type I**]{}. A study of the available factual material suggests that possibly a real scheme belongs to type I iff it is [*maximal*]{}, that is, it is not part of a larger real scheme of the same degree. This conjecture is true for $m\le 6$, and there is much to be said in its favour for $m>6$. There is an allusion to it in Klein: see \[4\], p.155 (=Klein 1922=Ges. Math. Abh. II [@Klein-Werke-II_1922]).” \[31.12.12\] Gabard’s guess: perhaps this conjecture of Klein-Rohlin follows from Ahlfors theorem translated in terms of total reality (intuitively having a total pencil, no real circuit can be added for otherwise Bézout would be corrupted, yet perhaps this is too naive, cf. our Sec.\[Klein-Rohlin-conj:sec\]) $\spadesuit$ [*Warning.*]{} p.788 of Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000] one reads: “[**Digression: real rational curves**]{}. As far as we know, the following problem is still open: is it possible to draw an irreducible real rational curve (or more precisely a connected component of it) of degree $q$ through any set of $3q-1$ real points in general position? In \[99\](=Rohlin 1978 [@Rohlin_1978]) the question is answered in the affirmative; however, the proof has never been published; possibly it contained a gap.”\]  V.A. Rohlin, [*New inequalities in the topology of real plane algebraic curves*]{}, Uspekhi Mat. Nauk. 14 (1980), 37–43; translation: Russian Math. Surveys ?? (198?), ??–??. \[$\spadesuit$\]  V.A. Rohlin, [*Two aspects of the topology of real algebraic curves*]{}, Proc. Leningrad Internat. Topology Conf., Nauka, Leningrad, 1983; (translation available?). \[$\spadesuit$ cited in Viro 1986/86 [@Viro_1986/86-Progress]\]  K. Rohn, [*Flächen vierter Ordnung hinsichtlich ihrer Knotenpunkte und ihrer Gestaltung*]{}, Preisschriften der Fürstlich Jablonowskischen Gesellschaft, Leipzig, 1886; also in Math. Ann. 29 (1887), 81–96. \[$\spadesuit$\]  K. Rohn, [*Die ebenen Kurven 6. Ordnung mit elf ovalen*]{}, Leipzig Ber. Dezember 1911. \[$\spadesuit$ cited in Petrovsky 1938 [@Petrowsky_1938] and considered there as the first rigorous proof of Hilbert’s announced theorem that an $M$-sextic cannot have all its $11$ ovals lying unnested. However Gudkov (e.g. in 1974 [@Gudkov_1974/74]) is more severe and does not consider Rohn’s proof as complete. $\spadesuit$ \[18.03.13\] perhaps nowadays the most expediting way to prove this Hilbert-Rohn theorem is via Rohlin’s formula for complex orientations, which proves more generally that any $M$-curve (of even degree) has some nesting provided its degree $m=2k\ge 6$. The first proof of this statement (and much more) goes really back to Petrovskii’s seminal inequalities of 1938, cf. Petrovskii 1933/38 [@Petrowsky_1938]\]  K. Rohn, [*Die Maximalzahl und Anordnung der Ovale bei der ebenen Kurve 6. Ordnung und bei der Fläche 4. Ordnung*]{}, Math. Ann. 73 (1913), 177–229. \[$\spadesuit$\]  H. Röhrl, [*Unbounded coverings of Riemann surfaces and extensions of rings of meromorphic functions*]{}, Trans. Amer. Math. Soc. 107 (1963), 320–346. \[$\spadesuit$ cited in Alling 1965 [@Alling_1965], and one may wonder about a connection with Ahlfors 1950, i.e. the “unbounded covering” in question (cf. definition on p.328) are probably related to circle maps, at least extended versions thereof where the target is not necessarily the unit disc of course Röhrl’s notion is quite standard, albeit the terminology is far from uniformized, cf. e.g. Ahlfors-Sario’s “complete covering surfaces” (in 1960=[@Ahlfors-Sario_1960 p.42] themselves patterned after Stoilow’s “total coverings” $\spadesuit$ alas, it does not seem that Röhrl reproves Ahlfors result (which would have been pleasant in view of Röhrl great familiarity with Meis’ work 1960 [@Meis_1960])\]  F. Ronga, [*Analyse réelle post-élémentaire*]{}, Presses polytechniques romandes, 1999. \[$\spadesuit$ cited for the picture Fig.\[ItenbergViroRiem:fig\]d\]  P.C. Rosenbloom, [*Quelques classes de problèmes extrémaux*]{}, Bull. Soc. Math. France 80 (1952), 183–215. \[$\spadesuit$ this worked is cited in Forelli 1979 [@Forelli_1979], where it is employed to derive another existence-proof of circle-maps with the same control upon the degree as in Ahlfors 1950 [@Ahlfors_1950]\]  M. Ross, [*The second variation of nonorientable minimal submanifolds*]{}, Trans. Amer. Math. Soc. 349 (1997), 3093–3104. \[$\spadesuit$ p.3097 criticizes the argument of Li-Yau 1982 [@Li-Yau_1982] for the Witt-Martens mapping $\spadesuit$ gives differential geometric application of it to (non-orientable) minimal surfaces\]  H.L. Royden, [*Harmonic functions on open Riemann surfaces*]{}, Trans. Amer. Math. Soc. 73 (1952), 40–94. 50 \[$\clubsuit$ this is, in substance, the author’s Thesis \[Harvard University, 1951\] (under Ahlfors) $\spadesuit$ it contains very deep material “sufficient condition for the hyperbolic type in term of a triangulation of the surface” (causing a great admiration by Pfluger, etc.), yet from our finitistic perspective the paper seems to contain little about the Ahlfors map, for this issue see rather the subsequent paper Royden 1962 [@Royden_1962]\]  H.L. Royden, [*Rings of meromorphic functions*]{}, Proc. Amer. Math. Soc. 9 (1958), 959–965. \[$\clubsuit$ this article is often credited by Alling to be the first employment of Ahlfors map as a technique to lift truths from the disc to more general finite bordered surfaces, e.g. in the Acknowledgements of Alling 1965 [@Alling_1965] or in Alling’s review of Stout 1965 [@Stout_1965] one reads: “The third technique is dependent on the existence of the Ahlfors map $P$ (=1950 [@Ahlfors_1950]), which maps a compact bordered Riemann surface $\overline R$, finite-to-one, onto $\overline U$. This gives rise to the algebraic approach, for the adjoint of $P$ is an isomorphism of $H_{\infty}(U)$ into $H_{\infty}(R)$, the extension being finite and very tractable. This approach was apparently first used by Royden 1958 \[=this entry=[@Royden_1958]\]. Later it was utilized extensively by the reviewer, who working independently of the author\[=Stout\], announced his extension of Carleson’s corona result to $R$ \[$\dots$\]”\]$\bigstar$$\bigstar$  H.L. Royden, [*The boundary values of analytic and harmonic functions*]{}, Math. Z. 78 (1962), 1–24. \[$\clubsuit$ re-prove the existence and properties of the Ahlfors function via Hahn-Banach, along the path of Read 1958 [@Read_1958_Acta]\]  L.A. Rubel, J.V. Ryff, [*The bounded weak-star topology and the bounded analytic functions*]{}, J. Funct. Anal. 5 (1970), 167–183. 47, 50 \[$\clubsuit$\]$\bigstar$$\bigstar$$\bigstar$NY(only-MR)  L.A. Rubel, [*Bounded convergence of analytic functions*]{}, Bull. Amer. Math. Soc. 77 (1971), 13–24. 47, 50 \[$\clubsuit$ p.18 the two works of Ahlfors 1947 [@Ahlfors_1947], 1950 [@Ahlfors_1950] are quoted in connection with the following problem about inner functions: “In the case of the general region $G$ \[supposed (cf. p.17) to support nonconstant bounded analytic functions and to enclose no removable singularities for all bounded analytic functions\], one would guess that the solution, known to exist, of any of several extremal problems would be inner, and consequently hypo-inner. For example, choose a point $z_0\in G$ and consider $f\in B_H(G)$ \[i.e. the space of bounded analytic function\] so that $\| f\|_{\infty}\le 1$ and $f(z_0)=0$, and maximize $\vert f'(z_0)\vert$. The extremal function is the so-called Ahlfors function, and in case $G$ is finitely connected, it is known \[2\](=Ahlfors 1947 [@Ahlfors_1947]), \[3\](=Ahlfors 1950 [@Ahlfors_1950]) to be inner.” $\spadesuit$ let us recall definitions (cf. p.17–18): a bounded analytic function on the disc $F\in B_H(D)$ is [*inner*]{} if $\|F\|_{\infty}\le 1$ and if its Fatou radial limit function $F^{\ast}(e^{i\theta})=\lim_{r\to 1} F(re^{i\theta})$ has unit modulus for almost all $\theta$ (w.r.t. usual arc length). It is said to be [*hypo-inner*]{} if the Fatou limit has unit modulus for a set of $\theta$ of positive measure. For a function on a general domain $G$, $f\in B_H(G)$, the notions of inner and hypo-inner are transposed via precomposition with the universal covering map $D \to G$. $\spadesuit$ now as to Rubel’s guess, it seems to be answered in the negative in Gamelin 1973 [@Gamelin_1973-BAMS p.1107], with details to be found in Gamelin 1974 [@Gamelin_1974-Shilov]\]  L.A. Rubel, [*Some research problems about algebraic differential equations*]{}, Trans. Amer. Math. Soc. 280 (1983), 43–52. \[$\clubsuit$ p.47 the Ahlfors function is mentioned as follows: “To prepare the way for the next problem, we shall define the [*Ahlfors function*]{}. If $G$ is a (presumably multiply connected) region and $z_0$ is a point in $G$, we define the Ahlfors function $\alpha_{z_0}$ with [*basepoint*]{} $z_0$ as the (unique)solution of the following extremal problem: (i) $\alpha(z_0)=0$, (ii) $\vert \alpha(z)\vert \le 1$ for all $z\in G$, (iii) $\alpha'(z_0)$ is as large as it can be for the class of functions satisfying (i) and (ii). In case $G$ is simply-connected, $\alpha_{z_0}$ becomes the Riemann map of $G$ onto $D$ that takes $z_0$ to $0$, with positive derivative there. [*Problem*]{} 11. [*Suppose $\alpha_{z_0}$ is hypotranscendental, and let $z_1\in G$ be another base point. Must $\alpha_{z_1}$ be hypotranscendental too?*]{}\]  W. Rudin, [*Some theorems on bounded analytic functions*]{}, Trans. Amer. Math. Soc. 78 (1955), 333–342. 47, 78 \[$\spadesuit$ new (simpler) proof of an (unpublished) theorem of Chevalley-Kakutani stating that a plane domain $B$ such that for each of its boundary-point $p$ there is a bounded analytic function on $B$ possessing at $p$ a singularity is determined (modulo a conformal transformation) by the ring of all bounded analytic functions on $B$ $\spadesuit$ the proof makes uses of general results of Ahlfors 1947 [@Ahlfors_1947], yet apparently no use is made of the Ahlfors function\]  W. Rudin, [*Analytic functions of class $H^p$*]{}, Trans. Amer. Math. Soc. 78 (1955), 46–66. 47 \[$\spadesuit$\]  W. Rudin, [*The closed ideals in an algebra of continuous functions*]{}, Canad. J. Math. 9 (1957), 426–434. \[$\spadesuit$ proof of an unpublished result of Beurling describing the ideal theory of the algebra $A(\overline \Delta)$ of continuous function on the closed disc analytic on its interior $\spadesuit$ for extensions of this Beurling-Rudin result to compact bordered surfaces, cf. Voichick 1964 [@Voichick_1964], Limaye’s Thesis 1968 and Stanton 1971 [@Stanton_1971] (who makes use of the Ahlfors map) $\spadesuit$ for an extension to non-orientable Klein surfaces (where no Ahlfors map are available!), see Alling-Limaye 1972 [@Alling-Limaye_1972]\]  W. Rudin, [*Pairs of inner functions on finite Riemann surfaces*]{}, Trans. Amer. Math. Soc. 140 (1969), 423–434. \[$\spadesuit$ inner function as a synonym of the (Ahlfors) circle maps\]  W. Rudin, [*Real and complex analysis*]{}, McGraw-Hill. \[$\spadesuit$\]  L. Rudolph, [*Some topologically locally-flat surfaces in the complex projective plane*]{}, Comment. Math. Helv. 59 (1984), 592–599. \[$\spadesuit$ locally-flat counterexamples to Thom’s conjecture (based upon work of Freedman) $\spadesuit$ p.593 contains the sharpest historical information I am aware of about the terminology “Thom conjecture”, namely: “Professor Thom has remarked (personal communication, November 19, 1982) that the conjecture perhaps more properly belongs to folklore.” $\spadesuit$ As far as I know the designation “Thom conjecture” appears first in Kirby’s problem list (1970) [@Kirby_1970--95]\]  R. Rüedy, [*Einbettungen Riemannscher Flächen in den dreidimensionalen euklischen Raum*]{}, Comment. Math. Helv. 43 (1968), 417–442. \[$\spadesuit$ p.417: “Flächen im Sinne der elementaren Differentialgeometrie können zu Riemannschen Flächen gemacht werden, indem man die iostheremen Parameter als lokale Koordinaten benutzt. Diese Struktur nennt man die [*natürliche*]{}, weil genau diese lokalen Darstellung winkeltreu sind.—F. Klein warf schon 1882 in seiner Schrift [*Über Riemanns Theorie der algebraischen Funktionen und ihrer Integrale*]{} das Problem auf, ob sich jede Riemannsce Fläche konform und bijectiv auf eine solche differentialgeometrische Fläche abbilden lasse.—Der Weg zu diesem überraschend schwierig zugänglichen Problem wurde durch die fundamentalen Arbeiten von Teichmüller geöffnet; aber erst um 1960 gelang der Beweis für den folgenden Satz:—[Einbettungssatz von Garsia.]{} [*Jede kompakte Riemannsche Fläche ist konform äquivalent zu einer differentialgeometrischen Fläche, die reelle-algebraisch im dreidimensionalen euklidischen Raum eingebettet ist.*]{}”\]  R. Rüedy, [*Embeddings of open Riemann surfaces*]{}, Comment. Math. Helv. 46 (1971), 214–225. \[$\spadesuit$ p.214: “In the final section of his famous thesis Riemann states that in his investigations the branched covering surfaces of the plane could be replaced by smooth orientable surfaces embedded in Euclidean $3$-space. \[…\] In his lectures Felix Klein emphasized the concept of viewing classical surfaces as Riemann surfaces, …. It was also he who asked in 1882 if every Riemann surface were conformally equivalent to a classical surface. \[F. Klein, [*Ges. math. Abh.,*]{} Bd.3 (Springer 1923), p.502 and p.635.\]—For a long time the only result in this direction were that every compact Riemann surface of genus zero is conformally equivalent to the sphere, every non-compact planar (schlichtartig) surface is conformally equivalent to a subregion of the plane, and a compact Riemann surface of genus $1$ is conformally equivalent to a ring surface provided its modulus is purely imaginary (see \[16\]=Weyl 1913/65 [@Weyl_1913], 3.Auflage).—The first result beyond these facts was obtained by Teichmüller in \[15\](=1944 [@Teichmueller_1944-Beweis-der-analytischen-Abhaengigkeit]), where he applied his theory of spaces of Riemann surfaces to the embedding problem. He could show that not all compact embedded surfaces of genus $1$ are conformally equivalent to ring surfaces. More important than this result was the method by which he obtained it: He deformed an embedded surface by moving each point along the normal line and studied the dependence of the modulus of the deformed surface on the deformation.—Around 1960 Garsia constructed a surprisingly large class of compact Riemann surfaces whose moduli could be determined (\[5\](=1960 [@Garsia_1960-Pacific]),\[6\](=1960 [@Garsia_1960-Rend.])). But he succeeded in answering Klein’s question in the affirmative for all compact Riemann surfaces only when he abandoned his beautiful models and embarked on Teichmüller’s road. His proof in \[7\](=Garsia-Rodemich 1961 [@Garsia-Rodemich_1961]) and \[8\](=Garsia 1961 [@Garsia_1961]) is an ingenious combination of Teichmüller’s ideas and results, constructions inspired by Nash’ isometric embeddings, and Brouwer’s fixed point theorem.—We will see in this paper that his methods are even strong enough to prove this theorem for noncompact surfaces too. \[…\], we may formulate our theorem as follows:—[Embedding theorem.]{} [*Every Riemann surface $R$ is conformally equivalent to a complete classical surface. A model can be constructed by deforming any topologically equivalent complete classical surface $X$ in the direction of the normals. [$X$ is complete, if $X$ is a closed subset of Euclidean space.]{}*]{}—A nontrivial corollary (due to R. Osserman) follows, if $R$ is the unit disc and $X={\Bbb C}$: For a suitable real-valued $C^{\infty}$-function $f$ the classical surface represented by $(x,y)\to(x,y,f(x,y))$, $x+iy\in{\Bbb C}$, is hyperbolic.”\]  R. Rüedy, [*Deformations of embedded Riemann surfaces*]{}, Ann. of Math. Studies 66, 1971. \[$\spadesuit$\]  S. Saitoh, [*The kernel functions of Szegö type on Riemann surface*]{}, Kodai Math. Sem. Rep. 24 (1972), 410–421. \[$\spadesuit$ Bergman kernel on compact bordered Riemann surfaces\]  S. Saitoh, [*The exact Bergman kernel and the kernels of Szegö*]{}, Pacific J. Math. 71 (1977), 545–557. \[$\spadesuit$ Bergman kernel on compact bordered Riemann surfaces\]  S. Saitoh, [*The Bergman norm and the Szegö norm*]{}, Trans. Amer. Math. Soc. 249 (1979), 261–279. \[$\spadesuit$ Bergman kernel on compact bordered Riemann surfaces\]  S. Saitoh, [*A characterization of the adjoint $L$-kernel of Szegö type*]{}, Pacific J. Math. 96 (1981), 489–493. \[$\spadesuit$ compact bordered Riemann surfaces, Green’s function and reproducing kernel\]  S. Saitoh, [*Theory of reproducing kernels and its applications*]{}, Pitman Res. Notes in Math Series 189, 1988. x+157pp. \[$\spadesuit$ reproducing kernel in the abstract united exposition of Aronszajn 1950 [@Aronszajn_1950], followed by a specialization to the case of multiply connected plane domains (esp. Garabedian’s $L$-kernel as the solution to an extremal problem for the Dirichlet integral)\]  S. Saitoh, [*Theory of reproducing kernels; applications to approximate solutions of bounded linear operator equations on Hilbert spaces*]{}, Amer. Math. Soc. Transl., 2010. \[$\spadesuit$ mentions the “Ahlfors function”\]  M. Sakai, [*On constants in extremal problems of analytic functions*]{}, Kodai Math. Sem. Report 21 (1969), 223–225. \[$\spadesuit$ p.223 seems to consider the problem of minimizing the Dirichlet integral $D[f]=\int\int_W df \cdot \overline{df^{\ast}}$ among the analytic functions $f$ on a Riemann surface $W$ normalized by $f(t)=0$ and $f'(t)=1$ (w.r.t. some local uniformizer) \[see also Schiffer-Spencer 1954 [@Schiffer-Spencer_1954]\] $\spadesuit$ alas nothing seems to be asserted about the range of the least area mapping (in particular we still wonder if it is a circle map as looks plausible in view of the simply-connected case treated in Bieberbach 1914 [@Bieberbach_1914])\]  T. Salvemini, [*Sulla rappresentazione conforme delle aree piane pluriconnesse su una superficie di Riemann di genere zero in cui sono siano eseguiti dei tagli paralleli*]{}, Ann. Scuola Norm. Super. Pisa (1) 16 (1930), 1–34. \[$\spadesuit$ just cited to mention that Schottky’s proof of PSM relied on a parameter count not completely justified at his time\]  M.V. Samo\[k\]hin, [*On some questions connected with the problem of existence of automorphic analytic functions with given modulus of boundary values*]{}, Mat. Sb. 111 (1980); English transl.: Math. USSR Sbornik 39 (1981), 501–518. \[$\spadesuit$ p.505 occurrence of the Ahlfors function as an example of non-constant function in $H^{\infty}$ whose Gelfand transform is unity on the Šilov boundary of $H^{\infty}$, p.509: “We used an Ahlfors function to “knock down” the growth of the function…”, p.512: another occurrence of the Ahlfors function\]  M.V. Samokhin, [*Cauchy’s integral formula in domains of arbitrary connectivity*]{}, Sb. Math. 191 (2000), 1215–1231. \[$\spadesuit$ From the Abstract: An example of a simply-connected domain with boundary of infinite length is constructed such that for fairly general functionals on $H^{\infty}$ no extremal function (including the Ahlfors function) can be represented as a Cauchy potential\]  D. Sarason, [*Representing measures for $R(X)$ and their Green’s functions*]{}, J. Funct. Anal. 7 (1971), 359–385. \[$\spadesuit$ some questions asked in the paper are answered in Nash 1974 [@Nash_1974]\]  L. Sario, [*A linear operator method on arbitrary Riemann surfaces*]{}, Trans. Amer. Math. Soc. 72 (1952), 281–295. 50 \[$\spadesuit$ perhaps first a general remark about Sario: to the best of my knowledge none of Sario’s papers (or books) works out a reproof of Ahlfors circle maps, albeit he is often gravitating around closely related or even more grandiose (i.e. foundational) paradigms. Quite ironically, much of the impulse and modernity along the Nevanlinna-Sario tradition finds its starting point in the Schwarz alternating method (which seemed outdated after Hilbert 1900 [@Hilbert_1900] “direct” resolution (=resuscitation) of the Dirichlet principle) $\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited in the bibliography yet apparently not within the text\]  L. Sario, [*Extremal problems and harmonic interpolation on open Riemann surfaces*]{}, Trans. Amer. Math. Soc. 79 (1955), 362–377. 50 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited on p.364 as follows: “Concerning extremal problems for differentials, the reader is referred to the comprehensive study \[1\](=1950 [@Ahlfors_1950]) by Ahlfors.” $\spadesuit$ “The ultimate purpose of the present paper is to study interpolation of harmonic and analytic functions on open Riemann surfaces $W$. We shall, however, first take a less restricted viewpoint and consider, in general, extremal problems on Riemann surfaces.” $\spadesuit$ the bulk of the paper is a reduction of a certain extremal problem over very general open Riemann surfaces to the special case of compact bordered surface (with analytic contours) via the usual exhaustion trick\]  L. Sario, [*Strong and weak boundary components*]{}, J. Anal. Math. 5 (1956/57), 389–398. \[$\spadesuit$ quoted in Reich-Warschawski 1960 [@Reich-Warschawski_1960] for another proof of Grötzsch’s extension to infinite connectivity of the Kreisbogenschlitztheorem\]  L. Sario, K. Oikawa, [*Capacity Functions*]{}, Grundlehren d. math. Wiss. 149, Springer, Berlin, 1969. 47, 50, 78 \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited at three places: pp.46, 110, 175, where this last citation come closest to our interest (but the discussion seems to be confined to plane regions $W$, cf. p.175 (top)). $\spadesuit$ We cite the relevant extract (p.175): “Concerning the quantity $c_B$, Schwarz’s lemma give us the unique function minimizing $M[F]$ if $W$ is simply-connected. The problem has not been solved completely for an arbitrary region $W$. However, Carleson \[1\](=1968 [@Carleson_1967-book]) established the uniqueness of the minimizing function if $c_B>0$. For a regular region $W$ (in which case $c_B>0$), further results have been obtained by Ahlfors \[1\](=1947 [@Ahlfors_1947]), \[2\](=1950 [@Ahlfors_1950]), Garabedian \[1\](=1949 [@Garabedian_1949]), and Nehari \[2\](=1951 [@Nehari_1951-survey-BAMS]), \[3\](=1952 [@Nehari_1952-BOOK]). For the function minimizing $M[F]$, they obtained a characterization which in particular implies that the function maps $W$ onto an $n$-sheeted disk of radius $1/c_B$, where $n$ is the connectivity of $W$; note that this property does not in turn characterize the minimizing function. Garabedian and Nehari further derived a relationship with Szegö’s kernel function (Szegö \[1\](=1921 [@Szego_1921]), Schiffer \[5\](=1950 [@Schiffer_1950-Duke])). However, we shall not go into a more detailed discussion of these interesting results.”\]  L. Sario, M. Nakai, [*Classification Theory of Riemann Surfaces*]{}, Grundlehren d. math. Wiss. 164, Springer, Berlin, 1970, 446 pp. 47, 50, 78 \[$\spadesuit$ cite the work Ahlfors 1950 [@Ahlfors_1950] in the bibliography (p.412), but not in the main body of the text (sauf erreur!) $\spadesuit$ p.452, the article by Kusunoki 1952 [@Kusunoki_1952] (where the Ahlfors map of a bordered surface is applied to the so-called “type problem”) is cited (and as far as I know this is the [*unique*]{} citation of Kusunoki’s work throughout the world literature). Alas, Kusunoki’s work does not seem to be quoted inside the main body of the text. $\spadesuit$ p.332: “The concept of harmonic measure was introduced by Schwarz \[1\](=Ges. math. Abh. 1890) and effectively used by Beurling \[1\](=1935 [@Beurling_1935-These]). Nevanlinna \[1\](=1934 [@Nevanlinna_1934]) coined the phrase “harmonic measure” and introduced the class of “nullbounded” surfaces characterized by the vanishing of the harmonic measure. That this class coincides with the class $O_G$ of “parabolic” surfaces was shown by Myrberg \[2\](=1933 [@Myrberg_1933]) for surfaces of finite genus.”\]  S. Scheinberg, [*Hardy spaces and boundary problems in one complex variables*]{}, Ph.D. Thesis, Princeton University, 1963. \[$\spadesuit$ includes a proof of the corona theorem on annuli, cf. also Stout 1965 [@Stout_1965]\]$\bigstar$  L. Schläfli, [*On the distribution of surfaces of the third order into species, in reference to the absence or presence of singular points, and the reality of their lines*]{}, Philos. Trans. Roy. Soc. London 153 (1863), 195–241. \[$\spadesuit$ the heaviest brain ever met ca. 1.9 kg?\]  E. Schmidt, [*Entwicklung willkürlicher Funktionen nach Systemen vorgeschriebener ???*]{}, Math. Ann. 63 (1907), 433–476. \[$\spadesuit$ quoted in Nehari 1955 [@Nehari_1955]\]  E. Schmidt, [*Zur Theorie der linearen und nichtlinearen Integralgleichungen*]{}, Math. Ann. 64 (1907), 161–174. \[$\spadesuit$ quoted in Bergman 1950 [@Bergman_1950]\]  M. Schiffer, [*Sur les domaines minima dans la théorie des transformations pseudo-conformes*]{}, C.R. Acad. Sci. Paris 207 (1938), 112–115. \[$\spadesuit$ quoted in Maschler 1956 [@Maschler_1956 p.506] for the issue that minimal domains satisfy the mean value property; thus perhaps if ranges of least area maps are minimal domains we may hope that by virtue of a theorem of XXX-Schiffer (cited in the introd. of Aharonov-Shapiro 1976 [@Aharonov-Shapiro_1976]) the least area map is a circle map \[02.08.12\] $\spadesuit$ also quoted in Bergman 1947 [@Bergman_1947 p.32] for the issue that for a proof of the partial result that for starlike domains the least area map effects the Riemann mapping upon the circle\]$\bigstar$$\bigstar$$\bigstar$  M. Schiffer, [*Sur un théorème de la représentation conforme*]{}, C.R. Acad. Sci. Paris 207 (1938), 520–522. 60, 78 \[$\spadesuit$ located via Reich-Warschawski 1960 [@Reich-Warschawski_1960], who cite the paper for another proof of Grötzsch’s extension 1929–1931 [@Groetzsch_1931] to infinite connectivity of the Kreisschlitzbereich mapping of Koebe 1918 [@Koebe_1918] $\spadesuit$ contains indeed a proof based upon an extremal problem of the circular slit map, yet the argument seems to depend upon the longer paper Schiffer 1937/38 [@Schiffer_1937/38]\]  M. Schiffer, [*A method of variation within the family of simple functions*]{}, Proc. London Math. Soc. (2) 44 (1937/38), 432–449. 60, 78 \[$\spadesuit$ principle of areas (Flächensatz) of Bieberbach-Faber $\spadesuit$ quotes Grötzsch 1930 and extends a result of Marty 1934\]  M. Schiffer, [*The span of multiply connected domains*]{}, Duke Math. J. 10 (1943), 209–216. 60, 78 $\bigstar$  M. Schiffer, [*The kernel function of an orthonormal system*]{}, Duke Math. J. 13 (1946), 529–540. 78 $\bigstar$ $\bigstar$$\bigstar$ \[$\spadesuit$ establish for domains an identity relating the Bergman kernel to the Green’s function\]  M. Schiffer, [*An application of orthonormal functions in the theory of conformal mapping*]{}, Amer. J. Math. 70 (1948), 147–156. 60, 78 \[$\spadesuit$ new derivation via the Bergman kernel of inequalities of Grunsky’s Thesis 1932, which were previously derived by variational methods\]  M. Schiffer, [*Various types of orthogonalization*]{}, Duke Math. J. 17 (1950), 329–366. $\bigstar$$\bigstar$$\bigstar$  M. Schiffer, [*Some recent developments in the theory of conformal mapping*]{}, Appendix to R. Courant, 1950 [@Courant_1950], 249–324. \[$\spadesuit$ an extremely readable survey of several trends in potential theory, including the Green-Dirichlet yoga, the kernel method and some of the allied extremal problems, plus the method of extremal length and schlicht functions\]  M. Schiffer, [*Variational methods in the theory of conformal mapping*]{}, Proc. Internat. Congr. Math., Cambridge, Mass., 1950, (1952), 233–240. 78 \[$\spadesuit$ survey of variational methods\]  M. Schiffer, D.C. Spencer, [*Functionals of Finite Riemann Surfaces*]{}, Princeton Mathematical Series, Princeton University Press, 1954. \[$\clubsuit$ \[noticed the 26.07.12\] on p.135 the authors consider the problem of the least-area map (normed at a point $q$)for a compact bordered Riemann surface $\clubsuit$ it would be extremely desirable to know if the extremal map is a circle map, and if it relates to the Ahlfors function described in Ahlfors 1950 [@Ahlfors_1950]\] M. Schiffer, [*Extremum problems and variational methods in conformal mapping*]{}, Proc. Internat. Congr. Math., Stockholm, 1958, 211–231. 78 \[$\spadesuit$ p.229 suggest a new proof (via Fredholm) of Schottky’s famous circular mapping (i.e. Kreisnormierung): details to be found in the next voluminous paper\]  M. Schiffer, [*Fredholm eigenvalues of multiply connected domains*]{}, Pacific J. Math. 9 (1959), 211–269. 78 \[$\spadesuit$ includes a new proof (via an extremum problem involving the Fredholm determinant) of the Schottky-Koebe Kreisnormierung; for yet another proof cf. the next item [@Schiffer-Hawley_1962]\]  M. Schiffer, N.S. Hawley, [*Connections and conformal mapping*]{}, Acta Math. 107 (1962), 175–274. 78 \[$\spadesuit$ p.183–189 includes yet another proof of the Schottky-Koebe Kreisnormierung (finite-connectivity) via an extremum problem of the Dirichlet type\]  M. Schiffer, [*Fredholm eigenvalues and conformal mapping*]{}, Rend. Mat. e Appl. (5) 22 (1963), 447–468. 78 \[$\spadesuit$ which mappings? the method must be the same as the previous item\]  M. Schiffer, G. Springer, [*Fredholm eigenvalues and conformal mapping of multiply connected domains*]{}, J. Anal. Math. 14 (1965), 337–378. 78  M. Schiffer, [*Half-order differentials on Riemann surfaces*]{}, J. SIAM Appl. Math. 14 (1966), 922–934. 60, 78 \[$\spadesuit$ summary of research joint with Hawley, $\spadesuit$ immediate generalization for the Bergman kernel for any closed Riemann surface to be found in Schiffer-Spencer 1954 [@Schiffer-Spencer_1954] $\spadesuit$ contour integration introduced by Riemann himself\]  M. Schiffman, [*The Plateau problem for non-relative minima*]{}, Ann. of Math. (2) 40 (1939), 834–854. \[$\spadesuit$ Seidel’s summary: the problem of mapping a region bounded by a simple closed curve with a continuously turning tangent is reduced to that of minimizing a functional, somewhat similar to that of Douglas (cf. Douglas 1931 [@Douglas_1931-Solution]). This functional has an electrostatic interpretation which may provide an effective mechanical method for the determination of conformal maps\]  M. Schiffman, [*Uniqueness theorems for conformal mapping of multiply connected domain*]{}, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 137–139. 78 \[$\spadesuit$ quoted in Bergman 1950 [@Bergman_1950]\]  P. Schmutz, [*Riemann surfaces with shortest geodesics of maximal length*]{}, Geom. Funct. Anal. 3 (1993), 564–631. \[$\spadesuit$\]  P. Schmutz-Schaller, [*Geometry of Riemann surfaces based on closed geodesics* ]{}, Bull. Amer. Math. Soc. 35 (1998), 193–214. \[$\spadesuit$... Extremal problems have been considered in similar contexts; see in particular Bollobás \[9\] for extremal graphs and Ahlfors \[3\] for extremal problems in conformal geometry. ...\]  A. Schönflies, [*Über gewisse geradlinig begrenzte Stücke Riemann’scher Flächen*]{}, Nachr. Akad. Wiss. Göttingen (1892), 257–267. $\bigstar$ \[$\spadesuit$ detected via AS60.\]  E. Scholz, [*Geschichte des Mannigfaltigkeitsbegriff von Riemann bis Poincaré*]{}. Birkäuser, 1980. \[$\spadesuit$\]  E. Scholz, [*The concept of manifold, 1850–1950.*]{} Chapter 2, in: History of Topology, 25–64. Elsevier, 1999 \[$\spadesuit$ p.26: “Also leading mathematicians like Cauchy and Gauss started to use geometrizing language in ${\Bbb R}^n$ in publications (Cauchy, 1847) or lecture courses (Gau[ß]{}, 1851/1917). Gauss, in his lecture courses, even used the vocabulary of [*$(n-k)$-dimensional manifolds (Mannigfaltigkeiten)*]{}, but still restricted in his context to affine subspaces of the $n$-dimensional real space (Gau[ß]{}, 1851/1917, pp.477ff.). There is no reason to doubt that Riemann got at least some vague suggestion of how to generalize the basic conceptual frame for geometry along these lines from Gauss and developed it in a highly independent way.” $\spadesuit$ p.36: “In geometric function theory divers authors contributed to a refined understanding of the rôle of topological concepts, in particular C. Neumann with his calculation of the connectivity of a Riemann surface from the winding orders of branch points \[Neumann 1865 [@Neumann_1865-Vorlesungen]\], Lüroth, Clebsch and Clifford with their normalized representation during the 1870-s for branched coverings of $P_1({\Bbb C})$, which represent a Riemann surface with given number of leaves, given loci and winding numbers of branch points.”\]  F. Schottky, [*Ueber die conforme Abbildung mehrfach zusammenhängender ebener Flächen*]{}, (Diss. Berlin 1875) Crelle J. für die Math. 83 (1877), 300–351. 60, 78 \[$\clubsuit$ after Riemann 1857/58 [@Riemann_1857_Nachlass], the first existence proof of the Ahlfors map in the planar case $\spadesuit$ contains in germ all type of mapping like the Circle mapping, the Kreisnormierung plus the parallel-slit maps $\spadesuit$ the only drawback is a certain confinement to planar regions, but this will be quickly relaxed in Klein 1882 [@Klein_1882] $\spadesuit$ regarding the rigor of proofs, the appreciation are rather random, compare Cecioni 1908 [@Cecioni_1908] and Grunsky 1978 who ascribes the first rigorous proof of the PSM to Cecioni\]  F. Schottky, [*Ueber eindeutige Functionen mit linearen Transformationen in sich. (Auszug aus einem Schreiben an Herrn F. Klein.)*]{} Math. Ann. 19 (1882), ?–?.  F. Schottky, [*Ueber eindeutige Functionen mit linearen Transformationen in sich,*]{} Math. Ann. 20 (1882), 293–300. 60 $\bigstar$  F. Schottky, [*Zur Frage: Haben die Klassenfunktionen Differentialgleichungen,*]{} Sitz.-Ber. Peu[ß]{}. Akad. Wiss., math.-phys. Kl. (1922), 414–423. \[$\spadesuit$ cited in Grunsky 1978 [@Grunsky_1978 p.197] as follows: “Considering meromorphic functions on $D$ with real boundary values (which he later called “Klassenfunktionen”, \[476\](=this entry); now they are called Schottky functions) and proving the existence of a real algebraic relation between any two of them, he disclosed an intimate relation between problems in multiply connected domains and the theory of algebraic functions. The most concise expression of this relation is the idea of the “Schottky double” of a multiply connected domain (or of any finite Riemann surface) with analytic boundary; this is a compact Riemann surface, gained by identifying boundary points of two replicas …”\] $\bigstar$  O. Schramm, [*Conformal uniformization and packings*]{}, Israel J. Math. 93 (1996), 399–428. \[$\spadesuit$ new proof of the Brandt-Harrington (1980 [@Brandt_1980] and 1982 [@Harrington_1982]) generalization of Koebe’s KNP via a topological method (mapping degree), plus the PSM (parallel slit maps) and some other gadgets\]  K. Schüffler, [*Zur Fredholmtheorie des Riemann-Hilbert-Operator*]{}, Arch. Math. 47 (1986), 359–366. \[$\spadesuit$ p.359: “Ausgehend von dem bekannten klassischen Riemann-Hilbert Randwertproblem \[8,S.181ff\](=Vekua 1963 [@Vekua_1963]) betrachten wir den Operator $RH\colon A^m(\Omega)\to H^{m-1/2}(\partial \Omega, {\Bbb R})$, $RH(f):={\rm Re}(\bar \lambda f)\vert_{\partial \Omega}$. \[…\] das Symbol “$A^m$” bezeichne den Sobolevraum $H^{m,2}$ der auf $\Omega$ holomorphen Funktionen, $m\ge 2$; die komplexwertige Funktion $\lambda$ sei nullstellenfrei (auf $\partial \Omega$) und o.E. glatt.—Es ist bekannt, da[ß]{} der Operator $RH$ für glattberandete, endliche Riemannsche Flächen ein Fredholmoperator ist. Sien Index hängt sowohl von der Topologie von $\Omega$ (Anzahl der Randkomponenten und Geschlecht) als vom “geometrischen Index, dem Argumentzuwachs $\kappa(\lambda)=\Delta \arg (\lambda)/ 2\pi \in {\Bbb Z}$ von $\lambda$ beim positiven Durchlaufen von $\partial \Omega$ ab (siehe \[8, S.189\]=Vekua 1963 [@Vekua_1963])” $\spadesuit$ \[17.10.12\] this seems connected to the Ahlfors map, by taking $\lambda$ its boundary restriction\]  H.A. Schwarz, [*Ueber einige Abbildungsaufgaben*]{}, Crelle J. für die Math. 70 (1869), 105–120. \[$\spadesuit$ introduces the principle of symmetry $\clubsuit$ solves special case of the RMT by hand\]  H.A. Schwarz, [*Zur Theorie der Abbildung*]{}, Züricher Vierteljahrsschrift (1869/70); also (theilweise umgearbeitet ca. 1890) in Ges. Abh. II, 108–132.  H.A. Schwarz, [*Ueber die Integration der partiellen Differentialgleichung $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$ für die Fläche eines Kreises*]{}, Züricher Vierteljahrsschrift (1870), 113–128; reprinted (or rather integrated) in the longer paper Schwarz 1872 [@Schwarz_1872]. \[$\spadesuit$ this entry is the first rigorous solution to the Dirichlet problem to have appeared in print (for the very special case of the disc) and via usage of the Poisson integral (occurring in several publications dated 1820–23–27–29–31–35) $\spadesuit$ see also Prym 1871 [@Prym_1871] for an essentially simultaneous resolution (which however turned out to have less impact on the future events)\]  H.A. Schwarz, [*Ueber einen Grenzübergang durch alternirendes Verfahren*]{}, Züricher Vierteljahrsschrift (1870), 272–286; also in Ges. Abh. II, 133–143.  H.A. Schwarz, [*Ueber die Integration der partiellen Differentialgleichung $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$ unter vorgeschriebenen Grenz- und Unstetigkeitsbedingungen*]{}, Berliner Monatsb. (1870), 767–795; or Ges. Abh. Bd. II, 144–171 \[$\spadesuit$ p.167–170 uniqueness of the conformal structure on the 2-sphere\]  H.A. Schwarz, [*Zur Integration der partiellen Differentialgleichung $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$*]{}, Crelle J. für die Math. 74 (1872), 218–253; or Ges. Abh. II, 175–210. A. Sebbar, Th. Falliero, [*Equilibrium points of Green’s function for the annulus and Eisenstein series*]{}, Proc. Amer. Math. Soc. 135 (2007), 313–328. \[$\spadesuit$ p.314: “By the classical Hopf’s lemma, the normal derivative of the Green’s function is positive on the boundary \[of a multi-connected domain\], and one may ask if there is a compact set \[in the domain\], independent of the pole, containing all the equilibrium points of the Green’s function.” $\spadesuit$ a positive answer to this problem is supplied by Solynin 2007 [@Solynin_2007]\]  B. Segre, [*Sui moduli delle curve poligonali, e sopra un complemento al teorema di esistenza di Riemann*]{}, Math. Ann. 100 (1928), 537–551.  W. Seidel, [*Bibliography of numerical methods in conformal mapping*]{}. In: [*Construction and Applications of Conformal Maps*]{}, Proc. of a Sympos. held on June 22–25 1949, Applied Math. Series [*18*]{}, 1952, 269–280. \[$\spadesuit$ a useful compilation of (old) conformal maps literature emphasizing the numerical methods, and out of which we borrowed several summaries\]  H.L. Selberg, [*Ein Satz über beschränkte endlichvieldeutige analytische Funktionen*]{}, Comment. Math. Helv. 9 (1937), 104–108. \[$\spadesuit$ quoted in Hayashi-Nakai 1988\]$\bigstar$$\bigstar$$\bigstar$\[CHECK\]  M. Seppälä, [*Teichmüller spaces of Klein surfaces*]{}, Ann. Acad. Sci. Fenn. Ser.AI Math. Dissertationes 15 (1978), 1–37. \[$\spadesuit$\]  M. Seppälä, [*Quotient of complex manifolds and moduli spaces of Klein surfaces*]{}, ?? (198?), ?–?. \[$\spadesuit$\]  M. Seppälä, R. Silhol, [*Moduli spaces for real algebraic curves and real abelian varieties*]{}, Math. Z. 201 (1989), 151–165. \[$\spadesuit$ modernization of Klein’s resp. Comessatti’s theories\]  M. Seppälä, [*Real algebraic curves in the moduli spaces of complex curves*]{}, Compos. Math. 74 (1990), 259–283. \[$\spadesuit$\]  M. Seppälä, [*Moduli spaces of stable real algebraic curves*]{}, Ann. Sci. Éc. Norm. Sup. 24 (1991), 519–544. \[$\spadesuit$\]  M. Seppälä, [*Computation of period matrices of real algebraic curves*]{}, Discr. Comput. Geom. (1994). \[$\spadesuit$ Abstract. In this paper we derive a numerical method which allows us to compute periods of differentials on a real algebraic curve with real points. This leads to an algorithm which can be implemented on a computer and can be used to study the Torelli mapping numerically.\]  F. Severi, [*Vorlesungen über algebraische Geometrie*]{}, Leipzig, Teubner, 1921. \[$\spadesuit$ p.159 re-proves the upper bound for the gonality of a complex curve (according to Segre 1928 [@Segre_1928]), but for the “modern standards” the first accepted proof is that of Meis 1960 [@Meis_1960] $\spadesuit$ contains a brief discussion of Klein’s theory of real algebraic curves $\spadesuit$ Anhang F also contains the complex case of Brusotti’s theorem (1921 [@Brusotti_1921]) on the independence of smoothing nodal curves $\spadesuit$ the same ideas where used in Harris’ proof on the irreducibility of the variety of plane curves of fixed degree and prescribed genus\]  F. Severi, [*Sul teorema di esistenza di Riemann*]{}, Rend. Circ. Mat. Palermo 46 (1922), 105–116.  G.B. Shabat, V.A. Voevodsky, [*Equilateral triangulations of Riemann surfaces and curves over algebraic number fields*]{}, Doklady SSSR 304 (1989), 265–268; Soviet Math. Dokl. 39 (1989), 38–41. \[$\spadesuit$ geometric translation of Belyi-Grothendieck’s theorem that a curve is defined over $\Qbar$ iff it ramifies only over 3 points of the sphere. Question: can one extend this to Ahlfors maps in the bordered case cf. Sec.\[sec:Belyi-Grothendieck\] for a pessimist answer, yet probably all real curves are to be integrated. So what about real Riemann surfaces with an equilateral triangulation invariant under complex conjugation. So the vertices occurs as ${\Bbb Q}$-ratioanl points? etc.\]$\bigstar$  G.B. Shabat, V.A. Voevodsky, [*Drawing curves over number fields*]{}, in: Grothendieck Festschrift, Birkhäuser.  I.R. Shafarevich, [*Basic Algebraic Geometry*]{}, NAuka, Moscow, 1972; English. transl., Die Grundlehren der math. Wiss. in Einzeldarstellungen, Bd.213, Springer-Verlag, Berlin, 1974. (many subsequent reeditions) \[$\spadesuit$ contain a lovely picture of a real elliptic curve (with two real circuits) acted upon by complex conjugation (I confess that this little picture is actually, besides some theory told by Felice Ronga and Daniel Coray, the very origin of my modest involvement with the topic of real algebraic curves)\]  C.S. Sheshadri, [*Space of unitary vector bundles on a compact Riemann surface*]{}, Ann. of Math. 85 (1967). \[$\spadesuit$\]  M. Shiba, K. Shibata, [*Singular hydrodynamical continuations of finite Riemann surfaces*]{}, Kyoto J. Math. (1985). \[$\spadesuit$ The present study arose, in close relationships to a series of our investigations \[16\],\[17\] and \[18\], from an attempt to embed an arbitrary open Riemann surface of finite genus into another closed Riemann surface of the same genus, so that the prolongation of the ...\]  G.E. Shilov, [*On rings of functions with uniform convergence*]{}, Ukrain. Mat. Z. 3 (1951), 404–411. \[$\spadesuit$\]$\bigstar$  V.V. Shokurov, [*The Noether-Enriques theorem on canonical curves*]{}, Mat. Sb. Nov. Ser. 86 (1971), 367–408; English transl., Math. USSR Sb. 15 (1972), 361–401. \[$\spadesuit$\]$\bigstar$  E.I. Shustin, [*The Hilbert-Rohn method and bifurcation of complicated singular points of curves of degree $8$*]{}, Uspekhi Mat. Nauk 38 (1983), 157–158; English transl., ??. \[$\spadesuit$\]$\bigstar$  E.I. Shustin, [*Gluing of singular algebraic curves*]{}, in: Methods of qualitative theory. Gorky Univ. Press, Gorky, 1985, 116–128 (Russian). \[$\spadesuit$\]$\bigstar$  E.I. Shustin, [*Independent removal of singular points and new $M$-curves of degree $8$*]{}, Uspekhi Mat. Nauk 40 (1985), ?–?; English transl., ??. \[$\spadesuit$\]$\bigstar$  E.I. Shustin, [*The Hilbert-Rohn method and smoothing of singular points of real algebraic curves*]{}, Dokl. Akad. Nauk SSSR 281 (1985), 33–36; English transl., Soviet Math. Doklady 31 (1985), 282–286. \[$\spadesuit$\]$\bigstar$  E.I. Shustin, [*Counterexamples to a conjecture of Rokhlin*]{}, Funkt. Anal. Prilozhen 19 (1985), 94–95; English transl., Funct. Anal. Appl. 19 (1985), 162–163. \[$\spadesuit$ counterexamples in degree 8 to Rohlin’s conjecture (type I iff maximal), based on earlier work by Polotovskii [@Polotovskii_1981] compare the discussion in Viro 1986/86 [@Viro_1986/86-Progress] $\spadesuit$ \[25.01.13\] this note also implies a counterexample to an Ansatz of Klein 1876 [@Klein_1876], to the effect that nondividing curves could always win a supplementary oval by crossing a solitary node. In fact Shustin’s note uses predominantly a Bézout-like obstruction for $M$-octics due to Viro 1983 [@Viro_1983/84-new-prohibitions] extending the one of Fiedler 1982/83 [@Fiedler_1982/83-Pencil]. In Shustin’s disproof, the counterexample is an $(M-2)$-curve, which (either itself or more likely one of its $(M-1)$-enlargements) is maximal but of type II. $\spadesuit$ \[25.01.13\] I do not know if Klein’s Ansatz has some chance to be true in degree $7$. $\spadesuit$ nor do I know if it could old for $(M-2)$-curves, or more generally all curves except possibly $(M-1)$-curves. Compare Orevkov’s remark in Sec.\[Orevkov:sec\], which seems to prompt that there is some open problem here.\]  E.I. Shustin, [*A new $M$-curve of degree $8$*]{}, Mat. Zametki 42 (1987), 180–186; English transl., Math. Notes 42 (1987), 606–610. \[$\spadesuit$ this is perhaps the paper to which Orevkov is referring to in Sec.\[Orevkov:sec\]\]  E.I. Shustin, [*Versal deformations in the space of a fixed degree curves*]{}, Funct. Anal. Appl. 21 (1987), 90–91. \[$\spadesuit$\]  E.I. Shustin, [*Smoothness and irreducibility of varieties of singular algebraic curves*]{}, in: Arithmetic and geometry of algebraic varieties. Saratov Univ. Press/Kuibyshev branch, Kuibyshev, 1989, 102–117. \[$\spadesuit$\]  E.I. Shustin, [*Geometry of discriminant and topology of algebraic curves*]{}, in: Proc. Internat. Congr. Math., Kyoto, Japan, 1990, Math. Soc. Japan. (1991), 559–567. \[$\spadesuit$ p.566: “\[…\] the complete description of discriminant in the space of plane real quartics curves and complete classification of inflexion point arrangements on these curves \[9\](=Gudkov 1988 [@Gudkov_1988-quartic]).” $\spadesuit$ p.566: “It should also be noted that there is $M$-curve of degree $8$, whose constructions does not satisfy conditions of Viro method and is based on Theorem 4 \[22\](=Shustin 1987 [@Shustin_1987/87-a-new-M-curve-of-deg-8]).”\]  E.I. Shustin, [*On manifolds of singular algebraic curves*]{}, Selecta Math. Soviet. 10 (1991), 27–37; this is the English transl., of the Russian original dating back to 1983. \[$\spadesuit$\]  R.J. Sibner, [*Uniformization of symmetric Riemann surfaces by Schottky groups*]{}, (Diss.) Trans. Amer. Math. Soc. 116 (1965), 79–85. 78 \[$\clubsuit$ new proofs of the Rückkehrschnitttheorem (retrosection theorem) and the Kreisnormierung=KNP via quasiconformal mappings techniques (Ahlfors-Bers)=Teichmüller modernized; as oft emphasized in our text (cf. Sec.\[sec:question\]) thiis might be the route through which one can hope to reprove the Ahlfors mapping via the original method of Klein (as cryptically asserted in Teichmüller 1941 [@Teichmueller_1941])\]  R.J. Sibner, [*Symmetric Fuchsian groups*]{}, Amer. J. Math. 90 (1968), 1237–1259. \[$\spadesuit$\]  R.J. Sibner, [*Remarks on the Koebe Kreisnormierungsproblem*]{}, Comment. Math. Helv. 43 (1968), 289–295. 78 \[$\clubsuit$ quasiconformal reduction of KNP: can every plane domain be deformed quasiconformally onto a circle domain? (still open today June 2012)\]  R.J. Sibner, [*An elementary proof of a theorem concerning infinitely connected domains*]{}, Proc. Amer. Math. Soc. 37 (1973), 459–461. 78 \[$\spadesuit$ simplifies by circumventing the usage of quasi-conformal techniques (normal family proof instead) an earlier proof of the fact that any domain of infinite connectivity admits a conformally equivalent model bounded by analytic contours (Jordan curves) $\spadesuit$ as probably just a matter of nomenclature it is not perfectly clear (to the writer) if this is obtained for all domains (as stated e.g. in Grunsky’s review (1978) [@Grunsky_1978 p.196] of this work) or if the assertion is only established in the case of countably many boundary components (cf. the parenthetical proviso on p.459 of [*opera cit.*]{}) $\spadesuit$ of course the real dream of Koebe (Kreisnormierung) would be that all these Jordan contours are ultimately circles!\]  , Ueber eine neue analytische Behandlungsweise der Brennpunkte. [*J. Reine Angew. Math. 64*]{} (1865), 175–182. \[$\spadesuit$\]  L. Siebenmann, *The Osgood-Schoenflies theorem revisited*, Russian Math. Surveys [ 60]{} (2005), 645–672. See also the online version available in the Hopf archive: http://hopf.math.purdue.edu/cgi-bin/generate?/Siebenmann/Schoen-02Sept2005 (from which a number of the editors misprints have been removed.)\[$\spadesuit$ contains a brilliant historical discussion of the contribution due to the complex analytic community (Osgood, Carathéodory) upon the so-called Schoenflies theorem about the bounding disc property of plane Jordan curves\]  C.L. Siegel, *Topics in Complex Function Theory*, Vols.I–III. John Wiley and Sons, Inc., New York, 1960, 1971, 1973. \[$\spadesuit$\]  J.-C. Sikorav, [*Proof that every torus with one hole can be properly holomorphically embedded in ${\Bbb C}^2$*]{}, preprint, October 1997 (unpublished). \[$\spadesuit$ self-explanatory title, and see Černe-Forstnerič for an extension of Sikorav’s result\]$\bigstar$  R. Silhol, [*Real algebraic surfaces*]{}, Lecture Notes in Math. 1392, Springer-Verlag, 1989. \[$\spadesuit$\]  R. Silhol, [*Compactifications of moduli spaces in real algebraic geometry*]{}, Invent. Math. (1992). \[$\spadesuit$ It is probably useful to begin this paper by explaining why an approach, specific to real algebraic geometry, is necessary for Moduli problems. We will only be concerned in this paper, with the moduli problems for curves and abelian varieties (but the remarks we ...\]$\bigstar$  R. Silhol, [*The Schottky problem for real genus 3 $M$-curves*]{}, Math. Z. 236 (2001), 841–881. \[$\spadesuit$\]$\bigstar$  R.R. Simha, [*The Carathéodory metric of the annulus*]{}, Proc. Amer. Math. Soc. 50 (1975), 162–166. 50, 78 \[$\spadesuit$ write down everything (Ahlfors function, Carathéodory metric) in the case of an annulus\]  S.O. Sinanjan, [*Approximation by polynomials in the mean with respect to area*]{}, Mat. Sbornik 82 (1970); English transl.: Math USSR Sbornik 11 (1970), 411–421. \[$\spadesuit$ p.416: “Let $\phi(z)$ be an Ahlfors $p$-function of the set $E$: $\gamma_p(E,\phi)=\gamma_p(E)$, $\phi \in A^p_E$. Such a function exists due to the compactness of the set $A^p_E$.” $\spadesuit$ p.420 one further occurrence of the Ahlfors function\]  D. Singerman, [*Automorphisms of compact non-orientable Riemann surfaces*]{}, Proc. London Math. Soc. (3) 10 (1969), 376–394. 78 \[$\spadesuit$ “Using the definition of a Riemann surface, as given for example by Ahlfors-Sario, one can prove that all Riemann surfaces are orientable. However by modifying their definition one can obtain structures on non-orientable surfaces. In fact non-orientable Riemann surfaces have been considered by Klein and Teichmüller amongst others. The problem we consider here is to look for the largest possible groups of automorphisms of compact non-orientable Riemann surfaces and we find that this throws light on the corresponding problem for orientable Riemann surfaces, which was first considered by Hurwitz \[1\](=1893 [@Hurwitz_1893-U-algebr-Gebilde-m-eind-Transf-in-sich]). He showed that the order of a group of automorphisms of compact orientable Riemann surface of genus $g$ cannot be bigger than $84(g-1)$. This bound he knew to be attained because Klein had exhibited a surface of genus $3$ which admitted $PSL(2,7)$ as its automorphism group, and the order of $PSL(2,7)$ is $168=84(3-1)$. More recently Macbeath \[5,3\](=1967 [@Macbeath_1967],=1961 [@Macbeath_1961]) and Lehner and Newman \[2\](=1967 [@Lehner-Newman_1967]) have found infinite families of compact orientable surfaces for which the Hurwitz bound is attained, and in this paper we shall exhibit some new families.”\]$\bigstar$$\bigstar$$\bigstar$  D. Singerman, [*Mirrors on Riemann surfaces*]{}, Contemp. Math. 184 (1995), 411–417. \[$\spadesuit$\]$\bigstar$  V. Singh, [*An integral equation associated with the Szegö kernel function*]{}, Proc. London Math. Soc. (3) 10 (1960), 376–394. 78 \[$\spadesuit$\]$\bigstar$$\bigstar$$\bigstar$  E.P. Smith, [*The Garabedian function of an arbitrary compact set*]{}, Pacific J. Math. 51 (1974), 289–300. 78 \[$\spadesuit$ Ahlfors function is mentioned in its usual connection with the analytic capacity (p.289, 290) $\spadesuit$ Gamelin’s summary in 1973 [@Gamelin_1973-BAMS]: “Recently E. Smith \[43\](=the present entry Smith 1974 [@Smith_1974]) settled a problem left open by S.Ya. Havinson \[26\](=Havinson 1961/64 [@Havinson_1961/64]), proving that if domains $D_n$ with analytic boundaries increase to an arbitrary domain $D$, then the Garabedian functions of the $D_n$ converge normally. The limiting function depends only on $D$ and on the point $z_0$, and it is accordingly called the Garabedian function of $D$. In order to study the subadditivity problem for analytic capacity, Smith had been led to investigate the dependence of the Szegö kernel on certain perturbations of domains with analytic boundaries. The result on the Garabedian function dropped out as a special dividend. There is now the problem of simplifying Smith’s proof, and of freeing the result from Hilbert space considerations, in order to extend the theorem to more general extremal problems. [Added in proof]{}. A simple proof, which still depends on Hilbert space considerations, has been given by N. Suita 1973 (=[@Suita_1973])”\]  A.Yu. Solynin, [*A note on equilibrium points of Green function*]{}, Proc. Amer. Math. Soc. 136 (2008), 1019–1021. \[$\spadesuit$ given a finitely connected planar domain, it is shown that there is a “universal” compactum (inside the domain) containing all critical points of the Green’s functions $G(z,t)$ whatever the location of the pole $t$ is (answering thereby a question of Sebbar-Falliero 2007 [@Sebbar-Falliero_2007]) $\clubsuit$ question (Gabard \[11.08.12\]): does this fantastic result extends to (compact) bordered surfaces $\spadesuit$ further there must be a minimum Solynin’s compactum $K$, what can be said about its shape, area, etc. $\spadesuit$ considering the example of a ring (annulus, say circular to simplify) it seems evident that upon dragging the pole around the hole the unique critical point of Green will rotate (being roughly located at the “antipode”), thus it seems that Solynin’s compactum will be a sub-annulus it this case $\spadesuit$ maybe in general the inclusion of $K$ into the domain is a homotopy equivalence\]  A.J. Sommese, [*Real algebraic spaces*]{}, Ann. Scuola Norm. Sup. Pisa 4 (1977), 599–612. \[$\spadesuit$\]  A. Speiser, [*Über symmetrische analytische Funktionen*]{}, Comment. Math. Helv. 16 (1944), 105–114. 60 \[$\spadesuit$ not symmetric in the “reality” sense of Felix Klein, so a priori no link with Ahlfors 1950 [@Ahlfors_1950]\]  G. Springer, [*Introduction to Riemann surfaces*]{}, Addison-Wesley, Reading, Mass., 1957, 307 pp. \[$\spadesuit$ contains a discussion of the Schottky double, the Prüfer surface, etc.\]  Ch.M. Stanton, [*The closed ideals in a function algebra*]{}, Trans. Amer. Math. Soc. 154 (1971), 289–300. 50 \[$\spadesuit$ a clear-cut application of the Ahlfors function (mapping) is given to a “bordered surface” extension of a “disc result” of Beurling (unpublished)—Rudin (1957) (telling that—in the function algebra $A(W)$ of functions analytic in the interior and continuous up to the boundary—[*every closed ideal is the closure of a principal ideal*]{}) $\spadesuit$ this extension was actually first derived by Voichick 1964 [@Voichick_1964], via the more complicated universal covering whose uniformizing map presents rather complicated boundary behavior $\spadesuit$ p.293, as Royden’s student, the author points out that Ahlfors result is re-proved in Royden 1962 [@Royden_1962] (this is not an isolated attitude, cf. Sec.\[dissident:sec\] for an exhaustive list) $\spadesuit$ p.289, the author remarks that similar use of Ahlfors’ theorem was initiated by Alling 1965 [@Alling_1965] and Stout (in the corona realm) $\spadesuit$ naive question of the writer \[08.08.12\]: is it reasonable to expect that the same Ahlfors-Alling lifting procedure conducts to an extension of Fatou’s theorem about existence of radial limits a.e. from the disc to a bordered surface: the notion of radiality is simple to define (orthogonality to the boundary), yet a function on the bordered surface does not descend to one on the disc via the Ahlfors branched covering (thus rather a method of localization is required, and the problem is surely well treated by several authors (e.g. Heins, Voichick 1964, Gamelin (localization of the corona), etc.) UPDATE \[12.09.12\]: see also Alling 1966 [@Alling_1966 p.345], who claims that Fatou is trivial to extend upon appealing to the Ahlfors map $\spadesuit$ \[08.08.12\] in the same vein it should be noted that the Ahlfors function shows some weakness for instance in the problem of solving the Dirichlet problem which in the disc-case can be cracked via the Poisson formula (H.A. Schwarz’s coinage) and one could hope to lift the solution to the bordered surface via the Ahlfors map. Alas, for given boundary values along the contours of the bordered surface there is no naturally defined procedure to descend the data along the boundary of the disc (implying a failure of the naive lifting trick). Consequently, the Dirichlet problem (for a bordered surface) lies somewhat deeper than the Ahlfors function, since it is probably well-known that the Ahlfors function may be derived from Dirichlet (or its close avatar the Green’s functions), see our Sec.\[Green:sec\] where we shall attempt to redirect to the first-hand sources implementations (Grunsky (planar case), Ahlfors, maybe Cecioni’s students, and Heins 1950 [@Heins_1950]).\]  Ch.M. Stanton, [*Bounded analytic functions on a class of open Riemann surfaces*]{}, Pacific J. Math. 59 (1975), 557–565. \[$\spadesuit$ p.559 uses the terminology Myrberg surface for a concept closely allied to the Ahlfors function in the sense of our circle maps\]  K. Stein, [*Topics on holomorphic correspondences*]{}, Rocky Mountain J. Math. 2 (1972), 443–463. \[$\spadesuit$ Ahlfors 1950 [@Ahlfors_1950] is cited on p.457: “By a theorem of Ahlfors \[1\](=1950 [@Ahlfors_1950]) there is always a meromorphic function $\varphi\colon \widehat{R_0} \to \overline{\Bbb C}$ \[from the double of a bordered surface to the sphere\] such that $R_0=\{ \xi \in \widehat{R_0} : \vert \varphi(\xi)\vert <1$; hence $R_0$ is a distinguished polyhedral domain in $\widehat{R_0}$.”\]  S. Stoïlow, [*Leçons sur les principes topologiques de la théorie des fonctions analytiques*]{}, Gauthier-Villars, Paris 1938. (Second edition in 1956,; Russian translation 1964.) \[$\spadesuit$ includes in particular a notion of “total Riemann covering”, defined by asking that any sequence tending to the boundary has an image tending to the boundary. This topological behaviour subsumes of course those of Ahlfors circle maps. $\spadesuit$ of course Stoïlow’s concept is also implicit in Radó 1922 [@Rado_1922-Z-Theorie-mehr], as one sees e.g. from Landau-Osserman’s account (1960 [@Landau-Osserman_1960]) $\spadesuit$ from Grunsky’s Review (JFM): “in die Definition der Mannigfaltigkeit wird dabei kein Abzählbarkeitsaxiom aufgenommen; es folgt der Beweis des Brouwerschen Satzes von der Invarianz des inneren Punktes nach [*Lebesgue-Sperner*]{}. \[$\dots$\] Zur Verdeutlichung dienen Beispiele nicht orientierbarer Fläche sowie ein von [*Prüfer*]{} stammendes Beispiel einer nicht triangulieren zweidimensionalen Mannigfaltigkeit (in der Formel Zeile 11 v.u. S.72 findet sich ein störender Druckfehler: \[$\dots$\]).” $\spadesuit$ In fact most relevant to our purpose (of the Ahlfors map) is Chap.VI of the book, which Grunsky (loc. cit.) summarizes as follows: “Ferner werden innere Abbildungen einer Riemannschen Fläche $R$ auf eine andere, $S$, betrachtet. Eine solche hei[ß]{}t eine totale Überdeckung von $S$ durch $R$, wenn jede Punktfolge aus $R$, die keine kompakte Teilfolge enthält (die “gegen den Rand strebt”) in eine ebensolche übergeht. Die Überdeckung ist dann auch vollständig, d.h. [*jeder*]{} Punkt von $S$ wird überdeckt, und au[ß]{}erdem auch jeder gleich oft.”\]  S. Stoïlow, [*Sur les surfaces de Riemann normalement exhaustibles et sur le théorème des disques pour ces surfaces*]{}, Compositio Math. 7 (1940), 428–435. \[$\spadesuit$\]  S. Stoïlow, [*Einiges über topologische Funktionentheorie auf nicht orientierbaren Flächen*]{}, Rev. Roumaine Math. Pures Appl. 19 (1974), 503–506. \[$\spadesuit$\]  E.L. Stout, [*Bounded holomorphic functions on finite Riemann surfaces*]{}, Trans. Amer. Math. Soc. 120 (1965), 255–285. 50 \[$\spadesuit$ on p.263 (and 272), Ahlfors 1950 [@Ahlfors_1950] is quoted as follows (without precise bound): “In order to establish our result, we shall need to make use of a result of Ahlfors \[1\](=Ahlfors 1950 [@Ahlfors_1950]). For an alternative proof, one may consult Royden \[15\](=Royden 1962 [@Royden_1962]). Theorem 3.1 [*There exists a function $P$ holomorphic on a neighborhood of $\bar R$ which maps $R$ onto the open unit disc in an one-to-one manner for some $n$ and which satisfies $\vert P \vert =1$ on $\partial R$.*]{}” $\spadesuit$ first it is evident that “one-to-one” is a misprint that should be read as “$n$-to-one” $\spadesuit$ the paper addresses primarily the corona problem (overlapping with Alling 1964 [@Alling_1964]) and the allied interpolation, notably an extension of the celebrated results of Carleson and Newman on interpolation sets for the disc (i.e. those subsets enjoying the property that every bounded complex-valued function on $E$ can be extended to a bounded analytic function on the disc)\]  E.L. Stout, [*On some algebras of analytic functions on finite open Riemann surfaces*]{}, Math. Z. 92 (1966), 366–379; with Corrections in: Math. Z. 95 (1967), 403–404. 50 \[$\spadesuit$ cite Ahlfors 1950 [@Ahlfors_1950] twice, on p.366: “Let $R$ be a finite open Riemann surface whose boundary $\Gamma$ consists of $N$ analytic, pairwise disjoint, simple closed curves. Let $\eta$ be an analytic mapping from $R$ onto $U$, the open unit disc which is holomorphic on a neighborhood of $\overline R$ and which is of modulus one on $\Gamma$. That such functions exists was first established by Ahlfors \[1\](=Ahlfors 1950 [@Ahlfors_1950]); another proof of their existence is in the paper \[12\](=Royden 1962 [@Royden_1962]).” Then on p.375: “Ahlfors \[1\] has shown that if $z_0, z_1$ are distinct points of $R$ (neither in $\Gamma$), then any solution of the extremal problem $\sup\{\vert f(z_0): f \textrm{ in } H_{\infty}[R], f(z_1)=0, \|f \|\le 1\}$ is an inner function in $A[R]$. Thus inner functions separate points on $R$. …” $\spadesuit$ quoted by Fedorov, for using “inner function” as a synonym of “circle map”\]  E.L. Stout, [*Interpolation on finite open Riemann surfaces*]{}, Proc. Amer. Math. Soc. 18 (1967), 274–278. 50 \[$\spadesuit$ p.274, Ahlfors 1950 is quoted as follows: “It is convenient to make use of an [*Ahlfors map*]{} for $R$, i.e., a function continuous on $\overline R$ and holomorphic in $R$ which is constantly of modulus one on $\Gamma$. The existence of such function was established by Ahlfors in \[1\](=Ahlfors 1950 [@Ahlfors_1950]); an alternative proof of their existence is in \[4\](=Royden 1962 [@Royden_1962])” $\spadesuit$ The Ahlfors map (and the machinery of uniformization) are again utilized to lift the characterization of interpolating sets for the disc (available from the celebrated results of Carleson, Newman, cf. also Hoffman 1962 [@Hoffman_1962]). The main theorem states that a subset $E\subset R$ of a finite open Riemann surface is an interpolating set for $R$ [*iff*]{} $\inf_{z\in E} d_R(z, E)>0 $, where $d_R(z,E):=\sup \{ \vert f(z) \vert : f\in H_{\infty}(R), f_{\vert E-\{z\}}=0, \|f\|_R \le 1 \}$. For convenience, recall that the subset $E$ is called an interpolation set for $R$ if every bounded complex-valued function on $E$ can be extended to a bounded analytic function on $R$.\]  E.L. Stout, [*Inner functions, doubles and special analytic polyhedra*]{}, Amer. J. Math. 94 (1972), 343–365. 50 \[$\spadesuit$ p.345 credits Heins 1950 [@Heins_1950] for another (beside Ahlfors’ 1950 [@Ahlfors_1950]) eleg[*r*]{}ant \[sic\] construction of inner functions on compact bordered surfaces\]  E. Study, W. Blaschke, [*Vorlesungen über ausgewählten Gegenstände der Geometrie*]{}, vol.2, Konforme Abbildung einfach zusammenhängender Bereiche, Teubner, Leipzig, 1912. \[$\spadesuit$ closely related to Carathéodory’s seminal study of the boundary behaviour of the Riemann map along an arbitrary Jordan curve and the more general theory of prime ends\]  A. Stray, [*Approximation by analytic functions which are uniformly continuous on a subset of their domain of definition*]{}, Amer. J. Math. 99 (1977), 787–800. \[$\spadesuit$ p.797 brief apparition of the Ahlfors function via cross-reference to Gamelin 1969 [@Gamelin_1969]\]  K. Strebel, [*Über das Kreisnormierungsproblem der konformen Abbildung*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. 101 (1951), 22 pp. 60, 78 \[$\diamondsuit$ Kurt Strebel is a student of R. Nevanlinna (who teached frequently in Zürich)\] $\bigstar$  K. Strebel, [*Über die konforme Abbildung von Gebieten unendlich hohen Zusammenhangs, (I. Teil)*]{}, Comment. Math. Helv. 27 (1952), 101–127 78 \[$\clubsuit$ partial results on the Kreisnormierung in infinite connectivity\]  K. Strebel, [*Ein Klassifizierungsproblem für Riemannsche Fläche vom Geschlecht 1* ]{}, Arch. Math. 48 (1987), 77–81. \[$\clubsuit$ p.77: “Herr K. Schüffler benötigt in seiner Arbeit \[2\] zur Theorie der Minimalflächen vom Geschlecht 1 den Satz, da[ß]{} [*jeder $p$-fach gelochte Torus auf einen ebensolchen mit kreisförmigen Löchern konform abgebildet werden kann, und da[ß]{} eine solche Abbildung durch diese geometrische Forderung im wesentlichen eindeutig bestimmt ist.*]{} Dabei wird der Torus durch die komplexe Ebene ${\Bbb C}$ modulo einer Translationsgruppe dargestellt, und die Kreisförmigkeit der Löcher ist ebenfalls in ${\Bbb C}$ gemeint.” $\spadesuit$ \[17.10.12\] one naturally wonders about higher genuses than one (where one must probably interpret the Kreisförmigkeit within the hyperbolic plane/disc), and it seems that such positive genus instances of the Kreisnormierung are also handled in Haas 1984 [@Haas_1984]\]  V. Strehl, [*Minimal transitive products of transpositions–the reconstruction of a proof by A. Hurwitz*]{}, Sem. Lothar. Combinat. 37 (1996), Art.B37c, 12pp. \[$\spadesuit$ modern reconstruction of Hurwitz’s count of the number of Riemann surfaces having prescribed ramification, cf. also Ekedahl-Lando-Shapiro-Vainshtein 2001 [@Ekedahl-Lando-Shapiro-Vainshtein_2001]\]  D.J. Struik, [*Outline of a history of differential geometry II*]{}, Isis 20 (1933), 161–191. \[$\spadesuit$ Gauss 1844 (and even F.T. Schubert) are credited for the nomenclature “conformal” as follows, p.164: “Of Gauss’ contribution to notation and nomenclature we mention the symbols $E,F,G,D,D',D''$ for what we now call the coefficients of the first and second fundamental differential form, and the word “conformal”. (6a)=footnote=(6a) In the first paper on higher geodesy, 1844: “ich werde daher dieselben conforme Abbildungen oder Übertragungen nennen, indem ich diesem sonst vagen Beiworte eine mathematisch scharf bestimmte Bedeutung beilege” \[Werke IV, p.262\]. The word is indeed, already used by F.T. Schubert, “De projectione sphaeroidis ellipticae geographica”, [*Nova Acta Petr.*]{}, p.130–146, see Cantor IV, p.575.”\]  T. Sugawa, [*Unified approach to conformally invariant metrics on Riemann surfaces*]{}, Proc. of the Second ISAAC Congress, Vol.2 (Fukuoka, 1999), 1117–1127, Int. Soc. Anal. Appl. Comput., 8, Kluwer Acad. Publ., Dordrecht, 2000. \[$\spadesuit$ the Ahlfors function is mentioned on p.5: “The quantity $c_R(p)$ is sometimes called the analytic capacity. An extremal function $f\colon R \to \Bbb D$ satisfying $\vert df \vert(p)=c_R(p)$ is usually called the [*Ahlfors function*]{} at $p$ and known to be unique up to unimodular constants (see \[4\](=Fisher 1983 [@Fisher_1983])). We remark that the condition $c_R(p)=0$ at some point $p$ need not imply that $c_R(p)=0$ at every point $p$ in the case that $R$ is non-planar. A counterexample was constructed by Virtanen \[13\](=Virtanen 1952 [@Virtanen_1952]) (see also \[10,X.2K\]=Sario-Oikawa 1969 [@Sario-Oikawa_1969]).” $\spadesuit$ the article as whole present an unified framework to the interplay between conformally invariant metrics and extremal problems emphasizing the contractive property of holomorphic maps (à la Schwarz-Pick-Ahlfors) $\spadesuit$ more precisely several metrics are presented culminating to their comparison as $$a\;\le \;s \!\!\!\!\buildrel{\rm AB50S69}\over{\le}\!\!\! c \le \begin{Bmatrix} \;\;\;\le \;\;r\buildrel{\rm Bu79}\over{\le} k \\ \buildrel{\rm HeSu72}\over{\le}\!\!\! b\;\;\; \le\;\, q \end{Bmatrix}\le h,$$ where $a$ stands for Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950], $s$ for span (or Schiffer!), $c$ for Carathéodory(-Reiffen) (or for analytic capacity), $r$ for Robin (or logarithmic capacity), $k$ for Kobayashi, $b$ for Bergman, $q$ for quadratic differentials (Grötzsch-Teichmüller!), $h$ for Hahn $\spadesuit$ the inequality AB50S69 is due to Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950] for the planar case and in general to Sakai 1969/70 [@Sakai_1969] $\spadesuit$ inequality Bu79 is due to Burbea 1979 [@Burbea_1979-Schwarzian] $\spadesuit$ inequality HeSu72 is due to Hejhal 1972 [@Hejhal_1972-Memoirs-AMS p.106] (case of finite bordered surface) and Suita 1972 [@Suita_1972] in general\]  T. Sugawa, [*An explicit bound for uniform perfectness of the Julia sets of rational maps*]{}, Math. Z. 238 (2001), 317–333. \[$\spadesuit$ the Ahlfors map is briefly mentioned as follows: “In fact, for a finitely connected planar domain $U$ whose boundary consists of Jordan curves, it is known that there exists a branched holomorphic covering map from $U$ onto the unit disk (e.g. the Ahlfors map). Thus $L_U$ cannot be estimated from below by only the data of $W$ (in this case $L_W=+\infty$).”\]  N. Suita, [*Capacities and kernels on Riemann surfaces*]{}, Arch. Rat. Mech. Anal. 46 (1972), 212–217. \[$\spadesuit$\]  N. Suita, [*On a metric induced by analytic capacity*]{}, Kodai Math. Sem. Report 25 (1973), 215–218. 78 \[$\spadesuit$ Ahlfors function à la Havinson 1961/64 [@Havinson_1961/64], i.e. for domains $D\notin O_{AB}$ (i.e. supporting nonconstant bounded analytic functions), analytic capacity and conformal metrics $\spadesuit$ the metric in question is also known as the Carathéodory metric (cf. e.g., Grunsky 1940 [@Grunsky_1940])\]  N. Suita, [*On a class of analytic functions*]{}, Proc. Amer. Math. Soc. 43 (1974), 249–250. 78 \[$\spadesuit$ p.249, the Ahlfors function is discussed as follows: “If $\Omega \notin O_{AB}$ \[i.e. $\Omega$ is a plane region having a nonconstant bounded analytic function\], there exist the extremal functions $A(z)$ which maximize $\vert f'(z_0)\vert$ in ${\frak B}_0$ \[the class of analytic functions $f$ such that $f(z_0)=0$ and $\vert f(z) \vert \le 1$\]. Those functions are called the [*Ahlfors functions*]{} which are unique save for rotations \[3\](=Havinson 1961/64 [@Havinson_1961/64]).” $\spadesuit$ the note includes a counterexample to an (erroneous) claim made by Ahlfors-Beurling 1950 [@Ahlfors-Beurling_1950] about the compactness of the class ${\frak E}_0$ of those analytic functions in a plane region $\Omega \notin O_{AB}$ vanishing at $z_0\in \Omega$ and such that $1/f$ omits a set of of values of area $\ge \pi$\]  N. Suita, [*On a metric induced by analytic capacity, II*]{}, Kodai Math. Sem. Report 27 (1976), 159–162. 47 \[$\spadesuit$ the Ahlfors function appears on p.160 and 161 $\spadesuit$ for a plane region $\Omega\notin O_{AB}$ (i.e. supporting nonconstant bounded analytic functions) it was known (Suita 1973 [@Suita_1973] via “making use of a supporting metric due to Ahlfors 1938”) that the curvature $\kappa(\zeta)$ of the metric $ds_B=c_B(\zeta) \vert d\zeta\vert$ induced by analytic capacity $c_B(\zeta)=\sup \vert f'(\zeta)\vert$ in the class of functions bounded-by-one (=stretching factor of the Ahlfors function at $\zeta$) is $\le -4$ $\spadesuit$ the present article rederives this estimate ($\kappa \le -4$) by a limiting/exhaustion argument reducing to the case of a regularly bounded finitely connected domain which is analyzed via Bergman’s method of minimal integrals, but making also extensive use of Garabedian’s sharp analysis (our opinion!) $\spadesuit$ the novelty of the present article is that the ‘Bergman-Garabedian method’ gives the “more precise estimation $\kappa(\zeta)<-4$” for regions with more than one contour $\spadesuit$ paraphrase (p.161): “the equality $\kappa(\zeta)=-4$ at one point $\zeta\in \Omega$ implies that $\Omega$ is conformally equivalent to the unit disc.” $\spadesuit$ \[23.09.12\] maybe it would be worth looking if Suita’s work extends to finite bordered surfaces (the problem being that quantity $\vert f'(\zeta)\vert$ depends on a local uniformizer), yet it seems that the theory is extensible (cf. e.g. Sugawa 1999/00 [@Sugawa_1999/00])\]  N. Suita, A. Yamada, [*On the Lu Qi-keng conjecture*]{}, Proc. Amer. Math. Soc. 59 (1976), 222–224. \[$\spadesuit$ “We shall give a complete answer to the Lu Qi-keng conjecture for finite Riemann surfaces. Our result is that every finite Riemann surface which is not simply-connected is never a Lu Qi-keng domain, i.e. the Bergman kernel $K(z,t)$ of it has zeros for suitable $t$’s.”\]  G. Szegö, [*Über orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Ebene gehören*]{}, Math. Z. 9 (1921), 218–270 78 \[$\spadesuit$ Szegö kernel representation of the Riemann mapping (p.245) $\spadesuit$ like Bergmann 1922 [@Bergman_1922] or Bochner 1922 [@Bochner_1922] it is confessed (p.249) that the method does not duplicate a new existence proof of the Riemann mapping (this had to wait upon Garabedian and Lehto 1949 [@Lehto_1949]) $\spadesuit$ what is the geometric interpretation (i.e. the allied extremal problem): answer of course it is just that of minimizing the integral $\int_C \vert f(z) \vert^2 ds$, where integration is taken along the contour $C$ of the domain (and $ds$ is its Bogenelement)\]  G. Szegö, [*Über die Randwerte einer analytischer Funktion*]{}, Math. Ann. 84 (1921), 232–244 \[$\spadesuit$\]  G. Szegö, [*Verallgemeinerung des ersten Bieberbachschen Flächensatzes auf mehrfach zusammenhängende Gebiete*]{}, Sitz.-Ber. Preu[ß]{}. Akad. d. Wiss., math.-phys. Kl. (1928), 477–481 78 \[$\spadesuit$ can we do the same on a Riemann surface? and relate this to a Bergman-style proof of the Ahlfors map?\] $\bigstar$$\bigstar$$\bigstar$  G. Szegö, [*Inequalities for certain eigenvalues of a membrane of given area*]{}, J. Rat. Mech. Anal. 3 (1954), 343–356 \[$\spadesuit$ one of the early implementation of the conformal transplantation method to vibratory/elasticity problem; for wide extensions cf. Hersch 1970 [@Hersch_1970], Yang-Yau 1980 [@Yang-Yau_1980] and Fraser-Schoen 2011 [@Fraser-Schoen_2011], the last article effecting the junction with the Ahlfors map\]  J. Tagamlizki, [*Zum allgemeinen Kreisnormierungsprinzip der konformen Abbildung*]{}, Ber. Verhandl. Sächs. Akad. Wiss., math.-phys. Kl. 95 (1943), 111–132. 78 $\bigstar$  M. Taniguchi, [*Bell’s result on, and representations of finitely connected planar domains*]{}, Some Japanese fonts 1352 (2004), 47–53. \[$\spadesuit$ survey of several results of Bell on the Ahlfors function and concludes by some questions about Bell representations, i.e. a certain family of canonical domains admitting an evident proper holomorphic map to the disc\] $\bigstar$  T.J. Tegtmeyer, A.D. Thomas, [*The Ahlfors map and Szegö kernel for an annulus*]{}, Rocky Mountain J. Math. 29 (1999), 709–723. \[$\spadesuit$ contains some lovely pictures of Ahlfors function in the case of an annulus\]  O. Teichmüller, [*Eine Verschärfung des Dreikreisesatzes*]{}, Deutsche Math. 4 (1939), 16–22. 78 \[$\spadesuit$ quoted (joint with Carlson 1938 [@Carlson_1938]) in Grunsky 1940 [@Grunsky_1940] as a forerunner of the extremal problem for bounded analytic functions $\diamondsuit$ Oswald Teichmüller (1913–1943) is formally a student of Hasse, but his interest shifted to function theory (presumably due to lectures held in Göttingen ca. 1935 by R. Nevanlinna) and then joined ca. 1937 Berlin where Bieberbach was located\]  O. Teichmüller, [*Extremale quasikonforme Abbildungen und quadratische Differentiale*]{}, Abh. Peu[ß]{}. Akad. Wiss. math.-naturw. Kl. 22 (1939), 1–197; also in the Collected Papers, 335–531. 60, 78 \[$\spadesuit$ discusses in details the Klein dictionary between symmetric surfaces and bordered Riemann surfaces through the [*Verdoppelung*]{} (=Schottky-Klein double) $\spadesuit$ discusses moduli in a way quite anticipated in Klein 1882 [@Klein_1882], modulo of course the usual Riemann-style heuristics\]  O. Teichmüller, [*Über Extremalprobleme der konformen Geometrie*]{}, Deutsche Math. 6 (1941), 50–77; also in Collected Papers, 554–581. 60, 78 \[$\clubsuit$ a mention is made (without proof and a cryptical unreferenced allusion to Klein) of a statement which could be interpreted as a forerunner of the Ahlfors circle map $\spadesuit$ despite long searches, the writer (Gabard) was unable—on the basis of printed evidence—to adhere conclusively to Teichmüller’s accreditation of the result to Klein, compare Sec.\[sec:Teichmueller\] for more tergiversations $\spadesuit$ the original Teichmüller text reads as follows (p.554–5): “Wir beschäftigen uns nur mit [**orientierten endlichen Riemannschen Mannigfaltigkeiten.**]{} Diese können als Gebiete auf geschlossenen orientierten Riemannschen Flächen erklärt werden, die von endlich vielen geschlossenen, stückweise analytischen Kurven begrenzt werden. Sie sind entweder geschlossen, also selbst geschlossene orientierte Riemannsche Flächen, die man sich endlichvielblättrig über eine $z$-Kugel ausgebreitet vorstellen darf, oder berandet. Im letzteren Falle, kann man sie nach Klein[^147] durch konforme Abbildung auf folgende Normalform bringen: ein endlichvielblättriges Flächenstück über der oberen $z$-Halbebene mit endlich vielen Windungspunkten, das durch Spiegelung an der reellen Achse eine symmetrische geschlossene Riemannsche Fläche ergibt; \[…\] —(So lä[ß]{}t sich z.B. jedes Ringgebiet, d.h. jede schlichtartige endliche Riemannsche Mannigfaltigkeit mit zwei Randkurven, konform auf eine zweiblättrige Überlagerung der oberen Halbebene mit zwei Verzweigungspunkte abbilden.)” $\spadesuit$ Another puzzle would be to know if Teichmüller’s text exerted some influence over Ahlfors subsequent findings (1950 [@Ahlfors_1950]). Possibly yes, but note the absence of cross-citation until Ahlfors-Sario 1960 [@Ahlfors-Sario_1960]. All this should by no mean palish the originality of Ahlfors achievement which looks substantially sharper by controlling the mapping degree.\]  O. Teichmüller, [*Beweis der analytischen Abhängigkeit des konformen Moduls einer analytischen Ringflächenschar von den Parametern*]{}, Deutsche Math. 7 (1944), 309–336; also in Collected Papers, ??–??. \[$\spadesuit$ quoted in Rüedy 1971 [@Ruedy_1971] as the technological forerunner of the Garsia embedding result\]  O. Teichmüller, [*Gesammelte Abhandlungen, Collected Papers*]{}, Herausgegeben von L.V. Ahlfors und F.W. Gehring, Springer Verlag, Berlin, 1982.  R. Thom, [*Sur l’homologie des variétés algébriques réelles*]{}, in: Differential and Combinatorial Topology, Symposium in honor of Marston Morse, Princeton Univ. Press, Princeton, N.J., 1965, 255–265. \[$\spadesuit$ cf. also a related work by J. Milnor 1964 [@Milnor_1964]\] W. Thomson (later Lord Kelvin), [*Sur une équation aux différences partielles qui se présente dans plusieurs questions de physique mathématique*]{}, J. Math. Pures Appl. 12 (1847), 493–496. \[$\spadesuit$ one of the early apparition of the Dirichlet principle, cf. also Green 1928 [@Green_1828], Gauss 1839 [@Gauss_1839], Kirchhoff 1850 [@Kirchhoff_1850], Riemann 1851–57–57 [@Riemann_1851; @Riemann_1857; @Riemann_1857-DP] and Dirichlet as edited by Grube 1876 [@Dirichlet_1840-1876]\] W. Thurston, [*The geometry and topology of $3$-manifolds*]{}, Princeton University Notes, Princeton, N.J., 1979. \[$\spadesuit$ circle packing theorem, cf. precise citations e.g. in He 1990 [@He_1990], i.e. especially Corollary 13.6.2 and Theorem 13.7.1 (circle packing theorem)\]  H. Tietz, [*Eine Normalform berandeter Riemannscher Flächen*]{}, Math. Ann. 129 (1955), 44–49. 50, 60, 78 \[$\clubsuit$ cite Ahlfors 1950 [@Ahlfors_1950] and Nehari 1950 [@Nehari_1950], then criticizes the arguments of the latter $\clubsuit$ seems to reprove a sort of circle map for bordered surfaces inspired by Ahlfors (but with the desideratum of schlichtness along the boundary), alas Tietz’s argument is criticized (and apparently destroyed) in Köditz-Timmann 1975 [@Koeditz-Timmann_1975] $\spadesuit$ Grunsky 1978 [@Grunsky_1978 p.198] also seems to approve the Köditz-Timmann critique for he cites the (present) paper Tietz 1955 [@Tietz_1955], but right after add the parenthetical proviso “(cf. \[266\])”, that is Köditz-Timmann $\spadesuit$ despite those defects the prose of the introduction is brilliant and worth quoting (especially as it emphasizes the historical rôle of Ahlfors 1950 [@Ahlfors_1950], note however that Tietz seems to neglect both the Italian works as well as the cryptical allusion in Teichmüller 1941 [@Teichmueller_1941]): “Die Existenz eindeutiger analytischer Funktionen auf Riemannschen Flächen[^148] bedeutet, da[ß]{} jede Klasse konformäquivalenter Riemannscher Flächen “realisiert” werden kann durch Überlagerungsflächen der Zahlenebene. Damit stellt sich die Frage nach besonders einfachen Realisierungen oder Normalformen[^149].—Das wichtigste Ergebnis zu dieser Frage ist der Riemannsche Abbildungssatz, der sie für einfach-zusammenhängende Riemannsche Flächen beantwortet[^150]. Einen Schritt weiter gehen die Schlitztheoreme, die von den topologischen Voraussetzungen des Riemannschen Abbildungssatzes nur die Schlichtartigkeit der Riemannschen Fläche beibehalten. Hierher gehört auch der Satz, da[ß]{} jede berandete schlichtartige Riemannsche Fläche einem mehrfach überdeckten Kreis mit geeigneten Verzweigungsschnitten, die den Rand nicht treffen, konformäquivalent ist[^151].—Die Frage nach kanonischen Riemannschen Flächen im Falle höheren Geschlechtes is erst in letzter Zeit von Herrn [Ahlfors]{} \[1\](=1950 [@Ahlfors_1950]) angeschnitten und von Herrn [Nehari]{} \[2\](=1950 [@Nehari_1950]) systematisch behandelt worden:—Herr [Ahlfors]{} zeigt, da[ß]{} jede berandete Riemannsche Fläche realisiert werden kann als mehrfach überdeckter Einheitskreis, während Herr [Nehari]{} die Schlitztheoreme auf diesen Fall überträgt[^152]. \[…\]—Es erscheint wünschenswert, eine Normalform für berandete Riemannschen Flächen zu besitzen, die—im Gegensatz zur [Ahlfors]{}schen—sicherstellt, da[ß]{} das Bild jeder einzelnen Randkurve schlicht über die Linie des Einheitskreises liegt. \[…\]” $\clubsuit$ Tietz concludes his paper (p.49) as follows: “Die selben Überlegungen, die zu unserem Abbildungssatz führten, ermöglichen auch einen neuen Existenzbeweis für die Ahlforsche Normalform, wiederum jedoch ohne eine Schranke für die Anzahl der benötigten Blätter zu ergeben.” so this would be another (weak) version of Ahlfors, alas it seems that Tietz’s arguments where the object of critics, cf. Köditz-Timmann 1975 [@Koeditz-Timmann_1975]\]  H. Tietz, [*Zur Realisierung Riemannscher Flächen*]{}, Math. Ann. 128 (1955), 453–458. 60 \[$\spadesuit$ with corrections in the next entry [@Tietz_1955_Berechtigung]\]  H. Tietz, [*Berechtigung der Arbeit “Zur Realisierung Riemannscher Flächen”*]{}, Math. Ann. 129 (1955), 453–458. 60 St. Timmann, [*Kompakte berandete Riemannsche Flächen*]{}, Diss. Hannover, 1969, 56 S. 78 \[$\spadesuit$ this entry is cited on the “critical” page 198 of Grunsky 1978 [@Grunsky_1978], according to which it gives a generalization to Riemann surfaces of the Bieberbach-Grunsky theorem (i.e. circle map in the planar case) $\spadesuit$ in particular, it could be the case that Timmann’s reproves the existence of an Ahlfors circle map, yet probably this is not the case\] $\bigstar$  X. Tolsa, [*Painlevé’s problem and the semiadditivity of analytic capacity*]{}, Acta Math. 190 (2003), 105–149. 47 \[$\spadesuit$ complete solutions of both problems of the title are given, the first being usually regarded as implicitly posed in Painlevé 1888 [@Painleve_1888] (albeit nobody was ever able to locate the precise place, see e.g. Rubel 1971 [@Rubel_1971] or Verdera 2004 [@Verdera_2004] for why) and the second emanated from Vitushkin’s advanced studies in the 1960’s $\spadesuit$ the introduction contain a historical sketch, from Ahlfors 1947 [@Ahlfors_1947], Vitushkin 1950’s to Murai 1988 [@Murai_1987], Melnikov 1995 [@Melnikov_1995] (curvature of measures), G. David 1998 [@David_1998] (solution of Vitushkin’s conjecture), etc.\]  G. Toumarkine, S. Havinson, [*Propriétés qualitatives des solutions des problèmes extrémaux de certains types*]{}, In: Fonctions d’une variable complexe. Problèmes contemporains. Paris 1962, p.73. \[$\spadesuit$ survey containing a quite complete bibliography\]  S. Treil, [*Estimates in the corona theorem and ideals of $H^{\infty}$: a problem of T. Wolff*]{}, J. Anal. Math. 87 (2002), 481–495. \[$\spadesuit$ improved lower estimates for the solution of the corona problem, but with still a large gap up the upper bound of Uchiyama 1980 (cf. esp. p.494)\]  C.L. Tretkoff, M.D. Tretkoff, [*Combinatorial group theory, Riemann surfaces and differential equations*]{}, In: [*Contribution to Group Theory*]{}, Contemp. Math. 33, 467–519. Amer. Math. Soc., Providence, 1984. \[$\spadesuit$\]$\bigstar$  A. Tromba, [*On Plateau’s problem for minimal surfaces of higher genus in ${\Bbb R}^n$*]{}, SFB 72-Preprint 580, Bonn, 1983. \[$\spadesuit$ doubts expressed about the validity of Douglas and Courant for the Plateau problem in the case of higher topological structure, compare Jost 1985 [@Jost_1985]\] $\bigstar$$\bigstar$$\bigstar$  A. Tromba, [*Dirichlet’s energy on Teichmüller’s moduli space and the Nielsen realization problem*]{}, Math. Z. 222 (1996), 451–464. \[$\spadesuit$\] $\bigstar$$\bigstar$  V.V. Tsanov, [*On hyperelliptic Riemann surfaces and doubly generated function algebras*]{}, C.R. Acad. Bulgare Sci. 31 (1978), 1249–1252. \[$\spadesuit$ quoted in Černe-Forstnerič 2002 [@Cerne-Forstneric_2002]\]$\bigstar$$\bigstar$  M. Tsuji, [*A simple proof of Bieberbach-Grunsky’s theorem*]{}, Comment. math. Univ. St. Paul 4 (1956), 29–32. 78 \[$\spadesuit$ Nehari’s review (in MR): “A new proof of the classical result that there exists a $(1,n)$ conformal mapping of a plane domain $D$ of connectivity $n$ onto the unit circle which carries a given point on each of the boundary components of $D$ into the same point of the unit circumference.”\] $\bigstar$  M. Tsuji, [*Potential theory in modern function theory*]{}, Tokyo, Maruzen, 1959. (Chelsea edition 1975.) $\bigstar$ 78 \[$\spadesuit$ contains apparently yet another proof of the Bieberbach-Grunsky theorem, perhaps the same as in the previous item\]$\bigstar$  A.W. Tucker, [*Branched and folded coverings*]{}, Bull. Amer. Math. Soc. 42 (1936), 859–862. \[$\spadesuit$\]$\bigstar$  G. Tumarkin, see Toumarkine. N.X. Uy, [*On Riesz transforms of bounded functions of compact support*]{}, Michigan Math. J. 24 (1977), 169–175. \[$\spadesuit$ p.170 the Ahlfors function (referenced via Gamelin’s book 1969 [@Gamelin_1969]) is involved in a theorem involving the Riesz transform\]  L. Vajsburd, A. Radul, [*Non-orientable strings*]{}, Comm. Math. Phys. 135 (1991), 413–420. \[$\spadesuit$ real algebraic (diasymmetric) curves as applied to string theory, more related refs. in Natanzon 1999 [@Natanzon_1999-Moduli-real-alg-surf.superanal-differ-spinors]\]  Ch. de la Vallée Poussin, [*Sur la représentation conforme des aires multiplement connexes*]{}, Ann. École Norm. (3) 47 (1930), 267–309 78  J. Verdera, [*Removability, capacity and approximation*]{}, in: Complex Potential Theory, NATO ASI Series, Kluwer Acad. Publ., Dordrecht, 1994, 419–473. \[$\spadesuit$\]  J. Verdera, [*The $L^2$ boundedness of the Cauchy integral and Menger curvature*]{}, Contemp. Math. 277 (2001), 139–158. \[$\spadesuit$\]  J. Verdera, [*Ensembles effaçables, ensembles invisibles et le problème du voyageur de commerce, ou comment l’analyse réelle aide l’analyse complexe*]{}, Gazette des Math. 101 (2004), 21–49 47 \[$\spadesuit$ a thorough survey about Painlevé null-sets including the following points: $\spadesuit$ Painlevé’s problem about searching a geometric characterization of null-sets (nobody ever found an explicit formulation in Painlevé’s writings, but Ahlfors 1947 [@Ahlfors_1947] may be considered as the father of the modern era (introduction of the analytic capacity and insistance upon pure geometric conditions) $\spadesuit$ Tolsa’s resolution (ca. February 2003) of Painlevé’s problem (via bilipchitzian invariance of analytic capacity) is mentioned $\spadesuit$ p.29: the Denjoy conjecture (i.e., a compactum of a rectifiable curve is a (Painlevé) null-set iff its length is zero). This conjecture was cracked by the seminal work of Calderón 1977 [@Calderon_1977] as was made explicit in a note of Marshall $\spadesuit$ the (Vitushkin)-Garnett 1970 [@Garnett_1970] example of the $1/4$-Cantor set is discussed: this has positive length (because a certain projection is a full segment) but is a null-set (removable) $\spadesuit$ this is used to motivate Besicovitch’s notion of “invisible sets”, i.e. those projecting to sets of zero-length along almost every angular direction $\spadesuit$ Vitushkin’s conjecture: a compactum of the plane is a null-set iff it is invisible (alas, there is counter-examples of Mattila, and Jones-Murai 1988 [@Jones-Murai_1988]), yet the direct sense is true if finite length (as follows from the Denjoy conjecture solved since Calderón), hence $\spadesuit$ weak Vitushkin conjecture (1967): among compacta of finite length, the null-sets coincide with the invisible sets. This was completed in G.David 1998 [@David_1998] upon combining a chain of contributions: Christ 1990, Mattila-Melnikov-Verdera 1996 [@Mattila-Melnikov-Verdera_1996] and Jones 1990\]  I.N. Vekua, [*Generalized analytic functions*]{}, Pergamon Press, Oxford, 1962. \[$\spadesuit$ an account of the theory of the Beltrami equation, with roots going back to Gauss, Korn, Lichtenstein, Morrey, Lavrentiev, Bojarski, Lehto, Ahlfors and Bers, etc.\]  I.N. Vekua, [*Verallgemeinerte analytische Funktionen*]{}, Berlin 1963 \[$\spadesuit$ Riemann-Hilbert problem on finite bordered Riemann surfaces (and the allied Fredholm theory), cf. also Koppelman 1959 [@Koppelman_1959] Schüffler 1986 [@Schueffler_1986]\]  H. Villat, [*Le problème de Dirichlet dans une aire annulaire*]{}, Rend. Circ. Mat. Palermo 33 (1912), 149 \[$\spadesuit$ a brief proof of Villat’s formula in Komatu (1945)\]  V. Vinnikov, [*Self-adjoint determinantal representations of real plane curves*]{}, Math. Ann. 296 (1993), 453–479. \[$\spadesuit$ a brilliant presentation of the theory of Klein-Weichold of real curves and simplified proof of results of Dubrovin-Natanzon, discuses complex orientations (à la Rohlin) $\spadesuit$ mentions the result that a real plane curve with a nest of maximal depth is dividing (via Rohlin 1978 [@Rohlin_1978 p.93]), whose argument can (in our opinion) can be slightly simplified as follows $\spadesuit$ given $C_m\subset {\Bbb P}^2$ a nonsingular curve of degree $m$ with a deep nest then projecting the curve from any point chosen in the innermost oval gives a morphism $C_m \to {\Bbb P}^1$ whose fibers over real points are totally real, hence there is an induced map between the imaginary loci and it follows that $C_m$ is dividing (just by using the fact that the image of a connected set is connected). q.e.d. (N.B.: this is exactly Rohlin’s argument except that we avoid the consideration of the canonical fibering ${\rm pr}\colon {\Bbb C} P^2-{\Bbb R} P^2 \to S^2$ envisaged by Rohlin) $\spadesuit$ p.478 mentions the result of Nuij 1968 [@Nuij_1968]: “any two real smooth plane curves of degree $n$ having a nest of of ovals of maximal depth are [*rigidly isotopic*]{} (i.e. belongs to the same component in the space of all real smooth plane curves of degree $n$)” $\spadesuit$ \[30.09.12\] I vaguely remember of a sharper question (result?) asking if the space of deeply nested curves is not even a (contractible) cell $\spadesuit$ \[02.10.12\] probably this question was rather asked for ovalless real curves, yet the idea (coming to me only today) is that the $\pi_1$ (fundamental group) of any chamber (=component of the complement of the discriminant hypersurface $D\subset \vert {\cal O}_{{\Bbb P}^2}(m) \vert=\vert mH\vert$ consisting of all singular curves) must act on the set of ovals of any fixed plane curves. Hence when there is no oval or a nest (not necessarily of maximal depth) then the induced (monodromy) permutation must be trivial and consequently there is no obstruction to the chamber having a simple topology. More generally this applies when there are several nests of different depths (then again nothing can be permuted). In contrast when there is collection of non-nested ovals (or two nests of the same depth) then there is no obstruction to there permutability (e.g. imagine a quartic with $4$ ovals resulting from the smoothing of two conics then by rotating the plane we can achieve a transitive permutation of cyclic type). But probably the monodromy group of this quartic is bigger. How large exactly? $\spadesuit$ a problem would be to count the number of component of $\vert mH\vert-D$ and if possible to describe the complex encoding the adjacency relation between the different chambers $\spadesuit$ of course in the general question of describing the monodromy of a given curve, one can exploit Rohlin’s idea of the complex orientation in the case where the curve is dividing, as the latter must probably be conserved during an isotopy-loop (up to reversion). If so then for the 4 ovals quartic we get an obstruction to there complete permutability, and the monodromy group is not the full symmetric group ${\frak S}_4$. Naively two ovals gyrate clock-wise and two anti-clock-wise (draw the complex orientations by doing sense preserving smoothings), yet since ${\Bbb R}P^2$ is nonorientable nothing is secure (i.e. the clockwise can continuously mutate in the anti-clock-wise)? (of course all this must be described somewhere with more care!) $\spadesuit$ as in Nuij’s result one can ask when the real scheme (Rohlin’s jargon) determine unambiguously the isotopy type (or what is the same a unique chamber). A naive (probably wrong) guess is that if the monodromy is trivial, then the chamber is unique\]  V. Vinnikov, [*Commuting operators and function theory on a Riemann surface*]{}, In: Holomorphic spaces (Berkeley 1995), MSRI Publications 33, 1998. 50 \[$\spadesuit$ p.468, Ahlfors 1950 is briefly cited as a mapping onto the upper half-plane, and is applied to problems of operator theory and maybe as well to a generalization of the Riesz-Nevanlinna-Smirnov factorization $\spadesuit$ compare optionally Havinson 1989/89[@Havinson_1989/89] where a similar desideratum was found to be difficult (and unsolved?) $\spadesuit$ from the abstract: “In the late 70’s M.S. Livsic has discovered that a pair of commuting nonselfadjoint operators in a Hilbert space, with finite nonhermitian ranks, satisfy a polynomial equation with constant (real) coefficients; …”, whence the link with real curves (à la Klein) and therefore with Ahlfors\]  O.Ya. Viro, [*Construction of multicomponent real algebraic surfaces*]{}, Dokl. Akad. Nauk SSSR 248 (1979), 279–282; English transl., Soviet Math. Dokl. 20 (1979), 991–995. \[$\spadesuit$\]  O.Ya. Viro, [*Construction of $M$-surfaces*]{}, Funkt. Anal. i Prilozhen. 13 (1979), 71–72; English Transl. in: ???. \[$\spadesuit$\]  O.Ya. Viro, [*Curves of degree $7$, curves of degree $8$, and the Ragsdale conjecture*]{}, Dokl. Akad. Nauk SSSR 254 (1980), 1305–1310; English Transl., ???. \[$\spadesuit$\]  O.Ya. Viro, [*Gluing of plane real algebraic curves and construction of curves of degree $6$ and $7$*]{}, in: Topology, Proc. Leningrad 1982, Lect. Notes in Math. 1060, 1984, 185–200. \[$\spadesuit$\]  O. Viro, [*Progress over the last five years in the topology of real algebraic varieties*]{}, Proc. Internat. Congr. of Mathematicians, Warsaw 1983, 525–611. \[$\spadesuit$ a more expanded version of the same material in Viro 1986/86 [@Viro_1986/86-Progress]\]  O. Viro, [*Real varieties with prescribed topological properties*]{}, Doct. Thesis, Leningrad Univ., 1983. \[$\spadesuit$ under the direction of V.A. Rohlin\]  O.Ya. Viro, [*??*]{}, Izv. Akad. Nauk SSSR, Ser. Mat. 47 (1983), 1135–1150; English Transl., ???. \[$\spadesuit$ cited in Shustin 1985/85 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin]\]  O.Ya. Viro, [*Real plane curves of degree $7$ and $8$: new prohibitions*]{}, Izv. Akad. Nauk SSSR, Ser. Mat. 47 (1983), 1135–1150; English Transl., Math. USSR Izv. 23 (1984), 409–422. \[$\spadesuit$ cited in Shustin 1985/85 [@Shustin_1985/85-ctrexpls-to-a-conj-of-Rohlin] for his counterexample to Rohlin’s maximality conjecture $\spadesuit$ another point of this paper is that it completes the isotopy classification of septics, thereby cracking the next case of Hilbert’s 16th problem. This is based on work of 1979 by Viro, where after a bunch of construction it remained him to prohibit the scheme $\langle J \sqcup 1 \langle 14 \rangle \rangle$. This was done using auxiliary curves of degree $2$ and the theory of complex orientations. The resulting classification involves $121=11^2$ real schemes (cf. e.g. Viro 1989/90 [@Viro_1989/90-Construction p.1124]). $\spadesuit$ \[24.01.13\] try at the occasion to draw the corresponding pyramid. Another idea try to prohibit Viro’s scheme $\langle J \sqcup 1 \langle 14 \rangle \rangle$ via CCC (collective contraction conjecture of empty ovals, cf. Sec.\[CCC:sec\]), yet looks difficult unless another idea appears $\spadesuit$ by Marin 1979 [@Marin_1979] (or Fiedler) we know that such schemes even when enriched by the type (or the stronger complex orientations) do not encodes unambigously the rigid-isotopy class, hence the rigid-isotopy classification is even much harder and probably still unsolved, compare e.g. Viro 2008 [@Viro_2008-From-the-16th-Hilb-to-tropical]. As a pure guessing (of Gabard) one could expect something like $512=2^9$ chambers???\]  O.Ya. Viro, [*Gluing of algebraic hypersurfaces, smoothing of singularities and construction of curves*]{}, Proc. Leningrad Int. Topology Conf., 1983, 149–197; English Transl., ???. \[$\spadesuit$ cited in Kharlamov-Viro 1988/91 [@Kharlamov-Viro_1988/91]\]  O. Viro, [*Progress in the topology of real algebraic varieties over the last six years*]{}, Uspekhi Mat. Nauk 41 (1986), 45–67; English Transl., Russian Math. Surveys 41 (1986), 55–82. \[$\spadesuit$ “Contents. Introduction 55 §1. Real algebraic curves as complex objects 57 §2. Numerical characteristics and encoding of schemes of curves 59 §3. Old restrictions on schemes of curves 60 §4. New restrictions on schemes of curves 63 §5. Klein’s assertion 67 §6. …”\]  O.Ya. Viro, [*Real algebraic plane curves: constructions with controlled topology*]{}, Alg. i Analiz 1 (1989), 1–73; English transl., Leningrad Math. J. 1 (1990), 1059–1134. \[$\spadesuit$ includes a complete solution of Hilbert’s 16th problem for $M$-curves up to orders $m\le 7$\]  O.Ya. Viro, [*Patchworking real algebraic varieties*]{}, preprint: http://www.math.uu.se/$\sim$oleg \[$\spadesuit$\]  O.Ya. Viro, S.Yu. Orevkov, [*Congruence modulo $8$ for real algebraic curves of degree $9$*]{}, Uspekhi Mat. Nauk 56 (2001), 137–138; English transl., Russian Math. Surveys 56 (2001), 770–771. \[$\spadesuit$\]  O.Ya. Viro, [*From the sixteenth Hilbert problem to tropical geometry*]{}, Japan. J. Math. 3 (2008), 185–214. \[$\spadesuit$ includes a complete solution of Hilbert’s 16th problem for $M$-curves up to orders $m\le 7$\]  K.I. Virtanen, [*Über die Existenz von beschränkten harmonischen Funktionen auf offenen Riemannschen Flächen*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. 75 (1950), 8 pp. 50, 60 \[cite Ahlfors 1950 [@Ahlfors_1950] in a footnote (p.6) as follows: “Zusatz b.d. Korr.: Die Extremalfunktion $\eta_n$ findet sich auch bei Ahlfors 1950 (=[@Ahlfors_1950]).” $\spadesuit$ yet this function is only a harmonic function, hence not the (analytic) Ahlfors map we are focused upon. In particular Virtanen’s paper does not reproves the existence of the Ahlfors maps, its main purpose being rather to establish the inclusion $O_{HB}\subset O_{HD}$ in the so-called classification theory of open Riemann surfaces\]  K.I. Virtanen, [*Über Extremalfunktionen auf offenen Riemannschen Flächen*]{}, Ann. Acad. Sci. Fenn. Ser. A. I. 141 (1952), 7 pp. 60 \[Ahlfors 1950 [@Ahlfors_1950] is cited maybe?\]  A.G. Vitushkin, [*Analytic capacity of sets and some of its properties*]{}, Dokl. Akad. Nauk SSSR 123 (1958), ?–?. (Russian) \[$\spadesuit$ cited in Melnikov 1967 [@Melnikov_1967] for the definition of the Ahlfors function\]  A.G. Vitushkin, [*Example of a set of positive length but zero analytic capacity*]{}, Dokl. Akad. Nauk SSSR 127 (1959), 246–249. (Russian) \[$\spadesuit$ compare also the (simplified) construction in Garnett 1970 [@Garnett_1970], who warns us that Vitushkin’s paper contains many typographical errors $\spadesuit$ the basic implication “zero analytic capacity whenever zero linear measure” is a classical theorem of Painlevé (cf. e.g. Ahlfors 1947 [@Ahlfors_1947 p.2], a simple application of Cauchy’s formula)\]  A.G. Vitushkin, [*Analytic capacity of sets in problems of approximation theory*]{}, Uspekhi Mat. Nauk 22 (1967), 141–199; English transl.: Russian Math. Surveys 22 (1967), 139–200. 47 \[$\spadesuit$ Ahlfors function appears on p.142 $\spadesuit$ formulation of the problem of the semi-additivity of analytic capacity solved (jointly with the older Painlevé problem on the geometric characterization of removable singularities) in Tolsa 2003 [@Tolsa_2003]\]  Vo Dang Thao, [*Über einige Flächeninhaltsformeln bei schlichtkonformer Abbildung von Kreisbogenschlitzgebieten*]{}, Math. Nachr. 74 (1976), 253–261. \[$\spadesuit$ cited in Alenicyn 1981/82 [@Alenicyn_1981/82]\] $\bigstar$$\bigstar$$\bigstar$  M. Voichick, [*Ideals and invariant subspaces of analytic functions*]{}, Trans. Amer. Math. Soc. 111 (1964), 493–512. \[$\spadesuit$ bounded analytic functions, nontangential boundary values (almost everywhere), inner function, Beurling’s invariant subspace theorem extended to finite Riemann surfaces (tools: Harnack’s principle, Fatou’s theorem, plus Read 1958 [@Read_1958_Acta] and Royden 1962 [@Royden_1962] (both direct descendants of Ahlfors 1950 [@Ahlfors_1950]), but the link if any is masked behind “une propice brume d’analyse fonctionnelle”) $\spadesuit$ similar work by Hasumi 1966 [@Hasumi_1966] $\spadesuit$ Voichick’s work also contains a “bordered” extension of the Beurling-Rudin description of closed ideals in the disc algebra, for which result Stanton 1971 [@Stanton_1971] proposes another route hinging on the use of the Ahlfors map\]  M. Voichick, L. Zalcman, [*Inner and outer functions on Riemann surfaces*]{}, Proc. Amer. Math. Soc. 16 (1965), 1200–1204. \[$\spadesuit$ factorization theory in the Hardy classes $H^p$ for finite bordered Riemann surfaces extending the classical theory (Hardy and the Riesz brothers) on the disc (antecedent by Parreau, Rudin 1955 [@Rudin_1955-class-Hp], and Royden 1962 [@Royden_1962]), inner function, Blaschke product, Green’s function, etc. $\spadesuit$ naively speaking one could hope that the Ahlfors function alone is a sufficient tool to lift the truth from the disc to the bordered surface, yet the implementation usually diverge slightly (here by using the universal covering to effect the reduction to the classical disc case)\]  M. Voichick, [*Extreme points of bounded analytic functions of infinitely connected regions*]{}, Proc. Amer. Math. Soc. 11 (1966), 83–86. 50, 78 \[$\spadesuit$ p.1369, cite Ahlfors 1950 [@Ahlfors_1950] for the existence of a negative harmonic function whose harmonic conjugate has prescribed periods $\spadesuit$ this page contains an acrobatical implementation of the usual yoga attempting to annihilate periods to ensure single-valuedness (hence quite close to Ahlfors’ existence-proof of a circle map) $\spadesuit$ p.1367: “It should be noted that Gamelin in \[2\](=to appear=and seems to have appeared under extended coauthoring, namely Gamelin-Voichick 1968 [@Gamelin-Voichick_1968]) characterized the extreme points of the unit ball of $H^{\infty}(R)$ when $R$ is a finite bordered Riemann surface.”\]  M. Voichick, [*Invariant subspaces on Riemann surfaces*]{}, Canad. J. Math. 18 (1966), 399–403. \[$\spadesuit$\]$\bigstar$$\bigstar$  V. Volterra, [*Sul Principo di Dirichlet*]{}, Palermo Rend. 11 (1897), 83–86. 60 \[$\spadesuit$\]  B.L. van der Waerden, [*Topologie und Uniformisierung der Riemannschen Flächen*]{}, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl. 93 (1941), 147–160. 60 \[$\spadesuit$ cf. also Carathéodory 1950 [@Caratheodory_1950] and ref. therein, esp. to Reichardt.\] $\bigstar$  B.L. van der Waerden, [*Einführung in die algebraische Geometrie*]{}, Die Grundlehren der math. Wiss. in Einzeldarstellungen, Bd.51, Springer-Verlag, Berlin, 1973. (Zweite Auflage of the 1939 original). \[$\spadesuit$ p.223, Riemann-Roch in der Brill-Noetherschen Fassung, etc.\]  W. von Dyck, [*Beiträge zur Analysis situs. I Aufsatz, Ein- und zwidimensionale Mannigfaltigkeiten*]{}, Math. Ann. (1888), 457–512. \[$\spadesuit$ contains an account of what was known at that time (nearly definitive results) on the topology of surfaces, as well a historical account of the theory of foliation. The sole possible forerunner of that period is Poincaré 1885, which in our opinion (albeit not perfectly organized) is sometimes more digest than Dyck’s account, especially when it comes to the “Poincaré” index formula, which can perhaps only be regarded as anticipated by masters like Cauchy, Gauss, Riemann, Kronecker\]  ?. von Staudt, [*Geometrie der Lage*]{}, ??. \[$\spadesuit$ often cited by early worker in the topology of real curves, for the notion of ovals and pseudo-line, i.e. isotopy classification of a circle in the real projective plane $\spadesuit$ so cited e.g. in Harnack 1876 [@Harnack_1876], Hilbert 1891 [@Hilbert_1891_U-die-rellen-Zuege]\]  J. Wahl, [*Deformations of plane curves with nodes and cusps*]{}, Amer. J. Math. 96 (1974), 529–577. \[$\spadesuit$ cited e.g. in Shustin 1990/91 [@Shustin_1990/91-Geom-of-discr-alg-curve]\]  R.J. Walker, [*Algebraic Curves*]{}. Dover Publications, Inc., New York, 1962; unabridged and corrected reprint of the work first published as Princeton Mathematical Series [*13*]{}. Princeton University Press, Princeton, N.J., 1950.\[$\spadesuit$ often cited e.g. by Gudkov\]  J.L. Walsh, [*The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions*]{}, Bull. Amer. Math. Soc. 35 (1929), 499–544. \[$\spadesuit$ quoted via Axler’s review (BAMS) of Fisher’s book, for the harmonic conjugate as generally multiple-valued with periods\]$\bigstar$$\bigstar$$\bigstar$  J.L. Walsh, [*Interpolation and functions analytic interior to the unit circle*]{}, Trans. Amer. Math. Soc. 34 (1932), 523–556. \[$\spadesuit$ Pick-Nevanlinna like still in the disc but see Heins 1975 [@Heins_1975] for an extension subsuming (in principle) the theory of the Ahlfors map\]$\bigstar$  J.L. Walsh, [*Approximation by polynomials in the complex domain*]{}, Mémorial des Sci. Math. 73 (1935), 1–72. \[$\spadesuit$ formulates a general formalism of best approximation which encloses as special cases the least area interpretation of the Riemann mapping of Bieberbach 1914 [@Bieberbach_1914], as well as generalizations of Julia, and many other workers including Kubota, Wirtinger, Kakey, F. Riesz (cf. esp. p.61) $\spadesuit$ further on p.64 it is emphasized that (at time) virtually nothing was known for multiply-connected regions (this had to wait over Grunsky, Ahlfors, etc.)\]  J.L. Walsh, [*On the shape of level curves of Green’s function*]{}, Amer. Math. Monthly 44 (1937), 202–213. \[$\spadesuit$\]$\bigstar$$\bigstar$$\bigstar$  J.L. Walsh, [*The critical points of linear combinations of harmonic functions*]{}, Bull. Amer. Math. Soc. ?? (1948), 196–205. 47 \[$\spadesuit$ p.196: “In various extremal problems of function theory the critical points of linear combinations of Green’s functions and harmonic measures are of significance (See for instance M. Schiffer 1946; L.V. Ahlfors 1947 [@Ahlfors_1947].) $\spadesuit$ p.205: “In connection with the methods we are using, a remark due to Bôcher (1904) is appropriate: “The proofs of the theorems which we have here deduced from mechanical intuition can readily be thrown, without essentially modifying their character, into purely algebraic form. The mechanical problem must nevertheless be regarded as valuable, for it suggests not only the theorems but also the method of proof.” ”\]  J.L. Walsh, [*The location of critical points*]{}, Amer. Math. Soc. Colloq. Publ. 34, 1950. \[$\spadesuit$ Chap.VII is quoted in Jones-Marshall 1985 [@Jones-Marshall_1985] “for more information on the location of the critical points” \[of the Green’s function\]\]  J.L. Walsh, [*Note on least-square approximation to an analytic function by polynomials, as measured by a surface integral*]{}, Proc. Amer. Math. Soc. 10 (1959), 273–279.  J.L. Walsh, [*History of the Riemann mapping theorem*]{}, Amer. Math. Monthly 80 (1973), 270–276. \[$\spadesuit$ a brilliant essay, which on p.273 mentions briefly the counterexamples to the “naive” Dirichlet principle cooked by Prym 1871 and Hadamard 1906 (the precise links are not given but are Prym 1871 [@Prym_1871] and Hadamard 1906 [@Hadamard_1906])\]  S. Warschawsky, [*Über einige Konvergenzsätze aus der Theorie der konformen Abbildung*]{}, Nachr. Ges. Wiss. Göttingen (1930), 344–369. 60 $\bigstar$  H. Weber, [*Note zu\[m\] Riemann’s Beweis des Dirichlet’schen Prinzips*]{}, J. Reine Angew. Math. 71 (1870), 29–39. 60 \[$\spadesuit$ an attempt is made to complete the reasoning of Riemann to establish the Dirichlet principle $\spadesuit$ this work is quoted in Ahlfors-Sario’s masterpiece [@Ahlfors-Sario_1960], but Weber’s work seems to be subjected to serious objections (according to Zaremba 1910 [@Zaremba_1910]) including the basic one of Weierstrass about the existence of a minimum value for the Dirichlet integral $\spadesuit$ further \[as our attempt to make Zaremba’s objections more explicit\] on p.30 (line 4) Weber makes the tacit assumption that he can find a function $u$ matching the boundary values and of [*finite*]{} Dirichlet integral: this is however violently attacked by the Hadamard 1906 [@Hadamard_1906] counterexample of a boundary data all of whose matching functions explode to infinite Dirichlet integral $\spadesuit$ a weaker result of this type was already obtained by Prym 1871 [@Prym_1871] who gave a continuous function on the unit-circumference whose harmonic extension to the disc (existence via e.g. Poisson) has infinite Dirichlet integral $\spadesuit$ can we characterize such exploding functions? Maybe in terms of wild oscillations (can a such be differentiable (probably recall the wild functions à la Köpcke–Denjoy, etc.), $C^1$ (=continuously derivable), etc.) $\diamondsuit$ H. Weber albeit not a direct student of Riemann, was regarded as one of the efficient successor (e.g. by Thieme, compare Elstrodt-Ulrich [@Elstrodt-Ullrich_1999]). Weber played a pivotal rôle (joint with Dedekind) in editing Riemann’s Werke (including the Nachlass [@Riemann_1857_Nachlass]), and replaced Clebsch who desisted from this task due to health problems\]  H. Weber, [*Lehrbuch der Algebra*]{}, Bd.I und II. Friedrich Vieweg und Sohn Verlag, Braunschweig, 1898/99. \[$\spadesuit$ Galois theory made in Germany, etc.\]  H. Weber, [*Lehrbuch der Algebra*]{}, Bd.III. Friedrich Vieweg und Sohn Verlag, Braunschweig, 1908. \[$\spadesuit$ Vorwort (p.VII): “Dagegen habe ich, einem mehrfach an mich herangetretenen Wunsche entschprechend, einen Abri[ß]{} der Theorie der algebraischen Funktionen auf arithmetischer Grundlage beigefügt, der sich im wesentlichen an die Abhandlung von Dedekind und mir im 92.Bande von Crelles Journal anschlie[ß]{}t, aber durch Anwendung der Theroie dr Funktionale, auf die ich im zweiten Bande der Theorie der algebraischen Zahlen gegründet habe, wie mir scheint, eine Vereinfachung erreicht.”\]  G. Weichold, [*Ueber symmetrischen Riemann’sche Flächen und die Periodicitätsmoduln der zugehörigen Abel’schen Normalintegrale erster Gattung*]{}, Z. Math. Phys. 28 (1883), 321–351. \[$\spadesuit$ exposes the theory of Klein’s symmetric surfaces in full detail (basing the topological study upon the Möbius-Jordan classification [@Jordan_1866]), and do some more subtle things with period matrices $\spadesuit$ this latter object is re-treated in Klein 1892 [@Klein_1892_Realitaet], and will influence the work of Comessatti 1924/26 [@Comessatti_1924/26] $\diamondsuit$ Guido Weichold was a student of Klein, who seems to have been strongly attracted to the topic of symmetric Riemann surfaces through Klein’s lectures. Apparently, Weichold did not pursued his research on this topic\]  K. Weierstrass, [*Über das sogenannte Dirichletsche Princip*]{}. In: Werke vol. 2, Mayer & Müller, 49–54, 1895. gelesen in der Königl. Akademie der Wissenschaften am 14. Juli 1870. \[$\spadesuit$ a little objection to the Dirichlet principle, yet with desastrous repercussions $\spadesuit$ resurrection by Hilbert 1900, etc. [@Hilbert_1900] $\diamondsuit$ Karl Weierstrass needs not to be introduced. Formally a student of Gudermann, he came across the problem of Jacobi inversion, but unfortunately never published his solution (probably being slightly devanced by Riemann 1857 in this respect). Of course as the whole Riemann approach was for a long time subjected to critics, it would have been of prior interest to know what can be achieved through the pure Weierstrass conceptions collapsing to a sort of arithmetics of power series\]  K. Weierstrass, [*Vorlesungen 1875/76*]{}. In: Werke, Bd.IV. \[$\spadesuit$ algebraic and Abelian functions\]  A. Weil, [*The field of definition of a variety*]{}, Amer. J. Math. 78 (1956), 509–524. \[$\spadesuit$\]  A. Weil, [*Modules des surfaces de Riemann*]{}, Sém. Bourbaki, Mai (1958). \[$\spadesuit$ Teichmüller et cie.\]  G.G. Weill, [*Reproducing kernels and orthogonal kernels for analytic differentials on Riemann surfaces*]{}, Pacific J. Math. 12 (1962), 729–767. \[$\spadesuit$ refers to Ahlfors-Sario 1960 [@Ahlfors-Sario_1960] for the Bergman kernel on Riemann surfaces, other source includes Schiffer-Spencer 1954 [@Schiffer-Spencer_1954] $\diamondsuit$ Weill is a student of Sario (Ph.D.) ca. 1962\]  H.F. Weinberger, [*An isoperimetric inequality for the $N$-dimensional free membrane problem*]{}, J. Rat. Mech. Anal. 5 (1956), 633–636. \[$\spadesuit$ inspired by Szegö, but starts to give a more topological argument but the existence of balanced test functions; culminate to Fraser-Schoen 2011 [@Fraser-Schoen_2011], where the junction with the Ahlfors map is made explicit\]  R. Weinstock, [*Inequalities for a classical eigenvalue problem*]{}, J. Rat. Mech. Anal. 3 (1954), 745–753. \[$\spadesuit$ inspired by Szegö, but Steklov eigenvalues; culminate in Fraser-Schoen 2011 [@Fraser-Schoen_2011], where the junction with the Ahlfors map is made explicit\]  G. Weiss, [*Complex methods in harmonic analysis*]{}, Amer. Math. Monthly 77 (1970), 465–474. \[$\spadesuit$\]$\bigstar$$\bigstar$$\bigstar$  J. Wermer, [*Function rings and Riemann surfaces*]{}, Ann. of Math. (2) 67 (1958), 45–71. \[$\spadesuit$\]$\bigstar$$\bigstar$$\bigstar$  J. Wermer, [*Rings of analytic functions*]{}, Ann. of Math. (2) 68 (1958), 550–561. \[$\spadesuit$\]$\bigstar$$\bigstar$$\bigstar$  J. Wermer, [*Analytic disks in maximal ideal spaces*]{}, Amer. J. Math. 86 (1964), 161–170. \[$\spadesuit$\]$\bigstar$$\bigstar$$\bigstar$  H. Weyl. [*Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen*]{}, Math. Ann. 71 (1912), 441–479. \[$\spadesuit$ the so-called Weyl’s (asymptotic) law asserting that one can hear the area of a drum $\spadesuit$ naive conjecture \[ca. Mai 2011\] can this Weyl’s law be employed as tool to prove the Gromov filling area conjecture (eventually in conjunction with an Ahlfors map to make the usual conformal transplantation of vibratory modes, cf. e.g. Fraser-Schoen 2011 [@Fraser-Schoen_2011] for the first implementation of Ahlfors’ circle maps in spectral theory)\]  H. Weyl. [*Die Idee der Riemannschen Fläche*]{}, B.G. Teubner, Leipzig und Berlin 1913.  H. Weyl. [*Ueber das Pick-Nevanlinnasche Interpolationsproblem und sein infinitesimales Analogon*]{}, Ann. of Math. (2) 36 (1935), 230–254. \[$\spadesuit$\]  H. Weyl. [*The method of orthogonal projection in potential theory*]{}, Duke Math. J. 7 (1940), 411–440. 60 $\bigstar$  H. Whitney, [*Complex analytic varieties*]{}, Addison Wesley Publ. Company, Reading, Mass. 1972. \[$\spadesuit$\]  H. Widom. [*Extremal polynomials associated with a system of curves in the complex plane*]{}, Adv. Math. 3 (1969), 127–232. \[$\spadesuit$\]$\bigstar$$\bigstar$  H. Widom, [*$H_p$ sections of vector bundles over Riemann surfaces*]{}, Ann. of Math. (2) 94 (1971), 304–324. 60 \[$\spadesuit$ the geometric quintessence of the paper seems to be Lemma 6 (p.320), created with apparently some helping hand from Royden, and amounting to prescribe (modulo $2\pi$) the periods of the conjugate differential of a superposition of (modified) Green’s functions $\spadesuit$ albeit Ahlfors 1950 [@Ahlfors_1950] is not directly cited, a certain technological “air de famille” transpires throughout the execution $\spadesuit$ alas, Widom’s argument (pp.320–1) seems to give only a poor control upon the number of poles $\zeta_k$ required, and is therefore unlikely to reprove Ahlfors 1950 [@Ahlfors_1950] by specializing to the trivial line bundle case $\spadesuit$ but of course, Widom do something quite grandiose and so the real depth of the work cannot be appreciated by focused comparison with Ahlfors 1950 [@Ahlfors_1950] $\spadesuit$ in particular Widom (re)discover a certain class of open Riemann surfaces (alias of Parreau-Widom) type which are characterized by a moderate growth of the Betti number during the cytoplasmic expansion generated by levels of the Green’s function, which turns out to be a very distinguished class of Riemann surfaces where paradigms like the corona, etc. extends reasonably\]  R.J. Wille, [*Sur la transformation intérieure d’une surface non orientable dans le plan projectif*]{}, Indagationes Math. 56 (1953), 63–65. \[$\spadesuit$ probably a nonorientable avatar of Stoïlow’s work, and maybe related to Witt 1934 [@Witt_1934]\]$\bigstar$$\bigstar$$\bigstar$  G. Wilson, [*Hilbert’s sixteenth problem*]{}, Topology 17 (1978), 53–73. \[$\spadesuit$ discusses Klein’s orthosymmetry (as dividing curves) and ask whether the dividing character of a real plane curve may be recognized by sole inspection of its real locus, p.67: “I do not know if one can tell whether or not $X$ divides by examining only the real part $X_{\Bbb R}\subset {\Bbb R}P^2$.” $\spadesuit$ Our partisan answer (compare Gabard 2004 [@Gabard_2004 p.7]) is a decided [*yes*]{}, posited by Ahlfors theorem $\spadesuit$ however this is pure existence theory and some algorithmic recipes still deserve to be implemented at the occasion. For related efforts cf. e.g. Kalla-Klein 2012 [@Kalla-Klein_2012]\]  A. Wiman, [*Über die reellen Züge der ebenen algebraischen Kurven*]{}, Math. Ann. 90 (1923), 222–228. \[$\spadesuit$\]  J. Winkelmann, [*Non-degenerate maps and sets*]{}, Math. Z. (2005), 783–795. 50 \[$\spadesuit$ \[27.09.12\] Ahlfors 1950 [@Ahlfors_1950] is cited, yet not within the main-body of the text, but its companion Bell 1992 [@Bell_1992-Book] is cited for the same purpose. In fact Winkelmann’s article only uses the planar case of the Ahlfors function, hence citing Ahlfors 1947 [@Ahlfors_1947] may have been more appropriate (yet recall that the latter article contains a little gap fixed in Ahlfors 1950 [@Ahlfors_1950 p.123, footnote]) $\spadesuit$ the author gives the following lovely application of the Ahlfors map of a plane bounded domain $\spadesuit$ call a holomorphic map [*dominant*]{} if it has dense image, and a complex manifold universally dominant (UDO) if it admits a dominant map to any irreducible complex space. The author shows first that the unit disc $\Delta$ is UDO (Cor.3, p.786), and via the Ahlfors function this implies more generally that any complex manifold admitting a nonconstant bounded analytic function (BAF) is UDO. Here are the details. $\spadesuit$ first if the complex manifold is UDO, then it dominates the unit disc $\Delta$, and so it carries a nonconstant BAF. Conversely, let $f\colon X \to {\Bbb C}$ be a nonconstant BAF then $f(X)$ is a bounded domain. (It is crucial here to assume $X$ connected, for $X$ the disjoint union of say two Riemann spheres carries a nonconstant BAF, yet fails to be UDO.) Now observe the following fact. [**Lemma.**]{} [*The Ahlfors map $f_a$ at $a$ of the bounded domain $G\ni a$ is dominant.*]{}—[*Proof.*]{} If not, then the map $f_a\colon G\to \Delta$ misses a little disc $D\subset \Delta$ not overlapping the origin (recall that $f_a(a)=0$). Since the identity map restricted to the ring $\Delta-\overline{D}$ is bounded-by-one (hence admissible in the extremal problem), it follows that the Ahlfors map for the ring centered at $0$, say $g_0$, has a derivative with modulus strictly larger than unity, i.e. $\vert g'_0(0)\vert > \vert (id)' (0) \vert=1$. But then the composed map $(g_0 \circ f_a)$ effects the stretching $\vert (g_0 \circ f_a )'(a) \vert=\vert (g_0'(f_a(a)) \cdot f_a'(a) \vert=\vert g_0'(0) \cdot f_a'(a) \vert> \vert f_a'(a) \vert$, violating the extremal property of $f_a$. q.e.d.—$\spadesuit$ At this stage it may be observed that the Ahlfors map of a bounded domain needs not be surjective. Consider indeed the unit disc $\Delta$ punctured at say $1/2$, then the Ahlfors function of $\Delta-\{1/2\}$ centered at $0$, denoted $f_0$, is the identity (up to a rotation). Indeed, since a (pointlike) puncture is a removable singularity for BAF any function admitted in the extremal problem extends analytically across the whole unpunctured disc. More generally, the Ahlfors map is insensitive to the puncturing of a removable singularity (alias Painlevé null sets), e.g. Cantor’s $1/4$-set described in Garnett 1970 [@Garnett_1970] $\spadesuit$ back to Winkelmann’s argument, the above lemma applied to $G:=f(X)$ gives a dominant map $f_a$ to the disc, hence a dominant map $X\to f(X)\to \Delta$. Summarizing: [*any complex manifold $X$ supporting a nonconstant BAF dominates the disc.*]{} $\spadesuit$ Perhaps one could try to improve this by using the surjectivity of the Ahlfors map for a domain of finite connectivity (without pointlike boundaries), assuming e.g. that $X$ has a finitely generated fundamental group $\pi_1$. Alas, this does not seem to imply automatically that $\pi_1(f(X))$ is of finite generation and we need of course to control the shape of the image $f(X)$, which has to be a finite region bounded by Jordan curves $\spadesuit$ finally since the disc $\Delta$ dominates any irreducible complex space $Y$ (of course the definition of the latter must be calibrated so as to avoid non-metric complex manifolds of Calabi-Rosenlicht of the Prüfer type, at least those specimens which are not separable), the composition $X\to f(X) \to \Delta \to Y$ yields the desired dominant map showing that $X$ is UDO. This completes Winkelmann’s proof.\]  W. Wirtinger, [*Untersuchungen über Thetafunktionen*]{}, Teubner, 1895.  W. Wirtinger, [*Algebraische Funktionen und ihre Integrale*]{}, Enzykl. d. math. Wiss. $2_2$ (1902), 115–175. 60  W. Wirtinger, [*Über die konforme Abbildung der Riemannschen Flächen durch Abelsche Integrale besonders bei $p=1,2$*]{}, Denkschr. Wien (1909), 22pp. 60  W. Wirtinger, [*Über eine Minimalaufgabe im Gebiete der analytischen Funktionen*]{}, Monatsh. Math. u. Phys. 39 (1932), 377–384. \[$\clubsuit$ quoted p.269 of Schiffer 1950 [@Schiffer_1950-Appendix-Courant] for a the first notice of a certain reproducing kernel property, also quoted in Bergman 1950 [@Bergman_1950] $\spadesuit$ poses (and solves via the Green’s function) the problem of the best analytic approximation $f$ in $L^2$-norm $\int\int_B \vert f- \Phi \vert^2 d \omega$ of a given continuous function $\Phi$\]  W. Wirtinger, [*Zur Theorie der konformen Abbildung mehrfach zusammenhängender ebener Flächen*]{}, Abh. Preu[ß]{}. Akad. Wiss. math.-nat. Kl. 4 (1942), 1–9. 60, 78 \[$\clubsuit$ reproves the theorem of Riemann-Schottky-Bieberbach-Grunsky(=RSBG), i.e. the schlicht(artig) case of the Ahlfors map, via algebraic functions (i.e. Riemann-(Roch) essentially)\]  E. Witt, [*Zerlegung reeller algebraischer Funktionen in Quadrate*]{}, J. Reine Angew. Math. 171 (1934), 4–11. \[$\clubsuit$ contains a sort of non-orientable version of the Riemann/Ahlfors map. Subsequent developments in Geyer 1964/67 [@Geyer_1964-67] and Martens 1978 [@Martens_1978]\]  E. Witten, [*Two dimensional gravity and intersection theory on the moduli space*]{}, Survey in Differential Geometry, Leigh Univ., 1991, 243–310. \[$\spadesuit$\]  J.J. Wolfart, [*The ’obvious’ part of Belyi’s theorem and Riemann surfaces with many automorphisms* ]{}, In: [*Geometric Galois Actions, 1*]{}, London Math. Soc. Lecture Note Series 242, 1997. \[$\clubsuit$\]  S. Wolpert, [*The length spectra as moduli for compact Riemann surfaces*]{}, Ann. of Math. (2) (1979). \[$\clubsuit$\]  D.V. Yakubovich, [*Real separated algebraic curves, quadrature domains, Ahlfors type functions and operator theory*]{}, J. Funct. Anal. 236 (2006), 25–58. 50 \[$\clubsuit$ contains also (after Alling-Greenleaf [@Alling-Greenleaf_1969]) a clear-cut formulation of the Klein-Ahlfors correspondence: i.e. a curve is dividing/separating iff it maps to the line in a totally real fashion (i.e. real fibres are entirely real)\]  A. Yamada, [*On the linear transformations of Ahlfors functions*]{}, Kōdai Math. J. 1 (1978), 159–169. 50 \[$\clubsuit$ evaluate the degree of the Ahlfors function at the Weierstrass points of a non-planar hyperelliptic membrane as taking the maximum value permissible, i.e. $r+2p=g+1$\]  A. Yamada, [*A remark on the image of the Ahlfors function*]{}, Proc. Amer. Math. Soc. 88 (1983), 639–642. \[$\spadesuit$ domains of infinite connectivity $\spadesuit$ p.639 (abstract extract): “By an example we show that the complement in the unit disc of the image of the Ahlfors function for $ \Omega $ and $ p$ can be a fairly general set of logarithmic capacity zero.”\]  A. Yamada, [*Ahlfors functions on Denjoy domains*]{}, Proc. Amer. Math. Soc. 115 (1992), 757–763. \[$\spadesuit$ domains of infinite connectivity $\spadesuit$ p.757: “The main result of our paper gives a necessary and sufficient condition for a subset of the unit disc to be the omitted set of the Ahlfors function $F$ for some maximal Denjoy domain and $\infty$ such that $F$ is a covering onto its image. As a corollary we give examples of omitted sets of Ahlfors functions that have positive logarithmic capacity.” $\spadesuit$ \[05.10.12\] if I don’t mistake Yamada’s example thus answers a question by Minda 1981 [@Minda_1981-image-Ahlfors-fct p.755] about knowing if the Ahlfors function can “omit an uncountable set”. (Recall indeed that sets of zero logarithmic capacity are stable under countable unions, cf. p.762, where Yamada refers to Tsuji 1959 [@Tsuji_1959-BOOK/Chelsea1975 p.57].)\]  A. Yamada, [*Ahlfors functions on compact bordered Riemann surfaces*]{}, J. Math. Soc. Japan 53 (2001), 261–283. 50 \[$\clubsuit$ establish a conjecture of Gouma 1998 [@Gouma_1998] to the effect that the Ahlfors degree of a hyperelliptic membrane centered outside the Weierstrass points always degenerates to the minimum value $2$\]  P.C. Yang, S.-T. Yau, [*Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds*]{}, Ann. Sc. Norm. Sup. di Pisa (4) 7 (1980), 55–63. \[$\spadesuit$ applies conformal branched covering of [*closed*]{} Riemann surfaces to the sphere and the trick of conformal transplantation to generate test functions yielding an estimate of the first three Laplace eigenvalues of a closed Riemann surface considered as a vibrating membrane. Inspiration Szëgo, Hersch 1970 [@Hersch_1970], but goes somewhat deeper as there is no fear of multi-sheetedness $\spadesuit$ for an adaptation of Yang-Yau’s method to bordered surfaces via the Ahlfors map see Fraser-Schoen 2011 [@Fraser-Schoen_2011], or some derived products like Gabard 2011 [@Gabard_2011] or Girouard-Polterovich 2012 [@Girouard-Polterovich_2012]\]  O. Yavuz, [*Invariant subspaces for Banach space operators with a multiply connected spectrum*]{}, Integr. Equ. Oper. Theory 58 (2007), 433–446. \[$\spadesuit$ p.439–440, the Ahlfors function (via Fisher’s book 1983 [@Fisher_1983]) is employed to extend a result on the existence of invariant subspaces for operators with a multiply-connected spectrum (previously known when the spectrum contained the unit-circle)\]  O. Yavuz, [*A reflexivity result concerning Banach space operators with a multiply connected spectrum*]{}, Integr. Equ. Oper. Theory 68 (2010), 473–485. \[$\spadesuit$ p.475–6, Ahlfors function via Fisher’s book 1983 [@Fisher_1983]\]  N.X. Yu, [*On Riesz transforms of bounded function of compact support*]{}, Michigan Math. J. 24 (1977), 169–175. \[$\spadesuit$ p.170, Ahlfors function via Carleson’s book 1967 [@Carleson_1967-book ChapterVIII]\]  L. Zalcman, [*Analytic capacity and rational approximation*]{}, Lecture Notes in Math. 50, Springer-Verlag, Berlin–New York, 1968. 50 \[$\spadesuit$\]  L. Zalcman, [*Analytic functions and Jordan arcs*]{}, Proc. Amer. Math. Soc. 19 (1968), 508. \[$\spadesuit$\]  L. Zalcman, [*Bounded analytic functions on domains of infinite connectivity*]{}, Trans. Amer. Math. Soc. 144 (1969), 241–270. \[$\spadesuit$\]  K. Zarankiewicz, [*Sur la représentation conforme d’un domaine doublement connexe sur un anneau circulaire*]{}, C.R. Acad. Sci. Paris 198 (1934), 1347–1349. \[$\spadesuit$ Seidel’s summary: method for the effective construction of the conformal mapping of a doubly connected domain upon a circular ring, via orthogonal systems (Bergman kernel) $\spadesuit$ consider (with Bergman \[no precise cross-ref.\]) the problem of maximizing the modulus of $f(t)$ among functions with $L^2$-norm bounded by $1$: $\int\int_B {\vert f(z)\vert}^2\le 1$\]  K. Zarankiewicz, [*Über ein numerisches Verfahren zur konformen Abbildung zweifach zusammenhängender Gebiete*]{}, Zeitschr. f. angew. Math. u. Mech. 14 (1934), 97–104. \[$\spadesuit$ Seidel’s summary: a detailed account is given of the method indicated in Zarankiewicz 1934 [@Zarankiewicz_1934], i.e. Bergman kernel style numerical device to compute the conformal map of a doubly connected domain $\spadesuit$ oft quoted e.g. in Lehto 1949 [@Lehto_1949], Bergman 1950 [@Bergman_1950]\]$\bigstar$$\bigstar$$\bigstar$  S. Zaremba, [*Sur le calcul numérique des fonctions demandées dans le problème de Dirichlet et le problème hydrodynamique*]{}, Bull. Inst. Acad. Sci. Cracovie (1908), 125–195. \[$\spadesuit$ quoted (e.g.) in Lions 2000/02 [@Lions_2000/02] as one of the very early apparition of the notion of reproducing kernel\]  S. Zaremba, [*Sur le principe de Dirichlet*]{}, Acta Math. ? (1910), 293–316. \[$\spadesuit$ Hadamard’s 1906 [@Hadamard_1906] counterexample to the Dirichlet principle is cited (but not the earlier one of Prym 1871 [@Prym_1871]) and further (p.294) asserts that Weber’s 1869/70 [@Weber_1870] attempt to consolidate Riemann’s proof is subjected to serious objections $\spadesuit$ unfortunately, Zaremba does not make explicit any objection, but it is implicit that he has in mind the Weierstrass critique (of a functional not achieving a minimum) and further Weber’s tacit assumption that the Dirichlet integral is finite is violently attacked by the Hadamard 1906 [@Hadamard_1906] counterexample of a boundary data all of whose matching functions have infinite Dirichlet integral (of course, Prym 1871 [@Prym_1871] is a sufficient torpedo to destroy completely Weber’s argumentation) $\spadesuit$ notice that Arzelà 1897 [@Arzela_1897] has to be counted as a forerunner of Hilbert’s triumph of all the difficulty of the question (in certain particular cases), and mentions the remarkable extensions due to B. Levi 1906 [@Beppo-Levi_1906], Fubini 1907 [@Fubini_1907] and Lebesgue 1907 [@Lebesgue_1907], while proposing to recover those results through a simpler method without loosing anything essential to their generality\]  R. Zarrow, [*Anticonformal automorphisms of compact Riemann surfaces*]{}, Proc. Amer. Math. Soc. 54 (1976), 162–164. \[$\spadesuit$ cf. little corrections in Costa 1996 [@Costa_1996-PAMS]\]  H.G. Zeuthen, [*Sur les formes différentes des courbes du quatrième ordre*]{}, Math. Ann. 7 (1874), 410–432. (+Tafel I, II, Fig.1,2,3,4,5) \[$\spadesuit$ a work who inspired much of Klein investigation $\spadesuit$ cite von Staudt (Geometrie der Lage) $\spadesuit$ uses the term “ovale” $\spadesuit$ p.411: “Une courbe du quatrième ordre ([*quartique*]{}) a, au plus, quatre branches externes l’une à l’autre, ou deux branches dont l’une se trouve dans la partie du plan interne à l’autre, et dans ce dernier cas la branche interne ne peut avoir des tangentes doubles ou d’inflexion.—Car s’il en était autrement on pourrait construire des coniques rencontrant la courbe en plus de 8 points, ou des droites la rencontrant en plus de 4 points.” (This is the sort of Bézout-type argument out of which will emerge the Harnack inequality 1876 [@Harnack_1876]), yet the full intrinsic grasp (especially the interpretation via Riemann surfaces) will be effected through Klein’s work 1876 [@Klein_1876] $\spadesuit$ p.412: “Nous appelons ici réelle toute courbe dont l’équation ne contient que des coefficients réels.” $\spadesuit$ p.428, cite Geiser and the yoga between cubic surface and quartic curves, which is instrumental in Klein 1876 to make the rigid-isotopy classification of quartic curves.\]  H.G. Zeuthen, [*Études des propriétés de situation des surfaces cubiques*]{}, Math. Ann. 8 (1874/75), 1–30. \[$\spadesuit$ also quoted by Klein 1876, as to complete the rigid-isotopy classification of quartics by reduction to the case of cubics (Schläfli 1863 [@Schlaefli_1863] and Klein 1873 [@Klein_1873-Uber-Flächen-dritter-Ordn])\]  M. Zhang, Y. Li, W. Zeng, X. Gu, [*Canonical conformal mapping for high genus surfaces with boundaries*]{}, Computers Graphics 36 (2012), 417–426. \[$\spadesuit$ completely in line with our present topic, and use high powered machinery like Koebe’s iteration and (Yau-Hamilton’s) Ricci flow for conformal theoretic purposes $\spadesuit$ can we adapt such algorithms to the (Ahlfors) circle map\]  V.A. Zmorovič, [*The generalization of the Schwarz formula for multiply connected domains*]{}, (Ukrainian) Dokl. Akad. Nauk Ukrain. SSR 7 (1962), 853–856. \[$\spadesuit$ quoted in Khavinson 1984 [@Khavinson-Dimitri_1984]\]$\bigstar$$\bigstar$  V.I. Zvonilov, [*Complex topological characteristics of real algebraic curves on surfaces*]{}, Funkt. Anal. Prilozhen. 16 (1982), 56–57; English transl. in Funct. Anal. Appl. 16 (1982), 202–204. \[$\spadesuit$\]$\bigstar$  V.I. Zvonilov, [*Complex orientations of real algebraic curves with singularities*]{}, Dokl. Akad. Nauk SSSR 268 (1983), 22–26; English transl., Soviet Math. Dokl. 27 (1983), 14–17. \[$\spadesuit$\]$\bigstar$  V.I. Zvonilov, [*Complex topological characteristics of real algebraic curves on a hyperboloid and an ellipsoid*]{}, Funkt. Anal. Prilozhen. ?? (1986), ?–?; English transl. in Funct. Anal. Appl. ?? (1986), ?–?. \[$\spadesuit$\]$\bigstar$  Zal68 L. Zalcman, [*Analytic capacity and rational approximation*]{}, Lecture Notes in Math. 50, Springer-Verlag, Berlin–New York, 1968. 50 \[$\spadesuit$\]  Zal68b L. Zalcman, [*Analytic functions and Jordan arcs*]{}, Proc. Amer. Math. Soc. 19 (1968), 508. \[$\spadesuit$\]  Zal69 L. Zalcman, [*Bounded analytic functions on domains of infinite connectivity*]{}, Trans. Amer. Math. Soc. 144 (1969), 241–270. \[$\spadesuit$\]  Zar34 K. Zarankiewicz, [*Sur la représentation conforme d’un domaine doublement connexe sur un anneau circulaire*]{}, C.R. Acad. Sci. Paris 198 (1934), 1347–1349. \[$\spadesuit$ Seidel’s summary: method for the effective construction of the conformal mapping of a doubly connected domain upon a circular ring, via orthogonal systems (Bergman kernel) $\spadesuit$ consider (with Bergman \[no precise cross-ref.\]) the problem of maximizing the modulus of $f(t)$ among functions with $L^2$-norm bounded by $1$: $\int\int_B {\vert f(z)\vert}^2\le 1$\]  Zar34b K. Zarankiewicz, [*Über ein numerisches Verfahren zur konformen Abbildung zweifach zusammenhängender Gebiete*]{}, Zeitschr. f. angew. Math. u. Mech. 14 (1934), 97–104. \[$\spadesuit$ Seidel’s summary: a detailed account is given of the method indicated in Zarankiewicz 1934 [@Zarankiewicz_1934], i.e. Bergman kernel style numerical device to compute the conformal map of a doubly connected domain $\spadesuit$ oft quoted e.g. in Lehto 1949 [@Lehto_1949], Bergman 1950 [@Bergman_1950]\]$\bigstar$$\bigstar$$\bigstar$  Zar08 S. Zaremba, [*Sur le calcul numérique des fonctions demandées dans le problème de Dirichlet et le problème hydrodynamique*]{}, Bull. Inst. Acad. Sci. Cracovie (1908), 125–195. \[$\spadesuit$ quoted (e.g.) in Lions 2000/02 [@Lions_2000/02] as one of the very early apparition of the notion of reproducing kernel\]  Zar10 S. Zaremba, [*Sur le principe de Dirichlet*]{}, Acta Math. ? (1910), 293–316. \[$\spadesuit$ Hadamard’s 1906 [@Hadamard_1906] counterexample to the Dirichlet principle is cited (but not the earlier one of Prym 1871 [@Prym_1871]) and further (p.294) asserts that Weber’s 1869/70 [@Weber_1870] attempt to consolidate Riemann’s proof is subjected to serious objections $\spadesuit$ unfortunately, Zaremba does not make explicit any objection, but it is implicit that he has in mind the Weierstrass critique (of a functional not achieving a minimum) and further Weber’s tacit assumption that the Dirichlet integral is finite is violently attacked by the Hadamard 1906 [@Hadamard_1906] counterexample of a boundary data all of whose matching functions have infinite Dirichlet integral (of course, Prym 1871 [@Prym_1871] is a sufficient torpedo to destroy completely Weber’s argumentation) $\spadesuit$ notice that Arzelà 1897 [@Arzela_1897] has to be counted as a forerunner of Hilbert’s triumph of all the difficulty of the question (in certain particular cases), and mentions the remarkable extensions due to B. Levi 1906 [@Beppo-Levi_1906], Fubini 1907 [@Fubini_1907] and Lebesgue 1907 [@Lebesgue_1907], while proposing to recover those results through a simpler method without loosing anything essential to their generality\]  Zeu74 H.G. Zeuthen, [*Sur les formes différentes des courbes du quatrième ordre*]{}, Math. Ann. 7 (1874), 410–432. (+Tafel I, II, Fig.1,2,3,4,5) \[$\spadesuit$ a work who inspired much of Klein investigation $\spadesuit$ cite von Staudt (Geometrie der Lage) $\spadesuit$ uses the term “ovale” $\spadesuit$ p.411: “Une courbe du quatrième ordre ([*quartique*]{}) a, au plus, quatre branches externes l’une à l’autre, ou deux branches dont l’une se trouve dans la partie du plan interne à l’autre, et dans ce dernier cas la branche interne ne peut avoir des tangentes doubles ou d’inflexion.—Car s’il en était autrement on pourrait construire des coniques rencontrant la courbe en plus de 8 points, ou des droites la rencontrant en plus de 4 points.” (This is the sort of Bézout-type argument out of which will emerge the Harnack inequality 1876 [@Harnack_1876]), yet the full intrinsic grasp (especially the interpretation via Riemann surfaces) will be effected through Klein’s work 1876 [@Klein_1876] $\spadesuit$ p.412: “Nous appelons ici réelle toute courbe dont l’équation ne contient que des coefficients réels.”\]  Zha12 M. Zhang, Y. Li, W. Zeng, X. Gu, [*Canonical conformal mapping for high genus surfaces with boundaries*]{}, Computers Graphics 36 (2012), 417–426. \[$\spadesuit$ completely in line with our present topic, and use high powered machinery like Koebe’s iteration and (Yau-Hamilton’s) Ricci flow for conformal theoretic purposes $\spadesuit$ can we adapt such algorithms to the (Ahlfors) circle map\]  Zmo62 V.A. Zmorovič, [*The generalization of the Schwarz formula for multiply connected domains*]{}, (Ukrainian) Dokl. Akad. Nauk Ukrain. SSR 7 (1962), 853–856. \[$\spadesuit$ quoted in Khavinson 1984 [@Khavinson-Dimitri_1984]\]$\bigstar$$\bigstar$  Alexandre Gabard Université de Genève Section de Mathématiques 2-4 rue du Lièvre, CP 64 CH-1211 Genève 4 Switzerland alexandregabard@hotmail.com [^1]: =Fusion in Klein’s prose when viewing all his work (and that of Sophus Lie) as being merely a Galois-Riemann synthesis. [^2]: Best example thereof, read Borsuk’s article ca. 1936 where a contractible compactum lacking the fixed-point property is presented. If you have just the boring (unreadable) formulas of Borsuk you understand nothing, but if you know the picture that the space in question is a crumpled-cube spiraling twice around itself as pictured by Bing, you start to believe why the fact holds true. [^3]: Source=H. Cremer, Erinnerungen an Paul Koebe, Jahresber. DMV, 1968, p.160. (Mitteilung von Heinrich Behnke). [^4]: For more historical details on the theory of quasiconformal mappings compare Ahlfors 1984 [@Ahlfors_1984-The-Joy] or Lehto 1998 [@Lehto_1998]. \[02.10.12\] Alas we were not as yet able to show any deep connection between the theory of Ahlfors circle maps and that of quasiconformal maps, yet it is not unlikely that such a connection is worth studying, more in Section \[sec:question\]. [^5]: Prose borrowed by Louis de Branges. [^6]: This nomenclature is used by Hajek, compare some arXiv preprints of the author joint with Gauld. [^7]: There is a letter form Jordan to Lebesgue saying roughly: “Perséverez dans vos recherches mathématiques, vous allez y éprouver beaucoup de plaisirs, mais il va vous falloir apprendre à y gouter seul, car en général les géomètres ne se lisent même pas entre eux-mêmes.” (quoted by pure memory, hence highly unreliable). [^8]: Apparently via H.A. Schwarz, compare Bieberbach 1925 [@Bieberbach_1925]. [^9]: Notice that Ahlfors never quote Teichmüller 1981 [@Teichmueller_1941], except in Ahlfors-Sario 1960 [@Ahlfors-Sario_1960], where also all the Italian works of Matildi 1945/48 [@Matildi_1945/48] and Andreotti 1950 [@Andreotti_1950] are cited. [^10]: Gabard micro-comment: Here the last edition of Riemann’s Werke contains a little misprint $F$ instead of the obvious $T$, not present e.g., in the French translation of Riemann by Laugel, Paris 1898. [^11]: Joke of Ivan Babenko, yet irritating the western auditors coming down from the “alpage”. [^12]: This is not explicitly specified in the paper, but is the (common) jargon in Klein surface theory, probably due to Alling-Greenleaf 1971 [@Alling-Greenleaf_1971]. [^13]: Of course this can hardly be taken seriously, in view of the messy nature of the present text! [^14]: Perhaps it would be more corrected to say “along” here. Compare in this respect Ahlfors, p.108, the text just preceding footnote 3) [^15]: Here our argument shorten slightly the prose of Ahlfors, hopefully without loosing in precision?! [^16]: In the combinatorial sense, by opposition to the topological sense. [^17]: Since Petrovskii 1938 [@Petrowsky_1938] it is customary to call an $M$-curve, any curve realizing Harnack’s bound $r\le g+1$ of 1876. [^18]: This concept is not really meaningful for $M$-curves. [^19]: \[28.03.13\] I would personally be much interested, if someone can guess more explicitly what Rohlin had in mind at this place! [^20]: This shows that Klein anticipated the phenomenon of total reality. [^21]: Alas the word “oval” is quite ambiguous, as it is either just a real component of the abstract curve, or sometimes used in the much more specific sense of a component of a plane curve which is null-homotopic (equivalently bounds a disc) in ${\Bbb R}P^2$. Whenever we use the term oval in this abstract sense, due to a lack of better synonym (in German there is a good one “Zug/Züge”), we write it “entre guillemet” (=inverted comma in English, according to my Dictionary). [^22]: This appellation is now a common joke in Geneva, based on a mixture of the writer’s name with themuch more eminent Paul Garabedian, the notorious student of Ahlfors, Schiffer, Bergman, who seems to have played a pivotal rôle in the ultimate shape of Ahlfors theorem, as published in 1950 [@Ahlfors_1950]. [^23]: \[30.03.13\] Strictly speaking I do not know how to proof this, but hope this to be a triviality of basic algebraic geometry. [^24]: Unpublished, but see Rohn 1911 [@Rohn_1911], 1913 [@Rohn_1913], yet not judged complete by Gudkov 1974. [^25]: Prose borrowed by Jack Milnor, when he speaks about non-metric manifold, cf. his preprint on foliated bundles. [^26]: I.e. identity. [^27]: Thom learned Sard from de Rham, cf. the 1954 Commentarii article. [^28]: \[29.03.13\] This is just a very special case of a more general satellite principle, cf. Sec.\[satellite-total-reality:sec\]. [^29]: Omit this bracketing for it is just to refer to Gudkov’s notation. [^30]: I.e. the present text as it was on the date of the 09.01.13, meanwhile pagination may have changed. [^31]: This is not perfectly true, as I exposed Ahlfors theorem to Oleg Viro during its last visit in Geneva (ca. 2010–11, his talk on fields of char 1, Connes, tropical geometry, etc.), yet probably my explanations where so obscure that Oleg immediately forgot about it. [^32]: =Vladimir Abramovich Rohlin, of course. [^33]: Sic, singular or plural? Not so important of course. [^34]: i.e. Lemma \[Klein-Marin:lem\], warning meanwhile the numbering may changes, but the one in this footnote is automatic (hence the right one) [^35]: “isotopie” of course. [^36]: Meanwhile this numbering may have changed into Quote \[Klein\_1876-niemals-isolierte:quote\]. [^37]: Not clear how to interpret this? Does it mean that Shustin’s claim is wrong, or simply that this scheme is an $(M-1)$-scheme. My question was whether this $(M_1)$-scheme is realized algebraically, of course. Yet, I admit that my question was a bit ill posed. [^38]: \[24.01.13\] On reading Viro’ survey 1989/90 [@Viro_1989/90-Construction p.1085, 2.5.H], one should easily locate the source for this assertion. References seems to be Viro 1983 [@Viro_1983/84-new-prohibitions], and the survey Viro 1986 [@Viro_1986/86-Progress]. As explained there (Viro 1989/90, p.1085) this is a prohibition not coming from topology, but from Bézout. In fact this result is mentioned again in Viro 1989/90 [@Viro_1989/90-Construction p.1126, 5.3.E], with the exact cross-reference as being Viro 1983 [@Viro_1983/84-new-prohibitions] [^39]: \[30.01.13\] This is another very interesting (inedited?) piece of information, as to my knowledge this never appeared under the (printed) pen of V.A. Rohlin. So here Viro tell us something very interesting not yet available in print (as far as I know). [^40]: \[30.01.13\] This is probably true yet this requires a more highbrow dissipation theory, than in Gudkov’s second existence proof which apart from the trick with Cremona uses only dissipation of ordinary nodes à la Brusotti 1921 (so-called $A_1$-singularities). [^41]: \[30.01.13\] This alas turned out to be quite foiled as the invisible part of the discriminant as real codimension 2. [^42]: \[26.03.13\] Alas as spotted by Th. Fiedler this turned out to be wrong as it was based on the tacit supposition that the Arnold surface is always orientable, compare Fiedler’s letter dated \[21.03.13\]. [^43]: \[29.03.13\] A nearby corroboration of Rohlin’s claim is now available in Le Touzé 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics]. [^44]: \[30.03.13\] This is not exactly Viro’s opinion, cf. his letter in Sec.\[e-mail-Viro:sec\]. [^45]: I presume this list is not exhaustive, as Shustin’s scheme above ought to be also realized? If I have well understood the former “It is.”. [^46]: This is Viro’s notation, and mean 6 ovals enveloped in one, and this thrice. So 21 ovals. [^47]: \[24.01.13\] The exact reference for this result is Viro 1983 [@Viro_1983/84-new-prohibitions] [^48]: Our (non-standard) terminology: V. Kharlamov explained us (cf. Sec.\[e-mail-Viro:sec\]) that this phenomenon was quite crucial a motivation when Morosov suspected some anomaly in Gudkov’s initial solution to Hilbert’s 16th problem along the lines expected by Hilbert. Compare for more Viatcheslav’s e-mail in Sec.\[e-mail-Viro:sec\], and his terminology “partner relationship”. \[01.04.13\] It would be interesting to know if the low Gudkov-schemes $\frac{5}{1}4$, and $\frac{5}{1}3$ can be constructed from their mirrors. [^49]: This conclusion actually holds true unconditionally as follows from Rohlin’s formula $2(\Pi^{+}-\Pi^{-})=r-k^2$. [^50]: We often commit an abuse of language, as we should say one chamber residual to the discriminant. Such an abuse is harmless like when speaking of the group of a knot, when it is really that of its complement. [^51]: After several permutation in this text, it is not clear anymore what was the “previous section”, but checking dates and contents this can be almost surely identified as Sec.\[application-of-CCC:sec\]. [^52]: I should acknowledge the assistance of my cousin Élias Boulé-Schneider for several discussions on such topics, like topographical discussions about Hilbert’s 16th. [^53]: We use “smooth” as an abridgement of nonsingular in the algebro-geometric sense, or if you prefer smoothness of the complexification, but not merely of the real locus. So a curve may look smooth while having imaginary conjugate singularities. Such a curve is not considered as smooth by us. [^54]: I borrow this jargon from Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000]. [^55]: Indeed, we remember well, cf. e.g. EDM=Encyclopaedic Dictionary of Mathematics 1968/87 [@EDM_1968/87 p.46, Art. 13 G] “\[…\] the form $f$ is anisotropic, i.e. the homogeneous equation $f=0$ has no solution other than zero in $k$”. Or cf. Serre’s “Cours d’arithmétique”, 1970–1977, who seems clever enough to avoid the jargon, yet speaks of isotropic for quadratic forms. I don’t know who coined the term (in arithmetics), maybe Minkowski, Hilbert, Weil? Ask a competent arithmetician. [^56]: Joke of Misha Gromov (yet another notorious student of V.A. Ro\[k\]hlin). [^57]: Student of E. Schmidt, himself student of D. Hilbert. So we are not to far apart form the 16th problem. [^58]: But widely anticipated by Poincaré, Tietze, Brouwer, and many others combinatorial topologists of the early 20th century. [^59]: Laboureur means nearly laborieux in French, and was Feldbau’s pseudonym to publish Comptes Rendus notes during the German occupation of France (World War II, 1939–45). Alas, it did not helped to save his life from the Nazi persecutions. Another notorious victim of the genocide soon afterwards was F. Hausdorff, 1944. Why so much dramas in the human history is a puzzle to each philosopher. Materialism, capitalism, caused by the ontological existential fears ought to be the cause of such disasters. We can only hope that the Riemann(=woman) surface will quickly lead us to stabler psychological comforts (immortality, and global resurrection as to repair such disasters). [^60]: Without this proviso it is false, e.g. the tangent bundle to the simply-connected Prüfer surface is not trivial, for otherwise the manifold could be given a Riemann metric tensor, hence be metricized. Compare Radó 1925 (publishing a contribution of Heinz Prüfer 1922), Calabi-Rosenlicht 1953, Spivak 1970 (Vol.I, Appendix of Differential Geometry), or ask Mathieu Baillif why. The latter’s e-mail is: labaffle(at)gmail.com [^61]: Peintre américain d’origine russe (Dvinsk 1903–New York 1970). Il est célèbre pour la formule d’abstraction chromatique qu’il a établie vers 1950. (Source=Larousse Dictionnary, 1991). [^62]: Read “chamber” if you prefer. [^63]: Read “isotopy classes”, if you like. [^64]: This seems to just mean non-singular curve, cf. footnote in GMA. [^65]: If you are called Felix Klein!, else it may be tricky especially if our previous theorem is right in which case Klein-Rohlin are false! But of course, it is more likely that Gabard missed something. \[06.04.13\] More seriously, it would be interesting if a detailed account of this direct proof (alluded to by Klein) has been worked out meanwhile. Some details are perhaps gleanable from Degtyarev-Kharlamov 2000 [@Degtyarev-Kharlamov_2000]. [^66]: This is a slight abuse of language to suit Russian jargon (coined by Petrovskii 1938). [^67]: \[08.04.13\] Compare with Le Touzé’s article 2013 [@Fiedler-Le-Touzé_2013-Totally-real-pencils-Cubics], where it is asserted that this was implicitly conjectured by Rohlin, in 1978. Recall that Le Touzé’s husband Fiedler is a direct student of Rohlin, and so this may also be based upon some oral tradition, in case Rohlin was too cautious to put crazy ideas on the paper. [^68]: Tartaglia, also known as Niccolo Fontana (1500?–1557) won in 1535 a mathematical contest by solving many different cubics, and gave his solution to Cardano (1501–1576), who published in 1539 “Artis Magnae” alias the “Great Art, or the Rules of Algebra”, where complex numbers are used in Cardano’s formula to express the real roots of cubics. [^69]: Again this claim is a mistake: an obstruction follows from Thom’s conjecture, meanwhile the theorem of Kronheimer-Mrowka 1994 [@Kronheimer-Mrowka_1994]. [^70]: \[08.03.13\] Fortunately this schemes is not realized. [^71]: \[06.03.13\] This special treatment can be dispensed as this scheme is prohibited by Thom conjecture, cf. Theorem \[Thom-Ragsdale:thm\], or more elementarily by Rohlin’s formula. [^72]: \[02.03.13\] This is a misconception of mine and Le Touzé’s statement is finer and so caricatural (or strong) as I misinterpreted her announcement, more discussion about this soon. However I still do not know whether the strong caricatural statement is wrong or not. [^73]: The official name of this author is Fiedler-Le Touzé, yet as well-known née Le Touzé and abridged as a such in the sequel. [^74]: This seems alas to be the bitter state of affairs as follows from a recent consultation (January 2013) of the leading experts (Viro, Marin, Kharlamov, Fiedler, Le Touzé, etc.); compare e-mails gathered in Sec.\[e-mail-Viro:sec\] [^75]: I agree, but the argument is nearly trivial in the sense that it just uses the fact that the image of a connected set is connected (Marin’s oral remark during my talk in Grenoble ca. 1999). Personally, I knew this argument since 1999 (arguing with pathes prior to Marin’s oral simplification of it). We cannot record if we rediscovered it independently of Rohlin 1978 (but do record that we may have found some indirect inspiration from Gross-Harris 1981, who treat the case of hyperelliptic curves $y^2=f(x)$, with $f(x)>0$ throughout). At any rate it is evident that Rohlin’s argument can be drastically simplified. Rohlin uses a lot a certain fibering while it is plain that it suffices to use the map to the equatorial (orthosymmetric) sphere, cf. e.g. Gabard 2006 [@Gabard_2006]. [^76]: My opinion was always that a positive answer should be a trivial consequence of Ahlfors theorem (cf. e.g. Gabard’s Thesis 2004 [@Gabard_2004 p.7]). However since Marin warned me in January 2013 (cf. Sec.\[e-mail-Viro:sec\]) it may be the case that the transition from the abstract to the embedded viewpoints is not so easy. Yet I am still confident that it holds true. The point is primarily a matter of projective algebraic geometry, namely the question if any abstract morphism on a concrete plane curve is induced by a pencil of ambient curves. This is either trivially true or trivially wrong, but alas I do not know the answer due to failing memory about the foundations of algebraic geometry. \[12.04.13\] Additionally, it can also be that sometimes imaginary basepoints have to be allowed, and so total reality really the mobile part of the pencil. We hope to be capable presenting this more clearly in the future, but see perhaps already (\[Le-Touzé-scholium-deg-6:lem\]). [^77]: Whether this is implicit or not is an interpretation-matter, unless of course some direct contact with Rohlin (e.g. via the husband) testifies such a conjecture of Rohlin. Again it could be the case that Ahlfors theorem nearly trivially implies this (novel) conjecture of Rohlin. Even if so in abstracto then the game is probably far from finished as one would like to get synthetic descriptions of the total pencils. It seems quite likely that this game can keep busy several generation of workers. \[12\] [^78]: \[12.04.13\] (Please skip this footnote, if you believe in capitalism).—We invented this exoplanet metaphor in 2004, as to sell our postdoc-research programme to an FNS-administrator (FNS=SNF=Schweizerische National Fond), specialized in astronomy (at some Geneva observatory). The success was very limited, no funding were ever obtained and much energy and time wasted for nothing. Some few weeks later another Swiss cooperative stole me 15’000 Euros of economies. Life then started to require environmental punch (nutrition in the containers, and other pleasant duties like bicycling the heavy nutriments over steep mountains). Can we develop a more tolerant science enrolling more people on less restrictive financial constraints, especially more modest retribution of the workers? Ahlfors is far from a hero in this respect (elitist attitude than looks much overdone in view of the little originality of his contributions to science, compare what he borrowed from Courant, Hurwitz, Riemann, Grötzsch, Teichmüller, etc.). The real question is of course: can we get rid off of capitalism, granting the fact that a sufficient motor of life is to reach immortality (for free and for all), as it was ever encoded in our genes since the amoebic morphogenesis. [^79]: \[27.03.13\] Meanwhile the simplest counterexample, I was able to find is the Itenberg-Viro curve constructed on Fig.\[Itenberg:fig\]. [^80]: \[21.03.13\] Find accurate references, by Thom, Cerf, Hirzebruch, Milnor, Wall, etc. I confess that I lack a precise reference. [^81]: That is the cautious Petrovskii version of Ragsdale’s estimates. [^82]: It seems to me that the estimates on $p$ follows from Thom’s conjecture, as explained in Lemma \[Thom-implies-one-half-of-Ragsdale:lem\] [^83]: Tarik Garidi (aus der Nordseeküste) is a well-known scientist in Geneva (student of Piron), specialized in anti-de-Sitter and notorious for having introduced a mass concept which can be negative-valued, like the signed difference $\Delta \Pi:=\Pi^+-\Pi^-$ of Rohlin. [^84]: \[18.03.13\] I think that (modest) Theorem \[Alsatian-schemes:thm\] below corrupts this belief of Th. Fiedler (who left the subject a long time ago), yet this does not jeopardize at all his invaluable help (and incredible memory!) in view of all the crucial corrections he took care to make on the present text. [^85]: Our notational trick is to denote with parenthesis the degree of the scheme, since without parenthesis e.g. in an $M$-scheme the magnitude in front is traditionally not the degree but the number of ovals of the scheme. [^86]: The usual notation is $n^-$ at least to be conform with Rohlin 1978 [@Rohlin_1978 p.86–87], but I keep it so to stay faithful to the message of Fiedler. [^87]: As far as I am informed the general coinage of this trichotomy is due to Felix Klein (in geometry) and Dubois-Reymond (in PDE’s). [^88]: From the Greek “poros”=“hole” (aping a bit Grothendieck’s “topos” or “topoi”). [^89]: It is crucial here to adopt the modern convention regarding the sign of Euler’s $\chi$. This is courtesy of Michel Kervaire, that turned out to be correct when looking at old texts, like perhaps Listing, von Dyck 1988, Poincaré 1885–1895, etc., where the opposite sign convention was used! [^90]: \[19.03.13\] The answer is no and follows from Petrovskii’s inequalities. [^91]: As we shall soon recall, since Thom/Kronheimer-Mrowka 1995 we may replace “nicht immer” by “never”!!! (provided $m\ge 6$) and $m$ even. \[17.03.13\] In fact this was known much earlier since Petrovskii 1938, cf. Lemma \[Hilbert’s-nesting-intuition:lem\] below. An elementary proof also follows from Rohlin’s formula as reminded in the same lemma. It would be interesting to say more on the odd degree case (again Petrovskii 1938 [@Petrowsky_1938] should suffice to corroboration “Hilbert thesis” for $m\ge 7$). [^92]: So-called because V.A. Rohlin was born in Bakou, from parents themselves coming from Odessa, and if I am not wrong in Geography Bakou belongs to Caucasus. [^93]: This French prose is borrowed by René Thom, in a letter to André Haefliger reproduced in part in a recent issue of L’Enseign. Math., ca. 2011–12. [^94]: I.e. the present text as it was on the date of the 09.01.13, meanwhile pagination may have changed. [^95]: This is not perfectly true, as I exposed Ahlfors theorem to Oleg Viro during its last visit in Geneva (ca. 2010–11, his talk on fields of char 1, Connes, tropical geometry, etc.), yet probably my explanations where so obscure that Oleg immediately forgot about it. [^96]: =Vladimir Abramovich Rohlin, of course. [^97]: Sic, singular or plural? Not so important of course. [^98]: i.e. Lemma \[Klein-Marin:lem\], warning meanwhile the numbering may changes, but the one in this footnote is automatic (hence the right one) [^99]: “isotopie” of course. [^100]: \[26.03.13\] This “evidently” sounds to me a bit sloppy, as the full credit for this remark goes to Viro. [^101]: Meanwhile this numbering may have changed into Quote \[Klein\_1876-niemals-isolierte:quote\]. [^102]: Not clear how to interpret this? Does it mean that Shustin’s claim is wrong, or simply that this scheme is an $(M-1)$-scheme. My question was whether this $(M_1)$-scheme is realized algebraically, of course. Yet, I admit that my question was a bit ill posed. [^103]: \[24.01.13\] On reading Viro’ survey 1989/90 [@Viro_1989/90-Construction p.1085, 2.5.H], one should easily locate the source for this assertion. References seems to be Viro 1983 [@Viro_1983/84-new-prohibitions], and the survey Viro 1986 [@Viro_1986/86-Progress]. As explained there (Viro 1989/90, p.1085) this is a prohibition not coming from topology, but from Bézout. In fact this result is mentioned again in Viro 1989/90 [@Viro_1989/90-Construction p.1126, 5.3.E], with the exact cross-reference as being Viro 1983 [@Viro_1983/84-new-prohibitions] [^104]: \[30.01.13\] This is another very interesting (inedited?) piece of information, as to my knowledge this never appeared under the (printed) pen of V.A. Rohlin. So here Viro tell us something very interesting not yet available in print (as far as I know). [^105]: \[30.01.13\] This is probably true yet this requires a more highbrow dissipation theory, than in Gudkov’s second existence proof which apart from the trick with Cremona uses only dissipation of ordinary nodes à la Brusotti 1921 (so-called $A_1$-singularities). [^106]: \[30.01.13\] This alas turned out to be quite foiled as the invisible part of the discriminant as real codimension 2. [^107]: \[26.03.13\] It turned that this is wrong, cf. Fiedler’s letter dated \[21.03.13\]. [^108]: \[26.03.13\] This is again a misconception of mine, since the estimate $\chi\le k^2$ of Theorem \[Thom-Ragsdale:thm\] was based on the erroneous supposition that the Arnold surface is always orientable, cf. again Fiedler’s letter dated \[21.03.13\]. [^109]: For the same reason as the previous footnote this comment is not pertinent anymore. [^110]: Presumably, the authors omit the rotational ambiguity. [^111]: Of course Ahlfors’ statement is somewhat stronger giving $r \le n\le r+2p$, where $r$ is the number of contours and $p$ the genus. [^112]: According to Havinson 2003/04 [@Havinson_2003/04-Erokhin1958], this terminology is due to Erokhin 1958: “In accordance with V.D. Erokhin’s proposal (1958), the quantity $\gamma(F)$ has been called the [*analytic capacity*]{} or the [*Ahlfors capacity*]{} since that time.” [^113]: Who exactly? candidates: Golusin, Havinson, Havin, Vitushkin, etc., but see also Nehari (alias Willi Weisbach) as early as 1950. Indeed, “Ahlfors’ extremal function” occurs already in Nehari’s survey 1950 [@Nehari_1951-survey-BAMS p.357], and “Ahlfors mapping” alone occurs in Nehari 1950 [@Nehari_1950 p.267]. This probably beats any Russian contribution, for one of the first text is Golusin 1952/57 [@Golusin_1952/57], where actually the term “Ahlfors function” is not employed. However Havinson torrential list of publication on the topic starts as early as 1949 [@Havinson_1949]. [^114]: Existence is ensured under the mild condition that the domain supports nonconstant bounded analytic functions. [^115]: A coinage of Carathéodory, cf. Carathéodory 1912 [@Caratheodory_1912]. [^116]: This seems to be a misprint, and should be “$n$ zeros” (\[27.09.12\]). Further it is tacitly assumed that the domain is bounded by Jordan curve, for pointlike punctures are removable singularities hence do not affect the Ahlfors function. To be concrete making $(n-1)$ punctures in the unit disc the domain reaches connectivity $n$ but its Ahlfors function is still the identity as if there were no punctures. [^117]: Again “$n$ times covered disk” sounds more correct. [^118]: This is indeed one of the fascinating difficulty also discussed in A. Mori 1951 [@Mori_1951] and Fedorov 1991 [@Fedorov_1991], who coins the lovely prose of a “rather opaque condition must be satisfied”. [^119]: Here there is maybe a wrong cross-reference and Myrberg 1933 [@Myrberg_1933] was rather understood? [^120]: Can one be more explicit? Hahn-Banach like in Read 1958 [@Read_1958_Acta] or Royden 1962 [@Royden_1962] or just something more in the realm of classical analysis. [^121]: Of course for this purpose it would have been enough to cite Bieberbach 1925 [@Bieberbach_1925]. [^122]: This is true modulo the possibility of the planar case (i.e. Harnack-maximal Schottky double). [^123]: Addition of Gabard, otherwise seems an abuse of notation. [^124]: This argument looks all right, yet it seems to the writer than one can easily dispense of the concept of orientability, by just using the separation effected by the existence of the map induced on imaginary loci, i.e. $X({\Bbb C})-X({\Bbb R})\to {\Bbb P}^1({\Bbb C})-{\Bbb P}^1({\Bbb R})$. [^125]: This is maybe a misprint and the “$K$” should be an $E$? [^126]: Of course the notation $P$ instead of $N$ could have been more appealing, yet Forelli had obviously to reserve the letter $P$ for “probability measures”, to enter soon the arena! So imagine the “$N$” standing for non-negative real parts (which is incidentally more correct if we let penetrate the boundary behavior in the game). [^127]: Of course behind both techniques there is the paradigm of compactness in suitable function spaces, first occurring as a such in the related Hilbert’s investigation on the Dirichlet principle (add maybe Arzelà-Vitali to be fair, cf. e.g. Zaremba 1910 [@Zaremba_1910]). So everything started to be solid after Hilbert 1900, and Montel 1907, etc. [^128]: This is German for belt (=ceinture) in French. [^129]: This is indeed quite trivial to see, if we know the Riemann(-Roch) inequality, cf. e.g. Gabard 2006 [@Gabard_2006]. [^130]: Of course any geometric topologist (or reasonable being) could find the writing $\partial \overline W$ semantically more precise, yet we follow Forelli’s alleged notation. [^131]: “Les anglaises c’est comme le pudding, elles ne bougent pas quand on fait l’amour.” (Joke from Sherbrooke, learned from Gaston Boulé). [^132]: Jargon of Ahlfors-Sario 1960 [@Ahlfors-Sario_1960 p.42], implying that the map covers each point the same number of times (counting properly by multiplicity); but of course inspired by Stïlow’s book 1938 [@Stoilow_1938-Lecons]. [^133]: Some specialists from Grenoble (especially Emmanuel Ferrand) told me (ca. 1999/00) that the idea of filling the membrane by the insides of the ovals truly goes back to Arnold, which is probably essentially correct, yet Rohlin’s full credit for effecting the lovely perturbation and counting things properly is surely not at all affected. [^134]: Gabard’s addition [^135]: Severi perhaps? Try also Hurwitz?? [^136]: Try also Nevanlinna. [^137]: \[26.03.13\] This is a misconception of Gabard, that was corrected in Fiedler’s letter dated \[21.03.13\]. [^138]: This prose is bracketed as it seems to be verbatim copied from Rüedy 1971 [@Ruedy_1971]. [^139]: It seems to be rather earlier!?? [^140]: No attempt to correct the English, since Gabard’s English is even more indigest than the everything what has been ever written. [^141]: Means “mouche” in Polish [^142]: Perhaps it would be better to say no accumulation point. [^143]: Read perhaps multiconnected to be more faithful to Riemann’s original text. [^144]: Easy to sharpen as Klein 1876 [@Klein_1876]. [^145]: This is especially true under the Russian perspective, yet in the West workers were a bit more universalist, e.g. Koebe 1907 [@Koebe_1907_UrAK], J. Douglas 1936 [@Douglas_1936-Some-new-results], Teichmüller 1939 [@Teichmueller_1939], Ahlfors 1950 [@Ahlfors_1950], Schiffer-Spencer 1954 [@Schiffer-Spencer_1954], etc. [^146]: Read “finite” to be more conventional. [^147]: Big challenge: find where? Possibly this is not to be found in Klein and Teichmüller (probably lacking a good library during the war time) sloppily extrapolated what he remembered from his Klein reading (namely reality of Riemann surfaces, yet as far as we know never the total reality of orthosymmetric curves). [^148]: If the surface is open this the non-trivial result of Behnke-Stein 1947/49 [@Behnke-Stein_1947/49]. [^149]: This jargon goes back to Weierstrass (vgl. etwa Schottky 1877 [@Schottky_1877]). [^150]: Maybe here one can pinpoint about a confusion with the uniformization of Klein-Poincaré-Koebe. [^151]: This is essentially the theorem of Bieberbach-Grunsky (with antecedent by Riemann and Schottky). [^152]: Compare maybe also Hilbert and Courant for similar works

--- abstract: 'We study a rotating two-component Bose-Einstein condensate in which an optically induced Josephson coupling allows for population transfer between the two species. In a regime where separation of species is favored, the ground state of the rotating system displays domain walls with velocity fields normal to them. Such a configuration looks like a vortex split into two halves, with atoms circulating around the vortex and changing their internal state as they cross the domain wall.' author: - 'Juan J.' - Fernando Sols - 'Víctor M.' title: 'Split vortices in optically coupled Bose-Einstein condensates' --- Vortex formation is generally viewed as an unequivocal signature of superfluid motion in atomic Bose-Einstein condensates. Both in one- and two-component systems, a number of vortex-like structures have been created and observed [@JILA; @vortex-exp] that confirm theoretical predictions [@vortex-theo; @freqs; @asym]. The Josephson effect between two weakly coupled condensates is another paradigm of superfluid transport that so far has received less attention [@JILA2; @Yale; @sols99]. In this article, we present a combined study of these two fundamental properties of superfluid systems: vortex and Josephson dynamics. We study the ground state properties of a rotating double condensate system in which the combined role of vortex formation and optical coupling between two different hyperfine states gives rise to a rich physical behavior. A crucial consequence of the internal Josephson coupling is the generation of an [*effective attraction*]{} between both atomic species due to the energy that atoms gain by choosing a symmetric mixture of the coupled internal states. Therefore, the most interesting physics is reached by combining this coupling with setups that otherwise favor species separation, such as a particular combination of atomic scattering lengths [@phase-separation; @walls] or a separation of the respective confining potentials [@JILA1]. In this type of setups we find that the effective attraction due to the optical coupling causes an increase in the thickness of the domain wall where the two components physically overlap. More important is the fact that, if we add rotation to a Josephson coupled condensate with separate domains, atoms can use the domain wall to mutate their internal state and shift between components in a continous way. Combining this persistent current in the inner space with a persistent current in real space, the double condensate may now easily create a vortex core for each component in the region where that component has a low density. Even for otherwise small values of the Josephson coupling, these [*split vortices*]{} support a net mass flow comparable to that of conventional vortices. The formation of these novel structures is less costly in terms of angular speed because the vortex of a given component is formed not within its own domain but in the opposite one, where its superfluid density is low and the cost in kinetic energy is therefore small. To study this type of vortex structure, we analyze first a rotating two-component condensate without optical coupling (i.e. with impenetrable domain walls) and show that its behavior is essentially that of a one-component system with a displaced axis of rotation. Then we show that vortex formation is strongly inhibited because of the ability of the system to gain angular momentum by merely distancing itself from the rotation axis. The picture changes qualitatively when a Raman coupling is introduced to permit coherent hopping between the two internal states. Because the flow of particles in a given state is no longer a conserved quantity, the circulation lines of a component can cross the domain wall with a concomitant decrease in their supporting superfluid density. This results in a global structure of two asymmetrical vortices where matter is efficiently transported both in real space and within the hyperfine doublet. *The model.-* In this paper we focus on double condensate systems such as those made of Rb in JILA [@JILA], but this time in rotating traps and with a permanent optical coupling between the species. In the rotating frame of reference [@asym] the Gross-Pitaevskii equations for the condensate wavefunctions read $$\begin{aligned} i\hbar\partial_{\tau}\Psi_1 &=& \left[H_1+{\textstyle \sum_j}U_{1j}|\Psi_j|^2\right]\Psi_1 - {\textstyle \frac{1}{2}} \hbar \Omega_R \Psi_2,\\ i\hbar\partial_{\tau}\Psi_2 &=& \left[H_2+{\textstyle \sum_j}U_{2j}|\Psi_j|^2\right]\Psi_2 - {\textstyle \frac{1}{2}} \hbar \Omega_R \Psi_1.\end{aligned}$$ Here $H_j$ correspond to the non-interacting Hamiltonians $H_{1,2} = -\frac{\hbar^2}{2m}\Delta + V({\bf r}-{\bf r}_{1,2}) - \hbar \Omega L_z + \hbar\bar\delta_{1,2}$, where $V({\bf r}) = \frac{1}{2}m\omega^2r^2$ is the trapping potential, ${\bf r}_j$ is the center of the trap for component $j$, and $\Omega$ is the angular speed. The mutual interactions $U_{ij}=4\pi\hbar^2a_{ij}/m$, are proportional to the $s$-wave scattering lengths $a_{ij}$. The last term in each equation models an optical coupling between species. This is a Josephson-type coupling that allows atoms to change their internal state. Since we focus on stationary configurations, the normalization of each wavefunction is fixed, $N_i = \int |\Psi_i(\mathbf{r})|^2 d\mathbf{r}$. Finally, $\hbar\bar\delta_i$ is a tunable energy splitting in the hyperfine space, which controls the population of each component. In this paper we restrict our attention to axially symmetric two-dimensional configurations which may describe pancake type traps [@2D]. In such traps the nonlinear parameter is affected by a corrective factor [@asym; @molina] and $N_1$ and $N_2$ are interpreted as effective number of particles. To ease the analysis we introduce dimensionless variables $\mathbf{x} = \mathbf{r}/a_0$ and $t = \tau/T$. With this change and $\psi_j(\mathbf{x},t) = N_j^{-1/2}\Psi_j(\mathbf{r},\tau)$, we write the equations for stationary states \[gpe\] $$\begin{aligned} \mu_1\psi_1 &=& \left[\mathcal{H}_1+\hbox{$\sum_j$}g_{1j}|\psi_j|^2\right]\psi_1 - \lambda \psi_2,\\ \mu_2\psi_2 &=& \left[\mathcal{H}_2+\hbox{$\sum_j$}g_{2j}|\psi_j|^2\right]\psi_2 - \lambda \psi_1,\end{aligned}$$ with rescaled Hamiltonians $$\label{ham2} \mathcal{H}_{1,2} = {\textstyle\frac{1}{2}}\left[-\Delta +(x - x_{1,2})^2 + y^2\right] + i\Omega \left(x\partial_y - y\partial_x\right).$$ In Eq. (\[gpe\]) the splittings $\bar\delta_i$ are included in the chemical potentials. The parameter $\lambda=2\Omega_R/\omega$ measures the intensity of the optical coupling. Although it is widely tunable, in our work it will be spatially uniform and at most of order unity. We have solved numerically Eq. (\[gpe\]), looking for the solutions that have lower energy, the so called ground states. Each of such solutions represent a stable, experimentally realizable configuration. We will consider two different scenarios in which the double condensate exhibits domain walls, and which give qualitatively similar results. The first case, which we call “setup B”, corresponds to a situation in which $g_{11}=g_{22}=N, g_{12}=2N,$ with a choice of $N=38$. In this case the inequality $g_{12}^2>g_{11}g_{22}$ is satisfied and the domains form spontaneously [@walls] with no need for trap separation nor splitting, i. e. $x_{1,2} = 0$, $\mu_1=\mu_2$. The second and most important scenario, which we call “setup A" corresponds to the case of $^{87}$Rb [@JILA] with parameter values $\left(\begin{smallmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{smallmatrix}\right) = \left(\begin{smallmatrix} 1 & 0.94 \\ 0.94 & 0.97 \end{smallmatrix}\right) \times \alpha N$ (i.e. $N_1=N_2=N$), and a typical effective value of $N=100$. This type of condensates has been studied in many previous experimental and theoretical works. For $^{87}$Rb, the inequality mentioned above is close to saturation [@walls] and a separation $x_1=-x_2=1$ ensures the formation of two different domains. The external splitting takes small values, $\mu_1-\mu_2\leq 0.01$, and it does not influence the results. *Rotation without Josephson coupling.-* Here we consider the case $\lambda=0$. The existence of domain walls implies that the motion of species is spatially constrained. Therefore, if we impose some angular speed to the traps containing the condensates, each cloud tends to slip tangentially to the domain wall. The density distribution is similar to the case without rotation, but now, due to the centrifugal force, the species separate a little and gain linear speed. Together with the deformation of the clouds, this mechanism permits the acquisition of a large amount of angular momentum without generating vortices \[Fig. \[fig-domains\](d)\], which only form at very high $\Omega$. In Fig. \[fig-domains\](a) we show the ground state of the rotating double condensate system for $\lambda=0$, showing a structure of domain walls that persists in the presence of rotation and prevents the formation of vortices, as revealed by the phase and circulation patterns of Fig. \[fig-domains\](b). From Fig. \[fig-domains\](c) it is evident that the both the mutual repulsion and the rotation of the trap increase the separation among species. The flow pattern presents a curvature which is too small \[Fig. \[fig-domains\](b)\], and the gain of angular momentum \[Fig. \[fig-domains\](d)\] is due instead to the displacement of the condensate away from the origin. To get a deeper understanding of this configuration it is useful to study the off-axis rotation of a single component condensate. It experiences a potential $V({\bf x},t) = \frac{1}{2}({\bf x}-{\bf r}_0(t))A(t)({\bf x}-{\bf r}_0(t))$ with $A(t)$ and $\mathbf{r}_0(t)$ rotating at speed $\Omega$ around a displaced axis, as depicted in Fig. \[fig:rot-trap\]. Thanks to a particular symmetry of the nonlinear Schrödinger equation with harmonic potential [@us-pre], any solution can be written as $$\label{solution} \psi({\bf x},t) = \phi({\bf x}-{\bf u},t) e^{im\dot{\bf u}\cdot {\bf x}/\hbar + f(t)}$$ where $\phi({\bf x}-{\bf u})$ is centered on the center of mass $\mathbf{u}$ and satisfies the Gross-Pitaevskii equation (\[gpe\]) for ${\bf r}_0=0$. In the case of radially symmetric traps, $A=\omega^2$, we find $(1-\Omega^2/\omega^2){\bf u}={\bf r}_0$. This tells us that the behavior of a condensate in an off-axis rotating trap is qualitatively similar to that of a condensate in a centered trap: Vortices nucleate at similar angular speeds and they all appear inside the condensed cloud. The difference is that now the condensate has an additional source of angular momentum due to its displacement with respect to the origin, $L_z \propto \Omega |u|^2$. These arguments, which are rigorous for the single condensate system, may be extended to the two-component condensate case where the overlap between clouds is small, meaning that vortices should appear in the center of the corresponding clouds even when the trap centers are displaced with respect to the axis of rotation. *Role of Josephson coupling.-* In order to explain the role of Josephson coupling let us first consider situation A. It is easy to show that the Josephson terms favor energetically the mixing of both species. In the limit of strong coupling, they coexist in space even when their traps are separated. This is most intuitively appreciated by inspecting the energy functional associated to Eq. (\[gpe\]) $$\begin{aligned} \label{energy} E[\psi_1,\psi_2] & = & \int \left[ \bar\psi_1 \mathcal{H}_1 \psi_1 + \bar\psi_2 \mathcal{H}_2 \psi_2\nonumber \right] \\ &+& \int \left[ \sum_{i,j =1,2} g_{ij} |\psi_j |^2 |\psi_i |^2 - \lambda \mathrm{Re}\left(\bar\psi_1 \psi_2\right) \right].\end{aligned}$$ The coupling term by itself is minimized with a solution such that $\arg\psi_1=\arg\psi_2$. The analogy with the off-axis rotation of a single condensate \[see Eq. (\[solution\])\] suggests the variational wavefunction $$\begin{aligned} \psi_1(x,y) &\propto& e^{-(x-u)^2/2-y^2/2} e^{iv y},\\ \psi_2(x,y) &\propto& e^{-(x+u)^2/2-y^2/2} e^{-iv y}.\end{aligned}$$ Substituting this ansatz into Eq. (\[energy\]) one gets an effective energy $$\label{energy-variational} E \sim {\textstyle\frac{1}{2}}(u-d)^2 + {\textstyle\frac{1}{2}}v^2 + \Omega u v - \lambda e^{-u^2} + g_{12}Ce^{-2u^2},$$ where $C$ is of order unity. Minimization with respect to the velocity $v$ leads to $v=-\Omega u$, and thus $\Omega$ favors separation. More importantly, it is clear from (\[energy-variational\]) that $\lambda$ effectively decreases the repulsion between condensates. Thus, the separation $u$ decreases with the strength of the optical coupling and becomes zero for a strong enough value of the Josephson coupling, $\lambda_c$. Both the repulsive interaction among bosons and the rotation of the trap, tend to inhibit mixture of species, thus increasing the value of $\lambda_c$. *Vortex formation.-* The structure of matter flow in the condensate suffers a drastic trasformation when the Josepson coupling is introduced. From numerical solutions of Eqs. (\[gpe\]) for setups A and B, we see that any nonzero coupling allows the formation of vortices \[Figs. \[fig-cyclotron\](a) and \[fig-lz-3d\]\]. These vortices involve a matter flow which is orthogonal to the wall separating both condensates \[Fig. \[fig-cyclotron\](b)\]. We note that the exact location of both vortex cores depends on the intensity of the coupling \[Fig. \[fig-cyclotron\](d)\]. The mathematical reason for this striking change is that, with a finite value of $\lambda$, the matter flow of each species is no longer conserved. Assuming a divergence free flow, the equation of continuity along the trajectory of a boson becomes $$\frac{\partial|\psi_1|^2}{\partial l}v_1 = -\lambda \mathrm{Re}(\bar\psi_1 \psi_2) = -\frac{\partial|\psi_2|^2}{\partial l}v_2.$$ Thus, as the current line is closed around the origin, there is an exchange of bosons among components. A typical boson flowing around the origin suffers a transformation from state $|1\rangle$ to $|2\rangle$ and viceversa as it completes a circle. Physically, there is a fundamental reason why bosons are transferred from one component to the other. For $\lambda = 0$, vortices must lay inside each condensed cloud, due to current conservation. However, for any nonzero coupling, vortices may appear [*outside*]{} the bulk of the clouds, at a variable distance from the origin. Placed at low density regions, the twist of the phase requires less kinetic energy $\int|\psi_i|^2(\nabla\arg\psi_i)^2$ and provides more angular momentum than the mere separation of clouds. In Figure \[fig-lz-3d\](a-b) we plot the angular momentum per particle as a function of the Josephson coupling and the angular speed. The cliffs on the surface are due to the nucleation of successive vortices. The separation of these vortices from the $x=0$ domain wall decreases very fast with increasing $\lambda$: Fig. \[fig-cyclotron\](d) shows that, already for $\lambda =0.05$, the vortices become visible. As $\lambda\rightarrow 0^+$ for fixed $\Omega$, vortices disappear continuously moving their core to infinity. To create vortices at $\lambda=0$, one needs to reach a much higher angular speed, $\Omega_c$, comparable to that of single component condensates [@freqs]. The upshot is that the optical coupling permits the formation of split vortices with important mass flow at lower angular velocities. *Internal Josephson dynamics.-* The Hamiltonian (5) may be written as that of a nonrigid pendulum [@sols99] with an effective interaction energy $E_c=\sum_{ij} g_{ij} \sigma_{ij} (-1)^{i+j}$, where $\sigma_{ij}\equiv \int |\psi_i|^2 |\psi_j|^2$, and an effective Rabi frequency $\omega_R=\Omega_R s_{12}$, with $s_{12} \equiv \int \bar{\psi_1}\psi_2$. The conclusion is that, for the setups considered here ($\lambda \alt 0.5$), the internal two-state dynamics lies in the collective Josephson regime ($2\omega_R/N \ll E_c \ll N\omega_R/2$), while in the JILA experiment [@JILA2], where no walls are formed, the same dynamics lies in the noninteracting Rabi limit ($E_c \ll 2\omega_R/N$). This work has been partially supported by Ministerio de Ciencia y Tecnología under grants BFM2000-0521 and PB96-0080-C02. [99]{} See e.g. A. Svidzinsky, A. L. Fetter, Jour. Phys. B [ **13**]{}, R135 (2001), and references therein. D. S. Hall, M. R. Matthews, J. R. Ensher, C. E. Wieman, E. A. Cornell, Phys. Rev. Lett. [**81**]{}, 1539 (1998). .

--- abstract: 'The paper presents a novel approach of spoofing wireless signals by using a general adversarial network (GAN) to generate and transmit synthetic signals that cannot be reliably distinguished from intended signals. It is of paramount importance to authenticate wireless signals at the PHY layer before they proceed through the receiver chain. For that purpose, various waveform, channel, and radio hardware features that are inherent to original wireless signals need to be captured. In the meantime, adversaries become sophisticated with the cognitive radio capability to record, analyze, and manipulate signals before spoofing. Building upon deep learning techniques, this paper introduces a spoofing attack by an adversary pair of a transmitter and a receiver that assume the generator and discriminator roles in the GAN and play a minimax game to generate the best spoofing signals that aim to fool the best trained defense mechanism. The output of this approach is two-fold. From the attacker point of view, a deep learning-based spoofing mechanism is trained to potentially fool a defense mechanism such as RF fingerprinting. From the defender point of view, a deep learning-based defense mechanism is trained against potential spoofing attacks when an adversary pair of a transmitter and a receiver cooperates. The probability that the spoofing signal is misclassified as the intended signal is measured for random signal, replay, and GAN-based spoofing attacks. Results show that the GAN-based spoofing attack provides a major increase in the success probability of wireless signal spoofing even when a deep learning classifier is used as the defense.' author: - Yi Shi - Kemal Davaslioglu - 'Yalin E. Sagduyu' title: Generative Adversarial Network for Wireless Signal Spoofing --- Introduction ============ Wireless communications is susceptible to adversaries due to the open and shared nature of wireless medium. Among various wireless attacks, the spoofing attack is launched by an adversary that aims to mimic a legitimate user in its transmissions. This attack may serve different purposes including primary user emulation (PUE) in cognitive radio networks, passing through signal authentication systems, and intrusion into protected networks. One common approach for wireless signal spoofing is recording a legitimate user’s transmission and replaying the signal later by potentially adjusting the transmit power. While this approach can represent various features in the signal at a high level, it may fall short of reliably mimicking combined waveform, channel, and device effects. In this context, machine learning provides automated means to authenticate signals by analyzing wireless signals and identifying anomalies. Enabled by recent advances in computational resources, deep learning can effectively process raw spectrum data and operate on latent representations, while analyzing high-dimensional spectrum dynamics that feature-based machine learning algorithms fail to achieve. Deep learning finds rich applications in wireless communications, including spectrum sensing [@Kemal2018] and modulation recognition [@OShea2016]. On the other hand, the adversary can also apply deep learning to launch wireless attacks. For example, an adversary may try to learn the underlying transmit behavior by training a deep neural network and effectively jam data transmissions [@Yi18:Jamming]. There are various security concerns regarding the safe use of machine learning algorithms. *Adversarial machine learning* [@AMLbook; @Sec2] studies learning in the presence of adversaries and aims to enable safe adoption of machine learning to the emerging applications such as wireless communications. For example, the adversary may manipulate input data into a machine learning classifier by jamming the sensing period [@Yi18:Poisoning]. Similarly, an adversary may launch an evasion attack by manipulating signals over the air so as to force a receiver in making wrong signal classification decisions [@LarssonAML; @Headley19; @Deniz19; @Silvija19]. In this paper, we introduce a *spoofing attack* motivated by adversarial machine learning. In particular, consider training a *generative adversarial network* (GAN) to spoof wireless signals as if they originate from intended users (legitimate or higher priority users such as primary users). GAN has been extensively applied to other domains such as computer vision and text analytics to generate synthetic data that is statistically similar to real data [@Goodfellow2014]. Recently, there have been efforts to apply GAN to wireless communications. The focus has been to augment the training data sets such as those used to train classifiers for spectrum sensing [@Kemal2018] and jamming [@Tugba2018]. In this paper, our goal is to train a GAN from an adversarial point of view to spoof wireless signals that cannot be reliably discriminated from intended signals. Different from applications in other domains such as computer vision, data in wireless medium is received through channel and (transmitter and receiver) hardware effects, and depends on transmitter-receiver positions that all need to be matched by the GAN. In this context, a receiver (assuming the role of a defender) aims to classify signal sources as intended user or not. This classification can be done by using a deep learning based classifier to analyze spectrum sensing results. Then the adversary launches a spoofing attack such that the classifier at the receiver incorrectly authenticates its transmissions as intended. As a starting point, we show that a deep learning-based classifier can distinguish signals of an intended transmitter $T$ from other randomly generated signals. Given that each device introduces its own phase shift and each channel has its own propagation gain and phase shift, such a classifier can be built at a receiver $R$ by collecting spectrum sensing results for signals from $T$ and other signals, and processing them to obtain a number of features for each sample. These features and labels (from $T$ or not) form the training data to build a classifier by deep learning. We show that such a classifier can successfully distinguish different signals. In particular, we can regard signals from other transmitter as a naive spoofing attack with random signals. This naive attack, as we expected, does not perform well. Its success probability (the probability that signals from other transmitter is classified as from $T$) is limited to only $7.89\%$. We then study the replay attack, where an adversary transmitter $A_T$ amplifies and forwards the previously received signal from $T$. This attack keeps some pattern of $T$ (but not the entire), and thus is better than using random signals. We show that the success probability increases to $36.2\%$ against a defender that uses a deep learning classifier. However, this probability is still much less than $50\%$, i.e., most of replay-based spoofing attacks based on amplifying and forwarding signals are still not successful. To launch a successful spoofing attack, the adversary transmitter $A_T$ needs to generate signals such that signals received from $A_T$ are statistically similar to signals received from $T$. The challenge is that $A_T$ does not have any knowledge on $T$’s waveform (in our case, characterized by the modulation scheme), its phase shift, as well as the channel between $T$ and $R$. In this paper, we introduce a GAN-based approach to capture all these effects from observed signals and generate spoofing signals without the need of prior knowledge (that may not be available at all due to unpredictable spectrum dynamics). The transmitter and its surrogate receiver (used only for training) are trained offline as the generator-discriminator pair of the GAN to generate the best spoofing signals that aim to fool the best-trained defense mechanism. The output is a signal generator that creates wireless signals with fake signatures to spoof signals. This generator is trained as a deep neural network to fool the optimized discriminator that is trained as another deep neural network. During the training, $A_T$ provides some flag to label its transmissions, and thus $A_R$ knows the true label. $A_R$ processes received signals, obtains features, and builds a discriminator to classify signals as from $T$ or not. The classification results are sent back to $A_T$ as feedback. Then $A_T$ updates its generator to generate better synthetic data, namely to increase classification error probability at $A_R$. Thus, $A_T$ and $A_R$ play a minimax game, which forms a GAN to improve both generator and discriminator. Once GAN converges, the generator at $A_T$ can generate high fidelity synthetic data (similar to real data) for spoofing attack. This approach inherently captures all waveform, channel, and device effects jointly. As a result, the success probability of spoofing attack increases to $76.2\%$, when the GAN-based approach is used. The rest of the paper is organized as follows. Section \[sec:related\] discusses related work. Section \[sec:scenario\] describes the system model. Section \[sec:classifier\] describes the pre-trained classifier to detect intended transmissions. Section \[sec:adversary\] describes and compares the replay and GAN-based spoofing attacks. Section \[sec:conclusion\] concludes the paper. Related Work {#sec:related} ============ There are different types of attacks on wireless communications in the literature [@Clancy08:CogSec]. In particular, attacks on spectrum sensing include spectrum sensing data falsification (SSDF) attack [@Penna; @Sagduyu2014], primary user emulation (PUE) attack [@PUE], eavesdropping [@Zou15:eavesdropping], and noncooperation [@Sagduyu09:noncoop]. Attacks on data transmission include jamming [@Sagduyu118:satellite] in form of a denial-of-service (DoS) attack [@DoS] with different levels of prior information [@Sagduyu11:jamming]. There are also attacks on higher layers, e.g., attacks on routing in the network layer [@Lu2017] and network flow inference attacks [@LuCliff2017]. Defense methods were developed to address these attacks. For example, an adaptive, jamming-resistant spectrum access protocol was proposed in [@JamRes] for cognitive radio ad hoc networks, where there are multiple channels that the secondary users can utilize. Jamming games between a cognitive user and a smart jammer was considered in [@UserCentric], where they individually determine their transmit powers. Launching and detecting spoofing attacks have been extensively studied [@Lichtman16:spoofing; @Gai17:spoofing; @Chen07:spoofing; @Sheng08:spoofing; @Sajjad18:spoofing]. Spoofing along with other attacks such as jamming and sniffing were assessed in [@Lichtman16:spoofing] A spoofing attack was designed in [@Gai17:spoofing] using optimal power distribution. In this paper, we optimize both power and phase shift for spoofing attack. Received signal strength (RSS) was used to in [@Chen07:spoofing; @Sheng08:spoofing] detect spoofing attack. In this paper, we use raw spectrum sensing (I/Q) samples. Recently, deep learning was also applied to detect spoofing attacks, e.g., CNN was used in [@Sajjad18:spoofing], while the use of deep learning in this paper is to optimize launching spoofing attacks. Wireless security finds rich applications of deep learning. Deep learning was applied to authenticate signals [@Saad2018], detect and classify jammers of different types [@Poor2018; @Wu2017], and control communications to mitigate jamming effects [@Poor2018; @UserCentric]. Jammers typically do not use machine learning techniques, e.g., [@Poor2018; @EnergyHarvestingCN]. Recently, there have been efforts to build deep learning-based jammers [@Yi18:Poisoning; @Yi18:Jamming]. Using wireless sensors, deep learning was also used to infer private information in analogy to exploratory attacks [@Liang18]. From a different perspective, GAN was applied to model wireless communication channels, e.g., [@Yang19:channel; @Oshea18:channel; @Oshea18:channel2]. In this paper, we use GAN to generate synthetic spoofing signals, which need to model not only channel effects but also device related effects such as phase shift as well as relative positions of transmitters and receivers from both attacker and defender sides. Adversarial deep learning was applied to launch evasion attacks by adding perturbations to the received signals and manipulating the input to the machine learning algorithm. [@LarssonAML; @Headley19; @Deniz19; @Silvija19] considered evasion attacks against modulation classifiers. In this paper, we consider a spoofing attack with the same final goal as the evasion attack. However, the classifier to fool in this paper does not only use modulation but also channel effects, device related effects such as phase shifts, and relative positions of transmitters and receivers of both attacker and defender. As a baseline, we consider the replay (amplify and forward) attack [@Kinnunen17:replay; @Hoehn16:replay] as a method of spoofing. However, although replay attack can keep some features in original signals, it is not very effective. The limits of replaying signals as the spoofing attack were assessed in [@Kinnunen17:replay]. Also, schemes to detect replay attacks were reported in [@Hoehn16:replay]. In this paper, our results show that GAN-based spoofing significantly outperforms the replay attack. System Model {#sec:scenario} ============ We consider a wireless communication environment with intended users (such as legitimate or high-priority) and others (such as the adversary). We assume that a pre-trained deep learning-based classifier is used at a receiver to predict whether a transmission is from an intended one, or not. The adversary aims to launch a spoofing attack such that its transmissions are classified as an intended one. Due to unique device properties (such as phase shift) and communication channel properties (such as channel gain), we show that random signal transmission by an adversary can be easily detected as an unintended transmission. Thus, the adversary needs to learn the pattern of intended transmissions and generate its transmissions following the same pattern for spoofing attack. Two adversaries collaborate for this purpose and act as a transmitter-receiver pair to train a GAN (see Fig. \[fig:gan\]). In particular, a generator is trained at the transmitter and a discriminator is trained at the receiver. In the spoofing attack (namely, test phase), only the generator at the transmitter is used. ![GAN structure.[]{data-label="fig:gan"}](gan.pdf){width="\columnwidth"} ![Network topology during the training process for spoofing attack.[]{data-label="fig:topology1"}](topology1.pdf){width="\columnwidth"} In this paper, we consider the scenario that there is an intended transmitter $T$, a receiver $R$ (that classifies if received signals are from $T$, or not), an adversary transmitter $A_T$, and an adversary receiver $A_R$. $T$ transmits with power $P=1000$. There is a device-related phase shift for $T$, which is unknown to adversaries $A_T$ and $A_R$. We assume a Rayleigh channel between any two nodes. $R$ has a deep learning-based classifier to classify whether signal is from $T$ or other transmitters, e.g., $A_T$. We consider three methods for spoofing attacks. - *Random signal attack:* $A_T$ performs random transmissions with power $P$. - *Replay attack:* $A_T$ amplifies and forwards previously received signal from $T$. Since $A_T$ does not have any knowledge on channel gains, it cannot optimally tune its power. Thus, $A_T$ uses power $P$ to amplify signals. - *GAN-based spoofing attack:* $A_T$ generates synthetic signals using GAN. The transmission power is up to $P$. The generation process of the GAN-based spoofing attack is as follows. The adversary receiver $A_R$ is placed close to $R$ such that channel from $T$ (or $A_T$) to $A_R$ is similar to channel $T$ (or $A_T$) to $R$. Thus, $A_R$ can receive similar signals as $R$ and if $A_R$ cannot distinguish signals from $T$ and $A_T$, $R$ cannot either. $A_T$ makes its transmissions with a flag such that $A_R$ knows these signals are from $A_T$ (i.e., the true label). $A_R$ builds a discriminator to classify signals from $T$ or $A_T$, and transmits the classification results to $A_T$ as feedback. Then $A_T$ builds a generator to improve its transmitted signals such that these signals are more similar to $T$’s signals in terms of resulting in larger classification error at $A_R$. This process continues several rounds until convergence. In this setting, $A_T$ and $A_R$ play a minimax game, which is exactly the GAN process (see Fig. \[fig:topology1\]). When the GAN converges, the generator at $A_T$ should be able to generate synthetic signals very similar to $T$’s signals, which are then used for spoofing attack (see Fig. \[fig:topology2\]). ![Network topology during the spoofing attack.[]{data-label="fig:topology2"}](topology2.pdf){width="\columnwidth"} The advantage of this attack is that the adversary does not need to assume any prior knowledge on $T$, which will be learned by $A_R$. Moreover, the adversary does not need to learn channel effect explicitly. Instead, channel effects such as phase shift and propagation gain are learned implicitly through collaboration of $A_T$ with $A_R$. The Classifier for Signal Authentication {#sec:classifier} ======================================== In this section, we describe the pre-trained classifier at $R$ that needs to identify whether a transmission is from $T$ or not. We assume that $T$ transmits data using the QPSK modulation. Note that other modulations can also be used without changing the algorithms (classifiers) developed in this paper. The transmit power is $P$ and the wireless channels are Rayleigh channel. $R$ uses limited sensing data, say $8$ bits of data. Under QPSK, there are four possible modulated signals, where each signal may have a different phase shift for $2$ bits. In addition, each device has its own phase shift. Denote $\theta_T$ as the phase shift of $T$, which is added on the QPSK signal’s phase shift. Note that other settings on the number of bits in sensing data and modulation will not affect the developed algorithms. $R$ trains a deep neural network classifier to analyze spectrum sensing results and identifies whether a transmission is from $T$ or not. This classifier is pre-trained using many samples with labels on whether a transmission is from $T$ or not. Each sample has four signals and each received signal is sampled $100$ times. Thus, there are $400$ features for each sample. As an example, for two bits $0$ and $0$, QPSK determines a phase shift $\frac{\pi}{4}$ for coded signal. Adding $\theta_T$ and a random channel phase shift $\theta_{T R}$ under the Rayleigh model, the received signal has phase shift $\frac{\pi}{4} + \theta_T + \theta_{T R}$. The $k$-th sample point, $0 \le k < 100$, has phase shift $\frac{\pi}{4} + \theta_T + \theta_{T R} + \frac{k \pi}{50}$. The received power is $g_{T R} P$, where $g_{T R}$ is a random channel gain under the Rayleigh model. The mean value of channel gain is $d^{-2}$, where $d$ is the distance between a transmitter and a receiver. Then the $k$-th sampled data is $$\begin{aligned} d_{TR}^k = g_{TR} P e^{j (\frac{\pi}{4} + \theta_T + \theta_{TR} + \frac{k \pi}{50})} \; .\end{aligned}$$ During the training, $T$ may send a flag to indicate its transmissions and this flag is used to label samples. After observing a certain period of time, $R$ collects a number of samples with labels to be used as training data to build a deep learning classifier. Once a classifier is built, $R$ uses it to predict signal labels (‘$T$’ or ‘not $T$’). For this algorithm, there may be two types of errors: - *Misdetection*. The signal is from $T$ but it is detected as from other transmitters. - *False alarm*. The signal is from other transmitters but it is detected as from $T$. $R$ aims to minimize the probability of both errors. Denote $e_{MD}$ and $e_{FA}$ as the probabilities of misdetection and false alarm at $R$, respectively. Then the objective is to minimize $\max\{ e_{MD}, e_{FA} \}$. For a given test data with $n$ samples and $n_T$ as the number of samples with signals from $T$, denote $n_{MD}$ as the number of misdetections and $n_{FA}$ as the number of false alarms. These error probabilities are calculated by $$\begin{aligned} e_{MD} = \frac{n_{MD}}{n_T} ,\; \:\: e_{FA} = \frac{n_{FA}}{n-n_T} \; .\end{aligned}$$ We use TensorFlow to build a deep learning classifier for $R$. In particular, we use the following deep neural network: - A feedforward neural network is trained with backpropagation function by using cross-entropy as the loss function. The structure of a feedforward neural network is shown in Figure \[fig:fnn\]. - Number of hidden layers is 3. - Number of neurons per hidden layer is 50. - Rectified linear unit (ReLU) is used as activation function at hidden layers. - Softmax is used as the activation function at output layer. - Batch size is 100. - Number of training steps is 1000. Note that $R$ can further optimize the hyperparameters (e.g., number of layers and number of neurons per layer) of its deep neural network. ![The structure of a feedforward neural network.[]{data-label="fig:fnn"}](FNN_pic.pdf){width="\columnwidth"} In the simulation setting, the location of $T$ is $(0,0)$, the location of $R$ is $(10,0)$, the location of $A_T$ is $(0,10)$, the location of $A_R$ is $(10,0.1)$ (see Figs. \[fig:topology1\] and \[fig:topology2\]), and the normalized transmit power at $B$ is $1000$. $R$ collects $1000$ samples, each with $400$ spectrum sensing results, and a label (‘$T$’ or ‘not $T$’) as training data and applies the classifier on another $1000$ samples to evaluate accuracy. We used the above deep neural network and tuned its parameters. We find that batch size $150$ can achieve better performance, while other parameters are unchanged. There are $504$ signals from $T$ and $496$ signals from other transmitters in the test data. Among them, $39$ signals from other transmitters are identified as from $T$ and $37$ signals from $T$ are identified as not from $T$. Thus, we obtain $e_{FA} = 39/496 = 7.86\%$, $e_{MD} = 37/504 = 7.34\%$. Both errors are small, showing that $R$ can reliably determine the signal labels. We can also regard this case as a naive spoofing attack, where the adversary uses random signals to attack. The success probability of this attack is only $7.86\%$. Spoofing Attacks {#sec:adversary} ================ There is an adversary transmitter $A_T$ that aims to mimic the transmitter $T$’s behavior such that $R$ classifies signals from $A_T$ as from $T$. $A_T$ does not have knowledge on $T$’s device related phase shift, or the channel between $T$ and $R$. Instead, an adversary receiver $A_R$ is placed close to $R$ to learn received signal pattern from $T$ and from $A_T$. In the replay attack, $A_R$ is not used. Replay Attack based on Amplifying and Forwarding Signals -------------------------------------------------------- We start with the replay attack based on simply amplifying and forwarding signals, i.e., $A_T$ records previously received signals from $T$, amplifies to power $P$ and forwards them to $R$. Assume that $\theta_{A_T}$ is the phase shift for $A_T$, $\theta_{ij}$ is the phase shift and $g_{ij}$ is a random channel gain for the Rayleigh channel from node $i$ to node $j$. $T$’s parameters are unknown to $A_T$. As an example, for two bits $0$ and $0$, QPSK determines a phase shift $\frac{\pi}{4}$. the received phase shift at $R$ is $\frac{\pi}{4} + \theta_{TA_T} + \theta_{A_T} + \theta_{A_T R}$ and the received power is $g_{A_T R} P$. The $k$-th sampled data then becomes $$\begin{aligned} d_{A_T R}^k = g_{A_TR} P e^{j (\frac{\pi}{4} + \theta_T + \theta_{TA_T} + \theta_{A_T} + \theta_{A_T R})} \; \label{eq:dar}.\end{aligned}$$ On the other hand, if the same signal is transmitted by $T$, the $k$-th sampled data is $$\begin{aligned} d_{TR}^k = g_{TR} P e^{j (\frac{\pi}{4} + \theta_T + \theta_{TR})} \; \label{eq:dtr}.\end{aligned}$$ We can see that both received power and phase shift in (\[eq:dar\]) and (\[eq:dtr\]) are different. However, a simple detector based on discriminating received power and/or phase shift cannot work. The reason is that a detector needs to make a classification based on limited data samples, e.g., when an intruder is authenticated at the physical layer. Due to random channel, the estimation of received power and phase shift cannot be accurate. As a result, if we set a small region around the actual value of $T$, we will end up with a large misdetection probability. On the other hand, if we set a large region, the false alarm probability will be large. Simulation results show that the replay attack is better than transmitting random signals, i.e., the success probability of spoofing is increased to $36.2\%$, which is much larger than the success probability $7.86\%$ if $A_T$ transmits random signals. This is because by amplifying and forwarding signals, only some signal pattern from $T$ is kept. However, as we showed above, signals from $T$ and forwarded by $A_T$ are different even for the same data. Thus, $R$ can still successfully classify most of spoofing signals. GAN-based Spoofing Attack ------------------------- We now consider the spoofing attack based on the GAN [@Goodfellow2014]. As the first step, $A_R$ collects $500$ signal samples from $T$ and $500$ signal samples from $A_T$, where $A_T$ can flag its transmissions such that $A_R$ can have ground truth. Each sample has coded data of $8$ bits under the QPSK modulation, i.e., $4$ signals. The sampling rate for a signal is $100$, and thus the total data (features) for a sample is $400$. $A_R$ then builds the first version of discriminator $D$ based on these data samples with the objective of minimizing classification error, i.e., $$\begin{aligned} \min_{D} \mathbb{E}_{\bm{z} \sim p_{\bm{z}}} [\log(1 - D(G(\bm{z})))] - \mathbb{E}_{\bm{x} \sim p_{data}} [\log(D(\bm{x}))] \; ,\end{aligned}$$ where $\bm{z}$ is a noise input to generator $G$ with a random distribution of $p_{\bm{z}}$ and $G(\bm{z})$ is the generator output and input data $\bm{x}$ has distribution $p_{data}$. On the other hand, $A_T$ collects the classification result from $A_R$, builds the first version of generator $G$ to generate synthetic data, and then transmits synthetic data to $A_R$. The objective of $A_T$ is maximizing $A_R$’s classification error, i.e., $$\begin{aligned} \max_{G} \mathbb{E}_{\bm{z} \sim p_{\bm{z}}} [\log(1 - D(G(\bm{z})))] - \mathbb{E}_{\bm{x} \sim p_{data}} [\log(D(\bm{x}))], \label{eq:obj1}\end{aligned}$$ where $D$ is the first version of discriminator. This process continues with updated $G$ and $D$. Both $G$ and $D$ have three hidden dense layers, each with $128$ neurons. They improve in each round until convergence. The entire process forms a GAN with a minimax game played between $A_T$ and $A_R$. $$\begin{aligned} \max_{G} \min_{D} \mathbb{E}_{\bm{z} \sim p_{\bm{z}}} [\log(1 - D(G(\bm{z})))] - \mathbb{E}_{\bm{x} \sim p_{data}} [\log(D(\bm{x}))] \; ,\end{aligned}$$ although traditionally GAN is running at one entity. Note that when $G$ is trained with the objective in (\[eq:obj1\]), the gradients of $G$ rapidly vanish, which makes the training of GAN very difficult. To address the vanishing gradient problem, we use $$\begin{aligned} \max_{G} \mathbb{E}_{\bm{z} \sim p_{\bm{z}}} [\log(1 - D(G(\bm{z})))]\end{aligned}$$ as the objective function at $G$ [@Goodfellow2014]. Once converged, $A_T$ can apply its generator to generate and transmit synthetic signals. After going through the wireless channel, the transmitted signals are received by $R$, which are statistically similar to the received signals from $T$. We use the same simulation setting as that in Section \[sec:classifier\]. The only difference is that signals from other transmitters is replaced by synthetic signals from $A_T$, where these synthetic signals are generated by $A_T$’s generator. For convergence, we check the maximum perturbation in $G$ and $D$ loss functions over the most recent 100 epochs of GAN training. When this perturbation drops below $5\%$ of current loss value, we terminate the GAN training. This way, the GAN is run only for $478$ epochs in this simulation setting. We find that the same classifier pre-trained at $R$, which works very well to discriminate signals from $T$ from random or replayed signals, cannot successfully identify synthetic signals generated by the GAN. The success probability of spoofing attack is increased to $76.2\%$, when the GAN is used by the adversary. Method of spoofing attack Success probability ------------------------------ --------------------- Random signal $7.89\%$ Replay (Amplify and forward) $36.2\%$ GAN-based spoofing $76.2\%$ : Success probability of spoofing attack by different methods.[]{data-label="table:spoofing"} Finally, we consider a more challenging case where $A_T$’s location is changed after the training process, e.g., $A_T$ moves from $(0,10)$. $A_T$ may request the collaboration of $A_R$ to retrain a GAN and use its updated generator for spoofing attack. If such update is not available, $A_T$ can still use its current generator to launch the attack. Table \[table:mobility\] shows results under different $A_T$ locations. We can see that as $A_T$ moves away from $T$, the attack success probability decreases. This is an expected result since the distribution of the received channel characteristics varies as the receiver moves to a different location. However, the attack success probability is still significantly higher than the one achieved by the replay attack. $A_T$’s location Success probability ------------------ --------------------- $(0,10)$ $76.2\%$ $(0,11)$ $65.2\%$ $(0,15)$ $61.0\%$ $(0,20)$ $56.2\%$ : The impact of $A_T$’s mobility on success probability of spoofing attack.[]{data-label="table:mobility"} Conclusion {#sec:conclusion} ========== We designed a novel approach of spoofing wireless signals by generating synthetic signals by the GAN. We considered the case that an adversary transmits synthetic signals such that they are misclassified as the intended ones. We first showed that if there is no attack, a pre-trained deep learning-based classifier can distinguish signals reliably. We then considered a simple spoofing mechanism such as the replay (amplify-and-forward) attack that can only keep some pattern of intended signals. Therefore, the success probability of replay attack against a deep learning-based classifier remains limited. In this paper, we designed a GAN-based spoofing attack that generates synthetic data that is transmitted by an adversary transmitter and distinguishes real and synthetic data at an adversary receiver. The minimax game between the adversary transmitter and receiver tunes both the generator and the discriminator. Then the signals generated by the GAN generator are transmitted for spoofing attack. The GAN-based spoofing attack provides a major improvement in attack success probability over the random signal and replay attacks even when the node locations change from training to test time. As the GAN opens us new opportunities to effectively spoof wireless signals, new defense mechanisms are called for as future work. Acknowledgments {#acknowledgments .unnumbered} =============== This effort is supported by the U.S. Army Research Office under contract W911NF-17-C-0090. The content of the information does not necessarily reflect the position or the policy of the U.S. Government, and no official endorsement should be inferred. [99]{} K. Davaslioglu and Y. E. Sagduyu, “Generative adversarial learning for spectrum sensing," *IEEE International Conference on Communications (ICC)*, Kansas City, MO, May 20–24, 2018. T. O’Shea, J. Corgan, and C. Clancy, “Convolutional radio modulation recognition networks," *International Conference on Engineering Applications of Neural Networks*, Aberdeen, United Kingdom, Sept. 2–5, 2016. Y. Shi, Y. E Sagduyu, T. Erpek, K. Davaslioglu, Z. Lu, and J. Li, “Adversarial deep learning for cognitive radio security: Jamming attack and defense strategies,” *IEEE International Conference on Communications (ICC) Workshop on Promises and Challenges of Machine Learning in Communication Networks*, Kansas City, MO, May 24, 2018. Y. Vorobeychik and M. Kantarcioglu, *Adversarial machine learning*, Morgan & Claypool, 2018. Y. Shi, Y. E. Sagduyu, K. Davaslioglu, and R. Levy, “Vulnerability detection and analysis in adversarial deep learning," *Guide to Vulnerability Analysis for Computer Networks and Systems - An Artificial Intelligence Approach*, Springer, Cham, 2018. Y. Shi, T. Erpek, Y. E. Sagduyu, and J. Li, “Spectrum data poisoning with adversarial deep learning,” *IEEE Military Communications Conference (MILCOM)*, 2018. M. Sadeghi and E.G. Larsson, “Adversarial attacks on deep-learning based radio signal classification," *IEEE Wireless Communications Letters*, 2018. B. Flowers, R. M. Buehrer, and W. C. Headley, “Evaluating adversarial evasion attacks in the context of wireless communications," *arXiv preprint, arXiv:1903.01563*, 2019. M. Z. Hameed, A. Gyorgy, and D. Gunduz, “Communication without interception: Defense against deep-learning-based modulation detection," *arXiv preprint, arXiv:1902.10674*, 2019. S. Kokalj-Filipovic and R. Miller, “Adversarial examples in RF deep learning: Detection of the attack and its physical robustness," *arXiv preprint, arXiv:1902.06044*, 2019. I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville A, and Y. Bengio, “Generative Adversarial Nets,” *Advances in Neural Information Processing Systems*, 2014. T. Erpek, Y. E. Sagduyu, and Y. Shi, “Deep learning for launching and mitigating wireless jamming attacks," *IEEE Transactions on Cognitive Communications and Networking*, vol. 5, no. 1, pp. 2–14, Mar. 2019. T. C. Clancy, and N. Goergen, “Security in cognitive radio networks: Threats and mitigation," *IEEE Conference on Cognitive Radio Oriented Wireless Networks and Communications (CrownCom)*, Singapore, May 15–17, 2008. F. Penna, Y. Sun, L. Dolecek, and D. Cabric, “Detecting and counteracting statistical attacks in cooperative spectrum sensing," *IEEE Transactions on Signal Processing*, vol. 60, no. 4, pp. 1806–1822, Apr. 2012. Y. E. Sagduyu, “Securing cognitive radio networks with dynamic trust against spectrum sensing data falsification," *IEEE Military Communications Conference (MILCOM)*, Baltimore, MD, Oct. 6–8, 2014. Z. Yuan, D. Niyato, H. Li, J.B. Song, and Z. Han, “Defeating primary user emulation attacks using belief propagation in cognitive radio networks," *IEEE Journal Selected Areas in Communications*, vol. 30, no. 10, pp. 1850–1860, Nov. 2012. Y. Zou, J. Zhu, L. Yang, Y.-C. Liang, and Y.-D. Yao, “Securing physical-layer communications for cognitive radio networks," *IEEE Communications Magazine*, vol. 53, no. 9, pp. 48–54, Sep. 2015. Y. E. Sagduyu, R. Berry, and A. Ephremides, “MAC games for distributed wireless network security with incomplete information of selfish and malicious user types," *IEEE International Conference on Game Theory for Networks (GameNets)*, 2009. Y. E. Sagduyu, Y. Shi, A. B. MacKenzie, and T. Hou, “Regret minimization for primary/secondary access to satellite resources with cognitive interference," *IEEE Transactions on Wireless Communications*, vol. 17, no. 5, pp. 3512–3523, May 2018. Y. E. Sagduyu and A. Ephremides, “A game-theoretic analysis of denial of service attacks in wireless random access," *Journal of Wireless Networks*, vol. 15, no. 5, pp. 651-666, 2009. Y. E. Sagduyu, R. Berry, and A. Ephremides, “Jamming games in wireless networks with incomplete information," *IEEE Communications Magazine*, vol. 49, no 8, pp. 112–118, Aug. 2011. Z. Lu, Y. E. Sagduyu, and J. Li, “Securing the backpressure algorithm for wireless networks," *IEEE Transactions on Mobile Computing*, vol. 16, no. 4, pp. 1136–1148, Apr. 2017. Z. Lu and C. Wang, “Enabling network anti-inference via proactive strategies: a fundamental perspective," *IEEE/ACM Transactions on Networking*, vol. 25, no. 1, pp. 43–55, Feb. 2017. Q. Wang, K. Ren, P. Ning, and S. Hu, “Jamming-resistant multiradio multichannel opportunistic spectrum access in cognitive radio networks,” *IEEE Transactions on Vehicular Technology*, vol. 65, no. 10, pp. 8331–8344, 2016. L. Xiao, J. Liu, Q. Li, N. B. Mandayam, and H. V. Poor, “User-centric view of jamming games in cognitive radio networks,” *IEEE Transactions on Information Forensics and Security*, vol. 10, no. 12, pp. 2578–2590, 2015. M. Lichtman, R.P. Jover, M. Labib, R. Rao, V. Marojevic, and J.H. Reed, “LTE/LTE-A jamming, spoofing, and sniffing: Threat assessment and mitigation," *IEEE Communications Magazine*, vol. 54, no. 4, pp. 54–61, 2016. K. Gai, M. Qiu, Z. Ming, H. Zhao, and L. Qiu, “Spoofing-jamming attack strategy using optimal power distributions in wireless smart grid networks," *IEEE Transactions on Smart Grid*, vol. 8, no. 5, pp. 2431–2439, Sept. 2017. Y. Chen, W. Trappe, and R.P. Martin, “Detecting and localizing wireless spoofing attacks," *IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks*, 2007. Y. Sheng, K. Tan, G. Chen, D. Kotz, and A. Campbell, “Detecting 802.11 MAC layer spoofing using received signal strength," *IEEE Conference on Computer Communications (INFOCOM)*, Phoenix, AZ, 2008. M. Sajjad, S. Khan, T. Hussain, K. Muhammad, A.K. Sangaiah, A. Castiglione, C. Esposito, and S.W. Baik, “CNN-based anti-spoofing two-tier multi-factor authentication system," *Pattern Recognition Letters*, 2018. A. Ferdowsi and W. Saad, “Deep learning for signal authentication and security in massive internet of things systems," *arXiv preprint, arXiv:1803.00916*, 2018. L. Xiao, D. Jiang, D. Xu, H. Zhu, Y. Zhang, and V. Poor, “Two-dimensional antijamming mobile communication based on reinforcement learning," *IEEE Transactions on Vehicular Technology*, vol. 67, no. 10, pp. 9499–9512, Oct. 2018. Z. Wu. Y. Zhao, Z. Yin, and H. Luo, “Jamming signals classification using convolutional neural network," *IEEE International Symposium on Signal Processing and Information Technology (ISSPIT)*, pp. 62–67, 2017. D. T. Hoang, D. Niyato, P. Wang, and D. I. Kim, “Performance analysis of wireless energy harvesting cognitive radio networks under smart jamming attacks,” *IEEE Transactions on Cognitive Communications and Networking*, vol. 1, no. 2, pp. 200–216, June 2015. Y. Liang, Z. Cai, J. Yu, Q. Han, and Y. Li, “Deep learning based inference of private information using embedded sensors in smart devices," *IEEE Network*, 2018. Y. Yang, Y. Li, W. Zhang, F. Qin, P. Zhu, and C.-X. Wang, “Generative-adversarial-network-based wireless channel modeling: Challenges and opportunities," *IEEE Communications Magazine,* vol. 57, no. 3, pp. 22–27, Mar. 2019. T.J. O’Shea, T. Roy, N. West, and B.C. Hilburn, “Physical Layer communications system design over-the-air using adversarial networks," *arXiv preprint, arXiv:1803.03145*, 2018. T.J. O’Shea, T. Roy, and N. West, “Approximating the void: Learning stochastic channel models from observation with variational generative adversarial networks," *arXiv preprint, arXiv:1805.06350*, 2018. T. Kinnunen, M. Sahidullah, H. Delgado, M. Todisco, N. Evans, J. Yamagishi, and K.A. Lee, “The ASVspoof 2017 challenge: Assessing the limits of replay spoofing attack detection," 2017. A. Hoehn and P. Zhang, “Detection of replay attacks in cyber-physical systems," *IEEE American Control Conference (ACC)*, 2016.

--- abstract: 'We investigate the parameter space of the Standard Model enhanced by a gauge singlet real scalar $S$. Taking into account all the theoretical and experimental constraints, we show the allowed parameter space for two different types of such singlet-enhanced Standard Model. For the first case, the scalar potential has an explicit $Z_2$-symmetry, and may lead to a dark matter candidate under certain conditions. For the second case, the scalar potential does not respect any $Z_2$. This is again divided into two subcategories: one where the Standard Model vacuum is stable, and one where it is unstable and can decay into a deeper minimum. We show how the parameters in the scalar potential control the range of validity of all these models. Finally, we show the effect of one-loop correction on the positions and depths of the minima of the potential.' --- [**[Potential of a singlet scalar enhanced Standard Model]{}**]{} [**Swagata Ghosh**]{},$^{a,}$[^1] [**Anirban Kundu**]{},$^{a,}$[^2] and [**Shamayita Ray**]{} $^{a,b,}$[^3] ${}^a$   [*[Department of Physics, University of Calcutta,\ 92 Acharya Prafulla Chandra Road, Kolkata 700009, India ]{}*]{} ${}^b$ [*[Laboratory for Elementary Particle Physics, Cornell University, Ithaca, NY 14853, USA]{}*]{} Introduction {#intro} ============ One of the minimalistic extensions of the Standard Model (SM) is that by one or more gauge singlet real (or complex) scalar field(s). Motivations to introduce a singlet scalar to the SM are, amongst others: (i) to provide a viable cold dark matter (CDM) candidate through Higgs portal models [@portal-dm], (ii) to make the electroweak phase transition a strong first-order one [@1107.5441; @ew-others], and (iii) to address the naturalness problem of the SM Higgs boson [@singlet-veltman]. Phenomenological aspects of such singlets have also been discussed in case of colliders [@barger-real; @barger-cmplx; @robens; @dawson; @collider-singlet], and in the context of electroweak precision constraints [@precision]. Such a singlet with a mass around 750 GeV may also be responsible for the recently observed excess in the diphoton channel [@atlas-750; @cms-750] if one adds vectorial fermions to the model. One often imposes a $Z_2$-symmetry on the scalar potential under which the SM particles are all even and the extra singlet(s) is(are) odd, which can make the lightest $Z_2$-odd particle a CDM candidate. In an analogous way to what happens in R-parity conserving supersymmetry and universal extra dimension models, a $Z_2$-symmetry on the scalar potential under which the singlet $S$ is odd, can in principle lead to the Higgs portal dark matter models where $S$ constitutes the dark matter. A necessary condition for this is zero (or infinitesimally small) vacuum expectation value (VEV) for $S$, so that it cannot mix with the SM doublet scalar $\Phi$. One must remember that this $Z_2$ is rather ad hoc, introduced just for the sake of having a CDM candidate. The nature of the tree-level scalar potential has also been discussed by several authors [@potential-tree]. The potential is more complicated than the SM one because of one extra field and the possibility that the CP-even neutral component of the SU(2) doublet $\Phi$, which will be denoted by $\phi$, and the gauge singlet scalar $S$ can both have nonzero VEVs. We denote these VEVs by $v$ and $v_s$ respectively. If there is only one minimum for $v_s$, one often uses the shift symmetry $S\to (S+\Delta)$ to ensure $v_s=0$, which in turn ensures a CDM candidate if the model is $Z_2$-symmetric. However, if there are more than one minima for $v_s$, there is no particular advantage in using the shift symmetry, except ensuring that $v_s=0$ is an extremum. In this paper, we investigate the nature of the tree-level potential of the SM extended by one real singlet scalar, which we call SM+S, with and without the $Z_2$-symmetry. We find the allowed parameter space for three different SM+S models: (1) the Higgs portal model [*i.e.*]{}, the $Z_2$-symmetric model where $S$ can be a CDM candidate, (2) the $Z_2$-symmetric model with no CDM candidate, and (3) the $Z_2$-asymmetric model. Obviously, the allowed parameter space has to be consistent with all theoretical and current experimental constraints. For the third case, $Z_2$ breaking is soft, coming from operators with mass dimension less than 4. Dimension-4 $Z_2$ breaking operators are forbidden from gauge and/or Lorentz symmetry. As we consider only one real singlet scalar, all the couplings are real in these models. Data from the Large Hadron Collider (LHC) essentially constrains the mixing between $S$ and $\phi$. The mixing angle $\theta$ is constrained to be so small that all SM+S models also satisfy the constraints coming from the oblique parameters. We will discuss it in detail later. The measurement of the $W$ boson mass also puts serious restrictions on the parameter space of SM+S [@1406.1043]. Apart from these experimental constraints, there are three other important constraints. Firstly, the potential has to be stable at all energy scales till one reaches the range of validity. This range is, in general, way below the Planck scale, apart from some exceptional choices of the parameters. Above this limit, either at least one of the couplings blow up, or the potential becomes unbounded from below along some direction in the field space. The second one is the existence of a minimum, either global or local, where $\bra\phi\ket=v=246$ GeV, which will be referred to as the electroweak (EW) vacuum. The third constraint comes from the stability of the EW vacuum; if there is another minimum deeper than the EW vacuum, the tunnelling lifetime should not be less than the age of the universe. We also study the effect of one-loop corrections to the potential. In general, the one-loop corrections are expected to be small compared to the tree-level potential, unless one looks along a flat direction. Even when both $S$ and $\Phi$ have non-zero mass terms, one can find a direction in the field space, at least for large values of the fields, following the prescriptions of Gildener and Weinberg [@gildener]. One should also choose the regularization scale properly. This choice, in principle, is arbitrary if one uses renormalization group (RG) improved couplings. On the other hand, one can always tune the scale so that the one-loop corrected EW vacuum still has $\bra\phi\ket = 246$ GeV [@indrani-4]. However, this makes the choice of the regularization scale dependent on the model parameters. The paper is arranged as follows. In , we give a brief outline of the singlet-enhanced SM, and discuss the constraints. discusses the $Z_2$-symmetric model where the CDM is allowed, and is on models with a non-zero singlet-doublet mixing. While these are all tree-level results, we discuss the one-loop corrections to the potential in . In , we summerize and conclude. The real singlet scalar enhanced SM {#model} =================================== Let us consider the most general potential for SM+S, the single real-scalar extended SM: V(,S) &=& -\^2\^- M\^2 S\^2 + (\^)\^2 + a\_1 \^S + a\_2 \^S\^2 + b\_1 S + b\_3 S\^3 + b\_4 S\^4, \[pot\] where $\Phi$ is the SM doublet and $S$ is a gauge singlet scalar field. We denote the CP-even neutral component of $\Phi$ by $\phi/\sqrt{2}$, and the VEVs are given as $\bra\phi\ket = v$, $\bra S \ket = v_s$. There are six new parameters in over and above those in the SM. Of them $a_2$, $M^2$, and $b_4$ respect the $Z_2$ symmetry of $S\to -S$, while $a_1$, $b_1$ and $b_3$ break it softly. Note that $\mu^2,M^2 > 0$ stand for wrong-sign mass terms in the potential. All the couplings are, of course, real, because we have only one real singlet $S$ in this model. The stability conditions are obtained from the requirement that the potential should not become negative along any direction of the field space, which gives &gt; 0, b\_4 &gt; 0, a\_2 + 2 &gt; 0, \[stability\] along the directions $S=0$, $\Phi =0$, and $\sqrt{\lambda}\Phi^\dag\Phi = \sqrt{b_4}S^2$ directions respectively. In case there is mixing between $\phi$ and $S$, the mass eigenstates $(h,s)$ are defined as h = + S , s = -+ S , \[mass-eigen\] where $\theta$ is the mixing angle. In terms of VEVs $v$ and $v_s$, the potential in becomes V(v,v\_s) = -12 \^2 v\^2 - M\^2 v\_s\^2 + 14v\^4 + 12 a\_1 v\^2 v\_s + 12 a\_2 v\^2 v\_s\^2 + b\_1 v\_s + b\_3 v\_s\^3 + b\_4 v\_s\^4, \[pot-v-vs-1\] and the extremization conditions are -\^2 v + v\^3 + a\_1 vv\_s + a\_2 vv\_s\^2 &=& 0, \[pot-min1-v\]\ 12 a\_1 v\^2 + a\_2 v\^2 v\_s + b\_1 - 2M\^2 v\_s + 3b\_3 v\_s\^2 + 4b\_4 v\_s\^3 &=& 0 . \[pot-min1-vs\] One can always apply the shift symmetry $S\to (S+\Delta)$ to , where $\Delta$ is some constant, since this shift in $S$ does not change the physics. We can use this freedom[^4] to remove one of the independent terms of the potential. Let us choose b\_1 = - a\_1 v\^2 . \[b1\] Use of in removes the linear terms in $S$ and simplifies the potential to V(v,v\_s) = -12 \^2 v\^2 - M\^2 v\_s\^2 + 14v\^4 + 12 a\_2 v\^2 v\_s\^2 + b\_3 v\_s\^3 + b\_4 v\_s\^4 . \[pot-v-vs\] The extremization conditions now guarantee one extremum line along $v=0$ and another along $v_s=0$ (because of the shift symmetry): v (-\^2 + v\^2 + a\_2 v\_s\^2) &=& 0, \[pot-min2-v\]\ v\_s (a\_2 v\^2 - 2M\^2 + 3b\_3 v\_s + 4b\_4 v\_s\^2 ) &=& 0. \[pot-min2-vs\] From Eqs.  and one finds that there are three extrema for $v_s$, and for each $v_s$ there are three extrema for $v$, depending on the existence of real solutions[^5]. For $v=0$, can be simplified to v\_s (- 2M\^2 + 3b\_3 v\_s + 4b\_4 v\_s\^2 ) = 0. \[pot-min3-vs-v0\] For the other two nonzero extrema in the $\Phi$-direction, one uses $v^2 = (\mu^2 - a_2 v_s^2)/ \lambda$ from and get v\_s ( v\_s\^2 + 3 b\_3 v\_s + ) = 0. \[pot-min3-vs\] Given the parameters of the potential, Eq. (\[pot-min3-vs\]) gives the condition for real non-zero solutions for $v_s$ along $v\ne0$. Note that for a $Z_2$-symmetric potential ($b_3=0$), the stability criteria ensure that $(a_2\mu^2/\lambda - 2M^2) < 0$. The condition that any extremum of $V(v,v_s)$ is a minimum, and not a maximum or saddle point, is &gt; 0, &gt; 0. \[cond-minima\] To get the mixing between the CP-even component of the SM Higgs doublet $\phi$ and the real scalar $S$, we expand the potential in about the VEVs of the fields. The terms quadratic in fields are given by V(,S) && \^2+ S\^2 + 2S\ && & S [M]{} S , where = 12( -[\^2]{} + 3 v\^2 + [a\_1]{} v\_s + [a\_2]{}v\_s\^2) & + a\_2 vv\_s + a\_2 vv\_s & -M\^2 + + 3 b\_3 v\_s + 6 b\_4 v\_s\^2 , \[massmat\] in its most general form. It should be noted that the mixing between $\phi$ and $S$ also depends on the parameter $a_1$, which does not appear in the minimization conditions in and , because of the choice in . If $M_1, M_2$ are the eigenvalues of ${\cal M}$, the masses of the physical states are given by m\_h = , m\_s = , \[mass\] and the mixing angle $\theta$ is the angle which parametrize the $2\times2$ rotation matrix that diagonalises ${\cal M}$. Zero VEV for at least one field ------------------------------- One instructive case is when one of the minima has either $v=0$ or $v_s=0$ or both. ### Minimum at $v=v_s=0$ {#v0-vs0} The point $v=v_s=0$ is an extremum where = -\^2 , = -2 M\^2 , = 0 , \[extreme0\] and will be a minimum if $\mu^2, M^2 < 0$ i.e. the mass terms are right-sign. From one can see for this case that $v$ will also have two non-zero real roots if $(a_2 v_s^2 - \mu^2) < 0$, which requires a large negative $a_2$, resulting in a large singlet-doublet mixing. ### Minimum at $v=0$, $v_s\ne0$ If there is a minimum for $v=0$, $\partial^2 V/\partial v \partial v_s = 2a_2 v v_s =0$, and the condition for minimum translates to &gt; 0      a\_2 v\_s\^2 -\^2 &gt; 0 ,    &gt; 0     8 b\_4 v\_s\^2 + 3 b\_3 v\_s &gt; 0. The first condition shows, from , that there is only one real solution for $v$ at $v=0$. As $b_4 > 0$ from stability criterion, the second condition yields interesting bounds. If $b_3 < 0$ and $v_s < 0$ or $b_3 >0$ and $v_s>0$, ${\partial^2 V}/{\partial v_s^2}$ is definitely positive. On the other hand, if $v_s > 0$ but $b_3 < 0$, there is a lower bound on $v_s$, v\_s &gt; . Similarly, if $b_3 > 0$ and $v_s < 0$, there is an upper bound on $v_s$: v\_s &lt; - . ### Minimum at $v\ne0$, $v_s=0$ For $v\not=0$ and $v_s=0$, the minima along $v$ are symmetric, at $v=\pm \sqrt{\mu^2/\lambda}$. This means $\mu^2 > 0$ for $v$ to be real and hence $\partial^2V/\partial v^2 = 2 \mu^2 > 0$. So the condition for minimum at $v\ne0$, $v_s=0$ reduces to $(a_2 v^2 - 2M^2) > 0$. Minimum at $v\not= 0$, $v_s \not=0$ {#v-vs-nonzero} ----------------------------------- The constraints for $v\ne0$, $v_s\ne0$ are obtained using the concavity condition in , along with the expression for the second derivatives, to be = 2(\^2 - a\_2 v\_s\^2) = 2 v\^2 , = 3 b\_3 v\_s + 8 b\_4 v\_s\^2 , = 2 a\_2 v v\_s . For the $Z_2$-symmetric potential, $b_3=0$, and hence the condition for a minimum at $v\ne0$,$v_s\not= 0$ simplifies to $4 \lambda b_4 > a_2^2$, which is nothing but one of the stability criteria for the potential as shown in . Further simplifications occur if we have only one minimum of the potential. In this case, it has to be at $|v|=246$ GeV, and hence solutions for $v_s\ne 0$ can be obtained in a straightforward way from : v\_s = . \[vs-sol2\] Thus, for the $Z_2$-symmetric case, the condition for a minimum at $v_s\not= 0$ is $a_2 v^2 - 2M^2 < 0$, as $b_4 > 0$ from stability criterion. For nonzero $b_3$, the condition is a\_2 v\^2 - 2 M\^2 &lt; 9 b\_3\^2 / 16 b\_4. One can easily have more than one minima with $v\not=0$ and $v_s\not=0$. However, if one minimum is at the origin, the second minimum at nonzero $v$ and $v_s$ requires $a_2$ to be large and negative, as discussed in Section \[v0-vs0\]. Such large values of $a_2$ are under severe kosh from the LHC data. A detailed discussion on the nature of the scalar potential related to the electroweak phase transition can be found in Ref. [@1107.5441]. LHC constraints --------------- If there is no mixing between $\phi$ and $S$, there are no constraints on $S$ coming from the LHC data, except that it cannot be so light ($<m_h/2$) that $h \to SS$ is allowed and the branching ratio is more than $34\%$ at 95% CL [@atlas-cms-combo]. Similarly, there is no constraint from electroweak precision observables as $S$ does not have any gauge coupling. If there is a mixing between $\phi$ and $S$ parametrized by an angle $\theta$, the number of events, which is just the production cross-section times the branching ratio, goes down by $\cos^2\theta$. Denoting the production cross-section times the decay width scaled to that in the SM by $\mu$, the ATLAS and CMS combined result shows = 1.09\^[+0.11]{}\_[-0.10]{}, from which we can put a limit of $\theta \leq 0.1$ at $1\sigma$, which we will use for our subsequent discussion. This helps us to avoid both LHC data and precision constraints at one stroke. Oblique parameters ------------------ Only the $T$ parameter may be significant in the small mixing case. In this model, the $T$ parameter is given by [@barger-real] T\^[[SM+S]{}]{} &=& -() { \^2.\ && . + \^2}, where $m_1 (\approx 125$ GeV) and $m_2$ are the two mass eigenstates, and $\theta$ is the mixing angle. The SM expression for $T$ can be found by putting $\theta=0$ and $m_1=m_h$. The quantity constrained by the electroweak fit, $\Delta T$, is given by [@pdg2014-ewreview] T = T\^[[SM+S]{}]{} - T\^[[SM]{}]{} = 0.01 0.12. $T$ is related with the $\rho$-parameter by $\rho-1=\alpha T$. For small mixing ($\theta \leq 0.1$), the constraints coming from the oblique parameters are not significant. Renormalization Group equations ------------------------------- We would also like to see how the couplings evolve with energy. The one-loop $\beta$-functions are [@Chakraborty:2012rb] 16\^2 \_&=& 12\^2 + 6g\_t\^2+ a\_2\^2 -32(g\_1\^2+3g\_2\^2) - 3 g\_t\^4 + (g\_1\^4 + 2 g\_1\^2 g\_2\^2 + 3 g\_2\^4),\ 16\^2 \_[b\_4]{} &=& 36 b\_4\^2 + a\_2\^2 ,\ 16\^2 \_[a\_2]{} &=& a\_2 ,\ 16\^2 \_[g\_t]{} &=& g\_t \[all-rge\] where $\beta_h\equiv dh/dt$, and $t \equiv \ln(Q^2/\mu^2)$. The $\beta$-functions for all gauge couplings are identical to that of the SM. For simplicity, we have put all the SM Yukawa couplings equal to zero except for that of the top quark. This hardly changes our conclusions. For the $Z_2$-asymmetric case, the trilinear couplings $a_1$ and $b_3$ also evolve: 16\^2 \_[a\_1]{} &=& a\_1(9+ 4 a\_2) + 6 a\_2 b\_3,\ 16\^2 \_[b\_3]{} &=& 2 a\_1 a\_2 + 36 b\_3 b\_4. \[z2-odd-rge\] Note that $\beta_{b_4}$ is always positive and hence can only increase, starting from a positive value. $\beta_\lambda$ also gets a positive contribution on top of the SM ones. These make the couplings blow up at a much lower scale than the Planck scale ($\sim 10^{19}$ GeV), unless one starts with very small values of $b_4$. Similarly, $\beta_{a_2}$ is proportional to $a_2$ itself and can lead to a blow-up for large $a_2$. For our analysis, we have taken the initial values of the couplings at the electroweak scale in such a way that the Higgs boson mass is correctly reproduced as $m_h\in [124:126]$ GeV. The threshold effects are taken at the singlet mass scale, however it has been seen that the final results are not very sensitive on the exact choice of this scale, and moreover, uncertainties coming from possible higher-loop contributions are larger compared to the uncertainties coming from the threshold corrections. The constraints on the parameter space are all obtained with the one-loop improved values of the couplings. The unstable vacuum case ------------------------ If there are two minima of the potential and the EW vacuum is shallower, the universe can tunnel down to the deeper vacuum. In such cases, the parameters must be chosen such that the lifetime of the shallower vacuum should be at least as large as the lifetime of the universe, which is about 13.7 billion years. ![(a) Left: Schematic diagram of an asymmetric double-well potential with a single field. Parameters $\epsilon$ and $\delta$ control the tunnelling lifetime. If there are more than one fields, one gets a multi-dimensional contour. (b) Right: Example of a two-dimensional contour for SM+S. The line shows the shortest path joining the two minima along which the tunnelling probability should be calculated. Notice that the path does not pass through the local maximum. Contour values denote the potential at that point in the unit $10^{10}$ GeV$^4$.[]{data-label="fig:metastable"}](fig1a.pdf "fig:"){width="7cm" height="5.8cm"} ![(a) Left: Schematic diagram of an asymmetric double-well potential with a single field. Parameters $\epsilon$ and $\delta$ control the tunnelling lifetime. If there are more than one fields, one gets a multi-dimensional contour. (b) Right: Example of a two-dimensional contour for SM+S. The line shows the shortest path joining the two minima along which the tunnelling probability should be calculated. Notice that the path does not pass through the local maximum. Contour values denote the potential at that point in the unit $10^{10}$ GeV$^4$.[]{data-label="fig:metastable"}](fig1b.pdf "fig:"){width="7cm" height="5.6cm"} To calculate the lifetime of the metastable state [@coleman], let us assume that the potential at the EW vacuum is zero (this can always be achieved by a constant shift), the true vacuum has a depth $-\epsilon$ ($\epsilon > 0$), and the height of the barrier with respect to the SM vacuum is $\delta$, as shown in . The decay width density of the universe, $\Gamma/V$, is given by = A (-B), where $A$ is a small pre-factor, and $B$ is estimated in the thin-wall approximation as [@barroso] B = ( )\^3. \[B-def\] This expression is true for a single field whose self-quartic coupling is given by $\lambda$. With more than one fields, there are several dimensionless couplings and the tunnelling path may not even go through under the hill. However, for SM+S, what we have done numerically is to find the two minima and then calculate the tunnelling along the straight line joining them, as shown in . This is effectively a single-field approximation, where the field is a combination of $h$ and $S$. The height $\delta$ is taken to be the maximum height above zero-level [*along this path*]{} and not the local maximum of the field space. The ratio $\delta/\epsilon$ should be greater than $0.1$ to make the shallower vacuum stable with respect to the lifetime of the universe. Keeping in mind of the necessary simplifications, we have chosen only those models for which this ratio is more than unity, and therefore the stability is assured. One-loop corrections to the potential ------------------------------------- The one-loop effective potential in the SM is given by V\_1(\_c) = \_i n\_i m\_i(\_c)\^4 ( - C\_i), where $i$ runs over $h,G,W,Z$ and $t$, and n\_i = 1,3,6,3,-12,    C\_i = 32,32,56,56,32, for $i=h,G,W,Z$ and $t$ respectively. The masses are field-dependent and depend on the classical minimum $\phi_c$. $Q$ is the arbitrary regularization scale. To be precise, the radiative corrections, being suppressed by the loop factor, are significant only along a flat or near-flat direction in the field space. If all the dimensionful parameters are zero, the theory becomes scale invariant. The minimization conditions in this case, namely, $v \left(\lambda v^2 + a_2 v_s^2 \right)= 0$ and $v_s \left( a_2 v^2 + 4 b_4 v_s^2\right) = 0$, yield $4\lambda b_4 = a_2^2$ as the consistency condition if neither $v$ nor $v_s$ vanishes. The last condition makes the determinant of the mass matrix equal to zero, ensuring a massless mode and hence a flat direction in the field space. If dimensionful couplings are present, there is in general no flat direction in the $\phi$-$S$ plane, and one expects the radiative corrections to the potential to have a small effect. Note that $4 \lambda b_4 = a_2^2$ is the limiting case of the stability condition. However, If there is a strong hierarchy between the two VEVs, the direction along the smaller-VEV field is almost flat. The choice of the regularization scale may change $v_s$ significantly, and if there is a significant singlet-doublet mixing, both the physical scalar masses may get affected. We will see the numerical estimates later. This also shows that putting $a_1 = - 2 a_2 v_s$ in does not lead to a CDM candidate, because such a fine-tuned relationship is not stable under radiative corrections. With two scalar fields, the form of the one-loop corrected potential is V(\_c,\_c) = V\_0(\_c,\_c) + V\_1(\_c,\_c), where $\phi_c$ and $\eta_c$ are the classical minima along $\phi$ and $S$ respectively, and $V_0$ is the tree-level potential written in terms of the classical minima. $V_1$ is the one-loop correction, given by V\_1(\_c,\_c) = V\_1(\_c) + m\_S\^4(\_c,\_c) . One should note that in case of nonzero mixing, the field-dependent mass of $h$ in $V_1(\phi_c)$ is a function of $\eta_c$ too. Thus, $V_1$ includes all the SM fields and the singlet. The regularization scale $Q$ is arbitrary, but one can choose it in such a way that the one-loop corrected minimum for $\Phi$ remains unchanged, in other words, $\phi_c$ falls at $v$. This keeps all the SM fermion and gauge boson masses invariant, and also keeps the Goldstone bosons massless; thus, we do not need to consider the Goldstone boson contributions for the effective potential. A detailed discussion of the procedure is given in Ref. [@indrani-4] for the two-Higgs doublet model potential. ${Z_2}$-symmetric potential with ${v_s=0}$ {#vs0} ========================================== Let us first go through the well-studied case of the $Z_2$-symmetric potential ($a_1=b_1=b_3=0$ in ), for the sake of completeness of this study. One can always get a minimum at $v_s=0$ by applying the shift symmetry $S\to S+\Delta$. This minimum should better be the only one, or at least the global one (a local one with lifetime more than the age of the universe will also do) if we want to have a CDM candidate in $S$, with mass of $\sqrt{a_2 v^2 - 2M^2}$. The points with $M^2 > 0$ need a very large and negative $a_2$ and are ruled out by the CDM spin-independent scattering cross-section limits. Thus, all such Higgs portal dark matter models must have a right-sign mass term ($M^2 < 0$) for $S$. This is true even for the narrow region of Higgs resonance, at about $m_S\approx m_h/2$. ![The range of validity as a function of $a_2$ (left) and $b_4$ (right), in the $Z_2$-symmetric case with the minimum at $v_s=0$.[]{data-label="fig:DMvalidity"}](fig3a.pdf "fig:"){width="8.5cm"} ![The range of validity as a function of $a_2$ (left) and $b_4$ (right), in the $Z_2$-symmetric case with the minimum at $v_s=0$.[]{data-label="fig:DMvalidity"}](fig3b.pdf "fig:"){width="8.5cm"} The only way for $S$ to interact with the SM sector is through the term $a_2 S^2 \Phi^\dag\Phi$ in , since $a_1 = b_1 = b_3 = 0$ in this case. The spin-independent CDM-nucleon scattering cross-section is given, in this scenario, by [@feng; @duerr] = , where $m_N, m_S$ and $m_h$ are the masses of the nucleon, $S$, and Higgs respectively. The matrix element for scattering is given by $f$, whose value is approximately $0.3$. If $a_2$ is very small, the CDM detection cross-section becomes small, but the annihilation rate goes down too, leaving more dark matter in the universe than is allowed, and thus leading to overclosure. If $a_2$ is large, the scattering cross-section of CDM with nucleons is also large and hence will be severely constrained by direct detection experiments, in particular LUX, which gives the best limits now [@dm-limits; @lux]. Thus, apart from the narrow Higgs resonance region, only a small wedge for the dark matter mass $M_{\rm DM} > 200$ GeV is still allowed, and we focus only on those models that provide $M_{\rm DM}$ in this range. In , we show the allowed regions as a function of $a_2$, every dot corresponds to a particular choice of parameters. For all these models $M^2 < 0$, which, by our definition of the potential, means a right-sign mass term for the singlet and no symmetry breaking in the $S$-direction. Technically, $M^2 > 0$ can also lead to a local minimum at $\bra S \ket = 0$, but if we take $a_2$ to be in the perturbative region, such models lead to low $M_{\rm DM}$ and are hence ruled out by the direct detection data. Depending on the values of $a_2$ and $b_4$, one can also check how far in the energy scale the singlet DM model remains valid. For this, we use the one-loop renormalization group (RG) equations[^6] and see where the couplings become non-perturbative and ultimately hit the Landau pole, or the potential becomes unstable. Our results are shown in for the two relevant parameters $a_2$ and $b_4$. Note that there is an upper limit on $b_4 \sim 0.4$ above which the model ceases to be valid even before 50 TeV. While there is no such limit for $a_2$, low-$a_2$ models have a smaller range of validity compared to medium-$a_2$ ($\sim 0.3$) models, where the validity can be as high as $10^{15}$ GeV. While the allowed region for each model depends on the exact values of the parameters chosen, some intuitive insights can be put forward. As there is no mixing, $\lambda$ must start from its SM value $\sim 0.13$. The only modification to its $\beta$-function comes from the $a_2^2$ term, so the range of validity increases with increasing $a_2$, provided $b_4$ is sufficiently small to start with and does not hit its Landau pole earlier ($b_4$ starts from a positive value, and always increases). So, for the low-$a_2$ regions, it is the vacuum stability that mostly controls the allowed range. After $a_2$ passes a certain value, and/or $b_4$ becomes large, the range is controlled by the blowing up of one or more of the couplings. As we have just shown, $a_2 < 0$ is already ruled out for this model. A similar study on the parameter space of Higgs portal dark matter models was performed recently in Ref. [@1509.01765]. There is always a chance that with improved measurements, the wedge region may go away. In the Higgs resonance region, the allowed values of $a_2$ are much smaller, and from , we see that such models cease to be valid at about $10^6$ GeV. This model, with addition of vectorlike fermions, may explain the recently observed resonance at 750 GeV. However, introduction of such fermions spoils the possibility of a Higgs portal dark matter, as the scalar decays through the fermion loops. On the other hand, there is a new constraint on the singlet mass, which narrows down the parameter space even further. The renormalization group equations also change, with new Yukawa couplings introduced, and may affect the stability of the potential. We will not discuss this extension any further here. Singlet-doublet mixing with $v_s\not=0$ {#vsne0} ======================================= $Z_2$-symmetric case {#z2symm_NoDM} -------------------- ![The allowed parameter space for $Z_2$-symmetric case with $v_s\not=0$.[]{data-label="fig:Z2noDM-aps"}](fig4a.pdf "fig:"){width="8.5cm"} ![The allowed parameter space for $Z_2$-symmetric case with $v_s\not=0$.[]{data-label="fig:Z2noDM-aps"}](fig4b.pdf "fig:"){width="8.5cm"} ![Range of validity for different values of $a_2$ and $\lambda$ in the $Z_2$-symmetric SM+S models with $v_s\not= 0$. The plot for $b_4$ is similar to that for $\lambda$. Large values of either $b_4$ or $\lambda$ limit the range of validity of the models, because of the nature of the RG equations. $a_2$ plays a subdominant role here.[]{data-label="fig:Z2noDM-val"}](fig5a.pdf "fig:"){width="8.5cm"} ![Range of validity for different values of $a_2$ and $\lambda$ in the $Z_2$-symmetric SM+S models with $v_s\not= 0$. The plot for $b_4$ is similar to that for $\lambda$. Large values of either $b_4$ or $\lambda$ limit the range of validity of the models, because of the nature of the RG equations. $a_2$ plays a subdominant role here.[]{data-label="fig:Z2noDM-val"}](fig5b.pdf "fig:"){width="8.5cm"} If the singlet field $S$ develops a nonzero VEV, [*i.e.*]{} $v_s\ne0$, the physical fields $h$ and $s$ become orthogonal combinations of $\phi$ and $S$, with the mixing angle constrained by the LHC data to be $\theta < 0.1$. Hence there is no CDM candidate in this model [^7]. The allowed parameter space is shown in for two distinct cases: $M^2 < 0$ (right-sign mass term for the singlet) and $M^2 > 0$ (wrong-sign mass term). We note the following characteristics: - For $M^2 < 0$, only negative values of $a_2$ are allowed. This follows from the condition $(a_2 v^2 - 2M^2) < 0$. Both positive and negative values of $a_2$ are allowed for $M^2 > 0$. - There is a correlation between $a_2$ and $\lambda$ for $M^2 < 0$. This follows from the $a_2^2/\lambda$ dependence of $v_s^2$ in . - Only small positive values of $\mu^2$ are allowed for $M^2 < 0$. This can be understood from the constraint $(a_2\mu^2/\lambda - 2M^2 )< 0$ coming from the stability criteria, as mentioned in Sec. \[model\]. It can be shown easily that for $M^2 < 0$, $a_2$ and $\mu^2$ have to be of opposite signs to have real $v_s$, and the only possibility to have a real $v$ as well is to have $\mu^2 >0$ and $a_2<0$. The magnitudes of $a_2$, $\mu^2$ and $M^2$ are, however, restricted by the Higgs mass and the mixing angle $\theta$. For $M^2 < 0$, there is no such constraint and $|\mu^2|$ can be large. The range of validity for the models is shown in . One can see that large values of $a_2$ or $\lambda$ necessarily mean a smaller range of validity, which follows from the nature of the RG equations. The coupling $b_4$ always increases, so one has to start with a sufficiently small value of $b_4$ not to hit the Landau pole. The other two couplings, $\lambda$ and $a_2$, can be controlled by the negative contributions coming from Yukawa or gauge couplings if they are sufficiently small to start with. Too small a value means instability setting in at a low scale, and too large a value means a quick blowing up of the couplings. Thus, an intermediate range, $a_2 \sim \lambda \sim 0.2$ is the region for maximum validity. We have explicitly checked that the mixing angle is always well within the LHC limit. One-loop corrections do not change the nature of the potential qualitatively, except changing the depth of the potential. However, the composition of the CP-even neutral scalars change with the one-loop corrections, because the ratio of $\phi_c$ and $\eta_c$, and hence the mixing angle $\theta$ of the mass matrix changes from its tree-level value. $Z_2$-asymmetric case {#z2asymm} --------------------- ![Allowed region for $\lambda$ and $a_2$, where the EW vacuum is the “global” (red) or the “local” (blue) minimum in $Z_2$-asymmetric SM+S models, to have the validity at least upto 50 TeV.[]{data-label="fig:noz2val"}](fig6a.pdf "fig:"){width="8.5cm"} ![Allowed region for $\lambda$ and $a_2$, where the EW vacuum is the “global” (red) or the “local” (blue) minimum in $Z_2$-asymmetric SM+S models, to have the validity at least upto 50 TeV.[]{data-label="fig:noz2val"}](fig6b.pdf "fig:"){width="8.5cm"} In this section, we focus on only those models that allow two non-zero minima for the potential, as discussed in Section 2.2. From we can see that for $b_3\ne0$ the two minima have unequal depths. We demand one of them to be the EW vacuum with $|v|=246$ GeV. If this is the deeper minimum, the universe is stable; if this is the shallower one, the universe can decay to the true vacuum and in that case the lifetime must at least be equal to the age of the universe. First, let us focus on the tree-level potential. In , we show the range of validity of these models for various choices of $a_2$ and $\lambda$. The trend is similar to what we have seen before in Section 4.1: large values of $\lambda$, $a_2$, or even $b_4$ make the couplings blow up at a relatively low scale. Note that $a_2 < 0$ ($a_2 > 0$) models tend to have a local (global) minimum at $v=246$ GeV, but there are exceptions. The range of validity of these models is almost the same as that of the $Z_2$-symmetric case, because the RG evolutions of the couplings are controlled by the dimensionless couplings. The allowed region for $a_2$, however, is bunched more towards $a_2\sim 0$. ![The dependence of the lifetime of the metastable vacuum on $b_3$ (left) and $\lambda$ (right).[]{data-label="fig:stable"}](fig7a.pdf "fig:"){width="8.5cm"} ![The dependence of the lifetime of the metastable vacuum on $b_3$ (left) and $\lambda$ (right).[]{data-label="fig:stable"}](fig7b.pdf "fig:"){width="8.5cm"} For models with a shallower minimum at the EW vacuum, one may also estimate the lifetime of the universe. This is bound to be a rough estimate as the path between the two minima need not pass through a local maximum or even a saddle point. Approximately, $B \geq 1$ (see ) leads to a metastable vacuum [@barroso] while $B \leq 1$ tends to make the universe unstable. Assuming the maximum of the quartic couplings to be of the order of unity, this results in an approximate bound of $\delta/\epsilon > 0.05$. In we show how the tunnelling lifetime depends on the parameter $b_3$ as a function of the ratio $\delta/\epsilon$ and $\lambda$ as a function of $B$. The dependence of $b_4$ is similar to that of $\lambda$. shows that the controlling parameter is $b_3$ because that creates the depth difference between the two minima. The smaller $b_3$ is, the larger is the lifetime of the metastable minimum. As a conservative estimate, we have taken the stability limit to be $\delta/\epsilon \geq 1$, which excludes the low $\delta/\epsilon$ and hence low $B$ values. Such excluded models are also shown in the right hand side plot of . The distribution of models does not depend much on the quartic couplings $b_4$ or $\lambda$. One loop corrections to the SM+S potential {#oneloop} ========================================== ![The potential profile as a function of $\phi_c$ (the classical minimum for the doublet-dominated field) and $\phi_s$ (the classical minimum for the singlet-dominated field) for the two models described in the text. The upper panel is for model 1 and the lower panel is for model 2. The profiles are drawn along the line joining the two minima. Note that the second minimum can change significantly and the shift depends on the choice of the regularization scale $Q$. []{data-label="fig:oneloop"}](fig8a.pdf "fig:"){width="8.5cm"} ![The potential profile as a function of $\phi_c$ (the classical minimum for the doublet-dominated field) and $\phi_s$ (the classical minimum for the singlet-dominated field) for the two models described in the text. The upper panel is for model 1 and the lower panel is for model 2. The profiles are drawn along the line joining the two minima. Note that the second minimum can change significantly and the shift depends on the choice of the regularization scale $Q$. []{data-label="fig:oneloop"}](fig8b.pdf "fig:"){width="8.5cm"}\ ![The potential profile as a function of $\phi_c$ (the classical minimum for the doublet-dominated field) and $\phi_s$ (the classical minimum for the singlet-dominated field) for the two models described in the text. The upper panel is for model 1 and the lower panel is for model 2. The profiles are drawn along the line joining the two minima. Note that the second minimum can change significantly and the shift depends on the choice of the regularization scale $Q$. []{data-label="fig:oneloop"}](fig8c.pdf "fig:"){width="8.5cm"} ![The potential profile as a function of $\phi_c$ (the classical minimum for the doublet-dominated field) and $\phi_s$ (the classical minimum for the singlet-dominated field) for the two models described in the text. The upper panel is for model 1 and the lower panel is for model 2. The profiles are drawn along the line joining the two minima. Note that the second minimum can change significantly and the shift depends on the choice of the regularization scale $Q$. []{data-label="fig:oneloop"}](fig8d.pdf "fig:"){width="8.5cm"} The effect of one-loop corrections on the tree-level potential is not very drastic. This is expected because both VEVs are nonzero and neither $\Phi$ nor $S$ direction is a flat one. The one-loop corrections, being perturbative in nature, are suppressed by the standard loop factor of $1/64\pi^2$ and are expected to be significant only if we look at a flat direction. We show the one-loop corrections for two models, the parameters at the electroweak scale are given in . Model 1 Model 2 ------------------------------- ------------------- ------------------- $a_1$ (GeV) $-20.8$ $90.1$ $a_2$ $0.40$ $0.22$ $b_3$ (GeV) $-58.6$ $79.3$ $b_4$ $0.26$ $0.33$ $\lambda$ $0.46$ $0.60$ $\mu^2$ (GeV$^2$) $4.58\times 10^5$ $1.16\times 10^5$ $M^2$ (GeV$^2$) $4.85\times 10^5$ $1.72\times 10^5$ Global min.: $(v, v_s)$ (GeV) $(247, 1035)$ $(245, -598)$ Local min.: $(v, v_s)$ (GeV) $(673,-786)$ $(362, 406)$ : Parameter values for the models for which the 1-loop corrections are shown in the .[]{data-label="table"} We choose the regularization scale $Q$ in such a way that even after the one-loop corrections, the EW vacuum stays at $v \approx 246$ GeV (so that all SM particles have their masses unaffected). This choice of regularization scale was motivated in Ref. [@indrani-4]. We show the effect of one-loop corrections in for these two models. While they both have the global minimum at $v \approx246$ GeV, the required regularization scales differ by more than one order of magnitude. The nature of change is similar for all models, and thus we do not expect an unstable vacuum model to become metastable (or vice versa) because of the one-loop corrections. However, one may note how much the global minimum has been lowered by the one-loop corrections for the second model. In fact, for all the models scanned, we have never found a switch from global to local minimum induced by the radiative corrections. Summary ======= The potential of SM+S, a real singlet enhanced SM, shows several interesting features. In this paper, we have investigated the parameter space for several types of SM+S: the Higgs portal dark matter models, the potential with an explicit $Z_2$-symmetry and having singlet-doublet mixing, the $Z_2$-asymmetric potential with a stable EW minimum, or the same with an unstable or metastable EW minimum. The general features can be summarized as follows. - Adding one more real singlet makes the potential less stable in general at a high energy. This happens because the renormalization group equations for the couplings tend to hit the Landau pole much below the Planck scale, more so if the starting values at the EW scale is large. All the quartic couplings, namely, $\lambda$, $b_4$, and $a_2$, have to be small at the EW scale to keep the model valid up to a high scale, as the $\beta$-functions are coupled. (However, such new scalar couplings are helpful to avoid the vacuum stability bound, coming from the negative pull caused by the large top Yukawa coupling. Again, such conclusions are not valid if there are more degrees of freedom, like vectorlike fermions.) Our numerical results are shown at one-loop but inclusion of higher-order corrections do not change the result qualitatively. In particular, if we start with a large value of either $\lambda$ or $b_4$, the model hits the Landau pole at a relatively low scale. This can be taken as a possible indication of some new physics taking over and one should consider the effect of higher-dimensional operators on the low-scale physics. - While there are some minor variations for the allowed range of parameters among different class of models ([*e.g.*]{} $a_2$), the overlap is significant, and so one has to determine all the couplings experimentally to know what class of SM+S it really is. This is, of course, an extremely challenging task, if not outright impossible; the determination of $b_4$ is apparently beyond the reach of any present or upcoming colliders unless there is a significant mixing between the singlet and the doublet and the Higgs self-couplings are determined with sufficient accuracy. For the prospect of determination of the singlet-doublet mixing angle in future colliders, we refer the reader to Ref. [@1407.5342]. - The tree-level results are robust enough as far as the metastability issue is concerned. This is expected as we are not looking along any flat direction. However, the one-loop corrections can change the position of the minima. **[Acknowledgements]{}** We thank Filippo Sala for pointing out a mistake in the first version of the paper. S.G. acknowledges the University Grants Commission, Government of India, for a research fellowship. A.K. acknowledges Department of Science and Technology, Government of India, and Council for Scientific and Industrial Research, Government of India, for extramural projects. S.R. likes to thank the Department of Science and Technology (DST), Government of India for the INSA-INSPIRE Faculty Fellowship. S.R. would also like to thank Prof. Yuval Grossman, LEPP, Cornell University for hosting the visit as a part of INSA-INSPIRE Faculty Fellowship, where a part of this work has been done. [99]{} V. Silveira and A. Zee, Phys. Lett. B [**161**]{}, 136 (1985);\ C. P. Burgess, M. Pospelov and T. ter Veldhuis, Nucl. Phys. B [**619**]{}, 709 (2001) \[arXiv:hep-ph/0011335\];\ J. McDonald, Phys. Rev. D [**50**]{}, 3637 (1994) \[arXiv:hep-ph/0702143\];\ J. M. Cline, K. Kainulainen, P. Scott and C. Weniger, Phys. Rev. D [**88**]{}, 055025 (2013) \[Phys. Rev. D [**92**]{}, 039906 (2015)\] \[arXiv:1306.4710 \[hep-ph\]\]. J. R. Espinosa, T. Konstandin and F. Riva, Nucl. Phys. B [**854**]{}, 592 (2012) \[arXiv:1107.5441 \[hep-ph\]\]. S. J. Huber and M. G. Schmidt, Nucl. Phys. B [**606**]{}, 183 (2001) \[hep-ph/0003122\];\ S. W. Ham, Y. S. Jeong and S. K. Oh, J. Phys. G [**31**]{}, 857 (2005) \[hep-ph/0411352\]. A. Kundu and S. Raychaudhuri, Phys. Rev. D [**53**]{}, 4042 (1996) \[arXiv:hep-ph/9410291\];\ A. Drozd, B. Grzadkowski and J. Wudka, JHEP [**1204**]{}, 006 (2012) \[arXiv:1112.2582 \[hep-ph\]\];\ A. Drozd, arXiv:1202.0195 \[hep-ph\];\ I. Chakraborty and A. Kundu, Phys. Rev. D [**87**]{}, 055015 (2013) \[arXiv:1212.0394 \[hep-ph\]\];\ F. Bazzocchi and M. Fabbrichesi, Phys. Rev. D [**87**]{}, 036001 (2013) \[arXiv:1212.5065 \[hep-ph\]\]. V. Barger, P. Langacker, M. McCaskey, M. J. Ramsey-Musolf and G. Shaughnessy, Phys. Rev. D [**77**]{}, 035005 (2008) \[arXiv:0706.4311 \[hep-ph\]\]. V. Barger, P. Langacker, M. McCaskey, M. Ramsey-Musolf and G. Shaughnessy, Phys. Rev. D [**79**]{}, 015018 (2009) \[arXiv:0811.0393 \[hep-ph\]\]. G. M. Pruna and T. Robens, Phys. Rev. D [**88**]{}, 115012 (2013) \[arXiv:1303.1150 \[hep-ph\]\];\ T. Robens and T. Stefaniak, Eur. Phys. J. C [**75**]{}, 104 (2015) \[arXiv:1501.02234 \[hep-ph\]\]. C. Y. Chen, S. Dawson and I. M. Lewis, Phys. Rev. D [**91**]{}, 035015 (2015) \[arXiv:1410.5488 \[hep-ph\]\]. A. Datta, A. Raychaudhuri, S. Raychaudhuri and S. Chakrabarti, Z. Phys. C [**72**]{}, 449 (1996) \[hep-ph/9510432\];\ A. Datta and A. Raychaudhuri, Phys. Rev. D [**57**]{}, 2940 (1998) \[hep-ph/9708444\];\ R. Schabinger and J. D. Wells, Phys. Rev. D [**72**]{}, 093007 (2005) \[hep-ph/0509209\];\ D. O’Connell, M. J. Ramsey-Musolf and M. B. Wise, Phys. Rev. D [**75**]{}, 037701 (2007) \[hep-ph/0611014\];\ V. Barger, P. Langacker and G. Shaughnessy, Phys. Rev. D [**75**]{}, 055013 (2007) \[hep-ph/0611239\];\ O. Bahat-Treidel, Y. Grossman and Y. Rozen, JHEP [**0705**]{}, 022 (2007) \[hep-ph/0611162\].\ D. Buttazzo, F. Sala and A. Tesi, JHEP [**1511**]{}, 158 (2015) \[arXiv:1505.05488 \[hep-ph\]\].\ F. Bojarski, G. Chalons, D. Lopez-Val and T. Robens, arXiv:1511.08120 \[hep-ph\]. S. I. Godunov, A. N. Rozanov, M. I. Vysotsky and E. V. Zhemchugov, arXiv:1503.01618 \[hep-ph\]. ATLAS Collaboration, ATLAS-CONF-2015-081. CMS Collaboration, CMS-PAS-EXO-15-004. S. Khan, S. Goswami and S. Roy, Phys. Rev. D [**89**]{}, 073021 (2014) \[arXiv:1212.3694 \[hep-ph\]\];\ N. Khan and S. Rakshit, Phys. Rev. D [**90**]{}, 113008 (2014) \[arXiv:1407.6015 \[hep-ph\]\];\ G. Chalons and F. Domingo, Phys. Rev. D [**86**]{}, 115024 (2012) \[arXiv:1209.6235 \[hep-ph\]\];\ N. Haba, H. Ishida, K. Kaneta and R. Takahashi, Phys. Rev. D [**90**]{}, 036006 (2014) \[arXiv:1406.0158 \[hep-ph\]\]. D. López-Val and T. Robens, Phys. Rev. D [**90**]{}, 114018 (2014) \[arXiv:1406.1043 \[hep-ph\]\]. E. Gildener and S. Weinberg, Phys. Rev. D [**13**]{}, 3333 (1976). I. Chakraborty and A. Kundu, Phys. Rev. D [**92**]{}, no. 9, 095023 (2015) \[arXiv:1508.00702 \[hep-ph\]\]. Talk by M. Pieri at LHCP 2015, St. Petersburg, Russia, Aug.-Sep. 2015, available at https://indico.cern.ch/event/389531/session/31/contribution/51/attachments/ 1147368/1650410/LHCHCP_MarcoPieri_fin_1.pdf See the review on Electroweak Model in K. A. Olive [*et al.*]{} \[Particle Data Group Collaboration\], Chin. Phys. C [**38**]{}, 090001 (2014). I. Chakraborty and A. Kundu, Phys. Rev. D [**87**]{}, no. 5, 055015 (2013) \[arXiv:1212.0394 \[hep-ph\]\]. S. R. Coleman, Phys. Rev. D [**15**]{}, 2929 (1977) \[Phys. Rev. D [**16**]{}, 1248 (1977)\];\ V. A. Rubakov, Classical theory of gauge fields, Princeton, USA: Univ. Pr. (2002). A. Barroso, P. M. Ferreira, I. P. Ivanov and R. Santos, JHEP [**1306**]{}, 045 (2013) \[arXiv:1303.5098 \[hep-ph\]\]. L. Feng, S. Profumo and L. Ubaldi, JHEP [**1503**]{}, 045 (2015) \[arXiv:1412.1105 \[hep-ph\]\]. M. Duerr, P. Fileviez Perez and J. Smirnov, arXiv:1508.04418 \[hep-ph\]. M. Ackermann [*et al.*]{} \[Fermi-LAT Collaboration\], Phys. Rev. D [**88**]{}, 082002 (2013) \[arXiv:1305.5597 \[astro-ph.HE\]\]. D. S. Akerib [*et al.*]{} \[LUX Collaboration\], Phys. Rev. Lett.  [**112**]{}, 091303 (2014) \[arXiv:1310.8214 \[astro-ph.CO\]\]. H. Han and S. Zheng, JHEP [**1512**]{}, 044 (2015) \[arXiv:1509.01765 \[hep-ph\]\]. S. Profumo, M. J. Ramsey-Musolf, C. L. Wainwright and P. Winslow, Phys. Rev. D [**91**]{}, no. 3, 035018 (2015) \[arXiv:1407.5342 \[hep-ph\]\]. [^1]: swgtghsh54@gmail.com [^2]: anirban.kundu.cu@gmail.com [^3]: shamayita.ray@gmail.com, sr643@cornell.edu [^4]: One can always remove the tadpole term if the potential is $Z_2$-symmetric. However, for a $Z_2$-asymmetric potential, there can be two minima with different values of $v$, and removal of the tadpole at one minimum does not ensure its removal at the other. [^5]: For any given value of $v_s$, there is one extremum at $v=0$. If has three real solutions, and has three real solutions for each of the solutions for $v_s$, there can be nine such extrema. Out of these nine extrema, three have $v=0$ and thus cannot be the EW vacuum. The two solutions for nonzero $v$ for a given solution of $v_s$ are symmetrically placed about $v=0$. [^6]: This gives a pretty good estimate, although two-loop results are available. [^7]: Such a model suffers from the usual domain wall problem. However, existence of multiple vacuum states in the universe is not yet ruled out, and in fact is a distinct possibility in several string theory motivated scenarios.

--- author: - Ali Sharif Razavian - Hossein Azizpour - | \ Atsuto Maki - Josephine Sullivan - Carl Henrik Ek - Stefan Carlsson bibliography: - 'egbib.bib' title: Persistent Evidence of Local Image Properties in Generic ConvNets --- Acknowledgment {#acknowledgment .unnumbered} ============== We would like to gratefully acknowledge the support of NVIDIA for the donation of multiple GPU cards for this research.

--- abstract: 'We investigate the microscopic origin of the ferromagnetic and antiferromagnetic spin exchange couplings in the quasi one-dimensional cobalt compound Ca$_3$Co$_2$O$_6$. In particular, we establish a local model which stabilizes a ferromagnetic alignment of the $S=2$ spins on the cobalt sites with trigonal prismatic symmetry, for a sufficiently strong Hund’s rule coupling on the cobalt ions. The exchange is mediated through a $S=0$ cobalt ion at the octahedral sites of the chain structure. We present a strong coupling evaluation of the Heisenberg coupling between the $S=2$ Co spins on a separate chain. The chains are coupled antiferromagnetically through super-superexchange via short O-O bonds.' author: - Raymond Frésard - Christian Laschinger - Thilo Kopp - Volker Eyert title: 'The Origin of Magnetic Interactions in Ca$_3$Co$_2$O$_6$' --- Recently there has been renewed interest in systems exhibiting magnetization steps. In classical systems such as CsCoBr$_3$ one single plateau is typically observed in the magnetization versus field curve at one third of the magnetization at saturation.[@Hida94] This phenomenon attracted considerable attention, and Oshikawa, Yamanaka and Affleck demonstrated that Heisenberg antiferromagnetic chains exhibit such magnetization plateaus when embedded in a magnetic field.[@Oshikawa97] These steps are expected when $N_c (S-m)$ is an integer, where $N_c$ is the number of sites in the magnetic unit cell, $S$ the spin quantum number, and $m$ the average magnetization per spin, which we shall refer to as the OYA criterion. The steps can be stable when chains are coupled, for instance in a ladder geometry. In that case the magnetic frustration is an important ingredient to their stability.[@Mila98] Plateaus according to the OYA criterion are also anticipated for general configurations, provided gapless excitations do not destabilize them.[@Oshikawa00] Indeed several systems exhibiting magnetization steps are now known;[@Shiramura98; @Narumi98] they all obey the OYA criterion, they are usually far from exhausting all the possible $m$ values, they all are frustrated systems, and they all can be described by an antiferromagnetic Heisenberg model. Related behavior has been recently found in other systems. For example, up to five plateaus in the magnetization vs. field curve have been observed in Ca$_3$Co$_2$O$_6$ at low temperature[@Aasland97; @Kageyama97; @Maignan00]. However there is to date no microscopic explanation to this phenomenon, even though the location of the plateaus is in agreement with the OYA criterion. Ca$_3$Co$_2$O$_6$ belongs to the wide family of compounds A’$_3$ABO$_6$, and its structure belongs to the space group R3c. It consists of infinite chains formed by alternating face sharing AO$_6$ trigonal prisms and BO$_6$ octahedra — where Co atoms occupy both A and B sites. Each chain is surrounded by six chains separated by Ca atoms. As a result a Co ion has two neighboring Co ions on the same chain, at a distance of $2.59$ Å, and twelve Co neighbors on the neighboring chains at distances $7.53$ Å  (cf. Fig. \[Fig:plane\]).[@Fjellvag96] Concerning the magnetic structure, the experiment points toward a ferromagnetic ordering of the magnetic Co ions along the chains, together with antiferromagnetic correlations in the buckling a-b plane.[@Aasland97] The transition into the ordered state is reflected by a cusp-like singularity in the specific heat at 25 K,[@Hardy03] — at the temperature where a strong increase of the magnetic susceptibility is observed. Here we note that it is particularly intriguing to find magnetization steps in a system where the dominant interaction is ferromagnetic. In order to determine the effective magnetic Hamiltonian of a particular compound one typically uses the Kanamori-Goodenough-Anderson (KGA) rules[@Goodenough]. Knowledge of the ionic configuration of each ion allows to estimate the various magnetic couplings. When applying this program to Ca$_3$Co$_2$O$_6$ one faces a series of difficulties specifically when one tries to reconcile the neutron scattering measurements that each second Co ion is non-magnetic. Even the assumption that every other Co ion is in a high spin state does not settle the intricacies related to the magnetic properties; one still has to challenge issues such as: i) what are the ionization degrees of the Co ions? ii) how is an electron transfered from one cobalt ion to a second? iii) which of the magnetic Co ions are magnetically coupled? iv) which mechanism generates a ferromagnetic coupling along the chains? These questions are only partially resolved by ab initio calculations. In particular, one obtains that both Co ions are in 3+ configurations.[@Whangbo03] Moreover both Co-O and direct Co-Co hybridizations are unusually large, and low spin and high spin configurations for the Co ions along the chains alternate.[@Eyert03] Our publication addresses the magnetic couplings, and in particular the microscopic origin of the ferromagnetic coupling of two Co ions through a non-magnetic Co ion. In view of the plethoric variety of iso-structural compounds,[@Stitzer01] the presented mechanism is expected to apply to many of these systems. We now derive the magnetic inter-Co coupling for Ca$_3$Co$_2$O$_6$ from microscopic considerations. The high-spin low-spin scenario confronts us with the question of how a ferromagnetic coupling can establish itself, taking into account that the high spin Co ions are separated by over 5 Å, linked via a non-magnetic Co and several oxygens. Let us first focus on the Co-atoms in a single Co-O chain of Ca$_3$Co$_2$O$_6$. As mentioned above the surrounding oxygens form two different environments in an alternating pattern. We denote the Co ion in the center of the oxygen-octahedron Co1, and the Co ion in the trigonal prisms Co2. The variation in the oxygen-environment leads to three important effects. First, there is a difference in the strength of the crystal field splitting, being larger in the octahedral environment. As a result Co1 is in the low spin state and Co2 in the high spin state. Second, the local energy levels are in a different sequence. For the octahedral environment we find the familiar $t_{2g}$–$e_g$ splitting, provided the axes of the local reference frame point towards the surrounding oxygens. The trigonal prismatic environment accounts for a different set of energy levels. For this local symmetry one expects a level scheme with $d_{3z^2-r^2}$ as lowest level, followed by two twofold degenerate pairs $d_{xy}$, $d_{x^2-y^2}$ and $d_{xz}$, $d_{yz}$. However, our LDA calculations[@Eyert03] show that the $d_{3z^2-r^2}$ level is actually slightly above the first pair of levels. Having clarified the sequence of the energy levels, we now turn to the microscopic processes which link the Co ions. Two mechanisms may be competing: either the coupling involves the intermediate oxygens, or direct Co-Co overlap is more important. Relying on electronic structure calculations, we may safely assume that the direct Co-Co overlap dominates.[@Eyert03] The identification of the contributing orbitals is more involved. Following Slater and Koster[@Slater54] one finds that only the $3z^2$-$r^2$ orbitals along the chains have significant overlap. However, we still have to relate the Koster-Slater coefficients and the coefficients for the rotated frame since the natural reference frames for Co1 and Co2 differ. On the Co2 atoms with the triangular prismatic environment the $z$-axis is clearly defined along the chain direction, and we choose the $x$ direction to point toward one oxygen. This defines a reference frame $S$. The $x$ and $y$ directions are arbitrary and irrelevant to our considerations. The octahedral environment surrounding the Co1 atoms defines the natural coordinate system, which we call $S'$. By rotating $S'$ onto $S$ one obtains the $3z^2$-$r^2$ orbital in the reference frame $S$ as an equally weighted sum of $x'y'$, $x'z'$, $y'z'$ orbitals in $S'$. The above observation that the only significant overlap is due to the $3z^2$-$r^2$ orbitals on both Co ions now translates into an overlap of the $3z^2$-$r^2$ orbital on high spin cobalt with all $t_{2g}$ orbitals on low spin cobalt. ![Typical hopping paths for a) ferromagnetic and b) antiferromagnetic ordering. The displayed ferromagnetic path is the only one for ferromagnetic ordering and has the highest multiplicity of all, ferromagnetic and antiferromagnetic. There are similar paths for antiferromagnetic ordering but with a Hund’s rule penalty and lower multiplicity. The path in (b) is unique for the antiferromagnetic case and has low energy but also low multiplicity. \[Fig:levelscheme\]](fig_1.eps){width=".47\textwidth"} We proceed with a strong coupling expansion to identify the magnetic coupling along the chain. This amounts to determine the difference in energy, between the ferromagnetic and antiferromagnetic configurations, to fourth order in the hopping, since this is the leading order to the magnetic interaction between the high spin Co ions. As explained above we only have to take into account the $3z^2$-$r^2$ level on Co2 and the $t_{2g}$ levels on Co1. In an ionic picture all $t_{2g}$ levels on Co1 are filled while the $d_{3z^2-r^2}$ level on Co2 is half-filled and we therefore consider hopping processes from the former to the latter. In the ferromagnetic configuration we include processes where two down spin electrons hop from Co1 to both neighboring Co2 and back again as displayed in Fig. \[Fig:levelscheme\]a. There are in total $3\times 2\times 2\times 2\times 2 = 48$ such processes. The intermediate spin state for Co1 is in agreement with Hund’s rule. The energy gain per path is given by: $${ E_f=\frac{t^4}{E_0^2} \; \left(3U-5J_{\rm{H}}+4 E_{\rm loc}(\Delta_{\rm{cf}},J_{\rm{H}},-1)\right)^{-1}}$$ with $${ E_0 = U-J_{\rm{H}}+2E_{\rm loc}(\Delta_{\rm{cf}},J_{\rm{H}},-2)}$$ $$\Delta_{\rm{cf}} = \Delta_{\rm{Co1}} + \frac{4}{10} \Delta_{\rm{Co2}}$$ and $${ E_{\rm loc}(\Delta_{\rm{cf}},J_{\rm{H}},l)}= {\frac{\Delta_{\rm{cf}}J_{\rm{H}}^2}{(\Delta_{\rm{cf}}- \frac{1}{2}lJ_{\rm{H}})(2\Delta_{\rm{cf}}+3J_{\rm{H}})}}$$ where $\Delta_{\rm{Co1}}$ and $\Delta_{\rm{Co2}}$ denote the crystal field splittings on Co1 and Co2, respectively. The Hund’s coupling is $J_{\rm{H}}$, assumed to be identical on both, Co1 and Co2, and $U$ denotes the local Coulomb repulsion. There are no further paths in this configuration, besides the one which twice iterates second order processes. In the antiferromagnetic case the situation is slightly more involved. Here three different classes of paths have to be distinguished. The first class, denoted $a1$ in the following, consists of hopping events of one up spin and one down spin electron from the same Co1 level. (There are $3\times 2\times 2 = 12$ such paths). The second class ($a2$) consists of hopping events of one down spin and one up spin electron from different Co1 levels (There are $3\times 2\times 2 \times 2 = 24$ such paths). The third class ($a3$), shown in Fig. \[Fig:levelscheme\]b, consists of hopping processes where one electron is hopping from Co1 to Co2 and then another electron is hopping from the other Co2 to the same Co1 and back again (There are $3\times 2= 6 $ such paths). In total this sums up to 42 paths in the antiferromagnetic configuration. Consequently, we have more ferromagnetic than antiferromagnetic exchange paths. However the energy gain depends on the path. For the classes $a1$ and $a2$, the intermediate Co1 state violates the Hund’s rule, and we identify an energy gain per path given by: $$\begin{aligned} \nonumber E_{a1}&=&\frac{t^4}{E_0^2}\left(3U-5J_{\rm{H}}+(4-\frac{6J_{\rm{H}}}{\Delta_{\rm{cf}}}) E_{\rm loc}(\Delta_{\rm{cf}},J_{\rm{H}},1)\right)^{-1}\\ E_{a2}&=&\frac{t^4}{E_0^2}\left(3U-2J_{\rm{H}}-\rm{F}(\Delta_{\rm{cf}},J_{\rm{H}})\right)^{-1}\end{aligned}$$ Here F is a positive function which is smaller than $J_{\rm{H}}$. The expression $3U-2J_{\rm{H}}-F$ is the lowest eigenvalue of $\langle i|H_{\rm Co}|j \rangle$ where the states $i$ and $j$ are all possible states on Co1 consistent with two of the $d$-orbitals filled and three empty. For the class $a3$ one observes that one does not need to invoke a Co1 ion with four electrons as an intermediate state, in contrast to all other processes we considered so far. We find the energy gain as: $$E_{a3}=\frac{t^4}{E_0^2\left(U+2J_{\rm H}\right)}\\$$ Altogether we obtain the difference in energy gain between the ferromagnetic and the antiferromagnetic configurations as: $$E^{\rm F} - E^{\rm AF}=48 E_f-24E_{a1}-12E_{a2}-6E_{a3}$$ The dependence of $E^{\rm F} - E^{\rm AF}$ on $J_{\rm H}$ for different values of the local interaction $U$ is shown in Fig. \[Fig:stability\]. Using $J_{\rm H}=0.6~\rm{eV}$,[@Laschinger03] $U=5.3~\rm{eV}$,[@Sawatzky91] $t=1.5~\rm{eV}$,[@Eyert03] $\Delta_{\rm{Co1}}=2.5~\rm{eV}$ and $\Delta_{\rm{Co2}}=1.5~\rm{eV}$,[@Eyert03] we obtain an estimate for the Heisenberg exchange coupling (for the Co2 spin $S=2$): $$J^{\rm F}=(E^{\rm F} - E^{\rm AF})/2S^2\approx 2 \rm{meV}$$ which is in reasonable agreement with the experimental transition temperature of 25 K. In this context one should realize that a one-dimensional chain does not support a true phase transition into the magnetic state. However, as the length $L$ of the chains is finite, a crossover into the ferromagnetic state may be observed when the correlation length is approximately $L$.[@chain] ![Energy gain $E^{\rm F} - E^{\rm AF}$ of a nearest neighbor Co2 ferromagnetic alignment with respect to an antiferromagnetic orientation as a function of the Hund’s coupling $J_{\rm H}$, for a typical parameter set of Hubbard $U$. The grey shaded area indicates the interval of $J_{\rm H}$ for which a Co2-high-spin Co1-low-spin configuration can be stabilized for the considered crystal field splittings.[@Laschinger03] []{data-label="Fig:stability"}](fig_2.eps){width="48.00000%"} To emphasize the importance of the chain geometry we now briefly discuss the hypothetical case where the $z$-axis of the octahedra corresponds to that of the prism. In this geometry there is only one orbital on each Co ion which contributes to the hopping processes. In this situation the process favoring ferromagnetism shown in Fig. \[Fig:levelscheme\]a does not exist, in contrast to the process $a3$ shown in Fig. \[Fig:levelscheme\]b, and the resulting coupling is therefore antiferromagnetic. In the large class of known isostructural compounds[@Stitzer01] the non-magnetic ion is not necessarily a Co ion. If the non-magnetic ion is in a $3d^2$ (or $4d^2$) configuration, the above argument applies, and the coupling is antiferromagnetic. If the configuration is $3d^4$, all the discussed electronic processes contribute, however with different multiplicities. Moreover, additional paths have to be considered for the antiferromagnetic case. They represent exchange processes through an empty orbital on the non-magnetic ion. As a result, the ferromagnetic scenario has fewer paths than the antiferromagnetic, and the coupling becomes antiferromagnetic. Correspondingly, a ferromagnetic interaction can only occur when all three orbitals on the nonmagnetic ion participate in the exchange process. Obviously the situation we consider differs from the standard 180 degree superexchange mechanism in many respects. With the investigation of the [*interchain*]{} magnetic interaction one first notices that each magnetic Co ion has twelve neighboring Co ions on different chains. However, as displayed in Fig. \[Fig:plane\], there is an oxygen bridge to only six neighbors, one per chain. Here the coupling $J^{\rm AF}$ results from the super-superexchange mechanism (with exchange via two oxygen sites), and it is antiferromagnetic. Since the Co-O hybridization is unusally large in this system, we expect the interchain magnetic coupling to be sufficiently strong to account for the observed antiferromagnetic correlations. From our previous considerations we introduce the minimal magnetic Hamiltonian: $$\begin{aligned} \label{Eq:Hamiltonian} H &=& \sum_{i,j} \left(J^{\rm F}_{i,j} {\vec S}_i \cdot {\vec S}_j + J^{\rm AF}_{i,j} {\vec S}_i \cdot {\vec S}_j \right) -D \sum_{i} S_{z,i}^2 \\ {\rm with}&& J^{\rm F}_{i,j} = \left\{ \begin{array}{ll} J^{\rm F}& \mbox{if ${\vec j} - \vec{i} = \pm 2 {\vec d}$}\\ 0& \mbox{otherwise} \end{array} \right. \nonumber \\ {\rm and} \nonumber\\ J^{\rm AF}_{i,j}\! &=& \!\! \left\{ \begin{array}{ll} J^{\rm AF}& \mbox{if ${\vec j} - \vec{i} = \pm ( {\vec a}+{\vec d}), \pm ( {\vec b}+{\vec d}), \pm ( {\vec c}+{\vec d}) $}\nonumber \\ 0& \mbox{otherwise.} \end{array} \right. \\ \nonumber\end{aligned}$$ Here we use the site vectors ${\vec a} = a (-1/2,\sqrt{3}/2,c/(12 a))$, ${\vec b} = a (-1/2,-\sqrt{3}/2,c/(12 a))$, ${\vec c} = a (1,0,c/(12 a))$, and ${\vec d} = {\vec a}+{\vec b}+{\vec c}$ where $a=9.06$ Å and $c=10.37$ Å are the lattice constants of the hexagonal unit cell. The Hamiltonian, Eq. (\[Eq:Hamiltonian\]), also includes a phenomenological contribution $D S_{z}^2$ which accounts for the anisotropy observed, for example, in the magnetic susceptibility.[@Kageyama97a; @Maignan00] ![ Projection of Ca$_3$Co$_2$O$_6$ in the (100) plane, with those oxygen atoms which are closest to the plane. Shadows indicate the shortest O-O bonds, along which super-superexchange processes take place. Yellow (red) large spheres denote Co1 (Co2) atoms, dark (light) small blue spheres oxygen atoms located above (below) the Co plane. \[Fig:plane\]](fig_3.eps){width="49.00000%"} The stability of the magnetization steps results from the magnetic frustration which is introduced through the antiferromagnetic interchain coupling. Indeed, the lattice structure suggests that this magnetic system is highly frustrated, since the chains are arranged on a triangular lattice. However investigating the Hamiltonian (\[Eq:Hamiltonian\]) reveals that the microscopic mechanism leading to frustration is more complex. It is visualized when we consider a closed path $TLRT'T$ — where the sites $T$ and $T'$ are next nearest-neighbor Co2 sites on the same chain and the sequence of sites $TLR$ is located on a triangle of nearest neighbor chains. One advances from $T$ to $L$ to $R$ to $T'$ on a helical path[@comment] formed by the oxygen bridges from Fig. \[Fig:plane\]. Since the structure imposes $T$ and $T'$ to be next nearest-neighbors, the frustration occurs independently of the sign of the intrachain coupling. In summary we established the magnetic interactions in an effective magnetic Hamiltonian for Ca$_3$Co$_2$O$_6$. It is a spin-2 Hamiltonian, with antiferromagnetic interchain coupling, and ferromagnetic intrachain interactions. The latter is obtained from the evaluation of all spin exchange paths between two high-spin Co2 sites through an intermediary low-spin Co1 site. This mechanism is particular to the geometry of the system as is the microscopic mechanism which leads to magnetic frustration. We expect that the discussed microscopic mechanisms also apply to other isostructural compounds, such as Ca$_3$CoRhO$_6$ and Ca$_3$CoIrO$_6$. We are grateful to A. Maignan, C. Martin, Ch. Simon, C. Michel, A. Guesdon, S. Boudin and V. Hardy for useful discussions. C. Laschinger is supported by a Marie Curie fellowship of the European Community program under number HPMT2000-141. The project is supported by DFG through SFB 484 and by BMBF (13N6918A). F. Hida, J. Phys. Soc. Jpn. [**63**]{}, 2359 (1994). M. Oshikawa [*et al.*]{}, Phys. Rev. Lett. [**78**]{}, 1984 (1997). F. Mila, Eur. Phys. J. B [**6**]{}, 201 (1998). M. Oshikawa, Phys. Rev. Lett. [**84**]{}, 1535 (2000). Y. Narumi [*et al.*]{}, Physica B [**246-247**]{}, 509 (1998). W. Shiramura [*et al.*]{}, J. Phys. Soc. Jpn. [**67**]{}, 1548 (1998). S. Aasland [*et al.*]{}, Solid State Commun. [**101**]{}, 187 (1997). H. Kageyama [*et al.*]{}, J. Phys. Soc. Jpn. [**66**]{}, 1607 (1997). A. Maignan [*et al.*]{}, Eur. Phys. J. B [**15**]{}, 657 (2000). H. Fjellv[å]{}g [*et al.*]{}, J. Sol. State Chem. [**124**]{}, 190 (1996). V. Hardy [*et al.*]{}, Phys. Rev. B [**68**]{}, 014424 (2003). J. B. Goodenough, [*Magnetism and the Chemical Bond*]{} (Interscience Publishers, John Wiley & and sons, New York, 1963) M. H. Whangbo [*et al.*]{}, Solid State Commun. [**125**]{}, 413 (2003). V. Eyert [*et al.*]{}, Preprint, unpublished (2003). K. E. Stitzer [*et al.*]{}, Opin. Solid State Mater. Sci. [**5**]{}, 535 (2001). J. C. Slater and G. F. Koster, Phys. Rev. [**94**]{}, 1498 (1954). C. Laschinger [*et al.*]{}, J. Magn. and Magnetic Materials, (in press, 2004). J. van Elp [*et al.*]{}, Phys. Rev. B [**44**]{}, 6090 (1991). Given the strong magnetic anisotropy of Ca$_3$Co$_2$O$_6$,[@Maignan00] which is phenomenologically included in Eq. (\[Eq:Hamiltonian\]), we expect that one can capture some physics of the system in the Ising limit. In that case, for a typical chain length of the order of one hundred sites, the correlation length extends to the system size for $T\simeq 0.8 J$ which is to be interpreted as the crossover temperature to the ferromagnetic state. Correspondingly, the spin susceptibility peaks at about $T/J\simeq0.8-1.2$ which we verified in exact diagonalization. Even with chain lengths of 10,000 Co sites, the crossover temperature is still in the given range. H. Kageyama [*et al.*]{}, J. Phys. Soc. Jpn. [**66**]{}, 3996 (1997). The positions of the considered sites are: $T$ is an arbitrary Co2 site on any chain, $L=T+{\vec b}+{\vec d}$, $R=L+{\vec c}+{\vec d}$ and $T'=R+{\vec a}+{\vec d}$.

--- abstract: 'We extend the definition of the Chen-Ruan cohomology ring to smooth, proper, tame, [Deligne-Mumford]{} stacks over fields of positive characteristic and prove that a modified version of the Frobenius action preserves the product.' address: 'Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T1Z2, Canada' author: - 'Michael A. Rose' bibliography: - 'xbib.bib' date: 'September 13, 2007' title: 'Frobenius Action on $\ell$-adic Chen-Ruan Cohomology' --- Introduction ============ The Weil conjectures describe a strong relationship between the arithmetic and topological properties of a smooth projective scheme $X$ over Spec(${\mathbb{Z}}$). In [@Weil-1956], Weil himself observed that the conjectures would follow from an appropriate cohomology theory for abstract schemes (analogous to singular cohomology for complex varieties), and in the 1960’s a great amount of work was done by Artin, Deligne, Grothendieck and others to develop *$\ell$-adic cohomology* for this purpose. In particular, by considering a smooth reduction $X_{{\mathbb{F}}_q}$ and applying the Lefschetz Trace Theorem to the geometric Frobenius morphism $F$ on ${\overline}{X} {\mathrel{\mathop:}=}X \times_{{\mathbb{F}}_q} {\overline}{{\mathbb{F}}}_q$ one obtains the fundamental equation $$\label{intro_eqn_1} \text{det}(1 - F^*t\ |\ H^*({\overline}{X}_{\acute{e}t}, {\mathbb{Q}}_l))\ =\ \text{exp}\ \sum_{r = 1}^{\infty}\ |X({\mathbb{F}}_{q^r})| \frac{t^r}{r}$$ where *det* denotes a graded determinant, and $l$ is coprime to $q$. Now we replace $X$ with a smooth [Deligne-Mumford]{} stack ${\mathcal{X}}$ over Spec(${\mathbb{Z}}$) (further hypothesis to be considered below), and one is naturally led to consider the Weil Conjectures on ${\mathcal{X}}$. Again, one focuses on finding an appropriate cohomology theory. On one hand, there is already a natural notion of $\ell$-adic cohomology in this setting, and several of the properties crucial to proving an analog of the Weil conjectures for stacks have been established: Poincaré Duality appeared in [@LO1; @LO2] while a Lefschetz Trace Theorem was established in [@Be-Lef]. On the other hand, while motivated by string theory, Chen and Ruan [@CR2] constructed a ring now bearing their names: the *Chen-Ruan cohomology ring*, $H^*_{CR}({\mathcal{X}}_{{\mathbb{C}}})$. (The authors actually defined the ring for an almost complex orbifold; the theory was developed for [Deligne-Mumford]{} stacks in [@AGV1; @AGV2; @AV1]). The goal of this article is to study the relationship between the Chen-Ruan cohomology ring and the arithmetic properties of ${\mathcal{X}}$. If ${\mathcal{X}}$ is now a smooth, tame, [Deligne-Mumford]{} ${\mathbb{F}}_q$-stack with projective coarse moduli scheme and ${\overline}{{\mathcal{X}}} {\mathrel{\mathop:}=}{\mathcal{X}}\times_{{\mathbb{F}}_q} {\overline}{{\mathbb{F}}}_q$, then we define the *$\ell$-adic Chen-Ruan cohomology ring* of ${\overline}{{\mathcal{X}}}$ denoted $H^*_{CR}({\overline}{{\mathcal{X}}}_{\acute{e}t}, {\mathbb{Q}}_l)$. Most of the technical requirements for this construction already appeared in [@AGV2]. Furthermore we construct an action of the arithmetic Frobenius on $H^*_{CR}({\overline}{{\mathcal{X}}}_{\acute{e}t}, {\mathbb{Q}}_l)$. (We use the arithmetic Frobenius as opposed to the geometric Frobenius merely to simplify the proofs. In the context of Artin stacks however, this distinction is crucial for convergence issues. See [@Be-Lef]). This latter result deserves comment. In general, $H^*_{CR}$ *is not functorial*. Let $F:{\overline}{{\mathcal{X}}} {\rightarrow}{\overline}{{\mathcal{X}}}$ denote the arithmetic Frobenius on ${\overline}{{\mathcal{X}}}$. While $F$ naturally induces a linear map ${\mathcal{I}}_{{\mathbf{\mu}}}(F)^*: H^*_{CR}({\overline}{{\mathcal{X}}}_{\acute{e}t}, {\mathbb{Q}}_l) {\rightarrow}H^*_{CR}({\overline}{{\mathcal{X}}}_{\acute{e}t}, {\mathbb{Q}}_l)$ (see Section \[Arithmetic\_Frobenius\] for a detailed definition), ${\mathcal{I}}_{{\mathbf{\mu}}}(F)^*$ is not a ring homomorphism. However, the main proposition of this article shows that a slight modification of ${\mathcal{I}}_{{\mathbf{\mu}}}(F)^*$ indeed preserves the product structure: \[main\] The *orbifold Frobenius morphism* $F_{orb}$ given by $$\begin{aligned} {1} F_{orb}: \ H^*_{CR}({\overline}{{\mathcal{X}}}_{\acute{e}t}, {\mathbb{Q}}_l) &\longrightarrow H^*_{CR}({\overline}{{\mathcal{X}}}_{\acute{e}t}, {\mathbb{Q}}_l) \notag \\ \alpha &\longmapsto q^{-\text{age}(\alpha)} \cdot {\mathcal{I}}_{{\mathbf{\mu}}}(F_{{\mathcal{X}}})^*(\alpha) \notag\end{aligned}$$ is an isomorphism of graded rings. Here, $\text{age}: H^*_{CR}({\overline}{{\mathcal{X}}}_{\acute{e}t}, {\mathbb{Q}}_l) {\rightarrow}{\mathbb{Q}}$ is a function appearing also in the grading on $H^*_{CR}({\overline}{{\mathcal{X}}}_{\acute{e}t}, {\mathbb{Q}}_l)$ described in Section \[l-adic\_cohomology\]. We study the arithmetic information contained by this Galois representation in Section \[zeta\] below. In particular, we include the analogues of equation (\[intro\_eqn\_1\]) with $H^*$ (resp. $F^*$) replaced by $H^*_{CR}$ (resp. $F^*_{CR}$), and we list a consequence of Yasuda’s proof [@Yas] of an additive version of the Crepant Resolution Conjecture. *Conventions.* Unless specified otherwise, assume all sheaves on a [Deligne-Mumford]{} stack ${\mathcal{X}}$ are defined on the étale site of ${\mathcal{X}}$. We fix an isomorphism of the Tate twist ${\mathbb{Q}}_l(1) \cong {\mathbb{Q}}_l$ as sheaves on ${\overline}{{\mathcal{X}}}$, inducing an isomorphism ${\mathbb{Q}}_l(r) \cong {\mathbb{Q}}_l$ for each $r$. Then for smooth ${\mathcal{X}}$ of dimension $n$, the duality theorem in [@LO2 Theorem 7.7] yields $H^i({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l) \cong H^{2n-i}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l)^{\vee}$. This isomorphism is used implicitly throughout. We define homology groups $H_i({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l) {\mathrel{\mathop:}=}H^{2n-i}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l)$. To improve the exposition proofs of several lemmas appear only in the appendix. Also, no attempt is made to make the statements as general as possible, and we work over explicit base fields ${\mathbb{F}}_q$ and ${\overline}{{\mathbb{F}}}_q$. The author would like to thank Lev Borisov for inspiration and for encouraging me to pursue this subject, Andrei Căldărau and Jordan Ellenberg for useful conversations, and Hsian-Hua Tseng and Tom Graber for pointing out the relevance of Yasuda’s work in this context. Arithmetic Frobenius and Inertia Stacks {#Arithmetic_Frobenius} ======================================= In this section, we review inertia stacks and the induced action of the arithmetic Frobenius on them. The notion of inertia stack is essential to the Gromov-Witten theory of stacks, and it’s $\ell$-adic cohomology will form the underlying vector space for main object of study in this article: the $\ell$-adic Chen-Ruan cohomology ring. Let ${\mathcal{X}}$ be a [Deligne-Mumford]{} stack over ${\mathbb{F}}_q$. Fix an algebraic closure ${\overline}{{\mathbb{F}}}_q \supset {\mathbb{F}}_q$ and denote ${\overline}{{\mathcal{X}}} {\mathrel{\mathop:}=}{\mathcal{X}}\times_{{{\mathbb{F}}}_q} {\overline}{{\mathbb{F}}}_q$. Let ${\overline}{{\mathbb{F}}}_q \xrightarrow{\phi} {\overline}{{\mathbb{F}}}_q$ denote the Frobenius morphism given by $\lambda {\rightarrow}\lambda^q$, and let $F_{{\overline}{{\mathcal{X}}}, q} {\mathrel{\mathop:}=}1_{{\mathcal{X}}} \times \ \text{Spec}(\phi)$  denote the *arithmetic Frobenius* morphism on ${\mathcal{X}}$. The subscripts will be dropped when no confusion arises. Let ${\mathbf{\mu}}_r {\mathrel{\mathop:}=}\text{Spec}({\mathbb{Z}}[t]/ \langle t^r - 1 \rangle)$ denote the group scheme over ${\mathbb{Z}}$ of $\text{r}^{th}$ roots of unity, and also denote by ${\mathbf{\mu}}_r$ the base change to ${\mathbb{F}}_q$ when the context is clear. Let $B{\mathbf{\mu}}_r {\mathrel{\mathop:}=}[\text{Spec}({\mathbb{F}}_q) / {\mathbf{\mu}}_r]$ denote the quotient stack corresponding to the trivial action of ${\mathbf{\mu}}_r$ on $\text{Spec}({\mathbb{F}}_q)$. \[inertia\_defn\] 1. We denote by ${\mathcal{I}}_{{\mathbf{\mu}}_r}({\mathcal{X}}) \equiv \text{\underline{Hom}}^{\text{rep}}_{\ {\mathbb{F}}_q}(B{\mathbf{\mu}}_r, {\mathcal{X}})$ the stack of representable 1-morphisms over ${\mathbb{F}}_q$ from $B{\mathbf{\mu}}_r$ to ${\mathcal{X}}$.\ 2. The *cyclotomic inertia stack* is given by $${\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}}) {\mathrel{\mathop:}=}\bigsqcup_r {\mathcal{I}}_{{\mathbf{\mu}}_r}({\mathcal{X}}).$$ ${\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})$ is a [Deligne-Mumford]{} stack and is smooth when ${\mathcal{X}}$ is tame and smooth (this follows from [@AGV2 Section 3]). \[inertia\_example\] Let $G {\rightarrow}\text{Spec }({\mathbb{F}}_q)$ be a finite étale group scheme acting on ${\mathbb{F}}_q$-scheme $X$, and consider the corresponding stack $[X/G]$. Assume that the action is tame or equivalently that the stack $[X/G]$ is tame. For simplicity, assume that ${\mathbb{F}}_q$ contains the $r^{th}$ roots of unity for each $r$ dividing the order of $G$. We then have ${\mathcal{I}}_{{\mathbf{\mu}}}([X/G]) \cong {\mathcal{I}}([X/G])$ the usual inertia stack, and thus $${\mathcal{I}}_{{\mathbf{\mu}}}([X/G]) = \bigsqcup_{(g)} [X^g / C(g)].$$ Here the union is over conjugacy classes of elements of $G({\mathbb{F}}_{q})$, $C(g)$ denotes the centralizer group scheme of $g$, and $X^g$ denotes the fixed subscheme of $g$. Let ${\mathcal{X}}$ and ${\mathcal{Y}}$ be ${\mathbb{F}}_q$-stacks. Let $f$ and $g$ be 1-morphisms from ${\mathcal{X}}$ to ${\mathcal{Y}}$, and let $\phi: f \rightrightarrows g$ be a 2-morphism. Then composition induces 1-morphisms ${\mathcal{I}}_{{\mathbf{\mu}}}(f)$ and ${\mathcal{I}}_{{\mathbf{\mu}}}(g)$, and a 2-morphism ${\mathcal{I}}_{{\mathbf{\mu}}}(\phi)$ making ${\mathcal{I}}_{{\mathbf{\mu}}}(-)$ into a 2-functor. Definition \[inertia\_defn\] and the above remarks clearly also apply to ${\overline}{{\mathbb{F}}}_q$-stacks. The map ${\overline}{{\mathbb{F}}}_q \xrightarrow{\phi} {\overline}{{\mathbb{F}}}_q$ then induces two morphisms on ${\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}})$ which agree by the following lemma. \[inertia\_compat\] 1. There is an equivalence $${\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}}\times_{{\mathbb{F}}_q} {\overline}{{\mathbb{F}}}_q) \xrightarrow{\cong} {\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}}) \times_{{\mathbb{F}}_q} {\overline}{{\mathbb{F}}}_q.$$ 2. We denote the latter simply by ${\overline}{{\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})}$. Under the identification above, the following functors are 2-isomorphic: $${\mathcal{I}}_{{\mathbf{\mu}}}(F_{{\overline}{{\mathcal{X}}}})\ \overset{\cong}{\rightrightarrows} \ F_{{\overline}{{\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})}}.$$ The proof proceeds exactly as in the proof of Lemma \[K\_compat\] appearing in the appendix. It is important to note that there are several variations of the definition of inertia stack appearing in the literature. In particular, to work in the most general setting one should consider the *rigidified* inertia stack [@AGV1; @AGV2]. However, since our goal is to study the Chen-Ruan cohomology (and hence only degree zero stable maps), Definition \[inertia\_defn\] above is preferred. $\ell$-adic Cohomology {#l-adic_cohomology} ====================== Let $l$ be coprime to $q$ and denote by ${\mathbb{Z}}_l$ (resp. ${\mathbb{Q}}_l$) the $\ell$-adic integers (resp. numbers). Let ${\mathcal{X}}$ be a proper, smooth, tame, [Deligne-Mumford]{} ${\mathbb{F}}_q$-stack with projective coarse moduli scheme, and let ${\overline}{{\mathcal{X}}} {\mathrel{\mathop:}=}{\mathcal{X}}\times_{{\mathbb{F}}_q} {\overline}{{\mathbb{F}}}_q$. \[coho\_defn\] The *orbifold $\ell$-adic cohomology* of ${\overline}{{\mathcal{X}}}$ is a graded ${\mathbb{Q}}_l$-algebra. As a vector space, it is given by $$\begin{aligned} {1} H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l)\ &{\mathrel{\mathop:}=}\ H^*({\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}}), {\mathbb{Q}}_l) \notag \\ &= \ \bigoplus_i \ (\ \underset{n}{\varprojlim}\ H^i({\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}}),\ {\mathbb{Z}}/ l^n{\mathbb{Z}})\ ) \underset{{\mathbb{Z}}_l}{\otimes} {\mathbb{Q}}_l. \notag\end{aligned}$$ Any morphism $g: {\overline}{{\mathcal{X}}} {\rightarrow}{\overline}{{\mathcal{Y}}}$ induces ${\mathcal{I}}_{{\mathbf{\mu}}}(g): {\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}}) {\rightarrow}{\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{Y}}})$. Since ${\mathbb{Z}}/ l^r{\mathbb{Z}}$ is a constant sheaf we have ${\mathcal{I}}_{{\mathbf{\mu}}}(g)^*({\mathbb{Z}}/ l^r{\mathbb{Z}}) \cong {\mathbb{Z}}/ l^r{\mathbb{Z}}$. Passing to the limit, we obtain a linear map on $H^*({\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}}), {\mathbb{Q}}_l)$. Corresponding push-forward maps on homology groups are then induced by duality. We now introduce the grading on $H^*({\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}}), {\mathbb{Q}}_l)$ using the notion of *age*. We include the definition at this stage because the definition itself is fairly direct. The motivation however arises later from the appearance of age in the Riemann-Roch theorem on curves [@AGV2 Thm. 7.2.1] (and hence also appears in the computation of the degree of the virtual fundamental class on the moduli stack of stable maps into ${\mathcal{X}}$). Since one goal of this article is to determine the effect of the Frobenius morphism on this virtual fundamental class, we proceed without assuming our base field is algebraically closed. For every $r$ fix an embedding ${\mathbf{\mu}}_r \hookrightarrow \mathbb{G}_m$. Then a group scheme morphism $\rho: {\mathbf{\mu}}_r {\rightarrow}\mathbb{G}_m$ over ${{\mathbb{F}}}_q$ is determined by an integer $0 \le k \le r-1$ with $\rho(g) = g^k$. Define $$\text{age}(\rho) {\mathrel{\mathop:}=}\frac{k}{r} \in {\mathbb{Q}}.$$ When $(q, r) = 1$, this function extends by linearity to a function on the representation ring of the group scheme. For any object $((B{\mathbf{\mu}}_r)_S \xrightarrow{f} {\mathcal{X}})$ in ${\mathcal{I}}_{{\mathbf{\mu}}_r}({\mathcal{X}})(S)$, each fiber of $f^*T_{{\mathcal{X}}}$ over $S$ gives a representation of ${\mathbf{\mu}}_r$, and we obtain a locally constant function on ${\mathcal{I}}_{{\mathbf{\mu}}_r}({\mathcal{X}})$. We obtain a well-defined locally constant function also denoted by $\text{age}$: $$\text{age}: {\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}}) {\rightarrow}{\mathbb{Q}}.$$ Finally, we may pull this function back to a function on ${\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}})$, and this induces a function $\text{age}: H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l) {\rightarrow}{\mathbb{Q}}$. The various uses of this notation will be clear from the context. The grading is then given by $$\label{grading} H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l) = \bigoplus_{i \in {\mathbb{Q}}} H_{CR}^{i}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l)$$ where $$\label{grading2} H_{CR}^{i}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l) {\mathrel{\mathop:}=}\bigoplus_{a + 2b = i} H^a(\text{age}^{-1}(b), {\mathbb{Q}}_l).$$ Note that the choices of embeddings ${\mathbf{\mu}}_r \hookrightarrow \mathbb{G}_m$ will change the age function on ${\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}})$. \[inertia\_remark\] Note that in general an automorphism of ${\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})$ *over* ${\mathcal{X}}$ will not preserve the age function. An essential example is the involution $i: {\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}}) {\rightarrow}{\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})$ given by precomposing each $(B{\mathbf{\mu}}_r)_S {\rightarrow}{\mathcal{X}}$ with the automorphism $(B{\mathbf{\mu}}_r)_S {\rightarrow}(B{\mathbf{\mu}}_r)_S$ induced by $g \mapsto g^{-1}: {\mathbf{\mu}}_r {\rightarrow}{\mathbf{\mu}}_r$. On the other hand, any automorphism of ${\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})$ *induced by an automorphism of ${\mathcal{X}}$* does preserve the age. An essential example is the arithmetic Frobenius on ${\overline}{{\mathcal{X}}}$. \[Yasuda\] In [@Yas], Yasuda defines the additive Chen-Ruan cohomology as in Definition \[coho\_defn\] above with the following exceptions. If we define a function on ${\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}})$ by $sht := age \circ i$ where $i$ is as in Remark \[inertia\_remark\], then Yasuda replaces the right side of (\[grading2\]) with $$\bigoplus_{a + 2b = i} H^a(\text{sht}^{-1}(b), {\mathbb{Q}}_l(-b)).$$ (Strictly speaking, Yasuda also works with the cohomology of coarse moduli space.) The Tate twist ${\mathbb{Q}}_l(-b)$ changes the weight as a Galois representation, and one motivation for this arises from Yasuda’s proof ([@Yas Cor. 4.9]) of an additive version of the Crepant Resolution Conjecture [@BG; @Ruan1; @Ruan2]. In the current paper, another motivation is found via the Galois action on the *ring* structure of Chen-Ruan cohomology defined in the section. Ring Structure ============== In this section we gather the results in [@BF], [@AV1], and [@AGV2] needed to define the ring structure. One subtlety is the cycle map from Chow groups to $\ell$-adic cohomology which requires the moduli stack to be smooth (see Remark \[cycle\]). Introduction {#introduction-1 .unnumbered} ------------ The ring structure we shall impose on $H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l)$ is motivated by quantum cohomology, and so we give a brief description. For simplicity consider a smooth projective scheme $Y$ over ${\mathbb{C}}$. The quantum cohomology of $Y$ is a deformation of $H^*(Y, {\mathbb{C}})$. If $H_2^+(Y,{\mathbb{Z}}) \subset H_2(Y, {\mathbb{Z}})$ denotes the classes generated by effective curves in $Y$, then the parameter space of the deformation is given by the semigroup algebra ${\mathbb{Q}}[H_2^+(Y, {\mathbb{Z}})]$. The product in this deformed ring requires integrals over the moduli stack of curve in $Y$ (more precisely over a compactification by stable maps due to Kontsevich [@Ko]). Finally, the original ring $H^*(Y, {\mathbb{C}})$ is recovered by setting the deformation parameters to zero (i.e. by only considering the moduli stack of constant stable maps). However, when $Y$ is replaced with a stack ${\mathcal{Y}}$, this limit of quantum cohomology does not agree with $H^*({\mathcal{Y}}, {\mathbb{C}})$. The new ring obtained is called the *Chen-Ruan cohomology ring*. In what follows we define the ring structure directly using the complex case as motivation. Stable maps {#stable-maps .unnumbered} ----------- Let ${\mathcal{X}}$ be a proper, smooth, tame, [Deligne-Mumford]{} stack over ${\mathbb{F}}_q$ with projective coarse moduli scheme $X$, and let ${\overline}{{\mathcal{X}}} {\mathrel{\mathop:}=}{\mathcal{X}}\times_{{\mathbb{F}}_q} {\overline}{{\mathbb{F}}}_q$. We define the moduli stack of stable maps into ${\mathcal{X}}$ as constructed in [@AV1]. A *balanced twisted stable n-pointed map* $({\mathcal{C}}\xrightarrow{f} {\mathcal{X}}_S, \{\Sigma_i\})$ is a commutative diagram of ${\mathbb{F}}_q$-stacks \_[i=1]{}\^n \_i & & & \^[f]{} & \_S\ & & \^ & &\ & & C & \^[|f|]{} & X\_S where 1. ${\mathcal{C}}$ is a proper [Deligne-Mumford]{} stack with coarse moduli space $C$ 2. $(C, \{\pi(\Sigma_i)\})$ is an n-pointed nodal curve 3. Over the node $\{xy = 0\}$ of $C$, ${\mathcal{C}}$ has étale chart $$[\{xy = 0\} / ({\mathbf{\mu}}_r)_{{\mathbb{F}}_q}]$$ where the action is given by $(x,y) \mapsto (\xi u, \xi^{-1} v)$ 4. Over a marked point $\pi(\Sigma_i)$ of $C$, ${\mathcal{C}}$ has étale chart $$[\mathbb{A}^1 / ({\mathbf{\mu}}_r)_{{\mathbb{F}}_q}]$$ where the action is given by $u \mapsto \xi u$ and $\Sigma_i$ is the substack defined by $u = 0$ 5. $\pi$ is an isomorphism away from the markings and nodes 6. $f$ is representable with $|f|$ the induced map on coarse moduli spaces 7. $|f|$ is stable in the sense of Kontsevich [@Ko]. We shall refer to the above merely as *stable maps*. For any $d,g \ge 0$, we say that $({\mathcal{C}}\xrightarrow{f} {\mathcal{X}}, \{\Sigma_i\})$ has degree $d$ and genus $g$ if $|f|$ does. After appropriately defining such objects over an arbitrary ${\mathbb{F}}_q$-scheme, one obtains the stack ${\mathcal{K}}_{g,n}({\mathcal{X}}, d)$, a proper stack of finite type over ${\mathbb{F}}_q$ with projective coarse moduli space [@AV1 Theorem 1.4.1]. Our purposes only require the case when $g = 0$, $d = 0$, and $n = 3$. We shall denote ${\mathcal{K}}_{0,3}({\mathcal{X}}, 0)$ simply by ${\mathcal{K}}({\mathcal{X}})$, and we have the following additional properties: \[K\_smooth\] ${\mathcal{K}}({\mathcal{X}}) = {\mathcal{K}}_{0,3}({\mathcal{X}}, 0)$ is a smooth [Deligne-Mumford]{} stack over ${\mathbb{F}}_q$. See the appendix. \[eval\] There exist evaluation morphisms over ${\mathbb{F}}_q$ denoted $e_i$ for $i = 1,2,3:$ $${\mathcal{K}}({\mathcal{X}}) \xrightarrow{e_i} {\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})$$ which applies to objects over ${\mathbb{F}}_q$ as $$({\mathcal{C}}\xrightarrow{f} {\mathcal{X}}, \{\Sigma_i\}) \mapsto (\Sigma_i \xrightarrow{f|_{\Sigma_i}} {\mathcal{X}}).$$ The proof of [@AGV1 Lemma 6.2.1] carries over to this context without change. The subtlety of this lemma lies in the definition of stable maps over more general base schemes. Virtual Classes {#virtual-classes .unnumbered} --------------- Finally, we consider integrating cohomology classes on this moduli space, and for this purpose we a need a fundamental class. However, even though ${\mathcal{K}}({\mathcal{X}})$ is smooth, the natural vector spaces holding obstructions to deforming stable maps may still be non-trivial. In [@BF], the authors define an *obstruction theory* to describe this phenomenon. They proceed to construct a *virtual fundamental class* in the Chow group of the moduli stack as a replacement of the usual fundamental class. In the case of interest in this article, these constructions have the following concrete description. Let $$\label{universal_diagram} \begin{diagram} \mathcal{U} & \rTo^{f} & {\mathcal{X}}\\ \dTo_{\pi} & & \\ {\mathcal{K}}({\mathcal{X}}) & & \\ \end{diagram}$$ be the universal curve and universal stable map to ${\mathcal{X}}$. Then we have the following lemma. \[obs\_theory\] 1. The natural map $$(R^{\bullet}\pi_* f^*T_{{\mathcal{X}}})^{\vee} \ \xrightarrow{\phi} \ \Omega^1_{{\mathcal{K}}({\mathcal{X}})/ {\mathcal{M}}_{0,3}^{tw}}$$ is a perfect relative obstruction theory with virtual dimension (denoted vdim) given by the locally constant function $$vdim = dim {\mathcal{X}}- \text{age}\circ e_1 - \text{age}\circ e_2 - \text{age}\circ e_3.$$ 2. $R^1\pi_* f^*T_{{\mathcal{X}}}$ is locally free (denote the locally constant rank by $r$), and the virtual fundamental class (denoted $[{\mathcal{K}}({\mathcal{X}})]^{vir}$) in $A_{vdim}({\mathcal{K}}({\mathcal{X}}))_{{\mathbb{Q}}}$ induced by $\phi$ is $$[{\mathcal{K}}({\mathcal{X}})]^{vir} = c_{r}(R^1\pi_*f^*T_{{\mathcal{X}}}).$$ See the appendix. The previous lemma will also hold if ${\mathcal{X}}$ is replaced by ${\overline}{{\mathcal{X}}}$. However, the statement over the non-algebraically closed field is essential in the proof of Proposition \[main\] (specifically Lemma \[F\_on\_vir\_class\]) where we compute the action of the Frobenius on $[{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{vir}$ in the Chow group. \[cycle\] Since ${\mathcal{K}}({\overline}{{\mathcal{X}}})$ is smooth, one can construct the cycle map $$A_*({\mathcal{K}}({\overline}{{\mathcal{X}}}))_{{\mathbb{Q}}} \otimes_{{\mathbb{Q}}} {\mathbb{Q}}_l \xrightarrow{cl} H_*({\mathcal{K}}({\overline}{{\mathcal{X}}}), {\mathbb{Q}}_l) = H^*({\mathcal{K}}({\overline}{{\mathcal{X}}}), {\mathbb{Q}}_l)$$ by proceeding as in [@Milne VI. Section 9] using the Gysin sequence in [@Be-Lef Cor. 2.1.3] and a slight refinement of the long exact sequence in Section 2.1 of \[ibid.\]. We denote the image of $[{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{vir}$ under $cl$ also by $[{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{vir}$. Ring structure {#ring-structure-1 .unnumbered} -------------- The ring structure can now be constructed formally on $H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l)$ just as in the original formulation [@CR2]. For $\alpha, \beta \in H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l)$ define $$\alpha \star \beta {\mathrel{\mathop:}=}i_*(e_3)_*(e_1^*\alpha \cup e_2^* \beta \cap [{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{vir})$$ where $i: {\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}}) {\rightarrow}{\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}})$ is the morphism induced by the isomorphisms $\lambda \mapsto \lambda^{-1}: {\mathbf{\mu}}_r {\rightarrow}{\mathbf{\mu}}_r$. \[assoc\] The operation $\star$ makes $H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l)$ into a (graded) commutative, associative ring with unity. The description of the boundary strata of ${\mathcal{K}}_{0,4}({\overline}{{\mathcal{X}}}, 0)$, and the proofs in Sections 5 and 6 of [@AGV2] apply to this context as well. Note that since we are only concerned with degree zero maps, there is no need to decompose ${\mathcal{K}}_{0,n}({\overline}{{\mathcal{X}}})$ via curve classes on the coarse moduli space of ${\overline}{{\mathcal{X}}}$. \[example\_ring\] Suppose $b$ is coprime to $q$ and ${\mathbb{G}}_m = \text{Spec }({\mathbb{F}}_q[t, \frac{1}{t}])$ acts on ${\mathbb{A}}^2$ with weights $1$ and $b$. Assume for simplicity that $b$ is prime and $F_q$ contains the $b^{th}$ roots of unity. The stack ${\mathcal{X}}{\mathrel{\mathop:}=}[({\mathbb{A}}^2 \backslash \{0\})/ {\mathbb{G}}_m]$ has étale neighborhoods $$\begin{aligned} {1} [{\mathbb{A}}^1/{\mathbf{\mu}}_b] &{\rightarrow}{\mathcal{X}}\notag \\ {\mathbb{A}}^1 &{\rightarrow}{\mathcal{X}}\notag\end{aligned}$$ whose induced map on coarse moduli spaces form a cover of ${\mathbb{P}}^1$, the coarse moduli scheme of ${\mathcal{X}}$. Here the ${\mathbf{\mu}}_b$ action is given by the character $\lambda \mapsto \lambda^b: {\mathbf{\mu}}_b {\rightarrow}{\mathbb{G}}_m$. We may then use Example \[inertia\_example\] above to compute $${\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}}) = {\mathcal{X}}\ \bigsqcup\ \sqcup_{i=1}^{b-1} B{\mathbf{\mu}}_b.$$ Let $A$ denote the copy of ${\overline}{B{\mathbf{\mu}}_b} \subset {\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}})$ with age equal to $\frac{1}{b}$. Denote also by $A$ a generator of $H^0(A, {\mathbb{Q}}_l) \subset H^0({\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}}), {\mathbb{Q}}_l)$. Then one has $$H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l) \cong {\mathbb{Q}}_l[A]/\langle A^{b+1} \rangle.$$ In particular, $A^b$ generates $H^2({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l) \subset H^2({\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}}), {\mathbb{Q}}_l)$ and has age equal to $0$. Frobenius Actions {#Frobenius_Actions} ================= Let ${\mathcal{X}}$, ${\overline}{{\mathcal{X}}}$ be as in the last section. The arithmetic Frobenius morphism $F_{{\overline}{{\mathcal{X}}}}$ on ${\overline}{{\mathcal{X}}}$ induces the morphism ${\mathcal{I}}_{{\mathbf{\mu}}}(F_{{\overline}{{\mathcal{X}}}}) = F_{{\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}})}$ on ${\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}})$. Thus we have an induced map on Chen-Ruan cohomology groups which we denote by $${\mathcal{I}}_{{\mathbf{\mu}}}(F_{{\overline}{{\mathcal{X}}}})^* : H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l) {\rightarrow}H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l).$$ Consider Example \[example\_ring\] above. ${\mathcal{I}}_{{\mathbf{\mu}}}(F_{{\overline}{{\mathcal{X}}}})^*$ acts on the orbifold $\ell$-adic cohomology by fixing $A^1, \cdots, A^{b-1}$ and sending $A^b$ to $q^{-1}A^b$. Thus ${\mathcal{I}}_{{\mathbf{\mu}}}(F_{{\overline}{{\mathcal{X}}}})^*$ does not preserve the ring structure. However by composing ${\mathcal{I}}_{{\mathbf{\mu}}}(F_{{\overline}{{\mathcal{X}}}})^*$ with the morphism sending $A^i \mapsto q^{-\text{age}(A^i)}A^i$ we indeed obtain a homomorphism of rings sending $A^i \mapsto q^{-\frac{i}{b}} A^i$. Proposition \[main\] from the introduction shows that this phenomenon holds in general. [Proposition \[main\]]{} The *orbifold Frobenius morphism* given by $$\begin{aligned} {1} F_{{\overline}{{\mathcal{X}}}, q, orb}: \ H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l) &\longrightarrow H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l) \notag \\ \alpha &\longmapsto q^{-\text{age}(\alpha)} \cdot {\mathcal{I}}_{{\mathbf{\mu}}}(F_{{\overline}{{\mathcal{X}}}, q})^*(\alpha) \notag\end{aligned}$$ is a homomorphism of graded rings. When no confusion arises, the subscripts ${\overline}{{\mathcal{X}}}$ and $q$ on $F$ may be dropped. \[Yasuda2\] Note that instead of twisting the natural map ${\mathcal{I}}_{{\mathbf{\mu}}}(F_{{\overline}{{\mathcal{X}}}})^*$ to obtain $F_{orb}$, one could twist the coefficients of the Chen-Ruan cohomology. For instance, if we redefined the Chen-Ruan cohomology groups by replacing the right side of (\[grading2\]) with $$\bigoplus_{a + 2b = i} H^a(\text{age}^{-1}(b), {\mathbb{Q}}_l(-b)),$$ then the natural map action of ${\mathcal{I}}_{{\mathbf{\mu}}}(F_{{\overline}{{\mathcal{X}}}})$ on these new groups would be a ring isomorphism. Before proving the proposition we make a few observations. First we consider the map on ${\mathcal{K}}({\overline}{{\mathcal{X}}})$ induced by $F_{{\overline}{{\mathcal{X}}}}$. For any ${\overline}{{\mathbb{F}}}_q$-scheme $S$, and any object \_[i=1]{}\^3 \_i & & & \^[f\_S]{} &\ & & & &\ & & S & \^ & (\_q) of ${\mathcal{K}}({\overline}{{\mathcal{X}}})(S)$, the commutative diagram \_[i=1]{}\^3 \_i & & & \^[F\_ f\_S]{} &\ & & & &\ & & S & \^[F\_[\_q]{} ]{} & (\_q) is clearly an object of ${\mathcal{K}}({\overline}{{\mathcal{X}}})$ once we check the stability condition. However, this is easy since $f_S$ has degree zero. The obvious map on morphisms then determines a functor we denote by ${\mathcal{K}}(F_{{\overline}{{\mathcal{X}}}})$. Note that ${\mathcal{K}}(F_{{\overline}{{\mathcal{X}}}})$ covers the Frobenius map on $\text{Spec}({\overline}{{\mathbb{F}}}_q)$. The following lemma compares ${\mathcal{K}}(F_{{\overline}{{\mathcal{X}}}})$ with the Frobenius morphism $F_{{\mathcal{K}}({\overline}{{\mathcal{X}}})}$. \[K\_compat\] 1. There is an equivalence $${\mathcal{K}}({\mathcal{X}}\times_{{\mathbb{F}}_q} {\overline}{{\mathbb{F}}}_q) \xrightarrow{\cong} {\mathcal{K}}({\mathcal{X}}) \times_{{\mathbb{F}}_q} {\overline}{{\mathbb{F}}}_q.$$ 2. We denote the latter simply by ${\overline}{{\mathcal{K}}({\mathcal{X}})}$. Under the identification above, the following functors are 2-isomorphic: $${\mathcal{K}}(F_{{\overline}{{\mathcal{X}}}})\ \overset{\cong}{\rightrightarrows} \ F_{{\overline}{{\mathcal{K}}({\mathcal{X}})}}.$$ See the appendix. These identifications allow us to determine the image of the virtual fundamental class under ${\mathcal{K}}(F_{{\overline}{{\mathcal{X}}}})$ in homology. \[F\_on\_vir\_class\] ${\mathcal{K}}(F_{{\overline}{{\mathcal{X}}}})_* [{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{vir} = q^{-vdim}[{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{vir}$ See the appendix. For simplicity, denote $F_{orb} {\mathrel{\mathop:}=}F_{{\overline}{{\mathcal{X}}}, q, orb}$, ${\mathcal{I}}(-) {\mathrel{\mathop:}=}{\mathcal{I}}_{{\mathbf{\mu}}}(-)$, and $F {\mathrel{\mathop:}=}F_{{\overline}{{\mathcal{X}}}, q}$. Let ${\mathcal{I}}({\overline}{{\mathcal{X}}}) = \bigsqcup_j {\mathcal{I}}({\overline}{{\mathcal{X}}})_j$ be a decomposition into connected components. Define $a_j {\mathrel{\mathop:}=}\text{age}({\mathcal{I}}({\overline}{{\mathcal{X}}})_j)$ and $\hat{a}_j {\mathrel{\mathop:}=}\text{age}(i({\mathcal{I}}({\overline}{{\mathcal{X}}})_j))$ where $i: {\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}}) {\rightarrow}{\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})$ is the isomorphism appearing in the definition of $\star$ above. To prove the proposition it suffices to check that $$\label{compat} F_{orb}(\alpha_1) \star F_{orb}(\alpha_2) = F_{orb}(\alpha_1 \star \alpha_2)$$ for any $\alpha_k \in H^*({\mathcal{I}}({\overline}{{\mathcal{X}}})_{j(\alpha_k)}, {\mathbb{Q}}_l) \subset H^*({\mathcal{I}}({\overline}{{\mathcal{X}}}), {\mathbb{Q}}_l)$ ($k = 1,2$). Apply ${\mathcal{I}}(F)^* {\mathcal{I}}(F)_*$ to the left side of (\[compat\]), where Poincaré duality isomorphisms have been suppressed. By the projection formula and since ${\mathcal{K}}(F)$ and ${\mathcal{I}}(F)$ commute, we then have $$\begin{aligned} {1} \label{eqn1} {\mathcal{I}}(F)^* {\mathcal{I}}(F)_* (&F_{orb}(\alpha_1) \star F_{orb}(\alpha_2)) = \notag \\ &= {\mathcal{I}}(F)^* i_* (e_3)_* {\mathcal{K}}(F)_* (q^{-a_1 - a_2} {\mathcal{K}}(F)^* e_1^* \alpha_1 \cup {\mathcal{K}}(F)^* e_2^* \alpha_2 \cap [{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{vir}) \notag \\ &= {\mathcal{I}}(F)^* i_* (e_3)_*(q^{-a_1 - a_2} e_1^* \alpha_1 \cup e_2^* \alpha_2 \cap {\mathcal{K}}(F)_* [{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{vir}) \notag \\ &= {\mathcal{I}}(F)^* i_* (e_3)_*(q^{-a_1 - a_2 - vdim} e_1^* \alpha_1 \cup e_2^* \alpha_2 \cap [{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{vir}).\end{aligned}$$ Now note that the operator ${\mathcal{I}}(F)^* {\mathcal{I}}(F)_*$ decomposes according to the following lemma whose proof is left to the reader: Let $1_j$ denote the denote both the fundamental class of ${\mathcal{I}}({\overline}{{\mathcal{X}}})_j$ and the operator given by taking cup product with $1_j$. Then $${\mathcal{I}}(F)^* {\mathcal{I}}(F)_* = \sum_j q^{-\text{dim} {\mathcal{I}}({\overline}{{\mathcal{X}}})_j} 1_j.$$ Thus (\[eqn1\]) implies $$\begin{aligned} {1} \label{eqn2} F_{orb}(\alpha_1) &\star F_{orb}(\alpha_2) = \notag \\ &= \sum_j 1_j\ q^{\text{dim} {\mathcal{I}}({\overline}{{\mathcal{X}}})_j} {\mathcal{I}}(F)^*i_* (e_3)_*(q^{-a_1 - a_2 - \text{vdim}} e_1^* \alpha_1 \cup e_2^* \alpha_2 \cap [{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{vir}) \notag \\ &= \sum_j 1_j\ F_{orb} i_* (e_3)_*(q^{-a_1 - a_2 - \text{vdim} + \text{dim} {\mathcal{I}}({\overline}{{\mathcal{X}}})_j + a_3} e_1^* \alpha_1 \cup e_2^* \alpha_2 \cap [{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{vir}).\end{aligned}$$ Now for each $j$, the $1_j$ in (\[eqn2\]) restricts the class to ${\mathcal{I}}({\overline}{{\mathcal{X}}})_j$. This contribution will not be changed if we replace $[{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{vir}$ by its restriction to $e_1^{-1}{\mathcal{I}}({\overline}{{\mathcal{X}}})_{j_1} \bigcap \ e_2^{-1}{\mathcal{I}}({\overline}{{\mathcal{X}}})_{j_2} \bigcap\ (i \circ e_3)^{-1}{\mathcal{I}}({\overline}{{\mathcal{X}}}_j)$ where $j_1$ (resp. $j_2$) is the index of the component of ${\mathcal{I}}({\overline}{{\mathcal{X}}})$ supporting $\alpha_1$ (resp. $\alpha_2$). On this locus of ${\mathcal{K}}({\overline}{{\mathcal{X}}})$, vdim is constant and equal to $\text{dim}\ {\overline}{{\mathcal{X}}} - a_1 - a_2 - \hat{a}_3$. Since for each $k$, $$a_k + \hat{a}_k = \text{dim}\ {\overline}{{\mathcal{X}}} - \text{dim} {\mathcal{I}}({\overline}{{\mathcal{X}}})_{j(a_k)},$$ the exponent of $q$ in (\[eqn2\]) is zero. Thus (\[eqn2\]) implies $$\begin{aligned} {1} F_{orb}(\alpha_1) \star F_{orb}(\alpha_2) &= \sum_j 1_j\ F_{orb} i_* (e_3)_*(e_1^* \alpha_1 \cup e_2^* \alpha_2 \cap [{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{vir}) \notag \\ &= F_{orb}(\alpha_1 \star \alpha_2), \notag\end{aligned}$$ and the proposition is proved. Orbifold Zeta Functions {#zeta} ======================= Let ${\mathcal{X}}$, ${\overline}{{\mathcal{X}}}$ be as in the last section. For simplicity assume in addition that ${\mathcal{X}}$ is *Gorenstein* condition: ${\mathcal{X}}$ has generically trivial isotropy, and for any geometric point $\xi \in {\mathcal{X}}({\overline}{{\mathbb{F}}}_q)$, the representation $$\text{Aut}(\xi) {\rightarrow}\text{GL}(T_{{\overline}{{\mathcal{X}}}}|_{\xi})$$ has determinant 1. The latter condition is equivalent to $\text{age}: {\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}}) {\rightarrow}{\mathbb{Q}}$ taking integer values. Thus $H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l)$ is ${\mathbb{Z}}$-graded. For a linear map $F: V {\rightarrow}V$ on a ${\mathbb{Z}}$-graded vector space we write $V = \oplus V_i$, $F = \oplus F_i$ and we denote $$\begin{aligned} {1} \text{det}\ (F\ |\ V)\ &=\ \prod_i \text{det}\ (F_i\ |\ V_i)^{(-1)^{i+1}} \notag \\ \text{Tr}\ (F\ |\ V)\ &=\ \sum_i (-1)^i\ \text{Tr}\ (F_i\ |\ V_i). \notag\end{aligned}$$ \[zeta\_defn\] The *orbifold cohomological zeta function* is given by $$\begin{aligned} {1} Z_{H^*_{CR}}({\mathcal{X}}, t)\ &{\mathrel{\mathop:}=}\ \text{det}\ (1 - F_{orb}\ t\ |\ H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l)) \notag \\ &=\ \text{exp}\ (\sum_{r=1}^{\infty} \text{Tr}(F_{orb}^{r}\ |\ H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l)) \frac{t^r}{r}). \notag\end{aligned}$$ One obtains a trace formula for $Z_{H^*_{CR}}({\mathcal{X}}, t)$ by applying the Lefschetz Trace Theorem of [@Be-Lef] to ${\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}})$. For a ${\mathbb{F}}_q$-scheme $S$, let $[{\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})(S)]$ denote the set of isomorphism classes of the groupoid ${\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})(S)$. For $\xi \in [{\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})(S)]$, let $\text{Aut}(\xi)$ denote the automorphism group of any representative of $\xi$. Finally, let ${\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}}) = \bigsqcup_i {\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})_i$ be a decomposition into connected components so that $\text{age}$ and dimension ($\text{dim}$) are constant on each ${\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})_i$. Then [@Be-Lef Theorem 3.1.2] yields $$\begin{aligned} {1} \label{trace_formula} \text{Tr}\ (F_{orb}\ |\ H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l))\ &=\ \sum_i \text{Tr}\ (F_{orb}|_{{\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}})_i}\ |\ H^*({\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}})_i, {\mathbb{Q}}_l)) \notag \\ &=\ \sum_i q^{-\text{age}({\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}})_i)} \text{Tr}\ (F^*_{{\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}})_i}\ |\ H^*({\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}})_i, {\mathbb{Q}}_l)) \notag \\ &=\ \sum_i q^{-\text{age}({\mathcal{I}}_{{\mathbf{\mu}}}({\overline}{{\mathcal{X}}})_i)} \sum_{\xi \in [{\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})_i({\mathbb{F}}_q)]} \frac{q^{-\text{dim}(\xi)}}{\# \text{Aut}(\xi)} \notag \\ &=\ \sum_{\xi \in [{\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})({\mathbb{F}}_q)]} \frac{q^{-\text{age}(\xi)-\text{dim}(\xi)}}{\# \text{Aut}(\xi)}.\end{aligned}$$ One is led to claim that the trace of $F_{orb}$ counts objects of\ $\bigcup_r \text{\underline{Hom}}^{\text{rep}}_{\ {\mathbb{F}}_q}(B{\mathbf{\mu}}_r, {\mathcal{X}})({\mathbb{F}}_q)$ counted with weights by the age and dimension. It is natural to ask if the trace of $F_{orb}$ counts some natural objects on ${\mathcal{X}}$ without weights. Recall that $F_{orb} = F_{q, orb}$ depends on the base field. Since $F^r_{q, orb} = F_{q^r, orb}$, equation (\[trace\_formula\]) yields a formula for the trace of each iterate of $F_{q, orb}$. We then obtain the following analog of (\[intro\_eqn\_1\]): $$\text{det}\ (1 - F_{orb}\ t\ |\ H^*_{CR}({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l))\ =\ \text{exp}\ \sum_{r = 1}^{\infty}\ ( \sum_{\xi \in [{\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})({\mathbb{F}}_{q^r})]} \frac{(q^r)^{-\text{age}(\xi)-\text{dim}(\xi)}}{\# \text{Aut}(\xi)} ) \frac{t^r}{r}.$$ We attempt to further interpret the arithmetic information contained by $Z_{H^*_{CR}}({\mathcal{X}}, t)$ motivated by the Crepant Resolution Conjecture [@BG; @Ruan1; @Ruan2]. A theorem of Yasuda [@Yas Cor. 4.9] gives the following result. Two smooth, proper stacks ${\mathcal{X}}_1$ and ${\mathcal{X}}_2$ are *K-equivalent* if there exists smooth, proper stack ${\mathcal{Y}}$ and proper, tame and birational maps $\pi_1$ and $\pi_2$ & & & &\ & \^[\_1]{} & & \^[\_2]{} &\ \_1 & & & & \_2\ such that $\pi_1^*K_{{\mathcal{X}}_1} \cong \pi_2^*K_{{\mathcal{X}}_2}$ where $K_{{\mathcal{X}}_i}$ is the canonical line bundle on ${\mathcal{X}}_i$. If ${\mathcal{X}}_1$ and ${\mathcal{X}}_2$ are K-equivalent, proper, smooth, tame [Deligne-Mumford]{} stacks (recall also the Gorenstein assumption made throughout this section), then $$Z_{H^*_{CR}}({\mathcal{X}}_1, t) = Z_{H^*_{CR}}({\mathcal{X}}_2, t).$$ We simply show that the orbifold zeta function defined above agrees with the natural zeta function built from the Galois representation in [@Yas Defn. 4.6] for which the statement holds [@Yas Cor. 4.9]. Following Remarks \[Yasuda\] and \[Yasuda2\] above, it suffices to show $$\begin{aligned} {1} \text{Tr}\ (\ {\mathcal{I}}(F)^* \ |\ \bigoplus_{a + 2b = j} &H^a(\text{age}^{-1}(b), {\mathbb{Q}}_l(-b))\ ) = \notag \\ &\text{Tr}\ (\ {\mathcal{I}}(F)^* \ |\ \bigoplus_{a + 2b = j} H^a((\text{age} \circ i)^{-1}(b), {\mathbb{Q}}_l(-b))\ ) \notag\end{aligned}$$ where $i: {\mathcal{I}}({\mathcal{X}}) {\rightarrow}{\mathcal{I}}({\mathcal{X}})$ is the involution. But this follows from $i \circ {\mathcal{I}}(F) = {\mathcal{I}}(F) \circ i$ which one can easily show. In particular, we see that the orbifold zeta function carries the arithmetic information of any crepant resolution (when one exists) of the coarse moduli scheme. \[Yasuda\_corollary\] Let ${\mathcal{X}}$ be a proper, smooth, tame [Deligne-Mumford]{} stack satisfying the hard Lefschetz condition with trivial generic stabilizer. Suppose $Y {\rightarrow}X$ is a crepant resolution of the coarse moduli scheme $X$ of ${\mathcal{X}}$, then $$Z_{H^*_{CR}}({\mathcal{X}}, t) = Z(Y,t)$$ where $Z(Y,t)$ is the classical zeta function. Yasuda also proves an analog of [@Yas Cor. 4.9] over the complex numbers, and this result was obtained independently by Lupercio and Poddar [@LP]. It is natural to associate to an orbifold, the zeta function of any crepant resolution (when one exists) of the coarse moduli space (see for example [@W page 9]). The above corollary then shows that this definition agrees with Definition \[zeta\_defn\] above in the special case when such a crepant resolution exists. Suppose 2 is coprime to $q$, and suppose ${\mathbb{G}}_m = \text{Spec }({\mathbb{F}}_q[t, \frac{1}{t}])$ acts on ${\mathbb{A}}^3$ with weights $1$, $1$, and $2$. The stack ${\mathcal{X}}{\mathrel{\mathop:}=}[{\mathbb{A}}^3 \backslash \{0\}/ {\mathbb{G}}_m]$ has étale neighborhood $[{\mathbb{A}}^2/ {\mathbf{\mu}}_2] {\rightarrow}{\mathcal{X}}$ where the action of ${\mathbf{\mu}}_2 \cong {\mathbb{Z}}/2{\mathbb{Z}}$ is given by the direct sum of two copies of the non-trivial character of ${\mathbf{\mu}}_2$. If $${\mathcal{X}}\xrightarrow{\pi_1} |{\mathcal{X}}|$$ denotes the morphism to the coarse moduli scheme, then $|{\mathcal{X}}|$ is the projective closure of ${\mathbb{A}}^2/ {\mathbf{\mu}}_2 \cong \text{Spec }({\mathbb{F}}_q[x,y,z]/\langle xy-z^2 \rangle)$. The canonical sheaf $K_{|{\mathcal{X}}|}$ is locally free and $\pi_1^* K_{|{\mathcal{X}}|} \cong K_{{\mathcal{X}}}$. Furthermore, the blow-down map $$Y {\mathrel{\mathop:}=}{\mathbb{P}}({\mathcal{O}}_{{\mathbb{P}}^1} \oplus {\mathcal{O}}_{{\mathbb{P}}^1}(1)) \xrightarrow{\pi_2} |{\mathcal{X}}|$$ is a resolution of singularities with $\pi_2^* K_{|{\mathcal{X}}|} \cong K_Y$. Thus the fibered product $Y \times_{|{\mathcal{X}}|} {\mathcal{X}}$ induces a $K$-equivalence between $Y$ and ${\mathcal{X}}$. On each space, the cohomology is generated by algebraic classes and so the Frobenius action is easily computed. Since $Y$ is a scheme, ${\mathcal{I}}_{{\mathbf{\mu}}}(Y) = Y$, $H^*_{CR}({\overline}{Y}, {\mathbb{Q}}_l) = H^*({\overline}{Y}, {\mathbb{Q}}_l)$, and $F_{orb} = F_Y^*$ is the map on cohomology induced by the usual arithmetic Frobenius morphism. The cohomology of $Y$ is well-known, and we have $$Z_{H^*_{CR}}(Y, t) = \frac{1}{(1-t)(1-q^{-1}t)^2(1-q^{-2}t)}.$$ For the cohomology of ${\mathcal{X}}$, we proceed as in Example \[example\_ring\] obtaining $${\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}}) = {\mathcal{X}}\ \sqcup B{\mathbf{\mu}}_2.$$ The substack $B{\mathbf{\mu}}_2$ has age 1 and $H^0({\overline}{B{\mathbf{\mu}}_2}, {\mathbb{Q}}_l)$ is fixed by $F_{orb}$. Thus $B{\mathbf{\mu}}_2$ contributes a factor of $\frac{1}{1-q^{-1}t}$ to $Z_{H^*_{CR}}({\mathcal{X}}, t)$. The substack ${\mathcal{X}}\subset {\mathcal{I}}_{{\mathbf{\mu}}}({\mathcal{X}})$ has age 0 and the action of $F_{orb}$ on $H^*({\overline}{{\mathcal{X}}}, {\mathbb{Q}}_l)$ agrees with the action of $F_{|{\overline}{{\mathcal{X}}}|}$ on $H^*(|{\overline}{{\mathcal{X}}}|, {\mathbb{Q}}_l)$. Thus ${\mathcal{X}}$ contributes a factor of $\frac{1}{(1-t)(1-q^{-1}t)(1-q^{-2}t)}$ to $Z_{H^*_{CR}}({\mathcal{X}}, t)$, and we have $$Z_{H^*_{CR}}({\mathcal{X}}, t) = \frac{1}{(1-t)(1-q^{-1}t)^2(1-q^{-2}t)}.$$ Appendix ======== Here we collect the proofs of several lemmas used above. [Lemma \[K\_smooth\]]{} ${\mathcal{K}}({\mathcal{X}}) = {\mathcal{K}}_{0,3}({\mathcal{X}}, 0)$ is a smooth [Deligne-Mumford]{} stack over ${\mathbb{F}}_q$. First, the moduli of stable maps to the coarse moduli scheme $X$ is given by ${\mathcal{K}}_{0,3}(X, 0) \cong {\overline}{{\mathcal{M}}}_{0,3} \times X \cong \text{Spec}({\mathbb{F}}_q) \times X$, and hence is a [Deligne-Mumford]{} stack over ${\mathbb{F}}_q$. Thus by [@AV1 Theorem 1.4.1], ${\mathcal{K}}({\mathcal{X}})$ is a [Deligne-Mumford]{} stack as well. To see that ${\mathcal{K}}({\mathcal{X}})$ is smooth, it is sufficient to prove smoothness of ${\mathcal{K}}({\overline}{{\mathcal{X}}})$. let ${\mathcal{K}}({\overline}{{\mathcal{X}}}) \xrightarrow{p} {\mathcal{M}}_{0,3}^{tw}$ be the forgetful functor to the ${\overline}{{\mathbb{F}}}_q$-stack of (not necessarily stable) genus zero twisted curves with three marked points (see [@Ol]). It follows from Remark 1.10 in \[ibid.\] that ${\mathcal{M}}_{0,3}^{tw}$ is smooth over ${\overline}{{\mathbb{F}}}_q$, thus it suffices to show that $p$ is smooth. By Proposition 17.10 and Corollary 17.9.2 in [@LMB], $p$ is smooth if and only if $\Omega^1_{{\mathcal{K}}({\overline}{{\mathcal{X}}})/ {\mathcal{M}}_{0,3}^{tw}}$ is locally free of finite rank. For this is suffices to show $R^0 \pi_* f^* T_{{\overline}{{\mathcal{X}}}}$ is locally free of finite rank where $\pi$ and $f$ are the universal curve and stable map respectively as in (\[universal\_diagram\]). Let $\text{Spec}({\overline}{{\mathbb{F}}}_q) \xrightarrow{p} {\mathcal{K}}({\overline}{{\mathcal{X}}})$ be a geometric point corresponding to the stable map $({\mathcal{C}}\xrightarrow{g} {\overline}{{\mathcal{X}}}, \{\Sigma_i\})$. Then to show $R^0 \pi_* f^* T_{{\overline}{{\mathcal{X}}}}$ is locally-free it suffices to show that $\text{dim} H^0({\mathcal{C}}, g^*T_{{\overline}{{\mathcal{X}}}})$ is locally constant as $p$ varies. For a tuple $\underline{b} = (b_1, b_2, b_3)$ of positive integers, let ${\mathcal{M}}_{0,3}^{tw}(\underline{b}) \subset {\mathcal{M}}_{0,3}^{tw}$ denote the locus of curves with isotropy group ${\mathbf{\mu}}_{b_i}$ at the $i^{th}$ marked point. The decomposition ${\mathcal{M}}_{0,3}^{tw} = \bigsqcup_{\underline{b}} {\mathcal{M}}_{0,3}^{tw}(\underline{b})$ [@Ol Section 5.4] induces a decomposition $${\mathcal{K}}({\overline}{{\mathcal{X}}}) = \bigsqcup_{\underline{b}} p^{-1}({\mathcal{M}}^{tw}_{0,3}(\underline{b})).$$ Note that since ${\mathcal{K}}({\overline}{{\mathcal{X}}})$ consists of degree zero maps, the image of $p$ is contained in the locus of smooth curves with stable coarse moduli space. Fix $\underline{b}$ for which $p^{-1}({\mathcal{M}}^{tw}_{0,3}(\underline{b}))$ is non-empty and let ${\mathcal{C}}$ be the *unique* domain curve of maps in $p^{-1}({\mathcal{M}}^{tw}_{0,3}(\underline{b}))$. Then we have a morphism of ${\overline}{{\mathbb{F}}}_q$-stacks $$p^{-1}({\mathcal{M}}^{tw}_{0,3}(\underline{b})) \xrightarrow{\Phi} \text{Pic}({\mathcal{C}})$$ sending $({\mathcal{C}}\times S \xrightarrow{f} {\overline}{{\mathcal{X}}}, \{\Sigma_i\})$ to $(f^*T_{{\overline}{{\mathcal{X}}}} {\rightarrow}{\mathcal{C}}\times S)$. ($\Phi$ is defined on morphisms in the obvious way). Since $\text{Pic}({\mathcal{C}})$ is discrete [@Ca1 Section 3.1], $\Phi$ is locally constant. Thus the functions $({\mathcal{C}}\times S \xrightarrow{f} {\overline}{{\mathcal{X}}}, \{\Sigma_i\}) \mapsto h^i({\mathcal{C}}, f^*T_{{\overline}{{\mathcal{X}}}})$ are locally constant as well. This proves the lemma. We note that over the complex numbers the smoothness of ${\mathcal{K}}({\mathcal{X}})$ is asserted in [@AGV2 Section 6.2]. [Lemma \[obs\_theory\]]{} 1. The natural map $$(R^{\bullet}\pi_* f^*T_{{\mathcal{X}}})^{\vee} \ \xrightarrow{\phi} \ \Omega^1_{{\mathcal{K}}({\mathcal{X}})/ {\mathcal{M}}_{0,3}^{tw}}$$ is a perfect relative obstruction theory with virtual dimension (denoted vdim) given by the locally constant function $$vdim = dim {\mathcal{X}}- \text{age}\circ e_1 - \text{age}\circ e_2 - \text{age}\circ e_3.$$ 2. $R^1\pi_* f^*T_{{\mathcal{X}}}$ is locally free (denote the locally constant rank by $r$), and the virtual fundamental class (denoted $[{\mathcal{K}}({\mathcal{X}})]^{vir}$) in $A_{vdim}({\mathcal{K}}({\mathcal{X}}))_{{\mathbb{Q}}}$ induced by $\phi$ is $$[{\mathcal{K}}({\mathcal{X}})]^{vir} = c_{r}(R^1\pi_*f^*T_{{\mathcal{X}}}).$$ Part (2) follows from [@BF Prop. 7.3] using Part(1), Lemma \[K\_smooth\] above, and [@Kresch Thm. 5.2.1]. (The fact that $R^1\pi_* f^*T_{{\mathcal{X}}}$ is locally free also follows from the proof of Lemma \[K\_smooth\] above). For Part (1), we proceed exactly as outlined in Section 4.5 of [@AGV2]. Finally for the virtual dimension, it suffices to compute $\chi({\mathcal{C}}, f^*T_{{\mathcal{X}}}) = \chi({\overline}{{\mathcal{C}}}, {\overline}{f}^*T_{{\mathcal{X}}})$ where $({\mathcal{C}}\xrightarrow{f} {\mathcal{X}}, \{\Sigma_i\})$ is a ${\mathbb{F}}_q$-point in ${\mathcal{K}}({\mathcal{X}})$, and $({\overline}{{\mathcal{C}}} \xrightarrow{{\overline}{f}} {\mathcal{X}}, \{\Sigma_i\})$ is the corresponding point in ${\overline}{{\mathcal{K}}({\mathcal{X}})}$. But then the formula follows from the Riemann-Roch theorem on curves [@AGV2 Thm. 7.2.1]. [Lemma \[K\_compat\]]{} 1. There is an equivalence $${\mathcal{K}}({\mathcal{X}}\times_{{\mathbb{F}}_q} {\overline}{{\mathbb{F}}}_q) \xrightarrow{\cong} {\mathcal{K}}({\mathcal{X}}) \times_{{\mathbb{F}}_q} {\overline}{{\mathbb{F}}}_q.$$ 2. Under the identification above, the following functors are 2-isomorphic: $${\mathcal{K}}(F_{{\overline}{{\mathcal{X}}}})\ \overset{\cong}{\rightrightarrows} \ F_{{\overline}{{\mathcal{K}}({\mathcal{X}})}}.$$ First we note that Part(1) appears in [@AV1 Prop. 5.2.1]. However, we include an elementary and explicit proof required for Part(2). Denote the structure map by $A: \text{Spec}({\overline}{{\mathbb{F}}_q}) {\rightarrow}\text{Spec}({\mathbb{F}}_q)$, and denote projection maps by $p$ (e.g. $p_{{\mathcal{X}}}: {\overline}{{\mathcal{X}}} {\rightarrow}{\mathcal{X}}$). For part (1), an object of ${\mathcal{K}}({\mathcal{X}}\times_{{\mathbb{F}}_q} {\overline}{{\mathbb{F}}}_q)$ over base ${\overline}{{\mathbb{F}}}_q$-scheme $S$ is given by a commutative diagram $$\label{d1} \begin{diagram} \sqcup_{i=1}^3 \Sigma_i & \rInto & {\mathcal{C}}& \rTo^{f_S} & {\overline}{{\mathcal{X}}} \\ & & \dTo^{\pi_S} & & \dTo \\ & & S & \rTo^{q} & \text{Spec}({\overline}{{\mathbb{F}}}_q) \end{diagram}$$ where we have suppressed the map on coarse moduli spaces. Define a functor $\phi$ by associating the pair consisting of the map $q$ together with the diagram given by composing $f_S$ (resp. $q$) in (\[d1\]) with $p_{{\mathcal{X}}}$ (resp. with $A$). It is clear that $p_{{\mathcal{X}}} \circ f_S$ is representable. To see that it is stable, note that the map on coarse moduli schemes induced by $p_{{\mathcal{X}}}$ is $p_{X}$, the projection onto $X$. Moreover, $p_{X}^* {\mathcal{O}}_{X}(1) \cong {\mathcal{O}}_{{\overline}{X}}(1)$ and so $f_S$ is stable if and only if $p_{{\mathcal{X}}} \circ f_S$ is stable. One can define $\phi$ on morphisms in the obvious way, and it is easy to see that indeed $\phi$ is a functor. For the reverse, an object of ${\mathcal{K}}({\mathcal{X}}) \times_{{\mathbb{F}}_q} {\overline}{{\mathbb{F}}}_q$ over base ${\overline}{{\mathbb{F}}}_q$-scheme $S$ is given by a commutative diagram $$\label{d2} \begin{diagram} \sqcup_{i=1}^3 \Sigma_i & \rInto & {\mathcal{C}}& \rTo^{f'_S} & {\mathcal{X}}\\ & & \dTo^{\pi_S} & & \dTo \\ & & S & \rTo^{q'} & \text{Spec}({\mathbb{F}}_q) \end{diagram}$$ where $q'$ factors through the structure map $q'': S {\rightarrow}\text{Spec}({\overline}{{\mathbb{F}}}_q)$ (i.e. $A \circ q'' = q'$). Then define $\psi$ (inverse of $\phi$) by sending (\[d2\]) to the diagram given by (\[d1\]) with $q$ replaced by $q''$ and $f_S$ replaced by the unique map induced by the pair $(q'' \circ \pi_S, f'_S)$. One can define $\phi$ on morphisms in the obvious way. It easy to check that indeed $\psi$ is a functor and that the pair $(\phi, \psi)$ gives the required equivalence. For the second part, we show that the following diagram is 2-commutative. [()]{} & \^[F\_[[()]{}]{}]{} & [()]{}\ \_ & & \_\ () & \^[(F\_)]{} & () Let $\eta$ denote the diagram (\[d1\]) in ${\mathcal{K}}({\overline}{{\mathcal{X}}})$ above. Since $F_{{\overline}{{\mathcal{K}}({\mathcal{X}})}} = 1_{{\mathcal{X}}} \times F_{\text{Spec}({\overline}{{\mathbb{F}}}_q)}$, we have that $\phi \circ F_{{\overline}{{\mathcal{K}}({\mathcal{X}})}} \circ \psi (\eta)$ is given by a diagram similar to (\[d2\]) with $q$ replaced by $F_{\text{Spec}({\overline}{{\mathbb{F}}}_q)} \circ q$ and $f_S$ replaced by the unique morphism induced by the pair $(F_{\text{Spec}({\overline}{{\mathbb{F}}}_q)} \circ q \circ \pi_S, p_{{\mathcal{X}}} \circ f_S)$. Denote this unique morphism by $\ast$. On the other hand, consider the commutative diagram \_S & \^[f\_S]{} & & \^[F\_]{} & & \^[p\_]{} &\ \^[\_S]{}& & & & & &\ S & \^[q]{} & (\_q) & \^[F\_[(\_q)]{}]{} & (\_q) & \^[A]{} & (\_q) with two right squares Cartesian and $F_{{\overline}{{\mathcal{X}}}} \circ p_{{\mathcal{X}}} = p_{{\mathcal{X}}}$. This diagram shows that $\ast$ is given by $F_{{\overline}{{\mathcal{X}}}} \circ f_S$. Thus we see that $\phi \circ F_{{\overline}{{\mathcal{K}}({\mathcal{X}})}} \circ \psi (\eta) = {\mathcal{K}}(F_{{\overline}{{\mathcal{X}}}})(\eta)$. One can further check that $\phi \circ F_{{\overline}{{\mathcal{K}}({\mathcal{X}})}} \circ \psi$ and ${\mathcal{K}}(F_{{\overline}{{\mathcal{X}}}})$ agree on morphism as well and this proves the lemma. [Lemma \[F\_on\_vir\_class\]]{} The equation $${\mathcal{K}}(F_{{\overline}{{\mathcal{X}}}})_* [{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{vir} = q^{-vdim}[{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{vir}$$ holds in $H_*({\mathcal{K}}({\overline}{{\mathcal{X}}}), {\mathbb{Q}}_l)$. Consider the Cartesian diagram () & \^[p\_[()]{}]{} & ()\ & &\ (\_q) & \^[A]{} & (\_q)\ where we identify ${\overline}{{\mathcal{K}}({\mathcal{X}})}$ with ${\mathcal{K}}({\overline}{{\mathcal{X}}})$ by Lemma \[K\_compat\]. The projection $p_{{\mathcal{K}}({\mathcal{X}})}$ induces a map on Chow groups $p_{{\mathcal{K}}({\mathcal{X}})}^*: A_*({\mathcal{K}}({\mathcal{X}}))_{{\mathbb{Q}}} {\rightarrow}A_*({\mathcal{K}}({\overline}{{\mathcal{X}}}))_{{\mathbb{Q}}}$. We first show that $p_{{\mathcal{K}}({\mathcal{X}})}^* [{\mathcal{K}}({\mathcal{X}})]^{\text{vir}} = [{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{\text{vir}}$. Let ${\mathcal{U}}_{{\mathcal{X}}}$ (resp. ${\mathcal{U}}_{{\overline}{{\mathcal{X}}}}$) denote the universal curves over ${\mathcal{K}}({\mathcal{X}})$ (resp. ${\mathcal{K}}({\overline}{{\mathcal{X}}})$). By [@AV1 Corollary 9.1.3], ${\mathcal{U}}_{{\mathcal{X}}}$ (resp. ${\mathcal{U}}_{{\overline}{{\mathcal{X}}}}$) is an open and closed substack of ${\mathcal{K}}_{0,4}({\mathcal{X}}, 0)$ (resp. ${\mathcal{K}}_{0,4}({\overline}{{\mathcal{X}}}, 0))$. Thus the proof of Lemma \[K\_compat\] identifies ${\mathcal{U}}_{{\overline}{{\mathcal{X}}}}$ with ${\mathcal{U}}_{{\mathcal{X}}} \times_{{\mathbb{F}}_q} {\overline}{{\mathbb{F}}}_q$, and we have the following diagram with Cartesian squares: & & & \^[p\_]{} & &\ & \^[f\_]{} & & & \^[f\_]{} &\ \_ & & \^[p\_[\_]{}]{} & \_ & &\ \^[\_]{} & & & \_[\_]{} & &\ () & & \^[p\_[()]{}]{} & (). & &\ Since $p_{{\mathcal{X}}}^*T_{{\mathcal{X}}} \cong T_{{\overline}{{\mathcal{X}}}}$ and $p_{{\mathcal{K}}({\mathcal{X}})}^* R^1(\pi_{{\mathcal{X}}})_* \cong R^1(\pi_{{\overline}{{\mathcal{X}}}})_* p_{{\mathcal{U}}_{{\mathcal{X}}}}^*$ a simple diagram chase gives $$\begin{aligned} {1} p_{{\mathcal{K}}({\mathcal{X}})}^* [{\mathcal{K}}({\mathcal{X}})]^{\text{vir}} &= c_{vdim}(R^1(\pi_{{\mathcal{X}}})_*f_{{\mathcal{X}}}^*T_{{\mathcal{X}}}) \notag \\ &= c_{vdim}(R^1(\pi_{{\overline}{{\mathcal{X}}}})_*f_{{\overline}{{\mathcal{X}}}}^*T_{{\overline}{{\mathcal{X}}}}) \notag \\ &= [{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{\text{vir}}. \notag\end{aligned}$$ Now fix a connected component ${\mathcal{K}}({\mathcal{X}})_0 \subset {\mathcal{K}}({\mathcal{X}})$ so that the virtual dimension (vdim) is constant. If the virtual class on ${\mathcal{K}}({\mathcal{X}})_0$ is given by $\sum n_i [V_i]$ where each $V_i$ is a substack of pure dimension vdim, then $[{\mathcal{K}}({\overline}{{\mathcal{X}}})]^{\text{vir}} = \sum n_i [{\overline}{V}_i]$. Thus we have ${\mathcal{K}}(F_{{\overline}{{\mathcal{X}}}})({\overline}{V}_i) = F_{{\mathcal{K}}({\overline}{{\mathcal{X}}})}({\overline}{V}_i) = {\overline}{V}_i$. Furthermore if $F_{{\mathcal{K}}({\overline}{{\mathcal{X}}})}^{\text{geo}}$ denotes the geometric Frobenius morphism, then we also have $F_{{\mathcal{K}}({\overline}{{\mathcal{X}}})}^{\text{geo}}({\overline}{V}_i) = {\overline}{V}_i$. Hence $(F_{{\mathcal{K}}({\overline}{{\mathcal{X}}})}^{\text{geo}})_*\ [{\overline}{V}_i] = q^{vdim} [{\overline}{V}_i]$ in the Chow group. But after passing to (co)homology the arithmetic and geometric Frobenius maps are inverses, and this proves the lemma.

--- abstract: | The discrete Fourier transform of the greatest common divisor $$\widehat{\pi}[a](m)=\sum_{k=1}^m \gcd{k}{m} \alpha_m^{ka},$$ with $\alpha_m$ a primitive $m$-th root of unity, is a multiplicative function that generalises both the gcd-sum function and Euler’s totient function. On the one hand it is the Dirichlet convolution of the identity with Ramanujan’s sum, $\widehat{\pi}[a]=\pi\ast c_\bullet(a)$, and on the other hand it can be written as a generalised convolution product, $\widehat{\pi}[a]=\pi \ast_a \phi$. We show that $\widehat{\pi}[a](m)$ counts the number of elements in the set of ordered pairs $(i,j)$ such that $i\cdot j \equiv a \mod m$. Furthermore we generalise a dozen known identities for the totient function, to identities which involve the discrete Fourier transform of the greatest common divisor, including its partial sums, and its Lambert series. author: - 'Peter H. van der Kamp' date: | Department of Mathematics and Statistics\ La Trobe University\ Victoria 3086, Australia\ title: | On the Fourier transform of\ the greatest common divisor --- Introduction ============ In [@WS] discrete Fourier transforms of functions of the greatest common divisor were studied, i.e. $$\widehat{h}[a](m) = \sum_{k=1}^m h(\gcd{k}{m})\alpha_m^{ka},$$ where $\alpha_m$ is a primitive $m$-th root of unity. The main result in that paper is the identity[^1] $\widehat{h}[a]=h\ast c_\bullet(a)$, where $\ast$ denoted Dirichlet convolution, i.e. $$\label{hffc} \widehat{h}[a](m) = \sum_{d\mid m} h(\frac{m}{d}) c_d(a),$$ and $$\label{Ram} c_m(a)=\sum_{\underset{\gcd{k}{m}=1}{k=1}}^m \alpha_m^{ka}$$ denotes Ramanujan’s sum. Ramanujan’s sum generalises both Euler’s totient function $\phi=c_\bullet(0)$ and the Möbius function $\mu=c_\bullet(1)$. Thus, identity (\[hffc\]) generalizes the formula $$\label{C} \sum_{k=1}^m h(\gcd{k}{m}) = (h \ast \phi)(m),$$ already known to Cesàro in 1885. The formula (\[hffc\]) shows that $\widehat{h}[a]$ is multiplicative if $h$ is multiplicative (because $c_\bullet(a)$ is multiplicative and Dirichlet convolution preserves multiplicativity). Here we will take $h=\pi$ to be the identity function (of the greatest common divisor) and study its Fourier transform. Obviously, as $\pi: \pi(n)=n$ is multiplicative, the function $\widehat{\pi}[a]$ is multiplicative, for all $a$. Two special cases are well-known. Taking $a=0$ we have $\widehat{\pi}[0]=\g$, where $$\label{g} \g(m)=\sum_{k=1}^m \gcd{k}{m}.$$ is known as Pillai’s arithmetical function or the gcd-sum function. Secondly, by taking $a=1$ in (\[hffc\]), we find that $\widehat{\pi}[1]=\pi \ast \mu$ equals $\phi$, by Möbius inversion of Euler’s identity $\phi \ast u = \pi$, where $u=\mu^{-1}$ is the unit function defined by $u(m)=1$. Let $\S am$ denote the set of ordered pairs $(i,j)$ such that $i\cdot j\equiv a \mod m$, the set of [*factorizations of $a$ modulo $m$*]{}. We claim that $\widehat{\pi}[a](m)$ counts its number of elements. Let us consider the mentioned special cases. - For given $i\in\{1,2,\ldots,m\}$ the congruence $i\cdot j \equiv 0 \mod m$ yields $$\frac{i}{\gcd im} j \equiv 0 \mod \frac{m}{\gcd im},$$ which has a unique solution modulo $m/\gcd im$, and so there are $\gcd im$ solutions modulo $m$. Hence, the total number of elements in $\S 0m$ is $\g(m)$. - The totient function $\phi(m)$ counts the number of invertible congruence classes modulo $m$. As for every invertible congruence class $i$ modulo $m$ there is a unique $j=i^{-1} \mod m$ such that $i\cdot j \equiv 1 \mod m$, it counts the number of elements in the set $\S 1m$. To prove the general case we employ a Kluyver-like formula for $\widehat{\pi}[a]$, that is, a formula similar to the formula for the Ramanujan sum function $$\label{klu} c_k(a)=\sum_{d\mid \gcd ak} d \mu(\frac{k}{d}).$$ attributed to Kluyver. Together the identities (\[hffc\]) and (\[klu\]) imply, cf. section \[grs\], $$\label{pham} \widehat{\pi}[a](m)=\sum_{d\mid \gcd am} d \phi(\frac{m}{d}),$$ and we will show, in the next section, that the number of factorizations of $a \mod m$ is given by the same sum. The right hand sides of (\[klu\]) and (\[pham\]) are particular instances of the so called generalized Ramanujan sums [@AA], and both formulas follow as consequence of a general formula for the Fourier coefficients of these generalised Ramanujan sums [@Tom; @Tom2]. In section \[grs\] we provide simple proofs for some of the nice properties of these sums. In particular we interpret the sums as a generalization of Dirichlet convolution. This interpretation lies at the heart of many of the generalised totient identities we establish in section \[gti\]. The number of factorizations of $a \mod m$ {#numset} ========================================== For given $i,m\in\N$, denote $g=\gcd im$. If the congruence $i\cdot j \equiv a \mod m$ has a solution $j$, then $g\mid a$ and $j\equiv i^{-1} a/g$ is unique mod $m/g$, so mod $m$ there are $g$ solutions. This yields $$\#\S am=\sum_{\underset{ \gcd im\mid a}{i=1}}^m \gcd im,$$ which can be written as $$\label{f} \#\S am=\sum_{d\mid a} \sum_{\underset{\gcd im=d}{i=1}}^m d$$ If $d\nmid m$ then the sum $$\sum_{\underset{\gcd im=d}{i=1}}^m 1$$ is empty. Now let $d\mid m$. The only integers $i$ which contribute to the sum are the multiples of $d$, $kd$, where $\gcd k{m/d}=1$. There are exactly $\phi(m/d)$ of them. Therefore the right hand sides of formulae (\[pham\]) and (\[f\]) agree, and hence $\#\S am=\widehat{\pi}[a](m)$. A historical remark, and generalised Ramanujan sums {#grs} =================================================== It is well known that Ramanujan was not the first who considered the sum $c_m(a)$. Kluyver proved his formula (\[klu\]) in 1906, twelve years before Ramanujan published the novel idea of expressing arithmetical functions in the form of a series $\sum_s a_s c_s(n)$ [@Ram]. It is not well known that Kluyver actually showed that $c_m(a)$ equals Von Sterneck’s function, introduced in [@VoS], i.e. $$\label{Hol} c_m(a)=\frac{\mu(\frac{m}{\gcd am})\phi(m)}{\phi(\frac{m}{\gcd am})}.$$ This relation is referred to in the literature as Hölder’s relation, cf. the remark on page 213 in [@AA]. However, Hölder published this relation thirty years after Kluyver [@Hol]. We refer to [@AA Theorem 2], or [@Tom Theorem 8.8] for a generalisation of (\[Hol\]). The so called generalized Ramanujan sums, $$\label{fag} f\ast_a g(m)=\sum_{d\mid\gcd am} f(d)g(\frac{m}{d}),$$ were introduced in [@AA]. The notation $\ast_a$ is new, the sums are denoted $S(a;m)$ in [@AA], $s_m(a)$ in [@Tom], and $S_{f,h}(a,m)$ in [@Tom2]. In the context of $r$-even functions [@LTPH] the sums are denoted $S_{f,g}(a)$, and considered as sequences of $m$-even functions, with argument $a$. We consider the sums as a sequence of functions with argument $m$, labeled by $a$. We call $f\ast_a g$ [**the ${\mathbf a}$-convolution of ${\mathbf f}$ and ${\mathbf g}$**]{}. The concept of $a$-convolution is a generalization of Dirichlet convolution as $f\ast_0 g=f\ast g$. As we will see below, the function $f\ast_a g$ is multiplicative (for all $a$) if $f$ and $g$ are, and the following inter-associative property holds, cf. [@Tom2 Theorem 4]. $$\label{assp} (f\ast_a g) \ast h = f\ast_a (g \ast h).$$ We also adopt the notation $f_a=\pi \ast_a f$, and call this [**the Kluyver, or ${\mathbf a}$-extension of ${\mathbf f}$**]{}. Thus, we have $f_0=\pi \ast f$, $f_1=f$, and formulas (\[klu\]) and (\[pham\]) become $c_m(a)=\mu_a(m)$, and $\widehat{\pi}[a]=\phi_a$, respectively. The identity function $I$, defined by $f\ast I=f$, is given by $I(k)=[k=1]$, where the Iverson bracket is, with $P$ a logical statement, $$[P]=\left\{\begin{array}{ll} 1 & \text{if } P, \\ 0 & \text{if } \text{not } P. \end{array} \right.$$ Let us consider the function $f\ast_a I$. It is $$f \ast_a I(k)=\sum_{d\mid\gcd ak} f(d)[d=k]=[k\mid a]f(k).$$ Since the function $k\rightarrow[k\mid a]$ is multiplicative, the function $f\ast_a I$ is multiplicative if $f$ is multiplicative. Also, we may write, cf. [@Tom2 eq. (9)], $$f\ast_a g(m)=\sum_{d\mid m} [d\mid a]f(d)g(\frac{m}{d}) = (f\ast_a I) \ast g (m),$$ which shows that $f\ast_a g$ is multiplicative if $f$ and $g$ are. Also, the inter-associativity property (\[assp\]) now easily follows from the associativity of the Dirichlet convolution, $$(f\ast_a g) \ast h = ((f\ast_a I) \ast g) \ast h = (f\ast_a I) \ast (g \ast h) = f\ast_a (g \ast h).$$ We note that the $a$-convolution product is neither associative, nor commutative. The inter-associativity and the commutativity of Dirichlet convolution imply that $$\begin{aligned} \label{ig} f_a \ast g = (f\ast g)_a = f \ast g_a.\end{aligned}$$ Formula (\[pham\]) states that the Fourier transform of the greatest common divisor is the Kluyver extension of the totient function. We provide a simple proof. [**Proof**]{} \[of (\[pham\])\] Employing (\[hffc\]), (\[klu\]) and (\[ig\]) we have $ \widehat{\pi}[a] = \pi \ast c_\bullet(a) = \pi \ast \mu_a = (\pi \ast \mu)_a = \phi_a. $ $\square$ The formula (\[pham\]) also follows as a special case of the following formula for the Fourier coefficients of $a$-convolutions, $$\label{ha} f\ast_a g(m)=\sum_{k=1}^m h_k(m) \alpha_m^{ka},\qquad h_k = g \ast_k \frac{f}{\pi},$$ given in [@AA; @Tom]. The formula (\[ha\]) combines with (\[hffc\]) and (\[klu\]) to yield a formula for functions of the greatest common divisor, $\bar{h}[k]: m\rightarrow h(\gcd km)$, namely $$\label{con} \bar{h}[k] = (h \ast \mu) \ast_k u.$$ [**Proof**]{} \[of (\[con\])\] The Fourier coefficients of $\widehat{h}[a](m)$ are $\bar{h}[k](m)$. But $\widehat{h}[a]=h \ast c_\bullet(a) = (\pi \ast_a \mu) \ast h = \pi \ast_a (\mu \ast h)$, and so, using (\[ha\]), the Fourier coefficients are also given by $(h \ast \mu) \ast_k u(m)$. $\square$ For a Dirichlet convolution with a Fourier transform of a function of the greatest common divisor we have $$\label{mt1} f\ast\widehat{g}[a] = \widehat{f\ast g}[a].$$ [**Proof**]{} \[of (\[mt1\])\] $f\ast\widehat{g}[a]=f\ast(g\ast\mu_a)=(f\ast g)\ast \mu_a= \widehat{f\ast g}[a]$ $\square$ Similarly, for an $a$-convolution with a Fourier transform of a function of the greatest common divisor, $$\label{mt2} f\ast_a \widehat{g}[b] = \widehat{f\ast_a g}[b].$$ [**Proof**]{} \[of (\[mt2\])\] $f\ast_a\widehat{g}[b]=f\ast_a(g\ast\mu_b)=(f\ast_a g)\ast \mu_b= \widehat{f\ast_a g}[b]$ $\square$ Generalised totient identities {#gti} ============================== The totient function is an important function in number theory, and related fields of mathematics. It is extensively studied, connected to many other notions and functions, and there exist numerous generalisation and extensions, cf. the chapter “The many facets of Euler’s totient” in [@SC]. The Kluyver extension of the totient function is a very natural extension, and it is most surprising it has not been studied before. In this section we generalise a dozen known identities for the totient function $\phi$, to identities which involve its Kluyver extension $\phi_a$, a.k.a. the discrete Fourier transform of the greatest common divisor. This includes a generalisation of Euler’s identity, the partial sums of $\phi_a$, and its Lambert series. The value of $\phi_a$ at powers of primes ----------------------------------------- We start by providing a formula for the value of $\phi_a$ at powers of primes. This depends only on the multiplicity of the prime in $a$. The formulae, with $p$ prime, $$\g(p^k) = (k + 1)p^k - kp^{k-1}, \qquad \phi(p^k)=p^k-p^{k-1},$$ of which the first one is Theorem 2.2 in [@Bro], generalise to $$\label{pp} \phi_a(p^k)=\left\{\begin{array}{ll} (p^k-p^{k-1})(l+1) & \text{if } l<k, \\ (k+1)p^{k}-kp^{k-1} & \text{if } l\geq k, \end{array} \right.$$ where $l$ is the largest integer, or infinity, such that $p^l\mid a$. [**Proof**]{} \[of (\[pp\])\] We have $$\begin{aligned} \phi_a(p^k)&=\sum_{d\mid \gcd {p^l}{p^k}} d \phi(\frac{p^k}{d})\notag\\ &=\sum_{r=0}^{\min(l,k)} p^r \phi(p^{k-r})\notag \\ &=\left\{\begin{array}{ll} \sum_{r=0}^l p^k-p^{k-1} & \text{if } l<k, \\ (\sum_{r=0}^{k-1} p^k-p^{k-1}) + p^k & \text{if } l\geq k, \end{array} \right.\end{aligned}$$ which equals (\[pp\]). Partial sums of $\phi_a/\pi$ ---------------------------- To generalise the totient identity $$\label{fti} \sum_{k=1}^n \frac{\phi(k)}{k} = \sum_{k=1}^n \frac{\mu(k)}{k} \lfloor \frac{n}{k} \rfloor.$$ to an identity for $\phi_a$ we first establish $$\label{ook} \sum_{k=1}^n \frac{f_0(k)}{k} = \sum_{k=1}^n \frac{f(k)}{k} \lfloor \frac{n}{k} \rfloor.$$ [**Proof**]{} \[of (\[ook\])\] Since there are $\lfloor n/d \rfloor$ multiples of $d$ in the range $[1,n]$ it follows that $$\sum_{k=1}^n \frac{f\ast \pi(k)}{k} = \sum_{k=1}^n \sum_{d\mid k} \frac{f(d)}{d} = \sum_{d=1}^n \frac{f(d)}{d} \lfloor \frac{n}{d} \rfloor.$$$\square$ As a corollary we obtain $$\label{cook} \sum_{k=1}^n \frac{f\ast_a g_0(k)}{k} = \sum_{k=1}^n \frac{f\ast_a g(k)}{k} \lfloor \frac{n}{k} \rfloor.$$ [**Proof**]{} \[of (\[cook\])\] Employing (\[assp\]) we find $$\sum_{k=1}^n \frac{f\ast_a (g \ast \pi)(k)}{k} = \sum_{k=1}^n \frac{(f\ast_a g) \ast \pi(k)}{k} = \sum_{k=1}^n \frac{f\ast_a g(k)}{k} \lfloor \frac{n}{k} \rfloor.$$$\square$ Now taking $f=\pi$ and $g=\mu$ in (\[cook\]) we find $$\label{g1} \sum_{k=1}^n \frac{\phi_a(k)}{k} = \sum_{k=1} ^n \frac{c_k(a)}{k} \lfloor \frac{n}{k} \rfloor.$$ Partial sums of $\g_a/\pi$ expressed in terms of $\phi_a$ --------------------------------------------------------- Taking $f=\pi$ and $g=\phi$ in (\[cook\]) we find $$\label{g2} \sum_{k=1}^n \frac{\g_a(k)}{k} = \sum_{k=1} ^n \frac{\phi_a(k)}{k} \lfloor \frac{n}{k} \rfloor.$$ Note that by taking either $a=0$ in (\[g1\]), or $a=1$ in (\[g2\]), we find an identity involving the totient function and the gcd-sum function, $$\label{g3} \sum_{k=1}^n \frac{\g(k)}{k} = \sum_{k=1} ^n \frac{\phi(k)}{k} \lfloor \frac{n}{k} \rfloor.$$ Partial sums of $\phi_a$ ------------------------ To generalise the totient identity, with $n>0$, $$\label{iop} \sum_{k=1}^n \phi(k) = \frac{1}{2}\left(1+\sum_{k=1}^n \mu(k) \lfloor \frac{n}{k} \rfloor^2\right),$$ we first establish $$\label{fid} \sum_{k=1}^n f_0 (k) = \frac{1}{2}\left( \sum_{k=1}^n f(k) \lfloor \frac{n}{k} \rfloor^2 + \sum_{k=1}^n f\ast u (k)\right).$$ [**Proof**]{} \[of (\[fid\])\] We have, by changing variable $k=dl$, $$\begin{aligned} \sum_{k=1}^n (2 f\ast \pi - f\ast u)(k) &= \sum_{k=1}^n \sum_{d\mid k} f(d) (\frac{2k}{d} -1) \\ &= \sum_{d=1}^n \sum_{l=1}^{\lfloor n/d \rfloor} f(d) (2l -1) \\ &= \sum_{d=1}^n f(d) \lfloor \frac{n}{d} \rfloor^2.\end{aligned}$$$\square$ Note that this gives a nice proof of (\[iop\]), taking $f=\mu$, as $\sum_{k=1}^n I(k)=[k>0]$. When $f=\mu_a$, then (\[ig\]) implies $f\ast \pi=\phi_a$, and $f\ast u= I_a$, and therefore as a special case of (\[fid\]) we obtain $$\label{g4} \sum_{k=1}^n \phi_a(k) = \frac{1}{2}\left( \sum_{k\mid a} k[k\leq n] +\sum_{k=1}^n c_k(a) \lfloor \frac{n}{k} \rfloor^2\right).$$ We remark that when $n\geq a$ we have $ \sum_{k\mid a} k[k\leq n] = \sigma(a), $ where $\sigma=\pi\ast u$ is the sum of divisors function. Generalisation of Euler’s identity ---------------------------------- Euler’s identity, $\phi\ast u=\pi$, generalises to $$\label{jkl} \sum_{d\mid m} \phi_a(d) = \tau(\gcd am)m,$$ where $\tau=u\ast u$ is the number of divisors function. [**Proof**]{} \[of (\[jkl\])\] We have $\phi_a\ast u=(\phi\ast u)_a=\pi_a$ where $$\label{ida} \pi_a(m) = \sum_{d\mid \gcd am} d \frac{m}{d} = m \tau(\gcd am).$$ $\square$ Partial sums of $\g_a$ expressed in terms of $\phi_a$ (and $\tau$) ------------------------------------------------------------------ If $f=\phi_a$, then $f\ast \pi=\g_a$, and (\[fid\]) becomes, using (\[jkl\]), $$\label{g5} \sum_{k=1}^n \g_a(k) = \frac{1}{2}\left( \sum_{k=1}^n \tau(\gcd ak)k +\sum_{k=1}^n \phi_a(k) \lfloor \frac{n}{k} \rfloor^2\right).$$ Four identities of Césaro ------------------------- According to Dickson [@Dick] the following three identities were obtained by Césaro: $$\begin{aligned} \sum_{d\mid n} d \phi(\frac{n}{d})&=\g(n), \label{c1}\\ \sum_{d\mid n} \frac{d}{n} \phi(d)&=\sum_{j=1}^n \frac{1}{\gcd jn}, \label{c2}\\ \sum_{d\mid n} \phi(d)\phi(\frac{n}{d})&=\sum_{j=1}^n\phi(\gcd jn). \label{c3}\\end{aligned}$$ Identity (\[c1\]), which is Theorem 2.3 in [@Bro], is obtained by taking $a=0$ in (\[pham\]), or $h=\pi$ in (\[C\]). It generalises to $$\label{p1} \sum_{d\mid n} d\phi_a(\frac{n}{d}) = \g_a(n).$$ [**Proof**]{} \[of (\[p1\])\] By taking $f=\phi$ and $g=\pi$ in (\[ig\]). $\square$ Identity (\[c2\]) is obtained by taking $h=1/\pi$ in (\[C\]) and generalises to $$\label{p2} \sum_{d\mid n} \frac{d}{n}\phi_a(d) = \sum_{j=1}^n \sum_{d\mid\gcd an} \frac{1}{\gcd j{\frac{n}{d}}},$$ [**Proof**]{} \[of (\[p2\])\] By taking $f=\phi$ and $g=1/\pi$ in (\[ig\]). $\square$ Identity (\[c3\]) is also a special case of (\[C\]), with $h=\phi$. It generalises to $$\label{p3} \sum_{d\mid n} \phi_a(d)\phi_b(\frac{n}{d})=\sum_{j=1}^n \sum_{d\mid\gcd an} \phi_b(\gcd j{\frac{n}{d}}).$$ [**Proof**]{} \[of (\[p3\])\] We have $$(\pi \ast_a \phi) \ast (\pi \ast_b \phi) = \pi \ast_a (\pi \ast_b (\phi\ast \phi)) = \pi \ast_a (\pi \ast_b \widehat{\phi}[0]) = \pi \ast_a \widehat{\phi_b}[0],$$ and evaluation at $m$ yields $$\sum_{d\mid\gcd am} d \sum_{j=1}^{m/d} \phi_b(\gcd j{\frac{m}{d}}) = \sum_{d\mid\gcd am} \sum_{j=1}^{m} \phi_b(\gcd j{\frac{m}{d}}).$$ $\square$ The more general identity (\[C\]) generalises to $$\label{gC} \sum_{k=1}^m h_a(\gcd km) = h\ast \phi_a (m).$$ Three identities of Liouville ----------------------------- Dickson [@Dick p.285-286] states, amongst many others identities that were presented by Liouville in the series [@Liou], the following $$\begin{aligned} \sum_{d\mid m} \phi(d) \tau(\frac{m}{d}) = \sigma(m), \label{l1}\\ \sum_{d\mid m} \phi(d) \sigma[n+1](\frac{m}{d}) = m\sigma[n](m), \label{l2}\\ \sum_{d\mid m} \phi(d) \tau(\frac{m^2}{d^2}) = \sum_{d\mid m} d\theta(\frac{m}{d}), \label{l3}\end{aligned}$$ where $\sigma[n]=\pi[n]\ast u$, $\pi[n](m)=m^n$, and $\theta(m)$ is the number of decompositions of $m$ into two relatively prime factors. All three are of the form $\phi\ast f = g$ and therefore they gain significance due to (\[C\]), thought Liouville might not have been aware of this. For example, (\[C\]) and (\[l1\]) combine to yield $$\sum_{k=1}^m \tau(\gcd km) = \sigma(m).$$ The three identities are easily proven by substituting $\tau=u\ast u$, $\sigma[n]=\pi[n]\ast u$, $\tau\circ\pi[2]=\theta\ast u$, $\phi=\mu\ast \pi$, and using $\mu\ast u=I$. They generalise to $$\begin{aligned} \sum_{d\mid m} \phi_a(d) \tau(\frac{m}{d}) = \sigma_a(m), \label{lg1}\\ \sum_{d\mid m} \phi_a(d) \sigma[n+1](\frac{m}{d}) = m u \ast_a \sigma[n](m), \label{lg2}\\ \sum_{d\mid m} \phi_a(d) \tau(\frac{m^2}{d^2}) = \sum_{d\mid m} d\tau(\gcd ad)\theta(\frac{m}{d}). \label{lg3}\end{aligned}$$ These generalisation are proven using the same substitutions, together with (\[ig\]), or for the latter identity, (\[assp\]) and (\[ida\]). One identity of Dirichlet ------------------------- Dickson [@Dick] writes that Dirichlet [@Dir], by taking partial sums on both sides of Euler’s identity, obtained $$\sum_{k=1}^n \lfloor \frac{n}{k} \rfloor \phi(k) = {n+1\choose 2}.$$ By taking partial sums on both sides of equation (\[jkl\]) we obtain $$\label{tyu} \sum_{k=1}^n \lfloor \frac{n}{k} \rfloor \phi_a(k) = \sum_{d\mid a} d {\lfloor \frac{n}{d} \rfloor +1\choose 2} .$$ [**Proof**]{} \[of (\[tyu\])\] Summing the left hand side of (\[jkl\]) over $m$ yields $$\sum_{m=1}^n \sum_{d \mid m} \phi_a(d) = \sum_{d=1}^n \lfloor \frac{n}{d} \rfloor \phi_a(d)$$ and summing the right hand side of (\[jkl\]) over $m$ yields $$\sum_{m=1}^n \tau(\gcd am)m = \sum_{m=1}^n \sum_{d\mid\gcd am} m = \sum_{d\mid a} \sum_{k=1}^{\lfloor n/d\rfloor} dk = \sum_{d\mid a} d\lfloor \frac{n}{d}\rfloor\left(\lfloor \frac{n}{d}\rfloor+1\right)/2.$$ $\square$ The Lambert series of $\phi_a$ ------------------------------ As shown by Liouville [@Liou], cf. [@Dick p.120], the Lambert series of the totient function is given by $$\sum_{m=1}^\infty \phi(m) \frac{x^m}{1-x^m} = \frac{x}{(1-x)^2}.$$ The Lambert series for $\phi_a$ is given by $$\label{lam} \sum_{m=1}^\infty \phi_a(m) \frac{x^m}{1-x^m} = p[a](x)\frac{x}{(1-x^a)^2},$$ where the coefficients of $p[a](x)=\sum_{k=1}^{2a} c[a](k)x^{k-1}$ are given by $c[a]=\pi_a \circ t[a]$, and $t[a]$ is the piece-wise linear function $t[a](n)=a-|n-a|$. The polynomials $p[a]$ seem to be irreducible over ${\mathbb Z}$ and their zeros are in some sense close to the $a$-th roots of unity, see Figures \[37\] and \[35\]. {width="6cm"} [**Proof**]{} \[of (\[lam\])\] Cesàro proved [@Dick] $$\sum_{n=1}^\infty f(n)\frac{x^n}{1-x^n} = \sum_{n=1}^\infty x^n \sum_{d\mid n} f(d),$$ cf. exercise 31 to chapter 2 in [@GF]. By substituting (\[jkl\]) in this formula we find $$\sum_{n=1}^\infty \phi_a(n)\frac{x^n}{1-x^n} = \sum_{n=1}^\infty x^n \tau(\gcd an) n.$$ Multiplying the right hand side by $(1-2x^a+x^{2a})$ yields $$\begin{aligned} &( \sum_{n=1}^\infty x^n \tau(\gcd an) n) -2 ( \sum_{n=a+1}^\infty x^n \tau(\gcd an) (n-a)) \\ &+( \sum_{n=1+2a}^\infty x^n \tau(\gcd an) (n-2a)) = \sum_{n=1}^\infty c[a](n) x^n,\end{aligned}$$ where $$c[a](n)=\left\{ \begin{array}{ll} \tau(\gcd an)n & 0<n\leq a, \\ \tau(\gcd an)(n-2(n-a))=\tau(\gcd an)(2a-n) & a<n\leq 2a, \\ \tau(\gcd an)(n-2(n-a)+n-2a)=0 & n>2a. \end{array} \right.$$ Rewriting, using (\[ida\]), the fact that $\gcd a{a+k}=\gcd a{a-k}$, and dividing by $x$, yields the result. $\square$ {width="6cm"} A series related to the Lambert series of $\phi_a$ -------------------------------------------------- Liouville [@Liou] also showed $$\sum_{m=1}^\infty \phi(m) \frac{x^m}{1+x^m} = (1+x^2)\frac{x}{(1-x^2)^2}.$$ We show that $$\label{mola} \sum_{m=1}^\infty \phi_a(m) \frac{x^m}{1+x^m} = q[a](x)\frac{x}{(1-x^{2a})^2},$$ where $q[a](x)=\sum_{k=1}^{4a} b[a](k) x^{k-1}$, with $$b[a]=h[a]\circ t[2a],\quad h[a](k)=\pi_a(k)-2[2\mid k]\pi_a(k/2).$$ At the end of this section we show that $1+x^2$ divides $q[a](x)$ if $a$ is odd. [**Proof**]{} \[of (\[mola\])\] As the left hand side of (\[mola\]) is obtained from the left hand side of (\[lam\]) by subtracting twice the same series with $x$ replaced by $x^2$, the same is true for the right hand side. Thus it follows that $ q[a](x)=p[a](x)(1+x^a)^2-2p[a](x^2)x, $ and hence, that $$b[a](k)=\left\{ \begin{array}{ll} \pi_a(k)-2[2\mid k]\pi_a(k/2)& k\leq a, \\ \pi_a(2a-k)+2\pi_a(k-a)-2[2\mid k]\pi_a(k/2) & a<k\leq 2a, \\ 2\pi_a(3a-k)+\pi_a(k-2a)-2[2\mid k]\pi_a(2a-k/2) & 2a<k\leq 3a, \\ \pi_a(4a-k)-2[2\mid k]\pi_a(2a-k/2) & 3a<k\leq 4a. \end{array} \right.$$ The result follows due to the identities $$\pi_a(2a-k)+2\pi_a(k-a)=\pi_a(k),\ \ 2\pi_a(3a-k)+\pi_a(k-2a)=\pi_a(4a-k),$$ which are easily verified using (\[ida\]). $\square$ We can express the functions $b[a]$ and $h[a]$ in terms of an interesting fractal function. Let a function $\kappa$ of two variables be defined recursively by $$\label{deka} \kappa[a](n)=\left\{ \begin{array}{ll} 0 & 2\mid n , 2\nmid a, \text{ or } n=0, \\ \kappa[a/2](n/2) & 2\mid n , 2\mid a, \\ \tau(\gcd an) & 2\nmid n. \end{array} \right.$$ The following properties are easily verified using the definition. We have $$\label{pr1} \kappa[a](2a+n)=\kappa[a](2a-n),$$ and, with $\gcd ab=1$, $$\label{pr2} \kappa[an](bn)=\left\{ \begin{array}{ll} 0 & 2 \mid b, \\ \alpha(n) & 2\nmid b, \end{array} \right.$$ where $\alpha$ denotes the number of odd divisors function, i.e. for all $k$ and odd $m$ $$\label{alp} \alpha(2^k m) = \tau(m).$$ Property (\[pr2\]) is a quite remarkable fractal property; from the origin in every direction we see either the zero sequence, or $\alpha$, at different scales. We claim that $$\label{e1} h[a]=\kappa[a]\pi$$ follows from (\[deka\]), (\[alp\]), and (\[ida\]). From (\[e1\]) and (\[pr1\]) we obtain $$\label{e2} b[a]=\kappa[a]t[2a].$$ We now prove that $1+x^2$ divides $q[a]$ when $a$ is odd. Noting that $b[a](2a+k)=b[a](2a-k)$ and, when $2\nmid a$, $b[a](2k)=0$, we therefore have $$\begin{aligned} q[a](x)&=\sum_{n=1}^{2a} b[a](2n-1)x^{2n-2} \\ &=\sum_{m=1}^{a} b[a](2a-2m+1)x^{2a-2m} + b[a](2a+2m-1)x^{2a+2m-2} \\ &=\sum_{m=1}^{a} b[a](2a-2m+1)x^{2a-2m}(1+x^{4m-2}),\end{aligned}$$ which vanishes at the points where $x^2=-1$. $\square$ Apart from the factor $1+x^2$ when $a$ is odd, the polynomial $q[a]$ seems to be irreducible over ${\mathbb Z}$ and its zeros are in some sense close to the $2a^{\text{th}}$ roots of $-1$ or, to the $(a+1)^{\text{st}}$ roots of unity, see Figure \[19\]. {width="6cm"} A perfect square ---------------- Our last identity generalises the faint fact that $\phi(1)=1$. We have $$\label{nsq} \sum_{a=1}^n \phi_a(n) = n^2.$$ [**Proof**]{} \[of (\[nsq\])\] For any lattice point $(i,j)$ in the square $[1,n]\times[1,n]$ the product $i\cdot j \mod n$ is congruent to some $a$ in the range $[1,n]$. $\square$ [**Acknowledgment**]{} This research has been funded by the Australian Research Council through the Centre of Excellence for Mathematics and Statistics of Complex Systems. [10]{} D.R. Anderson and T.M. Apostol, The evaluation of Ramanujan’s sum and generalizations, Duke Math. J. 20 (1953) 211-216. T.M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. T.M. Apostol, Arithmetical properties of generalized Ramanujan sums, Pacific J. Math. 41 (1972) 281-293. K.A. Broughan (2001), “The gcd-sum function”, Journal of Integer Sequences 4, Art. 01.2.2. L.E. Dickson, History of the theory of numbers, Carnegie Inst., Washington, D.C., 1919; reprinted by Chelsea, New York, 1952. G.L. Dirichlet, Über die Bestimmung der mittleren Werte in der Zahlentheorie, Abh. Akad. Wiss. Berlin, 1849; 78-8. Also in Werke, vol. 2, 1897, 60-64. O. Hölder, Zur Theorie der Kreisteilungsgleichung $K_m(x)=0$, Prace Matematyczno Fizyczne 43 (1936) 13-23. J.C. Kluyver, Some formulae concerning the integers less than $n$ and prime to $n$, in: KNAW, Proceedings 9 I, Amsterdam, (1906) 408-414. J. Liouville, Sur quelques séries et produits infinis, J. Math. Pures Appl. 2 (1857) 433-440. L. Tóth and P. Haukkanen, The discrete Fourier transform of r-even functions, Acta Univ. Sapientiae Math. 3 (2011) 5-25. S. Ramanujan, On Certain Trigonometric Sums and their Applications in the Theory of Numbers, Transactions of the Cambridge Philosophical Society 22 (1918) 259Ð276. Also in: Collected papers of Srinivasa Ramanujan, Ed. G.H. Hardy et al, Chelsea Publ. Comp., New York (1962) 179-199. J. Sandor and B. Crstici, Handbook of Number Theory II, (2004) Kluwer Acad. Publ., Dordrecht. R. Daublebsky Von Sterneck, Ein Analogon zur additiven Zahlentheorie, Sitzber, Akad. Wiss. Wien, Math. Naturw. Klasse, vol. I ll (Abt. IIa) (1902), 1567-1601. H. S. Wilf, Generatingfunctionology, Academic Press, 2nd edition, 1994. W. Schramm, The Fourier transform of functions of the greatest common divisor, Integers 8 (2008) A50. [^1]: Similar results in the context of $r$-even function were obtained earlier, see [@LTPH] for details.

--- abstract: | Current recommender systems exploit user and item similarities by collaborative filtering. Some advanced methods also consider the temporal evolution of item ratings as a global background process. However, all prior methods disregard the *individual evolution* of a user’s [*experience*]{} level and how this is expressed in the user’s [*writing*]{} in a review community. In this paper, we model the [*joint evolution*]{} of [*user experience*]{}, interest in specific [*item facets*]{}, [*writing style*]{}, and [*rating behavior*]{}. This way we can generate individual recommendations that take into account the user’s maturity level (e.g., recommending art movies rather than blockbusters for a cinematography expert). As only item ratings and review texts are observables, we capture the user’s experience and interests in a [*latent model*]{} learned from her reviews, vocabulary and writing style. We develop a generative HMM-LDA model to trace user evolution, where the Hidden Markov Model (HMM) traces her latent experience progressing over time — with solely user reviews and ratings as observables over [*time*]{}. The facets of a user’s interest are drawn from a Latent Dirichlet Allocation (LDA) model derived from her reviews, as a function of her (again latent) experience level. In experiments with five real-world datasets, we show that our model improves the rating prediction over state-of-the-art baselines, by a substantial margin. We also show, in a use-case study, that our model performs well in the assessment of user experience levels. author: - bibliography: - 'kdd14.bib' title: Item Recommendation with Evolving User Preferences and Experience ---

--- author: - 'Erich Poppitz and M. Erfan Shalchian T.' title: 'String tensions in deformed Yang-Mills theory' --- Introduction ============= Systematic ways to study the long-distance behaviour of nonabelian gauge theories, where nonperturbative phenomena set in—confinement, the generation of mass gap, and the breaking of chiral symmetries—are hard to come by. Up to date, there are only a few examples in continuum quantum field theory where theoretically-controlled analytic methods allow one to make progress. Many of those examples, such as Seiberg-Witten theory, require various amounts of supersymmetry and utilize its power. In the past 10 years, a new direction of research into nonperturbative dynamics, applicable to a wider class of gauge theories, not necessarily supersymmetric, has emerged [@Unsal:2007jx; @01]: the study of gauge theories compactified[^1] on $\R^{1,2} \times \S^1$. The control parameter is the size of the $\S^1$-circle L. When L is taken such that NL$\Lambda \ll 1$, where N is the number of colours of an SU(N) gauge theory and $\Lambda$ its dynamical scale, it allows—as we shall review here for the theory we study—for semiclassical weak-coupling calculability. It has led to new insight into a variety of nonperturbative phenomena and has spawned new areas of research. A comprehensive list of references is, at this point, too long to include here and we recommend the recent review article [@Dunne:2016nmc] instead. This paper studies confining strings in deformed Yang-Mills theory (dYM). dYM is a deformation of pure Yang-Mills theory, whose nonperturbative dynamics is calculable at small L. It is also believed that the dynamics is continuously connected to the large-L limit of $\R^4$, in particular that the theory exhibits confinement and has a nonzero[^2] mass gap for every size of $\S^1$. The confining mechanism in dYM is a generalization of the three dimensional Polyakov mechanism of confinement [@12], but owing to the locally four-dimensional nature of the theory many of its properties are quite distinct. As we further discuss, many features of dYM on $\R^3 \times \S^1$ can be traced back to the unbroken global center symmetry. {width="\textwidth"} [\[fig:0\]]{} The properties we set out to study here are the $N$-ality dependence of the string tensions and their behaviour in the large-N limit. Renewed motivation to study the large-N limit of dYM arose from a recent intriguing observation [@Cherman:2016jtu]: in the double-scaling limit L $\rightarrow 0$, N $\rightarrow \infty$, with fixed LN$\Lambda$, the four-dimensional theory on $\R^3 \times \S^1_{L \rightarrow 0}$ dynamically generates a latticized dimension whose size grows with N. This phenomenon has superficial similarities to T-duality in string theory and is not usually expected in quantum field theory. Originally, the emergence of a discretized dimension and its properties were studied in a $\R^3 \times \S^1$ compactification and double scaling limit of ${\cal{N}}=1$ super-Yang-Mills (sYM) theory. We show here that, as already observed in sYM, in dYM string tensions also stay finite in the large-N limit while the mass gap vanishes. Most of this rather long paper is devoted to a review of dYM and to a detailed explanation of the various methods we have developed; a guide to the paper is at the end of this Section. The expert reader interested in the physics and not in the technical details should proceed to our “Summary of results” Section \[summary\], and to the more extended discussion in Section \[sec:5\]. Summary of results {#summary} ------------------ Here we summarize our main results, concerning both the confining string properties and the technical tools developed for their study: 1. [*k-string tension ratios:*]{} In the regime of parameters studied in this work, in particular NL$\Lambda \ll 1$, the asymptotic string tensions in dYM depend only on the $N$-ality of the representation. We argue in Section \[sec:5\] that the lowest tension stable strings between sources of $N$-ality k are sourced by quarks with charges in the highest weight of the k-index antisymmetric representation, see (\[eq:54\]).[^3] Their tensions are hence referred to as the “k-string tensions.” Denoting by T$_{\text k}$ the k-string tension, on Fig. \[fig:0\] we show the ratio T$_{\text k}$/T$_1$ for SU(10), the largest group we studied numerically. The string tension ratio in dYM is compared to other known and much studied scaling laws, such as the Sine law and the Casimir law. It is clear from the figure that k-string tension ratios in dYM are different and do, instead, come closest to a less-known scaling, found long ago in the MIT Bag Model of the Yang-Mills vacuum: the “Square root of Casimir” scaling [@13]. In Section \[bagmodelsection\], we argue that the relation between the two is $$\label{dymscaling} \left(T_k \over T_1\right)_{\text{dYM}} \le \;\sqrt{ k (N-k) \over N}~,$$ where the r.h.s. is the square root of the ratio of quadratic Casimirs of the k-index antisymmetric representation and the fundamental representation. The reason behind the similarity is that the model assumptions of the MIT Bag, that inside the bag the QCD chromoelectric fields can be treated classically and that the vacuum abhors chromoelectric flux, are realized almost verbatim—albeit for the Cartan components only—by the calculable confinement in dYM. 2. [*Large-N limit and $1\over N$ corrections to string tensions:*]{} As already mentioned, string tensions stay finite at large N and fixed LN$\Lambda \ll 1$, as we show using various tools in Section \[sec:largeN\]. Further, as can be inferred qualitatively from Figure \[fig:0\], and quantitatively from the analysis of Section \[sec:largeN\], k-strings in dYM are not free at large N. We show that $$\begin{aligned} \label{dymscaling2} {\text{T}_2 \over \text{T}_1} &=& 1.347 \pm 0.001 + (-2.7 \pm 0.2) ({1 \over N})^2 + ...,\nonumber \\ {\text{T}_3 \over \text{T}_1} &=& 1.570 \pm 0.001 + (-7.5 \pm 0.2) ({1 \over N})^2 + ... ,\end{aligned}$$ instead of approaching the free-string values T$_\text{k}$ = kT$_1$. The large-N limit leading to the above behaviour is taken [*after*]{} the large-RT limit (RT is the Wilson loop area). As the discussion there shows, assuming large-N factorization does not always imply that k-strings are free and the way the large-RT and large-N limits are taken has to be treated with care, as we discuss in detail in Section \[sec:largeN\].[^4] 3. [*Comparing abelian confinements:*]{} We compare the properties of confining strings in dYM and in Seiberg-Witten theory [@Seiberg:1994rs], another four dimensional theory with calculable abelian confinement. We argue that the unbroken $\Z_N$ center symmetry in dYM has dramatic implications for the meson and baryon spectra. In particular there is a “baryon vertex” in dYM, leading to “Y”-type baryons, while only linear baryons exist in Seiberg-Witten theory [@15]. Thus, owing to the unbroken center symmetry, in many ways confinement in dYM is closer to the one in “real world” YM theory.[^5] For a discussion of these issues, see Section \[sec:compare\] and Figs. \[fig:dymstring\] and \[fig:SWstrings\]. 4. [*“Perturbative evaluation” of string tensions:*]{} A technical tool to calculate string tensions analytically is developed in Section \[sec:4\]. We call it “perturbative,” as it utilizes a resummed all-order expansion and, at every step, requires the use of only Gaussian integrals. This method serves as a check on the computationally very intensive numerical methods that were employed in the numerical study. It also allows the large-N limit to be taken analytically, subject to the limitations discussed above, and permits us to discuss the subtleties regarding the order of limits that lead to (\[dymscaling2\]). This method can be generalized to perform a path integral expansion about a saddle point boundary value problem (e.g. a transition amplitude in quantum mechanics) using perturbation theory (Gaussian integrals) only. Further applications of these tools is the subject of work in progress [@63]. Open issues for future studies ------------------------------ As already stressed, one of our motivations is to study the peculiar large-N limit of dYM confinement, similar to the large-N limit of sYM from ref. [@Cherman:2016jtu], which shows many intriguing features that (at least superficially) resemble stringy properties. We have not yet fully addressed this limit in dYM, as there is the upper bound on N discussed above. We believe that this restriction on N is technical and more work is required to remove it. Our study here also only briefly touched on the spatial structure of confining k-strings, noting that, upon increasing N, they become more “fuzzy” due to the decreasing mass of many of the dual photons, but retain a finite string tension due to the (also large) number of dual photons of finite mass. This spatial structure may have to do with their interacting nature and would be interesting to investigate further. Further, in this paper, we ignored the $\theta$-angle dependence of the $k$-strings. The topological angle dependence in Yang-Mills theory has received renewed recent attention, see e.g. [@Thomas:2011ee; @Unsal:2012zj; @Anber:2013sga; @Bhoonah:2014gpa; @Gaiotto:2017yup; @Tanizaki:2017bam; @Kikuchi:2017pcp; @Gaiotto:2017tne; @Anber:2017rch]. As seen in some of the aforementioned work, the corresponding physics in dYM is also very rich and worth of future studies. There are also the many intriguing observations of [@Aitken:2017ayq] on the nature of the dual photon, glueball, etc., bound state spectra in dYM (at arbitrary N) that await better understanding. Finally, there is the question about the (still conjectural) continuity of dYM from the calculable small $\Lambda$NL regime to the regime of large $\Lambda$NL. To this end, it would be desirable to study this theory on the lattice; for some lattice studies of related theories, see [@Cossu:2009sq; @Vairinhos:2011gv; @Bergner:2014dua; @Bergner:2015cqa]. Organization of this paper -------------------------- Section \[sec:2\] is devoted to a review of dYM theory.[^6] In Section \[sec:2.1\] we review how dYM theory on $\R^3\times \S^1$ avoids a deconfinement transition at small L. The perturbative spectrum of dYM is discussed in Section \[sec:2.2.1\] and the nonperturbative minimal action monopole-instanton solutions—in Section \[sec:2.2.2\]. The action of a dilute gas of monopoles is discussed in at length in Section \[2.3\], with emphasis on details that often not emphasized in the literature. The derivation of the string tension action, used to calculate the semiclassical string tensions is given in Section \[sec:2.4.1\]. ection \[numericsection\] is devoted to a numerical study of the k-string tensions in dYM. The action and its discretization are studied in Sections \[sec:3.1\] and \[sec:3.3\]. The minimization procedure, the numerical methods, and the error analysis are described in Section \[sec:minimize\]. The numerical results for the k-string tensions for gauge groups up to SU(10) are summarized in Table \[table:1\], see Section \[sec:3.4\]. ection \[sec:4\] presents an analytic perturbative procedure to calculate the string tensions. We begin by explaining the main ideas with fewer technical details. In Sections \[su2analyticsection\] and \[suNanalyticsection\] we give the detailed calculations for SU(2) and general SU(N) gauge groups, respectively. The results are tabulated in Appendix \[sec:C\], demonstrating the precision of this procedure, which also serves as a check on the numerical results of Section \[numericsection\]. ection \[sec:5\] contains the discussion of our results from various points of view, including relations to other models of confinement and the behaviour of k-strings in the large-N limit: In Section \[sec:5.1.1\] we argue that the lowest, among all weights of any given representation, semiclassical asymptotic string tensions in dYM depends only on $N$-ality $k$ of the representation and is the one obtained for quark sources with charges in the highest weight (and its $\Z_N$ orbit) of the $k$-index antisymmetric representation. In Section \[sec:compare\] we compare confinement in dYM with confinement in Seiberg-Witten theory and point out that the unbroken $\Z_N$ center symmetry in dYM is responsible for the major differences, which make abelian confinement in dYM closer—in many aspects—to confinement in the nonabelian regime. In Section \[bagmodelsection\] we point out the similarity, already discussed around eq. (\[dymscaling\]), of the k-string tension ratios in dYM to the ones in the MIT Bag Model and discuss the physical reasons. In Section \[sec:5.1.4\] we compare the k-string tension scaling laws to other scaling laws considered in various theoretical models. In Section \[sec:largeN\], we discuss the abelian large-N limit. The leading large-N terms in the k-string tension ratios, eq. (\[dymscaling2\]) above, are derived in \[sec:5.2.1\]. The fact that large-N factorization does not always imply that k-strings become free at large-N is discussed in Section \[sec:5.2.2\]. The analytic methods of Section \[sec:4\] prove indispensable in being able to track the importance of the way the large-N and large area limits are taken. Review of dYM theory {#sec:2} ==================== In this Section we will have a brief review of dYM theory. The emphasis is on topics usually not covered in detail the literature and on topics that will be needed for the rest of the paper. Confinement of charges in deformed Yang-Mills theory for all $\S^1$-circle sizes {#sec:2.1} --------------------------------------------------------------------------------- Consider four-dimensional Yang-Mills theory in the Euclidean formulation with one of its dimensions compactified on the circle: $$\label{eq:2.1} S = \int_{{\rm \R}^3 \times \S^1}d^4 x{1 \over 2g^2} \text{tr} F^2_{\mu\nu}(x)~.$$ We set the $\theta$-angle to zero in this paper, leaving the study of the $\theta$-dependence of strings’ properties for the future. Here, $T^a$ ($a=1, ..., N^2 -1$) refer to the Hermitean generators of the group $SU(N)$, $F_{\mu\nu} = F^a_{\mu\nu}T^a$, $\text{tr} (T^aT^b) = {1\over 2} \delta^{ab}$. The compactification circle $\S^1$ in pure Yang-Mills theory can either be considered as a spatial dimension of size $L$, or as a temporal one with $L = {1 / T}$ being the inverse temperature $T$. It is known, see e.g. [@04], that above a critical temperature $T_c = {1\over L_c}$ Yang-Mills theory loses confinement (i.e. the static potential between two heavy probe quarks no longer shows a linearly rising behaviour as a function of distance between the quarks). The transition from a confining to a non-confining phase, in theories with gauge groups that have a nontrivial center, is accompanied by the breaking of the center-symmetry.[^7] The critical size $L_c$ is approximately of order $\Lambda^{-1}$, with $\Lambda$ the $\overline{MS}$ strong scale of the theory. Different studies give an estimate of $200$ MeV $<T_c <300$ MeV for $SU(2)$ Yang-Mills theory in four dimensions. In what follows we shall deform Yang-Mills theory in a way that preserves confinement of charges for any circle size $L$. Due to asymptotic freedom the coupling constant is small at the compactification scale ${1 \over L}$ for small circle sizes $(L \ll \Lambda^{-1}$; as we argue below, the precise condition for $SU(N)$ gauge theories turns out to be $\Lambda L N \ll 1$). This deformation would enable us to have a model of confinement that we can study analytically in the limit of a small circle size $L$. The expectation value of the trace of the Polyakov loop, $P(\text{x}) = \text{tr} \;{\cal{P}}\text{exp}(-i \oint_{S^1} dx_4 \text{A}_4 (\text{x},x_4))$ (where ${\cal{P}}$ denotes path ordering) serves as an order parameter for confinement [@04]: \[eq:2.2\] &P()= 0     ,\ &P()0     . && On the other hand, the Polyakov loop is not invariant under a center-symmetry transformation and picks up a center element $z$, i.e. $U_zP(\text{x})U^{\dagger}_{z} = z P(\text{x})$, where we used the notation of Footnote \[center\]. Therefore for a center-symmetric vacuum $| 0 \rangle$ we have: \[eq:2.3\] 0 | P()|0= 0|U\^\_zU\_z P()U\^\_zU\_z|0 = z0 | P()|0 P() = 0 ,   z 1 ,&& indicating that a center-symmetric phase is a confined phase. In order to show that Yang-Mills theory deconfines at high temperatures we need to show that the expectation value of the Polyakov loop at high temperatures is nonzero. The Polyakov loop is gauge invariant and the eigenvalues of the holonomy $\Omega(\text{x}) = \text{P} \text{exp}(- i \oint_{S^1} dx_4 \text{A}_4 (\text{x},x_4))$ constitute its gauge invariant content ($P = \text{tr} \Omega$). At tree level the eigenvalues of the holonomy can take any value, as there is no potential for $\Omega$ in the classical Yang-Mills Lagrangian (\[eq:2.1\]). To find an effective potential for the eigenvalues of the holonomy at one-loop, we expand around a constant diagonal $A_4$ field and evaluate the one loop contribution to the effective potential by integrating out the quadratic terms of gauge and ghost fields [@01; @06], to find: \[eq:2.4\] V\_1\[\] = -[2\^2 L\^3]{}|\^n|\^2,     = (-iLA\_4) . From it can be seen that $V_1[\Omega]$ is minimized when $\Omega$ is an element of the centre of the gauge group, i.e. $\omega_N^k \;I$, with $I$ the unit matrix.[^8] This would imply $\langle P(\text{x})\rangle = \langle \text{tr}(\Omega)\rangle = N \omega_N^k \ne 0 $, indicating a deconfined center-symmetry broken phase of Yang-Mills theory at high temperatures, or small circle sizes $L$ (owing to asymptotic freedom, the small-$L$/high-$T$ regime is the one where the calculation leading to (\[eq:2.4\]) can be trusted). In order to change this picture and have a model of confinement at arbitrary small circle sizes $L$ we can add a deformation potential term to Yang-Mills theory [@01; @Myers:2007vc]: $$\label{eq:2.5} S = \int_{{\rm I\!R}^3 \times S^1}{1 \over 2g^2} \text{tr} F^2_{\mu\nu}(x) + \Delta S, \ \Delta S \equiv \int_{{\rm I\!R}^3}{1 \over L^3} P[\Omega(\text{x})], \ P[\Omega] \equiv {2 \over \pi^2} \overset{[N/2]}{\underset{n=1}{\sum}}{b_n \over n^4}|\textrm{tr}(\Omega^n)|^2~,$$ with $b_n$—sufficiently large and positive coefficients. The effect of the $\Delta S$ term is to dominate the gluonic and ghost potential in a way that the minimum of $V_1[\Omega] + {1 \over L^3}P[\Omega]$ occurs when $\text{tr}(\Omega^n) = 0$ for $n$ mod $N$ $\ne$ 0. This would imply $\langle P(\text{x})\rangle = \langle \text{tr}(\Omega)\rangle = 0$ and hence a confinement phase for deformed Yang-Mills theory at arbitrarily small circle sizes. The $\Delta S$ deformation term in would make the theory non-renormalizable. To have a well-behaved theory at high energies, the deformation can be considered as an effective potential term generated by some renormalizable dynamics, notably $n_f$ flavors of massive adjoint Dirac fermions with periodic boundary conditions along the $\S^1$. Following [@05] for conventions on Euclidean formulation of Dirac fermions we have: $$\label{eq:2.6} S_{dYM} = \int_{{\rm I\!R}^3 \times S^1}\{ {1 \over 2g^2} \text{tr}F^2_{\mu\nu}(x) -i \underset{i=1}{\overset{n_f}{\sum}}\bar{\psi}_i(\slashed{D}+m)\psi_i \}$$ The effective potential for the holonomy generated by the $n_f$ massive adjoint Dirac fermions is given by [@07; @08]: $$\label{eq:2.7} V_2[\Omega] = +{2\over \pi^2 L^3}\underset{n=1}{\overset{\infty}{\sum}}n_f(nLm)^2K_2(nLm){|\textrm{tr}\Omega^n|^2 \over n^4}~,$$ where $K_2$ is the modified Bessel function of the second kind. It has to be noted that in the deformed theory the compactified dimension $S^1$ can only be a spatial dimension since the heavy fermions satisfy periodic boundary conditions along this direction. There are two free parameters $n_f$ and $NLm$ in the effective potential .[^9] The beta function of $SU(N)$ Yang-Mills theory with $n_f$ flavours of Dirac fermions in the adjoint representation of the gauge group is, at the one loop level, $\beta(g) = - {g^3 \over (4 \pi)^2} ({11 \over 3}N - {4 \over 3}n_fN)$, hence to assure asymptotic freedom $n_f = 1 \ \text{or} \ 2$. If we allow for massive Majorana flavors, $n_f = 5/2$ is the maximum value. On the other hand, if we want the effective potential $V_2$ to dominate the gluonic potential $V_1$, $NLm$ should be of order 1 ($NLm \sim 1$; for larger values of $m$, the fermions decouple and the theory loses confinement at small $L$). To gain some intuition on how the coefficients of the potential $V_2$ behave let $c_n \equiv n_f ({n\over N}LmN)^2 K_2({n\over N}LmN)$. Choosing $n_f = 2$ ($n_f = 1$) and $NLm = 4$ ($NLm = 3$) gives $c_n \approx 4$ ($c_n \approx 2$) for $n/N \approx 0$, $c_n \approx 2$ ($c_n \approx 1.3$) for $n/N \approx 0.5$, $c_n \approx 0.56$ ($c_n \approx 0.55$) for $n/N \approx 1$ and $c_n$ approaching zero exponentially for $n/N > 1$. Minimizing[^10] the combined potential $V_1[\Omega] + V_2[\Omega]$ gives $\langle\Omega\rangle = \text{diag} (\omega_N^{N-1}, ..., \omega_N, 1)$ for odd $N$, and $\langle\Omega\rangle = e^{i {\pi \over N}} \text{diag} (\omega_N^{N-1}, ..., \omega_N, 1)$ for even $N$, which gives $\text{tr} \langle\Omega\rangle^n = 0$ for $n$ mod $N \neq 0$, hence, a confined phase for deformed Yang-Mills theory.[^11] Perturbative and non-perturbative content of dYM {#sec:2.2} ------------------------------------------------ ### Perturbative content {#sec:2.2.1} The eigenvalues of the holonomy $\Omega (\text{x}) = {\cal{P}}\text{exp}(i\oint d\text{x}_4 A_4(\text{x},\text{x}_4))$, are the only gauge invariant content of the gauge field component in the compact direction $A_4(x)$ and are invariant under any periodic gauge transformation. Working in a gauge such that the $A_4(x)$ field assumes these eigenvalues ($A_4(x) = -i {\text{ln}(\Lambda(\text{x})) / L}$, with $\Lambda(\text{x})$—the diagonal matrix of eigenvalues of $\Omega(\text{x})$) and expanding around a center-symmetric $vev$ $$\begin{aligned} \label{a4vev} A^{vev}_4 &=& {1 \over N L} \text{diag}(2\pi (N-1), ..., 2\pi, 0) \; \; \text{for odd}\; N, \nonumber \\ A^{vev}_4 &=& {1 \over N L} \text{diag}(2\pi (N-1) + \pi, ..., 3\pi, \pi)\;\;\text{for even} \; N, \end{aligned}$$ the perturbative particle content of dYM theory with action can be worked out by writing the second order Lagrangian of the modes expanded around the above center-symmetric $vev$. Clearly, the $vev$ of the “Higgs field” $A_4$ breaks the gauge symmetry $SU(N) \rightarrow U(1)^{N-1}$. The gauge fields associated with the non-compact direction can be written as: $$\label{eq:2.8} A_i(\text{x},\text{x}_4) = {\sqrt{2} }\; A_{i,0}(\text{x}) + {}\overset{+\infty}{\underset{k = - \infty} {\sum}^\prime} A_{i,k}(\text{x})\;\text{exp}(ik{2\pi \over L}\text{x}_4) \ \ ({\sum}^\prime \text{ over}\; k \neq 0) ~,$$ with $A_{i,k}(\text{x}) = A_{i,k1}(\text{x}) + iA_{i,k2}(\text{x}) = A^{\dagger}_{i,-k}(\text{x})$ in order to ensure reality of the $A_i$ fields. It turns out that the gauge-boson field content is a non-trivial one. We work out the quadratic Lagrangian in Appendix \[app:wboson\], by substituting (\[eq:2.8\]) in the action and expanding around (\[a4vev\]). We begin with a discussion of the abelian spectrum. The diagonal components of the gauge fields $A_i$ commute with the $vev$ $A_4^{vev}$. Hence, their zeroth Fourier modes along $\S^1$ correspond to massless 3d photons and their higher Fourier modes gain mass of ${2 \pi m \over L}$ where $m=1,2, ...\infty$ is the non-zero momentum in the compact direction. At tree level, the Lagrangian for the $N-1$ photons is simply the reduction of (\[eq:2.1\]) to the Cartan subalgebra of $SU(N)$. The leading-order coupling of the 3d $U(1)^{N-1}$ gauge theory is given by $g_3^2 = g^2/L$, where $g$ is the four-dimensional gauge coupling at the scale of the lightest $W$-boson mass, $m_W = {2\pi \over NL}$, see below.[^12] The physical components of the “Higgs” field $A_4(x)$ are diagonal and $x_4$ independent. They are massless at the classical level but gain mass of order at least $g \over \sqrt{N} L$ via the one loop effective potential $V_1 + V_2$ generated by quantum corrections.[^13] The massive adjoint Dirac fermions are also expanded in their Kaluza-Klein modes. Taking into account the effects of $A^{vev}_4$, it can be seen that there are massive Dirac fermions with masses $m + {2 \pi k \over L} + {2 \pi p \over LN}$ for $k = 0, 1, ...$ and $p = 0, 1, ..., N-1$. As we are interested in physics below the scale of the lightest fermion, we shall not present details of the massive fermion spectrum. Finally, the relation from Appendix \[app:wboson\] shows that there are $W$-bosons with masses $|{2 \pi m \over L} - {2\pi |l-k| \over NL}|$ and $|{2 \pi m \over L} + {2\pi |l-k| \over NL}|$ respectively for $m =0,1,2,...$ and $1\leq l < k \leq N$. The mass of the lightest $W$-boson is $m_W = {2 \pi \over NL}$. Clearly, below that scale, there are no fields charged under the unbroken $U(1)^{N-1}$ gauge group. Thus, the gauge coupling of the $N-1$ photons is frozen at the scale $O(m_W)$. The condition that the theory be weakly coupled, therefore, is that $m_W \gg \Lambda$, or $N L \Lambda \ll 1$. This is the semiclassically calculable regime that we study in this paper. In summary, the perturbative particle content of deformed Yang-Mills theory expanded around the center-symmetric $vev$ consists of $N-1$ photons (the diagonal Cartan components of the gauge fields, whose zero Fourier components along the $\S^1$ are massless), $N^2 - N$ massive gauge fields, charged under the $U(1)^{N-1}$ unbroken gauge symmetry, whose spectrum is given in (\[eq:2.13\]), $N-1$ massive eigenvalues of the holonomy, neutral under $U(1)^{N-1}$ and charged and uncharged massive Dirac fermions. ### Non-perturbative content: minimal action instanton solutions {#sec:2.2.2} Finite action Euclidean configurations of pure Yang-Mills theory on $\R^3 \times \S^1$ were studied in [@06]. It was shown that they are classified by their magnetic charge $q_\alpha$, Pontryagin index $p$, and asymptotics of the $\S^1$ holonomy $\Omega$ at infinity, which are related by the following formula: $$\label{eq:2.14} Q = p + {\text{ln} \mu_{\alpha} \over 2 \pi i}\; q_{\alpha}~.$$ Here, $Q = {1 \over 32 \pi^2}\int_{{ \R}^3 \times \S^1} d^4 x \;F_{\mu\nu}^a \widetilde{F}_{\mu\nu}^a$ is the topological charge with $\widetilde{F}_{\mu \nu} = {1\over2} \epsilon_{\mu \nu \alpha \beta} F_{\alpha \beta}$. In this Section, we use $\mu_{\alpha}$ with $\alpha = 0, ..., \kappa \leq N-1$ to label the distinct eigenvalues of the holonomy $\Omega (\text{x})$ at spatial infinity. Notice that, for finite action configurations, the eigenvalues of $\Omega$ are independent of the direction that we approach spatial infinity, and that, for the center-symmetric holonomy, $\kappa = N-1$ as all eigenvalues are distinct, given by the $N$ values of $e^{i L A_4^{vev}}$ with $A_4^{vev}$ of (\[a4vev\]): $\ln \mu_\alpha = {2 \pi i (N-1-\alpha) \over N}$ for odd $N$. The integer magnetic charges are denoted by $q_{\alpha}$, satisfy $\sum^{\kappa}_{\alpha = 0}q_{\alpha} = 0$, and will be explicitly defined further below, see paragraph after eq. (\[eq:2.21\]). The Pontryagin index $p$ is the winding number for mappings of $\S^3$ onto the full group $SU(N)$.[^14] It is expected that for any value of the quantities $p$, $q_{\alpha}$, and $\mu_{\alpha}$ there is a separate sector of finite action configurations, with the self-dual ($F_{\mu\nu} = \widetilde{F}_{\mu\nu}$) or anti-self dual ($F_{\mu\nu} = - \widetilde{F}_{\mu\nu}$) solution corresponding to the minimum action configuration in that sector. For self-dual or anti-self-dual solutions the topological charge is proportional to the value of the action. Therefore finding configurations with the minimal non-zero topological charge is equivalent to finding the minimal non-zero action configurations. Based on the values of $\mu_{\alpha} = \text{exp}(i {2 \pi (N-1 -\alpha) \over N})$ for $\alpha = 0, ..., N-1$ in a center-symmetric vacuum, and the fact that $p$ and $q_{\alpha}$ are integers with $\sum^{N-1}_{\alpha = 0}q_{\alpha} = 0$, it can be clearly seen that the minimal non-zero topological charge is $|Q| = {1 \over N}$. Configurations of minimal $Q = {1 \over N}$ would then correspond to the following values of $q_{\alpha}$ and $p$: \[eq:2.15\] &p = 0,  q\^i\_m = (0,..,0,,-1,0,...,0),        i = 1, ..., N-1,\ \[eq:2.16\] &p = 1,  q\^N\_m = (-1,0,... ,0, 1), where the $N$ components of the vector $q_m^i$ are the magnetic charges $q_\alpha$ corresponding to the $i$-th minimal action configuration. Minimal action configurations with Pontryagin indices and magnetic charges given in will be referred to as the $N-1$ $SU(N)$ BPS solutions and the minimal action configurations with Pontryagin number and magnetic charges from —as the Kaluza-Klein (KK) solution.[^15] The $\overline{\text{BPS}}$ (anti-BPS) and $\overline{\text{KK}}$ (anti-KK) configurations have the opposite sign for the magnetic charges and Pontryagin index and thus a negative topological charge $Q = -{1 \over N}$. In total this classification shows that there exist $2N$ minimum finite action non-trivial configurations. We will refer to these finite action configurations as the “non-perturbative content” of deformed Yang-Mills theory—because, as we shall see, it is these Euclidean configurations that lead to confinement of charges, to leading order in $NL\Lambda \ll 1$. In order to construct such configurations we start from the $SU(2)$ BPS and Kaluza-Klein monopoles and embed them in $SU(N)$. For the BPS solution, this can be done in $N-1$ different ways leading to the $N-1$ different configurations in and for the Kaluza-Klein monopole this can be done in only one way. The $SU(2)$ BPS monopole solution is given by [@23; @Diakonov:2009jq]: $$\label{eq:2.17} \begin{split} & A^a_4 = \mp n_a \nu \mathcal{P}(\nu r)\ \ , \ \ \ \ \ \ \ \ \ \ \ \ \ \mathcal{P}(y) = \text{coth}(y) - {1 \over y} \\ & A^a_i = \epsilon_{aij} n_j {1-\mathcal{A}(\nu r) \over r}\ \ , \ \ \ \ \ \ \ \mathcal{A}(y) = {y \over \text{sinh}(y)}~, \end{split}$$ where $n_a$ for $a = 1, 2, 3$ refer to the components of a unit vector in ${\rm I\!R}^3$ and $\nu$ is related to the eigenvalues of the holonomy at infinity. In what follows, as in the above relations, the upper sign always corresponds to the self-dual BPS solution and the lower one to the anti-self-dual $\overline{\text{BPS}}$ solution. The magnetic field strength $B_i ={1 \over 2} \epsilon_{ijk}F_{jk}$ of this solution is: $$\label{eq:2.18} B^a_i = (\delta_{i}^a - n^an_i) \nu^2 F_1(\nu r) + n^a n_i \nu^2 F_2(\nu r)$$ The functions $F_1$ and $F_2$ are given below in . In order to embed these solutions in $SU(N)$ and in a center-symmetric vacuum $A^{vev}_4$, we first make a gauge transformation that will make the $A_4$ component diagonal in colour space along $\tau^3 /2$. For this we solve for the equation $S_- \tau^a n_a S_-^{ \dagger} = -\tau^3$ for the BPS and $S_{+} \tau^a n_a S_+^{ \dagger} = \tau^3$ for the $\overline{\text{BPS}}$. This gives: $$\label{eq:2.19} \begin{split} & S_+(\theta, \phi) = \text{cos} { \theta \over 2} + i \tau^2 \text{cos} \ \phi \ \text{sin} {\theta \over 2} -i \tau^1 \text{sin} \ \phi \ \text{sin} { \theta \over 2} = e^{-i\phi \tau^3 / 2} e^{i\theta \tau^2 / 2} e^{i\phi \tau^3 / 2} \\ & S_-(\theta, \phi) = - \text{sin}{ \theta \over 2} \; \text{cos} \ \phi - i\; \text{sin}{ \theta \over 2} \;\text{sin} \ \phi \ \tau^3 + i\; \text{cos}{ \theta \over 2} \tau^2 = e^{i\phi \tau^3 / 2} e^{i (\theta + \pi) \tau^2 /2 } e^{i\phi \tau^3 / 2}~. \end{split}$$ After performing the gauge transformation $A_{\mu} \longrightarrow A^S_{\mu} = S A_{\mu} S^{\dagger} + iS \partial_{\mu} S^{\dagger}$ for $S = S_- \ \text{or} \ S_+$ we get: $$\label{eq:2.20} \begin{aligned} & A^S_4 = \nu \mathcal{P}(\nu r) {\tau^3 \over 2} \\ & A^S_r = 0 \\ & A^S_{\theta} = {\mathcal{A}(\nu r) \over 2r} (\pm \tau^1 \text{sin} \phi + \tau^2 \text{cos} \phi) \\ & A^S_{\phi} = {\mathcal{A}(\nu r) \over 2r} (\pm \tau^1 \text{cos} \phi - \tau^2 \text{sin} \phi) \pm \tau^3 {1 \over 2r} \text{tan} {\theta \over 2} ~, \end{aligned}$$ where $A^S_r = \hat{r}_i A^S_i$, $A^S_{\theta} = \hat{\theta}_i A^S_i$, $A^S_{\phi} = \hat{\phi}_i A^S_i$ are the components of $A^S_i$ along the unit vectors in spherical coordinates.[^16] It has to be noted that the $A^S_{\phi}$ solution shows a singular string along $\theta = \pi$. This is a gauge artifact and does not cause any problems for to satisfy the self-duality or anti-self-duality condition. In other words, the magnetic fields evaluated from are everywhere smooth functions of the spherical coordinates, as can be seen by finding the magnetic field strength in the stringy gauge: $$\label{eq:2.21} \begin{aligned} & B^S_r = \mp {\nu}^2 F_2(\nu r) \tau^3 /2 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ F_2(y) = {1 \over \text{sinh}^2 y } - {1 \over y^2} \\ & B^S_{\theta} = \nu^2 F_1(\nu r)/2 (\mp \tau^1 \text{cos} \phi + \tau^2 \text{sin} \phi) , \ \ \ \ \ \ \ \ F_1(y) = {1 \over \text{sinh} \ y }( {1 \over y} - \text{coth} \ y) \\ & B^S_{\phi} = \nu^2 F_1(\nu r)/2 (\pm \tau^1 \text{sin} \phi + \tau^2 \text{cos} \phi) ~. \end{aligned}$$ Using the diagonal components of the field $B^S$ at infinity, diag$(B^S)$, the magnetic charge vector for the $SU(2)$ BPS monopole solution, in the normalization of (\[eq:2.15\]) can now be defined[^17].by a surface integral at infinity, $(q_1, q_2) = \oint\limits_{S^2_\infty} d^2 \sigma \;\text{diag} (B_r^S)/(2 \pi) = (1,-1)$.[^18] There are $N-1$ $SU(2)$ Lie subalgebras, corresponding to the elements $a_{ii}$, $a_{ii+1}$, $a_{i+1i}$, $a_{i+1i+1}$ for $i = 1, ..., N-1$, along the diagonal of an $SU(N)$ Lie algebra matrix and we can embed an $SU(2)$ BPS monopole in each of them. Only these embedded BPS monopoles will have the lowest topological charge $|Q| = {1 \over N}$. We will illustrate the embedding for the top left $SU(2)$ Lie subalgebra. We simply place the $SU(2)$ solution , with $\nu = {2\pi \over NL}$, in the top left $SU(2)$ Lie subalgebra of an $SU(N)$ Lie algebra matrix with all other elements being zero. Next, in order to make the value of $A^{S,SU(N)}_4$ ($\equiv$ $SU(2)$ $A^S_4$ solution of embedded in $SU(N)$) at infinity the same as $A^{vev}_4$ of (\[a4vev\]), we add the matrix $\bar{A} ={1 \over NL} \text{diag}(2\pi(N-1) -\pi, 2\pi(N-1) -\pi, 2\pi(N-2),...,0)$ for odd $N$ and $\bar{A} ={1 \over NL} \text{diag}(2\pi(N-1) , 2\pi(N-1) , 2\pi(N-2) + \pi,...,\pi)$ for even $N$ to $A^{S,SU(N)}_4$. Similarly, BPS monopoles can be embedded in the remaining $N-2$ diagonal $SU(2)$ subalgebras of $SU(N)$. For the Kaluza-Klein solution[^19] we start from the BPS solution of in a vacuum where $\nu$ is replaced by $\nu \rightarrow {2 \pi \over L} - \nu$. To obtain the KK solution ($\overline{\text{KK}}$) we gauge transform the BPS solution of with $S_+$ (with $S_-$) using the upper sign (lower sign). Now the asymptotic behaviour of the $A_4$ field for both solutions is $(\nu - {2\pi \over L}) {\tau^3 \over 2}$. In order to make the asymptotics similar to the BPS $A^S_4$ field in , we perform an $x_4$-dependent gauge transformation $U(\text{x}_4) = \text{exp}(i {2 \pi \over L} \text{x}_4 {\tau^3 \over 2})$, which brings the asymptotics back to $\nu {\tau^3 \over 2}$. This gauge transformation gives a non-trivial $x_4$-dependence to the cores of the KK and $\overline{\text{KK}}$ solutions. Since the Pontryagin index $p$ of a KK monopole in relation is $p = 1$, in order to obtain the lowest topological non-zero charge (which is $|Q| = {1 \over N}$), the second term in should equal $-{N-1 \over N}$ therefore, as already discussed, there is only one way to embed an $SU(2)$ KK monopole in $SU(N)$ in a centre-symmetric $vev$ that would give the lowest action and that is to choose the $SU(2)$ subalgebra corresponding to the components $a_{11}, a_{1N}, a_{N1}, a_{NN}$ of an $SU(N)$ Lie algebra matrix (i.e. with $q_1 = - q_N = -1$, as per (\[eq:2.16\])). This was a brief summary of the non-perturbative solutions in dYM theory that are responsible for confinement of charges to leading order in the limit $NL \Lambda \rightarrow 0$. Action of a dilute gas of monopoles {#2.3} ----------------------------------- The action of the (anti-)self-dual solution embedded in $SU(N)$, with $\nu = 2\pi /NL$, is given by: \[eq:2.22\] S\_ = [2 L g\^2]{} \_[[IR]{}\^3]{} d\^3 (B\_i B\_i) = [8 L g\^2]{} \^\_0 d|[r]{} |[r]{}\^2 { [1 2]{} F\^2\_2(|[r]{}) + F\^2\_1(|[r]{})}= [4 L g\^2]{} = [8 \^2 g\^2 N]{}  ,&& where $\bar{r} = \nu r$, $r$ being the radial coordinate in spherical coordinates. Next, we calculate the action of two far-separated BPS solutions of embedded in $SU(N)$ and living in a center-symmetric vacuum $A^{vev}_4$. We embed the first monopole (second monopole) in the $i \text{-th}$ ($j \text{-th}$) subalgebra of $SU(N)$ along the diagonal for $1 \leq i, j \leq N-1$. We work in the limit $\nu^{-1} = {NL \over 2\pi} \ll r_0 \ll d$, where $d$ denotes the distance between the centers of the monopoles and $r_0$ is the radius of a two-sphere surrounding each monopole. In constructing far separated monopole solutions, first we need to mention how the monopoles are patched together. To patch the monopoles together, we use the string gauge and first subtract $A^{vev}_4$ from the $A_4$ component of each monopole solution. The resulting configuration will have an asymptotically vanishing behaviour at infinity for all its gauge field components. Now we simply add up the fields corresponding to the various monopole configurations, with their centres being separated by a large, in the precise sense defined above, distance $d$ from each other. At the end, we add $A^{vev}_4$ to obtain the final configuration (had we simply added the two monopole configurations at a large separation $d$, asymptotically the $A_4$ component of the two monopole configuration would be $2A^{vev}_4$; this way of construction avoids this double counting of $A^{vev}_4$). In calculating the action of far separated monopole configurations we only consider gauge invariant leading order terms in the self-energy and interaction energy of the monopoles.[^20] We write the fields as $A_{\mu} = A^{(1)}_{\mu} + A^{(2)}_{\mu}$, for $\mu =1,... , 4$. $A^{(1)}_{\mu}$ and $A^{(2)}_{\mu}$ refer to the contribution of the first and second monopole to the total $A_{\mu}$ field of the monopole configuration respectively. When $A_4$ appears in the commutator term of $F_{k4}$ the overall $A^{vev}_4$ is considered as part of $A^{(i)}_4$ in the two-sphere region of radius $r_0$ surrounding the $i$-th monopole for $i =1, 2$ and otherwise can be distributed in an arbitrary smooth way between $A^{(1)}_{4}$ and $A^{(2)}_{4}$ and for the $\partial_k A_4$ term in $F_{k4}$ the overall $A^{vev}_4$ vanishes and can be neglected. The total action of the far separated two-monopole configuration can be written as: $$\label{eq:2.23'} \begin{split} S^{\prime}_{2 \text{-monopole}} & = {L \over g^2} \int d^3 \text{x} \{ \text{tr} ( B_kB_k ) + \text{tr} ( F_{k4}F_{k4}) \} \\ & = \sum^{2}_{i=1}S^{(i)}_{\text{self-energy}} + S_{\text{inter.}, > r_0} + S_{\text{inter.}, < r_0} + S_{\text{non-gauge-invariant}} \end{split}$$ Each of the above terms in will be explained and evaluated below: 1. For the self-energy, we calculate the contribution of one monopole to the action neglecting the other monopole. Similar to we find: $$\label{eq:2.23} S^{(i)}_{\text{self-energy}} = {L \over g^2} \int d^3 \text{x} \{ \text{tr} ( B^{(i)}_kB^{(i)}_k ) + \text{tr} ( F^{(i)}_{k4}F^{(i)}_{k4}) \} = {8 \pi^2 \over g^2 N} + O(\text{exp}(-\nu r_0)) , \ \ \ \ i =1, 2$$ where $B_k^{(i)}$ and $F^{(i)}_{k4}$ refer to the magnetic and “electric” field[^21] of the $i$-th monopole respectively. The fact that the overall $A^{vev}_4$ is distributed between $A^{(1)}_{4}$ and $A^{(2)}_{4}$ outside their surrounding two-spheres of radius $r_0$ would make the monopole self-energies in to differ from by $O(\text{exp}(-\nu r_0))$. 2. The first contribution to the interaction between two monopoles at the classical level comes from the long range magnetic and electric fields of each monopole. This contribution comes from the region outside the two-spheres of radius $r_0$ surrounding each monopole (the second contribution comes from the long range electric influence that each monopole has on the other monopole inside their surrounded sphere of radius $r_0$, see next item). The magnetic and electric interactions beyond the two surrounding spheres can be evaluated as: \[eq:2.24\] S\_[, &gt; r\_0]{} &= [2L g\^2]{} d\^3 { ( B\^[(1)]{}\_kB\^[(2)]{}\_k ) + ( F\^[(1)]{}\_[k4]{}F\^[(2)]{}\_[k4]{}) }\ &= [2 L g\^2]{} d\^3 { [ - \_1 2| - \_1|\^3 ]{} q\^[i]{}\_[m1]{} q\^[j]{}\_[m2]{} + [ - \_1 2| - \_1|\^3 ]{} q\^[i]{}\_[e1]{} q\^[j]{}\_[e2]{} }\ &    + O([L \^[-2]{} g\^2 d\^3]{}),         d = |\_1 - \_2| && Here, $q^{i}_{m1} = q^{i}_{m}$ refers to the magnetic charge of the first monopole with $q^{i}_{m}$—an $N$-component charge vector given by relation . Similarly $q^{j}_{m2} = q^{j}_{m}$, $1 \leq i, j \leq N-1$. We substituted, from the Cartesian form of the radial terms proportional to ${1 \over r^2}$ in (\[eq:2.21\]), the magnetic field $B^{(1)}_{k,r^{-2}} \equiv \text{diag}(q^i_{m1} { \text{x}_k - \text{x}_{1k} \over 2|\text{x} - \text{x}_1|^3})$ and $B^{(2)}_{k,r^{-2}} \equiv \text{diag}(q^j_{m2} { \text{x}_k - \text{x}_{2k} \over 2|\text{x} - \text{x}_2|^3})$ (understood as a diagonal matrix with entries determined by the charge vectors $q_{m1}^i$ and $q_{m1}^j$ (\[eq:2.15\]) of the two monopoles) into the first line in (\[eq:2.24\]), and similarly for $F_{k4}$. We then replaced the trace of the product of these abelian matrices with an inner product over the vector of magnetic (or electric, i.e. scalar) charges corresponding to the diagonal elements of these abelian matrices. For a self-dual BPS solution, the electric charges are $q^i_{e1} = q^i_{m1}$ and $q^j_{e2} = q^j_{m2}$. The error term in comes from the inner product of the long range magnetic (or electric) field of one monopole with the term $\sim \hat{r}^{\text{other}} {1 \over \text{sinh}^2 \nu r}$ of magnetic (or electric) field of the other monopole in , integrated over the two-sphere of radius $r_0$ surrounding the other monopole, with $\hat{r}^{\text{other}}$ being the unit vector in the radial direction of the other monopole. After writing ${ \text{x} - \text{x}_i \over 2|\text{x} - \text{x}_i|^3} = -\nabla_{\text{x}} {1 \over 2|\text{x} - \text{x}_i|}$ for $i =1, 2$ in and integrating by parts with $\nabla^2_{\text{x}} {1 \over |\text{x} - \text{x}_i|} = -4 \pi \delta^3 (\text{x} - \text{x}_i)$ we get: $$\label{eq:2.25} S_{\text{inter.} > r_0} = {2 \pi L \over g^2 d} \;{q^i_{m1}\cdot q^j_{m2} } + {2 \pi L \over g^2 d}\; {q^i_{e1}\cdot q^j_{e2} } + \text{O}({L \nu^{-2} \over g^2d^3})~.$$ 3. The final Dirac-string independent contribution to the interaction between the two monopoles is the electric influence of the second monopole on the core of the first monopole[^22] There is also a similar electric influence from the first monopole on the core of the second one. It originates from the following cross terms in the action: $$\label{eq:2.26} S^{2-1}_{\text{inter.}, < r_0} = -{2 iL \over g^2} \int_{< r_0 , 1} d^3 \text{x} \text{tr} ([A^{(1)}_k,A^{(2)}_4] F^{(1)}_{k4}) - {L \over g^2} \int_{< r_0 , 1} d^3 \text{x} \text{tr} ([A^{(1)}_k,A^{(2)}_4]^2 ) ~.$$ The integration region is within a sphere of radius $r_0$ centered around the first monopole. The main contribution ($\sim 1/d$) in comes from the first integral. Since we excluded the overall $A^{vev}_4$ from $A^{(2)}_4$ within the two-sphere of radius $r_0$ around the first monopole, we have[^23] $A^{(2)}_4 \approx -{1 \over d} {{\tau^3_{(j)} \over 2}}$ ; one can check that there are no other (Dirac-string independent) terms that can contribute order $1/d$ interaction terms. We can work out the integrand of the first integral, using the $A_4^{(2)}$ asymptotics just given, as: $$\label{eq:2.27} \text{tr} ([A^{(1)}_k,A^{(2)}_4] F^{(1)}_{k4}) = \text{tr} ([F^{(1)}_{k 4}, A^{(1)}_k] A^{(2)}_4) = -{i \over 4d} ( F^{(1),1}_{k4} A^{(1),2}_k - F^{(1),2}_{k4} A^{(1),1}_k ) \; q^i_{e1} \cdot q^j_{e2} ~.$$ Here, $A^{(1),1}_k$ refers to the component of the $A^{(1)}_k$ along the first generator of the $i$-th $SU(2)$ subalgebra along the diagonal of an $SU(N)$ Lie algebra matrix (similar to $\tau^1 /2$ in $SU(2)$) and similarly for the others. The values of the fields $A^{(1)}_k$ and $F_{k4}^{(1)}$ can be read from relations and for a self-dual ($B_k = E_k$) solution. For the integral of the first term in (\[eq:2.26\]), we find $ \int_{< r_0, 1} d^3 \text{x} (F^{(1),1}_{k4} A^{(1),2}_k - F^{(1),2}_{k4} A^{(1),1}_k) = - 8 \pi \int\limits_{0}^{r_0 \nu} dy y {F_1(y) {\cal{A}}(y)} = 4 \pi ({\cal{A}}^2(0) - {\cal{A}}^2(r_0 \nu)) \simeq 4 \pi + \text{O}(e^{- 2\nu r_0})$, where we used $y F_1 = \partial_y {\cal{A}} $. Thus, going back to (\[eq:2.26\]), one obtains in total: $$\label{eq:2.28} S_{\text{inter.}, < r_0} = S^{2-1}_{\text{inter.}, < r_0} + S^{1-2}_{\text{inter.}, < r_0} = - {2\pi L \over g^2 d}\; q^i_{e1}\cdot q^j_{e2} - {2 \pi L \over g^2 d}\; q^j_{e2}\cdot q^i_{e1} + \text{O}({L \nu^{-1} \over g^2d^2})~.$$ The $\text{O}({L \nu^{-1} \over g^2d^2})$ error term comes from the evaluation of the second integral in and the error of the first integral in coming from the variation of $A_4^{(2)}$ from its value at the center of the sphere around the first monopole over the region of integration is of order $\text{O}({L \nu^{-2} \over g^2d^3})$ which we have neglected. Summing up the electric interactions in and we get $- {2\pi L \over g^2 d}\; q^i_{e2} \cdot q^j_{e1}$, which shows a negative potential for same-sign electric charges, hence an attractive electric force between the two monopoles with same electric charges. Since the electric interaction is mediated by the exchange of a massless (at the classical level) scalar field $A_4$, which is attractive for same sign charges, this is expected and was originally observed in [@25] using a slightly different approach. Although for simplicity we initially assumed that the solutions are BPS, eq.  is general, meaning that if we had done the same calculation in with two other monopoles (e.g. a KK and a BPS) we would have reached the same relation in , but with their appropriate electric charges replaced. 4. The $S_{\text{non-gauge-invariant}}$ term in consists of any term in the action that depends on the Dirac-string singularity (in it occurs at $\theta = \pi$) or on its orientation. These non-gauge-invariant terms are unphysical and will be neglected; we were careful to only evaluate contributions that are independent of the Dirac string or its orientation. To be more specific on this matter, we notice that the $A^S_{\phi}$ component in is singular at $\theta = \pi$. Considering the commutator term $[A^{(1)}_i,A^{(2)}_j]$ in $F_{ij}$ for two far separated monopoles at the location of the string of the first monopole, we realize that the ${\text{tan} \ \theta \over 2r} {\tau^3\over 2}$ term in $A^{S, (1)}_{\phi}$ for this monopole does not commute with terms proportional to $\tau^1$ or $\tau^2$ in the components $A^{S, (2)}_{\theta}$ or $A^{S, (2)}_{\phi}$ of the second monopole therefore (even though these terms would be exponentially suppressed outside the sphere of radius $r_0$ of the second monopole) for the action they would give a term proportional to $\int^{\pi}_0 d \theta \text{sin} (\theta) \text{tan}^2 {\theta \over 2}$ which is singular when integrated near $\theta = \pi$. Or, similar to the electric interaction inside the monopole cores as in , another contribution can be evaluated for the magnetic interaction coming from the term $\approx$ ${\text{tan} {\theta \over 2} \over d} {\tau^3 \over 2}$ of $A^{S,(2)}_{\phi}$ near the center of the first monopole which would depend on the orientation of the Dirac string of the second monopole. These contributions are unphysical and a more precise treatment of the interaction of far separated monopoles, as in [@25; @29; @30], does not involve any orientation dependent contributions, at least in the $d \nu \gg 1$ limit, but will only involve interactions similar to the gauge-invariant interaction terms $S_{\text{inter.}, > r_0} + S_{\text{inter.}, < r_0}$ evaluated above.[^24] summarize, using the relations , , and , the (Dirac-string-independent) action of two far separated monopole solutions in the limit $\nu^{-1} = {NL \over 2\pi} \ll r_0 \ll d$, with $|\text{x}_1 - \text{x}_2| = d$, can be summarized as: $$\label{eq:2.29} S_{2 - \text{monopoles}} = 2 \times {8 \pi^2 \over g^2 N} + {2 \pi L \over g^2 d}\; {q^i_{m1} \cdot q^j_{m2}} - {2 \pi L \over g^2 d} \;{q^i_{e1} \cdot q^j_{e2}} + \text{O}({L \nu^{-1} \over g^2d^2})~.$$ Although for simplicity we assumed that both solutions are BPS, the relation is general and applies to two arbitrary monopoles or anti-monopoles. Therefore the action of a dilute gas of $n^{(i)}$ monopoles of type $i$ and $\bar{n}^{(i)}$ anti-monopoles of type $i$ for $i = 1, ..., N$, referring to the KK monopole as the monopole of type $N$, with $n = \sum^{N}_{i = 1}({n}^{(i)} +\bar{n}^{(i)})$ their total number, is given by:[^25] $$\label{eq:2.30} S_{\text{monopole-gas}} = {8 \pi^2 \over g^2 N}n + S_{\text{int}, m} + S_{\text{int}, e} + \text{O}({n^{4/3} L \nu^{-1} \over g^2d^2})~.$$ In (\[eq:2.30\]), $S_{\text{int},m}$ ($S_{\text{int},e}$) is the sum of magnetic (“electric”) interaction terms similar to for every pair of monopoles in the gas: $$\label{eq:2.31} S_{\text{int},m} = {2\pi L \over g^2}\big{[} {1 \over 2} \sum_{ \underset{\text{dist. pairs}}{i, j, k_i, k_j} } {q^i_m \cdot q^j_m \over |r^{(i)}_{k_i} - r^{(j)}_{k_j}|} + {1 \over 2} \sum_{ \underset{\text{dist. pairs}}{i, j, \bar{k}_i, \bar{k}_j} } {\bar{q}^i_m \cdot \bar{q}^j_m \over |\bar{r}^{(i)}_{\bar{k}_i} - \bar{r}^{(j)}_{\bar{k}_j}|} + \sum_{{i, j, k_i, \bar{k}_j }} {q^i_m \cdot \bar{q}^j_m \over |r^{(i)}_{k_i} - \bar{r}^{(j)}_{\bar{k}_j}|} \big{]}~,$$ $$\label{eq:2.32} S_{\text{int},e} = - {2\pi L \over g^2} \big{[} {1 \over 2} \sum_{ \underset{\text{dist. pairs}}{i, j, k_i, k_j} } {q^i_e \cdot q^j_e \over |r^{(i)}_{k_i} - r^{(j)}_{k_j}|} + {1 \over 2} \sum_{ \underset{\text{dist. pairs}}{i, j, \bar{k}_i, \bar{k}_j} } {\bar{q}^i_e \cdot \bar{q}^j_e \over |\bar{r}^{(i)}_{\bar{k}_i} - \bar{r}^{(j)}_{\bar{k}_j}|} + \sum_{{i, j, k_i, \bar{k}_j }} {q^i_e \cdot \bar{q}^j_e \over |r^{(i)}_{k_i} - \bar{r}^{(j)}_{\bar{k}_j}|} \big{]}~,$$ with $1 \leq i, j \leq N$, $1 \leq k_i \leq n^{(i)}$ and $1 \leq \bar{k}_i \leq \bar{n}^{(i)}$. The summation is being performed over distinct pairs of monopole-monopole and anti-monopole-anti-monopole interactions and a factor of $1 \over 2$ has been included to cancel the double counting of pairs in these summations. Note that since the notion of anti-monopole is in regard to opposite magnetic charges, although $\bar{q}^j_m = -q^j_m$, this is not true for the electric charges, which satisfy $\bar{q}^j_e = q^j_e$. The reader should also be reminded that the electric interaction term will not be important for us in the quantum theory since it is gapped due to the one loop effective potential for the $A_4$ field and hence is of short range ($\sim { \nu^{-1}}$). Therefore in the next Section we will be only concerned with the magnetic interaction term . Derivation of the string tension action {#sec:2.4.1} --------------------------------------- The static quark-antiquark potential in a representation $r$ of the gauge group is determined by evaluating the expectation value of a rectangular Wilson loop of size $R\times T$ in representation $r$, and considering the leading exponential in the large Euclidean time $T$ limit [@04]: $$\label{eq:2.33} \underset{T\rightarrow \infty}{\lim} \langle W_r (R,T) \rangle = \underset{T\rightarrow \infty}{\lim} \langle \text{tr}_r ({\cal{P}}\text{exp}(\int_{R\times T}A_{\mu}dx^{\mu}))\rangle \ \sim \ \text{exp}(-V_r(R)T)~.$$ In confining gauge theories in the absence of string breaking effects the potential $V_r(R)$ has a linear behaviour $V_r(R) = \sigma_r R$ at large distances,[^26] where $\sigma_r$ is referred to as the string tension for quarks in the representation $r$. At intermediate distances ($\approx \Lambda^{-1}$), the string tension can have a dependence on the particular representation $r$, it is known that the asymptotic—a few $\Lambda^{-1}$ and more—string tension, because of colour screening by gluons, depends only on the $N$-ality $k$ of the representation $r$, hence asymptotically $\sigma_r$ is referred to as the $k$-string tension $\sigma_k$. In this Section we will be deriving an expression for the $k$-string tensions in dYM theory by evaluating . We want to calculate using the low energy degrees of freedom, to leading order in the limit of $NL \Lambda \rightarrow 0$: $$\label{eq:2.34} \langle W_r (R,T)\rangle = {\int [D\psi][D\bar{\psi}][D A] \text{tr}_r {\cal{P}} \text{exp}(i\oint_{R \times T} dx_{\mu} A^{\mu}) \text{exp}(-S_{dYM}) \over \int [D\psi][D\bar{\psi}][A] \text{exp}(-S_{dYM}) }~.$$ To evaluate the partition function $Z = \int [D\psi][D\bar{\psi}][DA]\text{exp}(-S_{dYM})$, we expand the action around the perturbative and non-perturbative minimum action configurations (the $2N$ minimum action monopole solutions discussed in Section \[sec:2.2.2\]), including the contribution of the approximate saddle points made up of far separated monopole configurations (dilute gas of monopoles), to second order and evaluate the functional determinant in these backgrounds, using the approximate factorization of determinants around widely separated monopoles. The result is the grand canonical partition function of a multi-component Coulomb gas [@01][^27]: $$\label{eq:2.35} Z = {Z_{\text{pert.}}} \underset{i=1} {\overset{N}{\Pi}} \{ \sum^{\infty}_{n^{(i)} = 0} {\zeta^{n^{(i)}} \over n^{(i)} !} \sum^{\infty}_{\bar{n}^{(i)} = 0} {\zeta^{\bar{n}^{(i)}} \over \bar{n}^{(i)} !} \int_{{\rm I\!R}^3}\underset{k=1} {\overset{n^{(i)}}{\Pi}} d\text{r}^{(i)}_k \int_{{\rm I\!R}^3}\underset{l=1} {\overset{\bar{n}^{(i)}}{\Pi}} d\bar{\text{r}}^{(i)}_l \} \text{exp}(- S_{\text{int},m})~,$$ where the product over $i$ implies the inclusion of the $N$ types of minimal action BPS and KK monopole-instantons (and anti-monopole-instantons) and the sum over $n^{(i)}, \bar{n}^{(i)}$ indicates that arbitrary numbers of such configurations with centers at $r_k^{(i)},\bar r_k^{(i)}$ are allowed. For any term in involving $n^{(i)}$ monopoles and $\bar{n}^{(i)}$ anti-monopoles for $i = 1, ..., N$, $S_{\text{int},m}$ is given by and the fugacity is: $$\label{eq:2.36} \zeta = C \text{e}^{-S_0} = \bar{A} {\bar{D}_f} \; m_W^3 (g^2(m_W) N)^{-2} \text{e}^{-8 \pi^2 /N g^2(m_W)},$$ similar to the expression for the fugacity derived in [@01]. The only difference is that now ${\bar{D}_f}$ the finite part of ${D_f \equiv {\text{det}^{n_f}(\slashed{D} + m) \over \text{det}^{n_f}(\slashed{D} + M )}}$, the Pauli-Villars regulated determinant of massive adjoint fermions, is replacing $\text{e}^{- \Delta S}$ in the expression for fugacity in [@01] (instead of the $\Delta S$ term in we now have massive adjoint fermions, of mass $m \sim m_W$, in ). $\bar{A}$ is a dimensionless and $N$-independent coefficient and the finite part of $D_f$ can be absorbed in its redefinition (after taking into account its effect on coupling renormalization; we omit any details on this). Consider now the following identity,[^28] where $\sigma$ denotes the $N$-component vector $(\sigma_1,$ $\sigma_2,...\sigma_N)$: $$\label{eq:2.37} \begin{split} & \int D[ \sigma] \;\text{exp}(-\int_{{\rm I\!R}^3} d^3\text{x} {1 \over 2} {g^2 \over 8 \pi^2 L}( \nabla \sigma )^2 ) \; \text{exp}( iq^i_m \cdot \sigma (\text{x}_1)) \;\text{exp}( iq^j_m \cdot \sigma(\text{x}_2)) = \\ & {Z_{\text{pert.}}} \times \; \text{exp} (-{2 \pi L \over g^2} {q^i_m\cdot q^j_m \over |\text{x}_1 - \text{x}_2|} ) \; \text{exp} (-2 \times {2 \pi L \over g^2} \lim_{|\text{x}| \rightarrow 0} {1 \over |\text{x}|})~. \end{split}$$ After regularizing the infinite self-energies ($\lim_{|\text{x}| \rightarrow 0} \{{1 \over |\text{x}|} -{\text{exp} (-\mu |\text{x}|) \over |\text{x}|} \} = \mu$) using the Pauli-Villars method, a typical term in , abbreviated t.t. below, using the analog of for $n$ monopoles, with $n^{(i)}$ monopoles and $\bar{n}^{(i)}$ anti-monopoles of each kind, can be written as: $$\label{eq:2.38} \begin{split} Z_{\text{t.t.}} & = \int D[ \sigma]\; \text{exp}(-\int_{{\rm I\!R}^3} d^3\text{x} {1 \over 2} {g^2 \over 8 \pi^2 L}( \nabla \sigma )^2 ) \; \times \\ & \underset{i=1} {\overset{N}{\Pi}} \{ {( \widetilde{N} \zeta)^{n^{(i)}} \over n^{(i)} !} { (\widetilde{N} \zeta )^{\bar{n}^{(i)}} \over \bar{n}^{(i)} !} \underset{k=1} {\overset{n^{(i)}}{\Pi}} \int_{{\rm I\!R}^3} d\text{r}^{(i)}_k \text{exp}( iq^i_m \cdot \sigma (\text{r}^{(i)}_k)) \underset{l=1} {\overset{\bar{n}^{(i)}}{\Pi}} \int_{{\rm I\!R}^3} d\bar{\text{r}}^{(i)}_l \text{exp}( i\bar{q}^i_m \cdot \sigma(\bar{\text{r}}^{(i)}_l) ) \}~, \end{split}$$ where $\tilde{N} = \text{exp} ( + {2 \pi L \over g^2} \mu )$. Before we continue, we pause to note that the scalar fields $(\sigma^1,...,\sigma^N)$ are the magnetic duals to the U(N) Cartan-subalgebra electric gauge fields, the so-called “dual photons.” For the purpose of the paragraph that follows, in order to elucidate the physical meaning of gradients of the $\sigma$ fields, we revert to Minkowski space. The duality relation, with $(+,-,-)$ metric, is $$\label{dualityrelation} F_{kl}^A = - {g^2 \over 2\sqrt{2} \pi L} \epsilon_{klm} \partial^m \sigma^A~,~~A=1,...,N~.$$ The kinetic term in the Minkowski space version of (\[eq:2.38\]) is nothing but a rewriting of the first (“magnetic”) term in Minkowski space version of the action (\[eq:2.23’\]) restricted to its Cartan subalgebra and considered for a U(N) gauge group via dual variables.[^29] In order to do this in a proper way consider the Minkowski space action of the 3-dimensional low energy theory in perturbation theory with the Bianchi identity imposed as a constraint via the auxiliary field $\sigma$ to eliminate gauge degrees of freedom: $$\label{bianchi} S = \int_{\mathbb{R}^{1,2}} \{ -{L \over 4 g^2}F^A_{kl}F^{A kl} + h \epsilon_{k l m } \partial^m F^{A kl}\sigma^A \}~, \;\;\;\;\;\; A =1, ...,N.$$ Integrating by parts the Lagrange multiplier term, completing the square of the $F^a_{kl}$ fields and integrating them out leaves an action only in terms of the dual fields $\sigma$: $S_{\text{dual}} = {2 \over L} h^2 g^2 \int_{\mathbb{R}^{1,2}} (\partial_k \sigma)^2 $. Demanding ${2 \over L} h^2 g^2 = {g^2 \over 2} {1 \over 8 \pi^2 L}$, the coefficient of the gradient of the $\sigma$ fields in , gives $h = 1/(4\sqrt{2}\pi)$. Varying the action with respect to $F_{kl}$, we obtain the duality relation . An immediate remark, relevant for the discussion in Section \[bagmodelsection\], is that the duality relation (\[dualityrelation\]) implies that spatial gradients of $\sigma$ represent perpendicular electric fields $\vec{E}^A$, i.e. $$E_i^A \equiv F_{i0}^A = {g^2 \over 2 \sqrt{2} \pi L} \epsilon_{ij} \partial_j \sigma^A~. \label{dualityelectric}$$ Returning to our main objective—obtaining the effective theory of the dYM vacuum—we sum over the contributions of all monopoles and antimonopoles in (\[eq:2.38\]), and find that the full partition function becomes: $$\label{eq:2.39} \hspace*{-1cm}Z = \int D[ \sigma] \text{exp}(-\int_{{\rm I\!R}^3} d^3\text{x} {1 \over 2} {g^2 \over 8 \pi^2 L}( \nabla \sigma )^2 ) \sum^{\infty}_{n = 0} { (\widetilde{N} \zeta )^{n} \over n!} \{ \sum^{N}_{i = 1} \int_{{\rm I\!R}^3} d^3\text{x}( e^{iq^i_m \cdot \sigma (\text{x})} + e^{i\bar{q}^i_m \cdot \sigma (\text{x})} ) \}^n~.$$ Thus, the final form of the dual photon action reads: $$\label{eq:2.40} Z = \int D[ \sigma] \text{exp}(-\int_{{\rm I\!R}^3} d^3\text{x} \{ {1 \over 2} {g^2 \over 8 \pi^2 L}( \nabla \sigma )^2 - \widetilde{\zeta}\; \sum^{N}_{i = 1} \text{cos}(q^i_m \cdot \sigma) \} )~,$$ where $\widetilde{\zeta} = 2 \tilde{N} \zeta$.[^30] Before working out the Wilson loop integral, we will derive the $NL \Lambda$ dependence of the fugacity and the dual photon mass, verify the dilute gas limit conjecture and discuss the hierarchy of scales in this theory. Using the one loop renormalization group invariant scale $\Lambda$ for $n_f = 1, 2$ flavours of Dirac fermions in the adjoint representation of the gauge group: $$\label{eq:2.41} \Lambda^{b_0} = \mu^{b_0} \; \text{exp}(- {8 \pi^2 \over N g^2}), \ \ \ \ \ \ \ \ b_0 = (11 - 4n_f)/3~,$$ we can determine the leading dependence of the fugacity on $NL\Lambda$. The Pauli-Villars scale used in (\[eq:2.37\]) should be thought of as the cutoff of the long-distance theory containing no charged excitations and should be taken below the scale of any charged excitation; for the sake of definiteness, we shall take $\mu \sim {g \sqrt{N} \over N L}$, the lowest eigenvalue of the holonomy fluctuations. Then, we can neglect $\tilde{N}$ compared to $\text{exp} (- S_0)$ in the small-$NL \Lambda$ limit.[^31] The one-loop massive fermion determinant contributes some calculable constant and renormalizes the coupling of the long-distance theory, as already accounted for in (\[eq:2.41\]). From and , neglecting any $\text{log}(NL \Lambda)$ (or, equivalently, $g^2 {N}$) dependence, for the leading $NL \Lambda$ dependence of the fugacity we obtain: $$\label{eq:2.42} \widetilde{\zeta} \sim \zeta \sim ({ 1 \over NL })^3 (NL)^{b_0} \Lambda^{b_0} = (NL\Lambda)^{b_0 - 3}\Lambda^{3}~.$$ The fugacity $\tilde{\zeta}$ or $\zeta$ is proportional to the monopole density $n_d$. For a gas with density $n_d$ the average distance between the particles in the gas is $\sim {1 \over n_d^{1/3}}$ and in order to verify the dilute gas conjecture we should have that this separation be much larger than the size of the monopoles, of order $NL$: $$\label{eq:2.43} d \sim {1 \over \zeta^{1/3} } \gg NL \longrightarrow (NL \Lambda)^{-{b_0\over 3}} \gg 1 \longrightarrow b_0 > 0 , \ \ n_f \leq 2~ .$$ Since we are working in the limit $NL \Lambda \rightarrow 0$, the condition will be satisfied if $b_0 > 0$, which is the same condition as the asymptotic freedom condition and gives $n_f \leq 2$ (or $n_f \leq 5/2$ if Majorana masses are considered instead). The mass of the dual photon can be read from . The coefficient of the quadratic term in the dual photon action, after expanding the cosine term and factoring out ${g^2 \over 8 \pi^2 L}$ is ${8 \pi^2 L \over 2g^2} \tilde{\zeta}$. We define the dual-photon mass scale $$\label{eq:2.44} m^2_{\gamma} = {8 \pi^2 NL \over Ng^2} \; \widetilde{\zeta} \sim (NL\Lambda)^{b_0-2} \;\Lambda^2 ~,$$ and note that it is of order the mass of the heaviest dual photon (the dual photon mass eigenvalues, after diagonalizing the quadratic term via a discrete Fourier transform, are $ m_\gamma \sin {\pi k \over N}$ with $k=1,...,N-1$). Thus, the hierarchy of scales in this theory can be summarized as: $$\label{eq:2.45} {\begin{split} m_W \sim {1 \over NL}\; &\gg \; m_H \sim {g \sqrt{N} \over NL} \sim \mu \; \gg {1 \over d} \sim ({1 \over NL \Lambda })^{1 - {b_0 \over 3}} \Lambda \; \gg \; m_{\gamma} \sim ({1 \over NL \Lambda})^{1 - {b_0 \over 2}} \Lambda~. \end{split}}$$ For $n_f = 1$, we have ${1 \over d} \gg \Lambda \gg m_{\gamma}$ , but for $n_f = 2$: ${1 \over d} \gg m_{\gamma} \gg \Lambda$; we stress again that the scale $\Lambda$ has no physical significance in the small-$L$ theory (except that to ensure weak coupling, we must have $m_W \gg \Lambda$). Now we will work out the Wilson loop integral. In the dilute gas limit ($NL\Lambda \rightarrow 0$) the leading contribution to the Wilson loop integral comes from the long distance abelian behaviour of the monopole gas far from the cores ($\sim NL$) of the monopoles. Therefore for a Wilson loop in the $\text{x}_1, \text{x}_2$ plane and representation $r$ with $N$-ality $k$ and for a typical monopole gas background involving $n$ monopoles we have: $$\label{eq:2.46} \begin{split} \hspace{-1cm}\{ \text{tr}_r \ \text{exp}( i \oint_{R\times T} A^c_m t_r^c d\text{x}^m) \}_{\text{typ. mon.}} & = \text{tr}_r \ \text{exp} (i \int_{S(R\times T)}\epsilon_{anm} \partial_n A^c_m t_r^c d\text{S}^a ) \\ & = \sum^{d(r)}_{j=1} \text{exp}( i \int_{S(R\times T)} d\text{x}_1d\text{x}_2 \overset{n}{\underset{i=1}{\sum}} \mu^j_r \cdot q^i_m\; {\text{R}^i_3 \over 2 |R^i - \text{x}|^{3}})~, \end{split}$$ where $c=1, ..., N-1$ labels the Cartan generators of $SU(N)$ in the representation $r$ and $\mu^j_r$ is it’s $j$-th weight. On the first line above, we used Gauss’ law to rewrite the Wilson loop integral as an integral of the magnetic flux through a surface $S$ spanning the loop, and on the second line, we replaced the magnetic field by the field of $n$ monopole-instantons at positions $R^i \in \R^3 $, $i=1,...,n$.[^32] Defining the solid angle $\eta(\text{x})$ that the Wilson loop is seen at from the point $\text{x} \in \R^3$, $\eta(\text{x})$ $\equiv$ $\int_{S(R\times T)} d\text{y}_1d\text{y}_2 {\text{x}_3 \over 2 |y - \text{x}|^{3}}$, we have for the contribution to the Wilson loop expectation value of an $n$-monopole configuration: $$\label{eq:2.47} \{ \text{tr}_r \ \text{exp}( i \oint_{R\times T} A_m d\text{x}^m) \}_{\text{typ. mon.}} = \sum^{d(r)}_{j=1} \text{exp}( i \sum^n_{i=1} \mu^j_r \cdot q^i_m \eta (\text{R}^i) )~.$$ Comparing with , we see that the effect of the Wilson loop insertion is to shift the $\sigma (\text{r}^{(i)}_k)$ field multiplying the magnetic charges in by $\mu^j_r \eta (\text{r}^{(i)}_k)$ (and similarly for $\sigma (\bar{\text{r}}^{(i)}_k)$ and the $\sigma(\text{x})$ field in ). Thus, shifting the $\sigma(\text{x})$ field by $\sigma(\text{x}) \rightarrow \sigma(\text{x}) - \mu^j_r \eta (\text{x})$ gives the final form of the expectation value of the Wilson loop in dYM theory to leading order in $NL\Lambda \rightarrow 0$, calculated using the low energy effective theory: $$\label{eq:2.48} \langle W_r(R,T)\rangle = \int D[ \sigma] \sum^{d(r)}_{j=1} \text{exp}(-\int_{{\rm I\!R}^3} d^3\text{x} \{ {1 \over 2} {g^2 \over 8 \pi^2 L}( \nabla \sigma - \mu^j_r \nabla \eta )^2 - \widetilde{\zeta} \sum^{N}_{i = 1} \text{cos}(q^i_m \cdot \sigma) \} ) / Z~.$$ The string tension action is given by making a saddle point approximation to . The Lagrange equations of motion for the contribution of the $j$-th weight of $r$ to the Wilson loop expectation value are: $$\label{eq:2.49} \nabla^2 \sigma_i = 2 \pi (\mu^j_r)_i \theta_A(\text{x}_1, \text{x}_2) \partial_3 \delta (\text{x}_3) + m^2_{\gamma} (\text{sin}(\sigma_i - \sigma_{i+1}) + \text{sin}(\sigma_i - \sigma_{i-1})) , \ \ \ i = 1, ..., N~,$$ where $\sigma_{0} \equiv \sigma_N$, $ \sigma_{N+1} \equiv \sigma_{1}$ and $\nabla^2 \eta (\text{x}) = 2 \pi\theta_A(\text{x}_1, \text{x}_2) \partial_3 \delta (\text{x}_3)$. $\theta_{A}(\text{x}_1, \text{x}_2)$ is one for $(\text{x}_1, \text{x}_2) \in A$ and zero for $(\text{x}_1, \text{x}_2) \notin A$, with $A$ being the area of the Wilson loop $R \times T$ in the $\text{x}_1, \text{x}_2$ plane. For large Wilson loops ($R, T \rightarrow \infty$) the saddle point configuration of is zero for regions outside the Wilson loop. The solution near the boundaries would be more complicated and gives a contribution proportional to the perimeter of the Wilson loop. The solution interior to the Wilson loop far from the boundaries would depend on $\text{x}_3$ only. The corresponding one-dimensional equation is: $$\label{eq:2.50} {\partial^2 \sigma_i \over \partial \text{x}_3 ^2} = 2 \pi (\mu^j_r)_i \theta_A(\text{x}_1, \text{x}_2) \partial_3 \delta (\text{x}_3) + m^2_{\gamma} (\text{sin}(\sigma_i - \sigma_{i+1}) + \text{sin}(\sigma_i - \sigma_{i-1})) , (\text{x}_1 , \text{x}_2) \in A~.$$ Eq.  represents a boundary value problem showing a discontinuity of $2 \pi (\mu^j_r)_i$ for the $\sigma_i$ $(i=1,...,N)$ fields at $\text{x}_3 = 0$. Therefore to leading order in $NL \Lambda \rightarrow 0$ (the saddle point approximation is valid in this limit) and $R, T \rightarrow \infty$ is $$\label{eq:2.51} \langle W_r(R,T)\rangle \sim \sum^{d(r)}_{j=1} \text{exp}(- T^j_r RT)~,$$ given by a sum over the exponential contributions of the different weights of the representation $r$. Sources in every weight have their own string tension, $T^j_r$, given by: $$\label{eq:52} T^j_r = \underset{\sigma(\text{x}_3)}{\text{min}}\int^{+\infty}_{-\infty} d\text{x}_3 \{{1\over {2L}}{g^2 \over {8\pi^2}}({\partial \sigma \over \partial \text{x}_3})^2+ \tilde{\zeta} \overset{N}{\underset{i=1}{\sum}}[1- \text{cos}(\sigma_i-\sigma_{i+1})]\} \Big |_{\Delta \sigma(0) =2 \pi \mu^j_r}~,$$ with $\Delta \sigma(0) \equiv \sigma(0^+) - \sigma(0^-)$. Notice that because the long-distance theory is abelian, within the abelian theory, we can insert static quark sources with charges given by any $\mu_r^j$, $j=1, ... , d(r)$. Clearly, this is not the case in the full theory, where the entire representation appears and color screening from gluons is operative. In our $N L \Lambda \rightarrow 0$ limit, the gauge group is broken, $SU(N) \rightarrow U(1)^{N-1}$, and the screening is due to the heavy off-diagonal $W$-bosons, which were integrated out to arrive at (\[eq:2.51\]). Thus, we expect that at distances $R$ such that $T_r^j R > O(m_W)$, $W$-bosons can be produced (as in the Schwinger pair-creation mechanism) causing the strings in representation $r$ with higher string tensions to decay to the string with lowest tension in $r$. Hence, we shall not study all $T^j_r$ tensions, but will focus only on the strings of lowest tension confining quarks in representation $r$.[^33] It is shown, in Appendix \[sec:B1\], that any representation of $SU(N)$ with $N$-ality $k$ ($= 1, ..., N-1$) contains the $k$-th fundamental weight, $\mu_k$ (given by below) as one of its weights. Furthermore, in Section \[sec:5.1.1\] it is shown that this weight would give the lowest string tension action among the other weights of that representation. Therefore reduces to: $$\label{eq:2.53} \langle W_r (R,T)\rangle \sim \text{exp}(- T_k RT)~,$$ up to pre-exponential factors and subdominant terms corresponding to the higher string tensions. The $k$-string tension $T_k$, defined by: $$\label{eq:54} T_k=\underset{\sigma(\text{x}_3)}{\text{min}}\int^{+\infty}_{-\infty} d\text{x}_3 \{{1\over {2L}}{g^2 \over {8\pi^2}}({\partial \sigma \over \partial \text{x}_3})^2+ \tilde{\zeta} \overset{N}{\underset{i=1}{\sum}}[1- \text{cos}(\sigma_i-\sigma_{i+1})]\} \Big |_{\Delta \sigma(0) =2 \pi \mu_k}$$ will be the object of our numerical and analytical studies in the rest of this paper. String tensions in dYM: a numerical study {#numericsection} ========================================= String tension action {#sec:3.1} --------------------- As derived in Section \[sec:2.4.1\] the “$k$-string tension action” is given by: $$\label{eq:3.1} T_k=\underset{\sigma(\text{z})}{\text{min}}\int^{+\infty}_{-\infty} d\text{z}\{{1\over {2L}}{g^2 \over 8\pi^2}({\partial \sigma \over \partial \text{z}})^2+ \tilde{\zeta}\; \overset{N}{\underset{j=1}{\sum}}[1- \text{cos}(\sigma_j-\sigma_{j+1})]\} \Big |_{\Delta \sigma(0) =2 \pi \mu_k}, \ \ \sigma_{N+1}\equiv \sigma_1~,$$ [where $\mu_k$, the fundamental weights of $SU(N)$, are obtained by solving the equation ${2\alpha_i \cdot \mu_k / \left | \alpha_i \right |^2}=\delta_{ik}$ [@09], and are given by:]{} $$\label{eq:3.2} \mu_k=({N - k \over N}, ..., \overset {\text{k-th}} {\widehat{{N-k \over N}}}, {-k \over N}, ..., {-k \over N}),\ \ \ \ \ \ 1\leq k \leq N-1,$$ Here $\alpha_i$ are the simple roots of $SU(N)$, given in their $N$-dimensional representation by: $$\label{eq:3.3} \alpha_i=(0,..,0,\overset {\text{i-th}}{\widehat{1}},-1,0,...,0), \ \ \ \ \ \ 1\leq i \leq N-1.$$ In deriving the relation for the $\mu_k$’s, it is assumed that the $\mu_k$’s and $\alpha_i$’s span the same subspace in ${\rm I\!R}^N$, orthogonal to the vector $(1,1,1,...,1,1)$. The string tension action will be minimized when the discontinuity $\Delta \sigma(0) = 2 \pi \mu_k$ is equally split between $\sigma (0^+)$ and $\sigma (0^-)$.[^34] Therefore the value of the action would also be equally split between the positive and negative $z$-axis and we can consider half the $k$-string action. Defining $m_{\gamma}^2 \equiv {8\pi^2 \over g^2} L \tilde{\zeta}$, the parameter-free form of half of is given by making the change of variable $z ={z' {1 \over \sqrt {2} m_{\gamma}}}$: \[eq:3.4\] & [ m\_ ]{} [T\_k 2]{} = \_[0]{}\^[+]{} d’{([f ’]{})\^2+\[1-(f\_j-f\_[j+1]{})\]}  ,\ & f(+ ) = 0, f(0) = \_k    f(’)=( [’ m\_]{})  . The equations of motion for $f$ are given by: $$\label{eq:3.5} {d^2f_j \over d\text{z}^2}={1\over 2} (\text{sin}(f_j-f_{j+1})+ \text{sin}(f_j-f_{j-1})),\ \ 1\leq j \leq N, \ f_0\equiv f_{N}, \ f_{N+1} \equiv f_{1}.$$ It is possible to solve the equations of motion directly for $SU(2)$ and $SU(3)$ and derive the exact value of .[^35] The solution for $SU(2)$ and its corresponding $\bar{T}_1$ value is: $$\label{eq:3.6} -f_2(\text{z}) = f_1(\text{z}) = 2 \text{ arctan}(\text{exp}(-\sqrt{2}\;\text{z}))\xrightarrow{\text{after inserting in \eqref{eq:3.4}}} \bar{T}_1 = 8/\sqrt{2}~.$$ Due to charge conjugation symmetry $\bar{T}_{k} = \bar{T}_{N-k}$ [@01] hence for $SU(3)$ $\bar{T}_1 = \bar{T}_2$ and therefore it suffices to solve the equations of motion and find the action for the first fundamental weight $\mu_1$ of $SU(3)$: $$\label{eq:3.7} -2f_2(\text{z}) = -2f_3(\text{z}) = f_1(\text{z}) = {8 \over 3} \text{ arctan}(\text{exp}(-\sqrt{3/2}\;\text{z}))\xrightarrow{\text{after inserting in \eqref{eq:3.4}}} \bar{T}_1 = 16/\sqrt{6}~>$$ In the following Sections, these exact values will be used as a check on our numerical methods. Discretization of the string tension action {#sec:3.3} ------------------------------------------- We will obtain the numerical value of the string tensions in deformed Yang-Mills theory by discretizing and minimizing the multivariable function obtained upon discretization. We set the boundary conditions at $f_j(0)=\pi (\mu_k)_j$ and $f_j(J)=0$ for $J > 0$. To discretize divide the interval $[0,J]$ into $m$ partitions and consider the following array of discretized variables: $$\label{eq:3.8} \{f_{jl}\,|\,\,\, 1\leq j \leq N\,,\,\,\,0\leq l \leq m\,,\,\,\, f_{j0}=\pi (\mu_k)_j\,,\, f_{jm}=0\}~.$$ We denote $\delta \text{z} = J/m$, and introduce the discretized functions $f_j^{(m)}$, linearly interpolated in every interval of width $\delta \text{z}$:[^36] $$\label{eq:3.9} f^{(m)}_j(\text{z})=f_{jh}+{f_{jh+1}-f_{jh}\over \delta \text{z}}( \text{z}-h\delta \text{z})\,, \,\,\, \text{z} \in [h\delta \text{z},(h+1)\delta \text{z}]\,,\, h = 0, ..., m-1~.$$ Inserting in and performing the $\text{z}$ integration, the linearly discretized action is: $$\label{eq:3.10} \begin{split} \bar{T}^{m,J}_k=\bar{T}^{m,J}_{k1}+\bar{T}^{m,J}_{k2}, \ &\bar{T}^{m,J}_{k1} = \underset{j,h}{\sum}{(f_{jh+1}-f_{jh})^2\over \delta \text{z}}, \\ &\bar{T}^{m,J}_{k2} = NJ-\delta \text{z} \underset{j,h}{\sum}{\text{sin}(f_{jh+1}-f_{j+1h+1})- \text{sin}(f_{jh}-f_{j+1h})\over{(f_{jh+1}-f_{j+1h+1})-(f_{jh}-f_{j+1h})}}~, \end{split}$$ where we split the discretized action into a kinetic ($\bar{T}^{m,J}_{k1}$) and potential ($\bar{T}^{m,J}_{k2}$)parts. Notice their different scalings with the width of partition $\delta \text{z}$ (we shall make use of this fact in Section \[bagmodelsection\] when discussing similarities with the MIT Bag Model). Minimization of the string tension action and error analysis {#sec:minimize} ------------------------------------------------------------ In order to obtain more accurate numerical results and have control over the minimization process, a systematic method is utilized for minimizing the multivariable function . For sufficiently small $\delta \text{z}$, $\bar{T}^{m,J}_k$ has a parabolic structure along the direction of any variable $f_{lp}$ (i.e.${\partial^2\bar{T}^{m,J}_k\over \partial f_{lp}^2}>0$). The second derivative of the 1st term in with respect to $f_{lp}$ is ${4\over \delta \text{z}}$ and the second derivative with respect to the 2nd term is at least[^37] $-{4\over 3}\delta \text{z}$, hence: $$\label{eq:3.11} {\partial^2\bar{T}^{m,J}_k\over \partial f_{jh}^2}> 0 \Longrightarrow {4\over \delta \text{z}}-{4\over 3}\delta \text{z} > 0 \Longrightarrow \delta \text{z} < \sqrt{3}~.$$ To minimize we assume $\delta z$ small enough in order to satisfy and have a parabolic structure along the direction of any variable. The string tension action and its discretized form are positive quantities with the extremum solution of being the minimum point of the action, therefore following the parabolas along the direction of any variable downward should lead us to this minimum point. In order to do this in a systematic way $R$ random points are generated in the $N\times (m-1)$ dimensional space of discretized variables and starting from each random point the multivariable function is *minimized to width* $w$ (i.e. it is minimized to a point such that moving $w$ in either direction along any variable $f_{lp}$ and keeping other variables fixed gives a higher value for the action). This process is continued by dividing $w$ in half and *minimizing to width* ${w\over 2}$ and further continued to *minimization to width* ${w\over 2^n}$ at the $n$-th step until the difference between the string tension value at step $n$ and $n-1$ is sufficiently small. Let $X_n$ denote the random variable for the value of the string tension at step $n$ obtained by this minimization process. The *minimization error* (i.e.$| \text{min}\bar{T}^{m,J}_k - \langle X_n\rangle |$) in minimizing the multivariable function is reduced to the desired accuracy if the following quantities are sufficiently small: \[eq:3.12\] i)  [\_n ]{} ,           ii)  |X\_n-X\_[n-1]{}| ,&& where $\sigma_n$ is the standard deviation of $X_n$ and $\langle X_n\rangle$ denotes the average of $X_n$. The same analysis described above is done for $2m$ number of partitions and the *discretization error* (i.e. $| \text{min}\bar{T}^{m,J}_k - \text{min} \bar{T}^{\infty,J}_k|$) of the string tensions is reduced to the desired accuracy if the difference between the string tension values obtained for $m$ number of partitions and $2m$ number of partitions is small enough. We consider the difference $|\text{min}\bar{T}^{m,J}_k - \text{min} \bar{T}^{2m,J}_k|$ as an estimate for the *discretization error* of the string tension values obtained for $2m$ number of partitions. The boundary value number $\text{z} = J$ is assumed large enough to ensure the *truncation error* (i.e. $|\text{min} \bar{T}^{\infty,J}_k - \text{min} \bar{T}^{\infty,\infty}_k(=\bar{T}_k)|$) is small enough. An upper bound estimate for the truncation error is given by \[sec:A1\]: $$\label{eq:3.13} |\text{min} \bar{T}^{\infty,J}_k-\bar{T}_k|<2| \text{min} \bar{T}^{\infty,J}_{k1} - \text{min}\bar{T}^{\infty,J}_{k2}|.$$ The total error estimate in minimizing is given by: $$\label{eq:3.14} \begin{split} \text{Total Error} & = \text{Min. E.} + \text{Dis. E.} + \text{Trunc. E.} \\ & = |\langle X_n\rangle -\langle X_{n-1}\rangle | +{\sigma_n \over \sqrt{R}}+|\text{min}\bar{T}^{m,J}_k- \text{min}\bar{T}^{2m,J}_k| \\ &+ 2|\text{min} \bar{T}^{\infty,J}_{k1} - \text{min}\bar{T}^{\infty,J}_{k2}|~. \end{split}$$ The analysis of the errors defined above is discussed at length in Appendix \[errorappendix\]. Numerical value of ${k}$-string tensions in dYM {#sec:3.4} ----------------------------------------------- The numerical values of obtained by the minimization method above with their corresponding errors are listed in Table \[table:1\] below.[^38] Since the minimum value in always lies below the numerical values obtained in a numerical minimization procedure the upper bound estimate for the error has been indicated with a minus sign only.\ [|c|c|c|c|c|c|c|c|c|c|]{} --------- --- $SU(N)$ k --------- --- : The numerical values, , of half $k$-string tensions for gauge groups ranging from $SU(2)$ to $SU(10)$. The upper bound estimate for error is $-0.006$.[]{data-label="table:1"} &1&2 &3&4&5&6&7&8 &9\ 2 & 5.6576&-&-&-&-&-&-&-&-\ 3 & 6.5326&6.5326&-&-&-&-&-&-&-\ 4&6.8583&8.0006&6.8583&-&-&-&-&-&-\ 5&7.0140&8.6602&8.6602&7.0140&-&-&-&-&-\ 6&7.1001&9.0168&9.5547&9.0168&7.1001&-&-&-&-\ 7&7.1526&9.2318&10.0744&10.0744&9.2318&7.1526&-&-&-\ 8&7.1868&9.3713&10.4051&10.7192&10.4051&9.3713&7.1868&-&-\ 9&7.2104&9.4670&10.6292&11.1455&11.1455&10.6292&9.4670&7.2104&-\ 10&7.2273&9.5355&10.7882&11.4434&11.6491&11.4434&10.7882&9.5355&7.2273\ The minimization method was carried out for $J=14.0$, $m=100$ and $m=200$ number of partitions and the initial width was $w=1$. The number of random points $R$ generated initially is $R=24$. The multivariable function for $m=100$ was minimized to width $w \over 2^n$ for $n=20$. For $m=200$, was minimized to step $n=20$ for $SU(2 \leq N \leq 7)$ and to step $n=22$ for $SU(8 \leq N \leq 10)$. The numerical values listed in Table \[table:1\] refer to the numbers obtained with $m=200$ number of partitions rounded to the fourth decimal. A comparison of the known analytical results for $SU(2)$ () and $SU(3)$ () half $k$-string tensions with the numerical results is made in Table \[table:2\]. Numerical value Analytical (exact) value --------- -------------------- -------------------------------- $SU(2)$ $5.6576_{- 0.006}$ ${8/\sqrt{2}}\approx 5.6569$ $SU(3)$ $6.5326_{-0.006}$ ${16/ \sqrt{6}}\approx 6.5320$ : $SU(2)$ & $SU(3)$ numerical and analytical half $k$-string tensions.[]{data-label="table:2"} he same minimization process and error analysis used to derive the $SU(2)$ and $SU(3)$ half $k$-string tensions was utilized for the higher gauge groups. A discussion of the results shown in Table \[table:1\], especially regarding the $k$-scaling of string tensions and the large-$N$ limit will be given in Section \[sec:5\]. String tensions in dYM: perturbative evaluation {#sec:4} =============================================== Here, we will rederive the half $k$-string tensions in Table \[table:1\] by a perturbative evaluation of the saddle point. We stress that this is not an oxymoron and that, indeed, we will be using (resummed) expansions and only Gaussian integrals to compute a nonperturbative effect. In order to explain the main ideas, we briefly summarize them now, in an attempt to divorce them from the many technical details given later. Our starting point is the partition function of the Wilson loop inserted dual photon action for a fundamental weight $\mu_k$ from Section \[sec:2.4.1\]: $$\label{eq:4.1} Z^{\eta} =\int [D\sigma] \text{exp}(-\int_{{\rm I\!R}^3} d^3\text{x}\{{1\over {2L}}{g^2 \over 8\pi^2}({\nabla \sigma - \nabla \eta \mu_k })^2- \tilde{\zeta} \;\overset{N}{\underset{j=1}{\sum}}\text{cos}(\sigma_j-\sigma_{j+1})\})~.$$ For a Wilson loop in the $\text{y}_1, \text{y}_2$ plane, $\eta(\text{x}) = \underset{\text{A}}{\int} \text{dy}_1\text{dy}_2 {\text{x}_3 \over 2|\text{x} -\text{y}|^3}$ with $\text{y} = (\text{y}_1,\text{y}_2, 0)$ and “A” stands for the area of the rectangular Wilson loop $R \times T$ where the integral is being evaluated. We will rewrite in a form appropriate for a perturbative evaluation. Defining ${1\over \beta} \equiv {\tilde{\zeta} \over m_{\gamma}^3}$, rescaling $\text{x}_l \rightarrow {1 \over m_{\gamma}} {\hat{\text{x}}_l}$, $\text{y}_l \rightarrow {1 \over m_{\gamma}} \hat{\text{y}}_l$ ($l = 1,2,3$) with $m^2_{\gamma} = {8 \pi^2 \over g^2} L \tilde{\zeta}$ and expanding the cosine term (neglecting the leading constant term) we have: $$\label{eq:4.222} Z^{\eta} =\int [D\sigma] \text{exp}(-{1 \over \beta}\int_{{\rm I\!R}^3} d^3\hat{\text{x}}\{{1\over 2}(\partial_l \sigma)^2 - (\mu_k)_j\partial_l \sigma_j \partial_l \eta + {1 \over 2} (\partial_l \eta)^2 {\mu_k}^2 + {1 \over 2} (\sigma_j-\sigma_{j+1})^2 -{1 \over 4!} (\sigma_j-\sigma_{j+1})^4 + ... \})~,$$ with $\sigma_{N+1} \equiv \sigma_1$ and an implicit sum over $j=1,..,N$ and $l = 1,2,3$. We note that based on and for the leading $NL \Lambda$ dependence of $\beta$ we have $\beta \sim (NL \Lambda)^{b_0 \over 2}$ with $b_0>0$ given by . Thus, in the regime of validity $NL \Lambda \rightarrow 0$ of the semiclassical expansion $\beta \rightarrow 0$ and the partition function (\[eq:4.222\]) can be evaluated using the saddle point approximation, which was done numerically in Section \[numericsection\] and analytically in this Section. We shall present details for both $SU(2)$, in Section \[su2analyticsection\], and $SU(N)$, in Section \[suNanalyticsection\] below, but begin by explaining the salient points of the analytic method here. For this purpose consider for an SU(2) gauge group. Following steps from equations to we obtain: $$\label{eq:4.71} Z^{\eta}_{g^4} =\int [Dg] \text{exp}(-\int_{{\rm I\!R}^3} d^3\hat{\text{x}}\{{1\over 2}(\partial_l g)^2 + {m^2 \over 2} g^2 + \beta \lambda g^4 + {1 \over 2}({b \over 2\pi})^2 (\partial_l \eta)^2 \}) \text{exp}(+ {b \over \sqrt{\beta} } \underset{\text{A}}{\int} d\hat{\text{x}}_1 d\hat{\text{x}}_2 \partial_3g)~.$$ The differences between (\[eq:4.71\]) and (\[eq:4.222\]) are that: [*i.*]{}) the integration variable, the single dual photon of $SU(2)$, is now called $g$, [*ii.*]{}) nonlinear terms higher than quartic are discarded in order to simply illustrate the procedure, [*iii.*]{}) arbitrary dimensionless constants are introduced: mass parameter $m$, quartic coupling $\lambda$ and the boundary coefficient $b$ in (\[eq:4.71\]). Naturally the values of $m, \lambda$ and $b$ are determined by the original action in (\[eq:4.222\]) (or see below equation ), but it is convenient to keep them general in order to organize the expansion. As we explain below, we perform a combined expansion in $\beta$ to all orders, and in $\lambda$ to any desired order.[^39] To explain the procedure, from it can be seen that the $\beta$ parameter is similar to an $\hbar$ parameter and in (\[eq:4.71\]) the fields have been rescaled by this parameter for a perturbative evaluation of the saddle point. A rescaling of fields by a parameter will not change how an expansion in that parameter behaves therefore the expansion in $\beta$ is similar to an $\hbar$ expansion. In the limit of an infinite Wilson loop the $\beta$ expansion will be organized in the following way: $$\label{eq:betaexpansion} e^{- \hat{R} \hat{T}\{ {1 \over \beta}S_0(\lambda) + f_{1\text{-loop}}(\lambda) + \beta f_{2\text{-loop}}(\lambda) + ... \} }~ ,$$ $S_0$ is the one-dimensional saddle-point action, $f_{1\text{-loop}}$, $f_{2\text{-loop}}$, etc correspond to the summation of the one loop, two loop, etc diagrams. For brevity we have only included a $\lambda$ dependence although generally they depend on $\lambda$, $m$ and $b$. In this Section we will be only concerned with the leading saddle point result of order $1 \over \beta$ in the exponent of . To carry this out we expand the Wilson loop exponent and the $g^4$ term in and look for connected terms of order $1 \over \beta$. These will be the terms that exponentiate to produce the saddle point action in . As an example consider the Wilson loop exponent expanded to second order: ${1 \over 2!}({b \over \sqrt{\beta} } \underset{\text{A}}{\int} d\hat{\text{x}}_1 d\hat{\text{x}}_2 \partial_3g)^2$. When evaluated using the free massive propagator it is a connected diagram of order $1 \over \beta$, combined with the evaluation of the non-connected terms ${1 \over 4!}({b \over \sqrt{\beta} } \underset{\text{A}}{\int} d\hat{\text{x}}_1 d\hat{\text{x}}_2 \partial_3g)^4$, etc it would exponentiate to produce the term $- {\hat{R} \hat{T} \over \beta}S_0(\lambda = 0)$ in the exponent of . The odd terms in the expansion of the Wilson loop exponent vanish due to an odd functional integral. Similarly higher order contributions in $\lambda$ to the saddle point action can be evaluated. The order $\lambda$ contribution comes from the exponentiation of the connected diagram involving one $g^4$ term and four Wilson loop terms which is of order $1 \over \beta$: $ - \beta \lambda \int_{{\rm I\!R}^3} d^3{\text{x}} g^4$ ${1 \over 4!}({b \over \sqrt{\beta}} \int d\text{x}_1 d\text{x}_2 \partial_3 g)^4$. Terms of order $\lambda^2$, etc in the expansion of the saddle point action in can also be evaluated perturbatively which would result in the perturbative expansion of the saddle point in $\lambda$ (or more precisely ${ \lambda b^2 \over m^2}$) as in . Clearly the large value of $1 \over \beta$ causes no problem for these exponentiations since the radius of convergence of an exponential function is infinite. At every order in this combined expansion, we are faced with the calculation of Gaussian integrals only—hence the “perturbative evaluation” in the title of this Section. In this paper, we compute the leading-order contributions to the Wilson loop expectation value that behave as $$\label{wilsonperturbative} e^{- {\hat{R} \hat{T}\over \beta}( a_1 + a_2 \lambda)}~,$$ where ${\hat{R} \hat{T}}$ is the dimensionless area of the Wilson loop, defined in Section \[su2analyticsection\], and $a_1$( $= S_0(\lambda = 0)$), $a_2$ are numerical coefficients that we compute (for $SU(2)$ we will evaluate a few higher order corrections as well). Setting $\lambda =0$ in (\[wilsonperturbative\]) corresponds to ignoring non-linearities and is equivalent to a calculation of the saddle point action using the Gaussian approximation for the dual photon action. This was previously done in the 3d Polyakov model in [@Antonov:2003tz; @Anber:2013xfa]. However, as noted in [@Anber:2013xfa] and also follows from our results, the neglect of nonlinearities introduces an order unity error in the string tension. On the other hand, incorporating even only the leading quartic nonlinearity and setting $\lambda$ equal to the value that follows from (\[eq:4.222\]) at the end of the calculation leads to a significantly better agreement with the exact analytic or numerical data. One explanation for this is that the value of the saddle point functions approach zero quickly from its boundary value at $\text{x}_3 = 0$ which for $SU(2)$ is $\pi$ and for higher gauge groups is less than $\pi$ therefore the non-linearities will be suppressed. We show this in great detail in Appendix \[sec:C\] for a wide range of $N$ and $k$. Here, to illustrate the utility of the method, in Table \[table:71\], we only list the results for the $k=1$-string tension for gauge groups $SU(2)$—$SU(10)$, obtained via the method explained above and keeping the quartic nonlinearity only. A look at Table \[table:71\] shows that the convergence to the numerical (or exact analytic, when available) result is evident.[^40] $SU(N)$ $a_1$ $a_2 \lambda$ $a_1 + a_2 \lambda$ Num. value --------- -------- --------------- --------------------- ------------------- 2 9.870 -2.029 7.841 8.000 [(exact)]{} 3 11.396 -2.343 9.053 9.238 [(exact)]{} 4 11.913 -2.396 9.517 9.699 5 12.150 -2.410 9.740 9.919 6 12.277 -2.415 9.862 10.041 7 12.355 -2.417 9.938 10.114 8 12.405 -2.418 9.987 10.163 9 12.439 -2.418 10.021 10.196 10 12.463 -2.418 10.045 10.221 : Comparison of $N$-ality $1$ $k$-string tensions for $SU(2 \leq N \leq 10)$, obtained using the perturbative method explained here—leading contribution $a_1$ plus first subleading $a_2 \lambda$, from eq. (\[wilsonperturbative\])—with the results of the numerical study. To avoid confusion, we note that the exact analytic values for $SU(2)$ and $SU(3)$ in the dimensionless units used here are $8$ and $9.238$, respectively which agree with the numerical values listed in Appendix \[sec:C\]; see also the end of Section \[su2analyticsection\].[]{data-label="table:71"} Our final comment is that, in principle, this approach would also allow one to compute corrections to the leading semiclassical result. In the case at hand, this would necessitate a more precise matching of the long-distance theory to the underlying gauge theory; needless to say, any detailed study of such corrections is left for future work. Evaluation for $\text{SU(2)}$ {#su2analyticsection} ----------------------------- We will first demonstrate the basic ideas of the method in the simpler case of $SU(2)$ which an analytic solution to the saddle point is available, hence a direct comparison can be made with the perturbative evaluation. Defining $g_1 \equiv (\sigma_1 - \sigma_2)/\sqrt{2}$ and $g_2 \equiv (\sigma_1 + \sigma_2)/\sqrt{2}$ with $\mu_1 = (0.5,-0.5)$, for $SU(2)$ reduces to: $$\label{eq:4.3} Z^{\eta} =\int [Dg] \text{exp}(-{1 \over \beta}\int_{{\rm I\!R}^3} d^3\hat{\text{x}}\{{1\over 2}(\partial_l g_1)^2 +{1\over 2}(\partial_l g_2)^2 - {1\over \sqrt{2}}\partial_l g_1 \partial_l \eta + {1 \over 4} (\partial_l \eta)^2 + {4 \over 2} (g_1)^2 -{8 \over 4!} (g_1)^4 + ... \})~.$$ From it is clear that $g_2$ only appears in the kinetic term hence can be neglected. In what follows we will neglect the higher order interactions and demonstrate how the method works for a $g_1^4$ interaction term only. We replace $g_1$ with $g$ and use general dimensionless parameters for the mass, the $g^4$ coupling constant and the coefficient of the Wilson loop terms: $$\label{eq:4.4} Z^{\eta}_{g^4} =\int [Dg] \text{exp}(-{1 \over \beta}\int_{{\rm I\!R}^3} d^3\hat{\text{x}}\{{1\over 2}(\partial_l g)^2 + {m^2 \over 2} g^2 +\lambda g^4 - {b \over 2 \pi}\partial_l g \partial_l \eta + {1 \over 2}({b \over 2\pi})^2 (\partial_l \eta)^2 \})~.$$ For later use we note that the corresponding values of $m$, $\lambda$ and $b$ in are $m=2$, $\lambda = -{8 \over 4!}$ and $b = \sqrt{2} \pi$. Integrating by parts the Wilson loop term (linear term in $\partial_l g$) with: $$\label{eq:4.5} \partial_l \partial_l \eta(\text{x}) = -2\pi \underset{\text{A}}{\int} \text{dy}_1\text{dy}_2 \partial_3 \partial^2 {1 \over 4 \pi |\text{x} -\text{y}|} = 2\pi \theta_A(\text{x}_1,\text{x}_2) \partial_3 \delta (\text{x}_3)$$ gives: $$\label{eq:4.6} Z^{\eta}_{g^4} =\int [Dg] \text{exp}(-{1 \over \beta}\int_{{\rm I\!R}^3} d^3\hat{\text{x}}\{{1\over 2}(\partial_l g)^2 + {m^2 \over 2} g^2 +\lambda g^4 + b \partial_3 \delta (\hat{\text{x}}_3) \theta_A(\hat{\text{x}}_1, \hat{\text{x}}_2) g + {1 \over 2}({b \over 2\pi})^2 (\partial_l \eta)^2 \})~,$$ where $\theta_A(\hat{\text{x}}_1,\hat{\text{x}}_2)$ is $1$ on the Wilson loop area and zero otherwise. To evaluate perturbatively rescale $g \rightarrow \sqrt{\beta} g$: $$\label{eq:4.7} Z^{\eta}_{g^4} =\int [Dg] \text{exp}(-\int_{{\rm I\!R}^3} d^3\hat{\text{x}}\{{1\over 2}(\partial_l g)^2 + {m^2 \over 2} g^2 + \beta \lambda g^4 + {1 \over 2}({b \over 2\pi})^2 (\partial_l \eta)^2 \}) \text{exp}(+ {b \over \sqrt{\beta} } \underset{\text{A}}{\int} d\hat{\text{x}}_1 d\hat{\text{x}}_2 \partial_3g)~.$$ In what follows, we will drop the $hat$ on $\text{x}$; due to the rescaling made earlier it should be remembered that we are working with dimensionless variables. We will first calculate the Wilson loop exponent using the quadratic terms (kinetic term + mass term) in . In expanding the exponential $\text{exp}(+ {b \over \sqrt{\beta} } \underset{\text{A}}{\int} \text{dx}_1\text{dx}_2 \partial_3g)$ the odd terms vanish due to an odd functional integral and the even terms will be organized in the form of an expansion of an exponent (hence they would exponentiate) therefore it would be sufficient to only evaluate the second order term:[^41] $$\label{eq:4.8} \big\langle {b^2 \over 2 \beta } \underset{\text{A A}}{\iint}' \partial_3g \partial'_3g \big\rangle_{0} = {b^2 \over 2 \beta } \underset{\text{A A}}{\iint}' \partial_3 \partial'_3 P(\text{x}-\text{x}') = {b^2 \over 2 \beta } \underset{\text{A A}}{\iint}' \partial_3 \partial'_3 {\text{exp}(-m|\text{x}-\text{x}'|) \over 4 \pi |\text{x}-\text{x}'|}, \ \text{at} \ \text{x}_3= \text{x}'_3 = 0~.$$ The last expression can be evaluated as: $$\label{eq:4.9} \underset{\text{A A}}{\iint}' \partial_3 \partial'_3 P(\text{x}-\text{x}') = \underset{\text{A A}}{\iint'} \{ (-\partial^2 + m^2)P(\text{x}-\text{x}') + (\partial^2_1+\partial^2_2)P(\text{x}-\text{x}') - m^2P(\text{x}-\text{x}') \}~.$$ For a Wilson loop in the $\text{x}_1, \text{x}_2$ plane we can bring the second term on the right hand side of on the boundaries of the Wilson loop using the identity: $$\label{eq:4.10} \begin{split} \underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} P(\text{x}-\text{x}') & = \{ \underset{\text{A A}}{\iint'} dS^l dS'^l \partial_n\partial'_n - \underset{\text{A A}}{\iint'} dS^l dS'^k \partial_k\partial'_l \} P(\text{x}-\text{x}') \\ & = - \underset{\text{A A}}{\iint'} d^2\text{x} d^2\text{x}' \{ \partial_1^2 + \partial_2^2 \} P(\text{x}-\text{x}')~. \end{split}$$ Here, $b(\text{A})$ stands for the boundary of the Wilson loop area. Using relations , , and noting that $P(\text{x}-\text{x}')$ is the Greens function of the operator $-\partial^2 + m^2$ we have: $$\label{eq:4.12} \big\langle {b^2 \over 2 \beta } \underset{\text{A A}}{\iint}' \partial_3g \partial'_3g \big\rangle_{0} ={b^2 \over 2 \beta } \{ \delta(0) \hat{R}\hat{T} - \underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} P(\text{x}-\text{x}') - {m^2} \underset{\text{A A}}{\iint'} P(\text{x} - \text{x}') \}~.$$ The subscript of zero of the expectation value refers to it being evaluated using the free theory Lagrangian (i.e. at $\lambda =0$). The first term and the infinite part of the second term on the right hand side of would cancel with the infinite parts of the ${1 \over 2} ({b \over 2 \pi})^2(\partial_l \eta)^2$ term in to give a finite perimeter law for the Wilson loop and the third term on the right hand side of would give rise to an area law in the large area limit. Evaluating the ${1 \over 2} ({b \over 2 \pi})^2(\partial_l \eta)^2$ term in using and similar methods used to evaluate gives: \[eq:4.13\] \_[[IR]{}\^3]{} d\^3[1 2]{}([b 2]{})\^2 (\_l )\^2 & = - [1 2]{}([b 2]{})\^2 \_[[IR]{}\^3]{} d\^3 () 2\_A(\_1,\_2) \_3 (\_3) = -[b\^2 2]{} \_A d\^2\_A d\^2 \^2\_3 [1 4 | -|]{}\ & = [b\^2 2]{} { (0) - \^[l k]{}[1 4 | -|]{} }  .&& Further, , and give: $$\label{eq:4.14'} \hspace{-0.7cm} {\{ Z^{\eta}_{g^4} \}_{\lambda =0} \over \{ Z^{\eta}_{g^4} \}_{b=\lambda =0} } = \text{exp}(- {1 \over \beta} \{ {m^2 b^2 \over 2} \underset{\text{A A}}{\iint'} P(\text{x} - \text{x}') + {b^2 \over 2} \underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} ( P(\text{x}-\text{x}') - {1 \over 4 \pi |\text{x} -\text{x}'|}) \} )~.$$ In the limit that the area of the Wilson loop goes to infinity the first term on the right hand side of can be evaluated explicitly. Consider a Wilson loop with $\hat{R} = \hat{T} \equiv a$. Rescaling $\text{x}_k \rightarrow a \text{x}_k$, $\text{x}^{\prime}_k \rightarrow a \text{x}^{\prime}_k$ for $k = 1, 2$ and considering the limit $a \rightarrow \infty$ we have: \[eq:4.11\] ’P(-’) & = a\^3 d\^2 d\^2’ [(-am|-’|) 4 |-’|]{}\ & = a\^3 d\^2 2\^\_0 rdr [(-am) 4 ]{}\ &= [12m]{}(-m|\_3 - ’\_3|) a\^2 = [R T 2 m]{} . To arrive at the result above, we noted that in the limit of a large Wilson loop area ($a \rightarrow \infty$) due to the exponential suppression the main contribution to the $d^2\text{x}'$ integral comes from a small circle with radius $r \sim {1 \over a}$ centered at the point $\text{x} = (\text{x}_1, \text{x}_2 , 0)$ in the Wilson loop. Therefore it can be seen that the exact value of the $d^2\text{x}'$ integral in this limit would be given when the $dr$ integral is evaluated from zero to infinity. This would imply that the $d^2\text{x}'$ integral is independent of $\text{x}$, hence the $d^2\text{x}$ integral would be a trivial one over a unit square. Using , in the limit of a large Wilson loop becomes: $$\label{eq:4.14} {\{ Z^{\eta}_{g^4} \}_{\lambda =0} \over \{ Z^{\eta}_{g^4} \}_{b=\lambda =0} } = \text{exp}(- {1 \over \beta} \{ {mb^2 \over 4} \hat{R}\hat{T} + {b^2 \over 2} \underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} ( P(\text{x}-\text{x}') - {1 \over 4 \pi |\text{x} -\text{x}'|}) \} )~.$$ contains an area law term (first term in the exponent) and a perimeter law term (last two terms in the exponent). We note that without the use of the perturbative saddle point method the evaluation of the perimeter law term, due to the complicated behaviour of the saddle point solution near the boundaries of the Wilson loop, would have been a difficult task. The perimeter law term in is a finite quantity and proportional to $\text{Log}(a)a$ (for $a = \hat{R} = \hat{T}$) hence negligible compared to the area law term in the limit $a \rightarrow \infty$. Due to this and the fact that our main focus in this Section is the area law term we will drop this term in what follows. Another point worth mentioning, as will be seen in what follows, is that in the limit of $a \rightarrow \infty$ only the area law term in will receive $\lambda$ corrections. The saddle point equation of motion of is given by: $$\label{eq:4.15} \partial^2 g = m^2 g + 4 \lambda g^3 + b \partial_3 \delta (\text{x}_3) \theta_A(\text{x}_1, \text{x}_2)~.$$ For large Wilson loops, far from the boundaries of the Wilson loop the saddle point solution to obeys . For regions outside the Wilson loop the solution is zero. For regions close to the boundaries the saddle point solution will be more complicated and gives the perimeter law contribution in . For regions interior to the Wilson loop far from the boundaries the solution only depends on $\text{x}_3$ with a discontinuity of $b$ at $\text{x}_3 = 0$ and gives the area law contribution in . The corresponding one dimensional problem is given by: $$\label{eq:4.16} {d^2 \over d\text{x}^2_3} h = m^2 h + 4 \lambda h^3 \ \ \ \ \ \ h(0^+) = {b / 2} , \ \ \ \ h(0^-) = -{b / 2}~.$$ The discontinuity should be equally split above and below the Wilson loop in order to give the lowest action. The solution to for $\lambda = 0$ is given by: $$\label{eq:4.17} h(\text{x}_3) = \left \{ \begin{array}{c} {b \over 2} \text{exp}(-m\text{x}_3) \ \ \ \text{for} \ \ \ \text{x}_3 > 0 \\ -{b \over 2} \text{exp}(m\text{x}_3) \ \ \ \text{for} \ \ \ \text{x}_3 < 0 \\ \end{array} \right .$$ The action of this solution is: $$\label{eq:4.18} S[h] = \int^{+\infty}_{-\infty} dx_3 \{ {1 \over 2} ({dh \over dx_3})^2 + {1 \over 2} m h^2 \} = {m \over 4}b^2 , \ \ \ \ \text{at} \ \ \lambda = 0~,$$ which is the same as the coefficient of the area law term in . This demonstrates the validity of the perturbative saddle point method in producing the corresponding action of the saddle point boundary value problem. In order to further verify this method we will evaluate the saddle point action for a nonzero $\lambda$ and compare it with the corresponding analytic solution. We expand the exponential of the $g^4$ term in . The order $\lambda$ term contracts with the fourth order term in the expansion of the Wilson loop exponent: $$\label{eq:4.19} \big\langle - \beta \lambda \int d^3\text{x} g^4 ({b \over \sqrt{\beta}})^4{1\over 4!} (\int_A d^2\text{y} \partial_{\text{y}_3} g)^4 \big\rangle_{0,C} = -{4! \over 4!} {\lambda \over \beta} b^4 \int d^3\text{x} \prod^{4}_{i=1} \int_A d^2\text{y}^i-\partial_{\text{x}_3} P(\text{x}-\text{y}^i)~.$$ The subscript $C$ refers to the connected contribution. It has to be reminded that we will only be interested in connected terms of order $1 \over \beta$ since these terms would exponentiate to produce the series expansion of the saddle point action in . In the limit of $a \rightarrow \infty$ (after rescaling the variables $\text{x}_k \rightarrow a \text{x}_k, \text{y}^i_k \rightarrow a \text{y}^i_k$ for $k =1, 2$) due to the exponential suppression the $\text{x}_1$ and $\text{x}_2$ components of $\text{x}$ will be restricted to the Wilson loop and the main contribution to the integrals would come from a small circle of radius $\sim {1 \over a}$ centred at $(\text{x}_1, \text{x}_2,0)$. Following similar steps as we have: \[eq:4.20\] \_A d\^2 (-\_[\_3]{} P(-)) & = -\_[\_3]{} \_A d\^2 P(-) -\_[\_3]{} [1 2m]{}(-m|\_3|)\ & = [(\_3) 2]{} (-m |\_3|)  .&& Then becomes: $$\label{eq:4.211} \big\langle - {\lambda \over \beta} {b^4 \over 4!} \int d^3\text{x} g^4 (\int_A d^2\text{y} \partial_{\text{y}_3} g)^4 \big\rangle_{0,C} = - {1 \over \beta}{\lambda b^4 \over 16m} \Bbbk_1 a^2 = - {1 \over \beta}{\lambda b^4 \over 32m} \hat{R}\hat{T}~,$$ with $\Bbbk_1$ given by: $$\label{eq:4.22} \Bbbk_1 \equiv \int^{+\infty}_{-\infty} d\text{x}_3 \text{E}(\text{x}_3)^4 = {1 \over 2}, \ \ \ \text{E}(\text{x}_3) \equiv \text{sign}(\text{x}_3) \text{exp}(- |\text{x}_3|)~.$$ We note that the $d \text{x}_1d\text{x}_2$ integral would be a trivial one over a unit square and hence only an integral over $d\text{x}_3$ would remain. In a similar way the $\lambda^2$ term can be evaluated. This term would contract with the sixth order term in the expansion of the Wilson loop exponent: \[eq:4.23\] (d\^3 g\^4)\^2 ([b ]{})\^6[16!]{} (\_A d\^2 \_[\_3]{} g)\^6 \_[0,C]{} = &[16 6! 2 6!]{} [\^2 ]{} b\^6 d\^3d\^3’ \^[3]{}\_[i=1]{} \_A d\^2\^i\_[\_3]{} P(-\^i)\ & P(-’)\^[3]{}\_[i=1]{} \_A d\^2’\^i\_[’\_3]{} P(’-’\^i)  .& Rescaling the variables ($\text{x}_k \rightarrow a \text{x}_k , ... $ for $k = 1, 2$), considering the limit $a \rightarrow \infty$, using and we have: $$\label{eq:4.24} \big\langle {\lambda^2 b^6 \over 2 \beta 6!} (\int d^3\text{x} g^4)^2(\int_A d^2\text{y} \partial_{\text{y}_3} g)^6 \big\rangle_{0,C} = {1 \over \beta}{8 \lambda^2 b^6 \over 128} {\Bbbk_2 \over m^3} a^2 = {1 \over \beta}{\lambda^2 b^6 \over 16} {1 \over 24 m^3} \hat{R}\hat{T}~,$$ with $\Bbbk_2$ given by: \[eq:4.25\] \_2 d\_3d’\_3 (\_3)\^3 (-|\_3-’\_3|) (’\_3)\^3 = [1 24]{} . Higher order terms in $\lambda$ can be calculated similarly. The $\lambda^n$ term contracts with the $2n + 2$ order term in the expansion of the Wilson loop exponent. For $n > 3$ there would be more than one way of contracting the connected diagrams hence the evaluation would be more complicated but possible in principle Now we will directly solve for the saddle point and compare the result with the above expressions. is the one dimensional problem of interest. This is the motion of a particle moving in a potential $V(h) = -({m^2 \over 2}h^2 +\lambda h^4)$. Therefore the total energy is a constant of motion ${1\over 2}({dh \over d\text{x}_3})^2 + V(h) = C$. The minimum action corresponds to when $C=0$ therefore ${1\over 2}({dh \over d\text{x}_3})^2 =-V(h)$: \[eq:4.29\] S\[h\] & = \^[+]{}\_[-]{} d\_3 { [1 2]{} ([dh d\_3]{})\^2 - V(h) } = 2\^[0]{}\_[b 2]{} [d\_3 dh]{} dh { - 2V(h) } = 2m \_[0]{}\^[b 2]{} dh h  .&& We expand the square root[^42] and evaluate the integral term by term. We also multiply the action by a negative sign to take into account the negative in the exponent. We find: \[eq:4.30\] -S\[h\] & = \_[0]{}\^[b 2]{} dh { -2mh + -[2m]{}h\^3 + (-1)\^n [2 \^n m\^[2n-1]{}]{} [(2n-3)!! n!]{}h\^[2n+1]{} }\ & = -m [b\^2 4]{} - [b\^4 32 m]{} +\ & [= -[mb\^2 4]{} { 1 + [1 2]{} [b\^2 4 m\^2]{} - } ]{} . && The order $\lambda$ and $\lambda^2$ terms in match with the coefficients of ${\hat{R}\hat{T} \over \beta}$ in and respectively. This further demonstrates the validity of the perturbative evaluation of the saddle point. [The verification to higher orders in $\lambda$ ($n \geq 3$) can be made if the corresponding diagrams are evaluated.]{} The higher order terms in ($g^6_1, g^8_1, ...$) and their cross terms with each other can also be evaluated similarly. Next we will compare the $SU(2)$ $k$-string result of the perturbative evaluation of the saddle point to next to leading term, with the exact result of $SU(2)$ $k$-string in Table \[table:2\] of Section \[sec:3.4\]. The exact $SU(2)$ saddle point area law is given by , : \[eq:4.32\] [ Z\^ Z\^[0]{} ]{} = (- T\_1 RT) = (-[2 |[T]{}\_1 ]{} ) = (- [8 ]{} ),  , ,  0 , where $R = {1 \over m_{\gamma}} \hat{R}$, $T = {1 \over m_{\gamma}} \hat{T}$, $\beta = {m^3_{\gamma} / \tilde{\zeta}} $ and, from Table \[table:2\] of Section \[sec:3.4\], $\bar{T}_1 = {8 \over \sqrt{2}}$. Using and the perturbative saddle point method gives: \[eq:4.33\] [ Z\^ Z\^[0]{} ]{} = (- [1 ]{}([mb\^2 4]{} + [b\^4 32m]{} + ... ) ) = (- [1 ]{}( 7.84 + ...  ) ), , ,  0  .&& From the comment below equation , the values of $m=2$, $\lambda = - {8 \over 4!}$ and $b = \sqrt{2} \pi$ have been replaced. This shows the convergence of the $SU(2)$ perturbative saddle point method result to the exact value obtained by a direct calculation of the saddle point. Evaluation for $\text{SU(N)}$ {#suNanalyticsection} ----------------------------- Having shown how the method works for $SU(2)$, in this Section we will evaluate the $k$-string tensions perturbatively to next to leading order for $SU(N)$. We start from (recall ): $$\label{eq:4.2333} Z^{\eta} =\int [D\sigma] \text{exp}(-{1 \over \beta}\int_{{\rm I\!R}^3} d^3\hat{\text{x}}\{{1\over 2}(\partial_l \sigma)^2 - (\mu_k)_j\partial_l \sigma_j \partial_l \eta + {1 \over 2} (\partial_l \eta)^2 {\mu_k}^2 + {1 \over 2} (\sigma_j-\sigma_{j+1})^2 -{1 \over 4!} (\sigma_j-\sigma_{j+1})^4 + ... \})~.$$ The mass term in can be diagonalized. Let $(\sigma_j-\sigma_{j+1})^2 = \sigma^{T} A \sigma$ (for $j = 1, ...,N$) where $A$ is the following $N \times N$ matrix for $N \geq 3$: $$\label{eq:4.34} A_{ij} = 2 \ \text{for} \ i=j \ , \ A_{ij} = -1 \ \text{for} \ |i - j| = 1 \ , \ A_{1N} = A_{N1} = -1 \ , \ A_{ij} = 0 \ \text{otherwise}~,$$ while for ${SU(2)}$: $$\label{eq:4.35} A_{11} = A_{22} = 2 \ \text{and} \ A_{12} = A_{21} = -2~.$$ The matrix $A$ is symmetric and can be diagonalized by an orthogonal transformation $D$, explicitly $D^TD = I$, $A = D \Lambda D^T$. $D = (v_1 \ v_2 \ ... \ v_N)$ with $v_q$ the eigenvectors of $A$. This gives $(\sigma_j-\sigma_{j+1})^2 = g^{T} \Lambda g$ with $g = D^T \sigma$. $A$ has an eigenvalue[^43] of zero corresponding to the eigenvector $v^{T}_N \equiv ({1 \over \sqrt{N}}, ... , {1 \over \sqrt{N}})$. The corresponding field $g_N = (\sigma_1 + ... + \sigma_N)/ \sqrt{N}$ would be a massless component which decouples from the rest of the fields and hence can be neglected. We will now express the higher order interaction terms in a form convenient for a perturbative expansion. For this define the matrix $B$ as follows: $$\label{eq:4.36} B_{ij} = 1 \ \text{for} \ i=j \ , \ B_{ij} = -1 \ \text{for} \ j = i + 1 \ , \ B_{ij} = 0 \ \text{otherwise}~.$$ Defining $h_q \equiv \sigma_q - \sigma_{q+1}$ for $q = 1, ..., N-1$ we have $BDg = B\sigma = \big( \begin{array}{c} h \\ \sigma_N \end{array} \big )$. Also define the top left $(N-1) \times (N-1)$ block of the matrix $BD$ as $K \equiv (BD)_{N-1 \times N-1}$. Since the first $N-1$ elements of the last column of $BD$ are zero[^44] this gives: $K_{pq} g_q = h_p \equiv \sigma_p - \sigma_{p+1}$ for $p = 1, ..., N-1$. Using the previous notations and definitions, can now be rewritten as: $$\label{eq:4.37} \begin{split} Z^{\eta} =\int [Dg] \text{exp}( &-{1 \over \beta}\int_{{\rm I\!R}^3} d^3{\text{x}}\{ {1\over 2}(\partial_l g_N)^2+{1\over 2}(\partial_l g_q)^2 + {1 \over 2} \Lambda_q g^2_q - (\mu_k)_j D_{jq}\partial_l g_q \partial_l \eta \\ & + {1 \over 2} (\partial_l \eta)^2 {\mu_k}^2 -{1 \over 4!}\underset{p} {\sum}(K_{pq}g_q)^4 -{1 \over 4!}(\underset{p} {\sum}K_{pq}g_q)^4 + ... \})~, \end{split}$$ where a summation over $p,q$ and $l$ is implicit. Note that $(\mu_k)_j D_{jN} = 0$, hence the massless mode $g_N$ completely decouples from the rest of the modes and interactions. Integrating by parts the Wilson loop term and rescaling $g_q \rightarrow \sqrt{\beta} g_q$, we cast it, similar to (\[eq:4.7\]), into a form appropriate for a perturbative evaluation of the saddle point: \[eq:4.38\] Z\^ =( &-\_[[IR]{}\^3]{} d\^3{ [12]{}(\_l g\_N)\^2+[12]{}(\_l g\_q)\^2 + [1 2]{} \_q g\^2\_q + [1 2]{} ([b\_q 2]{})\^2 (\_l )\^2\ & +(K\_[pq]{}g\_q)\^4 + (K\_[pq]{}g\_q)\^4 + ... })(+ [b\_q ]{} \_A d\_1d\_2 \_3 g\_q) . Here, we defined $b_q = 2\pi (\mu_k)_j D_{jq}$ and $\lambda = - {8 \over 4!}$. To evaluate the Wilson loop exponent using the quadratic terms, we follow steps similar to the ones leading from to . We obtain the analogue of for $SU(N)$: $$\label{eq:4.39'} {\{ Z^{\eta}_{g^4} \}_{\lambda =0} \over \{ Z^{\eta}_{g^4} \}_{b_q=\lambda =0} } = \text{exp}(- {1 \over \beta} \{ {\Lambda_q b_q^2 \over 2} \underset{\text{A A}}{\iint'} P_q(\text{x} - \text{x}') + {b_q^2 \over 2} \underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} ( P_q(\text{x}-\text{x}') - {1 \over 4 \pi |\text{x} -\text{x}'|}) \} )~,$$ where $P_q(\text{x}-\text{x}') = {\text{exp}(-\sqrt{\Lambda_q} |\text{x}-\text{x}'|) / (4 \pi |\text{x}-\text{x}'|} )$ and with an implicit summation over $q$. In the limit of a large Wilson loop area ($\hat{R}, \hat{T} \rightarrow \infty$), eq. , following a similar calculation as in , reduces to: $$\label{eq:4.39} {\{ Z^{\eta}_{g^4} \}_{\lambda =0} \over \{ Z^{\eta}_{g^4} \}_{b_q=\lambda =0} } = \text{exp}(- {1 \over \beta} \{ {\sqrt{\Lambda_q}b^2_q \over 4} \hat{R}\hat{T} + {b_q^2 \over 2} \underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} ( P_q(\text{x}-\text{x}') - {1 \over 4 \pi |\text{x} -\text{x}'|}) \} )~,$$ Using the explicit form of the eigenvectors given in Footnote \[eigenvectorfootnote\], we can analytically show that confining strings have finite tension in the large-N limit, despite the vanishing mass gap.[^45] In particular, for $k=1$ strings we find the infinite-N limit $a_1 = \lim\limits_{N \rightarrow \infty} \sum\limits_{q=1}^{N-1}{1\over 4} \sqrt{\Lambda_q}b^2_q = 4 \pi$, consistent with the results from Table \[table:71\]. For further comments on the large-N limit, see Sections \[sec:compare\] and \[sec:largeN\]. Appendix \[sec:appxproduct\] discusses the large-N limit of the string tensions in product representations and gives more details on the analytic calculations using the leading-order saddle point method of this Section. Next, we evaluate the leading corrections to this result for $SU(N)$ and compare the values obtained with the numerical results in Table \[table:1\]. The integrals we need to do are the generalization of (\[eq:4.19\]) from the $SU(2)$ calculation: \[eq:4.40\] &\_ - d\^3 { (K\_[pq]{}g\_q)\^4 + (K\_[pq]{}g\_q)\^4 } [14!]{}([b\_q ]{} \_A d\^2 \_[\_3]{} g\_q)\^4 \_[0,C]{}\ & = -[1 ]{}[4! 4!]{} [8]{}P\_[q\_1q\_2q\_3q\_4]{} { \^4\_[i=1]{} K\_[pq\_i]{}b\_[q\_i]{} + \^4\_[i=1]{} K\_[p\_iq\_i]{}b\_[q\_i]{} }  , && and a summation over $p$, $q_i$, $p_i$ ($i =1,2,3,4$) from $1, ..., N-1$ is implicit. The quantities $P_{q_1q_2q_3q_4}$ are given by: $$\label{eq:4.41} P_{q_1q_2q_3q_4} \equiv \int d^3 \text{x} \prod^4_{i=1} \int_A d^2\text{y}^i \partial_{\text{y}^i_3}P_{q_i}(\text{x} -\text{y}^i)~.$$ In the limit that $a \rightarrow \infty$ ($a = \hat{R} = \hat{T}$) using we have: $$\label{eq:4.42} P_{q_1q_2q_3q_4} \overset{a \rightarrow \infty}{=} {a^2 \over 16} \bar{P}_{q_1q_2q_3q_4} \ \ \text{with} \ \ \bar{P}_{q_1q_2q_3q_4} = \int^{+\infty}_{-\infty} d\text{x}_3 \prod^4_{i=1} \text{E}(\sqrt{\Lambda_{q_i}} \text{x}_3)~,$$ hence $\text{I}_{\lambda}$ becomes: $$\label{eq:4.43} \text{I}_{\lambda} \overset{a \rightarrow \infty} {=} -{1 \over \beta}{\lambda \over 128}\bar{P}_{q_1q_2q_3q_4} \{ \prod^4_{i=1} K_{pq_i}b_{q_i} + \prod^4_{i=1} K_{p_iq_i}b_{q_i} \} \hat{R}\hat{T}~,$$ where $K_{pq} = B_{pj}D_{jq}$ and $b_q = 2\pi (\mu_k)_j D_{jq}$ for $q, p = 1, ..., N-1$. The leading area law term in and its leading correction in need to be evaluated numerically with mathematical software. The results are summarized in Appendix \[sec:C\]; results for $k=1$ strings for $SU(3)-SU(10)$ were already shown in Table \[table:71\]; the inclusion of the leading order correction brings the numerical value closer to the numerical or analytic (for $SU(3)$) value. $\mathbf{N}$-ality dependence and large-$\mathbf{N}$ behaviour of $\mathbf{k}$-strings in dYM {#sec:5} ============================================================================================== In this Section we give a discussion on two main questions regarding the properties of $k$-string tensions: their $N$-ality dependence and large-$N$ behaviour in dYM theory. The $N$-ality of an irreducible representation of a gauge group $SU(N)$ refers to the number of boxes in the Young tableaux of the representation mod $N$ [@09] or the charge of the representation under the action of the center element $\text{exp}(-i{2 \pi \over N})\text{I}$ of the gauge group. It is believed that asymptotically the string tensions in a gauge theory depend only on the $N$-ality of that representation, see [@04]. This is due to the screening effect by gluons. A cloud of gluons would transform any charge in a representation with $N$-ality $k$ to its $k$-antisymmetric representation which carries the stable lowest energy k-string among different representations with the same $N$-ality $k$. We will argue that this is also true in dYM theory and show that the asymptotic string tensions will only depend on the $N$-ality $k$ of the representation $k$. The screening by gluons, in the framework of dYM theory, is due to the pair production of $W$-bosons, an effect (in principle) calculable using weak coupling semiclassical methods. We discuss qualitatively the role of the unbroken $\Z_N$ center symmetry in dYM for the confining string properties and contrast them to those in another theory with abelian confinement—Seiberg-Witten theory. We also derive an approximate analytic formula for k-string ratios in dYM theory for $N \sim 10$ and smaller and have a comparison with known scaling laws of k-string ratios. In regards to their large-$N$ behaviour we show that dYM $k$-string ratios favour even power corrections similar to the sine law scaling and derive the leading terms in the $1/N$ expansion of $k$-string ratios in dYM theory. At the end we will argue that at large $N$ $k$-strings are not necessarily free in gauge theories; in other words, $T$$_{\text{k}}$ can remain smaller than $kT$$_{\text{1}}$ in the large-$N$ limit. $N$-ality dependence -------------------- ### Asymptotic string tensions depend only on the $N$-ality of the representation {#sec:5.1.1} The expectation value of the Wilson loop for charges in a representation $r$ with $N$-ality $k$ of $SU(N)$ evaluated using the low energy effective theory in dYM theory in the limit of $\hat{R}, \hat{T} \rightarrow \infty$ and $\beta (= {m^3_{\gamma} \over \tilde{\zeta}})\rightarrow 0$ using and is given by: $$\label{eq:5.1} \langle W_r(R,T) \rangle = \sum^{d(r)}_{i = 1} \text{exp}(- T^i_rRT) = \sum^{d(r)}_{i = 1} \text{exp}(- {2 \bar{T}^i_{r} \over \sqrt{2} \beta}\hat{R}\hat{T})~,$$ where $d(r)$ refers to the dimension of the representation $r$ and $\bar{T}^i_{r}$ is given by a similar expression as but with $\mu_k$ replaced by the weight $\mu^i_r$ of representation $r$ with $N$-ality $k$: \[eq:5.2\] [|[T]{}\^i\_r]{}= \_[0]{}\^[+]{} d{([f ]{})\^2+\[1-(f\_j-f\_[j+1]{})\]},      f(+ ) = 0, f(0) = \_r\^i  .&& Expression is the sum of exponential of area laws. The leading exponential in the limit of large $\hat{T}$ and $\hat{R}$ would give the string tension for charges in representation $r$ with $N$-ality $k$. Any representation of $SU(N)$ with $N$-ality $k$ contains the fundamental weight $\mu_k$ as one of its weights (Appendix \[sec:B1\]). Therefore in order to show that string tensions would only depend on the $N$-ality of the representation $r$ of the group $SU(N)$ we have to show that the lowest string tension action in corresponds to boundary conditions dictated by the fundamental weight $\mu_k$ among all the weights $\mu_r^i$ ($i=1, ..., d(r)$) of the representation $r$. Any weight of a representation of $SU(N)$ can be obtained from the highest weight by lowering with the simple roots [@09] and as noted above any representation with $N$-ality $k$ contains $\mu_k$ as one of its weights. Therefore any weight of a representation $r$ with $N$-ality $k$ can be obtained from the fundamental weight $\mu_k$ by adding or subtracting the simple roots. We will now qualitatively (but convincingly) argue that adding or subtracting any simple root from $\mu_k$ would result in boundary conditions that would give a value for the minimum of the action which is equal to[^46] or more than the value obtained by boundary conditions of $\mu_k$. The saddle point solutions $f_j$ of for $\mu_r^i = \mu_k$ start from $\pi (\mu_k)_j$ at $z = 0$ and decrease or increase monotonically to zero at $z = + \infty$. From the form of $\mu_k$ given in (\[eq:3.2\]), one sees that there are two discontinuities, as a function of $j$, in the boundary conditions for $f_j$. These occur between $j=N$ and $j=1$, since $(\mu_k)_N = -k/N$ and $(\mu_k)_1 =1 - k/N$, and between $j=k$ and $j=k+1$, as $(\mu_k)_k = 1- k/N$ and $(\mu_k)_{k+1} = -k/N$. These two discontinuities in the boundary conditions make the corresponding terms in the cosine potential $1 - \text{cos}(f_N - f_1)$ and $1 - \text{cos}(f_k - f_{k+1})$ to start from 2 at $z = 0$ and reach $0$ at $z = + \infty$ (this is in contrast to all the other terms, which start from $0$ at $z=0$ and reach $0$ again at $z = \infty$). Thus, for boundary conditions given by $\mu_k$ we would summarize the behaviour of $f_j$’s as follows. For the kinetic term in we would have $k$ functions $f_1, ..., f_k$ that start from $\pi (1 - k/N)$ at $z = 0$ and reach $0$ at $z = + \infty$ and $N - k$ functions $f_{k+1}, ..., f_N$ that start from $- \pi k/N$ at $z = 0$ and reach $0$ at $z = + \infty$. For the potential term, since only the difference between the $f_j$’s enters the cosine, the $1 - \text{cos}(f_N - f_1)$ and $1 - \text{cos}(f_k - f_{k+1})$ terms start at 2 at $z = 0$ and reach 0 at $z= + \infty$, while the rest of the terms start from 0 at $z = 0$ and reach 0 at $z= + \infty$. Now let us ask how this picture would change if simple roots are added or subtracted from $\mu_k$. The picture of the kinetic term will either remain the same (k functions start from $\pi (1-k/N)$ and $N - k$ functions starting from $- \pi k/N$) or it would become worse (and thus increase the value of action) in a way that the boundaries values at $z=0$ become higher and result in an increase in the kinetic term (since it is the square of the derivative of the functions). The same is true for the cosine term—any addition or subtraction of the simple roots from the weight $\mu_k$ would either not change the picture of the cosine term or it would make it worse (increase the value of action) in a way that we would have more than 2 discontinuities in the boundaries that would result in more than two terms of the potential term having to start from 2 at $z = 0$, or we would still have two discontinuities but the boundary conditions would have become larger and the cosine terms corresponding to these discontinuities would oscillate between 2 and zero more than once. Both our numerical results and the simple variational ansatz of Section \[bagmodelsection\] confirm this picture. ### Comparing different abelian confinements: strings in dYM vs. softly-broken Seiberg-Witten theory {#sec:compare} The two most-studied examples where confinement of quarks becomes analytically calculable within quantum field theory are softly-broken Seiberg-Witten theory on $\R^4$ and QCD(adj) with massive or massless adjoint fermions on $\R^3 \times \S^1$. This paper is devoted to the study of dYM theory, which belongs to the second class, QCD(adj) with massive adjoint fermions. Semiclassical calculability in dYM is achieved, as mentioned many times, by taking the $N L \Lambda \ll 1$ limit. In both dYM and Seiberg-Witten theory confinement is “abelian:”[^47] the confining strings form in a regime where $W$-bosons are not relevant and the dynamics of confinement is described by a weakly-coupled abelian gauge theory. In Seiberg-Witten theory, this is the dual magnetic gauge theory on $\R^4$, while in dYM it is the long-distance theory on $\R^3\times \S^1$—the theory of the dual photons discussed at length in earlier Sections. In both cases, the confining dynamics involves magnetically charged—and thus nonperturbative from the point of view of the electric gauge theory—objects: the magnetic monopoles or dyons in Seiberg-Witten theory condense to break the magnetic gauge symmetry, while in dYM, the proliferation of monopole-instantons in the vacuum (which should not really be called “condensation,” the title of [@Unsal:2007jx] nothwithstanding) leads to the expulsion of electric flux.[^48] We shall see, in the next Section, that the physics of confinement in dYM has a flavor very similar to the picture of the QCD vacuum underlying the MIT Bag Model. Here, we want to stress two aspects in which dYM confinement is distinct from Seiberg-Witten theory that have not been much discussed in the literature: 1. The presence of a global unbroken $\Z_N$ (zero-form[^49]) center symmetry in dYM vs. the fact that the Weyl group in Seiberg-Witten theory is broken [@15]. The unbroken $\Z_N$ symmetry has implications for the “meson” and “baryon” spectra of the theory, as we explain further in this Section. 2. The abelian large-$N$ behaviour: confining string tensions remain finite in dYM in the large-$N$, fixed $\Lambda N L \ll 1$ limit. This is different from their behaviour in the analogous limit of Seiberg-Witten theory, where the string tensions vanish along with the mass gap [@15]. For further discussion, see Section \[sec:largeN\]. Here we concentrate on the first point above: the unbroken $\Z_N$ center symmetry in dYM. In the long-distance theory, in the $N$-dimensional basis of dual photons we are using, this symmetry appears as a clock symmetry, taking $\sigma_i \rightarrow \sigma_{i+1}$, with $N+1 \equiv 1$. In gauge-variant terms, the action of the $\Z_N$ center symmetry resembles that of an unbroken cyclic subgroup of the Weyl symmetry of $SU(N)$, as can be seen by noting that it cyclically interchanges the $N$ monopole-instantons associated with the simple and affine root of the Lie algebra.[^50] On the other hand, in Seiberg-Witten theory, the Weyl group is spontaneously broken, as pointed out long ago [@15]. The different global symmetry realization has interesting implications for the nature of confining strings in the two theories. To illustrate the differences it suffices to consider the confinement of fundamental quarks in $SU(3)$. In dYM theory, there are degenerate “mesons” composed of quarks (introduced as static sources) of the three different colors, of weights $\nu_1 = \mu_1$, $\nu_2 = \nu_1 - \alpha_1$, and $\nu_3=\nu_2 - \alpha_2$, respectively. These mesons are confined by distinct flux tubes related by the $\Z_N$ global symmetry action (the action of $\Z_N$ on the weights of the $SU(N)$ fundamental representation is to cyclically permute them). Furthermore, the fluxes carried by these three strings add up to zero, so one can form a “baryon,” where the “baryon vertex” is a junction of three strings (a domain wall junction), as illustrated on Figure \[fig:dymstring\]. ![Strings between static quarks of different colors (denoted by color circles) in $SU(3)$ dYM theory. [*Left panel:*]{} $Q_i \overline{Q}_i$ mesons in $SU(3)$ dYM are degenerate, due to the unbroken $Z_3$ center symmetry. There are three flux tubes carrying fluxes $\nu_i$ ($i=1,2,3$), one for fundamental quarks of each weight (color). [*Right panel:*]{} A “baryon vertex” in dYM is a 3-domain wall junction, which exists due to the vanishing total flux $\nu_1+\nu_2+\nu_3=0$. Similar structures persist for arbitrary number of colors in dYM theory.](dYMstrings.pdf){width="\textwidth"} [\[fig:dymstring\]]{} In contrast, in $SU(3)$ Seiberg-Witten theory, there are two $U(1)$ magnetic gauge groups broken by the monopole condensate, giving rise to two Abrikosov-Nielsen-Olesen (ANO) vortices. The flux of one vortex is proportional to $\mu_1$ and confines quarks in the highest weight of the fundamental representation. The other flux tube carries electric flux proportional to $\mu_2$ (the second fundamental weight of $SU(3)$) and confines quarks in the highest weight of the two-index antisymmetric representation (anti-quarks, for $SU(3)$). There is no third flux tube. The picture of “mesons” in $SU(3)$ Seiberg-Witten theory that results is shown on Figure \[fig:SWstrings\]: the lowest and highest weights of the fundamental quarks are confined by the two ANO flux tubes, while the middle-weight quark is confined by two flux tubes: one of flux $\mu_2$ and an anti-flux tube of flux $\mu_1$. The lack of a third flux tube becomes especially noticeable when baryons are considered: baryons in Seiberg-Witten theory are “linear molecules” only, as shown on Figure \[fig:SWstrings\]. This difference persists and becomes more pronounced for higher rank $SU(N)$ gauge groups.[^51] ![Strings in $SU(3)$ Seiberg-Witten theory. [*Left panel:*]{} $Q_i \overline{Q}_i$ mesons for different color quarks are non degenerate, due to existence of only two ANO flux tubes (denoted by lines with a single or double arrow) carrying electric fluxes $\mu_1$ and $\mu_2$ (notice that $\nu_2 = \mu_2-\mu_1$), respectively. [*Right panel:*]{} Only linear baryons exist in Seiberg-Witten theory. Similar pictures hold for any number of colors.](SWstrings.pdf){width="\textwidth"} [\[fig:SWstrings\]]{} As the $SU(3)$ example illustrates, the different symmetry realizations in dYM and Seiberg-Witten theory have implications for the spectrum of mesons and baryons. We shall not pursue this further here, but only note that in dYM one can add dynamical massive quarks and the meson, baryon (as well as glueball) spectra can be studied within weakly-coupled field theory, revealing many unusual and surprising features discussed in [@Aitken:2017ayq]. ### An approximate form of $ \text{k}$-string ratios and the MIT Bag Model {#bagmodelsection} Here, we shall derive a naive analytic upper bound for the half $k$-string tensions in Table \[table:1\] by approximating the integral in in a simple manner. We shall arrive at a simple $k$-string tension scaling law, which is in good agreement with the available data, as described further below. We shall also elaborate on the similarity between confinement in dYM and the MIT Bag Model of the Yang-Mills vacuum. We begin by repeating the dimensionless half $k$-string tension action (recall e.g. eq. (\[eq:5.2\])): \[eq:3.41\] [|[T\_k]{}]{}= \_[0]{}\^[+]{} d’{([ ’]{})\^2+ \_[j=1]{}\^N\[1-(f\_j-f\_[j+1]{})\]},    (+ ) = 0, (0) = \_k  .   Here $\vec{f}$ represents the $N$-dimensional vector of dual photon fields (whose components are summed explicitly in the second term; we omit the arrows in what follows) and the boundary conditions at the origin and at infinity are the ones appropriate for static sources in the highest weight of the $k$-index antisymmetric tensor representation. A simple variational ansatz for the half domain wall extremizing (\[eq:3.41\]) can be obtained by approximating the first term in the action as a linear function connecting the boundary value $\pi \mu_k$ at $z = 0$ to zero at a finite positive $z = J$. The second term is approximated by simply taking its value at $z=0$ (i.e. with $f=\pi \mu_k$) multiplied by $J$; in other words, the fields $f_i$ are taken in the vacuum (where the potential term in (\[eq:3.41\]) vanishes) outside a region of width $J$ which represents the thickness of the flux tube in our variational ansatz. As the form of $\mu_k$ and the potential term imply, for $f= \pi \mu_k$ only two terms in the sum of $N$ cosines contribute a factor of $2$ each, giving rise to second term in (\[eq:5.3\]), while the remaining $N-2$ terms do not contribute.[^52] Collecting everything, using the explicit form of the fundamental weight $\mu_k$ from (\[eq:3.2\]), we obtain the string tension as a function of the one variational parameter $J$, the flux tube thickness: $$\label{eq:5.3} \bar{T_k}^{naive}(x) = J\{ ({\pi {N-k \over N}\over J})^2k + ({\pi {k \over N}\over J})^2(N-k) \} + 4J = {\beta_k \over J} + 4J~,$$ where the parameter $$\beta_k \equiv (\pi {N-k \over N})^2k + (\pi {k \over N})^2(N-k) = \pi^2 {(N-k)k \over N} , \label{betak}$$ is proportional to the quadratic Casimir of the $k$-index antisymmetric tensor. Extremizing with respect to $J$ gives $J_{k, \text{ext}} = {\sqrt{\beta_k} \over 2}$. The value of the string tension at the extremum point is: $$\label{eq:5.4} \bar{T}_{k}^{\text{naive}} = 4\pi \sqrt{(N-k)k \over N}~.$$ Although the relation is only a naive upper bound estimate for the $k$-strings in Table \[table:1\], its ratio with the fundamental ($k=1$) $k$-string gives a good fit to the ratios of $k$-strings of Table \[table:1\]: $$\label{eq:5.5} {\bar{T}_{k}^{\text{naive}} \over \bar{T}_{1}^{\text{naive}}} = \sqrt{(N-k)k \over N-1}~.$$ The relation is, in fact, known as the “square root of the Casimir” scaling law for $k$-string ratios. It was first seen to arise in the MIT Bag Model of the QCD vacuum a long time ago [@13].[^53] As far as we are aware, dYM theory is the only known example where this “square root of Casimir” k-string scaling has been seen to arise within a controlled approximation in quantum field theory. We shall now discuss the physics behind (\[eq:5.3\]) and (\[eq:5.5\]) and will argue that the similarity of strings in dYM to those in the MIT Bag Model is not an accident. The first term on the r.h.s. of (\[eq:5.3\]) represents the gradient energy of the $\sigma$-field. Recall that the duality relation (\[dualityrelation\]) maps spatial gradients of the dual photon field to electric fields in the perpendicular direction (i.e. to electric flux going from the quark to the antiquark, which are here taken at infinite separation). Thus, the $\beta_k \over J$ term represents the electric field energy cost (per unit length) for a flux tube of thickness $J$. The coefficient $\beta_k$, the total electric flux, is determined by the sources—quarks in the $k$-index antisymmetric tensor representation—and is proportional to the quadratic Casimir of that representation, as in the classical MIT Bag Model of the confining string.[^54] Naturally, in order to minimize its energy, the electric flux tube wants to expand, i.e. maximize $J$—in a perturbative vacuum, the chromoelectric field would relax to the dipole field of the quarks. The second term on the r.h.s. of (\[eq:5.3\]), equal to $4 J$, represents the energy cost per unit length to “expelling the vacuum” and replacing it with electric flux in a region of width $J$. This term represents the “volume energy cost,” proportional to the bag constant parameter of the MIT Bag Model. In dYM, the vacuum is a monopole-antimonopole medium which abhors electric flux and wants to minimize $J$; the “bag constant” in dYM is not a model parameter, but is determined by the fugacity of monopole-instantons, ultimately fixed by the underlying gauge theory. The compromise between the two contributions to the energy results in $k$-strings of width $J_{k, \text{ext}} =\sqrt\beta_k/2$ and tensions given in (\[eq:5.4\]). As we already alluded to, the agreement between the dYM and MIT Bag Model $k$-string tensions is not accidental. In the MIT Bag Model, the major assumption is that the chromoelectric fields within the confines of the (presumably small) bag can be treated classically, owing to asymptotic freedom. The “bag constant" of the YM vacuum, characterizing its abhorrence of electric flux, is introduced as a model parameter. In dYM, both the classical treatment of the Cartan electric fields and the expulsion of electric flux are dynamical features arising from the judiciously chosen deformation of YM theory and are justified in the $N \Lambda L \ll 1$ limit. Finally, we note that while the physical picture in dYM is similar to that in the bag model, the “square root of Casimir” scaling of $k$-string tensions discussed here is not exact in dYM, as it results from a simple variational estimate. It is only an upper bound on the string tensions in dYM, see the following Section and, in particular, Figure \[fig:1\]. ### Comparison with known scaling laws {#sec:5.1.4} It is known that the asymptotic string tensions depend only on the $N$-ality $k$ of the representation of the confined charges, hence they are often referred to as the $k$-strings. Different models of confinement make different predictions for the ratios of $k$-string tensions. The main ones are the sine law and Casimir scaling. We also include the square root of Casimir scaling in the list below, due to its similarity with the $k$-string ratios in dYM theory for N $\sim$ 10 and smaller: \[eq:5.6\] & : [T\_kT\_1]{} = [()]{},\ & : [T\_kT\_1]{} = [k (N-k) N-1]{},\ & : [T\_kT\_1]{} =  .&& In field theory calculations, usually the corresponding $k$-string tension is calculated to leading order in a small parameter expansion. It has to be noted that the above relations correspond to the leading order result in that expansion and, in each case, are subject to corrections. ![Comparison of $SU(10)$ $k$-string ratios of with dYM $k$-string ratios, labeled by “dYM”, to other $k$-string tension laws. The Sine law labeled by “sin”, the Casimir scaling by “cas”, and scaling with the Square root of the Casimir scaling by “sqrtcas”. From the known theoretical models predicting different scalings of $k$-string tensions, the ones in dYM are closest to the MIT Bag Model “square root of Casimir” $k$-string tension law. There is a clear physical reason behind this similarity, explained in Section \[bagmodelsection\].](figure_1.png){width="\textwidth"} [\[fig:1\]]{} The Sine law is found in Seiberg-Witten theory [@15], in MQCD [@16], in three-dimensional SU(N) gauge theories with massless Dirac or Majorana fermions [@61], and in some AdS/CFT-inspired models [@17]. Casimir scaling of string tensions refers to the relation between string tensions $T_r/T_F = C_2(r)/C_2(F)$, where $C_2(r)$ and $C_2(F)$ are the quadratic Casimir of representation $r$ and the fundamental representation, respectively ($T_r$ denotes the string tension for charges in representation $r$). This relation can be derived from the “dimensional reduction” form of the Yang-Mills vacuum wave functional [@18], from the stochastic vacuum picture [@19], and from certain supersymmetric dual models [@20]. $SU(3)$ lattice simulations have shown scaling with the Casimir of the representation $C_2(r)$ with a good accuracy [@21]: it holds at intermediate distances ($\lessapprox$ 1 fm) but at larger distances (asymptotically) gluons screen the charges down to the representation of the same non-zero $N$-ality with the lowest dimensionality which carries the most stable lowest string tension—then $C_2(r)$ is replaced by the Casimir of the $k$-antisymmetric representation which leads to the Casimir scaling relation shown in ; notice however, that for $N=3$ $T_1 = T_2$. Lattice studies of $3$-dimensional YM theory seem to also favor Casimir scaling of $k$-string tensions ratios for gauge groups up to $SU(8)$ [@Bringoltz:2008nd], while studies of $4$-dimensional YM theory (for similar number of colors) appear to favor scaling in-between the sine and Casimir laws, see [@Lucini:2012gg] for references and discussion. The various $k$-string ratios shown in are compared with dYM $k$-string ratios for $SU(10)$ in Figure \[fig:1\]. It is clear from the figure that the square root of Casimir scaling shows most similarity with the dYM $k$-string ratios. This scaling arises in the MIT Bag Model of QCD [@13] and the reasons for the similarity was discussed in Section \[bagmodelsection\]. Large-$\text{N}$ behaviour {#sec:largeN} -------------------------- One feature of the abelian large-N limit in dYM was already mentioned: in the $N\rightarrow \infty$, fixed-$NL \Lambda \ll 1$ double scaling limit, the mass gap vanishes, but the string tensions stay finite. This large-N behaviour is quite different from a similar abelian large-N limit of Seiberg-Witten theory, where both the string tensions and mass gap vanish [@15]. Furthermore, as observed in [@Cherman:2016jtu], in the above double-scaling limit on $\R^3\times \S^1$, where the size of the dimension $L\rightarrow 0$ and the number of colors $N\rightarrow \infty$, with $NL$-fixed, in both super Yang-Mills and dYM, the infrared theory can be viewed as a theory “living” in an emergent latticized dimension, in a manner reminiscent of T-duality in string theory. This is a behaviour not quite expected of quantum field theory and clearly deserves a better understanding. The results of this paper show the nonvanishing of the string tension in dYM at large N. In the remainder of this Section, we study the leading large-N corrections to k-string ratios and their large-N behaviour, for a range of N that includes exponentially large values, but does not strictly extent to infinity. The reason for this restriction, already mentioned in Section \[summary\] of the Introduction, is that our analysis has neglected the fact that at large values of N, the virtual effects of the W-bosons become important, as there is a large number of them. In particular, W-boson loops induce mixing between the Cartan algebra photons (and hence between dual photons), which were not incorporated in our effective Lagrangian. Similar to the discussion of ref. [@Cherman:2016jtu] for sYM (using the calculations of refs. [@08; @Anber:2014sda]), we estimate that these mixing terms become important when $N$ becomes comparable to $N^* = 2 \pi e^{+ {24 \pi^2 \over (11 - 4 n_f)(N g^2)}}$. This exponentially large value of $N^*$ is the one that applies to massless adjoint QCD, and uses the computations in [@vito]. The corresponding calculation for dYM (adjoint QCD with massive adjoint flavors) has not yet been performed, but we expect the appearance of a similar exponentially large $N^*$. Studying the role of these corrections in dYM is an interesting task for future work, which will allow to further study the intriguing features of the abelian large-N limit. ### Leading large-$\text{N}$ terms {#sec:5.2.1} In this Section, we derive the leading large-N corrections to k-string ratios in dYM theory for T$_2$/T$_1$ and T$_3$/T$_1$. We will show that the k-string ratios in dYM theory favour even power corrections in $1\over N$. For this we add noise[^55] of order $0.0005$, the typical value of error of dYM k-string ratios[^56] to the exact k-string ratios of the Casimir scaling and sine law, whose scaling behaviour is known, and analyze them along with the k-string ratios in dYM theory. From Figures \[fig:2\] and \[fig:3\] it can clearly be seen that the coefficient of the linear correction term in dYM k-string ratios similar to the sine law is suppressed (whereas for the Casimir scaling law it is of same order) compared to the constant or the coefficient of the second order term therefore it can be concluded that dYM $k$-string ratios similar to the sine law disfavour a linear correction term and favour even power corrections. To find the leading term and leading correction term, we add noise of order $0.0005$ to the exact k-string ratios of the sine law for $\text{SU}(5\leq N \leq 10)$ to generate data with errors of order of the errors of the dYM data. Next we generate n = 1000 noised data for dYM and the sine law data with noise and make even power polynomial fits: $c_0 + c_2\text{x}^2 + ... + c_p\text{x}^{p}$ for $p = 2k, k \geq 0$, with $\text{x}=1/N$. The average and standard deviation of $c_0$ and $c_2$ give estimates for the values of these coefficients and their errors. We increase $p$ and make higher order polynomial fits until consistent results are reached. Tables \[table:3\] and \[table:4\] summarize the values obtained by this analysis. It can be seen that consistent results are obtained for $p = 6$ and $p = 8$ polynomial fits. In fact the values of the $p = 6$ column for the noised sine law data are in agreement with the exact coefficients of the $1 \over N$ expansion of the sine law k-string ratios as can be seen from . This is not limited to the sine law. Any other function with even power corrections shows a similar behaviour and the results for $c_0$ and $c_2$ coefficients for an even polynomial fit with $p=6$ would agree with the true values of the coefficients in its $1/N$ expansion (e.g. doing the same analysis for a cosine(x) function with $\text{x} = {1 \over N}$). So assuming dYM k-string ratios have only even power corrections, the values of the coefficients in the $p = 6$ column would be in agreement with the true values in dYM theory. The following relations summarize the leading large $N$ corrections in sine, Casimir, square root of Casimir and dYM k-string ratios: \[eq:5.7\] &: [(k[N]{}) / ([N]{}) ]{} = k + (k/6 - k\^3/6)\^2 ([1 N]{})\^2 + ...  ,\ &: [k(N -k) / (N-1)]{} = k + (k - k\^2) ([1 N]{}) + ...  ,\ &: = k\^[1 2]{} + [1 2]{}(k\^[1 2]{} - k\^[3 2]{}) ([1 N]{}) + ...  ,\ &: \_2 / \_1 = 1.347 0.001 + (-2.7 0.2) ([1 N]{})\^2 + ...  ,\ & \_3 / \_1 = 1.570 0.001 + (-7.5 0.2) ([1 N]{})\^2 + ...  .&& {width="\textwidth"} [\[fig:2\]]{} {width="\textwidth"} [\[fig:3\]]{} p=2 p=4 p=6 p=8 ------------------- --------------------- --------------------- ------------------- ------------------- $\text{c}_0$(sin) 1.9962 $\pm$ 0.0001 2.001 $\pm$ 0.0004 2.001 $\pm$ 0.001 1.998 $\pm$ 0.005 $\text{c}_2$(sin) -9.458 $\pm$ 0.006 -9.91 $\pm$ 0.03 -9.9 $\pm$ 0.2 -9 $\pm$ 1 $\text{c}_0$(dYM) 1.3482 $\pm$ 0.0001 1.3465 $\pm$ 0.0004 1.347 $\pm$ 0.001 1.347 $\pm$ 0.005 $\text{c}_2$(dYM) -2.822 $\pm$ 0.006 -2.65 $\pm$ 0.03 -2.7 $\pm$ 0.2 -3 $\pm$ 1 : $c_0$ and $c_2$ for even power polynomial fits of order p for $\text{T}_2 / \text{T}_1$ for noised ($\sim$ 0.0005) sine law data and dYM[]{data-label="table:3"} p=2 p=4 p=6 p=8 ------------------- --------------------- --------------------- ------------------- ------------------- $\text{c}_0$(sin) 2.9397 $\pm$ 0.0001 2.9984 $\pm$ 0.0004 3.000 $\pm$ 0.001 2.999 $\pm$ 0.005 $\text{c}_2$(sin) -33.334 $\pm$ 0.006 -39.17 $\pm$ 0.03 -39.4 $\pm$ 0.2 -39 $\pm$ 1 $\text{c}_0$(dYM) 1.5815 $\pm$ 0.0001 1.5682 $\pm$ 0.0004 1.570 $\pm$ 0.001 1.569 $\pm$ 0.005 $\text{c}_2$(dYM) -8.594 $\pm$ 0.006 -7.27 $\pm$ 0.03 -7.5 $\pm$ 0.2 -7 $\pm$ 1 : $c_0$ and $c_2$ for even power polynomial fits of order p for $\text{T}_3 / \text{T}_1$ for noised ($\sim$ 0.0005) sine law data and dYM[]{data-label="table:4"} As a short summary of this Section, we argued that k-strings in dYM are not free at large N, i.e. $T_k/T_1 \ne k$, and leading corrections to $T_k/T_1$ are of order $1/N^2$. In the next Section, we discuss some theoretical questions behind these findings. ### Comments on free $\text{k}$-strings and large-$\text{N}$ factorization {#sec:5.2.2} An often-discussed expected behaviour of $k$-strings at large $N$ is that they become free, meaning that the string tension with $N$-ality $k$ becomes $k$ times the tension of the fundamental $k=1$-string at large $N$ [@14; @10]. From the previous Section, in particular , it can be clearly seen that the k-string tensions in dYM theory show a different behaviour: $\underset{N \rightarrow \infty}{\text{lim}}\text{T}_2 = (1.347 \pm 0.001) \text{T}_1 < 2 \text{T}_1 $ and $\underset{N \rightarrow \infty}{\text{lim}}\text{T}_3 = (1.570 \pm 0.001) \text{T}_1 < 3 \text{T}_1$. The usual line of reasoning that leads to the conclusion that k-strings become free at large $N$ is based on large-$N$ factorization and assumes the commutativity of the large-$N$ and large Euclidean time $T$ limits. We will show that factorization and commutativity of limits should be treated more carefully.[^57] e first briefly review the usual arguments that lead to free k-strings at large $N$:\ A correlator of two gauge invariant operators A and B can always be written as a factorized expectation value plus a connected expectation value. In the lattice strong coupling expansion and in perturbation theory in gauge theories it is known that the leading term in the large $N$ limit is the factorized one [@11]. Assuming a normalization $\langle \text{AB} \rangle\sim O(1)$ we have: \[eq:5.8\] = + \_C ,      \~O(1),      \_C \~O([1 N\^2]{}) . && In particular, we will apply this formula to the expectation value of a Wilson loop in the product representation: \[eq:5.9\] & W\_ (U\_ ... U\_)= { (U\_ ... U\_) (U\_ ... U\_) }= W\_ W\_\ & W\_= W\_ W\_= W\_W\_+ W\_W\_\_C  .&& A subscript of a “square” (as in $W_{\square}$) refers to the fundamental representation. The product of the link matrices $U$ is being taken along a rectangular Wilson loop $R \times T$. To find the k-string tensions we take the large $T$ and $R$ limit and consider the leading exponential on the right hand side of . To consider the properties of the k-strings at large $N$ we also take the large $N$ limit. If the large $T$ and $R$ and large $N$ limits commute, then we can reverse the order of limits. Taking the large $N$ limit first makes the connected term vanish, then taking the large $T$ and $R$ limit we would find: $$\label{eq:5.10}\nonumber \langle W_{\square \otimes \square}\rangle \sim \langle W_{\square}\rangle\langle W_{\square} \rangle, ~ \langle W_{\square \otimes \square}\rangle \sim \text{exp}(- T_2 RT), ~\langle W_{\square}\rangle \sim \text{exp}(- T_1 RT) ~~ \Longrightarrow~ ~T_2 = 2 T_1,$$ i.e. the result that the $k=2$-string tension is twice the fundamental string tension. The line of reasoning represented above leads to the result that $k$ strings are “non-interacting” and would be correct if the large $T$ and $R$ and large $N$ limits commuted, which is not always true, as we discuss at length below (see [@62] for a discussion in a similar framework and [@Witten:1978qu] for a reminder that large-distance and large-$N$ limits’ non-commutativity has a long history). An important difference between the large area limit and large-$N$ limit relevant to their non-commutativity is the fact that the large area limit is taken in the same quantum field theory where as the large-$N$ limit is taken in different quantum field theories. To study the general properties of field theories at large $N$, e.g. large $N$ factorization, one takes the large $N$ limit first but to study the asymptotic k-string tensions at large $N$, due to the non-commutativity of the large area and large N limits, one should not take the large $N$ limit first. For any $SU(N)$ theory, the proper way to find asymptotic $k$-string tensions at large $N$ is to first solve for the $k$-string tensions at fixed $N$, this is done by taking the large area $(RT)$ limit and considering the leading exponential in this limit. Then, once the asymptotic $k$-string tensions are determined for each $N$ from the coefficient of the area term of the leading exponential, the large-$N$ limit of $k$-string tensions can be taken. The limits cannot be taken in reverse order as the leading exponential in the large area ($RT$) limit, which gives the $k$-string tension for the given value of $N$, can be suppressed in the large-$N$ limit compared to exponentials sub-leading in the large area limit. Let us first illustrate this important point in a toy example, similar to the way the non-commutativity of limits is realized in dYM. As the discussion of dYM is somewhat lengthy and slightly technical[^58] we prefer to first illustrate the result by the following example. Consider the function $$\label{acd} g(A,N) = \text{exp}(-T_{N} A) + {1 \over N^p}~ \text{exp}(-T^{\prime}_{N} A)~, ~{\rm with} ~ p>0,$$ where $A$ stands in for the area of the Wilson loop. Let the large-$N$ limits of $T_N$ and $T'_N$, $\underset{N \rightarrow \infty}{\text{lim}}T^{\prime}_N = T^{\prime}$, $\underset{N \rightarrow \infty}{\text{lim}}T_N = T$, be such that $T^{\prime} < T$. For any large but fixed $N$ the leading term in the large-$A$ limit is $g(A,N) \sim \text{exp}(-T^{\prime} A)$ and for any large but fixed $A$ the leading term in the large-$N$ limit is $g(A,N) \sim \text{exp}(-TA)$. Therefore, if one is interested in the leading exponential in the large-$A$ limit, one should not take the large-$N$ limit first, as this will make the second term in (\[acd\]), which is leading in the large area limit, vanish. One would then find $g(A,N) \sim \text{exp}(-T A)$, which is an incorrect result for the leading exponential in the large-$A$ limit. A similar behaviour happens in dYM as we discuss in detail further below, see discussion after (\[eq:5.13’\]). In what follows, we shall see that in the regime of parameters studied in this work, in particular in the framework of a bounded large-$N$ (see the comments in the beginning of Section \[sec:largeN\]), the leading exponential in the large $T$ and $R$ limits, which determines the k-string tensions, comes from the connected term, although it can be shown that for fixed $R$ and $T$ this term will be sub-leading in $N$ compared to the factorized term, similar to the toy example of eqn. (\[acd\]). First, we argue how this can be seen more explicitly from the results of Section \[sec:5.1.1\]. There, it was argued that the lowest string tension action in the product representation of $N$-ality $2$ corresponds to boundary conditions on the dual photon fields determined by the fundamental weight $\mu_2$ ($\bar{T}^i_{\square \otimes \square}$ in relation for $\mu^i_{\square \otimes \square} = \mu_2$) and for a fundamental representation of unit $N$-ality it corresponds to $\mu_1$ ($\bar{T}^i_{\square}$ in relation for $\mu^i_{\square} = \mu_1$). Hence the leading exponential of the factorized term in the limit of large $\hat{T}$ and $\hat{R}$ is $\text{exp}(-2\times{2\bar{T}_1 \over \sqrt{2} \beta } \hat{R}\hat{T}) $ and the leading exponential for the Wilson loop in the product representation is $\text{exp}(-{2 \bar{T}_2 \over \sqrt{2} \beta } \hat{R}\hat{T})$. For large values of $N$ from equation we have $\underset{N \rightarrow \infty}{\text{lim}}2\bar{\text{T}}_2 = (1.347 \pm 0.001) 2\bar{\text{T}}_1 <2 \times 2 \bar{\text{T}}_1 $. Clearly, the factorized term can never produce this leading exponential which should, therefore, come from the connected term. This result quoted above can also be obtained without referring to numerics, via the perturbative saddle point method developed in Section \[sec:4\], as shown in Appendix \[sec:appxproduct\].\ Next, we wish to verify the large $N$ factorization result in dYM and directly argue that, for a Wilson loop in the product representation the factorized term is leading and the connected term is sub-leading for large $N$. The discussion that we begin now becomes more transparent and explicit after reviewing the calculations of Appendix \[sec:appxproduct\]. Consider the expectation value of a Wilson loop in the product representation ${\square \otimes \square}$. Based on , for fixed but large $R$ and $T$ we have: $$\label{eq:5.13'} \langle W_{{\square \otimes \square}}(R,T) \rangle \sim \sum^{d({\square \otimes \square})}_{h = 1} \text{exp}(- T^h_{{\square \otimes \square}}RT)~,$$ where $d({\square \otimes \square}) = N^2$ refers to the dimension of the product representation. In words, the expectation value of the Wilson loop in the product representation is given, in the abelianized dYM theory, by a sum of decaying exponentials, one for each weight $h$ of the product representation, with string tension $T^h_{{\square \otimes \square}}$ corresponding to each weight. On the other hand, for a Wilson loop in the fundamental representation $\square$ we have: $$\label{eq:5.14'} \langle W_{\square}(R,T)\rangle \sim \sum^{d(\square)}_{i = 1} \text{exp}(- T^i_{\square}RT) = N \text{exp}(- T_1RT)~.$$ Similar to (\[eq:5.13’\]), this is a sum of decaying exponentials, one for every weight of the fundamental representation, with the only simplification ocurring because of the unbroken $\Z_N$ center symmetry, ensuring that the string tensions for all weights of the fundamental representation have the same value $T_1$. Now, let us study (\[eq:5.13’\]) in more detail. Our considerations from this point to eqn.  are more qualitative than quantitatively rigorous (although, as already mentioned, they can be justified in the leading order perturbative evaluation of the saddle point, see Appendix \[sec:appxproduct\]). They carry similar flavor to our argument of Section \[sec:5.1.1\] that strings sourced by quarks with charges in the highest weight of the $k$-index antisymmetric representation have the smallest string tension. However, we find the considerations below quite suggestive and intuitive, supporting the large-$N$ vs. large-$RT$ limit subtlety. The weights of the product representation are given by the sum of the weights of the fundamental representation in : $\mu^{h}_{{\square \otimes \square}} \equiv \mu^{(ij)}_{{\square \otimes \square}} = \mu^{i}_{\square} + \mu^{j}_{\square}$ for $1 \leq h \leq N^2$ and $1 \leq i, j \leq N$. These weights enter the boundary conditions of the string tension action . In what follows, we show that for large $N$ and for $|i - j| \gg 1$ and $|i - j| \ll N$ the string tensions of the product representation become approximately equal to two times the string tension of the fundamental representation at large $N$, i.e. $T^{h}_{\square \otimes \square} \equiv T^{(ij)}_{\square \otimes \square} \approx 2 T_1 $. As there are $O(N^2)$ such tensions, it will be concluded, after considering eq. , that $\langle W_{{\square \otimes \square}}(R,T) \rangle - \langle W_{\square}(R,T) \rangle \langle W_{\square}(R,T) \rangle < O(N^2)$ and therefore the connected term would be sub-leading in $N$. Consider the fundamental string tension $T^q_\square$ given by with $r = \square$ and $q$ denoting one of the weights of the fundamental representation. The weight $\mu_\square^q$ is given in . Recall, from Section \[sec:3.3\], that the dual photon configuration extremizing the action of a given string interpolates between a value at the origin given by $\pi \mu^q_\square$ and zero at infinity. Since the $p$-th component of $\mu_\square^q$ is $(\mu_{\square }^q)_p = -{1 \over N} + \delta^{pq}$, for large values of $N$ all the components of $\sigma_a$, $1 \leq a \leq N$ at $z = 0$ approach zero except for the $q$’th component which approaches $\pi$. The fact that one component, namely $\sigma_q$, differs in its boundary conditions from the rest would result in a non-zero action for $T^q_\square$, otherwise if all components had the same boundary conditions (e.g. $-\pi/N$) at $z=0$ they would be linear functions interpolating between $-\pi/N$ at $z=0$ and zero at $z=J$ and when $J$ is taken to infinity would result in a zero action. This suggests that the main contribution to $T^q_\square$ would come from the components of $\sigma_a$ near[^59] the $q$’th (also, see Appendix \[sec:appxproduct\]). Conversely, the components farther away from the $q$’th component would approach a linear configuration, similar to the case when all boundary conditions were the same, in order to minimize the action as much as possible and will have negligible effect on the value of the string tension action $T^q_\square$, with their contribution being suppressed by a power of $1/N$. A similar picture is true for $T^{(ij)}_{\square \otimes \square}$ with $\mu^h_{\square \otimes \square} = \mu^{(ij)}_{{\square \otimes \square}}$.[^60] Due to the $Z_N$ symmetry of the action without loss of generality we can take $i = [(N - \Delta_{ij}+1) /2]$ and $j = [(N + \Delta_{ij}+1)/2]$ with $\Delta_{ij} = |i - j| \neq 0$; the square brackets refer to the integer part. For large $N$, all components $(\mu^{(ij)}_{\square \otimes \square})_p$ approach zero, except for the $i$-th and $j$-th components, which approach $1$. Consider now the calculation of the string tension $T^{(ij)}_{\square \otimes \square}$ for large $N$ with $|i - j| \gg 1$ and $|i - j| \ll N$. The components $\sigma_a$, $1 \leq a \leq N$ interpolate between $\pi (\mu^{(ij)}_{\square \otimes \square})_a$ at the origin and zero at infinity and therefore for large values of $N$ all the components would approach zero at $z=0$ except for $\sigma_i$ and $\sigma_j$ which approach $\pi$. Similar to the picture above for $T^q_\square$ it can be seen that the components of the dual photon fields $\sigma_a$ near the $\sigma_i$ and $\sigma_j$ components would make the main contribution to the string tension $T^{(ij)}_{\square \otimes \square}$. The components farther away from the $i$’th and $j$’th components would approach a linear configuration, similar to the case when all boundary conditions are the same, in order to minimize the action as much as possible and will have negligible effect on the values of the string tension action $T^{(ij)}_{\square \otimes \square}$, with their contribution being suppressed by a power of $1/N$. Next, we divide the string tension action of $T^{(ij)}_{\square \otimes \square}$ into two parts, one part associated with the components $\sigma_a$ for $1 \leq a \leq N /2$ and one for $N/2 < a \leq N$; at large-$N$ and $|i - j| \gg 1$, $|i - j| \ll N$, these actions become independent of each other. In each part, the components closer to the $i$’th and $j$’th components of $\sigma_a$, which are relevant to the string tension value and their boundary conditions are similar to the components near the $q$’th component of $\sigma_a$ for $T^q_\square$ for $SU([N/2])$.[^61] For large $N$, these string tensions approach—as per our numerical results of Table \[table:71\] or from the analytic study, recall paragraph after —a nonzero value $T_1$ with their differences suppressed by a power of $1/N$. From this observation, we conclude that for large $N$ and for $|i - j| \gg 1$ and $|i - j| \ll N$, $T^{(ij)}_{\square \otimes \square} \approx 2T_1$. Clearly, there are $O(N^2)$ such string tensions at large $N$. On the other hand, the highest weight of the antisymmetric two index representation, $\mu_2$, see , which was argued and numerically found to give rise to the smallest $N$-ality two string tension, $T_2 < 2 T_1$, is obtained from $ \mu^{(ij)}_{{\square \otimes \square}}$, by taking $i = j+1$ (mod$N$). There are $O(N)$ such string tensions, including the $\Z_N$-center orbit of the highest weight of the antisymmetric two-index representation. We now combine the results of the previous two paragraphs to conclude, recalling , that at large $N$ $$\label{eq:abd} \langle W_{{\square \otimes \square}}(R,T) \rangle \sim \sum^{d({\square \otimes \square})}_{h = 1} \text{exp}(- T^h_{{\square \otimes \square}}RT)~ \approx O(N^2) \; e^{- 2 T_1 RT} + O(N) \;e^{- T_2 RT}.$$ The first term in the last expression above represents the contribution of the $O(N^2)$ string tensions of weights $ \mu^{(ij)}_{{\square \otimes \square}}$ with $|i - j| \gg 1$ and $|i - j| \ll N$. The second term is the contribution of the $O(N)$ $k=2$ strings in the $\Z_N$ orbit of the highest weight of the two-index antisymmetric representation. We now note that eqn.  exactly mirrors the situation described and discussed earlier in eqn. , showing the subtlety of taking the large-$N$ vs. large-$RT$ limit. See also Appendix \[sec:appxproduct\], where is recovered using the leading-order perturbative saddle point, evaluated analytically for large-$N$. The discussion in this Subsection demonstrates that in the framework of a bounded large $N$ studied in this work (recall the preamble[^62] of Section \[sec:largeN\]) large $N$ factorization would not necessarily imply free k-strings and the leading exponential in the large area limit can come from the connected term of the correlator of two Wilson loops in the fundamental representation that is sub-leading in $N$ compared to the factorized term. We remind the reader that although the connected term is sub-leading in $N$ it would still contribute to the k-string tensions at large $N$—since, as already stressed at the beginning of this Section, to find asymptotic k-string tensions, the large area limit must be taken first and the leading exponential in this limit should be considered. Thus, no matter how large $N$ is, the connected term, which contains the leading exponential in the large area limit, would contribute to the k-string tensions at large $N$. ### A comment on “holonomy-decorated” Wilson loops Here, we want to make a point which gives additional justification of our emphasis to study $k$-strings of minimal tensions, corresponding to quark sources of a particular weight (e.g. $ \mu^{(ij)}_{{\square \otimes \square}}$, with $i = j+1$ (mod$N$) for $k=2$). So far, we only considered gauge invariant Wilson loop operators without insertions of the Higgs field (holonomy). In the small-$L$ abelianized regime of dYM theory, one can isolate the contribution of individual components of the fundamental quarks by inserting powers of the holonomy inside the trace defining the Wilson loop. This gives rise to Wilson loops in $\R^3$ “decorated” by loops winding around the $\S^1$, similar to the construction of [@Cherman:2016jtu; @Aitken:2017ayq]. The construction of these loops shows that the abelian strings of different tensions (due to quarks of a single weight) in product representations are physical, i.e. they are created by gauge invariant operators. We now define a “decorated” Wilson loop as follows. The fundamental representation holonomy around the $\S_1$ is $$\label{eq:pol1} \Omega(x)_F = {\cal{P}} e^{i \oint\limits_{\S_1} A^a_4(x,x_4)t^a_F dx_4}~, \ \ \ \ \ \ a =1, ..., N-1.$$ The gauge invariant Wilson loop projecting on a single component of a quark field can then be written as $$\label{eq:wilson1} W^k_F ={\rm tr}_F {\cal P} \left[{1 \over N} \sum\limits_{p=1}^N \omega_N^{-(N-k)p} (\Omega(x)_F)^p\right] \; e^{i \int\limits_{\text{x}}^{\text{x}} A_\mu d\text{x}^\mu} ~,$$ where $\omega_N = e^{i {2 \pi\over N}}$ and the integral $\int\limits_x^x$ is taken along a large $RT$ contour in $\R^3$, broken up at the point $x$ where the Higgs field is inserted. In the center symmetric vacuum at weak coupling $\Omega$ can be replaced by its vacuum expectation value, $\langle \Omega \rangle$, given (for brevity, shown below only for odd $N$ and recalling (\[a4vev\])), by $$\label{evs12} \langle \Omega \rangle = {\rm diag}( \omega_N^{N-1}, \omega_N^{N-2},..., \omega_N, 1)~.$$ Hence, the holonomy insertion and discrete Fourier transform in project (the term in square brackets inside the trace in ) the Wilson loop to an abelian component corresponding to a source given by the $k$-th component (weight) of the fundamental quark (in the ordering of eigenvalues as in (\[evs12\])). Using , one can construct sources of various weights in product representations. Derivation of $\mathbf{W}$-boson spectrum {#app:wboson} ========================================= Consider two analogs of off-diagonal $SU(2)$ generators in $SU(N)$, namely $T_{(kl)}^1$ and $T_{(kl)}^2$ (analogs of $\tau^1/2$ and $\tau^2/2$ in $SU(2)$ respectively), $1 \leq k , l \leq N$, where $k \neq l$, refer to the row and column of the non-zero components of these generators. We will work out the quadratic [mass]{} terms associated with their corresponding gauge fields $A_i^1T_{(kl)}^1$ and $A_i^2T_{(kl)}^2$. The mass term comes from the $F^2_{i4}$ term in with $F_{i4} = \partial_i A_4 - \partial_4 A_i -i [A_i,A_4]$: $$\begin{aligned} \label{eq:2.9} \{ {1 \over 2g^2} \text{tr}F^2_{i 4}(x) \}_{\text{quad} , A_i^{1,2}} &=& {1 \over 2g^2}\{ \text{tr}(\partial_4 A^{1,2}_i)^2 - \text{tr}([A^{1,2}_i,A^{vev}_4])^2 \\ \nonumber && + 2i \text{tr}(\partial_4 A^{1,2}_i [A^{1,2}_i,A^{vev}_4]) \} ~,\end{aligned}$$ with $A_i^{1,2} = A_i^1T_{(kl)}^1 + A_i^2T_{(kl)}^2$. Noting that $[A^{1,2}_i,A^{vev}_4] = {2\pi |l-k| \over NL}i(A_i^2T_{kl}^1 - A_i^1T_{kl}^2)$, expanding each component in its Fourier modes using , and integrating over the compact $\text{x}_4$ direction we have: \[eq:2.10\] \^L\_0 d\_4 { [1 2g\^2]{} F\^2\_[i 4]{}(x) }\_[ quad, A\_i\^[1,2]{}]{} & = [L 2g\^2]{} {([2 m L]{})\^2( A\_[i,m]{}\^1A\_[i,m]{}\^[1 ]{} + A\_[i,m]{}\^2A\_[i,m]{}\^[2 ]{} )\ & + ([2|l-k| NL]{})\^2 ( A\_[i,m]{}\^1A\_[i,m]{}\^[1 ]{} + A\_[i,m]{}\^2A\_[i,m]{}\^[2 ]{} )\ & +2i [2|l-k| NL]{}[2 m L]{} (A\^[1 ]{}\_[i,m]{}A\^[2]{}\_[i,m]{} - A\^[2 ]{}\_[i,m]{}A\^[1]{}\_[i,m]{}) }  . && Expanding in real and imaginary parts of Fourier components we have: \[eq:2.11\] \^L\_0 d\_4 { [1 2g\^2]{} F\^2\_[i 4]{}(x) }\_[ quad,A\_i\^[1,2]{}]{} & = [[L 2g\^2]{} { ([2 m L]{})\^2 \[ (A\_[i,m1]{}\^1)\^2 + (A\_[i,m2]{}\^1)\^2 + (A\_[i,m1]{}\^2)\^2 + (A\_[i,m2]{}\^2)\^2 \] ]{}\ & [+4 [2|l-k| NL]{}[2 m L]{} ( A\^[1]{}\_[i,m2]{}A\^[2]{}\_[i,m1]{} - A\^[2 ]{}\_[i,m2]{}A\^[1]{}\_[i,m1]{})]{}\ &  , && with $A_{i,02}^1 = A_{i,02}^2 = 0$. The above mass terms can be diagonalized by defining the following fields: $$\label{eq:2.12} \begin{split} \bar{A}^1_{i,m} \equiv (A^{1}_{i,m1} + A^{2}_{i,m2}) / \sqrt{2}, \bar{A}^2_{i,m} \equiv (A^{1}_{i,m2} + A^{2}_{i,m1}) / \sqrt{2} \\ \bar{A}^3_{i,m} \equiv (A^{1}_{i,m2} - A^{2}_{i,m1}) / \sqrt{2} , \bar{A}^4_{i,m} \equiv (A^{1}_{i,m1} - A^{2}_{i,m2}) / \sqrt{2}~, \end{split}$$ leading to the quadratic Lagrangian for the off-diagonal components: $$\label{eq:2.13} \begin{split} \int^L_0 d\text{x}_4 \{ {1 \over 2g^2} \text{tr}F^2_{i 4}(x) \}_{ quad,A_i^{1,2}} = & {L \over 2g^2} \overset{+\infty}{\underset{m=0}{\sum}} \{ ({2 \pi m \over L} - {2\pi |l-k| \over NL} )^2 {[(\bar{A}^1_{i,m})^2 + (\bar{A}^3_{i,m})^2]} \\ & + ({2 \pi m \over L} + {2\pi |l-k| \over NL} )^2 {[ (\bar{A}^2_{i,m})^2 + (\bar{A}^4_{i,m})^2 ]} \}~. \end{split}$$ Relation shows that there are W-bosons $ {W^{\pm}_1 = (\bar{A}^1_{i,m} \pm i \bar{A}^3_{i,m})/\sqrt{2}}$ and $ {W^{\pm}_2 = (\bar{A}^4_{i,m} \pm i \bar{A}^2_{i,m}) / \sqrt{2}}$ with masses $|{2 \pi m \over L} - {2\pi |l-k| \over NL}|$ and $|{2 \pi m \over L} + {2\pi |l-k| \over NL}|$ respectively for $m =0,1,2,...$ and $1\leq l < k \leq N$. Error analysis {#errorappendix} ============== Truncation error {#sec:A1} ---------------- In this Section we will discuss relation . Consider at $m \rightarrow \infty$ and at its minimum solution: $$\label{eq:A1} \bar{T}^{m,J}_{k,\text{min}}=\bar{T}^{m,J}_{k1,\text{min}}+\bar{T}^{m,J}_{k2,\text{min}}~.$$ The only explicit dependence on $J$ in is through $\Delta z = {J / m}$. Extracting this explicit dependence and suppressing the indices we have: $$\label{eq:A2} H^J={H_1\over J} + JH_2 , \ \ \ \bar{T}^{m,J}_{k,\text{min}} \equiv H^J, \ \ \ \bar{T}^{m,J}_{k1,\text{min}} \equiv {H_1\over J}, \ \ \ \bar{T}^{m,J}_{k2,\text{min}} \equiv {JH_2}~.$$ Taking the derivative of $H^J$ with respect to $J$ gives: $$\label{eq:A3} {dH^J\over dJ}=-{H_1\over J^2} + H_2 + {1\over J}{\partial H_1\over \partial f_{jh}}{df_{jh}\over dJ} + J{\partial H_2\over \partial f_{jh}}{df_{jh}\over dJ}~.$$ Since the partial derivatives at the minimum solution vanish we have: $$\label{eq:A4} {dH^J\over dJ}={1\over J}(-{H_1\over J} + JH_2)~.$$ For $J_1 < J_2$ it can be shown that $H^{J_2} < H^{J_1}$. The minimum solution of $H^{J_i}$ is a path $P_i$ that connects the boundary point $\pi \mu_k$ at $\text{z} = 0$ to $0$ at $\text{z} = J_i$ for $i=1,2$ ( Section \[sec:3.3\]). If we extend path $P_1$ on the z-axis from $\text{z} = J_1$ to $\text{z} = J_2$ we would obtain a path $\tilde{P_1}$ that connects the boundary point $\pi \mu_k$ at $\text{z} = 0$ to $0$ at $\text{z} = J_2$. But the value of the action of the paths $P_1$ and $\tilde{P_1}$ is the same since the portion of the path $\tilde{P_1}$ that is on the z-axis gives zero action. On the other hand the action of $\tilde{P_1}$ should be higher than the action of $P_2$ since $P_2$ is the minimizing path of $H^{J_2}$ hence $H^{J_2} < H^{J_1}$. Due to this $ {dH^J / dJ < 0}$. From this gives ${H_1 / J} > JH_2$ and hence $\bar{T}^{m, J}_{k1,\text{min}} > \bar{T}^{m,J}_{k2,\text{min}}$ for all $0 < J < \infty$. Also we should have $\underset{{J \rightarrow \infty}}{\lim} {dH^J / dJ} = 0$ since $H^J$ is a decreasing function of $J$ and it is bounded from below ($\underset{{J \rightarrow \infty}}{\lim} {H^J} = \bar{T_k}$). This shows that $\bar{T}^{m,\infty}_{k1,\text{min}} = \bar{T}^{m,\infty}_{k2,\text{min}}$. Also in the limit of $J \rightarrow 0$ the relations $\underset{{J \rightarrow 0}}{\lim} \bar{T}^{m,J}_{k1,\text{min}} = \infty$ and $\underset{{J \rightarrow 0}}{\lim} \bar{T}^{m,J}_{k2,\text{min}} = 0$ can be easily verified from . This can also be seen from relation , $H_1$ and $H_2$ are finite quantities hence the previous limits follow. Summarizing the previous relations derived we have: $$\label{eq:A5} \begin{aligned} & \bar{T}^{m, J}_{k1,\text{min}} > \bar{T}^{m,J}_{k2,\text{min}},\ \ \bar{T}^{m, J}_{k,\text{min}} > \bar{T}^{m, \infty}_{k,\text{min}}, \ \ \ \ \ \ \ \ \ \ \ 0 \leq J < \infty \\ & \bar{T}^{m,0}_{k1,\text{min}} = \infty ,\ \ \ \ \ \ \ \ \bar{T}^{m,0}_{k2,\text{min}} = 0, \ \ \ \ \ \ \ \ \bar{T}^{m, \infty}_{k1,\text{min}} = \bar{T}^{m, \infty}_{k2,\text{min}} = {1\over 2} \bar{T}^{m, \infty}_{k,\text{min}}~. \end{aligned}$$ From it can be seen that $\bar{T}^{m,J}_{k2,min}$ starts from $0$ at $J=0$ and approaches ${1\over 2} \bar{T}^{m, \infty}_{k,\text{min}}$ at $J=\infty$. We conjecture that for $0<J<\infty$, $\bar{T}^{m,J}_{k2,\text{min}}<{1\over 2} \bar{T}^{m, \infty}_{k,\text{min}}$[^63]. Also from we have ${1\over 2} \bar{T}^{m, \infty}_{k,\text{min}} < {1\over 2} \bar{T}^{m, J}_{k,\text{min}}={1\over 2}(\bar{T}^{m, J}_{k1,\text{min}} + \bar{T}^{m,J}_{k2,\text{min}}) < \bar{T}^{m, J}_{k1,\text{min}}$ so we obtain the following inequalities: $$\label{eq:A6} \bar{T}^{m,J}_{k2,\text{min}} < {1\over 2} \bar{T}^{m, \infty}_{k,\text{min}} < {1\over 2} \bar{T}^{m, J}_{k,\text{min}} < \bar{T}^{m, J}_{k1,\text{min}}~.$$ From relation easily follows. Sample error calculation ------------------------ We have shown data for half $k$-string tensions of $SU(10)$, k = 5 in Tables \[table:5\] and \[table:6\] to perform a sample error calculation. m n $<T_{k1}>$ $\sigma_{k1} \over \sqrt{R}$ $<T_{k2}>$ $\sigma_{k2} \over \sqrt{R}$ $<T_{k}> $ $\sigma_{k} \over \sqrt{R}$ ----- ---- ------------ ------------------------------ ------------ ------------------------------ ------------ ----------------------------- -- 100 18 5.82434259 6.9E-05 5.82600776 6.2E-07 11.6503503 6.2E-07 100 19 5.82423417 1.4E-05 5.82610277 1.4E-05 11.6503369 1E-07 100 20 5.82420397 3.9E-06 5.82612949 3.9E-06 11.6503335 1.3E-08 200 20 5.82500582 1.20E-04 5.82415592 1.20E-04 11.6491617 6.90E-07 200 21 5.82486451 5.00E-05 5.82428283 5.00E-05 11.6491473 1.70E-07 200 22 5.82482467 2.30E-05 5.82431903 2.30E-05 11.6491437 4.00E-08 : $SU(10)$, $k=5$ sample data with error in the mean ($\sigma \over \sqrt{R}$)[]{data-label="table:5"} m n $<T_{k1}>$ $\Delta_n$ $<T_{k2}>$ $\Delta_n$ $<T_{k}> $ $\Delta_n$ ----- ---- ------------ ------------ ------------ ------------ ------------ ------------ -- 100 18 5.82434259 - 5.82600776 - 11.6503503 - 100 19 5.82423417 -(1E-04) 5.82610277 1E-04 11.6503369 -(1E-05) 100 20 5.82420397 -(3E-05) 5.82612949 3E-05 11.6503335 -(3E-06) 200 20 5.82500582 - 5.82415592 - 11.6491617 - 200 21 5.82486451 -(1E-04) 5.82428283 1E-04 11.6491473 -1E-05 200 22 5.82482467 -(4E-05) 5.82431903 4E-05 11.6491437 -(4E-06) : $SU(10)$, $k=5$ sample data with $\Delta_n=<X_n> - <X_{n-1}>$.[]{data-label="table:6"} **Sample minimization error calculation:**\ Sample calculation for $SU(10)$, $k = 5$ and $m = 100$:\  Min.  Error  for  $<T_{k2}> = |\Delta_{20}| + {\sigma_k \over \sqrt{R}}$ = 3E-05 + 3.9E-06 $\sim$ 3E-05 From Tables \[table:5\] and \[table:6\] and the sample calculation above it is clear that the minimization errors are of order $10^{-5}$ and less so they can be safely neglected in comparison to the discretization and truncation errors.\ \ **Sample discretization error calculation:**\ The difference between $<T_{k1}>$, $<T_{k2}>$ and $<T_{k}>$ for $m=100$ and $m=200$ is: $$\label{eq:diserror} \begin{split} &<T_k>_{200} - <T_k>_{100} = 11.6491437 - 11.6503335 = -0.0011898 \\ &<T_{k1}>_{200} - <T_{k1}>_{100} = 5.82482467 - 5.82420397 = 0.0006207 \\ &<T_{k2}>_{200} - <T_{k2}>_{100} = 5.82431903 - 5.82612949 = -0.00181046~. \end{split}$$ Hence we predict that the continuum value of the string tensions of $SU(10)$, $k = 5$ for $J=14.0$ would be: $$\label{eq:A.8} \begin{split} &<T_k> = 11.6491_{-0.001} \\ &<T_{k1}> = 5.8248^{+0.0006} \\ &<T_{k2}> = 5.8243_{- 0.0018} ~. \end{split}$$ **Sample truncation error calculation:**\ \ Relation gives an upper bound estimate for the truncation error. Based on : $$\label{eq:truncerror1} \begin{split} |<T_{k1}> - <T_{k2}>| \lessapprox 5.8248 + 0.0006 - (5.8243 - 0.0018) = 0.0029 \\ \Longrightarrow \text{Trunc.} \ E. \ \lessapprox 2\times 0.0029 = 0.0058~. \end{split}$$ Adding the truncation and discretization error and neglecting the minimization error we have: $$\label{eq:tot} \text{Total Error} = \text{Trunc. E.} + \text{Dis. E.} = 0.0058 + 0.001 = 0.0068 \approx 0.007~.$$ Hence we predict the value of the half string tension for $SU(10)$ and $k = 5$ is: $11.6491_{- 0.007}$. The errors obtained by this method for different half $k$-string tensions varied from 0.005 to 0.007 therefore we have considered the average as an upper bound estimate for the value of the error for all half $k$-string tensions and included it in Table \[table:1\]. It has to be noted that upper bound estimates for errors always overestimate the true value of the errors as can be seen from Table \[table:2\]. Derivation of group theory results ================================== Any representation of ${SU(N)}$ with ${N}$-ality ${1 \leq k \leq N-1}$ contains the fundamental weight ${\mu_k}$ as one of its weights {#sec:B1} -------------------------------------------------------------------------------------------------------------------------------------- The simple roots $\alpha_i$ and fundamental weights $\mu_k$ of $SU(N)$ are given by the following relations: $$\label{eq:B1} \begin{split} & \alpha_i=(0,..,0,\overset {\text{i-th}}{\widehat{1}},-1,0,...,0), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq N-1 \\ & \mu_k=({N - k \over N}, ..., \overset {\text{k-th}} {\widehat{{N-k \over N}}}, {-k \over N}, ..., {-k \over N}),\ \ \ \ \ \ 1\leq k \leq N-1 \end{split}$$ An arbitrary representation of $SU(N)$ with $N$-ality $1 \leq k \leq N-1$ can be represented by its highest weight $w_k$: \[eq:B2\] w\_k = h\_i \_i    h h\_1 + 2h\_2 + ... + (N-1)h\_[N-1]{} = mN + k ,  m, h\_i ,  m, h\_i 0 , && where $ {h_i \geq 0}$ are the $N-1$ Dynkin indices of the representation, which determine $k$, its $N$-ality, by the mod($N$) relation given above. The proof involves two steps. First we will prove the following lemma:\ **Lemma** $w_k = \mu_k + a_i \alpha_i$ for $a_i \in \mathbb{Z} \ \text{and} \ a_i \geq 0$ It can be easily seen that: $$\label{eq:B3} \mu_k = k \mu_1 - \beta_k \ \ \text{with} \ \ \beta_k = (k-1)\alpha_1 + (k-2)\alpha_2 + ... + \alpha_{k-1}, \ \ \beta_1 = 0$$ Hence $w_k$ can be written as: $$\label{eq:B4} w_k = (h_1 + 2h_2 + ... + (N-1)h_{N-1}) \mu_1 - (h_2 \beta_2 + ... + h_{N-1} \beta_{N-1}) \\$$ With knowing $N \mu_1 = (N-1) \alpha_1 + (N-2) \alpha_2 + ... + \alpha_{N-1}$ and we have: $$\label{eq:B5} \begin{split} h \mu_1 = (mN + k) \mu_1 = & \ \mu_k + (m(N-1) + k-1) \alpha_1 + ... + (m(N-(k-1)) + 1) \alpha_{k-1} \\ & + m(N-k) \alpha_{k} + ... + m \alpha_{N-1} \end{split}$$ Therefore using and , $w_k$ in can be written as: $$\label{eq:B6} w_k = \mu_k + b_i \alpha_i \ \ \ \ \text{with} \ \ \ \ b_i \in \mathbb{Z}$$ We need to show that $b_i$ is greater than or equal to zero. Lets assume the contrary. First lets assume $b_k < 0$: $$\label{eq:B7} \begin{split} &b_k < 0 \Longrightarrow b_k \leq -1 \\ &\eqref{eq:B6} \ \& \ \eqref{eq:B2} \Longrightarrow \alpha_k \cdot w_k = 2b_k + 1 - b_{k+1} - b_{k-1} = h_k \geq 0, \\ & b_k \leq -1 \ \& \ 2b_k + 1 - b_{k+1} - b_{k-1} \geq 0 \Longrightarrow b_{k+1} \leq b_k \ \text{or} \ b_{k-1} \leq b_k~. \end{split}$$ Lets assume $b_{k-1} \leq b_k$: $$\label{eq:B8} \begin{split} & \eqref{eq:B6} \ \& \ \eqref{eq:B2} \Longrightarrow \alpha_{k-1} \cdot w_k = 2b_{k-1} - b_{k} - b_{k-2} = h_{k-1} \geq 0, \\ & b_{k-1} \leq b_k \ \& \ 2b_{k-1} - b_{k} - b_{k-2} \geq 0 \Longrightarrow b_{k-2} \leq b_{k-1}~. \end{split}$$ Similarly, it can be concluded that $0 > b_k \geq b_{k-1} \geq b_{k-2} \geq b_{k-3} \geq ... \geq b_2 \geq b_1$. Here we will clearly have a contradiction since we have: $$\label{eq:B9} \begin{split} & \eqref{eq:B6} \ \& \ \eqref{eq:B2} \Longrightarrow \alpha_{1} \cdot w_k = 2b_{1} - b_{2} = h_{1} \geq 0 \\ & \text{But if} \ b_1 \leq b_2 < 0 \Longrightarrow 2b_1 - b_2 < 0~. \end{split}$$ Similarly, a contradiction occurs if it is assumed that $b_{k+1} \leq b_k$. Now, if any other $b_i < 0$ for $i \neq k$, similarly it can be argued that either $b_{i+1} \leq b_i$ or $b_{i-1} \leq b_{i}$ and concluded that $2b_1 - b_2 < 0$ or $2b_{N-1} - b_{N-2} < 0$ or $b_k \leq b_i < 0$, which would lead to contradictions similar to above. The next step of the proof is to show that given a highest weight $w_k$ of a representation with $N$-ality $k$, it is always possible to lower with the simple roots to obtain $\mu_k$. Given a weight $\mu$ of a representation of $SU(N)$, the master formula in [@09] can be applied: $$\label{eq:B91} {2\mu\cdot \alpha_i \over \alpha^2_i} = \mu\cdot \alpha_i = -(p_i - q_i)$$ Where $p_i \in \mathbb{Z} \ \text{and} \ p_i \geq 0$ is the number of times which we can raise $\mu$ with $\alpha_i$ and $q_i \in \mathbb{Z} \ \text{and} \ q_i \geq 0$ is the number of times which we can lower $\mu$ with $\alpha_i$. Based on the above **Lemma**, we have $w_k = \mu_k + a_i \alpha_i$. If $a_i = 0$ for all $i$ then the representation contains $\mu_k$ as one of its weights but if at least one is greater than zero then we will show that for some $\alpha_i$ which $a_i > 0$, $w_k \cdot \alpha_i > 0$ which would imply that $q_i > 0$ and hence $w_k$ can be lowered with some $\alpha_i$ which $a_i > 0$. Let $a_j = \text{Max} \{ a_i | 1 \leq i \leq N-1 \} $ for some $1 \leq j \leq N-1$. Since we assumed at least one $a_i$ is greater than zero then $a_j > 0$. If $j = 1$, $j = N-1$ or $j = k$ then $w_k \cdot \alpha_j = 2a_1 - a_2 > 0$, $w_k \cdot \alpha_j = 2a_{N-1} - a_{N-2} > 0$ or $w_k \cdot \alpha_j = 2a_k +1 - a_{k-1} - a_{k+1} > 0$ respectively, since we assumed that $a_j > 0$ and is the maximum among others. Otherwise if $a_j > a_{j+1}$ or $a_j > a_{j-1}$ then $w_k \cdot \alpha_j = 2a_j - a_{j+1} - a_{j-1} > 0$. But if $a_j = a_{j+1} = a_{j-1}$ then $w_k \cdot \alpha_j = 0$. In this case $a_{j-1}$ and $a_{j+1}$ are greater than zero and both maximum among other $a_i$. Hence we can repeat what we did for $a_j$ for $a_{j+1}$ or $a_{j-1}$ for a number of steps until $j+r$ in $a_{j+r}$ for an $r \neq 0$ becomes $j+r = 1$, $j+r = N-1$ or $j+r = k$ or we would have $a_{j+r} > a_{j+r+1}$ or $a_{j+r} > a_{j+r-1}$ which in that case $w_k$ can be lowered with $\alpha_{j+r}$. So we proved that if at least one $a_i$ is greater than zero then $w_k$ can always be lowered with some $\alpha_h$ which $a_h > 0$. Hence we continue this process until all the $\alpha_i$ are removed from the highest weight $w_k = \mu_k + a_i \alpha_i$ and we reach $\mu_k$. This shows that any representation of $SU(N)$ with $N$-ality $k$ contains $\mu_k$ as one of its weights. Derivation of {#sec:B2} -------------- To derive , we will work out the steps for the contribution of one monopole with magnetic charge $q^1_m$. Consider the long distance behaviour of for generators in the fundamental representation of the gauge group, located at $R \in \R^3$: $$\label{eq:B11} \begin{split} \{ \text{tr} \ \text{exp}( i \oint_{R\times T} A^c_m t_F^c d\text{x}^m) \}_{\text{1 mon.}} & = \text{tr} \ \text{exp} (i \int_{S(R\times T)}\epsilon_{anm} \partial_n A^c_m t_F^c d\text{S}^a ) \\ & = \text{tr} \ \text{exp}( i \int_{S(R\times T)} d\text{x}_1d\text{x}_2\; {\text{R}_3 \over 2 |R - \text{x}|^{3}} \;Q^1)~. \end{split}$$ Here $c = 1, ..., N-1$ labels the abelian generators of $SU(N)$, $Q^1 = \text{diag}(1,-1,0, ...,0)$, and we substituted the magnetic field of a monopole, converted to Cartesian coordinates. Next, we first write $Q^1$ as a linear combination of the abelian generators of the fundamental representation $Q^1 = \bar{V}_i t^i_F = \text{diag}(\bar{V}\cdot \bar{\mu}^1_F, ...,\bar{V}\cdot \bar{\mu}^N_F)$ for $i=1, ...,N-1$; here $\bar{\mu}^j_F$ for $j = 1, ...,N$ are the $(N-1)$-dimensional weight vectors of the fundamental representation of $SU(N)$ and $\bar{V}$ is an $(N-1)$-dimensional vector. In order to transform this to an arbitrary representation $r$ we replace $t^c_F$ by its corresponding generator in the representation $r$ and write $Q^1_r {\equiv} \bar{V}_i t^i_r = \text{diag}(\bar{V}\cdot \bar{\mu}^1_r, ...,\bar{V}\cdot\bar{\mu}^{d(r)}_r)$. In order to write this in the $N$-dimensional form of weights used in this work , we note that the weight vectors of the fundamental representation of $SU(N)$ in their $N$-dimensional form are: $$\label{eq:B12} \mu^j_{F} =(-{1 \over N}, ..., -{1 \over N}, \overset {\text{j-th}} {\widehat{1- {1 \over N}}}, - {1 \over N}, ..., - {1 \over N}),\ \ \ \ \ \ \ \ \ \ j = 1, ..., N~,$$ which can be easily verified by lowering the fundamental weight $\mu_1$ in with the simple roots. From , the $N$-dimensional form of $\bar{V}$, named $V$, can be determined by requiring: $Q^1 = \text{diag}(\bar{V}\cdot\bar{\mu}^1_F, ...,\bar{V}\cdot\bar{\mu}^N_F) = \text{diag}(V\cdot \mu^1_F, ...,V\cdot\mu^N_F)$, which gives $V = q^1_m = (1,-1,0, ...,0)$. Hence $Q^1_r$, using the $N$-dimensional form of weights, becomes $Q^1_r = \text{diag}(V\cdot\mu^1_r, ...,V\cdot \mu^{d(r)}_r) = \text{diag}(q^1_m\cdot\mu^1_r, ...,q^1_m\cdot\mu^{d(r)}_r)$. Therefore , for generators in an arbitrary representation $r$, becomes: $$\label{eq:B13} \begin{split} \{ \text{tr} \ \text{exp}( i \oint_{R\times T} A^c_m t_r^c d\text{x}^m) \}_{\text{1 mon.}} & = \text{tr} \ \text{exp} (i \int_{S(R\times T)}\epsilon_{anm} \partial_n A^c_m t_r^c d\text{S}^a ) \\ & = \overset{d(r)}{\underset{j = 1}{\sum}} \ \text{exp}( i \int_{S(R\times T)} d\text{x}_1d\text{x}_2 \; \mu^j_r \cdot q^1_m\; {\text{R}_3 \over 2 |R - \text{x}|^{3}})~. \end{split}$$ Perturbative saddle point $\mathbf{k}$-strings: leading order $\mathbf{+}$ leading correction {#sec:C} ============================================================================================= The following tables \[table:7\] - \[table:15\] compare the values of $k$-strings obtained from a perturbative saddle point calculation to their numerical values in Table \[table:1\]. The ”Leading” and ”Leading Corr.” column give values for the coefficient of $- {RT \over \beta }$ in and respectively. If $T$ represents a half $k$-string value in Table \[table:1\] and $T'$ its corresponding one in the ”Num. value” column, they are related by: $$T' = \text{ROUND}(\bar{T},3) \ \ \ \text{with} \ \ \ \bar{T} \equiv \text{ROUNDDOWN}(T,3)\times2/ \sqrt{2}$$ We multiply the half $k$-strings by $2$ to obtain the full $k$-string then we divide it by $\sqrt{2}$ to normalize them similar to the perturbative saddle point method $k$-strings as in and . There is a high chance that the numerical half $k$-strings in Table \[table:1\] will match the exact half $k$-strings rounded to the third decimal if they are rounded down to the 3rd decimal. This is due to the fact that the true value of half $k$-strings always lies below the values obtained in Table \[table:1\] and the true value of the error is of order $0.001$ or less as can be seen from Table \[table:2\].\ $SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum --------- --------- --------------- ------------------------ ------------ ------------------ 2 9.870 -2.029 7.841 8.000 0.159 3 11.396 -2.343 9.053 9.238 0.185 4 11.913 -2.396 9.517 9.699 0.182 5 12.150 -2.410 9.740 9.919 0.179 6 12.277 -2.415 9.862 10.041 0.179 7 12.355 -2.417 9.938 10.114 0.176 8 12.405 -2.418 9.987 10.163 0.176 9 12.439 -2.418 10.021 10.196 0.175 10 12.463 -2.418 10.045 10.221 0.176 : Comparison of $N$-ality $1$ $k$-strings for $SU(2 \leq N \leq 10)$[]{data-label="table:7"} $SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum --------- --------- --------------- ------------------------ ------------ ------------------ 3 11.396 -2.343 9.053 9.238 0.185 4 13.958 -2.870 11.088 11.314 0.226 5 15.018 -2.99 12.028 12.247 0.219 6 15.568 -3.029 12.539 12.751 0.212 7 15.891 -3.045 12.846 13.055 0.209 8 16.097 -3.052 13.045 13.253 0.208 9 16.237 -3.056 13.181 13.388 0.207 10 16.337 -3.058 13.279 13.485 0.206 : Comparison of $N$-ality $2$ $k$-strings for $SU(3 \leq N \leq 10)$ []{data-label="table:8"} $SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum --------- --------- --------------- ------------------------ ------------ ------------------ 4 11.913 -2.396 9.517 9.699 0.182 5 15.018 -2.990 12.028 12.247 0.219 6 16.449 -3.142 13.307 13.511 0.204 7 17.248 -3.197 14.051 14.247 0.196 8 17.746 -3.222 14.524 14.715 0.191 9 18.078 -3.234 14.844 15.032 0.188 10 18.311 -3.241 15.070 15.257 0.187 : Comparison of $N$-ality $3$ $k$-strings for $SU(4 \leq N \leq 10)$ []{data-label="table:9"} $SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum --------- --------- --------------- ------------------------ ------------ ------------------ 5 12.15 -2.410 9.740 9.919 0.179 6 15.568 -3.029 12.539 12.751 0.212 7 17.248 -3.197 14.051 14.247 0.196 8 18.237 -3.262 14.975 15.159 0.184 9 18.876 -3.292 15.584 15.761 0.177 10 19.317 -3.307 16.010 16.183 0.173 : Comparison of $N$-ality $4$ $k$-strings for $SU(5 \leq N \leq 10)$ []{data-label="table:10"} $SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum --------- --------- --------------- ------------------------ ------------ ------------------ 6 12.277 -2.415 9.862 10.041 0.179 7 15.891 -3.045 12.846 13.055 0.209 8 17.746 -3.222 14.524 14.715 0.191 9 18.876 -3.292 15.584 15.761 0.177 10 19.629 -3.326 16.303 16.474 0.171 : Comparison of $N$-ality $5$ $k$-strings for $SU(6 \leq N \leq 10)$ []{data-label="table:11"} $SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum --------- --------- --------------- ------------------------ ------------ ------------------ 7 12.355 -2.417 9.930 10.114 0.176 8 16.097 -3.052 13.045 13.253 0.208 9 18.078 -3.234 14.844 15.032 0.188 10 19.317 -3.307 16.010 16.183 0.173 : Comparison of $N$-ality $6$ $k$-strings for $SU(7 \leq N \leq 10)$ []{data-label="table:12"} $SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum --------- --------- --------------- ------------------------ ------------ ------------------ 8 12.405 -2.418 9.987 10.163 0.176 9 16.237 -3.056 13.181 13.388 0.207 10 18.311 -3.241 15.070 15.257 0.187 : Comparison of $N$-ality $7$ $k$-strings for $SU(8 \leq N \leq 10)$ []{data-label="table:13"} $SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum --------- --------- --------------- ------------------------ ------------ ------------------ 9 12.439 -2.418 10.021 10.196 0.175 10 16.337 -3.058 13.279 13.485 0.206 : Comparison of $N$-ality $8$ $k$-strings for $SU(9 \leq N \leq 10)$ []{data-label="table:14"} $SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum --------- --------- --------------- ------------------------ ------------ ------------------ 10 12.463 -2.418 10.045 10.221 0.176 : Comparison of $N$-ality $9$ $k$-strings for $SU(10)$ []{data-label="table:15"} .\ \ \ \ \ \ \ Large-$N$ limit of string tensions for product representations: a saddle point leading-order perturbative evaluation {#sec:appxproduct} ==================================================================================================================== Our starting point is . Recall that this equation gives the contribution to the expectation value of the Wilson loop of quarks of charges (weight) $\mu$, evaluated to leading order using the perturbative saddle point method. For convenience, we now reproduce the area-law part of that equation ($\hat{R}, \hat{T} \rightarrow \infty, \beta \rightarrow 0$): $$\label{eq:4.391} {\{ Z^{\eta}_{g^4} \}_{\lambda =0} \over \{ Z^{\eta}_{g^4} \}_{b_q=\lambda =0} } = \text{exp}(- {1 \over \beta} \{ {1 \over 4} {\sum\limits_{q=1}^{N-1}\sqrt{\Lambda_q}b^2_q} \hat{R} \hat{T} \} )~.$$ Here, $b_q \equiv 2\pi (\mu)_j D_{jq}$ for a representation of weight $\mu$; $(\mu)_j$ denotes the $j$-th component, $j=1,...N$, of the weight vector. Recall also that $\Lambda_q = 4 \sin^2 {\pi q \over N}$ is the dimensionless mass of the $q$-th dual photon and that the components of the matrix $D_{jq}$, $1 \le j \le N$, are $D_{jq} = \sqrt{2 \over N} \sin {2 \pi q j \over N} $, for $1 \le q < {N\over 2}$ and $D_{jq} = \sqrt{2 \over N} \cos {2 \pi q j \over N} $, for ${N\over 2} < q < {N}$; for brevity, we only give the values for odd $N$. The $q=N$ component of $D_{jq}$ does not contribute to . The main difference compared to the discussion in the main text is that we now consider also weights corresponding to product representations, for concreteness the ${\square \otimes \square}$ representation. Recall, from , that the expectation value of the Wilson loop in the product representation is given, in the abelianized regime of this paper, by a sum of exponentials, one for each weight of the product representation: $$\label{eq:abd1} \langle W_{{\square \otimes \square}}(R,T) \rangle \sim \sum^{d({\square \otimes \square})}_{h = 1} \text{exp}(- T^h_{{\square \otimes \square}}RT) = \sum^{d({\square \otimes \square})}_{h = 1} \text{exp}(- {1 \over \beta} \hat{T}^h_{{\square \otimes \square}}\hat{R}\hat{T}) ~.$$ Where we have also written it in its dimensionless form (recall the relations $R = \hat{R} / m_{\gamma}$, $T = \hat{T} / m_{\gamma}$, $\beta = m_{\gamma}^3/ \tilde{\zeta}$ from the comment below ). Comparing with $\hat{T}^h_{{\square \otimes \square}}$ to leading order (l.o.) is given by: $$\label{eq:apxe1} \hat{T}^h_{{\square \otimes \square},\text{l.o.}}= {1 \over 4} {\sum\limits_{q=1}^{N-1}\sqrt{\Lambda_q}b^2_q}~, ~~ b_q \equiv 2\pi \sum\limits_{j=1}^N (\mu^h)_j D_{jq} ~,$$ with $\mu^h$—the $h$-th weight of the ${\square \otimes \square}$ representation. The goal of this Appendix is to evaluate for all weights of the ${\square \otimes \square}$ product representation, to leading order in the analytic perturbative saddle point method and in the large-$N$ limit. We shall see that the leading-order analytic considerations support the findings discussed qualitatively after around Eqns. (\[acd\]) and of the main text of the behaviour of the product-representation Wilson loop at large $N$. We begin by noting that the weights of the ${\square \otimes \square}$ representation are labeled by two integers $a, b=1,...N$ (there are $N^2$ weights) and are given by $$\label{eq:weight} (\mu^h)_j \rightarrow (\mu^{ab})_j = \delta^a_j + \delta^b_j -{2 \over N} \approx \delta^a_j + \delta^b_j .$$ The last equality is valid for sufficiently large $N$. From , recalling that we consider odd-$N$, we find an explicit expression for the tension of strings sourced by quarks with weight $\mu^{ab}$, , of the product representation at leading order:[^64] \[eq:appxe2\] \^[ab]{}\_[,]{} = 4 && The sum in can be evaluated exactly for arbitrary $N$, but to illustrate our point it suffices to consider [*i.*]{}) the results of a numerical evaluation and [*ii.*]{}) the evaluation of at infinite $N$ by replacing the sum by an integral. We begin with a discussion of the numerical results for the $N^2$ product representation string tensions shown on Figure \[fig:productstrings\]. The $N^2$ string tensions for $N=21$ are evaluated numerically using . As the plot shows, most of the $N^2$ string tensions are of order $2 T_{1,\text{l.o.}}$, while $2N$ of them are approximately equal to the minimal value $T_{2,\text{l.o.}}$, and $N$ are equal to approximately $4 T_{1,\text{l.o.}}$. Clearly, this is conforming to the discussion in the main text, Section \[sec:5.2.2\].. {width="\textwidth"} [\[fig:productstrings\]]{} We can also evaluate in the infinite-$N$ limit by replacing the sum by an integral, for $a,b$ fixed, i.e., \[eq:appxe3\] [\^[ab]{}\_[,]{} 4 ]{} = \_[0]{}\^[[2]{}]{} d x x ( + )\^2 + \_[[2]{}]{}\^d x x ( + )\^2  . Thus, the product representation string tensions, normalized to the fundamental string tension (equal to $4\pi$ in the leading saddle point approximation), becomes in the large-N limit $$\label{eq:E.7} {\hat{T}^{ab}_{{\square \otimes \square},\text{l.o.}}\over 4 \pi} = 2 - {2\over 4 (a - b)^2-1}~.$$ Due to the $Z_N$ symmetry of the string tension action we expect to obtain the same tensions for $|a-b| = n$ and $|a-b| =N - n$ for $1 \leq n \leq [N/2]$. This symmetry is lost in due to the infinite $N$ limit, therefore it is best to use this relation for $|a-b| \leq [N/2]$ only at large N and for $|a-b| > [N/2]$ make the replacement $|a-b| \rightarrow N - |a-b|$ in . In the limit $|a-b|\gg 1$, this relation approaches the value of $2$, while for $|a-b|=1$, we obtain the value ${4\over 3} \approx 1.33$; the value of $4$ for $a=b$ is also obtained. This distribution of the $N^2$ string tensions in the infinite-$N$ limit is consistent with the numerical result shown for $N=21$ and with the general discussion of Section \[sec:5.2.2\]. At the end, we also acknowledge an additional subtlety one might be worried about. The calculation that led to eqn. —see as well as eqns. (\[eq:4.7\]–\[eq:4.14\]) which directly lead to it—assumes that $RT$ is larger than the inverse mass squared of all dual photons, including the lightest one. Thus, strictly speaking one expects to pertain to the order of limits we advocated for here: infinite area at fixed N, followed by $N \rightarrow \infty$ which is the proper order of limits necessary for calculating k-strings at large $N$. However for the discussion of large $N$ factorization in gauge theories the large $N$ limit is taken first. Now, if $RT$ is smaller than the mass of some dual photons, the area law due to these photons should be replaced with a perimeter law contribution. This remark is relevant because if the large-N limit is taken first, the masses of some dual photons vanish—recall that their masses are scale as $\sqrt{\Lambda_q} =2 \sin {\pi q \over N}$—and these dual photons do not lead to an area law. To take this into account, consider the integral and omit contributions of dual photons of (dimensionless) mass $ 2 \sin {\pi q\over N} = 2 \sin x < {1\over \sqrt{\hat R \hat T}}$, as they do not give rise to area law. Thus, the region of integration in , instead of $(0, {\pi \over 2})$ and $({\pi\over 2}, \pi)$, should be replaced by, respectively, $(\epsilon, {\pi \over 2})$ and $({\pi\over 2}, \pi-\epsilon)$, with $\epsilon \sim {1\over \sqrt{\hat{R} \hat{T}}}$. In the further large $\hat{R}\hat{T}$ limit, we have that $\epsilon \rightarrow 0$, showing that the contributions to the string tension of dual photons of mass vanishing at large-$N$ is negligible. Thus, we expect that if the order of limits is taken as described now ($N$ to infinity first, large area next), the factorization result analyzed above in terms of string tensions is recovered. The discussion of large-$N$ factorization above and in the 2nd half of Section \[sec:5.2.2\] was carried out in terms of string tensions, since its more explicit and intuitive and allows for a qualitative analysis of large-$N$ factorization in terms of the full saddle point as was done in Section \[sec:5.2.2\]. For this analysis, the large area limit had to be taken to isolate the area law contribution and find the string tensions, as done in the previous paragraph. However one can show the large-$N$ factorization result in a more general and abstract setting without the need to refer to any large area limits or expressions for string tensions. Consider eq. , which is a general expression for the saddle point at leading order, without reference to any large area limit: $$\label{eq:E9} {\{ Z^{\eta}_{g^4} \}_{\lambda =0} \over \{ Z^{\eta}_{g^4} \}_{b_q=\lambda =0} } = \text{exp}(- {1 \over \beta} \{ {\Lambda_q b^2_q \over 2} \underset{\text{A} \ \text{A}}{\iint} d^2 \text{x} d^2 \text{x}' P_q(\text{x} - \text{x}') + {b_q^2 \over 2} \underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} ( P_q(\text{x}-\text{x}') - {1 \over 4 \pi |\text{x} -\text{x}'|}) \} )~,$$ Using and noting that the integrals in are finite quantities and a function of $\hat{R}, \hat{T}$ and $\sqrt{\Lambda_q} =2 \sin {\pi q \over N}$ with $x \equiv {\pi q \over N}$, the large $N$ limit of the leading saddle point (s.p.) result in reduces to: $$\label{eq:E10} \hspace{-0.5cm} \text{s.p.}_{\square \otimes \square,\text{l.o.}} = \int^{\pi /2}_0 dx ( \sin{2 a x} + \sin{2 b x} )^2 F_{\hat{R}, \hat{T}}(\sin x) + \int^{\pi}_{\pi /2} dx ( \cos{2 a x} + \cos{2 b x} )^2 F_{\hat{R}, \hat{T}}(\sin x)~,$$ where $F_{\hat{R},\hat{T}}(\sin {\pi q \over N})$ is given by: $$\label{eq:E11} \hspace{-1mm} F_{\hat{R}, \hat{T}}(\sin {\pi q \over N}) \equiv 4 \pi {\Lambda_q} \underset{\text{A} \ \text{A}}{\iint} d^2 \text{x} d^2 \text{x}' P_q(\text{x} - \text{x}') + 4 \pi \underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} ( P_q(\text{x}-\text{x}') - {1 \over 4 \pi |\text{x} -\text{x}'|})$$ The expression corresponding to in the fundamental representation of $SU(N)$ is: $$\label{eq:E12} \hspace{-0.5cm} \text{s.p.}_{\square,\text{l.o.}} = \int^{\pi /2}_0 dx ( \sin{2 a x})^2 F_{\hat{R}, \hat{T}}(\sin x) + \int^{\pi}_{\pi /2} dx ( \cos{2 a x})^2 F_{\hat{R}, \hat{T}}(\sin x)$$ Making the change of variable $x \rightarrow \pi - x$ in the second integral of and , they can be simplified to: \[eq:E13\] &\_[,]{} = 2\^[/2]{}\_0 dx F\_[, ]{}(x) + 2\^[/2]{}\_[0]{} dx (2a x - 2b x) F\_[, ]{}(x) ,\ \[eq:E14\] & \_[,]{} = \^[/2]{}\_0 dx F\_[, ]{}(x) . When $|a-b| \gg 1$ (and $|a-b| \ll N$ if the discrete form of is considered for a finite but large $N$), the second integral in , due to the rapid oscillations of $\cos (2a x - 2b x)$, is near zero therefore $O(N^2)$ weights of the product representation give approximately twice the value of the fundamental representation string tension in the leading saddle point approximation from , which has the same value for all weights of the fundamental representation. Therefore relations and clearly show the large $N$ factorization result in dYM theory at leading order of the saddle point without any reference to a large area limit. Although the calculations in this Appendix were done at the leading order saddle point level the same ideas and methods can be applied to show large $N$ factorization regarding the corrections (as in ) to these leading order saddle point results. We thank Mohamed Anber and Aleksey Cherman for comments on the manuscript and Adi Armoni for many enlightening discussions of k-strings. We acknowledge support by a NSERC Discovery Grant. The computations for this paper were performed on the gpc supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto. [99]{} M. " Unsal, “Magnetic bion condensation: A New mechanism of confinement and mass gap in four dimensions,” Phys. Rev. D [**80**]{}, 065001 (2009) \[arXiv:0709.3269 \[hep-th\]\]. M. " Unsal and L. G. Yaffe, “Center-stabilized Yang-Mills theory: Confinement and large N volume independence,” Phys. Rev. D [**78**]{}, 065035 (2008) \[arXiv:0803.0344 \[hep-th\]\]. G. V. Dunne and M. " Unsal, “New Nonperturbative Methods in Quantum Field Theory: From Large-N Orbifold Equivalence to Bions and Resurgence,” Ann. Rev. Nucl. Part. Sci.  [**66**]{}, 245 (2016) \[arXiv:1601.03414 \[hep-th\]\]. A. M. Polyakov, “Quark Confinement and Topology of Gauge Groups,” Nucl. Phys. B [**120**]{}, 429 (1977). A. Cherman and E. Poppitz, “Emergent dimensions and branes from large-$N$ confinement,” Phys. Rev. D [**94**]{}, no. 12, 125008 (2016) \[arXiv:1606.01902 \[hep-th\]\]. K. Johnson and C. B. Thorn, “String-Like Solutions of the Bag Model,” Phys. Rev. D [**13**]{}, 1934 (1976). N. Seiberg and E. Witten, “Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory,” Nucl. Phys. B [**426**]{}, 19 (1994) Erratum: \[Nucl. Phys. B [**430**]{}, 485 (1994)\] \[hep-th/9407087\]. M. R. Douglas and S. H. Shenker, “Dynamics of SU(N) supersymmetric gauge theory,” Nucl. Phys. B [**447**]{}, 271 (1995) \[hep-th/9503163\]. M. M. Anber, E. Poppitz and T. Sulejmanpasic, “Strings from domain walls in supersymmetric Yang-Mills theory and adjoint QCD,” Phys. Rev. D [**92**]{}, no. 2, 021701 (2015) \[arXiv:1501.06773 \[hep-th\]\]. A. Cherman, T. Sch" afer and M. " Unsal, “Chiral Lagrangian from Duality and Monopole Operators in Compactified QCD,” Phys. Rev. Lett.  [**117**]{}, no. 8, 081601 (2016) \[arXiv:1604.06108 \[hep-th\]\]. K. Aitken, A. Cherman, E. Poppitz and L. G. Yaffe, “QCD on a small circle,” arXiv:1707.08971 \[hep-th\]. M. Erfan Shalchian T., “Perturbative formulation of a quantum mechanical transition amplitude,” in preparation. E. Thomas and A. R. Zhitnitsky, “Topological Susceptibility and Contact Term in QCD. A Toy Model,” Phys. Rev. D [**85**]{}, 044039 (2012) \[arXiv:1109.2608 \[hep-th\]\]. M. " Unsal, “Theta dependence, sign problems and topological interference,” Phys. Rev. D [**86**]{}, 105012 (2012) \[arXiv:1201.6426 \[hep-th\]\]. M. M. Anber, “$\Theta$ dependence of the deconfining phase transition in pure $SU(N_{c})$ Yang-Mills theories,” Phys. Rev. D [**88**]{}, no. 8, 085003 (2013) \[arXiv:1302.2641 \[hep-th\]\]. A. Bhoonah, E. Thomas and A. R. Zhitnitsky, “Metastable vacuum decay and $\theta$ dependence in gauge theory. Deformed QCD as a toy model,” Nucl. Phys. B [**890**]{}, 30 (2014) \[arXiv:1407.5121 \[hep-ph\]\]. D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, “Theta, Time Reversal, and Temperature,” JHEP [**1705**]{}, 091 (2017) \[arXiv:1703.00501 \[hep-th\]\]. Y. Tanizaki and Y. Kikuchi, “Vacuum structure of bifundamental gauge theories at finite topological angles,” JHEP [**1706**]{}, 102 (2017) \[arXiv:1705.01949 \[hep-th\]\]. Y. Kikuchi and Y. Tanizaki, “Global inconsistency, ’t Hooft anomaly, and level crossing in quantum mechanics,” arXiv:1708.01962 \[hep-th\]. D. Gaiotto, Z. Komargodski and N. Seiberg, “Time-Reversal Breaking in QCD$_4$, Walls, and Dualities in 2+1 Dimensions,” arXiv:1708.06806 \[hep-th\]. M. M. Anber and A. R. Zhitnitsky, “Oblique Confinement at $\theta\neq 0$ in weakly coupled gauge theories with deformations,” arXiv:1708.07520 \[hep-th\]. G. Cossu and M. D’Elia, “Finite size phase transitions in QCD with adjoint fermions,” JHEP [**0907**]{}, 048 (2009) \[arXiv:0904.1353 \[hep-lat\]\]. H. Vairinhos, “Phase transitions in center-stabilized lattice gauge theories,” PoS LATTICE [**2011**]{}, 252 (2011) \[arXiv:1111.0303 \[hep-lat\]\]. G. Bergner and S. Piemonte, “Compactified $ \mathcal{N}=1 $ supersymmetric Yang-Mills theory on the lattice: continuity and the disappearance of the deconfinement transition,” JHEP [**1412**]{}, 133 (2014) \[arXiv:1410.3668 \[hep-lat\]\]. G. Bergner, P. Giudice, G. M" unster and S. Piemonte, “Witten index and phase diagram of compactified $\mathcal N=1$ supersymmetric Yang-Mills theory on the lattice,” PoS LATTICE [**2015**]{}, 239 (2016) \[arXiv:1510.05926 \[hep-lat\]\]. J. Greensite, “An introduction to the confinement problem,” Lect. Notes Phys.  [**821**]{}, 1 (2011). D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, “Generalized Global Symmetries,” JHEP [**1502**]{}, 172 (2015) \[arXiv:1412.5148 \[hep-th\]\]. D. J. Gross, R. D. Pisarski and L. G. Yaffe, “QCD and Instantons at Finite Temperature,” Rev. Mod. Phys.  [**53**]{}, 43 (1981). J. C. Myers and M. C. Ogilvie, “New phases of SU(3) and SU(4) at finite temperature,” Phys. Rev. D [**77**]{}, 125030 (2008) \[arXiv:0707.1869 \[hep-lat\]\]. M. Shifman, “Advanced topics in quantum field theory: A lecture course,” Cambridge University Press, 2012. T. Misumi and T. Kanazawa, “Adjoint QCD on $\mathbb{R}^3\times S^1$ with twisted fermionic boundary conditions,” JHEP [**1406**]{}, 181 (2014) \[arXiv:1405.3113 \[hep-ph\]\]. M. M. Anber, E. Poppitz and B. Teeple, “Deconfinement and continuity between thermal and (super) Yang-Mills theory for all gauge groups,” JHEP [**1409**]{}, 040 (2014) \[arXiv:1406.1199 \[hep-th\]\]. M. M. Anber and L. Vincent-Genod, “Classification of compactified $su(N_c)$ gauge theories with fermions in all representations,” arXiv:1704.08277 \[hep-th\]. M. M. Anber and T. Sulejmanpasic, “The renormalon diagram in gauge theories on $ {\mathrm{\mathbb{R}}}^3\times {\mathbb{S}}^1 $,” JHEP [**1501**]{}, 139 (2015) \[arXiv:1410.0121 \[hep-th\]\]. V. Pellizzani, “Perturbative aspects of deformed Yang-Mills theories,” M.Sc. Thesis, University of Toronto and EPFL (April 2017, unpublished). M. M. Anber and E. Poppitz, “Microscopic Structure of Magnetic Bions,” JHEP [**1106**]{}, 136 (2011) \[arXiv:1105.0940 \[hep-th\]\]. D. Diakonov, “Topology and confinement,” Nucl. Phys. Proc. Suppl.  [**195**]{}, 5 (2009) \[arXiv:0906.2456 \[hep-ph\]\]. K. M. Lee and P. Yi, “Monopoles and instantons on partially compactified D-branes,” Phys. Rev. D [**56**]{}, 3711 (1997) \[hep-th/9702107\]. T. C. Kraan and P. van Baal, “Monopole constituents inside SU(n) calorons,” Phys. Lett. B [**435**]{}, 389 (1998) \[hep-th/9806034\]. N. S. Manton, “The Force Between ’t Hooft-Polyakov Monopoles,” Nucl. Phys. B [**126**]{}, 525 (1977). J. N. Goldberg, P. S. Jang, S. Y. Park and K. C. Wali, “Interactions Between ’t Hooft-Polyakov Monopoles,” Phys. Rev. D [**18**]{}, 542 (1978). L. O’Raifeartaigh, S. Y. Park and K. C. Wali, “On Magnetic Monopole Interactions,” Phys. Rev. D [**20**]{}, 1941 (1979). D. Simic and M. " Unsal, “Deconfinement in Yang-Mills theory through toroidal compactification with deformation,” Phys. Rev. D [**85**]{}, 105027 (2012) \[arXiv:1010.5515 \[hep-th\]\]. H. Georgi, “Lie algebras in particle physics,” Front. Phys.  [**54**]{}, 1 (1999). Chris Loken et al., “SciNet: Lessons Learned from Building a Power-efficient Top-20 System and Data Centre,” J. Phys.: Conf. Ser. 256, 012026 (2010). D. Antonov and L. Del Debbio, “k string tensions in the 3-d SU(N) Georgi-Glashow model,” JHEP [**0312**]{}, 060 (2003) \[hep-th/0311046\]. M. M. Anber, “The abelian confinement mechanism revisited: new aspects of the Georgi-Glashow model,” Annals Phys.  [**341**]{}, 21 (2014) \[arXiv:1308.0027 \[hep-th\]\]. E. Poppitz and M. " Unsal, “Seiberg-Witten and ’Polyakov-like’ magnetic bion confinements are continuously connected,” JHEP [**1107**]{}, 082 (2011) \[arXiv:1105.3969 \[hep-th\]\]. M. M. Anber and E. Poppitz, “On the global structure of deformed Yang-Mills theory and QCD(adj) on $ {\mathrm{\mathbb{R}}}^3\times {\mathbb{S}}^1 $,” JHEP [**1510**]{}, 051 (2015) \[arXiv:1508.00910 \[hep-th\]\]. A. Hanany, M. J. Strassler and A. Zaffaroni, “Confinement and strings in MQCD,” Nucl. Phys. B [**513**]{}, 87 (1998) \[hep-th/9707244\]. P. Hasenfratz and J. Kuti, “The Quark Bag Model,” Phys. Rept.  [**40**]{}, 75 (1978). C. P. Herzog and I. R. Klebanov, “On string tensions in supersymmetric SU(M) gauge theory,” Phys. Lett. B [**526**]{}, 388 (2002) \[hep-th/0111078\]. J. Greensite and S. Olejnik, “Dimensional Reduction and the Yang-Mills Vacuum State in 2+1 Dimensions,” Phys. Rev. D [**77**]{}, 065003 (2008) \[arXiv:0707.2860 \[hep-lat\]\]. A. Di Giacomo, H. G. Dosch, V. I. Shevchenko and Y. A. Simonov, “Field correlators in QCD: Theory and applications,” Phys. Rept.  [**372**]{}, 319 (2002) \[hep-ph/0007223\]. R. Auzzi and S. P. Kumar, “Non-Abelian k-Vortex Dynamics in N=1\* theory and its Gravity Dual,” JHEP [**0812**]{}, 077 (2008) \[arXiv:0810.3201 \[hep-th\]\]; M. A. C. Kneipp, “Colour superconductivity, Z(N) flux tubes and monopole confinement in deformed N=2\* superYang-Mills theories,” Phys. Rev. D [**69**]{}, 045007 (2004) \[hep-th/0308086\]. G. S. Bali, “Casimir scaling of SU(3) static potentials,” Phys. Rev. D [**62**]{}, 114503 (2000) \[hep-lat/0006022\]. B. Bringoltz and M. Teper, “Closed k-strings in SU(N) gauge theories: 2+1 dimensions,” Phys. Lett. B [**663**]{}, 429 (2008) \[arXiv:0802.1490 \[hep-lat\]\]. B. Lucini and M. Panero, “SU(N) gauge theories at large N,” Phys. Rept.  [**526**]{}, 93 (2013) \[arXiv:1210.4997 \[hep-th\]\]. J. Greensite, “The Confinement problem in lattice gauge theory,” Prog. Part. Nucl. Phys.  [**51**]{}, 1 (2003) \[hep-lat/0301023\]. A. Armoni and M. Shifman, “Remarks on stable and quasistable k strings at large N,” Nucl. Phys. B [**671**]{}, 67 (2003) \[hep-th/0307020\]. Y. Makeenko, “Methods of contemporary gauge theory,” Cambridge University Press, 2002. E. Shuryak and T. Sulejmanpasic, “The Chiral Symmetry Breaking/Restoration in Dyonic Vacuum,” Phys. Rev. D [**86**]{}, 036001 (2012) \[arXiv:1201.5624 \[hep-ph\]\]. A. Armoni, D. Dorigoni and G. Veneziano, “k-String Tension from Eguchi-Kawai Reduction,” JHEP [**1110**]{}, 086 (2011) \[arXiv:1108.6196 \[hep-th\]\]. J. Greensite, B. Lucini and A. Patella, “k-string tensions and the 1/N expansion,” Phys. Rev. D [**83**]{}, 125019 (2011) \[arXiv:1101.5344 \[hep-th\]\]. E. Witten, “Chiral Symmetry, the 1/N Expansion, and the SU(N) Thirring Model,” Nucl. Phys. B [**145**]{}, 110 (1978). [^1]: Hereafter, as most of our studies are Euclidean, we shall denote the spacetime manifold simply by $\R^3 \times \S^1$, but we use $\R^{1,2}\times \S^1$ here in order to stress that $\S^1$ is a spatial circle and the object of our study is not finite-temperature theory. [^2]: Apart for the large-N limit, see below. [^3]: There is a plethora of metastable strings that can also be studied using the tools developed here. An evaluation of their tensions and decay rates is left for future work. See Appendix E for a calculation of some metastable string tensions at leading order. [^4]: An important additional subtlety is that the values of N for which the relations (\[dymscaling2\]) have been derived, while numerically large, are bounded above by an exponentially large number $N \ll 2 \pi e^{c\over \lambda}$, where $\lambda \sim |\log \Lambda N L|^{-1}$ is the arbitrarily small ’t Hooft coupling and $c$ is an ${\cal{O}}(1)$ coefficient. Preliminary estimates suggest that the effect of the W-boson induced mixing on the string tensions (whose neglect is the source of the upper bound on N, see Section \[sec:largeN\]) will not qualitatively change the large-$N$ limit. However, we prefer to defer further discussion until the relevant calculations for dYM have been performed. [^5]: Some of these points were, without elaboration, made earlier in [@Anber:2015kea]. We also note that the glueball spectra in dYM, as well as the mesonic and baryonic spectra with quarks added as in [@Cherman:2016hcd], exhibit many intriguing properties and are the subject of the more quantitative recent study [@Aitken:2017ayq]. [^6]: The reader already familiar with dYM and interested in our numerical and analytic metnods can proceed to Sections \[numericsection\] and \[sec:4\] and the discussion in Section \[sec:5\]. [^7]: \[center\]Center symmetry transformations are global symmetries that can be loosely thought as “gauge” transformations periodic up to the centre of the gauge group. For example, for an $SU(2)$ gauge group, the center-symmetry transformation periodic up to the nontrivial $\Z_2$ center element $z=-1$ can be represented by $U_{-1}(\text{x},x_4) = \text{exp}(i {\pi \over L} x_4 \sigma_3)$, with $U_{-1}(\text{x},0) = - U_{-1}(\text{x},L)$ with $\sigma_3$ the third Pauli matrix and $x_4$—the $\S^1$ coordinate. See [@04] for a proper definition of center symmetry as a global symmetry on the lattice and [@Gaiotto:2014kfa] for a continuum point of view. [^8]: We defined $\omega_N = e^{ 2 \pi i \over N}$. [^9]: The $n_f = 1/2$ massless case leads to vanishing potential, as is clear by comparing the massless limit of (\[eq:2.7\]) with (\[eq:2.4\]). This case corresponds to the minimally supersymmetric Yang-Mills theory in four dimensions. [^10]: This has been explicitly performed for the above choices of parameter up to $SU(10)$ and with considering the effective potentials up to $n=20$. [^11]: General $SU(N)$ theories with semiclassically calculable dynamics at small-$L$ have been classified in [@Anber:2017pak]. [^12]: At subleading order, threshold corrections from the $W$-bosons cause the $N-1$ photons (and consequently, the dual photons) to mix. These mixing effects are expected to be similar to the ones in super-Yang-Mills [@08] and QCD(adj) [@Anber:2014sda; @vito]. They become important in the abelian large-$N$ limit [@Cherman:2016jtu], where dYM has a curious “emergent dimension” representation. The mixing between the $N-1$ photons is also expected to affect the $k$-string tensions in the abelian large-$N$ limit. In this paper, we have not taken these effects into account. [^13]: The $N$-dependence of the lightest $A_4$ shows that the mass scale of the holonomy fluctuations remains fixed in the abelian large-$N$ limit, where $g^2 N$ and $m_W \gg \Lambda$ remain fixed. [^14]: After any twist of $\Omega$ at infinity associated with the magnetic charges $q_{\alpha}$ is removed by a (singular) gauge transformation, the resulting field may be regarded as a mapping of compactified three space (or $\S^3$) onto the group $SU(N)$, leading to the familiar Pontryagin index. More details regarding the definitions of these quantities can be found in [@06]. [^15]: This terminology is adopted for historical reasons. In the limit when the mass of the physical holonomy fluctuations is neglected, both our BPS and KK solutions satisfy a BPS bound and can be found by solving first-order equations. [^16]: Our convention for spherical coordinates is $r(\text{sin} \ \theta \ \text{cos} \ \phi,\text{sin} \ \theta \ \text{sin} \ \phi,\text{cos} \ \theta) = (\text{x}_1,\text{x}_2,\text{x}_3 )$). [^17]: This definition applies to when the eigenvalues of the holonomy ${\Omega}$ at infinity are distinct. For the general definition of magnetic charges that would also apply to holonomies with degenerate eigenvalues at infinity refer to relation (B.6) in [@06] [^18]: A direct calculation of $Q$ for the $SU(2)$ BPS solution yields $Q=1/2$, thus verifying explicitly (\[eq:2.14\]) with $p=0$ and the appropriate expression for $\mu_\alpha$. [^19]: More details regarding this solution and its explicit form can be found in, e.g. [@23]. These “twisted" solutions were first found in [@Lee:1997vp; @Kraan:1998sn] using different techniques. [^20]: While we use energetics terminology, motivated by the electro-/magneto-static analogy, we clearly mean Euclidean action. Also by gauge-invariant terms we refer to any terms in the action of two far separated monopoles that are independent of the Dirac string (singularity of the solutions at $\theta = \pi$ in ) or its orientation. [^21]: At the classical level, the $A_4$-field, mediating the so-called “electric” interactions, is massless hence it is of long range. We stress that the term “scalar interaction” is the precise one, within the framework of spatial-$\S^1$ compactifications; for brevity, we continue calling these interactions “electric” and omit the quotation marks in what follows. Furthermore, as already explained, at the quantum level the $A_4$ field gains mass hence the electric interaction is short range and not important in the derivation of the string tension action. We only discuss the electric interaction here for the sake of mentioning some points not usually explicitly discussed with regard to the classical interaction of monopole-instantons. [^22]: For another discussion on the core interaction between dyons refer to [@60]. [^23]: $\tau^3_{(j)}$ refers to the $\tau^3$ Pauli matrix placed in the j-th Lie subalgebra of $SU(N)$ along the diagonal. [^24]: This also implies that a more precise treatment (as opposed to simply summing far separated monopoles) for the construction of far separated monopole solutions is required, in particular, one that will not involve any non-gauge-invariant contributions. The construction of the monopole gas by summing far separated monopole solutions is appealing due to its simplicity and the fact that its leading gauge-invariant interaction terms reproduce results consistent with the more accurate far separated solutions, as studied in [@25; @29; @30]. [^25]: The $n^{4/3}$ power in the last term is an attempt at a better than naive estimate of the error. Naively, one could imagine the correction scaling as $n^2$, with $d$ being the typical separation between monopoles, but it is clear that not all monopoles are separated by the same distance. Assuming a uniform distribution of monopoles, with $d$ the closest distance between a given monopole and its neighbors, one can arrive at the estimate given (one expects some power $n^p$ with $1 < p < 2$). Note also the fact that not all $N$ types of monopoles have classical interactions, is not taken into account in writing the last term in (\[eq:2.30\]). [^26]: At distances $R$ $\gtrapprox \Lambda^{-1}$ with $\Lambda$ being the strong scale of the theory. [^27]: [More details regarding the derivation of this partition function can be found in [@01]. In this Section we will use this partition function to derive the Wilson loop inserted dual photon action for the evaluation of the $k$-string tensions. $Z_{\text{pert.}}$ refers to the perturbative contribution of the effective dual photon action: $Z_{\text{pert.}} = { \int D[ \sigma] \;\text{exp}(-\int_{{\rm I\!R}^3} d^3\text{x} {1 \over 2} {g^2 \over 8 \pi^2 L}( \nabla \sigma )^2 )}$.]{} [^28]: For simplicity it has been written for only two insertions of $e^{i q \cdot \sigma}$. [^29]: See Footnote \[modes1\] for the relation between the $U(N)$ and $SU(N)$ Cartan fields and further comments on the duality. [^30]: \[modes1\]The remarks that follow are tangential to our exposition, but serve to convince the reader of the consistency of our dual action coefficients with charge quantization (our Eq. (\[eq:2.40\]) was derived solely by demanding that the long-distance interactions between monopole-instantons from Section \[2.3\] are correctly reproduced) and to correct minor typos in expressions that have appeared previously in the literature. Integrating the duality relation (\[dualityelectric\]), we obtain $${g^2 \over \sqrt{2} L}\; \; \oint_C { d \sigma^A \over 2 \pi} = \oint_C d \vec{n} \cdot \vec{E}^A, \label{dual12}$$ representing the fact that a static electric charge inside $C$ generates flux through $C$ ($\vec{n}$ is an outward unit normal to $C$) which duality relates to the $\sigma$-field monodromy around $C$. Eq. (\[dual12\]) implies that the $\sigma^A$ fields have periodicities determined by the fundamental electric charges. To find them, we begin with the relation between the $N$ Cartan field strengths $F^A_{kl}$ that first appeared in (\[dualityrelation\], \[bianchi\]) and the original $N-1$ $SU(N)$ fields $F^a_{kl}$ in (\[eq:2.1\]). The reader can convince themselves that it is given by $F^A_{kl} = {1 \over \sqrt{N}} F_{kl}^0 + \sum\limits_{a=1}^{N-1} F_{kl}^a \lambda^{a A}$, with $\lambda^{aA} \equiv (\theta^{aA} - a \delta^{a+1, A})/\sqrt{a(a+1)}$, where $\theta^{aA} = 1$ for $a \ge A$ and $\theta^{aA}=0$ otherwise. The spectator $U(1)$ field $F^0_{kl}$ is not coupled to dynamical sources. The relations $\sum\limits_{A=1}^N \lambda^{a A} \lambda^{b A} = \delta^{ab}$ and $ \sum\limits_{A=1}^N \lambda^{a A} =0$ help establish that $\sum\limits_{A=1}^N (F^A)^2 = \sum\limits_{a=1}^{N-1} (F^a)^2 + (F^0)^2$. A fundamental static charge is represented by the insertion of a static Wilson loop in the fundamental representation. Early on, see (\[eq:2.1\]), we stated that our fundamental representation generators are normalized as tr($T^a T^b) = \delta^{ab}/2$. Thus, using the definitions just made, it follows that the fundamental representation Cartan generators are $T^a = {\rm diag}(\lambda^{a 1},..., \lambda^{a N})/\sqrt{2}$. A fundamental static Wilson loop is then represented by insertions of $\int dt A_0^a(\vec{r},t) \lambda^{aA}/\sqrt{2}$ ($A$ labels the Wilson loop eigenvalues) in the path-integral action. Considering one of the eigenvalues of the Wilson loop (one component of the fundamental static quark), the corresponding electric flux is found by solving the static equation of motion, ${L \over g^2} \nabla^2 A_0^a = {\lambda^{a A}\over \sqrt{2}}\delta(\vec{r})$, thus $ \oint_{C_A} d \vec{n}\cdot \vec{E}^a = - \lambda^{a A}{g^2 \over \sqrt{2} L}$. From the earlier relations, we also have that $\vec{E}^a = \sum\limits_{A=1}^N \lambda^{a A} \vec{E}^A$, thus $\sum\limits_{B=1}^N \lambda^{a B} \oint_{C_A} d \vec{n} \cdot \vec{E}^B = {g^2 \over \sqrt{2} L} \lambda^{a A}$. Finally, from (\[dual12\]), this leads to $\sum\limits_{B=1}^N \lambda^{a B} \oint_{C_A} { d \sigma^B \over 2 \pi} = \lambda^{a A}$. It can be already seen, from the explicit form of $\lambda^{a A}$, that this relation implies that the periodicity (monodromies) of differences of $\sigma^A$’s have to be proportional to $2 \pi$. Even more explicitly, from the relation $\sum\limits_{a=1}^{N-1} \lambda^{a A} \lambda^{a B} = \delta^{AB} - {1 \over N}$, one finds that the monodromies of the dual photons are given by $2 \pi$ times the weights of the fundamental representation. This is consistent with the periodicities of the potential terms in (\[eq:2.39\]) and with the dual photon actions given in e.g. [@Simic:2010sv; @23; @08]. [^31]: We note that with this choice the effect of $\mu$ on $\tilde\zeta$ is comparable to the effect of finite $A_4$ mass on the classical monopole action, an effect that we have neglected throughout. Matching between the UV theory, valid at scales $\ge m_W$, and the IR theory, valid at scales $\ll m_W$, to better precision that has been attempted so far is needed to properly account for these effects. We also note that in the supersymmetric case, the only case where the determinants in the monopole-instanton backgrounds have actually been computed, where $\sigma$ is replaced by a chiral superfield and the monopole-instantons are “localized in superspace,” this ambiguity is absent [@08]—in super-Yang-Mills, divergent self energies of monopoles due to electric and magnetic charge cancel out in the analogue of (\[eq:2.37\]). [^32]: A more detailed derivation of is done in Appendix \[sec:B2\]. [^33]: The order of magnitude of the string tension is $g^2 N m_\gamma m_W$. Thus $W$-boson production takes place once $R \sim {\rm O}(1/(g^2 N m_\gamma))$ and Higgs production (recall $m_H \sim g \sqrt{N} m_W$) when $R \sim {\rm O}(1/(g \sqrt{N} m_\gamma))$. Notice that the values on the r.h.s., owing to small coupling, are much larger than the Debye screening length $1/m_\gamma$. [^34]: This can be seen as follows. Consider the boundary conditions $\sigma (0^+) = \pi \mu_k + \bar{\text{z}}$ and $\sigma (0^-) = - \pi \mu_k + \bar{\text{z}}$. We will show that the minimum of is when $\bar{\text{z}} = 0$. If $\sigma(\text{z})$ is an extremum solution of so is $\sigma(-\text{z})$ and $- \sigma(\text{z})$, therefore it can be seen that $\bar{\text{z}} = 0$ is an extremum point. It is a minimum since otherwise the kinetic term will increase if we make the magnitude of the boundaries larger than $\pi (\mu_k)_j$ on either side of $\text{z} = 0^+$ or $\text{z} = 0^-$. [^35]: That an ansatz with a single exponential works for $SU(3)$ is a consequence of the existence of only a single mass scale in the dual-photon theory, a fact that only holds for $N=2,3$. [^36]: The superscript $(m)$ indicates that this is the discretization with $m$ partitions of the interval. [^37]: Minimizing the second derivative of $\bar{T}^{m,J}_{k,2}$ with respect to $f_{lp}$, gives $-{\delta \text{z} \over 3}$ for each time the variable $f_{lp}$ appears in the sum over $j$ and $h$. Since it appears 4 times when replacing each variable $f_{jh+1}$, $f_{j+1h+1}$, $f_{jh}$ and $f_{j+1h}$ in the expression and the minimum value is the same for all 4 cases, this gives $-{4\over 3} \delta \text{z}$ for a lower bound on the 2nd derivative of the second term. [^38]: Numerical computations of the string tensions were performed on the gpc supercomputer at the SciNet HPC Consortium [@22]. Due to a high number of $k$-string calculations ($>1000$) with most of them involving minimization of multivariable functions with more than 500 variables, using a cluster that could perform many $k$-string computations at the same time in parallel was necessary. [^39]: As is evident from equation , the expansion parameter is $ {\lambda b^2\over 4 m^2}$; as discussed there, convergence of the perturbative expansion of the saddle point for a $g^4$ interaction term only requires that this parameter be less than $ {1/2}$. This condition is met in dYM theory, but not in QCD(adj) [@Anber:2015kea], for the choices of parameters following from the underlying action (In dYM from below equation ), although not strictly required since the full potential in both theories includes higher non-linearities and in taking these into account the perturbative series evaluation of the saddle point would be a convergent one. [^40]: This agreement can be further improved, as we have verified for $SU(2)$. Summing only contributions to (\[wilsonperturbative\]) to order $\lambda$ we obtain the value $7.84$ shown in Table \[table:71\]. Including the higher-order correction terms show an oscillatory convergence: Including the first order correction due to the $g_1^6$ term in gives $8.285$ and including order $\lambda^2$ of the quartic term expansion, we obtain $8.007$, to be compared with the exact value $8$. [^41]: We have to note that since we are summing to all orders such a perturbative expansion is justified although ${1 \over \sqrt{\beta}}$ becomes large as $\beta \rightarrow 0$. [^42]: \[footnotetaylor\]The Taylor series expansion of $\sqrt{1+ \text{x}}$ converges for $|\text{x}| < r = 1$. Evaluating $\text{x} = {2\lambda h^2 / m^2}$ for $h = {b / 2}$ with values of parameters from below equation gives $|\text{x}| = {\pi^2 / 12} < 1$ which lies within the radius of convergence. [As a reminder we mention that the condition $|\text{x}| = |{2\lambda h^2 / m^2}| < 1$ is not strictly required in dYM since the full potential is cosine which would allow for a wider range of these parameters.]{} [^43]: The diagonalization matrix $D$ has the effect of an $\Z_N$ Fourier transform and the eigenvalues of $A$ are $\Lambda_q = 4 \sin^2{ \pi q\over N}$, $q = 1, ..., N-1$ and $\Lambda_N = 0$. As discussed in [@Cherman:2016jtu], this is the spectrum of a latticized emergent dimension of $N$ sites. [^44]: \[eigenvectorfootnote\]Because $D= (v_1,...,v_N)$, where $v_q$, $q=1,...N-1$ are the eigenvectors of $A$ with eigenvalues $\Lambda_q = 4 \sin^2{ \pi q \over N}$ and $v_N ={1\over \sqrt{N}} (1,1,1...,1)^T$ is the zero eigenvector. For use below, the other $N-1$ eigenvectors, for brevity shown for odd $N$ only, with components $v_q^l$ are: $v_{q < {N\over 2}}^l = \sqrt{2\over N} \sin{2 \pi q l\over N}$ and $v_{ {N\over 2}<q<N}^l = \sqrt{2\over N} \cos{2 \pi q l\over N}$. [^45]: Note that the string tension remains fixed at large-N, despite the vanishing mass gap, as there is a number of dual photons of nonzero mass ($\sim m_\gamma$) whose flux is confined, as well as a number of dual photons approaching zero mass ($\sim m_\gamma/N$) whose flux spreads out. Thus the finite tension confining string in the gapless abelian large-N limit is a rather fuzzy object. We defer a further study until the large-N corrections, discussed in [@Cherman:2016jtu] for sYM, are better understood in the dYM case. [^46]: Degenerate string tensions will occur when the corresponding weights are related by the unbroken $\Z_N$ center symmetry. For example, for the fundamental representation all weights have the same string tension, see Section \[sec:compare\]. For higher $N$-ality representations, the dim($r$) weights fall into distinct $\Z_N$ orbits, each of which has degenerate string tensions. [^47]: This should not be taken to mean that the nonabelian nature of the theory is not relevant: on the contrary, it is crucial in both examples. [^48]: There are hints that the two confinement mechanisms are related, see [@Poppitz:2011wy]. [^49]: In the terminology of [@Gaiotto:2014kfa]. [^50]: In terms independent of the choice of basis vectors of the root lattice, the $\Z_N$ center acts on the dual photons ${\sigma}$ as the ordered product of Weyl reflections with respect to all simple roots, see [@Anber:2015wha]. [^51]: See [@16] for a description of confining strings in softly-broken Seiberg-Witten theory within its $M$-theory embedding. [^52]: As discussed in Section \[sec:5.1.1\], one of the qualitative reasons why charges $\mu_k$ are confined by strings of the lowest tension (for every representation) is that adding or subtracting any root from $\mu_k$ leads to higher “vacuum energy” cost. [^53]: Ref. [@13] studied a rotating string solution, but a simpler static one exists, see discussion below and ref. [@Hasenfratz:1977dt], which also contains a review of the physical picture underlying the MIT Bag Model of the Yang-Mills vacuum. [^54]: The classical chromoelectric flux of static sources in a given representation is proportional to the quadratic Casimir, see Section 3.3 in [@Hasenfratz:1977dt]. Also note that the “square root of Casimir” scaling is obtained in the Bag Model without surface tension and that introducing additional Bag Model parameters, e.g. bag surface tension, modifies the scaling with the Casimir of the representation. [^55]: Noise of order $\epsilon$ refers to a random fluctuation of order $\epsilon$ imposed on the data. The fluctuation can be a Gaussian, uniform, etc., distribution of width $\epsilon$ centred on the data point. We have used a uniform distribution. [^56]: We consider half of the upper bound estimate of the error in Table \[table:1\] ($-0.006/2 = -0.003$) as the value of error for k-strings. Hence for k-string ratios as a typical example we get: $T_2/T_1 = 8.0006_{-0.003}/6.8583_{-0.003} \approx 1.1666^{+ 0.0005}_{- 0.0004} $. The reader has to be reminded that an error of $-0.003$ is still a high confidence interval for the true value of k-strings. [^57]: For another discussion on the non-commutativity of the large-$N$ and large-$T$ limits refer to [@62]. [^58]: See further below the discussion of this Section (between eqs.  and ) as well as the explicit calculations in Appendix \[sec:appxproduct\]. [^59]: Note that due to the $Z_N$ symmetry of the $N$ components of $\sigma_a$, $1 \leq a \leq N$ can be considered similar to $N$ points on a circle corresponding to angles $\theta = {2 \pi a / N }$. The components near the $q$’th component are defined as the points (components) close to the $q$’th point on this circle. [^60]: In components, the weights of the product representation are $(\mu^{(ij)}_{\square \otimes \square})_p = \delta^i_p + \delta^j_p -2/N$. The $\Z_N$ symmetry acts as $\mu^{(ij)}_{\square \otimes \square} \rightarrow \mu^{(i+1({\rm mod} N),j+1({\rm mod} N))}_{\square \otimes \square}$, i.e. the $N^2$ weights of the product representation fall into $N$ $\Z_N$ orbits. [^61]: In this regard, notice that the components of $\mu^{(ij)}_{\square \otimes \square}$ with $i \ne j$ can also be written as $-1/(N/2)$ or $1- 1/(N/2)$, similar to the components of $\mu^i_\square$ for $SU([N/2])$ (when $N$ is odd the difference would be clearly negligible). [^62]: At larger values of $N$, as mentioned in the preamble of Section \[sec:largeN\], the virtual effects of the W-bosons become important which has not been taken into account in this work. We speculate, based on preliminary results, that with taking these effects into account the same picture, i.e. large $N$ factorization and interacting k-strings, persists at large $N$. [^63]: This behaviour has been verified in the numerical simulations up to $J=14$. It has also been verified for cases when an analytic solution is possible. For example expanding the cosine term and keeping only the quadratic term. For this case it would be possible to solve the saddle point analytically for a finite boundary condition at $\text{z} = J$ and see that $\bar{T}^{\infty,J}_{k2,min}$ starts from $0$ at $\text{z}=0$ and increases monotonically to ${1\over 2} \bar{T}^{\infty, \infty}_{k,\text{min}}$ at $\text{z} = \infty$. [^64]: To obtain , we noted that for large and odd $N$, for $1 \le q < {N\over 2}$: $$\label{eq:E.6} \begin{split} {b_q \over 2\pi} = \sum\limits_{j=1}^N (\mu^h)_j D_{jq} = \sqrt{{2 \over N}} \{ \sin{2 \pi a q \over N} + \sin{2 \pi b q \over N} - \sum\limits_{j=1}^N {2 \over N} \sin{2 \pi j q \over N} \} \approx \sqrt{{2 \over N}} \{ \sin{2 \pi a q \over N} + \sin{2 \pi b q \over N} + 0 \}, \end{split}$$ and used the fact that the last term in the first line of for large $N$ can be approximated by $\approx -{1 \over \pi}\int_0^{2 \pi}d y \sin q y = 0$. Also, a similar expression can be written for ${N\over 2} < q < N$.

--- abstract: 'Corrections to scaling in the 3D Ising model are studied based on non–perturbative analytical arguments and Monte Carlo (MC) simulation data for different lattice sizes $L$. Analytical arguments show the existence of corrections with the exponent $(\gamma-1)/\nu \approx 0.38$, the leading correction–to–scaling exponent being $\omega \le (\gamma-1)/\nu$. A numerical estimation of $\omega$ from the susceptibility data within $40 \le L \le 2048$ yields $\omega=0.25(33)$. It is consistent with the statement $\omega \le (\gamma-1)/\nu$, as well as with the value $\omega = 1/8$ of the GFD theory. We reconsider the MC estimation of $\omega$ from smaller lattice sizes to show that it does not lead to conclusive results, since the obtained values of $\omega$ depend on the particular method chosen. In particular, estimates ranging from $\omega =1.274(72)$ to $\omega=0.18(37)$ are obtained by four different finite–size scaling methods, using MC data for thermodynamic average quantities, as well as for partition function zeros. We discuss the influence of $\omega$ on the estimation of exponents $\eta$ and $\nu$.' author: - | J. Kaupužs$^{1,2}$ [^1] , R. V. N. Melnik$^3$, J. Rimšāns$^{1,2,3}$\ $^1$Institute of Mathematics and Computer Science, University of Latvia\ 29 Raina Boulevard, LV–1459 Riga, Latvia\ $^2$ Institute of Mathematical Sciences and Information Technologies,\ University of Liepaja, 14 Liela Street, Liepaja LV–3401, Latvia\ $^3$ The MS2 Discovery Interdisciplinary Research Institute,\ Wilfrid Laurier University, Waterloo, Ontario, Canada, N2L 3C5 title: | **Corrections to finite–size scaling in the 3D Ising model based on non–perturbative approaches\ and Monte Carlo simulations** --- **Keywords:** Ising model, corrections to scaling, non–perturbative methods, Feynman diagrams, Monte Carlo simulation Introduction {#intro} ============ The critical exponents of the three–dimensional (3D) Ising universality class have been a subject of extensive analytical as well as Monte Carlo (MC) studies during many years. The results of the standard perturbative renormalization group (RG) methods are well known [@Amit; @Ma; @Justin; @Kleinert; @PV]. An alternative analytical approach has been proposed in [@K_Ann01] and further analyzed in [@K2012], where this approach is called the GFD (Grouping of Feynman Diagrams) theory. A review of MC work till 2001 is provided in [@HasRev]. More recent papers are [@Has1; @Has2; @GKR11; @KMR_2011; @KMR_2013]. In this paper we will focus on the exponent $\omega$, which describes the leading corrections to scaling. A particular interest in this subject is caused by recent challenging non-perturbative results reported in [@K2012], showing that $\omega \le (\gamma-1)/\nu$ holds in the $\varphi^4$ model based on a rigorous proof of certain theorem. The scalar 3D $\varphi^4$ model belongs to the 3D Ising universality class with $(\gamma-1)/\nu \approx 0.38$. Therefore, $\omega$ is expected to be essentially smaller than the values of about $0.8$ predicted by standard perturbative methods and currently available MC estimations. The results in [@K2012] are fully consistent with the predictions of the alternative theoretical approach of [@K_Ann01], from which $\omega=1/8$ is expected. We have performed a Monte Carlo analysis of the standard 3D Ising model, using our data for very large lattice sizes $L$ up to $L=2048$, to clarify whether $\omega$, extracted from such data, can be consistent with the results of [@K2012] and [@K_Ann01]. Since our analysis supports this possibility, we have further addressed a related question how a decrease in $\omega$ influences the MC estimation of critical exponents $\eta$ and $\nu$. We have also tested different finite–size scaling methods of estimation $\omega$ from smaller lattice sizes to check whether such methods always give $\omega$ consistent with $0.832(6)$, as one can be expected from the references in [@Has1]. Models with the so-called improved Hamiltonians are often considered instead of the standard Ising model for a better estimation of the critical exponents [@Has1; @Has2]. The basic idea of this approach is to find such Hamiltonian parameters, for which the leading correction to scaling vanishes. However, this correction term has to be large enough and well detectable for the estimation of $\omega$. So, this idea is not very useful in our case. Analytical arguments {#sec:analytical} ==================== In [@K2012], the $\varphi^4$ model in the thermodynamic limit has been considered, for which the leading singular part of specific heat $C_V^{sing}$ can be expressed as $$C_V^{sing} \propto \xi^{1/\nu} \left( \int_{k<\Lambda'} [G({\bf k})- G^*({\bf k})] d {\bf k} \right)^{sing} \;, \label{eq:CVsing}$$ assuming the power–law singularity $\xi \sim t^{-\nu}$ of the correlation length $\xi$ at small reduced temperature $t \to 0$. Here $G({\bf k})$ is the Fourier–transformed two–point correlation function, and $G^*({\bf k})$ is its value at the critical point. This expression is valid for any positive $\Lambda' < \Lambda$, where $\Lambda$ is the upper cut-off parameter of the model, since the leading singularity is provided by small wave vectors with the magnitude $k = \mid {\bf k} \mid \to 0$ and not by the region $\Lambda' \le k \le \Lambda$. In other words, $C_V^{sing}$ is independent of the constant $\Lambda'$. The leading singularity of specific heat in the form of $C_V^{sing} \propto (\ln \xi)^{\lambda} \xi^{\alpha/\nu}$ and the two–point correlation function in the asymptotic form of $G({\bf k}) = \sum_{\ell \ge 0} \xi^{(\gamma - \theta_{\ell})/\nu} g_{\ell}(k \xi)$, $G^*({\bf k}) = \sum_{\ell \ge 0} b_{\ell} k^{(-\gamma + \theta_{\ell})/\nu}$ with $\theta_0=0$ and $\theta_{\ell}>0$ for $\ell \ge 1$ have been considered in [@K2012]. These expressions are consistent with the conventional scaling hypothesis, $g_{\ell}(k \xi)$ being the scaling functions. The exponent $\lambda$ is responsible for possible logarithmic correction in specific heat, whereas the usual power–law singularity is recovered at $\lambda=0$. According to the theorem proven in [@K2012], the two–point correlation function of the $\varphi^4$ model contains a correction with the exponent $\theta_{\ell} = \gamma + 1 -\alpha - d \nu$, if $C_V^{sing}$ can be calculated from (\[eq:CVsing\]), applying the considered here scaling forms, if the result is $\Lambda'$–independent, and if the condition $\gamma + 1 -\alpha - d \nu >0$ is satisfied for the critical exponents. Applying the known hyperscaling hypothesis $\alpha + d \nu = 2$, it yields $\theta_{\ell} = \gamma -1$ for $\gamma >1$. Apparently, the listed here conditions of the theorem are satisfied for the scalar 3D $\varphi^4$ model. Since the critical singularities are provided by long–wave fluctuations, the condition of $\Lambda'$–independence is generally meaningful. The assumption $\xi \sim t^{-\nu}$ (with no logarithmic correction) and the considered here scaling forms (with $\lambda =0$), as well as the relation $\gamma + 1 -\alpha - d \nu >0$ (or $\gamma >1$ according to the hyperscaling hypothesis) are correct for the scalar 3D $\varphi^4$ model, according to the current knowledge about the critical phenomena. The correction with the exponent $\theta_{\ell}$ corresponds to the one with $\omega_{\ell} =\theta_{\ell}/\nu$ in the finite–size scaling. In such a way, the discussed here analytical arguments predict the existence of a finite–size correction with the exponent $(\gamma-1)/\nu$ in the scalar 3D $\varphi^4$ model. As discussed in [@K2012], nontrivial corrections tend to be cancelled in the 2D Ising model, in such a way that only trivial ones with integer $\theta_{\ell}$ are usually observed. However, there is no reason to assume such a scenario in the 3D case. Therefore, the existence of corrections with the exponent $(\gamma-1)/\nu$ is expected in the 3D Ising model, since it belongs to the same universality class as the 3D $\varphi^4$ model. Because this correction is not necessarily the leading one, the prediction is $\omega \le \omega_{\mathrm{max}}$, where $\omega_{\mathrm{max}} =(\gamma-1)/\nu$ is the upper bond for the leading correction–to–scaling exponent $\omega$. Using the widely accepted estimates $\gamma \approx 1.24$ and $\nu \approx 0.63$ [@Justin] for the 3D Ising model, we obtain $\omega_{\mathrm{max}} \approx 0.38$. The prediction of the GFD theory [@K_Ann01] is $\gamma=5/4$, $\nu =2/3$ and, therefore, $\omega_{\mathrm{max}} = 0.375$. Thus, we can state that in any case $\omega_{\mathrm{max}}$ is about $0.38$. The value of $\omega$ is expected to be $1/8$ according to the GFD theory considered in [@K_Ann01; @K2012]. MC estimation of $\omega$ from finite–size scaling ================================================== The case of very large lattice sizes $L \le 2048$ {#sec:large} ------------------------------------------------- We have simulated the 3D Ising model on simple cubic lattice with periodic boundary conditions. The Hamiltonian $H$ of the model is given by $$H/T = - \beta \sum_{\langle ij \rangle} \sigma_i \sigma_j \;,$$ where $T$ is the temperature measured in energy units, $\beta$ is the coupling constant and $\langle ij \rangle$ denotes the pairs of neighboring spins $\sigma_i = \pm 1$. The MC simulations have been performed with the Wolff single cluster algorithm [@Wolff], using its parallel implementation described in [@KMR_2010]. An iterative method, introduced in [@KMR_2010], has been used here to find pseudocritical couplings $\widetilde{\beta}_c(L)$ corresponding to certain value $U=1.6$ of the ratio $U=\langle m^4 \rangle / \langle m^2 \rangle^2$, where $m$ is the magnetization per spin. We have evaluated by this method the susceptibility $\chi = L^3 \langle m^2 \rangle$ an the derivative $\partial Q /\partial \beta$ at $\beta = \widetilde{\beta}_c(L)$, where $Q=1/U$. The results for $16 \le L \le 1536$ are already reported in Tab. 1 of our earlier paper [@KMR_2011]. We have extended the simulations to lattice sizes $L=1728$ and $L=2048$, using approximately the same number of MC sweeps as for $L=1536$ in [@KMR_2011]. Thus, Tab. 1 of [@KMR_2011] can be now completed with the new results presented in Tab. \[tab1\] here. [|c|c|c|c|]{} ------------------------------------------------------------------------ L & $\widetilde \beta_c$ & $\chi/L^2$ & $10^{-3} \partial Q /\partial \beta$\ 2048 & 0.2216546252(66) & 1.1741(27) & 151.1(1.1)\ 1728 & 0.2216546269(94) & 1.1882(20) & 116.98(87)\ The exponent $\omega$ describes corrections to the asymptotic finite–size scaling. In particular, for the susceptibility at $\beta = \widetilde{\beta}_c(L)$ we have $$\chi \propto L^{2-\eta} \left( 1 + a L^{-\omega} + o \left( L^{-\omega} \right) \right) \;. \label{eq:chi}$$ We define the effective exponent $\eta_{\mathrm{eff}}(L)$ as the mean slope of the $-\ln \chi$ vs $\ln L$ plot, evaluated by fitting the data within $[L/2,2L]$. It behaves asymptotically as $\eta_{\mathrm{eff}}(L)=\eta + \mathcal{O} \left( L^{-\omega} \right)$. It has been mentioned in [@KMR_2011] that $\omega$ might be as small as $1/8$, since the plot of the effective exponent $\eta_{\mathrm{eff}}$ vs $L^{-1/8}$ looks rather linear for large lattice sizes (see Fig. 6 in [@KMR_2011]). This observation is confirmed also by the extended here data, as it can be seen from Fig. \[fig1\] (left). ![The $\eta_{\mathrm{eff}}$ vs $L^{-1/8}$ (left) and the $\Phi_2(L)$ vs $L^{-1/8}$ (right) plots. Straight lines show the linear fits for large enough lattice sizes $L$.[]{data-label="fig1"}](eta_eff.eps "fig:"){width="48.50000%"} ![The $\eta_{\mathrm{eff}}$ vs $L^{-1/8}$ (left) and the $\Phi_2(L)$ vs $L^{-1/8}$ (right) plots. Straight lines show the linear fits for large enough lattice sizes $L$.[]{data-label="fig1"}](chi_ratio.eps "fig:"){width="48.50000%"} An estimate of $\omega$ can be obtained by fitting the $\eta_{\mathrm{eff}}(L)$ data. Here we use a more direct method, which gives similar, but slightly more accurate results. We consider the ratio $\Phi_b(L) = b^{-4} \chi(bL)/\chi(L/b)$ at $\beta = \widetilde{\beta}_c(L)$, where $b$ is a constant. According to (\[eq:chi\]), $\Phi_b(L)$ behaves as $$\Phi_b(L) = A + B L^{-\omega} \label{eq:phi}$$ at $L \to \infty$, where $A = b^{-2 \eta}$ and $B = a b^{-2 \eta} \left( b^{-\omega} -b^{\omega} \right)$. The correction amplitude $B$ is larger for a larger $b$ value, whereas a smaller $b$ value allows us to obtain more data points for $\Phi_b(L)$. The actual choice $b=2$ is found to be optimal for our data. Like the $\eta_{\mathrm{eff}}(L)$ vs $L^{-1/8}$ plot, also the $\Phi_2(L)$ vs $L^{-1/8}$ plot can be well approximated by a straight line for large enough lattice sizes, as shown in Fig. \[fig1\]. Thus, $\omega$ could be as small as $1/8$. [|c|c|c|]{} ------------------------------------------------------------------------ $L_{\mathrm{min}}$ & $\omega$ & $\chi^2/\mathrm{d.o.f.}$\ 32 & 1.055(76) & 1.07\ 40 & 0.99(11) & 1.09\ 48 & 0.99(16) & 1.16\ 54 & 1.02(22) & 1.23\ 64 & 0.76(29) & 1.14\ 80 & 0.25(33) & 0.76\ 96 & 0.06(38) & 0.74\ 108 & 0.27(46) & 0.70\ 128 & 0.11(59) & 0.75\ We have fit the quantity $\Phi_2(L)$ to (\[eq:phi\]) within $L \in [L_{\mathrm{min}},1024]$ (estimated from the $\chi/L^2$ data within $L \in [L_{\mathrm{min}}/2,2048]$) to evaluate $\omega$. The results are collected in Tab. \[tab2\]. The estimated $\omega$ values are essentially decreased for $L_{\mathrm{min}} \ge 80$ as compared to smaller $L_{\mathrm{min}}$ values. Moreover, the quality of fits is remarkably improved in this case, i. e., the values of $\chi^2$ of the fit per degree of freedom ($\chi^2/\mathrm{d.o.f.}$) become smaller. Note that $L_{\mathrm{min}} = 80$ corresponds to the fit interval for $\Phi_2(L)$ in Fig. \[fig1\], where the data are well consistent with $\omega=1/8$. From a formal point of view, $\omega = 0.25(33)$ at $L_{\mathrm{min}} = 80$ can be considered as the best estimate from our data, since it perfectly agrees with the results for $L_{\mathrm{min}} > 80$ and has the minimal statistical error within $L_{\mathrm{min}} \ge 80$. The estimate $\omega = 0.06(38)$ at $L_{\mathrm{min}} =96$ most clearly shows the deviation below the usually accepted values at about $0.8$, e. g., $\omega = 0.832(6)$ reported in [@Has1]. Our estimation is fully consistent with the analytical arguments in Sec. \[sec:analytical\], since all our $\omega$ values for $L_{\mathrm{min}} \ge 80$ are smaller than $\omega_{\mathrm{max}} \approx 0.38$ and also well agree with $1/8$. Unfortunately, the statistical accuracy of this estimation is too low to rule out a possibility that the dropping of $\omega$ to smaller values at $L_{\mathrm{min}} \ge 80$ is caused by statistical errors in the data. However, the decrease in $\omega$ for large enough lattice sizes is strongly supported by the theorem discussed in Sec. \[sec:analytical\]. Note also that the recent MC analysis of the 2D $\varphi^4$ model [@KMR_14] is consistent with this theorem. These facts make our MC estimation plausible. Note that there exist many quantities, which scale asymptotically as $A + B L^{-\omega}$ with different values of coefficients $A$ and $B$ — see, e. g., [@Hasenbusch; @Has1], as well as the examples in the next section. In principle, all of them can be used to estimate $\omega$. However, it is possible that the leading correction term $B L^{-\omega}$ for a subset of such quantities is too small as compared to statistical errors and, therefore, it is not well detectable at large lattice sizes. It means that a correction with small $\omega$ of about $1/8$, probably, will not be detected by MC analysis in many cases, but this still does not imply that such a correction does not exist. Thus, it is sufficient to demonstrate clearly that such a correction exists in one of the cases. Our MC analysis shows that $\Phi_2(L)$ is an appropriate quantity where corrections, i. e., variations in $\Phi_2(L)$, are well detectable even for very large values of $L$. Moreover, it suggests that a correction with such small $\omega$ as $1/8$, very likely, exists here. Different estimates from the data for smaller lattice sizes {#sec:difomega} ----------------------------------------------------------- The quality of fits with $L_{\mathrm{min}} < 80$ is remarkably improved if 5 data points for the largest lattice sizes are discarded, i. e., if $\Phi_2(L)$ is fit within $L \in [L_{\mathrm{min}},432]$. Choosing also not too large values of $L_{\mathrm{min}}$, we obtain formally quite good (provided by good fits with sufficiently small $\chi^2/\mathrm{d.o.f.}$ values) and stable estimates from remarkably smaller lattice sizes than those in Sec. \[sec:large\]. These are presented in Tab. \[tab3\]. The estimate $\omega = 1.171(96)$ at $L_{\mathrm{min}}=32$ is accepted as the best one from this reduced data set, since it perfectly agrees with the results for $L_{\mathrm{min}}>32$ and has the smallest statistical error. [|c|c|c|]{} ------------------------------------------------------------------------ $L_{\mathrm{min}}$ & $\omega$ & $\chi^2/\mathrm{d.o.f.}$\ 32 & 1.171(96) & 0.77\ 40 & 1.17(14) & 0.83\ 48 & 1.29(22) & 0.84\ We have tested another finite–size scaling method. Based on our simulations discussed in [@KMR_2013], we have evaluated $U=U(L)$ at the pseudocritical coupling $\hat{\beta}_c(L)$, corresponding to the maximum of specific heat $C_V$. It scales as $$U(L) = {\cal A} + {\cal B} L^{-\omega} \label{eq:UL}$$ at large $L$. This method is similar in spirit to the one used by Hasenbusch for the 3D $\varphi^4$ model in [@Hasenbusch]. The only difference is that another pseudocritical coupling (corresponding to certain value of $Z_a/Z_p$, where $Z_p$ and $Z_a$ are partition functions for the lattice with periodic and antiperiodic boundary conditions) has been used in [@Hasenbusch]. We have found that our $U(L)$ data provide a good fit to (\[eq:UL\]) within $8 \le L \le 384$, where $L=384$ is similar to the maximal size $L=360$ simulated in [@Has1]. These data are listed in Tab. \[tab4\], and the fit results are presented in Tab. \[tab5\]. [|c|c|c|]{} ------------------------------------------------------------------------ L & $\hat{\beta}_c$ & $U$\ 384 & 0.22167526(52) & 1.1884(62)\ 320 & 0.22168192(69) & 1.1901(63)\ 256 & 0.22169312(76) & 1.1937(50)\ 192 & 0.2217149(10) & 1.1940(42)\ 160 & 0.2217347(14) & 1.1951(44)\ 128 & 0.2217742(16) & 1.1831(32)\ 96 & 0.2218366(24) & 1.1917(33)\ 80 & 0.2219002(32) & 1.1885(32)\ 64 & 0.2220057(42) & 1.1888(30)\ 48 & 0.2221987(58) & 1.1930(27)\ 40 & 0.2223761(76) & 1.1933(26)\ 32 & 0.222659(10) & 1.1983(26)\ 24 & 0.223195(12) & 1.2035(19)\ 20 & 0.223686(13) & 1.2051(16)\ 16 & 0.224443(15) & 1.2121(13)\ 12 & 0.225813(16) & 1.22147(93)\ 10 & 0.226903(18) & 1.23159(86)\ 8 & 0.228567(20) & 1.24474(64)\ [|c|c|c|]{} ------------------------------------------------------------------------ $L_{\mathrm{min}}$ & $\omega$ & $\chi^2/\mathrm{d.o.f.}$\ 8 & 1.247(73) & 0.87\ 10 & 1.31(12) & 0.90\ 12 & 1.24(16) & 0.94\ 16 & 1.46(29) & 0.95\ The estimate $\omega=1.247(73)$ at $L_{\mathrm{min}}=8$ seems to be the best one, as it has the smallest statistical error, a good fit quality, and it perfectly agrees with the results for $L_{\mathrm{min}}>8$. This value disagrees (the discrepancy is $5.7$ standard deviations) with the best estimate $\omega = 0.832(6)$ of [@Has1], obtained by a different finite–size scaling method. It well agrees with the other value $\omega = 1.171(96)$ reported here. Searching for a different method, we have evaluated the Fisher zeros of partition function from MC simulations by the Wolff single cluster algorithm, following the method described in [@GKR11]. The results for $4 \le L \le 72$ have been reported in [@GKR11]. We have performed high statistics simulations (with MC measurements after each $\max\{2,L/4\}$ Wolff clusters, omitting $10^6$ measurements from the beginning of each simulation run, and totally $5 \times 10^8$ measurements used in the analysis for each $L$) for $4 \le L \le 128$. Two different pseudo-random number generators, discussed and tested in [@KMR_2011], have been used to verify that the results agree within error bars of about one or, sometimes, two standard deviations. Considering $\beta = \eta + i \xi$ as a complex number, the results for the first Fisher zero $\mathrm{Re} \, u^{(1)} + i \, \mathrm{Im} \, u^{(1)}$ in terms of $u=\exp(-4 \beta)$ are reported in Tab. \[tabzero1\]. Our values are obtained, evaluating $R = \langle \cos(\xi E) \rangle_{\eta} + i \langle \sin(\xi E) \rangle_{\eta}$ (where $E$ is energy) by the histogram reweighting method and minimizing $\mid R \mid$ (see [@GKR11]). Reliable results are ensured by the fact that, for each $L$, the simulation is performed at the coupling $\beta_{\mathrm{sim}}$ which is close to $\mathrm{Re} \beta^{(1)}$ – see Tab. \[tabzero1\]. We have reached it by using the results of [@GKR11] and finite–size extrapolations. [|c|c|c|c|c|]{} ------------------------------------------------------------------------ $L$ & $\beta_{\mathrm{sim}}$ & $\mathrm{Re} \beta^{(1)}$ & $\mathrm{Re} \, u^{(1)}$ & $\mathrm{Im} \, u^{(1)}$\ 4 & 0.2327517 & 0.2327392(37) & 0.3842870(59) & -0.0877415(55)\ 6 & 0.228982187 & 0.2289856(28) & 0.3975550(44) & -0.0454038(44)\ 8 & 0.22674832 & 0.2267531(27) & 0.4027150(44) & -0.0285905(42)\ 12 & 0.224558048 & 0.2245557(17) & 0.4070191(28) & -0.0149314(25)\ 16 & 0.223560276 & 0.2235605(12) & 0.4088085(19) & -0.0094349(17)\ 24 & 0.22268819 & 0.22268780(72) & 0.4103176(12) & -0.0049422(11)\ 32 & 0.222317896 & 0.22231846(49) & 0.41094218(81) & -0.00312478(84)\ 48 & 0.22200815 & 0.22200835(26) & 0.41146087(43) & -0.00163982(49)\ 64 & 0.221880569 & 0.22188039(17) & 0.41167349(29) & -0.00103825(37)\ 96 & 0.2217737 & 0.22177375(12) & 0.41185008(21) & -0.00054521(19)\ 128 & 0.22173025 & 0.221730228(83) & 0.41192200(14) & -0.00034552(14)\ We have also estimated the second zeros for $L=4, 32, 64$ from different simulation runs – see Tab. \[tabzero2\]. [|c|c|c|c|c|]{} ------------------------------------------------------------------------ $L$ & $\beta_{\mathrm{sim}}$ & $\mathrm{Re} \beta^{(2)}$ & $\mathrm{Re} \, u^{(2)}$ & $\mathrm{Im} \, u^{(2)}$\ 4 & 0.2464072 & 0.246484(18) & 0.344470(27) & -0.143307(23)\ 32 & 0.22313686 & 0.223169(15) & 0.409529(25) & -0.004891(25)\ 64 & 0.222166355 & 0.2221781(85) & 0.411182(14) & -0.001616(13)\ Our results in Tabs. \[tabzero1\] and \[tabzero2\] are reasonably consistent with those of [@GKR11], but are more accurate and include larger lattice sizes. Like in [@GKR11], the results for the second zeros are much less accurate than those for the first zeros. Therefore only the latter ones are used here in the analysis, considering the ratios $\Psi_1(L) = \mathrm{Im} \, u^{(1)}(L) / (\mathrm{Re} \, u^{(1)}(L) - u_c )$ and $\Psi_2(L) = \mathrm{Im} \, u^{(1)}(L) / \mathrm{Im} \, u^{(1)}(L/2)$, which behave asymptotically as $A + B L^{-\omega}$ at $L \to \infty$. Here $u_c = \exp(-4 \beta_c)$ is the critical $u$ value, corresponding to the critical coupling $\beta_c$. The estimation of correction–to–scaling exponent $\omega$ from fits of $\Psi_1(L)$ to $A + B L^{-\omega}$ has been considered in [@GKR11], assuming the known approximate value $0.2216546$ of $\beta_c$. The use of $\Psi_2(L)$ instead of $\Psi_1(L)$ is another method, which has an advantage that it does not require the knowledge of the critical coupling $\beta_c$. However, a disadvantage is that the data for two sizes, $L$ and $L/2$, are necessary for one value of $\Psi_2(L)$. The values of $\Psi_1(L)$ and $\Psi_2(L)$ are listed in Tab. \[tabPsi\]. The standard errors of $\Psi_1(L)$ are calculated by the jackknife method [@MC], thus taking into account the statistical correlations between $\mathrm{Re} \, u^{(1)}$ and $\mathrm{Im} \, u^{(1)}$. As in [@GKR11], the errors due to the uncertainty in $\beta_c$ are ignored, assuming that $\beta_c = 0.2216546$ holds with a high enough accuracy. According to [@KMR_2011], this $\beta_c$ value, likely, is correct within error bars of about $\pm 3 \times 10^{-8}$. It justifies the actual estimation. [|c|c|c|]{} ------------------------------------------------------------------------ $L$ & $\Psi_1$ & $\Psi_2$\ 8 & 3.0638(15) & 0.325829(52)\ 12 & 2.9698(17) & 0.328858(64)\ 16 & 2.9136(18) & 0.330001(77)\ 24 & 2.8581(21) & 0.330994(92)\ 32 & 2.8289(22) & 0.33119(11)\ 48 & 2.7988(23) & 0.33180(12)\ 64 & 2.7814(24) & 0.33226(15)\ 96 & 2.7718(31) & 0.33248(15)\ 128 & 2.7691(33) & 0.33279(18)\ [|c|c|c|c|]{} ------------------------------------------------------------------------ $L_{\mathrm{max}}$ & $L_{\mathrm{min}}$ & $\omega$ & $\chi^2/\mathrm{d.o.f.}$\ & 8 & 0.807(27) & 1.01\ 64 & 12 & 0.903(60) & 0.22\ & 16 & 0.84(10) & 0.06\ & 8 & 0.872(21) & 3.12\ 128 & 12 & 0.997(42) & 1.21\ & 16 & 1.026(65) & 1.43\ The values of exponent $\omega$, extracted from the fits of $\Psi_1(L)$ to $A + B L^{-\omega}$ within $L \in [L_{\mathrm{min}},L_{\mathrm{max}}]$, are collected in Tab. \[tabPsi1fit\]. The results $\omega=0.903(60)$ for $L \in [12,64]$ and $\omega=0.84(10)$ for $L \in [16,64]$ agree within error bars with the results for similar fit intervals in [@GKR11], i. e., $\omega = 0.77(9)$ for $L \in [12,72]$ and $\omega = 0.63(16)$ for $L \in [16,72]$. However, the fits with $L_{\mathrm{max}}=128$ are preferable for a reasonable estimation of the asymptotic exponent $\omega$. The best estimate with $L_{\mathrm{max}}=128$ is $\omega = 0.997(42)$, obtained at $L_{\mathrm{min}}=12$. Indeed, this fit has an acceptable $\chi^2/\mathrm{d.o.f.}$ value and the result is well consistent with that for $L_{\mathrm{min}}=16$, where the statistical error is larger. It turns out that the estimated value of $\omega$ becomes larger when $L_{\mathrm{max}}$ is increased from $64$ to $128$. One of possible explanations, which is consistent with the data in Tab. \[tabPsi\], is such that the $\Psi_1(L)$ plot has a minimum near $L=128$ or at somewhat larger $L$ values. In this case the actual method is really valid only for remarkably larger lattice sizes. The results of the other method, using the ratio $\Psi_2(L)$ instead of $\Psi_1(L)$, are collected in Tab. \[tabPsi2fit\]. Here $L_{\mathrm{max}}=128$ is fixed and only $L_{\mathrm{min}}$ is varied. The standard errors of $\omega$ are calculated, taking into account that fluctuations in $\mathrm{Im} \, u^{(1)}$ are the statistically independent quantities. As we can see, the estimated exponent $\omega$ decreases with increasing of $L_{\mathrm{min}}$ in the considered range. Since $\chi^2/\mathrm{d.o.f.}$ is about unity for moderately good fits, the estimates $\omega=0.61(19)$ at $L_{\mathrm{min}}=16$ and $\omega=0.18(37)$ at $L_{\mathrm{min}}=24$ are acceptable. [|c|c|c|]{} ------------------------------------------------------------------------ $L_{\mathrm{min}}$ & $\omega$ & $\chi^2/\mathrm{d.o.f.}$\ 8 & 1.400(57) & 4.12\ 12 & 0.96(13) & 2.02\ 16 & 0.61(19) & 1.20\ 24 & 0.18(37) & 0.88\ Summarizing the results of this section, we conclude that three of the considered here methods give larger values of $\omega$ ($1.171(96)$, $1.274(72)$ and $0.997(42)$) than $\omega=0.832(6)$ reported in [@Has1], whereas the fourth method tends to give smaller values ($0.61(19)$ and $0.18(37)$). Thus, it is evident that the estimation of $\omega$ from finite–size scaling, using the data for not too large lattice sizes (comparable with $L \le 360$ in [@Has1] or $L \le 72$ in [@GKR11]) does not lead to conclusive results. Indeed, the obtained values depend on the particular method chosen and are varied from $1.274(72)$ to $0.18(37)$ in our examples. Influence of $\omega$ on the estimation of exponents $\eta$ and $\nu$ {#sec:influence} ===================================================================== Allowing a possibility that the correction–to–scaling exponent $\omega$ of the 3D Ising model is, indeed, essentially smaller than the commonly accepted values of about $0.8$, we have tested the influence of $\omega$ on the estimation of critical exponents $\eta$ and $\nu$ (or $1/\nu$). We have fit our susceptibility data at $\beta = \widetilde{\beta}_c(L)$ to the ansatz $$\chi = L^{2-\eta} \left( a_0 + \sum\limits_{k=1}^m a_k L^{-k \omega} \right) \label{eq:chiansatz}$$ with $m=1$ and $m=2$ to estimate $\eta$ at three fixed values of the exponent $\omega$, i. e., $\omega=0.8$, $\omega=0.38$ and $\omega=1/8$. The first one is very close to the known RG value $\omega=0.799 \pm 0.011$ [@Justin] and also is quite similar to a more recent RG estimate $\omega=0.782(5)$ [@PS08] and the MC estimate $\omega=0.832(6)$ of [@Has1]. The second value corresponds to the upper bound $\omega_{\mathrm{max}} \approx 0.38$ stated in Sec. \[sec:analytical\], and the third value $1/8$ is extracted from the GFD theory [@K_Ann01; @K2012]. There exist different corrections to scaling, but the two correction terms in (\[eq:chiansatz\]) are the most relevant ones at $L \to \infty$, as it can be seen from the analysis in [@KMR_2013; @Has1]. The results of the fit within $L \in [L_{\mathrm{min}},2048]$ depending on $\omega$, $m$ and $L_{\mathrm{min}}$ are shown in Tab. \[tab6\]. Similarly, we have fit our $\partial Q/\partial \beta$ data at $\beta = \widetilde{\beta}_c(L)$ to the ansatz $$\frac{\partial Q}{\partial \beta} = L^{1/\nu} \left( b_0 + \sum\limits_{k=1}^m b_k L^{-k \omega} \right) \label{eq:Qansatz}$$ and have presented the results in Tab. \[tab7\]. [|c|c|c|c|c|]{} ------------------------------------------------------------------------ $\omega$ & $m$ & $L_{\mathrm{min}}$ & $\eta$ & $\chi^2/\mathrm{d.o.f.}$\ & & 32 & 0.03617(45) & 0.93\ & 1 & 48 & 0.03562(59) & 0.89\ 0.8 & & 64 & 0.03563(76) & 0.97\ & & 32 & 0.03521(94) & 0.91\ & 2 & 48 & 0.0366(14) & 0.90\ & & 64 & 0.0384(18) & 0.86\ & & 32 & 0.04387(78) & 1.40\ & 1 & 48 & 0.0414(11) & 0.82\ 0.38 & & 64 & 0.0407(14) & 0.84\ & & 32 & 0.0342(32) & 0.97\ & 2 & 48 & 0.0408(45) & 0.86\ & & 64 & 0.0465(60) & 0.84\ & & 32 & 0.0656(15) & 1.90\ & 1 & 48 & 0.0589(22) & 0.85\ $1/8$ & & 64 & 0.0562(30) & 0.80\ & & 32 & 0.0106(16) & 1.51\ & 2 & 48 & 0.031(54) & 0.87\ & & 64 & 0.075(30) & 0.83\ [|c|c|c|c|c|]{} ------------------------------------------------------------------------ $\omega$ & $m$ & $L_{\mathrm{min}}$ & $1/\nu$ & $\chi^2/\mathrm{d.o.f.}$\ & & 32 & 1.5872(16) & 0.63\ & 1 & 48 & 1.5895(22) & 0.48\ 0.8 & & 64 & 1.5880(27) & 0.47\ & & 32 & 1.5914(34) & 0.56\ & 2 & 48 & 1.5854(49) & 0.46\ & & 64 & 1.5869(64) & 0.50\ & & 32 & 1.5873(29) & 0.63\ & 1 & 48 & 1.5913(40) & 0.49\ 0.38 & & 64 & 1.5880(52) & 0.47\ & & 32 & 1.598(12) & 0.61\ & 2 & 48 & 1.576(15) & 0.47\ & & 64 & 1.584(22) & 0.50\ & & 32 & 1.5878(84) & 0.63\ & 1 & 48 & 1.599(14) & 0.50\ $1/8$ & & 64 & 1.588(15) & 0.47\ & & 32 & 1.636(21) & 0.64\ & 2 & 48 & 1.525(50) & 0.47\ & & 64 & 1.56(37) & 0.50\ Considering the fits with only the leading correction to scaling included ($m=1$), one can conclude from Tab. \[tab6\] that the estimated critical exponent $\eta$ increases with decreasing of $\omega$, whereas the exponent $1/\nu$ in Tab. \[tab7\] is rather stable. The sub-leading correction to scaling ($m=2$) makes the estimated exponents $\eta$ and $1/\nu$ remarkably less stable for small $\omega$ values, such as $\omega=1/8$. The latter value is expected from the GFD theory [@K_Ann01; @K2012], so that the estimation at $\omega=1/8$ is self-consistent within this approach. In this case, the estimation of $\eta$ appears to be compatible with the theoretical value $\eta=1/8=0.125$ of [@K_Ann01], taking into account that the evaluated $\eta$ increases with $L_{\mathrm{min}}$. Moreover, the self-consistent estimation of $1/\nu$ is even very well consistent with $\nu = 2/3$ predicted in [@K_Ann01]. In particular, $1/\nu = 1.525(50)$ can be considered as the best estimate at $\omega=1/8$ and $m=2$ (it has the smallest $\chi^2/\mathrm{d.o.f.}$ value and much smaller statistical error than the estimate at $L_{\mathrm{min}}=64$), which well agrees with $1.5$. Thus, contrary to the statements in [@GKR11], the value $\nu=2/3$ of the GFD theory is not ruled out, since it is possible that $\omega$ has a much smaller value than $0.832(6)$ assumed in [@GKR11]. A question can arise about the influence of $\omega$ value on the estimation of critical exponents in the case of improved Hamiltonians [@Has1; @Has2]. It is expected that the leading corrections to scaling vanish in this case, and therefore the influence of $\omega$ is small. However, in the case if the asymptotic corrections to scaling are described by the exponent $\omega \le \omega_{\mathrm{max}} \approx 0.38$, as it is strongly suggested by the theorem discussed in Sec. \[sec:analytical\], the vanishing of leading corrections cannot be supported by the existing MC analyses of such models. Indeed, in these analyses the asymptotic corrections to scaling are not correctly identified (probably, because of too small lattice sizes) if $\omega \le \omega_{\mathrm{max}} \approx 0.38$, since one finds that $\omega \approx 0.8$. Comparison of recent results {#sec:compare} ============================ It is interesting to compare our MC estimates and those of [@Has1] with the most recent RG (3D expansion) values of [@PS08] cited in [@Has1]. Note that the estimates of $\omega$ in [@Has1] and [@PS08], i. e., $\omega=0.832(6)$ and $\omega=0.782(5)$, are clearly inconsistent within the claimed error bars. This discrepancy, however, can be understood from the point of view of our MC analysis, suggesting that the real uncertainty in the MC estimation of $\omega$ can be rather large. source method $\eta$ $\nu$ -------------- -------- ------------- ------------- this work MC 0.0384(18) 0.6302(25) Ref. [@Has1] MC 0.03627(10) 0.63002(10) Ref. [@PS08] 3D exp 0.0318(3) 0.6306(5) : Recent estimates of the critical exponents $\eta$ and $\nu$ from different sources. Our values correspond to $\omega=0.8$, $m=2$ and $L_{\mathrm{min}}=64$.[]{data-label="tab8"} The comparison of critical exponents $\eta$ and $\nu$ is provided in Tab. \[tab8\]. This comparison includes only some recent or relatively new results, since older ones are extensively discussed in [@Has1; @Has2; @HasRev]. Our values correspond to the fits within $L \in [64,2048]$ at $m=2$ and $\omega=0.8$. The choice of $\omega=0.8$ is reasonable here, since this value is close enough to the above mentioned estimates $\omega=0.832(6)$ and $\omega=0.782(5)$, and practically the same results are obtained if $\omega=0.8$ is replaced by any of these two values. According to the claimed statistical error bars, the estimates of [@Has1] seem to be extremely accurate. Note, however, that these estimates are extracted from much smaller lattice sizes ($L \le 360$) as compared to ours ($L \le 2048$). The values of $\nu$ in Tab. \[tab8\] are consistent with each other. The MC estimates of $\eta$ are consistent, as well. However, the recent RG value of [@PS08] appears to be somewhat smaller and not consistent within the error bars with the actual MC estimations, even if the assumed values of $\omega$ are about $0.8$, as predicted by the perturbative RG theory. In particular, the discrepancy with the MC value of [@Has1] is about $45$ standard deviations of the MC estimation or about $15$ error bars of the RG estimation. Recently, the conformald field theory (CFT) has been applied to the 3D Ising model [@Showk] to obtain very accurate values of the critical exponents, using the numerical conformal bootstrap method. The conformal–symmetry relations for the correlation functions, like (2.1) in [@Showk], are known to hold asymptotically in two dimensions, whereas their validity in 3D case can be questioned. Here “asymptotically” means that the limit $L/x \to \infty$, $\xi/x \to \infty$ and $a/x \to 0$ is considered, where $x$ is the actual distance, $a$ is the lattice spacing and $\xi$ is the correlation length. In other words, the conformal symmetry is expected to hold exactly for the asymptotic correlation functions on an infinite lattice ($L = \infty$) at the critical point ($\beta = \beta_c$). These asymptotic correlation functions are obtained by subtracting from the exact correlations functions (at $L=\infty$ and $\beta=\beta_c$) the corrections to scaling, containing powers of $a/x$. The existence of the conformal symmetry in the 3D Ising model has been supported by a non–trivial MC test in [@MCconf]. Apart from the assumption of the validity of (2.1) in [@Showk] for the 3D Ising model, the following hypotheses have been proposed: 1. There exists a sharp kink on the border of the two–dimensional region of the allowed values of the operator dimensions $\Delta_{\sigma} = (1+\eta)/2$ and $\Delta_{\epsilon} = 3 - 1/\nu$; 2. Critical exponents of the 3D Ising model correspond just to this kink. These hypotheses have been supported by the MC estimates of the exponents $\eta$, $\nu$ and $\omega$ in [@Has1]. However, the obtained in [@Showk] exponent $\omega=0.8303(18)$ is not supported by our MC value $\omega = 0.25(33)$, obtained from the susceptibility data for very large lattice sizes $L \le 2048$. Moreover, it does not satisfy the inequality $\omega \le (\gamma-1)/\nu$, following from the theorem discussed in Sec. \[sec:analytical\]. This apparent contradiction can be understood from the point of view that corrections to scaling are not fully controlled in the CFT. Indeed, the prediction for $\omega$ in this CFT is based on the assumption that $\omega = \Delta_{\epsilon'} -3$ holds, where $\Delta_{\epsilon'}$ is the dimension of an irrelevant operator in the conformal analysis of the asymptotic four–point correlation function. It means that corrections to scaling of the exact four–point correlation function are discarded (to obtain the asymptotic correlation function, as discussed before) and not included into the analysis. More recently, a modified conformal bootstrap analysis has been performed in [@xx], where the mentioned two hypotheses have been replaced with the hypothesis that the operator product expansion (OPE) contains only two relevant scalar operators. The results for the exponents $\Delta_{\sigma}$ and $\Delta_{\epsilon}$ (or $\eta$ and $\nu$) are consistent with those of [@Showk]. This consistency is not surprising, since both methods agree with the idea that the operator spectrum of the 3D Ising model is relatively simple, so that the true values of $\Delta_{\sigma}$ and $\Delta_{\epsilon}$ are located inside of a certain narrow region (as in [@xx]) or on its border (as in [@Showk]), where many operators are decoupled from the spectrum. Apparently, the analysis in [@xx] does not lead to a contradiction with the two relations $\omega \le (\gamma-1)/\nu$ (the theorem) and $\omega = \Delta_{\epsilon'} -3$, since only $\Delta_{\epsilon'}>3$ is assumed for the dimension $\Delta_{\epsilon'}$. Thus, both relations can be satisfied simultaneously, if the 3D Ising point in Fig. 1 of [@Showk] is located inside of the allowed region, rather than on its border. This possibility is supported by the behavior of the effective exponent $\eta_{\mathrm{eff}}$ in our Fig. \[fig1\]. It suggests that the asymptotic exponent $\eta$ could be larger than it is usually expected from MC simulations for relatively small lattice sizes, as in [@Has1]. Thus, it is important to make further refined estimations, based on MC data for very large lattice sizes, in order to verify the hypotheses proposed in [@Showk; @xx]. If the hypothesis (i) about the existence of a sharp kink is true, then this kink, probably, has a special meaning for the 3D Ising model. Its existence, however, is not evident. ![The values of $\Delta(\sigma)$, corresponding to the minimum (solid circles) and the “kink” (empty circles) in the plots of Fig. 7 in [@Showk] depending on $N^{-3}$. The dashed lines show linear extrapolations.[]{data-label="fig2"}](conf.eps){width="60.00000%"} According to the conjectures of [@Showk], such a kink is formed at $N \to \infty$, where $N$ is the number of derivatives included into the analysis. As discussed in [@Showk], it implies that the minimum in the plots of Fig. 7 in [@Showk] should be merged with the apparent “kink” at $N \to \infty$. This “kink” is not really sharp at a finite $N$. Nevertheless, its location can be identified with the value of $\Delta(\sigma)$, at which the second derivative of the plot has a local maximum. The minimum of the plot is slightly varied with $N$, whereas the “kink” is barely moving [@Showk]. Apparently, the convergence to a certain asymptotic curve is remarkably faster than $1/N$, as it can be expected from Fig. 7 and other similar figures in [@Showk]. In particular, we have found that the location of the minimum in Fig. 7 of [@Showk] is varied almost linearly with $N^{-3}$. We have shown it in Fig. \[fig2\] by solid circles, the position of the “kink” being indicated by empty circles. The error bars of $\pm 0.000001$ correspond to the symbol size. The results for $N=153, 190, 231$ are presented, skipping the estimate for the location of the “kink” at $N=153$, which cannot be well determined from the corresponding plot in Fig. 7 of [@Showk]. The linear extrapolations (dashed lines) suggest that the minimum, very likely, is moved only slightly closer to the “kink” when $N$ is varied from $N=231$ to $N = \infty$. The linear extrapolation might be too inaccurate. Only in this case a refined numerical analysis for larger $N$ values can possibly confirm the hypothesis about the formation of a sharp kink at $N \to \infty$. The results of both [@Showk] and [@xx] strongly support the commonly accepted 3D Ising values of the critical exponents $\eta$ and $\nu$. In particular, the estimates $\eta=0.03631(3)$ and $\nu = 0.62999(5)$ have been reported in [@Showk]. However, these estimates are obtained, based on certain hypotheses. If these hypotheses are not used, then the conformal bootstrap analysis appears to be consistent even with the discussed here GFD values $\eta=1/8$ and $\nu =2/3$. Indeed, the corresponding operator dimensions $\Delta_{\sigma} = (1+\eta)/2$ and $\Delta_{\epsilon} = 3 - 1/\nu$ lie inside of the allowed region in Fig. 1 of [@Showk]. The hypotheses (i) and (ii) can be questioned in view of the observations summarized in Fig. \[fig2\]. The hypothesis of [@xx] about the existence of just two relevant scalar operators might be supported by some physically–intuitive arguments. In particular, one needs to adjust two scalar parameters $P$ (pressure) and $T$ (temperature) to reach the critical point of a liquid–vapor system. A real support for this hypothesis is provided by the already known estimations of the critical exponents. Taking into account the non–perturbative nature of the critical phenomena, the most reliable estimates are based on non–perturbative methods, such as the Monte Carlo simulation. An essential point in this discussion is that the MC estimates can be remarkably changed, if unusually large lattices are considered, as it is shown in our current study. Summary and conclusions ======================= Analytical as well as Monte Carlo arguments are provided in this paper, showing that corrections to scaling in the 3D Ising model are described by a remarkably smaller exponent $\omega$ than the usually accepted values of about $0.8$. The analytical arguments in Sec. \[sec:analytical\], which are based on a rigorous proof of certain theorem, suggest that $\omega \le (\gamma -1)/\nu$ holds, implying that $\omega$ cannot be larger than $\omega_{\mathrm{max}}=(\gamma -1)/\nu \approx 0.38$ in the 3D Ising model. The analytical prediction of the GFD theory [@K_Ann01; @K2012] is $\omega=1/8$ in this case. Our MC estimation of $\omega$ from the susceptibility ($\chi$) data of very large lattices (Sec. \[sec:large\]) is well consistent with these analytical results. Numerical values, extracted from the $\chi$ data within $40 \le L \le 2048$ and $48 \le L \le 2048$ (or $\Phi_2(L)=2^{-4}\chi(2L)/\chi(L/2)$ data within $80 \le L \le 1024$ and $96 \le L \le 1024$) are $\omega=0.25(33)$ and $\omega = 0.06(38)$, respectively. Unfortunately, the statistical errors in $\omega$ are rather large. As discussed in [@KMR_14], our analytical predictions generally refer to a subset of $n$-vector models, where spin is an $n$-component vector with $n=1$ in two dimensions and $n \ge 1$ in three dimensions. Our recent MC analysis agrees with these predictions for the scalar ($n=1$) 2D $\varphi^4$ model [@KMR_14], where statistical errors are small enough. The 3D case with $n=2$ has been tested in [@K2012], based on accurate experimental data for specific heat in zero gravity conditions very close to the $\lambda$–transition point in liquid helium. The test in Sec. 4 of [@K2012] reveals some inconsistency of the data with corrections to scaling proposed by the perturbative RG treatments, indicating that these corrections decay slower, i. e., $\theta=\nu \omega$ is smaller than usually expected. This finding is consistent with the theorem discussed in Sec. \[sec:analytical\]. The mentioned here facts emphasize the importance of our MC analysis. Our proposed values of $\omega$ may seem to be incredible in view of a series of known results, yielding $\omega$ at about $0.8$ for the 3D Ising model. However, it is meaningful to reconsider these results from several aspects. - First of all, they disagree with non–perturbative arguments in the form of the rigorously proven theorem, discussed in Sec. \[sec:analytical\]. - This theorem states that $\omega \le (\gamma-1)/\nu$, whereas the perturbative RG estimates are essentially larger. In view of the recent analysis in [@K2012x] (see also the discussions in [@K2012]), this discrepancy can be understood as a failure of the standard perturbative RG methods. The actually discussed (Sec. \[sec:compare\]) discrepancy between the recent RG and MC estimates of the critical exponent $\eta$ also points to problems in the perturbative approach. Moreover, any perturbative method, also the high- and low-temperature series expansions, can fail to give correct results in critical phenomena, since it is not the natural domain of validity of the perturbation theory. - The previous MC estimations of $\omega$ are based on simulations of lattices not larger than $L \le 360$ in [@Has1]. We have clearly demonstrated in Sec. \[sec:difomega\] that the values obtained from finite–size scaling with such relatively small (as compared to $L \le 2048$ in our study) lattice sizes depend on the particular method chosen. For example, different estimates ranging from $\omega =1.274(72)$ to $\omega=0.18(37)$ are obtained here, which substantially deviate from the usually reported values between $0.82$ and $0.87$ (see [@Has1; @HasRev]). - Although the recent estimate $\omega=0.8303(18)$ of the conformal bootstrap method [@Showk] is inconsistent with $\omega \le (\gamma-1)/\nu$, the apparent contradiction can be understood and resolved, as discussed in Sec. \[sec:compare\]. Taking into account the possibility that the correction–to–scaling exponent $\omega$ can be remarkably smaller than the usually accepted values at about $0.8$, we have tested in Sec. \[sec:influence\] the influence of $\omega$ on the estimation of critical exponents $\eta$ and $\nu$. We have concluded that the effect is remarkable if $\omega$ is changed from $0.8$ to a much smaller value, such as $\omega=1/8$ of the GFD theory. In this case, the error bars strongly increase, and the estimation becomes compatible, or even well consistent, with the predictions of the GFD theory. In particular, the estimate $1/\nu = 1.525(50)$ agrees with the GFD theoretical value $1.5$. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to Slava Rychkov for the useful communications concerning the conformal bootstrap method. This work was made possible by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET:www.sharcnet.ca). The authors acknowledge the use of resources provided by the Latvian Grid Infrastructure. For more information, please reference the Latvian Grid website (http://grid.lumii.lv). R. M. acknowledges the support from the NSERC and CRC program. [100]{} D. J. Amit, *Field Theory, the Renormalization Group, and Critical Phenomena*, World Scientific, Singapore, 1984. S. K. Ma, *Modern Theory of Critical Phenomena*, W. A. Benjamin, Inc., New York, 1976. J. Zinn-Justin, *Quantum Field Theory and Critical Phenomena*, Clarendon Press, Oxford, 1996. H. Kleinert, V. Schulte-Frohlinde, *Critical Properties of $\phi^4$ Theories*, World Scientific, Singapore, 2001. A. Pelissetto, E. Vicari, Phys. Rep. 368 (2002) 549–727. J. Kaupužs, Canadian J. Phys. **9**, 373 (2012). M. Hasenbusch, Int. J. Mod. Phys. C [**12**]{}, 911 (2001). M. Hasenbusch, Phys. Rev. B 82 (2010) 174433. M. Hasenbusch, Phys. Rev. B 82 (2010) 174434. A. Gordillo-Guerro, R. Kenna, J. J. Ruiz-Lorenzo, J. Stat. Mech., P09019 (2011). J. Kaupužs, J. Rimšāns, R. V. N. Melnik, Ukr. J. Phys. 56, 845 (2011). J. Kaupužs, R. V. N. Melnik, J. Rimšāns, Commun. Comp. Phys. **14**, 355 (2013). J. Kaupužs, Ann. Phys. (Berlin) 10 (2001) 299–331. U. Wolff, Phys. Rev. Lett. 62 (1989) 361. J. Kaupužs, J. Rimšāns, R. V. N. Melnik, Phys. Rev. E **81**, 026701 (2010). J. Kaupužs, R. V. N. Melnik, J. Rimšāns, arXiv:1406.7491 \[cond-mat.stat-mech\] (2014) M. Hasenbusch, J. Phys. A: Math. Gen. **32**, 4851 (1999). M. E. J. Newman, G. T. Barkema, *Monte Carlo Methods in Statistical Physics* (Clarendon Press, Oxford, 1999). A. A. Pogorelov, I. M. Suslov, J. Exp. Theor. Phys. **106**, 1118 (2008). S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, arXiv:1403.4542 \[hep-th\] (2014), to appear in a special issue of J. Stat. Phys. M. Billó, M. Caselle, D. Gaiotto, F. Gliozzi, M. Meineri, R. Pelligrini, arXiv:1304.4110 \[hep-th\] (2013) F. Kos, D. Poland, D. Simmons-Duffin, arXiv:1406.4858 \[hep-th\] (2014) J. Kaupužs, Int. J. Mod. Phys. A 27, 1250114 (2012) [^1]: E–mail: `kaupuzs@latnet.lv`

--- abstract: 'Let $R$ be the free $\CC$-algebra on $x$ and $y$ modulo the relations $x^5=yxy$ and $y^2=xyx$ endowed with the $\ZZ$-grading $\deg x=1$ and $\deg y=2$. Let $\BB_3$ denote the blow up of $\CC\PP^2$ at three non-colliear points. The main result in this paper is that the category of quasi-coherent $\cO_{\BB_3}$-modules is equivalent to the quotient of the category of $\ZZ$-graded $R$-modules modulo the full subcategory of modules $M$ such that for each $m \in M$, $(x,y)^nm=0$ for $n \gg 0$. This reduces almost all representation-theoretic questions about $R$ to algebraic geometric questions about the del Pezzo surface $\BB_3$. For example, the generic simple $R$-module has dimension six. Furthermore, the main result combined with results of Artin, Tate, and Van den Bergh, imply that $R$ is a noetherian domain of global dimension three' address: ' Department of Mathematics, Box 354350, Univ. Washington, Seattle, WA 98195' author: - 'S. Paul Smith' title: 'A non-commutative homogeneous coordinate ring for the third del Pezzo surface' --- [^1] Introduction ============ We will work over the field of complex numbers. The surface obtained by blowing up $\PP^2$ at three non-colinear points is, up to isomorphism, independent of the points. It is called the [third del Pezzo surface]{} and we will denote it by $\BB_3$. Let $R$ be the free $\CC$-algebra on $x$ and $y$ modulo the relations $$\label{relations} x^5=yxy \qquad \hbox{and } \qquad y^2=xyx.$$ Give $R$ a $\ZZ$-grading by declaring that $$\deg x=1 \qquad \hbox{and } \qquad \deg y=2.$$ In this paper we prove there is a surprisingly close relationship between the non-commutative algebra $R$ and the third del Pezzo surface. This relationship can be exploited to obtain a deep understanding of the representation theory of $R$. \[thm.main1\] Let $R$ be the non-commutative algebra $\CC[x,y]$ with relations (\[relations\]). Let $\Gr R$ be the category of $\ZZ$-graded left $R$-modules. Then there is an equivalence of categories $$\Qcoh \BB_3 \equiv {{\Gr R}\over{\sT}}$$ where the left-hand side is the category of quasi-coherent $\cO_{\BB_3}$-modules and the right-hand side is the quotient category modulo the full subcategory $\sT$ consisting of those modules $M$ such that for each $m \in M$, $(x,y)^nm=0$ for $n \gg 0$. Theorem \[thm.main1\] is a consequence of the following result. \[thm.main2\] Let $R$ be the non-commutative algebra $\CC[x,y]$ with relations (\[relations\]). There is an automorphism $\s$ of $\BB_3$ having order six, and a line bundle $\cL$ on $\BB_3$ such that $R$ is isomorphic to the twisted homogeneous coordinate ring $$B(\BB_3,\cL,\s) : = \bigoplus_{n \ge 0} H^0(\BB_3,\cL_n)$$ where $$\cL_n:= \cL \otimes (\s^*)\cL \otimes \cdots \otimes (\s^*)^{n-1}\cL.$$ Results of Artin, Tate, and Van den Bergh now imply that $R$ is a 3-dimensional Artin-Schelter regular algebra and therefore has the following properties. Let $R$ be the algebra $\CC[x,y]$ with relations (\[relations\]). Then 1. $R$ is a left and right noetherian domain; 2. $R$ has global homological dimension 3; 3. $R$ is Auslander-Gorenstein and Cohen-Macaulay in the non-commutative sense; 4. the Hilbert series of $R$ is the same as that of the weighted polynomial ring on three variables of weights 1, 2, and 3; 5. $R$ is a finitely generated module over its center [@ST94 Cor. 2.3]; 6. $R^{(6)}:=\oplus_{n=0}^\infty R_{6n}$ is isomorphic to $\bigoplus_{n=0}^\infty H^0(\BB_3\cO(-nK_{\BB_3})$; 7. $\Spec R^{(6)}$ is the anti-canonical cone over $\BB_3$, i.e., the cone obtained by collapsing the zero section of the total space of the anti-canonical bundle over $\BB_3$. This close connection between $R$ and $\BB_3$ means that almost all aspects of the representation theory of $R$ can be expressed in terms of the geometry of $\BB_3$. We plan to address this question in another paper. The justification for calling $R$ a non-commutative homogeneous coordinate ring for $\BB_3$ is the similarity between the equivalence of categories in Theorem \[thm.main1\] and following theorem of Serre: > if $X \subset \PP^n$ is the scheme-theoretic zero locus of a graded ideal $I$ in the polynomial ring $S=\CC[x_0,\ldots,x_n]$ with its standard grading, and $A=S/I$, then there is an equivalence of categories $$\label{eq.Serre} > \Qcoh X \equiv {{\Gr A}\over{\sT}}$$ where the right-hand side is the quotient category of $\Gr A$, the category of graded $A$-modules, by the full subcategory $\sT$ of modules supported at the origin. The author is grateful to the following people: Amer Iqbal for directing the author to some papers in high energy physics where the algebra $R$ appears in a hidden form; Paul Hacking and Sándor Kovács for passing on some standard results about del Pezzo surfaces and algebraic geometry; and Darin Stephenson for telling the author that the algebra $R$ is an iterated Ore extension, and for providing some background and guidance regarding his two papers cited in the bibliography. The non-commutative algebra $R=\CC[x,y]$ with $x^5=yxy$ and $y^2=xyx$ {#sect.nc.ring} ===================================================================== The following result is a straightforward calculation. The main point of it is to show that $R$ has the same Hilbert series as the weighted polynomial ring on three variables of weights 1, 2, and 3. \[Stephenson\] \[prop.steph\] The ring $R:=\CC[x,y]$ with defining relations $$x^5=yxy \qquad \hbox{and} \qquad y^2=xyx$$ is an iterated Ore extension of the polynomial ring $\CC[w]$. Explicitly, if $\zeta$ is a fixed primitive $6^{\th}$ root of unity, then: 1. $R= \CC[w][z;\s][x;\tau,\d]$ where $\s \in \Aut \CC[z]$, $\tau \in \Aut \CC[w,z]$, and $\d$ is the $\tau$-derivation defined by $$\begin{aligned} \s(w)& =\zeta w, \\ \tau(w) &= - \zeta^2 w, \quad \tau(z) = \zeta z, \\ \d(w) & = z, \quad \d(z) =- w^2; \end{aligned}$$ 2. A set of defining relations of $R=\CC[z,w,x]$ is given by $$\begin{aligned} zw = & \zeta wz, \\ xw = & -\zeta^2wx+z , \\ xz = & \zeta zx- w^2;\end{aligned}$$ 3. $R$ has basis $\{ w^iz^jx^k \; | \; i,j,k \ge 0\}$; 4. $R$ is a noetherian domain; 5. The Hilbert series of $R$ is $(1-t)^{-1}(1-t^2)^{-1}(1-t^3)^{-1}$. Define the elements $$\begin{aligned} w: & = \, y-x^2 \\ z: & = \, xw + \zeta^2 wx \\ &= \, xy +\zeta^2 yx - \zeta x^3\end{aligned}$$ of $R$. Since $y$ belongs to the ring generated by $x$ and $w$, $\CC[x,y]=\CC[x,w]=\CC[x,w,z]$. It is easy to check that $$\label{steph.relns} zw=\zeta wz, \quad xw= z- \zeta^2 wx, \quad xz=\zeta zx - w^2.$$ Let $R'$ be the free algebra $\CC\langle w,x,z \rangle$ modulo the relations in (\[steph.relns\]). We want to show $R'$ is isomorphic to $R$. We already know there is a homomorphism $R' \to R$ and we will now exhibit a homomorphism $R \to R'$ by showing there are elements $x$ and $Y$ in $R'$ that satisfy the defining relations for $R$. Define the element $Y:=w+x^2$ in $R'$. A straightforward computation in $R'$ gives $$xwx-x^2w = w^2+wx^2$$ so $$Y^2= w^2+x^2w + wx^2 + x^4 = xwx+x^4=xYx.$$ In the next calculation we make frequent use of the fact that $1-\xi+\xi^2=0$. Deep breath... $$\begin{aligned} YxY& = (w+x^2)xw +wx^3 + x^5 \\ & = (w+x^2) (z-\zeta^2wx)+ \big[ wx^3 + x^5 \big] \\ & = x^2 z -\zeta^2 x^2wx + \big[ wz -\zeta^2 w^2x +wx^3 + x^5 \big] \\ & = x(\zeta zx -w^2) -\zeta^2 x(z-\zeta^2wx)x + \big[ wz -\zeta^2 w^2x +wx^3 + x^5 \big] \\ & =(\zeta-\zeta^2)xzx - xw^2 - \zeta xwx^2 + \big[ wz -\zeta^2 w^2x +wx^3 + x^5 \big] \\ & =(\zeta-\zeta^2)(\zeta zx -w^2)x - (z-\zeta^2wx) w -\zeta(z- \zeta^2 wx)x^2 \\ & \qquad \qquad + \big[ wz -\zeta^2 w^2x +wx^3 + x^5 \big] \\ & =(\zeta^2-\zeta^3)zx^2 -(\zeta-\zeta^2)w^2x - zw +\zeta^2 wxw -\zeta zx^2 - wx^3 \\ & \qquad \qquad + \big[ wz -\zeta^2 w^2x +wx^3 + x^5 \big] \\ & =(\zeta^2-\zeta^3-\zeta)zx^2 +\zeta^2 wxw + \big[ (1-\zeta) wz -\zeta w^2x + x^5 \big] \\ & = \zeta^2 wxw + \big[ (1-\zeta) wz -\zeta w^2x + x^5 \big] \\ & = \zeta^2 w(z-\zeta^2wx) + \big[ -\zeta^2 wz -\zeta w^2x + x^5 \big] \\ & = x^5. \end{aligned}$$ Since $YxY=x^5$, $R$ is isomorphic to $R'$. Hence $R$ is an iterated Ore extension as claimed. The other parts of the proposition follow easily. It is an immediate consequence of the relations that $x^6=y^3$. Hence $x^6$ is in the center of $R$. The del Pezzo surface $\BB_3$ ============================= Let $\BB_3$ be the surface obtained by blowing up the complex projective plane $\PP^2$ at three non-collinear points. We will write $E_1$, $E_2$, and $E_3$ for the exceptional curves associated to the blowup. The $-1$-curves on $\BB_3$ lie in the following configuration $$\label{six.lines} \qquad \UseComputerModernTips \xymatrix{ && & \save []+<0cm,0.1cm>*\txt<4pc>{$\scriptstyle{E_2}$} \restore \ar@{-}[dddlll] & \ar@{-}[dddrrr] \save []+<0cm,0.1cm>*\txt<4pc>{$\scriptstyle{E_3}$} \restore \\ & \save []+<-0.3cm,0cm>*\txt<4pc>{$\scriptstyle{L_1}$} \restore \ar@{-}[rrrrr] &&&&& \save []+<0.5cm,0cm>*\txt<4pc>{$\scriptstyle{X=0}$} \restore & \\ \save []+<-0.2cm,0.1cm>*\txt<4pc>{$\scriptstyle{Z=0}$} \restore &&&&&&& \save []+<0.3cm,0.1cm>*\txt<4pc>{$\scriptstyle{Y=0}$} \restore &&&&& \\ \save []+<-0.2cm,-0.1cm>*\txt<4pc>{$\scriptstyle{s=0}$} \restore &&&&&&& \save []+<0.3cm,-0.1cm>*\txt<4pc>{$\scriptstyle{u=0}$} \restore & \\ & \save []+<-0.3cm,0cm>*\txt<4pc>{$\scriptstyle{E_1}$} \restore \ar@{-}[rrrrr] &&&&& \save []+<0.5cm,0cm>*\txt<4pc>{$\scriptstyle{t=0}$} \restore \\ &&& \ar@{-}[uuulll] \save []+<0cm,-0.1cm>*\txt<4pc>{$\scriptstyle{L_3}$} \restore & \save []+<0cm,-0.1cm>*\txt<4pc>{$\scriptstyle{L_2}$} \restore \ar@{-}[uuurrr] }$$ where $L_1$, $L_2$, and $L_3$ are the strict transforms of the lines in $\PP^2$ spanned by the points that are blown up. (The labeling of the equations for the $-1$-curves will be justified shortly.) The union of the $-1$-curves is an anti-canonical divisor and is, of course, ample. The Picard group of $\BB_3$ {#sect.Pic} --------------------------- The Picard group of $\BB_3$ is free abelian of rank four. We will identify it with $\ZZ^4$ by using the ordered basis $$H,\; -E_2, \; -E_1, \; -E_3$$ where the $E_i$s are the exceptional curves over the points blown up and $H$ is the strict transform of a line in $\PP^2$ in general position, i.e., missing the points being blown up. Thus $$H=(1,0,0,0), \quad E_1=(0,0,-1,0), \quad E_2=(0,-1,0,0), \quad E_3=(0,0,0,-1).$$ The canonical divisor $K$ is $-3H+E_1+E_2+E_3$ so the anti-canonical divisor is $$-K=(3,1,1,1).$$ Cox’s homogeneous coordinate ring --------------------------------- By definition, Cox’s homogeneous coordinate ring [@Co] for a complete smooth toric variety is $$S:= \bigoplus_{[\cL] \in \Pic X} H^0(X,\cL).$$ For the remainder of this paper $S$ will denote Cox’s homogeneous coordinate ring for $\BB_3$. Let $X,Y,Z,s,t,u$ be coordinate functions on $\CC^6$. One can present $\BB_3$ as a toric variety by defining it as the orbit space $$\BB_3:={{\CC^6 -W}\over{(\CC^\times)^4}}$$ where the irrelevant locus, $W$, is the union of nine codimension two subspaces, namely $$\label{irrel.locus} \begin{array}{ccc} \begin{array}{c} X=t=0\\ Y=s=0\\ Z=u=0 \end{array} \qquad \begin{array}{c} X=Y=0\\ Y=Z=0\\ Z=X=0 \end{array} \qquad \begin{array}{c} s=t=0\\ u=t=0\\ s=u=0 \end{array} \end{array}$$ and $(\CC^\times)^4$ acts with weights $$\begin{array}{rrrrrr} X & \phantom{x} & 1 & 1 & 0 & 1 \\ Y & \phantom{x} & 1 & 0 & 1 & 1 \\ Z & \phantom{x} & 1 & 1 & 1& 0 \\ s & \phantom{x} & 0 & -1 & 0 & 0\\ t & \phantom{x} & 0 & 0 & -1 & 0 \\ u & \phantom{x} & 0 & 0 & 0 & -1 \end{array}.$$ Therefore $S$ is the $\ZZ^4$-graded polynomial ring $$S=\CC[X,Y,Z,s,t,u]$$ with the degrees of the generators given by their weights under the $(\CC^\times)^4$ action. It follows from Cox’s results [@Co Sect. 3] that $$\Qcoh \BB_3 \equiv {{\Gr(S,\ZZ^4)}\over{\sT}}$$ where $\Gr(S,\ZZ^4)$ is the category of $\ZZ^4$-graded $S$-modules and $\sT$ is the full subcategory consisting of all direct limits of modules supported on $W$. The labelling of the $-1$-curves the diagram (\[six.lines\]) is explained by the existence of the morphisms $$\UseComputerModernTips \xymatrix{ & \BB_3 \ar[dl] \ar[dr] \\ \PP^2 && \PP^2 } \qquad \UseComputerModernTips \xymatrix{ & \save []+<-0.2cm,0.2cm>*\txt<4pc>{$\text{ $(X,Y,Z,s,t,u) $}$} \restore \ar[dl] \ar[dr] \\ \save []+<-0.6cm,-0.2cm>*\txt<4pc>{$\text{$(Xsu,Ytu,Zst)$}$} \restore&& \save []+<+0.3cm,-0.2cm>*\txt<4pc>{$\text{$(YZt,XZs,XYu)$}$} \restore }$$ that collapse the $-1$-curves. An order six automorphism $\s$ of $\BB_3$ ----------------------------------------- The cyclic permutation of the six $-1$-curves on $\BB_3$ extends to a global automorphism of $\BB_3$ of order six. We now make this explicit. The category of graded rings consists of pairs $(A,\G)$ consisting of an abelian group $\G$ and a $\G$-graded ring $A$. A morphism $(f,\theta):(A,\G) \to (B,\Upsilon)$ consists of a ring homomorphism $f:A \to B$ and a group homomorphism $\theta:\G \to \Upsilon$ such that $f(A_i) \subset B_{\theta(i)}$ for all $i \in \G$. Let $\tau:S \to S$ be the automorphism induced by the cyclic permutation $$\label{defn.tau} \UseComputerModernTips \xymatrix{ X \ar[r] & u \ar[r] & Y \ar[r]_\tau & t \ar[r] & Z \ar[r] & s \ar@{}@/_1pc/[lllll] | {<} }$$ and let $\theta:\ZZ^4 \to \ZZ^4$ be left multiplication by the matrix $$\theta= \begin{pmatrix} 2 & -1 & -1 & -1 \\ 1 & -1 & -1 & 0 \\ 1 & 0 & -1 & -1 \\ 1 & -1 & 0 & -1 \end{pmatrix}.$$ Then $(\tau,\theta):(S,\ZZ^4) \to (S,\ZZ^4)$ is an automorphism in the category of graded rings. Because the irrelevant locus (\[irrel.locus\]) is stable under the action of $\tau$, $\tau$ induces an automorphism $\s$ of $\BB_3$. It follows from the definition of $\tau$ that $\s$ cyclically permutes the six $-1$-curves Since $(\tau,\theta)^6=\id_{(S,\ZZ^4)}$ the order of $\s$ divides six. But the action of $\s$ on the set of $-1$-curves has order six, so $\s$ has order six as an automorphism of $\BB_3$. Fix a primitive cube root of unity $\omega$. The left action of $\theta$ on $\ZZ^4=\Pic \BB^3$ has eigenvectors $$v_1=\begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \\ \end{pmatrix}, \quad v_2=\begin{pmatrix} 3 \\ 1 \\ 1 \\ 1 \\ \end{pmatrix}, \quad v_3=\begin{pmatrix} 0 \\ 1 \\ \omega \\ \omega^2 \\ \end{pmatrix}, \quad v_4=\begin{pmatrix} 0 \\ 1 \\ \omega^2 \\ \omega \\ \end{pmatrix},$$ with corresponding eigenvalues $-1$, $+1$, $\omega^2$, $\omega$. A twisted hcr for $\BB_3$ ========================= A sequence of line bundles on $\BB_3$ {#sect.Dn} ------------------------------------- We will blur the distinction between a divisor $D$ and the class of the line bundle $\cO(D)$ in $\Pic\BB_3$. We define a sequence of divisors: $D_0$ is zero; $D_1$ is the line $L_1$; for $n \ge 1$ $$D_n:=(1+\theta + \cdots + \theta^{n-1})(D_1).$$ We will write $\cL_n:=\cO(D_n)$. Therefore $$\cL_n = \cL_1 \otimes \s^* \cL_1 \otimes \ldots \otimes (\s^*)^{n-1} \cL_1.$$ For example, $$\begin{array}{cc} \begin{array}{l} \cO(D_1)=\cL_1 = \cO(1,1,0,1) \\ \cO(D_2)=\cL_2 = \cO(1,1,0,0) \\ \cO(D_3)=\cL_3 = \cO(2,1,1,1) \\ \cO(D_4)=\cL_4 = \cO(2,1,0,1) \\ \cO(D_5)=\cL_5 = \cO(3,2,1,1) \\ \cO(D_6)=\cL_6 = \cO(3,1,1,1) \\ \phantom{xyz} \\ \phantom{xyz} \end{array} \begin{array}{l} =\cO(L_1 )\\ =\cO(L_1 - E_3 )\\ =\cO(L_1 + L_2 - E_3) \\ =\cO(L_1 + L_2 -E_1 - E_3) \\ =\cO(L_1 + L_2 + L_3 -E_1 - E_3) \\ =\cO(L_1 + L_2 + L_3 -E_1 -E_2 - E_3) \\ =\cO( 3H -E_1 -E_2 - E_3)\\ =\cO(-K). \end{array} \end{array}$$ \[lem.Dn\] Suppose that $m \ge 0$ and $0 \le r \le 5$. Then $$D_{6m+r} = D_{r}-mK.$$ Since $\theta^6=1$, $$\begin{aligned} \sum_{i=0}^{6m+r-1}\theta^i & =(1+\theta+\cdots+\theta^5)\sum_{j=0}^{m-1}\theta^{6j} + \theta^{6m}(1+\theta + \cdots+\theta^{r-1}) \\ &=(1+\theta + \cdots+\theta^{r-1}) + m(1+\theta+\cdots+\theta^5)\end{aligned}$$ where the sum $1+\theta+\cdots +\theta^{r-1}$ is empty and therefore equal to zero when $r=0$. Therefore $D_{6m+r}= D_{r} + mD_6 = D_{r}-mK$, as claimed. Vanishing results ----------------- For a divisor $D$ on a smooth surface $X$, we write $$h^i(D):=\dim H^i(X,\cO_{X}(D)).$$ We need to know that $h^1(D)=h^2(D)=0$ for various divisors $D$ on $\BB_3$. If $D-K$ is ample, then the Kodaira Vanishing Theorem implies that $h^0(K-D)=h^1(K-D)=0$ and Serre duality then gives $h^2(D)=h^1(D)=0$. The notational conventions in section \[sect.Pic\] identify $\Pic \BB_3$ with $\ZZ^4$ via $$aH-cE_1-bE_2-dE_3 \equiv (a,b,c,d)$$ where $H$ is the strict transform of a line in $\PP^2$. The intersection form on $\BB_3$ is given by $$H^2=1, \qquad E_i.E_j=-\d_{ij}, \qquad H.E_i=0,$$ so the induced intersection form on $\ZZ^4$ is $$(a,b,c,d) \cdot(a',b',c',d')=aa'-bb'-cc'-dd'.$$ \[lem.van\] Let $D=(a,b,c,d) \in \Pic\BB_3\equiv \ZZ^4$. Suppose that $$\label{eq.ample2} (a+3)^2> (b+1)^2+(c+1)^2+(d+1)^2$$ and $$\label{eq.ample3} b,\, c,\, d >-1, \qquad \hbox{and} \qquad a+1>b+c,\, b+d, \, c+d.$$ Then $D-K$ is ample, whence $h^1(D)=h^2(D)=0$. The effective cone is generated by $L_1$, $L_2$, $L_3$, $E_1$, $E_2$, and $E_3$ so, by the Nakai-Moishezon criterion, $D-K$ is ample if and only if $(D-K)^2>0$ and $D.L_i>0$ and $D.E_i>0$ for all $i$. Now $D-K=(a+3,b+1,c+1,d+1)$, so $(D-K)^2>0$ if and only if (\[eq.ample2\]) holds and $(D-K).D'>0$ for all effective $D'$ if and only if (\[eq.ample3\]) holds. Hence the hypothesis that (\[eq.ample2\]) and (\[eq.ample3\]) hold implies that $D-K$ is ample. The Kodaira Vanishing Theorem now implies that $h^0(K-D)=h^1(K-D)=0$. Serre duality now implies that $h^2(D)=h^1(D)=0$. \[lem.h1D\] For all $n \ge 0$, $h^1(D_n)=h^2(D_n)=0$. The value of $D_n$ for $0 \le n \le 6$ is given explicitly in section \[sect.Dn\]. We also note that $D_7 = D_1+D_6=(4,2,1,2)$. It is routine to check that conditions (\[eq.ample2\]) and (\[eq.ample3\]) hold for $D=D_n$ when $n=0,2,3,4,5,6,7$. Hence $h^1(D_n)=h^2(D_n)=0$ when $n=0,2,3,4,5,6,7$. We now consider $D_1$ which is the $-1$-curve $X=0$. (Since $(D_1-K).D_1 = 0$, $D_1-K$ is not ample so we can’t use Kodaira Vanishing as we did for the other small values of $n$.) It follows from the exact sequence $0 \to \cO_{\BB_3} \to \cO_{\BB_3}(D_1) \to \cO_{D_1}(D_1) \to 0$ that $H^p(\BB_3,\cO_{\BB_3}(D_1)) \cong H^p(\BB_3,\cO_{D_1}(D_1))$ for $p=1,2$. However, $\cO_{D_1}(D_1)$ is the normal sheaf on $D_1$ and as $D_1$ can be contracted to a smooth point on the second del Pezzo surface, $\cO_{D_1}(D_1) \cong \cO_{D_1}(-1)$. But $D_1 \cong \PP^1$ so $H^p(\BB_3, \cO_{D_1}(D_1)) \cong H^p(\PP^1,\cO(-1))$ which is zero for $p=1,2$. It follows that $h^1(D_1)=h^2(D_1)=0$. Thus $h^1(D_n)=h^2(D_n)=0$ when $0 \le n \le 7$. We have also shown that $D_n-K$ is ample when $2 \le n \le 7$. We now argue by induction. Suppose $n \ge 8$ and $D_{n-6}-K$ is ample. Now $D_n-K=D_{n-6}-K -K$. Since a sum of ample divisors is ample, $D_n-K$ is ample. It follows that $h^1(D_n)=h^2(D_n)=0$. The twisted homogeneous coordinate ring $B(\BB_3,\cL,\s)$ --------------------------------------------------------- We assume the reader is somewhat familiar with the notion of twisted homogeneous coordinate rings. Standard references for that material are [@AV], [@ATV1], [@ATV2], and [@AZ]. The notion of a $\s$-ample line bundle [@AV] plays a key role in the study of twisted homogeneous coordinate rings. Because $\cL_6$ is the anti-canonical bundle and therefore ample, $\cL_1$ is $\s$-ample. This allows us to use the results of Artin and Van den Bergh in [@AV] to conclude that the twisted homogeneous coordinate ring $$B(\BB_3,\cL,\s) = \bigoplus_{n=0}^\infty B_n = \bigoplus_{n=0}^\infty H^0(\BB_3,\cL_n)$$ is such that $$\Qcoh \BB_3 \equiv {{\Gr B}\over{\sT}}$$ where $\sT$ is the full subcategory of $\Gr B$ consisting of those modules $M$ such that for each $m \in M$, $B_nm=0$ for $n \gg 0$. It then follows that $B$ has a host of good properties—see [@AV] for details. We will now compute the dimensions $h^0(D_n)$ of the homogeneous $B_n$ of $B$. We will show that $B$ has the same Hilbert series as the non-commutative ring $R$, i.e., the same Hilbert series as the weighted polynomial ring with weights 1, 2, and 3. As usual we write $\chi(D) =h^0(D)-h^1(D)+h^2(D)$. The Riemann-Roch formula is $$\chi(\cO(D))=\chi(\cO)+\hbox{${{1}\over{2}}$} D\cdot (D-K)$$ where $K$ denotes the canonical divisor. We have $\chi(\cO_{\BB_3})=1$ and $K_{\BB_3}^2= 6$. \[lem.h0D\] Suppose $0 \le r \le 5$. Then $$h^0(D_{6m+r})=\begin{cases} (m+1)(3m+r) & \text{if $r \ne 0$,} \\ 3m^2+3m+1 & \text{if $r=0$} \end{cases}$$ and $$\sum_{n=0}^\infty h^0(D_n) t^n ={{1}\over{(1-t)(1-t^2)(1-t^3)}} .$$ Computations for $1 \le r \le 5$ give $D_{r}^2=r-2$ and $D_{r}\cdot K= -r $. Hence $$\begin{aligned} \label{T.series} \chi(D_{6m+r}) &= 1 + \hbox{${{1}\over{2}}$} (D_{r}-mK)\cdot (D_{r}-(m+1) K) \\ &= 1 + \hbox{${{1}\over{2}}$} \big(D_r^2-(2m+1)D_r.K +6m(m+1)^2\big) \\ & = (3m+r)(m+1)\end{aligned}$$ for $m \ge 0$ and $1 \le r \le 5$. When $r=0$, $D_r=0$ so $$\chi(D_{6m}) = 3m^2+3m+1.$$ By Lemma \[lem.h1D\], $\chi(D_n)=h^0(D_n)$ for all $n \ge 0$ so it follows from the formula for $\chi(D_n)$ that $$\label{eq.chi.diff} h^0(D_{n+6})-h^0(D_n)=n+6$$ for all $n \ge 0$. To complete the proof of the lemma, it suffices to show that $h^0(D_n)$ is the coefficient of $t^n$ in the Taylor series expansion $$f(t):={{1}\over{(1-t)(1-t^2)(1-t^3)}} = \sum_{n=0}^\infty a_n t^n.$$ Because $$(1-t^6)f(t) = (1-t+t^2)(1-t)^{-2} = 1 + \sum_{n=1}^\infty nt^n,$$ it follows that $$\begin{aligned} (1-t^6)f(t) & = a_0+a_1t+\cdots+a_5t^5 + \sum_{n=0}^\infty (a_{n+6}-a_n)t^{n+6} \\ & = 1+t+2t+\cdots+5t^5 + \sum_{n=6}^\infty nt^n \\ & = 1+t+2t+\cdots+5t^5 + \sum_{n=0}^\infty (n+6)t^{n+6}.\end{aligned}$$ In particular, if $0 \le r \le 5$, $a_r=h^0(D_r)$. We now complete the proof by induction. Suppose we have proved that $a_i=h^0(D_i)$ for $i \le n+5$. By comparing the expressions in the Taylor series we see that $$a_{n+6}= a_n + (n +6)= h^0(D_n)+n+6= h^0(D_{n+6})$$ where the last equality is given by (\[eq.chi.diff\]). ### Remark. It wasn’t necessary to compute $\chi(D_n)$ in the previous proof. The proof only used the fact that $\chi(D_{n+6})-\chi(D_n)=n+6$ which can be proved directly as follows: $$\begin{aligned} \chi(D_{n+6})-\chi(D_n) &= \hbox{${{1}\over{2}}$} D_{n+6}\cdot(D_{n+6}-K) - \hbox{${{1}\over{2}}$} D_n\cdot(D_{n}-K) \\ & = \hbox{${{1}\over{2}}$} (D_{n+6}-D_n)\cdot(D_{n+6}+D_n -K) \\ &=-K\cdot(D_r-(m+1)K) \\ & = 6(m+1) - K\cdot D_r \\ &=n+6.\end{aligned}$$ The isomorphism $R \to B(\BB_3,\cL,\s)$ --------------------------------------- The ring $B$ has the following basis elements in the following degrees: $$\begin{array}{ccccccc} \begin{array}{l} \cO(1,1,0,1) \\ \cO(1,1,0,0) \\ \cO(2,1,1,1) \\ \cO(2,1,0,1) \\ \cO(3,2,1,1) \\ \cO(3,1,1,1) \end{array} \begin{array}{l} X \\ Xu \\ XYu\\ XYtu\\ XYZtu\\ XYZstu \end{array} \begin{array}{l} \\ Zt \\ YZt\\ YZt^2\\ YZ^2t^2\\ YZ^2st^2 \end{array} \begin{array}{l} \\ \\ XZs\\ XZst\\ XZ^2st\\ XZ^2s^2t \end{array} \begin{array}{l} \\ \\ \\ X^2su\\ X^2Zsu\\ X^2Zs^2u \end{array} \begin{array}{l} \\ \\ \\ \\ X^2Yu^2\\ X^2Ysu^2 \end{array} \begin{array}{l} \\ \\ \\ \\ \\ XY^2tu^2 \, \; Y^2Zt^2u \end{array} \end{array}$$ Although $B$ is a graded subspace of Cox’s homogeneous coordinate ring $S$, > [ *the multiplication in $B$ is not that in $S$.* ]{} The multiplication in $B$ is Zhang’s twisted multiplication [@Ztwist] with respect to the automorphism $\tau$ defined in (\[defn.tau\]): the product in $B$ of $a \in B_m$ and $b \in B_n$ is $$\label{eq.Z.tw.mult} a *_B b := a\tau^m(b).$$ To make it clear whether a product is being calculated in $B$ or $S$ we will write $x$ for $X$ considered as an element of $B$ and $y$ for $Zt$ considered as an element of $B$. Therefore, for example, $$\begin{aligned} x^5 & =X\tau(X)\tau^2(X)\tau^3(X)\tau^4(X)\tau^5(X) \\ & = XuYtZ \\ &= (Zt) Y(uX) \\ & = Zt\tau^2(X)\tau^3(Zt) \\ & = yxy\end{aligned}$$ and $$y^2=Zt\tau^2(Zt)=Zt(sX) = X(sZ)t = X\tau(zt)\tau^3(X)=xyx.$$ The following proposition is an immediate consequence of these two calculations. \[prop.R-&gt;B\] Let $R$ be the free algebra $\CC\langle x,y\rangle$ modulo the relations $x^5=yxy$ and $y^2=xyx$. Then there is a $\CC$-algebra homomorphism $$R=\CC[x,y] \to B(\BB_3,\cL,\s), \qquad x \mapsto X, \; y \mapsto Zt.$$ \[lem.low.degs\] The homomorphism in Proposition \[prop.R-&gt;B\] is an isomorphism in degrees $\le 6$.[^2] By Proposition \[prop.steph\], $R$ has Hilbert series $(1-t)^{-1}(1-t^2)^{-1}(1-t^3)^{-1}$, so the dimension of $R_n$ in degrees $1,2,3,4,5,6$ is $1,2,3,4,5,7$. The $n^{\th}$ row in the following table gives a basis for $B_n$, $1 \le n \le 6$. One proceeds down each column by multiplying on the right by $x$. There wasn’t enough room on a single line for $B_6$ so we put the last two entries for $B_6$ on a new line. $$\begin{array}{cccccc} \begin{array}{l} x=X \\ x^2=Xu \\ x^3=XYu\\ x^4=XYtu\\ x^5=XYZtu\\ x^6=XYZstu\\ \phantom{x} \end{array} \!\!\!\! \begin{array}{l} \\ y=Zt \\ yx=YZt\\ yx^2=YZt^2\\ yx^3=YZ^2t^2\\ yx^4=YZ^2st^2\\ \phantom{x} \end{array} \!\! \begin{array}{l} \\ \\ xy=XZs\\ y^2=XZst\\ y^2x=XZ^2st\\ y^2x^2=XZ^2s^2t\\ \phantom{x} \end{array} \!\! \begin{array}{l} \\ \\ \\ x^2y=X^2su\\ xy^2=X^2Zsu\\ xy^2x=X^2Zs^2u\\ yx^2y=Y^2Zt^2u \end{array} \!\! \begin{array}{l} \\ \\ \\ \\ x^3y=X^2Yu^2\\ x^2y^2=X^2Ysu^2\\ x^4y=XY^2tu^2 \end{array} \end{array}$$ The products involving $x$ and $y$ were made by using the formula (\[eq.Z.tw.mult\]) in the same way that it was used to show that $x^5=yxy$. \[lem.gend.gl.sects\] $\cL_2$ is generated by its global sections. A line bundle $\cL$ on a variety is generated by its global sections if and only if for each point $p$ on the variety there is a section of $\cL$ that does not vanish at $p$. In this case, $H^0(\BB_3,\cL_2)$ is spanned by $Xu$ and $Zt$. One can see from the diagram (\[six.lines\]) that the zero locus of $Xu$ does not meet the zero locus of $Zt$, so the common zero locus of $Xu$ and $Zt$ is empty. As a $\CC$-algebra, $B$ is generated by $B_1$ and $B_2$. It follows from the explicit calculations in Lemma \[lem.low.degs\] that the subalgebra of $B$ generated by $B_1$ and $B_2$ contains $B_m$ for all $m \le 6$. It therefore suffices to prove that the twisted multiplication map $B_2 \otimes B_n \to B_{n+2}$ is surjective for all $n \ge 5$. By definition, $B_2=H^0(\cL_2)$ and this has dimension two so, by Lemma \[lem.gend.gl.sects\], there is an exact sequence $0 \to \cN \to B_2 \otimes \cO_{\BB_3} \to \cL_2 \to 0$ for some line bundle $\cN$. In fact, $\cN \cong \cL_2^{-1}$. By definition, $\cL_{n+2}=\cL_2 \otimes \cM$ where $\cM \cong \cO(D_{n+2}-D_2)$, and the twisted multiplication map $B_2 \otimes B_n \to B_{n+2}$ is the ordinary multiplication map $$B_2 \otimes H^0(\cM) = H^0(\cL_2) \otimes H^0(\cM) \to H^0(\cL_2 \otimes \cM).$$ Applying $-\otimes \cM$ to the exact sequence $0 \to \cL_2^{-1} \to B_2 \otimes \cO_{\BB_3} \to \cL_2 \to 0$ and taking cohomology gives an exact sequence $$0 \to H^0(\cL_2^{-1} \otimes \cM) \to B_2 \otimes H^0(\cM) \to H^0(\cL_2 \otimes \cM) \to H^1(\cL_2^{-1} \otimes \cM).$$ Hence, if $h^1(\cL_2^{-1} \otimes \cM)=0$, then $B_2B_n=B_{n+2}$. But $$\cL_2^{-1} \otimes \cM \cong \cO(-D_2 +D_{n+2}-D_2)$$ so we need to show that $D_{6m+r}-2D_2-K$ is ample whenever $6m+r \ge 7$ and $0 \le r \le 5$. By Lemma \[lem.Dn\], $D_{6m+r}=D_r-mK$. We therefore need to check that conditions (\[eq.ample2\]) and (\[eq.ample3\]) in Lemma \[lem.van\] hold for the divisors $D$ in the following table: $$\begin{array}{lll} && D:=D_r-2D_2 - (m+1)K\\ r=0 \quad & m \ge 2 \quad & (3m+1,m-1,m+1,m+1) \\ r=1 \quad & m \ge 1 \quad& (3m+2,m,m+1,m+2) \\ r=2 \quad & m \ge 1 \quad & (3m+2,m,m+1,m+1) \\ r=3 \quad & m \ge 1 \quad& (3m+3,m,m+2,m+2) \\ r=4 \quad & m \ge 1 \quad & (3m+3,m,m+1,m+2) \\ r=5 \quad & m \ge 1 \quad & (3m+4,m+1,m+2,m+2). \end{array}$$ This is a routine though tedious task. \[thm.isom\] Let $R$ be the free algebra $\CC\langle x,y\rangle$ modulo the relations $x^5=yxy$ and $y^2=xyx$. Then the $\CC$-algebra homomorphism $$\Phi: R=\CC[x,y] \to B(\BB_3,\cL,\s), \qquad x \mapsto X, \; y \mapsto Zt,$$ is an isomorphism of graded algebras. By Lemma \[lem.low.degs\], $B_1$ and $B_2$ are in the image of $\Phi$. It follows that $\Phi$ is surjective because $B$ is generated by $B_1$ and $B_2$. But $\Phi(R_n) \subset B_n$, and $R$ and $B$ have the same Hilbert series, so $\Phi$ is also surjective. Consider $R^{(3)}\supset \CC[x^3,xy,yx]$. Since $\dim R_6=7=(\dim R_3)^2 -2$ there is a 2-dimensional space of quadratic relations among the elements $x^3$, $xy$, and $yx$. Hence $R^{(3)}$ is not a 3-dimensional Artin-Schelter regular algebra. The relations in the degree two component of $R^{(3)}$ are generated by $$(x^3)^2=(xy)^2=(yx)^2.$$ [12]{} M. Artin, J. Tate and M. Van den Bergh, Some algebras related to automorphisms of elliptic curves, in [*The Grothendieck Festschrift, Vol.1,*]{} 33-85, Birkhauser, Boston 1990. M. Artin, J. Tate and M. Van den Bergh, Modules over regular algebras of dimension 3, [*Invent. Math.,*]{} [**106**]{} (1991) 335-388. M. Artin and M. Van den Bergh, Twisted homogeneous coordinate rings, [*J. Alg.,*]{} [**133**]{} (1990) 249-271. M. Artin and J.J. Zhang, Noncommutative Projective Schemes, [*Adv. Math.*]{}, [**109**]{} (1994) 228-287. D.A. Cox, The homogeneous coordinate ring of a toric variety, [*J. Alg. Geom.,*]{} [**4**]{} (1995) 17-50. Serre, J.-P., Faisceaux algébriques cohérents, [*Ann. Math.,*]{} [**61**]{} (1955) 197-278. S.P. Smith and J.J. Zhang, Homogeneous coordinate rings in noncommutative algebraic geometry, In preparation. S.P. Smith and J. Tate, The centers of the 3- and 4-dimensional Sklyanin algebras, [*K-Theory*]{} [**8**]{} (1994), 19-63. D. R. Stephenson, Artin-Schelter Regular Algebras of Global Dimension Three, [*J.Alg.,*]{} [**183**]{} (1996)55-73. D. R. Stephenson, Algebras Associated to Elliptic Curves, [*Trans. Amer. Math. Soc.,*]{} [**349**]{} (1997) 2317-2340. J.J. Zhang, Twisted graded algebras and equivalences of graded categories, [*Proc. Lond. Math. Soc.,*]{} [**72**]{} (1996) 281-311. [^1]: The author was supported by the National Science Foundation, Award No. 0602347 [^2]: We will eventually prove that the homomorphism is an isomorphism in all degrees but the low degree cases need to be handled separately.